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2,877,628,088,954 | arxiv | \section{Introduction}
The asymmetry of particle emission with respect to the reaction plane
in heavy ion reactions has attracted increasing
interests of both experimentalists and theorists.
It has been found recently that the
azimuthal asymmetry of nucleons depends significantly on the bombarding
energy \cite{ollit98}. If the incident energy is comparable to the
Fermi energy, the attractive nuclear mean field dominates the dynamics.
Projectile and target form a rotating system
which preferentially emits nucleons in the reaction
plane \cite{tsang84}. At high energies,
a positive pressure develops in the overlap region of the colliding nuclei
(so-called fireball) as a result of individual nucleon-nucleon
collisions. The emission of the fireball nucleons can be
hindered by the less disturbed matter located in the reaction
plane (spectators). This shadowing leads to more nucleons emitted
in the direction out of the reaction plane than in the
plane \cite{gutbr89,basti97}.
However, at ultra-relativistic energies the shadowing by spectator
matter is suppressed since the spectators depart too fast from the
fireball region. Then the nucleons can escape freely from the fireball.
Since the fireball has a smaller size along the impact parameter,
the pressure gradient is larger in the reaction plane. This results in
an in-plane preference of the nucleon emission \cite{barre94,sorge97}.
The transition from an in-plane preference due to rotation to an
out-of-plane one caused by shadowing was observed
by the FOPI Collaboration to occur at an incident
energy of 80-120 A.MeV \cite{basti97}.
The second transition to an in-plane preference at high energies
takes place at 4-6 A.GeV \cite{ollit98,danie98}.
In this work we study the azimuthal asymmetry of $K^+$ and $K^-$ mesons
in heavy ion reactions at incident energies of 1-2 A.GeV. As
mentioned above, shadowing dominates these reactions and leads to
more nucleons out of the reaction
plane. The phenomenon of an out-of-plane
enhancement of particle emission at midrapidity is called "squeeze-out".
E.g., pions exhibit a clear out-of-plane preference
\cite{brill93,venem93} which is due to shadowing,
namely to scattering and absorption of the pions on the
nucleons \cite{bli94,bass95}.
It is of high interest to see if the same mechanism still holds for kaons.
Unlike a pion, which keeps the mass more or less unchanged in a nuclear
environment, a kaon might vary its properties dramatically
in the medium due to chiral symmetry restoration \cite{kapla86}.
The $K^+$- and $K^-$-nucleus potentials have observable effects
in heavy ion reactions: they can change the transverse flow of
kaons \cite{li95,wang97,bratk97,li98,cassing99}, the production
yields \cite{li98,cassing99,li97}, and may give rise to a
collective flow, i.e. a radial flow \cite{wang98}.
This paper is organised as follows: In section 2 we describe the kaon
dynamics within the framework of the
Quantum Molecular Dynamics (QMD) model. The kaon potential is based on
chiral perturbation theory, however, with
the space-like components of the baryon current neglected as usual.
We present the azimuthal asymmetry of $K^+$ mesons
in section 3, and the corresponding results for $K^-$
in section 4. Section 5 contains a detailed discussion of a covariant
treatment of the in-medium kaon dynamics. In section 6 the paper is
summarised.
\section{Kaon dynamics in heavy ion reactions}
We adopt the QMD model to describe the dynamics of heavy
ion reactions. $K^+$ and $K^-$ mesons are produced from baryon-baryon
collisions or pion-baryon collisions. The corresponding cross sections
for the $K^+$ production are taken from
Refs. \cite{sibirtsev,tsushima} for
the baryon-baryon channels and from \cite{tuebingen} for the pion-baryon
induced channels. The $K^-$ production is treated as described in
Ref. \cite{cassing99} with the corresponding elementary cross sections
given there. The present QMD model describes successfully a
large set of observables,
such as production cross sections and collective flow
patterns of nucleons, pions and kaons \cite{uma98,fuchs97,wang972}.
In particular, this model
describes well the experimental transverse flow of protons,
$K^+$ mesons as well as $\Lambda$ hyperons \cite{wang97,wang982}.
While a $K^+$ meson is hardly
absorbed in nuclear matter due to strangeness conservation,
a $K^-$ meson can be easily annihilated through the
reaction $K^-$ + N $\rightarrow$
$\pi$ + Y, where Y denotes a $\Sigma$ or a $\Lambda$ hyperon.
Both $K^+$ and $K^-$ mesons can scatter elastically with nucleons.
The $K^-$ absorption cross section is
larger than 50 mb for $K^-$ momenta below 0.2 GeV/c.
Compared to this value the elastic $K^+ N$ cross section
is relatively small ($\sigma_{K^+N}$ $\approx$ 10 mb) which results in a
long mean free path of $K^+$ mesons in nuclear matter.
In addition to scattering and absorption process, the strong interaction
of $K^+$ and $K^-$ mesons with a nuclear medium gives rise
to a mean field which acts on the kaons when they propagate
in the medium. Here both, the strong and the Coulomb potential
are included \cite{wang98}.
The $K^+$ and $K^-$ potentials are defined in the usual way as the
difference of the in-medium dispersion relation and the free one
\begin{equation}
U_K = \omega_K - \sqrt{m_K^2 + \vec{p}^2}.
\end{equation}
Starting from an $SU(3)_L$$\times$$SU(3)_R$ chiral Lagrangian as proposed by
Kaplan and Nelson \cite{kapla86}, one obtains the field equations for $K^+$
and $K^-$ mesons in mean field approximation \cite{li95},
\begin{equation}
[ \partial_{\mu}\partial^{\mu} \pm \frac{3i}{4f^2_{\pi}}j_{\mu}
\partial^{\mu} + ( m^2_K - \frac{\Sigma_{KN}}{f^2_{\pi}}\rho_s )]
\phi_{K^{\pm}}(x) = 0 \quad ,
\end{equation}
where $\rho_s$ is the scalar baryon density and $j_{\mu}$ the baryon
four-vector current. $f_{\pi}$ $\approx$ 93 MeV is the
pion decay constant. According to recent lattice QCD
calculations \cite{brown96} the kaon-nucleon sigma term
is taken to be $\Sigma_{KN}$ $\approx$ 450 MeV.
The term coupling to the baryon current
(so-called Weinberg-Tomozawa term) is of leading order
in the chiral expansion, while the sigma term (so-called Kaplan-Nelson term)
comes from the next order. The in-medium dispersion
relation reads \cite{li95,fuchs98}
\begin{equation}
\omega^2 = m^2_K + \vec{p}^2 - \frac{\Sigma_{KN}}{f^2_{\pi}}\rho_s
+ \frac{3}{4f^2_{\pi}}(\rho_B\omega - \vec{j} \cdot \vec{p}).
\end{equation}
Usually the spatial components $\vec{j}$ of the baryon current are
neglected \cite{li95,wang97,bratk97,li98,cassing99,li97}. This means
that the kaons are propagated in a static, momentum independent
potential. Since the resulting kaon dynamics
has been shown to agree well with experiments
concerning, e.g., the transverse flow \cite{li95,wang97,li97,li98}
we follow this conventional treatment.
However, one looses Lorentz covariance by the neglection
of the spatial components of the four-vector baryon current
\cite{fuchs98}. Thus, in Section 5 we will also discuss
the covariant treatment of the kaon dynamics which includes
the momentum dependence of the mean field in lowest order, i.e.
that part which arises from Lorentz boosts.
Brown and Rho pointed out \cite{brown962} that the range
term, which is of the same order in the
chiral expansion as the $\Sigma_{KN}$ term also
contributes to the mean field. Moreover, the pion decay constant
$f_{\pi}$ can be reduced in the medium due to the decreasing
quark condensate. For $K^+$ the range term and the medium modification of
the pion decay constant are taken into account.
At nuclear matter saturation density $\rho_0$ = 0.16 $fm^{-3}$
this potential agrees with the empirical knowledge obtained
from the impulse approximation to free $K^+$N scattering data
\cite{barne94}. The $K^-$ potential is similar to
that used in other works \cite{li95,bratk97,cassing99,li97}.
It is constructed according to eq.(1) and eq.(3),
however, with a smaller value of $\Sigma_{KN}$ = 350 MeV and
the $f_{\pi}$ taken as in free space. The range term is neglected.
This potential is less attractive than
that extracted from kaonic atoms: the former is about -100 MeV
at $\rho_0$ while the latter is about -185 $\pm$ 15 MeV
\cite{mille88}. However, such a less attractive $K^-$
potential seems to be necessary in order to reproduce
the experimental $K^-$ yield in heavy ion reactions at
1-2 A.GeV \cite{cassing99,li97}. This inconsistency is so far unsolved
(see also Section 5).
In Fig.1 we show the potentials for $K^+$ and $K^-$
at zero momentum as a function of the nuclear matter density.
In addition to the influence on the propagation,
the potentials can also change the production thresholds of the mesons
and thus the corresponding yields.
For $K^-$ the potential has been included in the threshold
throughout the present work in the same way as in Refs.
\cite{cassing99,li97}. The influence of the $K^+$ in-medium
potential on the yields is demonstrated in Fig.2. The reaction
considered is $C+C$ at 2.0 A.GeV under minimal bias conditions.
For the comparison to the corresponding KaoS data \cite{kaos99}
we applied a $\Theta_{\rm Lab} = 40^o \pm 4^o $ polar angular cut.
We distinguish three different cases: First a calculation without
any medium effects, a calculation where the potential is only
included in the propagation and finally a full
calculation where the potential is included in the threshold
as well as in the propagation. Without medium effects we are able
to reproduce the low $p_t$-region of the spectrum
but underpredict the high $p_t$ part. On the other hand, the potential
effect on the thresholds strongly suppresses the low energetic kaons.
If the potential also acts on the propagation it makes the spectrum
harder which is due to the repulsive forces. This can be clearly seen
from the calculation where the potential is only included the
propagation of the kaons. However, the uncertainty
in the description of the cross section
is still too large in order to draw definite conclusions on
the importance of medium effects. Although
our calculation is in reasonable agreement with the results
of other groups \cite{li97,cassing99}, there remain still
discrepancies which reflect the uncertainties in the theoretical
knowledge of the elementary production cross sections.
Thus, for $K^+$ we will neglect the effect of the potential on the
production thresholds in the following. This treatment is also
justified since we study collective flow phenomena which do not
depend sensitive on total yields.
\section{$K^+$ squeeze-out}
The strength of the azimuthal asymmetry can be quantified by the ratio
of the particle multiplicity emitted perpendicular to the reaction plane
over the multiplicity parallel to the plane
\begin{equation}
R_{out/in} = \frac{N(\phi=90^0) + N(\phi=270^0)}{N(\phi=0^0) + N(\phi=180^0)}.
\end{equation}
The azimuthal angles $\phi$ = $0^0$ and $180^0$ correspond to the positions
of target and projectile in the reaction plane, while $\phi$ = $\pm$$90^0$
denote the directions perpendicular to the plane.
A ratio $R_{out/in}$$>$1 means a preference of the particle emission
out of the reaction plane. An azimuthal distribution can be expressed
in terms of a Fourier series \cite{volos97}
\begin{equation}
\frac{dN}{d\phi} \sim C[ 1 + a_1cos(\phi)
+ a_2cos(2\phi) + a_3cos(3\phi) + ...].
\end{equation}
The dipole term arises from a collective sideward deflection of
the particles in the reaction plane ("transverse flow")
and does not contribute to the squeeze-out ratio $R_{out/in}$. In
a symmetric collision, the cos(3$\phi$) term vanishes. The squeeze-out ratio
is then determined only by the quadrupole component
\begin{equation}
R_{out/in} = \frac{1 - a_2}{1 + a_2}.
\end{equation}
The azimuthal asymmetry of the $K^+$ production in heavy ion
reactions has been first studied by Li et al. \cite{li96}.
However, in \cite{li96} the authors focused on
kaons emitted at projectile or target rapidity.
In the present study we will focus on midrapidity kaons which
should yield more precise information on the dense fireball.
In Fig.3 the $K^+$ multiplicity as a function of azimuthal
angle $\phi$ is shown for a semi-central (b=6 fm) $Au + Au$
reaction at 1 A.GeV incident energy. A transverse momentum cut of $P_T$
$>$ 0.2 GeV/c and a rapidity cut centred at midrapidity, i.e.
-0.2 $<$ $(Y/Y_{proj})^{cm}$ $<$ 0.2, have been applied.
The same cuts have been adopted by the KaoS Collaboration.
The KaoS data \cite{shin98} for the same reaction at
semi-central impact parameters (corresponding to 5 fm $<$ b $<$ 10 fm)
are also shown. We performed the QMD calculations for three
different scenarios:
(a) with full in-medium $K^+$ dynamics;\\
(b) the $K^+$ potential due to the strong interaction is neglected;\\
(c) in addition, the Coulomb potential is neglected.
It can be seen from Fig.3 that the full calculation leads to
an enhanced out-of-plane ($\phi$ = $90^0, 270^0$)
$K^+$ emission. The corresponding ratio $R_{out/in}$ = 1.5
is close to the experimental
value of $R_{out/in}$ = 1.7 \cite{shin98}. We would like to mention that
we also performed calculations for different impact parameters
ranging from b=3 to b=10 fm. From the impact parameter dependence of
the kaon production we found that it is reasonable to compare
the simulation at the single impact parameter b=6 fm
with the experimental data since the kaon multiplicity
decreases very fast with decreasing centrality. (A corresponding
calculation of the Stony Brook group \cite{shin98} has been
performed for the representative impact parameter b=7 fm.)
Comparing the cases (b) and (c) one clearly sees
that the $K^+$ out-of-plane preference
is mainly a result of the strong potential. The $K^+$ emission is
nearly azimuthally isotropic in the calculation with neither the strong
nor the Coulomb potential, while the Coulomb potential slightly increases
the out-of-plane abundance. However, the Coulomb force has only a minor
effect, and leads to an azimuthal asymmetry which is much
weaker than the experimental one. Generally our results are in good
agreement with those found by the Stony Brook group \cite{shin98}.
Fig.4 shows the $K^+$ $R_{out/in}$ ratio at midrapidity
as a function of the transverse momentum. In this figure we
also show a calculation where all
final-state interactions including $K^+ N$ rescattering,
have been neglected. The primordial $K^+$ mesons
exhibit a slight in-plane preference
which increases with the momentum. This behaviour reflects the
primordial in-plane $K^+$ flow which follows the flow pattern of the
production sources, i.e. the in-plane flow of the nucleons.
Rescattering enhances the out-of-plane emission of the
$K^+$ mesons and leads to a nearly isotropic azimuthal distribution.
Thus we observe a clear shadowing effect by the spectator matter.
However, the effect of $K^+$-nucleon scattering is much less pronounced
than that of the strong potential. The strong potential
enhances dramatically the out-of-plane emission
at transverse momentum between 0.2 GeV/c and 0.6 GeV/c. Now
the azimuthal asymmetry exhibits a complex momentum
dependence since the out-of-plane preference decreases
again at high momenta.
This momentum dependence is, however, understandable.
First of all, midrapidity $K^+$ mesons experience the strongest repulsion
in the dense fireball. The repulsive potential gradient
is larger in the direction perpendicular to the reaction plane
than parallel to the plane.
In the configuration where the fireball is combined with the spectators
the distance from the fireball centre to free space is much shorter
along the out-of-plane direction than in the reaction plane.
Consequently, the $K^+$ mesons are driven by the potential
gradient preferentially out of plane. With other words, in the reaction
plane the kaons are repelled by the spectator matter. Thus, the
potential acts similar as the elastic scattering and strongly enhances
the shadowing effect. There occurs, however, a difference
between the effect of the potential and the shadowing by simple
rescattering/absorption on spectator matter. From nucleons and pions
\cite{basti97,brill93,venem93} it is known that the shadowing
enhances the out-of-plane emission of high energetic particles, i.e.
one observes an increasing $R_{out/in}$ ratio with increasing
momentum. The same effect is observed in our calculation for
$K^-$ discussed in the next section. This is understandable
since in these cases the high energetic particles stem mostly from the
early phase of the reaction where the fireball and the spectators
are clearly developed. However, the $K^+$ squeeze-out shows a different
trend. Recent data \cite{shin98} indicate a constant $R_{out/in}$
ratio whereas the calculation including the $K^+$ potential shows
even a decreasing $R_{out/in}$ with increasing momentum.
To understand this behaviour one
has to keep in mind that the repulsive $K^+$ potential accelerates the
particles and makes the spectrum harder (see also Fig.2).
High energetic $K^+$ mesons which experience the acceleration by
the medium for a longer time span stem probably from the
later stages of the reaction. But this means that the fireball and the
spectators are washed out to more extent which results in less
shadowing and a more isotropic azimuthal distribution of the
particles. However, it appears that for a
complete understanding of the $K^+$
squeeze-out phenomenon, also its momentum impact parameter dependence,
further going studies seem to be necessary.
\section{$K^-$ squeeze-out}
$K^-$ mesons are strongly scattered or absorbed in the nuclear medium.
In the absence of an additional in-medium potential,
one expects a strong out-of-plane emission of midrapidity $K^-$ mesons
much like pions, since in both cases shadowing
by the spectators plays a dominant role.
In Fig.5 we present the azimuthal distribution of the $K^-$ mesons emitted at
midrapidity (-0.2 $<$ $(Y/Y_{proj})^{cm}$ $<$ 0.2) in the
reaction of $Au+Au$ at E/A = 1.8 A.GeV and b=8 fm.
A $P_T$ cut of $P_T$ $>$ 0.5 GeV/c has been used. Fig.6 shows the
corresponding $R_{out/in}$ as a function of the
transverse momentum. Let us first consider the calculations \
where the strong and Coulomb potentials are neglected.
Figs.5 and 6 show that the $K^-$ mesons are then
preferentially emitted out of the reaction
plane. The $R_{out/in}$ ratio increases with transverse momentum.
This $P_T$ dependence is very similar to that observed experimentally
for nucleons and pions \cite{basti97,brill93,venem93}.
In Figs.5 and 6 the results of the full calculation
are presented as well. It can be seen that the $K^-$
in-medium potential reduces dramatically the out-of-plane abundance
($\phi$ = $90^0$ and $270^0$), and leads thereby to a nearly
isotropic azimuthal emission. The $R_{out/in}$ ratio remains
more or less flat as a function of the transverse momentum.
We can understand the effect of the in-medium
potential acting on $K^-$ in a similar way as we have done for $K^+$.
The $K^-$ potential is more attractive
at higher densities. Consequently, the $K^-$ mesons feel
an attractive potential gradient as they move from the dense fireball
to free space. This potential gradient is larger in the out-of-plane direction
than in the in-plane direction when the fireball is still connected
with the spectators. Thus the potential now favours the $K^-$
emission in the reaction plane.
It is found in our calculations that the Coulomb potential
has the same tendency but with a much smaller magnitude.
From previous studies on nucleons and pions,
both experimental \cite{basti97,brill93,venem93}
and theoretical \cite{bli94,bass95}, it is already clear that
frequent rescattering and re-absorptions will lead to a pronounced
out-of-plane preference of particle emission and an
increasing squeeze-out ratio with increasing transverse momentum.
Therefore, the disappearance of the out-of-plane preference
up to transverse momentum of $P_t$ = 0.8 GeV/c
can serve as an unambiguous signal for the
attractive $K^-$ in-medium potential.
So far we have shown that the azimuthal asymmetry of $K^+$
and $K^-$ mesons in heavy ion reactions at 1-2 A.GeV is
sensitive to the in-medium potentials.
The considered beam energies justify the mechanism where shadowing,
namely rescattering and reabsorption by the spectators,
compete with the in-medium potentials in the evolution of an
kaon azimuthal asymmetry. In order to minimise the shadowing
effect shadowing, one can study higher beam energies,
i.e. above 6 A.GeV, where
an isolated fireball elongated
in the direction perpendicular to the reaction plane, rather than
a combination of the fireball and the spectators, is responsible for
the azimuthal asymmetry of particle emission at midrapidity.
At such high beam energies, shadowing plays a minor role, and therefore,
the azimuthal asymmetry of the midrapidity kaons yields a cleaner information
on the in-medium potentials.
\section{Covariant kaon dynamics}
We have mentioned that the kaon dynamics is described in the present study
in a non-covariant way since the spatial components of the baryon current
are neglected. Such a description is correct in nuclear matter at rest
but generally not in energetic heavy ion collisions.
Keeping the four-vector baryon current in the kaon field equation,
eq.(2), one can construct a covariant description of the kaon
dynamics in the nuclear medium \cite{fuchs98}. In order to do this,
it is convenient to write the kaonic vector potential as
\begin{equation}
V_\mu = \frac{3}{8f_\pi^2}j_\mu
\quad ,
\end{equation}
The effective mass of the kaons is defined by
\begin{equation}
m_k^* = \sqrt{ m_k^2 - \frac{\Sigma_{KN}}{f_\pi^2}\rho_s + V_{\mu}V^{\mu} }
\quad .
\end{equation}
The kaon field equation is then rewritten as
\begin{equation}
[ (\partial_{\mu} + iV_{\mu})^2 + m_k^* ] \phi_{K^+} (x) = 0,
\end{equation}
\begin{equation}
[ (\partial_{\mu} - iV_{\mu})^2 + m_k^* ] \phi_{K^-} (x) = 0.
\end{equation}
Thus the vector field is introduced by minimal coupling into
the Klein-Gordon equation. The effective mass defined by eq.(8) is
a Lorentz scalar and is equal for $K^+$ and $K^-$ mesons \cite{fuchs98}.
We should note that this definition of kaon effective mass is different
from that conventionally used in the non-covariant description
of the kaon dynamics. In the latter case the effective mass of the kaons
is defined as kaon in-medium energy at zero momentum. Introducing
an effective kinetic momentum
\begin{equation}
p_{\mu}^* = p_{\mu} \mp V_\mu
\end{equation}
for $K^+$ ($K^-$) mesons, the Klein-Gordon equations (eq.(9) and eq.(10))
reads in momentum space
\begin{equation}
[ {p^*}^2 - {m^*}^2 ] \phi_{K^{\pm}} (p) = 0.
\end{equation}
This is just the mass-shell condition for the quasi-particles carrying
the effective mass and kinetic momentum, i.e. the effective energy is
on-shell $E^*$ = $p_0^*$ = $\sqrt{ \vec{p^*}^2 + {m^*} ^2 }$.
These quasi-particles
can be treated like free particles. This yields in the test-particle
limit the covariant equations of motion for the kaons \cite{fuchs98}
\begin{eqnarray}
\frac{d \vec{q}}{dt} &=& \frac{\vec{p^*}}{E^*} \\
\frac{d \vec{p^*}}{dt} &=& - \frac{m_k^*}{E^*} \frac{\partial m_k^*}
{\partial \vec{q}} \mp \frac{\partial V^0}{\partial \vec{q}}
\mp \partial_t \vec{V}
\pm \frac{\vec{p^*}}{E^*}\times (\frac{\partial}{\partial \vec{q}}
\times \vec{V})
\end{eqnarray}
where the upper (lower) signs refer to $K^+$ ($K^-$) mesons. The last
term of eq.(14) provides a momentum dependent
force which is missing in the non-covariant description of the kaon dynamics.
Such a force is analogous to the Lorentz
force in electrodynamics, and is a genuine feature of relativistic
dynamics as soon as a vector field is involved. Since in the QMD
approach the equations of motion are formulated in terms of canonical
momenta $\vec{p}$ rather than kinetic momenta $\vec{p^*}$, it is
instructive to rewrite eq.(14)
\begin{equation}
\frac{d \vec{p}}{dt} = - \frac{m_k^*}{E^*} \frac{\partial m_k^*}
{\partial \vec{q}} \mp \frac{\partial V^0}{\partial \vec{q}}
\pm \vec{\beta} \cdot \frac{\partial \vec{V}}{\partial \vec{q}}
= -\frac{\partial}{\partial \vec{q}}U_K \pm \vec{\beta} \cdot \frac{
\partial \vec{V}}{\partial \vec{q}}
\quad ,
\end{equation}
with $\vec{\beta}$ = $\vec{p}^* / E^*$ the kaon velocity and
the potential $U_K$ given by eq.(1). Eq.(15) is the equation which
we have solved in the present work.
A study with use of a relativistic version of
the QMD model (RQMD) has shown that the inclusion of the
Lorentz-like force leads
for the reaction Ni+Ni at 1.93 A.GeV $(b\leq 4$fm) to a $K^+$
transverse flow which essentially follows the nucleon flow \cite{fuchs98}.
This result contradicts the experimental data \cite{ritma95},
since the latter show clearly different flow patterns for
nucleons and $K^+$. The QMD model used in the present
work yields a result similar to that of the covariant RQMD
model after the Lorentz-like force is included.
However, as shown in our previous work, the same QMD
model but in absence of the Lorentz-force enabled us
to reproduce very well the FOPI data \cite{ritma95}
of the in-plane $K^+$ flow. Other groups also got
agreement with the data using conventional kaon dynamics
\cite{li95,bratk97}.
In Fig.7 we show the azimuthal $K^+$ distribution
for the same reaction as in Fig.3, however, now including
the Lorentz-force like momentum dependence.
One finds again that this force destroys the agreement with
experiments: the calculation with the momentum dependent force yields
an isotropic $K^+$ emission with respect to the azimuthal angle $\phi$,
while the KaoS data show a remarkable out-of-plane preference.
Both, the in-plane flow and the azimuthal asymmetry are predominately
determined by the interactions of the kaons with the spectators. Due
to the non trivial relative motion between the kaons and the spectators,
the momentum dependent Lorentz-force plays a role in cancelling
the effect from the time-like component of the vector potential
which is the major source of the repulsion acting on the
$K^+$ mesons. Therefore we observe a $K^+$ in-plane flow
similar to the nucleon flow and an azimuthal isotropy of the
midrapidity $K^+$ mesons. In a recent publication \cite{wang98}
we demonstrated that, in heavy ion reactions
at a beam energy below the kaon production threshold in free
space (1.58 A.GeV for $K^+$ and 2.5 A.GeV for $K^-$ mesons),
the in-medium potentials lead to new collective motion in the radial
direction of midrapidity $K^+$ and $K^-$ mesons.
This collective flow results in a "shoulder-arm" or a "concave"
structure in the transverse mass spectrum of the $K^+$ or $K^-$
mesons, respectively.
Such "shoulder-arm" or "concave" structure obviously
differentiates the kaon spectrum from the standard Boltzmann
distribution. We called this collective motion kaon radial flow.
The kaon radial flow is primarily a consequence of
the interactions of the midrapidity
kaons with the fireball. Since the relative
motion between the kaons and the fireball is small, the radial flow
has been found to keep more or less unchanged after the Lorentz-like
force is included \cite{wangd}.
It is surprising that the conventional description of the in-medium kaon
dynamics, rather than the more consistent one including the full
four-vector baryon current, turns out to agree
with the phenomenology of kaon production in heavy ion
reactions. To understand this fact it is important
to realize that the kaon field equation, eq.(2), which is the same starting
point for the two treatments, consists only of the
two lowest-order contributions from the chiral expansion,
namely the Weinberg-Tomozawa and Kaplan-Nelson term.
At that level the mean field approximation gives rise to
only density dependent scalar and vector potentials for the kaons.
Although the consistent treatment is based on the same level, it
accounts for the additional momentum dependence which arises by
Lorentz covariance and thus addresses the momentum dependence of
the interaction of the kaons in the nuclear medium at lowest order.
The success of the conventional treatment where the
momentum dependence is neglected
seems to imply that higher order contributions
in the chiral expansion might lead to
cancellation effects.
Thus, one probably needs to take into account
in a covariant formalism not only the lowest-order terms but also higher-order
contributions, e.g. P-wave contributions of the kaon-nucleus interaction
arising from nucleon hole-hyperon
excitations. Such a P-wave interaction goes
beyond the Weinberg-Tomozawa and Kaplan-Nelson terms
and leads to a non trivial momentum dependence of the kaon
potential \cite{kolom95,lutz97,sibir98}.
Furthermore, the mean field approximation which is used to obtain
the field equations (2) is justified at
low nuclear matter densities, but might be
questionable at high densities. The kaon-nucleon and
nucleon-nucleon correlations will
start to play a more important role \cite{pandh95}.
The effect of the correlations can be
illustrated considering the low and high density limits. At very low
densities a kaon interacts many times with the same nucleon
before it encounters another one. Thus, the impulse approximation
is justified and the energy of a kaon interacting with nucleons is given
in terms of the kaon-nucleon scattering length $a_{KN}$ \cite{lutz97},
\begin{equation}
\omega_{Lenz} = m_K - \frac{2\pi}{m_K}(1+\frac{m_K}{m_N})a_{KN}\rho_B
\quad .
\end{equation}
This is the so-called Lenz potential. At high densities,
however, the K-N interaction has to be summed over
many nucleons which results in a potential of a Hartree type.
According to an estimation given in Ref. \cite{pandh95},
the Hartree potential is reduced compared to the Lenz potential
by a factor of 1.63 for $K^-$'s in nuclear matter.
The correlation effect for $K^+$'s should be smaller
than for $K^-$'s, since the $K^+$-nucleon interaction
is relatively weak compared to other hadron-nucleon interactions.
However, the onset of short-range correlations
is a general feature of the dynamics of dense matter. Such effects
have, e.g., been discussed concerning particle production in heavy
ion reactions \cite{wang95}.
In Ref. \cite{heise98} the transition from the Lenz
to the Hartree potential was investigated for
$K^-$ mesons in neutron matter. There it is found that correlations
reduce the $K^-$ potential already significantly at
densities below 3$\rho_0$ which could also provide a solution
of the disagreement between the $K^-$ potential extracted from
kaonic atoms (at $\rho \leq \rho_0$)
and the less attractive potential which has to be
used to obtain a reasonable description of the
experimental $K^-$ yields in heavy ion reactions
at 1-2 A.GeV \cite{li98} (at $\rho \leq 3\rho_0$).
\section{Conclusions}
In the present paper we investigated the azimuthal
asymmetry of midrapidity $K^+$ and $K^-$ mesons in
heavy ion reactions at beam energies of 1-2 A.GeV.
The in-medium kaon potential is constructed from the lowest-order
terms of the chiral expansion. The kaon dynamics
are described in a conventional way with a quasi-potential formalism where
the space-like components of the baryon current are neglected. This
description turned out to be rather successful in reproducing the
current experimental data on $K^+$ and $K^-$ production.
The present study finds that
the in-medium potentials of the $K^+$ and $K^-$ mesons play an important
role in the evolution of an azimuthal asymmetry. Due to the long mean free
path of the $K^+$ mesons in nuclear matter, $K^+$-nucleon scattering
processes are insufficient to understand the
pronounced out-of-plane emission of
midrapidity $K^+$ mesons observed by the KaoS Collaboration.
The $K^+$ potential drives the kaons out of the reaction plane
by a repulsive potential gradient. These kaons are repelled by the
spectators which leads to an additional shadowing. The repulsive
potential is necessary in order to understand the KaoS data. The
momentum dependence of the $K^+$ squeeze-out signal is, however,
more complex.
We observe a decreasing squeeze-out signal with increasing transverse
momentum. For a complete understanding of the $p_t$
dependence of the $K^+$ squeeze-out further-going theoretical
and experimental investigations seem to be necessary.
The $K^-$ potential, in contrast, suppresses
the $K^-$ emission out of the reaction plane
by an attractive potential. The effect is found to
counterbalance to a large extent the strong
shadowing by scattering and absorption.
This results in a nearly isotropic azimuthal $K^-$ emission.
Since it is clear from both experimental and theoretical
studies that frequent scattering and absorption give rise to a remarkable
out-of-plane preference of the particle emissions, the disappearance of
the squeeze-out can serve as a signal of
the $K^-$ in-medium potential.
In a covariant treatment, in particular with respect to the
vector potential which is proportional to the
four-vector baryon current, one finds that the agreement
with experiments concerning both, the
azimuthal asymmetry and the in-plane flow is destroyed.
Since the corresponding fields originate from the lowest-order
terms of the kaon-nucleon interaction we suggest to
include in future also higher-order contributions, in
particular to treat the momentum dependence of the kaon-nucleon
interaction more carefully.
{\bf Acknowledgments}
The authors would like to thank M. Lutz for valuable discussions.
|
2,877,628,088,955 | arxiv | \section{Introduction}
\label{Introduction}
Since their introduction in \cite{Kin43}, Horn clauses have shown to have good logic properties and have proven to be of importance for many disciplines, ranging from logic programming, abstract specification of data structures and relational data bases, to abstract algebra and model theory. However, the analysis of Horn clauses has been mainly restricted to the sphere of classical logic. For a good exposition of the most relevant results concerning Horn clauses in classical logic we refer to \cite{Hod93}, and to \cite{Mak87} for a good study of their importance in computer science.
The interest in continuous t-norm based logics since its systematization by H\'ajek \cite{Ha98} and the subsequent study of core fuzzy logics \cite{CiHa10} invite to a systematic development of a model theory of these logics (and of algebraizable non-classical logics in general). Cintula and H\'ajek raised the open question of characterizing theories of Horn clauses in predicate fuzzy logics \cite{CiHa10}. Our first motivation to study the Horn fragment of predicate fuzzy logics was to solve this open problem, the present article is a first contribution towards its solution.
Some authors have contributed to the study of Horn clauses over fuzzy logic. In \cite{Be02,Be03,BeVy05,BeVic06,BeVic06b,Vy15} B\v{e}lohl\'avek and Vychodil study fuzzy equalities, they work with theories that consist of formulas that are implications between identities with premises weighted by truth
degrees. They adopt Pavelka style: theories are fuzzy sets of formulas and they consider degrees
of provability of formulas from theories. Their basic structure of truth degrees is a complete
residuated lattice. The authors derive a Pavelka-style completeness theorem (degree of provability
equals degree of truth) from which they get some particular cases by imposing restrictions
on the formulas under consideration. As a particular case, they obtain completeness of fuzzy
equational logic. In different articles they study the main logical properties of varieties of algebras with fuzzy equalities. Taking a different approach, in a series of papers \cite{Ge01b, Ge01,Ge05}, Gerla proposes to base fuzzy control on fuzzy logic programming, and observes that the class of fuzzy Herbrand interpretations gives a semantics for fuzzy programs. Gerla works with a complete, completely distributive, lattice of truth-values. For a reference on fuzzy logic programming see \cite{Voj01, Ebra01}.
Several definitions of Horn clause have been proposed in the literature of fuzzy logics, but there is not a canonical one yet. Cintula and H\'ajek affirm that the elegant approach of \cite{BeVic06} is not the only possible one. In \cite{DuPra96}, Dubois and Prade discuss different possibilities of defining \emph{fuzzy rules} and they show how these different semantics can be captured in the framework of fuzzy set theory and possibility theory. Following all these works, our contribution is a first step towards a systematic model-theoretic account of Horn clauses in the framework introduced by H\'ajek in \cite{Ha98}. We introduce a basic definition of Horn clause over the predicate fuzzy logic MTL$\forall^m$ that extends the classical one in a natural way. In future work we will explore different generalizations of our definitions for expanded languages. Our approach differs from the one of B\v{e}lohl\'avek and Vychodil because we do not restrict to fuzzy equalities. Another difference is that, unlike these authors and Gerla, our structures are not necessarily over the same complete algebra, because we work in the general semantics of \cite{Ha98}.
In the present work we have focused on the study of \emph{free models of Horn clauses}. Free structures have a relevant role in classical model theory and logic programming. Admitting free structures make reasonable the concepts of \emph{closed-word assumption} for databases and \emph{negation as failure} for logic programming. These structures allow also a procedural interpretation for logic programs (for a reference see \cite{Mak87}). Free structures of a given class are minimal from an algebraic point of view, in the sense that there is a unique homomorphism from these structures to any other structure in the class. The free structures introduced here are \emph{term structures}, structures whose domains consist of terms or equivalence classes of terms of the language. In classical logic, term structures have been used to prove the satisfiability of a set of consistent sentences, see for instance \cite[Ch.5]{EbiFlu94}. Notorious examples of term structures are Herbrand models, they play an important function in the foundations of logic programming. Several authors have been studied Herbrand models in the fuzzy context (for a reference see \cite{Ge05,Voj01,Ebra01}), providing theoretical background for different classes of fuzzy expert systems. For a general reference on Herbrand Theorems for substructural logics we refer to \cite{CiMet13}
\smallskip
The present paper is an extension of the work presented in the 18th International Conference of the Catalan Association for Artificial Intelligence (CCIA 2015) \cite{CoDe15}. Our main original contributions are the following:
\begin{itemize}
\item Introduction of the notion of term structure associated to a theory over predicate fuzzy logics. If the theory consist of universal Horn formulas, we show that the associated term structure is a model of the theory (Theorem 2).
\item Existence of free models in fuzzy universal Horn classes of structures. In the case that the language has an equality symbol $\approx$ interpreted as a similarity, we prove the existence of models which are free in the class of reduced models of the theory (Theorem 1). In the case that the language has the crisp identity, the class has free models in the usual sense.
\item Consistent universal Horn theories over predicate fuzzy logics (that contains only the truth-constants $\overline{1}$ and $\overline{0}$) have classical models (Corollary \ref{classic}).
\item Introduction of Herbrand structures. We prove that every equality-free consistent universal Horn theory over predicate fuzzy logics have a Herbrand model (Corollary \ref{corollary H-model}).
\end{itemize}
The paper is organized as follows. Section 2 contains the preliminaries on predicate fuzzy logics. In Section 3 we introduce the definition of Horn clause over predicate fuzzy logics. In Section 4 we study the term structures associated to universal Horn theories. In Section 5 we introduce Herbrand structures for equality-free theories. Finally, there is a section devoted to conclusions and future work.
\section{Preliminaires}
\label{Preliminaires}
Our study of the model theory of Horn clauses is focused on the basic predicate fuzzy logic MTL$\forall^m$ and some of its extensions based on propositional core fuzzy logics in the sense of \cite{CiHa10}. The logic MTL$\forall^m$ is the predicate extension of the left-continuous t-norm based logic MTL introduced in \cite{EsGo01}, where MTL-algebras are defined as bounded integral commutative residuated lattices $(A,\sqcap,\sqcup,*,\Rightarrow,0,1)$, where $\sqcap$ and $\sqcup$ are respectively the lattice meet and join operations and $(\Rightarrow,*)$ is a residuated pair, satisfying the pre-linearity equation $(x\Rightarrow y)\sqcup(y\Rightarrow x)=1$ (for an exhaustive exposition of MTL-algebras, see \cite{NoEsGis05}). In addition, completeness of this logic with respect to MTL-algebras is proven in \cite[Th.1]{EsGo01}, and Jenei and Montagna shown that MTL is the logic of all left continuous t-norms and their residua \cite{JeMon02}. Now we present the syntax and semantics of predicate fuzzy logics and we refer to \cite[Ch.1]{CiHaNo11} for a complete and extensive presentation.
\begin{defi} [Syntax of Predicate Languages]
A \emph{predicate language} $\mathcal{P}$ is a triple $\left\langle Pred_{\mathcal{P}},Func_{\mathcal{P}},Ar_{\mathcal{P}} \right\rangle$, where $Pred_{\mathcal{P}}$ is a nonempty set of \emph{predicate symbols}, $Func_{\mathcal{P}}$ is a set of \emph{function symbols} (disjoint from $Pred_{\mathcal{P}}$), and $Ar_{\mathcal{P}}$ represents the \emph{arity function}, which assigns a natural number to each predicate symbol or function symbol. We call this natural number the \emph{arity of the symbol}. The predicate symbols with arity zero are called \emph{truth constants}, while the function symbols whose arity is zero are named \emph{individual constants} (\emph{constants} for short) or \emph{objects}.
\end{defi}
The set of $\mathcal{P}$-terms, $\mathcal{P}$-formulas and the notions of free occurrence of a variable, open formula, substitutability and sentence are defined as in classical predicate logic. From now on, when it is clear from the context, we will refer to $\mathcal{P}$-terms and $\mathcal{P}$-formulas simply as \emph{terms} and \emph{formulas}. A term $t$ is \emph{ground} if it has no variables. Throughout the paper we consider the equality symbol as a binary predicate symbol, not as a logical symbol, that is, the equality symbol is not necessarily present in all the languages and its interpretation is not fixed. From now on, let $L$ be a core fuzzy logic in a propositional language $\mathcal{L}$ that contains only the truth-constants $\overline{1}$ and $\overline{0}$ (for an extended study of core fuzzy logics, see \cite{CiHa10}).
\begin{defi} We introduce an axiomatic system for the predicate logic $L\forall^m$:
\begin{description}
\item[($\mathrm{P}$)]$\space\space$ $\space\space$ $\space\space$ Instances of the axioms of $L$ (the propositional variables are substituted for first-order formulas).
\item[($\forall 1$)]$\space\space$ $(\forall x)\varphi(x)\rightarrow\varphi(t)$, where the term $t$ is substitutable for $x$ in $\varphi$.
\item[($\exists1$)]$\space\space$ $\varphi(t)\rightarrow(\exists x)\varphi(x)$, where the term $t$ is substitutable for $x$ in $\varphi$.
\item[($\forall 2$)]$\space\space$ $(\forall x)(\xi\rightarrow\varphi)\rightarrow(\xi\rightarrow(\forall x)\varphi(x))$, where $x$ is not free in $\xi$.
\item[($\exists2$)]$\space\space$ $(\forall x)(\varphi\rightarrow\xi)\rightarrow((\exists x)\varphi\rightarrow\xi)$, where $x$ is not free in $\xi$.
\end{description}
The deduction rules of $L\forall^m$ are those of $L$ and the rule of generalization: from $\varphi$ infer $(\forall x)\varphi$. The definitions of proof and provability are analogous to the classical ones. We denote by $\Phi\vdash_{L\forall^m}\varphi$ the fact that $\varphi$ is provable in $L\forall^m$ from the set of formulas $\Phi$. For the sake of clarity, when it is clear from the context we will write $\vdash$ to refer to $\vdash_{L\forall^m}$. A set of formulas $\Phi$ is \emph{consistent} if $\Phi\not\vdash\overline{0}$.
\end{defi}
\begin{defi} [\textbf{Semantics of Predicate Fuzzy Logics}] \label{evaluation} Consider a predicate language $\mathcal{P}=\langle Pred_{\mathcal{P}}, Func_{\mathcal{P}}, Ar_{\mathcal{P}} \rangle$ and let \textbf{A} be an $L$-algebra. We define an $\textbf{A}$\emph{-structure} $\mathrm{\mathbf{M}}$ for $\mathcal{P}$ as the triple $\langle M, (P_M)_{P\in Pred}, (F_M)_{F\in Func} \rangle$, where $M$ is a nonempty domain, $P_{\mathrm{\mathbf{M}}}$ is an $n$-ary fuzzy relation for each $n$-ary predicate symbol, i.e., a function from $M^n$ to $\textbf{A}$, identified with an element of $\textbf{A}$ if $n=0$; and $F_{\mathrm{\mathbf{M}}}$ is a function from $M^n$ to $M$, identified with an element of $M$ if $n=0$. As usual, if $\mathrm{\mathbf{M}}$ is an $\textbf{A}$-structure for $\mathcal{P}$, an $\mathrm{\mathbf{M}}$-evaluation of the object variables is a mapping $v$ assigning to each object variable an element of $M$. The set of all object variables is denoted by $Var$. If $v$ is an $\mathrm{\mathbf{M}}$-evaluation, $x$ is an object variable and $a\in M$, we denote by $v[x\mapsto a]$ the $\mathrm{\mathbf{M}}$-evaluation so that $v[x\mapsto a](x)=a$ and $v[x\mapsto a](y)=v(y)$ for $y$ an object variable such that $y\not=x$. If $\mathrm{\mathbf{M}}$ is an $\textbf{A}$-structure and $v$ is an $\mathrm{\mathbf{M}}$-evaluation, we define the \emph{values} of terms and the \emph{truth values} of formulas in $M$ for an evaluation $v$ recursively as follows:
\begin{description}
\item $||x||^{\small{\textbf{A}}}_{\mathrm{\mathbf{M}},v}=v(x)$;
\item $||F(t_1,\ldots,t_n)||^{\small{\textbf{A}}}_{\mathrm{\mathbf{M}},v}=F_{\mathrm{\mathbf{M}}}(||t_1||^{\small{\textbf{A}}}_{\mathrm{\mathbf{M}},v},\ldots,||t_n||^{\small{\textbf{A}}}_{\mathrm{\mathbf{M}},v})$, for $F\in Func$;
\item $||P(t_1,\ldots,t_n)||^{\small{\textbf{A}}}_{\mathrm{\mathbf{M}},v}=P_{\mathrm{\mathbf{M}}}(||t_1||^{\small{\textbf{A}}}_{\mathrm{\mathbf{M}},v},\ldots,||t_n||^{\small{\textbf{A}}}_{\mathrm{\mathbf{M}},v})$, for $P\in Pred$;
\item $||c(\varphi_1,\ldots,\varphi_n)||^{\small{\textbf{A}}}_{\mathrm{\mathbf{M}},v}=c_{\textbf{A}}(||\varphi_1||^{\small{\textbf{A}}}_{\mathrm{\mathbf{M}},v},\ldots,||\varphi_n||^{\small{\textbf{A}}}_{\mathrm{\mathbf{M}},v})$, for $c\in\mathcal{L}$;
\item $||(\forall x)\varphi||^{\small{\textbf{A}}}_{\mathrm{\mathbf{M}},v}=inf\{||\varphi||^{\small{\textbf{A}}}_{\mathrm{\mathbf{M}},v[x\rightarrow a]}\mid a\in M\}$;
\item $||(\exists x)\varphi||^{\small{\textbf{A}}}_{\mathrm{\mathbf{M}},v}=sup\{||\varphi||^{\small{\textbf{A}}}_{\mathrm{\mathbf{M}},v[x\rightarrow a]}\mid a\in M\}$.
\end{description}
If the infimum or the supremum do not exist, we take the truth value of the formula as undefined. We say that an $\textbf{A}$-structure is \emph{safe} if $||\varphi||^{\small{\textbf{A}}}_{\mathrm{\mathbf{M}},v}$ is defined for each formula $\varphi$ and each $\mathrm{\mathbf{M}}$-evaluation $v$. \end{defi}
\noindent For a set of formulas $\Phi$, we write $||\Phi||^{\emph{\textbf{A}}}_{\mathrm{\mathbf{M}},v}=1$ if $||\varphi||^{\emph{\textbf{A}}}_{\mathrm{\mathbf{M}},v}=1$ for every $\varphi\in\Phi$. We say that $\langle\emph{\textbf{A}},\mathrm{\mathbf{M}}\rangle$ is a \emph{model of a set of formulas $\Phi$} if $||\varphi ||^{\emph{\textbf{A}}}_{\mathrm{\mathbf{M}},v}=1$ for any $\varphi\in\Phi$ and any \textbf{M}-evaluation $v$. We denote by $||\varphi||^{\emph{\textbf{A}}}_{\textbf{M}}=1$ that $||\varphi||^{\emph{\textbf{A}}}_{\textbf{M},v}=1$ for all \textbf{M}-evaluation $v$. We say that a formula $\varphi$ is \emph{satisfiable} if there exists a structure $\langle\emph{\textbf{{A}}},\textbf{M}\rangle$ such that $||\varphi||^{\emph{\textbf{A}}}_{\textbf{M}}=1$. In such case, we also say that $\varphi$ is \emph{satisfied by} $\langle\emph{\textbf{{A}}}, \textbf{M}\rangle$ or that $\langle\emph{\textbf{{A}}},\textbf{M}\rangle$ \emph{satisfies $\varphi$}. Unless otherwise stated, from now on \emph{\textbf{A}} denotes an MTL-algebra and we refer to \emph{\textbf{A}}-structures simply as \emph{structures}. \smallskip
Now we recall the notion of homomorphism between fuzzy structures.
\begin{defi} {\em \textbf{\cite[Definition 6]{DeGaNo14}}} \label{def:mapping structures}
$\space$ Let $\langle\textbf{A},\mathrm{\mathbf{M}}\rangle$ and $\langle\textbf{B},\mathrm{\mathbf{N}}\rangle$ be structures, $f$ be a mapping from $\textbf{A}$ to $\textbf{B}$ and $g$ be a mapping from $M$ to $N$. The pair $\langle f,g\rangle$ is said to be a \emph{homomorphism} from $\langle\textbf{A},\mathrm{\mathbf{M}}\rangle$ to $\langle\textbf{B},\mathrm{\mathbf{N}}\rangle$ if $f$ is a homomorphism of ${L}$-algebras and for every $n$-ary function symbol $F$ and $d_1,\ldots,d_n\in M$,
$$g(F_{\mathrm{\mathbf{M}}}(d_1,\ldots,d_n))=F_{\mathrm{\mathbf{N}}}(g(d_1),\ldots,g(d_n)) $$
\noindent and for every $n$-ary predicate symbol $P$ and $d_1,\ldots,d_n\in M$,
$$ \text{ \emph{(*) }} \text{If }P_{\mathrm{\mathbf{M}}}(d_1,\ldots,d_n)=1 \text{, then } P_{\mathrm{\mathbf{N}}}(g(d_1),\ldots,g(d_n))=1.$$
We say that a homomorphism $\langle f,g\rangle$ is \emph{strict} if instead of \emph{(*)} it satisfies the stronger condition: for every $n$-ary predicate symbol $P$ and $d_1,\ldots,d_n\in M$,
$$P_{\mathrm{\mathbf{M}}}(d_1,\ldots,d_n)=1 \text{ if and only if } P_{\mathrm{\mathbf{N}}}(g(d_1),\ldots,g(d_n))=1.$$
\noindent Moreover we say that $\langle f,g\rangle$ is an \emph{embedding} if it is a strict homomorphism and both functions $f$ and $g$ are injective. And we say that an embedding $\langle f,g\rangle$ is an \emph{isomorphism} if both functions $f$ and $g$ are surjective.
\end{defi}
\section{Horn clauses}
\label{Horn clauses}
In this section we present a definition of Horn clause over predicate fuzzy logics that extends the classical definition in a natural way. In classical predicate logic, a \emph{basic Horn formula} is a formula of the form $ \alpha_{1}\wedge\dotsb \wedge\alpha_{n}\rightarrow\beta$, where $n\in\mathbb{N}$ and $\alpha_1,\ldots,\alpha_n,\beta$ are atomic formulas. Now we extend these definitions to work with predicate fuzzy logics. Observe that there is not a unique way to extend them due to the fact that, in predicate fuzzy logic, we have different conjunctions and implications.
\begin{defi}[Basic Horn Formula]\label{strong basic}A \emph{basic Horn formula} is a formula of the form \begin{equation} \label{1}
\alpha_1\&\dotsb\&\alpha_n\rightarrow\beta \hfill
\end{equation}
where $n\in\mathbb{N}$, $\alpha_1,\ldots,\alpha_n, \beta$ are atomic formulas.
\end{defi}
The formula obtained by substitution in expression (\ref{1}) of the strong conjunction $\&$ by the weak conjunction $\wedge$ will be called \emph{basic weak Horn formula}. From now on, for the sake of clarity, we will refer to the basic weak Horn formulas as \emph{basic w-Horn formulas}.
Analogously to classical logic, disjunctive definitions of basic Horn formulas can be defined. Nevertheless, it is an easy exercise to check that, for predicate fuzzy logics, these disjunctive forms are not in general equivalent to the implicational ones that we have introduced here. Here we focus our analysis on the implicational Horn clauses and we leave for future work the study of the properties of disjunctive Horn clauses.
\begin{defi}
\label{qf Horn} A \emph{quantifier-free Horn formula} is a formula of the form \newline $\phi_1\&\dotsb\&\phi_m$ where $m\in\mathbb{N}$ and $\phi_i$ is a basic Horn formula for every $1\leq i\leq m$. If $\phi_i$ is a basic w-Horn formula for every $1\leq i\leq m$, we say that $\phi_1\wedge\dotsb\wedge\phi_m$ is a \emph{quantifier-free w-Horn formula}.
\end{defi}
From now on, whenever it is possible, we present a unique definition for both the strong and the weak version, we use the \emph{w-} symbol into parenthesis.
\begin{defi}\label{Horn}A \emph{(w-)Horn formula} is a formula of the form $Q\gamma$, where $Q$ is a (possibly empty) string of quantifiers $(\forall x),(\exists x)$... and $\gamma$ is a quantifier-free (w-)Horn formula. A \emph{(w-)Horn clause} (or \emph{universal (w-)Horn formula}) is a (w-)Horn formula in which the quantifier prefix (if any) has only universal quantifiers. A \emph{(w-)universal Horn theory} is a set of (w-)Horn clauses.
\end{defi}
Observe that, in classical logic, the formula $(\forall x)\varphi \wedge (\forall x) \psi$ is logically equivalent to $(\forall x)(\varphi \wedge \psi).$ This result can be used to prove that every Horn clause is equivalent in classical logic to a conjunction of formulas of the form $(\forall x_1) \ldots (\forall x_k)\varphi$, where $\varphi$ is a basic Horn formula. Having in mind these equivalences, it is easy to see that the set of all Horn clauses is recursively defined in classical logic by the following rules: \begin{itemize}
\item[1.] If $\varphi$ is a basic Horn formula, then $\varphi$ is a Horn clause;
\item[2.] If $\varphi$ and $\psi$ are Horn clauses, then $\varphi\wedge\psi$ is a Horn clause;
\item[3.] If $\varphi$ is a Horn clause, then $(\forall x)\varphi$ is a Horn clause.
\end{itemize}
In MTL$\forall^m$ we can deduce $(\forall x)\varphi \wedge (\forall x) \psi\leftrightarrow(\forall x)(\varphi \wedge \psi)$. This fact allows us to show that in MTL$\forall^m$, any w-Horn clause is equivalent to a weak conjunction of formulas of the form $(\forall x_1)\dotsb(\forall x_k)(\varphi)$ where $\varphi$ is a basic w-Horn formula. Thus, w-Horn clauses can be recursively defined in MTL$\forall^m$ as above. But it is not the case for the strong conjunction since $(\forall x)\varphi \& (\forall x) \psi\leftrightarrow(\forall x)(\varphi \& \psi)$ can not be deduced from MTL$\forall^m$ (we refer to \cite[Remark p.281]{EsGo01}). So the set of Horn clauses is not recursively defined in MTL$\forall^m$.
\section{Term structures associated to a set of formulas}
\label{Term structure associated to a set of formulas}
In this section we introduce the notion of term structure associated to a set of formulas over predicate fuzzy logics. We study the particular case of sets of universal Horn formulas and prove that the term structure associated to these sets of formulas is free. Term structures have been used in classical logic to prove the satisfiability of a set of consistent sentences, see for instance \cite[Ch.5]{EbiFlu94}. From now on we assume that we work in a language with a binary predicate symbol $\approx$ interpreted as a similarity. We assume also that the axiomatization of the logic $L\forall^m$ contains the following axioms for $\approx$.
\begin{defi} $\emph{\textbf{ \cite[Definitions 5.6.1, 5.6.5]{Ha98}}}$ \label{def similarity}
Let $\approx$ be a binary predicate symbol, the following are the axioms of similarity and congruence:
\begin{itemize}
\item[S1.] $(\forall x)x\approx x$
\item[S2.] $(\forall x)(\forall y)(x\approx y\rightarrow y\approx x$)
\item[S3.] $(\forall x)(\forall y)(\forall z)(x\approx y \& y\approx z\rightarrow x\approx z)$ \end{itemize}
\begin{itemize}
\item[C1.] For each $n$-ary function symbol $F$, \end{itemize} {\footnotesize
$(\forall x_1)\dotsb(\forall x_n)(\forall y_1)\dotsb(\forall y_n)(x_1\approx y_1\&\dotsb \& x_n\approx y_n\rightarrow F(x_1,\ldots,x_n)\approx F(y_1,\ldots,y_n))$
}
\begin{itemize}
\item[C2.] For each $n$-ary predicate symbol $P$, \end{itemize} {\footnotesize
$(\forall x_1)\dotsb(\forall x_n)(\forall y_1)\dotsb(\forall y_n)(x_1\approx y_1\&\dotsb \& x_n\approx y_n\rightarrow (P(x_1, \ldots, x_n)\leftrightarrow P(y_1,\ldots, y_n)))$ }
\end{defi}
\begin{defi}\label{relacio}
Let $\Phi$ be a set of formulas, we define a binary relation on the set of terms, denoted by $\sim$, in the following way: for every terms $t_1,t_2$,
\begin{center}
$t_1\sim t_2$ if and only if $\Phi\vdash t_1\approx t_2$.
\end{center}
\end{defi}
By using \cite[Prop.1(5)]{EsGo01}, it is easy to check that for every set of formulas $\Phi$, $\sim$ is an equivalence relation. From now on we denote by $\overline{t}$ the $\sim$-class of the term $t$. The next result, which states that $\sim$ is compatible with the symbols of the language, can be easily proven using the Axioms of Congruence of Definition \ref{def similarity}.
\begin{lemma} \label{f} Let $\Phi$ be a set of formulas. The relation $\sim$ has the following property: if for every $1\leq i\leq n$, $t_i\sim t'_i$, then
\begin{itemize}
\item[(i)] For any $n$-ary function symbol $F$, $F(t_1,\ldots,t_n)\sim F(t'_1,\ldots,t'_n)$,
\item[(ii)] For any $n$-ary predicate symbol $P$, \small{
$\Phi\vdash P(t_1,\ldots, t_n)$ \text{iff} $\Phi\vdash P(t'_1, \ldots, t'_n)$}
\end{itemize}
\end{lemma}
\smallskip
\begin{defi} [Term Structure] \label{structure}
Let $\Phi$ be a consistent set of formulas. We define the following structure $\langle\textbf{B},\mathrm{\mathbf{T}}^{\Phi}\rangle$, where $\textbf{B}$ is the two-valued Boolean algebra, $\mathrm{\mathbf{T}}^{\Phi}$ is the set of all equivalence classes of the relation $\sim$ and
\begin{itemize}
\item For any $n$-ary function symbol $F$,
$$F_{\mathrm{\mathbf{T}}^{\Phi}}(\overline{t}_1,\ldots,\overline{t}_n)=\overline{F(t_1,\ldots,t_n)}$$
\item For any $n$-ary predicate symbol $P$,
$$ P_{\mathrm{\mathbf{T}}^{\Phi}}(\overline{t}_1,\ldots,\overline{t}_n)=\begin{cases} 1, & \mbox{if } \Phi\vdash P(t_1,\ldots, t_n) \\ 0, & \mbox{otherwise } \end{cases} $$
\end{itemize}
We call $\langle\textbf{B},\mathrm{\mathbf{T}}^{\Phi}\rangle$ the \emph{term structure associated to $\Phi$}.
\end{defi}
Notice that for every $0$-ary function symbol $c$, $c_{\mathrm{\mathbf{T}}^{\Phi}}=\overline{c}$. By using Lemma \ref{f}, it is easy to prove that the structure $\langle\emph{\textbf{B}},\mathrm{\mathbf{T}}^{\Phi}\rangle$ is well-defined, because the conditions are independent from the choice of the representatives. Observe that, so defined, $\langle\emph{\textbf{B}},\mathrm{\mathbf{T}}^{\Phi}\rangle$ is a classical structure. The following lemma agrees with this classical character.
\begin{lemma} \label{crisp}
Let $\Phi$ be a consistent set of formulas. The interpretation of the $\approx$ symbol in the structure $\langle\textbf{B},\mathrm{\mathbf{T}}^{\Phi}\rangle$ is the crisp equality.
\end{lemma}
\begin{proof}
Let $t_1,t_2$ be terms. We have $\overline{t}_1=\overline{t}_2$ iff $t_1\sim t_2$ iff $\Phi\vdash t_1\approx t_2$ iff
$\overline{t_1}\approx_{\mathrm{\mathbf{T}}^{\Phi}} \overline{t_2}$ (this last step by Definition \ref{structure}). \end{proof}
\bigskip
Now we prove some technical lemmas that will allow us to show that the term structrure $\langle\emph{\textbf{B}},\mathrm{\mathbf{T}}^{\Phi}\rangle$ is free.
\begin{defi} \label{canonical}
Given a consistent set of formulas $\Phi$, let $e^{\Phi}$ be the following $\emph{\textbf{T}}^{\Phi}$-evaluation: $e^{\Phi}(x)=\overline{x}$. We call $e^{\Phi}$ the \emph{canonical evaluation of} $\langle\textbf{B},\mathrm{\mathbf{T}}^{\Phi}\rangle$.
\end{defi}
\begin{lemma} \label{terms and atomic formulas}
Let $\Phi$ be a consistent set of formulas, the following holds:
\begin{itemize}
\item[(i)] For any term $t$, $|| t ||^{\textbf{B}}_{\mathrm{\mathbf{T}}^{\Phi},e^{\Phi}}=\overline{t}$.
\item[(ii)] For any atomic formula $\varphi$, $|| \varphi||^{\textbf{B}}_{\mathrm{\mathbf{T}}^{\Phi},e^{\Phi}}=1$ if and only if $\Phi\vdash\varphi$.
\item[(iii)] For any atomic formula $\varphi$, $|| \varphi||^{\textbf{B}}_{\mathrm{\mathbf{T}}^{\Phi},e^{\Phi}}=0$ if and only if $\Phi\not \vdash\varphi$.
\end{itemize}
\end{lemma}
\begin{proof}
(i) By induction on the complexity of $t$ and Definitions \ref{structure} and \ref{canonical}. \newline
(ii) Let $P$ be an $n$-ary predicate symbol and $t_1,\ldots,t_n$ be terms, we have:
\medskip$\begin{array}{rr} ||P(t_1,\ldots,t_n)||^{\emph{\textbf{B}}}_{\mathrm{\mathbf{T}}^{\Phi},e^{\Phi}}=1 & \text{iff}
\\[2ex] P_{\mathrm{\mathbf{T}}^{\Phi}}(|| t_1 ||^{\emph{\textbf{B}}}_{\mathrm{\mathbf{T}}^{\Phi},e^{\Phi}},\ldots,|| t_n ||^{\emph{\textbf{B}}}_{\mathrm{\mathbf{T}}^{\Phi},e^{\Phi}})=1 & \text{iff}
\\[2ex] P_{\mathrm{\mathbf{T}}^{\Phi}}(\overline{t}_1,\ldots,\overline{t}_n)=1 & \text{iff}
\\[2ex] \Phi\vdash P(t_1,\ldots, t_n) & \end{array}$
\medskip \noindent The second equivalence is by (i) of the present Lemma, and the third one by Definition \ref{structure}. (iii) holds because $\langle\emph{\textbf{B}},\mathrm{\mathbf{T}}^{\Phi}\rangle$ is a classical structure. \end{proof}
\bigskip
Observe that, since terms are the smallest significance components of a first-order language, Lemma \ref{terms and atomic formulas} (ii) and (iii) can be read as saying that term structures are minimal with respect to atomic formulas. Intuitively speaking, the term structure picks up the positive atomic information associated to $\Phi$.
\smallskip
\begin{lemma} \label{generates} Let $\Phi$ be a consistent set of formulas. The set $\{\overline{x}\mid x\in Var\}$ generates the universe $T^{\Phi}$ of the term structure associated to $\Phi$.
\end{lemma}
\begin{proof}
Let $\overline{t(x_1,\ldots,x_n)}\in T^{\Phi}$. By Lemma \ref{terms and atomic formulas}, $$\overline{t(x_1,\ldots,x_n)}=||t(x_1,\ldots,x_n) ||^{\emph{\textbf{B}}}_{\mathrm{\mathbf{T}}^{\Phi},e^{\Phi}}$$ and by the semantics of predicate fuzzy logics (Definition \ref{evaluation}), \begin{center}
$||t(x_1,\ldots,x_n) ||^{\emph{\textbf{B}}}_{\mathrm{\mathbf{T}}^{\Phi},e^{\Phi}}=t_{T^{\Phi}}(||x_1||^{\emph{\textbf{B}}}_{\mathrm{\mathbf{T}}^{\Phi},e^{\Phi}},\ldots,||x_n ||^{\emph{\textbf{B}}}_{\mathrm{\mathbf{T}}^{\Phi},e^{\Phi}})=t_{\mathrm{\mathbf{T}}^{\Phi}}(\overline{x}_1,\ldots,\overline{x}_n)$. \end{center} \end{proof}
\bigskip
Term structures do not necessarily satisfy the theory to which they are associated. In classical logic, if it is the case, from an algebraic point of view, the minimality of the term structure is revealed by the fact that the structure is \emph{free}. A model of a theory is free if there is a unique homomorphism from this model to any other model of the theory. Free structures have their origin in category theory, as a generalization of free groups (for a definition of free structure in category theory, see \cite[Def. 4.7.17]{BaWe98}). Free structures are also named \emph{initial} in \cite[Def. 2.1 (i)]{Mak87}. In the context of computer science, they appeared for the first time in \cite{GoThWaWr75}.
The possibility given by fuzzy logic of defining the term structure associated to a theory using the similarity symbol $\approx$ leads us to a notion of free structure restricted to the class of reduced models of that theory, as we will prove in next theorem. Remember that \emph{reduced structures} are those whose Leibniz congruence is the identity. By \cite[Lemma 20]{De12}, a structure $\langle\emph{\textbf{A}},\mathrm{\mathbf{M}}\rangle$ is reduced iff it has the \emph{equality property} (EQP) (that is, for any $d,e\in M$,
$d\approx_{\mathrm{\mathbf{M}}} e$ iff $d=e$).
\begin{theorem} \label{initial model}
Let $\Phi$ be a consistent set of formulas with $|| \Phi||^{\textbf{B}}_{\mathrm{\mathbf{T}}^{\Phi},e^{\Phi}}=1$. Then, $\langle\textbf{B},\mathrm{\mathbf{T}}^{\Phi}\rangle$ is a free structure in the class of the reduced models of $\Phi$, i.e., for every reduced structure $\langle\textbf{A},\mathrm{\mathbf{M}}\rangle$ and every evaluation $v$ such that $|| \Phi||^{\textbf{A}}_{\mathrm{\mathbf{M}},v}=1$, there is a unique homomorphism $\langle f,g\rangle$ from $\langle\textbf{B},\mathrm{\mathbf{T}}^{\Phi}\rangle$ to $\langle\textbf{A},\mathrm{\mathbf{M}}\rangle$ such that for every $x \in Var$, $g(\overline{x})=v(x)$.
\end{theorem}
\begin{proof} Let $\langle\emph{\textbf{A}},\mathrm{\mathbf{M}}\rangle$ be a reduced structure and $v$ an $\textbf{M}$-evaluation such that $|| \Phi||^{\emph{\textbf{A}}}_{\mathrm{\mathbf{M}},v}=1$. Now let $f:\emph{\textbf{B}}\rightarrow\emph{\textbf{A}}$ be the identity and define $g$ by: $g(\overline{t})=|| t ||^{\emph{\textbf{A}}}_{\mathrm{\mathbf{M}},v}$ for every term $t$. We show that $\langle f,g\rangle$ is the desired homomorphism (for the definition of homomorphism see the Preliminaries section, Definition \ref{def:mapping structures}).
First let us check that $g$ is well-defined. Given terms $t_1,t_2$ with $\overline{t}_1=\overline{t}_2$, that is, $t_1\sim t_2$, by Definition \ref{relacio}, $\Phi\vdash t_1\approx t_2$. Then, since $|| \Phi||^{\emph{\textbf{A}}}_{\mathrm{\mathbf{M}},v}=1$, we have $|| t_1\approx t_2||^{\emph{\textbf{A}}}_{\mathrm{\mathbf{M}},v}=1$. But $\langle\emph{\textbf{A}},\mathrm{\mathbf{M}}\rangle$ is reduced, which by \cite[Lemma 20]{De12} is equivalent to have the EQP; therefore $|| t_1 ||^{\emph{\textbf{A}}}_{\mathrm{\mathbf{M}},v}=|| t_2||^{\emph{\textbf{A}}}_{\mathrm{\mathbf{M}},v}$, that is, $g(\overline{t_1})=g(\overline{t_2})$.
Now, let us see that $g$ is a homomorphism. Let $\overline{t}_1,\ldots,\overline{t}_n\in T^{\Phi}$ be terms and $F$ be an $n$-ary function symbol. By Definition \ref{structure}, we have that $$F_{\mathrm{\mathbf{T}}^{\Phi}}(\overline{t}_1,\ldots,\overline{t}_n)=\overline{F(t_1,\ldots,t_n)}$$ and then
$g(F_{\mathrm{\mathbf{T}}^{\Phi}}(\overline{t}_1,\ldots,\overline{t}_n))=g(\overline{F(t_1,\ldots,t_n)})=|| F(t_1,\ldots,t_n) ||^{\emph{\textbf{A}}}_{\mathrm{\mathbf{M}},v}= \newline =F_{\textbf{M}}(|| t_1||^{\emph{\textbf{A}}}_{\mathrm{\mathbf{M}},v},\ldots ,|| t_n ||^{\emph{\textbf{A}}}_{\mathrm{\mathbf{M}},v})= F_{\textbf{M}}(g(\overline{t}_1),\ldots,g(\overline{t}_n))$.
\bigskip
Let $P$ be an $n$-ary predicate symbol such that $P_{\mathrm{\mathbf{T}}^{\Phi}}(\overline{t}_1,\ldots,\overline{t}_n)=1$. By Definition \ref{structure}, $\Phi\vdash P(t_1,\ldots,t_n)$. Since $|| \Phi||^{\emph{\textbf{A}}}_{\mathrm{\mathbf{M}},v}=1$, we have $$|| P(t_1,\ldots,t_n)||^{\emph{\textbf{A}}}_{\mathrm{\mathbf{M}},v}=1$$ and then $P_{\mathrm{\mathbf{M}}}(|| t_1||^{\emph{\textbf{A}}}_{\mathrm{\mathbf{M}},v},\ldots ,|| t_n ||^{\emph{\textbf{A}}}_{\mathrm{\mathbf{M}},v})=1$, that is, $P_{\mathrm{\mathbf{M}}}(g(\overline{t}_1),\ldots,g(\overline{t}_n))=1$.
\medskip
Finally, since by Lemma \ref{generates} the set $\{\overline{x}\mid x\in Var\}$ generates the universe $T^{\Phi}$ of the term structure associated to $\Phi$, $\langle f,g\rangle$ is the unique homomorphism such that for every $x \in Var$, $g(\overline{x})=v(x)$. \end{proof}
\bigskip
Observe that in languages in which the similarity symbol is interpreted by the crisp identity, by using an analogous argument to the one in Theorem \ref{initial model}, we obtain that the term structure is free in all the models of the theory and not only in the class of reduced models.
\bigskip
To end this section we prove that the term structure associated to a universal Horn theory is a model of this theory. We have shown above in Section \ref{Horn clauses} that the set of Horn clauses is not recursively defined in MTL$\forall^m$. For that reason we will present here proofs that differ from the proofs of the corresponding results in classical logic, using induction on the rank of a formula instead of induction on the set of the (w-)Horn clauses. We introduce first the notion of \emph{rank of a formula} $\varphi$. Our definition is a variant of the notion of \emph{syntactic degree of a formula} in
\cite[Definition 5.6.7]{Ha98}).
\begin{description}
\item $rk(\varphi)=0$, if $\varphi$ is atomic;
\item $rk(\neg\varphi)=rk((\exists x)\varphi)=rk((\forall x)\varphi)=rk(\varphi)+1$;
\item $rk(\varphi\circ\psi)=rk(\varphi)+rk(\psi)$, for every binary propositional connective $\circ$.
\end{description}
\begin{lemma} \label{Horn substitucio}
Let $\varphi$ be a (w-)Horn clause where $x_1,\ldots,x_m$ are pairwise distinct free variables. Then, for every terms $t_1,\ldots,t_m$, $$\varphi (t_1,\ldots,t_m/x_1,\ldots,x_m)$$ is a (w-)Horn clause.
\end{lemma}
\begin{proof}
We prove it for the strong conjunction but the proof is analogous for the weak conjunction. By induction on $rk(\varphi)$.
\bigskip
\underline{Case $rk(\varphi)=0$}. If $\varphi$ is a basic Horn formula of the form $\psi_1\& \ldots \&\psi_n\rightarrow\psi$, it is clear that $\varphi (t_1,\ldots,t_m/x_1,\ldots,x_m)$ is still a basic Horn formula. In case that $\varphi=\phi_1\& \dotsb\&\phi_l$ is a conjunction of basic Horn formulas, note that $\varphi (t_1,\ldots,t_m/x_1,\ldots,x_m)$ has the same form as $\varphi$.
\bigskip
\underline{Case $rk(\varphi)=n+1$}. Assume inductively that for any Horn clause $\psi$ where $x_1,\ldots,x_m$ are pairwise distinct free variables in $\psi$ and whose rank is $n$, the formula $\psi (t_1,\ldots,t_m/x_1,\ldots,x_m)$ is a Horn clause. Let $\varphi$ be a Horn clause of rank $n+1$, then $\varphi$ is of the form $(\forall y)\psi$, where $\psi$ has rank $n$. Assume without loss of generality that and $y\not\in\{x_1,\ldots,x_m\}$, then$$[(\forall y)\psi](t_1,\ldots,t_m/x_1,\ldots,x_m)=(\forall y)[\psi(t_1,\ldots,t_m/x_1,\ldots,x_m)]$$ thus we can apply the inductive hypothesis to obtain the desired result. \end{proof}
\begin{theorem} \label{theorem Horn formulas}
Let $\Phi$ be a consistent set of formulas. For every (w-)Horn clause $\varphi$, if $\Phi\vdash\varphi$, then $|| \varphi||^{\textbf{B}}_{\mathrm{\mathbf{T}}^{\Phi},e^{\Phi}}=1$.
\end{theorem}
\begin{proof} We prove it for the strong conjunction but the proof is analogous for the weak conjunction. By induction on $rk(\varphi)$.
\bigskip
\underline{Case $rk(\varphi)=0$.} We can distinguish two subcases:
\medskip
1) If $\varphi=\psi_1\&\dotsb\&\psi_n\rightarrow\psi$ is a basic Horn formula, we have to show that $||\psi_1\&\dotsb\&\psi_n||^{\emph{\textbf{B}}}_{\mathrm{\mathbf{T}}^{\Phi},e^{\Phi}}\leq|| \psi||^{\emph{\textbf{B}}}_{\mathrm{\mathbf{T}}^{\Phi},e^{\Phi}}$. If $|| \psi||^{\emph{\textbf{B}}}_{\mathrm{\mathbf{T}}^{\Phi},e^{\Phi}}=1$, we are done. Otherwise, by Definition \ref{structure}, $\Phi\not \vdash \psi$. Consequently, since $\Phi\vdash \psi_1\&\dotsb\&\psi_n\rightarrow\psi$, $\Phi\not \vdash \psi_1\&\dotsb\&\psi_n$ and thus for some $1 \leq i \leq n$, $\Phi\not \vdash \psi_i$. By Lemma \ref{terms and atomic formulas} (ii), we have $|| \psi_i||^{\emph{\textbf{B}}}_{\mathrm{\mathbf{T}}^{\Phi},e^{\Phi}}=0$ and then $||\psi_1\&\dotsb\&\psi_n||^{\emph{\textbf{B}}}_{\mathrm{\mathbf{T}}^{\Phi},e^{\Phi}}=0$. Therefore, we can conclude $||\psi_1\&\dotsb\&\psi_n||^{\emph{\textbf{B}}}_{\mathrm{\mathbf{T}}^{\Phi},e^{\Phi}}\leq|| \psi||^{\emph{\textbf{B}}}_{\mathrm{\mathbf{T}}^{\Phi},e^{\Phi}}$. Note that if $n=0$, $\varphi$ is an atomic formula and the property holds by Lemma \ref{terms and atomic formulas} (ii).
\smallskip
2) If $\varphi=\psi_1\&\dotsb\&\psi_n$ is a conjunction of basic Horn formulas and $\Phi\vdash\varphi$, then for every $1 \leq i \leq n$, $\Phi\vdash \psi_i$. Thus, by 1), for every $1 \leq i \leq n$, $||\psi_i||^{\emph{\textbf{B}}}_{\mathrm{\mathbf{T}}^{\Phi},e^{\Phi}}=1$ and then $|| \varphi||^{\emph{\textbf{B}}}_{\mathrm{\mathbf{T}}^{\Phi},e^{\Phi}}=1$.
\bigskip
\underline{Case $rk(\varphi)=n+1$.}
\medskip
If $\varphi=(\forall x)\phi(x)$ is a Horn clause, where $rk(\phi(x))=n$ and $\Phi\vdash\varphi$, by Axiom \emph{$\forall 1$} of $L\forall^m$, for every term $t$, $\Phi\vdash\phi(t/x)$. Since by Lemma \ref{Horn substitucio}, $\phi(t/x)$ is also a Horn clause and $rk(\phi(t/x))=n$, we can apply the inductive hypothesis and hence for every term $t$, $||\phi(t/x)||^{\emph{\textbf{B}}}_{\mathrm{\mathbf{T}}^{\Phi},e^{\Phi}}=1$, that is, by Lemma \ref{terms and atomic formulas} (i), for every element $\overline{t}$ of the domain, $||\phi(x)||^{\emph{\textbf{B}}}_{\mathrm{\mathbf{T}}^{\Phi},e^{\Phi}(x\rightarrow\overline{t})}=1$. Therefore, we can conclude that $||(\forall x)\phi(x)||^{\emph{\textbf{B}}}_{\mathrm{\mathbf{T}}^{\Phi},e^{\Phi}}=1$.\end{proof}
\bigskip
Observe that the inverse direction of Theorem \ref{theorem Horn formulas} is not true. Assume that we work in G\"odel predicate fuzzy logic G$\forall$. Let $P$ be a $1$-ary predicate symbol, $\overline{c}$ be an individual constant, $\Phi=\{\neg(P\overline{c}\rightarrow\overline{0})\}$ and $\varphi=P\overline{c}\rightarrow\overline{0}$. Now we show that $|| \varphi||^{\emph{\textbf{B}}}_{\mathrm{\mathbf{T}}^{\Phi}}=1$, but $\Phi\not \vdash\varphi$. First, in order to show that $\Phi\not \vdash\varphi$, consider a G-algebra $\emph{\textbf{A}}$ with domain the real interval $[0,1]$ and a structure $\langle\emph{\textbf{A}},\mathrm{\mathbf{M}}\rangle$ such that $||P\overline{c}||^{\emph{\textbf{A}}}_{\mathrm{\mathbf{M}}}=0.8$, then we have that $||\Phi||^{\emph{\textbf{A}}}_{\mathrm{\mathbf{M}}}=1$ and $||P\overline{c}\to \overline{0}||^{\emph{\textbf{A}}}_{\mathrm{\mathbf{M}}}\neq1$ consequently $\Phi\not\vdash_GP\overline{c}\to \overline{0}$. Using the same structure we obtain also that $\Phi\not\vdash_GP\overline{c}$. Finally, since $\Phi\not\vdash_GP\overline{c}$, by Lemma \ref{terms and atomic formulas}, $||P\overline{c}||^{\emph{\textbf{B}}}_{\mathrm{\mathbf{T}}^{\Phi}}=0$ and then $||\varphi||^{\emph{\textbf{B}}}_{\mathrm{\mathbf{T}}^{\Phi}}=1$.
\bigskip
Remark that, as a corollary of Theorem \ref{theorem Horn formulas}, we have that the substructure of $\langle\emph{\textbf{B}},\mathrm{\mathbf{T}}^{\Phi}\rangle$ generated by the set of ground terms is also a model for all universal Horn sentences that are consequences of the theory. Another important corollary of Theorem \ref{theorem Horn formulas} is the following:
\begin{corollary} \label{classic} Every consistent set of (w-)Horn clauses without free variables has a classical model.
\end{corollary}
Observe that Corollary \ref{classic} is not true in general. The consistent sentence $\neg (\overline{1} \to Pa) \& \neg (Pa \to \overline{0})$ has no classical model.
\section{Herbrand Structures}
\label{Herbrand Structures}
In this section we introduce Herbrand structures for fuzzy universal Horn theories. They are a prominent form of term structures, specially helpful when dealing with sets of equality-free formulas (that is, formulas in which the symbol $\approx$ does not occur), the reason is that, as it is shown below in Lemma \ref{equality}, no non-trivial equations are derivable from a set of equality-free formulas. In classical logic, Herbrand structures have been used to present a simplified version of a term structure associated to a consistent theory \cite[Ch.11]{EbiFlu94}, and they have also a relevant role in the foundation of logic programming (see for instance \cite{DoPo10}). Regarding Herbrand structures in fuzzy logic programming, we refer to the works \cite{Ge05,Voj01,Ebra01}. Throughout this section we assume that the symbol $\approx$ is interpreted always as the crisp identity and that there is at least an individual constant in the language.
\begin{lemma} \label{equality} Let $\Phi$ be a consistent set of equality-free formulas, then for every terms $t_1,t_2$,
$$ \text{If }\Phi\vdash t_1\approx t_2, \text{ then }t_1=t_2.$$
\end{lemma}
\begin{proof}
Assume that $\Phi$ is a consistent set of equality-free formulas and $\Phi\vdash t_1\approx t_2$ for terms $t_1,t_2$ of the language. Since CL$\forall$ is an extension of MTL$\forall^m$, $\Phi\vdash t_1\approx t_2$ in CL$\forall$. Then, by the analogous classical result \cite[Ch. 11, Th. 3.1]{EbiFlu94}, we have $t_1=t_2$. \end{proof}
\begin{defi} [Herbrand Structure] \label{Herbrand structure}
The \emph{Herbrand universe of a predicate language} is the set of all ground terms of the language. A \emph{Herbrand structure} is a structure $\langle\textbf{A},\emph{\textbf{H}}\rangle$, where $\emph{\textbf{H}}$ is the Herbrand universe, and: \begin{itemize}
\item[] For any individual constant symbol $c$, $c_{\emph{\textbf{H}}}=c$.
\item[] For any $n$-ary function symbol $F$ and any $t_1,\ldots,t_n\in H$, \begin{center}
\smallskip
$F_{\emph{\textbf{H}}}(t_1,\ldots,t_n)=F(t_1,\ldots,t_n)$
\end{center}
\end{itemize}
\end{defi}
Observe that in Definition \ref{Herbrand structure} no restrictions are placed on the interpretations of the predicate symbols and on the algebra we work over. The canonical models $\langle\emph{\textbf{Lind}}_{T},\mathbf{CM}(T)\rangle$ introduced in {\cite[Def.9]{CiHa06} are examples of Herbrand structures. In these structures $\emph{\textbf{Lind}}_{T}$ is the Lindenbaum algebra of a theory $T$ and the domain of $\mathbf{CM}(T)$ is the set of individual constants (the language in \cite{CiHa06} do not contain function symbols). Now we introduce a particular case of Herbrand structure and we show that every consistent Horn clause without free variables has a model of this kind.
\begin{defi} [H-structure and H-model] \label{Herbrand boolean} Let $\overline{H}$ be the set of all \newline
equality-free sentences of the form $P(t_1,\ldots,t_n)$, where $t_1,\ldots,t_n$ are ground terms, $n\geq 1$ and $P$ is an $n$-ary predicate symbol. For every subset $H$ of $\overline{H}$, we define the Herbrand structure $\langle\textbf{B},\mathbf{N}^{\emph{H}}\rangle$, where $\textbf{B}$ is the two-valued Boolean algebra, the domain $\mathbf{N}^{\emph{H}}$ is the set of all ground terms of the language, the interpretation of the function symbols is as in every Herbrand structure and the interpretation of the predicate symbols is as follows: for every $n\geq 1$ and every $n$-ary predicate symbol $P$,
$$ P_{\mathrm{\mathbf{N}}^{\emph{H}}}(t_1,\ldots,t_n)=\begin{cases} 1, & \mbox{if } P(t_1,\ldots,t_n) \in H \\ 0, & \mbox{otherwise. } \end{cases} $$
We call this type of Herbrand structures \emph{H-structures}. If $\Phi$ is a set of sentences, we say that an \emph{H}-structure is an \emph{H-model} of $\Phi$ if it is a model of $\Phi$.
\end{defi}
\begin{proposition} \label{proposition} Let $\langle\textbf{A},\mathbf{M}\rangle$ be a structure and H be the set of all atomic equality-free sentences $\sigma$ such that $||\sigma||^{\textbf{A}}_{\emph{\textbf{M}}}=1$. Then, for every equality-free sentence $\varphi$ which is a (w-)Horn clause, if $||\varphi||^{\textbf{A}}_{\emph{\textbf{M}}}=1$, then $||\varphi||^{\textbf{B}}_{\emph{\textbf{N}}^{\emph{H}}}=1$, where $\langle\textbf{B},\emph{\textbf{N}}^{\emph{H}}\rangle$ is an \emph{H}-structure as in Definition \ref{Herbrand boolean}.
\end{proposition}
\begin{proof} We prove it for the strong conjunction but the proof is analogous for the weak conjunction. Assume that $\varphi$ is an equality-free sentence which is a Horn clause and $||\varphi||^{\emph{\textbf{A}}}_{\textbf{M}}=1$. We proceed by induction on the rank of $\varphi$
\bigskip
\underline{Case $rk(\varphi)=0$.} We distinguish two cases:
\bigskip
1) If $\varphi=\psi_1\&\dotsb\&\psi_n\rightarrow\psi$ is a basic Horn formula, we have to show that $||\psi_1\&\dotsb\&\psi_n||^{\emph{\textbf{B}}}_{\textbf{N}^{\text{H}}} \leq ||\psi||^{\emph{\textbf{B}}}_{\textbf{N}^{\text{H}}}$. If $||\psi||^{\emph{\textbf{B}}}_{\textbf{N}^{\text{H}}}=1$, we are done. Otherwise, by Definition \ref{Herbrand boolean}, $\psi\not \in$ H , and thus $||\psi||^{\emph{\textbf{A}}}_{\textbf{M}}\not = 1$. Therefore, since $||\varphi||^{\emph{\textbf{A}}}_{\textbf{M}}=1$, we have that $||\psi_1\&\dotsb\&\psi_n||^{\emph{\textbf{A}}}_{\textbf{M}}\not = 1$. Consequently for some $1 \leq i \leq n$, $||\psi_i||^{\emph{\textbf{A}}}_{\textbf{M}}\not = 1$, therefore $\psi_i\not \in$ H and $||\psi_i||^{\emph{\textbf{B}}}_{\textbf{N}^{\text{H}}} =0$ and then $||\psi_1\&\dotsb\&\psi_n||^{\emph{\textbf{B}}}_{\textbf{N}^{\text{H}}} =0$. Hence, $||\psi_1\&\dotsb\&\psi_n||^{\emph{\textbf{B}}}_{\textbf{N}^{\text{H}}} \leq ||\psi||^{\emph{\textbf{B}}}_{\textbf{N}^{\text{H}}}$.
\medskip
2) If $\varphi=\psi_1\&\dotsb\&\psi_n$ is a strong conjunction of basic Horn formulas, then by 1) we have that $||\psi_i||^{\emph{\textbf{A}}}_{\textbf{M}}=1$ implies $||\psi_i||^{\emph{\textbf{B}}}_{\textbf{N}^{\text{H}}}=1$, for each $i\in\{1,\ldots,n\}$. Thus, if $||\varphi||^{\emph{\textbf{A}}}_{\textbf{M}}=1$, then $||\varphi||^{\emph{\textbf{B}}}_{\textbf{N}^{\text{H}}}=1$.
\bigskip
\underline{Case $rk(\varphi)=n+1$.}
\bigskip
Let $\varphi=(\forall x)\phi(x)$ be a Horn clause with $rk(\phi(x))=n$. Since $||\varphi||^{\emph{\textbf{A}}}_{\textbf{M}}=1$, by Axiom \emph{$\forall 1$} of $L\forall^m$, for every ground term $t$, $||\phi(t/x)||^{\emph{\textbf{A}}}_{\textbf{M}}=1$. By Lemma \ref{Horn substitucio}, $\phi(t/x)$ is also a Horn clause, and since $rk(\phi(t/x))=n$, we can apply the inductive hypothesis and hence for every ground term $t$, $||\phi(t/x)|^{\emph{\textbf{B}}}_{\textbf{N}^{\text{H}}}=1$. Finally, since $\langle\emph{\textbf{B}},\mathbf{N}^{\text{H}}\rangle$ is a Herbrand structure, we have that for every element $t$ of its domain $||\phi(t/x)||^{\emph{\textbf{B}}}_{\textbf{N}^{\text{H}}}=1$, and consequently $||(\forall x)\phi(x)||^{\emph{\textbf{B}}}_{\textbf{N}^{\text{H}}}=1$. \end{proof}
\bigskip
Notice that Proposition \ref{proposition} does not assert that given a structure $\langle\emph{\textbf{A}},\textbf{M}\rangle$, $\langle\emph{\textbf{A}},\textbf{M}\rangle$ and $\langle\emph{\textbf{B}},\textbf{N}^{\text{H}}\rangle$ satisfy exactly the same equality-free sentences which are Horn clauses. Actually, this is not true. Let $\mathcal{P}$ be a predicate language with three monadic predicate symbols $P_1,P_2,P_3$ and one individual constant $c$. Suppose that $\emph{\textbf{A}}$ is the \L ukasiewicz algebra $[0,1]_{\text{\L}}$ and let $\langle\emph{\textbf{A}},\textbf{M}\rangle$ be a structure over $\mathcal{P}$ such that $||P_1(c)||^{\emph{\textbf{A}}}_{\textbf{M}}=1$, $||P_2(c)||^{\emph{\textbf{A}}}_{\textbf{M}}=0.9$ and $||P_3(c)||^{\emph{\textbf{A}}}_{\textbf{M}}=0.5$. Let $\varphi$ be $P_1(c)\& P_2(c)\rightarrow P_3(c)$, $\varphi$ is an equality-free sentence which is a Horn clause with $||P_1(c)\& P_2(c)\rightarrow P_3(c)||^{\emph{\textbf{A}}}_{\textbf{M}}=0.6$, but if we consider its associated H-structure, $\langle\emph{\textbf{B}},\textbf{N}^{\text{H}}\rangle$, we have H$=\{P_1(c)\}$ and thus $||P_1(c)\& P_2(c)\rightarrow P_3(c)||^{\emph{\textbf{B}}}_{\textbf{N}^{\text{H}}}=1$.
\begin{corollary} \label{corollary H-model} An equality-free sentence which is a (w-)Horn clause has a model if and only if it has an \emph{H}-model.
\end{corollary}
We can conclude here, in the same sense as in Corollary \ref{classic}, that every consistent equality-free sentence which is a (w-)Horn clause has a classical Herbrand model.
\section{Discussion, Conclusions and Future work}
\label{Conclusions}
The present paper is a first step towards a systematic study of universal Horn theories over predicate fuzzy logics from a model-theoretic perspective. We have proved the existence of free models in universal Horn classes of structures. In the future we will pay special attention to the study of possible characterizations of universal Horn theories in terms of the existence of these free models and its relevance for fuzzy logic programming.
Future work will be devoted also to the analysis of the logical properties of the different definitions of Horn clauses introduced so far in the literature of fuzzy logics, for instance see \cite{BeVic06, BeVic06b, Ma99}. It is important to underline here some differences between our work and some important related references. Our paper differs from the approaches of B\v{e}lohl\'avek and Vychodil and also the one of Gerla, due to mainly three reasons: it is not restricted to fuzzy equalities, it does not adopt the Pavelka-style definition of the Horn clauses and it does not assume the completeness of the algebra. Our choice is taken because it gives more generality to the results we wanted to obtain, even if in this first work our Horn clauses are defined very basically.
We take as a future task to explore how a Pavelka-style definition of Horn clauses in the framework developed by H\'ajek \cite{Ha98} could change or even improve the results we have obtained on free models. We will follow the broad approach taken in \cite[Ch.8]{CiHaNo11} about fuzzy logics with enriched languages. Finally we will study also quasivarieties over fuzzy logic, and closure properties of fuzzy universal Horn classes by using recent results on direct and reduced products over fuzzy logic like \cite{De12}. Our next objective is to solve the open problem of characterizing theories of Horn clauses in predicate fuzzy logics, formulated by Cintula and H\'ajek in \cite{CiHa10}.
\section*{Acknowledgments} We would like to thank the referees for their useful comments. This project has received funding from the European Union's Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie grant agreement No 689176 (SYSMICS project). Pilar Dellunde is also partially supported by the project RASO TIN2015-71799-C2-1-P (MINECO/FEDER) and the grant 2014SGR-118 from the Generalitat de Catalunya.
|
2,877,628,088,956 | arxiv | \section{Introduction}
Throughout this paper, $V$ denotes a vertex operator
algebra $(\oplus_{n=0}^{\infty} V_n,Y,\1,\bw)$
with central charge $c$ and $Y(v,z)=\sum v(n)z^{-n-1}$ denotes a vertex
operator of $v$. (Abusing the notation, we will also use it
for vertex operators of $v$ for $V$-modules.)
$o(v)$ denotes the grade-keeping operator of $v$,
which is given by $v(m-1)$ for $v\in V_m$ and defined by
extending it for all elements of $V$ linearly.
In particular, $o(\bw)$ equals to $L(0)=\bw(1)$ for
Virasoro element $\bw$ of $V$ and
$o(v)=v(0)$ for $v\in V_1$. \hfill \\
In order to simplify the situation, we assume that $\dim V_0=1$ so that
there is a constant $\langle v,u\rangle \in \C$
such that $v_1u=-\langle v,u\rangle\1$ for $v, u\in V_1$.
We call $V$ a rational vertex operator algebra in case each $V$-module
is a direct sum of simple modules.
Define $C_2(V)$ to be the subspace of $V$ spanned by elements $u(-2)v$
for $u,v\in V$. We say that $V$ satisfies {\it Condition $C_2$} if
$C_2(V)$ has finite codimension in $V$.
For a $V$-module $M$ with grading $M=\oplus M_m$, we define the
formal character as
\begin{equation}
\ch_qM=q^{-c/24}\sum \dim M_mq^m={\rm tr}_M q^{-c/24+L(0)}.
\end{equation}
In this paper, we consider these functions less formally by
taking $q$ to be the usual local parameter $q=q_{\tau}=e^{2\pi \imath \tau}$
at infinity in the upper half-plane
$$ \CH=\{\tau\in \C|\Im \tau>0\}. $$
Although it is often said that a vertex operator algebra (VOA)
is a conformal field theory with
mathematically rigorous axioms, the axioms of VOA do not
assume the modular invariance. However, Zhu \cite{Z} showed the
modular ($SL_2(\Z)$) invariance of the space
\begin{equation}\label{zhu}
<q_1^{|a_1|}\cdots q_n^{|a_n|}{\rm tr}_W Y(a_1,q_1)
\cdots Y(a_n,q_n)q^{L(0)-c/24} :\ W \mbox{ irr. $V$-mod}>
\end{equation}
for a rational VOA $V$ with central charge $c$ and $a_i\in V_{|a_i|}$
under Condition $C_2$, which
are satisfied by many known examples, where $q_j=q_{z_j}=e^{2\pi \imath z_j}$
and $|a_i|$ denotes the weight of $a_j$.
For example, the space
$$<\ch_q W :\ W \mbox{ irreducible $V$-modules}> $$
is $SL_2(\Z)$-invariant.
Recently, Dong, Li and Mason
extended the Zhu's idea and proved a modular invariance of the space
\begin{equation}
<{\rm tr}_U \phi^i q^{L(0)-c/24}: i\in \Z, \ U \mbox{ $\phi$-twisted modules}>
\end{equation}
by introducing the concept of $\phi$-twisted
modules for a finite automorphism $\phi$, see \cite{DLiM}.
An easiest example of
automorphism of VOA is given by a vector $v\in V_1$ as
$\phi=e^{2\pi \imath v(0)}$. Especially,
if the eigenvalues of $o(v)$ $(=v(0))$ on modules are in ${1\over n}\Z$, then
the order of $e^{2\pi \imath o(v)}$ is finite. So for a $V$-module $W$ and
$u, v\in V_1$, we will define
\begin{equation}
Z_W(v;u;\tau)={\rm tr}_W e^{2\pi\imath (o(v)-{\langle v,u\rangle \over 2})}
q^{L(0)+o(u)-(c+12\langle u,u\rangle )/24}
\end{equation}
and we call $Z_W(v;0;\tau)\eta(\tau)^c$ a theta-function of $W$, where
$u_1u=-\langle u,u\rangle\1$ and \\
$\eta(\tau)=q^{1/24}\prod_{n=1}^{\infty}(1-q^n)$ is
the Dedekind $\eta$-function.
It is worth noting that
$c+12\langle u,u\rangle $ is the central
charge of conformal element $\bw+L(-1)u$ and
$o(\bw+L(-1)u)$ is equal to $L(0)+o(u)$, see \cite{DLnM}.
For example,
let $V$ is a lattice VOA $V_{2\Z x}$ constructed
from a $1$-dimensional lattice $L=2\Z x$ with $\langle x,x\rangle=1$.
It has exactly four irreducible modules \cite{D}:
$$W_0=V_{2\Z x}, W_1=V_{(2\Z+{1\over 2})x},
W_2=V_{(2\Z+1)x},
W_3=V_{(2\Z-{1\over 2})x}. $$
Let $\theta_{h,k}(z,\tau)$ $\left(:=\sum_{n\in \Z}\exp(\pi \imath(n+k)^2\tau
+2\pi(n+k)(z+k))\right)$ be theta functions for $h,k=0,\frac{1}{2}$.
By the construction of lattice VOA (see \cite{FLM}),
it is easy to check
\begin{equation}
\theta_{h,k}(z,\tau)
=\eta(\tau)((\imath)^{4hk}Z_{W_{2h}}(zx(-1)\1;0;\tau)
+(-1)^k(\imath)^{4hk}Z_{W_{2+2h}}(zx(-1)\1;0;\tau))
\end{equation}
for $h,k=0,\frac{1}{2}$ and
their modular transformations
\begin{equation}
\theta_{h,k}(z/\tau,-1/\tau)
=(i)^{4hk}(-\imath \tau)^{\frac{1}{2}}\exp(\pi \imath z^2/\tau)\theta_{k,h}(z,\tau)
\end{equation}
are well known, (see \cite{M}).
In particular, there are constants $A_{k}^h\in \C$ such that
\begin{equation}
Z_{W_h}(zx(-1)\1;0;-1/\tau)
=\sum A_k^h Z_{W_h}(0;zx(-1)+\frac{1}{2} z^2,\tau).
\end{equation}
Namely, the modular transformations of
$Z_{W_h}(zx(-1)\1;0:\tau)$ are expressed by linear
combinations of $Z_{W_k}(u;v;\tau)$ of (ordinary) modules $W_k$, but not
twisted modules.
By this result, for an automorphism $\phi=e^{v(0)}$, we can expect to
obtain a modular transformation by using only the ordinary modules,
which offers
some information about twisted modules.
This is the motivation of this paper and we will actually show that the above
result is generally true, that is,
we will prove that the following modular transformation by using
Zhu's result (2). \hfill \\
\vspace{5mm}
\noindent
{\bf Main Theorem} \qquad
Let $V$ be a rational vertex operator algebra with
the irreducible modules $\{W_i:i=1,...,m \}$ and
$u,v\in V_1$.
Assume $v(0)v=v(0)u=u(0)v=u(0)u=0$ and $v(1)v,v(1)u,u(1)u\in {\C}\1$.
If $V$ satisfies Zhu's finite condition $C_2$,
then
\begin{equation}
\{ Z_{W_h}(v;u;\tau): h=1,...,m \}
\end{equation}
satisfy a modular invariance, i.e., for
$\al=\pmatrix{a&b\cr f&d}\in SL_2(\Z)$, there are constants
$A_{\al,k}^h$ (see Theorem 4.1) such that
\begin{equation}
Z_{W_h}(v;u;{a\tau+b\over f\tau+d})
=\sum_{k=1}^m A^h_{\al,k}Z_{W_k}(av+bu;fv+du;\tau).
\end{equation}
\section{VOA}
A {\it vertex operator algebra} is a $\Z$-graded vector space:
\begin{equation}
V=\oplus_{n\in \Z}V_n
\end{equation}
satisfying $\dim V_n<\infty$ for all $n$ and $V_n=0 \mbox{ for } n<<0$,
equipped with a linear map
\begin{eqnarray*}
&V \to &({\rm End} V)[[z,z^{-1}]] \hfill \\
&v \to &Y(v,z)=\sum_{n\in \Z}v(n)z^{-n-1}
\end{eqnarray*}
and with two distinguished vectors, {\it vacuum element} $\1\in V_0$ and
{\it conformal vector} $\bw\in V_2$ satisfying
the following conditions for $u,v\in V$: \hfill \\
\begin{eqnarray*}
&&u(n)v=0 \quad \mbox{ for $n$ sufficiently large}; \hfill \\
&&Y(\1,z)=1; \hfill \\
&&Y(v,z)\1\in V[[z]] \mbox{ and }\lim_{z\to 0}Y(v,z)\1=v; \hfill \\
&&(z-x)^NY(v,z)Y(u,x)=(z-x)^NY(u,x)Y(v,z) \quad
\mbox{ for $N$ sufficiently large},
\end{eqnarray*}
where $(z_1-z_2)^n\ (n\in \Z)$ are to be expanded in nonnegative integral
powers of second variable $z_2$;
\begin{equation}
[L(m),L(n)]=(m-n)L(m+n)+{1\over 12}(m3-m)\delta_{m+n,0}c
\end{equation}
for $m,n\in \Z$, where $L(m)=\omega(m+1)$ and $c$ is called
{\it central charge};
\begin{eqnarray*}
&&L(0)v=nv \mbox{ for }v\in V_n; \hfill \\
&&{d\over dz}Y(v,z)=Y(L(-1)v,z).
\end{eqnarray*}
This completes the definition. \hfill \\
We also have the notion of modules: \hfill \\
Let $(V,Y,\1,\omega)$ be a vertex operator algebra. A {\it weak} module $W$ of
$(V,Y,\1,\omega)$ is a $\C$-graded vector space:
\begin{equation}
W=\oplus_{n\in \C}W_n
\end{equation}
equipped with a linear map
\begin{eqnarray*}
&V \to &({\rm End}(W))[[z,z^{-1}]] \hfill \\
&v \to &Y^W(v,z)=\sum_{n\in \Z}v^W(n)z^{-n-1} \qquad (v_n\in End(W))
\end{eqnarray*}
satisfying the following conditions for $u,v\in V$ and $w\in W$: \hfill \\
For $v\in V, w\in M$, $v^W(m)w=0$ for $m>>0$. \hfill \\
$$ Y^W(\1,z)=1; $$
$$ L^W(0)w=nw \mbox{ for }w\in W_n, L^W(0)=\omega^W(1); $$
$$ {d\over dz}Y^W(v,z)=Y(L(-1)v,z) $$
and the following Jacobi idenity holds.
\begin{eqnarray*}
&&z_0^{-1}\delta({z_1-z_2\over z_0})Y^W(u,z_1)Y^W(v,z_2)
-z_0^{-1}\delta({z_2-z_1\over -z_0})Y^W(v,z_2)Y^W(u,z_1) \hfill \\
&&=z_2^{-1}\delta({z_1-z_0\over z_2})Y^W(Y(u,z_0)v,z_2).
\end{eqnarray*}
A weak module $W$ is called a {\it module} if
every finitely generated weak submodule $M=\oplus_{r\in \C} M_r$ of $W$
satisfies \hfill \\
(1) $\dim M_r<\infty$, \hfill \\
(2) $M_{r+n}=0 \mbox{ for $n\in \Z$ sufficiently large, }$ \hfill \\
for any $r\in \C$.
\section{Formal power series}
We use the notation $q$ and $q_z$ to denote $e^{2\pi \imath \tau}$ and
$e^{2\pi \imath z}$,
respectively.
In this paper, the formal power series
$$ P_2(q_z,q)=
(2\pi \imath )^2\sum_{n=1}^{\infty}({nq_z^n \over 1-q^n}
+{nq_z^{-n}q^n\over 1-q^n})$$
plays an essential role,
where ${1\over 1-q^n}$ is understood as $\sum_{i=0}^{\infty}q^i$.
The limit of $P_2(q_z,q)$ (which we still denote as $P_2(q_z,q)$) relates to
$\FP(z,\tau)$ by
$$ P_2(q_z,q)=\FP(z,\tau)+G_2(\tau), $$
where
$$G_2(\tau)={\pi^2\over 2}+\sum_{m\in \Z-\{0\}}\sum_{n\in \Z}
{1\over (m\tau+n)^2}$$
is the Eisenstein series and
$\FP(z,\tau)$ is the Weierstrass $\FP$-function
$$ \FP(z,\tau)={1\over z^2}+\sum_{(m,n)\not=(0,0)}
({1\over (z-m\tau-n)^2}-{1\over (m\tau+n)^2}). $$
It is known that
$$ G_2({a\tau+b\over f\tau+d})=(f\tau+d)^2G_2(\tau)-2\pi \imath f(f\tau+d)$$
and
$$ \FP({z\over f\tau+d},{a\tau+b\over f\tau+d})=(f\tau+d)^2\FP(z,\tau).$$
In particular,
\begin{equation} P_2({z\over f\tau+d},{a\tau+b\over f\tau+d})
=(f\tau+d)^2P_2(z,\tau)-2\pi \imath f(f\tau+d).
\end{equation}
In this paper, we will use variables $\{z_1,...,z_n\}$
and calculate the products of formal power series $P_2(q_{z_i-z_j},\tau)$.
In order to simplify notation, we will use
a transposition $(i,j)$ of symmetric groups $\Sigma_n$ on
$\Omega=\{1,...,n\}$. For $\{(i_{11},i_{12}),...,(i_{t1},i_{t2})\}$ with
$i_{s1}<i_{s2}$ and $i_{ab}\not=i_{cd}$ for $(a,b)\not=(c,d)$,
we view $\sigma=(i_{11},i_{12})\cdots (i_{t1},i_{t2})$ as an
involution (element of order $2$) of $\Sigma_n$ and
denote $\prod_{j=1}^t P_2(q_{z_{t2}-z_{t1}},\tau)$ by
$\prod_{i<\sigma(i)} P_2(q_{z_{\sigma(i)}-z_i},\tau)$.
Let $I(n)$ denote the set of all elements $g$ in $\Sigma_n$ with $g^2=1$.
For $\sigma\in \Sigma_n$, set
\begin{eqnarray}
&m(\sigma)=\{i\in \Omega| \sigma(i)\not=i\} \hfill \\
&f(\sigma)=\{i\in \Omega|\sigma(i)=i\}.
\end{eqnarray}
For $\sigma_1, \sigma_2\in \Sigma_n$, $\sigma_1, \sigma_2$ are
called to be {\it disjoint} if
$m(\sigma_1)\cap m(\sigma_2)=\emptyset$.
$\sigma_1+\cdots+\sigma_n=\sigma$ expresses that
$\{\sigma_1,...,\sigma_n\}$ are mutually disjoint and
the product $\sigma_1\cdots\sigma_n$ is equal to $\sigma$.
For $v\in V_1$, $v(0)$ acts on the finite dimensional homogeneous
subspaces $V_m$ and satisfies $[v(0), v(m)]=(v(0)v)(m)$ and
\begin{eqnarray*}
&v(0)\bw&=v(0)\bw(-1)\1=-[\bw(-1),v(0)]\1
=-\sum_{i=0}^{\infty}(-1)^i(\bw(i)v)(-1-\imath )\1 \hfill \\
&&=-(\bw(0)v)(-1)\1+(\bw(1)v)(-2)\1=-v(-2)\1+(w(1)v)(-2)\1=0.
\end{eqnarray*}
Therefore,
$e^{v(0)}=\sum_{n=0}^{\infty} {v(0)^n \over n!}$ is well defined
and satifies that
$e^{v(0)}\bw=\bw$ and
$e^{v(0)}(u_mw)=(e^{v(0)}u)_m(e^{v(0)}w)$. In particular,
$e^{v(0)}$ is an automorphism of $V$.
\begin{dfn}
For a $V$-module $W$ and $u,v\in V_1$, define
\begin{equation}
Z_W(v;u;\tau)
={\rm tr}_W e^{2\pi \imath (v(0)-\langle v,u\rangle/2)}
q^{(u(0)-\langle u,u\rangle/2)+L(0)-c/24}.
\end{equation}
We set $\theta_W(v,\tau)=Z_W(v;0;\tau)\eta(\tau)^c$ and call it
a theta-function of $W$, where \\
$\eta(\tau)=q^{1/24}\prod_{n=1}^{\infty}(1-q^n)$ is the
Dedekind $\eta$-function.
\end{dfn}
For example,
let $V$ be a lattice VOA $V_{2\Z x}$ associated with
a $1$-dimensional lattice $L=2\Z x$ with $\langle x,x\rangle=1$, then
$$W_0=V_{2\Z x}, W_1=V_{(2\Z+{1\over 2})x},
W_2=V_{(2\Z+1)x},
W_3=V_{(\Z-{1\over 2})x}$$
are the irreducible $V_{2\Z x}$-modules by \cite{D} and
$\langle x(-1)\1, x(-1)\1\rangle=-1$.
We then have
\begin{eqnarray*}
&\theta_{0,0}(z,\tau)&
=\sum_{n\in \Z}\exp(\pi \imath n^2\tau+2\pi \imath nz)\hfill \\
&&=\theta_{W_0}(zx(-1)\1,\tau)+\theta_{W_2}(zx(-1)\1,\tau)) \hfill \\
&\theta_{0,\frac{1}{2}}(z,\tau)&
=\sum\exp(\pi \imath n^2\tau+2\pi \imath n(z+\frac{1}{2}))\hfill \\
&&=\theta_{W_0}(zx(-1)\1,\tau)-\theta_{W_2}(zx(-1)\1,\tau)) \hfill \\
&\theta_{\frac{1}{2},0}(z,\tau)&=\sum\exp(\pi \imath (n+\frac{1}{2})^2\tau+2\pi \imath
(n+\frac{1}{2})z)\hfill \\
&&=\theta_{W_1}(zx(-1)\1,\tau)+\theta_{W_3}(zx(-1)\1,\tau)) \hfill \\
&\theta_{\frac{1}{2},\frac{1}{2}}(z,\tau)&=\sum\exp(\pi \imath (n+\frac{1}{2})^2\tau+2\pi \imath
(n+\frac{1}{2})(z+\frac{1}{2})) \hfill \\
&&=\imath\theta_{W_1}(zx(-1)\1,\tau)-\imath \theta_{W_3}(zx(-1)\1,\tau))
\end{eqnarray*}
and their modular transformations
\begin{eqnarray*}
&&\theta_{0,0}(z/\tau,-1/\tau)
=(-\imath \tau)^{\frac{1}{2}}\exp(\pi \imath z^2/\tau)\theta_{0,0}(z,\tau) \hfill \\
&&\theta_{0,\frac{1}{2}}(z/\tau,-1/\tau)
=(-\imath \tau)^{\frac{1}{2}}\exp(\pi \imath z^2/\tau)\theta_{1,0}(z,\tau) \hfill \\
&&\theta_{\frac{1}{2},0}(z/\tau,-1/\tau)
=(-\imath \tau)^{\frac{1}{2}}\exp(\pi \imath z^2/\tau)\theta_{0,1}(z,\tau) \hfill \\
&&\theta_{\frac{1}{2},\frac{1}{2}}(z/\tau,-1/\tau)
=-(-\imath \tau)^{\frac{1}{2}}\exp(\pi \imath z^2/\tau)\theta_{1,1}(z,\tau)
\end{eqnarray*}
are well known, (see \cite{M}).
Therefore, the modular transformations
$Z_{W_h}(zx(-1)\1;0:{-1\over \tau})$ are expressed by linear
combinations of $Z_{W_k}(u;v;\tau)$ of (ordinary) modules $W_k$, but not
twisted modules.
\section{Modular invariance}
In this section, we will prove a modular invariance.
Throughout this section, we assume : \hfill \\
(A1) $V=\oplus_{n=0}^{\infty}V_n$ is a rational VOA; \hfill \\
(A2) $\{W^1,...,W^m\}$ is the set of all irreducible $V$-modules; \hfill \\
(A3) fix $v_1,...,v_n\in V_1$ satisfying $v_r(0)v_j=0$ and
$v_r(1)v_j\in {\C}\1$ for any $r,j$. \hfill \\
By (A3), we have
$$[v_r(m),v_j(n)]=\sum {m\choose k}(v_r(k)v_j)(m+n-\imath )
=m(v_r(1)v_j)(m+n-1)
=\delta_{m,-n}m\langle -v_r,v_j\rangle, $$
where $v_r(1)v_j=\langle -v_r,v_i\rangle\1$.
For a grade-keeping endomorphism formal power series
$\psi\in {\rm End}(W)((q_{z_1},...,q_{z_n},q_{\tau}))$ of $W$, set
$$S_W(\psi;z_1,...,z_n,\tau)={\rm tr}_W\psi
Y(v_1,q_{z_1})\cdots Y(v_n,q_{z_n})q_{z_1}\cdots
q_{z_n}q^{L(0)-{c\over 24}}. $$
By the same argument as in the proof of Proposition 4.3.2 in \cite{Z},
we have the following:
\begin{prn}
Assume $[\psi, v_r(n)]=0$ for $r=1,...,n$. Then we have :
\begin{equation}
S_W(\psi;z_1,z_2,...,z_n,\tau)
=\!\sum_{\sigma\in I(n)}\!S_W\!\left(\psi\! \prod_{r\in f(\sigma)}
\!o(v_r)\prod_{j<\sigma(j)}\!
\left(\langle -v_j,v_{\sigma(j)}\rangle
{P_2(z_{\sigma(j)}z_j^{-1},\tau)\over (2\pi\imath)^2}\right);\tau\right),
\end{equation}
where $I(n)$ is the set of all elements $\sigma$ in the symmetric group
$\Sigma_n$ on $n$ point set $\Omega=\{1,2,...,n\}$ with $\sigma^2=1$
and $f(\sigma)$ denotes the
set of fixed points of $\sigma$.
\end{prn}
\par \vspace{3mm} \noindent {\bf [Proof]} \qquad For $k\in \Z$, we have
\begin{eqnarray*}
& &S_W(\psi v_1(k)q_{z_1}^{-k};z_2,...,z_n,\tau) \hfill \\
&=&\!\!{\rm tr}_W \psi v_1(k)q_{z_1}^{-k}Y(v_2,q_{z_2})\cdots
Y(v_n,q_{z_n})q_{z_1}\cdots q_{z_n}q^{L(0)-c/24} \hfill \\
&=&\!\!{\rm tr}_W \psi[v_1(k),Y(v_2,q_{z_2})\cdots Y(v_n,q_{z_n})]q_{z_1}^{-k}q_{z_2}
\cdots q_{z_n}q^{L(0)-c/24} \hfill \\
& &+{\rm tr}_W \psi Y(v_2,q_{z_2})\cdots Y(v_n,q_{z_n})v(k)q_{z_1}^{-k}q_{z_2}
\cdots q_{z_n}q^{L(0)-c/24} \hfill \\
&=&\!\!\sum_{j=2}^n\sum_{i\in \N}{k\choose i}q_{z_j}^{k-\imath }
{\rm tr}_W \psi Y(v_2,q_{z_2})\cdots Y(v_1(i)v_j,q_{z_j})\cdots
Y(v_n,q_{z_n})q_{z_1}^{-k}q_{z_2}\cdots q_{z_n}q^{L(0)-c/24} \hfill \\
&&+{\rm tr}_W \psi Y(v_2,q_{z_2})\cdots Y(v_n,q_{z_n})q_{z_1}^{-k}q_{z_2}
\cdots q_{z_n}q^{L(0)-c/24}v(k)q^{k} \hfill \\
&=&\!\!\sum_{j=2}^n kq_{z_j}^{k-1}q_{z_1}^{-k}{\rm tr}_W \psi Y(v_2,q_{z_2})\cdots
Y(\langle -v_j,v_1\rangle \1,q_{z_j})\cdots Y(v_n,q_{z_n})q_{z_2}\cdots
q_{z_n}q^{L(0)-c/24} \hfill \\
&&+{\rm tr}_W \psi v_1(k)q_{z_1}^{-k}Y(v_2,q_{z_2})\cdots Y(v_n,q_{z_n})q_{z_2}
\cdots q_{z_n}q^{L(0)-c/24}q^k \hfill \\
&=&\!\!\sum_{j=2}^n{\rm tr}_W \langle -v_j,v_1\rangle kq_{z_j-z_1}^k \psi
Y(v_2,q_{z_2})\cdots \widehat{Y(v_j,q_{z_{j}})}
\cdots Y(v_n,q_{z_n})q_{z_2}...\widehat{q_{z_j}}...
q_{z_n}q^{L(0)-c/24} \hfill \\
&&+{\rm tr}_W \psi v_1(k)q_{z_1}^{-k}Y(v_2,q_{z_2})\cdots Y(v_n,q_{z_n})q_{z_2}
\cdots q_{z_n}q^{L(0)-c/24}q^k \hfill \\
&=&\!\!\sum_{j=2}^n{\rm tr}_W \langle -v_j,v_1\rangle kq_{z_j-z_1}^k
\psi Y(v_2,q_{z_2})\cdots
\widehat{Y(v_j,q_{z_{j}})}\cdots Y(v_n,q_{z_n})
q_{z_2}\cdots \widehat{q_{z_{j}}}\cdots q_{z_n}q^{L(0)-c/24} \hfill \\
&&+{\rm tr}_W \psi v_1(k)q_{z_1}^{-k}Y(v,q_{z_2})\cdots Y(v,q_{z_n})q_{z_2}
\cdots q_{z_n}q^{L(0)-c/24}q^k \hfill \\
&=&\!\!\sum_{j=2}^n \langle -v_1,v_j\rangle kq_{z_j-z_1}^k
S(\psi;z_2,...,\widehat{z_{j}},...,z_n,\tau)
+S(\psi v_1(k)q_{z_1}^{-k};z_2,...,z_n,\tau)q^k,
\end{eqnarray*}
where $\widehat{q_{z_j}}$ means that we take off the term $q_{z_j}$.
Hence, for $k\not=0$, we have:
\begin{equation}
S_W(\psi v_1(k)q_{z_1}^{-k};z_2,...,z_n,\tau)
=\sum_{j=2}^n\langle -v_j,v_1\rangle {kq_{z_j-z_1}^{k} \over 1-q^k}
S_W(\psi;z_2,...,z_{j-1},z_{j+1},...,z_n,\tau).
\end{equation}
Therefore, we have:
\begin{eqnarray*}
&&\quad S_W(\psi;z_1,z_2,...,z_n,\tau)\hfill \\
&=&S_W(\psi v_1(0);z_2,...,z_n,\tau)
+\sum_{k\not=0}S_W(\psi v(k)q_{z_1}^{-k};z_2,...,z_n,\tau)\hfill \\
&=&S_W(\psi v_1(0);z_2,...,z_n,\tau)
+\sum_{k\not=0}\sum_{j=2}^n
\langle -v_j,v_1\rangle {kq_{z_j-z_1}^{k} \over 1-q^k}
S_W(\psi;z_2,...,z_{j-1},z_{j+1},...,z_n,\tau) \hfill \\
&=&S_W(\psi v_1(0);z_2,...,z_n,\tau)
\!-\!\sum_{j=2}^n\langle v_u,v_1\rangle
{P_2(q_{z_j-z_1},\tau) \over (2\pi\imath)^2}
S_W(\psi;z_2,...,z_{j-1},z_{j+1},...,z_n,\tau)
\end{eqnarray*}
By substituting $\psi v_1(0)$ into $\phi$
and repeating these steps, we have:
\begin{equation}
S_W(\psi;z_1,z_2,...,z_n,\tau)
=\sum_{\sigma\in I(n)}
\prod_{i<\sigma(i)}\left( \langle -v_i,v_{\sigma(i)}\rangle
{P_2(q_{z_{\sigma(i)}-z_i},\tau) \over
(2\pi\imath)^2}\right)
S_W(\psi \prod_{i\in f(\sigma)}v_i(0),\tau).
\end{equation}
\hfill \qed
The main result we quote from \cite{Z} is
\begin{thm}[Zhu]
Let $(V,Y,\1,\bw)$ be a rational VOA satisfying (A1) and (A2). Then
for any $\al=\pmatrix{a &b \cr f& d}\in SL_2(\Z)$, we have
\begin{equation}
S_{W_h}(1,{z_1\over f\tau+d},...,{z_n\over f\tau+d},{a\tau+b\over
f\tau+d})=(f\tau+d)^n\sum_{j=1}^m A^h_{\al,k}S_{W_k}(1;z_1,...,z_n,\tau)
\end{equation}
where the $A^h_{\al,k}$ are constants depending only on $\al,h,k$.
\end{thm}
To simplify notation, $S_k(\cdots)$, $\al(v)$ and
$d(\al)$ denote $S_{W_k}(\cdots)$, $(f\tau+d)o(v)$ and $f\tau+d$,
respectively.
Set
\begin{eqnarray*}
&D(r,j)=&\langle -v_r,v_j\rangle({1\over 2\pi\imath})^2
(d(\al))^2P_2(q_{z_r-z_j},\tau), \hfill \\
&E(r,j)=&\langle -v_r,v_j\rangle ((d(\al))^2P_2(q_{z_r-z_j},\tau)
-({1\over 2\pi \imath})f(d(\al))), \hfill \\
&D_{\sigma}=&\prod_{j<\sigma(j)}D(\sigma(j),j), \mbox{ and } \hfill \\
&E_{\sigma}=&\prod_{j<\sigma(j)}E(\sigma(j),j).
\end{eqnarray*}
\begin{lmm}
If $|m(\sigma)|=2p$, then
\begin{equation}
\sum_{\sigma_1+...+\sigma_t=\sigma}(-1)^{t}
E_{\sigma_t}\cdots E_{\sigma_2}E_{\sigma_1}
=(-1)^pE_{\sigma}.
\end{equation}
\end{lmm}
\par \vspace{3mm} \noindent {\bf [Proof]} \qquad We first note that
$E_{\sigma_t}\cdots E_{\sigma_2}E_{\sigma_1}=E_{\sigma}$.
Therefore, we have to count the number of $E_{\sigma}$ in the left side.
We will prove it by induction on $p$. If $p=1$, then it is trivial.
For $\sigma=(r_1,r_2)\cdots(r_{2p-1},r_{2p})$, the number of $\sigma_1$
with $\sigma_1+\cdots+\sigma_t=\sigma$ and $|m(\sigma_1)|=2r$
is ${p\choose r}$.
Therefore, by induction, we have:
\begin{equation}
\sum_{\sigma_1+...+\sigma_t=\sigma}(-1)^{t}
E_{\sigma_t}\cdots E_{\sigma_2}E_{\sigma_1}
=-\sum_{j=1}^p{p\choose j}(-1)^{p-j}E_{\sigma}
=-(-(-1)^p)E_{\sigma}=(-1)^pE_{\sigma}.
\end{equation}
\hfill \qed
For $\al=\pmatrix{a&b\cr f&d}\in SL_2(\Z)$, we have
\begin{lmm}
\begin{equation}
S_h(\prod_{r=1}^n o(v_r); {a\tau+b\over f\tau+d})
=\sum_{k=1}^m A_{\al,k}^h \sum_{\sigma\in I(n)} \prod_{j<\sigma(j)}
({f(d(\al))\over 2\pi\imath })\langle -v_j,v_{\sigma(j)}\rangle
S_k(\prod_{r\in f(\sigma)}\al(v_r);\tau).
\end{equation}
\end{lmm}
\par \vspace{3mm} \noindent {\bf [Proof]} \qquad
By Theorem 3.1 and Proposition 3.1, we have:
\begin{eqnarray*}
&&\sum_{\sigma\in I(n)}\left(\prod_{r<\sigma(r)}
\left(\langle -v_{\sigma(r)},v_r\rangle({1\over 2\pi\imath })^2
P_2(q_{{z_{\sigma(r)}-z_r \over
f\tau+d}}, {a\tau+b\over f\tau+d})\right)\right)
S_h(\prod_{j\in f(\sigma)}o(v_j),{a\tau+b\over f\tau+d}) \hfill \\
&=&S_h(1;{z_1\over f\tau+d},...,{z_n\over f\tau+d},{a\tau+b\over f\tau+d})\hfill \\
&=&(d(\al))^n\sum_{k=1}^mA^h_{\al,k}S_k(1;z_1,...,z_n,\tau) \hfill \\
&=&(d(\al))^n\sum_{k=1}^mA^h_{\al,k}\sum_{\sigma\in I(n)}
\left(\prod_{r<\sigma(r)}\left(\langle -v_r,v_{\sigma(r)}\rangle
({1\over 2\pi\imath })^2P_2(q_{z_{\sigma(r)}-z_r},\tau)\right)
\right)S_k(\prod_{j\in f(\sigma)}o(v_j);\tau).
\end{eqnarray*}
Hence, we have
\begin{eqnarray*}
&&S_h(\prod_{r=1}^n o(v_r); {a\tau+b\over f\tau+d}) \hfill \\
&=&\sum_{k=1}^mA^h_{\al,k}\sum_{\sigma\in I(n)}\left(\prod_{r<\sigma(r)}
\left(\langle -v_r,v_{\sigma(r)}\rangle (d(\al))^2
({1\over 2\pi\imath })^2 P_2(q_{z_{\sigma(r)}-z_r},\tau)\right)\right)
S_k(\prod_{j\in f(\sigma)}\al(v_j)),\tau) \hfill \\
&&-\sum_{1\not=\sigma\in I(n)}\left(\prod_{r<\sigma(r)}
\left(\langle -v_r,v_{\sigma(r)}\rangle
((d(\al))^2({1\over 2\pi\imath })^2 P_2(q_{z_{\sigma(r)}-z_r},\tau)-
({1\over 2\pi\imath }) f(d(\al)))\right)\right)
\times \hfill \\
&&\times S_h(\prod_{j\in f(\sigma)} o(v_j);{a\tau+b\over f\tau+d}) \hfill \\
&=&\sum_{k=1}^m A^h_{\al,k}\sum_{\sigma\in I(n)}\left(
\prod_{r<\sigma(r)}D(\sigma(r),r)\right)S_k(\prod_{j\in f(\sigma)}\al(v_j),
\tau) \hfill \\
&&-\sum_{1\not=\sigma\in I(n)}\left(\prod_{r<\sigma(r)}E(\sigma(r),r)\right)
S_h(\prod_{j\in f(\sigma)} o(v_j); {a\tau+b\over f\tau+d}) \hfill \\
&=&\sum_{k=1}^m A^h_{\al,k}\sum_{\sigma\in I(n)}D_{\sigma}
S_k(\prod_{j\in f(\sigma)}\al(v_j); \tau)
-\sum_{1\not=\sigma\in I(n)}E_{\sigma}
S_h(\prod_{r\in f(\sigma)} o(v_r); {a\tau+b\over f\tau+d}).
\end{eqnarray*}
Substituting the above equation into the last term, we have
\begin{eqnarray*}
&&S_h(\prod_{r=1}^n o(v_r); {a\tau+b\over f\tau+d}) \hfill \\
&=&\sum_{k=1}^m A^h_{\al,k}\sum_{\sigma\in I(n)}D_{\sigma}
S_k(\prod_{j\in f(\sigma)}\al(v_j);\tau) \hfill \\
&&-\sum_{1\not=\sigma\in I(n)}E_{\sigma}
\sum_{k=1}^m A^h_{\al,k}
\sum_{\sigma'\in I(n), m(\sigma')\cap m(\sigma)=\emptyset}
D_{\sigma'}S_k(\prod_{r\in f(\sigma'+\sigma)}\al(v_r); \tau) \hfill \\
&&+\sum_{1\not=\sigma\in I(n)}E_{\sigma}
\sum_{1\not=\sigma'\in I(n),m(\sigma')\cap m(\sigma)=\emptyset}
E_{\sigma'}S_h(\prod_{j\in f(\sigma)\cap f(\sigma')}o(v_j);
{a\tau+b\over f\tau+d}) \}.
\end{eqnarray*}
Repeating the above steps and by the equations
\begin{eqnarray*}
&&S_h(o(v);{a\tau+b\over f\tau+d})=\sum A^h_{\al,k}S_k(o(v);\tau)\hfill \\
&&S_h(1;{a\tau+b\over f\tau+d})=\sum A^h_{\al,k}S_k(1;\tau).
\end{eqnarray*}
in \cite{Z}, we have
\begin{eqnarray*}
&&S_h(\prod_{r=1}^n o(v_r);{a\tau+b\over f\tau+d}) \hfill \\
&&=\sum_{k=1}^m A^h_{\al,k}
S_k(\sum_{\sigma\in I(n)}D_{\sigma}\prod_{j\in f(\sigma)}\al(v_j);\tau)
\hfill \\
&&\quad -\sum_{1\not=\sigma\in I(n)}E_{\sigma}
S_h(\prod_{r\in f(\sigma)} o(v_r);{a\tau+b\over
f\tau+d}) \hfill \\
&&=\sum_{k=1}^m A^h_{\al,k}
\{\sum_{\sigma_1,...,\sigma_t\in I(n):|f(\sigma_2+...\sigma_t)|\leq n-2}(-1)^{t-1}
E_{\sigma_t}\cdots E_{\sigma_2}D_{\sigma_1}
S_k(\prod_{r\in \cap_{j=1}^t f(\sigma_j)}\al(v_r);\tau) \hfill \\
&&\mbox{(if $n$ is odd) } -\sum_{j=1}^n
\sum_{\sigma_1,...,\sigma_t\in I(n):f(\sigma_1+...+\sigma_t)
=\{j\}}(-1)^{t}
E_{\sigma_t}\cdots E_{\sigma_2}E_{\sigma_1}S_k(\al(v_j);\tau)\} \hfill \\
&&\mbox{(if $n$ is even) }
-\sum_{\sigma_1,...,\sigma_t\in I(n):f(\sigma_1+...+\sigma_t)=\emptyset}
(-1)^{t}
E_{\sigma_t}\cdots E_{\sigma_2}E_{\sigma_1}S_k(1;\tau)\} \hfill \\
&&\mbox{by Lemma 3.1} \hfill \\
&&=\sum_{k=1}^m A_{\al,k}^h\{ \sum_{\sigma\in I(n)}
\sum_{\sigma_1,\sigma_2\in I(n):\sigma_1+\sigma_2=\sigma, |f(\sigma_1)|\geq 2}
(-1)^{|\sigma_1|/2}E_{\sigma_1}D_{\sigma_2}
S_k(\prod_{r\in f(\sigma)}\al(v_r);\tau) \hfill \\
&&\mbox{(if $n$ is odd) }
-\sum_{j=1}^n\sum_{\sigma\in I(n): f(\sigma)=\{j\}}(-1)^{(n-1)/2}E_{\sigma}
S_k(\al(v_j);\tau)\} \hfill \\
&&\mbox{(if $n$ is even) }
-\sum_{\sigma\in I(n): f(\sigma)=\emptyset}(-1)^{n/2}E_{\sigma}S_k(1;\tau)\}
\hfill \\
&&=\sum_{k=1}^m A_{\al,k}^h\{\sum_{\sigma\in I(n)}
\prod_{j<\sigma(j)}({1\over 2\pi \imath})
(f(d(\al))\langle -v_j,v_{\sigma(j)}\rangle )^p
S_k(\prod_{r\in f(\sigma)}\al(v_r);\tau)\}
\end{eqnarray*}
\hfill \qed
\noindent
{\bf Theorem A}\qquad For $\al=\pmatrix{a&b\cr f&d}\in SL_2(\Z)$, we have
\begin{equation}
S_h(e^{o(v)};{a\tau+b\over f\tau+d})
=\sum_{k=1}^m A^h_{\al,k}S_k(e^{{\langle -v,v\rangle\over 2}
({1\over 2\pi \imath})f(d(\al))+(d(\al))o(v)}, \tau).
\end{equation}
In particular,
\begin{equation}
Z_{W_h}(v;0;{a\tau+b\over f\tau+d})=\sum_{k=1}^mA_{\al,k}^h
Z_{W_k}(dv;fv;\tau).
\end{equation}
\par \vspace{3mm} \noindent {\bf [Proof]} \qquad
Let's calculate the coefficient of
$(\langle -u,v \rangle({1\over 2\pi \imath}) f(d(\al)))^p(\al(v))^r$
in (4.2) by setting $n=2p+r$ and $v_1=\cdots=v_{n}=v$.
The number of involutions $\sigma$ with $|f(\sigma)|=r$ is
${2p+r\choose r}\times {(2p+r)!\over p!2^p}$.
Therefore, we have
\begin{eqnarray*}
&&S_h(e^{o(v)};{a\tau+b\over f\tau+d})
=S_h(\sum_{n=0}^{\infty}{1\over n!}\prod_{i=1}^n o(v);{a\tau+b\over f\tau+d})
=\sum_{n=0}^{\infty}{1\over n!}S_h(\prod_{i=1}^n o(v);
{a\tau+b\over f\tau+d}) \hfill \\
&&=\sum_{r,p\in \N}{1\over n!}\sum_{k=1}^mA^h_{\al,k}
\sum_{\sigma\in I(n)}(\langle -v,v\rangle
{f(d(\al))\over 2\pi\imath})^{|m(\sigma)|/2}S_k(
(\al(v))^{|f(\sigma)|};\tau) \hfill \\
&&=\sum_{k=1}^mA^h_{\al,k}
\sum_{r,p\in \N}{1\over (r+2p)!}{r+2p\choose r}{(2p)!\over p!2^p}
(\langle -v,v \rangle {f(d(\al))\over 2\pi\imath})^{|m(\sigma)|/2}
S_k( (\al(v))^{|f(\sigma)|};\tau) \hfill \\
&&=\sum_{k=1}^mA^h_{\al,k}
\sum_{r,p\in \N}{1\over (r+p)!}{r+p\choose p}
({\langle -v,v \rangle \over 2}{f(d(\al))\over 2\pi\imath})^{p}
S_k((\al(v))^{r};\tau) \hfill \\
&&=\sum_{k=1}^mA^h_{\al,k}
\sum_{n=0}^{\infty}\sum_{r=0}^n
{1\over n!}{n\choose p}
({\langle -v,v\rangle\over 2}
{f(d(\al))\over 2\pi\imath})^{n-r}S_k((\al(v))^{r};\tau) \hfill \\
&&=\sum_{r=1}^mA^h_{\al,k}S_k(\sum_{n=0}^{\infty}
{1\over n!}(({\langle -v,v\rangle \over 2}{f(d(\al))\over 2\pi\imath})
+\al(v))^n;\tau) \hfill \\
&&=\sum_{r=1}^mA^h_{\al,k}S_k(e^{{\langle -v,v\rangle \over 2}
{f(d(\al))\over 2\pi\imath}+\al(v)};\tau),
\end{eqnarray*}
since
\begin{equation}
{1\over (r+2p)!}{r+2p\choose r}{(2p)!\over p!2^p}={1\over (r+2p)!}
{(r+2p)!\over k!(2p)!}{(2p)!\over p!2^p}={1\over (r+p)!}{r+p\choose r}
{1\over 2^p}.
\end{equation}
\hfill \qed
Let's show the final version of our result. \hfill \\
\noindent
{\bf Main Theorem}\qquad
For $\al=\pmatrix{a&b\cr f&d}\in SL_2(\Z)$, we have
\begin{equation}
Z_{W_h}(v;u;{a\tau+b\over f\tau+d})=\sum_{k=1}^m A_{\al,k}^h
Z_{W_k}(dv+bu;fv+au;\tau),
\end{equation}
where $A_{\al,k}^h$ are the coefficients in the equations given by
Zhu \cite{Z}. \hfill \\
\par \vspace{3mm} \noindent {\bf [Proof]} \qquad
Fix $s,t\in \N$ so that $n=s+t$.
Assume that
$$v_1=v_2=...=v_{s},\ (\mbox{ say, }=v), \quad
u_{s+1}=\cdots =v_{s+t},\ (\mbox{ say, }=u). $$
For $p$, $q$, $r\in \Z$, set $k=s-2p-r\geq 0$, $h=t-2q-r\geq 0$.
In order to simplify notation, for $u,v\in V_1$ and $\sigma\in I(n)$,
we will use the following notation:
\begin{eqnarray*}
&&\alpha(-v,u)={f(d(\al))\over 2\pi \imath}
\langle -v,u\rangle, \hfill \\
&&\al(v)=(d(\al))o(v) \hfill \\
&&m11\sigma=|\{r\in \Omega| r<\sigma(r)\leq s\}|, \hfill \\
&&m12\sigma=|\{r\in \Omega| r\leq s<\sigma(r)\}|, \hfill \\
&&m22\sigma=|\{r\in \Omega| s<r<\sigma(i)\}|, \hfill \\
&&f1\sigma=|\{r\in \Omega| r=\sigma(r)\leq s\}| \mbox{ and} \hfill \\
&&f2\sigma=|\{r\in \Omega| s<i=\sigma(r)\}|.
\end{eqnarray*}
Then by lemma 4.2, we have
\begin{eqnarray*}
&&\mbox{}\quad S_h(o(v)^{s}o(u)^{t};{a\tau+b\over f\tau+d})\hfill \\
&&=\sum_{g=1}^m A_{\al,g}^h \sum_{\sigma\in I(s+t)}
\alpha(-v,v)^{m11\sigma}
\alpha(-v,u)^{m12\sigma}
\alpha(-u,u)^{m22\sigma}
S_g(\al(v)^{f1\sigma}\al(u)^{f2\sigma};\tau).
\end{eqnarray*}
Hence,
\begin{eqnarray*}
&&\mbox{}\quad S_h(o(v)^{s}o(u)^{t};{a\tau+b\over f\tau+d}) \hfill \\
&&=\sum_{g=1}^m A_{\al,g}^h
\sum_{r,p,q}{(2p+r+k)!(2q+r+h)! \over k!h!r!p!q!2^{p+q}}
\alpha(-v,v)^{p}
\alpha(-v,u)^{r}
\alpha(-u,u)^{q}
S_g(\al(v)^{k}\al(u)^{h};\tau).
\end{eqnarray*}
Expanding the exponential, we have:
\begin{eqnarray*}
&&S_h(e^{2\pi \imath(o(v)+\langle -v,u\rangle/2)
+2\pi\imath\tau(o(u)+\langle -u,u\rangle/2)};\tau) \hfill \\
&&={\rm tr}_{W_h}\sum_{n=0}^{\infty}{(2\pi\imath)^n\over n!}
(o(v)+\tau o(u)+{\langle -v,u\rangle \over 2}+\tau{\langle -u,u\rangle \over 2})^n
q^{L(0)-c/24} \hfill \\
&&={\rm tr}_{W_h}\sum_{\al,\be,\ga,\delta}
{(2\pi\imath)^{\al+\be+\ga+\de}\over \al!\be!\ga!\de!}
o(v)^{\al}(\tau o(u))^{\be}({\langle -v,u\rangle \over 2})^{\ga}
({\langle -u,u\rangle \over 2}\tau)^{\delta}q^{L(0)-c/24}.
\end{eqnarray*}
Hence, for $\al={-1\over \tau}$, we have:
\begin{eqnarray*}
&&S_h(e^{2\pi i(o(v)+\langle -v,u\rangle/2+\tau(o(u)+\langle -u,u\rangle /2))};{-1\over \tau}) \hfill \\
&&=\sum_{\al,\be,\ga,\delta}
{(2\pi \imath)^{\al+\be+\ga+\de}\over \al! \be! \ga! \delta!}
S_h(o(v)^{\al}({-1\over \tau} o(u))^{\be}
({\langle -v,u\rangle \over 2})^{\ga}({\langle -u,u\rangle\over 2})^{\delta}({-1\over \tau})^{\delta},
{-1\over \tau}) \hfill \\
&&=\sum_{\ga,\delta,\al,\be}
{(2\pi\imath)^{\al+\be+\ga+\de}\over \al!\be!\ga!\de!}
({\langle -u,u\rangle\over 2})^{\delta}( {-1\over \tau})^{\delta}({\langle -v,u\rangle\over 2} )^{\ga}
({-1\over \tau})^{\be}S_h(o(v)^{\al}o(u)^{\be};{-1\over \tau}) \hfill \\
&&=\sum_{\ga,\delta,\al,\be}
{(2\pi \imath)^{\al+\be+\ga+\de}\over \al!\be!\ga!\de!}
({\langle -u,u\rangle\over 2})^{\delta}( {-1\over \tau})^{\delta}({\langle -v,u\rangle\over 2} )^{\ga}
({-1\over \tau})^{\be}\sum_{g=1}^m A_{\al,g}^h
\sum_{p,q,r,h,k:2p+r+k=\al, 2q+r+h=\be} \hfill \\
&&{\al!\be!\over h!k!r!p!q!2^{p+q}}\times
S_g((\tau{\langle -v,v\rangle\over 4\pi\imath} )^p(\tau{\langle -v,u\rangle\over 4\pi\imath} )^r
(\tau{\langle -u,u\rangle\over 4\pi\imath} )^q(\tau o(v))^k(\tau o(u))^h,\tau) \hfill \\
&&=\sum_{\ga,\delta,p,q,r,h,k}{(2\pi \imath)^{(p+k+r+q+h+\ga+\de)}\over
\ga!\de!h!k!r!p!q!}
({-1\over \tau}({\langle -u,u\rangle \over 2}))^{\delta}({\langle -v,u\rangle \over 2})^{\ga}\times \hfill \\
&&\times \sum_{g=1}^m A_{\al,g}^hS_g(
(\tau{\langle -v,v\rangle\over 2} )^p(-\langle -v,u\rangle)^r({1\over \tau}{\langle -u,u\rangle\over 2})^q
(\tau o(v))^k(-o(u))^h;\tau) \hfill \\
&&=\sum_{g=1}^m A_{\al,g}^h S_g(
e^{2\pi\imath({-1\over \tau}\langle -u,u\rangle/2 +\langle -v,u\rangle /2+
\tau\langle -v,v\rangle/2 -\langle -v,u\rangle +{1\over \tau}\langle -u,u\rangle/2 +\tau o(v)-o(u)) };\tau) \hfill \\
&&=\sum_{g=1}^m A_{\al,g}^h S_g(
e^{2\pi\imath(-\langle -v,u\rangle /2+
\tau\langle -v,v\rangle/2 +\tau o(v)-o(u))};\tau) \hfill \\
&&=\sum_{g=1}^m A_{\al,g}^h S_g(
e^{2\pi\imath(o(-u)+\langle u,v\rangle/2)+2\pi\imath\tau(o(v)+\langle -v,v\rangle/2)};\tau)
\end{eqnarray*}
Namely,
\begin{equation}
Z_{W_h}(v;u;{-1\over \tau})=
\sum A_{\al,k}^h Z_{W_k}(-u;v;\tau).
\end{equation}
On the other hand,
\begin{equation}
Z_{W_h}(v;u;\tau+1)=Z_{W_h}(v+u;u;\tau).
\end{equation}
Therefore, for $\al=\pmatrix{a&b\cr f&d}$, we have
\begin{equation}
Z_{W_h}(v;u;{a\tau+b\over f\tau+d})=\sum_{k=1}^mA_{\al,k}^h Z_{W_k}(
dv+bu;fv+au;\tau).
\end{equation}
\hfill \qed
\noindent
{\bf Example}
\qquad Let $V$ be a lattice VOA $V_{2\Z x}$ with $\langle x,x\rangle=1$
and $W$ a module $V_{\Z x}$.
For $z\in \C$, set
$$zx=zx(-1)\1\in (V_{2\Z x})_1. $$
Then it is easy to check by the definition of lattice VOAs that
$${{\rm tr}} o(\1)e^{2\pi izx(0)}q^{L(0)-1/24}|_M
={1\over \eta(\tau)}\theta(\tau,z). $$
For $\al=\pmatrix{0&-1 \cr 1&0}$,
$${1\over \eta({-1\over \tau})}\theta({-1\over \tau},z)=\sum_M
\la_M {{\rm tr}}
o({\1})q^{L(0)+zx(0)-{1\over 24}+{z^2\over 2}}|_M, $$
where $M$ runs over the following modules
$$\{V_L, V_{x+L}, V_{\frac{1}{2} x+L}, V_{-\frac{1}{2} x+L} \}.$$
Taking $\tau=i$ and several $z$, we have the known formula:
$${1\over \eta({-1\over \tau})}\theta({-1\over \tau},z)
={1\over \eta(\tau)}\theta(\tau,\tau z)e^{\pi iz^2\tau}. $$
|
2,877,628,088,957 | arxiv | \section{Introduction}
\label{sec:introduction}
Graph learning~\cite{zhang2020deep,li2022curvature,fu2021ace} has gained popularity over the past years due to its versatility and success in representing graph data across a wide range of domains~\cite{nastase2015survey,sun2020pairwise,gao2021room,yu2022cross,chen2022reinforced}.
Graph Neural Networks (GNNs)~\cite{wu2020comprehensive,sun2021sugar} have been the ``battle horse'' of graph learning, which propagate the features on the graph by exchanging information between neighbors in a message-passing paradigm~\cite{gilmer2017neural}.
Due to the asymmetric and uneven topology, learning on graphs by GNNs suffers a specific imbalance problem, i.e., topology-imbalance.
Topology-imbalance~\cite{chen2021topology} is caused by the uneven position distribution of labeled nodes in the topology space, which is inevitable in real-world applications due to data availability and the labeling costs.
For example, we may only have information for a small group of users within a local community in social networks, resulting in a serious imbalance of labeled node positions.
The uneven position distribution of labeled nodes leads to uneven information propagation, resulting in the poor quality of learned representations.
Although the imbalance learning on graphs has attracted many research interests in recent years, most of them focus on the class-imbalance issue~\cite{park2021graphens,wang2021distance}, i.e., the imbalanced number of labeled nodes of each class.
The topology-imbalance issue is proposed recently and is still under-explored.
The only existing work, ReNode~\cite{chen2021topology}, provides an understanding of the topology-imbalance issue from the perspective of label propagation and proposes a sample re-weighting method.
However, ReNode takes the node topological boundaries as decision boundaries based on a homophily assumption, which does not work with real-world graphs.
The strong assumption leads to poor generalization and unsatisfied performance of ReNode (see Section~\ref{sec:cls_realworld}).
There are \textbf{two remaining questions}:
\textit{(1) Why does topology-imbalance affect the performance of graph representation learning?} and
\textit{(2) What kind of graphs are susceptible to topology-imbalance?}
To answer the above two questions, how to measure the influence of labeled nodes is the key challenge in handling topology-imbalance due to the complex graph connections and the unknown class labels for most nodes in the graph.
\begin{figure}
\centering
\includegraphics[width=0.8\linewidth]{figs/reaching-squashing.pdf}
\caption{Schematic diagram of under-reaching and over-squashing in the topology-imbalance issue.}
\label{fig:reach-squash}
\end{figure}
\textbf{New understanding for topology-imbalance. }
In this work, we provide a new understanding of the topology-imbalance issue from a global view of the supervision information distribution in terms of under-reaching and over-squashing:
\textbf{(1) Under-reaching}: the influence of labeled nodes decays with the topology distance~\cite{buchnik2018bootstrapped}, resulting in the nodes far away from labeled nodes lack of supervision information.
In Figure~\ref{fig:reach-squash}, the node $v_a$ cannot reach the valuable labeled node $v_c$ within the receptive field of the GNN model, resulting in the quantity of information it received is limited.
\textbf{(2) Over-squashing}: the supervision information of valuable labeled nodes is squashed when passing across the narrow path together with other useless information.
In Figure~\ref{fig:reach-squash}, the valuable supervision information of $v_b$ to $v_a$ is compressed into a vector together with the information of many nodes belonging to other classes, resulting in the quality of supervision information that $v_a$ received being poor.
Then we introduce two metrics (reaching coefficient and squashing coefficient) to give a quantitative analysis of the relation between the learning performance, label positions, and graph structure properties.
We further draw a conclusion that \textit{better reachability and lower squashing to labeled nodes lead to better classification performance for GNN models}.
\textbf{Present work. }
In light of the above analysis, we propose a \textbf{P}osition-\textbf{A}ware \textbf{ST}ructur\textbf{E} \textbf{L}earning method named \textbf{PASTEL}, which directly optimizes the information propagation path and solves the problem of topology-imbalance issue in essence.
The key insight of PASTEL~is to enable nodes within the same class to connect more closely with each other for more supervision information.
Specifically, we design a novel \textit{anchor-based position encoding mechanism} to capture the relative position between nodes and incorporate the position information into structure learning.
Then we design a \textit{class-wise conflict measure} based on the Group PageRank, which measures the influence from labeled nodes of each class and acts as a guide to increase the intra-class connectivity via adjusting edge weight.
The main contributions are as follows:
\begin{itemize}[leftmargin=*]
\item
We provide a new understanding of the topology-imbalance issue from the perspective of supervision information distribution in terms of under-reaching and over-squashing and provide two new quantitative metrics for them.
\item
Equipped with the proposed position encodings and class-wise conflict measure, PASTEL~can better model the relationships of node pairs and enhance the intra-class inductive bias by maximizing the label influence.
\item
Experimental results demonstrate that the proposed PASTEL~enjoys superior effectiveness and indeed enhances the GNN model’s power for in-the-wild extrapolation.
\end{itemize}
\section{Related Work}
\subsection{Imbalance Learning}
Imbalanced classification problems~\cite{sun2009classification,he2009learning} have attracted extensive research attention.
Most existing works~\cite{haixiang2017learning,lin2017focal} focus on the class-imbalance problem, where the model performance is dominated by the majority class.
The class-imbalance learning methods can be roughly divided into two types: data-level re-sampling and algorithm-level re-weighting.
\textbf{Re-sampling} methods re-sample~\cite{chawla2002smote,ando2017deep,xiaolong2019over} or augment data~\cite{park2021graphens} to balance the number of data for each class during the data selection phase.
\textbf{Re-weighting} methods~\cite{cao2019learning,cui2019class,ren2018learning} adjust different weights to different data samples according to the number of data during the training phase.
For the graph-specific topology-imbalance issue as mentioned in Section~\ref{sec:introduction}, directly applying these methods to the graph data fails to take the special topology properties into consideration.
ReNode~\cite{chen2021topology} is the first work for the graph topology-imbalance issue, which follows the paradigm of classical re-weighting methods.
Specifically, ReNode defines an influence conflict detection based metric and re-weights the labeled nodes based on their relative positions to class boundaries.
However, ReNode is limited by its homophily assumption and only has a slight performance improvement.
\textit{In this paper, PASTEL~alleviates topology-imbalance by learning a new structure that maximizes the intra-class label influence, which can be seen as ``label re-distribution'' in the topology space. }
\subsection{Graph Structure Learning}
Graph structure learning~\cite{zhu2021deep} learns an optimized graph structure for representation learning and most of them aim to improve the robustness~\cite{jin2020graph,zheng2020robust} of GNN models.
There are also some works~\cite{franceschi2019learning,chen2020iterative,topping2021understanding,chen2020label,sun2022graph} that utilize the structure learning to improve the graph representation quality.
As for the over-squashing problem, \cite{wang2021combining} assigns different weights to edges connected to two nodes of the same class for better representations.
However, \cite{wang2021combining} still fails with the issue of under-reaching.
SDRF~\cite{topping2021understanding} rewires edges according to the Ricci curvatures to solve the over-squashing problem by only considering topology properties.
Multiple measurements in existing structure learning works are leveraged for modeling node relations, including node features~\cite{zhao2021heterogeneous}, node degrees~\cite{jin2020graph}, node encodings~\cite{zhang2019hierarchical} and edge attributes~\cite{zheng2020robust}.
The node positions play an important role in generating discriminative representations~\cite{you2019position} and are seldom considered in structure learning.
\textit{In this work, we advance the structure learning strategy for the graph topology-imbalance issue and introduce a position-aware framework to better capture the nodes' underlying relations. }
\section{Alleviate Topology-Imbalance by Structure Learning}
In this section, we introduce \textbf{PASTEL}, a \textbf{P}osition-\textbf{A}ware \textbf{ST}ructur\textbf{E} \textbf{L}earning framework, to optimize the information propagation path directly and address the topology-imbalance issue in essence.
In light of the analysis in Section~\ref{sec:understand}, PASTEL~aims to learn a better structure that increases the intra-class label influence for each class and thus relieves the under-reaching and over-squashing phenomena.
The overall architecture of PASTEL~is shown in Figure~\ref{fig:framework}.
\subsection{Position-aware Structure Learning}
\label{sec:position_encoding}
To form structure with better intra-class connectivity, we use an anchor-based position encoding method to capture the topology distance between unlabeled nodes to labeled nodes.
Then we incorporate both the merits of feature information as well as topology information to learn the refined structure.
\textbf{Anchor-based Position Encoding. }
Inspired by the position in transformer~\cite{vaswani2017attention,shaw2018self}, we use an anchor-based position encoding method to capture the relative position of unlabeled nodes with respect to all the labeled nodes of the graph.
Since we focus on maximizing the reachability between unlabeled nodes and labeled nodes within the same class, we directly separate the labeled node set $\mathcal{V}_L$ into $C$ anchor sets $\{\mathcal{V}^{1}_L, \mathcal{V}^2_L, \cdots, \mathcal{V}^{C}_L\}$, where each subset $\mathcal{V}^c_L$ denotes the labeled nodes whose labels are $c$.
The class-wise anchor sets help distinguish the information from different classes rather than treating all the anchor nodes the same and ignoring the class difference as in~\cite{you2019position}.
Concretely, for any node $v_i$, we consider a function $\phi(\cdot, \cdot)$ which measures the position relations between $v_i$ and the anchor sets in graph $\mathcal{G}$.
The function can be defined by the connectivity between the nodes in the graph.
\begin{equation}
\mathbf{p}_i=\left(\phi\left(v_i, \mathcal{V}^1_L\right),\phi\left(v_i, \mathcal{V}^2_L\right),\cdots,\phi\left(v_i, \mathcal{V}^C_L\right)\right),
\end{equation}
where $\phi(v_i, \mathcal{V}^c_L)$ is the position encoding function defined by the connectivity between the node $v_i$ and the anchor set $\mathcal{V}^c_L$ in graph.
Here we choose $\phi(v_i, \mathcal{V}^c_L)$ to be the mean length of shortest path between $v_i$ and nodes in $\mathcal{V}^c_L$ if two nodes are connected:
\begin{equation}
\phi(v_i, \mathcal{V}^c_L)=\frac{\sum_{v_j \in \mathcal{N}_{c}(v_i)}|\mathcal{P}_{sp}(v_i,v_j)|}{|\mathcal{N}_{c}(v_i)|},
\end{equation}
where $\mathcal{N}_{c}(v_i)$ is the nodes connected with $v_i$ in $\mathcal{V}^c_L$ and $|\mathcal{P}_{sp}(v_i,v_j)|$ is the length of shortest path between $v_i$ and $v_j$.
Then we transform the position encoding into the $d_0$ dimensional space:
\begin{equation}
\label{eq:position_encoding}
\mathbf{h}^p_i=\mathbf{W}_{\phi}\cdot\mathbf{p}_i,
\end{equation}
where $\mathbf{W}_{\phi}$ is a trainable vector.
If two nodes have similar shortest paths to the anchor sets, their position encodings are similar.
\begin{figure}[t]
\centering
\includegraphics[width=0.95\linewidth]{figs/oneSBM-samelabels.pdf}
\caption{
Predictions of GCN with and the same labeled nodes and different graph structures.
}
\label{fig:samelabels}
\end{figure}
\begin{figure*}[t]
\centering
\includegraphics[width=\linewidth]{figs/framework.pdf}
\caption{
Overall architecture of PASTEL.
PASTEL~encodes the relative position between nodes with the labeled nodes as anchor sets \{$\mathcal{S}$\} and incorporates the position information with node features for structure learning.
For each pair of nodes, PASTEL~uses the class-wise conflict measure as the edge weights to learn a graph with better intra-class connectivity.
}
\label{fig:framework}
\end{figure*}
\textbf{Position-aware Metric Learning. }
After obtaining the position encoding, we use a metric function that accounts for both node feature information and the position-based similarities to measure the possibility of edge existence.
PASTEL~is agnostic to various similarity metric functions and we choose the widely used multi-head cosine similarity function here:
\begin{equation}
\label{eq:metric}
a^P_{ij}=\frac{1}{m}\sum^{m}_{h=1}\cos\left(\mathbf{W}_h \cdot \left(\mathbf{z}_i||\mathbf{h}^p_i\right), \mathbf{W}_h\cdot \left(\mathbf{z}_j||\mathbf{h}^p_j\right)\right),
\end{equation}
where $m$ is the number of heads, $\mathbf{W}_h$ is the weight matrix of the $h$-th head, $\mathbf{z}_i$ denotes the representation vector of node $v_i$ and $||$ denotes concatenation.
The effectiveness of the position-aware structure learning is evaluated in Section~\ref{sec:abla_position}.
\subsection{Class-wise Conflict Measure}
\label{sec:conflict}
We aim to increase the intra-class connectivity among nodes, thereby increasing the supervision information they received and their influence on each other.
Here we propose a class-wise conflict measure to guide what nodes should be more closely connected.
According to the inherent relation of GNNs with Label Propagation~\cite{wang2021combining,chen2021topology}, we use the \textit{Group PageRank}~\cite{chen2020distance} as a conflict measure between nodes.
Group PageRank (GPR) extends the traditional PageRank\cite{page1999pagerank} into a label-aware version to measure the supervision information from labeled nodes of each class.
Specifically, for class $c\in\{1,2,\cdots, C\}$, the corresponding GPR matrix is
\begin{equation}
\mathbf{P}^{gpr}(c) = (1-\alpha)\mathbf{A}'\mathbf{P}^{gpr}(c) + \alpha \mathbf{I}_c,
\end{equation}
where $\mathbf{A}'=\mathbf{A}\mathbf{D}^{-1}$,
$\alpha$ is the random walk restart probability at a random node in the group and $\mathbf{I}_c\in \mathbb{R}^{n}$ is the teleport vector:
\begin{equation}
\mathbf{I}^i_c = \left\{\begin{matrix}
\frac{1}{|\mathcal{V}^c_L|}, & if~y_i=c \\
0,& otherwise
\end{matrix}\right.
\end{equation}
where $|\mathcal{V}^c_L|$ is the number of labeled nodes with class $c$.
We calculate the GPR for each group individually and then concatenate all the GPR vectors to form a final GPR matrix $\mathbf{P}^{gpr}\in \mathbb{R}^{N\times C}$ as in~\cite{chen2020distance}:
\begin{equation}
\label{eq:gpr_matrix}
\mathbf{P}^{gpr}=\alpha\left(\mathbf{E}-\left(1-\alpha\right)\mathbf{A}'\right)^{-1}\mathbf{I}^*,
\end{equation}
where $\mathbf{E}$ is the unit matrix of nodes and $\mathbf{I}^*$ is the concatenation of $\{\mathbf{I}_c, c=1,2,\cdots,C\}$.
Under $\mathbf{P}^{gpr}$, node $v_i$ corresponds to a GPR vector $\mathbf{P}^{gpr}_{i}$ (the $i$-th row of $\mathbf{P}^{gpr}$), where the $c$-th dimension represents the the supervision influence of labeled nodes of class $c$ on node $v_i$.
The GPR value contains not only the global topology information but also the annotation information.
For each node pair nodes $v_i$ and $v_j$, we use the Kullback Leiber (KL) divergence of their GPR vectors to measure their conflict when forming an edge:
\begin{equation}
\kappa_{ij}=\operatorname{KL}\left(\mathbf{P}^{gpr}_{i}, \mathbf{P}^{gpr}_{j}\right).
\end{equation}
The distance of GPR vectors reflects the influence conflict of different classes when exchanging information.
We use a cosine annealing mechanism to calculate the edge weights by the relative ranking of the conflict measure:
\begin{equation}
\label{eq:conflict}
w_{ij}=\frac{1}{2}\left [-\cos{\frac{\operatorname{Rank}(\kappa_{ij})}{|\mathcal{V}|\times |\mathcal{V}|}*\pi}+1\right ],
\end{equation}
where $Rank(\cdot)$ is the ranking function according to the magnitude.
The more conflicting the edge is, the less weight is assigned to it.
With the class-wise conflict measure, we aim to learn a graph structure that makes the GPR vectors of nodes have ``sharp'' distributions focusing on their ground-truth classes.
Then $w_{ij}$ is used as the connection strength of edge $e_{ij}$, with the corresponding element $\tilde{a}^P_{ij}$ in the adjacency matrix being:
\begin{equation}
\label{eq:conflict_weight}
\tilde{a}^P_{ij}=w_{ij}\cdot a^P_{ij}.
\end{equation}
The effectiveness of the class-wise conflict measure is evaluated in Section~\ref{sec:abla_conflict} and the change of GPR vectors is shown in Section~\ref{sec:gpr_visual}.
\begin{algorithm}[!t]
\caption{The overall process of PASTEL}
\label{alg:training}
\LinesNumbered
\KwIn{Graph $\mathcal{G}=\{\mathcal{V},\mathcal{E}\}$ with node labels $\mathcal{Y}$; Number of heads $m$; Number of training epochs $E$; Structure fusing coefficients $\lambda_1, \lambda_2$; Loss coefficients $\beta_1, \beta_2, \beta_3$}
\KwOut{Optimized graph $\mathcal{G}^*=(\mathbf{A}^*, \mathbf{X})$, predicted label $\hat{\mathcal{Y}}$}
Parameter initialization;\\
\For{$e=1,2,\cdots,E$}{
\tcp{Learn position-aware graph structure}
Learn position encodings $\mathbf{h}^p_i$ $\gets$ Eq.~\eqref{eq:position_encoding};\\
Learn edge possibility $a^P_{ij}$ $\gets$ Eq.~\eqref{eq:metric};\\
Calculate the Group PageRank matrix $\mathbf{P}^{gpr}\gets$ Eq.~\eqref{eq:gpr_matrix};\\
Calculate the class-wise conflict measure $w_{ij}\gets$ Eq.~\eqref{eq:conflict};\\
Obtain position-aware structure $\mathbf{A}_P$ $\gets$ Eq.~\eqref{eq:positionA};\\
\tcp{Learn node representations}
Obtain the optimized structure $\mathbf{A}^*$ $\gets$ Eq.~\eqref{eq:A*};\\
Calculate representations and labels $\mathbf{Z}$, $\hat{\mathcal{Y}}$ $\gets$ Eq.~\eqref{eq:representation_predict};\\
\tcp{Optimize}
Calculate the losses $\mathcal{L}_{cls}\gets$ Eq.~\eqref{eq:loss_cls}, $\mathcal{L}_{smooth}\gets$Eq.~\eqref{eq:loss_smooth}, $\mathcal{L}_{con}\gets$Eq.~\eqref{eq:loss_connectivity}, and $\mathcal{L}_{spar}\gets$Eq.~\eqref{eq:loss_sparsity}; \\
Update model parameters to minimize $\mathcal{L}$ $\gets$ Eq.~\eqref{eq:loss}.
}
\end{algorithm}
\subsection{Learning with the Optimized Structure}
With the above structure learning strategy, we can obtain a position-aware adjacency $\mathbf{A}_{P}$ with maximum intra-class connectivities:
\begin{equation}
\label{eq:positionA}
\mathbf{A}_{P}=\{\tilde{a}^P_{ij}, i,j\in \{1,2,\cdots,N\}\}.
\end{equation}
The input graph structure determines the learning performance to a certain extent.
Since the structure learned at the beginning is of poor quality, directly using it may lead to non-convergence or unstable training of the whole framework.
We hence incorporate the original graph structure $\mathbf{A}$ and a structure in a node feature view $\mathbf{A}_N$ as supplementary to formulate an optimized graph structure $\mathbf{A}^*$.
Specifically, we also learn a graph structure $\mathbf{A}_{N}=\{a^N_{ij}, i,j\in \{1,2,\cdots,N\}\}$ in a node feature view with each element being:
\begin{equation}
a^{N}_{ij}=\frac{1}{m}\sum^{m}_{h=1}\cos\left(\mathbf{W}_h \cdot \left(\mathbf{x}_i||\mathbf{h}^{p_0}_i\right), \mathbf{W}_h\cdot \left(\mathbf{x}_j||\mathbf{h}^{p_0}_j\right)\right),
\end{equation}
where $\mathbf{x}_i$ is the feature vector of node $v_i$ and $\mathbf{h}^{p_0}_i$ is the position encoding with the original structure.
Then we can formulate an optimized graph structure $\mathbf{A}^*$ with respect to the downstream task:
\begin{equation}
\label{eq:A*}
\mathbf{A}^*=\lambda_1\mathbf{D}^{-\frac{1}{2}}\mathbf{A}\mathbf{D}^{-\frac{1}{2}}+\left(1-\lambda_1\right)\left(\lambda_2 f\left(\mathbf{A}_{N}\right)+\left(1-\lambda_2\right)f\left(\mathbf{A}_P\right)\right),
\end{equation}
where $f(\cdot)$ denotes the row-wise normalization function, $\lambda_1$ and $\lambda_2$ are two constants that control the contributions of original structure and feature view structure, respectively.
Here we use a dynamic decay mechanism for $\lambda_1$ and $\lambda_2$ to enable the position-aware structure $\mathbf{A}_P$ to play a more and more important role during training.
To control the quality of learned graph structure, we impose additional constraints on it following~\cite{kalofolias2016learn,chen2020iterative} in terms of smoothness, connectivity, and sparsity:
\begin{align}
\label{eq:loss_smooth}
\mathcal{L}_{smooth}&=\frac{1}{N^2}{\rm tr}\left(\mathbf{X}^{T}\mathbf{L}^*\mathbf{X}\right),\\
\label{eq:loss_connectivity}
\mathcal{L}_{con}&=\frac{1}{N}\mathbf{1}^{T}\log(\mathbf{A}^*\mathbf{1}),\\
\label{eq:loss_sparsity}
\mathcal{L}_{spar}&=\frac{1}{N^2}||\mathbf{A}^*||^{2}_{F},
\end{align}
where $\mathbf{L}^*=\mathbf{D}^*-\mathbf{A}^*$ is the Laplacian of $\mathbf{A}^*$ and $\mathbf{D}^*$ is the degree matrix of $\mathbf{A}^*$.
To speed up the computation, we extract a symmetric sparse non-negative adjacency matrix by masking off (i.e., set to zero) those elements in $\mathbf{A}^*$ which are smaller than a predefined non-negative threshold $a_0$.
Then $\mathcal{G}^*=(\mathbf{A}^*, \mathbf{X})$ is input into the $\operatorname{GNN-Encoder}$ for the node representations $\mathbf{Z}\in\mathbb{R}^{N\times d}$, predicted labels $\hat{y}$ and classification loss $\mathcal{L}_{cls}$:
\begin{equation}
\label{eq:representation_predict}
\mathbf{Z}=\operatorname{GNN-Encoder}(\mathbf{A}^*, \mathbf{X}),\hat{\mathcal{Y}}=\operatorname{Classifier}(\mathbf{Z}),
\end{equation}
\begin{equation}
\label{eq:loss_cls}
\mathcal{L}_{cls}=\operatorname{Cross-Entropy}(\mathcal{Y}, \hat{\mathcal{Y}}).
\end{equation}
The overall loss is defined as the combination of the node classification loss and graph regularization loss:
\begin{equation}
\label{eq:loss}
\mathcal{L}=\mathcal{L}_{cls}+\beta_1 \mathcal{L}_{smooth}+\beta_2 \mathcal{L}_{con}+\beta_3 \mathcal{L}_{spar}.
\end{equation}
The overall process of PASTEL~is shown in Algorithm~\ref{alg:training}.
\section{Understanding Topology-Imbalance}
In this section, we provide a new understanding of the topology-imbalance issue in terms of under-reaching and over-squashing.
Then we perform a quantitative analysis of the relations between them to answer two questions:
\noindent\textbf{Q1:} Why does topology-imbalance affect the performance of graph representation learning?
\noindent\textbf{Q2:} What kind of graphs are susceptible to topology-imbalance?
\subsection{Notations and Preliminaries}
Consider a graph $\mathcal{G}=\{\mathcal{V},\mathcal{E}\}$, where $\mathcal{V}$ is the set of $N$ nodes and $\mathcal{E}$ is the edge set.
Let $\mathbf{A}\in \mathbb{R}^{N\times N}$ be the adjacency matrix and $\mathbf{X}\in \mathbb{R}^{N\times d_0}$ be the node attribute matrix, where $d_0$ denotes the dimension of node attributes.
The diagonal degree matrix is denoted as $\mathbf{D}\in \mathbb{R}^{N\times N}$ where $\mathbf{D}_{ii}=\sum^{N}_{j=1}\mathbf{A}_{ij}$.
The graph diameter is denoted as $D_{\mathcal{G}}$.
Given the labeled node set $\mathcal{V}_L$and their labels $\mathcal{Y}_L$ where each node $v_i$ is associated with a label $y_i$, \textit{semi-supervised node classification} aims to train a node classifier $f_{\theta}:v\rightarrow\mathbb{R}^{C}$ to predict the labels $\mathcal{Y}_U$ of remaining nodes $\mathcal{V}_U=\mathcal{V} \setminus \mathcal{V}_L$, where $C$ denotes the number of classes.
we separate the labeled node set $\mathcal{V}_L$ into $\{\mathcal{V}^{1}_L, \mathcal{V}^2_L, \cdots, \mathcal{V}^{C}_L\}$, where $\mathcal{V}^i_L$ is the nodes of class $i$ in $\mathcal{V}_L$.
\subsection{Understanding Topology-Imbalance via Under-reaching and Over-squashing}
\label{sec:understand}
In GNNs, node representations are learned by aggregating information from valuable neighbors.
The quantity and quality of the information received by the nodes decide the expressiveness of their representations.
We perceive the imbalance of the labeled node positions affects the performance of GNNs for two reasons:
\noindent(1) \textbf{Under-reaching}:
The influence from labeled nodes decays with the topology distance~\cite{buchnik2018bootstrapped}, resulting in that the nodes far away from labeled nodes lack supervision information.
When the node can't reach enough valuable labeled nodes within the receptive field of the model, the quantity of information it received is limited.
\noindent(2) \textbf{Over-squashing}:
The receptive field of GNNs is exponentially-growing and all information is compressed into fixed-length vectors~\cite{alon2020bottleneck}.
The supervision information of valuable labeled nodes is squashed when passing across the narrow path together with other useless information.
\subsection{Quantitative Analysis}
\label{sec:quantitative}
To provide quantitative analysis for topology-imbalance, we propose two metrics for reachability and squashing.
First, we define a reaching coefficient based on the shortest path, which determines the minimum layers of GNNs to obtain supervision information:
\begin{myDef}[\textbf{Reaching coefficient}]
Given a graph $\mathcal{G}$ and labeled node set $\mathcal{V}_L$, the reaching coefficient $RC$ of $\mathcal{G}$ is the mean length of the shortest path from unlabeled nodes to the labeled nodes of their corresponding classes:
\begin{equation}
RC=\frac{1}{|\mathcal{V}_U|}\sum_{v_i\in \mathcal{V}_U}\frac{1}{|\mathcal{V}^{y_i}_L|}\sum_{v_j\in \mathcal{V}^{y_i}_L}\left (1-\frac{\log |\mathcal{P}_{sp}(v_i,v_j)|}{\log D_{\mathcal{G}}}\right ),
\end{equation}
where $\mathcal{V}^{y_i}_L$ denotes the nodes in $\mathcal{V}_L$ whose label is $y_i$, $\mathcal{P}_{sp}(v_i,v_j)$ denotes the shortest path between $v_i$ and $v_j$, and $|\mathcal{P}_{sp}(v_i,v_j)|$ denotes its length, and $D_{\mathcal{G}}$ is the diameter of graph $\mathcal{G}$.
Specifically, for the unconnected $v_i$ and $v_j$, we set the length of their shortest path as $D_{\mathcal{G}}$.
\end{myDef}
The reaching coefficient reflects how long the the distance when the GNNs passes the valuable information to the unlabeled nodes.
Note that $RC\in [0,1)$ and larger $RC$ means better reachability.
For the quantitative metric of over-squashing, we define a squashing coefficient using the Ricci curvature to formulate it from a geometric perspective.
The Ricci curvature~\cite{ollivier2009ricci} reflects the change of topology properties of the two endpoints of an edge, where the negative $Ric(v_i,v_j)$ means that the edge behaves locally as a shortcut or bridge and positive $Ric(v_i,v_j)$ indicates that locally there are more triangles in the neighborhood of $v_i$ and $v_j$~\cite{ni2015ricci,topping2021understanding}.
\begin{myDef}[\textbf{Squashing coefficient}]
Given a graph $\mathcal{G}$, the squashing coefficient $SC$ of $\mathcal{G}$ is the mean Ricci curvature of edges on the shortest path from unlabeled nodes to the labeled nodes of their corresponding classes:
\begin{equation}
SC=\frac{1}{|\mathcal{V}_U|}\sum_{v_i\in \mathcal{V}_U}\frac{1}{|\mathcal{N}_{y_i}(v_i)|}\sum_{v_j\in \mathcal{N}_{y_i}(v_i)}\frac{\sum_{e_{kt}\in \mathcal{P}_{sp}(v_i,v_j)} Ric(v_k, v_t)}{|\mathcal{P}_{sp}(v_i,v_j)|},
\end{equation}
where $\mathcal{N}_{y_i}(v_i)$ denotes the labeled nodes of class $y_i$ that can reach $v_i$, $Ric(\cdot,\cdot)$ denotes the Ricci curvature, and $|\mathcal{P}_{sp}(v_i,v_j)|$ denotes the length of shortest path between $v_i$ and $v_j$.
\end{myDef}
We leverage the \textit{Ollivier-Ricci curvature}~\cite{ollivier2009ricci} as $Ric(\cdot,\cdot)$ here:
\begin{equation}
Ric(v_k,v_t)=\frac{Wasserstein(mass_k, mass_t)}{d_{geo}(v_k,v_t)},
\end{equation}
where $Wasserstein(\cdot,\cdot)$ is the Wasserstein distance, $d_{geo}(\cdot,\cdot)$ is the geodesic distance function, and $mass_k$ is the mass distribution~\cite{ollivier2009ricci} of node $v_k$.
Note that $SC$ can be either positive or negative and larger $SC$ means lower squashing because the ring structures are more friendly for information sharing.
\begin{figure}[!htp]
\centering
\includegraphics[width=0.95\linewidth]{figs/oneSBM-samestructure.pdf}
\caption{Predictions of GCN with the same graph structure and different labeled nodes.}
\label{fig:samestructure}
\end{figure}
In Figure~\ref{fig:samestructure} and Figure~\ref{fig:samelabels}, we show the relation between the reaching coefficient $RC$, the squashing coefficient $SC$, and the classification accuracy.
The higher the accuracy, the darker and larger the corresponding scatter.
First, we analyze the performance of GCN when trained on the same graph structure but with different labeled nodes.
In Figure~\ref{fig:samestructure}, we generate a synthetic graph by the Stochastic Block Model (SBM)~\cite{holland1983stochastic} with 4 classes and 3,000 nodes.
We randomly sample some nodes as the labeled nodes 10 times and scatter the classification accuracy in Figure~\ref{fig:samestructure}.
We can observe that even for the same graph structure, the difference in positions of labeled nodes may bring up to 15.42\% difference in accuracy.
There is a significant positive correlation between the reaching coefficient, the squashing coefficient, and the model performance.
Then we analyze the performance of GCN when trained with the same labeled nodes but on different graph structures.
In Figure~\ref{fig:samelabels}, we set the labeled nodes to be the same and generate different structures between them by controlling the edge probability between communities in the SBM model.
We can observe that with the same supervision information, there is up to a 26.26\% difference in accuracy because of the difference in graph structures.
There is also a significant positive correlation between the reaching coefficient, the squashing coefficient, and the model performance.
When the graph shows better community structure among nodes of the same class, the node representations can be learned better.
\textbf{Therefore, we make the following conclusions: }
(1)
Topology-imbalance hurts the performance of graph learning in the way of under-reaching and over-squashing.
(for Q1)
(2) The proposed two quantitative metrics can effectively reflect the degree of topology-imbalance.
Graph with poor reachability (i.e., smaller $RC$) and stronger squashing (i.e., smaller $SC$) is more susceptible to topology-imbalance. (for Q2)
(3) Optimizing the graph structure can effectively solve the topology-imbalance issue.
The above conclusions provide the guideline for designing the framework of PASTEL, i.e., balance the supervision information distribution by learning a structure with better reachability and lower squashing.
\section{Experiment}
In this section, we first evaluate PASTEL\footnote{The code of PASTEL~is available at \url{https://github.com/RingBDStack/PASTEL}.} on both real-world graphs and synthetic graphs.
Then we analyze the main mechanisms of PASTEL~and the learned structure.
We mainly focus on the following research questions:
\begin{itemize}[leftmargin=*]
\item \textbf{RQ1.}
How does PASTEL~perform in the node classification task?
(Section~\ref{sec:classification})
\item \textbf{RQ2.}
How does the position encoding and the class-wise conflict measure influence the performance of PASTEL?
(Section~\ref{sec:abla})
\item \textbf{RQ3.}
What graph structure PASTEL~tend to learn? (Section~\ref{sec:structure})
\end{itemize}
\subsection{Experimental Setups}
\begin{table*}[]
\caption{Weighted-F1 score and Macro-F1 score (\% ± standard deviation) of node classification on real-world graph datasets. }
\label{tab:cls}
\resizebox{\linewidth}{!}{%
\centering
\begin{tabular}{cccccccccccccc}
\hline
&
&
\multicolumn{2}{c}{Cora} &
\multicolumn{2}{c}{Citeseer} &
\multicolumn{2}{c}{Photo} &
\multicolumn{2}{c}{Actor} &
\multicolumn{2}{c}{Chameleon} &
\multicolumn{2}{c}{Squirrel} \\ \hline
Backbone &
Model &
W-F1 &
M-F1 &
W-F1 &
M-F1 &
W-F1 &
M-F1 &
W-F1 &
M-F1 &
W-F1 &
M-F1 &
W-F1 &
M-F1 \\ \hline
\multirow{8}{*}{GCN} &
original &
79.4±0.9 &
77.5±1.5 &
66.3±1.3 &
62.2±1.2 &
85.4±2.8 &
84.6±1.3 &
21.8±1.3 &
20.9±1.4 &
30.5±3.4 &
30.5±3.3 &
21.9±1.2 &
21.9±1.2 \\
&
ReNode &
80.0±0.7 &
78.4±1.3 &
66.4±1.0 &
62.4±1.1 &
86.2±2.4 &
85.3±1.6 &
21.2±1.2 &
20.2±1.6 &
30.3±3.2 &
30.4±2.8 &
22.4±1.1 &
22.4±1.1 \\
&
AddEdge &
79.0±0.9&
77.0±1.4&
66.2±1.3&
62.2±1.3&
85.5±1.5&
86.1±1.8&
21.2±1.3&
20.3±1.5&
30.6±1.6&
30.4±1.7&
21.7±1.5&
21.7±1.5 \\
&
DropEdge &
79.8±0.8&
77.8±1.0&
66.6±1.4&
63.4±1.6&
86.8±1.7&
85.4±1.3&
22.4±1.0&
21.4±1.3&
30.6±3.5&
30.6±3.3&
22.8±1.2&
22.8±1.2 \\
&
SDRF &
82.1±0.8 &
80.6±0.8 &
69.6±0.4 &
66.6±0.3 &
> 5 days &
> 5 days &
> 5 days &
> 5 days &
39.1±1.2 &
39.0±1.2 &
> 5 days &
> 5 days \\
&
NeuralSparse &
81.7±1.4 &
80.9±1.4 &
{\ul 71.8±1.2} &
{\ul 69.0±1.0} &
{\ul 89.7±1.9} &
88.7±1.8 &
24.4±1.5 &
{\ul 23.6±1.6} &
44.9±3.0 &
44.9±2.8 &
28.1±1.8 &
28.1±1.8 \\
&
IDGL &
{\ul 82.3±0.6} &
{\ul 81.0±0.9} &
71.7±1.0 &
68.0±1.3 &
88.6±2.3 &
{\ul 88.8±1.4} &
{\ul 24.9±0.8} &
22.0±0.7 &
{\ul 55.4±1.8} &
{\ul 55.0±1.7} &
{\ul 28.8±2.3} &
{\ul 28.9±2.2} \\
&
\textbf{PASTEL} &
\textbf{82.5±0.3} &
\textbf{81.2±0.3} &
\textbf{72.9±0.8} &
\textbf{69.3±0.9} &
\textbf{91.4±2.7} &
\textbf{91.3±2.2} &
\textbf{26.4±1.0} &
\textbf{24.4±1.2} &
\textbf{57.8±2.4} &
\textbf{57.3±2.4} &
\textbf{37.5±0.6} &
\textbf{37.5±0.7} \\ \hline
\multirow{8}{*}{GAT} &
original &
78.3±1.5 &
76.4±1.7 &
64.4±1.7 &
60.6±1.7 &
88.2±2.9 &
86.2±2.6 &
21.8±1.2 &
20.9±1.1 &
29.9±3.5 &
29.9±3.1 &
20.5±1.4 &
20.5±1.4 \\
&
ReNode &
78.9±1.2 &
77.2±1.5 &
64.9±1.6 &
61.0±1.5 &
89.1±2.4 &
87.1±2.6 &
21.5±1.2 &
20.5±1.1 &
29.2±2.3 &
29.1±2.0 &
20.4±1.8 &
20.4±1.8 \\
&
AddEdge &
78.0±1.6&
76.2±1.6&
64.0±1.3&
60.2±1.3&
88.2±2.4&
86.2±2.5&
21.3±1.2&
20.3±1.1&
29.8±1.7&
29.6±1.5&
20.7±1.6&
20.7±1.6 \\
&
DropEdge &
78.7±1.3&
76.9±1.5&
64.5±1.4&
60.5±1.3&
88.9±1.9&
87.1±2.1&
22.9±1.2&
21.8±1.1&
30.3±1.6&
30.2±1.2&
21.2±1.5&
21.2±1.5 \\
&
SDRF &
77.9±0.7 &
75.9±0.9 &
64.9±0.6 &
{\ul 61.9±0.9} &
> 5 days &
> 5 days &
> 5 days &
> 5 days &
43.0±1.9 &
42.5±1.9 &
> 5 days &
> 5 days \\
&
NerualSparse &
{\ul 81.4±4.8} &
79.4±4.8 &
64.8±1.5 &
{\ul 61.9±1.3} &
{\ul 90.2±2.5} &
{\ul 88.0±2.3} &
{\ul 23.4±1.7} &
\textbf{22.4±1.5} &
45.6±2.1 &
45.5±1.8 &
{\ul 28.8±1.3} &
{\ul 28.8±1.3} \\
&
IDGL &
80.6±1.0 &
{\ul 79.7±0.9} &
{\ul 66.5±1.5} &
{\ul 61.9±1.9} &
89.9±3.1 &
87.7±2.6 &
22.4±1.5 &
21.8±1.2 &
{\ul 48.4±4.0} &
{\ul 47.8±3.1} &
27.0±2.6 &
27.0±2.6 \\
&
\textbf{PASTEL} &
\textbf{81.9±1.4} &
\textbf{80.7±1.2} &
\textbf{66.6±1.9} &
\textbf{62.0±1.7} &
\textbf{91.8±3.2} &
\textbf{89.4±2.9} &
\textbf{24.4±2.6} &
{\ul 22.1±2.6} &
\textbf{52.1±2.7} &
\textbf{52.5±2.8} &
\textbf{35.3±0.9} &
\textbf{35.3±0.8} \\ \hline
\multirow{8}{*}{APPNP} &
original &
80.6±1.6 &
79.3±1.2 &
66.5±1.5 &
62.3±1.5 &
89.3±1.6 &
86.3±1.7 &
21.1±1.5 &
20.7±1.1 &
35.3±4.0 &
35.0±3.8 &
23.1±1.6 &
23.1±1.6 \\
&
ReNode &
81.1±0.9 &
79.9±0.9 &
66.6±1.7 &
62.4±1.6 &
89.6±1.4 &
87.2±1.3 &
20.2±2.0 &
20.0±1.7 &
33.5±2.5 &
33.3±2.3 &
23.9±2.0 &
23.9±2.0 \\
&
AddEdge &
80.3±1.3&
78.8±1.1&
66.6±2.1&
62.5±2.1&
89.3±1.2&
86.4±1.2&
21.5±1.3&
20.7±1.4&
35.7±1.7&
35.4±1.2&
23.1±1.6&
23.2±1.7 \\
&
DropEdge &
80.9±1.4&
79.4±1.2&
66.7±2.0&
63.0±1.9&
90.0±1.2&
87.0±1.2&
{\ul 21.8±1.8}&
20.8±1.4&
36.0±1.7&
35.7±1.6&
23.3±1.7&
23.3±1.7 \\
&
SDRF &
80.7±0.9 &
79.1±0.8 &
{\ul 67.1±0.6} &
{\ul 63.1±0.8} &
> 5 days &
> 5 days &
> 5 days &
> 5 days &
36.5±2.1 &
35.8±2.1 &
> 5 days &
> 5 days \\
&
NerualSparse &
81.1±1.4 &
79.9±1.2 &
66.8±1.9 &
62.7±1.9 &
91.3±1.8 &
{\ul 89.4±1.6} &
{\ul 21.8±1.9} &
\textbf{21.4±1.5} &
39.1±2.9 &
38.7±2.8 &
28.3±1.5 &
28.3±1.5 \\
&
IDGL &
{\ul 81.3±0.9} &
\textbf{80.2±0.9} &
67.0±1.3 &
62.9±1.3 &
{\ul 91.6±1.3} &
88.6±2.2 &
21.4±2.4 &
20.1±2.4 &
{\ul 41.2±2.2} &
{\ul 40.6±2.6} &
{\ul 29.6±2.3} &
{\ul 29.7±2.2} \\
&
\textbf{PASTEL} &
\textbf{82.0±1.0} &
{\ul 80.0±0.9} &
\textbf{67.3±1.3} &
\textbf{63.2±1.5} &
\textbf{92.3±3.1} &
\textbf{89.9±2.5} &
\textbf{22.5±2.0} &
{\ul 20.9±2.1} &
\textbf{44.2±3.2} &
\textbf{43.8±3.4} &
\textbf{34.6±1.6} &
\textbf{34.6±1.6} \\ \hline
\multirow{8}{*}{GraphSAGE} &
original &
75.4±1.6 &
74.1±1.6 &
64.8±1.6 &
60.7±1.6 &
86.1±2.5 &
83.3±2.4 &
24.0±1.2 &
23.2±1.0 &
36.5±1.6 &
36.2±1.6 &
27.2±1.7 &
27.2±1.7 \\
&
ReNode &
76.4±0.9 &
75.0±1.1 &
65.4±1.7 &
61.2±1.7 &
86.5±1.7 &
84.1±1.7 &
23.7±1.2 &
22.8±1.0 &
36.4±1.9 &
36.1±1.9 &
27.7±1.8 &
27.7±1.8 \\
&
AddEdge &
75.2±1.2&
73.7±1.2&
65.0±1.4&
60.9±1.3&
86.1±2.8&
83.4±2.6&
23.8±1.7&
23.2±1.6&
36.5±1.5&
36.2±1.3&
26.9±2.1&
26.9±2.1 \\
&
DropEdge &
76.0±1.6&
74.5±1.6&
65.1±1.4&
60.9±1.4&
86.2±1.6&
83.5±1.4&
24.1±1.0&
23.3±0.9&
37.5±1.4&
37.2±1.4&
27.5±1.8&
27.5±1.8 \\
&
SDRF &
75.7±0.8 &
74.6±0.8 &
65.3±0.6 &
\textbf{61.4±0.6} &
> 5 days &
> 5 days &
> 5 days &
> 5 days &
41.5±2.6 &
41.6±2.7 &
> 5 days &
> 5 days \\
&
NerualSparse &
{\ul 79.7±1.8} &
77.8±1.6 &
64.7±1.4 &
61.1±1.3 &
89.1±5.4 &
{\ul 86.7±5.5} &
{\ul 25.1±1.2} &
\textbf{24.4±1.1} &
39.1±1.9 &
39.0±1.9 &
32.2±2.4 &
32.2±2.4 \\
&
IDGL &
79.2±0.9 &
{\ul 78.4±0.8} &
{\ul 65.6±0.9} &
{\ul 61.3±1.2} &
{\ul 90.0±1.0} &
86.3±1.3 &
24.0±2.6 &
22.4±2.7 &
{\ul 43.8±3.4} &
{\ul 43.0±3.2} &
{\ul 33.9±0.9} &
{\ul 33.9±0.8} \\
&
\textbf{PASTEL} &
\textbf{81.1±0.8} &
\textbf{79.8±0.7} &
\textbf{65.7±1.1} &
\textbf{61.4±1.4} &
\textbf{92.0±0.6} &
\textbf{89.0±1.0} &
\textbf{26.0±2.4} &
{\ul 23.6±2.7} &
\textbf{47.7±0.9} &
\textbf{46.9±0.9} &
\textbf{35.5±1.4} &
\textbf{35.5±1.4} \\ \hline
\end{tabular}
}
\end{table*}
\subsubsection{Datasets}
We conduct experiments on synthetic and real-world datasets to analyze the model's capabilities in terms of both graph theory and real-world scenarios.
The real-word datasets include various networks with different heterophily degrees to demonstrate the generalization of PASTEL.
Cora and Citeseer~\cite{sen2008cora} are citation networks.
Photo~\cite{shchur2018pitfalls} and and Actor~\cite{pei2020geom} are co-occurrence network.
Chameleon and Squirrel~\cite{rozemberczki2021multi} are page-page networks in Wikipedia.
Since we focus on the topology-imbalance issue in this work, we set the number of labeled nodes in each class to be 20.
\subsubsection{Baselines}
We choose representative GNNs as backbones including GCN~\cite{GCN}, GAT~\cite{velivckovic2017graph}, APPNP~\cite{klicpera2018predict}, and GraphSAGE~\cite{hamilton2017inductive}.
The most important baseline is ReNode~\cite{chen2021topology}, which is the only existing work for the topology-imbalance issue.
We also include some graph structure learning baselines to illustrate the specific effectiveness of PASTEL~for the topology-imbalance issue.
DropEdge~\cite{rong2019dropedge} randomly removes edges at each epoch as structure augmentation.
To evaluate the effect of increasing the reachability randomly, we use a adding edges method named AddEdge, whose adding strategy is similar to DropEdge.
SDRF~\cite{topping2021understanding} rewires edges according to their curvatures for the over-squashing issue.
NeuralSparse~\cite{zheng2020robust} removes potentially task-irrelevant edges for clearer class boundaries.
IDGL~\cite{chen2020iterative} updates the node representations and structure based on these representations iteratively.
\subsubsection{Parameter Settings}
For the GNN backbones, we set their depth to be 2 layers and adopt the implementations from the PyTorch Geometric Library in all experiments.
We set the representation dimension of all baselines and PASTEL~to be 256.
We re-implement the NeuralSparse~\cite{zheng2020robust} and SDRF~\cite{topping2021understanding} and the parameters of baseline methods are set as the suggested value in their papers or carefully tuned for fairness.
For DropEdge and AddEdge, we set the edge dropping/adding probability to 10\%.
For PASTEL, we set the number of heads $m=4$ and the random walk restart probability $\alpha=0.15$.
The structure fusing coefficients ($\lambda_1$ and $\lambda_2$) and the loss coefficients ($\beta_1$, $\beta_2$ and $\beta_3$) are tuned for each dataset.
\subsection{Evaluation (RQ1)}
\label{sec:classification}
\begin{table}[]
\caption{Weighted-F1 scores and improvements on graphs with different levels of topology-imbalance.}
\label{tab:LMH}
\resizebox{\linewidth}{!}{%
\centering
\begin{tabular}{lcccccc}
\hline
& \multicolumn{2}{c}{Cora-L} & \multicolumn{2}{c}{Cora-M} & \multicolumn{2}{c}{Cora-H} \\ \hline
&
\begin{tabular}[c]{@{}c@{}}$RC$\\ 0.4130\end{tabular} &
\begin{tabular}[c]{@{}c@{}}$SC$\\ -0.6183\end{tabular} &
\begin{tabular}[c]{@{}c@{}}$RC$\\ 0.4100\end{tabular} &
\begin{tabular}[c]{@{}c@{}}$SC$\\ -0.6204\end{tabular} &
\begin{tabular}[c]{@{}c@{}}$RC$\\ 0.4060\end{tabular} &
\begin{tabular}[c]{@{}c@{}}$SC$\\ -0.6302\end{tabular} \\ \cline{2-7}
& W-F1 (\%) & $\Delta$ (\%) & W-F1 (\%) & $\Delta$ (\%) & W-F1 (\%) & $\Delta$ (\%) \\ \hline
GCN & 80.9±0.9 & — & 78.8±0.8 & — & 77.5±1.0 & — \\
ReNode & 81.3±0.7 & $\uparrow$0.4 & 79.3±0.8 & $\uparrow$0.5 & 78.3±1.1 & $\uparrow$0.8 \\
SDRF & 81.0±0.7 & $\uparrow$0.1 & 78.9±0.8 & $\uparrow$0.1 & 77.9±0.7 & $\uparrow$0.4 \\
IDGL & 82.5±1.0 & $\uparrow$1.6 & 80.4±1.0 & $\uparrow$1.6 & 81.6±1.1 & $\uparrow$4.1 \\
\textbf{PASTEL} & \textbf{82.7±0.9} & $\uparrow$\textbf{1.8} & \textbf{81.0±0.9} & $\uparrow$\textbf{2.2} & \textbf{81.9±1.1} & $\uparrow$\textbf{4.4} \\ \hline
\end{tabular}
}
\end{table}
\subsubsection{PASTEL~for Real-world Graphs}
\label{sec:cls_realworld}
We compare PASTEL~with the baselines on several datasets on node classification.
The overall Weighted-F1 (W-F1) scores and the class-balance Macro-F1 (M-F1) scores on different backbones are shown in Table~\ref{tab:cls}.
The best results are shown in bold and the runner-ups are underlined.
PASTEL~shows overwhelming superiority in improving the performance of backbones on all datasets.
It demonstrates that PASTEL~is capable of learning better structures with a more balanced label distribution that reinforces the GNN models.
ReNode~\cite{chen2021topology} achieves fewer improvements on datasets of poor connectivity (e.g., CiteSeer) and even damages the performance of backbones on heterophilic datasets (e.g., Chameleon and Actor).
We think it's because ReNode~\cite{chen2021topology} detects conflicts by Personalized PageRank and fails to reflect the node topological position well when the graph connectivity is poor.
Besides, ReNode takes the topology boundary as the decision boundary, which is not applicable for heterophilic graphs.
AddEdge doesn't work in most cases, demonstrating that randomly adding edge is not effective in boosting the reachability.
The structure augmentation strategy should be carefully designed considering the node relations.
SDRF~\cite{topping2021understanding} can improve the performance, supporting our intuition that relieving over-squashing helps graph learning.
But SDRF is still less effective than PASTEL~because it only considers the topological properties rather than the supervision information.
Both NeuralSparse~\cite{zheng2020robust} and IDGL~\cite{chen2020iterative} show good performance among the baselines, showing the effectiveness of learning better structures for downstream tasks.
However, they are still less effective than PASTEL~which
takes the supervision information distribution into consideration.
\begin{table*}[!tp]
\caption{Weighted-F1 scores (\%) and improvements ($\Delta$) on synthetic SBM graphs with different community structures.}
\label{tab:SBM}
\resizebox{0.92\linewidth}{!}{%
\centering
\begin{tabular}{c|cc|cc|cc|cc|cc|cc|cc}
\hline
&
\multicolumn{2}{c|}{SBM-1} &
\multicolumn{2}{c|}{SBM-2} &
\multicolumn{2}{c|}{SBM-3} &
\multicolumn{2}{c|}{SBM-4} &
\multicolumn{2}{c|}{SBM-5} &
\multicolumn{2}{c|}{SBM-6} &
\multicolumn{2}{c}{SBM-7} \\ \hline
$p$ &
\multicolumn{2}{c|}{0.5000} &
\multicolumn{2}{c|}{0.5000} &
\multicolumn{2}{c|}{0.5000} &
\multicolumn{2}{c|}{0.5000} &
\multicolumn{2}{c|}{0.5000} &
\multicolumn{2}{c|}{0.5000} &
\multicolumn{2}{c}{0.5000} \\
$q$ &
\multicolumn{2}{c|}{0.0300} &
\multicolumn{2}{c|}{0.0100} &
\multicolumn{2}{c|}{0.0083} &
\multicolumn{2}{c|}{0.0071} &
\multicolumn{2}{c|}{0.0063} &
\multicolumn{2}{c|}{0.0056} &
\multicolumn{2}{c}{0.0050} \\ \hline
$RC$ &
\multicolumn{2}{c|}{0.4979} &
\multicolumn{2}{c|}{0.4984} &
\multicolumn{2}{c|}{0.4990} &
\multicolumn{2}{c|}{0.4994} &
\multicolumn{2}{c|}{0.5002} &
\multicolumn{2}{c|}{0.5004} &
\multicolumn{2}{c}{0.5009} \\
$SC$ &
\multicolumn{2}{c|}{0.0998} &
\multicolumn{2}{c|}{0.0999} &
\multicolumn{2}{c|}{0.1000} &
\multicolumn{2}{c|}{0.1001} &
\multicolumn{2}{c|}{0.1007} &
\multicolumn{2}{c|}{0.1017} &
\multicolumn{2}{c}{0.1144} \\ \hline
&
W-F1 &
$\Delta$ &
W-F1 &
$\Delta$ &
W-F1 &
$\Delta$ &
W-F1 &
$\Delta$ &
W-F1 &
$\Delta$ &
W-F1 &
$\Delta$ &
W-F1 &
$\Delta$ \\ \hline
GCN &
40.29 &
— &
42.37 &
— &
42.99 &
— &
44.13 &
— &
45.19 &
— &
45.21 &
— &
45.22 &
— \\
ReNode &
41.33 &
$\uparrow$1.04 &
42.40 &
$\uparrow$0.03 &
43.21 &
$\uparrow$0.22 &
44.56 &
$\uparrow$0.43 &
45.20 &
$\uparrow$0.01 &
45.08 &
$\downarrow$0.13 &
44.89 &
$\downarrow$0.33 \\
\textbf{PASTEL} &
\textbf{45.67} &
\textbf{$\uparrow$5.38} &
\textbf{57.61} &
\textbf{$\uparrow$15.24} &
\textbf{58.33} &
\textbf{$\uparrow$15.34} &
\textbf{60.29} &
\textbf{$\uparrow$16.16} &
\textbf{66.41} &
\textbf{$\uparrow$21.22} &
\textbf{66.45} &
\textbf{$\uparrow$21.24} &
\textbf{66.57} &
\textbf{$\uparrow$21.35} \\ \hline
\end{tabular}
}
\end{table*}
\subsubsection{PASTEL~under Different Levels of Topology-imbalance}
To further analyze PASTEL's ability in alleviating the topology-imbalance issue, we verify the PASTEL~under different levels of topology-imbalance.
We randomly sampled 1,000 training sets and calculate the reaching coefficient $RC$ and squashing coefficient $SC$ as introduced in Section~\ref{sec:understand}.
Then we choose 3 training sets with different levels of topology-imbalance according to the conclusion in Section~\ref{sec:quantitative} and we denote them as Cora-L, Cora-M, and Cora-H, according to the degree of topology imbalance.
Note that larger $RC$ means better reachability and larger $SC$ means lower squashing.
We evaluate PASTEL~and several baselines with the GCN as the backbone and show the dataset information, the Weighted-F1 scores, and their improvements ($\Delta$) over the backbones in Table~\ref{tab:LMH}.
The performance of node representation learning generally gets worse with the increase of the topology-imbalance degree of the dataset.
Both the node re-weighting method (i.e., ReNode~\cite{chen2021topology}) and the structure learning methods (i.e., IDGL~\cite{chen2020iterative}, SDRF~\cite{topping2021understanding} and PASTEL) can achieve more improvement with the increase of dataset topology-imbalance.
PASTEL~performs best on all datasets with different degrees of topology-imbalance and it can achieve up to 4.4\% improvement on the highly topology-imbalance dataset.
\subsubsection{PASTEL~for Synthetic Graphs}
\begin{figure}[t]
\centering
\includegraphics[width=0.95\linewidth]{figs/abla_position.pdf}
\caption{The impact of position encoding. }
\label{fig:abla_position}
\end{figure}
We generate 7 synthetic graph datasets with different community structures using the Stochastic Block Model (SBM) $\mathcal{G}(N,C,p,q)$~\cite{holland1983stochastic}, where the number of nodes $N=3000$, the number of community $C=6$, $p$ denotes the edge probability within a community and $q$ denotes the edge probability between communities.
We show the classification Weighted-F1 scores and improvements are shown in Table~\ref{tab:SBM}.
With a more clear community structure, the reaching coefficient $RC$ increases and the squashing coefficient $SC$ also increases, leading to the increase of GCN's performance, which agrees with the conclusion obtained in Section~\ref{sec:quantitative}.
ReNode shows unsatisfied performance in boosting the node classification.
PASTEL~can increase the classification weighted-F1 score by 5.38\%-21.35\% on SBM graphs with different community structures, showing superior effectiveness.
\subsection{Analysis of PASTEL~(RQ2)}
\label{sec:abla}
We conduct ablation studies for the two main mechanisms of PASTEL, position encoding and class-wise conflict measure.
\subsubsection{Impact of the Position Encoding}
\label{sec:abla_position}
We design an anchor-based position encoding mechanism in Section~\ref{sec:position_encoding}, which reflects the relative topological position to labeled nodes and further maximizes the label influence within a class.
To evaluate the effectiveness of position encoding, we compare PASTEL~with a variant \textbf{PASTEL~(w/o PE)}, which removes the position encoding and directly take the node features for metric learning in Eq.~\eqref{eq:metric}.
\begin{figure}[!tp]
\centering
\includegraphics[width=0.95\linewidth]{figs/abla_conflict.pdf}
\caption{The impact of class-wise conflict measure. }
\label{fig:abla_conflict}
\end{figure}
Here we use the GCN as the backbone.
As shown in Figure~\ref{fig:abla_position}, the structure learning strategy of PASTEL~contributes the most, which can achieve at most 25.5\% improvement in terms of Weighted-F1 score with only node features.
Although PASTEL~(w/o PE) effectively improves the performance of backbones to some extent, the position encoding still benefits learning better structure to relieve the topology-imbalance with 1.0\%-1.8\% improvements than PASTEL~(w/o PE).
\begin{figure*}[!htp]
\centering
\subfigure[{Original Graph. }]{
\begin{minipage}[t]{ 0.1875\linewidth}
\centering
\includegraphics[width=\linewidth]{figs/original_cora.png}
\label{fig:cora}
\end{minipage}
}
\subfigure[ReNode. ]{
\begin{minipage}[t]{ 0.1875\linewidth}
\centering
\includegraphics[width=\linewidth]{figs/renode.png}
\label{fig:renode}
\end{minipage}
}
\subfigure[SDRF. ]{
\begin{minipage}[t]{ 0.1875\linewidth}
\centering
\includegraphics[width=\linewidth]{figs/sdrf.png}
\label{fig:SDRF}
\end{minipage}
}
\subfigure[IDGL. ]{
\begin{minipage}[t]{ 0.1875\linewidth}
\centering
\includegraphics[width=\linewidth]{figs/idgl.png}
\label{fig:IDGL}
\end{minipage}
}
\subfigure[PASTEL. ]{
\begin{minipage}[t]{ 0.1875\linewidth}
\centering
\includegraphics[width=\linewidth]{figs/ours.png}
\label{fig:ours}
\end{minipage}
}
\centering
\caption{Structure visualization. (a) Original graph of Cora and learned graphs by (b) ReNode, (c) SDRF, (d) IDGL and (e) PASTEL. }
\label{fig:visual}
\end{figure*}
\subsubsection{Impact of the Class-wise Conflict Measure}
\label{sec:abla_conflict}
We designed a class-wise conflict measure in Section~\ref{sec:conflict} as edge weights to guide learning structures with better intra-class connectivity.
Here, we compare PASTEL~with its two variants to analyze the impact of the class-wise conflict measure:
(1) \textbf{PASTEL~(w/o CCM)}, which removes the class-wise conflict measure and directly takes the learned edge possibilities in Eq.~\eqref{eq:metric} as the edge weights.
(2) \textbf{PASTEL~(Totoro)}, which takes the Totoro metric introduced in ReNode~\cite{chen2021topology} as the conflict measure of nodes in Eq.~\eqref{eq:conflict_weight}.
Here we use the GCN as the backbone.
The comparison results are shown in Figure~\ref{fig:abla_conflict}.
On four datasets, PASTEL~consistently outperforms the other two variants.
Even without the conflict measure, PASTEL~(w/o CCM) still shows better performance than PASTEL~(Totoro), indicating the limitation of ReNode when capturing the relative topology positions without clear homophily structures.
\subsection{Analysis of Learned Structure (RQ3)}
\label{sec:structure}
We analyze the learned graph by PASTEL~in terms of visualization and structural properties.
\subsubsection{Structure Visualization}
In Figure~\ref{fig:visual}, we visualize the original graph of Cora and the graphs learned by ReNode~\cite{chen2021topology}, SDRF~\cite{topping2021understanding}, IDGL~\cite{chen2020iterative} and PASTEL~using \textit{networkx}.
For clarity, the edges are not shown.
The solid points denote the labeled nodes, the hollow points denote the unlabeled nodes, and the layout of nodes denotes their connectivities.
The node size in Figure~\ref{fig:renode} denotes the learned node weight in ReNode, and the solid lines and dashed lines in Figure~\ref{fig:SDRF} denote the added and deleted edges by SDRF, respectively.
As we can observe, ReNode gives more weights to nodes in the topology center of each class and SDRF tends to build connections between distant or isolated nodes.
Even though the structure learned by IDGL can make the nodes of a class close, there are still some overlapping and entangled areas between classes.
Benefiting from the position encoding and class-wise conflict measure, PASTEL~can obtain graph structure with clearer class boundaries.
\begin{table}[tp]
\caption{Properties and performance of the original graph and learned graphs of Cora.}
\label{tab:graph_property}
\resizebox{\linewidth}{!}{%
\centering
\begin{tabular}{cccccc}
\hline
& Original Graph & ReNode & SDRF & IDGL & \textbf{PASTEL} \\ \hline
$RC$ & 0.4022 & 0.4022 & 0.4686 & 0.5028 & 0.5475 \\
$SC$ & -0.6299 & -0.6299 & -0.4942 & -0.4069 & -0.3389 \\ \hline
W-F1 (\%) & 79.44 & 80.34 & 82.01 & 82.38 & \textbf{82.86} \\ \hline
\end{tabular}
}
\end{table}
\subsubsection{Change of $RC$ and $SC$}
We also show the reaching coefficient $RC$ and the squashing coefficient $SC$ of the above graphs in Figure~\ref{fig:visual} and the Weighted-F1 score learned on them in Table~\ref{tab:graph_property}.
Here we choose the GCN as the model backbone.
All of the structure learning methods (SDRF~\cite{topping2021understanding}, IDGL~\cite{chen2020iterative} and PASTEL) learn structures with larger reaching coefficient and larger squashing coefficient, leading the performance improvement of node classification.
This phenomenon supports our propositions in Section~\ref{sec:quantitative} again.
\subsubsection{Change of GPR Vector}
\label{sec:gpr_visual}
The class-wise conflict measure is calculated by the Group PageRank (GPR), which reflects the label influence of each class.
We randomly choose 10 nodes for each class in Cora and show their GPR vectors $\mathbf{P}^{gpr}_{i}$ in the original graph in Figure~\ref{fig:gpr1} and the learned graph in Figure~\ref{fig:gpr2}, respectively, where the color shade denotes the magnitude, $V_i$ denotes 10 nodes of class $i$ and $C_i$ denotes the $i$-th class.
In Figure~\ref{fig:gpr1}, the off-diagonal color blocks are also dark, indicating that the label influence of each class that nodes obtained from the original graph is still entangled to some extent, which could bring difficulties to the GNN optimization.
After the structure learning guided by the proposed class-wise conflict measure, Figure~\ref{fig:gpr2} exhibits 7 clear diagonal blocks and the gaps between the diagonal and off-diagonal block are widened, indicating that nodes can receive more supervision information of its ground-truth class.
We can further make a conclusion that the class-wise conflict measure plays an important role on giving guidance for more class connectivity orthogonality.
\begin{figure}[tp]
\centering
\subfigure[Original Graph.]{
\begin{minipage}[t]{ 0.43\linewidth}
\centering
\includegraphics[width=\linewidth]{figs/GPR1.pdf}
\label{fig:gpr1}
\end{minipage}
}
\subfigure[Learned Graph.]{
\begin{minipage}[t]{ 0.43\linewidth}
\centering
\includegraphics[width=\linewidth]{figs/GPR2.pdf}
\label{fig:gpr2}
\end{minipage}
}
\centering
\caption{Heat maps for the Group PageRank value of (a) the original graph and (b) the learned graph by PASTEL. }
\label{fig:gpr}
\end{figure}
\section{Conclusion}
We proposed a novel framework named PASTEL~for the graph topology-imbalance issue.
We provide a new understanding and two quantitative analysis metrics of topology-imbalance in the perspective of under-reaching and over-squashing, answering the questions that how topology-imbalance affects GNN's performance as well as what graphs are susceptible to it.
PASTEL~designs an anchor-based position encoding mechanism and a class-wise conflict measure to obtain structures with better in-class connectivity.
Comprehensive experiments demonstrate the potential and adaptability of PASTEL.
An interesting future direction is to incorporate the proposed two quantitative metrics into the learning process to address topology-imbalance more directly.
|
2,877,628,088,958 | arxiv | \section*{Methods}%
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\title{Non-thermal X-rays from Colliding Wind Shock Acceleration in the Massive Binary $\eta$ Carinae}
\author{Kenji Hamaguchi$^{1,2, *}$,
Michael F. Corcoran$^{1,3}$,
Julian M. Pittard$^{4}$,\\
Neetika Sharma$^{2}$,
Hiromitsu Takahashi$^{5}$,
Christopher M. P. Russell$^{6,7}$,\\
Brian W. Grefenstette$^{8}$,
Daniel R. Wik$^{9}$,
Theodore R. Gull$^{6}$,\\
Noel D. Richardson$^{10}$,
Thomas I. Madura$^{11}$, \&
Anthony F. J. Moffat$^{12}$
}
\date{\it Nature Astronomy 2 (2018) 731-736}
\begin{document}
\maketitle
\begin{affiliations}
\item CRESST II and X-ray Astrophysics Laboratory NASA/GSFC, Greenbelt, MD 20771, USA, $^{*}[email protected]
\item Department of Physics, University of Maryland, Baltimore County, 1000 Hilltop Circle, Baltimore, MD 21250, USA
\item The Catholic University of America, 620 Michigan Ave. N.E., Washington, DC 20064, USA
\item School of Physics and Astronomy, The University of Leeds, Woodhouse Lane, Leeds LS2 9JT, UK
\item Department of Physical Sciences, Hiroshima University, Higashi-Hiroshima, Hiroshima 739-8526, Japan
\item Astrophysics Science Division, NASA Goddard Space Flight Center, Greenbelt, MD 20771, USA
\item Instituto de Astrof\'{i}sica, Pontificia Universidad Cat\'{o}lica de Chile, Santiago, Chile
\item Space Radiation Lab, California Institute of Technology, Pasadena, CA 91125, USA
\item Department of Physics \& Astronomy, University of Utah, Salt Lake City, UT 84112, USA
\item Ritter Observatory, Department of Physics and Astronomy, The University of Toledo, Toledo, OH 43606-3390, USA
\item San Jose State University, Department of Physics \& Astronomy, One Washington Square, San Jose, CA, 95192-0106, USA
\item D\'epartement de physique and Centre de Recherche en Astrophysique du Qu\'ebec (CRAQ), Universit\'e de Montr\'eal, C.P. 6128, Canada
\end{affiliations}
\begin{abstract}
Cosmic-ray acceleration has been a long-standing mystery \cite{Koyama1995a,Morlino2012a}
and despite more than a century of study, we still do not have a complete census of acceleration mechanisms.
The collision of strong stellar winds in massive binary systems creates powerful shocks,
which have been expected to produce high-energy cosmic-rays through Fermi acceleration at the shock interface.
The accelerated particles should collide with stellar photons or ambient material,
producing non-thermal emission observable in X-rays and $\gamma$-rays \citep{Pittard2006,DeBecker2017}.
The supermassive binary star $\eta$~Car\ drives the strongest colliding wind shock in the solar neighborhood \citep{Corcoran2005,Groh2012}.
Observations with non-focusing high-energy observatories indicate
a high energy source near $\eta$~Car, but have been unable to conclusively identify $\eta$~Car\ as the source
because of their relatively poor angular resolution \citep{Leyder2008,Sekiguchi2009,Abdo2010}.
Here we present the first
direct focussing observations of the non-thermal source in the extremely hard X-ray band,
which is found to be spatially coincident with the star within several arc-seconds.
These observations show that the source of non-thermal X-rays varies with the orbital phase of the binary,
and that the photon index of the emission is similar to that derived through analysis of the $\gamma$-ray spectrum.
This is conclusive evidence that the high-energy emission indeed originates from non-thermal particles
accelerated at colliding wind shocks.
\end{abstract}
Strong shocks accelerate particles to cosmic-ray energies through Fermi acceleration.
Supernova remnants are well established as a source of cosmic rays in the Milky Way \citep{Koyama1995a,Morlino2012a},
but other sources may also contribute.
Massive, luminous hot stars drive powerful stellar winds through their UV radiation \citep{Castor1975}
and, in a massive binary system,
the collision of the stellar winds will produce strong shocks and thermal X-ray emission.
This wind-wind collision region may serve as an additional source of cosmic-ray particles.
Indeed, non-thermal radio emission from colliding wind binary systems is often detected
\citep{Dougherty2000a,DeBecker2013}, and has been directly imaged by high-spatial-resolution observations
\citep[e.g.,][]{Williams1997,Dougherty2005}.
The emission is interpreted as radio synchrotron emission from high energy non-thermal electrons.
These accelerated, non-thermal particles can also produce high energy X-ray and $\gamma$-ray photons
through inverse-Compton (IC) scattering of stellar UV photons or pion-decay after collision with ambient material.
However, the detection of high energy non-thermal X-ray and $\gamma$-ray emission from colliding wind binaries is currently very challenging, and the handful of reported detections remain controversial \citep[see, e.g.,][]{DeBecker2017}.
The best candidate massive binary system for detecting the high-energy non-thermal radiation produced by
a shock-accelerated population of high-energy particles is $\eta$~Car.
Eta Carinae is the most luminous binary in our Galaxy and the variable thermal X-ray emission
produced by the hot plasma ({\it kT}~$\sim$4$-$5~keV, $L_{\rm X}$ $\sim$10$^{35}$~{\rm ergs~s$^{-1}$}) in its colliding wind shock has been well studied
\citep[][and references therein]{Corcoran2017a}.
The primary is one of the most massive stars in our Galaxy \citep[$\gtrsim$100~$M_{\odot}$,][]{Hillier2001}
and drives a powerful wind \citep[$v\sim$ 420~{\rm km~s$^{-1}$}, {$\dot M$}~$\sim$8.5$\times$10$^{-4}$$M_{\odot}$~yr$^{-1}$,][]{Groh2012}.
The secondary is perhaps a massive star of O or Wolf-Rayet type, which has never been directly observed, though its wind properties
\citep[$\sim$3000~{\rm km~s$^{-1}$}, {$\dot M$}~$\sim$10$^{-5}$$M_{\odot}$~yr$^{-1}$,][]{Pittard2002}
have been deduced through analysis of its X-ray spectrum.
Variations across the electromagnetic spectrum from $\eta$~Car\ have shown that the system has a long-period orbit with high eccentricity
\citep[$e \sim$0.9, $P \sim$5.54 yrs,][]{Corcoran2005,Damineli2008}.
In extremely high energy X-rays (15$-$100~keV),
the {\it INTEGRAL}\ and {\it Suzaku}\ observatories claimed detection of a non-thermal source near $\eta$~Car\
\citep{Leyder2008,Leyder2010,Sekiguchi2009,Hamaguchi2014b},
but two more sensitive {\it NuSTAR}\ observations near periastron in 2014 did not confirm this \citep{Hamaguchi2016a}.
The {\it AGILE}\ and {\it Fermi}\ space observatories detected a GeV $\gamma$-ray source near $\eta$~Car\ \citep{Tavani2009,Abdo2010},
while the
HESS telescope detected a source of high-energy $\gamma$-ray emission \citep{Leser2017a} at energies up to 300 GeV.
The $\gamma$-ray spectrum shows two components, above and below 10~GeV.
Both components vary slowly with {$\eta$~Car}'s orbital phase \citep[e.g.,][]{Reitberger2015}.
The poor angular resolutions ($\gtrsim$10$'$) of these observations
meant that $\eta$~Car\ could not be conclusively confirmed as the source of the high-energy emission.
The {\it NuSTAR}\ X-ray observatory, launched in 2012,
provides for the first time focusing observations at energies up to 79~keV \citep{Harrison2013}.
We obtained 11 {\it NuSTAR}\ observations of $\eta$~Car\ around $\eta$~Car's last periastron passage in 2014
through 2015 and 2016, along with coordinated observations at energies between 0.3$-$12~keV with the {\it XMM-Newton}\ observatory \cite{Jansen2001}.
The {\it NuSTAR}\ image at the highest available energy in which the source can be detected above background (30$-$50~keV) shows,
for the first time, that even at these high energies the emission clearly arises
in the direction of and is well-centered on the position of $\eta$~Car\ (Figure~\ref{fig:image}).
The soft X-ray ($<$15~keV) spectra obtained by {{\it NuSTAR}} are characterized by thermal emission from plasma with a maximum
temperature of 4$-$5~keV (Figure~\ref{fig:spectra}),
which is consistent with the {\it XMM-Newton}\ spectra simultaneously obtained,
and previous analyses of {$\eta$~Car}'s thermal X-ray emission \citep[e.g.,][]{Hamaguchi2014a}.
However, the extremely hard ($\gtrsim$15~keV) X-ray emission seen in 2015 and 2016,
following {$\eta$~Car}'s periastron passage in 2014,
is significantly brighter and flatter in slope than the {\it kT}~$\sim$4$-$5~keV plasma emission in this energy range,
and is detected above background up to energies of 50~keV.
The spectrum obtained in 2014 March 31, which is 4 times brighter than the 2015 and 2016 spectra below 15~keV,
follows the {\it kT} $\sim$4.5~keV thermal emission spectrum up to 30~keV,
but it flattens above that energy and converges to the 2015 \& 2016 spectrum.
The other two observations obtained near the maximum of the thermal X-ray emission,
which occurs just prior to periastron passage (Figure~\ref{fig:flux}),
follow a similar trend in the hard band slope and converge to the 2015 \& 2016 spectrum in the same way.
This result confirms the {\it kT}~$\sim$4$-$5~keV thermal component variability with orbital phase seen previously,
but it reveals that the highest energy emission is characterized by a flat emission component that is nearly
constant outside periastron passage.
The {\it NuSTAR}\ spectrum, however, shows that
this hard flat component nearly disappears during the minimum of the {\it kT}~$\sim$4$-$5~keV thermal emission
near periastron passage.
This {\it kT}~$\sim$4$-$5~keV thermal X-ray minimum is believed to be
caused by orbital changes in the head-on wind collision
both geometrically (i.e., eclipse by the primary wind) and mechanically (decay of the collisional shock activity) \citep{Hamaguchi2014a}.
The decline of the hard, flat component along with {\it kT}~$\sim$4$-$5~keV thermal X-ray minimum,
as well as the positional coincidence of the extremely hard source with $\eta$~Car,
is conclusive proof that $\eta$~Car\ itself, and its colliding wind activity,
is the source of this flat high-energy X-ray component.
If the 30$-$50~keV emission is thermal in nature, it would require a temperature of {\it kT}~$\gtrsim$20 keV,
a temperature much higher than could be mechanically produced by the wind of either star.
Thus the hard flat source must be produced by non-thermal processes.
We characterize the spectrum using a simple power-law spectrum of the form $KE^{-\Gamma}$
(where $K$ is the flux normalization, $E$ the photon energy, and $\Gamma$ the photon index).
We minimized the systematic uncertainty of the instrumental and cosmic background
through a detailed background study.
Our analysis constrained $\Gamma$ to be less than 3.
Values of $\Gamma \sim$3 can be ruled out since the non-thermal emission would then contribute significantly
to the observed emission below $10$~keV
at phases away from periastron; this would cause a variation of the equivalent width of the strong thermal line from He-like iron at 6.7~keV with phase, which is not seen.
Therefore, the photon index has to be in the range $\Gamma \lesssim2$.
There are several non-thermal emission processes that the colliding wind activity can drive --- synchrotron emission,
synchrotron self-Compton, IC up-scattering of stellar photons, relativistic bremsstrahlung and pion-decay.
However, to match the observed flux at 50 keV, the synchrotron process would require electrons with
Lorentz factor $\gamma \sim$3$\times$10$^{6}$ for a reasonable magnetic field strength ($B \sim$ 1~Gauss),
which do not seem likely to exist given the expected strong IC cooling \citep[e.g.,][]{Pittard2006b}.
Pion-decay emission peaks at 67.5 MeV and is important only above $\sim$10 MeV, while relativistic bremsstrahlung emission
and synchrotron self-Compton are unlikely to match the emission from IC up-scattering \citep[e.g.,][]{Pittard2006}.
Furthermore, the value of $\Gamma \lesssim$2 we derived is typical of 1st order Fermi acceleration and
similar to the radio indices measured from another well-known massive colliding wind binary system, WR 140 \citep{Dougherty2005}.
Thus IC up-scattering is the most plausible mechanism to produce the non-thermal emission in the extremely hard X-ray band.
This result demonstrates the presence of a high-energy non-thermal X-ray source physically associated with $\eta$~Car\
and lends additional strong support to the idea that the $\gamma$-ray source is also physically associated with $\eta$~Car.
With the now established physical association between the {\it NuSTAR}\ and {\it Fermi}\ sources, it now makes sense to consider
a consistent model for both the X-ray and $\gamma$-ray emission.
The extremely hard X-ray component seen by {\it NuSTAR}\ smoothly connects to the soft GeV $\gamma$-ray spectrum at a power-law slope of $\Gamma \sim1.65$
(Figure~\ref{fig:spectra} {\it right}).
This component also shows similar flux variation to the soft GeV component \citep[Figure~\ref{fig:flux} {\it bottom},][]{Reitberger2015}.
These characteristics strongly suggest that
the non-thermal X-ray component seen by {\it NuSTAR}\ is the low-energy tail of the soft GeV $\gamma$-ray component
produced by the IC mechanism \citep{Abdo2010,Farnier2011}.
There would be no obvious connection between the $\gamma$-ray and hard X-ray emission
if the soft GeV $\gamma$-ray component originates from the pion decay process \citep{Ohm2015}.
Earlier {\it INTEGRAL}\ and {\it Suzaku}\ flux measurements of extremely high energy emission were 2$-$3 times larger
than our {\it NuSTAR}\ measurements \citep[Figure~\ref{fig:flux},][]{Leyder2010,Hamaguchi2014b},
but the soft GeV emission has not varied remarkably since the beginning of {{\it Fermi}}'s monitoring in 2008.
This discrepancy either indicates some
cycle-to-cycle variation in the non-thermal emission (which seems unlikely given the consistency of the {\it NuSTAR}\ and {\it Fermi}\
spectra), or that these earlier measurements have overestimated the intrinsic source flux
due to poorly determined backgrounds or other issues.
A puzzle is the lack of an increase in luminosity of this IC scattered component as the thermal plasma emission increases near periastron.
If the non-thermal electrons fill the wind colliding region,
the IC luminosity should be proportional to the product of
the number of non-thermal electrons and the intensity of the stellar UV,
and the product is also proportional to the thermal plasma luminosity for a constant temperature.
That this variation is not observed can be explained by the rapid cooling that the non-thermal electrons undergo due to IC scattering as the stars approach each other.
Because of this effect, the non-thermal electrons that are capable of producing 50 keV photons ( i.e. those with a Lorentz factor $\gamma \sim$200)
gradually exist only in a thin layer downstream of the shock \citep{Pittard2006b}, rather than filling the entire wind colliding region.
This process would decrease the number of non-thermal electrons near periastron
and produce a flat light curve toward the X-ray maximum.
By localizing the position of the high energy source to better than 5$''$, and by showing that the source varies in phase
with the lower-energy X-ray emission, our {\it NuSTAR}\ observations prove conclusively that $\eta$~Car\ is clearly a source of
non-thermal high-energy X-ray emission,
and connect the non-thermal X-rays to the soft GeV $\gamma$-ray source detected by {\it Fermi}.
This confirms that a colliding wind shock can accelerate particles to sub-TeV energies.
Since the colliding-wind shock occurs steadily, persistently, and predictably,
massive binary systems are potentially important systems for studying particle acceleration by the Fermi process in an astrophysical setting.
The emission we observe is consistent with IC upscattering of lower-energy stellar photons.
IC emission should also be accompanied by lower-energy synchrotron emission, which has not been detected.
However, synchrotron emission from $\eta$~Car\ would be difficult to detect because of strong thermal dust emission from the surrounding nebula,
and because a suitable high-spatial-resolution radio interferometer in the southern hemisphere is not yet available.
The Square Kilometer Array, which is under construction in South Africa,
may eventually detect this emission component from $\eta$~Car.
Although there are other massive binary systems with strong colliding wind shocks, such as WR~140,
only $\eta$~Car\ has been confirmed as a $\gamma$-ray source.
Studying the differences amongst these systems in their X-ray and $\gamma$-ray emission will help elucidate the particle acceleration mechanism.
\begin{addendum}
\item This research has made use of data obtained from the High Energy Astrophysics Science Archive
Research Center (HEASARC), provided by NASA's Goddard Space Flight Center.
This research has made use of NASA's Astrophysics Data System Bibliographic Services.
We appreciate Drs. M. Yukita, K. Madsen and M. Stuhlinger on helping resolve the {\it NuSTAR}\ and {\it XMM-Newton}\ data analysis.
K.H. is supported by the {\it Chandra}\ grant GO4-15019A, GO7-18012A, the {\it XMM-Newton}\ grant NNX15AK62G, NNX16AN87G, NNX17AE67G, NNX17AE68G,
and the ADAP grant NNX15AM96G.
C.M.P.R. was supported by an appointment to the NASA Postdoctoral Program at the Goddard Space Flight Center,
administered by Universities Space Research Association under contract with NASA.
A.F.J.M. is supported by NSERC (Canada) and FQRNT (Quebec).
\item[Author Contributions]
K.H. and M.F.C. led the project, from proposing and planning observations, analyzing the data to composing the manuscript.
J.M.P. constructed a theoretical model that explains the variation of the non-thermal component.
N.S. performed initial analysis of the {\it NuSTAR}\ data in 2015.
H.T analyzed and discussed {\it Fermi}\ data of $\eta$~Car.
C.M.P.R. performed theoretical simulations of {$\eta$~Car}'s thermal X-ray emission.
B.W.G. and D.R.W. discussed {\it NuSTAR}\ data analysis, especially the background characteristics.
T.R.G. worked for the observation planning.
T.R.G., N.D.R., T.I.M., and A.F.J.M. discussed the wind property of $\eta$~Car.
All authors reviewed the manuscript and discussed the work.
\item[Competing Interests] The authors declare that they have no
competing financial interests.
\item[Correspondence] Correspondence and requests for materials
should be addressed to K.H.~\\(email: [email protected]).
\end{addendum}
\begin{figure}
\includegraphics[width=15cm]{figure1.eps}
\caption{
\textbf{{\it NuSTAR}\ image contours of the $\eta$~Car\ field.}
The contours in a conventional X-ray band (5$-$10~keV, {\bf a}) and an extremely hard X-ray band (30$-$50 keV, {\bf b})
are produced from the {\it NuSTAR}\ observations on 2015 July 16 ($\phi_{\rm orb} =$0.17) and 2016 June 15 ($\phi_{\rm orb} =$0.34)
and overlaid on a true colour X-ray image of the same field taken with the {\it Chandra}\ X-ray observatory
during the soft X-ray minimum in 2009 \citep{Hamaguchi2014a}.
The contours are drawn at intervals of 10\% starting from the X-ray peak above background.
The {\it NuSTAR}\ images were aligned with the {\it Chandra}\ image by matching the peak of the thermal emission at $E <$10~keV
in the {\it NuSTAR}\ image with that of the {\it Chandra}\ image.
The 30$-$50 keV source centroid, which has an uncertainty of about 5$''$\ at 2$\sigma$,
is consistent with the centroid of the thermal, 5$-$10 keV source (i.e., $\eta$~Car).
Earlier measurements of extremely hard X-ray and $\gamma$-ray source positions are
constrained at an accuracy of $\sim$1$'$\ or larger \citep[e.g.,][]{Leyder2010,Reitberger2015}.
\label{fig:image}
}
\end{figure}
\begin{figure}
\includegraphics[width=15cm]{figure2.eps}
\caption{
\textbf{{\it NuSTAR}\ spectra in three characteristic orbital phases of $\eta$~Car\ and a comparison
to a {\it Fermi}\ $\gamma$-ray spectrum.}
{\bf a}, {\it NuSTAR}\ spectra obtained during the rise of the soft X-ray flux toward periastron on 2014 March 31 ({\it black}, $\phi_{\rm orb} =$0.94),
the soft X-ray minimum on 2014 August 11 ({\it orange}, $\phi_{\rm orb} =$0.005), and
after the soft X-ray flux recovery from the 2014 periastron event ({\it red}).
The last spectrum is co-added from two spectra
in 2015 July 16 ($\phi_{\rm orb} =$0.17) and 2016 June 15 ($\phi_{\rm orb} =$0.34), to increase the signal-to-noise.
The vertical axis shows the raw photon counts from the detector.
Error bars are shown at 1$\sigma$.
The cyan and green solid lines show emission of {\it kT}\ = 4.5 keV thermal plasma and a $\Gamma$ = 1.65 power-law,
which are convolved with the detector response, to give expected histograms of the detector counts at each energy.
The thin cyan spectrum is $\sim$4 times brighter than the thick cyan spectrum.
The excess from the {\it kT}\ = 4.5 keV thermal plasma emission below $\sim$6~keV mostly
originates from a lower temperature ({\it kT}\ $\sim$1.1~keV) component.
{\bf b}, {\it NuSTAR}\ spectrum on 2016 June 15 and a {\it Fermi}\ spectrum \citep{Abdo2010}
after correcting the detector response ({\it black})
compared to the best-fit spectral model, a $\Gamma$ = 1.65 power-law cut-off at 1.6 GeV ({\it red}).
\label{fig:spectra}
}
\end{figure}
\begin{figure}
\includegraphics[width=14cm]{figure3.eps}
\caption{
\textbf{Flux variations of the thermal and non-thermal X-ray components with orbital phase.}
{\bf a}, Binary orbital positions of the companion during the {\it NuSTAR}\ observations.
The periastron timing is not constrained better than $\approx$0.02 in phase, so that the actual positions
especially near periastron have large uncertainties.
The companion size is not to scale.
{\bf b}, {\it RXTE}\ and {\it Swift}\ light curves of $\eta$~Car\ between 2$-$10 keV since 1998 \citep{Corcoran2017a}.
The labels ``Minimum"/``Maximum" show the timings of the soft X-ray minimum/maximum discussed in the text.
{\bf c}, 30$-$50~keV X-ray flux of the flat, power-law component measured with {\it NuSTAR}\ between 2014$-$2016 ({\it blue}),
assuming a power-law photon index at 1.65.
The solid and dotted black horizontal lines are the best-fit flux and its 90\% confidence range of a power-law component
measured with {\it Suzaku}\ assuming the flux is constant throughout the orbit \citep{Hamaguchi2014b}.
The {\it INTEGRAL}\ \citep[{\it green diamond,}][]{Leyder2008,Leyder2010} and {\it Suzaku}\ measurements were
converted to 30$-$50 keV fluxes.
Error bars are shown at 2$\sigma$.
\label{fig:flux}
}
\end{figure}
\clearpage
\begin{center}
{\LARGE Methods}
\end{center}
\noindent{\large\bf {\it NuSTAR}\ Data}\\
\noindent{\bf Observations:}
{\it NuSTAR}\ has two nested Wolter I-type X-ray telescopes with a 2$\times$2 array of CdZnTe pixel detectors
in each focal plane module \citep[FPMA/FPMB, ][]{Harrison2013}.
These mirrors are coated with depth-graded multilayer structures and focus X-rays over a 3$-$79 keV bandpass.
They achieve an angular resolution of roughly 60$''$\ half power diameter \citep{Madsen2015a}.
The focal plane detectors are sensitive between 3$-$79~keV and cover a 12$'$\ {\it FOV}.
The energy resolution of the detectors is 400 eV below $\sim$40 keV, rising to $\sim$ 1 keV at 60 keV.
Stray light contamination is not an issue unless there are bright sources ($>$100~mCrab) within 1{$^{\circ}$}\ to 5{$^{\circ}$}\ of the target.
{\it NuSTAR}\ observed $\eta$~Car\ on 9 occasions and produced 11 datasets with different observation identifiers (ObsID).
Two datasets on
2014 March 31 (ObsIDs: 30002010002, 30002010003) and
2014 August 11 (ObsIDs: 30002010007, 30002010008)
were performed consecutively, but they have different ObsIDs due to small pointing offsets.
The list of the datasets is summarized in Supplementary Table~1.
We used the HEASoft package\footnote{https://heasarc.nasa.gov/lheasoft/}, version 6.20 or above,
to analyze the {\it NuSTAR}\ data.
\noindent{\bf Reduction and Accurate Measurement of the {\it NuSTAR}\ Background:}
Measuring the spectrum of $\eta$~Car\ at energies above 10 keV requires some care. At the lower end of this energy range, emission is significantly affected by the high-energy tail of {$\eta$~Car}'s thermal source at a temperature of $\sim4.5$~keV,
and which we were able to precisely measure using {\it XMM-Newton}\ X-ray spectra in the 2$-$10~keV energy range.
At higher X-ray energies, the thermal contribution is negligible
(except for a short interval during the 2$-$10~keV X-ray maximum just before periastron),
but instrumental and cosmic background components grow in importance.
Our analysis requires careful measurements of {$\eta$~Car}'s spectral shape above $\sim$25~keV,
where non-thermal emission exceeds {\it kT}~$\sim$4.5~keV thermal emission.
X-ray emission from $\eta$~Car\ in this energy band is weak and comparable to {\it NuSTAR}\ particle background.
Therefore, we maximized the source signal with respect to background by
i) removing high background intervals during each observation, and
ii) employing a small source region.
We then accurately estimated the background spectrum by utilizing the background estimate tool {\tt nuskybgd} \citep{Wik2014a}.
Background particle events of the {\it NuSTAR}\ detectors sometimes increase abruptly when {\it NuSTAR}\ is near the South Atlantic Anomaly (SAA).
After reviewing the background variation in each observation\footnote{http://www.srl.caltech.edu/NuSTAR\_Public/NuSTAROperationSite/SAA\_Filtering/SAA\_Filter.php},
we removed the high background intervals with the tool {\tt saacalc} using the option, {\tt saacalc=2 --saamode=optimized --tentacle=yes}.
In all observations with abrupt background increases, this option removed high background intervals,
by decreasing exposure times by $\lesssim$5\%.
This process significantly reduced background of NUS$_{\rm 160615}$ by $\approx$40\% between 30$-$60 keV.
For extracting source light curves and spectra from each dataset, we used a circular region with a 30$''$\ radius,
which includes $\sim$50\% of the X-ray photons of an on-axis point source.
Since the source region is comparable to the mirror point-spread-function (PSF) size and there is a positional offset
in the absolute coordinates and the coordinate systems between FPMA and FPMB by up to $\sim$10$''$,
we re-calibrated the absolute coordinates on each detector image frame from a two-dimensional image fit with a PSF\ image.
{\it Chandra}\ observations indicate that colliding wind emission from $\eta$~Car\ dominates the emission below 10~keV,
so that we measured the peak position of $\eta$~Car\ between 6$-$8~keV in each detector image
by an on-axis PSF\ with the {\it Chandra}\ software {\tt CIAO/Sherpa}.
Before each fit, the PSF\ image was rotated to consider the satellite roll angle.
We then measured the {\it NuSTAR}\ background from surrounding source-free regions using {\tt nuskybgd}.
This tool extracts spectra from specified source-free regions and fits them simultaneously for
known background components --- line and continuum particle background, cosmic X-ray background (CXB)
passing through the mirror (focused) and unblocked stray light in the detector (aperture), and solar X-rays
reflecting at the mast. For the $\eta$~Car\ data, we ignored the solar reflection component as it is very soft ($\lesssim$5~keV).
There are a few more components that we added in the {\tt nuskybgd} model for the $\eta$~Car\ data (see Supplementary Figure~2).
One is the Galactic Ridge X-ray Emission (GRXE).
As $\eta$~Car\ is located almost on the Galactic plane ($l, b$) = (287.6{$^{\circ}$}, $-$0.63{$^{\circ}$}),
GRXE from {\it kT}\ $\sim$6~keV thermal plasma is as strong as CXB at $\sim$7 keV \citep[e.g.,][]{Miyaji1998}.
This emission comes from both the mirror and opening between the mirror and focal plane modules (stray light) similar to the CXB.
The only difference is that GRXE is concentrated within $\sim$4{$^{\circ}$} (FWHM) from the Galactic plane \citep[e.g.,][]{Valinia1998},
while CXB is uniform on the sky.
Earlier measurements give good estimate of the two (focused \& aperture) CXB components and focused GRXE.
We thus measured the contribution of aperture GRXE contamination by fixing the parameters for the other sky background components.
For this measurement, we used 3 datasets obtained during the lowest soft X-ray flux phase
(NUS$_{\rm 140728}$, NUS$_{\rm 140811a}$, NUS$_{\rm 140811b}$)
since $\eta$~Car\ outshines the entire detector {\it FOV}\ outside the soft X-ray minimum.
X-ray emission from unresolved young stars in the Carina nebula is not negligible below $\sim$7~keV,
so that we fit the background spectra only above this energy range.
We assume the GRXE spectral shape is similar to that in \citep{Ebisawa2005}, which is measured for GRXE
at ($l, b$) = (28.5{$^{\circ}$}, 0.0{$^{\circ}$}), but we changed its normalization to match the GRXE flux at the $\eta$~Car\ position \citep{Miyaji1998}.
We extracted data from 4 source regions, each of which has 5.5$'$$\times$5.5$'$, each of which covers a detector (0, 1, 2, 3) on each module
(FPMA/FPMB), excluding areas around the bright hard X-ray sources, $\eta$~Car, WR~25, and HD~93250.
This analysis shows that the observed stray light flux is 82\% (FPMA) and 75\% (FPMB) of the expected stray light
if the GRXE has the same surface brightness as at ($l, b$) = (285{$^{\circ}$}, 0.0{$^{\circ}$}).
We fixed the GRXE contamination at these values for the rest of the background analysis.
These ratios may change with the satellite roll angle, but our conclusions should not be significantly affected
as the GRXE is negligible above 15~keV.
The other background component accounts for particle background variations between the detectors.
{\tt Nuskybgd} assumes that instrumental background is uniform between the detectors (0, 1, 2, 3), but
some {\it NuSTAR} $>$15~keV images of $\eta$~Car\ show small but significant fluctuations (see Supplementary Figure~1).
These fluctuations possibly originate from the sensitivity difference between the detectors (private comm. Kristin Madsen),
or Cen X-3 contamination through the detector light baffle.
In either case, these fluctuations can introduce up to $\sim$10\% normalization error at the $\eta$~Car\ position in some observations.
We therefore added a contamination component to the {\tt nuskybgd} model, an absorbed power-law model
({\tt TBabs $\times$ Power-law}) whose normalization was allowed to vary between the detectors;
the normalization for the detector with the lowest enhancement was fixed at zero.
We added this component to the background model for $\eta$~Car.
Using these constraints, we ran {\tt nuskybgd} to estimate background for all $\eta$~Car\ datasets.
Since we need a precise measurement of the background above 25~keV,
we used a larger region for each detector to increase the photon statistics
--- the region includes WR~25 and HD~93250, which have little flux above 15 keV ---
and excludes smaller areas around $\eta$~Car.
We fit the unbinned estimated background spectra above 15 keV up to 150~keV using Poisson statistics to give the best measurement
of the estimated background shape between 25$-$79~keV.
We then normalized the best-fit result for each $\eta$~Car\ spectrum.
The background subtracted spectrum and the corresponding simulated background spectrum for
each observation is shown in the Supplementary Figure~3.
Three spectra shown in Figure~2a are co-additions of the spectra
NUS$_{\rm 140331a}$ and NUS$_{\rm 140331b}$ ({\it black}),
NUS$_{\rm 150716}$ and NUS$_{\rm 160615}$ ({\it red}), and
NUS$_{\rm 140811a}$ and NUS$_{\rm 140811b}$ ({\it orange}).
For spectral fits, we add the normalized background model to the source model and
fit the source spectra using Poisson statistics.
\noindent{\bf Analysis: }
As described in the previous section, the absolute coordinates on each image have uncertainties of several arc-seconds.
For Figure~1, we shifted each detector image by pixel offsets measured with the PSF\ fits to 6$-$8~keV images
and combined them for each band.
We recalibrated the absolute coordinates based on the soft band image.
We smoothed the image with a Gaussian of $\sigma =$8~pixels to increase the photon statistics.
Supplementary Figure~1 also shows the entire field of view of the co-added {\it NuSTAR}\ images of
NUS$_{\rm 150716}$ and NUS$_{\rm 160615}$.
The X-ray spectrum of $\eta$~Car\ is complex with these components which contribute to the emission above 3 keV:
i) variable multi-temperature thermal components produced by the hot, shocked colliding wind plasmas;
ii) a weak, stable central constant emission (CCE) component, which probably originates from hot shocked gas
inside the cavity of the secondary star's wind, which was ejected in the last few orbital cycles;
iii) X-ray reflection from the bipolar Homunculus nebula;
iv) a power-law component with photon index $\Gamma \lesssim2$.
We included all these components in the spectral model, to determine the non-thermal flux variation with orbital phase.
Component i) varies slowly with the binary orbital motion.
Earlier spectral analyses of $\eta$~Car\ between 0.5$-$10~keV \citep[e.g.,][]{Hamaguchi2007b}
show that this component can be described with two-temperature components having {\it kT}\ $\sim$4.5 and $\sim$1.1~keV,
each of which suffers independent absorption.
The {\it NuSTAR}\ spectra cannot constrain parameters of the cool ({\it kT} $\sim$1.1 keV) component well
without sensitivity below 3~keV where the emission dominates.
We therefore fixed {\it kT}, elemental abundance and {\it N$_{\rm H}$}\ of the cool component
at 1.1~keV, 0.8 solar, 5$\times$10$^{22}$~{\rm cm$^{-2}$},
the best-fit values of the {\it XMM-Newton}\ EPIC spectra on 2015 July 16.
On the other hand, we allowed parameters of the hot component
({\it kT}, abundance, normalization and absorption) to vary in all spectral fits.
Component ii) probably originates from the collision of secondary stellar winds with the primary winds
ejected in early cycles \citep[e.g.,][]{Hamaguchi2007b,Madura2013,Russell2016a}.
This component can be seen in $\eta$~Car\ spectra only around the soft X-ray minimum and
it does not change significantly in the latest 3 minima (2003, 2009 and 2014).
This component cannot be observed during other orbital phases, but a theoretical simulation suggests
that it is stable outside of the minimum as well \citep{Russell2016a}.
Component iii) originates from the reflection of the colliding wind X-ray emission at the surrounding Homunculus bipolar nebula.
The variation follows the wind colliding emission from the central binary system, with light travel time-delay
by 88 days, on average \citep{Corcoran2004}.
This component is extended ($\sim$20$''$) and can be spatially resolved with {\it Chandra}.
This component is weaker than the CCE (Component ii) except for the Fe fluorescence at 6.4~keV.
We therefore fixed this component to the best-fit spectrum derived from the {\it Suzaku}\ observation
during the deep X-ray minimum phase in 2014 \citep{Hamaguchi2016a}.
The components (ii) + (iii) only contribute $\sim$10\% to the spectra after the recovery in 2015 and 2016,
and dominate during the X-ray minimum.
Component iv) is proved to be present from the {\it NuSTAR}\ observations in this paper.
It dominates emission above 30 keV, and does not vary significantly outside the soft X-ray minimum.
No spectra show the shape of this component below 30~keV clearly.
However, our measurement of the equivalent width of the He-like iron K line varies less than 10\%
through the orbit outside of the X-ray minimum.
This means that the non-thermal component is less than 10\% of the thermal continuum at 6.7 keV,
which constrains the photon index at $\Gamma<$2.
We choose $\Gamma =$1.65 for consistency between the {\it NuSTAR}\ and {\it Fermi}\ data,
but the conclusions we draw do not change significantly for $\Gamma \lesssim$2.
The absorption column for the power-law component is tied to that of the hot {\it kT}\ component.
This is based on the assumption that the non-thermal emission originates from the apex of the colliding wind region,
but changing this {\it N$_{\rm H}$}\ does not affect the fitting result for $\Gamma<$2.
We simultaneously fit unbinned $\eta$~Car\ spectra of both focal plane modules (FPMA, FPMB)
using the maximum likelihood method assuming Poisson statistics (c-stat in Xspec).
The normalizations of the spectral models between FPMA and FPMB are independently varied to consider small effective area calibration uncertainty.
The errors are estimated using Markov Chain Monte Carlo simulations (mcmc in Xspec).
The fitting results are shown in Figure~3 and Supplementary Table~2.
\noindent{\large\bf {\it XMM-Newton}\ Data}
\noindent{\bf Observations:}
{\it XMM-Newton}\ has three nested Wolter I-type X-ray telescopes \citep{Aschenbach2000} with
the European Photon Imaging Camera (EPIC) CCD detectors
(pn, MOS1 and MOS2) in their focal planes \citep[][]{Struder2001, Turner2001}.
They achieve a spatial resolution of 15$''$\ half power diameter
and an energy resolution of 150~eV at 6.4~keV\footnote{http://xmm-tools.cosmos.esa.int/external/xmm\_user\_support/documentation/uhb/XMM\_UHB.pdf}.
There are three {\it XMM-Newton}\ observations simultaneous with the {\it NuSTAR}\ observations,
two of which are reported in \citep{Hamaguchi2016a}.
In all observations,
the EPIC-pn and MOS1 observations were obtained in the small window mode with the thick filter to avoid photon pile-up and optical leakage,
though the EPIC-MOS1 data in XMM$_{\rm 140606}$ was still affected by photon pile-up.
The EPIC-MOS2 observations used the full window mode with the medium filter to monitor serendipitous sources around $\eta$~Car,
so that its $\eta$~Car\ data are significantly affected by photon pile-up and optical leakage and
thus provide no useful information about $\eta$~Car.
Fortunately, most of the {\it XMM-Newton}\ observations were obtained during periods of low particle background.
\noindent{\bf Analysis:}
We followed \citep{Hamaguchi2007b} for extracting {\it XMM-Newton}\ source spectra,
taking the $\eta$~Car\ source region from a 50$''$$\times$37.5$''$\ ellipse
with the major axis rotated from the west to the north at 30{$^{\circ}$}.
For background, we used regions with negligible emission from $\eta$~Car\ on the same CCD chip.
In addition, we limited the EPIC-pn background regions using nearly the same RAWY position of $\eta$~Car,
according to the {\it XMM-Newton}\ analysis guide\footnote{http://xmm.esac.esa.int/sas/current/documentation/threads/PN\_spectrum\_thread.shtml}.
The source did not show significant variation.
We assumed chi-square statistics for the {\it XMM-Newton}\ fits to the background-subtracted spectra.
The {\it XMM-Newton}\ spectra show multiple emission lines, notably from helium-like Fe K emission lines.
The Fe K emission line is shifted by $\sim$25~eV for both EPIC-pn and MOS1,
which corresponds to $v \sim$1100 {\rm km~s$^{-1}$}.
However, the simultaneous {\it NuSTAR}\ observation did not show such a shift,
and a {\it Chandra}\ HETG grating observation of $\eta$~Car\ obtained at a very similar orbital phase, but one cycle previously
(ObsID: 11017, 11992, 12064, 12065, Date: 2009 Dec 21$-$23, $\phi_{\rm obs}$ = 2.168)
gives only a small shift of $\sim$7~eV.
In addition, we saw a similar energy shift in {\it XMM-Newton}\ data obtained with the same observing mode in 2014.
The shift seen in the {\it XMM-Newton}\ spectra is probably due to an error in energy-scale calibration.
After adjusting the gain shift,
the {\it XMM-Newton}\ spectra of $\eta$~Car\ are successfully reproduced by a model with the cooler {\it kT}\ at 1.1 keV and hotter {\it kT}\ at 4.5 keV.
These temperatures are similar to those measured in early {\it XMM-Newton}\ observations \citep{Hamaguchi2007b}.
\noindent{\large\bf Theoretical Model for the Constancy of the Non-thermal Component}
If the non-thermal electrons fill the wind-colliding region,
the IC luminosity, $L_{\rm IC}$, should be proportional to the number of non-thermal electrons
($N_{\rm acc} \propto nV$, where $n$ and $V$ are respectively the number density of the thermal plasma in the wind colliding region
and the volume of the wind colliding region) and the intensity of the stellar UV ($U_{\rm UV}$).
Since $n$ and $U_{\rm UV}$ are both $\propto D^{-2}$, and $V \propto D^{3}$, we might expect $L_{\rm IC} \propto 1/D$,
where $D$ is the stellar separation.
Therefore, the $L_{\rm IC}$ should follow the same variation as the X-ray luminosity of the thermal plasma (i.e. 2$-$10~keV light curve in Figure~3b),
which also has the $1/D$ dependence valid for the adiabatic limit \citep{Stevens1992}.
That this variation is not observed can be explained by the rapid cooling that the non-thermal electrons undergo due to IC scattering as they flow downstream from the companion star's shock\footnote{
For particles to be accelerated the shocks must be collisionless and mediated by the magnetic field. This requires that the postshock thermal collision timescale must be longer than the ion gyroperiod. This is not satisfied at high densities \citep[see, e.g.,][]{Eichler1993}. Since the shocked luminous blue variable wind is highly radiative, its post-shock density is several orders of magnitude greater than the post-shock density of the companion's wind, and is not likely to be collisionless.}.
Rather than filling the entire wind colliding region, the non-thermal electrons which are capable of producing 50 keV photons
(those with a Lorentz $\gamma \sim$200) instead only exist in a thin layer downstream from the shock \citep{Pittard2006b}.
For reasonable values (e.g. $D$ = 10 au, $r_{\rm O}/D$ = 0.3, where $r_{\rm O}$ is the distance from the companion star to
the shock on the line-of-centres,
$L_{\rm UV}$ = 5$\times$10$^{6}$ {\it L$_{\odot}$}) the rate at which the non-thermal electrons lose energy due to IC scattering is $d{\gamma}/dt$ $\sim$10$^{-6}$~$\gamma^{2}$ s$^{-1}$
\citep[cf. Eq. 4 in][]{Pittard2006b}.
Hence it takes roughly 6000 second (= $t_{\rm cool}$) to cool from the expected maximum energy of the electrons at the shock ($\gamma \sim$ 10$^{5}$)
to $\gamma \sim$200.
During this time the electrons will have travelled downstream from the shock a distance of $d_{\rm cool} = v_{\rm ps} t_{\rm cool}$,
where $v_{\rm ps}$ is the post-shock wind velocity.
Using $v_{\rm ps} = v_{\rm wind O}/4$ (appropriate for the gas on the line-of-centres between the stars), the cooling length $d_{\rm cool} \sim 0.01~D$.
This sets the thickness of the region where non-thermal electrons are capable of producing 50 keV photons.
As the stars approach each other, IC cooling becomes stronger and stronger, and $d_{\rm cool}/D$ decreases.
Since $d\gamma/dt \propto D^{-2}$, $d_{\rm cool}/D \propto D$.
So rather than the volume of non-thermal emitting particles scaling as $D^{3}$, it instead scales as $D^{4}$ ($D^{2}$ from the surface area of the shock(s),
and $D^{2}$ from the cooling length).
Hence $L_{\rm IC}$ becomes independent of $D$, as is indeed observed outside of the minimum.
At some very large value of $D$, $d_{cool}$ will be large enough that the non-thermal electrons completely fill the volume of the wind colliding region,
at which point $L_{\rm IC}$ should scale as 1/$D$, as originally hypothesized.
However, this is likely to require a value for $D$ which far exceeds the apastron separation in $\eta$~Car.
If $\gamma>$200, non-thermal electrons are confined to only part of the wind-colliding region,
and a change in the spectral shape of the non-thermal emission with $D$ is not expected.
So this model naturally explains the constant intensity and spectral shape of the IC emission outside of the X-ray minimum.
\noindent{\large\bf Data Availability}
The raw data of the {\it NuSTAR}\ and {\it XMM-Newton}\ observations are available from the NASA HEASARC archive https://heasarc.gsfc.nasa.gov.
\renewcommand{\figurename}{Supplementary Figure}
\renewcommand{\tablename}{Supplementary Table}
\input{tab1.tex}
\clearpage
\input{tab2.tex}
\clearpage
\setcounter{figure}{0}
\begin{figure}[h]
\caption{
\textbf{Whole {\it NuSTAR}\ images of the $\eta$~Car\ field.} \label{fig:image_whole}}
\includegraphics[width=16.5cm]{whole_fov.eps}
The images between 5$-$10~keV ({\bf a}, log scale) and 30$-$50 keV ({\bf b}, linear scale)
are produced from the 2015 July 16 ($\phi_{\rm orb} =$0.17) and 2016 June 15 ($\phi_{\rm orb} =$0.34) observations.
Each image is smoothed with a Gaussian function of $\sigma$=8 pixels.
The bright X-ray source in the 5$-$10 keV image, Tr~14 Y442, is a young star in the Carina nebula, which had a giant flare
during the 2015 observation \citep{Hamaguchi2015a}.
\end{figure}
\clearpage
\begin{figure}[h]
\caption{
\textbf{Individual background components for the FPMA $\eta$~Car\ spectrum of the 2016 June 15 observation.}
\label{fig:spec_bgd}
}
\includegraphics[width=16.5cm]{spec160616A_bgd_comp.eps}
\end{figure}
\clearpage
\begin{figure}[h]
\caption{
\textbf{Background subtracted {\it NuSTAR}\ FPMA+FPMB spectrum of each observation.}
\label{fig:spec_indiv}
}
\includegraphics[width=15.5cm]{specBGD_page1.eps}
The solid grey line shows the background level.
\end{figure}
\clearpage
\setcounter{figure}{2}
\begin{figure}[h]
\begin{flushleft}
\caption{Continue.}
\end{flushleft}
\includegraphics[width=15.5cm]{specBGD_page2.eps}
\end{figure}
\clearpage
\input{ms_nature_wfig.bbl}
\end{document}
|
2,877,628,088,959 | arxiv | \section{Introduction}
The pion remains an important system in hadronic physics to trace signatures of QCD in observables. It emerges non-perturbatively as the lightest quark-antiquark ($q\bar q$) bound state and it is at the same time identified with the Goldstone boson associated with spontaneous chiral-symmetry breaking (S$\chi$SB). The non-perturbative dynamics underlying such hadronic systems have been addressed in various modern approaches, such as QCD simulations on the lattice~\cite{Edwards,Guo}, light-front formulations of quantum field theory~\cite{Brodsky:1997de,Carbonell:1998rj,Sales:1999ec}, as well as models based on the Dyson-Schwinger--Bethe-Salpeter (DSBS) approach and the mass gap equation~\cite{Bars:1977ud,Amer:1983qa,LeYaouanc:1983it,Bicudo:1989sh,Bicudo:1989si,Bicudo:1989sj,Nefediev:2004by,Alkofer:2000wg,Maris:2003vk,Fischer:2006ub,Rojas:2013tza}, which have contributed to an understanding of a wide range of meson and baryon phenomena. We use the Covariant Spectator Theory (CST)~\cite{Gro69}, which is another modern approach that implements S$\chi$SB through the famous Nambu--Jona-Lasinio mechanism, similarly to DSBS. In contrast to the latter, whose four-dimensional integral equations are usually treated in Euclidean space, the CST equations can be solved directly in physical Minkowski space.
Previous CST quark models~\cite{Gross:1991te,Gross:1994he,Savkli:1999me} and improved versions of them that are currently being developed~\cite{PhysRevD.89.016005} employ an interaction kernel that includes linear confinement in a covariant-generalized form. Such kernels satisfy the relativistic version of the property that the nonrelativistic linear potential (in coordinate space) vanishes at the origin.\footnote{A useful method to treat the singularities of linear-confining interaction kernels has been recently proposed in Ref.~\cite{PhysRevD.90.096003}.} This implies that the confinement interaction \emph{decouples} from the CST-Dyson equation (CST-DE) for the scalar part of the dressed quark propagator, as well as from the CST pion equation in the chiral limit~\cite{Gross:1991pk}.
At present, the precise Lorentz structure of the confining interaction is not known. Some approaches suggest that it has a large scalar component, and---although one lacks first-principle evidence for this---it is still quite important to study to what extent such confining forces can be made compliant with S$\chi$SB. In the previous CST models, a confining interaction was used that included non--chirally-symmetric spin structures, such as scalar terms, as they are not inconsistent with a massless solution of the pion equation in the chiral limit owing to the above discussed \lq\lq decoupling property.''
In the present work we follow a different strategy. We start from the most general Lorentz structure for the $q\bar q$ interaction and then determine the constraints imposed by S$\chi$SB, similarly to what has been done in Ref.~\cite{PhysRevD.47.1145}. It turns out that a CST model with scalar confinement, together with an equal-weighted pseudoscalar part satisfies the S$\chi$SB condition of the Adler consistency zero~\cite{Adler_PhysRev.137.B1022} in $\pi$-$\pi$ scattering in the chiral limit.
\section{Axial-vector Ward-Takahashi identity}
Chiral symmetry and its breaking is expressed through the axial-vector Ward-Takahashi identity (AV-WTI), which is---within the Gross-Riska prescription~\cite{Gro87,Gro93,Gro96} of dealing with strong quark form factors---given by
\begin{eqnarray}
P_\mu \Gamma^{5\mu}_{R}(p',p)+2m_0 \Gamma^5_R(p',p)=\tilde S^{-1} (p')\gamma^5+\gamma^5\tilde S^{-1} (p)
\equiv\Gamma^{A}_R (p^\prime,p) \,,\label{eq:AVWTI}
\end{eqnarray}
where $\Gamma^{5\mu}_R(p',p)$ and $\Gamma^5_R(p',p)$ are the (reduced) dressed axial-vector and pseudoscalar vertex functions, respectively, $m_0$ is the bare quark mass, $p$ and $p'$ are the incoming and outgoing quark momenta, respectively, $P=p'-p$ is the momentum flowing into the vertex, $\tilde S (p)$ is the (damped) dressed quark propagator as introduced in Ref.~\cite{PhysRevD.90.096008}, and $\Gamma^{A}_R (p^\prime,p)$ is the (reduced) dressed \lq\lq axial vertex''---a convenient combination of the dressed axial-vector and the pseudoscalar vertices. $\Gamma^{A}_R (p^\prime,p)$ is the solution of an inhomogeneous CST Bethe-Salpeter equation (CST-BSE),
\begin{eqnarray}
\Gamma^{A}_R (p^\prime,p)=\gamma^{A}_R (p^\prime,p)+\mathrm i
\int_{k0} \mathcal V_R(p-k)
\tilde S(k') \Gamma^{A}_R (k',k) \tilde S(k) \,, \qquad \label{eq:CSTBSEGammaA}
\end{eqnarray}
where $\gamma^{A}_R (p^\prime,p)$ is the (reduced) bare axial vertex, $\mathcal V_R(p-k)$ is the (reduced) covariant interaction kernel depending only on the four-momentum transfer $p-k=p'-k'$, and \lq\lq $k0$'' indicates the charge-conjugation invariant CST prescription for performing the $k_0$ contour integration~\cite{Savkli:1999me}. The most general structure of the linear-confining kernel, together with a vector--axial-vector remainder, is given by
\begin{eqnarray}
\mathcal V_R(p-k)= V_{LR}(p-k)\Big[\lambda_S ({\bf 1}\otimes {\bf 1})+\lambda_S (\gamma^5\otimes\gamma^5) +\lambda_V
( \gamma^\mu
\otimes \gamma_{\mu})+ \lambda_A (\gamma^5\gamma^{\mu} \otimes \gamma^5\gamma_{\mu})+
\frac{\lambda_T}{2}(\sigma^{\mu\nu}\otimes\sigma_{\mu\nu})\Big] \nonumber\\+V_{CR}(p-k)
\Big[\kappa_V(\gamma^\mu\otimes\gamma_\mu)+\kappa_A (\gamma^5\gamma^{\mu} \otimes \gamma^5\gamma_{\mu})\Big]
\,,\label{eq:kernel}
\end{eqnarray}
where $V_{LR}$ and $V_{CR}$ are the momentum-dependent parts of the linear-confining and remaining kernels, respectively, with $V_{LR}$ satisfying
\begin{eqnarray}
\int \frac{\mathrm d^3 k}{E_k} V_{LR} (p\pm\hat k)=0\,,
\label{eq:VLzero}
\end{eqnarray}
where $E_k=\sqrt{m^2+\vec k^2}$, $\hat k=(E_k,\vec k)$, and $m$ is the dressed quark mass. The corresponding weight parameters $\lambda_i$ and $\kappa_i$ [with $i=S$ (scalar), $P$ (pseudoscalar), $V$ (vector), $A$ (axial-vector), and $T$ (tensor)] are arbitrary constants, except that scalar and pseudoscalar parts in~(\ref{eq:kernel}) are \emph{equal-weighted}, i. e. $\lambda_S=\lambda_P$. For this kernel it has been shown~\cite{PhysRevD.90.096008} that the AV-WTI (\ref{eq:AVWTI}) together with the CST-BSE (\ref{eq:CSTBSEGammaA}) implies that $\tilde S (p)$ satisfies the CST-DSE,
\begin{eqnarray}
\tilde S^{-1} (p)=\tilde S_0^{-1} (p)-\mathrm i\int_{k0}
{\cal V}_R(p-k) \tilde S(k)\, ,
\label{eq:CST-DE2}
\end{eqnarray}
where $\tilde S_0$ is the (damped) bare quark propagator, which obeys the AV-WTI for an off-shell Ansatz of $\gamma^{A}_R (p^\prime,p)$ according to Gross and Riska. It turns out that $\gamma^{A}_R (p^\prime,p)$ vanishes in the chiral limit of vanishing bare quark mass, $m_0\rightarrow 0$, and vanishing vertex momentum, $P\rightarrow 0$. In this limit, the CST-BSE~(\ref{eq:CSTBSEGammaA}) becomes homogeneous and identical to the zero-mass pion CST equation for the (reduced) pion vertex function in the chiral limit, $\Gamma^\pi_{R\chi}$, which implies the relation
\begin{eqnarray}
\Gamma^\pi_{R\chi}(p,p)\propto\Gamma^{A}_{R\chi}(p,p)\,.
\label{eq:Gammachi}
\end{eqnarray}
Because of Eq.~(\ref{eq:VLzero}), only $\mathcal V_{CR}$ contributes to the chiral-limit pion equation and to the scalar part of the CST-DE~\eqref{eq:CST-DE2}, i.e. to dynamical quark mass generation. Therefore, the linear-confinement part $\mathcal V_{LR}$ that also includes scalar, pseudoscalar and tensor structures in our model, decouples from these equations. For the pion equation this is diagrammatically depicted in Fig.~\ref{fig:pionChL} and was proven in
Ref.~\cite{PhysRevD.89.016005}.
\begin{figure*}
\includegraphics[height=.18\textheight]{pionChL1}
\caption{(Color online) The decoupling of the linear-confining kernel from the pion CST-equation in the chiral limit. Each red or blue arrowed line denotes a dressed quark propagator.
}\label{fig:pionChL}
\end{figure*}
\section{$\pi$-$\pi$ scattering}
A stronger constraint for chiral symmetry than the previously discussed decoupling property of non--chirally-symmetric Lorentz structures is the Adler consistency zero of the $\pi$-$\pi$~scattering amplitude in the chiral limit. It has been realized earlier~\cite{PhysRevD.65.076008}, and also shown in CST~\cite{PhysRevD.90.096008}, that in order to obtain the Adler zero, it is essential to go beyond the impulse approximation in the scattering diagrams. In particular, for crucial cancellations in the amplitude to occur, it is unavoidable to include intermediate-state interactions to all orders through the complete quark-quark ladder sum. There are three types of contributions to the $\pi$-$\pi$~scattering amplitude, referred to as $O$, $Z$, and $X$ diagrams, as shown in Fig.~\ref{fig:pipiscattering}.
\begin{figure}
\includegraphics[height=.4\textheight]{pipiscattering_total}
\caption{The contributions to $\pi$-$\pi$~scattering. The orange boxes denote the unamputated quark-quark scattering amplitudes.}
\label{fig:pipiscattering}
\end{figure}
It has been proven in CST that the sums of the three diagrams in each row vanish separately in the chiral limit~\cite{PhysRevD.90.096008}, i.e.
\begin{eqnarray}
T_O+S_O-D_O\rightarrow0,\quad T_Z+S_Z-D_Z\rightarrow0,\quad T_X+S_X-D_X\rightarrow0,\label{eq:chlOZXterms}
\end{eqnarray}
which constitutes the Adler zero. In the proof of (\ref{eq:chlOZXterms}) an additional ladder sum at one pion vertex has been inserted in the $T$ and the $S$ diagrams of Fig.~\ref{fig:pipiscattering} through the \lq\lq spectral decomposition''. Then, use of Eq.~(\ref{eq:Gammachi}) was made in the chiral limit to replace $\Gamma^\pi_{R\chi}$ by $\Gamma^{A}_{R\chi}$, allowing the application of the AV-WTI~\eqref{eq:AVWTI} between two ladder sums to reduce these diagrams to terms that cancel exactly the $D$ terms~\cite{PhysRevC.67.035201}. For our kernel \eqref{eq:kernel}, which includes scalar, pseudoscalar, and tensor structures that do not anticommute with $\gamma^5$, additional terms show up. Because of the decoupling of the linear-confining kernel from the zero-mass pion equation (Fig.~\ref{fig:pionChL}), these terms, however, vanish in the chiral limit (for details, see Ref.~\cite{PhysRevD.90.096008}).
\section{Dressed quark mass function}
In Ref.~\cite{PhysRevD.89.016005} we proposed a model for the $q\bar q$ interaction with a kernel of the form~\eqref{eq:kernel}, where $\mathcal V_{CR}$ is taken as a covariant momentum-space $\delta$-function with a pure vector structure ($\kappa_V=1$), and $\mathcal V_{LR}$ is a mixed scalar-vector linear-confining interaction, with $\lambda_S=2$ and $\lambda_V=1$ (with all other weight parameters set to zero). For this particular mixing the confining kernel does not contribute to the scalar part of Eq.~\eqref{eq:CST-DE2}, i.e. to the dressed quark mass. This leads to a rather simple dynamically-dressed quark mass function that is entirely determined by $\mathcal V_{CR}$. The same mass function is obtained from a similar kernel, with $\lambda_S=1$, $\lambda_P=1$, and $\kappa_V=1$ (with all other weight parameters set to zero), which now fully complies with S$\chi$SB.
Our mass function $M(p^2)$, obtained from solving the CST-DE~\eqref{eq:CST-DE2}, involves three free parameters: The dressed quark mass $m_\chi$ in the chiral limit, a mass parameter $M_g$ from the strong quark form factors, and the strength $C$ of the $\delta$-function kernel. Two of them, $m_\chi$ and $M_g$, are fixed in the chiral limit by a fit of the mass function to the lattice QCD data~\cite{Bowman:2005vx} extrapolated to $m_0=0$, which gives $m_\chi=0.308~\mathrm{GeV}$ and $M_g=1.734~\mathrm{GeV}$. The mass function for different values of $m_0$ is then found by solving the corresponding on-shell constraint $M(p^2=m^2)=m$, with $C$ as a function of $m_0$ chosen appropriately to fit the lattice data. The mass function result reads
\begin{eqnarray}
M(p^2)= \left(m_\chi+12 m_0\right)\,h^2(m^2)h^2(p^2)+m_0\,,\label{eq:1}
\end{eqnarray}
where
\begin{eqnarray}
h(p^2)=\left(\frac{\Lambda_\chi^2-m_\chi^2}{\Lambda^2-p^2}\right)^2\, \label{eq:2}
\end{eqnarray}
are the strong quark form factors, with $\Lambda=m+M_g$ and $\Lambda_\chi=m_\chi+M_g$. The Euclidean-space lattice data are compared with our Minkowski-space results at negative $p^2$, as shown in Figure~\ref{fig:1}.
\begin{figure}[h!]
\includegraphics[height=6cm]{fig1.eps}
\caption{The dressed quark mass function $M(p^2)$ compared with lattice QCD data~\cite{Bowman:2005vx} for different bare quark masses $m_0$. The five mass function curves and lattice data sets ($m_0=0$ data are extrapolated) from bottom to top correspond to $m_0=0$ (black, blobs), $m_0=0.016$ GeV (blue, squares), $m_0=0.032$ GeV (green, diamonds), and $m_0=0.047$ GeV (red, triangles) and $m_0=0.063$ GeV (brown, inverted triangles).}\label{fig:1}
\end{figure}
\section{Electromagnetic pion form factor}
As a first test that our CST model gives sensible results we use the mass function~(\ref{eq:1}) (in the chiral limit) for the computation of the electromagnetic pion form factor, $F_\pi(Q^2,\mu)$, in relativistic impulse approximation (RIA)~\cite{Gross:1965zz,Arn77,Arn80,VO95}. The pion current in RIA---by keeping only the spectator quark propagator pole contribution of the triangle diagram---is given by
\begin{eqnarray}\label{eq:picurrentA}
F_\pi(Q^2,\mu)(P_++P_-)^\mu
=\mathrm e\,\int \frac{\mathrm d^3 k}{(2\pi)^3}\frac{m}{E_k}\,\mathrm {tr}\Big[
\bar\Gamma_R^\pi (-\hat k,p_+) \tilde S(p_+) j_R^\mu (p_+,p_-)\tilde S(p_-) \Gamma_R^\pi(p_-, -\hat k) \tilde\Lambda(-\hat k)\Big] \,
\end{eqnarray}
where $\mu$ is the pion mass, $P_\pm$ are the incoming and outgoing pion momenta, $p_\pm=P_\pm-\hat k$ are the off-shell quark momenta, $p_+-p_-=q$ with $Q^2=-q^2$ is the four-momentum transfer by the photon, $j^\mu_R$ is the reduced quark-photon vertex, and $\tilde\Lambda(\hat k)$ is the (appropriately-normalized) positive-energy Dirac projector. For the present simple calculation we adopt an approximated pion vertex function near the chiral limit, together with an Ansatz for the (reduced) dressed quark-photon vertex that is determined by the (vector) Ward-Takahashi identity,
\begin{eqnarray}
q_\mu j^\mu_R(p_+,p_-)=\tilde S^{-1}(p_-)-\tilde S^{-1}(p_+)\,,
\end{eqnarray}
which ensures gauge invariance and, in particular, pion-current conservation. At a latter stage and for more realistic calculations we will use dynamically-calculated pion vertex functions and dressed quark-photon vertices obtained from solving the homogeneous pseudoscalar and inhomogeneous vector CST-BSE's, respectively.
Remarkably, the result for $F_\pi(Q^2,\mu)$ in the present simple model is insensitive to the particular choice of the strong quark form factors $h$, but it depends on the pion mass $\mu$, in particular, at small $Q^2$. For sufficiently large $\mu$ (of the order of $m_\chi$ and larger), one expects the RIA to be a good approximation for the full triangle diagram of the pion form factor, not only at high but also at low $Q^2$~\cite{PhysRevD.89.016006}. In particular, the value $\mu$=0.42 GeV gives the best fit to the experimental data~\cite{PhysRevC.78.045203} over the full (spacelike) range of $Q^2$. At large $Q^2$ we find an interesting scaling behavior between the pion form factors calculated with different pion masses:
\begin{eqnarray}
F_\pi(Q^2,\lambda \mu)\stackrel{Q^2\gg\mu^2}{\simeq}\lambda^{2} F_\pi(Q^2,\mu)\,, \label{eq:scaling}
\end{eqnarray}
where $\lambda$ is a scaling parameter. In Fig.~\ref{fig:2} the large-$Q^2$ tail of the form factor calculated with the physical value $\mu=0.14$ GeV is scaled to fit the one calculated with $\mu=0.42$ GeV.
\begin{figure}
\includegraphics[height=5.1cm]{piff1}
\includegraphics[height=5cm]{piff2}
\caption{The pion form factor compared to the JLab data~\cite{PhysRevC.78.045203}.
The left panel shows the pion form factor $F_\pi (Q^2,\mu=0.14)$ when scaled with $\lambda^2=(0.42/0.14)^2$ (red dashed line) to fit the pion form factor $F_\pi (Q^2,\mu=0.42)$ (black solid line) together with the $\rho$-pole contribution (blue dotted line). The right panel shows the same pion form factors but scaled with $Q^2$. }
\label{fig:2}
\end{figure}
It should be emphasized that this is a very simple model for the pion form factor that, for instance, does not include explicit contributions from the $\rho$ meson. Such contributions should be important in the timelike region according to vector meson dominance, and thus we expect this model to fail in this region. Nevertheless, in the spacelike region our result exhibits the correct monopole behavior at large $Q^2$ and is in good agreement with the experimental data. This concludes the first qualitative study of our model, which shows that it is able to give sensible results for both the dressed quark mass function and the spacelike pion form factor, at the same time. It is clear that for a more quantitative study of the pion structure that also extends to timelike $q^2$, the solutions of the CST-BSE's are needed, with a dynamically-dressed quark current that includes the $\rho$-pole contribution.
As a long-term goal we plan to calculate transition form factors, to be used---together with the dressed quark currents---in the calculation of hadronic contributions to light-by-light scattering, such as pseudoscalar-meson pole and dressed-quark loop contributions. They are essential ingredients in precision calculations of the anomalous magnetic moment of the muon that could reveal new physics beyond the Standard Model.
\begin{theacknowledgments}
This work received financial support from Funda\c c\~ao para a Ci\^encia e a
Tecnologia (FCT) under Grants No.~PTDC/FIS/113940/2009 and No. CFTP-FCT (PEst-OE/FIS/U/0777/2013). The research leading to these results has received funding from the European Community's Seventh Framework Programme FP7/2007-2013 under Grant Agreement No.\ 283286. This work was also partially supported by Jefferson Science Associates, LLC, under U.S. DOE Contract No. DE-AC05-
06OR23177.
\end{theacknowledgments}
\bibliographystyle{aipproc}
|
2,877,628,088,960 | arxiv | \section{Introduction}
Recently, unmanned aerial vehicle (UAV) communication has received considerable research interests due to its flexible deployment and the dominance of line-of-sight links \cite{yongzengmaga,huizhao}. UAV can be deployed as a relay when there is no direct link between any two nodes. Specifically, Zeng \emph{et al.} \cite{yongzeng2016} first studied the trajectory and power allocation for UAV-relay systems. In \cite{yunfeichen}, the reliability of the UAV relay was analyzed in terms of outage probability and bit error rate.
On the other hand, ultra-reliable and low-latency communications (URLLC) have been regarded as one of the three important use cases in 5G \cite{Shafijsac}. For URLLC, a transmitter usually sends a short packet such as command signals or measurement data to a receiver, in contrast to conventional human-to-human communication where long packet is normally transmitted. Hence, a direct result of the Shannon's capacity based on the law of large numbers may not be applicable. In \cite{Polyanskiy2010IT}, Peter \emph{et al. } have derived the maximal coding rate for short-packet transmission, which is a complicated function of channel blocklength and SNR.
In this paper, we consider a two-dimensional UAV-enabled industrial automation scenario in Fig.~\ref{systemodel}, where a controller needs to send command messages to a distant robot that conducts an experiment in a multi-hazard area. For the safety of workers, shelters such as thick cement/metal walls are built between the robot and the controller. Hence, the channel gain between the controller and the robot is weak and negligible, and requires a UAV to fly above the shelter to assist the transmission between the controller and the robot. In \cite{cpan2019}, we studied the problem of jointly optimizing the blocklength and location for UAV-relay communication systems, where the decoding-and-forward (DF) protocol was considered. However, additional processing time is required for the DF mode, which may not be applicable to URLLC applications. Motivated by above, we jointly optimize power and location to minimize the decoding error probability, where the relay is operating under the AF mode without the signal processing delay. The decoding error probability under short blocklength is adopted. We first prove that the decoding error probability is a monotonically decreasing function the SNR. Then, two channel models are studied: free-space channel model and the 3-D channel model. For the first one, an iterative algorithm is proposed to obtain the suboptimal solution with low complexity, and the globally optimal solution is obtained in closed form when the SNR is extremely high. Simulation results show the performance advantages of our proposed algorithms.
\vspace{-0.2cm}
\section{System Model}\label{system}
\begin{figure}
\centering
\includegraphics[width=2.4in]{UAV.pdf}
\caption{UAV relay system for delivering URLLC services.}
\vspace{-0.7cm}
\label{systemodel}
\end{figure}
As shown in Fig.~\ref{systemodel}, we consider a two-dimensional UAV-enabled industrial automation scenario, where the UAV hovers at a location $(x,H)$ above the horizontal line with height $H$. The locations of the controller and the robot are $(0,0)$ and $(D,0)$. The packet size of the command information is $L$ bits, which should be completed within $T_{\rm{max}}$ seconds. Then, the overall blocklength is given by $M=BT_{\rm{max}}$ \cite{Durisi2016}, where $B$ is system bandwidth. The total transmission has two phases, i.e., the first one corresponds to the transmission from the controller to the UAV, while the second one is the transmission from the UAV to the robot. We assume that the UAV adopts the AF protocol, which simply amplifies and forwards the received signals to the robot. Hence, the blocklength for these two phases should be equal, e.g., $m_1=m_2=m\triangleq M/2$. The transmit power of the controller and the UAV are respectively $p_1$ and $p_2$.
The channel power gain from the controller to the UAV and from the UAV to the robot are denoted as $h_1$ and $h_2$, respectively. These channel gains depends on the height $H$ and horizontal distance $x$. In the first phase, the received signal at UAV is given by ${y_1} = \sqrt {{p_1}{h_1}} {x_1} + {n_1}$, where $x_1$ is the command signal transmitted by the controller with unit power, and $n_1$ is the received noise at the UAV that is normalized to unit. The amplification coefficient at the UAV is given by $G = \sqrt {{{{p_2}} \mathord{\left/
{\vphantom {{{p_2}} {\left( {{p_1}{h_1} + 1} \right)}}} \right.
\kern-\nulldelimiterspace} {\left( {{p_1}{h_1} + 1} \right)}}} $. In the second phase, the received signal at the robot is given by ${y_2} = \sqrt {{h_1}{h_2}{p_1}} G{x_1} + \sqrt {{h_2}} G{n_1} + {n_2}$, where $n_2$ is the noise power at the robot that is normalized to unit. Then, the received signal-to-noise ratio (SNR) at the robot is
\begin{equation}\label{sdWDE}
\gamma = \frac{{{h_1}{h_2}{p_1}{p_2}}}{{{h_2}{p_2} + {h_1}{p_1} +1}}.
\end{equation}
In \cite{Polyanskiy2010IT}, the packet error probability of the AF relay system in short blocklength region can be approximately as:
\begin{equation}\label{wearf}
{\varepsilon } = Q\left( {f\left( {{\gamma },{m},L} \right)} \right),
\end{equation}
where $f\left( {\gamma, m, L} \right) = \ln2\sqrt {\frac{m}{V(\gamma)}} \left( {{{\log }_2}(1 + \gamma) - \frac{L}{m}} \right)$, $V(\gamma)$ is the channel dispersion given by $V(\gamma)=1-(1+\gamma)^{-2}$ \cite{Polyanskiy2010IT}, and $Q\left( x \right)$ is the Gaussian $Q$-function.
In this paper, we aim to minimize ${\varepsilon }$ by optimizing the power allocation and the location of the UAV:
\begin{subequations}\label{initial-pro1}
\begin{align}
\mathop {\min }\limits_{\left\{ {{p_1},{p_2},x,H} \right\}} \;\;\;& {{ \varepsilon }}\\
{\rm{s.t.}}\;\;\;& d_1\le x\le d_2,\label{djreitjgot}\\
& H_{\min}\le H\le H_{\max}, \label{fjroigtjio}\\
&p_1+p_2 \le P_{T},\label{freocdsdi}\\
&p_1\ge 0, p_2\ge 0, \label{ofrrefetypo}\vspace{-0.1cm}
\end{align}
\end{subequations}
where constraint (\ref{djreitjgot}) and (\ref{fjroigtjio}) specifies the feasible flying region of the UAV, and $P_T$ is the total power limit.
Before solving Problem (\ref{initial-pro1}), we first provide the following lemmas.
\emph{\textbf{Lemma 1}}: The packet decoding error probability $\varepsilon $ is a decreasing function of SNR $\gamma$.
\emph{\textbf{Proof}}: \upshape Please refer to Appendix \ref{prooflemma1}. \hfill\rule{2.7mm}{2.7mm}
Then, Problem (\ref{initial-pro1}) can be equivalently formulated as
\begin{subequations}\label{inDAfefro1}
\begin{align}
\mathop {\max }\limits_{\left\{ {{p_1},{p_2},x,H} \right\}} \;\;\;& \gamma \\
{\rm{s.t.}}\;\;\;& (\ref{djreitjgot}), (\ref{fjroigtjio}), (\ref{freocdsdi}), (\ref{ofrrefetypo}).
\end{align}
\end{subequations}
Then, we have the following lemma.
\emph{\textbf{Lemma 2}}: The total power constraint in (\ref{freocdsdi}) holds with equality at the optimal solution.
\emph{\textbf{Proof}}: \upshape This can be proved by using contradiction, the details of which are omitted due to limited space. \hfill\rule{2.7mm}{2.7mm}
It is difficult to obtain the globally optimal solution of Problem (\ref{inDAfefro1}) because the power allocation are coupled with the location. In the following, we first consider the free-space channel model, and then we consider the more practical 3-D channel model.
\section{Free-Space Channel Model}\label{proformu}
In this section, we assume the channel is dominated by line-of-sight (LOS) component, and consider the free space channel model, i.e.,
\begin{equation}\label{edeaftg}
{h_1} = \frac{{{\beta _1}}}{{{H^2} + {x^2}}},\ {h_2} = \frac{{{\beta _2}}}{{{H^2} + {{(D - x)}^2}}},\vspace{-0.1cm}
\end{equation}
where $\beta _1$ and $\beta _2$ are channel power gains at a reference distance of $d=1$ meter for the two links, respectively. In this case, we fix the height $H$, and optimize the power and horizontal distance $x$. Then, Problem (\ref{inDAfefro1}) becomes
\begin{subequations}\label{nfhoij}
\begin{align}
\mathop {\max }\limits_{\left\{ {{p_1},{p_2},x} \right\}} \;\;\;& \gamma \\
{\rm{s.t.}}\;\;\;& (\ref{djreitjgot}), (\ref{freocdsdi}), (\ref{ofrrefetypo}).
\end{align}
\end{subequations}
In the following, we first consider the general case and solve the problem by using the block coordinate decent (BCD) method. Then, we consider the special case when the SNR is extremely high, where the globally optimal solution can be obtained.
\subsection{General Case-BCD method}
In the following, we decouple Problem (\ref{nfhoij}) into two subproblems, i.e., optimize power allocation with fixed $x$ and
vice versa. Then, iteratively solve these two subproblems until convergence.
\subsubsection{Power Allocation with Fixed $x$}\label{joijiy}
Given $x$, Problem (\ref{inDAfefro1}) can be transformed to the following subproblem:
\begin{subequations}\label{defrgtwss}
\begin{align}
\mathop {\max }\limits_{\left\{ {{p_1},{p_2}} \right\}} \;\;\;& \gamma\\
{\rm{s.t.}}\;\;\;& (\ref{freocdsdi}), (\ref{ofrrefetypo}).\vspace{-0.1cm}
\end{align}
\end{subequations}
By substituting $p_2=P_T-p_1$ into the expression of $\gamma$ in (\ref{sdWDE}) and performing some manipulations, $\gamma$ can be rewritten as:
\begin{equation*}
\gamma \!= \! - \!\frac{{{h_1}{h_2}}}{{{A^2}}}\left(\! {A{p_1} + B}\! \right)\! -\! \frac{{\frac{{{h_1}{h_2}B}}{A}\left( {\frac{B}{A}\! +\! {P_T}} \right)}}{{A{p_1} \!+\! B}}\! + \! \frac{{{h_1}{h_2}}}{A}\left(\! {\frac{{2B}}{A} \!+ \!{P_T}} \!\right),
\end{equation*}
where $A=h_1-h_2$ and $B=P_Th_2+1$. The second order derivative of $\gamma$ w.r.t. $p_1$ is calculated as
\[\frac{{{\partial ^2}\gamma }}{{\partial p_1^2}} = - \frac{{2{h_1}{h_2}B\left( {B + A{P_T}} \right)}}{{{{\left( {A{p_1} + B} \right)}^3}}}\]
which can be checked to be negative. Hence, $\gamma $ is a concave function and the optimal solution of Problem (\ref{defrgtwss}) can be derived as follows:
\begin{equation}\label{dfregh}
p_1^* = \frac{{\sqrt {B\left( {B + A{P_T}} \right)} - B}}{A},\ p_2^* = {P_T} - p_1^*.
\end{equation}
\vspace{-0.2cm}\subsubsection{Location Optimization with Fixed $p_1$ and $p_2$}
By substituting the expressions of $h_1$ and $h_2$ in (\ref{edeaftg}) into the expressions of $\gamma$, Problem (\ref{inDAfefro1}) is equivalent to
\begin{subequations}\label{joijouopi}
\begin{align}
\mathop {\min }\limits_{x} \;\;\;& {p_2}{\beta _2}{D_1(x)} + {p_1}{\beta _1}{D_2(x)} + {D_1(x)}{D_2(x)}\\
{\rm{s.t.}}\;\;\;& d_1\le x\le d_2,\vspace{-0.1cm}
\end{align}
\end{subequations}
where $D_1(x)=H^2+x^2$ and $D_2(x)=H^2+(D-x)^2$. Obviously, the objective function (OF) in Problem (\ref{joijouopi}) is a continuous function, and the globally optimal solution of Problem (\ref{joijouopi}) is among the locally optimal solutions and boundary points. By setting the first derivative of OF w.r.t. $x$ to zero, we have
\begin{equation}\label{johpipkww}
a{x^3} + b{x^2} + cx + d = 0,
\end{equation}
where $a = 4$, $b = - 6D$, $d = - 2\left( {D{H^2} + D{p_1}{\beta _1}} \right)$, and $c = 2\left( {{D^2} + 2{H^2} + {\beta _1}{p_1} + {\beta _2}{p_2}} \right)$.
Dividing equation (\ref{johpipkww}) by $a$ and substituting $x=t-b/3a$, we have
\begin{equation}\label{swdeartrsg}
{t^3} + \rho t + \kappa = 0,
\end{equation}
where $\rho = \frac{{3ac - {b^2}}}{{3{a^2}}}$ and $\kappa = \frac{{2{b^3} - 9abc + 27{a^2}d}}{{27{a^3}}}$.
Equality (\ref{swdeartrsg}) is a cubic equation. The equation may have only one real solution or three solutions, which depends on the conditions. Specifically, if $4{\rho ^3} + 27{\kappa ^2} > 0$ and $\rho < 0$, there is only one real solution, given by
\begin{equation}\label{jojytjhy}
{t_0} = - 2\frac{{\left| \kappa \right|}}{\kappa }\sqrt { - \frac{\rho }{3}} \cosh \left( {\frac{1}{3}{\rm{arcosh}}\left( {\frac{{ - 3\left| \kappa \right|}}{{2\rho }}\sqrt {\frac{{ - 3}}{\rho }} } \right)} \right),
\end{equation}
if $\rho > 0$, there is only one real solution, given by
\begin{equation}\label{kkhlol}
{t_0} = - 2\sqrt {\frac{\rho }{3}} \sinh \left( {\frac{1}{3}{\rm{arsinh}}\left( {\frac{{3\kappa }}{{2\rho }}\sqrt {\frac{3}{\rho }} } \right)} \right),
\end{equation}
otherwise, there are three real solutions given by
\begin{equation}\label{koiylyol}
\!\!\!{t_k} \!=\! 2\sqrt {\! - \frac{\rho }{3}}\! \cos \left(\! {\frac{1}{3}{\rm{arcos}}\left(\! {\frac{{3\kappa }}{{2\rho }}\sqrt {\frac{{ - 3}}{\rho }} }\! \right) \!-\! \frac{{2\pi k}}{3}} \!\right),k = 0,1,2.
\end{equation}
Once obtaining the real solution of (\ref{swdeartrsg}), set $x_0=t_0-b/3a$ for only one real solution, and $x_k=t_k-b/3a, k=0,1,2$ for three different real solutions. For the one real solution case, if $x_0$ is in the range of $[d_1,d_2]$, choose one from the set $\{d_1,d_2,x_0\}$ with the minimum OF of Problem (\ref{joijouopi}), otherwise, choose one from set $\{d_1,d_2\}$ with the best OF. For the three real solutions case, choose the solutions that fall within the range of $[d_1,d_2]$, which is denoted as $\cal S$. Then, choose the one from the set $\{d_1,d_2,\cal S\}$ with the best OF as the globally optimal solution.
Finally, the BCD method, which iterates between power allocation and location optimization, is applied to solve Problem (\ref{inDAfefro1}) for the general case. The details are omitted for simplicity.
\subsection{Special Case: $1 \ll h_ip_i, i=1,2$}
In this case, the SNR $\gamma$ can be approximated as
\begin{equation}\label{hggtiohg}
\gamma \approx \frac{{{h_1}{h_2}{p_1}{p_2}}}{{{h_2}{p_2} + {h_1}{p_1}}}\buildrel \Delta \over = {\tilde \gamma }.
\end{equation}
By substituting the expressions of ${h_1}$ and $h_2$ in (\ref{edeaftg}) into (\ref{hggtiohg}), ${\tilde \gamma }$ can be obtained as
\begin{equation}\label{wdefr}
\tilde \gamma =\frac{{{\beta _1}{\beta _2}{p_1}{p_2}}}{{{\beta _2}{p_2}\left( {{H^2} + {x^2}} \right) + {\beta _1}{p_1}\left( {{H^2} + {{\left( {D - x} \right)}^2}} \right)}}.
\end{equation}
Let us denote $ x_0 = \frac{{D{\beta _1}{p_1}}}{{{\beta _1}{p_1} + {\beta _2}{p_2}}}$. The optimal $x$ that maximizes $\tilde \gamma $ can be expressed as
\begin{equation}\label{dagtjuik}
x^* = \left\{ \begin{array}{l}
{x_0},\quad{\rm{if}}\ {d_1} \le {x_0} \le {d_2},\\
{d_1},\quad{\rm{if}}\ {x_0} \le {d_1},\\
{d_2},\quad{\rm{if}}\ {x_0} \ge {d_2}.
\end{array} \right.
\end{equation}
We consider the conditions in (\ref{dagtjuik}) case-by-case.
\subsubsection{Condition I: ${d_1} \le {x_0} \le {d_2}$}
By substituting the optimal $x^*=x_0$ into (\ref{wdefr}), Problem (\ref{inDAfefro1}) can be rewritten as
\begin{subequations}\label{convert-pro1}
\begin{align}
\mathop {\min }\limits_{\left\{ {{p_1\ge 0},{p_2\ge 0}} \right\}} \;\;\;& \frac{{{H^2}\left( {{\beta _1}{p_1} + {\beta _2}{p_2}} \right)}}{{{p_1}{p_2}}} + \frac{{{\beta _1}{\beta _2}{D^2}}}{{{\beta _1}{p_1} + {\beta _2}{p_2}}}\\
{\rm{s.t.}}\;\;\;& D{\beta _1}{p_1} \ge {\beta _1}{d_1}{p_1} + {\beta _2}{d_1}{p_2},\label{dewdewf}\\
& D{\beta _1}{p_1} \le {\beta _1}{d_2}{p_1} + {\beta _2}{d_2}{p_2},\label{kiukoi}\\
& p_1+p_2 = P_{T},\label{hjidhw}
\end{align}
\end{subequations}
where (\ref{hjidhw}) is due to Lemma 2.
In the following, we address Problem (\ref{convert-pro1}) by considering two cases: 1) $\beta _1=\beta _2$; 2) $\beta _1 \ne \beta _2$.
\emph{Case I: $\beta _1=\beta _2$}:
Problem (\ref{convert-pro1}) is equivalent to
\begin{subequations}\label{case1}
\begin{align}
\mathop {\max }\limits_{ {{p_1\ge 0}} } \;\;\;& p_1(P_{T}-p_1)\\
{\rm{s.t.}}\;\;\;& \frac{{{d_2}{P_T}}}{D} \ge {p_1} \ge \frac{{{d_1}{P_T}}}{D}.
\end{align}
\end{subequations}
Obviously, the optimal solution can be obtained as follows:
\begin{equation}\label{joihuo}
p_1^* = \left\{ \begin{array}{l}
\frac{{{d_2}{P_T}}}{D},\quad{\rm{ if }}\ {2d_2} \le D,\\
\frac{{{d_1}{P_T}}}{D},\quad{\rm{ if }}\ {2d_1} \ge D,\\
\frac{{{P_T}}}{2},\qquad{\rm{ otherwise}}.
\end{array} \right.
\end{equation}
Then, the optimal $p_2$ is given by $p_2^*=P_T-p_1^*$.
\emph{Case II: $\beta _1\ne \beta _2$}:
The closed-form solution cannot be obtained as Case I. However, we can obtain the globally optimal solution of Problem (\ref{convert-pro1}).
\emph{\textbf{Theorem 1}}: Problem (\ref{convert-pro1}) is a convex optimization problem.
\emph{\textbf{Proof}}: \upshape Obviously, the set of constraints in Problem (\ref{convert-pro1}) is linear. Hence, we only need to prove the convexity of the OF of Problem (\ref{convert-pro1}).
Denote OF of Problem (\ref{convert-pro1}) as function $f(p_1,p_2)$. Obviously, $f(p_1,p_2)$ is twice differentiable, and its Hessian matrix can be derived as
\begin{equation}\label{ewfrfre}
\!\!{\nabla ^2}f \!\!=\!\! \left[ {\begin{array}{*{20}{c}}
{\!\!\frac{{2{H^2}{\beta _2}}}{{p_1^3}} + \frac{{2\beta _1^3{\beta _2}{D^2}}}{{{{\left( {{\beta _1}{p_1} + {\beta _2}{p_2}} \right)}^3}}}}&{\frac{{2\beta _1^2\beta _2^2{D^2}}}{{{{\left( {{\beta _1}{p_1} + {\beta _2}{p_2}} \right)}^3}}}}\\
{\frac{{2\beta _1^2\beta _2^2{D^2}}}{{{{\left( {{\beta _1}{p_1} + {\beta _2}{p_2}} \right)}^3}}}}&{\frac{{2{H^2}{\beta _1}}}{{p_2^3}} + \frac{{2{\beta _1}\beta _2^3{D^2}}}{{{{\left( {{\beta _1}{p_1} + {\beta _2}{p_2}} \right)}^3}}}\!\!\!\!}
\end{array}} \right]
\end{equation}
and its determinant is checked to be strictly bigger than zero. In addition, both the diagonal elements are strictly positive. Hence, ${\nabla ^2}f$ is positive definite. The proof completes. \hfill\rule{2.7mm}{2.7mm}
The globally optimal solution can be obtained by using standard convex optimization algorithms such as interior-point method \cite{boyd2004convex}.
\vspace{-0.1cm}
\subsubsection{Condition II: ${x_0}<{d_1}$} By substituting $x^*=d_1$ into (\ref{wdefr}) and using Lemma 2, Problem (\ref{inDAfefro1}) can be rewritten as
\begin{subequations}\label{CEREFTRooi}
\begin{align}
\mathop {\max }\limits_{p_1} \;\;\;& \frac{{{\beta _1}{\beta _2}{p_1}({P_T} - {p_1})}}{{({\beta _1}{D_2} - {\beta _2}{D_1}){p_1} + {\beta _2}{D_1}{P_T}}}\\
{\rm{s.t.}}\;\;\;& 0 \le {p_1} \le p_1^{{\rm{up}}},\label{hsadwafrhw}
\end{align}
\end{subequations}
where ${D_1} = {H^2} + {d_1^2}$, ${D_2} = {H^2} + {\left( {D - d_1} \right)^2}$, and $p_1^{{\rm{up}}} = \frac{{{d_1}{\beta _2}{P_T}}}{{\left( {D - {d_1}} \right){\beta _1} + {d_1}{\beta _2}}}$.
We solve this problem by considering two cases: 1) ${\beta _1}{D_2} = {\beta _2}{D_1}$; 2) ${\beta _1}{D_2} \ne {\beta _2}{D_1}$.
\emph{Case I:${\beta _1}{D_2} = {\beta _2}{D_1}$}: The optimal solution of Problem (\ref{CEREFTRooi}) can be obtained as follows:
\begin{equation}\label{dWDEFR}
p_1^* = \left\{ \begin{array}{l}
\frac{{{P_T}}}{2},\ {\rm{if }}\ p_1^{{\rm{up}}} > \frac{{{P_T}}}{2}\\
p_1^{{\rm{up}}},\ {\rm{otherwise}}.
\end{array} \right.
\end{equation}
\emph{Case II: ${\beta _1}{D_2} \ne {\beta _2}{D_1}$}: The OF of Problem (\ref{CEREFTRooi}) can be rewritten as:
\begin{equation*}
\tilde \gamma = \frac{{{\beta _1}{\beta _2}}}{{{\beta _1}{D_2} \!-\! {\beta _2}{D_1}}}\left( { \!-\! \left( {{p_1}\! +\! E} \right) \!-\! \frac{{E\left( {E \!+\! {P_T}} \right)}}{{{p_1} \!+\! E}} \!+\! 2E \!+\!{P_T}} \right),
\end{equation*}
where $E$ is equal to $ \frac{{{\beta _2}{D_1}{P_T}}}{{{\beta _1}{D_2} - {\beta _2}{D_1}}}$. The second derivative of $ \tilde \gamma$ w.r.t. $p_1$ is given by
\begin{equation}\label{WEDfoilio}
\frac{{{\partial ^2}\tilde \gamma }}{{\partial p_1^2}} = - \frac{{2{\beta _1}{\beta _2}}}{{{\beta _1}{D_2} - {\beta _2}{D_1}}}\frac{{E\left( {E + {P_T}} \right)}}{{{{\left( {{p_1} + E} \right)}^3}}}
\end{equation}
which is proved to be negative. Hence, Problem (\ref{CEREFTRooi}) is a convex optimization problem. Define ${{\bar p}_1} \triangleq \sqrt {E\left( {E + {P_T}} \right)} - E$. The optimal solution is given by
\begin{equation}\label{dsacR}
p_1^* = \left\{ \begin{array}{l}
{{\bar p}_1},\ {\rm{if }}\ p_1^{{\rm{up}}} > {{\bar p}_1}\\
p_1^{{\rm{up}}},\ {\rm{otherwise}}.
\end{array} \right.
\end{equation}
\vspace{-0.2cm}
\subsubsection{Condition III: ${x_0}>{d_2}$}
The optimal solution in this case can be obtained by using the similar method as those in Condition II, the details of which are omitted here.
When the optimal solution for each condition is obtained, select one solution with the largest value of ${\tilde \gamma }$ as the globally optimal solution of Problem (\ref{inDAfefro1}).
\section{3-D Channel Model}\label{jojt}
In this section, we extend the free-space channel model to 3-D channel model proposed in \cite{al2014optimal}, where the impacts of blockage and shadowing are taken into account and is more practical than free-space channel model. In specific, the line-of-sight (LoS) probability is given by
\begin{equation}\label{dwerji}
\vspace{-0.2cm}
{P_{{\rm{LoS}}}} = \frac{1}{{1 + a\exp \left( { - b\left( {\theta - a} \right)} \right)}},
\end{equation}
where $a$ and $b$ are positive environment-related parameters and $\theta$ is the elevation angle between the UAV and the ground devices (controller or robot) as shown in Fig.~\ref{systemodel}. Some typical values of $a$ and $b$ can be found in Table I of \cite{bor2016efficient}. It is observed from (\ref{dwerji}) that the LoS probability increases with the elevation angle, which is reasonable as the probability that signal is blocked is decreasing when the height of UAV is increasing.
When the location of one UAV is given, the mean path loss is given by \cite{al2014optimal}:
\vspace{-0.1cm}
\begin{equation}\label{xswdff}
\vspace{-0.1cm}
L(\theta ,d) = \frac{A}{{1 + a\exp \left( { - b\left( {\theta - a} \right)} \right)}} + 20{\log _{10}}\left( {{d}} \right) + C,
\end{equation}
where $A$ and $C$ are constants given by $A=\eta _{{\rm{LoS}}}-\eta _{{\rm{NLoS}}}$ and $C=20{\log _{10}}\left( {\frac{{4\pi {f_c}}}{c}} \right) + {\eta _{{\rm{NLoS}}}}$, respectively. $d$ is the distance between the UAV and the ground devices (controller or robot). $\eta _{{\rm{LoS}}}$ and $\eta _{\rm{NLoS}}$ are the path loss (in dB) corresponding to the LoS and non-LoS (NLoS) links. In general, $\eta _{\rm{NLoS}}$ is larger than $\eta _{\rm{LoS}}$ due to the more severe attenuation associated with NLoS. $f_c$ is the central frequency point, $c$ is the light speed.
Based on the path loss model in (\ref{xswdff}), the normalized channel gains w.r.t. noise power are given by
\begin{equation}\label{jjgjngkk}
h_i=\tilde C_i{d_i^{ - 2}}{10^{ \frac{{\tilde A_i}}{{1 + a_i\exp \left( { - b_i(\theta_i - a_i)} \right)}}}}, i=1,2
\end{equation}
where $\tilde A_i = -\frac{{A_i}}{{10}}>0$ and $\tilde C_i ={{{{10}^{ - \frac{C_i}{{10}}}}} \mathord{\left/
{\vphantom {{{{10}^{ - \frac{C_i}{{10}}}}} {{\delta ^2}}}} \right.
\kern-\nulldelimiterspace} {{\delta ^2}}}$ with $\delta ^2$ denoting the noise power, and $\theta_i$ are given by
\begin{equation}\label{hyuik}
{\theta _1} = \arctan \left( {\frac{H}{x}} \right),{\theta _2} = \arctan \left( {\frac{H}{{D - x}}} \right).
\end{equation}
Similar to the free-space case, we also adopt the BCD algorithm to solve Problem (\ref{inDAfefro1}). When $x$ and $H$ are given, channel gains $h_1$ and $h_2$ are fixed. Then, the power allocation can be optimized by using the same method in Subsection \ref{joijiy}. In the following, we only focus on the optimization of height $H$ and horizontal distance $x$ when the other parameters are fixed.
\subsection{Optimization of $H$ with fixed $x$, $p_1$ and $p_2$}
When $x$, $p_1$ and $p_2$ are given, the SNR $\gamma(H)$ is a very complicated function of $H$. It is difficult to strictly prove the monotonically and convexity of this function. As in \cite{bor2016efficient} and \cite{cpan2019}, we graphically illustrate these properties in Fig.~\ref{checkH}, where we show the SINR $\gamma(H)$ versus $H$ with $x=100$ m. Four different scenarios are illustrated, and the corresponding parameters for each scenario are given in \cite{bor2016efficient}. It can be found from this figure that for each scenario, the SINR value first increases with height $H$ and then decreases with $H$. As a result, there exists only one maximum point for each scenario, denoted as $H^\star$. The value of $H^\star$ is the solution to the following equation:
\begin{equation}\label{hjik}
\frac{{d\gamma (H)}}{{dH}} = 0.
\end{equation}
Similar to \cite{bor2016efficient}, the bisection search method can be used to find the root of the above equation.
\begin{figure}
\centering
\includegraphics[width=2.4in]{checkH.pdf}
\caption{SINR $\gamma$ versus height $H$ when $x=100$ m.}
\vspace{-0.3cm}
\label{checkH}
\end{figure}
\subsection{Optimization of $x$ with fixed $H$, $p_1$ and $p_2$}
In Fig.~\ref{checkx}, we illustrate the SINR value versus the horizontal distance $x$ when $H=120$ m. The channel from the controller the UAV is assumed to be suburban environment. The SNR values when the channel from the UAV to the robot experiences various environments are shown in Fig.~\ref{checkx}. Similar to Fig.~\ref{checkH}, the SINR value also first increases with $x$ and then decreases with $x$, or always increases with $x$. Then, the bisection search method can be adopted to find the optimal solution.
\begin{figure}
\centering
\includegraphics[width=2.4in]{checkx.pdf}
\caption{SINR $\gamma$ versus the horizontal distance $x$ when $H=120$ m.}
\vspace{-0.3cm}
\label{checkx}
\end{figure}
\section{Simulation Results}\label{simlresult}
Simulation results are performed to check the performance of the proposed algorithms. The system parameters are set as $D=200$ m, $H=120$ m, $d_1=30$ m, $d_2=170$ m, $L=100$ bits, $M=80$, and $P_T=4$ Watt.
\subsection{Free-space Channel Model}
We first study the free-space channel model, where $\beta_1=50$ dB, $\beta_2=59$ dB.
\begin{figure}
\centering
\includegraphics[width=2.4in]{convergence.pdf}
\caption{Convergence behaviour.}
\vspace{-0.4cm}
\label{convergence}
\end{figure}
In Fig.~\ref{convergence}, we study the convergence behaviour of the iterative algorithm for the general case. It is shown that the proposed iterative algorithm converges rapidly and in general four iterations are sufficient for the algorithm to converge, which implies low complexity of our proposed algorithm.
\begin{figure}
\centering
\includegraphics[width=2.4in]{percompari.pdf}
\caption{Performance comparison, $H=120$ m.}
\vspace{-0.4cm}
\label{Perforcompare}
\end{figure}
In Fig.~\ref{Perforcompare}, we compare the performance of various algorithms, which include: 1) General case (`General-Case Alg.'); 2) High-SNR case (`High-SNR Alg.'); 3) Exhaustive search algorithm (`Exhaus. Search'); 4) Fixed location with $x=(d_1+d_2)/2$ (`Fixed-Loca.'); 5) Fixed power allocation with $p_1=p_2=P_T/2$ (`Fixed-Power'). We also compare the proposed algorithm for AF relay with the one for DF relay in \cite{cpan2019}. For the DF relay, the number of channel uses for the signal processing at the relay is denoted as DD \footnote{In general, the system bandwidth is fixed, and then the number of channel users can be interpreted as time duration.}. It is observed in Fig.~\ref{Perforcompare} that the proposed two algorithms significantly outperform the Fixed-Loca. algorithm and Fixed-Power algorithm, which confirms the benefits of our proposed algorithms. It is interesting to find that the proposed two algorithms have almost the same performance as the exhaustive search method. This may be due to the fact that the SNR in this example generally operates in a very high regime. When the signal processing delay DD is small (e.g. DD=10), the DF relay outperforms the AF relay, which means DF relay is a good option. On the other hand, when DD is large, the AF relay performs better than the DF relay when the number of channel blocklength $M$ is small. In this example, when DD=20, the performance of the AF relay is better than that of the DF relay when $M\le 85$. This means that it is beneficial to adopt the AF relay when the latency requirement is stringent, which is usually the case in URLLC applications. The reason is that when more time is used for signal processing, the left time for data transmission will be reduced, which decreases the reliability performance.
\subsection{3-D channel model}
\begin{figure}
\centering
\includegraphics[width=2.4in]{3Dperfor.pdf}
\caption{Performance comparison for 3-D channel models.}
\vspace{-0.4cm}
\label{Perforcompare3D}
\end{figure}
In Fig.~\ref{Perforcompare3D}, we study the performance of the algorithm developed in Section \ref{jojt} for the 3-D channel model.
The simulation parameters are set as follows: $H_{\min}=10$ m, $H_{\max}=200$ m, $d_1=20$ m, $d_2=200$ m, $f_c=2.5$ GHz, noise power is -93 dB. The channel model from the controller the UAV is assumed to be suburban environment. We study the performance when the channel from the UAV to the robot experiences various environments. The environment parameters are referred to \cite{bor2016efficient}. To study the importance of optimizing the height, we also show the performance when the height is fixed as $H=100$ m, which is denoted as `FixedH' in Fig.~\ref{Perforcompare3D}. It can be observed that the proposed joint optimization algorithm outperforms the the `FixedH' algorithm for various environments, and the performance gains increase with $M$, which confirms the importance of optimizing the height.
\vspace{-0.1cm}
\section{Conclusions}\label{conclu}
\vspace{-0.1cm}
This paper studied the joint power allocation and location optimization for UAV AF-relay system with URLLC requirements. Both the free-space channel and the 3-D channel are considered. For the free-space channel, the iterative algorithm was proposed for general case, and the closed-form solution was derived for the special case with high SNR. A low-complexity iterative algorithm was proposed for the 3-D channel model. Simulation results showed that the proposed algorithms can achieve the same performance as the exhaustive search method, and outperform the other algorithms such as fixed location or fixed power allocation.
\vspace{-0.2cm}
\numberwithin{equation}{section}
\begin{appendices}
\section{Proof of Lemma 1}\label{prooflemma1}
The first derivative of $\varepsilon$ w.r.t. $\gamma$ is
\begin{equation}\label{daef}
\varepsilon'= - \frac{1}{{\sqrt {2\pi } }}{e^{ - \frac{{f^2({\gamma})}}{2}}}f'(\gamma),
\end{equation}
where ${f}({\gamma })$ is short for function $f\left( {\gamma, m, L} \right)$ and $f'(\gamma)$ is the first derivative of ${f}({\gamma })$ w.r.t. $\gamma$ that is given by
\begin{eqnarray}
\!\!\! \!\!\!\!\!\! f'(\gamma) \!\!\! &\!= \!&\frac{{\sqrt m }}{{\sqrt {{{(1 + \gamma )}^2} - 1} }}\left( {1\! -\! \ln 2\frac{{{{\log }_2}(1 \!+\! \gamma )\! -\! \frac{L}{m}}}{{{{(1 + \gamma )}^2} - 1}}} \right) \label{deafrg}\\
&\ge& \frac{{\sqrt m }}{{\sqrt {{{(1 + \gamma )}^2} - 1} }}\left( {1 - \frac{{{{\ln }}(1 + \gamma )}}{{{{(1 + \gamma )}^2} - 1}}} \right).\label{jytijju}
\end{eqnarray}
Let $x=1+\gamma$ and thus $x\ge1$. Define function $g(x)$ as
\begin{equation}\label{dwfrfr}
g(x)\buildrel \Delta \over = \frac{{{{\ln }}(x )}}{{{{x }^2} - 1}}.
\end{equation}
The first derivative of $g(x)$ w.r.t. $x$ is given by
\begin{equation}\label{defrfregt}
g'(x) = \frac{{G(x)}}{{x{{({x^2} - 1)}^2}}}
\end{equation}
where $G(x) = {x^2} - 1 - 2{x^2}\ln x$. The first derivation of $G(x)$ w.r.t. $x$ is given by $G'(x)=-4x\ln(x)\le 0$ for $x\ge 1$. Hence, $G(x)$ is a decreasing function for $x\ge 1$ and thus $G(x)\le G(1)=0$ holds. Please note that the denominator of (\ref{defrfregt}) is positive, then $g'(x)\le 0$ holds for $x\ge 1$. Hence $g(x)$ is a decreasing function of $x$ and $g(x)\le g(1)$. By using the L'Hospital's rule, $g(1)$ can be calculated as $g(1)=1/2$. By plugging the inequality $g(x)\le 1/2$ into (\ref{jytijju}), we obtain
\begin{equation}\label{dwfrrf}
f'(\gamma)\ge \frac{{\sqrt m }}{2{\sqrt {{{(1 + \gamma )}^2} - 1} }}\ge 0.
\end{equation}
Hence, $\varepsilon'\le 0$, which completes the proof.
\end{appendices}
\
\
\vspace{-0.3cm}
\bibliographystyle{IEEEtran}
\vspace{-0.2cm}
|
2,877,628,088,961 | arxiv | \section{Introduction}
Analyzing the time-frequency localization of functions
is an important topic in harmonic analysis.
Quantitative results on this localization are usually formulated in terms of function spaces
such as Sobolev spaces, modulation spaces, or Wiener amalgam spaces.
An especially important space is the \emph{Feichtinger algebra} $S_0 = M^1$
\cite{FeichtingerNewSegalAlgebra,JakobsenNoLongerNewSegalAlgebra} which has numerous
remarkable properties; see, e.g., \cite[Section A.6]{ChristensenBook} for a compact overview.
Yet, in some cases it is preferable to work with more classical spaces like the Sobolev space
$H^1 (\R^d) = W^{1,2}(\R^d)$,
the weighted $L^2$-space $L_{1 + |x|}^2(\R^d) = \{ f : \R^d \to \CC : (1 + |x|) f(x) \in L^2 \}$,
or the space $\mathbb{H}^1(\R^d) = H^1 (\R^d) \cap L_{1 + |x|}^2(\R^d)$ which consists of all functions
$g \in L^2(\R^d)$ with finite uncertainty product
\begin{equation}\label{eqn:FUP}
\left(
\int_{\R^d}
|x|^2
\cdot |g(x)|^2
\, dx
\right)
\left(
\int_{\R^d}
|\omega|^2
\cdot |\widehat g(\omega)|^2
\,d \omega
\right)
< \infty .
\end{equation}
Certainly, one advantage of these classical spaces is that membership of a function in the space
can be decided easily.
We remark that all of these spaces fall into the scale of modulation spaces
(see Section~\ref{s:Invariance}).
In Gabor analysis, it is known (see e.g., \cite[Proposition~5.2.1]{GroechenigTFFoundations} and \cite[Theorem~12.3.2]{ChristensenBook}) that for a Gabor frame generated by a lattice, the canonical dual frame is again a Gabor system (over the same lattice), generated by the so-called \emph{dual window}.
An important question is what kind of time-frequency localization conditions
are inherited by the dual window.
Precisely, if $g \in L^2(\R^d)$ belongs to a certain ``localization Banach space'' $V$
and if $\Lambda \subset \R^{2d}$ is such that $(g,\Lambda)$ forms a Gabor frame for $L^2(\R^d)$,
then does the canonical dual window belong to $V$ as well?
A celebrated result in time-frequency analysis states that this is true for the
Feichtinger algebra $V = S_0(\R^d)$; see \cite{GroechenigLeinert} for separable lattices $\Lambda$
and \cite[Theorem~7]{BalanDensityOvercompletenessLocalization} for irregular sets $\Lambda$.
In the case of separable lattices, the question has been answered affirmatively
also for the Schwartz space $V = \mathcal S} \newcommand{\frakS}{\mathfrak S(\R)$ \cite[Proposition~5.5]{j}
and for the Wiener amalgam space $V = W(L^{\infty},\ell_v^1)$
with a so-called \emph{admissible weight} $v$; see \cite{ko}.
Similarly, the setting of the spaces $V = W(C_\alpha,\ell_v^q)$ (with the H\"{o}lder spaces $C_\alpha$)
is studied in \cite{w}---but except in the case $q = 1$, some additional assumptions
on the window function $g$ are imposed.
To the best of our knowledge, the question has not been answered for modulation spaces other than
$V = M^1_v$, and in particular, not for any of the spaces $V = H^1(\R^d)$, $V = L_{1 + |x|}^2(\R^d)$,
and $V = \mathbb{H}^1(\R^d)$.
In this note, we show that the answer is affirmative for all of these spaces:
\begin{thm}\label{t:main}
Let $V \in \{ H^1(\R^d), L_{1 + |x|}^2(\R^d), \mathbb{H}^1(\R^d) \}$.
Let $g \in V$ and let $\Lambda \subset \R^{2d}$ be a lattice such that
the Gabor system $(g, \Lambda)$ is a frame for $L^2(\R^d)$ with frame operator $S$.
Then the canonical dual window $S^{-1} g$ belongs to $V$.
Furthermore, $(S^{-1/2} g, \Lambda)$ is a Parseval frame for $L^2(\R^d)$ with $S^{-1/2} g \in V$.
\end{thm}
As mentioned above, the corresponding statement of Theorem~\ref{t:main} for $V = S_0(\R^d)$
with separable lattices $\Lambda$ was proved in \cite{GroechenigLeinert}.
In addition to several deeper insights, the proof given in \cite{GroechenigLeinert}
relies on a simple but essential argument showing that the frame operator $S = S_{\Lambda,g}$
maps $V$ boundedly into itself, which is shown in \cite{GroechenigLeinert}
based on Janssen's representation of $S_{\Lambda,g}$.
In our setting, this argument is not applicable, because---unlike in the case of $V = S_0(\R^d)$---%
there exist functions $g \in \mathbb{H}^1$ for which $(g,\Lambda)$ is not an $L^2$-Bessel system.
In addition, the series in Janssen's representation is not even guaranteed
to converge unconditionally in the strong sense for $\mathbb{H}^1$-functions,
even if $(g,\Lambda)$ is an $L^2$-Bessel system; see Proposition~\ref{prop:MainResult}.
To bypass these obstacles, we introduce for each space $V \in \{ H^1, L_{1 + |x|}^2, \mathbb{H}^1 \}$
the associated subspace $V_\Lambda$ consisting of all those functions $g \in V$
that generate a Bessel system over the given lattice $\Lambda$.
We remark that most of the existing works concerning the regularity of the (canonical) dual window
rely on deep results related to Wiener's $1/f$-lemma
on absolutely convergent Fourier series.
In contrast, our methods are based on elementary spectral theory
(see \Cref{s:spectra}) and on certain observations regarding the interaction
of the Gabor frame operator with partial derivatives; see Proposition~\ref{p:bounded}.
The paper is organized as follows:
Section~\ref{s:BesselVectors} discusses the concept of Gabor Bessel vectors
and introduces some related notions.
Then, in Section~\ref{s:Invariance}, we endow the space $V_\Lambda$
(for each choice $V \in \{ H^1, L_{1 + |x|}^2, \mathbb{H}^1 \}$)
with a Banach space norm and show that the frame operator $S$
maps $V_\Lambda$ boundedly into itself, provided that
the Gabor system $(g,\Lambda)$ is an $L^2$-Bessel system and that
the window function $g$ belongs to $V$.
Finally, we prove in Section~\ref{s:spectra} that for any $V \in \{ H^1, L_{1 + |x|}^2, \mathbb{H}^1 \}$
the spectrum of $S$ as an operator on $V$ coincides with the spectrum of $S$ as an operator on $L^2$.
This easily implies our main result, Theorem~\ref{t:main}.
\section{Bessel vectors}
\label{s:BesselVectors}
For $a,b\in\R^d$ and $f\in L^2(\R^d)$ we define the operators of translation
by $a$ and modulation by $b$ as
\[
T_a f(x) := f(x-a)
\quad \text{and} \quad
M_b f(x) := e^{2\pi ib\cdot x} \cdot f(x),
\]
respectively.
Both $T_a$ and $M_b$ are unitary operators on $L^2(\R^d)$
and hence so is the {\em time-frequency shift}
\[
\pi(a,b)
:= T_a M_b
= e^{-2\pi ia\cdot b} \, M_b T_a .
\]
The Fourier transform $\mathcal F} \newcommand{\frakF}{\mathfrak F$ is defined on $L^1(\R^d) \cap L^2(\R^d)$ by
$\mathcal{F} f (\xi) = \widehat{f}(\xi) = \int_{\R^d} f(x) e^{-2 \pi i x \cdot \xi} \, d x$
and extended to a unitary operator on $L^2(\R^d)$.
For $z = (z_1, z_2) \in \R^d \times \R^d \cong \R^{2d}$ and $f \in L^2(\R^d)$,
a direct calculation shows that
\begin{equation}\label{e:FTpi}
\mathcal F} \newcommand{\frakF}{\mathfrak F [\pi(z)f] = e^{-2\pi iz_1\cdot z_2}\cdot\pi(Jz)\widehat f,
\end{equation}
where
\
J = \mat 0{I}{-I}0.
\]
A (full rank) {\em lattice} in $\R^{2d}$ is a set of the form $\Lambda = A\Z^{2d}$,
where $A\in\R^{2d\times 2d}$ is invertible.
The volume of $\Lambda$ is defined by $\operatorname{Vol}(\Lambda) := |\!\det A|$
and its density by $d(\Lambda) := \operatorname{Vol}(\Lambda)^{-1}$.
The {\em adjoint lattice} of $\Lambda$ is denoted and defined by $\Lambda^\circ := JA^{-{\!\top}}\Z^{2d}$.
The Gabor system generated by a window function $g\in L^2(\R^d)$
and a lattice $\Lambda\subset\R^{2d}$ is given by
\[
(g,\Lambda) := \bigl\{ \pi(\lambda)g : \lambda\in\Lambda \bigr\}.
\]
We say that $g\in L^2(\R^d)$ is a {\em Bessel vector} with respect to $\Lambda$
if the system $(g,\Lambda)$ is a Bessel system in $L^2(\R^d)$,
meaning that the associated {\em analysis operator} $C_{\Lambda,g}$ defined by
\begin{equation}\label{eq:CoefficientOperator}
C_{\Lambda,g} f := \big(\<f,\pi(\lambda)g \>\big)_{\lambda\in\Lambda},
\qquad f \in L^2(\R^d) ,
\end{equation}
is a bounded operator from $L^2(\R^d)$ to $\ell^2(\Lambda)$.
We define
\[
\mathcal B} \newcommand{\frakB}{\mathfrak B_\Lambda
:= \big\{g\in L^2(\R^d) : (g,\Lambda)\text{ is a Bessel system}\big\} ,
\]
which is a dense linear subspace of $L^2(\R^d)$ because each Schwartz function
is a Bessel vector with respect to any lattice; see \cite[Theorem~3.3.1]{fz}.
It is well-known that $\mathcal B} \newcommand{\frakB}{\mathfrak B_\Lambda = \mathcal B} \newcommand{\frakB}{\mathfrak B_{\Lambda^\circ}$
(see, e.g., \cite[Proposition~3.5.10]{fz}).
In fact, we have for $g \in \mathcal B} \newcommand{\frakB}{\mathfrak B_\Lambda$ that
\begin{equation}\label{e:Cnorms}
\big\| C_{\Lambda^\circ,g} \big\|
= \operatorname{Vol}(\Lambda)^{1/2} \cdot \big\| C_{\Lambda,g} \big\| ;
\end{equation}
see \mbox{\cite[proof of Theorem~2.3.1]{k}}.
The {\em cross frame operator} $S_{\Lambda,g,h}$ with respect to $\Lambda$
and two functions $g,h\in\mathcal B} \newcommand{\frakB}{\mathfrak B_\Lambda$ is defined by
\[
S_{\Lambda,g,h} := C_{\Lambda,h}^*C_{\Lambda,g}.
\]
In particular, we write $S_{\Lambda,g} := S_{\Lambda,g,g}$ which is called
the {\em frame operator} of $(g,\Lambda)$.
The system $(g,\Lambda)$ is called a \emph{frame} if $S_{\Lambda,g}$ is bounded and boundedly invertible
on $L^2(\R^d)$, that is, if $A \operatorname{Id}_{L^2(\R^d)} \leq S_{\Lambda,g} \leq B \operatorname{Id}_{L^2(\R^d)}$
for some constants $0 < A \leq B < \infty$ (called the frame bounds).
In particular, a frame with frame bounds $A=B=1$ is called a \emph{Parseval frame}.
In our proofs, the so-called {\em fundamental identity of Gabor analysis}
will play an essential role.
This identity states that
\begin{equation}\label{e:fi}
\sum_{\lambda\in\Lambda}\<f,\pi(\lambda)g\>\<\pi(\lambda)\gamma,h\>
= d(\Lambda) \cdot \sum_{\mu\in\Lambda^\circ}
\<\gamma,\pi(\mu)g\>
\<\pi(\mu)f,h\>.
\end{equation}
It holds, for example, if $f, h \in M^1(\R^d) = S_0(\R^d)$ (the Feichtinger algebra)
and $g,\gamma\in L^2(\R^d)$; see \mbox{\cite[Theorem~3.5.11]{fz}}.
We will use the following version of the fundamental identity:
\begin{lem
The fundamental identity \eqref{e:fi} holds if $g,h\in\mathcal B} \newcommand{\frakB}{\mathfrak B_\Lambda$ or $f,\gamma\in\mathcal B} \newcommand{\frakB}{\mathfrak B_\Lambda$.
\end{lem}
\begin{proof}
In \cite[Subsection~1.4.1]{j2}, the claim is shown for separable lattices in $\R^2$.
Here, we provide a short proof for the general case.
If $g,h \in \mathcal B} \newcommand{\frakB}{\mathfrak B_{\Lambda}$, then Equation~\eqref{e:Cnorms}
shows that both sides of Equation~\eqref{e:fi} depend continuously on $f,\gamma \in L^2$.
Similarly, if $f,\gamma \in \mathcal B} \newcommand{\frakB}{\mathfrak B_{\Lambda}$ then both sides of Equation~\eqref{e:fi}
depend continuously on $g,h \in L^2$.
Therefore, and because $\mathcal B} \newcommand{\frakB}{\mathfrak B_\Lambda$ is dense in $L^2(\R^d)$,
it is no restriction to assume that $f, g, h, \gamma \in \mathcal B} \newcommand{\frakB}{\mathfrak B_\Lambda$.
Let $\Lambda = A\Z^{2d}$ and define the function
\[
G(x)
= \sum_{n\in\Z^{2d}}
\big\<f,\pi(A(n-x))g\big\>
\big\<\pi(A(n-x))\gamma,h\big\> ,
\quad x\in\R^{2d}.
\]
Writing $A x = ( (A x)_1, (A x)_2 ) \in \R^d \times \R^d$, a direct computation shows that
\[
\langle f, \pi(A(n-x)) g \rangle
= e^{2 \pi i (A x)_2 \cdot ( (A x)_1 - (A n)_1)}
\cdot \langle \pi (A x) f, \pi(A n) g \rangle .
\]
Therefore, and because of $g, \gamma \in \mathcal B} \newcommand{\frakB}{\mathfrak B_\Lambda$ and since $z \mapsto \pi(z) u$
is continuous on $\R^{2d}$ for each $u \in L^2(\R^d)$, the function $G$ is continuous.
Furthermore, we have
\begin{align*}
\sum_{n \in \Z^{2d}}
|\langle f, \pi(A(n-x)) g \rangle
\cdot \langle \pi( A(n-x)) \gamma, h \rangle|
& = \sum_{n \in \Z^{2 d}}
|\langle \pi(A x) f, \pi(A n) g \rangle|
\cdot |\langle \pi(A n) \gamma, \pi(A x) h \rangle| \\
& \leq \| C_{\Lambda,g} [\pi(A x) f] \|_{\ell^2}
\cdot \| C_{\Lambda,\gamma} [\pi(A x) h] \|_{\ell^2} \\
& \leq \| C_{\Lambda,g} \|
\cdot \| C_{\Lambda,\gamma} \|
\cdot \| \pi(A x) f \|_{L^2}
\cdot \| \pi(A x) h \|_{L^2} \\
& = \| C_{\Lambda,g} \|
\cdot \| C_{\Lambda,\gamma} \|
\cdot \| f \|_{L^2}
\cdot \| h \|_{L^2} ,
\end{align*}
which will justify the application of the dominated convergence theorem in the following
calculation.
Indeed, $G$ is $\Z^{2d}$-periodic and the $k$-th Fourier coefficient of $G$ (for $k \in \Z^{2d}$)
is given by
\begin{align*}
c_k
&= \int_{Q}
G(x) e^{-2\pi i k x}
\, dx
= \sum_{n \in \Z^{2d}}
\int_{Q}
\big\<f, \pi(A(n-x))g\big\>
\big\<\pi(A(n-x))\gamma,h\big\>
e^{2\pi ik(n-x)}
\,d x \\
&= \int_{\R^{2d}}
\big\<f,\pi(Ax)g\big\>
\big\<\pi(Ax)\gamma,h\big\>
e^{2\pi ikx}
\,dx
= \frac{1}{|\det A|}
\int_{\R^{2d}}
V_g f(y)
\overline{V_\gamma h(y)}
\cdot e^{2\pi i A^{-{\!\top}}k\cdot y}
\, d y \\
&= d(\Lambda)
\int_{\R^{2d}}
V_{\pi(z_k)g}[\pi(z_k)f](y)
\overline{V_\gamma h(y)}
\, d y
= d(\Lambda) \cdot \<\pi(z_k)f,h\> \<\gamma,\pi(z_k)g\>,
\end{align*}
where $Q := [0,1]^{2d}$, $z_k := -JA^{-{\!\top}}k\in\Lambda^\circ$,
and for $f_1, g_1 \in L^2(\R^d)$, $V_{g_1} f_1 (z) = \langle f_1, \pi(z) g_1 \rangle$
for $z \in \R^{2d}$ is the \emph{short-time Fourier transform} of $f_1$ with respect to $g_1$.
Here, we used the orthogonality relation for the short-time Fourier transform
(see \mbox{\cite[Theorem~3.2.1]{GroechenigTFFoundations}})
and the identity $V_{\pi(z)g}[\pi(z)f] = e^{2\pi i\<Jz,\cdot\>}\cdot V_gf$
(\cite[Lemma~1.4.4(b)]{k}).
Now, as also $f,g\in\mathcal B} \newcommand{\frakB}{\mathfrak B_\Lambda = \mathcal B} \newcommand{\frakB}{\mathfrak B_{\Lambda^\circ}$,
we see that $(c_k)_{k \in \Z^{2d}} \in \ell^1(\Z^{2d})$.
Since $G$ is continuous and $\Z^{2d}$-periodic,
this implies that the Fourier series of $G$ converges uniformly and coincides pointwise with $G$.
Hence,
\[
G(x) = \sum_{k\in\Z^{2d}}
c_k \, e^{2\pi ikx}
\quad \text{for all} \;\; x\in\R^{2d} ,
\]
and setting $x = 0$ yields the claim.
\end{proof}
\section{Certain subspaces of modulation spaces invariant under the frame operator}
\label{s:Invariance}
The $L^2$-Sobolev-space $H^1 (\R^d) = W^{1,2}(\R^d)$ is the space of all functions
$f \in L^2(\R^d)$ whose distributional derivatives
$\partial_jf := \frac{\partial f}{\partial x_j}$, $j \in \{ 1,\ldots,d \}$,
all belong to $L^2(\R^d)$.
We will frequently use the well-known characterization
$H^1 (\R^d) = \bigl\{ f \in L^2(\R^d) : (1+| \cdot |) \widehat{f} (\cdot) \in L^2 \bigr\}$
of $H^1(\R^d)$ in terms of the Fourier transform.
With the weight function $w : \R^d \to \R$, $x \mapsto 1 + |x|$,
we define the weighted $L^2$-space
$L_w^2 (\R^d) := \bigl\{ f : \R^d \to \CC \colon w (\cdot) f (\cdot) \in L^2 \bigr\}$
which is equipped with the norm $\| f \|_{L_w^2} := \| w \, f \|_{L^2}$.
It is then clear that $L_w^2 (\R^d) = \mathcal{F} [ H^1 (\R^d) ] = \mathcal{F}^{-1} [ H^1 (\R^d) ]$.
Finally, we define $\mathbb{H}^1(\R^d) = H^1 (\R^d) \cap L_w^2 (\R^d)$ which is the space
of all functions $f \in H^1(\R^d)$ whose Fourier transform $\widehat f$ also belongs to $H^1(\R^d)$.
Equivalently, $\mathbb{H}^1(\R^d)$ is the space of all functions $g \in L^2(\R^d)$
with finite uncertainty product \eqref{eqn:FUP}.
It is worth to note that each of the spaces $H^1 (\R^d)$, $L_w^2 (\R^d)$,
and $\mathbb{H}^1(\R^d)$ can be expressed as a modulation space
\(
M_m^2(\R^d)
= \{
f \in L^2(\R^d)
:
\int_{\R^{2d}}
| \<f,\pi(z)\varphi \> |^2 \, |m(z)|^2
\, dz
< \infty
\}
\)
for some weight function $m: \R^{2d} \rightarrow \CC$,
where $\varphi \in \mathcal{S} (\R^d) \backslash \{ 0 \}$ is any fixed function%
\footnote{The definition of $M_m^2$ is known to be independent of the choice of $\varphi$;
see e.g., \cite[Proposition~11.3.2]{GroechenigTFFoundations}.}, for instance a Gaussian.
Indeed, we have
\[
H^1(\R^d) = M_{m_1}^2(\R^d),\quad
L_w^2(\R^d) = M_{m_2}^2(\R^d),\quad
\text{and}\quad
\mathbb{H}^1(\R^d) = H^1 (\R^d) \cap L_w^2(\R^d) = M_{m_3}^2(\R^d),
\]
with $m_1(x,\omega) = 1+|\omega|$, $m_2(x,\omega) = 1+|x|$,
and $m_3(x,\omega) = \sqrt{1 + |x|^2 + |\omega|^2}$, respectively;
see \cite[Proposition 11.3.1]{GroechenigTFFoundations} and \cite[Corollary~2.3]{HeilTinaztepe}.
Our main goal in this paper is to prove for each of these spaces that if the window function $g$
of a Gabor frame $(g,\Lambda)$ belongs to the space, then so does the canonical dual window.
In this section, we will mostly concentrate on the space $H^1(\R^d)$,
since this will imply the desired result for the other spaces as well.
The corresponding result for the Feichtinger algebra $S_0(\R^d)$ was proved
in \cite{GroechenigLeinert} by showing the much stronger statement that the frame operator
maps $S_0(\R^d)$ boundedly into itself and is in fact boundedly invertible on $S_0(\R^d)$.
However, the methods used in \cite{GroechenigLeinert} cannot be directly transferred
to the case of a window function in $\mathbb{H}^1(\R^d)$ (or $H^1(\R^d)$),
since the proof in \cite{GroechenigLeinert} leverages
two particular properties of the Feichtinger algebra which are not shared by $\mathbb{H}^1(\R^d)$:
\begin{enumerate}
\item[(a)] Every function from $S_0(\R^d)$ is a Bessel vector with respect to any given lattice;
\item[(b)] The series in Janssen's representation of the frame operator converges strongly
(even absolutely in operator norm) to the frame operator
when the window function belongs to $S_0(\R^d)$.
\end{enumerate}
Indeed, it is well-known that $g\in L^2(\R)$ is a Bessel vector with respect to $\Z\times\Z$
if and only if the Zak transform of $g$ is essentially bounded
(cf.~\cite[Theorem~3.1]{BenedettoDifferentiationAndBLT}), but
\mbox{\cite[Example~3.4]{BenedettoDifferentiationAndBLT}} provides an example
of a function $g\in\mathbb{H}^1(\R)$ whose Zak transform is not essentially bounded;
this indicates that (a) does not hold for $\mathbb{H}^1(\R^d)$ instead of $S_0(\R^d)$.
Concerning the statement (b) for $\mathbb{H}^1(\R^d)$, it is easy to see
that if Janssen's representation converges strongly (with respect to some enumeration of $\Z^2$)
to the frame operator of $(g,\Lambda)$, then the frame operator must be bounded on $L^2(\R)$
and thus the associated window function $g$ is necessarily a Bessel vector.
Therefore, the example above again serves as a counterexample: namely,
the statement (b) fails for such a non-Bessel window functions $g\in\mathbb{H}^1(\R)$.
Even more, we show in the Appendix that there exist Bessel vectors $g\in\mathbb{H}^1(\R)$
for which Janssen's representation neither converges unconditionally
in the strong sense nor conditionally in the operator norm.
We mention that in the case of the Wiener amalgam space $W(L^{\infty},\ell_v^1)$
with an admissible weight $v$, the convergence issue was circumvented
by employing Walnut's representation instead of Janssen's
to prove the result for $W(L^{\infty},\ell_v^1)$ in \cite{ko}.
Fortunately, it turns out that establishing the corresponding result
for $V = H^1(\R^d)$, $L_w^2 (\R^d)$, and $\mathbb{H}^1(\R^d)$ only requires the invertibility
of the frame operator on a particular subspace of $V$.
Precisely, given a lattice $\Lambda \subset \R^{2d}$, we define
\[
H^1_\Lambda(\R^d) := H^1(\R^d)\cap\mathcal B} \newcommand{\frakB}{\mathfrak B_\Lambda,
\quad
\mathbb{H}_\Lambda^1(\R^d) := \mathbb{H}^1(\R^d)\cap\mathcal B} \newcommand{\frakB}{\mathfrak B_\Lambda ,
\quad \text{and} \quad
L_{w,\Lambda}^2(\R^d) := L_w^2(\R^d) \cap \mathcal B} \newcommand{\frakB}{\mathfrak B_{\Lambda} .
\]
We equip the first two of these spaces with the norms
\[
\|f\|_{H^1_\Lambda} := \|\nabla f\|_{L^2} + \|C_{\Lambda,f}\|_{L^2 \to \ell^2}
\qquad\text{and}\qquad
\|f\|_{\mathbb{H}^1_\Lambda} := \|\nabla f\|_{L^2} + \|\nabla\widehat f\|_{L^2} + \|C_{\Lambda,f}\|_{L^2 \to \ell^2},
\]
respectively, where
\[
\|\nabla f\|_{L^2}
:= \sum_{j=1}^d
\|\partial_jf\|_{L^2}
\]
and $C_{\Lambda, f}$ is the analysis operator defined in \eqref{eq:CoefficientOperator}.
Finally, we equip the space $L_{w,\Lambda}^2(\R^d)$ with the norm
\[
\NormWeightedBessel{f} := \| f \|_{L_w^2} + \| C_{\Lambda,f} \|_{L^2, \ell^2} ,
\qquad \text{where} \qquad
\| f \|_{L_w^2} := \| w \cdot f \|_{L^2} .
\]
We start by showing that these spaces are Banach spaces.
\begin{lem}\label{lem:BesselH1IsBanach}
For a lattice $\Lambda\subset\R^{2d}$, the spaces $H^1_\Lambda(\R^d)$, $L_{w,\Lambda}^2(\R^d)$,
and $\mathbb{H}^1_\Lambda(\R^d)$ are Banach spaces which are continuously embedded in $L^2(\R^d)$.
\end{lem}
\begin{proof}
We naturally equip the space $\mathcal B} \newcommand{\frakB}{\mathfrak B_\Lambda\subset L^2(\R^d)$ with the norm
$\|f\|_{\mathcal B} \newcommand{\frakB}{\mathfrak B_\Lambda} := \|C_{\Lambda,f}\|_{L^2\to\ell^2}$.
Then $(\mathcal B} \newcommand{\frakB}{\mathfrak B_\Lambda,\|\cdot\|_{\mathcal B} \newcommand{\frakB}{\mathfrak B_\Lambda})$ is a Banach space by \cite[Proposition~3.1]{HanLarson}.
Moreover, for $f\in\mathcal B} \newcommand{\frakB}{\mathfrak B_\Lambda$,
\begin{equation}
\|f \|_{L^2}
= \big\| C_{\Lambda, f}^* \, \delta_{0,0} \big\|_{L^2}
\leq \|C_{\Lambda,f}^*\|_{\ell^2\to L^2}
= \|f\|_{\mathcal B} \newcommand{\frakB}{\mathfrak B_\Lambda},
\label{eq:BesselEmbedsIntoL2}
\end{equation}
which implies that $\mathcal B} \newcommand{\frakB}{\mathfrak B_\Lambda\hookrightarrow L^2(\R^d)$.
Hence, if $(f_n)_{n\in\N}$ is a Cauchy sequence in $H^1_\Lambda(\R^d)$,
then it is a Cauchy sequence in both $H^1(\R^d)$
(equipped with the norm $\| f \|_{H^1} := \| f \|_{L^2} + \| \nabla f \|_{L^2}$) and in $\mathcal B} \newcommand{\frakB}{\mathfrak B_\Lambda$.
Therefore, there exist $f\in H^1(\R^d)$ and $g\in\mathcal B} \newcommand{\frakB}{\mathfrak B_\Lambda$
such that $\|f_n-f\|_{H^1}\to 0$ and $\|f_n-g\|_{\mathcal B} \newcommand{\frakB}{\mathfrak B_\Lambda}\to 0$ as $n\to\infty$.
But as $H^1(\R^d)\hookrightarrow L^2(\R^d)$ and $\mathcal B} \newcommand{\frakB}{\mathfrak B_\Lambda\hookrightarrow L^2(\R^d)$,
we have $f_n\to f$ and $f_n\to g$ also in $L^2(\R^d)$, which implies $f=g$.
Hence, $\|f_n - f\|_{H^1_\Lambda}\to 0$ as $n\to\infty$, which proves that $H^1_\Lambda(\R^d)$ is complete.
The proof for $L_{w,\Lambda}^2(\R^d)$ and $\mathbb{H}^1_\Lambda(\R^d)$ is similar.
\end{proof}
\begin{prop}\label{p:bounded}
Let $\Lambda\subset\R^{2d}$ be a lattice.
If $g,h\in H^1_\Lambda(\R^d)$, then $S_{\Lambda,g,h}$ maps $H^1_\Lambda(\R^d)$ boundedly into itself
with operator norm not exceeding $\|g\|_{H^1_\Lambda}\|h\|_{H^1_\Lambda}$.
For $f\in H^1_\Lambda(\R^d)$ and $j \in \{1,\ldots,d\}$ we have
\begin{align}\label{e:DSf}
\partial_j(S_{\Lambda,g,h}f)
&= S_{\Lambda,g,h}(\partial_jf) + d(\Lambda)\cdot C_{\Lambda^\circ,f}^* \, d_{j,\Lambda^\circ,g,h},
\end{align}
where $d_{j,\Lambda^{\circ},g,h} \in \ell^2(\Lambda^{\circ})$ is defined by
\begin{align}\label{e:de}
(d_{j,\Lambda^\circ,g,h})_{\mu}
:= \big\<\partial_jh,\pi(\mu)g\big\>
+ \big\<h,\pi(\mu)(\partial_jg)\big\>,\quad \mu\in\Lambda^\circ.
\end{align}
\end{prop}
\begin{proof}
Let $f\in H^1_\Lambda(\R^d)$ and set $u := S_{\Lambda,g,h}f$.
First of all, we have $u \in \mathcal B} \newcommand{\frakB}{\mathfrak B_\Lambda$.
Indeed, a direct computation shows that $S_{\Lambda,g,h}$ commutes with $\pi (\lambda)$
for all $\lambda\in\Lambda$, and that $S_{\Lambda, g, h}^\ast = S_{\Lambda, h, g}$,
which shows for $v \in L^2(\R^d)$ that
\[
(C_{\Lambda,u} v)_{\lambda}
= \< v, \pi(\lambda) u\>
= \< v,\pi(\lambda) S_{\Lambda,g,h} f \>
= \< S_{\Lambda,h,g} v, \pi(\lambda) f \>
= (C_{\Lambda,f} \circ S_{\Lambda,h,g} \, v)_{\lambda},
\]
and therefore
\begin{equation}\label{e:alledrei}
\| C_{\Lambda,u} \|
\leq \|S_{\Lambda,h,g}\| \cdot \|C_{\Lambda,f}\|
\leq \| C_{\Lambda,g} \| \cdot \| C_{\Lambda,h} \| \cdot \| C_{\Lambda,f} \| < \infty ,
\end{equation}
since $S_{\Lambda,h,g} = C_{\Lambda,g}^*C_{\Lambda,h}$.
We now show that $u \in H^1(\R^d)$.
To this end, note for $v \in H^1(\R^d)$, $a, b \in \R^d$, and $j \in \{1,\ldots,d\}$ that
\[
\partial_j (M_b v)
= 2\pi i \cdot b_j \cdot M_{b}v + M_{b}(\partial_jv)
\qquad\text{and}\qquad
\partial_j(T_av) = T_{a}(\partial_jv)
\]
and therefore
\[
\partial_j(\pi(z)v)
= 2\pi i \cdot z_{d+j}\cdot\pi(z)v + \pi(z)(\partial_jv).
\]
Hence, setting $c_{\lambda,j} := 2\pi i \cdot \lambda_{d+j}\cdot\<f,\pi(\lambda)g \>$ for $\lambda = (a,b)\in \Lambda$,
we see that
\begin{align}
\begin{split}\label{e:cmn}
c_{\lambda,j}
&= \<\partial_jf,\pi(\lambda)g\> + \<f,\pi(\lambda)(\partial_jg)\>.
\end{split}
\end{align}
In particular, $(c_{\lambda,j})_{\lambda\in\Lambda}\in\ell^2(\Lambda)$ for each $j\in\{1,\ldots,d\}$,
because $f,g\in\mathcal B} \newcommand{\frakB}{\mathfrak B_\Lambda$ and $\partial_j f , \partial_j g \in L^2$.
In order to show that $\partial_ju$ exists and is in $L^2(\R^d)$,
let $\phi\in C_c^\infty(\R^d)$ be a test function.
Note that $C_c^\infty(\R^d)\subset \mathcal B} \newcommand{\frakB}{\mathfrak B_\Lambda$.
Therefore, we obtain
\begin{align*}
-\big\<u,\partial_j\phi\big\>
&= - \!\!
\sum_{\lambda\in\Lambda}
\<f,\pi(\lambda)g\big\>
\big\<\pi(\lambda)h,\partial_j\phi\big\>
= \sum_{\lambda\in\Lambda}
\<f,\pi(\lambda)g\big\>
\big\<
2\pi i\lambda_{d+j}\cdot\pi(\lambda)h
+ \pi(\lambda)(\partial_jh)
,
\phi
\big\>\\
&= \sum_{\lambda\in\Lambda}
c_{\lambda,j}
\cdot \big\<\pi(\lambda)h,\phi\big\>
+ \sum_{\lambda\in\Lambda}
\big\<f,\pi(\lambda)g\big\>
\big\<\pi(\lambda)(\partial_jh),\phi\big\> \\
& \!\!\overset{\eqref{e:cmn}}{=}
\<S_{\Lambda,g,h}(\partial_jf),\phi\>
+ \sum_{\lambda\in\Lambda}
\<f,\pi(\lambda)(\partial_jg)\>
\<\pi(\lambda)h,\phi\>
+ \sum_{\lambda\in\Lambda}
\big\<f,\pi(\lambda)g\big\>
\big\<\pi(\lambda)(\partial_jh), \phi\big\> \\
& \!\!\overset{\eqref{e:fi}}{=}
\<S_{\Lambda,g,h}(\partial_jf),\phi\>
+ d(\Lambda) \sum_{\mu\in\Lambda^\circ}
\Big[
\big\<h,\pi(\mu)(\partial_jg)\big\>
+ \big\<\partial_jh,\pi(\mu)g\big\>
\Big]
\big\<\pi(\mu)f,\phi\big\> \\[-0.1cm]
&= \left\<
S_{\Lambda,g,h} (\partial_jf)
+ d(\Lambda) \sum_{\mu\in\Lambda^\circ}
\Big[
\big\<h,\pi(\mu)(\partial_jg)\big\>
+ \big\<\partial_jh,\pi(\mu)g\big\>
\Big]
\pi(\mu)f
\,,\,\phi
\right\> \\
&= \left\<
S_{\Lambda,g,h}(\partial_jf)
+ d(\Lambda)\cdot C_{\Lambda^\circ,f}^* \, d_j
\,,\phi
\right\>,
\end{align*}
with $d_j = d_{j,\Lambda^\circ,g,h}$ as in \eqref{e:de}.
Note that $d_j\in \ell^2(\Lambda^\circ)$ because $g,h\in\mathcal B} \newcommand{\frakB}{\mathfrak B_{\Lambda} = \mathcal B} \newcommand{\frakB}{\mathfrak B_{\Lambda^\circ}$
and ${\partial_j h , \partial_j g \in L^2}$.
Since $j\in\{1,\ldots,d\}$ is chosen arbitrarily, this proves that $u\in H^1(\R^d)$ with
\[
\partial_ju
= S_{\Lambda,g,h}(\partial_jf)
+ d(\Lambda)\cdot C_{\Lambda^\circ,f}^* \, d_j
\,\in\, L^2(\R^d)
\]
for $j \in \{1,\ldots,d\}$, which is \eqref{e:DSf}.
Next, recalling Equation \eqref{e:Cnorms} we get
\[
\|d_j\|_{\ell^2}
\leq \|C_{\Lambda^\circ,h}\| \cdot \|\partial_jg\|_{L^2}
+ \|C_{\Lambda^\circ,g}\| \cdot \|\partial_jh\|_{L^2}
= \operatorname{Vol}(\Lambda)^{1/2}
\big(
\|C_{\Lambda,h}\| \cdot \|\partial_jg\|_{L^2}
+ \|C_{\Lambda,g}\| \cdot \|\partial_jh\|_{L^2}
\big),
\]
and $\|C_{\Lambda^\circ,f}^*\| = \operatorname{Vol}(\Lambda)^{1/2}\|C_{\Lambda,f}\|$.
Therefore,
\begin{align*}
\|\partial_ju\|_{L^2}
\leq \|S_{\Lambda,g,h}\| \cdot \|\partial_jf\|_{L^2}
+ \big(
\|C_{\Lambda,h}\| \cdot \|\partial_jg\|_{L^2}
+ \|C_{\Lambda,g}\| \cdot \|\partial_jh\|_{L^2}
\big)
\, \|C_{\Lambda,f}\|.
\end{align*}
Hence, with \eqref{e:alledrei}, we see
\vspace*{-0.3cm}
\begin{align*}
\|S_{\Lambda,g,h}f\|_{H^1_\Lambda}
&= \|\nabla u\|_{L^2}
+ \|C_{\Lambda,u}\|
\leq \sum_{j=1}^d \|\partial_ju\|_{L^2}
+ \|C_{\Lambda,g}\| \cdot \|C_{\Lambda,h}\| \cdot \|C_{\Lambda,f}\| \\
&\leq \|S_{\Lambda,g,h}\| \!\cdot\! \|\nabla f\|_{L^2}
+ \big(
\|C_{\Lambda,h}\| \!\cdot\! \|\nabla g\|_{L^2}
\!+\! \|C_{\Lambda,g}\| \!\cdot\! \|\nabla h\|_{L^2}
\!+\! \|C_{\Lambda,g}\| \!\cdot\! \|C_{\Lambda,h}\|
\big)
\|C_{\Lambda,f}\| \\
&\leq \|C_{\Lambda,g}\| \cdot \|C_{\Lambda,h}\| \cdot \|\nabla f\|_{L^2}
+ \big(
\|\nabla g\|_{L^2} + \|C_{\Lambda,g}\|
\big)
\big(
\|\nabla h\|_{L^2}
+ \|C_{\Lambda,h}\|
\big)
\, \|C_{\Lambda,f}\| \\
&\leq \|g\|_{H^1_\Lambda}
\|h\|_{H^1_\Lambda}
\cdot \|f\|_{H^1_\Lambda},
\end{align*}
and the proposition is proved.
\end{proof}
\section{Spectrum and dual windows}
\label{s:spectra}
Let $X$ be a Banach space.
As usual, we denote the set of bounded linear operators from $X$ into itself by $\mathcal B} \newcommand{\frakB}{\mathfrak B(X)$.
The {\em resolvent set} $\rho(T)$ of an operator $T\in\mathcal B} \newcommand{\frakB}{\mathfrak B(X)$ is the set of all $z\in\CC$
for which $T-z := T-z I : X \to X$ is bijective.
Note that $\rho(T)$ is always open in $\CC$.
The {\em spectrum} of $T$ is the complement $\sigma(T) := \CC\backslash\rho(T)$.
The {\em approximate point spectrum} $\sigma_{{ap}}(T)$ is a subset of $\sigma(T)$
and is defined as the set of points $z\in\CC$ for which there exists a sequence
$(f_n)_{n\in\N}\subset X$ such that $\|f_n\|=1$ for all $n\in\N$
and $\|(T-z)f_n\|\to 0$ as $n\to\infty$.
By \mbox{\cite[Proposition~VII.6.7]{ConwayFA}} we have
\begin{equation}\label{e:partial_sap}
\partial\sigma(T)\,\subset\,\sigma_{{ap}}(T).
\end{equation}
\begin{lem}\label{l:spectra_eq}
Let $(\mathcal H} \newcommand{\frakH}{\mathfrak H, \| \cdot \|)$ be a Hilbert space, let $S\in\mathcal B} \newcommand{\frakB}{\mathfrak B(\mathcal H} \newcommand{\frakH}{\mathfrak H)$ be self-adjoint,
and let $X\subset\mathcal H} \newcommand{\frakH}{\mathfrak H$ be a dense linear subspace satisfying $S(X) \subset X$.
If $\|\cdot\|_X$ is a norm on $X$ such that $(X,\| \cdot \|_X)$ is complete
and satisfies $X\hookrightarrow\mathcal H} \newcommand{\frakH}{\mathfrak H$, then $A := S|_X\in\mathcal B} \newcommand{\frakB}{\mathfrak B(X)$.
If, in addition, $\sigma_{{ap}}(A)\subset\sigma(S)$, then $\sigma(A) = \sigma(S)$.
\end{lem}
\begin{proof}
The fact that $A \in \mathcal B} \newcommand{\frakB}{\mathfrak B(X)$ easily follows from the closed graph theorem.
Next, since $X \hookrightarrow \mathcal H} \newcommand{\frakH}{\mathfrak H$, there exists $C > 0$
with $\| f \| \leq C \, \| f \|_X$ for all $f \in X$.
Assume now that additionally $\sigma_{{ap}}(A) \subset \sigma(S)$ holds.
Note that $\sigma(S) \subset \R$, since $S$ is self-adjoint.
Since $\sigma(A)\subset\CC$ is compact, the value ${r := \max_{w\in\sigma(A)}|\!\Im w|}$ exists.
Choose $z\in\sigma(A)$ such that $|\!\Im z| = r$.
Clearly, $z$ cannot belong to the interior of $\sigma(A)$, and hence $z \in \partial \sigma(A)$.
In view of Equation~\eqref{e:partial_sap}, this implies $z \in \sigma_{{ap}}(A) \subset \sigma(S) \subset \R$,
hence $r = 0$ and thus $\sigma(A) \subset \R$.
Therefore, $\sigma(A)$ has empty interior in $\CC$, meaning $\sigma (A) = \partial \sigma (A)$.
Thanks to Equation~\eqref{e:partial_sap}, this means $\sigma(A) \subset \sigma_{{ap}}(A)$,
and hence $\sigma(A) \subset \sigma(S)$, since by assumption $\sigma_{{ap}}(A) \subset \sigma(S)$.
For the converse inclusion it suffices to show that $\rho(A)\cap\R\subset\rho(S)$.
To see that this holds, let $z\in\rho(A)\cap\R$ and denote by $E$ the spectral measure
of the self-adjoint operator $S$.
Since $\R \cap \rho(A) \subset \R$ is open, there are
$a, b \in \R$ and $\delta_0 > 0$ such that $z \in (a,b)$
and $[a-\delta_0, b+\delta_0] \subset \rho(A)$.
By Stone's formula (see, e.g., \cite[Thm.\ VII.13]{rs}),
the spectral projection of $S$ with respect to $(a,b]$ can be expressed as
\[
E((a,b])f
= \lim_{\delta\downarrow 0}\,
\lim_{\varepsilon\downarrow 0}
\frac{1}{2\pi i}
\int_{a+\delta}^{b+\delta}
\big[
(S-t-i\varepsilon)^{-1}f
-(S-t+i\varepsilon)^{-1}f
\big]
\,dt,
\qquad f\in\mathcal H} \newcommand{\frakH}{\mathfrak H,
\]
where all limits are taken with respect to the norm of $\mathcal H} \newcommand{\frakH}{\mathfrak H$.
Note for $w \in \CC \setminus \R$ that $w \in \rho(S) \subset \rho(A)$.
Furthermore, $A - w = (S - w)|_X$, which easily implies $(S - w)^{-1}|_X = (A - w)^{-1}$.
Hence, for $f \in X$,
\begin{align*}
\|E((a,b])f\|
&\leq \lim_{\delta\downarrow 0}\,
\lim_{\varepsilon\downarrow 0}
\frac{1}{2\pi}
\int_{a+\delta}^{b+\delta}
\big\|(S- t-i\varepsilon)^{-1}f-(S- t+i\varepsilon)^{-1}f\big\|
\,d t\\
&\leq C \cdot \lim_{\delta\downarrow 0}\,
\lim_{\varepsilon\downarrow 0}
\frac{1}{2\pi}
\int_{a+\delta}^{b+\delta}
\big\|(A- t-i\varepsilon)^{-1}f-(A- t+i\varepsilon)^{-1}f\big\|_{X}
\,d t \\
&= C \cdot \lim_{\delta\downarrow 0}
\frac{1}{2\pi}
\int_{a+\delta}^{b+\delta}
\lim_{\varepsilon\downarrow 0}
\big\|(A- t-i\varepsilon)^{-1}f-(A- t+i\varepsilon)^{-1}f\big\|_{X}
\,d t\\
&=0,
\end{align*}
since the map $\rho(A)\to X$, $z\mapsto (A-z)^{-1}f$ is analytic
and thus uniformly continuous on compact sets.
This implies $E((a,b])f = 0$ for all $f\in X$ and therefore $E((a,b]) = 0$
as $X$ is dense in $\mathcal H} \newcommand{\frakH}{\mathfrak H$.
But this means that $(a,b) \subset \rho(S)$ (see \cite[Prop.\ on p.\ 236]{rs})
and thus $z \in \rho(S)$.
\end{proof}
For proving the invertibility of $S_{\Lambda,g}$ on $H_{\Lambda}^1, L_{w,\Lambda}^2$,
and $\mathbb{H}^1_{\Lambda}$, we first focus on the space $H^1_\Lambda(\R^d)$.
Note that if $g \in H^1_\Lambda(\R^d)$, then $S_{\Lambda,g}$ maps $H^1_\Lambda(\R^d)$ boundedly into itself
by Proposition~\ref{p:bounded}.
For $g \in H^1_\Lambda(\R^d)$, we will denote the restriction of $S_{\Lambda,g}$ to $H^1_\Lambda(\R^d)$
by $A_{\Lambda,g}$; that is, $A_{\Lambda,g} := S_{\Lambda,g} |_{H^1_\Lambda(\R^d)} \in\mathcal B} \newcommand{\frakB}{\mathfrak B(H^1_\Lambda(\R^d))$.
\begin{thm}\label{t:spectra}
Let $\Lambda\subset\Z^{2d}$ be a lattice and let $g\in H^1_\Lambda(\R^d)$.
Then
\[
\sigma(A_{\Lambda,g}) = \sigma(S_{\Lambda,g}).
\]
\end{thm}
\begin{proof}
For brevity, we set $A := A_{\Lambda,g}$ and $S := S_{\Lambda,g}$.
Due to Lemma~\ref{l:spectra_eq}, we only have to prove that $\sigma_{{ap}}(A)\subset\sigma(S)$.
For this, let $z\in\sigma_{{ap}}(A)$.
Then there exists a sequence $(f_n)_{n\in\N} \subset H^1_\Lambda(\R^d)$ such that
$\|f_n\|_{H^1_\Lambda} = 1$ for all $n \in \N$ and $\|(A - z)f_n\|_{H^1_\Lambda}\to 0$ as $n \to \infty$.
The latter means that, for each $j \in \{1, \ldots, d\}$,
\begin{equation}\label{e:two}
\big\| \partial_j(S f_n) - z \cdot (\partial_j f_n) \big\|_{L^2}\to 0
\qquad\text{and}\qquad
\big\| C_{\Lambda,(S - z)f_n} \big\|\to 0.
\end{equation}
Suppose towards a contradiction that $z \notin \sigma(S)$.
Since $S$ is self-adjoint, this implies $\overline{z} \notin \sigma(S)$.
Furthermore, because $S$ is self-adjoint and commutes with $\pi(\lambda)$ for all $\lambda \in \Lambda$,
we see for $f \in \mathcal B} \newcommand{\frakB}{\mathfrak B_\Lambda$ that $C_{\Lambda,(S - z)f} = C_{\Lambda,f} \circ (S - \overline z)$
and hence $C_{\Lambda,f_n} = C_{\Lambda,(S-z)f_n} \circ (S - \overline z)^{-1}$,
which implies that $\|C_{\Lambda,f_n}\|\to 0$.
Hence, also $\|C_{\Lambda^\circ,f_n}\|\to 0$ as $n\to\infty$ (see Equation~\eqref{e:Cnorms}).
Now, by Equation~\eqref{e:DSf}, we have
\[
\partial_j(Sf_n) - z \cdot (\partial_j f_n)
= (S- z)(\partial_jf_n) + C_{\Lambda^\circ,f_n}^*d_j
\]
with some $d_j\in\ell^2(\Lambda^\circ)$ which is \emph{independent of} $n$.
Hence, the first limit in \eqref{e:two} combined with $\|C_{\Lambda^\circ,f_n}\|\to 0$
implies that $\|(S - z)(\partial_jf_n)\|_{L^2}\to 0$ and thus $\|\partial_jf_n\|_{L^2}\to 0$
as $n\to\infty$ for all $j \in \{1,\ldots,d\}$, since $z \notin \sigma(S)$.
Hence, $\|f_n\|_{H^1_\Lambda} = \sum_{j=1}^d \|\partial_j f_n\|_{L^2} + \|C_{\Lambda,f_n}\|\to 0$
as $n\to\infty$, in contradiction to $\|f_n\|_{H^1_\Lambda} = 1$ for all $n \in \N$.
This proves that, indeed, $\sigma_{{ap}}(A)\subset\sigma(S)$.
\end{proof}
We now show analogous properties to Proposition~\ref{p:bounded} and Theorem~\ref{t:spectra}
for $L_{w,\Lambda}^2(\R^d)$.
\begin{cor}\label{cor:WeightedSpaceSpectrum}
Let $\Lambda\subset\Z^{2d}$ be a lattice.
If $g, h \in L_{w,\Lambda}^2(\R^d)$, then $S_{\Lambda,g,h}$ maps $L_{w,\Lambda}^2(\R^d)$
boundedly into itself.
If $g = h$ and if $A^w_{\Lambda,g} := S_{\Lambda,g} |_{L_{w,\Lambda}^2(\R^d)} \in \mathcal B} \newcommand{\frakB}{\mathfrak B(L_{w,\Lambda}^2(\R^d))$
denotes the restriction of $S_{\Lambda,g}$ to $L_{w,\Lambda}^2(\R^d)$, then
\[
\sigma(A^w_{\Lambda,g}) = \sigma(S_{\Lambda,g}).
\]
\end{cor}
\begin{proof}
We equip the space $\mathcal B} \newcommand{\frakB}{\mathfrak B_{\Lambda} \subset L^2(\R^d)$ with the norm
${\| f \|_{\mathcal B} \newcommand{\frakB}{\mathfrak B_\Lambda} := \| C_{\Lambda,f} \|_{L^2 \to \ell^2}}$,
where we recall from Equation~\eqref{eq:BesselEmbedsIntoL2} that
$\| f \|_{L^2} \leq \| f \|_{\mathcal B} \newcommand{\frakB}{\mathfrak B_\Lambda}$.
Equation~\eqref{e:FTpi} shows that the Fourier transform
is an isometric isomorphism from $\mathcal B} \newcommand{\frakB}{\mathfrak B_{\Lambda}$ to $\mathcal B} \newcommand{\frakB}{\mathfrak B_{\widehat{\Lambda}}$,
where $\widehat{\Lambda} := J \Lambda$.
Furthermore, it is well-known (see for instance \mbox{\cite[Section~9.3]{FollandRealAnalysis}})
that the Fourier transform ${\mathcal{F} : L^2 \to L^2}$
restricts to an isomorphism of Banach spaces ${\mathcal{F} : L_{w}^2(\R^d) \to H^1(\R^d)}$,
where $H^1$ is equipped with the norm $\| f \|_{H^1} := \| f \|_{L^2} + \| \nabla f \|_{L^2}$.
Taken together, we thus see that the Fourier transform restricts to an isomorphism
$\mathcal{F} : L_{w,\Lambda}^2(\R^d) \to H_{\widehat{\Lambda}}^1(\R^d)$; here, we implicitly used that
${\| f \|_{H_{\widehat{\Lambda}}^1} \asymp \| f \|_{H^1} + \| f \|_{\mathcal B} \newcommand{\frakB}{\mathfrak B_{\widehat{\Lambda}}}}$,
which follows from $\| \cdot \|_{L^2} \leq \| \cdot \|_{\mathcal B} \newcommand{\frakB}{\mathfrak B_{\widehat{\Lambda}}}$.
Plancherel's theorem, in combination with Equation~\eqref{e:FTpi}
shows for $f \in L^2(\R^d)$ that
\begin{align*}
\mathcal{F} \bigl[S_{\Lambda,g,h} f\bigr]
\!=\! \sum_{\lambda\in\Lambda}
\Big\<\widehat f,\widehat{\pi(\lambda)g}\Big\>
\widehat{\pi(\lambda)h}
=\! \sum_{\lambda\in\Lambda}
\<\widehat f,\pi(J\lambda)\widehat g \,\>
\pi(J\lambda)\widehat h
=\! \sum_{\lambda\in\widehat\Lambda}
\<\widehat f,\pi(\lambda)\widehat g \,\>
\pi(\lambda)\widehat h
= S_{\widehat\Lambda,\widehat g,\widehat h} \widehat f.
\end{align*}
Since
\(
A_{\widehat{\Lambda},\widehat{g},\widehat{h}}
= S_{\widehat{\Lambda},\widehat{g},\widehat{h}}|_{H_{\widehat{\Lambda}}^1} :
H_{\widehat{\Lambda}}^1(\R^d) \to H_{\widehat{\Lambda}}^1(\R^d)
\)
is well-defined and bounded by Proposition~\ref{p:bounded}, the preceding calculation
combined with the considerations from the previous paragraph shows that
$A_{\Lambda,g,h}^w = S_{\Lambda,g,h}|_{L_{w,\Lambda}^2(\R^d)} : L_{w,\Lambda}^2(\R^d) \to L_{w,\Lambda}^2(\R^d)$
is well-defined and bounded, with
\[
A_{\Lambda,g,h}^w
= \mathcal{F}^{-1} \circ A_{\widehat{\Lambda},\widehat{g},\widehat{h}} \circ \mathcal{F} .
\]
Finally, if $g = h$, we see
\(
\sigma(A_{\Lambda,g,g}^w)
= \sigma(A_{\widehat{\Lambda},\widehat{g},\widehat{g}})
= \sigma(S_{\widehat{\Lambda},\widehat{g},\widehat{g}})
= \sigma(S_{\Lambda,g,g}) ,
\)
where the second step is due to Theorem~\ref{t:spectra}, and the final step used the identity
$S_{\Lambda,g,h} = \mathcal{F}^{-1} \circ S_{\widehat{\Lambda},\widehat{g},\widehat{h}} \circ \mathcal{F}$
from above.
\end{proof}
Finally, we establish the corresponding properties for
$\mathbb{H}^1_\Lambda(\R^d) = H^1_\Lambda(\R^d) \cap L_{w,\Lambda}^2(\R^d)$.
\begin{cor}\label{c:bounded_HH}
Let $\Lambda\subset\Z^{2d}$ be a lattice.
If $g,h\in\mathbb{H}^1_\Lambda(\R^d)$, then $S_{\Lambda,g,h}$ maps $\mathbb{H}^1_\Lambda(\R^d)$ boundedly into itself.
If $g=h$ and $\mathbb A_{\Lambda,g} := S_{\Lambda,g} |_{\mathbb{H}^1_\Lambda(\R^d)} \in\mathcal B} \newcommand{\frakB}{\mathfrak B(\mathbb{H}^1_\Lambda(\R^d))$
denotes the restriction of $S_{\Lambda,g}$ to $\mathbb{H}^1_\Lambda(\R^d)$, then
\begin{equation}
\sigma(\mathbb A_{\Lambda,g}) = \sigma(S_{\Lambda,g}).
\label{eq:FatHSpectrum}
\end{equation}
\end{cor}
\begin{proof}
From the definition of $\mathbb{H}_\Lambda^1$ and the proof of Corollary~\ref{cor:WeightedSpaceSpectrum}
it is easy to see that $\mathbb{H}_\Lambda^1 = H_\Lambda^1 \cap L_{w,\Lambda}^2(\R^d)$,
and $\| \cdot \|_{\mathbb{H}_\Lambda^1} \asymp \| \cdot \|_{H_\Lambda^1} + \NormWeightedBessel{\cdot}$.
Therefore, Proposition~\ref{p:bounded} and Corollary~\ref{cor:WeightedSpaceSpectrum} imply
that $S_{\Lambda,g,h}$ maps $\mathbb{H}_\Lambda^1(\R^d)$ boundedly into itself.
Lemma~\ref{l:spectra_eq} shows that to prove \eqref{eq:FatHSpectrum},
it suffices to show $\sigma_{{ap}}(\mathbb{A}_{\Lambda,g}) \subset \sigma(S_{\Lambda,g})$.
Thus, let ${z \in \sigma_{{ap}}(\mathbb{A}_{\Lambda,g})}$.
Then there exists $(f_n)_{n \in \N} \subset \mathbb{H}^1_\Lambda(\R^d)$ with $\|f_n\|_{\mathbb{H}_{\Lambda}^1}=1$
for all $n \in \N$ and $\|(\mathbb A_{\Lambda,g}-z)f_n\|_{\mathbb{H}^1_\Lambda}\to 0$ as $n\to\infty$.
Thus, $\|(A_{\Lambda,g}-z)f_n\|_{H^1_\Lambda}\to 0$
and ${\NormWeightedBessel{(A_{\Lambda,g}^w - z)f_n} \to 0}$ as $n\to\infty$.
Furthermore, there is a subsequence $(n_k)_{k \in \N}$ such that
$\lim_{k\to\infty} \| f_{n_k} \|_{H_\Lambda^1} > 0$
or $\lim_{k\to\infty} \NormWeightedBessel{f_{n_k}} > 0$.
Hence, $z \in \sigma(A_{\Lambda,g})$ or $z \in \sigma(A_{\Lambda,g}^w)$.
But Theorem~\ref{t:spectra} and Corollary~\ref{cor:WeightedSpaceSpectrum} show
$\sigma(A_{\Lambda,g}) = \sigma(A_{\Lambda,g}^w) = \sigma(S_{\Lambda,g})$.
We have thus shown $\sigma_{{ap}}(\mathbb{A}_{\Lambda,g}) \subset \sigma(S_{\Lambda,g})$,
so that Lemma~\ref{l:spectra_eq} shows $\sigma(\mathbb{A}_{\Lambda,g}) = \sigma(S_{\Lambda,g})$.
\end{proof}
The next proposition shows that any operator obtained from
$S_{\Lambda,g}$ through the holomorphic spectral calculus
(see \cite[Sections~10.21--10.29]{RudinFunctionalAnalysis} for a definition)
maps each of the spaces $H_{\Lambda}^1(\R^d)$, $L_{w,\Lambda}^2(\R^d)$,
and $\mathbb{H}_\Lambda^1(\R^d)$ into itself.
\begin{prop}\label{t:inverse_closed}
Let $\Lambda\subset\R^{2d}$ be a lattice,
let $V \in \{ H_\Lambda^1(\R^d), L_{w,\Lambda}^2(\R^d), \mathbb{H}_\Lambda^1(\R^d) \}$, and $g \in V$.
Then for any open set $\Omega \subset \CC$ with $\sigma(S_{\Lambda,g}) \subset \Omega$,
any analytic function $F : \Omega \to \CC$, and any $f \in V$, we have $F(S_{\Lambda,g})f \in V$.
\end{prop}
\begin{proof}
We only prove the claim for $V = H_{\Lambda}^1 (\R^d)$;
the proofs for the other cases are similar, using Corollaries~\ref{cor:WeightedSpaceSpectrum}
or \ref{c:bounded_HH} instead of Theorem~\ref{t:spectra}.
Thus, let $g\in H^1_\Lambda(\R^d)$ and set $S := S_{\Lambda,g}$ and $A := A_{\Lambda,g}$.
Let $f\in H^1_\Lambda(\R^d)$ and define
\[
h
= - \frac{1}{2\pi i}
\int_\Gamma
F(z) \cdot (A-z)^{-1}f
\,dz
\,\in\, H^1_\Lambda(\R^d),
\]
where $\Gamma \subset \Omega \setminus \sigma(S)$ is a finite set of closed rectifiable curves
surrounding $\sigma(S) = \sigma(A)$
(existence of such curves is shown in \cite[Theorem~13.5]{RudinRealAndComplexAnalysis}).
Note that the integral converges in $H^1_\Lambda(\R^d)$.
Since $H^1_\Lambda(\R^d)\hookrightarrow L^2(\R^d)$, it also converges
(to the same limit) in $L^2(\R^d)$ and hence, by definition of the holomorphic spectral calculus,
\[
F(S)f
= -\frac{1}{2\pi i}
\int_\Gamma
F(z) \cdot (S-z)^{-1}f
\,dz
= h\,\in\,H^1_\Lambda(\R^d).
\qedhere
\]
\end{proof}
Our main result (Theorem~\ref{t:main}) is now an easy consequence of Proposition~\ref{t:inverse_closed}.
\begin{proof}[Proof of Theorem~\ref{t:main}]
Using the fact that $S_{\Lambda,g}$ commutes with $\pi(\lambda)$ for all $\lambda \in \Lambda$,
it is easily seen that $(S_{\Lambda,g}^{-1} \, g,\Lambda)$ is the canonical dual frame of $(g,\Lambda)$
and that $(S_{\Lambda,g}^{-1/2} g, \Lambda)$ is a Parseval frame for $L^2(\R^d)$;
see for instance, \cite[Theorem 12.3.2]{ChristensenBook}.
Note that since $(g,\Lambda)$ is a frame for $L^2(\R^d)$,
we have $\sigma(S_{\Lambda,g}) \subset [A,B]$
where $0 < A \leq B < \infty$ are the optimal frame bounds for $(g,\Lambda)$.
Thus, we obtain $S_{\Lambda,g}^{-1} \, g \in V_\Lambda \subset V$
and $S_{\Lambda,g}^{-1/2} g \,\in\, V_\Lambda \subset V$
from Proposition~\ref{t:inverse_closed} with $F(z) = z^{-1}$ and $F(z) = z^{-1/2}$
(with any suitable branch cut; for instance, the half-axis $(-\infty,0]$),
respectively, on $\Omega = \bigl\{ x + i y : x \in ( \frac{A}{2}, \infty) , y \in \R\bigr\}$.
\end{proof}
Finally, we state and prove a version of \Cref{t:main} for Gabor frame \emph{sequences}.
For completeness, we briefly recall the necessary concepts.
Generally, a (countable) family $(h_i)_{i \in I}$ in a Hilbert space $\mathcal H} \newcommand{\frakH}{\mathfrak H$
is called a \emph{frame sequence}, if $(h_i)_{i \in I}$ is a frame for the subspace
$\mathcal H} \newcommand{\frakH}{\mathfrak H' := \overline{\operatorname{span}} \{ h_i \colon i \in I \} \subset \mathcal H} \newcommand{\frakH}{\mathfrak H$.
In this case, the frame operator
$S : \mathcal H} \newcommand{\frakH}{\mathfrak H \to \mathcal H} \newcommand{\frakH}{\mathfrak H, f \mapsto \sum_{i \in I} \langle f, h_i \rangle h_i$,
is a bounded, self-adjoint operator on $\mathcal H} \newcommand{\frakH}{\mathfrak H$, and $S|_{\mathcal H} \newcommand{\frakH}{\mathfrak H'} : \mathcal H} \newcommand{\frakH}{\mathfrak H' \to \mathcal H} \newcommand{\frakH}{\mathfrak H'$
is boundedly invertible; in particular, $\operatorname{ran} S = \mathcal H} \newcommand{\frakH}{\mathfrak H' \subset \mathcal H} \newcommand{\frakH}{\mathfrak H$ is closed,
so that $S$ has a well-defined \emph{pseudo-inverse} $S^{\dagger}$, given by
\[
S^{\dagger}
= (S|_{\mathcal H} \newcommand{\frakH}{\mathfrak H'})^{-1} \circ P_{\mathcal H} \newcommand{\frakH}{\mathfrak H'} : \quad
\mathcal H} \newcommand{\frakH}{\mathfrak H \to \mathcal H} \newcommand{\frakH}{\mathfrak H' ,
\]
where $P_{\mathcal H} \newcommand{\frakH}{\mathfrak H'}$ denotes the orthogonal projection onto $\mathcal H} \newcommand{\frakH}{\mathfrak H'$.
The \emph{canonical dual system} of $(h_i)_{i \in I}$ is then given by
$(h_i')_{i \in I} = (S^{\dagger} h_i)_{i \in I} \subset \mathcal H} \newcommand{\frakH}{\mathfrak H'$, and it satisfies
$\sum_{i \in I} \langle f, h_i \rangle h_i ' = \sum_{i \in I} \langle f, h_i' \rangle h_i = P_{\mathcal H} \newcommand{\frakH}{\mathfrak H'} f$
for all $f \in \mathcal H} \newcommand{\frakH}{\mathfrak H$.
Finally, in the case where $(h_i)_{i \in I} = (g,\Lambda)$ is a Gabor family with a lattice $\Lambda$,
it is easy to see that $S \circ \pi(\lambda) = \pi(\lambda) \circ S$ and
$\pi(\lambda) \mathcal H} \newcommand{\frakH}{\mathfrak H' \subset \mathcal H} \newcommand{\frakH}{\mathfrak H'$ for $\lambda \in \Lambda$, which implies
$P_{\mathcal H} \newcommand{\frakH}{\mathfrak H'} \circ \pi(\lambda) = \pi(\lambda) \circ P_{\mathcal H} \newcommand{\frakH}{\mathfrak H'}$,
and therefore $S^\dagger \circ \pi(\lambda) = \pi(\lambda) \circ S^\dagger$
for all $\lambda \in \Lambda$.
Consequently, setting $\gamma := S^\dagger g$, we have $S^\dagger (\pi(\lambda) g) = \pi(\lambda)\gamma$,
so that the canonical dual system of a Gabor frame sequence $(g,\Lambda)$ is
the Gabor system $(\gamma,\Lambda)$, where $\gamma = S^{\dagger} g$ is called the
\emph{canonical dual window} of $(g,\Lambda)$.
Our next result shows that $\gamma$ inherits the regularity of $g$,
if one measures this regularity using one of the three spaces $H^1, L_w^2$, or $\mathbb{H}^1$.
\begin{prop}\label{p:MainResultForGaborFrameSequences}
Let $V \in \{ H^1(\R^d), L_w^2(\R^d), \mathbb{H}^1(\R^d) \}$.
Let $\Lambda \subset \R^{2d}$ be a lattice and let $g \in V$.
If $(g,\Lambda)$ is a frame sequence, then the associated canonical dual window $\gamma$
satisfies $\gamma \in V$.
\end{prop}
\begin{proof}
The frame operator $S : L^2(\R^d) \to L^2(\R^d)$ associated to $(g,\Lambda)$
is non-negative and has closed range.
Consequently, there exist $\varepsilon > 0$ and $R > 0$ such that $\sigma(S) \subset \{ 0 \} \cup [\varepsilon,R]$;
see for instance \cite[Lemma~A.2]{QuantitativeSubspaceBL}.
Now, with the open ball $B_\delta(0) := \{ z \in \CC \colon |z| < \delta \}$, define
\[
\Omega
:= B_{\varepsilon/4} (0)
\cup \big\{
x + i y
\colon
x \in (\tfrac{\varepsilon}{2},2 R), y \in (-\tfrac{\varepsilon}{4}, \tfrac{\varepsilon}{4})
\big\}
\subset \CC ,
\]
noting that $\Omega \subset \CC$ is open, with $\sigma(S) \subset \Omega$.
Furthermore, it is straightforward to see that
\[
\varphi : \quad
\Omega \to \CC, \quad
z \mapsto \begin{cases}
0, & \text{if } z \in B_{\varepsilon/4} (0), \\
z^{-1}, & \text{otherwise}
\end{cases}
\]
is holomorphic.
Since the functional calculus for self-adjoint operators
is an extension of the holomorphic functional calculus,
\cite[Lemma~A.6]{QuantitativeSubspaceBL} shows that $S^{\dagger} = \varphi(S)$.
Finally, since $g \in V_\Lambda$, \Cref{t:inverse_closed} now shows that
$\gamma = S^{\dagger} g = \varphi(S) g \in V_\Lambda \subset V$ as well.
\end{proof}
|
2,877,628,088,962 | arxiv | \section{Introduction}
It's nice to work with well-sampled astronomical images.
A well-sampled image can be readily resampled to various scales,
orientations, or more complex geometries without loss of information.
Its spatial resolution is well-understood, permitting
a clear analysis of the relative contributions of information and noise.
Further, many image processing algorithms will only work on well-sampled data.
In some cases, however, it's not practical or even desirable to
obtain well-sampled images.
Given detectors with a finite number of pixels and significant
readout noise, one may prefer to trade-off resolution for
increased field size or photometric sensitivity.
Both considerations were central to the design of the {\it HST}
WFPC-1 and WFPC-2 cameras, to give examples of
instruments that produce undersampled astronomical images.
WFPC-2 in particular has generated the largest library of high-resolution
optical astronomical images to date, but ironically the severe undersampling
in the WFC system, and the still less than critical sampling of
the PC at all but the reddest wavelengths, limit the resolution
of {\it HST} observations as much as the telescope optics, themselves.
There is no magic that can undo the undersampling in a single image;
analysis of such data always requires respect for their peculiarities.
At the same time, it may be possible
to obtain additional observations with the same camera system
that contain information lost in the original images.
For example, if the camera can be offset by a fraction
of a pixel over a sequence of exposures or ``dithered,''
one can observe how the structure of objects in the image varies
with respect to their positions on the pixel-grid, and thus
recover details not contained in any single image.
This suggests that one might construct
a well-sampled super-image from a set of undersampled, but dithered images.
In general, when the size of a pixel is important with respect
to the intrinsic point-spread function (PSF), the image as observed is
\begin{equation}
I(x,y)=O(x,y)\ast P(x,y) \ast \Pi(x,y),
\end{equation}
where $O$ is the intrinsic projected appearance of the astronomical
field being imaged, $P$ is the PSF due to the telescope and camera
optics, and $\Pi$ is the spatial form of the pixel itself,
(which is often assumed to be a uniform square, although this need
not be the case), and $\ast$ means convolution.
Both $P$ and $\Pi$ limit the resolution of $I$ and thus implicitly specify
the minimum sampling requirements --- a dilemma if $\Pi$ is too
big, since it sets what the sampling really is, regardless of what's needed.
If the astronomical scene and camera are time-stable, however,
dithering the camera allows proper sampling of the field
{\it convolved with the pixel response} as well as the PSF, to be obtained.
If the camera is pointed on a fine and regular $n\times n$ grid
of sub-pixel steps, where $n$ is the number of substeps within
the original large pixel, then the images can be simply interleaved into
a super-image that has small pixels equal to the dither step-size.
If the step-size is small enough, the super-image will be critically sampled.
A simple way to view this is to consider an image consisting of the
astronomical field just convolved with the PSF due to the optics alone.
The sampling would be done on pixels equal to the size of the dither step,
chosen to be fine enough to ensure critical sampling.
The image is then blurred by the original pixel response.
Drawing every $n^{th}$ pixel in $x$ and $y$ clearly recreates
one of the dithered images as actually created by the camera.
Therefore, conversely interleaving the dithered images creates the well-sampled
super-image.
In practice, however, it may not be possible to step the camera in a
regular pattern. Sub-pixel dithers have
been used in many WFPC-2 programs, for example, but were often
not executed with enough precision to fall on a regular pattern;
simple interlacing of the image-set cannot be done in such cases.
This problem is critical for the {\it Hubble
Deep Field} (HDF) observations \markcite{hdf}(Williams et al.\ 1996).
A regular dither was specified, but did not actually occur.
To solve the problem of combining images with
an irregular dither pattern, a {\it Drizzle}-algorithm was developed
(\markcite{hdf}Williams et al. 1996; \markcite{driz}Fruchter \& Hook 1998)
that works by simply dropping or ``drizzling'' the pixels in any
single image onto a finer grid, offsetting the image by the actual
sub-pixel step obtained, slicing up its pixels as they fall on the finer grid.
The {\it Drizzle} algorithm worked well, producing the now famous well-sampled
full-color image of the HDF.
The {\it Drizzle} algorithm is appealing, as it is intuitive ---
one is just shifting and overlapping the images on a fine grid,
shrinking the original pixels small enough so as to minimize any blurring
associated with forcing the pixels into the new grid, but keeping them
big enough so that there are no ``holes'' of empty data in the new super image.
Further, because {\it Drizzle} works in the spatial domain, it's easy
to correct for cosmic ray events, hot pixels, or any other data missing
in any single images, as well as correcting for any geometric distortion.
Development of {\it Drizzle} represents a significant improvement in
the software available to astronomers for analyzing undersampled images,
and has greatly improved the recovery of information from {\it HST} images.
Despite the success of {\it Drizzle,} however, it
is frankly justified on intuitive rather than formal theoretical grounds,
and indeed depends on two {\it ad hoc} parameters, namely the spacing of the
super-image grid and the size of the pixels to be drizzled.
It also introduces its own blurring function, $\Pi',$ which {\it statistically}
is about the size of the super-image pixel; in detail, the actual resolution
for any object depends on how it falls with respect to the final grid.
Although $\Pi'$ in practice may be much smaller than $\Pi,$ it still may be
large compared to the PSF and introduce significant blurring in its own right.
These issues were indeed discussed in the context of the HDF, and
limit its deconvolution or interpretation of its power spectrum on the
finest scales.
In attempt to develop an algorithm that both mines better
resolution from the data, and stands on a solid theoretical foundation,
I present a method that reconstructs a super-image from an
arbitrary set of dithered observations with no-degradation of resolution.
This method is only a modest extension to two-dimensional
data of a method for recovering one-dimensional functions from
undersampled data presented by \markcite{fft}Bracewell (1978).
The method works by computing the Fourier transform of
the super-image as a linear combination of the transforms of the
individual images; the aliased components are eliminated algebraically.
I have also extended the method to estimate the
super-image when it is actually overdetermined
by the dithered observations.
None of this is particularly complex, and not surprizingly,
the professional image processing literature already
contains discussions of this method (see \markcite{tsai}Tsai
\& Huang 1984, or \markcite{kim}Kim et al. 1990).
However, given the strong interest in using dithers in the context
of {\it HST} imaging, I considered it worthwhile to present this paper
as a tutorial on the method of Fourier algebraic reconstruction
and explore its use in the context of {\it HST} observations.
\section{The Theory of Reconstructing an Image From Aliased Data-sets}
\subsection{The Sampling of a 1-D Function}
To understand how to reconstruct an image from undersampled data, I start
by considering the effects of sampling on a 1-D function, $f(x).$
For reconstruction to work, $f(x)$ must be band-limited, so that its
Fourier transform,
\begin{equation}
\overline{f(x)}=F(u)=\int_{-\infty}^\infty f(x)e^{-2\pi ixu}dx,
\end{equation}
is non-zero only for $-u_c<u<u_c,$ where $u_c$ is the critical frequency.
If $x$ is expressed in terms of pixels, then sampling at every integer
pixel is sufficient provided that $u_c<1/2.$
This can be understood by
considering the Fourier transform of the sampled function,
The sampling of $f(x)$ is equivalent to multiplying it by a
{\it shah}-function,
\begin{equation}
\rm III(ax)\equiv{1\over{\vert a\vert}}\sum_{n=-\infty}^{+\infty}
\delta\left(x-{n\over a}\right),
\end{equation}
where $a=1$ for the specific case of integer-pixel sampling.
The Fourier transform of the sampled function is then,
\begin{eqnarray}
\label{1Dsum}
\overline{f(x)\cdot\rm III(x)}&=&F(u)\ast\rm III(u),\nonumber \\
&=&{\displaystyle \sum_{n=-\infty}^{+\infty}F(u-n),}
\end{eqnarray}
where I have used the fact that
the transform of a shah-function is itself a shah-function.
As is well-known, the Fourier transform of a sampled function
is periodic, repeating over the entire frequency domain.
If $f(x)$ is band-limited, however, none of the copies or
{satellites} of $F(u)$ overlap.
The satellites are spaced at each integer-step in $u,$
but the requirement that $u_c<1/2,$ means that they also reach
zero before crossing over the midpoint of the interval (Figure \ref{fig:alias}).
\begin{figure}[thbp]
\plotone{fig1.ps}
\caption{This figure schematically shows the effects of sampling
on the Fourier power spectrum of a continuous 1-D function.
Sampling causes the power spectrum to be periodic, with the period
inversely proportional to the spatial sampling frequency.
When the function is well-sampled, the satellites occur at intervals
of $2u_c,$ or greater, where $u_c$ is the critical frequency, or
the highest frequency at which the intrinsic function has non-zero
power (upper graph). With coarser sampling, the function becomes
undersampled and the satellites begin to overlap. With $2\times$
undersampling (bottom graph) the satellites occur at every integer
multiple of $u_c.$ The total transform (dotted) is the sum over
all satellites and is severely aliased.}
\label{fig:alias}
\end{figure}
This condition is no longer obeyed when the sampling interval is
larger than each integer pixel step.
For example, if every other pixel is sampled, then,
\begin{eqnarray}
\overline{f(x)\cdot\rm III\left({x\over2}\right)}&
=&F(u)\ast\rm III(2u),\nonumber \\
&=&{\displaystyle 2\sum_{n=-\infty}^{+\infty}F\left(u-{n\over2}\right).}
\end{eqnarray}
The transformed shah-function now samples at every
half-integer step in the Fourier domain, causing strong
overlaps or {\it aliasing} between the satellites of $F(u)$
(Figure \ref{fig:alias}).
If $f(x)$ is unknown, the full extent of its transform cannot be
deduced from the aliased sample, which in turn means that the
sample is itself an incomplete representation of $f(x).$
\subsection{Recovery of a 1-D Function}
\markcite{fft}Bracewell (1978) shows that
a function can be recovered from collection of undersampled data-sets
given prior knowledge of $u_c$ (as might exist given a
detector pixel shape and optical point-spread function), provided
that the sampling among the various data-sets is interlaced by
some fraction of the sampling interval and that the basic sampling
interval is not too sparse compared to $u_c.$
Consider again the alternate pixel sample above (which I relabel
as $d_0(x)$). For the fundamental interval $-1/2<u<1/2,$
\begin{eqnarray}
D_0(u)&=&{\displaystyle
\overline{f(x)\cdot\rm III\left({x\over2}\right)},}\nonumber \\
&=&{\displaystyle {1\over 2}\left(F(u-{1\over2})+
F(u)+F(u+{1\over2})\right).}
\end{eqnarray}
Since I have specified that $F(u)$ is band-limited to $\vert u\vert<1/2,$
for $0\leq u<1/2,$
\begin{equation}
D_0(u)={1\over 2}\left(F(u-{1\over2})+F(u)\right),
\end{equation}
and for $-1/2<u<0,$
\begin{equation}
D_0(u)={1\over 2}\left(F(u)+F(u+{1\over2})\right).
\end{equation}
Now let there be a second data set that also samples $f(x)$ with
alternate pixel spacing, but spatially offset from the $d_0(x)$ samples
by some $x_0\neq2n$ (one might presume
$0<x_0<2,$ but this is not required).
The transform of the new data-set, $d_{x_0}(x),$ is
\begin{eqnarray}
D_{x_0}(u)&=&{\displaystyle
\overline{f(x)\cdot\rm III\left({x\over2}-x_0\right)},}\nonumber \\
&=&{\displaystyle
F(u)\ast\overline{\left(\rm III
\left({x\over2}\right)\ast\delta(x-x_0)\right)},}\nonumber \\
&=&{\displaystyle
F(u)\ast\left(\rm III\left(2u\right)\cdot e^{-2\pi i ux_0}\right).}
\end{eqnarray}
This reduces to
\begin{eqnarray}
D_{x_0}(u)&=&{1\over2}\left(F(u)+e^{-\pi i x_0}
F(u-{1\over2})\right),\qquad 0\leq u<1/2, \\
&=&{1\over2}\left(F(u)+e^{+\pi i x_0}F(u+{1\over2})\right),
\qquad -1/2<u<0.
\end{eqnarray}
Note that $d_{x_0}(x)$ is no less aliased than is $d_0(x),$
but since the overlap
portion has a differing phase, the transforms of the two samples
can be combined to solve for the transform of $f(x),$
\begin{eqnarray}
F(u)&=&{\displaystyle 2~{D_{x_0}(u)
-e^{-\pi i x_0}D_0(u)\over
1-e^{-\pi i x_0}},}\qquad 0\leq u<1/2, \\
&=&{\displaystyle 2~{D_{x_0}(u)-e^{+\pi i x_0}D_0(u)\over
1-e^{+\pi i x_0}},}\qquad -1/2< u<0.
\end{eqnarray}
In other words, one can reconstruct $f(x)$ exactly from two data-sets
offset from each other, each undersampled by a factor two.
Note that in the special case, where $x_0=1,$ $d_0(x)$
holds the even-numbered pixels and $d_{x_0}(x)$ holds the odd-numbered ones,
then
\begin{equation}
F(u)=D_{x_0}(u)+D_0(u),
\end{equation}
as would be expected, since the sum in equation (\ref{1Dsum}) can clearly
be separated this way. With exact interlacing, one can just add the
transforms of the two individual data-sets (provided that the transform
preserves their relative phases).
\subsection{Recovery of an Image}
This method can be directly generalized to the case of reconstructing
a 2-D image. The shah-function becomes a 2-D regular grid of
$\delta$-functions, and the two-dimensional Fourier transform
of an image is:
\begin{equation}
\overline{f(x,y)}=F(u,v)=\int_{-\infty}^\infty
\int_{-\infty}^\infty f(x,y)e^{-(2\pi ixu + 2\pi iyv)}dx~dy.
\end{equation}
If there is an observation $d_{{x_1},{y_1}}(x,y)$ that is factor of
two undersampled in both $x$ and $y$ (thus having 1/4 of the pixels
of the critically sampled image), and offset by $x_1,\ y_1$ from the
nominal grid defining $f(x,y),$
then in the domain $0\leq u<1/2,\ 0\leq v<1/2,$
\begin{eqnarray}
D_{{x_1},{y_1}}(u,v)&=&{1\over4}\bigg(F(u,v)+e^{-\pi i x_1}
F(u-{1\over2},v) \nonumber \\
&&+e^{-\pi i y_1}F(u,v-{1\over2})+e^{-\pi i (x_1 + y_1)}
F(u-{1\over2},v-{1\over2})\bigg).
\end{eqnarray}
There are analogous expressions in the other three
quadrants of the $u,v$ plane;
however, for real-valued images, half of the $u,v$ plane will simply
be the complex conjugate of the other half and thus need not be computed
(see Figure \ref{fig:2x2}).
As can be seen, with four data-sets, each having a unique offset in
$x$ {\it or} $y,$ it is again possible to eliminate the overlap
contributions. This requires solving a system of equations with
complex coefficients:
\begin{equation}
{1\over4}\left(\matrix{1&e^{-\pi i x_1}&e^{-\pi i y_1}&e^{-\pi i(x_1+y_1)}\cr
1&e^{-\pi i x_2}&e^{-\pi i y_2}&e^{-\pi i(x_2+y_2)}\cr
1&e^{-\pi i x_3}&e^{-\pi i y_3}&e^{-\pi i(x_3+y_3)}\cr
1&e^{-\pi i x_4}&e^{-\pi i y_4}&e^{-\pi i(x_4+y_4)}\cr}\right){\bf F}={\bf D},
\end{equation}
where ${\bf F}$ is a 4-vector holding $F(u,v)$ in the first position,
followed by the $u,$ $v,$ and lastly $u,v$ satellites, ${\bf D}$
is a 4-vector of the transforms of the 4 undersampled data-sets.
One can then invert this matrix to find
\begin{equation}
\label{comb}
F(u,v)=\sum_{n=1}^4c_nD_{{x_n},{y_n}}(u,v),
\end{equation}
where $c_n$ will be a complex coefficient.
\begin{figure}[thbp]
\plotfiddle{fig2.ps}{2.5in}{0}{100}{100}{-180}{-144}
\caption{This figure schematically shows the configuration of the Fourier
domain for reconstructing an image with $2\times2$ subsampling.
For real images, Fourier transforms need only be calculated
for the semi-plane with $0\leq u\leq1/2,$ $-1/2<v\leq1/2$
(this presumes that the $x$-axis transform is computed first),
where the frequencies are defined with respect to the pixels of the
reconstructed image. Each image in the observed set is aliased, and
has satellites at all integer multiples of $(u,v)=1/2$ in the Fourier
domain, with each satellite having significant power over $\Delta u=\pm1/2,$
and $\Delta v=\pm1/2$ about its central location. The figure
shows as heavy dots the central location of all satellites that overlap
with the fundamental transform centered at $(u,v)=0.$
Algebraic elimination of the satellites is done in two regions, marked
1 and 2; the satellites that contribute to a given region are those
at its corners.}
\label{fig:2x2}
\end{figure}
Solution for the second quadrant is analogous --- the phases
differ only in sign, being positive when the domain of the frequency
is negative.
As an example, for the special case of where the four data-sets
contain the exact interlaces of integer pixels in $x$ and $y,$
{\bf F} and {\bf D} are more simply related as:
\begin{equation}
{1\over4}\left(\matrix{\phantom{-}1&\phantom{-}1&\phantom{-}1&\phantom{-}1\cr
\phantom{-}1&-1&\phantom{-}1&-1\cr
\phantom{-}1&\phantom{-}1&-1&-1\cr
\phantom{-}1&-1&-1&\phantom{-}1\cr}\right) {\bf F}={\bf D},
\end{equation}
which has the solution, as expected of
\begin{equation}
F(u,v)=\sum_{n=1}^4D_{{x_n},{y_n}}(u,v).
\end{equation}
\subsection{Recovery of an Image Overdetermined by the Data}
Four images determine $F(u,v),$ exactly, but
if one actually has additional images available,
$F(u,v)$ is overdetermined, and a least squares solution is required.
This means finding the $F(u,v)$ that minimizes the norm
\begin{equation}
E=\Vert{\bf\Phi}{\bf F}-{\bf D}\,\Vert,
\label{norm}
\end{equation}
where, as above ${\bf\Phi}$ is the matrix of phases.
In this case, however, ${\bf\Phi}$ is now an $n\times4$ matrix,
\begin{equation}
{\bf\Phi}={1\over4}\left(
\matrix{1&e^{-\pi i x_1}&e^{-\pi i y_1}&e^{-\pi i(x_1+y_1)}\cr
1&e^{-\pi i x_2}&e^{-\pi i y_2}&e^{-\pi i(x_2+y_2)}\cr
\vdots&\vdots&\vdots&\vdots\cr
1&e^{-\pi i x_n}&e^{-\pi i y_n}&e^{-\pi i(x_n+y_n)}\cr}\right),
\end{equation}
where $n\geq4$ is the number of data-sets, and ${\bf D}$ is now a vector of
length $n$ holding the data-sets; ${\bf F}$ is still the same 4-vector.
Expanding equation (\ref{norm}) gives
\begin{eqnarray}
E^2&=&\left({\bf\Phi}{\bf F}-{\bf D}\,\right)^H
\left({\bf\Phi}{\bf F}-{\bf D}\,\right)\nonumber \\
&=&{\bf F}^H{\bf\Phi}^H{\bf\Phi}{\bf F}-{\bf F}^H
{\bf\Phi}^H{\bf D}-{\bf D}^H{\bf\Phi}{\bf F}+{\bf D}^H{\bf D},
\end{eqnarray}
where $H$ denotes the Hermitian (or complex-conjugate) transpose.
Minimizing $E$ implies
\begin{equation}
\label{F2D}
{\bf F}=\left({\bf\Phi}^H{\bf\Phi}\right)^{-1}{\bf\Phi}^H{\bf D}.
\end{equation}\par
In the case of an overdetermined situation, one might further
want to weight the observations differently.
For example, it may not be practical to obtain exposures of identical
length over the sequence of observations, or they may have variable
backgrounds. In this case, it's easy to generalize equation (\ref{F2D})
to include weighting, giving
\begin{equation}
{\bf F}=\left({\bf\Phi}^H{\bf W}^T{\bf W}{\bf\Phi}\right)^{-1}
{\bf\Phi}^H{\bf W}^T{\bf W}{\bf D},
\label{F2W}
\end{equation}
where ${\bf W}$ is an $n\times n$ matrix of weights and ${\bf W}^T$
is its transpose (the weights are real-valued).
${\bf W}$ can account for any covarience between the images,
but it is most likely to be diagonal on the presumption that the
individual images will probably be independent.
\subsection{Generalization to Higher Degress of Subsampling}
Double sampling is likely to be sufficient to remove modest aliasing,
but higher levels of subsampling may be required
when the undersampling is severe.
Generalization to finer levels of subsampling is straight forward,
if somewhat tedious. As the observed images become coarser with respect
to the reconstructed image, the aliased satellites become closer
together and overlap more severely. Algebraic elimination
of the satellites requires identifying
all satellites contributing power to a given location in the
Fourier domain. In practice, this means slicing the Fourier
domain into an increasingly large number of subsets.
Figure \ref{fig:3x3} sketches out the structure of the Fourier
domain for $3\times3$ subsampling.
In the $3\times3$ case, the Fourier domain is divided into six regions,
with nine differing satellites contributing to ${\bf F}$ in each one;
at least $n\geq9$ dithered
images will be required to find a solution, and
${\bf\Phi}$ is will now be an $n\times9$ matrix.
An important distinction between the $2\times2$ and $3\times3$ cases,
is that in the former, since the satellites are spaced exactly by
$u_c,$ only the six satellites that are visible within the
Fourier semi-domain need be considered. In the $3\times3$ case,
the satellites are separated only by multiples of $2u_c/3,$ thus
the first set of satellites with their {\it centers} actually falling
outside the semi-domain will still overlap with it.
\begin{figure}[htbp]
\plotfiddle{fig3.ps}{3.0in}{0}{100}{100}{-162}{-144}
\caption{As for the $2\times2$ case,
this figure schematically shows the configuration of the Fourier
domain for reconstructing an image with $3\times3$ subsampling.
Again, the Fourier transforms are calculated
only for the semi-plane with $0\leq u\leq1/2,$ $-1/2<v\leq1/2.$
Satellites now occur at all integer multiples of $(u,v)=1/3,$
but each satellite still has significant power over $\Delta u=\pm1/2,$
and $\Delta v=\pm1/2$ about its central location. The figure
shows as heavy dots the central location of all satellites that overlap
with the fundamental transform centered at $(u,v)=0.$
Algebraic elimination of the satellites is now done in six regions;
the satellites that contribute to a given region are the one at its center,
and the eight surrounding it.}
\label{fig:3x3}
\end{figure}
\section{Implementation of the Fourier Image Reconstruction}
\subsection{Data-set Requirements\label{sec:req}}
The present reconstruction method works
only if the data satisfies a number of conditions,
the most important of which is that the intrinsic image
structure remain constant over the extent of the dithered data-taking sequence.
The PSF should not vary significantly in time, or if the
dither steps are large, in space as well.
``Significantly'' in this context means variations on spatial scales
where the Fourier $S/N$ ratio is greater than unity;
bright point sources are more vulnerable to PSF-variations than
faint or more diffuse sources.
Bright noise spikes, hot pixels,
cosmic ray hits, or any other variable sources, must also be eliminated or
repaired prior to reconstruction.
A final obvious requirement is that reconstruction can work only
on the portions of the dither set in common to all images;
as the dither takes place, it is likely that a larger region of
the sky will be imaged than is present on any single image ---
subimages of the common overlap region must be isolated prior to reconstruction.
The mathematics of the Fourier reconstruction method do not strictly require
that the angular size of the pixels be constant over the extent
of any image, provided that the dither steps are small enough that
they can be regarded as constant over the complete area of the images.
Images that have variations in their pixel scale large enough so that
the amplitude of the dithers (in pixels) varies significantly over
the extent of the image must be processed in subsets small enough
that the dithers can be regarded as constant over the angular domain selected.
Lastly, the dithers must be translational only, with no rotation.
The reader familiar with {\it Drizzle} may object
that these requirements are too restrictive for many sets of dithered data.
{\it Drizzle} performs cosmic ray event and defect rejection,
as well as geometric rectification, when building a sub-sampled image.
{\it Drizzle} is thus attractive for the complete reduction of
panoramic data sets. This issue will be discussed further in $\S\ref{sec:sum},$
but I emphasize that the present approach is solely concerned with
the specific task of accurate reconstruction of a Nyquist-sampled image.
Geometric rectification or defect rejection are problems
that can be separated from the actual reconstruction algorithm;
the caveats presented above do not necessarily prevent use of the
present method if they can be addressed apart from the reconstruction task.
Two other requirements on the data set concern the pattern and measurement
of the dithers. Ideally, the fractional portion of the dither steps (that
is ignoring the integer number of pixels stepped over) should match
the nominal $2\times2$ or $3\times3$ equal sub-stepping patterns as
closely as possible; or if the problem is heavily overdetermined be
at least evenly spread over the area of a single pixel.
In this case, solution of equation (\ref{F2W}) will generate a set of complex
coefficients, $c_n$ of nearly equal power (presuming
equal weights).
Formally solutions can be calculated for any nondegenerate dither pattern;
however, as the dither pattern moves away from optimal, the
images will be combined unevenly, with heavy weight being placed on
those with less redundant positions.
For real images, this means that the relative noise contributed
by such images will be amplified compared to others in the dither set.
Noise properties of the reconstructed image will be discussed below;
in practice, excess amplification of noise is only important for large
departures from an ideal pattern.
Accurate measurement of the dither steps is required to construct
the $\Phi$ matrix. This may be done iteratively.
Initially one might use simple centroids of stars or other compact
objects within a given image to measure dither offsets.
Once a reconstructed image has been generated,
it can be cross-correlated with the individual images to refine the offsets;
permitting a more accurate reconstruction to be done in a second iteration.
\subsection{Computing the Reconstructed Image}
Given the prepared set of dithered images and measured dither steps,
computation of the reconstructed image can proceed.
In practice I have done this within the Vista image processing system,
making use of its native image arithmetic and Fourier routines,
augmenting it only with a new subroutine to construct $\Phi,$ and
then solve for and apply $c_n$ to the Fourier transform of a given image.
For each image, the first steps are to normalize it
to a common exposure level, and to then expand it into a sparse array,
spacing out the pixels by $2\times$ or $3\times$ as desired.
Each pixel in an input image then occupies one of the corners of
a cell of $2\times2$ or $3\times3$ new pixels in the expanded image,
with the other $n\times n-1$ pixels in each cell set to zero.
This actualizes each image as a sparse \rm III\ function;
one can see that for exact $n\times n$ dithers,
the other images would simply be interlaced at the vacant locations.
Once an image is expanded, its Fourier transform is computed;
a power spectrum at this stage clearly shows the aliased satellites.
The next step is to multiply the transform by $c_n$, remembering that
different coefficients must be used for the various regions
within the domain. The adjusted transform is then added to
the adjusted transforms of the other images.
The reconstructed image is the inverse transform of the complete sum.
One important caveat is that each the transform of each image must be
multiplied by a complex phase, $\exp\left(-2\pi Ki\left(x_j+y_j\right)\right),$
where $(x_j,y_j)$ is its spatial offset from the average of the other images,
and $K$ is the degree of subsampling. This is required
because the mathematics presented in the previous section presume
a two-dimensional coordinate system anchored to the sky, rather
than the grid of the detector.
In other words, as each image is expanded, initially its \rm III\ function
has identical coordinates to those in the other images, with the object
apparently moving with respect to the detector coordinate system.
This step resets the coordinate system to that of the sky,
correctly phasing the various \rm III\ functions of the dither set.
\subsection{Examples of Reconstructed Images}
Figures \ref{fig:pc_psf} and \ref{fig:wfc_psf} show PC and WFC PSFs
\begin{figure}[htbp]
\plotone{fig4.ps}
\caption{Reconstruction of the {\it HST} PC PSF with $2\times2$
subsampling is shown based on 20 dithered F555W images of a
star in $\omega$ Cen. The image at left shows a linear stretch of one
of the PSF images (selected to be nearly centered on a pixel).
The central image shows the reconstructed PSF with the same intensity stretch.
The last image is a logarithmic stretch (with dynamic range 3.5 in log units)
of the reconstructed PSF.}
\label{fig:pc_psf}
\end{figure}
\begin{figure}[hbtp]
\plotone{fig5.ps}
\caption{Reconstruction of the {\it HST} WFC PSF with $3\times3$
subsampling is shown based on 20 dithered F555W images of a
star in $\omega$ Cen. The image at left shows a linear stretch of one
of the PSF images (selected to be nearly centered on a pixel).
The central image shows the reconstructed PSF with the same intensity stretch.
The last image is a logarithmic stretch (with dynamic range 3.5 in log units)
of the reconstructed PSF.}
\label{fig:wfc_psf}
\end{figure}
reconstructed from a calibration program of 20 F555W dithered images of
a field within the $\omega$ Cen globular cluster.
The PC PSF was reconstructed with $2\times2$ subsampling, while
$3\times3$ subsampling was used for the WFC PSF.
The cores of the PSFs are now well resolved, and no ``boxy'' artifacts
are seen as can occur in {\it Drizzle} reconstructions
(\markcite{driz}Fruchter \& Hook 1998).
It's also worthwhile to note the strong blurring introduced by the
WFC pixel function, $\Pi,$ itself.
Again, the reconstruction does not recover the intrinsic PSF due to
the optics only, but the intrinsic PSF convolved with $\Pi.$
The PC PSF clearly has the sharper and rounder core, while the
center of the WFC PSF is strongly determined by the pixel shape.
Figure \ref{fig:power} shows the power spectra at various stages in the
reconstruction of the WFC PSF to illustrate
\begin{figure}[thbp]
\plotone{fig6.ps}
\caption{Power spectra are shown at various stages in the reconstruction
of the WFC PSF with $3\times3$ subsampling. The left image shows the
power spectrum of a single PSF image expanded as a sparse $\rm III\ $ function.
The low contrast of the minima between the bright peaks of the satellites
shows the effects of the severe aliasing in WFC images. The middle
image shows the spectrum of the penultimate reconstruction. At this
stage 19 of the 20 images have been combined and the flanking satellites
have been greatly reduced in power. The right image shows the power spectrum
of the final reconstructed PSF --- the partial combination shown in the
middles has now been completed by the addition of the last image. The
display scale is identical and logarithmic (with a range of $10^5$) for
all three spectra. The power spectra are shown for the full Fourier
domain for ease of visual interpretation, even though the transforms
are computed only in a semi-plane.}
\label{fig:power}
\end{figure}
\begin{figure}[thbp]
\plotone{fig7.ps}
\caption{The reconstruction of the center of NGC 1023 with $2\times2$
subsampling. Five F555W PC images were used. Four of the images
(shown at left) define an approximate $2\times2$ interlace pattern;
however, the offsets typically differed from the nominal 0.5 pixel
steps by $\sim0.1$ pixel (the fifth image falls within 0.1 pixel of one
of the four images shown). The stretch is linear.}
\label{fig:n1023}
\end{figure}
the algorithm concretely. The final combination of 20 images has
reduced the contribution of the aliased satellites by $\sim10^5.$
The final power spectrum also ratifies the strong contribution
of the WFC pixel to the total PSF. The shape of the spectrum is
clearly boxy; further, the central lobe is surrounded by a strong
zero, which would be expected in the power spectrum of a nearly
square and uniform pixel function.
Turning to more interesting objects,
Figure \ref{fig:n1023} shows the $2\times2$ reconstruction of the
nucleus of the early-type galaxy NGC 1023.
Unlike the situation for the PSFs, which were highly overdetermined,
only five dithered images were available for NGC 1023.
The dither pattern was close to a nominal exact interlace,
but the offsets typically differed from the nominal 0.5 pixel
by $\sim0.1$ pixel, thus the present method was required.
This galaxy has a particularly compact center (\markcite{l95}Lauer et al. 1995).
The present observations were obtained to observe its central structure
with the best resolution available --- reconstructing the image without
introducing additional blurring is thus critical.
The reconstructed image clearly shows the sharp compact nucleus of NGC 1023,
but is also smooth and free from artifact; indeed this image can now be
processed further with PSF deconvolution.
Lastly, I show a $2\times2$ reconstruction of a chain-galaxy at
$z=1.355$ (\markcite{cohen}Cohen et al. 1996) in the {\it Hubble Deep
Field} (Figure \ref{fig:hdf_gal}), along with a {\it Drizzle}
reconstruction.
\footnote{The {\it Drizzle} reconstruction shown was done with the
same image set, weights, and pixel grid used for the Fourier reconstruction,
and differs from the {\it Drizzle}-reconstructed image of the
same galaxy in the official release of the {\it HDF.}}
Superficially the two images look identical; the gross morphology
is not strongly dependent on the reconstruction algorithm.
\begin{figure}[thbp]
\plotone{fig8.ps}
\caption{Two reconstructions of a $z=1.355$ chain galaxy in
the {\it Hubble Deep Field} with $2\times2$
subsampling, based on 11 F450W WFC images. The left image
was done with the present Fourier method, while the image on the
right is a {\it Drizzle} reconstruction. The stretch is linear.}
\label{fig:hdf_gal}
\end{figure}
Detailed comparison shows, however, that the present reconstruction
is slightly sharper --- the peak of the brightest knot in the image
is $\sim7\%$ brighter, for example.
Matching the resolution of the {\it drizzled} image requires
smoothing the Fourier reconstruction with a Gaussian with $FWHM\approx1$
pixel (on the subsampled scale). The Fourier reconstruction
does appear to have more noise, but again this is due to the
smoothing inherent in the {\it Drizzle} algorithm.
The Fourier reconstruction can be smoothed, but
one of the nice things about having a well-sampled image is that
optimal filters can be used to improve its appearance.
A Weiner filter, for example, can be used to reject much of the
noise in the present image with little effect on its resolution;
an option that is not possible with aliased images.
A more general comparison of the present method to {\it Drizzle} is
complex, as the difference between the two depends on the dither
pattern, the size of the image set, choice of the
reconstructed pixel size, and the {\it Drizzle} drop size.
For example, when the dither pattern is close to an exact interlace,
{\it Drizzle} can be configured to produce a simple interlaced
reconstruction, while at the opposite end of the scale, {\it Drizzle}
can do simple ``shift-and-add'' reconstructions on the
original pixel scale, which implies highly significant smoothing.
In general, it appears from a number of
additional experiments that when a large image set is available,
{\it Drizzle} effectively smooths a perfect
reconstruction with a gaussian with width of about one pixel, as
in the HDF galaxy above.
For WFC PSFs, for example, the blurring can cause a 10\%\ reduction
in the flux of the central pixel.
This is not guaranteed, however; in one WFC PSF experiment with only four
nearly exactly interlaced images, {\it Drizzle} produced a result
that was apparently {\it sharper} than the Fourier reconstruction.
Close examination, however, showed that the {\it Drizzle} result
was still aliased; aliasing can cause features to be artificially
sharpened as well as broadened. Further comparison of the
Fourier method to {\it Drizzle} is thus best done in a context
specific to the scientific problem at hand.
\subsection{Noise in the Reconstructed Image}
As alluded to in $\S\ref{sec:req},$ the noise level in
the reconstructed image depends on how well the dither pattern
matches an ideal interlace pattern. For $N$ images, the solutions
presented in equations (\ref{F2D}) or (\ref{F2W}) reduce to a
set of complex coefficients $\{c_n\}$ relating $F(u,v)$ to the
data, as in the exact solution shown in equation (\ref{comb}).
On the presumption that the noise from image to image is uncorrelated,
then the average power in noise in the reconstructed image is simply
\begin{equation}
\label{noise_lev}
\eta_F=\left(\sum_{n=1}^Nc_n^* c_n\eta_n^2\right)^{1/2},
\end{equation}
where $\eta_n$ is the noise power in image $n.$
With a nearly ideal dither pattern (and equally-weighted
data), $(c_n^*c_n)^{1/2}\approx K^2/N,$
where $K$ is the degree of subsampling; the noise level is as
expected for the simple addition of $N$ images.
As the dither pattern becomes less ideal, however, unequal weight
is placed on the images, depending on the uniqueness of their positions.
Highly redundant images will have small coefficients, while
more isolated images contribute relatively higher power.
The linear combination of the images still produces an exact
solution for the reconstructed image, but because the noise is incoherent from
image to image it may be amplified in the final image, relative
to its level in the ideal case.
Equation (\ref{noise_lev}) allows the noise in the reconstructed
image to be calculated in advance for any particular dither pattern.
Figure \ref{fig:noise} gives shows how the noise level in the reconstructed
\begin{figure}[hbtp]
\plotone{fig9.ps}
\caption{Simulations of the relative amplification of noise as a function
of the departure of the dither pattern from a perfect interlace are shown.
The departure is parameterized as a normal distribution of random
offsets with the standard deviation specified in original pixels.
The solid curve and points show the case for when only four images
are used in the reconstruction. The dashed-line and open symbols
show the simulations done with nine input images.}
\label{fig:noise}
\end{figure}
image varies as the dither pattern moves away from the ideal interlace
for two examples of $2\times2$ subsampling. In these tests,
variations in the dither pattern were treated as random gaussian errors
about the exact interlace. For a given standard deviation of the
random offset, several simulated image reconstructions were computed.
For the example with only four images, there is no redundant information,
and the noise level depends strongly on the particulars of the dither
pattern once excursions from the exact interlace become large.
For nine images the reconstruction is more stable to
departures from the ideal pattern, the final noise level showing less
large excursions. The real importance of this demonstration, however,
is to show that the noise level rises only slowly above its ideal for
small errors in the dither pattern. Experience with WFPC-2,
for example, shows that typical dithering errors
($\lesssim0.1$ PC pixel)
will give results within the regime of modest noise amplification.
\section{Discussion and Summary\label{sec:sum}}
As noted in the introduction, my interest in the Fourier reconstruction method
presented here stemmed from a strong desire to avoid the random blurring,
$\Pi',$ that {\it Drizzle} may introduce into the reconstructed image.
The present method permits exact reconstruction of the superimage, with no
blurring at the Nyquist scale, nor requires any arbitrary decisions or
parameters to control the form of the reconstructed image.
One might object that the degree
of subsampling selected is such a parameter; however, it is really
specified by the intrinsic spatial scale of the Nyquist frequency.
A Nyquist-sampled image can be resampled at finer scales without loss
of information content or introduction of artifact --- images generated
at various subsampling scales past the Nyquist scale are essentially
equivalent representations of the image.
The present algorithm places several preconditions on the data,
thus it is worthwhile to consider 1) the optimal data-taking strategy for
reconstructing images from dithered data-sets, and 2) how to best perform
the related tasks of artifact rejection, geometric rectification, and so on.
The mathematics of the Fourier method strongly recommends selecting
a dither pattern that contains fractional offsets as close to the
ideal interlace pattern, itself. If a good dither pattern is realized,
little is demanded of the linear combination of the images --- one
is simply accounting for the slight errors in its execution. It
should be emphasized that the dither pattern can also contain
integer pixel offsets as well, as might be desired to eliminate
hot pixels, traps, blocked columns, and other fixed detector defects
as well as cosmic rays. A nearly ideal program for the present
algorithm would be to attempt a $2\times2$ subsampling interlace,
but taking multiple exposures at each dither step to allow for
cosmic ray rejection. This strategy clearly demands a rather large
data-set, which may not be feasible for programs lasting only
an orbit or two on {\it HST.} However, it presents no difficulties
for multi-orbit programs, where one will be obtaining a large number
of exposures in any case.
With regards to the second issue above, I have focused solely on
the problem of reconstructing a Nyquist-sampled image. Tasks that are
required before this stage include image registration and defect repair.
Tasks that might follow reconstruction include geometric rectification,
deconvolution, and filtering.
{\it Drizzle} is attractive in part because it is a complete package that
does many of these steps together within the familiar IRAF/STSDAS environment.
This said, however, I emphasize that many of the preliminary reduction
steps can be done independently of the Fourier reconstruction
algorithm --- these issues should not impede its use.
Indeed, one might use {\it Drizzle} for an initial reconstruction
to provide for defect rejection prior
to a second reconstruction cycle using the present algorithm.
Geometric rectification is simple in principle if one is working with
well-sampled images; the issue is generating such an image if geometric
distortions are important in the undersampled observations.
As noted earlier, if the dithers are small, scale changes across the
image may be unimportant; if variations in the local dither step over
the image domain are limited to a few percent of a pixel, then the
entire domain may be reconstructed, and then later rectified. If the
dither steps are large, however, the fractional pixel offsets may
vary significantly over the image, requiring the reconstruction to
be done in subsets of the domain and later patched together.
This may be unattractive for some problems requiring panoramic imaging,
but may be irrelevant if the primary objects of interest are compact
or occupy only small portions of the images.
While the Fourier reconstruction method presented here works only for
translational dithers, in passing, I note that the professional image processing
literature does contain algorithms related to be present one that can combine
undersampled images with more complex geometric interrelationships.
\markcite{gran}Granrath \& Lersch (1998) present an algorithm that
constructs a Nyquist-sampled image from an image set whose members
can be related to each other with affine transformations, i. e.,
the geometric transformations that include rotation, scale change, and shear,
as well as simple translations.
The Granrath \& Lersch algorithm constructs a ``projection-onto-convex-sets''
estimate that gives the best reproduction of the image set, in
contrast to the present method, which yields a closed-form solution
to the Nyquist image.
Methods of this sort may be of interest in cases where the image
does not meet the conditions required for the present Fourier method,
but precise treatment of the Nyquist-scale is still important.
In summary, the Fourier technique presented here may not be the first
choice to construct a Nyquist image when the geometrical relationships
among the image set are complex, or the dither pattern is strongly
non-optimal. Further, its resolution gains may appear to be superficially
modest. Regardless, there remains a class of {\it HST} imaging problems
that push right against the diffraction scale of the instrument.
This class includes crowded field stellar photometry, the nuclear
structure of galaxies --- particularly those with bright AGN, the
morphology of lensed QSOs, and so on.
This method allows clean access to the Nyquist scale
and should be of use for these problems and more.
\acknowledgments
I wish to thank Bobby Hunt, Christoph Keller, and Ken Mighell
for useful conversations.
\clearpage
|
2,877,628,088,963 | arxiv |
\subsection{Core}
The only method of class \Abs{Core} is the run method,
see figure \ref{fig:ABScore} for its annotated implemenation.
\subsection{Cache}
\input{AnnotatedABS/flush_abs}
\section{Introduction}
\label{sec:introduction}
\input introduction
\section{ABS: Actors with Cooperative Concurrency}
\label{sec:abs}
\input abs-intro
\section{The General Methodology}\label{sec:methodology}
\input methodology
\section{A TSS for Multicore Memory Systems}
\label{sec:cachememory}
\input multicore
\section{The ABS Model of the Multicore Memory System}
\label{sec:model}
\input abs-multi
\section{The Simulation Relation}
\label{sec: correctness}
\input correctness
\section{Related Work}
\label{sec:related}
\input related
\section{Conclusion}
\label{sec:conclusion}
\input conclusion
\bibliographystyle{plain}
\subsection*{Transition Rules for Cores}
Figure~
\ref{fig:local.sos.1} shows the transition rules for
the basic core instructions $\reads{r}$, $\readBs{r}$, $\writes{r}$, and $\writeBs{r}$.
\begin{figure}[h]
\centering
{
\footnotesize
\begin{displaymath}
\begin{array}[t]{c}
\ncondrule{PrRd$_1$}{
\quad \first{\levid} = true \quad \coreid{\levid} = c
\quad
\statusfunc{\memory}{n} \in \{\sh,\mo\}
}{
(c \bullet \hlc{\reads{n};\task}) \hsep (\levid \bullet \memory \bullet \datainst) \hist{: h}
\to
(c \bullet \hlc{\task}) \hsep (\levid \bullet \memory \bullet \datainst) \hist{: h; R(c, n)} }
\\[30pt]
\ncondrule{PrRd$_2$}{
\quad \first{\levid} = true \quad \coreid{\levid} = c
\quad
\statusfunc{\memory}{n} \in \{\inv,\bot\}
}{
(c \bullet \hlc{\reads{n};\task}) \hsep (\levid \bullet \hlcc{\memory \bullet \datainst})\hist{: h}
\to \\
(c \bullet \hlc{\readBs{n};\task}) \hsep (\levid \bullet \hlcc{\memory \setto{n}{\bot} \bullet \datainst+\fetchs{n}}) \hist{: h}
}
\\[30pt]
\ncondrule{PrRd$_3$}{
\first{\levid} = true \quad \coreid{\levid} = c
\quad
n \in \dom{\memory}
}{
(c \bullet \hlc{\readBs{n};\task}) \hsep (\levid \bullet \memory \bullet \datainst)\hist{: h}
\to
(c \bullet \hlc{\reads{n};\task}) \hsep (\levid \bullet \memory \bullet \datainst) \hist{: h; R(c, n)}
}
\\[30pt]
\ncondrule{PrWr$_1$}{
\first{\levid} = true \quad \coreid{\levid} = c
\quad
\statusfunc{\memory}{n} = \mo
}{
(c \bullet \hlc{\writes{n};\task}) \hsep (\levid \bullet \memory \bullet \datainst) \hist{: h}
\to
(c \bullet \hlc{\task}) \hsep (\levid \bullet \memory \bullet \datainst) \hist{: h; W(c, n)}
}
\\[30pt]
\ncondrule{PrWr$_2$}{
\first{\levid} = true \quad \coreid{\levid} = c
\quad
\statusfunc{\memory}{n} = \pr{k}{\sh}
}{
(c \bullet \hlc{\writes{n};\task}) \hsep (\levid \bullet \hlcc{\memory} \bullet \datainst)\hist{: h}
\toL{\red{\sendRX{n}}}
(c \bullet \hlc{\task}) \hsep (\levid \bullet \hlcc{\memory \setto{n}{\pr{k}{\mo}}} \bullet \datainst) \hist{: h; W(c, n)}
}
\\[30pt]
\ncondrule{PrWr$_3$}{
\first{\levid} = true \quad \coreid{\levid} = c
\quad
\statusfunc{\memory}{n} \in \{\inv,\bot\} }{
(c \bullet \hlc{\writes{n};\task}) \hsep (\levid \bullet \hlcc{ \memory \bullet \datainst})\hist{: h}
\to \\
(c \bullet \hlc{\writeBs{n};\task}) \hsep (\levid \bullet \hlcc{\memory \setto{n}{\bot} \bullet \datainst+\fetchs{n}}) \hist{: h}
}
\\[30pt]
\ncondrule{PrWr$_4$}{
\first{\levid} = true \quad \coreid{\levid} = c
\quad
n \in \dom{\memory}
}{
(c \bullet \hlc{\writeBs{n};\task})\hsep (\levid \bullet \memory \bullet \datainst) \hist{: h}
\to
(c \bullet \hlc{\writes{n};\task}) \hsep (\levid \bullet \memory \bullet \datainst) \hist{: h}
}
\end{array}
\end{displaymath}
\vspace{-10pt}
\caption{ \label{fig:local.sos.1} Transition rules for $\reads{r}$, $\readBs{r}$, $\writes{r}$, and $\writeBs{r}$.}
}
\end{figure}
\subsection*{Transition Rules for Caches}
These rules are further structured in terms of separate TSS's for the individual \emph{rst} instructions (Figures~ \ref{fig:local.sos.2}, \ref{fig:fetchBl}, \ref{fig:fetchW}, and \ref{fig:flush}).
\begin{figure}[h]
\centering
{
\footnotesize
\begin{displaymath}
\begin{array}[t]{c}
\ncondrule{LC-Hit$_1$}{
\levno{\levid'} = \levno{\levid} + 1 \quad \coreid{\levid} = \coreid{\levid'} \\
\selectfunc{\memory}{n} = m \quad n\neq m\quad
\memory(m) = \pr{k_i}{s} \quad \memory'(n) = \pr{k_j}{s'} \quad s' = \sh \lor s' = \mo
}
{
(\levid \bullet \hlc{\memory} \bullet \datainst+\fetchs{n})\hsep(\levid' \bullet \hlcc{\memory'} \bullet \datainst')
\to \\
(\levid \bullet \hlc{\memory[m\mapsto \bot,n\mapsto
\pr{k_j}{s'}]} \bullet \datainst) \hsep
(\levid' \bullet \hlcc{\memory' [n\mapsto \bot,m\mapsto \pr{k_i}{s}]} \bullet \datainst')
}
\\[40pt]
\ncondrule{LC-Hit$_2$}{
\levno{\levid'} = \levno{\levid} + 1 \quad \coreid{\levid} = \coreid{\levid'} \\ \selectfunc{\memory}{n} = n \quad
\memory'(n) = \pr{k_j}{s'} \quad s' = \sh \lor s' = \mo
}
{
(\levid \bullet \hlc{\memory} \bullet \datainst+\fetchs{n})\hsep(\levid' \bullet \hlcc{\memory'} \bullet \datainst')
\to \\
(\levid _i \bullet \hlc{\memory[n\mapsto \pr{k_j}{s'}]} \bullet \datainst)\hsep
(\levid' \bullet \hlcc{\memory' [n\mapsto \bot]} \bullet \datainst')
}
\\[40pt]
\ncondrule{LC-Miss}{
\levno{\levid'} = \levno{\levid} + 1 \quad \coreid{\levid} = \coreid{\levid'} \quad
\statusfunc{\memory'}{n} \in \{\inv,\bot\}
}
{
(\levid \bullet \memory \bullet \hlc{\datainst+\fetchs{n}})\hsep(\levid' \bullet \hlcc{\memory' \bullet \datainst'})
\to \\
(\levid \bullet \memory \bullet \hlc{\datainst+\blocks{n}})\hsep(\levid'\bullet \hlcc{\memory' \setto{n}{\bot} \bullet \datainst'+\fetchs{n}})
}
\\[40pt]
\ncondrule{LLC-Miss}{
\last{\levid} = true
}
{
(\levid \bullet \memory \bullet \hlc{\datainst+\fetchs{n}}) \toL{\red{\sendR{n}}}
(\levid \bullet \memory \bullet \hlc{\datainst+\blocks{n}})
}
\\[40pt]
\end{array}
\end{displaymath}
\vspace{-10pt}
\caption{Transition rules for $\fetchs{n}$.}
\label{fig:local.sos.2}
}
\end{figure}
\begin{figure}[h]
\centering
{
\footnotesize
\begin{displaymath}
\begin{array}[t]{c}
\ncondrule{FetchBl$_1$}{
\last{\levid} = true \quad \selectfunc{\memory}{n} = n
\quad
s = \statusfunc{\overline{\memory}}{n}
}{
(\levid \bullet \hlc{\memory\bullet \datainst+\blocks{n}}) \hsep \overline{\memory} \to
( \levid \bullet \hlc{\memory'\setto{n}{ \pr{k}{s}} \bullet \datainst
}) \hsep \overline{\memory}
}
\\[30pt]
\ncondrule{FetchBl$_2$}{
\last{\levid} = true \quad \selectfunc{\memory}{n} = n'
\quad n' \not = n \quad
\statusfunc{\memory}{n'} \not = \mo
\quad
s = \statusfunc{\overline{\memory}}{n}
}{
(\levid \bullet \hlc{\memory'\bullet \datainst+\blocks{n}}) \hsep \overline{\memory}
\to
(\levid \bullet \hlc{\memory'[n'\mapsto \bot, n \mapsto \pr{k}{s}] \bullet \datainst
}) \hsep \overline{\memory}
}
\\[30pt]
\ncondrule{FetchBl$_3$}{
\last{\levid} = true \quad \selectfunc{\memory}{ n} = n' \quad n' \not = n \quad
\statusfunc{\memory}{n'} = \mo \quad
}{
(\levid \bullet \memory \bullet \hlc{\datainst+\blocks{n}})
\to
(\levid \bullet \memory \bullet \hlc{\datainst+\flushs{n'} +\fetchwait{n}{n'}})
}
\\[30pt]
\ncondrule{LC-Fetch-Unblock}{
\levno{\levid'} = \levno{\levid} + 1 \quad \coreid{\levid} = \coreid{\levid'} \quad n \in \dom{\memory'}
}
{
(\levid \bullet \memory \bullet \hlc{\datainst+\blocks{n}})\hsep(\levid' \bullet \memory' \bullet \datainst')
\to \\
(\levid \bullet \memory \bullet \hlc{\datainst+\fetchs{n}})\hsep
(\levid'\bullet \memory' \bullet \datainst')
}
\end{array}
\end{displaymath}
}
\caption{Transition rules for $\blocks{n}$. }
\label{fig:fetchBl}
\end{figure}
\begin{figure}[h]
\centering
{
\footnotesize
\begin{displaymath}
\begin{array}[t]{c}
\ncondrule{FetchW}{
\last{\levid} = true \quad
\statusfunc{\memory}{n'} \neq \mo \quad
}{
(\levid \bullet \memory \bullet \hlc{\datainst+\fetchwait{n}{n'}})
\to
(\levid \bullet \memory \bullet \hlc{\datainst+\blocks{n}})
}
\\[30pt]
\end{array}
\end{displaymath}
}
\caption{Transition rule for $\fetchwait{n}{n'}$.
}
\label{fig:fetchW}
\end{figure}
\begin{figure}[t!]
\centering
{
\footnotesize
\begin{displaymath}
\begin{array}[t]{c}
\ncondrule{Flush$_1$}{
\statusfunc{\memory}{n} = \mo
}{
(\levid \bullet \hlc{\memory \bullet \datainst+\flushs{n}}) \hsep \hlcc{\overline{\memory}}
\to
(\levid \bullet \hlc{\memory\setto{n}{ \pr{k'}{\sh}} \bullet \datainst}) \hsep \hlcc{\overline{\memory}\setto{n}{ \pr{k'}{\sh}}}
}
\\[30pt]
\ncondrule{Flush$_2$}{
\statusfunc{\memory}{n} \neq \mo
}{
(\levid \bullet \memory \bullet \hlc{\datainst+\flushs{n}})
\to
(\levid \bullet \memory \bullet \hlc{\datainst})
}
\end{array}
\end{displaymath}
}
\caption{Transition rules for $\flushs{n}$.
}
\label{fig:flush}
\end{figure}
\subsection*{Transition Rules for Global Synchronization}
These rules are further structured in terms of a TSS
for labelled transitions (Figure~ \ref{fig:local.sos.3}) and a TSS of rules
for matching these labelled transitions (Figure~ \ref{fig:synch}).
\begin{figure}[h]
\centering
{
\footnotesize
\begin{displaymath}
\begin{array}[t]{c}
\ncondrule{Invalidate-One-Line}{
\statusfunc{\memory}{n}= \sh
}{
\levid \bullet \hlc{\memory} \bullet \datainst
\toL{\red{\recRX{n}}}
\levid \bullet \hlc{\memory\setto{n}{\inv}} \bullet \datainst
}
\qquad
\ncondrule{Ignore-Invalidate-One-Line}{
\statusfunc{\memory}{n} \in \{\inv,\bot\}
}{
\levid \bullet \memory \bullet \datainst
\toL{\red{\recRX{n}}}
\levid \bullet \memory \bullet \datainst
}
\\[30pt]
\ncondrule{Flush-One-Line}{
\statusfunc{\memory}{n} = \mo
}{
\levid \bullet \memory \bullet \hlc{\datainst}
\toL{\red{\recR{n}}}
\levid \bullet \memory \bullet \hlc{\datainst+\flushs{n}}
}
\qquad
\ncondrule{Ignore-Flush-One-Line}{
\statusfunc{\memory}{n} \not = \mo
}{
\levid \bullet \memory \bullet \datainst
\toL{\red{\recR{n}}}
\levid \bullet \memory \bullet \datainst
}
\end{array}
\end{displaymath}
\vspace{-10pt}
\caption{Labelled input transitions.}
\label{fig:local.sos.3}
}
\end{figure}
\begin{figure}[h]
\centering
{
\footnotesize
\begin{displaymath}
\begin{array}[t]{c}
\ncondrule{Synch-Dist}{
\cache_1\not\in\many\cache\quad
\many{\cache}
\toL{\red{\sendR{n}}}
\many{\cache'}
\quad
\cache_1
\toL{\red{\recR{n}}}
\cache'_2
}{
\many{\cache}\cup\{ \cache_1\}
\toL{\red{\sendR{n}}}
\many{\cache'}\cup \{ \cache_2\}
}
\quad
\ncondrule{Synch}{
\many{\cache}\hist{: H}
\toL{\red{\sendR{n}}}
\many{\cache'}\hist{: H'}
}{
\langle \many{\core} \hsep \many{\cache} \hsep \memory \rangle \hist{: H}
\to
\langle \many{\core} \hsep \many{\cache'} \hsep \memory \rangle \hist{: H'}}
\\[30pt]
\ncondrule{Synch-DistX}{
\cache_1\not\in \many{\cache} \quad
\core,\many{\cache}
\toL{\red{\sendRX{n}}}
\core', \many{\cache'}
\quad
\cache_1
\toL{\red{\recRX{n}}}
\cache_2
}{
\core, \many{\cache}\cup \{\cache_1\}
\toL{\red{\sendRX{n}}}
\core', \many{\cache'}\cup \{\cache_2\}
}
\qquad
\ncondrule{SynchX}{
\core\not\in \many{\core_1}\quad
\core, \many{\cache}\hist{: H}
\toL{\red{\sendRX{n}}}
\core', \many{\cache'}\hist{: H'}
}{
\langle \many{\core_1} \cup\{\core\} \hsep \many{\cache} \hsep \memory\hist{: H} \rangle \\
\to
\langle \many{\core_1}\cup\{\core'\} \hsep \many{\cache'} \hsep \memory\setto{n}{\pr{\_}{\inv}} \hist{: H'}\rangle }
\end{array}
\end{displaymath}
}
\caption{Transition rules for global synchronization/broadcast.
}
\label{fig:synch}
\end{figure}
\subsection{Synchronization Patterns}\label{sec:trans-patt}
We discuss encodings in ABS of a basic locking mechanism, atomic operations,
and a broadcast mechanism for global synchronization (using barriers).
\paragraph{Locks}
\begin{wrapfigure}{r}{0.32\textwidth}
\vspace{-12pt}
\begin{absexamplen}{numbers=none}
class Lock {
Bool unlocked = True;
Unit take_lock{
await unlocked;
unlocked = False;
}
Unit release_lock{
unlocked = True;
}
}
\end{absexamplen}
\vspace{-10pt}
\caption{\label{fig:buslock} Lock implementation in ABS using
await on Booleans.}
\vspace{-10pt}
\end{wrapfigure}
The basic mechanisms of asynchronous method calls and cooperative
scheduling in ABS can be explained by the simple code example of a
class \Abs{Lock} (Figure~\ref{fig:buslock}). It uses an \Abs{await}
statement on a Boolean condition to model a binary semaphore, which
enforces exclusive access to a common resource ``lock'',
modeled as an instance of the class \Abs{Lock}
(dynamically created by the execution of the expression \Abs{new}
\Abs{Lock}). More
specifically, execution of the \Abs{take\_lock}
method will be suspended by the \Abs{await unlocked} statement. This
statement \emph{releases the control}, allowing the scheduling of
other (enabled) processes within the \Abs{Lock} object. When the
local condition \Abs{unlocked} inside the \Abs{Lock} object has become
true, the generated \Abs{take\_lock} processes within the \Abs{Lock}
object will compete for execution. The scheduled process then will
terminate and return by setting \Abs{unlocked} to \Abs{False}.
In general, the \emph{suspension points} defined by
\Abs{await} statements define the granularity of interleaving of the
processes of an object. The statement \Abs{await
lock}!\Abs{take_lock()} will only suspend the process that issued the call (and release control in the caller object) until
\Abs{take\_lock} has returned.
In contrast, a \emph{synchronous} call \Abs{lock.take\_lock()} in ABS will generate a process for the execution of the \Abs{take\_lock()} method
by the \Abs{lock} object and block
(all the processes of) the caller object until the method returns.
\begin{wrapfigure}[7]{r}{0.32\textwidth}
\vspace{-26pt}
\begin{absexamplen}{numbers=none}
Bool TestandSet (/*input*/){
Bool fail = False;
switch /*test(input)*/$\;${
True => /*set*/;
False => fail = True;$\;$}
return fail;$\;$}
\end{absexamplen}
\vspace{-10pt}
\caption{\label{fig:TAS}Test and set pattern in ABS.}
\end{wrapfigure}
\paragraph{Atomic operations}
The interleaving model of concurrency of ABS allows for a simple and
high-level implementation of atomic operations. For example,
Figure~\ref{fig:TAS} shows a general ABS implementation of
test-and-set instructions~\cite{andrewsParallelProg}, where the
concurrency model guarantees that the local \Abs{/*test(input)*/} and
\Abs{/*set*/} instructions, assuming that they do not involve
suspension points, are not interleaved and thus can be thought of as
executed in a single atomic operation. In ABS test instructions can
be implemented using the \Abs{switch}-instruction, which evaluates an
expression that matches the resulting value against a pattern
\Abs{p} in the different branches. This instruction has been mainly
used to pattern match the ADTs used in the ABS program
discussed in this paper. In the
simplest case, this pattern can be replaced by an
\Abs{if}-\Abs{else}-instruction.
Instances of this atomic pattern can be observed in
Figures~\ref{fig:abs-aux}~and~\ref{fig:abs-swap}, in the methods
\Abs{remove_inv} and \Abs{swap}.
\paragraph{Broadcast synchronization}
Figure~\ref{fig:synccomp}a shows how broadcast synchronization in a
labelled TSS can be enforced simply by matching labels (an example is
detailed in Section~\ref{sec:cachememory}), thus abstracting from the
implementation details of the implicit multi-party synchroniser. On the other hand, in programming languges like ABS the multi-party label synchronization needs to
be programmed explicitly; Figure~\ref{fig:synccomp}b illustrates the
architecture of the ABS implementation
in Figure~\ref{fig:absglobalsync}.
\begin{figure}[h]
\centering
\begin{minipage}{0.45\textwidth}
\vspace{7pt}
\includegraphics[width=\textwidth]{figures/sossynch}\\[16pt]
\begin{scriptsize}%
a) Broadcast synchronization in a labelled TSS.
\end{scriptsize}
\end{minipage}
\quad
\begin{minipage}{0.5\textwidth}
\includegraphics[width=\textwidth]{figures/abssynch}
\begin{scriptsize}%
b) Broadcast synchronisation using an explicit synchroniser\\[-3pt]
and barriers in the ABS model.
\end{scriptsize}
\end{minipage}
\caption{\label{fig:synccomp}Broadcast synchronisation patterns in TSS and ABS.}
\end{figure}
\begin{figure}[h!]
\begin{absexamplesm}
Interface IBroadcast {
Bool broadcastSync(...);
Unit receiveSync (IBarrier start, IBarrier end, ...)}
Class Broadcast implements IBroadcast, ...{
Bool broadcastSync(...){
Bool signal=False;
await sync!lock();
if /*test*/ { sync.sendSync(this,...); /*set*/; signal=True; }
sync.release();
return signal
}
Unit receiveSync(IBarrier start, IBarrier end, ...) {
$\label{lst:syncpatt.endsync2}$ await start!synchronise();
/*some local computation*/;
end.synchronise(); }
...
}
Class Synchroniser (Set<IBroadcast> network) implements ISynchroniser {
Bool unlocked = True;
Unit lock(){ await unlocked; unlocked = False;$\;$}
Unit release(){unlocked = True;$\;$}
Unit sendSync(IBroadcast caller,...) {
Set<IBroadcast> receivers = remove(network,caller);
Int nrrecs= size(receivers);
IBarrier start = new Barrier(nrrecs);
IBarrier end = new Barrier(nrrecs+1);
foreach (receiver in receivers) { receiver!receiveSync(start,end,...); }
end.synchronise();}
}
class Barrier(Int participants) implements IBarrier {
Unit synchronise() { participants = participants - 1; await (participants == 0);$\;$}
}
\end{absexamplesm}
\caption{Global synchronisation pattern in ABS.}
\label{fig:absglobalsync}
\end{figure}
The class \Abs{Broadcast} serves as a template (or design pattern) for the implementation of a broadcast mechanism between its instances
which is specified by the interface \Abs{IBroadcast}.
The \Abs{broadcastSync} method encapsulates a synchronisation
protocol between \Abs{Broadcast} instances which uses the additional classes \Abs{Synchroniser} and
\Abs{Barrier}. This protocol consists of a synchronous call to the
method \Abs{sendSync} of an instance of the class
\Abs{Synchronise} (denoted by \Abs{sync}) which in turn asynchronously calls the method \Abs{receiveSync} of the
objects stored in the set \Abs{network} of \Abs{Broadcast} instances,
excluding the caller object executing the \Abs{broadcastSync} method.
We abstract from whether the \Abs{sync} object
is passed as parameter of the \Abs{broadcastSync} method
or part of the local state of any \Abs{Broadcast} instance.
The local computation specified by the \Abs{receiveSync} method by the objects
in \Abs{receivers} is synchronized by calls of the method
\Abs{synchronise} of the new instances \Abs{start} and \Abs{end}
of class \Abs{Barrier}.
That is, execution of this method by the \Abs{start} and \Abs{end} barriers synchronise the start and the termination of the
execution of the method \Abs{receiveSync} by the objects
in \Abs{receivers} and termination of the \Abs{sendSync} method itself.
This is achieved by a ``countdown'' of the number of objects
in \Abs{receivers} that have called the \Abs{synchronise} method
plus one, in case of the \Abs{end} barrier.
The \Abs{synchronise} method of the
\Abs{start} barrier is called asynchronously
(Line~\ref{lst:syncpatt.endsync2}) and introduces a release point in
order to avoid a deadlock that may arise when an object that has not
yet called the \Abs{synchronize} method of the \Abs{start} barrier is
blocked on a synchronous method call to an object that has already
invoked (synchronously) the \Abs{synchronize} method of the
\Abs{start} barrier. On the other hand, the corresponding call to the
\Abs{end} barrier is synchronous to ensure that all the objects
in \Abs{receivers} have completed their
local computations.
The additional synchronisation of the synchroniser object on the \Abs{end} barrier ensures that also the caller of the \Abs{sendSync} method is
blocked until all the local computations specified by the \Abs{receiveSync} method have been completed.
Objects in ABS are input enabled, so it is always possible to call a
method on an object. In our implementation, this scheme could give
rise to inconsistent states if several objects start the protocol in
parallel.
To ensure exclusive access to the synchroniser at the start of the
protocol, we add a lock to the synchroniser protocol, such that the
caller must take the lock before calling \Abs{sendSync} and release
the lock upon completion of the call. The resulting exclusive access
to the synchroniser guarantees that its message pool contains at most
one call to the method \Abs{sendSync}.
\subsection{Semantics}
ABS is a formally defined language \cite{johnsen10fmco}; in fact, its (operational)
semantics is defined by a TSS which allows us to reason formally about
the execution of ABS programs. The semantics of an ABS model can be
described by a transition relation between global configurations. A
global configuration is a (finite) set of object configurations. An
object configuration is a tuple of the form
$\langle \mathit{oid}, \sigma, p, Q\rangle$, where $\mathit{oid}$ denotes the unique
identity of the object, $\sigma$ assigns values to the instance
variables (fields) of the object, $p$ denotes the currently executing
process, and $Q$ denotes a set of (suspended) processes (the object's
``queue''). A process is a closure $(\tau,S)$ consisting of an
assignment $\tau$ of values to the local variables of the statement
$S$. We refer to \cite{johnsen10fmco} for the details of the TSS for
deriving transitions $G\rightarrow G'$ between global configurations
in ABS.
Although only one thread of control can
execute in an active object at any time, cooperative scheduling allows
different threads to interleave at explicitly declared points in the
code, i.e., the \Abs{await} statements.
When the currently executing process is suspended by an \Abs{await}
statement, another (enabled) process is scheduled. Access to an
object's fields is protected: any non-local (outside of the object)
read or write to fields happens explicitly via method calls so as to
mitigate race-conditions or the need for extensive use of explicit
mutual exclusion mechanisms (locks).
Since active objects only interact via method calls and processes are
scheduled non-deterministically, which provides an abstraction from
the order in which the processes are generated by method calls, the
ABS semantics satisfies the following global confluence property
(see also \cite{bezirgiannis19fase,tveito20fase}) that
allows to commute consecutive local computations steps of processes
which belong to \emph{different} objects.
\begin{theorem}[Global confluence]
\label{lemma:global.confluence}
For any two transitions $G_1\rightarrow G_2$ and $G_1\rightarrow G_3$
that describe execution steps of processes of different
objects, there exists a global configuration $G_4$ such that
$G_2\rightarrow G_4$ and $G_3\rightarrow G_4$.
\end{theorem}
An important consequence of the above global confluence property,
which underlies the main results of this paper, is that we can
restrict the global
interleaving between processes by reordering
the execution steps in an ABS computation.
In particular,
we can restrict the interleaving
semantics of the ABS model taking into account general semantic
properties of synchronous communication,
and the implementation of locks
and broadcast synchronization in ABS, as explained next.
Since a synchronous call of a method of \emph{another} object in ABS, blocks
all processes of the caller (object), the global confluence property
allows further restricting
the interleaving of the ABS
processes
so that the caller process is resumed
\emph{immediately} after the synchronous method invocation has
terminated.
It is worthwhile to note that in general we can \emph{not} assume that
a method that is called synchronously in ABS is also scheduled
\emph{immediately for execution} because this would discard execution
of other processes by the callee.
The global confluence property also allows abstracting from the
internal computation steps of the above ABS implementation of the
global (broadcast) synchronization pattern because it allows
scheduling the processes generated by the \emph{broadcast} method so
that execution of this method is not interleaved with any other processes.
We can formalize the above in terms of the following notion
of \emph{stable} configurations.
\begin{definition}[Stable configurations]
An object configuration is {stable} if the statement to be executed
denotes the termination of an \emph{ asynchronously} called method
(we assume a special runtime syntax which denotes such termination),
or it starts with a synchronous call \emph{to another object} or a
\Abs{await} statement. A global ABS configuration is {stable} if
all its object configurations are stable.
\end{definition}
Note that since synchronous self-calls are executed by inlining they
do not represent an interleaving point.
In the sequel $G \Rightarrow G'$ denotes the transition relation which
describes execution starting from a global stable configuration $G$ to
a next one $G'$ (without passing intermediate global stable
configurations). We distinguish the following three
cases:
\begin{enumerate}
\item
The transition $G \Rightarrow G'$ describes the \emph{local} execution of a method by a single object.
\item
The transition $G \Rightarrow G'$ describes the \emph{rendez-vous} between
the caller and callee of a synchronous method call in terms of the terminating execution of the called method, \emph{followed} by
the resumption of the suspended call.
\item
The transition $G \Rightarrow G'$ describes the effect of executing the \Abs{broadcast} method, which thus describes the \emph{global} synchronization of different objects.
\end{enumerate}
This coarse-grained interleaving
semantics of ABS forms the basis for the general methodology to prove
correctness of ABS implementations of TSS specifications, described
next.
\subsection{The Structural Correspondence}
\label{sec:struct-view}
\begin{figure}[t!]
\centering
\vspace{-10pt}
\includegraphics[trim={10pt 10 10 11},clip, width=\textwidth]{figures/classDiagram}
\caption{ \label{fig:classDiagram} Class diagram of the ABS model.}
\bigskip
\bigskip
\includegraphics[trim={10pt 10 10 11},clip, width=.8\textwidth]{figures/initialConfiguration}
\caption{ \label{fig:initialConfiguration} Object diagram of an initial configuration.}
\bigskip
\input{AnnotatedABS/ADT}
\caption{\label{fig:abs-ADT} Abstract data types of the model of the multicore memory system.}
\end{figure}
The runtime syntax of the
multicore TSS is represented in ABS by classes, user-defined datatypes
and type synonyms, outlined in
Figures~\ref{fig:classDiagram}--\ref{fig:abs-ADT}. An ABS
configuration consists of class instances to reflect the cores
with their corresponding cache hierarchies and the main memory.
Object identifiers guarantee unique names and object references are
used to capture how cores and caches are related. These references are
encoded in a one-to-one correspondence with the naming scheme of the
multicore TSS.
A core $\cid \bullet \task$ in the multicore TSS corresponds to an
instance of the class \Abs{Core} in ABS, where a field
\Abs{currentTask} of type \Abs{RstList} (as defined in
Figure~\ref{fig:abs-ADT}) represents the current list of runtime
statements . Each instance of the class \Abs{Core} further holds a
reference to the first level cache.
An important design decision we made is to
represent the runtime statements $\task$ (of a core in the multicore TSS)
as an ADT (see Figure ~\ref{fig:abs-ADT}).
A core in ABS then drives the simulation by processing these runtime statements which in general requires information about the first-level cache.
Alternatively, a core in ABS could delegate the processing of its runtime
statements by calling corresponding methods of the first-level cache.
However, this latter approach complicates the required callbacks.
A cache $\levid \bullet \memory \bullet \datainst$ in the multicore TSS
corresponds to an instance of class \Abs{Cache} with
a class parameter \Abs{nextLevel} which holds a reference to the next
level cache and a field \Abs{cacheMemory} which models the cache's
memory $\memory$ (of type \Abs{MemMap}, Figure~\ref{fig:abs-ADT}).
The multiset $\datainst$ of a cache's data instructions (see
Figure~\ref{fig:rts}) is represented by corresponding \emph{processes}
in the message pool of the cache object in ABS. If the value of
\Abs{nextLevel} is \Abs{Nothing}, then the object represents the last
level cache (in the multicore TSS, a predicate $\flag$ is used to
identify the last level).
In addition, the ABS implementation of the global
synchronisation with labels $\sendR{n}$ and $\sendRX{n}$ used in the
multicore TSS is based on the global synchronisation pattern
as described in Figure~\ref{fig:absglobalsync}.
However,
instead of distinguishing between these two labels by means of an additional parameter, we introduce two corresponding
broadcast interfaces:
¨
\begin{absexamplesm}
Interface IBroadcast {
Bool broadcast(...);
Unit receiveRd (IBarrier start, IBarrier end, ...)}
Interface IBroadcastX {
Bool broadcastX(...);
Unit receiveRdX (IBarrier start, IBarrier end, ...)}
\end{absexamplesm}
The class \Abs{Cache} then provides an implementation of both interfaces
following the template of the class \Abs{Broadcast} in Figure~\ref{fig:absglobalsync}.
The ABS class \Abs{Bus}, on the other hand, follows the template
of the \Abs{Synchroniser} class with the two versions
\Abs{sendRd} and \Abs{sendRdX} of the method \Abs{sendSync}.
The object diagram in Figure~\ref{fig:initialConfiguration} shows an
initial configuration corresponding to the one depicted in
Figure~\ref{fig:whitebox}.
\subsection{The Behavioural Correspondence}
\label{sec:behv-view}
We next discuss the ABS implementation of the transition rules
of the multicore TSS, and the ABS synchronization patterns
described in Section~\ref{sec:abs}. We
observe that the combination of \emph{asynchronous method calls} and
\emph{cooperative scheduling} in ABS is crucial because of the
\emph{interleaving} inherent in the multicore TSS, which requires that
objects are able to process other requests while executing a method in
a controlled way; e.g., caches need to flush memory blocks while
waiting for a fetch to succeed.
\subsubsection{The Annotated ABS Multicore Implementation}
\label{sec:behv-trans}
The classes \Abs{Core} and \Abs{Cache} pose the main implementation
challenges. Here we explain the implementation of the \Abs{run}
method (Figure~\ref{fig:ABScore}) of the class \Abs{Core} (which is
its only method) informally, in terms of its annotations (as
introduced in Section~\ref{sec:Anno}).
In Section~\ref{sec: correctness} we introduce a formal semantics
of these annotations as a high-level description of a simulation relation,
and prove the correctness of the class \Abs{Cache}.
The \Abs{run} method may generate {synchronous} calls to the auxiliary
methods in Figure~\ref{fig:abs-aux}.
The method \Abs{remove\_inv} instantiates the test-and-set pattern
of Figure \ref{fig:TAS}.
The method \Abs{broadcastX}
is an instance of the global synchronization pattern
described in Section~\ref{sec:abs}, Figure \ref{fig:absglobalsync}. The method \Abs{sendRdX} of the
global synchroniser \Abs{bus} asynchronously calls the method
\Abs{receiveRdX}, see Figure~\ref{fig:abs_receiveRdx}, of all caches
(except for the calling cache), using the barrier synchronization
described in Section~\ref{sec:abs}.
\begin{figure}[t]
\input{AnnotatedABS/runcore_abs}
\caption{\label{fig:ABScore} Annotated \Abs{run} method.}
\end{figure}
\begin{figure}[h!]
\input{AnnotatedABS/ABSAux}
\caption{\label{fig:abs-aux}Methods \Abs{getStatus}, \Abs{remove_inv}, and \Abs{broadcastX} of class \Abs{Cache}.}
\bigskip
\input{AnnotatedABS/receiveRdx_abs}
\caption{\label{fig:abs_receiveRdx} Annotated \Abs{receiveRdX} method.}
\end{figure}
Since the stable point at the beginning of the \Abs{run} method has
no associated annotation, by definition (see Section~\ref{sec:Anno}),
for \emph{any} path from the beginning to a next stable point (or to termination) there
does \emph{not} correspond a transition rule (of the multicore TSS).
For example, there is no transition rule corresponding to the case
that the run method terminates when \Abs{curentTask==Nil} (note that
because of the structural correspondence also the corresponding core
has no runtime statements \emph{rst} to execute). Similarly, there
are no transition rules corresponding to the execution of the code
from the beginning of the method to the synchronous calls to the
auxiliary methods~\Abs{remove\_inv} (Line~\ref{lst:run.removeRead})
and \Abs{getStatus} (Lines~\ref{lst:run.useReadl},
\ref{lst:run.useWrite}, \ref{lst:run.WriteBl.use}) of the first level
cache which, besides the pattern matching, only consists of the call
itself.
The condition of the annotation \Abs{removed==True} :
$\crule{\rn{PrRd}_2}$ (Line~\ref{lst:run.removeRead}) associated with
the synchronous call to the \Abs{remove\_inv} method describes the
path which leads from its execution and return via the
\Abs{then}-branch of the subsequent \Abs{if}-statement to the
termination of the run method (after it has called itself again
asynchronously). According to the annotation, the execution of this
path corresponds to the $\crule{\rn{PrRd}_2}$ transition rule:
$$
\ncondrule{PrRd$_2$}{
\quad \first{\levid} = true \quad \coreid{\levid} = c
\quad
\statusfunc{\memory}{n} \in \{\inv,\bot\}
}{
(c \bullet \hlc{\reads{n};\task}) \hsep (\levid \bullet \hlcc{\memory \bullet \datainst})\hist{: h}
\to \\
(c \bullet \hlc{\readBs{n};\task}) \hsep (\levid \bullet \hlcc{\memory \setto{n}{\bot} \bullet \datainst+\fetchs{n}}) \hist{: h}
}
$$
This rule handles the case when a core intends to read a memory block
with address~$n$, which is not found in the first level cache. The
core will then be blocked while waiting for the memory block to be
fetched either from the lower level caches or main memory. Note that
the condition as returned by the \Abs{remove\_inv} method signals that
the status of the address of the first level cache is undefined or
invalid.
On the other hand, the condition \Abs{removed==False} describes the
path which leads from its execution and return via the \Abs{else}-branch (Line~\ref{lst:run.PrRd1}), which
also leads to the termination of this invocation of the run method. According to the
annotation, the execution of this path corresponds to the
$\crule{\rn{PrRd}_1}$ transition rule:
$$
\ncondrule{PrRd$_1$}{
\quad \first{\levid} = true \quad \coreid{\levid} = c
\quad
\statusfunc{\memory}{n} \in \{\sh,\mo\}
}{
(c \bullet \hlc{\reads{n};\task}) \hsep (\levid \bullet \memory \bullet \datainst) \hist{: h}
\to
(c \bullet \hlc{\task}) \hsep (\levid \bullet \memory \bullet \datainst) \hist{: h; R(c, n)} }
$$
This rule covers the case when the memory block to be read by a core is
found in its first level cache. . Note that the condition as returned by
the \Abs{remove\_inv} method implies that the status of the address of
the first level cache
is either shared or modified.
Next we consider the annotation \Abs{status} != \Abs{Nothing} :
$\crule{\rn{PrRd}_3}$ of the synchronous call to the \Abs{getStatus}
method (Line~\ref{lst:run.useReadl}). Its condition describes the
execution path which leads from the execution and return of the called
\Abs{getStatus} method to termination of the \Abs{run} method via
the \Abs{then}-branch of the subsequent \Abs{if}-statement
(Line~\ref{lst:run.ifReadl}). According to the annotation, the
execution of this path corresponds to the $\crule{\rn{PrRd}_3}$
transition rule:
$$
\ncondrule{PrRd$_3$}{
\first{\levid} = true \quad \coreid{\levid} = c
\quad
n \in \dom{\memory}
}{
(c \bullet \hlc{\readBs{n};\task}) \hsep (\levid \bullet \memory \bullet \datainst)\hist{: h}
\to
(c \bullet \hlc{\reads{n};\task}) \hsep (\levid \bullet \memory \bullet \datainst) \hist{: h; R(c, n)}
}
$$
This rule unblocks the core from waiting when~$n$ (i.e., the block to
be read) is found in the first level cache.
On the other hand, there does not exist a transition rule which
corresponds to the execution path described by the condition
\Abs{status==Nothing}. This path leads from the execution of the called \Abs{getStatus} method directly to the
termination of the \Abs{run} method without an update of the (local)
state, e.g., \Abs{currentTask} is not updated. In other words, the
evaluation of the \Abs{readBl(n)} instruction in ABS involves
\emph{busy waiting} until the status returned by the first level cache
is defined. Alternatively, this could be implemented by calling
synchronously a method of the first level cache which simply executes
the statement \Abs{await lookup(cacheMemory,n)}!=\Abs{Nothing}.
The annotation of the synchronous call to the method
\Abs{getStatus} (Line~\ref{lst:run.WriteBl.use}) involves the rule
$$
\ncondrule{PrWr$_4$}{
\first{\levid} = true \quad \coreid{\levid} = c
\quad
n \in \dom{\memory}
}{
(c \bullet \hlc{\writeBs{n};\task})\hsep (\levid \bullet \memory \bullet \datainst) \hist{: h}
\to
(c \bullet \hlc{\writes{n};\task}) \hsep (\levid \bullet \memory \bullet \datainst) \hist{: h}
}
$$
This annotation is explained in a similar manner as the annotation of
the synchronous call to the \Abs{getStatus} method on
Line~\ref{lst:run.useReadl}. This rule unblocks the core from
waiting when~$n$ (i.e., the block to be written) is found in the
first level cache.
We consider next the annotation \Abs{status}==J\Abs{ust(Mo)} :
$\crule{\rn{PrWr}_1}$ of the synchronous call to the method
\Abs{getStatus} (Line~\ref{lst:run.useWrite}) . Its condition
describes the execution path which leads from the execution of the called \Abs{getStatus} method and subsequent
execution of the \Abs{switch} statement to termination of the
\Abs{run} method. According to the annotation, the execution of this
path corresponds to the $\crule{\rn{PrWr}_1}$ transition rule:
$$
\ncondrule{PrWr$_1$}{
\first{\levid} = true \quad \coreid{\levid} = c
\quad
\statusfunc{\memory}{n} = \mo
}{
(c \bullet \hlc{\writes{n};\task}) \hsep (\levid \bullet \memory \bullet \datainst) \hist{: h}
\to
(c \bullet \hlc{\task}) \hsep (\levid \bullet \memory \bullet \datainst) \hist{: h; W(c, n)}
}
$$
This rule allows a core to write to memory block~$n$ if the block is
found in a modified state in the first level cache. On the other
hand, in case the condition does not hold, according to the annotation
no transition rules correspond to the execution paths which lead from
the execution of the called \Abs{getStatus} method to
the next stable points, i.e., the synchronous calls to the methods
\Abs{broadcastX} and \Abs{remove\_inv}
(Lines~\ref{lst:run.Write.sh.broadcastX} and \ref{lst:run.Write.else},
respectively).
The condition of the annotation \Abs{res==true} :
$\crule{\rn{PrWr}_2/\rn{SynchX}}$ of the synchronous call to the
\Abs{broadcastX} method (Line~\ref{lst:run.Write.sh.broadcastX}) of
the first level cache describes the path which leads from the execution
of the \Abs{broadcastX} method, followed by the execution of the subsequent
if-statement to termination of the run method (after an update of
\Abs{currentTask} and calling the \Abs{run} method again
asynchronously). According to the annotation this path corresponds to
the global synchronization rule
$$
\ncondrule{SynchX}{
\core\not\in \many{\core_1}\quad
\core, \many{\cache}\hist{: H}
\toL{\red{\sendRX{n}}}
\core', \many{\cache'}\hist{: H'}
}{
\langle \many{\core_1} \cup\{\core\} \hsep \many{\cache} \hsep \memory\hist{: H} \rangle
\to
\langle \many{\core_1}\cup\{\core'\} \hsep \many{\cache'} \hsep \memory\setto{n}{\pr{\_}{\inv}} \hist{: H'}\rangle }
$$
where the second premise is generated by successive applications of the rule
$$
\ncondrule{Synch-DistX}{
\cache_1\not\in \many{\cache} \quad
\core,\many{\cache}
\toL{\red{\sendRX{n}}}
\core', \many{\cache'}
\quad
\cache_1
\toL{\red{\recRX{n}}}
\cache_2
}{
\core, \many{\cache}\cup \{\cache_1\}
\toL{\red{\sendRX{n}}}
\core', \many{\cache'}\cup \{\cache_2\}
}
$$
This latter rule itself is triggered by the following rules
$$
\ncondrule{PrWr$_2$}{
\first{\levid} = true \quad \coreid{\levid} = c
\quad
\statusfunc{\memory}{n} = \pr{k}{\sh}
}{
(c \bullet \hlc{\writes{n};\task}) \hsep (\levid \bullet \hlcc{\memory} \bullet \datainst)\hist{: h}
\toL{\red{\sendRX{n}}}
(c \bullet \hlc{\task}) \hsep (\levid \bullet \hlcc{\memory \setto{n}{\pr{k}{\mo}}} \bullet \datainst) \hist{: h; W(c, n)}
}
$$
$$
\ncondrule{Invalidate-One-Line}{
\statusfunc{\memory}{n}= \sh
}{
\levid \bullet \hlc{\memory} \bullet \datainst
\toL{\red{\recRX{n}}}
\levid \bullet \hlc{\memory\setto{n}{\inv}} \bullet \datainst
}
$$
and
$$
\ncondrule{Ignore-Invalidate-One-Line}{
\statusfunc{\memory}{n} \in \{\inv,\bot\}
}{
\levid \bullet \memory \bullet \datainst
\toL{\red{\recRX{n}}}
\levid \bullet \memory \bullet \datainst
}
$$
These rules together capture the broadcast mechanism for invalidation
in the multicore memory system. Rule \rn{PrWr$_2$}
corresponds to the case where a core writes to a memory block~$n$
that is marked as shared in its first level cache, which requires
broadcasting an invalidation message, $\sendRX{n}$,
to all the other caches. This is achieved by triggering the global
synchronization rules \rn{SynchX} and \rn{Synch-DistX}. While the
former identifies the core~$\core$
that broadcasts the invalidation message, the latter recursively
propagates the message, $\recRX{n}$,
to the other caches. Depending on the local status of memory
block~$n$
in the recipient cache, the recipient cache will either invalidate the
local copy of the block (\rn{Invalidate-One-Line}), or ignore the
message (\rn{Ignore-Invalidate-One-Line}).
To explain this application of the $\crule{\rn{SynchX}}$
rule, we have a closer look at the definition of the \Abs{broadcastX}
method. Its body involves an instance of the global synchronization
pattern (Figure~\ref{fig:absglobalsync}). As discussed in
Section~\ref{sec:abs}, because of the global confluence property, we
may assume that its execution is atomic, i.e., not interleaved with
any process that it has not generated. The synchronous call to the
\Abs{sendRdX} method of the bus generates asynchronous calls to the
\Abs{receiveRdX} method (Figure~\ref{fig:abs_receiveRdx}) of all
caches except the one that initiated the global bus synchronization.
Following the general global synchronization pattern
(Figure~\ref{fig:absglobalsync}), these method calls are synchronized
by a \Abs{start} and an \Abs{end} barrier. The two conditions of the
annotation at the beginning of the \Abs{receiveRdX} method describe
the two possible execution paths and their corresponding transition
rules $\crule{\rn{Invalidate-One-Line}}$
and $\crule{\rn{Ignore-Invalidate-One-Line}}$.
In case the condition \Abs{res==true} does not hold, according to the
annotation, no transition rule corresponds to the execution of the
\Abs{broadcastX} method. In this case the bus synchronization, as
invoked by the \Abs{broadcastX} method (Figure~\ref{fig:abs-aux}),
failed because the status of the address of the first level cache is
not shared anymore (as required by the $\crule{\rn{PrWr}_2}$
rule). Consequently, the processing of the \Abs{write(n)} instruction
itself fails and it will be processed again by the asynchronous self
call to the \Abs{run} method.
We conclude the informal explanation of the annotated \Abs{run} method
with the annotation\linebreak \Abs{removed==True}~:~$\crule{\rn{PrWr}_3}$ of the
synchronous call to the method \Abs{remove\_inv}
(Line~\ref{lst:run.Write.else}). Its condition describes the path
that corresponds to the transition rule:
$$
\ncondrule{PrWr$_3$}{
\first{\levid} = true \quad \coreid{\levid} = c
\quad
\statusfunc{\memory}{n} \in \{\inv,\bot\} }{
(c \bullet \hlc{\writes{n};\task}) \hsep (\levid \bullet \hlcc{ \memory \bullet \datainst})\hist{: h}
\to \\
(c \bullet \hlc{\writeBs{n};\task}) \hsep (\levid \bullet \hlcc{\memory \setto{n}{\bot} \bullet \datainst+\fetchs{n}}) \hist{: h}
}
$$
This rule handles the case when a core tries to write to a memory
block with address~$n$, which is either invalid or not found in the
first level cache. The core will then be blocked while the memory
block is fetched from the lower level cache or from the main memory.
On the other hand, according to the annotation, no transition rule
corresponds to the execution path that is described by the negation of
the condition. Note that this covers the case when the status
returned by \Abs{getStatus} (Line~\ref{lst:run.useWrite}) has changed;
i.e., the status of the memory block is no longer undefined or
invalid. As above, the \Abs{run} method terminates without having
successfully processed the \Abs{write(n)} task, which will be
evaluated again by the next asynchronous invocation of
the \Abs{run} method.
In the next section we show how to formally validate the annotations
in terms of a simulation relation.
\subsection{Bisimulation}
We briefly discuss how to extend Theorem~\ref{theorem:simulation-relation}
to a bisimulation between the transitive, reflexive closure of the transition
relation $\Rightarrow$ of the ABS multicore program and that of the transition relation $\rightarrow$ of the multicore TSS.
Such a bisimulation relation then allows to prove both \emph{safety} and
\emph{liveness} properties of the ABS multicore program in terms of the multicore TSS.
The following theorem states that the multicore TSS is simulated by the ABS program.
\begin{theorem}
\label{theorem:bisimulation}
Let $G$ be a {reachable} stable global configuration of the ABS
multicore program.
If $\alpha(G)\rightarrow \mbox{\it cf}$ then there exists a stable
global configuration $G'$ such that $\alpha(G')=\mbox{\it cf}$ and
$G\Rightarrow^* G'$.
\end{theorem}
\begin{proof}
{\rm
We sketch a proof of this theorem which is based on the correctness of the annotations as established in the proof of Theorem ~\ref{theorem:simulation-relation}.
The global structure of the proof of Theorem~\ref{theorem:bisimulation} however involves an analysis of the individual TSS rules.
All these rules are triggered by a \emph{dst} instruction.
For each such instruction we check statically for each stable point of the corresponding ABS method whether there exists a path to another stable
point (\emph{not necessarily the next one}) execution of which corresponds
to the TSS rule application.
As an example of this scheme we give an analysis of an application of the rule
$$
\ncondrule{LC-Miss}{
\levno{\levid'} = \levno{\levid} + 1 \quad \coreid{\levid} = \coreid{\levid'} \quad
\statusfunc{\memory'}{n} \in \{\inv,\bot\}
}
{
(\levid \bullet \memory \bullet \hlc{\datainst+\fetchs{n}})\hsep(\levid' \bullet \hlcc{\memory' \bullet \datainst'})
\to \\
(\levid \bullet \memory \bullet \hlc{\datainst+\blocks{n}})\hsep(\levid'\bullet \hlcc{\memory' \setto{n}{\bot} \bullet \datainst'+\fetchs{n}})
}
$$
in $\alpha(G)$, triggered by the \emph{dst}
instruction $\fetchs{n}$.
By definition of $\alpha$ there exists
a process instance
(either executing or suspended) of the \Abs{fetch} method with the address $n$ as the value of the its formal
parameter.
Further, the status of the address $n$ of the next level cache, denoted by the \Abs{nextCache} field of the cache object to which
this process belongs, is undefined or invalid.
We have the following straightforward analysis of the stable points of this process.
In case of the initial stable point
and the stable point associated with the call of the
\Abs{remove\_inv} method (Line~\ref{lst:fetch.nl.nothing.in} of the \Abs{fetch} method, Figure~\ref{fig:abs-fetch}), by definition of $\alpha$ there exists a computation $G\Rightarrow^* G'$
which involves in both cases the execution of the path from
Line~\ref{lst:fetch.nl.nothing.in}
to Line \ref{lst:fetch.nl.nothing.in.tfb}
As argued in the proof of Theorem ~\ref{theorem:simulation-relation}, this path corresponds to an application of the
$\crule{\rn{LC-Miss}}$ rule.
Note that execution of the path from Line~\ref{lst:fetch.begin}
to Line~\ref{lst:fetch.nl.nothing.in} corresponds to a
silent transition in the multicore TSS,
In case of the stable point associated with the call of the \Abs{swap} method
(Line ~\ref{lst:fetch.nl.sh.mo.swap}), by definition of $\alpha$ there exists a computation $G\Rightarrow^* G'$
which involves execution of the path which leads from the return of the
\Abs{swap} method via the \Abs{else}-branch of the subsequent
\Abs{if}-statement (note that \Abs{s == Nothing}) to termination of the
\Abs{fetch} method, \emph{followed} by execution of the path from the initial stable point of the \emph{newly} generated process instance of the \Abs{fetch} method
to Line \ref{lst:fetch.nl.nothing.in.tfb}. As argued above, this latter path corresponds to an application of the
$\crule{\rn{LC-Miss}}$ rule. Execution of the first path corresponds to a
silent transition in the multicore TSS (as argued in the proof of Theorem ~\ref{theorem:simulation-relation}).
}
\end{proof}
\subsection{Annotating ABS with TSS Rules}\label{sec:Anno}
For a general introduction of transition system specifications we refer to \cite{G19}.
The general methodology for the development of ABS implementations of
abstract TSSs is based on the coarse-grained interleaving described in
Section~\ref{sec:abs} (denoted by the transition relation
$\Rightarrow$): it allows focusing on the design of \emph{local,
sequential} code that implements the individual transition rules.
This is reflected by the following use of transition rules as a
\emph{specification formalism} of ABS code. A \emph{conditional
transition rule} $b:R$ consists of a local Boolean condition $b$ in
ABS and a name $R$ of a transition rule. We use sequences
$b_1:R_1; \ldots; b_n:R_n$ of conditional transition rules to annotate
\emph{stable points}. A stable point of a method definition denotes
either its body or a substatement of its body that starts with an
external synchronous call or an \Abs{await} statement. The idea is
that each $b_i$ is evaluated as a condition which identifies a
\emph{path} leading from the annotated stable point to a next one or
to termination. The execution of this path should correspond to the
application of the associated transition rule $R_i$. This
correspondence involves a simulation relation, described below.
A sequence $b_1:R_1; \ldots; b_n : R_n$ of conditional transition
rules is evaluated from left to right, that is, the first transition
rule from the left, the Boolean condition of which evaluates to true,
is applicable. The case that all Boolean conditions are false means
that there does not exist a transition rule for \emph{any} path from
the annotated stable point to a next one or to termination (in the simulation relation
all these paths would correspond to a ``silent'' transition). As a
special case, we stipulate that for \emph{any} path leading from a
stable point \emph{which has no associated annotation} to a next
stable point (or to termination) there does \emph{not} exist a corresponding transition
rule. The use of annotations in the ABS code of the multicore memory
system is shown in Section~\ref{sec:behv-view}.
\subsection{Correctness of the Implementation}\label{sec:method}
The correctness of the ABS implementation with respect to a given
TSS can be established by means of a simulation relation between the
transition system describing the semantics of the ABS implementation
and the transition
system describing the TSS.
The annotation of ABS code with (conditional) TSS rules provides a
high-level description of the simulation relation, describing which
rule(s) correspond with the execution of the ABS code from one stable
point to a next one (or to termination). Underlying this high-level description, we define a simulation relation between ABS configurations and the runtime
states of the
TSS.
This simulation
relation is defined as an abstraction function $\alpha$ which maps every stable
global ABS configuration $G$ to a behavioral equivalent
TSS configuration $\alpha(G)$ (see Section \ref{sec: correctness}).
We restrict the simulation relation to \emph{reachable} ABS
configurations. A configuration $G$ of the ABS
program is {reachable} if $G_0\Rightarrow^* G$, for some
\emph{initial} configuration $G_0$.
In an initial configuration of
the ABS multicore program all process queues are empty, and the only
active processes are those about to execute the run methods of the
cores.
This restriction allows to use some general properties of the
ABS semantics; e.g., upon return of a synchronous call, the local
state of the calling object has not changed.
We can now express that a ABS
program
is a correct implementation of a
TSS specification by proving that the following
theorem holds, given an abstraction function $\alpha$:
\begin{definition}[Correctness]
\label{theorem:simulation-relation-methodology}
Given an ABS program and a TSS, let $\alpha$ be an
abstraction function from configurations of the ABS program to TSS configurations. The ABS program is a correct implementation of
the TSS,
if for any reachable configuration $G$ and transition
$G\Rightarrow G'$ of the ABS program we have that $\alpha(G)=\alpha(G')$ or
$\alpha(G)\rightarrow\alpha(G')$.
\end{definition}
Because of the general
confluence property of the ABS semantics to prove that $\alpha$ is a simulation relation, it suffices to verify the
annotations of
methods
in terms of the abstraction function $\alpha$. The general idea is
that for each transition $G\Rightarrow G'$ which results from the
execution from one stable point to a next one (or to termination), we have to show that
$\alpha(G')$ results from $\alpha(G)$ by application of the enabled
TSS rule associated with the initial stable point. In case no TSS rule
is enabled, we have a ``silent'' step, that is,
$\alpha(G)=\alpha(G')$.
\subsection{A TSS of Multicore Memory Systems\label{sec.formalization}}
The multicore TSS describes the interactions between a core, caches,
and the main memory. It further includes labeled transitions to model
instantaneous broadcast. In general a model of the multicore TSS is a
\emph{transition system}. We refer to a model of the multicore TSS,
which is parametric in the number of cores and caches, also as a
Multicore Memory System (MMS, for short). The multicore
TSS~\cite{bijo17facs,bijo19scp} is shown to guarantee correctness
properties for data consistency and cache coherence (see,
e.g.,~\cite{CullerBook,Sorin:2011}), including the preservation of
program order in each core, the absence of data races, and that stale
data is never accessed.
We outline the main aspects of a simplified version of the multicore
TSS which allows focusing on the main challenges of a correct
distributed implementation. The runtime syntax is given in
Figure~\ref{fig:rts}. A configuration $\mathit{cf}$ is a tuple
consisting of a main memory $\memory$, cores $\many\core$, caches
$\many\cache$ (we abstract from the tasks to be scheduled).
A core~$(\cid \bullet \task)$ with identifier~$\cid$ executes \emph{runtime
statements}~$\task$. A cache $(\levid \bullet \memory \bullet \datainst )$
with identifier~$\levid$ has a local cache memory~$\memory$ and data
instructions~$\datainst$. We assume that the cache identifier $\levid$
encodes the $\cid$ of the core to which the cache belongs and its
level in the cache hierarchy. We use $\mathit{Status}_{\bot}$ to
denote the extension of the set $\{ \mo, \sh , \inv \}$ of status tags
with the undefined value $\bot$. Thus, a memory
$\memory: \mathit{Address} \rightarrow \mathit{Status}_{\bot}$ maps
addresses~$n$ to either a status tag $\status$ or to $\bot$ if the
memory block with address~$n$ is not found in $\memory$.
\emph{Data access patterns} $\stask$ model tasks consisting of finite
sequences of $\reads{n}$ and $\writes{n}$ operations to address $n$
(that is, we abstract from control flow operations for sequential
composition, non-deterministic choice, repetition, and task creation).
The empty access pattern is denoted $\varepsilon$. Cores execute runtime
statements $\task$, which extend~$\stask$ with $\readBs{n}$ and
$\writeBs{n}$ to block execution while waiting for data. Caches
execute \emph{data instructions} from a multiset $\datainst$ to fetch
or flush a memory block with address~$n$; here, $\fetchs{n}$ fetches a
memory block with address $n$, $\blocks{n}$ blocks execution while
waiting for data, $\fetchwait{n}{n'}$ waits for a memory block $n'$ to
be flushed before fetching $n$ (this is needed when the cache is
full), and $\flushs{n} $ flushes a memory block.
The connection between the main memory and the caches of the different
cores is modelled by an \emph{abstract communication medium} which
allows messages from one cache to be transmitted to the other caches
and to main memory in a parallel instantaneous broadcast.
Communication in the abstract communication medium is captured in the
TSS by label matching on transitions.
For any address $n$, an output of the form $ \sendR{n}$ or
$\sendRX{n}$ is broadcasted and matched by its dual of the form
$\recR{n}$ or $\recRX{n}$. The syntax of the model is further
detailed in~\cite{bijo17facs,bijo19scp}. For a complete overview of
the transition rules we refer to \ref{sec:TSS}.
In the next section, we will introduce these
rules incrementally when discussing their ABS implementation.
The following auxiliary functions are used in the transition
rules, given a cache identifier~$\levid$:
\begin{itemize}
\item $\coreid{\levid}$ returns the identifier of the core to which
the cache belongs;
\item $\levno{\levid}$ gives the level at which the cache is located
in the memory hierarchy;
\item $\first{\levid}$ is true when $\levno{\levid} =1$, otherwise
false;
\item $\last{\levid}$ is true when $\levno{\levid} =l$ where $l$ is
the number of caches in the hierarchy, otherwise false;
\item $\statusfunc{M}{n}$ returns the status of block $n$ in
memory~$M$ or $\bot$ if the block is not found in~$M$; and
\item $\selectfunc{\memory}{n}$ determines the address where a
block~$n$ should be placed in the cache, based on a cache
associativity (e.g., random, set associativity or direct map) and a
replacement policy (e.g., random or LRU).
\end{itemize}
|
2,877,628,088,964 | arxiv | \section{Introduction and main results}
~~~~In this paper we consider the following mean curvature equation
\begin{equation}\label{1.1}\begin{array}{l}
\mbox {div}(\frac{\nabla u}{\sqrt{1+|\nabla u|^2}})=H~~\mbox{in}~~\Omega,
\end{array}\end{equation}
with Neumann and Robin boundary conditions respectively, where $H$ is a positive constant, $\Omega$ is a bounded smooth convex domain in $\mathbb{R}^{n}(n\geq2)$.
Critical points of solutions to elliptic equations is a significant research topic. In some cases, properties of critical points of solutions are themselves the main concern. In other cases, theory of critical points provide an important tool in the study of properties of solutions. There are many known results about the study of critical points. In 1971 Makar-Limanov \cite{Makar-Limanov} investigated the Poisson equation with constant inhomogeneous term in a planar convex domain, and proved that $u$ has one unique critical point. In 1992 Alessandrini and Magnanini \cite{AlessandriniMagnanini1} considered the geometric structure of the critical set of a solution to semilinear elliptic equation in a nonconvex domain $\Omega,$ whose boundary is composed of finite simple closed curves. They deduced that the critical set is made up of finitely many isolated critical points. In 2012 Arango and G\'{o}mez \cite{ArangoGomez2} considered the critical points of the solutions for quasilinear elliptic equations with Dirichlet boundary condition in strictly convex domains and nonconvex domains respectively. If the domain is strictly convex and $u$ is a negative solution, they proved that the critical set has exactly one non-degenerate critical point. On the other hand, they obtained the similar results of the semilinear case for a planar annular domain, whose boundary has nonzero curvature. In 2017 Deng, Liu and Tian \cite{Deng2} investigate the geometric structure of interior critical points of solutions $u$ to a quasilinear elliptic equation with nonhomogeneous Dirichlet boundary conditions in a simply connected or multiply connected domain $\Omega$ in $\mathbb{R}^2$. They develop a new method to prove $\Sigma_{i = 1}^k {{m_i}}+1=N$ or $\Sigma_{i = 1}^k {{m_i}}=N,$ where $m_1,\cdots,m_k$ are the respective multiplicities of interior critical points $x_1,\cdots,x_k$ of $u$ and $N$ is the number of global maximum points of $u$ on $\partial\Omega$. All the above results involved the critical points of solutions to elliptic equations with Dirichlet boundary condition in the planar domains. In addition, a number of other authors have investigated this problem and some other related problems (see \cite{AlessandriniMagnanini2,Alessandrini1,Alessandrini2,Alessandrini3,Cecchini,Enciso1,Enciso2,Kawohl,Kraus,Lopez,Zhu}). However, the critical set $K$ has not been fully considered for some general cases, especially for higher dimensional spaces, Neumann and Robin boundary value problems.
For higher dimensional cases, there has few results about the critical points of solutions to elliptic equations. In 1998 Cabr\'{e} and Chanillo \cite{CabreChanillo}, under the assumption of the existence of a semi-stable solution, showed that the solution of Poisson equation $-\triangle u=f(u)$ has exactly one non-degenerate critical point in a smooth bounded convex domain of $\mathbb{R}^{n}(n\geq2)$. Recently, the authors \cite{Deng1} investigated the geometric structure of critical points of solutions to mean curvature equations with Dirichlet boundary condition over a smooth bounded domain $\Omega$ in $\mathbb{R}^n(n\geq 2).$
Concerning the Neumann and Robin boundary value problems, the critical points of solutions to elliptic equations seems to be less considered. In 1990, Sakaguchi \cite{Sakaguchi} proved that the solutions of Poisson equation with Neumann and Robin boundary conditions respectively exist exactly one critical point. The goal of this paper is to obtain some results about the critical set of solutions to mean curvature equation with Neumann and Robin boundary conditions respectively in bounded smooth convex domains $\Omega$ of $\mathbb{R}^{n}(n\geq2)$.
Throughout this paper, we shall suppose that $\Omega$ is a bounded smooth convex domain. Our theorems concerns only qualitative properties of the critical points of solutions to the prescribed constant mean curvature equation, i.e., the uniqueness and non-degeneracy. Hence we only need the hypothesis of the existence of solutions. The existence of solutions can be seen in \cite{Gilbarg,MaXu,PucciSerrin} etc. Our main results are as follows:
\begin{Theorem}\label{th1.1}
Let $\Omega$ be a bounded smooth convex domain in $\mathbb{R}^2.$ Suppose that $H$ is a positive constant and that $u$ is a solution of the following boundary value problem
\begin{equation}\label{1.2}
\left\{
\begin{array}{l}
{\rm div}(\frac{\nabla u}{\sqrt{1+|\nabla u|^2}})=H~~~\mbox{in}~\Omega,\\
\frac{\partial u}{\partial \overrightarrow{n}}=c ~~~\mbox{on}~ \partial\Omega,
\end{array}
\right.
\end{equation}
where $\overrightarrow{n}$ is the unit outward normal vector of $\partial\Omega$ and $c$ is a positive constant. Then $u$ has exactly one critical point $p$ in $\Omega$ and $p$ is a non-degenerate interior minimal point of $u$.
\end{Theorem}
\begin{Theorem}\label{th1.2}
Let $\Omega$ be a bounded smooth convex domain in $\mathbb{R}^2.$ Suppose that $H$ is a positive constant. Let $u$ be a solution of the following boundary value problem
\begin{equation}\label{1.3}
\left\{
\begin{array}{l}
{\rm div}(\frac{\nabla u}{\sqrt{1+|\nabla u|^2}})=H~~~\mbox{in}~\Omega,\\
\frac{\partial u}{\partial \overrightarrow{n}}+\alpha u=0 ~~~\mbox{on}~ \partial\Omega,
\end{array}
\right.
\end{equation}
where $\alpha$ is a positive constant. Then $u$ has exactly one critical point $p$ in $\Omega$ and $p$ is a non-degenerate interior minimal point of $u$.
\end{Theorem}
In addition, we will give partial results about the critical points of solutions in higher dimensional spaces.
\begin{Theorem}\label{th1.3}
Let $\Omega$ be a bounded smooth convex domain of rotational symmetry with respect to $x_n$ axis in $\mathbb{R}^n(n\geq3).$ Suppose that $u=u(|x'|,x_n)$ is an axisymmetric solution of equation (\ref{1.2}) or (\ref{1.3}), where $|x'|=\sqrt{x_1^2+x_2^2+\cdots+x_{n-1}^2}$. Then $u$ has exactly one critical point $p$ in $\Omega$ and $p$ is a non-degenerate interior minimal point of $u$.
\end{Theorem}
The rest of this paper is organized as follows. In Section 2, we introduce the local Chen \& Huang's comparison technique. In Section 3, firstly, we prove that every interior critical point of $u$ is non-degenerate. Then, by the strong maximum principle and Hopf lemma, we show that $u$ does not have maximum points in $\Omega$ and that $u$ cannot have minimal points on $\partial\Omega$. Moreover, we prove the sufficient and necessary condition for existence of the saddle points of $u$. In Section 4, firstly, we show the uniqueness of the interior minimal points of $u$ by continuity argument. Then, by the sufficient and necessary condition for existence of the saddle points and the non-degeneracy of interior critical points in Section 3, we prove the uniqueness of the interior critical points of $u$. In Section 5, our main idea is to study the projection of higher dimensional spaces onto two dimensional plane. So we need to consider the domains $\Omega$ of revolution formed by taking a strictly convex planar domain about one axis. We deduce that $u$ has exactly one critical point $p$ in a bounded smooth strictly convex domain of $\mathbb{R}^{n}(n\geq3)$ and $p$ is a non-degenerate interior minimal point of $u$.
\section{Local Chen and Huang's comparison technique}
~~~~In order to obtain the non-degeneracy and uniqueness of the critical points of $u$ in planar domains. In this section, we will recall the key local Chen \& Huang's comparison technique in \cite{ChenHuang}. For the sake of completeness, in our setting, we will give a complete proof of Lemma 1 in \cite{ChenHuang}.
\begin{Lemma}\label{le2.1}
Suppose that $u, v$ satisfy the same constant mean curvature equation (\ref{1.1}). Without loss of generality, we suppose that $u,v$ have a second order contact at $Z_0=({x_{1}}_0,{x_{2}}_0,u({x_{1}}_0,{x_{2}}_0))$ with $({x_{1}}_0,{x_{2}}_0)=(0,0).$ Then by changing coordinate $(x_1,x_2)$ into $(\xi, \eta)$ linearly, the difference $u-v$ around $(\xi, \eta)=(0,0)$ is given by
\begin{equation}\label{2.1}\begin{array}{l}
u-v={\rm Re}(\rho\cdot(\xi+\eta i)^k)+o((\xi^2+\eta^2)^{\frac{k}{2}}),
\end{array}\end{equation}
where $k\geq 3,$ $\rho$ is a complex number and $\xi+\eta i$ is the complex coordinate.
\end{Lemma}
\begin{proof}[Proof] Since $u, v$ satisfy the same constant mean curvature equation. Then we have
\begin{equation}\label{2.2}
\begin{split}
0&=(1+u_{x_1}^2+u_{x_2}^2)(u_{x_1x_1}+u_{x_2x_2})-(u_{x_1}^2u_{x_1x_1}+u_{x_2}^2u_{x_2x_2}+2u_{x_1}u_{x_2}u_{x_1x_2})-H(1+|\nabla u|^2)^{\frac{3}{2}}\\
&=(1+u_{x_2}^2)u_{x_1x_1}+(1+u_{x_1}^2)u_{x_2x_2}-2u_{x_1}u_{x_2}u_{x_1x_2}-H(1+|\nabla u|^2)^{\frac{3}{2}},
\end{split}
\end{equation}
and
\begin{equation}\label{2.3}
\begin{array}{l}
0=(1+v_{x_2}^2)v_{x_1x_1}+(1+v_{x_1}^2)v_{x_2x_2}-2v_{x_1}v_{x_2}v_{x_1x_2}-H(1+|\nabla v|^2)^{\frac{3}{2}}.
\end{array}
\end{equation}
Now we define $p(t),q(t),m(t),r(t),s(t)$ for $0\leq t\leq 1$ by
\begin{equation*}
\begin{array}{l}
p(t)=(1-t)v_{x_1x_1}+tu_{x_1x_1},~~q(t)=(1-t)v_{x_1x_2}+tu_{x_1x_2},~~m(t)=(1-t)v_{x_2x_2}+tu_{x_2x_2},\\
r(t)=(1-t)v_{x_1}+tu_{x_1},~~s(t)=(1-t)v_{x_2}+tu_{x_2},
\end{array}
\end{equation*}
and consider the following function
\begin{equation*}
\begin{array}{l}
W=W(p(t),q(t),m(t),r(t),s(t))=(1+s^2)p+(1+r^2)m-2rsq-H(1+r^2+s^2)^{\frac{3}{2}}.
\end{array}
\end{equation*}
Let $w=u-v,$ therefore by (\ref{2.2}) minus (\ref{2.3}) we have
\begin{equation*}
\begin{split}
0&=W(p(1),q(1),m(1),r(1),s(1))-W(p(0),q(0),m(0),r(0),s(0))=\int_0^1 W_t dt\\
&=a_{11}w_{x_1x_1}+2a_{12}w_{x_1x_2}+a_{22}w_{x_2x_2}+b_{1}w_{x_1}+b_{2}w_{x_2},
\end{split}
\end{equation*}
where
\begin{equation*}
\begin{array}{l}
a_{11}=\int_0^1(1+s^2)dt,~~a_{12}=-\int_0^1rsdt,~~a_{22}=\int_0^1(1+r^2)dt,\\
b_1=\int_0^1[2(rm-sq)-3H\sqrt{1+r^2+s^2}r]dt,\\
b_2=\int_0^1[2(sp-rq)-3H\sqrt{1+r^2+s^2}s]dt.\\
\end{array}
\end{equation*}
Then $w$ satisfies the following equation
\begin{equation*}
\begin{array}{l}
Lw:=a_{11}w_{x_1x_1}+2a_{12}w_{x_1x_2}+a_{22}w_{x_2x_2}+b_{1}w_{x_1}+b_{2}w_{x_2}=0,
\end{array}
\end{equation*}
where $a_{12}^2-a_{11}a_{22}<0$ ensures the ellipticity of the equation $Lw=0.$ Next, the rest of proof is same to that in \cite{ChenHuang}. We transform $(x_1,x_2)$ into $(\xi,\eta)$
such that $\xi(0,0)=0, \eta(0,0)=0$ and at $(0,0)$
\begin{equation}\label{2.4}
\begin{array}{l}
Lw=(\frac{\partial^2}{\partial \xi^2}+\frac{\partial^2}{\partial \eta^2}+b_1'\frac{\partial}{\partial \xi}+b_2'\frac{\partial}{\partial \eta})w=0.
\end{array}
\end{equation}
Since the coefficients of $Lw$ and $w$ itself are analytic in $(x_1,x_2)$ as well as in $(\xi,\eta),$ then we get the following Taylor expansion around $(\xi,\eta)=(0,0)$ of $Lw$:
\begin{eqnarray}\label{2.5}
Lw&=&\Big\{(1+\alpha_{11}\xi+\beta_{11}\eta+O(\xi^2+\eta^2))\frac{\partial^2}{\partial \xi^2}+(1+\alpha_{22}\xi+\beta_{22}\eta+O(\xi^2+\eta^2))\frac{\partial^2}{\partial \eta^2}\nonumber\\
&&+2(\alpha_{12}\xi+\beta_{12}\eta+O(\xi^2+\eta^2))\frac{\partial^2}{\partial \xi\partial\eta}+(\tau_1+\delta_1\xi+\lambda_1\eta+O(\xi^2+\eta^2))\frac{\partial}{\partial \xi}\\
&&+(\tau_2+\delta_2\xi+\lambda_2\eta+O(\xi^2+\eta^2))\frac{\partial}{\partial \eta}\Big\}w.\nonumber
\end{eqnarray}
By Theorem I in \cite{Bers}, we have
\begin{equation}\label{2.6}
\begin{array}{l}
w(\xi,\eta)=\sum_{j=0}^{\infty}P_{k+j}(\xi,\eta),
\end{array}
\end{equation}
where $P_k(\xi,\eta)$ is a non-zero homogeneous polynomial in $(\xi,\eta)$ of degree $k$. By the assumption of $u$ and $v$ have a second order contact at $(0,0),$ we have $k\geq 3.$ By (\ref{2.5}) and (\ref{2.6}), the equation (\ref{2.4}) yields
\begin{equation*}
\begin{array}{l}
0=(\frac{\partial^2}{\partial \xi^2}+\frac{\partial^2}{\partial \eta^2})P_k+\{\mbox{terms of order}\geq k-1\}.
\end{array}
\end{equation*}
By the uniqueness of the power expansion, we show that $P_k$ is a harmonic homogeneous polynomial. Then
\begin{equation}\label{2.7}
\begin{array}{l}
P_k(\xi,\eta)={\rm Re}(\rho\cdot(\xi+\eta i)^k),
\end{array}
\end{equation}
where $\rho$ is a complex number, then (\ref{2.6}) and (\ref{2.7}) imply (\ref{2.1}).
\end{proof}
\begin{Lemma}\label{le2.2
(see\cite[Lemma 2]{ChenHuang}) Suppose that $u=u(x_1,x_2)$ is a non-constant solution of the following homogeneous quasilinear elliptic equation
\[Lu=a_{11}u_{x_1x_1}+2a_{12}u_{x_1x_2}+a_{22}u_{x_2x_2}+b_{1}u_{x_1}+b_{2}u_{x_2}=0~~\mbox{in}~~\Omega,\]
where the coefficients ${a_{ij}} ~\mbox{and}~{b_i}~(i,j=1,2)$ are analytic. Then every interior critical point of $u$ is an isolated critical point.
\end{Lemma}
\begin{Remark}\label{re2.3
By the above two lemmas and the implicit function theorem, we can know that the nodal set $N\cap\Omega$ of $(u-v)$ consists of at least three smooth arcs intersecting at $(0,0)$ and dividing $\Omega$ into at least six sectors. Moreover, the nodal set $N$ of $(u-v)$ is globally a union of smooth arcs.
\end{Remark}
\section{The sufficient and necessary condition for existence of the saddle points}
~~~~In this section, firstly, we investigate the non-degeneracy of critical points in a planar bounded smooth convex domain $\Omega$ by using the local Chen \& Huang's comparison technique. Then we prove the sufficient and necessary condition for existence of the saddle points by using the geometric properties of approximate surfaces at the non-degenerate critical points.
\begin{Lemma}\label{le3.1}
Suppose that $u$ is a solution to (\ref{1.3}). Then $u<0$ in $\overline{\Omega}$ and $\frac{\partial u}{\partial \overrightarrow{n}}>0$ on $\partial \Omega$.
\end{Lemma}
\begin{proof}[Proof] According to the assumption of
\[\left\{ \begin{array}{l}
\mbox{div}(\frac{\nabla u}{\sqrt{1+|\nabla u|^2}})=\sum\limits_{i,j=1}^{n}
a_{ij}(\nabla u)\frac{\partial^2 u}{\partial x_i \partial x_j}=H >0~~ \mbox{in}~~\Omega, \\
\frac{\partial u}{\partial \overrightarrow{n}}+\alpha u=0 ~~~\mbox{on}~ \partial\Omega,
\end{array} \right.\]
where $a_{ij}=\frac{1}{\sqrt{1+|\nabla u|^2}}(\delta_{ij}-\frac{u_{x_i}u_{x_j}}{1+|\nabla u|^2})$. By the strong maximum principle, $u$ obtains its maximum on $\partial\Omega.$ In fact, by the positive definiteness of the matrix $A=(a_{ij})$, if $u$ obtains the maximum at $x_0\in \Omega,$ then $B=(D_{ij}u(x_0))$ is seminegative definite. Hence the matrix $AB$ is seminegative definite with a nonpositive trace, it implies that $\sum_{i,j=1}^{n}a_{ij}(\nabla u)\frac{\partial^2 u}{\partial x_i \partial x_j}\leq 0,$ which is a contradiction. Thus there exists a point $x_0\in \partial\Omega$ such that $u(x_0)=\max_{\overline{\Omega}}u.$ Suppose that $u(x_0)\geq 0.$ By Hopf lemma we have $\frac{\partial u(x_0)}{\partial \overrightarrow{n}}>0.$ This contradicts with the fact that $\frac{\partial u(x_0)}{\partial \overrightarrow{n}}+\alpha u(x_0)=0,$ thus $u<0~\mbox{in}~\overline{\Omega}.$ Therefore $\frac{\partial u}{\partial \overrightarrow{n}}=-\alpha u>0~\mbox{on}~\partial\Omega.$
\end{proof}
\begin{Lemma}\label{le3.2}
Let $u$ be a solution to (\ref{1.2}), or (\ref{1.3}). Then $u$ has at least one critical point in $\Omega.$
\end{Lemma}
\begin{proof}[Proof] Since $\frac{\partial u}{\partial \overrightarrow{n}}>0~\mbox{on}~\partial\Omega,$ the Hopf lemma implies that $u$ cannot have minimal points on $\partial\Omega$. On the other hand, since $u$ is an analytic non-constant function, therefore $u$ must obtain its minimum at some $p\in\Omega$ with $\nabla u(p)=0.$ Then $u$ has at least one point $p$ with $\nabla u(p)=0.$
\end{proof}
\begin{Lemma}\label{le3.3}
Let $u$ be a solution to (\ref{1.2}), or (\ref{1.3}). Then $u$ is a Morse function, i.e., the Gaussian curvature $K(p):=\det(D^2u(p))\neq 0$ for any critical point $p.$
\end{Lemma}
Morse and semi-Morse function are described in \cite{Bott,Cavicchioli}. In order to prove Lemma \ref{le3.3}, we need the following lemma.
\begin{Lemma}\label{le3.4}
For constant $H$ is from (\ref{1.1}) and any constant $h$, there exists a number $T$ $(0<T<\infty)$ such that the following initial value problem
\begin{equation}\label{3.1}\begin{array}{l}
\left\{ \begin{array}{l}
X''(t)=H(1+|X'(t)|^2)^{\frac{3}{2}}, ~~-T<t<T,\\
X(0)=h, \\
X'(0)=0,
\end{array} \right.
\end{array}\end{equation}
has a unique $C^{\infty}$-solution $X(t)$, which satisfies the following
\begin{equation}\label{3.2}
\begin{array}{l}
X(t)=X(-t),~~-T<t<T,
\end{array}
\end{equation}
\begin{equation}\label{3.3}
\begin{array}{l}
X(t)\geq h,~~-T<t<T,
\end{array}
\end{equation}
\begin{equation}\label{3.4}
\begin{array}{l}
X'(t)\geq 0,~~0\leq t<T.
\end{array}
\end{equation}
\end{Lemma}
\begin{proof}[Proof] Since the solution of problem (\ref{3.1}) is $X(t)=h+\frac{1}{H}(1-\sqrt{1-H^2t^2})$ for $|t|<T=\frac{1}{H}.$ So the results naturally hold.
\end{proof}
\begin{proof}[Proof of Lemma \ref{le3.3}] We set up the usual contradiction argument. Suppose that $p\in \Omega$ is a point such that $\nabla u(p)=0$ and the Gaussian curvature $K(p)=0.$ Without loss of generality, by using a suitable parallel translation and a rotation of coordinates, we may suppose that
\begin{equation}\label{3.5}
\begin{array}{l}
p=0~~\mbox{and}~~[D_{ij}u(0)]=\mbox{diag}[H,0].
\end{array}
\end{equation}
By Lemma \ref{le3.4} for $h=u(0),$ we get a unique solution to (\ref{3.1}), denote by $v.$ Let $v(x)~(=v(x_1,x_2))=X(x_1),$ thus $v$ satisfies
\begin{equation}\label{3.6}
\left\{
\begin{array}{l}
\mbox{div}(\frac{\nabla v}{\sqrt{1+|\nabla v|^2}})=H, ~~\mbox{in}~(-T,T)\times \mathbb{R},\\
\left[D_{ij}v(0)\right]= \mbox{diag}[H,0],\\
v(0)=u(0)=h~~\mbox{and}~~\nabla v(0)=\nabla u(0)=0.
\end{array}
\right.
\end{equation}
By (\ref{3.6}), we know that $(u-v)$ vanishes up to second order derivatives at 0. Moreover, we can know that $(u-v)$ is not identically zero. In fact, $v=v(x)=X(x_1).$ On the other hand, Lemma \ref{le3.1} shows that $\frac{\partial u}{\partial \overrightarrow{n}}>0,$ we can suppose that unit outward normal vector $\overrightarrow{n}=(0,1)$, then $\frac{\partial u}{\partial \overrightarrow{n}}=\nabla u\cdot \overrightarrow{n}=u_{x_2}>0.$ So $(u-v)$ is not identically zero. The unique continuation theorem of solutions for elliptic equations shows that $(u-v)$ never vanishes up to infinite order at 0. Using Lemma \ref{le2.1}, we get
\begin{equation}\label{3.7}
\begin{array}{l}
(u-v)(x)=P_k(x)+o(|x|^k)~~\mbox{as}~|x|\rightarrow 0
\end{array}
\end{equation}
for some integer $k\geq 3,$ where $P_k(x)$ is a homogeneous polynomial of degree $k$ and $P_k(x)$ is not identically zero. In addition, Lemma \ref{le2.2} shows that every interior critical point of $(u-v)$ is isolated. Furthermore, Remark \ref{re2.3} shows that the nodal sets of $(u-v)$ consist of $k$ smooth arcs in some neighborhood $U$ of the origin, and that all smooth arcs intersect at $(0,0)$ and divide $U$ into $2k (k\geq 3)$ sectors.
Firstly, we investigate the case of Neumann boundary condition (\ref{1.2}). In order to prove the result, we should divide the proof into two cases, i.e., $T$ is large enough and not large enough respectively. Consider
\begin{equation}\label{3.8}
\begin{array}{l}
I_+=\Big\{x\in \Omega\cap \{(-T,T)\times\mathbb{R}\};u(x)-v(x)>0\Big\},
\end{array}
\end{equation}
and
\begin{equation}\label{3.9}
\begin{array}{l}
I_-=\Big\{x\in \Omega\cap \{(-T,T)\times\mathbb{R}\};u(x)-v(x)<0\Big\}.
\end{array}
\end{equation}
Therefore, it follows from Lemma \ref{le2.1} and Remark \ref{re2.3} that
\begin{equation}\label{3.10}
\begin{array}{l}
\mbox{Both}~I_+ ~\mbox{and}~ I_- ~\mbox{have\ at\ least\ three\ components\ and\ each}\\
\mbox{of\ them\ meets\ the\ boundary} ~\partial( \Omega\cap \{(-T,T)\times\mathbb{R}\}).
\end{array}
\end{equation}
Case 1: If $T$ is large enough, i.e., $\Omega\cap \{(-T,T)\times\mathbb{R}\}=\Omega$. Since $\Omega$ is convex, by Lemma \ref{le3.1} and Lemma \ref{le3.4} we have that $\frac{\partial u}{\partial \overrightarrow{n}}>0$ and $v'(x_1)\geq 0, v''(x_1)\geq 0 ~\mbox{for}~ 0\leq x_1<T$, then we know that $\frac{\partial (u-v)}{\partial \overrightarrow{n}}$ preserves sign in some fixed arc and $\frac{\partial (u-v)}{\partial \overrightarrow{n}}$ changes sign alternatively in two adjacent arcs. The sign distribution for directional derivative of $(u-v)$ on $\partial\Omega$ is shown in Fig. 1.
\begin{center}
\includegraphics[width=5cm,height=3.7cm]{Fig1.png}\\
\scriptsize {\bf Fig. 1.}~~The sign distribution for directional derivative of $(u-v)$ on $\partial\Omega$.
\end{center}
Now we put
\begin{equation}\label{3.11}
\begin{array}{l}
\gamma_+=\Big\{x\in \partial\Omega;\frac{\partial (u-v)}{\partial \overrightarrow{n}}(x)>0\Big\},
\end{array}
\end{equation}
and
\begin{equation}\label{3.12}
\begin{array}{l}
\gamma_{-}=\Big\{x\in \partial\Omega;\frac{\partial (u-v)}{\partial \overrightarrow{n}}(x)<0\Big\}.
\end{array}
\end{equation}
Therefore, it never occurs that a component of $(I_-\cap\Omega)$ meets $\partial\Omega$ exclusively in $\gamma_+.$ Suppose by contradiction that $\gamma_1$ is a component of ($I_-\cap\Omega$) which meets $\partial\Omega$ exclusively in $\gamma_+.$ By Lemma \ref{le2.1}, we see that $(u-v)$ satisfies (\ref{3.7}), then the strong maximum principle implies that a negative minimum of $(u-v)$ in $\overline{\gamma_1}$ is attained at $p\in\gamma_+$ and $\frac{\partial (u-v)}{\partial \overrightarrow{n}}(p)\leq0$. This contradicts with the definition of $\gamma_+.$ By the same way, we know that it never occurs that a component of $(I_+\cap\Omega)$ meets $\partial\Omega$ exclusively in $\gamma_-.$ But these facts contradict with (\ref{3.10})
Case 2: If $T$ is not large enough, i.e., $\Omega\backslash \{\Omega\cap \{(-T,T)\times\mathbb{R}\}\neq \varnothing$. Choose a number $\widetilde{T}$ such that $\widetilde{T}<T,$ which is sufficiently near to $T.$ Set $\widetilde{\Omega}=(-\widetilde{T},\widetilde{T})\times\mathbb{R}.$ We only should replace $\Omega$ by $\Omega\cap\widetilde{\Omega}$ and we can use the same method of case 1.
Secondly, we consider the case of Robin boundary condition (\ref{1.3}). We can use the same method in the situation of Neumann boundary condition. Indeed, we select $\widetilde{T}$ with $\{v(x)=0\}=\{x_1=\pm \widetilde{T}\}$. Thus $\partial(\Omega\cap\widetilde{\Omega})$ consists of
$$\mbox{at\ most\ two\ components\ of}~ \Big\{\frac{\partial (u-v)}{\partial \overrightarrow{n}}+\alpha (u-v)>0\Big\}$$
and
$$\mbox{at\ most\ two\ components\ of}~ \Big\{\frac{\partial (u-v)}{\partial \overrightarrow{n}}+\alpha (u-v)<0\Big\}.$$
\end{proof}
\begin{Lemma}\label{le3.5}
Let $u$ be a solution to (\ref{1.2}), or (\ref{1.3}). Then $u$ does not have maximum points in $\Omega.$
\end{Lemma}
\begin{proof}[Proof] Since $\mbox {div}(\frac{\nabla u}{\sqrt{1+|\nabla u|^2}})=H>0$, the strong maximum principle implies that $u$ cannot obtain maximum in $\Omega.$ In fact, let operator $L$ be the mean curvature operator. Suppose that $u$ has maximum points in $\Omega$ and that $x_0$ is a maximum point, then $D_1u(x_0)=D_2u(x_0)=0, D_{11}u(x_0)\leq 0$, $D_{22}u(x_0)\leq 0$ and $Lu(x_0)=D_{11}u(x_0) + D_{22}u(x_0)\leq 0$. However, this is contradict with $Lu>0$ in $\Omega.$ This completes the proof of Lemma \ref{le3.5}.
\end{proof}
Next, we show the sufficient and necessary condition for existence of the saddle points.
\begin{Lemma}\label{le3.6}
Let $u$ be a solution to (\ref{1.2}), or (\ref{1.3}). Then $u$ has at least two minimal points in $\Omega$, if and only if there exists a point $p$ such that $\nabla u(p)=0$ and $K(p)<0.$
\end{Lemma}
\begin{proof}[Proof] (i) Firstly, we prove the section of ``if". Suppose that $p$ is a point such that $\nabla u(p)=0~\mbox{and}~K(p)<0.$ Therefore there exists an open neighborhood $U$ of $p$ in which the nodal sets of $(u-u(p))$ consist of at least two smooth arcs intersecting at $p$ and divides $U$ into at least four sectors.
Next, we consider the following super-level set
\begin{equation*}
\begin{array}{l}
U_+:=\{x\in \Omega|u(x)>u(p)\}.
\end{array}
\end{equation*}
Lemma \ref{le3.5} implies that each component of $U_+$ has to meet the $\partial\Omega.$ Therefore we know that the following sub-level set
\begin{equation*}
\begin{array}{l}
U_-:=\{x\in \Omega|u(x)<u(p)\}
\end{array}
\end{equation*}
has at least two components. By $\frac{\partial u}{\partial \overrightarrow{n}}>0,$ then $u$ has at least two minimal points in $\Omega$.
(ii) Secondly, we prove the section of ``only if". Since $\frac{\partial u}{\partial \overrightarrow{n}}>0~\mbox{on}~\partial\Omega$ and $\Omega$ is convex, so we can extend the solution $u$ to $\mathbb{R}^2$, denoting by
\begin{equation}\label{3.13}\begin{array}{l}
u(x)=u(y)+\mbox{dist}(x,y)\cdot\frac{\partial u}{\partial \overrightarrow{n}}(y) ,
\end{array}\end{equation}
where $y$ is the unique point on $\partial\Omega$ such that $\mbox{dist}(x,y)=\mbox{dist}(x,\Omega).$ Hence we know that $u\in C^1(\mathbb{R}^2)~\mbox{and}~\nabla u\neq 0~\mbox{in}~\mathbb{R}^2\setminus\Omega.$ Next, we consider the sub-level set $N_z=\{x\in \mathbb{R}^2|u(x)<z\}.$ Therefore we have that
\begin{equation}\label{3.14}\begin{array}{l}
\partial N_z~\mbox{has\ only\ one\ curve\ for\ sufficiently\ large}~z.
\end{array}\end{equation}
Now, we suppose by contradiction that $u$ has at least two minimal points and there does not exist point $p$ such that $\nabla u(p)=0~\mbox{and}~K(p)<0.$
By Lemma \ref{le3.3}, Lemma \ref{le3.5} and (\ref{3.13}), we know that each critical point of $u$ is a minimal point and $\nabla u\neq 0~\mbox{in}~\mathbb{R}^2\backslash \Omega$ respectively. On the other hand, Lemma \ref{le2.2} shows that every critical point is isolated and the number of critical points is finite. Then we suppose that there exists a finite sequence of minimal points of $u$, denotes by $\{p_1,p_2,\ldots,p_k\}$ such that
\begin{equation}\label{3.15}\begin{array}{l}
\nabla u(x)\neq 0~\mbox{for\ all}~x\in \mathbb{R}^2\backslash\{p_1,p_2,\ldots,p_k\}.
\end{array}\end{equation}
Let $z_0=\max\{u(p_i)|1\leq i\leq k\}.$ Therefore we know that the boundary $\partial N_z$ of the sub-level set $N_z$ is $C^1$ curve for $z>z_0$ and $\{\partial N_z\}$ is diffeomorphic to each other. According to the assumption, since $K(p_i)>0,$ then the approximate surface is an elliptic paraboloid in a neighborhood of critical point $p_i$ (If $K(p)<0,$ then the approximate surface is a hyperbolic paraboloid in a neighborhood of critical point $p$). The elliptic paraboloid and hyperbolic paraboloid as shown in Fig. 2 and Fig. 3, respectively.
\begin{center}
\includegraphics[width=6.6cm,height=4.5cm]{Fig2.png}\includegraphics[width=6.6cm,height=4.5cm]{Fig3.png}\\
\scriptsize {\bf Fig. 2.}~~The elliptic paraboloid. ~~~~~~~~~~~~~~~~~~~~~~~~~~~~{\bf Fig. 3.}~~The hyperbolic paraboloid.
\end{center}
If $z$ is near to $z_0,$ then $\{\partial N_z\}$ has at least two curves. This contradicts with the fact (\ref{3.14}). This completes the proof of Lemma \ref{le3.6} .
\end{proof}
\section{The proof of Theorem \ref{th1.1} and Theorem \ref{th1.2}}
~~~~In this section, firstly, we show the uniqueness of the interior minimal points of $u$ in $\Omega$ by continuity argument. Then, by the sufficient and necessary condition for existence of the saddle points and the non-degeneracy of interior critical points in Section 3, we prove the uniqueness of the critical points.
For $t$ ($t\in [0,1]$), we will prove the uniqueness of the interior minimal points of the solutions $v_t$ to the following problems:
\begin{equation}\label{4.1}\begin{array}{l}
\left\{ \begin{array}{l}
\mbox{div}(\frac{\nabla v }{\sqrt{1 +t^2|\nabla v|^2}}) = H, ~\mbox{in}~\Omega,\\
\frac{\partial v}{\partial \overrightarrow n} = c,~\mbox{on}~\partial\Omega,
\end{array} \right.~~\mbox{or}~~
\left\{ \begin{array}{l}
\mbox{div}(\frac{\nabla v}{\sqrt {1+t^2|\nabla v|^2}}) = H,~\mbox{in}~\Omega, \\
\frac{\partial v}{\partial \overrightarrow n} + \alpha v = 0,~\mbox{on}~\partial\Omega.
\end{array} \right.
\end{array}\end{equation}
Let $u_t=tv_t$ for $t>0,$ then $u_t$ respectively satisfy
\begin{equation}\label{4.2}\begin{array}{l}
\left\{ \begin{array}{l}
\mbox{div}(\frac{\nabla u}{\sqrt{1+|\nabla u|^2}}) =tH, ~\mbox{in}~\Omega,\\
\frac{\partial u}{\partial \overrightarrow n}=c,~\mbox{on}~\partial\Omega,
\end{array} \right.~~\mbox{or}~~
\left\{ \begin{array}{l}
\mbox{div}(\frac{\nabla u}{\sqrt {1+|\nabla u|^2}})=tH,~\mbox{in}~\Omega, \\
\frac{\partial u}{\partial \overrightarrow n}+\alpha u=0,~\mbox{on}~\partial\Omega.
\end{array} \right.
\end{array}\end{equation}
By the analysis of Lemma \ref{le3.2}, we know that the solution $v_t$ of (\ref{4.1}) has at least one minimal point in $\Omega.$ According to Lemma \ref{le3.3} and Lemma \ref{le3.6}, we can easily get the following lemmas for $v_t$.
\begin{Lemma}\label{le4.1}
For any $t\in(0,1].$ Let $v_t$ be a solution to (\ref{4.1}). Then $v_t$ is a Morse function, i.e., the Gaussian curvature $K_t(p):=\det(D^2v_t(p))\neq 0~\mbox{for\ any\ critical\ point}~p.$
\end{Lemma}
\begin{Lemma}\label{le4.2}
For any $t\in[0,1].$ Then $v_t$ has at least two minimal points, if and only if there exists a point $p$ such that $\nabla v_t(p)=0$ and $K_t(p)<0.$
\end{Lemma}
Next, we will prove that the Gaussian curvature of $v_t$ is positive at any critical point for the case of $t=0.$
\begin{Lemma}\label{le4.3}
Let $v_t$ be the solution of (\ref{4.1}). Suppose that $\nabla v_0(p)=0$ for some point $p\in \Omega.$ Then the Gaussian curvature $K_0(p)>0$ for any critical point $p$.
\end{Lemma}
\begin{proof}[Proof]
We set up the usual contradiction argument. Suppose that there exists interior critical point $p$ such that $K_0(p)\leq0$. Without loss of generality, by using a suitable parallel translation and a rotation of coordinates, we may suppose that
\[p=(0,0), [D_{ij}v_0(p)]=diag[\lambda_1,\lambda_2],~\mbox{where}~\lambda_1+\lambda_2=H>0,\lambda_1>0,\lambda_2\leq0.\]
By Lemma \ref{le2.1}, then the difference $v_0-q$ around $(x_1,x_2)=(0,0)$ is given by
$$v_0(x_1,x_2)-q(x_1,x_2)=P_k(x_1,x_2)+o((x_1^2+x_2^2)^{\frac{k}{2}}),$$
where $P_k(x_1,x_2)$ is a homogeneous harmonic polynomial of degree $k$ in $\Omega$, and $$q(x_1,x_2)=v_0(0,0)+\frac{1}{2}\lambda_1x_1^2+\frac{1}{2}\lambda_2x_2^2.$$
Firstly, we study the case of Neumann boundary condition in (\ref{4.1}). Next, we consider
\begin{equation*}
\begin{array}{l}
\widehat{I}_+=\big\{x\in \Omega;v_0(x)-q(x)>0\big\},
\end{array}
\end{equation*}
and
\begin{equation*}
\begin{array}{l}
\widehat{I}_-=\big\{x\in \Omega;v_0(x)-q(x)<0\big\}.
\end{array}
\end{equation*}
Since $v_0(x_1,x_2)-q(x_1,x_2)$ vanishes up to second order derivatives at $(0,0)$ and $P_k(x_1,x_2)$ is real analytic. Then it follows from Lemma \ref{le2.1} and Remark \ref{re2.3} that $k\geq 3$ and
\begin{equation}\label{4.3}
\begin{array}{l}
\mbox{Both}~\widehat{I}_+ ~\mbox{and}~ \widehat{I}_- ~\mbox{have\ at\ least\ three\ components}\\
\mbox{ and\ each\ of\ them\ meets\ the\ boundary} ~\partial\Omega.
\end{array}
\end{equation}
Now we set
\begin{equation}\label{4.4}
\begin{array}{l}
\widehat{\gamma}_+=\Big\{x\in \partial\Omega;\frac{\partial (v_0-q)}{\partial \overrightarrow{n}}(x)>0\Big\},
\end{array}
\end{equation}
and
\begin{equation}\label{4.5}
\begin{array}{l}
\widehat{\gamma}_{-}=\Big\{x\in \partial\Omega;\frac{\partial (v_0-q)}{\partial \overrightarrow{n}}(x)<0\Big\}.
\end{array}
\end{equation}
Since $\Omega$ is convex and $q(x_1,x_2)=v_0(0,0)+\frac{1}{2}\lambda_1x_1^2+\frac{1}{2}\lambda_2x_2^2$ with $\lambda_1+\lambda_2=H>0,\lambda_1>0,\lambda_2\leq0,$ then we know that $\widehat{\gamma}_+$ and $\widehat{\gamma}_{-}$ has at most two components on $\partial\Omega.$ The rest of the proof is same as the proof of Lemma \ref{le3.3}. This contradicts with (\ref{4.3}).
Secondly, we consider the case of Robin boundary condition in (\ref{4.1}). We can use the same method in the case of Neumann boundary condition. This completes the proof.
\end{proof}
Next we will show the uniqueness of the interior minimal points of $u$ in $\Omega$ by using the continuity argument.
\begin{Lemma}\label{le4.4}
For any $t\in[0,1].$ Then $v_t$ has exactly one minimal point in $\Omega.$
\end{Lemma}
\begin{proof}[Proof] We set $M=[0,1]$ and divide $M$ into two sets $M_1$ and $M_2$ as follows:
\begin{equation}\label{4.6}
\begin{array}{l}
M_1=\{t\in M; v_t~\mbox{has\ only\ one\ minimal\ point\ in}~\Omega\},
\end{array}
\end{equation}
and
\begin{equation}\label{4.7}
\begin{array}{l}
M_2=\{t\in M; v_t~\mbox{has\ more\ than\ two\ minimal\ points\ in}~\Omega\}.
\end{array}
\end{equation}
Then $M=M_1+M_2$ and $M_1\cap M_2=\varnothing.$ Lemma \ref{le4.2} and Lemma \ref{le4.3} imply that $0\in M_1$, i.e., $M_1\neq \varnothing.$
Now we show that $M_2$ is open in $M.$ That is, for any $t_\star\in M_2,$ there exists a constant $\varepsilon>0$ with $(t_\star-\varepsilon,t_\star+\varepsilon)\subset M_2.$ In fact, the follows from Lemma \ref{le4.1} and Lemma \ref{le4.2} and inverse function theorem that $v_t$ has as many critical points as $v_{t_\star}$ when $t$ is near $t_\star.$ Suppose by contradiction that there exists a sequence $\{t_k\}\in M_1$ such that $\{v_{t_k}\}$ has only one minimal point and $t_k\in (t_\star-\frac{1}{k},t_\star+\frac{1}{k})$ for some positive $t_\star\in M_2$. Then it follows from Lemma \ref{le4.1} and Lemma \ref{le4.2} that $v_{t_k}$ does not has the saddle points, i.e., $v_{t_k}$ has exactly one critical point. By Lemma \ref{le4.1} and continuity, we may take a subsequence $\{v_{t_{k_j}}\}$ of $\{v_{t_k}\}$ such that
\begin{equation}\label{4.8}
\begin{array}{l}
p_{k_j}\rightarrow p,~~\nabla v_{t_{k_j}}(p_{k_j})=0,~~K_{t_{k_j}}(p_{k_j})>0,~~\nabla v_{t_\star}(p)=0,~~K_{t_\star}(p)>0.
\end{array}
\end{equation}
Since $t_\star\in M_2,$ then there exists another point $q\in U(p)\subset \Omega$ and a sequence of point $\{q_{k_j}\}$ such that
\[q_{k_j}\rightarrow q,~~\nabla v_{t_{k_j}}(q_{k_j})\rightarrow \nabla v_{t_\star}(q)=0.\]
According to the uniqueness of the critical point of $v_{t_k},$ we can take a subsequence $\{v_{t_{k_j}}\}$ of $\{v_{t_k}\}$ such that $v_{t_{k_j}}$ are all monotone in the line $\gamma(p_{k_j},q_{k_j})$. Therefore there exists a sequence of points $\{z_{k_j};z_{k_j}\in \gamma(p_{k_j},q_{k_j})\}$ which satisfy
\begin{equation}\label{4.9}
\begin{array}{l}
|\nabla v_{t_{k_j}}(z_{k_j})|\leq |\nabla v_{t_{k_j}}(q_{k_j})|\rightarrow 0,~~|K_{t_{k_j}}(z_{k_j})|=\frac{|\nabla v_{t_{k_j}}(q_{k_j})|}{|p_{k_j}-q_{k_j}|}\rightarrow 0.
\end{array}
\end{equation}
By (\ref{4.9}) and continuity, then there should be a point $z\in \gamma (p,q)$ such that
\[\nabla v_{t_\star}(z)=0,~~K_{t_\star}(z)=0,\]
this contradicts with Lemma \ref{le4.1}, then we complete the proof which $M_2$ is open set in $M.$
On the other hand, we show that $M_2$ is closed in $M.$ In fact, let $\{t_i\}$ be a sequence of points in $M_2$ such that $t_i\rightarrow t_0~\mbox{as}~i\rightarrow \infty.$ Then Lemma \ref{le4.2} and the continuity argument imply that there exists a subsequence $\{t_j\}$ of $\{t_i\}$, a sequence $\{p_j\}$ and a point $p\in\overline{\Omega}$ such that
\begin{equation}\label{4.10}
\begin{array}{l}
p_j\rightarrow p~\mbox{as}~j\rightarrow\infty,~~\nabla v_{t_j}(p_j)=0,~\mbox{and}~K_{t_j}(p_j)<0.
\end{array}
\end{equation}
By (\ref{4.10}) and continuity, we have
\begin{equation}\label{4.11}
\begin{array}{l}
\nabla v_{t_0}(p)=0,~\mbox{and}~K_{t_0}(p)\leq 0.
\end{array}
\end{equation}
Since $\nabla v_{t_0}\neq 0~\mbox{on}~\partial\Omega,$ then we have $p\in \Omega.$ Hence it follows from Lemma \ref{le4.1}, Lemma \ref{le4.2}, Lemma \ref{le4.3} and (\ref{4.11}) that $t_0\in M_2.$ This shows that $M_2$ is closed in $M.$ Then $M_2$ must be $M$ or $\varnothing.$ Since $M_1\neq\varnothing,$ so $M_2=\varnothing~\mbox{and}~M_1=M.$ This completes the proof.
\end{proof}
\begin{proof}[Proof of Theorem \ref{th1.1} and Theorem \ref{th1.2}] By Lemma \ref{le3.3} and Lemma \ref{le3.5}, we know that the Gaussian curvature $K(p)\neq0$ for any critical point $p$ and solution $u$ does not have maximum points in $\Omega$. In addition, Lemma \ref{le3.6} shows that
\begin{equation}\label{4.12}\begin{array}{l}
\mbox{if}~\exists~ p\in \Omega~ \mbox{such\ that} ~\nabla u(p)=0~ \mbox{and}~ K(p)<0\Leftrightarrow~\sharp\{\mbox{minimal\ points\ of}~u\}\geq 2.
\end{array}\end{equation}
On the other hand, by Lemma \ref{le4.4}, we know that $u$ has exactly one minimal point in $\Omega.$ Therefore, $u$ does not have saddle points in $\Omega,$ this implies that $u$ has exactly one critical point $p$ in $\Omega$ and $p$ is a non-degenerate interior minimal point of $u$.
\end{proof}
\section{The proof of Theorem \ref{th1.3}}
~~~~~In this section, we investigate the geometric structure of critical point set $K$ of solutions to prescribed constant mean curvature equation with Neumann and Robin boundary conditions respectively in higher dimensional spaces.
\begin{proof}[Proof Theorem \ref{th1.3}] We divide the proof into three steps.
Step 1, we turn the mean curvature equation (\ref{1.1}) for $n$-dimension into the similar mean curvature equation for 2-dimension. Without loss generality, let $\Omega$ be a domain of revolution formed by taking a strictly convex planar domain in the $x_1,x_n$ plane with respect to the $x_n$ axis. In the sequel, $x=(x',x_n), x'=(x_1,\cdots,x_{n-1})$ and $r=\sqrt{x_1^2+\cdots+x_{n-1}^2}.$
By the assumptions, we have that the solution $u$ satisfies
\begin{equation}\label{5.1}\begin{array}{l}
u(x',x_n)=u(|x'|,x_n)\triangleq v(r,x_n)
\end{array}\end{equation}
and
\begin{equation}\label{5.2}\begin{array}{l}
\frac{\partial v}{\partial r}(r,x_n)>0$ $ $ for $ $ $ r\neq0.
\end{array}\end{equation}
From (\ref{5.2}), we can know that the critical points of $u$ lie on $x_n$ axis. Next, according to (\ref{5.1}), we have that
\begin{equation}\label{5.3}\begin{array}{l}
u_{x_n}(x',x_n)=v_{x_n}(r,x_n).
\end{array}\end{equation}
Moreover, we can deduce that $u_{x_n}$ satisfies the following equation
\[\sum\limits_{i,j=1}^{n}
a_{ij}(\nabla u)\frac{\partial^2 u_{x_n}}{\partial x_i \partial x_j}+\sum\limits_{i,j=1}^{n}
\frac{\partial a_{ij}(\nabla u) }{\partial x_n }\frac{\partial^2 u}{\partial x_i \partial x_j}=0~~~n\geq 3.\]
That is
\begin{equation}\label{5.4}\begin{array}{l}
\mathscr{L}u_{x_n}:=\sum\limits_{i,j=1}^{n}
a_{ij}(\nabla u)\frac{\partial^2 u_{x_n}}{\partial x_i \partial x_j}+\sum\limits_{i,j=1}^{n}\frac{\partial^2 u}{\partial x_i \partial x_j}
\frac{\partial a_{ij}(\nabla u)}{\partial x_n}=0,
\end{array}\end{equation}
where $a_{ij}(\nabla u)=\frac{1}{\sqrt{1+|\nabla u|^2}}(\delta_{ij}-\frac{u_{x_i}u_{x_j}}{1+|\nabla u|^2}),$ $\frac{\partial a_{ij}(\nabla u)}{\partial x_n }=\frac{1}{(1+|\nabla u|^2)^{3/2}}\big[(\frac{3u_{x_i}u_{x_j}}{1+|\nabla u|^2}-\delta_{ij})(\nabla u\cdot\nabla u_{x_n})-(u_{x_i}u_{x_nx_j}+u_{x_j}u_{x_nx_i})\big]$ is the first derivative term of $u_{x_n}$.
By the assumptions, the strict convexity of $\Omega$ and the Hopf lemma, we can know that $u_{x_n}$ vanishes precisely on the $(n-2)$ dimensional sphere given by
\[S=\{x_n=a\}\cap\partial\Omega,\]
for some $a\in \mathbb{R}.$ For convenience, we define the nodal set
\[\mathscr{N}=\{x\in \Omega|u_{x_n}(x)=0\}.\]
It is clear that all critical points of solution $u$ are contained in $\mathscr{N}$. Also from (\ref{5.3}), $\mathscr{N}$ is rotationally invariant about the $x_n$ axis.
Now we turn the mean curvature equation (\ref{1.1}) for $n$-dimension
\begin{equation*}
\begin{array}{l}
div(\frac{\nabla u}{\sqrt{1+|\nabla u|^2}})=H
\end{array}\end{equation*}
into the following similar mean curvature equation on 2-dimension
\[div(\frac{\nabla v}{\sqrt{1+|\nabla v|^2}})+\frac{1}{\sqrt{1+|\nabla v|^2}}\frac{n-2}{r}v_r=H,\]
that is
\begin{equation}\label{5.5}\begin{array}{l}
\sum\limits_{i,j=1}^{2}a_{ij}(\nabla v)v_{ij}+\frac{1}{\sqrt{1+|\nabla v|^2}}\frac{n-2}{r}v_r=H,
\end{array}\end{equation}
where $\nabla v=(\frac{\partial v}{\partial r},\frac{\partial v}{\partial x_n}),$ $a_{ij}=\frac{1}{\sqrt{1+|\nabla v|^2}}(\delta_{ij}-\frac{v_iv_j}{1+|\nabla v|^2})$ and $v_1=\frac{\partial v}{\partial r}, v_2=\frac{\partial v}{\partial x_n}.$
For any $\theta=(\theta_1,\theta_2)=(\cos \alpha,\sin \alpha)\in S^1,$ where $\alpha \in [0,\pi).$ We turn the quasilinear elliptic equation associated to $v$ into a linear elliptic equation associated to $w=v_{\theta}=\nabla v\cdot \theta.$ Firstly, we differentiate the equation (\ref{5.5}), then take inner product with $\theta.$ For convenience, we set $y=(y_1,y_2)=(x_n,r),$ hence we can get the following equation
\begin{equation}\label{5.6}\begin{array}{l}
L_{v}w+h_1(y)\frac{\partial w}{\partial y_1}+h_2(y)\frac{\partial w}{\partial y_2}+\frac{1}{(1+|\nabla v|^2)^{\frac{3}{2}}}\frac{n-2}{r}\Big[(1+v_{y_1}^2)\frac{\partial w}{\partial y_2}-v_{y_1}v_{y_2}\frac{\partial w}{\partial y_1}\Big] =\frac{1}{\sqrt{1+|\nabla v|^2}} \frac{n-2}{r^2}v_r \theta_2,
\end{array}\end{equation}
where
\[L_{v}w:=\sum\limits_{i,j=1}^{2}
a_{ij}(\nabla v)\frac{\partial^2 w}{\partial y_i \partial y_j}\]
and
\[h_{k}(y)=\sum\limits_{i, j=1}^{2} v_{y_i y_j}\frac{\partial a_{ij}}{\partial v_{y_k}},~~~k=1,2.\]
By (\ref{5.2}) and (\ref{5.6}), we deduce that
\begin{equation}\label{5.7}\begin{array}{l}
L_{v}w+h_1(y)\frac{\partial w}{\partial y_1}+h_2(y)\frac{\partial w}{\partial y_2}+\frac{1}{(1+|\nabla v|^2)^{\frac{3}{2}}}\frac{n-2}{r}\Big[(1+v_{y_1}^2)\frac{\partial w}{\partial y_2}-v_{y_1}v_{y_2}\frac{\partial w}{\partial y_1}\Big]\geq 0.
\end{array}\end{equation}
By (\ref{5.7}), so we can consider the result of projecting a graph onto $x_1, x_n$ plane (see Fig. 4).
\begin{center}
\includegraphics[width=8cm,height=3.5cm]{Fig4.png}\\
\scriptsize {\bf Fig. 4.}~~The graphic projection of higher dimensional space onto two dimensional plane.
\end{center}
Step 2, we show the uniqueness of critical points. This subsection is based on the results of Gidas, Ni and Nirenberg \cite{Gidas} and Caffarelli and Friedman \cite{Caffarelli}, the ideas of Payne \cite{Payne} and Sperb \cite{Sperb}. To prove that whenever critical set has exactly one point, since all critical points of $u$ are contained in $\mathscr{N}\cap\{x_2=\cdots=x_{n-1}=0\}$ and lie on the $x_n$ axis. The nodal set $\mathscr{N}=\{x\in \Omega | u_{x_n}(x)=0\}$ is rotationally invariant about the $x_n$ axis, formed by a set $N_2$ contained in the $x_1,x_n$ 2-dimensional plane rotation about the $x_n$ axis, by (\ref{5.5}), where $N_2$ can be seen as the projection of $\mathscr{N}$ in the $x_1,x_n$ 2-dimensional plane and $\mathscr{N}$ cannot enclose any subdomain of $\Omega$ (By Lemma 2.1 in \cite{Deng1}, $N_2$ cannot enclose any planar subdomain of $\Omega\cap\{x_2=\cdots=x_{n-1}=0\},$ where locally $N_2$ looks like the nodal set of some homogeneous polynomial in $x_1,x_n.$). Because $N_2$ is symmetric with respect to the $x_n$ axis and intersects the $x_n$ axis at exactly one point, hence we prove the uniqueness of critical points.
Step 3, we show the non-degeneracy of critical point. How to show that critical point $p$ is non-degenerate, we restatement that $u$ is rotationally symmetric with respect to $x_n$ axis and critical point $p$ lies on this axis. From (\ref{5.1}) and (\ref{5.2}), we have that $\{u_{x_k}=0\}=\{x_k=0\}\cap \Omega$ for all $1\leq k\leq n-1.$ Hence $u_{x_ix_j}(p)=0$ for any index $1\leq i<j\leq n,$ that is, $D^2u(p)$ is diagonal. By (\ref{5.2}), we can know that $u_{x_k}>0$ in domain $\mathscr{D}_k=\{x_k>0\}\cap \Omega$ for $1\leq k\leq n-1.$ Furthermore, in domain $\mathscr{D}_k$, $u_{x_k}$ satisfies
\begin{equation}\label{5.8}\begin{array}{l}
\mathscr{L}u_{x_k}=\sum\limits_{i,j=1}^{n}
a_{ij}(\nabla u)\frac{\partial^2 u_{x_k}}{\partial x_i \partial x_j}+\sum\limits_{i,j=1}^{n}\frac{\partial^2 u}{\partial x_i \partial x_j}
\frac{\partial a_{ij}(\nabla u)}{\partial x_k}=0,
\end{array}\end{equation}
where $\frac{\partial a_{ij}(\nabla u)}{\partial x_k }=\frac{1}{(1+|\nabla u|^2)^{3/2}}\big[(\frac{3u_{x_i}u_{x_j}}{1+|\nabla u|^2}-\delta_{ij})(\nabla u\cdot\nabla u_{x_k})-(u_{x_i}u_{x_kx_j}+u_{x_j}u_{x_kx_i})\big]$ is the first derivative term of $u_{x_k}$. According to the Hopf lemma, we deduce that $u_{x_kx_k}(p)>0$ for all $1\leq k\leq n-1,$ where critical point $p\in \partial \mathscr{D}_k.$
Finally, we recall that the function $u_{x_n}$ satisfies (\ref{5.4}). By the definition of $\mathscr{N}$, $u_{x_n}>0$ to one side of $\mathscr{N}.$ By applying the Hopf lemma to $u_{x_n}$ at $p\in \mathscr{N},$ we have that $u_{x_nx_n}(p)>0.$ So we prove that the Hessian matrix $D^2u(x)$ of $u$ is diagonal and positive definite at critical point $p$, hence $p$ is the unique critical point and $p$ is a non-degenerate interior minimal point of $u$. This completes the proof of Theorem \ref{th1.3}.
\end{proof}
|
2,877,628,088,965 | arxiv | \section{Introduction}
\label{sec: intro}
There is no doubt about the existence of dark matter in our universe thanks to astrophysical and cosmological observations ranging from galactic to cosmological scales. The nature of dark matter, however, still remains as one of the biggest mysteries of particle physics in spite of tremendous efforts to ascertain its nature since several decades. What we currently know is that the dark matter in our universe has an abundance of $\Omega_c h^2 = 0.1198 (15)$\,\cite{Ade:2015xua} at present and that it interacts gravitationally. The dark matter is expected to be some new particle which is electrically neutral and colorless. Among several candidates proposed so far, the Weakly Interacting Massive Particle (WIMP) is one of the most attractive candidates because of its simplicity and predictability. The thermal freeze-out mechanism of WIMP, which is nowadays called the WIMP miracle, naturally explains the abundance observed today when its mass is around the electroweak scale. This may open up a way to understand the origin of the electroweak symmetry breaking (EWSB). Moreover, we are hopeful to detect the WIMP, because some of the predicted interactions between the WIMP and the standard model (SM) particles provide a strong driving force of WIMP detections at the high energy colliders, underground laboratories (direct dark matter detections) and in cosmological/astrophysical observations (indirect dark matter detections).
We have, on the other hand, not obtained any conclusive signatures of the WIMP so far, and it pushes the celebrated WIMP
scenarios gradually to the corner. This fact, however, also means that the era of serious WIMP searches has begun, so that currently the most important questions are {\it To what extent can we constrain the WIMP models in the future?} and {\it What will then be the remaining unexplored regions in the WIMP parameter space for each of these models?} If the simplicity and predictability of the WIMP are getting lost, such theories will become less attractive. If not, we have to consider what kinds of experiments are required to cover the unconstrained parameter region. Several big experiments, such as multi-tons scale direct dark matter detections\,\cite{Malling:2011va, Kish:2014daa} and future lepton colliders\,\cite{Behnke:2013xla, CEPC-SPPCStudyGroup:2015csa, Gomez-Ceballos:2013zzn}, are being proposed, and it is worth figuring out what roles these experiments can play in this direction. The goal of this paper is to try and answer certain aspects of the aforementioned questions by performing a comprehensive analysis of the singlet-doublets WIMP model~\cite{ArkaniHamed:2006mb}.
We take a standard strategy to study such models based on the WIMP's quantum number. Once the spin of the WIMP is fixed, the WIMP field can always be written as a linear combination of colorless representations of the SM gauge group\footnote{A decomposition formula like Eq.~\ref{eq1} is always possible. Of course, if certain additional symmetries exist, one can extend this formula by including the additional representations. The only case which one must tread carefully is when the WIMP is vector-like. In this scenario, there will always be some additional gauge symmetry, and we should include the explicit form in our formula from the very beginning.}, \textit{viz.} SU(2)$_L \times$ U(1)$_Y$, which must involve the electrically neutral components:
\begin{align}
\label{eq1}
\text{WIMP}(x) = \sum_i\,z_i\,[\chi_i(x)]_\text{N.\,C.},
\end{align}
with `N.\,C.' indicating an electrically neutral component of the representation $\chi_i(x)$. The coefficient $z_i$ follows the sum rule $\sum_i |z_i|^2 = 1$ because of the canonical normalization of the field, and $|z_i|^2$ measures how much the representation $\chi_i(x)$ is involved as a component of the WIMP field. The most straightforward way for studying such WIMP scenarios is to write down an effective Lagrangian of the WIMP field involving all interactions which can be responsible for the WIMP miracle phenomenon, though it is not practical because we have to introduce infinitely many representations and interactions. Instead, we try to cover the theory space by an infinite number of small patches based on the quantum number of the WIMP and construct an appropriate effective Lagrangian in each patch. However, there are still infinitely many possibilities of the effective Lagrangian in each patch; we can always construct the Lagrangian with any desired complexity. We therefore take simplicity as a guiding principle to write down the Lagrangian in each patch.
The WIMP models can be divided into two categories: one in which the WIMP is described almost by a gauge eigenstate of the SU(2)$_L$ interaction, namely one of the $|z_i|^2$s is close to unity. The other is the one in which the WIMP is not described by a single SU(2)$_L$ gauge eigenstate. It indicates that none of the $|z_i|^2$s are close to unity. This is nowadays referred to as the well-tempered WIMP\,\cite{ArkaniHamed:2006mb}. The former category is further divided into many patches: when a SU(2)$_L$ singlet component dominates, the so called singlet WIMP is realized\,\cite{Silveira:1985rk, McDonald:1993ex, Jungman:1995df, Servant:2002aq, Kim:2008pp, Kakizaki:2006dz, Ellwanger:2009dp, Kanemura:2010sh, Djouadi:2011aa, Belanger:2010yx, Cline:2013gha, Matsumoto:2014rxa, Beniwal:2015sdl}. The bino-like neutralino in the minimal supersymmetric standard model (MSSM) and the Kaluza-Klein photon in the universal extra-dimension model (UED) are good examples. When the SU(2)$_L$ doublet components dominate,\footnote{In order to describe the fermionic doublet WIMP, we introduce a pair of SU(2)$_L$ doublet (Weyl) fermions whose hypercharges are opposite with each other, for it makes the theory free from anomaly.} the doublet WIMP is obtained\,\cite{Mizuta:1992qp, Ma:2006km, Barbieri:2006dq, Nagata:2014wma}. Examples of such a WIMP are the Higgsino-like neutralino in the MSSM and the inert Higgs doublet in some two Higgs doublet models (2HDMs). We have the triplet WIMP when a SU(2)$_L$ triplet component dominates\,\cite{Gherghetta:1999sw, Moroi:1999zb, Hisano:2006nn}, with a famous example being the wino-like neutralino in the MSSM. A WIMP which is described by a higher multiplet is nowadays called the minimal dark matter\,\cite{Cirelli:2005uq, Cirelli:2007xd}.
WIMP in the latter category, namely the well-tempered WIMP, is more complicated to be classified. A systematic method to decompose it into patches can be based on the number of representations that are participating in for describing the well-tempered WIMP. For example, it follows $|z_i|^2 + |z_j|^2 \simeq 1$ when this number is two with both $|z_i|^2$ and $|z_j|^2$ being not close to unity, $|z_i|^2 + |z_j|^2 + |z_k|^2 \simeq 1$ when the number is three, and so on. The first example, say the minimal composition in this paper, is usually adequate enough to grasp the typical nature of the well-tempered WIMP. Moreover, it is actually realized in most of the parameter regions of realistic new physics models, even though we should not deny interesting possibilities that WIMPs described by many representations can potentially show some new features that are not seen in the minimal composition. Here, it is worth pointing out how coannihilation regions are incorporated in this framework. This framework automatically involves coannihilations between components inside a SU(2)$_L$ multiplet and between particles which are mixed with each other in the well-tempered WIMP. On the other hand, other coannihilations, such as the one between WIMP and a particle which resides in a different SU(2)$_L$ multiplet and is not mixed with the WIMP, should be considered in each individual patch, even though such a coannihilation region usually does not appear in the minimal setup of the effective Lagrangian in each patch.
In this paper, we focus on a fermionic WIMP in the well-tempered regime. In the next section (Sec.\,\ref{sec: framework}) the well-tempered WIMP in general, and show that some non-trivial conditions are required in the minimal composition setup. After that, the simplest case of the minimal composition setup, which is called the singlet-doublets WIMP model in this paper, is presented.\footnote{Other examples of the minimal composition setup utilizing higher SU(2)$_L$ multiplets, such as the doublets-triplet WIMP model\,\cite{Dedes:2014hga} and the triplet-quadruplets WIMP model\,\cite{Tait:2016qbg}, have also been studied.} We discuss the so-called Higgs- and $Z$-boson-blind spot parameter regions of the model which are recently gaining attention because they evade severe direct dark matter detection constraints\,\cite{Cheung:2012qy}. The WIMP model has indeed been studied in the literature\,\cite{Mahbubani:2005pt, D'Eramo:2007ga, Cohen:2011ec, Cheung:2013dua, Calibbi:2015nha, Freitas:2015hsa, Fedderke:2015txa, NewOne}, but a complete analysis is still lacking. We want to show the current status of the model in its general parameter space, how it will change in the near future and what will be the parameter space remaining in the future if no WIMP signal is detected by then. Current experimental constraints and their expected future improvements are summarized in Sec.\,\ref{sec: constraints}, which involves those from dark matter relic density, direct dark matter detections, indirect dark matter detections and also those from high energy colliders. We also discuss the detailed physics behind these constraints. Following these, in Sec.\,\ref{sec: results}, we perform a numerical analysis to scan the parameter space of the WIMP model and figure out its (would-be) status at present, near future and future, respectively, by imposing the experimental constraints discussed in the previous section. Sec.\,\ref{sec: summary} is devoted to a summary of our findings and discusses the kind of experiments and observations required to explore the yet unconstrained parameter regions in the future. For the sake of completeness, we also put a detailed discussion on the blind spot regions in Appendix\,\ref{app: blind spots}, a supplementary explanation about a collider study of the well-tempered WIMP at the Large Hadron Collider (LHC) with high luminosity in Appendix\,\ref{app: lhc14}, and several figures in Appendix\,\ref{app: figures} which are obtained by scanning the model parameter space numerically and supplement the figures in the main text.
\section{Well-Tempered WIMP}
\label{sec: framework}
As was mentioned in the introduction, the WIMP is in general described by a linear combination of colorless representations of the SM gauge group $\chi_i$; WIMP $= \sum_i z_i\,[\chi_i]_{\rm N.\,C.}$ with $[\chi_i]_{\rm N.\,C.}$ being an electrically neutral component of $\chi_i$. We focus on a fermionic WIMP in the well-tempered regime, where all of the $|z_i|^2$ are not very close to one. In order to make the following discussion concrete, we define the regime as follows:
\begin{eqnarray}
R_i \equiv |z_i|^2 \leq 0.95, \qquad {\rm for~~all}~~i.
\label{eq: well-tempered condition}
\end{eqnarray}
Though the WIMP can potentially be composed of many representations $\chi_i$, it is usually described by a minimal composition in most of the realistic scenarios. We thus take the number of participating representations
describing the WIMP to be as small as possible.
Then, the representations for the well-tempered WIMP must follow several criteria. First, we must introduce at least two representations whose weak isospins differ by one half, because a large enough mixing between different representations calls for a Yukawa interaction with the SM Higgs field whose weak isospin is 1/2. Second, because one of the two representations, which has a half integer weak isospin, also carries a non-zero hypercharge so that it contains an electrically neutral component, its conjugate field must also be introduced in order to make the theory anomaly free. Finally, the other representation, which has an integer weak isospin, must have a Majorana mass term. Otherwise the WIMP becomes a Dirac particle giving a large scattering cross-section off a nucleus through a $Z$-boson exchange, which has already been ruled out by recent direct dark matter detection experiments. In summary, the hypercharge of the representation having an integer weak isospin must be zero, while those of the other representations having a half integer weak isospin are 1/2 and -1/2 due to the aforementioned reason.
We introduce the following three representations as a minimal composition: a representation having an integer weak isospin and no hypercharge, and two representations (which are conjugates of each other) having a half integer weak isospin and hypercharges of $\pm$1/2. The minimal composition is then expressed as follows:
\begin{align}
{\bf (2n - 1)}_0, \, {\bf (2n)}_{1/2}, \, {\bf (2n)}_{-1/2}
\qquad {\rm or} \qquad
{\bf (2n + 1)}_0, \, {\bf (2n)}_{1/2}, \, {\bf (2n)}_{-1/2},
\end{align}
with $n$ being an integer equal to or larger than one. The numbers in parentheses stand for the quantum number of the SU(2)$_L$ interaction, while the subscript is the hypercharge $Y$. Some examples of field contents in the minimal scenario are shown in Tab.~\ref{tab: minimal composition}. We will consider the simplest one as a case study of a WIMP in the well-tempered regime, where it involves a SU(2)$_L$ singlet Weyl fermion field with $Y = 0$ ($S$) and two SU(2)$_L$ doublet Weyl fermion fields with $Y = \pm 1/2$ ($D_1$ and $D_2$). We will clarify how severely the present dark matter search experiments put constraints on the model parameter space of the WIMP, and discuss some prospects to detect the WIMP in the (near) future.
\renewcommand{\arraystretch}{1.5}
\begin{table}[t]
\begin{center}
\begin{tabular}{lcccr} \hline
&Weyl Fermion & SU(2)$_L$ & SU(3)$_C$ & U(1)$_Y$ \\
\hline
\hline
\rowcolor[cmyk]{0,0,0.1,0}
Singlet-Doublets& $S$ & ${\bf 1}$ & ${\bf 1}$& $0$ \\
\rowcolor[cmyk]{0,0,0.1,0}
& $D_1$ & ${\bf 2}$ & ${\bf 1}$ & $1/2$ \\
\rowcolor[cmyk]{0,0,0.1,0}
& $D_2$ & ${\bf 2}$ & ${\bf 1}$ & $- 1/2$ \\
\hline
\rowcolor[cmyk]{0.1,0,0,0}
Doublets-Triplet & $D_1$ & ${\bf 2}$ & ${\bf 1}$ & $1/2$ \\
\rowcolor[cmyk]{0.1,0,0,0}& $D_2$ & ${\bf 2}$ & ${\bf 1}$ & $- 1/2$ \\
\rowcolor[cmyk]{0.1,0,0,0}& $T$ & ${\bf 3}$ & ${\bf 1}$& $0$ \\
\hline
Triplet-Quadruplets& $T$ & ${\bf 3}$ & ${\bf 1}$& $0$ \\
& $Q_1$ & ${\bf 4}$ & ${\bf 1}$ & $1/2$ \\
& $Q_2$ & ${\bf 4}$ & ${\bf 1}$ & $- 1/2$ \\
\hline
\end{tabular}
\caption{\it \small
Some examples of field contents in the minimal composition of a well-tempered WIMP.}
\label{tab: minimal composition}
\end{center}
\end{table}
\renewcommand{\arraystretch}{1}
\subsection{Singlet-Doublets WIMP Model}
\label{sec: sd}
The singlet-doublets model for the well-tempered WIMP, which involves three Weyl fermion fields, \textit{viz.} $S$, $D_1$ and $D_2$, is described by the following Lagrangian:
\begin{align}
{\cal L}_\text{SD}
&= {\cal L}_\text{kin}
- \com{ \frac{1}{2} M_S S S
+ M_D D_1 \cdot D_2
+ y_1 S D_1 \cdot \tilde H
+ y_2 S D_2 \cdot H
+ \text{H.c.}
},
\label{eq: sd model}
\end{align}
with ${\cal L}_\text{kin}$ being the kinetic term of the three new fermion fields, whose form will be explicitly mentioned later in the text. The Lagrangian involves all renormalizable interactions of the three fermion fields by taking the $Z_2$ odd property of the fields into account.\footnote{A $Z_2$ symmetry is implicitly imposed in the Lagrangian\,(\ref{eq: sd model}) in order to guarantee the stability of the WIMP. The three new particles are odd under this symmetry, while all of the SM particles are even under it.} The dot ($\cdot$) indicates the contraction of the SU(2)$_L$ indices via the anti-symmetric tensor $\epsilon_{ij}$. The SM Higgs field is denoted by $H$ with its quantum number ${\bf 2}_{1/2}$, while $\tilde H \equiv \epsilon H^\dag$ is its conjugate with the quantum number ${\bf 2}_{-1/2}$. We suppress all possible higher dimensional interactions that might come from integrating out some other new heavy particles in a fundamental theory, assuming these particles are enough heavier than the fermions in the above Lagrangian. The above model can thus be regarded as an effective theory describing physics around and below the mass scale of the three fermions.
We have four complex parameters $y_1$, $y_2$, $M_S$ and $M_D$. Three of their phases are rotated away by redefining the three Weyl fermion fields, and one phase remains as a physical one associated with the invariant $\phi \equiv \arg[ M_S M_D y_1^\ast y_2^\ast ]$. When the phase is neither $0$ nor $\pi$, the interactions of the model violate CP symmetry and contribute to the electric dipole moment of electron through Barr-Zee type diagrams\,\cite{Mahbubani:2005pt, D'Eramo:2007ga, Giudice:2005rz}, which have already been limited by recent experiments\,\cite{Baron:2013eja} and will be constrained even more severely in the near future\,\cite{Sakemi:2011zz, Kawall:2011zz, Kara:2012ay}. We thus take $\phi = 0$ or $\pi$ in our analysis to avoid such constraints. The model is hence characterized by four real parameters with one sign: $y_1$, $y_2$, $M_S$, $M_D$ and $\text{sign} \com{M_S M_D y_1 y_2 } = \pm 1$. We take the following convention to describe our results. First, $M_S$, $y_1$, $y_2 \geqslant 0$, while $M_D$ being both positive and negative to take the sign into account. Next, we parametrize the two Yukawa couplings as $y_1 = y \cos \theta$ and $y_2 = y \sin \theta$ with $y \geqslant 0$ and $0 \leqslant \theta \leqslant \pi/2$. The range of the angle $\theta$ is further restricted to be $\pi/4 \leqslant \theta \leqslant \pi/2$ because of the symmetry which leaves physics described by the model unchanged: $(y_1, y_2) \leftrightarrow (y_2 , y_1)$. Concerning the symmetry, it is worth noting that we can always rename the doublet fields as $(D_1, D_2) \leftrightarrow (\bar D_2, \bar D_1)$. In summary, we have four real parameters in the model: {\boldmath $M_S \geqslant 0$}, {\boldmath $M_D$}, {\boldmath $y \geqslant 0$} and {\boldmath $\pi/4 \leqslant \theta \leqslant \pi/2$} ({\boldmath $\tan \theta \geqslant 1$} or {\boldmath $0 \leqslant \cot \theta \leqslant 1$}).
It is here instructive to compare our model with the MSSM\footnote{
We must note here that the singlet-doublets model has overlap only with a particular limit of MSSM, where only binos and higgsinos lighter compared to the other SUSY breaking parameters.
}. In the MSSM, our singlet field corresponds to the Bino, while the doublet fields are the up- and down-type Higgsinos. Moreover, we have a correspondence between our model parameters and those of the MSSM: $M_S \leftrightarrow M_1$, $M_D \leftrightarrow \mu$, $\tan \theta \leftrightarrow \tan \beta$ and $y \leftrightarrow g'/\sqrt{2}$. Here, $M_1$, $\mu$, $g'$ and $\tan \beta$ are the supersymmetry breaking Bino mass, the supersymmetry invariant Higgsino mass, the U(1)$_Y$ gauge coupling and the ratio between the vacuum expectation values of two Higgs doubles introduced in the MSSM. The Difference thus appears at the strength of the Yukawa coupling; it is described by the gauge coupling $g'$ in the MSSM, while it is taken to be a free parameter in our model.
\subsection{Interactions in the Singlet-Doublets WIMP Model}
We have so far discussed the setup of the singlet-doublets WIMP model. In what follows, we consider the mass spectra and interactions predicted by the model.
\subsubsection{Mass spectra}
The Higgs field acquires the vacuum expectation value after the electroweak symmetry breaking as $\vev{H} = (0, v)^T/\sqrt{2}$ with $v \sim 246$\,GeV. As a result, the neutral components of the singlet and doublet fields are mixed with each other. Using the notation $D_1 = (D_1^+, D_1^0)^T$ and $D_2 = (D_2^0, D_2^-)^T$, the mass terms of the neutral components are
\begin{align}
{\cal L}_N = - \frac{1}{2} (S, D_1^0, D_2^0)\,M_N
\begin{pmatrix} S \\ D_1^0 \\ D_2^0 \end{pmatrix} + \text{H.c.},
\qquad
M_N \equiv \begin{pmatrix} M_S & -y_1 v/\sqrt{2} & y_2 v/\sqrt{2} \\
-y_1 v/\sqrt{2} & 0 & -M_D \\ y_2 v/\sqrt{2} & -M_D & 0 \end{pmatrix},
\label{eq: mass matrix}
\end{align}
while those of the charged components are simply given by {\boldmath ${\cal L}_C = - M_D D_1^+ D_2^- + \text{\bf H.c.}$} Since $M_N$ is a real symmetric matrix, it can be diagonalized by an orthogonal matrix $O_N$. In order to make all mass eigenvalues positive, we consider an unitary matrix $U_N \equiv O_N\,\Phi$ instead of $O_N$, where $\Phi = {\rm diag} (\eta_1, \eta_2, \eta_3)$ with each component taking a value 1 or $i$ according to the eigenvalue of $O_N^T M_N O_N$. Then, the mass eigenvalues and eigenstates are
\begin{align}
U_N^T M_N U_N = {\rm diag} (M_{N1}, M_{N2}, M_{N3}),
\qquad
\tilde{N} \equiv U_N^\dag (S, D_1^0, D_2^0)^T,
\end{align}
where we adopt the convention $M_{N3} \geqslant M_{N2} \geqslant M_{N1} \geqslant 0$. Here, it is useful to introduce the parameters $R_S$ and $R_D$ in order to quantify how the singlet and doublet components are mixed with each other in the lightest $Z_2$ odd state described by the field $\tilde{N}_1$:
\begin{align}
R_S \equiv |(U_N)_{11}|^2, \qquad
R_D \equiv |(U_N)_{21}|^2 + |(U_N)_{31}|^2,
\label{eq: mixing parameters}
\end{align}
where $R_S + R_D = 1$ is guaranteed by the unitarity of $U_N$, or in other words, the normalization of the lightest $Z_2$ odd field. As was already stated in Eq.\,(\ref{eq: well-tempered condition}), we consider the parameter region satisfying $R_S, R_D \leqslant 0.95$ in order to focus on the WIMP in the well-tempered regime.
\subsubsection{Interactions}
For the sake of convenience, instead of the two component spinor notation used so far, we use the following four component notation for the interactions of the model (\ref{eq: sd model}):
\begin{align}
N_i \equiv (\{\tilde{N}_i^\dag\}_\alpha,\,\tilde{N}_i^{\dot \alpha})^T, \qquad
C \equiv (\{D_2^{-\,\dag}\}_\alpha,\,\{D^+_1\}^{\dot \alpha})^T.
\end{align}
Using the four component spinors, their kinetic terms are simply given by canonical forms {\boldmath $(1/2)\,\bar{N}_i\,(i\slashed{\partial} - M_{N_i})\,N_i$} and {\boldmath $\bar{C}\,(i\slashed{\partial} - M_D)\,C$}. On the other hand, the interactions of the model after the electroweak symmetry breaking are summarized as follows:
\begin{align}
{\cal L}_{\rm int} =&
-h\,\bar{N}_i\,[y_{ij} P_L + y_{ij}^\ast P_R]\,N_j
-\bar{N}_i\,\slashed{Z}\,[g_{ij} P_L - g_{ij}^\ast P_R]\,N_j
+e\,\bar{C}\,\slashed{A}\,C
+\frac{g}{2c_W}\,(2c_W^2 - 1)\,\bar{C}\,\slashed{Z}\,C \nonumber \\
&
+\frac{g}{\sqrt{2}}\,\bar{C}\,\slashed{W}^\dag\,[(U_N)_{2i} P_L - (U_N^\ast)_{3i} P_R]\,N_i
+\frac{g}{\sqrt{2}}\,\bar{N}_i\,\slashed{W}\,[(U_N^\dag)_{i2} P_L - (U_N^T)_{i3} P_R]\,C,
\end{align}
where $h$, $\slashed{Z} = Z_\mu \gamma^\mu$, $\slashed{A} = A_\mu \gamma^\mu$ and $\slashed{W} = W_\mu \gamma^\mu$ are the Higgs boson, photon, $Z$ and $W$ boson fields, respectively, with $P_{L/R} = (1/2)\,(1 \mp \gamma_5)$ being the chirality projection operator, while $e$, $g$ and $c_W = \cos \theta_W$ are the electromagnetic coupling, the weak gauge coupling and the Weinberg angle, respectively. The couplings $y_{ij}$ and $g_{ij}$ are defined as
\begin{align}
y_{ij} &\equiv \frac{1}{\sqrt{2}} [-(U_N^T)_{i1} y_1 (U_N)_{2j} + (U_N^T)_{i1} y_2 (U_N)_{3j}],
\label{eq: def y11} \\
g_{ij} &\equiv \frac{g}{4 c_W} [(U_N^\dag)_{i2} (U_N)_{2j} - (U_N^\dag)_{i3} (U_N)_{3j}].
\label{eq: def g11}
\end{align}
\subsubsection{Blind spots}
\label{subsubsec: blind spots}
It is instructive to discuss the couplings $y_{11}$ and $g_{11}$ in some details, because they play important roles in the WIMP phenomenology developed in the following sections.
We first consider the coupling $y_{11}$, which represents the interaction strength between the WIMP and the Higgs boson, and leads to the spin-independent scattering of the WIMP off a nucleus. After some calculations (see Appendix\,\ref{app: higgs blind spot}), its explicit form reads
\begin{align}
y_{11} = y_{11}^\ast =
- \frac{y^2\,v\,(\eta_1^2 M_D \sin 2\theta + M_{N1})}
{2M_D^2 + y^2 v^2 + 4 \eta_1^2 M_S M_{N1} - 6 M_{N1}^2}.
\label{eq: y11}
\end{align}
Since we assumed the CP invariance in the model, the Yukawa coupling $y_{ii}$ becomes real ($y_{ii} = y_{ii}^\ast$), so that all the neutral particles $N_i$s have their own scalar interactions ($h\,\bar{N}_i N_i$). On the other hand they do not have the pseudoscalar ones ($i h\,\bar{N}_i \gamma_5 N_i$).
This coupling $y_{11}$ has already been severely constrained by several present direct dark matter detection experiments, as will be shown in the next section. The parameter region with $y_{11} \simeq 0$ is thus preferred, which is nowadays known to be {\bf the Higgs blind spot region}\,\cite{Cheung:2012qy}. The condition $y_{11} = 0$ holds when
\begin{align}
M_{N1} = - \eta_1^2 M_D \sin 2 \theta.
\label{eq: higgs blind spot}
\end{align}
Moreover, it turns out that the WIMP mass becomes $M_{N1} = M_S$ and $\eta_1 = 1$ when this condition holds (see again Appendix\,\ref{app: higgs blind spot}). As a result, the blind spot region appears when the mass parameter of the charged particle $C$ is negative, namely $M_D < 0$. The other neutral particles $N_2$ and $N_3$ have masses $M_{N2} = M_{N3} = (M_D^2 + y^2 v^2 /2)^{1/2}$. The mixing parameter $R_S$ in Eq.\,(\ref{eq: mixing parameters}) is given by {\boldmath $R_S = (M_D^2 - M_{N1}^2)/(M_D^2 - M_{N1}^2 + y^2 v^2/2)$} in this region, so that the condition in Eq.\,(\ref{eq: higgs blind spot}) can be accommodated with the one in Eq.\,(\ref{eq: well-tempered condition}).
Next we consider the coupling $g_{11}$, which gives the axial gauge coupling of the WIMP to the $Z$-boson and leads to the spin-dependent scattering between the WIMP and a nucleus. Its explicit from is obtained by using the same method for $y_{11}$ (see Appendix\,\ref{app: z blind spot}):
\begin{align}
g_{11} = g_{11}^\ast =
- \frac{y^2\,v\,m_Z\,\cos 2\theta}
{2\,(2M_D^2 + y^2 v^2 + 4 \eta_1^2 M_S M_{N1} - 6 M_{N1}^2)},
\label{eq: g11}
\end{align}
which again gives a real coupling ($g_{ii} = g_{ii}^\ast$), so that all the neutral particles $N_i$s have their own axial vector current interactions ($\bar{N}_i \slashed{Z} \gamma_5 N_i$), while they never have the vector current ones ($\bar{N}_i \slashed{Z} N_i$), as expected from the Majorana nature of the neutral particles $N_i$s.
This coupling is being gradually limited by recent direct dark matter detection experiments
and it will be more constrained in the near future if no signals are detected. Thus, the region with $g_{11} \sim 0$, which is called {\bf the $Z$-boson blind spot region}\,\cite{Cheung:2012qy}, will be preferred. According to the explicit form in Eq.\,(\ref{eq: g11}), the coupling $g_{11}$ is suppressed when $\theta \sim \pi/4$. Eventually, the regions satisfying both the blind spot conditions,
Eqs.\,\eqref{eq: higgs blind spot} and $\theta \sim \pi/4$, will be preferred if no signals are detected.
As a result, all of the particles $N_1$, $N_2$, $N_3$ and $C$ tend to be degenerate. It is worth noticing here that a very degenerate spectrum leads to $R_S < 0.05$ as can be seen in the $R_S$ formula in the previous paragraph, and it contradicts with the condition for the well-tempered WIMP.
\section{Experimental Constraints}
\label{sec: constraints}
In this section, we discuss the existing experimental constraints used to constrain the model parameter region of the singlet-doublets WIMP model. Moreover, we also discuss some expected constraints which are likely to be obtained in the (near) future in dark matter search experiments, and figure out which part of the regions allowed by the existing constraints will be explored there. We employ the profile-likelihood method\,\cite{Rolke:2004mj} to search for the region with high probability, in which various experimental constraints are incorporated in the form of the likelihood function $L$ with their statistical and systematical uncertainties. The likelihood function that we adopt is constructed by four components:
\begin{eqnarray}
L[M_S, M_D, y, \cot \theta] =
L_{\rm CS}[M_S, \cdots] \times
L_{\rm DD}[M_S, \cdots] \times
L_{\rm CL}[M_S, \cdots] \times
L_{\rm ID}[M_S, \cdots],
\end{eqnarray}
where these component likelihood functions $L_{\rm CS}$, $L_{\rm DD}$, $L_{\rm CL}$ and $L_{\rm ID}$ are constructed based on experimental results obtained from dark matter cosmology, direct dark matter detections, collider experiments and indirect dark matter detections, respectively. We evaluate the likelihood function, $L[M_S, M_D, y, \cot \theta]$, numerically using the so-called MultiNest sampling algorithm\,\cite{Feroz:2008xx}. In what follows, we will discuss all of the component functions in some details together with the physics behind them.
\subsection{Dark Matter Relic Density}
\label{subsec: CS}
We adopt the following likelihood function $L_{\rm CS}$, which is taken as a Gaussian:
\begin{eqnarray}
L_{\rm CS}[M_S, M_D, y, \cot \theta] \propto
\theta(\Omega_{\rm obs} - \Omega_{\rm th})
+\exp\left[-\frac{(\Omega_{\rm th} - \Omega_{\rm obs})^2}{2\,\sigma_{\rm obs}^2}\right]
\theta(\Omega_{\rm th} - \Omega_{\rm obs})\,,
\label{eq: LCS}
\end{eqnarray}
where $\Omega_{\rm obs} = 0.1198/h^2$ ($h$ is the normalized Hubble constant) is the cosmological dark matter parameter observed by the PLANCK experiment\,\cite{Ade:2015xua}, while $\sigma_{\rm obs} = 0.0015/h^2$ is the error associated with the observation. We implicitly assume thermal equilibrium abundance of the WIMP as an initial condition, and compute $\Omega_{\rm th}$ using {\tt micrOMEGAs}\,\cite{Belanger:2010gh, Belanger:2008sj, Belanger:2006is} with the input model file for {\tt CalcHEP}\,\cite{Belyaev:2012qa} generated by {\tt FeynRules}\,\cite{Christensen:2008py, Alloul:2013bka}.
The observed dark matter density is considered as an upper bound in the above likelihood function. The most interesting part of the constraint is, of course, the one satisfying the WIMP miracle, namely $\Omega_{\rm th} \simeq \Omega_{\rm obs}$. However, it is often a practice to consider a non-thermal WIMP production in addition to the thermal one. An example is the late time decay of some heavy particle into WIMP, such as a gravitino/moduli decay into a neutralino in the MSSM\,\cite{Moroi:2013sla, Fujii:2001xp}, which gives an additional contribution to the WIMP abundance today. The other example is the late time entropy production, which dilutes the thermally produced WIMP abundance $\Omega_{\rm th}$.\footnote{One should be careful in considering such a scenario as it also dilutes the baryon asymmetry of the universe.} Though such non-thermal mechanisms are not described in the WIMP model, they can exist in some other sectors which do not affect the WIMP phenomenology except the one related to the WIMP abundance.
If we consider the former non-thermal production, the thermally produced abundance $\Omega_{\rm th}$ is required to be less than $\Omega_{\rm obs}$, for the non-thermal production also gives a positive contribution to the WIMP abundance today. The WIMP is then required to have a stronger interaction to the SM particles than the case without the non-thermal production, because $\Omega_{\rm th}$ is inversely proportional to the WIMP annihilation cross-section. On the other hand, if we consider the latter non-thermal production, the opposite situation arises; the WIMP is required to have a weaker interaction. In this paper, we only consider the case with $\Omega_{\rm th} \leq \Omega_{\rm obs}$, allowing the possible existence of the former non-thermal production, because it gives a lower limit on some couplings between the WIMP and the SM particles (while other constraints in the following subsections give upper limits on the couplings) and still allows us to discuss non-trivial WIMP phenomenology including the WIMP miracle region $\Omega_{\rm th} \simeq \Omega_{\rm obs}$.\footnote{If we consider the possible existence of the latter non-thermal production, the WIMP which does not have any interaction to the SM particles at all is allowed without conflicting all the WIMP constraints.} The WIMP is therefore assumed to have the correct relic density $\Omega_{\rm obs}$ in the present universe even if the set of the model parameters gives $\Omega_{\rm th} < \Omega_{\rm obs}$.
In order to obtain the correct dark matter relic density observed today, the WIMP must have interactions with the SM particles with sufficient strength. It usually requires some special mechanisms, and thus gives a stringent constraint on the parameter space. It is not difficult to imagine that the following regions are allowed by the constraint:
\begin{itemize}
\setlength{\itemsep}{0cm}
\item Higgs boson and $Z$-boson resonance regions.
\item Coannihilation region with degenerate $N_1$ and $C$/$N_2/N_3$.
\item Region in which $N_1$ has a high doublet fraction (smaller $R_S$).
\item Blind spot region with a large Yukawa coupling $y$.
\end{itemize}
The WIMP annihilation in the resonant regions is enhanced when its mass is close to half of the Higgs or the $Z$-boson mass\,\cite{Mahbubani:2005pt, D'Eramo:2007ga, Calibbi:2015nha, Drees:1992am, Hamaguchi:2015rxa}. In the coannihilation region, there is always a process with a large annihilation cross-section: thanks to the weak charge of a coannihilating particle. This fact is also true for the region in which the WIMP has a large doublet fraction. According to the result of the Higgsino dark matter in the MSSM\,\cite{Mizuta:1992qp, Cirelli:2007xd}, the WIMP mass can be as large as 1\,TeV in both regions. The blind spot region allows us to take large Yukawa couplings while avoiding stringent constraints from the direct dark matter detection experiments. On the other hand, the coupling between the WIMP and the $Z$-boson is not severely constrained at present, for the sensitivity of the detections utilizing a spin-dependent scattering is still low. Thus, the WIMP can efficiently annihilate into top quarks by a $Z$-boson exchange. As a result, the constraint from the dark matter relic density will be avoided even when the WIMP mass is larger than 1\,TeV.
\subsection{Dark matter direct detection}
\label{subsec: DD}
The WIMP is scattered off a nucleon in a spin-independent manner by exchanging a Higgs boson and spin-dependently through a $Z$-boson exchange. Spin-independent WIMP scatterings off a proton and a neutron take place at almost the same rate due to small iso-spin violation, while spin-dependent ones do not. Since no conclusive evidence of a dark matter signal has yet been obtained, we adopt the following likelihood function assuming null signal with its central value being fixed to zero:
\begin{eqnarray}
L_{\rm DD}[M_S, M_D, y, \cot \theta] \propto
\exp\left[-\frac{1}{2}
\left\{
\frac{\sigma_{\rm SI}^2}{\delta\sigma_{\rm SI}^2 + \tau_{\rm SI}^2} +
\frac{\sigma_{\rm SDp}^2}{\delta\sigma_{\rm SDp}^2 + \tau_{\rm SDp}^2} +
\frac{\sigma_{\rm SDn}^2}{\delta\sigma_{\rm SDn}^2 + \tau_{\rm SDn}^2}
\right\}
\right].
\end{eqnarray}
Here, $\sigma_i$ is the scattering cross-section predicted by the model, while $\delta \sigma_i$ is an experimental upper limit at $90\%$\,C.L. divided by $1.64$. Theoretical uncertainty from hadron matrix elements required to compute $\sigma_i$ is also introduced, and is estimated to be $\tau_i = 0.2\,\sigma_i$.\footnote{This uncertainty has been obtained by varying the hadron matrix elements inside {\tt micrOMEGAs}.} Other theoretical uncertainties exist, which are from the local velocity distribution and the local mass density of dark matter. The former one is small compared to those from the matrix elements when the WIMP mass is large enough\,\cite{Catena:2011kv}. On the other hand, the latter one may not be small. Fortunately, all experimental limits are derived assuming that the mass density is 0.3\,GeV/cm$^3$, which is lower than the one obtained using the recent Milky Way mass model\,\cite{Nesti:2013uwa}, and hence we do not take this uncertainty into account.
The most stringent constraint on the spin-independent scattering off a nucleon is from the LUX experiment\,\cite{Akerib:2013tjd}, while those on the spin-dependent scatterings off a proton and a neutron are from PICO-60\,\cite{Amole:2015pla} and LUX\,\cite{Akerib:2016lao} experiments, respectively. The constraints will be updated by the XENON1T experiment\,\cite{Aprile:2012zx, Cushman:2013zza,Aprile:2015uzo} in the near future if no dark matter signals are detected. Moreover, the LZ experiment\,\cite{Malling:2011va, Cushman:2013zza} will eventually update the limits on the spin-independent scattering and the spin-dependent scattering off a neutron, and the PICO250 experiment\,\cite{Cushman:2013zza} will do the same job on the spin-dependent scattering off a proton. We will use these projected limits to investigate how efficiently the future experiments can explore the parameter space of the WIMP model.
The spin-independent and spin-dependent scatterings of the WIMP are controlled by the couplings $y_{11}$ in Eq.\,(\ref{eq: y11}) and $g_{11}$ in Eq.(\ref{eq: g11}), respectively. The coupling $y_{11}$ has already been limited as $y_{11} \leqslant$ 0.043\,(0.076) when the WIMP mass is 300\,GeV\,(1\,TeV). This limit puts a strong constraint on the Yukawa coupling $y$ in general, and, as a result, it forces the coupling $g_{11}$ to be as small as ${\cal O}(10^{-4})$.\footnote{This fact has been numerically confirmed using Monte Carlo data, which we discuss in section\,\ref{sec: results}.} Thus the only possible interactions of the WIMP to the SM particles with sufficient strengths are the weak interactions with the other new particles $C$, $N_2$ and $N_3$, if the WIMP has a high doublet fraction. On the other hand, the severe limit on the coupling $g_{11}$ disappears when the model parameters reside inside the Higgs blind spot region. In this case, even though the present spin-dependent direct detection experiments have directly put a limit on $g_{11}$ as $g_{11} \leqslant$ 0.034\,(0.06) for the WIMP mass of 300\,GeV\,(1\,TeV), the WIMP can still annihilate into top quarks efficiently through a $Z$-boson exchange. This is because its annihilation cross-section is boosted by the longitudinal component of this $Z$-boson such that $\sigma v \simeq 3 g_{11}^2 g^2 m_t^2 / (8\pi c_W^2 m_Z^4)$ with $m_t$ and $m_Z$ being the top quark and the $Z$-boson masses, respectively. Thus, even a WIMP heavier than 1\,TeV can survive in the Higgs blind spot region, as will be shown in the next section. Constraints on the couplings $y_{11}$ and $g_{11}$ will become much more severe: $y_{11} \leqslant$ 0.01\,(0.016), $g_{11} \leqslant$ 0.005\,(0.009) in the near future and $y_{11} \leqslant$ 0.002\,(0.004), $g_{11} \leqslant$ 0.0016\,(0.003) in the future for a WIMP mass of 300\,GeV\,(1\,TeV), if no dark matter signals are detected. Then, the $Z$-boson blind spot region will also be favored due to the direct severe limits on the coupling $g_{11}$, and only the region satisfying both the Higgs and the $Z$-boson blind spot conditions, namely the coannihilation region as shown in the previous section, will survive.
\subsection{Dark matter indirect detection}
\label{subsec: ID}
A number of attempts are now being made to detect dark matter indirectly. Observing cosmic-ray species, such as positrons and anti-protons\,\cite{Adriani:2008zr, Adriani:2010rc, Aguilar:2013qda, Accardo:2014lma, Aguilar:2014mma}, are the well-known ones, where the dark matter signals are expected to be detected as anomalous excesses of cosmic-ray fluxes. The observations are, however, also known to receive large systematic uncertainties originating in the cosmic-ray propagation inside our galaxy and the estimation of the background cosmic-ray fluxes\,\cite{Giesen:2015ufa, Kappl:2015bqa, Evoli:2015vaa}. The uncertainties can be avoided if we utilize neutrinos, even though its detection efficiency is still too low to put a strong limit\,\cite{Abbasi:2011eq, Aartsen:2013dxa, Aartsen:2014hva, Aartsen:2015xej}. On the other hand, observing gamma-rays, in particular those from Milky Way satellites called dwarf spheroidal galaxies (dSphs), are regarded as the most effective way to detect dark matter\,\cite{Ahnen:2016qkx, Ackermann:2015zua, Drlica-Wagner:2015xua}. The systematic uncertainties associated with a dark matter distribution inside each dSph, however, still remains\,\cite{Martinez:2013els, GeringerSameth:2011iw, Bonnivard:2015vua, Bonnivard:2015xpq, Bonnivard:2015tta, Ullio:2016kvy, Hayashi:2016kcy}, so that its detection capability at present is not strong enough if we take these into account.
Among several indirect dark matter detections, the one utilizing the cosmic microwave background (CMB) currently allows us to put a robust limit on dark matter annihilation\,\cite{Chen:2003gz, Padmanabhan:2005es, Zhang:2006fr, Mapelli:2006ej, Slatyer:2009yq, Galli:2009zc, Cirelli:2009bb, Slatyer:2015jla, Kawasaki:2015peu, Slatyer:2015kla, Kanzaki:2009hf, Galli:2013dna}. This is because the universe at the recombination epoch is described by the linear density perturbation, so that all systematic uncertainties mentioned above, which are caused mainly by the non-linearity of the perturbation, can be avoided. The universe at the epoch was made up of the thermal plasma among photons, electrons, protons, neutral hydrogen and helium. An efficient annihilation of dark matter injecting energetic particles into the plasma affects the recombination history of the universe, resulting in the boost of the residual ionization fraction.\footnote{The dark matter annihilation also affects the thermal history of the Big Bang nucleosynthesis\,\cite{Kawasaki:2015yya}.} This effect is imprinted in the CMB spectrum, and precise observations enable us to detect the effect even if it is caused by a small change of the fraction. Since such an effect has not yet been detected, we put a constraint on the singlet-doublets WIMP model by adopting the following likelihood function:
\begin{eqnarray}
L_{\rm ID}[M_S, M_D, y, \cot \theta] \propto
\theta[{\rm UL}\,(M_{N1}) - \langle \sigma v \rangle],
\end{eqnarray}
where $\theta[x]$ is the Heaviside step function, while $\langle \sigma v \rangle$ is the annihilation cross-section predicted by the model, which is evaluated by taking its thermal average at the temperature of the recombination epoch. The upper limit on the cross-section at $90\%$ confidence level is obtained as ${\rm UL}\,(M_{N1}) = 1.453 \times 10^{-27}\,(M_{N1}/1\,{\rm GeV})^{1.05}$\,cm$^3$/s\,\cite{Slatyer:2015jla} based on the PLANCK result.\,\cite{Ade:2015xua}. This limit will give a constraint on the parameter region with a light WIMP having a high doublet fraction, because such a WIMP has a large self-annihilation cross-section with a large enough number density at the recombination epoch.
The constraint on the annihilation cross-section will be upgraded by several factors in the future\,\cite{Madhavacheril:2013cna}, and a heavier WIMP with a high doublet fraction can thus be explored. WIMP with a mass of ${\cal O}(1)$\,TeV seems, however, difficult to be searched for by this detection. Such a heavy WIMP region can potentially be covered by the indirect detection utilizing gamma-ray observations of dSphs when dark matter profiles inside dSphs are precisely evaluated by accumulating enough kinematical data of the galaxies\,\cite{Bhattacherjee:2014dya}.
\subsection{Dark matter searches at colliders}
\label{subsec: CL}
Recent and future colliders, such as the Large Electron Positron Collider (LEP), the Tevatron, the LHC, the International Linear Collider (ILC)\,\cite{Behnke:2013xla}, the Circular Electron Positron Collider (CEPC)\,\cite{CEPC-SPPCStudyGroup:2015csa} and the Future Circular Collider of Electrons and Positrons (FCC-ee)\,\cite{Gomez-Ceballos:2013zzn}, are designed to look for signatures of physics beyond the standard model (BSM). The LEP and Tevatron have not detected any conclusive BSM signatures. The LHC is now on a hunt for such signatures, and the hunt will be the most important task at future colliders (ILC, CEPC and FCC-ee). We consider several constraints on the singlet-doublets WIMP model obtained by collider experiments performed so far, and discuss some prospects of looking at the signatures of this model in the (near) future.
\subsubsection{Invisible Z-boson decay}
When $m_{N1} < m_Z/2$, the $Z$-boson can decay into a pair of WIMPs. This process should be regarded as a part of the invisible decay width of the $Z$-boson ($\Gamma_Z^{\rm Inv}$) at collider experiments. The measurement of the total decay width of the $Z$-boson ($\Gamma_Z^{\rm Tot}$) at the LEP was very precise, so that it gives an upper limit on the width as $\Gamma_{Z}^{\rm Inv} \leqslant$ 2\,MeV at 90\% confidence level\,\cite{ALEPH:2005ab}. We thus adopt the following likelihood function to involve this limit:
\begin{eqnarray}
L_{\rm InvZ}[M_S, M_D, y, \cot \theta] =
\exp\left[-\frac{(\Gamma_Z^{\rm Inv})^2}{2\,(2\,{\rm MeV}/1.64)^2}\right],
\end{eqnarray}
where the invisible decay width $\Gamma_Z^{\rm Inv}$ is computed within the singlet-doublets WIMP model. At first glance, this constraint seems very severe and puts a stringent limit on the model parameter space, in particular, on the parameter region where the WIMP coupling to the $Z$-boson is not suppressed. This is, however, not true, for spin-dependent direct dark matter detections are already putting more stringent limit on the coupling, and this trend will be more strengthened in the (near) future if no WIMP signal is detected. On the other hand, when the WIMP is much lighter than 10\,GeV, the sensitivities of the direct detection experiments are very weak, so that the invisible width search can play a dominant role to put limits on the parameter space. Such a light WIMP region is, however, being limited by other dark matter searches: the new particle searches at the LEP already kill the region when the WIMP has a significant doublet fraction, while the WIMP annihilation cross-section in the region is too small to satisfy the constraint from the dark matter relic abundance when the WIMP is close to an
SU(2)$_L$ singlet.
\subsubsection{Invisible Higgs decay}
When the WIMP is lighter than half of the Higgs mass, the Higgs boson can decay into a pair of the WIMPs, which is, this time, regarded as the invisible decay width of the Higgs boson ($\Gamma_h^{\rm Inv}$). The invisible decay branching ratio is being constrained by a global fit of Higgs data at the LHC, which leads to an upper limit on the ratio as ${\rm Br}\,(h \to N_1 N_1) \leq 0.24$ at 90\% confidence level\,\cite{Giardino:2013bma}. We thus adopt the following likelihood function:
\begin{eqnarray}
L_{\rm InvH}[M_S, M_D, y, \cot \theta] =
\exp\left[-\frac{{\rm Br}\,(h \to N_1 N_1)^2}{2\,(0.24/1.64)^2}\right].
\end{eqnarray}
Using the invisible decay width $\Gamma_h^{\rm Inv}$ computed from the singlet-doublets WIMP model, the branching ratio is defined as ${\rm Br}\,(h \to N_1 N_1) \equiv \Gamma_h^{\rm Inv}/[\Gamma^{\rm SM}_h + \Gamma_h^{\rm Inv}]$, where the total decay width of the Higgs boson within the SM framework is denoted by $\Gamma^{\rm SM}_h \simeq 4.08$\,MeV\,\cite{Heinemeyer:2013tqa} with the Higgs boson mass being $m_h = 125.09$\,GeV\,\cite{Aad:2015zhl}. This constraint puts a limit on the WIMP coupling to the Higgs boson. However, as in the case of the $Z$-boson invisible width, this coupling has already been severely constrained by spin-independent direct dark matter detections. As a result, the invisible Higgs width measurement will not play an important role to put a limit on the model parameter space.
\subsubsection{Electroweak precision measurements}
We inevitably introduce particles having electroweak interactions in the singlet-doublets WIMP model, so that oblique corrections\,\cite{Peskin:1991sw} might be a useful tool to detect such a WIMP indirectly even if it cannot be produced directly at lepton colliders (see the next subsubsection for more details). As is usual, the so-called $S$ and $T$ parameters can play important roles in the WIMP model: the $S$ parameter measures the size of the weak isospin breaking but it turns out to give a loose constraint on the model due to small field degrees of freedom. On the other hand, a severe constraint may come from the $T$ parameter, which measures the size of the custodial symmetry breaking, because it is proportional to the difference of the Yukawa couplings $y_1^2 - y_2^2$ and can thus be sizable for a larger $y$ and a smaller $ \cot \theta$. With the use of Monte Carlo data discussed in the next section, the $T$ parameter, however, turns out to be always smaller than 0.1 after all the other constraints are imposed. This value is well below the current upper bound on the $T$ parameter\,\cite{Baak:2014ora}.
As was mentioned in section\,\ref{subsec: DD}, if no new physics signals are detected, all the parameter regions of the WIMP model will shrink into the coannihilation region in the (near) future due to severe constraints from direct dark matter detections. In fact, it leads to a larger $\cot \theta$ for a larger Yukawa coupling $y$. As a result, contributions to the $T$ parameter from the WIMP is less and less significant. After all, observing the oblique corrections, namely the electroweak precision measurements, is not (will not be) an ideal way to detect the WIMP at present (future), and hence we do not include it in our analysis.
\subsubsection{New particle searches at lepton colliders}
New particles having electroweak interactions can be easily searched for at lepton colliders. The singlet-doublets WIMP model predicts an electrically charged (Dirac) particle $C^\pm$ and three electrically neutral (Majorana) particles $N_1$, $N_2$ and $N_3$.
The charged particle $C$ can be pair produced at the colliders by exchanging a photon and a $Z$-boson in the $s$-channel, and its annihilation cross-section is always large thanks to its electroweak quantum numbers. The charged particle can therefore be detected irrespective of its decay mode whenever its production is kinematically allowed.\footnote{Mono-photon signals can be expected even if the mass difference $m_C - m_{N_1}$ is small\,\cite{Abdallah:2003xe, Chen:1995yu}.} Since conclusive signals of new particle productions were not obtained at the LEP, we have a lower limit on the mass of the charged particle $M_D$. Thus in the next section, we perform a numerical scan of the model parameter space in the range {\boldmath \bf $M_D \geq$ 100\,GeV}\,\cite{Abdallah:2003xe}.
On the other hand, the neutral particles are pair produced by exchanging only a $Z$-boson in the $s$-channel at lepton ($e^+e^-$) colliders. Its signal strength thus depends on how a large coupling to the $Z$-boson the neutral particle pair has. Since the charged particle production has already put a constraint on the parameter space, it is expected that the neutral particle productions in the singlet-doublets WIMP model can be important in the following three cases: First one is that all charged and neutral particles are light enough to be produced at the colliders. The charged particle pair production however puts a more stringent constraint than those of the neutral particle pair. Next one is that doublet-like neutral particles are lighter than the singlet one. It however leads to the same result as the previous case, for the charged particle is also as light as the light neutral particles. The last case is opposite to the second case, where a singlet neutral particle is lighter than the doublet ones. Since the $N_1$-$N_1$-$Z$ coupling is already constrained by the spin-dependent direct dark matter detection experiments, the relevant process is $e^-e^+ \to N_1 N_2 (N_3)$. The coupling of $N_1$ to the $Z$-boson with $N_2$ or $N_3$ is, however, suppressed in this case because of the singlet nature of $N_1$
, so that the LEP experiment does not put any significant limit on the model. There may be a more subtle spectrum for the neutral particles. It however turns out that by generating Monte Carlo data as shown in the next section, all the cross-sections of the neutral particle pair productions are always below 1\,pb after imposing all other constraints. Thus, the LEP experiment does not put any constraint on the neutral particle pair production in the singlet-doublet WIMP model, and hence we do not include it in our analysis.
Though lepton colliders do not play an important role to put a constraint on the singlet-doublets WIMP model at present, these can potentially be important in the (near) future. As was already mentioned in previous sections, if no WIMP signals are detected in the (near) future from all kinds of dark matter search experiments, then only the coannihilation region remains as the unexplored one with the WIMP mass being larger than a few hundred GeV. The WIMP in such a region can hardly be detected at both direct and indirect dark matter detections, while future lepton colliders such as the ILC, CEPC and FCC-ee can search for it at least through the charged particle pair production if the center of mass energy of the collisions are large enough.
\subsubsection{New particle searches at hadron colliders}
\label{subsubsec: hadron colliders}
New particles having electroweak interactions are also searched for at the LHC. One of the prominent examples is the pair production of charginos and neutralinos in the framework of supersymmetry. This search can be recast to the singlet-doublets WIMP model thanks to the analogy between this WIMP model and the MSSM, and thus can put constraint on the parameter space. Among various pair production channels, that of a charged particle $C$ and a heavier neutral particle $N_2$ or $N_3$ seems the best one, for it gives a rather clean signal, namely the three leptons plus missing transverse energy final state, even in the messy environment of the hadron collider. The full process reads
\begin{equation}
p p \to C\,N_{2,\,3} \to (W^{(*)}\,N_1)\,(Z^{(*)}\,N_1) \to 3 \ell + \slashed{E}_T.
\end{equation}
Here, the superscript `$(*)$' indicates that the $W$ and $Z$-bosons can go off-shell as well. We include the off-shell $W$ and $Z$-bosons as well because we hope to have lesser backgrounds in this region since the on-shell $Z$-production can be vetoed away.
It has been shown in Ref.\,\cite{Calibbi:2015nha} that the corresponding ATLAS search at 8\,TeV\,\cite{Aad:2014nua} is capable of covering the region, $m_{N_2}$, $m_{N_3}$, $m_C$ $\lesssim$ 270\,GeV and $m_{N_1} \lesssim$ 75\,GeV, assuming that the average of the branching fractions Br$\,(N_2 \to Z N_1)$ and Br\,$(N_3 \to Z N_1)$ is at least 60\%. We thus perform a thorough study to investigate how this search puts constraints on our available model parameter space. However, it is shown not to have any significant power to rule out the parameter space that has already been limited by other dark matter constraints. For the current model, the charged particle, $C$ will always decay to one of the neutral particles and
an on- or off-shell $W$ boson because these are the dominant decay modes. The branching ratios are
largely independent of the mass differences between the charged and neutral particles. However, for a small
mass difference, the cut efficiencies for various cuts will begin to decrease because of a dearth of phase space, for
instance the lepton $p_T$ will peak at smaller values and hence the $p_T(\ell_1,\ell_2,\ell_3) > 50$ GeV cut
will yield a considerably smaller efficiency. After computing the $3 \ell + \slashed{E}_T$ cross-section, we find that
this is not enough to put any significant constraints on the parameter space as the cross-section varies between
$10^{-4}$ fb and a few times $10^{-3}$ fb before any detector level cuts. The $C N_2$ pair production cross-section can however be as large as 0.1 fb at 8 TeV (and can be as large as 1 fb for the 14 TeV run) but the branching ratio suppresses the final cross-sections to significantly smaller values. We must also note that, we are working in the parameter region allowed by XENON1T. Hence we find that after implementing {\tt CheckMATE}\,\cite{Drees:2013wra, deFavereau:2013fsa, Cacciari:2011ma, Cacciari:2008gp, Read:2002hq} for this ATLAS analysis, one cannot exclude a single point of the parameter space allowed by other constraints. The ChecKMATE analysis includes all the five main signal regions and the sub-regions which have been optimised following Ref.~\cite{Aad:2014nua}. For completeness, we include a plot (Fig.~\ref{fig:3lMET-eps-converted-to.pdf}) showing the signal cross-section in the $3 \ell + \slashed{E}_T$ final state as a function of $m_C-m_{N_2}$ for the 8 TeV and 14 TeV cases.
\begin{figure}[t]
\centering
\includegraphics[width=0.7\textwidth]{./figs/n2cha_3l_MET_8_14.pdf}
\caption{\sl \small Parton level cross-section for $p p \to C N_2 \to 3 \ell + \slashed{E}_T$}
\label{fig:3lMET-eps-converted-to.pdf}
\end{figure}
As mentioned before, the $3 \ell +\slashed{E}_T$ cross-section in the parameter region allowed by the direct
detection constraints in the (near) future, will be suppressed mainly because the average mass of the particles $C$ and
$N_{2,3}$ is large. Also when they are nearly degenerate with $N_1$, the cut-efficiency of the signal is reduced because
of softer leptons produced from off-shell $W$ bosons and hence unable to satisfy certain cuts demanding larger values for
the $p_T$ of the leptons. We also perform ATLAS's prediction study for the high-luminosity run at 14\,TeV (See Appendix\,\ref{app: lhc14}). However it is also shown not to have any significant power to exclude the parameter space.\footnote{
Ref.\,\cite{Hamaguchi:2015rxa} plays a complementary role to our study, which also includes the singlet-like regime with $R_s < 0.05$. It has been shown that, for the case of the singlet-like regime with a supersymmetric $Z$- and Higgs-resonant neutralino dark matter, the ATLAS search at 14\,TeV\,\cite{ATL-PHYS-PUB-2014-010} can be a strong probe. In another complementary study, Ref.~\cite{vanBeekveld:2016hbo} has shown in the context of pMSSM that the $3\ell + \slashed{E}_T$ channel can have constraining power for $m_{\textrm{NLSP}} - m_{\textrm{LSP}} \sim (5-50)$ GeV.
} With the proposed 33\,TeV High-energy LHC (HE-LHC) or the 100\,TeV Very Large Hadron Collider (VLHC) in the future, the signal cross-sections can be enhanced. However, one has to deal with very large backgrounds and a full analysis at such colliders is beyond the scope of this paper. Hence, at least for the singlet-doublets WIMP model, the direct detection constraints play a stronger role than those from the LHC, and we do not consider constraints from new particle searches at the LHC in our analysis.
\section{Scanning results}
\label{sec: results}
We are now in a position to present our scanning results to quantitatively display which parameter regions of the singlet-doublets WIMP model survive at present, will be covered in the near future, and will be left over in the future. The strategy of our numerical scanning is as follows: Engaging with {\tt MultiNest\,v2.18}\,\cite{Feroz:2008xx} of 20000 living points, a stop tolerance factor of $10^{-4}$, and an enlargement factor (reduction) parameter of 0.8, we perform several random scans in the following parameter space,
\begin{eqnarray}
10\,{\rm GeV} & \leq\,\,\,\,M_S\,\,\,\,\leq & 2\,{\rm TeV}, \nonumber \\
100\,{\rm GeV} & \leq\,\,|M_D|\,\,\leq & 2\,{\rm TeV}, \nonumber \\
0 & \leq\,\,\,\,\,\,y\,\,\,\,\,\,\leq &1, \nonumber\\
0 & \leq\,\cot\theta\,\leq & 1.
\label{eq: domain}
\end{eqnarray}
The doublet mass parameter $M_D$ can either be positive or negative. Some of the regions such as the blind spot regions or the resonance regions have very low prior probabilities to be accessed if only flat or log prior distributions are adopted. To obtain a better coverage, similar to a Bayesian approach, we update our priors, namely, we start with one log and one flat prior scans, but later adjust our scan range and prior distribution based on the previous likelihood distribution. In the end, we obtain in total more than $10^6$ points out of which for only $6 \times 10^5$ points $\chi^2 \equiv -2\ln\,L[M_S, M_D, y, \cot\theta] < 50$ have been recorded and used for our figures. Our minimal $\chi^2$ value turns out to be almost zero. Below, we present our scanning results for the present, the near future, and the future.
\subsection{Present status}
\label{subsec: present}
\begin{figure}[t]
\centering
\subfloat[\label{subfig: MN_MD_P}] { \includegraphics[width=0.48\textwidth]{./figs/MN_MD_P-eps-converted-to.pdf} }
\subfloat[\label{subfig: MN_RS_P}] { \includegraphics[width=0.48\textwidth]{./figs/MN_RS_P-eps-converted-to.pdf} }
\caption{\sl \small Present constraints on the WIMP parameter space in the singlet-doublets WIMP model.}
\label{fig: present}
\end{figure}
The present status of the singlet-doublets WIMP model can be seen in Fig.\,\ref{fig: present}, where the 1$\sigma$\,(2$\sigma$) contour is shown with the yellow (blue) band. The left and right panels (Figs.\,\ref{subfig: MN_MD_P} and \ref{subfig: MN_RS_P}) show a viable model parameter space at present in the planes of $(M_{N1}, M_D)$ and $(M_{N1}, R_S)$, respectively (Comprehensive figures showing the whole viable parameter space after imposing present experimental constraints are provided in Appendix\,\ref{app: figures}.). The most important feature is an asymmetry in Fig.\,\ref{subfig: MN_MD_P} with respect to the sign of the doublet mass parameter, $M_D \leftrightarrow - M_D$, owing to the stringent constraints form spin-independent direct dark matter detections. As was mentioned in the previous section, these constraints can be avoided if the model parameters lie in the Higgs blind spot region, and the regions appears only when $M_D < 0$. Below, we will look into the figure in more details.
The mass parameter $M_D$, whose absolute value gives the mass of the new charged particle $C$, is limited to be $|M_D| > 100$\,GeV due to the LEP constraint. There are two regions when $M_D > 100$\,GeV. One of these comes from the coannihilation region with $M_{N1} \sim M_D$. This mass degeneracy between the WIMP and the particle $C$ is required to satisfy the constraint from the dark matter relic abundance and also the well-tempered condition in Eq.\,\eqref{eq: well-tempered condition}, because the size of the Yukawa couplings are always limited when $M_D > 100$\,GeV. The upper bound on the mass parameter $M_D \lesssim 1$\,TeV in this coannihilation region can be understood by an analogy with the Higgsino-like dark matter in the MSSM. The other one is the Higgs/$Z$-boson resonance region with $M_{N1} < 100$\,GeV. The WIMP here is singlet-like because $M_D > 100$\,GeV, as can also be seen in Fig.\,\ref{subfig: MN_RS_P}. The upper bound on the mass parameter $M_D \lesssim 800$\,GeV comes from the well-tempered condition given in Eq.\,\eqref{eq: well-tempered condition}.
On the other hand, an additional parameter region becomes available when $M_D < -100$\,GeV, owing to the Higgs blind spot condition, which allows large Yukawa couplings. This region appears when $M_{N1}$ is larger than the top quark mass, where the annihilation of the WIMP into a pair of top quarks is boosted by the longitudinal component of the $s$-channel exchanged $Z$-boson, as discussed in Sec.\,\ref{subsec: DD}. It also takes a role to relax the mass degeneracy between the WIMP and the charged particle $C$. Here also, the lower bound on $M_D$ comes from the condition in Eq.\,\eqref{eq: well-tempered condition}. In the resonance region with $M_D < -100$\,GeV, a slightly lighter WIMP mass is allowed compared to the one with $M_D > 100$\,GeV, because larger Yukawa couplings help the total WIMP annihilation to be sufficiently large, opening up a lighter WIMP mass region slightly away from the pole.
Here, we also comment on the role of indirect dark matter detections. As can be seen in Fig.\,\ref{subfig: MN_RS_P}, the constraint from the CMB observation excludes some parameter regions; a characteristic spike around $M_{N1} \sim 120$\,GeV and $R_S \lesssim 0.2$. This fact means that the indirect detection does not play an important role for the WIMP when $R_S$ larger than 0.2. This is because the s-channel (velocity unsuppressed) annihilation of the well-tempered WIMP into SM particles (e.g. weak gauge bosons) is suppressed when $R_S$ increases. Moreover, even if $R_S$ is small enough, the constraint becomes less significant when the WIMP is heavier, for the number density of the WIMP in the present universe decreases.\footnote{The spike structure of the region with $M_{N1} \sim 120$\,GeV and $R_S \lesssim 0.2$ is because of this reason and the threshold behavior of the s-channel (velocity unsuppressed) annihilation process, $N_1 N_1 \to W^+ W^-$.}
\subsection{Near future prospects}
\label{subsec: near future}
\begin{figure}[t]
\centering
\subfloat[\label{subfig: MN_MD_NF}] { \includegraphics[width=0.48\textwidth]{./figs/MN_MD_NF-eps-converted-to.pdf} }
\subfloat[\label{subfig: MN_RS_NF}] { \includegraphics[width=0.48\textwidth]{./figs/MN_RS_NF-eps-converted-to.pdf} }
\caption{\sl \small Potential constraints on the WIMP parameter space in the near future in the singlet-doublets WIMP model.}
\label{fig: near future}
\end{figure}
The expected viable model parameter space after the XENON1T experiment is shown in Fig.\,\ref{fig: near future}, assuming that the experiment will not observe any dark matter signals (Comprehensive figures showing whole viable parameter space after imposing the XENON1T constraints are provided in Appendix\,\ref{app: figures}.). The crucial difference from the present status in Fig.\,\ref{fig: present} is that the Higgs/$Z$-boson resonance region and the Higgs blind spot region seem to disappear in Fig\,\ref{subfig: MN_MD_NF}. This is because not only the spin-independent but also the spin-dependent direct dark matter detections will be much improved as mentioned in Sec.\,\ref{subsec: DD}, and hence these will put very severe limits on the singlet-doublets WIMP model. As a result, only the coannihilation regions in both $M_D > 100$\,GeV and $M_D < -100$\,GeV seem to survive. Note that even if the blind spot region disappears, the coannihilation region, which is realized in both $M_D > 100$\,GeV and $M_D < -100$\,GeV regions, can survive, for the coannihilating particle such as $C$ has a large enough annihilation cross-section, as discussed in Sec.\,\ref{subsec: CS}.
The importance of the spin-dependent direct dark matter detection can be seen from Fig.\,\ref{subfig: MN_RS_NF}. It is indeed confirmed that the resonant region will be excluded. On the other hand, the region with $M_{N1} \gtrsim 1$\,TeV still survives. The region with such a large WIMP mass is realized when the coupling between the WIMP and the $Z$-boson ($g_{11}$) is large enough, and such a large coupling is realized only in the Higgs blind spot region. This fact means that some part of the blind spot regions turns out to survive yet in the near future even if any WIMP signal is not detected until then. Because the coupling $g_{11}$ can be directly constrained by the spin-dependent cross-section measurement, the region will be eventually excluded in future if no WIMP signal is detected. (See also Fig.\,\ref{appfig: Y_COT_NF}, where the region with large Yukawa couplings $y$ is pushed to the $Z$-boson blind spot region, namely $\cot \theta \to1$.).
Here, it is also worth pointing out that all the new particles are degenerate in the coannihilation region, and hence it is very hard to discover their signals at hadron colliders due to the small mass splitting between $N_1$ and $C/N_{2,3}$, as discussed in Sec.\,\ref{subsubsec: hadron colliders}.
\subsection{Future prospects}
\label{subsec: future prospects}
\begin{figure}[t]
\centering
\subfloat[\label{subfig: MN_MD_F}] { \includegraphics[width=0.48\textwidth]{./figs/MN_MD_F-eps-converted-to.pdf} }
\subfloat[\label{subfig: MN_RS_F}] { \includegraphics[width=0.48\textwidth]{./figs/MN_RS_F-eps-converted-to.pdf} }
\caption{\sl \small Future prospects on the WIMP in the singlet-doublets WIMP model.}
\label{fig: future}
\end{figure}
Future prospects on the singlet-doublets WIMP model after the LZ and the PICO250 experiments are shown in Fig.\,\ref{fig: future}. The properties look similar to those from the near future constraints, even though the constraints from the spin-independent and the spin-dependent direct dark matter detections are much more severe. As a result, almost all the viable model parameter space shrinks to the coannihilation regions. The remaining region with $M_D < 100$\,GeV is the one which almost satisfies both the Higgs and the $Z$-boson blind spot conditions simultaneously; $M_{N1} \simeq M_S \simeq - M_D$ and $\cot \theta \simeq 1$ (See also Fig.\,\ref{appfig: future} in Appendix\,\ref{app: figures} for further confirmation.). It can also be seen from Fig.\,\ref{subfig: MN_MD_F} that the small coannihilation region with $M_D > 100$\,GeV, which is outside the Higgs-blind spot, will survive, even though an extreme tuning of the mass splitting is required. Therefore, in the future, the singlet-doublets WIMP model will be constrained alone by the future direct dark matter detections to the corner of the coannihilation region, if no signals are detected.
\section{Summary and Discussion}
\label{sec: summary}
We have investigated the current status and the (near) future prospects of a fermionic WIMP in the well-tempered regime, particularly focusing on the simplest case of the minimal composition setup, namely the singlet-doublets WIMP model. It then turns out that a viable model parameter space at present can be classified into the following regions:
\begin{itemize}
\setlength{\itemsep}{0cm}
\item[(i)] Higgs boson and $Z$-boson resonance regions.
\item[(ii)] Coannihilation region with degenerate $N_1$ and $C$/$N_2/N_3$.
\item[(iii)] Region in which $N_1$ has a high doublet fraction (smaller $R_S$).
\item[(iv)] Blind spot region with a large Yukawa coupling $y$.
\end{itemize}
Owing to the stringent constraint from the LUX experiment on the spin-independent cross-section for a WIMP scattering, the coupling between the WIMP and the Higgs boson is already constrained, and hence it is pushing the viable model parameter space towards the Higgs-blind spot region, even though other regions still survive. A low mass region with a high doublet fraction is somewhat constrained by the CMB measurement, which was adopted as a robust indirect dark matter constraint in our analysis.
In the near future, significant improvements on the spin-independent and the spin-dependent WIMP scattering cross-sections are expected from the XENON1T experiment. The WIMP couplings to both the Higgs and the $Z$-bosons will be severely constrained if the experiment does not observe any dark matter signals. Then, the viable model parameter region tends to shrink to those regions where both the Higgs and the $Z$-boson blind spot criteria are satisfied. This trend will be strengthened in the future, when the LZ and the PICO250 experiments update their constraints. The viable parameter space will then shrink to the one where the regions (ii) and (iv) overlap, or the small one where only region (ii) is satisfied without satisfying the Higgs blind spot condition.
It is important to consider what kinds of experiments have sufficient capability to cover the leftover regions, assuming that none of the future direct dark matter detection experiments unfortunately observe any dark matter signals. Hadron colliders such as the LHC seem difficult to do it even if their luminosities are high enough, for all the new particles are highly degenerate in mass in the leftover regions. The WIMP in these regions may have a significant doublet fraction as can be seen from Fig.\,\ref{subfig: MN_RS_F}. Indirect dark matter detections may probe such WIMPs if systematic errors associated with astrophysical uncertainties are within good control. However, if the WIMP has a large singlet fraction, the indirect detections do not work at all, for the WIMP annihilation is severely $p$-wave suppressed in the present universe. In such a case, future lepton colliders such as the ILC can be useful in exploring the leftover region directly (via the pair production of the charged particle $C$, etc.) and indirectly (via radiative corrections to SM processes, etc.\,\cite{Harigaya:2015yaa}).
\vskip 0.5cm
\noindent
{\bf Acknowledgments}\\[0.1cm]
\noindent
This work is supported by the Grant-in-Aid for Scientific research from the Ministry of Education, Science, Sports, and Culture (MEXT), Japan No. 26104009 and 26287039 (S. M.), as well as by the World Premier International Research Center Initiative (WPI), MEXT, Japan (S. M., K. M. and Y. S. T.). The work of K.M. is supported in part by a JSPS Research Fellowships for Young Scientists. S. B acknowledges the support of the Indo French LIA THEP (Theoretical high Energy Physics) of the CNRS. S. B. acknowledges the hospitality of IPMU where the idea for this work was conceived, and also acknowledges the cluster facility at the Harish-Chandra Research Institute\,\url{http://www.hri.res.in/cluster/}. We also thank Daniel Schmeier for technical help regarding CheckMATE.
\newpage
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2,877,628,088,966 | arxiv |
\section{Summary and Conclusion}
Based on the success of cross-modality transfer approaches such as IMUTube \cite{kwon2020imutube}, in this paper we explored to what extent virtual IMU data benefit HAR systems that tackle activities, which are based on more subtle body (parts) movements.
IMUTube was originally designed to support HAR systems that target coarse movement activities, such as locomotion.
Our exploration unveils two important aspects that are relevant for the broader HAR community:
\textit{i)} The subtlety of motions in activities is quantifiable in video data -- through our newly introduced Motion Subtlety Index (MSI), which correlates with the eventual downstream activity recognition accuracy on IMU data;
\textit{ii)}
Our analysis identified a range of subtle movement activities for which IMUTube is beneficial--most notably, and somewhat surprisingly, eating-- and another range of subtle movement activities, for which it is not.
We were able to systematically, and objectively assess how the addition of virtual IMU data benefits general HAR systems and showed how the \textit{a-priori} calculation of MSI values \textit{on videos} can be used to effectively guide the application of systems like IMUTube.
In cases where the addition of IMUTube-generated virtual IMU data is not beneficial for downstream HAR accuracy, it is most likely that the underlying 3D human motion tracking techniques cannot capture enough movement detail, for example, the detailed hand and wrist coordination as it is common for subtle movements.
As a result, the generated virtual IMU data is essentially only adding noise, which--not surprisingly--has a detrimental effect on the overall system effectiveness.
\input{figure/msi_f1_17cls}
Overall, the MSI-based assessment of subtlety of activity movements and, correlated to that, the prediction of downstrem HAR effectiveness is of substantial practical value for practitioners in planning resource allocation when using cross-modality systems like IMUTube.
Such \textit{a-priori} analysis is of growing interest \cite{hiremath2020deriving} as it allows non-HAR experts to make informed decisions on what it will take to set up an effective HAR system \textit{before} actually embarking on such an endeavor.
Finally, it is worth mentioning that the study presented in this paper is limited to wrist motions and a handful of human activities.
As such, more research and follow-up studies are encouraged for broader generalizability, specifically to:
\textit{i)} explore the use of MSI and virtual IMU data for other on-body sensor locations and wider range of human activities that involves subtle motions; and
\textit{ii)} to resolve the challenges in related techniques in tracking subtle motions for computer vision, graphics and biomechanics research communities.
The work presented in this paper represents the first step in this direction.
\section{Conclusion}
IMUTube system was designed to capture substantial motions in underlying activities for converting human activity video into virtual IMU dataset, and was demonstrated to be extremely effective for activities with coarse movements, such as locomotion or gym exercises.
However, human activities in daily livings in fact involve much subtle motions in many context, such as eating, cooking, or washing hands that are important to assess the quality of life.
In this work, we studied the use of virtual IMU data with respect to such subtle motions in daily livings to understand the working and failing scenario of virtual IMU dataset.
To assess the utility of virtual IMU data in the context of subtle motions, we proposed a novel metric, \textit{Motion Subtlety Index} (MSI), to quantify the subtlety of motions in human activity videos that was highly correlated with actual amount of motion captured in real IMU dataset.
With MSI, we devised a system that can effectively select the segments of the video that will derive virtual IMU data with similar level of motion that reflects most probable signals captured with real IMU data.
The proposed system demonstrated that, when only sparse real IMU data is available, matching the motion subtlety of real and virtual IMU data for model training is critical to improve classification performances.
More importantly, our analysis identified the uncanny valley with respect to motion subtlety when IMUTube system miserably fails and even harms the classification performances.
IMUTube turned out to be useful when underlying human activiy involved either nearly motionless or large motions.
Any activity in between them are more likely not to benefit from IMUtube system.
We consider the major suspect of IMUTube failure is in the current limitation of 3D human motion tracking techniques, that cannot capture detailed hand and wrist coordination for subtle movements.
Overall, quantifying subtlety of the motion through MSI will provide a practical guideline to understand the degree of performance gain that can be expected from virtual IMU data in relation to particular human activities of interest, which helps for allocating time and resources for the systems like IMUTube.
Although this study takes an important step toward the application of virtual IMU data for human activities with subtle motions, it is limited to wrist motions and handful of human activities.
Therefore, our findings calls for the new research opportunity 1) to explore the use of MSI and virtual IMU data for various on-body sensor locations and wider range of human activities that involves subtle motions, and 2) to resolve the challenges in related techniques in tracking subtle motions for computer vision, graphics and biomechanics research communities.
\section{Conclusion}
In this work, we studied the use of virtual IMU data with respect to subtle motions in daily livings to understand the working and failing scenario of virtual IMU dataset.
To quantify the subtlety of motions in human activity videos, we proposed a novel metric, \textit{Motion Subtlety Index} (MSI), and measured the subtlety of motions involved in various daily activities, which is then associated with the changes in classification performances coming from using virtual IMU dataset.
Our analysis identified the sweet spot of MSI ($0.35 \leq MSI \leq 3.33$) where adding virtual IMU data will more likely to improve the classification performance than not.
Furthermore, we analyzed how the selection of virtual IMU data according to MSI value impacted classification performance on each activity.
This revealed that activities with lower MSI preferred virtual IMU data with lower MSI values for training, and similarly for the activities with high MSIs.
Matching MSI of virtual and real IMU data could significantly improve classification performance compared to naively using all virtual IMU data available.
However, these success came with certain limitations.
Performance improvements were visible only when the activity MSI fell into the range of sweet spot, and the model started to ignore virtual IMU data as more real IMU data becomes available.
We consider the major suspect of IMUTube failure is in the current limitation of 3D human motion tracking techniques, that cannot capture detailed hand and wrist coordination for subtle movements.
Overall, quantifying subtlety of the motion through MSI will provide a practical guideline to understand the degree of performance gain that can be expected from virtual IMU data in relation to particular human activities of interest.
Knowing the falling point can help planning the resource allocation when using the systems like IMUTube for developing activity classifier.
Although this study takes an important step toward the application of virtual IMU data for human activities with subtle motions, it is limited to wrist motions and handful of human activities.
Therefore, our findings calls for the new research opportunity 1) to explore the use of MSI and virtual IMU data for various on-body sensor locations and wider range of human activities that involves subtle motions, and 2) to resolve the challenges in related techniques in tracking subtle motions for computer vision, graphics and biomechanics research communities.
\subsection{First Analysis}
The results of our initial experimental evaluation suggest that the addition of virtual IMU, generated using the vanilla IMUTube system, is only of limited effectiveness \textit{on average} for the task of recognizing activities of daily living that only exhibit subtle body (part) movements.
However, the addition of IMUTube-generated virtual IMU data was very effective for the analysis of eating activities.
In what follows, we dive deeper into the analysis, which resembles the aforementioned detailed ``stress test'', which will provide us with a deeper, objective understanding \textit{when}, i.e., for which activities IMUTube is beneficial, and when not.
Aiming for concrete guidelines for practitioners who want to use IMUTube-like systems for their HAR deployments, we utilize the newly introduced MSI score as a means for quantitative assessments and guidance.
\subsection{MSI-Based Assessment of IMUTube's Utility for HAR}
\input{figure/per_class_f1_real_100}
To understand for which activity virtual IMU data was most useful, \autoref{fig:per_class_f1} shows the detailed classification performance changes for each class in the daily activities recognition task when adding virtual IMU data (from curated videos) to model training.
Performance changes of up to more than $\pm 5\%$ absolute (F1 score) can be observed.
Improvements can be seen for activities like flipping, reading, washing hands, bed making, driving, and cutting.
This suggest that virtual IMU data does have value for activities with only subtle body (part) movements.
Aiming for an objective, up-front assessment--based on \textit{a-priori} analysis of the source videos--we now explore how the eventual classification performance for individual activities correlates to the video-based subtlety quantification using our MSI index.
Having a \textit{cut-off} threshold MSI value would work as a practical means for practitioners to help deciding whether or not to put resources and time into generating virtual IMU data using a system like IMUTube.
\autoref{fig:msi_cutoff} plots the changes in classification performance (``Change in F1'' -- y-axis) in relation to MSI values calculated for the underlying activity videos (``MSI'' -- x-axis).
The MSI value for each class (cMSI) represents the mode of the MSI distribution computed from all curated videos for each class using Kernel Density Estimation.
As we increase the MSI cut-off values (x-axis), we included activity classes with more subtle classes for classification task.
Then, we computed the classification performance changes coming from using virtual IMU data at each MSI cut-off value.
For easier interpretation, we fit a spline curve (\textcolor{green}{green}) to find an approximative decision boundary between activities that benefit from additional virtual IMU data (left of the boundary), or not (right of it).
\input{figure/msi_f1_cutoff}
\subsection{MSI-Driven Control of Virtual IMU Data Generation}
We additionally conducted an experiment for the case of all targeting all 17 activities but generating--and using--virtual IMU data only for those classes for which the MSI value was determined to be below the cut-off value, i.e., they are predicted to be north of the decision boundary in Fig.\ \ref{fig:msi_cutoff}.
It is worth reminding that the MSI values are calculated on the videos \textit{prior} to actually generating virtual IMU data using IMUTube.
\autoref{fig:msi_cutoff_17cls} shows the strong negative correlation ($r=-0.85$, $p\leq0.001$) (Pearson) between the MSI cut-off values and changes in F1 score introduced by using virtual IMU data for the activity classes with cMSI below the MSI cut-off value.
The zero-crossing of the linear line fit for all data points was $MSI=0.89$, which was very similar to the previous experiment.
This strongly supports that MSI can provide a reference to gauge the benefit of using virtual IMU data when classifying activities with subtle motions.
Overall, this study demonstrates that the proposed MSI provides quantifiable approach to pinpoint when IMUTube will fail when it comes to subtle activities.
Activity classes beyond $0.9\leq MSI$ seems to have very subtle and complex motions in hand and wrist movements that are very difficult for IMUTube to capture due to the current limitations of state-of-the-art human motion tracking techniques~\cite{desmarais2021review}.
\subsection{Activity-specific Effectiveness of Virtual IMU Data}
\subsubsection{Real vs. Virtual IMU data}
\subsection{Challenges in Video Quality regarding Motion Tracking}
\subsection{Quantifying the subtlety of Activity from Videos}
\subsection{Subtlety of Activity vs. Effectiveness of Virtual IMU Data}
\section{Discussion}
Our experiment results demonstrated that employing MSI and using virtual IMU data with motion subtlety that corresponds to the activity class can help improve model performance when only sparse real IMU data is available.
Here, we study in-depth for the effect of MSI and real IMU data size for classifying activities with subtle motions.
\subsection{MSI vs. Classification Performance}
\label{sec:msi_f1_all}
\input{figure/msi_vs_f1}
\autoref{fig:msi_f1} explores the changes in classification performance according to the MSI values of the virtual IMU dataset used for model training.
To effectively show the role of MSI, we used 20\% of real IMU data for each task as virtual IMU data were more useful when real IMU data was sparse.
MSI value (x-axis) was used as a upper threshold value to include virtual IMU data for model training.
For example, when MSI threshold was 7.5, all virtual IMU data with MSI below 7.5 was used for the model training.
For cut-off values, we picked 10 to 100 percentile values in entire virtual IMU dataset.
We measured the changes in F1 score (y-axis) compared to our baseline model which were trained only with real IMU dataset.
Also, we used virtual IMU data from the curated video dataset in this analysis.
For 17 activity classes overall in daily activity classification (Left), the optimal MSI value was the 20 percentile of the entire virtualIMU dataset.
This is consistent with hypothesis that for classifying activities with subtle motions are preferred.
As MSI value has increased, the F1 score converged to the point not adding any merit for the model.
\hyeok{For Eating activity, I am waiting for the results from ZiKang}
\input{figure/msi_vs_f1_per_class}
Seeing that (\hl{hypothetically}) classification performance is significantly affected by MSI, we investigated further into how MSI values affects per-class F1 score in daily activity classification task.
We used 20\% of real IMU data similarly to the previous analysis.
\autoref{fig:msi_f1_per_class} shows four example activities for classification performance according to MSI value thresholds.
The analysis shows that the optimal MSI value depends on the activity.
Interestingly, low MSI activities (Eat with hand and Driving Automatic) preferred lower MSI values, mid MSI activities(flipping) and high MSI activities preferred higher MSI values, which alings with our hypothesis that the model benefits more when the provided virtual IMU data capture the quantity of motions reflecting the underlying activities.
\input{figure/low_mid_high_real}
\subsection{Low/Mid/High MSI Classes vs. Real IMU Scale}
\label{sec:msi_group_real_size}
Observing that the optimal MSI values are very specific to subtlety of motions of the involved activity, we hypothesized that the impact of virtual IMU data will increase even in the presence of large real IMU dataset, if activities with similar motion levels are classified together with virtual IMU data also having similar level of motions.
To validate this hypothesis, we explored the classification performance when classification is performed only within the groups of low, mid, and high MSI activities.
Also,we changed the amount of real IMU data for model training at the same time.
We independently trained low, mid, high MSI activity groups with virtual IMU data with low, mid, high MSI values, respectively.
For each case, we also compared the classification performance when using all virtual IMU dataset.
\autoref{fig:low_mid_high_real} demonstrates F1 score for each cases also including when using all activity classes combined with the proposed method.
When classifying with only low-MSI activities (left upper), the low-MSI virtual IMU data was especially effective $7+$ F1 score increase even when the 50\% of real IMU data was used.
When there is little to no motion, It is feasible for IMUTube to capture such activities reliably.
Intuitively speaking, those activities will be reflected in almost no movements in 3D motion tracking information that is transferred to virtual IMU dataset.
For high MSI activity classification (left bottom), only using high-MSI virtual IMU data significantly outperformed naively using all virtual IMU data for model training for all cases.
This reflects that choosing appropriate virtual IMU data through MSI is important, but with the limited success.
As soon as more than 30\% of real IMU was used, the effect of virtual IMU data was negligible.
Even for high MSI activities here, they are still in the domain of \textit{subtle motions}, which is difficult to track with state-of-the-art 3D motion tracking pipeline that roughly estimated joint locations.
For mid MSI activity classification (right upper), the effect of virtual IMU data was trivial or even degrading the model performance even when using MSI.
This exactly demonstrates the breakdown points of IMUTube system.
When the activity is nearly motionless but has still very subtle motions at the same time in the point of motion tracking view, it is the uncanny valley where IMUTube system completely fails to capture quality virtual IMU dataset.
All activities combined (right bottom), the proposed method could benefit from virtual IMU data with MSI only up to 40\% of real IMU dataset.
We suspect the inclusion of mid-MSI motions have suppressed the gain from either low or high MSI motions.
Overall, this study demonstrates the working and failing scenario of IMUTube.
IMUTube either prefers the activity is nearly motionless or with larger motions, which closely follows current limitation of state-of-the-art human motion tracking techniques~\cite{desmarais2021review}.
When the motion quantity of underlying activity falls into somewhere in between, then IMUTube is likely to fall out.
Our study shows usefulness of the proposed MSI, which can tell if an activity of interest falls into this uncanny valley for IMUTube or not.
\section{Discussion}
\hyeok{Need to update all analysis with the results when using all real IMU data.}
The earlier experiment results demonstrated that the utility of virtual IMU to heavily depend on the amount of motion involved in activity classes.
Virtual IMU data was very effective when capturing various arm motions in eating activities, whereas it was having difficulty capturing subtle hand and wrist motions in other daily activities.
In this section, we further investigate the relationship between the motion subtlety in activities and classification performance through our proposed MSI.
\subsection{MSI vs. Classification Performance}
\label{sec:msi_f1_all}
\input{figure/per_class_f1_real_100}
\input{figure/msi_f1_spline}
To understand for which activity virtual IMU data was most useful, \autoref{fig:per_class_f1} shows the detailed classification performance changes for each class in daily activity classification task by adding virtual IMU dataset for model training.
The baseline model is trained with 20\% of real IMU data.
Performance change had a wide range of variability ($\pm 10$ absolute F1-score).
Performance gain was significant (absolute 5+\%) for activities like wiping, washing hands, reading, flipping, playing guitar.
Whereas significant loss (absolute 5-\%) was observed for drawing, writing, playing violin, and cutting activities.
We studied the relationship between the MSI score for each activity with the changes in F1 score.
\autoref{fig:msi_f1_spline} shows the spline curve fitting on the data points for the MSI mode and the corresponding F1 score changes of each class introduced by using virtual IMU data.
As representative MSIs for each activity, we picked mode MSI from the distribution of MSI for each class.
This analysis shows that there exists a range of MSI values ($0.35 \leq MSI \leq 3.33$) that are more likely to increase the model performance (2.5+\%, \textcolor{red}{red line}) by using virtual IMU dataset.
Extremely low subtlety of motions ($MSI < 0.35$) such as drawing or writing were very difficult for IMUTube to extract useful virtual IMU data, and the mid range ($3.33 < MSI < 7.46$) or high range ($7.46 < MSI$) MSI activities showed mixed results but with overall expectation with only marginal or negative impact on model performance when using virtual IMU dataset.
To much of our surprise, this finding suggested that there exists a sweet spot of MSI that can still make IMUTube useful even for the activities with subtle motions.
\subsection{MSI for Virtual IMU Data Selection}
\input{figure/msi_vs_f1}
We further assess the how does the class-specific classification performance varies according to MSI value used in daily activity classification task.
\autoref{fig:msi_f1} explores the changes in classification performance according to the MSI values of the virtual IMU dataset used for model training.
To effectively show the role of MSI, we used 20\% of real IMU data for each task as virtual IMU data were more useful when real IMU data was sparse.
MSI value (x-axis) was used as a upper threshold value to include virtual IMU data for model training.
For example, when MSI threshold was 7.5, all virtual IMU data with MSI below 7.5 was used for the model training.
For cut-off values, we picked 10 to 100 percentile values in entire virtual IMU dataset.
We measured the changes in F1 score (y-axis) compared to our baseline model which were trained only with real IMU dataset.
Also, we used virtual IMU data from the curated video dataset in this analysis.
For 17 activity classes overall in daily activity classification (Left), the optimal MSI value was in the range of $0.35 \leq MSI \leq 3.33$, which was consistent with the earlier annalysis.
As MSI value has increased, the F1 score converged to the point not adding any merit for the model.
\hyeok{For Eating activity, I am waiting for the results from ZiKang}
\input{figure/msi_vs_f1_per_class}
Seeing that classification performance is significantly affected by MSI, we investigated further into how MSI values affects per-class F1 score in daily activity classification task.
We used 20\% of real IMU data similarly to the previous analysis.
\autoref{fig:msi_f1_per_class} shows four example activities for classification performance according to MSI value thresholds.
The analysis shows that the optimal MSI value depends on the kind of activity.
Interestingly, low MSI activities (Eat with hand and Driving Automatic) preferred lower MSI threshold values, mid (flipping) and high MSI activities preferred higher MSI threshold values to improve classification performance.
This may indicate that the model benefits more when the provided virtual IMU data capture the quantity of motions reflecting the underlying activities.
\subsection{Low/Mid/High MSI Classes vs. Real IMU Scale}
\label{sec:msi_group_real_size}
Observing that the optimal MSI values for each activity are very specific to subtlety of motions of involved activity, we hypothesized that the impact of virtual IMU data will increase, if virtual IMU data provided for each activity had similar level of motion subtlety as in the corresponding target activity.
To validate this hypothesis, we categorized each activity in to low, mid, and high MSI groups according to mode MSI values of each activity.
The low MSI group included reading, driving automatic, drawing, eating, playing guitar, the mid MSI group included playing violin, writing, driving manual, flipping, playing piano, cutting components, wiping, and the high MSI group included bed making, shower, sweeping, washing hands, washing dishes.
For activities in each MSI group, we assigned virtual IMU data with the similar amount of motion subtlety according to the MSI values.
Virtual IMU data from all classes are categorized into low, mid, and high MSI groups ($MSI_{low} < 3$, $3 \leq MSI_{mid} <6$, and $6 \leq MSI_{high}$) according to our observation in \autoref{fig:msi_f1_spline}.
For model training, we only used virtual IMU data with $MSI_{low}$ for low MSI group activities, $MSI_{mid}$ for mid MSI group activities, and $MSI_{high}$ for high MSI group activities.
\input{figure/low_mid_high_real}
\autoref{fig:low_mid_high_real} shows the effect of matching motion subtlety between real and virtual IMU data for model training.
We independently trained low (upper left), mid (upper right), high (bottom left) MSI activity groups, and also, trained for all activity classes combined (bottom right).
For each case, we also compared the classification performance when naively using all virtual IMU dataset.
We additionally changed the amount of real IMU data for model training (x-axis) to better understand the impact of real IMU dataset.
When classifying with only low-MSI activities (left upper), the low-MSI virtual IMU data was especially effective $7+$ F1 score increase even when the 50\% of real IMU data was used.
Consistent with previous analysis, this range of motion subtlety was most effective region for IMUTube.
For mid MSI activity classification (right upper), the effect of virtual IMU data was trivial or even degrading the model performance even when using MSI.
This exactly demonstrates the breakdown points of IMUTube system.
For high MSI activity classification (left bottom), only using high-MSI virtual IMU data significantly outperformed naively using all virtual IMU data for model training for all cases.
This reflects that choosing appropriate virtual IMU data through MSI is important, but with the limited success.
As soon as more than 30\% of real IMU was used, the effect of virtual IMU data was negligible.
Although they fall into high MSI groups in our analysis, they are still the activities with \textit{subtle motions}, which is difficult to track with state-of-the-art 3D motion tracking pipeline that roughly estimated joint locations.
All activities combined (right bottom), the proposed method could benefit from virtual IMU data with MSI only up to 40\% of real IMU dataset.
We suspect the inclusion of mid-MSI motions have suppressed the gain from either low or high MSI motions.
Overall, this study demonstrates the working and failing scenario of IMUTube when it comes to subtle activities.
Activity classes beyond $3\leq MSI$ seems to have subtle yet very complex motions in hand and wrist movements that are very difficult for IMUTube to capture, which closely follows current limitation of state-of-the-art human motion tracking techniques~\cite{desmarais2021review}.
Also, as more real IMU data become available, the model starts to ignore virtual IMU data.
Therefore, amount of real IMU data is another important factor to consider when choosing to use IMUTube for subtle activities.
\section{Experiment}
In this study, we look at potential changes in the accuracy (F1 score) of human activity recognition models trained using virtual IMU data when activities with more subtle movements are taregted (quantified using our newly proposed MSI measure).
We first give an overview of our workflow that utilizes IMUTube.
Then we will discuss the benchmark datasets and video sources used for our study.
\iffalse
In our experiments, we will analyze the effectiveness of MSI for quantifying the subtlety of movements in various daily activities through conduct correlation study between MSI and rMSI. \thomas{No! rMSI has no proven relevance for any practical outcome. At the end of the day, it is the F1 scores on HAR tasks that matter.}
Then, we compare recognition performance for daily activity classification tasks using virtual IMU data MSI and rMSI with and without.
Here, we introduce our experiment settings and the benchmark datasets used for annalysis.
\fi
\subsection{HAR with Virtual IMU Data from Wrist Sensors}
We followed the approach used in the original IMUTube experiments \cite{kwon2020imutube}.
We extracted virtual IMU data for wrist sensors from video datasets, which are calibrated with (a small amount of) real IMU data used for training.
Real IMU data is subsampled to 25 Hz to match the sampling rate of virtual IMU data extracted from videos.
Both virtual and real IMU data are segmented into fixed length analysis frames that contain consecutive readings using the sliding window technique.
The choice of window size depends on the particular classification task of interest.
Then, features are extracted from both real and virtual analysis windows, which are used to train a classification back-end.
In line with previous IMUTube studies, we employ the Random Forest classifier.
Lastly, the trained model is tested on unseen real IMU data.
For assessing motion subtlety, MSI is derived from the video sequence that corresponds to each virtual IMU data segment.
As baseline, we use the model that was trained only using real IMU data, i.e., the training sets as they were provided by the individual datasets, which is in line with previous work.
Identical to previous IMUTube explorations, we then train a model using both real and virtual IMU data, which is used to assess the effectiveness of virtual IMU data for HAR tasks.
Model effectiveness is evaluated through F1 scores.
As explained below, virtual IMU datasets are extracted from curated or in-the-wild video datasets.
\input{tables/dataset}
\subsection{Benchmark Real IMU Dataset}
For studying subtle motions in daily activities, we use two benchmark datasets for wrist sensors: daily activity classification~\cite{inproceedings} and eating activity classification~\cite{thomaz2015practical}.
\autoref{tab:dataset} shows statistics of the dataset used in our experiments.
\subsubsection{Daily Activities}
We used HAD-AW dataset \cite{inproceedings} as our benchmark dataset, which consists of 31 activity classes from subtle to coarse motions collected using an Apple Watch.
Out of a total of 31 activity classes, we selected 17 activity classes that are based on more subtle movements of the body (parts):
\textit{
playing violin,
playing piano,
playing guitar,
driving automatic,
driving manual,
reading,
writing,
eating,
cutting components,
washing dishes,
washing hands,
showering,
sweeping,
wiping,
drawing,
flipping,
bed-making}.
\subsubsection{Eating}
We used the large public dataset from Thomaz \emph{et al}. \cite{thomaz2015practical} that contains wrist-recorded IMU data from both in-lab and in-the-wild eating settings.
The \textit{Lab-20} dataset was collected from 21 participants in the lab covering both eating and non-eating activities for approximately 31 minutes period on average.
The eating moments involve,
eating with fork \& knife,
hand,
and spoon.
While non-eating activities include
watching trailer, conversation,
brush teeth,
place a phone call etc., all being subtle hand motions.
For in-the-wild dataset, we used the \textit{Wild-7} dataset \cite{thomaz2015practical}, which was collected from seven participants.
Out of a total of 31 hours and 28 minutes of data, 2 hours and 8 minutes were labeled as eating activities.
\subsection{Curated Video Data Set}
For assessing the feasibility of using virtual IMU data for recognizing subtle motion activities, we first collected virtual IMU data from a well curated video dataset, Kinetics-400 \cite{kay2017kinetics}, where all videos were manually trimmed down to clips of 10 seconds to contain specific activity classes.
\subsubsection{Daily Activities}
\input{figure/daily_activity}
For the first set of experiments we used the well curated Kinetics-400 video dataset \cite{kay2017kinetics}, and focused on 17 activity classes that are based on subtle body (parts) movements (as assessed by us).
In total, we generated a virtual IMU dataset with approximately 50\% of the size of the corresponding, original real IMU dataset.
As shown in \autoref{fig:daily_activity_data_proportion}, the virtual IMU samples are imbalanced, as many video clips are removed due to low quality and noisy scenes for extracting virtual IMU for wrist movement. Hence, the duration of the extracted virtual IMU dataset is shorter than the combined duration of the video clips.
\subsubsection{Eating}
From the Kinetics-400 dataset, we collected 417 video clips for the eating class, which were labeled as one of 10 eating-related classes (eating: burger, cake, carrots, chips, doughnuts, hotdog, ice cream, nachos, spaghetti, watermelon).
The number of eating instances in the extracted virtual IMU dataset is approximately 6\% of that in the Lab-20 dataset.
\subsection{In-the-Wild Video Dataset}
YouTube houses one of the largest collection of uncurated videos that are coarsely labeled using Google's search algorithm, providing a virtually unlimited source of videos for any activity.
As shown in \autoref{fig:eating}, videos from such repositories may contain scenes that do not capture movements related to eating, i.e., scenes that IMUTube can automatically filter out with Adaptive Video Selection modules (yellow box;\autoref{fig:imutube}).
We downloaded videos from YouTube using activity names as search queries. Returned videos are sorted according to relevance.
\subsubsection{Daily Activities}
We downloaded one video for each class, which was on average approximately 8 minutes long (per activity).
Based on this, we generated virtual IMU data that are in total approximately 16\% the size of the baseline, real IMU dataset.
\subsubsection{Eating}
We downloaded two YouTube videos, where the first video is 22 minutes long, and the second one has a duration of 17 minutes.
From those two videos combined, we could collect 11\% of the amount of eating data samples compared to Lab-20 dataset.
\input{figure/eating}
\subsection{Experiment Settings and Hyperparameters}
\subsubsection{Daily Activities}
We applied 5-fold stratified cross-validation.
We could not test leave-one-user-out cross validation because the public released version of the HAD-AW dataset is incomplete, so many activities were not performed by the same users.
After calibrating the virtual IMU data with the HAD-AW dataset, all real and virtual IMU data were segmented using a sliding window procedure with frame length of 3 seconds and step size of 1.5 seconds (identical to previous work \cite{inproceedings}).
We then extracted ECDF features \cite{hammerla2013preserving} from each analysis frame. The extracted ECDF features are then used to train a Random Forest classifier.
We report the average of the macro F1 score from all folds for three runs.
\subsubsection{Eating}
We used the Lab-20 dataset as training set and Wild-7 as testing set.
Accordingly, virtual IMU data was calibrated with the Lab-20 dataset.
Identical to previous eating studies \cite{thomaz2015practical}, we applied a sliding window based segmentation with a window size of 6 seconds, and 50\% overlap between consecutive analysis frames.
For each frame we then extracted mean, variance, skewness, kurtosis and root mean square features.
The predictions from the trained Random Forest (RF) model were fed into a DBSCAN clustering to aggregate eating moments (identical to previous work \cite{thomaz2015practical}).
We report binary F1-scores from the average of three runs on Wild-7 dataset.
\subsection{IMUTube with MSI}
\input{tables/dataset}
\thomas{Coming back to my thoughts on the introduction: Why don't we lead with the analysis of subtlety of the movements underlying activities of daily living? That would make one fine, strong contribution on its own and put the experiments (and the findings!) on a solid foundation. Thoughts?}
Evaluating IMUTube requires benchmarking different datasets that cover a variety of \textbf{fine motion} \thomas{fine or subtle?} activities along with video datasets that can be used to extract the required virtual IMU data.
We mainly studied classification tasks on two sets of representative activities with fine motions using wrist sensors: 1) Eating~\cite{thomaz2015practical}, and 2) daily activities~\cite{inproceedings}.
In what follows, we discuss benchmark datasets and virtual IMU data used to analyze the effectiveness IMUTube on such subtle motions in activities.
\autoref{tab:dataset} shows statistics of the dataset used in our experiments.
\thomas{See my comment above re leading with the index. Also: This is too eating-centric again, no?}
\thomas{We should also fork out the details of the overall processing pipeline, which is not data related. So, have a separate subsection on the classification backend, training procedure, IMUTube configuration (calibration and such). We need to provide as much detail here as we can, such that the results can be interpreted well (and be trusted!).}
\thomas{I am also missing some sort of an overview on the experiment protocol: What do we do with the individual datasets? How do we feed them into the IMUTube system? Any parameter optimization? How do we measure effectiveness? We can cover all this in one leading, separate subsection where we describe all details independent of the actual datasets -- that comes later, and can then be much more streamlined and compressed. We could have a workflow figure here that illustrates the overall procedure. We do have the space.}
\subsection{Eating Recognition}
\input{figure/eating}
\subsubsection{Benchmark Real IMU Dataset}
For eating detection, we used a large public dataset from Thomaz~\emph{et al}.~\cite{thomaz2015practical} that contains both in-lab and in-the-wild real IMU data for benchmarking IMUTube.
This choice was motivated by previous work \cite{thomaz2015practical,bedri2015wearable, bedri2017earbit, bedri2020fitbyte}, which focused on develop eating moment recognition solutions in controlled settings that can also work well in-the-wild.
The \textit{Lab-20} dataset was collected from 21 participants between the ages of 20 and 43 covering both eating and non-eating activities for approximately 31 minutes period on average.
The activities contained a mix of eating and non-eating moments with no time constraints on participants. The eating moments involve,
eating with fork \& knife,
hand,
and spoon
while non-eating activities include
watching trailer, conversation,
brush teeth,
place a phone call etc., all being \textit{fine hand} motions.
For in-the-wild experiments, we used the \textit{Wild-7} dataset \cite{thomaz2015practical}, which was collected from seven participants between the ages of 21 and 29 throughout the day using wrist sensors and wearable cameras.
Out of a total of 31 hours and 28 minutes of data, 2 hours and 8 minutes were labeled as eating activities.
Similar to previous work \cite{thomaz2015practical}, we applied a DBSCAN \cite{ester1996density} based analysis approach to determine eating moments.
\thomas{Given that we only use that dataset, we may want to compress this presentation here a bit.}
\subsubsection{Curated Video Dataset}
For assessing the feasibility, we first collected virtual IMU data from well curated video dataset, Kinetics-400 \cite{kay2017kinetics}, where all videos were manually trimmed down to clips of 10 seconds to contain specific activity classes.
In our experiments, we used 417 video clips for the eating class, which were labeled as one of 10 eating-related classes (eating: burger, cake, carrots, chips, doughnuts, hotdog, ice cream, nachos, spaghetti, watermelon).
The extracted virtual IMU data contained approximately 6\% more eating instances compared to Lab-20 dataset.
\subsubsection{In-the-wild Video Dataset}
YouTube houses one of the largest collection of uncurated videos that are coarsely labeled using Google's search algorithm, providing a virtually unlimited source of videos for any activity.
We download two YouTube videos that were labeled as "eating" (through our search query and brief thumbnail inspection for the returned results):
The first video is 22 minutes long, whereas the second one has a duration of 17 minutes.
As shown in \autoref{fig:eating}, videos from online repositories contain scenes that do not involve human eating motions, i.e., scenes that IMUTube~\cite{kwon2021approaching} can automatically filter out with Adaptive Video Selection modules (yellow box;\autoref{fig:imutube}).
However, we could already collect 11\% more eating data samples compared to eating instances in Lab-20 dataset from those two videos combined.
\subsubsection{Experiment Protocol}
For our eating detection experiment, we use Lab-20 dataset as training set and Wild-7 as testing set.
Our baseline model is trained only with real IMU data from Lab-20 dataset.
Our proposed model is trained with both real and virtual IMU dataset, where virtual IMU dataset is either from curated or in-the-wild video dataset.
the collected virtual IMU dataset is weakly labeled as search keyword 'eating' was used for labels.
Following the standard process in IMUTube~\cite{kwon2020imutube}, virtual IMU data was calibrated with training real IMU data from the lab (Lab-20) for model training.
For model training and evaluation, we sub-sampled the real IMU data to 25 Hz to match the sampling rate of the virtual IMU dataset extracted from videos.
Following previous eating studies \cite{thomaz2015practical}, we applied sliding window segmentation with 6 second window size and 50\% overlap between consecutive analysis frames, and extracted mean, variance, skewness, kurtosis and root mean square features from each frame.
The choice of window length was kept such that the whole eating activity could be completed within this time window
We trained Random Forest (RF) model and reported binary F1-scores from the average of three runs on on Wild-7 dataset.
\thomas{See my comment above on a separate subsection that covers the recognition backend detals.}
\subsection{Recognition of Daily Activities with Subtle Motions}
\subsubsection{Benchmark Real IMU Dataset}
Many daily activities include far more subtle motions compared to eating activities.
To further test IMUTube systems on such fine activities, we used HAD-AW dataset \cite{inproceedings} as our benchmark dataset, which consists of 31 activity classes, including bed-making, flipping pancakes, and showering, in total that is collected from a smart watch.
The volunteers' age ranges from 25 to 55 years old, weights from 55 kg to 95 kg and have representation from both males and females. The data is collected using an Apple Watch worn by the candidates and have 160 samples on average per activity.
Not all of 31 activity classes are fine activities.
Thus, we selected 17 activity classes for our experiments:
playing violin,
playing piano,
playing guitar,
driving automatic,
driving manual,
reading,
writing,
eating,
cutting components,
washing dishes,
washing hands,
showering,
sweeping,
wiping,
drawing,
flipping,
bed-making.
\subsubsection{Curated Video Dataset}
\input{figure/daily_activity}
For the feasibility test, we collected well curated video dataset from Kinetics-400 dataset for 17 activity classes used for experiments.
In total, we collected virtual IMU dataset with approximately 50\% of the size of real IMU dataset from HAD-AW dataset.
As shown in \autoref{fig:daily_activity_data_proportion}, the virtual IMU samples are imbalanced due to many video clips posing challenges to extract virtual IMU for wrist movements.
IMUTube system automatically filters the scenes that are challenging for effective motion tracking, which has risk of collecting low-quality virtual IMU data, such as scenes including zoomed-in focus on hand movements not showing entire arm movements or occlusions on wrist motions due to viewing angles.
\subsubsection{In-the-wild YouTube Video Dataset}
\hyeok{TK-Waiting for Zikang to get the data and results.}
\subsubsection{Experiment Protocol}
For the experiment with HAD-AW dataset, the real IMU datasets were downsampled to 25 Hz to match the sampling rate with virtual IMU datasets from videos.
We applied 5-fold stratified cross-validation.
we could not test leave-one-user-out cross validation, as some activities were only performed by a single user.
All real and virtual IMU data were segmented with sliding window of 3-second analysis frames with step size of 1.5-second following the previous work~\cite{inproceedings}.
We extracted ECDF features~\cite{hammerla2013preserving} from each analysis frame.
Our baseline model was the model trained only with real IMU dataset, and proposed model was the model trained with both real and virtual IMU datasets.
We used Random forest model for all experiments and reported mean F1 score average from three runs from all cases.
When we use virtual IMU dataset, we calibrated virtual IMU dataset with the training real IMU data at each fold likewise the standard protocol used in IMUTube.
\section{Experiment}
\thomas{See my other comment. I am very confused why the entire logic has been put upside down now.}
In our experiments, we will analyze the effectiveness of MSI for quantifying the subtlety of movements in various daily activities through conduct correlation study between MSI and rMSI. \thomas{No! rMSI has no proven relevance for any practical outcome. At the end of the day, it is the F1 scores on HAR tasks that matter.}
Then, we compare recognition performance for daily activity classification tasks using virtual IMU data MSI and rMSI with and without.
Here, we introduce our experiment settings and the benchmark datasets used for annalysis.
\subsection{Classification with Virtual IMU Data, MSI, and rMSI}
We followed the approach used in the original IMUTube experiments \cite{kwon2020imutube}.
We extracted virtual IMU data for wrist sensors from video datasets, which are calibrated with (the small amounts of) real IMU data used for training.
Real IMU data is subsampled to 25 Hz to match the sampling rate of virtual IMU data from videos.
Both virtual and real IMU data are segmented into analysis frames, for example 4-seconds window, using sliding window operation.
The choice of window size depends on the particular activity of interest
For calculating MSI, we segment video sequence where each virtual IMU analysis frame is originated from.
Then, optical flow~\cite{farneback2003two} and 2D pose~\cite{fang2017rmpe} is estimated from video segments, which are used to calculate MSI for each virtual IMU analysis frame.
rMSI is directly calculated from the segmented real IMU analysis frames.
Based on the calculated rMSI and MSI, the virtual IMU analysis frames are selected for model training according to low/mid/high motion subtlety for involved activity classes.
Then, reatures are extracted from both real and virtual analysis windows, which is used to train a classification backend model, where we use Random Forest in this study.
Lastly, the trained model is then tested on unseen real IMU dataset.
Our baseline models are 1) the model trained only with real IMU data and 2) the model trained with both real and virtual IMU dataset.
Our proposed model additionally uses MSI and rMSI when trained with both real and virtual IMU dataset.
Virtual IMU datasets are either from curated or in-the-wild video dataset as explained below.
\input{tables/dataset}
\subsection{Benchmark Real IMU Dataset}
In this work, we do not consider activities with inherently coarse movements, such as locomotion or exercises, that involves nearly full-body motion signals, which is already well-studied in the previous work~\cite{kwon2020imutube,kwon2021approaching}.
For studying subtle motions in daily activities, we use two benchmark datasets for wrist sensors: daily activity classification~\cite{inproceedings} and eating activity classification~\cite{thomaz2015practical}.
\autoref{tab:dataset} shows statistics of the dataset used in our experiments.
\subsubsection{Daily Activity}
We used HAD-AW dataset \cite{inproceedings} as our benchmark dataset, which consists of 31 activity classes from subtle to coarse motions collected using an Apple Watch.
Out of all of 31 activity classes, we selected 17 activity classes with subtle motions for our experiments:
playing violin,
playing piano,
playing guitar,
driving automatic,
driving manual,
reading,
writing,
eating,
cutting components,
washing dishes,
washing hands,
showering,
sweeping,
wiping,
drawing,
flipping,
bed-making.
\subsubsection{Eating Activity}
We used a large public dataset from Thomaz~\emph{et al}.~\cite{thomaz2015practical} that contains both in-lab and in-the-wild real IMU data for benchmarking IMUTube.
The \textit{Lab-20} dataset was collected from 21 participants in the lab covering both eating and non-eating activities for approximately 31 minutes period on average.
The eating moments involve,
eating with fork \& knife,
hand,
and spoon.
While non-eating activities include
watching trailer, conversation,
brush teeth,
place a phone call etc., all being subtle hand motions.
For in-the-wild datset, we used the \textit{Wild-7} dataset \cite{thomaz2015practical}, which was collected from seven participants.
Out of a total of 31 hours and 28 minutes of data, 2 hours and 8 minutes were labeled as eating activities.
\subsection{Curated Video Dataset}
For assessing the feasibility for using virtual IMU data for subtle motions, we first collected virtual IMU data from well curated video dataset, Kinetics-400 \cite{kay2017kinetics}, where all videos were manually trimmed down to clips of 10 seconds to contain specific activity classes.
\subsubsection{Daily Activity}
\input{figure/daily_activity}
We collected well curated video dataset from Kinetics-400 dataset for 17 activity classes used for experiments.
In total, we collected virtual IMU dataset with approximately 50\% of the size of real IMU dataset from real IMU dataset.
As shown in \autoref{fig:daily_activity_data_proportion}, the virtual IMU samples are imbalanced, as many video clips are removed due to the challenging scenes for extracting virtual IMU for wrist movement.
\subsubsection{Eating Activity}
We used 417 video clips for the eating class, which were labeled as one of 10 eating-related classes (eating: burger, cake, carrots, chips, doughnuts, hotdog, ice cream, nachos, spaghetti, watermelon).
The extracted virtual IMU data contained approximately 6\% more eating instances compared to Lab-20 dataset.
\subsection{In-the-wild Video Dataset}
YouTube houses one of the largest collection of uncurated videos that are coarsely labeled using Google's search algorithm, providing a virtually unlimited source of videos for any activity.
As shown in \autoref{fig:eating}, videos from online repositories contain scenes that do not involve human eating motions, i.e., scenes that IMUTube~\cite{kwon2021approaching} can automatically filter out with Adaptive Video Selection modules (yellow box;\autoref{fig:imutube}).
We downloaded videos from YouTube using activity class as search keywords, where queried videos are sorted according to relevance.
\subsubsection{Daily Activity}
We downloaded one video for each class, which were approximately 8 minutes of video clips on average per activity.
We could collect approximately 16\% size of virtual IMU data compared to real IMU benchmark dataset.
\subsubsection{Eating Activity}
We download two YouTube videos, where the first video is 22 minutes long, and the second one has a duration of 17 minutes.
From those two videos combined, we could collect 11\% more eating data samples compared to eating instances in Lab-20 dataset.
\input{figure/eating}
\subsection{Hyperparameters}
Here we explain specific settings and hyperparmeters we used for dailty activity and eating recognition tasks.
\subsubsection{Daily Activity}
We applied 5-fold stratified cross-validation.
we could not test leave-one-user-out cross validation, as some activities were only performed by a single user.
All real and virtual IMU data were segmented with sliding window of 3-second analysis frames with step size of 1.5-second following the previous work~\cite{inproceedings}.
We extracted ECDF features~\cite{hammerla2013preserving} from each analysis frame.
Threshold values when using MSI and rMSI were $TH^V_L=3$, $TH^V_H=6$, $TH^R_L=0.05$, and $TH^R_H=0.1$.
Among 17 classes, the low MSI group included reading, driving automatic, drawing, eating, playing guitar, the mid MSI group included playing violin, writing, driving manual, flipping, playing piano, cutting components, wiping, and the high MSI group included bed making, shower, sweeping, washing hands, washing dishes.
We reported mean F1 score average from three runs from all cases.
\subsubsection{Eating Activity}
We use Lab-20 dataset as training set and Wild-7 as testing set.
Accordingly, virtual IMU data was calibrated with Lab-20 dataset.
Following previous eating studies \cite{thomaz2015practical}, we applied sliding window segmentation with 6 second window size and 50\% overlap between consecutive analysis frames, and extracted mean, variance, skewness, kurtosis and root mean square features from each frame.
The predictions from the trained Random Forest (RF) model was applied with DBSCAN clustering to determine eating moments with the best hyperparameter from previous work \cite{thomaz2015practical}, where a minimum number of intake gestures is two (minPts=2) and a distance measure given as a temporal neighborhood was 80 (eps=80).
Threshold values when using MSI and rMSI were $TH^V=4.5$ and $TH^R=0.06$.
We reported binary F1-scores from the average of three runs on on Wild-7 dataset.
\section{Introduction}
The effectiveness of supervised learning methods for deriving human activity recognition systems (HAR) for wearables depends heavily on the availability of curated, i.e., annotated datasets \cite{chen2021sensecollect}.
One major issue with current machine learning solutions in the field is the paucity of labeled datasets.
Annotating sensor data in HAR is expensive, often privacy invasive, and often prone to errors or has other practical limitations \cite{kwon2019handling,jiang2021research,cilliers2020wearable}.
Recently, systems like IMUTube \cite{kwon2020imutube} have been introduced that tackle the aforementioned problem by generating virtual IMU data from videos
to increase the size of training datasets that can be used for model training.
IMUTube was designed to recognize substantial body movements through estimating the movements of 3D body keypoints from single-view videos.
The system has been successfully validated with virtual IMU data for wrist sensors for locomotion and gym exercise activities, i.e., targeting activities that are based on substantial body movements (or of parts thereof).
\input{figure/imutube}
With the previous success of systems like IMUTube (Fig.\ \ref{fig:imutube}) for recognizing human activities with underlying coarse motions, the next step now is to explore to what extent such approaches generalize to activities with more subtle body movements.
Many important activities of daily living are more subtle, exhibiting much lesser motion differences between and more variance within activities, e.g.,
eating, holding (and talking on) a phone, or washing dishes, etc~\cite{thomaz2015practical}.
As such, it is desirable to understand how substantial the body (part) movements need to be for systems like IMUTube to be of practical value.
The work presented in this paper targets exactly that aforementioned problem:
We develop a measure for the quantitative assessment of the subtlety of movements and correlate it to the effectiveness of HAR systems that have been derived using virtual IMU training data generated using IMUTube on unconstrained videos of target activities.
As such, we aim to find the ``breaking point'' of IMUTube with respect to when the motions assigned with activities of interest are too subtle for the system to be beneficial for designing HAR systems.
This analysis allows us to formulate concrete guidelines for HAR practitioners who plan to employ cross-modality transfer systems for their HAR model development efforts.
We hypothesize that the amount of human motion in a video sequence can be captured by measuring local pixel movements in the vicinity of the target on-body sensor locations.
We define a novel metric--the \textit{Motion Subtlety Index} (MSI)--that measures the subtlety of motion of human activities performed in a video sequence with respect to the particular on-body sensor locations of interest by using optical flow and pose estimation methods.
Our experimental evaluation on a range of activities of daily living shows that the MSI extracted from human activity videos is highly correlated to the
the eventual recognition accuracy of HAR systems that were derived using virtual IMU data extracted from videos.
As such, the MSI is an excellent proxy that can be used for the \textit{a-priori} prediction of the potential effectiveness of cross-modal transfer approaches.
With this new measure, we are now in the position to ``stress-test'' the IMUTube system.
We study its effectiveness for virtual wrist sensors on a much more extensive range than in the original publications when the system was first introduced.
We focus on activities with more subtle motions than the coarse motions studied previously, including .
washing hands, playing instruments, driving, and so on, which are essential activities when it comes to assessing an individuals' quality of life \cite{inproceedings}.
Our experiment shows that activities with more subtle motions can benefit from virtual IMU data extracted from videos.
In the second part of our study we focus specifically on an eating detection task \cite{thomaz2015practical} based on wrist-worn IMUs, which has important applications in health assessments \cite{morshed2022food,konttinen2019depression,navarro2021effects,van2018causes}.
On this task, IMUTube showed surprising effectiveness for improving recognition accurcy with gains of 8.4\% absolute when using curated video data and of 5.9\% absolute (both F1-scores) when using unconstrained video data as they were retrieved through a keyword-based search on YouTube.
Such performance improvements are significant because IMUTube was designed with coarse, i.e., significant body motion in mind, contrasting with the subtle movements in daily activities.
The contributions of this paper are two-fold:
\begin{enumerate}
\item By correlating activity recognition accuracy, measured in F1-score, and MSI, our study allows us to identify when IMUTube ``reaks''. i.e., to assess the application range of cross-modality transfer approaches for developing HAR systems.
\item Through the newly introduced quantification of the subtlety of human movements and its correlation to the eventual effectiveness of HAR systems that were derived based on virtual IMU data, we can draw conclusions about application cases for systems like IMUTube and, essentially, map out the landscape for IMUTube applications in practical scenarios.
\end{enumerate}
Our results shall encourage the HAR community to investigate further the use of cross-modality transfer approaches such as IMUTube for various complex and subtle human activities.
\section{Quantifying Motion Subtlety in Human Activity Video}
The IMUTube system was originally designed to capture substantial body (part) movements in human activity videos -- information, which is subsequently transferred to \textit{virtual} IMU data.
In this work, we explore how such a system can be used when studying human activities with only \textit{subtle} underlying movements.
Essentially, we aim to find the ``breaking point'' of IMUTube-like approaches, i.e., what the limits of its applicability are -- beyond the originally studied coarse movement activities.
We define subtle motions in human activities as those movements that involve one or two body parts moving in the very limited range of distances.
For example, activities such as washing hands, writing, cooking, eating, and so on, involve only hand or arm movements that are much smaller in range compared to activities that are based on whole-body involvement, such as sports or gym exercises.
\input{figure/motion_subtlety}
3D human motion tracking is at the heart of IMUTube.
Although state-of-the-art 3D human motion tracking techniques are effective at capturing whole body movements, they are still limited at capturing subtle movements that only involve local body parts \cite{desmarais2021review}.
We consider the IMUTube system as upper bounded by the amount of motion that motion tracking techniques can capture in videos of human activities.
Therefore, we hypothesize that IMUTube will start to struggle to generate useful virtual IMU data as the motions involved in underlying activity become more subtle.
With this in mind, it is important to identify early on for which activities IMUTube may ``break'' before actually allocating resources and time to generate virtual IMU data and develop activity recognition model~\cite{hiremath2020deriving}.
For an objective quantification of the subtlety of movements involved in an activity of interest, we introduce a novel metric, namely \textit{Motion Subtlety Index} (MSI).
This metric measures the motion subtlety of activities from videos, which supports the aforementioned early decision making whether using IMUTube is beneficial or not.
Specifically, we combine optical flow \cite{zhai2021optical}
and 2D pose estimation \cite{dubey2022comprehensive} from a video sequence to quantify the pixel-level motion quantity in the local area near the targeted on-body sensor location.
\autoref{fig:msi} illustrates how the MSI is calculated for an exemplary video segment that captures a sequence of a writing activity with the wrist as the target location for the virtual IMU sensor.
For each frame in a video segment with $T$ frames, we first compute the optical flow and estimate 2D poses.
The estimated optical flow at each pixel and time is normalized according to frame size to take account for different resolution of videos available, $(u^t_i, v^t_i) \rightarrow (u^t_i/H, v^t_i/W)$, where $(u^t_i, v^t_i)$ are vertical and horizontal optical flow at pixel $i$ and time $t$ and $(H, W)$ are height and width of frame size of the video.
Next, at each frame, we calculate the average magnitude of the normalized optical flow at the local patch, $K\times K$, in the neighborhood of the wrist keypoint location, which is automatically detected by our 2D pose estimation procedure~\cite{fang2017rmpe}.
To take account of varying resolution of video frames, the patch size is 2\% of the larger dimension of the frame, $K=0.02\times max(H, W)$.
\begin{equation}
MSI_t = \frac{1}{N} \sum_{-\frac{K}{2}\geq i,j \geq \frac{K}{2}} \sqrt{(u^t_{n + i})^2 + (v^t_{n + j})^2},
\end{equation}
where $u^t_n$ and $v^t_n$ are vertical and horizontal components of the normalized optical flow measurements from the keypoint location at time $t$, and $N=K \times K$.
The MSI for the analysis window is then computed as the exponential of negative standard deviation of $MSI_{1\cdots T}=[MSI_1, MSI_2, \cdots, MSI_T]$ :
\begin{equation}
MSI = e^{-w\times std(MSI_{1\cdots T})}
\end{equation}
We used $w=100$ to account for the minimal difference in MSI where $std(MSI_{1\cdots T}) \approx 0$.
Overall, MSI captures the motion information recorded around the on-body sensor location for underlying activities in the given video sequence.
A smaller MSI means more significant motions are involved with ongoing activities, whereas a larger MSI indicates more subtle movements. Note that computing the MSI for any given video segment takes far less time and resources than extracting the virtual IMU data from the video segment using IMUTube.
\iffalse
\input{figure/overall_system}
\subsubsection{Motion Subtlety in Real IMU Data}
When it comes to quantifying subtlety of motions in real IMU dataset, we can directly compute \textit{real Motion Subtlety Index} (rMSI) from captured signals.
Given a sequence of accelerometry signal in the analysis frame, $X^{T\times 3}$, where $T$ is the duration of real IMU data, we first remove the gravity component, $\hat{X}_t = X_t - X_g$, which is the average of accelerometry signal, $X_g = \frac{1}{T}\sum_{0 \leq t \leq T} X_t$.
Then, the magnitude of acceleration is computed, $\bar{X}_t = \sqrt{\hat{x}^2_t + \hat{y}^2_t + \hat{z}^2_t}$, where $\hat{X}_t = [\hat{x}, \hat{y}, \hat{z}]$.
Lastly, rMSI is defined as the standard deviation of the magnitude of accelerometry signal without gravity components, $rMSI = std([\bar{X}_1, \bar{X}_2, \cdots, \bar{X}_T])$.
Overall, high rMSI value indicates larger motions involved for performing the on-going activity.
\thomas{I do not understand why we need the MSI on IMU?}
\subsection{IMUTube with Subtle Motions}
We hypothesize that activities with subtle motions can be further categorized into more, mildly, or less subtle activities, and that virtual IMU data is most useful when it is extracted from video sequence performing similar level of subtle motion as in target activity. \thomas{Where does that come from? I don't think it makes much sense. All we set out to is to quantify the subtlety and, then, to correlate that quantitative measure to eventual HAR accuracy outcomes. So, this sort of speculation here is, imho, not helpful, if not hurting.}
The proposed method is aimed to match the subtlety of motion captured in virtual and real IMU data for the model training so that potentially irrelevant virtual IMU data for particular activity class is avoided for using. \thomas{First time I hear about this ... Does not make much sense to me.}
Given activity classes of interest, each activity are categorized into three different amount (low, mid, high) of motions involved in performing activities.
Once rMSI is computed for all the analysis frames for real IMU data, the mode of rMSI per class, rMSI$_c$, is derived from the distribution of rMSI for each class to represent motion subtlety for each class.
Then, each class is categorized to low, mid, and high rMSI values using the threshold values, $Th^R_L$ and $TH^R_H$, for low and high rMSI, respectively.
We empirically selected $Th^R_L$ and $TH^R_H$ as 33 and 66 percentiles of the collection of rMSI$_c$ from all classes.
low, mid, and high rMSI classes are determined through $rMSI_c < Th^R_L$, $Th^R_L \leq rMSI_c \leq Th^R_H$, and $Th^R_H < rMSI_c$, respectively.
We also categorize low, mid, and high MSI for virtual IMU dataset.
From the collection of MSIs, $MSI^i$, from all virtual IMU data frames, we also select 33 and 66 percentiles, $Th^V_L$ and $Th^V_H$, as cut-off values to distinguish low, mid, and high MSI.
low, mid, and high MSI windows are determined through $MSI^i < Th^V_L$, $Th^V_L \leq MSI^i \leq Th^V_H$, and $Th^V_H < MSI^i$, respectively.
Now that we have categorized virtual IMU analysis frames that belong to low, mid, and high MSI, and categorized activity classes that belongs to low, mid, and high rMSI$_c$, we use this categorization for training the activity recognition model.
Specifically, for each class, we only use virtual IMU data that belongs to same category of motion subtlety as the activity class.
For example, if an activity class is categorized as low rMSI$_c$, then we only use virtual IMU data sample of the same class that is categorized as low MSI.
This is done similarly for other activities that is categorized as mid and high motion subtlety.
The illustration of entire process is shown in \autoref{fig:overall_system}.
For the case with binary classification, we use similar approach but with low and high categorization of the two activities and virtual IMU analysis window.
For low and high activity class, single threshold value for rMSI$_c$ and MSI, $Th^R$ for and $Th^V$, respectively.
$Th^R$ is median of two rMSI$_c$ to put each class into either low or high motion classes.
$Th^V$ is median of all MSI samples from both classes to categorize each virtual IMU analysis frame into either low or high motion classes.
\thomas{Sorry, but I do not understand the purpose of this entire section B. What we had in mind was to come up with a quantitative measure for the subtlety of movements belonging to activities -- to be calculated on videos. With that measure, we do have a characterization of a task (activity recognition) in the video domain. We then go ahead and translate a bunch of activity videos into virtual IMU data -- using IMUTube -- and train downstream HAR classifiers as we did before. We then get F1 scores for all those recognition tasks (so, look at class-specific F1 scores) and are now--post mortem--able to correlate those F1 scores back to the MSI scores (again, computed on videos). Our analysis can then tell us: for which activities does IMUTube still work. More systematically: Up to what threshold of MSI (on videos!) can we expect IMUTube to still work. So, for future application cases one would go ahead and calculate MSI values on a bunch of videos BEFORE even doing the whole IMUTube application -- which is quite a bit of effort. As such, we will be able to provide guidance on when to use IMUTube -- and when not -- through the systematic analysis of MSI as evidenced through the experiments in this paper that correlate MSI (on video) to downstream F1 scores on actual HAR experiments that were run using virtual IMU data as generated by IMUTube. So, our original goal is still to look for when IMUTube breaks but now with the additional underlying systematic approach of actually measuring the subtlety!}
\fi
\section{Method}
\subsection{Quantifying Motion Subtlety in Human Activity}
The IMUTube system was originally designed to capture substantial body (part) movements in human activity videos -- information, which is subsequently transferred to \textit{virtual} IMU data.
In this work, we aim to study how such system can be used when studying human activities with only \textit{subtle} underlying movements.
Essentially, we aim to explore what the ``breaking point'' of IMUTube-like approaches us, i.e., what the limits of its applicability--beyond the originally studied coarse movement activities--are.
We define subtle motions in human activities as those movements that involve one or two body parts moving in the very limited range of distances.
For example, activities such as washing hands, writing, cooking, eating, and so on, involve only hand or arm movements that are much smaller in range compared to activities that are based on whole-body involvement, such as sports or gym exercises.
\input{figure/motion_subtlety}
\subsubsection{Motion Subtlety in Virtual IMU Data}
\thomas{Hold on, folks. Aren't we measuring the MSI on video data? So, why is the focus here on virtual IMU data?}
To objectively define the amount of motion subtlety in virtual IMU data, we propose \textit{Motion Subtlety Index} (MSI).
We measure motion subtlety of virtual IMU data from original source video dataset.
Specifically for human motion in the scene, 2D pose estimation is widely used to capture how much joint keypoints moved between subsequent frames~\cite{dubey2022comprehensive}.
However, pose estimation is limited in quantifying very detailed pixel-level motions, as 2d keypoints are only approximated locations of human joint locations.
This applies similarly to the virtual IMU dataset, as virtual IMU data is derived from pose estimation method so that quantifying motion subtlety from virtual IMU data directly is not sufficient for pixel-level motion subtlety.
On the other hand, conventional approaches to measure the any detailed movements in a video sequence is through the use of optical flow~\cite{zhai2021optical}.
Optical flow measures the displacement of pixels between subsequent frames, which tells the amount of motion in the local regions in each frame.
Therefore, we combine optical flow and 2D pose estimation from video sequence to quantify the pixel-level motion quantity, Motion Subtlety index (MSI) in other words.
\thomas{See my comment above: This should be about videos! Our motivation is to have some quantification that allows us to estimate the value virtual IMU data may have for HAR down the line BEFORE we do all the virtual IMU data extraction and the HAR modeling. In the language we used earlier on: we want to quantify the subtlety of human movements in videos and correlate it to the effectiveness of IMUTube-based HAR systems. So, this paragraph needs to be revised please and be made more clear. (next are fine) }
\autoref{fig:msi} illustrates how the MSI is calculated for an exemplary video that captures a sequence of writing activity with the wrist as the target location for the virtial IMU sensor.
For each of the sequence of $T$ video frames, we first compute the optical flow and estimate 2D poses.
Next, we calculate the average magnitude (over the entire sequence) of optical flow at the local patch, $5\times 5$, in the neighborhood of the wrist keypoint location, which is automatically detected by our 2D pose estimation procedure:
\begin{equation}
MSI_t = \frac{1}{N} \sum_{-2\geq i,j \geq 2} \sqrt{(v_{k + i})^2 + (u_{k + j})^2},
\end{equation}
where $v$ and $u$ are vertical and horizontal optical flow measurements from keypoint location, $(v_k, u_k)$, and $N=5 \times 5 = 25$ at time $t$.
The MSI for the analysis window is then computed as the standard deviation of $MSI_t$ across all analysis frames:
\begin{equation}
MSI = std([MSI_1, MSI_2, \cdots, MSI_T])
\end{equation}
MSI captures the amount of motion information recorded around the on-body sensor location for underlying activities in the given video sequence.
A smaller MSI means more subtle motions are involved with on-going activities, whereas a larger MSI indicates more coarse movements.
\input{figure/overall_system}
\subsubsection{Motion Subtlety in Real IMU Data}
When it comes to quantifying subtlety of motions in real IMU dataset, we can directly compute \textit{real Motion Subtlety Index} (rMSI) from captured signals.
Given a sequence of accelerometry signal in the analysis frame, $X^{T\times 3}$, where $T$ is the duration of real IMU data, we first remove the gravity component, $\hat{X}_t = X_t - X_g$, which is the average of accelerometry signal, $X_g = \frac{1}{T}\sum_{0 \leq t \leq T} X_t$.
Then, the magnitude of acceleration is computed, $\bar{X}_t = \sqrt{\hat{x}^2_t + \hat{y}^2_t + \hat{z}^2_t}$, where $\hat{X}_t = [\hat{x}, \hat{y}, \hat{z}]$.
Lastly, rMSI is defined as the standard deviation of the magnitude of accelerometry signal without gravity components, $rMSI = std([\bar{X}_1, \bar{X}_2, \cdots, \bar{X}_T])$.
Overall, high rMSI value indicates larger motions involved for performing the on-going activity.
\thomas{I do not understand why we need the MSI on IMU?}
\subsection{IMUTube with Subtle Motions}
We hypothesize that activities with subtle motions can be further categorized into more, mildly, or less subtle activities, and that virtual IMU data is most useful when it is extracted from video sequence performing similar level of subtle motion as in target activity. \thomas{Where does that come from? I don't think it makes much sense. All we set out to is to quantify the subtlety and, then, to correlate that quantitative measure to eventual HAR accuracy outcomes. So, this sort of speculation here is, imho, not helpful, if not hurting.}
The proposed method is aimed to match the subtlety of motion captured in virtual and real IMU data for the model training so that potentially irrelevant virtual IMU data for particular activity class is avoided for using. \thomas{First time I hear about this ... Does not make much sense to me.}
Given activity classes of interest, each activity are categorized into three different amount (low, mid, high) of motions involved in performing activities.
Once rMSI is computed for all the analysis frames for real IMU data, the mode of rMSI per class, rMSI$_c$, is derived from the distribution of rMSI for each class to represent motion subtlety for each class.
Then, each class is categorized to low, mid, and high rMSI values using the threshold values, $Th^R_L$ and $TH^R_H$, for low and high rMSI, respectively.
We empirically selected $Th^R_L$ and $TH^R_H$ as 33 and 66 percentiles of the collection of rMSI$_c$ from all classes.
low, mid, and high rMSI classes are determined through $rMSI_c < Th^R_L$, $Th^R_L \leq rMSI_c \leq Th^R_H$, and $Th^R_H < rMSI_c$, respectively.
We also categorize low, mid, and high MSI for virtual IMU dataset.
From the collection of MSIs, $MSI^i$, from all virtual IMU data frames, we also select 33 and 66 percentiles, $Th^V_L$ and $Th^V_H$, as cut-off values to distinguish low, mid, and high MSI.
low, mid, and high MSI windows are determined through $MSI^i < Th^V_L$, $Th^V_L \leq MSI^i \leq Th^V_H$, and $Th^V_H < MSI^i$, respectively.
Now that we have categorized virtual IMU analysis frames that belong to low, mid, and high MSI, and categorized activity classes that belongs to low, mid, and high rMSI$_c$, we use this categorization for training the activity recognition model.
Specifically, for each class, we only use virtual IMU data that belongs to same category of motion subtlety as the activity class.
For example, if an activity class is categorized as low rMSI$_c$, then we only use virtual IMU data sample of the same class that is categorized as low MSI.
This is done similarly for other activities that is categorized as mid and high motion subtlety.
The illustration of entire process is shown in \autoref{fig:overall_system}.
For the case with binary classification, we use similar approach but with low and high categorization of the two activities and virtual IMU analysis window.
For low and high activity class, single threshold value for rMSI$_c$ and MSI, $Th^R$ for and $Th^V$, respectively.
$Th^R$ is median of two rMSI$_c$ to put each class into either low or high motion classes.
$Th^V$ is median of all MSI samples from both classes to categorize each virtual IMU analysis frame into either low or high motion classes.
\thomas{Sorry, but I do not understand the purpose of this entire section B. What we had in mind was to come up with a quantitative measure for the subtlety of movements belonging to activities -- to be calculated on videos. With that measure, we do have a characterization of a task (activity recognition) in the video domain. We then go ahead and translate a bunch of activity videos into virtual IMU data -- using IMUTube -- and train downstream HAR classifiers as we did before. We then get F1 scores for all those recognition tasks (so, look at class-specific F1 scores) and are now--post mortem--able to correlate those F1 scores back to the MSI scores (again, computed on videos). Our analysis can then tell us: for which activities does IMUTube still work. More systematically: Up to what threshold of MSI (on videos!) can we expect IMUTube to still work. So, for future application cases one would go ahead and calculate MSI values on a bunch of videos BEFORE even doing the whole IMUTube application -- which is quite a bit of effort. As such, we will be able to provide guidance on when to use IMUTube -- and when not -- through the systematic analysis of MSI as evidenced through the experiments in this paper that correlate MSI (on video) to downstream F1 scores on actual HAR experiments that were run using virtual IMU data as generated by IMUTube. So, our original goal is still to look for when IMUTube breaks but now with the additional underlying systematic approach of actually measuring the subtlety!}
\section{Related Work}
In this work, we analyze to what extent virtual IMU data from wrist sensors are of value for deriving recognition systems that target activities with underlying subtle motions.
In what follows, we briefly explain the virtual sensor generation method through IMUTube, on which our work is based, before summarizing previous work in classifying fine motion activities in daily life.
\subsection{IMUTube: Virtual IMU Data from Videos}
IMUTube was proposed to tackle the challenges that come with the collection of labeled wearable data such as privacy, accuracy, but also practical and logistical obstacles.
The system automatically converts large-scale video datasets into virtual IMU data that can be used for training sensor-based HAR systems \cite{kwon2020imutube, kwon2021approaching,kwon2021complex}.
As shown in \autoref{fig:imutube}, IMUTube resembles a processing pipeline that incorporates computer vision, graphics, and machine learning models to extract virtual IMU data from human activity videos for various on-body locations.
The system consists of three main parts:
\textit{i)} adaptive video selection;
\textit{ii)} 3D human motion tracking; and
\textit{iii)} virtual IMU data extraction and calibration.
When querying YouTube with activity keywords, for example, eating, the returned videos may contain:
\textit{i)} many sequences that are irrelevant to human activities, such as intros and outros; or
\textit{ii)} video sequences that are very challenging to track human motions accurately.
To identify video segments that will lead to high-quality virtual IMU data, the adaptive video selection module actively filters video sequences with irrelevant frames, noisy poses, occlusions, or extreme foreground and background camera motions.
The remainder of the videos selected from this process is then processed to estimate 3D joint rotation and global motion for extracting 3D motion information of human activities.
Virtual IMU data is then extracted using the 3D motion information.
The extracted virtual IMU data is calibrated using a small amount of target real IMU data to overcome the inevitable gap between the distributional characteristics of the two domains.
IMUTube was mainly designed, deployed and validated for human activities with underlying coarse motions, such as locomotion or gym exercises.
Yet, many real-world activities are based on only subtle body (part) movements and, hence, in this paper we explore how far the application range of approaches like IMUTube reaches.
\subsection{Sensor-Based Recognition of Daily Activities}
To assess well-being or quality of life for individuals, understanding activities in daily routines is important. \cite{kushner2013lifestyle,phillips2020lifestyle}
For example, to assess the--potentially sedentary--lifestyle of individuals, early works in human activity recognition with wearable sensors (HAR) focused on detecting locomotion activities, including sitting or walking \cite{stisen2015smart,sztyler2016body,zhang2012usc}.
After achieving success in recognizing locomotion activities, more recent work explored the application of wearable-based human activity recognition to fine-grained daily activities that provide more context for how users are situated.
Due to the popularity of commodity wrist devices, many works have studied fine-grained activity recognition with wrist sensors.
Laput and Harrison~\cite{laput2019sensing} demonstrated that fine-grained hand activities, such as writing or typing on a keyboard, can be detected by accelerometry data from wrist motions.
Moschetti~\emph{et al}.~\cite{moschetti2017daily} used wrist and index finger motion data to recognize eating, drinking, and brushing teeth.
Ashri~\emph{et al}.~\cite{inproceedings} have explored the use of wrist motion sensors to recognize daily activities such as driving, getting dressed, or house cleaning.
Other work showed that wrist or arm sensors could successfully classify multiple gym exercises~\cite{morris2014recofit,koskimaki2017myogym,koskimaki2014recognizing,um2017exercise}.
With the increasing attention to eating activities for measuring risks in physical and mental health~\cite{konttinen2019depression,navarro2021effects,van2018causes}, some work has focused on designing HAR models that are specific to eating detection using werables~\cite{thomaz2015practical,bedri2017earbit,bedri2020fitbyte,bedri2015wearable}.
Eating as an activity is a very subtle motion.
It involves repeated hand movement to the mouth, which varies vastly depending on the eating habits and the food consumed.
One of the critical challenges in a fully automated food intake monitoring system is to identify whether the eating has happened or not within a given time frame.
Past works have tried to solve this problem using training an eating moment recognition method on carefully curated in-lab datasets and then testing upon semi-controlled wild datasets\cite{thomaz2015practical,morshed2020real}.
While previous methods for recognizing daily activities were shown to be effective, they are still limited by the amount of lab data available for training.
For such activities with subtle motions, collecting large-scale labeled datasets is pivotal for improving the generalizability of the trained model.
\subsection{Eating Detection}
One of the primary examples of fine motion activity is eating~\cite{thomaz2015practical,bedri2017earbit,bedri2020fitbyte,bedri2015wearable}.
It involves repeated hand movement to the mouth, which varies vastly depending on the eating habits and food consumed.
There are also lots of similar movements in daily living that can create false positives for such detection methods.
\thomas{Trying not run in circles here, but: Why do we have a full section solely dedicated to eating detection? I thought we wanted to be broader? Has our focus been changed back to eating again?}
Amongst many daily activities, detecting eating activity has gained much interest in wearable-based human activity recognition community for its direct connection with severe health conditions, such as depression, anxiety, or obesity~\cite{konttinen2019depression,navarro2021effects,van2018causes}.
Eating detection is a mature field and many studies have been conducted in which a range of wrist based sensing and analysis systems have been utilized \cite{bedri2017earbit,bedri2015wearable,bi2017toward}.
Eating as an activity is a very subtle motion. It involves repeated hand movement to the mouth which varies vastly depending on the eating habits and the food consumed.
Many studies \cite{althubaiti2016information, bedri2017earbit, bedri2015wearable, bi2017toward, morshed2020real} have proposed many wrist based wearable systems to automatically monitor food consumption yet most of them are either based on special custom systems or are tested in a controlled environment. While other monitoring wearable try to match one of the actions that occur during food intake; e.g., hand movement while bringing the food to the mouth, biting and chewing, or swallowing \cite{dong2013detecting}. However, there are lots of proxy movements in daily living that can create false positives for such detection methods.
One of the key challenges in a fully automated food intake monitoring system is to identify whether the eating has happened or not within a given time frame. Past works have tried to solve this problem using training an eating moment recognition method on carefully curated in-lab datasets and then testing upon semi-controlled wild datasets \cite{thomaz2015practical}.
While these methods have shown robust accuracies, they are still limited by the amount of lab data available to them for training. Our work illustrates a potential solution to this problem in the form of an existing work IMUTube \cite{kwon2020imutube}, while also understanding various caveats of its working.
\thomas{Structure is ok -- if we agree on actually having a dedicated eating section in here (I thought we wanted to broaden the scope? Let's discuss.) -- but it needs serious proofreading and copy-editing. Grammar is a bit all over the place.}
\section{Results}
We first show overall classification performance when using virtual IMU data, as discussed earlier, for classifying activities with subtle motions. We then study how virtual IMU data impacted classification performance for each activity.
\autoref{tab:results} shows classification results for both daily activities, and eating recognition tasks for the models trained only with real IMU data, and for those that additionally use virtual IMU data either from curated or in-the-wild dataset.
\subsection{Daily Activity Classification}
\input{tables/results_no_msi_no_20}
Using additional virtual IMU data resulted in similar classification performance when compared to the case where only real IMU data was used.
We consider that the model relied heavily on real IMU data to capture very detailed movement patterns for the classification tasks as virtual IMU data could not capture the characteristics of those subtle motions from the video data.
The result shows the \textit{breaking point} of IMUTube as using virtual IMU data no longer improves classification performance and can actually degrade model performance. We see that the model used in-the-wild virtual IMU dataset performed better than the model that used the curated dataset. This is because the in-the-wild dataset contained less data than the curated dataset, resulting in less impact to the F1 score compared to the baseline model.
In the later analysis, we will quantify failing scenario of IMUTube according to the MSI analysis for daily activities
\subsection{Eating Moment Recognition}
Much to our surprise, the addition of virtual IMU data--generated using the unmodified IMUTube system, which, again, was originally designed for the analysis of coarse motion activities--was very effective for eating specific activity classification.
Different from the daily activities recognition task, adding virtual IMU data significantly improved the performance of the downstream model.
The baseline model using only real IMU data achieved 71.5\% F1-score, and our model, trained including the curated virtual IMU data, improved the result to 79.94\%.
For the in-the-wild-scenario we observe improvements of 5.9\% to 77.45\%.
We suspect this improvement came from the virtual IMU data containing wide range of fine-grained eating motions collected from videos for varying utensils or food types that involves large arm motions.
The dataset for the daily activities classification task only included "eating a sandwich" for eating. Upon examining videos of people eating a sandwich, we find that most of the body movements involve the head. The wrist and arm movements are limited compared to other kinds of eating activities, such as eating with a fork \& knife
\section{Results}
\input{figure/msi_corr}
\subsection{Correlation Analysis between MSI and rMSI}
Here, we first validate how the proposed MSI correlates with the motion subtlety actually measured from real IMU dataset, rMSI.
From 17 activity classes included in the benchmark daily activity datasets~\cite{inproceedings}, we collect the mode of MSI distribution for each class samples, MSI$_c$, from curated video dataset.
The modes of rMSI distribution for each class are collected directly from real IMU datasets.
\autoref{fig:msi_corr} shows the correlation analysis between MSI and rMSI.
Pearson correlation analysis demonstrated statistically strong correlation between two values ($r=0.746$, $p\leq 0.006$) across 17 activities validating that MSI can effectively quantify subtle motions from human activities in the video.
\subsection{Daily Activity Classification}
\autoref{tab:results} shows the classification performance measured in F1 score for daily activity and eating recognition tasks.
\subsubsection{Effect of Real IMU Data Scale}
We have simulated two cases for the application of virtual IMU data for the activities with subtle motions: the availability of small and large scale real IMU dataset.
For the cases with small IMU dataset, we used 20\% of real IMU data available in the training datasets, and for the cases with large scale real IMU dataset, we used all (100\%) real IMU dataset available.
The use of virtual IMU data was partially successful with those two cases.
Virtual IMU data could only improve the classification performance statistically significantly when small real IMU data was available.
For daily activity classification, compared to when only using 20\% of real IMU dataset, using curated and in-the-wild videos, relatively improved 4.2\% and 2.3\% F1 scores, respectively.
As expected, model trained with virtual IMU data from curated video outperformed that from in-the-wild video, which was consistent with the previous studies~\cite{kwon2021approaching}.
When we had sufficient number of real IMU dataset, the effect of using virtual IMU data was only marginal statistically showing very similar performances compared to real IMU dataset.
We consider that the model relied heavily on real IMU data to capture very detailed movement patterns for the classification tasks as the movements in virtual IMU data were either too simplistic to capture discriminative patterns.
IMUTube is designed to capture substantional movements using 3D joint motion tracking techniques, which tracks approximated estimation of joint locations that is limited in capturing subtle movements in the wrist.
\subsubsection{Effect of Using MSI}
The proposed method could statistically significantly improve the model by additionally quantifying the motion subtlety index for both virtual and real IMU data to select most effective virtual IMU data segements for model traning.
When small real IMU dataset is used, using MSI could improve relative 3$\%$ and \hl{X\%} when using virtual IMU data from curated and in-the-wild videos, respectively.
Similarly, when sufficiently large real IMU dataset was available, using MSI for being selectively in choosing virtual IMU data, could improve relative 2\% and \hl{X\%} of F1 score for daily activity classification tasks.
\input{tables/results}
\subsection{Eating Moment Recognition}
\subsubsection{Effect of Real IMU Data Scale}
\hyeok{Waiting for 20\% results from Zikang}
Much to our surprise, virtual IMU data was very effective for eating specific activity classification task.
Even when using all real IMU dataset, unlikely from the daily activity classification task, adding virtual IMU data have significantly improved the performance of the model.
The baseline model using only real IMU data achieved 71.5\% F1-score, and our model trained with the curated virtual IMU dataset improved that to 79.94\%.
The in-the-wild-scenario increased the recognition performance by 5.9\% to 77.45\%.
We suspect this improvement came from the virtual IMU data containing wide range of fine-grained eating motions collected from videos for varying utensils or food types.
(For the case of daily activity dataset, we only collected eating video for "eating sandwich" class, which were more limited in sample size and variations in motions)
\subsubsection{Effect of Using MSI}
\hyeok{Waiting for Zikang's results}
\section{Results}
\input{tables/results_no_msi}
Before discussing the classification performance changes with respect to MSI, we first demonstrate the classification results for daily activity and eating recognition tasks with the standard approach used with virtual IMU dataset. \thomas{Why?}\hyeok{To first to show that IMUTube fails for these classes}
Here, we additionally study the effect of virtual IMU data regarding the availability of real IMU dataset. \thomas{Why? That was done in previous papers.}\hyeok{So, using virtual IMU dataset was only hurting when using 100\% real IMU data. We wanted to find the point where at least virtual IMU data gives benefit and analyze there.}
In \autoref{tab:results}, we reported classification results either using 20\% or 100\% of real IMU dataset available in each dataset as we could observe different behaviors.
More detailed analysis on the effect of real IMU data scale is conducted in \autoref{sec:msi_group_real_size}.
\subsection{Daily Activity Classification}
This dataset includes activities that invovles very subtle motions such as writing, reading, or drawing activities in daily living.
For such activity, the use of virtual IMU data was partially successful, where virtual IMU data could only improve the classification performance statistically significantly when small real IMU data was available.
For daily activity classification, compared to when only using 20\% of real IMU dataset, using curated and in-the-wild videos, relatively improved 4.2\% and 2.3\% F1 scores, respectively.
As expected, model trained with virtual IMU data from curated video outperformed that from in-the-wild video, which was consistent with the previous studies~\cite{kwon2021approaching}.
In \autoref{sec:msi_f1_all}, we discuss more in detail how F1 score changes according to MSI values for the virtual IMU data used for model training.
When we had sufficient number of real IMU dataset, using virtual IMU data demonstrated to be statistically insignificant and resulted in very similar performances compared to real IMU dataset.
We consider that the model relied heavily on real IMU data to capture very detailed movement patterns for the classification tasks as the movements in virtual IMU data were either too simplistic to capture discriminative patterns.
IMUTube is designed to capture substantial movements using 3D joint motion tracking techniques, which tracks approximated estimation of joint locations that is limited in capturing subtle movements in the wrist.
\subsection{Eating Moment Recognition}
\hyeok{Waiting for 20\% results from Zikang}
Much to our surprise, virtual IMU data was very effective for eating specific activity classification task.
Even when using all real IMU dataset, differently from the daily activity classification task, adding virtual IMU data have significantly improved the performance of the model.
The baseline model using only real IMU data achieved 71.5\% F1-score, and our model trained with the curated virtual IMU dataset improved that to 79.94\%.
The in-the-wild-scenario increased the recognition performance by 5.9\% to 77.45\%.
We suspect this improvement came from the virtual IMU data containing wide range of fine-grained eating motions collected from videos for varying utensils or food types.
Daily activity classification dataset only included "eating sandwich" class, which were more limited in hand and arm motions compared to other kinds of eating activities.
|
2,877,628,088,967 | arxiv |
\section{Introduction}\label{sec:introduction}
The HTML \texttt{<canvas>}\xspace is used to display high-quality graphics in web applications, and is particularly useful for web games (i.e., \texttt{<canvas>}\xspace games)~\cite{parisi2012webgl, parisi2014programming, konstantinidis2016moving, yogya2014comparison, fulton2013html5}.
HTML5 \texttt{<canvas>}\xspace games are receiving growing attention from industry~\cite{google2020gamesnacks, goodboy2020playco}, but it is challenging to automatically test \texttt{<canvas>}\xspace games, as widely used web testing techniques and tools do not work for the \texttt{<canvas>}\xspace~\cite{macklon2022taxonomy}.
Many commonly used web testing tools leverage the Document Object Model (DOM) to drive test automation, but as demonstrated in Figure~\ref{fig:intro}, \texttt{<canvas>}\xspace graphics are not represented in the DOM.
\begin{figure}[t]
\centering
\setlength{\fboxsep}{0pt}%
\begin{subfigure}[t]{\linewidth}
\centering
\fbox{\includegraphics[width=\textwidth]{images/sample_frame.png}}
\caption{Screenshot of our test \texttt{<canvas>}\xspace game}
\label{fig:sampleframe}
\end{subfigure}
\hfill
\begin{subfigure}[t]{0.975\linewidth}
\begin{lstlisting}[language=html,basicstyle=\ttfamily\footnotesize,frame=single]
<!DOCTYPE html>
<html>
<head>
<style> body { margin: 0; display: flex; } </style>
<script src="main.js"></script>
</head>
<body>
<canvas width="1280px" height="720px"></canvas>
</body>
</html>
\end{lstlisting}
\caption{HTML code for our test \texttt{<canvas>}\xspace game}
\end{subfigure}
\caption{The graphics of a \texttt{<canvas>}\xspace game are represented as a bitmap, and not in the DOM, while the game's source code resides in the script \texttt{main.js}.}
\label{fig:intro}
\end{figure}
To overcome this challenge, snapshot testing has become the industry standard approach to visual testing for \texttt{<canvas>}\xspace applications, as it does not rely on the DOM, but instead relies on screenshots of the web application.
Snapshot testing targets visual bugs, i.e., bugs that are related to the graphics of the application, by automatically comparing oracle screenshots with screenshots that are recorded during the execution of a test case.
However, as we discuss in Section~\ref{subsec:snapshottesting}, snapshot testing cannot deal with the dynamic nature of \texttt{<canvas>}\xspace games, as this dynamism causes variation between the screenshots that is hard to account for automatically.
Because of the technical challenges of \texttt{<canvas>}\xspace testing, and the inherent difficulties of testing games~\cite{politowski2020dataset, murphy2014cowboys, stacey2009temporal, petrillo2009went, lewis2010went, kamienski2021empirical}, \texttt{<canvas>}\xspace games are mostly tested manually.
Manual testing requires large amounts of manual time and effort, limiting the amount of bugs that quality assurance (QA) analysts can realistically discover and report.
\begin{figure*}[t]
\centering
\setlength{\fboxsep}{0pt}%
\begin{subfigure}[t]{0.3\linewidth}
\centering
\fbox{\includegraphics[width=\textwidth]{images/intro_a.png}}
\caption{Oracle screenshot}
\label{fig:intro_a}
\end{subfigure}
\hfill
\begin{subfigure}[t]{0.3\linewidth}
\centering
\fbox{\includegraphics[width=\textwidth]{images/intro_bug.png}}
\caption{Test screenshot}
\label{fig:intro_b}
\end{subfigure}
\hfill
\begin{subfigure}[t]{0.3\linewidth}
\centering
\fbox{\includegraphics[width=\textwidth]{images/intro_compare_bug.png}}
\caption{Image difference (in pink)}
\label{fig:intro_compare_bug}
\end{subfigure}
\caption{Two screenshots from our test game. In the test screenshot, the viking character is missing a log on his shoulders (injected visual bug \hyperref[tab:bugdetection]{\texttt{S4}} in Table~\ref{tab:bugdetection}, \textit{viking animation not updating}). However, as observed in the third screenshot, the in-game randomness causes a larger difference between the screenshots than the bug itself.}
\label{fig:intro_sample}
\end{figure*}
Therefore, we propose an automated approach for the visual testing of \texttt{<canvas>}\xspace games.
Rather than (manually) curating oracle screenshots, we use game assets (see Section~\ref{subsec:canvasgames}) to automatically generate visual test oracles during the test, and automatically compare these oracle assets with individual objects on a screenshot of the \texttt{<canvas>}\xspace.
Our approach leverages the game's internal representation of objects on the \texttt{<canvas>}\xspace, i.e., the \texttt{<canvas>}\xspace objects representation (COR), to decompose screenshots of the \texttt{<canvas>}\xspace into individual object images.
We evaluated our approach by injecting 24 unique visual bugs from 4 different bug types (state, appearance, layout, rendering) as defined by Macklon et al.~\cite{macklon2022taxonomy} into a custom \texttt{<canvas>}\xspace test game.
Our approach performed automated visual comparisons of the oracle assets and the rendered objects using four similarity metrics: percentage overlap, mean squared error, structural similarity, and embedding similarity.
We compared our approach with a baseline approach that is the industry standard, i.e., snapshot testing.
We found that when using mean squared error, structural similarity, or embedding similarity as the similarity metric, our approach achieves an accuracy of 100\% for the 24 injected bugs in our test game, compared to an accuracy of 44.6\% with the baseline approach.
The main contributions of our paper are as follows:
\begin{itemize}
\item We designed 24 synthetic visual bugs to evaluate automated testing approaches for \texttt{<canvas>}\xspace games, and we confirmed with an industrial partner that these bugs were representative of the bugs found in real \texttt{<canvas>}\xspace games.
\item We created a testbed for evaluating visual testing techniques for \texttt{<canvas>}\xspace games, i.e., a test \texttt{<canvas>}\xspace game, which includes a non-buggy version and a buggy version of the game containing the 24 synthetic bugs.
\item We extensively evaluated combinations of four widely-used similarity metrics for automatically detecting visual bugs in \texttt{<canvas>}\xspace games.
\item For reproducibility, we open-sourced our testbed and visual bugs dataset at the following link: \url{https://github.com/asgaardlab/canvas-visual-bugs-testbed}.
\item A live version of our test \texttt{<canvas>}\xspace game is available at the following link:
\url{https://asgaardlab.github.io/canvas-visual-bugs-testbed/game}.
\end{itemize}
The remainder of our paper is structured as follows.
Section~\ref{sec:background} discusses background information.
Section~\ref{sec:relatedwork} discusses related work.
Section~\ref{sec:ourapproach} presents our approach.
Section~\ref{sec:experiments} details our experiment setup.
Section~\ref{sec:results} presents our results.
Section~\ref{sec:threats} contains threats to validity.
Section~\ref{sec:conclusion} is the conclusion to our paper.
\section{Background}\label{sec:background}
In this section, we give background information about HTML5 \texttt{<canvas>}\xspace games and snapshot testing.
\subsection{HTML5 \texttt{<canvas>}\xspace games}\label{subsec:canvasgames}
By combining the high-quality graphics of the \texttt{<canvas>}\xspace with browser events, such as mouse clicks, game developers can create complete games that run in a web browser.
\subsubsection*{Open-source frameworks}
It is difficult to integrate the \texttt{<canvas>}\xspace with other parts of a web application~\cite{macklon2022taxonomy}, and so \texttt{<canvas>}\xspace \emph{frameworks} are used to ease the development of \texttt{<canvas>}\xspace games.
There exist several free and open-source (FOSS) \texttt{<canvas>}\xspace frameworks that are widely-used to develop \texttt{<canvas>}\xspace games.
For example, \texttt{PixiJS} and \texttt{Phaser} receive much attention from game developers, as indicated by the high number of forum posts related to each framework on the HTML5 Game Devs~\cite{html5gamedevs:0} and Stack Overflow~\cite{almansoury2020investigating} forums.
Such \texttt{<canvas>}\xspace frameworks typically provide a custom internal representation of objects on the \texttt{<canvas>}\xspace, i.e., a \texttt{<canvas>}\xspace objects representation (COR), which can be manipulated by developers to easily create animations on the \texttt{<canvas>}\xspace.
For example, in \texttt{PixiJS}, the COR is termed \emph{scene graph}, and has a tree structure.
\subsubsection*{Assets}
A common way to integrate graphics into a video game is using source images (\emph{assets}) that are used to display objects in the game.
For \texttt{<canvas>}\xspace games, assets are loaded by the web application client from some file server through web requests, like any other image in a web application.
However, assets are not rendered as image (\texttt{<img>}\xspace) elements on a web page, but instead are used as source bitmaps that are displayed on the \texttt{<canvas>}\xspace bitmap.
\subsection{Snapshot testing} \label{subsec:snapshottesting}
Snapshot testing, e.g., using \texttt{Percy}, is the industry standard for visually testing web applications~\cite{ricca2021ai}.
Visual testing is used to target visual bugs; visual bugs are mismatches between actual and expected visual properties in the graphics of a software application~\cite{issa2012visual}.
Traditional snapshot testing typically involves comparing screenshots of the web application from the same test across different runs, after some change(s) to the source code (e.g., a pull request).
To perform traditional snapshot testing, first a set of oracle screenshots that have been collected during a test run must be manually verified, and then new test screenshots can be automatically collected and compared at a later time using an image comparison algorithm.
If a screenshot does not pass the image comparison check, that screenshot (or test case) is flagged for manual review.
Figure~\ref{fig:intro_sample} shows how most of the visual differences between the oracle and test screenshots occur due to random elements of the \texttt{<canvas>}\xspace game, which are desired functionality, rather than the injected visual bug.
It is difficult to distinguish between visual bugs and intended functionality for \texttt{<canvas>}\xspace games when using snapshot testing.
This problem can lead to many false positives, increasing the manual workload (due to oracle re-verification) and reducing the benefit of using snapshot testing as an automated testing approach.
Therefore, the industry-standard approach for snapshot testing is far from ideal for testing many \texttt{<canvas>}\xspace applications, particularly \texttt{<canvas>}\xspace games.
\section{Related Work}\label{sec:relatedwork}
In this section, we discuss related work on \texttt{<canvas>}\xspace testing, visual web and GUI testing, and visual game testing.
\subsection{\texttt{<canvas>}\xspace testing}
Macklon et al.~\cite{macklon2022taxonomy} analyzed open source projects on GitHub that utilize the \texttt{<canvas>}\xspace, and proposed a taxonomy of \texttt{<canvas>}\xspace bugs.
They showed that the most frequently reported bugs in the open source projects are visual bugs, i.e., bugs that are related to the graphics of an application.
Their findings emphasize that research on \texttt{<canvas>}\xspace testing is at an early stage and has many opportunities, and that visual bugs are a primary concern for \texttt{<canvas>}\xspace testing.
Only one prior study has investigated testing methods for the \texttt{<canvas>}\xspace.
Bajammal and Mesbah propose an approach to enable DOM-based testing of the \texttt{<canvas>}\xspace by leveraging traditional computer vision techniques to detect objects on the \texttt{<canvas>}\xspace, and subsequently augment the DOM with a representation of those objects~\cite{bajammal2018web}.
They report high accuracy in detecting objects on the \texttt{<canvas>}\xspace that should not be present (similar to visual bug \hyperref[tab:bugdetection]{\texttt{S6}} in Table~\ref{tab:bugdetection}), however any other type of overlapping visual bug on the \texttt{<canvas>}\xspace would pose challenges for their visual inference algorithm.
In contrast, we evaluate our approach on 24 unique bugs from 4 visual bugs types, and find that our approach shows strong performance for catching a wide variety of visual bugs that are representative of bugs found in real-world \texttt{<canvas>}\xspace games.
\subsection{Visual web and GUI testing}
As previously outlined, existing automated web testing techniques and tools do not work for the \texttt{<canvas>}\xspace, but prior research has also indicated that \texttt{<canvas>}\xspace bugs overlap with visual bugs found in graphical user interfaces (GUIs) and generic web applications~\cite{macklon2022taxonomy}.
We refer to the survey of computer vision applications in software engineering by Bajammal et al.~\cite{bajammal2020survey} and the grey literature review of AI-based test automation techniques by Ricca et al.~\cite{ricca2021ai} for an overview of visual testing for GUIs and web apps.
Here, we only discuss related work that was not covered in the survey by Bajammal et al.~\cite{bajammal2020survey}.
Several prior studies have proposed the use of visual analysis to assist in automated testing methods for web applications.
Yandrapally and Mesbah~\cite{yandrapally2021fragment} proposed a method to automatically detect near-duplicate states in web applications by comparing fragments of a web page instead of entire screenshots.
They decomposed the DOM along with screenshots and performed automatic structural and visual comparisons between automatically inferred web page states.
Bajammal and Mesbah~\cite{bajammal2021semantic} automatically inferred the semantic role of regions in a web page and automated the testing of web accessibility requirements.
In another work by Bajammal and Mesbah~\cite{bajammal2021page}, they combined visual analysis with DOM attributes to improve automated web page segmentation, which can assist with bug localization.
These works focus on segmenting and testing the structure of web pages, i.e., what is represented in the DOM, but as previously explained, the contents of the \texttt{<canvas>}\xspace are not represented in the DOM, meaning these approaches cannot be used to automatically catch visual bugs in \texttt{<canvas>}\xspace games.
Several prior studies proposed the use of computer vision to leverage the visual aspect of a software application in an automated testing process.
Mazinanian et al.~\cite{mazinanian2021style} automatically predicted actionable elements on a web page through a supervised deep learning approach.
White et al.~\cite{white2019improving} proposed a supervised deep learning approach and automatically identified GUI components to improve the coverage of random testing.
Xue et al.~\cite{xue2022learning} proposed a supervised deep learning approach to assist in performing record-and-replay GUI testing in a mobile or web application.
Mozgovoy and Pyshkin~\cite{mozgovoy2017unity} used template matching to recognize objects and GUI elements in a screenshot of a mobile game, which allow test assertions to be made against the visual content of the game.
Ye et al.~\cite{ye2021empirical} proposed a similar GUI widget detection approach for mobile games, in which they instrumented the source code of a mobile game to automatically extract samples of GUI widgets, and subsequently trained a supervised deep learning model for GUI widget detection.
Visual bugs would interfere with the GUI element identification methods in the aforementioned works, while our approach instead targets visual bugs in \texttt{<canvas>}\xspace games without training any new models.
Zhao et al. proposed the use of unsupervised deep learning methods to detect anomalous GUI animations, which requires only several ground truth samples of a correct GUI animation to detect the anomalous animations~\cite{zhao2020seenomaly}.
Given the dynamic nature of \texttt{<canvas>}\xspace games, it would be extremely challenging to collect ground truth samples of all correct animations in a \texttt{<canvas>}\xspace game, which does not solve the problems posed by snapshot testing.
We avoid this problem in our approach by automatically generating visual test oracles during the test.
\subsection{Visual game testing}
Given that the \texttt{<canvas>}\xspace is often used to build web games, we provide an overview of visual testing in video games.
Automated methods for graphics glitch detection in video games have been proposed in prior work.
Nantes et al. propose a semi-automated approach to detect shadow glitches in a video game using traditional computer vision techniques~\cite{nantes2008framework}.
However, in our work we propose a fully automated approach to detect a wider range of visual bugs that are relevant to \texttt{<canvas>}\xspace games.
Other studies have utilized relatively recent advancements in deep learning to detect graphics glitches in video games~\cite{davarmanesh2020automating, ling2020using, chen2021glib} or to leverage the visual aspect of video games for sprite and event extraction~\cite{kim2020synthesizing, smirnov2021marionette, luo2018player, luo2019making, taesiri2022clip}.
However, these methods all require either significant manual effort to prepare the data or in-house machine learning expertise to train and fine-tune the models, or they target only a limited set of visual bugs (when compared to the four types evaluated in our paper).
As our approach utilizes a pre-trained model, it requires only very basic applied machine learning knowledge, and it does not require much data preparation.
\section{Our Approach}\label{sec:ourapproach}
In this section, we present our approach for automatically detecting visual bugs in \texttt{<canvas>}\xspace games.
Figure~\ref{fig:overview} shows an overview of the steps of our approach.
\subsection{Collecting data}\label{subsec:collectingdata}
We begin by automatically instrumenting the rendering loop of the \texttt{<canvas>}\xspace game with our custom code to collect snapshots and assets.
Each snapshot contains a screenshot of the \texttt{<canvas>}\xspace and a respective \texttt{<canvas>}\xspace objects representation (COR) from the same point in time.
For each snapshot, we automatically collect a screenshot and its respective COR.
Figure~\ref{fig:database} illustrates what a COR contains in our approach.
A COR is used by a \texttt{<canvas>}\xspace game to determine how to render game objects to the \texttt{<canvas>}\xspace, such as the player character, background layers, and projectiles.
Each object in the COR has properties such as position, size, and rotation.
While performing a snapshot, we prevent new animation frames from being rendered, and save a frozen copy of the COR along with a synchronized screenshot of the current animation frame (as rendered to the \texttt{<canvas>}\xspace).
Although our snapshot operation briefly prevents the rendering of a few new frames, it does not necessarily interrupt the main game loop (depending upon how a game is implemented).
As described in Section~\ref{subsec:canvasgames} of our paper, assets in \texttt{<canvas>}\xspace games are served through web requests, and so we created a custom crawler to collect assets based on the resource URLs of objects in a \texttt{<canvas>}\xspace game.
As can be seen in Figure~\ref{fig:database}, game objects are linked to their respective assets in the COR, meaning associating a game object with its respective asset is straightforward.
\begin{figure}[!t]
\centering
\includegraphics[width=\linewidth]{images/overview.pdf}
\caption{Overview of our approach (shown with visual bug \hyperref[tab:bugdetection]{\texttt{L4}} in Table~\ref{tab:bugdetection}, \emph{Viking has wrong rotation}).}
\label{fig:overview}
\end{figure}
\begin{figure*}[!t]
\centering
\includegraphics[width=\linewidth]{images/class-diagram.pdf}
\caption{Unified modeling language (UML) class diagram for a \texttt{<canvas>}\xspace objects representation (COR).}
\label{fig:database}
\end{figure*}
\subsection{Preprocessing images}\label{subsec:preprocessingassets}
For each snapshot, we leverage the COR to automatically generate oracle assets and extract object images for comparison.
Figure~\ref{fig:preprocessing} shows our automated image processing pipeline.
Below, we detail our preprocessing steps for oracle assets and object images.
We automatically preprocess game assets to generate oracle assets during the execution of a test using the following process:
\begin{enumerate}
\item Apply transformations to the asset as specified in the COR. For example, crop, scale, tile, and/or rotate the asset.
\item Paste the asset onto a blank image that is the same size as the \texttt{<canvas>}\xspace. The paste location is determined by the COR, and will match the location of the game object in the screenshot if no bugs are present.
\item Generate an image mask from the pasted asset (i.e., the result from the previous step) and save for later masking operations.
\item For any overlapping objects, apply their saved masks over top of the pasted asset. Figure~\ref{fig:masking} shows an example of what this might look like.
\item Crop the pasted asset.
\end{enumerate}
We automatically decompose screenshots into a set of individual object images according to the following process:
\begin{enumerate}
\item Apply the background mask, i.e., the mask generated from the object's respective asset.
\item Apply the foreground masks, i.e., the masks that were generated from assets belonging to overlapping objects.
\item Crop the object image out of the screenshot.
\end{enumerate}
After preprocessing, we have a set of image pairs, with each pair containing an oracle asset and object image, which should be exactly the same if no visual bugs are present.
\begin{figure*}[!t]
\centering
\includegraphics[width=\linewidth]{images/preprocessing.pdf}
\caption{Automated image preprocessing pipeline (shown with visual bug \hyperref[tab:bugdetection]{\texttt{S1}}, \emph{player-character is invisible}).}
\label{fig:preprocessing}
\end{figure*}
\begin{figure*}[!t]
\setlength{\fboxsep}{0pt}%
\centering
\hfill
\begin{subfigure}[t]{0.24\linewidth}
\centering
\fbox{\includegraphics[width=\textwidth]{images/mask-ss.png}}
\caption{Input screenshot}
\label{fig:mask_input}
\end{subfigure}
\hfill
\begin{subfigure}[t]{0.24\linewidth}
\centering
\fbox{\includegraphics[width=\textwidth]{images/mask-bg.png}}
\caption{The background mask}
\label{fig:mask_bg}
\end{subfigure}
\hfill
\begin{subfigure}[t]{0.24\linewidth}
\centering
\fbox{\includegraphics[width=\textwidth]{images/mask-fg.png}}
\caption{One of the foreground masks}
\label{fig:mask_fg}
\end{subfigure}
\hfill
\begin{subfigure}[t]{0.24\linewidth}
\centering
\fbox{\includegraphics[width=\textwidth]{images/mask-object.png}}
\caption{Output object image}
\label{fig:mask_output}
\end{subfigure}
\hfill
\caption{A background mask and all overlapping foreground masks are applied to isolate object images in our approach.}
\label{fig:masking}
\end{figure*}
\subsection{Detecting visual bugs}
For each pair of oracle asset and object image, we use an image similarity metric to automatically perform a visual comparison of the images.
Our approach relies on a threshold for this similarity metric to decide if a visual test case should pass or fail.
This threshold should be defined empirically and for each game, as different games may have different levels of in-game randomness that could affect the similarity metric.
\section{Experiment Setup}\label{sec:experiments}
In this section, we describe our experiment setup for evaluating the performance of our approach and the baseline approach for automatically detecting visual bugs.
We selected snapshot testing as the baseline approach for comparison with our approach, as snapshot testing is the industry standard approach for detecting visual bugs in \texttt{<canvas>}\xspace applications.
\subsection{The test game}
To evaluate our approach, we created a custom \texttt{<canvas>}\xspace game using the \texttt{PixiJS}\footnote{\href{https://pixijs.com/}{https://pixijs.com/}} library and a freely available asset pack\footnote{\href{https://raventale.itch.io/parallax-background}{https://raventale.itch.io/parallax-background}}.
\texttt{PixiJS} is a popular free and open-source \texttt{<canvas>}\xspace rendering library for creating 2D animations with the \texttt{<canvas>}\xspace.
Figure~\ref{fig:testgame} shows the state machine diagram for our test game.
Our test game is a so-called catching game, i.e., a game in which projectiles are randomly thrown for the player to catch.
The test game contains a variety of animations, including animated sprites, rotating sprites, and background tiling sprites.
The test game was designed to be played at a resolution of 720p, with a maximum frame rate of $60$ FPS.
\subsection{The test case}
We wrote an automated test case for our \texttt{<canvas>}\xspace game.
In our test case, the game was automatically opened in a browser window with size $1280px\times720px$.
Next, the game was started through an automated user click, and then the player-character was moved back and forth across the screen with automated mouse movements until the player lost a life (after which, the test case ended).
During each test case execution, 10 snapshots were taken.
\subsection{Injected visual bugs} \label{subsec:injectedbugs}
We evaluated the performance of the approaches by injecting visual bugs into our test game.
To target bugs that are relevant to \texttt{<canvas>}\xspace games, we used the taxonomy of \texttt{<canvas>}\xspace bugs constructed by Macklon et al.~\cite{macklon2022taxonomy}, and verified with an industrial partner that our injected bugs were relevant to industrial \texttt{<canvas>}\xspace games.
In Table~\ref{tab:visualbugtypes}, we provide each visual bug type and an example description of a bug of that type as defined in the taxonomy of \texttt{<canvas>}\xspace bugs.
For each of the four visual bug types defined in the taxonomy of \texttt{<canvas>}\xspace bugs, we injected six different bugs, with some primarily affecting foreground objects, and others primarily affecting background objects.
In total, we injected 24 visual bugs. Figure~\ref{fig:bug_samples} shows four example instances of visual bugs we injected into the test game, while Table~\ref{tab:bugdetection} provides detailed descriptions of each injected bug.
We injected most of the visual bugs by altering an asset during test execution, and then replaced it with the non-bugged (original) asset at the preprocessing stage of our approach.
We injected most of the visual bugs this way because real visual bugs can be very complex and difficult to reproduce~\cite{macklon2022taxonomy}.
Although our injected visual bugs had a different root cause than real visual bugs on the \texttt{<canvas>}\xspace, we confirmed with an industrial partner that the visual effects were similar to visual bugs found in real \texttt{<canvas>}\xspace games, meaning that our injected visual bugs were suitable for evaluating our approach.
\begin{table}[t]
\centering
\caption{Visual bug types found in \texttt{<canvas>}\xspace applications~\cite{macklon2022taxonomy}.}
\begin{tabular*}{\linewidth}{l@{\extracolsep{\fill}}l}
\toprule
\textbf{Type} & \textbf{Example Description} \\
\midrule
State & Object visible but should be invisible.\\
Appearance & Object has incorrect colour.\\
Layout & Object has incorrect position, size, layer, etc.\\
Rendering & Object is distorted, blurry, or contains artifacts.\\
\bottomrule
\end{tabular*}
\label{tab:visualbugtypes}
\end{table}
\begin{figure}[!t]
\centering
\setlength{\fboxsep}{0pt}%
\begin{subfigure}[t]{0.46\linewidth}
\centering
\fbox{\includegraphics[width=\textwidth]{images/RB3.png}}
\caption{Rendering bug \hyperref[tab:bugdetection]{\texttt{R3}} in Table~\ref{tab:bugdetection}. \textit{Viking and logs are blurred.}}
\label{fig:rendering}
\end{subfigure}
\hfill
\begin{subfigure}[t]{0.46\linewidth}
\centering
\fbox{\includegraphics[width=\textwidth]{images/LB5.png}}
\caption{Layout bug \hyperref[tab:bugdetection]{\texttt{L5}} in Table~\ref{tab:bugdetection}. \textit{Trees are in the wrong layer.}}
\label{fig:layout}
\end{subfigure}
\hfill
\begin{subfigure}[t]{0.46\linewidth}
\centering
\fbox{\includegraphics[width=\textwidth]{images/SB5.png}}
\caption{State bug \hyperref[tab:bugdetection]{\texttt{S5}} in Table~\ref{tab:bugdetection}. \textit{Fallen log animation is not updating.}}
\label{fig:state}
\end{subfigure}
\hfill
\begin{subfigure}[t]{0.46\linewidth}
\centering
\fbox{\includegraphics[width=\textwidth]{images/AB4.png}}
\caption{Appearance bug \hyperref[tab:bugdetection]{\texttt{A4}} in Table~\ref{tab:bugdetection}. \textit{Logs are a different colour.}}
\label{fig:appearance}
\end{subfigure}
\caption{Sample instances of our injected visual bugs.}
\label{fig:bug_samples}
\end{figure}
\begin{figure*}[t]
\centering
\includegraphics[width=\linewidth]{images/statemachine.pdf}
\caption{Finite state machine diagram for our test \texttt{<canvas>}\xspace game.}
\label{fig:testgame}
\end{figure*}
\subsection{Similarity metrics} \label{subsec:similaritymetrics}
In our experiments we compared images using four similarity metrics: percentage overlap, mean squared error, structural similarity, and embedding similarity.
\subsubsection*{Percentage overlap (PCT)}
We selected percentage overlap as a similarity metric because it is the simplest method for calculating the similarity of two images, and is used in industry-standard tools such as \texttt{Percy}.
We calculated the PCT for a pair of images by calculating the percentage of pixels that exactly match between the two images.
\subsubsection*{Mean squared error (MSE)}
We selected mean squared error as a similarity metric because it is widely used in image processing as an image quality index~\cite{wang2002universal, eskicioglu1995image}, i.e., to measure degradation between original and reconstructed images.
The MSE actually captures the amount of difference (i.e., lower is better) instead of similarity (i.e., higher is better).
We calculated the mean squared error for a pair of images using the \texttt{scikit-image}\footnote{\href{https://scikit-image.org/}{https://scikit-image.org/}} library.
\subsubsection*{Structural similarity (SSIM)}
We selected structural similarity as a similarity metric because it is intended to be complementary to mean squared error as an image quality index~\cite{wang2004image}.
We used the \texttt{scikit-image} library to calculate structural similarity for each pair of images.
\subsubsection*{Embedding similarity (ESIM)}
Our fourth and final metric, embedding similarity, was the similarity of two images when represented as embeddings of an image classification model, i.e., a deep learning vision model.
Embeddings are the vision model's inner-layer representation(s) of an image, i.e., the feature representations of the image before the classification layer.
We implemented embedding similarity in our experiments by encoding the images as the embeddings of the final convolutional layer of the \texttt{ResNet-50} model pre-trained on the ImageNet dataset~\cite{he2015deep}.
These image embeddings had a feature volume of $(2048,7,7)$.
We selected the pre-trained \texttt{ResNet-50} model as the \texttt{ResNet} architecture is widely used for transfer learning applications~\cite{rezende2017malicious, chen2020unblind, luo2019making}.
We extracted the embeddings of the final convolutional layer of the \texttt{ResNet-50} model, as is done in prior work~\cite{taesiri2020video, fazli2021under}.
To calculate the similarity of the embeddings, we selected cosine similarity, a widely used similarity metric~\cite{zhang2018unreasonable, viggiato2021identifying, viggiatousing, taesiri2022clip}.
We used the pre-trained model in inference mode, meaning we did not have to perform any data labelling, training, or fine-tuning, i.e., we used the model out-of-the-box.
We performed inference with the pre-trained \texttt{ResNet-50} model on an NVIDIA Titan RTX graphics card.
We loaded the model from the \texttt{torchvision}\footnote{\href{https://pytorch.org/vision/stable/index.html}{https://pytorch.org/vision/stable/index.html}} library and used the \texttt{PyTorch}\footnote{\href{https://pytorch.org/}{https://pytorch.org/}} library to calculate cosine similarity.
\subsection{Empirical threshold selection}\label{subsubsec:thresholds}
We empirically selected a single threshold for each similarity metric used in each approach to decide whether a test case is buggy.
To empirically determine the thresholds, we calculated the similarities of all image pairs for 10 repetitions of test data with no bugs injected (i.e., with non-buggy snapshots), and took the overall lowest (or highest for MSE) similarity score for each metric as our thresholds. Hence, we chose the thresholds to yield zero false positives, as false positives result in a wasted effort from the game developer's perspective (as they need to investigate the false positive).
\subsection{Evaluating the experiments}
Here we describe the methods we used to evaluate our experiments.
\subsubsection*{Statistical significance and effect sizes}
We used the Mann-Whitney U test~\cite{mann1947test} to determine if the populations of similarity scores were statistically significantly different.
The Mann-Whitney U test is a non-parametric test that compares two distributions of unrelated populations to determine how much the populations statistically overlap, with some probability $p$.
Generally, a $p$ value of less than $0.05$ indicates that the populations display a statistically significant difference, as a very low $p$ value indicates it is very unlikely that two populations are statistically similar.
To better understand the results of the Mann-Whitney U test, we also calculated Cliff's delta~\cite{cliff1993dominance} to determine the extent to which the populations of buggy and non-buggy similarity scores were different per metric.
To interpret the Cliff's delta values ($d$), we used the thresholds provided by Romano et al.~\cite{romano2006exploring} to determine the effect sizes, as done in prior work~\cite{kamienski2021empirical}.
The thresholds used were as follows:
\begin{math}
\text{Effect size} =
\begin{cases}
\text{negligible} &\quad\text{if}\quad |d|\leq0.147 \\
\text{small} &\quad\text{if}\quad 0.147 < |d| \leq 0.33 \\
\text{medium} &\quad\text{if}\quad 0.33 < |d| \leq 0.474 \\
\text{large} &\quad\text{if}\quad 0.474 < |d| \leq 1\\
\end{cases}
\end{math}
\subsubsection*{Accuracy}
Our choice of threshold selection (Section~\ref{subsubsec:thresholds}) meant that it was only possible for there to be true positive (TP) and false negative (FN) cases in our results for visual bug detection.
Therefore, the best choice of evaluation metric was accuracy, which was calculated as follows:
$\mathit{accuracy}\text{\space}=\frac{(\#\text{\space}\mathit{true}\text{\space}\mathit{positives})}{(\#\text{\space}\mathit{true}\text{\space}\mathit{positives})\text{\space}+\text{\space}(\#\text{\space}\mathit{false}\text{\space}\mathit{negatives})}$
\section{Results}\label{sec:results}
In this section, we present our experimental results for automatically detecting visual bugs with our approach and the baseline approach.
When using MSE, SSIM, or ESIM as the similarity metric, we find that our approach achieves an accuracy of $100\%$ for our 24 injected visual bugs, compared to an accuracy of $44.6\%$ with the baseline approach (with PCT as the similarity metric).
Our results show that our approach is much more effective for automatically detecting visual bugs in \texttt{<canvas>}\xspace-based applications than the baseline approach (traditional snapshot testing).
\begin{figure*}[t]
\centering
\includegraphics[width=\linewidth]{images/boxplots.pdf}
\caption{Boxplots of similarity scores for each evaluated similarity metric in each approach. Our selected thresholds are indicated by the grey dotted lines. Similarity scores in the greyed-out ranges are classified as non-buggy by each approach.}
\label{fig:boxplots}
\end{figure*}
\subsection{Similarity scores}
Figure~\ref{fig:boxplots} shows the distributions of the similarity scores for each of the evaluated similarity metrics, with the minimum similarity for normal snapshots providing the thresholds for bug detection, as described in Section~\ref{subsubsec:thresholds}.
For each distribution, scores above the set threshold (within the greyed-out areas) would be accepted as within the normal range, whereas scores below the threshold would indicate a visual bug is present.
While the distributions are significantly different for all similarity metrics, the effect sizes show that there is a lot of overlap between the metrics when using the snapshot testing approach.
As a result, a threshold is much harder to select for snapshot testing, and there will always be a trade-off between precision and recall.
For our approach, the effect size (Table~\ref{tab:mwucliff}) is much larger indicating there is far less overlap between the distributions, allowing us to choose better-performing thresholds (which can also be observed from Figure~\ref{fig:boxplots}).
Clearly, there is much more overlap between normal and buggy cases when using snapshot testing than when using our approach with any of the similarity metrics.
\subsection{Bug detection}
Table~\ref{tab:bugdetection} shows the results for bug detection with each evaluated approach and similarity metric.
Our approach achieves a considerably higher accuracy (with any of our four evaluated similarity metrics) than snapshot testing.
In particular, our approach shows exciting potential for detecting visual bugs when MSE, SSIM, or ESIM is used as the similarity metric.
Our results indicate that only a single similarity metric is needed for our approach -- combining several metrics does not improve our overall results.
\begin{table}[t]
\centering
\caption{Mann-Whitney U test and Cliff's delta results.}
\begin{tabular}{@{\extracolsep{\fill}}llc@{\extracolsep{\fill}}c}
\toprule
& & \multicolumn{1}{c}{\textbf{Mann-Whitney U test}} & \multicolumn{1}{c}{\textbf{Cliff's delta}} \\
& \textbf{Metric} & \multicolumn{1}{c}{\textbf{Significant difference}} & \multicolumn{1}{c}{\textbf{Effect size}} \\
\midrule
\textbf{Snapshot} & \textbf{PCT} & \textit{yes} & \textit{small} \\
\textbf{Testing} & \textbf{MSE} & \textit{yes} & \textit{medium} \\
& \textbf{SSIM} & \textit{yes}& \textit{small} \\
\midrule
\textbf{Our} & \textbf{PCT} & \textit{yes}& \textit{medium} \\
\textbf{Approach} & \textbf{MSE} & \textit{yes} & \textit{medium} \\
& \textbf{SSIM} & \textit{yes}& \textit{medium} \\
& \textbf{ESIM} & \textit{yes}& \textit{large} \\
\bottomrule
\end{tabular}
\label{tab:mwucliff}
\end{table}
\begin{table*}[t]
\centering
\caption{Number of repetitions (out of 10) each visual bug was detected for each approach with each similarity metric. As detailed in Section~\ref{subsec:similaritymetrics}, we use the following similarity metrics: percentage overlap (PCT), mean squared error (MSE), structural similarity (SSIM), and embedding similarity (ESIM).}
\label{tab:bugdetection}
\begin{tabular*}{\linewidth}{@{\extracolsep{\fill}}l@{\extracolsep{\fill}}l@{\extracolsep{\fill}}lrrrrrrr}
\toprule
& & & \multicolumn{3}{l}{\textbf{Snapshot Testing}} & \multicolumn{4}{l}{\textbf{Our Approach}} \\
\textbf{Type} & \textbf{Key} & \textbf{Bug Description} & \multicolumn{1}{l}{\textbf{PCT}} & \multicolumn{1}{l}{\textbf{MSE}} & \multicolumn{1}{l}{\textbf{SSIM}} & \multicolumn{1}{l}{\textbf{PCT}} & \multicolumn{1}{l}{\textbf{MSE}} & \multicolumn{1}{l}{\textbf{SSIM}} & \multicolumn{1}{l}{\textbf{ESIM}} \\ \midrule
\multicolumn{1}{l}{\multirow{6}{*}{\rotatebox{90}{State}}} & \texttt{S1} & \textit{Viking is invisible.} & \cellcolor[HTML]{EFF9F4}1 & \cellcolor[HTML]{FFFFFF}0 & \cellcolor[HTML]{FFFFFF}0 & \cellcolor[HTML]{57BB8A}10 & \cellcolor[HTML]{57BB8A}10 & \cellcolor[HTML]{57BB8A}10 & \cellcolor[HTML]{57BB8A}10 \\
& \texttt{S2} & \textit{A background hill is invisible.} & \cellcolor[HTML]{79C9A2}8 & \cellcolor[HTML]{BCE4D1}4 & \cellcolor[HTML]{9BD7B9}6 & \cellcolor[HTML]{FFFFFF}0 & \cellcolor[HTML]{57BB8A}10 & \cellcolor[HTML]{57BB8A}10 & \cellcolor[HTML]{57BB8A}10 \\
& \texttt{S3} & \textit{Ship is invisible.} & \cellcolor[HTML]{57BB8A}10 & \cellcolor[HTML]{57BB8A}10 & \cellcolor[HTML]{57BB8A}10 & \cellcolor[HTML]{57BB8A}10 & \cellcolor[HTML]{57BB8A}10 & \cellcolor[HTML]{57BB8A}10 & \cellcolor[HTML]{57BB8A}10 \\
& \texttt{S4} & \textit{Viking animation is not updating.} & \cellcolor[HTML]{EFF9F4}1 & \cellcolor[HTML]{FFFFFF}0 & \cellcolor[HTML]{DEF2E8}2 & \cellcolor[HTML]{57BB8A}10 & \cellcolor[HTML]{57BB8A}10 & \cellcolor[HTML]{57BB8A}10 & \cellcolor[HTML]{57BB8A}10 \\
& \texttt{S5} & \textit{Fallen log animation is not updating.} & \cellcolor[HTML]{8AD0AE}7 & \cellcolor[HTML]{CDEBDC}3 & \cellcolor[HTML]{ABDDC5}5 & \cellcolor[HTML]{57BB8A}10 & \cellcolor[HTML]{57BB8A}10 & \cellcolor[HTML]{57BB8A}10 & \cellcolor[HTML]{57BB8A}10 \\
& \texttt{S6} & \textit{Button should be hidden.} & \cellcolor[HTML]{ABDDC5}5 & \cellcolor[HTML]{57BB8A}10 & \cellcolor[HTML]{79C9A2}8 & \cellcolor[HTML]{FFFFFF}0 & \cellcolor[HTML]{57BB8A}10 & \cellcolor[HTML]{57BB8A}10 & \cellcolor[HTML]{57BB8A}10 \\
\midrule
\multicolumn{1}{l}{\multirow{6}{*}{\rotatebox{90}{Appearance}}} & \texttt{A1} & \textit{Viking has the wrong beard colour.} & \cellcolor[HTML]{EFF9F4}1 & \cellcolor[HTML]{BCE4D1}4 & \cellcolor[HTML]{BCE4D1}4 & \cellcolor[HTML]{57BB8A}10 & \cellcolor[HTML]{57BB8A}10 & \cellcolor[HTML]{57BB8A}10 & \cellcolor[HTML]{57BB8A}10 \\
& \texttt{A2} & \textit{Entire viking has the wrong colour.} & \cellcolor[HTML]{DEF2E8}2 & \cellcolor[HTML]{8AD0AE}7 & \cellcolor[HTML]{CDEBDC}3 & \cellcolor[HTML]{57BB8A}10 & \cellcolor[HTML]{57BB8A}10 & \cellcolor[HTML]{57BB8A}10 & \cellcolor[HTML]{57BB8A}10 \\
& \texttt{A3} & \textit{Viking and logs are grey-scaled.} & \cellcolor[HTML]{CDEBDC}3 & \cellcolor[HTML]{FFFFFF}0 & \cellcolor[HTML]{CDEBDC}3 & \cellcolor[HTML]{57BB8A}10 & \cellcolor[HTML]{57BB8A}10 & \cellcolor[HTML]{57BB8A}10 & \cellcolor[HTML]{57BB8A}10 \\
& \texttt{A4} & \textit{Logs have the wrong colour.} & \cellcolor[HTML]{CDEBDC}3 & \cellcolor[HTML]{EFF9F4}1 & \cellcolor[HTML]{BCE4D1}4 & \cellcolor[HTML]{57BB8A}10 & \cellcolor[HTML]{57BB8A}10 & \cellcolor[HTML]{57BB8A}10 & \cellcolor[HTML]{57BB8A}10 \\
& \texttt{A5} & \textit{The ship's sail has the wrong colour.} & \cellcolor[HTML]{68C296}9 & \cellcolor[HTML]{CDEBDC}3 & \cellcolor[HTML]{8AD0AE}7 & \cellcolor[HTML]{57BB8A}10 & \cellcolor[HTML]{57BB8A}10 & \cellcolor[HTML]{57BB8A}10 & \cellcolor[HTML]{57BB8A}10 \\
& \texttt{A6} & \textit{A background bunny has the wrong colour.} & \cellcolor[HTML]{CDEBDC}3 & \cellcolor[HTML]{CDEBDC}3 & \cellcolor[HTML]{ABDDC5}5 & \cellcolor[HTML]{FFFFFF}0 & \cellcolor[HTML]{57BB8A}10 & \cellcolor[HTML]{57BB8A}10 & \cellcolor[HTML]{57BB8A}10 \\
\midrule
\multicolumn{1}{l}{\multirow{6}{*}{\rotatebox{90}{Layout}}} & \texttt{L1} & \textit{Ship is in the wrong location.} & \cellcolor[HTML]{57BB8A}10 & \cellcolor[HTML]{57BB8A}10 & \cellcolor[HTML]{57BB8A}10 & \cellcolor[HTML]{57BB8A}10 & \cellcolor[HTML]{57BB8A}10 & \cellcolor[HTML]{57BB8A}10 & \cellcolor[HTML]{57BB8A}10 \\
& \texttt{L2} & \textit{Viking is in the wrong location.} & \cellcolor[HTML]{DEF2E8}2 & \cellcolor[HTML]{DEF2E8}2 & \cellcolor[HTML]{DEF2E8}2 & \cellcolor[HTML]{57BB8A}10 & \cellcolor[HTML]{57BB8A}10 & \cellcolor[HTML]{57BB8A}10 & \cellcolor[HTML]{57BB8A}10 \\
& \texttt{L3} & \textit{Background clouds are in the wrong location.} & \cellcolor[HTML]{57BB8A}10 & \cellcolor[HTML]{ABDDC5}5 & \cellcolor[HTML]{CDEBDC}3 & \cellcolor[HTML]{57BB8A}10 & \cellcolor[HTML]{57BB8A}10 & \cellcolor[HTML]{57BB8A}10 & \cellcolor[HTML]{57BB8A}10 \\
& \texttt{L4} & \textit{Viking has the wrong rotation.} & \cellcolor[HTML]{DEF2E8}2 & \cellcolor[HTML]{FFFFFF}0 & \cellcolor[HTML]{DEF2E8}2 & \cellcolor[HTML]{57BB8A}10 & \cellcolor[HTML]{57BB8A}10 & \cellcolor[HTML]{57BB8A}10 & \cellcolor[HTML]{57BB8A}10 \\
& \texttt{L5} & \textit{Background trees are in the wrong layer.} & \cellcolor[HTML]{79C9A2}8 & \cellcolor[HTML]{8AD0AE}7 & \cellcolor[HTML]{57BB8A}10 & \cellcolor[HTML]{57BB8A}10 & \cellcolor[HTML]{57BB8A}10 & \cellcolor[HTML]{57BB8A}10 & \cellcolor[HTML]{57BB8A}10 \\
& \texttt{L6} & \textit{Logs have the wrong rotation.} & \cellcolor[HTML]{DEF2E8}2 & \cellcolor[HTML]{EFF9F4}1 & \cellcolor[HTML]{EFF9F4}1 & \cellcolor[HTML]{57BB8A}10 & \cellcolor[HTML]{57BB8A}10 & \cellcolor[HTML]{57BB8A}10 & \cellcolor[HTML]{57BB8A}10 \\
\midrule
\multicolumn{1}{l}{\multirow{6}{*}{\rotatebox{90}{Rendering}}} & \texttt{R1} & \textit{Viking and logs are very distorted.} & \cellcolor[HTML]{DEF2E8}2 & \cellcolor[HTML]{EFF9F4}1 & \cellcolor[HTML]{DEF2E8}2 & \cellcolor[HTML]{57BB8A}10 & \cellcolor[HTML]{57BB8A}10 & \cellcolor[HTML]{57BB8A}10 & \cellcolor[HTML]{57BB8A}10 \\
& \texttt{R2} & \textit{Viking and logs are slightly distorted.} & \cellcolor[HTML]{DEF2E8}2 & \cellcolor[HTML]{EFF9F4}1 & \cellcolor[HTML]{CDEBDC}3 & \cellcolor[HTML]{57BB8A}10 & \cellcolor[HTML]{57BB8A}10 & \cellcolor[HTML]{57BB8A}10 & \cellcolor[HTML]{57BB8A}10 \\
& \texttt{R3} & \textit{Viking and logs are blurred.} & \cellcolor[HTML]{DEF2E8}2 & \cellcolor[HTML]{FFFFFF}0 & \cellcolor[HTML]{FFFFFF}0 & \cellcolor[HTML]{57BB8A}10 & \cellcolor[HTML]{57BB8A}10 & \cellcolor[HTML]{57BB8A}10 & \cellcolor[HTML]{57BB8A}10 \\
& \texttt{R4} & \textit{Background trees covered in patches.} & \cellcolor[HTML]{8AD0AE}7 & \cellcolor[HTML]{79C9A2}8 & \cellcolor[HTML]{79C9A2}8 & \cellcolor[HTML]{FFFFFF}0 & \cellcolor[HTML]{57BB8A}10 & \cellcolor[HTML]{57BB8A}10 & \cellcolor[HTML]{57BB8A}10 \\
& \texttt{R5} & \textit{Background bushes have artifacts.} & \cellcolor[HTML]{DEF2E8}2 & \cellcolor[HTML]{FFFFFF}0 & \cellcolor[HTML]{DEF2E8}2 & \cellcolor[HTML]{FFFFFF}0 & \cellcolor[HTML]{57BB8A}10 & \cellcolor[HTML]{57BB8A}10 & \cellcolor[HTML]{57BB8A}10 \\
& \texttt{R6} & \textit{Background beach has tearing.} & \cellcolor[HTML]{ABDDC5}5 & \cellcolor[HTML]{FFFFFF}0 & \cellcolor[HTML]{8AD0AE}7 & \cellcolor[HTML]{FFFFFF}0 & \cellcolor[HTML]{57BB8A}10 & \cellcolor[HTML]{57BB8A}10 & \cellcolor[HTML]{57BB8A}10 \\
\midrule
\textbf{Accuracy} & & & \cellcolor[HTML]{B5E1CB}\textbf{44.6\%} & \cellcolor[HTML]{C7E9D8}\textbf{33.3\%} & \cellcolor[HTML]{B5E1CB}\textbf{44.6\%} & \cellcolor[HTML]{81CCA8}\textbf{75.0\%} & \cellcolor[HTML]{57BB8A}\textbf{100.0\%} & \cellcolor[HTML]{57BB8A}\textbf{100.0\%} & \cellcolor[HTML]{57BB8A}\textbf{100.0\%} \\
\bottomrule
\end{tabular*}
\end{table*}
\subsection{Execution duration}
To better understand the performance of our approach, we timed the executions of our approach and the baseline approach.
Our approach took considerably longer (3 additional seconds per snapshot) to run than the baseline approach.
However, the accuracy of the baseline approach indicates that it is not a very useful one in practice.
The bulk of time in our approach is spent preprocessing the images, whereas calculating the similarities is relatively quick, with the exception of SSIM.
In practice, we would not have to compute SSIM, because MSE, SSIM, and ESIM provide similar accuracy in our experiments.
\section{Threats to Validity}\label{sec:threats}
\subsubsection*{Construct validity}
Our results may be biased towards the set of visual bugs that we injected in our experiments.
However, our injected visual bugs covered all four visual bug types that are relevant to the \texttt{<canvas>}\xspace, as defined in prior work~\cite{macklon2022taxonomy}.
While the visual bugs we injected may not have the same cause as real visual bugs found in \texttt{<canvas>}\xspace applications, the visual effects are the same; each injected visual bug was designed to resemble a real world example.
To mitigate the threat of injecting unrealistic bugs we also confirmed with an industrial partner that our injected bugs were representative of real visual bugs in industrial \texttt{<canvas>}\xspace games.
There are different possible choices of image comparison metrics for snapshot testing, the baseline approach used in our experiments.
We selected PCT as an image comparison metric because \texttt{Percy}, a widely used snapshot testing tool, uses a threshold-based image comparison metric that is similar to PCT.
We also empirically evaluated MSE and SSIM for snapshot testing, and determined that neither were better than PCT for snapshot testing.
\subsubsection*{Internal validity}
In our approach, we utilized the \texttt{<canvas>}\xspace objects representation (COR) combined with the game assets to automatically generate visual test oracles (i.e., oracle assets).
Our approach therefore assumes that no bugs originate in these parts of the game.
It is fair to assume that the assets provide accurate baselines for comparison with the rendered game objects on the \texttt{<canvas>}\xspace.
In addition, it is fair to assume that the COR can be used to generate test oracles for detecting visual bugs, as any bug present in the COR would not be a visual bug.
A threat to internal validity is our choice of background fill colour when applying masks during image preprocessing in our experiments.
A fill colour must be selected to fill the blank space that results from the masking operations.
To address this threat, we ran our experiments with three different fill colours in 8-bit RGBA format: (0, 0, 0, 255), (255, 255, 255, 255), and (255, 0, 0, 255).
We empirically determined that for our test game, changing the fill colour for masking only affected our results with embedding similarity (ESIM), indicating that our use of ESIM may not be appropriate due to our extensive preprocessing.
A threat to internal validity is related to how we handle assets that have partial transparency in our experiments.
In our experiments, we chose to remove all partial transparency (i.e., make it fully transparent), as we empirically determined that this choice provided the best performance.
However, this means that we may miss some visual bugs that (primarily) affect the partially transparent areas of an object on the \texttt{<canvas>}\xspace.
More work is required to better handle assets with partial transparency when generating masks.
\subsubsection*{External validity}
Our approach has only been evaluated with a single \texttt{<canvas>}\xspace game.
Thresholds for detecting visual bugs with our approach will most likely differ on a per-game basis, and not all games may have as clear similarity thresholds as those found for our test game.
Therefore, automatically setting the thresholds to detect visual bugs may not be as effective for other \texttt{<canvas>}\xspace games, meaning manual adjustment may be required.
We designed our test game with the randomness and a variety of animations that were inspired by the visual styles and effects of industrial \texttt{<canvas>}\xspace games.
However, future studies should investigate further how different styles of games impact the performance of our approach.
Our approach is evaluated only a 2D \texttt{<canvas>}\xspace game that was built with the \texttt{PixiJS} \texttt{<canvas>}\xspace rendering framework.
More work is required to understand how well our approach works for other 2D \texttt{<canvas>}\xspace games, 3D \texttt{<canvas>}\xspace games, and non-\texttt{<canvas>}\xspace games.
In our experiments we leverage an existing \texttt{<canvas>}\xspace objects representation (COR) provided by \texttt{PixiJS}.
If a COR is not available in a \texttt{<canvas>}\xspace game, our approach would not work for that game.
Similarly, in our experiments we leverage existing \texttt{<canvas>}\xspace game assets to generate visual test oracles during the test, but if a \texttt{<canvas>}\xspace game does not use assets for its graphics rendering, then our approach would not work for that game.
Some types of graphics (e.g., skeletal animations, particle effects) are not in our test game, and are therefore not accounted for in the implementation of our approach.
More work is required to understand how our method performs when implemented for other types of graphics that are common in 2D \texttt{<canvas>}\xspace games.
\section{Conclusion}\label{sec:conclusion}
In this paper, we present a novel approach for automatically detecting visual bugs in \texttt{<canvas>}\xspace games.
By leveraging the \texttt{<canvas>}\xspace objects representation (COR), we are able to automatically generate oracle assets for comparison with isolated object images (as rendered to the \texttt{<canvas>}\xspace) and detect a wide variety of visual bugs in \texttt{<canvas>}\xspace games.
We found that our approach far outperforms the current industry standard approach (traditional snapshot testing) for automatically detecting visual bugs in \texttt{<canvas>}\xspace games.
We evaluated four similarity metrics with our approach, and found that mean squared error (MSE), structural similarity (SSIM), and embedding similarity (ESIM) each provided an accuracy of $100\%$ for our 24 injected visual bugs.
An implementation of our approach and our testbed is available at the following link: \href{https://github.com/asgaardlab/canvas-visual-bugs-testbed}{https://github.com/asgaardlab/canvas-visual-bugs-testbed}.
\section*{Acknowledgments}
The research reported in this article has been supported by Prodigy Education and the Natural Sciences and Engineering Research Council of Canada under the Alliance Grant project ALLRP 550309.
\balance
\printbibliography
\end{document}
|
2,877,628,088,968 | arxiv | \section{Introduction}
Over the past decade, several influential studies have demonstrated how machine learning (ML) -- when applied to digital data sources -- can provide accurate, timely, and granular estimates of wealth and poverty \cite{blumenstock2015predicting,jean2016combining,steele_mapping_2017,blumenstock2018estimating,burke_using_2020,yeh2020using,chi_microestimates_2022}.
These ML-based estimates, which can be produced at a fraction of the cost of traditional survey-based methods of data collection, can fill in data gaps in poor, fragile, and conflict-affected regions where recent surveys do not exist \cite{jerven2013poor}. And increasingly, the ML-based estimates are being used to inform critical policy decisions, including the targeting of humanitarian aid \cite{aiken2022machine} and the allocation of social assistance \cite{smythe_geographic_2022,gentilini2021cash}.
Despite the important role ML-based poverty estimates now play in development policy and disaster response, very little consideration has been given to the sampling strategy used to collect the ground truth labels used to train the ML models. Instead, the vast majority of poverty prediction models are trained on publicly-available household surveys that were collected for a different purpose (typically, to provide nationally representative demographic and health statistics).
These standardized surveys typically use (stratified) random sampling to
reduce measurement error \citep{howes1998does}. However, sampling training data according to these strategies may not optimize predictive power for a machine learning algorithm subject to a budget constraint for data collection. In an \textit{adaptive sampling} strategy, each incremental round of data acquisition is informed by an aspect of the performance of the model trained on data seen in previous rounds of data collection. In this paper, we investigate whether adaptive sampling strategies for label data collection improve the performance of machine learning models trained to predict poverty from digital data sources, focusing on model accuracy and fairness.
\section{Application Context}
This work builds on an emergency cash-transfer program run by the government of Togo that used poverty estimates from digital data sources to prioritize the distribution of emergency mobile money payments in response to the COVID-19 pandemic (described in more detail in Appendix \ref{additional_background} and \citet{aiken2022machine}). In designing the program, the government sought to distribute cash to 60,000 of the country's poorest individuals. At the time, however, the government did not have any traditional poverty data with which to determine eligibility. Instead, the Togolese government turned to satellite imagery analysis to identify the country's poorest districts, and then used machine learning models using features derived from subscribers' mobile phone records to identify the poorest subscribers in eligible districts for humanitarian aid. The ground truth labels for the wealth prediction models came from surveys conducted in Togo prior to the program's launch. An evaluation of the program's targeting found that relative to alternative poverty targeting methods (such as geographic or occupation-based targeting), the phone-based approach was most effective at targeting cash transfers to the poorest people in the country \cite{aiken2022machine}.
In this paper, we study whether using an adaptive strategy for sampling the poverty labels obtained from the phone survey in Togo could have improved the accuracy of the trained machine learning model. Our results speak most directly to recent efforts to estimate wealth and poverty from mobile phone data. More broadly, these results can inform the larger community of researchers and policymakers interested in estimating social and demographic characteristics from digital data in contexts where traditional data are unavailable or out-of-date.
\section{Methods}
We conduct experiments to simulate various data acquisition strategies using an in-person household survey conducted in Togo in 2018 ($N$ = 4,595) with a stratified random sampling strategy \footnote{While the survey was stratified by region and urban/rural designation, our experiment treats this survey as if it were sampled uniform-at-random.}. The survey collected information on daily per-capita consumption for each household interviewed (measured in USD/day), which we use as our ground truth measure of poverty. We match the household survey to anonymized features derived from call data records provided by the nation’s two cellular network operators. After partitioning the survey-labelled features into a label pool ($N_{\small{\textrm{LP}}}$ = 0.75$N$ = 3,446) and a holdout validation set ($N_{\small{\textrm{V}}}$ = 0.25$N$ = 1,149), we train random forest regressor models \cite{scikit-learn} on subsets of the label pool of incrementally increasing size to simulate how the resulting model performs throughout the data acquisition process. The model's performance is evaluated against the holdout set across all dataset sizes.
The baseline of acquiring data by uniform random sampling is compared against two adaptive data acquisition approaches, in which each data point still remaining in the label pool is assigned a heuristic sampling weight. In the \textit{query-by-committee} strategy, this weight is proportional to the variance of the outputs of the individual decision trees within the random forest regressor; unlabeled points for which the ensemble's constituent models disagree the most are upweighted in the next round of data acquisition. An alternative is \textit{margin-based uncertainty sampling}, in which an independent logistic regression classifier is trained on the same features to predict whether a given individual falls below a relevant poverty threshold; points whose probability of classification is closer to 50\% are upweighted in order to gain more information on points whose classification is close to random. Query-by-committee and uncertainty sampling are two standard methods for active learning for model accuracy \citep{settles2009active}. They have both been shown to substantially improve model performance subject to a budget constraint for data collection in small and relatively low-dimensional datasets unrelated to the poverty prediction task we consider in this paper \citep{settles2009active, settles2008analysis, mccallumzy1998employing, korner2006multi}.
Inspired by literature on active learning for fairness outcomes, \cite{abernethy2020active, cai2022adaptive}, we also consider adaptively constructing the training dataset by increasing the sampling weights for the points from the demographic categories with the lowest accuracy (\textit{accuracy-weighted}), highest mean-squared-error (\textit{MSE-weighted}), or highest exclusion/inclusion error rates (\textit{disparity-weighted}). In these group-based strategies, each member of the demographic category is assigned the same weight, but each group's weight differs by group-specific model performance in each round of simulated data collection.
Focusing on these demographic category weights also allows us to assess aspects of fairness of the model during the data collection process, as measured by minimum group accuracy, maximum group loss, or total inclusion-exclusion errors. These goals are independent of increasing the model's predictive power: even if deviation from uniform-at-random sampling provides no benefits in increasing accuracy or reducing MSE, there may be benefits in model fairness. In this investigation, the demographic category is geographic residence in each of the major admin-1 regions of Togo (depicted in more detail in Figure \ref{fig:togo_map}), inspired by previous analysis indicating there may exist slight geographic bias to the phone-based models \cite{aiken2022machine}.
We run 50 simulations for each strategy, with incremental dataset sizes $S_t$ logarithmically spaced between 0 and the full label pool size to better illustrate the effects of different strategies at smaller sample sizes. Within each simulation, the newly sampled points are appended to the previously-seen dataset ($S_t - S_{t-1}$ new data points are seen at time $t$ rather than $S_t$ points sampled memorylessly). Our code for the experiments is available on \href{https://github.com/satejsoman/cdr-active-learning}{GitHub}.
\section{Results}
We present the results of each of our sampling strategies, comparing individual level active learning methods (query-by-committee and uncertainty sampling) and group-level active learning methods (accuracy, MSE, and disparity weighting) to a baseline of uniform-at-random sampling.
In Figure \ref{fig:point_sampling_metrics}, we report the average performance of each sampling method across 50 simulations, with shaded regions representing boostrapped 95\% confidence intervals. Evaluating across a number of standard machine learning accuracy metrics (Spearman $\rho$, mean-square-error, AUROC, accuracy, precision, and recall).\footnote{Poverty predictions are continuous and produced with regression models; we binarize predictions using the international poverty line of US\$1.90/day to obtain binary classification metrics.}, it is clear that neither model-based nor group-weighting strategies provide substantial gains to overall model performance above uniform at random sampling. With some strategies, such as the query by committee approach, we do see some gains to model precision, but these are on the order of 1\% and are offset by reduction in recall (which is a more important metric for aid prioritization, since a model with lower recall will mistakenly exclude more recipients who should receive aid).
\begin{figure}[th]
\centering
\includegraphics[width=0.75\textwidth]{figs/point_sampling_metrics.pdf}
\caption{Comparison of model-based data acquisition strategies on key machine learning performance metrics as dataset size incrementally grows.}
\label{fig:point_sampling_metrics}
\end{figure}
Our group-based sampling strategies rely on upweighting groups (regions of Togo) for which our model performs worse. We evaluate these strategies based on the same metrics as above, and three additional fairness-related metrics calculated for the major geographic regions in Togo. In Figure \ref{fig:group_sampling_metrics}, we plot minimum group accuracy, maximum group loss, and a summary statistic for targeting inclusion/exclusion errors (absolute demographic deviation; ADD, explained in more detail in Appendix \ref{fairness_metrics_explained}). Mean values are plotted, with shaded regions representing bootstrapped 95\% confidence intervals. Again, we observe negligible differences in fairness between our active learning approaches and the uniform-at-random baseline sampling strategy.
Additional experiments in Appendix \ref{app:additional_experiments} detail similar results when using a feature set of reduced dimensionality and a hyperparameter tuning procedure across dataset sizes.
\begin{figure}[ht]
\centering
\includegraphics[width=\textwidth]{figs/group_sampling_metrics.pdf}
\caption{Comparison of group weighting strategies on standard machine learning metrics, as well as fairness-oriented metrics such as minimum group-level accuracy, maximum group-level mean-square-error, and total inclusion/exclusion errors (ADD).}
\label{fig:group_sampling_metrics}
\end{figure}
\section{Discussion}
Our results indicate that a well-implemented, uniformly-sampled survey may be close to optimal as a source of training data for machine learning models that aim to predict poverty levels from digital trace data. In particular, neither the model-based data acquisition strategies nor the group characteristic weighting approaches we considered showed appreciable gains on accuracy or fairness metrics over simple uniform random sampling.
This result has important implications for the growing research literature -- and increasing number of policy applications -- that rely on estimates of poverty predicted from digital trace data. Nearly all such estimates have been generated from ``found'' data, i.e., survey data that were collected for another purpose, and which were subsequently re-used as labels for training a poverty prediction algorithm \cite[e.g.,][]{blumenstock2015predicting,jean2016combining,steele_mapping_2017,blumenstock2018estimating,burke_using_2020,yeh2020using,chi_microestimates_2022}. Our hypothesis in initiating this analysis was that such re-use was sub-optimal; if the poverty prediction use case was known \textit{ex ante}, we expected there to be performance gains from a more strategic approach to obtaining training data. However, our results indicate that --- at least for the adaptive approaches that we tested -- no such gains exist.
Whether this is a general finding, or one that is specific to our particular data context (i.e., mobile phone data from Togo), is an area of future work that we are continuing to explore. We also see several new directions in which to extend this analysis. For instance, all of the simulations described here assume an identical labeling cost for all data points. This uniform-cost model is appropriate in settings where the cost of interviewing an individual is the same regardless of their physical location in the country -- as is the case with phone-based surveys. By contrast, many settings involve highly heterogeneous sampling costs -- such as household surveys where some households are easily accessed and others are remote. In such settings, factoring in the costs of traveling to survey locations and conducting surveys may provide more realistic strategies for collecting training labels in household surveys. In the current investigation, in which groups are defined geographically, group-specific sampling is especially cost-infeasible because where the group to be sampled may change in every round of incremental data acquisition. In continued work, we plan to investigate cost-sensitive data acquisition strategies, and generate theoretical explanations for the lack of gains in any of the non-uniform sampling strategies.
\newpage
\begin{ack}
This material is based upon work supported by the National Science Foundation under Grant No. DGE- 2125913.
\end{ack}
\bibliographystyle{unsrtnat}
|
2,877,628,088,969 | arxiv | \section{Constraints over a larger parameter range}
\label{sec:supplemental_constraints}
Supplemental~\Cref{fig:bounds_large} generalizes \cref{fig:bounds} in the main text to a larger perspective. We note the attractive possibility of a new analysis of SN 1987A data probing the gap between our timing constraints and prior cooling constraints.
\begin{figure}[hbtp]
\centering
\includegraphics[width=\columnwidth]{plots/bounds_large_annotated.pdf}
\caption{Same as \cref{fig:bounds}, but for a larger range of parameters, now showing important constraints from $Z$-boson decay~\cite{Brdar:2020nbj} and SN 1987A cooling~\cite{Kachelriess:2000qc, Farzan:2002wx, Heurtier:2016otg}. In addition, we show subdominant constraints from present IceCube HESE data~\cite{IceCube:2020wum} (obtained with the code \texttt{nuSIProp} and following Ref.~\cite{Esteban:2021tub}), SN 1987A shock revival, and SN 1987A neutrino propagation~\cite{Shalgar:2019rqe, Kolb:1987qy}.
}
\label{fig:bounds_large}
\end{figure}
\clearpage
\newpage
\section{Flavor-dependent interactions}
\label{sec:supplemental_flavor}
Supplemental~\Cref{fig:bounds_flavor} generalizes \cref{fig:bounds} in the main text to flavor-dependent interactions, covering the simple cases where $\nu$SI appears in just one of the three flavor sectors. This is based on new formalism, developed below, where we also discuss flavor structures that could avoid our limit.
Though we leave a dedicated study of flavor-dependent $\nu$SI effects for future work, we note some expectations. Because we consider length scales much larger than the oscillation lengths~\cite{Dighe:1999bi, Dasgupta:2016dbv, Capozzi:2022slf} (including matter effects inside the envelope of the progenitor star, typically larger than the neutrino ball), decoupling naturally happens in the mass-eigenstate basis. For SN 1987A, only $\bar{\nu}_e$, which are dominantly composed of $\bar{\nu}_1$ and $\bar{\nu}_2$, were detected. Thus, we generically estimate $\ell_\mathrm{FS}$ by setting the optical depths of \emph{both} $\bar{\nu}_1$ and $\bar{\nu}_2$ to $\tau_\mathrm{FS}$, i.e.~(see~\cref{eq:tau}),
\begin{equation}
\begin{cases}
\displaystyle{\sum_i \left\langle N_\nu^i \sigma_{\nu\nu}^{1i}\right\rangle \frac{3}{4\pi \ell_\mathrm{FS}^2} = \tau_\mathrm{FS}} \\
\displaystyle{\sum_i \left\langle N_\nu^i \sigma_{\nu\nu}^{2i}\right\rangle \frac{3}{4\pi \ell_\mathrm{FS}^2} = \tau_\mathrm{FS}}
\end{cases} \, .
\label{eq:tau_FS_flavor}
\end{equation}
This means that strong $\nu$SI could evade our limits if only one or none of these mass eigenstates is affected. In this scenario, the constraints from $K$ decay would still apply. Also, the supernova would generate at least two neutrino signals with very different timing and flavor content, a dramatic signature that could be tested with future data.
In \cref{eq:tau_FS_flavor}, $N_\nu^i$ is the number of $\bar{\nu}_i$ in the ball (see the discussion in the preamble of the S.M.), and $\sigma_{\nu \nu}^{ji}$ is the cross section for $\bar{\nu}_j \bar{\nu}_i$ interactions. This is dominated by the s-channel contribution~\cite{Esteban:2021tub},
\begin{equation}
\sigma_{\nu \nu}^{ji} = |g_{ij}|^2 \sum_{k, l} |g_{kl}|^2\frac{1}{16\pi} \frac{s}{(s-M_\phi^2)^2+M_\phi^2 \Gamma^2} \, ,
\end{equation}
with $g_{ij}$ the elements of the coupling matrix in the mass basis (i.e., $\mathcal{L}_{\nu\mathrm{SI}} = -1/2 \, g_{ij} \bar{ \nu}_i \nu_j$), and $\Gamma = \sum_{k,l} |g_{kl}|^2 \,M_{\phi}/16 \pi$. In the resonant regime,
\begin{equation}
\sigma_{\nu \nu}^{ji, \, \mathrm{res}} = |g_{ij}|^2 \, \pi \delta(s-M_\phi^2) \, .
\end{equation}
\begin{figure}[b]
\centering
\includegraphics[width=\textwidth]{plots/bounds_flavor_annotated.pdf}
\caption{Same as \cref{fig:bounds}, but for $\nu$SI restricted to individual flavors, i.e., assuming $g_{ee}$, $g_{\mu \mu}$ or $g_{\tau \tau}$ to be the only non-zero component. The ``Moderately Interacting neutrino (MI$\nu$) solution'' (hatched) has been argued to improve the fit to present Cosmic Microwave Background data, but it only remained viable if $\nu$SI were restricted to $\nu_\tau$~\cite{Kreisch:2019yzn}.}
\label{fig:bounds_flavor}
\end{figure}
Assuming that the supernova emits comparable amounts of neutrinos and antineutrinos of all flavors, $N_\nu^i \sim N_\nu / 6$. Then, our flavor-independent limits on $g$ translate into limits on $g_\mathrm{eff} \equiv \min\left(\sqrt{\sum_j |g_{1j}|^2}, \, \sqrt{\sum_j |g_{2j}|^2} \right)$ in the resonant regime, and on $g_\mathrm{eff} \equiv \min\left(\sqrt[4]{\sum_j |g_{1j}|^2 \sum_{kl} |g_{kl}|^2}, \, \sqrt[4]{\sum_j |g_{2j}|^2 \sum_{kl} |g_{kl}|^2} \right)$ in the non-resonant regime. The expressions are simpler in the $\nu$SI interaction eigenbasis $\{\nu_a, \nu_b, \nu_c\}$, where our limits on $g$ translate into limits on
\begin{equation}
g_\mathrm{eff} \equiv
\begin{cases}
\min \left( \sqrt{\sum_a g_a^2 |V_{a1}|^2}, \, \sqrt{\sum_a g_a^2 |V_{a2}|^2} \right) & \text{in the resonant regime} \\
\min \left( \sqrt[4]{\sum_a g_a^2 |V_{a1}|^2 \sum_b g_b^2}, \, \sqrt[4]{\sum_a g_a^2 |V_{a2}|^2 \sum_b g_b^2} \right) & \text{in the non-resonant regime}
\end{cases} \, ,
\label{eq:g_eff_flavor}
\end{equation}
where $\{g_a, g_b, g_c\}$ are the eigenvalues of $g_{ij}$, and $V$ is the matrix linking $\nu$SI interaction eigenstates to mass eigenstates as ${\nu_a = \sum_i V_{ai} \nu_i}$. With this expression and the mixing angles from Ref.~\cite{Esteban:2020cvm}, we obtain the limits in Suppl.~\cref{fig:bounds_flavor} (for the matrix elements that depend on $\delta_\mathrm{CP}$, we vary this parameter between 0 and $2\pi$ and we conservatively take the largest $g_\mathrm{eff}$).
Finally, we note that flavor-dependent $\nu$SI could introduce an effective $\nu {\emph - }\nu$ potential affecting flavor oscillations. However, for scalar $\nu$SI this is a correction to the neutrino mass $\sim n_\nu (g^2/M_\phi^2) (m_\nu/E_\nu)$~\cite{Bergmann:1999rz, Ge:2018uhz}. The suppression factor for ultrarelativistic neutrinos $m_\nu/E_\nu \sim 10^{-8}$ guarantees that this correction is small.
\clearpage
\newpage
\section{Computation of the neutrino optical depth}
\label{sec:supplemental_tau}
Here we compute the average $\nu$SI neutrino optical depth given in \cref{eq:tau}. Since we aim for factor-two precision, we conservatively consider a neutrino that travels a distance $\ell$ (the radius of the ball) instead of $2\ell$ (the diameter of the ball). We also assume a uniform background density $n_\nu = N_\nu / (4 \pi \ell^3 / 3)$, neglecting $\mathcal{O}(1)$ variations due to the expansion of the ball and its conversion into a shell. The general expression then reads
\begin{equation}
\tau(\ell) = \frac{1}{6} N_\nu \left(\frac{4 \pi}{3} \ell^3\right)^{-1} \ell \int \mathrm{d}E_1 \, \mathrm{d}E_2 \, \mathrm{d}\Omega_1 \, \mathrm{d}\Omega_2 \, f(E_1, \hat{\Omega}_1) f(E_2, \hat{\Omega}_2) \, (1 - \cos \theta_{12}) \, \sigma_{\nu \nu}(E_1, E_2, \cos \theta_{12}) \, .
\label{eq:tau_integral}
\end{equation}
The factor $\frac{1}{6}$ takes into account that we consider s-channel scattering and flavor-independent $\nu$SI: as discussed in the preamble of the S.M., a neutrino of a given flavor only scatters with another neutrino of the same flavor (and analogously for antineutrinos) and we assume that supernovae emit comparable amounts of neutrinos and antineutrinos of all three flavors. In \cref{eq:tau_integral}, $N_\nu \sim |\Delta E_{\rm b}| / \langle E_\nu \rangle$ is the total number of neutrinos in the ball, $E_1$ and $E_2$ are the energies of the incoming neutrinos, $\Omega_1$ and $\Omega_2$ are the solid angles covered by their directions of motion, $\theta_{12}$ is the angle between their momenta, $f(E, \hat{\Omega})$ is the neutrino energy and angular distribution, $(1- \cos \theta_{12})$ is the M{\o}ller factor~\cite{moller_1945, 1975ctf..book.....L}, and $\sigma_{\nu \nu}$ is the $\nu$SI cross section.
We assume that neutrinos move isotropically. While this is not strictly true, particularly close to the edges of the expanding neutrino ball, directional effects average when considering interactions among neutrinos from the entire ball. For generality, we take a pinched Maxwell-Boltzmann neutrino energy distribution~\cite{Keil:2002in, Tamborra:2012ac},
\begin{equation}
f(E, \hat{\Omega}) = \frac{1}{4 \pi}\frac{(\beta + 1)^{\beta + 1}}{\Gamma(\beta + 1)} \frac{E^\beta}{\langle E_\nu \rangle^{\beta +1}} e^{-(\beta+1) \frac{E}{\langle E_\nu \rangle}} \, ,
\label{eq:energy_distribution}
\end{equation}
with $\Gamma$ being Euler's function; and $\beta$ a ``pinching'' parameter that controls the width of the distribution, where $\beta=2$ corresponds to a Maxwell-Boltzmann distribution, $\beta=2.3$ to a Fermi Dirac distribution with no chemical potential, and other values give other distributions. Supernova simulation outputs are well described by $\beta \in [2, 4]$~\cite{Tamborra:2012ac, Keil:2002in}. As we show below, our results are robust to changes in $\beta$, so in the main text we assume $\beta=2$.
We consider the s-channel scattering cross section in \cref{eq:csec}. Including other scattering channels would increase both the cross section and the number of target neutrinos (although random walks in all directions may not be guaranteed; see \cref{sec:supplemental_isotropization}). This may lead to improved constraints, particularly for light mediators (see, e.g., Ref.~\cite{Esteban:2021tub}).
For simplicity, we separate our calculations into three different kinematic regimes:
\begin{itemize}
\item $M_\phi \ll \sqrt{s}$: here,
%
\begin{equation}
\sigma_{\nu \nu} = \frac{g^4}{16 \pi} \frac{1}{s} = \frac{g^4}{32 \pi} \frac{1}{E_1 E_2 (1 - \cos \theta_{12})} \, ,
\label{eq:csec_nres_SM}
\end{equation}
%
and \cref{eq:tau_integral} gives
%
\begin{equation}
\tau = \frac{3}{128 \pi^2} \left(\frac{1+\beta}{\beta}\right)^2 \, \frac{|\Delta E_{\rm b}|}{6 \langle E_\nu \rangle} \, \frac{g^4}{\langle E_\nu \rangle^2} \, \frac{1}{\ell^2} \, .
\label{eq:tau_lowM}
\end{equation}
%
As described in the main text, we estimate $\ell_\mathrm{FS}$ as a function of $g$ and $M_\phi$ by taking $\tau(\ell_\mathrm{FS}) \equiv \tau_\mathrm{FS}$. Conversely, given $\ell_\mathrm{FS}$ and $\tau_\mathrm{FS}$, the sensitivity to $g$ is given by
%
\begin{equation}
g \sim 1.9 \times 10^{-2} \,\left(\frac{\tau_{\rm FS}}{10}\right)^{1/4} \left(\frac{\ell_{\rm FS}/c}{30 \, \mathrm{s}}\right)^{1/2}
\left(\frac{|\Delta E_{\rm b}|}{3 \times 10^{53} \, \mathrm{ergs}}\right)^{-1/4} \left(\frac{\langle E_\nu \rangle}{10 \, \mathrm{MeV}}\right)^{3/4} \left(\frac{(1+\beta)/\beta}{3/2}\right)^{-1/2} \, .
\label{eq:sens_lowM}
\end{equation}
\item $M_\phi \gg \sqrt{s}$: here,
%
\begin{equation}
\sigma_{\nu \nu} = \frac{g^4}{16 \pi} \frac{s}{M_\phi^4} = \frac{g^4}{8 \pi} \frac{E_1 E_2 (1 - \cos \theta_{12})}{M_\phi^4} \label{eq:csec_nres_LM}
\end{equation}
and \cref{eq:tau_integral} gives
%
\begin{equation}
\tau = \frac{1}{8 \pi^2} \, \frac{|\Delta E_{\rm b}|}{6 \langle E_\nu \rangle} \, g^4 \frac{\langle E_\nu \rangle^2}{M_\phi^4} \, \frac{1}{\ell^2} \, .
\label{eq:tau_largeM}
\end{equation}
The sensitivity to $g$ is given by
%
\begin{equation}
g \sim 0.15 \,\left(\frac{\tau_{\rm FS}}{10}\right)^{1/4} \left(\frac{\ell_{\rm FS}/c}{30 \, \mathrm{s}}\right)^{1/2} \left(\frac{M_\phi}{100 \, \mathrm{MeV}}\right)
\left(\frac{\Delta E_{\rm b}}{3 \times 10^{53} \, \mathrm{ergs}}\right)^{-1/4} \left(\frac{\langle E_\nu \rangle}{10 \, \mathrm{MeV}}\right)^{-1/4} \, .
\label{eq:sens_largeM}
\end{equation}
%
\item $M_\phi \sim \sqrt{s}$: here,
%
\begin{equation}
\sigma_{\nu \nu} = g^2 \pi \,\delta(s-M_\phi^2) = g^2 \pi \, \delta(2 E_1 E_2 [1 - \cos \theta_{12}] - M_\phi^2) \, ,
\label{eq:csec_res}
\end{equation}
%
and \cref{eq:tau_integral} gives
%
\begin{equation}
\tau = \frac{3}{8} \left(\frac{(1+\beta)^{1+\beta}}{2^{\beta/2} \Gamma(1+\beta)}\right)^2 \, \frac{|\Delta E_{\rm b}|}{6 \langle E_\nu \rangle} \, \frac{g^2}{M_\phi^2} \, \frac{1}{\ell^2} \, \mathcal{F}_\beta \left( \frac{M_\phi}{\langle E_\nu \rangle} \right) \, ,
\label{eq:taures_beta}
\end{equation}
%
with
%
\begin{equation}
\mathcal{F}_\beta (x) \equiv x^{2+2\beta} \int_0^2 \mathrm{d}q \, q^{-\beta} K_0\left(\sqrt{\frac{2}{q}}(1+\beta) x\right) \, , \label{eq:F_beta}
\end{equation}
%
and $K_0$ the modified Bessel function. For a given $\ell_\mathrm{FS}$, the sensitivity to $g$ is given by
%
\begin{equation}
\begin{split}
g \sim 5.7 \times 10^{-5} \, & \left(\frac{\tau_{\rm FS}}{10}\right)^{1/2} \left(\frac{\ell_{\rm FS}/c}{30 \, \mathrm{s}}\right) \left(\frac{M_\phi}{10 \, \mathrm{MeV}}\right) \left(\frac{|\Delta E_{\rm b}|}{3 \times 10^{53} \, \mathrm{ergs}}\right)^{-1/2} \left(\frac{\langle E_\nu \rangle}{10 \, \mathrm{MeV}}\right)^{1/2}
\left[ \frac{\mathcal{F}_\beta(M_\phi/\langle E_\nu \rangle)}{\mathcal{F}_2(1)} \right]^{-1/2} \\ & \left( \frac{2^{\beta/2} \Gamma(1+\beta) / (1+\beta)^{1+\beta}}{4/27} \right) \, .
\label{eq:sens_resonance}
\end{split}
\end{equation}
%
For $\beta=2$, $\mathcal{F}_2(x) = x^5 \,K_1(3x)/3$, and \cref{eq:taures_beta,eq:sens_resonance} correspond to \cref{eq:taures_analytic,eq:gres} in the main text.
\end{itemize}
\Cref{eq:sens_largeM,eq:sens_lowM,eq:sens_resonance} give our sensitivity for any mediator mass in \cref{fig:bounds} and Suppl.~\cref{fig:bounds_large}. In the transitions between the different kinematic regimes, we conservatively take the largest optical depth instead of adding them up.
Supplemental~\Cref{fig:bounds_assumptions} shows that our limit is mostly insensitive to our assumptions. Changing $\langle E_\nu \rangle$ modifies the mediator mass for which resonant scattering is most frequent, $M_\phi \sim \langle E_\nu \rangle$; the cross section for that mediator mass, ${\sigma \propto 1/M_\phi^2 \sim 1/\langle E_\nu \rangle^2}$; as well as the number of neutrinos in the ball, $N_\nu \propto |\Delta E_{\rm b}| / \langle E_\nu \rangle$. These effects do not affect our limit by more than a factor $\sim 2$ even if we allow $\langle E_\nu \rangle$ to change by $\sim 50\%$. Our result is thus insensitive to small variations in the neutrino energy due to the temperature evolution of the supernova, gravitational redshift of neutrinos as they leave the PNS, or hierarchies in the average energies of the different flavors. Changing the pinching parameter $\beta$ modifies the low-energy and high-energy tails of the neutrino energy distribution, which induces minor differences on the resonant production of mediators with masses $M_\phi \neq \langle E_\nu \rangle$. Finally, even if we are extremely conservative and assume that decoupling begins as soon as $\tau$ drops below $\sim 100$, i.e., setting $\tau_\mathrm{FS} = 100$ instead of $10$, our limit on $g$ only changes by a factor $\sqrt{100/10} \sim 3$ (or $\sqrt[4]{100/10} \sim 1.8$ outside the resonant regime). Changing the total neutrino energy $|\Delta E_{\rm b}|$ is numerically equivalent to changing $\tau_\mathrm{FS}$, as our limit depends on $\tau_\mathrm{FS}/|\Delta E_{\rm b}|$.
\begin{figure}[hbtp]
\centering
\includegraphics[width=\textwidth]{plots/bounds_assumptions.pdf}
\caption{Conservative sensitivity in \cref{fig:bounds} for different assumptions, as labeled. }
\label{fig:bounds_assumptions}
\end{figure}
Supplemental~\Cref{fig:bounds_time} shows that in the burst-outflow case the supernova neutrino signal duration is extremely large if the coupling is above our conservative limit. We compute the signal duration conservatively assuming that the neutrino ball starts decoupling at optical depth $\tau_\mathrm{FS} = 10$. For a mediator mass $M_\phi \sim 10 \, \mathrm{MeV}$ and a coupling that saturates $K$-decay bounds, the signal duration is ${\sim 1 \, \mathrm{hour}}$; and even greater for some interactions affecting cosmology.
\begin{figure}[b]
\centering
\includegraphics[width=0.60\textwidth]{plots/bounds_time.pdf} \caption{Supernova neutrino signal duration for different $\nu$SI parameters, assuming the burst-outflow case. Other limits and scales are as in \cref{fig:bounds}.}
\label{fig:bounds_time}
\end{figure}
The results above also demonstrate that our limit is robust against $2 \nu \rightarrow 4 \nu$ processes. These take place only for the largest couplings~\cite{Shalgar:2019rqe} (the yellow region labeled ``SN 1987A shock'' in Suppl.~\cref{fig:bounds_large}), and they reduce the average neutrino energy $\langle E_\nu \rangle$. For $M_\phi \lesssim 70 \, \mathrm{MeV}$, such large couplings imply a supernova neutrino signal duration ${\gtrsim 1 \, \mathrm{hour}}$ (see~Suppl.~\cref{fig:bounds_time}), very strongly excluded no matter what value of $\langle E_\nu \rangle$ is used. For $M_\phi \gtrsim 70 \, \mathrm{MeV}$, our limit scales as ${g \sim \langle E_\nu \rangle^{-1/4}}$, see~\cref{eq:sens_largeM} and Suppl.~\cref{fig:bounds_assumptions}, and it is therefore insensitive to the precise value of $\langle E_\nu \rangle$.
\clearpage
\newpage
\section{Random walks induced by frequent scattering}
\label{sec:supplemental_isotropization}
Supplemental~\Cref{fig:def_total} shows that frequent scattering establishes random motion in all directions, which in the burst-outflow case leads to the formation of a homogeneous neutrino ball, as explained in the main text. Conversely, when scattering is rare, neutrinos are not significantly deflected by $\nu$SI, and the neutrino ball becomes a shell (see~\cref{fig:macro_cartoon,fig:micro_cartoon}). For the mediator masses in our region of interest in \cref{fig:bounds}, $\nu$SI have not fully decoupled yet when $\tau \sim 10$. Thus, setting the transition period at $\tau_\mathrm{FS}=10$ leads to a conservative underestimate of the signal duration. (See Suppl.~\cref{fig:bounds_FS_isotropic} for the impact for a wider mediator mass range.)
\begin{figure}[hbtp]
\centering
\includegraphics[width=0.68\textwidth]{plots/Avg_net_deflection_Nscatt.pdf}
\caption{Average deflection of an initially forward neutrino ensemble after many resonant scatterings with an isotropic background, for different $M_{\phi}/\langle E_{\nu} \rangle$ (see below for non-resonant scattering). When $\langle \cos \theta_\mathrm{total} \rangle\sim 0$, $\nu$SI randomize the directions of motion.}
\label{fig:def_total}
\end{figure}
Intuitively, $\nu$SI quickly establish random motion in all directions because for the scalar-mediated s-channel scattering we consider, the distribution of the outgoing neutrino directions is isotropic in the center of mass (CM) frame. Furthermore, the CM boost is
\begin{equation}
\gamma_\mathrm{CM} = \frac{1}{\sqrt{1 - \beta_\mathrm{CM}^{\,2}}} = \frac{1 + E_1 / E_2}{2 \sqrt{E_1/E_2}} \cdot \left( {\rm sin}\,\frac{\theta_{12}}{2} \right)^{-1} \,,
\label{eq:gamma_c}
\end{equation}
which is typically $\mathcal{O}(1)$ as $E_1$ and $E_2$, the energies of the incoming neutrinos, are drawn from similar distributions; and the scattering angle $\theta_{12}$ is not generally small. Thus, the outgoing distribution in the laboratory frame is also quite isotropic.
For increasing $M_\phi/\langle E_\nu \rangle$, neutrinos move in all directions with less scatterings. There are several ways of understanding this. From energy and momentum conservation, producing a heavier mediator requires a larger scattering angle $\theta_{12}$. This leads to a large momentum transfer, and thus to a large deflection. Alternatively, larger $\theta_{12}$ decreases the CM boost (see~\cref{eq:gamma_c}), making the outgoing neutrinos more isotropic. Finally, one can also check that for resonant scattering the CM boost can be written as ${\gamma_\mathrm{CM}=(E_1+E_2)/\sqrt{s} = (E_1+E_2)/M_\phi}$. Larger $M_\phi$ thus leads to smaller CM boosts and more isotropic scattering.
\begin{figure}[hbtp]
\centering
\includegraphics[width=0.62\textwidth]{plots/Isotropization.pdf}
\caption{Distribution of directions of an initially forward neutrino ensemble after many resonant scatterings with an isotropic background, for different $M_\phi/\langle E_{\nu}\rangle$. The total deflection angle, $\theta_{\rm total}$, is measured with respect to the initial direction.}
\label{fig:def_dist}
\end{figure}
Supplemental \Cref{fig:def_dist} explicitly shows that the distribution of neutrino directions becomes isotropic after scattering. For large $M_\phi$, the momentum transfer in a resonant scattering is large as noted above, leading to a backwards-peaked distribution for the first scatterings (see~top panel in the figure).
\begin{figure}[hbtp]
\centering
\includegraphics[width=0.62\textwidth]{plots/Avg_deflection_Nscatt.pdf}
\caption{Average {\it individual} deflection by each resonant scattering, for different $M_{\phi}/\langle E_\nu\rangle$.}
\label{fig:def_each_scatt}
\end{figure}
Supplemental \Cref{fig:def_each_scatt} shows, for completeness, that the average deflection by each \emph{individual} scattering is sizable. It rapidly converges to a constant, as scattering processes tend to reach equilibrium. Also, deflections are bigger for larger $M_{\phi}/\langle E_{\nu} \rangle$ as discussed above.
To generate Suppl.~\cref{fig:def_total,fig:def_dist,fig:def_each_scatt}, we perform numerical simulations following these steps:
\begin{enumerate}
\item We start with an ensemble of test neutrinos with energies $\{E_1\}$, randomly drawn from a Maxwell-Boltzmann distribution with average energy $\langle E_{\nu} \rangle$.
%
\item Each test neutrino with energy $E_1$ scatters with a background neutrino with energy $E_2$ and direction $\cos \theta_{12}$. As the probability density for this scattering to happen is proportional to the optical depth in \cref{eq:tau_integral},
\begin{equation}
\mathcal{P}(E_2, \cos \theta_{12} | E_1) \propto \,f(E_2,\hat{\Omega}_2) \,(1 - \cos \theta_{12}) \,\sigma_{\nu\nu}(E_1,E_2,\cos \theta_{12}) \, ,
\end{equation}
we draw $E_2$ and $\cos \theta_{12}$ from this distribution. Plugging the cross sections~(\ref{eq:csec_nres_LM}), (\ref{eq:csec_nres_SM}), and (\ref{eq:csec_res}); and setting $f(E_2, \hat{\Omega}_2)$ to an isotropic Maxwell-Boltzmann distribution with average energy $\langle E_\nu \rangle$, we have
%
\begin{itemize}
\item $M_\phi \ll \sqrt{s}$:
\begin{equation}
\mathcal{P}(E_2, \cos \theta_{12} | E_1) \propto E_2 \, \exp\left[ - \frac{3 E_2}{\langle E_\nu \rangle}\right] \,;
\label{eq:dist_SM}
\end{equation}
%
\item $M_\phi \gg \sqrt{s}$:
\begin{equation}
\mathcal{P}(E_2, \cos \theta_{12} | E_1) \propto (1 - \cos\theta_{12})^2 \, E_2^3 \, \exp\left[ - \frac{3 E_2}{\langle E_\nu \rangle}\right] \,;
\label{eq:dist_LM}
\end{equation}
%
\item $M_\phi \sim \sqrt{s}$ (resonant regime):
\begin{align}
& \mathcal{P}(E_2, \cos \theta_{12} | E_1) \propto E_2^2 \, \exp \left[- \frac{3 E_2}{\langle E_\nu \rangle} \right]\, \delta\left( E_2 -\frac{M_{\phi}^2}{2E_1 (1 - \cos\theta_{12})}\right) \nonumber
\\
& \qquad\qquad \propto (1 - \cos\theta_{12})^{-2} ~ {\rm exp}\left[\frac{- 3 M_{\phi}^2}{2 E_1 \langle E_{\nu} \rangle (1 - \cos\theta_{12} )} \right] \, \delta\left( E_2 -\frac{M_{\phi}^2}{2E_1 (1 - \cos\theta_{12})}\right)
\label{eq:dist_res}
\end{align}
\end{itemize}
%
with $-1 \leq \cos \theta_{12} \leq 1$ and $E_2 > 0$.
%
\item In each scattering, the angles between the test neutrino direction and the outgoing neutrino directions, $\Delta \theta$, are given by kinematics
\begin{equation}
\cos \Delta \theta = \frac{1}{\gamma_{\rm CM} (1 \pm \beta_{\rm CM} \,\cos\vartheta_c )} \left[ \pm~ {\rm sin}\,\theta_{1c} \,{\rm sin}\,\vartheta_c\, \cos\varphi_c + \gamma_{\rm CM}\, \cos\theta_{1c} (\beta_{\rm CM} \pm \cos\vartheta_c) \right] \, ,
\label{eq:u_13_14}
\end{equation}
where $\cos\vartheta_c$ and $\varphi_c$ are the outgoing polar and azimuth angle in the CM frame, respectively; and $\theta_{1c}$ is the angle between the test neutrino momentum and the boost direction of the CM,
\begin{equation}
\cos\theta_{1c} = \frac{E_1/E_2 + \cos\theta_{12}}{\sqrt{ \left( E_1 /E_2 + \,\cos\theta_{12}\right)^2 + {\rm sin}^2\,\theta_{12}}}\,.
\end{equation}
We draw $\cos\vartheta_c$ and $\varphi_c$ from uniform distributions over the intervals $[-1,\,1]$ and $[0,\,2\pi)$, respectively, as s-channel scattering is isotropic in the CM. This is not the case for other scattering channels~\cite{Esteban:2021tub}. To be conservative, we set the individual deflection by each scattering to the smallest value among the two possibilities in \cref{eq:u_13_14}.
%
\item We repeat steps $2$ and $3$ to simulate many scatterings.
The incoming test neutrino energies in the $n$-th scattering are set to the outgoing neutrino energies in the $(n-1)$-th scattering; that is, from kinematics,
\begin{equation}
E_1^{\,(n)} \equiv \frac{E_1^{\,(n-1)}+E_2^{\,(n-1)}}{2} \, ( 1 \pm \beta_{\rm CM}^{\,(n-1)} \,\cos\vartheta_c^{\,(n-1)}) \, ,
\end{equation}
where the two values correspond to the two possible angles in \cref{eq:u_13_14}. To be consistent, we choose the energy corresponding to the smallest $\Delta \theta$.
%
\item The \emph{total} deflection with respect to the initial direction after the $n$-th scattering is given by~\cite{Goudsmit:1940zza}
\begin{equation}
\cos \theta_\mathrm{total}^{(n)} = \cos \theta_\mathrm{total}^{(n-1)} \cos \Delta \theta^{(n)} + \sin \theta_\mathrm{total}^{(n-1)} \sin \Delta \theta^{(n)} \cos \phi^{(n)} \, ,
\label{eq:total_defl}
\end{equation}
with $\Delta \theta^{(n)}$ the individual deflection by the $n$-th scattering, \cref{eq:u_13_14}; and $\phi^{(n)}$ the relative azimuth deflection by the $n$-th scattering, uniformly distributed over $[0, \,2\pi)$.
Recursively applying \cref{eq:total_defl}, we obtain the total deflection angle after many scatterings, $\theta_\mathrm{total}$.
\end{enumerate}
The results in Suppl.~\cref{fig:def_total,fig:def_dist,fig:def_each_scatt} are obtained with an ensemble of $10^4$ test neutrinos.
Supplemental~\Cref{fig:bounds_FS_isotropic} shows that ${\tau_\mathrm{FS} = 10}$ is a good approximation to the more realistic condition of decoupling happening when $\nu$SI do not randomize neutrino directions. The dashed line corresponds to setting $\tau_\mathrm{FS}^2$ to the number of scatterings below which directions are not randomized (computed using the simulation above). We consider directions not to be randomized if the relative difference between the amount of forward-moving neutrinos ($\cos \theta_{\rm total}>0$) and backward-moving neutrinos ($\cos \theta_{\rm total}<0$) is bigger than 10\%.
Supplemental~\Cref{fig:nres_SM_def,fig:nres_LM_def} show our results of our simulations for non-resonant scattering, which are insensitive to $M_{\phi}/\langle E_{\nu} \rangle$ (see~\cref{eq:dist_LM,eq:dist_SM}).
\begin{figure}[hbtp]
\centering
\includegraphics[width=0.75\textwidth]{plots/bounds_tau_isotropization_large.pdf}
\caption{Conservative sensitivity in Suppl.~\cref{fig:bounds_large} for different choices of $\tau_\mathrm{FS}$ (note the different Y axis range). In our primary region of interest, above BBN limits, our choice in the main text is shown to be always conservative.}
\label{fig:bounds_FS_isotropic}
\end{figure}
\begin{figure}[hbtp]
\centering
\includegraphics[width=0.61\textwidth]{plots/nres_SM_def.pdf}
\caption{Same as Suppl.~\cref{fig:def_total,fig:def_dist,fig:def_each_scatt}, but for non-resonant scattering with $M_{\phi} \ll \sqrt{s}$.}
\label{fig:nres_SM_def}
\end{figure}
\begin{figure}[hbtp]
\centering
\includegraphics[width=0.61\textwidth]{plots/nres_LM_def.pdf}
\caption{Same as Suppl.~\cref{fig:def_total,fig:def_dist,fig:def_each_scatt}, but for non-resonant scattering with $M_{\phi} \gg \sqrt{s}$.}
\label{fig:nres_LM_def}
\end{figure}
\clearpage
\newpage
\section{Energy dependence of the decoupling condition}
\label{sec:supplemental_energy}
Here we assess the impact of the energy dependence of the $\nu$SI cross section, \cref{eq:csec}. In the main text and the derivations in \cref{sec:supplemental_tau}, we obtain the size of the neutrino ball at decoupling, $\ell_\mathrm{FS}$, by requiring the \emph{average} optical depth of all neutrinos to be $\tau_\mathrm{FS}$ (see~\cref{eq:tau_integral}).
\begin{figure}[hbtp]
\centering
\includegraphics[width=0.9\textwidth]{plots/tau_energy.pdf}
\caption{Resonant optical depth as a function of neutrino energy, divided by its average value, for different mediator masses. For very small mediator masses, $\tau/\langle\tau\rangle$ is independent of $M_\phi$.}
\label{fig:tau_energy}
\end{figure}
Supplemental~\Cref{fig:tau_energy} shows that a large fraction of neutrinos have optical depths $\tau$ close to the average $\langle \tau \rangle$. We compute $\tau$ assuming resonant scattering; and we average over the target neutrino distribution, which smooths out the strongly energy-dependent resonant cross section ${\sigma_\mathrm{res} \propto \delta(s-M_\phi^2) = \delta(2 E_1 E_2 [1-\cos \theta_{12}] - M_\phi^2)}$. The figure also illustrates the rich phenomenology of energy-dependent $\nu$SI. If $M_\phi$ deviates from $\langle E_\nu \rangle$, part of the neutrinos will free-stream earlier ($\ell_\mathrm{FS} \propto \tau_{\rm FS}^{-2}$). In the burst-outflow case, this would generate both a long and a short signal with different characteristic energies.
These results do not affect our conservative SN 1987A sensitivity. Kam-II and IMB observed neutrinos with energies between $\sim 8 \, \mathrm{MeV}$ and $\sim 40 \, \mathrm{MeV}$: to be consistent with data, low-energy \emph{and} high-energy neutrinos must have $\ell_\mathrm{FS}/c \lesssim \ell_0/c \sim 10 \, \mathrm{s}$. On top of that, if we increase $\tau_\mathrm{FS}$ and decrease $\ell_\mathrm{FS}^2$ by the same factor, the limit stays the same (see~\cref{eq:gres}): we could as well define our conservative analysis by $\ell_\mathrm{FS}/c = 15 \, {\rm s}$ (still incompatible with data, see~\cref{fig:data}) and $\tau_\mathrm{FS} = 40$, guaranteeing a large optical depth when decoupling begins across different energies.
\begin{figure}[hbtp]
\centering
\includegraphics[width=0.63\textwidth]{plots/tau_energy_non_resonant.pdf}
\caption{Same as Suppl.~\cref{fig:tau_energy}, but for non-resonant scattering.}
\label{fig:tau_energy_off_resonant}
\end{figure}
Supplemental~\Cref{fig:tau_energy_off_resonant} shows that the picture is similar for non-resonant scattering. In this case the cross section has a weaker dependence on neutrino energy, hence the smoother optical depth.
\clearpage
\newpage
\section{Dirac neutrinos}
\label{sec:supplemental_Dirac}
Here we generalize our results to also cover Dirac neutrinos. The simple $\nu$SI Lagrangian for Majorana neutrinos $\mathcal{L}_{\nu\mathrm{SI}} = -1/2 \, g \bar{\nu} \nu \phi$ admits two generalizations for Dirac neutrinos
\begin{align}
\mathcal{L}^1_{\nu\mathrm{SI}, \, \mathrm{D}} & = - g \overline{\nu^c} \nu \phi + \mathrm{h.c.} \, ,
\label{eq:nuSI_Lagrangian_Dirac_1} \\
\mathcal{L}^2_{\nu\mathrm{SI}, \, \mathrm{D}} & = - g \bar{\nu} \nu \phi \, ,
\label{eq:nuSI_Lagrangian_Dirac_2}
\end{align}
with $\nu^c$ the charge conjugate of the neutrino field. In the first case, the mediator has lepton number $L_\phi=-2$~\cite{Burgess:1993xh}; whereas in the second case $L_\phi=0$.
The largest sensitivity of supernova neutrinos to $\nu$SI comes from resonant scalar production. For the model in \cref{eq:nuSI_Lagrangian_Dirac_1}, lepton number conservation enforces this process to happen via $\nu \nu \rightarrow \phi^*$ or $\bar{\nu} \bar{\nu} \rightarrow \phi$. For the model in \cref{eq:nuSI_Lagrangian_Dirac_2}, in turn, it must happen via $\nu \bar{\nu} \rightarrow \phi$. As mentioned in the preamble of the S.M., angular momentum conservation in the center of mass frame requires the initial particles to have the same helicity. Since supernova neutrinos are ultrarelativistic and are produced via Standard Model interactions, neutrinos are left-handed and antineutrinos right-handed. Thus, for the first model resonant scattering takes place and our results in the resonant regime directly apply up to $\mathcal{O}(1)$ factors in the cross section~\cite{Esteban:2021tub}.
For the second model, resonant scattering is only possible if a large population of right-handed neutrinos (and left-handed antineutrinos) gets built up. This can happen via non-resonant t- and u-channel scattering, which flip the helicity of the interacting particles~\cite{Esteban:2021tub}. We estimate the frequency of these interactions by computing their associated mean free path inside the PNS. Neutrinos diffuse inside the PNS over timescales $\sim 1 \, \mathrm{s}$, as discussed in the main text. That is, they travel a total distance $d \sim c \cdot 1 \, \mathrm{s} \sim 3 \times 10^{10} \, \mathrm{cm}$. The neutrino number density is $n_\nu \sim 10^{36} \, \mathrm{cm}^{-3}$, corresponding to the thermal equilibrium density with an average energy $\langle E_\nu^\mathrm{core} \rangle \sim 100 \, \mathrm{MeV}$. Thus, the ratio between the traveled distance and the non-resonant $\nu$SI mean free path is
\begin{equation}
\frac{d}{\lambda_\text{non-res}} = d \, n_\nu \, \sigma_\text{non-res} \sim 5 \times 10^4 \cdot \left( \frac{g}{5 \times 10^{-5}} \right)^4 \left( \frac{10 \, \mathrm{MeV}}{M_\phi} \right)^2 \, ,
\end{equation}
where we approximate the non-resonant scattering cross section as $\sigma_\text{non-res} \sim g^4 / (4 \pi M_\phi^2)$. This ratio is very large even for the smallest couplings of our conservative analysis (see~\cref{fig:bounds}), ensuring a large number of scatterings and thus that a large population of right-handed neutrinos and left-handed antineutrinos gets built up. Thus, our conservative analysis holds.
For lower couplings, such as the lowest end of our estimated sensitivity ($g \sim 2 \times 10^{-6}$), $d / \lambda_\mathrm{non-res} \sim 0.1$. That is, a small but non-negligible population of right-handed neutrinos and left-handed antineutrinos will get gradually built up. A more involved calculation taking into account all interaction channels (including, e.g., double-scalar production or scalar radiation in neutrino-nucleus interactions) and the time-dependent abundance of right-handed neutrinos is thus needed to understand the full $\nu$SI sensitivity of a dedicated analysis in this model.
\clearpage
\newpage
\section{Relativistic hydrodynamics of a neutrino fluid}
\label{sec:supplemental_hydro}
Here we describe the hydrodynamic equations that govern the evolution of neutrinos with strong $\nu$SI, and we derive the \emph{burst outflow} and the \emph{wind outflow}.
As discussed in the main text, in our region of interest the self-scattering mean free path of neutrinos is initially tiny, on the $\mu$m scale --- many orders of magnitude smaller than the size of the PNS and any other relevant length scale. Hence, the behavior of the neutrino ball is described by relativistic hydrodynamics of a perfect fluid.
Outside the PNS, the fluid equations follow from energy and momentum conservation~\cite{Weinberg:1972kfs},
\begin{equation}
\nabla_{\alpha} T^{\alpha \beta} = 0 \,, \label{eq:EM_cons}
\end{equation}
with $T^{\alpha \beta}$ the energy-momentum tensor and $\nabla_\alpha$ the covariant derivative. The former can be related to the neutrino energy density $\tilde{\rho}$ and pressure $\tilde{P}$ in the comoving frame, i.e., the frame where the fluid is locally at rest,
\begin{equation}
T^{\alpha \beta} = \tilde{P}\, g^{\alpha\beta} + (\tilde{\rho} + \tilde{P}) \, U^{\alpha}U^{\beta} = \frac{1}{3}\tilde{\rho}\, g^{\alpha\beta} + \frac{4}{3}\tilde{\rho} \, U^{\alpha}U^{\beta} \, , \label{eq:EM_tensor}
\end{equation}
with $g^{\alpha \beta}$ the metric tensor, $U^\alpha$ the four-velocity of the fluid, and for a relativistic fluid $\tilde{P} = \tilde{\rho}/3$. $U^\alpha$ is related to the fluid bulk velocity $\vec{v}$ as
\begin{align}
U^0 & = \gamma \, , \\
U^{i} & = \gamma v^i \,,
\end{align}
with $\gamma = (1-|\vec{v}|^2)^{-1/2}$ the fluid bulk Lorentz factor.
As we only consider number-conserving $2 \rightarrow 2$ scattering, the total number of neutrinos is also conserved,
\begin{equation}
\nabla_{\alpha} (\tilde{n} \, U^{\alpha}) = 0 \label{eq:n_cons_1}
\end{equation}
with $\tilde{n}$ the neutrino number density in the comoving frame, related to the laboratory-frame number density $n$ by $n = \gamma \tilde{n}$.
\Cref{eq:EM_cons,eq:n_cons_1} describe the flow of tightly coupled ultrarelativistic neutrinos in the absence of external forces. If we further assume spherical symmetry, the energy-momentum tensor components are given by
\begin{align}
T^{00} & = \tilde{\rho}\gamma^2 \left(1 + \frac{1}{3}v^2 \right) \,,
\\
T^{0r} & = \frac{4}{3} \tilde{\rho}\gamma^2 v \,,
\\
T^{rr} & = \tilde{\rho}\gamma^2 \left(\frac{1}{3} + v^2 \right) \,,
\\
T^{\vartheta \vartheta} & = \frac{1}{3} \tilde{\rho} r^{-2} \, , \\
T^{\varphi\varphi} & = \frac{1}{3} \tilde{\rho} r^{-2} \sin^{-2} \vartheta \,.
\end{align}
Inserting the proper Christoffel symbols to compute the covariant derivatives in spherical coordinates, \cref{eq:EM_cons,eq:n_cons_1} read
\begin{align}
& \frac{\partial}{\partial t} \left[ \tilde{\rho}\gamma^2 \left( 1 + \frac{1}{3}v^2\right) \right] + \frac{1}{r^2} \frac{\partial}{\partial r} \left( r^2 \frac{4}{3}\, \tilde{\rho}\gamma^2 v \right) = 0 \,, \label{eq:E_cons}
\\
& \frac{\partial}{\partial t} \left( \frac{4}{3} \,\tilde{\rho}\gamma^2 v \right) + \frac{1}{r^2} \frac{\partial}{\partial r} \left[ r^2 \tilde{\rho}\gamma^2 \left(\frac{1}{3} + v^2 \right) \right] - \frac{2}{3r} \tilde{\rho} = 0 \,, \label{eq:M_cons}
\\
& \frac{\partial n}{\partial t} + \frac{1}{r^2}\frac{\partial}{\partial r} \left(r^2 n v \right) = 0 \,. \label{eq:n_cons_3}
\end{align}
\Cref{eq:E_cons,eq:M_cons,eq:n_cons_3} are our starting point to derive the different supernova neutrino outflows with strong $\nu$SI.
The \emph{burst outflow} happens when a uniform neutrino fluid undergoes free expansion in vacuum. Given uniform initial conditions for a neutrino ball of size $\ell_0$ whose edge is expanding at the speed of light, i.e., $n(r, t=0) = n_0 H(\ell_0 - r)$, $\partial_t n(r, t=0) = \dot{n}_0 H(\ell_0 - r)$, and $v(r=\ell_0, t=0) = 1$, with $H$ the Heaviside step function; we obtain the following solution to \cref{eq:E_cons,eq:M_cons,eq:n_cons_3}
\begin{align}
n(r, t) & = n_0 \left( \frac{\ell_0}{\ell(t)} \right)^3 H(\ell(t) - r) \, , \label{eq:homologous_n}\\
v(r, t) & = \frac{r}{\ell(t)} \, , \label{eq:homologous_v}
\end{align}
with $\ell(t) = \ell_0 + t$. This is the homologous expansion of a homogeneous neutrino ball described in the main text.
We have checked with the \texttt{PLUTO} hydrodynamics code~\cite{Mignone_2007} that a variety of smooth initial density and velocity profiles evolve after a short time as \cref{eq:homologous_n,eq:homologous_v}. This is consistent with the behavior found in similar studies of free expansions of relativistic gases~\cite{1980Ap&SS..72..447Y}. Intuitively, the form of the velocity profile can be understood by integrating \cref{eq:n_cons_3} from $r$ to $\ell$,
\begin{equation}
\ell^2 \, n(\ell,\,t) \, v(\ell,\,t) - r^2\, n(r,\,t) \,v(r,\,t) = - \int_r^{\ell} {\rm d}r' \, r'^2 \,\frac{\partial n}{\partial t} \,. \label{eq:n_cons_4}
\end{equation}
This equation is the integral form of the continuity equation, reflecting number conservation in a spherical shell. If both $n$ and $\partial n/\partial t$ are approximately uniform and different from 0, we obtain $v(r) \propto r$. Finally, we note that \cref{eq:homologous_n} has a sharp discontinuity at $r=\ell$, where \cref{eq:homologous_v} indicates an expansion velocity exactly equal to $c$. Hence, close to the edge the solution must change to a smooth transition to $n=0$ and $v \neq c$. This can also be seen from the laboratory-frame energy density profile associated to the homologous expansion, ${T^{00}(r, t) \propto \frac{3 + (r/\ell(t))^2}{[1 - (r/\ell(t))^2]^3}}$, whose integral over the ball diverges. In our \texttt{PLUTO} simulations we observe a smooth transition to vacuum outside the ball, a result also found in Ref.~\cite{1980Ap&SS..72..447Y}. The details of this transition do not affect our results, as we aim for factor-two precision and we conservatively compute the optical depth for a neutrino traveling a distance $\ell$, hence our results are not sensitive to how the edge of the ball expands.
Alternatively, the {\it wind outflow} of a neutrino fluid occurs when we look for a steady-state solution to the equations of hydrodynamics. Such solutions have been invoked in fireball models of gamma-ray bursts~\cite{Piran:1993jm} and in Parker's solar-wind model~\cite{1965SSRv....4..666P}. If we drop the time derivative terms in \cref{eq:E_cons,eq:M_cons,eq:n_cons_3}, we obtain
\begin{align}
\frac{\partial}{\partial r} \left( r^2 \frac{4}{3}\, \tilde{\rho}\gamma^2 v \right) & = 0 \,, \label{eq:E_cons_wind}
\\
\frac{1}{r^2} \frac{\partial}{\partial r} \left[ r^2 \tilde{\rho}\gamma^2 \left(\frac{1}{3} + v^2 \right) \right] - \frac{2}{3r} \tilde{\rho} & = 0 \,, \label{eq:M_cons_wind}
\\
\frac{\partial}{\partial r} \left(r^2 n v \right) & = 0 \,. \label{eq:n_cons_wind}
\end{align}
Combining \cref{eq:E_cons_wind,eq:M_cons_wind} gives a first-order ordinary differential equation for $v(r)$
\begin{equation}
\frac{1}{4}\frac{{\rm d} v}{{\rm d} r} \left( \frac{1}{v^2} - 3 \right) + \frac{1}{2r\gamma^2 v} = 0 \, , \label{eq:EM_cons_steady}
\end{equation}
whose solution is
\begin{equation}
r \sqrt{v (1-v^2)} = {\rm constant} \,. \label{eq:sonic}
\end{equation}
$\sqrt{v (1-v^2)}$ is a concave function, with a maximum at $v=1/\sqrt{3}$, that must decrease as $r$ increases, so depending on the boundary conditions there are two possible solutions. If $v < 1/\sqrt{3}$, the fluid decelerates as $r$ increases and, at large $r$, $v(r) \propto r^{-2}$. If $v > 1/\sqrt{3}$ the fluid accelerates as $r$ increases and, at large $r$, $\gamma(r) \propto r$ --- i.e., $1-v(r) \propto 1/r^2$. From \cref{eq:n_cons_wind,eq:E_cons_wind}, the former implies constant density and pressure at $r \rightarrow \infty$ and it is hence unphysical in the absence of external pressure; whereas the latter implies $n(r) \propto r^{-2}$ at large $r$ and no pressure at $r \rightarrow \infty$. This corresponds to the wind outflow described in the main text. In the wind outflow, the bulk velocity quickly approaches $c$, individual neutrinos move radially, and the arguments that make bursts increase the duration of the supernova neutrino signal do not apply as described in the main text.
Supplemental~\Cref{fig:outflows} shows the density and velocity profiles for both outflow cases. The wind outflow is the only physical steady-state outflow, requires $v > 1/\sqrt{3}$ everywhere, and cannot be continued down to $r=0$ where \cref{eq:sonic} has no solution. Hence, it requires a boundary condition at finite $r$ with $v > 1/\sqrt{3}$. For supernova neutrinos, this condition must be set at the PNS, where neutrino production and scattering with baryons modify \Cref{eq:EM_cons,eq:n_cons_1}. To gain insight, below we investigate the outflow close to the edge of the PNS in a simplified model. More detailed investigation beyond the scope of this work is needed to fully understand the problem.
\begin{figure}[hbtp]
\centering
\includegraphics[width=0.75\textwidth]{plots/burst_outflow.pdf}
\includegraphics[width=0.75\textwidth]{plots/wind_outflow.pdf}
\caption{Laboratory-frame density $n(r)$ and expansion velocity $v$ profiles for the burst-outflow and wind-outflow cases. The wind outflow is time-independent. \emph{Unlike burst outflows, the wind outflow is inhomogeneous and expands with bulk velocities very close to $c$}.}
\label{fig:outflows}
\end{figure}
As a first approximation to the physics close to the edge of the PNS, we allow for momentum transfer between the neutrino fluid and the baryons. We model this by introducing a bulk neutrino momentum loss with a rate inversely proportional to the time between neutrino-nucleon collisions, i.e., the neutrino-nucleon mean free path $\lambda_{\nu N}$. We set the net energy transfer between the neutrinos and baryons to zero, in keeping with the steady state assumption. (The technical assumption is that the net neutrino heating or cooling of the baryons in the layer of interest is negligible compared to the neutrino luminosity. This assumption will be violated in the deep interior of the PNS, but it is a reasonable first approximation in the outer layers since they have a small fraction of the overall heat capacity.) This modifies \cref{eq:EM_cons} as
\begin{align}
\nabla_{\alpha} T^{\alpha t} &= 0 \, , \label{eq:drag_1}\\
\frac{1}{T^{0r}}\nabla_{\alpha} T^{\alpha r} & = - \frac{1}{\lambda_{\nu N}(r)} \, . \label{eq:drag_2}
\end{align}
If we look for a steady-state solution to \cref{eq:drag_1,eq:drag_2}, we obtain following the same steps as above
\begin{equation}
\frac{1}{4}\frac{{\rm d} v}{{\rm d} r} \left( \frac{1}{v^2} - 3 \right) + \frac{1}{2 r \gamma^2 v} = \frac{1}{\lambda_{\nu N}(r)} \, . \label{eq:drag_3}
\end{equation}
Outside the PNS $1/\lambda_{\nu N}=0$, recovering \cref{eq:EM_cons_steady} and, if $v > 1/\sqrt{3}$, the wind outflow. Inside the PNS $\lambda_{\nu N} \ll r$, and \cref{eq:drag_3} gives $v \rightarrow 1/(4\tau)$, where $\tau = (R-r)/\lambda_{\nu N}$ is the neutrino-nucleon optical depth into the PNS. As we go down into the PNS, within a few optical depths $v$ becomes $\ll 1$. This solution also has an energy density that increases linearly with $\tau$, which is similar to the case of radiative diffusion \cite{1960ratr.book.....C}. As we go outward, in order to match with the wind outflow, $v=1/\sqrt{3}$ must be crossed continuously. This can only happen if $2 r \gamma^2 v = \lambda_{\nu N}$, i.e. for $r=\lambda_{\nu N}(r)/\sqrt{3}$, at the edge of the PNS where the neutrino mean free path is of the order of the PNS size.
Supplemental~\Cref{fig:wind} shows the solutions to \cref{eq:drag_3}, assuming $\lambda_{\nu N} = R/50$ inside the PNS and $1/\lambda_{\nu N}=0$ outside (the behavior is qualitatively similar for other values of $1/\lambda_{\nu N} \gg 1/R$), where $R$ is the PNS radius. The sonic point at the edge of the PNS corresponds to the only point where the solution can be single-valued, continuous, and cross $v=1/\sqrt{3}$. The solid orange line is the only continuous steady-state outflow with no pressure at $r \rightarrow \infty$, it therefore corresponds to the \emph{wind outflow}. In it, the fluid must rapidly accelerate at the edge of the PNS, from $v \simeq 0.14$ to $v=1/\sqrt{3} \simeq 0.58$ in the last neutrino-nucleon mean free path (from $r/R=0.98$ to $1.0$). Acceleration outside the PNS is more gradual.
These results imply that the wind outflow requires unique conditions \emph{inside and outside the PNS}. Further work is needed to understand if generic initial conditions relax to the wind solution, the associated timescales, and the impact on supernova physics. In the main text, we outline several ideas that may lead to new observables.
\begin{figure}[hbtp]
\centering
\includegraphics[width=0.6\textwidth]{plots/wind.pdf}
\caption{Steady-state outflows \emph{outside and inside} the PNS. The horizontal axis is arcsinh-stretched to show details near $r\approx R$. \emph{The wind outflow is the only steady-state solution inside and outside the PNS, and it must have $v=1/\sqrt{3}$ at the edge}.}
\label{fig:wind}
\end{figure}
\end{document}
|
2,877,628,088,970 | arxiv | \section{Introduction}
In this paper we are concerned with the following steady transport equation with inflow boundary condition:
\begin{equation} \label{sys}
\begin{array}{lcr}
\sigma + U \cdot \nabla \sigma = H & \textrm{in} & \Omega,\\
\sigma = \sigma_{in} & \textrm{on} & \Gamma_{in}.
\end{array}
\end{equation}
Here, $\Omega$ is a bounded domain in $\mathbb{R}^2$,
$\sigma$ is the unknown function and $U$ is a given vector field with normal component nonvanishing at some parts
of the boundary $\Gamma = \partial \Omega$.
Let us denote
\begin{equation} \label{ubdr}
U \cdot n(x) = d(x),
\end{equation}
where $n$ is the outward normal to $\Gamma$. Then $\Gamma$
is divided into inflow part $\Gamma_{in}$,
outflow part $\Gamma_{out}$ and remaining part $\Gamma_0$ defined as
\begin{equation} \label{bdr}
\begin{array}{c}
\Gamma_{in} = \{ x \in \Gamma: d(x)<0 \},\\
\Gamma_{out} = \{ x \in \Gamma: d(x)>0 \},\\
\Gamma_0 = \{ x \in \Gamma: d(x)=0 \}.
\end{array}
\end{equation}
Supposing that $H \in W^s_p(\Omega)$ and $\sigma_{in} \in W^s_p(\Gamma_{in})$, where $W^s_p$ is fractional order Sobolev space equipped with
Slobodetskii norm (we recall the definitions below), we show existence of a solution $\sigma \in W^s_p(\Omega)$
of the system \eqref{sys} under certain assumption on $\Omega$, which turns out to be quite general and apply
to wide class of domains.
Before we formulate the assumptions and our main result more precisely let us recall some known
results concerning steady transport equations of the type of \eqref{sys}.
One of natural and important applications of equation \eqref{sys}$_1$ is analysis of stationary compressible
Navier-Stokes system. Namely, using the concept of effective viscous flux we can eliminate divergence of
the velocity from the continuity equation which is thus reduced to steady transport equation of a type \eqref{sys}$_1$.
This approach was applied first in \cite{BdV2} in the context of regular solutions in bounded domains
and later applied widely in the 90's, let us mention in particular
\cite{NoPa} for an important improvement enabling to treat a problem exterior domain.
The equation \eqref{sys}$_1$ itself has also been a subject of research.
Let us mention here results concerning regular solutions for small
data in bounded \cite{BdV1} and exterior (\cite{No1}, \cite{No2}) domains. A stronger result on regularity in Sobolev spaces
has been shown in \cite{GiTar}. Weak solutions for $f \in L_p, 1 \leq p \leq \infty$, $f \in \textrm{BMO}$
and $f \in {\cal H}^1$ are studied in \cite{Bae}.
We shall underline that all above results concern the case of normal component of the velocity vanishing
on the boundary. Admission of flow across the boundary,
which is necessary if we want to apply above described approach to stationary compressible
Navier-Stokes system with inflow/outflow conditions,
results in substantial mathematical difficulties
in the investigation of equation \eqref{sys}$_1$. Namely, as the equation is solved along characteristics
determined by the velocity field we have to prescribe the density on the inflow part, that is we
obtain the boundary condition \eqref{sys}$_2$.
The above mentioned difficulties are related to the singularity which arises
around the points where the characteristics
become tangent to the boundary. We will denote a set of these points, called briefly singularity points, by $\Gamma_s$. With the definition
\eqref{bdr} we have
\begin{equation} \label{def_gamma_s}
\Gamma_s = \overline{\Gamma_0} \cap (\overline{\Gamma_{in}} \cup \overline{\Gamma_{out}}).
\end{equation}
The influence of boundary singularity can be seen in \cite{Kw1} where the system \eqref{sys}
is applied in context of strong solutions to the steady compressible Navier-Stokes equations
with inflow boundary conditions.
The authors obtain a solution $u \in W^2_p$, $\rho \in W^1_p$ where $u$ is the velocity of the fluid
and $\rho$ is the density. The result is subject to a constraint on the boundary around the singularity
points
which means roughly that the boundary must behave like a polynomial of order not higher then $2$,
or, in other words, that the curvature of the boundary in the singularity points must
be strictly positive, which means that the characteristics cannot enter the domain too flatly.
There is also a limitation on the integrability of the derivatives of the solution
$2<p<3$. It is remarkable that similar constraints on the boundary and integrability are obtained in
\cite{PS}. These constraints motivates research towards better understanding of the singularity
in the system \eqref{sys}.
In \cite{TP} analogous problem with inhomogeneous slip boundary condition on the velocity
is considered in a rectangle where a problem of singularity does not appear and solutions
in the same class are obtained without limitation on $p$. The system \eqref{sys}
is solved there with a technique of elliptic regularization. In \cite{PMTP} the result
is generalized for a cylindrical domain, this time \eqref{sys} is solved with a Lagrange-type
transformation which replaces the term $u \cdot \nabla \sigma$ with a single derivative.
Finally let us mention a recent result \cite{Ber} where the system \eqref{sys} is studied in context of weak
solutions with inflow condition prescribing the normal component of
$\sigma U$ instead of $\sigma$. Existence of a solution in $L_2$ is shown for right hand side in $L_2$
and a divergence free vector field $U \in H^1$. The condition on the domain is Lipschitz continuity
without additional constraints in singularity points.
In this paper we make a step towards better understanding of boundary singularity in system \eqref{sys}
working with strong solutions. Motivated by above described limitation in $W^1_p$ solutions
we look for solutions in slightly more general class. Namely, assuming $H \in W^s_p$ which
is a fractional order Sobolev space with Slobodetskii norm defined below, we show existence
of a solution $\sigma \in W^s_p$ with $s$ sufficiently small and $p$ large enough to ensure
$sp>2$ which yields $\sigma \in L_\infty$.
Our solution is hence 'slightly worse' than a $W^1_p$ solution,
for this price we obtain existence in a wide class of domains,
in particular piecewise analytical domains fits our assumptions.
In order to define our solutions we need a weak formulation of \eqref{sys}.
\begin{df} \label{def_sol}
By a $W^s_p$-solution to \eqref{sys} we mean a function $\rho \in W^s_p(\Omega)$ such that
\begin{equation} \label{weak}
\begin{array}{c}
\int_\Omega \sigma (\phi - U \cdot \nabla \phi - \phi {\rm div}\,U) \,dx
= -\int_{\Gamma_{in}} d \phi \sigma_{in} \,dS + \int_{\Omega} H \phi \,dx
\quad \forall \phi \in C^1(\overline{\Omega}): \phi|_{\Gamma_{out}} = 0.
\end{array}
\end{equation}
\end{df}
We make the following assumptions on the velocity field $U$:
\begin{equation} \label{straight}
U = [u, 0], \quad u \in W^1_{\infty}, \quad u \geq c >0.
\end{equation}
The assumptions \eqref{straight} are quite
strong and definitely require a comment.
The point is that \eqref{straight} leads to quite simple proof of the \emph{a priori}
estimate where we see easily why the approach in fractional order spaces is natural
for our problem. Therefore, introducing this assumption can be considered as a starting point
of investigation of the problem \eqref{sys} in Sobolev-Slobodetskii spaces, which is a novelty approach.
Extension of our result for more general fields $U$ leads to serious complications and the nature of
singularity becomes hidden by many technicalities. It will be a subject of our forthcoming paper.
{\bf Functional spaces.} We use standard Sobolev spaces $W^k_p$ with natural $k$, which consist
of functions with the weak derivatives up to order $k$ in $L_p(\Omega)$, for the definition we refer
for example to \cite{Ad}.
However, in view of Definition \ref{def_sol}, most important for us are Sobolev-Slobodetskii spaces
$W^s_p$ with fractional $s$.
For the sake of completeness we recall the
definition here. By $W^s_p(\Omega)$ we denote the space of functions for which the norm:
\begin{equation} \label{wsp}
\|f\|_{W^s_p(\Omega)} = \|f\|_{L_p(\Omega)}+\big( \iint_{\Omega^2} \frac{ |f(x) - f(y)|^p }{|x-y|^{2+sp}} \,dxdy \big)^{1/p}
\end{equation}
is finite.
It is possible and will be convenient for us to express the $W^s_p$ - norm with a sort of fractional derivatives.
To this end let us denote the cuts of $\Omega$:
$$
\Omega_1(a) = \Omega \cap \{x_1=a\}, \quad \Omega_2(b) = \Omega \cap \{x_2=b\}.
$$
Furthermore let us define
\begin{equation} \label{def_ux_ox}
\ux^*={\rm inf}\{\ux(x_2): (\ux(x_2),x_2) \in \Gamma_{in}\}, \quad
\ox^*={\rm sup}\{\ox(x_2): (\ox(x_2),x_2) \in \Gamma_{out}\}.
\end{equation}
An equivalent norm in $W^s_p(\Omega)$ is defined in the following way:
\begin{fa}
In $W^s_p$ we have an equivalent norm:
\begin{equation} \label{wsp2}
\|f\|_{W^s_p(\Omega)}^* = \|f\|_{L_p(\Omega)}+\|f\|_{W^s_p,x_1(\Omega)} + \|f\|_{W^s_p,x_2(\Omega)},
\end{equation}
where
\begin{equation} \label{wsp2_1}
\|f\|_{W^s_p,x_1(\Omega)} = \big( \int_a^b dx_2 \int_{\Omega_2(x_2)} dx_1 \int_{\Omega_2(x_2)} \frac{|f(y_1,x_2)-f(x_1,x_2)|^p}{|x_1-y_1|^{1+sp}} dy_1 \big)^{1/p}
\end{equation}
and
\begin{equation} \label{wsp2_2}
\|f\|_{W^s_p,x_2(\Omega)} = \big( \int_{\ux^*}^{\ox^*} dx_1 \int_{\Omega_1(x_1)} dx_2 \int_{\Omega_1(x_1)} \frac{|f(x_1,y_2)-f(x_1,x_2)|^p}{|x_2-y_2|^{1+sp}} dy_2 \big)^{1/p}.
\end{equation}
\end{fa}
\noindent
The limits of integration in outer integrals come from the definitions \eqref{def_in_out}
and \eqref{def_ux_ox}.
The proof of equivalence of norms \eqref{wsp} and \eqref{wsp2}
can be found in \cite{Tr}.
Let us recall also a particular case of imbedding theorem for fractional order spaces,
which will be crucial for our estimates:
\begin{fa}
Let $f \in W^s_p(\Omega)$ with $sp>d$ where $d$ is the space dimension.
Then $f \in L_\infty(\Omega)$ and
\begin{equation} \label{imbed}
\|f\|_{L_\infty(\Omega)} \leq C \|f\|_{W^s_p(\Omega)}.
\end{equation}
\end{fa}
{\bf The domain. General case.} Our result holds for a wide class of domains where the inflow and
outflow parts are defined in a natural way. A general setting is to consider inflow and outflow described as
\begin{equation} \label{def_in_out}
\begin{array}{c}
\Gamma_{in}=\{(\underline{x_1}(x_2),x_2): x_2 \in (a,b)\},\\
\Gamma_{out}=\{(\overline{x_1}(x_2),x_2): x_2 \in (a,b)\},\\
\end{array}
\end{equation}
with singularity points given by
$$
\Gamma_s = \bigcup_{1 \leq i \leq m} \{(\ux(x_2),x_2),x_2=k_{in}^i\} \; \cup \;
\bigcup_{1 \leq j \leq n} \{(\ox(x_2),x_2),x_2=k_{out}^j\}
$$
where
$$
a=k^1_{in} < \ldots < k^m_{in}=b, \quad a=k^1_{out} < \ldots < k^n_{out}=b.
$$
Taking into account the definition of singularity points \eqref{def_gamma_s}
and assumption \eqref{straight}, around the singularity points we must have
$$
{\rm lim}_{x_2 \to k_{in}^{i-}} |\ux'(x_2)| = {\rm lim}_{x_2 \to k_{in}^{i+} } |\ux'(x_2)| = \infty,
$$$$
{\rm lim}_{x_2 \to k_{out}^{i-}} |\ox'(x_2)| = {\rm lim}_{x_2 \to k_{out}^{i+}} |\ox'(x_2)| = \infty.
$$
Furthermore we assume
\begin{equation} \label{def_sing3}
\begin{array}{c}
{\rm lim}_{x_2 \to a^+} \underline{x_1}(x_2) = c_1, \quad {\rm lim}_{x_2 \to b^-} \underline{x_1}(x_2) = c_2,\\
{\rm lim}_{x_2 \to a^+} \overline{x_1}(x_2) = d_1, \quad {\rm lim}_{x_2 \to b^-} \overline{x_1}(x_2) = d_2,
\end{array}
\end{equation}
with $c_i \leq d_i$. Then we have
\begin{equation}
\Gamma_0 = (c_1,d_1) \times \{a\} \cup (c_2,d_2) \times \{b\} \cup \Gamma_s.
\end{equation}
{\bf The domain. Simple representative case.}
As we shall see, the proof relies essentially on investigation of the behaviour of the boundary near the singularity points.
Provided that around each singularity point condition \eqref{flat} below is satisfied,
it is enough to consider a following simple domain with
inflow and outflow parts of the boundary given by \eqref{def_in_out}, \eqref{def_sing3} with
$a=c_i=d_i=0$
and
\begin{equation} \label{def_sing2}
\begin{array}{c}
{\rm lim}_{x_2 \to 0^+} \underline{x_1}'(x_2) = {\rm lim}_{x_2 \to b^-} \overline{x_1}'(x_2)= -\infty,\\
{\rm lim}_{x_2 \to 0^+} \overline{x_1}'(x_2) = {\rm lim}_{x_2 \to b^-} \underline{x_1}'(x_2)= +\infty.
\end{array}
\end{equation}
Then we have two singularity points:
$$
\Gamma_s=\Gamma_0=\{(0,0),(0,b)\}.
$$
We assume further that these are the only singularity points, that is, there are no singularity points
'inside' $\Gamma_{in}$ and $\Gamma_{out}$.
Similar domain is considered in \cite{Kw1}, hence our choice facilitates a comparison of the results
and techniques with above mentioned paper.
Around each singularity point $x_2$
is given as a function of $x_1$. We assume that this function satisfies the following constraint:
\begin{equation} \label{flat}
|x_2(x_1) - x_2(y_1)| \geq C |x_1-y_1|^r \quad \textrm{for some} \quad r>1.
\end{equation}
This condition is crucial in our proof and it deserves a comment.
Condition \eqref{flat} means that the boundary around the singularity points
is allowed to behave in particular like a polynomial of arbitrary degree.
It is worth to compare our constraint with \cite{Kw1} where analogous condition
is required with $r=2$, therefore our restriction is much more general.
The condition (\ref{flat}) is quite technical, however it applies for a wide class of functions,
in particular it holds true if the boundary around the singularity points is an analytic function.
It is quite basic result but we show it for the sake of completeness.
\begin{lem}
Assume that $x_2$ is an analytic function of $x_1$ around the singularity points.
Then (\ref{flat}) holds.
\end{lem}
\noindent
\emph{Proof.}
By \eqref{def_sing2} we have $x_2'(x_1)=0$ at the singularity points. Therefore
it is enough to show that if $f: \mathbb{R} \to \mathbb{R}$ is analytic in some $[-r,r]$,
$f(0)=f'(0)=0$ and $f \neq 0$ then
\begin{equation}
|f(x)| \geq C |x|^N \quad \textrm{for} \quad x \in (-l,l)
\end{equation}
for some $C>0$, $N \geq 2$ and $l<r$ sufficiently small.
Since $f \neq 0$ and $f$ is analytic, we must have $f^{(n)}(0) \neq 0$ for some $n \geq 2$.
Let $f^{(k)}(0)$ be the first derivative not vanishing in $0$. Then we have
$$
f(x) = \frac{f^{(k)}(0)}{k!} x^k + R^{k+1}(x),
$$
where $|R^{k+1}(x)| \leq M |x|^{k+1}$ for $x \in (-r,r)$.
Hence
$$
|f(x)| \geq \Big|\frac{f^k(0)}{k!}\Big| |x|^k-M|x|^{k+1} = \Big( \frac{f^k(0)}{k!} - M|x|\Big) |x|^k. \square
$$
{\bf Main result.} We are now ready to formulate our main result.
\begin{tw} \label{main}
Assume that the boundary of $\Omega$ around the singularity points satisfy
\eqref{flat} for some $C>0$ and $N \in \mathbb{N}$.
Assume $U$ satisfy the assumptions \eqref{straight}. Assume further that
$H \in W^s_p(\Omega)$ and $\sigma_{in} \in W^s_p(\Gamma_{in})$ for $s,p$ such that
$\frac{1}{r}>s>\frac{2}{p}$ where $r$ is the exponent from \eqref{flat}.
Then there exists a solution $\sigma$ to \eqref{sys} such that
\begin{equation} \label{est_trans}
\|\sigma\|_{W^s_p(\Omega)} \leq C [\|H\|_{W^s_p(\Omega)} + \|\sigma_{in}\|_{W^s_p(\Gamma_{in})}].
\end{equation}
\end{tw}
\section{Proof of Theorem \ref{main}}
As we are concerned with linear system, the core of the proof is in appropriate estimates.
The following proposition gives \emph{a priori} estimate in $W^s_p$ for a solution of
\eqref{sys}.
\begin{prop}
Assume $\Omega$, $H$, $U$ and $\sigma_{in}$ satisfy the assumptions of Theorem \ref{main}.
Let $\sigma$ be a sufficiently regular solution to the equation \eqref{sys}. Then
\eqref{est_trans} holds.
\end{prop}
\noindent
\emph{Proof.} First of all, let us notice that it is enough to show \eqref{est_trans}
for the solution of the equation
\begin{equation} \label{sys2}
\begin{array}{c}
\tilde \sigma_{x_1} = \tilde H \in W^s_p(\Omega), \nonumber \\[3pt]
\tilde \sigma|_{\Gamma_{in}} =\tilde \sigma_{in} \in W^s_p(\Gamma_{in}) .
\end{array}
\end{equation}
Indeed, if we define
\begin{displaymath}
V(x)=\int_{\ux(x_2)}^{x_1}\frac{1}{u(s,x_2)}\,ds, \quad \tilde \sigma = e^V \sigma,
\end{displaymath}
then we have
\begin{equation} \label{dx1}
\partial_{x_1}\tilde \sigma = V_{x_1} e^V\sigma + e^V\sigma_{x_1}.
\end{equation}
Since $V_{x_1}=\frac{1}{u}$, multiplying \eqref{dx1} by $u$ we get
$$
u\tilde \sigma_{x_1} = [\sigma + u\sigma_{x_1}]e^V
=He^V,
$$
so
\begin{equation}
\tilde \sigma_{x_1} = \frac{He^V}{u}=:\tilde H.
\end{equation}
Now due to boundedness of $\Omega$ and assumptions on $u$ we have $\frac{e^V}{u} \in W^1_{\infty}$ and
$0<M_1 \leq \frac{e^V}{u}\leq M_2$ in $\Omega$, therefore
\begin{equation}
C_1 \|\frac{e^V}{u} f\|_{W^s_p} \leq \|f\|_{W^s_p} \leq C_2 \|\frac{e^V}{u} f\|_{W^s_p}
\end{equation}
for $f \in W^s_p(\Omega)$, which shows equivalence of estimates for \eqref{sys2} and \eqref{sys}. $\square$
Let us proceed with the proof of Theorem 1 for \eqref{sys2} (skipping tildes for simplicity).
For convenience we use the norm $\|\cdot\|_{W^s_p}^*$ (\ref{wsp2}), hence
it is enough to find the bounds on the norms $\|\sigma\|_{L_p}$,
$\|\sigma\|_{W^s_p,x_1}$ and $\|\sigma\|_{W^s_p,x_2}$ defined in (\ref{wsp2_1}) and (\ref{wsp2_2}).
The idea is to express pointwise values of $\sigma$ with integrals of $H$ along the characteristics
of \eqref{sys}$_1$, that is, straight lines due to assumption \eqref{straight}.
In fact, to operate with pointwise values we need sufficiently smooth functions, therefore we
consider smooth approximations of $\sigma$ and $H$ and use standard density argument.
We start with estimate for $\|\sigma\|_{L_p}$. We have
$$
\sigma(x_1,x_2)=\sigma_{in}(\ux(x_2),x_2)+\int_{\ux(x_2)}^{x_1} H (t,x_2)dt,
$$
therefore we directly get
\begin{equation} \label{e0}
\|\sigma\|_{L_p(\Omega)} \leq C [\|\sigma_{in}\|_{L_p(\Gamma_{in})}+\|H\|_{L_p(\Omega)}].
\end{equation}
Now we consider $\|\sigma\|_{W^s_p,x_2(\Omega)}$. For convenience denote $h:=y_2-x_2$.
We can assume that $\ux(y_2)<\ux(x_2)$, otherwise we interchange $\ux(y_2)$ and $\ux(x_2)$ in the integrals.
We write
$$
\sigma(x_1,x_2+h) - \sigma(x_1,x_2) = \sigma_{in}(\ux(x_2+h))-\sigma_{in}(\ux(x_2)) +
$$$$
\int_{\underline{x_1}(x_2+h)}^{x_1} H(t,x_2+h) \,dt -
\int_{\underline{x_1}(x_2)}^{x_1} H(t,x_2) \,dt
$$$$
= \sigma_{in}(\ux(x_2+h))-\sigma_{in}(\ux(x_2)) +
\int_{\underline{x_1}(x_2)}^{x_1} [H(t,x_2+h) - H(t,x_2)] \,dt \
+ \int_{\underline{x_1}(x_2+h)}^{\underline{x_1}(x_2)} H(t,x_2+h) \,dt =
$$$$
=: I_0 + I_1 + I_2,
$$
Hence $|\sigma(x_1,x_2+h) - \sigma(x_1,x_2)|^p \sim |I_0|^p + |I_1|^p + |I_2|^p$.
We omit the limits of integration w.r.t. $x_1$ and $x_2$ since they do not play role in the computations.
Concerning the limits of $h=y_2-x_2$, these depends on $x$, but what is important is integrability around
$0$, therefore we assume without loss of generality $h>0$ and integrate
with respect to $h$ from $0$ to some $\delta>0$.
First of all, notice that we have $dx_2 \leq dS(x_2)$ where $dS(x_2)$ is the boundary measure at $\Gamma_{in}$.
Therefore
\begin{align} \label{e10}
\int dx_1 \int dx_2 \int_0^{\delta} \frac{|I_0|^p}{|h|^{1+sp}} \,dh & \leq
\int dx_1 \iint_{\Gamma_{in}^2} \frac{ |\sigma_{in}(\ux(y_2))-\sigma_{in}(\ux(x_2))|^p }{|x_2-y_2|^{1+sp}} dS(x_2)dS(y_2) \nonumber \\[5pt]
& \leq C(\Omega) \|\sigma_{in}\|_{W^s_p(\Gamma_{in})}.
\end{align}
Next, by Jensen inequality we have
$$
\int dx_1 \int dx_2 \int_0^\delta \frac{|I_1|^p}{|h|^{1+sp}} \,dh = \int dx_1 \int dx_2 \int_0^\delta dh \frac{| \int_{\underline{x_1}(x_2)}^{x_1} [H(t,x_2+h) - H (t,x_2)] \,dt |^p}{|h|^{1+sp}}
$$$$
\leq \int dx_1 \int dx_2 \int_0^{\delta} dh \frac{ |x_1-\underline{x_1}(x_2)|^{p-1} \int_{\underline{x_1}(x_2)}^{x_1} |H(t,x_2+h)-H(t,x_2)|^p \,dt } {|h|^{1+sp}}
$$$$
\leq C(\Omega) \int dx_1 \int dx_2 \int_{\underline{x_1}(x_2)}^{x_1} \int_0^{\delta} \frac{|H(t,x_2+h)-H(t,x_2)|^p}{|h|^{1+sp}} \,dh\,dt
$$$$
\leq C(\Omega) \int dx_1 \int dx_2 \int_{\underline{x_1}(x_2)}^{\overline{x_1}(x_2)} \int_0^{\delta} \frac{|H(t,x_2+h)-H(t,x_2)|^p}{|h|^{1+sp}} \,dh\,dt.
$$
which yields
\begin{equation} \label{e11}
\int dx_1 \int dx_2 \int_0^\delta \frac{|I_1|^p}{|h|^{1+sp}} \,dh \leq C(\Omega)\|H\|_{W^s_p,x_2}^p.
\end{equation}
Note that the above estimate did not involve any assumption on the boundary. The assumption \eqref{flat}
will play role in the second part of the estimate. Namely, for $I_2$ we have
$$
\int dx_1 \int dx_2 \int_0^{\delta} \frac{|I_2|^p}{|h|^{1+sp}}\,dh =
\int dx_1 \int dx_2 \int_0^{\delta} \frac{ \big|\int_{\underline{x_1}(x_2+h)}^{\underline{x_1}(x_2)} H(t,x_2+h)^p \,dt\big| }{|h|^{1+sp}} \,dh
$$$$
\leq \int dx_1 \int dx_2 \int_0^{\delta} dh \frac{ |\underline{x_1}(x_2)-\underline{x_1}(x_2+h)|^{p-1} \int_{\underline{x_1}(x_2+h)}^{\underline{x_1}(x_2)}|H(t,x_2+h)|^p \,dt}{|h|^{1+sp}}
$$$$
\leq \int dx_1 \int dx_2 \int_0^{\delta} dh \frac{ \|H\|_{L_{\infty}}^p |\underline{x_1}(x_2)-\underline{x_1}(x_2+h)|^p }{|h|^{1+sp}}.
$$
Now condition (\ref{flat}) implies that
\begin{equation} \label{flat_inv}
|\underline{x_1}(x_2) - \underline{x_1}(x_2+h)| \leq C |h|^\epsilon
\end{equation}
where $\epsilon=\frac{1}{r}$, therefore
$$
\int dx_1 \int dx_2 \int_0^{\delta} \frac{|I_2|^p}{|h|^{1+sp}}\,dh \leq
\|H\|_{L_\infty}^p \int dx_1 \int dx_2 \int_0^\delta |h|^{-1+p(\epsilon-s)}.
$$
The last integral is finite provided that $s<\epsilon$.
By the imbedding theorem we get
\begin{equation}
\int dx_1 \int dx_2 \int_0^{\delta} \frac{|I_2|^p}{|h|^{1+sp}}\,dh \leq C \|H\|_{W^s_p}^p, \label{e12}
\end{equation}
provided that $s<\epsilon$ where $\epsilon$ is the exponent from (\ref{flat_inv}).
On the other hand we require $s>\frac{2}{p}$.
Therefore we can take $p>2r$ where $r$ is from \eqref{flat} and
$\frac{1}{r}>s>\frac{2}{p}$.
From \eqref{e10}, \eqref{e11} and \eqref{e12} we conclude
\begin{equation} \label{e1}
\|\sigma\|_{W^s_p,x_2(\Omega)} \leq C [\|H\|_{W^s_p(\Omega)} + \|\sigma_{in}\|_{W^s_p(\Gamma_{in})}].
\end{equation}
It remains to estimate the fractional norm with respect to $x_1$. The estimate is more direct
and does not involve any assumptions on the geometry of the boundary. We have
$$
\sigma(x_1+h,x_2)-\sigma(x_1,x_2) = \int_{x_1}^{x_1+h} H(t,x_2) \,dt.
$$
Now
$$
\int dx_1 \int dx_2 \int_0^{\delta} dh \frac{ |\int_{x_1}^{x_1+h} H(t,x_2) dt|^p }{|h|^{1+sp}} \leq
$$$$
\leq \int dx_1 \int dx_2 \int_0^\delta dh \frac{|h|^{p-1}\int_{x_1}^{x_1+h} |H(t,x_2)|^p \,dt}{|h|^{1+sp}}
\leq C \|H\|_{L_\infty}^p \int_0^\delta |h|^{-1+p(1-s)} \,dh.
$$
The last integral is finite for $s<1$. We see that
\begin{equation} \label{e2}
\|\sigma\|_{W^s_p,x_1} \leq C \|H\|_{W^s_p}
\end{equation}
for $H \in W^s_p$ where $1>s>\frac{2}{p}$, what is clearly weaker assumption then in the previous estimate.
Combining \eqref{e0}, \eqref{e1} and \eqref{e2} we arrive at \eqref{est_trans}. $\square$
\emph{Proof of Theorem 1.} As we deal with a linear system, once we have shown the estimate \eqref{est_trans}
the rest follows from a standard density argument. Namely, we consider a sequence of smooth functions
$$
H_\epsilon \to H \quad \textrm{in} \quad W^s_p(\Omega).
$$
Let $\sigma_{\epsilon}$ be a solution to \eqref{sys} with $H_{\epsilon}$.
In particular it satisfies the weak formulation \eqref{weak} with $H_{\epsilon}$.
As $\sigma_{\epsilon}$
is bounded in $W^s_p$, it converges weakly in $W^s_p$ up to a subsequence (still denoted by $\sigma_{\epsilon}$)
to some $\sigma \in W^s_p$. We have to show that all the integrals in the weak formulation convergence
but this is obvious due to weak convergence as each integral can be understood as a functional on $W^s_p$.
We conclude that $\sigma$ is a solution to \eqref{sys}. $\square$
\smallskip
\noindent
{\bf Acknowledgements.} The work was supported by polish NCN grant UMO-2014/14/M/ST1/00108.
|
2,877,628,088,971 | arxiv | \section{Introduction}
Graph representation has been broadly studied in information extraction, relational representation, and multi-modality data fusion~\citep{RN1,RN2,RN3}. The rich topological and spatial characteristics of graphs essentially uncover differential relations among individual graph elements~\citep{RN4}. In medical image analysis, the diverse shape, anatomy, and appearance information provide a key data source to characterize the interactions among the diagnostic region of interests (ROIs) and reveal disease status~\citep{RN5}. Therefore, image-based graph modeling and inference can deepen our understanding of the complex relational patterns hidden in disease tissue regions. The surge of graph convolutional networks (GCNs), a branch of deep learning characterized by graph-level model development, has brought a new wave of information fusion techniques through their widespread applications in medical imaging, from disease classification~\citep{RN6}, tumor segmentation~\citep{RN7}, to patient outcome prediction~\citep{RN8}.
Graph convolutional networks (GCNs) explore the heterogeneous graph data via a series of graph-level convolution, sampling, and enabling the model inference on both graph node attributes and relational structures~\citep{RN4}. The development of GCNs extends conventional graph embedding methods (e.g., Deepwalk~\citep{RN9} and node2vec~\citep{RN10}) primarily on generating a low-dimensional graph representation without considering node attributes. To enable a synergistic analysis with medical imaging, GCNs demonstrate their multifaceted advances on feature extraction, data fusion, and interpretation ability. First, GCNs extract multi-scale spatial relations by characterizing inter- and intra-interactions between different tissue regions, which are vital to understanding disease developmental mechanisms~\citep{RN1}. Further, GCNs present a strong fusion capability to handle heterogeneous cross-modality data of both imaging and non-imaging data. The cross-modality analysis is of substantial interest since it can broaden our understanding of disease mechanisms beyond the scope of single modality data. For instance, the brain networks naturally present the relationship between neurons, and the fusion with clinical records can provide auxiliary benefits for brain disease analysis~\citep{RN11}. Similarly, an integrative analysis of multi-omics profiles and imaging patterns promises to discover novel image-to-genome associations for cancer biomarker discovery~\citep{RN12,RN13}. Finally, GCNs provide a possibility of outcome interpretation by capturing the structural dynamics of complex graphs. The model outcomes can visualize both node distribution and subgraph connectivity derived from the entire graph representation. Taken together, GCNs demonstrate the potential to analyze the abundant amount of graph-level information that is crucial to advance medical imaging understanding and inform decision making in clinics.
A general pipeline for utilizing GCNs in medical imaging is shown in Fig.~\ref{fig1}, highlighting key components of multi-modality imaging and clinical data, graph representation frameworks, and downstream clinical applications. To provide a guideline to foster cross-disciplinary research in the field of GCNs and medical imaging, the major contributions of this survey can be summarized as follows:
\begin{enumerate}
\item We outline current state-of-the-art GCNs that are widely used in medical image analysis. We summarize the key understanding of their concepts, architectures, and trade-offs of the network architecture design to advance graph and medical imaging research.
\item We highlight the convergence of graph-driven studies in radiological imaging, histopathological imaging, and other imaging modalities. We organize them in a unified taxonomy from the graph construction approaches and their downstream clinical tasks.
\item We offer insights into the image-to-graph transformation that is vital to determine the success of GCNs, including the definition of graph components and different graph construction metrics. This review provides a key reference for researchers to explore the fast-growing synergy between graph architecture and medical imaging components.
\item Emerging opportunities and future directions are discussed in image-based GCNs and their extensions across multiple disciplines. These insights can greatly expand the scope of advancing GCNs in medical imaging and related data-driven medical studies.
\end{enumerate}
\begin{figure}[!t]
\centering
\includegraphics[scale=.23]{Figures/fig1.png}
\caption{A general pipeline for utilizing GCNs in medical image analysis.(a) Medical image analysis data. Multi-modality medical imaging and other non-image data can be jointly considered for GCN modeling and analysis. (b) Graph representation learning. The image-graph transformation pipeline includes node selection, node attribute extraction, and edge construction. For different types of medical images, we aim to design a variety of task-specific transformation strategies. (c) Graph convolutional networks framework. The input of GCNs is the constructed data-rich graphs based on image contents. The GCNs architecture contains input, hidden, and output layers to allow information extraction and inference. (d) Clinical tasks. We review a broad range of tasks with clinical relevance that incorporate disease detection, segmentation, and outcome prediction.}
\label{fig1}
\end{figure}
\section{Methodology of graph convolutional networks}
The architecture of graph convolutional networks (GCNs) essentially addresses the cyclic mutual dependencies with weight parameters in each network layer~\citep{RN3}. The graph convolutional layer updates graph representations by aggregating node information from their neighborhoods. Also, the edge weights and connections will be updated in specified GCNs applications. Conceptually, GCNs could broadly fall into two categories including spectral-based and spatial-based GCNs. First, the spectral-based graph convolutions are defined in the spectral domain based on the graph Fourier transformation~\citep{RN3}, which can be regarded as an analogy of the signal Fourier transform in 1-D space. Second, the spatial-based graph convolutions are defined in the spatial domain that the aggregations of node representations come from the collective information of neighboring nodes. Also, we discuss important graph pooling modules as downsampling strategies to reduce the size of graph representation~\citep{RN1}, which can critically alleviate issues of overfitting, permutation invariance, and computational complexity in the development of graph neural networks. In later sections, we define a graph as $G = (V, E)$, where $V$ is the graph node and $E$ is the edge between nodes. For graph representation learning, we use H to donate the hidden state vector of nodes.
\subsection{Spectral graph convolutional networks}
Spectral-based graph convolutional networks are derived from the field of graph signal processing, where the spectral-based convolutional operators are defined in the spectral domain~\citep{RN1}. Theoretically, a graph signal $x$ will be transformed to the spectral domain by a graph Fourier transform $\mathcal{F}$ before the convolution operation. In this way, the spectral-based graph convolutions can be computed by taking the inverse Fourier transform of the multiplication between two Fourier transformed graph signals~\citep{RN5}. Then the resulting signal is transformed back by the inverse graph Fourier transform $\mathcal{F}^{-1}$. These transformations are defined as:
\begin{equation}
\mathcal{F}(x)= U^{T}x
\end{equation}
\begin{equation}
\mathcal{F}^{-1}(x) = Ux
\end{equation}
$U$ is the matrix of eigenvectors of the normalized graph Laplacian matrix $L = I_{N}-D^{-\frac{1}{2}} AD^{-\frac{1}{2}}$, where $I_{N}$ is the normalized identity matrix, $D$ is a node degree matrix and $A$ is the adjacency matrix, which represents the connectivity between every two nodes. $L$ has the property of being real symmetric positive semidefinite. With this property, the normalized Laplacian matrix can be factorized as $L = U \Lambda U^{T},$ where $\Lambda$is a diagonal matrix of all the eigenvalues. According to the graph fourier transformation, the input graph signal $x$ with a filter $g \in R^{n}$ is defined as:
\begin{equation}
g\star x =\mathcal{F}^{-1)}(\mathcal{F}(g) \odot \mathcal{F}(x)) = U(U^{T}g\odot U^{T}x)
\end{equation}
where $\odot$ denotes the element-wise product, $U^{T}g$ is a filter in the spectral domain. If we simplify the filter by a learnable diagonal matrix $g_{\theta}= diag(U^{T}g)$, then the spectral graph convolution can be simplified as:
\begin{equation}
g_{\theta}\odot x =Ug{\theta}U^{T}x
\end{equation}
The majority of spectral-based graph convolutional networks are based on the above definitions, and the design of filter $g_{\theta}$ determines the various performance of individual approaches. Normally, the spectral-based graph convolutional network designs the convolution operation in the Fourier domain by computing the eigen-decomposition of the graph Laplacian~\citep{RN3}. They assume that the filter $g_{\theta}=\Theta_{i,j}^{(k)}$ is a set of learnable parameters and considers graph signals with multiple channels. Due to the eigen-decomposition of the Laplacian matrix, any perturbation to a graph can result in changes of eigenbasis~\citep{RN3}. The learned filters are domain dependent with a poor graph structure generalization. Also, eigen-decomposition has a high computational complexity that is unfavorable for large-scale data processing. To overcome the limitations, especially the computational complexity, Chebyshev spectral CNN (ChebNet)~\citep{RN14} used K-polynomial filters to achieve a good localization in the vertex domain by integrating the node features within the K-hop neighborhood, i.e., $g_{\theta}=\sum_{i=0}^{k}\theta_{i}T_{i}\bar{L}$, where $\bar{L} = \frac{2}{\lambda_{max}}L=I_{N}$, $\lambda_{max}$ denotes the largest eigenvalue of L. The range of the eigenvalues in $\bar{L}$ is $[-1,1]$. The Chebyshev polynomials are defined recursively as $T_{i}(x) = 2xT_{i-1}(x)-T_{i-2}(x)$ with $T_{0}(x)=1$ and $T_{1}(x)=x$. The convolution operation can be written as:
\begin{equation}
g_{\theta} \star x = \sum_{i=0}^{k}k\theta_{i}T_{i}\bar{L}x
\end{equation}
For a similar purpose of improving computational efficiency, CayleyNet~\citep{RN15} applies the Cayley polynomials that are parametric rational functions to capture narrow frequency bands. The spectral graph convolution operation is defined as:
\begin{equation}
g_{\theta} \star x =c_{\theta} x + 2Re{\sum_{j=1}^{r}c_{j}(hL-iI)^{-j)}x}
\end{equation}
Where $Re(\cdotp)$ returns the real part of a complex number, $c_{0}$is a real coefficient, $c_j$ is a complex coefficient, $i$ is the imaginary number, and $h$ is the parameter that controls the spectrum of a Cayley filter. ChebNet could be regarded as a special case of CayleyNet via the use of the Chebyshev polynomial approximation to reduce the computational complexity.
A notable variant of ChebNet for further simplifying the computational complexity, which truncates the Chebyshev polynomial to the first-order approximation that the central node only considers its 1-hop neighboring nodes~\citep{RN16}. The approach simply filters in (5) with $i=1$ and $\lambda_{max}=2$ to alleviate the problem of overfitting:
\begin{equation}
g_{\theta} \star x = \sum_{i=0}^{k}k\theta_{i}T_{i}\bar{L}x \approx \theta_{0}x + \theta_{1}(L - I_{N})x = \theta_{0}x - \theta_{1}D^{-\frac{1}{2}} AD^{-\frac{1}{2}}x
\end{equation}
To restrain the number of parameters and avoid overfitting, GCN further assumes that $\theta = \theta_{0} = \theta_{1}$ so that $g_{\theta} = \theta(I_{N} + D^{-\frac{1}{2}} AD^{-\frac{1}{2}})$. To solve the exploding or vanishing gradient problem in (7): $I_{N} + D^{-\frac{1}{2}} AD^{-\frac{1}{2}} \rightarrow \bar{D}^{-\frac{1}{2}}\bar{A}\bar{D}^{-\frac{1}{2}}$, with $ A = A + I_{N}$ and $\bar{D_{ij}} = \sum_{j}A_{ij}$. The propagation layer of GCN is defined as:
\begin{equation}
H = \bar{D}^{-\frac{1}{2}}\bar{A}\bar{D}^{-\frac{1}{2}}X\Theta
\end{equation}
where $X \in R^{N \times F}$is the input matrix, $\Theta \in R^{N \times F^{\prime}}$ is the parameter and $H \in R^{N \times F^{\prime}}$ is the output matrix. $F$ and $F^{\prime}$ are the dimensions of the input and the output, respectively.
Recent research findings demonstrate the improvement of GCN’s feasibility and consistency on graph models. The adaptive graph convolution network (AGCN)~\citep{RN17} could construct and learn a residual graph Laplacian matrix for each sample in the batch through a learnable distance function that takes two nodes’ features as inputs. The residual graph Laplacian matrix leads to achieving high-level performance in public graph-structured datasets. In addition, the dual graph convolutional network (DGCN)~\citep{RN18} explores the perspective of augmenting the graph Laplacian as AGCN~\citep{RN17}. DGCN jointly considers the local consistency and global consistency on graphs through two convolutional networks. The first convolutional network is the same as (8), while the second network replaces the adjacency matrix with the positive pointwise mutual information (PPMI) matrix.
Spectral-based graph convolutional networks have a solid theoretical foundation derived from graph signals theories. Despite efforts to overcome the computation complexity, the generalization power of spectral-based GCNs is limited as opposed to the broad usage of spatial-based approaches below. Currently, the spectral-based methods train the filters on the fixed graph structure, making the trained filters unable to apply to a new graph with different structures. However, the graph structures can dramatically vary in both size and connectivity in practical applications~\citep{RN17}. The generalization power across different tasks and the high computation complexity become the major hurdles to developing spectral-based graph convolutional networks.
\subsection{Spatial graph convolutional networks}
The spatial graph convolutional operation essentially focuses on aggregating and updating node representation by propagating node information along edges~\citep{RN3}. The aggregation strategy can directly improve the generalization power of dealing with different structured graphs by aggregating the information from neighboring nodes and updating the center node representation.
The message-passing neural network (MPNN)~\citep{RN19} represents a general framework of spatial-based GCNs~\citep{RN3}. The key forward propagation strategy of MPNN is passing the information between nodes through edges directly. As defined in the propagation function below, MPNN runs $T$ steps message-passing iterations so that the information could be propagated between nodes. Notably, GraphSAGE~\citep{RN20} is a general inductive framework which generates embeddings by sampling and aggregating features from a node’s local neighborhood. GraphSAGE leverages node feature information to efficiently generate node embeddings for previously unseen data~\citep{RN20}.
The propagation rule follows:
\begin{equation}
h_{N(v)}^{k} = AGGREGATE_{k}({h_{u}^{k-1}),\forall u \in N(v)})
\end{equation}
\begin{equation}
h_{v}^{k} =\sigma(W^{k} \cdot CONCAT(h_{v}^{k-}),h_{N(v)}^{k}
\end{equation}
Where AGGREGATE is an aggregator function that could aggregate information from node neighbors. Three types of aggregators are utilized in GraphSAGE, including mean aggregator, LSTM aggregator, and pooling aggregator. $W^{k}$is a set of weight matrices that are used to propagate information from different layers. CONCAT is the concatenated operation. Interestingly, GraphSAGE with a mean aggregator can be considered as an inductive version of GCN. To further identify the graph structures that cannot be distinguished by GraphSAGE~\citep{RN21}, Graph Isomorphism Network (GIN)~\citep{RN22} is a maximally powerful architecture to distinguish the isomorphism graph. As proved in GIN~\citep{RN22}, the injective aggregation update maps node neighborhoods to different feature vectors so that the isomorphism graph can be distinguished. To achieve the injectivity of the AGGREGATE, sum-pooling is applied in GIN. The AGGREGATE and COMBINE steps are integrated as follows:
\begin{equation}
h_{v}^{(k)} = MLP^{(k)}((1+ \epsilon ^{(k)}) \cdot h_{v}^{(k-1)} + \sum_{u \in N(v)}h_{u}^{(k-1)})
\end{equation}
MLP is a multi-layer perceptron that could represent the composition of functions.
The attention mechanism has been increasingly applied in spatial-based GCNs models for various sequence-based approaches\linebreak ~\citep{RN22, RN1}. Several key works are attempting to utilize attention mechanisms on graphs. Different from the design of spectral and spatial convolutional operations, the attention-based convolutional operations assign different weights for neighbors to stabilize the learning process and thus alleviate noise effects. A benefit of attention mechanisms is that they allow for dealing with variable-sized inputs, and focusing on the most relevant parts of the input to make decisions~\citep{RN22}. Graph Attention Network (GAT)~\citep{RN22} proposes a computationally efficient graph attentional layer which leverages self-attention and multi-head attention mechanisms. The GAT layer is parallelizable across all nodes in the entire graph while allowing for assigning different importance weights to different (degree) nodes in different size neighborhoods, and does not depend on knowing the entire graph structure. The coefficients computed by the attention mechanism and the propagation of GAT is formulated as:
\begin{equation}
\alpha_{ij} = \frac{exp(LeakyReLU(\alpha^{T}[Wh_{i} ||Wh_{j}]))}{\sum_{k \in N_{i}}exp(LeakyReLU(\alpha^{T}[Wh_{i} ||Wh_{k}])}
\end{equation}
\begin{equation}
h_{i}^{\prime} = \sigma(\sum_{k \in N_{i}} \alpha_{ij} W h_{j})
\end{equation}
where $\alpha$ and and $W$ are weight vectors, and $||$ is the concatenation operation.
Furthermore, GAT leverages multi-head attention~\citep{RN23} to stabilize the learning process of self-attention (13), which can be written as:
\begin{equation}
h_{i}^{\prime} = \prod_{k=1}^{k}(\sum_{k \in N_{i}}\alpha_{ij}Wh{j})
\end{equation}
\begin{equation}
h_{i}^{\prime} = \sigma(\frac{1}{k}\sum_{k \in N_{i}} \sum_{j \in {N_{i}}}\alpha_{ij}^{k} W_{k} h_{j})
\end{equation}
where $\alpha_{ij}^{k}$ are normalized attention coefficients computed by the k-th attention mechanism. GAT achieved significant improvement in both transductive tasks and inductive tasks, especially in the inductive task (e.g., protein-protein interaction dataset), GAT improved the micro-averaged F1 scores by 20.5\% compared to the best GraphSAGE result.
In summary, spatial-based convolutional graph operations follow a neighborhood aggregation strategy, where we can iteratively update the representation of a node by aggregating representations of its neighbors. After k iterations of aggregation, a node’s representation captures the structural information within its k-hop network neighborhood. The rapid development of spatial-based GCNs has displayed their computational efficiency, graph-structure flexibility, and potential generalization across tasks while compared with spectral-based GCNs~\citep{RN3}. First, spatial-based GCNs tend to be more efficient than spectral-based GCNs because they directly perform convolutions in the graph domain via node information propagation. Thus spatial-based GCNs do not have to perform eigenvector computation or handle the whole graph computation simultaneously. Second, spatial-based models are flexible to handle multi-sourced graph inputs via the convenient aggregation function~\citep{RN3}. These graph inputs can be prepared as edge inputs ~\citep{RN24, RN25, RN26, RN27, RN28}, directed graphs~\citep{RN29, RN30}, signed graphs~\citep{RN31}, and heterogeneous graphs~\citep{RN32, RN33}. Third, spatial-based models perform graph convolutions locally on each node where network weights can be efficiently generalized across different nodes and graph structures. Therefore, spatial-based models have been shown to achieve superior performance on both transductive (e.g., semi-supervised learning) and inductive (e.g., the traditional supervised learning) tasks with flexibility on graph structures.
\subsection{Graph pooling mechanisms}
Graph pooling is a key strategy to address the computational challenges derived from graph convolutional operations~\citep{RN34}. Pooling operations reduce the size of a graph representation while preserving valuable structural information. Typically, graph pooling layers are located after graph convolutional layers and work as a down-sampling strategy. Graph pooling can be categorized into global and hierarchical graph poolings as shown in Fig.~\ref{fig2}.
\begin{figure}[!t]
\centering
\includegraphics[scale=.15]{Figures/fig2.png}
\caption{Graph pooling mechanism. (a) Global graph pooling. The function of global graph pooling is to flatten the node representations to a graph representation. In node representation, each node will include multiple dimensions of node attributes. After utilizing the global graph pooling, the most representative feature will be selected as the node attribute in graph representation. (b) Clustering graph pooling. Clustering-based poolings offer an efficient means to find strongly-connected communities on a graph. The nodes in the same clusters are represented by a new cluster node representation. (c) Sorting graph pooling. Sorting-based pooling updates the node representation by sorting the nodes attributes or edges weights. Both (b) and (c) are hierarchical pooling operations that refine the node representation to gain model robustness and improve computation efficiency.}
\label{fig2}
\end{figure}
Global pooling operation aggregates the node representations via simple flattening procedures such as summing, averaging, or maxing the node embeddings that are widely used in graph classification tasks~\citep{RN34}. Further, a global sorting pooling~\citep{RN35} sorts the node features in a descending order based on their last feature channel and the k-largest nodes form the updated graph representation of the global sorting pooling layer. Also, global attention pooling~\citep{RN36} acts as a soft attention mechanism that decides relevant nodes to the current graph-level task. Such global-wise pooling strategies, also known as readout layers, are often used to generate graph-level representation based on the previous node representations.
Hierarchical pooling operation is designed to refine the node representation by down-sampling strategies and overcome model overfitting. Hierarchical pooling strategies could be further categorized into two types including clustering-based and sorting-based methods. In clustering methods, spectral clustering (SC) offers an efficient means to find strongly-connected communities on a graph. SC can be used in GCNs to implement pooling operations that aggregate nodes especially belonging to the same cluster~\citep{RN37}. However, the expense of eigen-decomposition of the Laplacian and the generalization of SC strategies remain yet to be explicitly addressed. Alternatively, a graph clustering approach~\citep{RN37} formulates a continuous relaxation of the normalized min-cut problem and trains GCNs to compute cluster assignments. Spatial-based clustering strategies are proposed to achieve a higher computation efficiency compared with spectral-based clustering strategies. For example, DIFFPooling~\citep{RN38} is a differentiable graph pooling strategy that can generate hierarchical representations of graphs and can be combined with various GCNs architectures in an end-to-end fashion~\citep{RN38}. The key design of DIFFPooling is to learn a differentiable soft cluster assignment for nodes at each GCNs layer and mapping nodes to a set of clusters, which then forms the coarsened input for the next GCNs layer. In sorting-based methods, they focus on updating the node representation by sorting the nodes and edges depending on their attributes or weights. TopKPooling and SAGPooling shared a similar idea on the node sorting by their attention scores~\citep{RN39, RN40, RN41, RN42}. These poolings are designed to help select the top-kth nodes to summarize the entire graph for further feature computations. Notably, TopKPooling and SAGPooling can drop the node during model training to improve the computation efficiency and thus overcome the model overfitting. From the graph edge perspective, EdgePooling~\citep{RN43, RN44} is an inspiring example that could drop edges and merge nodes by sorting all edge scores and successively choosing the useful edges with the highest score whose two nodes have not yet been part of a contracted edge.
\subsection{Trade-offs in the design of GCNs architectures }
To optimize the performance of graph network models, there are multiple trade-offs between the network architecture and the corresponding model performance. The ability of information collection and the strategy of effective aggregation are crucial factors for measuring the performance of GCNs models. Intuitively, a deeper architecture corresponds to a larger receptive field, which can collect more auxiliary information towards enhanced performance of GCNs. However, the performance might decrease when layers go deeper to evolve larger receptive fields in real applications~\citep{RN25}. Such performance deterioration could be attributed to the over-smoothing of node representation with an increased architecture depth. In other words, the repeated and mixed message aggregation can lead to node representations of inter-classes indistinguishable~\citep{RN45}. It is commonly seen that the over-smoothing issue always occurs in the nodes with a dense connection with other nodes (e.g., the core of the graph) that could rapidly aggregate information in the entire graph. In contrast, the node in the tree part (e.g., leaves of the tree) could only include a very small fraction of information of all nodes with a small number of GCNs layers. To improve the GCNs model performance, it is necessary to overcome the graph over-smoothing phenomena and achieve informative node representation. For example, the study~\citep{RN46} implemented a co-training and self-training scheme with a smoothness regularizer term and adaptive edge optimization~\citep{RN45} to alleviate the over-smoothing problem. Co-training a GCN with the random walk model can explore the global graph topology. Further, self-training a GCN could exploit feature extraction capability to overcome its localized limitation. Informative node representation via the jumping knowledge network (JK-Net)~\citep{RN47} tends to demonstrate compelling performance on graph computing efficiency and alleviate overfitting. Notably, the idea of layer-aggregation across layers helps select the most informative nodes and reduce the overfitting issue, and the LSTM-attention could further identify the useful neighborhood ranges. Inspired by the architecture of JK-Net, Deep adaptive graph neural network (DAGCNs)~\citep{RN25} developed an adaptive score calculation scheme for each layer, which could balance the information from both local and global neighborhoods for each node. Both JK-Net and DAGCNs aim to find a trade-off between accuracy performance and the size of receptive fields by adaptively adjusting the information from local and global neighborhoods. For the design of network architecture, we expect additional efforts to overcome the over-fitting issues while keeping a flexible architecture to explore more meaningful information in the context of disease detection and diagnosis.
\section{Development of GCNs in medical imaging}
\subsection{Radiological image analysis}
Over the past decades, multi-modality radiological images have been routinely utilized in abnormality segmentation~\citep{RN48, RN49}, detection~\citep{RN50, RN51}, and patient outcome classification~\citep{RN52, RN53}. In this section, we discuss the growing body of GCNs studies applied to radiological analysis~\citep{RN54, RN55, RN56}, including magnetic resonance imaging (MRI), Computed Tomography (CT), and X-ray imaging. The combination of GCNs and radiological imaging promises to reflect the interaction among tissue regions and provide an intuitive means to fuse the morphological and topological-structured features among key image regions to advance modeling, interpretation, and outcome prediction. We here discuss the representative neuroimaging research and other related studies to highlight the usefulness of GCNs across different radiological imaging modalities and clinical tasks.
\subsubsection{Neuroimaging}
In neuroimaging, multi-modality MRI is a useful diagnostic technique by providing high-quality three-dimensional (3D) images of brain structures with detailed structural information~\citep{RN57}. Conceptually, multi-modality MRI data can be categorized into functional MRI (fMRI), structural MRI (sMRI), and diffusion MRI (DMRI). The fMRI measures brain activity and detects the changes in blood oxygenation and blood flow in response to neural activity~\citep{RN58}. The sMRI translates the local differences in water content into different shades of gray that serve to outline the shapes and sizes of the brain’s various subregions~\citep{RN59}. The DMRI is a magnetic resonance imaging technique in which the contrast mechanism is determined by the microscopic mobility of water molecules~\citep{RN60}. All these imaging modalities provide vital diagnostic support for neurological disease analysis because they can capture anatomical, structural, and diagnosis-informative features in neurology. Therefore, the overarching goal is to develop useful graph network models to define, explore, and interpret interactions of brain neurons and tissues. The detailed process of utilizing GCNs in the neuro-imaging analysis is illustrated in Fig.~\ref{fig3}.
To analyze the complex brain region connectivity and interaction, a brain graph representation can intuitively portray human brain organization, neurological disorders, and associated clinical diagnosis. Conventionally, the human brain could be modeled into a brain biological network containing nodes (e.g., region of interests) and edges among brain network nodes. The edges could be determined by brain signals or the real fiber connection. Yet these biologically-defined networks are often unable to faithfully capture neurological disorders and outcomes of patients~\citep{RN61}. To overcome this challenge, it is encouraged to leverage informative image-based features to considerably enrich graph node attributes. Comprehensive graph representation can integrate multiple types of information (e.g., image features, human brain signals, and clinical data) to greatly expand the knowledge base of brain dynamics and potentially provide auxiliary clinical diagnosis assistance. The use of GCNs here can be helpful to augment the architecture of human brain networks and has achieved remarkable progress in explaining the functional abnormality from the network mechanism~\citep{RN62}. In particular, GCNs are able to consider the functional or structural relations among brain regions together with image-based features that are beyond the scope of the conventional CNN-based methods~\citep{RN63,RN64,RN65}. The CNN-based model is merely viewed as a feature extractor for disease representation without consideration of structure information of the brain. For example, the deep 3-D convolutional neural network architecture was not unable to capture underlying structure information for Alzheimer's disease classification using brain MRI scans~\citep{RN61}. By contrast, the convergence of GCNs methods and MRI provide an alternative means to characterize the architecture of human brain networks and has achieved outstanding progress in brain abnormality explanation~\citep{RN62}.
The graph representations can be divided into functional and structural brain connectivity graphs based on the definitions of the graph components. First, graph nodes are regions of interest (ROI) as defined in MRI. ROI definition is commonly done through the anatomical parcellation of the Montreal neurological institute (MNI) using sMRI and fMRI data~\citep{RN66,RN67,RN68}. Second, graph edges are determined by the physical connectivity (e.g., the fiber tracts) of nodes in structural brain networks while calculated from the signal series analysis in functional brain networks. We therefore discuss insights of functional and structural brain connectivity graph developments below.
\begin{figure}[!t]
\centering
\includegraphics[scale=.12]{Figures/fig3.png}
\caption{The framework of developing GCNs in neuro-imaging analysis. Multi-modality MRIs are firstly converted into graph structure which is determined by the region of interest in terms of real human brain signals or fiber connectivity (e.g., node and edge definitions). Through graph-level model development and inference, we highlight numerous image-based analysis and diagnosis of diseases in neurology.}
\label{fig3}
\end{figure}
The human brain functional connectivity denotes the functional relations between specific human brain areas and functional brain graphs can represent estimates of interactions among time series of neuronal activity~\citep{RN62}. In functional brain networks, the nodes are defined as brain parcellation ROIs while the node attributes could be hand-crafted features or correlation measurements between nodes. The edges are created through the node correlations between different regions. For example, the GCNs framework achieved high-level performance in classifying Autism spectrum disorders (ASD) and healthy controls (HC) using task-functional magnetic resonance imaging (task-fMRI) through the appropriate ROI definition~\citep{RN69}. Their model consists of a message-passing neural network (MPNN)~\citep{RN19} as convolutional layers that is invariant to graph symmetries~\citep{RN69}. Furthermore, Top-k poolings~\citep{RN39} is able to downsample the node to achieve a higher computation efficiency while preserving a meaningful graph delineation. For a similar ASD and HC classification, another study~\citep{RN70} used GAT~\citep{RN22} as the convolutional layer incorporated with Top-k~\citep{RN39, RN40} and SAGE~\citep{RN42} pooling for achieving node importance scores. They also introduced two distance losses to enhance the distinguish among the nodes. Also, a group-level consistency loss is added to force the node importance scores to be similar for different input instances in the first pooling layer. Inspired by metric learning, a siamese graph convolutional neural network (s-GCN) is proposed for the ASD and HC classification purpose~\citep{RN54}, where samples were collected from Autism Brain Imaging Data Exchange (ABIDE)~\citep{RN71} database and UK Biobank~\citep{RN72}. The graph metric learning method essentially utilized GCN layer~\citep{RN16} in a siamese network~\citep{RN73}. Two types of graph construction methods are designed as the input of the model, such as spatial and functional graphs which determine nodes by ROIs, meanwhile, the KNN algorithm is utilized for both spatial and functional graph construction. The inputs for s-GCN are two same structure spatial or functional graphs with different signals (e.g., rows or columns of functional matrix). Notably, the spatio-temporal graph could be used to analyze the functional dependency between different brain regions and the information in the temporal dynamics of brain activity simultaneously. The spatio-temporal graph convolutional network is utilized to analyze the Blood-oxygen-Level-dependent (BOLD) signal of resting-state fMRI (rs-fMRI) for human age and sex prediction~\citep{RN74}. Also, studies~\citep{RN75, RN76} analyze the BOLD signal of fMRI for accurate detection of cognitive state changes of the human brain by presenting a dynamic graph learning approach to generate an ensemble of subject-specific dynamic graph embeddings. The brain networks are able to disentangle cognitive events more accurately than using the raw BOLD signals. The functional graph is meaningful to reflect the average functional connection strength between pairs of brain regions within a population. Generally, Pearson’s correlation is a useful strategy to construct functional connectivity matrix and define the node attributes.
The human brain's structural connectivity in vivo can be captured by structural and diffusion MRI~\citep{RN77, RN78}, and structural brain graphs could represent anatomical wiring diagrams~\citep{RN62}. Similar to the definition of nodes in functional connectivity networks, the nodes in structural connectivity networks are defined as a region of interests (ROIs), which are parceled from the brain based on structural and diffusion MRI.. Clinically, the structural brain connectivity represents the structural associations of altered neuronal elements, including both the morphometric alternation and accurate anatomical connectivity as seen in imaging. In the complex brain networks, structural brain connectivity assesses to white amount projections bond cortical and subcortical regions~\citep{RN79}. The edges indicate the actual neural fiber connections between different brain regions. For example, a stack architecture design of combining a heterogeneous GCN model~\citep{RN55} with an efficient adaptive pooling scheme~\citep{RN38} is able to predict the clinical score of Parkinson’s disease (PD) and HC using diffusion-weighted MRI (DWI) on Parkinson Progression Marker Initiative (PPMI) ~\citep{RN80}. To construct the graph structure from DWI, nodes are defined by the ROIs in the brain while three whole-brain probabilistic tractography algorithms are able to determine different brain structural. The node attributes corresponding to rows in the human brain network are defined as features. Another novel framework is developed to explore graph structure in the q-space by representing DMRI data and utilizing graph convolutional neural networks to estimate tissue microstructure~\citep{RN81}. This approach is capable of not only reducing the data acquisition time but also accelerating the estimation procedure of tissue microstructure. The nodes of the weighted graphs are sets of points on a manifold. Also, the adjacency weights are defined between two nodes using Gaussian kernels, accounting for differences in gradient directions and diffusion weightings. The q-space signal measurements are represented by using the constructed graph that encodes the geometric structure of q-space sampling points. A residual ChebNet~\citep{RN14} can learn the mapping between sparsely sampled q-space data and high-quality estimates of microstructure indices.
Beyond single-modality MRI analysis, multi-modality image data analysis emerged as active research areas for GCNs modeling. Multi-modality MRI data analysis is able to deepen our understanding of disease diagnosis from different data aspects. In neuroimaging, the structural connectivity in sMRI reflects the anatomical pathways of white matter tracts connecting different regions, whereas the functional connectivity in fMRI encodes the correlation between the activity of brain regions. A unique advantage of multi-modality MRI data analysis is that they have incorporated complementary information from different modalities simultaneously. The multimodal data fusion can be implemented by two types of strategy: (1) constructing the original graphs directly using the partial information from functional and structural brain connectivity; (2) constructing the original functional and structural graphs separately and updating the graph representations by computing and fusing the two-side information. For the first fusion strategy, the study~\citep{RN82} introduced an edge-weighted graph attention network (EGAT)~\citep{RN22} with a diffPooling~\citep{RN38} to classify Bipolar disorder (BP) and HC from sMRI and fMRI in cerebral cortex analysis. Also, the framework of Siamese community-preserving graph convolutional network (SCP-GCN)~\citep{RN83} is able to learn the structural and functional joint embedding of brain networks on two public datasets (i.e., Bipolar and HIV dataset~\citep{RN83}) for brain disease classification. Especially, siamese architecture can exploit pairwise similarity learning of brain networks to guide the learning process to alleviate the data scarcity problem~\citep{RN83}. Ninety cerebral regions are selected as nodes for both structural (e.g., Diffusion Tensor Image (DTI)) and functional (e.g., fMRI) networks, and the node attribute is determined by the functional connectivity between nodes corresponding to fMRI. The edge connectivity is determined by the DTI via a series of preprocessing (distortion correction, noise filtering, repetitive sampling from the distributions of principal diffusion directions for each voxel). To preserve the community property of brain networks, the design of a community loss presents its usefulness to minimize the intra-community loss and maximize the intercommunity loss. For the second fusion strategy, the study~\citep{RN84} fused information from multimodal brain networks on rs-fMRI and dMRI for age prediction. After constructing the original functional and structural brain networks separately, the study reconstructed the positive and negative connections in the functional networks depending on structural networks' information. They utilized a multi-stage graph convolutional layer, motivated by GAT~\citep{RN22} and ResGCN~\citep{RN85}, for structural network edge feature update and class classification. Then, the edge feature and class are utilized to update the functional networks for age prediction. Also, for liver lesion segmentation, a mutual information-based graph co-attention module~\citep{RN86, RN87} is proposed by extracting modality-specific features from T1-weighted images (T1WI) and establishing the regional correspondence from T2-weighted images (T2WI) simultaneously. In this study, they constructed two separate graphs for either T1WI or T2W2; meanwhile, they used GCN~\citep{RN16} to propagate representations of all nodes. The mutual information-based graph co-attention module updates the T1WI-based node representation by selectively accumulating information from node features from T2WI-based node representation. The fused node representation is re-projected and added to the original feature for the final segmentation.
Furthermore, GCNs extend to allow the multi-modality integration between MRI and non-imaging data for analyzing complex disease patterns. For example, an Edge-Variational GCN (EV-GCN)~\citep{RN11} could automatically integrate imaging data (e.g. fMRI data) with non-imaging data (e.g. age, gender and diagnostic words) in populations for uncertainty-aware disease prediction. They constructed weighted graphs via an edge-variational population graph modeling strategy. In the weighted graphs, the graph nodes are ROIs and the node attributes are features extracted from histology and fMRI images. It is particularly notable that the weight of the edge is achieved by a learnable function of their non-imaging measurements. The proposed Monte-Carlo edge dropout randomly drops a fraction of edges in the constructed graphs to reduce overfitting and increase the graph sparsity. In addition, two similar studies~\citep{RN88, RN89} constructed the sparse graph by combining the information from the functional connectivity (e.g., rs-fMRI), structural connectivity (e.g., DTI), and demographic records (e.g., gender and age) for mild cognitive impairment detection and classification. In these studies, they constructed functional and structural brain networks for each subject (e.g., image). Then, they defined each subject as graph node. For the first study~\citep{RN88}, they concatenated the feature from functional and structural connectivity as vertices features. Also, they calculate the feature and phenotypic information similarity to constructed graph edges. Further, they utilized GCN~\citep{RN16} incorporated with the random walk algorithm to enhance the detection performance. For the second study~\citep{RN89}, it constructed graphs for the functional and structural connectivity separately. Beyond the similarity evaluation of subjects (e.g., vertices) feature and non-image phenotypic information, this study also determined the edge by connecting nodes belong to the same receptive field class directly. Further, they used the constructed graph to pretrain GCN to update graphs and GCN for the final disease deterioration prediction.
\subsubsection{X-ray and CT imaging}
Extensive studies have also utilized GCNs in X-ray and Computed Tomography (CT) images for disease analysis~\citep{RN56, RN90, RN91}. Different from MRI data, CT images are able to reflect the vessel skeleton information that could assist a variety of clinical tasks. For example, chest CT scans can assist with arteries-veins separations that are of great clinical relevance for chest abnormality detection~\citep{RN90}. The graph was constructed of the voxels on the skeletons resulting in a vertex set and their connections in an adjacency matrix. The skeletons are extracted from chest CT scans by vessel segmentation and skeletonization. In this study~\citep{RN90}, GCN layers can extract and learn connectivity information. The one-degree (direct) neighbors were considered and the vertices attributes were extracted by CNN model to consider the local image information. In addition, chest CT scans together with GCN is able to assist airway semantic segmentation, which refers to the segmentation of airway from background and dividing it into anatomical segments for lung lobe analysis~\citep{RN92}. Also, a prototype-based GCN framework~\citep{RN93} provided a means for airway anomaly detection to aid in lung disease diagnosis. The GCN layers calculated the initial anomaly score for every node, while the prototype-based detection algorithm computed the entire graph's anomaly score. Another study~\citep{RN94} utilized radiotherapy CT (RTCT) and PET, which is registered to the RTCT for lymph node gross tumor volume (GTV\_LN) detection. In this study, the 3D CNN extracted instance-wise visual features while the GNN model analyzed the inter-LN relationship. The feature fusion of CNN and GNN boosted the GTV\_LN detection performance. Furthermore, the study~\citep{RN95} utilized GCN~\citep{RN16} on cone-beam computed tomography (CBCT) images for craniomaxillofacial (CMF) landmark localization, which is important for designing treatment plans of reconstructive surgery. They utilized an attention feature extraction network for localizing landmarks and generating attention features for the graph construction. In addition, the study~\citep{RN56} proposed an end-to-end hybrid network to train a CNN and GAT network to leverage both advanced feature learning and inter-class feature representations on Chest-Xray 14 dataset~\citep{RN96}. To utilize the image sequencing information, they determine each image from the same patient as a vertice of a graph and the extracted features are the attributes of vertices. Furthermore, they leverage non-imaging meta-data, such as clinical information, to construct edges between the vertices. After constructing the graph and updating the graph representation with GAT, they combine the CNN extracted features with graph representation by skip-connectivity to achieve hybrid representation. The motivation of generating hybrid representation is to improve the distinction between samples. Furthermore, CT images and non-image clinical information could be analyzed jointly for Lymph node metastasis (LNM) prediction. The study~\citep{RN97} proposed a co-graph convolutional layer consisting of Con-GAT~\citep{RN22} and Corr-GAT~\citep{RN98} layers to achieve the node’s new representation by weighted averaging its neighboring nodes and measuring the weight score by feature difference-based correlation. Due to the pandemic of COVID-19, GCNs have also been utilized in disease detection. GraphCovidNet ~\citep{RN91} utilized GIN for COVID-19 detection on both CT and X-ray images. The graph is used to depict the outline of an object (e.g., organ) in the image. First, they applied edge detection to determine the edge outline. Then, the graph nodes are defined by the pixel having a grayscale intensity value greater than or equal to 128, which implies nodes reside only on the prominent edges of the edge image. The node attribute consists of the grayscale intensity of the corresponding pixel. An edge exists between the two nodes which represent neighboring pixels in the original image. For example, the GCN-based model~\citep{RN99} extractes node information hierarchically towards both diagnosis and prognosis for COVID-19 patients. Their distance-aware pooling, including graph-based clustering and feature pooling, is able to aggregate node information on the dense graph effectively. Also, the proposed model could coarsely localize the most informative slices for CT scans to provide the interpretability for better clinical decision-making.
Table.~\ref{table1} summarizes a variety of graph construction methods and GCNs application in radiologic image analysis. Compared to conventional methods, GCNs methods for the analysis of brain networks have the possibility of combining image-based features with the conventional brain networks.
\begin{longtable}{|p{2.7cm}|p{1.6cm}|p{1.6cm}|p{10.5cm}|}
\caption{\label{table1}Summary of GCNs in radiologic image analysis}\\
\hline
Method & Category & Input & Graph Construction\\
\hline
\endhead
\multicolumn{4}{r}{\footnotesize Continue on the next page
\endfoot
\endlastfoot
\multirow{2}{*}{\cite{RN69}} & \multirow{2}{*}{\makecell{Single\\modality}} & \multirow{2}{*}{Task-fMRI} & Nodes are ROIs and node attributes are hand-craft features.\\
\cline{4-4}
& & & Edges are determined by region-to-region correlations and edge attributes are the values of Pearson correlation and partial correlation among nodes.\\
\hline
\multirow{2}{*}{\cite{RN54}} & \multirow{2}{*}{\makecell{Single\\modality}} & \multirow{2}{*}{fMRI} & Nodes are ROIs and node attributes are rows/column of the functional connectivity matrix\\
\cline{4-4}
& & & Spatial graph: a KNN graph based on spatial coordinates of the ROI.\par Functional graph: a KNN graph based on correlation distance between all ROI pairs.\\
\hline
\multirow{2}{*}{\cite{RN70}} & \multirow{2}{*}{\makecell{Single\\modality}} & \multirow{2}{*}{fMRI} & Nodes are ROIs and node attributes are determined by Pearson correlation coefficient among nodes.\\
\cline{4-4}
& & & Edge is determined by Edge-set and edge attributes are determined by Partial correlation coefficient among nodes.\\
\hline
\multirow{2}{*}{\cite{RN74}} & \multirow{2}{*}{\makecell{Single\\modality}} & \multirow{2}{*}{rs-fMRI} & Spatial graph: nodes are ROIs at the same time point and node attributes are determined by the average BOLD signal of the ROI i at time point t.
\par
Temporal graph: the same ROI at the different time points and node attributes are determined by the functional affinity between ROIs. The functional affinity is calculated by the magnitude of correlation between the concatenated average BOLD time series.
\\
\cline{4-4}
& & & Spatio graph: connect all nodes at the same time point.\par
Temporal graph: connect the corresponding node to the node of the same ROI at the proceeding time point. \\
\hline
\multirow{2}{*}{\makecell{\cite{RN75}\\ \cite{RN76}}} & \multirow{2}{*}{\makecell{Single\\modality}} & \multirow{2}{*}{fMRI} & Nodes are defined by ROIs (e.g., Brain regions), and the node attributes are hand-craft features.\\
\cline{4-4}
& & & Edges are determined by Edge-set and edge attributes are the statistical correlations of BOLD signals among nodes\\
\hline
\newpage
\multirow{2}{*}{\cite{RN55}} & \multirow{2}{*}{\makecell{Single\\modality}} & \multirow{2}{*}{DWI} & Node are ROIs, and the node attributes are the rows/columns of the connectivity matrix.\\
\cline{4-4}
& & & Edges are determined by the whole-brain probabilistic tractography algorithm\\
\hline
\multirow{2}{*}{\cite{RN81}} & \multirow{2}{*}{\makecell{Single\\modality}} & \multirow{2}{*}{DMRI} & Nodes are the points on the manifold.\\
\cline{4-4}
& & & Edges are constructed between two nodes when edge attributes are larger than 0, and edge attributes are determined by the differences in gradient directions and diffusion weighting calculated by two Gaussian kernels in q-space.\\
\hline
\multirow{2}{*}{\cite{RN90}} & \multirow{2}{*}{\makecell{Single\\modality}} & \multirow{2}{*}{CT images} & Nodes are the connectivity of voxels, and node attributes are extracted by CNN model.\\
\cline{4-4}
& & & Edges are constructed between the voxels on the skeletons.\\
\hline
\multirow{2}{*}{\cite{RN92}} & \multirow{2}{*}{\makecell{Single\\modality}} & \multirow{2}{*}{CT images} & Airway nodes: nodes are the segmented airway regions, and node attributes are extracted by CNN model. \par Landmark nodes: nodes are the detected landmark positions, and node attributes are extracted by CNN model.
\\
\cline{4-4}
& & & 1.Internal graph: the connectivity between nodes and their KNN neighbors in Euclidean space. \par 2.External graph: the airway structural prior information to connect two types of nodes.
\\
\hline
\multirow{2}{*}{\cite{RN93}} & \multirow{2}{*}{\makecell{Single\\modality}} & \multirow{2}{*}{CT images} & Nodes are the segmented airway regions, and node attributes are extracted by CNN model and the anatomical airway structure.
\\
\cline{4-4}
& & & The edges are determined by the airway structure.
\\
\hline
\multirow{2}{*}{\cite{RN94}} & \multirow{2}{*}{\makecell{Single\\modality}} & \multirow{2}{*}{\makecell{RTCT and\\PET images}} & Nodes are the $GTV_{LN}$ candidates, and node attributes are extracted by CNN model.
\\
\cline{4-4}
& & & The edges are determined by calculating the Euclidean distance among the boundary voxels of the tumor mask.
\\
\hline
\multirow{2}{*}{\cite{RN95}} & \multirow{2}{*}{\makecell{Single\\modality}} & \multirow{2}{*}{\makecell{CBCT\\images}} & Nodes are the landmarks, and node attributes are extracted by CNN model.
\\
\cline{4-4}
& & & The edge between landmarks is determined by whether the two landmarks are in the same anatomical regions .
\\
\hline
\multirow{2}{*}{\cite{RN99}} & \multirow{2}{*}{\makecell{Single\\modality}} & \multirow{2}{*}{CT images} & Nodes are the slices of CT image, and node attributes are extracted by CNN model and wavelet decomposition.
\\
\cline{4-4}
& & & The graph is densely connected graph, and edge attributes are calculated by cosine similarity among node features.
\\
\hline
\multirow{2}{*}{\cite{RN82}} & \multirow{2}{*}{\makecell{multi\\modality}} & \multirow{2}{*}{\makecell{fMRI \\and sMRI}} & Nodes are ROIs, and node attributes are seven anatomical features and four functional connectivity statistic features.
\\
\cline{4-4}
& & & The graph is densely connected graph, and edge attributes are calculated by the Pearson correlation-induced similarity..
\\
\hline
\multirow{2}{*}{\cite{RN83}} & \multirow{2}{*}{\makecell{multi\\modality}} & \multirow{2}{*}{\makecell{fMRI \\and DTI}} & Nodes are ROIs, and node attributes are rows/column of connectivity matrix.
\\
\cline{4-4}
& & & The edges are determined by region-to-region correlations.
\\
\hline
\multirow{2}{*}{\cite{RN84}} & \multirow{2}{*}{\makecell{multi\\modality}} & \multirow{2}{*}{\makecell{rs-fMRI \\and dMRI}} & Nodes are ROIs.
\\
\cline{4-4}
& & & The edges are determined by edge-set. For functional brain network, the edge attributes are the correlation of fMRI signals between nodes. For structural brain network, the edge attributes are the probability of fiber tractography between nodes.
\\
\hline
\multirow{2}{*}{\makecell{\cite{RN86}\\\cite{RN87}}} & \multirow{2}{*}{\makecell{multi\\modality}} & \multirow{2}{*}{\makecell{T1WI \\and T2WI}} & Nodes are a group of features in the partial region of the original regular grid coordinates, and node attributes are extracted by CNN model.
\\
\cline{4-4}
& & & The graph is a fully-connected graph.
\\
\hline
\newpage
\multirow{2}{*}{\cite{RN11}}&\multirow{2}{*}{\makecell{multi\\modality}} & \multirow{2}{*}{\makecell{fMRI and\\clinical data}} & Nodes are images (e.g., subjects), and node attributes are the concatenated local weighted clustering coefficient features from functional and structural connectivity.
\\
\cline{4-4}
& & & The edges are determined by calculating the similarity between the node features and incorporates the phenotypic information.
\\
\hline
\multirow{2}{*}{\cite{RN88}} & \multirow{2}{*}{\makecell{multi\\modality}} & \multirow{2}{*}{\makecell{rs-fMRI,\\DTI, gender\\ and age}} & Nodes are images (e.g., subjects), and node attributes are the the concatenated local weighted clustering coefficient features from functional and structural connectivity.
\\
\cline{4-4}
& & & The edges are determined by the similarity between the node features and incorporates the phenotypic information.
\\
\hline
\multirow{2}{*}{\cite{RN89}} & \multirow{2}{*}{\makecell{multi\\modality}} & \multirow{2}{*}{\makecell{rs-fMRI,\\DTI, gender\\ and\\acquisition\\equipment}} & Nodes are images (e.g., subjects), and node attributes are the recursive feature elimination extracted features from functional and structural connectivity.
\\
\cline{4-4}
& & & The edges are determined by:\par
1. The similarity between the node features and incorporates the phenotypic information.\par
2. Connected nodes that belong to the same receptive field.
\\
\hline
\multirow{2}{*}{\cite{RN56}} & \multirow{2}{*}{\makecell{multi\\modality}} & \multirow{2}{*}{\makecell{X-ray\\images\\and\\meta-data}} & Nodes are images (e.g., patients), and node attributes are extracted by CNN model.
\\
\cline{4-4}
& & & The edges are determined by non-image meta-data.
\newline
\\
\hline
\multirow{2}{*}{\cite{RN97}} & \multirow{2}{*}{\makecell{multi\\modality}} & \multirow{2}{*}{\makecell{CT images\\and\\non-image\\data}} & Nodes are ROIs, and node attributes are the concatenation feature consists of CNN extracted features and non-imaging clinical information.
\newline
\\
\cline{4-4}
& & & The graph is fully connected graph.
\\
\hline
\multirow{2}{*}{\cite{RN91}} & \multirow{2}{*}{\makecell{multi\\modality}} & \multirow{2}{*}{\makecell{CT and\\X-ray\\images}} & Nodes are pixels, and node attributes are grayscale intensity of the pixel.
\\
\cline{4-4}
& & & The edges are determined by the neighborhood relationship between pixels.
\newline
\\
\hline
\end{longtable}
\subsection{Histopathological image analysis }
The growth of digitalized histopathological images presents a valuable resource to support rapid and accurate clinical decision making. The high-resolution whole slide image (WSI) contains rich tissue characteristics including patterns of cell nuclei, glands, and lymphocytes~\citep{RN100, RN101}. Extensive pathological characteristics of tissue and cell interactions can be evidently observed that are not available in other clinical image data. For instance, lymphocytic infiltration of cancer status can be deduced only from histopathology imagery~\citep{RN102}. These pathological patterns can be used to build the biological graph networks that can inform disease status and thus discern predictive imaging biomarkers. Overall, we recognize that GCNs analysis is uniquely positioned to address key issues of histopathological applications, including data annotation, tissue connections, global-local information diagnostic fusion, and model prediction performance in challenging settings.
Developments of GCNs have brought remarkable advances into computational histopathology including label efficiency and multi-scale context representation. First, graph structure provides a reasonable choice to represent the entire slide in terms of tissue content connectivity. Such entire-slide graph representation can avoid fine-grained patch-wise label annotation. Since we know that patch-level labeling is highly time-intensive, even impossible, to include all ranges of tumor patches annotated by human experts. Second, graph structural representation can capture multi-scale contexts considering both global and local image-wise features towards enhanced prediction of disease outcomes. Third, graph structural representation builds upon the interaction among spatially-separated tiles that enables a more flexible and comprehensive receptive field. Such advances are analogous to the workflow of human experts that we consider tumor environment, tissue contents, and their interactions, rather than single tumor tiles, to diagnose tissue status of patients.
Because the high-resolution histopathological image does not present a natural form of graph structure, efficient graph representation becomes a vital factor for model development and optimization. Current graph construction in histopathology can be broadly categorized into patch-based and cell-based methods. First, patch-based graph construction methods aim to enable information extraction by considering the entire micro-environment (e.g., the cells and tissues), where comprehensive tissue micro-environment and cell dynamics can be captured. In these patch-based methods, graph nodes are defined as the selected patches determined by ROIs in the histopathological image. The associated node attributes can be extracted by standard feature extractors (e.g., ResNet18 or VGG16). Graph edges are defined as the connectivity between nodes, which is determined by the feature or coordinate distance between two nodes. A smaller distance means a higher probability of connectivity. The connectivity between nodes could determine an adjacency matrix to represent the entire topological structure of the graph. Although the definition of primitive graph components (e.g., node and edge) are conceptually similar, most patch-based graph construction methods have different settings for node attributes and edge construction. As opposed to the patch-based graphs, cell-based graph methods emphasize the possible biological significance derived from histopathology. Cell-based graph construction methods aim to model the relationship between different cells and the micro-environment (e.g., tissues or vessels) utilizing graph-based features~\citep{RN103}. In a cell graph, the detected and segmented nuclei or cell clusters are considered as nodes. The node attribute is defined as the combination of image-wised features, such as features extracted by CNN models, and the hand-crafted feature, such as the number or the size of nuclei, the average RGB value of nucleus, gray level co-occurrence matrix features, VGG19 features, and the number of neighbors of a nucleus~\citep{RN104}. According to the assumption that adjacent cells are more likely to interact~\citep{RN103}, the edge between the nodes can be determined via Delaunay triangulation~\citep{RN105} or the K-nearest-neighbour method~\citep{RN106}, which could evaluate whether two cells (nodes) belong to the same cluster. The cells in the same cluster are more likely to have an edge between them. Despite a good performance on clinical classification tasks, these approaches cannot work well in capturing the diagnostic and prognostic information from the surrounding micro-environments (e.g., tissues and vessels). Meanwhile, constructing cell-centered graphs highly depends on cell detection accuracy. It is notable that constructing a cell-based graph and subsequent graph computing need an excessive computational complexity. The process of utilizing GCNs in histopathological image analysis is shown in Fig.~\ref{fig4}. We outline several areas of clinical interest for GCNs in histopathology below.
\begin{figure}[!t]
\centering
\includegraphics[scale=.22]{Figures/fig4.pdf}
\caption{The illustration of the computational framework of GCNs in histopathological imagery. (a) Overall steps of image-based graph convolutional network framework. (b) Image preprocessing. The high-resolution images are normally split into manageable small-sized tumor tiles. We primarily focus on tumor tile processing and analysis for GCN development. (c) Image-based graph transformation. The transformation between image and graph-structured data is vital according to different tasks. Both patch-based and cell-based graphs can be established for downstream tasks. (d) GCNs computation and outcome interpretation. The inputs of GCNs are the constructed image-based graphs. The outcome interpretation of GCNs includes both node- and edge-wise findings to enable a multi-dimensional interpretation of outcomes.}
\label{fig4}
\end{figure}
\subsubsection{Tumor segmentation}
Accurate tumor segmentation in histopathology is designed to assist pathologists for improving workflow efficiency of clinical diagnosis~\citep{RN7}. Graph-based segmentation approaches can incorporate both local and global inter-tissue-region relations to build contextualized segmentation and thus improve the overall performance. For example, SEGGINI performs semantic segmentation of images by constructing tissue-graph representation and performing weakly-supervised segmentation via node classification by using weak multiplex annotations, i.e., inexact and incomplete annotations, in prostate cancer~\citep{RN7}. In this study, they defined graph nodes by superpixels merging based on channel-wise color similarity of superpixels at higher magnification. The node attribute is determined by the spatial and morphological features of the merged node (e.g., the merged superpixel). The spatial feature is computed by normalizing superpixel centroids by the image size and the morphological feature is extracted by a pre-trained MobileNetV2~\citep{RN107}. They defined the edges by constructing a region adjacency graph (RAG)~\citep{RN108} from the spatial connectivity of superpixels. The local and global connection of tissue details creates an alternative avenue for pixel-level segmentation evaluation that draws a contrast to other conventional convolutional-based tumor segmentation approaches ~\citep{RN109, RN110, RN111}. Another study~\citep{RN112} proposed an end-to-end framework that utilizes an unsupervised pretrained CNN to extract tile features and generate dynamics superpixels for graph construction, while using GCN for predicting the final segmentation map. In this study, the dynamics superpixels can be viewed as a key bridge between CNN and the GCN model, which are generated according to the CNN feature extraction.
\subsubsection{Cancer tissue classification}
Cancer subtype classification is crucial in clinical image analysis that can impact patient stratification, outcome assessment, and treatment development~\citep{RN6, RN113}. GCNs have been extensively studied in cancer subtype classification due to their unique ability to explore the relational features among tissue sub-regions (e.g., patches or cells). Patch-based graph construction approaches are intuitive to build a bridge between image features and graph structure. Conceptually, patches are defined as nodes and node attributes are extracted patch features, including CNN-based extracted and hand-crafted features. The edges are typically determined by the Euclidean distance of nodes. For example, the combination of ChebNet~\citep{RN14} and GraphSage~\citep{RN20} presents its usefulness for classifying lung cancer subtypes in histopathological images~\citep{RN6} via patch selection. All patches in the tissue region are grouped into multiple classes, and a portion of all clustered patches (e.g., 10\%) are randomly selected within each class. Also, a simplified graph construction process~\citep{RN6} can be useful to leverage all patch information. The global context among patches is considered while using a fully connected graph to represent the connection among nodes. Global pooling layers (e.g., global attention, max, and sum poolings) are able to generate graph representations for analyzing cancer classification. In particular, global attention pooling~\citep{RN36} provides strong interpretability to determine which nodes are relevant to the current graph-level classification tasks. In colorectal cancer histopathology, ChebNet~\citep{RN14} shows its predictive power in lymph node metastasis (LNM) prediction~\citep{RN113}. Interestingly, a combination model of a variational autoencoder and generative adversarial network (VAE-GAN)~\citep{RN114} is utilized to train as a feature extractor to decode the latent representations closer to their original data space. Further, the pixel-based graph construction could be understood as a variant of patch-based approaches. The study~\citep{RN115} developed a group quadratic graph convolutional network for breast tissue and grade classification on pixel-based graph representation. The proposed model reduces the redundant node (e.g., pixel) feature, selects superior fusion feature, and enhances the representation ability of the graph convolutional unit by the pixel-based graph analysis.
As opposed to patch-based approaches, cell-based graph construction is under a key assumption that cell-cell interactions are the most salient points of information~\citep{RN116}. A common example is to define the detected nuclei as nodes~\citep{RN104} and while the overall node attributes are aggregated by concatenating multiple types of features (see Table2). The graph edge is determined by thresholding the Euclidean distance between nodes. In addition, the cell graph convolutional network~\citep{RN103} presents a generalized framework for grading colorectal cancer histopathological images based on the combination of GraphSage~\citep{RN20}, JK-Net~\citep{RN47}, and Diffpooling~\citep{RN38}. The edge between two nuclei is determined by a fixed distance while the maximum degree of each node is set to k corresponding to its k-nearest neighbors. Sharing a similar cell-graph construction strategy and graph component definition with~\citep{RN103}, a GIN-based~\citep{RN21} framework is designed for breast cancer subtype classification~\citep{RN117}. In addition, the clinical interpretation is provided by a cell-graph explainer that is inspired by a previous graph explainer~\citep{RN118}, a post-hoc interpretability method based on graph pruning optimization. The cell-graph explainer is able to prune the redundant graph components, such as the nodes that could not provide enough information in the decision making, and define the resulting subgraph as the explanation. Another cell graph application of cancer classification~\citep{RN119} is built on top of robust spatial filtering (RSF)~\citep{RN32}, where RSF combined with attention mechanisms to rank the graph vertices in their relative order of importance, providing visualizable results on breast cancer and prostate cancer classification.
To leverage the advantages of patch- and cell-based graphs simultaneously, the model integration can provide additional auxiliary benefits by capturing detailed nuclei and micro-environment tissue information. A hierarchical cell-to-tissue graph neural network (HACT-Net)~\citep{RN120} is an example to consist of a low-level cell-based graph (e.g., cell-graph), a high-level patch-based graph (e.g., tissue-graph), and a hierarchical-cell-to-tissue representation for breast carcinoma subtype classification. For the cell-based graph, they defined nuclei as graph nodes that are detected by the pre-trained Hover-Net~\citep{RN50, RN121, RN122}. For the patch-based graph, they determined graph nodes and their attributes by creating non-overlapping homogeneous superpixels and their features. The edges are constructed by a region adjacency graph~\citep{RN108} using the spatial centroids of the super-pixels. Overall, such a joint analysis across histopathological scales leads to enhanced performance for cancer subtype classification.
Cancer staging and grade classification is also of clinical significance that comprises tumor tissue and nodal (e.g., tumor and lymph nodes) staging~\citep{RN116}. Patch-based graph construction strategy is commonly used in tumor staging classification in terms of graph attention~\citep{RN8}. Also, graph topological feature extraction is useful in colon cancer tumor stage prediction with well interpretation~\citep{RN116}. In particular, they utilized the Mapper~\citep{RN123} to project high-dimensional graph representation to a lower-dimensional space, summarizing higher-order architectural relationships between patch-level histological information to provide more favorable interpretations for histopathologists. Further, for liver fibrosis stage classification, the study~\citep{RN124} proposed a patch-based graph structure together with the GCN attention layer to analyze the spatial organization of the fibrosis patterns. They use the KNN algorithm to cluster the tiles to select regions of high collagen content as the centroid node for graph construction. The proposed pipeline allows for the separation of fibers in the slide into localized fibrosis patterns and the individual regions can be inspected by a pathologist~\citep{RN124}. Also, cell-based graph construction strategy is useful for cancer grade classification. For example, for prostate cancer grade classification~\citep{RN125} in tissue micro-array, GraphSage~\citep{RN20} learns the global distribution of cell nuclei, cell morphometry, and spatial features without requiring pixel-level annotation. In this study, the cell nuclei are the node of the graph while the three types of features consist of the node attribute, including the morphological feature (e.g., the area, roundness, eccentricity, convexity, orientation for each of the nucleus.), texture feature (e.g., the dissimilarity, homogeneity, energy, and ASM based on the obtained grey level co-occurrence matrix), and contrastive predictive coding features.
\subsubsection{Survival prediction}
Survival analysis is a long-standing clinical task to determine the prognostic likelihood of patients~\citep{RN113, RN126}. Both cell- and patch-based approaches can be considered to capture survival sensitive information of patients. For instance, the graph convolutional neural network with attention learning has shown to achieve a good performance on the survival prediction in colorectal cancer~\citep{RN127}. Tumor tiles are defined as nodes and node attributes are extracted by the VGG16. Graph edges are constructed by thresholding the Euclidean distances between node attributes. After constructing the graph, they used the ChebNet~\citep{RN14} framework for survival analysis on the histopathological images. With a similar definition of graph components, another study~\citep{RN128} designed a patch-based graph construction strategy in the Euclidean space. They utilized the DeepGCN~\citep{RN85} and global attention layer to boost the survival prediction performance and provided the interpretability across five cancer types. An integrated framework~\citep{RN129} extracted morphological features from histology images using CNNs and from the constructed cell-based graph using GraphSage~\citep{RN20}, and also genomic (mutations, CNV, RNA-Seq) features using SNNs. A fusion of these deep features using the Kronecker Product is of great interest for accurate survival outcome prediction. In addition, cell-based and patch-based graphs can be further unified to allow a trade-off between efficiency and granularity~\citep{RN130}. They used GAT or prostate cancer survival prediction using WSIs. Notably, a self-supervised learning method is proposed to pretrain the model, yielding improved performance over trained-from-scratch counterparts. For cell-based graphs, they use a Mask R-CNN~\citep{RN131} for nuclei segmentation and define an eight-pixel width of the ring-like neighborhood region around each nucleus as its cytoplasm area. The nuclear morphometry features and visual texture features (intensity, gradient, and Haralick features) have made substantial contributions for both nuclear and cytoplasm region representations respectively. Despite these advances, uncertainty remains for exploring definitive roles of cell-level and patch-level characteristics with regard to overall survival likelihood of patients.
\subsubsection{Molecular biomarker prediction}
Image-based molecular biomarker prediction is promising to deepen our understanding of cancer biology across data modalities. Enormous efforts are gaining momentum to explore multiple image-to-genome associations in cancer research~\citep{RN132, RN133, RN134} . The feature-enhanced graph network (FENet)~\citep{RN12} leverages histopathological-based graph structure to predict key molecular outcomes in colon cancer. Through the spatial measurement of tumor patches, the image-to-graph transformation illustrates its unique value in predicting key genetic mutations. In particular, the use of GIN~\citep{RN21} layer and jumping knowledge structure are useful to aggregate and update the patch embedding information. Alternatively, the cell-based construction method is considerable for cancer biomarker prediction~\citep{RN135}. HoverNet~\citep{RN122} is a popular choice for nuclei segmentation to support cell graph construction. Next, the agglomerative clustering~\citep{RN136} is utilized to group spatially neighboring nuclei into clusters. These clusters can be defined as graph nodes and the node attribute is determined by the standard deviation of nuclei sizes. Meanwhile the edges are constructed by using Delauney triangulation based on the geometric coordinates of cluster centers with a maximum distance connectivity threshold. Both cell- and patch-based approaches contribute to the integration of histopathology and genome as more biological data become accessible. We recognize that graph-based models can offer an efficient means to measure the cross-modality differences, which requires careful inputs on graph construction, model layer architectures, proper design of feature extraction for achieving improved performance of molecular outcome prediction.
Overall, Table.~\ref{tab2} summarizes the category, type of tasks, and the graph-structure construction strategies. In this chapter, we have discussed novel perspectives for computational histopathological image analysis. In particular, GCNs-based methods provide a novel perspective to consider tumor heterogeneity in histopathological image analysis. Despite multiple challenges, the evolving capacity of current graph construction strategies (edge, node, and node attributes) makes it possible to address a variety of clinical tasks using histopathological images.
\begin{longtable}{|p{2.9cm}|p{1.2cm}|p{2cm}|p{9.5cm}|}
\caption{\label{tab2}Summary of GCNs in histopathological image analysis}\\
\hline
Method & Category & Tasks & Graph Construction\\
\hline
\endhead
\multicolumn{4}{r}{\footnotesize Continue on the next page
\endfoot
\endlastfoot
\multirow{2}{*}{\cite{RN7}} & \multirow{2}{*}{\makecell{Patch\\based}} & \multirow{2}{*}{Segmentation} & Nodes are superpixels, and node attributes are normalizing superpixel centroids by image size and pre-trained MobileNetV2 extracted features.\\
\cline{4-4}
& & & The graph is region adjacency graph.\\
\hline
\multirow{2}{*}{\cite{RN112}} & \multirow{2}{*}{\makecell{Patch\\based}} & \multirow{2}{*}{Segmentation} & Nodes are superpixels, and node attributes are extracted by fully-convolutional network.\\
\cline{4-4}
& & & The edges are determined by the spatial adjacent neighbors, and edge attributes are determined by the similarity between statistical histogram features.\\
\hline
\multirow{2}{*}{\cite{RN113}} & \multirow{2}{*}{\makecell{Patch\\based}} & \multirow{2}{*}{\makecell{LNM\\ Prediction}} & Nodes are image patches, and node attributes are extracted features and closer the feature space to the original one by VAE-GAN.
\\
\cline{4-4}
& & & The edges are determined by calculating the Euclidean distance of node attributes\\
\hline
\multirow{2}{*}{\cite{RN6}} & \multirow{2}{*}{\makecell{Patch\\based}} & \multirow{2}{*}{\makecell{Cancer\\ Type/subtype \\classification}} & Nodes are image patches, and node attributes are extracted by DenseNet.
\\
\cline{4-4}
& & & The graph is fully connected graph.\newline\\
\hline
\multirow{2}{*}{\cite{RN116}} & \multirow{2}{*}{\makecell{Patch\\based}} & \multirow{2}{*}{\makecell{Tumor Stage\\Prediction}} & Nodes are image patches, and node attributes are extracted features of images patch.
\\
\cline{4-4}
& & & The edges are determined by the spatial relationship between nodes.\\
\hline
\multirow{2}{*}{\cite{RN8}} & \multirow{2}{*}{\makecell{Patch\\based}} & \multirow{2}{*}{\makecell{TNM Stage\\Prediction}} & Nodes are image patches, and node attributes are texture feature extraction.
\\
\cline{4-4}
& & & The edges are determined by the connectivity between nodes and their KNN neighbors in a fixed threshold.\\
\hline
\multirow{2}{*}{\cite{RN115}} & \multirow{2}{*}{\makecell{Pixel\\based}} & \multirow{2}{*}{\makecell{Cancer tissue \\and grade \\classification}} & Nodes are image pixels, and node attributes are extracted by ResNet18.
\\
\cline{4-4}
& & & The edges are determined by the connectivity between nodes and their KNN neighbors.\\
\hline
\multirow{2}{*}{\cite{RN124}} & \multirow{2}{*}{\makecell{Patch\\based}} & \multirow{2}{*}{\makecell{Cancer tissue \\and grade \\classification}} & Nodes are image patches, and node attributes are extracted by ResNet18.
\\
\cline{4-4}
& & & The edges are determined by the connection between patch and the centroid node of each dense collagen region. The edge attribute is encoded by the Euclidean distance from each tile to its corresponding centroid.\\
\hline
\multirow{2}{*}{\cite{RN125}} & \multirow{2}{*}{\makecell{Cell\\based}} & \multirow{2}{*}{\makecell{TMA\\Grade \\classification}} & Nodes are nuclei, and node attributes are morphological, texture, and contrastive predictive coding features.
\\
\cline{4-4}
& & & The edge is determined by nodes and their KNN neighbors.\\
\hline
\newpage
\multirow{2}{*}{\cite{RN104}} & \multirow{2}{*}{\makecell{Cell\\based}} & \multirow{2}{*}{\makecell{Cancer \\type/subtype\\classification}} & Nodes are nuclei, and node attributes are concatenated by multiple types of features, including average RGB value, gray level co-occurrence matrix features, VGG19 features, and the number of neighbors of a nucleus.
\\
\cline{4-4}
& & & The edge connections are determined by thresholding the Euclidean distance between nuclei.\\
\hline
\multirow{2}{*}{\cite{RN103}} & \multirow{2}{*}{\makecell{Cell\\based}} & \multirow{2}{*}{\makecell{Cancer \\type/subtype\\classification}} & Nodes are nuclei, and node attributes are 16 hand-craft features and 17 nuclear descriptors.
\\
\cline{4-4}
& & & The edges are determined by the connectivity between nodes and their KNN neighbors.\\
\hline
\multirow{2}{*}{\cite{RN117}} & \multirow{2}{*}{\makecell{Cell\\based}} & \multirow{2}{*}{\makecell{Cancer \\type/subtype\\classification}} & Nodes are nuclei, and node attributes are 16 hand-crafted features.
\\
\cline{4-4}
& & & The edges are determined by thresholding the kNN graph by removing edges longer than a specified distance.\\
\hline
\multirow{2}{*}{\cite{RN119}} & \multirow{2}{*}{\makecell{Cell\\based}} & \multirow{2}{*}{\makecell{Cancer \\type/subtype\\classification}} & Nodes are nuclei, and node attributes are concatenated edge and vertex features of nodes.
\\
& & & The edges are determined by the Euclidean distance between nuclei\\
\hline
\multirow{4}{*}{\cite{RN120}} & \multirow{2}{*}{\makecell{Patch\\based}} & \multirow{4}{*}{\makecell{Cancer \\type/subtype\\classification}} & Nodes are superpixel, and node attributes are features of superpixels.
\\
\cline{4-4}
& & & The graph is region adjacency graph.
\\
\cline{2-2}
\cline{4-4}
& \multirow{2}{*}{\makecell{Cell\\based}}& & Nodes are nuclei, and node attributes are hand-craft features.\\
\cline{4-4}
& & & The edges are determined by the connectivity between nodes and their KNN neighbors.\\
\hline
\multirow{2}{*}{\cite{RN127}} & \multirow{2}{*}{\makecell{Patch\\based}} & \multirow{2}{*}{\makecell{Survival\\prediction}} & Nodes are patches, and node attributes are extracted by VGG16.
\\
\cline{4-4}
& & & The edges are determined by the Euclidean distances between node attributes.\\
\hline
\multirow{2}{*}{\cite{RN128}} & \multirow{2}{*}{\makecell{Patch\\based}} & \multirow{2}{*}{\makecell{Survival\\prediction}} & Nodes are patches, and node attributes are extracted by ResNet50.
\\
\cline{4-4}
& & & The edges are determined by the Euclidean distances between node coordinates.\\
\hline
\multirow{2}{*}{\cite{RN129}} & \multirow{2}{*}{\makecell{Cell\\based}} & \multirow{2}{*}{\makecell{Survival\\prediction}} & Nodes are patches, and node attributes are hand-craft and contrastive predictive coding features.
\\
\cline{4-4}
& & & The edges are determined by the connectivity between nodes and their KNN neighbors.\\
\hline
\multirow{4}{*}{\cite{RN130}} & \multirow{2}{*}{\makecell{Patch\\based}} & \multirow{4}{*}{\makecell{Cancer \\type/subtype\\classification}} & Nodes are patches, and node attributes are image features and cell-based graph representation.
\\
\cline{4-4}
& & & The edges are determined by the connectivity between nodes and their KNN neighbors.
\\
\cline{2-2}
\cline{4-4}
& \multirow{2}{*}{\makecell{Cell\\based}}& & Nodes are nuclei, and node attributes are nuclear morphometry features and imaging features (including intensity, gradient and Haralick features.\\
\cline{4-4}
& & & The edges are determined by the connectivity between nodes and their KNN neighbors.\\
\hline
\multirow{2}{*}{\cite{RN12}} & \multirow{2}{*}{\makecell{Patch\\based}} & \multirow{2}{*}{\makecell{Biomarker\\Prediction}} & Nodes are patches, and node attributes are extracted by ResNet18.
\\
\cline{4-4}
& & & The edges are determined by thresholding the Euclidean distance between node coordinates.\\
\hline
\multirow{2}{*}{\cite{RN135}} & \multirow{2}{*}{\makecell{Cell\\based}} & \multirow{2}{*}{\makecell{Biomarker\\Prediction}} & Nodes are the geometric center of nuclei cluster, and node attributes are determined by the count of the six nuclei types and the standard deviation of nuclear sizes.
\\
\cline{4-4}
& & & The edges are determined by the Delauney triangulation between cluster center with a maximum distance threshold.\\
\hline
\end{longtable}
\section{Other image-based applications}
GCNs have demonstrated their analytical ability in alternative medical image disciplines to facilitate structural analysis of disease diagnosis (e.g., eye disease and skin lesion), surgery scene understanding, and Bone Age Assessment. For instance, GCNs have been studied in dermatology and eye-related diseases, involving retinal, fundus, and fluorescein angiography (FA) images~\citep{RN126,RN137, RN138, RN139}. Similar to radiological and histopathological images, patch-based graph construction strategies are widely used in the above image domains. GCNs have shown to be valuable to learn the vessel shape structures and local appearance for vessel segmentation in retinal images~\citep{RN137}. Also, GCNs were applied to the artery and vein classification by using both fundus images and corresponding fluorescein angiography (FA) images~\citep{RN138}. With a designed graph U-Nets architecture~\citep{RN40}, the high-level connectivity of vascular structures can be learned from node clustering in the node pooling layers. In addition, a study~\citep{RN140} proposed a framework that combines the CNN and ResGCN~\citep{RN85} model to enhance the segmentation performance of fetal head on ultrasound images. Furthermore, GCNs show their power in differential diagnosis of skin conditions using clinical images~\citep{RN139}. This problem is formulated as a multi-label classification task over 80 conditions when only incomplete image labels are available. The label incompleteness is addressed by combining a classification network with a graph convolutional network that characterizes label co-occurrence~\citep{RN139}. Each clinical image is defined as a graph node and the connectivity between two nodes is determined by domain knowledge of skin condition by board-certified dermatologists. It is noteworthy that edge connection is made by inputs from human experts that two dermatologists provide overlapped differential diagnoses groups and connect an edge when two labels appear in at least one differential group by both dermatologists. In addition, a cell-based graph analysis~\citep{RN126} combines multiple types of GCNs with graph poolings, including GIN~\citep{RN20}, GraphSage~\citep{RN21}, and GCN~\citep{RN16} for survival prediction of gastric cancer using immunohistochemistry (mIHC) images. The graph nodes are determined by six antibodies of PanCK, CD8, CD68, CD163, Foxp3, and PD-L1, which were used as annotation indicators for six different types of cells. The node attributes are determined by cell locations, types, and morphological features. The edges are constructed by the maximum effective distance between immune and tumor cells, which is equivalent to 40 pixels in the magnification of this study. Furthermore, a surgery scene graph analysis~\citep{RN141} utilized GraphSage to predict surgical interactions between instruments and surgical regions of interest. In addition, GCNs provide the potential for automatic bone age assessment and ROI score prediction on hand radiograph~\citep{RN142}. This study proposed an anatomy-based group convolution block to predict the ROI scores by processing the local features of ROIs. Also, they presented a dual graph-based attention module to compute the patient-specific attention and context attention for ROI score prediction.
\begin{longtable}{|p{2.7cm}|p{2.2cm}|p{2cm}|p{9cm}|}
\caption{\label{tab3}Summary of GCNs in other image analysis}\\
\hline
Method & Image Type & Tasks & Graph Construction\\
\hline
\endhead
\multicolumn{4}{r}{\footnotesize Continue on the next page
\endfoot
\endlastfoot
\multirow{2}{*}{\cite{RN126}} & \multirow{2}{*}{\makecell{Immuno-\\histochemistry\\(mIHC) images}} & \multirow{2}{*}{\makecell{Survival\\prediction}} & Nodes are cell, and node attributes are determined by the cell locations, optical features of stained cells, and morphology features.\\
\cline{4-4}
& & & The edges are determined by the Euclidean distances between nodes, and edge attributes is calculated by the follow equation:
\begin{equation}
\frac{40}{disttance between cells}
\end{equation}
It will be set to 0 while no interaction between nodes.
\\
\hline
\multirow{2}{*}{\cite{RN137}} & \multirow{2}{*}{Retinal image} & \multirow{2}{*}{\makecell{Vessel\\segmentation}} & Nodes are pixels with maximum vessel probability, and node attributes are extracted by CNN.\\
\cline{4-4}
& & & The edges are determined by thresholding the Geodesic distance between nodes.
\\
\hline
\newpage
\multirow{2}{*}{\cite{RN138}} & \multirow{2}{*}{\makecell{Fundus images\\and \\corresponding\\fluorescein \\angiography \\images}} & \multirow{2}{*}{\makecell{The artery \\ and vein \\classification}} & Nodes are vessel pixels with in $N\times N$ local patches, and node attributes are extracted by graph U-nets.
\newline
\\
\cline{4-4}
& & & The edges are constructed with existing vessel pixels within an $N\times N$ local patch.
\newline
\\
\hline
\multirow{2}{*}{\cite{RN140}} & \multirow{2}{*}{\makecell{Fundus images \\ ultrasound \\images}} & \multirow{2}{*}{\makecell{Optic disc \\and cup \\segmentation\\ and fetal head \\segmentation}} & Nodes are the object boundaries that are divided into N vertices with the same interval, and node attributes are extracted by CNN.\\
\cline{4-4}
& & & The edges are determined by every two consecutive vertices on the boundary and the center vertex are connected to form a triangle.
\\
\hline
\multirow{2}{*}{\cite{RN139}} & \multirow{2}{*}{\makecell{Skin clinical\\images}} & \multirow{2}{*}{\makecell{Skin condition\\classification}} & Nodes are images, and node attributes are extracted by CNN.\\
\cline{4-4}
& & & The edges are connected when two labels appear in at least one differential group by both dermatologists, and edge attributes is calculated by the follow equation:
\begin{equation}
\frac{C(i,j)}{C(i) + C(j)}
\end{equation}
$C(i,j)$ is the number of images have two label at same time. $C(i)$ or ${C(j)}$ is the number of images in class $i$ or $j$.
\\
\hline
\multirow{2}{*}{\cite{RN141}} & \multirow{2}{*}{\makecell{Surgery scene\\images}} & \multirow{2}{*}{\makecell{Surgery scene\\understanding}} & Nodes are surgical tools and defective tissues , and node attributes are extracted by CNN with label smoothing.\\
\cline{4-4}
& & & The edges are determined by the interactions between nodes.
\\
\hline
\multirow{2}{*}{\cite{RN142}} & \multirow{2}{*}{\makecell{Hand radiograph}} & \multirow{2}{*}{\makecell{Bone age \\assessment}} & Nodes are ROIs , and node attributes are extracted by CNN.\\
\cline{4-4}
& & & The edges are determined by the natural connections among ROIs and the full connections among the ROIs in the same anatomy group.
\\
\hline
\end{longtable}
\section{Discussion and future direction}
The rapid growth of GCNs~\citep{RN12, RN113, RN127} and their extensions have been increasingly utilized for processing, integrating, and analyzing multi-modality medical imaging and other types of biological data. We here discuss several future research directions and common challenges to advance the research in medical image analysis and related research fields. We particularly outline key aspects of importance, including GCN model interpretation, the value of pre-training model, evaluation pipeline, large-scale benchmark, and emerging technical insights.
\subsection{Interpretability}
The interpretation of GCNs is of heightened interest to make the outcome understandable, ensure model validity, and enhance clinical relevance. In our focus, a well-designed interpretation framework of GCNs is expected to provide the explanation and visualization for both image-wise and graph components understanding. Such an interpretative ability can be highly attractive to clinicians in the process of diagnosing regions of interest in histopathology, enabling an understanding of spatial and regional interactions from graph structures~\citep{RN12}. As demonstrated, three metrics are useful to design and understand the interpretation capability of GCNs~\citep{RN143}: (1) Fidelity refers to the importance of classification as measured by the impact of node attributes, (2) Contrastively points to the significance with respect to different classes, and (3) Sparsity reflects the sparseness level on a graph. These metrics can help generate and measure the valuable heat maps of graph nodes given their attributes. Representative approaches include gradient class activation mapping (Grad-CAM), contrastive excitation backpropagation (c-EB), and contrastive gradient (CG)~\citep{RN144}. Further, we recognize that emerging studies have explored the specified interpretation strategy for GCNs~\citep{RN117, RN118}. For instance, an ROI-selection pooling layer (R-pool)~\citep{RN143} highlights the node importance for predicting neurological disorders by removing noisy nodes to realize a dimension reduction of the entire graph. Rather than node-level feature interpretation, additional efforts will be greatly needed on interpreting the relational information in graphs. GnnExplainer~\citep{RN118} is an example to leverage the recursive neighborhood-aggregation scheme to identify graph pathways as well as node feature information passing along the edges. The design of GnnExplainer is appealing to visualize the detailed cell-graph structure and provide class-specific interpretation for breast cancer~\citep{RN117}. As a result, we strongly emphasize that the interpretation process considers an in-depth joint understanding of the clinical task, graph model architecture, and model performance.
\subsection{Model pretraining}
Pretraining GCNs aims to train a model on the tasks with a sufficient amount of data and labels and finetune the model into downstream tasks. Pre-trained GCNs can serve as a foundation model to improve the generalization power when the size of the training set is often limited in medical imaging~\citep{RN145}. The pretraining workflow of GCNs typically includes the model training rules, hyper-parameter settings, and constructed-graph augmentation strategies. A key pretraining scheme for a graph-level task is to reconstruct the vertex adjacency information (e.g. GraphSAGE) without hurting intrinsic structural information~\citep{RN146}. We offer several compelling directions of pretraining strategy to improve GCNs model robustness and their utility in different tasks. First, the graph-wise augmentation strategies have a large room to facilitate the pretraining of graphs. For instance, the out-of-distribution samples can be analyzed via node-level and graph-level augmentations~\citep{RN145, RN146}. Second, exploring label-efficient models (e.g., unsupervised or self-supervised learning) in conjunction with pretraining strategies~\citep{RN146} could greatly alleviate the labeling shortage. Notable studies~\citep{RN145, RN146, RN147} have achieved good performance in downstream tasks while leveraging the graph-based pretraining strategies. Considering the above directions, a self-supervised learning framework for GCNs pretraining~\citep{RN146} demonstrates that graph-wise augmentation strategies are useful to address the graph data heterogeneity. The pretraining is performed through maximizing the agreement between two augmented views of the same graph via performing node dropping, edge perturbation, attribute masking, and subgraph selection. Notably, only a small partition of graph components will be changed, meanwhile the semantic meaning of the graph has been preserved. Such a strategy brings graph data diversity that is greatly needed for building robust pre-trained GCN models. Taken together, the research on pretraining GCNs and their practical impact is only to start and will continue to make progress on downstream image-related clinical tasks.
\subsection{Evaluation of graph construction strategies }
The evaluation of graph construction in medical imaging is vital because the associated graph construction could significantly affect the model performance and the interpretation of outcomes. The general graph construction methods used ROIs (e.g., image patches or brain neurons) as graph nodes and the node attributes are obtained by standard feature extractors (e.g., ResNet18). In addition, edges represent the connections between nodes which could be determined by the Euclidean distance between node features, or the connections between ROIs which are determined by patch coordinates or the actual neural fiber connections. Currently, graph construction strategies are applied in different tasks and a generalized graph construction evaluation strategy is not explicitly developed yet. It is even more difficult to determine which kind of graph construction is generalizable for task-specific medical image analysis because of various datasets and graph construction metrics. Also, developing a generalized graph construction evaluation strategy is necessary for GCNs to better process medical image data across multiple modalities because the model performance is highly related to the quality of constructed graph-structured data. The benchmarking framework~\citep{RN158, RN149} has rigorously evaluated the performance of graph neural networks on medium-scale datasets and demonstrates its usefulness for analyzing message-passing capability in GCNs. Also, a comparison strategy among multiple GCNs~\citep{RN133} can address the issues of reproducibility and replicability. Following the graph evaluation~\citep{RN132, RN133}, we need to define statistical distinctions to ensure the performance of GCNs. For example, it is helpful for model training and human understanding if the graph structure and feature distribution differences between positive and negative patient samples are significantly different.
\subsection{Real-world large-scale graph benchmark}
Despite the remarkable effort on standardization of medical imaging cohorts, the high-quality, large-scale graph-defined benchmark has not been readily available for AI model evaluation, especially in medical image analysis. Open Graph Benchmark (OGB) exemplifies the initiative that contains a diverse set of real-world benchmark datasets (e.g., protein, drug, and molecular elements) to facilitate scalable and reproducible graph machine learning research~\citep{RN150}. The number of graphs and nodes in each graph are both massive in OGB. Even small-scale OGB graphs can have more than 100 thousand nodes or more than 1 million edges. This comprehensive dataset in various domains can be viewed as a baseline to support the GCNs’ development and comparison. Related works have been explored on graphs including a chemistry dataset with 2 million graphs and a biology dataset with 395K graphs~\citep{RN145}. As seen in OGB development, there are challenges to collecting and processing suitable medical image datasets and constructing meaningful graphs following the image-to-graph transformation. First, we need to collect a large number of medical images across multiple centers to ensure data diversity. It is also essential to provide detailed annotation information for collected datasets on the image-level region of interest. Second, graph-wise statistics is important to allow measurement of graph-level dynamics. Notable graph metrics~\citep{RN151}, such as the average node degree, clustering coefficient, closeness centrality, and betweenness centrality, can be used to assess graph characteristics and help determine unique graph structures. For instance, the average node degree calculates the average degree of the neighborhood of each node to delineate the connectivity between nodes to their neighbors. The clustering coefficient measures how many nodes in the graph tend to cluster together. Closeness centrality highlights nodes that can easily access other nodes. Third, we must carefully design image-graph components, such as the definition of graph nodes in different types of graphs that are vital to downstream clinical tasks. Finally, the real impact of pre-trained foundation models on large-scale graph-wise datasets still needs to be explored. While using pretraining GCNs to improve data-efficiency issues in medical image analysis, the models can be well-trained on the large-scale graph-wise dataset and adapt into specific tasks, even with a limited size of downstream data.
\subsection{Technological advancements}
The rapid development of deep learning is bringing novel perspectives to address the challenges of graph-based image analysis. The transformer architecture~\citep{RN23}, emphasizing the use of a self-attention mechanism to explore long-range sequential knowledge, emerges to improve the model performance in a variety of natural language processing (NLP)~\citep{RN152} and computer vision tasks~\citep{RN153, RN154}. A graph-wise transformer can be effectively considered to capture both local and global contexts, thus holding the promise to overcome the limitation of spatial-temporal graph convolutions. For example, graph convolutional skeleton transformers integrate both dynamical attention and global context, as well as local topology structure in GCNs~\citep{RN155} while the spatial transformer attention module discovers the global correlations between the bone-connected and the approximated connected joints of graph topology. In medical image analysis, the combination of GCNs and transformer models can be favored to process 3D MRI sequences to boost the model prediction performance, where GCNs explore the topological features while Transformers could model the temporal relationship among MRI sequences. In the meantime, self-supervised learning strategy is emerging in graph-driven analysis with limited availability of imaging data. Notably, self-supervised learning (SSL) provides a means to pretrain a model with unlabeled data, followed by fine-tuning the model for a downstream task with limited annotations~\citep{RN156}. Contrastive learning (CL), as a particular variant of SSL, introduces a contrastive loss to enforce representations to be close for similar pairs and far for dissimilar pairs~\citep{RN156, RN157}. Another technique to address the limitation of data labeling is the advent of self-training learning to generate pseudo-label for model retraining and optimization~\citep{RN158}. A self-training method for MRI segmentation has shown the potential solution for cross-scanner and cross-center data analytical tasks~\citep{RN158}. Also, the teacher-student framework is another type of self-training, which trains a good teacher model with labeled data to annotate the unlabeled data, and finally, the labeled data and data with pseudo-labels can jointly train a student model~\citep{RN131}. Overall, both self-supervised learning and self-training strategies can be utilized in GCNs model training to potentially improve the model performance and overcome the annotation and data scale challenges.
\section{Conclusion}
We have witnessed a growing trend of graph convolutional networks applied to medical image analysis over the past few years. The convergence of GCNs, medical imaging data, and other clinical data, brings advances into outcome interpretation, disease understanding, and novel insights into data-driven model assessment. These breakthroughs, together with data fusion ability, local and global feature inference, and model training efficiency, lead to a wide range of downstream applications across clinical imaging fields. Nevertheless, the development of benchmark graph-based medical datasets is yet to be established. Consistency and validity of graph construction strategy in medical imaging are greatly needed in future research. Recent technological advances can be considered to enhance and optimize GCNs in addressing challenging problems. We hope that the gleaned insights of this review will serve as a guideline for researchers on graph-driven deep learning across medical imaging disciplines and will inspire continued efforts on data-driven biomedical research and healthcare applications.
\bibliographystyle{model2-names.bst}\biboptions{authoryear}
|
2,877,628,088,972 | arxiv | \section{Introduction and
Motivation}\label{introduction-and-motivation}}
Maternal mortality is an important indicator of the health and
development of a country. It is explicitly linked to child health and
development outcomes, and is strongly associated with women's education,
rights and access to adequate healthcare (McAlister and Baskett 2006;
Alvarez et al. 2009; Muldoon et al. 2011; Mbizvo and Say 2012; Alkema et
al. 2016; Briozzo et al. 2016). Reducing maternal deaths is a public
health priority, with Sustainable Development Goal (SDG) 3.1 aiming to
reduce the global maternal mortality ratio (MMR) to less than 70 deaths
per 100,000 live births by the year 2030 (UNDESA 2020). Over the past
several decades, substantial progress towards this goal has been made,
with the global estimate of maternal deaths in 2017 being 35\% lower
than deaths in 2000 (295,000 compared to 451,000) (WHO et al. 2019).
However, substantial disparities exist across countries; in particular,
Sub-Saharan African countries account for around 66\% of all maternal
deaths worldwide.
An important part of understanding differences in levels and trends in
overall mortality is determining the main causes of maternal death.
Understanding the main causes of death would help to guide how best to
improve conditions in regions of the world that are making relatively
slow progress towards the 2030 goals, offering direct insight into how
resources could be most effectively allocated to reduce overall
mortality. As such, the aim of this project was to estimate, for all
regions in the world, the proportion of maternal deaths due to major
causes, including hemorrhage, sepsis, hypertension, embolism, abortion,
indirect causes and other direct causes.
If data existed on every maternal death and the underlying and
associated causes, then it would be a simple tabulation exercise to
calculate the maternal cause of death distributions for regions
worldwide. However, as is the case with most other population statistics
and indicators, the degree of data availability on maternal deaths, and
the quality of such data, varies substantially across countries. One of
the main sources of data on deaths are civil registration and vital
statistics (CRVS) systems, which aim to record all vital events (such as
births and deaths) for a population. While well-functioning CRVS systems
exist in most high-income countries, many lower- and middle-income
countries (LMIC) have no such systems in place. Indeed, only 26\% of the
world's population lives in countries with complete registration of
deaths (CDC 2015). Countries that lack CRVS systems are concentrated in
regions where the overall mortality burden is highest, such as
Sub-Saharan Africa and Southern Asia. In such countries, other data
sources on maternal mortality may be available, such as data from
surveys, or local-level administrative records from hospitals or
community health centers. Such data are generally of lower quality and
unlikely to be representative of the broader population of interest.
Data issues are further exacerbated when considering the estimation of
causes of death. Even if data exist on the total number of maternal
deaths in a population, we may not have information on some or all of
the causes of maternal death. In a particular population or country, the
causes of maternal death reported may vary by year or data source. In
addition, the quality of reporting of causes of death varies
substantially across populations, even within CRVS systems, and is
dependent on factors such as country-level practices and policies, and
physician and health official training (Messite and Stellman 1996;
Salanave et al. 1999; Eriksson et al. 2013; MacDorman et al. 2016).
While cause of death information has a standardized classification
system --- the International Classification of Diseases (ICD), currently
in its 10th revision (ICD-10) --- it is not always the case that deaths
are reported according to ICD-10 classifications, particularly when data
are sourced from surveys or single institution studies.
Data quality and sparsity issues are particularly pertinent in the case
of maternal deaths compared to deaths at other ages (for example, child
mortality). Although the MMR is much higher than is acceptable in many
countries, the absolute number of maternal deaths relative to the number
of live births (the measure of exposure to risk) is relatively small. In
addition, an important cause of maternal death --- deaths due to
abortion --- are substantially under-reported in many countries, due to
definition, cultural and legality issues (Gerdts, Vohra, and Ahern 2013;
Gerdts et al. 2015; Abouchadi, Zhang, and De Brouwere 2018). All these
data characteristics make the goal of estimating a maternal cause of
death distribution particularly challenging.
In this paper we present a Bayesian hierarchical modeling framework to
estimate maternal cause of death distributions in contexts of varying
data availability and quality. The model estimates a set 14
sub-categories within 7 main categories of maternal death, for each
country of interest. The model accounts for varying data quality and
coverage, thereby allowing for many different types of data sources to
be included to inform estimates. The modeling framework can be easily
adapted to estimate maternal cause of death distributions in a variety
of different populations and time periods, in varying data availability
contexts.
The remainder of the paper is structured as follows. We first briefly
discuss the use of Bayesian modeling in global health estimation in
general, and specific efforts to model causes of death. We then outline
the goals of estimation and introduce the 14 different categories that
make up the cause of death distributions. Section \ref{section_data}
outlines the data sources and availability and how these were processed
into the main cause of death categories. Section \ref{section_model}
describes in detail the modeling framework and computational aspects. We
then illustrate some key features of the model through illustrative
results, and present results from validation exercises in Section
\ref{section_results}. Finally, we discuss possible extensions to the
modeling framework and future work in Section \ref{section_discussion}.
\hypertarget{background}{%
\section{Background}\label{background}}
Statistical modeling frameworks used in demographic and global health
estimation have increasingly shifted to using Bayesian methods over the
past decade. The introduction of Bayesian methods by the United Nations
Population Division to estimate world population trends from 2010
(Raftery, Alkema, and Gerland 2014) paved the way for other UN
organizations to follow suit. Many of the global health indicators
related to mortality, health, fertility, and family planning are modeled
using Bayesian methods (Alkema and New 2014; Alkema et al. 2017;
Alexander and Alkema 2018; Cahill et al. 2018; Wakefield et al. 2019).
The majority of models used in these contexts are Bayesian hierarchical
frameworks, with a combination of systematic components capturing
theoretical relationships or relationships with key covariates, and a
temporal component which allows for trends over time to be smoothed and
projected forward. For example, Cahill et al. (2018) presents the Family
Planning Estimation Model (FPEM), a country-level model of contraceptive
use rates among married women of reproductive age from 1990-2020. The
model is based on a logistic curve, which assumes that adoption of
contraception is expected to start slowly, speed up, then slow down
before reaching an asymptote. Specific parameters of the curve are
data-driven and modeled hierarchically, with deviations from the
expected rate of change modeled with an AR(1) process.
Another example is the `Bmat' model, discussed in Alkema et al. (2017),
which describes the estimation method used by the United Nations
Maternal Mortality Estimation Interagency Group (MMEIG) to produce
country-level trends in the all-cause maternal mortality ratio (MMR).
The model consists of a hierarchical model which models the non-HIV/AIDS
MMR as a function of the general fertility rate, average number of
skilled attendants at birth, and gross domestic product. In addition,
country-time-specific deviations are modeled as an ARMA(1,1) process,
allowing for noisy observed trends to be smoothed and projected forward.
The Bmat model is particularly relevant to this work, as the estimates
of all-cause maternal mortality are used as the `envelope' for our
cause-specific estimates, meaning that we constrain the total number of
maternal deaths to be consistent with those produced by the MMEIG.
In general, Bayesian hierarchical models are particularly suited to
problems of estimating global health indicators for a set of multiple
populations with varying data quality and availability. For example, the
use of informative priors, based on substantive and theoretical
knowledge of the process being modeled, is useful to obtain estimates in
populations where the level of missing data are high. In addition,
hierarchical structures allow for information about mortality and other
health trends to be shared across similar populations that may have
varying amounts of data availability. Finally, Bayesian models aid in
the combination of multiple data sources with varying types of data
quality.
\hypertarget{estimation-of-cause-specific-mortality}{%
\subsection{Estimation of cause-specific
mortality}\label{estimation-of-cause-specific-mortality}}
In terms of efforts to model and estimate specific causes of death, and
with multiple causes being estimated in the same model (that is, the
estimation of cause of death distributions), the existing literature
mostly focuses on estimating cause-specific child mortality or maternal
mortality. This is probably partly driven by research agendas in the
Millennium Development (MDG) and Sustainable Development Goal (SDG)
eras, of which two important indicators are child and maternal health
(as part of SDG 3).
A range of different statistical approaches have been used to model
cause of death distributions. For example, Liu et al. (2016) uses a
multinomial logistic regression to model cause proportions of under-five
child mortality, while making various data adjustments to account for
varying data quality and underlying cause profiles. The resulting cause
proportions are then applied to the `envelope' (all-cause) estimates
produced by the methodology described in Alkema and New (2014) to get
death counts. Other approaches to modeling causes of death tend to model
causes separately and then rescale or perform post-hoc adjustments to
ensure the sum of the deaths by causes adds up to some predefined total.
In particular, published estimates of causes specific deaths produced as
part of the Global Burden or Disease Study (Naghavi et al. 2017; Wang et
al. 2017) rely on a multi-stage ensemble modeling technique which models
each cause of death in a similar but separate way. In contrast,
Schumacher et al. (2020) present a model that allows for cause- and
age-specific child mortality to be estimated from sample registration
data, with cause-specific and total deaths being estimated all within
the one framework.
Say et al. (2014) present estimates of maternal deaths by cause based on
a Bayesian hierarchical framework which models the underlying
proportions of death as a multivariate logistic Normal distribution.
Assumptions about data quality and coverage are based on the source of
the data, and HIV/AIDS maternal deaths are modeled separately. The model
we propose in this paper improves on Say et al. (2014) in several ways;
in particular, we explicitly account for the type of data,
down-weighting data sources of lower quality, and include a much larger
data set of observations of causes of maternal deaths, including those
from subnational sources, single-causes studies, and potentially
multiple data sources for the same country-year.
\hypertarget{goals-of-estimation}{%
\section{\texorpdfstring{Goals of estimation
\label{section_goals}}{Goals of estimation }}\label{goals-of-estimation}}
A maternal death is defined by the World Health Organization as `the
death of a woman while pregnant or within 42 days of termination of
pregnancy, irrespective of the duration and site of the pregnancy, from
any cause related to or aggravated by the pregnancy or its management
but not from accidental or incidental causes' (WHO 2020). We are
interested in obtaining estimates of the proportion of total maternal
deaths in a particular country and year by each of the 7 main cause
categories (as defined below), including additional breakdowns of 3 main
categories into sub-cause categories. Once estimates of the cause of
death distribution for each country-year are obtained, they are
aggregated across both geographic and time dimensions to obtain
estimates of cause of death distribution by SDG region for a specific
time period of interest (2009-2017).
\hypertarget{main-cause-of-death-categories}{%
\subsection{\texorpdfstring{Main cause of death categories
\label{subsection_causes}}{Main cause of death categories }}\label{main-cause-of-death-categories}}
There are many potential cause of death group classifications that we
could consider estimating. Motivated by previous work and priorities of
health policy agendas (Say et al. 2014), we estimate the proportion of
maternal deaths from each of the following seven cause categories:
\begin{itemize}
\tightlist
\item
Abortion (ABO)
\item
Embolism (EMB)
\item
Hypertension (HYP)
\item
Hemorrhage (HEM)
\item
Sepsis (SEP)
\item
Other direct causes (DIR)
\item
Other indirect causes (IND)
\end{itemize}
Of particular note are maternal deaths related to HIV/AIDS, which are
encompassed in the indirect causes category. However, due to the
substantially different epidemiological profile of this death category,
as well as vastly different trends over time, we follow previous work on
mortality estimation (Alkema and New 2014; Say et al. 2014; Alkema et
al. 2017) in restricting our goal to be estimating the proportion of
non-HIV/AIDS maternal deaths by cause. Estimates of HIV/AIDS maternal
deaths produced elsewhere (UNAIDS 2017; WHO et al. 2019) are then
incorporated into the final cause of death distributions (as part of the
indirect cause group). See Section \ref{section_hiv} for more details.
\hypertarget{cause-of-death-subcategories}{%
\subsection{\texorpdfstring{Cause of death subcategories
\label{subsection_subcauses}}{Cause of death subcategories }}\label{cause-of-death-subcategories}}
In addition to the 7 main cause of death categories listed above, we are
interested in estimating further sub-cause breakdowns of hemorrhage,
sepsis and other direct cause groups. For hemorrhage (HEM) and sepsis
(SEP) deaths, we estimate the following breakdowns based on the timing
of death:
\begin{itemize}
\tightlist
\item
Ante-partum (ANT)
\item
Intra-partum (INT)
\item
Post-partum (POS)
\end{itemize}
For other direct maternal deaths (DIR), we further break down the
category into four sub-categories:
\begin{itemize}
\tightlist
\item
Anesthesia (ANE)
\item
Obstructed labour (OBS)
\item
Obstetric trauma (OBT)
\item
Other causes (OTH)
\end{itemize}
Thus, a total of 14 separate causes of death categories are estimated.
Figure \ref{fig:data_diagram} summarizes the classification of data
available on maternal deaths, and the cause of death categories to be
estimated. Maternal deaths are first classified as either deaths due to
HIV/AIDS or non-HIV/AIDS. Of the non-HIV/AIDS deaths, these can be
further classified into one of the 7 main cause of death categories. For
deaths due to hemorrhage (HEM), sepsis (SEP), and other direct causes
(DIR), we further classify into one of 10 sub-cause categories. In each
stage of the classification process, if no further information is
available, these deaths are excluded from the analysis.
\begin{figure}
\scalebox{0.8}{
\linespread{1}
\begin{tikzpicture}
\node (mat) at (9, 6) [rectangle, draw = black, fill = white] {maternal deaths};
\node (nonaids) at (9, 3.5) [rectangle, draw = black, fill = white] {non-HIV/AIDS};
\node (aids) at (18, 3.5) [rectangle, draw = black, fill = white] {HIV/AIDS};
\draw[thick, ->] (mat.south) -- (nonaids.north);
\draw[thick, ->] (mat.south) -- (aids.north);
\draw[red, thick, ->] (nonaids.south) -- (9, .5);
\node(nfi1) at (3, 4.5) [rectangle, dashed, draw = black, fill = none, align = center] { no further \\information};
\draw[dashed, ->] (mat.south) -- (nfi1.north);
\node(nfi2) at (5, 1.5) [rectangle, dashed, draw = black, fill = none, align = center] {no further \\information};
\draw[dashed, ->] (nonaids.south) -- (nfi2.north);
\node at (1.5, 0.8) [text = red] {main category distribution};
\fill [red!30] (-1, -.5) rectangle (19, .5);
\node (ABO) at (0, 0) [rectangle, draw=black, fill=white] {ABO};
\node (DIR) at (3, 0) [rectangle, draw=black, fill=white] {DIR};
\node (EMB) at (6, 0) [rectangle, draw=black, fill=white] {EMB};
\node (HEM) at (9, 0) [rectangle, draw=black, fill=white] {HEM};
\node (HYP) at (12, 0) [rectangle, draw=black, fill=white] {HYP};
\node (SEP) at (15, 0) [rectangle, draw=black, fill=white] {SEP};
\node (IND) at (18, 0) [rectangle, draw=black, fill=white] {IND};
\draw[thick, ->] (aids.south) -- (IND.north);
\node at (3.5, -5.5) [text = orange] {DIR subdistribution};
\fill [orange!30] (2.6, -1) rectangle (4.4, -5);
\node (DIRobs) at (3.5, -1.5) [rectangle, draw=black, fill = white] {DIR\textsubscript{obs}};
\node (DIRane) at (3.5, -2.5) [rectangle, draw=black, fill = white] {DIR\textsubscript{ane}};
\node (DIRobt) at (3.5, -3.5) [rectangle, draw=black, fill = white] {DIR\textsubscript{obt}};
\node (DIRoth) at (3.5, -4.5) [rectangle, draw=black, fill = white] {DIR\textsubscript{oth}};
\draw[orange, thick, ->] (DIR.south) -- (3.5, -1);
\node(nfi3) at (1, -2) [rectangle, dashed, draw = black, fill = none, align = center] {no further \\information};
\draw[dashed, ->] (DIR.south) -- (nfi3.north);
\node at (9.5, -4.5) [text = OliveGreen] {HEM subdistribution};
\fill [OliveGreen!30] (8.5, -1) rectangle (10.5, -4);
\node (HEMante) at (9.5, -1.5) [rectangle, draw=black, fill = white] {HEM\textsubscript{ante}};
\node (HEMintra) at (9.5, -2.5) [rectangle, draw=black, fill = white] {HEM\textsubscript{intra}};
\node (HEMpost) at (9.5, -3.5) [rectangle, draw=black, fill = white] {HEM\textsubscript{post}};
\draw[OliveGreen, thick, ->](HEM.south) -- (9.5, -1);
\node(nfi4) at (7, -2) [rectangle, dashed, draw = black, fill = none, align = center] {no further \\information};
\draw[dashed, ->] (HEM.south) -- (nfi4.north);
\node at (15.5, -4.5) [text = blue] {SEP subdistribution};
\fill [blue!30] (14.5, -1) rectangle (16.5, -4);
\node (SEP_ante) at (15.5, -1.5) [rectangle, draw = black, fill = white] {SEP\textsubscript{ante}};
\node (SEP_intra) at (15.5, -2.5) [rectangle, draw = black, fill = white] {SEP\textsubscript{intra}};
\node (SEP_post) at (15.5, -3.5) [rectangle, draw = black, fill = white] {SEP\textsubscript{post}};
\draw[blue, thick, ->] (SEP.south) -- (15.5, -1);
\node(nfi5) at (13, -2) [rectangle, dashed, draw = black, fill = none, align = center] {no further \\information};
\draw[dashed, ->] (SEP.south) -- (nfi5.north);
\end{tikzpicture}
}
\caption{Organization of maternal deaths into analysis categories. Non-HIV/AIDS deaths are classified into 7 main cause of death categories, of which hemorrhage (HEM), sepsis (SEP), and other direct causes (DIR) are classified further. After estimation of the main distribution using only non-HIV/AIDS observations, externally obtained estimates of HIV/AIDS are added to the other indirect causes (IND) category to give the complete distribution.}
\label{fig:data_diagram}
\end{figure}
\hypertarget{data}{%
\section{\texorpdfstring{Data \label{section_data}}{Data }}\label{data}}
\hypertarget{overview-of-data-sources}{%
\subsection{Overview of data sources}\label{overview-of-data-sources}}
Data on maternal deaths by cause come from three main sources: Civil
Registration and Vital Statistics (CRVS) systems; `grey literature,'
which refers to government reports, technical reports and other
non-peer-reviewed publications; and `studies,' which were the result of
a large-scale systematic literature review. CRVS systems and grey
literature provide national-level data. Data from studies, on the other
hand, can be from a number of different geographic levels. While some
may be at the national level, many refer to subnational areas, for
example Administrative 1 level (ADM1, state/province), Administrative 2
level (county), or lower. Indeed, in many cases, studies may only report
observation from a single hospital or group of hospitals or other health
institutions.
\hypertarget{overview-of-data-availability}{%
\subsection{Overview of data
availability}\label{overview-of-data-availability}}
\begin{table}[!h]
\caption{\label{tab:data_table_1}Breakdown of data availability by SDG region and source.}
\centering
\resizebox{\linewidth}{!}{
\begin{tabular}[t]{>{\raggedright\arraybackslash}p{4cm}>{\raggedleft\arraybackslash}p{2cm}>{\raggedleft\arraybackslash}p{2cm}>{\raggedleft\arraybackslash}p{2cm}>{\raggedleft\arraybackslash}p{2cm}>{\raggedleft\arraybackslash}p{2cm}>{\raggedleft\arraybackslash}p{2cm}>{\raggedleft\arraybackslash}p{2cm}}
\toprule
\multicolumn{3}{c}{ } & \multicolumn{4}{c}{Number of observed country-years by data sources} & \multicolumn{1}{c}{ } \\
\cmidrule(l{3pt}r{3pt}){4-7}
SDG Region & Number of countries & Number of observed country-years & CRVS & Grey Literature & Studies ADM1 or higher & Studies Below ADM1 & Prop. country-years missing 2+ causes\\
\midrule
\cellcolor{gray!6}{Central and Southern Asia} & \cellcolor{gray!6}{12} & \cellcolor{gray!6}{122} & \cellcolor{gray!6}{31} & \cellcolor{gray!6}{16} & \cellcolor{gray!6}{2} & \cellcolor{gray!6}{73} & \cellcolor{gray!6}{0.533}\\
Europe and Northern America & 36 & 260 & 252 & 8 & 0 & 0 & 0.738\\
\cellcolor{gray!6}{Northern Africa and Western Asia} & \cellcolor{gray!6}{18} & \cellcolor{gray!6}{91} & \cellcolor{gray!6}{81} & \cellcolor{gray!6}{6} & \cellcolor{gray!6}{0} & \cellcolor{gray!6}{4} & \cellcolor{gray!6}{0.538}\\
Oceania excl. Australia and New Zealand & 3 & 7 & 4 & 1 & 0 & 2 & 0.714\\
\cellcolor{gray!6}{Sub-Saharan Africa} & \cellcolor{gray!6}{28} & \cellcolor{gray!6}{151} & \cellcolor{gray!6}{18} & \cellcolor{gray!6}{20} & \cellcolor{gray!6}{15} & \cellcolor{gray!6}{98} & \cellcolor{gray!6}{0.642}\\
\addlinespace
Latin America and the Caribbean & 31 & 218 & 207 & 10 & 0 & 1 & 0.339\\
\cellcolor{gray!6}{Australia and New Zealand} & \cellcolor{gray!6}{2} & \cellcolor{gray!6}{26} & \cellcolor{gray!6}{16} & \cellcolor{gray!6}{10} & \cellcolor{gray!6}{0} & \cellcolor{gray!6}{0} & \cellcolor{gray!6}{0.846}\\
Eastern and South-Eastern Asia & 12 & 75 & 46 & 9 & 2 & 18 & 0.467\\
\bottomrule
\end{tabular}}
\end{table}
Over the period of interest (2009--2017), at least one observation of
cause-specific maternal mortality were available for 142 of the 183 UN
member countries. Table \ref{tab:data_table_1} provides a breakdown of
the available data by SDG region. The majority of available country-year
observations, roughly 70\%, come from CRVS data. Most notably these are
observations from Europe and Northern America or Latin America and the
Caribbean. In other regions, data from the grey literature and studies
play a particularly important role. For instance, as Table
\ref{tab:data_table_1} demonstrates, the vast majority of available data
for Sub-Saharan Africa and Central and Southern Asia come from various
subnational studies.
It is worth noting that it is not uncommon for a country-year to report
maternal mortality counts for only a subset of the cause categories that
we are interested in estimating. As illustrated in the rightmost column
of Table \ref{tab:data_table_1}, the majority of the country-year
observations in each region have two or more causes missing.
\hypertarget{classification-into-cause-of-death-groups-and-subgroups}{%
\subsection{Classification into cause of death groups and
subgroups}\label{classification-into-cause-of-death-groups-and-subgroups}}
Where possible, deaths are assigned into one of the 14 cause categories
listed in Section \ref{subsection_causes} and \ref{subsection_subcauses}
above. This is straightforward when deaths are classified according to
ICD-10 codes, which is the case for the majority of data (encompassing
all CRVS and some studies and grey literature data). The assignment of
ICD-10 codes to each of the 14 cause categories is summarized in
Appendix \ref{appendix_icd10}.
However, for the studies and grey literature data, classification into
cause categories is more complex. In many studies the cause of death is
referred to using a free text description, rather than an ICD-10 code.
Sometimes the free text can be clearly translated into an ICD-10 code or
be identified as one of the broader level ICD-10 cause groups defined by
the WHO (WHO 2012) and, hence, designated into one of the seven main
cause categories. Other times, the free text describes more than a
single ICD-10 code or cause group associated with a number of maternal
deaths. In such cases we rely on the clinician's expert interpretation
of the correct proportion of maternal deaths corresponding to each code
or cause group. Cases where the cause of death could not be assigned
(for example, instances where only the likely organ system causing the
death could be identified from free text) were excluded from the
analysis.
\hypertarget{other-data-sources}{%
\subsection{Other data sources}\label{other-data-sources}}
In addition to data on causes of maternal death, we make use of a number
of other data sources in the model. We use annual all-cause female
mortality estimates from the UN World Population Prospects (WPP) 2019
edition to assess the coverage, and subsequently the `usability,' of
death data observed by country (UNPD 2019). We also employ the UN
Maternal Mortality Estimation Inter-Agency Group's (MMEIG) 2019
estimates on maternal mortality to obtain an estimate for the total
number of maternal deaths, and the number of maternal deaths due to
HIV/AIDS (WHO et al. 2019).
Additional information on data adjustments related to the treatment of
zero reported deaths, HIV/AIDS deaths, and multi-year studies are
detailed in Appendix \ref{appendix_data_adjustments}.
\hypertarget{model}{%
\section{\texorpdfstring{Model
\label{section_model}}{Model }}\label{model}}
\hypertarget{overview}{%
\subsection{Overview}\label{overview}}
We begin with grouping the observed non-HIV/AIDS cause-specific death
counts for each country into 7 main cause of death categories, and we
model these as coming from a multinomial distribution with 7 categories.
The relative proportions in the multinomial distribution then depend
primarily on regional and country-level differences. Additionally, we
include an error term applied only to lower quality data to allow for
deviation from the assumed true country mean to lessen the impact of
potentially misrepresentative observations.
As detailed in Section \ref{section_goals}, subdistributions for certain
categories are also calculated. The DIR, HEM, and SEP counts are further
divided into separate multinomial distributions representing more
granular classifications of cause of death. Similar to the main cause
distribution, these are modeled as the sum of regional and country-level
differences.
The true main cause of death distribution and the subcause distributions
for each country are calculated using the estimated regional and a
weighting of the estimated country-level differences. In order to
reflect increased uncertainty in countries where data coverage is low,
the weight given to the country's specific estimated term depends on the
coverage of the available data in that country.
\hypertarget{notation}{%
\subsection{Notation}\label{notation}}
For \(i = 1, \dots, N\), let
\(\mathbf{y_i} = (y_{i,1}, \dots, y_{i,7})\) be the observed
non-HIV/AIDS maternal deaths by cause, where \(y_{i,j}\) is the number
of deaths in the \(i\)th observation attributed to cause group \(j\),
for \(j \in \{\text{ABO, EMB, HEM, SEP, DIR, IND, HYP}\}.\) Define the
total number of observed non-HIV/AIDS maternal deaths for the \(i\)th
observation to be \(d_i\).
Point estimates from the MMEIG of the HIV/AIDS-omitted total number of
maternal deaths in a country-year \(ct\) are denoted as
\(\mathring{d}_{ct}\).
\hypertarget{model-for-observed-cause-proportions}{%
\subsection{Model for observed cause
proportions}\label{model-for-observed-cause-proportions}}
Deaths in each observation are modeled using a multinomial distribution.
The log-ratios of proportions are taken to be the sum of an intercept,
region effect, and country effect. However, for observations from CRVS
or studies data, an additional adjustment is applied to allow for
deviation from the true country mean. This adjustment allows us to
partially mitigate the impact of potentially misrepresentative
observations that arise from lower quality data.
For observation \(i\), the multinomial proportions
\((p_{i,1}, \dots, p_{i, 7})\) are modeled as \begin{equation*}
\begin{gathered}
\mathbf{y_i} \sim \text{Multinomial}(d_i, \mathbf{p_i}) \\
\mathbf{p_i} = (p_{i,1}, \dots, p_{i, 7}) \\
\log \left(\frac{p_{i,j}}{p_{i,7}}\right) = \beta_{0,j} + \beta_{r(c(i)), j} + u_{c(i),j} + q_{i,j}, \\
\end{gathered}
\end{equation*} where \(c(i)\) refers to country of observation \(i\),
and \(r(c(i))\) corresponds to the region of that country. The log-ratio
of proportions for category \(j\) relative to category 7 (HYP) is taken
to be the sum of an intercept term \(\beta_{0, j}\), a region effect
\(\beta_{r(c(i)), j}\), a country effect \(u_{c(i),j}\) and a data
quality adjustment term \(q_{i,j}\).
The region effects are pooled with a global variance term. For all
regions \(r = 1, \dots, R\), \begin{equation*}
\begin{gathered}
\beta_{r,j} \sim \text{Normal} (0, \sigma_{\beta}^2) \\
\sigma_{\beta} \sim \text{Normal} (0, 1^2).
\end{gathered}
\end{equation*} The regions \(r = 1, \dots, R\) are the same as those
used by Say et al. (2014), which broadly aim to group countries that are
believed to be epidemiologically similar. This region classification is
distinct from the SDG region system used for aggregating and reporting
results. More details on the modeling region classification are given in
Appendix \ref{appendix_regions}.
\hypertarget{multinomial-likelihood-and-missing-values}{%
\subsubsection{Multinomial likelihood and missing
values}\label{multinomial-likelihood-and-missing-values}}
In an ideal situation where counts for all categories are recorded, each
\(\mathbf{y_i}\) is treated as a 7-category multinomial observation with
probabilities \((p_{i, 1}, \dots, p_{i, 7})\). For observation \(i\),
let the ratio of proportion \(j\) to the reference category be denoted
\(g_{ij}\). That is, \[
\frac{p_{i,j}}{p_{i,7}} = g_{i,j} = \exp(\beta_{0,j} + \beta_{r(c(i)), j} + u_{c(i),j} + q_{i,j}).
\] The probabilities can then be expressed \[
p_{i,j} = \frac{g_{i,j}}{\sum_{k=1}^7 g_{i,k}},
\] and the corresponding multinomial likelihood \(L_M\)for all \(N\)
observations is \begin{equation}
\label{eq:full-multinomial}
L_M = \prod_{i=1}^N \prod_{j=1}^7 p_{i,j}^{y_{i,j}} = \prod_{i=1}^N \prod_{j=1}^7 \left(\frac{g_{i,j}} {\sum_{k=1}^7 g_{i,k} } \right)^{y_{ij}}.
\end{equation}
However, as stated in Section \ref{section_data}, there are observations
for which certain categories' counts are considered to be missing. In
such cases where we believe an apparent zero count is some category
\(k\) is unreliable, we wish to treat \(y_{i, k}\) as unknown instead.
We can accomplish this by treating the observation as a multinomial
observation with a reduced number of categories. For example, if \(j=1\)
(ABO) is missing for observation \(i\), then the likelihood contribution
for that observation would instead be \[
\prod_{j=2}^7 \grave{p}_{i,j}^{y_{i,j}} = \prod_{j=2}^7 \left(\frac{g_{i,j}} {\sum_{k=2}^7 g_{i,k} } \right)^{y_{ij}}.
\] where the probabilities \((\grave{p}_{2,j}, \dots, \grave{p}_{i,7})\)
are the original probabilities rescaled to sum to 1.
An appropriately reduced multinomial is used for every observation with
any combination of missing categories.
\hypertarget{country-specific-effect-and-correlations-across-causes}{%
\subsubsection{Country-specific effect and correlations across
causes}\label{country-specific-effect-and-correlations-across-causes}}
We allow for the possibility of correlations between death categories in
countries' cause of death distributions. Intuitively, because of
co-morbidities and common, systematic underlying factors affecting
maternal health, we would expect certain death categories to co-occur.
In order to capture correlations in countries' cause of death
distributions, the country effects \(u_{c,j}\) are modeled as
multivariate normal with a common 6x6 covariance matrix \(\Sigma\). We
decompose \(\Sigma\) into its correlation matrix \(\Omega\) and diagonal
matrices of variance terms,
\(\Sigma = \text{diag}(\mathbf{v})\, \Omega \, \text{diag} (\mathbf{v})\).
An \(\text{LKJ}(1)\) prior is used for \(\Omega\) (Lewandowski,
Kurowicka, and Joe 2009; Stan Development Team 2019) . For each country
\(c = 1, \dots, C\), \begin{equation*}
\begin{gathered}
(u_{c,1}, \dots, u_{c, 6}) \sim \text{MVN} (\mathbf{0}, \Sigma) \\
\Sigma = \text{diag}(\mathbf{v}) \Omega \text{diag} (\mathbf{v}) \\
\Omega \sim \text{LKJ}(1) \\
\mathbf{v} \sim \text{Normal} (0, 3).
\end{gathered}
\end{equation*}
\hypertarget{data-quality-adjustment}{%
\subsubsection{\texorpdfstring{Data-quality adjustment
\label{subsection_model_dataquality}}{Data-quality adjustment }}\label{data-quality-adjustment}}
Even where cause of death data are available, the data may be
misrepresentative due to systematic deficiencies and biases in data
collection processes. In our model, we classify observations into 4 data
quality `types', and incorporate this information into the model by
including an extra error term for observations from lower quality types,
partially mitigating the impact of these observations on the country
means.
The data quality type classification of each observation depends on the
source and estimated coverage of the observation. Type 1 observations
consist only of observations arising from grey literature. These are
considered to be of the highest quality, and no additional adjustment is
applied to these data.
Observations collected from studies are assigned to be Type 4, the
lowest quality, since the included studies may from subnational regions,
and also may not aim to be a comprehensive surveying of all causes of
maternal death.
Observations from CRVS data can be classified as Type 2, 3, or 4,
depending on the estimated coverage of the data. We follow a similar
approach to Say et al. (2014) in defining a `usability' index, which is
a function of the presence of ill-defined deaths, the coverage of the
observed number of deaths, and the presence of contributory-cause
misclassification.
For observations where the number of observed maternal deaths is no more
than 5, the usability index is calculated using the proportion of
ill-defined deaths denoted \(p_{i}^{\text{ill}}\), and the all-cause
(maternal and otherwise) female death coverage \(C_i\) of the
observation. This is calculated as the ratio of all-cause death count in
the CRVS system to the all-cause death count estimate from WPP 2019
(UNPD 2019). In particular, the usability for CRVS observation \(i\),
\(\nu_i\) is calculated as \[
\nu_i = \frac{d_i}{\mathring{d}_{ct(i)}} (1 - p_{i}^{\text{ill}}).
\] If there are more than 5 maternal deaths in an observation, then we
also consider the proportion \(p_{i}^{\text{contr}}\) of maternal deaths
attributed to contributory causes. Maternal deaths should always be
classified with a main ICD-10 code that is an underlying cause of death,
not a contributory cause of death. If contributory causes are listed as
the main cause of death, then this suggests CRVS systems of a lower
quality. In particular, where there are more than 5 maternal deaths
observed, \(\nu_i\) is calculated as \[
\nu_i = \frac{d_i}{\mathring{d}_{ct(i)}} (1 - p_{i}^{\text{ill}}) (1 - p_{i}^{\text{contr}})
\] The usability index \(\nu_i\) is then used to classify CRVS data into
types 2, 3, or 4 as follows:
\begin{itemize}
\tightlist
\item
the observation is classified as type 2 if \(\nu_i > 85\%\) and is one
of three consecutive years with \(\nu_i > 60 \%\),
\item
type 3 if \(65\% < u_i \leq 85\%\) and is one of three consecutive
years with \(u_i > 60 \%\), and
\item
type 4 otherwise.
\end{itemize}
\begin{table}
\caption{\label{tab:unnamed-chunk-2}Summary of data quality type classification. Highest quality data is classified as type 1, and lowest quality data is classified as type 4. Data quality is assessed based on data source and a calculated usability index.}
\centering
\fontsize{10}{12}\selectfont
\begin{tabular}[t]{rl}
\toprule
Type & Sources\\
\midrule
\cellcolor{gray!6}{1} & \cellcolor{gray!6}{most grey literature observations}\\
2 & some grey lterature observations; high usability CRVS\\
\cellcolor{gray!6}{3} & \cellcolor{gray!6}{medium usability CRVS}\\
4 & low usability CRVS; studies\\
\bottomrule
\end{tabular}
\end{table}
For observations arising from type 2, 3, or 4 data sources, a data
adjustment term \(q_{i,j}\) is individually realized from a Normal
distribution for each observation, with an estimated variance term
dependent on the type of data source. For each observation \(i\) and
cause \(j\), \begin{equation*}
\begin{gathered}
q_{i, j}
\begin{cases}
= 0 & \text{if type}(i) = 1 \\
\sim \text{Normal} (0, \sigma^2_{\text{type}(i)}) & \text{if type}(i) = 2, 3, 4
\end{cases} \\
\sigma_{\text{type}} \sim \text{Normal} (0, 0.25^2).
\end{gathered}
\end{equation*}
\hypertarget{estimation-of-true-proportions}{%
\subsection{Estimation of true
proportions}\label{estimation-of-true-proportions}}
Although some observations may have large death counts, they may only
represent a fraction of the true number of maternal deaths in that
country-year, which we take to be the MMEIG point estimate
\(\mathring{d}_{ct}\). In these cases where a large fraction of the
deaths are unrecorded or unclassified, we may have an unduly high degree
of certainty in our estimate of \(p_{i,j}\). We therefore apply a
weighting scheme to better reflect our uncertainty where data coverage
is low. For country \(c\), let \(w(c) \in [0,1]\) be the maximum
coverage among its observations. If \(ct(i)\) refers to the country-year
of observation \(i\), then \[
w(c) := \max_{i:\,c(i) = c} \frac{d_i}{\mathring{d}_{ct(i)}}.
\] The weight \(w(c)\) is then used to regulate the contribution of the
estimated country effect \(u_{c,j}\). For countries with only
low-coverage observations (\(w(c)\) is low), the country-specific
information is considered incomplete, and the estimate should therefore
lean more heavily towards the region mean rather than the observed
country mean. The estimate of the true HIV/AIDS-omitted cause of death
distribution is calculated as \begin{equation*}
\begin{gathered}
\log \left(\frac{p^\ast_{c,j}}{p^\ast_{c,7}}\right) = \beta_{0,j} + \beta_{r(c), j} + w_{c} \cdot u_{c,j} + (1-w_{c}) \cdot \tilde{u}_{c, j} \\
\tilde{\mathbf{u}}_{c} \overset{\text{RNG}}{\sim} \text{MVN} (0, \Sigma),
\end{gathered}
\end{equation*} where \(\tilde{\mathbf{u}}_c\) is a new generic
realization of the country effect.
\hypertarget{incorporating-hivaids-deaths}{%
\subsection{\texorpdfstring{Incorporating HIV/AIDS deaths
\label{section_hiv}}{Incorporating HIV/AIDS deaths }}\label{incorporating-hivaids-deaths}}
Deaths from HIV/AIDS are treated separately. Recall that
\(\mathring{d}_{ct}\) denotes the HIV/AIDS-omitted MMEIG estimate of
maternal deaths in country-year \(ct\). Let
\(\mathring{d}_{ct, \text{HIV}}\) denote the MMEIG point estimate of the
number of HIV/AIDS deaths in country-year \(ct\), and let
\(\mathring{p}_{ct, \text{HIV}} = \mathring{d}_{ct, \text{HIV}} / ( \mathring{d}_{ct, \text{HIV}} + \mathring{d}_{ct})\)
denote the MMEIG-estimated proportion of maternal deaths attributed to
HIV/AIDS.
Similarly, let
\(\hat{p}_{i, \text{HIV}} = d_{i, \text{HIV}} / (d_i + d_{i, \text{HIV}})\),
where \(d_i\) and \(d_{i, \text{HIV}}\) denote the analogous quantities
from observation \(i\).
Define
\(\sigma_{\text{HIV}} = \text{sd}(\hat{p}_{i, \text{HIV}} - \mathring{p}_{ct(i), \text{HIV}})\).
Then \begin{equation*}
\begin{gathered}
p'_{ct, \text{HIV}} \overset{\text{RNG}}{\sim} \text{Normal} (\mathring{p}_{ct, \text{HIV}}, (\mathring{p}_{ct, \text{HIV}} \cdot \sigma_{\text{HIV}})^2) T(0, 1) \\
d'_{ct, \text{HIV}} = p'_{ct, \text{HIV}} \cdot (d_{ct, \text{HIV}} + \mathring{d}_{ct}). \\
\end{gathered}
\end{equation*}
We then add the HIV/AIDS deaths \(d'_{ct, HIV}\) to the to the IND
group, and recalculate proportions to obtain the final
HIV/AIDS-inclusive country-year distributions
\((p'_{ct, 1}, \dots, p'_{ct, 7})\). We do this by converting the
HIV/AIDS-omitted proportions \(p^\ast_{c,j}\) into counts
\(d^\ast_{ct,j}\), adding the estimated HIV/AIDS counts appropriately,
and rescaling these counts into proportions: \begin{equation*}
\begin{gathered}
d^\ast_{ct, j} = p^\ast_{c,j} \cdot \mathring{d}_{ct} \\
d'_{ct, j} =
\begin{cases}
d^\ast_{ct, j} + d'_{i, \text{HIV}} & \text{if } j=\text{IND}\\
d^\ast_{ct, j} & \text{otherwise} \\
\end{cases} \\
p'_{ct,j} = \frac{d'_{ct,j}}{\sum_{l=1}^7 d'_{ct,l}}.
\end{gathered}
\end{equation*}
\hypertarget{calculating-regional-and-global-cause-of-death-distributions}{%
\subsection{Calculating regional and global cause of death
distributions}\label{calculating-regional-and-global-cause-of-death-distributions}}
Regional cause of death distributions are obtained by aggregating
country death counts. For (SDG) regions \(h = 1, \dots, H\) (distinct
from regions \(r = 1, \dots, R\) used above in the model), the
countries' HIV/AIDS-inclusive counts \(d'_{ct, j}\) are aggregated
accordingly to obtain regional counts for each cause \[
d'_{h, j} = \sum_{h(c) = h} d'_{ct, j}.
\] The counts are then normalized to give proportions \[
p'_{h,j} = \frac{d'_{h, j}}{\sum_{l=1}^7 d'_{h, l}}.
\]
Similarly, global estimates are obtained by aggregating regional death
counts and normalizing \begin{equation*}
\begin{gathered}
d'_{\text{global}, j} = \sum_{h=1}^H d'_{h, j} \\
p'_{\text{global},j} = \frac{d'_{\text{global}, j}}{\sum_{l=1}^7 d'_{\text{global}, l}}
\end{gathered}
\end{equation*}
\hypertarget{subcause-distribution-estimation}{%
\subsection{Subcause distribution
estimation}\label{subcause-distribution-estimation}}
Within the cause groups HEM, SEP, and DIR, we are interested in a finer
classification of cause of death. For each of these three main
categories \(j\), let \(K_j\) denote the number of subcategories. These
categories can be further subdivided as follows.
\begin{enumerate}
\def\arabic{enumi}.{\arabic{enumi}.}
\tightlist
\item
Within the hemorrhage (HEM) category, deaths can be classified as
antepartum (HEM\textsubscript{ante}), intrapartum
(HEM\textsubscript{intra}), and postpartum (HEM\textsubscript{post})
hemorrhage. \(K_{\text{HEM}} = 3\).
\item
The sepsis (SEP) category can similarly be divided into antepartum
(SEP\textsubscript{ante}), intrapartum (SEP\textsubscript{intra}), and
postpartum sepsis (SEP\textsubscript{post}). \(K_{\text{SEP}} = 3\)
\item
The direct cause (DIR) category can be divided into
(DIR\textsubscript{obs}), (DIR\textsubscript{ane}),
(DIR\textsubscript{obt}), and other direct causes
(DIR\textsubscript{oth}). \(K_{\text{DIR}} = 4\).
\end{enumerate}
The subcategory estimation procedure is similar to the main categories'
procedure, with some key differences. First, there is no data quality
error term (\(q\)) in the estimation of the log ratio of proportions.
Second, in the estimation of true proportions, we modify the weights
that balance the estimated country effect \(u\) and the new realization
of the country effect, \(\tilde{u}\). In the main category model, we use
the \(w_c\) and \(1-w_c\), where \(w_c\) is the defined as the maximum
coverage of any single country-year observation for country \(c\). This
was to reflect the uncertainty that arises from not observing up to a
proportion of \(1-w_c\) of the maternal deaths in a country-year. An
analogous statistic for the subcategory distributions is not readily
available, since we don't know specifically the coverage of deaths in
some category \(j\). For simplicity, we continue to use \(w_c\) for the
purpose of reflecting unobserved deaths in the HEM, SEP, and DIR
subdistributions. However, an additional complication is that more
granular information is unavailable for some observations. For instance,
some deaths may be attributed to hemorrhage but no further information
is given about the timing, and therefore do not hold useful information
about the subdistribution.
To account for this the subcategory model, we use the weight
\(z_{c, j} w_c\) and \(1 - z_{c, j} w_c\), where \(z_{c, j}\) is defined
as the maximum proportion, among observations for country \(c\), of
deaths in category \(j\) that have subcategory information available.
\hypertarget{implementation-and-computation}{%
\subsection{Implementation and
Computation}\label{implementation-and-computation}}
\hypertarget{equivalent-poisson-likelhood}{%
\subsubsection{Equivalent Poisson
Likelhood}\label{equivalent-poisson-likelhood}}
We implement the multinomial model using an equivalent Poisson
likelihood. The Poisson implementation speeds up computation, and easily
allows for arbitrary combinations of missing values. We rely on a result
presented by Ghosh, Zhang, and Mukherjee (2006) which states that the
multinomial likelihood in Equation (\ref{eq:full-multinomial}) is
equivalent to the Poisson likelihood \[
L_P = \prod_{i=1}^N \prod_{j = 1}^7 (g_{ij} \exp \phi_i )^{y_{ij}} \exp (- g_{ij} \exp \phi_i),
\] where \(\phi_i\) have independent improper priors \(p(\phi_i)\).
Note that if, for each observation \(i\), some arbitrary subset of
categories \(J_i \subset \{1, \dots, 7\}\) is missing, and the desired
reduced multinomial is across the categories
\(\{ 1, \dots, 7\} \backslash J_i\), then the likelihood reduces to \[
L_P = \prod_{i=1}^N \prod_{j \in \{1, \dots, 7\} \backslash J_i} (g_{ij} \exp \phi_i )^{y_{ij}} \exp (- g_{ij} \exp \phi_i).
\] in which the unused terms in \(J_i\) can simply be omitted from the
likelihood while the other terms remain unchanged.
In practical terms, this means we treat non-missing count \(y_{ij}\) as
coming from a Poisson distribution, \[
y_{ij} \sim \text{Poisson} (g_{ij} \cdot \exp\phi_i),
\] omitting missing counts from the likelihood evaluation. Importantly,
this means we do not have to rescale the probabilities to 1 for the
reduced multinomial likelihood, which would require recalculation of the
denominator \((\sum_{j \not \in K_i} g_{ij})\) for each observation.
\hypertarget{computation-in-stan}{%
\subsubsection{Computation in Stan}\label{computation-in-stan}}
Posterior samples were obtained using Hamiltonian Monte Carlo
implemented in Stan via the \texttt{cmdstanr} R package version 0.2.0,
with 4 parallel chains of 6000 warmup iterations and 4000 sampling
iterations (Stan Development Team 2019; Gabry and Češnovar 2020).
Standard checks for \(\hat{R}\) and effective sample size were
performed. This computation was enabled in part by resources provided by
Compute Ontario (\url{https://computeontario.ca/}) and ComputeCanada
(\url{https://www.computecanada.ca/}).
\hypertarget{results}{%
\section{\texorpdfstring{Results
\label{section_results}}{Results }}\label{results}}
In this section we illustrate some results of the estimation process at
the regional level and also for the three case-study countries. We also
present results of model validation exercises.
\hypertarget{results-by-region}{%
\subsection{Results by region}\label{results-by-region}}
Figure \ref{fig:fig_region} illustrates the estimates and 95\% credible
intervals for the maternal cause of death distributions (showing the
seven main causes of death categories) for the seven SDG regions. The
results show substantial differences in the both the cause of death
distributions by region and also the uncertainty around the resulting
estimates.
The estimates for the Europe and North America, and Australia and New
Zealand regions, which both encapsulate high income countries, are
fairly similar. In particular, the proportion of deaths due to indirect
causes is relatively high, which is expected given these countries are
in the late stage of the epidemiological transition (Nair,
Nelson-Piercy, and Knight 2017). Additionally, the uncertainty around
the estimates in these regions is relatively low, which is a consequence
of the large amount of data available in these countries. The Latin
American and the Caribbean region also has a relatively high proportion
of indirect deaths and low uncertainty, but additionally has a high
proportion of deaths due to hypertension.
In other regions, such as in Africa and Asia, the proportion of deaths
due to hemorrhage is much higher, which partly reflects lack of access
to high quality health care and skilled attendants at birth (Prata et
al. 2011; Montgomery et al. 2014; Maduka and Ogu 2020). Larger
uncertainty intervals are mostly due to a lack of data available, apart
from in Oceania where the uncertainty is mostly driven by small
population sizes.
\begin{figure}
\centering
\includegraphics{fig/sdg_geofacet.pdf}
\caption{\label{fig:fig_region}Estimates and 95\% credible intervals for
the maternal cause of death distributions for SDG regions}
\end{figure}
\hypertarget{case-studies}{%
\subsection{Case studies}\label{case-studies}}
Figure \ref{fig:fig_case_studies} illustrates the observed and estimated
cause of death distributions for the three case-study countries under
three different model set-ups, to illustrate the effect of different
model components. Country A is a country with relatively low maternal
mortality levels, with good-quality CRVS data available. Country B is a
country with relatively high maternal mortality but only has data
available from subnational studies. Country C has CRVS data with some
issues, in particular there are a high number of mis-classified `other
direct' maternal deaths. In addition, Country C also has one
high-quality data observation available from a maternal mortality
surveillance study.
In each of the graphs, the dots represent the observed proportions by
cause sequentially over 2009--2017. If the particular data source of
interest contained sequential years then the dots are joined together
with a line (for example, CRVS data in countries A and C). Note that the
multiple proportions equal to one observed in country B refer to
single-case studies, i.e.~studies where only one cause of death was
reported.
The columns of the figure show the implied estimates for three different
modeling set-ups. The first, shown on in the left-hand column shows the
results from a model with no quality adjustment and no weighting to
account for under-coverage. The middle column shows the results based on
a model with a quality adjustment but no weighting. And finally, the
right-hand column shows the results from a model with both the quality
and coverage adjustments, which is equivalent to the model described
above.
There are several observations to note. Firstly, the estimates for
Country A do not change substantially across the three model
alternatives. This reflects the fact that Country A's CRVS system is
considered of good quality and high coverage.
Secondly, looking at Country B, the addition of the quality term (from
the left to middle column) increases the uncertainty around the
estimates only slightly, after accounting for the study data being of
relatively poor quality. However, the addition of the weighting (from
the middle to right column) changes both the point estimates and
uncertainty intervals substantially. This is a consequence of the severe
under-coverage of the available data in Country B. Although there is one
relatively large study available in Country B with approximately 1400
deaths reported, this is still substantially lower than the estimated
66000 total number of maternal deaths here in 2015. As a consequence,
the final estimates for this country are weighted more heavily towards
its regional distribution (possibly away from its own observed
proportions), and the uncertainty around the estimates is much larger.
Finally, Country C is an example where the CRVS data have known issues,
but available data is supplemented with high-quality surveillance data
(`grey literature'). Going from left to right, the inclusion of a data
quality term has a substantial effect on the point estimates, as the
estimates are pulled to be more in line with the grey literature (in
particular for other direct deaths). The final step of weighting has
minimal impact as the coverage of both these data sources is relatively
high.
\begin{figure}
\centering
\includegraphics{fig/case_studies.pdf}
\caption{\label{fig:fig_case_studies}Observed proportions (shown as
points), and estimates (shown as bars) with 95\% credible intervals for
three case study countries, using three different model set-ups. The
final model, shown in the right-most column, accounts for data coverage
and data quality.}
\end{figure}
\hypertarget{validation-sensitivity-to-data-exclusion}{%
\subsection{Validation: sensitivity to data
exclusion}\label{validation-sensitivity-to-data-exclusion}}
To assess the sensitivity of the cause of death distribution estimates,
we performed a series of validation exercises leaving out certain types
of data. Firstly, we were interested in sensitivity to exclusion of all
data from studies, as these data represented a large amount of
additional information compared to previous efforts (Say et al. 2014).
Additionally, we assessed the change in the estimates when data from
high-income countries were left out. These countries have relatively
large amounts of data available and therefore run the risk of being
overly influential in the hierarchical model set-up. Finally, we
assessed the sensitivity of estimates to an exclusion of 20\% of the
data of each type (CRVS, studies and grey literature).
Table \ref{tab:validation_table} shows the mean absolute difference
between estimates by cause across SDG regions when the model was rerun
excluding any information from studies. Similar results for the other
two exercises (leaving high-income country data out and 20\% of the data
out) are shown in Appendix \ref{appendix_validation}. In general,
differences in estimates are small across all causes and regions,
particularly in Europe and North America, Australia and New Zealand, and
Latin America and the Carribean. In these regions the availability of
CRVS data is high and little information comes from studies. The mean
absolute differences are by far largest in Sub-Saharan Africa, with the
mean difference in embolism, hemorrhage and sepsis proportions being at
least 0.02. The relatively larger differences in Sub-Saharan Africa are
expected, given the lack of other data available in this region means
the data from studies is quite influential.
\begin{table}
\caption{\label{tab:validation_table}Mean absolute difference in estimated country proportions, leaving out observations from studies}
\centering
\resizebox{\linewidth}{!}{
\begin{tabular}[t]{l>{\raggedleft\arraybackslash}p{3cm}>{\raggedleft\arraybackslash}p{3cm}>{\raggedleft\arraybackslash}p{3cm}>{\raggedleft\arraybackslash}p{3cm}>{\raggedleft\arraybackslash}p{3cm}>{\raggedleft\arraybackslash}p{3cm}>{\raggedleft\arraybackslash}p{3cm}>{\raggedleft\arraybackslash}p{3cm}>{}p{3cm}}
\toprule
Cause & Central and Southern Asia & Europe and Northern America & Northern Africa and Western Asia & Oceania excl. Australia and New Zealand & Sub-Saharan Africa & Latin America and the Caribbean & Australia and New Zealand & Eastern and South-Eastern Asia\\
\midrule
\cellcolor{gray!6}{ABO} & \cellcolor{gray!6}{0.009} & \cellcolor{gray!6}{0.001} & \cellcolor{gray!6}{0.001} & \cellcolor{gray!6}{0.007} & \cellcolor{gray!6}{0.007} & \cellcolor{gray!6}{0.001} & \cellcolor{gray!6}{0.001} & \cellcolor{gray!6}{0.004}\\
DIR & 0.007 & 0.002 & 0.003 & 0.008 & 0.016 & 0.003 & 0.002 & 0.009\\
\cellcolor{gray!6}{EMB} & \cellcolor{gray!6}{0.006} & \cellcolor{gray!6}{0.003} & \cellcolor{gray!6}{0.005} & \cellcolor{gray!6}{0.012} & \cellcolor{gray!6}{0.027} & \cellcolor{gray!6}{0.002} & \cellcolor{gray!6}{0.001} & \cellcolor{gray!6}{0.008}\\
HEM & 0.009 & 0.002 & 0.006 & 0.010 & 0.021 & 0.002 & 0.003 & 0.011\\
\cellcolor{gray!6}{SEP} & \cellcolor{gray!6}{0.014} & \cellcolor{gray!6}{0.002} & \cellcolor{gray!6}{0.003} & \cellcolor{gray!6}{0.011} & \cellcolor{gray!6}{0.020} & \cellcolor{gray!6}{0.002} & \cellcolor{gray!6}{0.003} & \cellcolor{gray!6}{0.008}\\
\addlinespace
IND & 0.005 & 0.003 & 0.004 & 0.010 & 0.010 & 0.003 & 0.003 & 0.009\\
\cellcolor{gray!6}{HYP} & \cellcolor{gray!6}{0.008} & \cellcolor{gray!6}{0.002} & \cellcolor{gray!6}{0.003} & \cellcolor{gray!6}{0.009} & \cellcolor{gray!6}{0.017} & \cellcolor{gray!6}{0.002} & \cellcolor{gray!6}{0.001} & \cellcolor{gray!6}{0.007}\\
\bottomrule
\end{tabular}}
\end{table}
\hypertarget{discussion}{%
\section{\texorpdfstring{Discussion
\label{section_discussion}}{Discussion }}\label{discussion}}
In this paper we presented a Bayesian hierarchical framework to estimate
maternal cause of death distributions at the global, regional and
national levels. A total of 14 cause groups are estimated, encompassed
within the 7 main categories of causes of death: abortion, embolism,
hemorrhage, hypertension, sepsis, other direct causes and indirect
causes. The framework allows for data from various sources to be
combined, and accounts for the fact that we usually do not have data on
all cause of death categories. The model pools information across
regions, accounts for correlation between cause groups, and includes
adjustments for data quality and coverage of the observed deaths,
utilizing information on total maternal mortality and all-cause
mortality more broadly. We illustrated the model on three cases of
varying data availability and quality. While our motivation was to
estimate causes of maternal death, the framework could easily be adapted
and applied to estimate cause of death distributions for other key
population sub-groups, such as under-five mortality or premature
mortality.
The model framework has several advantages over previous efforts to
estimate causes of maternal death. Firstly, while previous efforts chose
the one `best data source' available for each country, we account for
data quality in the model, which in turn allows data from multiple
sources for a particular country to be included in the model (Say et al.
2014). Secondly, we account for under-coverage of available data by
weighting the final estimate by the observed number of total maternal
deaths. This means the final estimate for a particular country is
informed not only by the available data but also mortality trends in the
broader region. Finally, the model accounts for missing observations of
causes of death and allows for studies where only a single cause of
death is observed to be included in the estimation process.
There are, however, some limitations to our approach. One of the main
limitations is that we do not adjust for non-representativeness or bias
of data sources. For example, many of the smaller studies that were
included where CRVS data do not exist come from single institutions or a
group of healthcare institutions from residential areas that are
relatively urban. However, women included in these studies may not be
representative of the broader population. In addition, causes of
maternal death that are observed in healthcare facilities are likely to
be different to causes of maternal death that happen in community
centers or if the death occurs at home. While we include a data quality
term in the model, this only adds uncertainty to the data, rather than
adjusting the proportions of a particular cause up or down. In addition,
the re-weighting of estimates based on under-coverage implicitly assumes
the data we have is a representative sample of the broader population,
and also assumes cause of death distributions are similar within
geographical regions. In practice, is it usually very difficult or
impossible to know the types of women captured within a study (in terms
of their age, socioeconomic status, pre-existing conditions, etc) and
how this differs from the broader population, and thus how it could be
post-stratified to be more representative. Future work will aim to
explore options to better account for non-representative samples, and
include adjustments for countries where possible.
A second limitation is that we do not account for the uncertainty in
cause of death classifications. Much of the data that came from studies
had to be translated into one of the cause of death categories from
`free text' descriptions of cause. In some cases it may have not
necessarily been clear which cause of death category observations belong
to (there may be one or two possibilities). In these cases we split
deaths equally based on possible categorizations and did not account for
the relative likelihood or observing one cause over another.
Reducing maternal mortality continues to be an integral part of the
Sustainable Development Goal agenda. The latest report from the MMEIG on
maternal mortality suggests that many countries are not on track to
reach the goal of less than 70 deaths per 100,000 live births by 2030.
In particular, the 10 countries with the highest MMRs in 2017 have seen
a stagnation or slowing of the annual rate of reduction of maternal
mortality, and therefore remain at the greatest risk (WHO et al. 2019).
Understanding the underlying causes of maternal death is necessary to
evaluate progress and aid resource allocation. The importance of
reducing maternal mortality as a public health priority justifies
continued efforts to improve not only estimates but also the data
collected on causes of death.
\hypertarget{funding-acknowledgement}{%
\section*{Funding acknowledgement}\label{funding-acknowledgement}}
\addcontentsline{toc}{section}{Funding acknowledgement}
This work received funding from the UNDP-UNFPA-UNICEF-WHO-World Bank
Special Programme of Research, Development and Research Training in
Human Reproduction (HRP), a cosponsored programme executed by the World
Health Organization (WHO).
\newpage
|
2,877,628,088,973 | arxiv | \section{INTRODUCTION}
In the standard picture of structure formation, within the framework
of Cold Dark Matter (CDM) models, small subgalactic clumps are formed
first in CDM halos. Such building blocks of normal galaxies in local
universe grow hierarchically into more massive galaxies through
galaxy mergers and subsequent star formation. Ly$\alpha$ emitters
(LAEs) at high-$z$ universe are considered to be building blocks
because of their small stellar masses, young ages, and low
metallicities inferred from their broadband spectral energy
distributions (e.g., Chary et al. 2005; Gawiser et al. 2006; Nilsson
et al. 2007, 2009; Finkelstein et al. 2008; Ono et al. 2010a,~b; Yuma
et al. 2010; Acquaviva et al. 2011; Guaita et al. 2011; Vargas et
al. 2014). Since they are important population as a probe of galaxy
formation in the young universe as well as a probe of cosmic
reionization, much effort has been paid to search them (e.g., Cowie
\& Hu 1998; Rhoads et al. 2000; Ouchi et al. 2005, 2008, 2010;
Taniguchi et al. 2005; Shimasaku et al. 2006; Gronwall et al. 2007;
Murayama et al. 2007; Shioya et al. 2009; Kashikawa et al. 2011).
The redshift of the most distant LAE has now reached beyond $z = 7$
(Ono et al. 2012; Shibuya et al. 2012; Finkelstein et al. 2013), at
which cosmic reionization has not been completed yet.
However, it is still unclear in what physical conditions a galaxy is
observed as an LAE, which has intense Ly$\alpha$ emission. This is
mainly because Ly$\alpha$ is a resonance line of neutral hydrogen;
that is, mean free path of Ly$\alpha$ photon in interstellar medium
(ISM) is significantly short and hence it experiences enormous number
of scattering by neutral hydrogen before escaping from its host
galaxy. The multiple scattering makes Ly$\alpha$ extremely
vulnerable to dust attenuation. This is consistent with the
observational results for the LAEs in both nearby and high-$z$
universe which have revealed that the Ly$\alpha$ escape fraction
depends clearly on dust extinction, although the escape fraction does
not follow the expected one for a simple attenuation (Atek et
al. 2009, 2014; Kornei et al. 2010; Hayes et al. 2011, 2014).
Theoretical studies have also been executed, in which Ly$\alpha$
radiative transfer code is coupled with cosmological numerical
simulation in order to examine the Ly$\alpha$ escape fraction in
realistic ISM condition for high-$z$ LAE (e.g., Laursen \&
Sommer-Larsen 2007; Laursen et al. 2009a,~b; Zheng et al. 2010;
Yajima et al. 2012a,~b). These theoretical studies predict that ISM
clumpiness and morphology have a strong impact on Ly$\alpha$ escape
fraction and that clumpy and dusty ISM is favored for Ly$\alpha$ to
escape (Yajima et al. 2012b; Laursen et al. 2013; Duval et al. 2014;
Gronke \& Dijkstra 2014). Moreover, such clumpy and dusty ISM is
also found to be favored to reproduce the observed statistical
properties of LAEs (Kobayashi et al. 2007, 2010).
In such context, observational studies for the size and morphology of
high-$z$ LAEs have been widely conducted by using the Advanced Camera
for Surveys (ACS) on-board the \textit{Hubble Space Telescope}
(\textit{HST}) because these properties give us insights on how LAEs
are assembled and how their intense star formation events are
triggered (e.g., Stanway et al. 2004; Rhoads et al. 2005; Venemans et
al. 2005; Pirzkal et al. 2007; Overzier et al. 2008; Bond et
al. 2009, 2012; Taniguchi et al. 2009; Vanzella et al. 2009;
Finkelstein et al. 2011; Law et al. 2012; Malhotra et al. 2012;
Mawatari et al. 2012; Chonis et al. 2013; Jiang et al. 2013; Hagen et
al. 2014; Shibuya et al. 2014). It has been found that most of the
high-$z$ LAEs have small sizes of 0\farcs 1--0\farcs 2 in rest-frame
ultraviolet (UV) continuum, which remain almost constant in the
redshift range of $z \sim 2$--6 (Malhotra et al. 2012; Hagen et
al. 2014). This is against the hypothesis that LAE is simply a
subset of Lyman-break galaxy (LBG) population, which present a clear
redshift evolution of size in rest-frame UV continuum (e.g., Ono et
al. 2013).
In this paper, we examine the morphological properties of the 61 LAEs
at $z = 4.86$ selected by Shioya et al. (2009; hereafter S09) in the
Cosmic Evolution Survey (COSMOS) field (Scoville et al. 2007a),
providing one of the largest samples of LAEs in a large contiguous
field. Since F814W-band imaging taken with the \textit{HST}/ACS is
available for the COSMOS field (Scoville et al. 2007b; Koekemoer et
al. 2007), the sizes and morphologies of the LAEs in the COSMOS field
can be investigated in detail. In this paper, we present our
detailed analysis of ACS images of the LAE sample of S09.
We use a standard cosmology with $\Omega_M = 0.3$, $\Omega_\Lambda =
0.7$, and $H_0 = 70~\mathrm{km~s^{-1}~Mpc^{-1}}$. Under the adopted
cosmological parameters, the angular scale of $1^{\prime\prime}$
corresponds to the physical scale of 6.37~kpc at $z = 4.86$.
Throughout this paper, we use magnitudes in the AB system.
\section{OBSERVATIONAL DATA AND ACS COUNTERPARTS OF
LAEs}\label{sec:data}
In S09, 79 LAE candidates at $ 4.83 < z < 4.89$ have been carefully
selected from optical imaging with the narrow-band filter, NB711
($\lambda_\mathrm{c} = 7126$~{\AA}, $\Delta \lambda = 73$~{\AA}; see
Figure~\ref{fig:filters}), and broad-band filters from $B$ to
$z^\prime$ taken for the entire $1.95~\mathrm{deg^2}$ area of the
COSMOS field using the Suprime-Cam (Miyazaki et al. 2002) on the
Subaru Telescope (Kaifu et al. 2000; Iye et al. 2004). Details of
the Subaru observations and data processing are described by
Taniguchi et al. (2007) and Capak et al. (2007). Among 79 LAEs, 13
LAEs have spectroscopic information and all of them are confirmed as
$z \approx 4.86$ (P. Capak et al. 2015, in preparation), verifying
the effectiveness of our selection method\footnote{Although a
follow-up spectroscopy has also been performed for 5 additional LAEs,
their spectroscopic redshifts have not been determined because of low
data quality (see Table~\ref{tab:z4p9LAE}).}.
\begin{figure}
\epsscale{1.15}
\plotone{f1}
\caption{Transmission curves for the filters related to our
analysis. The blue curve represents the transmission curve for the
\textit{HST}/ACS F814W-band, while the magenta, green, and red
curves are the transmissions for the Subaru/Suprime-Cam NB711--,
$i^\prime$--, and $z^\prime$--bands, respectively. The effects of
the CCD sensitivity, the atmospheric transmission, and the
transmission of the telescope and the instrument are taken into
account for each transmission curve. Model spectrum of a LAE at $z
= 4.86$ with a rest-frame Ly$\alpha$ equivalent width
($\mathrm{EW_0}$) of 30~{\AA} is also plotted by black
curve. \label{fig:filters}}
\end{figure}
\begin{figure}
\epsscale{1.15}
\plotone{f2}
\caption{Spatial distribution of our sample of 79 LAEs at $z = 4.86$
selected in S09. In the whole COSMOS field of
$1.95~\mathrm{deg}^2$, the \textit{HST}/ACS images are available for
$1.64~\mathrm{deg}^2$ indicated by the solid gray line. The gray
shaded regions represent the areas masked out for detection. The 18
LAEs outside the \textit{HST}/ACS field are shown by black open
circles. Among the remaining 61 LAEs in the \textit{HST}/ACS field,
7 LAEs undetected in the ACS images are represented by green crosses
and 46 (8) LAEs with single (double) component(s) are shown by red
filled (blue double) circles. The ID \# in S09 is labeled for
reference.\label{fig:map}}
\end{figure}
\begin{deluxetable*}{lll}
\tablecaption{SExtractor configuration for the HST/ACS F814W-band
detection \label{table:SExParam}}
\tablehead{
\colhead{Parameter} & \colhead{Value} & \colhead{Comment}
}
\startdata
DETECT\_THRESH & 1.1 & Detection threshold in sigma\\
DETECT\_MINAREA & 25 & Minimum number of pixels above threshold\\
FILTER\_NAME & gauss\_3.0\_7x7.conv & Name of the filter for detection\\
DEBLEND\_NTHRESH & 64 & Number of deblending sub-thresholds\\
DEBLEND\_MINCONT & 0.015 & Minimum contrast parameter for deblending\\
PHOT\_AUTOPARAMS & 2.5, 0.5 & MAG\_AUTO parameters: Kron factor and minimum radius\\
BACK\_SIZE & 64 & Background mesh size\\
BACK\_FILTERSIZE & 3 & Background filter size\\
BACKPHOTO\_TYPE & GLOBAL & Photometry background subtraction type
\enddata
\end{deluxetable*}
The \textit{HST}/ACS F814W-band data ($\lambda_\mathrm{c} =
8333$~{\AA}, $\Delta \lambda = 2511$~{\AA}; see
Figure~\ref{fig:filters}) is available for a part of the COSMOS
field, $1.64~\mathrm{deg^2}$ ($\approx 84$\% of the COSMOS field), as
shown in Figure~\ref{fig:map}. In our analysis, we use the official
COSMOS ACS image (Scoville et al. 2007b; Koekemoer et al. 2007),
Version 2.0. The ACS data were processed to $0\farcs
03~\mathrm{pixel}^{-1}$ images. We find that ACS imaging data are
available for 61 LAEs out of 79 LAE samples selected by S09. The
remaining 18 LAEs are not covered by the ACS field or are on the edge
of the ACS field. Spatial distribution of all 79 LAEs in the COSMOS
field is shown in Figure~\ref{fig:map}. Our data analysis procedure
for ACS data are similar to those in Taniguchi et al. (2009), in
which the official COSMOS ACS image Version 1.3 with the pixel scale
of $0\farcs 05~\mathrm{pixel}^{-1}$ was utilized. The source
detection of the LAEs in the \textit{HST}/ACS image was carried out
with their weight map using SExtractor (Bertin \& Arnouts 1996). The
fundamental parameters of the SExtractor's configuration are shown in
Table~\ref{table:SExParam}, which are also basically similar to those
used in Taniguchi et al. (2009) and modified slightly for the
$0\farcs 03~\mathrm{pixel}^{-1}$ images. Note that these parameters
are determined by the tradeoff between detecting fainter
objects/components and avoiding noise effects such as the false
detection or the noise confusion.
\begin{deluxetable}{lllrr}
\tablecaption{COSMOS $z = 4.86$ LAE Sample \label{table:Sample}}
\tablewidth{0pt}
\tablehead{
\multicolumn{3}{c}{LAE Sample} & \colhead{Number of LAEs} &
\colhead{Spectroscopic}\\
\multicolumn{4}{c}{} & \colhead{Confirmation}
}
\startdata
\multicolumn{3}{l}{In the ACS/F814W-band field} & 61 & 13 \vspace{1mm}\\
& \multicolumn{2}{l}{ACS/F814W-band detected} & 54 & 12 \vspace{1mm}\\
& & \hspace{4mm}Single component & 46 & 10 \vspace{1mm}\\
& & \hspace{4mm}Double component & 8 & 2 \vspace{1mm}\\
& \multicolumn{2}{l}{ACS/F814W-band undetected} & 7 & 1 \vspace{1mm}\\
\multicolumn{3}{l}{Out of the ACS/F814W-band field} & 18 & 0 \vspace{1mm}\\
\cline{1-5}\vspace{-2mm}\\
\multicolumn{3}{c}{Total} & 79 & 13
\enddata
\end{deluxetable}
Among the 61 LAEs in the ACS field, we find the ACS counterparts of
the 54 LAEs detected near the LAE positions defined in NB711--band
images (i.e., separation of $\le 1^{\prime\prime}$). Any sources are
not detected near the LAE positions for the remaining seven LAEs.
While most of the ACS counterparts consist of single component, eight
LAEs among the 54 ACS-detected LAEs have double components in the ACS
images within separation of $\le 1^{\prime\prime}$ from the LAE
positions, providing the double-component LAE fraction,
$f_\mathrm{double}$, of $8 / 54 = 14.8$\%. The numbers of the total
sample, both the ACS-detected and undetected LAEs, are summarized in
Table~\ref{table:Sample}.
Separations between each component in the 8 double-component LAEs are
found to be 0\farcs 36--0\farcs 98 (the mean is 0\farcs 63). For
these double-component LAEs, the mean offset of the ACS centroids
from the NB711--band centroids is found to be 0\farcs 39, which is
larger than the NB711--band pixel scale of $0\farcs
15~\mathrm{pixel}^{-1}$. This is possibly because these double ACS
components are unresolved in the NB711--band images taken by the
Subaru telescope, in which the mean half-light radius of unsaturated
stars is 0\farcs 25, and their NB711--band positions can be close to
their flux-weighted centroid. On the other hand, for the 46
single-component LAEs, the mean offset between the ACS F814W- and
NB711--band centroids is 0\farcs 16, which is comparable to the pixel
scale of the NB711--band images. We note that, in the following
analysis, the double ACS components in each one of the 8
double-component LAEs are treated as sub-components in a single
object at first and the morphological properties of the objects are
measured. The results with each of the components in the double
component systems treated separately as different objects with close
angular separation are presented in Section~\ref{subsec:double}.
As shown in Figure~\ref{fig:map}, the ACS-detected (filled and double
circles) and ACS-undetected LAEs (crosses) seem to be distributed
randomly in the whole ACS field. Therefore, their distributions may
not be affected by large-scale inhomogeneity of the ACS data quality
(e.g., edges of the field). It should be noted that the LAE \#20 is
on the edge of the ACS field as shown in Figure~\ref{fig:map}. While
the LAE \#20 seems to have a double-component ACS source, we do not
include it in our sample of the ACS-detected LAEs since the ACS data
quality is highly doubtful.
\begin{figure}
\epsscale{1.1}
\plotone{f3}
\caption{Thumbnails of 8 LAEs with double-component ACS F814W-band
sources. North is up and east is left. Each panel has a size of
$5^{\prime\prime} \times 5^{\prime\prime}$. Red ellipses
overplotted on the NB711-- and $z'$--band images are half light
ellipses of the detected LAE counterparts in the ACS image, while
yellow ellipses are those of the individual components in each LAE
counterpart detected by SExtractor. \label{fig:ThumbnailsDouble}}
\end{figure}
\begin{figure*}
\epsscale{1.1}
\plotone{f4a}
\caption{Same as Figure~\ref{fig:ThumbnailsDouble} but for 46 LAEs
with single component ACS source. \label{fig:ThumbnailsSingle}}
\end{figure*}
\setcounter{figure}{3}
\begin{figure*}
\epsscale{1.1}
\plotone{f4b}
\caption{(Continued.)}
\end{figure*}
\setcounter{figure}{3}
\begin{figure}
\epsscale{1.1}
\plotone{f4c}
\caption{(Continued.)}
\end{figure}
\begin{figure}
\epsscale{1.1}
\plotone{f5}
\caption{Same as Figure~\ref{fig:ThumbnailsDouble} but for 7 LAEs
without ACS sources. \label{fig:ThumbnailsNot}}
\end{figure}
\begin{turnpage}
\begin{deluxetable*}{lcccccccrrrc}
\tabletypesize{\scriptsize}
\tablecaption{\textit{ACS} F814W-band Properties for the 61 LAEs at
$z = 4.86$ with ACS Data \label{tab:z4p9LAE}}
\tablewidth{\linewidth}
\tablehead{
\colhead{ID\tablenotemark{a}} &
\colhead{$I_{814}$\tablenotemark{b}} &
\colhead{$a_\mathrm{HL}$\tablenotemark{c}} &
\colhead{$R_\mathrm{HL}$\tablenotemark{d}} &
\colhead{$\epsilon (I_{814})$\tablenotemark{e}} &
\colhead{$NB711$\tablenotemark{f}} &
\colhead{$a_\mathrm{HL} (\mathrm{NB711})$\tablenotemark{g}} &
\colhead{$i^\prime$\tablenotemark{f}} &
\colhead{$z^\prime$\tablenotemark{f}} &
\colhead{$L(\mathrm{Ly\alpha})$\tablenotemark{h}} &
\colhead{$\mathrm{EW_0}$\tablenotemark{i}} &
\colhead{$z_\mathrm{spec}$\tablenotemark{j}}\\
\colhead{} &
\colhead{(mag)} &
\colhead{(arcsec)} &
\colhead{(arcsec)} &
\colhead{} &
\colhead{(mag)} &
\colhead{(arcsec)} &
\colhead{(mag)} &
\colhead{(mag)} &
\colhead{($10^{42}~\mathrm{erg~s^{-1}}$)} &
\colhead{({\AA})} &
\colhead{}
}
\startdata
\multicolumn{12}{c}{8 ACS-detected LAEs with Double Components}\vspace{1mm}\\
\cline{1-12}\vspace{-1mm}\\
11 & $25.52 \pm 0.06$ & $0.49 \pm 0.02$ & $0.32 \pm 0.02$ & $0.76 \pm 0.01$ & $24.55 \pm 0.18$ & $0.51$ & $25.73 \pm 0.42$ & $24.98 \pm 0.24$ & $ 5.6 \pm 1.3$ & $18 \pm 6$ & $-99.0$\\
12 & $24.11 \pm 0.03$ & $0.44 \pm 0.02$ & $0.34 \pm 0.01$ & $0.47 \pm 0.02$ & $23.64 \pm 0.08$ & $1.02$ & $24.16 \pm 0.07$ & $24.31 \pm 0.13$ & $ 9.5 \pm 1.2$ & $16 \pm 3$ & 4.850\\
19 & $25.06 \pm 0.04$ & $0.55 \pm 0.02$ & $0.37 \pm 0.02$ & $0.79 \pm 0.01$ & $24.18 \pm 0.13$ & $0.47$ & $24.90 \pm 0.10$ & $24.82 \pm 0.17$ & $ 6.4 \pm 1.1$ & $18 \pm 4$ & \nodata\\
21 & $25.05 \pm 0.04$ & $0.56 \pm 0.04$ & $0.24 \pm 0.01$ & $0.77 \pm 0.01$ & $23.96 \pm 0.12$ & $1.05$ & $25.26 \pm 0.12$ & $24.83 \pm 0.18$ & $10.1 \pm 1.3$ & $28 \pm 6$ & \nodata\\
30 & $23.87 \pm 0.01$ & $0.31 \pm 0.00$ & $0.25 \pm 0.01$ & $0.54 \pm 0.01$ & $23.50 \pm 0.07$ & $0.56$ & $24.17 \pm 0.05$ & $24.13 \pm 0.09$ & $11.7 \pm 1.1$ & $17 \pm 2$ & \nodata\\
37 & $25.21 \pm 0.06$ & $0.34 \pm 0.02$ & $0.23 \pm 0.01$ & $0.64 \pm 0.02$ & $24.08 \pm 0.13$ & $0.76$ & $24.59 \pm 0.07$ & $24.31 \pm 0.11$ & $ 5.8 \pm 1.2$ & $10 \pm 2$ & \nodata\\
59 & $25.47 \pm 0.07$ & $0.59 \pm 0.05$ & $0.32 \pm 0.03$ & $0.81 \pm 0.02$ & $24.52 \pm 0.17$ & $0.54$ & $25.52 \pm 0.14$ & $26.03 \pm 0.47$ & $ 5.4 \pm 1.1$ & $45 \pm 26$ & \nodata\\
68 & $25.02 \pm 0.02$ & $0.25 \pm 0.01$ & $0.13 \pm 0.00$ & $0.65 \pm 0.01$ & $24.06 \pm 0.12$ & $0.86$ & $24.65 \pm 0.07$ & $25.12 \pm 0.22$ & $ 6.5 \pm 1.1$ & $24 \pm 7$ & 4.798\vspace{1mm}\\
\cline{1-12}\vspace{-2mm}\\
\multicolumn{12}{c}{46 ACS-detected LAEs with Single Component}\vspace{1mm}\\
\cline{1-12}\vspace{-1mm}\\
1 & $26.20 \pm 0.10$ & $0.17 \pm 0.01$ & $0.13 \pm 0.01$ & $0.49 \pm 0.04$ & $24.86 \pm 0.26$ & $0.40$ & $26.54 \pm 0.32$ & $25.95 \pm 0.46$ & $ 4.8 \pm 1.2$ & $ 38 \pm 22$ & $-99.0$\\
3 & $26.79 \pm 0.11$ & $0.11 \pm 0.01$ & $0.10 \pm 0.01$ & $0.13 \pm 0.11$ & $24.46 \pm 0.20$ & $0.37$ & $26.38 \pm 0.31$ & $> 26.64$ & $ 7.1 \pm 1.3$ & $>105$ & \nodata\\
5 & $26.27 \pm 0.07$ & $0.10 \pm 0.01$ & $0.10 \pm 0.01$ & $0.13 \pm 0.07$ & $23.43 \pm 0.07$ & $0.53$ & $25.63 \pm 0.15$ & $25.61 \pm 0.29$ & $19.0 \pm 1.1$ & $109 \pm 34$ & 4.839\\
9 & $27.30 \pm 0.10$ & $0.08 \pm 0.01$ & $0.06 \pm 0.00$ & $0.37 \pm 0.09$ & $24.22 \pm 0.13$ & $0.58$ & $26.02 \pm 0.21$ & $> 26.64$ & $ 8.8 \pm 1.1$ & $> 130$ & \nodata\\
13 & $26.97 \pm 0.07$ & $0.09 \pm 0.00$ & $0.08 \pm 0.00$ & $0.24 \pm 0.08$ & $24.71 \pm 0.20$ & $0.44$ & $25.97 \pm 0.19$ & $25.67 \pm 0.31$ & $ 5.0 \pm 1.1$ & $ 30 \pm 12$ & \nodata\\
15 & $25.73 \pm 0.06$ & $0.12 \pm 0.01$ & $0.12 \pm 0.01$ & $0.06 \pm 0.06$ & $24.07 \pm 0.12$ & $1.02$ & $25.18 \pm 0.11$ & $25.43 \pm 0.27$ & $ 8.7 \pm 1.2$ & $ 42 \pm 13$ & \nodata\\
16 & $27.26 \pm 0.16$ & $0.09 \pm 0.01$ & $0.08 \pm 0.02$ & $0.13 \pm 0.14$ & $24.46 \pm 0.17$ & $0.60$ & $25.43 \pm 0.13$ & $25.87 \pm 0.39$ & $ 5.4 \pm 1.1$ & $ 39 \pm 19$ & \nodata\\
17 & $27.09 \pm 0.16$ & $0.12 \pm 0.02$ & $0.11 \pm 0.01$ & $0.13 \pm 0.16$ & $24.74 \pm 0.22$ & $0.59$ & $26.76 \pm 0.40$ & $> 26.64$ & $ 5.6 \pm 1.2$ & $> 83$ & \nodata\\
22 & $27.00 \pm 0.11$ & $0.09 \pm 0.01$ & $0.08 \pm 0.01$ & $0.21 \pm 0.08$ & $24.51 \pm 0.17$ & $0.90$ & $26.36 \pm 0.24$ & $> 26.64$ & $ 6.8 \pm 2.8$ & $> 101$ & \nodata\\
23 & $25.33 \pm 0.04$ & $0.23 \pm 0.00$ & $0.16 \pm 0.01$ & $0.56 \pm 0.02$ & $24.41 \pm 0.15$ & $0.42$ & $24.94 \pm 0.08$ & $24.73 \pm 0.15$ & $ 4.5 \pm 1.0$ & $ 11 \pm 3$ & \nodata\\
24 & $26.54 \pm 0.10$ & $0.12 \pm 0.01$ & $0.11 \pm 0.01$ & $0.24 \pm 0.06$ & $24.49 \pm 0.18$ & $0.62$ & $26.12 \pm 0.18$ & $> 26.64$ & $ 6.7 \pm 1.2$ & $> 98$ & 4.845\\
25 & $26.78 \pm 0.11$ & $0.10 \pm 0.01$ & $0.08 \pm 0.01$ & $0.27 \pm 0.08$ & $24.38 \pm 0.17$ & $0.80$ & $25.04 \pm 0.10$ & $24.49 \pm 0.13$ & $ 5.1 \pm 1.3$ & $ 10 \pm 3$ & \nodata\\
26 & $26.18 \pm 0.07$ & $0.11 \pm 0.01$ & $0.10 \pm 0.00$ & $0.20 \pm 0.05$ & $24.46 \pm 0.17$ & $0.42$ & $26.12 \pm 0.21$ & $> 26.64$ & $ 6.8 \pm 1.1$ & $> 101$ & \nodata\\
27 & $26.62 \pm 0.07$ & $0.15 \pm 0.01$ & $0.11 \pm 0.00$ & $0.49 \pm 0.05$ & $24.38 \pm 0.15$ & $0.78$ & $25.88 \pm 0.18$ & $25.73 \pm 0.35$ & $ 7.3 \pm 1.1$ & $ 47 \pm 19$ & \nodata\\
29 & $26.70 \pm 0.04$ & $0.08 \pm 0.00$ & $0.07 \pm 0.00$ & $0.20 \pm 0.05$ & $23.87 \pm 0.10$ & $0.65$ & $26.82 \pm 0.39$ & $> 26.64$ & $13.4 \pm 1.1$ & $> 199$ & \nodata\\
31 & $25.98 \pm 0.07$ & $0.14 \pm 0.01$ & $0.12 \pm 0.01$ & $0.26 \pm 0.05$ & $24.58 \pm 0.18$ & $0.65$ & $26.13 \pm 0.22$ & $25.48 \pm 0.31$ & $ 6.1 \pm 1.1$ & $ 31 \pm 12$ & \nodata\\
33 & $26.17 \pm 0.08$ & $0.13 \pm 0.01$ & $0.11 \pm 0.01$ & $0.33 \pm 0.05$ & $24.64 \pm 0.21$ & $0.51$ & $26.13 \pm 0.26$ & $> 26.64$ & $ 5.6 \pm 1.3$ & $> 83$ & \nodata\\
34 & $24.61 \pm 0.03$ & $0.15 \pm 0.00$ & $0.13 \pm 0.00$ & $0.27 \pm 0.02$ & $23.56 \pm 0.08$ & $0.70$ & $24.78 \pm 0.08$ & $25.14 \pm 0.21$ & $14.0 \pm 1.2$ & $ 52 \pm 12$ & \nodata\\
38 & $25.82 \pm 0.06$ & $0.13 \pm 0.01$ & $0.11 \pm 0.01$ & $0.29 \pm 0.05$ & $24.04 \pm 0.11$ & $0.51$ & $26.37 \pm 0.22$ & $> 26.64$ & $11.0 \pm 1.1$ & $> 163$ & 4.873\\
39 & $25.94 \pm 0.12$ & $0.20 \pm 0.02$ & $0.16 \pm 0.03$ & $0.59 \pm 0.04$ & $24.21 \pm 0.19$ & $0.87$ & $26.05 \pm 0.19$ & $> 26.64$ & $ 8.8 \pm 1.6$ & $> 131$ & $-99.0$\\
40 & $27.45 \pm 0.15$ & $0.09 \pm 0.01$ & $0.09 \pm 0.01$ & $0.04 \pm 0.16$ & $24.76 \pm 0.20$ & $0.51$ & $26.05 \pm 0.17$ & $26.26 \pm 0.55$ & $ 4.8 \pm 1.0$ & $ 50 \pm 34$ & 4.818\\
41 & $24.97 \pm 0.04$ & $0.09 \pm 0.00$ & $0.09 \pm 0.00$ & $0.04 \pm 0.04$ & $23.39 \pm 0.07$ & $0.47$ & $24.88 \pm 0.08$ & $25.62 \pm 0.36$ & $17.7 \pm 1.2$ & $103 \pm 41$ & 4.830\\
42 & $26.05 \pm 0.07$ & $0.18 \pm 0.01$ & $0.13 \pm 0.01$ & $0.51 \pm 0.04$ & $23.82 \pm 0.09$ & $0.75$ & $25.37 \pm 0.11$ & $25.58 \pm 0.30$ & $12.1 \pm 1.0$ & $ 67 \pm 22$ & \nodata\\
43 & $26.41 \pm 0.11$ & $0.13 \pm 0.01$ & $0.11 \pm 0.01$ & $0.43 \pm 0.07$ & $24.49 \pm 0.16$ & $0.79$ & $25.83 \pm 0.16$ & $25.59 \pm 0.31$ & $ 6.1 \pm 1.1$ & $ 34 \pm 13$ & \nodata\\
44 & $25.63 \pm 0.04$ & $0.12 \pm 0.00$ & $0.11 \pm 0.00$ & $0.24 \pm 0.04$ & $23.91 \pm 0.10$ & $9999.0$ & $24.93 \pm 0.10$ & $25.16 \pm 0.21$ & $ 9.4 \pm 1.1$ & $ 36 \pm 9$ & \nodata\\
45 & $27.07 \pm 0.17$ & $0.17 \pm 0.04$ & $0.12 \pm 0.02$ & $0.56 \pm 0.08$ & $24.62 \pm 0.16$ & $0.49$ & $26.01 \pm 0.16$ & $26.54 \pm 0.83$ & $ 5.5 \pm 1.0$ & $ 74 \pm 69$ & 4.865\\
46 & $25.76 \pm 0.05$ & $0.10 \pm 0.01$ & $0.09 \pm 0.00$ & $0.17 \pm 0.05$ & $24.65 \pm 0.17$ & $0.61$ & $25.48 \pm 0.11$ & $25.27 \pm 0.23$ & $ 4.5 \pm 1.0$ & $ 19 \pm 6$ & 4.865\\
48 & $26.65 \pm 0.09$ & $0.07 \pm 0.00$ & $0.07 \pm 0.00$ & $0.07 \pm 0.07$ & $23.95 \pm 0.11$ & $0.52$ & $26.09 \pm 0.19$ & $> 26.64$ & $11.7 \pm 1.1$ & $> 173$ & \nodata\\
50 & $25.99 \pm 0.06$ & $0.09 \pm 0.00$ & $0.09 \pm 0.00$ & $0.02 \pm 0.06$ & $23.69 \pm 0.09$ & $0.74$ & $26.22 \pm 0.25$ & $26.34 \pm 0.65$ & $15.4 \pm 1.2$ & $172 \pm 143$ & \nodata\\
54 & $25.35 \pm 0.03$ & $0.08 \pm 0.00$ & $0.07 \pm 0.00$ & $0.07 \pm 0.03$ & $23.45 \pm 0.07$ & $0.64$ & $25.42 \pm 0.13$ & $25.44 \pm 0.29$ & $18.3 \pm 1.1$ & $ 90 \pm 28$ & \nodata\\
55 & $25.09 \pm 0.05$ & $0.17 \pm 0.01$ & $0.13 \pm 0.01$ & $0.40 \pm 0.03$ & $24.50 \pm 0.15$ & $0.75$ & $25.21 \pm 0.10$ & $25.22 \pm 0.23$ & $ 4.8 \pm 1.0$ & $ 19 \pm 6$ & 4.830\\
56 & $25.50 \pm 0.05$ & $0.19 \pm 0.01$ & $0.15 \pm 0.01$ & $0.54 \pm 0.02$ & $24.45 \pm 0.16$ & $0.71$ & $25.42 \pm 0.11$ & $26.08 \pm 0.50$ & $ 5.6 \pm 1.1$ & $ 49 \pm 31$ & $-99.0$\\
57 & $26.91 \pm 0.10$ & $0.12 \pm 0.01$ & $0.09 \pm 0.01$ & $0.42 \pm 0.07$ & $24.67 \pm 0.21$ & $0.33$ & $26.53 \pm 0.33$ & $25.54 \pm 0.30$ & $ 5.9 \pm 1.2$ & $ 32 \pm 12$ & \nodata\\
58 & $25.79 \pm 0.06$ & $0.15 \pm 0.01$ & $0.13 \pm 0.01$ & $0.29 \pm 0.04$ & $23.90 \pm 0.12$ & $0.85$ & $25.72 \pm 0.15$ & $25.50 \pm 0.31$ & $11.8 \pm 1.3$ & $ 61 \pm 21$ & 4.840\\
60 & $25.89 \pm 0.06$ & $0.14 \pm 0.01$ & $0.13 \pm 0.01$ & $0.13 \pm 0.06$ & $24.14 \pm 0.14$ & $0.70$ & $25.43 \pm 0.13$ & $25.28 \pm 0.24$ & $ 8.5 \pm 1.2$ & $ 36 \pm 10$ & \nodata\\
61 & $26.84 \pm 0.10$ & $0.14 \pm 0.01$ & $0.12 \pm 0.01$ & $0.33 \pm 0.07$ & $24.80 \pm 0.22$ & $0.31$ & $26.09 \pm 0.22$ & $25.71 \pm 0.34$ & $ 4.6 \pm 1.1$ & $ 29 \pm 13$ & \nodata\\
63 & $25.79 \pm 0.04$ & $0.10 \pm 0.00$ & $0.08 \pm 0.00$ & $0.31 \pm 0.03$ & $23.68 \pm 0.09$ & $0.55$ & $24.31 \pm 0.06$ & $23.91 \pm 0.08$ & $ 9.2 \pm 1.2$ & $ 11 \pm 2$ & \nodata\\
64 & $25.99 \pm 0.09$ & $0.15 \pm 0.01$ & $0.12 \pm 0.01$ & $0.40 \pm 0.05$ & $24.27 \pm 0.14$ & $0.75$ & $25.09 \pm 0.10$ & $25.30 \pm 0.28$ & $ 6.1 \pm 1.1$ & $ 26 \pm 9$ & \nodata\\
65 & $25.24 \pm 0.06$ & $0.14 \pm 0.01$ & $0.12 \pm 0.01$ & $0.21 \pm 0.04$ & $24.41 \pm 0.16$ & $0.61$ & $25.06 \pm 0.10$ & $24.81 \pm 0.16$ & $ 5.0 \pm 1.1$ & $ 14 \pm 4$ & \nodata
\enddata
\end{deluxetable*}
\end{turnpage}
\setcounter{table}{2}
\begin{turnpage}
\begin{deluxetable*}{lcccccccrrrc}
\tabletypesize{\scriptsize}
\tablecaption{(Continued.)}
\tablewidth{\linewidth}
\tablehead{
\colhead{ID\tablenotemark{a}} &
\colhead{$I_{814}$\tablenotemark{b}} &
\colhead{$a_\mathrm{HL}$\tablenotemark{c}} &
\colhead{$R_\mathrm{HL}$\tablenotemark{d}} &
\colhead{$\epsilon (I_{814})$\tablenotemark{e}} &
\colhead{$NB711$\tablenotemark{f}} &
\colhead{$a_\mathrm{HL} (\mathrm{NB711})$\tablenotemark{g}} &
\colhead{$i^\prime$\tablenotemark{f}} &
\colhead{$z^\prime$\tablenotemark{f}} &
\colhead{$L(\mathrm{Ly\alpha})$\tablenotemark{h}} &
\colhead{$\mathrm{EW_0}$\tablenotemark{i}} &
\colhead{$z_\mathrm{spec}$\tablenotemark{j}}\\
\colhead{} &
\colhead{(mag)} &
\colhead{(arcsec)} &
\colhead{(arcsec)} &
\colhead{} &
\colhead{(mag)} &
\colhead{(arcsec)} &
\colhead{(mag)} &
\colhead{(mag)} &
\colhead{($10^{42}~\mathrm{erg~s^{-1}}$)} &
\colhead{({\AA})} &
\colhead{}
}
\startdata
66 & $26.84 \pm 0.10$ & $0.13 \pm 0.01$ & $0.10 \pm 0.01$ & $0.34 \pm 0.08$ & $24.88 \pm 0.26$ & $0.44$ & $26.84 \pm 0.42$ & $> 26.64$ & $ 4.9 \pm 1.3$ & $> 72$ & \nodata\\
69 & $27.57 \pm 0.12$ & $0.07 \pm 0.01$ & $0.06 \pm 0.01$ & $0.15 \pm 0.15$ & $24.49 \pm 0.15$ & $0.58$ & $25.20 \pm 0.10$ & $25.79 \pm 0.39$ & $ 4.5 \pm 1.0$ & $31 \pm 15$ & 4.854\\
70 & $25.09 \pm 0.03$ & $0.13 \pm 0.00$ & $0.10 \pm 0.00$ & $0.34 \pm 0.03$ & $23.50 \pm 0.07$ & $0.47$ & $24.67 \pm 0.07$ & $24.88 \pm 0.17$ & $14.6 \pm 1.1$ & $43 \pm 8$ & \nodata\\
71 & $25.27 \pm 0.05$ & $0.15 \pm 0.01$ & $0.13 \pm 0.00$ & $0.22 \pm 0.04$ & $24.32 \pm 0.14$ & $0.52$ & $24.87 \pm 0.08$ & $24.66 \pm 0.14$ & $ 4.9 \pm 1.1$ & $12 \pm 3$ & \nodata\\
72 & $25.51 \pm 0.08$ & $0.30 \pm 0.01$ & $0.22 \pm 0.01$ & $0.50 \pm 0.03$ & $24.35 \pm 0.15$ & $0.54$ & $25.59 \pm 0.15$ & $> 26.64$ & $ 7.0 \pm 1.1$ & $> 104$ & \nodata\\
77 & $26.56 \pm 0.11$ & $0.09 \pm 0.01$ & $0.08 \pm 0.01$ & $0.21 \pm 0.07$ & $24.70 \pm 0.22$ & $0.37$ & $26.19 \pm 0.25$ & $> 26.64$ & $ 5.4 \pm 1.2$ & $> 81$ & \nodata\\
78 & $25.60 \pm 0.05$ & $0.15 \pm 0.01$ & $0.13 \pm 0.01$ & $0.24 \pm 0.04$ & $23.25 \pm 0.06$ & $0.55$ & $25.05 \pm 0.10$ & $24.83 \pm 0.16$ & $21.2 \pm 1.2$ & $59 \pm 10$ & \nodata\vspace{1mm}\\
\cline{1-12}\vspace{-1mm}\\
\multicolumn{12}{c}{7 ACS-undetected LAEs}\vspace{1mm}\\
\cline{1-12}\vspace{-1mm}\\
10 & \nodata & \nodata & \nodata & \nodata & $24.80 \pm 0.22$ & $0.41$ & $27.04 \pm 0.61$ & $> 26.64$ & $5.5 \pm 1.2$ & $> 82$ & $-99.0$\\
14 & \nodata & \nodata & \nodata & \nodata & $24.22 \pm 0.13$ & $1.06$ & $26.99 \pm 0.58$ & $> 26.64$ & $9.6 \pm 1.1$ & $> 142$ & \nodata\\
18 & \nodata & \nodata & \nodata & \nodata & $24.10 \pm 0.12$ & $1.64$ & $25.23 \pm 0.11$ & $25.25 \pm 0.22$ & $8.2 \pm 1.1$ & $34 \pm 9$ & \nodata\\
47 & \nodata & \nodata & \nodata & \nodata & $24.70 \pm 0.19$ & $0.60$ & $26.60 \pm 0.27$ & $26.49 \pm 0.64$ & $5.7 \pm 1.0$ & $74 \pm 60$ & 4.840\\
49 & \nodata & \nodata & \nodata & \nodata & $24.93 \pm 0.33$ & $0.35$ & $26.72 \pm 0.38$ & $25.84 \pm 0.44$ & $4.6 \pm 1.6$ & $33 \pm 20$ & \nodata\\
51 & \nodata & \nodata & \nodata & \nodata & $24.64 \pm 0.21$ & $0.46$ & $25.56 \pm 0.16$ & $25.67 \pm 0.36$ & $4.5 \pm 1.3$ & $27 \pm 13$ & \nodata\\
67 & \nodata & \nodata & \nodata & \nodata & $24.16 \pm 0.13$ & $0.99$ & $26.16 \pm 0.20$ & $> 26.64$ & $9.5 \pm 1.1$ & $> 141$ & \nodata
\enddata
\tablecomments{(a) The LAE ID given in Shioya et al.~(2009). (b)
SExtractor's MAG\_AUTO magnitude and its $1\sigma$ error. (c)
Half-light major radius and its $1\sigma$ error measured on ACS
F814W-band images. (d) Half-light radius and its $1\sigma$ error
measured on ACS F814W-band images. (e) Ellipticity and its
$1\sigma$ error measured on ACS F814W-band images. (f)
$3^{\prime\prime}$ diameter aperture magnitude and its $1\sigma$
error. (g) Half-light major radius measured on NB711--band images.
The entry of 9999.0 for the LAE \#44 means that its size estimation
is impossible because of the presence of a close bright contaminant.
(h) Ly$\alpha$ line luminosity and its $1\sigma$ error. (i)
Rest-frame Ly$\alpha$ EW and its $1\sigma$ error. Note that these
values are different from $\mathrm{EW_0}$ listed in Table~1 in S09
by a factor of 0.83 because of an error (see Erratum of S09). (j)
Spectroscopic redshift. The entry of $-99.0$ means that redshift is
not determined whereas follow-up spectroscopy is performed.}
\end{deluxetable*}
\end{turnpage}
We show the thumbnails of the 61 LAEs in the ACS F814W-band images
together with their Subaru NB711--, $i^\prime$--, and
$z^\prime$--band images in Figure~\ref{fig:ThumbnailsDouble} (8
ACS-detected LAEs with double-component),
Figure~\ref{fig:ThumbnailsSingle} (46 ACS-detected LAEs with
single-component), and Figure~\ref{fig:ThumbnailsNot} (7
ACS-undetected LAEs). In these figures, the detected ACS sources
identified as LAE counterparts are indicated by red ellipses on the
NB711--, $i^\prime$--, and $z^\prime$--band images. For the
double-component LAEs shown in Figure~\ref{fig:ThumbnailsDouble}, the
individual ACS sources detected are also overlayed by yellow ellipses
on the NB711--, $i^\prime$--, and $z^\prime$--band images.
The total magnitude ($I_{814}$), circularized half-light radius
($R_\mathrm{HL}$), half-light major radius ($a_\mathrm{HL}$), and
ellipticity ($\epsilon$) are measured for each detected source with
SExtractor on the original ACS F814W-band image (i.e., not on the
smoothed image). We cannot use the profile fitting which is usually
used to estimate the radius and ellipticity because it is not obvious
whether or not the profile fitting can estimate intrinsic radius and
ellipticity well for very faint sources like our sources which is
fainter than previous studies. The ellipticity is defined as
$\epsilon = 1 - b / a$, where $a$ and $b$ are the major and minor
radii, respectively. We adopt SExtractor's MAG\_AUTO, MAGERR\_AUTO,
and FLUX\_RADIUS with PHOT\_FLUXFRAC of 0.5 as $I_{814}$, error of
$I_{814}$ and $R_\mathrm{HL}$, respectively. In order to obtain
half-light major radius $a_{\rm HL}$, we modified the code for
growth-curve measurement (growth.c) in SExtractor so that the
half-light radius is measured with elliptical apertures which have
the same ellipticity and position angle derived from the second-order
moments by the SExtractor rather than circular apertures. For the
double-component LAEs as single sources, these properties are
evaluated using both of SExtractor and IDL. The Errors of
$a_\mathrm{HL}$ and $R_\mathrm{HL}$ are based on the magnitude error.
The errors of ellipticity is based on local background noise
fluctuation.
These photometric properties of the ACS data are listed in
Table~\ref{tab:z4p9LAE}. Note that the 3$\sigma$ limiting magnitude
of the F814W-band images is 27.4~mag in a $1^{\prime\prime}$ diameter
aperture. All magnitudes are corrected for the Galactic extinction
of $A_\mathrm{F814W} = 0.035$ (Capak et al. 2007). In
Table~\ref{tab:z4p9LAE}, we also list the photometric properties of
the LAE candidates from S09. The 3$\sigma$ limiting magnitudes
within a $3^{\prime\prime}$ diameter aperture in the NB711--,
$i^\prime$--, and $z^\prime$--band images are 25.17, 26.49, and
25.45, respectively.
\section{MORPHOLOGICAL PROPERTIES}
The ACS counterparts of the LAEs look differently from object to
object as shown in Figures~\ref{fig:ThumbnailsDouble} and
\ref{fig:ThumbnailsSingle}. Here we examine first which emission the
ACS F814W-band image probes, Ly$\alpha$ line or UV stellar continuum.
Then we present the morphological properties of the 54 ACS-detected
LAEs measured on the ACS F814W-band image, that is, half-light radius
$R_\mathrm{HL}$, half-light major radius $a_\mathrm{HL}$, and
ellipticity $\epsilon$.
\subsection{What Do ACS F814W-band Images Probe?}\label{subsec:ACS}
As presented in Figure~\ref{fig:filters}, the transmission curve of
the F814W-band filter covers both Ly$\alpha$ line emission and
rest-frame UV continuum emission at wavelengths of $\sim
1200$--1640~{\AA} from a source at $z = 4.86$. Which emission do
ACS F814W-band images mainly probe?
\begin{figure*}
\plotone{f6}
\caption{Distribution of the 62 ACS-detected LAEs in the
$I_{814}$--$NB711$ (left), $I_{814}$--$i^\prime$ (middle), and
$I_{814}$--$z^\prime$ planes (right). For the double-component
LAEs represented by blue double circles, sum of $I_{814}$ of each
component is adopted in this plot. The dotted lines represent the
equality of $I_{814}$ and $NB711$, $i^\prime$, or $z^\prime$. The
$3\sigma$ limiting magnitude of ACS in a $1^{\prime\prime}$
diameter aperture, 27.4~mag, is shown by the horizontal dashed
line, while those of NB711--, $i^\prime$--, and $z^\prime$--band
images in a $3^{\prime\prime}$ diameter aperture, 25.17, 26.49, and
25.45~mag, are shown by the vertical dashed lines, respectively.
In the middle panel, the solid line represents the best-fit linear
relation between $I_{814}$ and $i^\prime$: $I_{814} = 0.88(i^\prime
- 29.0) + 29.0$. In the right panel, the LAEs fainter than the
$1\sigma$ limiting magnitude of $z^\prime$ are located at $z^\prime
= z^\prime(1\sigma) = 26.64$~mag with
arrows. \label{fig:I814vsNB711}}
\end{figure*}
The detected emission in the F814W-band filter seems to be primarily
from rest-frame UV continuum rather than Ly$\alpha$ line emission.
This is clearly exhibited by the presence of a positive correlation
between $I_{814}$ and $i^\prime$, which are close to $I_{814} \sim
i^\prime$, as shown in the middle panel of
Figure~\ref{fig:I814vsNB711}. The linear correlation coefficient is
estimated to be $r = 0.72$. It is similar for $z^\prime$, which
linear correlation coefficient is $r = 0.61$\footnote{In this
calculation, the LAEs with $z^\prime \ge z^\prime(1\sigma) =
26.64$~mag are excluded.}, while dispersion from $I_{814} =
z^\prime$ relation is more significant. On the other hand, the
correlation between $I_{814}$ and $NB711$ appears to be poorer
compared with the correlations between $I_{814}$ and $i^\prime$ or
$z^\prime$ (its linear correlation coefficient is $r = 0.56$). This
result is consistent with the facts that most LAEs have
observer-frame EW much smaller than $\Delta \lambda$ of the F814W
(i.e., 2511~{\AA}) and that the wavelength of the NB711 band which
is almost blue edge of the wavelength coverage of the F814W-band
filter (see Figure~\ref{fig:filters}).
Therefore, we can conclude that the ACS F814W-band images primarily
probe rest-frame UV continuum emission from young massive stars in
the LAEs at $z = 4.86$.
\subsection{Size: Half-light Radius and Half-light Major
Radius}\label{subsec:size}
Then we analyze the sizes of our LAE sample in the ACS F814W-band
images, that is, half-light radius $R_\mathrm{HL}$ and half-light
major radius $a_\mathrm{HL}$. We emphasize that these measured
sizes should be considered as the extent of the young star-forming
regions in the LAEs and they do not necessarily reflect the stellar
mass distribution since the F814W-band images mainly prove their
rest-frame UV continuum emissions at wavelengths of $\sim
1200$--1640~{\AA} as presented in
Section~\ref{subsec:ACS}\footnote{The Wide Fields Camera 3 (WFC3)
F160W-band images ($\lambda_c = 15,369$~{\AA} and $\Delta \lambda =
2683$~{\AA}) are also available only in a limited part of the COSMOS
field (210~$\mathrm{arcmin^2} \approx 3\%$ of the COSMOS field),
which are taken by the Cosmic Assembly Near-infrared Deep
Extragalactic Legacy Survey (CANDELS; Grogin et al. 2011; Koekemoer
et al. 2011). Although the F160W-band images can prove our LAE
samples at the slightly longer rest-frame wavelengths of $\sim
2390$--2850~{\AA}, only two single-component LAEs of \#46 and \#55
are covered in the CANDELS/COSMOS field; therefore, we do not show
the morphological properties of our LAE sample in the F160W-band
images in this paper. We just comment that, while their sizes in
the F160W-band images are larger than those in the F814W-band
images, the differences of the sizes between these two-band images
are consistent the differences of the PSF sizes and pixel scales.}.
Note that the measured half-light radii of $\approx 4600$
unsaturated stars with $I_{814} = 20$--22~mag and $\mathrm{FWHM} \le
4~\mathrm{pix}$ ($= 0\farcs 12$) in the ACS F814W-band images,
$R_\mathrm{PSF}$, are typically 0\farcs 07; we adopt this angular
scale as the ``PSF size'' of the ACS F814W-band images\footnote{We
also measure FWHMs of the same stars and obtain a typical FWHM of
0\farcs 1, which is consistent with the average PSF FWHM reported by
Koekemoer et al. (2007). Note that the half-light radius
$R_\mathrm{PSF}$ is smaller than the measured FWHMs of stars by a
factor of 2 in the case that the PSF is completely described by
Gaussian profile. The actual PSF is different from a Gaussian
profile and hence the ratio of $\mathrm{FWHM} / R_\mathrm{PSF}$ can
be different from 2. Since a confusion of $R_\mathrm{PSF}$ and FWHM
for the term of ``PSF size'' is seen in a non-negligible number of
literatures, we emphasized that a particular attention should be
paid to which of $R_\mathrm{PSF}$ or FWHM the ``PSF size''
indicates.} in our analysis.
\begin{figure}
\plotone{f7}
\caption{Distributions of the 54 ACS-detected LAEs in the
$R_\mathrm{HL}$--$I_{814}$ (top) and $a_\mathrm{HL}$--$I_{814}$
(bottom) planes. In the main panels, the LAEs with single- and
double-components in the ACS F814W-band images are represented by
the red filled and blue double circles, respectively. For the 8
double-component LAEs, the size and magnitude of the double ACS
components measured as a single object are plotted. The ACS
sources with $I_{814} \ge 26$~mag are shown by the small symbols.
The $3\sigma$ limiting magnitude in a $1^{\prime\prime}$ diameter
aperture and PSF half-light radius derived from stars, 27.4~mag and
0\farcs 07, are also shown by the vertical and horizontal dashed
lines, respectively. Note that we show the same value of
$R_\mathrm{PSF}$ both in the top and bottom panels since the PSF is
found to have negligibly small ellipticity (i.e.,
$\epsilon_\mathrm{PSF} < 0.03$). The solid curves in the bottom
panel indicate the 50\% detection completenesses for exponential
disk objects with the input ellipticities of $\epsilon^\mathrm{in}
= 0.0$, 0.4, and 0.8 estimated by a Monte Carlo simulation. The
same curve with $\epsilon^\mathrm{in} = 0.0$ is also depicted by
the solid curve in the top panel. The distributions shown in the
main panels are projected onto the two side panels where histograms
of $I_{814}$, $R_\mathrm{HL}$, and $a_\mathrm{HL}$ are
displayed. \label{fig:size-I814}}
\end{figure}
Figure~\ref{fig:size-I814} shows the distributions of the 54
ACS-detected LAEs in the $R_\mathrm{HL}$--$I_{814}$ and
$a_\mathrm{HL}$--$I_{814}$ planes. It is found that the ACS
magnitudes $I_{814}$ of the LAEs are widely distributed in
23.87--27.57~mag with the mean value of $\langle I_{814} \rangle =
25.99 \pm 0.11$~mag. Both sizes of $R_\mathrm{HL}$ and
$a_\mathrm{HL}$ of the LAEs are also found to be widely distributed
in $R_\mathrm{HL} = 0\farcs 06$--0\farcs 37 and $a_\mathrm{HL} =
0\farcs 07$--0\farcs 59 with mean values of $\langle R_\mathrm{HL}
\rangle = 0\farcs 133 \pm 0\farcs 010$ and $\langle a_\mathrm{HL}
\rangle = 0\farcs 175 \pm 0\farcs 017$. Their distributions are
similar with each other, having a concentration at small sizes and
an elongated tail toward large sizes. As shown in
Figure~\ref{fig:size-I814}, both distributions of $R_\mathrm{HL}$
and $a_\mathrm{HL}$ are not concentrated around the means but show a
clear separation between the single- and double-component LAEs; the
latters have larger sizes than the formers typically. Moreover, all
single-component LAEs are found to have sizes of $\lesssim 0\farcs
3$; if the double-component LAEs are excluded, the mean half-light
major radius becomes 0\farcs 13.
Most of the ACS sources have $a_\mathrm{HL} > R_\mathrm{HL}$,
implying that they have non-zero ellipticities. Since
$a_\mathrm{HL}$ is generally considered to be a more appropriate
measure of size than $R_\mathrm{HL}$ for such sources having
non-zero ellipticities, we adopt $a_\mathrm{HL}$ as the fiducial
size of the individual ACS source in the following, rather than
$R_\mathrm{HL}$.
In Figure~\ref{fig:size-I814}, in order to see the effect of
limiting surface brightness in our ACS data, we also plot the
50\% detection completeness limits for faint extended sources in the
ACS F814W-band images estimated via performing Monte Carlo
simulations; the details of our Monte Carlo simulations are
described in Appendix~\ref{subsec:MC_single}. As the resultant 50\%
detection completeness is found to depend on the input ellipticity
$\epsilon^\mathrm{in}$, we show the 50\% detection completeness
limits for $\epsilon^\mathrm{in} = 0.0$, 0.4, and 0.8 in the bottom
panel of Figure~\ref{fig:size-I814}. This simulation suggests that,
in our ACS images, extended objects may suffer from the effect of
limiting surface brightness if they have small ellipticities and are
fainter than $I_{814} \sim 26$~mag, which is close to the mean
magnitude for the ACS sources. There are found to be a
non-negligible fraction of the ACS sources (i.e., $9 / 54 = 16.7\%$)
in the domain where the detection completeness limit for
$\epsilon^\mathrm{in} = 0.0$ is below 50\%. Hence, the number
fraction of the LAEs having extended ACS sources can be larger than
the observed one.
\subsection{Ellipticity}\label{subsec:ellipticity}
\begin{figure}
\plotone{f8}
\caption{Distribution of the 54 ACS sources in the
$\epsilon$--$a_\mathrm{HL}$ plane. The symbols are the same as
those in Figure~\ref{fig:size-I814}. The vertical dashed line
indicates the PSF size of the ACS F814W-band images derived from
stars. The ID of a double-component LAE which is an outlier in the
positive relationship between ellipticity and $a_\mathrm{HL}$
(i.e., \#12) is labeled for reference. We also label the IDs of
two double-component LAEs (i.e., \#30 and \#59) for reference. The
distribution in the main panel is projected onto the two side
panels where histograms of $a_\mathrm{HL}$ and $\epsilon$ are
displayed. \label{fig:e-aHL-total}}
\end{figure}
The measured ellipticities of the 54 ACS sources are widely
distributed from 0.02 (i.e., almost round-shape) to 0.81 (i.e.,
elongated- or ellipsoidal-shape) as shown in
Figure~\ref{fig:e-aHL-total} and Table~\ref{tab:z4p9LAE}. It is
found that the double-component LAEs tend to have larger
ellipticities than the single-component LAEs; at $\epsilon > 0.6$,
all sources are the double-component LAEs.
We also find a strong positive correlation between $\epsilon$ and
$a_\mathrm{HL}$ as presented in Figure~\ref{fig:e-aHL-total} (its
Spearman's rank-order correlation coefficient $\rho$ and Kendall's
$\tau$ are $\rho = 0.74$ and $\tau = 0.58$, respectively); larger
LAEs have more elongated shapes. Moreover, all ACS sources larger
than 0\farcs 2 have elongated morphologies ($\epsilon \gtrsim 0.5$),
except for a double-component LAE at $a_\mathrm{HL} \sim 0\farcs 44$
and $\epsilon \sim 0.47$ (i.e., the LAE \#12 as labeled in
Figure~\ref{fig:e-aHL-total}). In other words, there is no LAE with
large size and round shape, that is, $a_\mathrm{HL} \gtrsim 0\farcs
2$ and $\epsilon \lesssim 0.45$. It should be emphasized that,
since such large round-shaped galaxies can be detected if they are
bright enough (i.e., $I_{814} \lesssim 26$~mag) as shown in
Figure~\ref{fig:size-I814} (see also Figure~\ref{fig:MC-detecomp}),
the absence of such galaxies can be considered as real result, not
suffered by selection bias against them.
It is possible that measuring the sizes and ellipticities of the
double-component LAEs as single sources makes the correlation
strengthen. This is because their sizes and ellipticities are found
to be well correlated with the separations between the two
components (see Tables~\ref{tab:z4p9LAE} and \ref{tab:EachDouble})
in the sense that the LAE with larger separation has larger size and
ellipticity as a system with two components. In the following
section, we re-measure the sizes and ellipticities of individual ACS
sources in the double-component LAEs separately and re-examine the
correlation with size and ellipticity for the resultant quantities.
\subsection{Size and Ellipticity of Individual ACS Component and
Their Correlation}\label{subsec:double}
As described in Section~\ref{sec:data}, the 8 ACS-detected LAEs are
found to consist of the double components with close angular
separation (i.e., $\lesssim 1^{\prime\prime}$) in the ACS images.
We have shown their morphological properties measured as single
systems with double components in the previous
Sections~\ref{subsec:size} and \ref{subsec:ellipticity}. However,
the distributions of single- and double-component LAEs in both size
and ellipticity are found to be clearly different from each other;
the double-component LAEs have systematically larger sizes and
ellipticities than the single-component LAEs as shown in
Figure~\ref{fig:e-aHL-total}. Moreover, as described in the
previous section, the sizes and ellipticities of the
double-component LAEs are found to be well correlated with the
angular separation between the two components. These findings may
indicate that, for the 8 double-component LAEs, the morphological
properties of individual ACS components should be measured
separately so that they might be similar to those of the
single-component LAEs.
\begin{figure}
\plotone{f9}
\caption{Same as Figure~\ref{fig:e-aHL-total} but for the
distribution in which the double ACS components in the 8
double-component LAEs are plotted separately. The components of
the single- and double-component LAEs are shown as red filled and
blue open circles, respectively. The double-component LAEs with
$I_{814} \ge 26$~mag are shown by the small symbols. The IDs of
the 5 double-component LAEs which are outliers in the positive
relationship between ellipticity and $a_\mathrm{HL}$ are labeled
for reference.\label{fig:e-aHL-sep}}
\end{figure}
\begin{deluxetable*}{lcccccc}
\tabletypesize{\scriptsize}
\tablecaption{\textit{ACS} F814W-band Properties for the Individual
Components in the 8 Double-component LAEs at $z = 4.86$ with ACS
Data \label{tab:EachDouble}}
\tablehead{
\colhead{ID \# \tablenotemark{a}} &
\colhead{$I_{814}$\tablenotemark{b}} &
\colhead{$a_\mathrm{HL}$\tablenotemark{c}} &
\colhead{$R_\mathrm{HL}$\tablenotemark{d}} &
\colhead{$\epsilon (I_{814})$\tablenotemark{e}} &
\colhead{$r_\mathrm{sep}$\tablenotemark{f}}\\
\colhead{} &
\colhead{(mag)} &
\colhead{(arcsec)} &
\colhead{(arcsec)} &
\colhead{} &
\colhead{(arcsec)}
}
\startdata
11a & $26.18 \pm 0.08$ & $0.15 \pm 0.01$ & $0.13 \pm 0.01$ & $0.20 \pm 0.06$ & $0.59 \pm 0.01$\\
11b & $26.37 \pm 0.10$ & $0.20 \pm 0.01$ & $0.15 \pm 0.01$ & $0.49 \pm 0.04$ & \nodata \\
12a & $25.43 \pm 0.07$ & $0.21 \pm 0.01$ & $0.19 \pm 0.01$ & $0.17 \pm 0.06$ & $0.47 \pm 0.01$\\
12b & $24.49 \pm 0.03$ & $0.26 \pm 0.01$ & $0.23 \pm 0.01$ & $0.23 \pm 0.03$ & \nodata \\
19a & $25.86 \pm 0.07$ & $0.14 \pm 0.01$ & $0.12 \pm 0.01$ & $0.25 \pm 0.05$ & $0.70 \pm 0.01$\\
19b & $25.76 \pm 0.07$ & $0.15 \pm 0.01$ & $0.13 \pm 0.01$ & $0.32 \pm 0.04$ & \nodata \\
21a & $26.72 \pm 0.12$ & $0.13 \pm 0.02$ & $0.11 \pm 0.02$ & $0.29 \pm 0.09$ & $0.98 \pm 0.01$\\
21b & $25.32 \pm 0.04$ & $0.13 \pm 0.00$ & $0.13 \pm 0.00$ & $0.06 \pm 0.04$ & \nodata \\
30a & $24.56 \pm 0.02$ & $0.22 \pm 0.00$ & $0.19 \pm 0.01$ & $0.30 \pm 0.02$ & $0.36 \pm 0.00$\\
30b & $24.69 \pm 0.01$ & $0.10 \pm 0.00$ & $0.09 \pm 0.00$ & $0.15 \pm 0.02$ & \nodata \\
37a & $25.94 \pm 0.09$ & $0.15 \pm 0.01$ & $0.13 \pm 0.01$ & $0.26 \pm 0.07$ & $0.41 \pm 0.01$\\
37b & $25.98 \pm 0.10$ & $0.14 \pm 0.01$ & $0.13 \pm 0.01$ & $0.26 \pm 0.07$ & \nodata \\
59a & $26.39 \pm 0.11$ & $0.21 \pm 0.02$ & $0.18 \pm 0.02$ & $0.30 \pm 0.11$ & $0.53 \pm 0.01$\\
59b & $26.08 \pm 0.10$ & $0.23 \pm 0.02$ & $0.19 \pm 0.02$ & $0.31 \pm 0.08$ & \nodata \\
68a & $25.17 \pm 0.02$ & $0.09 \pm 0.00$ & $0.09 \pm 0.00$ & $0.06 \pm 0.03$ & $0.97 \pm 0.01$\\
68b & $27.23 \pm 0.15$ & $0.17 \pm 0.02$ & $0.13 \pm 0.02$ & $0.39 \pm 0.10$ & \nodata
\enddata
\tablecomments{(a) The LAE ID given in Shioya et al.~(2009). (b)
SExtractor's MAG\_AUTO magnitude and its $1\sigma$ error. (c)
Half-light major radius and its $1\sigma$ error measured on ACS
F814W-band images. (d) Half-light radius and its $1\sigma$ error
measured on ACS F814W-band images. (e) Ellipticity and its
$1\sigma$ error measured on ACS F814W-band images. (f) Angular
separation between the two components in a double-component LAE and
its $1\sigma$ error.}
\end{deluxetable*}
Figure~\ref{fig:e-aHL-sep} shows the resultant distribution in
which, even for the double-component LAEs, both of $\epsilon$ and
$a_\mathrm{HL}$ of the individual ACS components are measured
separately using SExtractor with the same parameters shown in
Table~\ref{table:SExParam}. The morphological properties for the
individual components of the 8 double-component LAEs as well as
their angular separations are listed in Table~\ref{tab:EachDouble}.
Compared with the distributions shown in
Figure~\ref{fig:e-aHL-total}, the distribution of the
double-component LAEs becomes similar to that of the
single-component LAEs, while there seem to be five outliers at
$a_\mathrm{HL} \sim 0\farcs 21$--0\farcs 26 and $\epsilon \sim
0.17$--0.31; the outliers are the LAEs \#12a, \#12b, \#30a, \#59a,
and \#59b as labeled in Figure~\ref{fig:e-aHL-sep}. As shown in
Figure~\ref{fig:ThumbnailsDouble} and Table~\ref{tab:EachDouble},
the morphological properties of these outliers may be affected by
the other component of a pair because of the close angular
separation $r_\mathrm{sep}$, which is characterized by
$r_\mathrm{sep} \lesssim 2.5a_\mathrm{HL}$. On the other hand,
those of other double-component LAEs are found to be characterized
by $r_\mathrm{sep} \gtrsim 3a_\mathrm{HL}$ and hence they could not
be affected by the other component.
Figure~\ref{fig:e-aHL-sep} also shows that the positive correlation
between $\epsilon$ and $a_\mathrm{HL}$ still exists, while it
becomes weaker ($\rho = 0.64$ and $\tau = 0.46$) compared with the
correlation shown in Figure~\ref{fig:e-aHL-total} ($\rho = 0.74$ and
$\tau = 0.58$). If we consider that, as usually do, the LAE
consists of thin disk and the ellipticity of ACS source reflects the
inclination angle to its disk, the existence of such correlation and
the absence of the sources with large $a_\mathrm{HL}$ and small
$\epsilon$ are unnatural. We will discuss their origin(s) in
Section~\ref{subsec:dis-mor}.
\section{DISCUSSION}
\subsection{Comparison of Size and
Ellipticity with Those in The Literatures}\label{subsec:comparison}
As shown in Section~\ref{subsec:size}, the single-component LAEs are
found to be widely distributed in $a_\mathrm{HL}$ of rest-frame UV
continuum from 0\farcs 07 to 0\farcs 30 ($\langle a_\mathrm{HL}
\rangle = 0\farcs 13$), which correspond to the physical sizes of
0.45~kpc and 1.90~kpc (0.83~kpc for the mean) at $z = 4.86$. These
measured sizes are quantitatively consistent with the previous
measurements for the sizes in rest-frame UV continuum of the LAEs at
$z \sim 2$--6 compiled in Malhotra et al. (2012; see also Hagen et
al. 2014 for more recent observational results of the size
measurements for the LAEs at $z = 1.9$--3.6). Therefore, our size
measurements for the LAEs at $z = 4.86$ provide a further support to
the result of Malhotra et al. (2012), that is, the sizes of LAEs in
rest-frame UV continuum do not show redshift evolution in $z \sim
2$--6.
In contrast, the sizes of the double-component LAEs are
systematically larger than those of the previous measurements;
$a_\mathrm{HL}$ of the double-component LAEs ranges from 0\farcs 25
to 0\farcs 59 with a mean of 0\farcs 44 (see
Figure~\ref{fig:e-aHL-total}). On the other hand, their sizes are
found to be also consistent with those in the literature if their
sizes of the individual ACS components are adopted; as shown in
Table~\ref{tab:EachDouble}, $a_\mathrm{HL}$ of the individual
components in the double-component LAEs are in the range of 0\farcs
09--0\farcs 26 with a mean of 0\farcs 17 (see
Figure~\ref{fig:e-aHL-sep}). Therefore, it seems to be more natural
that the individual ACS components in the double-component LAEs are
typical LAEs and that the double-component LAEs are interacting
and/or merging galaxies compared to the interpretation that they are
sub-components (e.g., star-forming clumps) in an LAE. We will
discuss these interpretations of our ACS sources further in
Section~\ref{subsec:Implication}.
In terms of $\epsilon$, we also presented in
Figure~\ref{fig:e-aHL-sep} (i.e., the case that the ACS components
in the double-component LAEs are treated separately) that the
distribution of the ACS sources in $\epsilon$ shows a peak around
the mean ellipticity of $\langle \epsilon \rangle = 0.27$ and has
long tails toward both smaller and larger ellipticities in the range
of $\epsilon = 0.02$--0.59. This distribution in $\epsilon$ is
found to be quite similar to the previous observational estimates
for the LAEs at $z \sim 2.2$ (Shibuya et al. 2014) and $z \sim 3.1$
(Gronwall et al. 2011). Although we found the positive correlation
between $\epsilon$ and $a_\mathrm{HL}$ as shown in
Figure~\ref{fig:e-aHL-sep}, such correlation has not been
investigated so far; therefore, we do not have any previous results
that can be compared with ours. It is still unclear whether or not
such correlation is seen among LAEs at different redshifts and how
it evolves with redshift. Nevertheless, we will discuss the
origin(s) of the positive correlation in
Section~\ref{subsec:dis-mor}.
\subsection{Implication for the Sizes of the ACS-undetected LAEs}
\begin{figure}
\plotone{f10}
\caption{Frequency distributions of $i^\prime$--band magnitudes for
the 54 ACS-detected (\textit{top}) and 7 ACS-undetected LAEs
(\textit{bottom}). The $3\sigma$ limiting magnitude of
$i^\prime$--band images in a $3^{\prime\prime}$ diameter aperture,
26.49~mag, is also represented by the vertical dashed
line. \label{fig:ipDist}}
\end{figure}
Among the 61 LAEs with the ACS F814W-band imaging data, 7 LAEs are
not detected in the ACS images. Here we try to estimate the
half-light radii of these ACS-undetected LAEs using the correlation
between $I_{814}$ and $i^\prime$ found for the ACS-detected LAEs
(see Figure~\ref{fig:I814vsNB711}) and the $i^\prime$--band magnitude
distribution of the ACS-undetected LAEs.
As shown in Figure~\ref{fig:ipDist}, while the ACS-undetected LAEs
are found to be at fainter part in the $i^\prime$--band magnitude
distribution compared with the ACS-detected LAEs, most of the
ACS-undetected LAEs have similar $i^\prime$--band magnitudes to those
of the ACS-detected LAEs. Considering the result of $I_{814}
\approx i^\prime$ found for the ACS-detected LAEs, the
ACS-undetected LAEs with similar $i^\prime$--band magnitudes to the
ACS-detected LAEs ought to be detected if they are compact and have
small $R_\mathrm{HL}$. Therefore, the results of their
non-detection in the ACS images imply that the surface brightnesses
of the 7 ACS-undetected LAEs are too low to be detected; that is,
even if they are bright enough to be detected in $I_{814}$, they
cannot be detected in ACS image in the case that they are spatially
extended significantly as discussed in Section~\ref{subsec:size}
(see the 50\% detection completeness shown in the top panel of
Figure~\ref{fig:size-I814}). Therefore, large $R_\mathrm{HL}$ can
be expected for the ACS-undetected LAEs.
We can estimate the half-light radii of the ACS-undetected LAEs as
follows. First, we evaluate the expected $I_{814}$-band magnitude
from $i^\prime$--band magnitude, $I_{814} (i^\prime)$, using the
best-fit linear relation between $I_{814}$- and $i^\prime$--band
magnitudes for the ACS-detected LAEs: $I_{814}(i^\prime) = 0.88
(i^\prime - 29.0) + 29.0$. This is motivated by the result that
$I_{814}$ is well correlated with $i^\prime$ as described in
Section~\ref{subsec:ACS}. Providing $i^\prime = 25.23$--27.04~mag
for the ACS-undetected LAEs, we obtain $I_{814}(i^\prime) =
25.7$--27.3~mag. Then, as the ACS-undetected LAEs are expected to
be in the domain on the $R_\mathrm{HL}$--$I_{814}$ plane, where
detection completeness is low, a lower-limit of $R_\mathrm{HL}$ for
the ACS-undetected LAEs can be estimated from $I_{814}(i^\prime)$
and the curve in the $R_\mathrm{HL}$--$I_{814}$ plane at which
detection completeness for exponential disk objects with the input
ellipticity of $\epsilon^\mathrm{in} = 0.0$ is 50\% (see
Figure~\ref{fig:size-I814}). As a result, the expected half-light
radii of the ACS-undetected LAEs are $R_\mathrm{HL} \gtrsim 0\farcs
07$--0\farcs 32. This result may imply that there are some LAEs
with very large $R_\mathrm{HL}$ among the ACS-undetected LAEs.
\subsection{Comparison of the Sizes in Ly$\alpha$ and UV Continuum}
\begin{figure}
\plotone{f11}
\caption{Distribution of the 60 LAEs with ACS F814W--band imaging
data and with measured $a_\mathrm{HL} (\mathrm{NB711})$ in the
$a_\mathrm{HL} (\mathrm{NB711})$--$NB711$ plane. The 7
ACS-undetected LAEs are shown by green crosses and the 45 (8)
ACS-detected LAEs with single (double) component(s) are represented
by red filled (blue double) circles. The $3\sigma$ limiting
magnitude in a $3^{\prime\prime}$ diameter aperture and the PSF
half-light radius of NB711--band images, 25.17~mag and 0\farcs 25,
respectively, are also indicated by the vertical dotted and
horizontal dashed lines. \label{fig:FWHMvsNB711}}
\end{figure}
In Section~\ref{subsec:size}, we found that the ACS-detected LAEs
have a wide range of $a_\mathrm{HL}$ in the ACS F814W-band images
from 0\farcs 07 to 0\farcs 59 ($\langle a_\mathrm{HL}
(\mathrm{F814W}) \rangle = 0\farcs 175 \pm 0\farcs
017$\footnote{Only in this Section and Figure~\ref{fig:RHLvsFWHM},
in order to avoid confusion with $a_\mathrm{HL} (\mathrm{NB711})$,
we refer the half-light major radius in the ACS F814--band image as
$a_\mathrm{HL} (\mathrm{F814W})$ rather than $a_\mathrm{HL}$ used in
the other parts of this paper.}). These angular scales correspond
to the physical scales of 0.45--3.8~kpc ($1.11\pm 0.11$~kpc for the
mean) at $z = 4.86$. As shown in Figure~\ref{fig:FWHMvsNB711}, the
half-light major radii in the NB711--band images, $a_\mathrm{HL}
(\mathrm{NB711})$, of the 61 LAEs with ACS data are also found to
widely distribute in 0\farcs 31--1\farcs 64, corresponding to the
physical scales of 1.97--10.45~kpc at $z = 4.86$. Since the PSF
half-light radius of NB711--band images is 0\farcs 25\footnote{Note
that the PSF FWHM of the NB711--band images is estimated to be
0\farcs 79 (Shioya et al. 2009).}, most LAEs are significantly
extended in Ly$\alpha$ emissions. This result is consistent with
previous studies (e.g., Taniguchi et al. 2005, 2009, 2015; Malhotra
et al. 2012; Mawatari et al. 2012; Momose et al. 2014).
It is interesting to examine the relation between $a_\mathrm{HL}
(\mathrm{NB711})$ and $a_\mathrm{HL} (\mathrm{F814W})$ for the
ACS-detected LAEs. As shown in Figure~\ref{fig:RHLvsFWHM},
$a_\mathrm{HL} (\mathrm{NB711})$ is systematically larger than
$a_\mathrm{HL} (\mathrm{F814W})$ except for three double-component
LAEs with $a_\mathrm{HL} (\mathrm{NB711}) / a_\mathrm{HL}
(\mathrm{F814W}) \approx 1$ (i.e., the LAEs \#11, \#19, and \#59
have the ratios of 1.04, 0.85, and 0.96, respectively). The ratio
of $a_\mathrm{HL} \mathrm{(NB711)} / a_\mathrm{HL} (\mathrm{F814W})$
is widely distributed from $\approx 1$ to $\approx 10$. This result
may be a consequence of the non-detection of extended UV continuum
in the ACS images since extended sources are difficult to be
detected as discussed in Section~\ref{subsec:size}. Nevertheless,
the large ratio of $a_\mathrm{HL} \mathrm{(NB711)} / a_\mathrm{HL}
(\mathrm{F814W})$ can be a real feature for high-$z$ LAEs. In this
case, it is suggested that the compact star-forming regions in the
LAEs (i.e., $\lesssim 0\farcs 30$ or $\lesssim 1.9$~kpc) observed by
the ACS F814W-band images ionize the surrounding gas, which emits
spatially extended Ly$\alpha$ (i.e., $\gtrsim 0\farcs 3$ or $\gtrsim
1.9$~kpc) detected in the NB711-band images.
\begin{figure}
\plotone{f12}
\caption{Distribution of the 53 ACS sources with measured
$a_\mathrm{HL} (\mathrm{NB711})$ in the $a_\mathrm{HL}
\mathrm{(NB711)}$--$a_\mathrm{HL} \mathrm{(F814W)}$ plane. Symbols
are the same as those in Figure~\ref{fig:e-aHL-total}. The PSF
sizes of ACS F814W-- and NB711--band images are presented by
vertical and horizontal dashed lines, respectively. The dotted
lines represent $a_\mathrm{HL} \mathrm{(NB711)} / a_\mathrm{HL}
\mathrm{(F814W)} = 1$ (lower) and 5 (upper) as labeled. The IDs of
the 3 outliers located at $a_\mathrm{HL} (\mathrm{NB711}) /
a_\mathrm{HL} (\mathrm{F814W}) \approx 1$ are also labeled for
reference. \label{fig:RHLvsFWHM}}
\end{figure}
\subsection{Origin of the Correlation between Ellipticity and
Size}\label{subsec:dis-mor}
As described in Section~\ref{subsec:ellipticity}, the ACS sources
show a strong positive correlation between ellipticity $\epsilon$
and half-light major radius $a_\mathrm{HL}$, that is, larger ACS
sources have more elongated morphologies
(Figure~\ref{fig:e-aHL-total}). As shown in
Section~\ref{subsec:double} and Figure~\ref{fig:e-aHL-sep}, while
the correlation is found to be strengthened by our measurements of
$\epsilon$ and $a_\mathrm{HL}$ for the two ACS sources in individual
double-component LAEs collectively, the correlation still exists if
$\epsilon$ and $a_\mathrm{HL}$ are measured for the two ACS sources
in double-component LAEs separately.
\begin{figure*}
\plotone{f13a}
\plotone{f13b}
\plotone{f13c}
\caption{Distributions of the artificial sources in the
$\epsilon$--$a_\mathrm{HL}$ plane. 1000 artificial sources with
the exponential light profile, each of which is assumed to consist
of a single component, are generated by a Monte Carlo simulation
and prepared for each set of input parameters, that is,
$I_{814}^\mathrm{in} = 24.0$--28.0~mag, $a_\mathrm{HL}^\mathrm{in}
= 0\farcs 03$--0\farcs 36, and $\epsilon^\mathrm{in} = 0.0$--0.9
(see Appendix~\ref{subsec:MC_single} for details). The resultant
distributions are plotted separately for the artificial sources
with the input ellipticities $\epsilon^\mathrm{in}$ of 0.0 (top),
0.4 (middle), and 0.8 (bottom), as well as those with the output
magnitudes $I_{814}^\mathrm{out}$ as labeled in each panel. The
grayscale represents the number of the artificial sources in the
linear scale from 0 (white) to $n_\mathrm{max}$ (black), where
$n_\mathrm{max}$ is the maximum number of the artificial sources in
a grid and different among panels. Note that these grayscales are
evaluated only from the detected artificial sources. The vertical
dashed lines indicate the PSF half-light radius $R_\mathrm{PSF}$
and the horizontal dotted lines represent $\epsilon^\mathrm{in}$.
The observed 62 ACS sources are overlaid with the same symbols as
those in Figure~\ref{fig:e-aHL-sep}. \label{fig:MCsingle}}
\end{figure*}
Here, we examine the possibilities that the observed correlation
between $\epsilon$ and $a_\mathrm{HL}$ is (1) an
``\textit{apparent}'' correlation caused by deformation effects for
a single source (e.g., pixelization, PSF broadening, and shot noise)
and (2) the ``\textit{intrinsic}'' correlation originated from
blending with unresolved double or multiple sources through Monte
Carlo simulations. We consider the correlation between $\epsilon$
and $a_\mathrm{HL}$ for the individual ACS sources of the
double-component LAEs (i.e., the correlation shown in
Figure~\ref{fig:e-aHL-sep}) since it is more natural interpretation
as described in Section~\ref{subsec:double}.
\subsubsection{Apparent Correlation Caused by Deformation
Effects}\label{subsubsec:SingleDeform}
As shown in the previous section, our ACS sources are typically
very compact and faint. In general, the sizes and ellipticities of
compact sources whose angular scales are comparable to the pixel
scale can be modified because of the pixelization of the digital
images depending on their places on the pixels. However, since the
ACS images we used have sufficiently large PSF half-light radius of
$R_\mathrm{PSF} = 0\farcs 07$ compared to the pixel scale of
0\farcs 03, such pixelization seems not to affect the morphological
parameters of the detected sources, which are definitely affected
by PSF broadening. The sizes and ellipticities of faint sources
whose surface brightnesses are comparable to the surface brightness
limit of imaging data also tend to be modified by shot noise; that
is, if a bright pixel contaminated by shot noise appears near a
compact faint source, they could be blended with each other and
detected as a single elongated source via our source detection
using SExtractor. The angular separation between the
noise-contaminated pixel and source is translated into the size and
ellipticity of the detected source. Therefore, a positive
correlation between $\epsilon$ and $a_\mathrm{HL}$ is naturally
expected to emerge even if source has intrinsically compact and
perfectly round shape.
\begin{figure*}
\plotone{f14}
\caption{Same as Figure~\ref{fig:MCsingle}, but for the result of
a Monte Carlo simulation for the artificial sources with various
$\epsilon^\mathrm{in}$ uniformly distributed in
0.0--0.9. \label{fig:MCsingle_all}}
\end{figure*}
In order to examine these deformation effects from PSF broadening
and shot noise on the distribution in the
$\epsilon$--$a_\mathrm{HL}$ plane, we perform the Monte Carlo
simulation which is the same as the one done to estimate the
detection completeness (see Appendix~\ref{subsec:MC_single} for the
details). Figure~\ref{fig:MCsingle} shows the resultant
distributions of the detected artificial sources with
$\epsilon^\mathrm{in} = 0.0$ (top), 0.4 (middle), and 0.8 (bottom)
in the $\epsilon^\mathrm{out}$--$a_\mathrm{HL}^\mathrm{out}$ plane.
As expected, the deformation effects are found to produce a
correlation between $\epsilon$ and $a_\mathrm{HL}$ which is similar
to the observed correlation, although the sources intrinsically
distribute in the
$\epsilon^\mathrm{in}$--$a_\mathrm{HL}^\mathrm{in}$ plane
uniformly.
How do these deformation effects produce such an apparent
correlation between $\epsilon$ and $a_\mathrm{HL}$? The PSF
broadening significantly affects the shapes of the detected sources
with small sizes, in the sense that their measured ellipticities
$\epsilon^\mathrm{out}$ converge on the PSF ellipticity,
$\epsilon_\mathrm{PSF} \approx 0$, regardless of
$I_{814}^\mathrm{in}$ and $\epsilon^\mathrm{in}$ of them. Since
this effect becomes less important for larger sources with
sufficiently bright surface brightnesses, their
$\epsilon^\mathrm{out}$ are expected to be reproduced as
$\epsilon^\mathrm{out} \sim \epsilon^\mathrm{in}$. Therefore, if
the sources have non-zero $\epsilon^\mathrm{in}$, a correlation
between $\epsilon^\mathrm{out}$ and $a_\mathrm{HL}^\mathrm{out}$
emerges, as shown in the two left-most panels for
$\epsilon^\mathrm{in} = 0.4$ and 0.8 of Figure~\ref{fig:MCsingle}.
On the other hand, the effects of shot noise can be easily seen in
the apparent positive correlation between $\epsilon^\mathrm{out}$
and $a_\mathrm{HL}^\mathrm{out}$ for the sources with
$\epsilon^\mathrm{in} = 0.0$; the ellipticities of the detected
sources increase with $a_\mathrm{HL}^\mathrm{out}$ although their
input ellipticities are exactly zero. The emergence of the
correlation can be interpreted via a combination of lower detection
completenesses and larger influences of noise-contaminated pixel
for the sources with lower surface brightnesses, that is, those
with larger sizes and/or smaller ellipticities (see
Figure~\ref{fig:MC-detecomp}). These effects are more significant
for the sources with fainter magnitudes of $I_{814}^\mathrm{out}$
and hence the slopes of the correlation become steeper for fainter
sources. For the sources with $I_{814}^\mathrm{out} > 26$~mag,
since the effects of shot noise are dominant, the correlation does
not depend on $\epsilon^\mathrm{in}$ significantly as shown in the
two right-most panels of Figure~\ref{fig:MCsingle}.
As a combination of these deformation effects, the detected
artificial sources distribute similar to the observed distribution
in the $\epsilon^\mathrm{out}$--$a_\mathrm{HL}^\mathrm{out}$ plane
as shown in Figure~\ref{fig:MCsingle_all}, although they are
uniformly distributed in the
$\epsilon^\mathrm{in}$--$a_\mathrm{HL}^\mathrm{in}$ plane. This
result may indicate that the observed correlation between
$\epsilon$ and $a_\mathrm{HL}$ is apparent one caused by the
deformation effects. The dispersion of $\epsilon^\mathrm{out}$ for
a given $a_\mathrm{HL}^\mathrm{out}$ is predicted to be larger for
the sources with bright magnitudes of $I_{814}^\mathrm{out}$. This
is because the distributions of the brighter sources in the
$\epsilon^\mathrm{out}$--$a_\mathrm{HL}^\mathrm{out}$ plane do
depend on $\epsilon^\mathrm{in}$ and those of the fainter sources
do not. Our simulation suggests that, in order to reproduce the
distributions of the LAEs with relatively bright (i.e., $I_{814}
\approx 25$--26~mag) and large sizes and ellipticities (i.e.,
$a_\mathrm{HL} \approx 0\farcs 20$ and $\epsilon \approx 0.45$),
intrinsically large ellipticities (i.e., $\epsilon^\mathrm{in}
\gtrsim 0.8$) are required. We note that the observed distribution
can be reproduced even better if the artificial sources have a
Gaussian distribution peaked at $(a_\mathrm{HL}^\mathrm{in},\
\epsilon^\mathrm{in}) \sim (0\farcs 15,\ 0.3)$.
\subsubsection{Intrinsic Correlation Originated from Blending with
Unresolved Double/Multiple Sources}\label{subsubsec:DoubleDeform}
\begin{figure*}
\plotone{f15}
\caption{Same as Figure~\ref{fig:MCsingle}, but for the results of
a Monte Carlo simulation in the case that a pair of two identical
objects is detected as a single-blended source. 5000 sources
which consists of two identical objects with
$a_\mathrm{HL}^\mathrm{in} = 0\farcs 07$ and $\epsilon^\mathrm{in}
= 0.0$ are generated in each set of input parameters, that is,
$I_{814}^\mathrm{in} = 25.5$--28.0~mag ($\Delta
I_{814}^\mathrm{in} = 0.5$~mag) and angular separation of 0\farcs
03--0\farcs 30 ($\Delta r_\mathrm{sep} = 0\farcs 03$). Note that
the grayscales are evaluated only from the artificial sources
detected as single sources (i.e., the sources detected as double
detached sources are neglected). The solid curves show the
expected correlation between $\epsilon^\mathrm{out}$ and
$a_\mathrm{HL}^\mathrm{out}$ through the separation between two
identical objects $r_\mathrm{sep}$, which is represented as
Equations~(\ref{eq-ExpectedCorr1}) and (\ref{eq-ExpectedCorr2}),
with $a_\mathrm{HL}^\mathrm{in} = 0\farcs 10$ (left) and 0\farcs
15 (right). \label{fig:MCdouble}}
\end{figure*}
Another possible origin of the positive correlation between
$\epsilon$ and $a_\mathrm{HL}$ is \textit{blending} with unresolved
double or multiple sources. Let us consider a simplified situation
where two identical round-shaped objects with half-light radius of
$a_\mathrm{HL}^\mathrm{in}$ are located closely with a separation
of $r_\mathrm{sep}$. If these objects are blended as a single
elongated source because of their small angular separation (i.e.,
$r_\mathrm{sep} \lesssim 2 a_\mathrm{HL}^\mathrm{in}$), its
half-light major radius of $a_\mathrm{HL}^\mathrm{out}$ and
ellipticity of $\epsilon^\mathrm{out}$ can be roughly parameterized
with $a_\mathrm{HL}^\mathrm{in}$ and $r_\mathrm{sep}$ as
\begin{eqnarray}
a_\mathrm{HL}^\mathrm{out} & \approx & \left(r_\mathrm{sep} +
2a_\mathrm{HL}^\mathrm{in}\right) / 2, \label{eq-ExpectedCorr1}\\
\epsilon^\mathrm{out} & \approx &
r_\mathrm{sep} / \left(r_\mathrm{sep} +
2a_\mathrm{HL}^\mathrm{in} \right).
\label{eq-ExpectedCorr2}
\end{eqnarray}
Since both $a_\mathrm{HL}^\mathrm{out}$ and $\epsilon^\mathrm{out}$
are found to increase with $r_\mathrm{sep}$, a positive correlation
between these two quantities will emerge even though each component
has a perfectly round shape. Under this interpretation, the
observed correlation contains useful information that the LAEs may
consist of two or more components with small angular separation.
In this case, the correlation can be considered as an intrinsic one
not an apparent one described in the previous subsection.
In order to examine whether this interpretation results in a
similar distribution in the $\epsilon$--$a_\mathrm{HL}$ plane to
the observed one quantitatively, we perform Monte Carlo simulations
whose details are described in Section~\ref{subsec:MC_double}. The
resultant distribution of the artificial sources in the
$\epsilon^\mathrm{out}$--$a_\mathrm{HL}^\mathrm{out}$ plane are
shown in Figure~\ref{fig:MCdouble}. Since the distributions for
the sources with $I_{814}^\mathrm{out} > 26$~mag are completely
determined by the effects of shot noise, the distributions for the
double-component sources with $I_{814}^\mathrm{out} > 26$~mag are
similar to those for the single-component sources with
$I_{814}^\mathrm{out} > 26$~mag shown in Figures~\ref{fig:MCsingle}
and \ref{fig:MCsingle_all}. On the other hand, the distributions
for the double-component sources with $I_{814}^\mathrm{out} <
26$~mag are different from those for the single-component sources.
And these distributions appear to be consistent with the expected
distributions from the simplified situation shown by the solid
curves in Figure~\ref{fig:MCdouble}. While the observed
distribution of the LAEs with $I_{814} < 26$~mag is reproduced
well, the LAEs with relatively bright (i.e., $I_{814} \approx
25$--26~mag) and large sizes and ellipticities (i.e.,
$a_\mathrm{HL} \approx 0\farcs 20$ and $\epsilon \approx 0.45$) are
failed to be reproduced. This is because $\epsilon^\mathrm{out}$
cannot be much larger than 0.5 via such a blending of two identical
sources since the angular separation should be smaller than $\sim
2a_\mathrm{HL}^\mathrm{in}$ in order to be detected as a
single-blended source. However, these LAEs may also be reproduced
if non-zero intrinsic ellipticities are adopted. Moreover, the
observed distribution of the LAEs are reproduced even better if the
artificial sources with $\epsilon^\mathrm{in} = 0$ have a Gaussian
distribution peaked at $(a_\mathrm{HL}^\mathrm{in},\
r_\mathrm{sep}) \sim (0\farcs 10,\ 0\farcs 15)$.
We note that, in our simulation, only the artificial sources with
large separation (i.e., $\gtrsim 0\farcs 3$) are well resolved into
two detached sources. This result explain the observed results
that the double-component LAEs have angular separation of
$r_\mathrm{sep} > 0\farcs 36$ as shown in
Table~\ref{tab:EachDouble} and that the single-component LAEs have
$a_\mathrm{HL} < 0\farcs 3$. In this interpretation, some of the
46 single-component LAEs may contain double or multiple components
with close angular separations. Moreover, some of the components
in the 8 double-component LAEs can be further resolved into compact
components; that is, they can be regarded as multiple-component
LAEs which consists of three or more components. Therefore, the
double-component fraction in our sample could be as high as
$\approx 100$\%.
\subsection{Dependence of Ly$\alpha$ Line EW and Luminosity on
Size}\label{subsec:EW+LLya-Size}
\begin{figure}
\plotone{f16}
\caption{Distribution in the $\mathrm{EW_0}$--$a_\mathrm{HL}$ (top)
and $L (\mathrm{Ly\alpha})$--$a_\mathrm{HL}$ planes (bottom). The
filled circles with upward arrows in the top panel indicate the
lower limit of $\mathrm{EW_0}$. The vertical dashed line indicates
the PSF size of ACS image of 0\farcs 11. The boxes with error bars
represent the median values of $\mathrm{EW_0}$ (top) and $L
(\mathrm{Ly\alpha})$ (bottom) for the ACS counterparts in each bin
of $a_\mathrm{HL}$, in which the same number of the counterparts
enters. For the LAEs with lower limit of $\mathrm{EW_0}$, we use
their $\mathrm{EW_0}$ lower-limit values to evaluate the median
values. On the other hand, the crosses with error bars in the top
panel show the median values of $\mathrm{EW_0}$ for the case that
the LAEs with lower limit of $\mathrm{EW_0}$ are neglected.
\label{fig:EW+LLya-aHL}}
\end{figure}
It has been reported observationally that the high-$z$ LAEs and LBGs
exhibit anti-correlation between size measured in rest-frame UV
continuum and rest-frame Ly$\alpha$ EW, that is, the galaxies with
large $\mathrm{EW_0}$ tend to have smaller sizes (e.g., Law et
al. 2012; Vanzella et al. 2009; Pentericci et al. 2010; Shibuya et
al. 2014; see also Bond et al. 2012 against these results). As
shown in the top panel of Figure~\ref{fig:EW+LLya-aHL}, our LAE
sample at $z = 4.86$ also present such anti-correlation between
$\mathrm{EW_0}$ and $a_\mathrm{HL}$. However, this result seems to
depend on the treatment of the LAEs with lower limits of
$\mathrm{EW_0}$. If they are included to calculate the
binned-median values of $\mathrm{EW_0}$ using their lower limits of
$\mathrm{EW_0}$, the anti-correlation between $\mathrm{EW_0}$ and
$a_\mathrm{HL}$ is clearly seen as shown by the boxes with
error-bars in the top panel of Figure~\ref{fig:EW+LLya-aHL}. On the
other hand, if they are completely neglected to calculate the
robustly determined binned-median values, the anti-correlation
disappears. Therefore, in order to conclude whether or not the
anti-correlation between $\mathrm{EW_0}$ and $a_\mathrm{HL}$ does
exist, a deeper imaging data of the broadband which is used to
determine $\mathrm{EW_0}$ (i.e., Subaru $z^\prime$ band for our LAE
sample) is required. Our LAE sample does not show strong
correlation between $L (\mathrm{Ly\alpha})$ and $a_\mathrm{HL}$ as
shown in the bottom panel of Figure~\ref{fig:EW+LLya-aHL}, while the
dynamic range of $L (\mathrm{Ly\alpha})$ is only a factor of $\sim
5$ and the maximum $L (\mathrm{Ly\alpha})$ seems to decrease as
$a_\mathrm{HL}$ increases.
Shibuya et al. (2014) found that, for their sample of the LAEs at $z
\sim 2.2$, merger fraction decreases at large $\mathrm{EW_0}$. If
we consider the double-component LAEs as merging galaxies, as
discussed in Section~\ref{subsec:Implication}, the same trend is
also seen in our sample as shown in Figure~\ref{fig:EW+LLya-aHL};
all double-component LAEs have $\mathrm{EW_0} < 50$~{\AA}. Same
trend hold for $L(\mathrm{Ly\alpha})$. However, as presented in
Section~\ref{subsec:dis-mor}, we cannot rule out the possibility
that the single-component LAEs are merging galaxies. We note that,
although the trend seen in the $\mathrm{EW_0}$--$a_\mathrm{HL}$
plane has been usually interpreted as the absence of the galaxies
with large stellar mass (i.e., large in size) and large
$\mathrm{EW_0}$, the trend is consistent with the model in which the
galaxy merger and/or close encounter will activate Ly$\alpha$
emission. This is because the single-component LAEs can contain the
galaxies with much smaller separations than the double-component
LAEs and because galaxy pairs with smaller separations can result in
more enhanced star formation as found in the nearby universe using
the Sloan Digital Sky Survey (Patton et al. 2013). Moreover, based
on this scenario, since the single-component LAEs can contain both
of the galaxies with short and long elapsed times from galaxy
merger/interaction which activates Ly$\alpha$ emission, the median
values of $\mathrm{EW_0}$ and $L (\mathrm{Ly\alpha})$ may not depend
on the separation. This expectation is also consistent with the
observed distributions of the LAEs shown in
Figure~\ref{fig:EW+LLya-aHL}.
\subsection{Implication for Star Formation in the LAEs at $z =
4.86$}\label{subsec:Implication}
We detected 54 counterparts in the ACS images for our LAEs at $z =
4.86$ in the COSMOS field. While 8 of them have double component
with the angular separations of 0\farcs 36--0\farcs 98 (i.e.,
2.3--6.2~kpc at $z = 4.86$), the magnitudes and morphologies of
individual components were found to be similar to those of the other
46 single-component LAEs (see Figure~\ref{fig:e-aHL-sep} and
Tables~\ref{tab:z4p9LAE} and \ref{tab:EachDouble}) and the typical
LAEs in the literature (e.g., Malhotra et al. 2012; Hagen et
al. 2014). This result indicates that the double-component LAEs are
interacting and/or merging galaxies with close separation, that is,
the projected separation is comparable to or not larger than ten
times of the size of a galaxy, $r_\mathrm{sep} / a_\mathrm{HL} \sim
1$--10.
Moreover, as shown in Section~\ref{subsec:dis-mor} through our Monte
Carlo simulations, the observed positive correlation between
$\epsilon$ and $a_\mathrm{HL}$ for our ACS-detected LAEs may
indicate that both of the single-component LAEs and the individual
components in the double-component LAEs consist of unresolved
components with close separation of $r_\mathrm{sep} \lesssim 0\farcs
3$ (i.e., $\lesssim 1.9$~kpc at $z = 4.86$), while another
interpretation for the observed correlation (e.g., apparent
correlation caused by the deformation effects such as PSF broadening
and shot noise) was still possible. Our Monte Carlo simulation also
indicates that a typical size of individual component is $\sim
0\farcs 10$--0\farcs 15 (i.e., $\sim 0.64$--0.96~kpc at $z = 4.86$).
Since the observed wavelength of the ACS F814W-band corresponds to
rest-frame UV wavelength of $\sim 1200$--1640~{\AA} at $z = 4.86$,
the ACS components are considered to be young star-forming regions.
Therefore, the small separation suggests the following two cases:
(1) the individual component is a large star-forming region in an
extended galaxy and star-formation activity in the LAEs occurs in a
clumpy fashion or (2) individual component in an ACS source is a
compact star-forming galaxy and the LAEs are the galaxies in close
encounter and/or merger. In order to distinct the above two
interpretations, deeper imaging data at longer wavelength with
similar or higher spatial-resolution than our ACS F814W-band data is
inevitable. If diffuse and faint underlying component which is
surrounding the two (or multiple) components is detected and it does
not show any signatures of galaxy interaction/merger, the clumpy
star-formation in a galaxy will be confirmed.
\begin{figure}
\plotone{f17}
\caption{Same as Figure~\ref{fig:EW+LLya-aHL}, but for distribution
in the $a_\mathrm{HL} (\mathrm{NB711})$--$\epsilon$ (top) and
$\mathrm{EW_0}$--$\epsilon$ planes (bottom). The horizontal dotted
line in the top panel indicates the PSF size of NB711--band image
of 0\farcs 25. \label{fig:Epsilon-FWHM}}
\end{figure}
In the interpretation of clumpy star-formation in a disk-like
galaxy, as usually observed in high-$z$ galaxies (e.g., Elmegreen et
al. 2009; F{\"o}rster Schreiber et al. 2011; Murata et al. 2014;
Tadaki et al. 2014), the ellipticity of a source may be an intrinsic
property related to the viewing angle of the disk. That is, large
ellipticity implies that its viewing angle is close to edge-on and
that stellar disk lies in the elongated direction. If we consider
that Ly$\alpha$ is emitted in directions perpendicular to the disk,
as predicted by the recent theoretical studies for Ly$\alpha$ line
transfer (e.g., Verhamme et al. 2012; Yajima et al. 2012b), the
pitch angle of Ly$\alpha$ emission will be at right angles to that
of UV continuum. Moreover, in such case, it is also expected that
the size in Ly$\alpha$ emission $a_\mathrm{HL} (\mathrm{NB711})$
shows a positive correlation with ellipticity measured in rest-frame
UV continuum because Ly$\alpha$ emitting region in bipolar
directions perpendicular to the disk can be viewed in longer
distance if the viewing angle of the disk is closer to edge-on, that
is, larger ellipticity. However, as presented in the top panel of
Figure~\ref{fig:Epsilon-FWHM}, we do not find such positive
correlation between $a_\mathrm{HL} (\mathrm{NB711})$ and ellipticity
for the 54 ACS-detected LAEs. Furthermore, as shown in the bottom
panel of Figure~\ref{fig:Epsilon-FWHM}\footnote{Note that,
considering the strong positive correlation between $a_\mathrm{HL}$
and $\epsilon$ shown in Figure~\ref{fig:e-aHL-total}, this plot is
qualitatively identical to the distribution in the
$\mathrm{EW_0}$--$a_\mathrm{HL}$ plane shown in the top panel of
Figure~\ref{fig:EW+LLya-aHL}.}, the observed distribution of the
LAEs in the $\mathrm{EW_0}$--$\epsilon$ plane seems not to be
quantitatively consistent with the interpretation of clumpy
star-formation in a disk-like galaxy, where $\mathrm{EW_0}$ is
expected to decrease significantly toward edge-on direction (i.e.,
larger ellipticity) via radiative transfer effects for Ly$\alpha$
resonance photons (e.g., Verhamme et al. 2012; Yajima et al. 2012b);
this result is consistent with Shibuya et al. (2014). Therefore,
the interpretation of clumpy star-formation in a disk-like galaxy
seems not to be preferred for our LAE sample. This conclusion can
be reinforced with the absence of the ACS source with large size and
round shape; if there are multiple clumpy star-forming regions in a
disk-like galaxy, some of such galaxies will be viewed from face-on,
resulting in large size and round shape. We emphasize again that
this result is not affected by a selection bias against them if they
are bright enough (i.e., $I_{814} \lesssim 26$~mag) as shown in
Figures~\ref{fig:size-I814} and \ref{fig:MC-detecomp}.
On the other hand, the interpretation of merger and/or interaction
is broadly consistent with these observed results. The correlation
between $\epsilon$ and $a_\mathrm{HL}$ can be reproduced by blending
with double (or multiple) sources with close separations as shown in
Section~\ref{subsubsec:DoubleDeform} through our Monte Carlo
simulations. The anti-correlation between the maximum value of
$\mathrm{EW_0}$ or $L (\mathrm{Ly\alpha})$ and $a_\mathrm{HL}$ is
also expected if the single-component LAEs are the merging and/or
interacting galaxies with close separations and if Ly$\alpha$
emissions are activated in such situation as described in
Section~\ref{subsec:EW+LLya-Size}. Moreover, the observed results
that the median values of $\mathrm{EW_0}$ and $L
(\mathrm{Ly\alpha})$ do not depend on $a_\mathrm{HL}$ are also
consistent with this merger interpretation as shown in
Section~\ref{subsec:EW+LLya-Size}. Therefore, the interpretation of
merging and/or interacting galaxies seems to be more feasible for
our LAE samples.
\section{CONCLUSIONS}
We have examined the morphological properties of 61 LAEs at $z =
4.86$ based on the \textit{HST}/ACS imaging in the F814W-band filter,
which are originally selected in the COSMOS field by S09. Our main
results and conclusions are summarized below.
\begin{enumerate}
\item While the ACS counterparts of 7 LAEs are not detected, 62 ACS
sources are detected with $I_{814} \lesssim 28$~mag for the
remaining 54 LAEs. Of the 54 LAEs with ACS sources, 8 LAEs
have double ACS components and 46 LAEs have single component.
\item For the double-component 8 LAEs, the angular separation
between two components are found to be 0\farcs 36--0\farcs 98
($= 2.3$--6.2~kpc at $z = 4.86$) with a mean separation of
0\farcs 63 ($= 4.0$~kpc). The angular separation is
sufficiently large compared to the PSF size of ACS image,
$R_\mathrm{PSF} = 0\farcs 07$, which is the reason why they
are separately detected.
\item Comparing ACS F814W-band magnitude $I_{814}$ with Suprime-Cam
NB711--, $i^\prime$--, and $z^\prime$--band magnitudes, we find
that the ACS F814W-band image probes rest-frame UV continuum
rather than Ly$\alpha$ line (Figure~\ref{fig:I814vsNB711}).
We observe the extent of star-forming regions in our LAE
sample at $z = 4.86$ via the F814W-band filter.
\item All of 62 ACS sources have small spatial sizes of
$a_\mathrm{HL} \sim 0\farcs 07$--0\farcs 30 ($=
0.45$--1.9~kpc) as shown in Figure~\ref{fig:e-aHL-sep}. Their
mean size is 0\farcs 14
($= 0.89$~kpc), which is consistent
with the previous measurements for the size in rest-frame UV
continuum of the LAEs at $z \sim 2$--6 in the literatures.
\item The measured ellipticities of the 62 ACS sources are widely
distributed in $\epsilon = 0.02$--0.59 and a positive
correlation between $\epsilon$ and $a_\mathrm{HL}$
(Figure~\ref{fig:e-aHL-sep}). It is evident even if we
exclude the faint ACS sources with $I_{814} > 26$~mag.
Moreover, the absence of the large (i.e., $a_\mathrm{HL}
\gtrsim 0\farcs 2$) sources with almost round shape (i.e.,
$\epsilon \lesssim 0.2$) is also found.
\item The 7 ACS-undetected LAEs are expected to have low surface
brightnesses so that they are undetected in our ACS images.
We estimate their half-light radii from Suprime-Cam
$i^\prime$--band magnitudes of $i^\prime = 25.23$--27.04~mag
(Figure~\ref{fig:ipDist}) to be $R_\mathrm{HL} \gtrsim 0\farcs
07$--0\farcs 32.
\item All ACS sources have significantly smaller sizes in UV
continuum than those in Ly$\alpha$ lines probed by NB711--band
(Figure~\ref{fig:RHLvsFWHM}). The size ratios of
$a_\mathrm{HL} (\mathrm{NB711}) / a_\mathrm{HL}
(\mathrm{F814W})$ are widely distributed in the range of
$\approx 1$--10.
\item The observed positive correlation between $\epsilon$ and
$a_\mathrm{HL}$ can be interpreted by either (1) an apparent
one caused by the deformation effects such as the PSF
broadening and shot noise or (2) an intrinsic one originated
from blending with unresolved double or multiple sources.
These are proved through our Monte Carlo simulations, which
reproduce the observed correlations as presented in
Figures~\ref{fig:MCsingle_all} and \ref{fig:MCdouble} for the
former and latter interpretations, respectively.
\item Both Ly$\alpha$ EW and luminosity of LAEs do not show strong
dependencies on sizes in rest-frame UV continuum
(Figure~\ref{fig:EW+LLya-aHL}). Moreover, there are no LAEs
with double ACS components at large $\mathrm{EW_0}$ and
$L(\mathrm{Ly\alpha})$. These results are consistent with the
model in which galaxy merger and/or close encounter will
activate Ly$\alpha$ emissions.
\item The 8 double-component LAEs are considered to be merger and/or
interacting galaxies since the angular separations between
components are significantly larger than the sizes of each
component although we cannot completely reject the possibility
that their underlying (disk) component is missed by its
faintness and they are single object with multiple
star-forming knot. The absence of the ACS sources with large
sizes and small ellipticities (Figures~\ref{fig:e-aHL-total}
and \ref{fig:e-aHL-sep}), the anti-correlation between
$\mathrm{EW_0}$ or $L (\mathrm{Ly\alpha})$ and $a_\mathrm{HL}$
(Figure~\ref{fig:EW+LLya-aHL}), and the absence of the
correlation between $\epsilon$ and $a_\mathrm{HL}
(\mathrm{NB711})$ (Figure~\ref{fig:Epsilon-FWHM}) suggest the
possibility that a significant fraction of 46 single-component
LAEs are also merger/interacting galaxies with a very small
separation. In order to decipher which interpretation is
adequate for our LAE sample, further observation with high
angular resolution at the wavelengths which are longer than
the Balmer/4000~{\AA} break in rest frame (i.e., $\gtrsim
2.3~\mu$m in observer frame for our LAE sample at $z = 4.86$)
will be required.
\end{enumerate}
\acknowledgements
We would like to thank both the Subaru and HST staff for their
invaluable help, all members of the COSMOS team, Tsutomu T. Takeuchi
at Nagoya university for his help in running our Monte Carlo
simulations using his computers, and Shinki Oyabu at Nagoya university
for providing valuable suggestions/comments. We would also like to
thank the anonymous referees for his/her useful comments. This work
is based on observations taken by the CANDELS Multi-Cycle Treasury
Program with the NASA/ESA HST, which is operated by the Association of
Universities for Research in Astronomy, Inc., under NASA contract
NAS5-26555. This work was in part financially supported by JSPS
(15340059 and 17253001).
|
2,877,628,088,974 | arxiv | \section{Introduction}
The quest for artificial materials with a strong magnetic, as well as electric, response at optical frequencies, has fueled the rich and rapidly expanding field of metamaterials~\cite{Zheludev12}. Thin, effectively two-dimensional (2D), layers of such metamaterials, known as metasurfaces, can impart an abrupt phase shift on transmitted or reflected light, allowing for unconventional beam shaping over subwavelength distances~\cite{Yu14,Luo18}. An important example of a metasurface is the Huygens' surface, based on Huygens' principle, that every point acts as an ideal source of forward propagating waves~\cite{Huygens,Love1901}. By engineering crossed electric and magnetic dipoles, a physical implementation of Huygens' fictitious sources can be realized, providing full transmission with arbitrary $2\pi$ phase allowing extreme control and manipulation of light~\cite{Pfeiffer13,Decker15,Yu15,Chong15,Shalaev15}.
The use of artificial metamaterials for these applications is due to the restriction that most natural materials interact weakly with magnetic fields at optical frequencies, to the extent that the magnetic response can be considered negligible. However, arrays of atoms with a dominant electric dipole transition can have \emph{collective} excitations which interact strongly with the magnetic light component~\cite{Ballantine20Huygens,Alaee20}. As the ability to use the electric dipole response of regular ultrathin 2D planar arrays of atoms to control and manipulate light has already been explored in many contexts~\cite{Jenkins2012a,Bettles2017,Perczel2017a,Plankensteiner2017,Facchinetti16,Bettles2016,Yoo2016,Asenjo-Garcia2017a,Mkhitaryan18,Orioli19,Shahmoon,Javanainen19,
Bettles20,Parmee2020}, this opens the way to combining both electric and magnetic degrees of freedom. Subwavelength arrays of atoms can operate at a single-photon quantum level~\cite{Guimond2019,Williamson2020b,Cidrim20,
Ballantine20ant,Zhang21,Cardoner21} and are increasingly experimentally achievable~\cite{Rui2020,Glicenstein2020,Endres16,Barredo18}. Indeed, recent experiments have already investigated the cooperative response of a single layer of atoms, and found characteristic spectral narrowing below the fundamental quantum limit for a single atom~\cite{Rui2020}.
Here we illustrate how an atomic Huygens' surface~\cite{Ballantine20Huygens} can be used for novel beam-shaping and optical manipulation applications. Collective coherent, uniform electric-dipole and magnetic-dipole excitations of atomic arrays have different fundamental reflection properties, corresponding to electric and magnetic mirrors, for which in the latter case the standard $\pi$ phase shift of the reflected beam is absent.
We show how superpositions of such collective excitations can be used to create a nearly reflection-less Huygens' surface to engineer the wavefront of the light, so that the atomic array acts as an ultrathin flat lens, an optical sorter, or a converter to different orbital angular momentum (OAM) states of light. The simulations of the atomic array produce diffraction-limited focusing of light with very short wavelength-scale focal lengths. The optical sorting is achieved by steering the beam's propagation in desired directions using the atoms with negligible reflection.
\section{Engineering optically active magnetism}
\subsection{Atom-light interaction}
\begin{figure}[htbp]
\centering
\includegraphics[width=\columnwidth]{./finalfigs/fig1rs.pdf}
\vspace{-0.6cm}
\caption{An atomic Huygens' surface with strong magnetic response at optical frequencies.
(a) The atomic array consists of a 2D square lattice in the $yz$ plane. Each site further consists of a square unit cell of four atoms, forming an atomic bilayer. (b) Uniform polarization on each atom leads to an effective electric dipole moment $\vec{d}$ from the unit cell. (c) Azimuthal polarization leads to a net zero electric dipole moment, but a perpendicular magnetic dipole moment $\vec{m}$.
}
\label{fig:fig1}
\end{figure}
We consider a rectangular lattice in the $yz$ plane, at $x=0$, with spacing $d_y$, $d_z$. Every lattice site consists of a unit cell of four atoms displaced by $\pm(a_x/2)\unitvec{x}\pm (a_y/2) \unitvec{y}$, with a $\ket{J=0}\rightarrow\ket{J^\prime=1,m=\sigma}$ transition. Such a geometry, illustrated in Fig.~\ref{fig:fig1}, could be realized as a bilayer optical lattice~\cite{Koepsell20,Gall21}, with a double-well superlattice in the $y$ direction with two minima at $\pm(a_y/2)\unitvec{y}$ in every period $d_y$, or by optical tweezers~\cite{Barredo18}. Each atom is driven by the coherent incident laser field $\boldsymbol{\mathbf{\cal E}}={\cal E}(\vec{r}) e^{ikx}\unitvec{e}_y$ (all field components and amplitudes here and in the following refer to the slowly varying, positive frequency components with the rapid variations $\sim\exp{(i\Omega t)}$ at the laser frequency $\Omega$ filtered out), as well as by the scattered field from all other atoms. In the limit of a low-intensity coherent laser-drive the atoms behave as classical linear coupled dipoles, i.e.\ as a set of damped, driven, coupled, harmonic oscillators, as described in detail in Refs.~\cite{Lee16,Morice1995a,Ruostekoski1997a,
Sokolov2011}. The origin of these dipoles are the quantum-mechanical electronic transitions between atomic orbitals. Due to the selection rules, only resonant transitions corresponding to electric dipoles can be relevant, while other multipoles, including magnetic dipoles, are negligible. The total electric field amplitude is then the sum of the incident field amplitude and the scattered field from each atom acting as a point electric dipole~\cite{Jackson}, and from the resulting coupled equations the light field can be solved
exactly~\cite{Lee16}. For subwavelength spacing, the light-mediated long-range interactions between the atoms are strong due to recurrent multiple scattering where a photon is scattered more than once by the same atom.
The steady-state of the system in the low light-intensity limit is described by the set of polarization amplitudes $\mathcal{P}_{\sigma}^{(j)}$, where
the dipole moment of atom $j$ is $\vec{d}_j=\mathcal{D}\sum_\sigma \mathcal{P}_\sigma^{(j)}\unitvec{e}_\sigma$, and $\mathcal{D}$ and $\unitvec{e}_\sigma$ are the reduced dipole matrix element and the relevant unit vector, respectively~\cite{Lee16}. An identical formalism, without drive, describes the decay of a single-photon excitation in the absence of an incident laser field, where $\mathcal{P}_\sigma^{(j)}$ then represents the probability amplitude of the excitation on level $\sigma$ of atom $j$~\cite{SVI10,Ballantine20ant}.
The response of the array to incident light can be understood in terms of the collective excitation eigenmodes $\vec{v}_n$ of the radiatively coupled atoms, with eigenvalues $\delta_n+i\upsilon_n$, where $\delta_n$ is the collective line shift and $\upsilon_n$ is the collective linewidth, which can differ dramatically from the linewidth $\gamma$ of a single atom~\cite{Jenkins_long16}. It is useful to define the biorthogonality occupation measure of the collective eigenmode $\vec{v}_j$ as~\cite{Facchinetti16}
\begin{equation}
L_j=\frac{\abst{\vec{v}_j^T\vec{b}}}{\sum_k\abst{\vec{v}_k^T\vec{b}}},
\end{equation}
where $\vec{b}_{\sigma+3j-1}=\mathcal{P}_\sigma^{(j)}$ denote the polarization amplitudes in vector form.
To understand the collective excitations of the entire array, we first consider an isolated unit cell of four atoms. For $a\lesssim \lambda$, the scattered field from an individual unit cell can be well described by a multipole decomposition in terms of single electric and magnetic dipoles, quadrupoles, etc., at its center. Each unit cell in isolation exhibits collective eigenmodes, owing to the light-mediated coupling between the four atoms. The eigenmode shown in Fig.~\ref{fig:fig1}(b), consists of all atomic dipoles oscillating in phase and pointing in one direction leading to an effective electric dipole moment. While this collective mode replicates the electric dipole moment of individual atoms, radiative excitations with, e.g., magnetic properties, not present in individual atoms, can also be engineered by utilizing more complex collective excitation eigenmodes.
A second eigenmode, shown in Fig.~\ref{fig:fig1}(c), consists of an arrangement of four atoms at the corners of a rectangle, representing a net zero electric dipole, but a perpendicular magnetic dipole. The orientations of quantum-mechanical atomic transitions in these four point-like discrete atoms, each of which generates an electric dipole,
approximate a circular loop of a continuous azimuthal electric polarization density. In this collective eigenmode of Fig.~\ref{fig:fig1}(c), the electric dipoles at each point due to the electronic transitions therefore leads to an equivalent circulating current around the center producing a magnetic dipole moment, analogous to a classical continuous distribution of oscillating charges on a ring. This subwavelength current loop is close to indistinguishable in the far-field from a fundamental magnetic dipole~\cite{Ballantine20Huygens,Alaee20}.
Strong light-mediated interactions between different unit cells then lead to collective radiative excitations of the whole lattice that synthesize effective electric and magnetic dipole arrays. In particular, the lattice in Fig.~\ref{fig:fig1}(a) has two collective excitation eigenmodes of interest. One is a uniform repetition of coherent, in-phase, electric dipole moments oriented along the $y$ direction on each unit cell, and has a collective linewidth comparable with a single atom~\cite{Facchinetti18}. The other is a similarly coherent, uniform, excitation of magnetic dipole moments oriented along the $z$ direction, which has a significantly narrower linewidth for large lattices.
\subsection{Magnetic Mirror}
Full reflection from an array of dipoles can occur when $\vec{E}_s^{(+)}=-\boldsymbol{\mathbf{\cal E}}$, where $\vec{E}_s^{(+)}$ ($\vec{E}_s^{(-)}$) is the scattered field in the forward (backward) direction, leading to destructive interference in the transmitted light and a standing wave in the backward direction. As seen in Fig.~\ref{fig:fig1}(b), the electric dipole collective excitation eigenmode has polarization amplitude that is symmetric around $x=0$, and so $\vec{E}_s^{(-)}=\vec{E}_s^{(+)}$.
Defining the complex reflection and transmission amplitudes,
\begin{equation}
r=\frac{\unitvec{e}_y\cdot\vec{E}_s^{(-)}}{\unitvec{e}_y\cdot\boldsymbol{\mathbf{\cal E}}},\quad t=\frac{\unitvec{e}_y\cdot\left[\boldsymbol{\mathbf{\cal E}} + \vec{E}_s^{(+)}\right]}{\unitvec{e}_y\cdot\boldsymbol{\mathbf{\cal E}}},
\end{equation}
respectively, gives $r=-1$ when $t=0$. While here we consider a collective mode of effective dipoles formed on each unit cell, this result is well-known for uniform arrays of coherently oscillating electric dipoles~\cite{ Tretyakov,Abajo07,Moitra2015,CAIT}, including equivalent single layers of atoms~\cite{Bettles2016,Facchinetti16,Shahmoon}. The $r=-1$ condition leads to a node of the standing wave at the array, equivalent to reflection from a perfect electrical conductor. Indeed, reflection from the electric-dipole collective excitation of a single-layer atomic array with subwavelength spacing has now been observed in experiments~\cite{Rui2020}.
An alternative condition, with $r=1$, occurs for reflection from magnetic dipole excitations in metal~\cite{Sievenpiper99}, dielectric~\cite{Ginn12,Liu14,Lin16}, and atomic structures~\cite{Alaee20}. For the magnetic dipole excitation shown in Fig.~\ref{fig:fig1}(c), the $y$ component of polarization is anti-symmetric in $x$. For the uniform, coherent, excitation of a large lattice with all dipoles oscillating in phase, the scattered field in the forward and backward directions depends only on this in-plane component, and $\vec{E}_s^{(-)}=-\vec{E}_s^{(+)}$. Hence, when the magnetic dipole lattice displays full reflection it is with $r=1$, and an antinode at the plane of the mirror. This distinction means that an emitter placed in the near-field will interfere constructively with its image, rather than destructively as for an electric mirror~\cite{Liu14}.
\begin{figure}[htbp]
\centering
\includegraphics[width=\columnwidth]{./finalfigs/fig2rs.pdf}
\vspace{-0.6cm}
\caption{ Collective resonance of a magnetic mirror.
(a) Magnitude (left axis) and phase (right axis) of reflection coefficient $r$ showing $r\approx 1$ close to magnetic resonance. (b) Occupation $L$ of magnetic dipole mode and $L^\prime$ of all other modes, normalized such that $\sum_j L_j=1$ at the center of the resonance. Gaussian beam with ${\cal E}(0)=1$, beam waist radius $w_0=6.4\lambda$, incident on $25\times 25 \times 4$ lattice with $d_y=d_z=0.7\lambda$, $a_x=a_y=0.15 \lambda$.
}
\label{fig:mm2}
\end{figure}
Figure~\ref{fig:mm2}(a) shows the magnitude and phase of the reflection amplitude for an incident Gaussian beam, as the detuning $\Delta=\Delta_\sigma^{(j)}=\Omega-\omega^{(j)}_\sigma$ of the laser frequency from the identical atomic resonance $\omega_\sigma^{(j)}$ of each level $\sigma$ on atom $j$, is tuned through the resonance of the collective magnetic excitation. While the magnetic dipole mode is very subradiant for large lattices, it can be well excited by an incident plane wave or Gaussian because the $\exp{(ikx)}$ rapid phase variance in the $x$ direction has an overlap with the antisymmetric polarization amplitudes of the mode. We find the numerical value $r\approx 0.99\exp{(0.1i)}$ on resonance, close to the expected value $r= 1$. This full reflection is a result of the collective nature of the excitation, and depends on the uniform, in-phase, coherent nature of the magnetic dipole oscillations, as well as the symmetry of the individual dipole radiation. The occupation $L$ of the collective excitation eigenmode corresponding most closely to this ideal uniform magnetic dipole excitation is shown in Fig.~\ref{fig:mm2}(b), along with the occupation of all other modes. Although the magnetic mode is dominant, it is the small contribution from other modes, with different symmetry, which leads to a minor deviation from $r=1$.
\begin{figure}[htbp]
\centering
\includegraphics[width=\columnwidth]{./finalfigs/fig3rs.pdf}
\vspace{-0.6cm}
\caption{Magnetic and electric mirror formed by a bilayer of atoms. Spatial variation of $|\vec{E}|^2/|{\cal E}(0)|^2$ in $z=0$ plane,
resulting from interference between incoming Gaussian beam and reflected light from (a) magnetic mirror, when the incident light is resonant with the collective magnetic dipole mode, showing antinodes at integer and half integer distances from the lattice corresponding to $r\approx 1$, and (b) on resonance with the electric dipole mode with $r\approx -1$. Parameters as in Fig.~\ref{fig:mm2}.
}
\label{fig:mm1}
\end{figure}
In Fig.~\ref{fig:mm1} the spatial dependence of the intensity of the total (incident plus scattered) light is shown at the frequency of the collective magnetic mode, and at the collective electric dipole mode resonance. As well as almost complete reflection, a clear $\lambda/4$ shift in the positions of the peaks of the resulting standing wave is evident.
\section{Huygens' surface}
We superpose the collective electric-dipole and magnetic-dipole excitations to form a nearly reflection-less Huygens' surface that controls the phase of the transmitted light to engineer its wavefront. A Huygens' surface is a physical implementation of
Huygens' principle~\cite{Huygens}, which states that every point in a propagating wave acts as a source of further forward-propagating waves. While electric and magnetic dipoles both scatter light forwards and backwards with equal amplitude, a crossed electric and magnetic dipole can lead to destructive interference in the backward direction and constructive interference in the forward direction, providing a physical realization of Huygens' fictitious sources~\cite{Love1901,Schelkunoff36}.
The principle of how simultaneous excitation of both modes can lead to full transmission, with arbitrary phase, is illustrated in Fig.~\ref{fig:hexp}. In Fig.~\ref{fig:hexp}(a), the individual response of either of the collective excitation eigenmodes (electric or magnetic) to an incident field $\boldsymbol{\mathbf{\cal E}}$ is shown, where the arrows display the forward-scattered field $\vec{E}_s^{(+)}$ and the total transmitted field $\vec{E}=\boldsymbol{\mathbf{\cal E}}+ \vec{E}_s^{(+)}$ in the complex plane for some particular detuning from resonance of the incident laser frequency, and the circles illustrate the range of possible values these fields can take as the detuning varies. As resonance is approached the scattered field grows originally at a phase $\pi/2$ to the incident field, reaching a maximum amplitude when $\vec{E}_s^{(+)}=-\boldsymbol{\mathbf{\cal E}}$, and then falling off as the phase approaches $3\pi/2$, tracing out the blue circle. The total transmitted field $\vec{E}$ meanwhile, starts equal to $\boldsymbol{\mathbf{\cal E}}$ far from resonance, then decreases to zero where the phase shifts from $\pi/2$ to $-\pi/2$, before returning to $\boldsymbol{\mathbf{\cal E}}$, tracing out the orange circle. The result is $-\pi/2 < \mathrm{arg}(t)<\pi/2$, and $|t|\leq 1$, with $|t|=0$ on resonance.
\begin{figure}[htbp]
\centering
\includegraphics[width=\columnwidth]{./finalfigs/fig4rs.pdf}
\vspace{-0.6cm}
\caption{
Principle of Huygens' surface of atoms. Possible complex values for the forward scattered field $\vec{E}_s^{(+)}$ and the total transmitted field $\vec{E}=\boldsymbol{\mathbf{\cal E}} + \vec{E}_s^{(+)}$ due to an incident field $\boldsymbol{\mathbf{\cal E}}$ for (a) a uniform electric dipole excitation and (b) crossed electric and magnetic dipoles at each site. Circles show range of possible values traced out as the detuning is varied. (c) Numerical calculation of scattered field from plane wave (${\cal E}=1$) incident on an atomic Huygens' surface as the detuning $\langle\Delta\rangle$ of the laser from the average single-atom resonance is varied. The phase of the scattered light (dashed line, right axis) ranges from $\pi/2$ to $3\pi/2$ as a function of the laser frequency, while the magnitude (solid line, left axis) can be almost double the input field. (d) Magnitude (left axis) and phase (right axis) of total forward-propagating field showing high transmission over the full $2\pi$ range. (c,d) Are for $20\times 20\times 4$ lattice with $d_y=d_z=0.8\lambda$, $a_x=0.12\lambda$, $a_y=0.11\lambda$.
}
\label{fig:hexp}
\end{figure}
For the simultaneous excitation of the uniform collective modes corresponding to crossed magnetic and electric dipoles, illustrated in Fig.~\ref{fig:hexp}(b), we write $\vec{E}_s^{(\pm)}=\vec{E}_{s,d}^{(\pm)}+\vec{E}_{s,m}^{(\pm)}$ where $\vec{E}_{s,d}^{(\pm)}$ ($\vec{E}_{s,m}^{(\pm)}$) is the contribution from the electric (magnetic) dipoles. Again, away from resonance, $\vec{E}_s^{(+)}=0$ and $t=1$. As the resonance is approached, however, the two contributions lead to double the amplitude, and on resonance $\vec{E}_{s,m}^{(+)}=\vec{E}_{s,d}^{(+)}=-\boldsymbol{\mathbf{\cal E}}$, giving $t=-1$, after which the scattered field decreases again to zero. In between the phase of $\vec{E}$ varies over the full $2\pi$ phase range while the symmetry condition $\vec{E}_{s,d}^{(-)}=-\vec{E}_{s,m}^{(-)}$ ensures $r=0$ as expected~\cite{Love1901}. The destructive interference of the reflected fields, which is a pre-condition for a Huygens' surface leading to full transmission, is known as the Kerker effect~\cite{Kerker83,Liu18}.
The numerically calculated scattered light from the atomic array in Fig.~\ref{fig:hexp}(c) shows that indeed the scattered field approaches twice the magnitude of the incident field. While the phase of the scattered field is constrained between $\pi/2$ and $3\pi/2$, as is the case for a single uniform excitation, this increased magnitude means that the total field in Fig.~\ref{fig:hexp}(d) covers a full $2\pi$ phase range while the transmission $|t|=|\vec{E}|/|\mathcal{E}|$ remains close to one. We find from a multipole decomposition that at the center of the resonance the normalized electric dipole moment from a single unit cell is $0.55$, while the magnetic dipole moment is $0.44$, with the remaining contribution from quadrupole or higher moments.
To simultaneously excite the electric and magnetic collective excitation eigenmodes, we take the level shifts $\Delta_\sigma^{(j)}$ to vary independently between atomic levels within a unit cell, while keeping them identical between each unit cell. The relative level shifts, along with the unit cell size and lattice constants, are numerically optimized to maximize transmission, and here in the studied examples vary between $\pm 4\gamma$.
The level shifts could be achieved by ac Stark shifts~\cite{gerbier_pra_2006} of standing waves, with similar periodicity as the unit cells of the lattice, such that the pattern is repeated on each cell, while for a slow variation across the array also microwave or magnetic fields can be suitable.
The result of the numerical optimization we employ here can be understood as inducing a coupling between the superradiant collective mode of uniform electric dipoles on each unit cell, with $\upsilon=1.3\gamma$, and the subradiant collective mode of uniform magnetic dipoles, with $\upsilon=0.1\gamma$. For a single atom, although the incident light directly only drives the atomic dipoles along the $y$ direction, relative level shifts of the $m=\pm1$ states lead to coupling between $x$ and $y$ polarization amplitudes. Then, over the entire lattice, repeated patterns of varying level shifts can be used to engineer an effective coupling between collective modes of the array, allowing modes to be occupied even if they are not directly driven~\cite{Facchinetti16,Facchinetti18,Ballantine20Huygens,Ballantine20Toroidal}.
The maximum magnitude of scattered field in Fig.~\ref{fig:hexp}(c) occurs when the incident light is on resonance with the magnetic dipole mode, which as a result is highly occupied with $L=0.88$. The resulting scattered field from the magnetic dipoles again approximately cancels the incident field, with $\vec{E}_{s,m}^{(+)}\sim -\boldsymbol{\mathbf{\cal E}}$, as in the magnetic mirror case shown in Fig.~\ref{fig:mm1}. However, the coupling also leads to a small occupation, $L=0.08$, of the off-resonance electric dipole mode. The smaller occupation of this mode is compensated by the larger linewidth, leading to a similar magnitude of scattered field with $\vec{E}_{s,d}^{(+)}\sim \vec{E}_{s,m}^{(+)}\sim -\boldsymbol{\mathbf{\cal E}}$, and $\vec{E}_{s,d}^{(-)}\sim -\vec{E}_{s,m}^{(-)}$. The result is close to full transmission with a $\pi$ phase shift. As the occupation of the electric dipole mode is a direct result of coupling from the magnetic mode, the occupation of both modes falls away on either side of the magnetic resonance, and the total scattered field decreases to zero.
\section{Beam focusing}
\begin{figure}[htbp]
\centering
\includegraphics[width=\columnwidth]{./finalfigs/fig5rs.pdf}
\vspace{-0.6cm}
\caption{
Beam focusing by ultrathin atomic lens at a focal length $f=10\lambda$. (a) $|\vec{E}|^2/|{\cal E}|^2$ from plane wave ${\cal E}=1$ incident on lattice at $x=0$. (b) Focal spot along $y$ axis at $x=10\lambda$, $z=0$ and (c) along $x$ axis at $y,z=0$. Focal spot has FWHM $0.77\lambda$ in plane and $4.0\lambda$ along $x$ axis. Lattice parameters as in Fig.~\ref{fig:hexp}. Atomic positions are illustrated by black dots.
}
\label{fig:focus}
\end{figure}
We show how an ultrathin flat atomic array, acting as a Huygens' surface, can achieve tight diffraction-limited focusing in a propagation distance of a few wavelengths. The transmitted phase profile is chosen to be equivalent to a lens, namely~\cite{Aieta12,Khorasaninejad16}
\begin{equation}
\phi = \left(\frac{2\pi}{\lambda}\right)\left(f-\sqrt{f^2+\rho^2}\right),
\end{equation}
where $f$ is the focal length of the lens and $\rho=\sqrt{y^2+z^2}$, but because of the abrupt phase change at the array, this is achieved in a vastly smaller propagation distance than a solid lens. Here, the phase profile is implemented by adding a slowly-varying spatially dependent shift to all levels, in the range $\pm\gamma$, such that the locally transmitted phase varies as predicted by Fig.~\ref{fig:hexp}(d). In free space the spot size is limited by the Abbe-Rayleigh diffraction limit,
$
\lambda/\left(2\sin{\theta}\right),
$
where $\theta$ is the maximum angle of incoming rays, such that $\tan{\theta} = L/(2f)$ for an array of side length $L$. An example is shown in Fig.~\ref{fig:focus}, for $f=10\lambda$. The resulting spot size has a subwavelength full-width-at-half-maximum (FWHM) of $0.78\lambda$, approximately equal to the diffraction limit in this case. Focusing close to the diffraction limit can be achieved for a range of focal lengths, down to $f\approx\lambda$. More complex phase variations could, in principle, be used to correct aberrations and further limit the focal spot size as has been achieved in plasmonic and dielectric metalenses~\cite{Liu20}.
\section{Beam steering}
\begin{figure}[htbp]
\centering
\includegraphics[width=\columnwidth]{./finalfigs/fig6rs.pdf}
\vspace{-0.6cm}
\caption{
Beam steering by $15^\circ$ by using an atomic Huygens' surface. Real part of electric field $\mathrm{Re}(E_y)/|{\cal E}(0)|$ for incident Gaussian beam being steered by atomic lattice at $x=0$. Incident beam has ${\cal E}(0)=1$ and beam waist radius $w_0=6.4\lambda$. $20\times 20 \times 4$ lattice with $d_y=0.82\lambda$, $d_z=0.65\lambda$, $a_x=0.11 \lambda$, $a_y=0.1\lambda$.
}
\label{fig:steering}
\end{figure}
\begin{figure*}[t!]
\centering
\includegraphics[width=\textwidth]{./finalfigs/fig7rs.pdf}
\vspace{-0.6cm}
\caption{Converting an optical vortex-free beam to beams with orbital angular momentum (OAM).
$|\vec{E}|/|{\cal E}(0)|$ (left of each pair) and $\mathrm{arg}(E_y)$ (right) for several OAM vortex beams. (a,b) $l=1$, (c,d) $l=-2$, (e,f) superposition of $l=1$ and $l=-1$, in $yz$ plane at $x=6\lambda$. Gaussian beam with ${\cal E}(0)=1$ and beam waist radius $w_0=6.4\lambda$ is incident on $20\times 20 \times 4$ lattice with parameters as in Fig.~\ref{fig:hexp}.
}
\label{fig:oam}
\end{figure*}
We use an atomic array to produce beam steering, a resource in controlling the flow of light, by redirecting an incoming beam in different directions. This is achieved by adding an additional level shift gradient, which varies linearly across the lattice, such that the phase of the transmitted light also varies linearly~\cite{Pfeiffer13,Ni12,Liu17bs}.
To steer the beam by an angle $\theta$ away from the normal requires the phase profile $\phi = \alpha z$ where
$\alpha = 2\pi\sin{\theta}/\lambda$.
For discrete dipoles with spacing $d_z$ in the $z$ direction, this phase profile is unique modulo $2\pi$ if $|\sin{\theta}| < (\lambda/d_z)-1$.
An example is shown in Fig.~\ref{fig:steering} where an incident Gaussian beam is transmitted at an angle of $15^\circ$ to the $x$ axis. Beam steering from atoms could be used to sort light for different subsequent stages, e.g., as part of a modular quantum information architecture. In contrast to fabricated plasmonic or dielectric systems, the detuning gradient can in principle be varied in-situ, changing the steering angle and allowing for the dynamic sorting and redirection of different light beams. Ultrathin atomic lattices thus offer a major advantage for beam steering, combined with focusing and other beam shaping, allowing multiple independent steps with separation on the order of wavelengths, all without losses or fabrication inconsistencies.
\section{Orbital angular momentum}
Metasurfaces can be used to generate vortex beams with OAM~\cite{Mehmood16,Zhang18}. The angular momentum of paraxial beams of light can be separated into spin angular momentum, depending on its polarization, and OAM, depending on the spatial variation of the field. Figure~\ref{fig:oam}(a-d) shows the creation of vortex beams with a phase winding $\exp{(il\phi)}$, achieved by an additional angle-dependent level shift, for integer $l=1,-2$, corresponding to a quantized OAM $l\hbar$ per photon. The incoming light, with no angular momentum, has been imparted with quantized OAM, with a corresponding torque on the atomic array. The integer quantization of OAM provides a larger alphabet for quantum logic than typical two-state bases, a useful resource for quantum computation~\cite{Mair01}. Particularly important for quantum applications is the ability to form coherent superpositions of different OAM values. This can be achieved by matching the phase to that of the desired transmitted beam, with an example shown in Fig.~\ref{fig:oam}(e,f). While in all cases the incoming light has an intensity maximum at the center, destructive interference leads to a vortex beam with zero intensity at the beam center, and along lines of phase discontinuities. Despite the change in intensity along the beam axis, the total integrated transmission remains high, with e.g.~$|t|^2\approx 0.97$ for the $l=1$ case.
\section{Conclusion}
Strong light-mediated interactions can be used to synthesize optical magnetism in arrays of atoms. The flexibility this provides in designing collective radiative excitations opens new avenues for the control and manipulation of light, with the potential for rapid technological as well as fundamental progress. These collective excitations can be engineered by varying the atomic level shifts, leading to very different responses to incident light. Here we have shown how an atomic lattice can be used for such versatile wavefront engineering as diffraction-limited focusing, optical sorting using beam-steering, and OAM mode conversion. Atomic arrays offer advantages over plasmonic or dielectric platforms, including the absence of absorptive loss and fabrication inhomogeneities, as well as the great flexibility to operate at the quantum limit, at the same time when the quest for developing nanophotonic quantum technologies for metasurfaces is becoming one of the key challenges in the research field~\cite{Solntsev2020}. The ultrathin nature of the arrays described here and the adaptability in wavefront engineering allows for several stages to be combined for advanced wavefront shaping and modular optical processing.
\begin{funding}
We acknowledge financial support from the UK EPSRC (Grant Nos. EP/S002952/1, EP/P026133/1).
\end{funding}
|
2,877,628,088,975 | arxiv | \section{Introduction}
In \cite{FuMa:81,Ful:84}, Fulton and MacPherson define for any scheme $X$ a
graded cohomology ring $A^*_{\op}(X)$ which equals the classical
intersection ring when $X$ is non-singular. An element of
$A^*_{\op}(X)$ is a collection of operations on Chow groups of
$X$-schemes compatible with basic operations in intersection theory.
The product structure is given by composition. By construction there
is a pullback of operational rings $A^*_{\op}(Y) \to A^*_{\op}(X)$
for any morphism $X \to Y$.
For an arbitrary singular scheme elements of $A^*_{\op}(X)$ do not
have natural interpretations in terms of algebraic cycles,
although Kimura \cite{Kim:92} showed how $A^*_{\op}(X)$ can be related to the
intersection ring of a resolution of singularities of $X$.
Despite the formal structure, there is a class of singular varieties
where the operational Chow groups are readily computable. Specifically, if $X$
is a complete linear variety then Totaro \cite{Tot:14} proved that the
pairing
$$A^k_{\op}(X) \times A_k(X) \to A_0(X) = \ZZ\; , (c,\alpha)\mapsto c \cap \alpha$$
is perfect so $A^k_{\op}(X) = \Hom(A_k(X), \ZZ)$, where $A_k(X)$ denotes the classical Chow group of $k$-dimensional cycles.
When $X$ is a complete toric variety Fulton and Sturmfels
\cite{FuSt:97} gave an explicit description of the operational
product in terms of Minkowski weights. This was generalized by Payne
\cite{Pay:06} who showed that the $T$-equivariant operational Chow ring of
a toric variety $X$
can be identified with the ring of integral piecewise polynomial functions on the fan
of $X$.
The Cox construction expresses any toric variety as a good quotient
$(\mathbb{A}^n \setminus B)/G$ where $G$ is a diagonalizable group. In
\cite{EdSa:17} the first author and Satriano showed that if $X = Z/G$
is the good quotient of a smooth variety by a linearly reductive group
then the rational operational Chow ring $A^*_{\op}(X)_\mathbb{Q}$ naturally
embeds in the equivariant Chow ring $A^*_G(Z)_\mathbb{Q}$. Moreover, the image
of an element in $A^k_{\op}(X)$ is represented by a class of the
form $\sum c_i [Z_i]$ where the $Z_i \subset Z$ are codimension-$k$
$G$-invariant subvarieties of $Z$ which are saturated with respect to
the quotient map $\pi \colon Z \to X$.
By the \'etale slice theorem \cite{Lun:73} the local model for the
good quotient of a smooth variety at a closed orbit $Gx \subset Z$ is
the quotient $V \to V/G_x$ where $V = T_{x,Z}/T_{x,Gx}$ is the normal
space to the orbit $Gx$ at $x$, and $G_x$ is the stabilizer of
$x$. Therefore, a natural problem is to compute the operational Chow
rings of good quotients $V/G$ where $V$ is a representation of a
linearly reductive group $G$. Examples of such quotients are affine
toric varieties associated to maximal dimensional
strongly convex polyhedral cones. In
\cite{Ric:19, EdRi:19} it is shown that $A^*(X) = \ZZ$ for any affine
toric variety $X$, and likewise that $\op K^0(X) =\ZZ$ where $\op K^0$
is the operational $K$-theory defined by Anderson and Payne
\cite{AnPa:15}.
The purpose of this paper is to show that stronger results hold. The
contravariant functors $A^*_{\op}$ and $\op K^0$ are both
$\mathbb{A}^1$-homotopy invariants. The cone theorem (Theorem
\ref{thm.cone}) implies the vanishing of such functors on a large
class of naturally occurring varieties including affine toric varieties
and quotients of representations of reductive groups.
Since quotients of the form $V/G$ are also the local models for good moduli
spaces of Artin stacks \cite{AHR:15} we conclude with a discussion of how the
vanishing of $A^*_{\op}(V/G)$ relates to questions about the image of
$A^*_{\op}(X)$ in $A^*{\mathcal X}$ when $X$ is the good moduli space of a
smooth algberaic stack ${\mathcal X}$.
{\bf Dedication.} It is a pleasure to dedicate this work to William Fulton on the occasion of his 80th birthday.
\section{Homotopy invariant functors}
Fix a ground field $k$ and let ${\cat Sch}/k$ denote the category of $k$-schemes
of finite type.
\begin{defn}
A homotopy invariant functor is a contravariant functor $H \colon
{\cat Sch}/k \to {\cat Ab}$ such that the pullback $H(X) \stackrel{\pi^*} \to H(X \times
\mathbb{A}^1)$ is an isomorphism for all $X$ in ${\cat Sch}/k$.
Likewise if $G$ is an algebraic group then a $G$-homotopy invariant
functor is a contravariant functor $H^G \colon {\cat Sch}^G/k \to {\cat Ab}$ such
that for any $G$-scheme $X$ the pullback $H^G(X) \stackrel{\pi^*} \to H^G(X \times
\mathbb{A}^1)$ is an isomorphism, where the action of $G$ on $\mathbb{A}^1$ is trivial.
\end{defn}
In this paper we focus on several homotopy invariant functors -- the
operational Chow cohomology ring defined in \cite{FuMa:81} and
\cite[Chapter 17]{Ful:84} as well as its equivariant counterpart
defined in \cite{EdGr:98} and the (equivariant) operational
$K$-theory defined by Anderson and Payne in \cite{AnPa:15}.
\subsection{Chow cohomology}
Let $X$ be a scheme. Following \cite{Ful:84}, let $A_k(X)$ denote the group of dimension
$k$ cycle classes modulo rational equivalence, and if $X$ is
equidimensional of dimension $n$ we let $A^k(X)$ denote the group of $(n-k)$-dimensional
cycle classes modulo rational equivalence. If $X$ is
smooth and equidimensional, then the intersection product on $A_k(X)$ as constructed in
\cite[Chapter 6.1]{Ful:84} makes $A^*(X)$ into a commutative, graded
ring.
For general schemes \cite[Chapter 17]{Ful:84} defines a graded operational
Chow cohomology ring
$A^*_{\op}(X) := \oplus_{k \geq 0} A^k_{\op}(X)$: an
element $c\in A^k_{\op}(X)$ is a collection of homomorphisms of groups:
\[ c^{(k)}_g: A_pX' \rightarrow A_{p-k}X'\]
for every morphism $g:X'\rightarrow X$ which are
compatible with respect to proper pushforward, and pullbacks
along flat morphisms and regular embeddings (see \cite[Definitions 17.1 and 17.3]{Ful:84}). The
product is given by composition and turns $A^*_{\op}(X)$ into
a graded ring called the $\textbf{Chow
cohomology ring}$ of $X$. Moreover, if
$X$ has a resolution of singularities (e.g. if the characteristic of
the ground field is zero or if $X$ is a toric variety) then
$A^*_{\op}(X)$ is known to be commutative.
If $X$ is smooth, then \cite[Corollary
17.4]{Ful:84} proves that the Poincar\'e duality map $A^k_{\op}(X)
\rightarrow A^k(X) = A_{n-k}(X)$ is an isomorphism of rings where
the intersection product agrees with the product given by composition.
\begin{rem}
In \cite[Chapter 17]{Ful:84} the Chow cohomology
ring is also denoted $A^*(X)$ without the inclusion of the subscript
'op'.
\end{rem}
\begin{prop}
Chow cohomology is an $\mathbb{A}^1$-homotopy invariant functor.
\end{prop}
\begin{proof}
Consider the pullback $\pi^* \colon A^*_{\op}(X) \to A^*_{\op}(X \times \mathbb{A}^1)$ where $\pi \colon X \times \mathbb{A}^1 \to X$ is the projection.
First note that injectivity of $\pi^*$ is a formal consequence of the functoriality; the composition $X \stackrel{\iota} \to X \times \mathbb{A}^1 \stackrel{\pi} \to X$
is the identity, where $\iota(x) = (x,0)$.
Suppose that $c \in A^k_{\op}(X \times \mathbb{A}^1)$. We wish to
show that $c = \pi^* d$ for some $d \in A^k_{\op}(X)$.
Given a morphism $Y \stackrel{f} \to X$ let
$g = f \times \id \colon Y \times \mathbb{A}^1 \to X \times \mathbb{A}^1$.
Since the flat pullback $\pi^* \colon A_*(Y) \to A_*(Y \times \mathbb{A}^1)$
is an isomorphism and this isomorphism is compatible with other operations
on Chow homology we can define a class $d \in A^*_{\op}(X)$ such
that $\pi^*d = c$ by the formula
$d_f(\alpha) = (\pi^*)^{-1}c_g(\pi^*\alpha)$.
\end{proof}
\subsubsection{Equivariant Chow cohomology}
An equivariant version of operational Chow cohomology was defined in
\cite{EdGr:98}. An element $c \in A^k_{\op, G}(X)$ is a collection operations on equivariant Chow groups $c_f \colon
A_*^G(X') \to A_{*-k}^GG(X')$ for every equivariant morphism $X'
\stackrel{f} \to X$ compatible with equivariant proper pushforward and
equivariant flat maps and equivariant regular embeddings maps. \cite[Corollary 2]{EdGr:98} states
that if $X$ admits a resolution of singularities then
$A^k_{\op,G}(X)$ can be identified with the operational Chow
group $A^k_{\op}(X_G)$ where $X_G$ is an algebraic space of the
form $X \times^G U$. Here $U$ is an open set in a representation $V$ of
$G$ on which $G$ acts freely and $\codim (V \setminus U) >
k$. It follows from this identification that the equivariant Chow
cohomology groups $A^k_{{\op},G}(X)$ enjoy all of the formal
properties of ordinary operational Chow cohomology. In particular, the
functor $A^*_{{\op},G}$ is a homotopy invariant functor on the
category of schemes or algebraic spaces with a $G$-action.
\subsection{Operational $K$-theory}
Following \cite{AnPa:15}, if $X$ is a scheme
we denote by $K_0(X)$ the Grothendieck group of coherent sheaves, and
$K^0(X)$ the Grothendieck group of perfect complexes. If $X$ has an ample
family of line bundles, then $K^0(X)$ is the same as the naive Grothendieck
group of vector bundles.
For any scheme $X$, Anderson and Payne define the {\bf operational
$K$-theory} $\op K^0(X)$ of $X$ as follows. An element $c \in \op
K^0(X)$ is a collection of operators $c_f \colon K_0(X') \to K_0(X')$
indexed by morphisms $X' \stackrel{f} \to X$ compatible with proper
pushforward, flat pullback and pullback along regular embeddings.
For any scheme $X$, there is a canonical map $\op K^0(X) \to K_0(X)$ given
by $c \mapsto c_{\id_X}({\mathcal O}_X)$. If $X$ is smooth, then
\cite[Corollary 4.5]{AnPa:15} states that this map is an isomorphism.
\begin{thm}\cite[Theorem 1.1] {AnPa:15}
$\op K^0$ is a homotopy invariant functor.
\end{thm}
\subsubsection{Equivariant operational $K$-theory}
Anderson and Payne also define the equivariant operational $K$-theory
ring as the ring of operations on the equivariant Grothendieck group of
coherent sheaves, $K_0^G(X)$. Since the equivariant Grothendieck
group is an $\mathbb{A}^1$-homotopy invariant the proof of \cite[Theorem 1.1
]{AnPa:15} goes through and we conclude that $\op K^0_G$ is a homotopy
invariant functor. If $G = T$ is a torus and $X$ is smooth,
then Anderson and Payne also
prove that $\op K^0_T(X)$ can be identified with $K_0^T(X)$.
\section{The cone theorem}
Fix a base scheme $X$ of finite type defined over a field $k$.
\begin{defn}
An $X$-cone is a scheme of the form $C = {\Spec}_X S$
where
$S = \oplus_{n=0}^\infty S_i$ is a finitely generated
graded ${\mathcal O}_X$-algebra such that $S_0 = {\mathcal O}_X$.
(Note that we do not require that $S$ be locally generated in degree one.)
More generally, if $G$ is an algebraic group and $X$ is a $G$-scheme then we say that $C = \Spec S$ is a $G$-cone
if the $S_i$ are sheaves of $G$-${\mathcal O}_X$ modules and multiplication
of local sections is $G$-equivariant.
\end{defn}
The inclusion $S_0 \to S$
defines a projection $\rho \colon C \to X$ and the identification
of $S_0 = S/S^+$ defines an inclusion $\iota \colon X \to C$. Clearly,
$\rho \circ \iota = \id_X$.
The key property of homotopy invariant functors is the following
cone theorem.
\begin{thm}{\cite[cf. Exercise IV.11.5]{Wei:13}} \label{thm.cone}
The pullbacks $\rho^*$ and $\iota^*$ are inverses. In particular
$H(X) = H(C)$. Likewise if $H^G$ is an $G$-homotopy invariant
functor and $C =\Spec S$ is a $G$-cone then $H^G(X) = H^G(Y)$.
\end{thm}
\begin{proof}
We give the proof in the non-equivariant case as the proof in
the equivariant case is identical.
Since $\rho \circ \iota = \id_X$, we know that $\iota^* \circ \rho^* \colon X \to X$ is the identity. In particular, $\rho^*$ is injective. Thus
it suffices to prove that $\rho^* \circ \iota^* \colon C \to C$ is an
isomorphism.
Since $C$ is a cone over $X = \Spec S_0$,
the map of graded rings $S \to S[t]$, sending $S_i$ to $t^i S_i$.
defines an $\mathbb{A}^1$ action
$\sigma \colon C \times \mathbb{A}^1 \to C$
with fixed scheme $X= \Spec S_0$.
Let $s_t \colon C \to C$ be the map $x \mapsto tx$. For $t \neq 0$,
$s_t$ is an isomorphism with inverse $s_{t^{-1}}$ and $s_0 \colon
C \to C$ is the composition $\iota \circ \rho$. The map
$s_t$ is itself a composition
$$C \stackrel{i_t} \to C \times \mathbb{A}^1 \stackrel{\sigma} \to C$$
where $i_t \colon X \hookrightarrow X \times \mathbb{A}^1$ is the inclusion
$x \mapsto (x,t)$.
Now if $\pi \colon X \times \mathbb{A}^1 \to X$ is the projection,
then for any $t$, $\pi \circ i_t = \id$. Since $\pi^*$ is assumed to be an isomorphism, $i_t^*$ must also be an isomorphism for any $t$. Since, for
$t \neq 0$ the composite $s_t = \sigma \circ i_t$ is an isomorphism
we see that $\sigma^*$ must also be an isomorphism. Hence,
$s_0^* = (\sigma \circ i_0)^*$ is an isomorphism.
But $s_0= (\iota \circ \rho)$, so $(\iota \circ \rho)^*$ is an
isomorphism as claimed.
\end{proof}
\begin{exa}
Let $X \subset Y$ be a closed subscheme and let $C_XY$ be the normal
cone of $X$ in $Y$. Theorem \ref{thm.cone} implies that the pullback
$A^*_{\op}(X) \to A^*_{\op}(C_XY)$ is an isomorphism. In particular if $X$ is
smooth then $A^*_{\op}(C_XY)$ is identified with the Chow ring of $X$. If
the closed embedding $X \hookrightarrow Y$ is a regular embedding (for example if
$X$ and $Y$ are both smooth) then $C_XY$ is the
normal bundle to $X$ in $Y$ and this identification follows from the usual
homotopy invariance of operational Chow rings.
\end{exa}
\subsection{Cone theorem for bivariant groups}
The operational Chow and $K$-theory rings defined by Fulton--MacPherson
and Anderson--Payne are part of a more general construction
associated to the covariant (for proper morphisms) functors $K_0$ and
$A_*$. Given a morphism of schemes $Y \to X$ the bivariant
Chow group $A^k_{\op}(Y \to X)$ is the graded abelian group consisting
of a collection of operators $c_f \colon A_*(Y') \to A_{* -k}*(X')$
for each morphism $X' \stackrel{f} \to X$ compatible with
proper pushforward and flat pullbacks and pullback and pullback along regular
embeddings, where $Y' = Y \times_X X'$.
The group $\op K^0(Y \to X)$ is defined analogously but the compatibility
is with proper pushforward, flat pullback and pullback along regular embeddings.
The groups $A^*_{\op}(Y \to X)$ and $\op K^0(Y \to X)$ are contravariant functors
on the category whose objects are morphisms of schemes $Y \to X$
and whose morphisms are cartesian diagrams
$\begin{array}{ccc}
Y' & \to & X'\\
\downarrow & & \downarrow\\ Y & \to & X
\end{array}
$\\
It is easy to show that pullback along the diagram
$\begin{array}{ccc}
Y \times \mathbb{A}^1 & \to & X \times \mathbb{A}^1\\ \downarrow & & \downarrow\\
Y & \to & X
\end{array}
$\\
induces isomorphisms $A^*_{\op}(Y \to X) \to
A^*_{\op}(Y \times \mathbb{A}^1 \to X \times \mathbb{A}^1)$,\\ $\op K^0(Y \to X) \to
\op K^0(Y \times \mathbb{A}^1 \to X \times \mathbb{A}^1)$.
As a corollary we obtain a cone isomorphism theorem for these bivariant
groups.
\begin{cor}
Given a morphism, $Y \to X$ and cone $C \to X$ let $C_Y \to Y$
be the cone obtained by base change. Then the pullbacks
$A^*_{\op}(Y \to X) \to A^*_{\op}(C_Y \to C)$
and $\op K^0(Y \to X) \to \op K^0(C_Y \to C)$ are isomorphisms.
\end{cor}
\section{Affine toric varieties}
\begin{thm}
If $X = X(\sigma)$ is an affine toric variety defined by a strongly convex
rational cone $\sigma$ in a lattice $N$, then for any homotopy invariant
functor $H$ on ${\cat Sch}/k$, $H(X) = H(T_0)$ where
$T_0$ is an algebraic torus.
\end{thm}
\begin{proof}
Since $X(\sigma)$ is an affine toric variety the proof of
\cite[Proposition 3.3.9]{CLS:11} shows that we can decompose
$X = X(\overline{\sigma}) \times T_0$ where $T_0$ is a torus
and $\overline{\sigma}$ is a full dimensional cone. Since
$\overline{\sigma}$ is full-dimensional the semi-group
$S_{\overline{\sigma}}$ is generated in positive degree, so
$R=k[X(\overline{\sigma})]$ is a positively graded ring with
$R_0 = k$.
Hence $S = k[X(\sigma)] = k[T_0] \otimes_k R$ is a positively graded ring
with $S_0 = k[T_0]$. Hence by the cone theorem,
$H(X) = H(T_0)$
\end{proof}
\begin{cor} \label{cor.toric}
If $X$ is an affine toric variety then $A^0(X) = \ZZ$ and
$A^k_{\op}(X) = 0$ for $k > 0$.
Likewise, $\op K^0(X) = \ZZ$.
\end{cor}
\begin{proof}
By the theorem we know that $A^*_{\op}(X) = A^*_{\op}(T_0)$
and $\op K^0(X) = \op K^0(T_0)$. Since a torus is an open
subset of $\mathbb{A}^n$, $A^k_{\op}(T_0) = A^k(T_0) = 0$ if $k > 0$
and $A^0(X) = \ZZ$ for any $X$. Likewise, $\op K^0(X) =
\op K^0(T_0) =K_0(T_0) = \ZZ$.
\end{proof}
\begin{exa} When $X$ is a complete toric variety
then we know that $A^k_{\op}(X) = \Hom(A_k(X), \ZZ)$. However, for
non-complete toric variety this result fails. For example
if $X = \mathbb{A}^1$ then $A_0(X) = 0$ but $A^0(X) = \ZZ$. Another example
is to let $\sigma$ denote the cone generated by
$\{(1,0,1),(0,-1,1),(-1,0,1),(0,1,1)\}$ in $\RR^3$.
One can compute that $A_2(X(\sigma)) = \ZZ/2 \oplus \ZZ$. Thus,
$\Hom(A_2(X(\sigma)), \ZZ) = \ZZ$ but by
Corollary \ref{cor.toric}, $A^2_{\op}(X(\sigma)) = 0$.
\end{exa}
\begin{exa}
In the equivariant case we have an analogous result for the $T$-equivariant
operational Chow ring and $K$-theory. To simplify the notation
we assume that the cone $\sigma$ is full dimensional.
\begin{cor}
If $X$ is an affine toric variety associated to a full dimensional cone $\sigma$
then $A^*_{\op, T}(X) = \Sym(X(T))$ and $\op K^0_T(X) = R(T)$.
Here $\Sym(X(T))$ is the polynomial algebra generated by the character group
of $T$ and $R(T)$ is the representation ring of $T$.
\end{cor}
\begin{proof}
In this case $S_0 = \Spec k$, so $A^*_{\op, T}(X) = A^*_T(\pt) =\Sym(X(T))$
by \cite[Section 3.2]{EdGr:98} and $\op K_T^0(X) =K_T^0(\pt) =R(T)$.
\end{proof}
\end{exa}
\subsection{An alternative proof of the vanishing of Chow cohomology and operational $K$-theory on affine toric varieties}
In \cite{Ric:19, EdRi:19} a more involved proof that $A^*_{\op}(X) = \op K^0(X) = \ZZ$
when $X$ is an affine toric variety is given. The proof of
both of these statements
rests on the fact that both $A^*_{\op}(X)$ and $\op K^0(X)$ satisfy the
following descent property for proper surjective morphisms.
\begin{quote}If $X' \to X$ is a proper surjective morphism and if $H$ denotes either functor
$A^*_{\op}$ or $\op K^0$ then the sequence
$$0 \to H(X)\otimes \mathbb{Q} \to H(X')\otimes \mathbb{Q} \stackrel{p_1^* - p_2^*} \to H(X' \times_X X') \otimes \mathbb{Q}$$ is
exact where $p_1, p_2$ are the two projections $X' \times_X X' \to X'$.
\end{quote}
This descent property does not hold for arbitrary homotopy
invariant functors -- for example it need not hold for the functor $\op K^0_{G}$ when
$G$ is not a torus. However when the descent property holds for a functor
$H$, it can be used
as a tool to calculate $H$ on singular schemes \cite{Kim:92, AnPa:15}.
\section{Operational Chow rings of good moduli spaces}
The goal of this section is to explain how the cone theorem for homotopy
invariant functors can shed light on questions about the structure
of the operational Chow ring for quotients of smooth varieties and, more generally, good moduli spaces of smooth Artin stacks.
\subsection{Strong cycles on good moduli spaces of Artin stacks}
Let $G$ be a linearly algebraic group acting on a scheme $X$. We say
that a scheme $Y$ equipped with a $G$-invariant morphism $p \colon X \to Y$
is a {\em good quotient} if $p$ is affine and $(p_*{\mathcal O}_X)^G = {\mathcal O}_Y$. The basic example is
the quotient $X^{ss} \to X^{ss}/G$ where $X^{ss}$ is the set of
semi-stable points (with respect to a choice of linearization) for the action of a linearly reductive group on a projective
variety $X$. This definition was extended to Artin stacks by Alper.
\begin{defn}[{\cite[Definition 4.1]{Alp:13}}]
Let ${\mathcal X}$ be an Artin stack and let $X$ be an algebraic space. We say
that $X$ is a {\em good moduli space of ${\mathcal X}$} if there is a morphism
$\pi \colon {\mathcal X} \to X$ such that
\begin{enumerate}
\item $\pi$ is {\em cohomologically affine} meaning that the pushforward functor $\pi_*$
on the category of quasi-coherent ${\mathcal O}_{\mathcal X}$-modules is exact.
\item $\pi$ is {\em Stein} meaning that the natural map ${\mathcal O}_X \to \pi_* {\mathcal O}_{\mathcal X}$ is an isomorphism.
\end{enumerate}
\end{defn}
\begin{rem}
If ${\mathcal X} = [Z/G]$ where $G$ is a linearly reductive algebraic group
then the statement that $X$ is a good moduli space for ${\mathcal X}$ is
equivalent to the
statement that $X$ is the good quotient of $Z$ by $G$.
\end{rem}
\begin{defn}[{\cite{EdRy:17}}] \label{def.stablegms}
Let ${\mathcal X}$ be an Artin stack with good moduli space $X$ and
let $\pi \colon {\mathcal X} \to X$ be the good moduli space morphism. We say that a closed point $x$
of ${\mathcal X}$ is
{\em
stable} if $\pi^{-1}(\pi(x)) = x$ under the induced map of
topological spaces $|{\mathcal X}| \to |X|$. A closed point $x$ of ${\mathcal X}$ is {\em
properly stable} if it is stable and the stabilizer of $x$ is finite.
We say ${\mathcal X}$ is stable (resp.~properly stable) if there is a good moduli
space $\pi \colon {\mathcal X}\to X$ and the set of stable (resp.~properly stable) points is non-empty. Likewise we say that $\pi$ is a stable (resp.~properly stable) good moduli space morphism.
\end{defn}
\begin{rem} \label{remark:ps}
Again this definition is modeled on GIT. If $G$ is a linearly reductive group
and $X^{ss}$ is the set of semistable points for a linearization of the
action of $G$ on a projective variety $X$ then a (properly) stable point
of $[X^{ss}/G]$ corresponds to a (properly) stable orbit in the sense of GIT.
The stack $[X^{ss}/G]$ is stable if and only if $X^{s} \neq \emptyset$. Likewise
$[X^{ss}/G]$ is properly stable if and only if $X^{ps} \neq \emptyset$. As is the case for GIT quotients, the set of stable (resp. properly stable points)
is open \cite{EdRy:17}.
\end{rem}
\begin{defn} \cite{EdSa:18, EdSa:17}
\label{def:strong}
Let ${\mathcal X}$ be an irreducible Artin stack with stable good moduli space $\pi \colon {\mathcal X} \to X$. A
closed integral substack ${\mathcal Z} \subseteq {\mathcal X}$ is {\em strong} if $\codim_{\mathcal X}{\mathcal Z}
=\codim_X\pi({\mathcal Z})$ and ${\mathcal Z}$ is saturated with respect to $\pi$, i.e.~$\pi^{-1}(\pi({\mathcal Z})) = {\mathcal Z}$ as stacks. We say ${\mathcal Z}$ is {\em topologically strong} if
$\codim_{\mathcal X}{\mathcal Z}=\codim_X\pi({\mathcal Z})$ and $\pi^{-1}(\pi({\mathcal Z}))_{red} = {\mathcal Z}$.
\end{defn}
Let $A^*_{\tst}({\mathcal X}/X)$ be the subgroup of $A^*({\mathcal X})$ generated by topologically strong cycles and $A^*_{\st}({\mathcal X}/X)$ be the subgroup generated by by strong cycles.
(Here the Chow group $A^*({\mathcal X})$ is the Chow group defined by Kresch \cite{Kre:99}.
When ${\mathcal X} = [Z/G]$ it can be identified with the equivariant Chow group
$A^*_G(Z)$ of \cite{EdGr:98}.)
The main result of \cite{EdSa:17} is the following theorem which says that if
$X$ is smooth then
any operational class $c \in A^k(X)$ can be represented by a codimension
$k$-cycle on the stack ${\mathcal X}$ which is saturated with respect to the good moduli space morphism ${\mathcal X} \to X$.
\begin{thm} \cite[Theorem 1.1]{EdSa:17}
Let ${\mathcal X}$ be a properly stable smooth Artin stack with good moduli space
${\mathcal X} \stackrel{\pi} \to X$. Then there is a pullback $\pi\colon A^*_{\op}(X)_{\mathbb{Q}} \to A^*({\mathcal X})_{\mathbb{Q}}$ which is injective and
factors through the subgroup $A^*_{\tst}({\mathcal X})_\mathbb{Q}$.
\end{thm}
\cite[Example 3.24]{EdSa:17} shows that not every topologically
strong cycle is in the image of $A^*_{\op}(X)$. However, \cite[Theorem 1.7c]{EdSa:17} states that any strong lci cycle on ${\mathcal X}$ is
in the image of $A^*_{\op}(X)_{\mathbb{Q}}$. (A cycle $\sum_i a_i [{\mathcal Z}_i]$
is lci if the ${\mathcal Z}_i$ are closed substacks of ${\mathcal X}$ such that the inclusion
${\mathcal Z}_i \hookrightarrow {\mathcal X}$ is an lci morphism.)
This leads to a number of successively weaker questions about
the operational Chow rings of good moduli spaces of smooth Artin stacks.
They can be viewed as analogues for quotients of smooth varieties by reductive
groups
of Conjectures 2 and 3 of \cite{MaVe:16}.
\begin{question} \label{ques.one}
Is the image of $A^*_{\op}(X)_\mathbb{Q}$ contained in the subgroup
of $A^*({\mathcal X})_\mathbb{Q}$ generated by strong lci cycles?
\end{question}
\begin{question} \label{ques.two}
Is the image of $A^*_{\op}(X)_\mathbb{Q}$ equal to the subring of $A^*({\mathcal X})_\mathbb{Q}$
generated by strong lci cycles?
\end{question}
\begin{question} \label{ques.three}
Is $A^*_{\op}(X)_\mathbb{Q}$ generated by Chern classes of perfect complexes
on $X$?
\end{question}
\begin{rem}
Note that Question \ref{ques.three} is an analogue of the question raised
by Anderson and Payne about the surjectivity of the map $K^0(X) \to
\op K^0(X)$ where $K^0(X)$ is the Grothendieck group of perfect complexes.
Anderson and Payne prove that for 3-dimensional complete toric varieties this map is in fact surjective. By comparison in \cite{EdSa:17} the authors prove that
$A^*(X)_\mathbb{Q} = A^*_{\st}({\mathcal X}/X)_\mathbb{Q}$ and in the case of 3-dimensional toric varieties \cite{Ric:19} shows that $A^*(X)_\mathbb{Q}$ is generated by strong lci cycles.
\end{rem}
Given the relation between the operational Chow ring and strong cycles leads to the following additional question.
\begin{question}
Is $A^*_{\st}({\mathcal X}/X)$ (resp. $A^*_{tst}({\mathcal X}/X)$) a subring of $A^*({\mathcal X})$; i.e.,
is the product of strong (resp. topologically strong) cycles strong?
\end{question}
\subsection{The cone theorem and local models for good moduli}
The \'etale slice theorem of Alper, Hall and Rydh \cite{AHR:15} states if ${\mathcal X}$
is a smooth stack
then at a closed point $x$ of ${\mathcal X}$ the good moduli morphism ${\mathcal X} \to X$ is \'etale locally isomorphic to the quotient $[V/G_x] \to \Spec k[V]^{G_x}$
where $G_x$ is the inertia group of $x$ in ${\mathcal X}$ and $V$ is a representation
of $G_x$. Thus we may view stacks of the form $[V/G]$ with their
good moduli spaces $\Spec k[V]^G$ as the ``affine models''
of smooth stacks with good moduli spaces.
The cone theorem has the following corollary.
\begin{cor} \label{cor.invring}
Let $V$ be a representation of a reductive group $G$
and let $X = \Spec k[V]^G$ be the quotient. Then for any homotopy invariant
functor $H(X) = H(\Spec k)$.
In particular, $A^*_{\op}(X) = \ZZ$ and $\op K^0(X) = \ZZ$.
\end{cor}
\begin{proof}
Since $G$ acts linearly on $V$ the action of $G$ on $k[V]$
preserves the natural grading. Hence the invariant ring $k[V]^G$ is
also graded. Thus by Theorem \ref{thm.cone} $H(X) = H(\Spec k)$
since $k$ is the 0-th graded piece of $k[V]^G$.
\end{proof}
Combining Corollary \ref{cor.invring} with \cite[Theorem 1.7c]{EdSa:17} yields the following result.
\begin{cor}
If ${\mathcal Z} \subset {\mathcal X} = [V/G]$ is a proper closed substack which is strongly regularly embedded then $[{\mathcal Z}]$ is torsion in $A^*({\mathcal X}) = A^*_G(\pt)$. In particular
if $A^*_G(\pt)$ is torsion free (for example if $G$ is torus or $\GL_n$)
then $[{\mathcal Z}] = 0$.
\end{cor}
Note that any integral strong divisor ${\mathcal D}$ on ${\mathcal X} = [V/G]$ is necessarily defined by a single
$G$-invariant equation. In this case ${\mathcal O}({\mathcal D})$ is
an equivariantly trivial line bundle so $[{\mathcal D}] = c_1({\mathcal O}({\mathcal D})) = 0$ in the equivariant Chow ring $A^*_G(V)$. More generally,
if an integral substack ${\mathcal Z} \subset [V/G]$ is a complete intersection of strong divisors, then its class in $A^*_G(V)$ is also 0. In particular it shows that
in the ``local case'' the image of $A^*_{\op}(X)$ is contained in the
subgroup generated by strong global complete intersections.
This leads to the following question.
\begin{question}
Are there examples of strong integral substacks ${\mathcal Z} \subset [V/G]$
such that $[{\mathcal Z}] \neq 0$ in $A^*([V/G])_\mathbb{Q}$?
\end{question}
\subsection{Acknowledgments} The authors are grateful to Sam Payne and Angelo Vistoli for helpful comments.
\bibliographystyle{amsmath}
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|
2,877,628,088,976 | arxiv | \section{Introduction\label{s:intro}}
Pulsars are often depicted as relatively simple objects: a highly
magnetized, fast rotating neutron star (NS) emitting radiation along its poles.
Most emission models start from this basic picture, in principle common
to all isolated pulsars.
The rotation of the magnetic dipole induces an electric field near the
polar caps in vacuum, which is then short circuited by an e$^\pm$ pair
plasma (Ruderman \& Sutherland 1975), or in space-charge limited flow,
which is screened at a pair formation front (Arons \& Scharlemann 1979).
Models seek then suitable regions of particle acceleration.
Although the radio emission is thought to originate in the polar regions
of the magnetosphere (e.g Michel 1991 and references therein),
the high-energy emission is now thought to
originate from the outer magnetosphere, with the original polar cap
concept (Daugherty \& Harding 1996) having been superseded by outer-gap
(Cheng, Ho \& Ruderman 1986; Romani 1996) and slot-gap (Arons 1983, Muslimov \& Harding 2004) models.
The picture that has emerged is that radiation is emitted in a
continuous, very broad, spectral range, with curvature radiation
producing most of the GeV emission (Romani 1996) and synchrotron and/or
Compton scattering by cascade products producing the near-infrared to
soft {$\gamma$-ray}\ emission (Takata \& Chang 2007, Harding et al. 2008).
The high energy emission properties of pulsars were revealed in the 1990s when use of the EGRET experiment on the \textit{Compton Gamma-Ray Observatory} (e.g. Thompson, 2001 and references therein) led to the multiwavelength spectral characterization of half a dozen {$\gamma$-ray}\ pulsars, including the discovery of the radio-quiet pulsar Geminga (Halpern \& Holt 1992), the second brightest steady GeV source in the {$\gamma$-ray}\ sky.
Amongst the EGRET legacy was a sample of 170 unidentified sources, 74 of which are at $|b|<10^{\circ}$ (Hartmann et al. 1999).
It has now been found that $\approx 43$ EGRET unidentified sources have counterparts in the {\sl Fermi}\ Large Area Telescope (LAT) first year catalog (Abdo et al. 2010b; 1FGL) and a large fraction of those lying on the Galactic plane turned out to be pulsars (Abdo et al. 2008, 2009a, 2009b, 2010a, 2010b), a result anticipated by several authors (Yadigaroglu \& Romani 1995, Cheng \& Zhang 1998, Harding \& Muslimov 2005).
In fact, it is very plausible that many more radio-loud or radio-quiet
pulsars are hidden in the unidentified Galactic LAT sources, although estimates are highly dependent on details of the different emission mechanisms.
At this point it is still unclear what makes some {$\gamma$-ray}\ pulsars radio-loud and some radio-quiet.
Understanding the distinct properties of the individual sources will surely lead to a better understanding of models for the emission mechanisms, for example, the connection between the radio and the near-infrared to the {$\gamma$-ray}\ component.
The source 2CG 078+2 was one of about twenty {$\gamma$-ray}\ sources detected by
the {\em COS-B} satellite 30 years ago (Swanenburg et al. 1981).
The source is in the line of sight towards the supernova remnant G78.2+2.1.
The remnant comprises a 1-degree-diameter, circular, radio shell with
two bright and broad opposing arcs on its rim (Higgs, Landecker \& Roger
1977, Wendker, Higgs \& Landecker 1991).
The remnant has a kinetic distance of about 1.5 kpc (Landecker, Roger,
and Higgs 1980; Green 2009) and is estimated to have an age of 5400 yr
(Sturner \& Dermer 1995).
2CG 078+2 is often cited as $\gamma$-Cygni due to its proximity to the
second magnitude foreground star (m$_{\rm V}$ = 2.2, spectral type F8Iab) that lies on
the eastern edge of the remnant although there is no other physical
association between the supernova remnant (SNR) and the star.
A small HII region, located close to the star, forms the $\gamma$-Cygni
nebula.
The {$\gamma$-ray}\ source, next cataloged as 3EG J2020+4017 (Hartmann et al.
1999), was suspected to be either extended emission from the
SNR or a pulsar formed in the supernova event (Sturner \& Dermer 1996).
Soon after its June 2008 launch, the {\sl Fermi}\ {$\gamma$-ray}\ Space Telescope
highlighted the discovery of a radio-quiet pulsar in the CTA-1 SNR (Abdo et al. 2008) as a first light result.
This was followed by the discovery of twenty-three other {$\gamma$-ray}\ pulsars
using blind search techniques (Abdo et al. 2009b, Saz Parkinson et al.
2010), among them PSR J2021+4026\ lying within the EGRET error circle of 3EG
J2020+4017.
PSR J2021+4026\ is a 265-ms pulsar with a spin-down age of 77 kyr and a total spin-down luminosity of 1.1$\times 10^{35}$ erg/s.
Interestingly, most of the pulsars discovered by the LAT with blind searches have not been seen at other wavelengths.
In fact, only three of the 26 discovered to date have been detected in radio (Camilo et al. 2009; Abdo et~al. 2010c)
Even before the {\sl Fermi}\ discovery of pulsations, and because the EGRET
{$\gamma$-ray}\ properties of the source matched those of known {$\gamma$-ray}-pulsars (hard,
steady and in the Galactic plane), the search for the radio and
X-ray counterpart of this source began (Brazier et al. 1996; Becker et
al. 2004, Weisskopf et al. 2006).
No radio pulsar is associated with PSR J2021+4026\ with searches having now been
conducted down to $100 \mu$Jy at 1665 MHz (Trepl et al. 2010),
40~$\mu$Jy at 820 MHz (Becker et al. 2004), and to $11 \mu$Jy at 2 GHz (Ray et al. 2011).
There is an extended radio source (GB6 J2021+4026) in the vicinity which appears to be positioned more or less symmetrically (see Figure 8 of Trepl et al. 2010) about our best position for the X-ray counterpart but there is no evidence that it is associated with the {$\gamma$-ray}\ pulsar.
We have previously (Becker et al. 2004, Weisskopf et al. 2006) performed
{\sl Chandra}\ observations (ObsIDs 3856 \& 5533) aimed at different portions
of the $\gamma$-Cygni field and these pointings were based on the best
known {$\gamma$-ray}\ positions available at the time.
The latter {\sl Chandra}\ observation, ObsID~5533, overlapped with about 3/4 of the {\sl current}
Fermi-LAT 99\%-confidence positional error circle and detected several
potential X-ray counterparts, including the source designated as "S21" as
reported in Weisskopf et al. (2006).
Subsequently, Trepl et al. (2010) reanalyzed the available {\sl Chandra}\ data and also
used XMM-Newton data to search for a counterpart.
They identify source 2XMM~J202131+402645,
at virtually the same location as the {\sl Chandra}\ source ``S21'', as the likely counterpart because
the X-ray source position fell within the $\sim$4\arcmin-radius
Fermi-LAT 95\%-confidence positional error circle at the time (0FGL, Abdo et al. 2009b).
We note that the most recent Fermi-LAT error circle radius (2FGL, Abdo et al. 2011)
is 0.6\arcmin\ (\S3) and no longer includes ``S21''.
We re-evaluate the situation using both the greatly improved LAT source localization
and the position measured from timing of PSR J2021+4026.
The current work describes a new observation that completes the {\sl Chandra}\ coverage of the field.
As a result of these observations, we conclude that the source originally
labelled ``S21'' by Weisskopf et al. (2006), 2XMM~J202131+402645 by Trepl et al. (2010),
and source \#20 in the new observation,
remains the most probable X-ray counterpart to PSR J2021+4026.
Moreover, we also show that the X-ray source is dominated by thermal,
not powerlaw, emission.
In this regard, it is interesting to compare this source with Geminga,
one of the best studied radio-quiet {$\gamma$-ray}\ pulsars, which is older (340
ky) and less luminous ($\sim3\times 10^{34}$ erg/s) than PSR J2021+4026\ but has a similar period and
period derivative.
\S\ref{s:analysis_image} describes the analysis of the X-ray image; \S\ref{s:analysis_spectrum}
describes the analysis of the X-ray spectrum, especially of the X-ray source \#20; \S\ref{s:geminga} compares the X-ray spectral properties of source \#20 to Geminga and CTA 1; \S\ref{s:analysis_timing} describes new Fermi-LAT pulsar timing of PSR J2021+4026; \S\ref{s:optical} describes our search for an optical counterpart; and \S\ref{s:other_sources} discusses properties of the other {\sl Chandra}\ sources in the field.
We provide summary conclusions in \S\ref{s:summary}.
\section{{\sl Chandra}\ X-Ray Observations and Data Analysis \label{s:xray_obs}}
We obtained a 56-ks {\sl Chandra}\ observation (ObsID 11235, 2010 August 27) using the Advanced CCD Imaging Spectrometer (ACIS).
Here we report the data taken with the back-illuminated CCD ACIS S3 in faint, timed-exposure mode with 3.141-s frame time.
Background levels were nominal throughout the observation.
Standard {\sl Chandra}\ X-ray Center (CXC) processing provided accurate aspect determination.
Starting with level-1 event lists, we reprocessed the data using the CIAO~v4.2 tool {\tt acis\_process\_events} to remove pixel randomization in order to improve the on-axis point spread function (PSF), thus enhancing the source-detection efficiency and positional accuracy.
In searching for sources, we utilized events in pulse-invariant channels 35-550 corresponding
to 0.5 to 8.0 keV.
To verify the {\sl Chandra}\ position accuracy, we compared the pointing parameters (RA, Dec, roll) given by the {\sl Chandra}\ data-processing to that using the 19 2MASS sources (Section~\ref{s:oi_2mass}) with high probability X-ray counterparts.
For these calculations we used $0.075\arcsec$ per axis for the 2MASS position
uncertainty\footnote{http://www.ipac.caltech.edu/2mass/releases/allsky/doc/explsup.html}.
Assuming no change to the {\sl Chandra}\ parameters yielded an acceptable positional fit for these 19
sources with $\chi^2$\ of 24 for 38 degrees of freedom.
Allowing the three pointing parameters to vary marginally improved $\chi^2$\ to 21 for 35 degrees of freedom and would imply the following corrections: $(0.14\pm0.10)\arcsec$ (RA), $(-0.04\pm0.08)\arcsec$ (Dec), and $(43\pm78)\arcsec$ (roll).
However, these corrections, even allowing for errors, being negligible we did not correct the {\sl Chandra}\ positions.
\subsection{X-Ray Image Analysis \label{s:analysis_image}}
We searched for point-like X-ray sources employing techniques described in Tennant (2006), using a circular-Gaussian approximation to the PSF and setting the signal-to-noise (S/N) threshold for detection to 2.4.
The resulting background-subtracted point-source detection limit is about 7 counts, with fewer than 1 accidental detection expected over the field.
Based upon tests on {\sl Chandra}\ deep fields, this approach finds all X-ray sources in those fields down to 10 counts, which we thus regard as the completeness limit.
Figure~\ref{f:new_field} shows the ACIS-S3 image with a small circle at the position of each {\sl Chandra}-detected source.
Table~\ref{t:data_x} tabulates the X-ray properties of the 44 {\sl Chandra}-detected sources, with the source number in column 1.
Columns~2--5 give, respectively, right ascension RA, declination Dec, extraction radius $\theta_{\rm ext}$, and approximate number of X-ray counts $C_{x}$ detected from the source.
Column~6 lists the single-axis RMS error $\sigma_{x}=[(\sigma_{\rm PSF}^2/C_{x}) + \sigma^2_{\rm sys}]^{1/2}$ in the X-ray-source position, where $\sigma_{\rm PSF}$ is the dispersion of the circular Gaussian that approximately matches the PSF at the source location and $\sigma_{\rm sys}$ is a systematic error.
Uncertainties in the plate scale\footnote{See
http://cxc.harvard.edu/mta/ASPECT/aca\_plate\_scale/} imply $\sigma_{\rm sys}\approx 0\farcs13$:
To be conservative, we set $\sigma_{\rm sys}=0\farcs2$ (per axis).
Column~7 gives the radial uncertainty $\epsilon_{99}=3.03\: \sigma_{x}$ in the X-ray position~---~i.e., $\chi^{2}_{2}=9.21=3.03^2$ corresponds to 99\% confidence on 2 degrees of freedom~---~for inclusion of the true source position.
Columns~8 \& 9 are color ratios defined and discussed in \S~\ref{s:analysis_spec}.
In view of the spin-down age and energetics of PSR J2021+4026, the possibility exists that a pulsar wind nebula (PWN)
may also be present in the X-ray image of the field.
We searched for moderately extended sources in the field and identified three features of interest.
Two are located near the edge of the S3 CCD which increases the likelihood of their being false
positives. The third feature lies $\sim$6\arcsec\ west of source \#20.
Simultaneously fitting the combination of this feature and source \#20 to circular Gaussians (plus
a constant background) results in a Gaussian width of 2.2\arcsec\ and a total of 22 X-ray events
within this extended feature.
We also compared the spatial distribution of events in and around source \#20 to a model of the
{\sl Chandra}/ACIS PSF valid for its location relative to the aimpoint and its characteristic energy using
the PSF library psflib v4.1 (CALDB v4.2). Source \#20 is consistent with being pointlike.
\subsection{X-Ray Spectral Analysis \label{s:analysis_spectrum}}
We used the XSPEC (v.12.5) spectral-fitting package (Arnaud, 1996) to perform spectral fits to the 44 X-ray point sources. We treat source \#20 separately in \S \ref{s:s20_spectrum}.
Data were binned to obtain at least 10 counts per spectral bin before background subtraction.
The background was determined after masking off a circular region around each of the 44 detected X-ray sources corresponding to a circle of radius 25 times the uncertainty listed in column 6 of Table~\ref{t:data_x} and then averaging over the remaining pixels.
Individual response matrix and ancillary response files appropriate to each source position were
created for this analysis using the {\tt mkacisrmf} and {\tt mkarf} tools available in CIAO version 4.2.
\subsubsection{The X-Ray Spectrum of Source \#20\label{s:s20_spectrum}}
A sufficient number of counts were detected from source \#20 to perform more extensive spectral
analysis.
Figure~\ref{f:s20_spectrum} shows the X-ray spectrum of source \#20 with background subtracted.
Data were again binned to obtain at least 10 counts per bin before background subtraction.
The background is less than 3\% of the signal plus background from the region that includes source \#20.
We begin our spectral analysis by first considering single-component spectral models (Table~\ref{t:smodels}) with a multiplicative absorption component: an absorbed power law ({\tt powerlaw} in XSPEC); an absorbed black body ({\tt bbodyrad}); and three different absorbed neutron star atmosphere models,\footnote{A NS radius of 12.996~km and mass of 1.358~M$_{\odot}$ is assumed throughout for purposes of computing effects of gravitational redshift in the neutron star atmosphere models} namely, {\tt nsa} (Pavlov et al. 1995),
{\tt nsmax-1260} and {\tt nsmax-130190} (Ho, Potekhin \& Chabrier 2008).
For computing absorption, we utilized abundances (XSPEC's {\tt wilm}) from Wilms, Allen, \& McCray (2000) with cross-sections ({\tt vern}) from Verner et al.(1993) and allowed for interstellar extinction by grains using the model ({\tt tbabs}) of Wilms, Allen, \& McCray (2000).
All of these models provide statistically adequate fits to the data in the 0.5-8.0 keV range (Table~\ref{t:smodels}).
(We note that so did a fit to an absorbed {\tt mekal} model.)
From the {\sc Hi} in the Galaxy (Dickey \& Lockman, 1990), one infers a column density $N_{\rm H} \approx 1.4\times 10^{22}$~cm$^{-2}$ through the Galaxy in the direction of source \#20, implying that values below this should be expected if there is no circumstellar absorption, and this is indeed the case (Table~\ref{t:smodels}).
Next we posit that the very steep power law index of almost 5 is not physical, but indicative of a soft, thermal component.
The {\tt bbodyrad} model's normalization,
$(R^2_{\rm km}/D^2_{10{\rm kpc}})=0.80$, where $R_{\rm km}$ is the radius of the emitting area in km and $D_{10{\rm kpc}}$ is the distance to the source in units of 10~kpc,
corresponds to an emitting area of $R_{\rm km}=0.13$, assuming the association of the pulsar with G78.2+2.1 at a nominal distance of $D_{10{\rm kpc}}=0.15$.
This is much smaller than a typical NS radius, but not very different than the standard polar cap radius,
$\approx R_{NS} sin \theta \approx R_{NS}^{3/2} (2\pi/cP)^{1/2}$ which for $P=0.265$~s and $R_{NS}=13$~km is 0.42~km.
Conversely, if we assume the emission comes from a NS with a characteristic radius of 13 km, then the {\tt bbodyrad} norm implies a source distance of $\approx145$ kpc, well outside the Galaxy.
This distance estimate drops for the different NS-atmosphere models, e.g. down to 13 kpc (Table~\ref{t:smodels}), still somewhat distant to remain within the Galaxy.
In addition, the temperature estimates of the (cooling) NS, range from $\log(T_{\infty})\geq 6.5$ to $6.1$, with the precise value depending on the model (Table~\ref{t:smodels}).
These estimates are consistent with those expected for a pulsar of an estimated age somewhere between 5400 and 77000 years, depending of course on the equation of state, composition of the heat-blanketing envelope, and the degree of superfluidity in the star's core.
(See Yakovlev et al. 2010 and references therein for recent details on the subject of cooling NSs.)
We then ask whether or not combining a power law with the other models is indicated by the data.
Table~\ref{t:dmodels} tabulates the change in $\chi^2$, the f-statistic, and the probability that combining models has significantly improved the quality of the fit.
The table also tabulates he derived spectral parameters.
In all cases combining various thermal models with a power law does improve the quality of the fit with a confidence better than 2-sigma, but not 3-sigma.
Moreover, these two-component models allow a wide latitude for the uncertainties of the best-fit parameters and hence these parameters are not as constrained as one might wish.
This follows from the fact that all of the single models (Table ~\ref{t:smodels}) themselves provide statistically adequate fits.
Thus, the 3-sigma contours for any two-component model allows for one or the other of the component models to have a zero norm -- e.g. a power law component is not completely {\em required} by the data at this level of significance.
Keeping this proviso in mind, we continue to examine the two-component models.
Certainly the physical interpretation of the data is perhaps more sensible when both components are introduced, especially in light of the strong {$\gamma$-ray}\ emission which cannot arise from any single-component ``thermal'' model spectrum.
In addition, both the inferred NS temperatures and distances implied by the two-component models (Table~\ref{t:dmodels}), with the possible exception of the {\tt bbodyrad+powerlaw} model,
are consistent with what one might expect for a young cooling neutron star within the galaxy.
\subsubsection{Discussion of the Spectrum of Source \#20 \label{s:spectrum_discussion}}
The considerations of the single-component models in the previous section lead one to conclude that we are perhaps seeing a hot spot of size comparable to the polar cap rather than thermal emission from the entire surface of the neutron star and with a temperature higher than one expects from cooling of a neutron star at an age $> 5000$ years.
It is thus possible that the bulk of the emission comes from heating of the polar cap by backflowing accelerated particles.
The expected luminosity and temperature from heating by positrons produced by curvature radiation of primaries in a space-charge limited flow model is $L_+ \sim 10^{31}\,\rm erg\,s^{-1}$ and $\log(T_+) \simeq 6.3$ (Harding \& Muslimov 2001).
This luminosity is deposited over an area roughly that of the polar cap and radiated on a timescale less then the heat diffusion timescale across field lines to other areas of the NS.
The temperature $T_+$ is close to that determined from the {\tt bbodyrad} and {\tt nsa} model fits allowing for the gravitational redshift.
It is therefore quite possible that the X-ray emission from PSR J2021+4026 is dominated by polar cap heating.
On the other hand,
many well-studied neutron stars that exhibit both X-ray and {$\gamma$-ray}\ emission have composite X-ray spectra, showing non-thermal power law magnetospheric emission and/or hot polar cap emission in addition to the lower-temperature full surface thermal emission (e.g. Geminga, PSR B0665+14, PSR B1055-52, see De Luca et al., 2005).
While the present data quality do not demand such two-component models, the inferred distances from such two-component fits, e.g., $\sim 6$~kpc for the NS atmosphere models (Table ~\ref{t:dmodels}) become reasonable for plausible stellar radii assuming full surface emission. This distance is larger than
the kinematic distance to the SNR~78.2$+$2.1 of 1.5~kpc but is comparable to the distance to the
Cygnus arm at Galactic longitude $\sim$78\arcdeg.
Thus, it is possible that a two-component model, with full-surface cooling and magnetospheric power law emission present at lower levels, might be needed to accurately describe the emission
physics.
In practice, decomposing such complex X-ray spectra requires good statistics and phase-resolved X-ray spectroscopy.
For a target this faint, extremely long observations or next-generation X-ray satellites are clearly required.
\subsection{Comparison of the Spectrum of \#20 to Geminga and CTA 1 \label{s:geminga}}
There are similarities and differences between the X-ray spectrum of PSR J2021+4026\ and two of the other radio-quiet {$\gamma$-ray}\ pulsars with detected X-ray emission, Geminga and CTA 1.
First PSR J2021+4026, like Geminga (Jackson and Halpern, 2005) and CTA 1 (Caraveo et al. 2010), may also be characterized by two spectral components, a thermal component and power law.
In Geminga, however, the power law component begins to dominate above about 0.5 keV and $\log (T_{\infty})$ is about 5.7.
For PSR J2021+4026\ the power law dominates above 2.5 keV and $\log (T_{\infty})$ is higher, as one would expect as PSR J2021+4026, based on its spin-down age, is younger.
The spectral indices for the power law components for source \#20 and Geminga are not dissimilar, but one needs to recognize the large uncertainty in the measurements reported here.
Another, possibly important, spectral difference between the two X-ray spectra is that, in the case of Geminga, the blackbody component gives an emission radius that is plausible for a NS radius.
This is not so for PSR J2021+4026.
In this case, the neutron star atmosphere models seem to yield more physically reasonable parameters than the {\tt bbodyrad+powerlaw} model.
If we assume that the younger and hotter star still has an atmosphere while the older Geminga does not, then these results are sensible.
Finally, there is a weak extended emission feature near source \#20 that may be indicative of a PWN. If so, then it extends no more than 0.04 to 0.17~pc from source \#20 (assuming a distance of 1.5 to 6.0~kpc, respectively) and contributes $\sim$7\% of the X-ray counts detected from source \#20 and its surroundings.
Emission associated with the Geminga PWN has a similar extent and contributes ~10\% of the non-thermal X-ray flux of the pulsar, but only about 1\% of the total flux (Pavlov et~al.\ 2010).
For CTA 1, the measured temperature, powerlaw index and emission radius are $\log (T_{\infty}) = 6.08$, $\Gamma = 1.3$ and $r = 0.64$ km for a {\tt powerlaw+bbodyrad} model, and $\log (T_{\infty})= 5.78$, $\Gamma = 1.25$ and $r = 4.92$ km for {\tt powerlaw+nsa}, with slightly lower $\chi^2$ for the former (Caraveo et al. 2010).
In both cases, the emitting radius is significantly smaller than a standard NS radius.
Thus, similar to PSR J2021+4026, CTA-1 shows a possible heated polar cap component, with a temperature very close to the model prediction of $\log (T_{\infty}) \simeq 6.2$ (Harding \& Muslimov 2001), however, the emitting radius in this case is a factor of 2.5 larger than the polar cap radius.
CTA 1 does not show evidence for a cool component, with the upper limits making it unusually cool for its age.
\section{{\sl Fermi}-LAT Localization and Timing Analysis\label{s:analysis_timing}}
The {\sl Fermi}-LAT normally localizes {$\gamma$-ray}\ sources using the incident {$\gamma$-ray}\ photon directions.
The LAT has a point spread function that is strongly energy dependent, with a resolution of about 0.8$^\circ$ at 1 GeV.
For a bright source, however, localization with arcminute accuracy is possible.
The source in the second LAT catalog (Abdo et al. 2011) that corresponds to PSR J2021+4026\ is 2FGL J2021.5+4026.
The catalog position for this source is R.A. 20$^h$21$^m$34$^s$.1, Decl.\ $+$40\arcdeg26\arcmin28\arcsec\ (J2000), with a 95\% confidence radius of 0.60\arcmin.
For pulsars, one can use timing techniques better to localize the source, independent of the photon direction measurements, as described in Ray et al. (2011).
The position determination from timing of this pulsar is hampered by the large contribution from rotational instabilities common in young pulsars and manifest as ``timing noise''.
Therefore, we have taken two different approaches to try to confirm the association between PSR J2021+4026\ and source \#20.
For this analysis, we used {$\gamma$~rays}\ detected by the LAT from 2008 Aug 4 to 2011 Mar 12, selecting only those within $0.8^{\rm o}$ from the previous best position (Ray et al. 2011), and with energies greater than 400 MeV.
These ``cuts'' were chosen to maximize the significance of the pulsation.
We chose only photons belonging to the most restrictive ``diffuse'' class according to the ``Pass~6'' instrument response functions (see Atwood et al. 2009), which have the lowest background contamination. Furthermore, we selected only photons with a zenith angle of $<105^{\rm o}$, to reduce contamination due to secondary-atmospheric {$\gamma$~rays}.
As a first test, we evaluate the significance of the {$\gamma$-ray}\ pulsations by assuming the pulsar is at each of the candidate X-ray source locations seen in our Chandra observation (Table ~\ref{t:data_x}).
For each candidate location, we transform arrival times to the barycenter using the X-ray position and then use the {\tt prepfold} routine from the PRESTO pulsar package (Ransom, Eikenberry \& Middleditch 2002) to find the frequency ($f$), and its first and second derivatives, $\dot{f}$ and $\ddot{f}$, which maximize the statistical significance of the pulsation.
Figure~\ref{f:pulse_timing} shows the results from this exploratory search, where it is clear that source \#20 gives the maximum significance for pulsation using this algorithm.
This indicates that of the possible X-ray sources in the field, source \#20 is the most likely X-ray counterpart.
Next, we use pulsar timing to fit for the position of the pulsar, as described in Ray et al. (2011).
We measured 55 times of arrival (TOAs) based on 22-day integrations spanning the data set described above.
The typical uncertainty on each TOA measurement was 4.7 ms.
Using \textsc{Tempo2} (Hobbs et al. 2006), we fit the TOAs to a timing model including $f$, $\dot{f}$, and $\ddot{f}$.
With only these terms in the model, we observe very large residuals and the $ \chi^2 $ of the fit is very poor.
This poor model fit means that the statistical errors in the fitted right ascension and declination reported by \textsc{Tempo2} are unreliable.
To get an estimate of the systematic error in the position fit, we use the following procedure.
We added 5 so-called ``WAVE'' terms to the timing model to account for the timing noise using harmonically-related sinusoids (Hobbs et al. 2004).
We then perform a fine scan over a positional grid around the location of source \#20.
Holding the position fixed at each grid point, we fit for the spin and WAVE parameters.
The grid position with the lowest resulting $\chi^2$ for the fit is R.A. 20$^h$21$^m$29$^s$.683, Decl.\ $+$40\arcdeg26\arcmin54.61\arcsec\ (J2000).
This new timing position is $10\arcsec$ away from the one reported in Ray et al. (2011) which was based on 14 fewer months of data.
Based on the $\chi^2$ map over the grid, we estimate the 95\% confidence region
of the new timing position to be an ellipse of dimensions 26\arcsec$\times$10\arcsec, as shown in Figure~\ref{f:snapshot}.
The separation between the position of source \#20 and the refined timing analysis position
obtained here is $14.7\arcsec$ and source \#20 lies outside the 95\% confidence region.
A precise evaluation of systematic timing errors is complicated by the (erratic) timing behavior of the pulsar itself and the relatively short data span with respect to the 1-year modulation that is introduced by an incorrect position.
That is, the position can be perturbed by any component of the timing noise that appears to be a 1-year sinusoid.
The magnitude of this effect is difficult to estimate because we have just one realization of the stochastic timing noise process in our data.
To see the potential contribution, we have plotted an estimate of the timing noise contributions in Figure~\ref{f:tnoise}.
Here, we assume that the parameters $f$ and $\dot{f}$ are dominated by the secular spindown of the pulsar, while any higher order frequency derivatives and all of the WAVE parameters are dominated by timing noise.
Clearly, this is a red-noise stochastic process.
Note that a position error of 10$\arcsec$ will introduce a sinusoidal term with a 1-year period and an amplitude of 24 ms.
To get another estimate of the systematic error, we fit a timing model including the pulsar position to three overlapping segments of data, the first half of our data span, the middle half, and the last half. These three fitted positions are separated by 7--9$\arcsec$, giving another estimate of the systematic error resulting from timing noise.
Based on these considerations, we adopt $10\arcsec$ as an estimate of the systematic error in the timing position. Combining, in quadrature, this systematic error estimate with the error estimated in Ray et~al. (2011) results in the smaller, 10.3\arcsec\ radius, circular 95\% confidence region depicted in Figure~\ref{f:snapshot}. Source \#20 lies within this region.
In summary, when using pulsar timing to derive a position at the few arcsec level it is important to allow for low frequency (year timescale) timing noise.
We have done this two ways: First, by adding WAVE terms to the solution and allowing these terms to vary when deriving an error in our grid search.
Second, by time slicing the data and looking at how the derived position changes.
As seen in Figure~\ref{f:snapshot}, each of these methods gave similar sized error regions that overlap.
\section{Search for an Optical Counterpart to the X-Ray Source\label{s:optical}}
As reported in Weisskopf et al (2006) and repeated in \S\ref{s:oi}, there are no cataloged optical counterparts for source \#20.
This is not surprising as, with the exception of the $m\approx 16$ Crab, most optically detected pulsars have magnitudes $\geq 25$.
In addition the field of PSR J2021+4026\ is crowded and optical observations are further hampered by the presence of the 8$^{th}$ magnitude star BD+39 4152 (=V405 Cyg) one arcmin away to the east.
We present here observations of the field taken on October 31, 2008 with the OPTIC orthogonal frame-transfer camera on the Kit Peak National Observatory, 3.6m, Wisconsin, Yale, Indiana, \& NOAO (hence WIYN) telescope.
The OPTIC camera, with a 10$^\prime$ field and plate scale 0.141$^{\prime\prime}$/pixel, allows improved image quality
through ``Orthogonal Transfer'' (OT) rapid electronic guiding following motions of a reference star (Tonry et al. 2004).
We used BD+39 4152 itself as the guide star and were able to correct at 50\,Hz, collecting $3\times180$\,s dithered exposures in $r^\prime$ and $i^\prime$.
These frames were subject to standard calibrations, except for the flat field
frames which were assembled by applying image shifts matching those of the
OT guiding during the individual science exposures.
The resulting image stacks have final PSF widths of 0.87$^{\prime\prime}$ ($r^\prime$) and 0.62$^{\prime\prime}$ ($i^\prime$) near the guide star; the PSF width increases by $\sim 30$\% toward the edge of the frame.
We estimate that the frame is aligned to the {\sl Chandra}\ coordinates with $<0.2^{\prime\prime}$ precision.
Figure~\ref{f:WIYN_r2_i2} shows a portion of the OPTIC frames in $r^\prime$ and $i^\prime$ centered near the position of source \#20.
Note the secondary reflections of BD+39 4152 and the strong scattering background, especially in $i^\prime$.
Magnitudes were corrected to the SDSS photometry scale using observations of the calibration star Ru 149F (Smith et al. 2002).
We measured the fluxes of the faintest detectable stellar sources in the
vicinity of our target position and used these to estimate upper limits
($\sim 95$\% confidence) on the undetected optical flux for source \#20
of $i^\prime > 23.0$ mag (the sensitivity is severely limited by scattered flux) and $r^\prime > 25.2$~mag.
Some of the diffuse emission toward the right (west) of the $r^\prime$ image is part of larger scale filamentary structure visible over several arcminutes.
This is likely H$\alpha$/[NII] associated with the $\gamma$~Cygni\ SNR itself.
We note that this remnant has been poorly studied in the optical.
Mavromatakis (2003) described extensive diffuse line emission over
a $\sim 1^\circ$ region, but found little filamentary emission and was not able to detect the very faint filaments seen in our data.
No corresponding X-ray structure is seen, supporting the claim in Mavromatakis (2003) that the $\gamma$~Cygni\ remnant is dominated by low velocity shocks.
Deep narrow band imaging to trace this structure could be useful in testing the connection, if any, between PSR J2021+4026 and the $\gamma$~Cygni\ SNR.
\section{The Other 43 X-Ray Sources in the Field \label{s:other_sources}}
\subsection{Spectral Analysis\label{s:analysis_spec}}
There is a drop in the {\sl Chandra}\ energy response above the mirror coating's iridium-M edges ($\approx\!2$~keV), thus any source with a substantial fraction of its detected photons above 2~keV is indicative of a very hard spectrum.
Figure~\ref{f:hratio} shows the X-ray color-color diagram for the 21 sources that have more than 15 source counts.
The diagram comprises 3 bands: S (soft) covering 0.5 to 1.0 keV; M (medium) covering 1.0 to 2.0 and H (hard) 2 to 8 keV with T (total) simply the sum of S, M and H.
The color ratios that comprise Figure~\ref{f:hratio} are given in columns 8 \& 9 of Table ~\ref{t:data_x}.
The X coordinate in Figure~\ref{f:hratio} (H-S)/T measures how hard the spectrum is and the y coordinate (M/T) measures how centrally peaked the spectrum is.
Positive source counts requires data points to be inside the triangle, but background subtraction causes a few to appear slightly outside.
Sources with (H-S)/T greater than 0.5 are spectrally very hard and are likely background AGNs shining through the galactic plane.
Only one of this group of sources, \#42, has a cataloged optical counterpart (Table~\ref{t:data_oi}).
Sources with negative values of (H-S)/T likely have thermal spectra and are plausibly lightly absorbed foreground stars.
Note that source \#20 has the highest fraction of counts in the 1-2 keV
band.
Two X-ray sources, other than source \#20, have sufficient counts to warrant a spectral analysis: \#3, the brightest in the field; and \#32.
\subsubsection{The X-Ray Spectrum of Source \#3\label{s:s3_spectrum}}
We first fit the source~\#3 data using absorbed {\tt powerlaw}, {\tt bbodyrad}, and {\tt mekal} models.
None of these models provided acceptable fits to the data, $\chi^2$\ being 48.3, 57.2, 70.8, respectively, for 32 degrees of freedom.
The only two-component+ model that provided an acceptable fit was a two-temperature {\tt mekal} model with $\chi^2$\ of 23.4 for 30 degrees of freedom, further indicating that this is a foreground star.
The results of our spectral analysis, following the procedures discussed in \S\ref{s:s20_spectrum}, are in Table~\ref{t:s3_spectrum}.
\subsubsection{The X-Ray Spectrum of Source \#32\label{s:s32_spectrum}}
The data for source \#32 are well-fit by an absorbed powerlaw model ($\chi^2$\ of 15.7 for 19 degrees of freedom), but, as with source \#20, the powerlaw index is very steep being 3.4. In this case, however, neither a single-temperature mekal model nor a {\tt bbodyrad} model provide as compelling fits ($\chi^2$\ of 46.9 \& 23.9 respectively), although the latter is statistically acceptable.
There is simply too much uncertainty to firmly classify this source on the basis of the X-ray spectroscopy alone.
\subsection{Temporal Variability\label{s:analysis_var}}
The general paucity of counts also precludes a sensitive time-variability analysis for almost all these sources.
Nonetheless, one of the three X-ray-brightest sources, \#32, shows evidence for a significant temporal variation, suggestive of stellar coronal emission.
The existence of both a likely 2MASS and USNO candidate counterpart (Table~\ref{t:data_oi}) reinforces this interpretation in which case simple spectral models might not be expected to fit the data, as we have seen (\S\ref{s:s32_spectrum})
\subsection{Candidate Catalog Optical and Near-Infrared Counterparts\label{s:oi}}
We searched for candidate optical counterparts to the detected X-ray sources.
We used HEASARC's {\tt BROWSE}\footnote{See
http://heasarc.gsfc.nasa.gov/db-perl/W3Browse/w3browse.pl.}
feature to search for cataloged objects within the 99\%-confidence radius ($\epsilon_{99}$) of X-ray source positions in Table~\ref{t:data_x}.
Table~\ref{t:data_oi} tabulates results of a cross correlation of the X-ray positions of the {\sl Chandra}-detected sources (column 1)
with optical sources (columns~2--7) in the USNO-B1.0 catalog (Monet et al. 2003) and with near-infrared sources (columns~8--12) in the 2MASS catalog (Skrutskie et al. 2006).
\subsubsection{USNO-B1.0\label{s:oi_usno}}
For fourteen (14) X-ray sources, we found a USNO-B1 (optical) source within the 99\%-confidence radius $\epsilon_{99}$ of the {\sl Chandra}\ position (Table~\ref{t:data_x}).
Table~\ref{t:data_oi} columns~2--4 list, respectively, the USNO-B1 right ascension RA, declination Dec, and RMS positional error $\sigma_{o}$ in the form ($\sigma_{o}({\rm RA})$, $\sigma_{o}({\rm Dec})$).
Column~5 gives the angular separation $\delta_{ox}$ between optical and X-ray positions; column~6, the I-band magnitude.
Column~7 estimates the probability $p_o(\delta_{ox}, {\rm I})$ for a chance coincidence within the observed separation of an object as bright or brighter than the I magnitude of the optical candidate.
We determined this probability from the I-magnitude distribution of the 893 USNO sources within $6\arcmin$ (slightly larger than the 8$\times$8\arcmin\ {\sl Chandra}\ field of view)
of the X-ray pointing direction.
We designate a potential optical counterpart to an X-ray source as a `strong candidate' only if the sample impurity---i.e., probability of chance coincidence---$p_o(\delta_{ox}, {\rm I})<1\%$.
All the candidate USNO-B1 sources satisfy this criterion.
\subsubsection{2MASS\label{s:oi_2mass}}
For nineteen (19) X-ray sources, we found a 2MASS (near-infrared) source within the 99\%-confidence radius $\epsilon_{99}$ of the {\sl Chandra}\ position (Table~\ref{t:data_x}).
Table~\ref{t:data_oi} columns~8 and 9 list, respectively, the 2MASS right ascension RA and declination Dec, each with an RMS positional error $\sigma_{i}\approx 0\farcs080$.
Column~10 gives the angular separation $\delta_{ix}$ between near-infrared and X-ray positions; column~11, the K$_{s}$-band magnitude.
Column~12 estimates the probability $p_i(\delta_{ix}, {\rm K}_{s})$ for a chance coincidence within the observed separation of an object as bright or brighter than the K$_{s}$ magnitude of the infrared candidate.
We determined this probability from the $K_s$-magnitude distribution of the 1188 2MASS sources within $6\arcmin$ of the X-ray pointing direction.
We designate a potential near-infrared counterpart to an X-ray source as a `strong candidate' only if the sample impurity---i.e., probability of chance coincidence---$p_i(\delta_{ix}, {\rm K}_{s})<1\%$.
Eighteen (18) sources satisfy this criterion, the exception being source \#44.
Note that the 2MASS set of 18 strong candidate counterparts includes 13 of the 14 USNO-B1 set of strong candidates (\S\ref{s:oi_usno}).
Table~\ref{t:2MASS_colors} tabulates the 2MASS near-infrared photometry (columns~2--7) of the 18 strong-candidate optical (visible--near-infrared) counterparts to {\sl Chandra}-detected X-ray sources.
Examination of the near-infrared color--color diagram for all 2MASS sources within 6\arcmin\ of the pointing direction indicates those that are strong-candidate counterparts to X-ray sources are distributed as the field sources---i.e., as reddened main-sequence stars.
Although most Galactic-plane 2MASS objects are normal stars, the objects identified with the X-ray sources need not be normal stars:
For example, the X-ray emission may originate in an accreting compact companion.
Figure ~\ref{f:2MASS_colors} shows the X-ray hardness ratio of the 9 brighter X-ray sources versus the infrared color J-H of the corresponding strong 2MASS counterparts.
The x-ray-soft and bluer infrared sources in the lower left corner of the figure are likely foreground stars.
The x-ray-hard and reddened sources in the upper right hand corner
may be X-ray binaries and/or AGN.
\subsubsection{WIYN observations\label{s:wiyn}}
The images we describe in Section~\ref{s:optical} allow us to measure or limit the optical magnitudes of several other X-rays sources.
A few sources were not covered in all sub-frames of the image stack and so were measured from individual exposures.
Three additional optical counterparts to the X-ray sources were detected in the low exposure guide sector allowing improved frame registration.
Table~\ref{t:omags} gives the detected magnitudes and upper limits.
\section{Further Discussion and Summary\label{s:summary}}
Using the {\sl Chandra}\ X-ray Observatory, we continued our search (Becker et al. 2004, Weisskopf et al. 2006) for possible X-ray counterparts to the intriguing {$\gamma$-ray}\ source now known as PSR J2021+4026.
We found 44 X-ray sources in a field centered on the PSR J2021+4026\ position, located along the line-of-sight toward the $\gamma$-Cyg SNR.
Only one of these sources, \#20, can reasonably and with high confidence be taken as the X-ray counterpart to the {$\gamma$-ray}\ source.
There are a number of reasons supporting this conclusion.
First and foremost, our X-ray source \#20, is only 14.7\arcsec\ distant from the best-fitting {$\gamma$-ray}\ timing position.
In addition, this separation is within the combined statistical and systematic errors on that position.
There are also no other X-ray sources detected within 66\arcsec\ of the {$\gamma$-ray}\ source position to a {\sl Chandra} source-detection limit of $\sim$10$^{-15}$ erg~cm$^{-2}$~s$^{-1}$ in the 0.5$-$8.0 keV bandpass
making it highly unlikely that any of the other X-ray sources in the field are candidate counterparts.
Furthermore, the spectrum of source \#20 has a shape consistent with soft ($\log (T_{\infty}) \sim 6.0$ to $6.5$) thermal emission as expected from a young neutron star though perhaps somewhat higher than expected from the spin-down age of 77000~yr estimated for PSR J2021+4026.
There is also a hint of extended diffuse X-ray emission in the vicinity of source \#20 that
may be an associated PWN.
With source \#20 as the counterpart, we infer $F_\gamma/F_{\rm X} \sim 1.1 \times 10^4$, not atypical of young isolated neutron stars (e.g., Becker 2009).
If source \#20 is not the X-ray counterpart, the flux ratio is at least 30$\times$ larger,
which would be substantially larger
than the observed ratio for other {$\gamma$-ray}\ pulsars.
A similar argument using the optical data also supports source \#20 as the counterpart: our r$^\prime >25.2$ limit implies a lower limit of $F_{\rm X}/F_{\rm V} \approx 250$, with some uncertainty due to extinction.
This is already larger than the maximum value for X-ray binaries ($\sim$15) or BL Lacs ($\sim$100) and is approaching typical values for isolated neutron stars ($\sim 10^{3-4}$) (e.g. Schwope et al. 1999).
Thus, based on the X-ray/optical evidence alone, source \#20 is likely an isolated neutron star and is the likely counterpart for PSR J2021+4026.
Finally, the X-ray spectrum has a shape consistent with the soft thermal emission expected from a young neutron star.
This emission likely represents a fraction of the stellar surface heated by back-flowing particles generated by magnetospheric activity (eg. Harding \& Muslimov 2001).
At present the fitted parameters suggest that this thermal component implies a relatively large distance, $\approx 6$ kpc, incompatible with an association with SNR G78+2.1.
However, the fits also indicate a complex spectrum with at least two components; much higher S/N data will be needed to extract strong spectral constraints.
Of course, a heated polar cap suggests that sensitive observations should also be able to detect X-ray pulsations at the 265~ms spin period, the definitive test of the counterpart's association.
\acknowledgements
The work of MCW, DAS, RFE, SLO, and AFT is supported by the {\sl Chandra}\ Program.
The {\sl Chandra}\ data was obtained in response to proposal number 11500575
by the Chandra X-ray Observatory Center, which is operated by the Smithsonian Astrophysical Observatory for and on behalf of the National Aeronautics Space Administration under contract NAS8-03060. Support for this work was also provided to PSP in response to this proposal through Chandra Award Number GO0-11086A issued by the Chandra X-ray Observatory Center.
The work of RWR was supported in part by NASA grant NNX08AW30G.
The \textit{Fermi} LAT Collaboration acknowledges generous ongoing support
from a number of agencies and institutes that have supported both the
development and the operation of the LAT as well as scientific data analysis.
These include the National Aeronautics and Space Administration and the
Department of Energy in the United States, the Commissariat \`a l'Energie Atomique
and the Centre National de la Recherche Scientifique / Institut National de Physique
Nucl\'eaire et de Physique des Particules in France, the Agenzia Spaziale Italiana
and the Istituto Nazionale di Fisica Nucleare in Italy, the Ministry of Education,
Culture, Sports, Science and Technology (MEXT), High Energy Accelerator Research
Organization (KEK) and Japan Aerospace Exploration Agency (JAXA) in Japan, and
the K.~A.~Wallenberg Foundation, the Swedish Research Council and the
Swedish National Space Board in Sweden.
Additional support for science analysis during the operations phase is gratefully
acknowledged from the Istituto Nazionale di Astrofisica in Italy and the Centre National d'\'Etudes Spatiales in France.
Our analyses utilized software tools provided by the Chandra X-ray Center (CXC) in the application package CIAO and from the High-Energy Astrophysics Science Archive Research Center (HEASARC, operated by the NASA Goddard Space Flight Center, Greenbelt MD, and by the Smithsonian Astrophysical Observatory, Cambridge MA).
We also thank Mike Wolff for a careful critique of the manuscript.
|
2,877,628,088,977 | arxiv | \subsection{Results}
{\bf System size and sampling scheme.} We employ simulation cells containing up to $256$ hydrogen atoms and use a novel MD scheme with friction in the NVT ensemble (see Supplementary Note 1, Supplementary Fig. 1), for simulation times of few picoseconds, long enough to have well converged results on the pressure, internal energy, and the radial pair distribution function $g(r)$.
Notice also that our approximation to consider QMC in its simplest variational formulation (see Methods), i.e., variational Monte Carlo (VMC), is already quite satisfactory because the much more computationally expensive diffusion Monte Carlo (DMC) can correct the pressure only by few GPa's (see Fig.\ref{fig:finitesize}a)
which is not relevant for the present accuracy in the phase diagram, rather size effects seem to be much more important (see Methods).
In this regard we also performed a DMC-MD on a much smaller system with 54 protons and we have verified that, apart from an overall shift in the total electronic energy, VMC and DMC dynamics give quantitatively the same results for the pressure and the $g(r)$ (see Supplementary Fig. 2, Supplementary Note 2). This implies that the forces, evaluated at the VMC level, are already accurate to drive the dynamics to the correct equilibrium distribution.
\begin{figure}[h!]
\begin{center}
\includegraphics[scale=0.40]{2}
\caption{ {\bf Accuracy and finite size effects.} ({\bf a}) Pressure as a function of the system sizes $N = $64, 128, and 256 at a density
$r_s=1.22$ (the Wigner-Seitz radius $r_s$ is defined as $V/N= 4/3 \pi (r_s a_0)^3$where $V$ is the volume, $N$ the number of ions, and $a_0$
is the Bohr radius.) near the transition at 600 K. The Diffusion Monte Carlo (DMC) value, obtained from
20 equilibrated ($N=256$) configurations generated by the Variational Monte Carlo (VMC) dynamics, is also plotted (red square). ({\bf b}) Finite size scaling of the condensation energy gain at $r_s= 1.28$ and 600 K. The condensation energy gain becomes negligible in the infinite size limit (see Methods). ({\bf c}) Dissociation energy curves for the $H_2$ molecule for different methods, QMC at the VMC level and with the same wavefunction variational ansatz employed in the dynamics, DFT with PBE or HSE DFs\cite{revdft}, and the exact curve obtained with full configuration interaction (CI) method\cite{corongiu}.}
\label{fig:finitesize}
\end{center}
\end{figure}
\begin{figure}[h!]
\begin{center}
\includegraphics[scale=0.525]{3}
\caption{{\bf Ergodicity of the simulations.} ({\bf a}) Energy $E$ and ({\bf b}) pressure $P$ as a function of the ionic steps
during a Langevin dynamics simulation at $r_s=1.28$ and $T=600$ K.
Red squares refer to a simulation whose starting configuration is an atomic fluid, while black circles correspond to a molecular
initial configuration (see inset {\bf c} for the radial pair distribution functions of these two configurations). After a short equilibration,
energy, pressure, and radial pair distribution function (inset {\bf d}) converge to the same values.
}
\label{fig:ergo}
\end{center}
\end{figure}
{\bf Ergodicity of QMC simulations.} In our approach, we cannot address directly the issue of metallicity or insulator behavior but can assess possible first order transition by large scale simulations, where any discontinuity of correlation functions with varying temperature or pressure should be fairly evident and clear. Considering that even an insulator with a true gap should also exhibit residual activated conductivity at finite temperatures, a metal-insulator transition at finite temperatures can be experimentally characterized only by a jump (usually by several orders of magnitude) of the conductivity. Hence, without a first order transition, namely, without discontinuous jumps in the correlation functions upon a smooth variation of temperature or pressure, we cannot define a true metal-insulator transition at finite temperatures, otherwise it is rather a crossover.
In order to assess that our QMC method is capable of characterizing correctly a phase transition, we carefully check for the possible lack of thermalization near the phase transition by repeating the simulations with very different starting ionic configurations at the same thermodynamic point.
Moreover, the functional form of the trial wave function is flexible enough to correctly describe both the paired and the dissociated state (see Methods), and therefore our approach is expected to be particularly accurate even for the LLT. A complete equilibration is reached within the QMC framework since no hysteresis effects occur in all the range of temperatures studied (see Fig.~\ref{fig:ergo} and Supplementary Fig. 3).
\begin{figure}[h!]
\begin{center}
\includegraphics[scale=0.40]{4}
\caption{{\bf Liquid-liquid transition at 2300 K.} ({\bf a}) Pressure as a function of the density. A clear plateau is visible around $r_s=1.31 - 1.32$,
indicating the first order transition. This evidence is also supported by the discontinuous change with $r_s$ in the radial pair distribution function $g(r)$ (inset {\bf b}). ({\bf c}) Average lifetime of pairs as a function of pressure. A pair is defined here as a couple of ions which are nearest neighbors and whose distance $r$ is smaller than a cutoff $r_c= 1.70$ a.u. The shape of this curve is qualitatively similar for every reasonable choice of $r_c$ although its amplitude may slightly vary. In the insets ({\bf d}) and ({\bf e}), radial pair distribution functions for two different pressures ($P$) in the molecular fluid. The higher the pressure, the smaller is the molecular peak and more coordination shells appear in the long range tail. }
\label{fig:2400}
\end{center}
\end{figure}
{\bf Characterizing the first-order transition. }
To identify the LLT, we trace four isotherms in the range 600 - 2300 K, looking for a possible singular behaviour of the pressure and the radial pair distribution function $g(r)$, in a wide range of density (see Supplementary Note 3, Supplementary Fig. from 4 to 11 ). We indeed find a relatively small discontinuity, which appears to be clear also at the highest temperature considered (see Fig.~\ref{fig:2400}a and b). A similar first-order behavior is also found by looking at the pressure as a function of temperature at fixed density (see Supplementary Fig. 12), i.e. by crossing the LLT vertically, along the isochor having density $r_s$=1.28. Notice that, close to the transition a fully molecular phase is not stable, as a large fraction of pairs is found to be already dissociated (Fig.~\ref{fig:2400}b). Our results are summarized in the $P$-$T$ phase diagram shown in Fig.~\ref{fig:phase}. We should note here that, in our calculations, we neglect nuclear quantum effects. However, this approximation should slightly affect the location of the transition, as it was shown that the inclusion of the zero point motion shifts the LLT toward smaller pressures only by about 40 GPa~\cite{morales,mcmahon}.
In the high pressure phase diagram, our results suggest the existence of a mixed -- although mainly molecular -- liquid, surrounding the solid IV mixed molecular atomic phase (see Fig.~\ref{fig:phase}). We have studied the average lifetime of the pairs formed by nearest neighboring hydrogen ions. As shown in Fig.~\ref{fig:2400}c, it exhibits a clear jump with varying pressure at 2300K, supporting the location of the LLT at 375 GPa. We also notice a precursor drop of the lifetime at around 150 GPa, which corresponds to the onset of the dissociation. This value of 150 GPa is consistent with the pressure range where a drastic but continuous drop of the resistivity is observed in the molecular phase\cite{weir}. In order to better characterize the LLT, we study in Fig.~\ref{fig:2400}d and e the dissociation fraction and the long range behavior of $g(r)$ for two fluid configurations at pressures much smaller than the true first order transition point. Nevertheless, a qualitative change in the behavior of these quantities is evident even within the same phase. Remarkably, not only the dissociation fraction increases with the pressure but also the number of oscillations in the long range $g(r)$ tail becomes larger, both features being very similar to what is observed in the high pressure phase. By taking also into account that, at non zero temperature, a finite and large conductivity can be activated, it is clear that a rather sharp variation of physical quantities can occur much before the first order transition.
\subsection{Discussion}
In conclusion, we have reported the first description of the dissociation transition in liquid hydrogen by {\it ab-initio} simulation based on QMC method with fairly large number of atoms. The main outcomes of our study are summarized as follows: i) the transition, which appears to be first-order, occurs at substantially higher pressures than the previous {\it ab-initio} predictions based on DFT. ii) Employing QMC simulations with large number of atoms is essential because the stability of the molecular phase is otherwise underestimated. iii) The first order character is evident also at the highest temperatures, suggesting that even at these temperatures
this transition is not a crossover. iv) the shape of the LLT boundary is rather unusual and becomes a vertical line in the $P$-$T$ phase diagram for $T < 1100$ K.
By assuming that also at lower temperatures no solid phase emerges, the dissociation pressure should remain almost temperature independent. Therefore, even by considering an upper limit of 100 GPa shift to lower pressures, due to proton quantum effects not included here, we predict that experiments should be done at least above 500 Gpa to realize the Wigner and Huntington dream of hydrogen atomic metallization.
\section{Methods}
{\bf Accuracy of QMC methods.}
In this work we employ the QMC approach for electronic properties. In the simplest formulation, the correlation between electrons is described by the so called Jastrow factor $J$ of the following general form
\begin{equation}
\label{eq:jastrowsimple}
J = \prod \limits_{i<j} e^{ u ({\bm r}_i,{\bm r}_j) }
\end{equation}
where $\bm r_i$ and $\bm r_j$ are electron positions and $u$ is a two-electron function to be determined variationally. It is enough to apply this factor to a single Slater determinant to remove the energetically expensive contributions of too close electrons occupying the same atomic orbital and to obtain for instance the correct dissociation limit for the $H_2$ molecule (see Fig. \ref{fig:finitesize}.c), and essentially exact results for the benchmark hydrogen chain model~\cite{stella}. Starting from this Jastrow correlated ansatz, an important projection scheme has been developed - the diffusion Monte Carlo (DMC) - that allows an almost exact description of the correlation energy, with a full {\it ab-initio} many-body approach, namely by the direct solution of the Schr\"odinger equation. Unfortunately, QMC is much more expensive than DFT, and so far its application has been limited to small number of atoms~\cite{morales}.
{\bf Variational wavefunction.}
For the calculation of the electronic energy and forces we use a trial correlated wavefunction of the form \begin{equation}
J |SD\rangle
\end{equation}
The determinantal part $|SD\rangle $ (Slater determinant) is constructed starting from $N/2$ molecular orbitals, $N$ being the total number of electrons, while the Jastrow part can be written as $J = J_1 J_2 J_3$. The Jastrow factor takes into account the electronic correlation between electrons and is conventionally split
into a homogeneous interaction $J_2$ depending on the relative distance between two electrons i.e., a two-body term as in Eq.\ref{eq:jastrowsimple}, and a non homogeneous contribution depending also on the electron-ion distance, included in the one-body $J_1$ and three-body $J_3$. The exact functional form of these components is given in Ref.\cite{marchi}. Both the Jastrow functions and the determinant of molecular orbitals are expanded in a gaussian localized basis. The optimization of the molecular orbitals is done simultaneously with the correlated Jastrow part. We performed a systematic reduction of the basis set in order to minimize the total number of variational parameters. Indeed for the present accuracy for the phase diagram a small $2s$/atom basis set is sufficient as a larger $3s1p$/atom basis set only improves the total energy of $<1$ mH/atom and leaves substantially unchanged the radial pair distribution function, i.e. the atomic(molecular) nature of the fluid (see Supplementary Fig. 13). The value for the LLT critic pressure is not significantly affected (for the present accuracy in the phase diagram) by the choice of the basis (see Supplementary Fig. 14), rather the difference between QMC and DFT with PBE functional is already evident for a system of 64 atoms.
We now address an issue that, in our opinion, can be extremely important in the context of hydrogen metallization. Close to a metal-insulator transition a resonating valence bond scenario is possible, and may give rise to unconventional superconductivity\cite{pwa}, namely a superconductivity stabilized without the standard BCS electron-phonon mechanism. In order to study this interesting possibility, we have calculated the energy gain obtained by replacing the Slater determinant with a BCS type of wave function, both with the same form of the Jastrow factor. This quantity is known as the condensation energy~\cite{marchi}, and is non zero in the thermodynamic limit when the variational wave function represents a superconductor. Though we neglect quantum effects on protons and we have not systematically studied this issue for all densities, this VMC condensation energy (see Fig.~\ref{fig:finitesize}b), computed by considering about 20 different independent samples at $r_s$=1.28 and T=600 K, is very small and decreases very quickly with $N$ by approaching zero in the thermodynamic limit ($1/N=0$). This result at least justifies the use of a simpler Slater determinant wave function, and shows that, the quality of the wave function can be hardly improved by different, in principle more accurate, variational ansatzes. In this way, a straightforward reduction of the number of variational parameters is possible, by exploiting also the fact that matrix elements connecting localized orbitals above a threshold distance $r_{\rm max}$ do not contribute significantly to the energy. Indeed, as we have systematically checked in several test cases (see Supplementary Fig. 15 and Supplementary Table 1), it is enough to consider $r_{\rm max}$= 4 a.u. to have essentially converged results for the molecular orbitals, implying a drastic reduction of the variational space (from $\simeq 40000$ parameters to $\simeq 5000$ in a 256 hydrogen system).
{\bf Finite size effects.}
All the results for the LLT presented here refer to a cubic supercell at the $\Gamma-$point (see Supplementary Table 2) with the largest affordable number of atoms (256) in order to be as close as possible to the thermodynamic limit. Indeed, even tough the pressure seems to converge with the size of the simulation cell, the molecular (atomic) nature of the liquid is very sensitive to the number of atoms N. This issue was previously reported in Refs. \cite{bonev2,desj} and cannot be removed with a better \emph{k}-point sampling, because this will be equivalent to enforce a fictitious periodicity to a liquid phase. In particular, the N=64 supercell simulations, even with \emph{k}-point sampling, strongly favour the dissociated liquid in both DFT and QMC frameworks (see Supplementary Fig. 16). Thus the critical LLT density is severely underestimated by employing supercells smaller than N=256, which is now considered a standard size in DFT simulations of liquids. The main reason of this effect is the structural frustration, requiring the use of much larger supercells, whose dimension L has to exceed the correlation length of the liquid. A possible rule of thumb consists in checking that the $g(r)$ is smoothly approaching its asymptotic value 1 at $r=L$. Our evidences support the conclusion that a failure in dealing with the finite size effects will result in a severe underestimation of the LLT critical densities. DFT-MD simulations were performed using the QuantumEspresso code\cite{qe}.
{\bf Sampling the canonical ensamble with Langevin dynamics.}
In this study, we sample the canonical equilibrium distribution for the ions by means of a second order generalized Langevin equation, as introduced in Ref.~\onlinecite{attaccalite}. The major advantage of this technique consists in the efficient control of the target temperature even in the presence of noisy QMC forces. Here we improve upon this scheme devising a numerical integrator which is affected by a smaller time step error (see Supplementary Note 1). We adopt the ground state Born-Oppenheimer approximation, namely the variational parameters, defining our wave function, are all consistently optimized at each iteration of our MD. Therefore the electronic entropy has been neglected in all our calculations. However we have carefully checked that this entropy contribution is clearly neglegible in the relevant temperature range studied (see Supplementary Note 4).
As well known, hysteresis is usually found by using local updates in simple Monte Carlo schemes, that can not be therefore reliable to determine the phase boundary of a first order transition. Our method, based on an advanced second order MD with friction, is instead powerful enough that very different phases can be reached during the simulation, with time scales that remain accessible for feasible computations. Indeed, we have also experienced a spontaneous solidification in a pressure and temperature range where the solid phase is expected to be stable (see Fig.~\ref{fig:phase} and Supplementary Note 5, Supplementary Fig. 17 for details).
Though this effect has been observed in a much smaller system (64 hydrogen atoms), we believe that, after the inclusion of proton quantum effects, the present method can also shed light on understanding low temperature solid phases, that remain still highly debated and controversial in recent years~\cite{pick}.
\section{Acknowledgements}
We acknowledge G. Bussi, S. Scandolo, F. Pederiva, S. Moroni, and S. De Gironcoli for useful discussions and support by MIUR, PRIN 2010-2011.
Computational resources are provided by PRACE Project Grant 2011050781 and K computer at RIKEN Advanced Institute for
Computational Science (AICS).
\section{Author contributions}
G.M. and S.S. designed the research
and performed the QMC and DFT calculations. All authors conceived the project, partecipated in the discussion of the results and in the writing
of the paper.
|
2,877,628,088,978 | arxiv | \section*{Acknowledgments}
We thank K. Ono, K. Fukaya, M. Akaho, K. Cieliebak and U. Frauenfelder
for useful comments and suggestions.
We also thank H. Hofer for pointing out a lack of a condition in the
definition of grouped multisection which we implicitly used in the proof
of the construction of its extension.
This work was supported by Grant-in-Aid for JSPS Research Fellow.
\input{SFT-bibliography}
Research Institute for Mathematical Sciences, Kyoto University, Kyoto, Japan
\\
[email protected]
\end{document}
\section{Theory of Kuranishi structure}
\label{theory of Kuranishi structure}
We use the theory of Kuranishi structure for the construction of symplectic field theory.
This theory was developed by Fukaya and Ono in \cite{FO99},
and it is a useful tool to perturb the given equation and get algebraic information of the
moduli space of the solutions.
A neighborhood of each point of the moduli space is usually expressed
as the zero set of a Fredholm map between Banach spaces
or the quotient of the zero set by a group actin.
Since this map is not always transverse to zero, in order to get some
algebraic information, we need to perturb the map and make it transverse to zero.
To get information of the moduli space, the most important thing is
perturbing these maps in a compatible way.
The theory of Kuranishi structure is a scheme to manipulate this compatibility problem.
In this section, we recall this theory and introduce new notions of
pre-Kuranishi space and weakly good coordinate system.
First we explain roughly about what is Kuranishi structure and how we construct it.
As mentioned above, a neighborhood of each point of the moduli space
is expressed as the zero set of a Fredholm map between Banach spaces
or its quotient by a group action.
Adding a finite dimensional vector space to the domain of each Fredholm map
and extending the map to this product space,
we first make each map transverse to zero.
Then the original zero set is the zero set of the projection map from the new zero set
to the added finite dimensional space.
This implies that a neighborhood of each point of the moduli space is expressed
as a zero set of a smooth section of an finite dimensional vector bundle or orbibundle.
(This expression is called a Kuranishi neighborhood.)
This expression depends on the choice of the additional finite dimensional space,
but if one additional space is a subspace of another additional space,
then the former orbibundle can be naturally embedded in the latter.
Kuranishi structure is, roughly speaking, a collection of Kuranishi neighborhoods
with the relation of this kind of embeddings.
\subsection{Orbibundle}
\label{section of orbibundle}
In this subsection, we explain about orbibundle.
First we explain our notation of corners of manifold.
For an open subset $V \subset [0, \infty)^n$, we define the interior of
the corner of codimension $k$
\[
\mathring{\partial}^k V = \{ (x_j) \in [0, \infty)^n;
\# \{j; x_j = 0\} = k\}
\]
and the boundary $\partial V = \bigcup_{k \geq 1} \mathring{\partial}^k V$.
For each point $x \in \mathring{\partial}^k V$,
we define the normal space $T^\bot_x \mathring{\partial}^k V
= T_x \mathbb{R}^n / T_x \mathring{\partial}^k V$.
We say a smooth map $\phi$ from an open subset $V \subset [0, \infty)^n$
to another $V' \subset [0, \infty)^{n'}$ is an embedding if
$\phi$ is the restriction of some embedding from an open subset of $\mathbb{R}^n$ to $\mathbb{R}^{n'}$,
$\phi(\mathring{\partial}^k V) \subset \mathring{\partial}^k V'$ for each $k \geq 0$, and
the differential $\phi_{\ast x} : T^\bot_x \mathring{\partial}^k V
\to T^\bot_{\phi(x)} \mathring{\partial}^k V'$ is an isomorphism
for each $k \geq 1$ and $x \in \mathring{\partial}^k V$.
The definition of the diffeomorphism is similar.
Using these definitions, we define manifold with corners and
embedding between two manifolds.
We also explain our definition of
clean intersection.
Let $(M_\alpha)_{\alpha \in A}$ be a locally finite family of
submanifolds of a manifold $M$.
We say $(M_\alpha)_{\alpha \in A}$ intersect cleanly if
for any $p \in M$, there exist a coordinate $U_p \cong \mathbb{R}^m$ of $p \in M$
such that submanifolds $M_\alpha \cap U_p$
coincides with the subspace $\{(t_i) \in \mathbb{R}^m; t_i = 0 \text{ for } i \in I_{\alpha, p}\}$
for some $I_{\alpha, p} \subset \{1, \dots, m\}$
for all $\alpha \in A$ such that $p \in M_\alpha$.
Note that if there exists a smooth function $f_\alpha$ on $M_\alpha$
for each $\alpha \in A$ and they coincide on their intersections,
then we can extend them to a smooth function on $M$ if we shrink $M_\alpha$
to their arbitrary relatively compact subsets.
(In the case of orbifolds,
we say a family of embeddings intersects cleanly
if all of their lifts intersect cleanly.)
First we show the following elementary lemma for the definition of orbifold and
orbibundle.
\begin{lem}\label{effective action}
Let $V$ and $V'$ be connected manifolds (with or without corners).
Suppose a finite group $G'$ acts on $V'$ effectively.
Then the following hold true.
\begin{enumerate}[label=\normalfont(\roman*)]
\item
If two submersions $\phi, \psi : V \to V'$ induce the same map
$\overline{\phi}= \overline{\psi} : V \to V'/G'$, then
there exists a unique $h\in G' $ such that $\psi = h \phi : V\to V'$.
\item
Assume that two embeddings $\phi, \psi : V \to V'$ induce the same map
$\overline{\phi}= \overline{\psi} : V \to V'/G'$.
If $\phi(V) = \psi(V)$, $\phi(V) \subset V'$ is $G' $-invariant and
the $G'$-action on $\phi (V)$ is effective,
then there exists a unique $h\in G' $ such that $\psi = h \phi : V\to V'$.
\end{enumerate}
\end{lem}
\begin{proof}
(i)
First we claim that the differentials $D\phi$ and $D\psi$ induce
the same map $\overline{D\phi} = \overline{D\psi} : TV \to TV'/G'$.
For any curve $l$ in $V$ there exist some $t_j\to 0$ and $h \in G' $
such that $\phi (l(t_j)) = h \psi (l(t_j))$.
Hence $D\phi(l(0)) \dot{l}(0) = h D\psi(l(0)) \dot{l}(0)$, which implies the claim.
Next we show that for any $p\in V$, there exists unique $h_p \in G' $ such that
$D\psi(p) = h_p D\phi(p)$.
Uniqueness is a consequence of the effectiveness of the action.
The first claim $\overline{D\phi} = \overline{D\psi}$ implies that
\[
T_{\phi(p)} V' = \bigcup _{g \in G'} \{v \in T_{\phi (p)} V';
gv = D\psi(p) D\phi(p)^{-1} v \}.
\]
(For each $v \in T_{\phi (p)} V'$,
$D\psi(p) D\phi(p)^{-1} v \subset T_{\psi(p)} V'$ is an affine space
which is contained in the orbit $G' v$.
Hence it is a point in $G' v$.)
Since the right hand side of this equation is a finite union of subspaces of $T_{\phi(p)} V'$,
one of them coincides with the whole space.
In other words, there exists some $h_p \in G'$ such that $D\psi(p) = h_p D\phi(p)$.
Since the dimension of the space
\[
\{ v \in T_{\phi (p)} V' ; gv = D\psi(p) D\phi(p)^{-1} v \}
\]
is upper semi-continuous with respect to $p$ for each $g \in G'$,
the uniqueness of $h_p \in G'$ and the connectedness of $V$ imply that
$h = h_p \in G'$ does not depend on $p \in V$.
Therefore $\psi = h \phi$ on $V$.
(ii)
Apply (i) to $\phi, \psi : V \to \phi (V)=\psi (V)$.
\end{proof}
\begin{rem}
\label{effective action for composition of embedding and submersion}
In (ii), we assumed that $\phi$ and $\psi$ are embedding.
However, it is clear that the same holds if $\phi$ and $\psi$ are
submersions to their image $\phi(V) = \psi(V)$.
\end{rem}
\begin{defi}
An orbichart $\mathcal{V} = (V, \pi_V, \mathcal{V})$ consists of
a connected manifold $V$ (with or without corners), a topological space $\mathcal{V}$ and
a continuous map $\pi_V : V \to \mathcal{V}$ such that
\begin{itemize}
\item
there exists some finite group $G_V $ acting smoothly and effectively on $V$
\item
$\pi_V$ induces a homeomorphism $\pi_V : V/G_V \stackrel{\cong}{\to} \mathcal{V}$
\item
if $\mathring{\partial}^k V \neq \emptyset$, then $G_V$ acts effectively
on each connected component of $\mathring{\partial}^k V$.
\end{itemize}
\end{defi}
Lemma \ref{effective action} implies that the image of $G_V$ in $\mathrm{Aut}\,V$ is
$\mathrm{Aut}_{\mathcal{V}} V := \{ g : V \stackrel{\cong}{\to} V ; \pi_V g = \pi_V \}$.
We always use $G_V = \mathrm{Aut}_{\mathcal{V}} V$ in this paper.
For a connected open subset $\mathcal{U} \subset \mathcal{V} $, let $U\subset V$ be a connected
component of $\pi_V^{-1}(\mathcal{U})$.
Then $\mathcal{U} = (U, \pi_V|_{U}, \mathcal{U}) $ is an orbichart, and this does not depend on the choice
of the connected component.
Note that $G_U \subsetneq G_V$ in general.
\begin{defi}
We say a continuous map $\varphi : \mathcal{V} \to \mathcal{V}'$ between two orbicharts is smooth if
there exists a smooth map $\phi : V \to V'$ such that $\pi_{V'} \phi = \varphi \pi_V$
on $V$.
We call $\phi$ a lift of $\varphi$.
\[
\begin{tikzcd}
&V \ar{d}{\pi_V} \ar{r}{\phi} &V' \ar{d}{\pi_{V'}} \\
&\mathcal{V} \ar{r}{\varphi} & \mathcal{V}'
\end{tikzcd}
\]
\end{defi}
\begin{defi}\label{smooth map}
An embedding $\varphi : \mathcal{V} \to \mathcal{V}'$ is an injective smooth map such that
there exists a connected neighborhood $\mathcal{U}'$ of $\varphi (V) \subset \mathcal{V}'$
which satisfies the following conditions:
The lift $\phi : V \to U'$ of $\varphi : \mathcal{V} \to \mathcal{U}'$ is an embedding,
$\phi (V) \subset U' $ is $G_{U'}$-invariant and
$G_{U'}$ acts on $\phi (V)$ effectively.
If in addition $\phi(V) \subset U'$ is open, then we say $\varphi$ is an open embedding.
(This is the case where $\dim V = \dim V'$.)
\end{defi}
\begin{rem}
In the above definition, we cannot always take $\mathcal{U}' = \mathcal{V}'$ since
$\phi (V) \subset V'$ is not always $G_{V'}$-invariant.
We also note that Lemma \ref{effective action} implies $G_V \cong G_{U'}$.
\end{rem}
\begin{defi}
An orbibundle chart $(\mathcal{V}, \mathcal{E}) = ( (V,\pi_V, \mathcal{V}) , (E, \pi_E , \mathcal{E}) , \tilde{\pi} , \pi)$
consists of
\begin{itemize}
\item
topological spaces $\mathcal{V} , \mathcal{E}$
\item
a vector bundle $\tilde{\pi} : E \to V$ over a connected manifold $V$
\item
continuous maps $\pi_V : V \to \mathcal{V}, \pi_E : E \to \mathcal{E}, \pi : \mathcal{E} \to \mathcal{V} $
\end{itemize}
which satisfy the following conditions:
\begin{itemize}
\item
There exists a finite group $G$ acting smoothly and effectively on $V$ and $E$
such that
\begin{itemize}
\item
$\tilde{\pi} : E \to V$ is $G$-equivariant
\item
$\pi_V$ and $\pi_E$ are $G$-equivariant, and they induce homeomorphisms
$\pi_V : V/G \to \mathcal{V}$ and $\pi_E : E/G \to \mathcal{E}$
\item
if $\mathring{\partial}^k V \neq \emptyset$, then $G$ acts effectively on
every connected component of $\mathring{\partial}^k V$
\end{itemize}
\item
The following diagram is commutative.
\[
\begin{tikzcd}
&E \ar{d}{\tilde{\pi}} \ar{r}{\pi_E} &\mathcal{E} \ar{d}{\pi} \\
&V \ar{r}{\pi_{V}} & \mathcal{V}
\end{tikzcd}
\]
\end{itemize}
\end{defi}
Note that we can take $G = \mathrm{Aut}_{\mathcal{V}} V $.
Note also that $\mathcal{V}$ and $ \mathcal{E}$ are orbichart themselves.
For a connected open subspace $\mathcal{U} \subset \mathcal{V}$,
\[
(\mathcal{U}, \mathcal{E} | _{\mathcal{U}}) = ((U, \pi_V|_U, \mathcal{U}),
(E|_U, \pi_E |_{E|_U}, \mathcal{E} | _{\mathcal{U}}), \tilde{\pi} |_{E|_U}, \pi |_{\mathcal{E} |_{\mathcal{U}}})
\]
is also an orbibundle chat.
\begin{defi}
A bundle map $(\varphi , \hat\varphi) : (\mathcal{V}, \mathcal{E}) \to (\mathcal{V}', \mathcal{E}')$ between
two orbibundle charts is a pair of continuous maps
$\varphi : \mathcal{V} \to \mathcal{V}'$ and $\hat\varphi : \mathcal{E} \to \mathcal{E}'$ such that
there exist some smooth bundle map
$(\phi,\hat\phi) : (V,E) \to (V',E')$ which makes the following diagram commutative.
\[
\begin{tikzcd}
&E \ar{r}{\hat\phi} \ar{d} \ar{rrd} &E' \ar{d} \ar{rrd} & & \\
&V \ar{r}{\phi} \ar{rrd} &V'\ar{rrd} &\mathcal{E} \ar{r}[swap]{\hat\varphi} \ar{d} & \mathcal{E}' \ar{d} \\
& &&\mathcal{V} \ar{r}[swap]{\varphi} &\mathcal{V}'
\end{tikzcd}
\]
\end{defi}
\begin{defi}
We say a bundle map $(\varphi , \hat\varphi) : (\mathcal{V}, \mathcal{E}) \to (\mathcal{V}', \mathcal{E}')$ is
an embedding if $\varphi : \mathcal{V} \to \mathcal{V}' $ is an embedding and
the restriction of $\hat \phi : E \to E' $ to each fiber is injective.
In this case, $\hat\varphi : \mathcal{E} \to \mathcal{E}' $ is also an embedding between two orbicharts.
We say $(\varphi, \hat\varphi)$ is an open embedding if in addition $\varphi$ is
an open embedding and $\hat\phi$ is an isomorphism on each fiber.
\end{defi}
\begin{defi}\label{orbibundle}
An orbibundle $(\mathcal{V}, \mathcal{E}) = (\mathcal{V}, \mathcal{E}, \pi)$ consists of
Hausdorff spaces $\mathcal{V}$, $\mathcal{E}$ and a continuous map $\pi : \mathcal{E} \to \mathcal{V}$
which satisfies the following conditions.
\begin{itemize}
\item
For each $x \in \mathcal{V}$, there exists a neighborhood $\mathcal{V}_x \subset \mathcal{V}$ such that
$(\mathcal{V}_x,\allowbreak \mathcal{E}|_{\pi^{-1}(\mathcal{V}_x)},\allowbreak \pi|_{\mathcal{E}|_{\pi^{-1}(\mathcal{V}_x)}})$ has a structure of
orbibundle chart.
We define $\mathcal{E}_x = \mathcal{E}|_{\pi^{-1}(\mathcal{V}_x)}$.
We always assume $\pi_{V_x}^{-1}(x) \subset V_x$ is one point and
$G_{V_x} = \mathop{\mathrm{Aut}}\nolimits_{\mathcal{V}_x}V_x$ fixes this point.
We denote this point $\pi_{V_x}^{-1}(x) \subset V_x$ by $x\in V_x$.
We always assume that $x \in \mathring{\partial}^k V_x$ for the largest $k \geq 0$
such that $\mathring{\partial}^k V_x \neq \emptyset$.
\item
For each $y \in \mathcal{V}_x$, if we shrink the neighborhood $\mathcal{V}_y$, the inclusion map
$(\mathcal{V}_y, \mathcal{E}_y) \hookrightarrow (\mathcal{V}_x, \mathcal{E}_x)$ is an open embedding of orbibundle chart.
\end{itemize}
\end{defi}
\begin{defi}
An embedding $(\varphi, \hat \varphi) : (\mathcal{V}, \mathcal{E}, \pi) \to (\mathcal{V}', \mathcal{E}', \pi')$ of an orbibundle is
a pair of continuous maps $\varphi : \mathcal{V} \to \mathcal{V}'$ and $\hat \varphi : \mathcal{E} \to \mathcal{E}'$
such that
\begin{itemize}
\item
$\pi' \circ \hat \varphi = \varphi \circ \pi : \mathcal{E} \to \mathcal{V}'$
\item
for each $x \in \mathcal{V}$, $(\varphi_x, \hat \varphi_x)
:= (\varphi, \hat \varphi)|_{(\mathcal{V}_x, \mathcal{E}_x)} : (\mathcal{V}_x, \mathcal{E}_x)
\to (\mathcal{V}_{\varphi(x)}, \mathcal{E}_{\varphi(x)})$ is an embedding of an orbibundle chart
if we shrink $\mathcal{V}_x$.
\end{itemize}
\end{defi}
\begin{defi}
Let $(\mathcal{V}, \mathcal{E}) = (\mathcal{V}, \mathcal{E}, \pi)$ be an orbibundle.
A smooth section $s : \mathcal{V} \to \mathcal{E}$ is a continuous map such that
$\pi \circ s = \mathrm{id}_{\mathcal{V}}$ and the restriction of $s$ on each $\mathcal{V}_x$ is
a smooth map between orbicharts $\mathcal{V}_x$ and $\mathcal{E}_x$.
Note that the lift of $s$ on $V_x$ is unique and it is a $G_{V_x}$-equivalent section of
$(V_x, E_x)$.
We also denote this $G_{V_x}$-equivalent section by $s : V_x \to E_x$.
\end{defi}
\begin{defi}
We say a bundle map $(\varphi, \hat \varphi) : (\mathcal{V}, \mathcal{E}) \to (\mathcal{V}', \mathcal{E}')$ between two
orbibundle charts is a submersion if for its lift $(\phi, \hat \phi)$,
$\phi : V \to V'$ is a submersion and
the restriction of $\hat \phi$ to each fiber is an isomorphism.
Note that Lemma \ref{effective action} implies that
there exists a homomorphism $\rho_\phi : G_V \to G_{V'}$ such that
$\phi \circ g = \rho_\phi(g) \circ \phi$.
\end{defi}
Finally we consider fiber product.
Let $(\mathcal{V}, \mathcal{E})$ be an orbibundle chart and $\varphi : \mathcal{V} \to Y$ be a submersion
to a manifold $Y$.
Then for any submanifold $Z \subset Y$,
$(\varphi^{-1}(Z), \mathcal{E}|_{\varphi^{-1}(Z)})$ is an orbibundle chart
(or a disjoint union of orbibundle charts if $\varphi^{-1}(Z)$ is disconnected).
We note that this satisfies the assumption of the effective group action.
Indeed, the $G_V$-action on $\pi_V^{-1}(\varphi^{-1}(Z))
= \phi^{-1}(Z)$ is effective, where $\phi : V \to Y$ is a lift of $\varphi$.
For the construction of SFT (in particular for Bott-Morse case),
we also need to treat fiber products over orbifolds.
\begin{defi}
\label{def of fiber product of orbibundle over orbifold}
Let $\mathcal{W} = (W, \pi_W, \mathcal{W})$ be an orbibundle chart and
$K \subset \mathcal{W}$ be an embedded simplicial complex.
We assume that there exists a regular $G_W$-complex
$L \subset W$ (see \cite{Bre72} for regular complex) and an isomorphism
$\psi : L/G_W \cong K$ such that $\psi \circ \pi_L = \pi_W$
on $L \subset W$, where $\pi_L : L \to L/G_W$ is the quotient map.
Let $\varphi$ be a submersion from an orbichart $\mathcal{V}$ to $\mathcal{W}$ and assume that
for any point $p \in V$, the stabilizer $G_p \subset G_V$ of $p$ acts on
a neighborhood of $p$ in $\pi_V^{-1}(\varphi^{-1}(\varphi(\pi_V(p))))$ effectively.
Then $\varphi^{-1}(K) = (\phi^{-1}(L), \pi_V|_{\phi^{-1}(L)}, \varphi^{-1}(K))$
(or its connected components) are not orbicharts in a strict sense because
$\phi^{-1}(L)$ is not a manifold,
but for each connected component $\phi^{-1}(L)_0$ of $\phi^{-1}(L)$,
the group $\{g \in G_V; g\phi^{-1}(L)_0 = \phi^{-1}(L)_0\}$ acts effectively on it.
We regard each connected component of $(\varphi^{-1}(K), \mathcal{E}|_{\varphi^{-1}(K)})$
as an orbibundle chart.
Using this kind of orbibundle charts,
we define the fiber product of an orbibundle with a simplex in an orbibundle.
We say a section (or a multisection) of $(\varphi^{-1}(K), \mathcal{E}|_{\varphi^{-1}(K)})$
is smooth if its lift (or its branches) are the restrictions of some smooth sections
defined on a neighborhood of $\phi^{-1}(L) \subset V$.
\end{defi}
In application, we sometimes need to use the following notion of essential submersion.
In the following definition, we regard usual orbibundle charts $(\mathcal{V}, \mathcal{E})$ as
a fiber product of $(\mathcal{V}, \mathcal{E})$ with the $0$-dimensional simplex in a point.
\begin{defi}
\label{def of essential submersion}
Let $(\mathcal{V}_i, \mathcal{E}_i)$ be orbibundle charts for $i = 1,2$, and
let $(\varphi_i^{-1}(K_i), \allowbreak \mathcal{E}_i|_{\varphi_i^{-1}(K_i)})$ be their fiber products
with embedded simplicial complexes $K_i \subset \mathcal{W}_i$
as in Definition \ref{def of fiber product of orbibundle over orbifold}.
As in Definition \ref{def of fiber product of orbibundle over orbifold},
let $\phi_i$ be the lifts of $\varphi_i$, and
let $L_i \subset W_i$ be the regular $G_W$-complex
such that $L_i / G_W \cong K_i$.
An essential submersion $(\varphi, \hat \varphi)$ from
$(\varphi_1^{-1}(K_1), \mathcal{E}_1|_{\varphi_1^{-1}(K_1)})$ to
$(\varphi_2^{-1}(K_2), \mathcal{E}_2|_{\varphi_2^{-1}(K_2)})$ is
a smooth bundle map from $(\mathcal{V}_1, \mathcal{E}_2)$ to $(\mathcal{V}_2, \mathcal{E}_2)$ whose lifts
$(\phi, \hat \phi)$ satisfy the following conditions.
\begin{itemize}
\item
The image $\varphi(\varphi_1^{-1}(K_1))$
is contained in $\varphi_2^{-1}(K_2)$.
\item
For any $k_1, k_2 \geq 0$ and any simplices $s_1$ of $L_1$ and $s_2$ of $L_2$,
\[
A = \phi^{-1}\bigl(\mathring{\partial}^{k_2} V_2 \cap \phi_2^{-1}(\mathop{\mathrm{Int}}\nolimits s_2)\bigr)
\cap (\mathring{\partial}^{k_1} V_1 \cap \phi_1^{-1}(\mathop{\mathrm{Int}}\nolimits s_1))
\subset V_1
\]
is a submanifold of $V_1$,
$\phi|_A : A \to \mathring{\partial}^{k_2} V_2 \cap \phi_2^{-1}(\mathop{\mathrm{Int}}\nolimits s_2)$ is
a submersion.
\item
The restriction of $\hat \phi$ to each fiber is an isomorphism.
\end{itemize}
For fiber products $(\varphi_i^{-1}(K_i), \mathcal{E}_i|_{\varphi_i^{-1}(K_i)})$
of orbibundles $(\mathcal{V}_i, \mathcal{E}_i)$ with simplicial complexes $K_i \subset \mathcal{W}_i$,
an essential submersion $(\varphi, \hat \varphi)$ from
$(\varphi_1^{-1}(K_1), \allowbreak \mathcal{E}_1|_{\varphi_1^{-1}(K_1)})$ to
$(\varphi_2^{-1}(K_2), \mathcal{E}_2|_{\varphi_2^{-1}(K_2)})$ is
a smooth bundle map from a neighborhood $(\mathcal{N}_1, \mathcal{E}_1|_{\mathcal{N}_1})$
of $\varphi_1^{-1}(K_1)$ to $(\mathcal{V}_2, \mathcal{E}_2)$
whose restriction to each orbibundle chart is an essential submersion
in the above sense.
\end{defi}
\begin{eg}
The map $f : [0, \infty)^k \times \mathbb{R} \to [0, \infty) \times \mathbb{R}$ defined by
\[
f((s_i)_{1 \leq i \leq k}, t) = (s_1 \cdots s_k, t)
\]
is an essential submersion if we regard $[0, \infty)^k \times \mathbb{R}$ and
$[0, \infty) \times \mathbb{R}$ as orbibundle charts of rank $0$ without group action.
The map $h : [0, \infty) \times \mathbb{R} \to [0, \infty)^2$ defined by
\[
h(\hat \rho, b) = (\hat \rho, \hat \rho e^b)
\]
is also an essential submersion.
\end{eg}
\begin{rem}
The map $h$ in the above example maps the point $(0, 0)$
of the corner of codimension $1$ to the point $(0, 0)$ of the corner of
codimension $2$.
Note that there does not exist a submersion which maps a point of the corner
of codimension $k$ to a point of the corner of codimension $l > k$.
Therefore the notion of essential submersion is crucial to understand
the structure of moduli space of disconnected holomorphic buildings.
See Section \ref{Kuranishi of disconnected buildings}.
\end{rem}
We also use the following generalization.
In Definition \ref{def of fiber product of orbibundle over orbifold},
we assumed that $L \subset W$ is a regular $G_W$-complex.
Instead, let $L \subset W$ be a $G_W$-invariant Euclidean cell complex.
(We do not assume that $L / G_W \subset W$ is a Euclidean cell complex.)
Then we can similarly define the fiber product
$(\varphi^{-1}(L / G_W), \allowbreak \mathcal{E}|_{\varphi^{-1}(L / G_W)})$.
In this case, we read simplices in Definition \ref{def of essential submersion}
as cells of the Euclidean cell complexes.
For example, for an embedded simplicial complex $K \subset \mathcal{W}$,
the fiber product with $\prod^N K / \mathfrak{S}_N \subset \prod^N \mathcal{W} / \mathfrak{S}$
is defined as the above generalization.
We do not use any subdivision of $\prod^N K$ in this case.
\subsection{Multisections}
In this paper, we use a different definition of multisection.
Perturbed multisection in Definition \ref{sum of section and grouped multisection}
plays the role of multisection in \cite{FO99}.
\begin{defi}
\label{multisection for chart}
A multisection $s = (s^\omega)_{\omega \in \Omega}$
of an orbibundle chart $(\mathcal{V} , \mathcal{E})$ is a family of
smooth sections $s^\omega : V \to E$ ($\omega \in \Omega$) indexed by
a finite $G_V$-set $\Omega$ such that
$s^{g \omega} = g_{\ast} s^\omega$ for any $\omega \in \Omega$
and $g \in G_V$.
\end{defi}
\begin{defi}\label{grouped multisection for chart}
A grouped multisection $\boldsymbol{\epsilon}
= (\epsilon^\omega)_{\omega \in \coprod_j \Omega_j}$ of an orbibundle chart
$(\mathcal{V}, \mathcal{E})$ is a multisection of $(\mathcal{V}, \mathcal{E})$ whose index set $\Omega$ has
a decomposition $\Omega = \coprod_j \Omega_j$ preserved by the action of
$G_V$, that is, for any $g \in G_V$ and $j$,
$g \Omega_j$ coincides with some $\Omega_{j'}$.
We define a family of sections $\epsilon_j = (\epsilon^\omega)_{\omega \in \Omega_j}$
for each $j$,
and we also denote the grouped multisection by
$\boldsymbol{\epsilon} = \{\epsilon_j\}_j$ as a set of such families.
We define the support of each $\epsilon_j$ by
$\mathop{\mathrm{supp}}\nolimits(\epsilon_j) = \bigcup_{\omega \in \Omega_j} \mathop{\mathrm{supp}}\nolimits(\epsilon^\omega)
\subset V$.
For a grouped multisection, we also impose the condition
$\mathop{\mathrm{supp}}\nolimits(\epsilon_j) \neq \emptyset$ for all $j$.
(This is for consistency with the definition of restriction below.)
\end{defi}
\begin{defi}
For a connected open subset $\mathcal{U} \subset \mathcal{V}$, the restriction of a grouped multisection
$\boldsymbol{\epsilon} = (\epsilon^\omega)_{\omega \in \coprod_j \Omega_j}$
of $(\mathcal{V}, \mathcal{E})$ to $(\mathcal{U}, \mathcal{E}|_{\mathcal{U}})$ is defined by
\[
\boldsymbol{\epsilon}|_{\mathcal{U}}
= (\epsilon^\omega|_U)_{\omega \in \coprod_{j \in I_U} \Omega_j}
\]
where $I_U = \{j ; \mathop{\mathrm{supp}}\nolimits(\epsilon_j) \cap U \neq \emptyset \}$.
\end{defi}
\begin{eg}
Let $(\mathcal{V}, \mathcal{E})$ be an orbibundle chart and
let $\epsilon : V \to E$ be a smooth section.
Then its average $\mathop{\mathrm{Av}}\nolimits \epsilon = (g_{\ast} \epsilon)_{g \in G_V}$ is a multisection.
\end{eg}
\begin{eg}
For finite number of grouped multisections $\boldsymbol{\epsilon}^k$ of $(\mathcal{V}, \mathcal{E})$,
their union $\coprod_k \boldsymbol{\epsilon}^k$ is also a grouped multisection.
In particular, for finite number of non-zero multisections
$\epsilon_j = (\epsilon^\omega)_{\omega \in \Omega_j}$,
$\boldsymbol{\epsilon} = \{\epsilon_j\}$ is a grouped multisection.
(We cannot always assume that each $\epsilon_j$ is a multisection,
that is, each $\Omega_j$ is not $G_V$-invariant in general.
We need the general case for the induced multisection of the quotient of the product
of the same pre-Kuranishi spaces. See Section \ref{compatible perturbed multisection})
\end{eg}
\begin{defi}
\label{sum of section and grouped multisection}
For a smooth section $s$ and a grouped multisection $\boldsymbol{\epsilon}
= (\epsilon^\omega)_{\omega \in \coprod_j \Omega_j}$ of an orbibundle chart $(\mathcal{V}, \mathcal{E})$,
their sum is defined by the multisection $s + \boldsymbol{\epsilon}
= (s + \sum_j \epsilon^{\omega_j})_{(\omega_j) \in \prod_j \Omega_j}$
with the product index set $\prod_j \Omega_j$.
We call a multisection of this form a perturbed multisection.
\end{defi}
We will construct a perturbed multisection of a pre-Kuranishi space by the sum
$s + \boldsymbol{\epsilon}$ of the given smooth section $s$ and
a grouped multisection $\boldsymbol{\epsilon}$.
Hence it is enough to define compatibility condition of grouped multisection
$\boldsymbol{\epsilon}$ with embedding instead of the multisection
$s + \boldsymbol{\epsilon}$.
\begin{defi}
\label{(varphi, hat varphi)-relation for chart}
Let $(\varphi, \hat \varphi) : (\mathring{\mathcal{V}}, \mathring{\mathcal{E}}) \to (\mathcal{V}, \mathcal{E})$
be an embedding between two orbibundle charts.
We say a grouped multisection $\boldsymbol{\mathring{\epsilon}}
= (\mathring{\epsilon}^\omega)_{\omega \in \coprod_j \mathring{\Omega}_j}$ of
$(\mathring{\mathcal{V}}, \mathring{\mathcal{E}})$ and
$\boldsymbol{\epsilon}
= (\epsilon^\omega)_{\omega \in \coprod_j \Omega_j}$ of $(\mathcal{V}, \mathcal{E})$
are $(\varphi, \hat \varphi)$-related if
there exists an injection $\nu^\phi : \coprod_j \mathring{\Omega}_j
\to \coprod_j \Omega_j$ for each lift $(\phi, \hat \phi)$ of $(\varphi, \hat \varphi)$
and they satisfy the following conditions:
\begin{itemize}
\item
$\nu^\phi$ maps each $\mathring{\Omega}_j$ to some $\Omega_{j'}$
bijectively.
\item
$\epsilon^{\nu^\phi(\omega)} \circ \phi = \hat \phi \circ \mathring{\epsilon}^\omega$
for each $\omega \in \coprod_j \mathring{\Omega}_j$.
\item
$\epsilon^{\nu^\phi(\omega)} = 0$ on a neighborhood of
$\phi(\mathring{V})$ for any $\omega \in \coprod_j \Omega_j \setminus
\nu^\phi(\coprod_j\mathring{\Omega}_j)$.
\item
For any connected open subset $\mathring{U} \subset \mathring{V}$ and $j$,
if $\mathring{\epsilon}^\omega|_{\mathring{U}} = 0$ for all
$\omega \in \mathring{\Omega}_j$,
then $\epsilon^{\nu^\phi(\omega)} = 0$ on a neighborhood of
$\phi(\mathring{U})$ for all $\omega \in \mathring{\Omega}_j$.
\item
$\nu^{g \phi \mathring{g}} = g \circ \nu^\phi \circ \mathring{g}$
for any $g \in G_V$ and $\mathring{g} \in G_{\mathring{V}}$.
\end{itemize}
\end{defi}
\begin{defi}
\label{grouped multisection for orbibundle}
Let $(\mathcal{V}, \mathcal{E})$ be an orbibundle.
A grouped multisection $\boldsymbol{\epsilon} = (\mathcal{B}, \boldsymbol{\epsilon}_\mathcal{U},
\nu_{\mathcal{U}_2, \mathcal{U}_1}^\phi)$ of $(\mathcal{V}, \mathcal{E})$
consists of the following.
$\mathcal{B} = \{\mathcal{U}\}$ is a set of connected open subsets of $\mathcal{V}$ such that
each $(\mathcal{U}, \mathcal{E}|_{\mathcal{U}})$ is an orbibundle chart and
if $\mathcal{U} \in \mathcal{B}$ then every connected open subset of $\mathcal{U}$ is contained in $\mathcal{B}$.
For each $\mathcal{U} \in \mathcal{B}$,
$\boldsymbol{\epsilon}_\mathcal{U} = (\epsilon^\omega_\mathcal{U})_{\omega \in \coprod_j \Omega_j^U}$
is a grouped multisection of $(\mathcal{U}, \mathcal{E}|_{\mathcal{U}})$.
For each pair $\mathcal{U}_1, \mathcal{U}_2 \in \mathcal{B}$ such that
$\mathcal{U}_1 \subset \mathcal{U}_2$ and a lift $\phi : U_1 \to U_2$
of the inclusion map $\mathcal{U}_1 \hookrightarrow \mathcal{U}_2$, there exists an injective map
$\nu_{\mathcal{U}_2, \mathcal{U}_1}^\phi : \coprod_j \Omega_j^{U_1} \to \coprod_j \Omega_j^{U_2}$
which satisfy the following conditions:
\begin{itemize}
\item
$\nu_{\mathcal{U}_2, \mathcal{U}_1}^\phi$ maps each $\Omega_j^{U_1}$ to some $\Omega_{j'}^{U_2}$
bijectively.
\item
$\epsilon^{\nu_{\mathcal{U}_2, \mathcal{U}_1}^\phi(\omega)}_{\mathcal{U}_2} \circ \phi
= \hat \phi \circ \epsilon_{\mathcal{U}_1}^\omega$
for any $\omega \in \coprod_j \Omega_j^{U_1}$, where
$\hat \phi$ is the lift of $\hat \varphi$ uniquely determined by $\phi$.)
\item
$\epsilon^{\omega'}_{\mathcal{U}_2} \circ \phi =0$
for any $\omega' \in \coprod_j \Omega_j^{U_2} \setminus
\nu_{\mathcal{U}_2, \mathcal{U}_1}^\phi(\coprod_j \Omega_j^{U_1})$.
\item
$\nu_{\mathcal{U}_2, \mathcal{U}_1}^{g_2 \circ \phi \circ g_1} = g_2 \circ \nu_{\mathcal{U}_2, \mathcal{U}_1}^\phi \circ g_1$
for any $g _1\in G_{U_1}$ and $g_2 \in G_{U_2}$.
\item
$\nu_{\mathcal{U}_3, \mathcal{U}_2}^{\phi_{3,2}} \circ \nu_{\mathcal{U}_2, \mathcal{U}_1}^{\phi_{2,1}}
= \nu_{\mathcal{U}_3, \mathcal{U}_1}^{\phi_{3,2} \circ \phi_{2,1}}$ for any triple $\mathcal{U}_1, \mathcal{U}_2, \mathcal{U}_3 \in \mathcal{B}$
such that $\mathcal{U}_1 \subset \mathcal{U}_2 \subset \mathcal{U}_3$
and lifts $\phi_{2,1} : U_1 \to U_2$ and $\phi_{3,2} : U_2 \to U_3$.
\end{itemize}
\end{defi}
\begin{rem}
We do not define a multisecton of an orbibundle.
(Definition \ref{multisection for chart} is the definition of a multisection of
an orbibundle chart, and Definition \ref{grouped multisection for orbibundle} is
the definition of a grouped multisection of an orbibundle.)
We construct a grouped multisection of an orbibundle,
and for each orbibundle chart, we use the perturbed multisection
$s + \boldsymbol{\epsilon}$ of Definition \ref{sum of section and grouped multisection}.
\end{rem}
\begin{eg}
In general, a grouped multisection $\boldsymbol{\epsilon}
= (\mathcal{B}, \boldsymbol{\epsilon}_\mathcal{U}, \nu_{\mathcal{U}_2, \mathcal{U}_1}^\phi)$ of an orbibundle $(\mathcal{V}, \mathcal{E})$
does not have a global grouped multisection $\boldsymbol{\epsilon}_\mathcal{V}$.
Namely, even if $(\mathcal{V}, \mathcal{E})$ itself is an orbibundle chart,
there does not exist a grouped multisection $\boldsymbol{\epsilon}_\mathcal{V}$
of an orbibundle chart $(\mathcal{V}, \mathcal{E})$
(in the sense of Definition \ref{grouped multisection for chart})
whose restrictions to $\mathcal{U}$ coincide with $\boldsymbol{\epsilon}_\mathcal{U}$
for all $\mathcal{U} \in \mathcal{B}$.
For example, let $f : \mathbb{R} \to \mathbb{R}$ be a periodic smooth function
of period $4\pi$, and let $\chi : \mathbb{R}_{\geq 0} \to \mathbb{R}$ be a smooth function
whose support is contained in $[1/2, 1] \subset \mathbb{R}_{\geq 0}$.
Then $F(r \cos \theta, r \sin \theta) = \chi(r) f(\theta)$ defines
a grouped multisection of the trivial orbibbundle of rank $1$ on $\mathbb{R}^2$
(without group action).
(We define the decomposition of the index sets so that
the indices for the two branches of $F$ constitute one group.)
However, it cannot be represented by a grouped multisection of
the trivial orbibundle chart on $\mathbb{R}^2$.
We also note that on a neighborhood of $(0,0) \in \mathbb{R}^2$,
it is represented by the grouped multisection whose index set
is the empty set.
\end{eg}
\begin{defi}
\label{(varphi, hat varphi)-relation for orbibundles}
For an embedding $(\varphi, \hat \varphi) : (\mathring{\mathcal{V}}, \mathring{\mathcal{E}}) \to
(\mathcal{V}, \mathcal{E})$ between two orbibundles, we say a grouped multisection
$\boldsymbol{\mathring{\epsilon}} = (\mathring{\mathcal{B}},
\boldsymbol{\mathring{\epsilon}}_{\mathring{\mathcal{U}}},
\mathring{\nu}_{\mathring{\mathcal{U}}_2, \mathring{\mathcal{U}}_1}^\phi)$ of
$(\mathring{\mathcal{V}}, \mathring{\mathcal{E}})$ and
$\boldsymbol{\epsilon} = (\mathcal{B}, \boldsymbol{\epsilon}_\mathcal{U}, \nu_{\mathcal{U}_2, \mathcal{U}_1}^\phi)$ of
$(\mathcal{V}, \mathcal{E})$ are $(\varphi, \hat \varphi)$-related if
the following conditions hold.
For any $\mathring{\mathcal{U}} \in \mathring{\mathcal{B}}$ and $\mathcal{U} \in \mathcal{B}$ such that
$(\varphi, \hat \varphi)$ defines an embedding
$(\mathring{\mathcal{U}}, \mathring{\mathcal{E}}|_{\mathring{U}}) \to (\mathcal{U}, \mathcal{E}|_{\mathcal{U}})$, and its lift
$(\phi, \hat \phi)$,
there exists an injective map $\nu_{\mathcal{U}, \mathring{\mathcal{U}}}^\phi
: \coprod_j \mathring{\Omega}_j^{\mathring{U}} \to \coprod_j \Omega_j^{U}$
which satisfies the following conditions:
\begin{itemize}
\item
$\nu_{\mathcal{U}, \mathring{\mathcal{U}}}^\phi$ maps each $\mathring{\Omega}_j^{\mathring{U}}$
to some $\Omega_{j'}^{U}$ bijectively.
\item
$\epsilon^{\nu_{\mathcal{U}, \mathring{\mathcal{U}}}^\phi(\omega)}_{\mathcal{U}} \circ \phi
= \hat \phi \circ \mathring{\epsilon}_{\mathring{\mathcal{U}}}^\omega$
for any $\omega \in \coprod_j \mathring{\Omega}_j^{\mathring{U}}$.
\item
$\epsilon^{\omega'}_{\mathcal{U}} = 0$ on a neighborhood of $\phi(\mathring{U})$
for any $\omega' \in \coprod_j \Omega_j^{U} \setminus
\nu_{\mathcal{U}, \mathring{\mathcal{U}}}^\phi(\coprod_j \mathring{\Omega}_j^{\mathring{U}})$.
\item
$\nu_{\mathcal{U}, \mathring{\mathcal{U}}}^{g \circ \phi \circ \mathring{g}}
= g \circ \nu_{\mathcal{U}, \mathring{\mathcal{U}}}^\phi \circ \mathring{g}$
for any $g \in G_U$ and $\mathring{g} \in G_{\mathring{U}}$.
\item
$\nu_{\mathcal{U}_3, \mathcal{U}_2}^{\phi_{3,2}} \circ \nu_{\mathcal{U}_2, \mathring{\mathcal{U}}_2}^\phi \circ
\nu_{\mathring{\mathcal{U}}_2, \mathring{\mathcal{U}}_1}^{\mathring{\phi}_{2,1}}
= \nu_{\mathcal{U}_3, \mathring{\mathcal{U}}_1}^{\phi_{3,2} \circ \phi \circ \mathring{\phi}_{2,1}}$
for any $\mathring{\mathcal{U}}_1 \subset \mathring{\mathcal{U}}_2 \in \mathring{\mathcal{B}}$,
$\mathcal{U}_2 \subset \mathcal{U}_3 \in \mathcal{B}$
such that $(\varphi, \hat \varphi)$ defines an embedding
$(\mathring{\mathcal{U}}_2, \mathring{\mathcal{E}}|_{\mathring{U}_2}) \to (\mathcal{U}_2, \mathcal{E}|_{\mathcal{U}_2})$
and lifts $\mathring{\phi}_{2,1} : \mathring{U}_1 \to \mathring{U}_2$,
$\phi : \mathring{U}_2 \to U_2$ and $\phi_{3,2} : U_2 \to U_3$.
\end{itemize}
More precisely, in the above case,
we say $\boldsymbol{\mathring{\epsilon}}$ and $\boldsymbol{\epsilon}$ are
$(\varphi, \hat \varphi)$-related by
$(\nu_{\mathcal{U}, \mathring{\mathcal{U}}}^\phi)_{(\mathcal{U}, \mathring{\mathcal{U}})}$.
\end{defi}
\begin{defi}
\label{compatibility of (varphi, hat varphi)-relations}
Let $(\varphi^{j,i}, \hat \varphi^{j,i}) : (\mathcal{V}^i, \mathcal{E}^i) \to (\mathcal{V}^j, \mathcal{E}^j)$
be embeddings of orbibundles for $1 \leq i < j \leq 3$ such that
$(\varphi^{3,1}, \hat \varphi^{3,1})
= (\varphi^{3,2}, \hat \varphi^{3,2}) \circ (\varphi^{2,1}, \hat \varphi^{2,1})$.
Let $\boldsymbol{\epsilon}^i = (\mathcal{B}^i, \boldsymbol{\epsilon}^i_{\mathcal{U}^i},
\nu_{\mathcal{U}^i_2, \mathcal{U}^i_1}^\phi)$ be a grouped multisection of $(\mathcal{V}^i, \mathcal{E}^i)$
for each $i = 1,2,3$, and assume that
$\boldsymbol{\epsilon}^i$ and $\boldsymbol{\epsilon}^j$ are
$(\varphi^{j,i}, \hat \varphi^{j,i})$-related by $(\nu_{\mathcal{U}^j, \mathcal{U}^i}^\phi)_{(\mathcal{U}^j, \mathcal{U}^i)}$
for $1 \leq i < j \leq 3$.
We say these relations are compatible if
$\nu_{\mathcal{U}^3, \mathcal{U}^2}^{\phi^{3,2}} \circ \nu_{\mathcal{U}^2, \mathcal{U}^1}^{\phi^{2,1}}
= \nu_{\mathcal{U}^3,\mathcal{U}^1}^{\phi^{3,2} \circ \phi^{2,1}}$ for any
$\mathcal{U}^i \in \mathcal{B}^i$ such that
$(\varphi^{j,i}, \hat \varphi^{j,i})$ defines an embedding
$(\mathcal{U}^i, \mathcal{E}^i|_{\mathcal{U}^i}) \to (\mathcal{U}^j, \mathcal{E}^j|_{\mathcal{U}^j})$
for all $1 \leq i < j \leq 3$, and
lifts $(\phi^{j,i}, \hat \phi^{j,i}) : (U^i, E^i|_{U^i}) \to (U^j, E^j|_{U^j})$
for $(i,j) = (1,2), (2,3)$.
\end{defi}
We always assume the above compatibility condition for compositions of
embeddings.
Next we consider the extension of grouped multisection for
an embedding of an orbibundle.
\begin{lem}
\label{extension of grouped multisection for embedding}
Let $(\varphi, \hat \varphi) : (\mathring{\mathcal{V}}, \mathring{\mathcal{E}}) \to (\mathcal{V}, \mathcal{E})$
be an embedding between two orbibundles.
For any grouped multisection $\boldsymbol{\mathring{\epsilon}} = (\mathring{\mathcal{B}},
\boldsymbol{\mathring{\epsilon}}_{\mathring{\mathcal{U}}},
\mathring{\nu}_{\mathring{\mathcal{U}}_2, \mathring{\mathcal{U}}_1}^\phi)$ of
$(\mathring{\mathcal{V}}, \mathring{\mathcal{E}})$ and its arbitrary relatively
compact open subset $\mathring{\mathcal{V}}' \Subset \mathring{\mathcal{V}}$,
we can construct a grouped multisection
$\boldsymbol{\epsilon}$ of $(\mathcal{V}, \mathcal{E})$ which is $(\varphi, \hat \varphi)$-related to
$\boldsymbol{\mathring{\epsilon}}|_{\mathring{\mathcal{V}}'}$.
\end{lem}
\begin{proof}
Let $(\mathring{\mathcal{V}}_\alpha, \mathring{\mathcal{E}}_\alpha)_{\alpha \in \mathcal{A}}$ and
$(\mathcal{V}_\alpha, \mathcal{E}_\alpha)_{\alpha \in \mathcal{A}}$ be finite number of orbibundle charts of
$(\mathring{\mathcal{V}}, \mathring{\mathcal{E}})$ and $(\mathcal{V}, \mathcal{E})$ respectively such that
$\mathring{\mathcal{V}}_\alpha \in \mathring{\mathcal{B}}$,
$\{\mathring{\mathcal{V}}_\alpha\}_{\alpha \in \mathcal{A}}$ covers the closure of $\mathring{\mathcal{V}}'$,
and $(\varphi, \hat \varphi)$ defines an embedding of
$(\mathring{\mathcal{V}}_\alpha, \mathring{\mathcal{E}}_\alpha)$ to $(\mathcal{V}_\alpha, \mathcal{E}_\alpha)$.
We fix a lift $(\phi_\alpha, \hat \phi_\alpha)$ of this embedding for each $\alpha \in \mathcal{A}$.
Replacing $\mathcal{V}_\alpha$ with a smaller connected open neighborhood of
$\varphi(\mathring{\mathcal{V}}_\alpha)$ if necessary,
we may assume that this lift defines an isomorphism of the automorphism group of
$(\mathring{\mathcal{V}}_\alpha, \mathring{\mathcal{E}}_\alpha)$ and that of $(\mathcal{V}_\alpha, \mathcal{E}_\alpha)$.
Take compact subsets $\mathring{K}_\alpha \subset \mathring{\mathcal{V}}_\alpha$
such that $\bigcup_{\alpha \in \mathcal{A}} \mathop{\mathrm{Int}}\nolimits \mathring{K}_\alpha \supset \mathring{\mathcal{V}}'$.
We can take finite orbibundle charts
$(\mathring{\mathcal{V}}_\kappa, \mathring{\mathcal{E}}_\kappa)_{\kappa \in \mathcal{K}}$ of
$(\mathring{\mathcal{V}}, \mathring{\mathcal{E}})$ and subsets $A_\kappa \subset \mathcal{A}$ ($\kappa \in \mathcal{K}$)
such that
$\bigcup_{\kappa \in \mathcal{K}} \mathring{\mathcal{V}}_\kappa \Supset \mathring{\mathcal{V}}'$ and
$\mathring{\mathcal{V}}_\kappa \Subset \bigcap_{\alpha \in A_\kappa} \mathring{\mathcal{V}}_\alpha
\setminus \bigcup_{\beta \in \mathcal{A} \setminus A_\kappa} \mathring{K}_\beta$.
For each $\kappa \in \mathcal{K}$, let $(\mathcal{V}_\kappa, \mathcal{E}_\kappa)$ be an orbibundle chart of
$(\mathcal{V}, \mathcal{E})$ such that
$\mathcal{V}_\kappa \Subset \bigcap_{\alpha \in A_\kappa} \mathcal{V}_\alpha
\setminus \bigcup_{\beta \in \mathcal{A} \setminus A_\kappa} \varphi(\mathring{K}_\beta)$ and
$(\varphi, \hat \varphi)$ defines an embedding of
$(\mathring{\mathcal{V}}_\kappa, \mathring{\mathcal{E}}_\kappa)$ to $(\mathcal{V}_\kappa, \mathcal{E}_\kappa)$.
We fix a lift of this embedding $(\phi_\kappa, \hat \phi_\kappa)$
for each $\kappa \in \mathcal{K}$ and assume that this
lift defines an isomorphism between their automorphism groups.
For each pair $\kappa_1, \kappa_2 \in \mathcal{K}$ such that
$\mathring{\mathcal{V}}_{\kappa_1} \cap \mathring{\mathcal{V}}_{\kappa_2} \neq \emptyset$,
let $\{\mathring{\mathcal{V}}_{\kappa_1, \kappa_2, \gamma}\}_{\gamma}$
be the connected components of
the intersection $\mathring{\mathcal{V}}_{\kappa_1} \cap \mathring{\mathcal{V}}_{\kappa_2}$.
Similarly, for each triple $\kappa_1, \kappa_2, \kappa_3 \in \mathcal{K}$ such that
$\mathring{\mathcal{V}}_{\kappa_1} \cap \mathring{\mathcal{V}}_{\kappa_2} \cap
\mathring{\mathcal{V}}_{\kappa_3} \neq \emptyset$,
let $\{\mathring{\mathcal{V}}_{\kappa_1, \kappa_2, \kappa_3, \gamma}\}_{\gamma}$
be the connected components of
the intersection $\mathring{\mathcal{V}}_{\kappa_1} \cap \mathring{\mathcal{V}}_{\kappa_2}
\cap \mathring{\mathcal{V}}_{\kappa_3}$.
For each $\mathring{\mathcal{V}}_{\kappa_1, \kappa_2, \gamma}$,
let $(\mathcal{V}_{\kappa_1, \kappa_2, \gamma}, \mathcal{E}_{\kappa_1, \kappa_2, \gamma})$ be
an orbibundle chart contained in the intersection
$\mathcal{V}_{\kappa_1} \cap \mathcal{V}_{\kappa_2}$ such that
$(\varphi, \hat \varphi)$ defines an embedding of
$(\mathring{\mathcal{V}}_{\kappa_1, \kappa_2, \gamma},
\mathring{\mathcal{E}}_{\kappa_1, \kappa_2, \gamma})$
to $(\mathcal{V}_{\kappa_1, \kappa_2, \gamma}, \mathcal{E}_{\kappa_1, \kappa_2, \gamma})$
and its lift $(\phi_{\kappa_1, \kappa_2, \gamma}, \hat \phi_{\kappa_1, \kappa_2, \gamma})$
defines an isomorphism between their automorphism groups.
Similarly, for each $\mathring{\mathcal{V}}_{\kappa_1, \kappa_2, \kappa_3, \gamma}$, we define
$(\mathcal{V}_{\kappa_1, \kappa_2, \kappa_3, \gamma},
\mathcal{E}_{\kappa_1, \kappa_2, \kappa_3, \gamma})$
and $(\phi_{\kappa_1, \kappa_2, \kappa_3, \gamma},
\hat \phi_{\kappa_1, \kappa_2, \kappa_3, \gamma})$.
We assume that $\mathcal{V}_{\kappa_1, \kappa_2, \kappa_3, \gamma}$ is contained in
$\mathcal{V}_{\kappa_1, \kappa_2, \gamma_{1,2}} \cap \mathcal{V}_{\kappa_2, \kappa_3, \gamma_{2,3}}
\cap \mathcal{V}_{\kappa_1, \kappa_3, \gamma_{1,3}}$
if $\mathring{\mathcal{V}}_{\kappa_1, \kappa_2, \kappa_3, \gamma}$ is contained in
$\mathring{\mathcal{V}}_{\kappa_1, \kappa_2, \gamma_{1,2}} \cap
\mathring{\mathcal{V}}_{\kappa_2, \kappa_3, \gamma_{2,3}}
\cap \mathring{\mathcal{V}}_{\kappa_1, \kappa_3, \gamma_{1,3}}$.
For each $\kappa \in \mathcal{K}$ and $\alpha \in \mathcal{A}$,
we fix a lift $(\phi_{\alpha, \kappa}^\circ, \hat \phi_{\alpha, \kappa}^\circ)$
of the inclusion map from $(\mathring{\mathcal{V}}_\kappa, \mathring{\mathcal{E}}_\kappa)$ to
$(\mathring{\mathcal{V}}_\alpha, \mathring{\mathcal{E}}_\alpha)$.
Then we can define a lift $(\phi_{\alpha, \kappa}, \hat \phi_{\alpha, \kappa})$
of the inclusion map from $(\mathcal{V}_\kappa, \mathcal{E}_\kappa)$ to
$(\mathcal{V}_\alpha, \mathcal{E}_\alpha)$ by the condition
$(\phi_{\alpha, \kappa}, \hat \phi_{\alpha, \kappa}) \circ
(\phi_\kappa, \hat \phi_\kappa) = (\phi_\alpha, \hat \phi_\alpha)
\circ (\phi_{\alpha, \kappa}^\circ, \hat \phi_{\alpha, \kappa}^\circ)$.
Similarly, we fix a lift
$(\phi_{\alpha, (\kappa_1, \kappa_2, \gamma)}^\circ,
\hat \phi_{\alpha, (\kappa_1, \kappa_2, \gamma)}^\circ)$
of embedding from $(\mathring{\mathcal{V}}_{\kappa_1, \kappa_2, \gamma},
\mathring{\mathcal{E}}_{\kappa_1, \kappa_2, \gamma})$ to
$(\mathring{\mathcal{V}}_\alpha, \mathring{\mathcal{E}}_\alpha)$,
define the lift $(\phi_{\alpha, (\kappa_1, \kappa_2, \gamma)},
\hat \phi_{\alpha, (\kappa_1, \kappa_2, \gamma)})$
of embedding from
$(\mathcal{V}_{\kappa_1, \kappa_2, \gamma}, \mathcal{E}_{\kappa_1, \kappa_2, \gamma})$ to
$(\mathcal{V}_\alpha, \mathcal{E}_\alpha)$,
and so on.
For each $\alpha \in \mathcal{A}$,
we independently construct a grouped multisection
$\boldsymbol{\epsilon}_{\mathcal{V}_\alpha} = (\epsilon_{\mathcal{V}_\alpha}^\omega)_{\omega \in \coprod_j
\mathring{\Omega}_j^{\mathring{V}_\alpha}}$ of $(\mathcal{V}_\alpha, \mathcal{E}_\alpha)$
which is $(\varphi, \hat \varphi)$-related to
$\boldsymbol{\mathring{\epsilon}}_{\mathring{\mathcal{V}}_\alpha}$.
We use the same index set for $\boldsymbol{\epsilon}_{\mathcal{V}_\alpha}$
as that of $\boldsymbol{\mathring{\epsilon}}_{\mathring{\mathcal{V}}_\alpha}$,
and assume that $\nu^{\phi_\alpha} = \mathrm{id}$
in Definition \ref{(varphi, hat varphi)-relation for chart}.
Shrinking $\mathcal{V}_\kappa$, $\mathcal{V}_{\kappa_1, \kappa_2, \gamma}$ and
$\mathcal{V}_{\kappa_1, \kappa_2, \kappa_3, \gamma}$ to smaller neighborhoods of
$\varphi(\mathring{\mathcal{V}}_\kappa)$, $\varphi(\mathring{\mathcal{V}}_{\kappa_1, \kappa_2, \gamma})$
and $\varphi(\mathring{\mathcal{V}}_{\kappa_1, \kappa_2, \kappa_3, \gamma})$ respectively
if necessary, we may assume the following conditions on $\epsilon_{\mathcal{V}_\alpha}^\omega$.
For each $j$, if $\mathring{\epsilon}_{\mathring{\mathcal{V}}_\alpha}^\omega \circ
\phi_{\alpha, \kappa}^\circ = 0$ for all
$\omega \in \mathring{\Omega}_j^{\mathring{V}_\alpha}$, then
$\epsilon_{\mathcal{V}_\alpha}^\omega \circ \phi_{\alpha, \kappa} = 0$
for all $\omega \in \mathring{\Omega}_j^{\mathring{V}_\alpha}$.
Similarly, if $\mathring{\epsilon}_{\mathring{\mathcal{V}}_\alpha}^\omega \circ
\phi_{\alpha, (\kappa_1, \kappa_2, \gamma)}^\circ = 0$
for all $\omega \in \mathring{\Omega}_j^{\mathring{V}_\alpha}$ then
$\epsilon_{\mathcal{V}_\alpha}^\omega \circ \phi_{\alpha, (\kappa_1, \kappa_2, \gamma)} = 0$
for all $\omega \in \mathring{\Omega}_j^{\mathring{V}_\alpha}$,
and if $\mathring{\epsilon}_{\mathring{\mathcal{V}}_\alpha}^\omega \circ
\phi_{\alpha, (\kappa_1, \kappa_2, \kappa_3 \gamma)}^\circ = 0$
for all $\omega \in \mathring{\Omega}_j^{\mathring{V}_\alpha}$ then
$\epsilon_{\mathcal{V}_\alpha}^\omega \circ
\phi_{\alpha, (\kappa_1, \kappa_2, \kappa_3, \gamma)} = 0$
for all $\omega \in \mathring{\Omega}_j^{\mathring{V}_\alpha}$.
We note that these conditions do not depend on the choice of the lifts
$\phi_{\alpha, \kappa}^\circ$, $\phi_{\alpha, (\kappa_1, \kappa_2, \gamma)}^\circ$
or $\phi_{\alpha, (\kappa_1, \kappa_2, \kappa_3, \gamma)}^\circ$.
Since $\{\mathcal{V}_\kappa\}_{\kappa \in \mathcal{K}}$ covers the closure of
$\varphi(\mathring{\mathcal{V}}')$,
we can construct open subsets $\mathcal{V}'_\kappa \Subset \mathcal{V}_\kappa$
such that $\mathcal{V}'_{\kappa_1} \cap \mathcal{V}'_{\kappa_2}$ is contained in
the union of $\mathcal{V}_{\kappa_1, \kappa_2, \gamma}$,
$\mathcal{V}'_{\kappa_1} \cap \mathcal{V}'_{\kappa_2} \cap \mathcal{V}'_{\kappa_3}$ is contained in
the union of $\mathcal{V}_{\kappa_1, \kappa_2, \kappa_3, \gamma}$, and
$\{\mathcal{V}'_\kappa\}_{\kappa \in \mathcal{K}}$ covers the closure of $\varphi(\mathring{\mathcal{V}}')$.
Let $\{\chi_\alpha\}_\alpha$ be a family of smooth functions on $\mathcal{V}$ such that
$\varphi^{-1}(\mathop{\mathrm{supp}}\nolimits \chi_\alpha) \subset \mathring{K}_\alpha$
and $\sum_\alpha \chi_\alpha \equiv 1$ on $\varphi(\mathring{\mathcal{V}}')$.
We assume that $\mathcal{V}_\kappa \cap \mathop{\mathrm{supp}}\nolimits \chi_\beta = \emptyset$
for all $\kappa \in \mathcal{K}$ and $\beta \in \mathcal{A} \setminus A_\kappa$.
We also assume that
$\bigcup_{\alpha \in \mathcal{A}} \mathop{\mathrm{supp}}\nolimits \chi_\alpha \subset \bigcup_{\kappa \in \mathcal{K}} \mathcal{V}'_\kappa$.
Then we define grouped multisections $\boldsymbol{\epsilon}_{\mathcal{U}}$
for all connected open subsets contained in some $\mathcal{V}'_\kappa$
and connected open subsets which do not intersect with
$\bigcup_{\alpha \in \mathcal{A}} \mathop{\mathrm{supp}}\nolimits \chi_\alpha$.
For the latter, we define $\boldsymbol{\epsilon}_{\mathcal{U}}$
by zero (the grouped multisection whose index set is the empty set).
For the former, we define $\boldsymbol{\epsilon}_{\mathcal{U}}$ as follows.
First we define a grouped multisection $\boldsymbol{\epsilon}_{\mathcal{V}_\kappa}
= (\epsilon_{\mathcal{V}_\kappa}^\omega)_{\omega \in \coprod_j
\mathring{\Omega}^{\mathring{V}_\kappa}_j}$
of $(\mathcal{V}_\kappa, \mathcal{E}_\kappa)$ for each $\kappa \in \mathcal{K}$ by
\[
\epsilon_{\mathcal{V}_\kappa}^\omega = \sum_{\alpha \in A_\kappa}
\chi_\alpha \, \phi_{\alpha, \kappa}^\ast
\epsilon_{\mathcal{V}_\alpha}^{\nu_{\alpha, \kappa}(\omega)},
\]
where $\nu_{\alpha, \kappa} = \nu_{\mathring{\mathcal{V}}_\alpha, \mathring{\mathcal{V}}_\kappa}
^{\phi_{\alpha, \kappa}^\circ}$.
For each connected open subset $\mathcal{U}$ contained in some $\mathcal{V}'_\kappa$,
we fix one of such $\kappa \in \mathcal{K}$, and define its grouped multisection
$\boldsymbol{\epsilon}_\mathcal{U}$
by the restriction of the multisection $\boldsymbol{\epsilon}_{\mathcal{V}_\kappa}$.
Namely, we fix a lift $(\phi_{\kappa, U}, \hat \phi_{\kappa, U})$ of the inclusion map and
define $\boldsymbol{\epsilon}_\mathcal{U}
= (\epsilon_\mathcal{U}^\omega)_{\omega \in \coprod_{j \in I_{\kappa, U}}
\mathring{\Omega}^{\mathring{V}_\kappa}_j}$ by
$\epsilon_\mathcal{U}^\omega = \phi_{\kappa, U}^\ast \epsilon_{\mathcal{V}_\kappa}^\omega$,
where $I_{\kappa, U} = \{j; \phi_{\kappa, U}^\ast \epsilon_{\mathcal{V}_\kappa}^\omega \neq 0$
for some $\omega \in \mathring{\Omega}^{\mathring{\mathcal{V}}_\kappa}_j\}$.
We need to construct $\nu_{\mathcal{U}_2, \mathcal{U}_1}^{\phi_{U_2, U_1}}$ for pairs $\mathcal{U}_1 \subset \mathcal{U}_2$
and lifts $(\phi_{U_2, U_1}, \hat \phi_{U_2, U_1})$ of the inclusion $\mathcal{U}_1 \hookrightarrow \mathcal{U}_2$.
Assume that the grouped multisections of $\mathcal{U}_1$ and $\mathcal{U}_2$ are
defined by using $\kappa_1$ and $\kappa_2$ respectively.
In particular, $\mathcal{U}_1 \subset \mathcal{V}'_{\kappa_1} \cap \mathcal{V}'_{\kappa_2}$ is contained in
$\mathcal{V}_{\kappa_1, \kappa_2, \gamma}$ for some $\gamma$.
Fix a lift $(\phi_{(\kappa_1, \kappa_2, \gamma), U_1},
\hat \phi_{(\kappa_1, \kappa_2, \gamma), U_1})$ of the inclusion
from $(\mathcal{U}_1, \mathcal{E}|_{\mathcal{U}_1})$ to $(\mathcal{V}_{\kappa_1, \kappa_2, \gamma},
\mathcal{E}_{\kappa_1, \kappa_2, \gamma})$, and define
$g^{\kappa_1}_{(\kappa_1, \kappa_2, \gamma), U_1} \in G_{V_{\kappa_1}}$ by
\[
\phi_{\kappa_1, U_1}
= g^{\kappa_1}_{(\kappa_1, \kappa_2, \gamma), U_1} \circ
\phi_{\kappa_1, (\kappa_1, \kappa_2, \gamma)} \circ
\phi_{(\kappa_1, \kappa_2, \gamma), U_1}.
\]
First we show that $\coprod_{j \in I_{\kappa_1, U_1}}
\mathring{\Omega}^{\mathring{V}_{\kappa_1}}_j$ is contained in the image of
\begin{equation}
\nu_{\mathring{\mathcal{V}}_{\kappa_1}, \mathring{\mathcal{V}}_{\kappa_1, \kappa_2, \gamma}}
^{g^{\kappa_1}_{(\kappa_1, \kappa_2, \gamma), U_1} \circ
\phi_{\kappa_1, (\kappa_1, \kappa_2, \gamma)}^\circ}.
\label{nu g phi}
\end{equation}
For each $j \in I_{\kappa_1, U_1}$,
there exists some $\omega \in \mathring{\Omega}_j^{\mathring{V}_{\kappa_1}}$
such that $\phi_{\kappa_1, U_1}^\ast \epsilon_{\mathcal{V}_{\kappa_1}}^\omega \neq 0$.
Hence
\begin{align*}
&(g^{\kappa_1}_{(\kappa_1, \kappa_2, \gamma), U_1} \circ
\phi_{\kappa_1, (\kappa_1, \kappa_2, \gamma)})^\ast \epsilon_{\mathcal{V}_{\kappa_1}}^\omega \\
&= \sum_{\alpha \in A_{\kappa_1}}
\chi_\alpha (\phi_{\alpha, \kappa_1} \circ
g^{\kappa_1}_{(\kappa_1, \kappa_2, \gamma), U_1} \circ
\phi_{\kappa_1, (\kappa_1, \kappa_2, \gamma)})^\ast
\epsilon_{\mathcal{V}_\alpha}^{\nu_{\alpha, \kappa_1}(\omega)}
\end{align*}
is nonzero.
This implies that
some $(\phi_{\alpha, \kappa_1} \circ
g^{\kappa_1}_{(\kappa_1, \kappa_2, \gamma), U_1} \circ
\phi_{\kappa_1, (\kappa_1, \kappa_2, \gamma)})^\ast
\epsilon_{\mathcal{V}_\alpha}^{\nu_{\alpha, \kappa_1}(\omega)}$ is nonzero.
Since $\phi_{\alpha, \kappa_1} \circ
g^{\kappa_1}_{(\kappa_1, \kappa_2, \gamma), U_1} \circ
\phi_{\kappa_1, (\kappa_1, \kappa_2, \gamma)}$ is a lift of the
open embedding $\mathcal{V}_{\kappa_1, \kappa_2, \gamma} \hookrightarrow \mathcal{V}_\alpha$,
the assumption of $\boldsymbol{\epsilon}_{\mathcal{V}_\alpha}$ implies that
\[
(\phi_{\alpha, \kappa_1}^\circ \circ
g^{\kappa_1}_{(\kappa_1, \kappa_2, \gamma), U_1} \circ
\phi_{\kappa_1, (\kappa_1, \kappa_2, \gamma)}^\circ)^\ast
\mathring{\epsilon}_{\mathring{\mathcal{V}}_\alpha}^{\nu_{\alpha, \kappa_1}(\omega')}
= (g^{\kappa_1}_{(\kappa_1, \kappa_2, \gamma), U_1} \circ
\phi_{\kappa_1, (\kappa_1, \kappa_2, \gamma)}^\circ)^\ast
\mathring{\epsilon}_{\mathring{\mathcal{V}}_{\kappa_1}}^{\omega'}
\]
is also nonzero for some $\omega' \in \mathring{\Omega}^{\mathring{V}_{\kappa_1}}_j$.
This implies that $\mathring{\Omega}^{\mathring{V}_{\kappa_1}}_j$
is contained in the image of (\ref{nu g phi}).
Hence $\coprod_{j \in I_{\kappa_1, U_1}} \mathring{\Omega}^{\mathring{V}_{\kappa_1}}_j$
is contained in the image of (\ref{nu g phi}).
We define $g^{\kappa_2}_{U_2, U_1} \in G_{\mathcal{V}_{\kappa_2}}$ by
\[
\phi_{\kappa_2, U_2} \circ \phi_{U_2,U_1}
= g_{U_2, U_1}^{\kappa_2} \circ \phi_{\kappa_2, (\kappa_1, \kappa_2, \gamma)} \circ
\phi_{(\kappa_1, \kappa_2, \gamma), U_1},
\]
and define $\nu_{\mathcal{U}_2, \mathcal{U}_1}^{\phi_{U_2, U_1}}
: \coprod_{j \in I_{\kappa_1, U_1}} \mathring{\Omega}^{\mathring{V}_{\kappa_1}}_j
\to \coprod_{j \in I_{\kappa_2, U_2}} \mathring{\Omega}^{\mathring{V}_{\kappa_2}}_j$
by
\[
\nu_{\mathcal{U}_2, \mathcal{U}_1}^{\phi_{U_2, U_1}}
= \nu_{\mathring{\mathcal{V}}_{\kappa_2}, \mathring{\mathcal{V}}_{\kappa_1, \kappa_2, \gamma}}
^{g_{U_2, U_1}^{\kappa_2}
\circ \phi_{\kappa_2, (\kappa_1, \kappa_2, \gamma)}^\circ}
\circ (\nu_{\mathring{\mathcal{V}}_{\kappa_1}, \mathring{\mathcal{V}}_{\kappa_1, \kappa_2, \gamma}}
^{g^{\kappa_1}_{(\kappa_1, \kappa_2, \gamma), U_1} \circ
\phi_{\kappa_1, (\kappa_1, \kappa_2, \gamma)}^\circ})^{-1}.
\]
We need to check that this satisfies the conditions of $\nu_{\mathcal{U}_2, \mathcal{U}_1}^{\phi_{U_2, U_1}}$.
First we check the condition
\begin{equation}
\epsilon^{\nu_{\mathcal{U}_2, \mathcal{U}_1}^{\phi_{U_2, U_1}}(\omega)}_{\mathcal{U}_2} \circ \phi_{U_2, U_1}
= \hat \phi_{U_2, U_1} \circ \epsilon_{\mathcal{U}_1}^\omega
\label{(phi, hat phi)-relation for extension}
\end{equation}
for $\omega \in \coprod_{j \in I_{\kappa_1, U_1}}
\mathring{\Omega}^{\mathring{V}_{\kappa_1}}_j$.
This equation also implies that the image of $\nu_{\mathcal{U}_2, \mathcal{U}_1}^{\phi_{U_2, U_1}}$
is indeed contained in $\coprod_{j \in I_{\kappa_2, U_2}}
\mathring{\Omega}^{\mathring{V}_{\kappa_2}}_j$.
Define $\hat \omega \in \coprod_j
\mathring{\Omega}^{\mathring{V}_{\kappa_1, \kappa_2, \gamma}}_j$
by
\[
\omega = \nu_{\mathring{\mathcal{V}}_{\kappa_1}, \mathring{\mathcal{V}}_{\kappa_1, \kappa_2, \gamma}}
^{g^{\kappa_1}_{(\kappa_1, \kappa_2, \gamma), U_1} \circ
\phi_{\kappa_1, (\kappa_1, \kappa_2, \gamma)}^\circ}(\hat \omega).
\]
Then by definition,
\[
\nu_{\mathcal{U}_2, \mathcal{U}_1}^{\phi_{U_2, U_1}}(\omega)
= \nu_{\mathring{\mathcal{V}}_{\kappa_2}, \mathring{\mathcal{V}}_{\kappa_1, \kappa_2, \gamma}}
^{g_{U_2, U_1}^{\kappa_2}
\circ \phi_{\kappa_2, (\kappa_1, \kappa_2, \gamma)}^\circ}(\hat \omega).
\]
Therefore
\begin{align}
&\phi_{U_2, U_1}^\ast \epsilon^{\nu_{\mathcal{U}_2, \mathcal{U}_1}^{\phi_{U_2, U_1}}(\omega)}_{\mathcal{U}_2} \notag \\
&= (\phi_{\kappa_2, U_2} \circ \phi_{U_2, U_1})^\ast
\epsilon_{\mathcal{V}_{\kappa_2}}^{\nu_{\mathcal{U}_2, \mathcal{U}_1}^{\phi_{U_2, U_1}}(\omega)} \notag \\
&= (g_{U_2, U_1}^{\kappa_2} \circ \phi_{\kappa_2, (\kappa_1, \kappa_2, \gamma)} \circ
\phi_{(\kappa_1, \kappa_2, \gamma), U_1})^\ast
\epsilon_{\mathcal{V}_{\kappa_2}}^{\nu_{\mathring{\mathcal{V}}_{\kappa_2},
\mathring{\mathcal{V}}_{\kappa_1, \kappa_2, \gamma}}^{g_{U_2, U_1}^{\kappa_2}
\circ \phi_{\kappa_2, (\kappa_1, \kappa_2, \gamma)}^\circ}(\hat \omega)} \notag \\
&= (\phi_{\kappa_2, (\kappa_1, \kappa_2, \gamma)} \circ
\phi_{(\kappa_1, \kappa_2, \gamma), U_1})^\ast
\epsilon_{\mathcal{V}_{\kappa_2}}^{\nu_{\mathring{\mathcal{V}}_{\kappa_2},
\mathring{\mathcal{V}}_{\kappa_1, \kappa_2, \gamma}}^{
\phi_{\kappa_2, (\kappa_1, \kappa_2, \gamma)}^\circ}(\hat \omega)} \notag \\
&= \sum_{\alpha \in A_{\kappa_2}}
\chi_\alpha (\phi_{\alpha, \kappa_2} \circ
\phi_{\kappa_2, (\kappa_1, \kappa_2, \gamma)} \circ
\phi_{(\kappa_1, \kappa_2, \gamma), U_1})^\ast
\epsilon_{\mathcal{V}_\alpha}^{\nu_{\mathring{\mathcal{V}}_\alpha,
\mathring{\mathcal{V}}_{\kappa_1, \kappa_2, \gamma}}^{\phi_{\alpha, \kappa_2}^\circ
\circ \phi_{\kappa_2, (\kappa_1, \kappa_2, \gamma)}^\circ}(\hat \omega)}.
\label{phi ast epsilon U2}
\end{align}
Since $\chi_\alpha|_{\mathcal{U}_1} = 0$
for $\alpha \in A_{\kappa_2} \setminus A_{\kappa_1}$,
the terms for $\alpha \in A_{\kappa_2} \setminus A_{\kappa_1}$ are zero.
For each $\alpha \in A_{\kappa_1} \cap A_{\kappa_2}$,
we define $g^\alpha \in G_{V_\alpha}$ by
\[
\phi_{\alpha, \kappa_2}^\circ
\circ \phi_{\kappa_2, (\kappa_1, \kappa_2, \gamma)}^\circ
= g^\alpha \circ \phi_{\alpha, \kappa_1}^\circ \circ
g^{\kappa_1}_{(\kappa_1, \kappa_2, \gamma), U_1} \circ
\phi_{\kappa_1, (\kappa_1, \kappa_2, \gamma)}^\circ.
\]
Then
\begin{align*}
&\phi_{\alpha, \kappa_2} \circ
\phi_{\kappa_2, (\kappa_1, \kappa_2, \gamma)} \circ
\phi_{(\kappa_1, \kappa_2, \gamma), U_1} \\
&= g^\alpha \circ \phi_{\alpha, \kappa_1} \circ
g^{\kappa_1}_{(\kappa_1, \kappa_2, \gamma), U_1} \circ
\phi_{\kappa_1, (\kappa_1, \kappa_2, \gamma)} \circ
\phi_{(\kappa_1, \kappa_2, \gamma), U_1} \\
&= g^\alpha \circ \phi_{\alpha, \kappa_1} \circ \phi_{\kappa_1, U_1}.
\end{align*}
Hence (\ref{phi ast epsilon U2}) is equal to
\begin{align*}
&\sum_{\alpha \in A_{\kappa_1}}
\chi_\alpha (g^\alpha \circ \phi_{\alpha, \kappa_1} \circ \phi_{\kappa_1, U_1})^\ast
\epsilon_{\mathcal{V}_\alpha}^{
\nu_{\mathring{\mathcal{V}}_\alpha, \mathring{\mathcal{V}}_{\kappa_1, \kappa_2, \gamma}}
^{g^\alpha \circ \phi_{\alpha, \kappa_1}^\circ \circ
g^{\kappa_1}_{(\kappa_1, \kappa_2, \gamma), U_1} \circ
\phi_{\kappa_1, (\kappa_1, \kappa_2, \gamma)}^\circ}(\hat \omega)}\\
&= \sum_{\alpha \in A_{\kappa_1}}
\chi_\alpha (\phi_{\alpha, \kappa_1} \circ \phi_{\kappa_1, U_1})^\ast
\epsilon_{\mathcal{V}_\alpha}^{
\nu_{\mathring{\mathcal{V}}_\alpha, \mathring{\mathcal{V}}_{\kappa_1, \kappa_2, \gamma}}
^{\phi_{\alpha, \kappa_1}^\circ \circ
g^{\kappa_1}_{(\kappa_1, \kappa_2, \gamma), U_1} \circ
\phi_{\kappa_1, (\kappa_1, \kappa_2, \gamma)}^\circ}(\hat \omega)} \\
&= \epsilon_{\mathcal{U}_1}^{
\nu_{\mathring{\mathcal{V}}_{\kappa_1}, \mathring{\mathcal{V}}_{\kappa_1, \kappa_2, \gamma}}
^{g^{\kappa_1}_{(\kappa_1, \kappa_2, \gamma), U_1} \circ
\phi_{\kappa_1, (\kappa_1, \kappa_2, \gamma)}^\circ}(\hat \omega)}\\
&= \epsilon_{\mathcal{U}_1}^\omega.
\end{align*}
Hence (\ref{(phi, hat phi)-relation for extension}) holds for
$\omega \in \coprod_{j \in I_{\kappa_1, U_1}}
\mathring{\Omega}^{\mathring{V}_{\kappa_1}}_j$.
Next we check the condition
$\epsilon^{\omega}_{\mathcal{U}_2} \circ \phi_{U_2, U_1} = 0$
for $\omega \in \coprod_{j \in I_{\kappa_2, U_2}}
\mathring{\Omega}^{\mathring{V}_{\kappa_2}}_j$ not contained in the
image of $\nu_{\mathcal{U}_2, \mathcal{U}_1}^{\phi_{U_2, U_1}}$.
If $\omega = \nu_{\mathring{\mathcal{V}}_{\kappa_2}, \mathring{\mathcal{V}}_{\kappa_1, \kappa_2, \gamma}}
^{g_{U_2, U_1}^{\kappa_2}
\circ \phi_{\kappa_2, (\kappa_1, \kappa_2, \gamma)}^\circ}(\hat \omega)$
for some $\hat \omega \in \coprod_j
\mathring{\Omega}^{\mathring{V}_{\kappa_1, \kappa_2, \gamma}}_j$,
then the same argument as above implies that
\[
\phi_{U_2, U_1}^\ast \epsilon^{\omega}_{\mathcal{U}_2}
= \phi_{\kappa_1, U_1}^\ast
\epsilon_{\mathcal{V}_{\kappa_1}}^{
\nu_{\mathring{\mathcal{V}}_{\kappa_1}, \mathring{\mathcal{V}}_{\kappa_1, \kappa_2, \gamma}}
^{g^{\kappa_1}_{(\kappa_1, \kappa_2, \gamma), U_1} \circ
\phi_{\kappa_1, (\kappa_1, \kappa_2, \gamma)}^\circ}(\hat \omega)},
\]
and this is zero because $\nu_{\mathring{\mathcal{V}}_{\kappa_1},
\mathring{\mathcal{V}}_{\kappa_1, \kappa_2, \gamma}}
^{g^{\kappa_1}_{(\kappa_1, \kappa_2, \gamma), U_1} \circ
\phi_{\kappa_1, (\kappa_1, \kappa_2, \gamma)}^\circ}(\hat \omega)
\notin \coprod_{j \in I_{\kappa_1, U_1}}
\mathring{\Omega}^{\mathring{V}_{\kappa_1}}_j$.
If $\omega$ is not contained in the image of
$\nu_{\mathring{\mathcal{V}}_{\kappa_2}, \mathring{\mathcal{V}}_{\kappa_1, \kappa_2, \gamma}}
^{g_{U_2, U_1}^{\kappa_2}
\circ \phi_{\kappa_2, (\kappa_1, \kappa_2, \gamma)}^\circ}$,
then
\[
(\phi_{\alpha, \kappa_2}^\circ \circ
g_{U_2, U_1}^{\kappa_2} \circ \phi_{\kappa_2, (\kappa_1, \kappa_2, \gamma)}^\circ)^\ast
\mathring{\epsilon}_{\mathring{\mathcal{V}}_\alpha}
^{\nu_{\mathring{\mathcal{V}}_\alpha, \mathring{\mathcal{V}}_{\kappa_2}}
^{\phi_{\alpha, \kappa_2}^\circ}(\omega)}
= (g_{U_2, U_1}^{\kappa_2} \circ \phi_{\kappa_2, (\kappa_1, \kappa_2, \gamma)}^\circ)^\ast
\mathring{\epsilon}_{\mathring{\mathcal{V}}_\alpha}^{\omega}
= 0,
\]
and this (and the same equations for the other indices $\omega$ in the same
index group) imply
\[
(\phi_{\alpha, \kappa_2} \circ
g_{U_2, U_1}^{\kappa_2} \circ \phi_{\kappa_2, (\kappa_1, \kappa_2, \gamma)})^\ast
\epsilon_{\mathcal{V}_\alpha}^{\nu_{\mathring{\mathcal{V}}_\alpha, \mathring{\mathcal{V}}_{\kappa_2}}
^{\phi_{\alpha, \kappa_2}^\circ}(\omega)}
= 0
\]
by the assumption of $\boldsymbol{\epsilon}_{\mathcal{V}_\alpha}$.
Hence
\begin{align*}
\phi_{U_2, U_1}^\ast \epsilon_{\mathcal{U}_2}^\omega
&= (g_{U_2, U_1}^{\kappa_2} \circ \phi_{\kappa_2, (\kappa_1, \kappa_2, \gamma)} \circ
\phi_{(\kappa_1, \kappa_2, \gamma), U_1})^\ast
\epsilon_{\mathcal{V}_{\kappa_2}}^\omega \\
&= \sum_{\alpha \in A_{\kappa_2}} \chi_\alpha (\phi_{\alpha, \kappa_2} \circ
g_{U_2, U_1}^{\kappa_2} \circ \phi_{\kappa_2, (\kappa_1, \kappa_2, \gamma)} \circ
\phi_{(\kappa_1, \kappa_2, \gamma), U_1})^\ast
\epsilon_{\mathcal{V}_\alpha}^{\nu_{\mathring{\mathcal{V}}_\alpha, \mathring{\mathcal{V}}_{\kappa_2}}
^{\phi_{\alpha, \kappa_2}^\circ}(\omega)} \\
&= 0.
\end{align*}
Finally we check the condition about composition.
For a triple $\mathcal{U}_1 \subset \mathcal{U}_2 \subset \mathcal{U}_3$ and lifts
$(\phi_{U_2, U_1}, \hat \phi_{U_2, U_1})$, $(\phi_{U_3, U_2}, \hat \phi_{U_3, U_2})$
of the inclusion maps, we prove that
\[
\nu_{\mathcal{U}_2, \mathcal{U}_1}^{\phi_{U_2, U_1}} \circ \nu_{\mathcal{U}_2, \mathcal{U}_1}^{\phi_{U_2, U_1}}
= \nu_{\mathcal{U}_3, \mathcal{U}_1}^{\phi_{U_3, U_2} \circ \phi_{U_2, U_1}}.
\]
Since $\mathcal{U}_1$ is contained in
$\mathcal{V}'_{\kappa_1} \cap \mathcal{V}'_{\kappa_2} \cap \mathcal{V}'_{\kappa_3}$,
$\mathcal{U}_1 \subset \mathcal{V}_{\kappa_1, \kappa_2, \kappa_3, \gamma'}$ for some $\gamma'$.
Fix a lift $(\phi_{(\kappa_1, \kappa_2, \kappa_3, \gamma'), U_1}, \allowbreak
\hat \phi_{(\kappa_1, \kappa_2, \kappa_3, \gamma'), U_1})$ of the inclusion
from $(\mathcal{U}_1, \mathcal{E}|_{\mathcal{U}_1})$ to $(\mathcal{V}_{\kappa_1, \kappa_2, \kappa_3, \gamma'}, \allowbreak
\mathcal{E}_{\kappa_1, \kappa_2, \kappa_3, \gamma'})$, and define
$g^{\kappa_1}_{(\kappa_1, \kappa_2, \kappa_3, \gamma'), U_1} \in G_{\mathcal{V}_{\kappa_1}}$ by
\[
\phi_{\kappa_1, U_1}
= g^{\kappa_1}_{(\kappa_1, \kappa_2, \kappa_3, \gamma'), U_1} \circ
\phi_{\kappa_1, (\kappa_1, \kappa_2, \kappa_3, \gamma')} \circ
\phi_{(\kappa_1, \kappa_2, \kappa_3, \gamma'), U_1}.
\]
By the same argument as above, $\coprod_{j \in I_{\kappa_1, U_1}}
\mathring{\Omega}^{\mathring{V}_{\kappa_1}}_j$ is contained in the image of
\[
\nu_{\mathring{\mathcal{V}}_{\kappa_1}, \mathring{\mathcal{V}}_{\kappa_1, \kappa_2, \kappa_3, \gamma'}}
^{g^{\kappa_1}_{(\kappa_1, \kappa_2, \kappa_3, \gamma'), U_1} \circ
\phi_{\kappa_1, (\kappa_1, \kappa_2, \kappa_3, \gamma')}^\circ}.
\]
Hence it is enough to prove
\begin{align*}
&\nu_{\mathcal{U}_2, \mathcal{U}_1}^{\phi_{U_2, U_1}} \circ \nu_{\mathcal{U}_2, \mathcal{U}_1}^{\phi_{U_2, U_1}} \circ
\nu_{\mathring{\mathcal{V}}_{\kappa_1}, \mathring{\mathcal{V}}_{\kappa_1, \kappa_2, \kappa_3, \gamma'}}
^{g^{\kappa_1}_{(\kappa_1, \kappa_2, \kappa_3, \gamma'), U_1} \circ
\phi_{\kappa_1, (\kappa_1, \kappa_2, \kappa_3, \gamma')}^\circ} \\
&= \nu_{\mathcal{U}_3, \mathcal{U}_1}^{\phi_{U_3, U_2} \circ \phi_{U_2, U_1}} \circ
\nu_{\mathring{\mathcal{V}}_{\kappa_1}, \mathring{\mathcal{V}}_{\kappa_1, \kappa_2, \kappa_3, \gamma'}}
^{g^{\kappa_1}_{(\kappa_1, \kappa_2, \kappa_3, \gamma'), U_1} \circ
\phi_{\kappa_1, (\kappa_1, \kappa_2, \kappa_3, \gamma')}^\circ},
\end{align*}
which follows from the conditions of $\nu$'s of
the grouped multisection $\boldsymbol{\mathring{\epsilon}}$.
By construction, this grouped multisection $\boldsymbol{\epsilon}$
is $(\varphi, \hat \varphi)$-related to
$\boldsymbol{\mathring{\epsilon}}|_{\mathring{\mathcal{V}}'}$.
\end{proof}
We note that we can apply the same argument for more general cases.
Let $(\varphi_i, \hat \varphi_i) : (\mathring{\mathcal{V}}_i, \mathring{\mathcal{E}}_i) \to (\mathcal{V}, \mathcal{E})$
be embeddings of orbibundles and $\boldsymbol{\mathring{\epsilon}}_i$ be
grouped multisections of $(\mathring{\mathcal{V}}_i, \mathring{\mathcal{E}}_i)$.
Assume that these embeddings intersect cleanly and
the grouped multisections are compatible on the intersections.
Then for any relatively compact subsets $\mathring{\mathcal{V}}'_i \Subset \mathring{\mathcal{V}}_i$,
we can also construct a grouped multisection of $(\mathcal{V}, \mathcal{E})$
which is $(\varphi_i, \hat \varphi_i)$-related to
$\boldsymbol{\mathring{\epsilon}}_i|_{\mathring{\mathcal{V}}'_i}$ for all $i$.
For a submersion, we can define the pull back of a grouped multisection.
\begin{defi}
Let $\boldsymbol{\epsilon} = (\epsilon^\omega)_{\omega \in \coprod_j \Omega_j}$
be a grouped multisection of an orbibundle chart $(\mathcal{V}, \mathcal{E})$.
Let $(\varphi, \hat \varphi)$ be a submersion from another orbibundle chart
$(\mathcal{V}', \mathcal{E}')$ to $(\mathcal{V}, \mathcal{E})$, and $(\phi, \hat \phi)$ be its lift.
Then we can define the pull back $\varphi^\ast \boldsymbol{\epsilon}$ by
$\varphi^\ast \boldsymbol{\epsilon}
= (\phi^\ast \epsilon^\omega)_{\omega \in \coprod_{j \in I} \Omega_j}$,
where $I = \{j; \mathop{\mathrm{supp}}\nolimits \epsilon_j \cap \phi(V') \neq \emptyset\}$.
We define the $G_{V'}$-action on $\coprod_{j \in I} \Omega_j$ by
the homomorphism $\rho_\phi : G_{V'} \to G_V$ associated to $\phi$.
The pull back of a grouped multisection of an oribibundle by a submersion is
defined by the pull backs for its orbibundle charts.
\end{defi}
We can also define the pull back of a grouped multisection by
an essential submersion.
\subsection{Pre-Kuranishi structure and construction of its perturbed multisection}
We introduce the notion of pre-Kuranishi structure.
This is essentially equivalent to the usual Kuranishi structure in the sense that
we can obtain a Kuranishi structure from a pre-Kuranishi structure and
in application, when we construct a Kuranishi structure,
we usually construct a pre-Kuranishi structure implicitly.
(See Remark \ref{pre-Kuranishi and Kuranishi}.)
However, for a pre-Kuranishi space, we can define weakly good coordinate system,
which is more compatible with product than good coordinate system.
\begin{defi}
Let $X$ be a compact Hausdorff space.
A pre-Kuranishi structure on $X$ consists of the following data
$(\widetilde{X},\allowbreak \mu,\allowbreak (\mathcal{W}_x, \mathcal{E}_x, s_x, \widetilde{\psi}_x),\allowbreak
(\varphi_{x, y}, \hat \varphi_{x, y}))$:
\begin{itemize}
\item
$\widetilde{X}$ is a Hausdorff space,
and $\mu : \widetilde{X} \to X$ is a locally-homeomorphic surjection such that
$\# \mu^{-1}(p)$ $(p \in X)$ is bounded.
\item
Each $(\mathcal{W}_x, \mathcal{E}_x, s_x, \widetilde{\psi}_x)$ is a Kuranishi neighborhood of
$x \in \widetilde{X}$.
Namely, $(\mathcal{W}_x, \mathcal{E}_x)$ is an orbibundle, $s_x : \mathcal{W}_x \to \mathcal{E}_x$ is a smooth section,
and $\widetilde{\psi}_x : s_x^{-1}(0) \hookrightarrow \widetilde{X}$ is a homeomorphism
onto a neighborhood of $x \in \widetilde{X}$.
We assume that $\psi_x = \mu \circ \widetilde{\psi}_x : s_x^{-1}(0) \hookrightarrow X$ is also
a homeomorphism onto a neighborhood of $\mu(x)$.
Hence $(\mathcal{W}_x, \mathcal{E}_x, s_x, \psi_x)$ is a Kuranishi neighborhood of $\mu(x)$.
\item
For each $p \in X$, $\mu^{-1}(p)$ has a partial order such that any two elements
$x, y \in \mu^{-1}(p)$ have a unique supremum $x \vee y \in \mu^{-1}(p)$.
Furthermore we assume that $\vee$ is continuous in the following sense:
If $x' \in \widetilde{\psi}_x(s_x^{-1}(0))$, $y' \in \widetilde{\psi}_y(s_y^{-1}(0))$ and
$z' \in \widetilde{\psi}_{x \vee y}(s_{x \vee y}^{-1}(0))$ satisfy
$\mu(x') = \mu(y') = \mu(z')$, then $z' = x' \vee y'$.
Note that this implies the continuity of the partial order, that is,
if $x \geq y$ then $x' \geq y'$ for any $x' \in \widetilde{\psi}_x(s_x^{-1}(0))$ and
$y' \in \widetilde{\psi}_y(s_y^{-1}(0))$ such that $\mu(x') = \mu(y')$.
\item
For each $p \in \psi_x(s_x^{-1}(0))$, define $p_x$ by the unique point in
$\widetilde{\psi}_x(s_x^{-1}(0))$ such that $\mu(p_x) = p$.
We sometimes denote the point $\widetilde{\psi}_x^{-1}(p_x) \in \mathcal{W}_x$
by the same symbol $p_x$.
\item
For any points $x, y \in \widetilde{X}$, if there exists a point
$p \in \psi_x(s_x^{-1}(0)) \cap \psi_y(s_y^{-1}(0))$ such that $p_x \geq p_y$,
then there exists an open neighborhood $\mathcal{W}_{x, y} \subset \mathcal{W}_y$ of
$\psi_y^{-1}(\psi_x(s_x^{-1}(0)))$ and an embedding $(\varphi_{x, y}, \hat\varphi_{x, y}) :
(\mathcal{W}_{x, y}, \mathcal{E}_y|_{\mathcal{W}_{x, y}}) \to (\mathcal{W}_x, \mathcal{E}_x)$ which satisfy the following conditions:
\begin{itemize}
\item
The following diagrams are commutative.
\[
\begin{tikzcd}
\mathcal{E}_y|_{\mathcal{W}_{x, y}} \ar{r}{\hat\varphi_{x, y}} & \mathcal{E}_x\\
\mathcal{W}_{x, y} \ar{u}{s_y} \ar{r}{\varphi_{x, y}} & \mathcal{W}_x \ar{u}{s_x}
\end{tikzcd}
\quad
\begin{tikzcd}
s_y^{-1}(0) \cap \mathcal{W}_{x, y} \ar{r}{\psi_y} \ar[hook]{d}{\varphi_{x, y}}& X\\
s_x^{-1}(0) \ar{ru}[swap]{\psi_x}&
\end{tikzcd}
\]
\item
The vertical differential
\[
d^\bot s_x : \frac{T_{p_x}W_x}{(\phi_{x, y})_\ast T_{p_y}W_y} \stackrel{\cong}{\to}
\frac{(E_y)_{p_y}}{\hat\phi_{x, y} (E_x)_{p_x}}
\]
is an isomorphism for each point $p \in \psi_x(s_x^{-1}(0)) \cap \psi_y(s_y^{-1}(0))$,
where $(\phi_{x, y}, \hat \phi_{x, y})$ is a lift of $(\varphi_{x, y}, \hat \varphi_{x, y})$.
(More precisely, $(\phi_{x, y}, \hat \phi_{x, y})$ is a lift of the restriction of
$(\varphi_{x, y}, \hat \varphi_{x, y})$ to orbibundle charts of $(\mathcal{W}_y, \mathcal{E}_y)$
and $(\mathcal{W}_x, \mathcal{E}_x)$ which contains $p_y$ and $p_x$ respectively.)
\item
For $x, y, z \in \widetilde{X}$, if there exists a point
$p \in \psi_x(s_x^{-1}(0)) \cap \psi_y(s_y^{-1}(0)) \cap \psi_z(s_z^{-1}(0))$
such that $p_x \geq p_y \geq p_z$, then
\[
(\varphi_{x, y}, \hat\varphi_{x, y}) \circ (\varphi_{y, z}, \hat\varphi_{y, z})
= (\varphi_{x, z}, \hat\varphi_{x, z})
\]
on some neighborhood $\mathcal{W}_{x, y, z} \subset \mathcal{W}_{x, z} \cap \varphi_{y, z}^{-1}(\mathcal{W}_{x, y})$
of $\psi_z^{-1}(\psi_x(s_x^{-1}(0)) \cap \psi_y(s_y^{-1}(0)))$.
\item
If $p_x > p_y$, then the embedding is not invertible, that is, $\dim \mathcal{W}_x > \dim \mathcal{W}_y$.
\item
(separating condition)\\
For any points $a \in s_y^{-1}(0)$ and $b \in s_x^{-1}(0)$, if $\psi_y(a) \neq \psi_x(b)$,
then there exist some neighborhood $\mathcal{U}_a \subset \mathcal{W}_y$ of $a$ and
$\mathcal{U}_b \subset \mathcal{W}_x$ of $b$ such that $\mathcal{U}_a \cap \varphi_{x, y}^{-1}(\mathcal{U}_b) = \emptyset$.
(This condition is not essential because it always holds true if we replace
$\mathcal{W}_x$, $\mathcal{W}_y$ and $\mathcal{W}_{x, y}$ with their relatively compact subsets.)
\end{itemize}
\end{itemize}
Note that for two points $x, y \in \widetilde{X}$ such that
$\widetilde{\psi}_x(s_x^{-1}(0)) \cap \widetilde{\psi}_y(s_y^{-1}(0)) \neq \emptyset$,
$(\varphi_{x, y}, \hat \varphi_{x, y})$ is an open embedding
since $p_x = p_y$ for any point $p \in \psi_x(s_x^{-1}(0)) \cap \psi_y(s_y^{-1}(0))$.
The Hausdorff space $X$ endowed with a pre-Kuranishi structure is called
a pre-Kuranishi space.
We say $X$ is $n$-dimensional if $\dim \mathcal{W}_x - \dim \mathcal{E}_x = n$ for all $x \in \widetilde{X}$.
For two points $x, y \in \widetilde{X}$, we say $x \unrhd y$ if there exists some point
$p \in \psi_x(s_x^{-1}(0)) \cap \psi_y(s_y^{-1}(0))$ such that $p_x \geq p_y$.
Note that by assumption, this condition is independent of the choice of the point
$p \in \psi_x(s_x^{-1}(0)) \cap \psi_y(s_y^{-1}(0))$.
We also note that this is not a partial order.
Indeed, $x \unrhd y$ and $y \unrhd z$ do not imply $x \unrhd z$ in general.
\end{defi}
\begin{rem}
\label{we can shrink Wx}
We sometimes shrink each $\mathcal{W}_x$ to a smaller open neighborhood $\mathring{\mathcal{W}}_x$ of
$\widetilde{\psi}_x^{-1}(x) \subset \mathcal{W}_x$.
(For example, see Remark \ref{shrink W_x for submersion}.)
Then we replace $\mathcal{W}_{x, y}$ and $\mathcal{W}_{x, y, z}$ with
$\mathring{\mathcal{W}}_{x, y} = \mathring{\mathcal{W}}_y \cap \varphi_{x, y}^{-1}(\mathring{\mathcal{W}}_x)$
and $\mathring{\mathcal{W}}_{x, y, z} = \mathcal{W}_{x, y, z} \cap \mathring{\mathcal{W}}_{x, z} \cap
\varphi_{y, z}^{-1}(\mathring{\mathcal{W}}_{x, y})
= \mathcal{W}_{x, y, z} \cap \mathring{\mathcal{W}}_z \cap \varphi_{y, z}^{-1}(\mathring{\mathcal{W}}_y) \cap
\varphi_{x, z}^{-1}(\mathring{\mathcal{W}}_x)$ respectively.
However, once we construct a weakly good coordinate system, we should not
shrink $\mathcal{W}_x$ any more.
Similarly, we sometimes replace $\widetilde{X}$ to its open subset which satisfies
the same conditions.
\end{rem}
\begin{rem}\label{pre-Kuranishi and Kuranishi}
We can construct a Kuraishi structure from the above pre-Kuranishi structure
as follows.
For a compact subset $\widehat{X} \subset \widetilde{X}$ such that
$\mu(\widehat{X}) = X$, define a compact subset
$\widehat{X}^+ \subset \widetilde{X}$ by
\[
\widehat{X}^+ = \{ x_1 \vee x_2 \vee \dots \vee x_k; x_1, x_2, \dots, x_k \in \widehat{X},
\mu(x_1) = \mu(x_2) = \dots = \mu(x_k)\}.
\]
Then for each $p \in X$, $\mu^{-1}(p) \cap \widehat{X}^+ = \{x_i\}$ has a maximal point
$x_0$.
Take an open subset $\mathcal{W}_p \subset \mathcal{W}_{x_0}$ such that
\[
\psi_{x_0}(s_{x_0}^{-1}(0) \cap \mathcal{W}_p) \cap \mu\biggl(\widehat{X}^+ \setminus
\bigcup_{x_i \in \mu^{-1}(p) \cap \widehat{X}^+} \widetilde{\psi}_{x_i}(s_{x_i}^{-1}(0))\biggr)
= \emptyset.
\]
Then $(\mathcal{W}_p, \mathcal{E}_{x_0}|_{\mathcal{W}_p}, s_{x_0}, \psi_{x_0})$ and restrictions of
$(\varphi_{x, y}, \hat \varphi_{x, y})$ defines a Kuranihsi structure of $X$
in the sense of \cite{FO99}.
\end{rem}
For the construction of each embedding $(\varphi_{x, y}, \hat \varphi_{x, y})
: (\mathcal{W}_{x, y}, \mathcal{E}_y|_{\mathcal{W}_{x, y}}) \to (\mathcal{W}_x, \mathcal{E}_x)$, we usually use the following
easy argument in application.
\begin{lem}
Let $(\mathcal{W}_x, \mathcal{E}_x, s_x, \psi_x)$ and $(\mathcal{W}_y, \mathcal{E}_y, s_y, \psi_y)$ be Kuranishi
neighborhoods of $x$ and $y$ in a compact Hausdorff space $X$ respectively.
Assume that for each point $q \in \psi_y^{-1}(\psi_x(s_x^{-1}(0)))$,
there exist an open neighborhood $\mathcal{W}_y^q \subset \mathcal{W}_y$ of
$\psi_y^{-1}(q)$ and an embedding $(\varphi^q, \hat \varphi^q)
: (\mathcal{W}_y^q, \mathcal{E}_y|_{\mathcal{W}_y^q}) \to (\mathcal{W}_x, \mathcal{E}_x)$.
Assume that each $(\varphi^q, \hat \varphi^q)$ satisfies
$\hat \varphi^q \circ s_y = s_x \circ \varphi^q$ on $\mathcal{W}_y^q$,
$\psi_x \circ \varphi^q = \psi_y$ on $s_y^{-1}(0) \cap \mathcal{W}_y^q$ and
the condition of the vertical differential of $s_x$.
We also assume that for any two points
$q, q' \in \psi_y^{-1}(\psi_x(s_x^{-1}(0)))$,
if $s_y^{-1}(0) \cap \mathcal{W}_y^q \cap \mathcal{W}_y^{q'} \neq \emptyset$,
then $(\varphi^q, \hat \varphi^q)$ and $(\varphi^{q'}, \hat \varphi^{q'})$
coincide on some neighborhood $\mathcal{W}_y^{q, q'}$ of
$s_y^{-1}(0) \cap \mathcal{W}_y^q \cap \mathcal{W}_y^{q'}$.
Then we can construct an open neighborhood $\mathcal{W}_{x, y} \subset \mathcal{W}_y$ of
$\psi_y^{-1}(\psi_x(s_x^{-1}(0)))$ and
an embedding $(\varphi_{x, y}, \hat \varphi_{x, y}) : (\mathcal{W}_{x, y}, \mathcal{E}_y|_{\mathcal{W}_{x, y}})
\to (\mathcal{W}_x, \mathcal{E}_x)$ which coincides with $(\varphi^q, \hat \varphi^q)$
on a neighborhood of $\psi_y^{-1}(q)$ for all
$q \in \psi_y^{-1}(\psi_x(s_x^{-1}(0)))$ and
which satisfies $\hat \varphi_{x, y} \circ s_y = s_x \circ \varphi_{x, y}$
on $\mathcal{W}_{x, y}$, $\psi_x \circ \varphi_{x, y} = \psi_y$ on $s_y^{-1}(0) \cap \mathcal{W}_{x, y}$ and
the condition of the vertical differential of $s_x$.
\end{lem}
\begin{proof}
Let $\psi_y^{-1}(\psi_x(s_x^{-1}(0)) \cap \psi_y(s_y^{-1}(0))) = \bigcup_j K_j$ be
a locally finite covering by compact subsets such that
$K_j \subset \mathcal{W}_y^{q_j}$ for some
$q_j \in \psi_y^{-1}(\psi_x(s_x^{-1}(0)) \cap \psi_y(s_y^{-1}(0)))$ for each $j$.
Let $\mathring{\mathcal{W}}_y^{q_j} \subset \mathcal{W}_y^{q_j}$ be open neighborhoods of $K_j$
such that
$\mathring{\mathcal{W}}_y^{q_j} \cap \mathring{\mathcal{W}}_y^{q'_j} = \emptyset$ if
$K_j \cap K_{j'} = \emptyset$ and
$\mathring{\mathcal{W}}_y^{q_j} \cap \mathring{\mathcal{W}}_y^{q'_j} \subset \mathcal{W}_y^{q_j, q'_j}$
if $K_j \cap K_{j'} \neq \emptyset$.
Define $\mathcal{W}_{x, y} = \bigcup_j \mathring{\mathcal{W}}_y^{q_j}$ and
define $(\varphi_{x, y}, \hat \varphi_{x, y}) : (\mathcal{W}_{x, y}, \mathcal{E}_y|_{\mathcal{W}_{x, y}})
\to (\mathcal{V}_x, \mathcal{E}_x)$ by the union of $(\varphi^q, \hat \varphi^q)|_{\mathring{\mathcal{W}}_y^{q_j}}$.
Then it is well-defined and it is a required embedding.
\end{proof}
Although we may construct a good coordinate system from the Kuranishi structure
obtained in Remark \ref{pre-Kuranishi and Kuranishi} as in \cite{FO99}, in this paper,
we directly construct a good coordinate system from pre-Kuranishi structure.
\begin{defi}
A totally ordered cover of a pre-Kuranishi space $X$ is an open subset
$\mathcal{Y} \subset \widetilde{X}$ such that
$\mu(\mathcal{Y}) = X$ and each fiber $\mathcal{Y} \cap \mu^{-1}(p)$ ($p \in X$) is totally ordered.
\end{defi}
Note that if an open subset $\mathcal{Y}' \subset \mathcal{Y}$ satisfies $\mu(\mathcal{Y}') = X$, then
$\mathcal{Y}'$ is also a totally ordered cover.
The following is our good coordinate system.
\begin{defi}
A good coordinate system of a pre-Kuranishi space $X$ is a family of finite pairs
$(x, \mathcal{V}_x)_{x \in P}$ of points $x \in \widetilde{X}$ and open neighborhoods
$\mathcal{V}_x \subset \mathcal{W}_x$ of $\widetilde{\psi}_x^{-1}(x)$ which satisfies the following conditions.
For two points $x, y \in P$ such that $x \unrhd y$, we define
$\mathcal{V}_{x, y} = \mathcal{V}_y \cap \varphi_{x, y}^{-1}(\mathcal{V}_x)$.
Then $\mathcal{V}_x$ and $\mathcal{V}_{x, y}$ satisfy the following conditions:
\begin{enumerate}[label=$(\arabic*)^{\mathrm{G}}$]
\item
\label{good P totally ordered}
$\bigcup_{x \in P} \widetilde{\psi}_x(\mathcal{V}_x \cap s_x^{-1}(0)) \subset \widetilde{X}$ is
a totally ordered cover.
\item
\label{good (x, y, z)-relation}
For any $x, y, z \in P$, if there exists some point $p \in \psi_x(\mathcal{V}_x \cap s_x^{-1}(0))
\cap \psi_y(\mathcal{V}_y \cap s_y^{-1}(0)) \cap \psi_z(\mathcal{V}_z \cap s_z^{-1}(0))$ such that
$p_x \geq p_y \geq p_z$, then
\begin{equation}
\varphi_{x, y}(\mathcal{V}_{x, y}) \cap \varphi_{x, z}(\mathcal{V}_{x, z}) \subset \varphi_{x, z}(\mathcal{W}_{x, y, z})
\label{good (x, y, z)-overlap}
\end{equation}
and
\begin{equation}
\mathcal{V}_{y, z} \cap \varphi_{y, z}^{-1}(\mathcal{V}_{x, y}) \subset \mathcal{W}_{x, y, z}.
\label{good (x, y, z)-composition}
\end{equation}
\item
\label{good (x, y, z)-no relation}
For any $x, y, z \in P$ such that $\psi_x(\mathcal{V}_x \cap s_x^{-1}(0)) \cap
\psi_y(\mathcal{V}_y \cap s_y^{-1}(0)) \cap \psi_z(\mathcal{V}_z \cap s_z^{-1}(0)) = \emptyset$,
\begin{itemize}
\item
if $x \unrhd y$ and $x \unrhd z$, then
$\varphi_{x, y}(\mathcal{V}_{x, y}) \cap \varphi_{x, z}(\mathcal{V}_{x, z}) = \emptyset$,
\item
if $x \unrhd y$ and $y \unrhd z$, then
$\mathcal{V}_{x, y} \cap \varphi_{y, z}(\mathcal{V}_{y, z}) = \emptyset$, and
\item
if $x \unrhd z$ and $y \unrhd z$, then
$\mathcal{V}_{x, z} \cap \mathcal{V}_{y, z} = \emptyset$.
\end{itemize}
\end{enumerate}
\end{defi}
Condition \ref{good P totally ordered} implies that for any $x, y \in P$,
if $\psi_x(\mathcal{V}_x \cap s_x^{-1}(0)) \cap \psi_y(\mathcal{V}_y \cap s_y^{-1}(0)) \neq \emptyset$
and $\dim \mathcal{V}_x \geq \dim \mathcal{V}_y$, then $x \unrhd y$.
Hence there exists an embedding
$(\varphi_{x, y}, \hat \varphi_{x, y}) : (\mathcal{V}_{x, y}, \mathcal{E}_y|_{\mathcal{V}_{x, y}}) \to
(\mathcal{V}_x, \mathcal{E}_x|_{\mathcal{V}_x})$.
Therefore if we fix a total order $\preceq$ of $P$ such that
$\dim \mathcal{V}_y \leq \dim \mathcal{V}_x$ if $y \leq x$, then
our good coordinate system is essentially the same as that of \cite{FO99}.
We can construct a good coordinate system from a totally ordered cover
as follows.
\begin{lem}\label{good coordinate from totally ordered cover}
Assume that a totally ordered cover $\mathcal{Y} \subset \widetilde{X}$ is given.
Then for any compact subset $\mathcal{K} \subset \mathcal{Y}$,
there exists a good coordinate system $(x, \mathcal{V}_x)_{x \in P}$ such that
$\mathcal{K} \subset \bigcup_{x \in P} \widetilde{\psi}_x(\mathcal{V}_x \cap s_x^{-1}(0)) \subset \mathcal{Y}$.
\end{lem}
\begin{proof}
We may assume that $\mu(\mathcal{K}) = X$.
Choose finite points $P = \{x\} \subset \widetilde{X}$ and compact subsets
$\mathcal{K}_x \subset s_x^{-1}(0)$ such that
$\mathcal{K} \subset \bigcup_{x \in P} \widetilde{\psi}_x(\mathcal{K}_x) \subset \mathcal{Y}$.
We claim that if we choose a sufficiently small open neighborhood $\mathcal{V}_x \subset \mathcal{W}_x$ of
$\mathcal{K}_x$ for each $x \in P$ then $(x, \mathcal{V}_x)_{x \in P}$ is a good coordinate system.
First, it is clear that Condition \ref{good P totally ordered} holds if
$\mathcal{V}_x \subset\mathcal{W}_x$ ($x \in P$) are sufficiently small so that
$\widetilde{\psi}_x(\mathcal{V}_x \cap s_x^{-1}(0)) \subset \mathcal{Y}$.
For Condition \ref{good (x, y, z)-relation}, first we note that for any two points
$x, y \in P$ such that $x \unrhd y$, if we choose sufficiently small neighborhood
$\mathcal{V}_x$ and $\mathcal{V}_y$ of $\mathcal{K}_x$ and $\mathcal{K}_y$ respectively then
$\mathcal{V}_{x, y} = \mathcal{V}_y \cap \varphi_{x, y}^{-1}(\mathcal{V}_x)$ is contained in an arbitrary small
neighborhood of $\mathcal{K}_y \cap \psi_y^{-1}(\psi_x(\mathcal{K}_x))$.
This can be proved as follows.
Let $\mathcal{V}_x^k$ and $\mathcal{V}_y^k$ be decreasing sequences of relatively compact neighborhoods
of $\mathcal{K}_x$ and $\mathcal{K}_y$ such that $\bigcap_k \overline{\mathcal{V}_x^k} = \mathcal{K}_x$ and
$\bigcap_k \overline{\mathcal{V}_y^k} = \mathcal{K}_y$ respectively.
Then $\mathcal{V}_{x, y}^k = \mathcal{V}_y^k \cap \varphi_{x, y}^{-1}(\mathcal{V}_x^k)$ is a decreasing sequence of
relatively compact neighborhoods of $\mathcal{K}_y \cap \psi_y^{-1}(\psi_x(\mathcal{K}_x))$ such that
$\bigcap_k \overline{\mathcal{V}_{x, y}^k} = \mathcal{K}_y \cap \psi_y^{-1}(\psi_x(\mathcal{K}_x))$.
Indeed, for $a \in \bigcap_k \overline{\mathcal{V}_{x, y}^k}$, there exists a sequence
$a_k \in \mathcal{V}_y^k \cap \varphi_{x, y}^{-1}(\mathcal{V}_x^k)$ converging to
$a \in \bigcap_k \overline{\mathcal{V}_y^k} = \mathcal{K}_y \subset s_y^{-1}(0)$.
Taking subsequence, we may assume that $\varphi_{x, y}(a_k) \in \mathcal{V}_x^k$ converges to
some point $b \in \bigcap_k \overline{\mathcal{V}_x^k} = \mathcal{K}_x \subset s_x^{-1}(0)$.
Then the last condition of pre-Kuranishi space (separating condition) implies that
$\psi_y(a) = \psi_x(b)$.
Hence $a$ is contained in $\mathcal{K}_y \cap \psi_y^{-1}(\psi_x(\mathcal{K}_x))$.
Therefore $\mathcal{V}_{x, y}^k$ is a decreasing sequence of
relatively compact neighborhoods of $\mathcal{K}_y \cap \psi_y^{-1}(\psi_x(\mathcal{K}_x))$ such that
$\bigcap_k \overline{\mathcal{V}_{x, y}^k} = \mathcal{K}_y \cap \psi_y^{-1}(\psi_x(\mathcal{K}_x))$,
which implies that we can make $\mathcal{V}_{x, y}^k$ be an arbitrary small neighborhood of
$\mathcal{K}_y \cap \psi_y^{-1}(\psi_x(\mathcal{K}_x))$.
Consider any triple $x, y, z \in P$ such that there exists some point
$p \in \psi_x(\mathcal{K}_x) \cap
\psi_y(\mathcal{K}_y) \cap \psi_z(\mathcal{K}_z)$ such that $p_x \geq p_y \geq p_z$.
The above argument implies that
if we choose small $\mathcal{V}_x$, $\mathcal{V}_y$ and $\mathcal{V}_z$,
then
\[
\varphi_{x, z}^{-1}(\varphi_{x, y}(\mathcal{V}_{x, y}) \cap \varphi_{x, z}(\mathcal{V}_{x, z}))
= \mathcal{V}_z \cap \varphi_{x, z}^{-1}(\mathcal{V}_x) \cap \varphi_{x, z}^{-1}(\varphi_{x, y}(\mathcal{V}_y))
\]
and
\[
\mathcal{V}_{y, z} \cap \varphi_{y, z}^{-1}(\mathcal{V}_{x, y})
= \mathcal{V}_z \cap \varphi_{y, z}^{-1}(\mathcal{V}_y) \cap \varphi_{y, z}^{-1} (\varphi_{x, y}^{-1}(\mathcal{V}_x))
\]
are contained in an arbitrary small neighborhood
of $\mathcal{K}_z \cap \psi_z^{-1}(\psi_x(\mathcal{K}_x) \cap \psi_y(\mathcal{K}_y))$.
In particular, we may assume that they are contained in $\mathcal{W}_{x, y, z}$.
Then Condition \ref{good (x, y, z)-relation} holds for the triples $(x, y, z)$ such that
$p_x \geq p_y \geq p_z$ for some point $p \in \psi_x(\mathcal{K}_x) \cap
\psi_y(\mathcal{K}_y) \cap \psi_z(\mathcal{K}_z)$.
We may also assume that for any $x, y, z \in P$,
if $\psi_x(\mathcal{K}_x) \cap \psi_y(\mathcal{K}_y) \cap \psi_z(\mathcal{K}_z) = \emptyset$ then
$\psi_x(\mathcal{V}_x \cap s_x^{-1}(0)) \cap \psi_y(\mathcal{V}_y \cap s_y^{-1}(0)) \cap
\psi_z(\mathcal{V}_z \cap s_z^{-1}(0)) = \emptyset$.
Then Condition \ref{good (x, y, z)-relation} holds for all triples $(x, y, z)$.
We can also prove that Condition \ref{good (x, y, z)-no relation} holds if
$\mathcal{V}_x$, $\mathcal{V}_y$ and $\mathcal{V}_z$ are sufficiently small similarly.
Hence we can construct a required good coordinate system.
\end{proof}
We can construct a totally ordered cover by the following lemma.
\begin{lem}\label{totally ordered cover}
Let $\mu : \widetilde{X} \to X$ be a locally homeomorphic and
surjective comtinuous map between Hausdorff spaces.
Assume that for each $p \in X$, $\mu^{-1}(p)$ has a partial order $\leq$ which satisfies
the following conditions:
\begin{itemize}
\item
each $\mu^{-1}(p)$ has a maximum.
\item
$\leq$ is continuous in the following sense:
For any $x, y \in \widetilde{X}$ such that $\mu(x) = \mu(y)$, if $x \leq y$, then
there exist open neighborhoods $U_x \subset \widetilde{X}$ and
$U_y \subset \widetilde{X}$ of $x$ and $y$ respectively such that
$x' \leq y'$ for any $x' \in U_x$ and $y' \in U_y$ such that $\mu(x') = \mu(y')$.
\end{itemize}
We also assume that there exists an integral-valued continuous function
$l : \widetilde{X} \to \mathbb{Z}$ such that $l(x) < l(y)$ if $x < y$.
Then for any compact subset $L \subset X$, there exists an open subset $V \subset
\widetilde{X}$ such that $\mu(V) \supset L$ and each nonempty fiber
$V \cap \mu^{-1}(p)$ $(p \in \mu(V))$ is totally ordered.
\end{lem}
\begin{cor}\label{totally ordered cover for pre-Kuranishi space}
Any pre-Kuranishi space has a totally ordered cover.
\end{cor}
\begin{proof}[Proof of Corollary \ref{totally ordered cover for pre-Kuranishi space}]
Apply Lemma \ref{totally ordered cover} to
$l(x) = \dim \mathcal{V}_x$ ($x \in \widetilde{X}$) and $L = X$.
Then $\mathcal{Y} = V$ is a totally ordered cover.
\end{proof}
\begin{proof}[Proof of Lemma \ref{totally ordered cover}]
For each $p \in L$, let $l(p)$ be the maximal value of $l$ on $\mu^{-1}(p)$.
Define $L_{\leq l} = \{p \in L; l(p) \leq l\}$ for each $l \in \mathbb{Z}$.
Note that it is compact.
For each $l \in \mathbb{Z}$, define an open subset $\widetilde{X}_l = \{x \in \widetilde{X};
l(x) = l\}$.
By the induction in $l$, we construct open subsets $V_l \Subset \widetilde{X}_l$
such that $V_{\leq l} = \bigcup_{k \leq l} V_k$ satisfies
$L_{\leq l} \subset \mu(V_{\leq l})$ and each fiber of $\mu|_{V_{\leq l}} : V_{\leq l} \to X$
is totally ordered.
Then $V = \bigcup_l V_l$ satisfies the conclusion of the claim.
First we consider the minimal $l$ such that $L_{\leq l} \neq \emptyset$.
Note that the restriction of $\mu$ to $\widetilde{X}_l \cap \mu^{-1}(L_{\leq l})$ is
injective.
For each $p \in \widetilde{X}_l \cap \mu^{-1}(L_{\leq l})$, let $U_p \subset \widetilde{X}_l$
be an open neighborhood of $p$ such that $\mu|_{U_p}$ is injective.
Since we can separate $\mu(p)$ and $L_{\leq l} \setminus \mu(U_p)$ by open sets,
there exist an open neighborhood $V_p \Subset U_p$ of $p$ and an open neighborhood
$W_p \subset \widetilde{X}_l$ of $\widetilde{X}_l \cap \mu^{-1}(L_{\leq l}) \setminus U_p$
such that $\mu(V_p) \cap \mu(W_p) = \emptyset$.
Choose finite points $p_i \in \widetilde{X}_l \cap \mu^{-1}(L_{\leq l})$ so that
$V_{p_i}$ covers $\widetilde{X}_l \cap \mu^{-1}(L_{\leq l})$.
Then the restriction of $\mu$ to the open neighborhood $V_l = (\bigcup_i V_{p_i}) \cap
\bigcap_i (U_{p_i} \cup W_{p_i})$ of $\widetilde{X}_l \cap \mu^{-1}(L_{\leq l})$ is injective.
Indeed, if $p \in V_{p_i}$ and $q \in U_{p_i} \cup W_{p_i}$ satisfy $\mu(p) = \mu(q)$,
then $q \notin W_{p_i}$ by the definition of $V_{p_i}$ and $W_{p_i}$.
Hence both of $p$ and $q$ is contained in $U_{p_i}$, which implies that
$p = q$ since $\mu|_{U_{p_i}}$ is injective.
Therefore the restriction of $\mu$ to $V_l$ is injective.
Next we assume that we have already constructed required open subsets
$V_k \Subset \widetilde{X}_k$ for $k < l$.
Namely, we assume that $L_{\leq k} \subset \mu(V_{\leq k})$ for $k < l$
and that each fiber of $\mu|_{V_{\leq l-1}} : V_{\leq l-1} \to X$ is totally ordered.
We construct $V_l \subset \widetilde{X}_l$ as follows.
Since $A_l = \widetilde{X}_l \cap \mu^{-1}(L_{\leq l} \setminus \mu(V_{\leq l-1}))$ consists
of maximums, the restriction of $\mu$ to $A_l$ is injective.
For each $p \in A_l$, let $U_p \subset \widetilde{X}_l$ be an open neighborhood of
$p$ which makes $\mu|_{U_p}$ injective and the following condition hold true:
If $q \in U_p$ and $r \in V_{\leq l-1}$ satisfy $\mu(q) = \mu(r)$, then $q \geq r$.
(This condition holds if $U_p$ is sufficiently small because $p \geq r$
for any $r \in \overline{V_{\leq l-1}}$ such that $\mu(p) = \mu(r)$.)
As in the case of minimal $l$, we define open subsets $V_p$ and $W_p$ for each
$p \in A_l$, and choose finite points $p_i \in A_l$ such that
$V_{p_i}$ covers $A_l$.
Then the restriction of $\mu$ to $V_l = (\bigcup_i V_{p_i}) \cap
\bigcap_i (U_{p_i} \cup W_{p_i})$ is injective, and
if $q \in V_l$ and $r \in V_{\leq l-1}$ satisfy $\mu(q) = \mu(r)$ then $q \geq r$.
Hence this $V_l$ is a required open subset.
\end{proof}
Good coordinate system is enough for the construction of the virtual fundamental chain
of one Kuranishi space, but it is not closed under product operation.
One way which was used before to overcome this problem is that
first we construct a new Kuranishi space from each good coordinate system and
reconstruct a good coordinate system of the product of the new Kuranishi spaces.
(However, this gives rise to another problem about compatibility with the various orders
of product of more than two spaces.)
Instead, we introduce a new notion of weakly good coordinate system,
which is more compatible with product.
This is defined by using the following cover of $X$ instead of a totally ordered cover.
\begin{defi}
A meet-semilattice cover of a pre-Kuranishi space $X$ is an open subset
$\mathcal{Y} \subset \widetilde{X}$ which satisfies $\mu(\mathcal{Y}) = X$ and the following conditions
for each $p \in X$:
\begin{enumerate}[label=$(\arabic*)^{\mathrm{M}}$]
\item
\label{clean intersection for meet-semilattice cover}
For any $x \in \mu^{-1}(p)$,
$\{\varphi_{x, y}(\mathcal{W}_{x, y}); y \leq x, y \in \mathcal{Y} \cap \mu^{-1}(p)\}$
intersect cleanly on a neighborhood of $\widetilde{\psi}_x^{-1}(x) \in \mathcal{W}_x$.
\item
\label{wedge existence for meet-semilattice cover}
For any two points $y, z \in \mathcal{Y} \cap \mu^{-1}(p)$, there exists some point
$w \in \mathcal{Y} \cap \mu^{-1}(p)$ such that
$w \leq y$, $w \leq z$, and
$\varphi_{y \vee z, w}(\mathcal{W}_{y \vee z, w})$ contains the intersection
$\varphi_{y \vee z, y}(\mathcal{W}_{y \vee z, y}) \cap \varphi_{y \vee z, z}(\mathcal{W}_{y \vee z, z})$
in a neighborhood of $\widetilde{\psi}_{y \vee z}^{-1}(y \vee z)$.
\end{enumerate}
(We do not assume that $x \in \mathcal{Y}$ or $y \vee z \in \mathcal{Y}$ in the above conditions.)
\end{defi}
Note that for a meet-semilattice cover $\mathcal{Y}$ and two points
$y, z \in \mathcal{Y} \cap \mu^{-1}(p)$, the point $w \in \mathcal{Y} \cap \mu^{-1}(p)$ which satisfies
Condition \ref{wedge existence for meet-semilattice cover} is unique.
This is easily seen as follows.
If there exist two points $w_1, w_2 \in \mathcal{Y} \cap \mu^{-1}(p)$ satisfying this
condition, then the images of $\varphi_{y \vee z, w_1}$ and $\varphi_{y \vee z, w_2}$
coincides in a neighborhood of $\widetilde{\psi}_{y \vee z}^{-1}(y \vee z)$.
Hence the images of $\varphi_{w_1 \vee w_2, w_1}$ and
$\varphi_{w_1 \vee w_2, w_2}$ also coincide in a neighborhood of
$\widetilde{\psi}_{w_1 \vee w_2}^{-1}(w_1 \vee w_2)$.
Condition \ref{wedge existence for meet-semilattice cover} for
$w_1, w_2 \in \mathcal{Y} \cap \mu^{-1}(p)$ implies that there exists some
$v \in \mathcal{Y} \cap \mu^{-1}(p)$ such that $v \leq w_1$, $v \leq w_2$ and
the image of $\varphi_{w_1 \vee w_2, v}$ coincides with those of
$\varphi_{w_1 \vee w_2, w_1}$ and $\varphi_{w_1 \vee w_2, w_2}$.
Hence $\varphi_{w_1, v}$ and $\varphi_{w_2, v}$ are diffeomorphisms,
which implies $w_1 = w_2 = v$.
We denote the unique point $w$ for a pair $y, z \in \mathcal{Y} \cap \mu^{-1}(p)$
by $y \wedge z$.
We also note that $\wedge$ is continuous, that is,
for any two points $y, z \in \mathcal{Y} \cap \mu^{-1}(p)$, there exist neighborhoods
$\mathcal{U}_y$, $\mathcal{U}_z$ and $\mathcal{U}_{y \wedge z}$ of $y$, $z$ and $y \wedge z$
in $\mathcal{Y}$ respectively such that for any $y' \in \mathcal{U}_y$, $z' \in \mathcal{U}_z$ and
$w' \in \mathcal{U}_{y \wedge z}$, if $\mu(y') = \mu(z') = \mu(w')$ then
$w' = y' \wedge z'$.
\begin{defi}
\label{def of weakly good coordinate system}
A weakly good coordinate system of a pre-Kuranishi space $X$ is a family of finite pairs
$(x, \mathcal{V}_x)_{x \in P}$ of points $x \in \widetilde{X}$ and open neighborhoods
$\mathcal{V}_x \subset \mathcal{W}_x$ of $\widetilde{\psi}_x^{-1}(x)$ which satisfies
the following conditions.
For two points $x, y \in P$ such that $x \unrhd y$, we define
$\mathcal{V}_{x, y} = \mathcal{V}_y \cap \varphi_{x, y}^{-1}(\mathcal{V}_x)$.
Then $\mathcal{V}_x$ and $\mathcal{V}_{x, y}$ satisfy the following conditions:
\begin{enumerate}[label=$(\arabic*)^{\mathrm{W}}$]
\item
\label{P meet-semilattice}
$\bigcup_{x \in P} \psi_x(\mathcal{V}_x \cap s_x^{-1}(0))$ is a meet-semilattice cover of $X$.
\item
\label{(x, y, z)-relation}
For any $x, y, z \in P$, if there exists some point $p \in \psi_x(\mathcal{V}_x \cap s_x^{-1}(0))
\cap \psi_y(\mathcal{V}_y \cap s_y^{-1}(0)) \cap \psi_z(\mathcal{V}_z \cap s_z^{-1}(0))$ such that
$p_x \geq p_y \geq p_z$, then
\begin{equation}
\varphi_{x, y}(\mathcal{V}_{x, y}) \cap \varphi_{x, z}(\mathcal{V}_{x, z}) \subset \varphi_{x, z}(\mathcal{W}_{x, y, z})
\label{(x, y, z)-overlap}
\end{equation}
and
\begin{equation}
\mathcal{V}_{y, z} \cap \varphi_{y, z}^{-1}(\mathcal{V}_{x, y}) \subset \mathcal{W}_{x, y, z}.
\label{(x, y, z)-composition}
\end{equation}
\item
\label{(x, y, z)-no relation}
For any $x, y, z \in P$ such that $\psi_x(\mathcal{V}_x \cap s_x^{-1}(0)) \cap
\psi_y(\mathcal{V}_y \cap s_y^{-1}(0)) \cap \psi_z(\mathcal{V}_z \cap s_z^{-1}(0)) = \emptyset$,
\begin{itemize}
\item
if $x \unrhd y$ and $x \unrhd z$, then
$\varphi_{x, y}(\mathcal{V}_{x, y}) \cap \varphi_{x, z}(\mathcal{V}_{x, z}) = \emptyset$,
\item
if $x \unrhd y$ and $y \unrhd z$, then
$\mathcal{V}_{x, y} \cap \varphi_{y, z}(\mathcal{V}_{y, z}) = \emptyset$, and
\item
if $x \unrhd z$ and $y \unrhd z$, then
$\mathcal{V}_{x, z} \cap \mathcal{V}_{y, z} = \emptyset$.
\end{itemize}
\item
\label{clean intersection for weakly good coordinate system}
For any $x, y_i \in P$ such that $x \unrhd y_i$,
$(\varphi_{x, y_i}(\mathcal{V}_{x, y_i}))_i$ intersect cleanly.
\item
\label{(x, y, z, w)-relation}
For any $x, y, z \in P$, if there exists some point $p \in \psi_x(\mathcal{V}_x \cap s_x^{-1}(0))
\cap \psi_y(\mathcal{V}_y \cap s_y^{-1}(0)) \cap \psi_z(\mathcal{V}_z \cap s_z^{-1}(0))$ such that
$p_x \geq p_y$ and $p_x \geq p_z$,
then
there exists finite points $w_j \in P$ such that
$y \unrhd w_j$, $z \unrhd w_j$ and
\begin{equation}
\varphi_{x, y}(\mathcal{V}_{x, y}) \cap \varphi_{x, z}(\mathcal{V}_{x, z}) \subset
\bigcup_j \varphi_{x, w_j}(\mathcal{V}_{w_j} \cap \mathcal{W}_{x, y, w_j} \cap \mathcal{W}_{x, z, w_j})
\label{(x, y, z, w)-overlap}
\end{equation}
\end{enumerate}
\end{defi}
Condition \ref{(x, y, z)-relation} and \ref{(x, y, z)-no relation} are the same
with those for good coordinate system.
We also note that in Condition \ref{(x, y, z, w)-relation},
if $p_y \geq p_z$, then (\ref{(x, y, z, w)-overlap}) for $\{w_j\} = \{z\}$
follows from (\ref{(x, y, z)-overlap}).
(We read $\mathcal{W}_{x, z, z}$ as $\mathcal{W}_{x, z}$.)
Similarly to Lemma \ref{good coordinate from totally ordered cover},
we can prove the following.
\begin{lem}
\label{weakly good coordinate from meet-semilattice}
Assume that a meet-semilattice cover $\mathcal{Y} \subset \widetilde{X}$ is given.
Then for any compact subset $\mathcal{K} \subset \mathcal{Y}$,
there exists a weakly good coordinate system $(x, \mathcal{V}_x)_{x \in P}$ such that
$\mathcal{K} \subset \bigcup_{x \in P} \widetilde{\psi}_x(\mathcal{V}_x \cap s_x^{-1}(0)) \subset \mathcal{Y}$.
\end{lem}
\begin{proof}
The proof is similar to Lemma \ref{good coordinate from totally ordered cover}, but
for Condition \ref{(x, y, z, w)-relation}, we need to construct $\mathcal{V}_x$
by the induction in $\dim \mathcal{W}_x$ as follows.
We may assume that $\mu(\mathcal{K}) = X$.
Since we can replace $\mathcal{K}$ with the compact set $\{x_1 \wedge \dots \wedge x_k;
x_i \in \mathcal{K}\}$,
we may also assume that $\mathcal{K}$ is closed under $\wedge$.
Choose finite points $P = \{x\} \subset \widetilde{X}$ and compact subsets
$\mathcal{K}_x \subset s_x^{-1}(0)$ such that $\mathcal{K} = \bigcup_{x \in P} \widetilde{\psi}_x(\mathcal{K}_x)$.
We construct neighborhoods $\mathcal{V}_x$ of $\mathcal{K}_x$ in $\mathcal{W}_x$ by the induction in $\dim \mathcal{W}_x$.
As we saw in the proof of Lemma \ref{good coordinate from totally ordered cover},
the conditions other than \ref{clean intersection for weakly good coordinate system} and
\ref{(x, y, z, w)-relation} hold if each $\mathcal{V}_x$ is sufficiently
small.
First we consider Condition \ref{clean intersection for weakly good coordinate system}.
This condition holds if $\mathcal{V}_x$ and $\mathcal{V}_{y_i}$ are sufficiently small neighborhood of
$\mathcal{K}_x$ and $\mathcal{K}_{y_i}$ respectively because $\widetilde{\psi}_x(\mathcal{K}_x)$ and
$\widetilde{\psi}_{y_i}(\mathcal{K}_{y_i})$ are contained
in a meet-semilattice cover $\mathcal{Y}$.
Next we consider Condition \ref{(x, y, z, w)-relation}.
As in the proof of Lemma \ref{good coordinate from totally ordered cover}, we may
assume that for any triple $x, y, z \in P$,
if $\psi_x(\mathcal{K}_x) \cap \psi_y(\mathcal{K}_y) \cap \psi_z(\mathcal{K}_z) = \emptyset$, then
$\psi_x(\mathcal{V}_x \cap s_x^{-1}(0)) \cap \psi_y(\mathcal{V}_y \cap s_y^{-1}(0)) \cap
\psi_z(\mathcal{V}_z \cap s_z^{-1}(0)) = \emptyset$.
Since the case where $p_y \leq p_z$ or $p_y \geq p_z$ is contained
in Condition \ref{(x, y, z)-relation}, we may assume otherwise.
In particular, the dimension of the intersection
$\varphi_{x, y}(\mathcal{V}_{x, y}) \cap \varphi_{x, z}(\mathcal{V}_{x, z})$ is less than those of
$\mathcal{W}_y$ or $\mathcal{W}_z$.
Let $l \geq 0$ be arbitrary and assume that $\mathcal{V}_w$ for all $w \in P$ such that
$\dim \mathcal{W}_w < l$ are given.
Consider Condition \ref{(x, y, z, w)-relation}
for a triple $x, y, z \in P$ such that $\min(\dim \mathcal{W}_y, \dim \mathcal{W}_z) = l$.
Since $\mathcal{K}$ is closed under $\wedge$, there exists finite points $w_j \in P$
such that $y \unrhd w_j$, $z \unrhd w_j$, and
$\{\varphi_{x, w_j}(\mathcal{V}_{w_j} \cap \mathcal{W}_{x, y, w_j} \cap \mathcal{W}_{x, z, w_j})\}_j$ covers
$\mathcal{K}_x \cap \psi_x^{-1}(\psi_y(\mathcal{K}_y) \cap \psi_z(\mathcal{K}_z))$.
Hence if $\mathcal{V}_x$ for $x \in P$ such that $\dim W_x \geq l$ are sufficiently small
neighborhoods of $\mathcal{K}_x$, then Condition \ref{(x, y, z, w)-relation}
for $x, y, z \in P$ such that $\min(\dim \mathcal{W}_y, \dim \mathcal{W}_z) = l$ holds true.
Therefore we can construct neighborhoods $\mathcal{V}_x$ of $\mathcal{K}_x$ in $\mathcal{W}_x$ which satisfy
the conditions of weakly good coordinate system by the induction in $\dim \mathcal{W}_x$.
\end{proof}
\begin{defi}
Let $(x, \mathcal{V}_x)_{x \in P}$ be a weakly good coordinate system of
a pre-Kuranishi space $X$.
A grouped multisection $\boldsymbol{\epsilon} = (\boldsymbol{\epsilon}_x)_{x \in P}$ of
$(x, \mathcal{V}_x)_{x \in P}$ is a family of grouped multisections $\boldsymbol{\epsilon}_x$ of
orbibundles $(\mathcal{V}_x, \mathcal{E}_x|_{\mathcal{V}_x})$
which satisfies the following compatibility condition:
For any $x, y \in P$, if there exists some $p \in \psi_x(s_x^{-1}(0) \cap \mathcal{V}_x) \cap
\psi_y(s_y^{-1}(0) \cap \mathcal{V}_y)$ such that $p_x \geq p_y$, then
$\boldsymbol{\epsilon}_x$ and $\boldsymbol{\epsilon}_y|_{\mathcal{V}_{x, y}}$ are
$(\varphi_{x, y}, \hat \varphi_{x, y})$-related.
We emphasize that each $\boldsymbol{\epsilon}_x$ is a grouped multisection of
an orbibundle $(\mathcal{V}_x, \mathcal{E}_x|_{\mathcal{V}_x})$, and we do not assume that it is a grouped
multisection of an orbibundle chart.
\end{defi}
The following was proved in \cite{FO99} for the case of good coordinate system.
\begin{lem}
\label{construction of grouped multisection}
For a weakly good coordinate system $(x, \mathcal{V}_x)_{x \in P}$ of a pre-Kuranishi space $X$,
shrinking $\mathcal{V}_x$ slightly if necessary, we can construct a grouped multisection
$(\boldsymbol{\epsilon}_x)_{x \in P}$ which satisfies the following transversality
condition:
For any orbibundle chart $(\mathcal{V}, \mathcal{E})$ in $(x, \mathcal{V}_x)$,
every branch of the multisection $s_x|_{\mathcal{V}} + \boldsymbol{\epsilon}_x|_{\mathcal{V}}$
is transverse to the zero section of $E$, and
its restriction to each corner of $V$ is also transverse to the zero section.
Furthermore, we can take an arbitrarily $C^\infty$-small grouped multisection.
\end{lem}
\begin{proof}
Fix a total order $\preceq$ of $P$ such that $\dim \mathcal{V}_y \leq \dim \mathcal{V}_x$ if
$y \preceq x$.
We construct the grouped multisection $\boldsymbol{\epsilon}_x$ by the induction in
$x \in P$ with respect to this order.
For the minimum $x \in P$, shrinking $\mathcal{V}_x$ if necessary,
we may assume that the orbibundle $(\mathcal{V}_x, \mathcal{E}_x|_{\mathcal{V}_x})$
is covered by finite number of orbibundle charts $(\mathcal{V}_{x, j}, \mathcal{E}_{x, j})$.
Take a smooth function $\chi_{x, j} \geq 0$ on $\mathcal{V}_{x, j}$ whose support in $\mathcal{V}_x$ is
contained in $\mathcal{V}_{x, j}$ for each $j$ such that $\{\{\chi_{x, j} > 0\}\}_j$ covers $\mathcal{V}_x$.
For each $j$, we take a smooth section $\epsilon^0_{x, j}$ of $E_{x, j} \to V_{x, j}$ and
define a multisection $\epsilon_{x, j}$ of $(\mathcal{V}_{x, j}, \mathcal{E}_x|_{\mathcal{V}_{x, j}})$
by $\epsilon_{x, j} = \mathop{\mathrm{Av}}\nolimits(\chi_{x, j} \epsilon^0_{x, j})$.
Define a grouped multisection of $(\mathcal{V}_x, \mathcal{E}_x|_{\mathcal{V}_x})$ by the union
$\boldsymbol{\epsilon}_x = \coprod_j \epsilon_{x, j}$.
Sard's theorem implies that we can choose smooth sections $\epsilon^0_{x, j}$
so that every branch of $s_x + \boldsymbol{\epsilon}_x$ is transverse to
the zero section.
Assume that the grouped multisections $\boldsymbol{\epsilon}_y$ for $y \in P$
less than $x \in P$ are given.
We construct the grouped multisection $\boldsymbol{\epsilon}_x$ as follows.
First we check that $(\varphi_{x, y}, \hat \varphi_{x, y})$-relations compatibly
define $\boldsymbol{\epsilon}_x$ on $\bigcup_{y \prec x, y \unlhd x}
\varphi_{x, y}(\mathcal{V}_{x, y}) \subset \mathcal{V}_x$.
Let $y, z \in P$ be two points such that $y, z \prec x$, $y \unlhd x$, $z \unlhd x$
and $\varphi_{x, y}(\mathcal{V}_{x, y}) \cap \varphi_{x, z}(\mathcal{V}_{x, z}) \neq \emptyset$.
Condition \ref{(x, y, z)-no relation} implies that
$\psi_x(\mathcal{V}_x \cap s_x^{-1}(0)) \cap \psi_y(\mathcal{V}_y \cap s_y^{-1}(0)) \cap
\psi_z(\mathcal{V}_z \cap s_z^{-1}(0)) \neq \emptyset$.
$y \unlhd x$ and $z \unlhd x$ implies that $p_x \geq p_y$ and $p_x \geq p_z$
for any point $p \in \psi_x(\mathcal{V}_x \cap s_x^{-1}(0))
\cap \psi_y(\mathcal{V}_y \cap s_y^{-1}(0)) \cap \psi_z(\mathcal{V}_z \cap s_z^{-1}(0))$.
If $p_y \geq p_z$, then Condition \ref{(x, y, z)-relation} implies
\[
\varphi_{x, y}(\mathcal{V}_{x, y}) \cap \varphi_{x, z}(\mathcal{V}_{x, z}) \subset \varphi_{x, z}(\mathcal{W}_{x, y, z}).
\]
Hence the grouped multisection $\boldsymbol{\epsilon}_y$ on
$\varphi_{x, y}^{-1}(\varphi_{x, y}(\mathcal{V}_{x, y}) \cap \varphi_{x, z}(\mathcal{V}_{x, z}))$ is defined
by $(\varphi_{y, z}, \hat \varphi_{y, z})$-relation with $\boldsymbol{\epsilon}_z$.
Therefore $(\varphi_{x, y}, \hat \varphi_{x, y})$-relation and
$(\varphi_{x, z}, \hat \varphi_{x, z})$-relation are compatible on the intersection
$\varphi_{x, y}(\mathcal{V}_{x, y}) \cap \varphi_{x, z}(\mathcal{V}_{x, z})$.
Next we consider the case where $p_y \not \geq p_z$ and $p_y \not \leq p_z$.
Condition \ref{(x, y, z, w)-relation} implies that
there exists finite points $w_j \in P$ such that
$y \unrhd w_j$, $z \unrhd w_j$ and
\[
\varphi_{x, y}(\mathcal{V}_{x, y}) \cap \varphi_{x, z}(\mathcal{V}_{x, z}) \subset
\bigcup_j \varphi_{x, w_j}(\mathcal{V}_{w_j} \cap \mathcal{W}_{x, y, w_j} \cap \mathcal{W}_{x, z, w_j}).
\]
$p_y \not \geq p_z$ and $p_y \not \leq p_z$ imply that
$\dim \mathcal{V}_{w_j} < \min(\dim \mathcal{V}_y, \dim \mathcal{V}_z)$.
In particular, $w_j \prec y, z$.
The above inclusion implies that the grouped multisection $\boldsymbol{\epsilon}_y$
on $\varphi_{x, y}^{-1}(\varphi_{x, y}(\mathcal{V}_{x, y}) \cap \varphi_{x, z}(\mathcal{V}_{x, z}))$ is defined
by $(\varphi_{y, w_j}, \hat \varphi_{y, w_j})$-relations with $\boldsymbol{\epsilon}_{w_j}$,
and $\boldsymbol{\epsilon}_z$ on
$\varphi_{x, z}^{-1}(\varphi_{x, y}(\mathcal{V}_{x, y}) \cap \varphi_{x, z}(\mathcal{V}_{x, z}))$ is defined by
$(\varphi_{z, w_j}, \hat \varphi_{z, w_j})$-relations with $\boldsymbol{\epsilon}_{w_j}$.
Hence $(\varphi_{x, y}, \hat \varphi_{x, y})$-relation and
$(\varphi_{x, z}, \hat \varphi_{x, z})$-relation are compatible on the intersection.
Therefore, $(\varphi_{x, y}, \hat \varphi_{x, y})$-relations compatibly
define $\boldsymbol{\epsilon}_x$ on $\bigcup_{y \prec x, y \unlhd x}
\varphi_{x, y}(\mathcal{V}_{x, y})$.
We also note that Condition \ref{clean intersection for weakly good coordinate system}
implies that $(\varphi_{x, y}(\mathcal{V}_{x, y}))_{y \prec x, y \unlhd x}$ intersect cleanly.
Next we extend $\boldsymbol{\epsilon}_x$ defined on this subset of $\mathcal{V}_x$ to its
neighborhood.
We may shrink $\mathcal{V}_y$ ($y \prec x$) slightly if necessary for the smooth extension.
Using a smooth function $\chi \geq 0$ on $\mathcal{V}_x$ which satisfies $\chi \equiv 1$
on a small neighborhood of this subset and whose support is contained in a slightly larger
neighborhood, we may assume that the support of $\boldsymbol{\epsilon}_x$ is contained
in a small neighborhood of $\bigcup_{y \prec x, y \unlhd x} \varphi_{x, y}(\mathcal{V}_{x, y})$.
By the assumption of pre-Kuranishi space, for any $y \preceq x$,
the vertical differentials
\[
d^\bot s_x : \frac{T_{p_x}V_x}{(\phi_{x, y})_\ast T_{p_y}V_y} \stackrel{\cong}{\to}
\frac{(E_y)_{p_y}}{\hat\phi_{x, y} (E_x)_{p_x}}
\]
are isomorphisms for any points $p_x \in s_x^{-1}(0)$ and $p_y \in s_y^{-1}(0)$ such that
$\psi(p_x) = \psi(p_y)$.
Hence if $\boldsymbol{\epsilon}_x$ is sufficiently $C^1$-small, then
the transversality conditions for $\boldsymbol{\epsilon}_y$ ($y \prec x$) imply
that $\boldsymbol{\epsilon}_x$ also satisfies the transversality condition
on a neighborhood of $\bigcup_{y \prec x, y \unlhd x} \varphi_{x, y}(\mathcal{V}_{x, y})$.
On the complement of a neighborhood of $\bigcup_{y \prec x, y \unlhd x}
\varphi_{x, y}(\mathcal{V}_{x, y})$, as in the case of minimal $x \in P$, we take finite number of
orbibundle charts and their grouped multisections,
and add them to $\boldsymbol{\epsilon}_x$ (take the union).
Then the constructed $\boldsymbol{\epsilon}_x$ satisfies the transversality condition
and $(\varphi_{x, y}, \hat \varphi_{x, y})$-relations for all $y \prec x$.
\end{proof}
Next we consider the triangulation of the zero set of the perturbed multisection.
First we explain some notations about simplicial complex.
For a simplicial complex $K$ and its subset $A \subset K$,
we denote by $\mathrm{St}(A, K)$ the minimal subcomplex of $K$ which contains all simplices
intersecting with $A$.
If $K$ is embedded in a space $X$, then for a subset $U \subset X$,
we denote by $K|_U$ the subcomplex consisting of the simplices contained in $U$.
\begin{defi}
For a smooth section $s$ and a grouped multisection
$\boldsymbol{\epsilon} = (\epsilon^\omega)_{\omega \in \coprod_j \Omega_j}$ of
an orbibundle chart $(\mathcal{V}, \mathcal{E})$,
an embedding of simplicial complex
$K = (K^{(\omega_j)})_{(\omega_j) \in \prod_j \Omega_j}$
to the zero set of $s + \boldsymbol{\epsilon}
= (s + \sum_j \epsilon^{\omega_j})_{(\omega_j) \in \prod_j \Omega_j}$
is a family of embeddings of simplicial complexes
$K^{(\omega_j)} \hookrightarrow \{s + \sum_j \epsilon^{\omega_j} = 0\}$ such that
$K^{g \cdot (\omega_j)} = g \cdot K^{(\omega_j)}$ for all $g \in G_V$.
For a subset $A \subset \mathcal{V}$, we define
\[
\mathrm{St}(A, K) = (\mathrm{St}(\pi_V^{-1}(A), K^{(\omega_j)}))_{(\omega_j) \in \prod_j \Omega_j}
\]
and
\[
K|_A = (K^{(\omega_j)}|_{\pi_V^{-1}(A)})_{(\omega_j) \in \prod_j \Omega_j}.
\]
For a subset $B \subset \mathcal{V}$, we say $K$ covers $B$ if
each $K^{(\omega_j)}$ contains $\{s + \sum_j \epsilon^{\omega_j} = 0\} \cap B$
for all $(\omega_j) \in \prod_j \Omega_j$.
For a connected open subset $\mathcal{U} \subset \mathcal{V}$, fix a connected component
$U \subset \pi_V^{-1}(\mathcal{U})$ and regard $\mathcal{U} = (U, \pi_V|_U, \mathcal{U})$ as an orbichart.
Let $\boldsymbol{\epsilon}|_{\mathcal{U}} = (\epsilon^\omega|_U)_{\omega \in \coprod_{j \in I_U}
\Omega_j}$ be the restriction of the grouped multisection $\boldsymbol{\epsilon}$,
where $I_U = \{j; \mathop{\mathrm{supp}}\nolimits (\epsilon^\omega)_{\omega \in \Omega_j} \cap
U \neq \emptyset\}$.
Let $K_{\mathcal{U}} = (K_U^{(\omega_j)})_{(\omega_j) \in \prod_{j \in I_U} \Omega_j}$
be an embedding of simplicial complex
to the zero set of $s|_{\mathcal{U}} + \boldsymbol{\epsilon}|_{\mathcal{U}}$.
Choose $g_k \in G_V$ so that $\pi_V^{-1}(\mathcal{U}) = \coprod_k g_k U$.
We say $K$ is equivalent to $K_\mathcal{U}$ if
$K^{(\omega_j)} = \bigcup_k g_k K^{\overline{g_k^{-1} (\omega_j)}}_U$ for all
$(\omega_j) \in \prod_j \Omega_j$,
where $\overline{(\omega_j)} \in \prod_{j \in I_U} \Omega_j$ is the image of
$(\omega_j) \in \prod_j \Omega_j$ by the projection $\prod_j \Omega_j \to
\prod_{j \in I_U} \Omega_j$.
\end{defi}
\begin{defi}
Let $(\boldsymbol{\epsilon}_x)_{x \in P}$ be a grouped multisection of a weakly good
coordinate system $(x, \mathcal{V}_x)_{x \in P}$ of a pre-Kuranishi space $X$
which satisfies the transversality condition in Lemma
\ref{construction of grouped multisection}.
Let $(\mathcal{U}_\tau, \mathcal{E}_x|_{\mathcal{U}_\tau})_{\tau \in T_x}$ be a finite family of orbibundle charts of
each orbibundle $(\mathcal{V}_x, \mathcal{E}_x)$.
Let $\mathring{\mathcal{U}}_\tau \Subset \mathcal{U}_\tau$ be their relatively compact open subsets,
and define $\mathring{\mathcal{V}}_x = \bigcup_{\tau \in T_x} \mathring{\mathcal{U}}_\tau$.
For each $\tau \in T_x$, let
$K_\tau = (K_\tau^{(\omega_j)})_{(\omega_j) \in \prod_j \Omega_{\tau, j}}$ be
an embedding of simplicial complex to the zero set of
$s_x|_{\mathcal{U}_\tau} + \boldsymbol{\epsilon}_x|_{\mathcal{U}_\tau}$.
We say $(\mathcal{U}_\tau, \mathring{\mathcal{U}}_\tau, K_\tau)_{x \in P, \tau \in T_x}$
is a triangulation of the zero set of $(s_x + \boldsymbol{\epsilon}_x)_{x \in P}$ if
the following conditions are satisfied:
\begin{enumerate}[label=$(\arabic*)^{T}$]
\item
\label{each K}
For each $\tau \in T_x$, $K_\tau$ covers $\mathring{\mathcal{U}}_\tau$, and
$K_\tau = \mathrm{St}(\mathring{\mathcal{U}}_\tau, K_\tau)$.
\item
\label{x intersection}
For any $x \in P$ and two indices $\tau, \tau' \in T_x$,
there exists a subset $T_{\tau, \tau'} \subset T_x$ such that
$\mathring{\mathcal{U}}_\tau \cap \mathring{\mathcal{U}}_{\tau'} = \bigcup_{\tau'' \in T_{\tau, \tau'}}
\mathring{\mathcal{U}}_{\tau''}$ and
$\mathcal{U}_{\tau''} \subset \mathcal{U}_\tau \cap \mathcal{U}_{\tau'}$ for all $\tau'' \in T_{\tau, \tau'}$.
\item
\label{K contains K}
For any $\tau, \tau' \in T_x$ such that
$\mathring{\mathcal{U}}_\tau \subset \mathring{\mathcal{U}}_{\tau'}$ and $\mathcal{U}_\tau \subset \mathcal{U}_{\tau'}$,
$\mathrm{St}(\mathring{\mathcal{U}}_\tau, K_{\tau'})$ is equivalent to $K_\tau$.
\item
\label{x y intersection}
For any two points $x, y \in P$ such that $x \unrhd y$ and any $\tau \in T_x$,
$\tau' \in T_y$, there exists a subset $T_{\tau, \tau'} \subset T_y$ such that
$\mathring{\mathcal{U}}_{\tau'} \cap \varphi_{x, y}^{-1}(\mathring{\mathcal{U}}_\tau)
= \bigcup_{\tau'' \in T_{\tau, \tau'}} \mathring{\mathcal{U}}_{\tau''}$ and
$\mathcal{U}_{\tau''} \subset \mathcal{U}_{\tau'} \cap \varphi_{x, y}^{-1}(\mathcal{U}_\tau)$ for all
$\tau'' \in T_{\tau, \tau'}$.
\item
\label{x y same}
For any two points $x, y \in P$ such that $x \unrhd y$ and any $\tau \in T_y$,
if $\mathcal{U}_\tau \subset \mathcal{V}_{x, y}$, then there exists some $\tau' \in T_x$ such that
$\mathcal{U}_\tau = \varphi_{x, y}^{-1}(\mathcal{U}_{\tau'})$ and
$\mathring{\mathcal{U}}_\tau = \varphi_{x, y}^{-1}(\mathring{\mathcal{U}}_{\tau'})$.
Furthermore, we assume that
the automorphism group of $\mathcal{U}_\tau$ and $\mathcal{U}_{\tau'}$ are isomorphic.
\item
\label{K coincides K}
For any any $\tau \in T_y$ and $\tau' \in T_x$ in Condition \ref{x y same},
let $\phi_{\tau', \tau}$ be a lift of $\varphi_{x, y}|_{\mathcal{U}_\tau} : \mathcal{U}_\tau \to \mathcal{U}_{\tau'}$.
Then $K_\tau = (K_\tau^\omega)_{\omega \in \coprod_j \Omega_{\tau, j}}$ and
$K_{\tau'} = (K_{\tau'}^\omega)_{\omega \in \coprod_j \Omega_{\tau', j}}$
satisfy $K_{\tau'}^{\nu^{\phi_{\tau', \tau}}_{\mathcal{U}_{\tau'}, \mathcal{U}_\tau}(\omega)}
= \phi_{\tau', \tau}(K_\tau^\omega)$ for all $\omega \in \coprod_j \Omega_{\tau, j}$.
\item
\label{again weakly good}
$(x, \mathring{\mathcal{V}}_x)_{x \in P}$ is also a weakly good coordinate system.
\end{enumerate}
\end{defi}
We note that in Condition \ref{x y intersection}, $\mathcal{U}_{\tau''}$ is contained in
$\mathcal{V}_{x, y} = \mathcal{V}_y \cap \varphi_{x, y}^{-1}(\mathcal{V}_x)$.
Hence Condition \ref{x y same} implies that there exists some $\tau''' \in T_x$ such that
$\mathcal{U}_{\tau''} = \varphi_{x, y}^{-1}(\mathcal{U}_{\tau'''})$ and
$\mathring{\mathcal{U}}_{\tau''} = \varphi_{x, y}^{-1}(\mathring{\mathcal{U}}_{\tau'''})$.
\begin{lem}
Let $(\boldsymbol{\epsilon}_x)_{x \in P}$ be a grouped multisection of a weakly good
coordinate system $(x, \mathcal{V}_x)_{x \in P}$
which satisfies the transversality condition in Lemma
\ref{construction of grouped multisection}.
Then we can construct a triangulation
$(\mathcal{U}_\tau, \mathring{\mathcal{U}}_\tau, K_\tau)_{x \in P,\tau \in T_x}$ of the zero set of
$(s_x + \boldsymbol{\epsilon}_x)_{x \in P}$.
\end{lem}
\begin{proof}
First we construct open subsets $\mathring{\mathcal{U}}_\tau \Subset \mathcal{U}_\tau \subset \mathcal{V}_x$
($\tau \in T_x$) which satisfy Condition \ref{x intersection}, \ref{x y intersection},
\ref{x y same} and \ref{again weakly good}.
First we take relatively compact open subsets $\mathring{\mathcal{V}}_x \Subset \mathcal{V}_x$
such that $(x, \mathring{\mathcal{V}}_x)_{x \in P}$ is also a weakly good coordinate system.
(We can shrink weakly good coordinate system slightly.)
Let $(\mathcal{U}_\tau, \mathcal{E}_x|_{\mathcal{U}_\tau})_{\tau \in T_x}$ be a family of orbibundle charts of
$(\mathcal{V}_x, \mathcal{E}_x)$ which covers the closure of $\mathring{\mathcal{V}}_x$, and
let $\mathring{\mathcal{U}}_\tau \Subset \mathcal{U}_\tau$ be relatively compact open subsets
such that $\mathring{\mathcal{V}}_x = \bigcup_{\tau \in T_x} \mathring{\mathcal{U}}_\tau$.
We can easily make Condition \ref{x intersection} hold by adding connected components
$\mathcal{U}_{\tau''}$ of $\mathcal{U}_\tau \cap \mathcal{U}_{\tau'}$ which intersect $\mathring{\mathcal{U}}_\tau \cap
\mathring{\mathcal{U}}_{\tau'}$ to $(\mathcal{U}_\tau)_{\tau \in T_x}$ and defining $\mathring{\mathcal{U}}_{\tau''}
= \mathcal{U}_{\tau''} \cap \mathring{\mathcal{U}}_\tau \cap \mathring{\mathcal{U}}_{\tau'}$
for each pair $\tau, \tau' \in T_x$.
Similarly, we can make Condition \ref{x y intersection} and \ref{x y same}
hold by adding appropriate open subsets to $(\mathcal{U}_\tau)_{\tau \in T_x}$ and
$(\mathring{\mathcal{U}}_\tau)_{\tau \in T_x}$.
It is easy to check that these do not break Condition \ref{again weakly good}.
Hence we can construct open subsets $\mathring{\mathcal{U}}_\tau \Subset \mathcal{U}_\tau
\subset \mathcal{V}_x$ ($\tau \in T_x$) which satisfy Condition
\ref{x intersection}, \ref{x y intersection}, \ref{x y same} and \ref{again weakly good}.
We can construct embeddings of simplicial complexes $K_\tau$
($\tau \in \bigcup_{x \in P} T_x$) which satisfy Condition \ref{each K},
\ref{K contains K} and \ref{K coincides K} similarly to the case of usual
triangulation of smooth manifold.
\end{proof}
For the definition of virtual fundamental chain, we need an orientation of the
pre-Kuranishi space $X$ and a strong continuous map from $X$ to a topological space.
\begin{defi}
A strong continuous map $f = (f_x)_{x \in \widetilde{X}}$ from a pre-Kuranishi space $X$
to a topological space $Y$ is a family of continuous maps $f_x : \mathcal{W}_x \to Y$
($x \in \widetilde{X}$) such that
$f_x \circ \varphi_{x, y} = f_y$ on $\mathcal{W}_{x, y}$ for all $x, y \in \widetilde{X}$ such that
$x \unrhd y$.
For a strong continuous map $f = (f_x)_{x \in \widetilde{X}}$, we define
continuous maps $\widetilde{f} : \widetilde{X} \to Y$ and $f : X \to Y$ by
the conditions $\widetilde{f} \circ \widetilde{\psi}_x = f_x$ and
$f \circ \psi_x = f_x$ on $s_x^{-1}(0)$ for all $x \in \widetilde{X}$.
If $Y$ is a smooth manifold and each $f_x$ are smooth, then we call
$f$ a strong smooth map.
\end{defi}
\begin{defi}
We say an orbibundle chart $(\mathcal{V}, \mathcal{E})$ is orientable if $\det TV \otimes_\mathbb{R} \det E^\ast$
is orientable and the $G_V$-action preserves the orientation.
In this case, an orientation of $(\mathcal{V}, \mathcal{E})$ is a homotopy type of isomorphism
$\Phi : \det TV \stackrel{\cong}{\to} \det E$.
We say an orbibundle is oriented if orientations of its orbibundle charts are given and
they coincide on the intersections.
\end{defi}
\begin{defi}
A pre-Kuranishi space $X$ is oriented if $(\mathcal{W}_x, \mathcal{E}_x)$ are oriented
for all $x \in \widetilde{X}$ and they satisfy the following compatibility condition:
For any $x, y \in \widetilde{X}$ and any point $p \in \psi_x(s_x^{-1}(0)) \cap
\psi_y(s_y^{-1}(0))$ such that $p_x \geq p_y$, let $(\mathcal{W}_{x, p}, \mathcal{E}_{x, p})$ and
$(\mathcal{W}_{y, p}, \mathcal{E}_{y, p})$ be orbibundle charts of $(\mathcal{W}_x, \mathcal{E}_x)$ and $(\mathcal{W}_y, \mathcal{E}_y)$
which contain $p_x$ and $p_y$ respectively such that
$\mathcal{W}_{y, p} \subset \varphi_{x, y}^{-1}(\mathcal{W}_{x, p})$.
Then the condition is that there exists a family of orientations of
$T_{p_y}W_{y, p}$, $T_{p_x}W_{x, p}$, $T_{p_x}W_{x, p} / (\phi_{x, y})_\ast T_{p_y}W_{y, p}$
$E_{x, p}|_{p_x}$, $E_{y, p}|_{p_y}$ and $E_{x, p}|_{p_x} / \hat \phi_{x, y} E_{y, p}|_{p_y}$
which makes the following isomorphisms preserve the orientations:
\begin{gather*}
T_{p_x}W_{x, p} \cong T_{p_y}W_{y, p} \oplus
T_{p_x}W_{x, p} / (\phi_{x, y})_\ast T_{p_y}W_{y, p} \\
E_{x, p}|_{p_x} \cong E_{y, p}|_{p_y} \oplus E_{x, p}|_{p_x} / \hat \phi_{x, y} E_{y, p}|_{p_y} \\
\Phi_y : \det T_{p_y}W_{y, p} \cong \det E_{y, p}|_{p_y} \\
\Phi_x : \det T_{p_x}W_{x, p} \cong \det E_{x, p}|_{p_x} \\
d^\bot s_x : T_{p_x}W_{x, p} / (\phi_{x, y})_\ast T_{p_y}W_{y, p} \cong
E_{x, p}|_{p_x} / \hat \phi_{x, y} E_{y, p}|_{p_y}
\end{gather*}
\end{defi}
\begin{defi}
Let $(\mathcal{V}, \mathcal{E})$ be an oriented orbibundle chart whose orientation is defined by
$\Phi : \det TV \cong \det E$.
Let $s$ be its smooth section, $\boldsymbol{\epsilon} =
(\epsilon^\omega)_{\omega \in \prod_j \Omega_j}$ be its grouped multisection,
and $K = (K^{(\omega_j)})_{(\omega_j) \in \prod_j \Omega_j}$ be an embedding of
simplicial complex to the zero set of $s + \boldsymbol{\epsilon}$.
For a continuous map $f$ from $\mathcal{V}$ to a topological space $Y$,
we define a singular chain $f_{\#}(K)$ in $Y$ by
\[
f_{\#}(K) = \frac{1}{\# G_V \cdot \prod_j \# \Omega_j}
\sum_{(\omega_j) \in \prod_j \Omega_j} \sum_{\Delta \in (K^{(\omega_j)})^{\text{top}}}
\pm f_{\#}(\Delta),
\]
where the sum $\sum_{\Delta \in (K^{(\omega_j)})^{\text{top}}}$ is taken over
all top-dimensional simplices $\Delta$ of $K^{(\omega_j)}$, and
the sign $\pm$ of each $\Delta$ is defined as follows.
The sign is $+$ if the isomorphism
\[
T_q |\Delta| \oplus E|_q \cong T_q V
\]
given by a split of the exact sequence
\[
0 \to T_q |\Delta| \to T_q V
\xrightarrow{d^\bot (s + \sum_j \epsilon^{(\omega_j)})} E|_q \to 0
\]
preserves the orientations for all $q \in \Delta$, where
the relation of the orientations of $T_q V$ and $E|_q$ are defined by
the isomorphism $\Phi$.
Note that if there exist a connected open subset $\mathcal{U} \subset \mathcal{V}$ and
an embedding of simplicial complex $K_{\mathcal{U}}$ of the zero set of
$s|_\mathcal{U} + \boldsymbol{\epsilon}|_\mathcal{U}$ which is equivalent to $K$,
then $f_{\#}(K) = f_{\#}(K_\mathcal{U})$.
\end{defi}
Let $f = (f_x)_{x \in \widetilde{X}}$ be a strong continuous map
from an oriented pre-Kuranishi space $X$ to a topological space $Y$.
Assume that a grouped multisection
$\boldsymbol{\epsilon} = (\boldsymbol{\epsilon}_x)_{x \in P}$ of
a weakly good coordinate system $(x, \mathcal{V}_x)_{x \in P}$ of $X$ and
a triangulation $(\mathcal{U}_\tau, \mathring{\mathcal{U}}_\tau, K_\tau)_{x \in P, \tau \in T_x}$ of
the zero set of $(s_x + \boldsymbol{\epsilon}_x)_{x \in P}$ are given.
For $x, y \in P$, $p \in \mathcal{V}_x$ and $q \in \mathcal{V}_y$,
we say $p$ and $q$ are equivalent ($p \sim q$) if there exist some $z \in P$ and
$r \in \mathcal{V}_z$ such that $x \unrhd z$, $y \unrhd z$,
$p = \varphi_{x, z}(r)$ and $q = \varphi_{y, z}(r)$.
This is indeed an equivalence relation because $(x, \mathcal{V}_x)_{x \in P}$ is
a weakly good coordinate system.
Define sets
\[
(s + \boldsymbol{\epsilon})^{-1}(0)|_{\mathcal{U}_\tau}
= \pi_{U_\tau}\biggl(\bigcup_{(\omega_j) \in \prod_j \Omega_j}
\{s_x + \sum_j \epsilon_x^{\omega_j} = 0\} \cap U_\tau\biggr)
\subset \mathcal{U}_\tau
\]
and
\[
(s + \boldsymbol{\epsilon})^{-1}(0) = \bigcup_{x \in P, \tau \in T_x}
(s + \boldsymbol{\epsilon})^{-1}(0)|_{\mathcal{U}_\tau} / \sim.
\]
Let $\pi : (s + \boldsymbol{\epsilon})^{-1}(0)|_{\mathcal{U}_\tau} \hookrightarrow
(s + \boldsymbol{\epsilon})^{-1}(0)$ be the quotient map.
Then the assumption of $(\mathcal{U}_\tau, \mathring{\mathcal{U}}_\tau, K_\tau)_{x \in P, \tau \in T_x}$
implies that for any subsets $A_1, B_1 \subset \mathring{\mathcal{U}}_{\tau_1}$
and $A_2, B_2 \subset \mathring{\mathcal{U}}_{\tau_2}$ such that $\pi(A_1) = \pi(A_2)$
and $\pi(B_1) = \pi(B_2)$,
the singular chains $f_{\#}(\mathrm{St}(A_1, K_{\tau_1})|_{B_1})$ and
$f_{\#}(\mathrm{St}(A_2, K_{\tau_2})|_{B_2})$ coincide.
Note that if the grouped multisection $(\boldsymbol{\epsilon}_x)_{x \in P}$
is sufficiently small, then $(s + \boldsymbol{\epsilon})^{-1}(0)$ is covered by
$\{\pi(\mathring{\mathcal{U}}_\tau)\}_{\tau \in \bigcup_{x \in P} T_x}$.
(We need to assume that $(\boldsymbol{\epsilon}_x)_{x \in P}$ is sufficiently small.
Otherwise the zeros of the perturbed multisections leak from our open covering.)
Fix an order to the finite set $\bigcup_{x \in P} T_x$, and write it as
$\bigcup_{x \in P} T_x = \{\tau_k\}_{k = 1, 2, \dots}$.
Choose arbitrary subsets $A_k \subset \mathring{\mathcal{U}}_{\tau_k}$ such that
$\bigcup_k \pi(A_k) = (s + \boldsymbol{\epsilon})^{-1}(0)$, and
define $B_k = \mathcal{U}_{\tau_k} \setminus \pi^{-1}(\bigcup_{l < k} \pi(A_l))$.
Then we define the virtual fundamental chain $f_\ast(X)$ by
\[
f_\ast(X) = \sum_k f_{\#}(\mathrm{St}(A_k, K_{\tau_k})|_{B_k}).
\]
This is independent of the order of $\bigcup_{x \in P} T_x$ and
the choice of the subsets $A_k$.
In the case where the dimension of $X$ is zero,
we usually use the trivial strong continuous map to a point.
In this case, we regard the virtual fundamental chain as a rational number.
There is another way to represent the virtual fundamental chain of a pre-Kuranishi
space using differential forms.
For a strong smooth map $f = (f_x)_{x \in \widetilde{X}}$ from $X$ to a manifold
$Y$ and $h = (h_x)_{x \in \widetilde{X}}$ from $X$ to an oriented manifold $Z$,
we represent the virtual fundamental chain as a linear map
$(h_! \circ f^\ast)_X : \Omega(Y) \to \Omega(Z)$ as follows.
If $Z$ is a point, then this map $(h_! \circ f^\ast)_X : \Omega(Y) \to \mathbb{R}$
is the dual representation of the virtual fundamental chain $f_\ast(X)$.
In this case, we often denote the value of this map at $\theta \in \Omega(Y)$
by $\int_{X} f^\ast \theta$.
Let $(x, \mathcal{V}_x)_{x \in P}$ be a weakly good coordinate system of
a pre-Kuranishi space $X$, and let $\beta_x : \mathcal{V}_x \to \mathbb{R}$ be a smooth function
with compact support for each $x \in P$.
Define $\mathcal{Y} = \bigcup_{x \in P} \widetilde{\psi}_x(\mathcal{V}_x \cap s_x^{-1}(0))$.
Note that for any $p \in X$, $\mu^{-1}(p) \cap \mathcal{Y}$ has the unique minimum
$p^\mathcal{Y}_{\min}$ since $\mathcal{Y}$ is a meet-semilattice cover.
We say $(\beta_x)_{x \in P}$ is a partition of unity subordinate to $(x, \mathcal{V}_x)_{x \in P}$
if for any $p \in X$,
$\sum_{x \in P, x \unrhd p^\mathcal{Y}_{\min}} \beta_x \circ \varphi_{x, p^\mathcal{Y}_{\min}} \equiv 1$
on a neighborhood of $\psi_{p^\mathcal{Y}_{\min}}^{-1}(p)$ in $\mathcal{W}_{p^\mathcal{Y}_{\min}}$.
Let $\boldsymbol{\epsilon} = (\boldsymbol{\epsilon}_x)_{x \in P}$ be
a grouped multisection of $(x, \mathcal{V}_x)_{x \in P}$ which satisfies the transversality
condition in Lemma \ref{construction of grouped multisection}.
We assume that the restriction of $h$ to the zero set of each branch of
the multisections $s_x + \boldsymbol{\epsilon}_x$ is submersive.
We can construct such a perturbed multisection if $Z$ is a point.
(In general, we need to use continuous family of multisections.
See Section \ref{continuous family of multisections}.)
We further assume that $\boldsymbol{\epsilon}$ is sufficiently small so that
$\sum_{z \in P, z \unrhd x} \beta_z \circ \varphi_{z, x} = 1$ on
$(s_x + \boldsymbol{\epsilon}_x)^{-1}(0) \cap (\mathcal{V}_x)_{\min}$ for any $x \in P$,
where $(s_x + \boldsymbol{\epsilon}_x)^{-1}(0) \subset \mathcal{V}_x$ is the set of points
at which one of the branchs of the multisection $s_x + \boldsymbol{\epsilon}_x$
takes zero, and $(\mathcal{V}_x)_{\min} \subset \mathcal{V}_x$ is the set of points $q \in \mathcal{V}_x$ such that
there do not exist any $y \in P$ such that $x \unrhd y$, $\dim \mathcal{W}_x > \dim \mathcal{W}_y$
and $q \in \varphi_{x, y}(\mathcal{V}_{x, y})$.
For each $x \in P$, we take finite orbibundle charts $(\mathcal{V}_\tau, \mathcal{E}_\tau)_{\tau \in T_x}$
of $(\mathcal{V}_x, \mathcal{E}_x)$ and smooth functions $\beta_\tau : \mathcal{V}_\tau \to \mathbb{R}$
with compact support such that $\beta_x = \sum_{\tau \in T_x} \beta_\tau$.
Then for each differential form $\theta \in \Omega(Y)$,
$(h_! \circ f^\ast)_X \theta \in \Omega(Z)$ is defined by
\begin{align}
&(h_! \circ f^\ast)_X \theta \notag \\
&= \sum_{x \in P, \tau \in T_x}
\frac{\sum_{(\omega_j) \in \prod_j \Omega_{\tau, j}}
\Bigl(h_x|_{\{s_\tau^{(\omega_j)} = 0\}}\Bigr)_{\textstyle !}\,
\Bigl(\beta_\tau \cdot (f_x|_{V_\tau})^\ast
\theta|_{\{s_\tau^{(\omega_j)} = 0\}}\Bigr)}
{\# G_{V_\tau
\cdot \prod_j \# \Omega_{\tau, j}},
\label{virtual fundamental chain by forms}
\end{align}
where $s_\tau = s_x|_{\mathcal{V}_\tau}$, $\boldsymbol{\epsilon}_x|_{\mathcal{V}_\tau}
= (\epsilon^\omega)_{\omega \in \coprod_j \Omega_{\tau, j}}$,
$s_\tau^{(\omega_j)} = s_\tau + \sum_j \epsilon_\tau^{(\omega_j)}$, and
$(h_x|_{\{s_\tau^{(\omega_j)} = 0\}})_!$ is integration
along fiber for the fibration
$h_x : \{s_\tau^{(\omega_j)} = 0\} \to Z$.
In our convention, the orientation of the fiber $F$ is defined by
$T_{h_x(p)}Z \oplus T_pF = T_p\{s_\tau^{(\omega_j)} = 0\}$ at each point $p \in F$.
It is easy to check that $(h_! \circ f^\ast)_X \theta$ is independent of the choice of
the partition of unity $(\beta_x)_{x \in P}$ and functions $\beta_\tau$.
If $Z$ is non-orientable, then instead of a compatible family of orientations
$\Phi_x : \det TW_x \cong \det E_x$, we assume that a compatible family of
isomorphisms $\widetilde{\Phi}_x : \det TW_x \cong
h_x^\ast \mathcal{O}_Z \otimes \det E_x$
is given, where $\mathcal{O}_Z = \det TZ$ is the orientation bundle of $Z$.
Then we can define the orientation of the fiber of
each $h_x : \{s_\tau^{(\omega_j)} = 0\} \to Z$ and define
$(h_! \circ f^\ast)_X \theta : \Omega(Y) \to \Omega(Z)$ similarly.
\subsection{Compatible perturbed multisections}
\label{compatible perturbed multisection}
In application, we need to construct perturbed multisections of moduli spaces
which respect their algebraic properties.
In this section, we consider the compatibility of the perturbed multisections
of various pre-Kuranishi spaces.
\subsubsection{The boundary of a pre-Kuranishi space}
First, we consider compatibility of the grouped multisection of a pre-Kuranishi space
and the grouped multisection of its boundary.
\begin{defi}
For a pre-Kuranishi space $X$ with corners, we define the boundary
$\partial X \subset X$ by the set of points $p \in X$ such that
for any $x \in \mu^{-1}(p)$, $\widetilde{\psi}_x^{-1}(x)$ is contained in the boundary of
$\mathcal{W}_x$.
(This condition is independent of the choice of $x \in \mu^{-1}(p)$.)
The restriction of the pre-Kuranishi structure of $X$ defines
the pre-Kuranishi structure of $\partial X$.
\end{defi}
For a weakly good coordinate system $(x, \mathcal{V}_x)_{x \in P}$ of $X$,
$(x, \partial \mathcal{V}_x)_{x \in P \cap \mu^{-1}(\partial X)}$ is a weakly good coordinate
system of $\partial X$.
(We note that $\partial \mathcal{V}_x = \emptyset$ for $x \in P \setminus \mu^{-1}(\partial X)$
because $\partial \mathcal{W}_x = \emptyset$ by assumption.)
Conversely, for a weakly good coordinate system
$(x, \mathcal{V}^{\partial X}_x)_{x \in P^{\partial X}}$ of $\partial X$,
we can construct a weakly good coordinate system
$(x, \mathcal{V}_x)_{x \in P}$ of $X$ such that
$(x, \partial \mathcal{V}_x)_{x \in P \cap \mu^{-1}(\partial X)}
= (x, \mathcal{V}^{\partial X}_x)_{x \in P^{\partial X}}$
by the following lemma
and Lemma \ref{weakly good coordinate from meet-semilattice}.
\begin{lem}
\label{extension of meet-semilattice cover}
For any meet-semilattice cover $\mathcal{Y}^{\partial X}$ of $\partial X$,
there exists a meet-semilattice cover $\mathcal{Y}$ of $X$ such that
$\mathcal{Y} \cap \mu^{-1}(\partial X) = \mathcal{Y}^{\partial X}$.
\end{lem}
\begin{proof}
First we construct open neighborhood $\mathcal{Y}^{N(\partial X)}$ of
$\mathcal{Y}^{\partial X} \subset \widetilde{X}$ such that
$\mathcal{Y}^{N(\partial X)} \cap \partial \widetilde{X} = \mathcal{Y}^{\partial X}$ and
$\mathcal{Y}^{N(\partial X)} \subset \widetilde{X}$ satisfies the conditions of
meet-semilattice cover other than the covering condition $\mu(\mathcal{Y}) = X$
as follows.
Let $\mathcal{Y}^{N(\partial X)}_{(0)} \subset \widetilde{X}$ be an open neighborhood of
$\mathcal{Y}^{\partial X}$ such that $\mathcal{Y}^{N(\partial X)}_{(0)} \cap \partial \widetilde{X}
= \mathcal{Y}^{\partial X}$.
We construct a decreasing sequence of open neighborhoods
$\mathcal{Y}^{N(\partial X)}_{(k)} \subset \mathcal{Y}^{N(\partial X)}_{(0)}$ ($k \geq 0$)
of $\mathcal{Y}^{\partial X}$ which satisfies the following conditions,
where $\mathcal{Y}^{N(\partial X)}_{(k), l} = \{x \in \mathcal{Y}^{N(\partial X)}_{(k)}; \dim \mathcal{W}_x = l\}$
for each $l \geq 0$.
\begin{enumerate}[label=(\roman*)]
\item
$\mathcal{Y}^{N(\partial X)}_{(k + 1)} \subset \mathcal{Y}^{N(\partial X)}_{(k)}$ for all $k \geq 0$.
\item
$\mathcal{Y}^{N(\partial X)}_{(k), l} = \mathcal{Y}^{N(\partial X)}_{(l), l}$ for $k \geq l$.
\item
\label{clean intersection for k and (k-1)}
For any $k \geq 1$, $m \geq 2$ and
$y_1, \dots, y_m \in \mathcal{Y}^{N(\partial X)}_{(k)}$ such that $\mu(y_i) = \mu(y_1)$
for all $1 \leq i \leq m$, define $x = y_1 \vee \dots \vee y_m \in \widetilde{X}$.
Then $(\varphi_{x, y_i}(\mathcal{W}_{x, y_i}))_{1 \leq i \leq m}$ intersect cleanly on
a neighborhood of $\widetilde{\psi}_x^{-1}(x) \subset \mathcal{W}_x$.
Furthermore, there exists some $w \in \mathcal{Y}^{N(\partial X)}_{(k-1)}$
such that $\mu(w) = \mu(y_1)$, $w \leq y_i$ for all $1 \leq i \leq m$,
and the intersection of $(\varphi_{x, y_i}(\mathcal{W}_{x, y_i}))_{1 \leq i \leq m}$
is contained in the image of $\varphi_{x, w}$ on a neighborhood of
$\widetilde{\psi}_x^{-1}(x) \subset \mathcal{W}_x$.
\end{enumerate}
Then $\mathcal{Y}^{N(\partial X)} = \bigcup_{l \geq 0} \mathcal{Y}^{N(\partial X)}_{(l), l}
\subset \widetilde{X}$ satisfies the required conditions.
We construct $\mathcal{Y}^{N(\partial X)}_{(k)} \subset \mathcal{Y}^{N(\partial X)}_{(0)}$
by the induction in $k \geq 1$.
Assume that we have already constructed $\mathcal{Y}^{N(\partial X)}_{(k)}$ for $k < k_0$.
Define $\widehat{\mathcal{Y}} = \mathcal{Y}^{N(\partial X)}_{(k_0-1)}$.
Let $\{\mathcal{U}_j\}_j$ be a finite open covering of $X$ such that
each $\widehat{\mathcal{Y}}|_{\overline{\mathcal{U}_j}}$ can be decomposed into its
finite disjoint open subsets
$\coprod_a \widehat{\mathcal{Y}}^{j, a}$ such that
each $\mu|_{\widehat{\mathcal{Y}}^{j, a}}$ is injective.
We say $p \in X$ is a good point if
Condition \ref{clean intersection for k and (k-1)} is satisfied for
$k = k_0$, $\mathcal{Y}^{N(\partial X)}_{(k_0)} = \widehat{\mathcal{Y}}$ and any points
$y_1, \dots, y_m \in \widehat{\mathcal{Y}} \cap \mu^{-1}(p)$.
Note that every point in $\partial X$ is good.
We shrink $\widehat{\mathcal{Y}}$ on each $\overline{\mathcal{U}_j}$ by the induction in $j$
to make every point in $\overline{\mathcal{U}_j}$ a good point by the following argument.
We shrink $\widehat{\mathcal{Y}}|_{\overline{\mathcal{U}_j}}$ by the induction in $m_0 \geq 2$
so that Condition \ref{clean intersection for k and (k-1)} is satisfied
for $k = k_0$, $2 \leq m \leq m_0$ and
any points $y_1, \dots, y_m \in \widehat{\mathcal{Y}}|_{\overline{\mathcal{U}_j}}$
such that $\mu(y_i) = \mu(y_1)$ as follows.
Assume that Condition \ref{clean intersection for k and (k-1)} is satisfied
for $k = k_0$, $2 \leq m \leq m_0 - 1$ and
any points $y_1, \dots, y_m \in \widehat{\mathcal{Y}}|_{\overline{\mathcal{U}_j}}$.
For each sequence $a_1, \dots, a_{m_0}$,
let $C_{a_1, \dots, a_{m_0}} \subset \bigcap_{1 \leq i \leq m_0} \mu(\widehat{\mathcal{Y}}^{j, a})$
be the set of points $p \in \bigcap_{1 \leq i \leq m_0} \mu(\widehat{\mathcal{Y}}^{j, a_i})$ such
that Condition \ref{clean intersection for k and (k-1)} does not hold for
$k = k_0$, $m = m_0$ and
$y_i = (\mu|_{\widehat{\mathcal{Y}}^{j, a_i}})^{-1}(p)$ ($1 \leq i \leq m_0$).
Then $C_{a_1, \dots, a_{m_0}}$ is closed in the relative topology of
$\bigcap_{1 \leq i \leq m_0} \mu(\widehat{\mathcal{Y}}^{j, a})$.
(Its complement is open.)
Furthermore, $C_{a_1, \dots, a_{m_0}}$ does not contain any good point.
Fix a distance of $X$ and
define $(\widehat{\mathcal{Y}}^{j, a_i})^\ast \subset \widehat{\mathcal{Y}}^{j, a_i}$ by the interior of
\begin{align}
\bigl(\widehat{\mathcal{Y}}^{j, a_i} \setminus \mu^{-1}(C_{a_1, \dots, a_{m_0}})\bigr)
\cup \{x \in \widehat{\mathcal{Y}}^{j, a_i}&; \mathop{\mathrm{dist}}\nolimits(\mu(x), \mu(\widehat{\mathcal{Y}}^{j, a_i}) \cap \partial X)
\notag \\
& < \max_{i' \neq i} \mathop{\mathrm{dist}}\nolimits(\mu(x), \mu(\widehat{\mathcal{Y}}^{j, a_{i'}}) \cap \partial X)\}.
\label{in order to shrink Y}
\end{align}
(We read $\mathop{\mathrm{dist}}\nolimits(\mu(x), \emptyset)$ as $\mathop{\mathrm{dist}}\nolimits(\mu(x), \emptyset) = \infty$.)
We claim that
$(\widehat{\mathcal{Y}}^{j, a_i})^\ast$ contains $\widehat{\mathcal{Y}}^{j, a_i} \cap \partial \widetilde{X}$.
This is proved as follows.
For any point $x \in \widehat{\mathcal{Y}}^{j, a_i} \cap \partial \widetilde{X}$,
if $x \in \bigcap_{i'} \widehat{\mathcal{Y}}^{j, a_{i'}} \cap \partial \widetilde{X}$, then
$x$ is contained in the open subset $\bigcap_{i'} \widehat{\mathcal{Y}}^{j, a_{i'}} \setminus
\mu^{-1}(C_{a_1, \dots, a_{m_0}})$,
which is contained in (\ref{in order to shrink Y}).
If $x \notin \widehat{\mathcal{Y}}^{j, a_{i'}}$ for some $i' \neq i$,
then $x$ is contained in the open subset
\[
\{x \in \widehat{\mathcal{Y}}^{j, a_i}; \mathop{\mathrm{dist}}\nolimits(\mu(x), \mu(\widehat{\mathcal{Y}}^{j, a_i}) \cap \partial X)
< \max_{i' \neq i} \mathop{\mathrm{dist}}\nolimits(\mu(x), \mu(\widehat{\mathcal{Y}}^{j, a_{i'}}) \cap \partial X)\},
\]
which is contained in (\ref{in order to shrink Y}).
Hence
$(\widehat{\mathcal{Y}}^{j, a_i})^\ast$ contains $\widehat{\mathcal{Y}}^{j, a_i} \cap \partial \widetilde{X}$,
and we may replace $\widehat{\mathcal{Y}}^{j, a_i}$ with
$(\widehat{\mathcal{Y}}^{j, a_i})^\ast$.
Then $C_{a_1, \dots, a_{m_0}}$ becomes the empty set.
We apply the same argument for all sequences $a_1, \dots, a_{m_0}$.
Then Condition \ref{clean intersection for k and (k-1)} is satisfied
for $k = k_0$, $2 \leq m \leq m_0$ and
any points $y_1, \dots, y_m \in \widehat{\mathcal{Y}}|_{\overline{\mathcal{U}_j}}$
such that $\mu(y_i) = \mu(y_1)$.
Hence the induction in $m_0$ works and
we can shrink $\widehat{\mathcal{Y}}|_{\overline{\mathcal{U}_j}}$ so that
Condition \ref{clean intersection for k and (k-1)} is satisfied
for $k = k_0$, $m \geq 2$ and
any points $y_1, \dots, y_m \in \widehat{\mathcal{Y}}|_{\overline{\mathcal{U}_j}}$
such that $\mu(y_i) = \mu(y_1)$.
Therefore, the induction in $j$ also works, and
we obtain an open neighborhood
$\widehat{\mathcal{Y}} \subset \mathcal{Y}^{N(\partial X)}_{(k_0-1)}$ of $\mathcal{Y}^{\partial X}$
such that Condition \ref{clean intersection for k and (k-1)} holds for
$k = k_0$, $\mathcal{Y}^{N(\partial X)}_{(k_0)} = \widehat{\mathcal{Y}}$ and any points
$y_1, \dots, y_m \in \widehat{\mathcal{Y}}$ such that $\mu(y_i) = \mu(y_1)$.
Then we define $\mathcal{Y}^{N(\partial X)}_{(k_0)} = \widehat{\mathcal{Y}}$.
Note that if $\mathcal{Y}^{N(\partial X)}_{(k), l} = \mathcal{Y}^{N(\partial X)}_{(l), l}$ for
$l \leq k \leq k_0 -1$, then it also holds for $l \leq k = k_0$
by the above construction.
Hence we can construct a decreasing sequence of open neighborhoods
$\mathcal{Y}^{N(\partial X)}_{(k)} \subset \mathcal{Y}^{N(\partial X)}_{(0)}$ ($k \geq 0$)
of $\mathcal{Y}^{\partial X}$ inductively, and
$\mathcal{Y}^{N(\partial X)} = \bigcup_{l \geq 0} \mathcal{Y}^{N(\partial X)}_{(l), l}
\subset \widetilde{X}$ satisfies the conditions of
meet-semilattice cover other than the covering condition $\mu(\mathcal{Y}) = X$.
Take an open subset $\mathring{X} \subset X$ such that
$\mathring{X} \cup \mu(\mathcal{Y}^{N(\partial X)}) =X$ and
$\overline{\mathring{X}} \cap \partial X = \emptyset$.
Let $N_0(\partial X)$ be an open neighborhood of $\partial X \subset X$ such that
$N_0(\partial X) \cap \mathring{X} = \emptyset$.
We construct a totally ordered cover $\mathring{Y}$ of $X$ such that
$\mathring{Y} \subset \mu^{-1}(\mathring{X}) \cup \mathcal{Y}^{N(\partial X)}$.
Then $\mathcal{Y} = \mathring{Y} \cup (\mathcal{Y}^{N(\partial X)} \cap \mu^{-1}(N_0(\partial X)))$
is a required meet-semilattice cover of $X$.
(Since $\mathring{Y} \cap \mu^{-1}(N_0(\partial X))$ is contained in $\mathcal{Y}^{N(\partial X)}$,
$\mathcal{Y}$ satisfies the conditions of meet-semilattice cover.)
We explain the construction of the totally ordered cover $\mathring{Y}$ of $X$.
Applying Lemma \ref{totally ordered cover} to the map
$\m
: \mathcal{Y}^{N(\partial X)} \cap \mu^{-1}(X \setminus \mathring{X})
\to X \setminus \mathring{X}$ and the opposite partial order
``$\preccurlyeq$''$=$``$\geq$'',
we get a totally ordered cover $\mathcal{Y}^{X \setminus \mathring{X}}$ of
$X \setminus \mathring{X}$ contained in $\mathcal{Y}^{N(\partial X)}$.
Then we can apply Lemma \ref{totally ordered cover} to the map
$\mu
: \mu^{-1}(\mathring{X}) \cup \mathcal{Y}^{X \setminus \mathring{X}}
\to X$
and the partial order $\leq$, and we get a totally ordered cover $\mathring{Y}$ of $X$
such that
$\mathring{Y} \subset \mu^{-1}(\mathring{X}) \cup \mathcal{Y}^{X \setminus \mathring{X}}
\subset \mu^{-1}(\mathring{X}) \cup \mathcal{Y}^{N(\partial X)}$.
Hence we can construct a required meet-semilattice cover.
\end{proof}
By the argument used for the proof of Lemma
\ref{extension of grouped multisection for embedding},
we can extend a grouped multisection of
$(x, \partial \mathcal{V}_x)_{x \in P \cap \mu^{-1}(\partial X)}$ to
a grouped multisection of $(x, \mathcal{V}_x)_{x \in P}$ if we shrink $\mathcal{V}_x$ slightly.
(The only difference of this extension and the extension proved in
Lemma \ref{extension of grouped multisection for embedding}
is whether the rank of the obibundle changes or not,
which has nothing to do with the construction of the extension.)
\subsubsection{Pull back by submersion}
Next we define the pull back of the perturbed multisection
by a submersion from a pre-Kuranishi space to another.
First we define the submersion between pre-Kuranishi spaces.
\begin{defi}
Let $X^k$ ($k = 1,2$) be two pre-Kuranishi spaces with pre-Kuranishi structures
$(\widetilde{X}^k, \mu^k, (\mathcal{W}^k_x, \mathcal{E}^k_x, s^k_x, \widetilde{\psi}^k_x),
(\varphi^k_{x, y}, \hat \varphi^k_{x, y}))$.
A submersion $f = (f, \widetilde{f}, (\varphi^f_x, \hat \varphi^f_x))$
from $X^1$ to $X^2$ consists of
continuous maps $f : X^1 \to X^2$ and $\widetilde{f} : \widetilde{X}^1 \to \widetilde{X}^2$
such that $\widetilde{f} \circ \mu^1 = \mu^2 \circ f$, and
submersions $(\varphi^f_x, \hat \varphi^f_x)$ ($x \in \widetilde{X}^1$) from
$(\mathcal{W}^1_x, \mathcal{E}^1_x)$ to $(\mathcal{W}^2_{\widetilde{f}(x)}, \mathcal{E}^2_{\widetilde{f}(x)})$
which satisfy the following conditions:
\begin{enumerate}[label=$(\arabic*)^{\mathrm{S}}$]
\item
\label{isom of pos for submersion}
For each $p \in X^1$, $\widetilde{f}|_{(\mu^1)^{-1}(p)} : (\mu^1)^{-1}(p) \cong
(\mu^2)^{-1}(f(p))$
is an isomorphism of partially ordered sets.
\item
$s^2_{\widetilde{f}(x)} \circ \varphi^f_x = \hat \varphi^f_x \circ s^1_x$ on $\mathcal{W}^1_x$
and $\widetilde{\psi}^2_{\widetilde{f}(x)} \circ \varphi^f_x
= \widetilde{f} \circ \widetilde{\psi}^1_x$ on $(s^1_x)^{-1}(0)$ for all $x \in \widetilde{X}^1$.
\item
\label{compatibility of submersion with embedding}
For any $x, y \in \widetilde{X}^1$, if $x \unrhd y$, then
$\varphi^f_y(\mathcal{W}^1_{x, y}) \subset \mathcal{W}^2_{\widetilde{f}(x), \widetilde{f}(y)}$ and
\begin{equation}
\label{submersion embedding compatibility equation}
(\varphi^f_x, \hat \varphi^f_x) \circ (\varphi^1_{x, y}, \hat \varphi^1_{x, y})
= (\varphi^2_{\widetilde{f}(x), \widetilde{f}(y)},
\hat \varphi^2_{\widetilde{f}(x), \widetilde{f}(y)})
\circ (\varphi^f_y, \hat \varphi^f_y)
\end{equation}
on $\mathcal{W}^1_{x, y}$.
\end{enumerate}
\end{defi}
\begin{rem}
\label{shrink W_x for submersion}
In the above definition, we assume that each $(\varphi^f_x, \hat \varphi^f_x)$
is defined on the whole of $(\mathcal{W}^1_x, \mathcal{E}^1_x)$.
Hence for the construction of a submersion $f$, we sometimes need
to shrink each $\mathcal{W}^1_x$ as we explained in Remark \ref{we can shrink Wx}.
Similarly, we sometimes need to replace $\widetilde{X}^1$ to
its appropriate open subset.
(See Remark \ref{quotient map is a submersion} for example.)
\end{rem}
\begin{rem}
In Condition \ref{compatibility of submersion with embedding},
we consider the composition of a submersion and an embedding.
In (\ref{submersion embedding compatibility equation}),
the lifts of the right hand side satisfy the same conclusion as that of
Lemma \ref{effective action} as we saw in
Remark \ref{effective action for composition of embedding and submersion}.
Although we cannot apply this lemma to the left hand side a priori,
(\ref{submersion embedding compatibility equation}) implies that
their lifts also satisfy the same property.
\end{rem}
\begin{defi}
\label{compatible weakly good coordinate systems for submersion}
Let $(x_1, \mathcal{V}^1_{x_1})_{x_1 \in P^1}$ and $(x_2, \mathcal{V}^2_{x_2})_{x_2 \in P^2}$ be
weakly good coordinate systems of pre-Kuranishi spaces $X^1$ and $X^2$ respectively.
We say these are compatible with respect to the submersion $f$ if
they satisfy the following conditions:
\begin{enumerate}[label=(\roman*)]
\item
\label{meet-semilattice cover compatibility}
The meet-semilattice covers $\mathcal{Y}^i = \bigcup_{x_i \in P^i}
\widetilde{\psi}^i_{x_i}((s^i_{x_i})^{-1}(0) \cap \mathcal{V}^i_{x_i}))$ ($i = 1,2$) satisfy
$\widetilde{f}(\mathcal{Y}^1) \subset \mathcal{Y}^2$.
\item
\label{V_{x_1} subset V_{x_2}}
For any $x_1 \in P^1$ and $x_2 \in P^2$,
if $\widetilde{f}(x_1) \in \widetilde{\psi}^2_{x_2}((s^2_{x_2})^{-1}(0) \cap \mathcal{V}^2_{x_2}))$,
then $\varphi^f_{x_1}(\mathcal{V}^1_{x_1}) \subset
(\varphi^2_{x_2, \widetilde{f}(x_1)})^{-1}(\mathcal{V}^2_{x_2})$.
\item
\label{composition compatibility for submersion}
For any $y_1, x_1 \in P^1$ and $y_2, x_2 \in P^2$,
if $\widetilde{f}(x_1) \in \widetilde{\psi}^2_{x_2}((s^2_{x_2})^{-1}(0) \cap \mathcal{V}^2_{x_2}))$,
$\widetilde{f}(y_1) \in \widetilde{\psi}^2_{y_2}((s^2_{y_2})^{-1}(0) \cap \mathcal{V}^2_{y_2}))$, and
there exists some $p \in \psi_{x_1}((s^1_{x_1})^{-1}(0) \cap \mathcal{V}^1_{x_1}) \cap
\psi_{y_1}((s^1_{y_1})^{-1}(0) \cap \mathcal{V}^1_{y_1})$ such that $p_{x_1} \geq p_{y_1}$,
then
\[
\varphi_{y_1}^f(\mathcal{V}^1_{x_1, y_1}) \subset \mathcal{W}^2_{x_2, \widetilde{f}(x_1), \widetilde{f}(y_1)}
\cap \mathcal{W}^2_{x_2, y_2, \widetilde{f}(y_1)}.
\]
(Note that the previous condition implies that
$\widetilde{f}(p_{x_1}) \in \widetilde{\psi}^2_{x_2}((s^2_{x_2})^{-1}(0) \cap \mathcal{V}^2_{x_2}))$,
$\widetilde{f}(p_{y_1}) \in \widetilde{\psi}^2_{y_2}((s^2_{y_2})^{-1}(0) \cap \mathcal{V}^2_{y_2}))$ and
$\widetilde{f}(p_{x_1}) \geq \widetilde{f}(p_{y_1})$.)
\end{enumerate}
\end{defi}
If $(x_1, \mathcal{V}^1_{x_1})_{x_1 \in P^1}$ and $(x_2, \mathcal{V}^2_{x_2})_{x_2 \in P^2}$ are
compatible with respect to the submersion $f$,
then for a grouped multisection $\boldsymbol{\epsilon}^2
= (\boldsymbol{\epsilon}^2_{x_2})_{x_2 \in P^2}$ of $(x_2, \mathcal{V}^2_{x_2})_{x_2 \in P^2}$,
we can define a grouped multisection $\boldsymbol{\epsilon}^1
= (\boldsymbol{\epsilon}^1_{x_1})_{x_1 \in P^1}$ of $(x_1, \mathcal{V}^1_{x_1})_{x_1 \in P^1}$
as follows.
For each $x_1 \in P^1$, Condition \ref{meet-semilattice cover compatibility}
implies that there exists some $x_2 \in P^2$ such that
$\widetilde{f}(x_1) \in \widetilde{\psi}^2_{x_2}((s^2_{x_2})^{-1}(0) \cap \mathcal{V}^2_{x_2}))$, and
Condition \ref{V_{x_1} subset V_{x_2}} implies that
$\varphi^f_{x_1}(\mathcal{V}^1_{x_1}) \subset
(\varphi^2_{x_2, \widetilde{f}(x_1)})^{-1}(\mathcal{V}^2_{x_2})$.
We define $\boldsymbol{\epsilon}^1_{x_1}$ by the pull back of
$\boldsymbol{\epsilon}^2_{x_2}$ by
$(\varphi^2_{x_2, \widetilde{f}(x_1)}, \hat \varphi^2_{x_2, \widetilde{f}(x_1)}) \circ
(\varphi^f_{x_1}, \hat \varphi^f_{x_1})$.
We need to check that $\boldsymbol{\epsilon}^1
= (\boldsymbol{\epsilon}^1_{x_1})_{x_1 \in P^1}$ satisfies
$(\varphi^1_{x_1, y_1}, \hat \varphi^1_{x_1, y_1})$-relations.
It is enough to prove the following claim:
If $y_1, x_1 \in P^1$ and $y_2, x_2 \in P^2$ satisfy
$\widetilde{f}(x_1) \in \widetilde{\psi}^2_{x_2}((s^2_{x_2})^{-1}(0) \cap \mathcal{V}^2_{x_2}))$,
$\widetilde{f}(y_1) \in \widetilde{\psi}^2_{y_2}((s^2_{y_2})^{-1}(0) \cap \mathcal{V}^2_{y_2}))$, and
there exists some $p \in \psi_{x_1}((s^1_{x_1})^{-1}(0) \cap \mathcal{V}^1_{x_1}) \cap
\psi_{y_1}((s^1_{y_1})^{-1}(0) \cap \mathcal{V}^1_{y_1})$ such that $p_{x_1} \geq p_{y_1}$,
then the pull backs
$(\varphi^2_{x_2, \widetilde{f}(x_1)} \circ \varphi^f_{x_1})^\ast
\boldsymbol{\epsilon}^2_{x_2}$ and
$(\varphi^2_{y_2, \widetilde{f}(y_1)} \circ \varphi^f_{y_1})^\ast
\boldsymbol{\epsilon}^2_{y_2}|_{\mathcal{V}^1_{x_1, y_1}}$ are
$(\varphi_{x_1, y_1}, \hat \varphi_{x_1, y_1})$-related.
Note that in particular, the case where $y_1 = x_1$ and $y_2 \neq x_2$ implies that
the definition of $\boldsymbol{\epsilon}^1_{x_1}$ does not depend on the
choice of $x_2$.
We can prove this claim as follows.
Condition \ref{compatibility of submersion with embedding} and
\ref{composition compatibility for submersion} imply that
\begin{align*}
&(\varphi^2_{x_2, \widetilde{f}(x_1)}, \hat \varphi^2_{x_2, \widetilde{f}(x_1)}) \circ
(\varphi^f_{x_1}, \hat \varphi^f_{x_1}) \circ
(\varphi^1_{x_1, y_1}, \hat \varphi^1_{x_1, y_1}) \\
&= (\varphi^2_{x_2, \widetilde{f}(x_1)}, \hat \varphi^2_{x_2, \widetilde{f}(x_1)}) \circ
(\varphi^2_{\widetilde{f}(x_1), \widetilde{f}(y_1)},
\hat \varphi^2_{\widetilde{f}(x_1), \widetilde{f}(y_1)}) \circ
(\varphi^f_{y_1}, \hat \varphi^f_{y_1}) \\
&= (\varphi^2_{x_2, \widetilde{f}(y_1)}, \hat \varphi^2_{x_2, \widetilde{f}(y_1)}) \circ
(\varphi^f_{y_1}, \hat \varphi^f_{y_1}) \\
&= (\varphi^2_{x_2, y_2}, \hat \varphi^2_{x_2, y_2}) \circ
(\varphi^2_{y_2, \widetilde{f}(y_1)}, \hat \varphi^2_{y_2, \widetilde{f}(y_1)})
\circ (\varphi^f_{y_1}, \hat \varphi^f_{y_1})
\end{align*}
on $(\mathcal{V}^1_{x_1, y_1}, \mathcal{E}^1_{y_1}|_{\mathcal{V}^1_{x_1, y_1}})$.
This and $(\varphi^2_{x_2, y_2}, \hat \varphi^2_{x_2, y_2})$-relation of
$\boldsymbol{\epsilon}^2_{x_2}$ and $\boldsymbol{\epsilon}^2_{y_2}$ imply
that $(\varphi^2_{x_2, \widetilde{f}(x_1)} \circ \varphi^f_{x_1})^\ast
\boldsymbol{\epsilon}^2_{x_2}$ and
$(\varphi^2_{y_2, \widetilde{f}(y_1)} \circ \varphi^f_{y_1})^\ast
\boldsymbol{\epsilon}^2_{y_2}|_{\mathcal{V}^1_{x_1, y_1}}$ are
$(\varphi_{x_1, y_1}, \hat \varphi_{x_1, y_1})$-related.
Therefore $\boldsymbol{\epsilon}^1
= (\boldsymbol{\epsilon}^1_{x_1})_{x_1 \in P^1}$ is a well-defined grouped multisection
of $(x_1, \mathcal{V}^1_{x_1})_{x_1 \in P^1}$.
We call this grouped multisection the pull back of $\boldsymbol{\epsilon}^2$ by the
submersion $f$, and denote it by $f^\ast \boldsymbol{\epsilon}^2$.
Next we claim that for a meet-semilattice cover $\mathcal{Y}^2 \subset \widetilde{X}^2$
of $X^2$, $\mathcal{Y}^1 = \widetilde{f}^{-1}(\mathcal{Y}^2)$ is a meet-semilattice cover of $X^1$.
To see this, first we note the following fact.
For any $x, y \in \widetilde{X}^1$ such that $x \unrhd y$,
\begin{equation}
\dim \mathcal{W}_x^1- \dim \mathcal{W}_y^1 = \dim \mathcal{W}_{\widetilde{f}(x)}^2 - \dim \mathcal{W}_{\widetilde{f}(y)}^2.
\label{dimension relation for inverse cover}
\end{equation}
This is because $\mathop{\mathrm{rank}}\nolimits \mathcal{E}_x^1 = \mathop{\mathrm{rank}}\nolimits \mathcal{E}_{\widetilde{f}(x)}^2$,
$\mathop{\mathrm{rank}}\nolimits \mathcal{E}_y^1 = \mathop{\mathrm{rank}}\nolimits \mathcal{E}_{\widetilde{f}(y)}^2$,
$\dim \mathcal{W}_x^1 - \mathop{\mathrm{rank}}\nolimits \mathcal{E}_x^1 = \dim \mathcal{W}_y - \mathop{\mathrm{rank}}\nolimits \mathcal{E}_y^1$ and
$\dim \mathcal{W}_{\widetilde{f}(x)}^2 - \mathop{\mathrm{rank}}\nolimits \mathcal{E}_{\widetilde{f}(x)}^2
= \dim \mathcal{W}_{\widetilde{f}(y)}^2 - \mathop{\mathrm{rank}}\nolimits \mathcal{E}_{\widetilde{f}(y)}^2$.
Equation (\ref{dimension relation for inverse cover}) and
submersiveness of $\varphi^f_x$ imply that
\begin{align}
(\phi^f_x)_\ast &: T_{\phi^1_{x, y}(z)} W^1_x / (\phi^1_{x, y})_\ast T_z W^1_{x, y} \notag \\
& \quad
\stackrel{\cong}{\to} T_{\phi^2_{x, y}(\phi^f_y(z))} W^2_{\widetilde{f}(x)}
/ (\phi^2_{\widetilde{f}(x), \widetilde{f}(y)})_\ast T_{\phi^f_y(z)}
W^2_{\widetilde{f}(x), \widetilde{f}(y)}
\label{transversality for embeddings}
\end{align}
for any $z \in \mathcal{W}^1_{x, y}$, where
$\phi^f_x$, $\phi^f_y$, $\phi^1_{x, y}$ and $\phi^2_{\widetilde{f}(x), \widetilde{f}(y)}$
are lifts of $\varphi^f_x$, $\varphi^f_y$, $\varphi^1_{x, y}$ and
$\varphi^2_{\widetilde{f}(x), \widetilde{f}(y)}$ respectively for
appropriate orbibundle charts of neighborhoods of $z$ and its images
such that $\phi^f_x \circ \phi^1_{x, y} = \phi^2_{\widetilde{f}(x), \widetilde{f}(y)}
\circ \phi^f_y$.
This implies that
for any point $p \in \psi^1_x((s^1_x)^{-1}(0)) \cap \psi^1_y((s^1_y)^{-1}(0))$,
$\varphi^1_{x, y}(\mathcal{W}^1_{x, y})$ coincides with
$(\varphi^f_x)^{-1}(\varphi^2_{\widetilde{f}(x), \widetilde{f}(y)}
(\mathcal{W}^2_{\widetilde{f}(x), \widetilde{f}(y)}))$ on a neighborhood of
$(\psi^1_x)^{-1}(p) \in \mathcal{W}^1_x$.
Therefore, Condition \ref{clean intersection for meet-semilattice cover} for
$\mathcal{Y}^1 = \widetilde{f}^{-1}(\mathcal{Y}^2)$ follows from that for $\mathcal{Y}^2$.
Next we check Condition \ref{wedge existence for meet-semilattice cover} for $\mathcal{Y}^1$.
For any $p \in X^1$ and any two points $y, z \in \mathcal{Y}^1 \cap (\mu^1)^{-1}(p)$,
Condition \ref{isom of pos for submersion} implies that there exists a unique
$w \in \mathcal{Y}^1 \cap (\mu^1)^{-1}(p)$ such that
$\widetilde{f}(w) = \widetilde{f}(y) \wedge \widetilde{f}(z)$.
Then $w$ satisfies the condition of $y \wedge z$.
Hence $\mathcal{Y}^1$ is a meet-semilattice cover of $X^1$.
For a weakly good coordinate system $(x_2, \mathcal{V}^2_{x_2})_{x_2 \in P^2}$ of $X^2$,
define
$\mathcal{Y}^2 = \bigcup_{x_2 \in P^2} \widetilde{\psi}^2_{x_2}(\mathcal{V}^2_{x_2} \cap (s^2_{x_2})^{-1}(0))$.
Then the above argument implies that $\mathcal{Y}^1 = \widetilde{f}^{-1}(\mathcal{Y}^2)$ is
a meet-semilattice cover of $X^1$.
Therefore Lemma \ref{weakly good coordinate from meet-semilattice} implies that
for any compact subset $K \subset \mathcal{Y}^1$, we can construct a weakly good coordinate
system $(x_1, \mathcal{V}^1_{x_1})_{x_1 \in P^1}$ of $X^1$ which is compatible with
$(x_2, \mathcal{V}^2_{x_2})_{x_2 \in P^2}$ and $K \subset
\bigcup_{x_1 \in P^1} \widetilde{\psi}^1_{x_1}(\mathcal{V}^1_{x_1} \cap
(s^1_{x_1})^{-1}(0)) \subset \mathcal{Y}^1$.
(Condition \ref{V_{x_1} subset V_{x_2}} holds if each $\mathcal{V}_{x_1}^1$ is sufficiently small, and
Condition \ref{composition compatibility for submersion} holds
if each $\mathcal{V}_{x_1}^1$ is a sufficiently small neighborhood of
$(s_{x_1}^1)^{-1}(0) \cap \mathcal{V}_{x_1}^1$.)
Then for a grouped multisection $\boldsymbol{\epsilon}^2$ of
$(x_2, \mathcal{V}^2_{x_2})_{x_2 \in P^2}$,
we can define its pull back $f^\ast \boldsymbol{\epsilon}^2$
as a grouped multisection of $(x_1, \mathcal{V}^1_{x_1})_{x_1 \in P^1}$.
\begin{rem}
In the above argument, we can replace submersion with essential submersion
because the pull back of a grouped multisection by an essential submersion
also satisfies the transversality condition in
Lemma \ref{construction of grouped multisection}.
Note that even for an essential submersion,
(\ref{transversality for embeddings}) is an isomorphism
since
\[
T_{\phi^1_{x, y}(z)} W^1_x / (\phi^1_{x, y})_\ast T_z W^1_{x, y}
\cong T_{\phi^1_{x, y}(z)} \mathring{\partial}^k W^1_x
/ (\phi^1_{x, y})_\ast T_z \mathring{\partial}^k W^1_{x, y}
\]
and
\begin{align*}
&T_{\phi^2_{x, y}(w)} W^2_{\widetilde{f}(x)}
/ (\phi^2_{\widetilde{f}(x), \widetilde{f}(y)})_\ast T_w W^2_{\widetilde{f}(x), \widetilde{f}(y)} \\
&\cong T_{\phi^2_{x, y}(w)} \mathring{\partial}^{k'} W^2_{\widetilde{f}(x)}
/ (\phi^2_{\widetilde{f}(x), \widetilde{f}(y)})_\ast T_w \mathring{\partial}^{k'}
W^2_{\widetilde{f}(x), \widetilde{f}(y)}
\end{align*}
for any $z \in \mathring{\partial}^k \mathcal{W}^1_{x, y}$ and
$w \in \mathring{\partial}^{k'} \mathcal{W}^2_{\widetilde{f}(x), \widetilde{f}(y)}$.
(The case of of an essential submersion between fiber products with simplicial complexes
is also similar.
See the next section for the definition of the fiber product of a pre-Kuranishi space
with a simplicial complex.)
\end{rem}
\subsubsection{Product of pre-Kuranishi spaces}
\label{Product of pre-Kuranishi spaces}
Next we define the product of pre-Kuranishi spaces.
The definition of the product of different pre-Kuranishi spaces,
the product of their weakly good coordinate systems and
the product of their grouped multisections are straightforward.
In application, we need to consider the quotient of the product of the same
pre-Kuranishi spaces by the permutation.
In this case, we need to get rid of the products of Kuranishi neighborhoods
which is not compatible with the permutation action.
\begin{defi}
Let $(\widetilde{X}, \mu, (\mathcal{W}_x, \mathcal{E}_x, s_x, \widetilde{\psi}_x),
(\varphi_{x, y}, \hat \varphi_{x, y}))$
be the pre-Kuranishi structure of a compact Hausdorff space $X$.
We assume that $\dim \mathcal{W}_x > 0$ for each $x \in \widetilde{X}$.
Then for each $N \geq 2$, we say a pre-Kuranishi structure
$(\widetilde{X}^{\# N}, \mu^{\# N},
(\mathcal{W}^{\# N}_x, \mathcal{E}^{\# N}_x, s^{\# N}_x, \widetilde{\psi}^{\# N}_x),
(\varphi^{\# N}_{x, y}, \hat \varphi^{\# N}_{x, y}))$ of
$(\prod^N X) / \mathfrak{S}_N$ is compatible
with that of $X$ if the following conditions hold:
\begin{itemize}
\item
$\widetilde{X}^{\# N}$ is an open subset of $\bigl(\prod^N \widetilde{X}\bigr)
/ \mathfrak{S}_N$ defined by
\[
\widetilde{X}^{\# N} = \{(x_i)_{1 \leq i \leq N} \in \bigl(\prod^N \widetilde{X}\bigr)
/ \mathfrak{S}_N; x_i = x_j \text{ if } \mu(x_i) = \mu(x_j) \}.
\]
\item
$\mu^{\# N}$ is the restriction of the product of $\mu$ to $\widetilde{X}^{\# N}$.
\item
For any two elements $x = (x_i), y = (y_i) \in (\mu^{\# N})^{-1}(p)$ in a fiber,
we reorder the sequences so that $\mu(x_i) = \mu(y_i)$ for all $i$.
Then $y \leq x$ if and only if $y_i \leq x_i$ for all $i$.\item
For each $x = (x_i) \in \widetilde{X}^{\# N}$, the following hold.
\begin{itemize}
\item
$\mathcal{W}^{\# N}_x$ is a neighborhood of $x$ in
$\prod_i \mathcal{W}_{x_i} / (\mathfrak{S}_N)_{(x_i)}$,
where $(\mathfrak{S}_N)_{(x_i)} \subset \mathfrak{S}_N$ is the stabilizer of $(x_i)$.
\item
$\mathcal{E}^{\# N}_x$, $s^{\# N}_x$ and $\widetilde{\psi}^{\# N}_x$ are the restriction of the
product of $\mathcal{E}_{x_i}$, $s_{x_i}$ and $\widetilde{\psi}_{x_i}$ respectively to $\mathcal{W}^{\# N}_x$.
\end{itemize}
\item
For any $x = (x_i), y = (y_i) \in \widetilde{X}^{\# N}$ such that $x \unrhd y$,
$(\varphi^{\# N}_{x, y}, \hat \varphi^{\# N}_{x, y})$
are the restrictions of the products of $(\varphi_{x_i, y_i}, \hat \varphi_{x_i, y_i})$
to a neighborhood $\mathcal{W}_{x, y}$ of $\psi_y^{-1}(\psi_x(s_x^{-1}(0)))$.
\end{itemize}
\end{defi}
We note that in the above definition, the action of $(\mathfrak{S}_N)_{(x_i)}$ on
$\prod_i \mathcal{W}_{x_i}$ is effective because of the assumption $\dim \mathcal{W}_{x_i} > 0$.
\begin{defi}
Let $(x, \mathcal{V}_x)_{x \in P}$ and $((x_i), \mathcal{V}^{\# N}_{(x_i)})_{(x_i) \in P^{\# N}}$ be
weakly good coordinate systems of pre-Kuranishi spaces $X$ and
$\prod^N X / \mathfrak{S}_N$ respectively.
We say these are compatible if the following conditions hold.
\begin{enumerate}[label=(\roman*)]
\item
For any $x = (x_i) \in P^{\# N}$, there exist some $\widehat{x}_i \in P$
such that $x_i \in \widetilde{\psi}_{\widehat{x}_i}(s_{x_i}^{-1}(0) \cap \mathcal{V}_{x_i})$ for all $i$.
\item
Let $x = (x_i) \in P^{\# N}$ and $\widehat{x}_i \in P$ be points such that
$x_i \in \widetilde{\psi}_{\widehat{x}_i}(s_{x_i}^{-1}(0) \cap \mathcal{V}_{x_i})$ and
$\widehat{x}_i = \widehat{x}_j$ if $x_i = x_j$.
For such a pair $((x_i), (\widehat{x}_i))$, we impose the condition
$\mathcal{V}^{\# N}_{(x_i)} \subset \prod_i \varphi_{\widehat{x}_i, x_i}^{-1}(\mathcal{V}_{\widehat{x}_i})
/ (\mathfrak{S}_N)_{(x_i)}$.
\item
Let $x = (x_i), y = (y_i) \in P^{\# N}$ and
$\widehat{x}_i, \widehat{y}_i \in P$ be points such that both of $((x_i), (\widehat{x}_i))$
and $((y_i), (\widehat{y}_i))$ are the pairs in the previous condition.
Assume that there exists some
$p = (p_i) \in \psi^{\# N}_x((s_x^{\# N})^{-1}(0) \cap \mathcal{V}^{\# N}_x)
\cap \psi^{\# N}_y((s_y^{\# N})^{-1}(0) \cap \mathcal{V}^{\# N}_y)$ such that $p_x \geq p_y$.
We reorder $p_i$, $x_i$ and $y_i$ so that
\begin{align*}
(p_i) &\in (\prod_i \psi_{x_i})(\pi_x^{-1}((s_x^{\# N})^{-1}(0) \cap \mathcal{V}^{\# N}_x)) \\
& \quad
\cap (\prod_i \psi_{y_i})(\pi_y^{-1}((s_y^{\# N})^{-1}(0) \cap \mathcal{V}^{\# N}_y))
\end{align*}
and $(p_i)_{x_i} \geq (p_i)_{y_i}$,
where $\pi_x : \prod_i \mathcal{W}_{x_i} \to \prod_i \mathcal{W}_{x_i} / (\mathfrak{S}_N)_{(x_i)}$ and
$\pi_y : \prod_i \mathcal{W}_{y_i} \to \prod_i \mathcal{W}_{y_i} / (\mathfrak{S}_N)_{(y_i)}$ are the projections.
Then
\[
\pi_y^{-1}(\mathcal{V}^{\# N}_{x, y}) \subset \prod_i (\mathcal{W}_{\widehat{x}_i, x_i, y_i} \cap
\mathcal{W}_{\widehat{x}_i, \widehat{y}_i, y_i}).
\]
\end{enumerate}
If $(x, \mathcal{V}_x)_{x \in P}$ and $((x_i), \mathcal{V}^{\# N}_{(x_i)})_{(x_i) \in P^{\# N}}$ are
compatible weakly good coordinate systems, then
for a grouped multisection $\boldsymbol{\epsilon} = (\boldsymbol{\epsilon}_x)$ of
$(x, \mathcal{V}_x)_{x \in P}$, we can define a grouped multisection $\boldsymbol{\epsilon}^{\# N}
= (\boldsymbol{\epsilon}_{(x_i)})$ of $((x_i), \mathcal{V}^{\# N}_{(x_i)})_{(x_i) \in P^{\# N}}$
by the restriction of
$\coprod_i \pi_i^\ast(\boldsymbol{\epsilon}_{\widehat{x}_i})$, where
$\widehat{x}_i \in P$ are the points in the above condition, and
$\pi_i^\ast(\boldsymbol{\epsilon}_{\widehat{x}_i})
= (\pi_i^\ast \epsilon^\omega)_{\omega \in \Omega_{x_i, j}}$ is a family of
sections of $\prod_i E_{x_i} \to \prod_i V_{x_i}$ defined by the pull backs of the
sections $(\epsilon^\omega)_{\omega \in \Omega_{x_i, j}}$
by the projection $\pi_i : \prod_{i'} V_{x_{i'}} \to V_{x_i}$.
As in the case of pull back by submersion,
$(\varphi_{x, y}, \hat \varphi_{x, y})$-relations of
$\boldsymbol{\epsilon} = (\boldsymbol{\epsilon}_x)$ and the above conditions
imply $(\varphi_{(x_i), (y_i)}, \hat \varphi_{(x_i), (y_i)})$-relations of
$\boldsymbol{\epsilon}^{\# N} = (\boldsymbol{\epsilon}_{(x_i)})$.
\end{defi}
Note that for a meet-semilattice cover $\mathcal{Y} \subset \widetilde{X}$ of
a pre-Kuranishi space $X$,
$\mathcal{Y}^{\# N} = \prod^N \mathcal{Y} / \mathfrak{S}_N \cap \widetilde{X}^{\# N}$ is
a meet-semilattice cover of $X^N / \mathfrak{S}_N$.
Indeed, for any two points $(x_i), (y_i) \in \mathcal{Y}^{\# N} \cap (\mu^{\# N})^{-1}(p)$,
if we reorder the sequences so that $\mu(x_i) = \mu(y_i)$ for all $i$, then
$(x_i) \wedge (y_i) = (x_i \wedge y_i)$.
For a weakly good coordinate system $(x, \mathcal{V}_x)_{x \in P}$ of $X$,
$\mathcal{Y} = \bigcup_{x \in P} \widetilde{\psi}_x(\mathcal{V}_x \cap s_x^{-1}(0))$ is
a meet-semilattice cover of $X$ by definition.
Hence Lemma \ref{weakly good coordinate from meet-semilattice} implies that
for any compact subset $K \subset \mathcal{Y}^{\# N}$,
we can construct a weakly good coordinate system
$((x_i), \mathcal{V}^{\# N}_{(x_i)})_{(x_i) \in P^{\# N}}$ of $\prod^N X / \mathfrak{S}_N$
which is compatible with $(x, \mathcal{V}_x)_{x \in P}$
and which satisfies
\[
K \subset \bigcup_{(x_i) \in P^{\# N}}
\widetilde{\psi}^{\# N}_{(x_i)}(\mathcal{V}^{\# N}_{(x_i)} \cap (s^{\# N}_{(x_i)})^{-1}(0))
\subset \mathcal{Y}^{\# N}.
\]
Hence a grouped multisection $\boldsymbol{\epsilon} = (\boldsymbol{\epsilon}_x)$ for
$(x, \mathcal{V}_x)_{x \in P}$ defines a grouped multisection $\boldsymbol{\epsilon}^{\# N}
= (\boldsymbol{\epsilon}_{(x_i)})$ for $((x_i), \mathcal{V}^{\# N}_{(x_i)})_{(x_i) \in P^{\# N}}$
as above.
\begin{rem}
\label{quotient of product pre-Kuranishi by subgroup}
For any subgroup $\Gamma \subset \mathfrak{S}_N$, we can
similarly define the compatibility conditions of
a pre-Kuranishi structure of $(\prod^N X) / \Gamma$ with that of $X$.
\end{rem}
\begin{rem}
\label{quotient map is a submersion}
The quotient map
$\prod^N X \to (\prod^N X) / \mathfrak{S}_N$ is a submersion if
we replace $\widetilde{\prod^N X} = \prod^N \widetilde{X}$ with its open subset
$\{(x_i)_{1 \leq i \leq N} \in \prod^N \widetilde{X}; x_i = x_j \text{ if } \mu(x_i) = \mu(x_j) \}$
(and shrink the Kuranishi neighborhoods of $\prod^N X$).
Similarly, for a subgroup $\Gamma \subset \mathfrak{S}$,
we can make the quotient map
$(\prod^N X) / \Gamma \to (\prod^N X) / \mathfrak{S}_N$
a submersion.
\end{rem}
Next we consider fiber product of pre-Kuranishi spaces.
\begin{defi}
\label{def of fiber product of pre-Kuranishi}
Let $f = (f_x)_{x \in \widetilde{X}}$ be a strong continuous map from
a pre-Kuranishi space $X$ to a smooth manifold $Y$ such that
each $f_x : \mathcal{W}_x \to Y$ is a smooth submersion.
Then for a submanifold $Z \subset Y$, the pre-Kuranishi structure
$(\widetilde{X}', \mu', (\mathcal{W}'_x, \mathcal{E}'_x, s'_x, \widetilde{\psi}'_x),
(\varphi'_{x, y}, \hat \varphi'_{x, y}))$ of $f^{-1}(Z) \subset X$
is defined by $\widetilde{X}' = \mu^{-1}(f^{-1}(Z))$, $\mu' = \mu|_{\widetilde{X}'}$,
$\mathcal{W}'_x = f_x^{-1}(Z)$, $\mathcal{E}'_x = \mathcal{E}_x|_{\mathcal{W}'_x}$, $s'_x = s_x|_{\mathcal{W}'_x}$, $\widetilde{\psi}'
= \widetilde{\psi}|_{(s'_x)^{-1}(0)}$,
$\mathcal{W}'_{x, y} = \mathcal{W}'_y \cap\varphi_{x, y}^{-1}(\mathcal{W}'_x)$ and
$(\varphi'_{x, y}, \hat \varphi'_{x, y})
= (\varphi_{x, y}, \hat \varphi_{x, y})|_{\mathcal{W}'_{x, y}}$.
\end{defi}
Similarly to Lemma \ref{extension of meet-semilattice cover},
for a meet-semilattice cover $\mathcal{Y}^Z$ of $f^{-1}(Z)$, we can construct
a meet-semilattice cover $\mathcal{Y}$ of $X$ such that
$\mathcal{Y} \cap \widetilde{f}^{-1}(Z) = \mathcal{Y}^Z$.
Hence for a weakly good coordinate system $(x, \mathcal{V}^Z_x)_{x \in P^Z}$ of $f^{-1}(Z)$,
we can construct a weakly good coordinate system
$(x, \mathcal{V}_x)_{x \in P}$ of $X$ such that $P \cap \widetilde{f}^{-1}(Z) = P^Z$,
$\mathcal{V}_x \cap f_x^{-1}(Z) = \mathcal{V}^Z_x$ for $x \in P^Z$, and
$\mathcal{V}_x \cap f_x^{-1}(Z) = \emptyset$ for $x \in P \setminus P^Z$.
Furthermore, if a grouped multisection of $(x, \mathcal{V}^Z_x)_{x \in P^Z}$ is given, then
we can extend it to a grouped multisection of $(x, \mathcal{V}_x)_{x \in P}$
if we shrink $\mathcal{V}_x$ slightly.
In the above definition of fiber product, $Z$ is a submanifold of a manifold $Y$.
We also consider the case of a simplicial complex in an orbifold.
\begin{defi}\label{def of fiber product of pre-Kuranishi over orbifold}
Let $K \subset \mathcal{Z}$ be an embedded simplicial complex in a smooth orbifold $\mathcal{Z}$.
We assume that for any point $p \in K$, $\mathrm{St}(p, K)$ is contained in an orbichart
$\mathcal{Z}_p = (Z_p, \pi_{Z_p}, \mathcal{Z}_p)$ of $\mathcal{Z}$.
Define $G_p = \mathop{\mathrm{Aut}}\nolimits_{\mathcal{Z}_p} Z_p$.
We assume that there exists a regular $G_p$-complex
$L \subset Z_p$ and an isomorphism $\varphi : L/G_p \cong \mathrm{St}(p, K)$ such that
$\varphi \circ \pi_L = \pi_{Z_p}$ on $L \subset Z_p$,
where $\pi_L : L \to L/G_p$ is the quotient map.
Let $f = (f_x)_{x \in \widetilde{X}}$ be a strong continuous map from
a pre-Kuranishi space $X$ to $\mathcal{Z}$ such that each $f_x : \mathcal{W}_x \to \mathcal{Z}$
is a smooth submersion.
We assume that for each point $x \in \widetilde{X}$, $G_{W_x}$ acts effectively on
$\pi_{W_x}^{-1}(f_x^{-1}(\widetilde{f}(x))) \subset W_x$.
Then we can define the pre-Kuranishi structure
$(\widetilde{X}', \mu', (\mathcal{W}'_x, \mathcal{E}'_x, s'_x, \widetilde{\psi}'_x),
(\varphi'_{x, y}, \hat \varphi'_{x, y}))$ of $f^{-1}(K) \subset X$
similarly as in the case of Definition \ref{def of fiber product of pre-Kuranishi},
whose orbibundle charts are defined as in
Definition \ref{def of fiber product of orbibundle over orbifold}.
\end{defi}
We need to modify various definitions for the fiber product
$f^{-1}(K) \subset X$ with simplicial complex as follows.
A meet-semilattice cover of $f^{-1}(K) \subset X$ is
an open subset $\mathcal{Y} \subset \widetilde{X}$ which satisfies
$\mu(\mathcal{Y}) \supset f^{-1}(K)$ and
the conditions of a meet-semilattice cover of $X$
other than the condition $\mu(\mathcal{Y}) = X$.
A weakly good coordinate system of $f^{-1}(K) \subset X$
is a family of finite pairs $(x, \mathcal{V}_x)_{x \in P}$ of points $x \in \widetilde{f}^{-1}(K)$
and open neighborhoods $\mathcal{V}_x \subset \mathcal{W}_x$ of $\widetilde{\psi}_x^{-1}(x)$
which satisfies the following conditions:
\begin{enumerate}[label=(\roman*)]
\item
\label{P and simplex}
For any $x \in P$ and simplex $s$ of $K$,
if $x \notin f^{-1}(s)$ then $f_x(\mathcal{V}_x) \cap s = \emptyset$.
\item
$\bigcup_{x \in P} \psi_x(\mathcal{V}_x \cap s_x^{-1}(0))$ is a meet-semilattice cover of $f^{-1}(K)$
instead of Condition \ref{P meet-semilattice}.
\item
Condition \ref{(x, y, z)-relation} to \ref{(x, y, z, w)-relation}
in Definition \ref{def of weakly good coordinate system} is satisfied.
\item
In Condition \ref{(x, y, z, w)-relation},
if $x, y, z \in P$ is contained in $\widetilde{f}^{-1}(K_0)$ for a subcomplex
$K_0 \subset K$, then
the condition still holds even if we impose the condition
$w_j \in P \cap \widetilde{f}^{-1}(K_0)$.
\end{enumerate}
The first and last condition imply that $(x, \mathcal{V}_x)_{x \in P \cap \widetilde{f}^{-1}(K_0)}$
is also a weakly good coordinate system of $f^{-1}(K_0) \subset X$
for any subcomplex $K_0 \subset K$.
For a weakly good coordinate system $(x, \mathcal{V}_x)_{x \in P}$ of $f^{-1}(K) \subset X$,
similarly to Lemma \ref{construction of grouped multisection},
shrinking $\mathcal{V}_x$ slightly if necessary, we can construct a grouped multisection of
$(\boldsymbol{\epsilon}_x)_{x \in P}$ which satisfies the
following transversality condition:
For any orbibundle chart $(\mathcal{V}, \mathcal{E})$ in $(x, \mathcal{V}_x)$ and any
lift $\tilde t \subset V$ of a simplex $t$ of $K$,
the restriction of each branch of the multisection
$s_x|_{\mathcal{V}} + \boldsymbol{\epsilon}_x|_{\mathcal{V}}$
to $V \cap f_x^{-1}(\tilde t)$ is transverse to the zero section of $E$,
and the same holds for each corner of $\mathcal{V}$ instead of $\mathcal{V}$.
(In the case of the generalization of the fiber product with
a Euclidean cell complex, we read $\tilde t$ as its cell.)
\begin{rem}
\label{meaning of triangulation}
Even in the case where $K = \mathcal{Y}$ is a triangulation of $\mathcal{Y}$,
the fiber product $f^{-1}(K)$ is meaningful because
the transversality condition imposed on the grouped multisection of
$f^{-1}(K)$ is stronger than that of $X$.
\end{rem}
We note that for the construction of grouped multisection,
we use the same inductive extension of grouped multisections as in the usual case, and
we do not need to use any induction in the dimension of the simplex of $K$.
As to the extension from a subcomplex $K_0 \subset K$ to $K$,
we can prove the following.
First, if a meet-semilattice cover $\mathcal{Y}^0$ of $f^{-1}(K_0)$ is given, then
we can construct a meet-semilattice cover $\mathcal{Y}$ of $f^{-1}(K)$ such that
$\mathcal{Y}|_{N(f^{-1}(K_0))} = \mathcal{Y}^0|_{f^{-1}(K_0)}$ for some neighborhood $N(f^{-1}(K_0))$
of $f^{-1}(K_0) \subset X$.
We can prove this by the argument of Lemma \ref{extension of meet-semilattice cover}.
(In Lemma \ref{extension of meet-semilattice cover},
first we need to extend the given meet-semilattice cover of $\partial X$ to
its neighborhood, but in this case, $\mathcal{Y}^0$ is already defined on a neighborhood of
$f^{-1}(K_0)$.
Hence if we read $\mathcal{Y}^{N(\partial X)}$ in the proof of
Lemma \ref{extension of meet-semilattice cover} as $\mathcal{Y}^0$,
then the last two paragraph of its proof is the proof of the extension in this case.)
Next, if a weakly good coordinate system $(x, \mathcal{V}^{K_0}_x)_{x \in P^0}$ of $f^{-1}(K_0)$
is given, then we can construct a weakly good coordinate system
$(x, \mathcal{V}_x)_{x \in P}$ of $f^{-1}(K)$ such that
$P^0 = P \cap f^{-1}(K_0)$ and for each $x \in P^0$,
$\mathcal{V}_x \subset \mathcal{V}^{K_0}_x$ is a neighborhood of
$\mathcal{V}^{K_0}_x \cap f_x^{-1}(K_0)$.
Furthermore, if a grouped multisection
$(\boldsymbol{\epsilon}^{K_0}_x)_{x \in P \cap \widetilde{f}^{-1}(K_0)}$
of $(x, \mathcal{V}^{K_0}_x)_{x \in P^0}$ is given,
then shrinking $\mathcal{V}_x$ slightly if necessary,
we can construct a grouped multisection
$(\boldsymbol{\epsilon}^K_x)_{x \in P}$ of $(x, \mathcal{V}_x)_{x \in P}$ such that
the restrictions of $\boldsymbol{\epsilon}^{K_0}_x$ and $\boldsymbol{\epsilon}^K_x$
to a neighborhood of $\mathcal{V}_x \cap f_x^{-1}(K_0)$ coincide for all $x \in P^0$.
This is due to Condition \ref{P and simplex} for $(x, \mathcal{V}_x)_{x \in P}$.
\begin{eg}\label{fiber prod of pre-Kuranishi with diagonal of orbifold}
Let $f = (f_x)_{x \in \widetilde{X}}$ be a strong continuous map
from a pre-Kuranishi space $X$ to an orbifold $\mathcal{Y}$ such that
each $f_x$ is a smooth submersion.
Assume that for each point $x \in \widetilde{X}$, the
dimension of $\pi_{W_x}^{-1}(f_x^{-1}(\widetilde{f}(x))) \subset W_x$ is $>0$
if it is not an empty set, and $G_{W_x}$ acts effectively on it.
Then we can define fiber product
$(f \times f)^{-1}(\Delta_\mathcal{Y} / \mathfrak{S}_2) \subset (X \times X) / \mathfrak{S}_2$
by regarding the diagonal
$\Delta_\mathcal{Y} / \mathfrak{S}_2 \subset (\mathcal{Y} \times \mathcal{Y}) / \mathfrak{S}_2$
as an embedded simplicial complex.
The pre-Kuranishi structure depends on the choice of the triangulation of
$\Delta_\mathcal{Y} / \mathfrak{S}_2$ in the following sense.
If we change the triangulation, then the transversality condition imposed on its
grouped multisection also changes.
\end{eg}
\subsubsection{Multi-valued partial submersions}
\label{multi-valued partial submersions}
First consider the following two trivial examples of the construction of
compatible sections.
\begin{eg}
Let $(V, E)$ be a vector bundle and let $V_i \subset V$ be
finite number of submanifolds which intersect cleanly.
Assume that a smooth section $s_i : V_i \to E|_{V_i}$ is given for each $i$, and
that $s_i$ and $s_j$ coincide on the intersection $V_i \cap V_j$ for all $i, j$.
Then we can construct a smooth section $s : V \to E$ whose restriction to
$V_i$ coincides with $s_i$ for all $i$.
\end{eg}
\begin{eg}
Let $N \geq 1$ be an integer.
We denote a decomposition $\{1, \dots, N\} = \coprod_k A_k$ of integers
by $A = (A_k)$.
We say $g \in \mathfrak{S}_N$ preserves the decomposition $A$
if $g$ maps each $A_k$ to some $A_{k'}$ bijectively.
For each decomposition $A$, let $G_A \subset \mathfrak{S}_N$ be
the group of permutations of $\{1, \dots, N\}$ which preserve $A$.
For two decompositions $A = (A_k)$ and $B = (B_k)$,
we say $A \geq B$ if $B$ is a refinement of $A$, that is,
if each $B_k$ is contained in some $A_{k'}$.
Note that $A \geq B$ does not imply $G_A \subset G_B$ in general.
We denote the discrete decomposition $\{1, \dots, N\} = \coprod_{i=1}^N \{i\}$ by $1^N$.
This is the minimum with respect to this partial order of decompositions.
Let $(V, E)$ be a vector bundle such that $\dim V > 0$,
and define an orbibundle charts $(\mathcal{V}_A, \mathcal{E}_A)$ by
$\mathcal{V}_A = (\prod_N V)/G_A$ and $\mathcal{E}_A = (\prod_N E) / G_A$
for each decomposition $A = (A_k)$.
For each pair $A > B$, let $(\varphi^{B, A} ,\hat \varphi^{B, A})$ be
the multi-valued map from $(\mathcal{V}_A, \mathcal{E}_A)$ to $(\mathcal{V}_B, \mathcal{E}_B)$
induced by the identity map of $(\prod_N V, \prod_N E)$.
Namely, $\varphi^{B, A} \subset \mathcal{V}_A \times \mathcal{V}_B$ is the image of
the diagonal set by the quotient map $(\prod_N V) \times (\prod_N V)
\to \mathcal{V}_A \times \mathcal{V}_B$, and
$\hat \varphi^{B, A} \subset \mathcal{E}_A \times \mathcal{E}_B$ is similar.
Let $A_0$ be an arbitrary decomposition, and assume that
a smooth section $s_A$ of $(\mathcal{V}_A, \mathcal{E}_A)$ is given for each $A < A_0$ and
they satisfy $s_B \circ \varphi^{B, A} = \hat \varphi^{B, A} \circ s_A$ for
all pairs $A > B$ such that $A < A_0$.
Then we can construct a smooth section $s_{A_0}$ of $(\mathcal{V}_{A_0}, \mathcal{E}_{A_0})$
such that $s_A \circ \varphi^{A, A_0} = \hat \varphi^{A, A_0} \circ s_{A_0}$ for
all $A < A_0$.
This is because $\varphi^{1^N, A} \circ \varphi^{A, A_0} = \varphi^{1^N, A_0}$
and $\varphi^{1^N , A_0}$ is single-valued.
More directly, $s_{A_0} = (\varphi^{1^N , A_0})^\ast s_{1^N}$ is the required section.
\end{eg}
In application, we need to consider the combination of the above two examples
in the case of pre-Kuranishi spaces.
Although the case of pre-Kuranishi spaces is also essentially nothing more than
the above two trivial examples,
we explain this case in details since it looks complicated
if we do not give precise definitions.
In this section, we define a compatible system of multi-valued partial submersions
and explain how to construct a compatible family of grouped multisections.
\begin{defi}
\label{smooth multi-valued partial map}
Let $\mathcal{V} = (V, \pi_V, \mathcal{V})$ and $\mathcal{V}'_i = (V'_i, \pi_{V'_i}, \mathcal{V}'_i)$ be
(at most countable number of) orbicharts.
A smooth multi-valued partial map $\varphi : \mathcal{V} \to \coprod_i \mathcal{V}'_i$
is a closed subset $\varphi \subset \mathcal{V} \times \coprod_i \mathcal{V}'_i$ such that
there exists a set of smooth maps $\{\phi : \mathcal{D}(\phi) \to \coprod_i V'_i\}$
which satisfies the following conditions.
We call each $\phi$ a lift of $\varphi$.
\begin{itemize}
\item
The domain $\mathcal{D}(\phi)$ of each lift $\phi$ is a closed connected submanifold of $V$
(with or without corners), and $\phi : \mathcal{D}(\phi) \to \coprod_i V'_i$ is a smooth map.
\item
Assume that the image of $\phi$ is contained in $V'_i$.
Then for any $g \in G_V$ and $g' \in G_{V'_i}$, $g' \circ \phi \circ g
: \mathcal{D}(g' \circ \phi \circ g) = g^{-1} \mathcal{D}(\phi) \to V'_i$ is also a lift of $\varphi$.
\item
$\{\mathcal{D}(\phi)\}$ is locally finite in $V$, and
they intersect cleanly.
\item
For two lifts $\phi$ and $\phi'$, if there exists a point $p \in \mathcal{D}(\phi) \cap \mathcal{D}(\phi')$
such that $\phi(p) = \phi'(p)$,
$T_p \mathcal{D}(\phi) \subset T_p \mathcal{D}(\phi')$ and
$(\phi_\ast)_p = (\phi'_\ast)_p|_{T_p \mathcal{D}(\phi)}$ on $T_p \mathcal{D}(\phi)$,
then $\phi = \phi'$.
\item
$(\pi_V \times \coprod_i \pi_{V'_i})^{-1}(\varphi) \subset V \times \coprod_i V'_i$
coincides with the union of the graphs of the lifts $\{\phi\}$.
\end{itemize}
\end{defi}
\begin{lem}
In the above definition, the set of lifts
$\{\phi : \mathcal{D}(\phi) \to \coprod_i V'_i\}$ is uniquely determined by
a closed subset $\varphi \subset \mathcal{V} \times \coprod_i \mathcal{V}'_i$ if it exists.
\end{lem}
\begin{proof}
Let $D\varphi \subset TV \times \coprod_i TV'_i$ be the
set of tangent vectors which can be represented by
some smooth curves of $V \times \coprod_i V'_i$ contained in
$(\pi_V \times \coprod_i \pi_{V'_i})^{-1}(\varphi)$.
$D\varphi$ coincides with the union of the tangent bundles of
the graphs of the lifts $\phi$.
For each point $p \in V \times \coprod_i V'_i$,
the fiber $(D\varphi)_p$ at $p$ is the union of some subspaces $(E_{p, j})_j$ of
$(TV \times \coprod_i TV'_i)_p$.
We assume that this decomposition is irredundant, that is,
there does not exist two indices $i \neq j$ such that $E_{p, i} \subset E_{p, j}$.
Each $E_{p, j}$ defines a point of the union of the Grassmann bundles
$G(TV \times \coprod_i TV'_i) = \coprod_k G_k(TV \times \coprod_i TV'_i)$
of $TV \times \coprod_i TV'_i$ over all dimensions.
Let $G(D\varphi)$ be the union of these points.
By assumption, $G(D\varphi)$ coincides with
the disjoint union of the closed submanifolds defined by the graphs of the lifts $\phi$.
Hence the lifts $\phi$ are uniquely determined by the connected components of
the submanifold $G(D\varphi)$ of $G(TV \times \coprod_i TV'_i)$.
\end{proof}
\begin{defi}
Let $(\mathcal{V}, \mathcal{E})$ and $(\mathcal{V}'_i, \mathcal{E}'_i)$ be orbibundle charts.
A multi-valued partial submersion
$(\varphi, \hat \varphi) : (\mathcal{V}, \mathcal{E}) \to \coprod_i (\mathcal{V}'_i, \mathcal{E}'_i)$
is a pair of
smooth multi-valued partial maps $\varphi : \mathcal{V} \to \coprod_i \mathcal{V}'_i$
and $\hat \varphi : \mathcal{E} \to \coprod_i \mathcal{E}'_i$ such that
there is a one-to-one correspondence between their lifts $\{\phi\}$ and
$\{\hat \phi\}$, and each pair $(\phi, \hat \phi)$
is a bundle map $(\mathcal{D}(\phi), \mathcal{E}|_{\mathcal{D}(\phi)}) \to \coprod_i (V'_i, E'_i)$
which satisfies the following conditions:
The underlying map $\phi : \mathcal{D}(\phi) \to \coprod_i V'_i$ is a submersion,
and the restriction of $\hat \phi$ to each fiber is an isomorphism.
We call each pair $(\phi, \hat \phi)$ a lift of $(\varphi, \hat \varphi)$.
\end{defi}
We do not define the composition of two arbitrary multi-valued partial submersions.
Instead, we define the following compatible system.
\begin{defi}
\label{compatible system of multi-valued partial submersions for charts}
Let $(\mathcal{V}^a, \mathcal{E}^a)_{a \in A}$ be a finite family of orbibundle charts whose
index set $A$ has a partial order.
For each pair $a, b \in A$ such that $a > b$,
let $(\varphi^{b, a}, \hat \varphi^{b, a}) : (\mathcal{V}^a, \mathcal{E}^a) \to (\mathcal{V}^b, \mathcal{E}^b)$ be
a multi-valued partial submersion.
We say $((\mathcal{V}^a, \mathcal{E}^a)_{a \in A}, (\varphi^{b, a}, \hat \varphi^{b, a})_{a > b \in A})$
is a compatible system of multi-valued partial submersions if
the following hold.
\begin{itemize}
\item
For any $a \in A$,
$\coprod_{b < a} (\varphi^{b, a}, \hat \varphi^{b, a}) : (\mathcal{V}^a, \mathcal{E}^a) \to
\coprod_{b < a} (\mathcal{V}^b, \mathcal{E}^b)$ is also a multi-valued partial submersion.
This means that the condition of clean intersection of the
domains of the lifts $\{\mathcal{D}(\phi) \subset V^a\}$
holds not only independently for each $b < a$ but also
for the union over all $b < a$.
\item
For any lifts $(\phi_1^{b_1, a}, \hat \phi_1^{b_1, a})$ and
$(\phi_2^{b_2, a}, \hat \phi_2^{b_2, a})$ of
$(\varphi_1^{b_1, a}, \hat \varphi_1^{b_1, a})$ and
$(\varphi_1^{b_1, a}, \hat \varphi_1^{b_1, a})$ respectively,
if $\mathcal{D}(\phi_1^{b_1, a}) \cap \mathcal{D}(\phi_2^{b_2, a}) \neq \emptyset$,
then there exists some $c \in A$ such that $c \leq b_1, b_2$ and
some lifts $(\phi_1^{c, b_1}, \hat \phi_1^{c, b_1})$,
$(\phi_2^{c, b_2}, \hat \phi_2^{c, b_2})$
and $(\phi^{c, a}, \hat \phi^{c, a})$ of
$(\varphi_1^{c, b_1}, \hat \varphi_1^{c, b_1})$,
$(\varphi_2^{c, b_2}, \hat \varphi_2^{c, b_2})$ and
$(\varphi^{c, a}, \hat \varphi^{c, a})$ respectively
which satisfy the following conditions:
\[
\mathcal{D}(\phi_1^{b_1, a}) \cap \mathcal{D}(\phi_2^{b_2, a})
= \mathcal{D}(\phi_1^{c, b_1} \circ \phi_1^{b_1, a})
= \mathcal{D}(\phi_2^{c, b_2} \circ \phi_2^{b_2, a})
= \mathcal{D}(\phi^{c, a}),
\]
and
\[
(\phi_1^{c, b_1}, \hat \phi_1^{c, b_1}) \circ (\phi_1^{b_1, a} \hat \phi_1^{b_1, a})
= (\phi_2^{c, b_2}, \hat \phi_2^{c, b_2}) \circ (\phi_2^{b_2, a}, \hat \phi_2^{b_2, a})
= (\phi^{c, a}, \hat \phi^{c, a}).
\]
(If $c = b_1$, then we read $(\varphi_1^{c, b_1}, \hat \varphi_1^{c, b_1})$
as the identity map.
Hence its lift is an element of the automorphism group $G_{V^c}$.)
\end{itemize}
\end{defi}
\begin{defi}
Let $(\mathcal{V}, \mathcal{E})$ and $(\mathcal{V}', \mathcal{E}')$ be orbibundles.
A multi-valued partial submersion $(\varphi, \hat \varphi) : (\mathcal{V}, \mathcal{E}) \to (\mathcal{V}', \mathcal{E}')$
is a pair of closed subsets
$\varphi \subset \mathcal{V} \times \mathcal{V}'$ and
$\hat \varphi \subset \mathcal{E} \times \mathcal{E}'$
which satisfies the following conditions.
\begin{itemize}
\item
For any point $x \in \mathcal{V}$, $\varphi(x) \subset \mathcal{V}'$ consists of finite points,
where $\varphi(x)$ is defined by $\varphi \cap (\{x\} \times \mathcal{V}')
= \{x\} \times \varphi(x)$.
\item
For any point $x \in \mathcal{V}$ such that $\varphi(x) \neq \emptyset$,
let $(\mathcal{V}'_i, \mathcal{E}'_i)$ be a finite family of disjoint orbibundle charts of $(\mathcal{V}', \mathcal{E}')$
which covers $\varphi(x)$.
Then there exists some orbibundle chart $(\mathcal{V}_x, \mathcal{E}_x)$ of a neighborhood of
$x \in \mathcal{V}$ such that $(\mathcal{V}'_i, \mathcal{E}'_i)$ covers $\varphi(\mathcal{V}_x)$ and
$(\varphi \cap (\mathcal{V}_x \times \coprod_i \mathcal{V}'_i),
\hat \varphi \cap (\mathcal{E}_x \times \coprod_i \mathcal{E}'_i))$ is a multi-valued partial submersion
from $(\mathcal{V}_x, \mathcal{E}_x)$ to $\coprod_i (\mathcal{V}'_i, \mathcal{E}'_i)$.
\end{itemize}
\end{defi}
Note that for any compact subset $K \subset \mathcal{V}$,
$\varphi(K) = \{x' \in \mathcal{V}'; (x, x') \in \varphi \text{ for some } x \in K\} \subset \mathcal{V}'$
is compact.
This is because in Definition \ref{smooth multi-valued partial map},
we assumed that $\mathcal{D}(\phi)$ is closed for each lift $\phi$.
\begin{defi}
\label{compatible system of multi-valued partial submersions for orbibundles}
Let $(\mathcal{V}^a, \mathcal{E}^a)_{a \in A}$ be a finite family of orbibundles whose
index set $A$ has a partial order.
For each pair $a, b \in A$ such that $a > b$,
let $(\varphi^{b, a}, \hat \varphi^{b, a}) : (\mathcal{V}^a, \mathcal{E}^a) \to (\mathcal{V}^b, \mathcal{E}^b)$ be
a multi-valued partial submersion.
We say $((\mathcal{V}^a, \mathcal{E}^a)_{a \in A}, (\varphi^{b, a}, \hat \varphi^{b, a})_{a > b \in A})$
is a compatible system of multi-valued partial submersions if
the following holds.
\begin{itemize}
\item
For any $a > b > c \in A$,
$\varphi^{c, b} \circ \varphi^{b, a} \subset \varphi^{c, a}$
and $\hat \varphi^{c, b} \circ \hat \varphi^{b, a} \subset \hat \varphi^{c, a}$.
\item
For each point $x_0 \in \mathcal{V}^{a_0}$, we define a partial order of
$\bigcup_{a \leq a_0} \varphi^{a, a_0}(x_0)$ as follows.
For $x \in \varphi^{a, a_0}(x_0)$ and $y \in \varphi^{b, a_0}(x_0)$,
$x > y$ if $a > b$ and $y \in \varphi^{b, a}(x)$.
Then for each $x \in \varphi^{a, a_0}(x_0)$,
there exists an orbibundle chart $(\mathcal{V}^x, \mathcal{E}^x)$ of a neighborhood of $x \in \mathcal{V}^a$
such that the restrictions of $(\varphi^{b, a}, \hat \varphi^{b, a})$
define multi-valued partial submersions $(\varphi^{y, x}, \hat \varphi^{y, x})
: (\mathcal{V}^x, \mathcal{E}^x) \to (\mathcal{V}^y, \mathcal{E}^y)$ for all pairs
$x > y \in \bigcup_{a \leq a_0} \varphi^{a, a_0}(x_0)$, and
\[
((\mathcal{V}^x, \mathcal{E}^x)_{x \in \bigcup_{a \leq a_0} \varphi^{a, a_0}(x_0)},
(\varphi^{y, x}, \hat \varphi^{y, x})_{x > y \in \bigcup_{a \leq a_0} \varphi^{a, a_0}(x_0)})
\]
is a compatible system of multi-valued partial submersions in the sense of
Definition \ref{compatible system of multi-valued partial submersions for charts}.
($\mathcal{V}^x$ may depend on $x_0$.)
\end{itemize}
\end{defi}
We also define $(\varphi, \hat \varphi)$-relation of grouped multisections for
a multi-valued partial submersion $(\varphi, \hat \varphi)$
between orbibundles.
The definition is the same as
Definition \ref{(varphi, hat varphi)-relation for orbibundles}.
For a compatible system of multi-valued partial submersions
$((\mathcal{V}^a, \mathcal{E}^a)_{a \in A}, (\varphi^{b, a}, \hat \varphi^{b, a})_{a > b \in A})$,
we also always assume the compatibility condition similar to
Definition \ref{compatibility of (varphi, hat varphi)-relations}.
The proof of the following lemma is straightforward.
\begin{lem}
\label{extension of grouped multisection for multi-valued partial submersions}
Let $((\mathcal{V}^a, \mathcal{E}^a)_{a \in A}, (\varphi^{b, a}, \hat \varphi^{b, a})_{a > b \in A})$
be a compatible system of multi-valued partial submersions.
Let $a_0 \in A$ be an arbitrary index and assume that
grouped multisections $\boldsymbol{\epsilon}_a$ of $(\mathcal{V}^a, \mathcal{E}^a)$
are given for $a < a_0$.
We also assume that $\boldsymbol{\epsilon}_a$ and $\boldsymbol{\epsilon}_b$
are $(\varphi^{b, a}, \hat \varphi^{b, a})$-related for $a > b \in \mathcal{A}$ such that
$a < a_0$.
Then we can construct a grouped multisection $\boldsymbol{\epsilon}_{a_0}$
of $(\mathcal{V}^{a_0}, \mathcal{E}^{a_0})$ which is $(\varphi^{a, a_0}, \hat \varphi^{a, a_0})$-related
to $\boldsymbol{\epsilon}_a$ for all $a < a_0$.
\end{lem}
\begin{defi}
Let $X^k$ ($k = 1,2$) be two pre-Kuranishi spaces with pre-Kuranishi structures
$(\widetilde{X}^k, \mu^k, (\mathcal{W}^k_x, \mathcal{E}^k_x, s^k_x, \widetilde{\psi}^k_x),
(\varphi^k_{x, y}, \hat \varphi^k_{x, y}))$.
A multi-valued partial submersion
$f = (f, \widetilde{f}, (\mathcal{W}^f_{\widehat{x}| x}, \varphi^f_{\widehat{x}, x},
\hat \varphi^f_{\widehat{x}, x}))$
from $X^1$ to $X^2$ consists of the following.
$f \subset X^1 \times X^2$ and
$\widetilde{f} \subset \widetilde{X}^1 \times \widetilde{X}^2$ are closed subsets
such that $(\mu^1 \times \mu^2)(\widetilde{f}) = f$.
For each $(x, \widehat{x}) \in \widetilde{f}$,
$\mathcal{W}^f_{\widehat{x}| x} \Subset \mathcal{W}^2_{\widehat{x}}$ is an open neighborhood of
$(\widetilde{\psi}^2_{\widehat{x}})^{-1}(\widehat{x})$, and
$(\varphi^f_{\widehat{x}, x}, \hat \varphi^f_{\widehat{x}, x})
: (\mathcal{W}^1_{x}, \mathcal{E}^1_{x}) \to (\mathcal{W}^f_{\widehat{x}| x}, \mathcal{E}^2_{\widehat{x}}|_{\mathcal{W}^f_{\widehat{x}| x}})$
is a multi-valued partial submersion.
We impose the following conditions on them:
\begin{enumerate}[label=$(\arabic*)^{\mathrm{MP}}$]
\item
For any $(p, q) \in f$ and $\widetilde{p} \in (\mu^1)^{-1}(p)$,
there exists a unique $\widetilde{q} \in (\mu^2)^{-1}(q)$
such that $(\widetilde{p}, \widetilde{q}) \in \widetilde{f}$.
Furthermore, this defines an isomorphism
\[
\widetilde{f}|_{(\mu^1)^{-1}(p) \times (\mu^2)^{-1}(q)} : (\mu^1)^{-1}(p) \cong
(\mu^2)^{-1}(q)
\]
between partially ordered sets.
\item
\label{properness of tilde f}
For any compact subset $K \subset \widetilde{X}^2$,
$\widetilde{f} \cap (\widetilde{X}^1 \times K)$ is compact.
\item
For any $x \in \widetilde{X}^1$, closed subsets
$\{\psi^2_{\widehat{x}}((s^2_{\widehat{x}})^{-1}(0) \cap \overline{\mathcal{W}^f_{\widehat{x}| x}})
\subset X^2; \widehat{x} \in \widetilde{f}(x)\}$ are disjoint.
\item
\label{varphi condition for multi-valued partial submersion}
For any $(x, \widehat{x}) \in \widetilde{f}$,
$s^2_{\widehat{x}} \circ \varphi^f_{\widehat{x}, x}
= \hat \varphi^f_{\widehat{x}, x} \circ s^1_{x}$ as a subset
of $\mathcal{W}^1_x \times \mathcal{E}^2_{\widehat{x}}|_{\mathcal{W}^f_{\widehat{x}| x}}$.
Furthermore, for any $x \in \widetilde{X}^1$,
$\coprod_{\widehat{x} \in \widetilde{f}(x)}
(\widetilde{\psi}^2_{\widehat{x}} \circ \varphi^f_{\widehat{x}, x})
= \widetilde{f} \circ \widetilde{\psi}^1_{x}$
as a subset of $(s^1_{x})^{-1}(0) \times \widetilde{X}^2$.
\item
For any $x \in \widetilde{X}^1$ and $(y, \widehat{y}) \in \widetilde{f}$
such that $x \unrhd y$, define
$\widetilde{f}(x)_{(y, \widehat{y})} =
\{\widehat{x} \in \widetilde{f}(x); q_{\widehat{x}} \geq q_{\widehat{y}}
\text{ for some } q \in
\psi_{\widehat{x}}((s^2_{\widehat{x}})^{-1}(0) \cap \mathcal{W}^f_{\widehat{x}| x}) \cap
\psi_{\widehat{y}}((s^2_{\widehat{y}})^{-1}(0) \cap \mathcal{W}^f_{\widehat{y}| y})\}$.
Then the following conditions hold:
\begin{itemize}
\item
$\varphi^f_{\widehat{y}, y}(\mathcal{W}^1_{x, y})$ is contained in
$\bigcup_{\widehat{x} \in \widetilde{f}(x)_{(y, \widehat{y})}}
(\varphi^2_{\widehat{x}, \widehat{y}})^{-1}(\mathcal{W}^f_{\widehat{x}| x})$.
Furthermore,
$\{\varphi^f_{\widehat{y}, y}(\mathcal{W}^1_{x, y}) \cap
(\varphi^2_{\widehat{x}, \widehat{y}})^{-1}(\mathcal{W}^f_{\widehat{x}| x});
\widehat{x} \in \widetilde{f}(x)_{(y, \widehat{y})}\}$ are disjoint.
\item
For any $\widehat{x} \in \widetilde{f}(x)_{(y, \widehat{y})}$,
\begin{equation}
(\varphi^f_{\widehat{x}, x}, \hat \varphi^f_{\widehat{x}, x}) \circ
(\varphi^1_{x, y}, \hat \varphi^1_{x, y})
= (\varphi^2_{\widehat{x}, \widehat{y}}, \hat \varphi^2_{\widehat{x}, \widehat{y}}) \circ
(\varphi^f_{\widehat{y}, y}, \hat \varphi^f_{\widehat{y}, y})
\end{equation}
as pairs of smooth multi-valued partial maps
from $(\mathcal{W}^1_{x, y}, \mathcal{E}^1_y|_{\mathcal{W}^1_{x, y}})$ to
$(\mathcal{W}^f_{\widehat{x}| x}, \mathcal{E}^2_{\widehat{x}}|_{\mathcal{W}^f_{\widehat{x}| x}})$.
\end{itemize}
\end{enumerate}
\end{defi}
In relation to Condition \ref{properness of tilde f},
we note that for a compact subset $K \subset \widetilde{X}^1$,
$\widetilde{f} \cap (K \times \widetilde{X}^2)$ is compact.
This is because each $\varphi^f_{\widehat{x}, x}$ maps compact sets to
compact sets as we noted above.
It implies that for any closed subset $A \subset \widetilde{X}^2$,
$\widetilde{f}^{-1}(A) = \{x \in \widetilde{X}^1; (x, \widehat{x}) \in \widetilde{f}
\text{ for some } \widehat{x} \in A\} \subset \widetilde{X}^1$ is also closed.
In particular, $\widetilde{f}^{-1}(\widetilde{X}^2) \subset \widetilde{X}^1$
is closed.
\begin{defi}
Let $(x_1, \mathcal{V}^1_{x_1})_{x_1 \in P^1}$ and $(x_2, \mathcal{V}^2_{x_2})_{x_2 \in P^2}$ be
weakly good coordinate systems of pre-Kuranishi spaces $X^1$ and $X^2$ respectively.
We say these are compatible with respect to the multi-valued partial submersion $f$ if
they satisfy the following conditions:
\begin{enumerate}[label=$(\arabic*)^{\mathrm{CW}}$]
\item
\label{compatibility of meet-semilattice covers}
The meet-semilattice covers $\mathcal{Y}^i = \bigcup_{x_i \in P^i}
\widetilde{\psi}^i_{x_i}((s^i_{x_i})^{-1}(0) \cap \mathcal{V}^i_{x_i}))$ ($i = 1,2$) satisfy
$\widetilde{f}(\mathcal{Y}^1) \subset \mathcal{Y}^2$.
($\widetilde{f}(\mathcal{Y}^1) \subset \widetilde{X}^2$ is defined by
$\widetilde{f}(\mathcal{Y}^1) = \{\widehat{x} \in \widetilde{X}^2; (x, \widehat{x}) \in \widetilde{f}
\text{ for some } x \in \mathcal{Y}^1\}$.)
\item
For any $x_1 \in P^1$, if
$\widetilde{\psi}^1_{x_1}((s^1_{x_1})^{-1}(0) \cap \mathcal{V}^1_{x_1}) \cap
\widetilde{f}^{-1}(\widetilde{X}^2) \neq \emptyset$,
then $x_1 \in \widetilde{f}^{-1}(\mathcal{Y}^2)$ and
$\widetilde{f}(x_1) \subset \mathcal{Y}^2$.
Furthermore, for any $y_1, x_1 \in P^1$ such that
$y_1 \notin \widetilde{f}^{-1}(\mathcal{Y}^2)$ and $x_1 \in \widetilde{f}^{-1}(\mathcal{Y}^2)$,
if there exists some $p \in \psi^1_{x_1}((s^1_{x_1})^{-1}(0) \cap \mathcal{V}^1_{x_1})
\cap \psi^1_{y_1}((s^1_{y_1})^{-1}(0) \cap \mathcal{V}^1_{y_1})$ such that
$p_{y_1} \leq p_{x_1}$, then $\varphi^1_{x_1, y_1}(\mathcal{V}^1_{x_1, y_1}) \subset \mathcal{V}^1_{x_1}$
does not intersect with the domain of $\varphi^f_{\widehat{x}_1, x_1}$
for any $\widehat{x}_1 \in \widetilde{f}(x_1)$.
\item
\label{(x1, widehat x1, x2)-composition}
For any $x_1 \in P^1$, $\widehat{x}_1 \in \widetilde{X}^2$ and $x_2 \in P^2$,
if $(x_1, \widehat{x}_1) \in \widetilde{f}$ and
$\widehat{x}_1 \in \widetilde{\psi}^2_{x_2}((s^2_{x_2})^{-1}(0) \cap \mathcal{V}^2_{x_2}))$,
then $\varphi^f_{\widehat{x}_1, x_1}(\mathcal{V}^1_{x_1}) \subset
(\varphi^2_{x_2, \widehat{x}_1})^{-1}(\mathcal{V}^2_{x_2})$.
\item
\label{composition compatibility for multi-valued partial submersion}
For any $y_1, x_1 \in P^1$ and $y_2, x_2 \in P^2$,
if there exist some $p \in \psi_{x_1}((s^1_{x_1})^{-1}(0) \cap \mathcal{V}^1_{x_1}) \cap
\psi_{y_1}((s^1_{y_1})^{-1}(0) \cap \mathcal{V}^1_{y_1})$ such that
$p_{x_1} \geq p_{y_1}$, then
the following holds
for any $\widehat{y}_1 \in \widetilde{f}(y_1) \cap
\widetilde{\psi}^2_{y_2}((s^2_{y_2})^{-1}(0) \cap \mathcal{V}^2_{y_2})$ and
$\widehat{x}_1 \in \widetilde{f}(x_1)_{(y, \widehat{y})} \cap
\widetilde{\psi}^2_{x_2}((s^2_{x_2})^{-1}(0) \cap \mathcal{V}^2_{x_2})$.
\[
\varphi^f_{\widehat{y}_1, y_1}(\mathcal{V}^1_{x_1, y_1}) \cap
(\varphi^2_{\widehat{x}_1, \widehat{y}_1})^{-1}(\mathcal{W}^f_{\widehat{x}_1| x_1})
\subset
\mathcal{W}^2_{x_2, \widehat{x}_1, \widehat{y}_1} \cap
\mathcal{W}^2_{x_2, y_2, \widehat{y}_1}
\]
\end{enumerate}
\end{defi}
\begin{defi}
Let $(x_1, \mathcal{V}^1_{x_1})_{x_1 \in P^1}$ and $(x_2, \mathcal{V}^2_{x_2})_{x_2 \in P^2}$ be
weakly good coordinate systems of pre-Kuranishi spaces $X^1$ and $X^2$ respectively
which are compatible with respect to the multi-valued partial submersion $f$.
Let $(\boldsymbol{\epsilon}^1_{x_1})_{x_1 \in P^1}$ and
$(\boldsymbol{\epsilon}^2_{x_2})_{x_2 \in P^2}$ be grouped multisections of
$(x_1, \mathcal{V}^1_{x_1})_{x_1 \in P^1}$ and $(x_2, \mathcal{V}^2_{x_2})_{x_2 \in P^2}$ respectively.
We say these grouped multisections are compatible with respect to $f$ if
for any $x_1 \in P^1$, $\widehat{x}_1 \in \widetilde{X}^2$ and $x_2 \in P^2$
such that $(x_1, \widehat{x}_1) \in \widetilde{f}$ and
$\widehat{x}_1 \in \widetilde{\psi}^2_{x_2}((s^2_{x_2})^{-1}(0) \cap \mathcal{V}^2_{x_2}))$,
$\boldsymbol{\epsilon}_{x_1}$ and $\boldsymbol{\epsilon}_{x_2}$ are
$(\varphi^2_{x_2, \widehat{x}_1} \circ \varphi^f_{\widehat{x}_1, x_1},
\hat \varphi^2_{x_2, \widehat{x}_1} \circ \hat \varphi^f_{\widehat{x}_1, x_1})$-related.
(Note that the triple $(x_1, \widehat{x}_1, x_2)$ is that
in Condition \ref{(x1, widehat x1, x2)-composition}.
Hence $(\varphi^2_{x_2, \widehat{x}_1} \circ \varphi^f_{\widehat{x}_1, x_1},
\hat \varphi^2_{x_2, \widehat{x}_1} \circ \hat \varphi^f_{\widehat{x}_1, x_1})$ is
a multi-valued partial submersion from $(\mathcal{V}^1_{x_1}, \mathcal{E}^1_{x_1}|_{\mathcal{V}^1_{x_1}})$ to
$(\mathcal{V}^2_{x_2}, \mathcal{E}^2_{x_2}|_{\mathcal{V}^2_{x_2}})$.)
\end{defi}
For a single multi-valued partial submersion $f$, in general,
we cannot construct a weakly good coordinate system
$(x_1, \mathcal{V}^1_{x_1})_{x_1 \in P^1}$ of $X^1$ which is compatible with
a given weakly good coordinate system $(x_2, \mathcal{V}^2_{x_2})_{x_2 \in P^2}$ of $X^2$.
Similarly, even if $(x_1, \mathcal{V}^1_{x_1})_{x_1 \in P^1}$ and $(x_2, \mathcal{V}^2_{x_2})_{x_2 \in P^2}$
are compatible weakly good coordinate systems of $X^1$ and $X^2$ respectively,
we cannot construct a grouped multisection
$(\boldsymbol{\epsilon}^1_{x_1})_{x_1 \in P^1}$ of $(x_1, \mathcal{V}^1_{x_1})_{x_1 \in P^1}$
which is compatible with a given grouped multisection
$(\boldsymbol{\epsilon}^2_{x_2})_{x_2 \in P^2}$ of $(x_2, \mathcal{V}^2_{x_2})_{x_2 \in P^2}$.
What we can say in general is the following.
\begin{lem}
\label{compatible pull backs for multi-valued partial submersion}
Assume that $(x_1, \mathcal{V}^1_{x_1})_{x_1 \in P^1}$ and $(x_2, \mathcal{V}^2_{x_2})_{x_2 \in P^2}$
are compatible weakly good coordinate systems of $X^1$ and $X^2$, and
let $(\boldsymbol{\epsilon}^2_{x_2})_{x_2 \in P^2}$ be
a grouped multisection of $(x_2, \mathcal{V}^2_{x_2})_{x_2 \in P^2}$.
For any $x_1 \in P^1$, $\widehat{x}_1 \in \widetilde{X}^2$
such that $(x_1, \widehat{x}_1) \in \widetilde{f}$,
let $P^2(\widehat{x}_1) \subset P^2$ be the set of points $x_2 \in P^2$ such that
$\widehat{x}_1 \in \widetilde{\psi}^2_{x_2}((s^2_{x_2})^{-1}(0) \cap \mathcal{V}^2_{x_2}))$.
For each $x_2 \in P^2(\widehat{x}_1)$, let
$(\varphi^2_{x_2, \widehat{x}_1})^\ast \boldsymbol{\epsilon}^2_{x_2}$ be the pull back of
$\boldsymbol{\epsilon}^2_{x_2}$ by
\begin{align*}
&(\varphi^2_{x_2, \widehat{x}_1}, \hat \varphi^2_{x_2, \widehat{x}_1})
|_{(\varphi^2_{x_2, \widehat{x}_1})^{-1}(\mathcal{V}^2_{x_2})} \\
&: ((\varphi^2_{x_2, \widehat{x}_1})^{-1}(\mathcal{V}^2_{x_2}),
\mathcal{E}^2_{\widehat{x}_1}|_{(\varphi^2_{x_2, \widehat{x}_1})^{-1}(\mathcal{V}^2_{x_2})})
\to (\mathcal{V}^2_{x_2}, \mathcal{E}^2_{x_2}|_{\mathcal{V}^2_{x_2}}).
\end{align*}
Then its restriction to a neighborhood of $\varphi^f_{\widehat{x}_1, x_1}(\mathcal{V}^1_{x_1})$
does not depend on $x_2 \in P^2(\widehat{x}_1)$.
More precisely, their restrictions to
\[
\bigcap_{x_2, y_2 \in P^2(\widehat{x}_1)} \mathcal{W}^2_{x_2, y_2, \widehat{x}_1}
\cap \bigcap_{x_2 \in P^2(\widehat{x}_1)} (\varphi^2_{x_2, \widehat{x}_1})^{-1}(\mathcal{V}^2_{x_2})
\]
coincides.
\end{lem}
\begin{proof}
First note that
Condition \ref{composition compatibility for multi-valued partial submersion}
for $y_1 = x_1$, $\widehat{y}_1 = \widehat{x}_1$ and
$y_2, x_2 \in P^2(\widehat{x}_1)$ implies that
$\varphi^f_{\widehat{x}_1, x_1}(\mathcal{V}^1_{x_1}) \subset
\bigcap_{x_2, y_2 \in P^2(\widehat{x}_1)} \mathcal{W}^2_{x_2, y_2, \widehat{x}_1}$.
Hence the claim follows from $(\varphi^2_{x_2, y_2}, \hat \varphi^2_{x_2, y_2})$-relation
of $(\boldsymbol{\epsilon}^2_{x_2})_{x_2 \in P^2}$.
\end{proof}
\begin{defi}
Let $(X^\alpha)_{\alpha \in \mathcal{A}}$ be finite number of pre-Kuranishi spaces,
and assume that its index set $\mathcal{A}$ has a partial order.
For each pair $\alpha, \beta \in \mathcal{A}$ such that $\alpha > \beta$,
let $f^{\beta, \alpha} = (f^{\beta, \alpha}, \widetilde{f}^{\beta, \alpha},
(\varphi^{f^{\beta, \alpha}}_{x_2, x_1}, \hat \varphi^{f^{\beta, \alpha}}_{x_2, x_1}))$
be a multi-valued partial submersion from $X^\alpha$ to $X^\beta$.
We say $((X^\alpha)_{\alpha \in \mathcal{A}}, (f^{\beta, \alpha})_{\alpha > \beta})$
is a compatible system of
multi-valued partial submersions if it satisfies the following conditions:
\begin{itemize}
\item
For any triple $\alpha > \beta > \gamma \in \mathcal{A}$,
$f^{\gamma, \beta} \circ f^{\beta, \alpha} \subset f^{\gamma, \alpha}$
as a subset of $X^\alpha \times X^\gamma$
and $\widetilde{f}^{\gamma, \beta} \circ \widetilde{f}^{\beta, \alpha} \subset
\widetilde{f}^{\gamma, \alpha}$ as a subset of $\widetilde{X}^\alpha \times
\widetilde{X}^\gamma$.
\item
For each $x_0 \in \widetilde{X}^{\alpha_0}$,
we define a partial order of
$\bigcup_{\alpha < \alpha_0} \widetilde{f}^{\alpha, \alpha_0}(x_0)$ as follows.
For $x \in \widetilde{f}^{\alpha, \alpha_0}(x_0)$ and $y \in f^{\beta, \alpha_0}(x_0)$,
$x > y$ if $\alpha > \beta$ and $y \in \widetilde{f}^{\beta, \alpha}(x)$.
Then
\[
(\varphi^{f^{\beta, \alpha}}_{y, x}, \hat \varphi^{f^{\beta, \alpha}}_{y, x})
: (\mathcal{W}^\alpha_x, \mathcal{E}^\alpha_x) \to
(\mathcal{W}^{f^{\beta, \alpha}}_{y|x}, \mathcal{E}^\beta|_{\mathcal{W}^{f^{\beta, \alpha}}_{y|x}})
\hookrightarrow (\mathcal{W}^\beta_y, \mathcal{E}^\beta_y)
\]
for $x \in \widetilde{f}^{\alpha, \alpha_0}(x_0)$ and
$y \in \widetilde{f}^{\beta, \alpha_0}(x_0)$
such that $x > y$ in $\bigcup_{\alpha < \alpha_0} \widetilde{f}^{\alpha, \alpha_0}(x_0)$
constitute a compatible system of multi-valued partial submersions
in the sense of
Definition \ref{compatible system of multi-valued partial submersions for orbibundles}.
\end{itemize}
\end{defi}
Note that the above condition implies that
$\bigcup_{\alpha < \alpha_0} \widetilde{f}^{\alpha, \alpha_0}(x_0)$ with the above
partial order has a unique minimum for each $x_0 \in \widetilde{X}^{\alpha_0}$.
\begin{defi}
Let $((X^\alpha)_{\alpha \in \mathcal{A}}, (f^{\beta, \alpha})_{\alpha > \beta})$
be a compatible system of multi-valued partial submersions,
and let $(x_\alpha, \mathcal{V}^\alpha_{x_\alpha})_{x_\alpha \in P^\alpha}$ be a weakly good
coordinate system of $X^\alpha$ for each $\alpha \in \mathcal{A}$.
We say these weakly good coordinate systems
$(x_\alpha, \mathcal{V}^\alpha_{x_\alpha})_{x_\alpha \in P^\alpha}$ are compatible if
for any $\alpha > \beta \in \mathcal{A}$,
$(x_\alpha, \mathcal{V}^\alpha_{x_\alpha})_{x_\alpha \in P^\alpha}$ and
$(x_\beta, \mathcal{V}^\beta_{x_\beta})_{x_\beta \in P^\beta}$ are
compatible with respect to $f^{\beta, \alpha}$.
\end{defi}
For a compatible system of multi-valued partial submersions
$((X^\alpha)_{\alpha \in \mathcal{A}}, \allowbreak (f^{\beta, \alpha})_{\alpha > \beta})$,
we can construct a compatible family of weakly good coordinate systems
$(x_\alpha, \mathcal{V}^\alpha_{x_\alpha})_{x_\alpha \in P^\alpha}$ of $X^\alpha$
for all $\alpha \in \mathcal{A}$ as follows.
First we claim that we can construct meet-semilattice covers
$\mathcal{Y}^\alpha \subset \widetilde{X}^\alpha$ of $X^\alpha$ for all $\alpha \in \mathcal{A}$
such that
\begin{equation}
\mathcal{Y}^\alpha \cap (\widetilde{f}^{\beta, \alpha})^{-1}(\widetilde{X}^\beta)
= (\widetilde{f}^{\beta, \alpha})^{-1}(\mathcal{Y}^\beta)
\label{compatible meet-semilattice covers for multi-valued partial submersions}
\end{equation}
for all pairs $\alpha, \beta \in \mathcal{A}$ such that $\alpha > \beta$.
First we note that if
(\ref{compatible meet-semilattice covers for multi-valued partial submersions})
is satisfied for pairs $\alpha > \beta \in \mathcal{A}$ such that $\alpha < \alpha_0$, then
$\widetilde{f}^{\alpha, \alpha_0}(x_0) \subset \mathcal{Y}^\alpha$
for any $x_0 \in (\widetilde{f}^{\alpha, \alpha_0})^{-1}(\mathcal{Y}^\alpha)$.
This is proved as follows.
$x_0 \in (\widetilde{f}^{\alpha, \alpha_0})^{-1}(\mathcal{Y}^\alpha)$ implies that
$(x_0, x) \in \widetilde{f}^{\alpha, \alpha_0}$ for some $x \in \mathcal{Y}^\alpha$.
Let $x_{\min} \in \widetilde{f}^{\alpha_1, \alpha_0}(x_0)$ be the unique minimum of
$\bigcup_{\alpha < \alpha_0} \widetilde{f}^{\alpha, \alpha_0}(x_0)$.
Then $\{x_{\min}\} = \widetilde{f}^{\alpha_1, \alpha}(x)$,
which implies
\[
x \in \mathcal{Y}^\alpha \cap (\widetilde{f}^{\alpha_1, \alpha})^{-1}(\widetilde{X}^{\alpha_1})
= (\widetilde{f}^{\alpha_1, \alpha})^{-1}(\mathcal{Y}^{\alpha_1}).
\]
Since $\widetilde{f}^{\alpha_1, \alpha}(x) = \{x_{\min}\}$ consists of one point,
this implies that $x_{\min} \in \mathcal{Y}^{\alpha_1}$.
Since any point $x' \in \widetilde{f}^{\alpha, \alpha_0}(x_0)$ satisfies
$(x', x_{\min}) \in \widetilde{f}^{\alpha_1, \alpha}$,
this implies that
$x' \in (\widetilde{f}^{\alpha_1, \alpha})^{-1}(\mathcal{Y}^{\alpha_1}) \subset \mathcal{Y}^\alpha$.
Hence $\widetilde{f}^{\alpha, \alpha_0}(x_0) \subset \mathcal{Y}^\alpha$
for any $x_0 \in (\widetilde{f}^{\alpha, \alpha_0})^{-1}(\mathcal{Y}^\alpha)$.
We also note that this implies that
$(\widetilde{f}^{\alpha, \alpha_0})^{-1}(\mathcal{Y}^\alpha)$ is open
in the relative topology of $(\widetilde{f}^{\alpha, \alpha_0})^{-1}(\widetilde{X}^\alpha)$.
We can construct such meet-semilattice covers
by the same argument as the proof of Lemma \ref{extension of meet-semilattice cover}.
Namely, we can extend
$\bigcup_{\beta < \alpha} (\widetilde{f}^{\beta, \alpha})^{-1}(\mathcal{Y}^\beta)
\subset \widetilde{X}^\alpha$ to the cover of $X^\alpha$
by the induction in $\alpha \in \mathcal{A}$.
Similarly, we can construct decreasing sequences of meet-semilattice covers
$\mathcal{Y}^\alpha_k \subset \widetilde{X}^\alpha$ ($\alpha \in \mathcal{A}$, $k \geq 1$) such that
$\mathcal{Y}^\alpha_{k+1} \Subset \mathcal{Y}^\alpha_k$ and
\[
\mathcal{Y}^\alpha_k \cap (\widetilde{f}^{\beta, \alpha})^{-1}(\widetilde{X}^\beta)
= (\widetilde{f}^{\beta, \alpha})^{-1}(\mathcal{Y}^\beta_k)
\]
for $k \geq 1$ and all pairs $\alpha, \beta \in \mathcal{A}$ such that $\alpha > \beta$.
(This is because Condition \ref{properness of tilde f} implies that
$(\widetilde{f}^{\beta, \alpha})^{-1}(\mathcal{Y}^\beta_{k+1}) \Subset
(\widetilde{f}^{\beta, \alpha})^{-1}(\mathcal{Y}^\beta_k)$.)
We attach an integer $k_\alpha \geq 1$ for each $\alpha \in \mathcal{A}$
so that $k_\alpha > k_\beta$ if $\alpha > \beta$.
Then by the induction in $\alpha \in \mathcal{A}$, we can
construct a compatible family of weakly good coordinate systems
$(x_\alpha, \mathcal{V}^\alpha_{x_\alpha})_{x_\alpha \in P^\alpha}$ of $X^\alpha$
for $\alpha \in \mathcal{A}$ such that
$\mathcal{Y}^\alpha_{k_\alpha + 1} \Subset \bigcup_{x_\alpha \in P^\alpha}
\widetilde{\psi}_{x_\alpha}((s^\alpha_{x_\alpha})^{-1}(0) \cap \mathcal{V}^\alpha_{x_\alpha})
\subset \mathcal{Y}^\alpha_{k_\alpha}$.
(We can apply the argument of
Lemma \ref{weakly good coordinate from meet-semilattice}.
Only non-trivial condition is
Condition \ref{compatibility of meet-semilattice covers},
but it is satisfied if
we construct $(x_\alpha, \mathcal{V}^\alpha_{x_\alpha})_{x_\alpha \in P^\alpha}$ inductively
so that they satisfy
$\mathcal{Y}^\alpha_{k_\alpha + 1} \Subset \bigcup_{x_\alpha \in P^\alpha}
\widetilde{\psi}_{x_\alpha}((s^\alpha_{x_\alpha})^{-1}(0) \cap \mathcal{V}^\alpha_{x_\alpha})
\subset \mathcal{Y}^\alpha_{k_\alpha}$ for all $\alpha \in \mathcal{A}$.
The other conditions hold if each $\mathcal{V}_{x_\alpha}^\alpha$ is sufficiently small
and each $\mathcal{V}_{x_\alpha}^\alpha$ is a sufficiently small neighborhood of
$(s_{x_\alpha}^\alpha)^{-1}(0) \cap \mathcal{V}_{x_\alpha}^\alpha$.)
\begin{defi}
Let $((X^\alpha)_{\alpha \in \mathcal{A}}, (f^{\beta, \alpha})_{\alpha > \beta})$
be a compatible system of multi-valued partial submersions,
and let $(x_\alpha, \mathcal{V}^\alpha_{x_\alpha})_{x_\alpha \in P^\alpha}$ be a
compatible family of weakly good coordinate systems of $X^\alpha$
for $\alpha \in \mathcal{A}$.
For each $\alpha \in \mathcal{A}$,
let $(\boldsymbol{\epsilon}^\alpha_{x_\alpha})_{x_\alpha \in P^\alpha}$ be
a grouped multisection of $(x_\alpha, \mathcal{V}^\alpha_{x_\alpha})_{x_\alpha \in P^\alpha}$.
We say these grouped multisections are compatible if
for any $\alpha > \beta \in \mathcal{A}$,
$(\boldsymbol{\epsilon}^\alpha_{x_\alpha})_{x_\alpha \in P^\alpha}$ and
$(\boldsymbol{\epsilon}^\beta_{x_\beta})_{x_\beta \in P^\beta}$ are
compatible with respect to $f^{\beta, \alpha}$.
\end{defi}
\begin{prop}
Let $((X^\alpha)_{\alpha \in \mathcal{A}}, (f^{\beta, \alpha})_{\alpha > \beta})$
be a compatible system of multi-valued partial submersions,
and let $(x_\alpha, \mathcal{V}^\alpha_{x_\alpha})_{x_\alpha \in P^\alpha}$ be a
compatible family of weakly good coordinate systems of $X^\alpha$
for $\alpha \in \mathcal{A}$.
Let $\alpha_0 \in \mathcal{A}$ be an arbitrary index and
assume that a compatible family of grouped multisections
$(\boldsymbol{\epsilon}^\alpha_{x_\alpha})_{x_\alpha \in P^\alpha}$
for $\alpha < \alpha_0$ are given.
Then we can construct a grouped multisection
$(\boldsymbol{\epsilon}^{\alpha_0}_{x_{\alpha_0}})_{x_{\alpha_0} \in P^{\alpha_0}}$
of $(x_{\alpha_0}, \mathcal{V}^{\alpha_0}_{x_{\alpha_0}})_{x_{\alpha_0} \in P^{\alpha_0}}$
which is compatible with
$(\boldsymbol{\epsilon}^\alpha_{x_\alpha})_{x_\alpha \in P^\alpha}$ $(\alpha < \alpha_0)$.
\end{prop}
\begin{proof}
We saw in Lemma \ref{compatible pull backs for multi-valued partial submersion}
that for any $x_{\alpha_0} \in P^{\alpha_0}$ and $\widehat{x}_{\alpha_0}^{(\alpha, k)}
\in \widetilde{X}^\alpha$ such that
$(x_{\alpha_0}, \widehat{x}_{\alpha_0}^{(\alpha, k)}) \in \widetilde{f}^{\alpha, \alpha_0}$,
the pull back $(\varphi^\alpha_{x_{(\alpha, k)}, \widehat{x}_{\alpha_0}^{(\alpha, k)}})^\ast
\boldsymbol{\epsilon}^\alpha_{x_{(\alpha, k)}}$ does not depend on
$x_{(\alpha, k)} \in P^\alpha(\widehat{x}_{\alpha_0}^{(\alpha, k)})$
on some neighborhood $\mathcal{U}_{\widehat{x}_{\alpha_0}^{(\alpha, k)}| x_{\alpha_0}}$ of
$\varphi^{f^{\alpha, \alpha_0}}_{\widehat{x}_{\alpha_0}^{(\alpha, k)}, x_{\alpha_0}}
(\mathcal{V}^{\alpha_0}_{x_{\alpha_0}})$.
Shrinking $\mathcal{U}_{\widehat{x}_{\alpha_0}^{(\alpha, k)}| x_{\alpha_0}}$ if necessary,
we may assume that
\[
\varphi^{f^{\beta, \alpha}}_{\widehat{x}_{\alpha_0}^{(\beta, k')},
\widehat{x}_{\alpha_0}^{(\alpha, k)}}(\mathcal{U}_{\widehat{x}_{\alpha_0}^{(\alpha, k)}| x_{\alpha_0}})
\subset \mathcal{U}_{\widehat{x}_{\alpha_0}^{(\beta, k')}| x_{\alpha_0}}
\]
for $(\widehat{x}_{\alpha_0}^{(\alpha, k)}, \widehat{x}_{\alpha_0}^{(\beta, k')})
\in \widetilde{f}^{\beta, \alpha}$.
Define a partial order of
$\bigcup_{\alpha \leq \alpha_0} \widetilde{f}^{\alpha, \alpha_0}(x_{\alpha_0})$ by
the condition that
$\widehat{x}_{\alpha_0}^{(\alpha, k)} > \widehat{x}_{\alpha_0}^{(\beta, k')}$ if
$(\widehat{x}_{\alpha_0}^{(\alpha, k)}, \widehat{x}_{\alpha_0}^{(\beta, k')})
\in \widetilde{f}^{\beta, \alpha}$.
Then
\[
((\mathcal{U}_{\widehat{x}_{\alpha_0}^{(\alpha, k)}| x_{\alpha_0}})
_{\widehat{x}_{\alpha_0}^{(\alpha, k)} \in
\bigcup_{\alpha \leq \alpha_0} \widetilde{f}^{\alpha, \alpha_0}(x_{\alpha_0})},
(\varphi^{f^{\beta, \alpha}}_{\widehat{x}_{\alpha_0}^{(\beta, k')},
\widehat{x}_{\alpha_0}^{(\alpha, k)}},
\hat \varphi^{f^{\beta, \alpha}}_{\widehat{x}_{\alpha_0}^{(\beta, k')},
\widehat{x}_{\alpha_0}^{(\alpha, k)}})
_{\widehat{x}_{\alpha_0}^{(\alpha, k)} > \widehat{x}_{\alpha_0}^{(\beta, k')}})
\]
is a compatible system of multi-valued partial submersions in the sense of
Definition \ref{compatible system of multi-valued partial submersions for orbibundles},
where $\mathcal{U}_{x_{\alpha_0}| x_{\alpha_0}} = \mathcal{V}^{\alpha_0}_{x_{\alpha_0}}$.
We claim that $(\varphi^\alpha_{x_{(\alpha, k)}, \widehat{x}_{\alpha_0}^{(\alpha, k)}})^\ast
\boldsymbol{\epsilon}^\alpha_{x_{(\alpha, k)}}$ are compatible
with respect to this system.
This implies that using
Lemma \ref{extension of grouped multisection for multi-valued partial submersions}
and \ref{extension of grouped multisection for embedding},
we can construct a compatible grouped multisection
$(\boldsymbol{\epsilon}^{\alpha_0}_{x_{\alpha_0}})_{x_{\alpha_0} \in P^{\alpha_0}}$
of $(x_{\alpha_0}, \mathcal{V}^{\alpha_0}_{x_{\alpha_0}})_{x_{\alpha_0} \in P^{\alpha_0}}$.
The above claim is proved as follows.
For any $\widehat{x}_{\alpha_0}^{(\alpha, k)}, \widehat{x}_{\alpha_0}^{(\beta, k')}
\in \bigcup_{\alpha \leq \alpha_0} \widetilde{f}^{\alpha, \alpha_0}(x_{\alpha_0})$ such that
$(\widehat{x}_{\alpha_0}^{(\alpha, k)}, \widehat{x}_{\alpha_0}^{(\beta, k')}) \in
\widetilde{f}^{\beta, \alpha}$ and any
$x_{(\alpha, k)} \in P^\alpha(\widehat{x}_{\alpha_0}^{(\alpha, k)})$,
there exists some $\widehat{x}_{(\alpha, k)}^{(\beta, k')} \in
\widetilde{f}^{\beta, \alpha}(x_{(\alpha, k)})$ such that
\begin{equation}
\widehat{x}_{\alpha_0}^{(\beta, k')} \in
\widetilde{\psi}^\beta_{\widehat{x}_{(\alpha, k)}^{(\beta, k')}}
\Bigl(\bigl(s^\beta_{\widehat{x}_{(\alpha, k)}^{(\beta, k')}}\bigr)^{-1}(0) \cap
\varphi^{f^{\beta, \alpha}}_{\widehat{x}_{(\alpha, k)}^{(\beta, k')}, x_{(\alpha, k)}}
(\mathcal{V}^\alpha_{x_{(\alpha, k)}})\Bigr).
\label{(alpha, k) (beta, k') relation}
\end{equation}
This is because Condition \ref{varphi condition for multi-valued partial submersion}
for $f^{\beta, \alpha}$ implies
\[
\coprod_{\widehat{x}_{(\alpha, k)}^{(\beta, k')}
\in \widetilde{f}^{\beta, \alpha}(x_{(\alpha, k)})}
(\widetilde{\psi}^\beta_{\widehat{x}_{(\alpha, k)}^{(\beta, k')}}
\circ \varphi^{f^{\beta, \alpha}}_{\widehat{x}_{(\alpha, k)}^{(\beta, k')}, x_{(\alpha, k)}})
= \widetilde{f}^{\beta, \alpha} \circ \widetilde{\psi}^\alpha_{x_{(\alpha, k)}}
\]
and $\widetilde{\psi}^\alpha_{x_{(\alpha, k)}}
((s^\alpha_{x_{(\alpha, k)}})^{-1}(0) \cap \mathcal{V}^\alpha_{x_{(\alpha, k)}})$
contains $\widehat{x}_{\alpha_0}^{(\alpha, k)}$.
Note that (\ref{(alpha, k) (beta, k') relation}) and
Condition \ref{(x1, widehat x1, x2)-composition} for $f^{\beta, \alpha}$ imply
$P^\beta(\widehat{x}_{(\alpha, k)}^{(\beta, k')}) \subset
P^\beta(\widehat{x}_{\alpha_0}^{(\beta, k')})$.
Choose one point $x_{(\beta, k')} \in P^\beta(\widehat{x}_{(\alpha, k)}^{(\beta, k')})$.
It is enough to prove that
$(\varphi^\alpha_{x_{(\alpha, k)}, \widehat{x}_{\alpha_0}^{(\alpha, k)}})^\ast
\boldsymbol{\epsilon}^\alpha_{x_{(\alpha, k)}}$
on $\mathcal{U}_{\widehat{x}_{\alpha_0}^{(\alpha, k)}| x_{\alpha_0}}$ and
$(\varphi^\beta_{x_{(\beta, k')}, \widehat{x}_{\alpha_0}^{(\beta, k')}})^\ast
\boldsymbol{\epsilon}^\beta_{x_{(\beta, k')}}$
on $\mathcal{U}_{\widehat{x}_{\alpha_0}^{(\beta, k')}| x_{\alpha_0}}$ are
$(\varphi^{f^{\beta, \alpha}}_{\widehat{x}_{\alpha_0}^{(\beta, k')},
\widehat{x}_{\alpha_0}^{(\alpha, k)}},
\hat \varphi^{f^{\beta, \alpha}}_{\widehat{x}_{\alpha_0}^{(\beta, k')},
\widehat{x}_{\alpha_0}^{(\alpha, k)}})$-related,
and this follows from the assumption that
$\boldsymbol{\epsilon}^\alpha_{x_{(\alpha, k)}}$ and
$\boldsymbol{\epsilon}^\beta_{x_{(\beta, k')}}$ are
$(\varphi^\beta_{x_{(\beta, k')}, \widehat{x}_{(\alpha, k)}^{(\beta, k')}} \circ
\varphi^{f^{\beta, \alpha}}_{\widehat{x}_{(\alpha, k)}^{(\beta, k')}, x_{(\alpha, k)}},
\hat \varphi^\beta_{x_{(\beta, k')}, \widehat{x}_{(\alpha, k)}^{(\beta, k')}} \circ
\hat \varphi^{f^{\beta, \alpha}}_{\widehat{x}_{(\alpha, k)}^{(\beta, k')},
x_{(\alpha, k)}})$-related.
\end{proof}
We can treat the case of the fiber products $h^{-1}(K) \subset X$
of pre-Kuranishi spaces $X$ with simplicial complexes $K \subset \mathcal{W}$
by strong smooth submersions $h = (h_x)_{x \in \widetilde{X}}$
and essential submersions between them.
In this case, multi-valued partial essential submersion is defined on
a neighborhood of $h^{-1}(K) \subset X$, and we read the every compatibility
conditions as the conditions on a neighborhood of the fiber products.
Then we can also apply the same argument in this case.
\begin{rem}
Since we need to assume that grouped multisections are sufficiently small,
we cannot treat an infinite family of pre-Kuranishi spaces as a compatible
family of multi-valued partial submersions.
In application, we usually construct invariants for finite subfamilies of
pre-Kuranishi spaces and construct the invariant of the infinite family as a limit.
Hence it is enough to assume that grouped multisections are sufficiently small
for the construction of the invariant of a fixed subfamily and
the proof of its invariance.
\end{rem}
\subsection{Continuous family of multisections}
\label{continuous family of multisections}
First consider the following example.
Let $X$ be a $0$-dimensional pre-Kuranishi space and
$f = (f_x)_{x \in \widetilde{X}}$ be a strong smooth map from $X$ to a manifold $Y$.
We want to construct a perturbed multisection of $X$ such that
the induced perturbed multisection of $(f \times f)^{-1}(\Delta_Y) \subset X \times X$
also satisfies the transversal condition, but
it is impossible unless the perturbed multisection do not take zero or
$\dim Y = 0$.
To treat such a case, we use continuous family of multisections and make
the restriction of $f_x$ to every branch of the perturbed multisection submersive.
Continuous family of multisection were used in \cite{FOOO10} and
\cite{FOOO11}.
We recall its definition in our setting.
\begin{defi}
For an orbibundle chart $(\mathcal{V}, \mathcal{E})$,
let $D$ be a finite-dimensional oriented open disk, and
consider the pull back bundle $p_V^\ast E$ by the projection $p_V : V \times D \to V$.
Let $s^\omega : V \times D \to p_V^\ast E$ ($\omega \in \Omega$)
be a family of smooth sections indexed by a finite $G_V$-set $\Omega$
such that $s^{g\omega} = g_\ast s^\omega$ for any $\omega \in \Omega$ and
$g \in G_V$.
Let $\alpha$ be a top-dimensional form on $D$ with compact support
such that $\int_D \alpha = 1$.
We call such a triple $\epsilon = (D, (s^\omega)_{\omega \in \Omega}, \alpha)$
a continuous family of multisections of $(\mathcal{V}, \mathcal{E})$.
We also define the version of grouped multisection similarly as follows.
A continuous family of grouped multisections
$\boldsymbol{\epsilon} = (D_j, (\epsilon^\omega)_{\omega \in \coprod_j \Omega_j},
\alpha_j)$ of an orbibundle chart $(\mathcal{V}, \mathcal{E})$
consists of the following.
$(D_j)_{j = 1, \dots, k}$ are finite number of finite dimensional oriented open disks,
and for each $j$, $s^\omega : V \times D_j \to p_V^\ast E$ ($\omega \in \Omega_j$) is
a family of smooth sections.
Each $\alpha_j$ is a top-dimensional form $\alpha_j$ on $D_j$ with compact support
which satisfies $\int_{D_j} \alpha_j = 1$.
We assume that there is an $G_V$-action on $\coprod_{1 \leq j \leq k} \Omega_j$
which preserves the decomposition and assume that
if $g \in G_V$ maps $\Omega_j$ to $\Omega_{j'}$, then
$D_j = D_{j'}$ and $\alpha_j = \alpha_{j'}$.
We also assume that the smooth sections satisfy
$s^{g\omega} = (g \times 1_{D_j})_\ast s^\omega$ for any $\omega \in \Omega_j$ and
$g \in G_V$.
For each $j$, we define $\epsilon_j = (D_j, (\epsilon^\omega)_{\omega \in \Omega_j},
\alpha_j)$ and also denote the family of grouped multisections by $\boldsymbol{\epsilon}
= \{\epsilon_j\}$.
\end{defi}
We define the support of each $\epsilon_j$ by
$\mathop{\mathrm{supp}}\nolimits(\epsilon_j) = \bigcup_{j \in \Omega_j} p_V(\mathop{\mathrm{supp}}\nolimits(\epsilon^\omega)) \subset V$.
For a connected open subset $\mathcal{U} \subset \mathcal{V}$, the restriction of
a family of grouped multisections $\boldsymbol{\epsilon}
= (D_j, (\epsilon^\omega)_{\omega \in \coprod_j \Omega_j}, \alpha_j)$ of
$(\mathcal{V}, \mathcal{E})$ to $(\mathcal{U}, \mathcal{E}|_{\mathcal{U}})$ is defined by
\[
\boldsymbol{\epsilon}|_\mathcal{U} = ((D_j)_{j \in I_U},
(\epsilon^\omega|_{U \times D_j})_{\omega \in \coprod_{j \in I_U} \Omega_j},
(\alpha_j)_{j \in I_U}),
\]
where $I_U = \{j; \mathop{\mathrm{supp}}\nolimits(\epsilon_j) \cap U \neq \emptyset\}$.
We can similarly define $(\varphi, \hat \varphi)$-relation of continuous families of
grouped multisections for an embedding $(\varphi, \hat \varphi)$ between
orbibundle charts, and pull back of a continuous family of grouped multisections
by a submersion.
For a smooth section $s$ and a continuous family of grouped multisections
$\boldsymbol{\epsilon} = (D_j, (\epsilon^\omega)_{\omega \in \coprod_j \Omega_j},
\alpha_j)$ of an oribibundle chart, we define their sum
by the continuous family of multisections
\[
s + \boldsymbol{\epsilon}
= \bigl(\prod_j D_j, \bigl(s + \sum_j \epsilon^{\omega_j}\bigr)
_{(\omega_j) \in \prod_j \Omega_j},
\alpha_1 \wedge \dots \wedge \alpha_k\bigr).
\]
Let $f = (f_x)_{x \in \widetilde{X}}$ be a strong smooth map from a pre-Kuranishi space
$X$ to a manifold $Y$ such that each $f_x : \mathcal{V}_x \to Y$ is submersive.
Then for a weakly good coordinate system $(x, \mathcal{V}_x)_{x \in P}$ of $X$,
similarly to Lemma \ref{construction of grouped multisection},
shrinking $\mathcal{V}_x$ slightly if necessary,
we can construct a continuous family of grouped multisections
$\boldsymbol{\epsilon} = (\boldsymbol{\epsilon}_x)_{x \in P}$ for $(x, \mathcal{V}_x)_{x \in P}$
which satisfies the following transversality condition:
For any orbibundle chart $(\mathcal{V}, \mathcal{E})$ in $(\mathcal{V}_x, \mathcal{E}_x)$,
every branch of the multisection $s|_\mathcal{V} + \boldsymbol{\epsilon}_x|_\mathcal{V}$ is
transverse to the zero section, and the restriction of $f_x$ to its zero set is
submersive.
Furthermore, the same holds for the restriction to the corners of $V$.
For a continuous family of perturbed multisections,
it is not suitable to represent the virtual fundamental chain as a singular chain.
Instead, for strong smooth maps $f = (f_x)_{x \in \widetilde{X}}$ from $X$
to a manifold $Y$ and $h = (h_x)_{x \in \widetilde{X}}$ from $X$ to a manifold $Z$,
we represent the virtual fundamental chain as a linear map
$(h_! \circ f^\ast)_X : \Omega(Y) \to \Omega(Z)$.
This map is defined as follows.
As in the usual case, we take a partition of unity $(\beta_x)_{x \in P}$
subordinate to $(x, \mathcal{V}_x)_{x \in P}$, finite number of orbibundle charts
$(\mathcal{V}_\tau, \mathcal{E}_\tau)_{\tau \in T_x}$ of $(\mathcal{V}_x, \mathcal{E}_x)$ and smooth functions
$\beta_\tau : \mathcal{V}_\tau \to \mathbb{R}$ with compact support such that
$\beta_x = \sum_{\tau \in T_x} \beta_\tau$.
Then for each differential form $\theta \in \Omega(Y)$,
$(h_! \circ f^\ast)_X \theta \in \Omega(Z)$ is defined by
\begin{align*}
&(h_! \circ f^\ast)_X \theta \\
&= \sum_{\substack{x \in P \\ \tau \in T_x}}
\frac{\sum_{(\omega_j) \in \prod_j \Omega_{\tau, j}}
\Bigl(h_x|_{\{s_\tau^{(\omega_j)} = 0\}}\Bigr)_{\textstyle !}\,
\Bigl(\beta_\tau \cdot (f_x|_{V_\tau})^\ast
\theta \wedge \alpha_1 \wedge \dots \wedge \alpha_k|_{\{s_\tau^{(\omega_j)} = 0\}}\Bigr)}
{\# G_{V_\tau
\cdot \prod_j \# \Omega_{\tau, j}}
\end{align*}
instead of Equation (\ref{virtual fundamental chain by forms}).
In this equation, we define the orientation of the fiber of
$h_x|_{\{s_\tau^{(\omega_j)} = 0\}}$ similarly to the usual case
using the following orientation of $V_x \times \prod_j D_j$.
The orientation of $V_x \times \prod_j D_j$ is defined by
$T(V_x \times \prod_j D_j) = (-1)^{(\sum_j \dim D_j) \mathop{\mathrm{rank}}\nolimits E_x}
T V_x \oplus \bigoplus_j T D_j$.
\section{Introduction}
The aim of this paper is to provide a construction of symplectic field theory (SFT).
SFT is a theory of contact manifolds and symplectic manifolds
with cylindrical ends proposed by Eliashberg, Givental and Hofer in \cite{EGH00}.
It is a generalization of contact homology and Gromov-Witten invariant, and it is
constructed by counting the number of appropriate pseudo-holomorphic
curves in the symplectization of a contact manifold or a symplectic manifold
with cylindrical ends.
In general, we need perturbation to obtain transversality of moduli spaces
of pseudo-holomorphic curves,
and it was a difficult problem to carry out perturbation
with compatibility conditions required for the construction of the algebras.
To give a concrete and transparent proof of the construction,
Hofer, Wysocki and Zehnder
developed the theory of polyfold (\cite{HWZ07I}-\cite{HWZ10II}).
However, they have not yet published a complete proof of
the construction of SFT.
There were various other attempts to overcome this difficulty in special cases.
For example, cylindrical contact homology of some three-dimensional contact manifolds
was constructed by Bao and Honda \cite{BH15} and Hutchings and Nelson \cite{HN15}.
Recently, Contact homology was constructed by Pardon \cite{Pa15}
and Bao and Honda \cite{BH16} independently.
However, the general SFT has not yet been fully constructed.
The main result of this paper is construction of SFT in full generality.
\begin{thm}\label{main result}
For each closed contact manifold $(Y, \xi)$ and each finite subset
$\overline{K}^0 \subset H_\ast(Y, \mathbb{Q})$, we can define
SFT cohomology $H^\ast_{\mathrm{SFT}}(Y, \xi, \overline{K}^0)$,
rational SFT cohomology $H^\ast_{\mathrm{RSFT}}(Y, \xi, \overline{K}^0)$ and
contact homology $H^\ast_{\mathrm{CH}}(Y, \xi, \overline{K}^0)$
as invariants of $(Y, \xi, \overline{K}^0)$.
\end{thm}
In fact, we construct generating functions
defined in \cite{EGH00} for contact manifolds and symplectic manifolds with
cylindrical ends and prove all of their properties explained in \cite{EGH00}.
We also deal with Bott-Morse case (see Section \ref{asymptotic estimates} for
the definition of the Bott-Morse condition).
Some easy cases of Bott-Morse case was studied by Bourgeois in \cite{Bo02}.
We use the chain complex of triangulation of the space of periodic orbits instead of
Morse chain complex used in \cite{Bo02}.
Constructing SFT by a Bott-Morse contact form, we can calculate
the SFT cohomology of a contact manifold with $S^1$-action generated
by the Reeb vector field. For example, we can prove the following.
\begin{thm}\label{H vanishes}
Assume that $(Y, \xi)$ admits a contact form $\lambda$ whose Reeb flow defines
a locally free $S^1$-action on $Y$.
We also assume that all cycles in $\overline{K}^0$ are $S^1$-invariant.
Let $\overline{P}$ be the space of non-parametrized periodic orbits.
Then $H^\ast_{\mathrm{SFT}}(Y, \xi, \overline{K}^0)$ is the algebra
generated by $H_\ast(\overline{P}; \mathbb{R})$, $H^\ast_c (\overline{P}; \mathbb{R})$
and the variables $t_x$ $(x \in \overline{K}^0)$, $\hbar$ with the product defined by the
following commutative relations:
all variables are super-commutative except
\[
[p_c, q_\alpha] =\langle c, \alpha \rangle \hbar
\]
for all $c \in H_\ast(\overline{P}; \mathbb{R})$ and $\alpha \in H^\ast_c (\overline{P}; \mathbb{R})$,
where we denote the elements corresponding to $c$ or $\alpha$ by
$p_c$ or $q_\alpha$.
\end{thm}
We use the Kuranishi theory of Fukaya and Ono.
It is one of the general techniques to overcome the transversality problem
and it was first used
in \cite{FO99} for the construction of Gromov-Witten invariant and
Hamiltonian Floer Homology of symplectic manifolds.
We mainly follow the argument of \cite{FO99}.
We explain the new features of this paper briefly.
First we recall the general way to construct a Kuranishi neighborhood of a point in
a moduli space.
For example, consider a point $p = (\hat \Sigma, z, u)$ in the moduli space of
stable curves in a closed symplectic manifold $(M, \omega)$ with a compatible almost
complex structure $J$.
For simplicity, assume that the domain curve $(\hat \Sigma, z)$ is stable
and the automorphism group of $p$ is trivial.
Let $X$ be the deformation space of the domain curve $(\hat \Sigma, z)$.
For each $a = (\hat \Sigma_a, z_a) \in X$,
we construct a approximate solution $u_a$
of $J$-holomorphic equation, and consider the equation as a Fredholm map $F_a$
from $W^{1,p}(\hat \Sigma_a, u_a^\ast TM)$ to $L^p(\hat \Sigma_a,
{\textstyle\bigwedge}^{0,1} T^\ast \hat \Sigma_a \otimes_\mathbb{C} u_a^\ast TM)$, where $p > 2$.
We construct a finite vector space $E$ and a family of linear maps
$\lambda_a : E \to L^p(\hat \Sigma_a,
{\textstyle\bigwedge}^{0,1} T^\ast \hat \Sigma_a \otimes_\mathbb{C} u_a^\ast TM)$ which makes
each Fredholm map $F_a^+ = F_a \oplus \lambda_a :
W^{1,p}(\hat \Sigma_a, u_a^\ast TM) \oplus E \to L^p(\hat \Sigma_a,
{\textstyle\bigwedge}^{0,1} T^\ast \hat \Sigma_a \otimes_\mathbb{C} u_a^\ast TM)$ transverse to
zero.
Define $V = \bigcup_{a \in X} F_a^{-1}(0)$.
Then the zero set of the projection $s : V \to E$ is a neighborhood of $p$.
Roughly speaking, $(V, E, s)$ defines a Kuranishi neighborhood of $p$.
\begin{figure}
\centering
\includegraphics[width= 250pt]{Fig_Sigma0_z0.png}
\caption{$(\Sigma^0, z^0)$}\label{(Sigma0, z0)}
\includegraphics[width= 250pt]{Fig_Sigma1_z1.png}
\caption{$(\Sigma^1, z^1)$}\label{(Sigma1, z1)}
\end{figure}
For the construction of SFT, we count the $J$-holomorphic curves
in the symplectization $Y \times \mathbb{R}$ of a closed contact manifold $Y$.
Hence we consider the case of $M = Y \times \mathbb{R}$.
For example, consider the holomorphic building $(\Sigma^0, z^0, u^0)$
whose domain curve $(\Sigma^0, z^0)$ is as in Figure \ref{(Sigma0, z0)}.
(Holomoprhic buildings are the elements in the compactification of the space of
$J$-holomorphic curves. See Section \ref{space of holomorphic buildings}
for its definition.)
In the neighborhood of its domain curve $(\Sigma^0, z^0)$,
there is a curve like $(\Sigma^1, z^1)$ in Figure \ref{(Sigma1, z1)}.
However, we cannot consider the equation of $J$-holomorphic curves for
the curves like $(\Sigma^1, z^1)$ since they do not have floor structure.
This problem happens because we only consider the deformation of the domain curve
and ignore the deformation of the target space.
Therefore in this case, we need to use not the deformation space $X$
of the domain curve
but the space which parametrizes the deformation of the domain curve and
the deformation of the target space $\mathbb{R} \times Y$ simultaneously.
In Section \ref{construction of Kuranishi}, we define such a parameter space, and
construct an approximate solution and a Fredholm map for each of its points.
For the construction of the counterpart of chain homotopy in SFT, we need to treat
the space of $J$-holomoprhic curves in $1$-parameter family of symplectic manifolds
with cylindrical ends.
For a disjoint curve, we need to use the perturbation induced by the perturbations
for the connected components.
In the case of $1$-parameter family, this implies that the zero set of
the perturbed section for a disjoint curve is the fiber product of those
for the connected components over the parameter space.
However, in general, we cannot make the projections from the zero sets of the perturbed
sections for the connected components to the parameter space submersive,
which implies that the induced section
for the disjoint curve does not satisfy the transversality condition.
To overcome this problem, we use continuous family of perturbations.
(See Section \ref{continuous family of multisections} for its definition.)
It is a technique used in \cite{FOOO10} and \cite{FOOO11}.
Roughly speaking, instead of counting the number of zeros of the perturbed section,
we consider the perturbed section of the product of the moduli space with some
finite vector space and use the average of the number of zeros over the vector space.
If we use the product with appropriate vector spaces, then we can make the projection
from the zero sets to the parameter space submersive.
In the theory of Kuranishi structure, the smoothness of the Kuranishi structre is
one of its difficult part.
If we restrict on $0$- and $1$-dimensional Kuranishi spaces, then
often we do not need to consider the smoothness, but to use continuous
family of multisections, we cannot avoid this problem.
The difficulty is due to the fact that we need to use different Banach spaces for
different domain curves.
If the diffeomorphism type of the domain curve does not change, then the smoothness is
easy to prove since we can use the same Banach space by using diffeomorphisms.
However, if the diffeomorphism type changes, then we cannot identify the Banach spaces.
Hence we need to define artificially the smooth structure and prove the smoothness
of maps in Kuranishi theory (embeddings and evaluation maps).
Fukaya, Oh, Ohta and Ono treated this problem briefly in \cite{FOOO09II}, and
they explained the details of the argument in \cite{FOOO16}.
The key point is the following elementary fact:
If a continuous function $f$ on $\mathbb{R}$ is continuously differentiable on the complement of
a point, and the differential has a limit at this point, then $f$ is continuously
differentiable on the whole of $\mathbb{R}$.
In particular, we can prove the smoothness of $f$
if we check that the norm of its differentials converge to zero at this point.
This implies that it is enough to prove the convergence of the differentials at the strata
where the diffeomorphism type of the domain curve changes.
They proved the convergence
by estimating approximating solutions appearing in Newton's method.
We also prove the smoothness by estimating the limit of the differentials,
but we prove these estimates by
using the estimates of the implicit functions which define the solutions.
Using an appropriate family of identifications of the domain curves,
we estimate the differentials of the implicit functions
by direct calculation (Lemma \ref{estimates of implicit function 0} and
Corollary \ref{estimates of implicit function}).
Once we get the estimates of the implicit functions,
we can prove the estimates of the norm of the differentials of solutions
by Proposition \ref{asymptotic phi} and Corollary \ref{asymptotic estimates of phi}.
Another new feature of this paper is an improvement of the Kuranishi theory.
In the usual Kuranishi theory, the notion of good coordinate system is not
compatible with the product.
Hence usually, for the product space, we need to reconstruct the Kuranishi structures
of the factors from the good coordinate systems and
again construct a good coordinate system from the product of
the new Kuranishi structures.
Furthermore, we need to take care of the order of the product for the product of
more than two factors.
To avoid these complexities, we introduce the new notions of pre-Kuranishi structure and
its weakly good coordinate system.
Roughly speaking, a good coordinate system uses a total order,
but a weakly good coordinate system uses a meet-semilattice.
Similarly to meet-semilattice,
weakly good coordinate system is compatible with product, and we can directly use
their product for the product space.
This simplifies the construction of the algebra.
(See Section \ref{theory of Kuranishi structure}
for pre-Kuranishi sturcture and weakly good coordinate system.)
Finally, we explain about Bott-Morse case.
Bourgeois used Morse function on the space of periodic orbit, but instead,
we triangulate the space of periodic orbit and use the chain complex of the
simplicial complex.
Using this chain complex, we treat the most general case where
bad orbits appear as a subcomplex of the space of periodic orbits.
To construct the algebras by counting intersection numbers with simplices,
we need to use correction terms which correspond to cascades in \cite{Bo02}.
Since the algebra of SFT is more complicated than that of Contact homology,
the correction terms are also complicated.
Hence we need to solve algebraic equations to define appropriate correction terms.
(See Section \ref{algebra for correction}.)
In Bott-Morse case, we need to use the fiber product of pre-Kuranishi spaces
over an orbifold.
For example, we need to consider the fiber products with the diagonal
$\Delta_{\overline{P}}$ in $\overline{P} \times \overline{P}$,
where $\overline{P}$ is the space of non-parametrized periodic orbits.
We treat $\Delta_{\overline{P}}$ not as a suborbifold of $\overline{P} \times \overline{P}$
but as a simplicial complex in $\overline{P} \times \overline{P}$.
(See Definition \ref{def of fiber product of pre-Kuranishi over orbifold} and
Example \ref{fiber prod of pre-Kuranishi with diagonal of orbifold}.)
Although the fiber product of Kuranishi spaces over a manifold was treated before,
this paper is the first which treats the case of orbifold.
We briefly explain the outline of this paper.
First we investigate the local behavior and asymptotic behavior of pseudo-holomorphic
curves in Section \ref{local asymptotic estimates}.
Using them, we define the topology of the moduli space of holomorphic buildings and
prove its topological properties in Section \ref{space of holomorphic buildings}.
Next in Section \ref{theory of Kuranishi structure}, we recall the general theory of
Kuranishi structure and introduce the notions of pre-Kuranishi structure and
its weakly good coordinate system.
In Section \ref{construction of Kuranishi}, we construct a basic pre-Kuranishi structure
of the moduli space of holomorphic buildings.
In Section \ref{fiber prod}, we construct various fiber products of the basic pre-Kuranishi
spaces and construct their compatible multisections.
In this section, we also explain about how to treat the bad orbits.
Defining the orientations of the fiber products, we construct their virtual fundamental
chains, and using them, we construct the algebra.
In Section \ref{case of X} to \ref{composition},
we consider the cases of a symplectic manifold with cylindrical ends,
its $1$-parameter version and the composition of two symplectic cobordisms.
Using them, we prove that the algebras are invariants of contact manifolds in
Section \ref{independence}.
Finally in Section \ref{S^1 action}, we consider the calculation of the SFT cohomology of
contact manifolds with the $S^1$-action generated by the Reeb vector field.
\section{Local estimates and asymptotic estimates}\label{local asymptotic estimates}
Let $(Y,\lambda,J)$ be a triple which consists of
a closed ($2n-1$)-dimensional manifold $Y$ and a contact form $\lambda$,
and a compatible almost complex structure $J$ of $\xi = \mathop{\mathrm{Ker}}\nolimits \lambda$.
$\xi$ has a symplectic structure given by $d\lambda$, and compatibility of $J$
means $d\lambda(\cdot, J \cdot)$ is a hermitian metric on $\xi$.
We denote by $R_\lambda$ the Reeb vector field of $\lambda$, which is defined by
$\lambda(R_\lambda) =1$ and $i_{R_\lambda}d\lambda = 0$.
We say a loop $\gamma : S^1 \to Y$ is a periodic orbit of period $L = L_\gamma > 0$
if it satisfies $\partial_t \gamma(t) = L R_\lambda(\gamma(t))$.
We note that the period of a periodic orbit $\gamma$ can be expressed as
$L = \int_{S^1} \gamma^\ast \lambda$.
We denote the space of all periodic orbits by $P = P_Y \subset C^\infty(S^1, Y)$.
$S^1 = \mathbb{R} / \mathbb{Z}$ acts on $P$ by $(s \cdot \gamma) (t) = \gamma(t + s)$.
$\overline{P} = P/S^1$ is the space of non-parametrized periodic orbits.
For $L>0$, we denote by $P_L \subset P$ the subspace of periodic orbits with period $L>0$.
It is sometimes convenient to define $P_L$ for $L\leq 0$ by
\[
P_L = \{ \gamma \in C^\infty (S^1, Y); \partial_t \gamma - LR_\lambda(\gamma) = 0 \}.
\]
For example, $P_0 = Y$ is the space of constant loops.
(However, we do not count these loops as periodic orbits.)
Let $\hat Y := \mathbb{R} \times Y$ be the symplectization of $Y$.
The coordinate of its $\mathbb{R}$-component is denoted by $\sigma$.
We can extend the complex structure $J$ of $\xi$ to an almost complex structure of
$\hat Y$ by $J (\partial_\sigma) = R_\lambda$, which we still denote by $J$.
The hermitian metric $g$ of $T \hat Y$ is then defined by
$g(\cdot, \cdot) = (d\sigma \wedge \lambda + d\lambda) ( \cdot, J \cdot)$.
In this paper, we construct algebras by counting $J$-holomorphic curves
in manifolds of this type.
Note that if $u : \Sigma \to \hat Y$ is a $J$-holomorphic curve,
then its $\mathbb{R}$-translations $o_{\sigma_0} \circ u : \Sigma \to \hat Y$ are also
$J$-holomorphic, where $o_{\sigma_0} : \mathbb{R} \times Y \to \mathbb{R} \times Y$ ($\sigma_0 \in \mathbb{R}$)
are the translation maps
defined by $o_{\sigma_0}(\sigma, y) = (\sigma + \sigma_0, y)$.
Other symplectic manifolds we consider in this paper are symplectic manifolds
with cylindrical ends. (Sometimes these are called contact ends.)
A symplectic manifold $(X, \omega)$ has cylindrical ends if
there exist contact manifolds $(Y^\pm, \lambda^\pm)$, and
$X$ can be decomposed as
$X = (-\infty, 0] \times Y^- \cup Z \cup [0, \infty) \times Y^+$,
where $Z$ is a compact manifold with boundary $\partial Z = Y^- \coprod Y^+$,
and the symplectic form satisfies
$\omega|_{(-\infty, 0] \times Y^-} = d(e^\sigma \lambda^-)$
and $\omega|_{[0, \infty) \times Y^+} = d(e^\sigma \lambda^+)$.
An almost complex structure $J$ on $X$ is said to be compatible if
$\omega(\cdot, J \cdot)$ is a hermitian metric and the restriction of $J$ on $(-\infty, 0] \times Y^-$ and $[0, \infty) \times Y^+$ are obtained by some complex structures on $\xi^-$ and $\xi^+$ respectively as above.
Two energies of a $J$-holomorphic map $u : (\Sigma, j ) \to (\hat Y, J)$ from
a Riemann surface $(\Sigma, j)$ to $\hat Y$ are defined as follows.
One is
\[
E_{\hat \omega}(u) = \int_{\Sigma} u^\ast d\lambda
\]
and the other is
\[
E_{\lambda}(u) = \sup_{I \subset \mathbb{R}} \frac{1}{|I|}\int_{(\sigma\circ u)^{-1}(I)} u^\ast (d\sigma \wedge \lambda),
\]
where the sup is taken over all intervals $I\subset \mathbb{R}$, and $|I|$ is the length of $I$.
The original energy introduced by Hofer in \cite{Ho93} was
\[
\sup \{\int_\Sigma u^\ast d(\varphi \lambda); \varphi \in C^\infty(\mathbb{R}, [1/2, 1]), \varphi'\geq 0 \}.
\]
This is equivalent to $E_{\hat \omega}(u) + E_{\lambda}(u)$ up to constant factors.
We define the norm of the differential $du(z)$ by
\[
|du(z)|^2 = \frac{|du(z)\zeta|_g^2 + |du(z)j \zeta|_g^2}{|\zeta|_h^2},
\]
where $h$ is a hermitian metric on $\Sigma$ and $\zeta$ is a non-zero vector of $T_z\Sigma$.
This does not depend on $\zeta$ (but depends on $h$).
If $u$ is $J$-holomorphic, then
$\int_\Sigma |du|^2 \mathrm{vol} = \int_\Sigma u^\ast (d\sigma \wedge \lambda + d\lambda)$.
Decomposing the tangent space $T \hat Y$ as $T \hat Y = \mathbb{R} \partial_\sigma \oplus \mathbb{R} R_\lambda \oplus \xi$, we denote the $\xi$-component of $du$ by $d^\xi u$.
Then $E_{\hat \omega}$-norm of $u$ coincides with
$||d^\xi u||_{L^2}^2 = \int_{\Sigma} |d^\xi u|^2 \mathrm{vol}_{\Sigma}$.
The energies of a $J$-holomorphic map $u : (\Sigma, j) \to (X, J)$ are defined as follows.
One is
\[
E_{\hat \omega}(u) = \int_{\Sigma} u^\ast \hat\omega,
\]
where $\hat \omega$ is a (discontinuous) $2$-form defined by
$\hat\omega|_Z = \omega$, $\hat\omega|_{(-\infty,0] \times Y^-}= d\lambda^-$
and $\hat\omega|_{[0, \infty) \times Y^+} = d\lambda^+$.
Note that the integral is invariant by homotopy of $u$ with compact support
(or relative to the boundary $\partial \Sigma$).
The other energy is
\begin{align*}
E_{\lambda}(u) &= \max\biggl\{\sup_{I \subset (-\infty,0]} \frac{1}{|I|}
\int_{u^{-1}(I \times Y^-)} u^\ast (d\sigma \wedge \lambda^-),\\
&\quad \hphantom{\max\biggl(}
\sup_{I \subset [0, \infty)} \frac{1}{|I|}\int_{u^{-1}(I \times Y^+)}
u^\ast (d\sigma \wedge \lambda^+)\biggr\}.
\end{align*}
\subsection{Local estimates}
The local estimates of $J$-holomorphic curves given in this subsection are not new
and have been already written in various forms.
(See \cite{Ho93} for example.)
However, for the convenience of the subsequent sections,
we state and prove them.
We use the following notation.
For non-negative functions $A$ and $B$,
$A \lesssim B$ means there exists a constant $C>0$ such that $A \leq CB$.
$A \sim B$ means $A \lesssim B$ and $B \lesssim A$.
\begin{lem}\label{L^infty bound}
For any $C_0 > 0$, there exist $\delta >0$ and $C_1>0$ such that
any $J$-holomorphic map $u : B_r(0) \to \hat Y$
($B_r(0) \subset \mathbb{C}$ is a ball with radius $r>0$)
with energies $E_\lambda (u) \leq C_0$ and $E_{\hat \omega}(u) \leq \delta$
satisfies $r |du(0)| \leq C_1$.
\end{lem}
\begin{proof}
If this did not hold, there would exist a constant $C_0 >0$, a sequence $\delta_k \to 0$
and $J$-holomorphic maps $u_k : B_{r_k}(0) \to \hat Y$ such that
$E_\lambda(u_k) \leq C_0$, $E_{\hat \omega} (u_k) \leq \delta_k$
and $r_k |du_k(0)| \to \infty$.
The lemma below implies that we may assume
$\sup_{B_{r_k}(0)} |du_k(0)| \leq 2 |du_k(0)|$ by changing the center of the ball.
Rescaling the domain if necessary, we may assume $|du_k(0)| =1$.
In this case, the assumption implies $r_k \to \infty$.
Further we may assume $\sigma \circ u_k (0) = 0$ by $\mathbb{R}$-translation.
Then some subsequence of $u_k$ uniformly converges to a $J$-holomorphic map
$u_\infty : \mathbb{C} \to \hat Y$ such that
$|du_\infty(0)| = 1$, $E_\lambda(u_\infty) \leq C_0$ and $E_{\hat \omega} (u_\infty) = 0$.
$E_{\hat \omega} (u_\infty) = 0$ implies that the image of $du_\infty$ is contained
in the integrable subbundle $\mathbb{R} \partial_\sigma \oplus \mathbb{R} R_\lambda \subset T \hat Y$.
Hence the image of $u_\infty$ is contained in one of its leaves.
Each leaf is written as the image of a $J$-holomorphic map
$\Phi : \mathbb{C} \to \hat Y$ given by $\Phi(s + \sqrt{-1}t) = (s, \tilde\gamma(t))$,
where $\tilde \gamma : \mathbb{R} \to Y$ is an integral curve of $R_\lambda$.
Hence $u_\infty$ has a lift $\tilde u_\infty : \mathbb{C} \to \mathbb{C}$ such that
$|d\tilde u_\infty(0)| = 1$ and $u_\infty = \Phi \circ \tilde u_\infty$.
$E_\lambda(u_\infty) \leq C_0$ implies
\begin{align*}
\int_{\tilde u_\infty^{-1}(I \times \mathbb{R})} \tilde u_\infty^\ast(ds \wedge dt)
&= \int_{\tilde u_\infty^{-1}(I \times \mathbb{R})} |d\tilde u_\infty|^2 dsdt \\
&\leq C_0 |I| < \infty
\end{align*}
for any interval $I \subset \mathbb{R}$, which is a contradiction since any non-constant
holomorphic function on $\mathbb{C}$ takes all values except at most one value.
\end{proof}
\begin{lem}[\cite{HV92}]
Let $W$ be a complete metric space, and let $\varphi : W \to \mathbb{R}_{\geq 0}$ be
a continuous non-negative function.
For any $x_0 \in W$ and $r_0>0$,
there exist a point $x_1 \in B_{2r_0}(x_0)$ and $0 < r_1 < r_0$ such that
\[
\sup_{B_{r_1}(x_1)} \varphi \leq 2 \varphi(x_1) \text{ and }
r_0 \varphi(x_0) \leq r_1 \varphi(x_1).
\]
\end{lem}
\begin{lem}\label{preannulus}
For any $C_0 >0$, $l \geq 1$ and $\epsilon >0$,
there exist some $\delta >0 $, $A >0$ and $L_0 >0$ such that
any $J$-holomorphic map
$u : [-A, T + A] \times S^1 \to \hat Y$ ($T \geq 0$ is arbitrary) with energies
$E_\lambda (u) \leq C_0$ and $E_{\hat \omega}(u) \leq \delta$ satisfies
\[
\mathop{\mathrm{dist}}\nolimits_{C^l(S^1, \hat Y)} (o_{-\sigma_s} \circ u(s, \cdot), \bigcup_{|L| \leq L_0} P_L) < \epsilon \text{ for all } s \in [0,T],
\]
where $\sigma_s = \sigma(u(s,0))$, and we regard $P_L$ as a subset of $C^l(S^1, \hat Y)$ by the embedding
$Y = \{ 0 \} \times Y \hookrightarrow \hat Y$.
\end{lem}
\begin{proof}
Let $L_0 = 2C_1$ be the double of the constant of Lemma \ref{L^infty bound}.
Note that Lemma \ref{L^infty bound} implies that if $A> \frac{1}{2}$ then
$|du|_{L^\infty([-A + 1/2, T + A - 1/2] \times S^1)} \leq L_0$.
It is enough to prove the claim for $T=0$.
If this lemma did not hold,
there would exist some sequences $A_k \to \infty$ and $\delta_k \to 0$,
some constant $\epsilon >0$, and a sequence of $J$-holomorphic maps
$u_k : [-A_k, A_k] \times S^1 \to \hat Y$ such that
$E_\lambda(u_k) \leq C_0$, $E_{\hat \omega}(u_k) \leq \delta_k$ and
$\mathop{\mathrm{dist}}\nolimits_{C^l(S^1, \hat Y)} (o_{-\sigma_k} \circ u_k(0, \cdot), \bigcup_{|L| \leq L_0} P_L)
\geq \epsilon$.
We may assume $\sigma_k = \sigma(u_k(0,0)) = 0$.
Then a subsequence of $u_k$ uniformly converges to
a $J$-holomorphic map $u_\infty : \mathbb{R} \times S^1 \to \hat Y$ such that
$E_{\hat \omega}(u_\infty) = 0$ and $|du_\infty|_{L^\infty(\mathbb{R} \times S^1)} \leq L_0$.
We can deduce as follows that there exists some constant $|L|\leq L_0$ and
some periodic orbit $\gamma \in P_L$ such that
$u_\infty(s,t) = (L_\gamma s, \gamma(t))$, which contradicts the assumption
$\mathop{\mathrm{dist}}\nolimits_{C^l(S^1, \hat Y)} (u_k(0, \cdot) \bigcup_{|L| \leq L_0} P_L) \geq \epsilon$.
As in the proof of Lemma \ref{L^infty bound},
there exists an integral curve $\tilde \gamma : \mathbb{R} \to Y$ such that
the image of $u_\infty$ is contained in the image of the $J$-holomorphic map
$\Phi : \mathbb{C} \to \hat Y$ given by $\Phi(s + \sqrt{-1} t) = (s, \tilde \gamma (t))$.
If $u_\infty$ has a lift $\tilde u_\infty : \mathbb{R} \times S^1 \to \mathbb{C}$, then
$|d \tilde u_\infty|_{L^\infty(\mathbb{R} \times S^1)} < \infty$ implies $u_\infty$ is
a constant map.
(This is the case of $L = 0$.)
If $u_\infty$ does not have such a lift, then there exists $L \neq 0 \in \mathbb{R}$ such that
$u_\infty$ has a lift
\begin{align*}
\tilde u_\infty : \mathbb{R} \times S^1 \to \mathbb{C}/L\sqrt{-1} &\cong \mathbb{R} \times S^1\\
(Ls + \sqrt{-1} Lt) & \leftrightarrow (s, t)
\end{align*}
such that $(\tilde u_\infty)_\ast = 1$ on $\pi_1(\mathbb{R} \times S^1)$.
Since $\tilde u_\infty : \mathbb{R} \times S^1 \to \mathbb{R}\times S^1$ is a $J$-holomorphic map
such that $\sigma \tilde u_\infty(0,0) = 0$, this implies
$\tilde u_\infty(s,t) = (s, t + \theta)$ for some $\theta \in S^1$.
Hence $u_\infty(s,t) = (L s, \gamma(t))$,
where $\gamma(t) = \tilde \gamma(L(t + \theta)) : S^1 \to Y$.
The inequality $|du_\infty|_{L^\infty([0,A]\times S^1)} \leq L_0$ implies $|L| \leq L_0$.
\end{proof}
\begin{cor}\label{first annulus}
For any $C_0 >0$ and $\epsilon >0$,
there exist some $\delta >0 $, $A >0$ and $L_0 >0$ such that
for any $0\leq T\leq \infty$ and any $J$-hoomorphic map
$u : [-A, T + A] \times S^1 \to \hat Y$ with energies
$E_\lambda (u) \leq C_0$ and $E_{\hat \omega}(u) \leq \delta$,
there exists some $|L| \leq L_0$ such that $P_{L} \neq \emptyset$ and
\[
||\partial_t u - L R_\lambda(u) ||_{L^\infty ([0,T] \times S^1)} \leq \epsilon.
\]
\end{cor}
The case of a symplectic manifold $X = (-\infty, 0] \times Y^- \cup Z \cup [0, \infty) \times Y^+$ with cylindrical ends is similar.
\begin{lem}\label{L^infty bound for X}
For any $C_0 > 0$, there exist $\delta >0$ and $C_1>0$ such that
any $J$-holomorphic map $u : B_r(0) \to X$
with energies $E_\lambda (u) \leq C_0$ and $E_{\hat \omega}(u) \leq \delta$
satisfies $r |du(0)| \leq C_1$.
\end{lem}
\begin{proof}
If the claim did not hold, there would exist a constant $C_0 >0$, a sequence $\delta_k \to 0$ and
$J$-holomorphic maps $u_k : B_{r_k}(0) \to X$ such that
$E_\lambda(u_k) \leq C_0$, $E_{\hat \omega} (u_k) \leq \delta_k$ and
$r_k |du_k(0)| \to \infty$.
We may assume
$\sup_{B_{r_k}(0)} |du_k(0)| \leq 2 |du_k(0)|$.
Rescaling the domain if necessary, we may assume $|du_k(0)| =1$.
In this case, the assumption implies $r_k \to \infty$.
Lemma \ref{L^infty bound} implies there exists a constant $R>0$ such that
every $u_k(B_R(0))$ intersects with $Z$.
Hence some subsequence of $u_k$ uniformly converges to a $J$-holomorphic map
$u_\infty : \mathbb{C} \to X$ such that
$|du_\infty(0)| = 1$, $E_\lambda(u_\infty) \leq C_0$ and $E_{\hat \omega} (u_\infty) = 0$.
Since $du_\infty|_{u_\infty^{-1}(Z)} \equiv 0$,
if the image of $u$ intersects with the interior of $Z$,
unique continuation theorem implies $u_\infty$ is a constant map,
which is a contradiction.
On the other hand, if the image of $u_\infty$ does not intersect with the interior of $Z$,
the same argument as in Lemma \ref{L^infty bound} leads to a contradiction.
\end{proof}
\begin{lem}\label{preannulus for X}
For any $C_0 >0$, $l \geq 1$ and $\epsilon >0$,
there exist some $\delta >0 $, $A >0$ and $L_0 >0$ such that
any $J$-holomorphic map
$u : [-A, T + A] \times S^1 \to X$ with energies
$E_\lambda (u) \leq C_0$ and $E_{\hat \omega}(u) \leq \delta$ satisfies
\[
\mathop{\mathrm{dist}}\nolimits_{C^l(S^1, X)} (u(s, \cdot), (-\infty, 0] \times \bigcup_{|L| \leq L_0} (P_{Y^-})_L \cup Z
\cup [0,\infty) \times \bigcup_{|L| \leq L_0} (P_{Y^+})_L) < \epsilon
\]
for all $s \in [0,T]$,
where we regard each point $(\sigma, \gamma) \in (-\infty, 0] \times
\bigcup_{|L| \leq L_0} (P_{Y^-})_L$ as a loop
$(\sigma, \gamma(t)) \in C^l(S^1, (-\infty, 0] \times Y^-) \subset C^l(S^1, X)$,
each $x\in Z$ as a constant loop in $C^l(S^1, X)$,
and each $(\sigma, \gamma) \in [0, \infty) \times \bigcup_{|L| \leq L_0} (P_{Y^+})_L$
as a loop $(\sigma, \gamma(t)) \in C^l(S^1, [0, \infty) \times Y^+) \subset C^l(S^1, X)$.
\end{lem}
\begin{proof}
Let $L_0 = 2C_1$ be the double of the constant of Lemma \ref{L^infty bound for X}.
Let $A_0>0$ be the constant of Lemma \ref{preannulus} for $\hat Y^\pm$.
Then the claim holds if $u([-A_0, T + A_0] \times S^1)$ does not intersect with the interior of $Z$ by Lemma \ref{preannulus}.
It is enough to prove the claim for $T = 0$.
If it did not hold,
there would exist some sequences $A_k \to \infty$ and $\delta_k \to 0$, some constant
$\epsilon >0$, and a sequence of $J$-holomorphic maps
$u_k : [-A_k, A_k] \times S^1 \to \hat Y$ such that
$E_\lambda(u_k) \leq C_0$, $E_{\hat \omega}(u_k) \leq \delta_k$ and
\[
\mathop{\mathrm{dist}}\nolimits_{C^l(S^1, X)} (u_k(s, \cdot), (-\infty, 0] \times \bigcup_{|L| \leq L_0} (P_{Y^-})_L \cup
Z \cup [0,\infty) \times \bigcup_{|L| \leq L_0} (P_{Y^+})_L) \geq \epsilon.
\]
Since each $u_k([-A_0, A_0] \times S^1)$ intersects with $Z$,
a subsequence of $u_k$ uniformly converges to
a $J$-holomorphic map $u_\infty : \mathbb{R} \times S^1 \to X$ such that
$E_{\hat \omega}(u_\infty) = 0$ and $|du_\infty|_{L^\infty(\mathbb{R} \times S^1)} \leq L_0$.
Since $du_\infty|_{u_\infty^{-1}(Z)} \equiv 0$,
if the image of $u_\infty$ intersects with the interior of $Z$,
unique continuation theorem implies $u_\infty$ is a constant map, which is a contradiction.
On the other hand, if the image of $u_\infty$ does not intersect with the interior of $Z$,
then the same argument as in Lemma \ref{preannulus} leads to a contradiction.
\end{proof}
\begin{rem}
In the above Lemma,
$(-\infty, 0] \times \bigcup_{0 < |L| \leq L_0} (P_{Y^-})_L$,
$X = (-\infty, 0] \times P_0^- \cup Z \cup [0, \infty) \times P_0^+$ and
$[0, \infty) \times \bigcup_{0 < |L| \leq L_0} (P_{Y^+})_L$ are disjoint closed subsets.
Hence if $\epsilon >0$ is sufficiently small,
then it is independent of $s \in [0, T]$ which of these three
$u|_{\{s\} \times S^1}$ is close to.
\end{rem}
The following lemmas are well known.
See \cite{Gr85} or \cite{Hu97} for example.
\begin{lem}[Removal of Singularities]
Any $J$-holomorphic map $u : D \setminus 0 \to \hat Y$
{\rm(}or $u : D \setminus 0 \to X${\rm)}
with $||du||_{L^2} < \infty$ can be extended uniquely to a $J$-holomorphic map
$u : D \to \hat Y$ {\rm(}or $u : D \to X$ respectively{\rm)}.
\end{lem}
\begin{lem}[Monotonicity Lemma]\label{monotonicity lemma}
There exist some $r_0>0$ and $C>0$ such that
for any compact Riemann surface $\Sigma$ with or without boundary,
any non-constant $J$-holomorphic map $u : \Sigma \to \hat Y$
{\rm(}or $u : \Sigma \to X${\rm)},
any point $z_0\in \mathop{\mathrm{Int}}\nolimits \Sigma$ and any $0\leq r\leq r_0$, the following holds true.
If $u(\partial \Sigma) \cap B_r(u(z_0)) = \emptyset$ then
\[
||du||_{L^2(u^{-1}(B_r(z_0)))}^2 \geq C r^2.
\]
\end{lem}
\begin{lem}\label{L^infty diam}
For any disc $D_0 \Subset D$, there exist $\delta>0$ and
$C>0$ such that any $J$-holomorphic curve $u: D \to \hat Y$
{\rm(}or $u : D \to X${\rm)}
with $\mathrm{diam} \, u(D) \leq \delta$ satisfies
\[
||du||_{L^\infty(D_0)} \leq C \mathrm{diam} \, u(D).
\]
Similarly, if a $J$-holomorphic curve $u_0 : D \to \hat Y$
{\rm(}or $u_0 : D \to X${\rm)} is given,
then there exist $\delta > 0$ and $C >0$ such that for any $J$-holomorphic curve
$u: D \to \hat Y$ {\rm(}or $u : D \to X$ respectively{\rm)},
if $\mathop{\mathrm{dist}}\nolimits_{L^\infty(D)}(u, u_0) \leq \delta$ then
\[
||du - du_0||_{L^\infty(D_0)} \leq C \mathop{\mathrm{dist}}\nolimits_{L^\infty(D)} (u, u_0).
\]
\end{lem}
\subsection{Asymptotic estimates}\label{asymptotic estimates}
To obtain asymptotic estimates of the ends of $J$-holomorphic curves, we need to
assume that the contact form satisfies the following condition.
Recall that $P \subset C^\infty(S^1, Y)$ is the space of (parametrized) periodic orbits
of the Reeb flow of $(Y, \lambda)$.
Let $\mathrm{ev}_t : P \to Y$ be the evaluation map at $t \in S$ defined by
$\mathrm{ev}_t \gamma = \gamma(t)$.
\begin{defi}\label{def of Bott-Morse}
For each periodic orbit $\gamma \in P$, we define an $L^2$ self-adjoint operator
$A_\gamma : W^{1,2}(S^1, \gamma^\ast T \hat Y) \to
L^2(S^1, \gamma^\ast T \hat Y)$ by
\[
A_\gamma \xi = J(\gamma) (\nabla_t \xi - L_\gamma \nabla_\xi R_\lambda(\gamma)),
\]
where we regard $\gamma$ as an element of
$C^\infty(S^1, \{0 \} \times Y) \subset C^\infty (S^1, \hat Y)$.
We say $(Y, \lambda)$ satisfies the Bott-Morse condition
(or $(Y, \lambda)$ is Bott-Morse)
if $P \subset C^\infty(S^1 ,Y)$ is a countable union of closed manifolds,
and every operator $A_\gamma$ satisfies
$\mathop{\mathrm{Ker}}\nolimits A_\gamma = \mathbb{R} \partial_\sigma \oplus T_\gamma P$.
This condition can be stated by using the linearization of the Reeb flow
$\varphi_t^\lambda : Y \to Y$ as
\[
\mathop{\mathrm{Ker}}\nolimits ((\varphi^{\lambda}_{L_\gamma})_\ast -1 : T_{\gamma(0)} Y \to T_{\gamma(0)} Y)
= T_{\gamma(0)} \mathrm{ev}_0 P_{L_\gamma}
\]
for all periodic orbits $\gamma \in P$.
Note that the Bott-Morse condition implies that
each $P_{\leq L_0} = \coprod_{0< L \leq L_0} P_L$ consists of
finite closed manifolds.
We say $(Y, \lambda)$ satisfies the Morse condition
if it satisfies the Bott-Morse condition and $\overline{P}$ consists of discrete points.
Note that in this case, $\dim \mathop{\mathrm{Ker}}\nolimits A_\gamma = 2$ for all $\gamma \in P$.
\end{defi}
The above definition of Bott-Morse condition is more natural than that given
in \cite{Bo02} and \cite{BEHWZ03}.
(Their definition assumes another condition.)
In this paper, we always assume $(Y,\lambda)$ is Bott-Morse.
Under this condition,
we can prove more strict estimates on the curves appearing in
Corollary \ref{first annulus}.
\begin{prop}\label{second annulus}
Let $L \in \mathbb{R}$ be a constant such that $P_L \neq \emptyset$.
Then there exist constants $\epsilon > 0$, $\kappa> 0$ and $C>0$ such that
the following holds true.
For any $0 < T \leq \infty$ and any $J$-holomorphic map
$u : [0,T] \times S^1 \to \hat Y$ such that
$||\partial_t u - L R_\lambda(u) ||_{L^\infty([0,T] \times S^1)} \leq \epsilon$,
there exists $(b, \gamma) \in \mathbb{R} \times P_L$ such that
\[
\mathop{\mathrm{dist}}\nolimits (u(s,t), (Ls + b, \gamma(t)) )
\leq C (e^{-\kappa s} + e^{-\kappa (T-s)})
|| \partial_t u - L R_\lambda(u) ||_{L^\infty([0,T] \times S^1)}
\]
on $[0, T] \times S^1$.
\end{prop}
A similar estimate was proved in \cite{BEHWZ03} under their Bott-Morse condition.
If $T= \infty$ and $L >0$, we say $u$ is positively asymptotic to
a periodic orbit $\gamma \in P_L$.
If $T= \infty$ and $L<0$, we say $u$ is negatively asymptotic to a periodic orbit
$\gamma(-t) \in P_{|L|}$.
In this case, using a biholomorphism $(s,t) \mapsto (-s, -t)$,
we usually consider $u$ as a $J$-holomorphic map
$u : (-\infty, 0]\times S^1 \to \hat Y$ such that
$\lim_{s \to -\infty} u(s,t) = \gamma(-t)$.
This proposition and Corollary \ref{first annulus} imply the following.
\begin{cor}\label{third annulus}
For any constants $C_0 > 0$ and $\epsilon >0$,
there exist $\delta > 0$, $\kappa > 0$, $A > 0$ and $L_0 > 0$
such that the following holds true.
For any $0 \leq T \leq \infty$ and any $J$-holomorphic curve
$u : [-A, T + A] \times S^1 \to \hat Y$ with energies
$E_\lambda(u) \leq C_0$ and $E_{\hat \omega}(u) \leq \delta$,
there exists $L \in \mathbb{R}$ and $(b, \gamma) \in \mathbb{R} \times P_L$ such that
$|L| \leq L_0$ and
\[
\mathop{\mathrm{dist}}\nolimits ( u(s,t), (Ls + b, \gamma(t)) ) \leq \epsilon ( e^{-\kappa s} + e^{-\kappa (T - s)})
\]
for all $(s, t) \in [0, T] \times S^1$.
\end{cor}
\begin{rem}
The proof below implies that the constant $\kappa > 0$ in Proposition
\ref{second annulus}
can be taken arbitrary close to the minimum of the absolute values of the non-zero
eigenvalues of $A_\gamma$ ($\gamma \in P_L$).
(Instead, we need to take small $\epsilon > 0$.)
Note that in Corollary \ref{third annulus}, $L_0 > 0$ is determined by $C_0 > 0$ and
$\epsilon > 0$, and is independent of $\delta > 0$, $\kappa > 0$, and $A > 0$.
Therefore, also in Corollary \ref{third annulus}, the constant $\kappa > 0$ can be taken
arbitrary close to the minimum of the absolute values of the non-zero eigenvalues of
$A_\gamma$ ($\gamma \in P_{\leq L_0}$).
\end{rem}
To prove the above proposition, we need to rewrite the equation of $J$-holomorphic
curves in a neighborhood of a periodic orbit.
For each coordinate $\phi : B_\epsilon^m(0) \hookrightarrow P_L$ of $P_L$,
we take a family of open embeddings
$\psi_t : B_\epsilon^m(0) \times B^{2n-1-m}(0) \hookrightarrow Y$ ($t \in S^1$)
such that $\psi_t(x,0) = \mathrm{ev}_t \phi(x)$ for all $x\in B_\epsilon^m(0)$.
(The existence of such a family is due to the orientability of $Y$.)
First we show that if $\eta : S^1 \to \hat Y$ is a loop such that
$\eta(0) = (\sigma, \psi_0 (x,y))$,
then
\begin{gather}
|y| \lesssim ||\partial_t \eta - L R_\lambda(\eta)||_{L^\infty(S^1)}\label{y estimate}\\
\mathop{\mathrm{dist}}\nolimits_{C^1(S^1, \hat Y)}(\eta(t), (\sigma, \gamma(t)))
\lesssim ||\partial_t \eta - L R_\lambda(\eta)||_{L^\infty(S^1)}\label{C^1 estimate}
\end{gather}
where $\gamma(t) = \mathrm{ev}_t \phi(x)$.
(\ref{y estimate}) is because
\begin{align*}
|y| &\sim \mathop{\mathrm{dist}}\nolimits(\pi_Y \circ \eta(0), \mathrm{ev}_0 P_L)\\
&\lesssim \mathop{\mathrm{dist}}\nolimits (\pi_Y \circ \eta(0), \varphi^\lambda_L (\pi_Y \circ \eta(0)))
\text{ (by the Bott-Morse condition)} \\
&\lesssim ||\partial_t (\varphi^\lambda_{-Lt}(\pi_Y \circ \eta(t)))||_{L^\infty(S^1)}\\
&\lesssim ||\partial_t \eta - L R_\lambda(\eta)||_{L^\infty(S^1)},
\end{align*}
where $\pi_Y : \hat Y = \mathbb{R} \times Y \to Y$ is the projection.
(\ref{C^1 estimate}) is because
\begin{align*}
\mathop{\mathrm{dist}}\nolimits_{C^1(S^1, \hat Y)}(\eta(t), (\sigma, \gamma(t)))
&\leq \mathop{\mathrm{dist}}\nolimits_{C^1(S^1, \hat Y)}(\eta(t), (1 \times \varphi^\lambda_{Lt}) \circ \eta(0))\\
&\quad + \mathop{\mathrm{dist}}\nolimits_{C^1(S^1, \hat Y)}((1 \times \varphi^\lambda_{Lt}) \circ \eta(0),
(\sigma, \gamma(t)))\\
&\sim
\mathop{\mathrm{dist}}\nolimits_{C^1(S^1, \hat Y)}((1 \times \varphi^\lambda_{-L_t}) \circ \eta(t), \eta(0))\\
&\quad + \mathop{\mathrm{dist}}\nolimits_{\hat Y}(\eta(0), (\sigma, \gamma(0)))\\
&\lesssim ||\partial_t ((1 \times \varphi^\lambda_{-Lt}) \circ \eta (t))||_{L^\infty(S^1,
\hat Y)} + |y|\\
&\lesssim||\partial_t \eta - L R_\lambda(\eta)||_{L^\infty(S^1)}.
\end{align*}
Define a family of smooth maps $\hat\psi_{s,t} : \mathbb{R} \times B_\epsilon^m(0) \times B^{2n-1-m}(0) \hookrightarrow \mathbb{R} \times Y$
($(s,t) \in \mathbb{R} \times S^1$) by
$\hat\psi_{s,t} (\sigma, x, y) = (L s + \sigma, \psi_t(x, y))$.
Assume a smooth map $u : I \times S^1 \to \hat Y$ satisfies
$\pi_Y \circ u (I \times \{ t \} ) \subset \mathrm{Im} \psi_t$ for all $t \in S^1$.
Then $u$ can be written as
$u(s,t) = \hat \psi_{s,t} (v(s,t))$, where
$v : I \times S^1 \to \mathbb{R} \times B_\epsilon^m(0) \times B^{2n-1-m}(0)$
is a smooth function.
We regard $N_0 = \mathbb{R} \oplus \mathbb{R}^m \oplus 0^{2n - m - 1} \subset \mathbb{R}^{2n}$ as a subspace of $W^{1,2}(S^1, \mathbb{R}^{2n})$
consisting of constant functions.
Then (\ref{C^1 estimate}) implies that there exists $z^0_s \in N_0$ for each $s\in I$ such that
\begin{equation}
||v|_{\{s\} \times S^1} - z^0_s||_{W^{1,2}(S^1)} \lesssim ||\partial_t u - L R_\lambda(u)||_{L^\infty (\{s\} \times S^1)}\label{pre N_1 estimate}
\end{equation}
The equation
\[
(\partial_s u - L\partial_\sigma) + J(u) (\partial_t u - L R_\lambda(u)) = 0
\]
of $J$-holomorphic curve for $u$
is equivalent to the following equation of $v$.
\begin{align*}
\partial_s v + ((\hat\psi_{s,t})_\ast)^{-1}& J(\hat\psi_{s,t}(v)) (\hat\psi_{s,t})_\ast \partial_t v\\
&+ ((\hat\psi_{s,t})_\ast)^{-1} J(\hat\psi_{s,t}(v)) (\partial_t \hat\psi_{s,t}(v) - L R_\lambda(\hat\psi_{s,t}(v))) = 0
\end{align*}
Note that this equation is also $\mathbb{R}$-translation invariant, that is,
if $v$ is a solution of the equation then $v(s,t) + (b,0)$ also satisfies the equation for any $b\in \mathbb{R}$.
We regard the solution $v$ as a map
$v : I \to C^\infty(S^1, \mathbb{R} \times B_\epsilon^m(0) \times B^{2n-1-m}(0))
\allowbreak (\subset C^\infty(S^1, \mathbb{R}^{2n}))$.
Then the above equation has the following form.
\[
\partial_s v + F(v) = 0,
\]
where $F : W^{1,2}(S^1, \mathbb{R}^{2n}) \to L^2(S^1, \mathbb{R}^{2n})$ is a smooth Fredholm map
(more precisely, the domain of $F$ is an open neighborhood of
$0 \in W^{1,2}(S^1, \mathbb{R}^{2n})$)
which satisfies the following conditions:
\begin{itemize}
\item
$F$ maps $W^{k+1,2}(S^1, \mathbb{R}^{2n})$ to $W^{k,2}(S^1, \mathbb{R}^{2n})$ ($k \geq 0$).
\item
$F(v + \sigma) = F(v)$ for any $\sigma \in \mathbb{R} \oplus 0^m \subset N_0$.
\item
For any $z \in \mathbb{R} \times B_\epsilon^m(0) \subset N_0$, $F$ satisfies
$F(z)=0$ and $\mathop{\mathrm{Ker}}\nolimits DF(z) = N_0$
(This is exactly the Bott-Morse condition.)
\item
There exists a family of inner product $(g_t)_{t \in S}$ of the vector space $\mathbb{R}^{2n}$
which makes the operator $A = DF(0) : W^{1,2}(S^1, \mathbb{R}^{2n}) \to L^2(S^1, \mathbb{R}^{2n})$
$L^2$ self-adjoint.
(In this case, $g_t$ is the pull back of $g$ by $(1 \times \psi_t)_\ast$
at $0 \in \mathbb{R} \times B_\epsilon^m(0) \times B^{2n-1-m}(0)$.)
\end{itemize}
In the following, we denote by $\langle \cdot, \cdot \rangle$ and $|\cdot|$
the inner product and the norm of $L^2(S^1, \mathbb{R}^{2n})$ given by $g_t$ ($t \in S^1$)
respectively.
The norm of $W^{1,2}(S^1, \mathbb{R}^{2n})$ is equivalent to $|v^0| + |A v^1|$.
First note that (\ref{pre N_1 estimate}) implies
\begin{equation}
|A v(s)| \lesssim ||\partial_t u - L R_\lambda(u)||_{L^\infty (\{s \} \times S^1)}.
\label{A v}
\end{equation}
Next we estimate
\begin{align*}
\partial_s^2 \langle A v, A v \rangle
&= 4 \langle A^2 v, A^2 v \rangle
+ 6 \langle A (F(v) - DF(0) v), A^2 v \rangle \\
&\quad + 2|A (F(v) - DF(0) v)|^2
+ 2\langle \pi_1 (DF(v) - DF(0)) F(v), A^2 v \rangle.
\end{align*}
Let $\pi_{\mathbb{R}^m}$ be the second projection of $N_0 = \mathbb{R} \oplus \mathbb{R}^m$.
In the above equation,
\begin{align*}
|A(F(v) - DF(0)v)| &\lesssim (|\pi_{\mathbb{R}^m} v^0| + |A v|) |A^2 v|\\
|\pi_1 (DF(v) - DF(0))F(v)| &\lesssim (|\pi_{\mathbb{R}^m} v^0| + |A v|) |A^2 v|
\end{align*}
because
\begin{align*}
A(F(v) - DF(0)v)
&= A(F(\pi_{\mathbb{R}^m} v^0 + v^1) - F(\pi_{\mathbb{R}^m} v^0) - DF(\pi_{\mathbb{R}^m} v^0)v^1)\\
&\quad +A((DF(\pi_{\mathbb{R}^m} v^0) - DF(0))v^1)\\
&= A\int_0^1\int_0^1 D^2F(\pi_{\mathbb{R}^m} v^0 + \tau_1 \tau_2 v^1) \tau_1 v^1 \cdot v^1
d\tau_1 d\tau_2\\
&\quad + A \int_0^1 D^2F(\tau \pi_{\mathbb{R}^m} v^0) (\pi_{\mathbb{R}^m} v^0) \cdot v^1 d\tau,
\end{align*}
\begin{align*}
&\pi_1 (DF(v) - DF(0))F(v)\\
&= \pi_1 (DF(\pi_{\mathbb{R}^m} v^0 + v^1) - DF(0))(F(v) - F(v^0))\\
&= \pi_1 \int_0^1 D^2F(\tau_1 (\pi_{\mathbb{R}^m} v^0 + v^1)) d\tau_1 (\pi_{\mathbb{R}^m} v^0 + v^1)
\cdot \int_0^1 DF(v^0 + \tau_2 v^1) v^1 d\tau_2,
\end{align*}
and $D^2F$ satisfies
\[
||(D^2F)(v)\xi \cdot \eta||_{W^{k, 2}(S^1, \mathbb{R}^{2n})} \lesssim
\sum_{\substack{i, j \geq 1 \\ i + j = k + 2}}
||\xi||_{W^{i, 2}(S^1, \mathbb{R}^{2n})} ||\eta||_{W^{j, 2}(S^1, \mathbb{R}^{2n})}
\]
for all $k \geq 0$.
(This is because $F$ is a differential operator.)
Therefore, if $||\pi_{\mathbb{R}^m} v^0||_{L^\infty(I, N_0)}$ and
$||A\tilde v||_{L^\infty(I, L^2(S^1, \mathbb{R}^{2n}))}$ are sufficiently small
(this assumption is satisfied if $B_\epsilon^m(0)$ and
$||\partial_t u - L R_\lambda(u)||_{L^\infty(I\times S^1)}$ are sufficiently small),
then there exists $\epsilon \ll 1$ such that
\begin{align*}
\partial_s^2 \langle A v, A v \rangle
&\geq 4 |A^2 v|^2 - C (|\pi_{\mathbb{R}^m} v^0| + |A v|) |A^2 v|^2\\
&\geq (4- \epsilon) |A^2 v|^2\\
&\geq (4- \epsilon ) \kappa_0^2 |A v|^2
\end{align*}
for all $s \in I$, where $\kappa_0 > 0$ is the minimum of the absolute values of
the non-zero eigenvalues of $A$.
Therefore the lemma below (Lemma \ref{absolute annulus}) implies that if $I = [0,T]$ then
\begin{equation}\label{N_1 annulus}
|A v (s)| ^2 \leq ( e^{-\sqrt{4 - \epsilon} \kappa_0 s} + e^{-\sqrt{4 - \epsilon} \kappa_0 (T-s)}) ||A v||^2_{L^\infty(I, L^2(S^1, \mathbb{R}^{2n}))}.
\end{equation}
In particular,
\begin{equation}
||A v||_{L^1(I, L^2(S^1, \mathbb{R}^{2n}))}
\lesssim ||A v||_{L^\infty(I, L^2(S^1, \mathbb{R}^{2n}))} \label{L^1 Av}
\end{equation}
is an estimate uniform with respect to $|I|$.
Since $|\pi_0 F(v(s))| \lesssim |A v^1(s)|$,
the equation $\partial_s v^0 + \pi_0 F (v) = 0$ implies
\begin{equation}
|\partial_s v^0(s)| \lesssim |Av^1(s)|. \label{N_0 norm}
\end{equation}
(\ref{A v}), (\ref{L^1 Av}) and (\ref{N_0 norm}) implies
\begin{align}
||\partial_s v^0||_{L^1(I, N_0)} &\lesssim ||A v||_{L^1(I, L^2(S^1, \mathbb{R}^{2n}))}\notag \\
&\lesssim ||Av||_{L^\infty(I, L^2(S^1, \mathbb{R}^{2n}))} \notag \\
&\lesssim ||\partial_t u - L R_\lambda (u)||_{L^\infty(I \times S^1)}\label{N_0 variation}
\end{align}
Using the above argument, now we prove Proposition \ref{second annulus}.
\begin{proof}[Proof of Proposition \ref{second annulus}]
Suppose $\epsilon>0$ is sufficiently small and that
a $J$-holomorphic map $u : [0,T] \times S^1 \to \hat Y$ satisfies
$||\partial_t u - L R_\lambda(u) ||_{L^\infty([0,T] \times S^1)} \leq \epsilon$.
There exists a coordinate $\phi$ of $P_L$ such that
$\pi_Y u ([0,T] \times \{t\}) $ is contained in the image of $\psi_t$ for all $t \in S^1$
since inequality (\ref{N_0 variation}) implies the variation of $v^0$ on $[0,T]$ is small.
Equalities (\ref{A v}), (\ref{N_1 annulus}) and (\ref{N_0 norm}) imply that
for $z = v^0(T/2) \in N_0$,
\begin{align*}
|v^0 (s) - z| &= \int_{T / 2}^s |\partial_s v^0| |ds|\\
&\lesssim (e^{-\frac{1}{2}\sqrt{4 - \epsilon} \kappa_0 s}
+ e^{-\frac{1}{2}\sqrt{4 - \epsilon} \kappa_0 (T-s)})
|| \partial_t u - L R_\lambda(u) ||_{L^\infty([0,T] \times S^1)}.
\end{align*}
(\ref{A v}) and (\ref{N_1 annulus}) imply
\[
||A v||_{L^2(S^1, \mathbb{R}^{2n})} \lesssim (e^{-\frac{1}{2}\sqrt{4 - \epsilon} \kappa_0 s} + e^{-\frac{1}{2}\sqrt{4 - \epsilon} \kappa_0 (T-s)})|| \partial_t u - L R_\lambda(u) ||_{L^\infty([0,T] \times S^1)}.
\]
Combining the above two inequalities, we see
\[
||v (s) - z||_{W^{1,2}(S^1, \mathbb{R}^{2n})} \lesssim (e^{-\kappa s} + e^{-\kappa (T-s)})
|| \partial_t u - L R_\lambda(u) ||_{L^\infty([0,T] \times S^1)},
\]
where $\kappa = \frac{1}{2}\sqrt{4 - \epsilon} \kappa_0$.
Therefore, if $(b, \gamma) \in \mathbb{R} \times P_L$ corresponds to $z$,
that is, $z = (b,x) \in \mathbb{R} \times B^m_\epsilon(0)$ and $\phi(x) = \gamma \in P_L$,
then
\[
\mathop{\mathrm{dist}}\nolimits (u(s,t), (Ls + b, \gamma(t)) )
\lesssim (e^{-\kappa s} + e^{-\kappa (T-s)})|| \partial_t u - L R_\lambda(u) ||_{L^\infty([0,T] \times S^1)}.
\]
\end{proof}
\begin{lem}\label{absolute annulus}
If a $C^2$-function $f : [a,b] \to \mathbb{R}$ satisfies $f''(s) \geq \kappa^2 f(s)$ then
\[
f(s) \leq e^{-\kappa (s-a)} f(a)_+ + e^{-\kappa (b-s)} f(b)_+,
\]
where $f(s)_+ = \max (f(s) , 0)$.
\end{lem}
\begin{proof}
Since $g(s) = f(s) - (e^{-\kappa (s-a)} f(a)_+ + e^{-\kappa (b-s)} f(b)_+)$ also satisfies
$g''(s) \geq \kappa^2 g(s)$, we may assume $f(a) \leq 0$ and $f(b)\leq 0$.
If $f$ attained a positive value at some point $s_1$,
then there would exist some $a < s_0 < s_1$ such that
$f(s_0) > 0$ and $f'(s_0) >0$.
However this and the assumption $f''(s) \geq \kappa^2 f(s)$
would imply $f$ is monotone increasing on $s \geq s_0$,
which contradict the assumption $f(b) \leq 0$.
\end{proof}
The case of a symplectic manifold $X$ with cylindrical ends is covered by
Proposition \ref{second annulus}, Corollary \ref{third annulus},
and the following propositions.
\begin{prop}\label{second annulus for X}
There exist constants $\epsilon > 0$, $\kappa> 0$ and $C>0$ such that
the following holds true.
For any $0 < T \leq \infty$ and any $J$-holomorphic map
$u : [0,T] \times S^1 \to X$ such that
$||\partial_t u||_{L^\infty([0,T] \times S^1)} \leq \epsilon$,
there exists a point $x \in X$ such that
\[
\mathop{\mathrm{dist}}\nolimits (u(s,t), x ) \leq C (e^{-\kappa s} + e^{-\kappa (T-s)})
||\partial_t u||_{L^\infty([0,T] \times S^1)}.
\]
on $[0, T] \times S^1$.
\end{prop}
The proof of this proposition is the same as that of Proposition \ref{second annulus}.
\begin{cor}\label{third annulus for X}
For any constants $C_0 > 0$ and $\epsilon >0$,
there exist $\delta > 0$, $\kappa > 0$, $A > 0$ and $L_0 > 0$
such that the following holds true.
For any $0 \leq T \leq \infty$ and any $J$-holomorphic curve
$u : [-A, T + A] \times S^1 \to X$ with energies
$E_\lambda(u) \leq C_0$ and $E_{\hat \omega}(u) \leq \delta$,
one of the following two occurs:
\begin{itemize}
\item
There exists a point $x \in X$ such that
\[
\mathop{\mathrm{dist}}\nolimits ( u(s,t), x ) \leq \epsilon ( e^{-\kappa s} + e^{-\kappa (T - s)})
\]
for all $(s, t) \in [0, T] \times S^1$.
\item
There exists $L \neq 0 \in \mathbb{R}$ and $(b, \gamma) \in \mathbb{R} \times P_L$ such that
$|L| \leq L_0$ and
\[
\mathop{\mathrm{dist}}\nolimits ( u(s,t), (Ls + b, \gamma(t)) ) \leq \epsilon ( e^{-\kappa s} + e^{-\kappa (T - s)})
\]
for all $(s, t) \in [0, T] \times S^1$.
\end{itemize}
\end{cor}
\section{The space of holomorphic buildings}\label{space of holomorphic buildings}
In this section, we study the compactification of the space of $J$-holomorphic curves
in the symplectization of a contact manifold or a symplectic manifold with cylindrical ends.
Compactification was studied by Bourgeois, Eliashberg, Hofer, Wysocki and Zehnder
in \cite{BEHWZ03},
and the curves appeared in the compactified space are called holomorphic buildings.
First we recall about holomorphic buildings, and next we explain the topology
of the compactified space.
For the later use, we adopt a different definition of the topology.
This would be the same as that of \cite{BEHWZ03},
but we prove the compactness and Hausdorff property independently.
\subsection{The case of the symplectization}
First we consider holomorphic buildings for the symplectization $\hat Y = \mathbb{R} \times Y$.
The domain curve of a holomorphic building is constructed as follows.
Let $(\check \Sigma, z \cup (\pm \infty_i))$ be a marked semistable curve
or a disjoint union of marked semistable curves.
$z = (z_i)$ and $(\pm \infty_i)$ are sequences of marked points.
See \cite{FO99} for the definition of marked semistable curve.
Assume that an integer $i(\alpha) \in \{1, 2, \dots, k\}$ is attached to each irreducible
component $\check \Sigma_\alpha$ of $\check\Sigma$
(we call this integer the floor of $\check\Sigma_\alpha$) and
\begin{itemize}
\item
the difference of the floors of any adjacent two components is $\leq 1$,
\item
the floor of the component which contains some of the marked points $-\infty_i$ is $1$
(the lowest floor) and
\item
the floor of the component which contains some of the marked points $+\infty_i$ is $k$
(the highest floor).
\end{itemize}
We can construct a new curve from $\check \Sigma$ by oriented blow up.
Oriented blow up is a local deformation defined as follows.
Oriented blow up at $0 \in D = \{ z \in \mathbb{C}; |z|<1\}$ is
\[
\widetilde{D} = \{ (z,\theta) \in D \times S^1; z = |z|\theta\},
\]
and
oriented blow up at a nodal point $(0, 0) \in D\cup D = \{ (x,y) \in D\times D; xy =0\}$
by $\varphi \in S^1$ is
\[
D\widetilde{\cup}_\varphi D = \{ (x, \theta_x, y, \theta_y) \in \widetilde{D} \times
\widetilde{D}; xy= 0, \theta_x \theta_y = \varphi\}.
\]
$S^1 = \{(0, \theta); \theta \in S^1 \} \subset \widetilde{D}$ is called limit circle,
and $S^1 = \{(0, \theta_x, 0, \theta_y); \theta_x \theta_y = \varphi \}
\subset D\widetilde{\cup}_\varphi D$ is called joint circle.
These two circles are collectively called imaginary circles.
The domain curve $(\Sigma, z)$ of a holomorphic building is obtained by oriented blow up
of $(\check \Sigma, z)$ at the points $\pm \infty_i$ and all the nodal points
which join two components with different floors by some $\varphi \in S^1$.
We regard the curve $\Sigma$ as a topological space, and the complement of
its imaginary circles as an open smooth curve with a complex structure.
The topological space $\Sigma$ is compact.
Note that there exists a surjection $\Sigma \to \check \Sigma$ which collapses
the imaginary circles.
For each irreducible component $\check \Sigma_\alpha$ of $\check \Sigma$,
we denote its inverse image by $\Sigma_\alpha \subset \Sigma$ and call it
an irreducible component of $\Sigma$.
We say that the marked curve $(\Sigma, z)$ is connected if $\Sigma$ is connected
as a topological space, that is, if it is constructed from one semistable curve
(not from a disjoint union of several semistable curves).
We emphasize the difference between the notion of irreducible component and
connected component.
For example, two irreducible components of $\Sigma$ connected by a joint circle
are considered to be in the same connected component.
\begin{figure}
\centering
\includegraphics[width= 350pt]{Fig_hatSigma_z.png}
\caption{$(\hat \Sigma, z \cup (\pm\infty_i))$}\label{(hatSigma, z)}
\end{figure}
\begin{figure}
\centering
\includegraphics[width= 350pt]{Fig_Sigma_z.png}
\caption{$(\Sigma, z)$}\label{(Sigma, z)}
\end{figure}
\begin{defi}
A holomorphic building $(\Sigma, z, u, \phi)$ for $\hat Y$ consists of
\begin{itemize}
\item
a marked curve $(\Sigma, z)$ obtained from some marked semistable curve
$(\check \Sigma, z \cup (\pm \infty_i))$ (or a union of marked semistable curves)
with a floor structure and some blowing up parameters $\varphi \in S^1$ as above,
\item
a continuous map $u : \Sigma \to (\overline{\mathbb{R}}_1 \cup \overline{\mathbb{R}}_2 \cup \dots
\cup \overline{\mathbb{R}}_k) \times Y$,
where $\overline{\mathbb{R}} = \{-\infty\} \cup \mathbb{R} \cup \{ + \infty\}$ is a compactification of
$\mathbb{R}$ (homeomorphic to an closed interval) and we identify
$+ \infty \in \overline{\mathbb{R}}_i$ and $-\infty \in \overline{\mathbb{R}}_{i+1}$,
and
\item
a family of coordinates $\phi_{\pm\infty_i} : S^1 = \mathbb{R}/ \mathbb{Z} \stackrel{\cong}{\to} S^1_{\pm\infty_i}$ of
limit circles, where $S^1_{\pm\infty_i}$ is the limit circle corresponding to
$\pm\infty_i \in \check \Sigma$.
\end{itemize}
which satisfy the following conditions:
\begin{itemize}
\item
$u(\Sigma_\alpha \setminus \coprod_{\text{imaginary circles}} S^1) \subset \mathbb{R}_{i(\alpha)} \times Y$ for each component $\Sigma_\alpha$.
\item
$u|_{\Sigma_\alpha \setminus \coprod S^1} : \Sigma_\alpha \setminus \coprod S^1 \to \mathbb{R}_{i(\alpha)} \times Y$ is $J$-holomorphic.
\item
$E_\lambda(u) <\infty$ and $E_{\hat \omega}(u) <\infty$,
where these energies are defined by
\begin{align*}
E_\lambda(u) &= \max_{1 \leq i \leq k} \sup_{I \subset \mathbb{R}_i} \frac{1}{I}
\int_{(\sigma\circ u)^{-1}(I)} u^\ast (d\sigma \wedge \lambda), \\
E_{\hat \omega}(u) &= \int_{\Sigma} u^\ast d\lambda.
\end{align*}
\item
$u$ is positively asymptotic to a periodic orbit $\gamma_{+\infty_i}
= \pi_Y \circ u \circ \phi_{+\infty_i} \in P$ at each $S^1_{+\infty_i}$,
and negatively asymptotic to a periodic orbit $\gamma_{-\infty_i}
= \pi_Y \circ u \circ \phi_{-\infty_i} \in P$ at each $S^1_{-\infty_i}$.
At every joint circle, $u$ is positively asymptotic to a periodic orbit
on the side of lower floor and negatively asymptotic to the same periodic orbit
on the side of higher floor.
\item
For each component $\check\Sigma_\alpha$, if $u|_{\Sigma_\alpha}$ is a constant map,
then $2g_\alpha + m_\alpha \geq 3$,
where $g_\alpha$ is the genus of $\check\Sigma_\alpha$ and $m_\alpha$ is the sum of
the numbers of marked points and imaginary circles in $\Sigma_\alpha$ and
nodal points which join $\Sigma_\alpha$ with the other components.
\item
An irreducible component is called a trivial cylinder if it is isomorphic to
$\overline{\mathbb{R}} \times S^1$ without any special points such that
the restriction of $u$ on this component is written as
$u(s, t) = (L_\gamma s + b, \gamma(t))$ for some $b\in \mathbb{R}$ and $\gamma \in P$.
The other irreducible components are called nontrivial components.
We assume that for each $i \in \{1,2, \dots, k\}$,
$i$-th floor $u^{-1}(\overline{\mathbb{R}}_i \times Y) \subset \Sigma$ contains
nontrivial components.
(We do not assume the same condition
for each floor of each connected component of $\Sigma$.)
\end{itemize}
We call $k$ the height of $(\Sigma, z, u, \phi)$.
\end{defi}
We say two holomorphic buildings $(\Sigma, z, u, \phi)$ and $(\Sigma', z', u', \phi')$ are
isomorphic if there exist
\begin{itemize}
\item
a biholomorphism $\varphi : \Sigma' \to \Sigma$
(this means $\varphi$ is a homeomorphism which maps each imaginary circle of $\Sigma'$
to a imaginary circle of $\Sigma$ and is biholomorphic on the outside of these circles)
and
\item
an $\mathbb{R}$-translation $\theta : \overline{\mathbb{R}}_1 \cup \overline{\mathbb{R}}_2 \cup \dots \cup
\overline{\mathbb{R}}_k \to \overline{\mathbb{R}}_1 \cup \overline{\mathbb{R}}_2 \cup \dots \cup \overline{\mathbb{R}}_k$
(this means $\theta$ is a map such that $\theta(\overline{\mathbb{R}}_i) \subset \overline{\mathbb{R}}_i$
and $\theta|_{\overline{\mathbb{R}}_i} (s)= s+ a_i$ for some $a_i\in \mathbb{R}$)
\end{itemize}
such that
\begin{itemize}
\item
$\varphi(z'_i) = z_i$ for all $i$,
\item
$u' = (\theta \times 1) \circ u \circ \varphi$, and
\item
$\varphi \circ \phi'_{\pm\infty_i} = \phi_{\pm\infty_i}$ for all $\pm\infty_i$.
\end{itemize}
We denote the space of all connected holomorphic buildings by
$\overline{\mathcal{M}}^0 = \overline{\mathcal{M}}^0(Y, \lambda, J)$,
and the space of all holomorphic buildings without trivial buildings by
$\overline{\mathcal{M}} = \overline{\mathcal{M}}(Y, \lambda, J)$,
where a trivial building in $(\Sigma, z, u, \phi)$ is a connected component of $\Sigma$
which consists of trivial cylinders only.
First we define the topology of $\overline{\mathcal{M}}^0$.
It is enough to define the neighborhoods of each point
$p_0 = (\Sigma_0, z_0, u_0, \phi_0) \in \overline{\mathcal{M}}^0$.
We consider a fibration $(\widetilde{P} \to \widetilde{X}, Z)$ consisting of
some deformations of the domain curve $(\Sigma_0, z_0)$, and
construct a map $\Psi : \widetilde{P} \to \widetilde{P}_0$.
Then the neighborhood of $p_0$ is defined by the set of holomorphic buildings whose
domain curves appear as a fiber $\widetilde{P}_a$ of $\widetilde{P}$ and
which are close to $u_0 \circ \Psi|_{\widetilde{P}_a}$ in $L^\infty$-norm modulo
$\mathbb{R}$-gluings.
Now we explain the details.
First we add marked points $z^+_0$ to $(\Sigma_0, z_0)$ to make
$(\Sigma_0, z_0\cup z_0^+)$ stable,
where $z_0 \cup z_0^+$ is a sequence of marked points obtained by placing the sequence
$z_0^+$ after $z_0$, and stableness of $(\Sigma_0, z_0\cup z_0^+)$ means that
the curve $(\check\Sigma_0, z_0\cup z^+_0 \cup (\pm\infty_i))$ is a stable curve.
The local universal family $(\widetilde{P} \to \widetilde{X}, Z \cup Z^+)$ of
$(\Sigma_0, z_0 \cup z_0^+)$ is defined by the oriented blow up of the local universal
family $(\check P \to \check X, Z \cup Z^+ \cup (Z_{\pm \infty_i}))$ of
the stable curve $(\check\Sigma_0, z_0\cup z^+_0 \cup (\pm\infty_i))$ at
$Z_{\pm \infty_i}$ and the set of nodal points corresponding to the nodal points of
$\check \Sigma_0$ which are blown up in $\Sigma_0$.
Oriented blow up of the local universal family is defined as follows.
For each nodal point of $\check \Sigma_0$, the fibration $\check P \to \check X$ is
locally equivalent to
\begin{align*}
N = D^{m-1} \times D \times D &\to D^{m-1} \times D = \check X,\\
(z, x, y) &\mapsto (z, xy)
\end{align*}
where $(0, 0) \in D^{m-1} \times D = \check X$ is the point corresponding to
the curve $(\check\Sigma_0, z_0\cup z^+_0 \cup (\pm\infty_i))$, and
the nodal point of $\check\Sigma_0$ is $(0, 0, 0) \in N$.
Then the oriented blow up at the set of nodal points $D^{m-1} \times \{(0, 0)\}$ is
defined by
\begin{align*}
\widetilde{N} = D^{m-1} \times \widetilde{D} \times \widetilde{D}
&\to D^{m-1} \times \widetilde{D} = \widetilde{X}.\\
(z, (x, \theta_x), (y, \theta_y)) &\mapsto (z, (xy, \theta_x \theta_y))
\end{align*}
For each marked point $\pm\infty_i$ of
$(\check\Sigma_0, z_0\cup z^+_0 \cup (\pm\infty_i))$, the fibration
$\check P \to \check X$ is locally equivalent to
\begin{align*}
N = D^m \times D &\to D^m = \check X,\\
(z, w) &\mapsto z
\end{align*}
where $0 \in D^m = \check X$ is the point corresponding to the curve
$(\check\Sigma_0, z_0\cup z^+_0 \cup (\pm\infty_i))$, and
$Z_{\pm\infty_i}(z) = (z, 0)$ is the section of marked point corresponding
to the marked point $\pm\infty_i$.
Then the oriented blow up at $Z_{\pm\infty_i}$ is defined by
\begin{align*}
\breve N = D^m \times \widetilde{D} &\to D^m = \widetilde{X}.\\
(z, (w, \theta_w)) &\mapsto z
\end{align*}
We take a discontinuous map $\Psi : \widetilde{P} \to \widetilde{P}_0$ (or a continuous
map which is defined on the complement of some codimension one subset) which satisfies
the following conditions:
\begin{itemize}
\item
$\Psi|_{\widetilde{P}_0} = \mathrm{id}$
\item
For each nodal point of $\Sigma_0$, we fix a neighborhood $\check N \subset
\widetilde{P}$ such that the restriction of the fibration $\widetilde{P} \to \widetilde{X}$
to $\check N$ is equivalent to
\begin{align*}
\check N = A \times D \times D &\to A \times D = \widetilde{X},\\
(a, x, y) &\mapsto (a, xy)
\end{align*}
where $A$ is some complex manifold or its oriented blow up, and $(0, 0) \in A \times D
= \widetilde{X}$ is the point corresponding to the curve $(\Sigma_0, z_0\cup z_0^+)$.
Then the restriction of $\Psi$ to $\check N$ is given by
\[
\Psi (a, x, y) = \begin{cases}
(0, x, 0) \in A \times D \times D \text{ if } |x| \geq |y|\\
(0, 0, y) \in A \times D \times D \text{ if } |y| \geq |x|
\end{cases}.
\]
Note that this is not well defined at the codimension one subset $\{|x| = |y|\}$.
\item
For each joint circle of $\Sigma_0$, we fix its neighborhood $\widetilde{N} \subset
\widetilde{P}$ such that the restriction of the fibration $\widetilde{P} \to \widetilde{X}$
to $\widetilde{N}$ is equivalent to
\begin{align*}
\widetilde{N} = A \times \widetilde{D} \times \widetilde{D} &\to A \times \widetilde{D}
= \widetilde{X},\\
(a, (x, \theta_x), (y, \theta_y)) &\mapsto (a, (xy, \theta_x \theta_y))
\end{align*}
where $(0, 0, e^{2\pi \sqrt{-1} \cdot 0}) \in \widetilde{X}$ is the point corresponding to
the curve $(\Sigma_0, z_0\cup z_0^+)$.
Then the restriction of $\Psi$ to $\widetilde{N}$ is given by
\[
\Psi(a, (x, \theta_x), (y, \theta_y)) = \begin{cases}
(0, (x, \theta_x), (0, \theta_x^{-1})) \text{ if } |x| \geq |y|\\
(0, (0, \theta_y^{-1}), (y, \theta_y)) \text{ if } |y| \geq |x|
\end{cases}.
\]
Note that if we rewrite the above fibration by the isomorphism
$\widetilde{D} \cong [-\infty, 0) \times S^1 \cong (0, \infty] \times S^1$
given by $(e^{2\pi(s + \sqrt{-1} t)}, e^{2\pi\sqrt{-1} t}) \leftrightarrow (s, t)
\leftrightarrow (-s, -t)$
as
\begin{align*}
\widetilde{N} = A \times ((0, \infty] \times S^1) \times ([-\infty, 0) \times S^1)
&\to A \times ((0, \infty] \times S^1) = \widetilde{X},\\
(a, (s_x, t_x), (s_y, t_y)) &\mapsto (a, (s_x - s_y, t_x - t_y))
\end{align*}
then $\Psi|_{\widetilde{N}}$ is expressed as
\[
\Psi(a, (s_x, t_x), (s_y, t_y)) = \begin{cases}
(0, (s_x, t_x), (-\infty, - t_x)) \text{ if } |s_x| \geq |s_y|\\
(0, (+\infty, -t_y), (s_y, t_y)) \text{ if } |s_y| \geq |s_x|
\end{cases}.
\]
\item
For each $+ \infty$-limit circle $S_{+\infty}^1$ of $\Sigma_0$, we fix its neighborhood
$\breve N_{+\infty_i} \subset \widetilde{P}$ such that the restriction of the fibration
$\check P \to \check X$ is locally equivalent to
\begin{align*}
\breve N_{+\infty_i} = A \times ((0, \infty] \times S^1)&\to A = \widetilde{X},\\
(a, s, t) &\mapsto a
\end{align*}
where $0 \in A = \widetilde{X}$ is the point corresponding to the curve
$(\Sigma_0, z_0\cup z_0^+)$.
Then the restriction of $\Psi$ to $\breve{N}_{+\infty_i}$ is given by
\[
\Psi(a, s, t) = (0, s, t).
\]
\item
For each $-\infty$-limit circle $S_{-\infty_i}^1$ of $\Sigma_0$,
we also fix its neighborhood
$\breve N_{-\infty_i} \subset \widetilde{P}$ similarly, and we assume that
the restriction of $\Psi$ to $\breve N_{-\infty_i}$ is given similarly.
\item
$\Psi$ is smooth on the complement $\widetilde{P} \setminus
(\bigcup_{\text{nodal points}} \check N \cup \bigcup_{\text{joint circles}} \widetilde{N}
\cup \bigcup_{\text{limit circles}} \breve N_{\pm\infty_i})$.
\item
$\Psi$ is continuous at the joint $\bigcup \partial \check N \cup \bigcup \partial
\widetilde{N} \cup \bigcup \partial \breve N_{\pm\infty_i}$.
\end{itemize}
A map $\theta : \overline{\mathbb{R}}_1 \sqcup \overline{\mathbb{R}}_2 \sqcup \dots \sqcup
\overline{\mathbb{R}}_k \to \overline{\mathbb{R}}_1 \cup \overline{\mathbb{R}}_2 \cup \dots \cup \overline{\mathbb{R}}_l$
is called an $\mathbb{R}$-gluing if there exist a surjection $\mu : \{1,2,\dots, k\} \to
\{1,2,\dots, l\}$ and constants $c_i \in \mathbb{R}$ ($i = 1,2, \dots, k$) such that
\begin{itemize}
\item
if $i \leq j$ then $\mu(i) \leq \mu(j)$,
\item
$\theta(\overline{\mathbb{R}}_i) = \overline{\mathbb{R}}_{\mu(i)}$, and
\item
$\theta|_{\overline{\mathbb{R}}_i} (s) = s + c_i \ (\in \overline{\mathbb{R}}_{\mu(i)})$.
\end{itemize}
For each $\mathbb{R}$-gluing $\theta$, let
$\theta \times 1
: (\overline{\mathbb{R}}_1 \sqcup \overline{\mathbb{R}}_2 \sqcup \dots \sqcup \overline{\mathbb{R}}_k) \times Y
\to (\overline{\mathbb{R}}_1 \cup \overline{\mathbb{R}}_2 \cup \dots \cup \overline{\mathbb{R}}_l) \times Y$
be the product with the identity map on $Y$.
For each neighborhood $U \subset \widetilde{X}$ of $0 \in \widetilde{X}$ and
each constant $\epsilon>0$,
we define a subset $\mathcal{W}_{p_0}(U, \epsilon) = \mathcal{W}_{p_0}(U, \epsilon, \Psi)
\subset \overline{\mathcal{M}}^0$ as follows.
$(\Sigma, z, u, \phi) \in \overline{\mathcal{M}}^0$ belongs to $\mathcal{W}_{p_0}(U, \epsilon)$ if
there exist a point $a\in U$, an isomorphism $(\Sigma, z) \cong (\widetilde{P}_a, Z(a))$
and an $\mathbb{R}$-gluing $\theta$ such that
\begin{equation}
\mathop{\mathrm{dist}}\nolimits_{L^\infty}(u, (\theta \times 1) \circ u_0 \circ \Psi|_{\widetilde{P}_a}) < \epsilon
\label{L^infty comparison of maps}
\end{equation}
and
\begin{equation}
\mathop{\mathrm{dist}}\nolimits_{L^\infty(S^1)}(\pi_{S^1}^{\breve N_{\pm\infty_i}} \circ \phi_{\pm\infty_i},
\phi_{0, \pm\infty_i}) < \epsilon,
\label{L^infty comparison of coordinates of S^1}
\end{equation}
where the left hand side of (\ref{L^infty comparison of maps}) is the essential sup of
$\mathop{\mathrm{dist}}\nolimits(u(z), (\theta \times 1) \circ u_0 \circ \Psi (z))$ over $\widetilde{P}_a$, and
in (\ref{L^infty comparison of coordinates of S^1}),
\[
\pi_{S^1}^{\breve N_{\pm\infty_i}} : \breve N_{\pm\infty_i} \supset \widetilde{X} \times
\{\pm\infty\} \times S^1 \to \{0\} \times \{\pm\infty\} \times S^1
\]
are the projections.
We define a neighborhood of $p_0$ as a subset of $\overline{\mathcal{M}}^0$ which contains
$\mathcal{W}_{p_0}(U, \epsilon)$ for some $U \subset \widetilde{X}$ and $\epsilon >0$.
First we prove that this definition of neighborhood is independent of the choice of
$\check N$, $\widetilde{N}$, $\breve N_{\pm\infty}$ and $\Psi$.
Let $(\check N', \widetilde{N}', \breve N'_{\pm\infty}, \Psi')$ be another choice.
We claim that for any $\epsilon > 0$, there exists a neighborhood
$U \subset \widetilde{X}$ such that for any $a \in U$ and any $\mathbb{R}$-gluing $\theta$,
\begin{equation}
\mathop{\mathrm{dist}}\nolimits_{L^\infty} ((\theta \times 1) \circ u_0 \circ \Psi'|_{\widetilde{P}_a},
(\theta \times 1) \circ u_0 \circ \Psi|_{\widetilde{P}_a})
< \epsilon + \Delta((\theta \times 1) \circ u_0 \circ \Psi|_{\widetilde{P}_a})
\label{difference}
\end{equation}
and
\begin{equation}
\mathop{\mathrm{dist}}\nolimits_{L^\infty(U \times \{\pm\infty\} \times S^1)}
(\pi_{S^1}^{\breve N'_{\pm\infty_i}}, \pi_{S^1}^{\breve N_{\pm\infty_i}}) < \epsilon,
\label{S^1 difference}
\end{equation}
where $\Delta((\theta \times 1) \circ u_0 \circ \Psi|_{\widetilde{P}_a})$
is the maximum of the differences of the limits of
$(\theta \times 1) \circ u_0 \circ \Psi|_{\widetilde{P}_a}$ on the both sides
at the discontinuous codimension one subset.
First we prove that these inequalities imply the independence of the choice of
$\check N$, $\widetilde{N}$, $\breve N_{\pm\infty_i}$ and $\Psi$.
For any $(\Sigma, z, u) \in \mathcal{W}_{p_0}(U, \epsilon, \Psi)$,
there exist a point $a\in U$, an isomorphism $(\Sigma, z) \cong (\widetilde{P}_a, Z(a))$
and an $\mathbb{R}$-gluing $\theta$ such that
\[
\mathop{\mathrm{dist}}\nolimits_{L^\infty}(u, (\theta \times 1) \circ u_0 \circ \Psi|_{\widetilde{P}_a}) < \epsilon
\]
and
\[
\mathop{\mathrm{dist}}\nolimits_{L^\infty}(\pi_{S^1}^{\breve N_{\pm\infty_i}} \circ \phi_{\pm\infty_i},
\phi_{0, \pm\infty}) < \epsilon.
\]
Since
\[
\Delta((\theta \times 1) \circ u_0 \circ \Psi|_{\widetilde{P}_a})
\leq 2 \mathop{\mathrm{dist}}\nolimits_{L^\infty}(u, (\theta \times 1) \circ u_0 \circ \Psi|_{\widetilde{P}_a})
< 2 \epsilon,
\]
inequality (\ref{difference}) implies
\[
\mathop{\mathrm{dist}}\nolimits_{L^\infty} ((\theta \times 1) \circ u_0 \circ \Psi'|_{\widetilde{P}_a},
(\theta \times 1) \circ u_0 \circ \Psi|_{\widetilde{P}_a}) < 3 \epsilon,
\]
hence
\[
\mathop{\mathrm{dist}}\nolimits_{L^\infty}(u, (\theta \times 1) \circ u_0 \circ \Psi'|_{\widetilde{P}_a}) < 4 \epsilon.
\]
On the other hand, inequality (\ref{S^1 difference}) implies
\[
\mathop{\mathrm{dist}}\nolimits_{L^\infty}(\pi_{S^1}^{\breve N'_{\pm\infty_i}} \circ \phi_{\pm\infty_i},
\phi_{0, \pm\infty_i})
< 2 \epsilon.
\]
Therefore $\mathcal{W}_{p_0}(U, \epsilon, \Psi) \subset \mathcal{W}_{p_0}(U, 4 \epsilon, \Psi')$,
which implies the independence of $\check N$, $\widetilde{N}$, $\breve N_{\pm\infty_i}$
and $\Psi$.
The above inequalities ((\ref{difference}) and (\ref{S^1 difference})) are proved as follows.
First we need to observe the correspondence of the coordinates of $\check N$ and
$\check N'$.
Since $\{ x = 0 \}$ and $\{ y = 0 \} \subset \check N$ corresponds to $\{ x' = 0 \}$ and
$\{ y' = 0 \} \subset \check N'$ respectively,
\begin{align*}
x' &= C x (1 + O(a, x, y))\\
y' &= C' y (1 + O(a, x, y))
\end{align*}
for some constants $C$ and $C'$.
Hence in the coordinate of $\check N$,
$\Psi'$ is written as
\[
(a, x, y) \mapsto \begin{cases}
(0, x (1 + O(a, x)), 0) \text{ if } |x'| \geq |y'| \\
(0, 0, y (1 + O(a, y))) \text{ if } |y'| \geq |x'|
\end{cases}.
\]
Since $u_0$ is continuous near each nodal point,
there exists a neighborhood $\check N^\circ \subset \check N$ of the nodal point
such that
\[
\mathop{\mathrm{dist}}\nolimits_{L^\infty(\check N^\circ)} (u_0 \circ \Psi'|_{\check N^\circ},
u_0 \circ \Psi|_{\check N^\circ}) < \epsilon.
\]
Hence
\[
\mathop{\mathrm{dist}}\nolimits_{L^\infty} ((\theta \times 1) \circ u_0 \circ \Psi'|_{\check N^\circ},
(\theta \times 1) \circ u_0 \circ \Psi|_{\check N^\circ}) < \epsilon
\]
for any $\mathbb{R}$-gluing $\theta$.
Next we consider the neighborhoods $\widetilde{N}$ and $\widetilde{N}'$
of each joint circle of $\Sigma_0$.
As in the case of $\check N$ and $\check N'$,
the correspondence of these two coordinates is
\begin{align*}
(s_x', t_x') &= C + (s_x, t_x) + O(a, e^{-2\pi s_x}, e^{2\pi s_y}) \\
(s_y', t_y') &= C' + (s_y, t_y) + O(a, e^{-2\pi s_x}, e^{2\pi s_y})
\end{align*}
for some constants $C, C' \in \mathbb{R} \times S^1$.
Hence in the coordinate of $\widetilde{N}$,
$\Psi'$ is written as
\begin{align*}
&(a, (s_x, t_x), (s_y, t_y))\\
&\mapsto \begin{cases}
(0, (s_x, t_x) + O(a, e^{-2\pi s_x}), (-\infty, -t_x) + O(a, e^{-2\pi s_x}))
\text{ if } |s_x'| \leq |s_y'| \\
(0, (\infty, -t_y) + O(a, e^{2\pi s_y}), (s_y, t_y) + O(a, e^{2\pi s_y}))
\text{ if } |s_y'| \leq |s_x'|
\end{cases}
\end{align*}
We decompose $\widetilde{N}$ into the following four pieces and prove the inequality
for each piece.
\begin{align*}
A_1 &= \{ |s_x| \leq |s_y| \} \cap \{ |s_x'| \leq |s_y'| \} \\
A_2 &= \{ |s_x| \geq |s_y| \} \cap \{ |s_x'| \geq |s_y'| \} \\
A_3 &= \{ |s_x| \geq |s_y| \} \cap \{ |s_x'| \leq |s_y'| \} \\
A_4 &= \{ |s_x| \leq |s_y| \} \cap \{ |s_x'| \geq |s_y'| \}
\end{align*}
First we consider the pieces $A_1$ and $A_2$.
The above expression of $\Psi'$ implies that
there exists a neighborhood $\widetilde{N}^\circ \subset \widetilde{N}$ of the joint circle
such that
\[
\mathop{\mathrm{dist}}\nolimits_{L^\infty} (u_0 \circ \Psi'|_{\widetilde{N}^\circ \cap A_i},
u_0 \circ \Psi|_{\widetilde{N}^\circ \cap A_i}) < \epsilon
\]
for $i = 1, 2$.
Hence for any $\mathbb{R}$-gluing $\theta$,
\[
\mathop{\mathrm{dist}}\nolimits_{L^\infty} ((\theta \times 1) \circ u_0 \circ \Psi'|_{\widetilde{N}^\circ \cap A_i},
(\theta \times 1) \circ u_0 \circ \Psi|_{\widetilde{N}^\circ \cap A_i}) < \epsilon.
\]
Next we consider the piece $A_3$.
For any constant $C > 0$, if $U \subset \widetilde{X}$ is a sufficiently small neighborhood
of $0 \in \widetilde{X}$, then
$|s_x|, |s_x'|, |s_y|, |s_y'| \geq C$ on $\widetilde{N}^\circ|_U \cap A_3$
since $|s_x' - s_x|$ and $|s_y' - s_y|$ are bounded on $\widetilde{N}|_U \cap A_3$.
Applying Corollary \ref{third annulus} to $u_0$, we see that
\begin{align*}
u_0(s_x, t_x) &= (Ls_x + b_x, \gamma(t_y)) + O(e^{-\delta s_x})\\
u_0(s_y, t_y) &= (Ls_y + b_y, \gamma(t_y)) + O(e^{\delta s_y})
\end{align*}
for some $\gamma \in P_L$ and $b_x, b_y \in \mathbb{R}$.
Assume that $U$ is sufficiently small so that
$|O(e^{-\delta s_x})| \leq \epsilon$ and $|O(e^{\delta s_y})| \leq \epsilon$ on
$\widetilde{N}|_U \cap A_3$ in the above equation.
Then for any $z = (a, (s_x, t_x), (s_y, t_y)) \in \widetilde{N}|_U \cap A_3$,
\begin{align*}
&\mathop{\mathrm{dist}}\nolimits((\theta \times 1) \circ u_0 \circ \Psi'(z),
(\theta \times 1) \circ u_0 \circ \Psi(z))\\
&\leq \mathop{\mathrm{dist}}\nolimits((\theta \times 1) \circ u_0 \circ \Psi'(z),
(\theta \times 1) \circ u_0 (s_y, t_y)) \\
&\quad +
\mathop{\mathrm{dist}}\nolimits((\theta \times 1) \circ u_0 (s_y, t_y), (\theta \times 1)(L s_y + b_y, \gamma(t_y)))\\
&\quad + \mathop{\mathrm{dist}}\nolimits((\theta \times 1) \circ u_0 (s_x, t_x),
(\theta \times 1)(L s_x + b_x, \gamma(t_x)))\\
&\quad + \mathop{\mathrm{dist}}\nolimits((\theta \times 1)(L s_x + b_x, \gamma(t_x)),
(\theta \times 1)(L s_y + b_y, \gamma(t_y)))\\
&< \epsilon + \epsilon + \epsilon \\
& \quad
+ \mathop{\mathrm{dist}}\nolimits_{L^\infty(\{|s_x| = |s_y|\})}((\theta \times 1)(L s_x + b_x, \gamma(t_x)),
(\theta \times 1)(L s_y + b_y, \gamma(t_y)))\\
&< 3\epsilon + 2\epsilon + \Delta((\theta \times 1) \circ u_0 \circ \Psi|_{\widetilde{P}_a}).
\end{align*}
Similarly, for any $z = (a, (s_x, t_x), (s_y, t_y)) \in \widetilde{N}|_U \cap A_4$,
\[
\mathop{\mathrm{dist}}\nolimits ((\theta \times 1) \circ u_0 \circ \Psi'(z), (\theta \times 1) \circ u_0 \circ \Psi(z))
< 5\epsilon + \Delta((\theta \times 1) \circ u_0 \circ \Psi|_{\widetilde{P}_a}).
\]
For each limit circle of $\Sigma_0$,
it is easy to see that there exists a neighborhood $\breve N_{\pm\infty_i}^\circ \subset
\breve N_{\pm\infty_i}$ of the circle such that
\[
\mathop{\mathrm{dist}}\nolimits_{L^\infty} (u_0 \circ \Psi'|_{\breve N_{\pm\infty_i}^\circ},
u_0 \circ \Psi|_{\breve N_{\pm\infty_i}^\circ}) < \epsilon.
\]
Finally, we consider the complement of the neighborhoods $\check N^\circ$
$\widetilde{N}^\circ$ and $\breve N^\circ_{\pm\infty_i}$.
If $U \subset \widetilde{X}$ is sufficiently small neighborhood of $0 \in \widetilde{X}$,
then the continuity of $\Psi$ and $\Psi'$ on
$\widetilde{P}_U \setminus (\bigcup \check N^\circ \cup \bigcup \widetilde{N}^\circ
\cup \bigcup \breve N_{\pm\infty_i}^\circ)$ implies that
\[
\mathop{\mathrm{dist}}\nolimits_{L^\infty}(u_0 \circ \Psi'|_{\widetilde{P}_U \setminus (\bigcup \check N^\circ \cup
\bigcup \widetilde{N}^\circ \cup \bigcup \breve N_{\pm\infty_i}^\circ)},
u_0 \circ \Psi|_{\widetilde{P}_U \setminus (\bigcup \check N^\circ \cup
\bigcup \widetilde{N}^\circ \cup \bigcup \breve N_{\pm\infty_i}^\circ)}) < \epsilon.
\]
These estimates prove inequality (\ref{difference}).
Inequality (\ref{S^1 difference}) can be easily checked.
Next we prove the definition of neighborhood does not depend on the choice of
the additional marked points $z^+_0$.
It is enough to compare with another sequence of marked points $z^{++}_0$
which contains $z^+_0$.
We may assume that the local universal family
$(\widetilde{P}^{++} \to \widetilde{X}^{++}, Z \cup Z^{++})$
of $(\Sigma_0, z_0 \cup z^{++}_0)$
has the form $\widetilde{P}^{++} = \widetilde{P} \times D^k$ and
$\widetilde{X}^{++} = \widetilde{X} \times D^k$,
where $D^k$ is the parameter space which determines the value of
$Z^{++} \setminus Z^+$,
and that $Z \cup Z^+$ is independent of $D^k$.
Then we can take $\check N^{++} = \check N \times D^k$,
$\widetilde{N}^{++} = \widetilde{N} \times D^k$ and
$\breve N^{++}_{\pm\infty_i} = \breve N_{\pm\infty_i} \times D^k$
as neighborhoods of nodal points and imaginary circles.
Hence we can take $\Psi^{++} = \Psi \circ \pi_{\widetilde{P}} : \widetilde{P}^{++}
\to \widetilde{P}^{++}_0 = \widetilde{P}_0$,
which implies the definitions of the neighborhood coincide.
Finally, we prove that these neighborhood systems define a topology.
It is enough to prove the following claim:
For each $p_1 \in \mathcal{W}_{p_0}(U, \epsilon)$,
there exists a neighborhood of $p_1$ contained in
$\mathcal{W}_{p_0}(U, \epsilon)$.
This implies not only the well-definedness of the topology but also that
each $\mathcal{W}_{p_0}(U, \epsilon)$ is open.
For each $p_1 = (\Sigma_1, z_1, u_1, \phi_1) \in \mathcal{W}_{p_0}(U, \epsilon)$,
there exist a point $a_1\in U$, an isomorphism
$(\Sigma_1, z_1) \cong (\widetilde{P}_{a_1}, Z(a_1))$
and an $\mathbb{R}$-gluing $\theta_1$ such that
\[
\mathop{\mathrm{dist}}\nolimits_{L^\infty}(u_1, (\theta_1 \times 1) \circ u_0 \circ \Psi|_{\widetilde{P}_{a_1}})
< \epsilon
\]
and
\[
\mathop{\mathrm{dist}}\nolimits_{L^\infty(S^1)}(\pi_{S^1}^{\breve N_{\pm\infty_i}} \circ \phi_{1, \pm\infty_i},
\phi_{0, \pm\infty_i}) < \epsilon.
\]
Let $\epsilon_1 > 0$ be a small constant such that
\[
\mathop{\mathrm{dist}}\nolimits_{L^\infty}(u_1, (\theta_1 \times 1) \circ u_0 \circ \Psi|_{\widetilde{P}_{a_1}})
< \epsilon - 2\epsilon_1
\]
and
\[
\mathop{\mathrm{dist}}\nolimits_{L^\infty}(\pi_{S^1}^{\breve N_{\pm\infty_i}} \circ \phi_{1, \pm\infty_i},
\phi_{0, \pm\infty_i}) < \epsilon - \epsilon_1.
\]
We use the additional marked points $z_1^+$ of $(\Sigma_1, z_1)$ which correspond to
$Z^+(a_1)$ under the above isomorphism $(\Sigma_1, z_1) \cong
(\widetilde{P}_{a_1}, Z(a_1))$.
Then the local universal family of $(\Sigma_1, z_1 \cup z_1^+)$ is the restriction of
$(\widetilde{P} \to \widetilde{X}, Z \cup Z^+)$ to a neighborhood $U'_1 \subset
\widetilde{X}$ of $a_1$.
Then for the definition of the neighborhoods of $p_1$,
we can take a discontinuous map $\Psi^1 : \widetilde{P}|_{U'_1} \to \widetilde{P}_{a_1}$
which satisfies the following conditions:
\begin{itemize}
\item
$\Psi|_{\widetilde{P}_{a_1}} \circ \Psi^1|_{\check N}
= \Psi|_{\check N} : \check N \to \widetilde{P}_0$
for the neighborhood $\check N$ of each nodal point of $\Sigma_1$.
\item
$\Psi|_{\widetilde{P}_{a_1}} \circ \Psi^1|_{\widetilde{N}}
= \Psi|_{\widetilde{N}} : \widetilde{N} \to \widetilde{P}_0$
for the neighborhood $\widetilde{N}$ of each joint circle of $\Sigma_1$.
\item
On the neighborhood of each limit circle of $\Sigma_1$, $\Psi^1$ is defined by
using the same coordinate of $\breve N_{\pm\infty_i}$ as that for $\Psi$.
\item
Let $\mathcal{D} \subset \widetilde{P}$ be the codimension one subset consisting of nodal points,
imaginary circles and discontinuous points of $\Psi$.
Then, $\Psi^1$ preserves $\mathcal{D}$.
\end{itemize}
Since $u_0$ is continuous on $\widetilde{P}_0 \setminus \mathcal{D}$,
the above assumption of $\Psi^1$ implies that
if $U_1 \subset U'_1$ is sufficiently small, then
\[
\mathop{\mathrm{dist}}\nolimits_{L^\infty} (u_0 \circ \Psi|_{\widetilde{P}_{a_1}} \circ \Psi^1|_{\widetilde{P}_{U_1}}, u_0 \circ \Psi|_{\widetilde{P}_{U_1}}) < \epsilon_1.
\]
For any $p = (\Sigma, z, u, \phi) \in \mathcal{W}_{p_1}(U_1, \epsilon_1)$,
there exist a point $a \in U_1$, an isomorphism $(\Sigma, z) \cong (\widetilde{P}_a, Z(a))$
and an $\mathbb{R}$-gluing $\theta$ such that
\[
\mathop{\mathrm{dist}}\nolimits_{L^\infty}(u, (\theta \times 1) \circ u_1 \circ \Psi^1|_{\widetilde{P}_a}) < \epsilon_1
\]
and
\[
\mathop{\mathrm{dist}}\nolimits_{L^\infty(S^1)}(\pi_{S^1}^{\breve N_{\pm\infty_i}} \circ \phi_{\pm\infty_i},
\phi_{1, \pm\infty_i}) < \epsilon_1.
\]
Hence
\begin{align*}
&\mathop{\mathrm{dist}}\nolimits_{L^\infty}(u, (\theta \times 1) \circ (\theta_1 \times 1) \circ u_0 \circ
\Psi|_{\widetilde{P}_a})\\
&\leq \mathop{\mathrm{dist}}\nolimits_{L^\infty}(u, (\theta \times 1) \circ u_1 \circ \Psi^1|_{\widetilde{P}_a}) \\
&\quad + \mathop{\mathrm{dist}}\nolimits_{L^\infty}((\theta \times 1) \circ u_1 \circ \Psi^1|_{\widetilde{P}_a},
(\theta \times 1) \circ (\theta_1 \times 1) \circ u_0 \circ \Psi|_{\widetilde{P}_{a_1}}
\circ \Psi^1|_{\widetilde{P}_a})\\
&\quad + \mathop{\mathrm{dist}}\nolimits_{L^\infty}((\theta \times 1) \circ (\theta_1 \times 1) \circ u_0
\circ \Psi|_{\widetilde{P}_{a_1}} \circ \Psi^1|_{\widetilde{P}_a}, (\theta \times 1)
\circ (\theta_1 \times 1) \circ u_0 \circ \Psi|_{\widetilde{P}_a})\\
&\leq \mathop{\mathrm{dist}}\nolimits_{L^\infty}(u, (\theta \times 1) \circ u_1 \circ \Psi^1|_{\widetilde{P}_a})
+ \mathop{\mathrm{dist}}\nolimits_{L^\infty}(u_1, (\theta_1 \times 1) \circ u_0 \circ \Psi|_{\widetilde{P}_{a_1}})\\
&\quad + \mathop{\mathrm{dist}}\nolimits_{L^\infty}(u_0 \circ \Psi|_{\widetilde{P}_{a_1}} \circ
\Psi^1|_{\widetilde{P}_a}, u_0 \circ \Psi|_{\widetilde{P}_a})\\
&< \epsilon,
\end{align*}
and
\begin{align*}
&\mathop{\mathrm{dist}}\nolimits_{L^\infty(S^1)}(\pi_{S^1}^{\breve N_{\pm\infty_i}} \circ \phi_{\pm\infty_i},
\phi_{0, \pm\infty_i})\\
&\leq
\mathop{\mathrm{dist}}\nolimits_{L^\infty(S^1)}(\pi_{S^1}^{\breve N_{\pm\infty_i}|_{U_1}} \circ \phi_{\pm\infty_i}, \phi_{1, \pm\infty_i})
+ \mathop{\mathrm{dist}}\nolimits_{L^\infty(S^1)}(\pi_{S^1}^{\breve N_{\pm\infty_i}} \circ \phi_{1, \pm\infty_i}, \phi_{0, \pm\infty_i})\\
& < \epsilon.
\end{align*}
These imply $p \in \mathcal{W}_{p_0}(U, \epsilon)$, which proves the claim.
Next we prove the topological properties of
$\overline{\mathcal{M}}^0 = \overline{\mathcal{M}}^0(Y, \lambda, J)$ along the similar lines in the case of
Gromov-Witten theory in \cite{FO99}. $\overline{\mathcal{M}}^0$ is decomposed as
$\overline{\mathcal{M}}^0 = \coprod_{g, \mu, L^-, L^+} \overline{\mathcal{M}}^0_{g, \mu}(L^-,L^+)$,
where $\overline{\mathcal{M}}^0_{g, \mu}(L^-,L^+)$ is the space of holomorphic buildings with
genus $g$ and $\mu$ marked points such that
$\sum L_{\gamma_{-\infty_i}} = L^-$ and $\sum L_{\gamma_{+\infty_i}} = L^+$.
(The genus of blown up curve $\Sigma$ is by definition the genus of $\check \Sigma$.)
First we show that we have a nice way to add marked points to the domain curves.
\begin{lem}\label{nice marked points}
Let $\epsilon > 0$ and $\delta_0 > 0$ be arbitrary small constants,
and let $(\Sigma, z, u, \phi) \in \overline{\mathcal{M}}^0_{g, \mu}(L^-, L^+)$ be an arbitrary
holomorphic building.
We regard $u : \Sigma \to (\overline{\mathbb{R}}_1 \cup \overline{\mathbb{R}}_2 \cup \dots \cup
\overline{\mathbb{R}}_k) \times Y$
not as an equivalence class by $\mathbb{R}$-translations but as a map.
Then there exist a closed subset $I \subset \mathbb{R}_1 \cup \mathbb{R}_2 \cup \dots \cup \mathbb{R}_k$
and additional marked points $z^+$ of $\Sigma$ which satisfy the following conditions:
\begin{itemize}
\item
$I$ is a finite union of intervals in the form $[l, l+1] \subset \mathbb{R}_i$ {\rm(}$l \in \mathbb{Z}${\rm)}.
\item
The length of $I$ and the number of additional marked points are bounded by
some constant determined by $g$, $\mu$, $L^-$, $L^+$, $\epsilon$ and $\delta_0$.
\item
$(\Sigma, z \cup z^+)$ is stable.
\item
There exists a constant $A_1 > 0$ depending only on $g$, $\mu$, $L^-$, $L^+$,
$\epsilon$ and $\delta_0$ such that
if $[-A_1, T + A_1] \times S^1 \subset \Sigma$ does not contain any marked points $z \cup z^+$,
then one of the following two holds true.
\begin{enumerate}[label=\normalfont(\arabic*)]
\item
$u([0, T] \times S^1) \subset I \times Y$ and $\mathrm{diam} \, u([0, T] \times S^1) \leq 20 \epsilon$.
\item
$\sigma \circ u ([0, T] \times S^1)$ is contained in the $\frac{1}{3}$-neighborhood of
the complement of $I \subset \mathbb{R}_1 \cup \mathbb{R}_2 \cup \dots \cup \mathbb{R}_k$,
and there exist $L \in \mathbb{R}$ and $(b, \gamma) \in \mathbb{R} \times P_L$ such that
\[
\mathop{\mathrm{dist}}\nolimits ( u(s,t), (Ls + b, \gamma(t)) ) \leq \epsilon ( e^{-\kappa s} + e^{-\kappa (T - s)})
\]
on $[0, T] \times S^1$.
\end{enumerate}
In particular,
for any disc $D \subset \Sigma$ such that
$D \setminus 0$ does not contain any marked points,
$\mathrm{diam} \, u(\{ z\in D; |z| \leq e^{-2\pi A_1}\}) \leq 20 \epsilon$.
\item
$\sigma \circ u(z \cup z^+) \subset \mathbb{R}_1 \cup \mathbb{R}_2 \cup \dots \cup \mathbb{R}_k$ is contained
in the $\frac{1}{3}$-neighborhood of $I$.
\item
Each connected component of $u^{-1}(I \times Y)$ either
contains at least one point of $z \cup z^+$
or is contained in the inverse image of the $\frac{1}{3}$-neighborhood of
the complement of $I$ by $\sigma \circ u$.
\item
For the $\frac{1}{3}$-neighborhood $J$ of each connected component of the complement
of $I$,
$E_{\hat \omega}(u|_{u^{-1}(J \times Y)}) \leq \delta_0$.
\end{itemize}
\end{lem}
\begin{proof}
First we see the energy bound:
$E_\lambda(u) \leq L^+$ and $E_{\hat \omega}(u) = L^+ - L^-$
for any $(\Sigma, z, u, \phi) \in \overline{\mathcal{M}}^0_{g, \mu}(L^-,L^+)$.
The former is because for any interval $I \subset \mathbb{R}_i$,
\begin{align*}
\frac{1}{|I|} \int_{u^{-1}(I\times Y)} u^\ast (d\sigma \wedge \lambda)
&= \int u^\ast d\varphi \wedge \lambda\\
&= \int u^\ast d(\varphi \lambda) - \int u^\ast (\varphi d\lambda)\\
&\leq L^+,
\end{align*}
where $\varphi : \mathbb{R}_1 \cup \mathbb{R}_2 \cup \dots \cup \mathbb{R}_k \to \mathbb{R}$ is defined by
\[
\varphi(\sigma) = \int_{-\infty}^\sigma \frac{1}{|I|} 1_I (\sigma') d\sigma' \quad \text{on } \mathbb{R}_i,
\]
$\varphi \equiv 0$ on $\mathbb{R}_j$ ($j < i$)
and $\varphi \equiv 1$ on $\mathbb{R}_j$ ($j > i$), and the last inequality is because
$u^\ast(\varphi d\lambda) \geq 0$ by the equation of $J$-holomorphic curves.
Proof of the latter equation $E_{\hat \omega}(u) = L^+ - L^-$ is straightforward.
Next we prove the number of irreducible components of $\check \Sigma$ is bounded by
some constant depending only on $g$, $\mu$, $L^-$ and $L^+$.
Note that if $E_{\hat\omega} (u|_{\Sigma_\alpha}) > 0$, then
$E_{\hat\omega} (u|_{\Sigma_\alpha}) \geq \min (\sum_i L^+_i - \sum_j L^-_j) \ (>0)$,
where the minimum is taken over all pairs of families of periods $(L^+_i)_i$ and $(L^-_j)_j$
such that $\sum_j L^-_j < \sum_i L^+_i \leq L^+$.
Hence the number of the components $\Sigma_\alpha$ on which $u$ have non-zero
$E_{\hat\omega}$-energies is bounded.
If $E_{\hat\omega} (u|_{\Sigma_\alpha}) = 0$ and $2g_\alpha + m_\alpha < 3$, then
$\Sigma_\alpha$ does not contain any marked points and $(\Sigma_\alpha, u)$ is a trivial cylinder.
We can see it by the following consideration:
\begin{itemize}
\item
If the number of imaginary circles in $\Sigma_\alpha$ were zero,
then $u$ would be a constant map since every closed $J$-holomorphic curve
in $\hat Y$ is a constant map.
However, this contradict the definition of holomorphic building.
\item
The number of imaginary circles in $\Sigma_\alpha$ cannot be one since $E_{\hat\omega}(u|_{\Sigma_\alpha}) = 0$.
\item
If the number of imaginary circles in $\Sigma_\alpha$ is two,
then $g_\alpha = 0$ and $\Sigma_\alpha$ does not contain any marked points or nodal points.
Hence $\Sigma_\alpha \cong \overline{\mathbb{R}} \times S^1$ and $(\Sigma_\alpha, u)$ is a
trivial cylinder.
\end{itemize}
Therefore the number of the nontrivial components $\Sigma_\alpha$ such that
$2 g_\alpha + m_\alpha < 3$ is bounded.
Since the number of the limit circles is bounded, this implies that
the number of the components $\Sigma_\alpha$ such that
$2 g_\alpha + m_\alpha \geq 3$ is also bounded.
This is due to the equality
\[
2g + \mu + \, \text{(the number of the limit circles)} \, -2 = \sum_\alpha (2g_\alpha + m_\alpha -2)
\]
and the fact that trivial cylinders do not contribute to the sum on
the right hand side of the equation.
Therefore the number of the nontrivial components is bounded.
In particular, the height $k$ of the $J$-holomorphic building $(\Sigma, z, u)$ is bounded.
Let $S \subset \Sigma$ be the union of the trivial cylinders of $(\Sigma, z, u)$.
Then each connected component of $S$ consists of at most $(k-1)$ trivial cylinders
and it shares a joint circle with some nontrivial component.
Since the number of the joint circles contained in the nontrivial components is bounded,
it implies that the number of the trivial cylinders is also bounded.
Hence the number of the irreducible components of $\check \Sigma$ is bounded.
Therefore, the number of marked points we need to add to $(\Sigma, z)$ in order to
make $(\Sigma, z \cup z^+)$ stable is bounded.
Assuming that $(\Sigma, z \cup z^+)$ is stable,
we further add marked points $z^{++}$ as follows.
We may assume that $\epsilon < \min(\frac{1}{60}, \frac{1}{24} L_{\min} )$,
where $L_{\min}$ is the minimal period of periodic orbits.
Let $\delta > 0$, $\kappa > 0$, $A > 0$ and $L_0$ be the constant of
Corollary \ref{third annulus} for $C_0 = L^+$ and the given $\epsilon > 0$.
We may assume that $\delta \leq \delta_0$.
First, let $I \subset \mathbb{R}_1 \cup \mathbb{R}_2 \cup \dots \cup \mathbb{R}_k$ be a finite union of intervals
$[l, l+1] \subset \mathbb{R}_i$ ($l \in \mathbb{Z}$)
such that
\begin{itemize}
\item
$E_{\hat\omega} (u|_{u^{-1}(J \times Y)}) \leq \delta$ for the
$\frac{1}{3}$-neighborhood $J$ of each connected component of
the complement of $I \subset \mathbb{R}_1 \cup \mathbb{R}_2 \cup \dots \cup \mathbb{R}_k$, and
\item
$\sigma \circ u(z \cup z^+) \subset I$.
\end{itemize}
We may assume that the length of $I$ is bounded by some constant depending only on
$E_{\hat\omega} (u)$, $\delta$ and the number of marked points $z \cup z^+$.
Let $\bigcup_\alpha B_\alpha^1 \supset I \times Y$ be a finite covering by open balls
with radius $\epsilon$,
where the distance of $\mathbb{R} \times Y$ is given by
$\mathop{\mathrm{dist}}\nolimits((\sigma, y), (\sigma', y'))^2 = |\sigma - \sigma'|^2 + \mathop{\mathrm{dist}}\nolimits_Y(y, y')^2$
for some distance $\mathop{\mathrm{dist}}\nolimits_Y$ of $Y$.
We may assume that the number of open balls is bounded by some constant depending
on the length of $I$ and $\epsilon$.
For each $B_\alpha^1$, let $B_\alpha^2$ be the concentric ball with radius $2\epsilon$.
We may assume that $\sigma(B_\alpha^2) \subset \mathbb{R}_1 \cup \mathbb{R}_2 \cup \dots
\cup \mathbb{R}_k$ is contained in the $\frac{1}{3}$-neighborhood of $I$
since $4\epsilon < \frac{1}{3}$.
Then for each connected component of $u^{-1}(B_\alpha^2)$ which contains some points
of $u^{-1}(B_\alpha^1)$, we choose one of these points in $u^{-1}(B_\alpha^1)$
as an additional marked point.
Then the number of the additional marked points is bounded since
\begin{itemize}
\item
if a connected component $\Omega$ of $u^{-1}(B_\alpha^2) \subset \Sigma$
contains a point $z \in u^{-1}(B_\alpha^1)$,
then $u(\partial \Omega) \cap B_\epsilon(u(z)) = \emptyset$,
hence Lemma \ref{monotonicity lemma} implies
$|du|_{L^2(\Omega)}$ is larger than some positive constant depending on $\epsilon$,
and
\item
the total energy on $u^{-1}(B_\alpha^2)$ is bounded by
$|du|_{L^2(u^{-1}(B_\alpha^2))}^2 \leq E_{\hat \omega}(u) + 4\epsilon E_\lambda (u)$.
\end{itemize}
We rewrite $z^+ \cup z^{++}$ as $z^+$.
We claim that this is the required additional marked points.
The only non-trivial condition is the condition about annuli.
Define $A_1 = (2 A + 2) \cdot \lceil \frac{E_{\hat\omega} (u)}{\delta} \rceil$.
(Recall that $A > 0$ is the constant of Corollary \ref{third annulus}.)
First we claim that for each annulus $[0, A_1] \times S^1 \subset \Sigma$,
there exist $s_0 \in [A, A_1 -A]$, $L\in \mathbb{R}$ and $(b, \gamma) \in \mathbb{R} \times P_L$
such that
\[
\mathop{\mathrm{dist}}\nolimits( u(s,t), (Ls + b, \gamma(t))) \leq 2\epsilon \quad \text{on } [s_0 -1, s_0 +1] \times S^1.
\]
This is proved as follows.
Decompose $[0, A_1] = \bigcup [(2A + 2) i, (2A + 2) (i + 1)]$
into $\lceil \frac{E_{\hat\omega} (u)}{\delta} \rceil$ pieces of
intervals with length $2A + 2$.
Then one of them $[(2A + 2) i, (2A + 2) (i + 1)]$ satisfies
$E_{\hat\omega} (u|_{[(2A + 2) i, (2A + 2) (i + 1)] \times S^1}) \leq \delta$.
Hence Corollary \ref{third annulus} implies $s_0 = (2A + 2) i + A + 1$ satisfies the above condition.
Now we assume $[-A_1, T + A_1] \times S^1 \subset \Sigma$ does not contain any
marked points and prove that the required condition holds true.
The above claim implies that there exist $s_1 \in [-A_1 + A, -A]$,
$s_2 \in [T + A, T + A_1 -A]$, $L_i\in \mathbb{R}$ and $(b_i, \gamma_i) \in \mathbb{R} \times P_{L_i}$
($i = 1,2$) such that
\begin{equation}
\mathop{\mathrm{dist}}\nolimits( u(s,t), (L_i s + b_i, \gamma_i (t))) \leq 2\epsilon \quad \text{on } [s_i -1, s_i +1] \times S^1 \label{thin}
\end{equation}
for each $i = 1, 2$.
In particular, this implies $\mathrm{diam} \, \sigma \circ u( \{s_i\} \times S^1) \leq 4 \epsilon$
for each $i = 1, 2$.
For each $z \in [s_1, s_2] \times S^1$ such that $u(z) \in I \times Y$,
there exists some $\alpha$ such that $u(z) \in B_\alpha^1$.
Then $B_\alpha^2$ intersects with $u( \partial ([s_1, s_2] \times S^1) )$ since
$[s_1, s_2] \times S^1$ does not contain any marked points.
(If they did not intersect, then the connected component of $u^{-1}(B_\alpha^2)$
containing $z$ would be contained in $[s_1, s_2] \times S^1$.)
Therefore $u(z)$ is contained in the $3 \epsilon$-neighborhood of
$u(\partial [s_1, s_2] \times S^1)$.
We separate the argument into the following two cases.
\begin{enumerate}[label=\normalfont(\arabic*)]
\item
$\sigma\circ u([s_1, s_2] \times S^1) \subset I$
\item
$\sigma\circ u([s_1, s_2] \times S^1) \not\subset I$
\end{enumerate}
In the first case, $u([s_1, s_2] \times S^1)$ is contained in
the $3 \epsilon$-neighborhood of $u(\{s_1\} \times S^1) \cup u(\{s_2\} \times S^1)$.
Since the diameter of the $3 \epsilon$-neighborhood of each
$\sigma \circ u(\{s_i\} \times S^1)$ is
$\leq 4\epsilon + 2 \cdot 3 \epsilon \leq 10\epsilon$,
it implies $\mathrm{diam} \, \sigma \circ u([s_1, s_2] \times S^1) \leq 20 \epsilon$.
Then $L_i = 0$ ($i = 1, 2$) because if not, (\ref{thin}) implies that
the diameter of $\sigma \circ u ([s_1, s_1 + 1] \times S^1)$ or
$\sigma \circ u ([s_2 - 1, s_2] \times S^1)$ is $\geq L_{\min} -4\epsilon > 20 \epsilon$.
Therefore (\ref{thin}) implies that
$\mathrm{diam} \, u(\{s_i\} \times S^1) \leq 4 \epsilon$ ($i = 1,2$).
Hence $\mathrm{diam} \, u([s_1, s_2] \times S^1) \leq 20\epsilon$ because
$u([s_1, s_2] \times S^1)$ is contained in
the $3 \epsilon$-neighborhood of $u(\{s_1\} \times S^1) \cup u(\{s_2\} \times S^1)$.
In the second case, $\sigma \circ u ([s_1, s_2] \times S^1)$ is contained in
the $20 \epsilon$-neighborhood of the complement of $I$
because it is covered by the complement of $I$ and the $3 \epsilon$-neighborhood of
$u(\{s_1\} \times S^1) \cup u(\{s_2\} \times S^1)$.
Since $20\epsilon < \frac{1}{3}$, it is contained in the $\frac{1}{3}$-neighborhood of
a connected component of the complement of $I$, which implies
$E_{\hat\omega} (u|_{[s_1, s_2] \times S^1}) \leq \delta$.
Since $[-A, T + A] \subset [s_1, s_2]$,
Corollary \ref{third annulus} implies
there exists $L \in \mathbb{R}$, $(b, \gamma) \in \mathbb{R} \times P_L$ and $\kappa > 0$ such that
\[
\mathop{\mathrm{dist}}\nolimits( u(s,t), (Ls + b, \gamma(t))) \leq \epsilon (e^{-\kappa s} + e^{- \kappa(T-s)}) \quad \text{on } [0, T] \times S^1.
\]
\end{proof}
\begin{cor}\label{cor nice marked points}
In Lemma \ref{nice marked points},
we can replace the condition of annuli with the following stronger condition:
\begin{itemize}
\item
If $[-A_1, T + A_1] \times S^1 \subset \Sigma$ does not contain any marked points $z \cup z^+$,
then there exist $L \in \mathbb{R}$ and $(b, \gamma) \in \mathbb{R} \times P_L$ such that
\[
\mathop{\mathrm{dist}}\nolimits ( u(s,t), (Ls + b, \gamma(t)) ) \leq \epsilon ( e^{-\kappa s} + e^{-\kappa (T - s)})
\]
on $[0, T] \times S^1$.
Furthermore, if $L \neq 0$, then
$\sigma \circ u ([0, T] \times S^1)$ is contained in the $\frac{1}{3}$-neighborhood of
the complement of $I$.
\end{itemize}
\end{cor}
\begin{proof}
This is because if the diameter of $u([-1, T +1] \times S^1)$ is sufficiently small,
then Lemme \ref{L^infty diam} implies $|du|_{L^\infty([0, T] \times S^1)}$ is also small,
and we can apply Proposition \ref{second annulus} on $[0, T] \times S^1$.
\end{proof}
\begin{prop}\label{second countable}
$\overline{\mathcal{M}}^0(Y, \lambda, J)$ is second countable.
\end{prop}
\begin{proof}
It is enough to prove that each $\overline{\mathcal{M}}^0_{g, \mu}(L^-, L^+)$ is second countable.
Basically, this is because Lemma \ref{nice marked points} implies
that $\overline{\mathcal{M}}^0_{g, \mu}(L^-, L^+)$ is covered by a countable family of
open subsets consisting of equicontinuous maps.
To explain the details, first we need a preliminary consideration.
Let $(\Sigma, z, u, \phi) \in \overline{\mathcal{M}}^0_{g}(L^-, L^+)$ be a holomorphic building
with a stable domain curve $(\Sigma, z)$
(the number of marked points may be larger than $\mu$).
Let $(\widetilde{P} \to \widetilde{X}, Z)$ be its local universal family.
Let $R \subset \widetilde{X}$ be the subset of the points whose fibers have
the same number of nodal points and imaginary circles as that of $(\Sigma, z)$.
Take a discontinuous map $\Psi : \widetilde{P} \to \widetilde{P}_0$ as in the definition of
the topology of $\overline{\mathcal{M}}^0(Y, \lambda, J)$. ($0 \in \widetilde{X}$ is the point
whose fiber is isomorphic to $(\Sigma, z)$.)
We may assume that for each $a\in R$,
\[
\Psi|_{\widetilde{P}_a \setminus \coprod S^1} : \widetilde{P}_a \setminus
\coprod_{\text{joint circles}} S^1 \to \widetilde{P}_0 \setminus
\coprod_{\text{joint circles}} S^1
\]
is a homeomorphism.
Hence when we regard $(\widetilde{P} \to \widetilde{X}, Z)$
as the local universal family of $(\widetilde{P}_a, Z(a))$,
we can use $\Psi^a = (\Psi|_{\widetilde{P}_a})^{-1} \circ \Psi : \widetilde{P}
\to \widetilde{P}_a$.
For each open subset $U \subset \widetilde{X}$,
we define $\Xi (U)$ as the set of pairs $(a, u)$
each of which consists of a point $a \in U$ and a holomorphic building
$u : \widetilde{P}_a \to
(\overline{\mathbb{R}}_1 \cup \overline{\mathbb{R}}_2 \cup \dots \cup \overline{\mathbb{R}}_k) \times Y$
which is contained in $\overline{\mathcal{M}}^0_{g}(L^-, L^+)$ and which satisfies the
following condition:
If $[-A_1, T + A_1] \times S^1 \subset \widetilde{P}_a$ does not contain any marked
points $Z(a)$, then there exist $L \in \mathbb{R}$ and $(b, \gamma) \in \mathbb{R} \times P_L$
such that
\[
\mathop{\mathrm{dist}}\nolimits ( u(s,t), (Ls + b, \gamma(t)) ) \leq \epsilon ( e^{-\kappa s} + e^{-\kappa (T - s)})
\]
on $[0, T] \times S^1$.
Define $\Xi_R(U) = \{ (a,u) \in \Xi(U); a \in U \cap R \}$.
For each $\epsilon >0$ and $(a_0, u_0) \in \Xi_R(U)$,
let $\widetilde{\mathcal{W}}_{(a_0, u_0)} (U, \epsilon)$ be the space of points
$(a,u) \in \Xi (U)$ such that
\[
\mathop{\mathrm{dist}}\nolimits_{L^\infty}(u, (\theta \times 1) \circ u_0 \circ \Psi^{a_0}|_{\widetilde{P}_a})
< \epsilon
\]
for some $\mathbb{R}$-gluing $\theta$.
First we prove that for any $\epsilon >0$ and any two points
$(a_0, u_0)$, $(a_1, u_1) \in \Xi_R(U)$,
if $(a_0, u_0) \in \widetilde{\mathcal{W}}_{(a_1, u_1)}(U, \epsilon)$,
then $\widetilde{\mathcal{W}}_{(a_1, u_1)}(U, \epsilon) \subset
\widetilde{\mathcal{W}}_{(a_0, u_0)}(U, 2\epsilon)$.
Since $(a_0, u_0) \in \widetilde{\mathcal{W}}_{(a_1, u_1)}(U, \epsilon)$,
there exists an $\mathbb{R}$-translation
$\theta_0 : \overline{\mathbb{R}}_1 \cup \overline{\mathbb{R}}_2 \cup \dots \cup \overline{\mathbb{R}}_k
\to \overline{\mathbb{R}}_1 \cup \overline{\mathbb{R}}_2 \cup \dots \cup \overline{\mathbb{R}}_k$ such that
\[
\mathop{\mathrm{dist}}\nolimits_{L^\infty} (u_0, (\theta_0 \times 1) \circ u_1 \circ \Psi^{a_1}|_{\widetilde{P}_{a_0}})
< \epsilon.
\]
For any $(a, u) \in \widetilde{\mathcal{W}}_{(a_1, u_1)}(U, \epsilon)$,
there exists an $\mathbb{R}$-gluing $\theta : \overline{\mathbb{R}}_1 \sqcup \overline{\mathbb{R}}_2 \sqcup
\dots \sqcup \overline{\mathbb{R}}_k \to \overline{\mathbb{R}}_1 \cup \overline{\mathbb{R}}_2 \cup \dots \cup
\overline{\mathbb{R}}_l$ such that
\[
\mathop{\mathrm{dist}}\nolimits_{L^\infty} (u, (\theta \times 1) \circ u_1 \circ \Psi^{a_1}|_{\widetilde{P}_a})
< \epsilon.
\]
Since $a_0$ and $a_1$ are contained in $R$,
$\Psi^{a_0} = (\Psi^{a_1}|_{\widetilde{P}_{a_0}})^{-1} \circ \Psi^{a_1}
: \widetilde{P} \to \widetilde{P}_{a_0}$.
Hence
\begin{align*}
&\mathop{\mathrm{dist}}\nolimits_{L^\infty} (u, (\theta \circ \theta_0^{-1} \times 1) \circ u_0 \circ
\Psi^{a_0}|_{\widetilde{P_a}})\\
&\leq \mathop{\mathrm{dist}}\nolimits_{L^\infty} (u, (\theta \times 1) \circ u_1 \circ \Psi^{a_1}|_{\widetilde{P}_a})\\
& \quad + \mathop{\mathrm{dist}}\nolimits_{L^\infty} ((\theta \times 1) \circ u_1 \circ \Psi^{a_1}|_{\widetilde{P}_a},
(\theta \circ \theta_0^{-1} \times 1) \circ u_0 \circ
(\Psi^{a_1}|_{\widetilde{P}_{a_0}})^{-1} \circ \Psi^{a_1}|_{\widetilde{P}_a})\\
& \leq \mathop{\mathrm{dist}}\nolimits_{L^\infty} (u, (\theta \times 1) \circ u_1 \circ \Psi^{a_1}|_{\widetilde{P}_a})
+ \mathop{\mathrm{dist}}\nolimits_{L^\infty} (u_1, (\theta_0 \times 1)^{-1} \circ u_0 \circ
(\Psi^{a_1}|_{\widetilde{P}_{a_0}})^{-1})\\
&< 2\epsilon,
\end{align*}
which implies $\widetilde{\mathcal{W}}_{(a_1, u_1)}(U, \epsilon) \subset
\widetilde{\mathcal{W}}_{(a_0, u_0)}(U, 2\epsilon)$.
We can choose a countable points $(a_i, u_i) \in \Xi_R(U)$
such that $\{\widetilde{\mathcal{W}}_{(a_i, u_i)}(U, \epsilon)\}_i$ covers $\Xi_R(U)$
for any $\epsilon >0$ because the assumption of the holomorphic buildings in $\Xi(U)$
implies the equicontinuity.
Then for any $(a, u) \in \Xi_R(U)$ and $\epsilon >0$,
there exists $(a_i, u_i)$ such that
$(a, u) \in \widetilde{\mathcal{W}}_{(a_i, u_i)}(U, \epsilon) \subset
\widetilde{\mathcal{W}}_{(a, u)}(U, 2\epsilon)$
Let $\{U_j\}$ be a countable open basis of the union of the base spaces of
the universal families and we choose the above $\{(a_i^{(j)}, u_i^{(j)})\}_i$ for each $U_j$.
For each $(a_i^{(j)}, u_i^{(j)})$, we fix a family of coordinates of limit circles
$\phi_{\pm\infty_l} : S^1 \to S^1_{a_i^{(j)}, \pm\infty_l}$.
Let $\mathring{Z}(a_i^{(j)}) \subset Z(a_i^{(j)})$ be an arbitrary subsequence whose
cardinality is $\mu$.
Then
\[
(\widetilde{P}_{a_i^{(j)}}, \mathring{Z}(a_i^{(j)}), u_i^{(j)},
(\phi_{\pm\infty_l} + \frac{c_l}{2^n})) \quad (n \in \mathbb{N}, 1 \leq c_l \leq 2^n)
\]
is a countable family of holomorphic buildings in $\overline{\mathcal{M}}^0_{g, \mu}(L^-, L^+)$.
Let $p_k^{(i, j)}$ ($k \in \mathbb{N}$) be such holomorphic buildings for each $(a_i^{(j)}, u_i^{(j)})$
and all choices of the subsequence $\mathring{Z}(a_i^{(j)}) \subset Z(a_i^{(j)})$.
We claim that $\{\mathcal{W}_{p_k^{(i, j)}}(U_j, 2^{-l}) \}_{i, j, k, l \in \mathbb{N}}$ is a countable basis of
$\overline{\mathcal{M}}^0_{g, \mu}(L^-, L^+)$.
This is proved as follows.
For any $p = (\Sigma, z, u) \in \overline{\mathcal{M}}^0_{g, \mu}(L^-, L^+)$,
we can choose additional marked points $z^+ \subset \Sigma$
as in Corollary \ref{cor nice marked points}.
Let $(\widetilde{P} \to \widetilde{X}, Z \cup Z^+)$ be the local universal family of
$(\Sigma, z \cup z^+)$.
Then for any neighborhood $\mathcal{N}$ of $p$,
there exists $U_j$ and $\epsilon \in \{2^{-l} \}$ such that
$\mathcal{W}_p(U_j, 2\epsilon)$ is contained in $\mathcal{N}$.
Note that we may assume that the point $a \in U_j$ whose fiber is isomorphic to
$(\Sigma, z \cup z^+)$ is contained in $R$.
Choose $(a_i^{(j)}, u_i^{(j)}) \in \Xi_R(U_j)$ such that
\[
(a, u) \in \widetilde{\mathcal{W}}_{(a_i^{(j)}, u_i^{(j)})} (U_j, \epsilon) \subset
\widetilde{\mathcal{W}}_{(a, u)}(U_j, 2\epsilon).
\]
This implies that there exists a holomorphic
building $p_k^{(i, j)} \in \overline{\mathcal{M}}^0_{g, \mu}(L^-, L^+)$ such that
\[
p \in \mathcal{W}_{p_k^{(i, j)}} (U_j, \epsilon) \subset \mathcal{W}_p(U_j, 2 \epsilon) \subset \mathcal{N}.
\]
Therefore $\{\mathcal{W}_{p_k^{(i, j)}}(U_j, 2^{-l}) \}_{i, j, k, l \in \mathbb{N}}$ is a countable basis of
$\overline{\mathcal{M}}^0_{g, \mu}(L^-, L^+)$.
\end{proof}
\begin{prop}\label{compactness}
Each $\overline{\mathcal{M}}^0_{g, \mu}(L^-, L^+)$ is compact.
\end{prop}
\begin{proof}
Since $\overline{\mathcal{M}}^0_{g, \mu}(L^-,L^+)$ is second countable,
in order to prove its compactness, it is enough to prove that any sequence
$p_i = (\Sigma_i, z_i, u_i, \phi_i) \in \overline{\mathcal{M}}^0_{g, \mu}(L^-,L^+)$ contains
a subsequence which converges to a point in $\overline{\mathcal{M}}^0_{g, \mu}(L^-,L^+)$.
Let $I_i \subset \mathbb{R}_1 \cup \mathbb{R}_2 \cup \dots \cup \mathbb{R}_{k_i}$ and $z_i^+ \subset \Sigma_i$
be the pair of closed subset and additional marked points given by Corollary
\ref{cor nice marked points} for sufficiently small $\epsilon > 0$ and $\delta_0 > 0$.
Passing to a subsequence,
we may assume the following:
\begin{itemize}
\item
The number of the additional marked points is independent of $i$.
\item
$(\check \Sigma_i, z_i \cup z_i^+ \cup (\pm \infty_i))$ converges to a stable curve
$(\check \Sigma, z \cup z^+ \cup (\pm\infty_i))$ in the moduli space of
marked stable curves.
\end{itemize}
Let $(\Sigma', z \cup z^+)$ be the oriented blow up of $(\check \Sigma, z \cup z^+)$
at $\pm \infty_i$ and nodal points of $\check \Sigma$ corresponding to joint circles in
$\Sigma_i$ by appropriate $\varphi$'s ($\in S^1$).
Let $(\widetilde{P}' \to \widetilde{X}', Z \cup Z^+)$ be the local universal family of
$(\Sigma', z \cup z^+)$, and let $0 \in \widetilde{X}'$ be the point
whose fiber is isomorphic to $(\Sigma', z \cup z^+)$.
Choosing appropriate $\varphi$'s, we may assume
there exists a sequence $x'_i \in \widetilde{X}'$ converging to $0 \in \widetilde{X}'$
such that $(\widetilde{P}'_{x'_i}, Z(x'_i) \cup Z^+(x'_i)) \cong (\Sigma_i, z_i \cup z_i^+)$.
Let $\Psi' : \widetilde{P}' \to \widetilde{P}'_0$ be the discontinuous map used for the
definition of the topology of $\overline{\mathcal{M}}^0(Y, \lambda, J)$.
We may assume $\Psi'$ maps marked points $Z \cup Z^+$ to $Z(0) \cup Z^+(0)$.
Define a map
\begin{align*}
&v_i = u_i \circ (\Psi'|_{\widetilde{P}'_{x'_i} \setminus \coprod S^1})^{-1} :
\Sigma' (\cong \widetilde{P}_0)
\supset \Psi'(\widetilde{P}'_{x'_i} \setminus \coprod S^1) \\
&\hphantom{v_i = u_i \circ (\Psi'|_{\widetilde{P}'_{x'_i} \setminus \coprod S^1})^{-1} :
\Sigma' (\cong \widetilde{P}_0) \supset}
\to (\overline{\mathbb{R}}_1 \cup \overline{\mathbb{R}}_2 \cup \dots \cup \overline{\mathbb{R}}_{k_i}) \times Y
\end{align*}
for each $i$.
Let $q^j$ be the new nodal points in $\Sigma'$.
(Namely, neighborhoods of $q^j$ correspond to annuli in $\Sigma_i$.)
Then the annulus condition of Lemma \ref{nice marked points} (or Corollary
\ref{cor nice marked points}) implies that
on any connected compact subset of $\Sigma' \setminus (\coprod S^1 \cup \{q^j\})$,
a subsequence of $v_i$ converges to a $J$-holomorphic map $v_\infty$
if we change each $v_i$ by $\mathbb{R}$-translation.
Let $q \in \Sigma'$ be one of new nodal points.
Recall that the restriction of the fibration $\widetilde{P}' \to \widetilde{X}'$ to the
neighborhood $\check N \subset \widetilde{P}$ of $q$ is equivalent to
\begin{align*}
\check N = A \times D \times D &\to A \times D = \widetilde{X}'\\
(a, x, y) &\mapsto (a, xy)
\end{align*}
and $\Psi'|_{\check N}$ is defined by
\[
\Psi'(a, x, y) =\begin{cases}
(0, x, 0) \quad \text{if } |x| \geq |y|\\
(0, 0, y) \quad \text{if } |y| \geq |x|
\end{cases}
\]
We may assume that $\check N$ does not contain any marked points.
Assume that $x_i' = (a_i, \allowbreak e^{2\pi(-\rho_i + \sqrt{-1} \varphi_i)})$.
Then $\rho_i \to \infty$ as $i \to \infty$,
and $\check N \cap \widetilde{P}'_{x'_i} \cong
[0, \rho_i] \times S^1 \cup [-\rho_i, 0] \times S^1$,
where $\{\rho_i\} \times S^1 \subset [0, \rho_i] \times S^1$ and
$\{-\rho_i\} \times S^1 \subset [-\rho_i, 0] \times S^1$ are identified by
$(\rho_i, t_x) \sim (-\rho_i, t_y)$ if $t_y -t_x = \varphi_i$.
Since $\check N$ does not contain any marked points,
there exist $L \in \mathbb{R}$ and $(b_i, \gamma_i) \in \mathbb{R} \times P_L$ such that
\begin{align*}
\mathop{\mathrm{dist}}\nolimits (u_i(s,t), (Ls + b_i, \gamma_i(t))) &\leq 2\epsilon e^{-\kappa |s|} \quad \text{on }
[A_1, \rho_i] \times S^1,\\
\mathop{\mathrm{dist}}\nolimits (u_i(s,t), (Ls + b_i + 2 L \rho_i, \gamma_i(t + \varphi_i))) & \leq 2 \epsilon
e^{-\kappa |s|} \quad
\text{on } [-\rho_i, -A_1] \times S^1.
\end{align*}
We may assume that $L$ is nonnegative and independent of $i$.
If $L = 0$, then $\gamma_i$ is a sequence of points in $Y$,
and its subsequence converges to a point of $Y$.
Hence a subsequence of $v_i$ uniformly converges to a $J$-holomorphic map $v_\infty$
on a neighborhood of this nodal point in $\Sigma'$ if we change each $v_i$ by
$\mathbb{R}$-translation,
where uniform convergence means that the $L^\infty$-distance between
$v_i$ and $v_\infty$ on the intersection of the domain of $v_i$ and the neighborhood of
the nodal point converges to zero.
If $L > 0$,
then a subsequence of $\varphi_i$ converges to some $\varphi \in S^1$.
We blow up these nodal points $q$ of $\Sigma'$ by $\varphi$'s and denote the new curve
by $\Sigma$.
Then it is easy to see that there exists a $J$-holomorphic map $v_\infty$
on a neighborhood $\widetilde{N}^\circ$ of each of these new joint circles to
$(\mathbb{R} \cup \mathbb{R}) \times Y$
such that for each $i$, there exists an $\mathbb{R}$-gluing $\theta_i : \overline{\mathbb{R}} \sqcup
\overline{\mathbb{R}} \to \mathbb{R}$ such that $L^\infty$-distance of $u_i$ and
$(\theta_i \times 1) \circ v_\infty \circ \Psi|_{\widetilde{P}_{x_i}}$ converges to zero
as $i$ goes to infinity,
where $(\widetilde{P} \to \widetilde{X}, Z \cup Z^+)$ is the local universal family
of the blown up curve $(\Sigma, z \cup z^+)$,
$\Psi$ is the discontinuous map for this local universal family,
and $x_i \in \widetilde{X}$ is the point whose fiber is isomorphic to
$(\Sigma_i, z_i \cup z_i^+)$ for each $i$.
Let $\Sigma \setminus \coprod_{\text{imaginary circles}} S^1 = \mathring{\Sigma}^\nu$
be the decomposition into the connected components.
We have already seen that on each closure $\Sigma^\nu
= \overline{\mathring{\Sigma}^\nu}$,
$v_i$ converges to a $J$-holomorphic curve $v_\infty$ if we change each $v_i$ by
$\mathbb{R}$-translation.
(But these $\mathbb{R}$-translations may depend on $\Sigma^\nu$.)
We may assume that each $\Sigma^\nu$ contains some marked points by the following
argument. For each $\Sigma^\nu$ which does not contain any marked points,
we take a holomorphic section $Z' : \widetilde{X} \to \widetilde{P}$ which intersects with
$\Sigma^\nu$,
and we add $z'_i = Z'(x_i) \subset \widetilde{P}_{x_i} \cong \Sigma_i$ as
an additional marked point for each $i$.
Let $I_i^{++}$ be the union of intervals $[k, k+ 1] \subset \mathbb{R}_1 \cup \mathbb{R}_2 \cup \dots
\cup \mathbb{R}_k$ which contains $\sigma \circ u_i (z'_i)$.
We further add marked points $z_i^{++}$ to $\Sigma_i$ as in Lemma
\ref{nice marked points}, that is,
take a finite covering of $I_i^{++} \times Y$ by open balls $B_\alpha^1$
with radius $\epsilon$ and add a marked point for each connected component of
$u_i^{-1}(B_\alpha^2)$ which contains a point of $u^{-1}(B_\alpha^1)$.
We can do the same argument using $(\Sigma_i, z_i \cup z_i^+ \cup z'_i \cup z_i^{++})$
instead of $(\Sigma_i, z_i \cup z_i^+)$, and we get a curve
$(\Sigma^{++}, z \cup z^+ \cup z' \cup z^{++})$ instead of $(\Sigma, z \cup z^+)$.
Then it is clear that $(\Sigma, z \cup z^+ \cup z')$ is obtained by collapsing all unstable
component of $(\Sigma^{++}, z \cup z^+ \cup z')$.
We claim that each connected component of $\Sigma^{++} \setminus
\coprod_{\text{imaginary circles}} S^1$ contains some marked points.
This can be seen as follows.
First we show that every irreducible component $\Sigma^{++}_\alpha$ of $\Sigma^{++}$
which contains at least one imaginary circle and will be collapsed when we forget marked
points $z^{++}$ is a cylinder with at least one additional marked point $z^{++}$ and without
any marked points $z \cup z^+ \cup z'$ or any nodal points.
Such a component $\Sigma^{++}_\alpha$ is either a closed disc or a cylinder,
but the former cannot be occur because if it did, then
$(\Sigma^{++}_\alpha, z \cup z^+)$ would be a closed disc $\mathbb{C} \cup S^1_\infty$
with at most one marked point,
hence the annulus condition for the marked points $z_i \cup z^+_i$ in Lemma
\ref{nice marked points} would imply the diameter of the image of $\Sigma^{++}_\alpha$
by $v_\infty$ is $\leq 2\epsilon$, which is a contradiction.
(We assume that $2\epsilon$ is smaller than the minimal diameter of periodic orbits.)
Hence $\Sigma_\alpha^{++}$ is a cylinder which does not contain any marked points
$z_i \cup z_i^+$ or any nodal points, which implies that $\Sigma^{++}_\alpha$ contains
at least one additional marked point $z^{++}$.
Using this, we can prove each connected component of
$\Sigma^{++} \setminus \coprod S^1$ contains some marked points $z \cup z^+ \cup z'
\cup z^{++}$.
Indeed, if one connected component of $\Sigma^{++} \setminus \coprod S^1$ did not
contain any marked points, then its closure does not collapse to a imaginary circle in
$(\Sigma, z \cup z^+ \cup z')$ and the corresponding component of
$\Sigma \setminus \coprod S^1$ would not contain any marked points
$z \cup z^+ \cup z'$, but this contradicts the choice of $z'$.
Therefore, rewriting $I_i \cup I_i^{++}$ as $I_i$, and $z_i^+ \cup z'_i \cup z^{++}$ as $z^+_i$,
we may assume each connected component $\mathring{\Sigma}^\nu$ of
$\Sigma \setminus \coprod S^1$ contains at least one marked point.
Let $I_i = I_i^1 \cup I_i^2 \cup \dots \cup I_i^{l_i}$ be the decomposition into connected
components for each $i$.
We define an equivalence relation on the set $\{I_i^a\}_{1 \leq a \leq l_i}$
for sufficiently large $i$ as follows
and use these equivalence classes as floors.
Let $\tilde I$ be the $\frac{1}{3}$-neighborhood of $I$ for each interval $I$.
First note the following:
\begin{itemize}
\item
For each $\Sigma^\nu$, $\mathrm{diam} \, \sigma \circ v_i((z\cup z^+) \cap \Sigma^\nu)$ is
bounded uniformly with respect to $i$.
This can be seen by covering a path from one marked point to another by a finite
number of discs in $\Sigma^\nu$ and using the annulus condition for these discs.
Therefore there exists a constant $C>0$ such that if
$v_i((z\cup z^+) \cap \Sigma^\nu)$ intersects with both of
$\tilde I_i^a$ and $\tilde I_i^b$ then $\mathop{\mathrm{dist}}\nolimits (\tilde I_i^a, \tilde I_i^b) \leq C$.
\item
If $\Sigma^\nu$ and $\Sigma^{\nu'}$ are connected by joint circles in $\Sigma$,
then there exist $a_i \in \{1,2,\dots, l_i\}$ for all large $i$ such that
\begin{itemize}
\item
$v_i((z\cup z^+) \cap \Sigma^\nu)$ intersects with $\tilde I_i^{a_i}$, and
$v_i((z\cup z^+) \cap \Sigma^{\nu'})$ intersects with $\tilde I_i^{a_i + 1}$
(or the condition in which the order of $\nu$ and $\nu'$ is changed is satisfied) and
\item
$\mathop{\mathrm{dist}}\nolimits (\tilde I_i^{a_i}, \tilde I_i^{a_i +1}) > 2C$ and
$\mathop{\mathrm{dist}}\nolimits (\tilde I_i^{a_i}, \tilde I_i^{a_i +1}) \to \infty$ as $i \to \infty$.
\end{itemize}
This is because of the asymptotic behavior of $v_i$ on a neighborhood of a joint circle.
\item
For any $\tilde I_i^a$ and $\tilde I_i^b$ ($a < b$),
either of the following two occurs:
\begin{itemize}
\item
There exists $\Sigma^\nu$ such that $v_i((z\cup z^+) \cap \Sigma^\nu)$ intersects with
both of $\tilde I_i^a$ and $\tilde I_i^b$.
\item
There exist $a \leq c < b$ and a pair $\Sigma^\nu$ and $\Sigma^{\nu'}$ connected by
joint circles in $\Sigma$ such that $v_i((z\cup z^+) \cap \Sigma^\nu)$ intersects with
$\tilde I_i^c$ and $v_i((z\cup z^+) \cap \Sigma^{\nu'})$ intersects with $\tilde I_i^{c + 1}$.
\end{itemize}
This is proved as follows.
Since $\Sigma$ is connected, it is easy to see that there exist two marked points
$w_i^a, w_i^b \in z \cup z^+$ such that $v_i(w_i^a) \in \tilde I_i^a$,
$v_i(w_i^b) \in \tilde I_i^b$ and a path $\ell$ in $\Sigma$ from $w_i^a$ to $w_i^b$
such that $v_i(\ell)$ does not intersect with $\tilde I_i^{a-1}$ or $\tilde I_i^{b+1}$.
If $\ell$ intersects with some joint circles then the latter holds and
otherwise the former holds.
\end{itemize}
Therefore, for sufficiently large $i$ and any $\tilde I_i^a$ and $\tilde I_i^b$,
either $\mathop{\mathrm{dist}}\nolimits (\tilde I_i^a, \tilde I_i^b) \leq C$ or $\mathop{\mathrm{dist}}\nolimits (\tilde I_i^a, \tilde I_i^b) > 2C$.
Hence we can define the equivalence relation $\sim$ on the set of intervals
$\{\tilde I_i^1, \tilde I_i^2, \dots, \tilde I_i^{l_i}\}$ by
$\tilde I_i^a \sim \tilde I_i^b$ if $\mathop{\mathrm{dist}}\nolimits (\tilde I_i^a, \tilde I_i^b) \leq C$,
and the set of the equivalent classes has a natural total order.
Fix one large $i$.
Then we can define the floor of each $\Sigma^\nu$ as the equivalence class of
$\tilde I_i^a$ with which $\sigma \circ v_i((z\cup z^+) \cap \Sigma^\nu)$ intersects.
Then for any two components $\Sigma^\nu$ and $\Sigma^{\nu'}$ connected by
some joint circles in $\Sigma$, which of the two has a higher floor is independent of
the choice of $i$ and the difference is one.
Hence we have defined the floor structure of $\Sigma$ independently of $i$.
For each $i$ and floor $j \in \{1,2,\dots l\}$ represented by $I_i^a$,
take one point $b_i^j$ of $I_i^a$.
Define an $\mathbb{R}$-gluing $\theta_i : \overline{\mathbb{R}}_1 \sqcup \overline{\mathbb{R}}_2 \sqcup \dots
\sqcup \overline{\mathbb{R}}_l \to \overline{\mathbb{R}}_1 \cup \overline{\mathbb{R}}_2 \cup \dots \cup
\overline{\mathbb{R}}_{l_i}$ by $\theta_i (0_j) = b_i^j$.
Then it is easy to see that a subsequence of $(\theta_i \times 1)^{-1} \circ u_i \circ
(\Psi|_{\widetilde{P}_{a_i}})^{-1}$
converges to a $J$-holomorphic map
$u_\infty : \Sigma \to
(\overline{\mathbb{R}}_1 \cup \overline{\mathbb{R}}_2 \cup \dots \cup \overline{\mathbb{R}}_l) \times Y$, that is,
\[
\mathop{\mathrm{dist}}\nolimits_{L^\infty} (u_i, (\theta_i \times 1) \circ u_\infty \circ \Psi|_{\widetilde{P}_{a_i}})
\to 0
\]
as $i \to \infty$.
Finally, passing to a subsequence, we may assume that the sequence
$\pi_{S^1}^{\breve N_{\pm\infty_i}} \circ \phi_{i, \pm\infty_j} : S^1 \to S^1_{\pm\infty_j}$
converges to a family of coordinates $\phi_{\pm\infty_j} : S^1 \to S^1_{\pm\infty_j}$.
The constructed curve $(\Sigma, z, u_\infty, \phi)$ often has unstable components or
floors which consist of trivial cylinders.
Hence we first collapse the unstable components of $(\Sigma, z, u_\infty, \phi)$
(the components $\Sigma_\alpha$ on which $u_\infty$ is constant
and $2g_\alpha + m_\alpha <3$).
Next we collapse all the floors which consist of trivial cylinders.
Then it is clear that $(\Sigma, z_i, u_i, \phi_i)$ converges to this holomorphic building in
the topology of $\overline{\mathcal{M}}^0_{g, \mu}(L^-,L^+)$.
\end{proof}
\begin{prop}\label{Hausdorffness}
$\overline{\mathcal{M}}^0$ is Hausdorff.
\end{prop}
\begin{proof}
The proof is the same as the case of Gromov-Witten theory in \cite{FO99}.
Assuming a sequence $(\Sigma_i, z_i, u_i, \phi_i)\in \overline{\mathcal{M}}^0_{g, \mu}(L^-,L^+)$
converges to two points $(\Sigma, z, u, \phi)$ and $(\Sigma', z', u', \phi')$
in the topology of $\overline{\mathcal{M}}^0_{g, \mu}(L^-,L^+)$,
we prove that these two points coincide.
Let $z^+ \subset \Sigma$ be additional points which make $(\Sigma, z \cup z^+)$ stable,
and let $(\widetilde{P} \to \widetilde{X}, Z \cup Z^+)$ be the local universal family of
$(\Sigma, z \cup z^+)$.
Then by the definition of the topology,
there exists a sequence of points $x_i \to 0 \in \widetilde{X}$ and
a sequence of $\mathbb{R}$-gluings $\theta_i$ such that
$(\Sigma_i, z_i) \cong (\widetilde{P}_{x_i}, Z(x_i))$,
\[
\mathop{\mathrm{dist}}\nolimits_{L^\infty} (u_i, (\theta_i \times 1) \circ u \circ \Psi|_{\widetilde{P}_{x_i}}) \to 0
\]
and
\[
\mathop{\mathrm{dist}}\nolimits_{L^\infty(S^1)}(\pi_{S^1}^{\widetilde{N}_{\pm\infty_j}} \circ \phi_{i, \pm\infty_j},
\phi_{\pm\infty_j}) \to 0.
\]
Define additional marked points $z_i^+ = Z^+(x_i) \subset \Sigma_i$.
Similarly, let ${z'}^+ \subset \Sigma'$ be additional points which make
$(\Sigma', z' \cup {z'}^+)$ stable, and let $(\widetilde{P}' \to \widetilde{X}', Z' \cup {Z'}^+)$
be the local universal family of $(\Sigma', z' \cup {z'}^+)$.
Then there exists a sequence of points $x'_i \to 0 \in \widetilde{X}'$
and a sequence of $\mathbb{R}$-gluings $\theta'_i$ such that
$(\Sigma'_i, z'_i) \cong (\widetilde{P}'_{x'_i}, Z'(x'_i))$,
\[
\mathop{\mathrm{dist}}\nolimits_{L^\infty} (u_i, (\theta'_i \times 1) \circ u \circ \Psi'|_{\widetilde{P}'_{x'_i}}) \to 0
\]
and
\[
\mathop{\mathrm{dist}}\nolimits_{L^\infty(S^1)}(\pi_{S^1}^{\breve N_{\pm\infty_j}} \circ \phi_{i, \pm\infty_j},
\phi'_{\pm\infty_j}) \to 0.
\]
Define additional marked points ${z'}_i^+ = {Z'}^+(x'_i) \subset \Sigma_i$.
We may assume $\pi_Y \circ u (z^+)$ and $\pi_Y \circ u ({z'}^+)$ do not share any points.
Then $z_i^+$ and ${z'}_i^+$ are disjoint for large $i$.
Starting a holomorphic building $(\Sigma_i, z_i, u_i)$ with additional marked points
$z_i^+ \cup {z'}_i^+$, we further add marked points $z_i^{++}$ by the procedure we
explained in the proof of Proposition \ref{compactness}.
Passing to a subsequence if necessary, there exists a holomorphic building
$(\Sigma'',z \cup z^+ \cup {z'}^+ \cup z^{++}, u'')$ which satisfies the following condition.
Let $(\widetilde{P}'' \to \widetilde{X}'', Z \cup Z^+ \cup {Z'}^+ \cup Z^{++})$
be the local universal family of
$(\Sigma'', z \cup z^+ \cup {z'}^+ \cup z^{++})$.
Then there exists a sequence of points $x''_i \to 0 \in \widetilde{X}''$
and a sequence of $\mathbb{R}$-gluings $\theta''_i$ such that
$(\Sigma_i, z_i \cup z_i^+ \cup {z'}_i^+ \cup z_i^{++}) \cong (\widetilde{P}''_{x''_i},
Z(x''_i) \cup Z^+(x''_i) \cup {Z'}^+(x''_i) \cup Z^{++}(x''_i))$,
\[
\mathop{\mathrm{dist}}\nolimits_{L^\infty} (u_i, (\theta''_i \times 1) \circ u'' \circ \Psi''|_{\widetilde{P}_{x''_i}}) \to 0
\]
and
\[
\mathop{\mathrm{dist}}\nolimits_{L^\infty(S^1)}(\pi_{S^1}^{\breve N_{\pm\infty_i}} \circ \phi_{i, \pm\infty_j},
\phi''_{\pm\infty_j}) \to 0.
\]
Since the space of stable curves are Hausdorff and the forgetful map is continuous,
the stabilization of $(\Sigma'', z \cup z^+)$ is $(\Sigma, z \cup z^+)$.
Since the forgetful map $(\widetilde{P}'', \widetilde{X}'') \to (\widetilde{P}, \widetilde{X})$
maps $x''_i$ to $x_i$, $u$ and $\phi$ are the maps induced by $u''$ and $\phi''$.
Since the same is true for $(\Sigma', z', u')$, the two holomorphic buildings
$(\Sigma, z, u)$ and $(\Sigma', z', u')$ coincide.
\end{proof}
We also use the following quotient space
$\widehat{\mathcal{M}}^0(Y, \lambda, J) = \overline{\mathcal{M}}^0(Y, \lambda, J) / \sim$.
This space is obtained by ignoring the coordinates of limit circles and the order of
marked points and limit circles, that is, in $\widehat{\mathcal{M}}^0(Y, \lambda, J)$,
we identify two holomorphic buildings $(\Sigma, z, u, \phi)$ and $(\Sigma', z', u', \phi')$
if there exist a biholomorphism $\varphi : \Sigma' \to \Sigma$ and an $\mathbb{R}$-translation
$\theta$ such that $\varphi(\{z'_i\}) = \{z_i\}$ (that is, $\varphi$ maps $\{z'_i\}$ to
$\{z_i\}$ as a set) and $u' = (\theta \times 1) \circ u \circ \varphi$.
Hence we may write a point of $\widehat{\mathcal{M}}^0(Y, \lambda, J)$ as $(\Sigma, z, u)$,
where $z$ is a set of points of $\Sigma$.
$\widehat{\mathcal{M}}^0$ is also second countable and Hausdorff, and each
$\widehat{\mathcal{M}}^0_{g, \mu}(L^-, L^+)$ is compact because
$\widehat{\mathcal{M}}^0$ is a quotient space of a subspace of $\overline{\mathcal{M}}^0$
by a compact group locally.
Recall that $\overline{\mathcal{M}} = \overline{\mathcal{M}}(Y, \lambda, J)$ is the space of
all (possibly disconnected) holomorphic buildings without trivial buildings.
This space is decomposed by the number of the connected component of the domain
curve.
We can define the topology of each of them similarly and prove the second countability,
compactness and Hausdorff property as $\overline{\mathcal{M}}^0$.
The compactness is stated as follows, where the genus $g$ of
a disconnected holomophic building $(\Sigma, z, u, \phi)$ is defined
by $g = 1 - \frac{1}{2} \chi(\check \Sigma) \in \mathbb{Z}$ ($\chi(\check \Sigma)$ is
the Euler number of the curve $\check \Sigma$).
\begin{prop}\label{compactness for disconnected}
For any $g_0 \in \mathbb{Z}$, $\mu_0 \geq 1$ and $L^+_0 \in \mathbb{R}$,
\[
\bigcup_{\substack{-\infty < g \leq g_0 \\ \mu \leq \mu_0 \\ L^- \leq L^+ \leq L^+_0}}
\overline{\mathcal{M}}_{g, \mu}(L^-, L^+)
\]
is compact.
\end{prop}
\begin{proof}
It is enough to prove that the number of the connected components of the domain
curve of a holomorphic building in the above space is bounded by some constant
depending only on $g_0$, $\mu_0$ and $L^+_0$.
The number of the connected components which have $+\infty$-limit circles is bounded,
and so are the number of the components with marked points.
Since the other components are constant maps, each of them have genus $\geq 2$.
Therefore, the number of them is also bounded.
(Note that the genus of the curve is $g = 1 + \sum_i (g_i - 1)$, where $g_i$ are the
genera of connected components.)
\end{proof}
We define the quotient space $\widehat{\mathcal{M}} = \overline{\mathcal{M}} / \sim$ similarly.
\subsection{The case of manifolds with cylindrical ends}
\label{holomorphic buildings for X}
Next we consider the holomorphic buildings for a symplectic manifold $X$ with cylindrical
ends.
In this case, floor takes values in $\{ -k_-, -k_- +1, \dots, k_+\}$.
\begin{defi}
A holomorphic building $(\Sigma, z, u, \phi)$ for $X$ consists of
\begin{itemize}
\item
a marked curve $(\Sigma, z)$ which is obtained from
a union of marked semistable curves
$(\check \Sigma, z \cup (\pm \infty_i))$ with a floor structure,
\item
a continuous map $u : \Sigma \to (\overline{\mathbb{R}}_{-k_-} \cup \dots \cup
\overline{\mathbb{R}}_{-1}) \times Y^- \cup \overline{X} \cup (\overline{\mathbb{R}}_1 \cup \dots \cup
\overline{\mathbb{R}}_{k_+}) \times Y^+$ and
\item
a family of coordinates $\phi_{\pm\infty_i} : S^1 \to S^1_{\pm\infty_i}$ of limit circles
\end{itemize}
which satisfy the following conditions:
\begin{itemize}
\item
If $i(\alpha) <0$ then
$u(\Sigma_\alpha \setminus \coprod S^1) \subset \mathbb{R}_{i(\alpha)} \times Y^-$, and
$u|_{\Sigma_\alpha \setminus \coprod S^1} : \Sigma_\alpha \setminus \coprod S^1 \to \mathbb{R}_{i(\alpha)} \times Y$ is $J$-holomorphic
\item
If $i(\alpha) = 0$ then
$u(\Sigma_\alpha \setminus \coprod S^1) \subset X$, and
$u|_{\Sigma_\alpha \setminus \coprod S^1} : \Sigma_\alpha \setminus \coprod S^1 \to X$ is $J$-holomorphic
\item
If $i(\alpha) >0$ then
$u(\Sigma_\alpha \setminus \coprod S^1) \subset \mathbb{R}_{i(\alpha)} \times Y^+$, and
$u|_{\Sigma_\alpha \setminus \coprod S^1} : \Sigma_\alpha \setminus \coprod S^1 \to \mathbb{R}_{i(\alpha)} \times Y^+$ is $J$-holomorphic
\item
$E_\lambda(u) <\infty$ and $E_{\hat \omega}(u) <\infty$,
where these energies are defined by
\begin{align*}
E_\lambda(u) &= \max \biggl\{ \sup_{I \subset \mathbb{R}_{-k_-} \cup \dots \cup \mathbb{R}_{-1} \cup (-\infty, 0]} \frac{1}{|I|} \int_{u^{-1}(I \times Y^-)} u^\ast (d\sigma \wedge \lambda^-),\\
&\hphantom{= \max \biggl\{} \sup_{I \subset [0, \infty) \cup \mathbb{R}_1 \cup \dots \cup \mathbb{R}_{k_+}} \frac{1}{|I|} \int_{u^{-1}(I \times Y^+)} u^\ast (d\sigma \wedge \lambda^+) \biggr\}\\
E_{\hat \omega}(u) &= \int_{u^{-1}(X)} u^\ast \hat \omega
+ \int_{u^{-1}((\overline{\mathbb{R}}_{-k_-} \cup \dots \cup \overline{\mathbb{R}}_{-1}) \times Y^-)} u^\ast d\lambda^-\\
&\quad + \int_{u^{-1}((\overline{\mathbb{R}}_1 \cup \dots \cup \overline{\mathbb{R}}_{k_+}) \times Y^+)} u^\ast d\lambda^+.
\end{align*}
\item
$u$ is positively asymptotic to a periodic orbit $\gamma_{+\infty_i} = \pi_Y \circ u \circ \phi_{+\infty_i} \in P_{Y^+}$ at each $S^1_{+\infty_i}$,
and negatively asymptotic to a periodic orbit $\gamma_{-\infty_i} = \pi_Y \circ u \circ \phi_{-\infty_i} \in P_{Y^-}$ at each $S^1_{-\infty_i}$.
At every joint circle, $u$ is positively asymptotic to a periodic orbit
on the side of lower floor and negatively asymptotic to the same periodic orbit
on the side of higher floor.
\item
For each component $\hat\Sigma_\alpha$, if $u|_{\Sigma_\alpha}$ is a constant map,
then $2g_\alpha + m_\alpha \geq 3$.
\item
For each $i \neq 0$, $i$-th floor $u^{-1}(\overline{\mathbb{R}}_i \times Y^\pm) \subset
\Sigma$ contains nontrivial components.
\end{itemize}
\end{defi}
We say two holomorphic buildings $(\Sigma, z, u, \phi)$ and $(\Sigma', z', u', \phi')$ are
isomorphic if
there exist a biholomorphism $\varphi : \Sigma' \to \Sigma$
and a pair of $\mathbb{R}$-translations $\theta^- : \overline{\mathbb{R}}_{-k_-} \cup \dots \cup
\overline{\mathbb{R}}_{-1} \to \overline{\mathbb{R}}_{-k_-} \cup \dots \cup \overline{\mathbb{R}}_{-1}$ and
$\theta^+ : \overline{\mathbb{R}}_1 \cup \dots \cup \overline{\mathbb{R}}_{k_+}
\to \overline{\mathbb{R}}_1 \cup \dots \cup \overline{\mathbb{R}}_{k_+}$
such that
\begin{itemize}
\item
$\varphi(z'_i) = z_i$ for all $i$,
\item
$u' = (\theta \times 1) \circ u \circ \varphi$,
where
\begin{align*}
&\theta \times 1 : (\overline{\mathbb{R}}_{-k_-} \cup \dots \cup \overline{\mathbb{R}}_{-1})
\times Y^- \cup \overline{X} \cup (\overline{\mathbb{R}}_1 \cup \dots \cup \overline{\mathbb{R}}_{k_+})
\times Y^+ \\
&\hphantom{\theta \times 1 :}
\to
(\overline{\mathbb{R}}_{-k_-} \cup \dots \cup \overline{\mathbb{R}}_{-1}) \times Y^- \cup \overline{X}
\cup (\overline{\mathbb{R}}_1 \cup \dots \cup \overline{\mathbb{R}}_{k_+}) \times Y^+
\end{align*}
is defined by $(\theta \times 1)|_{\overline{X}} = \mathrm{id}_{\overline{X}}$ and
$(\theta \times 1)|_{\overline{\mathbb{R}_i} \times Y^\pm} = \theta^\pm \times 1$, and
\item
$\varphi \circ \phi'_{\pm\infty_i} = \phi_{\pm\infty_i}$ for all $\pm\infty_i$.
\end{itemize}
Note that the $0$-th floor of a holomorphic building may be empty.
We regard the empty curve, that is, the holomoprphic curve whose domain is
the empty set, as a disconnected holomorphic building for $X$, but we do not regard
it as a connected holomorphic building.
The genus of the empty curve is defined by $1$ ($= 1 - \frac{1}{2} \chi(\emptyset)$).
We denote the space of all holomorphic buildings for $X$ by
$\overline{\mathcal{M}}(X, \omega, J)$, and the space of connected ones
by $\overline{\mathcal{M}}^0(X, \omega, J)$.
The neighborhoods of each point
$p_0 = (\Sigma_0, z_0, u_0, \phi_0) \in \overline{\mathcal{M}}^0(X, \omega, J)$ is defined as follows.
As in the case of $\overline{\mathcal{M}}^0(Y, \lambda, J)$, first we add marked points $z_0^+$ to
$(\Sigma_0, z_0)$ to make $(\Sigma_0, z_0 \cup z_0^+)$ stable.
Let $(\widetilde{P} \to \widetilde{X}, Z \cup Z^+)$ be the local universal family of
$(\Sigma_0, z_0 \cup z_0^+)$.
For a pair $\theta = (\theta^-, \theta^+)$ of $\mathbb{R}$-gluings
$\theta^- : \overline{\mathbb{R}}_{-k_-} \sqcup \dots \sqcup \overline{\mathbb{R}}_0
\to \overline{\mathbb{R}}_{-l_-} \cup \dots \cup \overline{\mathbb{R}}_0$ and
$\theta^+ : \overline{\mathbb{R}}_0 \sqcup \dots \sqcup \overline{\mathbb{R}}_{k_+}
\to \overline{\mathbb{R}}_0 \cup \dots \cup \overline{\mathbb{R}}_{l_+}$,
we define a map
\begin{align*}
&(\theta \times 1) : (\overline{\mathbb{R}}_{-k_-} \sqcup \dots \sqcup \overline{\mathbb{R}}_{-1})
\times Y^- \sqcup \overline{X} \sqcup (\overline{\mathbb{R}}_1 \sqcup \dots \sqcup
\overline{\mathbb{R}}_{k_+}) \times Y^+ \\
&\hphantom{(\theta \times 1) :}
\to (\overline{\mathbb{R}}_{-l_-} \cup \dots \cup \overline{\mathbb{R}}_{-1}) \times Y^- \cup \overline{X}
\cup (\overline{\mathbb{R}}_1 \cup \dots \cup \overline{\mathbb{R}}_{l_+}) \times Y^+
\end{align*}
by
\begin{itemize}
\item
$(\theta \times 1)|_{\overline{X}} = \mathrm{id}$
\item
$(\theta \times 1)|_{\overline{\mathbb{R}} \times Y^\pm} = \theta^\pm \times 1$
if $\mu(i) \neq 0$.
(Recall $\mu$ is defined by $\theta^\pm(\overline{\mathbb{R}}_i) = \overline{\mathbb{R}}_{\mu(i)}$.)
\item
For each $i<0$ such that $\mu(i) = 0$,
$(\theta \times 1)(\sigma, y) = ( \min(\theta(\sigma), 0), y) \in (-\infty, 0] \times Y^- \subset X$
\item
For each $i>0$ such that $\mu(i) = 0$,
$(\theta \times 1)(\sigma, y) = ( \max(\theta(\sigma), 0), y) \in [0,\infty) \times Y^+ \subset X$
\end{itemize}
For a neighborhood $U \subset \widetilde X$ and $\epsilon>0$,
$\mathcal{W}_{p_0}(U, \epsilon) \subset \overline{\mathcal{M}}^0(X, \omega, J)$ is defined
as follows.
$(\Sigma, z, u) \in \overline{\mathcal{M}}^0(X, \omega, J)$ belongs to $\mathcal{W}_{p_0}(U, \epsilon)$ if
there exist a point $a\in U$, an isomorphism $(\Sigma, z) \cong (\widetilde{P}_a, Z(a))$
and a pair of $\mathbb{R}$-gluings $\theta = (\theta^-, \theta^+)$ such that
\[
\mathop{\mathrm{dist}}\nolimits_{L^\infty}(u, (\theta \times 1) \circ u_0 \circ \Psi|_{\widetilde{P}_a}) < \epsilon
\]
and
\[
\mathop{\mathrm{dist}}\nolimits_{L^\infty(S^1)}(\pi_{S^1}^{\breve N_{\pm\infty_i}} \circ \phi_{\pm\infty_i},
\phi_{0, \pm\infty_i}) < \epsilon.
\]
We define a neighborhood of $p_0$ as a subset of $\overline{\mathcal{M}}^0(X, \omega, J)$
which contains some $\mathcal{W}_{p_0}(U, \epsilon)$.
This defines the topology of $\overline{\mathcal{M}}^0(X, \omega, J)$ similarly
to the case of $\overline{\mathcal{M}}^0(Y, \lambda, J)$.
Define a closed two form $\widetilde{\omega}$
on $X = (-\infty,0] \times Y^- \cup Z \cup [0,\infty) \times Y^+$ by
$\widetilde{\omega}|_Z = \omega$,
$\widetilde{\omega}|_{(-\infty, 0] \times Y^-} = d(\varphi \lambda^-)$ and
$\widetilde{\omega}|_{[0, \infty) \times Y^+} = d(\varphi \lambda^+)$,
where $\varphi :\mathbb{R} \to \mathbb{R}_{\geq 0}$ is a smooth function with compact support
such that $\varphi(0) = 1$.
Then $\overline{\mathcal{M}}^0(X, \omega, J)$ is decomposed as
$\overline{\mathcal{M}}^0(X, \omega, J) = \coprod \overline{\mathcal{M}}^{0, e}_{g, \mu}(L^-, L^+)$,
where $\overline{\mathcal{M}}^{0, e}_{g, \mu}(L^-, L^+)$ consists of holomorphic buildings
$(\Sigma, z, u, \phi)$ with genera $g$ and $\mu$ marked points such that
$\sum L_{\gamma_{-\infty_i}} = L^-$, $\sum L_{\gamma_{+\infty_i}} = L^+$
and $\int_{u^{-1}(X)} u^\ast \widetilde{\omega} = e$.
(This is independent of the choice of the function $\varphi$.)
Note that $(\Sigma, z, u, \phi) \in \overline{\mathcal{M}}^{0, e}_{g, \mu}(L^-, L^+)$ satisfies
\begin{align}
E_\lambda (u) &\leq \max(e + L^+, L^+)
\label{E lambda estimate}\\
E_{\hat \omega} (u) &= e + (L^+ - L^-)
\label{E hat omega estimate}
\end{align}
(\ref{E lambda estimate}) is because
\begin{itemize}
\item
for any interval $I \subset \mathbb{R}_i$ ($i < 0$) or $I \subset (-\infty, 0]$,
\begin{align*}
\frac{1}{|I|}\int_{u^{-1}(I \times Y^-)} u^\ast (d\sigma \wedge \lambda^-)
&= \int u^\ast (d\varphi^- \wedge \lambda^-)\\
&= \int_{u^{-1}(\hat Y^-)} u^\ast d(\varphi^- \lambda^-)
- \int_{u^{-1}(\hat Y^-)} u^\ast(\varphi^- d\lambda^-)\\
&\leq \int_{u^{-1}(\hat Y^-)} u^\ast d(\varphi^- \lambda^-)\\
&= \int_{u^{-1}((-\infty, 0] \times Y^-)} u^\ast d(\varphi \lambda^-)\\
&\leq \int_{u^{-1}((-\infty, 0] \times Y^-)} u^\ast d(\varphi \lambda^-)
+ \int_{u^{-1}(Z)} u^\ast \omega\\
& \quad + \int_{u^{-1}(\hat Y^+)} u^\ast d\lambda^+\\
&= \int_{u^{-1}(X)} u^\ast \widetilde{\omega}
+ \int_{u^{-1}(\hat Y^+)} u^\ast d((1-\varphi)\lambda^+)\\
&= e + L^+,
\end{align*}
where $\hat Y^- = (\overline{\mathbb{R}}_{-k_-} \cup \dots \cup \overline{\mathbb{R}}_{-1}
\cup (-\infty, 0]) \times Y^-$,
$\hat Y^+ = ([0, \infty) \cup \overline{\mathbb{R}}_1 \cup \dots \cup \overline{\mathbb{R}}_{k_+})
\times Y^+$,
and $\varphi^-$ is defined by
\[
\varphi^-(\sigma, y) = \int_{-\infty}^\sigma \frac{1}{|I|} 1_I(\sigma') d\sigma'
\quad \text{on } \mathbb{R}_i \times Y^-
\]
$\varphi^- \equiv 0$ on $\mathbb{R}_j \times Y^-$ ($j < i$),
and $\varphi^- \equiv 1$ on $\mathbb{R}_j \times Y^-$ ($j < i$), $X$, and $\mathbb{R}_j \times Y^+$,
\item
for any interval $I \subset \mathbb{R}_i$ ($i > 0$) or $I \subset [0, \infty)$,
\begin{align*}
\frac{1}{|I|}\int_{u^{-1}(I \times Y^+)} u^\ast (d\sigma \wedge \lambda^+)
&= \int u^\ast (d\varphi^+ \wedge \lambda^+)\\
&= \int_{u^{-1}(\hat Y^+)} u^\ast d(\varphi^+ \lambda^+)
- \int_{u^{-1}(\hat Y^+)} \varphi^+ d\lambda^+\\
&\leq L^+,
\end{align*}
where $\varphi^+$ is defined by
\[
\varphi^+(\sigma) = \int_{-\infty}^\sigma \frac{1}{|I|} 1_{I}(\sigma') d\sigma'
\quad \text{on } \mathbb{R}_i,
\]
$\varphi^+ \equiv 0$ on $\mathbb{R}_j \times Y^-$, $X$, and $\mathbb{R}_j \times Y^+$ for ($j < i$),
and $\varphi^+ \equiv 1$ on $\mathbb{R}_j \times Y^+$ ($j > i$).
\end{itemize}
Proof of (\ref{E hat omega estimate}) is straightforward.
As in the case of $\hat Y$, we have a nice way to add marked points
to the domain curves.
\begin{lem}
Let $\epsilon > 0$ and $\delta_0 > 0$ be arbitrary small constants,
and let $(\Sigma, z, u, \phi) \in \overline{\mathcal{M}}^{0, e}_{g, \mu}(L^-, L^+)$ be an arbitrary
holomorphic building.
Then there exist closed subsets
$I^- \subset \mathbb{R}_{-k_-} \cup \dots \cup \mathbb{R}_{-1} \cup (-\infty, 0]$
and $I^+ \subset [0, \infty) \cup \mathbb{R}_1 \cup \dots \cup \mathbb{R}_{k_+}$,
and additional marked points $z^+$ of $\Sigma$ which satisfy the following conditions:
\begin{itemize}
\item
Both of $I^\pm$ are finite unions of intervals in the form $[l, l+1] \subset \mathbb{R}_i$
{\rm(}$l \in \mathbb{Z}${\rm)}.
\item
The lengths of $I^\pm$ and the number of additional marked points are bounded by
some constant determined by $g$, $\mu$, $L^-$, $L^+$, $e$, $\epsilon$ and $\delta_0$.
\item
$(\Sigma, z \cup z^+)$ is stable.
\item
There exists a constant $A_1 > 0$ depending only on $g$, $\mu$, $L^-$, $L^+$, $e$,
$\epsilon$ and $\delta_0$ such that
if $[-A_1, T + A_1] \times S^1 \subset \Sigma$ does not contain any marked points
$z \cup z^+$, then one of the following two holds true.
\begin{enumerate}[label=\normalfont(\arabic*)]
\item
$u([0, T] \times S^1) \subset I^- \times Y^- \cup Z \cup I^+ \times Y^+$ and
$\mathrm{diam} \, u([0, T] \times S^1) \leq 20 \epsilon$.
\item
$u ([0, T] \times S^1)$ is contained in $J^- \times Y^-$ or $J^+ \times Y^+$,
where $J^-$ is the $\frac{1}{3}$-neighborhood of
the complement of $I^- \subset \mathbb{R}_{-k_-} \cup \dots \cup \mathbb{R}_{-1} \cup (-\infty, 0]$,
and $J^+$ is the $\frac{1}{3}$-neighborhood of
the complement of $I^+ \subset [0, \infty) \cup \mathbb{R}_1 \cup \dots \cup \mathbb{R}_{k_+}$.
Moreover, in this case,
there exist $L \in \mathbb{R}$ and $(b, \gamma) \in \mathbb{R} \times P_L^\pm$ such that
\[
\mathop{\mathrm{dist}}\nolimits (u(s,t), (Ls + b, \gamma(t))) \leq \epsilon (e^{-\kappa s} + e^{-\kappa (T - s)})
\]
on $[0, T] \times S^1$.
\end{enumerate}
In particular,
for any disc $D \subset \Sigma$ such that
$D \setminus 0$ does not contain any marked points,
$\mathrm{diam} \, u(\{ z\in D; |z| \leq e^{-2\pi A_1}\}) \leq 20 \epsilon$.
\item
$u(z \cup z^+)$ is contained in the $\frac{1}{3}$-neighborhood of
$I^- \times Y^- \cup Z \cup I^+ \times Y^+$.
\item
Each connected component of $u^{-1}(I^- \times Y^- \cup Z \cup I^+ \times Y^+)$ either
contains at least one point of $z \cup z^+$
or is contained in the inverse image of the $\frac{1}{3}$-neighborhood of
the complement of $I^- \times Y^- \cup Z \cup I^+ \times Y^+$ by $u$.
\item
For the $\frac{1}{3}$-neighborhood $\widetilde{J}$ of each connected component of
the complement of $I^- \times Y^- \cup Z \cup I^+ \times Y^+$,
$E_{\hat \omega}(u|_{u^{-1}(\widetilde{J})}) \leq \delta_0$.
\end{itemize}
\end{lem}
\begin{proof}
First we show the number of irreducible components of $\check \Sigma$ is bounded by
some constant depending only on $g$, $\mu$, $L^-$, $L^+$ and $e$.
Define an energy $E'(u)$ by
\begin{align*}
E'(u) &= \int_{u^{-1}(Z)} u^\ast \omega
+ \int_{u^{-1}([-1, 0] \times Y^-)} u^\ast (d\sigma \wedge \lambda^- + d\lambda^-)\\
&\quad + \int_{u^{-1}([0, 1] \times Y^+)} u^\ast (d\sigma \wedge \lambda^+
+ d\lambda^+).
\end{align*}
This is bounded by
$E'(u) \leq E_{\hat \omega}(u) + 2 E_\lambda (u)$.
If $u$ is non-constant on $\Sigma_\alpha$ and $u(\Sigma_\alpha)$ intersects with $Z$,
then Lemma \ref{monotonicity lemma} implies
$E'(u|_{\Sigma_\alpha})$ is larger than some positive constant independent of $u$.
This implies the number of the irreducible components of $\Sigma$ on which $u$ is not
a constant map and whose image by $u$ intersects with $Z$ is bounded.
Hence as in the case of $\hat Y$,
the number of the nontrivial components $\Sigma_\alpha$ such that
$2g_\alpha + m_\alpha <3$ is bounded.
Therefore the height of $(\Sigma, z, u)$ is bounded,
which implies the number of trivial cylinders is also bounded.
Therefore the number of marked points we need to add to $(\Sigma, z)$ in order to
make $(\Sigma, z \cup z^+)$ stable is bounded. Assuming $(\Sigma, z \cup z^+)$ is
stable, we further add marked points $z^{++}$ as follows.
Let $\delta > 0$, $\kappa > 0$, $A > 0$ and $L_0$ be the constant of
Corollary \ref{third annulus for X} for $C_0 = e + 2L^+$ and the given $\epsilon > 0$.
We may assume $\delta \leq \delta_0$.
Let $I^- \subset \mathbb{R}_{-k_-} \cup \dots \cup \mathbb{R}_{-1} \cup (-\infty, 0]$
and $I^+ \subset [0, \infty) \cup \mathbb{R}_1 \cup \dots \cup \mathbb{R}_{k_+}$ be finite unions of
intervals $[l, l+1] \subset \mathbb{R}_i$ such that
\begin{itemize}
\item
$E_{\hat\omega} (u|_{u^{-1}(\widetilde{J})}) \leq \delta$
for the $\frac{1}{3}$-neighborhood $\widetilde{J}$ of each connected component of
the complement of $I^- \times Y^- \cup Z \cup I^+ \times Y^+$, and
\item
$u(z \cup z^+) \subset I^- \times Y^- \cup Z \cup I^+ \times Y^+$.
\end{itemize}
We may assume the lengths of $I^\pm$ are bounded by some constants
depending only on
$E_{\hat \omega}(u)$, $\delta$ and the number of marked points.
Let $\bigcup_\alpha B_\alpha^1 \supset I^- \times Y^- \cup Z \cup I^+ \times Y^+$
be a finite covering by open balls
with radius $\epsilon$.
We may assume the number of open balls is bounded by some constant depending
on the length of $I$ and $\epsilon$.
For each $B_\alpha^1$, let $B_\alpha^2$ be the concentric ball with radius $2\epsilon$.
Using this finite covering, we add marked points $z^{++}$ as in the proof of
Lemma \ref{nice marked points}.
Then by the similar argument, we can easily see that $I^\pm$ and
the additional marked points $z^+ \cup z^{++}$ satisfy the required conditions.
\end{proof}
Using the above lemma, we can prove the following propositions
similarly to the case of $\overline{\mathcal{M}}^0(Y, \lambda, J)$.
\begin{prop}
$\overline{\mathcal{M}}^0(X, \omega, J)$ is second countable.
\end{prop}
\begin{prop}
For any $g$, $\mu$, $e_0$, $L^-$ and $L^+$,
$\bigcup_{e \leq e_0} \overline{\mathcal{M}}^{0, e}_{g, \mu}(L^-, L^+)$ is compact.
\end{prop}
\begin{prop}
$\overline{\mathcal{M}}^0(X, \omega, J)$ is Hausdorff.
\end{prop}
Similarly to the case of symplectization $\hat Y$, we use the quotient space
\[
\widehat{\mathcal{M}}^0(X, \omega, J) = \overline{\mathcal{M}}^0(X, \omega, J) / \sim
\]
obtained by ignoring the coordinates of limit circles and the order of marked points
and limit circles.
We also define the quotient space $\widehat{\mathcal{M}}(X, \omega, J)
= \overline{\mathcal{M}}(X, \omega, J) / \sim$ similarly.
$\overline{\mathcal{M}}(X, \omega, J) / \sim$ and $\widehat{\mathcal{M}}(X, \omega, J)$
are also second countable and Hausdorff.
The compactness is stated as follows.
\begin{prop}
For any $g_0 \in \mathbb{Z}$, $\mu_0 \geq 0$, $L^1_0 \geq 0$ and $L^2_0 \geq 0$,
\[
\bigcup_{\substack{-\infty < g \leq g_0 \\ \mu \leq \mu_0 \\ e + L^+ \leq L^1_0 \\
L^+ \leq L^2_0}} \overline{\mathcal{M}}^e_{g, \mu} (L^-, L^+)
\]
is compact.
\end{prop}
The proof of the above proposition is almost same with that of Proposition
\ref{compactness for disconnected}.
In this case, in order to prove the boundedness of the number of the connected
components, we use the fact that the energy of a non-constant closed
$J$-holomorphic curve in $X$ is bounded below by some positive constant.
\section{Construction of pre-Kuranishi structure}\label{construction of Kuranishi}
In this section, we explain the way to construct
a pre-Kuranishi structure of $\widehat{\mathcal{M}} = \widehat{\mathcal{M}}(Y, \lambda, J)$.
This is the basis of the construction of pre-Kuranishi structures of other various spaces
such as fiber products in Section \ref{fiber prod}.
First we explain the Banach spaces we use.
Let $\Sigma_0$ be the domain curve of a holomorphic building $(\Sigma_0, z, u_0) \in
\widehat{\mathcal{M}}$, and let $\{\mu\}$ and $\{\pm\infty_i\}$ be the indices of
its joint circles and limit circles respectively.
Define positive constants $\delta_{0, \mu}$ and $\delta_{0, \pm\infty_i}$ by the
minimal nonzero absolute value of eigenvalues of $A_{\gamma_\mu}$ and
$A_{\gamma_{\pm\infty_i}}$ respectively,
where $\gamma_\mu$ and $\gamma_{\pm\infty_i}$ are the periodic orbits on
the corresponding imaginary circles of $\Sigma$.
(See Definition \ref{def of Bott-Morse} for the definition of the operator $A_\gamma$
for each periodic orbit $\gamma$.)
For a sequence of positive constants
$\delta = ((\delta_\mu)_{\mu}, (\delta_{\pm\infty_i})_{\pm\infty_i})$
such that $\delta_\mu < \delta_{0, \mu}$ and $\delta_{\pm\infty_i}
< \delta_{0, \pm\infty_i}$,
we use the Banach spaces $L_\delta^p(\Sigma)$ and $W_\delta^{1, p}(\Sigma)$
defined as follows.
Fix some coordinate $([0, \infty] \cup [-\infty, 0]) \times S^1$ of a neighborhood
$N_\mu$ of each joint circle $S_\mu^1$ of $\Sigma$,
and some coordinate $[0, \infty] \times S^1$ or $[-\infty, 0] \times S^1$ of
a neighborhood $N_{\pm\infty_i}$ of each limit circle $S^1_{\pm\infty_i}$,
and fix a volume form of $\Sigma$ such that its restriction to these neighborhoods
coincide with the usual Lebesgue measure $ds \wedge dt$.
(On a neighborhood $D\cup D$ of each nodal point,
we use usual volume form of $D$.)
Then $L_\delta^p$-norm of $\xi$ is defined by
\begin{align*}
||\xi||^p_{L_\delta^p} = &\int_{\Sigma \setminus (\coprod N_\mu \sqcup \coprod
N_{\pm\infty_i})} |\xi|^p \mathrm{vol}
+ \sum_\mu \int_{N_\mu} |e^{\delta_\mu |s|}\xi(s,t)|^p ds\wedge dt\\
&+ \sum_{\pm\infty_i}
\int_{\substack{[0, \infty] \times S^1\\ \text{or} \\ [-\infty, 0] \times S^1}}
|e^{\delta_{\pm\infty_i} |s|}\xi(s,t)|^p ds\wedge dt
\end{align*}
The Sobolev space $W^{1, p}_\delta(\Sigma)$ is the space of continuous functions
(or continuous sections) $\xi$ on $\Sigma$ whose $W^{1, p}_\delta$-norms
\begin{align*}
||\xi||^p_{W_\delta^{1, p}} = &\int_{\Sigma \setminus (\coprod N_\mu \sqcup \coprod
N_{\pm\infty_i})} (|\xi|^p + |\nabla \xi|^p) \mathrm{vol}\\
&+ \sum_\mu \int_{N_\mu} (|e^{\delta_\mu |s|}\xi|^p + |e^{\delta_\mu |s|} \partial_s \xi|^p
+ |e^{\delta_\mu |s|} \partial_t \xi|^p) ds\wedge dt\\
&+ \sum_{\pm\infty_i}
\int_{\substack{[0, \infty] \times S^1\\ \text{or} \\ [-\infty, 0] \times S^1}}
(|e^{\delta_{\pm\infty_i} |s|}\xi|^p + |e^{\delta_{\pm\infty_i} |s|} \partial_s \xi|^p
+ |e^{\delta_{\pm\infty_i} |s|} \partial_t \xi|^p) ds\wedge dt
\end{align*}
are finite.
For each holomorphic building $(\Sigma, z, u_0)$,
$\widetilde{W}^{1, p}_\delta(\Sigma, u_0^\ast T \hat Y)$
is the space of continuous sections $\xi$ of $u_0^\ast T \hat Y = \mathbb{R} \oplus
(\pi_Y \circ u_0)^\ast TY$ such that
\[
\xi = \xi_0 + \sum_\mu \beta_\mu v_\mu + \sum_{\pm\infty_i} \beta_{\pm\infty_i}
v_{\pm\infty_i}
\]
for some $\xi_0 \in W^{1, p}_\delta(\Sigma, u_0^\ast T \hat Y)$,
$v_\mu \in \mathop{\mathrm{Ker}}\nolimits A_{\gamma_\mu}$ and $v_{\pm\infty_i} \in
\mathop{\mathrm{Ker}}\nolimits A_{\gamma_{\pm\infty_i}}$, where $\beta_\mu$ is a smooth function which is $1$
on a neighborhood of $\mu$-th joint circle and whose support is contained in
its slightly larger neighborhood for each $\mu$, and $\beta_{\pm\infty_i}$ is a smooth
function which is $1$ on a neighborhood of the limit circle $S^1_{\pm\infty_i}$ and whose
support is contained in its slightly large neighborhood for each $\pm\infty_i$.
In the above equation, we regard $v_\mu$ as a section defined on
$([0, \infty] \cup [-\infty, 0] ) \times S^1$ by $v_\mu(s, t) = v_\mu(t)$, where
we fix a trivialization of $u_0^\ast T \hat Y$
on $([0, \infty] \cup [-\infty, 0]) \times S^1$.
The meaning of $v_{\pm\infty_i}$ is similar.
The definition of $\widetilde{W}^{1, p}_\delta(\Sigma, u_0^\ast T \hat Y)$ does not
depend on the choice of $\beta_\mu$ and $\beta_{\pm\infty_i}$.
As a Banach space, we regard $\widetilde{W}^{1, p}_\delta(\Sigma, u_0^\ast T \hat Y)$
as a direct sum of $W^{1, p}_\delta(\Sigma, u_0^\ast T \hat Y)$, $\mathop{\mathrm{Ker}}\nolimits A_{\gamma_\mu}$
and $\mathop{\mathrm{Ker}}\nolimits A_{\gamma_{\pm\infty_i}}$.
For a family of deformations of $\Sigma$, we need to use an appropriate family of
norms to obtain a uniform estimate.
This family of norms are used only for the construction of a Kuranishi neighborhood of
a holomorphic building $(\Sigma, z, u)$ and we do not need to assume that the
norm of a curve $\Sigma'$ as a deformation of $\Sigma$ coincides with that
used for the construction of a Kuranishi neighborhood of a holomorphic building
whose domain curve is $\Sigma'$.
Let $\Sigma'$ be a curve obtained from $\Sigma$ by replacing the neighborhood
$([0, \infty] \cup [-\infty, 0]) \times S^1$ of $S^1_\mu$ by $([0, \rho_\mu] \cup
[-\rho_\mu, 0]) \times S^1$ and the neighborhood $D \cup D$ of each nodal point
$q_\nu$ by $\{(x, y) \in D \times D; xy = \zeta_\nu\}$ for some $(\rho_\mu, \zeta_\nu)$.
Then the $L^p_\delta$-norm of $L^p_\delta(\Sigma')$ is defined by
\begin{align*}
||\xi||^p_{L_\delta^p} = &\int_{\Sigma \setminus (\coprod N_\mu \sqcup \coprod
N_{\pm\infty_i})} |\xi|^p \mathrm{vol}
+ \sum_\mu \int_{([0, \rho_\mu] \cup [-\rho_\mu, 0]) \times S^1}
|e^{\delta_\mu |s|}\xi(s,t)|^p ds\wedge dt\\
&+ \sum_{\pm\infty_i}
\int_{\substack{[0, \infty] \times S^1\\ \text{or} \\ [-\infty, 0] \times S^1}}
|e^{\delta_{\pm\infty_i} |s|}\xi(s,t)|^p ds\wedge dt,
\end{align*}
where the volume form on $\{(x, y) \in D \times D; xy = \zeta_\nu\}$ is defined by
$\frac{\sqrt{-1}}{2}dx \wedge d\bar x$ on $\{|x| \geq |y|\}$ and
$\frac{\sqrt{-1}}{2}dy \wedge d\bar y$ on $\{|y| \geq |x|\}$.
The norm of $W^{1, p}_\delta(\Sigma')$ is defined similarly.
The norm of $\widetilde{W}^{1, p}_\delta(\Sigma', u_0^\ast T \hat Y)$ is defined by
\begin{align*}
||\xi||_{\widetilde{W}^{1, p}_\delta}
= \inf \{& ||\xi_0||_{W^{1, p}_\delta(\Sigma')} + \sum_\mu ||v_\mu||_{\mathop{\mathrm{Ker}}\nolimits A_{\gamma_\mu}}
+ \sum_{\pm\infty_i} ||v_{\pm\infty_i}||_{\mathop{\mathrm{Ker}}\nolimits A_{\gamma\infty_i}};\\
&\xi = \xi_0 + \sum_\mu \beta_\mu v_\mu
+ \sum_{\pm\infty_i} \beta_{\pm\infty_i} v_{\pm\infty_i},\\
&\xi_0 \in W^{1, p}_\delta(\Sigma', u_0^\ast T \hat Y), v_\mu \in \mathop{\mathrm{Ker}}\nolimits A_{\gamma_\mu},
v_{\pm\infty_i} \in \mathop{\mathrm{Ker}}\nolimits A_{\gamma_{\pm\infty_i}}\}.
\end{align*}
In Section \ref{construction of nbds}, we explain the construction of
a Kuranishi neighborhood of a point in $\widehat{\mathcal{M}}(Y, \lambda, J)$, assuming
sufficient data including an additional vector space are given.
To construct a Kuranishi neighborhood by inverse function theorem of Banach spaces,
we need to prove the linearized gluing lemma, which is proved in Section
\ref{linearized gluing}.
In Section \ref{smoothness}, we prove the smoothness of Kuranishi neighborhood, and
in Section \ref{embed}, we consider the embedding of Kuranishi neighborhoods and
prove its smoothness.
In Section \ref{Kuranishi of disconnected buildings}, we consider the relation of
the Kuranishi neighborhoods of $\widehat{\mathcal{M}}^0$ and $\widehat{\mathcal{M}}$.
Finally in Section \ref{global construction},
we construct a global Kuranishi structure of $\widehat{\mathcal{M}}$.
\subsection{Construction of Kuranishi neighborhoods}\label{construction of nbds}
First we explain a way to construct a Kuranishi neighborhood of a point
$p_ 0= (\Sigma_0, z, u_0) \in \widehat{\mathcal{M}}(Y, \lambda, J)$.
The construction is based on the implicit function theorem (or inverse function theorem)
for Banach spaces of functions (or sections of some vector bundles) on deformed curves
of $\Sigma_0$.
Since the Banach space changes if the domain curve changes, we need to apply
the implicit function theorem for each deformed curve.
Using appropriate norms for these Banach spaces, we can apply the implicit function
theorem for them uniformly, and get a Kuranishi neighborhood as an (at least) continuous
fibration over the parameter space of the deformation of the domain curve.
We prove in Section \ref{smoothness} that this fibration is actually smooth in some sense,
and in Section \ref{embed}, we prove the smoothness of the embedding between
two Kuranishi neighborhoods.
We fix an order $z = (z_i)$ of the marked points.
As we have explained, to define a Kuranishi neighborhood,
we need an additional vector space which makes the Fredholm map transverse to zero.
Such an additional vector space is given as the following data
$(p_0^+, S, E^0, \lambda)$:
(These are given in Section \ref{global construction}.)
\begin{figure}
\centering
\includegraphics[width= 350pt]{Fig_Sigma_z_z+.png}
\caption{$(\Sigma_0, z \cup z^+)$}\label{(Sigma,z,z+)}
\includegraphics[width= 350pt]{Fig_checkSigma_z_z+_pminfty.png}
\caption{$(\check\Sigma_0, z \cup z^+ \cup (\pm\infty_i))$}
\label{(checkSigma,z,z+,pminfty)}
\includegraphics[width= 350pt]{Fig_hatSigma_z_z+_pminfty.png}
\caption{$(\hat \Sigma_0, z \cup z^+ \cup (\pm\infty_i))$}\label{(hatSigma,z,z+,pminfty)}
\end{figure}
\begin{itemize}
\item
$p_0^+ = (\Sigma_0, z \cup z^+, u_0)$ is a curve obtained by adding marked points
on the nontrivial components of $\Sigma_0$.
We assume all unstable components of $(\Sigma_0, z \cup z^+)$ are trivial cylinders
of $p_0$.
We assume that $G_0 = \mathop{\mathrm{Aut}}\nolimits(\Sigma_0, z, u_0)
:= \{g \in \mathop{\mathrm{Aut}}\nolimits (\Sigma_0); g (\{z_i\}) = \{z_i\}, u_0 \circ g = u_0\}$ preserves $z^+$
as a set, that is, $G_0$ acts on $z^+ = (z_i^+)$ as a symmetric group.
\item
$S \subset Y$ is a finite union of codimension-two submanifolds such that
$\pi_Y \circ u_0$ intersects with $S$ at $z^+$ transversely.
(We do not assume the transversality of the other intersections of $\pi_Y \circ u_0$
with $S$.)
This will be used to kill the excessive dimension of deformation
due to the additional marked points $z^+$.
\item
Let $(\hat \Sigma_0, z \cup z^+ \cup (\pm\infty_i))$ be the stabilization of
$(\check \Sigma_0, z \cup z^+ \cup (\pm\infty_i))$, and
let $(\hat P \to \hat X, Z \cup Z^+ \cup Z_{\pm\infty_i})$ be its local universal family.
Since $G_0$ acts on $\hat \Sigma_0$ preserving $z$, $z^+$ and $\{\pm\infty_i\}$ as
sets, it also acts on $\hat P$ by the universal property of $\hat P$.
Then $E^0$ is a finite dimensional $G_0$-vector space
and $\lambda : E^0 \to C^\infty(\hat P \times Y, {\textstyle\bigwedge}^{0, 1} V^\ast \hat P
\otimes (\mathbb{R} \partial_\sigma \oplus TY))$ is a $G_0$-equivariant linear map
which satisfies the following conditions:
($V^\ast \hat P$ is the dual of the vertical tangent space $V \hat P \subset T \hat P$
of $\hat P$.)
\begin{itemize}
\item
For each $h \in E^0$, the projection of the support of $\lambda(h)$ to $\hat P$
does not intersect with the nodal points of $\hat P$ or $Z_{\pm\infty_i}$.
\item
Let $E^0 \to C^\infty(\Sigma_0 \times Y, {\textstyle\bigwedge}^{0, 1} T^\ast\Sigma_0 \otimes
(\mathbb{R} \partial_\sigma \oplus TY))$ be the pullback of $\lambda$ by the composition of the
blowing down $\Sigma_0 \to \check \Sigma_0$ and the forgetful map
$(\check \Sigma_0, z \cup z^+) \stackrel{\cong}{\to} (\hat P_0, Z(0) \cup Z^+(0))$.
This pull back is also denoted by $\lambda$.
Then we assume that for a sufficiently small $\delta > 0$, the linear map
\begin{align*}
&D_{p_0}^+ : \widetilde{W}_\delta^{1, p}(\Sigma_0, u_0^\ast T \hat Y) \oplus E^0\\
&\to
L_\delta^p(\Sigma_0, {\textstyle\bigwedge}^{0, 1}T^\ast \Sigma_0 \otimes u_0^\ast T \hat Y)
\oplus
\bigoplus_{\text{limit circles}} \mathop{\mathrm{Ker}}\nolimits A_{\gamma_{\pm\infty_i}} / (\mathbb{R} \partial_\sigma
\oplus \mathbb{R} R_\lambda)\\
&\quad \oplus \bigoplus_{z_i} T_{\pi_Y \circ u_0(z_i)} Y
\end{align*}
{\abovedisplayskip=-5pt
\begin{align*}
&(\xi, h) \mapsto (D_{p_0} \xi(z) + \lambda(h)(z, \pi_Y \circ u_0(z)), \\
&\hphantom{(\xi, h) \mapsto (}
\sum_j \langle\xi|_{S^1_{\pm\infty_i}}, \eta_j^{\pm\infty_i}\rangle
\eta_j^{\pm\infty_i}, \pi_Y \circ \xi (z_i))
\end{align*}}
is surjective,
where $D_{p_0}$ is the linearization of the equation of $J$-holomorphic maps,
that is,
\[
D_{p_0} \xi = \nabla \xi + J(u_0) \nabla \xi j + \nabla_\xi J(u_0) du_0 j,
\]
and $\{\eta_j^{\pm\infty_i}\}_j$ is an orthonormal basis of
the orthogonal complement of $\mathbb{R} \partial_\sigma \oplus \mathbb{R} R_\lambda$ in
$\mathop{\mathrm{Ker}}\nolimits A_{\gamma_{\pm\infty_i}}$ for each $\pm\infty_i$.
\end{itemize}
\end{itemize}
The above data are given in the form which respects the $\mathbb{R}$-translation invariance.
However, to describe the Kuranishi neighborhood, we further need to fix the following
temporally data $(z^{++}, S', \hat R_i)$ which break the $\mathbb{R}$-translation invariance:
(The Kuranishi neighborhood constructed finally does not depend on these temporally
data. See Section \ref{embed}.)
\begin{itemize}
\item
$z^{++} = (z^{++}_i) \subset \Sigma$ are additional marked points which make
$(\Sigma_0, z \cup z^+ \cup z^{++})$ stable.
We assume $G_0$-action preserves $z^{++}$ as a set.
\item
$S' \subset (\mathbb{R}_1 \cup \dots \cup \mathbb{R}_k) \times Y$ is a codimension-two submanifold
such that $u_0$ intersects with $S'$ at $z^{++}$ transversely.
\item
For each $1 \leq i \leq k$,
let $\hat R_i = (\hat R_{i, l})_{1 \leq l \leq m_i}$ be a family of holomorphic sections
$\hat R_{i, l} : \hat X \to \hat P$ such that
$\sigma_i \circ u_0(\hat R_{i, l}(0)) = 0$, where $\sigma_i$ is the coordinate of $\mathbb{R}_i$,
and $\hat R_i$ is $G_0$-invariant as a family.
($G_0$ may permute $\{ \hat R_{i, l} \}_l$.)
We assume $\hat R_{i, l}$ do not intersect with nodal points or $Z_{\pm\infty_i}$.
Let $(\widetilde{P} \to \widetilde{X}, Z \cup Z^+ \cup Z^{++})$ be the local universal
family of $(\Sigma_0, z \cup z^+ \cup z^{++})$.
Then each $\hat R_{i, l}$ induces a section
$\widetilde{R}_{i, l} : \widetilde{X} \to \widetilde{P}$
which makes following diagram commutative.
\[
\begin{tikzcd}
\widetilde{P} \ar{r}{\mathop{\mathfrak{forget}}\nolimits}& \hat P\\
\widetilde{X} \ar{u}{\widetilde{R}_{i, l}} \ar{r}{\mathop{\mathfrak{forget}}\nolimits}& \hat X \ar{u}{\hat R_{i, l}}
\end{tikzcd}
\]
These families $\widetilde{R}_{i, l}$ are used to kill the $\mathbb{R}$-translations
by imposing the conditions $\sum_l \sigma_i \circ u(\widetilde{R}_{i, l}) = 0$.
The fact that $\widetilde{R}_{i, l}$ are induced by sections $\hat R_{i, l}$ of
$\hat P \to \hat X$ is important to define smooth embeddings in Section \ref{embed}.
\end{itemize}
The pullback $E^0 \to C^\infty(\widetilde{P} \times Y, {\textstyle\bigwedge}^{0, 1}V^\ast
\widetilde{P} \otimes (\mathbb{R} \partial_\sigma \oplus TY))$
of $\lambda$ by $\widetilde{P} \to \hat P$ is also denoted by $\lambda$.
Using the above data, we construct a Kuranishi neighborhood of $p_0$.
First we explain a convenient way to express curves close to $p_0$.
We separate the domain $\Sigma_0$ into several parts, namely,
neighborhoods of nodal points, neighborhoods of joint circles,
neighborhood of limit circles and the rest.
The local universal family $(\widetilde{P} \to \widetilde{X}, Z \cup Z^+ \cup Z^{++})$
can be described as follows.
Let $N_0 \subset \Sigma_0$ be a neighborhood of nodal points and imaginary circles
such that
\begin{itemize}
\item
$N_0 \cong \coprod_{l_0} (D \cup D) \sqcup \coprod_{l_1} (D \widetilde{\cup} D) \sqcup
\coprod_{l_2} \widetilde{D}$
\item
all marked points and $\widetilde{R}_i(0)$ are contained in $\Sigma_0 \setminus N_0$
\item
the support of $\lambda(h)$ is also contained in $\Sigma_0 \setminus N_0$
for each $h \in E^0$.
\end{itemize}
Let $\mathcal{J}_0$ be a finite dimensional complex manifold which consists of
holomorphic structures of $\check \Sigma_0$ near the original one $j_0$ such that
the restriction of any $j \in \mathcal{J}_0$ to $N_0$ coincides with $j_0$.
If we choose an appropriate $\mathcal{J}_0$, then we may regard $\widetilde{X}$ as
a neighborhood $\widetilde{X} \subset \mathcal{J}_0 \times D^{l_0} \times
\widetilde{D}^{l_1}$ of $(j_0, 0, (0, 0))$,
where $(\zeta_\nu)_{\nu =1}^{l_0} \in D^{l_0}$ are the parameters of deformation of
the neighborhood of nodal points, and
$(\zeta_\mu = \rho_\mu^{2\pi}e^{2\pi \sqrt{-1}\varphi_\mu},
e^{2\pi \sqrt{-1}\varphi_\mu})_{\mu = 1}^{l_1} \in \widetilde{D}^{l_1}$
are the parameters of deformation of the neighborhood of joint circles.
We sometimes denote the parameters $(\zeta_\mu, e^{2\pi \sqrt{-1} \varphi_\mu}) \in
\widetilde{D}$ by $(\rho_\mu, \varphi_\mu) \in [0, 1) \times S^1$.
More precisely, for each $a = (j, (\zeta_\nu)_{1 \leq \nu \leq l_0},
(\rho_\mu, \varphi_\mu)_{1 \leq \mu \leq l_1}) \in \widetilde{X}$,
the fiber $\widetilde{P}_a$ at $a$ has the following form.
\begin{align*}
\widetilde{P}_a =& (\Sigma_0 \setminus N_0)\\
&\cup \coprod_{\nu =1}^{l_0} \{(x, y) \in D \times D; xy = \zeta_\nu\}\\
&\cup \coprod_{\mu = 1}^{l_1} \{((s_x, t_x), (s_y, t_y))
\in [-1, \infty] \times S^1 \times [-\infty, 1] \times S^1;\\
&\quad\quad\quad\quad s_y - s_x = \log \rho_\mu,\ t_y -t_x = \varphi_\mu\}\\
&\cup \coprod_{l_2^-} [-\infty, 0] \times S^1 \cup \coprod_{l_2^+} [0, \infty] \times S^1
\end{align*}
($l_2^\pm$ are the number of $\pm$-limit circles respectively.)
Namely, $\widetilde{P}_a$ is obtained from $\Sigma$ by replacing the neighborhood
$D \cup D$ of the $\nu$-th nodal point with
$\{(x, y) \in D \times D; xy = \zeta_\nu\}$,
and the neighborhood $D \widetilde{\cup} D \cong
([-1, \infty] \cup_{\infty = -\infty} [-\infty, 1]) \times S^1$
of the $\mu$-th joint circle with
\begin{align*}
\widetilde{N}_a^\mu
&= \{((s_x, t_x), (s_y, t_y)) \in [-1, \infty] \times S^1 \times [-\infty, 1] \times S^1;\\
&\quad \quad s_y - s_x = \log \rho_\mu,\, t_y -t_x = \varphi_\mu\}.
\end{align*}
The complex structure of $\widetilde{P}_a$ is defined by $j$ on
$\Sigma_0 \setminus N_0$,
and the usual complex structure of the other parts.
(The complex structure is defined on the complement of the imaginary circles.)
The sections of marked points $Z_i$, $Z^+_i$, and $Z^{++}_i$ are defined by
the constant maps $Z_i \equiv z_i$, $Z^+_i \equiv z^+_i$ and $Z^{++}_i \equiv z^{++}_i$
($\in \Sigma_0 \setminus N_0$).
The above expression of $\widetilde{P}$ can be easily obtained by the local structure
of universal family of stable curves. (See \cite{RS06} for example.)
We identify $\widetilde{N}_a^\mu$ with
\[
([-1, - {\textstyle\frac{1}{2}} \log \rho_\mu]
\cup_{-\frac{1}{2} \log \rho_\mu = \frac{1}{2} \log \rho_\mu}
[{\textstyle\frac{1}{2}} \log \rho_\mu, 1]) \times S^1_\mu
\]
by
\begin{align}
[-1, - {\textstyle\frac{1}{2}} \log \rho_\mu] \times S^1_\mu &\to \widetilde{N}_a^\mu
\notag\\
(s, t) & \mapsto (s_x, t_x) = (s, t - {\textstyle\frac{1}{2}} \chi(s) \varphi_\mu)
\label{left coordinate near joint}\\
[{\textstyle\frac{1}{2}} \log \rho_\mu, 1] \times S^1_\mu &\to \widetilde{N}_a^\mu
\notag\\
(s', t') & \mapsto (s_y, t_y) = (s, t + {\textstyle\frac{1}{2}} \chi(-s) \varphi_\mu)
\label{right coordinate near joint}
\end{align}
where $\chi : \mathbb{R} \to \mathbb{R}_{\geq 0}$ is a smooth function such that
$\chi|_{(-\infty, -1]} \equiv 0$ and $\chi|_{[0, \infty)} \equiv 1$.
Let $j_{\varphi_\mu}$ be the complex structure on $([-1, - \frac{1}{2}\log \rho_\mu]
\cup [\frac{1}{2}\log \rho_\mu, 1]) \times S^1_\mu$ defined by the pull back of the
usual complex structure on $\widetilde{N}_a^\mu$ by the above identification.
We note that $-j_{\varphi_\mu} \partial_t
= \partial_s + \frac{1}{2} \varphi_\mu \chi'(s) \partial_t$
on $[-1, - \frac{1}{2}\log \rho_\mu] \times S^1_\mu$, and
$-j_{\varphi_\mu} \partial_t
= \partial_s + \frac{1}{2} \varphi_\mu \chi'(-s) \partial_t$
on $[\frac{1}{2} \log \rho_\mu, 1] \times S^1_\mu$.
Under this identification, we define the $L_\delta^p$-norms of the function spaces
of $\widetilde{P}_a$ as we explained before this section.
We use a parameter space $\mathring{X}$ which reflects the fact that
the splitting of $\hat Y$ occurs simultaneously with the deformation of
the domain curve.
$\mathring{X} \subset \widetilde{X} \times \prod_{\text{joint circles}} \mathbb{R}_\mu$
is a submanifold defined as follows.
Let $M_i$ be the set of joint circles between the $i$-th floor and the $(i+1)$-th floor.
Then $(a, (b_\mu)_\mu) \in \widetilde{X} \times \prod_{\text{joint circles}} \mathbb{R}_\mu$
belongs to $\mathring{X}$ if
$-L_\mu \log \rho_\mu + b_\mu \in (-\infty, \infty]$ is independent of $\mu \in M_i$ for
each $i = 1,2, \dots, k-1$,
where $L_\mu$ is the period of $\gamma_\mu$.
This implies in particular, whether $\rho_\mu = 0$ or not is independent of
$\mu \in M_i$ for each $i$,
and if $\rho_\mu \neq 0$ then $b_\mu$ is determined by $a \in \widetilde{X}$ and
one of $b_\mu$ for each $i$.
If we use an appropriate smooth structure of $\widetilde{X}$ (see Section
\ref{smoothness}), then $\mathring{X}$ is indeed a smooth submanifold of
$\widetilde{X} \times \prod_{\text{joint circles}} \mathbb{R}_\mu$.
For each $(a, b) \in \mathring{X}$, we define an equivalence relation $\sim_{a, b}$ of
$\overline{\mathbb{R}}_1 \sqcup \overline{\mathbb{R}}_2 \sqcup \dots \sqcup \overline{\mathbb{R}}_k$ by
identifying $s \in \overline{\mathbb{R}}_i$ and $s' \in \overline{\mathbb{R}}_{i+1}$ if $\rho_\mu \neq 0$
and $s - s' = -L_\mu \log \rho_\mu + b_\mu$ for some (and all) $\mu \in M_i$, and
identifying $+\infty \in \overline{\mathbb{R}}_i$ and $-\infty \in \overline{\mathbb{R}}_{i + 1}$
if $\rho_\mu = 0$.
Let $0_i \in (\overline{\mathbb{R}}_1 \sqcup \overline{\mathbb{R}}_2 \sqcup \dots \sqcup
\overline{\mathbb{R}}_k)/ \sim_{a, b}$ be the point corresponds to the zero in $\overline{\mathbb{R}}_i$.
If $\mu \in M_i$ and $\rho_\mu \neq 0$, then $b_\mu$ satisfies
\[
0_{i+1} - 0_i = -L_\mu \log \rho_\mu + b_\mu.
\]
If $\rho_\mu = 0$, the maps $u$ corresponding to the parameter $b_\mu$ will be
related to $b_\mu$ by
\begin{align}
b_\mu &= \lim_{s \to \infty} (\sigma \circ u|_{[0, \infty) \times S_\mu^1} (s, t)
- (0_i + L_\mu s))\notag\\
&\quad - \lim_{s \to -\infty} (\sigma \circ u|_{(-\infty, 0] \times S_\mu^1}(s, t)
- (0_{i+1} + L_\mu s)). \label{asymptotic parameter}
\end{align}
We call $b_\mu$ asymptotic parameters.
\begin{rem}
Before starting to construct a Kuranishi neighborhood, we calculate the virtual dimension
of the Kuranishi neighborhood of $p_0 = (\Sigma_0, z, u_0)
\in \widehat{\mathcal{M}}(Y, \lambda, J)$ and check that it coincides with the expected dimension.
Readers may skip this calculation since we do not use it for the construction of
Kuranishi neighborhood.
First, the dimension of the parameter space $\mathring{X}$ is
$\dim \mathring{X} = \dim \widetilde{X} + (k - 1)$, where $k$ is the height of $p_0$.
For each $(a, b) \in \mathring{X}$, we regard the equation of $J$-holomorphic curves
as a Fredholm map, whose index coincides with that of
the linearization $D_{p_0} : \widetilde{W}_\delta^{1, p}(\Sigma_0, u_0^\ast T \hat Y)
\to L_\delta^p(\Sigma_0, {\textstyle\bigwedge}^{0, 1}T^\ast \Sigma_0 \otimes u_0^\ast T \hat Y)$.
Since we need to kill the dimension of additional marked points $z^+ \cup z^{++}$
and the dimension ($= k$) corresponding to $\mathbb{R}$-translations,
the virtual dimension $m$ of the Kuranishi neighborhood
(that is, $\dim \mathcal{V} - \dim \mathcal{E}$ of the Kuranishi neighborhood $(\mathcal{V}, \mathcal{E}, s, \psi)$) is
\begin{align*}
m &= \dim \mathring{X} + \mathop{\mathrm{ind}}\nolimits D_{p_0} - 2 (\# z^+ + \# z^{++}) - k \\
&= (\dim \widetilde{X} - 2 (\# z^+ + \# z^{++})) + \mathop{\mathrm{ind}}\nolimits D_{p_0} - 1.
\end{align*}
Next we check the relation of the virtual dimension of $p_0$ and
those of its parts.
Assume that we can construct $p_0$ from finite number of
holomorphic buildings $p_\kappa = (\Sigma_\kappa, z_\kappa, u_\kappa)$
and finite number of trivial cylinders by jointing pairs of limit circles to
joint circles and jointing pairs of marked points to nodal points.
(For example, let $\{p_\kappa\}$ be the restrictions of $p_0$ to
the irreducible components $\Sigma_\alpha$ which are not trivial cylinders.)
Let $l_{\text{trivial}}$ be the number of trivial cylinders, and let
$l_{\text{nodal}}$ and $l_{\text{joint}}$ be the number of new nodal points and
new joint circles respectively.
It is easy to check that
\begin{align*}
&(\dim \widetilde{X} - 2 (\# z^+ + \# z^{++})) \\
&= \sum_\kappa (\dim \widetilde{X}_\kappa - 2 (\# z_\kappa^+ + \# z_\kappa^{++}))
- 2 l_{\text{trivial}} + 2 l_{\text{nodal}} + 2 l_{\text{joint}}.
\end{align*}
The index of $D_{p_0}$ and those of $D_{p_\kappa}$ are related by
\[
\mathop{\mathrm{ind}}\nolimits D_{p_0} = \sum_\kappa \mathop{\mathrm{ind}}\nolimits D_{p_\kappa} + 2 l_{\text{trivial}}
- \sum_{\star} \dim \mathop{\mathrm{Ker}}\nolimits A_{\gamma_\mu} - 2n l_{\text{nodal}}
\]
where the sum $\star$ is taken over new joint circles $\{S^1_\mu\}$
and each $\gamma_\mu$ is the periodic orbit on $S^1_\mu$.
The term $- \dim \mathop{\mathrm{Ker}}\nolimits A_{\gamma_\mu}$ in the above equation is due to
the fact that the Sobolev space $\widetilde{W}_\delta^{1, p}(\Sigma_0, u_0^\ast T \hat Y)$
contains one vector space $\mathop{\mathrm{Ker}}\nolimits A_{\gamma_\mu}$ for each joint circle
while the direct sum of the Sobolev spaces for $\{p_\kappa\}$ and limit circles
contains a pair of $\mathop{\mathrm{Ker}}\nolimits A_{\gamma_\mu}$ for each pair of limit circles.
For simplicity, assume Morse condition.
Then the above equations imply
\[
m - \sum_\kappa m_\kappa
= \#\{\kappa\} - 1 + (2 - 2n) l_{\text{nodal}},
\]
where each $m_\kappa$ is the virtual dimension of $p_\kappa$.
For example, this equation implies that if we divide a holomoprhic building into two parts
by a gap of floor, then the virtual dimension of the entire holomorphic building
is larger than the sum of the virtual dimensions of the two by one.
Similarly, the virtual dimension of disjoint holomoprhic building is
larger than the sum of the virtual dimensions of its connected components.
These coincide with the expected relations indeed.
\end{rem}
Now for each $(a, b) \in \mathring{X}$, we construct an approximate solution
$u_{a, b} : \widetilde{P}_a \to (\overline{\mathbb{R}}_1 \sqcup \overline{\mathbb{R}}_2 \sqcup \dots \sqcup
\overline{\mathbb{R}}_k)/ \sim_{a, b} \times Y$
and a map $\Phi_{a, b} : u_{a, b}^\ast T \hat Y \to (\overline{\mathbb{R}}_1 \sqcup
\overline{\mathbb{R}}_2 \sqcup \dots \sqcup \overline{\mathbb{R}}_k)/ \sim_{a, b} \times Y$.
They will satisfy the following conditions:
\begin{itemize}
\item
$u_{a, b}|_{\Sigma_0 \setminus N_0} = u_0|_{\Sigma_0 \setminus N_0}$
\item
The restriction of $\Phi_{a, b}$ to the zero section coincides with $u_{a, b}$,
that is, $\Phi_{a, b}(z, 0) = u_{a, b}(z)$ for all $z \in \widetilde{P}_a$.
\item
The vertical differential of $\Phi_{a, b}$ at the zero section is the identity map of
$u_{a, b}^\ast T \hat Y$.
\item
The restriction of $\Phi_{a, b}$ on $u_0^\ast T \hat Y|_{\Sigma_0 \setminus N_0}$
does not depend on $(a, b) \in \mathring{X}$.
\end{itemize}
First we consider the neighborhood of $\nu$-th nodal point.
Let $\phi^\nu : B_\epsilon^{2n}(0) \to \mathbb{R} \times Y$ be a coordinate centered
at the image of the nodal point by $u_0$.
Define $v_0^\nu : D \cup D \to B_\epsilon^{2n}(0)$ by
\[
u_0|_{(D \cup D)_\nu}(x,y) = \phi^\nu (v_0^\nu (x,y)).
\]
For each $(a, b) \in \mathring{X}$, define a piecewise smooth map $v_{a, b}^\nu :
N_{a, b}^\nu = \{(x,y) \in D \times D; xy = \zeta_\nu \} \to B_\epsilon^{2n}(0)$ by
\[
v_{a, b}^\nu(x,y) = \begin{cases}
v_0^\nu(\frac{r-\sqrt{|\zeta_\nu|}}{1-\sqrt{|\zeta_\nu|}} e^{\sqrt{-1} \theta}, 0)
\text{ if } x = r e^{\sqrt{-1}\theta} \text{ and } r \geq \sqrt{|\zeta_\nu|}\\
v_0^\nu(0, \frac{r-\sqrt{|\zeta_\nu|}}{1-\sqrt{|\zeta_\nu|}} e^{\sqrt{-1} \theta})
\text{ if } y = r e^{\sqrt{-1}\theta} \text{ and } r \geq \sqrt{|\zeta_\nu|}
\end{cases}.
\]
Define piecewise smooth maps $u_{a, b} : N_{a, b}^\nu \to \hat Y$ and
$\Phi_{a, b} : N_{a, b}^\nu \times \mathbb{R}^{2n} \to \hat Y$ by
\begin{align*}
u_{a, b}(x,y) &= \phi^\nu (v_{a, b}^\nu (x,y))\\
\Phi_{a, b}(x, y, \xi) &= \phi^\nu (v_{a, b}^\nu (x,y) + \xi).
\end{align*}
We identify $N_{a, b}^\nu \times \mathbb{R}^{2n}$ and $u_{a, b}^\ast T \hat Y|_{N_{a, b}^\nu}$
by the differential of $\Phi_{a, b}$ at the zero section $N_{a, b}^\nu \times \{0 \}$,
and consider $\Phi_{a, b}$ as a map $u_{a, b}^\ast T \hat Y|_{N_{a, b}^\nu} \to \hat Y$.
Next we consider the neighborhood of $\mu$-th joint circle.
Define $b_\mu^{0, \mathrm{left}},\allowbreak b_\mu^{0, \mathrm{right}},\allowbreak b_\mu^0 \in \mathbb{R}$ by
\begin{align*}
u_0|_{[-1,\infty)_\mu \times S^1}(s,t)
&= (L_\mu s + b_\mu^{0, \mathrm{left}}, \gamma_\mu(t)) + o(1)\\
u_0|_{(-\infty, +1]_\mu \times S^1}(s,t)
&= (L_\mu s + b_\mu^{0, \mathrm{right}}, \gamma_\mu(t)) + o(1)\\
b_\mu^0 &= b_\mu^{0, \mathrm{left}} - b_\mu^{0, \mathrm{right}}.
\end{align*}
Let $\phi^\mu : B^{m_\mu}_\epsilon (0) \to P$ be a coordinate centered at
$\gamma_\mu \in P$ for each $\mu$.
We take a family of open embeddings $\psi^\mu_t : B^{m_\mu}_\epsilon (0) \times
B^{2n-1-m_\mu}(0) \to Y$ ($t \in S^1$) such that
$\psi^\mu_t(x,0) = \mathrm{ev}_t \phi^\mu(x)$ for all $x \in B^{m_\mu}_\epsilon (0)$
as in Section \ref{asymptotic estimates}.
Define families of open embeddings $\hat \psi^{\mu, \mathrm{left}}_{s,t},
\hat \psi^{\mu, \mathrm{right}}_{s,t} : \mathbb{R} \times B^{m_\mu}_\epsilon (0) \times
B^{2n-1-m_\mu}(0) \to \mathbb{R} \times Y$ ($(s,t) \in \mathbb{R} \times S^1$) by
\begin{align*}
\hat \psi^{\mu, \mathrm{left}}_{s,t} (\sigma, (x,y))
&= (L_\mu s + b_\mu^{0, \mathrm{left}} + \sigma, \psi^\mu_t(x,y))\\
\hat \psi^{\mu, \mathrm{right}}_{s,t} (\sigma, (x,y))
&= (L_\mu s + b_\mu^{0, \mathrm{right}} + \sigma, \psi^\mu_t(x,y)).
\end{align*}
Define smooth functions $v_0^{\mu, \mathrm{left}} : [-1, \infty] \times S^1
\to \mathbb{R} \times B^{m_\mu}_\epsilon (0) \times B^{2n-1-m_\mu}(0)$ and
$v_0^{\mu, \mathrm{right}} : [-\infty, +1] \times S^1 \to \mathbb{R} \times
B^{m_\mu}_\epsilon (0) \times B^{2n-1-m_\mu}(0)$ by
\begin{align*}
u_0|_{[-1,\infty]_\mu \times S_\mu^1} (s,t)
&= \hat \psi^{\mu, \mathrm{left}}_{s,t} (v_0^{\mu, \mathrm{left}} (s,t))\\
u_0|_{[-\infty,+1]_\mu \times S_\mu^1} (s,t)
&= \hat \psi^{\mu, \mathrm{right}}_{s,t} (v_0^{\mu, \mathrm{right}} (s,t)).
\end{align*}
For each $\mu$, fix a constant $0 < \kappa_\mu < \delta_{0, \mu}$.
($\delta_{0, \mu}$ is the minimal nonzero absolute value of eigenvalues of
$A_{\gamma_\mu}$.)
Recall that we have identified $\widetilde{N}_a^\mu$ with
$([-1, -\frac{1}{2} \log \rho_\mu] \cup [\frac{1}{2} \log \rho_\mu, 1]) \times S^1_\mu$
by (\ref{left coordinate near joint}) and (\ref{right coordinate near joint}).
For each $(a,b) \in \mathring{X}$ and $\mu$, define
$v_{a,b}^{\mu, \mathrm{left}} : [-1, -\frac{1}{2} \log \rho_\mu] \times S^1
\to \mathbb{R} \times B^{m_\mu}_\epsilon (0) \times B^{2n-1-m_\mu}(0)$ and
$v_{a,b}^{\mu, \mathrm{right}} : [\frac{1}{2} \log \rho_\mu, 1] \times S^1
\to \mathbb{R} \times B^{m_\mu}_\epsilon (0) \times B^{2n-1-m_\mu}(0)$ by
\begin{align*}
v_{a,b}^{\mu, \mathrm{left}} (s,t) &=\begin{cases}
v_0^{\mu, \mathrm{left}} (s, t) &\text{if } s \in [-1, 0]\\
v_0^{\mu, \mathrm{left}} \biggl( {\displaystyle - \frac{1}{\kappa_\mu} \log
\biggl(\frac{e^{-\kappa_\mu s} - \rho_\mu^{\kappa_\mu/2}}
{1 - \rho_\mu^{\kappa_\mu/2}} \biggr), t }\biggr)
&\text{if } s \in [0, -\frac{1}{2} \log \rho_\mu]
\end{cases},\\
v_{a,b}^{\mu, \mathrm{right}} (s,t) &=\begin{cases}
v_0^{\mu, \mathrm{right}} (s, t) &\text{if } s \in [0, 1]\\
v_0^{\mu, \mathrm{right}} \biggl( {\displaystyle \frac{1}{\kappa_\mu} \log
\biggl(\frac{e^{\kappa_\mu s} - \rho_\mu^{\kappa_\mu/2}}
{1 - \rho_\mu^{\kappa_\mu/2}}\biggr), t }\biggr)
&\text {if } s \in [\frac{1}{2} \log \rho_\mu, 0]
\end{cases}.
\end{align*}
Then piecewise smooth maps $u_{a, b} : ([-1, -\frac{1}{2} \log \rho_\mu] \cup
[\frac{1}{2} \log \rho_\mu, 1]) \times S^1 \to
(\overline{\mathbb{R}}_i \cup \overline{\mathbb{R}}_{i + 1}) / \sim_{a, b} \times Y$ and
$\Phi_{a, b} : ([-1, -\frac{1}{2} \log \rho_\mu] \cup [\frac{1}{2} \log \rho_\mu, 1])
\times S^1 \times \mathbb{R}^{2n} \to (\overline{\mathbb{R}}_i \cup \overline{\mathbb{R}}_{i + 1}) / \sim_{a, b}
\times Y$ are defined by
\begin{equation*}
u_{a, b}(s, t)
= \begin{cases}
o_{\frac{1}{2}\chi(s) (b_\mu - b_\mu^0)} \circ
\hat \psi^{\mu, \mathrm{left}}_{s, t}
(v_{a,b}^{\mu, \mathrm{left}}(s,t)) \in \overline{\mathbb{R}}_i \times Y\\
\hspace{180pt}
\text{if } s \in [-1, -\frac{1}{2} \log \rho_\mu] \\
o_{-\frac{1}{2}\chi(-s) (b_\mu - b_\mu^0)} \circ
\hat \psi^{\mu, \mathrm{right}}_{s, t}
(v_{a,b}^{\mu, \mathrm{right}}(s,t)) \in \overline{\mathbb{R}}_{i + 1} \times Y\\
\hspace{180pt}
\text{if } s \in [\frac{1}{2} \log \rho_\mu, 1]
\end{cases}
\end{equation*}
and
\begin{equation*}
\Phi_{a, b}(s, t, \xi)
= \begin{cases}
o_{\frac{1}{2}\chi(s) (b_\mu - b_\mu^0)} \circ
\hat \psi^{\mu, \mathrm{left}}_{s, t}
(v_{a,b}^{\mu, \mathrm{left}}(s,t) + \xi) \in \overline{\mathbb{R}}_i \times Y\\
\hspace{170pt}
\text{if } s \in [-1, -\frac{1}{2} \log \rho_\mu] \\
o_{-\frac{1}{2}\chi(-s) (b_\mu - b_\mu^0)} \circ
\hat \psi^{\mu, \mathrm{right}}_{s, t}
(v_{a,b}^{\mu, \mathrm{right}}(s,t) + \xi) \in \overline{\mathbb{R}}_{i + 1} \times Y\\
\hspace{170pt}\text{if } s \in [\frac{1}{2} \log \rho_\mu, 1],
\end{cases}
\end{equation*}
where $o_c : \overline{\mathbb{R}} \times Y \to \overline{\mathbb{R}} \times Y$ is the translation map
of the $\overline{\mathbb{R}}$-factor defined by $o_c(\sigma, y) = (\sigma + c, y)$,
and $\chi : \mathbb{R} \to \mathbb{R}_{\geq 0}$ is a smooth function such that
$\chi|_{(-\infty, 0]} = 0$ and $\chi|_{[0, \infty)} = 1$.
We identify $([-1, -\frac{1}{2} \log \rho_\mu] \cup [\frac{1}{2} \log \rho_\mu, 1])
\times S^1 \times \mathbb{R}^{2n}$ and
$u_{a, b}^\ast T \hat Y|_{([-1, -\frac{1}{2} \log \rho_\mu] \cup
[\frac{1}{2} \log \rho_\mu, 1]) \times S^1}$
by the differential of $\Phi_{a, b}$ at the zero section $([-1, -\frac{1}{2} \log \rho_\mu]
\cup [\frac{1}{2} \log \rho_\mu, 1]) \times S^1 \times \{0 \}$,
and consider $\Phi_{a, b}$ as a map
$u_{a, b}^\ast T \hat Y|_{([-1, -\frac{1}{2} \log \rho_\mu] \cup
[\frac{1}{2} \log \rho_\mu, 1]) \times S^1}
\to (\overline{\mathbb{R}}_i \cup \overline{\mathbb{R}}_{i + 1}) / \sim_{a, b} \times Y$.
Next we consider the neighborhood of each limit circle.
Since this region does not change by $(a, b) \in \mathring{X}$,
we can use $u_{a, b} = u_0$ as an
approximate solution.
Assume this circle is $+ \infty$-limit circle $S_{+ \infty_i}^1$.
(The case of $-\infty$-limit circle is similar.)
Let $\phi^{+\infty_i} : B^{m_{+\infty_i}}_\epsilon (0) \to P$ be a coordinate centered at
$\gamma_{+\infty_i} \in P$, and take a family of open embeddings
$\psi^{+\infty_i}_t : B^{m_{+\infty_i}}_\epsilon (0) \times B^{2n-1-m_{+\infty_i}}(0) \to Y$
($t \in S^1$) such that $\psi^{+\infty_i}_t(x,0) = \mathrm{ev}_t \phi^{+\infty_i}(x)$ for all
$x \in B^{m_{+\infty_i}}_\epsilon (0)$ as in the previous case.
We define a smooth map $v_0^{+\infty_i} : [0, \infty] \times S^1 \to \mathbb{R} \times
B^{m_{+\infty_i}}_\epsilon (0) \times B^{2n-1-m_{+\infty_i}}(0)$ by
\[
u_0|_{[0,\infty] \times S^1} (s,t)
= (1 \times \psi^{+\infty_i}_t) (v_0^{+\infty_i} (s,t)).
\]
Then a smooth map $\Phi_{a, b} : [0, \infty] \times S^1 \times \mathbb{R}^{2n} \to \hat Y$
is defined by
\[
\Phi_{a, b}(s, t, \xi) = (1 \times \psi^{+\infty_i}_t) (v_0^{+\infty_i} (s,t) + \xi).
\]
(This does not depend on $(a, b) \in \mathring{X}$.)
Finally, we consider the rest $\Sigma_0 \setminus N_0$. Since $u_{a, b}|_{\partial N_0}
= u_0|_{\partial N_0}$, we can define a piecewise smooth map
$u_{a, b} : \widetilde{P}_a \to (\overline{\mathbb{R}}_1 \sqcup \overline{\mathbb{R}}_2 \sqcup \dots
\sqcup \overline{\mathbb{R}}_k)/ \sim_{a, b} \times Y$
by $u_{a, b}|_{\Sigma_0 \setminus N_0} = u_0|_{\Sigma_0 \setminus N_0}$.
Note that the restriction of $\Phi_{a, b}$ to $u_0^\ast T \hat Y|_{\partial N_0}$
does not depend on $(a, b)$.
Therefore, we can take a smooth extension
$\Phi : u_0^\ast T \hat Y|_{\Sigma_0 \setminus N_0} \to (\overline{\mathbb{R}}_1 \sqcup
\overline{\mathbb{R}}_2 \sqcup \dots \sqcup \overline{\mathbb{R}}_k)/ \sim_{a, b} \times Y$
which is independent of $(a, b) \in \mathring{X}$ and satisfies the desired conditions,
that is,
\begin{itemize}
\item
the restriction of $\Phi$ to the zero section coincides with $u_0$, and
\item
the vertical differential of $\Phi$ at the zero section is the identity map of
$u_0^\ast T \hat Y$.
\end{itemize}
We will give a differentiable structure to a neighborhood
\begin{align*}
\hat V \subset \bigcup_{(a, b) \in \mathring{X}} \{(a, b)\} \times
\{&(\xi, h) \in \widetilde{W}_\delta^{1, p}(\widetilde{P}_a; u_{a, b}^\ast T \hat Y)
\times E^0;\\
&d(\Phi_{a, b}(\xi)) + J d(\Phi_{a, b}(\xi)) j + h_{a, b}(z, \Phi_{a, b}(\xi)) = 0\}
\end{align*}
of $(0, b^0, 0,0)$ later, where $h_{a, b}$ is the restriction of $\lambda(h)$ to
$\widehat{P}_a \times Y$.
Then $G_0$ acts on $\hat V$ smoothly, and
a $G_0$-equivariant section $s^0 : \hat V \to \mathbb{R}^k \oplus
\bigoplus_{z_\beta^{++}} \mathbb{R}_i^2$ defined by
\[
s^0(a, b, \xi, h) = (\sigma_i \circ \Phi_{a, b}(\xi)(\widetilde{R}_i(a)),
p' \circ \Phi_{a, b}(\xi)(Z_\beta^{++}(a)))
\]
is a smooth submersion, where each
$\sigma_i \circ \Phi_{a, b}(\xi)(\widetilde{R}_i(a))$ is the abbreviation of
\[
\frac{1}{m_i} \sum_{l=1}^{m_i} \sigma_i \circ \Phi_{a, b}(\xi)(\widetilde{R}_{i, l}(a)),
\]
and $p'$ is a smooth submersion from a neighborhood of $S'$ to $\mathbb{R}^2$ such that
$S' = \{p' = 0\}$.
Let $V = \{s^0 = 0\} \subset \hat V$ be the zero set,
and consider the finite dimensional vector space
$E = E^0 \oplus \bigoplus_{z_\alpha^+} \mathbb{R}_\alpha^2$ as a trivial vector bundled on $V$.
Define a smooth section $s : V \to E$ by
\[
s(a, b, \xi, h) = (h, p \circ \pi_Y \circ \Phi_{a, b}(\xi)(Z_\alpha^+(a))),
\]
where $p$ is a smooth submersion from a neighborhood of $S \subset Y$ to $\mathbb{R}^2$
such that $S = \{p= 0\}$.
Since the zero set of $s$ consists of holomorphic buildings,
we can define a continuous map $\psi : \{s = 0\}/G_0 \to \widehat{\mathcal{M}}^0(Y, \lambda, J)$.
Finally we will prove that this map is a homeomorphism onto a neighborhood of $p$.
Now we start to define a differentiable structure of $\hat V$.
To do so, we express this set as a zero set of a Fredholm map between
Banach spaces.
To define a Fredholm map, first we rewrite the equation of $(\xi, h)$.
Note that the equation
\begin{equation}
d(\Phi_{a, b}(\xi))(z) + J(\Phi_{a, b}(\xi)(z)) d(\Phi_{a, b}(\xi))(z) j_z
+ h_{a, b}(z, \Phi_{a, b}(\xi)(z)) = 0 \label{xi eq}
\end{equation}
is equivalent to the equation of $J$-holomorphic curve on $N_0$ since $h_{a, b}$
vanishes on $N_0 \times Y$.
On $\{x \in D ; |x| \geq \sqrt{|\zeta_\nu|}\} \subset \{(x, y) \in D \times D;
x y = \zeta_\nu\}$ or
$\{y \in D ; |y| \geq \sqrt{|\zeta_\nu|}\} \subset \{(x, y) \in D \times D;
x y = \zeta_\nu\}$, $\Phi_{a, b}(\xi)$ is $J$-holomorphic if and only if
\begin{equation}
\partial_r (v_{a, b}^\nu + \xi) + \frac{1}{r}\widetilde{J}^\nu(v_{a, b}^\nu + \xi)
\partial_\theta (v_{a, b}^\nu + \xi) = 0, \label{node eq}
\end{equation}
where $\widetilde{J}^\nu = (\phi^\nu)^\ast J$ is the pull back of $J$,
and $(r, \theta)$ is the polar coordinate of $x$ or $y$ respectively.
On $[-1, -\frac{1}{2} \log \rho_\mu] \times S^1$,
$\Phi_{a, b}(\xi)$ is $J$-holomoprhic if and only if
\begin{multline*}
\hat\psi_\ast \partial_s (v_{a,b}^{\mu, \mathrm{left}} + \xi)
+ \Bigl(L_\mu + \frac{1}{2} \chi'(s) (b_\mu - b_\mu^0)\Bigr) \partial_\sigma \\
+\Bigl(\frac{1}{2} \varphi_\mu \chi'(s)
+ J(\hat\psi(v_{a,b}^{\mu, \mathrm{left}} + \xi))\Bigr)
((\partial_t \hat\psi) (v_{a,b}^{\mu, \mathrm{left}} + \xi)
+ \hat\psi_\ast \partial_t (v_{a,b}^{\mu, \mathrm{left}} + \xi))= 0
\end{multline*}
since $-j_{\varphi_\mu} \partial_t = \partial_s + \frac{1}{2} \varphi_\mu \chi'(s) \partial_t$.
This can be written as
\begin{multline}
\partial_s (v_{a,b}^{\mu, \mathrm{left}} + \xi)
+ \widetilde{J}^\mu_t(v_{a,b}^{\mu, \mathrm{left}} + \xi)
\partial_t (v_{a,b}^{\mu, \mathrm{left}} + \xi)
+ f^\mu_t(v_{a,b}^{\mu, \mathrm{left}} + \xi)\\
+ \frac{1}{2} (b_\mu -b_\mu^0) \chi'(s) \partial_\sigma
+ \frac{1}{2} \varphi_\mu \chi'(s) (g^\mu_t(v_{a,b}^{\mu, \mathrm{left}} + \xi)
+ \partial_t(v_{a,b}^{\mu, \mathrm{left}} + \xi)) =0, \label{+ joint circle eq}
\end{multline}
where $\widetilde{J}^\mu_t = (1 \times \psi^\mu_t)^\ast J$ and
\begin{align*}
f^\mu_t(\sigma, y) &= (1 \times \psi^\mu_t)_\ast^{-1} J(\psi^\mu_t(y))
(\partial_t \psi^\mu_t(y) - L R_\lambda(y))\\
g^\mu_t(\sigma, y) &= (1 \times \psi^\mu_t)_\ast^{-1} \partial_t \psi_t(y).
\end{align*}
In particular, on $[0, -\frac{1}{2} \log \rho_\mu] \times S^1
\subset [-1, \frac{1}{2} \log \rho_\mu] \times S^1$,
this equation can be written as
\[
\partial_s (v_{a,b}^{\mu, \mathrm{left}} + \xi)
+ \widetilde{J}^\mu_t(v_{a,b}^{\mu, \mathrm{left}} + \xi)
\partial_t (v_{a,b}^{\mu, \mathrm{left}} + \xi)
+ f^\mu_t(v_{a,b}^{\mu, \mathrm{left}} + \xi) = 0.
\]
We note that $f^\mu_t : B^{m_\mu}_\epsilon(0) \times B^{2n-1-m_\mu}(0) \to \mathbb{R}^{2n}$
satisfies $f^\mu_t|_{B^{m_\mu}_\epsilon(0) \times \{0\}} \equiv 0$.
Similarly, on $[\frac{1}{2} \log \rho_\mu, 1] \times S^1$, $\Phi_{a, b}(\xi)$ is
$J$-holomorphic if and only if
\begin{align}
&\partial_s (v_{a,b}^{\mu, \mathrm{right}} \! + \xi)
+ \widetilde{J}^\mu_t(v_{a,b}^{\mu, \mathrm{right}} \! + \xi)
\partial_t (v_{a,b}^{\mu, \mathrm{right}} \! + \xi)
+ f^\mu_t(v_{a,b}^{\mu, \mathrm{right}} \! + \xi)
\notag\\
&+ \frac{1}{2} (b_\mu -b_\mu^0) \chi'(-s) \partial_\sigma
+ \frac{1}{2} \varphi_\mu \chi'(-s) (g^\mu_t(v_{a,b}^{\mu, \mathrm{right}} \! + \xi)
+ \partial_t(v_{a,b}^{\mu, \mathrm{right}} \! + \xi)) \notag\\
& =0. \label{- joint circle eq}
\end{align}
On the neighborhood $[0, \infty] \times S^1$ of the limit circle $S_{+\infty_i}^1$
or on the neighborhood $[-\infty, 0] \times S^1$ of the limit circle $S_{-\infty_i}^1$,
$\Phi_{a, b}(\xi)$ is $J$-holomorphic if and only if
\begin{equation}
\partial_s (v_0^{\pm\infty_i} + \xi)
+ \widetilde{J}^{\pm\infty_i}_t(v_0^{\pm\infty_i} + \xi) \partial_t (v_0^{\pm\infty_i} + \xi)
+ f^{\pm\infty_i}_t (\pi_Y (v_0^{\pm\infty_i} + \xi)) = 0, \label{limit circle eq}
\end{equation}
where $f^{\pm\infty_i}_t : B^m_\epsilon(0) \times B^{2n-1-m}(0) \to \mathbb{R}^{2n}$ ($t\in S^1$)
is a smooth function which satisfies $f^\mu_t|_{B^m_\epsilon(0) \times \{0\}} \equiv 0$.
Now we define a Fredholm map
\begin{align*}
&F^{(a,b)} : \widetilde{W}^{1,p}_\delta( \widetilde{P}_a, u_{a, b}^\ast T \hat Y) \oplus E^0\\
&\to L^p(\Sigma_0 \setminus N_0, {\textstyle\bigwedge}^{0,1} T^\ast \Sigma_0 \otimes_\mathbb{C}
u_0^\ast T \hat Y)\\
&\quad \oplus \bigoplus_\nu (L^p (\{ x \in D; |x| \geq \sqrt{|\zeta_\nu|}\}, \mathbb{R}^{2n})
\oplus L^p (\{ y \in D; |y| \geq \sqrt{|\zeta_\nu|}\}, \mathbb{R}^{2n}))\\
&\quad \oplus \bigoplus_\mu (L^p_\delta ([-1, -{\textstyle\frac{1}{2}}\log \rho_\mu]
\times S^1, \mathbb{R}^{2n})
\oplus L^p_\delta ([{\textstyle\frac{1}{2}} \log \rho_\mu, +1] \times S^1, \mathbb{R}^{2n}))\\
&\quad \oplus \bigoplus_{+\infty_i} L^p_\delta([0, \infty] \times S^1, \mathbb{R}^{2n})
\oplus \bigoplus_{-\infty_i} L^p_\delta([-\infty, 0] \times S^1, \mathbb{R}^{2n})
\end{align*}
by the left hand sides of the above equations (\ref{xi eq}), (\ref{node eq}),
(\ref{+ joint circle eq}), (\ref{- joint circle eq}) and (\ref{limit circle eq}), that is,
its $L^p(\Sigma_0 \setminus N_0)$-component is defined by
\[
d(\Phi(\xi))(z) + J(\Phi(\xi)(z)) d(\Phi(\xi))(z) j_z + h_{a, b}(z, \Phi(\xi)(z)),
\]
its $L^p (\{ x \in D; |x| \geq \sqrt{|\zeta_\nu|}\})$-component is defined by
\[
\partial_r (v_{a, b}^\nu + \xi) + \frac{1}{r}\widetilde{J}^\nu(v_{a, b}^\nu + \xi)
\partial_\theta (v_{a, b}^\nu + \xi),
\]
and so on.
We abbreviate the range of this Fredholm map as
$L_\delta^p(\widetilde{P}_a, {\textstyle\bigwedge}^{0, 1} T^\ast \widetilde{P}_a \otimes
u_{a, b}^\ast T \hat Y)$.
We also define a Fredholm map
\[
F^{(a,b) +} : \widetilde{W}^{1,p}_\delta( \widetilde{P}_a, u_{a, b}^\ast T \hat Y) \oplus E^0
\to L_\delta^p(\widetilde{P}_a, {\textstyle\bigwedge}^{0, 1} T^\ast \widetilde{P}_a \otimes
u_{a, b}^\ast T \hat Y) \oplus \mathop{\mathrm{Ker}}\nolimits DF^{(0, b^0)}_{(0, 0)}
\]
by
\[
F^{(a,b) +} (\xi, h) = \bigl(F^{(a,b)}(\xi,h),
\sum_i (\langle \xi, \xi_i \rangle_{L^2(\Sigma_0 \setminus N_0)}
+ \langle h, h_i \rangle_{E^0}) \cdot x_i\bigr),
\]
where $\{ x_i = (\xi_i, h_i) \}$ is a orthonormal basis of $\mathop{\mathrm{Ker}}\nolimits DF^{(0, b^0)}_{(0, 0)}$
with the inner product given by
\[
\langle (\xi, h), (\xi', h') \rangle = \langle \xi, \xi' \rangle_{L^2(\Sigma_0 \setminus N_0)}
+ \langle h, h' \rangle_{E^0}
\]
for some inner product of $E^0$.
In order to apply the implicit function theorem to $F^{(a, b)}$,
or apply inverse function theorem to $F^{(a, b)+}$, we need to check their properties.
First we need to show that $F^{(a, b)}(0, 0)$ is small for any $(a, b) \in \mathring{X}$
sufficiently close to $(0, b^0) \in \mathring{X}$.
(This is equivalent to say that $u_{(a, b)}$ is close to the solution.)
Note that $F^{(a, b)}(0, 0)$ is zero on $\Sigma_0 \setminus N_0$ and
the neighborhoods of limit circles since these regions are independent of
$(a, b) \in \mathring{X}$.
Recall that $\delta_{0, \mu} > 0$ and $\delta_{0, \infty_i}$ are the minimal nonzero
absolute value of eigenvalues of $A_{\gamma_\mu}$ and $A_{\gamma_{\pm\infty_i}}$
respectively.
Assume that a sequence of positive constants
$\delta = ((\delta_\mu)_\mu, (\delta_{\pm\infty_i})_{\pm\infty_i})$ satisfies
$\delta_{\mu} < \delta_{0, \mu}$ and $\delta_{\pm\infty_i} < \delta_{\pm\infty_i}$.
We abbreviate this condition by $\delta < \delta_0$.
For such a sequence of positive constant $\delta$ and a constant $p > 2$,
we use the $L^p_\delta$-norm or $W^{1, p}_\delta$-norm on $\widetilde{P}_a$ as
a deformation of the curve $\Sigma_0$ explained before.
Let $\delta'_0 = ((\delta'_{0, \mu})_\mu, (\delta'_{0, \pm\infty_i})_{\pm\infty_i})$ be
an arbitrary sequence of positive constants such that $\delta < \delta'_0 < \delta_0$.
\begin{lem}\label{estimates of F(0)}
For any $0 < \delta < \delta'_0 < \delta_0$
and $p > 2$, there exists a constant $C>0$ such that
for any $(a,b) \in \mathring{X}$ sufficiently close to $(0, b^0)$,
the following inequalities hold true.
\begin{align*}
||F^{(a,b)}(0,0)|_{[-1, 0] \times S_\mu^1}||_{L^p}
&\leq C (|\varphi_\mu| + |b_\mu - b_\mu^0|)\\
||F^{(a,b)}(0,0)|_{[0, -\frac{1}{2} \log \rho_\mu] \times S_\mu^1}||_{L^p_{\delta_\mu}}
&\leq C \rho_\mu^{\min(\kappa_\mu, \delta'_{0, \mu} - \delta_\mu)/2}
(-\log \rho_\mu)^{1/p} \\
||F^{(a,b)}(0,0)|_{\{(x, y) \in N^\nu_{(a, b)}; |x| \geq \sqrt{|\zeta_\nu|} \}}||_{L^p}
&\leq C |\zeta_\nu|^{1/p}
\end{align*}
\end{lem}
\begin{proof}
First we estimate the $L^p$-norm of $F^{(a,b)}(0,0)|_{[-1, 0] \times S_\mu^1}$.
The equation
\[
0 = F^{(0, b^0)}(0,0)|_{[-1, 0] \times S_\mu^1}
= \partial_s v_0^{\mu, \text{left}}
+ \widetilde{J}^\mu_t(v_0^{\mu, \text{left}}) \partial_t v_0^{\mu, \text{left}}
+ f^\mu_t(v_0^{\mu, \text{left}})
\]
implies
\begin{align*}
&F^{(a, b)}(0, 0)|_{[-1, 0] \times S_\mu^1}\\
&= F^{(a, b)}(0, 0)|_{[-1, 0] \times S_\mu^1} - F^{(0, b^0)}(0,0)|_{[-1, 0] \times S_\mu^1}\\
&= \frac{1}{2} (b_\mu -b_\mu^0) \chi'(s) \partial_\sigma
+ \frac{1}{2} \varphi_\mu \chi'(s) (g^\mu_t(v_{a,b}^{\mu, \mathrm{left}})
+ \partial_t v_{a,b}^{\mu, \mathrm{left}}).
\end{align*}
The first inequality is clear from this equation.
Next we estimate the $L^p_\delta$-norm of
$F^{(a,b)}(0,0)|_{[0, -\frac{1}{2} \log \rho_\mu] \times S^1}$.
Since $\kappa_\mu < \delta_{0, \mu}$, we may assume that
$\delta'_{0, \mu} > \kappa_\mu$.
We omit the subscript $\mu$ of $\kappa_\mu$, $\rho_\mu$ and so on.
First we note that
\begin{align}
&F^{(a,b)}(0,0)|_{[0, -\frac{1}{2} \log \rho] \times S^1}(s, t) \notag\\
&= \biggl(1 + \frac{\rho^{\kappa/2}}{e^{-\kappa s} - \rho^{\kappa/2}}\biggr)
\partial_s v_0^{\mu, \text{left}}(\tilde s, t) \notag \\
&\quad \ + \widetilde{J}^\mu_t(v_0^{\mu, \text{left}}(\tilde s, t))
\partial_t v_0^{\mu, \text{left}}(\tilde s, t)
+ f_t^\mu(v_0^{\mu, \text{left}}(\tilde s, t)),
\label{F joint circle}
\end{align}
where
\[
\tilde s = -\frac{1}{\kappa} \log
\biggl(\frac{e^{-\kappa s} - \rho^{\kappa/2}}
{1 - \rho^{\kappa/2}}\biggr).
\]
Substituting
\[
\biggl( -\frac{1}{\kappa} \log
\biggl(\frac{e^{-\kappa s} - \rho^{\kappa/2}}
{1 - \rho^{\kappa/2}}\biggr), t \biggr)
\]
for $(s, t)$ in the equation
\[
0 = F^{(0,0)}(0,0)|_{[0,\infty) \times S^1}
= \partial_s v_0^{\mu, \text{left}}
+ \widetilde{J}^\mu_t(v_0^{\mu, \text{left}}) \partial_t v_0^{\mu, \text{left}}
+ f_t^\mu(v_0^{\mu, \text{left}}),
\]
and subtracting it from (\ref{F joint circle}),
we obtain
\begin{align*}
&F^{(a,b)}(0,0)|_{[0, -\frac{1}{2} \log \rho] \times S^1}\\
&= \frac{\rho^{\kappa/2}}{e^{-\kappa s} - \rho^{\kappa/2}}
(\partial_s v_0^{\mu, \text{left}})
\biggl( -\frac{1}{\kappa} \log
\biggl(\frac{e^{-\kappa s} - \rho^{\kappa/2}}
{1 - \rho^{\kappa/2}}\biggr), t \biggr).
\end{align*}
Recall that Proposition \ref{second annulus} implies $|v_0^{\mu, \text{left}}(s,t)|,
|\partial_s v_0^{\mu, \text{left}}(s,t)| \lesssim e^{-\delta'_0 s}$.
Hence
\begin{align*}
&\int_0^{-\frac{1}{2} \log \rho}
\biggl( \frac{\rho^{\kappa/2}}{e^{-\kappa s} - \rho^{\kappa/2}}
\biggl|(\partial_s v_0^{\mu, \text{left}})
\biggl( -\frac{1}{\kappa} \log
\biggl(\frac{e^{-\kappa s} - \rho^{\kappa/2}}
{1 - \rho^{\kappa/2}}\biggr), t \biggr)\biggr| e^{\delta s} \biggr)^p ds\\
&\lesssim \int_0^{-\frac{1}{2} \log \rho}
\biggl( \frac{\rho^{\kappa/2}}{e^{-\kappa s} - \rho^{\kappa/2}}
\biggl(\frac{e^{-\kappa s} - \rho^{\kappa/2}}
{1 - \rho^{\kappa/2}}\biggr)^{\! \delta'_0/\kappa}
e^{\delta s}\biggr)^p ds\\
& = \frac{\rho^{p\kappa/2}}{(1 - \rho^{\kappa/2})^{p\delta'_0/\kappa}}
\int_0^{-\frac{1}{2} \log \rho}
\bigl((e^{-\kappa s} - \rho^{\kappa/2})^{\delta'_0/\kappa - 1} e^{\delta s}\bigr)^p ds\\
& \leq \frac{\rho^{p\kappa/2}}{(1 - \rho^{\kappa/2})^{p\delta'_0/\kappa}}
\int_0^{-\frac{1}{2} \log \rho} e^{-p(\delta'_0 - \delta - \kappa) s} ds\\
& \lesssim \rho^{p\min(\kappa, \delta'_0 - \delta)/2} (-\log \rho).
\end{align*}
This is the proof of the second inequality.
Finally we estimate the $L^p$-norm of
$F^{(a,b)}(0,0)|_{\{x \in D; |x| \geq \sqrt{|\zeta_\nu|}\}}$,
where we denote a point
$(x, y) \in \{(x, y) \in N^\nu_{(a, b)}; |x| \geq \sqrt{|\zeta_\nu|} \}$
by $x \in \{x \in D; |x| \geq \sqrt{|\zeta_\nu|}\}$.
We abbreviate $\zeta_\nu$ to $\zeta$ and define $\rho = \sqrt{|\zeta|}$.
First note that
\begin{align}
&F^{(a,b)}(0,0)|_{\{x \in D; |x| \geq \rho\}}(re^{\sqrt{-1} \theta}) \notag\\
&= \frac{1}{1-\rho} (\partial_r v_0^\nu)
\Bigl(\frac{r-\rho}{1-\rho} e^{\sqrt{-1} \theta}\Bigr) \notag\\
&\quad + \frac{1}{r}\widetilde{J}^\nu
\Bigl(v_0^\nu\Bigl(\frac{r-\rho}{1-\rho} e^{\sqrt{-1} \theta}\Bigr) \Bigr)
\partial_\theta
\Bigl(v_0^\nu\Bigl(\frac{r-\rho}{1-\rho} e^{\sqrt{-1} \theta}\Bigr) \Bigr).
\label{F node}
\end{align}
We also note that $F^{(0,b^0)}(0,0)|_{\{ x \in D\}} = 0$ implies
\begin{equation}
\widetilde{J}^\nu(v_0^\nu(re^{\sqrt{-1} \theta}))
\partial_\theta v_0^\nu(re^{\sqrt{-1} \theta})
= - r \partial_r v_0^\nu(re^{\sqrt{-1} \theta}).
\label{node del theta}
\end{equation}
Substituting $((r - \rho) / (1 - \rho), \theta)$ for
$(r, \theta)$ in (\ref{node del theta}),
and substitute it into (\ref{F node}),
we obtain
\[
F^{(a,b)}(0,0)|_{\{x \in D; |x| \geq \rho\}}
= \frac{\rho}{(1-\rho) r} (\partial_r v_0^\nu)
\Bigl(\frac{r-\rho}{1-\rho} e^{\sqrt{-1} \theta}\Bigr).
\]
Since $|\partial_r v_0^\nu|$ is bounded on $\{ x \in D\}$, this implies
\[
||F^{(a,b)}(0,0)|_{\{x \in D; |x| \geq \rho\}}||_{L^p}
\lesssim \frac{\rho}{1 - \rho}
\biggl(\int_{\rho}^1 r^{-p} rdr \biggr)^{1/p}
\lesssim \rho^{2 / p} = |\zeta|^{1/p}
\]
\end{proof}
Next we need to prove the differential $D F^{(a,b) +}_{(\xi,h)}$ is uniformly invertible for
any $(a,b) \in \mathring{X}$ sufficiently close to $(0, b^0)$ and any
$(\xi,h) \in \widetilde{W}^{1,p}_\delta(\widetilde{P}_a, u_{a, b}^\ast T \hat Y) \oplus E^0$
sufficiently close to $(0,0)$.
Since the assumption of the surjectivity of $D_{p_0}^+$ implies that
$D F^{(0,0) +}_{(0,0)}$ is invertible, the case of $(\xi, h) = (0,0)$ is
Lemma \ref{linearized gluing lemma} in Section \ref{linearized gluing}.
The general case is a consequence of the following lemma,
which can be proved easily by direct calculations.
\begin{lem}\label{D^2F}
For any $\delta < \delta_0$, there exists a constant $C>0$ such that
for any $(a,b) \in \mathring{X}$ sufficiently close to $(0, b^0)$ and any
$(\xi,h) \in W^{1,p}_\delta( \widetilde{P}_a, (u_{a, b}^\ast T \hat Y)) \oplus E^0$
sufficiently close to $(0,0)$, the following inequalities hold true.
{\belowdisplayskip=0pt
\begin{multline*}
||DF^{(a,b) +}_{(\xi, h)} (\hat \xi, \hat h) - DF^{(a,b) +}_{(0,0)}
(\hat \xi, \hat h)||_{L^p_\delta([-1, -\frac{1}{2} \log \rho_\mu] \times S^1)}\\
\leq C (||\xi||_\infty ||\hat \xi||_{\widetilde{W}^{1,p}_\delta}
+ ||\xi||_{\widetilde{W}^{1,p}_\delta} ||\hat \xi||_\infty)
\end{multline*}
\begin{multline*}
||DF^{(a,b) +}_{(\xi, h)} (\hat \xi, \hat h) - DF^{(a,b) +}_{(0,0)} (\hat \xi, \hat h)||_{L^p
(\{ x \in D; |x| \geq \sqrt{|\zeta^\nu|}\})}\\
\leq C (||\xi||_\infty ||\hat \xi||_{W^{1,p}} + ||\xi||_{W^{1,p}} ||\hat \xi||_\infty)
\end{multline*}
}
\begin{multline*}
||DF^{(a,b) +}_{(\xi, h)} (\hat \xi, \hat h) - DF^{(a,b) +}_{(0,0)} (\hat \xi, \hat h)||_{L^p
(\Sigma_0 \setminus N_0)}\\
\leq C (||\xi||_\infty (||\hat \xi||_{W^{1,p}} + |\hat h|_{E^0})
+ (||\xi||_{W^{1,p}} + |h|_{E^0}) ||\hat \xi||_\infty)
\end{multline*}
\end{lem}
Therefore by the inverse function theorem, there exists some $\epsilon>0$ and $C>0$
such that for any $(a,b) \in \mathring{X}$ sufficiently close to $(0,b^0)$,
there exists a smooth map
\[
\phi^{a,b} : \mathop{\mathrm{Ker}}\nolimits DF^{(0, b^0)}_{(0, 0)} \supset B_\epsilon(0) \to B_C(0) \subset
\widetilde{W}^{1,p}_\delta (\widetilde{P}_a, u_{a, b}^\ast T \hat Y) \oplus E^0
\]
such that for any $(\xi, h) \in B_C(0)$ and $x \in B_\epsilon(0)$,
\begin{equation}
F^{(a,b) +}(\xi, h) = (0, x) \text{ if and only if } (\xi, h) = \phi^{a,b}(x).
\label{def of phi}
\end{equation}
Note that $\mathop{\mathrm{Ker}}\nolimits DF^{(0, b^0)}_{(0, 0)}$ does not depend on $p$ or $\delta$.
Although $\epsilon > 0$ may depend on $p$ and $\delta$ since so do the estimates,
$\phi^{a, b}$ does not depend on $p$ or $\delta$ on the intersection of the domains
since $\phi^{a, b}$ is defined by (\ref{def of phi}).
Shrinking $\mathring{X}$, we define $\hat V = \mathring{X} \times B_\epsilon(0)$ and
regard this space as a subspace of
\[
\bigcup_{(a, b) \in \mathring{X}} \{(a, b)\} \times
C^\infty(\widetilde{P}_a, (\overline{\mathbb{R}}_1 \cup
\overline{\mathbb{R}}_2 \cup \dots \cup \overline{\mathbb{R}}_k)
/\!\! \sim_{a, b} \! \times Y) \times E^0
\]
by
\[
(a, b, x) \mapsto (a, b, \Phi_{a, b}(\xi_x), h_x)
\]
where $(\xi_x, h_x) = (\xi_{(a, b, x)}, h_{(a, b, x)}) = \phi^{a, b}(x)$.
Define a map $s^0 : \hat V \to \mathbb{R}^k \oplus \bigoplus_{z_\beta^{++}} \mathbb{R}^2$ by
\begin{equation}
s^0(a, b, x) = (\sigma \circ \Phi_{a, b}(\xi_x)(\widetilde{R}_i(a)),
p' \circ \Phi_{a, b}(\xi_x)(Z_\beta^{++}(a))) \label{s^0}
\end{equation}
as we have already explained.
We will prove in Section \ref{smoothness} that
if we give a nice differentiable structure to the space $\mathring{X}$,
and give the product smooth structure to $\hat V = \mathring{X} \times B_\epsilon(0)$,
then
\begin{align*}
\hat V &\hookrightarrow \mathring{X} \times C^l(\Sigma_0 \setminus N_0,
(\mathbb{R}_1 \cup \mathbb{R}_2 \cup \dots \cup \mathbb{R}_k) \times Y) \times E^0\\
(a, b, x) &\mapsto (a, b, \Phi_{a, b}(\xi_x)|_{\Sigma_0 \setminus N_0}, h_x)
\end{align*}
is a smooth embedding for any $l \geq 1$.
In particular, $s^0$ is smooth.
Furthermore, the assumption of the surjectivity of $D_{p_0}^+$ implies that
$s^0$ is a submersion.
Define $V = \{s^0 = 0\} \subset \hat V$.
Then the map $s : V \to E = E^0 \oplus \bigoplus_{z_\alpha^+} \mathbb{R}^2$ defined by
\[
s(a, b, x) = (h, p \circ \pi_Y \circ \Phi_{a, b}(\xi_x)(Z_\alpha(a))
\]
is also smooth.
If we fix a family of coordinates $(\phi_{\pm\infty_i})$ of limit circles of $\Sigma_0$,
then the map
\begin{align*}
V &\to \prod_{z_i} Y \times \prod_{\pm\infty_i} P\\
(a, b, x) &\mapsto (\pi_Y \circ \Phi_{a, b}(\xi_x)(z_i), \pi_Y \circ
\Phi_{a, b}(\xi_x)|_{S^1_{\pm\infty_i}}
\circ \phi_{\pm\infty_i})
\end{align*}
is also smooth.
The assumption of the surjectivity of $D_{p_0}^+$ implies that this map is transverse
to the product directions of tangents of the $S^1$-actions on $P$,
that is, its differentials are transverse to $0 \oplus \bigoplus_{\pm\infty_i} \mathbb{R} R_\lambda
\subset \bigoplus_{z_i} TY \oplus \bigoplus_{\pm\infty_i} TS^1$.
It is easy to see that for any $(a, b, x) \in V$ and $\mu \in M_i$ such that
$\kappa_\mu = 0$, the asymptotic parameter $b_\mu$ and the map
$u = \Phi_{a, b}(\xi_x)$ satisfies equation (\ref{asymptotic parameter}).
Assuming the smoothness, we define an orbibundle $(\mathcal{V}, \mathcal{E})$ by
$\mathcal{V} = V / G_0$ and $\mathcal{E} = E/G_0$, where we regard $E$ as a trivial vector bundle
(with non-trivial $G_0$ action) on $V$.
We denote the induced smooth section $\mathcal{V} \to \mathcal{E}$ by $s$,
and define $\psi : \mathcal{V} \supset s^{-1}(0) \to \widehat{\mathcal{M}}^0$ by
$\psi(a, b, x) = (\widetilde{P}_a, Z(a), \Phi_{a, b}(\xi_x))$.
First we prove that $(\mathcal{V}, \mathcal{E}, s, \psi)$ is a Kuranishi neighborhood
of $p_0 \in \widehat{\mathcal{M}}(Y, \lambda, J)$ if $V$ is sufficiently small,
that is, we prove the following proposition.
\begin{prop}\label{psihomeo}
$\psi : \mathcal{V} \supset s^{-1}(0) \to \widehat{\mathcal{M}}$ is an homeomorphism onto
a neighborhood of $p_0 \in \widehat{\mathcal{M}}$ if $V$ is replaced by a small neighborhood
of $(0, b^0, 0)$.
\end{prop}
To prove this proposition,
first we prove a lemma about additional marked points.
To state the lemma, it is convenient to introduce the space
$\widehat{\mathcal{M}}' = \overline{\mathcal{M}} / \sim$.
This is the quotient space of $\overline{\mathcal{M}}$ obtained by ignoring the coordinates of
limit circles (but respecting the order of the limit circles and the
marked points).
Points of $\widehat{\mathcal{M}}'$ is written as $(\Sigma, z, u, \iota^\pm)$,
where we regard $z = (z_i)$ as a sequence, and $\iota^\pm$ are bijections from the
index set $\{\pm\infty_i\}$ to the set of $\pm\infty$-limit circles of $\Sigma$.
Let $p_0 = (\Sigma_0, z_0, u_0, \iota^\pm_0) \in \widehat{\mathcal{M}}'(Y, \lambda, J)$ be an
arbitrary curve.
Adding marked points to $p_0$,
we get a curve $p_0^+ = (\Sigma_0, z_0 \cup z_0^+, u_0, \iota^\pm_0)$
such that all nontrivial components of $(\Sigma, z_0 \cup z_0^+)$ are stable.
(Nontrivial components are the components which do not correspond to
the trivial cylinders in $p_0$.)
We assume that there exists a finite union of codimension-two submanifolds
$S \subset Y$ such that $\pi_Y \circ u_0$ intersects $S$ at $z_0^+$ transversely.
(We do not assume the transversality of the other intersections.)
We fix an order of $z_0^+ = (z^+_{0, i})_{i \in \Lambda}$ and regard it as a sequence of
additional marked points.
Assume that $G'_0 = \mathop{\mathrm{Aut}}\nolimits'(\Sigma_0, z_0, u_0, \iota^\pm_0)
= \{g \in \mathop{\mathrm{Aut}}\nolimits(\Sigma); u_0 \circ g = u_0, g(z_i) = z_i, g \iota^\pm_0 = \iota^\pm_0\}$
preserves $z_0^+$ as a set.
This implies $G'_0$ acts on the index set $\Lambda$ by
$z^+_{0, g \cdot i} = g^{-1} (z^+_{0, i})$.
Then the following lemma holds.
\begin{lem}\label{additional marked points and lifts}
If $U_0 \subset \widehat{\mathcal{M}}'$ is a sufficiently small neighborhood of $p_0$,
then there exists an open neighborhood $U_0^+ \subset \widehat{\mathcal{M}}'$ of
$p_0^+$ such that the following holds true.
For any $p = (\Sigma, z, u, \iota^\pm) \in U_0$, there exists a sequence of
additional marked points $z^+ = (z^+_i)_{i \in \Lambda} \subset \Sigma$ such that
$p^+ = (\Sigma, z \cup z^+, u, \iota^\pm) \in U_0^+$ and $\pi_Y \circ u(z^+) \subset S$.
Furthermore, for each $p \in U_0$, $G'_0$ acts on the set of such points $\{ p^+ \}$
transitively, where $G'_0$-action is defined by
\[
g \cdot (\Sigma, z \cup (z^+_i)_{i \in \Lambda}, u, \iota^\pm)
= (\Sigma, z \cup (z^+_{g^{-1} \cdot i})_{i \in \Lambda}, u, \iota^\pm).
\]
\end{lem}
We call each $p^+ \in U_0^+$ a lift of $p \in U_0$.
\begin{proof}
Since $\pi_Y \circ u$ intersects with $S$ at $z_i^+$ transversely,
the existence of such a sequence of additional marked points $z^+$ is clear
for each point $p$ in a neighborhood of $p_0$.
(We use the fact that if a $J$-holomorphic curve $v$ on a disc
$D = \{ z \in C; |z| \leq 1\}$ is sufficiently close to a given $J$-holomoprhic curve $u$
on $D$ in $L^\infty$-norm, then $v$ is close to $u$ in $C^\infty$-topology on
$\{ z\in \mathbb{C}; |z| \leq 1/2\}$.)
We need to prove that the $G'_0$-aciton on $\{p^+\}$ is transitive
for any point $p$ sufficiently close to $p_0$.
Suppose not.
Then there exists a sequence $p_k = (\Sigma_k, z_k, u_k, \iota^\pm_k) \in \widehat{\mathcal{M}}'$
converging to $p_0$ and sequences of additional marked points
$z_k^+ = (z_{k, i}^+)$ and ${z'}_k^+ = ({z'}_{k, i}^+) \subset \Sigma_k$ such that
$\pi_Y \circ u_k (z_{k, i}^+), \pi_Y \circ u_k ({z'}_{k, i}^+) \in S$ and
both of $p_k^+ = (\Sigma_k, z_k \cup z_k^+, u_k, \iota^\pm_k)$ and
${p'}_k^+ = (\Sigma_k, z_k \cup {z'}_k^+, u_k, \iota^\pm_k)$ converge to $p_0^+$,
but there is no $g \in G'_0$ such that ${z'}^+_{k, i} = z^+_{k, g \cdot i}$.
Take additional marked points $z_0^{++}$ of $(\Sigma_0, z_0 \cup z_0^+)$ to make
$(\Sigma_0, z_0 \cup z_0^+ \cup z_0^{++})$ stable, and
let $(\widetilde{P} \to \widetilde{X}, Z \cup Z^+ \cup Z^{++})$ be
the local universal family of $(\Sigma_0, z_0 \cup z_0^+ \cup z_0^{++})$.
Then by the definition of the topology,
there exist sequences $a_k, a'_k \in \widetilde{X}$ converging to $0$ and isomorphisms
\begin{align*}
\varphi_k &: (\widetilde{P}_{a_k}, Z(a_k) \cup Z^+(a_k)) \stackrel{\cong}{\to} (\Sigma_k, z_k \cup z_k^+),
\\
\varphi'_k &: (\widetilde{P}_{a'_k}, Z(a'_k) \cup Z^+(a'_k)) \stackrel{\cong}{\to} (\Sigma_k, z_k \cup
{z'}_k^+)
\end{align*}
which preserve the order of the limit circles,
and $\mathbb{R}$-gluings $\theta_k, \theta'_k : \overline{\mathbb{R}}_1 \sqcup \overline{\mathbb{R}}_2 \sqcup
\dots \sqcup \overline{\mathbb{R}}_l \to \overline{\mathbb{R}}_1 \cup \overline{\mathbb{R}}_2 \cup \dots
\cup \overline{\mathbb{R}}_{l_k}$ such that
\begin{align*}
\mathop{\mathrm{dist}}\nolimits_{L^\infty} &(u_k \circ \varphi_k, (\theta_k \times 1) \circ u_0 \circ
\Psi|_{\widetilde{P}_{a_k}}) \to 0,\\
\mathop{\mathrm{dist}}\nolimits_{L^\infty} &(u_k \circ \varphi'_k, (\theta'_k \times 1) \circ u_0 \circ
\Psi|_{\widetilde{P}_{a'_k}}) \to 0
\end{align*}
as $k \to \infty$.
Hence the biholomoprhisms $\phi_k = (\varphi'_k)^{-1} \circ \varphi_k
: \widetilde{P}_{a_k} \stackrel{\cong}{\to} \widetilde{P}_{a'_k}$ satisfy
\[
\mathop{\mathrm{dist}}\nolimits_{L^\infty}((\theta_k \times 1) \circ u_0 \circ \Psi|_{\widetilde{P}_{a_k}},
(\theta'_k \times 1) \circ u_0 \circ \Psi|_{\widetilde{P}_{a'_k}} \circ \phi'_k)
\to 0 \quad \text{as } k \to \infty.
\]
Note that $\phi_k(Z_i(a_k)) = Z_i(a'_k)$ for all $i$, which implies
$\Psi|_{\widetilde{P}_{a'_k}} \circ \phi_k \circ (\Psi|_{\widetilde{P}_{a_k}})^{-1}$
preserve marked points $Z(0)$.
Similarly, it preserves the order of the limit circles.
Hence a subsequence of
$\Psi|_{\widetilde{P}_{a'_k}} \circ \phi_k \circ (\Psi|_{\widetilde{P}_{a_k}})^{-1}$ converges to
a biholomorphism $g \in \mathop{\mathrm{Aut}}\nolimits' (\Sigma_0, z_0, u_0, \iota^\pm_0)$ on any compact subset
of the complement of nodal points and imaginary circles.
(First we see that
$\Psi|_{\widetilde{P}_{a'_k}} \circ \phi_k \circ (\Psi|_{\widetilde{P}_{a_k}})^{-1}$
converges to a biholomorphism on all nontrivial components, and then
we see the difference of $\theta_k$ and $\theta'_k$ goes to zero as $k \to \infty$,
which implies $\Psi|_{\widetilde{P}_{a'_k}} \circ \phi_k \circ (\Psi|_{\widetilde{P}_{a_k}})^{-1}$
converges to a biholomoprhism on trivial cylinders.)
Therefore
\[
\mathop{\mathrm{dist}}\nolimits (\Psi|_{\widetilde{P}_{a'_k}} \circ \phi_k \circ
(\Psi|_{\widetilde{P}_{a_k}})^{-1} (Z_i^+(0)), Z^+_{g^{-1} \cdot i}(0)) \to 0 \quad \text{as }
k \to \infty,
\]
which implies
\[
\mathop{\mathrm{dist}}\nolimits (\phi_k(Z_i^+(a_k)), Z_{g^{-1} \cdot i}^+(a'_k)) \to 0 \quad \text{as } k \to \infty.
\]
Since $\pi_Y \circ u_k \circ \varphi'_k = \pi_Y \circ u_k \circ \varphi_k \circ \phi_k^{-1}$
intersects transversely with $S$ at both of $\phi_k(Z_i^+(a_k))$ and
$Z_{g^{-1} \cdot i}^+ (a'_k)$,
it implies that $\phi_k(Z_i^+(a_k)) = Z_{g^{-1} \cdot i}^+(a'_k)$ for all large $k$.
Hence $z_{k, i}^+ = {z'}_{k, g^{-1} \cdot i}^+$, which is a contradiction.
\end{proof}
Next, for the proof of the continuity of $\psi$,
we consider the following approximate solutions centered at each point
$(a, b, x) \in \hat V$.
For $(a', b') \in \mathring{X}$ close to $(a, b)$,
we define $\xi_{(a, b, x)}^{(a', b')} \in
\widetilde{W}^{1, p}_\delta(\widetilde{P}_{a'}, u_{a', b'}^\ast T \hat Y)$ as follows.
We may assume that
$\rho'_\nu \neq 0$ for all $\nu$ such that $\rho_\nu \neq 0$, and that
$\rho'_\mu \neq 0$ for all $\mu$ such that $\rho_\mu \neq 0$.
Recall that $\phi^{a, b}(x) = (\xi_{(a, b, x)}, h_{(a, b, x)})$.
On $\Sigma_0 \setminus N_0$, we define
$\xi_{(a, b, x)}^{(a', b')}|_{\Sigma_0 \setminus N_0}
= \xi_{(a, b, x)}|_{\Sigma_0 \setminus N_0}$.
Similarly, we define $\xi_{(a, b, x)}^{(a', b')}|_{[0, \infty) \times S_{+\infty_i}^1}
= \xi_{(a, b, x)}|_{[0, \infty) \times S_{+\infty_i}^1}$ on $[0, \infty) \times S_{+\infty_i}^1$
and $\xi_{(a, b, x)}^{(a', b')}|_{(-\infty, 0] \times S_{-\infty_i}^1}
= \xi_{(a, b, x)}|_{(-\infty, 0] \times S_{-\infty_i}^1}$ on $(-\infty, 0] \times S_{-\infty_i}^1$.
On $[-1, 0] \times S_\mu^1 \subset[-1, -\frac{1}{2} \log \rho'_\mu] \times S_\mu^1$,
define $\xi_{(a, b, x)}^{(a', b')}(s, t) = \xi_{(a, b, x)}(s, t)$.
On $[0, -\frac{1}{2} \log \rho'_\mu] \times S_\mu^1$,
define $\xi_{(a, b, x)}^{(a', b')}(s', t') = \xi_{(a, b, x)}(s, t')$,
where $s$ is defined by
\begin{equation}
\frac{e^{- \kappa_\mu s'} - (\rho'_\mu)^{\kappa_\mu / 2}}
{1 - (\rho'_\mu)^{\kappa_\mu / 2}}
= \frac{e^{- \kappa_\mu s} - \rho_\mu^{\kappa_\mu / 2}}
{1 - \rho_\mu^{\kappa_\mu / 2}}.
\label{def of function s}
\end{equation}
We define similarly $\xi_{(a, b, x)}^{(a', b')}$ on
$[\frac{1}{2} \log \rho'_\mu, 1] \times S^1_\mu$.
For each $\nu$,
define $\rho_\nu$, $\varphi_\nu$, $\rho'_\nu$ and $\varphi'_\nu$ by
$\zeta_\nu = \rho_\nu^2 e^{2 \sqrt{-1} \varphi_\nu}$
and $\zeta'_\nu = (\rho'_\nu)^2 e^{2 \sqrt{-1} \varphi'_\nu}$.
On $N_{a', b'}^{\nu} = \{(z', w') \in D \times D; z'w' = \zeta'_\mu\}$,
define $\xi_{(a, b, x)}^{(a', b')}(z', w') = \xi_{(a, b, x)}(z, w)$,
where $(z, w) \in \{(z, w) \in D \times D; zw = \zeta_\mu\}$
is defined as follows.
If $|z'| \geq \rho'_\nu$ and $z' = r' e^{\sqrt{-1} \theta'}$
then $z = r e^{\sqrt{-1} \theta}$,
and if $|w'| \geq \rho'_\nu$ and $w' = r' e^{\sqrt{-1} \theta'}$
then $w = r e^{\sqrt{-1} \theta}$,
where $r$ and $\theta$ are defined by
\[
\frac{r' - \rho'_\nu}{1 - \rho'_\nu} = \frac{r - \rho_\nu}{1 - \rho_\nu},
\quad \theta' = \theta + (\varphi'_\nu - \varphi_\nu) \beta_\nu(r),
\quad \beta_\nu(r) = \max\Bigl( \frac{2 \rho_\nu - r}{\rho_\nu}, 0\Bigr).
\]
(If $\rho_\nu = 0$, we define $\beta_\nu = 0$.)
Then $(\xi_{(a, b, x)}^{(a', b')}, h_{(a, b, x)})$ satisfies the following estimates.
\begin{lem}\label{estimates for continuity}
For any $0 < \delta < \delta'_0 < \delta_0$, $p > 2$ and $(a, b, x) \in \hat V$,
there exists a constant $C>0$ such that
for any $(a',b') \in \mathring{X}$ sufficiently close to $(a, b)$,
\begin{align*}
&||F^{(a', b')}(\xi_{(a, b, x)}^{(a', b')}, h_{(a, b, x)})|_{[-1, 0] \times S_\mu^1}||_{L^p}
\leq C (|\varphi'_\mu - \varphi_\mu| + |b'_\mu - b_\mu|), \\
&||F^{(a', b')}(\xi_{(a, b, x)}^{(a', b')}, h_{(a, b, x)})|_{[0, -\frac{1}{2} \log \rho'_\mu]
\times S_\mu^1}||_{L^p_{\delta_\mu}}\\
&\hspace{90pt} \leq
\begin{cases}
C (\rho'_\mu)^{\min (\kappa_\mu, \delta'_{0, \mu} - \delta_\mu)/2}
(-\log \rho'_\mu)^{1/p}, & \text{if } \rho_\mu = 0\\
C |\rho'_\mu - \rho_\mu| & \text{if } \rho_\mu > 0
\end{cases},\\
&||F^{(a', b')}(\xi_{(a, b, x)}^{(a', b')}, h_{(a, b, x)})|_{\{(z, w) \in N^\nu_{(a', b')};
|z| \geq \rho'_\nu \}}||_{L^p} \leq C |\zeta'_\nu - \zeta_\nu|^{1/p}.
\end{align*}
\end{lem}
\begin{proof}
The estimate of $F^{(a', b')}(\xi_{(a, b, x)}^{(a', b')}, h_{(a, b, x)})|_{[-1, 0] \times S_\mu^1}$
is similar to Lemma \ref{estimates of F(0)}.
Since $F^{(a, b)}(\xi_{(a, b, x)}, h_{(a, b, x)})|_{[-1, 0] \times S_\mu^1} = 0$,
\begin{align*}
&F^{(a', b')}(\xi_{(a, b, x)}^{(a', b')}, h_{(a, b, x)})|_{[-1, 0] \times S_\mu^1}\\
&= F^{(a', b')}(\xi_{(a, b, x)}^{(a', b')}, h_{(a, b, x)})|_{[-1, 0] \times S_\mu^1}
- F^{(a, b)}(\xi_{(a, b, x)}, h_{(a, b, x)})|_{[-1, 0] \times S_\mu^1}\\
&= \frac{1}{2} (b'_\mu -b_\mu) \chi'(s) \partial_\sigma\\
&\quad + \frac{1}{2} (\varphi'_\mu - \varphi_\mu) \chi'(s)
(g^\mu_t(v_0^{\mu, \mathrm{left}} + \xi_{(a, b, x)})
+ \partial_t(v_0^{\mu, \mathrm{left}} + \xi_{(a, b, x)})).
\end{align*}
The first inequality follows from this equation.
Next we consider the second inequality.
If $\rho_\mu = 0$, then the proof is similar to Lemma \ref{estimates of F(0)}.
(The proof is obtained by
replacing $v_0^{\nu, \text{left}}$ and $\rho_\mu$ in the proof of
Lemma \ref{estimates of F(0)} with $v_0^{\nu, \text{left}}$ and $\rho'_\mu$
respectively.)
Consider the case of $\rho_\mu > 0$.
We abbreviate the subscript $\mu$.
First note that
\begin{align}
&F^{(a', b')} (\xi_{(a, b, x)}^{(a', b')}, h_{(a, b, x)})|_{[0, -\frac{1}{2} \log \rho'] \times S^1}
(s', t) \notag\\
&= \biggl(1 + \biggl(\frac{1 - \rho^{\kappa/2}}
{1 - (\rho')^{\kappa/2}}
(\rho')^{\kappa/2} - \rho^{\kappa/2}\biggr)
e^{\kappa s}\biggr)
\partial_s \bigl(v_{a,b}^{\mu, \mathrm{left}} + \xi_{(a, b, x)}\bigr)(s, t) \notag\\
&\quad + \widetilde{J}^\mu_t\bigl(
\bigl(v_{a,b}^{\mu, \mathrm{left}} + \xi_{(a, b, x)}\bigr)(s, t)\bigr)
\partial_t \bigl(v_{a,b}^{\mu, \mathrm{left}} + \xi_{(a, b, x)}\bigr)(s, t) \notag\\
&\quad + f^\mu_t \circ \pi_Y\bigl(
\bigl(v_{a,b}^{\mu, \mathrm{left}} + \xi_{(a, b, x)}\bigr)(s, t)\bigr),
\label{F' joint circle}
\end{align}
where $s$ is a function of $s'$ defined by (\ref{def of function s}).
Subtracting the equation
\begin{align*}
0 &= F^{(a, b)} (\xi_{(a, b, x)}, h_{(a, b, x)})|_{[0, -\frac{1}{2} \log \rho] \times S^1}(s,t)\\
&= \partial_s \bigl(v_{a,b}^{\mu, \mathrm{left}} + \xi_{(a, b, x)}\bigr)(s, t)\\
&\quad + \widetilde{J}^\mu_t\bigl(
\bigl(v_{a,b}^{\mu, \mathrm{left}} + \xi_{(a, b, x)}\bigr)(s, t)\bigr)
\partial_t \bigl(v_{a,b}^{\mu, \mathrm{left}} + \xi_{(a, b, x)}\bigr)(s, t) \\
&\quad + f^\mu_t \circ \pi_Y\bigl(
\bigl(v_{a,b}^{\mu, \mathrm{left}} + \xi_{(a, b, x)}\bigr)(s, t)\bigr)
\end{align*}
from (\ref{F' joint circle}), we obtain
\begin{align*}
&F^{(a', b')} (\xi_{(a, b, x)}^{(a', b')}, h_{(a, b, x)})|_{[0, -\frac{1}{2} \log \rho'] \times S^1}
(s', t) \\
&= \biggl(\frac{1 - \rho^{\kappa/2}}
{1 - (\rho')^{\kappa/2}}
(\rho')^{\kappa/2} - \rho^{\kappa/2}\biggr) e^{\kappa s}
\partial_s \bigl(v_{a,b}^{\mu, \mathrm{left}} + \xi_{(a, b, x)}\bigr)(s, t).
\end{align*}
Since $e^{\kappa s} \partial_s \bigl(v_{a,b}^{\mu, \mathrm{left}} + \xi_{(a, b, x)}\bigr)$
is a bounded function,
\[
||F^{(a', b')}(\xi_{(a, b, x)}^{(a', b')}, h_{(a, b, x)})|_{[0, -\frac{1}{2} \log \rho'_\mu]
\times S_\mu^1}||_{L^p_{\delta_\mu}} \lesssim |\rho' - \rho|,
\]
which prove the second inequality.
Finally, we consider the third inequality.
If $\rho_\nu = 0$, then the proof is similar to Lemma \ref{estimates of F(0)}.
We consider the case $\rho_\nu \neq 0$.
First note that
\begin{align}
&F^{(a', b')} (\xi_{(a, b, x)}^{(a', b')}, h_{(a, b, x)})|_{\{z \in D; |z| \geq \rho_\nu\}}
(r' e^{\sqrt{-1} \theta'}) \notag\\
&= \frac{1 - \rho_\nu}{1 - \rho'_\nu}
\Bigl((\partial_r v_{a, b}^\nu)(r e^{\sqrt{-1} \theta'})
+ (\partial_r \xi_{(a, b, x)})(r e^{\sqrt{-1} \theta}) \notag\\
& \quad \hphantom{\frac{1 - \rho_\nu}{1 - \rho'_\nu} \Bigl(}
+ \frac{\varphi'_\nu - \varphi_\nu}{\rho_\nu}
1_{[\rho_\nu, 2 \rho_\nu]}(r) (\partial_\theta \xi_{(a, b, x)})(r e^{\sqrt{-1} \theta})\Bigr)
\notag\\
&\quad
+ \widetilde{J}^\nu
\bigl(v_{a, b}^\nu(r e^{\sqrt{-1} \theta'}) + \xi_{(a, b, x)}(r e^{\sqrt{-1} \theta})\bigr)
\notag\\
&\quad \quad \cdot
\frac{(\partial_\theta v_{a, b}^\nu)(r e^{\sqrt{-1} \theta'})
+ (\partial_\theta \xi_{(a, b, x)})(r e^{\sqrt{-1} \theta})}{r'}.
\label{F' nodal point}
\end{align}
Subtracting
\begin{align*}
0 &= F^{(a, b)}(\xi_{(a, b, x)}, h_{(a, b, x)})|_{\{z \in D; |z| \geq \rho_\nu\}} \\
&= (\partial_r v_{a, b}^\nu)(r e^{\sqrt{-1} \theta})
+ (\partial_r \xi_{(a, b, x)})(r e^{\sqrt{-1} \theta}) \\
&\quad + \widetilde{J}^\nu
\bigl(v_{a, b}^\nu(r e^{\sqrt{-1} \theta}) + \xi_{(a, b, x)}(r e^{\sqrt{-1} \theta})\bigr)
\frac{(\partial_\theta v_{a, b}^\nu)(r e^{\sqrt{-1} \theta})
+ (\partial_\theta \xi_{(a, b, x)})(r e^{\sqrt{-1} \theta})}{r}
\end{align*}
from (\ref{F' nodal point}), we obtain
\begin{align*}
&F^{(a', b')} (\xi_{(a, b, x)}^{(a', b')}, h_{(a, b, x)})|_{\{z \in D; |z| \geq \rho_\nu\}}
(r' e^{\sqrt{-1} \theta'}) \\
&= ((\partial_r v_{a, b}^\nu)(r e^{\sqrt{-1} \theta'})
- (\partial_r v_{a, b}^\nu)(r e^{\sqrt{-1} \theta})) \\
&\quad + \frac{\rho'_\nu - \rho_\nu}{1 - \rho'_\nu}
\Bigl((\partial_r v_{a, b}^\nu)(r e^{\sqrt{-1} \theta'})
+ (\partial_r \xi_{(a, b, x)})(r e^{\sqrt{-1} \theta})\Bigr) \\
&\quad + \frac{1 - \rho_\nu}{1 - \rho'_\nu}
\cdot \frac{\varphi'_\nu - \varphi_\nu}{\rho_\nu}
1_{[\rho_\nu, 2 \rho_\nu]}(r) (\partial_\theta \xi_{(a, b, x)})(r e^{\sqrt{-1} \theta}) \\
&\quad +
\bigl(\widetilde{J}^\nu
\bigl(v_{a, b}^\nu(r e^{\sqrt{-1} \theta'}) + \xi_{(a, b, x)}(r e^{\sqrt{-1} \theta})\bigr)
- \widetilde{J}^\nu
\bigl(v_{a, b}^\nu(r e^{\sqrt{-1} \theta}) + \xi_{(a, b, x)}(r e^{\sqrt{-1} \theta})\bigr)
\bigr) \\
&\quad \quad \cdot
\frac{(\partial_\theta v_{a, b}^\nu)(r e^{\sqrt{-1} \theta'})
+ (\partial_\theta \xi_{(a, b, x)})(r e^{\sqrt{-1} \theta})}{r'} \\
&\quad + \widetilde{J}^\nu
\bigl(v_{a, b}^\nu(r e^{\sqrt{-1} \theta}) + \xi_{(a, b, x)}(r e^{\sqrt{-1} \theta})\bigr) \\
&\quad \quad \cdot
\frac{(\partial_\theta v_{a, b}^\nu)(r e^{\sqrt{-1} \theta'})
- (\partial_\theta v_{a, b}^\nu)(r e^{\sqrt{-1} \theta})}{r'} \\
&\quad + \Bigl( \frac{1}{r'} - \frac{1}{r} \Bigr)
\widetilde{J}^\nu
\bigl(v_{a, b}^\nu(r e^{\sqrt{-1} \theta}) + \xi_{(a, b, x)}(r e^{\sqrt{-1} \theta})\bigr) \\
&\quad \quad \cdot
\bigl((\partial_\theta v_{a, b}^\nu)(r e^{\sqrt{-1} \theta})
+ (\partial_\theta \xi_{(a, b, x)})(r e^{\sqrt{-1} \theta})\bigr).
\end{align*}
Hence it is easy to check that
\[
||F^{(a', b')} (\xi_{(a, b, x)}^{(a', b')}, h_{(a, b, x)})|_{\{z \in D; |z| \geq \rho_\nu\}}||_{L^p}
\lesssim |\rho'_\nu - \rho_\nu| + |\varphi'_\nu - \varphi_\nu|,
\]
and this inequality implies the claim.
\end{proof}
Note that the $\mathop{\mathrm{Ker}}\nolimits DF^{(0, b^0)}_{(0, 0)}$-factor of
$F^{(a', b') +}(\xi_{(a, b, x)}^{(a', b')}, h_{(a, b, x)})$ coincides with that of
$F^{(a, b) +}(\xi_{(a, b, x)}, h_{(a, b, x)})$.
Hence the above lemma implies that
\[
\bigl|\bigl|F^{(a', b') +}\bigl(\xi_{(a, b, x)}^{(a', b')}, h_{(a, b, x)}\bigr) - (0, x')\bigr|\bigr|
_{L^p_\delta \oplus \mathop{\mathrm{Ker}}\nolimits DF^{(0, b^0)}_{(0, 0)}} \to 0
\]
as $(a', b', x') \to (a, b, x)$.
Therefore
\begin{equation}
\bigl|\bigl| \phi^{(a', b')}(x')- \bigl(\xi_{(a, b, x)}^{(a', b')}, h_{(a, b, x)}\bigr)\bigr|\bigr|
_{\widetilde{W}^{1, p}_\delta(\widetilde{P}_{a'}, u_{a', b'}^\ast T \hat Y) \oplus E^0}
\to 0
\label{continuity of phi}
\end{equation}
as $(a', b', x') \to (a, b, x)$.
This implies the continuity of $\psi : \mathcal{V} \supset s^{-1}(0) \to
\widehat{\mathcal{M}}$.
Now we prove Proposition \ref{psihomeo}.
\begin{proof}[Proof of Proposition \ref{psihomeo}]
We have just proved the continuity of $\psi$.
Next we prove the injectivity.
Assume that the image of two points $(a, b, x), (a', b', x') \in \{ s = 0 \}$
($ \subset V$) coincide,
that is, the two holomorphic buildings $(\widetilde{P}_a, Z(a), \Phi_{a, b}(\xi_x))$
and $(\widetilde{P}_{a'}, Z(a'), \Phi_{a', b'}(\xi_{x'}))$ are the same point in $\widehat{\mathcal{M}}$.
We prove that these two points coincide in $V / G_p$.
Since $\widehat{\mathcal{M}}$ is a quotient space of $\widehat{\mathcal{M}}'$,
we may assume that these two holomorphic buildings also coincide in $\widehat{\mathcal{M}}'$
by replacing $(a', b', x')$ with $h \cdot (a', b', x')$ for some $h \in G_0$.
If $V$ is sufficiently small, then Lemma \ref{additional marked points and lifts} implies
that there exist an isomorphism
$\varphi : (\widetilde{P}_a, Z(a)) \stackrel{\cong}{\to} (\widetilde{P}_{a'}, Z(a'))$,
an $\mathbb{R}$-translation $\theta$, and $g \in G'_0$ such that
$\Phi_{a', b'}(\xi_{x'}) \circ \varphi = (\theta \times 1) \circ \Phi_{a, b}(\xi_x)$,
$\varphi \circ \iota^\pm = \iota^\pm$ and $\varphi(Z^+_i(a)) = Z^+_{g^{-1} \cdot i}(a')$.
Hence the isomorphism $(\hat P_a, Z(a)) \stackrel{\cong}{\to} (\hat P_{a'}, Z(a'))$
induced by $\varphi$ coincides with the restriction of $g : \hat P \to \hat P$.
Therefore $\varphi$ preserves $\widetilde{R}_i$ as a family,
which implies $\theta = \mathrm{id}$.
From this, we can see that $\varphi$ maps $Z^{++}_i(a)$ to $Z^{++}_{g^{-1} \cdot i}(a')$
because these points are contained in the inverse image of $S'$ by
$\Phi_{a, b}(\xi_x)$ and $\Phi_{a', b'}(\xi_{x'})$ respectively.
Hence $\varphi : (\widetilde{P}_a, Z(a)) \stackrel{\cong}{\to} (\widetilde{P}_{a'}, Z(a'))$ coincides with
the restriction of $g : \widetilde{P} \to \widetilde{P}$,
which implies $(a', b', x') = g \cdot (a, b, x)$.
Therefore $\psi : \{ s = 0 \}/G_0 \to \widehat{\mathcal{M}}$ is injective.
Finally we prove that the image of $\psi$ contains a neighborhood of $p$.
Assume contrary, that is, assume that there exists a sequence
$(\Sigma_k, z_k, u_k) \in \widehat{\mathcal{M}} \setminus \psi(\{ s = 0 \} /G_p)$ convergent to
$p_0 = (\Sigma_0, z, u_0)$.
We may assume that $(\Sigma_k, z_k, u_k, \iota^\pm_k) \in \widehat{\mathcal{M}}'$ converges to
$p_0 = (\Sigma, z, u_0, \iota^\pm_0) \in \widehat{\mathcal{M}}'$.
Let $(\Sigma_k, z_k \cup z_k^+, u_k)$ be the lift of $(\Sigma_k, z_k, u_k)$ for each $k$.
Then there exist a sequence $a_k \to 0 \in \widetilde{X}$, biholomorphisms
$(\Sigma_k, z_k \cup z_k^+) \cong (\widetilde{P}_{a_k}, Z(a_k) \cup Z^+(a_k))$
and $\mathbb{R}$-gluings $\theta_k$ such that
\[
\mathop{\mathrm{dist}}\nolimits_{L^\infty} (u_k, (\theta_k \times 1) \circ u_0 \circ \Psi|_{\widetilde{P}_{a_k}}) \to 0
\quad \text{as } k \to \infty.
\]
We may assume that $\theta_k$ maps $0 \in \mathbb{R}_i$ to
$\sigma \circ u_k(\widetilde{R}_i(a_k))$.
Let $z^{++}_k \subset \Sigma_k$ be the points corresponding to $Z^{++}(a_k) \subset
\widetilde{P}_{a_k}$. Changing $a_k$ slightly if necessary,
we may assume $u_k(z^{++}_k) \subset (\theta_k \times 1)(S')$.
Define $(b^k_\mu) \in \prod_{i = 1}^{k - 1} \bigoplus_{\mu \in M_i} \mathbb{R}$ as follows:
\begin{itemize}
\item
For each $\mu$ such that $\rho^k_\mu = 0$,
we define $b_\mu^{k, \mathrm{left}}$ and $b_\mu^{k, \mathrm{right}}$, and $b^k_\mu$ by
\begin{align*}
\sigma \circ u_k|_{[-1,\infty) \times S^1_\mu}(s_x,t_x)
&= \theta_k(0_i) + L_\mu s_x + b_\mu^{k, \mathrm{left}} + O(1)\\
\sigma \circ u_k|_{(-\infty,+1] \times S^1_\mu}(s_y,t_y)
&= \theta_k(0_{i + 1}) + L_\mu s_y + b_\mu^{k, \mathrm{right}} + O(1)\\
b^k_\mu = b_\mu^{k, \mathrm{left}} - b_\mu^{k, \mathrm{right}}.
\end{align*}
\item
For each $\mu$ such that $\rho^k_\mu \neq 0$,
we define $b^k_\mu$ by
\[
\sigma \circ u_k(\widetilde{R}_{i + 1}(a_k)) - \sigma u_k(\widetilde{R}_i(a_k))
= -L_\mu \log \rho_\mu + b^k_\mu.
\]
\end{itemize}
Then $b^k_\mu \to b^0_\mu$ as $k \to \infty$.
(Note that in the former case, the asymptotic estimates of the term
$O(1)$ is uniform with respect to $k$.)
Replacing each map $u_k$ with its appropriate $\mathbb{R}$-translation,
we may assume $\mathop{\mathrm{dist}}\nolimits_{L^\infty} (u_k, u_{a_k, b_k}) \to 0$.
Then there exists a section $\xi_k$ of $u_{a_k, b_k}^\ast T \hat Y$ for each $k$
such that $||\xi_k||_\infty \to 0$ as $k \to \infty$ and $u_k = \Phi_{a_k, b_k}(\xi_k)$.
To prove $|| \xi_k ||_{\widetilde{W}^{1,p}_\delta} \to 0$, we consider the following equations.
\begin{align*}
F^{(a_k, b^k)+} (\xi_k, 0) =& F^{(a_k, b^k)+}(0, 0) + DF^{(a_k, b^k)+}_{(0, 0)}(\xi_k, 0)\\
&+ \int_0^1\bigl( DF^{(a_k, b^k)+}_{(\lambda \xi_k, 0)} - DF^{(a_k, b^k)+}_{(0, 0)} \bigr)
(\xi_k, 0) d\lambda
\end{align*}
In the above equations,
\begin{align*}
&||F^{(a_k, b^k)+} (\xi_k, 0)||_{L_\delta^p \oplus \mathop{\mathrm{Ker}}\nolimits DF^{(0, b^0)}_{(0, 0)}}
= ||F^{(a_k, b^k)+} (\xi_k, 0)||_{\mathop{\mathrm{Ker}}\nolimits DF^{(0, b^0)}_{(0, 0)}} \to 0,\\
&||F^{(a_k, b^k)+} (0, 0)||_{L_\delta^p \oplus \mathop{\mathrm{Ker}}\nolimits DF^{(0, b^0)}_{(0, 0)}}
\lesssim |a_k| + |b^k - b^0| \to 0,\\
&||DF^{(a_k, b^k)+}_{(0, 0)} (\xi_k, 0)||_{L_\delta^p \oplus \mathop{\mathrm{Ker}}\nolimits DF^{(0, b^0)}_{(0, 0)}}
\geq \epsilon ||\xi_k||_{\widetilde{W}^{1,p}_\delta} \quad \quad \text{for some } \epsilon>0,
\\
&\Bigl|\Bigl|\int_0^1\bigl( DF^{(a_k, b^k)+}_{(\lambda \xi_k, 0)}
- DF^{(a_k, b^k)+}_{(0, 0)} \bigr)
(\xi_k, 0) d\lambda\Bigr|\Bigr|_{L_\delta^p \oplus \mathop{\mathrm{Ker}}\nolimits DF^{(0, b^0)}_{(0, 0)}}
\leq ||\xi_k||_\infty ||\xi_k||_{\widetilde{W}^{1,p}_\delta}
\end{align*}
by Lemma \ref{D^2F}.
These imply $|| \xi_k ||_{\widetilde{W}^{1,p}_\delta} \to 0$.
Hence $(\Sigma_k, z_k, u_k)$ is contained in the image of $\psi$ for large $k$,
which is a contradiction.
Therefore the image of $\psi$ contains a neighborhood of $p$.
Since $\{s=0 \} /G_0$ is locally compact and $\widehat{\mathcal{M}}$ is Hausdorff,
$\psi$ is a homeomorphism onto a neighborhood of $p_0$.
\end{proof}
Therefore $(\mathcal{V}, \mathcal{E}, s, \psi)$ is a Kuranishi neighborhood of $p_0$.
In this paper, we sometimes denote the Kuranishi neighborhood by the $5$-tuple
$(V, E, s, \psi, G_0)$.
Sometimes we write a point of $V$ as a $4$-tuple $(\Sigma, z, u, h)$ consisting of
a curve $\Sigma$, its marked points $z$, a map $u$ and a vector $h \in E^0$
which satisfy the equation $du + J(u)du + \lambda(h) = 0$.
\subsection{Linearized gluing lemma}\label{linearized gluing}
In this section, we prove the linearized gluing lemma
(Lemma \ref{linearized gluing lemma} below), which was
used in the previous section to prove the invertibility of $DF^{(a, b) +}_{(0, 0)}$.
Let $\Sigma$ be a domain curve of a holomorphic building,
and let $E \to \Sigma$ be a complex vector bundle of rank $n$.
Assume that on a neighborhood $N_0 \subset \Sigma$ of nodal points and
imaginary circles,
a trivialization $E|_{N_0} \cong N_0 \times \mathbb{C}^n$ is given.
$N_0$ is the union of $D \cup D$, $([0, \infty] \cup [-\infty, 0]) \times S^1$,
$[0, \infty] \times S^1$ and $[-\infty, 0] \times S^1$.
Assume that an elliptic operator $D_0$ on $E$ has the same symbol as $\overline{\partial}$,
and on the neighborhood $([0, \infty] \cup [-\infty, 0]) \times S^1$ of
each joint circle $S_\mu^1$, $D_0$ has the form
\[
D_0 \xi = \partial_s \xi + J_0 \partial_t \xi + S_\mu(s, t) \xi,
\]
where $S_\mu(s, t) : ([0, \infty] \cup [-\infty, 0]) \times S^1 \to \mathop{\mathrm{gl}}\nolimits(2n, \mathbb{R})$ is
a continuous matrix-valued function such that
$S_\mu(t) : = S_\mu(\pm\infty, t) : S^1 \to \mathop{\mathrm{gl}}\nolimits(2n, \mathbb{R})$ is a loop of symmetric matrices.
Also on the neighborhood $[0, \infty] \times S_{+\infty_i}^1$ or
$[-\infty, 0] \times S_{-\infty_i}^1$ of each limit circle $S_{\pm\infty_i}^1$,
$D_0$ has the form
\[
D_0 \xi = \partial_s \xi + J_0 \partial_t \xi + S_{\pm\infty_i}(s, t) \xi,
\]
where $S_{\pm\infty_i}$ are continuous matrix-valued functions on
$[0, \infty] \times S_{+\infty_i}^1$ or $[-\infty, 0] \times S_{-\infty_i}^1$ such that
$S_{\pm\infty_i}(t) := S_{\pm\infty_i}(\pm\infty, t) : S^1 \to \mathop{\mathrm{gl}}\nolimits(2n, \mathbb{R})$ are loop of
symmetric matrices.
We further assume that there exist a family of positive constants
$\delta_1 = ((\delta_{1, \mu})_\mu, (\delta_{1, \pm\infty_i})_{\pm\infty_i})$
and a constant $C > 0$ such that
\begin{align*}
|S_\mu(s, t) - S_\mu(t)| &\leq C e^{-\delta_{1, \mu} |s|}
\text{ for } s \in [0, \infty] \cup [-\infty, 0]\\
|S_{\pm\infty_i}(s, t) - S_{\pm\infty_i}(t)| &\leq C e^{-\delta_{1, \pm\infty_i} |s|}
\text{ for } s \in [0, \infty] \text{ (or } s\in [-\infty ,0])
\end{align*}
Let $\delta_0 = ((\delta_{0, \mu})_\mu, (\delta_{0, \pm\infty_i})_{\pm\infty_i})$
be the family of positive constants consisting of the minimal
non-zero absolute values of eigenvalues of
\[
A_\mu = J_0 \partial_t + S_\mu(t) : W^{1,2}(S^1, \mathbb{R}^{2n}) \to L^2(S^1,\mathbb{R}^{2n})
\]
and
\[
A_{\pm\infty_i} = J_0 \partial_t + S_{\pm\infty_i}(t) : W^{1,2}(S^1, \mathbb{R}^{2n}) \to
L^2(S^1,\mathbb{R}^{2n}).
\]
Let $\delta = ((\delta_\mu)_\mu, (\delta_{\pm\infty_i})_{\pm\infty_i})$ be an arbitrary
sequence of constants such that $\delta < \delta_0$ and $\delta < \delta_1$,
and let $2 < p < \infty$ be an arbitrary constant.
We define the $L_\delta^p$-norm on $[0, \infty] \times S^1$ or
$[-\infty, 0] \times S^1$ by $||\xi||_{L_\delta^p} = ||e^{\delta |s|} \xi||_{L^p}$,
using the usual Lebesgue measures of
$[0, \infty) \times S^1$ or $(-\infty, 0] \times S^1$.
Using the trivialization of $E|_{N_0}$, we define the $L_\delta^p$-space by
\begin{align*}
L_\delta^p(\Sigma, {\textstyle\bigwedge}^{0, 1}T^\ast \Sigma \otimes E)
&= L_\delta^p(\Sigma_0 \setminus N_0, {\textstyle\bigwedge}^{0, 1}T^\ast \Sigma \otimes E)\\
&\quad \oplus \bigoplus_\nu L^p(D \cup D ,\mathbb{C}^n)\\
&\quad \oplus
\bigoplus_\mu L_{\delta_\mu}^p(([0, \infty] \cup [-\infty, 0]) \times S^1, \mathbb{C}^n)\\
&\quad \oplus \bigoplus_{+\infty_i} L_{\delta_{+\infty_i}}^p([0, \infty] \times S^1, \mathbb{C}^n)\\
&\quad \oplus \bigoplus_{-\infty_i} L_{\delta_{-\infty_i}}^p([-\infty, 0] \times S^1, \mathbb{C}^n).
\end{align*}
We define a Banach space $\widetilde{W}_\delta^{1, p}(\Sigma, E)$ by
\begin{align*}
\widetilde{W}_\delta^{1, p}(\Sigma, E)
=&\ \{\xi = \xi_0 + \sum_\mu \beta_\mu v_\mu
+ \sum_{\pm\infty_i} \beta_{\pm\infty_i} v_{\pm\infty_i} \in C(\Sigma, E);\\
&\quad \quad \xi_0 \in W^{1,p}_\delta (\Sigma, E),
v_\mu \in \mathop{\mathrm{Ker}}\nolimits A_\mu, v_{\pm\infty_i} \in \mathop{\mathrm{Ker}}\nolimits A_{\pm\infty_i}\},
\end{align*}
where for each $\mu$, $\beta_\mu$ is a smooth function which is $1$
on some neighborhood of $\mu$-th joint circle and whose support is contained
in its slightly larger neighborhood,
and $\beta_{\pm\infty_i}$ is a similar function for each $\pm\infty_i$.
Each $v_\mu \in \mathop{\mathrm{Ker}}\nolimits A_\mu$ is regarded as a section
$v_\mu(s, t) = v_\mu(t) : ([0, \infty] \cup [-\infty, 0]) \times S^1 \to \mathbb{C}^n$,
and the meaning of the above $v_{\pm\infty_i}$ is similar for each $\pm\infty_i$.
Then we can regard $D_0$ as a linear operator
$D_0 : \widetilde{W}_\delta^{1, p}(\Sigma, E) \to
L_\delta^p(\Sigma,\allowbreak {\textstyle\bigwedge}^{0, 1}T^\ast \Sigma \otimes E)$.
For each $(\zeta, r) = (\zeta_\nu, r_\mu)
\in D^{l_0} \times (1, \infty]^{l_1}$,
a new curve $\Sigma_{(\zeta, r)}$ is constructed from $\Sigma$ by
replacing the neighborhood $D \cup D$ of the $\nu$-th nodal point with
\[
N^\nu_{\zeta_\nu} = \{(x, y) \in D \times D; xy = \zeta_\nu\},
\]
and replacing the neighborhood $D \widetilde{\cup} D$ of the $\mu$-th joint circle
with
\[
N^\mu_{r_\mu} = \{((s_x, t_x), (s_y, t_y)) \in [0, \infty] \times S^1 \times
[-\infty, 0] \times S^1;
s_y - s_x = -2r, t_y = t_x\}.
\]
$E$ induces a complex vector bundle on $\Sigma_{(\zeta, r)}$.
(We use the trivialization of $E|_{N_0}$.)
We also denote this vector bundle by $E$.
$L^p$-norm on $N^\nu_{\zeta_\nu}$ is defined by the measure
$\frac{\sqrt{-1}}{2} dx \wedge d\bar x$ on $\{(x, y) \in N^\nu_{\zeta_\nu}; |x| \geq |y|\}$
and the measure $\frac{\sqrt{-1}}{2} dy \wedge d\bar y$ on
$\{(x, y) \in N^\nu_{\zeta_\nu}; |y| \geq |x|\}$.
$L_\delta^p$-norm on $N^\mu_{r_\mu}$ is defined by
\[
||\xi||_{L_\delta^p}
= ||e^{\delta |s|} \xi||_{L^p([0, r_\mu] \times S^1)}
+ ||e^{\delta |s|} \xi||_{L^p([-r_\mu, 0] \times S^1)}.
\]
We define a Banach space $\widetilde{W}_\delta^{1, p}(\Sigma_{(\zeta, r)}, E)$ by
\begin{align*}
\widetilde{W}_\delta^{1, p}(\Sigma_{(\zeta, r)}, E)
=&\ \{\xi = \xi_0 + \sum_\mu \beta_\mu v_\mu
+ \sum_{\pm\infty_i} \beta_{\pm\infty_i} v_{\pm\infty_i} \in C(\Sigma, E);\\
&\quad \quad \xi_0 \in W^{1,p}_\delta (\Sigma_{(\zeta, r)}, E), v_\mu \in
\mathop{\mathrm{Ker}}\nolimits A_\mu,
v_{\pm\infty_i} \in \mathop{\mathrm{Ker}}\nolimits A_{\pm\infty_i}\},
\end{align*}
where $\beta_\mu$ and $\beta_{\pm\infty_i}$ are defined by regarding the curve
$\Sigma_{(\zeta, r)}$ as a curve constructed by patching the subsets
$\Sigma_0 \setminus N_0$, $\{x \in D; |x| \geq \sqrt{|\zeta_\nu|}\}$,
$\{y \in D; |y| \geq \sqrt{|\zeta_\nu|}\}$,
$[0, r_\mu] \times S_\mu^1$,
$[-r_\mu, 0] \times S_\mu^1$,
$[0, \infty] \times S_{+\infty_i}^1$ and $[-\infty, 0] \times S_{-\infty_i}^1$ of $\Sigma$.
The norm of $\widetilde{W}_\delta^{1, p}(\Sigma_{(\zeta, \kappa)}, E)$ is defined by
\begin{align*}
||\xi||_{\widetilde{W}_\delta^{1,p}(\Sigma_{(\zeta, \kappa)})}
= \inf\Bigl\{&\Bigl|\Bigl|\xi- \sum_\mu \beta_\mu v_\mu
- \sum_{\pm\infty_i} \beta_{\pm\infty_i} v_{\pm\infty_i}\Bigr|\Bigr|_{W^{1,p}_\delta}
+ \sum_\mu ||v_\mu||_{\mathop{\mathrm{Ker}}\nolimits A_\mu}\\
&+ \sum_{\pm\infty_i} ||v_{\pm\infty_i}||_{\mathop{\mathrm{Ker}}\nolimits A_{\pm\infty_i}};
v_\mu \in \mathop{\mathrm{Ker}}\nolimits A_\mu, v_{\pm\infty_i} \in \mathop{\mathrm{Ker}}\nolimits A_{\pm\infty_i} \Bigr\}.
\end{align*}
Regarding $\Sigma_{(\zeta, r)}$ as the curve constructed by patching
the subsets of $\Sigma$, we define the linear operator
$D_{(\zeta, r)} : \widetilde{W}_\delta^{1, p}(\Sigma_{(\zeta, r)}, E) \to
L_\delta^p(\Sigma_{(\zeta, r)}, {\textstyle\bigwedge}^{0, 1} T^\ast \Sigma_{(\zeta, r)}
\otimes E)$ from $D_0$.
(The coefficient of the operator is discontinuous in general.)
Let $\lambda : \mathbb{R}^N \to L_\delta^p(\Sigma, {\textstyle\bigwedge}^{0, 1}T^\ast \Sigma \otimes E)$
be a linear map which makes
\[
D_0 \oplus \lambda : \widetilde{W}_\delta^{1, p}(\Sigma, E) \oplus \mathbb{R}^N \to
L_\delta^p(\Sigma, {\textstyle\bigwedge}^{0, 1}T^\ast \Sigma \otimes E)
\]
surjective.
We assume the support of $\lambda$ is contained in $\Sigma_0 \setminus N_0$.
Then $\lambda$ induces a map
$\lambda_{(\zeta, r)} : \mathbb{R}^N \to L_\delta^p(\Sigma_{(\zeta, r)}, {\textstyle\bigwedge}^{0, 1}T^\ast \Sigma_{(\zeta, r)} \otimes E)$.
We prove the surjectivity of
\[
D_{(\zeta, r)} \oplus \lambda_{(\zeta, r)} : \widetilde{W}_\delta^{1, p}(\Sigma_{(\zeta, r)}, E)
\oplus \mathbb{R}^N \to L_\delta^p(\Sigma_{(\zeta, r)}, {\textstyle\bigwedge}^{0, 1}T^\ast \Sigma_{(\zeta, r)}
\otimes E)
\]
for sufficiently small $(\zeta, r^{-1})$.
Let $\{(\xi_k, h_k)\}$ be a orthonormal basis of $\mathop{\mathrm{Ker}}\nolimits (D_0 \oplus \lambda)$,
where the inner product of $\mathop{\mathrm{Ker}}\nolimits (D_0 \oplus \lambda)$ is defined by
\[
\langle (\xi, h), (\xi', h') \rangle
= \langle \xi, \xi' \rangle_{L^2(\Sigma_0 \setminus N_0)} +\langle h, h' \rangle_{\mathbb{R}^N}
\]
\begin{lem}\label{linearized gluing lemma}
There exists some constant $C>0$ such that for any sufficiently small $(\zeta, r^{-1})$,
\begin{align}
&||\xi||_{\widetilde{W}_\delta^{1,p}(\Sigma_{(\zeta, r)})} + |h|_{\mathbb{R}^N} \notag \\
&\ \leq C \bigl( ||D_{(\zeta, r)} \xi + \lambda_{(\zeta, r)} h||
_{L_\delta^p(\Sigma_{(\zeta, r)})}
+ \sum_k |\langle \xi,\xi_k \rangle_{L^2(\Sigma_0 \setminus N_0)}
+ \langle h,h_k \rangle_{\mathbb{R}^N} | \bigr)
\label{inequality of linearized gluing lemma}
\end{align}
\end{lem}
\begin{proof}
We may assume $D_0 = \overline{\partial}$ on some neighborhood of nodal points, and
$S_\mu(s, t) = s_\mu(t)$ for sufficiently large $|s|$ for all $\mu$ because
the Sobolev embedding $||\xi_0||_{L^\infty} \lesssim ||\xi_0||_{W^{1, p}}$ is uniform
with respect to small $(\zeta, r^{-1})$.
It is enough prove the inequality for
$\xi \in \widetilde{W}_\delta^{1, p}(\Sigma_{(\zeta, r)}, E) \cap
C^\infty(\Sigma_{(\zeta, r)}, E)$.
We construct a section $\tilde \xi \in \widetilde{W}_\delta^{1, p}(\Sigma, E)$
from $\xi$, and apply the inequality
\begin{align}
&||\tilde \xi||_{\widetilde{W}_\delta^{1,p}(\Sigma)} + |h|_{\mathbb{R}^N} \notag\\
&\ \leq C \bigl( ||D_0 \tilde \xi + \lambda h||_{L_\delta^p(\Sigma)}
+ \sum_k |\langle \tilde \xi,\xi_k \rangle_{L^2(\Sigma_0 \setminus N_0)}
+ \langle h,h_k \rangle_{\mathbb{R}^N}| \bigr)
\label{eq of tilde xi}
\end{align}
followed from the surjectivity of $D_0 \oplus \lambda$ to $(\tilde \xi, h)$.
From this inequality, we will derive the required inequality for $(\xi, h)$.
Define $\tilde \xi|_{\Sigma_0 \setminus N_0} = \xi|_{\Sigma_0 \setminus N_0}$.
We also define $\tilde \xi = \xi$ on the neighborhood of limit circles of $\Sigma$.
Next we consider the neighborhood of the $\nu$-th nodal point.
On $N^\nu_{\zeta_\nu}$, let
\begin{align*}
\xi|_{\{|x| = \sqrt{|\zeta_\nu|}\}}
&= \sum_k a^{(\nu)}_k x^k \in L^2(\{|x|=\sqrt{|\zeta_\nu|}\}, \mathbb{C}^n)\\
\xi|_{\{|y| = \sqrt{|\zeta_\nu|}\}}
&= \sum_k b^{(\nu)}_k y^k \in L^2(\{|y|=\sqrt{|\zeta_\nu|}\}, \mathbb{C}^n)
\end{align*}
be the Fourier expansions.
Note that $b_k^{(\nu)} = a_{-k}^{(\nu)} \zeta^{-k}$.
In particular, $a_0^{(\nu)} = b_0^{(\nu)}$.
Then $\tilde \xi|_{D \cup D}$ is defined by
\begin{align*}
\tilde \xi(x, 0) &= \begin{cases}
\xi(x) - \rho_{\zeta_\nu}(x) \sum_{k < 0} a_k^{(\nu)} x^k
& \text{for } \sqrt{|\zeta_\nu|} \leq |x| \leq 1\\
\sum_{k \geq 0} a_k^{(\nu)} x^k & \text{for } |x| \leq \sqrt{|\zeta_\nu|}
\end{cases}\\
\tilde \xi(0, y) &= \begin{cases}
\xi(y) - \rho_{\zeta_\nu}(y) \sum_{k < 0} b_k^{(\nu)} y^k
& \text{for } \sqrt{|\zeta_\nu|} \leq |y| \leq 1\\
\sum_{k \geq 0} b_k^{(\nu)} y^k & \text{for } |y| \leq \sqrt{|\zeta_\nu|}
\end{cases},
\end{align*}
where $\rho_\zeta$ is defined as follows.
Let $\rho : \mathbb{R}_{\geq 0} \to \mathbb{R}_{\geq 0}$ be a smooth function such that
$\rho|_{[0, 1]} = 1$ and $\mathop{\mathrm{supp}}\nolimits \rho \subset [0, 2]$, and fix a constant
$0 < \alpha < \frac{1}{2}$.
Then $\rho_\zeta$ is defined by $\rho_\zeta(z) = \rho(\frac{z}{|\zeta|^\alpha})$.
We note that
$\tilde \xi|_{D \cup D} \in W^{2, 2}(D \cup D) \subset W^{1, p}(D \cup D)$
because $\sum_k a^{(\nu)}_k x^k \in C^\infty(\{|x|=\sqrt{|\zeta_\nu|}\}, \mathbb{C}^n)
\subset W^{\frac{3}{2}, 2}(\{|x|=\sqrt{|\zeta_\nu|}\}, \mathbb{C}^n)$ and
$\sum_k b^{(\nu)}_k y^k \in C^\infty(\{|y|=\sqrt{|\zeta_\nu|}\}, \mathbb{C}^n)
\subset W^{\frac{3}{2}, 2}(\{|y|=\sqrt{|\zeta_\nu|}\}, \mathbb{C}^n)$.
Next we consider the neighborhood of the $\mu$-th joint circle.
On $N^\mu_{r_\mu}$, let
\begin{align*}
\xi|_{\{s = r_\mu\} \subset [0, r_\mu] \times S^1}
&= \sum_k a^{(\mu)}_k e^{-\lambda^{(\mu)}_k r_\mu} \phi^{(\mu)}_k(t)
\in L^2(S^1, \mathbb{R}^{2n})\\
\xi|_{\{s = -r_\mu\} \subset [-r_\mu, 0] \times S^1}
&= \sum_k b^{(\mu)}_k e^{\lambda^{(\mu)}_k r_\mu} \phi^{(\mu)}_k(t)
\in L^2(S^1, \mathbb{R}^{2n})
\end{align*}
be expansions by the eigenvectors $\phi^{(\mu)}_k$ of $A_\mu$,
where $\lambda^{(\mu)}_k$ is the eigenvalue corresponding to $\phi^{(\mu)}_k$.
Since $\{s = r_\mu\} \subset [0, r_\mu] \times S^1$ and
$\{s = -r_\mu\} \subset [-r_\mu, 0] \times S^1$ are the same circle,
$b^{(\mu)}_k = e^{-2 \lambda^{(\mu)}_k r_\mu} a^{(\mu)}_k$.
In particular, $b^{(\mu)}_k = a^{(\mu)}_k$ if $\lambda_k=0$.
Then $\tilde \xi|_{([0, \infty] \cup [-\infty, 0]) \times S_\mu^1}$ is defined by
\begin{align*}
\tilde \xi|_{[0, \infty] \times S^1} (s, t) &= \begin{cases}
\xi(s, t) - \chi_{r_\mu}(s) \sum_{\lambda_k^{(\mu)} < 0}
a_k^{(\mu)} e^{- \lambda_k^{(\mu)} s}
\phi_k^{(\mu)}(t) & 0 \leq s \leq r_\mu\\
\sum_{\lambda_k^{(\mu)} \geq 0} a_k^{(\mu)} e^{- \lambda_k^{(\mu)} s} \phi_k^{(\mu)}(t)
& r_\mu \leq s \leq \infty
\end{cases},\\
\tilde \xi|_{[-\infty, 0] \times S^1} (s, t) &= \begin{cases}
\sum_{\lambda_k^{(\mu)} \leq 0} b_k^{(\mu)} e^{- \lambda_k^{(\mu)} s} \phi_k^{(\mu)}(t)
& -\infty \leq s \leq -r_\mu\\
\xi(s, t) - \chi_{r_\mu}(s) \sum_{\lambda_k^{(\mu)} > 0}
b_k^{(\mu)} e^{-\lambda_k^{(\mu)} s}
\phi_k^{(\mu)}(t) & -r_\mu \leq s \leq 0
\end{cases},
\end{align*}
where $\chi : \mathbb{R}_{\geq 0} \to \mathbb{R}_{\geq 0}$ is a smooth function such that
$\chi|_{[0, 1/3]} = 0$ and $\chi|_{[2/3, \infty)} = 1$, and
$\chi_r : \mathbb{R}_{\geq 0} \to \mathbb{R}_{\geq 0}$ is defined by
$\chi_r(s) = \chi(\frac{s}{r})$.
It is easy to see that $\tilde \xi|_{([0, \infty] \cup [-\infty, 0]) \times S_\mu^1} \in
\widetilde{W}^{1, p}_\delta(([0, \infty] \cup [-\infty, 0]) \times S_\mu^1, \mathbb{C}^n)$.
We assume that $\zeta_\nu$ are sufficiently small and $r_\mu$ are sufficiently large
so that
\begin{itemize}
\item
$D_0 = \overline{\partial}$ on $\{(x, y) \in N^\nu_{\zeta_\nu}; |x| \leq 2 |\zeta_\nu|^\alpha
\text{ or } |y| \leq 2|\zeta_\nu|^\alpha\}$, and
\item
$S_\mu(s, t) = S_\mu(t)$ on $[\frac{1}{3} r_\mu, \infty] \times S_\mu^1 \cup
[-\infty, -\frac{1}{3} r_\mu] \times S_\mu^1$.
\end{itemize}
For each $\mu$, define $v_\mu \in \mathop{\mathrm{Ker}}\nolimits A_\mu$ by
\[
v_\mu = \sum_{\lambda_k^{(\nu)} = 0} a_k^{(\mu)} \phi_k^{(\mu)}(t).
\]
We also define $v_{\pm\infty_i} \in \mathop{\mathrm{Ker}}\nolimits A_\mu$ for limit circles $\pm\infty_i$ by
the condition
\[
\xi - \sum_\mu \beta_\mu v_\mu
- \sum_{\pm\infty_i} \beta_{\pm\infty_i} v_{\pm\infty_i}
\in W_\delta^{1,p}(\Sigma_{(\zeta, r)}, E).
\]
We can easily check the following inequalities, where $C > 0$ is some constant and
$0 < \epsilon \leq 1$ is arbitrary.
($C > 0$ does not depend on $\epsilon$.)
\begin{align}
&C \Bigl|\Bigl|\tilde \xi - \sum_\mu \beta_\mu v_\mu
- \sum_{\pm\infty_i} \beta_{\pm\infty_i} v_{\pm\infty_i}\Bigr|\Bigr|
_{W_\delta^{1,p}(\Sigma_0)} \notag \\
&\geq \epsilon \Bigl(\Bigl|\Bigl|\xi - \sum_\mu \beta_\mu v_\mu
- \sum_{\pm\infty_i} \beta_{\pm\infty_i} v_{\pm\infty_i}\Bigr|\Bigr|
_{W_\delta^{1,p}(\Sigma_{(\zeta, r)})} \notag \\
&\quad \hphantom{\epsilon \Bigl(}
- \sum_\nu \Bigl|\Bigl|\rho_{\zeta_\nu}(x) \sum_{k < 0} a_k^{(\nu)} x^k\Bigr|\Bigr|
_{W^{1,p}(\sqrt{|\zeta_\nu|} \leq |x| \leq 1)} \notag \\
&\quad \hphantom{\epsilon \Bigl(}
- \sum_\nu \Bigl|\Bigl|\rho_{\zeta_\nu}(y) \sum_{k < 0} b_k^{(\nu)} y^k\Bigr|\Bigr|
_{W^{1,p}(\sqrt{|\zeta_\nu|} \leq |y| \leq 1)} \notag \\
&\quad \hphantom{\epsilon \Bigl(}
- \sum_\mu \Bigl|\Bigl|\chi_{r_\mu}(s) \sum_{\lambda_k^{(\mu)} < 0} a_k^{(\mu)}
e^{- \lambda_k^{(\mu)} s}
\phi_k^{(\mu)}(t)\Bigr|\Bigr|
_{W_{\delta_\mu}^{1,p}([0, r_\mu] \times S^1)} \notag \\
&\quad \hphantom{\epsilon \Bigl(}
- \sum_\mu \Bigl|\Bigl|\chi_{r_\mu}(s) \sum_{\lambda_k^{(\mu)} > 0} b_k^{(\mu)}
e^{-\lambda_k^{(\mu)} s}
\phi_k^{(\mu)}(t)\Bigr|\Bigr|
_{W_{\delta_\mu}^{1,p}([-r_\mu, 0] \times S^1)}
\Bigr) \notag \\
&\quad
+ \sum_\nu \Bigl|\Bigl|\sum_{k \geq 0} a_k^{(\nu)} x^k
\Bigr|\Bigr|
_{W^{1,p}(|x| \leq \sqrt{|\zeta_\nu|})}
+ \sum_\nu \Bigl|\Bigl|\sum_{k \geq 0} b_k^{(\nu)} y^k
\Bigr|\Bigr|
_{W^{1,p}(|y| \leq \sqrt{|\zeta_\nu|})} \notag \\
&\quad
+ \sum_\mu \Bigl|\Bigl|\sum_{\lambda_k^{(\mu)} > 0} a_k^{(\mu)}
e^{- \lambda_k^{(\mu)} s} \phi_k^{(\mu)}(t)
\Bigr|\Bigr|
_{W_{\delta_\mu}^{1,p}([r_\mu, \infty] \times S^1)} \notag \\
&\quad
+ \sum_\mu \Bigl|\Bigl|\sum_{\lambda_k^{(\mu)} < 0} b_k^{(\mu)}
e^{- \lambda_k^{(\mu)} s} \phi_k^{(\mu)}(t)
\Bigr|\Bigr|
_{W_{\delta_\mu}^{1,p}([-\infty, -r_\mu] \times S^1)}
\label{tilde xi geq xi}
\end{align}
\begin{align}
&||D_0 \tilde \xi + \lambda h||_{L_\delta^p(\Sigma)} \notag \\
&\leq ||D_{(\zeta, r)} \xi + \lambda_{(\zeta, r)} h||_{L_\delta^p(\Sigma_{(\rho, r)})}
\notag \\
&\quad
+ C\Bigl(\sum_\nu |\zeta_\nu|^{-\alpha}
\Bigl|\Bigl|\sum_{k < 0} a_k^{(\nu)} x^k\Bigr|\Bigr|_{L^p
(|\zeta_\nu|^\alpha \leq |x| \leq 2 |\zeta_\nu|^\alpha)} \notag \\
&\quad \hphantom{+ C\Bigl( }
+ \sum_\nu |\zeta_\nu|^{-\alpha} \Bigl|\Bigl|\sum_{k < 0} b_k^{(\nu)} y^k\Bigr|\Bigr|_{L^p
(|\zeta_\nu|^\alpha \leq |y| \leq 2 |\zeta_\nu|^\alpha)} \notag \\
&\quad \hphantom{+ C\Bigl( }
+ \sum_\mu \frac{1}{r_\mu} \Bigl|\Bigl|\sum_{\lambda_k^{(\mu)} < 0} a_k^{(\mu)}
e^{-\lambda_k^{(\mu)} s} \phi_k^{(\mu)}(t)\Bigr|\Bigr|
_{L_{\delta_\mu}^p([0, r_\mu] \times S^1)}
\notag \\
&\quad \hphantom{+ C\Bigl( }
+ \sum_\mu \frac{1}{r_\mu} \Bigl|\Bigl|\sum_{\lambda_k^{(\mu)} > 0} b_k^{(\mu)}
e^{-\lambda_k^{(\mu)} s} \phi_k^{(\mu)}(t)\Bigr|\Bigr|
_{L_{\delta_\mu}^p([-r_\mu, 0] \times S^1)}
\Bigr)
\label{D tilde xi leq D xi}
\end{align}
Apply inequality (\ref{eq of tilde xi}) to $\tilde \xi \in
\widetilde{W}^{1, p}_\delta(\Sigma_0)$,
and use (\ref{tilde xi geq xi}) for sufficiently small $\epsilon > 0$,
(\ref{D tilde xi leq D xi}) and the following two lemmas
(Lemma \ref{disk cancellation inequality} and \ref{cylinder cancellation inequality}).
Then we can easily see that there exists some constant $C > 0$ such that
for any sufficiently small $(\zeta, r^{-1})$,
\begin{align*}
&\Bigl|\Bigl|\xi - \sum_\mu \beta_\mu v_\mu
- \sum_{\pm\infty_i} \beta_{\pm\infty_i} v_{\pm\infty_i}\Bigr|\Bigr|
_{W_\delta^{1,p}(\Sigma_{(\zeta, r)})} \\
&+ \sum_\mu ||v_\mu||_{\mathop{\mathrm{Ker}}\nolimits A_\mu}
+ \sum_{\pm\infty_i} ||v_{\pm\infty_i}||_{\mathop{\mathrm{Ker}}\nolimits A_{\pm\infty_i}}
+ |h|_{\mathbb{R}^N} \\
&\ \leq C \bigl( ||D_{(\zeta, r)} \xi + \lambda_{(\zeta, r)} h||
_{L_\delta^p(\Sigma_{(\zeta, r)})}
+ \sum_k |\langle \xi,\xi_k \rangle_{L^2(\Sigma_0 \setminus N_0)}
+ \langle h,h_k \rangle_{\mathbb{R}^N} | \bigr).
\end{align*}
We do not use any estimates of
$a_k^{(\nu)}$, $b_k^{(\nu)}$, $a_k^{(\mu)}$ or $b_k^{(\mu)}$ by $\xi$.
Lemma \ref{disk cancellation inequality} and \ref{cylinder cancellation inequality}
imply that these terms cancel each other.
(\ref{inequality of linearized gluing lemma}) follows from the above inequality.
\end{proof}
\begin{lem}
\label{disk cancellation inequality}
For any $2<p<\infty$ and $0 < \alpha < \frac{1}{2}$, there exists some $C>0$ such that
for any $\zeta \in D$ and any two sequences $(a_k)_{k \in \mathbb{Z}}$ and $(b_k)_{k \in \mathbb{Z}}$
such that $b_k = a_{-k} \zeta^{-k}$, the following inequalities hold true.
\[
\Bigl|\Bigl|\sum_{k<0}a_k z^k\Bigr|\Bigr|_{W^{1,p}(\sqrt{|\zeta|}\leq |z| \leq 1)}
\leq C \Bigl|\Bigl|\sum_{k\geq0} b_k z^k\Bigr|\Bigr|_{W^{1,p}
(|z|\leq \sqrt{|\zeta|})}
\]
\[
|\zeta|^{-\alpha} \Bigl|\Bigl|\sum_{k<0} a_k z^k\Bigr|\Bigr|_{L^p
(|\zeta|^\alpha \leq |z| \leq 2|\zeta|^\alpha)}
\leq C |\zeta|^{(1-\frac{2}{p})(1-2\alpha)} \Bigl|\Bigl|\sum_{k\geq0} b_k z^k
\Bigr|\Bigr|_{W^{1,p}(|z|\leq \sqrt{|\zeta|})}
\]
\end{lem}
\begin{proof}
Put $f(z) = \sum_{k\geq 0} b_k z^k$.
Then $\sum_{k<0} a_k z^k = f\bigl(\frac{\zeta}{z}\bigr) - b_0$.
Therefore $\frac{d}{dz}(\sum_{k<0} a_k z^k) = f'\bigl(\frac{\zeta}{z}\bigr) \bigl( -
\frac{\zeta}{z^2}\bigr)$.
This implies for any $0 < \alpha \leq \frac{1}{2}$,
\begin{align*}
\int_{|\zeta|^\alpha \leq |z| \leq 1} \biggl|\frac{d}{dz}\Bigl(\sum_{k<0} a_k z^k\Bigr)
\biggr|^p |dz|^2
&= \int_{|\zeta| \leq |w| \leq |\zeta|^{1-\alpha}} |f'(w)|^p
\biggl(\frac{|w|^2}{|\zeta|}\biggr)^{p-2} |dw|^2\\
&\leq
|\zeta|^{(p-2)(1-2\alpha)} \Bigl|\Bigl|\sum_{k\geq 0} b_k z^k\Bigr|\Bigr|_{W^{1,p}
(|z|\leq \sqrt{|\zeta|})}^p.
\end{align*}
On the other hand,
Poincar\'{e}'s inequality on $S^1$ implies
\begin{align*}
\Bigl|\Bigl|\sum_{k<0} a_k z^k\Bigr|\Bigr|_{L^p(|\xi|^\alpha \leq |x| \leq \rho)}^p
&= \int_{|\zeta|^\alpha}^\rho \int_0^{2\pi} \Bigl|\sum_{k<0}a_k r^k e^{\sqrt{-1}k\theta}
\Bigr|^p rdrd\theta\\
&\leq C \int_{|\zeta|^\alpha}^\rho\int_0^{2\pi}
\Bigl|\sum_{k<0}ka_kr^k e^{\sqrt{-1}k\theta}\Bigr|^p rdrd\theta\\
&= C \int_{|\zeta|^\alpha}^\rho\int_0^{2\pi}
\Bigl|\sum_{k<0}ka_kr^{k-1} e^{\sqrt{-1}(k-1)\theta}\Bigr|^p r^{p+1} drd\theta\\
&\leq C \rho^p \biggl|\biggl|\frac{d}{dz}\Bigl(\sum_{k<0} a_k z^k\Bigr)
\biggr|\biggr|_{L^p(|\xi|^\alpha \leq |z|\leq \rho)}^p
\end{align*}
for $\rho = 2|\zeta|^\alpha$ or $1$.
The first of the claimed inequalities is proved by substituting $\alpha = \frac{1}{2}$ and
$\rho = 1$, and the second is proved by substituting $\rho = 2|\zeta|^\alpha$.
\end{proof}
\begin{lem}
\label{cylinder cancellation inequality}
Let $(\phi_k)$ be a family of $W^{1, 2}$-functions on $S^1$.
Let $\delta > 0$ be a positive constant and $(\lambda_k)$ be a sequence of
real numbers such that $\lambda_k < -\delta$.
Then for any $1 < p < \infty$, $r > 0$ and any two sequences $(a_k)$ and $(b_k)$
such that $b_k = e^{-2\lambda_k r} a_k$, the following inequality holds true.
\[
\Bigl|\Bigl|\sum_{\lambda_k<0}a_k e^{-\lambda_k s} \phi_k(t)\Bigr|\Bigr|_{W_\delta^{1,p}
([0,r] \times S^1)}
\leq
\Bigl|\Bigl|\sum_{\lambda_k<0} b_k e^{-\lambda_k s} \phi_k(t)\Bigr|\Bigr|_{W_\delta^{1,p}
((-\infty,-r] \times S^1)}
\]
\end{lem}
\begin{proof}
The $L^p_\delta$-norm is estimated by
\begin{align*}
\int_0^r \Bigl|\sum_{\lambda_k<0}a_k e^{-\lambda_k s} \phi_k(t)\Bigr|^p
e^{p\delta s} dsdt
&= \int_{-2r}^{-r} \Bigl|\sum_{\lambda_k<0} b_k e^{-\lambda_k s} \phi_k(t)\Bigr|^p
e^{p\delta (s + 2r)} dsdt\\
&\leq \int_{-2r}^{-r} \Bigl|\sum_{\lambda_k<0} b_k e^{-\lambda_k s} \phi_k(t)
\Bigr|^p e^{-p\delta s} dsdt.
\end{align*}
Similarly, we can estimate
\[
\int_0^r \Bigl|\partial_s\Bigl(\sum_{\lambda_k<0}a_k e^{-\lambda_k s}
\phi_k(t)\Bigr)\Bigr|^p e^{p\delta s} dsdt
\]
and
\[
\int_0^r \Bigl|\partial_t\Bigl(\sum_{\lambda_k<0}a_k e^{-\lambda_k s}
\phi_k(t)\Bigr)\Bigr|^p e^{p\delta s} dsdt
\]
by the corresponding terms for $b_k$.
\end{proof}
\begin{rem}\label{uniform regularity}
The same argument implies that interior regularity property of $D_{(\zeta, r)}$ is
uniform with respect to small $(\zeta, r)$ on a neighborhood of a nodal point or
a imaginary circle.
\end{rem}
\subsection{Smoothness of Kuranishi neighborhoods}
\label{smoothness}
In this section, we prove that if we give $\widetilde{X}$ a stronger differential
structure and give the product differential structure to
$\hat V = \mathring{X} \times B_\epsilon(0) \subset \mathring{X} \times
\mathop{\mathrm{Ker}}\nolimits DF^{(0, b_0)}_{(0,0)}$, then
\begin{align*}
\hat V &\hookrightarrow \mathring{X} \times C^l(\Sigma_0 \setminus N_0,
(\mathbb{R}_1 \cup \mathbb{R}_2 \cup \dots \cup \mathbb{R}_k) \times Y) \times E^0\\
(a, b, x) &\mapsto (a, b, \Phi_{a, b}(\xi_x)|_{\Sigma_0 \setminus N_0}, h_x)
\end{align*}
is a smooth embedding for any $l$.
More precisely, we prove that for any $N \geq 1$, we can chose
a stronger differentiable structure of $\widetilde{X}$ such that the map
is of class $C^N$.
Note that we have already proved the continuity of the above map by
(\ref{continuity of phi}).
We also note that once we prove that this is a smooth embedding for $l = 1$,
then it follows that for any $l \geq 1$
and any $\widetilde{N}_0 \supset N_0$,
\[
\hat V \hookrightarrow \mathring{X} \times C^l(\Sigma_0 \setminus \widetilde{N}_0, (\mathbb{R}_1 \cup \mathbb{R}_2 \cup
\dots \cup \mathbb{R}_k) \times Y) \times E^0
\]
is also a smooth embedding provided that $\widetilde{N}_0$ does not cover any
irreducible components of $\Sigma_0$.
First we explain about the strong differential structure of $\widetilde{X}$.
It is based on the following Lemma.
\begin{lem}
Let $V \subset \mathbb{C}^n$ be an open set and $D \subset \mathbb{C}$ be a disk.
Assume a holomorphic function $f(w, \zeta_1, \dots, \zeta_l) : V \times D^l \to \mathbb{C}$
satisfies $\{(w, \zeta); f(w, \zeta) = 0\} = \bigcup_i \{ \zeta_i = 0\}$.
Define $\varphi_\alpha (r e^{\sqrt{-1} \theta}) = r^\alpha e^{\sqrt{-1} \theta}$
for $\alpha \geq 1$.
Then $\varphi_\alpha^{-1} \circ f(w, \varphi_\alpha (\zeta_1), \dots,
\varphi_\alpha (\zeta_l)) : V \times D^l \to \mathbb{C}$ is of class $C^{\lfloor \alpha \rfloor}$.
(If $\alpha = 2 N + 1$ for some $N \in \mathbb{Z}_{\geq 0}$ then it is real analytic.)
\end{lem}
\begin{proof}
There exists a holomorphic function $g : V \times D^l \to \mathbb{C} \setminus 0$ such that
$f(w,\zeta) = \zeta_1^{k_1} \dots \zeta_l^{k_l} g(w,\zeta)$ for some $k_i \geq 1$.
Then
\[
\varphi_\alpha^{-1}\bigl(f(w,\varphi_\alpha (\zeta_1), \dots,
\varphi_\alpha (\zeta_l))\bigr)
= \zeta_1^{k_1} \dots \zeta_l^{k_l} \varphi_\alpha^{-1}
\bigl(g(w,\varphi_\alpha (\zeta_1), \dots, \varphi_\alpha (\zeta_l))\bigr),
\]
where we have used $\varphi_\alpha (ab) =\varphi_\alpha(a) \varphi_\alpha(b)$.
Since $\varphi_\alpha$ is of class $C^{\lfloor \alpha \rfloor}$ and
$\varphi_\alpha^{-1} : \mathbb{C}\setminus 0 \to \mathbb{C}\setminus 0$ is real analytic,
$\varphi_\alpha^{-1} \circ f(w, \varphi_\alpha (\zeta_1), \dots,
\varphi_\alpha (\zeta_l))$ is of class $C^{\lfloor \alpha \rfloor}$.
\end{proof}
For any $\alpha \gg 0$ and $\beta \gg 0$,
a new differential structure of $\widetilde{X}$ is defined by
the coordinate
\[
\widetilde{X} \subset \mathcal{J}_0 \times D^{l_0} \times \widetilde{D}^{l_1}
\to \mathcal{J}_0 \times D^{l_0} \times ([0, 1] \times S^1)^{l_1}
\]
{\abovedisplayskip=0pt
\[
(j, (\zeta_\nu = \rho_\nu^2 e^{2\sqrt{-1} \varphi_\nu})_\nu,
(
\rho_\mu^{2\pi} e^{2\pi \sqrt{-1} \varphi_\mu})_\mu)
\mapsto (j, (\hat \zeta_\nu = \hat \rho_\nu^2 e^{2\sqrt{-1} \varphi_\nu})_\nu,
(\hat \rho_\mu, \varphi_\mu)_\mu)
\]}
defined by $\rho_\nu = \hat \rho_\nu^\alpha$ and
$\rho_\mu = \hat \rho_\mu^{\beta_\mu}$,
where $\beta_\mu = L_\mu^{-1} \beta$ ($L_\mu = L_{\gamma_\mu}$ is
the period of the periodic orbit on $S_\mu^1$).
The above lemma implies that this differential structure is independent of
the local description of the universal family $(\widetilde{P} \to \widetilde{X},
Z \cup Z^+ \cup Z^{++})$ given by a decomposition of $\Sigma_0$
since in any description, $\{\zeta_\nu\}$ consists of the curves which have
$\nu$-th nodal point and is preserved by the coordinate change.
The reason why we use the indeces $\beta_\mu = L_\mu^{-1} \beta$ depending on
$\mu$ is to make $\mathring{X} \subset \widetilde{X} \times \prod_\mu \mathbb{R}_\mu$
a submanifold.
(Recall that $\mathring{X}$ is defined by the condition that
$\rho_\mu^{L_\mu} e^{- b_\mu}$ does not depend on $\mu \in M_i$ for each
$i = 1, 2, \dots, k-1$.)
We fix large constants $\alpha \geq 1$ and $\beta > 0$, and use the differential
structure of $\widetilde{X}$ defined by the same $\alpha$ and $\beta$ for all
Kuranishi neighborhoods of $\widehat{\mathcal{M}}$ for each pre-Kuranishi structure of
$\widehat{\mathcal{M}}$.
Let $\mathring{X} = \coprod_{\Pi, \Pi'} \mathring{X}_{\Pi, \Pi'}$
be the decomposition defined by
\begin{align*}
\mathring{X}_{\Pi, \Pi'}
= \{ (a,b) \in \mathring{X};\, & \rho_\mu \neq 0 \text{ for all } \mu \in M_i
\text{ if and only if } i \in \Pi\\
& \zeta_\nu \neq 0 \text{ if and only if } \nu \in \Pi'\},
\end{align*}
where $\Pi \subset \{1, 2, \dots, k-1\}$ and
$\Pi'$ is a subset of nodal points of $\Sigma_0$.
We prove the differentiability of $\phi$ on each
$\mathring{X}_{\Pi, \Pi'} \times B_\epsilon(0)$ and
investigate its behavior near the boundary.
Fix one point $(a,b) \in \mathring{X}_{\Pi, \Pi'}$ and consider another point
$(\tilde a, \tilde b) \in \mathring{X}_{\Pi, \Pi'}$ close to $(a,b)$.
To investigate the behavior of the differential, we identify $\widetilde{P}_{\tilde a}$
and $\widetilde{P}_a$ by the piecewise smooth map $\Psi$ defined as follows.
On each $[-1, 0] \times S^1_\mu \subset [-1, -\frac{1}{2}\log \rho_\mu] \times
S^1_\mu$,
\begin{align*}
\Psi : [-1, 0] \times S^1_\mu &\to [-1, 0] \times S^1_\mu\\
(s,t) &\mapsto (\tilde s, \tilde t) = (s, t)
\end{align*}
is given by the identity map,
and on each $[0, -\frac{1}{2}\log \rho_\mu] \times S^1_\mu \subset
[-1,-\frac{1}{2}\log \rho_\mu] \times S^1_\mu$,
\begin{align*}
\Psi : [0, - {\textstyle \frac{1}{2}} \log \rho_\mu] \times S^1_\mu
&\to [0, - {\textstyle \frac{1}{2}} \log \tilde \rho_\mu] \times S^1_\mu\\
(s,t) &\mapsto (\tilde s, \tilde t)
\end{align*}
is defined by
\[
\frac{e^{-\kappa_\mu \tilde s} -
\tilde \rho_\mu^{\kappa_\mu/2}}{1 - \tilde \rho_\mu^{\kappa_\mu/2}}
= \frac{e^{-\kappa_\mu s} - \rho_\mu^{\kappa_\mu/2}}{1 - \rho_\mu^{\kappa_\mu/2}},
\quad \tilde t = t.
\]
$\Psi$ on each $[\frac{1}{2} \log \rho_\mu, 1] \times S^1$ is defined similarly.
For simplicity of notation, we denote
$[0, -\frac{1}{2}\log \rho_\mu] \times S^1_\mu \cup
[\frac{1}{2} \log \rho_\mu, 0] \times S^1_\mu$
by $N^\mu_{a, b}$.
Recall the definition of the approximate solutions and note that
$\Psi$ satisfies $v^\mu_{\tilde a, \tilde b} \circ \Psi = v^\mu_{a, b}$ on $N^\mu_{a, b}$.
On each $N^\nu_{a, b}$,
\begin{align*}
N^\nu_{a, b} = \{(x,y)\in \overline{D}\times\overline{D};xy=\zeta_\nu\}
&\to \{(\tilde x,\tilde y)\in \overline{D}\times\overline{D};\tilde x\tilde y=\tilde\zeta_\nu\}
=N^\nu_{\tilde a, \tilde b} \\
(x,y)&\mapsto (\tilde x,\tilde y)
\end{align*}
is defined by
\begin{itemize}
\item
$\tilde x= \tilde r e^{\sqrt{-1}\tilde\theta}$ if $|x| \geq \sqrt{|\zeta_\nu|}$
and $x = r e^{\sqrt{-1}\theta}$
\item
$\tilde y= \tilde r e^{\sqrt{-1}\tilde\theta}$ if $|y| \geq \sqrt{|\zeta_\nu|}$
and $y = r e^{\sqrt{-1}\theta}$
\end{itemize}
where $\tilde r$ and $\tilde \theta$ is defined by
\[
\frac{\tilde r - \tilde \rho_\nu}{1 - \tilde \rho_\nu}
= \frac{r - \rho_\nu}{1 - \rho_\nu}, \quad \quad
\tilde \theta = \theta + \beta_\nu(r) (\tilde \varphi_\nu - \varphi_\nu),
\quad \beta_\nu(r)
=\max \biggl(\frac{2 \rho_\nu - r}{\rho_\nu}, 0 \biggr),
\]
where $\rho_\nu$, $\varphi_\nu$, $\tilde \rho_\nu$ and $\tilde \varphi_\nu$
are defined by $\zeta_\nu = \rho_\nu^2 e^{2 \sqrt{-1} \varphi_\nu}$ and
$\tilde \zeta_\nu = \tilde \rho_\nu^2 e^{2 \sqrt{-1} \tilde \varphi_\nu}$.
On $\Sigma_0 \setminus N_0$,
$\Psi|_{\Sigma_0 \setminus N_0} = \mathrm{id}$.
Then under this identification, we consider $F^{(\tilde a, \tilde b)}$ as a map
\begin{align*}
&F^{(\tilde a,\tilde b)} : \widetilde{W}^{1,p}_\delta( \widetilde{P}_a, u_{a, b}^\ast T \hat Y)
\oplus E^0\\
&\to L^p(\Sigma_0 \setminus N_0, {\textstyle\bigwedge}^{0,1} T^\ast \Sigma_0 \otimes_\mathbb{C}
u_0^\ast T \hat Y)\\
&\quad \oplus \bigoplus_\mu (L^p_\delta ([-1, - {\textstyle \frac{1}{2}} \log \rho_\mu]
\times S^1, \mathbb{R}^{2n})
\oplus L^p_\delta ([{\textstyle \frac{1}{2}} \log \rho_\mu, 1] \times S^1, \mathbb{R}^{2n}))\\
&\quad \oplus \bigoplus_{+\infty_i} L^p_\delta([0, \infty] \times S^1, \mathbb{R}^{2n})
\oplus \bigoplus_{-\infty_i} L^p_\delta([-\infty, 0] \times S^1, \mathbb{R}^{2n})\\
&\quad \oplus \bigoplus_\nu (L^p (\{ x \in D; |x| \geq \sqrt{|\zeta_\nu|}\}, \mathbb{R}^{2n})
\oplus L^p (\{ y \in D; |y| \geq \sqrt{|\zeta_\nu|}\}, \mathbb{R}^{2n})).
\end{align*}
On each $[-1, 0] \times S^1_\mu \subset [-1, -\frac{1}{2} \log \rho_\mu] \times
S^1_\mu$,
\begin{align*}
F^{(\tilde a,\tilde b)} (\xi, h)
&=
\partial_s (v_{a,b}^{\mu, \mathrm{left}} + \xi)
+ \widetilde{J}^\mu_t(v_{a,b}^{\mu, \mathrm{left}} + \xi)
\partial_t (v_{a,b}^{\mu, \mathrm{left}} + \xi)
+ f^\mu_t(v_{a,b}^{\mu, \mathrm{left}} + \xi)\\
& \quad + \frac{1}{2} (b_\mu -b_\mu^0) \chi'(s) \partial_\sigma
+ \frac{1}{2} \tilde \varphi_\mu \chi'(s) (g^\mu_t(v_{a,b}^{\mu, \mathrm{left}} + \xi)
+ \partial_t(v_{a,b}^{\mu, \mathrm{left}} + \xi)).
\end{align*}
On each $[0, -\frac{1}{2} \log \rho_\mu] \times S^1_\mu \subset
[-1, -\frac{1}{2} \log \rho_\mu] \times S^1_\mu$,
\begin{align*}
F^{(\tilde a,\tilde b)} (\xi, h)
&= \biggl(1 + \biggl(\frac{1 - \rho_\mu^{\kappa_\mu/2}}
{1 - \tilde \rho_\mu^{\kappa_\mu/2}}
\tilde \rho_\mu^{\kappa_\mu/2} - \rho_\mu^{\kappa_\mu/2}\biggr)
e^{\kappa_\mu s}\biggr)
\partial_s (v_{a,b}^{\mu, \mathrm{left}} + \xi)\\
&\quad + \widetilde{J}^\mu_t(v_{a,b}^{\mu, \mathrm{left}} + \xi)
\partial_t (v_{a,b}^{\mu, \mathrm{left}} + \xi)
+ f^\mu_t(\pi_Y(v_{a,b}^{\mu, \mathrm{left}} + \xi)).
\end{align*}
On each $N^{\nu, \text{left}}_{a, b} = \{x \in D; |x| \geq \sqrt{|\zeta_\nu|}\}
\subset N^\nu_{a, b}$,
\begin{align*}
F^{(\tilde a,\tilde b)} (\xi, h)
=& \frac{1 - \rho_\nu}{1 - \tilde \rho_\nu}
\biggl((\partial_r v_{a, b}^\nu)_{\tilde \varphi_\nu} + \partial_r \xi
+ \frac{\tilde \varphi_\nu - \varphi_\nu}{\rho_\nu}
1_{\{\rho_\nu \leq r_\nu \leq 2 \rho_\nu\}} \partial_\theta \xi
\biggr)\\
&+ \widetilde{J}^\nu((v_{a, b}^\nu)_{\tilde \varphi_\nu} + \xi)
\frac{(\partial_\theta v_{a, b}^\nu)_{\tilde \varphi_\nu} + \partial_\theta \xi}{\tilde r}
\end{align*}
where
$w_{\tilde \varphi_\nu}(r e^{\sqrt{-1} \theta})
= w(r e^{\sqrt{-1}(\theta + \beta_\nu(r) (\tilde \varphi_\nu - \varphi_\nu))})$
for $w = \partial_r v_{a, b}^\nu$, $v_{a, b}^\nu$ or $\partial_\theta v_{a, b}^\nu$.
By the same equations, we can define $F^{(\tilde a,\tilde b)}$ for all
$(\tilde a, \tilde b) \in \widetilde{X}_{\Pi, \Pi'} \times \prod_{\mu} \mathbb{R}$
close to $(a, b)$,
where $\widetilde{X}_{\Pi, \Pi'} \subset \widetilde{X}$ is defined
as $\mathring{X}_{\Pi, \Pi'} \subset \mathring{X}$, that is,
\begin{align*}
\widetilde{X}_{\Pi, \Pi'}
= \{ a \in \widetilde{X};\, & \rho_\mu \neq 0 \text{ if and only if } \mu \in
\bigcup_{i \in \Pi} M_i,\\
& \zeta_\nu \neq 0 \text{ if and only if } \nu \in \Pi'\}.
\end{align*}
In the following lemma, we regard $\tilde a \in \widetilde{X}_{\Pi, \Pi'}$ and
$\tilde b \in \prod_{\mu} \mathbb{R}$ as independent parameters by extending
$F^{(\tilde a,\tilde b)}$ to
$(\tilde a, \tilde b) \in \widetilde{X}_{\Pi, \Pi'} \times \prod_{\mu} \mathbb{R}$
as above and estimate the derivatives at $(a, b)$.
We note that
$\partial_{\tilde \rho_\mu}^k F^{(a, b)}
= \partial_{\tilde \rho_\mu}^k F^{(\tilde a, \tilde b)}|_{(\tilde a, \tilde b) = (a, b)}$
vanishes on the complement of $N_{a, b}^\mu$ for $k > 0$, and
$\partial_{\tilde \rho_\nu}^k \partial_{\tilde \varphi_\nu}^l F^{(a, b)}$ vanishes
on the complement of $N_{a, b}^\nu$ for $(k, l) \neq (0, 0)$.
\begin{lem}\label{estimates of implicit function 0}\
\begin{enumerate}[label=\normalfont(\roman*)]
\item
For any $0 < \delta_\mu \leq \delta'_\mu < \kappa_\mu$,
$2 < p < \infty$, $i\in \Pi$, $\mu \in M_i$ and $k > 0$,
there exists some constant $C>0$ and $c_0 > 0$
such that the following hold for $||\xi||_{\widetilde{W}^{1,p}_{\delta'}} \leq c_0$.
\begin{align*}
||(\partial_{\tilde \rho_\mu}^k F^{(a, b)})(\xi, h)||_{L_{\delta_\mu}^p(N_{a, b}^\mu)}
&\leq C \rho_\mu^{(\delta'_\mu - \delta_\mu)/2 - k},\\
||(D \partial_{\tilde \rho_\mu}^k F^{(a, b)})_{(\xi, h)}
(\hat \xi, \hat h)||_{L_{\delta_\mu}^p(N_{a, b}^\mu)}
&\leq C \rho_\mu^{(\delta'_\mu - \delta_\mu)/2 - k}
||\hat \xi||_{\widetilde{W}_{\delta'_\mu}^{1, p}(N_{a, b}^\mu)},\\
D^m \partial_{\tilde \rho_\mu}^k F^{(a, b)} &\equiv 0 \quad (m \geq 2).
\end{align*}
\item
For any $2 < p \leq q < \infty$,
$\nu \in \Pi'$ and any $(m, k, l)$ such that $(k, l) \neq (0, 0)$,
there exists some constants $C > 0$ and $c_0 > 0$ such that
the following holds for $||\xi||_{W^{1, q}(N_{a, b}^\nu)} \leq c_0$.
\begin{align*}
||(D^m \partial_{\tilde \rho_\nu}^k \partial_{\tilde \varphi_\nu}^l F^{(a, b)})_{(\xi, h)}
(\hat \xi^{(m)}, \hat h^{(m)}) (\hat \xi^{(m-1)}, \hat h^{(m-1)}) \dots
(\hat \xi^{(1)}, \hat h^{(1)})||_{L^p(N_{a, b}^\nu)}\\
\leq C \rho_\nu^{2/p - 2/q - k} \prod_{i = 1}^m ||\hat \xi^{(i)}||_{W^{1,q}(N_{a, b}^\nu)}.
\end{align*}
\item
For any $2 < p < \infty$, $0< \delta < \delta_0$,
and any multi-index $(m, k_j, (l_\mu), k_b = (k_{b_\mu}))$,
there exists some constants $C > 0$ and $c_0 > 0$ such that
the following holds for $||\xi||_{\widetilde{W}^{1,p}_\delta} + |h|_E \leq c_0$.
\begin{align*}
||(D^m \partial_{\tilde \jmath}^{k_j} \partial_{(\tilde \varphi_\mu)}^{(l_\mu)}
\partial_{\tilde b}^{k_b} F^{(a, b)})_{(\xi,h)}
(\hat \xi^{(m)}\!, \hat h^{(m)}) (\hat \xi^{(m-1)}\!, \hat h^{(m-1)})\! \dots \!
(\hat \xi^{(1)}\!, \hat h^{(1)})||_{L^p_\delta(\widetilde{P}_a)}\\
\leq C \prod_{i = 1}^m (||\hat \xi^{(i)}||_{\widetilde{W}^{1,p}_\delta(\widetilde{P}_a)}
+ |\hat h^{(i)}|_E).
\end{align*}
\end{enumerate}
(See Appendix \ref{diff notation} for our notation of differential.)
\end{lem}
\begin{rem}\label{independent}
Note that
$\partial_{\tilde \rho_\mu} \partial_{\tilde \rho_{\mu'}}F^{(a, b)} (\xi, h) = 0$
if $\mu \neq \mu'$.
Similarly, the differential of $F$ with respect to two parameters
which correspond to different pieces of the curve vanishes.
Hence the above lemma is enough for the estimate of the differentials of $F$.
\end{rem}
\begin{proof}
(i)
We use a change of variable $\mathring{\rho}_\mu = (\tilde \rho_\mu)^{\kappa_\mu/2}$.
Then on $[0, -\frac{1}{2} \log \rho_\mu] \times S^1$,
\[
\partial_{\mathring{\rho}_\mu}^k F^{(\tilde a,\tilde b)}(\xi, h)|_{(\tilde a,\tilde b) = (a,b)}
= k ! (1 - \rho_\mu^{\kappa_\mu/2})^{-k} e^{\kappa_\mu s}
\partial_s (v_{a, b}^{\mu, \text{lef}} + \xi).
\]
Since $|\partial_s v_0^{\mu, \text{lef}}(s, t)| \lesssim e^{- \delta'_{0, \mu} s}$
on $[0, \infty) \times S^1$ for any $\kappa_\mu < \delta'_{0, \mu} < \delta_{0, \mu}$,
\[
\partial_s v_{a, b}^{\mu, \text{lef}}(s, t)
= \frac{e^{-\kappa_\mu s}}{e^{-\kappa_\mu s} - \rho_\mu^{\kappa_\mu / 2}}
\partial_s v_0^{\mu, \text{lef}}\biggl(- \frac{1}{\kappa_\mu} \log
\biggl(\frac{e^{-\kappa_\mu s} - \rho_\mu^{\kappa_\mu / 2}}
{1 - \rho_\mu^{\kappa_\mu / 2}}\biggr), t
\biggr)
\]
satisfies
\begin{align*}
\bigl|\partial_s v_{a, b}^{\mu, \text{lef}}\bigr| e^{\delta'_\mu s}
&\lesssim \frac{e^{-\kappa_\mu s}}{e^{-\kappa_\mu s} - \rho_\mu^{\kappa_\mu / 2}}
\cdot \biggl(\frac{e^{-\kappa_\mu s} - \rho_\mu^{\kappa_\mu / 2}}
{1 - \rho_\mu^{\kappa_\mu / 2}}\biggr)^{\delta'_{0, \mu} / \kappa_\mu}
e^{\delta'_\mu s} \\
&= e^{-(\kappa_\mu - \delta'_\mu) s} \cdot
\frac{\bigl(e^{-\kappa_\mu s} - \rho_\mu^{\kappa_\mu / 2}\bigr)
^{\delta'_{0, \mu} / \kappa_\mu - 1}}
{\bigl(1 - \rho_\mu^{\kappa_\mu / 2}\bigr)^{\delta'_{0, \mu} / \kappa_\mu}} \\
&\lesssim e^{-(\kappa_\mu - \delta'_\mu) s},
\end{align*}
which implies that
$||\partial_s v_{a, b}^{\mu, \text{lef}}||_{L_{\delta'_\mu}^p([0, -\frac{1}{2} \log \rho_\mu]
\times S^1)} \lesssim 1$.
Hence the assumption
$||\partial_s \xi||_{L_{\delta'_\mu}^p([0, -\frac{1}{2} \log \rho_\mu] \times S^1)}
\lesssim 1$ and
\[
e^{\kappa_\mu s} |\partial_s (v_{a, b}^{\mu, \text{lef}} + \xi)| e^{\delta_\mu s}
\leq \rho_\mu^{(\delta'_\mu - \delta_\mu - \kappa_\mu)/2}
\cdot |\partial_s (v_{a, b}^{\mu, \text{lef}} + \xi)| e^{\delta'_\mu s}
\]
imply
\begin{align*}
||\partial_{\mathring{\rho}_\mu}^k F^{(\tilde a,\tilde b)}(\xi, h)|_{(\tilde a,\tilde b) = (a,b)}||_{
L_{\delta}^p(\widetilde{P}_a)}
&\lesssim \rho_\mu^{(\delta'_\mu - \delta_\mu)/2 - \kappa_\mu/2},\\
||D \partial_{\mathring{\rho}_\mu}^k F^{(\tilde a,\tilde b)}_{(\xi, h)}
(\hat \xi, \hat h)|_{(\tilde a,\tilde b) = (a,b)}||_{L_{\delta}^p(\widetilde{P}_a)}
&\lesssim \rho_\mu^{(\delta'_\mu - \delta_\mu)/2 - \kappa_\mu/2}
||\hat \xi||_{\widetilde{W}_{\delta'_\mu}^{1, p}(N^\mu_{a, b})},\\
D^2 \partial_{\mathring{\rho}_\mu}^k F^{(\tilde a,\tilde b)}_{(\xi, h)} &\equiv 0.
\end{align*}
The claim follows from these inequalities because $\partial_{\tilde \rho_\mu}
= \frac{\kappa_\mu}{2}(\tilde \rho_\mu)^{\kappa_\mu/2 - 1}
\partial_{\mathring{\rho}_\mu}$.
(ii)
We can easily check the following equations by direct calculation
($\natural_{m, l}$ is defined below):
\begin{align*}
\partial_{\tilde \rho_\nu}^k F^{(a,b)}(\xi, h)|_{N_{a, b}^{\nu, \text{left}}}
&= k ! \frac{1}{(1 - \rho_\nu)^k}(\partial_r v_{a, b}^\nu + \partial_r \xi)\\
&\quad + (-1)^k k ! \frac{1}{r^{k + 1}}
\Bigl(\frac{1 - r}{1 - \rho_\nu} \Bigr)^k
\widetilde{J}^\nu(v_{a, b}^\nu + \xi)(\partial_\theta v_{a, b}^\nu + \partial_\theta \xi) \\
&= k ! \frac{1}{(1 - \rho_\nu)^k}(\partial_r v_{a, b}^\nu + \partial_r \xi)\\
&\quad + (-1)^k k ! \frac{1}{r^{k + 1}}
\Bigl(\frac{1 - r}{1 - \rho_\nu} \Bigr)^k \natural_{(0, 0)} \hspace{50pt} (k \geq 1)
\end{align*}
{\abovedisplayskip=0pt
\begin{align*}
\partial_{\tilde \rho_\nu}^k \partial_{\tilde \varphi_\nu}
F^{(a, b)}(\xi, h)|_{N_{a, b}^{\nu, \text{left}}}
&= k ! \frac{1}{(1-\rho_\nu)^k} \Bigl(\beta_\nu \cdot \partial_r \partial_\theta
v_\zeta^\nu + \frac{1_{\{\rho_\nu \leq r_\nu \leq 2 \rho_\nu\}}}{\rho_\nu}
\partial_\theta \xi \Bigr)\\
&\quad + (-1)^k k ! \frac{\beta_\nu}{r^{k + 1}}
\Bigl(\frac{1 - r}{1 - \rho_\nu}\Bigr)^k
\natural_{(0, 1)} \hspace{50pt} (k \geq 0)
\end{align*}}
{\abovedisplayskip=0pt
\begin{align*}
\partial_{\tilde \rho_\nu}^k \partial_{\tilde \varphi_\nu}^l
F^{(a, b)}(\xi, h)|_{N_{a, b}^{\nu, \text{left}}}
&= k ! \frac{\beta_\nu^l}{(1 - \rho_\nu)^k} \partial_r \partial_\theta^l v_{a, b}^\nu \\
&\quad + (-1)^k k ! \frac{\beta_\nu^l}{r^{k + 1}}
\Bigl(\frac{1 - r}{1 - \rho_\nu}\Bigr)^k \natural_{(0, l)} \quad (k \geq 0, l \geq 2)
\end{align*}}
{\abovedisplayskip=-5pt
\begin{align*}
(D \partial_{\tilde \rho_\nu}^k F^{(a, b)})_{(\xi, h)}(\hat \xi, \hat h)|
_{N_{a, b}^{\nu, \text{left}}}
&= k ! \frac{1}{(1 - \rho_\nu)^k} \partial_r \hat \xi \\
&\quad + (-1)^k k ! \frac{1}{r^{k + 1}}
\Bigl(\frac{1 - r}{1 - \rho_\nu}\Bigr)^k \natural_{(1, 0)} \quad (k \geq 1)
\end{align*}}
{\abovedisplayskip=0pt
\begin{align*}
(D \partial_{\tilde \rho_\nu}^k \partial_{\tilde \varphi_\nu}
F^{(a, b)})_{(\xi, h)}(\hat \xi, \hat h)|_{N_{a, b}^{\nu, \text{left}}}
&= k ! \frac{1}{(1 - \rho_\nu)^k} \frac{1_{\{\rho_\nu \leq r \leq 2\rho_\nu\}}}{\rho_\nu}
\partial_\theta \hat \xi\\
&\quad + (-1)^k k ! \frac{\beta_\nu}{r^{k + 1}}
\Bigl(\frac{1 - r}{1 - \rho_\nu}\Bigr)^k \natural_{(1, 1)} \quad (k \geq 0)
\end{align*}}
{\abovedisplayskip=-5pt
\begin{align*}
&(D^m \partial_{\tilde \rho_\nu}^k \partial_{\tilde \varphi_\nu}^l
F^{(a, b)})_{(\xi, h)}(\hat \xi^{(m)},
\hat h^{(m)}) (\hat \xi^{(m - 1)}, \hat h^{(m - 1)}) \dots
(\hat \xi^{(1)}, \hat h^{(1)})|_{N_{a, b}^{\nu, \text{left}}}\\
&\hspace{175pt}
= (-1)^k k ! \frac{\beta_\nu^l}{r^{k + 1}}
\Bigl(\frac{1 - r}{1 - \rho_\nu}\Bigr)^k \natural_{(m, l)}\\
&\hspace{130pt} (m \geq 2, (k, l) \neq (0, 0) \text{ or } m = 1, k \geq 0, l \geq 2)
\end{align*}}
In the above equations,
$\natural_{(m,l)}$ is a sum of terms in the following forms:
{\belowdisplayskip=0pt
\begin{multline*}
(D^{\alpha_1 + m -1} \widetilde{J}^\nu)_{v_{a, b}^\nu + \xi} \cdot
\partial_\theta^{j_1}v_{a, b}^\nu \cdot
\partial_\theta^{j_2} v_{a, b}^\nu \dots \partial_\theta^{j_{\alpha_1}} v_{a, b}^\nu
\cdot \hat \xi^{(m)} \cdot \hat \xi^{(m-1)} \dots \hat \xi^{(1)}\\
(\alpha_1 \geq 1, j_1 + j_2 + \dots + j_{\alpha_1} = l + 1)
\end{multline*}}
{\abovedisplayskip=5pt
\begin{multline*}
(D^{\alpha_1 + m} \widetilde{J}^\nu)_{v_{a, b}^\nu + \xi} \cdot
\partial_\theta^{j_1} v_{a, b}^\nu \cdot
\partial_\theta^{j_2} v_{a, b}^\nu \dots \partial_\theta^{j_{\alpha_1}} v_{a, b}^\nu
\cdot \hat \xi^{(m)} \cdot \hat \xi^{(m - 1)} \dots \hat \xi^{(1)} \cdot
\partial_\theta \xi\\
(\alpha_1 \geq 0, j_1 + j_2 + \dots + j_{\alpha_1} + 1 = l +1)
\end{multline*}}
{\abovedisplayskip=-6pt
\begin{multline*}
(D^{\alpha_1 + m -1} \widetilde{J}^\nu)_{v_{a, b}^\nu + \xi} \cdot
\partial_\theta^{j_1} v_{a, b}^\nu
\cdot
\partial_\theta^{j_2} v_{a, b}^\nu \dots \partial_\theta^{j_{\alpha_1}} v_{a, b}^\nu
\cdot \hat \xi^{(m)} \cdot \hat \xi^{(m - 1)} \stackrel{\alpha_2}{\breve \dots}
\hat \xi^{(1)} \cdot \partial_\theta \hat \xi^{(\alpha_2)}\\
(\alpha_1 \geq 1, 1 \leq \alpha_2 \leq m,
j_1 + j_2 + \dots + j_{\alpha_1} + 1 = l + 1)
\end{multline*}
}
We prove the case where $p < q$. The case where $p = q$ is easier.
Define $2 < s < \infty$ by $\frac{1}{s} = \frac{1}{p} - \frac{1}{q}$.
First note that
\[
\biggl( \int_{\rho_\nu \leq |x| \leq 1} |r^{- (l + 1)} \natural_{(m, l)}|^q r dr d\theta
\biggr)^{\textstyle \frac{1}{q}}
\lesssim
||\hat \xi^{(m)}||_{1, q}|| \hat \xi^{(m-1)}||_{1, q} \dots ||\hat \xi^{(1)}||_{1, q}.
\]
This is because
\begin{itemize}
\item
$||r^{-j} \partial_\theta^j v_{a, b}^\nu||_\infty \lesssim
||r^{-j} \partial_\theta^j v_0^\nu||_\infty < \infty$ are uniformly bounded
with respect to small $\zeta_\nu$,
\item
$||\hat \xi^{(i)}||_{L^\infty(N_{a, b}^{\nu, \text{left}})} \lesssim ||\hat \xi^{(i)}||_{W^{1,q}
(N_{a, b}^{\nu, \text{left}})}$ uniformly with respect to small $\zeta_\nu$, and
\item
$||r^{-1} \partial_\theta \hat \xi^{(i)}||_{L^q(N_{a, b}^{\nu, \text{left}})} \lesssim
||\hat \xi^{(i)}||_{W^{1, q}(N_{a, b}^{\nu, \text{left}})}$ uniformly with respect to
small $\zeta_\nu$.
\end{itemize}
We note that
$||r^{-j} \partial_r \partial_\theta^j v_{a, b}^\nu||_\infty
\lesssim ||r^{-j} \partial_r \partial_\theta^j v_0^\nu||_\infty < \infty$.
These imply
\begin{flalign*}
&||\partial_{\tilde \rho_\nu}^k F^{(a, b)}(\xi, h)||_{L^p}&\\
&\lesssim
\biggl(\int_{\rho_\nu \leq |x| \leq 1} (|\partial_r v_{a, b}^\nu|^p + |\partial_r \xi|^p)
r dr d\theta\biggr)^{\textstyle \frac{1}{p}}
\! + \biggl(\int_{\rho_\nu \leq |x| \leq 1} r^{-kp} |r^{-1} \natural_{(0, 0)}|^p
r dr d\theta\biggr)^{\textstyle \frac{1}{p}}\\
&\lesssim
||\partial_r v_{a, b}^\nu||_\infty + ||\partial_r \xi||_{L^q(N_{a, b}^\nu)}\\
&\quad +
\biggl(\int_{\rho_\nu}^{1} r^{-ks + 1} dr\biggr)^{\textstyle \frac{1}{s}}
\biggl(\int_{\rho_\nu \leq |x| \leq 1}
|r^{-1} \natural_{(0, 0)}|^q r dr d\theta\biggr)^{\textstyle \frac{1}{q}}\\
&\lesssim
\rho_\nu^{{\textstyle \frac{2}{s}} - k}
\hspace{50pt} (k \geq 1),
\end{flalign*}
\begin{flalign*}
&||\partial_{\tilde \rho_\nu}^k \partial_{\tilde \varphi_\nu} F^{(a, b)}(\xi, h)||_{L^p}&\\
&\lesssim
\biggl(\int_{\rho_\nu \leq |x| \leq 2\rho_\nu} r^p |r^{-1} \partial_r \partial_\theta
v_{a, b}^\nu|^p
r dr d\theta\biggr)^{\textstyle \frac{1}{p}}\\
&\quad +
\frac{1}{\rho_\nu} \biggl(\int_{\rho_\nu \leq |x| \leq 2\rho_\nu} r^p|r^{-1}
\partial_\theta \xi|^p
r dr d\theta\biggr)^{\textstyle \frac{1}{p}}\\
&\quad +
\biggl( \int_{\rho_\nu \leq |x| \leq 2\rho_\nu} r^{(1 - k)p} |r^{-2} \natural_{(0, 1)}|^p
r dr d\theta \biggr)^{\textstyle \frac{1}{p}}\\
&\lesssim
\biggl(\int_{\rho_\nu}^{2\rho_\nu} r^{p + 1} dr \biggr)^{\textstyle \frac{1}{p}}
|r^{-1} \partial_r \partial_\theta v_{a, b}^\nu|_\infty\\
&\quad +
\frac{1}{\rho_\nu} \biggl(\int_{\rho_\nu}^{2\rho_\nu} r^{s + 1} dr
\biggr)^{\textstyle \frac{1}{s}}
\biggl(\int_{\rho_\nu \leq |x| \leq 2\rho_\nu} |r^{-1} \partial_\theta \xi|^q r dr d\theta
\biggr)^{\textstyle \frac{1}{q}}\\
&\quad +
\biggl(\int_{\rho_\nu}^{2\rho_\nu} r^{(1 - k)s + 1} dr \biggr)^{\textstyle \frac{1}{s}}
\biggl(\int_{\rho_\nu \leq |x| \leq 2\rho_\nu} |r^{-2} \natural_{(0, 1)}|^q r dr d\theta
\biggr)^{\textstyle \frac{1}{q}}\\
&\lesssim
\rho_\nu^{\textstyle \frac{2}{s}} + \rho_\nu^{{\textstyle \frac{2}{s}} + 1 - k}\\
&\lesssim
\rho_\nu^{{\textstyle \frac{2}{s}} - (k - 1)_+} \hspace{50pt} (k \geq 0),
\end{flalign*}
\begin{flalign*}
&||\partial_{\tilde \rho_\nu}^k \partial_{\tilde \varphi_\nu}^l F^{(a, b)}(\xi, h)||_{L^p}&\\
&\lesssim
\biggl(\int_{\rho_\nu \leq |x| \leq 2\rho_\nu} r^{lp} |r^{-l} \partial_r \partial_\theta^l
v_{a, b}^\nu|^p r dr d\theta \biggr)^{\textstyle \frac{1}{p}}\\
&\quad +
\biggl(\int_{\rho_\nu \leq |x| \leq 2\rho_\nu} r^{(l - k)p} |r^{-(l + 1)} \natural_{(0, l)}|^p
r dr d\theta \biggr)^{\textstyle \frac{1}{p}}\\
&\lesssim
\biggl(\int_{\rho_\nu}^{2\rho_\nu} r^{lp + 1} dr \biggr)^{\textstyle \frac{1}{p}}
|r^{-l} \partial_r \partial_\theta^l v_{a, b}^\nu|_\infty\\
&\quad +
\biggl(\int_{\rho_\nu}^{2\rho_\nu} r^{(l - k)s + 1} dr \biggr)^{\textstyle \frac{1}{s}}
\biggl(\int_{\rho_\nu \leq |x| \leq 2\rho_\nu} |r^{-(l + 1)} \natural_{(0, 1)}|^q r dr d\theta
\biggr)^{\textstyle \frac{1}{q}}\\
&\lesssim
\rho_\nu^{{\textstyle \frac{2}{s}} + l - k} \hspace{30pt} (k \geq 0, l \geq 2),
\end{flalign*}
\begin{flalign*}
&||(D \partial_{\tilde \rho_\nu}^k F^{(a, b)})_{(\xi, h)} (\hat \xi, \hat h)||_{L^p}&\\
&\lesssim
\biggl(\int_{\rho_\nu \leq |x| \leq 1} |\partial_r \hat \xi|^p r dr d\theta
\biggr)^{\textstyle \frac{1}{p}}
+ \biggl(\int_{\rho_\nu \leq |x| \leq 1} r^{-kp} |r^{-1} \natural_{(1, 0)}|^p
r dr d\theta \biggr)^{\textstyle \frac{1}{p}}\\
&\lesssim
||\hat \xi||_{1,q}
+ \biggl(\int_{\rho_\nu}^{1} r^{-ks + 1} dr \biggr)^{\textstyle \frac{1}{s}}
\biggl(\int_{\rho_\nu \leq |x| \leq 1} |r^{-1} \natural_{(1, 0)}|^q r dr d\theta
\biggr)^{\textstyle \frac{1}{q}}\\
&\lesssim
\rho_\nu^{{\textstyle \frac{2}{s}} - k} ||\hat \xi||_{1, q} \hspace{20pt} (k \geq 1),
\end{flalign*}
\begin{flalign*}
&||(D \partial_{\tilde \rho_\nu}^k \partial_{\tilde \varphi_\nu} F^{(a, b)})_{(\xi, h)}
(\hat \xi, \hat h)||_{L^p}\\
&\lesssim
\frac{1}{\rho_\nu} \biggl(\int_{\rho_\nu \leq |x| \leq 2\rho_\nu} r^p |r^{-1} \partial_\theta
\hat \xi|^p r dr d\theta\biggr)^{\textstyle \frac{1}{p}}\\
&\quad +
\biggl(\int_{\rho_\nu \leq |x| \leq 2\rho_\nu} r^{-(k - 1)p} |r^{-2} \natural_{(1, 1)}|^p
r dr d\theta
\biggr)^{\textstyle \frac{1}{p}}\\
&\lesssim
\rho_\nu^{-1} \biggl(\int_{\rho_\nu}^{2\rho_\nu} r^{s + 1} dr\biggr)^{\textstyle \frac{1}{s}}
\biggl(\int_{\rho_\nu \leq |x| \leq 2\rho_\nu} |r^{-1} \partial_\theta \hat \xi|^q
r dr d\theta \biggr)^{\textstyle \frac{1}{q}}\\
&\quad +
\biggl(\int_{\rho_\nu}^{2\rho_\nu} r^{-(k - 1)s + 1} dr \biggr)^{\textstyle \frac{1}{s}}
\biggl(\int_{\rho_\nu \leq |x| \leq 2\rho_\nu} |r^{-2} \natural_{(1, 1)}|^q r dr d\theta
\biggr)^{\textstyle \frac{1}{q}}\\
&\lesssim
\rho_\nu^{{\textstyle \frac{2}{s}}-(k - 1)_+} ||\hat \xi||_{1, q} \hspace{20pt} (k \geq 0),
\hspace{200pt}
\end{flalign*}
\begin{flalign*}
&||(D^m \partial_{\tilde \rho_\nu}^k \partial_{\tilde \varphi_\nu}^l F^{(a, b)})_{(\xi, h)}
(\hat \xi^{(m)}, \hat h^{(m)}) (\hat \xi^{(m - 1)}, \hat h^{(m - 1)}) \dots
(\hat \xi^{(1)}, \hat h^{(1)})||_{L^p}&\\
&\lesssim
\biggl( \int_{\rho_\nu \leq |x| \leq 2\rho_\nu} r^{(l - k)p} |r^{-(l + 1)} \natural_{(m, l)}|^p
r dr d\theta \biggr)^{\textstyle \frac{1}{p}}\\
&\lesssim
\biggl(\int_{\rho_\nu}^{2\rho_\nu} r^{(l - k)s + 1} dr \biggr)^{\textstyle \frac{1}{s}}
\biggl(\int_{\rho_\nu \leq |x| \leq 2\rho_\nu} |r^{-(l + 1)} \natural_{(m, l)}|^q r dr d\theta
\biggr)^{\textstyle \frac{1}{q}}\\
&\lesssim
\rho_\nu^{{\textstyle \frac{2}{s}} + (l - k)}
||\hat \xi^{(m)}||_{1, q}||\hat \xi^{(m - 1)}||_{1, q} \dots ||\hat \xi^{(1)}||_{1, q}
\quad (m \geq 2, l > 0 \\
&\hphantom{\lesssim
\rho_\nu^{{\textstyle \frac{2}{s}} + (l - k)}
||\hat \xi^{(m)}||_{1, q}||\hat \xi^{(m - 1)}||_{1, q} \dots ||\hat \xi^{(1)}||_{1, q}
\quad (}
\text{ or } m = 1, k \geq 0, l \geq 2).
\end{flalign*}
\begin{flalign*}
&||(D^m \partial_{\tilde \rho_\nu}^k F^{(a, b)})_{(\xi, h)}
(\hat \xi^{(m)}, \hat h^{(m)}) (\hat \xi^{(m - 1)}, \hat h^{(m - 1)}) \dots
(\hat \xi^{(1)}, \hat h^{(1)})||_{L^p}&\\
&\lesssim
\biggl( \int_{\rho_\nu \leq |x| \leq 1} r^{(l - k)p} |r^{-(l + 1)} \natural_{(m, l)}|^p
r dr d\theta \biggr)^{\textstyle \frac{1}{p}}\\
&\lesssim
\biggl(\int_{\rho_\nu}^1 r^{-ks + 1} dr \biggr)^{\textstyle \frac{1}{s}}
\biggl(\int_{\rho_\nu \leq |x| \leq 1} |r^{-(l + 1)} \natural_{(m, l)}|^q r dr d\theta
\biggr)^{\textstyle \frac{1}{q}}\\
&\lesssim
\rho_\nu^{{\textstyle \frac{2}{s}} - k}
||\hat \xi^{(m)}||_{1, q}||\hat \xi^{(m - 1)}||_{1, q} \dots ||\hat \xi^{(1)}||_{1, q}
\hspace{50pt} (m \geq 2, k > 0)
\end{flalign*}
These inequalities prove the claim.
(iii)
It is straightforward to prove this case using the equalities
\begin{align*}
\partial_{\tilde \varphi_\mu} F^{(\tilde a, \tilde b)}(\xi, h)|_{(\tilde a, \tilde b) = (a, b)}
&= \frac{1}{2} \chi'(s) (g^\mu_t(v_{a, b}^{\mu, \mathrm{left}} + \xi)
+ \partial_t(v_{a, b}^{\mu, \mathrm{left}} + \xi))\\
\partial_{\tilde \varphi_\mu}^2 F^{(\tilde a,\tilde b)}(\xi, h) &= 0
\end{align*}
and
\begin{align*}
\partial_{\tilde b_\mu} F^{(\tilde a,\tilde b)}(\xi, h)|_{(\tilde a,\tilde b) = (a,b)}
&= \frac{1}{2} \chi'(s) \partial_\sigma\\
\partial_{\tilde b_\mu}^2 F^{(\tilde a,\tilde b)}(\xi, h) &= 0
\end{align*}
on $[-1, 0] \times S^1_\mu$.
\end{proof}
For each $i \in \Pi$, we fix a index $\mu_i \in M_i$.
Then a coordinate of $\mathring{X}_{\Pi, \Pi'}$ is given by
$(j, (b_\mu)_{\mu}, (\rho_{\mu_i})_{i \in \Pi}, (\varphi_\mu)_{\mu},
(\rho_\nu^{2\pi} e^{2\pi\sqrt{-1} \varphi_\nu})_{\nu \in \Pi'})$.
Note that in this coordinate,
$\rho_\mu = \rho_{\mu_i}^{L_{\mu_i} / L_\mu} e^{(b_\mu - b_{\mu_i}) / L_\mu}$
for any $\mu \in M_i$ ($i \in \Pi$).
We rewrite the above lemma in this coordinate and get the following corollary.
(The meaning of $\partial_{\mu_i}$ and $\partial_b$ in the following corollary
are different from that in Lemma \ref{estimates of implicit function 0}.)
\begin{cor}\label{estimates of implicit function}\
\begin{enumerate}[label=\normalfont(\roman*)]
\item
For any $0 < \delta_\mu \leq \delta'_\mu < \kappa_\mu$, $2 < p < \infty$,
$i \in \Pi$, $k \neq 0$ and multi-index $k_b = (k_{b_\mu})$,
there exists some constant $C>0$ and
$c_0 > 0$ such that if $||\xi||_{\widetilde{W}^{1,p}_{\delta'}} \leq c_0$, then
\begin{align*}
||(\partial_{\rho_{\mu_i}}^k \partial_b^{k_b} F^{(a, b)})(\xi, h)||
_{L_{\delta}^p(\widetilde{P}_a)}
&\leq C \rho_{\mu_i}^{L_{\mu_i} \tilde \delta_i /2 - k},\\
||(D \partial_{\rho_{\mu_i}}^k \partial_b^{k_b} F^{(a, b)})_{(\xi, h)}
(\hat \xi, \hat h)||_{L_\delta^p(\widetilde{P}_a)}
&\leq C \rho_{\mu_i}^{L_{\mu_i} \tilde \delta_i /2 - k}
||\hat \xi||_{\widetilde{W}_{\delta'}^{1, p}(\bigcup_{\mu \in M_i} N_{a, b}^\mu)},\\
D^m \partial_{\rho_{\mu_i}}^k \partial_b^{k_b} F^{(a, b)}
&\equiv 0 \quad (m \geq 2),
\end{align*}
where $\tilde \delta_i = \min\{(\delta'_\mu - \delta_\mu) / L_\mu;
\mu \in M_i\}$.
\item
For any $2 < p \leq q < \infty$,
$\nu \in \Pi'$ and $(m, k, l)$ such that $(k, l) \neq (0, 0)$,
there exists some constants $C > 0$ and $c_0 > 0$ such that
if $||\xi||_{W^{1, q}(N_{a, b}^\nu)} \leq c_0$, then
\begin{align*}
||(D^m \partial_{\rho_\nu}^k \partial_{\varphi_\nu}^l F^{(a, b)})_{(\xi, h)}
(\hat \xi^{(m)}, \hat h^{(m)}) (\hat \xi^{(m-1)}, \hat h^{(m-1)}) \dots
(\hat \xi^{(1)}, \hat h^{(1)})||_{L^p(N_{a, b}^\nu)}\\
\leq C \rho_\nu^{2/p - 2/q - k} \prod_{i = 1}^m ||\hat \xi^{(i)}||_{W^{1,q}(N_{a, b}^\nu)}.
\end{align*}
\item
For any $2 < p < \infty$, $0< \delta < \delta_0$,
and multi-index $(m, k_j, (l_\mu), k_b = (k_{b_\mu}))$,
there exists some constants $C > 0$ and $c_0 > 0$ such that
if $||\xi||_{\widetilde{W}^{1,p}_\delta} + |h|_E \leq c_0$, then
\begin{align*}
||(D^m \partial_j^{k_j} \partial_{(\varphi_\mu)}^{(l_\mu)} \partial_b^{k_b}
F^{(a, b)})_{(\xi,h)}
(\hat \xi^{(m)}\!, \hat h^{(m)}) (\hat \xi^{(m-1)}\!, \hat h^{(m-1)})\! \dots \!
(\hat \xi^{(1)}\!, \hat h^{(1)})||_{L^p_\delta(\widetilde{P}_a)}\\
\leq C \prod_{i = 1}^m (||\hat \xi^{(i)}||_{\widetilde{W}^{1,p}_\delta(\widetilde{P}_a)}
+ |\hat h^{(i)}|_E).
\end{align*}
\end{enumerate}
\end{cor}
Let $U \subset \mathring{X}_{\Pi, \Pi'}$ be a neighborhood of $(a, b)$, and
regard the family of smooth maps
\[
\phi^{\tilde a,\tilde b} : \mathop{\mathrm{Ker}}\nolimits D_0 \supset B_\epsilon(0)
\to \widetilde{W}^{1,p}_\delta(\widetilde{P}_a, u_{a, b}^\ast T \hat Y) \times E^0
\]
as a map
\begin{equation}
\phi : U \times B_\epsilon(0) \to
\widetilde{W}^{1,p}_\delta(\widetilde{P}_a, u_{a, b}^\ast T \hat Y) \times E^0.
\label{phi 1}
\end{equation}
We estimate the derivative of $\phi$ at $(a, b, x) \in U \times B_\epsilon(0)$.
As we have already mentioned, the domain of $\phi^{a, b}$ or $\epsilon > 0$
may depend on $2 < p < \infty$ and $0 < \delta < \delta_0$.
Hence in the following Proposition, we need to assume that $(a, b, x)$ is
sufficiently close to $(0, b^0, 0)$ for given $p$, $q$, $\delta$ and $\delta'$
to guarantee that $(a, b, x)$ is contained in the domains of various $\phi$.
\begin{prop}\label{asymptotic phi}
For any $2 < p < q$, $0 < \delta_\mu < \delta'_\mu < \kappa_\mu$ and any multi-index
$(k_x, k_j, k_b, (k_{\mu_i})_{i \in \Pi}, (l_\mu)_\mu, (k_\nu)_{\nu \in \Pi'},
(l_\nu)_{\nu \in \Pi'})$, there exists some constant $C > 0$ such that
\begin{multline*}
||\partial_x^{k_x} \partial_j^{k_j} \partial_b^{k_b} \partial_{(\rho_{\mu_i})}^{(k_{\mu_i})}
\partial_{(\varphi_\mu)}^{(l_\mu)} \partial_{(\rho_\nu)}^{(k_\nu)}
\partial_{(\varphi_\nu)}^{(l_\nu)} \phi(a, b, x)||_{\widetilde{W}_\delta^{1,p}
(\widetilde{P}_a, u_{a, b}^\ast T \hat Y) \times E^0}\\
\leq C \prod_{\substack{i \\ k_{\mu_i} \neq 0}}
\rho_{\mu_i}^{L_{\mu_i} \tilde \delta_i /2 - k_{\mu_i}}
\prod_{\substack{\nu \\ (k_\nu, l_\nu) \neq (0,0)}} \rho_\nu^{(2/p - 2/q) - k_\nu}
\end{multline*}
for any $(a, b, x) \in \mathring{X}_{\Pi, \Pi'} \times B_\epsilon(0)$ sufficiently close
to $(0, b^0, 0)$.
Furthermore, if $(k_{\nu_0}, l_{\nu_0}) = (0,0)$ then
\begin{multline*}
||\partial_x^{k_x} \partial_j^{k_j} \partial_b^{k_b} \partial_{(\rho_{\mu_i})}^{(k_{\mu_i})}
\partial_{(\varphi_\mu)}^{(l_\mu)} \partial_{(\rho_\nu)}^{(k_\nu)}
\partial_{(\varphi_\nu)}^{(l_\nu)} \phi(a, b, x)||_{\widetilde{W}^{1,q}
(N_{a, b}^{\nu_0})}\\
\leq C \prod_{\substack{i \\ k_{\mu_i} \neq 0}}
\rho_\mu^{L_{\mu_i} \tilde \delta_i /2 - k_{\mu_i}}
\prod_{\substack{\nu \\ (k_\nu, l_\nu) \neq (0,0)}} \rho_\nu^{(2/p - 2/q) - k_\nu}
\end{multline*}
for any $(a, b, x) \in \mathring{X}_{\Pi, \Pi'} \times B_\epsilon(0)$ sufficiently close
to $(0, b^0, 0)$.
\end{prop}
\begin{proof}
We prove the claim by the induction in $|k_x| + |k_j| + |k_b|
+ |(k_{\mu_i})| + |(l_\mu)| + |(k_\nu)| + |(l_\nu)|$.
The case $(k_x, k_j, k_x, (k_{\mu_i}), (l_\mu), (k_\nu), (l_\nu)) = (0, \dots, 0)$ is obvious.
Differentiating the equation $F^{(\tilde a,\tilde b) +} (\phi(\tilde a, \tilde b, x)) = (0, x)$
of smooth functions on a fixed curve $\widetilde{P}_a$
by $\partial_x^{k_x} \partial_{\tilde \jmath}^{k_j} \partial_{\tilde b}^{k_b}
\partial_{(\tilde \rho_{\mu_i})}^{(k_{\mu_i})} \partial_{(\tilde \varphi_\mu)}^{(l_\mu)}
\partial_{(\tilde \rho_\nu)}^{(k_\nu)} \partial_{(\tilde \varphi_\nu)}^{(l_\nu)}$,
we obtain an equation of the following form.
\begin{align}
&(DF^{(a, b) +})_{\phi(a, b, x)}
\partial_x^{k_x} \partial_j^{k_j} \partial_b^{k_b} \partial_{(\rho_{\mu_i})}^{(k_{\mu_i})}
\partial_{(\varphi_\mu)}^{(l_\mu)} \partial_{(\rho_\nu)}^{(k_\nu)}
\partial_{(\varphi_\nu)}^{(l_\nu)} \phi \notag\\
&+
\sum_{\star_1}
(D^{m} \partial_{\rho_{\nu_0}}^{k_{\nu_0}}\partial_{\varphi_{\nu_0}}^{l_{\nu_0}}
F^{(a, b) +})_{\phi(a, b, x)}
(\hat\xi^{(m)},\hat h^{(m)}) \dots (\hat\xi^{(1)},\hat h^{(1)}) \notag\\
&+
\sum_{\star_2}
(D^{m} \partial_{\rho_{\mu_{i_0}}}^{k_{\mu_{i_0}}} \partial_b^{k'_b}
F^{(a, b) +})_{\phi(a, b, x)}
(\hat\xi^{(m)},\hat h^{(m)}) \dots (\hat\xi^{(1)},\hat h^{(1)})
\notag\\
&+
\sum_{\star_3}
(D^{m} \partial_x^{k'_x} \partial_j^{k'_j} \partial_b^{k'_b}
\partial_{(\rho_\mu)}^{(k'_\mu)} \partial_{(\varphi_\mu)}^{(l'_\mu)}
\partial_{(\rho_\nu)}^{(k'_\nu)} \partial_{(\varphi_\nu)}^{(l'_\nu)}
F^{(a, b) +})_{\phi(a, b, x)}
(\hat\xi^{(m)},\hat h^{(m)}) \dots (\hat\xi^{(1)},\hat h^{(1)}) \notag\\
&=0,\label{good bad}
\end{align}
where each $(\hat\xi^{(l)},\hat h^{(l)})$ is some derivative of $\phi$, and
the sum of the indices of differentials which appear in each term is equal to
$(k_x, k_j, k_b, (k_{\mu_i}), (l_\mu), (k_\nu), (l_\nu))$;
in the sum $\star_1$,
$(k_{\nu_0}, l_{\nu_0}) \neq (0,0)$ and
each $(\hat\xi^{(l)},\hat h^{(l)})$ is some differential of $\phi$
by $\partial_x$, $\partial_j$, $\partial_b$, $\partial_{\rho_{\mu_i}}$,
$\partial_{\varphi_\mu}$, $\partial_{\rho_{\nu}}$ and $\partial_{\varphi_{\nu}}$
except $\partial_{\rho_{\nu_0}}$ and $\partial_{\varphi_{\nu_0}}$;
in the sum $\star_2$, $k_{\mu_{i_0}} \neq 0$, $m = 0, 1$ and
each $(\hat\xi^{(l)},\hat h^{(l)})$ is some differential of $\phi$
by $\partial_x$, $\partial_j$, $\partial_b$, $\partial_{\rho_{\mu_i}}$,
$\partial_{\varphi_{\mu}}$, $\partial_{\rho_\nu}$ and $\partial_{\varphi_\nu}$
except $\partial_{\rho_{\mu_{i_0}}}$;
in the sum $\star_3$, if $(k_\nu, l_\nu) \neq (0,0)$ then
$(k'_\nu, l'_\nu) \neq (k_\nu, l_\nu)$, and if $k_{\mu_i} \neq 0$ then
$k'_{\mu_i} < k_{\mu_i}$.
(As we have noted in Remark \ref{independent}, for example, if $k'_\nu \neq 0$ and
$k'_{\mu_i} \neq 0$, then this term vanishes.)
Corollary \ref{estimates of implicit function} (ii) and the assumption of the induction
(the second inequality) imply that the $L^p$-norm of each term
in the sum $\star_1$ is bounded by
\begin{align*}
&||(D^{m}\partial_{\rho_{\nu_0}}^{k_{\nu_0}}\partial_{\varphi_{\nu_0}}^{l_{\nu_0}}
F^{(a, b) +})_{\phi(a, b, x)}
(\hat\xi^{(m)},\hat h^{(m)}) \dots (\hat\xi^{(1)},\hat
h^{(1)})||_{L^p(N_{a, b}^\nu)}\\
&\lesssim
\rho_{\nu_0}^{2/p - 2/q-k_{\nu_0}} \prod_i ||\hat\xi^{(i)}||_{W^{1,q}(N_{a, b}^{\nu_0})}\\
&\lesssim
\prod_{\substack{i \\ k_{\mu_i} \neq 0}}
\rho_\mu^{L_{\mu_i} \tilde \delta_i /2 - k_{\mu_i}}
\prod_{\substack{\nu \\ (k_{\nu},l_{\nu})\neq (0,0)}} \rho_{\nu}^{2/p - 2/q-k_{\nu}}
\end{align*}
since $\partial_{\rho_{\nu}}$ or $\partial_{\varphi_{\nu}}$ appears in some
$(\hat\xi^{(l)},\hat h^{(l)})$
for each $\nu \neq \nu_0$ such that $(k_{\nu},l_{\nu}) \neq (0,0)$,
$\partial_{\rho_{\mu_i}}$ appears in some $(\hat\xi^{(l)},\hat h^{(l)})$ for each $i$
such that $k_{\mu_i} \neq 0$,
and neither $\partial_{\rho_{\nu_0}}$ nor $\partial_{\varphi_{\nu_0}}$ appears.
Next we consider the sum $\star_2$.
For each $i_0$, define a sequence of positive constants
$\delta'' = ((\delta''_\mu)_\mu, (\delta''_{\pm\infty_i})_{\pm\infty_i})$
by $\delta''_\mu = \delta'_\mu$ for $\mu \in M_{i_0}$, $\delta''_\mu = \delta_\mu$
for $\mu \notin M_{i_0}$, and $\delta''_{\pm\infty_i} = \delta_{\pm\infty_i}$.
Then Corollary \ref{estimates of implicit function} (i) and the assumption of the induction
(the first inequality) imply that the $L_{\delta}^p$-norm of the terms
with $m = 1$ in the sum $\star_2$ is bounded by
\begin{align*}
&||(D \partial_{\rho_{\mu_{i_0}}}^{k_{\mu_{i_0}}} F^{(a, b) +})_{\phi(a, b, x)}
(\hat\xi, \hat h)||_{L_\delta^p(\widetilde{P}_a)}\\
&\lesssim
\rho_{\mu_{i_0}}^{L_{\mu_{i_0}} \tilde \delta_{i_0} /2 - k_{\mu_{i_0}}}
||\hat \xi||_{\widetilde{W}_{\delta''}^{1, p}(\widetilde{P}_a)}\\
&\lesssim
\prod_{\substack{i \\ k_{\mu_i} \neq 0}}
\rho_{\mu_i}^{L_{\mu_i} \tilde \delta_i /2 - k_{\mu}}
\prod_{\substack{\nu \\ (k_\nu,l_\nu)\neq (0,0)}} \rho_\nu^{2/p - 2/q-k_\nu}
\end{align*}
since $\partial_{\rho_\nu}$ or $\partial_{\varphi_\nu}$ appears in some
$(\hat\xi^{(l)},\hat h^{(l)})$
for each $\nu$ such that $(k_\nu,l_\nu)\neq (0,0)$,
$\partial_{\rho_{\mu_i}}$ appears in some $(\hat\xi^{(l)},\hat h^{(l)})$ for each
$i \neq i_0$ such that $k_{\mu_i} \neq 0$,
and $\partial_{\rho_{\mu_{i_0}}}$ does not appear.
If the terms with $m = 0$ appear in the sum $\star_2$, then
$\partial_x^{k_x} \partial_j^{k_j} \partial_b^{k_b} \partial_{(\rho_{\mu_i})}^{(k_{\mu_i})}
\partial_{(\varphi_\mu)}^{(l_\mu)} \partial_{(\rho_\nu)}^{(k_\nu)}
\partial_{(\varphi_\nu)}^{(l_\nu)} = \partial_{\rho_{\mu_{i_0}}}^{k_{\mu_{i_0}}}$, and
Corollary \ref{estimates of implicit function} (i) implies
\[
||\partial_{\rho_{\mu_{i_0}}}^{k_{\mu_{i_0}}} F^{(a, b) +} (\phi(a, b, x))||_{
L_\delta^p(\widetilde{P}_a)} \lesssim
\rho_{\mu_{i_0}}^{L_{\mu_{i_0}} \tilde \delta_{i_0} /2 - k_{\mu_{i_0}}}.
\]
Similarly, Corollary \ref{estimates of implicit function} (i) for $\delta' = \delta$,
(ii) for $q = p$, (iii) and the assumption of the induction (the first inequality) imply
\begin{align*}
&||(D^{m} \partial_x^{k'_x} \partial_j^{k'_j} \partial_b^{k'_b}
\partial_{(\rho_{\mu_i})}^{(k'_{\mu_i})} \partial_{(\varphi_\mu)}^{(l'_\mu)}
\partial_{(\rho_\nu)}^{(k'_\nu)} \partial_{(\varphi_\nu)}^{(l'_\nu)}
F^{(a, b) +})_{\phi(a, b, x)}
(\hat\xi^{(m)}\!,\hat h^{(m)}) \!\dots\! (\hat\xi^{(1)}\!,\hat h^{(1)})||_{L^p_\delta
(\widetilde{P}_a)}\\
&\lesssim
\prod_{\substack{i \\ k'_{\mu_i} \neq 0}} \rho_{\mu_i}^{- k'_{\mu_i}}
\prod_{\substack{\nu \\ (k'_\nu, l'_\nu) \neq (0,0)}} \rho_\nu^{-k'_\nu}
\prod_l (||\hat\xi^{(l)}||_{\widetilde{W}^{1,p}_\delta(\widetilde{P}_a)} + |\hat h^{(l)}|_E)\\
&\lesssim
\prod_{\substack{i \\ k_{\mu_i} \neq 0}}
\rho_\mu^{L_{\mu_i} \tilde \delta_i /2 - k_{\mu_i}}
\prod_{\substack{\nu \\ (k_\nu,l_\nu)\neq (0,0)}} \rho_\nu^{2/p - 2/q-k_\nu}
\end{align*}
since $\partial_{\rho_\nu}$ or $\partial_{\varphi_\nu}$ appears in some
$(\hat\xi^{(l)},\hat h^{(l)})$ for each $\nu$ such that $(k_\nu,l_\nu)\neq (0,0)$,
and $\partial_{\rho_{\mu_i}}$ appears in some $(\hat\xi^{(l)},\hat h^{(l)})$ for each $i$
such that $k_{\mu_i} \neq 0$.
Since $(DF^{(a, b) +}_{\phi(a, b, x)})^{-1}$ is uniformly bounded,
these estimates imply the first inequality of the claim.
Next we prove the second inequality.
If $(k_{\nu_0},l_{\nu_0})=(0,0)$, then the restriction of equation (\ref{good bad})
to $L^p(N_{a, b}^{\nu_0})$ is
\begin{align}
&(DF^{(a, b)})_{\phi(a, b, x)}
\partial_x^{k_x} \partial_j^{k_j} \partial_b^{k_b} \partial_{(\rho_{\mu_i})}^{(k_{\mu_i})}
\partial_{(\varphi_\mu)}^{(l_\mu)} \partial_{(\rho_\nu)}^{(k_\nu)}
\partial_{(\varphi_\nu)}^{(l_\nu)} \phi \notag\\
&+
\sum (D^{m} F^{(a, b)})_{\phi(a, b, x)}(\hat\xi^{(m)},\hat h^{(m)})
\dots (\hat\xi^{(1)},\hat h^{(1)}) =0. \label{near node}
\end{align}
Sobolev embedding and the first inequality of the claim imply
\begin{align*}
&||\partial_x^{k_x} \partial_j^{k_j} \partial_b^{k_b} \partial_{(\rho_{\mu_i})}^{(k_{\mu_i})}
\partial_{(\varphi_\mu)}^{(l_\mu)} \partial_{(\rho_\nu)}^{(k_\nu)}
\partial_{(\varphi_\nu)}^{(l_\nu)} \phi||_{L^q(N_{a, b}^{\nu_0})}\\
&\lesssim
||\partial_x^{k_x} \partial_j^{k_j} \partial_b^{k_b} \partial_{(\rho_{\mu_i})}^{(k_{\mu_i})}
\partial_{(\varphi_\mu)}^{(l_\mu)} \partial_{(\rho_\nu)}^{(k_\nu)}
\partial_{(\varphi_\nu)}^{(l_\nu)} \phi||_{W^{1,p}(N_{a, b}^{\nu_0})}\\
&\lesssim
\prod_{\substack{i \\ k_{\mu_i} \neq 0}}
\rho_\mu^{L_{\mu_i} \tilde \delta_i /2 - k_{\mu_i}}
\prod_{\substack{\nu \\ (k_\nu,l_\nu)\neq (0,0)}} \rho_\nu^{2/p - 2/q-k_\nu}.
\end{align*}
Corollary \ref{estimates of implicit function} (iii) and the assumption of the induction
(the second inequality) imply
\begin{align*}
&||(D^{m} F^{(a, b)})_{\phi(a, b, x)}(\hat\xi^{(m)},\hat h^{(m)})
\dots (\hat\xi^{(1)},\hat h^{(1)})||_{L^q(N_{a, b}^{\nu_0})}\\
&\lesssim
\prod_{\substack{i \\ k_{\mu_i} \neq 0}}
\rho_\mu^{L_{\mu_i} \tilde \delta_i /2 - k_{\mu_i}}
\prod_{\substack{\nu \\ (k_\nu,l_\nu)\neq (0,0)}} \rho_\nu^{2/p - 2/q-k_\nu}.
\end{align*}
Hence $W^{1,q}$ regularity property of the elliptic operator
$(DF^{(a, b)})_{\phi(a, b, x)}$ in (\ref{near node})
implies the second inequality.
(Note that the regularity property of $(DF^{(a, b)})_{\phi(a, b, x)}$
is uniform with respect to small $\zeta_\nu$.
See Remark \ref{uniform regularity}.)
\end{proof}
Next we regard the family of smooth maps
\[
\phi^{a, b} : \mathop{\mathrm{Ker}}\nolimits D_0 \supset B_\epsilon(0)
\to \widetilde{W}^{1,p}_\delta(\widetilde{P}_a, u_{a, b}^\ast T \hat Y) \times E^0
\]
as a map
\begin{align}
\phi : \mathring{X} \times B_\epsilon(0) &\to
W^{1,p}(\Sigma_0 \setminus N_0, (\mathbb{R}_1 \sqcup \mathbb{R}_2 \sqcup \dots
\sqcup \mathbb{R}_k) \times Y) \times E^0 \notag\\
((a,b), x) &\mapsto (\Phi(z, \xi_{a, b, x}(z))|_{\Sigma_0 \setminus N_0}, h_{a, b, x}),
\label{phi 2}
\end{align}
where $\xi_{a, b, x}$ and $h_{a, b, x}$ is defined by
$\phi^{a,b}(x) = (\xi_{a, b, x}, \xi_{a, b, x})$.
For each $i = 1, \dots, k-1$, fix a index $\mu_i \in M_i$.
Then a coordinate of $\mathring{X} \subset \widetilde{X} \times \prod_\mu \mathbb{R}_\mu$
is given by $(j, (b_\mu), (\rho_{\mu_i})_i, (\varphi_\mu)_\mu,
(\varphi_\nu^{2\pi} e^{2\pi\sqrt{-1} \varphi_\nu})_\nu)$.
For a neighborhood $U \subset \mathring{X}_{\Pi, \Pi'}$ of each point
$(a, b) \in \mathring{X}_{\Pi, \Pi'}$, the restriction of (\ref{phi 2})
to $U \times B_\epsilon(0)$ is the composition of the map (\ref{phi 1})
and the projection
$\widetilde{W}^{1,p}_\delta(\widetilde{P}_a, u_{a, b}^\ast T \hat Y) \times E^0
\to W^{1,p}(\Sigma_0 \setminus N_0, (\mathbb{R}_1 \sqcup \mathbb{R}_2 \sqcup \dots
\sqcup \mathbb{R}_k) \times Y) \times E^0$.
Furthermore, the norm of this projection is uniform with respect to $(a, b)$.
Therefore, Proposition \ref{asymptotic phi} implies
\begin{multline*}
||\partial_x^{k_x} \partial_j^{k_j} \partial_b^{k_b} \partial_{(\rho_{\mu_i})}^{(k_{\mu_i})}
\partial_{(\varphi_\mu)}^{(l_\mu)} \partial_{(\rho_\nu)}^{(k_\nu)}
\partial_{(\varphi_\nu)}^{(l_\nu)} \phi(a, b, x)||_{W^{1,p}(\Sigma_0 \setminus N_0,
(\mathbb{R}_1 \sqcup \mathbb{R}_2 \sqcup \dots \sqcup \mathbb{R}_k) \times Y) \times E^0}\\
\leq C \prod_{\substack{i \\ k_{\mu_i} \neq 0}}
\rho_{\mu_i}^{L_{\mu_i} \tilde \delta_i/2 - k_{\mu_i}}
\prod_{\substack{\nu \\ (k_\nu, l_\nu) \neq (0,0)}} \rho_\nu^{(2/p - 2/q) - k_\nu},
\end{multline*}
where $\tilde \delta_i = \min\{(\delta'_\mu - \delta_\mu) /L_\mu;
\mu \in M_i\}$.
The same estimate holds true for any Sobolev norm $W^{k, p}$
or $C^l$-norm instead of $W^{1, p}$ if we change the constant $C > 0$
because of elliptic regularity.
Since these estimates hold true for
arbitrary $2 < p < q < \infty$ and $0 < \delta \leq \delta' < \delta_0$
such that $0 < \delta_\mu \leq \delta'_\mu < \kappa_\mu$ if we shrink
the domain of $\phi$, the following corollary holds true.
Define $\tilde \delta_{0, i} = \min\{ \kappa_\mu / L_\mu; \mu \in M_i\}$ for each $i$.
We regard $\phi$ as a map
\[
\phi : \mathring{X} \times B_\epsilon(0) \to
C^l(\Sigma_0 \setminus N_0, (\mathbb{R}_1 \sqcup \mathbb{R}_2 \sqcup \dots
\sqcup \mathbb{R}_k) \times Y) \times E^0.
\]
\begin{cor}\label{asymptotic estimates of phi}
For any $l \geq 1$, $0 < \epsilon < 1$,
$0 < \tilde \delta'_{0, i} < \tilde \delta_{0, i}$,
$(\Pi, \Pi')$ and any multi-index $(k_x, \allowbreak k_j, \allowbreak k_b, \allowbreak (k_{\mu_i})_{i \in \Pi}, \allowbreak
(l_\mu)_\mu, (k_\nu)_{\nu \in \Pi'}, (l_\nu)_{\nu \in \Pi'})$,
there exists some constant $C > 0$ such that
\begin{multline*}
||\partial_x^{k_x} \partial_j^{k_j} \partial_b^{k_b} \partial_{(\rho_{\mu_i})}^{(k_{\mu_i})}
\partial_{(\varphi_\mu)}^{(l_\mu)} \partial_{(\rho_\nu)}^{(k_\nu)}
\partial_{(\varphi_\nu)}^{(l_\nu)} \phi(a, b, x)||_{C^l(\Sigma_0 \setminus N_0,
(\mathbb{R}_1 \sqcup \mathbb{R}_2 \sqcup \dots \sqcup \mathbb{R}_k) \times Y) \times E^0}\\
\leq C \prod_{\substack{i \\ k_{\mu_i} \neq 0}}
\rho_\mu^{L_{\mu_i} \tilde \delta'_{0, i}/2 - k_{\mu_i}}
\prod_{\substack{\nu \\ (k_\nu, l_\nu) \neq (0,0)}} \rho_\nu^{\epsilon - k_\nu}
\end{multline*}
for all $(a, b, x) \in \mathring{X}_{\Pi, \Pi'} \times B_\epsilon(0)$
sufficiently close to $(0, b^0, 0)$.
\end{cor}
Recall that we give a strong differential structure to
$\widetilde{X}$ determined by fixed constants $\alpha$ and $\beta$,
and give $\hat V$ the product smooth structure.
(See the beginning of this section.)
\begin{cor}
For any $N$, $\phi$ is of class $C^N$ if $\alpha$ and $\beta$ are sufficiently large.
\end{cor}
\begin{proof}
If we change the coordinate $\rho_\mu$ and $\rho_\nu$ to $\hat \rho_\mu$ and
$\hat \rho_\nu$ respectively by
$\rho_\mu^{L_\mu} = (\hat \rho_\mu)^\beta$ and
$\rho_\nu = (\hat \rho_\nu)^\alpha$,
then the previous corollary implies that for any $l \geq 1$ and $0 < \epsilon < 1$,
\begin{align*}
&\Bigl|\Bigl|\partial_x^{k_x} \partial_j^{k_j} \partial_b^{k_b}
\partial_{(\hat \rho_{\mu_i})}^{(k_{\mu_i})}
\partial_{(\varphi_\mu)}^{(l_\mu)} \partial_{(\hat \rho_\nu)}^{(k_\nu)}
\Bigl(\prod_\nu \frac{1}{\hat \rho_\nu^{l_\nu}}\Bigr)\partial_{(\varphi_\nu)}^{(l_\nu)}
\phi(a, b, x)\Bigr|\Bigr|_{C^l(\Sigma_0 \setminus N_0, (\mathbb{R}_1 \sqcup \mathbb{R}_2 \sqcup
\dots \sqcup \mathbb{R}_k) \times Y) \times E^0}\\
&\lesssim \prod_{\substack{i \\ k_{\mu_i} \neq 0}}
(\hat \rho_{\mu_i})^{\beta \tilde \delta'_{0, i} / 2 - k_{\mu_i}}
\prod_{\substack{\nu \\ (k_\nu, l_\nu) \neq (0,0)}}
(\hat \rho_\nu)^{\epsilon \alpha - (k_\nu + l_\nu)}.
\end{align*}
If $\alpha$ and $\beta$ are sufficiently large, then
$\beta \tilde \delta'_{0, i} / 2 - N > 0$ and
$\epsilon \alpha - N > 0$.
Hence the claim follows from the fact that if a continuous function $f$ on a manifold
$U$ is continuously differentiable on the complement of a submanifold $S \subset U$
and the limit of its differential on $S$ is zero, then $f$ is continuously differentiable
on the entire space $U$.
\end{proof}
Since we use the same coordinates for the neighborhoods of limit circles
of $\widetilde{P}_a$ as those of $\Sigma_0$,
the above argument also implies that for any limit circle $S^1_{\pm\infty_i}$
of $\Sigma_0$,
\begin{align*}
\hat V &\to P\\
(a, b, x) &\mapsto \pi_Y \circ \Phi_{a, b}(\xi_x) \circ \phi_{\pm\infty_i}
\end{align*}
is smooth if we fix a coordinate $\phi_{\pm\infty_i} : S^1 \stackrel{\cong}{\to} S^1_{\pm\infty_i}$.
Similarly,
\begin{align*}
\hat V &\to \mathbb{R}\\
(a, b, x) &\to
\lim_{s \to \infty}(\sigma \circ \Phi_{a, b}(\xi_x)|_{[0, \infty) \times S^1_{+\infty_i}}(s, t)
- (0_{k_0} + L_{+\infty_i} s))
\end{align*}
and
\begin{align*}
\hat V &\to \mathbb{R}\\
(a, b, x) &\to
\lim_{s \to -\infty}(\sigma \circ \Phi_{a, b}(\xi_x)|_{(-\infty, 0] \times S^1_{-\infty_i}}(s, t)
- (0_1 + L_{-\infty_i} s))
\end{align*}
are smooth since
\begin{align*}
&\lim_{s \to \infty}(\sigma \circ \Phi_{a, b}(\xi_x)|_{[0, \infty) \times S^1_{+\infty_i}}(s, t)
- (0_{k_0} + L_{+\infty_i} s))\\
& = \pi_{\mathbb{R}_{+\infty_i}} \xi_x
+ \lim_{s \to \infty}(\sigma \circ u_0|_{[0, \infty) \times S^1_{+\infty_i}}(s, t)
- (0_{k_0} + L_{+\infty_i} s))
\end{align*}
and
\begin{align*}
&\lim_{s \to -\infty}(\sigma \circ \Phi_{a, b}(\xi_x)|_{(-\infty, 0] \times S^1_{-\infty_i}}(s, t)
- (0_1 + L_{-\infty_i} s))\\
& = \pi_{\mathbb{R}_{-\infty_i}} \xi_x
+ \lim_{s \to -\infty}(\sigma \circ u_0|_{(-\infty, 0] \times S^1_{-\infty_i}}(s, t)
- (0_1 + L_{-\infty_i} s)),
\end{align*}
where $\pi_{\mathbb{R}_{\pm\infty_i}} : \widetilde{W}^{1, p}(\widetilde{P}_a; u_{a, b}^\ast T \hat Y)
\to \mathbb{R}$ is the projection to $\mathbb{R}\partial_\sigma \subset \mathop{\mathrm{Ker}}\nolimits A_{\pm\infty_i}$.
\subsection{Embedding of Kuranishi neighborhoods}\label{embed}
In this section, we explain the way to construct an embedding of
a Kuranishi neighborhood $(V_1, E_1, s_1, \psi_1, G_1)$ to
another $(V_2, E_2, s_2, \psi_2, G_2)$.
Assume that $\psi_1(s_1^{-1}(0))$ and $\psi_2(s_2^{-1}(0))$ share a point
$q_0 \in \widehat{\mathcal{M}}$. We also assume that the additional marked points
$z_1^+$ for $(V_1, E_1, s_1, \psi_1, G_1)$ is a subsequence of $z_2^+$ for
$(V_2, E_2, s_2, \psi_2, G_2)$, and $E_1$ is a subspace of $E_2$ at $q_0$.
We do not assume any relationship between the additional temporary data
$(z_1^{++}, S'_1, \hat R^1_i)$ used for the description of $(V_1, E_1, s_1, \psi_1, G_1)$ and
$(z_2^{++} = (z_{2, i}^{++}), S'_2, \hat R^2_i)$ for $(V_2, E_2, s_2, \psi_2, G_2)$.
More precisely, we assume the following conditions:
\begin{itemize}
\item
For each $l = 1, 2$,
a Kuranishi neighborhood $(V_l, E_l, s_l, \psi_l)$
of a point $p_l = (\Sigma_l, z_l, u_l) \in \widehat{\mathcal{M}}(Y, \lambda, J)$ is defined
by the data $(p_l^+ = (\Sigma_l, z_l \cup z_l^+, u_l), S_l, E^0_l, \lambda_l)$
and the additional data $(z_l^{++} = (z_{l, i}^{++}), S'_l, \hat R^l_i)$.
Let $(\hat P_l \to \hat X_l, Z_l \cup Z_l^+ \cup Z_{\pm\infty_i})$ be the local universal
family of the stabilization $(\hat \Sigma_l, z_l \cup z_l^+ \cup (\pm\infty_i))$ of
the blown down curve of $(\Sigma_l, z_l \cup z_l^+)$, and
$(\widetilde{P}_l \to \widetilde{X}_l, Z_l \cup Z_l^+ \cup Z_l^{++})$ be the local universal
family of $(\Sigma_l, z_l \cup z_l^+ \cup z_l^{++})$.
\item
We assume $S_1 \subset S_2$. (We do not assume any correspondence between $S'_1$
and $S'_2$.)
\item
$q_0 = (\Sigma_0, z_0, u_0) \in \widehat{\mathcal{M}}$ is a point in the intersection
$\psi_1(s_1^{-1}(0)) \cap \psi_2(s_2^{-1}(0))$.
Hence there exist $(a^1_0, b^1_0, x^1_0) \in V_1$ and $(a^2_0, b^2_0, x^2_0) \in V_2$
such that $q_0 = \psi_1(a^1_0, b^1_0, x^1_0) = \psi_2(a^2_0, b^2_0, x^2_0)$.
We assume that there exist $\mathbb{R}$-translations
$\theta_1^0 : (\overline{\mathbb{R}}_1 \sqcup \overline{\mathbb{R}}_2 \sqcup \dots \sqcup
\overline{\mathbb{R}}_{k_1})/\sim_{a^1_0, b^1_0}
\to \overline{\mathbb{R}}_1 \cup \overline{\mathbb{R}}_2 \cup \dots \cup
\overline{\mathbb{R}}_{k_0}$,
$\theta_2^0 : (\overline{\mathbb{R}}_1 \sqcup \overline{\mathbb{R}}_2 \sqcup \dots \sqcup
\overline{\mathbb{R}}_{k_2})/\sim_{a^2_0, b^2_0}
\to \overline{\mathbb{R}}_1 \cup \overline{\mathbb{R}}_2 \cup \dots \cup
\overline{\mathbb{R}}_{k_0}$ and
an isomorphism
\[
\Xi_0 : ((\widetilde{P}_1)_{a^1_0}, Z_1(a^1_0)) \stackrel{\cong}{\to} ((\widetilde{P}_2)_{a^2_0}, Z_2(a^2_0))
\]
such that $(\theta_2^0 \times 1) \circ \Phi_{a^2_0, b^2_0}(\xi_{x^2_0}) \circ \Xi_0 =
(\theta_1^0 \times 1) \circ \Phi_{a^1_0, b^1_0}(\xi_{x_0^1})$.
\item
$\Xi_0$ maps the marked points $Z^+_1(a^1_0)$ to a subsequence $Z^+_{2|1}(a^2_0)$
of $Z^+_2(a^2_0)$.
\item
We denote by $\hat a^l \in \hat X_l$ the image of $a^l \in \widetilde{X}_l$
by the natural map $\widetilde{X}_l \to \hat X_l$.
Let $\hat U_1 \subset \hat X_1$ and $\hat U_2 \subset \hat X_2$ be small
neighborhoods of $\hat a^1_0$ and $\hat a^2_0$ respectively,
and let $\Theta : \hat P_2|_{\hat U_2} \to \hat P_1|_{\hat U_1}$ be the forgetful map
such that
\begin{itemize}
\item
it maps $Z \cup Z^+_{2|1} \cup Z_{\pm\infty_i}$ to $Z \cup Z^+_1 \cup Z_{\pm\infty_i}$,
\item
its underlying map $\hat U_2 \to \hat U_1$ maps $\hat a^2_0$ to $\hat a^1_0$, and
the isomorphism $\Theta|_{(\hat P_2)|_{\hat a^2_0}} : (\hat P_2)|_{\hat a^2_0} \cong
(\hat P_1)|_{\hat a^1_0}$ coincides with the map induced by $\Xi_0^{-1}$.
\end{itemize}
Let $\Theta^\ast \lambda_1 : E^0_1 \to C^\infty(\hat P_2 \times Y,
{\textstyle\bigwedge}^{0, 1} V^\ast \hat P_2 \otimes (\mathbb{R} \partial_\sigma \oplus TY))$
be the pull back of $\lambda_1$
by $\Theta$.
Then we assume that $E^0_1$ is embedded in $E^0_2$
as an $\mathop{\mathrm{Aut}}\nolimits(\Sigma_0, z_0, u_0)$-vector space,
and $\Theta^\ast \lambda_1 = \lambda_2|_{E^0_1}$.
(Note that we may regard $\mathop{\mathrm{Aut}}\nolimits(\Sigma_0, z_0, u_0)$ as a subgroup of
$\mathop{\mathrm{Aut}}\nolimits(\Sigma_i, z_i, u_i)$ for each $i = 1, 2$.)
\end{itemize}
Under the above assumption, we define an $\mathop{\mathrm{Aut}}\nolimits(\Sigma_0, z_0, u_0)$-equivariant
embedding $\phi$ of a neighborhood
$V^0_1$ of $(a^1_0, b^1_0, x^1_0) \in V_1$ to $V_2$ which makes the following diagrams
commutative.
\begin{equation}
\begin{tikzcd}
E_1 \ar[hookrightarrow]{r}{} & E_2\\
V_1^0 \ar{u}{s_1} \ar{r}{\phi} & V_2 \ar{u}{s_2}
\end{tikzcd}
\quad
\begin{tikzcd}
V_1^0 \cap s_1^{-1}(0) \ar{r}{\phi} \ar{dr}{\psi_1} & s_2^{-1}(0) \ar{d}{\psi_2}\\
&\widehat{\mathcal{M}}
\end{tikzcd}
\label{embed commute}
\end{equation}
We regard $V_1$ as a submanifold of
$\mathring{X}_1 \times C^{l_1}(\Sigma_1 \setminus N_1,
(\mathbb{R}_1 \cup \mathbb{R}_2 \cup \dots \cup \mathbb{R}_{k_1}) \times Y) \times E^0_1$
(see Section \ref{smoothness})
and write its point as $(a^1, b^1, u^1, h^1)$, where $(a^1, b^1) \in \mathring{X}_1 \subset
\widetilde{X}_1 \times \prod_\mu \mathbb{R}$, $u^1 \in C^{l_1}(\Sigma_1 \setminus N_1,
(\mathbb{R}_1 \cup \mathbb{R}_2 \cup \dots \cup \mathbb{R}_{k_1}) \times Y)$ and $h^1 \in E^0_1$.
Similarly, we write a point of $V_2$ as $(a^2, b^2, u^2, h^2)
\in \mathring{X}_2 \times C^{l_2}(\Sigma_2 \setminus N_2,
(\mathbb{R}_1 \cup \mathbb{R}_2 \cup \dots \cup \mathbb{R}_{k_1}) \times Y) \times E^0_2$.
We may assume $l_2 \ll l_1$ (since $l_1$, $l_2$ can be taken arbitrary).
The point $q_0$ corresponds to
$(a^1, b^1, u^1, h^1) = (a^1_0, b^1_0, \Phi_{a^1_0, b^1_0}(\xi_{x_0^1}), 0)$
and $(a^2, b^2, u^2, h^2) = (a^2_0, b^2_0, \Phi_{a^2_0, b^2_0}(\xi_{x^2_0}), 0)$.
The embedding $(a^1, b^1, u^1, h^1) \mapsto (a^2, b^2, u^2, h^2)$ is defined
by the following steps.
First, $h^2$ is the image of $h^1$ by the inclusion map $E^0_1 \hookrightarrow E^0_2$.
This map is obviously smooth.
Next, prior to defining $a^2 \in \widetilde{X}_2$, we define $\hat a^2 \in \hat U_2$
which should be the image of $a^2$ by the natural map $\widetilde{X}_2 \to \hat X_2$.
$\hat a^2 \in \hat U_2$ is the point in the inverse image of $\hat a^1$ by
$\hat U_2 \to \hat U_1$ (the underlying map of $\Theta$) such that
\[
(\pi_Y \circ u^1) \circ (\pi_1|_{(\widetilde{P}_1)_{a^1}})^{-1} \circ
\Theta|_{(\hat P_2)_{\hat a^2}} (Z^+_2(\hat a^2)) \subset S_2,
\]
where $\pi_1 : \widetilde{P}_1 \to \hat P_1$ is the composition of the blow down and
the forgetful map.
Since $\hat U_2 \to \hat U_1$ is a submersion and its fiber is the product of
neighborhoods of the points $Z^+_2(\hat a^2_0) \setminus Z^+_{2|1}(\hat a^2_0)$
in $\Sigma_2 \setminus N_2$, $\hat a^2$ is a smooth function of $(a^1, b^1, u^1, h^1)$.
We denote the sequence of points $(\pi_1|_{(\widetilde{P}_1)_{a^1}})^{-1} \circ
\Theta|_{(\hat P_2)_{\hat a^2}} (Z^+_2(\hat a^2))
\subset (\widetilde{P}_1)_{a^1}$ by $\mathcal{Z}^+_2
= \mathcal{Z}^+_2(a^1, u^1)$.
Define an $\mathbb{R}$-gluing $\theta = \theta_{(a^1, b^1, u^1)}: \overline{\mathbb{R}}_1 \sqcup
\overline{\mathbb{R}}_2 \sqcup \dots \sqcup \overline{\mathbb{R}}_{k_2} \to (\overline{\mathbb{R}}_1 \sqcup
\overline{\mathbb{R}}_2 \sqcup \dots \sqcup \overline{\mathbb{R}}_{k_1}) / \sim_{a^1, b^1}$ by
\[
\theta(0_i) = \sigma \circ u^1 \circ (\pi_1|_{(\widetilde{P}_1)_{a^1}})^{-1} \circ
\Theta|_{(\hat P_2)|_{\hat a^2}} (\hat R^2_i(\hat a^2)),
\]
and let
$\mathcal{Z}^{++}_2 = \mathcal{Z}^{++}_2(a^1, b^1, u^1)
\subset (\widetilde{P}_1)_{a^1}$ be the points near
$\Xi_0^{-1} (Z^{++}_2(\hat a^2_0)) \subset (\widetilde{P}_1)_{a^1_0}
\subset \widetilde{P}_1$ such that
$u^1 (\mathcal{Z}^{++}_2) \subset (\theta \times 1) (S'_2)$.
We assume that $Z^{++}_2(\hat a^2_0) \subset \Sigma_0$ is contained in
$\Sigma_1 \setminus N_1 \subset \Sigma_0$.
Then $\mathcal{Z}^{++}_2$ is a smooth function of $(a^1, b^1, u^1)
\in \widetilde{X}_1 \times
C^{l_1}(\Sigma_1 \setminus N_1, (\mathbb{R}_1 \cup \mathbb{R}_2 \cup \dots \cup \mathbb{R}_{k_1}) \times Y)$.
Let $\widetilde{U}_2 \subset \widetilde{X}_2$ be a neighborhood of $a^2_0$ and let
$a^2 \in \widetilde{U}_2$ be the point over $\hat a^2$ such that
there exists an isomorphism $\Xi_{(a^1, b^1, u^1, h^1)} : (\widetilde{P}_1)_{a^1}
\cong (\widetilde{P}_2)_{a^2}$ which maps
$Z(a^1)$, $\mathcal{Z}^+_2$ and $\mathcal{Z}^{++}_2$ to
$Z(a^2)$, $Z^+_2(a^2)$ and $Z^{++}_2(a^2)$ respectively.
Then $a^2$ is a smooth function of $(a^1, b^1, u^1, h^1)$.
In fact, the map $\Xi : V^0_1 \times_{\widetilde{X}_1} \widetilde{P}_1
\to \widetilde{P}_2$ is smooth because it is the composition of
\begin{itemize}
\item
the map from $V^0_1 \times_{\widetilde{X}_1} \widetilde{P}_1$
to the local universal family $\widetilde{P}_3$ of
$(\Sigma_0, z_0 \cup Z^+_2(a^2_0) \cup Z^{++}_1(a^1_0) \cup Z^{++}_2(a^2_0))$
which maps the marked points $Z$, $\mathcal{Z}^+_2$, $Z^{++}_1$ and
$\mathcal{Z}^{++}_2$ to the corresponding marked points of $\widetilde{P}_3$, and
\item
the forgetful map $\widetilde{P}_3$ to $\widetilde{P}_2$.
\end{itemize}
(We assume that $Z^+_2(a^2_0), Z^{++}_1(a^1_0), Z^{++}_2(a^2_0) \subset \Sigma_0$
are disjoint temporarily.)
We define $u^2 \in C^{l_2}(\Sigma_2 \setminus N_2,
(\mathbb{R}_1 \cup \mathbb{R}_2 \cup \dots \cup \mathbb{R}_{k_2}) \times Y)$ by
\[
u^2 = (\theta_{(a^1, b^1, u^1, h^1)} \times 1)^{-1} \circ u^1 \circ
(\Xi_{(a^1, b^1, u^1, h^1)})^{-1},
\]
where we assume $\Xi_{(a^1, b^1, u^1, h^1)}(N_1) \subset N_2$.
Then this is a smooth function of $(a^1, b^1, u^1, h^1)$ (since $l_2 \ll l_1$).
Then it is easy to see that
$\sigma_i \circ u^2 (\widetilde{R}_i^2(a^2)) = 0$ and
$u^2(Z^{++}_2(a^2)) \subset S'_2$.
Finally, we define the asymptotic parameter $b^2_\mu$.
First we recall the relationship between the parameter $b_\mu^l$ and the map
$u^l$ ($l = 1, 2$).
We denote the index set of joint circles of $\Sigma_l$
between the $j$-th floor and the $(j+1)$-th floor by $M^l_j$.
If $\mu \in M^l_j$ and $\rho_\mu^l \neq 0$, then
\[
b^l_\mu = (\theta_l \circ \sigma \circ u^l (\widetilde{R}^l_{j+1})
- \theta_l \circ \sigma \circ u^l (\widetilde{R}^l_j)) + L_\mu \log \rho^l_\mu,
\]
where $\theta_l : \overline{R}_1 \sqcup \overline{R}_2 \sqcup \dots
\sqcup \overline{R}_{k_1} \to \overline{R}_1 \cup \overline{R}_2 \cup \dots \cup
\overline{R}_k$ is an $\mathbb{R}$-gluing which induces an $\mathbb{R}$-translation
$(\overline{R}_1 \sqcup \overline{R}_2 \sqcup \dots \sqcup \overline{R}_{k_1})
/ \sim_{a^l, b^l} \to \overline{R}_1 \cup \overline{R}_2 \cup \dots \cup \overline{R}_k$.
If $\rho^l_\mu = 0$, then
\begin{align*}
b^l_\mu &= \lim_{s \to \infty} \bigl(\theta_l \circ \sigma \circ
u^l|_{[0, \infty) \times S^1_\mu}(s, t)
- \theta_l \circ \sigma \circ u^l(\widetilde{R}^l_j(a^l)) - L_\mu s \bigr)\\
&\quad - \lim_{s \to -\infty} \bigl(\theta_l \circ \sigma \circ
u^l|_{(-\infty, 0] \times S^1_\mu}(s, t)
- \theta_l \circ \sigma \circ u^l(\widetilde{R}^l_{j + 1}(a^l)) - L_\mu s \bigr)
\end{align*}
Since $u^1$ and $u^2$ represent the same curve, we may assume
$\theta_2 = \theta_1 \circ \theta_{(a^1, b^1, u^1, h^1)}$.
Assume $\mu \in M^2_j$ corresponds to $\iota(\mu) \in M^1_i$.
If $\rho^2_\mu \neq 0$, then
\begin{align}
b^2_\mu &= b^1_{\iota(\mu)} +
\bigl(\theta_1 \circ \sigma \circ u^1 \circ
(\Xi_{(a^1, b^1, u^1, h^1)})^{-1} (\widetilde{R}^2_{i + 1}(a^2))
- \theta_1 \circ \sigma \circ u^1 (\widetilde{R}^1_{j + 1}(a^1))\bigr) \notag\\
&\quad - \bigl(\theta_1 \circ \sigma \circ u^1 \circ
(\Xi_{(a^1, b^1, u^1, h^1)})^{-1}(\widetilde{R}^2_i(a^2))
- \theta_1 \circ \sigma \circ u^1 (\widetilde{R}^1_j(a^1))\bigr) \notag\\
& \quad + L_\mu (-\log \rho^1_{\iota(\mu)} + \log \rho^2_\mu) \notag\\
&= b^1_{\iota(\mu)} +
\bigl( \sigma \circ u^1 \circ (\Xi_{(a^1, b^1, u^1, h^1)})^{-1} (\widetilde{R}^2_{i + 1}(a^2))
- \sigma \circ u^1 (\widetilde{R}^1_{j + 1}(a^1))\bigr) \notag\\
&\quad - \bigl(\sigma \circ u^1 \circ (\Xi_{(a^1, b^1, u^1, h^1)})^{-1}
(\widetilde{R}^2_i(a^2)) - \sigma \circ u^1 (\widetilde{R}^1_j(a^1))\bigr) \notag\\
& \quad + L_\mu (-\log \rho^1_{\iota(\mu)} + \log \rho^2_\mu).
\label{b for nonzero kappa}
\end{align}
If $\rho^2_\mu = 0$, then
\begin{align}
b^2_\mu &= b^1_{\iota(\mu)} \notag\\
&\quad + \lim_{s \to \infty} \bigl(\theta_1 \circ \sigma \circ u^1 \circ
(\Xi_{(a^1, b^1, u^1, h^1)})^{-1}|_{[0, \infty) \times S^1_\mu}(s, t) \notag\\
&\quad \hphantom{+ \lim_{s \to \infty} \bigl(}
- \theta_1 \circ \sigma \circ u^1|_{[0, \infty) \times S^1_{\iota(\mu)}}(s, t) \bigr)
\notag\\
&\quad - \lim_{s \to -\infty} \bigl(\theta_1 \circ \sigma \circ u^1 \circ
(\Xi_{(a^1, b^1, u^1, h^1)})^{-1}|_{[0, \infty) \times S^1_\mu}(s, t) \notag\\
&\quad \hphantom{- \lim_{s \to -\infty} \bigl(}
- \theta_1 \circ \sigma \circ u^1|_{[0, \infty) \times S^1_{\iota(\mu)}}(s, t) \bigr)
\notag\\
&\quad + (\theta_1 \circ \sigma \circ u^1 \circ
(\Xi_{(a^1, b^1, u^1, h^1)})^{-1}(\widetilde{R}^2_{j+1}(a^2))
- \theta_1 \circ \sigma \circ u^1(\widetilde{R}^1_{i+1}(a^1))) \notag\\
& \quad - (\theta_1 \circ \sigma \circ u^1 \circ
(\Xi_{(a^1, b^1, u^1, h^1)})^{-1}(\widetilde{R}^2_j(a^2))
- \theta_1 \circ \sigma \circ u^1(\widetilde{R}^1_i(a^1))) \notag\\
&= b^1_{\iota(\mu)} +
\lim_{s \to \infty} L_\mu (p_1 \circ (\Xi_{(a^1, b^1, u^1, h^1)})^{-1}
|_{[0, \infty) \times S^1_\mu}(s, t) - s) \notag\\
&\quad - \lim_{s \to -\infty} L_\mu (p_1 \circ (\Xi_{(a^1, b^1, u^1, h^1)})^{-1}
|_{[0, \infty) \times S^1_\mu}(s, t) -s) \notag\\
&\quad + (\sigma \circ u^1 \circ (\Xi_{(a^1, b^1, u^1, h^1)})^{-1}(\widetilde{R}^2_{j+1}(a^2))
- \sigma \circ u^1(\widetilde{R}^1_{i+1}(a^1))) \notag\\
& \quad - (\sigma \circ u^1 \circ (\Xi_{(a^1, b^1, u^1, h^1)})^{-1}(\widetilde{R}^2_j(a^2))
- \sigma \circ u^1(\widetilde{R}^1_i(a^1))), \label{b for zero kappa}
\end{align}
where $p_1(s, t) = s$ is the projection,
and we have used the asymptotic behavior of $u^1$ near the joint circle
$S^1_{\iota(\mu)}$ for the last equality.
We define $b^2_\mu \in \mathbb{R}$ by (\ref{b for nonzero kappa}) and
(\ref{b for zero kappa}).
It is clear that this is a smooth function of $(a^1, b^1, u^1, h^1)$
at $\rho^2_\mu \neq 0$.
We need to prove the smoothness at $\rho^2_\mu = 0$.
We note that we may assume that if the function $\rho^2_\mu =
\rho^2_\mu(a^1, b^1, u^1, h^1)$ can take zero,
then $\Sigma_0$ has a joint circle corresponding to $\mu$.
To prove the smoothness of $b^2_\mu$, we need to study the map $\Xi$.
First we claim that there exists a smooth function
$f : V^0_1 \to \mathbb{C}^\ast = \mathbb{C} \setminus 0$ such that
\begin{equation}
(\rho^2_\mu)^{2\pi} e^{\sqrt{-1} \varphi_\mu^2}
= (\rho^1_{\iota(\mu)})^{2\pi} e^{\sqrt{-1} \varphi_{\iota(\mu)}^1} f(a^1, b^1, u^1, h^1).
\label{bf eq}
\end{equation}
To prove this claim,
recall that $\Xi$ is the composition of the map $\Xi^1 : V^0_1
\times_{\widetilde{X}_1} \widetilde{P}_1 \to \widetilde{P}_3$
and the forgetful map $\Xi^2 : \widetilde{P}_3 \to \widetilde{P}_2$,
where $(\widetilde{P}_3 \to \widetilde{X}_3, Z_3 \cup Z_3^+ \cup Z_{3, 1}^{++}
\cup Z_{3, 2}^{++})$ is the local universal family of $(\Sigma_0, z_0 \cup Z^+_2(a^2_0)
\cup Z^{++}_1(a^1_0) \cup Z^{++}_2(a^2_0))$.
Since $((\widetilde{P}_3)_0, Z_3(0) \cup Z_3^+(0) \cup Z_{3, 1}^{++}(0))$ is stable and
isomorphic to $((\widetilde{P}_1)_{a^1_0}, Z_1(a^1_0) \cup Z_1^+(a^1_0) \cup
Z_1^{++}(a^1_0))$, we may assume that there exists a neighborhood $U^0_1 \subset
\widetilde{X}_1$ of $a^1_0$ such that $(\widetilde{P}_3 \to \widetilde{X}_3, Z_3 \cup
Z_3^+ \cup Z_{3, 1}^{++} \cup Z_{3, 2}^{++})$ is isomorphic to the product of
$(\widetilde{P}_1|_{U^0_1} \to U^0_1, Z_1 \cup Z_1^+ \cup Z_1^{++})$ and
the parameter space $D^m$ for the marked point $Z_{3, 2}^{++}$.
We can use the coordinate of $\widetilde{X}_3$ defined by the isomorphism
$\widetilde{X}_3 \cong U^0_1 \times D^m \subset \widetilde{X}_1 \times D^m$
Let $S^1_\mu$ be a joint circle of $\Sigma_0 \cong (\widetilde{P}_3)_0$.
Let $S_{\iota_1(\mu)}^1$ and $S_{\iota_2(\mu)}^1$ be the corresponding joint circles
of $\Sigma_1$ and $\Sigma_2$ respectively.
Since the forgetful map $\Xi^2 : \widetilde{P}_3 \to \widetilde{P}_2$ is induced by its
blow down and it is a holomorphic map, there exists a smooth function
$f'_\mu : \widetilde{X}_3 \to \mathbb{C}^\ast$ such that
\[
(\rho^2_{\iota_2(\mu)})^{2\pi} e^{2\pi \sqrt{-1} \varphi^2_{\iota_2(\mu)}}
= (\rho^1_{\iota_1(\mu)})^{2\pi} e^{2\pi \sqrt{-1} \varphi^1_{\iota_1(\mu)}}
\cdot f'_\mu (a^3)
\]
for all $a^3 \in \widetilde{X}_3$, where
$(\rho^1_{\iota_1(\mu)}, \varphi^1_{\iota_1(\mu)})$ is a part of the coordinate
of $a^3 \in \widetilde{X}_3$ under the isomorphism
$\widetilde{X}_3 \cong U^0_1 \times D^m \subset \widetilde{X}_1 \times D^m$, and
$(\rho^2_{\iota_2(\mu)}, \varphi^2_{\iota_2(\mu)})$ is a part of the coordinate of
$\widetilde{X}_2$ at $\Xi^2(a^3)$.
Since the underlying map of $\Xi^1$ is smooth, the claim follows, that is,
there exists a smooth function
$f : V^0_1 \to \mathbb{C}^\ast = \mathbb{C} \setminus 0$ which satisfies equation (\ref{bf eq}).
Similarly, there exists smooth maps $f_\mu^{\text{left}}, f_\mu^{\text{right}} :
V^0_1 \times_{\widetilde{X}_2} \widetilde{P}_2 \to \mathbb{C}^\ast$ such that
if
\begin{align*}
\Xi_{(a^1, b^1, u^1, h^1)}|_{[0, \infty) \times S^1_{\iota_1(\mu)}}
(s_1^{\text{left}}, t_1^{\text{left}}) &= (s_2^{\text{left}}, t_2^{\text{left}})
\in [0, \infty) \times S^1_{\iota_2(\mu)},\\
\Xi_{(a^1, b^1, u^1, h^1)}|_{(-\infty, 0] \times S^1_{\iota_1(\mu)}}
(s_1^{\text{right}}, t_1^{\text{right}}) &= (s_2^{\text{right}}, t_2^{\text{right}})
\in (-\infty, 0] \times S^1_{\iota_2(\mu)},
\end{align*}
then
\begin{align*}
e^{-2\pi (s_2^{\text{left}} + \sqrt{-1} t_2^{\text{left}})}
&= e^{-2\pi (s_1^{\text{left}} + \sqrt{-1} t_1^{\text{left}})}
\cdot f_\mu^{\text{left}} (s_2^{\text{left}}, t_2^{\text{left}}, a^1, b^1, u^1, h^1),\\
e^{2\pi (s_2^{\text{right}} + \sqrt{-1} t_2^{\text{right}})}
&= e^{2\pi (s_1^{\text{right}} + \sqrt{-1} t_1^{\text{right}})}
\cdot f_\mu^{\text{right}} (s_2^{\text{right}}, t_2^{\text{right}}, a^1, b^1, u^1, h^1).
\end{align*}
Note that $f_\mu$, $f_\mu^{\text{left}}$ and $f_\mu^{\text{right}}$ satisfy
\[
f_\mu(a^1, b^1, u^1, h^1)
= f_\mu^{\text{left}}(s_2^{\text{left}}, t_2^{\text{left}},a^1, b^1, u^1, h^1)
f_\mu^{\text{right}}(s_2^{\text{right}}, t_2^{\text{right}}, a^1, b^1, u^1, h^1)
\]
if $(s_2^{\text{left}}, t_2^{\text{left}})$ and $(s_2^{\text{right}}, t_2^{\text{right}})$
denote the same point of $(\widetilde{P}_1)_{a^1}$.
In particular, if $\kappa^2_{\iota_2(\mu)} = 0$, then
\[
|f_\mu(a^1, b^1, u^1, h^1)| = \lim_{s \to \infty}
|f_\mu^{\text{left}}(s, t,a^1, b^1, u^1, h^1)|
\lim_{s \to -\infty}
|f_\mu^{\text{right}}(s, t, a^1, b^1, u^1, h^1)|.
\]
We can rewrite the formula of $b^2_{\iota_2(\mu)}$ by using the function $f_\mu$
as follows.
If $\rho^2_{\iota_2(\mu)} \neq 0$, then
\begin{align}
b^2_\mu &= b^1_\mu +
\bigl( \sigma \circ u^1 \circ (\Xi_{(a^1, b^1, u^1, h^1)})^{-1} (\widetilde{R}^2_{i + 1}(a^2))
- \sigma \circ u^1 (\widetilde{R}^1_{j + 1}(a^1))\bigr) \notag\\
&\quad - \bigl(\sigma \circ u^1 \circ (\Xi_{(a^1, b^1, u^1, h^1)})^{-1}
(\widetilde{R}^2_i(a^2)) - \sigma \circ u^1 (\widetilde{R}^1_j(a^1))\bigr)\notag\\
& \quad + L_\mu \cdot \frac{1}{2\pi} \log |f_\mu(a^1, b^1, u^1, h^1)|.
\label{b for any kappa}
\end{align}
If $\rho^2_{\iota_2(\mu)} = 0$, then
\[
L_\mu (p_1 \circ (\Xi_{(a^1, b^1, u^1, h^1)})^{-1}
|_{[0, \infty) \times S^1_\mu}(s, t) - s)
= \frac{1}{2\pi} \log |f_\mu^{\text{left}}(s, t, a^1, b^1, u^1, h^1)|,
\]
and
\[
L_\mu (p_1 \circ (\Xi_{(a^1, b^1, u^1, h^1)})^{-1}
|_{[0, \infty) \times S^1_\mu}(s, t) - s)
= - \frac{1}{2\pi} \log |f_\mu^{\text{right}}(s, t, a^1, b^1, u^1, h^1)|.
\]
Therefore, equation (\ref{b for any kappa}) also holds in this case.
Hence $b^2_{\iota_2(\mu)}$ is a smooth function of $(a^1, b^1, u^1, h^1)$.
Next we prove that the differential of
$\phi : (a^1, b^1, u^1, h^1) \mapsto (a^2, b^2, u^2, h^2)$ is injective everywhere.
It is enough to construct a smooth inverse from an open subset of the submanifold
$\{(a^2, b^2, u^2, h^2) \in V^2; h^2 \in E^0_1\}$ of $V^2$ to $V^1$.
We can construct this map by the same way as $\phi$.
Hence $\phi$ is indeed an embedding.
It is obvious that diagrams (\ref{embed commute}) are commutative for this $\phi$.
Hence it is the required embedding.
So far we have made some assumptions about the position of
the additional marked points $Z^{++}_1(a^1_0)$ or $Z^{++}_2(a^2_0)$.
(For example, we have assumed that $Z^{++}_2(a^2_0) \subset \Sigma_0$ is contained
in $\Sigma_1 \setminus N_1 \subset \Sigma_0$.)
We can remove these assumption because
two Kuranishi neighborhoods of the same point $p_1$ defined by using
the same data $(p_1^+ = (\Sigma_1, z_1 \cup z_1^+, u_1), S_1, E^0_1, \lambda_1)$,
different additional data $(\hat R^1_i, \widetilde{R}^1_i, Z^{++}_{1, i}, S'_1)
\neq (\hat R^2_i, \widetilde{R}^2_i, Z^{++}_{2, i}, S'_2)$ and
different decompositions of the curve $\Sigma_1$ into parts are isomorphic
by the same argument as above.
Note that the smoothness of $\mathop{\mathrm{Aut}}\nolimits(\Sigma_0, z, u_0)$-action on
a Kuranishi neighborhood of $p_0 = (\Sigma_0, z, u_0)$ also
follows from the above argument because the group action is also a kind of embedding.
\begin{rem}\label{natural projection from hat V to V}
Recall that for a Kuranishi neighborhood $(V, E, s, \psi, G)$,
$V$ is a submanifold of $\hat V = \mathring{X} \times B_\epsilon(0) \subset
\mathring{X} \times C^l(\Sigma_0 \setminus N_0,
(\mathbb{R}_1 \cup \mathbb{R}_2 \cup \dots \cup \mathbb{R}_k) \times Y) \times E^0$.
We can define a natural projection $\hat V \to V : (a', b', u', h') \mapsto (a, b, u, h)$
by a similar way to the above embedding.
It satisfies $h = h'$ and that for each $(a', b', u', h')$,
there exists a biholomorphism $\varphi : \widetilde{P}_{a'} \to \widetilde{P}_a$ and
an $\mathbb{R}$-translation $\theta$ such that
$\varphi$ maps $Z(a')$ and $Z^+(a')$ to $Z(a)$ and $Z^+(a)$ as sequences respectively
and $(\theta \times 1) \circ u \circ \varphi = u'$.
\end{rem}
\subsection{A Kuranishi neighborhood of a disconnected holomorphic building
and those of its connected components}
\label{Kuranishi of disconnected buildings}
In Section \ref{construction of nbds}, we constructed a Kuranishi neighborhood of
a point $p \in \widehat{\mathcal{M}}$ for data $(p^+, S, E^0, \lambda)$ and additional
data $(z^{++}, S', \hat R_j)$.
In Section \ref{embed}, we saw that the Kuranishi neighborhood is determined by
the data $(p^+, S, E^0, \lambda)$ and independent of the
additional data $(z^{++}, S', \hat R_j)$.
To obtain algebraic information of the moduli space,
the data $(p^+, S, E^0, \lambda)$ for a disconnected holomorphic building
should be given by the product of the data
for the connected components, but the additional data $(z^{++}, S', \hat R_j)$ can be
taken independently.
We emphasize that we do not construct a Kuranishi neighborhood from the Kuranishi
neighborhoods of the connected components as a quotient of their product, but
we construct it independently through the same procedure.
Instead, under the above assumption for the data $(p^+, S, E^0, \lambda)$,
we study the relationship between the Kuranishi neighborhood of
a disconnected holomorphic building and those of its connected components.
Let $p'_0 = (\Sigma'_0, z'_0, u'_0) \in \widehat{\mathcal{M}}$ be an arbitrary point and
let $\Sigma'_1, \Sigma'_2, \dots, \Sigma'_N$ be the connected components of
$\Sigma'_0$.
Let $p_i = (\Sigma_i, z_i, u_i)$ be the holomorphic building obtained by collapsing
the floors of $(\Sigma'_i, z'_0|_{\Sigma'_i}, u'_0|_{\Sigma'_i})$ consisting of trivial cylinders.
We call a map $\theta : \overline{\mathbb{R}}_1 \sqcup \overline{\mathbb{R}}_2 \sqcup \dots \sqcup
\overline{\mathbb{R}}_k \to \overline{\mathbb{R}}_1 \cup \overline{\mathbb{R}}_2 \cup \dots \cup
\overline{\mathbb{R}}_l$ an $\mathbb{R}$-compressing if there exist a map $\mu : \{1, 2, \dots, k\} \to
\{\frac{1}{2}, 1, \frac{3}{2}, \dots, l, l + \frac{1}{2}\}$ and constants $c_i \in \mathbb{R}$ ($i \in
\mu^{-1}(\mathbb{Z})$) such that
\begin{itemize}
\item
the image of $\mu$ contains $\{1, 2, \dots, l\}$,
\item
if $i \leq j$ then $\mu(i) \leq \mu(j)$,
\item
if $\mu(i) \in \mathbb{Z}$ then $\theta(\overline{\mathbb{R}}_i) = \overline{\mathbb{R}}_{\mu(i)}$ and
$\theta|_{\overline{\mathbb{R}}_i}(s) = s + c_i$, and
\item
if $\mu(i) \notin \mathbb{Z}$ then $\theta(\overline{\mathbb{R}}_i) = \{+\infty_{\lfloor \mu(i) \rfloor}\}
= \{-\infty_{\lceil \mu(i) \rceil}\}$.
\end{itemize}
As in the previous section, we assume the following conditions on the Kuranishi
neighborhoods of $p'_0$ and $p_i$:
\begin{itemize}
\item
Kuranishi neighborhoods $(V_i, E_i, s_i, \psi_i, G_i)$
of $p_i = (\Sigma_i, z_i, u_i) \in \widehat{\mathcal{M}}^0$ is defined by the data
$(p_i^+ = (\Sigma_i, z_i \cup z_i^+, u_i), S_i, E^0_i, \lambda_i)$
and the additional data $(z_i^{++} = (z_{i, i}^{++}), S'_i, \hat R^i_j)$.
Let $(\hat P_i \to \hat X_i, Z_i \cup Z_i^+ \cup Z_{\pm\infty_i})$ be the local universal
family of the stabilization $(\hat \Sigma_i, z_i \cup z_i^+ \cup (\pm\infty_i))$ of
the blown down curve of $(\Sigma_i, z_i \cup z_i^+)$, and
$(\widetilde{P}_i \to \widetilde{X}_i, Z_i \cup Z_i^+ \cup Z_i^{++})$ be the local universal
family of $(\Sigma_i, z_i \cup z_i^+ \cup z_i^{++})$.
\item
A Kuranishi neighborhood $(V_0, E_0, s_0, \psi_0, G_0)$
of $p'_0 = (\Sigma'_0, z'_0, u'_0) \in \widehat{\mathcal{M}}$ is defined by the data
$({p'_0}^+ = (\Sigma'_0, z'_0 \cup \bigcup_i z_i^+, u'_0), S_0 = \bigcup_i S_i,
E^0_0 = \bigoplus_i E^0_i, \lambda_0 = \bigoplus_i \lambda_i)$
and the additional data $({z'_0}^{++} = ({z'_{0, j}}^{++}), S'_0, \hat R^0_i)$.
We define ${z'_0}^+ = \bigcup_i z_i^+$.
Let $(\hat P'_0 \to \hat X'_0, Z'_0 \cup {Z'_0}^+ \cup Z_{\pm\infty_i})$ be the local
universal family of the stabilization $(\hat \Sigma'_0, z'_0 \cup {z'_0}^+ \cup (\pm\infty_i))$
of the blown down curve of $(\Sigma'_0, z'_0 \cup {z'_0}^+)$, and
$(\widetilde{P}'_0 \to \widetilde{X}'_0, Z'_0 \cup {Z'_0}^+ \cup {Z'_0}^{++})$
be the local universal family of $(\Sigma'_0, z'_0 \cup {z'_0}^+ \cup {z'_0}^{++})$.
\item
Let $\widetilde{P}'_0 = \coprod_i \widetilde{P}'_i$ be the decomposition into the connected
components corresponding to the decomposition $\Sigma'_0 = \coprod_i \Sigma'_i$.
We define $Z'_i = Z'_0 \cap \widetilde{P}'_i$, ${Z'_i}^+ = {Z'_0}^+ \cap \widetilde{P}'_i$ and
${Z'_i}^{++} = {Z'_0}^{++} \cap \widetilde{P}'_i$.
We assume that for each $i$, there exists a map
\[
\Xi_{0, i} : ((\widetilde{P}'_i)_0, Z'_i(0) \cup {Z'_i}^+(0))
\to ((\widetilde{P}_i)_0, Z_i(0) \cup Z_i^+(0))
\]
which collapses the floors consisting of trivial cylinders and which satisfies
$u_i \circ \Xi_{0, i} = (\theta_0 \times 1) \circ u_0|_{\Sigma'_0}$
on $\Sigma'_i \cong (\widetilde{P}'_i)_0$ for some $\mathbb{R}$-compressing
$\theta_0 : \overline{\mathbb{R}}_1 \sqcup \overline{\mathbb{R}}_2 \sqcup \dots \sqcup
\overline{\mathbb{R}}_{k_0} \to \overline{\mathbb{R}}_1 \cup \overline{\mathbb{R}}_2 \cup \dots \cup
\overline{\mathbb{R}}_{k_i}$.
\end{itemize}
Under theses assumptions, we prove that there exists a natural map
$\phi = (\phi_i) : V_0 \to \prod_i V_i$ which satisfies the following conditions:
\begin{itemize}
\item
The following diagram is commutative.
\[
\begin{tikzcd}
E_0 \ar{r}{} & E_i\\
V_0 \ar{u}{s_0} \ar{r}{\phi_i} & V_i \ar{u}{s_i}
\end{tikzcd}
\]
\item
For any $p \in s_0^{-1}(0)$,
the curve obtained by collapsing trivial floors of the $i$-th connected component of the
curve corresponding to the point $p$ is isomorphic to the curve corresponding to
$\phi_i(p) \in s_i^{-1}(0)$.
\item
For arbitrary integers $l_i \leq 1$ ($i = 1, \dots, N$), let $V^{(l_i)}_i \subset V_i$ be
the submanifold consisting of height-$l_i$ curves.
(This coincides with $\mathring{\partial}^{l_i - 1} V_i$
in Section \ref{section of orbibundle}.)
Then each $\phi^{-1}(\prod_i V_i^{(l_i)}) \subset V_0$ is a union
of the interiors of corners of $V_0$, and
$\phi|_{\phi^{-1}(\prod_i V_i^{(l_i)})} : \phi^{-1}(\prod_i V_i^{(l_i)})
\to \prod_i V_i^{(l_i)}$ is submersive on each of them.
We say that $\phi$ is essentially submersive if it satisfies this condition.
\end{itemize}
Fixing $i_0$, we construct a essential submersion $\phi_{i_0} : V_0 \to V_{i_0}$ as follows.
As in the previous section, we write a point of $V_0$ as $(a^0, b^0, u^0, h^0)$,
where $(a^0, b^0) \in \mathring{X}'_0 \subset \widetilde{X}'_0 \times \prod_\mu \mathbb{R}$,
$u^0 \in C^{l_0}(\Sigma'_0 \setminus N'_0, (\mathbb{R}_1 \cup \mathbb{R}_2 \cup \dots \cup \mathbb{R}_{k_0})
\times Y)$ and $h^0 = (h^0_i) \in E^0_0 = \bigoplus_i E^0_i$.
Similarly, a point of $V_{i_0}$ is written as
$(a^{i_0}, b^{i_0}, u^{i_0}, h^{i_0}) \in \widetilde{X}_{i_0} \times \prod_\mu \mathbb{R} \times
C^{l_{i_0}}(\Sigma_{i_0} \setminus N_{i_0}, (\mathbb{R}_1 \cup \mathbb{R}_2 \cup \dots \cup \mathbb{R}_{k_{i_0}})
\times Y) \times E^0_{i_0}$.
We may assume $l_{i_0} \ll l_0$.
The essential submersion $(a^0, b^0, u^0, h^0) \mapsto (a^{i_0}, b^{i_0}, u^{i_0}, h^{i_0})$
is defined by the following steps as in the previous section.
First, $h^{i_0} \in E^0_{i_0}$ is defined by $h^{i_0} = h^0_{i_0}$.
Next we note that $\hat X'_0 = \prod_i \hat X_i$ is a product
(but $\widetilde{X}'_0$ is not).
Hence we can define $\hat a^{i_0} \in \hat X_{i_0}$ by the $i_0$-th component
of $\hat a^0 = (a^0_i) \in \prod_i \hat X_i$.
Let $\pi_{i_0} : (\widetilde{P}'_{i_0}, Z'_{i_0} \cup {Z'_{i_0}}^+)
\to (\hat P_{i_0}, Z_{i_0} \cup Z_{i_0}^+)$ be the composition of the blow down and the
forgetful map, and define
\[
s_j = \sigma \circ u^0 \circ (\pi_{i_0}|_{(\widetilde{P}'_{i_0})_{\hat a^0}})^{-1}
(\hat R^{i_0}_j(\hat a^{i_0})) \in \mathbb{R}_1 \cup \mathbb{R}_2 \cup \dots \cup \mathbb{R}_{k_0}.
\]
Let $\theta = \theta_{(a^0, b^0, u^0, h^0)} : \overline{\mathbb{R}}_1 \sqcup \overline{\mathbb{R}}_2
\sqcup \dots \sqcup \overline{\mathbb{R}}_{k_0} \to \overline{\mathbb{R}}_1 \cup \overline{\mathbb{R}}_2
\cup \dots \cup \overline{\mathbb{R}}_{k_{i_0}}$ be the $\mathbb{R}$-compressing defined by the
following conditions:
\begin{itemize}
\item
If $s_j \in \mathbb{R}_i$ then $\theta(\overline{\mathbb{R}}_i) = \overline{\mathbb{R}}_j$ and
$\theta|_{\overline{\mathbb{R}}_i}(s) = s - s_j$.
\item
If $\mathbb{R}_i$ does not contain any $s_j$ then $\theta$ maps $\overline{\mathbb{R}}_i$ to some
$\infty$-point. More precisely, if $s_j \in \bigcup_{l < i} \mathbb{R}_l$ and $s_{j+1} \notin
\bigcup_{l \leq i} \mathbb{R}_l$ then $\theta(\overline{\mathbb{R}}_i) = \{+\infty_j\} \subset \overline{\mathbb{R}}_j$.
\end{itemize}
Let $\mathcal{Z}^{++} = \mathcal{Z}^{++}(a^0, b^0, u^0, h^0) \subset
(\widetilde{P}'_{i_0})_{a^0}$ be the sequence of points in a neighborhood of
$\Xi_{0, i_0}^{-1}(Z^{++}_{i_0}(0)) \subset (\widetilde{P}'_{i_0})_0 \subset \widetilde{P}'_{i_0}$
defined by $(\theta \times 1) \circ u^0 (\mathcal{Z}^{++}) \subset S'_{i_0}$.
Let $\Xi : V_0 \times_{\widetilde{X}'_0} \widetilde{P}'_{i_0} \to \widetilde{P}_{i_0}$ be the
natural map which preserves fibers and
which maps $Z'_{i_0}$, ${Z'_{i_0}}^+$ and $\mathcal{Z}^{++}$ to
$Z_{i_0}$, $Z_{i_0}^+$ and $Z_{i_0}^{++}$ respectively.
(The restriction of $\Xi$ to each fiber is the map collapsing trivial floors.)
Let $a^2$ be the image of $(a^0, b^0, u^0, h^0)$ by the underlying map $V_0 \to
\widetilde{X}_{i_0}$.
Define $u^{i_0} \in C^l(\Sigma_{i_0} \setminus N_{i_0}, (\mathbb{R}_1 \sqcup \mathbb{R}_2 \sqcup \dots
\sqcup \mathbb{R}_{k_{i_0}}) \times Y)$ by
\[
u^{i_0} = (\theta_{(a^0, b^0, u^0, h^0)} \times 1) \circ u^0 \circ
(\Xi_{(a^0, b^0, u^0, h^0)})^{-1},
\]
where $\Xi_{(a^0, b^0, u^0, h^0)}$ is the restriction of $\Xi$ to the
fiber at $(a^0, b^0, u^0, h^0) \in V_0$.
Finally, we define the asymptotic parameters $b^{i_0}_\mu$.
We denote the index set of the joint circles of $\Sigma'_0$ between the $j$-th floor
and the $(j+1)$-th floor by $M^0_j$, and
the index set of the joint circles of $\Sigma_{i_0}$ between
the $j$-th floor and the $(j+1)$-th floor by $M^{i_0}_j$.
For each $\mu \in M^{i_0}_{j'}$,
let $S_{\mu_j}^1, S_{\mu_{j+1}}^1, \dots, S_{\mu_{j+m}}^1 \subset \Sigma'_{i_0}
\subset \Sigma'_0$ be the joint circles of $\Sigma'_0$ which collapse to $S_\mu^1$
by $\Sigma'_{i_0} \to \Sigma_{i_0}$, where we assume $\mu_{j+l} \in M^0_{j+l}$.
First we consider the case of $\rho^{i_0}_\mu \neq 0$.
Note that $\rho^0_{\mu_{j+l}} \neq 0$ in this case.
Since $b^{i_0}_\mu$ and $b^0_{\mu_{j+l}}$ should satisfy
\begin{align*}
- L_\mu \log \rho^{i_0}_\mu + b^{i_0}_\mu
&= \sigma \circ u^{i_0}(\widetilde{R}^{i_0}_{j'+1}(a^{i_0}))
- \sigma \circ u^{i_0}(\widetilde{R}^{i_0}_{j'}(a^{i_0}))\\
- L_\mu \log \rho^0_{\mu_{j+l}} + b^0_{\mu_{j+l}}
&= \sigma \circ u^0(\widetilde{R}^0_{j+l+1}(a^0))
- \sigma \circ u^0(\widetilde{R}^0_{j+l}(a^0)),
\end{align*}
we define $b^{i_0}_\mu$ by
\begin{align}
b^{i_0}_\mu &= (b^0_{\mu_j} + b^0_{\mu_{j+1}} + \dots + b^0_{\mu_{j+m}}) \notag\\
&\quad
+ \bigl(\sigma \circ u^0 \circ (\Xi_{(a^0, b^0, u^0, h^0)})^{-1}
(\widetilde{R}^{i_0}_{j'+1}(a^{i_0}))
- \sigma \circ u^0 (\widetilde{R}^0_{j+m+1}(a^0))\bigr) \notag\\
&\quad
- \bigl(\sigma \circ u^0 \circ (\Xi_{(a^0, b^0, u^0, h^0)})^{-1}(\widetilde{R}^{i_0}_{j'}(a^{i_0}))
- \sigma \circ u^0 (\widetilde{R}^0_{j+m}(a^0))\bigr) \notag\\
&\quad
+ L_\mu(-\log \rho^0_{\mu_j} - \dots - \log \rho^0_{\mu_{j+m}}
+ \log \rho^{i_0}_\mu).
\label{b^{i_0} for nonzero kappa}
\end{align}
Next we consider the case of $\rho^{i_0}_\mu = 0$.
Then there exist some $1 \leq c \leq d \leq m$ such that
$\rho^0_{\mu_{j+c}} = 0$, $\rho^0_{\mu_{j+d}} = 0$ and
$\rho^0_{\mu_{j+l}} \neq 0$ for $1 \leq l < c$ and $d < l \leq m$.
Then $b^0_{\mu_{j+l}}$ satisfies
\[
b^0_{\mu_{j+l}} = \bigl(\sigma \circ u^0(\widetilde{R}^0_{j+l+1}(a^0))
- \sigma \circ u^0(\widetilde{R}^0_{j+l}(a^0))\bigr) + L_\mu \log \rho^0_{\mu_{j+l}}
\]
for $1 \leq l < c$ and $d < l \leq m$, and
\begin{align*}
b^0_{\mu_{j+l}}
&= \lim_{s \to \infty}\bigl(\sigma \circ u^0|_{[0, \infty) \times S^1_{\mu_{j+l}}}(s, t)
- \sigma \circ u^0(\widetilde{R}^0_{j+l}(a^0)) - L_\mu s \bigr)\\
& \quad
- \lim_{s \to -\infty}\bigl(\sigma \circ u^0|_{(-\infty, 0] \times S^1_{\mu_{j+l}}}(s, t)
- \sigma \circ u^0(\widetilde{R}^0_{j+l+1}(a^0)) - L_\mu s \bigr)
\end{align*}
for $l = c, d$.
Hence
\begin{align}
&b^0_{\mu_j} + \dots + b^0_{\mu_{j+c}} \notag\\
&=
\lim_{s \to \infty} \bigl(\sigma \circ u^0|_{[0, \infty) \times S^1_{\mu_{j+c}}}(s, t)
- \sigma \circ u^0(\widetilde{R}^0_j(a^0)) \notag\\
&\quad \hphantom{\lim_{s \to \infty} \bigl(}
- L_\mu \bigl(s - \log \rho^0_{\mu_j} - \dots
- \log \rho^0_{\mu_{j+c-1}}\bigr)\bigr) \notag\\
& \quad
- \lim_{s \to \infty} \bigl(\sigma \circ u^0|_{(-\infty, 0] \times S^1_{\mu_{j+c}}}(-s, t)
- (\sigma \circ u^0(\widetilde{R}^0_{j+c+1}(a^0)) - L_\mu s)\bigr)
\label{b of 0 to c}
\end{align}
and
\begin{align}
&b^0_{\mu_{j+d}} + \dots + b^0_{\mu_{j+m}} \notag\\
&=
- \lim_{s \to -\infty} \bigl(\sigma \circ u^0|_{(-\infty, 0] \times S^1_{\mu_{j+d}}}(s, t)
- \sigma \circ u^0(\widetilde{R}^0_{j+m+1}(a^0)) \notag\\
&\quad \hphantom{- \lim_{s \to -\infty} \bigl(}
- L_\mu \bigl(s + \log \rho^0_{\mu_{j+d+1}}
+ \dots + \log \rho^0_{\mu_{j+m}}\bigr) \bigr) \notag\\
& \quad
+ \lim_{s \to \infty} \bigl(\sigma \circ u^0|_{[0, \infty) \times S^1_{\mu_{j+d}}}(s, t)
-(\sigma \circ u^0(\widetilde{R}^0_{j+d}(a^0)) + L_\mu s)\bigr)
\label{b of d to m}
\end{align}
Assume that we use the decomposition of the trivial cylinder of
$(\Sigma'_0, z'_0 \cup {z'_0}^+, u_0)$ between $S^1_{\mu_{j + l}}$ and $S^1_{\mu_{j+l+1}}$
given by
\[
\overline{\mathbb{R}} \times S^1 = (-\infty, 1] \times S^1_{\mu_{j+l}}
\cup
[1, T_{j+l+1} -1] \times S^1 \cup [-1, \infty) \times S^1_{\mu_{j+l+1}}
\]
for the definition of the coordinate of $\widetilde{P}'_0$,
where we identify $\{1\} \times S^1$ and $\{T_{j+l+1} -1\} \times S^1$
with $\{1\} \times S^1_{\mu_{j+l}}$ and $\{-1\} \times S^1_{\mu_{j+l+1}}$ respectively,
and we consider the sections of the additional marked points ${Z'_{i_0}}^{++}$ as functions
to $[1, T_{j+l+1} -1] \times S^1$ instead of deforming the complex structure of
$[1, T_{j+l+1} -1] \times S^1$.
(Other cases can be covered by this case and the embeddings argued in the previous
section.)
First we assume $c < d$.
$u^0$ is trivial on the trivial cylinders between $S^1_{\mu_{j+c}}$ and
$S^1_{\mu_{j+d}}$, and the above assumption on the coordinate of $\widetilde{P}'_0$
implies that the natural coordinate of trivial cylinders and
the coordinates of $[0, \infty) \times S^1_{\mu_{j + l}}$ or
$(-\infty, 0] \times S^1_{\mu_{j + l}}$ coincide up to translation.
Therefore the following equations hold true.
\begin{multline}
\lim_{s \to -\infty} \bigl(\sigma \circ u^0|_{(-\infty, 0] \times S^1_{\mu_{j+c}}}(s, t)
- \sigma \circ u^0(\widetilde{R}^0_{j+c+1}(a^0)) - L_\mu s \bigr) \\
= - \bigl(\sigma \circ u^0(\widetilde{R}^0_{j+c+1}(a^0))
- \sigma \circ u^0|_{(-\infty, 0] \times S^1_{\mu_{j+c}}}(0, t)\bigr) \label{c trivial}
\end{multline}
\begin{multline}
\lim_{s \to \infty} \bigl(\sigma \circ u^0|_{[0, \infty) \times S^1_{\mu_{j+d}}}(s, t)
-\sigma \circ u^0(\widetilde{R}^0_{j+d}(a^0)) - L_\mu s \bigr) \\
= - \bigl(\sigma \circ u^0(\widetilde{R}^0_{j+d}(a^0))
- \sigma \circ u^0|_{[0, \infty) \times S^1_{\mu_{j+d}}}(0, t)\bigr). \label{d trivial}
\end{multline}
Similarly, for any $c < l < d$, whether $\rho^0_{\mu_{j+l}} = 0$ or not,
\begin{align}
b^0_{\mu_{j+l}}
&= \bigl(\sigma \circ u^0(\widetilde{R}^0_{j+l+1}(a^0))
- \sigma \circ u^0|_{(-\infty, 0] \times S^1_{\mu_{j+l}}}(0, t)\bigr) \notag\\
& \quad - \bigl(\sigma \circ u^0(\widetilde{R}^0_{j+l}(a^0))
- \sigma \circ u^0|_{[0, \infty) \times S^1_{\mu_{j+l}}}(0, t)\bigr).
\label{middle trivial}
\end{align}
Therefore equations (\ref{b of 0 to c}) to (\ref{middle trivial}) imply
\begin{align}
&b^0_{\mu_j} + \dots + b^0_{\mu_{j+m}} \notag\\
&= \lim_{s \to \infty} \bigl(\sigma \circ u^0|_{[0, \infty) \times S^1_{\mu_{j+c}}}(s, t)
- \sigma \circ u^0(\widetilde{R}^0_j(a^0)) \notag\\
& \hphantom{= \lim_{s \to \infty} \bigl(}
- L_\mu \bigl(s - \log \rho^0_{\mu_j} - \dots - \log \rho^0_{\mu_{j+c-1}}\bigr)
\bigr) \notag\\
& \quad
- \lim_{s \to -\infty}\bigl(\sigma \circ u^0|_{(-\infty, 0] \times S^1_{\mu_{j+d}}}(s, t)
- \sigma \circ u^0(\widetilde{R}^0_{j+m+1}(a^0)) \notag\\
& \hphantom{\quad - \lim_{s \to -\infty}\bigl(}
- L_\mu \bigl(s + \log \rho^0_{\mu_{j+d+1}} + \dots + \log \rho^0_{\mu_{j+m}}\bigr)
\bigr) \notag\\
& \quad
+ \sum_{c \leq l < d} \bigl(\sigma \circ u^0|_{[0, \infty) \times S^1_{\mu_{j+l+1}}}(0, t)
- \sigma \circ u^0|_{(-\infty, 0] \times S^1_{\mu_{j+l}}}(0, t)\bigr). \label{b sum}
\end{align}
It is easy to see that this equation also holds for the case of $c = d$.
The assumption on the coordinate of $\widetilde{P}'_0$ implies that the last terms
of (\ref{b sum}) are
\[
\sigma \circ u^0|_{[0, \infty) \times S^1_{\mu_{j+l+1}}}(0, t)
- \sigma \circ u^0|_{(-\infty, 0] \times S^1_{\mu_{j+l}}}(0, t)
= L_\mu T_{j+l+1}.
\]
Since $b^{i_0}_\mu$ is related to $u^{i_0}$ or $u^0$ by
\begin{align*}
b^{i_0}_\mu &= \lim_{s \to \infty} \bigl(\sigma \circ u^{i_0}|_{[0, \infty) \times S^1_\mu}(s, t)
- \sigma \circ u^{i_0}(\widetilde{R}^{i_0}_{j'}(a^{i_0})) - L_\mu s \bigr)\\
& \quad - \lim_{s \to -\infty} \bigl(\sigma \circ u^{i_0}|_{(-\infty, 0] \times S^1_\mu}(s, t)
- \sigma \circ u^{i_0}(\widetilde{R}^{i_0}_{j'+1}(a^{i_0})) - L_\mu s \bigr)\\
&= \lim_{s \to \infty} \bigl(\sigma \circ u^0 \circ
(\Xi_{(a^0, b^0, u^0, h^0)})^{-1}|_{[0, \infty) \times S^1_\mu}(s, t)\\
&\quad \hphantom{\lim_{s \to \infty} \bigl(}
- \sigma \circ u^0 \circ (\Xi_{(a^0, b^0, u^0, h^0)})^{-1}(\widetilde{R}^{i_0}_{j'}(a^{i_0}))
- L_\mu s \bigr)\\
& \quad - \lim_{s \to -\infty} \bigl(\sigma \circ u^0 \circ
(\Xi_{(a^0, b^0, u^0, h^0)})^{-1}|_{(-\infty, 0] \times S^1_\mu}(s, t)\\
&\quad \hphantom{- \lim_{s \to -\infty} \bigl(}
- \sigma \circ u^0 (\Xi_{(a^0, b^0, u^0, h^0)})^{-1}(\widetilde{R}^{i_0}_{j'+1}(a^{i_0}))
- L_\mu s \bigr)
\end{align*}
$b^{i_0}_\mu$ should satisfies
\begin{align}
b^{i_0}_\mu &= (b^0_{\mu_j} + \dots + b^0_{\mu_{j+m}}) \notag\\
& \quad
+ (\sigma \circ u^0 \circ (\Xi_{(a^0, b^0, u^0, h^0)})^{-1}(\widetilde{R}^{i_0}_{j'+1}(a^{i_0}))
- \sigma \circ u^0 (\widetilde{R}^0_{j+m+1}(a^0))) \notag\\
& \quad
- (\sigma \circ u^0 \circ (\Xi_{(a^0, b^0, u^0, h^0)})^{-1}(\widetilde{R}^{i_0}_{j'}(a^{i_0}))
- \sigma \circ u^0(\widetilde{R}^0_j(a^0))) \notag\\
& \quad
+ \lim_{s \to \infty}\bigl(\sigma \circ u^0 \circ
(\Xi_{(a^0, b^0, u^0, h^0)})^{-1}|_{[0, \infty) \times S^1_\mu}(s, t) \notag\\
& \quad \hphantom{+ \lim_{s \to \infty}\bigl(}
- \sigma \circ u^0|_{[0, \infty) \times S^1_{\mu_{j+c}}}
(s + \log \rho^0_{\mu_j} + \dots + \log \rho^0_{\mu_{j+c-1}}, t)\bigr) \notag\\
& \quad
- \lim_{s \to -\infty}\bigl(\sigma \circ u^0 \circ
(\Xi_{(a^0, b^0, u^0, h^0)})^{-1}|_{(-\infty, 0] \times S^1_\mu}(s, t) \notag\\
&\quad \hphantom{- \lim_{s \to -\infty}\bigl(}
- \sigma \circ u^0|_{(-\infty, 0] \times S^1_{\mu_{j+d}}}
(s - \log \rho^0_{\mu_{j+d+1}} - \dots - \log \rho^0_{\mu_{j+m}}, t)\bigr) \notag\\
& \quad
- L_\mu \sum_{c < l \leq d}T_{j+l} \notag\\
& =
(b^0_{\mu_j} + \dots + b^0_{\mu_{j+m}}) \notag\\
& \quad
+ (\sigma \circ u^0 \circ (\Xi_{(a^0, b^0, u^0, h^0)})^{-1}(\widetilde{R}^{i_0}_{j'+1}(a^{i_0}))
- \sigma \circ u^0 (\widetilde{R}^0_{j+m+1}(a^0))) \notag\\
& \quad
- (\sigma \circ u^0 \circ (\Xi_{(a^0, b^0, u^0, h^0)})^{-1}(\widetilde{R}^{i_0}_{j'}(a^{i_0}))
- \sigma \circ u^0(\widetilde{R}^0_j(a^0))) \notag\\
& \quad
+ \lim_{s \to \infty}
L_\mu(p_1 \circ (\Xi_{(a^0, b^0, u^0, h^0)})^{-1}|_{[0, \infty) \times S^1_\mu}(s, t)
\notag\\
& \quad \hphantom{+ \lim_{s \to \infty}L_\mu (}
- s - \log \rho^0_{\mu_j} - \dots - \log \rho^0_{\mu_{j+c-1}}) \notag\\
& \quad
- \lim_{s \to -\infty}
L_\mu(p_1 \circ (\Xi_{(a^0, b^0, u^0, h^0)})^{-1}|_{(-\infty, 0] \times S^1_\mu}(s, t)
\notag\\
&\quad \hphantom{- \lim_{s \to -\infty}L_\mu (}
- s + \log \rho^0_{\mu_{j+d+1}} + \dots + \log \rho^0_{\mu_{j+m}}) \notag\\
& \quad
- L_\mu \sum_{c < l \leq d}T_{j+l}
\label{b^{i_0} for zero kappa}
\end{align}
We define $b^{i_0}_\mu$ by the above formula (\ref{b^{i_0} for zero kappa}).
It is clear that $b^{i_0}_\mu$ is a smooth function of $(a^0, b^0, u^0, h^0)$
at $\rho^{i_0}_\mu \neq 0$.
We need to prove the smoothness at $\rho^{i_0}_\mu = 0$.
To prove the smoothness, we study the map $\Xi$.
As in the previous section,
we claim that there exists a smooth function $f_\mu : V_0 \to \mathbb{C}^\ast$ such that
\begin{align}
&(\rho^{i_0}_\mu)^{2\pi} e^{2\pi \sqrt{-1} \varphi^{i_0}_\mu)} \notag \\
&= \Bigl(\prod_{l=0}^m (\rho^0_{\mu_{j+l}})^{2\pi}\Bigr)
e^{2\pi (\sum_{l=0}^m
\sqrt{-1} \varphi^0_{\mu_{j+l}} - \sum_{l=1}^m T_{j+l})}
\cdot f_\mu(a^0, b^0, u^0, h^0). \label{bbf eq}
\end{align}
This can be proved as follows.
Let $(\widetilde{P}''_{i_0} \to \widetilde{X}''_{i_0}, Z''_{i_0})$ be the local universal family
of $(\Sigma'_{i_0}, z_{i_0} \cup z_{i_0}^+ \cup {z'_{i_0}}^{++} \cup z_{i_0}^{++})$.
Let ${z'''_{i_0}}^{++}$ be the points in ${z'_{i_0}}^{++} \cup z_{i_0}^{++}$ not contained in
the trivial floors of $(\Sigma'_{i_0}, z_{i_0}, u_0|_{\Sigma'_{i_0}})$,
and let $(\widetilde{P}'''_{i_0} \to \widetilde{X}'''_{i_0}, Z'''_{i_0})$ be the local universal
family of $(\Sigma_{i_0}, z_{i_0} \cup z_{i_0}^+ \cup {z'''_{i_0}}^{++})$.
Since the fiber of the center of $\widetilde{P}''_{i_0}$ is isomorphic to
$\widetilde{P}'_{i_0}$, $\widetilde{P}''_{i_0}$ is isomorphic to the product of
$\widetilde{P}'_{i_0}$ and a parameter space $D^M$ for additional marked points
corresponding to $z_{i_0}^{++}$.
Similarly, $\widetilde{P}'''_{i_0}$ is isomorphic to the product of $\widetilde{P}_{i_0}$
and a parameter space for additional marked points corresponding to
${z'''_{i_0}}^{++} \setminus z_{i_0}^{++}$.
By the assumption of the coordinate of $\widetilde{P}'_0$,
it is easy to see that if $(\rho'''_\mu, \varphi'''_\mu)$ is an appropriately chosen
parameter of $\widetilde{P}'''_{i_0}$ for the deformation of a neighborhood of
the joint circle $S^1_\mu$, then the following holds true under the natural map
$\widetilde{P}''_{i_0} \to \widetilde{P}'''_{i_0}$,
where we use the coordinate of $\widetilde{P}''_{i_0}$ given by
$\widetilde{P}''_{i_0} \cong \widetilde{P}'_{i_0} \times D^M$.
\[
(\rho'''_\mu)^{2\pi} e^{2\pi \sqrt{-1} \varphi'''_{\mu}}
= \Bigl(\prod_{l=0}^m (\rho^0_{\mu_{j+l}})^{2\pi}\Bigr)
e^{2\pi (\sum_{l=0}^m
\sqrt{-1} \varphi^0_{\mu_{j+l}} - \sum_{l=1}^m T_{j+l})}
\]
Since $\widetilde{P}'''_{i_0}$ is isomorphic to the product of $\widetilde{P}_{i_0}$ and
some parameter space, there exists a smooth map $f' : \widetilde{X}'''_{i_0} \to \mathbb{C}^\ast$
such that
\[
(\rho^{i_0}_\mu)^{2\pi} e^{2\pi \sqrt{-1} \varphi^{i_0}_{\mu}}
= (\rho'''_\mu)^{2\pi} e^{2\pi \sqrt{-1} \varphi'''_{\mu}} \cdot f'(a''')
\]
Therefore, there exists a smooth map $f : V_0 \to \mathbb{C}^\ast$ which satisfies
equation (\ref{bbf eq}).
Similarly, there exist smooth maps $f_\mu^{\text{left}}, f_\mu^{\text{right}}
: V_0 \times_{\widetilde{X_{i_0}}} \widetilde{P}_{i_0} \to \mathbb{C}^\ast$ such that
if $\rho^0_{\mu_{j+l}} \neq 0$ for $1 \leq l < c$ and $d < l \leq m$,
and $\rho^0_{\mu_{j+c}} = 0$ and $\rho^0_{\mu_{j+d}} = 0$, and
\begin{align*}
\Xi_{(a^0, b^0, u^0, h^0)}|_{[0, \infty) \times S^1_{\mu_{j+c}}}
(s_0^{\text{left}}, t_0^{\text{left}}) &= (s_{i_0}^{\text{left}}, t_{i_0}^{\text{left}}),\\
\Xi_{(a^0, b^0, u^0, h^0)}|_{(-\infty, 0] \times S^1_{\mu_{j+d}}}
(s_0^{\text{right}}, t_0^{\text{right}}) &= (s_{i_0}^{\text{right}}, t_{i_0}^{\text{right}}),
\end{align*}
then
\begin{align*}
e^{2\pi (s_{i_0}^{\text{left}} + \sqrt{-1} t_{i_0}^{\text{left}})}
&= e^{2\pi (s_0^{\text{left}} + \sqrt{-1} t_0^{\text{left}})}
\Bigl(\prod_{0 \leq l < c} (\rho^0_{\mu_{j+l}})^{2\pi} \Bigr)
e^{2\pi \sqrt{-1} \sum_{0 \leq l < c} \varphi^0_{\mu_{j+l}}}\\
&\quad \cdot e^{-2\pi \sum_{0 < l \leq c} T_{j+l}}
f_\mu^{\text{left}} (s_{i_0}^{\text{left}}, t_{i_0}^{\text{left}}, a^0, b^0, u^0, h^0),
\end{align*}
and
\begin{align*}
e^{2\pi (s_{i_0}^{\text{right}} + \sqrt{-1} t_{i_0}^{\text{right}})}
&= e^{2\pi (s_0^{\text{right}} + \sqrt{-1} t_0^{\text{right}})}
\Bigl(\prod_{d < l \leq m} (\rho^0_{\mu_{j+l}})^{2\pi} \Bigr)
e^{2\pi \sqrt{-1}\sum_{d < l \leq m}
\varphi^0_{\mu_{j+l}}}\\
& \quad \cdot e^{-2\pi \sum_{d < l \leq m} T_{j+l}}
f_\mu^{\text{right}} (s_{i_0}^{\text{right}}, t_{i_0}^{\text{right}}, a^0, b^0, u^0, h^0).
\end{align*}
Furthermore, $f_\mu$, $f_\mu^{\text{left}}$ and $f_\mu^{\text{right}}$ satisfy
\begin{align*}
f_\mu(a^0, b^0, u^0, h^0) &= \lim_{s_{i_0}^{\text{left}} \to \infty}
f_\mu^{\text{left}}(s_{i_0}^{\text{left}}, t_{i_0}^{\text{left}}, a^0, b^0, u^0, h^0)\\
& \quad \cdot \lim_{s_{i_0}^{\text{right}} \to -\infty}
f_\mu^{\text{right}}(s_{i_0}^{\text{right}}, t_{i_0}^{\text{right}}, a^0, b^0, u^0, h^0).
\end{align*}
Therefore, $b^{i_0}_\mu$ satisfies
\begin{align}
b^{i_0}_\mu
& =
(b^0_{\mu_j} + \dots + b^0_{\mu_{j+m}}) \notag\\
& \quad
+ (\sigma \circ u^0 \circ (\Xi_{(a^0, b^0, u^0, h^0)})^{-1}(\widetilde{R}^{i_0}_{j'+1}(a^{i_0}))
- \sigma \circ u^0 (\widetilde{R}^0_{j+m+1}(a^0))) \notag\\
& \quad
- (\sigma \circ u^0 \circ (\Xi_{(a^0, b^0, u^0, h^0)})^{-1}(\widetilde{R}^{i_0}_{j'}(a^{i_0}))
- \sigma \circ u^0(\widetilde{R}^0_j(a^0))) \notag\\
& \quad
+ L_\mu \cdot \frac{1}{2\pi} \log|f_\mu(a^0, b^0, u^0, h^0)|
- L_\mu \sum_{0 < l \leq m} T_{j+l} \label{b^{i_0} eq}
\end{align}
in both cases.
Hence $b^{i_0}_\mu$ is a smooth function of $(a^0, b^0, u^0, h^0)$.
It is easy to check that the constructed map
$(a^0, b^0, u^0, h^0) \to (a^{i_0}, b^{i_0}, u^{i_0}, h^{i_0})$ is the required essential
submersion.
For example, the essential submersiveness of $\phi : V_0 \to \prod_i V_i$ is seen
as follows.
By the coordinate change (i.e. changing the center $p'$ of the Kuranishi neighborhood),
it is enough to prove that
$\phi|_{\phi^{-1}(\prod_i V_i^{(k_i)})} : \phi^{-1}(\prod_i V_i^{(k_i)}) \to \prod_i V_i^{(k_i)}$
is submersive.
($V_i^{(k_i)} \subset V_i$ are the corners of the highest codimension.)
It is clear that $\phi^{-1}(\prod_i V_i^{(k_i)})$ is a union of corners of $V_0$,
and each of them is defined by
$\{\rho_\mu = 0; \text{ for all } \mu \in \bigcup_{j \in I} M^0_j\}$ for some
$I \subset \{1, 2, \dots, k_0 - 1\}$.
Then $b^0_\mu \in \mathbb{R}$ ($\mu \in \bigcup_{j \in I} M_i$) are independent parameters
in $\mathring{X}$ since $-L_\mu \log \rho^0_\mu + b^0_\mu = \infty$ for all $b^0_\mu$.
For each $\mu \in M^{i_0}_{j'}$, let $S^1_{\mu_j}, S^1_{\mu_{j+1}}, \dots, S^1_{\mu_m}
\subset \Sigma'_{i_0} \subset \Sigma'_0$ be the joint circles which collapse to
$S^1_\mu$ by $\Sigma'_{i_0} \to \Sigma_{i_0}$ as above.
Assume $\mu_j \in M^0_j$.
Then there exists some $j+l \in \{j, j+1, \dots, j+m\}$ such that $j+l \in I$.
Since the derivative of $b^{i_0}_\mu$ by $b^0_{\mu_{j+l}}$ does not vanish by
(\ref{b^{i_0} eq}), it is easy to check
that $\phi|_{\phi^{-1}(\prod_i V_i^{(k_i)})}$ is submersive.
Since $E_0$ is the direct sum of $E_i$ ($1 \leq i \leq N$),
grouped multisections of $(V_i, E_i)$ define a grouped multisection of $(V_0, E_0)$
by the pull back of the product grouped multisection by the essential submersion.
(We assume that the grouped multisection of $(V_i, E_i)$ and $(V_j, E_j)$ coincide
if $p_i = p_j$.)
We note that $\dim V_0 = \dim \prod_{i=1}^N V_i - (N - 1) > \dim \prod_{i=1}^N V_i$
if $N > 1$.
\subsection{Construction of global structure}\label{global construction}
In this section, we construct a global pre-Kuranishi structure of $\widehat{\mathcal{M}}$.
As we explained in Section \ref{embed},
a Kuranishi neighborhood of each point $p \in \widehat{\mathcal{M}}$ is determined
by the data $(z^+, S, E^0, \lambda)$.
Hence construction of a pre-Kuranishi structure of $\widehat{\mathcal{M}}$
is equivalent to constructing a Hausdorff space $\mathcal{X}$ with a locally homeomorphic
surjection $\mu : \mathcal{X} \to \widehat{\mathcal{M}}$ and giving such data for each point of $\mathcal{X}$.
In Section \ref{embed}, $S$ is a codimension-two submanifold of $Y$ and
$z^+$ is a finite subset of the domain curve,
but for the construction of global structure,
it is convenient to use a finite set $\mathcal{S} = \{S\}$ of codimension-two
submanifolds of $Y$ and a finite family $z^+ = (z^S)_{S \in \mathcal{S}}$ of finite subsets
of the domain curve indexed by $\mathcal{S}$ instead.
(We assume that $\pi_Y \circ u$ intersects each $S$ at $z^S$ transversely.)
First we introduce three versions of the space of holomorphic buildings
$\widehat{\mathcal{M}}_{\mathcal{S}}$, $\widehat{\mathcal{M}}_{\mathcal{S}, A}$ and
$\widehat{\mathcal{M}}_{o, \mathcal{S}, A}$.
We will realize the Hausdorff space $\mathcal{X}$ as a subspace of
$\widehat{\mathcal{M}}_{o, \mathcal{S}, A}$.
Let $\mathcal{S} = \{S\}$ be a finite set of codimension-two submanifolds of $Y$.
A point $(\Sigma, z, z^S, z^A, z^o, u)$ of $\widehat{\mathcal{M}}_{o, \mathcal{S}, A}$ consists of
a holomorphic building $(\Sigma, z, u) \in \widehat{\mathcal{M}} = \widehat{\mathcal{M}}(Y, \lambda, J)$,
finite subsets $z^S \subset \Sigma$ ($S \in \mathcal{S}$),
a finite subset $z^A \subset \Sigma$ and
a finite subset $z^o \subset \Sigma$ which satisfy the following conditions:
\begin{itemize}
\item
$\pi_Y \circ u$ intersects $S$ at $z^S$ transversely for each $S \in \mathcal{S}$.
\item
$z^S, z^A, z^o \subset \Sigma$ are disjoint, do not contain any special points
of $(\Sigma, z, u)$ and any points of the imaginary circles of $\Sigma$
and the trivial cylinders of $(\Sigma, z, u)$.
\item
All non-trivial components (i.e. irreducible components other than trivial cylinders) of
$(\Sigma, z, u)$ are stable in $(\Sigma, z, z^S)$.
\end{itemize}
$z^S$ are used to make the domain curve stable,
$z^A$ is used to control the automorphism group of the domain curve, and
$z^o$ is a mark which tells us the additional vector space $E^0$ we used
for the construction of the Kuranishi neighborhood.
In other words, $z^o$ is used to realize the space $\mathcal{X}$ as a subspace of
$\widehat{\mathcal{M}}_{o, \mathcal{S}, A}$.
Two points $(\Sigma, z, z^S, z^A, z^o, u)$ and $(\Sigma', z', (z')^S, (z')^A, (z')^o, u')$ are
the same point if there exists a biholomorphism $\varphi : \Sigma \to \Sigma'$ and
an $\mathbb{R}$-translation $\theta$ such that $\varphi(z) = z'$, $\varphi(z^S) = (z')^S$
for all $S \in \mathcal{S}$, $\varphi(z^A) = (z')^A$,
$\varphi(z^o) = (z')^o$ and $u' \circ \varphi = (\theta \times 1) \circ u$.
The topology of $\widehat{\mathcal{M}}_{o, \mathcal{S}, A}$ is defined as a quotient space of
a subspace of $\overline{\mathcal{M}}(Y, \lambda, J)$
(locally, it is the quotient by the $S^1$-actions on the coordinates of limit circles
and the symmetric group of the sets $z$, $z^S$ ($S \in \mathcal{S}$), $z^A$, $z^o$ and
the set of limit circles).
$\widehat{\mathcal{M}}_{\mathcal{S}, A}$ consists of points $(\Sigma, z, z^S, z^A, u)$,
and $\widehat{\mathcal{M}}_{\mathcal{S}}$ consists of points $(\Sigma, z, z^S, u)$.
We may regard them as the subspaces of $\widehat{\mathcal{M}}_{o, \mathcal{S}, A}$ defined by
$z^o = \emptyset$ and $(z^o, z^S) = (\emptyset, \emptyset)$ respectively.
If $\mathcal{S}' \supset \mathcal{S}$, we regard $\widehat{\mathcal{M}}_{o, \mathcal{S}, A}$
as a subspace of $\widehat{\mathcal{M}}_{o, \mathcal{S}', A}$.
The forgetful map $\mathop{\mathfrak{forget}}\nolimits_{\mathcal{S}, A} : \widehat{\mathcal{M}}_{\mathcal{S}, A} \to
\widehat{\mathcal{M}}$ is defined by forgetting the points $z^S$ and $z^A$.
Similarly, we define
$\mathop{\mathfrak{forget}}\nolimits_{o, \mathcal{S}, A} : \widehat{\mathcal{M}}_{o, \mathcal{S}, A} \to \widehat{\mathcal{M}}$.
For two holomorphic buildings
$p^k = (\Sigma^k, z^k, (z^k)^S, (z^k)^A, (z^k)^o, u^k) \in \widehat{\mathcal{M}}_{o, \mathcal{S}, A}$
($k=1,2$), we say $p^1 \leq p^2$ if $(z^2)^S$, $(z^2)^A$ and $(z^2)^o$
are $\mathop{\mathrm{Aut}}\nolimits(\Sigma^2, z^2, u^2)$-invariant and
there exists a biholomorphism $\Sigma^1 \cong \Sigma^2$ such that
under this biholomorphism,
$p_1$ is obtained from $p^2$ by forgetting some
subsets of $(z^2)^S$, $(z^2)^A$ and $(z^2)^o$.
(The forgetful map from $p^2$ to $p^1$ does not collapse any components.)
We also define two versions of the space of stable curves
$\overline{\mathcal{M}}^\mathrm{D}_{\mathcal{S}}$ and $\overline{\mathcal{M}}^\mathrm{D}_{\mathcal{S}, A}$ as follows.
A point $(\hat \Sigma, z, z^S, z^A)$ of $\overline{\mathcal{M}}^\mathrm{D}_{\mathcal{S}, A}$
consists of a semistable curve $\hat \Sigma$ and finite disjoint subsets $z$,
$z^S$ ($S \in \mathcal{S}$) and $z^A$ such that
they do not contain any nodal points and the automorphism group of
$(\hat \Sigma, z, z^S)$ is finite.
Similarly, $\overline{\mathcal{M}}^\mathrm{D}_{\mathcal{S}}$ consists of points
$(\hat \Sigma, z, z^S)$ which satisfy the same conditions.
There is another forgetful map $\mathop{\mathfrak{forget}}\nolimits_u : \widehat{\mathcal{M}}_{\mathcal{S}, A} \to
\overline{\mathcal{M}}^\mathrm{D}_{\mathcal{S}, A}$
defined by forgetting the map $u$, blowing down
joint circles to nodal points, blowing down limit circles and add these points to
marked points $z$, and stabilizing (collapsing all components corresponding to trivial
cylinders).
For example, $\mathop{\mathfrak{forget}}\nolimits_u$ maps $(\Sigma, z, z^S, z^A, u)$ in Figure \ref{(Sigma,z,zS,zA)} to
$(\hat \Sigma, z, z^S, z^A)$ in Figure \ref{(hatSigma,z,zS,zA)}.
\begin{figure}
\centering
\includegraphics[width= 350pt]{Fig_Sigma_z_zS_zA.png}
\caption{$p = (\Sigma, z, z^S, z^A, u)$}\label{(Sigma,z,zS,zA)}
\includegraphics[width= 350pt]{Fig_hatSigma_z_zS_zA.png}
\caption{$(\hat \Sigma, z, z^S, z^A) = \mathop{\mathfrak{forget}}\nolimits_u(p)$}\label{(hatSigma,z,zS,zA)}
\end{figure}
$(\mathcal{S}, A)$-forgetful map $f$ from
$p \in \overline{\mathcal{M}}^\mathrm{D}_{\mathcal{S}, A}$
to $q \in \overline{\mathcal{M}}^\mathrm{D}_{\mathcal{S}, A}$ is a forgetful map
$f : p \to q$ obtained by forgetting some subsets of $z^S$ and $z^A$, and
stabilizing the curve.
Fix an arbitrary large constant $L_{\max} > 0$, and
let $\delta_0 > 0$ be a positive constant such that
$4 \delta_0$ is less than the minimal period of periodic orbits and
$2 \delta_0$ is less than the minimal difference $L^+ - L^-> 0$ of two periods
$L^-, L^+$ of periodic orbits such that $L^- < L^+ \leq L_{\max}$.
Let $\widehat{\mathcal{M}}^{\leq L_{\max}} \subset \widehat{\mathcal{M}}$ be the subspace of
holomorphic buildings the sums of the periods of whose $+\infty$-limit circles
are $\leq L_{\max}$.
We say a holomorphic building $(\Sigma, z, u) \in \widehat{\mathcal{M}}$ is of type
$\theta = (g, k, E_{\hat \omega})$ if the genus of $\Sigma$ is $g$,
the total number of marked points and limit circles is $k$, and
$E_{\hat \omega}(u) = E_{\hat \omega}$.
For each triple $\theta = (g, k, E_{\hat \omega})$,
we define $\widetilde{e}(\theta) = \widetilde{e}_{\delta_0}(\theta)$ by
$\widetilde{e}_{\delta_0}(\theta) = 5(g-1) + 2k + E_{\hat \omega} / \delta_0$.
For a holomorphic building $p \in \widehat{\mathcal{M}}$ of type $\theta$, we
define $\widetilde{e}(p) = \widetilde{e}(\theta)$.
Then $\widetilde{e}$ satisfies the following:
\begin{itemize}
\item
$\widetilde{e}(p) \geq 1$
for any holomorphic building $p \in \widehat{\mathcal{M}}^{\leq L_{\max}}$.
\item
For any holomorphic building $p \in \widehat{\mathcal{M}}^{\leq L_{\max}}$,
any subset $C_0$ of its nodal points and any subset $C_1$ of the gaps of floors,
replace the nodal points in $C_0$ and the joint circles in the gaps in $C_1$
of $p$ to pairs of marked points and pairs of
limit circles respectively, and let $p'_i$ ($1 \leq i \leq N$) be their non-trivial connected
components. (They are connected holomorphic buildings of height-one without
nodal points.)
Then $\widetilde{e}(p) \geq \sum_i \widetilde{e}(p'_i)$, and
the inequality is strict if $C_0 \neq \emptyset$ or $C_1 \neq \emptyset$.
In particular, $\widetilde{e}(p) > \widetilde{e}(p'_i)$ for all $i$ if $N > 1$.
\end{itemize}
The second property is easy to check.
(We recall that $\widehat{\mathcal{M}}$ consists of holomorphic buildings without trivial buildings,
where a trivial building is a connected component which consists of trivial cylinders only.)
We check the first property.
By the second property, it is enough to check the property for connected height-one
holomorphic buildings.
If the domain curve is stable, then $\widetilde{e} \geq 1$ is clear.
If the domain curve is unstable, then since $E_{\hat \omega} > 0$,
$k$ must be $\geq 1$.
Hence $(g, k) = (0,1)$ or $(0,2)$.
If $(g, k) = (0,1)$, then $E_{\hat \omega}$ is greater than or equal to the minimal
period of periodic orbit.
If $(g, k) = (0,2)$, then $E_{\hat \omega}$ is greater than or equal to the minimal
difference of two periods of periodic orbits.
Hence in both cases, $\widetilde{e} \geq 1$ by the definition of $\delta_0 > 0$.
For each triple $\theta = (g, k, E_{\hat \omega})$, let
$\widehat{\mathcal{M}}^{\leq L_{\max}}_\theta \subset \widehat{\mathcal{M}}$ be the subspace of
holomorphic buildings of type $\theta$ such that
the sums of the periods of $+\infty$-limit circles are $\leq L_{\max}$.
We also define $\widehat{\mathcal{M}}^{\leq L_{\max}}_{\mathcal{S}, A, \theta}
= \mathop{\mathfrak{forget}}\nolimits_{\mathcal{S}, A}^{-1}(\widehat{\mathcal{M}}^{\leq L_{\max}}_\theta)$.
Note that for any constants $C \geq 0$,
$\widehat{\mathcal{M}}^{\leq L_{\max}}_{\leq C}
= \bigcup_{\widetilde{e}(\theta) \leq C} \widehat{\mathcal{M}}^{\leq L_{\max}}_\theta$
is compact.
For each triple $\theta = (g, k, E_{\hat \omega})$,
let $\overline{\mathcal{M}}^\mathrm{D}_{\mathcal{S}, A, \theta} \subset
\overline{\mathcal{M}}^\mathrm{D}_{\mathcal{S}, A}$ be the subspace of stable curves
whose genus and the number of marked points are $g$ and $k$ respectively.
We regard the spaces for different $E_{\hat \omega}$ as disjoint spaces,
and regard the forgetful map $\mathop{\mathfrak{forget}}\nolimits_u$ as a map from
$\widehat{\mathcal{M}}^{\leq L_{\max}}_{\mathcal{S}, A, \theta}$ to
$\overline{\mathcal{M}}^\mathrm{D}_{\mathcal{S}, A, \theta}$.
For a point $\hat p \in \overline{\mathcal{M}}^\mathrm{D}_{\mathcal{S}, A, \theta}$,
we define $\theta(\hat p) = \theta$.
For each point $\hat p = (\hat \Sigma, z, z^S, z^A) \in
\overline{\mathcal{M}}^\mathrm{D}_{\mathcal{S}, A, \theta}$, we define an integer $l(\hat p)$ by
$l(\hat p) = 3(g - 1) + \# z + \sum_{S \in \mathcal{S}} \# z^S + \# z^A$,
where $g$ is the genus of $\hat \Sigma$.
For a point $p \in \widehat{\mathcal{M}}_{\mathcal{S}, A}$, we define $l(p) = l(\mathop{\mathfrak{forget}}\nolimits_u(p))$.
For each $l \geq 0$, let
$\overline{\mathcal{M}}^\mathrm{D}_{\mathcal{S}, A, \theta, l} \subset
\overline{\mathcal{M}}^\mathrm{D}_{\mathcal{S}, A, \theta}$ be the subspace of curves $\hat p$
such that $l(\hat p) = l$, and $\widehat{\mathcal{M}}^{\leq L_{\max}}_{\mathcal{S}, A, \theta, l}
\subset \widehat{\mathcal{M}}^{\leq L_{\max}}_{\mathcal{S}, A, \theta}$ be the subspace of
holomorphic buildings $p$ such that $l(p) = l$.
Let $(\hat p, E^0, \lambda)$ be a triple of a stable curve
$\hat p = (\hat \Sigma, z, z^S, z^A, z^o) \in \overline{\mathcal{M}}^\mathrm{D}_{o, \mathcal{S}, A}$,
a finite-dimensional $\mathop{\mathrm{Aut}}\nolimits(\hat p)$-vector space $E^0$, and
an $\mathop{\mathrm{Aut}}\nolimits(\hat p)$-equivariant linear map
\[
\lambda : E^0 \to C^\infty(\hat \Sigma \times Y, {\textstyle\bigwedge}^{0, 1} T^\ast \hat \Sigma
\otimes_\mathbb{C} (\mathbb{R} \partial_\sigma \oplus T Y)).
\]
We call such a triple $(\hat p, E^0, \lambda)$ a stable curve with perturbation
parameters.
We say two stable curves with perturbation parameters $(\hat p_k, E^0_k, \lambda_k)$
($k =1,2$) are isomorphic if there exist an isomorphism $f : \hat p_1 \to \hat p_2$
and an isomorphism $\hat \phi_f : E^0_1 \to E^0_2$ which is $\mathop{\mathrm{Aut}}\nolimits(\hat p_1)$-equivariant
with respect to the isomorphism $\rho_f : \mathop{\mathrm{Aut}}\nolimits(\hat p_1) \to \mathop{\mathrm{Aut}}\nolimits(\hat p_2)$
associated to $f$, and they make the following diagram commutative.
\[
\begin{tikzcd}
E^0_1 \ar{r}{\lambda_1} \ar{d}{\hat \phi_f} &
C^\infty(\hat \Sigma_1 \times Y, {\textstyle\bigwedge}^{0, 1} T^\ast \hat \Sigma_1
\otimes_\mathbb{C} (\mathbb{R} \partial_\sigma \oplus T Y)) \\
E^0_2 \ar{r}{\lambda_2} &
C^\infty(\hat \Sigma_2 \times Y, {\textstyle\bigwedge}^{0, 1} T^\ast \hat \Sigma_2
\otimes_\mathbb{C} (\mathbb{R} \partial_\sigma \oplus T Y))
\ar{u}{f^\ast}
\end{tikzcd}
\]
A holomorphic building with perturbation parameters $(p, E^0, \lambda)$ is a triple
of a holomorphic building $p = (\Sigma, z, z^S, z^A, z^o, u) \in
\widehat{\mathcal{M}}_{o, \mathcal{S}, A}$, a finite-dimensional $\mathop{\mathrm{Aut}}\nolimits(p)$-vector space $E^0$,
and an $\mathop{\mathrm{Aut}}\nolimits(p)$-equivariant linear map
\[
E^0 \to C^\infty(\hat \Sigma \times Y, {\textstyle\bigwedge}^{0, 1} T^\ast \hat \Sigma
\otimes_\mathbb{C} (\mathbb{R} \partial_\sigma \oplus T Y)),
\]
where $(\hat \Sigma, z, z^S, z^A, z^o) = \mathop{\mathfrak{forget}}\nolimits_u(\Sigma, z, z^S, z^A, z^o, u)$
is the stabilization of the blow down of the domain curve.
We say two such triples $(p_k, E^0_k, \lambda_k)$ ($k=1,2$) are isomorphic if
there exist an isomorphism $f : p_1 \to p_2$ and
an $\mathop{\mathrm{Aut}}\nolimits(p_1)$-equivariant isomorphism $\hat \phi_f : E^0_1 \to E^0_2$ such that
$\lambda_1 = f^\ast \circ \lambda_2 \circ \hat \phi_f$.
For an arbitrary constant $C \geq 0$,
we will construct a space $\mathcal{X} = \mathcal{X}_{\leq C}
= \bigcup_{\widetilde{e}(\theta) \leq C} \mathcal{X}_{\theta}$ consisting of
holomorphic buildings with perturbation parameters
which satisfies the following conditions:
\begin{enumerate}
\item
For each $(p, E^0_p, \lambda_p) \in \mathcal{X}_{\theta}$,
$p$ is contained in $\widehat{\mathcal{M}}^{\leq L_{\max}}_{o, \mathcal{S}, A, \theta}$.
\item
For any $p = (\Sigma, z, u) \in \widehat{\mathcal{M}}^{\leq L_{\max}}_{\theta}$,
there exists $(p^+, E^0, \lambda) \in \mathcal{X}_{\theta}$
such that $p = \mathop{\mathfrak{forget}}\nolimits_{o, \mathcal{S}, A}(p^+)$.
Furthermore, $\mathop{\mathfrak{forget}}\nolimits_{o, \mathcal{S}, A} : \mathcal{X}_{\theta} \to
\widehat{\mathcal{M}}^{\leq L_{\max}}_{\theta}$ is locally homeomorphic.
\item
For any $(p = (\Sigma, z, z^S, z^A, z^o, u), E^0_p, \lambda_p) \in \mathcal{X}$, let
\[
E^0_p \to C^\infty(\Sigma \times Y, {\textstyle\bigwedge}^{0,1} T^\ast \Sigma \otimes_\mathbb{C}
(\mathbb{R} \partial_\sigma \oplus T Y))
\]
be the pull back of $\lambda_p$ by the forgetful map $p \to \mathop{\mathfrak{forget}}\nolimits_u(p)$ and
also denote it by the same symbol $\lambda_p$.
Then the linear map
\begin{align}
&\widetilde{W}^{1, q}_\delta(\Sigma, u^\ast T\hat Y) \oplus E^0_p \notag\\
&\to L^p_\delta(\Sigma, {\textstyle\bigwedge}^{0, 1} T^\ast \Sigma \otimes_\mathbb{C} u^\ast T \hat Y)
\oplus \bigoplus_{\text{limit circles}} \mathop{\mathrm{Ker}}\nolimits A_{\gamma_{\pm\infty_i}}
/ (\mathbb{R} \partial_\sigma \oplus \mathbb{R} R_\lambda) \notag\\
&\quad \oplus \bigoplus_{z_i \in z} T_{\pi_Y \circ u(z_i)} Y \notag\\
&(\xi, h)
\mapsto (D_p \xi + \lambda_p(h)(\cdot, \pi_Y \circ u(\cdot)), \notag \\
&\hphantom{(\xi, h) \mapsto (}
\sum_j \langle \xi|_{S^1_{\pm\infty_i}}, \eta^{\pm\infty_i}_j \rangle \eta^{\pm\infty_i}_j,
\pi_Y \circ \xi(z_i))
\label{X surjective map}
\end{align}
is surjective, where $D_p$ is the linearization of the equation of $J$-holomorphic maps,
and $\{\eta^{\pm\infty_i}\}_j$ is an orthonormal basis of the orthogonal complement of
$\mathbb{R} \partial_\sigma \oplus \mathbb{R} R_\lambda$ in $\mathop{\mathrm{Ker}}\nolimits A_{\gamma_{\pm\infty_i}}$
for each $\pm\infty_i$.
\item
\label{X embedding relation}
If two points $(p_k^+ = (\Sigma, z, z^{S, k}, z^{A, k}, z^{o, k}, u), E^0_k, \lambda_k) \in \mathcal{X}$
($k=1,2$) over the same holomorphic building $p = (\Sigma, z, u) \in \widehat{\mathcal{M}}$ satisfy
$z^{S, 2} \supset z^{S, 1}$, $z^{A, 2} \supset z^{A, 1}$ and $z^{o, 2} \supset z^{o, 1}$
(that is, if $p_1^+ \leq p^+_2$),
then $E^0_2$ contains $E^0_1$ as a subspace, and the restriction of $\lambda_2$
coincides with $\lambda_1$.
\item
For any $p = (\Sigma, z, z^S, z^A, z^o, u) \in \mathcal{X}$, $z^S$, $z^A$ and $z^o$ are
$\mathop{\mathrm{Aut}}\nolimits(\mathop{\mathfrak{forget}}\nolimits_{o, \mathcal{S}, A}(p))$-invariant.
\item
$\mathcal{X}$ is embedded in $\widehat{\mathcal{M}}_{o, \mathcal{S}, A}$.
In fact, we add marked point $z^o$ to distinguish $E^0$ and $\lambda$.
\item
\label{X vee existence}
For any two points
$(p_k^+ = (\Sigma, z, z^{S, k}, z^{A, k}, z^{o, k}, u), E^0_k, \lambda_k) \in \mathcal{X}$
($k=1,2$) for the same holomorphic building $p = (\Sigma, z, u) \in \widehat{\mathcal{M}}$,
there exists some
$(p_3^+ = (\Sigma, z, z^{S, 3}, z^{A, 3}, z^{o, 3}, u), E^0_3, \lambda_3)
\in \mathcal{X}$ such that $z^{S, 3} = z^{S, 1} \cup z^{S, 2}$, $z^{A, 3} = z^{A, 1} \cup z^{A, 2}$
and $z^{o, 3} = z^{o, 1} \cup z^{o, 2}$.
(In the definition of pre-Kuranishi structure, $p_3^+$ will be the unique supremum
$p_1^+ \vee p_2^+$.)
\item
\label{X decomposition into parts}
$\mathcal{X}$ satisfies the following compatibility condition with respect to the decomposition of
a holomorphic building into parts:
For any point $p = (\Sigma, z, z^S, z^A, z^o, u) \in \widehat{\mathcal{M}}_{o, \mathcal{S}, A}$,
replace all nodal points and all joint circles with pairs of marked points and
pairs of limit circles respectively,
and let $p'_i \in \widehat{\mathcal{M}}_{o, \mathcal{S}, A}$ ($i = 1, \dots, k$) be its
connected components other than trivial cylinders.
(Each $p'_i$ is a connected height-one holomorphic building without nodal points.)
Then $(p, E^0, \lambda) \in \mathcal{X}$ for some $E^0$ and $\lambda$
if $z^S$, $z^A$ and $z^o$ are $\mathop{\mathrm{Aut}}\nolimits((\mathop{\mathfrak{forget}}\nolimits_{o, \mathcal{S}, A}(p'_i))_i)$-invariant
and $(p'_i, E^0_i, \lambda_i) \in \mathcal{X}$ for some $E^0_i$ and $\lambda_i$ for all $i$.
($\mathop{\mathrm{Aut}}\nolimits((\mathop{\mathfrak{forget}}\nolimits_{o, \mathcal{S}, A}(p'_i))_i)$ is the automorphism group
of $\coprod_i \mathop{\mathfrak{forget}}\nolimits_{o, \mathcal{S}, A}(p'_i)$.)
Furthermore, $E^0$ is isomorphic to the direct sum of $E^0_i$ and
the restriction of $\lambda$ to $E^0$ coincides with the pull back of $E^0_i$ by
the forgetful map.
\end{enumerate}
We define a pre-Kuranishi structure of $\widehat{\mathcal{M}}^{\leq L_{\max}}_{\theta}$ by
$\mathcal{X}_{\theta} \subset
\widehat{\mathcal{M}}^{\leq L_{\max}}_{o, \mathcal{S}, A, \theta}$ and
a locally homeomorphic surjection
$\mathop{\mathfrak{forget}}\nolimits_{o, \mathcal{S}, A} : \mathcal{X}_{\theta} \to \widehat{\mathcal{M}}^{\leq L_{\max}}_{\theta}$.
To define a Kuranishi neighborhood of $\mathop{\mathfrak{forget}}\nolimits_{o, \mathcal{S}, A}(p)$ for
each $(p, E^0, \lambda) \in \mathcal{X}$, we need to extend
$\lambda$ to a local universal family of $\mathop{\mathfrak{forget}}\nolimits_u(p)$.
Hence we also construct a space of stable curves with perturbation parameters
which gives a neighborhood of the domain curves of holomorphic buildings in $\mathcal{X}$
in a sense. (See Lemma \ref{good family of additional vector spaces} for details.)
Condition \ref{X embedding relation} will imply that for any two $p^+_k \in \mathcal{X}$ ($k = 1,2$)
for the same holomorphic building $p \in \widehat{\mathcal{M}}$, if $p^+_1 \leq p^+_2$, then
we can define the embedding of the Kuranishi neighborhood of $p$ defined by
the data associated to $p^+_1$ to that defined by the data associated to $p^+_2$.
Furthermore, Condition \ref{X vee existence} imply the existence of
the unique supremum of any two points in the same fiber of
$\mathop{\mathfrak{forget}}\nolimits_{o, \mathcal{S}, A} : \mathcal{X}_{\theta} \to \widehat{\mathcal{M}}^{\leq L_{\max}}_{\theta}$.
If we ignore the algebraic structure of $\widehat{\mathcal{M}}$ such as the fiber product
structure, then we do not need Condition \ref{X decomposition into parts} and
the construction is easy.
To explain the idea, first we explain this easy version of the construction of $\mathcal{X}$ briefly.
We cover $\widehat{\mathcal{M}}^{\leq L_{\max}}_{\theta}$ by open subsets $\mathcal{U}_i$
($i = 1, \dots, N$) and for each $i$,
choose a family $\mathcal{S}_i =\{S\}$ of codimension-two submanifolds
of $Y$, and add the inverse images $(\pi_Y \circ u)^{-1}(S)$ to each holomorphic buildings
$p = (\Sigma, z, u) \in \mathcal{U}_i$ as the marked points $(z^i)^S$.
If we choose an appropriate family $\mathcal{S}_i$, then the $\pi_Y \circ u$ is transverse
to all submanifolds $S \in \mathcal{S}_i$ and all irreducible components of
$(\Sigma, z, (z^i)^S)$ other than trivial cylinders of $(\Sigma, z, u)$ are stable
for all $p = (\Sigma, z, u) \in \mathcal{U}_i$.
Assume that for each $i$, there exists a local universal family
$(\hat P^i \to \hat X^i, Z^i, (Z^i)^S)$ which contains $\mathop{\mathfrak{forget}}\nolimits_u(\Sigma, z, (z^i)^S, u)$
as fibers for all $p = (\Sigma, z, u) \in \mathcal{U}_i$.
Choose finite dimensional vector space $E^0_i$ and linear map
$\lambda_i : E^0_i \to C^\infty(\hat P^i \times Y, {\textstyle\bigwedge}^{0,1} V^\ast \hat P^i
\otimes_\mathbb{C} (\mathbb{R} \partial_\sigma \oplus TY))$
which makes the linear map (\ref{X surjective map}) surjective for all $p \in \mathcal{U}_i$.
Then we can define $\mathcal{X}_{\theta}$ by the space of holomorphic buildings with
perturbation parameters $(p^+ = (\Sigma, z, \bigcup_{i \in I} (z^i)^S, u),
\bigoplus_{i \in I} E^0_i, \bigoplus_{i \in I} \lambda_i)$
for holomorphic buildings $p = (\Sigma, z, u) \in \widehat{\mathcal{M}}$ and non empty subsets
$I \subset \{1, \dots, N\}$ such that $p \in \mathcal{U}_i$ for all $i \in I$.
For each point $(p^+ = (\Sigma, z, \bigcup_{i \in I} (z^i)^S, u),
\bigoplus_{i \in I} E^0_i, \bigoplus_{i \in I} \lambda_i)$, we associate the Kuranishi
neighborhood of $p = (\Sigma, z, u)$ defined by the direct sum of the pull backs of
$\lambda_i$ by the $(\mathcal{S}, A)$-forgetful maps for all $i \in I$.
To realize $\mathcal{X}$ as a subspace of $\widehat{\mathcal{M}}_{o, \mathcal{S}}$,
we choose a family of sections $(Z^i)^o = ((Z^i)^o_j)$ of $\hat P^i \to \hat X^i$
for each $i$ and add the union of the values of $(Z^i)^o$ for all $i \in I$ to
each $p^+ = (\Sigma, z, \bigcup_{i \in I} \bigcup_{i \in I} (z^i)^S, u)$
as marked points $z^o$.
This is the outline of the construction of $\mathcal{X}$ in the case where
we ignore the algebraic structure of $\widehat{\mathcal{M}}$.
For Condition \ref{X decomposition into parts}, we need to extend the linear maps
$\lambda$ for decomposable holomorphic buildings
given as the union of those associated for the parts
to their neighborhoods in a compatible way.
To compare the linear maps associated to different points in $\mathcal{X} \subset
\widehat{\mathcal{M}}_{o, \mathcal{S}, A}$ for the same holomorphic building,
we need to assume that these points are related by
$(\mathcal{S}, A)$-forgetful maps.
Hence first we construct the part of the marked points
$z^S$ and $z^A$ which enables us to compare the linear mas $\lambda$.
For any constant $C \geq 0$, we construct a finite set
$\mathcal{S} = \{S\}$ of codimension-two submanifolds of $Y$ and subsets
\[
\mathcal{V}_{\theta, l} \subset \mathcal{U}_{\theta, l} \subset
\widehat{\mathcal{M}}^{\leq L_{\max}}_{\mathcal{S}, A, \theta, l}
\]
and
\[
\mathcal{U}_{\theta, l}^\mathrm{D} \subset \overline{\mathcal{M}}^\mathrm{D}_{\mathcal{S}, A, \theta, l}
\]
for all triples $\theta$ such that $\widetilde{e}(\theta) \leq C$ and $l \geq 0$
which satisfy the following conditions, and call a family
$(\mathcal{S}, \mathcal{V}_{\theta, l}, \mathcal{U}_{\theta, l}, \mathcal{U}_{\theta, l}^\mathrm{D})$
a domain curve representation of
$\widehat{\mathcal{M}}^{\leq L_{\max}}_{\leq C}$.
\begin{enumerate}[label=$(\arabic*)^{\mathrm{D}}$]
\item
\label{(S, A)-covering}
For any $p \in \widehat{\mathcal{M}}^{\leq L_{\max}}_{\theta}$,
there exist some $l \geq 0$ and $p^+ \in \mathcal{V}_{\theta, l}$
such that $\mathop{\mathfrak{forget}}\nolimits_{\mathcal{S}, A}(p^+) = p$.
\item
\label{D-neighborhood}
The image of $\mathcal{U}_{\theta, l}$ by
$\mathop{\mathfrak{forget}}\nolimits_u$ is contained in $\mathcal{U}_{\theta, l}^\mathrm{D}$.
Furthermore, there exists an open neighborhood $\mathcal{W}_{\theta, l}
\subset \widehat{\mathcal{M}}^{\leq L_{\max}}_{\mathcal{S}, A, \theta, l}$ of the closure of
$\mathcal{U}_{\theta, l}$ such that
\[
\mathcal{U}_{\theta, l} = \{p \in \mathcal{W}_{\theta, l};
\mathop{\mathfrak{forget}}\nolimits_u(p) \in \mathcal{U}_{\theta, l}^\mathrm{D}\}.
\]
If $\mathcal{U}_{\theta, l}$ and $\mathcal{U}_{\theta, l}^\mathrm{D}$ satisfy this condition, then
we say $\mathcal{U}_{\theta, l}^\mathrm{D}$ is a D-neighborhood of $\mathcal{U}_{\theta, l}$.
\item
\label{open sets and inclusions}
$\mathcal{V}_{\theta, l}$ is open in the relative topology of $\mathcal{U}_{\theta, l}$,
and $\mathcal{V}_{\theta, l} \Subset \mathcal{U}_{\theta, l}$.
\item
\label{l max}
For any $\theta$, there exists some $l_{\theta}^{\max} \geq 0$ such that
$\mathcal{U}_{\theta, l} = \emptyset$ and $\mathcal{U}^\mathrm{D}_{\theta, l} = \emptyset$ for all
$l > l_{\theta}^{\max}$.
\item
\label{Z^A section}
For each point $\hat p \in \mathcal{U}_{\theta, l}^{\mathrm{D}}$,
there exist a local universal family
$(\hat P \to \hat X, Z, Z^S)$ of $\mathop{\mathfrak{forget}}\nolimits_A(\hat p)$ and
an $\mathop{\mathrm{Aut}}\nolimits(\hat p)$-invariant family of smooth sections
$Z^A = (Z^A_j)$ of $\hat P \to \hat X$ such that
\[
\{(\hat P_a, Z(a), Z^S(a), Z^A(a)); a \in \hat X\} / \mathop{\mathrm{Aut}}\nolimits(\hat p)
\]
is a neighborhood of $\hat p$ in $\mathcal{U}_{\theta, l}^{\mathrm{D}}$.
(We assume that $Z^A_j$ are disjoint.)
We call $(\hat P \to \hat X, Z, Z^S, Z^A)$ a local representation of a
neighborhood of $\hat p$ in $\mathcal{U}_{\theta, l}^\mathrm{D}$.
We note that we may regard $Z^A$
as an $\mathop{\mathrm{Aut}}\nolimits(\hat p)$-equivariant section of $(\prod^{\# z^A} \hat P)_{\hat X}
/ \mathfrak{S}_{\# z^A} \to \hat X$,
where $(\prod^{\# z^A} \hat P)_{\hat X}$ is the fiber product
over $\hat X$, and $\mathfrak{S}_{\# z^A}$ acts on it as permutations.
\item
\label{no collapse for (S, A)-forgetful map}
For any $\theta$, $l \geq l' \geq 0$,
$\hat p \in \mathcal{U}_{\theta, l}^\mathrm{D}$ and $\hat q \in \mathcal{U}_{\theta, l'}^\mathrm{D}$,
if there exists an $(\mathcal{S}, A)$-forgetful map $f$ from $\hat p$ to $\hat q$,
then $f$ does not collapse any component of $\hat p$.
(Namely, $\hat p$ is a curve obtained by adding some marked points to $\hat q$.)
\item
\label{Z^A pull back}
Under the same assumption,
let $(\hat P \to \hat X, Z, Z^S, Z^A)$ be the local representation of a neighborhood of
$\hat p$ in $\mathcal{U}_{\theta, l}^\mathrm{D}$, and
$(\hat P' \to \hat X', Z', (Z')^S, (Z')^A)$ be that of $\hat q$ in $\mathcal{U}_{\theta, l'}^\mathrm{D}$.
Shrink $\hat X$ and $\hat X'$ if necessary, and let $(\phi, \hat \phi)$ be the unique
forgetful map from $(\hat P \to \hat X, Z, Z^S)$ to $(\hat P' \to \hat X', Z', (Z')^S)$
whose restriction to the central fiber coincides with $f$.
Then the pull back of $(Z')^A$ by $(\phi, \hat \phi)$ is contained in $Z^A$ as a subfamily.
\item
\label{decomposition into parts U DM}
For any $\theta = (g, k, E_{\hat \omega})$,
$\hat p \in \mathcal{U}_{\theta, l}^\mathrm{D}$ and subset $\mathcal{N}$ of its nodal
points, replace each nodal point in $\mathcal{N}$ with a pair of marked points
(we regard the new marked points as points in the set $z$), and
let $\hat p'_i$ $(1 \leq i \leq N)$ be its connected components or an arbitrary
decomposition into unions of its connected components.
Let $g'_i$ and $k'_i$ be the genus and the number of marked points $z$ of
each $\hat p'_i$ respectively.
Then there exist some $E_{\hat \omega}^i \geq 0$ such that
$E_{\hat \omega} = \sum_i E_{\hat \omega}^i$ and
$\hat p'_i \in \mathcal{U}_{\theta'_i, l(\hat p'_i)}^\mathrm{D}$ for all $i$,
where $\theta'_i = (g'_i, k'_i, E_{\hat \omega}^i)$.
\item
\label{decomposition into parts U}
$\mathcal{U}_{\theta, l}$ satisfies
the following conditions about decomposition of a holomorphic building into parts.
\begin{itemize}
\item
For any $p \in \mathcal{U}_{\theta, l}$ and any decomposition $p_i$ ($1 \leq k$)
into unions of its connected components, let $p'_i$ be the holomorphic buildings
obtained by collapsing trivial floors (floors consisting of trivial cylinders).
Then $p'_i \in \mathcal{U}_{\theta(p'_i), l(p'_i)}$ for all $i$.
\item
For any $p \in \mathcal{U}_{\theta, l}$ and any gap between floors, let $p_1$ and $p_2$
be the holomorphic buildings obtained by dividing $p$ at this gap.
Then $p'_i \in \mathcal{U}_{\theta(p'_i), l(p'_i)}$ for $i = 1, 2$.
\item
For any $p \in \mathcal{U}_{\theta, l}$ and any subset of its nodal points,
the holomorphic building $p'$ obtained by replacing these nodal points to pairs of
marked points is contained in $\mathcal{U}_{\theta(p'), l(p')}$.
\end{itemize}
\item
\label{decomposition into parts V}
For each $p \in \widehat{\mathcal{M}}^{\leq L_{\max}}_{\mathcal{S}, A, \theta, l}$,
replace all nodal points and joint circles of $p$ to pairs of marked points and
pairs of limit circles respectively
(we regard the new marked points as points in the set $z$), and
let $p'_i$ $(1 \leq i \leq k)$ be their non-trivial connected components.
Then $p \in \mathcal{V}_{\theta, l}$ if and only if $p'_i \in \mathcal{V}_{\theta(p'_i), l(p'_i)}$ for all $i$.
\item
\label{existence of minimum}
For any $p = (\Sigma, z, u) \in \widehat{\mathcal{M}}^{\leq L_{\max}}_{\theta}$
and subsets $(z^k)^S \subset \Sigma$ ($S \in \mathcal{S}$) and $(z^k)^A \subset
\Sigma$ ($k =1,2$), if each $p^k = (\Sigma, z, (z^k)^S, (z^k)^A, u)$ is contained in
$\mathcal{U}_{\theta, l(p^k)}$, then
$p^3 = (\Sigma, z, (z^1)^S \cap (z^2)^S, (z^1)^A \cap (z^2)^A, u)$ is contained in
$\mathcal{U}_{\theta, l(p^3)}$.
Furthermore, $(z^1)^S \cup (z^2)^S$ $(S \in \mathcal{S})$ and $(z^1)^A \cup (z^2)^A$
are disjoint.
(This means that we can define a holomorphic building
$(\Sigma, z, (z^1)^S \cup (z^2)^S, (z^1)^A \cup (z^2)^A, u) \in
\widehat{\mathcal{M}}_{\mathcal{S}, A}$, but we do not assume that it is contained in some
$\mathcal{U}_{\theta, l}$.)
\item
\label{stably unique forgetful map DM}
For any $\hat p = (\hat \Sigma, z, z^S, z^A) \in \mathcal{U}_{\theta, l}^\mathrm{D}$ and
any subsets $(z^1)^S, (z^2)^S \subset z^S$ $(S \in \mathcal{S})$ and
$(z^1)^A, (z^2)^A \subset z^A$, if
each $\hat p^i = (\hat \Sigma, z, (z^i)^S, (z^i)^A)$ is contained in
$\mathcal{U}_{\theta, l(\hat p^i)}^\mathrm{D}$, then
$\hat p^3 = (\hat \Sigma, z, (z^1)^S \cap (z^2)^S, (z^1)^A \cap (z^2)^A)$
is stable and it is also contained in $\mathcal{U}_{\theta, l(\hat p^3)}^\mathrm{D}$.
\end{enumerate}
Note that Condition \ref{decomposition into parts U DM},
\ref{decomposition into parts U} and \ref{decomposition into parts V} are
conditions about one triple $\theta$ and other triples $\theta'$ such that
$\widetilde{e}(\theta') < \widetilde{e}(\theta)$,
and the others are conditions for each $\theta$.
We will prove the existence of a domain curve representation of
$\widehat{\mathcal{M}}^{\leq L_{\max}}_{\leq C}$ in Lemma
\ref{existence of a domain curve representation}.
First we prove that we can shrink $\mathcal{U}_{\theta, l}$ and $\mathcal{U}_{\theta, l}^\mathrm{D}$
preserving $\mathcal{V}_{\theta, l}$.
\begin{lem}
\label{shrinking and conditions}
Let $C \geq 0$ be an arbitrary constant, and assume that
a domain curve representation
$(\mathcal{S}, \mathcal{V}_{\theta, l}, \mathcal{U}_{\theta, l}, \mathcal{U}^\mathrm{D}_{\theta, l})$ of
$\widehat{\mathcal{M}}^{\leq L_{\max}}_{\leq C}$ is given.
Then we can construct open subsets
\[
\mathcal{V}_{\theta, l} \Subset \mathring{\mathcal{U}}_{\theta, l} \Subset \mathcal{U}_{\theta, l}
\quad (\widetilde{e}(\theta) = C, l \geq 0)
\]
and
\[
\mathring{\mathcal{U}}_{\theta, l}^\mathrm{D} \Subset \mathcal{U}_{\theta, l}^\mathrm{D} \quad
(\widetilde{e}(\theta) = C, l \geq 0)
\]
such that if we replace $\mathcal{U}_{\theta, l}$ and $\mathcal{U}_{\theta, l}^\mathrm{D}$
for $\widetilde{e}(\theta) = C$ in the family
$(\mathcal{S}, \mathcal{V}_{\theta, l}, \mathcal{U}_{\theta, l}, \mathcal{U}^\mathrm{D}_{\theta, l})$
with $\mathring{\mathcal{U}}_{\theta, l}$ and $\mathring{\mathcal{U}}_{\theta, l}^\mathrm{D}$
respectively, it still satisfies the conditions of a domain curve representation.
\end{lem}
\begin{proof}
The nontrivial conditions are Condition \ref{existence of minimum} and
\ref{stably unique forgetful map DM}.
We construct $\mathcal{V}_{\theta, l}$, $\mathcal{U}_{\theta, l}$ and $\mathcal{U}^\mathrm{D}_{\theta, l}$ ($l \geq 0$)
for each triple $\theta$ such that $\widetilde{e}(\theta) = C$.
First we consider Condition \ref{stably unique forgetful map DM}.
Consider the following condition for subsets
$\widehat{B}_{\theta, l}^\mathrm{D}, B_{\theta, l}^\mathrm{D} \subset \mathcal{U}_{\theta, l}^\mathrm{D}$:
\begin{itemize}
\item[\ref{stably unique forgetful map DM}$\:\!\!^+$]
For $l^1, l^2 < l^0$,
$\hat p = (\hat \Sigma, z, z^S, z^A) \in \widehat{B}_{\theta, l^0}^\mathrm{D}$ and
subsets $(z^1)^S, (z^2)^S \subset z^S$ $(S \in \mathcal{S})$ and
$(z^1)^A, (z^2)^A \subset z^A$, if each $\hat p^i = (\hat \Sigma, z, (z^i)^S, (z^i)^A)$ is
contained in $\widehat{B}_{\theta, l^i}^\mathrm{D}$ and
$\hat p^3 = (\hat \Sigma, z, (z^1)^S \cap (z^2)^S, (z^1)^A \cap (z^2)^A)$
does not coincide with $\hat p^1$ or $\hat p^2$, then $\hat p^3$ is contained
in $B_{\theta, l^3}^\mathrm{D}$ for some $l^3 < \min(l^1, l^2)$.
\end{itemize}
Condition \ref{stably unique forgetful map DM} is equivalent to
this condition for $\widehat{B}_{\theta, l}^\mathrm{D} = B_{\theta, l}^\mathrm{D}
= \mathcal{U}_{\theta, l}^\mathrm{D}$.
By the decreasing induction in $l \leq l_{\theta}^{\max}$, we construct
open neighborhoods
$\mathring{\mathcal{U}}_{\theta, l}^\mathrm{D} \Subset \mathcal{U}_{\theta, l}^\mathrm{D}$ of
the closure of $\mathop{\mathfrak{forget}}\nolimits_{\mathcal{S}, A}(\mathcal{V}_{\theta, l})$
so that for any $l_0$, Condition \ref{stably unique forgetful map DM}$\:\!\!^+$ holds for
$l^k > l_0$ ($k = 0,1,2$) and
\[
\widehat{B}_{\theta, l}^\mathrm{D} = \overline{\mathring{\mathcal{U}}_{\theta, l}^\mathrm{D}}
\quad (l > l_0),
\quad
B_{\theta, l}^\mathrm{D} = \begin{cases}
\mathring{\mathcal{U}}_{\theta, l}^\mathrm{D} & l \geq l_0 \\
\mathcal{U}_{\theta, l}^\mathrm{D} & l < l_0
\end{cases}.
\]
For $l = l_{\theta}^{\max}$, we may choose arbitrary open neighborhood
$\mathring{\mathcal{U}}_{\theta, l}^\mathrm{D} \Subset \mathcal{U}_{\theta, l}^\mathrm{D}$ of the closure of
$\mathop{\mathfrak{forget}}\nolimits_{\mathcal{S}, A}(\mathcal{V}_{\theta, l})$.
Assume that $\mathring{\mathcal{U}}_{\theta, l}^\mathrm{D}$ for $l > l_0$ are given.
Define $K_{\theta, l_0}^\mathrm{D} \subset \mathcal{U}_{\theta, l_0}^\mathrm{D}$
by the smallest subset such that the above condition holds for
$l^0, l^1, l^2 > l_0$ and
\[
\widehat{B}_{\theta, l}^\mathrm{D} = \overline{\mathring{\mathcal{U}}_{\theta, l}^\mathrm{D}}
\quad (l > l_0),
\quad
B_{\theta, l}^\mathrm{D} = \begin{cases}
\mathring{\mathcal{U}}_{\theta, l}^\mathrm{D} & l > l_0\\
K_{\theta, l_0}^\mathrm{D} & l = l_0\\
\mathcal{U}_{\theta, l}^\mathrm{D} & l < l_0
\end{cases}.
\]
Namely, $\hat p^3 \in \mathcal{U}_{\theta, l_0}^\mathrm{D}$ is contained in
$K_{\theta, l_0}^\mathrm{D}$ if
there exists some $l^0 > l_0$, $\hat p = (\hat \Sigma, z, z^S, z^A) \in
\overline{\mathring{\mathcal{U}}_{\theta, l^0}^\mathrm{D}}$ and
subsets $(z^i)^S \subset z^S$ $(i = 1,2$, $S \in \mathcal{S})$ and
$(z^i)^A \subset z^A$ such that
each $\hat p^i = (\hat \Sigma, z, (z^i)^S, (z^i)^A)$ is contained in
$\overline{\mathring{\mathcal{U}}_{\theta, l^i}^\mathrm{D}}$ for some $l_0 < l^1, l^2 < l^0$
and $\hat p^3$ is isomorphic to
$(\hat \Sigma, z, (z^1)^S \cap (z^2)^S, (z^1)^A \cap (z^2)^A)$.
It is easy to check that this is a compact subset of $\mathcal{U}_{\theta, l_0}^\mathrm{D}$.
Hence an open subset $\mathring{\mathcal{U}}_{\theta, l_0}^\mathrm{D} \Subset
\mathcal{U}_{\theta, l_0}^\mathrm{D}$ such that
$K_{\theta, l_0}^\mathrm{D} \cup \overline{\mathop{\mathfrak{forget}}\nolimits_{\mathcal{S}, A}(\mathcal{V}_{\theta, l})} \subset
\mathring{\mathcal{U}}_{\theta, l_0}^\mathrm{D}$ satisfies the required condition.
Therefore we can construct open subsets $\mathring{\mathcal{U}}_{\theta, l}^\mathrm{D}$
by the decreasing induction in $l \leq l_{\theta}^{\max}$.
Condition \ref{existence of minimum} is also similar.
Namely, we consider the following condition for
subsets $\widehat{B}_{\theta, l} \subset \mathcal{U}_{\theta, l}$
and $B_{\theta, l} \subset \mathcal{U}_{\theta, l}$:
\begin{itemize}
\item[\ref{existence of minimum}$\:\!\!^+$]
For any $0 \leq l^1, l^2 \leq l^{\theta}_{\max}$,
$p = (\Sigma, z, u) \in \widehat{\mathcal{M}}^{\leq L_{\max}}_{\theta}$,
and subsets $(z^k)^S \subset \Sigma$ ($S \in \mathcal{S}$) and $(z^k)^A \subset
\Sigma$ ($k =1,2$), if each $p^k = (\Sigma, z, (z^k)^S, (z^k)^A, u)$ is contained in
$\widehat{B}_{\theta, l^k}$ and
$p^3 = (\Sigma, z, (z^1)^S \cap (z^2)^S, (z^1)^A \cap (z^2)^A, u)$ does not
coincides with $p^1$ or $p^2$,
then $p^3$ is contained in $B_{\theta, l^3}$ for some $l^3 < \min(l^1, l^2)$.
Furthermore, $(z^1)^S \cup (z^2)^S$ ($S \in \mathcal{S}$)
and $(z^1)^A \cup (z^2)^A$ are disjoint.
\end{itemize}
Condition \ref{existence of minimum} is equivalent to this condition for
$\widehat{B}_{\theta, l} = B_{\theta, l} = \mathcal{U}_{\theta, l}$.
We construct open subsets $\mathring{\mathcal{U}}_{\theta, l}$
which make this condition holds for $l^1, l^2 > l_0$ and
\[
\widehat{B}_{\theta, l} = \overline{\mathring{\mathcal{U}}_{\theta, l}} \quad (l > l_0),
\quad
B_l = \begin{cases}
\mathring{\mathcal{U}}_{\theta, l} & l \geq l_0\\
\mathcal{U}_{\theta, l} & l < l_0
\end{cases}
\]
for all $l_0$ by the decreasing induction in $l \leq l_{\theta}^{\max}$.
Then as in the previous case, if $\mathring{\mathcal{U}}_{\theta, l}$ for $l > l_0$ are given,
define $K_{\theta, l_0} \subset \mathcal{U}_{\theta, l_0}$ by the smallest subset
which makes Condition \ref{existence of minimum}$\:\!\!^+$ hold for $l^1, l^2 > l_0$ and
\[
\widehat{B}_{\theta, l} = \overline{\mathring{\mathcal{U}}_{\theta, l}} \quad (l > l_0),
\quad
B_l = \begin{cases}
\mathring{\mathcal{U}}_{\theta, l} & l > l_0\\
K_{\theta, l} & l = l_0 \\
\mathcal{U}_{\theta, l} & l < l_0
\end{cases},
\]
then $K_{\theta, l_0}$ is a compact subset contained in $\mathcal{U}_{\theta, l}$.
Hence any open neighborhood
$\mathring{\mathcal{U}}_{\theta, l} \Subset \mathcal{U}_{\theta, l}$ of
$K_{\theta, l} \cup \overline{\mathcal{V}_{\theta, l}}$ satisfies the condition.
Therefore if we choose appropriate $\mathring{\mathcal{U}}_{\theta, l}^\mathrm{D}$
and define $\mathring{\mathcal{U}}_{\theta, l}$ by Condition \ref{D-neighborhood}, then
Condition \ref{existence of minimum} also holds for $\mathring{\mathcal{U}}_{\theta, l}$.
\end{proof}
Let $\widehat{\mathcal{M}}^\triangle \subset \widehat{\mathcal{M}}$ and
$\widehat{\mathcal{M}}^\triangle_{\mathcal{S}, A} \subset \widehat{\mathcal{M}}_{\mathcal{S}, A}$
be the subspaces of decomposable holomorphic buildings, that is, the subspaces of
disjoint holomorphic buildings and holomorphic buildings with nodal points or joint circles.
Similarly, let $\overline{\mathcal{M}}^{\mathrm{D}, \triangle}_{\mathcal{S}, A} \subset
\overline{\mathcal{M}}^\mathrm{D}_{\mathcal{S}, A}$ be the subspace of disjoint stable curves and
stable curves with nodal points.
\begin{lem}
\label{extension from the decomposable}
Let $C \geq 0$ be an arbitrary constant, and assume that
a domain curve representation
$(\mathcal{S}, \mathcal{V}_{\theta, l}, \mathcal{U}_{\theta, l}, \mathcal{U}^\mathrm{D}_{\theta, l})$ of
\[
\widehat{\mathcal{M}}^{\leq L_{\max}}_{< C} = \bigcup_{\widetilde{e}(\theta) < C}
\widehat{\mathcal{M}}^{\leq L_{\max}}_\theta
\]
is given.
We also assume that subsets
\[
\mathcal{V}_{\theta, l}^\triangle \subset \mathcal{U}_{\theta, l}^\triangle \subset
\widehat{\mathcal{M}}^{\leq L_{\max}, \triangle}_{\mathcal{S}, A, \theta, l}
\quad (\widetilde{e}(\theta) = C, l \geq 0)
\]
and
\[
\mathcal{U}_{\theta, l}^{\mathrm{D}, \triangle} \subset
\overline{\mathcal{M}}^{\mathrm{D}, \triangle}_{\mathcal{S}, A, \theta, l}
\quad (\widetilde{e}(\theta) = C, l \geq 0)
\]
are given and they satisfy the conditions of domain curve representation.
More precisely, they satisfy the conditions obtained by replacing
$\mathcal{U}_{\theta, l}$, $\mathcal{U}_{\theta, l}^\mathrm{D}$, $\widehat{\mathcal{M}}$ and so on
with the counterparts with superscript $\triangle$.
Condition \ref{Z^A section} for $\mathcal{U}_{e^1_0, e^2_0, l}^{\mathrm{D}, \triangle}$ is read as follows:
For a local universal family $(\hat P \to \hat X, Z, Z^S)$ of
$\mathop{\mathfrak{forget}}\nolimits_A(\hat p)$, let $\hat X^\triangle \subset \hat X$ be the subset of points whose
fiber are disconnected or whose fiber has nodal points.
Then Condition \ref{Z^A section} for $\mathcal{U}_{\theta, l}^{\mathrm{D}, \triangle}$ is that
there exists an $\mathop{\mathrm{Aut}}\nolimits(\hat p)$-invariant family of sections $Z^A$ of
$\hat P|_{\hat X^\triangle} \to \hat X^\triangle$ such that
\[
\{(\hat P_a, Z(a), Z^S(a), Z^A(a)); a \in \hat X^\triangle\} / \mathop{\mathrm{Aut}}\nolimits(\hat p)
\]
is a neighborhood of $\hat p$ in $\mathcal{U}_{\theta, l}^{\mathrm{D}, \triangle}$.
In Condition \ref{no collapse for (S, A)-forgetful map}, we read $\mathcal{U}_{\theta, l}^\mathrm{D}$
$($or $\mathcal{U}_{\theta', l'}^\mathrm{D}$$)$ for $\widetilde{e}(\theta) = C$
as $\mathcal{U}_{\theta, l}^{\mathrm{D}, \triangle}$ $($or $\mathcal{U}_{\theta', l'}^{\mathrm{D}, \triangle}$$)$.
Then we can construct subsets
\[
\mathcal{V}_{\theta, l}^1 \subset \mathcal{U}_{\theta, l}^1 \subset \mathcal{U}_{\theta, l}^2 \subset
\widehat{\mathcal{M}}^{\leq L_{\max}}_{\mathcal{S}, A, \theta, l} \quad
(\widetilde{e}(\theta) = C, l \geq 0)
\]
and
\[
\mathcal{U}_{\theta, l}^{1, \mathrm{D}} \subset \mathcal{U}_{\theta, l}^{2, \mathrm{D}} \subset
\overline{\mathcal{M}}^\mathrm{D}_{\mathcal{S}, A, \theta, l}
\quad (\widetilde{e}(\theta) = C, l \geq 0)
\]
which satisfy the following conditions:
\begin{itemize}
\item
The closure of $\mathcal{U}_{\theta, l}^{1, \mathrm{D}}$ is contained in
$\mathcal{U}_{\theta, l}^{2, \mathrm{D}}$.
\item
$\mathcal{V}_{\theta, l}^1 \cap \widehat{\mathcal{M}}_{\mathcal{S}, A}^\triangle
= \mathcal{V}_{\theta, l}^\triangle$,
$\mathcal{U}_{\theta, l}^2 \cap \widehat{\mathcal{M}}_{\mathcal{S}, A}^\triangle \subset
\mathcal{U}_{\theta, l}^\triangle$, and
$\mathcal{U}_{\theta, l}^{2, \mathrm{D}} \cap \overline{\mathcal{M}}_{\mathcal{S}, A}^{\mathrm{D}, \triangle}
\subset \mathcal{U}_{\theta}^{\mathrm{D}, \triangle}$.
\item
For each $k= 1,2$, $\mathcal{V}_{\theta, l} = \mathcal{V}_{\theta, l}^1$,
$\mathcal{U}_{\theta, l} = \mathcal{U}_{\theta, l}^k$ and $\mathcal{U}_{\theta, l}^\mathrm{D} = \mathcal{U}_{\theta, l}^{k, \mathrm{D}}$
for $\widetilde{e}(\theta) = C$
and the given $\mathcal{V}_{\theta', l}$, $\mathcal{U}_{\theta', l}$
and $\mathcal{U}_{\theta', l}^\mathrm{D}$ for $\widetilde{e}(\theta') < C$
satisfy the conditions of domain curve representation of
$\widehat{\mathcal{M}}_{\leq C}$ other than Condition \ref{(S, A)-covering}.
\end{itemize}
\end{lem}
\begin{proof}
We consider each triple $\theta$ such that $\widetilde{e}(\theta) = C$.
By the same argument used for the proof of Lemma \ref{shrinking and conditions},
we see that we can take an open neighborhood
$\mathring{\mathcal{U}}_{\theta, l}^\triangle \Subset \mathcal{U}_{\theta, l}^\triangle$ of
the closure of $\mathcal{V}_{\theta, l}^\triangle$ and
an open subset $\mathring{\mathcal{U}}_{\theta, l}^{\mathrm{D}, \triangle} \Subset
\mathcal{U}_{\theta, l}^{\mathrm{D}, \triangle}$ such that
$\mathcal{V}_{\theta, l}^\triangle$, $\mathring{\mathcal{U}}_{\theta, l}^\triangle$ and
$\mathring{\mathcal{U}}_{\theta, l}^{\mathrm{D}, \triangle}$ also satisfy the assumption.
We may assume that Condition \ref{stably unique forgetful map DM}$\:\!\!^+$ holds for
$\widehat{B}^\mathrm{D}_{\theta, l}
= \overline{\mathring{\mathcal{U}}_{\theta, l}^{\mathrm{D}, \triangle}}$ and
$B^\mathrm{D}_{\theta, l} = \mathring{\mathcal{U}}_{\theta, l}^{\mathrm{D}, \triangle}$,
and Condition \ref{existence of minimum}$\:\!\!^+$ holds for
$\widehat{B}_{\theta, l} = \overline{\mathring{\mathcal{U}}^\triangle_{\theta, l}}$ and
$B_{\theta, l} = \mathring{\mathcal{U}}^\triangle_{\theta, l}$.
Choose finite points $\hat p^i \in \mathcal{U}_{\theta, l}^{\mathrm{D}, \triangle}$ ($i \in I_l$)
for each $l$, let $(\hat P^i \to \hat X^i, Z^i,\allowbreak (Z^i)^S)$ be
a local universal family of each $\mathop{\mathfrak{forget}}\nolimits_A(\hat p^i)$ and
let $(Z^i)^A|_{(\hat X^i)^\triangle}$ be the family of section for each $i \in I_l$
so that
$\overline{\mathring{\mathcal{U}}_{\theta, l}^\triangle}$ are covered by
$\{((\hat X^i)^\triangle, (Z^i)^A|_{(\hat X^i)^\triangle})\}_{i \in I_l}$.
Namely, every stable curve in $\overline{\mathring{\mathcal{U}}_{\theta, l}^\triangle}$
appears as some fiber of the families obtained by adding the marked points
$(Z^i)^A|_{(\hat X^i)^\triangle}$ to the local universal families.
Shrinking each $\hat X^i$ if necessary (keeping the covering condition),
we construct an $\mathop{\mathrm{Aut}}\nolimits(\hat p^i)$-invariant extension $(Z^i)^A = ((Z^i)^A_j)_j$ of
$(Z^i)^A|_{(\hat X^i)^\triangle}$ to $\hat X^i$ and
an $\mathop{\mathrm{Aut}}\nolimits(\hat p^i)$-invariant open neighborhood $W^i = \coprod_j W^i_j$ of the value of
$(Z^i)^A = ((Z^i)^A_j)_j$ for each $i$ which satisfy the following condition:
\begin{itemize}
\item[$(\ast)$]
For any $l_0 \geq l$, $i_0 \in I_{l_0}$, $i \in I_l$,
$a \in \hat X^{i_0}$, $b \in \hat X^i$ and an $\mathcal{S}$-forgetful map
\[
\varphi : (\hat P^{i_0}_a, Z^{i_0}(a), (Z^{i_0})^S(a)) \to (\hat P^i_b, Z^i(b), (Z^i)^S(b)),
\]
if $\varphi^{-1}((Z^i)^A(b)) \subset W^{i_0}|_a$
then $\varphi^{-1}((Z^i)^A(b)) \subset (Z^{i_0})^A(a)$.
\end{itemize}
We construct such extensions and neighborhoods of their values by the induction in $l$
as follows.
Let $l = l_{\min}$ be the minimum such that
$\mathcal{U}_{\theta, l}^{\mathrm{D}, \triangle} \neq \emptyset$.
We fix an order of $I_{l_{\min}}$ and construct $(Z^i)^A$ and $W^i$ by the induction
in $i \in I_{l_{\min}}$.
First for the minimal $i \in I_{l_{\min}}$,
we construct an $\mathop{\mathrm{Aut}}\nolimits(\hat p^i)$-invariant extension $(Z^i)^A$ of
$(Z^i)^A|_{(\hat X^i)^\triangle}$. Since we may regard $(Z^i)^A$
as an $\mathop{\mathrm{Aut}}\nolimits(\hat p^i)$-equivariant section of the fiber product
$(\prod^{\# z^A} \hat P^i)_{\hat X^i} / \mathfrak{S}_{\# z^A} \to \hat X^i$,
we can extend $(Z^i)^A|_{(\hat X^i)^\triangle}$ to a neighborhood of
$(\hat X^i)^\triangle$.
Replacing $\hat X^i$ with a small neighborhood of $(\hat X^i)^\triangle$,
we may assume that $(Z^i)^A = (Z^i)^A_j$ is defined on $\hat X^i$.
Let $W^i = \coprod_j \mathring{W}^i_j$ be an $\mathop{\mathrm{Aut}}\nolimits(\hat p^i)$-invariant neighborhood
of $(Z^i)^A(\hat X^i) = \coprod_j (Z^i)^A_j(\hat X^i)$.
Assume that we have constructed $(Z^i)^A$ and $W^i$ for $i < i_0$
which satisfy Condition $(\ast)$.
First we construct an $\mathop{\mathrm{Aut}}\nolimits(\hat p^{i_0})$-invariant open neighborhood $W^{i_0}$
of the value of $(Z^{i_0})^A|_{(\hat X^{i_0})^\triangle}$ which satisfies the following
conditions:
\begin{enumerate}[label=\normalfont(\roman*)]
\item
\label{two with the same l}
For any $i < i_0$, $a \in \hat X^{i_0}$, $b_1, b_2 \in \hat X^i$ and isomorphisms
\[
\varphi_k : (\hat P^{i_0}_a, Z^{i_0}(a), (Z^{i_0})^S(a))
\cong (\hat P^i_{b_k}, Z^i(b_k), (Z^i)^S(b_k))
\quad (k = 1, 2),
\]
if $\varphi_k^{-1}((Z^i)^A(b_k)) \subset W^{i_0}|_a$ for $k = 1, 2$,
then there exists some $g \in \mathop{\mathrm{Aut}}\nolimits(\hat p^i)$ such that
$b_2 = g b_1$ and $\varphi_2 = g \circ \varphi_1$.
In particular, $\varphi_1^{-1}((Z^i)^A(b_1)) = \varphi_2^{-1}((Z^i)^A(b_2))$.
\item
\label{three with the same l}
For any $i_1, i_2 < i_0$, $a \in \hat X^{i_0}$, $b_k \in \hat X^{i_k}$ ($k = 1, 2$) and
isomorphisms
\[
\varphi_k : (\hat P^{i_0}_a, Z^{i_0}(a), (Z^{i_0})^S(a))
\cong (\hat P^{i_k}_{b_k}, Z^{i_k}(b_k), (Z^{i_k})^S(b_k))
\quad (k =1, 2),
\]
if $\varphi_k^{-1}((Z^{i_k})^A(b_k)) \subset W^{i_0}|_a$ for $k = 1, 2$, then the isomorphism
\[
\varphi = \varphi_2 \circ \varphi_1^{-1}
: (\hat P^{i_1}_{a_1}, Z^{i_1}(a_1), (Z^{i_1})^S(a_1))
\cong (\hat P^{i_2}_{a_2}, Z^{i_2}(a_2), (Z^{i_2})^S(a_2))
\]
satisfies $\varphi((Z^{i_1})^A(b_1)) \subset W^{i_2}|_{b_2}$.
Note that by Condition $(\ast)$, this implies that
$(Z^{i_2})^A(b_2) = \varphi((Z^{i_1})^A(b_1))$.
In particular, $\varphi_1^{-1}((Z^{i_1})^A(b_1))$ coincides with
$\varphi_2^{-1}((Z^{i_2})^A(b_2))$.
\end{enumerate}
Note that in Condition \ref{two with the same l} for $a \in (\hat X^{i_0})^\triangle$,
if $W^{i_0}|_a$ is a sufficiently small neighborhood of $(Z^{i_0})^A(a)$, then
the condition $\varphi_k^{-1}((Z^i)^A(b_k)) \subset W^{i_0}|_a$ implies that
$\varphi_k^{-1}((Z^i)^A(b_k)) = (Z^{i_0})^A(a)$.
Hence $g = \varphi_2 \circ \varphi_1^{-1}$ maps $(Z^i)^A(b_1)$ to $(Z^i)^A(b_2)$,
which implies $g \in \mathop{\mathrm{Aut}}\nolimits(\hat p^i)$.
Therefore Condition \ref{two with the same l} for general $a \in \hat X^{i_0}$ also holds
if $W^{i_0}$ is a sufficiently small neighborhood of the values of
$(Z^{i_0})^A|_{(\hat X^{i_0})^\triangle}$.
Similarly, in Condition \ref{three with the same l} for $a \in (\hat X^{i_0})^\triangle$,
if $W^{i_0}|_a$ is a sufficiently small neighborhood of $(Z^{i_0})^A(a)$, then
the condition $\varphi_k^{-1}((Z^i)^A(b_k)) \subset W^{i_0}|_a$ implies that
$\varphi_k^{-1}((Z^i)^A(b_k)) = (Z^{i_0})^A(a)$.
Hence $\varphi = \varphi_2 \circ \varphi_1^{-1}$ maps $(Z^{i_1})^A(b_1)$ to
$(Z^{i_2})^A(b_2) \subset W^{i_2}|_{b_2}$.
It implies that Condition \ref{three with the same l} for general $a \in \hat X^{i_0}$
holds if $W^{i_0}$ is a sufficiently small neighborhood of the values of
$(Z^{i_0})^A|_{(\hat X^{i_0})^\triangle}$.
For each point $a \in \hat X^{i_0}$ such that there exists some $i < i_0$,
$b \in \hat X^i$ and isomorphism
\[
\varphi : (\hat P^{i_0}_a, Z^{i_0}(a), (Z^{i_0})^S(a)) \cong (\hat P^i_b, Z^i(b), (Z^i)^S(b))
\]
such that $\varphi^{-1}((Z^i)^A(b)) \subset W^{i_0}|_a$, we define
$(Z^{i_0})^A(a) = \varphi^{-1}((Z^i)^A(b))$.
The above conditions of $W^{i_0}$ implies that this definition is independent of the choice
of $i$ and $b \in \hat X^i$ if they exist.
Shrinking $\hat X^i$ ($i < i_0$) slightly if necessary for smooth extension, we extend
$(Z^{i_0})^A$ to a neighborhood of $(\hat X^{i_0})^\triangle \subset \hat X^{i_0}$.
Replacing $\hat X^{i_0}$ to a neighborhood of
$(\hat X^{i_0})^\triangle \subset \hat X^{i_0}$, we assume that $(Z^{i_0})^A$ is defined
on whole of $\hat X^{i_0}$ and its value is contained in $W^{i_0}$.
Next we consider the general $l$.
Assume that we have already constructed the extensions for $l < l_0$.
We fix an order of $I_{l_0}$ and construct $(Z^i)^A$ and $W^i$ by the induction
in $i \in I_{l_0}$.
Assume that $(Z^i)^A$ and $W^i$ for $i < i_0$ are given.
As in the case of minimal $l$,
first we construct an $\mathop{\mathrm{Aut}}\nolimits(\hat p^i)$-invariant open neighborhood $W^{i_0}$
of the value of $(Z^i)^A|_{(\hat X^{i_0})^\triangle}$ which satisfies
Condition \ref{two with the same l}, \ref{three with the same l} and
the following condition:
\begin{enumerate}[label=\normalfont(\roman*)]
\setcounter{enumi}{2}
\item
\label{stably unique S-forget}
For any $a \in \hat X^{i_0}$, $l_1, l_2 < l_0$, $i_k \in I_{l_k}$ ($k = 1, 2$),
$b_k \in \hat X^{i_k}$ ($k = 1, 2$) and $\mathcal{S}$-forgetful maps
\[
\varphi_k : (\hat P^{i_0}_a, Z^{i_0}(a), (Z^{i_0})^S(a))
\to (\hat P^{i_k}_{b_k}, Z^{i_k}(b_k), (Z^{i_k})^S(b_k))
\quad (k =1,2),
\]
if $\varphi_k^{-1}((Z^{i_k})^A(b_k)) \subset W^{i_0}|_a$ for $k = 1,2$,
then there exist some $l_3 \leq \min(l_1, l_2)$, $i_3 \in I_{l_3}$,
$b_3 \in \hat X^{i_3}$ and $\mathcal{S}$-forgetful maps
\[
\psi_k : (\hat P^{i_k}_{b_k}, Z^{i_k}(b_k), (Z^{i_k})^S(b_k))
\to (\hat P^{i_3}_{b_3}, Z^{i_3}(b_3), (Z^{i_3})^S(b_3))
\quad (k = 1,2)
\]
which satisfy $\psi_1 \circ \varphi_1 = \psi_2 \circ \varphi_2$ and
the following condition:
For any triple $(j, j_1, j_2)$ such that $\varphi_k^{-1}((Z^{i_k})^A_{j_k}(b_k))
\subset W^i_j|_a$ for $k =1,2$, there exists some $j_3$ such that
$\psi_k^{-1}((Z^{i_3})^A_{j_3}(b_3)) \subset W^{i_k}_{j_k}|_{b_k}$ for $k = 1,2$.
By Condition $(\ast)$, this implies that
$(Z^{i_k})^A_{j_k}(b_k) = \psi_k^{-1}((Z^{i_3})^A_{j_3}(b_3))$ for $k = 1,2$.
In particular, $\varphi_1^{-1}((Z^{i_1})^A_{j_1}(b_1)) = \varphi_2^{-1}((Z^{i_2})^A_{j_2}(b_2))$.
\end{enumerate}
Note that in the above condition, if $a \in (\hat X^{i_0})^\triangle$ and
$W^{i_0}|_a$ is a sufficiently small neighborhood of $(Z^{i_0})^A(a)$, then the condition
$\varphi_k^{-1}((Z^{i_k})^A(b_k)) \subset W^{i_0}|_a$ implies that
$\varphi_k^{-1}((Z^{i_k})^A(b_k)) \subset (Z^{i_0})^A(a)$.
Hence Condition \ref{stably unique forgetful map DM} for
$\mathring{\mathcal{U}}_{\theta, l}^{\mathrm{D}, \triangle}$ implies that
there exist some $l_3 \leq \min(l_1, l_2)$, $i_3 \in I_{l_3}$,
$b_3 \in \hat X^{i_3}$ and $\mathcal{S}$-forgetful maps
\[
\psi_k : (\hat P^{i_k}_{b_k}, Z^{i_k}(b_k), (Z^{i_k})^S(b_k))
\to (\hat P^{i_3}_{b_3}, Z^{i_3}(b_3), (Z^{i_3})^S(b_3))
\quad (k = 1,2)
\]
such that $\psi_1 \circ \varphi_1 = \psi_2 \circ \varphi_2$ and
\[
\varphi_1^{-1}((Z^{i_1})^A(b_1)) \cap \varphi_2^{-1}((Z^{i_2})^A(b_2))
= (\psi_1 \circ \varphi_1)^{-1}((Z^{i_3})^A(b_3)).
\]
Therefore Condition \ref{stably unique S-forget} holds for $a \in (\hat X^{i_0})^\triangle$
if $W^{i_0}$ is a sufficiently small neighborhood of $(Z^{i_0})^A((\hat X^{i_0})^\triangle)$.
Hence it also holds if $a \in \hat X^{i_0}$ is contained in a small neighborhood of
$(\hat X^{i_0})^\triangle$.
Therefore Condition \ref{stably unique S-forget} holds for general $a \in \hat X^{i_0}$
if $W^{i_0}$ is sufficiently small.
The construction of $(Z^{i_0})^A$ is similar to the case of minimal $l$, but in this case,
some part of $(Z^{i_0})^A$ is determined by the pull backs of $(Z^i)^A$ ($i \in I_l, l < l_0$)
as follows.
For $a \in \hat X^{i_0}$, $l < l_0$, $i \in I_l$, $b \in \hat X^i$ and
an $\mathcal{S}$-forgetful map
\[
\varphi : (\hat P^{i_0}_a, Z^{i_0}(a), (Z^{i_0})^S(a)) \to (\hat P^i_b, Z^i(b), (Z^i)^S(b))
\]
such that $\varphi^{-1}((Z^i)^A(b)) \subset W^{i_0}|_a$,
we define $(z^{i_0})^A(a)_{b, \varphi} = \varphi^{-1}((Z^i)^A(b))$.
For each $a \in \hat X^{i_0}$, we define $(z^{i_0})^A(a)$ by the union of
$(z^{i_0})^A(a)_{b, \varphi}$ over the above pairs $(b, \varphi)$.
We need to construct the extension $(Z^{i_0})^A$ which contains $(z^{i_0})^A$
as a subfamily.
Condition \ref{stably unique S-forget} implies that $\varphi^{-1}((Z^i)^A(b)) \cap
W^{i_0}_j|_a$ consists of at most one point for each $j$, and this point is
independent of $(b, \varphi)$ if it exists.
Hence $(z^{i_0})^A(a) \cap W^{i_0}_j|_a$ consists of at most one point for each $j$.
It is clear that $(z^{i_0})^A$ is $\mathop{\mathrm{Aut}}\nolimits(\hat p^{i_0})$-invariant.
Hence shrinking $\hat X^i$ for $i \in I_l$ ($l < l_0$) and $i \in I_{l_0}$ such that
$i < i_0$ if necessary for smooth extension,
we can construct an $\mathop{\mathrm{Aut}}\nolimits(\hat p^{i_0})$-invariant extension $(Z^{i_0})^A$ of
$(Z^{i_0})^A|_{(\hat X^{i_0})^\triangle}$ to a neighborhood of $(\hat X^{i_0})^\triangle$
which contains $(z^{i_0})^A$ as a subfamily.
Replacing $\hat X^{i_0}$ with a small neighborhood of $(\hat X^{i_0})^\triangle$,
we get an extension $(Z^{i_0})^A$ on $\hat X^{i_0}$ such that
$(Z^{i_0})^A(\hat X^{i_0}) \subset W^{i_0}$.
Therefore the induction works and we can construct
$\mathop{\mathrm{Aut}}\nolimits(\hat p^i)$-invariant extensions $(Z^i)^A = ((Z^i)^A_j)_j$ of
$(Z^i)^A|_{(\hat X^i)^\triangle}$ to $\hat X^i$ and
$\mathop{\mathrm{Aut}}\nolimits(\hat p^i)$-invariant open neighborhoods $W^i = \coprod_j W^i_j$
of the values of $(Z^i)^A = ((Z^i)^A_j)_j$ which satisfy Condition $(\ast)$.
Now we construct $\mathcal{V}_{\theta, l}^1$, $\mathcal{U}_{\theta, l}^k$ and
$\mathcal{U}_{\theta, l}^{k, \mathrm{D}}$ ($k = 1, 2$) as follows.
First we define $\widehat{\mathcal{U}}_{\theta, l}^\mathrm{D} \subset
\overline{\mathcal{M}}^\mathrm{D}_{\mathcal{S}, A, \theta, l}$
by the union of the sets of stable curves
\[
\{(\hat P^i_a, Z^i(a), (Z^i)^S(a), (Z^i)^A(a));
a \in \hat X^i\}
\]
over $i \in I_l$.
We construct $\mathcal{U}_{\theta, l}^{2, \mathrm{D}}$ as a subset of
$\widehat{\mathcal{U}}_{\theta, l}^\mathrm{D}$.
For each $l$, let $\mathcal{W}_{\theta, l} \subset
\widehat{\mathcal{M}}^{\leq L_{\max}}_{\mathcal{S}, A, \theta, l}$ be
an open neighborhood of the closure of $\mathcal{U}_{\theta, l}^\triangle$ such that
\[
\mathcal{U}_{\theta, l}^\triangle = \{p \in \mathcal{W}_{\theta, l};
\mathop{\mathfrak{forget}}\nolimits_{\mathcal{S}, A}(p) \in \mathcal{U}_{\theta, l}^{\mathrm{D}, \triangle}\}.
\]
We construct small neighborhoods $\mathcal{U}_{\theta, l}^{2, \mathrm{D}} \subset
\widehat{\mathcal{U}}_{\theta, l}^\mathrm{D}$ of $\mathring{\mathcal{U}}_{\theta, l}^{\mathrm{D}, \triangle}$
such that they satisfy Condition \ref{no collapse for (S, A)-forgetful map} and
\ref{stably unique forgetful map DM}, and
\[
\mathcal{U}_{\theta, l}^2 = \{p \in \mathcal{W}_{\theta, l};
\mathop{\mathfrak{forget}}\nolimits_{\mathcal{S}, A}(p) \in \mathcal{U}_{\theta, l}^{2, \mathrm{D}}\}
\]
satisfy Condition \ref{existence of minimum} as follows.
(We also assume that $\mathcal{U}_{\theta, l}^{2, \mathrm{D}}$ is sufficiently small so that
$\mathcal{W}_{\theta, l}$ is still an open neighborhood of the closure of
$\mathcal{U}_{\theta, l}^2$.)
Since $\overline{\mathring{\mathcal{U}}_{\theta, l}^{\mathrm{D}, \triangle}}$ ($l \geq 0$)
satisfy Condition \ref{no collapse for (S, A)-forgetful map}, sufficiently small
neighborhoods $\mathcal{U}_{\theta, l}^{2, \mathrm{D}}$ of $\mathring{\mathcal{U}}_{\theta, l}^{\mathrm{D}, \triangle}$
also satisfy the same condition.
For Condition \ref{existence of minimum} and \ref{stably unique forgetful map DM},
we construct $\mathcal{U}_{\theta, l}^{2, \mathrm{D}}$ by the (usual increasing) induction in $l$ so that
for any $l_0$,
Condition \ref{stably unique forgetful map DM}$\:\!\!^+$ holds
for $l^1, l^2 < l^0$ such that
$\min(l^1, l^2) \leq l_0$ and
\[
\widehat{B}_{\theta, l}^\mathrm{D} = \begin{cases}
\overline{\mathring{\mathcal{U}}_{\theta, l}^{\mathrm{D}, \triangle}}& l > l_0\\
\overline{\mathcal{U}_{\theta, l}^{2, \mathrm{D}}} & l \leq l_0
\end{cases},
\quad
B_{\theta, l}^\mathrm{D} = \mathcal{U}_{\theta, l}^{2, \mathrm{D}} \quad (l < l_0),
\]
and Condition \ref{existence of minimum}$\:\!\!^+$ holds for $l^1, l^2$ such that
$\min(l^1, l^2) \leq l_0$ and
\[
\widehat{B}_{\theta, l} = \begin{cases}
\overline{\mathring{\mathcal{U}}_{\theta, l}^\triangle}& l > l_0\\
\overline{\mathcal{U}_{\theta, l}^2} & l \leq l_0
\end{cases},
\quad
B_{\theta, l}^\mathrm{D} = \mathcal{U}_{\theta, l}^2 \quad (l < l_0).
\]
The induction works because of the following reason.
Assume that $\mathcal{U}_{\theta, l}^{2, \mathrm{D}}$ for $l < l_0$ are given and they satisfy the
above conditions.
We prove that if $\mathcal{U}_{\theta, l_0}^{2, \mathrm{D}} \subset \widehat{\mathcal{U}}_{\theta, l_0}^\mathrm{D}$ is
a sufficiently small open neighborhood of
$\overline{\mathring{\mathcal{U}}_{\theta, l_0}^{\mathrm{D}, \triangle}}$, then
Condition \ref{stably unique forgetful map DM}$\:\!\!^+$ holds for
$l^1, l^2 < l^0$ such that $\min(l^1, l^2) \leq l_0$ and
\[
\widehat{B}_{\theta, l}^\mathrm{D} = \begin{cases}
\overline{\mathring{\mathcal{U}}_{\theta, l}^{\mathrm{D}, \triangle}}& l > l_0\\
\overline{\mathcal{U}_{\theta, l}^{2, \mathrm{D}}} & l \leq l_0
\end{cases},
\quad
B_{\theta, l}^\mathrm{D} = \mathcal{U}_{\theta, l}^{2, \mathrm{D}} \quad (l < l_0).
\]
First we consider this condition for $l^1 = l^2 = l_0 < l^0$.
Assume that this condition does not hold for any small open neighborhood
$\mathcal{U}_{\theta, l_0}^{2, \mathrm{D}}$ of $\overline{\mathring{\mathcal{U}}_{\theta, l_0}^{\mathrm{D}, \triangle}}$.
Then there exists a sequence of stable curves
$\hat p_k = (\hat \Sigma_k, z_k, z_k^S, z_k^A) \in
\overline{\mathring{\mathcal{U}}_{\theta, l^0}^{\mathrm{D}, \triangle}}$
and subsets $(z^1_k)^S, (z^2_k)^S \subset z_k^S$ $(S \in \mathcal{S})$ and
$(z^1_k)^A, (z^2_k)^A \subset z_k^A$ such that
two sequences $(\hat p^i_k)_{k \in \mathbb{N}} = (\hat \Sigma_k, z, (z^i_k)^S, (z^i_k)^A)_{k \in \mathbb{N}}$
converge to points in $\overline{\mathring{\mathcal{U}}_{\theta, l_0}^{\mathrm{D}, \triangle}}$,
but none of $\hat p^3_k
= (\hat \Sigma_k, z, (z^1_k)^S \cap (z^2_k)^S, (z^1_k)^A \cap (z^2_k)^A)$
is not contained in $\bigcup_{l < l_0} \mathcal{U}_{\theta, l}^{2, \mathrm{D}}$.
Taking a subsequence, assume that $\hat p_k$ converges to a stable curve
$\hat p_\infty = (\hat \Sigma_\infty, z_\infty, z_\infty^S, z_\infty^A) \in
\overline{\mathring{\mathcal{U}}_{\theta, l^0}^{\mathrm{D}, \triangle}}$.
We may assume that there exists subsets
$(z^1_\infty)^S, \allowbreak (z^2_\infty)^S \subset z_\infty^S$ $(S \in \mathcal{S})$ and
$(z^1_\infty)^A, \allowbreak (z^2_\infty)^A \subset z_\infty^A$ such that
each $(\hat p^i_k)_{k \in \mathbb{N}}$ converges to a stable curve
$\hat p^i_\infty
= (\hat \Sigma_\infty, z, (z^i_\infty)^S, (z^i_\infty)^A)
\in \overline{\mathring{\mathcal{U}}_{\theta, l_0}^{\mathrm{D}, \triangle}}$ for $i = 1,2$.
(This is because of Condition \ref{no collapse for (S, A)-forgetful map}
for $\overline{\mathring{\mathcal{U}}_{\theta, l}^{\mathrm{D}, \triangle}}$.)
Hence Condition \ref{stably unique forgetful map DM}$\:\!\!^+$ for
\[
\widehat{B}_{\theta, l}^\mathrm{D} = \begin{cases}
\overline{\mathring{\mathcal{U}}_{\theta, l}^{\mathrm{D}, \triangle}}& l \geq l_0\\
\overline{\mathcal{U}_{\theta, l}^{2, \mathrm{D}}} & l < l_0
\end{cases},
\quad
B_{\theta, l}^\mathrm{D} = \mathcal{U}_{\theta, l}^{2, \mathrm{D}} \quad (l < l_0-1),
\]
implies that
$\hat p^3_\infty = (\hat \Sigma_\infty, z, (z^1_\infty)^S \cap (z^2_\infty)^S,
(z^1_\infty)^A \cap (z^2_\infty)^A)$ is contained in
$\bigcup_{l < l_0} \mathcal{U}_{\theta, l}^{2, \mathrm{D}}$.
Since $(\hat p^3_k)_{k \in \mathbb{N}}$ converges to $\hat p^3_\infty$,
this contradicts to the openness of $\bigcup_{l < l_0} \mathcal{U}_{\theta, l}^{2, \mathrm{D}}$.
The other cases such as $l^1, l^2 < l^0 = l_0$ are similar.
Condition \ref{existence of minimum}$\:\!\!^+$ is also similar.
Hence we can construct open neighborhoods $\mathcal{U}_{\theta, l}^{2, \mathrm{D}}$
of $\mathring{\mathcal{U}}_{\theta, l}^{\mathrm{D}, \triangle}$ which satisfy
Condition \ref{no collapse for (S, A)-forgetful map}, \ref{existence of minimum}
and \ref{stably unique forgetful map DM}.
Next we construct $\mathcal{U}_{\theta, l}^{1, \mathrm{D}}$ by the same way as
$\mathcal{U}_{\theta, l}^{2, \mathrm{D}}$ under the condition $\mathcal{U}_{\theta, l}^{1, \mathrm{D}} \Subset
\mathcal{U}_{\theta, l}^{2, \mathrm{D}}$, and define $\mathcal{U}_{\theta, l}^1$ by
\[
\mathcal{U}_{\theta, l}^1 = \{p \in \mathcal{W}_{\theta, l};
\mathop{\mathfrak{forget}}\nolimits_{\mathcal{S}, A}(p) \in \mathcal{U}_{\theta, l}^{1, \mathrm{D}}\}.
\]
Finally, we take open subsets
$\mathcal{V}_{\theta, l}^1 \Subset \mathcal{U}_{\theta, l}^1$ such that $\mathcal{V}_{\theta, l}^1 \cap
\widehat{\mathcal{M}}_{\mathcal{S}, A}^\triangle = \mathcal{V}_{\theta, l}^\triangle$.
Then these $\mathcal{V}_{\theta, l}^1$, $\mathcal{U}_{\theta, l}^k$ and $\mathcal{U}_{\theta, l}^{k, \mathrm{D}}$
($k = 1,2$) are the required subsets.
\end{proof}
Now we explain the construction of a domain curve representation.
\begin{lem}\label{existence of a domain curve representation}
For any constant $C \geq 0$, there exists a domain curve representation of
$\widehat{\mathcal{M}}^{\leq L_{\max}}_{\leq C}$.
\end{lem}
\begin{proof}
First we claim that in general, for a holomorphic building
$p = (\Sigma, z, u) \in \widehat{\mathcal{M}}$ whose domain curve is irreducible and
which has nonzero $E_{\hat \omega}$-energy,
there exist a finite set $\mathcal{S} = \{S\}$ of codimension-two small disks in $Y$
and an open subset $U \subset \widehat{\mathcal{M}}_{\mathcal{S}}$ such that
\begin{itemize}
\item
$p \in \mathop{\mathfrak{forget}}\nolimits_{\mathcal{S}}(U)$,
\item
the restriction of
$\mathop{\mathfrak{forget}}\nolimits_u : \widehat{\mathcal{M}}_{\mathcal{S}} \to \overline{\mathcal{M}}^\mathrm{D}_{\mathcal{S}}$
to $U$ is injective, and
\item
$\mathop{\mathrm{Aut}}\nolimits(\mathop{\mathfrak{forget}}\nolimits_u(q)) = \mathop{\mathrm{Aut}}\nolimits(q)$ for all $q \in U$.
\end{itemize}
This is proved as follows.
Since the critical points of $\pi_Y \circ u$ are discrete,
we can add marked points by the intersections with codimension-two disks in $Y$
to make the domain curve stable.
Namely, we can choose a finite set $\mathcal{S} = \{S\}$ of codimension-two disks
in $Y$ and a point $p^+ = (\Sigma, z, z^S, u) \in \widehat{\mathcal{M}}_{\mathcal{S}}$ such that
$z^S$ is $\mathop{\mathrm{Aut}}\nolimits(p)$-invariant,
$\mathop{\mathfrak{forget}}\nolimits_{\mathcal{S}} p^+ = p$ and
$\mathop{\mathrm{Aut}}\nolimits(\mathop{\mathfrak{forget}}\nolimits_u(p^+)) = \mathop{\mathrm{Aut}}\nolimits(p^+)$ ($= \mathop{\mathrm{Aut}}\nolimits(p)$).
Let $D_u : \widetilde{W}^{1, p}_\delta(\Sigma, u^\ast T \hat Y) \to
L^p_\delta(\Sigma, {\textstyle\bigwedge}^{0, 1} T\ast \Sigma \otimes_\mathbb{C} T \hat Y)$ be the
linearization of the equation of $J$-holomorphic curve at $u$ and
consider the linear operator
\begin{align*}
D^+_u : \widetilde{W}^{1, p}_\delta(\Sigma, u^\ast T \hat Y)
&\to L^p_\delta(\Sigma, {\textstyle\bigwedge}^{0, 1} T^\ast \Sigma \otimes_\mathbb{C} T \hat Y) \\
& \quad \oplus
\bigoplus_{z \in z^S, S \in \mathcal{S}} T_{\pi_Y \circ u(z)} Y/T_{\pi_Y \circ u(z)}S
\oplus \mathbb{R}
\end{align*}
defined by $D^+_u \xi = (D_u \xi, (\pi_Y)_\ast\xi(z), \sigma_\ast \xi (R))$,
where $R \in \Sigma$ is an arbitrary fixed point and $\sigma : \hat Y \to \mathbb{R}$ is
the projection. If it is injective, then
$\mathop{\mathfrak{forget}}\nolimits_u : \widehat{\mathcal{M}}_{\mathcal{S}} \to
\overline{\mathcal{M}}^\mathrm{D}_{\mathcal{S}}$ is injective on a neighborhood of $p^+$.
We can choose $\mathcal{S} = \{S\}$ which makes $D^+_u$ injective because
for any vector $\xi \in \mathop{\mathrm{Ker}}\nolimits D_u$ other than $\xi = c \partial_\sigma$ ($c \in \mathbb{R}$ is
a constant), there does not exist any non-empty open subset of $\Sigma$ on which
$\pi_{TY} \xi$ vanishes.
Therefore for any $p \in \widehat{\mathcal{M}}$, we can construct finite number of
disks $\mathcal{S} = \{S\}$ in $Y$ and
an open subset $U \subset \widehat{\mathcal{M}}_{\mathcal{S}}$
which satisfy the above conditions.
We note the following fact:
For each disk $S \in \mathcal{S}$, let $S \times \mathbb{R}^2 \subset Y$ be
its tubular neighborhood. Then for any small $x \in \mathbb{R}^2$,
$\mathcal{S}^x = \{S \times \{x\}; S \in \mathcal{S}\}$ also satisfies
the same condition.
Namely, there exists an open subset $U^x \subset \widehat{\mathcal{M}}_{\mathcal{S}^x}$
such that $p \in \mathop{\mathfrak{forget}}\nolimits_{\mathcal{S}}(U^x)$,
the restriction of
$\mathop{\mathfrak{forget}}\nolimits_u|_{U^x} : U^x \to \overline{\mathcal{M}}^\mathrm{D}_{\mathcal{S}^x}$ is injective, and
$\mathop{\mathrm{Aut}}\nolimits(\mathop{\mathfrak{forget}}\nolimits_u(q)) = \mathop{\mathrm{Aut}}\nolimits(q) = \mathop{\mathrm{Aut}}\nolimits(\mathop{\mathfrak{forget}}\nolimits_{\mathcal{S}^x}(q))$ for all $q \in U^x$.
We may assume that $\mathop{\mathfrak{forget}}\nolimits_{\mathcal{S}^x}(U^x) = \mathop{\mathfrak{forget}}\nolimits_{\mathcal{S}}(U)$
for all sufficiently small $x \in \mathbb{R}^2$.
We construct
\[
\mathcal{V}_{\theta, l} \subset \mathcal{U}_{\theta, l} \subset
\widehat{\mathcal{M}}^{\leq L_{\max}}_{\mathcal{S}, A, \theta, l}
\]
and
\[
\mathcal{U}_{\theta, l}^\mathrm{D} \subset \overline{\mathcal{M}}^\mathrm{D}_{\mathcal{S}, A, \theta, l}
\]
for $\widetilde{e}(\theta) \leq C$ and $l \geq 0$
by the induction in $\widetilde{e}(\theta)$.
For each triple $\theta = (g, k, E_{\hat \omega})$ with minimal $\widetilde{e}(\theta)$,
$\widehat{\mathcal{M}}^{\leq L_{\max}}_{\theta}$ consists of connected height-one
holomorphic buildings without nodal points.
First we consider the case of $E_{\hat \omega} > 0$.
In this case, first we construct a finite set
$\mathcal{S} = \{S\}$ of codimension-two submanifolds of $Y$ and
open subsets $\mathring{U}_{\theta} \subset
\widehat{\mathcal{M}}^{\leq L_{\max}}_{\mathcal{S}, \theta}$ which satisfy
the following conditions:
(Recall that $\widehat{\mathcal{M}}^{\leq L_{\max}}_{\mathcal{S}, \theta}$ is the subspace of
$\widehat{\mathcal{M}}^{\leq L_{\max}}_{\mathcal{S}, A, \theta}$ defined by $z^A = \emptyset$.)
\begin{enumerate}[label=(\roman*)]
\item
\label{circ U injective}
$\mathop{\mathfrak{forget}}\nolimits_u|_{\mathring{U}_{\theta}}
: \mathring{U}_{\theta} \to \overline{\mathcal{M}}^\mathrm{D}_{\mathcal{S}, \theta}$ is injective,
and $\mathop{\mathrm{Aut}}\nolimits(\mathop{\mathfrak{forget}}\nolimits_u(p)) = \mathop{\mathrm{Aut}}\nolimits(p)$ for any $p \in \mathring{U}_{\theta}$.
Furthermore, for any $p, q \in \mathring{U}_{\theta}$, if there exists
an $(\mathcal{S}, A)$-forgetful map from $\mathop{\mathfrak{forget}}\nolimits_u(p)$ to $\mathop{\mathfrak{forget}}\nolimits_u(q)$, then
$p \geq q$.
\item
$\widehat{\mathcal{M}}^{\leq L_{\max}}_{\theta}$ is covered by the image of
$\mathring{U}_{\theta}$ by $\mathop{\mathfrak{forget}}\nolimits_{\mathcal{S}}$.
\item
\label{z^S disjoint circ U}
For any two holomorphic buildings
$p^i = (\Sigma, z, z^{S, i}, u) \in \mathring{U}_{\theta}$ ($i = 1, 2$) such that
$\mathop{\mathfrak{forget}}\nolimits_{\mathcal{S}}(p^1) = \mathop{\mathfrak{forget}}\nolimits_{\mathcal{S}}(p^2) = (\Sigma, z, u)$,
the following holds:
\begin{itemize}
\item
For any two different submanifolds $S_1 \neq S_2 \in \mathcal{S}$,
$z^{S_1, 1}$ and $z^{S_2, 2}$ are disjoint in $\Sigma$.
\item
For any $S \in \mathcal{S}$, if $z^{S, 1} \neq \emptyset$ and
$z^{S, 2} \neq \emptyset$ then $z^{S, 1} = z^{S, 2}$.
\end{itemize}
\end{enumerate}
We can construct such submanifolds $\mathcal{S} = \{S\}$ and
open subsets $\mathring{U}_{\theta}$ as follows.
The claim we proved in the above implies the following:
There exists an open covering $\{V_i\}$ of $\widehat{\mathcal{M}}^{\leq L_{\max}}_{\theta}$,
and for each $V_i$, there exist an infinite family of finite sets
$\mathcal{S}_i^x = \{S^x_i\}_{S_i \in \mathcal{S}_i}$ ($x \in \mathbb{R}^2$) of
codimension two disks of $Y$ and a family of open subsets
$U^x_i \subset \widehat{\mathcal{M}}^{\leq L_{\max}}_{\mathcal{S}^x, \theta}$
such that $\mathop{\mathfrak{forget}}\nolimits_u|_{U^x_i} : U^x_i \to \overline{\mathcal{M}}^\mathrm{D}_{\mathcal{S}^x}$ is injective,
$\mathop{\mathrm{Aut}}\nolimits(\mathop{\mathfrak{forget}}\nolimits_u(p)) = \mathop{\mathrm{Aut}}\nolimits(p) = \mathop{\mathrm{Aut}}\nolimits(\mathop{\mathfrak{forget}}\nolimits_{\mathcal{S}^x}(p))$ for all $p \in U^x_i$,
and $\mathop{\mathfrak{forget}}\nolimits_{\mathcal{S}^x}(U^x_i) = V_i$.
Furthermore, $\{S^x_i; x \in \mathbb{R}^2\}$ are disjoint for each $i$.
We choose finite numbers $x_i^1, x_i^2, \dots, x_i^{N_i} \in \mathbb{R}^2$
and open subsets $\mathring{U}_i^k \subset U_i^{x_i^k}$ which satisfy
the following conditions,
where we abbreviate $\mathcal{S}_i^{x_i^k}$ as $\mathcal{S}_i^k$:
\begin{itemize}
\item
$\{\mathop{\mathfrak{forget}}\nolimits_{\mathcal{S}_i^k}(\mathring{U}_i^k)\}_{i, k}$ covers
$\widehat{\mathcal{M}}^{\leq L_{\max}}_{\theta}$
\item
For any $p = (\Sigma, z, z^S, u) \in U_i^k$ and $p' = (\Sigma, z, (z')^S, u) \in U_j^l$,
if $\mathop{\mathfrak{forget}}\nolimits_{\mathcal{S}_i^k}(p) = \mathop{\mathfrak{forget}}\nolimits_{\mathcal{S}_j^l}(p')$, then
$z^S$ and $(z')^{S'}$ are disjoint in $\Sigma$ for any two different submanifolds
$S \neq S' \in \coprod_{i, k} \mathcal{S}_i^k$,
\end{itemize}
Then $\mathcal{S} = \coprod_{i, k} \mathcal{S}_i^k$ and
$\mathring{U}_{\theta} = \bigcup_{i, k} \mathring{U}_i^k$ satisfy
Condition \ref{circ U injective} to \ref{z^S disjoint circ U}.
We explain how to choose such numbers $x_i^k \in \mathbb{R}^2$ and open subsets
$\mathring{U}_i^k \subset U_i^{x_i^k}$.
Take open subsets $\mathring{V}_i \Subset V_i$ which cover
$\widehat{\mathcal{M}}^{\leq L_{\max}}_{\theta}$.
We construct $x_i^k \in \mathbb{R}^2$ and $\mathring{U}_i^k \subset U_i^{x_i^k}$
by the induction in $i$ so that $\{\mathop{\mathfrak{forget}}\nolimits_{\mathcal{S}_i^k}(\mathring{U}_i^k)\}_k$
covers $\mathring{V}_i$ for each $i$ as follows.
We assume that $x_i^k \in \mathbb{R}^2$ and $\mathring{U}_i^k \subset U_i^{x_i^k}$ are given
for $i < i_0$, and we construct those for $i = i_0$.
For each $p = (\Sigma, z, u) \in V_{i_0}$, let $A_p \subset \Sigma$ be the subset of the
points which appear in some $z^S$ for $p^ + = (\Sigma, z, z^S, u) \in \mathring{U}_i^k$
($i < i_0$) such that $\mathop{\mathfrak{forget}}\nolimits_{\mathcal{S}_i^k}(p^+) = p$.
Let $N > 0$ be a constant larger than $\# A_p$ for any $p \in V_{i_0}$.
Choose arbitrary points $x_{i_0}^1, \dots, x_{i_0}^N \in \mathbb{R}^2$.
Then for any $p \in V_{i_0}$, there exists at least one $x_{i_0}^k$ such that
for the point $p^+ = (\Sigma, z, z^S, u) \in U_{i_0}^{x_{i_0}^k}$ such that
$\mathop{\mathfrak{forget}}\nolimits_{\mathcal{S}_{i_0}^k}(p^+) = p$,
each $z^S$ is disjoint with $A_p$.
This is because $S^x_{i_0}$ for $x \in \mathbb{R}^2$ are disjoint.
Hence we can construct open subsets $\mathring{U}_{i_0}^k \subset U_{i_0}^{x_{i_0}^k}$
such that their images by $\mathop{\mathfrak{forget}}\nolimits_{\mathcal{S}_{i_0}^k}$ cover $\mathring{V}_i$
and for any $p^+ = (\Sigma, z, z^S, u) \in \mathring{U}_{i_0}^k$ and
$p = \mathop{\mathfrak{forget}}\nolimits_{\mathcal{S}_{i_0}^k}(p^+)$, each $z^S$ is disjoint with $A_p$.
Therefore we can construct a finite set
$\mathcal{S} = \{S\}$ of codimension-two submanifolds of $Y$ and
open subsets $\mathring{U}_{\theta} \subset
\widehat{\mathcal{M}}^{\leq L_{\max}}_{\mathcal{S}, \theta}$ which satisfy Condition
\ref{circ U injective} to \ref{z^S disjoint circ U}.
Let $U_{\theta} \subset \widehat{\mathcal{M}}_{\mathcal{S}}$ be the set of
holomorphic buildings $p = (\Sigma, z, z^S, u)$ such that there exist subsets
$z^{S, i} \subset z^S$ ($i = 1, \dots, k$, $S \in \mathcal{S}$) such that
$p_i = (\Sigma, z, z^{S, i}, u) \in \mathring{U}_{\theta}$ and $z^S = \bigcup_i z^{S, i}$.
Note that the assumption on $\mathcal{S}$ implies that for any finite holomorphic
buildings $p^i = (\Sigma, z, z^{S, i}, u) \in \mathring{U}_{\theta}$ such that
$\mathop{\mathfrak{forget}}\nolimits_{\mathcal{S}}(p^i) = \mathop{\mathfrak{forget}}\nolimits_{\mathcal{S}}(p^1)$,
we can define a holomorphic building $p = (\Sigma, z, z^S, u)$ by
$z^S = \bigcup_i z^{S, i}$.
(Namely, $z^S$ and $z^{S'}$ are disjoint for $S \neq S'$.)
$U_{\theta}$ satisfies the following conditions:
\begin{enumerate}[label=(\roman*)']
\item
$\mathop{\mathfrak{forget}}\nolimits_{\mathcal{S}}|_{U_{\theta}} : U_{\theta}
\to \widehat{\mathcal{M}}^{\leq L_{\max}}_{\theta}$ is surjective.
\item
\label{U DM rep}
$\mathop{\mathfrak{forget}}\nolimits_u|_{U_{\theta}} : U_{\theta} \to \overline{\mathcal{M}}^\mathrm{D}_{\mathcal{S}, \theta}$
is injective, and $\mathop{\mathrm{Aut}}\nolimits(\mathop{\mathfrak{forget}}\nolimits_u(p)) = \mathop{\mathrm{Aut}}\nolimits(p) = \mathop{\mathrm{Aut}}\nolimits(\mathop{\mathfrak{forget}}\nolimits_{\mathcal{S}}(p))$ for all $p \in U_{\theta}$.
Furthermore, for any $p, q \in U_{\theta}$, if there exists
an $(\mathcal{S}, A)$-forgetful map from $\mathop{\mathfrak{forget}}\nolimits_u(p)$ to $\mathop{\mathfrak{forget}}\nolimits_u(q)$, then
$p \geq q$.
\item
For any $p, q \in U_{\theta}$ such that $\mathop{\mathfrak{forget}}\nolimits_{\mathcal{S}}(p)
= \mathop{\mathfrak{forget}}\nolimits_{\mathcal{S}}(q)$, there exists some $r \in U_{\theta}$ such that
$\mathop{\mathfrak{forget}}\nolimits_{\mathcal{S}}(r) = \mathop{\mathfrak{forget}}\nolimits_{\mathcal{S}}(p)$, $r \geq p$ and $r \geq q$.
(For $p = (\Sigma, z, z^S, u)$ and $q = (\Sigma, z, (z')^S, u)$,
$r = (\Sigma, z, z^S \cup (z')^S, u)$ satisfies these conditions.)
\end{enumerate}
The last condition implies that we can apply Lemma \ref{totally ordered cover} for
$\mathop{\mathfrak{forget}}\nolimits_{\mathcal{S}}|_{U_{\theta}} : U_{\theta}
\to \widehat{\mathcal{M}}^{\leq L_{\max}}_{\theta}$.
It implies that there exists some open subset $V_{\theta} \subset U_{\theta}$
which satisfies the following conditions:
$\mathop{\mathfrak{forget}}\nolimits_{\mathcal{S}}(V_{\theta}) = \widehat{\mathcal{M}}^{\leq L_{\max}}_{\theta}$, and
if $p, q \in V_{\theta}$ satisfy $\mathop{\mathfrak{forget}}\nolimits_{\mathcal{S}}(p) = \mathop{\mathfrak{forget}}\nolimits_{\mathcal{S}}(q)$,
then $p \leq q$ or $p \geq q$.
Take open subsets $\mathcal{V}_{\theta} \Subset \mathcal{U}_{\theta} \Subset V_{\theta}$ such that
$\mathop{\mathfrak{forget}}\nolimits_{\mathcal{S}}(\mathcal{V}_{\theta}) = \widehat{\mathcal{M}}^{\leq L_{\max}}_{\theta}$.
Define $\mathcal{V}_{\theta, l} = \mathcal{V}_{\theta} \cap
\widehat{\mathcal{M}}^{\leq L_{\max}}_{\mathcal{S}, \theta, l}$ and $\mathcal{U}_{\theta, l}
= \mathcal{U}_{\theta} \cap \widehat{\mathcal{M}}^{\leq L_{\max}}_{\mathcal{S}, \theta, l}$.
It is clear that they satisfy Condition \ref{existence of minimum}.
We construct open neighborhoods
$\mathcal{U}_{\theta, l}^\mathrm{D} \subset \overline{\mathcal{M}}^{\mathrm{D}}_{\mathcal{S}, \theta, l}$
of $\mathop{\mathfrak{forget}}\nolimits_u(\mathcal{U}_{\theta, l})$,
which satisfy Condition \ref{D-neighborhood} and the following condition stronger than
\ref{stably unique forgetful map DM} as follows.
(In this case, $z^A = \emptyset$.)
\begin{itemize}
\item[\ref{stably unique forgetful map DM}$^{\mathrm{S}}$]
For any
$l \geq 0$, $\hat p = (\hat \Sigma, z, z^S, z^A) \in \mathcal{U}_{\theta, l}^\mathrm{D}$ and
any subsets $(z^i)^S \subset z^S$ $(i = 1,2$, $S \in \mathcal{S})$ and
$(z^i)^A \subset z^A$ ($i=1,2$), if each $\hat p^i = (\hat \Sigma, z, (z^i)^S, (z^i)^A)$ is
contained in $\mathcal{U}_{\theta, l(\hat p^i)}^\mathrm{D}$, then
$(z^1)^S \subset (z^2)^S$ and $(z^1)^A \subset (z^2)^A$, or
$(z^2)^S \subset (z^1)^S$ and $(z^2)^A \subset (z^1)^A$.
\end{itemize}
If we replace $\mathcal{U}_{\theta, l}^\mathrm{D}$ in this condition with $\mathop{\mathfrak{forget}}\nolimits_u(V_{\theta, l})$,
then it holds.
Indeed, for any $p = (\Sigma, z, z^S, u) \in V_{\theta}$ and subsets
$(z^i)^S \subset z^S$, if $\mathop{\mathfrak{forget}}\nolimits_u((\Sigma, z, (z^i)^S, u)) = \mathop{\mathfrak{forget}}\nolimits_u(p^i)$ for some
$p^i \in V_{\theta} \subset U_{\theta}$, then $p^i \leq p$ by
Condition \ref{U DM rep}, which implies that $p^i = (\Sigma, z, (z^i)^S, u)$.
Since $\mathop{\mathfrak{forget}}\nolimits_{\mathcal{S}}(p^i) = \mathop{\mathfrak{forget}}\nolimits_{\mathcal{S}}(p)$ for $i = 1,2$,
$p^1 \leq p^2$ or $p^2 \leq p^1$ by the property of $V_{\theta}$,
which implies that $(z^1)^S \subset (z^2)^S$ or $(z^2)^S \subset (z^1)^S$.
Hence the above condition holds for $\mathop{\mathfrak{forget}}\nolimits_u(V_{\theta, l})$.
Therefore if $\mathcal{U}_{\theta, l}^\mathrm{D} \subset \overline{\mathcal{M}}^{\mathrm{D}}_{\mathcal{S}, \theta, l}$
is a sufficiently small neighborhood of $\mathop{\mathfrak{forget}}\nolimits_u(\mathcal{U}_{\theta, l})$, then
they satisfy the condition.
It is easy to check that the subsets $\mathcal{V}_{\theta, l}$, $\mathcal{U}_{\theta, l}$ and
$\mathcal{U}_{\theta, l}^\mathrm{D}$ satisfy Condition \ref{(S, A)-covering} to
\ref{stably unique forgetful map DM}.
Next we consider the case of $e^2 = 0$.
In this case, for any $p = (\Sigma, z, u) \in \widehat{\mathcal{M}}^{\leq L_{\max}}_{\theta}$,
the domain curve $(\Sigma, z)$ is already stable.
We take finite points $\hat p^i \in \overline{\mathcal{M}}^\mathrm{D}_{\theta}$,
a local universal family $(\hat P^i \to \hat X^i, Z^i)$ of each $\hat p^i$
and an $\mathop{\mathrm{Aut}}\nolimits(\hat p^i)$-invariant family of disjoint smooth sections
$(Z^i)^A = ((Z^i)^A_j)$ of each $\hat P^i \to \hat X^i$ which satisfy the following
conditions:
\begin{itemize}
\item
$\overline{\mathcal{M}}^\mathrm{D}_{\theta}$ is covered by $\hat X^i$, that is, every stable curve in
$\overline{\mathcal{M}}^\mathrm{D}_{\theta}$ appears some fiber of the local universal families.
\item
For any $i \neq i'$, $a \in \hat X^i$, $a' \in \hat X^{i'}$ and isomorphism
$\varphi : (\hat P^i_a, Z^i(a)) \cong (\hat P^{i'}_{a'}, Z^{i'}(a'))$, $\varphi((Z^i)^A(a))$ and
$(Z^{i'})^A(a')$ are disjoint.
\end{itemize}
For the construction of the family of smooth sections,
we note that if we take a smooth section $(Z^i)^A_1$ of $\hat P^i \to \hat X^i$
whose values are contained in the open subset of the points of trivial stabilizer,
then the $\mathop{\mathrm{Aut}}\nolimits(\hat p^i)$-orbit of $(Z^i)^A_1$ is an $\mathop{\mathrm{Aut}}\nolimits(\hat p^i)$-invariant family of
disjoint smooth sections.
Let $U^\mathrm{D} \subset \overline{\mathcal{M}}^\mathrm{D}_{\mathcal{S}, A, \theta}$ be the set of
stable curves $(\hat \Sigma, z, z^A)$ ($z^S = \emptyset$) such that there exist
finite points $a_k \in \hat X^{i_k}$ ($k = 1, \dots, N$) and isomorphisms
$\varphi_k : (\hat P^{i_k}_{a_k}, Z^{i_k}(a_k)) \to (\hat \Sigma, z)$ such that
$z^A = \bigcup_k \varphi_k((Z^{i_k})^A(a_k))$.
Applying Lemma \ref{totally ordered cover} to the forgetful map $\mathop{\mathfrak{forget}}\nolimits_A$
from $U^\mathrm{D}$ to $\overline{\mathcal{M}}^\mathrm{D}_{\theta}$, we obtain an open subset
$V^\mathrm{D} \subset U^\mathrm{D}$ which satisfies the following conditions:
$\mathop{\mathfrak{forget}}\nolimits_A(V^\mathrm{D}) = \overline{\mathcal{M}}^\mathrm{D}_{\theta}$, and
if $\hat p, \hat q \in V^\mathrm{D}$ satisfy $\mathop{\mathfrak{forget}}\nolimits_A(\hat p) = \mathop{\mathfrak{forget}}\nolimits_A(\hat q)$,
then $\hat p \leq \hat q$ or $\hat p \geq \hat q$.
Take an open subset $\mathcal{U}_{\theta}^\mathrm{D} \subset V^\mathrm{D}$ such that
$\mathop{\mathfrak{forget}}\nolimits_{\mathcal{S}, A}(\mathcal{U}_{\theta}^\mathrm{D}) = \overline{\mathcal{M}}^\mathrm{D}_{\theta}$ and
define $\mathcal{U}_{\theta, l}^\mathrm{D} = \mathcal{U}_{\theta}^\mathrm{D} \cap \overline{\mathcal{M}}^\mathrm{D}_{\theta, l}$.
Define subsets $\mathcal{U}_{\theta, l} \subset
\widehat{\mathcal{M}}^{\leq L_{\max}}_{\mathcal{S}, A, \theta, l}$ by
$\mathcal{U}_{\theta, l} = \mathop{\mathfrak{forget}}\nolimits_u^{-1}(\mathcal{U}_{\theta, l}^\mathrm{D})$, and
take their relatively compact open subsets $\mathcal{V}_{\theta, l} \Subset \mathcal{U}_{\theta, l}$
whose images by $\mathop{\mathfrak{forget}}\nolimits_A$ cover $\widehat{\mathcal{M}}^{\leq L_{\max}}_{\theta}$.
Then they satisfy Condition \ref{(S, A)-covering} to \ref{stably unique forgetful map DM}.
For the construction of $\mathcal{U}^\mathrm{D}_{\theta, l}$ for the general triples $\theta$,
we add the following condition:
\begin{itemize}
\item[$(\natural)$]
For any $\hat p \in \mathcal{U}_{\theta, l}^\mathrm{D}$ and
$\hat q \in \mathcal{U}_{\theta', l'}^\mathrm{D}$,
if there exists an $(\mathcal{S}, A)$-forgetful map from $\hat p$ to $\hat q$,
then $E_{\hat \omega} \geq E_{\hat \omega}'$,
where $\theta = (g, k, E_{\hat \omega})$ and $\theta' = (g, k, E_{\hat \omega}')$.
\end{itemize}
We will construct $\mathcal{U}^\mathrm{D}_{\theta, l}$ so that
they also satisfy this condition.
We consider the general triple $\theta = (g, k, E_{\hat \omega})$.
Assume that $\mathcal{V}_{\theta', l}$, $\mathcal{U}_{\theta', l}$ and $\mathcal{U}_{\theta', l}^\mathrm{D}$ for $\theta'$
such that $\widetilde{e}(\theta') < \widetilde{e}(\theta)$ are given.
Define $\mathcal{U}_{\theta, l}^{\mathrm{D}, \triangle} \subset
\overline{\mathcal{M}}_{\mathcal{S}, A, \theta, l}^{\mathrm{D}, \triangle}$ by the largest subset
which satisfy Condition \ref{decomposition into parts U DM}.
Namely, $\hat p \in \overline{\mathcal{M}}^\mathrm{D}_{\mathcal{S}, A, \theta, l}$ is contained in
$\mathcal{U}_{\theta, l}^{\mathrm{D}, \triangle}$ if the following condition holds:
For any set $\mathcal{N}$ of its nodal
points, replace each nodal point in $\mathcal{N}$ with a pair of marked points, and
let $\hat p'_i$ $(1 \leq i \leq N)$ be its connected components or an arbitrary
decomposition into unions of its connected components.
Let $g_i$ and $k_i$ be the genus and the number of marked points of each $\hat p'_i$.
Then there exist some $E_{\hat \omega}^i \geq 0$ such that
$E_{\hat \omega} = \sum_i E_{\hat \omega}^i$ and
$\hat p'_i \in \mathcal{U}_{\theta'_i, l(\hat p'_i)}^\mathrm{D}$ for all $i$, where
$\theta'_i = (g_i, k_i, E_{\hat \omega}^i)$.
Similarly, we define
$\mathcal{U}_{\theta, l}^\triangle \subset \widehat{\mathcal{M}}_{\mathcal{S}, A, \theta, l}^\triangle$
by the largest subset which satisfy Condition \ref{decomposition into parts U},
and define
$\mathcal{V}_{\theta, l}^\triangle \subset \widehat{\mathcal{M}}_{\mathcal{S}, A, \theta, l}^\triangle$
by Condition \ref{decomposition into parts V}.
Then they satisfy the assumption of Lemma \ref{extension from the decomposable}.
We check that $\mathcal{U}_{\theta, l}^{\mathrm{D}, \triangle}$ ($l \geq 0$) satisfy Condition
\ref{stably unique forgetful map DM}.
For a stable curve $\hat p = (\hat \Sigma, z, z^S, z^A) \in \mathcal{U}^\mathrm{D}_{\theta, l}$ and
subsets $(z^1)^S, (z^2)^S \subset z^S$ ($S \in \mathcal{S}$) and
$(z^1)^A, (z^2)^A \subset z^A$, assume that each
$\hat p^j = (\hat \Sigma, z, (z^j)^S, (z^j)^A)$ is contained in
$\mathcal{U}^\mathrm{D}_{\theta, l(\hat p^j)}$.
We prove that
$\hat p^3 = (\hat \Sigma, z, (z^1)^S \cap (z^2)^S, (z^1)^A \cap (z^2)^A)$
is contained in $\mathcal{U}^\mathrm{D}_{\theta, l(\hat p^3)}$.
Let $\mathcal{N}$ be an arbitrary set of nodal points of $\hat \Sigma$.
Replace each nodal point of $\hat \Sigma$ in $\mathcal{N}$
with a pair of marked points, and decompose the curve into
arbitrary unions of connected components.
For $\hat p$ and $\hat p^j$ ($j=1,2,3$),
let $\{\hat p'_i\}_{1 \leq i \leq k}$ and $\{(\hat p^j)'_i\}_{1 \leq i \leq k}$ be
the obtained decomposition respectively.
Let $(E_{\hat \omega})_i \geq 0$ be non-negative numbers such that
$E_{\hat \omega} = \sum_i (E_{\hat \omega})_i$ and
$\hat p'_i \in \mathcal{U}_{\theta'_i l(\hat p'_i)}^\mathrm{D}$ for all $i$, where
$\theta'_i = (g'_i, k'_i, (E_{\hat \omega})_i)$.
Similarly, let $(E_{\hat \omega})^j_i$ ($j =1,2$) be pairs for $(\hat p^j)'_i$.
Since there exists an $(\mathcal{S}, A)$-forgetful map from $\hat p'_i$ to
$(\hat p^j)'_i$, Condition $(\natural)$ implies that
$(E_{\hat \omega})_i \geq (E_{\hat \omega})^j_i$ for all $i$.
Hence $(E_{\hat \omega})_i = (E_{\hat \omega})^j_i$ for all $i$.
Therefore, Condition \ref{stably unique forgetful map DM} for each $\theta'_i$
implies that each $(\hat p^3)'_i$ is contained in $\mathcal{U}^\mathrm{D}_{\theta'_i, l^3_i}$,
where $l^3_i = l((\hat p^3)'_i)$.
Therefore $\hat p^3$ is contained in $\mathcal{U}^\mathrm{D}_{\theta, l(\hat p^3)}$.
The other conditions in the assumption of Lemma
\ref{extension from the decomposable}
are easy to check.
Hence there exist subsets
\[
\mathcal{V}_{\theta, l}^1 \subset \mathcal{U}_{\theta, l}^1 \subset \mathcal{U}_{\theta, l}^2 \subset
\widehat{\mathcal{M}}^{\leq L_{\max}}_{\mathcal{S}, A, \theta, l} \quad
(l \geq 0)
\]
and
\[
\mathcal{U}_{\theta, l}^{1, \mathrm{D}} \subset \mathcal{U}_{\theta, l}^{2, \mathrm{D}} \subset
\overline{\mathcal{M}}^\mathrm{D}_{\mathcal{S}, A, \theta, l}
\quad (l \geq 0)
\]
which satisfy the conditions in Lemma \ref{extension from the decomposable}.
Since $\mathcal{U}^{\mathrm{D}, \triangle}_{\theta, l}$ ($l \geq 0$) satisfy
Condition $(\natural)$ (with the other $\mathcal{U}^\mathrm{D}_{\theta', l'}$),
we may assume that $\mathcal{U}_{\theta, l}^{2, \mathrm{D}}$ ($l \geq 0$) also satisfy this condition.
We consider separately the cases where $E_{\hat \omega} > 0$ or not.
First we consider the case where $E_{\hat \omega} > 0$.
Since $\widehat{\mathcal{M}}^{\leq L_{\max}}_{\theta} \setminus \mathop{\mathfrak{forget}}\nolimits_{\mathcal{S}, A}
(\bigcup_l \mathcal{V}^1_{\theta, l})$
consists of connected height-one holomorphic buildings without nodal points,
by the same argument as in the case of triples $\theta$
with minimal $\widetilde{e}(\theta)$,
we obtain a finite set $\mathcal{S}^+ = \{S\}$ of codimension-two submanifolds of $Y$
and open subsets $\mathring{U}_{\theta} \subset
\widehat{\mathcal{M}}^{\leq L_{\max}}_{\mathcal{S}^+, \theta}$
which satisfy the following conditions:
\begin{itemize}
\item
$\mathop{\mathfrak{forget}}\nolimits_u|_{\mathring{U}_{\theta}} : \mathring{U}_{\theta} \to
\overline{\mathcal{M}}^\mathrm{D}_{\mathcal{S}^+, \theta}$ is injective,
and $\mathop{\mathrm{Aut}}\nolimits(\mathop{\mathfrak{forget}}\nolimits_u(p)) = \mathop{\mathrm{Aut}}\nolimits(p)$ for any $p \in \mathring{U}_{\theta}$.
Furthermore, for any $p, q \in \mathring{U}_{\theta}$, if there exists
an $(\mathcal{S}^+, A)$-forgetful map from $\mathop{\mathfrak{forget}}\nolimits_u(p)$ to $\mathop{\mathfrak{forget}}\nolimits_u(q)$, then
$p \geq q$.
\item
$\widehat{\mathcal{M}}^{\leq L_{\max}}_{\theta} \setminus \mathop{\mathfrak{forget}}\nolimits_{\mathcal{S}, A}
(\bigcup_l \mathcal{V}^1_{\theta, l})$ is covered by the image of
$\mathring{U}_{\theta}$ by $\mathop{\mathfrak{forget}}\nolimits_{\mathcal{S}^+}$.
\item
For any two holomorphic buildings
$p^i = (\Sigma, z, z^{S, i}, u) \in \mathring{U}_{\theta}$ ($i = 1, 2$) such that
$\mathop{\mathfrak{forget}}\nolimits_{\mathcal{S}^+}(p^1) = \mathop{\mathfrak{forget}}\nolimits_{\mathcal{S}^+}(p^2) = (\Sigma, z, u)$,
the following holds:
\begin{itemize}
\item
For any two different submanifolds $S_1 \neq S_2 \in \mathcal{S}^+$,
$z^{S_1, 1}$ and $z^{S_2, 2}$ are disjoint in $\Sigma$.
\item
For any $S \in \mathcal{S}^+$, if $z^{S, 1} \neq \emptyset$ and
$z^{S, 2} \neq \emptyset$ then $z^{S, 1} = z^{S, 2}$.
\end{itemize}
\item
$\mathcal{S}^+$ and $\mathcal{S}$ do not share the same submanifolds of $Y$.
\item
For any $p^1 = (\Sigma, z, z^{S, 1}, u) \in \mathring{U}_{\theta}$ and
$p^2 = (\Sigma, z, z^{S, 2}, z^A, u) \in \mathcal{U}_{\theta, l}^2$ such that
$\mathop{\mathfrak{forget}}\nolimits_{\mathcal{S}^+}(p^1) = \mathop{\mathfrak{forget}}\nolimits_{\mathcal{S}, A}(p^2)$,
$z^{S_1, 1}$, $z^{S_2, 2}$ and $z^A$ are disjoint for any $S_1 \in \mathcal{S}^+$ and
$S_2 \in \mathcal{S}$.
\end{itemize}
We add $\mathcal{S}^+$ to $\mathcal{S}$, and denote the union by $\mathcal{S}$
in what follows.
Let $U_{\theta} \subset \widehat{\mathcal{M}}^{\leq L_{\max}}_{\mathcal{S}, \theta}$
be the set of holomorphic buildings $p = (\Sigma, z, z^S, u)$ such that
there exist some subsets $z^{S, i} \subset z^S$ ($i = 1, \dots, k$) such that
$p_i = (\Sigma, z, z^{S, i}, u) \in \mathring{U}_{\theta}$ and
$z^S = \bigcup_i z^{S, i}$.
We note that
\begin{equation}
\widehat{\mathcal{M}}^{\leq L_{\max}}_{\theta} = \mathop{\mathfrak{forget}}\nolimits_{\mathcal{S}}(U_{\theta})
\cup \mathop{\mathfrak{forget}}\nolimits_{\mathcal{S}, A}(\bigcup_l \mathcal{V}^1_{\theta, l}).
\label{U V^1 cover}
\end{equation}
Define
\[
U^3_{\theta} = U_{\theta} \setminus
\mathop{\mathfrak{forget}}\nolimits_{\mathcal{S}}^{-1}(\mathop{\mathfrak{forget}}\nolimits_{\mathcal{S}, A}(\bigcup_l \overline{\mathcal{U}_{\theta, l}^1}))
\]
and apply Lemma \ref{totally ordered cover} for the locally homeomorphic map
$\mathop{\mathfrak{forget}}\nolimits_{\mathcal{S}}|_{U^3_{\theta}} : U^3_{\theta} \to
\widehat{\mathcal{M}}^{\leq L_{\max}}_{\theta}$
and a compact subset $\widehat{\mathcal{M}}^{\leq L_{\max}}_{\theta} \setminus
\mathop{\mathfrak{forget}}\nolimits_{\mathcal{S}, A}(\bigcup_l \mathcal{U}^2_{\theta, l})$.
Then we obtain an open subset $V^3_{\theta} \Subset U^3_{\theta}$ such that
\[
\widehat{\mathcal{M}}^{\leq L_{\max}}_{\theta} = \mathop{\mathfrak{forget}}\nolimits_{\mathcal{S}}(V^3_{\theta})
\cup \mathop{\mathfrak{forget}}\nolimits_{\mathcal{S}, A}(\bigcup_l \mathcal{U}^2_{\theta, l})
\]
and if
$p, q \in V^3_{\theta}$ satisfy $\mathop{\mathfrak{forget}}\nolimits_{\mathcal{S}}(p) = \mathop{\mathfrak{forget}}\nolimits_{\mathcal{S}}(q)$
then $p \leq q$ or $q \leq p$.
We note that
\[
\mathop{\mathfrak{forget}}\nolimits_{\mathcal{S}}(V^3_{\theta}) \cap
\mathop{\mathfrak{forget}}\nolimits_{\mathcal{S}, A}(\bigcup_l \mathcal{U}_{\theta, l}^1)= \emptyset
\]
by the definition of $U^3_{\theta}$.
We define $U^4_{\theta} \subset U_{\theta}$ by the open subset of
holomorphic buildings $p \in U_{\theta}$ such that $p \geq q$
for any $q \in \overline{V^3_{\theta}}$
such that $\mathop{\mathfrak{forget}}\nolimits_{\mathcal{S}}(p) = \mathop{\mathfrak{forget}}\nolimits_{\mathcal{S}}(q)$.
Note that $\mathop{\mathfrak{forget}}\nolimits_{\mathcal{S}}(U^4_{\theta}) = \mathop{\mathfrak{forget}}\nolimits_{\mathcal{S}}(U_{\theta})$
since $U_{\theta}$ is closed under the union of $z^S$.
Hence (\ref{U V^1 cover}) implies that
\begin{equation}
\widehat{\mathcal{M}}^{\leq L_{\max}}_{\theta} = \mathop{\mathfrak{forget}}\nolimits_{\mathcal{S}}(U^4_{\theta})
\cup \mathop{\mathfrak{forget}}\nolimits_{\mathcal{S}, A}(\bigcup_l \mathcal{V}^1_{\theta, l}).
\label{4, 1 cover}
\end{equation}
We also define
$U^2_{\theta} \subset \widehat{\mathcal{M}}^{\leq L_{\max}}_{\mathcal{S}, A, \theta}$
by the set of holomorphic buildings $p = (\Sigma, z, \allowbreak z^S, \allowbreak z^A, \allowbreak u)$
which satisfy the following conditions:
\begin{itemize}
\item
$z^S$ and $z^A$ are $\mathop{\mathrm{Aut}}\nolimits(\Sigma, z, u)$-invariant.
\item
There exist subsets $z^{S, i} \subset z^S$ and $z^{A, i} \subset z^A$ ($i = 1, \dots, k$)
such that $p_i = (\Sigma, z, z^{S, i}, z^{A, i}, u) \in \mathcal{U}_{\theta, l(p_i)}^2$,
$z^S = \bigcup_i z^{S, i}$ and $z^A = \bigcup_i z^{A, i}$.
\item
$p \geq q$ for any $q \in \bigcup_l \overline{\mathcal{U}^1_{\theta, l}}$ such that
$\mathop{\mathfrak{forget}}\nolimits_{\mathcal{S}, A}(p) = \mathop{\mathfrak{forget}}\nolimits_{\mathcal{S}, A}(q)$.
\end{itemize}
Then $\mathop{\mathfrak{forget}}\nolimits_{\mathcal{S}, A}(U^2_{\theta})
= \mathop{\mathfrak{forget}}\nolimits_{\mathcal{S}, A}(\bigcup_l \mathcal{U}^2_{\theta, l})$, which implies that
\begin{equation}
\widehat{\mathcal{M}}^{\leq L_{\max}}_{\theta} = \mathop{\mathfrak{forget}}\nolimits_{\mathcal{S}}(V^3_{\theta})
\cup \mathop{\mathfrak{forget}}\nolimits_{\mathcal{S}, A}(U^2_\theta)
\label{3, 2 cover}
\end{equation}
Let $U^{2+4}_{\theta} \subset \widehat{\mathcal{M}}^{\leq L_{\max}}_{\mathcal{S}, A, \theta}$
be the set of holomorphic buildings $p = (\Sigma, z, z^S, z^A, u)$ such that
there exist some holomorphic buildings
$p_2 = (\Sigma, z, z^{S, 2}, z^{A, 2}, u) \in U^2_{\theta}$ and
$p_4 = (\Sigma, z, z^{S, 4}, u) \in U^4_{\theta}$ such that
$\mathop{\mathfrak{forget}}\nolimits_{\mathcal{S}, A}(p_i) = \mathop{\mathfrak{forget}}\nolimits_{\mathcal{S}, A}(p)$ ($i = 2,4$),
$z^S = z^{S, 2} \sqcup z^{S, 4}$ and $z^A = z^{A, 2}$.
Then (\ref{4, 1 cover}) and (\ref{3, 2 cover}) imply that
\[
\mathop{\mathfrak{forget}}\nolimits_{\mathcal{S}, A}(U^{2+4}_{\theta})
= \mathop{\mathfrak{forget}}\nolimits_{\mathcal{S}, A}(U^2_{\theta}) \cap
\mathop{\mathfrak{forget}}\nolimits_{\mathcal{S}}(U^4_{\theta})
\]
covers
\begin{equation}
\widehat{\mathcal{M}}^{\leq L_{\max}}_{\theta} \setminus
\bigl(\mathop{\mathfrak{forget}}\nolimits_{\mathcal{S}, A}\bigl(\bigcup_l \mathcal{V}^1_{\theta, l}\bigr) \cup
\mathop{\mathfrak{forget}}\nolimits_{\mathcal{S}}(V^3_{\theta})\bigr).
\label{V^1 V^3 complement}
\end{equation}
Furthermore, $\mathop{\mathfrak{forget}}\nolimits_u$ is injective on $U^{2+4}_{\theta}$.
We apply Lemma \ref{totally ordered cover} to the locally homeomorphic map
$\mathop{\mathfrak{forget}}\nolimits_{\mathcal{S}, A}|_{U^{2+4}_{\theta}}
: U^{2+4}_{\theta} \to \widehat{\mathcal{M}}^{\leq L_{\max}}_{\theta}$ and
the compact subset (\ref{V^1 V^3 complement}).
Then we obtain an open subset $V^{2+4}_{\theta} \subset U^{2+4}_{\theta}$
such that if $p, q \in V^{2+4}_{\theta}$ satisfy
$\mathop{\mathfrak{forget}}\nolimits_{\mathcal{S}, A}(p) = \mathop{\mathfrak{forget}}\nolimits_{\mathcal{S}, A}(q)$ then $p \leq q$ or $q \leq p$,
and
\[
\mathop{\mathfrak{forget}}\nolimits_{\mathcal{S}, A}\bigl(\bigcup_l \mathcal{V}^1_{\theta, l} \cup V^{2+4}_{\theta}
\cup V^3_{\theta}\bigr) = \widehat{\mathcal{M}}_{\theta}.
\]
Take open subsets
$\mathcal{V}_{\theta}^{2+4} \Subset \mathcal{U}_{\theta}^{2+4} \subset V_{\theta}^{2+4}$
and $\mathcal{V}_{\theta}^3 \Subset \mathcal{U}_{\theta}^3 \subset V_{\theta}^3$ such that
\[
\mathop{\mathfrak{forget}}\nolimits_{\mathcal{S}, A}(\bigcup_l \mathcal{V}^1_{\theta, l} \cup \mathcal{V}^{2+4}_{\theta}
\cup \mathcal{V}^3_{\theta}) = \widehat{\mathcal{M}}_{\theta},
\]
and define
$\mathcal{V}_{\theta, l}^k = \mathcal{V}_{\theta}^k \cap \widehat{\mathcal{M}}_{\mathcal{S}, A, \theta, l}$
and $\mathcal{U}_{\theta, l}^k = \mathcal{U}_{\theta}^k \cap \widehat{\mathcal{M}}_{\mathcal{S}, A, \theta, l}$
for $k \in \{2+4, 3\}$ and $l$.
Then
\[
\mathcal{V}_{\theta, l} = \mathcal{V}_{\theta, l}^1 \cup \mathcal{V}_{\theta, l}^{2+4} \cup \mathcal{V}_{\theta, l}^3
\]
and
\[
\mathcal{U}_{\theta, l} = \mathcal{U}_{\theta, l}^1 \cup \mathcal{U}_{\theta, l}^{2+4} \cup \mathcal{U}_{\theta, l}^3
\]
satisfy Condition \ref{(S, A)-covering}, \ref{open sets and inclusions},
\ref{decomposition into parts U}, \ref{decomposition into parts V} and
\ref{existence of minimum}.
Condition \ref{existence of minimum} is due to the following properties of
$\mathcal{U}_{\theta, l}^k$ ($k \in \{1, 2+4, 3\}$).
\begin{itemize}
\item
Each $\mathcal{U}_{\theta, l}^k$ ($k \in \{1, 2+4, 3\}$) satisfies
Condition \ref{existence of minimum}.
\item
$\mathop{\mathfrak{forget}}\nolimits_{\mathcal{S}, A}(\bigcup_l \mathcal{U}_{\theta, l}^1)$ and
$\mathop{\mathfrak{forget}}\nolimits_{\mathcal{S}, A}(\bigcup_l \mathcal{U}_{\theta, l}^3)$ do not intersect.
\item
For any $p \in \mathcal{U}_{\theta, l}^{2+4}$ and $q \in \mathcal{U}_{\theta, l}^1 \cup \mathcal{U}_{\theta, l}^3$,
if $\mathop{\mathfrak{forget}}\nolimits_{\mathcal{S}, A}(p) = \mathop{\mathfrak{forget}}\nolimits_{\mathcal{S}, A}(q)$, then $p \geq q$.
\end{itemize}
Finally, we construct $\mathcal{U}_{\theta, l}^\mathrm{D} \subset
\overline{\mathcal{M}}^\mathrm{D}_{\mathcal{S}, A, \theta, l}$ as follows.
First for each point $p = (\Sigma, z, z^S, z^A, u) \in \mathcal{U}_{\theta, l}^{2+4}$,
there exists a decomposition $z^S = z^{S, 2} \sqcup z^{S, 4}$
such that $p_2 = (\Sigma, z, z^{S, 2}, z^A, u) \in U_{\theta}^2$ and
$p_4 = (\Sigma, z, z^{S, 4}, u) \in U_{\theta}^4$ by definition.
By the definition of $U_{\theta}^2$, there exist some $p_{2, i}
= (\Sigma, z, z^{S, 2, i}, z^{A, 2, i}, u) \in \mathcal{U}_{\theta, l(p_{2, i})}^2$
such that $\mathop{\mathfrak{forget}}\nolimits_{\mathcal{S}, A}(p_{2, i}) = \mathop{\mathfrak{forget}}\nolimits_{\mathcal{S}, A}(p_2)$,
$z^{S, 2} = \bigcup_i z^{S, 2, i}$ and $z^A = \bigcup_i z^{A, 2, i}$.
Condition \ref{Z^A section} for $\mathcal{U}_{\theta, l}^{2, \mathrm{D}}$ implies that
for each $i$, there exist a local universal family
$(\hat P^i \to \hat X^i, Z^i, (Z^i)^S)$ of $\mathop{\mathfrak{forget}}\nolimits_A(\mathop{\mathfrak{forget}}\nolimits_u(p_{2, i}))$ and
an $\mathop{\mathrm{Aut}}\nolimits(\mathop{\mathfrak{forget}}\nolimits_u(p_{2, i}))$-invariant family of smooth sections
$(Z^i)^A = ((Z^i)^A_j)$ of $\hat P^i \to \hat X^i$ such that
\[
\{(\hat P^i_a, Z^i(a), (Z^i)^S(a), (Z^i)^A(a)); a \in \hat X^i\} / \mathop{\mathrm{Aut}}\nolimits(\mathop{\mathfrak{forget}}\nolimits_u(p_{2, i}))
\]
is a neighborhood of $\mathop{\mathfrak{forget}}\nolimits_u(p_{2, i})$ in $\mathcal{U}_{\theta, l}^{2, \mathrm{D}}$.
Let $(\hat P^p \to \hat X^p, Z^p, (Z^p)^S)$ be a local universal family of
$\mathop{\mathfrak{forget}}\nolimits_A(\mathop{\mathfrak{forget}}\nolimits_u(p))$ and define an $\mathop{\mathrm{Aut}}\nolimits(\mathop{\mathfrak{forget}}\nolimits_u(p))$-invariant family of
sections $(Z^p)^A$ of $\hat P^p \to \hat X^p$ by the union of the pull backs of
$(Z^i)^A$ by the forgetful maps.
We define $\mathcal{W}_{\theta, l}^{2+4, \mathrm{D}} \subset
\overline{\mathcal{M}}^\mathrm{D}_{\mathcal{S}, A, \theta, l}$ by the union of
\[
\{(\hat P^p_a, Z^p(a), (Z^p)^S(a), (Z^p)^A(a)); a \in \hat X^p\} / \mathop{\mathrm{Aut}}\nolimits(\mathop{\mathfrak{forget}}\nolimits_u(p))
\]
over $p \in \mathcal{U}_{\theta, l}^{2+4}$.
Then $\mathcal{W}_{\theta, l}^{2+4, \mathrm{D}}$ satisfy Condition \ref{Z^A section} instead of
$\mathcal{U}_{\theta, l}^\mathrm{D}$.
We construct $\mathcal{U}_{\theta, l}^\mathrm{D} \subset
\overline{\mathcal{M}}^\mathrm{D}_{\mathcal{S}, A, \theta, l}$ as the union of
$\mathcal{U}_{\theta, l}^{k, \mathrm{D}}$ ($k \in \{1, 2+4, 3\}$).
We construct $\mathcal{U}_{\theta, l}^{2+4, \mathrm{D}}$ ($l \geq 0$) as open neighborhoods of
$\mathop{\mathfrak{forget}}\nolimits_{\mathcal{S}, A}(\mathcal{U}_{\theta, l}^{2+4})$ in $\mathcal{W}_{\theta, l}^{2+4, \mathrm{D}}$
which satisfy Condition \ref{stably unique forgetful map DM}$^{\mathrm{S}}$ and
which are D-neighborhoods of $\mathcal{U}_{\theta, l}^{2+4}$.
$\mathcal{U}_{\theta, l}^{3, \mathrm{D}}$ are also constructed as open neighborhoods of
$\mathop{\mathfrak{forget}}\nolimits_{\mathcal{S}}(\mathcal{U}_{\theta, l}^3)$ in $\overline{\mathcal{M}}^\mathrm{D}_{\mathcal{S}, \theta}$
which satisfy Condition \ref{stably unique forgetful map DM}$^{\mathrm{S}}$ and
which are D-neighborhoods of $\mathcal{U}_{\theta, l}^3$.
($\mathcal{U}_{\theta, l}^1$ ($l \geq 0$) have been already constructed.)
Since $\mathop{\mathfrak{forget}}\nolimits_{\mathcal{S}, A}(\mathcal{U}_{\theta, l}^k)$ ($k = 2+4, 3$) satisfy condition
\ref{stably unique forgetful map DM}$^{\mathrm{S}}$,
sufficiently small open neighborhoods satisfy the condition.
Furthermore, if these open neighborhoods are sufficiently small, then they also satisfy
the following condition:
For $\hat p = (\hat \Sigma, z, z^S, z^A) \in \mathcal{U}_{\theta, l(p)}^{2+4, \mathrm{D}}$
and any subsets $(z^i)^S \subset z^S$ $(i = 1,2$, $S \in \mathcal{S})$ and
$(z^i)^A \subset z^A$, if
each $\hat p^i = (\hat \Sigma, z, (z^i)^S, (z^i)^A)$ is contained in
$\mathcal{U}_{\theta, l(\hat p^i)}^{1, \mathrm{D}}$ or $\mathcal{U}_{\theta, l(\hat p^i)}^{3, \mathrm{D}}$, then
$(z^1)^S \subset (z^2)^S$ and $(z^1)^A \subset (z^2)^A$, or
$(z^2)^S \subset (z^1)^S$ and $(z^2)^A \subset (z^1)^A$.
Then $\mathcal{U}_{\theta, l}^\mathrm{D} = \bigcup_{k \in \{1, 2+4, 3\}} \mathcal{U}_{\theta, l}^{k, \mathrm{D}}$
($l \geq 0$) satisfy Condition \ref{stably unique forgetful map DM}.
It is easy to check that $\mathcal{U}_{\theta, l}^\mathrm{D}$ ($l \geq 0$) satisfy the other conditions.
Hence we can construct the required subsets $\mathcal{V}_{\theta, l}$, $\mathcal{U}_{\theta, l}$ and
$\mathcal{U}_{\theta, l}^\mathrm{D}$ by the induction.
\end{proof}
Assume that a domain curve representation
$(\mathcal{S}, \mathcal{V}_{\theta, l}, \mathcal{U}_{\theta, l}, \mathcal{U}^\mathrm{D}_{\theta, l})$ of
$\widehat{\mathcal{M}}^{\leq L_{\max}}_{\leq C}$ is given.
Next we construct spaces $\mathcal{X}_{\theta}$ of holomorphic buildings
with perturbation parameters.
At the same time, we construct spaces $\mathcal{V}_{\alpha, \theta, l}$,
$\mathcal{U}_{\alpha, \theta, l}$ of holomorphic buildings with perturbation parameters and
sets $\mathcal{U}_{\alpha, \theta, l}^\mathrm{D}$ of stable curves with perturbation parameters
indexed by a finite index set $\mathcal{A} = \{\alpha\}$.
We call a family $(\mathcal{X}_{\theta}, \mathcal{V}_{\alpha, \theta, l}, \mathcal{U}_{\alpha, \theta, l},
\mathcal{U}^\mathrm{D}_{\alpha, \theta, l})$ Kurainshi data if it satisfies the following conditions:
\begin{enumerate}[label=$(\arabic*)^{\mathrm{K}}$]
\item
We may regard $\mathcal{V}_{\alpha, \theta, l}$ and $\mathcal{U}_{\alpha, \theta, l}$ as subspaces of
$\widehat{\mathcal{M}}^{\leq L_{\max}}_{o, \mathcal{S}, A, \theta, l}$ by
$(p, E^0_p, \lambda_p) \mapsto p$ for each $\alpha$.
Similarly, we may regard $\mathcal{U}_{\alpha, \theta, l}^\mathrm{D}$ as a subspace of
$\overline{\mathcal{M}}^\mathrm{D}_{o, \mathcal{S}, A, \theta, l}$ for each $\alpha$.
\item
\label{D-neighborhood o}
There exists an open neighborhood $\mathcal{W}_{\alpha, \theta, l}
\subset \widehat{\mathcal{M}}_{o, \mathcal{S}, A, \theta, l}$ of the closure of
$\mathcal{U}_{\alpha, \theta, l}$
such that
\[
\mathcal{U}_{\alpha, \theta, l} = \{p \in \mathcal{W}_{\alpha, \theta, l};
\mathop{\mathfrak{forget}}\nolimits_u(p) \in \mathcal{U}_{\alpha, \theta, l}^\mathrm{D}\}
\]
as a space of holomorphic buildings.
Furthermore, for each $p \in \mathcal{U}_{\alpha, \theta, l}$,
the associated vector space $E^0_p$ and $\lambda_p$ are defined by
$E^0_p = E^0_{\mathop{\mathfrak{forget}}\nolimits_u(p)}$ and $\lambda_p = \lambda_{\mathop{\mathfrak{forget}}\nolimits_u(p)}$.
In this case, we say that $\mathcal{U}_{\alpha, \theta, l}^\mathrm{D}$ is
a D-neighborhood of $\mathcal{U}_{\alpha, \theta, l}$.
\item
\label{forget o relation}
$\mathop{\mathfrak{forget}}\nolimits_o(\mathcal{V}_{\alpha, \theta, l})$, $\mathop{\mathfrak{forget}}\nolimits_o(\mathcal{U}_{\alpha, \theta, l})$ and
$\mathop{\mathfrak{forget}}\nolimits_o(\mathcal{U}_{\alpha, \theta, l}^\mathrm{D})$ are contained in
$\mathcal{V}_{\theta, l}$, $\mathcal{U}_{\theta, l}$ and $\mathcal{U}_{\theta, l}^\mathrm{D}$ respectively.
\item
$\mathcal{V}_{\alpha, \theta, l}$ is open in the relative topology of $\mathcal{U}_{\alpha, \theta, l}$,
and $\mathcal{V}_{\alpha, \theta, l} \Subset \mathcal{U}_{\alpha, \theta, l}$.
\item
The number of marked points $z^o$ of each holomorphic building
in $\mathcal{U}_{\alpha, \theta, l}$ or stable curve in $\mathcal{U}^\mathrm{D}_{\alpha, \theta, l}$
depends only on $\alpha$.
\item
\label{Z^o section with lambda}
For each point $\hat p \in \mathcal{U}_{\alpha, \theta, l}^\mathrm{D}$,
let $(\hat P \to \hat X, Z, Z^S, Z^A)$ be the local representation of a
neighborhood of $\mathop{\mathfrak{forget}}\nolimits_o(\hat p)$ in $\mathcal{U}_{\theta, l}^\mathrm{D}$.
If we shrink $\hat X$ then there exists an $\mathop{\mathrm{Aut}}\nolimits(\hat p)$-invariant family of smooth
sections $Z^o = (Z^o_j)$ of $\hat P \to \hat X$ such that
\[
\{(\hat P_a, Z(a), Z^S(a), Z^A(a), Z^o(a)); a \in \hat X\} / \mathop{\mathrm{Aut}}\nolimits(\hat p)
\]
is a neighborhood of $\hat p$ in $\mathcal{U}_{\alpha, \theta, l}^{\mathrm{D}}$.
Furthermore,
there exists an $\mathop{\mathrm{Aut}}\nolimits(\hat p)$-equivariant linear map
$\tilde \lambda_{\hat p} : E^0_{\hat p} \to
C^\infty(\hat P \times Y, {\textstyle\bigwedge}^{0, 1} V^\ast \hat P
\otimes_\mathbb{C} (\mathbb{R} \partial_\sigma \oplus TY))$ which satisfy the following conditions:
\begin{itemize}
\item
For each $h \in E^o_p$, the projection of the support of $\tilde \lambda_{\hat p}(h)$ to
$\hat P$ does not intersect with the nodal points or marked points $Z$.
\item
For any $a \in \hat X$, $\hat q \in \mathcal{U}_{\alpha, \theta, l}^\mathrm{D}$ and
isomorphism $f : (\hat P_a, Z(a),\allowbreak Z^S(a),\allowbreak Z^A(a), Z^o(a)) \to \hat q$,
there exists an isomorphism $\hat \phi_f : E^0_{\hat p} \to E^0_{\hat q}$ such that
the restriction of $\tilde \lambda_{\hat p}$ to $\hat P_a \times Y$
coincides with $f^\ast \circ \lambda_{\hat q} \circ \hat \phi_f$.
\end{itemize}
We call $(\hat P \to \hat X, Z, Z^S, Z^A, Z^o, E^0_{\hat p}, \tilde \lambda_{\hat p})$
a local representation of a neighborhood of $\hat p$ in $\mathcal{U}_{\alpha, \theta, l}^\mathrm{D}$.
\item
Any $(\mathcal{S}, A)$-forgetful map from
$\hat p \in \mathcal{U}_{\alpha, \theta, l}^\mathrm{D}$ to
$\hat q \in \mathcal{U}_{\alpha, \theta, l'}^\mathrm{D}$
does not collapse any component of $\hat p$.
(This condition follows from Condition \ref{forget o relation} and
Condition \ref{no collapse for (S, A)-forgetful map} of domain curve representation.)
\item
For any $(\mathcal{S}, A)$-forgetful map $f$ from
$\hat p \in \mathcal{U}_{\alpha, \theta, l}^\mathrm{D}$ to $\hat q \in \mathcal{U}_{\alpha, \theta, l'}^\mathrm{D}$,
there exists an isomorphism $\hat \phi_f : E^0_{\hat p} \cong E^0_{\hat q}$ such that
$\lambda_{\hat p} = f^\ast \circ \lambda_{\hat q} \circ \hat \phi_f$.
Furthermore, for another $(\mathcal{S}, A)$-forgetful map $h$ from $\hat q$ to
$\hat r \in \mathcal{U}^\mathrm{D}_{\alpha, \theta, l''}$, $\hat \phi_{h \circ f}$ coincides with
the composition of $\hat \phi_{\hat h}$ and $\hat \phi_{\hat f}$.
\item
\label{(Z^o, E, lambda) pull back relation}
For any $l \geq l'$,
$\hat p \in \mathcal{U}_{\alpha, \theta, l}^\mathrm{D}$ and $\hat q \in \mathcal{U}_{\alpha, \theta, l'}^\mathrm{D}$,
if there exists an $(\mathcal{S}, A)$-forgetful map $f$ from $\hat p$ to $\hat q$,
then the following condition holds true:
Let $(\hat P \to \hat X, Z, Z^S, Z^A, Z^o, E^0_{\hat p}, \tilde \lambda_{\hat p})$ be
a local representation of a neighborhood of $\mathop{\mathfrak{forget}}\nolimits_A(\hat p)$
in $\mathcal{U}_{\alpha, \theta, l}^\mathrm{D}$, and
$(\hat P' \to \hat X', Z', (Z')^S, (Z')^A, (Z')^o, E^0_{\hat q}, \tilde \lambda_{\hat q})$ be
that for $\hat q$.
Shrink $\hat X$ and $\hat X'$ if necessary, and
let $(\phi, \hat \phi)$ be the unique forgetful map
from $(\hat P \to \hat X, Z, Z^S)$ to $(\hat P' \to \hat X', Z', (Z')^S)$
whose restriction to the central fiber coincides with $f$.
Then the pull back of $(Z')^o$ by $(\phi, \hat \phi)$ coincides with $Z^o$,
and $\tilde \lambda_{\hat p}$ coincides with
the pull back of $\tilde \lambda_{\hat q}$ by $(\phi, \hat \phi)$
under the identification $\hat \phi_f : E^0_{\hat p} \cong E^0_{\hat q}$.
\item
\label{decomposition into parts U DM o}
For any $(\hat p, E^0_{\hat p}, \lambda_{\hat p}) \in \mathcal{U}_{\alpha, \theta, l}^\mathrm{D}$
and any subset $\mathcal{N}$ of the nodal points of $\hat p$,
replace each nodal point in $\mathcal{N}$ with a pair of marked points
(we regard the new marked points as points in the set $z$), and
let $\hat p'_i$ $(1 \leq i \leq k)$ be its connected components or an arbitrary
decomposition into unions of its connected components.
Let $g'_i$ and $k'_i$ be the genus and the number of marked points $z$ of each
$\hat p'_i$ respectively.
Then there exist some $E_{\hat \omega}^i \geq 0$ such that
$E_{\hat \omega} = \sum_i E_{\hat \omega}^i$ and the following holds:
Only one of $\hat p'_i$ contains marked points $z^o$,
the support of $\lambda_{\hat p}(h)$ is contained in this component
for all $h \in E^0_{\hat p}$, and
$(\hat p'_i, E^0_{\hat p}, \lambda_{\hat p})$ is contained in
$\mathcal{U}_{\alpha, \theta'_i, l(\hat p'_i)}^\mathrm{D}$, where $\theta'_i = (g'_i, k'_i, E_{\hat \omega}^i)$.
Furthermore, the other $\hat p'_i$ are contained in
$\mathcal{U}_{\theta'_i, l(\hat p'_i)}^\mathrm{D}$.
\item
\label{decomposition into parts U o}
$\mathcal{U}_{\alpha, \theta, l}$ satisfy the following conditions
about decomposition into parts:
\begin{itemize}
\item
For any $p \in \mathcal{U}_{\alpha, \theta, l}$ and any decomposition $p_i$ ($1 \leq k$)
into unions of its connected components, let $p'_i$ be the holomorphic buildings
obtained by collapsing trivial floors (floors consisting of trivial cylinders).
Then only one of $p'_i$ contains marked points $z^o$, and it is contained in
$\mathcal{U}_{\alpha, \theta(p'_i), l(p'_i)}$.
Furthermore, the others are contained in $\mathcal{U}_{\theta(p'_i), l(p'_i)}$.
\item
For any $p \in \mathcal{U}_{\alpha, \theta, l}$ and any gap between floors, let $p_1$ and $p_2$
be the holomorphic buildings obtained by dividing $p$ at this gap.
Then one of $p'_i$ $(i = 1,2)$ is contained in $\mathcal{U}_{\alpha, \theta(p'_i), l(p'_i)}$ and
the other is contained in $\mathcal{U}_{\theta(p'_i), l(p'_i)}$.
\item
For any $p \in \mathcal{U}_{\alpha, \theta, l}$ and any subset of its nodal points,
the holomorphic building $p'$ obtained by replacing these nodal points to pairs of
marked points is contained in $\mathcal{U}_{\alpha, \theta(p'), l(p')}$.
\end{itemize}
\item
\label{decomposition into parts V o}
For each $p \in \widehat{\mathcal{M}}^{\leq L_{\max}}_{o, \mathcal{S}, A, \theta, l}$,
replace all nodal points and joint circles of $p$ to pairs of marked points and
pairs of limit circles respectively
(we regard the new marked points as points in the set $z$), and
let $p'_i$ $(1 \leq i \leq k)$ be the stabilizations of its non-trivial
connected components.
Then $p \in \mathcal{V}_{\alpha, \theta, l}$ if and only if one of $p'_i$ $(1 \leq i \leq k)$ is
contained in $\mathcal{V}_{\alpha, \theta(p'_i), l(p'_i)}$ and the others are contained in
$\mathcal{V}_{\theta(p'_i), l(p'_i)}$.
\item
\label{existence of minimum o}
For any $\alpha \in \mathcal{A}$, $p = (\Sigma, z, u) \in \widehat{\mathcal{M}}^{\leq L_{\max}}_{\theta}$
and subsets $(z^k)^S, (z^k)^A, z^o \subset \Sigma$ ($S \in \mathcal{S}, k =1,2)$,
if each $p^k = (\Sigma, z, (z^k)^S, (z^k)^A, z^o, u)$ is contained in
$\mathcal{U}_{\alpha, \theta, l(p^k)}$, then
$p^3 = (\Sigma, z, (z^1)^S \cap (z^2)^S, (z^1)^A \cap (z^2)^A, z^o, u)$ is contained
in $\mathcal{U}_{\alpha, \theta, l(p^3)}$.
\item
\label{stably unique forgetful map DM o}
For any $\hat p = (\hat \Sigma, z, z^S, z^A, z^o) \in \mathcal{U}_{\alpha, \theta, l}^\mathrm{D}$ and
any subsets $(z^1)^S, (z^2)^S \subset z^S$ ($S \in \mathcal{S}$) and
$(z^1)^A, (z^2)^A \subset z^A$, if
each $\hat p^i = (\hat \Sigma, z, (z^i)^S, (z^i)^A, z^o)$ is contained in
$\mathcal{U}_{\alpha, \theta, l(\hat p^i)}^\mathrm{D}$, then
$\hat p^3 = (\hat \Sigma, z, (z^1)^S \cap (z^2)^S, (z^1)^A \cap (z^2)^A, z^o)$
is also contained in $\mathcal{U}_{\alpha, \theta, l(\hat p^3)}^\mathrm{D}$.
\item
\label{z^o disjoint}
For any $p = (\Sigma, z, u) \in \widehat{\mathcal{M}}^{\leq L_{\max}}_{\theta}$
and subsets $(z^1)^S, (z^2)^S \subset \Sigma$ ($S \in \mathcal{S}$),
$(z^1)^A, (z^2)^A \subset \Sigma$
and $(z^1)^o, (z^2)^o \subset \Sigma$, if $(z^1)^o \cap (z^2)^o \neq \emptyset$ and
each $p^k = (\Sigma, z, (z^k)^S, (z^k)^A, (z^k)^o, u)$ is contained in
$\mathcal{U}_{\alpha^k, \theta, l(p^k)}$ for some $\alpha^k \in \mathcal{A}$, then
$\alpha^1 = \alpha^2$ and $(z^1)^o = (z^2)^o$.
\item
\label{def of X}
Each $\mathcal{X}_{\theta}$ is determined by $(\mathcal{V}_{\alpha, \theta, l})_{\alpha \in \mathcal{A}, l \geq 0}$
as a subset of $\widehat{\mathcal{M}}^{\leq L_{\max}}_{o, \mathcal{S}, A, \theta}$ as follows.
$p = (\Sigma, z, z^S, z^A, z^o, u) \in
\widehat{\mathcal{M}}^{\leq L_{\max}}_{o, \mathcal{S}, A, \theta}$ is contained in $\mathcal{X}_{\theta}$
if it satisfies the following conditions:
\begin{enumerate}[label=(\alph*)]
\item
$z^S$, $z^A$ and $z^o$ are $\mathop{\mathrm{Aut}}\nolimits(\mathop{\mathfrak{forget}}\nolimits_{o, \mathcal{S}, A}(p))$-invariant as sets.
Furthermore,
replace all nodal points and joint circles of $p$ to pairs of marked points and
pairs of limit circles respectively
(we regard the new marked points as points in the set $z$), and
let $p'_i$ $(1 \leq i \leq k)$ be the stabilizations of its non-trivial
connected components as in Condition \ref{decomposition into parts V o}.
Then $z^S$, $z^A$ and $z^o$ are $\mathop{\mathrm{Aut}}\nolimits((\mathop{\mathfrak{forget}}\nolimits_{o, \mathcal{S}, A}(p'_i))_i)$-invariant.
(The latter stronger condition is related to Remark \ref{quotient map is a submersion}
in Section \ref{Product of pre-Kuranishi spaces}.
We need this condition to construct a map corresponding to
the decomposition by gaps of floors,)
\item
\label{(o, S, A)-union}
There exist subsets $z^{S, i} \subset z^S$, $z^{A, i} \subset z^A$, $z^{o, i} \subset z^o$
and indices $\alpha_i \in \mathcal{A}$ $(i = 1, \dots, k)$ such that
$p_i = (\Sigma, z, z^{S, i}, z^{A, i}, z^{o, i}, u) \in \mathcal{V}_{\alpha_i, \theta, l(p_i)}$ for all $i$,
$z^S = \bigcup_i z^{S, i}$, $z^A = \bigcup_i z^{A, i}$ and $z^o = \bigcup_i z^{o, i}$.
\item
The linear map (\ref{X surjective map}) is surjective for the vector space
$E^0_p$ and linear map $\lambda_p$ defined in the next condition.
\end{enumerate}
\item
\label{def of (E, lambda) of X}
For each $p = (\Sigma, z, z^S, z^A, z^o, u) \in \mathcal{X}_{\theta}$,
the associated vector space $E^0_p$ and linear map $\lambda_p$ are defined as follows.
First note that in \ref{(o, S, A)-union} of the above condition,
if $z^{o, i} \cap z^{o, i'} \neq \emptyset$, then $\alpha_i = \alpha_{i'}$ and
$z^{o, i} = z^{o, i'}$ by Condition \ref{z^o disjoint}.
Choose a subset $I \subset \{1, \dots, k\}$ such that $z^o = \coprod_{i \in I} z^{o, i}$,
and fix forgetful maps from $\hat p = \mathop{\mathfrak{forget}}\nolimits_u(p)$ to $\hat p_i = \mathop{\mathfrak{forget}}\nolimits_u(p_i)$
for each $i \in I$.
Then $E^0_p$ is the direct sum of $E^0_{\hat p_i}$, and $\lambda_p$ is the sum of
the pull backs of $\lambda_{\hat p_i}$ by the forgetful map $\hat p \to \hat p_i$.
This definition is independent of the choice of $p_i$ and $I$ by
Condition \ref{(Z^o, E, lambda) pull back relation} and \ref{existence of minimum o}.
\item
\label{(o, S, A)-covering}
For each triple $\theta$, the subspace
$\mathcal{X}_{\theta} \subset \widehat{\mathcal{M}}^{\leq L_{\max}}_{o, \mathcal{S}, A, \theta}$
defined by Condition \ref{def of X} satisfies
$\mathop{\mathfrak{forget}}\nolimits_{o, \mathcal{S}, A}(\mathcal{X}_{\theta}) = \widehat{\mathcal{M}}^{\leq L_{\max}}_{\theta}$.
\end{enumerate}
We can prove the following lemmas similarly to Lemma \ref{shrinking and conditions}
and \ref{extension from the decomposable} respectively.
\begin{lem}
\label{shrinking and conditions o}
Let $(e^1_0, e^2_0)$ be an arbitrary constant, and assume that
Kurainshi data $(\mathcal{X}_{\theta}, \mathcal{V}_{\alpha, \theta, l}, \mathcal{U}_{\alpha, \theta, l},
\mathcal{U}^\mathrm{D}_{\alpha, \theta, l})$ for $\widehat{\mathcal{M}}_{\leq C}$ are given.
Then we can construct open subsets
\[
\mathcal{V}_{\alpha, e^1_0, e^2_0, l} \Subset \mathring{\mathcal{U}}_{\alpha, e^1_0, e^2_0, l} \Subset
\mathcal{U}_{\alpha, e^1_0, e^2_0, l} \quad
(l \geq 0)
\]
and
\[
\mathring{\mathcal{U}}_{\alpha, e^1_0, e^2_0, l}^\mathrm{D} \Subset \mathcal{U}_{\alpha, e^1_0, e^2_0, l}^\mathrm{D}
\quad (l \geq 0)
\]
such that if we replace $\mathcal{U}_{\alpha, e^1_0, e^2_0, l}$ and
$\mathcal{U}_{\alpha, e^1_0, e^2_0, l}^\mathrm{D}$ in the family
$(\mathcal{X}_{\theta}, \mathcal{V}_{\alpha, \theta, l},\allowbreak \mathcal{U}_{\alpha, \theta, l},\allowbreak
\mathcal{U}^\mathrm{D}_{\alpha, \theta, l})$ with $\mathring{\mathcal{U}}_{\alpha, e^1_0, e^2_0, l}$ and
$\mathring{\mathcal{U}}_{\alpha, e^1_0, e^2_0, l}^\mathrm{D}$ respectively,
it still satisfies the conditions of Kuranishi data.
\end{lem}
\begin{proof}
Since only nontrivial conditions are Condition \ref{existence of minimum o} and
\ref{stably unique forgetful map DM o}, we can prove the claim similarly to
Lemma \ref{shrinking and conditions}.
\end{proof}
\begin{lem}
\label{extension from the decomposable o}
Assume that Kurainshi data $(\mathcal{X}_{\theta}, \mathcal{V}_{\alpha, \theta, l},
\mathcal{U}_{\alpha, \theta, l}, \mathcal{U}^\mathrm{D}_{\alpha, \theta, l})$ of
$\widehat{\mathcal{M}}_{< C}$ are given.
We also assume that spaces
\[
\mathcal{V}_{\alpha, \theta, l}^\triangle \subset \mathcal{U}_{\alpha, \theta, l}^\triangle
\quad (\alpha \in \mathcal{A}, \widetilde{e}(\theta) = C, l \geq 0)
\]
of holomorphic buildings with perturbation parameters and spaces
\[
\mathcal{U}_{\alpha, \theta, l}^{\mathrm{D}, \triangle} \quad (\alpha \in \mathcal{A},
\widetilde{e}(\theta) = C, l \geq 0)
\]
of stable curves with perturbation parameters are given and
they satisfy the conditions of Kuranishi data if we replace
$\widehat{\mathcal{M}}$ and $\overline{\mathcal{M}}^\mathrm{D}$ with $\widehat{\mathcal{M}}^\triangle$ and
$\overline{\mathcal{M}}^{\mathrm{D}, \triangle}$ respectively.
Then we can construct spaces
\[
\mathcal{V}_{\alpha, \theta, l} \Subset \mathcal{U}_{\alpha, \theta, l}
\quad (\alpha \in \mathcal{A}, \widetilde{e}(\theta) = C, l \geq 0)
\]
of holomorphic buildings with perturbation parameters and spaces
\[
\mathcal{U}_{\alpha, \theta, l}^\mathrm{D}
\quad (\alpha \in \mathcal{A}, \widetilde{e}(\theta) = C, l \geq 0)
\]
of stable curves with perturbation parameters which satisfy
$\mathcal{V}_{\alpha, \theta, l} \cap \widehat{\mathcal{M}}_{o, \mathcal{S}, A}^\triangle
= \mathcal{V}_{\alpha, \theta, l}^\triangle$,
$\mathcal{U}_{\alpha, \theta, l} \cap \widehat{\mathcal{M}}_{o, \mathcal{S}, A}^\triangle \subset
\mathcal{U}_{\alpha, \theta, l}^\triangle$,
$\mathcal{U}_{\alpha, \theta, l}^\mathrm{D} \cap \overline{\mathcal{M}}_{o, \mathcal{S}, A}^{\mathrm{D}, \triangle}
\subset \mathcal{U}_{\alpha, \theta}^{\mathrm{D}, \triangle}$ and
the conditions of Kuranishi data other than Condition \ref{(o, S, A)-covering}.
\end{lem}
\begin{proof}
For each $\alpha \in \mathcal{A}$, we construct the extensions of $Z^o$
as in the case of $Z^A$ in Lemma \ref{extension from the decomposable}.
In this case, we also construct the extensions of $\lambda$
at the same time by the same induction.
Their construction is also similar.
\end{proof}
Now we explain the construction of Kurainshi data.
\begin{lem}\label{good family of additional vector spaces}
There exist Kurainshi data
$(\mathcal{X}_{\theta}, \mathcal{V}_{\alpha, \theta, l}, \mathcal{U}_{\alpha, \theta, l},
\mathcal{U}^\mathrm{D}_{\alpha, \theta, l})$ of $\widehat{\mathcal{M}}^{\leq L_{\max}}_{\leq C}$
for any domain curve representation
$(\mathcal{S}, \mathcal{V}_{\theta, l}, \mathcal{U}_{\theta, l}, \mathcal{U}^\mathrm{D}_{\theta, l})$.
\end{lem}
\begin{proof}
We construct Kurainshi data by the induction in $\widetilde{e}(\theta)$.
For each triple $\theta$ with minimal $\widetilde{e}(\theta)$,
we take finite open subsets $U_\alpha \subset \mathcal{U}_{\theta, l_\alpha}$ and
$U_\alpha^\mathrm{D} \subset \mathcal{U}_{\theta, l_\alpha}^\mathrm{D}$ ($\alpha \in \mathcal{A}$)
such that
\begin{itemize}
\item
each $U_\alpha^\mathrm{D}$ is a D-neighborhood of $U_\alpha$,
\item
$U_\alpha^\mathrm{D}$ is covered by a local representation
$(\hat P^\alpha \to \hat X^\alpha, Z^\alpha, (Z^\alpha)^S, (Z^\alpha)^A)$
of a neighborhood a point $\hat p^\alpha$ in $\mathcal{U}^\mathrm{D}_{\theta, l_\alpha}$
for some $l_\alpha$, and
\item
$\{\mathop{\mathfrak{forget}}\nolimits_{o, \mathcal{S}, A}(U_\alpha)\}_{\alpha \in \mathcal{A}}$
covers $\overline{\mathcal{M}}^{\leq L_{\max}}_{\theta}$.
\end{itemize}
For each $\alpha$, we construct an $\mathop{\mathrm{Aut}}\nolimits(\hat p^\alpha)$ vector space $E^0_\alpha$
and an $\mathop{\mathrm{Aut}}\nolimits(\hat p^\alpha)$-equivariant linear map
\[
\lambda_\alpha : E^0_\alpha \to C^\infty(\hat P^\alpha \times Y,
{\textstyle\bigwedge}^{0, 1} V^\ast P^\alpha \otimes_\mathbb{C} (\mathbb{R} \partial_\sigma \oplus TY))
\]
such that for any $p \in U_\alpha$, $E^0_p = E^0_\alpha$ and
the restriction of $\lambda_\alpha$ to the fiber isomorphic to $\mathop{\mathfrak{forget}}\nolimits_u(p)$ make
the linear map (\ref{X surjective map}) surjective.
We assume that for each $h \in E^o_\alpha$,
the projection of the support of $\tilde \lambda_\alpha(h)$
to $\hat P^\alpha$ does not intersect with the nodal points or marked points $Z$.
We also construct an $\mathop{\mathrm{Aut}}\nolimits(\hat p^\alpha)$-invariant family of section $(Z^\alpha)^o
= ((Z^\alpha)^o_j)$ of $\hat P^\alpha \to \hat X^\alpha$ for each $\alpha \in \mathcal{A}$.
Then we define $\mathcal{U}_{\alpha, \theta}^\mathrm{D} = \mathcal{U}_{\alpha, \theta, l_\alpha}^\mathrm{D}$
by the space of stable curves $\hat p \in \mathop{\mathfrak{forget}}\nolimits_o^{-1}(U^\mathrm{D}_\alpha)$ such that
$\hat p$ is isomorphic to $(\hat P^\alpha_a, Z^\alpha(a), (Z^\alpha)^S(a),
(Z^\alpha)^A(a), (Z^\alpha)^o(a))$
for the point $a \in \hat X^\alpha$ such that
$(\hat P^\alpha_a, Z^\alpha(a), (Z^\alpha)^S(a), (Z^\alpha)^A(a))$ is isomorphic
to $\mathop{\mathfrak{forget}}\nolimits_o(\hat p)$.
For such a stable curve $\hat p$, we define $E^0_{\hat p}$ and
$\lambda_{\hat p}$ by $E^0_{\hat p} = E^0_\alpha$ and the restriction of
$\lambda_\alpha$ respectively, and regard $\mathcal{U}^\mathrm{D}_{\alpha, \theta, l_\alpha}$
as a space of stable curve with perturbation parameters.
Similarly, we define the space of holomorphic buildings $\mathcal{U}_{\alpha, \theta, l_\alpha}$
by $\mathcal{U}_{\alpha, \theta, l_\alpha} = \mathop{\mathfrak{forget}}\nolimits_o^{-1}(U_\alpha) \cap
\mathop{\mathfrak{forget}}\nolimits_u^{-1}(\mathcal{U}_{\alpha, \theta, l_\alpha}^\mathrm{D})$,
and for each $p \in \mathcal{U}_{\alpha, \theta, l_\alpha}$,
For $l \neq l_\alpha$, we define $\mathcal{U}^\mathrm{D}_{\alpha, \theta, l} = \emptyset$ and
$\mathcal{U}_{\alpha, \theta, l} = \emptyset$.
It is clear that we can choose the family of sections $(Z^\alpha)^o$ so that
Condition \ref{z^o disjoint} holds.
We take open subsets $\mathcal{V}_{\alpha, \theta, l} \Subset \mathcal{U}_{\alpha, \theta, l}$
such that $\{\mathop{\mathfrak{forget}}\nolimits_{o, \mathcal{S}, A}(V_{\alpha, \theta, l_\alpha})\}_{\alpha \in \mathcal{A}}$
covers $\widehat{\mathcal{M}}^{\leq L_{\max}}_{\theta}$,
and define $\mathcal{X}_{\theta}$ by Condition \ref{def of X} and \ref{def of (E, lambda) of X}.
Next we consider the general triple $\theta$.
We assume that $\mathcal{V}_{\alpha, \theta', l}$, $\mathcal{U}_{\alpha, \theta', l}$,
$\mathcal{U}^\mathrm{D}_{\alpha, \theta', l}$ and $\mathcal{X}_{\theta'}$
for $\widetilde{e}(\theta') < \widetilde{e}(\theta)$ are already constructed and
construct those for $\theta$.
Define $\mathcal{U}^{\mathrm{D}, \triangle}_{\alpha, \theta, l} \subset
\overline{\mathcal{M}}^{\mathrm{D}, \triangle}_{o, \mathcal{S}, A, \theta, l}$ by the largest space
which satisfies Condition \ref{decomposition into parts U DM o},
$\mathcal{U}^\triangle_{\alpha, \theta, l} \subset
\widehat{\mathcal{M}}^{\leq L_{\max}, \triangle}_{o, \mathcal{S}, A, \theta, l}$
by the largest space which satisfies Condition \ref{decomposition into parts U o}, and
$\mathcal{V}^\triangle_{\alpha, \theta, l}$ by Condition \ref{decomposition into parts V o}.
Then they satisfy the assumption of Lemma \ref{extension from the decomposable o}.
Hence we obtain spaces
\[
\mathcal{V}_{\alpha, \theta, l} \Subset \mathcal{U}_{\alpha, \theta, l}
\quad (\alpha \in \mathcal{A}, l \geq 0)
\]
of holomorphic buildings with perturbation parameters and spaces
\[
\mathcal{U}_{\alpha, \theta, l}^\mathrm{D}
\quad (\alpha \in \mathcal{A}, l \geq 0)
\]
of stable curves with perturbation parameters which satisfy the conclusion of
Lemma \ref{extension from the decomposable o}.
Define $\mathcal{X}^1_{\theta}$ for these spaces
$(\mathcal{V}_{\alpha, \theta, l})_{\alpha \in \mathcal{A}, l \geq 0}$ by Condition \ref{def of X}
and \ref{def of (E, lambda) of X}.
Then its image by $\mathop{\mathfrak{forget}}\nolimits_{o, \mathcal{S}, A}$ contains a neighborhood of
$\widehat{\mathcal{M}}^{\leq L_{\max}, \triangle}_{\theta}$.
For the complement $\widehat{\mathcal{M}}^{\leq L_{\max}}_{\theta} \setminus
\mathop{\mathfrak{forget}}\nolimits_{o, \mathcal{S}, A}(\mathcal{X}^1_{\theta})$,
we use the same argument as in the case of minimal $\widetilde{e}(\theta)$.
Namely, we construct spaces $\mathcal{U}_{\alpha', \theta, l}$ of holomorphic buildings
with perturbation parameters and
spaces $\mathcal{U}^\mathrm{D}_{\alpha', \theta, l}$ of stable curves with perturbation parameters
indexed by another finite index set $\mathcal{A}' = \{\alpha'\}$ which satisfy the following
conditions:
\begin{itemize}
\item
Each $\mathcal{U}_{\alpha', \theta, l}^\mathrm{D}$ is a D-neighborhood of
$\mathcal{U}_{\alpha', \theta, l}$.
\item
For each $(p = (\Sigma, z, z^S, z^A, z^o, u), E^0_p, \lambda_p) \in
\mathcal{U}_{\alpha', \theta, l}$, $z^S$, $z^A$ and $z^o$ are $\mathop{\mathrm{Aut}}\nolimits(\Sigma, z, u)$-invariant.
\item
For any $(p, E^0_p, \lambda_p) \in \mathcal{U}_{\alpha', \theta, l}$,
the linear map (\ref{X surjective map}) is surjective.
\item
$\{\mathop{\mathfrak{forget}}\nolimits_{o, \mathcal{S}, A}(\mathcal{U}_{\alpha', \theta, l})\}_{\alpha' \in \mathcal{A}'}$ covers
$\widehat{\mathcal{M}}^{\leq L_{\max}}_{\theta} \setminus
\mathop{\mathfrak{forget}}\nolimits_{o, \mathcal{S}, A}(\mathcal{X}^1_{\theta})$.
\end{itemize}
Take open subsets $\mathcal{V}_{\alpha', \theta, l} \Subset \mathcal{U}_{\alpha', \theta, l}$
such that $\{\mathop{\mathfrak{forget}}\nolimits_{o, \mathcal{S}, A}(\mathcal{V}_{\alpha', \theta, l})\}_{\alpha' \in \mathcal{A}'}$
covers $\widehat{\mathcal{M}}^{\leq L_{\max}}_{\theta} \setminus
\mathop{\mathfrak{forget}}\nolimits_{o, \mathcal{S}, A}(\mathcal{X}^1_{\theta})$, and
define the space $\mathcal{X}_{\theta}$ of holomorphic buildings
with perturbation parameters for
$(\mathcal{V}_{\alpha, \theta, l})_{\alpha \in \mathcal{A} \cup \mathcal{A}', l\geq 0}$
by Condition \ref{def of X} and \ref{def of (E, lambda) of X}.
Then $(\mathcal{X}_{\theta}, (\mathcal{V}_{\alpha, \theta, l}, \mathcal{U}_{\alpha, \theta, l},
\mathcal{U}^\mathrm{D}_{\alpha, \theta, l})_{\alpha \in \mathcal{A} \cup \mathcal{A}'})$ is Kurainshi data
of $\widehat{\mathcal{M}}_{\leq C}$.
\end{proof}
For Kuranishi data $(\mathcal{X}_{\theta}, \mathcal{V}_{\alpha, \theta, l}, \mathcal{U}_{\alpha, \theta, l},
\mathcal{U}^\mathrm{D}_{\alpha, \theta, l})$ of $\widehat{\mathcal{M}}^{\leq L_{\max}}_{\leq C}$,
we define the pre-Kuranishi structure
\[
(\mathcal{X}_{\theta}, \mathop{\mathfrak{forget}}\nolimits_{o, \mathcal{S}, A}, (\mathcal{W}_x, \mathcal{E}_x, s_x, \widetilde{\psi}_x),
(\varphi_{x, y}, \hat \varphi_{x, y}))
\]
of each $\widehat{\mathcal{M}}^{\leq L_{\max}}_{\theta}$ as follows.
For each $p = (\Sigma, z, u) \in \widehat{\mathcal{M}}^{\leq L_{\max}}_{\theta}$ and
two points
\[
p^+_i = (\Sigma, z, (z^i)^S, (z^i)^A, (z^i)^o, u) \in
\mathcal{X}_{\theta} \cap \mathop{\mathfrak{forget}}\nolimits_{o, \mathcal{S}, A}^{-1}(p) \quad (i = 1, 2)
\]
in the same fiber, we define
$p^+_1 \vee p^+_2 \in \mathcal{X}_{\theta}$ by
\[
p^+_1 \vee p^+_2 = (\Sigma, z, (z^1)^S \cup (z^2)^S, (z^1)^A \cup (z^2)^A,
(z^1)^o \cup (z^2)^o, u).
\]
For each point $p = (\Sigma, z, z^S, z^A, z^o, u) \in \mathcal{X}_{\theta}$,
the Kuranishi neighborhood $(\mathcal{W}_p, \mathcal{E}_p, s_p, \widetilde{\psi}_p)$ of
$\mathop{\mathfrak{forget}}\nolimits_{o, \mathcal{S}, A}(p)$ is defined as follows.
By Condition \ref{def of X},
there exist subsets $z^{S, i} \subset z^S$, $z^{A, i} \subset z^A$, $z^{o, i} \subset z^o$
and indices $\alpha_i \in \mathcal{A}$ $(i = 1, \dots, k)$ such that
$p_i = (\Sigma, z, z^{S, i}, z^{A, i}, z^{o, i}, u) \in \mathcal{V}_{\alpha_i, \theta, l(p_i)}$ for all $i$,
$z^S = \bigcup_i z^{S, i}$, $z^A = \bigcup_i z^{A, i}$ and $z^o = \bigcup_i z^{o, i}$.
As in Condition \ref{def of (E, lambda) of X},
choose a subset $I \subset \{1, \dots, k\}$ such that $z^o = \coprod_{i \in I} z^{o, i}$,
and fix forgetful maps $f_i$ from $\hat p = \mathop{\mathfrak{forget}}\nolimits_u(p)$ to $\hat p_i = \mathop{\mathfrak{forget}}\nolimits_u(p_i)$
for each $i \in I$.
Let $(\hat P \to \hat X, Z, Z^S)$ be the local universal family of
$\mathop{\mathfrak{forget}}\nolimits_u(\mathop{\mathfrak{forget}}\nolimits_{o, A}(p))$.
We define an $\mathop{\mathrm{Aut}}\nolimits(p)$-equivariant linear map $\widetilde{\lambda}_p : E^0_p \to
C^\infty(\hat P \times Y; {\textstyle\bigwedge}^{0, 1} V^\ast \hat P \otimes_\mathbb{C}
(\mathbb{R} \partial_\sigma \oplus T Y))$ by the sum of the pull backs of $\lambda_{\hat p_i}$
by the forgetful maps from $(\hat P \to \hat X, Z, Z^S)$ to the local universal families
of $\hat p_i$ whose restrictions to the central fiber coincide with $f_i$.
Then the Kuranishi neighborhood $(\mathcal{W}_p, \mathcal{E}_p, s_p, \widetilde{\psi}_p)$ of
$\mathop{\mathfrak{forget}}\nolimits_{o, \mathcal{S}, A}(p)$ is constructed by the argument in
Section \ref{construction of nbds} using this
$\mathop{\mathrm{Aut}}\nolimits(p)$-equivariant linear map $\widetilde{\lambda}_p$.
For any $p = (\Sigma, z, u) \in \widehat{\mathcal{M}}^{\leq L_{\max}}_{\theta}$
and any two points
\[
p^+_i = (\Sigma, z, (z^i)^S, (z^i)^A, (z^i)^o, u) \in
\mathcal{X}_{\theta} \cap \mathop{\mathfrak{forget}}\nolimits_{o, \mathcal{S}, A}^{-1}(p),
\]
in the same fiber, $p^+_1 \leq p^+_2$ means that $(z^1)^S \subset (z^2)^S$,
$(z^1)^A \subset (z^2)^A$ and $(z^1)^o \subset (z^2)^o$,
which implies that $E^0_{p^+_1}$ is a subspace of $E^0_{p^+_2}$ and
the restriction of $\widetilde{\lambda}_{p^+_2}$ to $E^0_{p^+_1}$ is the pull back of
$\widetilde{\lambda}_{p^+_1}$ by the forgetful map.
Hence the embedding of the Kuranishi neighborhood
$(\mathcal{W}_{p^+_1}, \mathcal{E}_{p^+_1}, s_{p^+_1}, \widetilde{\psi}_{p^+_1})$ to
$(\mathcal{W}_{p^+_2}, \mathcal{E}_{p^+_2}, s_{p^+_2}, \widetilde{\psi}_{p^+_2})$ is defined by the argument
in Section \ref{embed}.
More generally, for any two points $x, y \in \mathcal{X}_{\theta}$,
if there exists some $r \in \psi_x(s_x^{-1}(0)) \cap \psi_y(s_y^{-1}(0))$ such that
$r_x \leq r_y$, where $r_x = \widetilde{\psi}_x^{-1}(\psi_x(r))$ and
$r_y = \widetilde{\psi}_y^{-1}(\psi_y(r))$,
then we can define the embedding of $(\mathcal{W}_x, \mathcal{E}_x, s_x, \widetilde{\psi}_x)$
to $(\mathcal{W}_y, \mathcal{E}_y, s_y, \widetilde{\psi}_y)$ by the argument in that section.
It is straightforward to check that each
\[
(\mathcal{X}_{\theta}, \mathop{\mathfrak{forget}}\nolimits_{o, \mathcal{S}, A}, (\mathcal{W}_x, \mathcal{E}_x, s_x, \widetilde{\psi}_x),
(\varphi_{x, y}, \hat \varphi_{x, y}))
\]
satisfies the other conditions of pre-Kuranishi structure.
Furthermore, they satisfy the compatibility conditions with respect to the fiber product
structure corresponding to the decomposition of holomorphic buildings into parts
and essential submersions corresponding to the decomposition
of holomorphic buildings into their connected components.
(We will consider these compatibility conditions in Section \ref{fiber prod} in details.)
\subsection{Decomposition by floor structure}
\label{floor decomposition}
A holomorphic building in the boundary $\partial \widehat{\mathcal{M}}$ is of height $k > 1$
and it can be decomposed into
the $[1, k_1]$-th floor part and the $[k_1+1, k]$-th floor part for each $1 \leq k_1 < k$.
In this section, we see the relation of the Kuranishi neighborhood of the whole
holomorphic building to those of these two parts.
First we define a space $\widehat{\mathcal{M}}^{\diamond 2}$ as follows.
Its point $((\Sigma^i, z^i, u^i)_{i=1,2}, M^{1,2})$ consists of
two holomorphic buildings $(\Sigma^i, z^i, u^i)$ ($i=1,2$) and
a set $M^{1,2} = \{(S^1_{+\infty_l}, S^1_{-\infty_l})\}$ of pairs of $+\infty$-limit circles
$S^1_{+\infty_l}$ of $(\Sigma^1, z^1, u^1)$ and $-\infty$-limit circles $S^1_{-\infty_l}$
of $(\Sigma^2, z^2, u^2)$ such that
the pairs in $M^{1,2}$ do not share the same limit circles.
Two points $((\Sigma^i, z^i, u^i)_{i=1,2}, M^{1,2})$ and
$(((\Sigma')^i, (z')^i, (u')^i)_{i=1,2}, \allowbreak (M')^{1,2})$ are the same point if there exist
isomorphisms $\varphi^i : \Sigma^i \cong (\Sigma')^i$ and $\mathbb{R}$-translations
$\theta^i$ such that $\varphi^i(z^i) = (z')^i$,
$u^i = (\theta^i \times 1) \circ (u')^i \circ \varphi^i$ and
$(\varphi_1, \varphi_2)$ maps $M^{1,2}$ to $(M')^{1,2}$.
The pre-Kuranishi structure of $\widehat{\mathcal{M}}^{\diamond 2}$ is induced by
that of $(\widehat{\mathcal{M}} \times \widehat{\mathcal{M}}) / \mathfrak{S}_2$
since the only local difference of them is
the automorphism group.
Let $\widehat{\mathcal{M}}^{\diamond 2}_{l_{i, -}, l_{1,2}, l_{i, +}} \subset
\widehat{\mathcal{M}}^{\diamond 2}$ be the subspace of points
$((\Sigma^i, z^i, u^i)_{i=1,2}, M^{1,2})$
such that the number of pairs in $M^{1,2}$ is $l_{1,2}$ and
the number of $\pm\infty$-limit circles of $(\Sigma^i, z^i, u^i)$ which do not appear
in $M^{1,2}$ is $l_{i, \pm}$.
Let
\[
\Psi_{1, 2} : \widehat{\mathcal{M}}^{\diamond 2}_{l_{i, -}, l_{1,2}, l_{i, +}}
\to (\overline{P} \times \overline{P})^{l_{1,2}} / \mathfrak{S}_{l_{1,2}}
\]
be the continuous map which maps each point $((\Sigma^i, z^i, u^i)_{i=1,2}, M^{1,2})$
to the point $(\pi_Y \circ u^1|_{S^1_{+\infty_l}}, \pi_Y \circ u^2|_{S^1_{-\infty_l}})$.
Let $\Delta_{\overline{P}} \subset \overline{P} \times \overline{P}$ be the diagonal.
Since $\Psi_{1,2}$ is realized as a strong smooth map,
$\Psi_{1,2}^{-1}(\Delta_{\overline{P}}^{l_{1,2}} / \mathfrak{S}_{l_{1,2}})$
has a pre-Kuranishi structure.
We study about the map from $\partial \widehat{\mathcal{M}}$ to
$\Psi_{1,2}^{-1}(\Delta_{\overline{P}}^{l_{1,2}} / \mathfrak{S}_{l_{1,2}})$
defined by the decomposition by a gap of floors.
Since the decomposition depends on the choice of the gap,
this map is multi-valued.
We study about the relation of the Kuranishi neighborhoods of a point
$(\Sigma, z, u) \in \partial \widehat{\mathcal{M}}$ and
that of one of its image $((\Sigma^i, z^i, u^i)_{i=1,2}, M^{1,2})$ by the multi-valued map
$\partial \widehat{\mathcal{M}} \to
\Psi_{1,2}^{-1}(\Delta_{\overline{P}}^{l_{1,2}} / \mathfrak{S}_{l_{1,2}})$.
Assume that each $(\Sigma^i, z^i, u^i)$ is of height $k_i$.
Let $(V^i, E^i, s^i, \psi^i, G^i)$ be the Kuranishi neighborhood of $(\Sigma^i, z^i, u^i)$
defined by the data $((z^i)^+, S^i, E_i^0, \lambda^i)$ and additional data
$((z^i)^{++}, (S^i)', \hat R^i_j)$ for each $i=1,2$.
We consider the Kuranishi neighborhood
$(V, E, s, \psi, G)$ of $(\Sigma, z, u)$ in $\partial \widehat{\mathcal{M}}$
defined by the data $((z^1)^+ \cup (z^2)^+, S^1 \cup S^2, E_1^0 \oplus E_2^0,
\lambda^1 \oplus \lambda^2)$
and the additional data $((z^1)^{++}, (S^1)' \cup (S^2)', (\hat R^1_j, \hat R^2_j))$.
Fix a coordinate of each joint circle between the $k_1$-th floor and $(k_1 + 1)$-th floor
of $(\Sigma, z, u)$.
These define the coordinates of limit circles of $(\Sigma^1, z^1, u^1)$
and $(\Sigma^2, z^2, u^2)$ which appears in $M^{1,2}$.
Since the curves in each $V^i$ are constructed by patching parts of the curve
$\Sigma^i$, we can define a smooth map
\begin{align*}
\Upsilon:
V^1 \times V^2 &\to \prod_{(S^1_{+\infty_l}, S^1_{-\infty_l}) \in M^{1,2}} (P \times P)\\
((a^1, b^1, u^1, h^1), (a^2, b^2, u^2, h^2)) &\mapsto (\pi_Y \circ u^1|_{S^1_{+\infty_l}},
\pi_Y \circ u^2 \circ \phi|_{S^1_{-\infty_l}})
\end{align*}
by using these coordinates.
Let $I_\epsilon \subset \mathbb{R}$ be a small neighborhood of $0 \in \mathbb{R}$ and
define $I_\epsilon \cdot \Delta_{P} = \{(\gamma, t \cdot \gamma) \in P \times P;
\gamma \in P, t \in I_\epsilon\}$.
For each point in $\Upsilon^{-1}(I_{\epsilon} \cdot \Delta_{P})$,
we can define a (perturbed) holomorphic building by jointing each pair of limit circles
in $M^{1,2}$ by using the coordinates twisted by some $t_l \in I_\epsilon$.
In particular, we can define a continuous map $\psi$ from the zero set of
$(s^1 \oplus s^2)|_{(\Upsilon^{-1}(I_{\epsilon} \cdot \Delta_{P})}$ to $\widehat{\mathcal{M}}$.
Then
$(\Upsilon^{-1}(I_{\epsilon} \cdot \Delta_{P}), E^1 \oplus E^2, s^1 \oplus s^2, \psi, G)$
is isomorphic to a part of the Kuranishi neighborhood
$(V, E, s, \psi, G)$ of $(\Sigma, z, u)$ in $\partial \widehat{\mathcal{M}}$.
($V$ is a boundary of a manifold with corners, and it is a union of manifolds.
More precisely, one of them corresponding to the decomposition at the gap
between the $k_1$-th floor and the $(k_1 + 1)$-th floor
is isomorphic to
$(\Upsilon^{-1}(I_{\epsilon} \cdot \Delta_{P}), E^1 \oplus E^2, s^1 \oplus s^2, \psi, G)$.)
Indeed, we can define a map
\begin{align*}
\Upsilon^{-1}(I_{\epsilon} \cdot \Delta_{P}) &\to V\\
((a^1, b^1, u^1, h^1), (a^2, b^2, u^2, h^2)) &\mapsto (a^0, b^0, u^0, h^0)
\end{align*}
by
$h^0 = (h^1, h^2) \in E^0_1 \oplus E^0_2$,
$a^0 = (a^1, a^2, (0, t_l)_l) \in \widetilde{X} = \widetilde{X}^1 \times \widetilde{X}^2
\times \widetilde{D}^{l_{1,2}}$
($\widetilde{D}^{l_{1,2}}$ is the parameter space for the deformation near
the joint circles between $k_1$-th floor and $(k_1+1)$-th floor),
$u^0 = u^1 \cup u^2$,
$b^0_\mu = b^1_\mu$ for $\mu \in \bigcup_{1 \leq j < k_1} M_j = M^1$,
$b^0_\mu = b^2_\mu$ for $\mu \in \bigcup_{k_1 < j < k_1 + k_2} M_j = M^2$ and
\begin{align*}
b^0_\mu
&= \lim_{s \to \infty} (\sigma \circ u^1|_{[0, \infty) \times S^1_{+\infty_l}}(s, t)
- (0_{k_{i_0}} + L_\mu s))\\
&\quad - \lim_{s \to -\infty} (\sigma \circ u^2|_{(-\infty, 0] \times S^1_{-\infty_l}}(s, t)
- (0_0 + L_\mu s))
\end{align*}
for $\mu = (S^1_{+\infty_l}, S^1_{-\infty_l}) \in M_{k_1} \cong M^{1, 2}$.
As we explained in the last of Section \ref{smoothness}, $b^0_\mu$
($\mu \in M_{k_1}$) are smooth function of $((a^1, b^1, u^1, h^1), (a^2, b^2, u^2, h^2))$.
Hence this map is a diffeomorphism and it defines an isomorphism of
$(\Upsilon^{-1}(I_{\epsilon} \cdot \Delta_{P}), E^1 \oplus E^2, s^1 \oplus s^2, \psi, G)$
and its image in $(V, E, s, \psi, G)$.
The above isomorphism implies that the Kuranishi neighborhood of each point
in $\partial \widehat{\mathcal{M}}$ and those of its image by the map
$\partial \widehat{\mathcal{M}} \to
\Psi_{1,2}^{-1}(\Delta_{\overline{P}}^{l_{1,2}} / \mathfrak{S}_{l_{1,2}})$ are the same
modulo automorphism group.
In particular, the map $\partial \widehat{\mathcal{M}} \to
\Psi_{1,2}^{-1}(\Delta_{\overline{P}}^{l_{1,2}} / \mathfrak{S}_{l_{1,2}})$
is a multi-valued partial submersion between pre-Kuranishi spaces.
\section{Fiber products}\label{fiber prod}
The pre-Kuranishi spaces considered in Section \ref{construction of Kuranishi}
are the spaces of holomorphic buildings without any conditions on periodic orbits
on limit circles $S^1_{\pm\infty_i}$.
For the construction of the algebra, we need to use the fiber products of such
Kuranishi spaces with $\overline{P}$ and $Y$.
More precisely, we use the fiber products of $\overline{\mathcal{M}}$ with the lifts of simplices in
$\overline{P}$ to $P$, and we need to perturb the section so that the induced
multisections on the fiber products are independent of the choice of these lifts.
To construct the virtual fundamental chain, we also need to define the orientations of
Kuranishi spaces.
We cannot define the orientations of $\widehat{\mathcal{M}}$ or $\overline{\mathcal{M}}$, but it is enough
to define the orientation of the fiber products we use.
The fiber products with simplices in $Y$ and the lifts of simplices in $\overline{P}$ to $P$
are orientable provided that interiors of these simplices in $\overline{P}$ do not contain
bad orbits.
In the general Bott-Morse case, it is not enough to count the intersection numbers
with simplices in $\overline{P}$, and we need to add correction terms,
which are equivalent to counting cascades in \cite{Bo02}.
This is because the chain which represents the diagonal in Poincar\'e duality is different
from the genuine diagonal in chain level.
These correction terms appear in every Bott-Morse theory if we construct
the algebra by the intersection numbers of the moduli spaces with simplices.
However, since algebraic structure of SFT is more complicated than that of usual Morse
theory, to define the correction terms, we need to solve some algebraic equations.
First we explain the bad orbits in Section \ref{bad orbits}, and
in Section \ref{fiber prod with simpleces}, we explain the fiber products
of $\overline{\mathcal{M}}$ which we use for the construction of the algebra.
In Section \ref{construction of a multisection}, we construct
a family of perturbed multisections of fiber products of
another space $\widehat{\mathcal{M}}^\diamond$
which satisfies appropriate compatibility conditions, and we use
the induced multisections for the fiber products of $\overline{\mathcal{M}}$.
Next in Section \ref{fiber prod and orientation}, we explain the orientations
of the fiber products.
In Section \ref{algebra for correction},
we define the correction terms, and finally in Section \ref{construction of algebra},
we recall the algebra of SFT and explain how to define the algebra by
the virtual fundamental chains of our fiber products.
\subsection{Bad orbits and local coefficients}\label{bad orbits}
Before considering the fiber products of the space of holomorphic buildings,
first we explain about bad orbits.
In Morse case, it is well known that bad orbits should not count as the generators of
the chain complex.
However, in our Bott-Morse case, bad orbits appear as a closed subset of $\overline{P}$.
Hence we need to explain how to treat these bad orbits.
First we define bad orbits.
It is related to orientations of the following $\overline{\partial}$-operators associated
to periodic orbits.
For each $\gamma \in P \subset C^\infty(S^1, Y)$,
fix one trivialization $\gamma^\ast T \hat Y \cong \mathbb{C}^n$.
Let
\begin{align*}
\mathring{D}^+_\gamma :&\ W_\delta^{1, p}((-\infty, 0] \times S^1 \cup D_\infty, \gamma^\ast T \hat Y \cup \mathbb{C}^n)\\
&\to L_\delta^p((-\infty, 0] \times S^1, \gamma^\ast T \hat Y) \oplus
L^p(D_\infty, {\textstyle\bigwedge}^{0, 1}T^\ast D_\infty \otimes \mathbb{C}^n)\\
D^+_\gamma :&\ \widetilde{W}_\delta^{1, p}((-\infty, 0] \times S^1 \cup D_\infty, \gamma^\ast T \hat Y \cup \mathbb{C}^n)\\
&\to L_\delta^p((-\infty, 0] \times S^1, \gamma^\ast T \hat Y) \oplus
L^p(D_\infty, {\textstyle\bigwedge}^{0, 1}T^\ast D_\infty \otimes \mathbb{C}^n)\\
\mathring{D}^-_\gamma :&\ W_\delta^{1, p}(D_0 \cup [0, \infty) \times S^1, \mathbb{C}^n \cup \gamma^\ast T \hat Y)\\
&\to L^p(D_0, {\textstyle\bigwedge}^{0, 1}T^\ast D_\infty \otimes \mathbb{C}^n) \oplus
L_\delta^p([0, \infty) \times S^1, \gamma^\ast T \hat Y)\\
D^-_\gamma :&\ \widetilde{W}_\delta^{1, p}(D_0 \cup [0, \infty) \times S^1, \mathbb{C}^n \cup \gamma^\ast T \hat Y)\\
&\to L^p(D_0, {\textstyle\bigwedge}^{0, 1}T^\ast D_\infty \otimes \mathbb{C}^n) \oplus
L_\delta^p([0, \infty) \times S^1, \gamma^\ast T \hat Y)
\end{align*}
be $\overline{\partial}$-type linear operators such that
\[
\mathring{D}^+_\gamma \xi = D^+_\gamma \xi = \partial_s \xi + J(\gamma)(\nabla_t \xi - L_\gamma \nabla_\xi R_\lambda(\gamma))
\]
on $(-\infty, 0] \times S^1$ and
\[
\mathring{D}^-_\gamma \xi = D^-_\gamma \xi = \partial_s \xi + J(\gamma)(\nabla_t \xi - L_\gamma \nabla_\xi R_\lambda(\gamma))
\]
on $[0, \infty) \times S^1$, where
$D_\infty = \{ z \in \mathbb{C} \cup \{\infty\}; |z| \geq 1\}$, $D_0 = \{ z \in \mathbb{C}; |z| \leq 1\}$,
and we identify $\{0\} \times S^1$ with $\partial D_\infty$ or $\partial D_0$
by $(0, t) \leftrightarrow e^{2\pi \sqrt{-1} t}$.
(The above $\widetilde{W}^{1, p}_\delta$ is defined by
$\widetilde{W}^{1, p}_\delta = W^{1, p}_\delta \oplus \mathop{\mathrm{Ker}}\nolimits A_\gamma$ as in Section
\ref{construction of Kuranishi})
Adding finite-dimensional complex vector spaces to the domain vector spaces
if necessary, we assume the above operators are surjective.
We consider the orientations of these types of operators, that is, the orientation of their kernels.
Since $\overline{\partial}$-type operators of each type are connected linearly
(that is, two operators $D$ and $D'$ can be connected by a family of operators
$t D + (1-t) D'$ ($t \in [0, 1]$)),
we can define a consistent orientation of these operators for each type.
Furthermore, changing the trivialization of $\gamma^\ast T \hat Y$ is equivalent to
gluing a $\overline{\partial}$-operator of a holomorphic bundle on $\mathbb{C} P^1$ to the operators.
Since a $\overline{\partial}$-operator of a holomorphic bundle has the complex orientation,
an orientation of one $\mathring{D}^+_\gamma$ defines the compatible orientations of
all operators of type $\mathring{D}^+_\gamma$ for each $\gamma \in P$.
Therefore, we can consider an orientation of $\mathring{D}^+_\gamma$ without fixing
particular trivialization of $\gamma^\ast T \hat Y$ or an additional complex vector space.
Let $\S^D$ be the local system of orientation of $\mathring{D}^+_\gamma$ on $P$,
and let $\S^{\lsuperscript{D}{t}}$ be the local system of orientation of
$\mathring{D}^-_\gamma$ on $P$.
We say $\gamma \in \overline{P}$ is a bad orbit if
$\S^D$ is not trivial on $\pi_P^{-1}(\gamma) \subset P$.
Let $\overline{P}^{\text{bad}} \subset \overline{P}$ be the subset of bad orbits.
Similarly, let $\overline{P}^{^t\text{bad}} \subset \overline{P}$ be the set of points
$\gamma \in \overline{P}$ such that $\S^{\lsuperscript{D}{t}}$ is not trivial on
$\pi_P^{-1}(\gamma)$.
By the assumption of $K$, $\overline{P}^{\text{bad}}$ and $\overline{P}^{^t\text{bad}}$ are
subcomplexes of $\overline{P}$.
\begin{rem}
Let $\gamma_0$ be a simple periodic orbit, and $\gamma = \gamma_0^{2^k m}$ be its
$2^k m$-multiple, where $m \geq 1$ is an odd integer.
Then $\gamma$ is a bad orbit if and only if $k \geq 1$ and
$\mathop{\mathrm{ind}}\nolimits \mathring{D}^+_{\gamma^2_0} - \mathop{\mathrm{ind}}\nolimits \mathring{D}^+_{\gamma_0}$ is odd.
Similarly, $\gamma$ belongs to $\overline{P}^{^t\text{bad}}$ if and only if $k \geq 1$ and
$\mathop{\mathrm{ind}}\nolimits D^+_{\gamma^2_0} - \mathop{\mathrm{ind}}\nolimits D^+_{\gamma_0}$ is odd.
Note that the index of the operator $\mathring{D}^+_\gamma$ is determined by the
Conley Zehnder index of $\gamma$ and $\dim T_\gamma P / TS^1$ as follows.
Fix one trivialization $\gamma^\ast \xi \cong \mathbb{C}^{n-1}$, which induces a trivialization
$\gamma^\ast T \hat Y \cong (\mathbb{R} \partial_\sigma \oplus \mathbb{R} R_\lambda) \oplus
\gamma^\ast \xi \cong \mathbb{C}^n$.
We define the Conley Zehnder index $\mathop{\mathrm{CZ\mathchar`-ind}}\nolimits \gamma$ of $\gamma$ by
the Conley Zehnder index of the path $\{\varphi^\lambda_t\}_{t \in [0, L_\gamma]}$
of symplectic matrices under the above trivialization of $\gamma^\ast \xi$.
(See \cite{RS93} for the definition of Conley Zehnder index of a path of symplectic
matrices.)
Then it is easy to see that
\[
\mathop{\mathrm{ind}}\nolimits \mathring{D}^+_\gamma = (n-1) - \mathop{\mathrm{CZ\mathchar`-ind}}\nolimits \gamma - \frac{1}{2} \dim T\gamma P/ TS^1.
\]
Similarly, the index of the operator $D^+_\gamma$ is
\[
\mathop{\mathrm{ind}}\nolimits D^+_\gamma = (n+1) - \mathop{\mathrm{CZ\mathchar`-ind}}\nolimits \gamma + \frac{1}{2} \dim T_\gamma P / TS^1.
\]
\end{rem}
\begin{eg}
We give an example where bad orbits appear as a subcomplex of $\overline{P}$.
This example was given by Bourgeois in \cite{Bo02}.
Let $K = \mathbb{R}^2 / G$ be a Kulein bottle, where $G$ is a group of diffeomorphisms of $\mathbb{R}^2$
generated by $(x, y) \mapsto (x + 1, 1 - y)$ and $(x, y) \mapsto (x, y + 1)$.
We equip $K$ with the flat metric $dx \otimes dx + dy \otimes dy$, and regard its unit
tangent bundle $S(TK) \cong S(T^\ast K)$ as a contact manifold by the Liouville form.
Then the Reeb flow is the geodesic flow.
$\overline{P}_2$ contains a component
\[
\{\gamma_{y}(t) = ((t, y), (1, 0)) : [0, 2]/ \{0, 2\} \to S(TK); y \in [0, 1/2]\},
\]
which is homeomorphic to the interval $[0, 1/2]$.
It contains two multiple orbits $\gamma_0$ and $\gamma_{1/2}$, and the others are
simple.
It is easy to check that the index of the operators $\mathring{D}^+$ for the two are even
and those for $\gamma_0|_{[0, 1]}$ and $\gamma_{1/2}|_{[0, 1]}$ are odd.
Hence these two orbits are bad orbits.
\end{eg}
Let $f : K \to \overline{P}$ be an ordered triangulation.
(``ordered'' means the set of the vertices has a total order.)
For each point $p \in K$, let $d$ be the multiplicity of the periodic orbit corresponding
to $p$.
Then we assume that there exists a regular $\mathbb{Z}/d$-complex $L$
(see \cite{Bre72} for regular complex),
an isomorphism $\varphi : L / (\mathbb{Z}/d) \cong \mathrm{St}(p, K)$ and
a smooth $\mathbb{Z}/d$-equivariant embedding $\tilde f : L \to P$ such that
$f \circ \varphi \circ \pi_L = \pi_P \circ \tilde f : L \to \overline{P}$,
where $\pi_L : L \to L / (\mathbb{Z}/d)$ is the quotient map.
(Note that locally $\pi_P : P \to \overline{P}$ can be written as
$S^1 \times_{\mathbb{Z} / d} W \to W / (\mathbb{Z}/d)$ for some $\mathbb{Z}/d$-manifold $W$.
Hence a $\mathbb{Z} / d$-equivariant triangulation $\check f : L \to W$ defines an embedding
$\tilde f : L \to S^1 \times_{\mathbb{Z}/d} W$ by $\tilde f(x) = [0, \check f (x)]$.)
Let $K^2 \to \overline{P} \times \overline{P}$ be an Euclidean cell decomposition
which is a refinement of $\{s \times t; s, t \in K\}$ and
which contains $\Delta_\ast K = \{\Delta_\ast s; s \in K\}$ and
$\rho_\ast K = \{\partial_{p+1} \dots \partial_n s \times \partial_0 \dots \partial_{p-1} s;
s \in K, 0 \leq p \leq n = \dim s\}$ as subcomplexes.
The chain complex $C_\ast(\overline{P} \times \overline{P})$ is defined by
using this Euclidean cell decomposition as a CW decomposition.
Let $K^0 = (x)$ be a finite sequence of smooth cycles in $Y$.
We denote their cohomologies by $\overline{K}^0 = (\overline{x})$
($\overline{x} \in H_\ast (Y, \mathbb{Q})$).
We will use the generators $c \theta^D_c$ of the relative chain complex
$C_\ast(\overline{P}, \overline{P}^{\text{bad}}; \S^D \otimes \mathbb{Q})$ of
ordered simplicial complex, or the generators $(c \theta^D_c)^\ast$ of the cochain complex
with compact support, where $\S^D$ is the induced local system on
$\overline{P} \setminus \overline{P}^{\text{bad}}$.
($(c \theta^D_c)^\ast$ is the cochain which takes one at $c \theta^D_c$ and
which vanishes at the other simplices.)
The $\mathbb{Z}/2$-degree of the above chain complex is defined by
\[
|c \theta^D_c| = \dim c + |\theta^D_c|,
\]
where $|\theta^D_c|$ is the index of the operator $\mathring{D}^+_\gamma$
($\gamma \in |c|$), and
its boundary operator is defined by
\[
\partial (c \theta^D_c) = (\partial c) \theta^D_c.
\]
Note that local system $\S^D$ is not well-defined on $\overline{P}^{\text{bad}}$,
but the above relative chain complex is well-defined.
We construct algebra by counting some intersection numbers with the moduli spaces
and simplices in $\overline{P}$.
Hence we need Poincar\'e duality.
In particular, we need a local system of the orientation of $\overline{P}$.
However, in general, the local orientation of $\overline{P}$ is not well defined.
We treat this as follows.
Let $\S^{\overline{P}}$ be the local system of the orientation of $TP / TS^1$ on $P$,
where $TS^1$ is the tangent of the $S^1$-action on $P$.
We say $\gamma \in \overline{P}$ is a non-orientable point
if $\S^{\overline{P}}$ is not trivial on $\pi_P^{-1}(\gamma)$.
Let $\overline{P}^{\text{no}} \subset \overline{P}$ be the set of non-orientable points.
It is also a subcomplex of $\overline{P}$.
Then $\S^{\overline{P}}$ induces a local system on
$\overline{P} \setminus \overline{P}^{\text{no}}$.
For each top-dimensional simplex $\zeta \in K$
(the top-dimension depends on the connected component of $\overline{P}$),
let $m_\zeta$ be the multiplicity of the periodic orbits in $\mathop{\mathrm{Int}}\nolimits \zeta$ (it is constant on
$\mathop{\mathrm{Int}}\nolimits \zeta$).
$m_\zeta$ depends only on the connected component of $P$ containing $|\zeta|$.
Let $\tilde \zeta \hookrightarrow P$ be a lift of $\zeta$.
Then the orientation of $TP/ TS^1$ defined by the orientation of $\tilde \zeta$
induces a section $\theta^{\overline{P}}_\zeta$ of $\S^{\overline{P}}$ on $\mathop{\mathrm{Int}}\nolimits \zeta$.
This section is independent of the choice of the lift $\tilde \zeta$.
We call a chain
\[
[\overline{P}] = \sum_\zeta \frac{1}{m_\zeta} \zeta \theta^{\overline{P}}_\zeta
\in C_{\dim P - 1}(\overline{P}, \overline{P}^{\text{no}}; \S^{\overline{P}} \otimes \mathbb{Q})
\]
the fundamental cycle of $\overline{P}$, where the sum is taken over all top-dimensional
simplices of $K$.
As usual, this is a cycle in the relative chain complex.
Before considering cap products with the fundamental chain,
we see the relation of the orientations of the operators $\mathring{D}^\pm_\gamma$,
$D^\pm_\gamma$ and that of the tangent space $T_\gamma P / TS^1$.
First recall that
\[
\mathop{\mathrm{Ker}}\nolimits A_\gamma \cong \mathbb{R} \oplus T_\gamma P
\cong (\mathbb{R} \oplus TS^1) \oplus T_\gamma P / TS^1.
\]
We denote the kernel of a surjective operator $D$ on a curve
(or the kernel of the surjective operator obtained by adding a finite-dimensional complex
vector space to the domain of a non-surjective operator $D$) by $[D]$.
The fiber product
\[
[D^-_\gamma] \underset{\mathop{\mathrm{Ker}}\nolimits A_\gamma}{\times} [D^+_\gamma]
= [D^-_\gamma] \underset{(\mathbb{R} \oplus TS^1) \oplus T_\gamma P / TS^1}{\times}
[D^+_\gamma]
\]
is equivalent to the kernel of a $\overline{\partial}$-operator on a complex
vector bundle over $\mathbb{C} P^1$ by gluing.
Hence it has the complex orientation.
The space $[\mathring{D}^-_\gamma] \oplus [\mathring{D}^+_\gamma]$ is a subspace
of the above fiber product,
and its quotient space is isomorphic to $(\mathbb{R} \oplus TS^1) \oplus T_\gamma P / TS^1$.
Therefore, if orientations of $[\mathring{D}^+_\gamma]$ and $T_\gamma P / TS^1$ are
given, we can define the orientation of $[\mathring{D}^-_\gamma]$ so that the orientation
of the above fiber product defined by
\[
[\mathring{D}^-_\gamma] \oplus (\mathbb{R} \oplus TS^1) \oplus T_\gamma P / TS^1 \oplus
[\mathring{D}^+_\gamma]
\]
coincides with the complex orientation.
To define Poincar\'e dual, first we recall the definition of cap product
without local coefficient. (Our definition is a bit different from the usual one.)
For a $p$-cochain $\alpha$ and a simplex $\zeta$ of dimension $n$,
our cap product $\zeta \cap \alpha$ is defined by
\[
\zeta \cap \alpha = \partial_{n- p + 1} \partial_{n - p + 2} \dots \partial_n \zeta \langle \partial_0 \partial_1 \dots \partial_{n - p-1} \zeta, \alpha \rangle.
\]
\begin{rem}
Under this definition, the following equation holds true.
For any $p$-cochain $\alpha$ and $n$-chain $\zeta$,
\[
\partial (\zeta \cap \alpha) = \partial \zeta \cap \alpha + (-1)^{n - p} \zeta \cap \delta \alpha
\]
\end{rem}
For each cochain $\alpha = (c \theta^D_c)^\ast \in
C^\ast (\overline{P}, \overline{P}^{\text{bad}}; \S^D \otimes \mathbb{Q})$,
we define the chain $[\overline{P}] \cap \alpha \in
C_\ast(\overline{P}, \overline{P}^{^t\text{bad}}; \S^{\lsuperscript{D}{t}} \otimes \mathbb{Q})$
as follows.
For each top-dimensional simplex $\zeta$ in $K$, let $\tilde \zeta \subset P$ be its lift.
If $c = \partial_0 \partial_1 \dots \partial_{n - p - 1} \zeta$ for some $p$,
then we can extend the orientation $\theta^D_c$ of $\S^D$ on
$\partial_0 \partial_1 \dots \partial_{n - p-1} \tilde \zeta$ to that on $\tilde \zeta$.
Then $\theta^D_c$ and $\theta^{\overline{P}}_{\zeta}$ define the orientation
$\theta^{\lsuperscript{D}{t}}_{c, \zeta}$ of $\S^{\lsuperscript{D}{t}}$ on $\tilde \zeta$
as above.
If $\partial_{n- p + 1} \partial_{n - p + 2} \dots \partial_n \zeta$ is not contained in
$\overline{P}^{^t\text{bad}}$, then $\theta^{\lsuperscript{D}{t}}_{c, \zeta}$ defines
the orientation of $\S^{\lsuperscript{D}{t}}$ on
$\partial_{n- p + 1} \partial_{n - p + 2} \dots \partial_n \zeta$.
We define $[\overline{P}] \cap \alpha$ by the linear combination of the cap products
\[
(\zeta \theta^{\overline{P}}_\zeta) \cap (c \theta^D_c)^\ast
= \theta^{\lsuperscript{D}{t}}_{c, \zeta} (\zeta \cap c^\ast) .
\]
We define the boundary operator of $C_\ast(\overline{P}, \overline{P}^{^t\text{bad}};
\S^{\lsuperscript{D}{t}} \otimes \mathbb{Q})$ by
\[
\partial (\theta_\eta^{\lsuperscript{D}{t}} \eta) = (-1)^{|\theta_\eta^{\lsuperscript{D}{t}}|}
\theta_\eta^{\lsuperscript{D}{t}} \partial \eta,
\]
where $|\theta_\eta^{\lsuperscript{D}{t}}|$ is the index of the operator
$\mathring{D}_\gamma^-$ ($\gamma \in |\eta|$).
Similarly, the boundary operator of $C_\ast
(\overline{P} \times \overline{P}, \overline{P}^{^t\text{bad}} \times \overline{P} \cup
\overline{P} \times \overline{P}^{\text{bad}};
p_1^\ast \S^{\lsuperscript{D}{t}} \otimes p_2^\ast \S^D \otimes \mathbb{Q})$
is defined by
\[
\partial (\theta_\eta^{\lsuperscript{D}{t}} \eta \theta_\eta^D)
= (-1)^{|\theta_\eta^{\lsuperscript{D}{t}}|} \theta_\eta^{\lsuperscript{D}{t}}
(\partial \eta) \theta_\eta^D.
\]
Let $\Delta : \overline{P} \to \overline{P} \times \overline{P}$ be the diagonal map.
We define a cycle $\Delta_\ast [\overline{P}]$ of
\[
C_{\dim P - 1}
(\overline{P} \times \overline{P}, \overline{P}^{^t\text{bad}} \times \overline{P} \cup
\overline{P} \times \overline{P}^{\text{bad}};
p_1^\ast \S^{\lsuperscript{D}{t}} \otimes p_2^\ast \S^D \otimes \mathbb{Q})
\]
by
\[
\Delta_\ast [\overline{P}] = \sum \frac{1}{m_\zeta} \theta^{\lsuperscript{D}{t}}_\zeta
(\Delta_\ast \zeta) \theta^D_\zeta,
\]
where the sum is taken over all top-dimensional simplices of $K$ not contained in
$\overline{P}^{\text{bad}}$,
$\theta^D_\zeta$ is an arbitrary fixed orientation of $p_2^\ast \S^D$ on
$\mathop{\mathrm{Int}}\nolimits \Delta_\ast \zeta$, and $\theta^{\lsuperscript{D}{t}}_\zeta$ is the orientation of
$p_1^\ast \S^{\lsuperscript{D}{t}}$ defined by $\theta^D_\zeta$ and
$\theta^{\overline{P}}_\zeta$
as above.
This definition is independent of the choice of $\theta^D_\zeta$.
For each simplex $\zeta \in K$ of dimension $n$, we define
a chain $\rho_\ast \zeta$ in $\overline{P} \times \overline{P}$ by
\[
\rho_\ast \zeta = \sum_{0 \leq p \leq n} \partial_{p+1} \dots \partial_n \zeta
\times \partial_0 \dots \partial_{p+1} \zeta.
\]
This corresponds to the image of $\Delta_\ast \zeta$ by Alexander Whitney map
$C_\ast(\overline{P} \times \overline{P})
\to C_\ast(\overline{P}) \otimes C_\ast(\overline{P})$.
(Recall that $K^2$ is not a simplicial somplex but a Euclidean cell complex
which contains $\partial_{p+1} \dots \partial_n \zeta
\times \partial_0 \dots \partial_{p+1} \zeta$.
For the transversality condition, it is convenient not to subdivide these products
because if we subdivide the complex, then we need to make the zero set of
perturbed mulitisection transverse to the new simplices of less dimension.)
We define a cycle $\rho_\ast [\overline{P}]
\in C_{\dim P -1} (\overline{P} \times \overline{P},
\overline{P}^{^t\text{bad}} \times \overline{P} \cup
\overline{P} \times \overline{P}^{\text{bad}};
p_1^\ast \S^{\lsuperscript{D}{t}} \otimes p_2^\ast \S^D \otimes \mathbb{Q})$
by
\[
\rho_\ast [\overline{P}] = \sum \frac{1}{m_\zeta} \theta^{\lsuperscript{D}{t}}_\zeta
(\rho_\ast \zeta) \theta^D_\zeta.
\]
For later use, we remark that $\rho_\ast [\overline{P}]$ can be written as
\[
\rho_\ast [\overline{P}] = \sum_c ([\overline{P}] \cap (c \theta^D_c)^\ast) \otimes
c \theta^D_c,
\]
where the sum is taken over all simplices $c$ in $K$ which are not contained in
$\overline{P}^{\text{bad}}$.
Let $\epsilon_\ast : C_\ast(\overline{P}) \to C_\ast(\overline{P} \times \overline{P})$
be the natural linear map such that
$\rho_\ast - \Delta_\ast = \partial \circ \epsilon_\ast + \epsilon_\ast \circ \partial$,
and define a chain
$\epsilon_{\overline{P}} \in C_{\dim P} (\overline{P} \times \overline{P},
\overline{P}^{^t\text{bad}} \times \overline{P} \cup
\overline{P} \times \overline{P}^{\text{bad}};
p_1^\ast \S^{\lsuperscript{D}{t}} \otimes p_2^\ast \S^D \otimes \mathbb{Q})$ by
$\epsilon_{\overline{P}} = \epsilon_\ast [\overline{P}]$.
Then it satisfies
\[
(\rho_\ast - \Delta_\ast) [\overline{P}] = \partial \epsilon_{\overline{P}}.
\]
This chain will be used for the definition of the correction terms.
We define
$\mathring{K}^2 \subset K^2$ by the minimal subcomplex which contains
$\Delta_\ast s$, $\rho_\ast s$ and $\epsilon_\ast s$ for all $s \in K$.
\subsection{Fiber products with simpleces}\label{fiber prod with simpleces}
First we define a Hausdorff space
$\overline{\mathcal{M}}^m_{((l_{i, j}), (l_{i, \pm}), (\mu_i))}$
for each family of non-negative integers
$((l_{i, j})_{1 \leq i < j \leq m}, (l_{i, \pm})_{1 \leq i \leq m}, (\mu_i)_{1 \leq i \leq m})$
as follows.
(We can equip it with a natural pre-Kuranishi structure, but it is not necessary.)
Its point $(\Sigma_i, z_i, u_i, \phi_i)_{1 \leq i \leq m}$ is a sequence of
holomorphic buildings $(\Sigma_i, z_i, u_i, \phi_i) \in \overline{\mathcal{M}}$ with the following
index sets of limit circles.
The index set of $+\infty$-limit circles of $\Sigma_i$ is
\[
\{+\infty^{i, +\infty}_l; 1 \leq l \leq l_{i, +}\} \sqcup
\coprod_{j = i+1}^m \{+\infty^{i, j}_l; 1 \leq l \leq l_{i, j}\}
\]
and the index set of $-\infty$-limit circles of $\Sigma_i$ is
\[
\{-\infty^{i, -\infty}_l; 1 \leq l \leq l_{i, -}\} \sqcup
\coprod_{j = 1}^{i-1} \{-\infty^{i, j}_l; 1 \leq l \leq l_{j, i}\}.
\]
$\overline{\mathcal{M}}^m_{((l_{i, j}), (l_{i, \pm}), (\mu_i))}$
is locally isomorphic to the product $\prod^m \overline{\mathcal{M}}$.
(An isomorphism is determined if we fix a family of bijections between the above
index sets of limit circles and the usual index sets $\{\pm\infty^i_l\}$.)
Note that we respect the indices of limit circles.
Hence even if we change the indices $+\infty^{i, j}_l$ to $+\infty^{i, j}_{g \cdot l}$ and
$-\infty^{j, i}_l$ to $-\infty^{j, i}_{g \cdot l}$ for the same $g \in \mathfrak{S}_{l_{i, j}}$,
we distinguish the obtained curve from the original one
(unless $g$ can be extended to the automorphisms of $(\Sigma_i, z_i, u_i, \phi_i)$).
We define the genus of a point $(\Sigma_i, z_i, u_i, \phi_i)_{1 \leq i \leq m} \in
\overline{\mathcal{M}}^m_{((l_{i, j}), (l_{i, \pm}), (\mu_i))}$ by
\[
g = 1 + \sum_{i = 1}^m (g_i - 1) + \sum_{1 \leq i < j \leq m} l_{i, j},
\]
where $g_i$ is the genus of $\Sigma_i$.
(This is the genus of the curve obtained by gluing joint circles $S^1_{+\infty^{i, j}_l}$ and
$S^1_{-\infty^{j, i}_l}$ for all pairs $(+\infty^{i, j}_l, -\infty^{j, i}_l)$.)
Note that there exists a natural continuous map
\begin{align*}
&\overline{\mathcal{M}}^m_{((l_{i, j}), (l_{i, \pm}), (\mu_i))}
\to \prod_{1 \leq i < j \leq m} (P \times P)^{l_{i, j}}
\times \prod_{1 \leq i \leq m} P^{l_{i, -}}
\times \prod_{1 \leq i \leq m} Y^{\mu_i}
\times \prod_{1 \leq i \leq m} P^{l_{i, +}}\\
&(\Sigma_i, z_i, u_i, \phi_i)_i \mapsto ((\pi_Y \circ u_i \circ \phi_{+\infty^{i, j}_l},
\pi_Y \circ u_i \circ \phi_{-\infty^{j, i}_l}),\\
&\hphantom{(\Sigma_i, z_i, u_i, \phi_i)_i \mapsto (}
\pi_Y \circ u_i \circ \phi_{+\infty^{i, +\infty}_l},
\pi_Y \circ u_i(z_{i, l}), \pi_Y \circ u_i \circ \phi_{-\infty^{i, -\infty}_l})
\end{align*}
We consider the fiber products with respect to this continuous map.
We consider the following family of sequences
$((\hat \epsilon^{i, j}_l), (\hat c^i_l), (x^i_l), (\hat \eta^i_l))$
of simplices with local coefficients.
$(\hat \epsilon^{i, j}_l = \theta^{\lsuperscript{D}{t}}_{\epsilon^{i, j}_l} \epsilon^{i, j}_l
\theta^D_{\epsilon^{i, j}_l})_{1 \leq i < j \leq m, 1 \leq l \leq l_{i, j}}$
is a sequence of products of
\begin{itemize}
\item
cells $\epsilon^{i, j}_l$ in $\mathring{K}^2$ which are not contained in
$\overline{P}^{^t\text{bad}} \times \overline{P} \cup
\overline{P} \times \overline{P}^{\text{bad}}$, and
\item
orientations $\theta^{\lsuperscript{D}{t}}_{\epsilon^{i, j}_l}$ of
$p_1^\ast \S^{\lsuperscript{D}{t}}$
and $\theta^D_{\epsilon^{i, j}_l}$ of $p_2^\ast \S^D$ on $\mathop{\mathrm{Int}}\nolimits \epsilon^{i, j}_l$.
\end{itemize}
Take a lift $\tilde \epsilon^{i, j}_l \hookrightarrow P \times P$ for each $\epsilon^{i, j}_l$, and
define $\breve \epsilon^{i, j}_l = \theta^{\lsuperscript{D}{t}}_{\epsilon^{i, j}_l}
\tilde \epsilon^{i, j}_l \theta^D_{\epsilon^{i, j}_l}$.
$(\hat c^i_l = c^i_l \theta^D_{c^i_l})_{1 \leq l \leq l_{i, -}}$ ($1 \leq i \leq m$)
is a sequence of products of
\begin{itemize}
\item
simplices $c^i_l$ in $K$ which are not contained in $\overline{P}^{\text{bad}}$, and
\item
orientations $\theta^D_{c^i_l}$ of $\S^D$ on $\mathop{\mathrm{Int}}\nolimits c^i_l$.
\end{itemize}
For each $c^i_l$, we take its lift $\tilde c^i_l \hookrightarrow P$ and
define $\breve c^i_l = \tilde c^i_l \theta^D_{c^i_l}$.
$(x^i_1, x^i_2, \dots, x^i_{\mu_i})_{i = 1, 2, \dots, m}$
is a sequence of cycles in $K^0$.
$(\hat \eta^i_l = \theta^{\lsuperscript{D}{t}}_{\eta^i_l} \eta^i_l)_{1 \leq l \leq l_{i, +}}$
($1 \leq i \leq m$)
is a sequence of products of
\begin{itemize}
\item
simplices $\eta^i_l$ in $K$ which are not contained in $\overline{P}^{^t\text{bad}}$, and
\item
orientations $\theta_{\eta^i_l}^{\lsuperscript{D}{t}}$ of $\S^{\lsuperscript{D}{t}}$
on $\mathop{\mathrm{Int}}\nolimits \eta^i_l$.
\end{itemize}
For each $\hat \eta^i_l$, we take its lift $\tilde \eta^i_l \hookrightarrow P$ and
define $\breve \eta^i_l = \theta^{\lsuperscript{D}{t}}_{\eta^i_l} \tilde \eta^i_l$.
Then for such a family of sequences $((\breve \epsilon^{i, j}_l), (\breve c^i_l),
(x^i_l), (\breve \eta^i_l))$, we define a closed subspace
\begin{align*}
\overline{\mathcal{M}}^m_{((\breve \epsilon^{i, j}_l), (\breve c^i_l),
(x^i_l), (\breve \eta^i_l))}
&\subset \overline{\mathcal{M}}^m_{((l_{i, j}), (l_{i, \pm}), (\mu_i))}
\end{align*}
as the fiber product with
\[
\prod \tilde \epsilon^{i, j}_l \times \prod \tilde c^i_l \times \prod x^i_l \times
\prod \tilde \eta^i_l
\subset \prod (P \times P)^{l_{i, j}}
\times \prod P^{l_{i, -}}
\times \prod Y^{\mu_i}
\times \prod P^{l_{i, +}}.
\]
The pre-Kuranishi structure of the above fiber product is defined as follows.
For a point $(\Sigma_i, z_i, u_i, \phi_i)_{1 \leq i \leq m}
\in \overline{\mathcal{M}}^m_{((\breve \epsilon^{i, j}_l),
(\breve c^i_l), (x^i_l), (\breve \eta^i_l))}$,
let $(V^i, E^i, s^i, \psi^i)$ be the Kuranishi neighborhood of each
$(\Sigma_i, z_i, u_i) \in \widehat{\mathcal{M}}$.
(This is not a Kuranishi neighborhood of $\overline{\mathcal{M}}$ but of $\widehat{\mathcal{M}}$.)
Since the limit circles of each curve in $V^i$ are identified with the limit circles of
$\Sigma_i$ by construction, it is meaningful to say that a coordinate of a limit circle
of a curve corresponding to the point $(a'_i, b'_i, u'_i, h'_i)$ in $V^i$ is close to that of
$(\Sigma_i, z_i, u_i, \phi_i)$.
It is clear that if there exists a family of
coordinates $(\phi'_i)_{+\infty^{i, +\infty}_l}$ close to
$(\phi_i)_{+\infty^{i, +\infty}_l}$ such that
$\pi_Y \circ u'_i \circ (\phi'_i)_{+\infty^{i, +\infty}_l} \in |\tilde \eta^i_l|$ ($\subset P$),
then such a family is unique.
(Furthermore, if the restriction of $\pi_Y \circ u'_i$ to the $+\infty$-limit circle
corresponding to $\eta^i_l$ is contained in $|\eta^i_l|$ ($\subset \overline{P}$),
then there exists a coordinate $(\phi'_i)_{+\infty^{i, +\infty}_l}$ close to $(\phi_i)_{+\infty^{i, +\infty}_l}$
or its rotation by some element of $\mathbb{Z}/d \subset S^1$ such that
$\pi_Y \circ u'_l \circ (\phi'_i)_{+\infty^{i, +\infty}_l} \in |\tilde \eta^i_l|$,
where $d$ is the multiplicity of $\gamma^{i, +}_l$.)
The same is true for the coordinates of the limit circles corresponding to
$c^i_l$ or $\epsilon^{i, j}_l$.
Let
\[
(V^1 \times V^2 \times \dots \times V^m)_{((\breve \epsilon^{i, j}_l), (\breve c^i_l),
(x^i_j), (\breve \eta^i_l))}
\subset V^1 \times V^2 \times \dots \times V^m
\]
be the submanifold consisting of the families of curves which have families of
coordinates of
their limit circles close to that of $(\Sigma_i, z_i, u_i, \phi_i)_{1 \leq i \leq m}$ such that
the periodic orbits on the $\pm\infty$-limit circles are contained in the corresponding
$\tilde \epsilon^{i, j}_l$, $\tilde c^i_j$ and $\tilde \eta^i_l$,
and $\pi_y \circ u'_i$ takes a value in $x^i_l$ at each marked point $z_{i, l}$.
This submanifold can be regarded as a fiber product of
$V^1 \times V^2 \times \dots \times V^m$ with
the product of $(I_\delta \times I_\delta) \cdot \tilde \epsilon^{i, j}_l$,
$I_\delta \cdot \tilde c^i_l$, $x^i_l$ and $I_\delta \cdot \tilde \eta^i_l$,
where $I_\delta \subset S^1$ is a small neighborhood of $0 \in S^1$.
Then a Kuranishi neighborhood of $(\Sigma_i, z_i, u_i, \phi_i)_{1 \leq i \leq m}$ is defined by
this submanifold, the restrictions of the product vector bundle
$E = E^1 \times E^2 \times \dots \times E^m$, its section
$s = s^1 \times s^2 \times \dots \times s^m$,
a finite group $G = \prod_i \mathop{\mathrm{Aut}}\nolimits(\Sigma_i, z_i, u_i, \phi_i)$, and
the map
\[
\psi : s^{-1}(0)/G \to \overline{\mathcal{M}}^m
_{((\breve \epsilon^{i, j}_l), (\breve c^i_l), (x^i_l), (\breve \eta^i_l))}
\]
induced by the product $\psi^1 \times \psi^2 \times \dots \times \psi^m$ and
the coordinates of the limit circles
defined by the above argument.
Note that the pre-Kuranishi spaces for other lifts of $c^i_l$, $\eta^i_l$ or $\epsilon^{i, j}_l$
are naturally isomorphic to the above pre-Kuranishi space.
We need to construct their perturbed multisections which are independent of the choice
of the lifts.
We construct the perturbed multisections of the above fiber products
as pull backs by submersions to the fiber products of $\widehat{\mathcal{M}}$
in the next section.
Since these submersions forget the coordinates of limit circles,
the pull backs will be independent of the choice of the lifts of simpleces.
\subsection{Construction of a family of multisections}
\label{construction of a multisection}
In this section, we define fiber products of $\widehat{\mathcal{M}}$ and
construct their grouped multisections under appropriate compatibility conditions.
Recall that in Section \ref{global construction},
we construct a pre-Kuranishi structure of
$\widehat{\mathcal{M}}^{\leq L_{\max}}_{\leq C}$ only for fixed constants
$L_{\max} > 0$ and $C >0$.
We cannot treat the whole noncompact space $\widehat{\mathcal{M}}$ as a pre-Kuranishi
space.
Hence we need to read $\widehat{\mathcal{M}}$ below as its compact subset
$\widehat{\mathcal{M}}^{\leq L_{\max}}_{\leq C}$ for some $L_{\max} > 0$ and $C >0$.
We also read the other spaces as their similar compact subsets.
First we define the space $\widehat{\mathcal{M}}^\diamond$.
Its point $((\Sigma^\alpha, z^\alpha, u^\alpha)_{\alpha \in A}, M^{\mathrm{rel}})$
consists of finite number of connected holomorphic buildings
$(\Sigma^\alpha, z^\alpha, u^\alpha)$ and
a set $M^{\mathrm{rel}} = \{(S^1_{+\infty_l}, S^1_{-\infty_l})\}$ of pairs of
their $+\infty$-limit circle $S^1_{+\infty_l}$ and $-\infty$-limit circle $S^1_{-\infty_l}$
which satisfies the following conditions:
\begin{itemize}
\item
Any two pairs in $M^{\mathrm{rel}}$ do not share the same limit circles.
\item
Let $M^{\alpha, \alpha'} \subset M^{\mathrm{rel}}$ be the subset of pairs
$(S^1_{+\infty_l}, S^1_{-\infty_l})$ such that $S^1_{+\infty_l}$ is a $+\infty$-limit circle of
$\Sigma^\alpha$ and $S^1_{-\infty_l}$ is a $-\infty$-limit circle of $\Sigma^{\alpha'}$.
Then there does not exist a sequence
$\alpha_0, \alpha_1, \dots, \alpha_k = \alpha_0 \in A$
such that $M^{\alpha_i, \alpha_{i+1}} \neq \emptyset$ for all $i$.
\end{itemize}
(We emphasize again the meaning of connectedness of a holomorphic building.
The domain curve $\Sigma^\alpha$ of each holomorphic building
$(\Sigma^\alpha, z^\alpha, u^\alpha)$ is a compact space which contains
joint circles and limit circles.
The connectedness of $(\Sigma^\alpha, z^\alpha, u^\alpha)$ means that of
$\Sigma^\alpha$, and it does not imply that $(\Sigma^\alpha, z^\alpha, u^\alpha)$
is of height-one or that its restriction to each floor is connected.
We also note that the height of $(\Sigma^\alpha, z^\alpha, u^\alpha)$
may depend on $\alpha$, and that the total
$((\Sigma^\alpha, z^\alpha, u^\alpha)_{\alpha \in A}, M^{\mathrm{rel}})$
does not have a floor structure.)
Two points $((\Sigma^\alpha, z^\alpha, u^\alpha)_{\alpha \in A}, M^{\mathrm{rel}})$ and
$(((\Sigma')^{\alpha'}, (z')^{\alpha'}, (u')^{\alpha'})_{\alpha' \in A'}, (M')^{\mathrm{rel}})$ are
the same point if there exist a bijection $\nu : A \to A'$,
isomorphisms $\varphi^\alpha : \Sigma^\alpha \to (\Sigma')^{\nu(\alpha)}$,
and $\mathbb{R}$-translations $\theta^\alpha$ such that
$\varphi^\alpha(z^\alpha) = (z')^{\nu(\alpha)}$,
$u^\alpha = (\theta^\alpha \times 1) \circ (u')^{\nu(\alpha)} \circ \varphi^\alpha$
and the family of isomorphisms $\varphi^\alpha$ maps $M^{\mathrm{rel}}$
to $(M')^{\mathrm{rel}}$.
Forgetting $M^{\mathrm{rel}}$ defines a forgetful map from
$\widehat{\mathcal{M}}^\diamond$ to $\bigcup_N \prod^N (\widehat{\mathcal{M}}^0) / \mathfrak{S}_N$,
where $\widehat{\mathcal{M}}^0 \subset \widehat{\mathcal{M}}$ is the space of connected holomorphic
buildings.
Since the only local difference of these two spaces are automorphism group,
$\widehat{\mathcal{M}}^\diamond$ has the natural pre-Kuranishi structure which makes this
forgetful map a submersion.
For subsets $A_1, A_2 \subset A$, we define
$M^{A_1, A_2} = \bigcup_{\alpha_1 \in A_1, \alpha_2 \in A_2} M^{\alpha_1, \alpha_2}$.
We say a point $((\Sigma^\alpha, z, u^\alpha)_{\alpha \in A}, M^{\mathrm{rel}})
\in \widehat{\mathcal{M}}^\diamond$ is disconnected if
there exists a decomposition $A = A_1 \sqcup A_2$ such that
$M^{A_1, A_2} = M^{A_2, A_1} = \emptyset$.
Otherwise we say it is connected.
We denote the space of connected points of $\widehat{\mathcal{M}}^\diamond$ by
$(\widehat{\mathcal{M}}^\diamond)^0$.
Decomposition into connected components defines a map $\widehat{\mathcal{M}}^\diamond
\to \bigcup_N (\prod^N (\widehat{\mathcal{M}}^\diamond)^0) / \mathfrak{S}_N$.
Let $\Upsilon : \widehat{\mathcal{M}}^\diamond \to
(\prod (\overline{P} \times \overline{P})) / \mathfrak{S} \times
\prod \overline{P} / \mathfrak{S} \times \prod Y / \mathfrak{S}$
be the continuous map which maps each point
$((\Sigma^\alpha, z^\alpha, u^\alpha)_{\alpha \in A}, M^{\mathrm{rel}})
\in \widehat{\mathcal{M}}^\diamond$
to
\begin{align*}
((\pi_Y \circ u|_{S^1_{+\infty_l}}, \pi_Y \circ u|_{S^1_{-\infty_l}})
_{(S^1_{+\infty_l}, S^1_{-\infty_l}) \in M^{\mathrm{rel}}},
(\pi_Y \circ u|_{S^1_{\pm\infty}})_{S^1_{\pm\infty} \notin M^{\mathrm{rel}}}, \quad \\
(\pi_Y \circ u(\bigcup_\alpha z^\alpha))),
\end{align*}
where we denote the union of $u^\alpha$ by $u$, and
$S^1_{\pm\infty} \notin M^{\mathrm{rel}}$ means that the limit circle $S^1_{\pm\infty}$
is not contained in any pairs in $M^{\mathrm{rel}}$.
It is realized as a strong smooth map.
(The number of the product of the target space depends on the components of
$\widehat{\mathcal{M}}^\diamond$.)
Define the fiber product $(\widehat{\mathcal{M}}^\diamond, \mathring{K}^2, K,K^0) \subset
\widehat{\mathcal{M}}^\diamond$ by $(\widehat{\mathcal{M}}^\diamond, \mathring{K}^2, K,K^0)
= \Upsilon^{-1}(\prod \mathring{K}^2 / \mathfrak{S} \times \prod K / \mathfrak{S}
\times \prod K^0 / \mathfrak{S})$.
(Although $\prod K / \mathfrak{S}$ coincides with
the entire space $\prod P / \mathfrak{S}$,
the fiber product with $\prod K / \mathfrak{S}$ has a meaning.
It implies that we require the stronger transversality condition
for its grouped multisection. See Remark \ref{meaning of triangulation}.
Fiber product with $\prod K^0 / \mathfrak{S} \subset \prod Y / \mathfrak{S}$
is slightly an abuse of notation.
A simplicial complex is a set of simplices, but $K^0$ is a set of smooth cycles, and
we do not assume that they are embedded in $Y$.
The meaning is also that we require that the perturbed multisection
is transverse to the zero section even if we restrict it to the fiber product with
the cycles in $K^0$.)
Define a multi-valued partial submersion $\Xi : \widehat{\mathcal{M}}^\diamond \to
\widehat{\mathcal{M}}^\diamond$ by
\[
\Xi(((\Sigma^\alpha, z^\alpha, u^\alpha)_{\alpha \in A}, M^{\mathrm{rel}}))
= \{((\Sigma^\alpha, z^\alpha, u^\alpha)_{\alpha \in A},
\mathring{M}^{\mathrm{rel}}); \mathring{M}^{\mathrm{rel}} \subsetneq M^{\mathrm{rel}}\}.
\]
(The maps between their Kuranishi neighborhoods are also similarly defined.)
Let $\Xi^\circ$ be the restriction of
$\Xi \subset \widehat{\mathcal{M}}^\diamond \times \widehat{\mathcal{M}}^\diamond$ to
the set of points
\[
(((\Sigma^\alpha, z^\alpha, u^\alpha)_{\alpha \in A}, M^{\mathrm{rel}}),
((\Sigma^\alpha, z^\alpha, u^\alpha)_{\alpha \in A}, \mathring{M}^{\mathrm{rel}}))
\]
such that $((\Sigma^\alpha, z^\alpha, u^\alpha)_{\alpha \in A}, M^{\mathrm{rel}})
\in (\widehat{\mathcal{M}}^\diamond, \mathring{K}^2, K,K^0)$ and
$(\pi_Y \circ u|_{S^1_{+\infty_l}}, \pi_Y \circ u|_{S^1_{-\infty_l}})
\in \rho_\ast K$
for all $(S^1_{+\infty_l}, S^1_{-\infty_l}) \in M^{\mathrm{rel}} \setminus
\mathring{M}^{\mathrm{rel}}$.
Then $\Xi^\circ$ is a multi-valued partial essential submersion
from $(\widehat{\mathcal{M}}^\diamond, \mathring{K}^2, K,K^0)$ to itself.
\begin{rem}
\label{simple restriction of Xi is not essentially submersive}
Note that the restriction of $\Xi$ to $(\widehat{\mathcal{M}}^\diamond, \mathring{K}^2, K,K^0)$
is not a multi-valued partial essential submersion from
$(\widehat{\mathcal{M}}^\diamond, \mathring{K}^2, K,K^0)$ to itself.
This is because $\mathring{K}^2$ is finer than the product $K \times K$
on the outside of $\rho_\ast K$.
This is an important point in Bott-Morse case.
To explain this point, we consider the following easy example.
Let $s_i : \Delta^n \to \mathbb{R}^l$ ($i = 1,2$) be sections of the
trivial vector bundle on the $n$-simplex $\Delta^n$ whose restriction to each of
it faces is transverse to
the zero section.
These transversality conditions of $s_i$ do not imply that
the restriction of $(s_1 + s_2) : \Delta^n \times \Delta^n \to
\mathbb{R}^l \oplus \mathbb{R}^l$ to the diagonal is transverse to the zero section.
However, the restrictions to the cells in
$\rho_\ast \Delta^n
= \{\partial_{p+1} \dots \partial_n \Delta^n \times \partial_0 \dots \partial_{p-1} \Delta^n;
0 \leq p \leq n\}$
are transverse to the zero section.
Hence we use the induced section $s_1 + s_2$ only on $\rho_\ast \Delta^n$.
\end{rem}
We also define a multi-valued partial essential submersion
\[
\Lambda : (\partial \widehat{\mathcal{M}}^\diamond, \mathring{K}^2, K,K^0)
\to (\widehat{\mathcal{M}}^\diamond, \mathring{K}^2, K,K^0)
\]
as follows.
For each point
$((\Sigma^\alpha, z^\alpha, u^\alpha)_{\alpha \in A}, M^{\mathrm{rel}}) \in
(\partial \widehat{\mathcal{M}}^\diamond, \mathring{K}^2, K,K^0)$
and each $\alpha \in A$, let $k_\alpha \geq 1$ be the height of
$(\Sigma^\alpha, z^\alpha, u^\alpha)$.
Let $C = \{(\alpha_i, k_i)\}$ be an arbitrary non-empty set of pairs
$(\alpha_i, k_i) \in A \times \mathbb{N}$ such that $1 \leq k_i < k_{\alpha_i}$.
For all pairs $(\alpha_i, k_i) \in C$,
we replace all joint circles in the gap between the $k_i$-th floor and
the $(k_i + 1)$-th floor of $(\Sigma^{\alpha_i}, z^{\alpha_i}, u^{\alpha_i})$
with pairs of limit circles,
and let $(\Sigma^{\alpha'}, z^{\alpha'}, u^{\alpha'})_{\alpha' \in A'}$ be the
stabilization of the connected components of the new holomorphic buildings.
(Stabilization means we collapse all trivial floors of each connected component.)
$\Lambda : (\partial \widehat{\mathcal{M}}^\diamond, \mathring{K}^2, K,K^0)
\to (\widehat{\mathcal{M}}^\diamond, \mathring{K}^2, K,K^0)$ is the multi-valued partial
submersion which maps each point
$((\Sigma^\alpha, z^\alpha, u^\alpha)_{\alpha \in A}, M^{\mathrm{rel}}) \in
(\partial \widehat{\mathcal{M}}^\diamond, \mathring{K}^2, K,K^0)$
to the above points $(\Sigma^{\alpha'}, z^{\alpha'}, u^{\alpha'})_{\alpha' \in A'}$
for all $C = \{(\alpha_i, k_i)\}$.
(The maps between their Kuranishi neighborhoods are also similarly defined.)
For each point $p = ((\Sigma^\alpha, z^\alpha, u^\alpha)_{\alpha \in A},
M^{\mathrm{rel}}) \in (\widehat{\mathcal{M}}^\diamond, \mathring{K}^2, K,K^0)$,
we define
$\widetilde{e}(p) = \widetilde{e}_{\delta_0}(p)
= \sum_\alpha \widetilde{e}_{\delta_0}(\theta_\alpha)
+ \frac{1}{2} \# M^{\mathrm{rel}}$, where each
$\theta_\alpha$ is the type of $(\Sigma^\alpha, z^\alpha, u^\alpha)$,
and $\# M^{\mathrm{rel}}$ is the number of pairs.
($\# M^{\mathrm{rel}}$ is not the number of limit circles contained in the pairs in
$M^{\mathrm{rel}}$.
Recall that $\widetilde{e}_{\delta_0}(\theta) = 5(g-1) + 2k + E_{\hat \omega} / \delta_0$
for $\theta = (g, k, E_{\hat \omega})$, where
$g$ is the genus, $k$ is the total number of marked points and limit circles,
and $E_{\hat \omega}$ is the $E_{\hat \omega}$-energy.
See Section \ref{global construction} for the definition
and properties of
$\widetilde{e}_{\delta_0}(\theta_\alpha)$.)
Note that the maps $\Xi^\circ$ and $\Lambda$ decrease $\widetilde{e}$.
Hence if we decompose $(\widehat{\mathcal{M}}^\diamond, \mathring{K}^2, K,K^0)$ by
$\widetilde{e}$, then
$\Xi^\circ$ and $\Lambda$ constitute a compatible system of
multi-valued partial submersions.
(See Section \ref{multi-valued partial submersions} for the definition of
a compatible system of multi-valued partial submersions.)
We can construct the perturbed multisections of
$(\widehat{\mathcal{M}}^\diamond, \mathring{K}^2, K,K^0)$
which satisfy the following conditions:
\begin{itemize}
\item
Let $((\widehat{\mathcal{M}}^\diamond)^0, \mathring{K}^2, K, K^0) \subset
(\widehat{\mathcal{M}}^\diamond, \mathring{K}^2, K,K^0)$ be the subset of connected points.
Its grouped multisection induces that of
\[
\bigcup_N (\prod^N ((\widehat{\mathcal{M}}^\diamond)^0, \mathring{K}^2, K, K^0)
/ \mathfrak{S}_N.
\]
Then the grouped multisection of $(\widehat{\mathcal{M}}^\diamond, \mathring{K}^2, K,K^0)$
coincides with its pull back by the submersion
\[
(\widehat{\mathcal{M}}^\diamond, \mathring{K}^2, K,K^0) \to
\bigcup_N (\prod^N ((\widehat{\mathcal{M}}^\diamond)^0, \mathring{K}^2, K, K^0)
/ \mathfrak{S}_N
\]
defined by decomposition into connected components.
\item
The grouped multisections of
$(\widehat{\mathcal{M}}^\diamond, \mathring{K}^2, K,K^0)$
are compatible with respect to the compatible system of
multi-valued partial essential submersions defined by $\Xi^\circ$ and $\Lambda$.
\end{itemize}
Let
$\overline{\mathcal{M}}^m_{((\breve \epsilon^{i, j}_l), (\breve c^i_l), (x^i_l), (\breve \eta^i_l))}
\to (\widehat{\mathcal{M}}^\diamond, \mathring{K}^2, K,K^0)$
be the natural map defined by
decomposing each holomorphic building $(\Sigma_i, z_i, u_i, \phi_i)$
into its connected components (and stabilizing them),
forgetting the order of the sequence of holomorphic buildings,
the coordinates of limit circles and the order of marked points
and limit circles.
$M^{\mathrm{rel}}$ is defined by the set of pairs of limit circles
corresponding to the pairs $(S^1_{+\infty^{i,j}_l}, S^1_{-\infty^{j,i}_l})$.
We define the perturbed multisection of each
$\overline{\mathcal{M}}^m_{((\breve \epsilon^{i, j}_l), (\breve c^i_l), (x^i_l), (\breve \eta^i_l))}$
by the pull back by this natural submersion.
We emphasize the following point.
Although in Section \ref{fiber prod with simpleces},
we only consider the fiber products with simpleces or cells not contained
in $\overline{P}^{\text{bad}}$, $\overline{P}^{^t \text{bad}}$ or
$\overline{P}^{^t\text{bad}} \times \overline{P} \cup
\overline{P} \times \overline{P}^{\text{bad}}$, in this section,
we construct the perturbed multisections of the fiber products
of $\widehat{\mathcal{M}}^\diamond$ with all simpleces or cells.
We cannot ignore the bad orbits for the construction of the compatible family of
perturbed multisections, but for the construction of the algebra, we only use
the orientable fiber products.
\begin{rem}
In the above construction of grouped multisection, we do not
consider any compatibility condition with respect to
the decompositions at nodal points.
This is just because we do not use this kind of compatibility condition
for the construction of the algebras in this paper.
We can construct a grouped multisection with these compatibility conditions
if we have an appropriate formulation of invariants or algebras
which respect the decompositions at nodal points.
To regard the nodal point as a fiber product over $Y$ by the
maps of evaluation at the marked points, we need to use the
grouped multisection obtained as the fiber product.
As we saw in Remark \ref{simple restriction of Xi is not essentially submersive},
the fiber product of grouped multisections do not satisfy the
transversality condition in general.
There are two ways to solve this problem.
One is the use of continuous family of grouped multisections.
See Section \ref{continuous family of multisections} for details.
(See also Section \ref{fiber product and orientation for homotopy}.
We use this technique for the the case of a $1$-parameter family of
symplectic manifolds with cylindrical ends.)
The other is to use the same argument as the decomposition
by the gaps of floors as above.
In this case, we use the approximation of the diagonal,
and we need to use the correction terms for the difference of
this approximation and the genuine diagonal similar to
those we construct in Section \ref{algebra for correction}.
\end{rem}
\subsection{The orientations of fiber products}\label{fiber prod and orientation}
In this section, we define the orientations of the fiber products
$\overline{\mathcal{M}}^m_{((\breve \epsilon^{i, j}_l), (\breve c^i_l), (x^i_l), (\breve \eta^i_l))}$.
For calculation of orientations, it is convenient to treat these pre-Kuranishi spaces
as fiber products not with $I_\delta \cdot \tilde c^i_l$ but with ``manifold''
$(\mathbb{R} \times S^1) \cdot \tilde c^i_l \times \theta^D_{c^i_l}$.
First we define the orientation of the parameter space
$\mathring{X} \subset \widetilde{X} \times \prod \mathbb{R}_\mu$
used for the construction of the Kuranishi neighborhoods.
For each $i = 1, 2, \dots, k-1$,
we fix one joint circle $S^1_{\mu_i}$ between the $i$-th floor and the $(i+1)$-th
floor.
First we consider the orientation at a point
$(a, b) \in \mathring{X}$ such that $\rho_\mu \neq 0$ for all joint circles $S^1_\mu$.
In this case, we can use $(b_{\mu_i})_i$ and a chart of $\widetilde{X}$ as a chart of $\mathring{X}$
on a neighborhood of this point.
We define the orientation of $\mathring{X}$ by this chart
\[
(b_{\mu_1}, b_{\mu_2}, \dots, b_{\mu_{k-1}}, a)
\in \mathbb{R} \times \mathbb{R} \times \dots \times \mathbb{R} \times \widetilde{X},
\]
where the orientation of each $\mathbb{R}$ is the positive orientation, and the orientation of
$\widetilde{X}$ is the complex orientation
(the orientation induced by the complex orientation of the blow down space).
At a general point $(a, b) \in \mathring{X}$, the orientation of $\mathring{X}$ is defined
as follows.
We can write a point of $\widetilde{X}$ as $a = ((\rho_\mu, \varphi_\mu)_\mu, a^0)$,
where each $(\rho_\mu, \varphi_\mu) \in [0, 1) \times S^1$
is the parameter of the deformation of
the neighborhood of the joint circle $S^1_\mu$, and $a^0$ is the other parameters.
Note that the parameter $a^0$ is a complex parameter.
Then we can use
\[
(-\rho_{\mu_1}, -\rho_{\mu_2}, \dots, -\rho_{\mu_{k-1}},
(b_\mu, \varphi_\mu)_{\mu}, a^0)
\]
as a chart of $\mathring{X}$.
(More precisely, each $\rho_\mu$ should be replaced with its appropriate multiple.
See Section \ref{smoothness}.)
The orientation of $\mathring{X}$ is defined by this chart.
(The order of $\mu$ of $(b_\mu, \varphi_\mu)_{\mu}$ is independent of the orientation
since each $(b_\mu, \varphi_\mu)$ is even dimensional.)
It is easy to see that the two definition of the orientation coincide.
It is also easy to see that the orientation is independent of the choice of $\mu_i$.
To define the orientation of $(V^1 \times V^2 \times \dots \times
V^m)_{((\breve \epsilon^{i, j}_l), (\breve c^i_l), (x^i_j), (\breve \eta^i_l))}$,
we need to see the relations of its tangent space and other various vector spaces.
Recall that $T \hat V^i = T\mathring{X}^i \oplus \mathop{\mathrm{Ker}}\nolimits D^i$ for each $1 \leq i \leq k$,
where
\[
D^i : \widetilde{W}^{1, p}_\delta (\Sigma_i, u_i^\ast T \hat Y) \oplus E_i^0 \to
L_\delta^p (\Sigma_i, {\textstyle\bigwedge}^{0, 1} T^\ast \Sigma_i \otimes u_i^\ast T \hat Y)
\]
is the differential $DF^{(0, b^0)}_{(0, 0)}$ in Section \ref{construction of nbds}
for $(\Sigma_i, z_i, u_i)$.
The fiber product
\[
\mathcal{C}^i = [D^{i, -}] \underset{\mathop{\mathrm{Ker}}\nolimits A^{i, -}}{\times} [D^i]
\underset{\mathop{\mathrm{Ker}}\nolimits A^{i, +}}{\times} [D^{i, +}]
\]
is equivalent to the kernel of a $\overline{\partial}$-operator on a complex vector bundle over
a closed semistable curve by gluing, where
\begin{align*}
[D^{i, -}] &= \prod_{1 \leq j < i, 1 \leq l \leq l_{j, i}} [D^-_{\gamma_{-\infty^{i, j}_l}}]
\times \prod_{1 \leq l \leq l_{i, -}} [D^-_{\gamma_{-\infty^{i, -\infty}_l}}],\\
\mathop{\mathrm{Ker}}\nolimits A^{i, -} &= \prod_{1 \leq j < i, 1 \leq l \leq l_{j, i}}
\mathop{\mathrm{Ker}}\nolimits A_{\gamma_{-\infty^{i, j}_l}}
\times \prod_{1 \leq l \leq l_{i, -}} \mathop{\mathrm{Ker}}\nolimits A_{\gamma_{-\infty^{i, -\infty}_l}},\\
\mathop{\mathrm{Ker}}\nolimits A^{i, +} &= \prod_{i < j \leq m, 1 \leq l \leq l_{i, j}}
\mathop{\mathrm{Ker}}\nolimits A_{\gamma_{+\infty^{i, j}_l}}
\times \prod_{1 \leq l \leq l_{i, +}} \mathop{\mathrm{Ker}}\nolimits A_{\gamma_{+\infty^{i, +\infty}_l}},\\
[D^{i, +}] &= \prod_{i < j \leq m, 1 \leq l \leq l_{i, j}} [D^+_{\gamma_{+\infty^{i, j}_l}}]
\times \prod_{1 \leq l \leq l_{i, +}} [D^+_{\gamma_{+\infty^{i, +\infty}_l}}].
\end{align*}
The vector space $[\mathring{D}^{i, -}] \oplus [\mathring{D}^{i, +}]$
is a subspace of $\mathcal{C}^i$, and its quotient space is isomorphic to $[D^i]$,
where
\begin{align*}
&[\mathring{D}^{i, -}] = \prod_{1 \leq j < i, 1 \leq l \leq l_{j, i}}
[\mathring{D}^-_{\gamma_{-\infty^{i, j}_l}}] \times
\prod_{1 \leq l \leq l_{i, -}} [\mathring{D}^-_{\gamma_{-\infty^{i, -\infty}_l}}],\\
&[\mathring{D}^{i, +}] = \prod_{i < j \leq m, 1 \leq l \leq l_{i, j}}
[\mathring{D}^+_{\gamma_{+\infty^{i, j}_l}}] \times
\prod_{1 \leq l \leq l_{i, +}} [\mathring{D}^+_{\gamma_{+\infty^{i, +\infty}_l}}].
\end{align*}
We fix splittings $\mathcal{C}^i \to [\mathring{D}^-_{\gamma_{-\infty^{i, j}_l}}]$,
$\mathcal{C}^i \to [\mathring{D}^-_{\gamma_{-\infty^{i, -\infty}_l}}]$,
$\mathcal{C}^i \to [\mathring{D}^+_{\gamma_{+\infty^{i, j}_l}}]$
and $\mathcal{C}^i \to [\mathring{D}^+_{\gamma_{+\infty^{i, +\infty}_l}}]$.
Then the tangent space of
$(V^1 \times V^2 \times \dots \times V^m)_{((\breve \epsilon^{i, j}_l),
(\breve c^i_l), (x^i_j), (\breve \eta^i_l))}$
is isomorphic to the kernel of the map from
\[
(T\mathring{X}^1 \times \mathcal{C}^1) \oplus (T\mathring{X}^2 \times \mathcal{C}^2) \oplus \dots
\oplus (T\mathring{X}^m \times \mathcal{C}^m)
\]
to the direct sum of the following vector spaces:
\begin{itemize}
\item
$T \hat Y / (\mathbb{R}\partial_\sigma \oplus T x^i_l)$
\item
$[\mathring{D}^-_{\gamma_{-\infty^{i, -\infty}_l}}] \oplus
\bigl(\mathop{\mathrm{Ker}}\nolimits A_{\gamma_{-\infty^{i, -\infty}_l}} / (\mathbb{R} \oplus T S^1 \oplus T \tilde c^i_l)\bigr)$
\item
$\bigl(\mathop{\mathrm{Ker}}\nolimits A_{\gamma_{+\infty^{i, +\infty}_l}} / (\mathbb{R} \oplus T S^1 \oplus T \tilde \eta^i_l)\bigr)
\oplus [\mathring{D}^+_{\gamma_{+\infty^{i, +\infty}_l}}]$
\item
$[\mathring{D}^+_{\gamma_{+\infty^{i, j}_l}}] \oplus
\bigl((\mathop{\mathrm{Ker}}\nolimits A_{\gamma_{+\infty^{i, j}_l}} \oplus \mathop{\mathrm{Ker}}\nolimits A_{\gamma_{-\infty^{i, j}_l}}) /
(\mathbb{R} \oplus T S^1 \oplus T \tilde \epsilon^{i, j}_l \oplus \mathbb{R} \oplus T S^1)\bigr)
\oplus [\mathring{D}^-_{\gamma_{-\infty^{i, j}_l}}]$
\item
$\mathbb{R}^{k_i} \oplus \bigoplus_{z_{i, \beta}^{++}} \mathbb{R}^2$ (the range of the map $s^0$)
\end{itemize}
Now we explain the definition of the orientations.
For the convenience, we assume all $E^0$ are complex vector spaces and
$\lambda^0$ are complex linear map. (We can always assume this condition.)
First we define the orientation of the vector space
\[
\mathcal{W}^i = T\mathring{X}^i \times \mathcal{C}^i / (\mathbb{R}^{k_i} \oplus \bigoplus_{z_{i, \beta}^{++}} \mathbb{R}^2)
\]
by
\[
\bigl(\mathbb{R}^{k_i} \oplus \bigoplus_{z_{i, \beta}^{++}} \mathbb{R}^2\bigr) \oplus \mathcal{W}^i = T\mathring{X}^i \times \mathcal{C}^i,
\]
where the orientation of $\mathbb{R}^{k_i}$ is the product of the positive orientation of $\mathbb{R}$,
and its order is
\[
(\sigma_1 \circ \Phi_{a, b}(\xi) (R_1), \sigma_2 \circ \Phi_{a, b}(\xi) (R_2), \dots, \sigma_{k_i} \circ \Phi_{a, b}(\xi) (R_{k_i}));
\]
and the orientation of each $\mathbb{R}^2 \cong T^{\bot^Y}\!\! S = T \hat Y / T (\mathbb{R} \times S)$
is the complex orientation defined by the almost complex structure of $\hat Y$.
We note that the dimension of each $\mathcal{W}^i$ is odd.
Next we recall the definition of the orientation of fiber product.
We use only fiber products from right.
Let $f$ be a surjective linear map from an oriented vector space $V$ to another oriented
vector space $W$, and let $A \subset W$ be an oriented subspace.
Define the orientation of $\mathop{\mathrm{Ker}}\nolimits f$ by $\mathop{\mathrm{Ker}}\nolimits f \oplus W = V$.
Then the orientation of the fiber product $V_A = \{v \in V; f(v) \in A\}$ is defined by
$V_A = \mathop{\mathrm{Ker}}\nolimits f \oplus A$.
We deform each curve $\Sigma_i$, preserving a neighborhood of limit circles,
to a curve which consists of inner caps $[-\infty, 0] \times S^1 \cup D_\infty$ of
$-\infty$-limit circles, inner caps $D_0 \cup [0, \infty] \times S^1$ of
$+\infty$-limit circles, and a semistable curve $\mathring{\Sigma}_i$.
Each inner cap is connected to $\mathring{\Sigma}_i$ by a nodal point.
We also deform the linear operator $D^i$ to a $\overline{\partial}$-type linear operator
which coincide with
\begin{itemize}
\item
$D^+_{\gamma_{-\infty^{i, j}_l}}$ on the inner cap of $S^1_{-\infty^{i, j}_l}$,
\item
$D^+_{\gamma_{-\infty^{i, -\infty}_l}}$ on the inner cap of $S^1_{-\infty^{i, -\infty}_l}$,
\item
$D^-_{\gamma_{+\infty^{i, j}_l}}$ on the inner cap of $S^1_{+\infty^{i, j}_l}$, and
\item
$D^-_{\gamma_{+\infty^{i, +\infty}_l}}$ on the inner cap of $S^1_{+\infty^{i, +\infty}_l}$.
\end{itemize}
We denote the deformed linear operator by $\widetilde{D}^i$
In the definition of $\mathcal{C}$ and $\mathcal{W}^i$, we replace $D^i$ with $\widetilde{D}^i$,
and we get vector spaces $\widetilde{\mathcal{C}}$ and $\widetilde{\mathcal{W}}^i$.
Restriction to the inner cups defines linear maps from
$[\widetilde{D}^i]$ to $[D^+_{\gamma_{-\infty^{i, j}_l}}]$, $[D^+_{\gamma_{-\infty^{i, -\infty}_l}}]$,
$[D^-_{\gamma_{+\infty^{i, j}_l}}]$ and $[D^-_{\gamma_{+\infty^{i, +\infty}_l}}]$.
Therefore there exist linear maps from $\widetilde{\mathcal{C}}^1 \oplus
\widetilde{\mathcal{C}}^2 \oplus \dots \oplus \widetilde{\mathcal{C}}^m$
to the following vector spaces.
\begin{align*}
&[D^-_{\gamma_{-\infty^{i, j}_l}}] \underset{\mathop{\mathrm{Ker}}\nolimits A_{\gamma_{-\infty^{i, j}_l}}}{\times}
[D^+_{\gamma_{-\infty^{i, j}_l}}],
&&[D^-_{\gamma_{-\infty^{i, -\infty}_l}}] \underset{\mathop{\mathrm{Ker}}\nolimits A_{\gamma_{-\infty^{i, -\infty}_l}}}{\times}
[D^+_{\gamma_{-\infty^{i, -\infty}_l}}],
\\
&[D^-_{\gamma_{+\infty^{i, j}_l}}] \underset{\mathop{\mathrm{Ker}}\nolimits A_{\gamma_{+\infty^{i, j}_l}}}{\times}
[D^+_{\gamma_{+\infty^{i, j}_l}}],
&&[D^-_{\gamma_{+\infty^{i, +\infty}_l}}] \underset{\mathop{\mathrm{Ker}}\nolimits A_{\gamma_{+\infty^{i, +\infty}_l}}}{\times}
[D^+_{\gamma_{+\infty^{i, +\infty}_l}}].
\end{align*}
They define linear maps from $\widetilde{\mathcal{W}}^1 \oplus \widetilde{\mathcal{W}}^2
\oplus \dots \oplus \widetilde{\mathcal{W}}^m$ to the same vector spaces.
The above vector spaces have the complex orientations.
We regard the vector space
\[
\mathbb{R} \oplus TS^1 \oplus T \tilde c^i_l \oplus [\mathring{D}^+_{\gamma_{-\infty^{i, -\infty}_l}}]
\]
as a subspace of
\[
[D^-_{\gamma_{-\infty^{i, -\infty}_l}}] \underset{\mathop{\mathrm{Ker}}\nolimits A_{\gamma_{-\infty^{i, -\infty}_l}}}{\times}
[D^+_{\gamma_{-\infty^{i, -\infty}_l}}]
\]
by regarding $\mathbb{R} \oplus TS^1 \oplus T \tilde c^i_l \subset \mathbb{R} \oplus TP \cong
\mathop{\mathrm{Ker}}\nolimits A_{\gamma_{-\infty^{i, -\infty}_l}}$
as a subspace of the above space by a right inverse of the surjection
\[
[D^-_{\gamma_{-\infty^{i, -\infty}_l}}] \underset{\mathop{\mathrm{Ker}}\nolimits A_{\gamma_{-\infty^{i, -\infty}_l}}}{\times}
[D^+_{\gamma_{-\infty^{i, -\infty}_l}}] \to \mathop{\mathrm{Ker}}\nolimits A_{\gamma_{-\infty^{i, -\infty}_l}}.
\]
We define its orientation
by the direct sum of the complex orientation of $\mathbb{R} \oplus TS^1 = T(\mathbb{R} \times S^1)$,
the orientation of the ordered simplicial complex $\tilde c^i_l$,
and the orientation $\theta^D_{c^i_l}$ of $[\mathring{D}^+_{\gamma_{-\infty^{i, -\infty}_l}}]$.
We define the orientations of the following spaces similarly.
\[
[\mathring{D}^-_{\gamma_{+\infty^{i, +\infty}_l}}] \oplus \mathbb{R} \oplus TS^1 \oplus T \tilde \eta^i_l
\subset [D^-_{\gamma_{+\infty^{i, +\infty}_l}}] \underset{\mathop{\mathrm{Ker}}\nolimits A_{\gamma_{+\infty^{i, +\infty}_l}}}{\times}
[D^+_{\gamma_{+\infty^{i, +\infty}_l}}],
\]
\vspace{-12pt}
\begin{align*}
&[\mathring{D}^-_{\gamma_{+\infty^{i, j}_l}}] \oplus \mathbb{R} \oplus T S^1 \oplus T \tilde \epsilon^{i, j}_l \oplus \mathbb{R} \oplus T S^1 \oplus [\mathring{D}^+_{\gamma_{-\infty^{j, i}_l}}]\\
&\subset
([D^-_{\gamma_{+\infty^{i, j}_l}}] \underset{\mathop{\mathrm{Ker}}\nolimits A_{\gamma_{+\infty^{i, j}_l}}}{\times}
[D^+_{\gamma_{+\infty^{i, j}_l}}]) \\
&\quad \oplus
([D^-_{\gamma_{-\infty^{j, i}_l}}] \underset{\mathop{\mathrm{Ker}}\nolimits A_{\gamma_{-\infty^{j, i}_l}}}{\times}
[D^+_{\gamma_{-\infty^{j, i}_l}}]).
\end{align*}
For each marked point $z_{i, l}$, the orientation of $T x^i_l \oplus \mathbb{R} \subset
T \hat Y$ is defined by the orientation of the cycle $x^i_l$
and the positive orientation of $\mathbb{R}$.
The orientation of $T \hat Y$ is defined by the complex orientation.
The tangent space of
$(V^1 \times V^2 \times \dots \times V^m)_{((\breve \epsilon^{i, j}_l), (\breve c^i_l),
(x^i_j), (\breve \eta^i_l))}$
is isomorphic to the fiber product
\begin{equation}
(\widetilde{\mathcal{W}}^1 \oplus \widetilde{\mathcal{W}}^2 \oplus \dots \oplus \widetilde{\mathcal{W}}^m)_\star,
\label{ori}
\end{equation}
where
\begin{align*}
\star &= \bigoplus_{(i, j), l}
([\mathring{D}^-_{\gamma_{+\infty^{i, j}_l}}] \oplus \mathbb{R} \oplus T S^1 \oplus
T \tilde \epsilon^{i, j}_l \oplus \mathbb{R} \oplus T S^1 \oplus
[\mathring{D}^+_{\gamma_{-\infty^{j, i}_l}}])\\
&\quad \oplus \bigoplus_{i, l}
(\mathbb{R} \oplus TS^1 \oplus T \tilde c^i_l \oplus [\mathring{D}^+_{\gamma_{-\infty^{i, -\infty}_l}}])
\oplus \bigoplus_{i, l}
(\mathbb{R}\partial_\sigma \oplus T x^i_l)\\
&\quad \oplus \bigoplus_{i, l}
([\mathring{D}^-_{\gamma_{+\infty^{i, +\infty}_l}}] \oplus \mathbb{R} \oplus TS^1 \oplus T \tilde \eta^i_l).
\end{align*}
We give the above space the orientation of fiber product.
(The order of each direct sum $\bigoplus$ is the lexicographic order.)
This orientation and the complex orientation of the obstruction bundle
$E = E^1 \oplus E^2 \oplus \dots \oplus E^m$ define the orientation of the fiber product
as a pre-Kuranishi space.
(The orientation of the zero set of the perturbed multisection on $(V, E)$ is defined by
deleting the vector space $E$ from the tangent space of $V$.)
We note that the parity of the dimension of $\widetilde{\mathcal{W}}^1 \oplus \widetilde{\mathcal{W}}^2
\oplus \dots \oplus \widetilde{\mathcal{W}}^m$ is equal to the parity of $m$
since each $\widetilde{\mathcal{W}}^i$ has odd dimension.
We need to check that this orientation is compatible with the embeddings of Kuranishi
neighborhoods.
First we note that in (\ref{ori}), $\star$ is independent of the Kuranishi neighborhood.
Hence it is enough to compare the orientations of each $\mathcal{W}$ (or $\widetilde{\mathcal{W}}$).
For the convenience of the computation of orientations,
we may change the definition of the map $s^0 : \hat V \to \mathbb{R}^k \oplus
\bigoplus_{z_\beta^{++}} \mathbb{R}^2$ from (\ref{s^0}) to
\[
s^0(a, b, x) = (\sigma_i, p' \circ (\theta_{\sigma_i}^{-1} \times 1) \circ
\Phi_{a, b}(\xi_x)(Z_\beta^{++}(a)))
\]
where $\sigma_i = \sigma \circ \Phi_{a, b}(\xi_x)(\widetilde{R}_i(a))$,
and each $\theta_{\sigma_i} : \mathbb{R} \to \mathbb{R}$ is defined by
$\theta_{\sigma_i}(s) = s + \sigma_i$.
(Note that $V = \{s^0 = 0\}$ does not change.)
Let $C \subset T_{(0, b^0, 0)} \hat V$ be the tangent space of $\mathbb{R}$-translations.
Then the restriction of the differential of $s^0$ gives an isomorphism
$C \cong \mathbb{R}^k \oplus 0 \subset \mathbb{R}^k \oplus \bigoplus_{z_\beta^{++}} \mathbb{R}^2$, and
the differentials of $s$ or the evaluation maps at marked points or limit circles
vanish on $C$.
We note that under this identification $C \cong \mathbb{R}^k$,
the restriction of the differential of $(b_{\mu_1}, \dots, b_{\mu_{k-1}})$ to $C$ is
$(t_1, t_2, \dots, t_k) \mapsto (t_1 - t_2, t_2 - t_3, \dots, t_{k-1} - t_k) : C \cong \mathbb{R}^k \to
\mathbb{R}^{k-1}$.
It is clear that the definition of the orientation does not depend on the
choice of the family of sections $\hat R_j$ or the choice of
the coordinates of the neighborhoods of joint circles of $\Sigma_0$
used for the definition of the asymptotic parameters $b_\mu$.
We consider the situation discussed in Section \ref{embed},
and compare the orientation of $\mathcal{W}_1$ for $(V_1, E_1, s_1, \psi_1)$
and $\mathcal{W}_2$ for $(V_2, E_2, s_2, \psi_2)$ at $q_0$.
First we consider the case where $q_0 = p_1 = p_2$.
We may assume that we have used the same coordinates of the neighborhoods
of joint circles of $\Sigma_1 = \Sigma_2$ for the definition of the asymptotic
parameters $b^1_\mu$ and $b^2_\mu$.
The compatibility of the orientation in this case is essentially
because all the differences have the complex orientations.
To see this more precisely,
it is convenient to consider the following intermediate Kuranishi neighborhood.
Let $(\widetilde{P}_3 \to \widetilde{X}_3, Z)$ be the local universal family of
$(\Sigma_1, z_1 \cup z_2^+ \cup z_1^{++} \cup z_2^{++})$
and let $\lambda_3 : E_3^0 = E_2^0 \to C^\infty(\widetilde{P}_3 \times Y,
{\textstyle\bigwedge}^{0, 1} V^\ast \widetilde{P}_3 \otimes (\mathbb{R} \partial_\sigma \oplus TY))$
be the pull back of $\lambda_2$.
Let $\widetilde{R}^{3, 1}_j, \widetilde{R}^{3, 2}_j : \widetilde{X}_3 \to \widetilde{P}_3$
be the pull backs of $\widetilde{R}^1_j, \widetilde{R}^2_j$.
Then, using the parameter space $\mathring{X}_3 \subset \widetilde{X}_3 \times
\coprod_{\text{joint circles}} \mathbb{R}_\mu$ and $\lambda_3$,
we can construct a Kuranishi neighborhood of $p_1 = p_2$,
where in this case, for the definition of $V_3 = \{s^0_3 = 0\}$,
we use $s_3^0 : \hat V_3^0 \to \mathbb{R}^k \oplus \bigoplus_{z_{1, \beta^{++}}} \mathbb{R}^2
\oplus \bigoplus_{z_{2, \beta^{++}}} \mathbb{R}^2$
defined by
\begin{align*}
s_0^3(a, b, x) &= (\sigma^1_i, p' \circ (\theta_{\sigma^1_i}^{-1} \times 1) \circ
\Phi_{a, b}(\xi_x)(Z_{1, \beta}^{++}(a)),\\
&\quad\quad p' \circ (\theta_{\sigma^2_i}^{-1} \times 1) \circ
\Phi_{a, b}(\xi_x)(Z_\beta^{++}(a))),
\end{align*}
where $\sigma^l_i = \sigma \circ \Phi_{a, b}(\xi_x)(\widetilde{R}^{3, l}_i(a))$ for $l = 1, 2$.
It is clear that we can define the embeddings from $V_1$ and $V_2$ to $V_3$ as
in Section \ref{embed}.
Then $T_{(0, b^0, 0)} \hat V_3 = T_{(0, b^0, 0)} \hat V_1 \oplus F$,
where $F$ is the tangent of the parameters for the additional marked points
$(z_2^+ \setminus z_1^+) \cup z_2^{++}$.
It is clear that the projection of the restriction of $s_0^3 \times s^3$
gives an isomorphism of complex vector spaces
\[
F \stackrel{\cong}{\to} \bigoplus_{z_{2, \beta^{++}}} \mathbb{R}^2 \oplus
\bigoplus_{z^+ \in z_2^+ \setminus z_1^+} \mathbb{R}^2.
\]
(Recall that the complex orientations of $\mathbb{R}^2 \cong T \hat Y / TS'_2$ or
$\mathbb{R}^2 \cong T \hat Y / T(\mathbb{R} \times S_2)$
are defined by the almost complex structure of $\hat Y$.)
This implies that the pair of vector spaces $(TV_3, E_3)$ is isomorphic to
$(TV_1 \oplus F', E_1 \oplus F')$ for some complex vector space $F'$.
Similar condition is satisfied for the embedding $V_2 \hookrightarrow V_3$.
Therefore, the embedding preserves the orientation.
Next we consider the case where $q_0 = p_2$ and all the data for
the construction of $(V_2, E_2, s_2, \psi_2)$ are the restriction of those for
$(V_1, E_1, s_1, \psi_1)$.
We assume that the $i$-th floor and the $(i+1)$-th floor of $p_1$
are glued into one floor in $p_2$ and the others are not.
We may assume that $(\hat R^2_1, \dots, \hat R^2_{k_2})
= (\hat R^1_1, \stackrel{i}{\check \dots}, \hat R^1_{k_1})$.
Let $C_l \subset T\hat V_l$ be the tangent space of $\mathbb{R}$-translations
for each $l =1, 2$.
Then it is easy to see that $C_1 \cong C_2 \oplus \mathbb{R}$ and the sign of
$b^1_{\mu_i}$ and $\sigma^1_i$
($= \sigma \circ \Phi^1_{a, b}(\xi_x)(\widetilde{R}^1_i(a))$ at $(a, b, x) \in \hat V_1$)
coincide on this $\mathbb{R}$.
Since the orientation of $\mathcal{W}_l$ are defined by subtracting
vector space $(\mathbb{R}^{k_l} \oplus \bigoplus_{z_l^{++}} \mathbb{R}^2)$ from
$T\mathring{X}_l \times \mathcal{C}_l$ and the subtractions are from the left,
this implies that the embedding preserves the orientation.
The general case is covered by the combination of the above two cases.
Furthermore, the orientation is independent of the choice of the lifts of $c^i_l$,
$\eta^i_l$ and $\epsilon^{i, j}_l$ under the natural isomorphism.
Hence we may denote the above fiber product pre-Kuranishi space by
$\overline{\mathcal{M}}^m_{((\hat \epsilon^{i, j}_l), (\hat c^i_l),
(x^i_l), (\hat \eta^i_l))}$.
The algebras of SFT are constructed by the virtual fundamental chains of
the zero-dimensional component of these fiber product pre-Kuranishi spaces, and
the algebraic properties of them are proved by the equation corresponding to
the boundary of the one-dimensional component of the fiber products.
First we study the boundary of $\overline{\mathcal{M}}_{((\hat c_l), (x_l), (\hat \eta_l))}$.
It consists of several parts, and some of them are due to the splitting in $\mathbb{R}$-direction,
and the others are due to the boundaries of the simplices $c_l$ and $\eta_l$.
We consider the former.
For each Kuranishi neighborhood $(V, E, s, \psi)$ of $\hat \mathcal{M}$,
each of these parts corresponds to the subspace
$\{\rho_\mu = 0; \text{ for some (and all) } \allowbreak \mu \in M_i\}$ of $V$.
($M_i$ is the set of the indices of joint circles between the $i$-th floor and the $(i+1)$-th
floor.)
We note that the normal direction is $\kappa_i = - L_\mu \log \rho_\mu + b_\mu$
($\mu \in M_i$) and $\kappa_i^{-1} = 0$ defines the boundary.
As we have seen in Section \ref{construction of a multisection}
(related to the second compatibility condition of the multisection),
the curve corresponding to each zero of the multisection in this boundary is determined
by two curves and a family of diffeomorphisms between some of their limit circles.
Assume that a $+ \infty$-limit circle $S^1_{+\infty_{(1, 2), l}} \subset \Sigma_1$
and a $-\infty$-limit circle $S^1_{-\infty_{(2, 1), l}} \subset \Sigma_2$ are identified
by a diffeomorphism $\phi_{(1, 2), l} : S^1_{+\infty_{(1, 2), l}} \to S^1_{-\infty_{(2, 1), l}}$.
$\phi_{(1, 2), l}$ is determined by a pair of the coordinates $\phi_{+\infty_{(1, 2), l}} :
S^1 \to S^1_{+\infty_{(1, 2), l}}$ of $S^1_{+\infty_{(1, 2), l}}$ and
$\phi_{-\infty_{(2, 1), l}} : S^1 \to S^1_{-\infty_{(2, 1), l}}$ of $S^1_{-\infty_{(2, 1), l}}$
such that $\pi_Y \circ u_1 \circ \phi_{+\infty_{(1, 2), l}} = \pi_Y \circ u_2 \circ
\phi_{-\infty_{(2, 1), l}}$ in $P$.
(Namely, this pair corresponds to
$\phi_{(1, 2), l} = \phi_{-\infty_{(2, 1), l}} \circ \phi_{+\infty_{(1, 2), l}}^{-1}$.)
For any $g \in S^1 \subset \mathop{\mathrm{Aut}}\nolimits S^1$, $(\phi_{+\infty_{(1, 2), l}} \circ g,
\phi_{-\infty_{(2, 1), l}} \circ g)$ and $(\phi_{+\infty_{(1, 2), l}}, \phi_{-\infty_{(2, 1), l}})$
correspond to the same diffeomorphism.
Assume that $\pi_Y \circ u_1|_{S^1_{\phi_{+\infty_{(1, 2), l}}}}$
($= \pi_Y \circ u_2|_{S^1_{\phi_{-\infty_{(2, 1), l}}}}$) is contained in $\mathop{\mathrm{Int}}\nolimits \zeta$ for
some top-dimensional simplex $\zeta$ of $\overline{P}$.
(Note that this assumption is satisfied if we restrict to the case of the boundary of the
one-dimensional component of $\overline{\mathcal{M}}_{((\hat c_l), (x_l), (\hat \eta_l))}$.
This is due to the first condition of the multisection in Section
\ref{construction of a multisection}.)
Let $\tilde \zeta \subset P$ be a lift of $\zeta$.
Then we can choose a pair of the coordinates $(\phi_{+\infty_{(1, 2), l}},
\phi_{-\infty_{(2, 1), l}})$ such that $\pi_Y \circ u_1 \circ \phi_{+\infty_{(1, 2), l}} \in
\tilde \zeta$.
For each diffeomorphism $\phi_{(1, 2), l} : S^1_{+\infty_{(1, 2), l}} \to S^1_{-\infty_{(2, 1), l}}$,
the number of such representatives is $m_\zeta$, where $m_\zeta$ is the multiplicity of
the periodic orbits in $\mathop{\mathrm{Int}}\nolimits \zeta$.
(The number of different diffeomorphisms $S^1_{+\infty_{(1, 2), l}} \to
S^1_{-\infty_{(2, 1), l}}$ is also $m_\zeta$.)
Define a chain $\widetilde{\Delta}_{\overline{P}}$ (not a cycle) in $P \times P$ by
\[
\widetilde{\Delta}_{\overline{P}} = \sum \frac{1}{m_\zeta}
\theta^{\lsuperscript{D}{t}}_{\tilde \zeta}
(\Delta_\ast \tilde \zeta) \theta^D_{\tilde \zeta},
\]
where the sum is taken over all top-dimensional simplices of $K$, including the simplices
contained in $\overline{P}^{\text{bad}}$.
As in the definition of $\Delta_\ast [\overline{P}]$,
$\theta^D_{\tilde \zeta}$ is an arbitrary fixed orientation of $p_2^\ast \S^D$ on
$\mathop{\mathrm{Int}}\nolimits \Delta_\ast \tilde \zeta$, and $\theta^{\lsuperscript{D}{t}}_{\tilde \zeta}$ is
the orientation of $p_1^\ast \S^{\lsuperscript{D}{t}}$ defined by
$\theta^D_{\tilde \zeta}$ and $\theta^{\overline{P}}_{\tilde \zeta}$.
Then by the above argument, the part of the boundary of the zero-dimensional
component of $\overline{\mathcal{M}}_{((\breve c_l), (x_l), (\breve \eta_l))}$ corresponding to
the splitting in $\mathbb{R}$-direction is the zero-dimensional component of
\[
- \sum (-1)^\ast \overline{\mathcal{M}}^2_{
(e^{\widetilde{\Delta}_{\overline{P}}}, (\breve c^i_l), (x^i_l), (\breve \eta^i_l))}
\]
where the sum is taken over all decompositions
\[
\{\breve c_l\} = \{\breve c^1_l\} \sqcup \{\breve c^2_l\},
\quad \{x_l\} = \{x^1_l\} \sqcup \{x^2_l\},
\quad \{\breve \eta_l\} = \{\breve \eta^1_l\} \sqcup \{\breve \eta^2_l\}
\]
as sets, and
the order of each $(\breve c^i_l)_l$ is defined by the order of $(\breve c_l)_l$.
The orders of $(x^i_l)_l$ or $(\breve \eta^i_l)_l$ are similar.
$\ast$ is the weighted sign of the permutation
\[
\begin{pmatrix}
(\breve c^1_l)_l (\breve c^2_l)_l
& (x^1_l)_l (x^2_l)_l & (\breve \eta^1_l)_l (\breve \eta^2_l)_l\\
(\breve c_l)_l & (x_l)_l & (\breve \eta_l)_l
\end{pmatrix},
\]
where weighted sign is defined as follows.
The weighted sign of the transposition $(a, b)$ is defined by
$\deg a \cdot \deg b \in \mathbb{Z} / 2$, where
the degree is defined by $\deg c \theta^D_c = \deg (c \theta^D_c)^\ast
= \dim c + \dim [\mathring{D}^+_\gamma]$ ($\gamma \in |c|$) and
$\deg s = \mathop{\mathrm{codim}}\nolimits_Y |s|$.
The weighted sign of a general permutation is defined by the product of the weighted sign
of the transpositions whose product coincides with the permutation.
$e^{\widetilde{\Delta}_{\overline{P}}} = 1 + \widetilde{\Delta}_{\overline{P}}
+ \frac{1}{2} (\widetilde{\Delta}_{\overline{P}}, \widetilde{\Delta}_{\overline{P}}) + \cdots$
is the exponential.
We claim that the virtual fundamental chain of the zero-dimensional component of the
above pre-Kuranishi space does not change if we replace
$\widetilde{\Delta}_{\overline{P}}$
with the sum taken over the top-dimensional simplices $\zeta$ of $K$ not contained
in $\overline{P}^{\text{bad}}$.
This is because if $\pi_Y \circ u_1 \circ \phi_{+\infty_{(1, 2), l}} \in \mathop{\mathrm{Int}}\nolimits \tilde \zeta$
and $\zeta$ is contained in $\overline{P}^{\text{bad}}$, then the curve obtained by
the pair of coordinates $(\phi_{+\infty_{(1, 2), l}}, \phi_{-\infty_{(2, 1), l}} \circ
g_{1 / m_\zeta})$ instead of $(\phi_{+\infty_{(1, 2), l}}, \phi_{-\infty_{(2, 1), l}})$
($g_{1 / m_\zeta} \in \mathop{\mathrm{Aut}}\nolimits S^1$ is the translation by $1 / m_\zeta$)
is also a zero of the multisection, but its orientation is opposite.
Similarly, in the parts of the boundary of $\overline{\mathcal{M}}_{((\hat c_l), (x_l), (\hat \eta_l))}$
due to the boundaries of the simplices of $c_l$ and $\eta_l$,
the parts of the boundaries of $c_l$ and $\eta_l$ contained in $\overline{P}^{\text{bad}}$
do not affect the virtual fundamental chain.
Therefore,
\begin{align}
0 &= [\partial \overline{\mathcal{M}}_{((\hat c_l), (x_l), (\hat \eta_l))}]^0 \notag\\
&= - [\overline{\mathcal{M}}_{\partial ((\hat c_l), (x_l), (\hat \eta_l))}]^0
- \sum (-1)^\ast \bigl[\overline{\mathcal{M}}^2_{
(e^{\Delta_\ast [\overline{P}]}, (\hat c^i_l), (x^i_l), (\hat \eta^i_l))}\bigr]^0,
\label{boundary of M}
\end{align}
where
$[\cdot]^0$ denotes the virtual fundamental chain of the zero-dimensional component,
and $\partial((\hat c_l), (x_l), (\hat \eta_l))$ is defined by
\begin{align*}
&\partial((\hat c_l), (x_l), (\hat \eta_l))\\
&= \sum_j (-1)^{\sum_{l < j} |\hat c_l|}
((\hat c_1, \dots, \partial \hat c_j, \dots, \hat c_{l_-}), (x_l), (\hat \eta_l))\\
&\quad + \sum_j (-1)^{\sum_l |\hat c_l| + \sum_l |x_l|^\bot + \sum_{l < j} |\hat \eta_l|}
((\hat c_l), (x_l), (\hat \eta_1, \dots, \partial \hat \eta_j, \dots, \hat \eta_{l_+})),
\end{align*}
where $|x|^\bot = \mathop{\mathrm{codim}}\nolimits_Y x$.
Similarly, it is easy to see that for any $((\hat c_l), (x_l), (\hat \eta_l))$ and
$(\hat \epsilon^{i, j}_l)$,
\begin{align}
0 &= \sum_{\star_m} (-1)^\ast
\bigl[\partial (\overline{\mathcal{M}}^m_{((\hat \epsilon^{i, j}_l), (\hat c^i_l), (x^i_l), (\hat \eta^i_l))})
\bigr]^0
\notag\\
& = (-1)^m \sum_{\star_m} (-1)^\ast
\bigl[\overline{\mathcal{M}}^m_{\partial ((\hat \epsilon^{i, j}_l), (\hat c^i_l), (x^i_l), (\hat \eta^i_l))}
\bigr]^0 \notag\\
& \quad + \sum_{\substack{1 \leq i_0 \leq m \\ \star_{m+1}}} (-1)^{\ast + i_0}
\bigl[\overline{\mathcal{M}}^{m+1}_{((e^{\Delta_\ast [\overline{P}]})^{i_0, i_0+1} \cup
(\tau_{i_0}\hat \epsilon^{i, j}_l), (\hat c^i_l), (x^i_l), (\hat \eta^i_l))}\bigr]^0,
\label{boundary of MM}
\end{align}
where the sum of $\star_m$ is taken over all decompositions
\[
\{\hat c_l\} = \coprod_i \{\hat c^i_l\}, \quad \{x_l\} = \coprod_i \{x^i_l\},
\quad \{\hat \eta_l\} = \coprod_i \{\hat \eta^i_l\}
\]
as sets, and
$\ast$ is the weighted sign of the permutation
\[
\begin{pmatrix}
(\hat c^1_l)_l & \cdots & (\hat c^m_l)_l
& (x^1_l)_l & \cdots & (x^m_l)_l & (\hat \eta^1_l)_l & \cdots & (\hat \eta^m_l)_l\\
&(\hat c_l)_l&&&(x_l)_l&&&(\hat \eta_l)_l&
\end{pmatrix}.
\]
$\tau_{i_0}\hat \epsilon^{i, j}_l$ is defined by
\[\tau_{i_0} a^{i, j} = \begin{cases}
a^{i+1, j+1} & i_0 < i < j\\
a^{i_0, j+1} + a^{i_0 + 1, j+1} & i = i_0 < j\\
a^{i, j+1} & i < i_0 < j\\
a^{i, i_0} + a^{i, i_0 + 1} &i < j = i_0\\
a^{i, j} &i < j < i_0
\end{cases},
\]
where $a^{i, j}$ means the fiber product with $a$ at a $+\infty$-limit circle of
$i$-th holomorphic building and a $-\infty$-limit circle of $j$-th holomorphic building.
Unfortunately, equation (\ref{boundary of M}) is not the equation for
$\overline{\mathcal{M}}_{((\hat c_l), (x_l), (\hat \eta_l))}$'s in the Bott Morse case
since the second term cannot be written as a function of
$\overline{\mathcal{M}}_{((\hat c_l), (x_l), (\hat \eta_l))}$'s.
(The diagonal $\Delta_\ast [\overline{P}]$ cannot be written as a linear combination of
products of simplices in $K$.)
To obtain a meaningful equation, we add correction terms to
$\overline{\mathcal{M}}_{((\hat c_l), (x_l), (\hat \eta_l))}$ as follows.
The addition of these correction terms are equivalent to count the cascades
in \cite{Bo02}.
Let $(\hat c_l)$ be a family of chains in
$C_\ast(\overline{P}, \overline{P}^{\text{bad}}; \S^D \otimes \mathbb{Q})$,
let $(x_l)$ be a family of simplices in $K^0$, and let $(\alpha_l)$ be a family of cochains in
$C^\ast(\overline{P}, \overline{P}^{\text{bad}}; \S^D \otimes \mathbb{Q})$ with compact supports.
Then for such a family $((\hat c_l), (x_l), (\alpha_l))$, we define a pre-Kuranishi space
$\overline{\mathcal{M}}((\hat c_l), (x_l), (\alpha_l))$
(or a linear combination of pre-Kuranishi spaces) by
\begin{align*}
\overline{\mathcal{M}}((\hat c_l), (x_l), (\alpha_l))
&= \overline{\mathcal{M}}_{((\hat c_l), (x_l), ([\overline{P}] \cap \alpha_l))}\\
&\quad + \sum_{m = 2}^\infty \sum_{\star_m} (-1)^\ast
\overline{\mathcal{M}}^m_{(F_m,
(\hat c^i_l), (x^i_l), ([\overline{P}] \cap \alpha^i_l))},
\end{align*}
where the sum of $\star_m$ is taken over all decompositions
\[
\{\hat c_l\} = \coprod_i \{\hat c^i_l\}, \quad \{x_l\} = \coprod_i \{x^i_l\},
\quad \{\alpha_l\} = \coprod_i \{\alpha^i_l\}
\]
as sets, and
the order of each $(\hat c^i_l)_l$ is defined by the order of $(\hat c_l)_l$.
The orders of $(x^i_l)_l$ or $(\alpha^i_l)_l$ are similar.
$\ast$ is the weighted sign of the permutation
\[
\begin{pmatrix}
(\hat c^1_l)_l & \cdots & (\hat c^m_l)_l
& (x^1_l)_l & \cdots & (x^m_l)_l & (\alpha^1_l)_l & \cdots & (\alpha^m_l)_l\\
&(\hat c_l)_l&&&(x_l)_l&&&(\alpha_l)_l&
\end{pmatrix}.
\]
$(F_m)_{m \geq 2}$ is an appropriate family of linear combinations of
\[
((\rho_\ast [\overline{P}])^{i, j}, \dots, (\rho_\ast [\overline{P}])^{i, j},
\epsilon_{\overline{P}}^{i, j}, \dots, \epsilon_{\overline{P}}^{i, j},
(\Delta_\ast [\overline{P}])^{i, j}, \dots, (\Delta_\ast [\overline{P}])^{i, j})_{1 \leq i < j \leq m}
\]
defined in the next section.
The first term
$\overline{\mathcal{M}}_{(\hat c_l), (x_l), ([\overline{P}] \cap \alpha_l)}$
is the main term, and
the second is for the correction of the difference between
$[\overline{P}]$ and $\rho_\ast[\overline{P}]$.
Note that if $(\Sigma_i, z_i, u_i, \phi_i)_{1 \leq i \leq m}$ is in the zero set of the perturbed
multisection of the zero-dimensional component of
$\overline{\mathcal{M}}^m_{(F_m, (\hat c^i_l), (x^i_l), ([\overline{P}] \cap \alpha^i_l))}$,
then each $\Sigma_i$ is connected.
This is because the multisection of $\overline{\mathcal{M}}^m
_{(F_m, (\hat c^i_l), (x^i_l), ([\overline{P}] \cap \alpha^i_l))}$
is the pull back of that of
$(\widehat{\mathcal{M}}^\#, \mathring{K}^2)^{\diamond m}
_{(\bar F_m, (\bar c^i), (\bar x^i), ([\overline{P}] \cap \bar \alpha^i))}$,
and its dimension is $< 0$ if some $\Sigma_i$ is disconnected.
In particular, the genus of each $\Sigma_i$ is $\geq 0$.
Since the total number of $(\rho_\ast [\overline{P}])^{i, j}$,
$\epsilon_{\overline{P}}^{i, j}$ and $(\Delta_\ast [\overline{P}])^{i, j}$ ($1 \leq i < j \leq m$)
contained in each term of $F_m$ is $\geq m - 1$
(in fact, the number of $\epsilon_{\overline{P}}$ is $m - 1$),
the genera of the sequences of curves $(\Sigma_i, z_i, u_i, \phi_i)_{1 \leq i \leq m}$
corresponding to the zeros of the multisection of
the zero-dimensional component of $\overline{\mathcal{M}}^m
_{(F_m, (\hat c^i_l), (x^i_l), ([\overline{P}] \cap \alpha^i_l))}$ are $\geq 0$.
\begin{rem}\label{connected or not}
We do not know whether or not we can choose $(F_m)_{m \geq 2}$ so that
all sequences of holomorphic buildings in the zero-dimensional part of
the fiber products $\overline{\mathcal{M}}^m
_{(F_m, (\hat c^i_l), (x^i_l), ([\overline{P}] \cap \alpha^i_l))}$ are connected
(in the appropriate sense).
However, for the construction of the algebras, it is enough to show that their genera are
$\geq 0$.
\end{rem}
In the next section, we prove that if we choose an appropriate family $(F_m)_{m \geq 2}$,
then the following equation holds true.
\begin{align}
0 &= [\partial \overline{\mathcal{M}}((\hat c_l), (x_l), (\alpha_l))]^0 \notag\\
&=
- [\overline{\mathcal{M}}\bigl(\partial((\hat c_l), (x_l), (\alpha_l))\bigr)]^0 \notag\\
&\quad + \sum_{\blacklozenge} (-1)^{\ast}
\frac{1}{k !} [\overline{\mathcal{M}}((\hat c^1_l), (x^1_l), (\alpha^1_l) \cup (\hat d_1^\ast,
\hat d_2^\ast, \dots, \hat d_k^\ast))]^0\notag\\
&\hspace{70pt}
\times
[\overline{\mathcal{M}}((\hat d_k, \hat d_{k-1}, \dots, \hat d_1) \cup (\hat c^2_l), (x^2_l),
(\alpha^2_l))]^0
\label{boundary formula}
\end{align}
$\partial((\hat c_l), (x_l), (\alpha_l))$ is defined by
\begin{align*}
&\partial((\hat c_l), (x_l), (\alpha_l))\\
&= \sum_j (-1)^{\sum_{l < j} |\hat c_l|}
((\hat c_1, \dots, \partial \hat c_j, \dots, \hat c_{l_-}), (x_l), (\alpha_l))\\
&\quad + \sum_j (-1)^{\sum_l |\hat c_l| + \sum_l |x_l|^\bot + \sum_{l < j} |\alpha_l|}
((\hat c_l), (x_l), (\alpha_1, \dots, \partial \alpha_j, \dots, \alpha_{l_+})),
\end{align*}
where $\partial \alpha$ is defined by
$\partial \alpha = (-1)^{|\alpha|} \delta \alpha = (-1)^{|\alpha|} \alpha \circ \partial$.
The sum $\blacklozenge$ of the last term is taken over
all decompositions
\[
\{\hat c_l\} = \{\hat c^1_l\} \sqcup \{\hat c^2_l\}, \quad \{x_l\} = \{x^1_l\} \sqcup \{x^2_l\},
\quad \{\alpha_l\} = \{\alpha^1_l\} \sqcup \{\alpha^2_l\}
\]
as sets, $k \geq 0$, and all sequences of simplices $d_l$ of $K$ not contained in
$\overline{P}^{\text{bad}}$. (We fix $\theta^D_{d}$ for each simplex $d$ and define $\hat d
= d \theta^D_{d}$.)
The sign $\ast$ of the last term is the weighted sign of the permutation
\[
\begin{pmatrix}
(\hat c^1_l)_l \ (x^1_l)_l \ (\alpha^1_l)_l \ (\hat c^2_l)_l \ (x^2_l)_l \ (\alpha^2_l)_l\\
(\hat c_l)_l \quad (x_l)_l \quad (\alpha_l)_l
\end{pmatrix}.
\]
For the proof of equation (\ref{boundary formula}), we use the fact
\[
\sum_d ([\overline{P}] \cap (\hat d)^\ast) \otimes \hat d = \rho_\ast [\overline{P}].
\]
\subsection{Construction of the correction terms}
\label{algebra for correction}
In this section, we construct $(F_m)_{\geq 2}$ used for the definition of the correction
terms in $\overline{\mathcal{M}}((\hat c_l), (x_l), (\alpha_l))$, and prove the equation
(\ref{boundary formula}).
For that sake, we consider an algebra modeled on the splitting of holomorphic buildings.
For $m \geq 2$, let $A_m = \bigoplus_{n=0}^{\frac{m(m-1)}{2}} A_m^n$ be the $\mathbb{Z}$-graded
super-commutative algebra with coefficient $\mathbb{Q}$ generated by the variables
$\rho_{(e_i, e_j)}$, $\Delta_{(e_i, e_j)}$ and $\epsilon_{(e_i, e_j)}$ ($1 \leq i < j \leq m$),
where the $\mathbb{Z}$-grading is defined by
$\dim \rho_{(e_i, e_j)} = \dim \Delta_{(e_i, e_j)} = 0$ and $\dim \epsilon_{(e_i, e_j)} = 1$.
$\rho$, $\Delta$ and $\epsilon$ are variables corresponding to
$\rho_\ast [\overline{P}]$, $\Delta_\ast [\overline{P}]$, and
$\epsilon_{\overline{P}}$ respectively.
In particular, the parity of the dimension of a monomial in $A_m$ coincides with
that of the corresponding product of simplices.
(We call $n$ dimension in order to distinguish it from the degree $m$.)
We sometimes use the following notation:
$x_{(\sum_i a_i e_i, \sum_i b_i e_i)} = \sum_{i, j} a_i b_j x_{(e_i, e_j)}$, where $x$ is $\rho$,
$\Delta$ or $\epsilon$.
For $m = 1$, we define $A_1 = \mathbb{Q}$.
For each $m \geq 2$, the differential $\partial' : A_m \to A_m$ is defined by
$\partial' \epsilon_{(a, b)} = (-1)^m (\rho_{(a, b)} - \Delta_{(a, b)})$ and
$\partial' \rho_{(a, b)} = \partial' \Delta_{(a, b)} = 0$.
For $m = 1$, we define $\partial' = 0 : \mathcal{A}_1 \to \mathcal{A}_1$.
We define homomorphisms $\tau_i : A_m \to A_{m+1}$ ($1 \leq i \leq m$, $m \geq 2$) by
$\tau_i(x_{(a, b)}) = x_{(\hat \tau_i(a), \hat \tau_i(b))}$,
where each $\hat \tau_i$ is defined by
\[
\hat \tau_i (e_j) = \begin{cases}
e_j &j < i\\
e_i + e_{i + 1} &j = i\\
e_{j + 1} & j > i
\end{cases}.
\]
For example,
\[
\tau_2(\Delta_{(e_1, e_2)} \epsilon_{(e_2, e_3)})
= (\Delta_{(e_1, e_2)} + \Delta_{(e_1, e_3)}) (\epsilon_{(e_2, e_4)} + \epsilon_{(e_3, e_4)}).
\]
For $m = 1$, we define $\tau_1 = \mathrm{id}_{\mathbb{Q}}$.
For $i > m$, we define $\tau_i = 0 : A_m \to A_{m+1}$.
We also define homomorphisms $\Box : A_m \otimes A_{m'} \to A_{m + m'}$
($m, m' \geq 1$) by
\[
\Box(f \otimes g) = (-1)^{(m - 1)m'} f \cdot
\exp (\rho_{(\sum_{1\leq i \leq m} e_i, \sum_{m + 1 \leq j \leq m + m'} e_j)})
\cdot g^{+ m}
\]
where $g^{+ m}$ is the image of $g$ by the homomorphism $A_{m'} \to A_{m + m'}$
defined by $x_{(e_i, e_j)} \mapsto x_{(e_{i + m}, e_{j + m})}$.
For example, if $m = 2$ and $m' = 2$, then
\begin{align*}
&\Box(\Delta_{(e_1, e_2)} \otimes \rho_{(e_1, e_2)} \epsilon_{(e_1, e_2)})\\
&= \Delta_{(e_1, e_2)} \rho_{(e_3, e_4)} \epsilon_{(e_3, e_4)}
\exp (\rho_{(e_1, e_3)} + \rho_{(e_1, e_4)} + \rho_{(e_2, e_3)} + \rho_{(e_2, e_4)}).
\end{align*}
Define a linear subspace $\Ddot A_m \subset A_m$ as follows.
(It is not an ideal.)
For each $1 \leq i \leq m-1$ and each monomial
\[
f = x^{(1)}_{(a_1, b_1)} x^{(2)}_{(a_2, b_2)} \dots x^{(n)}_{(a_n, b_n)},
\]
(each $x^{(j)}$ is $\rho$, $\Delta$ or $\epsilon$)
such that $(a_j, b_j) \neq (e_i, e_{i + 1})$ for all $1 \leq j \leq n$,
we define a monomial
\[
f^{(e_i, e_{i + 1})} = x^{(1)}_{(a'_1, b'_1)} x^{(2)}_{(a'_2, b'_2)} \dots x^{(n)}_{(a'_n, b'_n)}
\]
by permuting $e_i$ and $e_{i + 1}$ appearing in $\{a_j, b_j\}$.
$\Ddot A_m \subset A_m$ is the subspace spanned by $f + f^{(e_i, e_{i + 1})}$
for all such pairs of $i$ and $f$.
Define $\mathcal{A}_m = A_m / \Ddot A_m$.
It is not an algebra, but the following maps are well defined.
(Namely, the corresponding maps on $A_m$ or $A_m \otimes A_{m'}$
induce the following maps.)
\begin{align*}
\partial' &: \mathcal{A}_m \to \mathcal{A}_m & (m \geq 1)\\
\sum_{i = 1}^m (-1)^i e^{\Delta_{(e_i, e_{i + 1})}} \tau_i &: \mathcal{A}_m \to \mathcal{A}_{m + 1}
& (m \geq 1)\\
\Box &: \mathcal{A}_m \otimes \mathcal{A}_{m'} \to \mathcal{A}_{m + m'} & (m, m' \geq 1)
\end{align*}
The well-definedness of the first and the third maps are easy to see.
The well-definedness of the second is proved as follows.
If $f \in A_m$ does not contain any $x_{e_{i_0}, e_{i_0 + 1}}$ ($x = \rho, \Delta, \epsilon$),
then
\begin{align*}
&\sum_{i = 1}^m (-1)^i e^{\Delta_{(e_i, e_{i + 1})}} \tau_i (f + f^{(e_{i_0}, e_{i_0 + 1})})\\
& = \sum_{i \neq i_0, i_0 + 1} (-1)^i
\bigl((e^{\Delta_{(e_i, e_{i + 1})}} \tau_i f)
+ (e^{\Delta_{(e_i, e_{i + 1})}} \tau_i f)^{(\hat \tau_i(e_{i_0}), \hat \tau_i(e_{i_0 + 1}))}\bigr)\\
& \quad + (-1)^{i_0} e^{\Delta_{(e_{i_0}, e_{i_0 + 1})}} \tau_{i_0} (f + f^{(e_{i_0}, e_{i_0 + 1})})\\
& \quad + (-1)^{i_0 + 1} e^{\Delta_{(e_{i_0 + 1}, e_{i_0 + 2})}} \tau_{i_0 + 1}
(f + f^{(e_{i_0}, e_{i_0 + 1})}).
\end{align*}
The sum of the last two terms of the right hand side is an element of $\ddot A_{m+1}$
since
\[
\bigl(e^{\Delta_{(e_{i_0}, e_{i_0 + 1})}} \tau_{i_0} f \bigr)^{(e_{i_0 + 1}, e_{i_0 + 2})}
= \bigl(e^{\Delta_{(e_{i_0 + 1}, e_{i_0 + 2})}} \tau_{i_0 + 1}(f^{(e_{i_0}, e_{i_0 + 1})})
\bigr)^{(e_{i_0}, e_{i_0 + 1})}
\]
and
\[
\bigl(e^{\Delta_{(e_{i_0}, e_{i_0 + 1})}} \tau_{i_0}(f^{(e_{i_0}, e_{i_0 + 1})})
\bigr)^{(e_{i_0 + 1}, e_{i_0 + 2})}
= \bigl(e^{\Delta_{(e_{i_0 + 1}, e_{i_0 + 2})}} \tau_{i_0 + 1} f \bigr)^{(e_{i_0}, e_{i_0 + 1})}.
\]
Hence $\sum_{i = 1}^m (-1)^i e^{\Delta_{(e_i, e_{i + 1})}} \tau_i : \mathcal{A}_m \to \mathcal{A}_{m+1}$
is well defined.
Let $\mathcal{A} = (\bigoplus_{m = 1}^\infty \mathcal{A}_m^{m - 1})^\wedge$ be the completion
with respect to the degree $m$.
We also define $\mathcal{A}' = (\bigoplus_{m = 2}^\infty \mathcal{A}_m^{m - 2})^\wedge$.
In this section, we prove that the map $\mathcal{A} \to \mathcal{A}'$ defined by
\[
F \mapsto \partial' F + \sum_i (-1)^i e^{\Delta{(e_i, e_{i + 1})}} \tau_i F + \Box (F \otimes F)
\]
has a zero $F = F_1 + F_2 + \dots \in \mathcal{A}$ such that
$F \equiv 1 \in \mathcal{A} / (\bigoplus_{m = 2}^\infty \mathcal{A}_m^{m - 1})^\wedge \cong \mathcal{A}_1$.
Equation (\ref{boundary formula}) holds for such a zero $F$
if we replace the variables $\rho_{(e_i, e_j)}$, $\epsilon_{(e_i, e_j)}$ and $\Delta_{(e_i, e_j)}$
with $(\rho_\ast [\overline{P}])^{i, j}$, $\epsilon_{\overline{P}}^{i, j}$
and $(\Delta_\ast [\overline{P}])^{i, j}$ respectively.
This can be seen as follows.
Equation (\ref{boundary of MM}) implies that
for any $F_m \in A_m^{m-1}$ and $((\hat c_l), (x_l), (\alpha_l))$,
\begin{align*}
&\sum_{\star_m} (-1)^\ast \partial' \bigl(
\overline{\mathcal{M}}^m_{(F_m,
(\hat c^i_l), (x^i_l), ([\overline{P}] \cap \alpha^i_l))}\bigr)\\
&= - \sum_{\star_m} (-1)^\ast
\overline{\mathcal{M}}^m_{(F_m,
\partial((\hat c^i_l), (x^i_l), ([\overline{P}] \cap \alpha^i_l)))}\\
&\quad + \sum_{\star_m} (-1)^\ast
\overline{\mathcal{M}}^m_{(\partial' F_m,
(\hat c^i_l), (x^i_l), ([\overline{P}] \cap \alpha^i_l))}\\
&\quad + \sum_{\star_{m + 1}} (-1)^\ast
\overline{\mathcal{M}}^{m+1}_{(
\sum_i (-1)^i e^{\Delta{(e_i, e_{i + 1})}} \tau_i F_m,
(\hat c^i_l), (x^i_l), ([\overline{P}] \cap \alpha^i_l))},
\end{align*}
On the other hand,
for any $((\hat c^i_l)_{1 \leq i \leq m + m'}, (x^i_l)_{1 \leq i \leq m + m'},
(\alpha^i_l)_{1 \leq i \leq m + m'})$,
$F_m \in A_m^{m-1}$ and $F_{m'} \in A_{m'}^{m'-1}$,
\begin{align*}
&\sum (-1)^\ast \frac{1}{k !} [
\overline{\mathcal{M}}^m_{(F_m,
(\hat c^i_l)_{i = 1}^m, (x^i_l)_{i = 1}^m,
([\overline{P}] \cap \alpha^i_l)_{i = 1}^m \cup ((\hat d_1^\ast)^m,
\dots, (\hat d_k^\ast)^m))}]^0\\
&\hphantom{\sum (-1)^\ast \frac{1}{k !}}
\cdot [
\overline{\mathcal{M}}^{m'}_{(F_{m'},
((\hat d_k)^1, \dots, (\hat d_1)^1) \cup
(\hat c^{i+m}_l)_{i = 1}^{m'},
(x^{i+m}_l)_{i = 1}^{m'},
([\overline{P}] \cap \alpha^{i+m}_l)_{i = 1}^{m'})}]^0\\
&= [\overline{\mathcal{M}}^{m+ m'}_{(- \Box
(F_m \otimes F_{m'}),
(\hat c^i_l)_{i = 1}^{m + m'}, (x^i_l)_{i = 1}^{m + m'},
([\overline{P}] \cap \alpha^i_l)_{i = 1}^{m + m'})}]^0,
\end{align*}
where the sum is taken over all $k \geq 0$ and all sequences of simplices
$d_l \in K$ such that $d_l \not \subset \overline{P}^{\text{bad}}$, and
$\ast$ is the weighted sign of the permutation
\[
\begin{pmatrix}
(\hat c^i_l)_{1 \leq i \leq m} \ (x^i_l)_{1 \leq i \leq m} \ (\alpha^i_l)_{1 \leq i \leq m}
\ (\hat c^{i+m}_l)_{1 \leq i \leq m'} \ (x^{i+m}_l)_{1 \leq i \leq m'}
\ (\alpha^{i+m}_l)_{1 \leq i \leq m'}\\
(\hat c^i_l)_{1 \leq i \leq m + m'} \quad (x^i_l)_{1 \leq i \leq m + m'} \quad
(\alpha^i_l)_{1 \leq i \leq m + m'}
\end{pmatrix}.
\]
These equations imply that equation (\ref{boundary formula}) holds for a zero $F$.
The quotient space $\mathcal{A}_m = A_m / \Ddot A_m$ corresponds to the fact that
we can permute the $i$-th holomorphic building and $(i + 1)$-th holomorphic building
in $(\overline{\mathcal{M}} \times \dots \times \overline{\mathcal{M}})_{((l_{i, j}), (l_{i, \pm}), (\mu_i))}$
if $l_{i, i+1} = 0$.
Note that the homology of $\partial' : \mathcal{A}_m^\ast \to \mathcal{A}_m^\ast$ is zero
at $\ast \neq 0$. This is because that K\"unneth formula implies that
the homology of $A_m^\ast \cong (A_2^\ast)^{\otimes \frac{m(m-1)}{2}}$ is zero
at $\ast \neq 0$, and there exists a splitting $T : \mathcal{A}_m^\ast \to A_m^\ast$.
The splitting $T$ is defined as follows.
For a monomial
\[
f = x^{(1)}_{(a_1, b_1)} x^{(2)}_{(a_2, b_2)} \dots x^{(n)}_{(a_n, b_n)},
\]
we define a subgroup $\mathfrak{S}_f \subset \mathfrak{S}_m$ by
\[
\mathfrak{S}_f = \{\sigma \in \mathfrak{S}_m; \sigma(a_j) < \sigma(b_j) \text{ for all } j\}.
\]
Then $Tf$ is defined by
\[
Tf = \frac{1}{\# \mathfrak{S}_f} \sum_{\sigma \in \mathfrak{S}_f} \mathop{\mathrm{sign}}\nolimits \sigma \cdot
x^{(1)}_{(\sigma(a_1), \sigma(b_1))} x^{(2)}_{(\sigma(a_2), \sigma(b_2))} \dots
x^{(n)}_{(\sigma(a_n), \sigma(b_n))}.
\]
Starting with $F_1 = 1 \in \mathcal{A}_1$,
we inductively construct $F_{\leq m} = F_1 + \dots + F_m \in \bigoplus_{l = 1}^m
\mathcal{A}_l^{l - 1}$ such that
\begin{equation}
\partial' F_{\leq m} + \sum_i (-1)^i e^{\Delta{(e_i, e_{i + 1})}} \tau_i F_{\leq m-1}
+ \Box (F_{\leq m-1} \otimes F_{\leq m-1}) \equiv 0 \label{F_m eq}
\end{equation}
in $\mathcal{A}' / (\bigoplus_{l = m + 1}^\infty \mathcal{A}_l^{l - 2})^\wedge$.
First we define $F_{\leq 2} = F_1 + F_2 \in \mathcal{A}_1 \oplus \mathcal{A}_2^1$ by
\begin{align*}
F_{\leq 2} = 1 - \frac{1}{k !} \sum_{k = 1}^\infty
(&\underbrace{\epsilon_{(e_1, e_2)} \Delta_{(e_1, e_2)} \cdots \Delta_{(e_1, e_2)}}_k\\
&+
\underbrace{\rho_{(e_1, e_2)} \epsilon_{(e_1, e_2)} \Delta_{(e_1, e_2)} \cdots
\Delta_{(e_1, e_2)}}_k\\
&+ \dots +
\underbrace{\rho_{(e_1, e_2)} \cdots \rho_{(e_1, e_2)} \epsilon_{(e_1, e_2)}}_k)
\end{align*}
It is easy to check that this satisfies equation (\ref{F_m eq}) for $m = 2$.
Next assuming that we have already constructed
$F_{\leq m-1} \in \bigoplus_{l = 1}^m \mathcal{A}_l^{l-1}
$, we need to prove that
there exists a required $F_{\leq m}$ ($m \geq 3$).
Since $\partial'$ is exact at $n \geq 1$,
it is enough to show that
\begin{equation}
\partial' \Bigl(
\sum_i (-1)^i e^{\Delta{(e_i, e_{i + 1})}} \tau_i F_{\leq m-1} + \Box (F_{\leq m-1} \otimes F_{\leq m-1})
\Bigr)
\equiv 0 \label{A closed}
\end{equation}
in $(\bigoplus_{l = 3}^\infty \mathcal{A}_l^{l-3})^\wedge / (\bigoplus_{l = m}^\infty \mathcal{A}_l^{l-3})^\wedge$.
Since $F_{\leq m-1} = F_1 + \dots + F_{m-1}$ satisfies
\[
\partial' F_{\leq m-1} + \sum_i (-1)^i e^{\Delta{(e_i, e_{i + 1})}} \tau_i F_{\leq m-1} + \Box (F_{\leq m-1} \otimes F_{\leq m-1}) \equiv 0
\]
in $\mathcal{A}' / (\bigoplus_{l = m}^\infty \mathcal{A}_l^{l - 2})^\wedge$,
we see that
\begin{align*}
&\partial' \Bigl(
\sum_i (-1)^i e^{\Delta{(e_i, e_{i + 1})}} \tau_i F_{\leq m-1} + \Box (F_{\leq m-1} \otimes F_{\leq m-1})
\Bigr)\\
&=
\sum_i (-1)^{i + 1} e^{\Delta{(e_i, e_{i + 1})}} \tau_i \partial' F_{\leq m-1}
+ \Box \Bigl(\partial' F_{\leq m-1} \otimes \sum_{1 \leq j \leq m - 1} (-1)^j F_j\Bigr)\\
&\quad - \Box (F_{\leq m-1} \otimes \partial' F_{\leq m-1})\\
&=
\sum_i (-1)^i e^{\Delta{(e_i, e_{i + 1})}} \tau_i \Bigl(
\sum_j (-1)^j e^{\Delta{(e_j, e_{j + 1})}} \tau_j F_{\leq m-1} + \Box (F_{\leq m-1} \otimes F_{\leq m-1})
\Bigr)\\
&\quad - \Box \Bigl(\Bigl(\sum_i (-1)^i e^{\Delta{(e_i, e_{i + 1})}} \tau_i F_{\leq m-1}
+ \Box (F_{\leq m-1} \otimes F_{\leq m-1}) \Bigr)\\
&\hphantom{\quad - \Box \Bigl(\Bigl(}
\otimes \sum_{1 \leq j \leq m - 1} (-1)^j F_j\Bigr)\\
&\quad +\Box \Bigl(F_{\leq m-1} \otimes \Bigl(\sum_i (-1)^i e^{\Delta{(e_i, e_{i + 1})}} \tau_i F_{\leq m-1}
+ \Box (F_{\leq m-1} \otimes F_{\leq m-1})\Bigr)\Bigr).
\end{align*}
By direct calculation, it is easy to see that the following equations hold true.
{\belowdisplayskip= 0pt
\[
\Bigl(\sum_i (-1)^i e^{\Delta{(e_i, e_{i + 1})}} \tau_i\Bigr) \circ
\Bigl(\sum_j (-1)^j e^{\Delta{(e_j, e_{j + 1})}} \tau_j\Bigr) = 0,
\]
}
\begin{multline*}
\sum_i (-1)^i e^{\Delta{(e_i, e_{i + 1})}} \tau_i \Box (f \otimes g)
- \Box \Bigl(\Bigl(\sum_i (-1)^i e^{\Delta{(e_i, e_{i + 1})}} \tau_i f\Bigr) \otimes (-1)^{\deg g} g\Bigr)\\
+ \Box \Bigl(f \otimes \Bigl(\sum_i (-1)^i e^{\Delta{(e_i, e_{i + 1})}} \tau_i g\Bigr)\Bigr) = 0,
\end{multline*}
\[
\Box (f \otimes \Box (g \otimes h)) - \Box ( \Box (f \otimes g) \otimes (-1)^{\deg h} h) = 0.
\]
Therefore
\[
\partial' \Bigl(
\sum_i (-1)^i e^{\Delta{(e_i, e_{i + 1})}} \tau_i F_{\leq m-1} + \Box (F_{\leq m-1} \otimes F_{\leq m-1})
\Bigr)
\equiv 0
\]
in $\bigoplus_{l = 3} \mathcal{A}_l^{l-3} / \bigoplus_{l = m} \mathcal{A}_l^{l-3}$,
and we can construct a required
$F_{\leq m} = F_1 + \dots + F_m \in \bigoplus_{l = 1}^m \mathcal{A}_l^{l-1}$.
\begin{rem}
In fact, we do not need to use $\mathcal{A}_m$, and we can replace $\mathcal{A}_m$ with $A_m$.
However, for the construction of the correction terms for $X$
in Section \ref{correction terms for X}, we need to use a counterpart of $\mathcal{A}_m$.
\end{rem}
\subsection{Construction of the algebras}\label{construction of algebra}
Using the virtual fundamental chains of the $0$-dimensional components of
the pre-Kuranishi spaces in the previous section,
we construct the algebra of symplectic field theory.
We mainly follow the construction explained in \cite{EGH00}.
First we consider general SFT.
We do not consider the $H_2(Y; \mathbb{Z})$-grading or the $H_1(Y; \mathbb{Z})$-grading for simplicity.
(See the above paper for these gradings.)
For each simplex $c$ of $K$ not contained in $\overline{P}^{\text{bad}}$,
we fix an orientation $\theta^D_c$ and define $\hat c = c \theta^D_c$.
We use the following variables:
$q_{\hat c^\ast}$ and $p_{\hat c}$ for each simplex $c$ of $K$ not contained in $\overline{P}^{\text{bad}}$,
$t_x$ for each cycle $x$ of $K^0$, and $\hbar$.
The $\mathbb{Z}/2$-degrees of these variables are defined by
$|q_{\hat c^\ast}| = |p_{\hat c}| = \dim c + \mathop{\mathrm{ind}}\nolimits \mathring{D}_\gamma^+$
($\gamma \in |c|$), $|t_x| = \mathop{\mathrm{codim}}\nolimits_Y x$ and $|\hbar| = 0$.
We define the energies of these variables by
$e(q_{\hat c^\ast}) = L_\gamma$ and $e(p_{\hat c}) = - L_\gamma$ for each $c$,
where $\gamma \in |c|$ is an arbitrary periodic orbit and $L_\gamma$ is its period, and
$e(t_x) = e(\hbar) = 0$.
The algebra $\mathcal{W}_Y = \mathcal{W}_{(Y, \lambda, K_Y, \overline{K}_Y^0)}$ is defined as follows.
Its elements are formal series
\[
\sum_{(\hat c_i^\ast), (\hat c'_i)}
f_{(\hat c_i^\ast), (\hat c'_i)}(t, \hbar) q_{\hat c_1^\ast} q_{\hat c_2^\ast} \dots
q_{\hat c_{k_q}^\ast} p_{\hat c'_1} p_{\hat c'_2} \dots p_{\hat c'_{k_p}},
\]
where $f_{(\hat c_i^\ast), (\hat c'_i)}(t, \hbar) \in \mathbb{R}[[t, \hbar]]$ are formal series
of the variables $t_x$ and $\hbar$, and the infinite sum is taken over
all pairs of sequences $(\hat c_i)$ and $(\hat c'_i)$ with the following Novikov condition:
for any $C \geq 0$, the number of the terms with $\sum_i e(p_{\hat c'_i}) \geq -C$
is finite.
(This is equivalent to the condition that for each sequence $(\hat c'_i)$,
all but finite sequences $(\hat c_i^\ast)$ satisfy $f_{(\hat c_i^\ast), (\hat c'_i)} = 0$.)
We sometimes use the following notation:
for a linear combination $\sum_i r_i \hat c_i$, we define
$p_{\sum_i r_i \hat c_i} = \sum_i r_i p_{\hat c_i}$.
We use the similar notation for variables $q$ and $t$.
The associative product $\circ$ of $\mathcal{W}_Y$ is defined by the following commutative
relations: all variables are super-commutative except
\[
[p_{\hat c}, q_\alpha]
= p_{\hat c} \circ q_\alpha - (-1)^{|p_{\hat c}| \cdot |q_\alpha|} q_\alpha \circ p_{\hat c}
= \langle \hat c, \alpha \rangle \hbar.
\]
We often omit the symbol $\circ$ and denote the product $f \circ g$ by $fg$.
For each $\kappa \geq 0$, we define a submodule $\mathcal{W}_Y^{\leq \kappa} \subset \mathcal{W}_Y$
by imposing the condition $\sum_i e(q_{\hat c_i^\ast}) + \sum_i e(p_{\hat c'_i})
\leq \kappa$.
(This condition is stronger than the Novikov condition.)
For each triple $(C_0, C_1, C_2)$,
we define a submodule $I^{\leq \kappa}_{C_0, C_1, C_2} \subset \mathcal{W}_Y^{\leq \kappa}$ by
\begin{align*}
I^{\leq \kappa}_{C_0, C_1, C_2} &= \Bigl\{\sum
a_{(x_i), (\hat c_i^\ast), (\hat c'_i), g}\ t_{x_1} \dots t_{x_{k_t}}
q_{\hat c_1^\ast} \dots q_{\hat c_{k_q}^\ast}
p_{\hat c'_1} \dots p_{\hat c'_{k_p}} \hbar^g \in \mathcal{W}_Y^{\leq \kappa};\\
&\quad \quad a_{(x_i), (\hat c_i^\ast), (\hat c'_i), g} = 0
\text{ for all } ((x_i)_{i = 1}^{k_t}, (\hat c_i^\ast),_{i = 1}^{k_q} (\hat c'_i)_{i = 1}^{k_p}, g)
\text{ such that}\\
&\quad \quad k_t \leq C_0,\, \widetilde{g} \leq C_1 \text{ and }
\sum e(p_{\hat c'_i}) \geq - C_2 \Bigr\},
\end{align*}
where
\[
\widetilde{g} = g + \frac{1}{2}(k_t + k_q + k_p)
- \frac{\sum_i e(q_{\hat c_i^\ast}) + \sum_j e(p_{\hat c'_j})}
{L_{\min}}.
\]
($L_{\min}$ is the minimal period of the periodic orbits of $R_\lambda$.)
We note that
\[
\mathcal{W}_Y \cong \varprojlim_{C_2} \varinjlim_{\kappa} \varprojlim_{C_0, C_1}
\mathcal{W}_Y^{\leq \kappa} / I^{\leq \kappa}_{C_0, C_1, C_2}.
\]
The multiplication of $\mathcal{W}_Y$ defines the maps
\[
\mathcal{W}_Y^{\leq \kappa_1} / I^{\leq \kappa_1}_{C_0, C_1 + \kappa_2 L_{\min}^{-1},
C_2 + \kappa_2}
\times \mathcal{W}_Y^{\leq \kappa_2} / I^{\leq \kappa_2}_{C_0, C_1 + \kappa_1 L_{\min}^{-1}, C_2}
\to \mathcal{W}_Y^{\leq \kappa_1 + \kappa_2} / I^{\leq \kappa_1 + \kappa_2}_{C_0, C_1, C_2}.
\]
Let $(\hbar^{-1} \mathcal{W}_Y^{\leq 0})^+ \subset \hbar^{-1} \mathcal{W}_Y^{\leq 0}$
be the submodule defined by
\begin{align*}
&(\hbar^{-1} \mathcal{W}_Y^{\leq 0})^+ \\
&= \bigl\{\sum
a_{(x_i), (\hat c_i^\ast), (\hat c'_i), g}\ t_{x_1} \dots t_{x_{k_t}}
q_{\hat c_1^\ast} \dots q_{\hat c_{k_q}^\ast}
p_{\hat c'_1} \dots p_{\hat c'_{k_p}} \hbar^g \in \hbar^{-1} \mathcal{W}_Y^{\leq 0};
\widetilde{g} \geq 0\bigr\},
\end{align*}
and $(\hbar^{-1} \mathcal{W}_Y^{\leq 0})^+_{C_0, C_1, C_2} \subset (\hbar^{-1} \mathcal{W}_Y^{\leq 0})^+$
be the submodule defined by
\begin{align*}
&(\hbar^{-1} \mathcal{W}_Y^{\leq 0})^+_{C_0, C_1, C_2} \\
&= \Bigl\{\sum
a_{(x_i), (\hat c_i^\ast), (\hat c'_i), g}\ t_{x_1} \dots t_{x_{k_t}}
q_{\hat c_1^\ast} \dots q_{\hat c_{k_q}^\ast}
p_{\hat c'_1} \dots p_{\hat c'_{k_p}} \hbar^g \in (\hbar^{-1} \mathcal{W}_Y^{\leq 0})^+;\\
&\quad \quad a_{(x_i), (\hat c_i^\ast), (\hat c'_i), g} = 0
\text{ for all } ((x_i)_{i = 1}^{k_t}, (\hat c_i^\ast),_{i = 1}^{k_q} (\hat c'_i)_{i = 1}^{k_p}, g)
\text{ such that}\\
&\quad \quad k_t \leq C_0,\, \widetilde{g} \leq C_1 \text{ and }
\sum e(p_{\hat c'_i}) \geq - C_2 \Bigr\}
\end{align*}
for each triple $(C_0, C_1, C_2)$.
If we fix a triple $(\overline{C}_0, \overline{C}_1, \overline{C}_2)$,
then, choosing a compatible family of perturbations $\mathcal{B}$ of the multisections
of finite number of pre-Kuranishi spaces and
using their virtual fundamental chains,
we can define the generating function
$\mathcal{H} = \mathcal{H}_{(Y, \lambda, K_Y, K_Y^0, K_Y^2, J, \mathcal{B})}
= \hbar^{-1} \sum_g \mathcal{H}_g \hbar^g \in
(\hbar^{-1} \mathcal{W}_Y^{\leq 0})^+
/ (\hbar^{-1} \mathcal{W}_Y^{\leq 0})^+_{\overline{C}_0, \overline{C}_1, \overline{C}_2}$ by
\[
\mathcal{H}_g = \sum_{k_q, k_t, k_p \geq 0} \frac{1}{k_q !k_t ! k_p !}
\bigl[\overline{\mathcal{M}}_g(\underbrace{\mathbf{q}, \dots, \mathbf{q}}_{k_q};
\underbrace{\mathbf{t}, \dots, \mathbf{t}}_{k_t};
\underbrace{\mathbf{p}, \dots, \mathbf{p}}_{k_p})\bigr]^0,
\]
where $\mathbf{q} = \sum_c q_{\hat c^\ast} \hat c$,
$\mathbf{t} = \sum_x t_x x$ and
$\mathbf{p} = \sum_c p_{\hat c} \hat c^\ast$ are formal series.
We need to check that
$\mathcal{H}$ is indeed an element of $(\hbar^{-1} \mathcal{W}_Y^{\leq 0})^+$, that is, every
holomorphic building satisfies
\[
\sum_j L_{\gamma_{+\infty_j}} - \sum_i L_{\gamma_{-\infty_i}} \geq 0
\]
and
\[
\tilde g = g + \frac{1}{2}(k_t + k_q + k_p)
+ \frac{\sum_j L_{\gamma_{+\infty_j}} - \sum_i L_{\gamma_{-\infty_i}}}
{L_{\min}} \geq 1,
\]
where $g$ is its genus, $k_t$, $k_q$ and $k_p$ are the numbers of its marked points,
$-\infty$-limit circles, and $+\infty$-limit circles respectively, and
$L_{\gamma_{\pm\infty_i}}$ are the periods of the periodic orbits on its limit circles.
The former is because the left hand side is the $E_{\hat \omega}$-energy.
The latter is proved as follows.
First note that $\tilde g - 1$ is additive with respect to disjoint union or
gluing at limit circles.
Hence it is enough to prove the case of a connected holomorphic building of height one.
Assume that there exists a connected holomorphic building $(\Sigma, z, u, \phi)$
of height one such that $\tilde g < 1$.
Since $\tilde g < 1$ implies $g = 0$ and $k_t \leq 1$, $u$ is not a constant map.
Since $\sigma \circ u$ cannot attain a maximum at the interior, it implies that
$k_p \geq 1$.
Therefore $\tilde g < 1$ implies $k_q = 0$ and $k_p = 1$.
However, this implies
\[
\frac{\sum_j L_{\gamma_{+\infty_j}} - \sum_i L_{\gamma_{-\infty_i}}}
{L_{\min}}
= \frac{L_{\gamma_{+\infty_1}}}
{L_{\min}}
\geq 1,
\]
which contradict the assumption $\tilde g < 1$.
Therefore $\mathcal{H}$ is an element of $(\hbar^{-1} \mathcal{W}_Y^{\leq 0})^+$.
We also note that $\mathcal{H}$ has the odd degree.
Define a differential $\delta : \mathcal{W}_Y \to \mathcal{W}_Y$ by
$\delta q_\alpha = q_{\delta \alpha}$,
$\delta t_x = 0$,
$\delta p_{\hat c} = (-1)^{1 + |\hat c|} p_{\partial \hat c}$ and $\delta \hbar = 0$.
(Note that this is well defined, that is, $\delta [p_{\hat c}, q_\alpha] = 0$.)
Note the following equations:
\[
\sum_c \delta q_{\hat c^\ast} \hat c = \sum_c q_{\hat c^\ast} \partial \hat c, \quad
\sum_c \delta p_{\hat c} \hat c^\ast = \sum_c p_{\hat c} \partial \hat c^\ast.
\]
(Recall that we have defined $\partial \alpha$ by
$\partial \alpha = (-1)^{|\alpha|} \delta \alpha$ for a cochain $\alpha$.)
We also define the differential $\delta$ on $\hbar^{-1} \mathcal{W}_Y$ similarly.
Then equation (\ref{boundary formula}) implies
\begin{equation}
\delta \mathcal{H} - \mathcal{H} \circ \mathcal{H} = 0
\label{main eq}
\end{equation}
in $(\hbar^{-1} \mathcal{W}_Y^{\leq 0})^+
/ (\hbar^{-1} \mathcal{W}_Y^{\leq 0})^+_{\overline{C}_0, \overline{C}_1, \overline{C}_2}$.
For each four-tuple $(\kappa, C_0, C_1, C_2)$ such that
$\overline{C}_0 \geq C_0$, $\overline{C}_1 \geq C_1 + \frac{\kappa}{L_{\min}}$
and $\overline{C}_2 \geq C_2 + \kappa$,
define a linear map $D_Y = D_{(Y, \lambda, K_Y, K_Y^0, K_Y^2, J, \mathcal{B})}
: \mathcal{W}_Y^{\leq \kappa} / I^{\leq \kappa}_{C_0, C_1, C_2} \to
\mathcal{W}_Y^{\leq \kappa} / I^{\leq \kappa}_{C_0, C_1, C_2}$ by
\[
D_Y f = \delta f - [\mathcal{H}, f].
\]
Then $D_Y$ is a differential, that is,
\begin{align}
D_Y^2 &= 0 \label{D_Y^2}\\
D_Y(fg) &= (D_Y f) g + (-1)^{|f|} f D_Y g \label{D_Y diff alg}
\end{align}
(\ref{D_Y^2}) is a consequence of (\ref{main eq}).
(\ref{D_Y diff alg}) holds if the multiplications are well defined.
Namely, for $f \in \mathcal{W}_Y^{\leq \kappa_1}
/ I^{\leq \kappa_1}_{C_0, C_1 + \kappa_2 L_{\min}^{-1}, C_2 + \kappa_2}$
and $g \in \mathcal{W}_Y^{\leq \kappa_2}
/ I^{\leq \kappa_2}_{C_0, C_1 + \kappa_1 L_{\min}^{-1}, C_2}$,
(\ref{D_Y diff alg}) holds in $\mathcal{W}_Y^{\leq \kappa_1 + \kappa_2}
/ I^{\leq \kappa_1 + \kappa_2}_{C_0, C_1, C_2}$.
We denote the homology of the chain complex
$(\mathcal{W}_Y^{\leq \kappa} / I^{\leq \kappa}_{C_0, C_1, C_2}, D_Y)$
by $H^\ast(\mathcal{W}_Y^{\leq \kappa} / I^{\leq \kappa}_{C_0, C_1, C_2}, D_Y)
= \mathop{\mathrm{Ker}}\nolimits D_Y / \mathop{\mathrm{Im}}\nolimits D_Y$.
We will prove that the homology
\[
H^\ast(\mathcal{W}_{(Y, \lambda, K_Y, \overline{K}_Y^0)}^{\leq \kappa}
/ I^{\leq \kappa}_{C_0, C_1, C_2}, D_{(Y, \lambda, K_Y, K_Y^0, K_Y^2, J, \mathcal{B})})
\]
is independent of the choice of $(K_Y, K_Y^0, K_Y^2, J, \mathcal{B})$ in Section \ref{independence}
(Lemma \ref{short concordance for Y}).
Therefore we can define the limit
\begin{align*}
&H^\ast(\mathcal{W}_{(Y, \lambda, \overline{K}_Y^0)}, D_{(Y, \lambda, \overline{K}_Y^0)})\\
&= \varprojlim_{C_2} \varinjlim_{\kappa} \varprojlim_{C_0, C_1}
H^\ast(\mathcal{W}_{(Y, \lambda, K_Y, \overline{K}_Y^0)}^{\leq \kappa}
/ I^{\leq \kappa}_{C_0, C_1, C_2}, D_{(Y, \lambda, K_Y, K_Y^0, K_Y^2, J, \mathcal{B})}).
\end{align*}
(\ref{D_Y diff alg}) implies that this is an algebra.
We will also prove that this is independent of the choice of the contact form $\lambda$
of the contact manifold $(Y, \xi)$ in Section \ref{independence}.
\begin{rem}
We can use the spectral sequence defined by the filtration given by the energy
$\sum e(q_{\hat c_i^\ast}) + \sum e(p_{\hat c'_i})$ for each
$H^\ast(\mathcal{W}_Y^{\leq \kappa} / I^{\leq \kappa}_{C_0, C_1, C_2}, D_Y)$
since $\mathcal{W}_Y^{\leq \kappa} / I^{\leq \kappa}_{C_0, C_1, C_2}$ is finite dimensional.
\end{rem}
Next we briefly explain the construction of rational symplectic field theory.
Define a super-commutative algebra $\mathcal{P}_Y = \mathcal{P}_{(Y, \lambda, K_Y,
\overline{K}^0_Y)}$ by $\mathcal{P}_Y = \mathcal{W}_Y|_{\hbar = 0}$.
It is regarded as a quotient of $\mathcal{W}_Y$.
Its (graded) Poisson structure is defined by
\begin{align*}
\{f, g\} &= (\hbar^{-1} [f, g])|_{\hbar = 0}\\
&= \sum_c \biggl(\frac{\overleftarrow{\partial} f}{\partial p_{\hat c}}
\frac{\overrightarrow{\partial} g}{\partial q_{\hat c^\ast}}
- (-1)^{|f| |g|} \frac{\overleftarrow{\partial} g}{\partial p_{\hat c}}
\frac{\overrightarrow{\partial} f}{\partial q_{\hat c^\ast}}\biggr),
\end{align*}
where $\overrightarrow{\partial}$ and $\overleftarrow{\partial}$ are
differential from left and right respectively.
It is easy to check that it is indeed a Poisson structure, that is, it satisfies the following
equations:
\begin{align*}
\{f, gh\} &= \{f, g\} h + (-1)^{|f| |g|} g\{f, h\},\\
\{g, f\} &= -(-1)^{|f| |g|} \{f, g\},\\
\{\{f, g\}, h\} &= \{f, \{g, h\}\} - (-1)^{|f| |g|} \{g, \{f, h\}\}.
\end{align*}
The differential $\delta : \mathcal{P}_Y \to \mathcal{P}_Y$ is defined similarly
to the case of $\mathcal{W}_Y$.
For each $\kappa \geq 0$, we define a submodule $\mathcal{P}^{\leq \kappa}_Y
\subset \mathcal{P}_Y$ by imposing the condition
$\sum_i e(q_{\hat c_i^\ast}) + \sum_i e(p_{\hat c'_i}) \leq \kappa$.
For each triple $(\kappa, C_0, C_2)$,
We define a submodule $I^{\leq \kappa}_{C_0, C_2} \subset
\mathcal{P}_Y^{\leq \kappa}$ by
\begin{align*}
I^{\leq \kappa}_{C_0, C_2} &= \Bigl\{\sum
a_{(x_i), (\hat c_i^\ast), (\hat c'_i)}\ t_{x_1} \dots t_{x_{k_t}}
q_{\hat c_1^\ast} \dots q_{\hat c_{k_q}^\ast}
p_{\hat c'_1} \dots p_{\hat c'_{k_p}} \in \mathcal{P}_Y^{\leq \kappa};\\
&\quad \quad a_{(x_i), (\hat c_i^\ast), (\hat c'_i)} = 0
\text{ for all } ((x_i)_{i = 1}^{k_t}, (\hat c_i^\ast),_{i = 1}^{k_q} (\hat c'_i)_{i = 1}^{k_p})
\text{ such that}\\
&\quad \quad k_t \leq C_0 \text{ and }
\sum e(p_{\hat c'_i}) \geq - C_2 \Bigr\}.
\end{align*}
In this case the following holds true.
\[
\mathcal{P}_Y \cong \varprojlim_{C_2} \varinjlim_{\kappa} \varprojlim_{C_0}
\mathcal{P}_Y^{\leq \kappa} / I^{\leq \kappa}_{C_0, C_2}.
\]
Note that the Poisson bracket induces the maps
\[
\mathcal{P}_Y^{\leq \kappa_1} / I^{\leq \kappa_1}_{C_0, C_2 + \kappa_2}
\times \mathcal{P}_Y^{\leq \kappa_2} / I^{\leq \kappa_2}_{C_0, C_2 + \kappa_1}
\to \mathcal{P}_Y^{\leq \kappa_1 + \kappa_2} / I^{\leq \kappa_1 + \kappa_2}_{C_0, C_2}.
\]
More generally, the Poisson bracket induces the maps
\begin{align}
&(\mathcal{P}_Y^{\leq \kappa^\circ_1} + I^{\leq \kappa_1}_{C_0, C_2})
/ I^{\leq \kappa_1}_{C_0, C_2 + \kappa^\circ_2} \times
(\mathcal{P}_Y^{\leq \kappa^\circ_2} + I^{\leq \kappa_2}_{C_0, C_2})
/ I^{\leq \kappa_1}_{C_0, C_2 + \kappa^\circ_1} \notag\\
&\hspace{180pt}
\to \mathcal{P}_Y^{\leq \kappa_1 + \kappa_2} / I^{\leq \kappa_1 + \kappa_2}_{C_0, C_2}
\label{Poisson bracket for fiber product}
\end{align}
for $\kappa^\circ_i \leq \kappa_i$ ($i = 1,2$).
Note that for $\kappa^\circ \leq \kappa$ and $C^\circ \leq C$,
$(\mathcal{P}_Y^{\leq \kappa^\circ} + I^{\leq \kappa}_{C_0, C^\circ_2})
/ I^{\leq \kappa}_{C_0, C_2}$ is the fiber product of
$\mathcal{P}_Y^{\leq \kappa^\circ} / I^{\leq \kappa^\circ}_{C_0, C^\circ_2}$
and $\mathcal{P}_Y^{\leq \kappa} / I^{\leq \kappa}_{C_0, C_2}$ over
$\mathcal{P}_Y^{\leq \kappa} / I^{\leq \kappa}_{C_0, C^\circ_2}$.
Equation (\ref{main eq}) implies that $\mathcal{H}_0 \in \mathcal{P}_Y^{\leq 0}
/ I^{\leq 0}_{\overline{C}_0, \overline{C}_2}$ satisfies
\begin{equation}
\delta \mathcal{H}_0 - \frac{1}{2} \{\mathcal{H}_0, \mathcal{H}_0\} = 0.
\label{main eq for rational}
\end{equation}
in $\mathcal{P}_Y^{\leq 0} / I^{\leq 0}_{\overline{C}_0, \overline{C}_2}$.
For each triple $(\kappa, C_0, C_2)$ such that $\overline{C}_0 \geq C_0$,
$\overline{C}_2 \geq C_2 + \kappa$,
define a linear map $d_Y = d_{(Y, \lambda, K_Y, K_Y^0, K_Y^2, J, \mathcal{B})}
: \mathcal{P}_Y^{\leq \kappa} / I^{\leq \kappa}_{C_0, C_2}
\to \mathcal{P}_Y^{\leq \kappa} / I^{\leq \kappa}_{C_0, C_2}$ by
\[
d_Y f = \delta f - \{\mathcal{H}_0, f\} \ (= D_Y f|_{\hbar = 0}).
\]
Then $d_Y$ satisfies the following.
\begin{align}
d_Y^2 &= 0, \label{d_Y^2}\\
d_Y(fg) &= (d_Y f) g + (-1)^{|f|} f d_Y g \label{d_Y Leibnitz}\\
d_Y \{f, g\} &= \{d_Y f, g\} + (-1)^{|f|} \{f, d_Y g\}.
\label{d_Y Poisson}
\end{align}
(\ref{d_Y^2}) is due to (\ref{main eq for rational}).
(\ref{d_Y Leibnitz}) and (\ref{d_Y Poisson}) hold if the multiplications or
Poisson brackets are well defined.
We denote the cohomology of the complex
$(\mathcal{P}_Y^{\leq \kappa} / I^{\leq \kappa}_{C_0, C_2}, d_Y)$
by $H^\ast(\mathcal{P}_Y^{\leq \kappa} / I^{\leq \kappa}_{C_0, C_2}, d_Y)$.
We remark that $(\mathcal{P}_Y^{\leq \kappa} / I^{\leq \kappa}_{C_0, C_2}, d_Y)$
can be regarded as a quotient of the chain complex of
general symplectic cohomology by the ideal $(\hbar)$.
As in the case of general SFT, we will define rational SFT cohomology as a limit
\begin{align*}
&H^\ast(\mathcal{P}_{(Y, \lambda, \overline{K}^0_Y)}, d_{(Y, \lambda, \overline{K}^0_Y)}) \\
&= \varprojlim_{C_2} \varinjlim_{\kappa} \varprojlim_{C_0}
H^\ast(\mathcal{P}_{(Y, \lambda, K_Y, \overline{K}^0_Y)}^{\leq \kappa}
/ I^{\leq \kappa}_{C_0, C_2}, d_{(Y, \lambda, K_Y, K_Y^0, K_Y^2, J, \mathcal{B})}).
\end{align*}
Finally, we consider the construction of contact homology.
We use the super-commutative algebra $\mathcal{A}_Y = \mathcal{A}_{(Y, \lambda, K_Y, \overline{K}^0_Y)}$
defined by $\mathcal{A}_Y = \mathbb{R}[[t]](q)$.
Its elements are written as
\[
\sum_{(\hat c_i)} f_{(\hat c_i)}(t) q_{\hat c_1^\ast} \dots q_{\hat c_{k_q}^\ast},
\]
where $f_{(\hat c_i)}(t) \in \mathbb{R}[[t]]$ are formal series of the variables $t_x$
and the sum is a finite sum.
For each $\kappa \geq 0$, we define a submodule $\mathcal{A}_Y^{\leq \kappa} \subset
\mathcal{A}_Y$ by
\[
\mathcal{A}_Y^{\leq \kappa} = \{ \sum_{(\hat c_i)} f_{(\hat c_i)}(t) q_{\hat c_1^\ast} \dots
q_{\hat c_{k_q}^\ast} \in \mathcal{A}_Y; \sum_i |e(q_{\hat c_i^\ast})| \leq \kappa \text{ if }
f_{(\hat c_i)}(t) \neq 0\}.
\]
For each $C_0 \geq 0$, we also define a submodule $I^{\leq \kappa}_{C_0}
\subset \mathcal{A}^{\leq \kappa}_Y$ by
\[
I^{\leq \kappa}_{C_0}
= \{\sum_{(x_i), (\hat c_i^\ast)} a_{(x_i); (\hat c_i^\ast)} t_{x_1} \dots t_{x_{k_t}}
q_{\hat c_1^\ast} \dots q_{\hat c_{k_q}^\ast} \in \mathcal{A}_Y^{\leq \kappa};
a_{(x_i); (\hat c_i^\ast)} = 0 \text{ for } k_t \leq C_0\}.
\]
Let
\[
\widehat{\mathcal{H}}_0 = \sum_{c}
\frac{\overleftarrow{\partial} \mathcal{H}_0}{\partial p_{\hat c}} \biggr|_{p = 0}
\cdot p_{\hat c} \in \mathcal{P}_Y^{\leq 0} / I^{\leq 0}_{\overline{C}_0, \overline{C}_2}
\]
be the homogeneous component of degree $1$ with respect to the variables $p_{\hat c}$.
Then equation (\ref{main eq for rational}) implies
\begin{equation}
\delta \widehat{\mathcal{H}}_0
- \frac{1}{2} \{\widehat{\mathcal{H}}_0, \widehat{\mathcal{H}}_0\} = 0
\label{main eq for contact}
\end{equation}
in $\mathcal{P}_Y^{\leq 0} / I^{\leq 0}_{\overline{C}_0, \overline{C}_2}$
because $\partial_{q_{\hat c^\ast}} (\mathcal{H}_0|_{p = 0}) = 0$ implies
$\{ \cdot, \mathcal{H}_0|_{p = 0} \} = 0$.
For each pair $(\kappa, C_0)$ such that $\overline{C}_0 \geq C_0$ and
$\overline{C}_2 \geq \kappa$,
define a linear map $\partial_Y = \partial_{(Y, \lambda, K_Y, K_Y^0, K_Y^2, J, \mathcal{B})}
: \mathcal{A}_Y^{\leq \kappa} / I^{\leq \kappa}_{C_0} \to
\mathcal{A}_Y^{\leq \kappa} / I^{\leq \kappa}_{C_0}$ by
\[
\partial_Y f = \delta f - \{\widehat{\mathcal{H}}_0, f\} \ (= d_Y f|_{p = 0}).
\]
Then it satisfies the following equations:
\begin{align}
\partial_Y^2 &= 0, \label{partial_Y^2} \\
\partial_Y (fg) &= (\partial_Y f) g + (-1)^{|f|} f \partial_Y g.
\label{partial diff alg}
\end{align}
((\ref{partial_Y^2}) is due to (\ref{main eq for contact}) and
((\ref{partial diff alg}) is satisfied if the multiplication is well defined.))
As in the other two theory, we define the contact homologies by their limit
\[
H^\ast(\mathcal{A}_{(Y, \lambda, \overline{K}^0_Y)}, \partial_{(Y, \lambda, \overline{K}^0_Y)})
= \varinjlim_\kappa \varprojlim_{C_0}
H^\ast(\mathcal{A}_{(Y, \lambda, K_Y, \overline{K}^0_Y)}^{\leq \kappa}
/ I^{\leq \kappa}_{C_0}, \partial_{(Y, \lambda, K_Y, K_Y^0, K_Y^2, J, \mathcal{B})}).
\]
\section{The case of holomorphic buildings for $X$}
\label{case of X}
In this section, we construct the algebras for a symplectic manifold $X$ with
cylindrical ends.
First we explain the construction of a pre-Kuranishi structure of the space
of holomorphic buildings for $X$ in Section \ref{Kuranishi for X}.
In Section \ref{fiber prod for X},
we construct the perturbed multisections of its fiber products.
We construct the correction terms for $X$ in Section \ref{correction terms for X}
and finally we construct the algebras for $X$ in Section \ref{algebra for X}
\subsection{Construction of pre-Kuranishi spaces for $X$}
\label{Kuranishi for X}
In this section, we construct a pre-Kuranishi structure of
$\widehat{\mathcal{M}}^X = \widehat{\mathcal{M}}(X, \omega, J)$.
The construction is almost the same as the case of the symplectization $\hat Y$.
First we explain the construction of a Kuranishi neighborhood of a point
$p_0 \in \widehat{\mathcal{M}}(X, \omega, J)$.
As in the case of $\hat Y$, we assume the following data
$(p_0^+, S = (S_X, S_{Y^\pm}), E^0, \lambda)$ are given:
\begin{itemize}
\item
$p_0^+ = (\Sigma_0, z \cup z^+, u_0)$ is a curve obtained by adding marked points
on the nontrivial components of $\Sigma_0$.
We assume all unstable components of $(\Sigma_0, z \cup z^+)$ are trivial cylinders
of $p_0$, and $G_0 = \mathop{\mathrm{Aut}}\nolimits(\Sigma_0, z, u_0)$ preserves $z^+$ as a set.
\item
$S_X \subset X$ and $S_{Y^\pm} \subset Y^\pm$ are finite unions of codimension-two
submanifolds such that
$u_0$ intersects with $(-\infty, 0) \times S_{Y^-} \cup S \cup (0, \infty) \times S_{Y^+}$
at $z^+ \cap \bigcup_{i(\alpha) = 0} \Sigma_\alpha$ transversely,
$\pi_{Y^-} \circ u_0$ intersects with $S_{Y^-}$ at
$z^+ \cap \bigcup_{i(\alpha) < 0} \Sigma_\alpha$ transversely, and
$\pi_{Y^+} \circ u_0$ intersects with $S_{Y^+}$ with at
$z^+ \cap \bigcup_{i(\alpha) > 0} \Sigma_\alpha$ transversely.
\item
Let $(\hat \Sigma_0, z \cup z^+ \cup (\pm\infty_i))$ be the stabilization of
$(\check \Sigma_0, z \cup z^+ \cup (\pm\infty_i))$, and
let $(\hat P \to \hat X, Z \cup Z^+ \cup Z_{\pm\infty_i})$ be its local universal family.
$G_0$ acts on $\hat P$ by the universal property.
Then we assume $E^0$ is a finite dimensional $G_0$-vector space,
and $\lambda = (\lambda_X, \lambda_{Y^-}, \lambda_{Y^+})$ is a family of
$G_0$-equivariant linear maps
$\lambda_X : E^0 \to C^\infty(\hat P \times X; {\textstyle\bigwedge}^{0, 1}V^\ast \hat P \otimes TX)$
and $\lambda_{Y^\pm} : E^0 \to
C^\infty(\hat P \times Y^\pm; {\textstyle\bigwedge}^{0, 1}V^\ast \hat P \otimes (\mathbb{R} \partial_\sigma \oplus TY^\pm))$
which satisfies following conditions:
\begin{itemize}
\item
$\lambda_X$ and $\lambda_{Y^\pm}$ are related by
\begin{align*}
&\lambda_X(h)|_{\hat P \times ((-\infty, -T] \times Y^-)}(z, \sigma, y)
= \lambda_{Y^-}(h)(z, y)\\
&\lambda_X(h)|_{\hat P \times ([T, \infty) \times Y^+)}(z, \sigma, y) = \lambda_{Y^+}(h)(z, y)
\end{align*}
for some $T \geq 0$.
\item
For each $h \in E^0$, the projections of the support of $\lambda_X(h)$ or
$\lambda_{Y^\pm}(h)$ do not intersect with the nodal points of $\hat P$ or $Z_{\pm\infty_i}$.
\item
Let $E^0 \to C^\infty(\Sigma_0, {\textstyle\bigwedge}^{0, 1} T^\ast \Sigma_0 \otimes u_0^\ast TX)$
be the linear map defined by the pullbacks of $\lambda_X$ and $\lambda_{Y^\pm}$ by
the composition of the blowing down $\Sigma_0 \to \check \Sigma_0$ and the forgetful
map $(\check \Sigma_0, z \cup z^+) \stackrel{\cong}{\to} (\hat P_0, Z(0) \cup Z^+(0))$,
where $u_0^\ast TX$ is the vector bundle on $\Sigma_0$ defined by
$(u_0|_{\bigcup_{i(\alpha) = 0} \Sigma_\alpha})^\ast TX$,
$(u_0|_{\bigcup_{i(\alpha) < 0} \Sigma_\alpha})^\ast T \hat Y^-$ and
$(u_0|_{\bigcup_{i(\alpha) > 0} \Sigma_\alpha})^\ast T \hat Y^+$.
Then we assume that for a sufficiently small $\delta > 0$,
the linear map
\begin{align*}
&D_{p_0}^+ : \widetilde{W}_\delta^{1, p}(\Sigma_0, u_0^\ast TX) \oplus E^0\\
&\to
L_\delta^p(\Sigma_0, {\textstyle\bigwedge}^{0, 1}T^\ast \Sigma \otimes u_0^\ast TX)
\oplus
\bigoplus_{\text{limit circles}} \mathop{\mathrm{Ker}}\nolimits A_{\gamma_{\pm\infty_i}} / (\mathbb{R} \partial_\sigma \oplus
\mathbb{R} R_\lambda)\\
&\quad \oplus \bigoplus_{z_i \in \bigcup_{i(\alpha) \gtrless 0} \Sigma_\alpha}
T_{\pi_{Y^\pm} \circ u_0(z_i)} Y^\pm
\oplus \bigoplus_{z_i \in \bigcup_{i(\alpha) = 0} \Sigma_\alpha} T_{u_0(z_i)} X\\
&(\xi, h) \mapsto (D_{p_0} \xi + \lambda(h),
\sum_j \langle \xi|_{S^1_{\pm\infty_i}}, \eta^{\pm\infty_i}_j \rangle \eta^{\pm\infty_i}_j,
\pi_{Y^\pm} \circ \xi (z_i), \xi(z_i))
\end{align*}
is surjective,
where $D_{p_0}$ is a linearization of the equation of the $J$-holomorphic maps,
that is,
\[
D_{p_0} \xi = \nabla \xi + J(u_0) \nabla \xi j + \nabla_\xi J(u_0) du_0 j,
\]
and each $\{\eta^{\pm\infty_i}_j\}_j$ is an orthonormal basis of
the complement of $\mathbb{R} \partial_\sigma \oplus \mathbb{R} R_\lambda$ in
$\mathop{\mathrm{Ker}}\nolimits A_{\gamma_{\pm\infty_i}}$.
\end{itemize}
\end{itemize}
We fix the following temporally data $(z^{++}, S', (\hat R_i)_{i \neq 0})$:
\begin{itemize}
\item
$z^{++} = (z^{++}_i) \subset \Sigma$ are additional marked points which make
$(\Sigma_0, z \cup z^+ \cup z^{++})$ stable.
We assume that $G_0$-action preserves $z^{++}$ as a set.
\item
$S' \subset (\mathbb{R}_{-k_-} \cup \dots \cup \mathbb{R}_{-1}) \times Y^-
\cup (\mathbb{R}_1 \cup \dots \cup \mathbb{R}_k) \times Y^+$ is a codimension-two submanifold
such that $u_0$ intersects with $S'$ at $z^{++}$ transversely.
\item
For each $-k_- \leq i \leq -1$ or $1 \leq i \leq k_+$,
let $\hat R_i = (\hat R_{i, l})_{1 \leq l \leq m_i}$ be a family of holomorphic sections
$\hat R_{i, l} : \hat X \to \hat P$ such that
$\sigma_i \circ u_0(\hat R_{i, l}(0)) = 0$, where $\sigma_i$ is the coordinate of $\mathbb{R}_i$,
and $\hat R_i$ is $G_0$-invariant as a family.
We assume $\hat R_i$ do not intersect with nodal points or $Z_{\pm\infty_i}$.
Let $(\widetilde{P} \to \widetilde{X}, Z \cup Z^+ \cup Z^{++})$ be the local universal family
of $(\Sigma_0, z \cup z^+ \cup z^{++})$.
Then each $\hat R_{i, l}$ induces a section $\widetilde{R}_{i, l} : \widetilde{X} \to \widetilde{P}$
which makes following diagram commutative.
\[
\begin{tikzcd}
\widetilde{P} \ar{r}{\mathop{\mathfrak{forget}}\nolimits}& \hat P\\
\widetilde{X} \ar{u}{\widetilde{R}_{i, l}} \ar{r}{\mathop{\mathfrak{forget}}\nolimits}& \hat X \ar{u}{\hat R_{i, l}}
\end{tikzcd}
\]
We use these families of sections $\widetilde{R}_{i, l}$ to kill the $\mathbb{R}$-translations
by imposing the conditions $\sum_l \sigma_i \circ u(\widetilde{R}_{i, l}) = 0$ ($i \neq 0$).
\end{itemize}
The pullbacks $E^0 \to C^\infty(\widetilde{P} \times X, {\textstyle\bigwedge}^{0, 1} V^\ast \widetilde{P}
\otimes TX)$ of $\lambda_X$ and $E^0 \to C^\infty(\widetilde{P} \times Y^\pm,
{\textstyle\bigwedge}^{0, 1} V^\ast \widetilde{P} \otimes (\mathbb{R} \partial_\sigma \oplus TY^\pm))$ of $\lambda_{Y^\pm}$
by $\widetilde{P} \to \hat P$ are also denoted by $\lambda_X$ and
$\lambda_{Y^\pm}$ respectively.
Using the above data, we can construct a smooth Kuranishi neighborhood of $p_0$ as in
Section \ref{construction of nbds}.
The main differences are the following two.
One is that the range of $s^0$ does not contain the factor
to kill the $\mathbb{R}$-translation of $0$-th floor.
The other is about smoothness.
The former does not have any difficulty but the latter do.
The difficulty is that in general, the constants $\alpha$ and $\beta$ for $Y^-$
which determine the differential structure of the parameter space of deformation of
the domain curve and the target space are different from those for $Y^+$.
We explain about this issue in the next section.
The definition of the embedding of a Kuranishi neighborhood to another is also similar.
We can construct an essential submersion from a Kuranishi neighborhood of a
disconnected holomorphic building to the product of those of its connected components.
A holomorphic building for $X$ is also decomposed by its floor structure, and the
relation between the Kuranishi neighborhood of the whole holomorphic building and
the Kuranishi neighborhoods of the parts is similar to the case of $Y$.
Next we consider the construction of a global pre-Kuranishi structure.
Similarly to the case of symplectization $\hat Y$,
we construct a domain curve representation
of the space of holomorphic buildings and Kuranishi data.
The main differences are as follows.
First, instead of a set $\mathcal{S}$ of codimension-two submanifolds of $Y$,
we use a triple $(\mathcal{S}_X, \mathcal{S}_{Y^-}, \mathcal{S}_{Y^+})$ consisting of
sets of codimension two submanifolds of $X$, $Y^-$ and $Y^+$ respectively.
Hence for example, we use the space
$\widehat{\mathcal{M}}^X_{(\mathcal{S}_X, \mathcal{S}_{Y^\pm})}$
of points $(\Sigma, z,
(z^S)_{S \in \mathcal{S}_X \cup \mathcal{S}_{Y^-} \cup \mathcal{S}_{Y^+}}, u)$ which
satisfy the following conditions:
\begin{itemize}
\item
For each $S \in \mathcal{S}_X$, $z^S$ is contained in the $0$-th floor and
$u$ intersects $S$ at $z^S$ transversely,
\item
For each $S \in \mathcal{S}_{Y^-}$, $z^S$ is contained in the union of
the negative floors and the inverse image of $(-\infty, 0] \times Y^- \subset X$ by $u$,
and $u$ intersects $\mathbb{R} \times S$ at $z^S$ transversely
\item
For each $S \in \mathcal{S}_{Y^+}$, $z^S$ is contained in the union of
the positive floors and the inverse image of $[0, \infty) \times Y^+ \subset X$ by $u$,
and $u$ intersects $\mathbb{R} \times S$ at $z^S$ transversely.
\end{itemize}
Similarly, we use a triple $(\lambda_X, \lambda_{Y^-}, \lambda_{Y^+})$ instead of
$\lambda$.
Second, instead of fixing one constant $L_{\max}$, we fix two constants
$L^1_{\max}$ and $L^2_{\max}$ and consider the subspace
$\widehat{\mathcal{M}}^{X, \leq (L^1_{\max}, L^2_{\max})} \subset\widehat{\mathcal{M}}^X$ consisting of
holomorphic buildings $(\Sigma, z, u)$ such that
$e + L^+ \leq L^1_{\max}$ and $L^+ \leq L^2_{\max}$,
where $e = \int_{u^{-1}(X)} u^\ast \widetilde{\omega}$.
(See Section \ref{holomorphic buildings for X} for the definition of $\widetilde{\omega}$
and recall the estimates (\ref{E lambda estimate}) and (\ref{E hat omega estimate}).)
Let $\delta_0 >0$ be a constant which satisfies the following conditions:
\begin{itemize}
\item
$6 \delta_0$ is less than the minimal $E_{\hat \omega}$-energy of
a non-constant $J$-holomorphic sphere in $X$.
\item
$4 \delta_0$ is less than the minimal $E_{\hat \omega}$-energy of
a holomorphic plain in $X$ whose $E_{\lambda}$-energy is
$\leq \max(L^1_{\max}, L^2_{\max})$.
\item
$2 \delta_0$ is less than the minimal $E_{\hat \omega}$-energy of
a holomorphic cylinder in $X$ whose $E_{\lambda}$-energy is
$\leq \max(L^1_{\max}, L^2_{\max})$.
\item
$\delta_0$ is less than the minimal $E_{\hat \omega}$-energy of
a non-constant $J$-holomorphic torus in $X$.
\end{itemize}
Then for a triple $\theta = (g, k, E_{\hat \omega})$, we define
$\widetilde{e}(\theta) = \widetilde{e}_{\delta_0}(\theta)
= 5(g-1) + 2k + E_{\hat \omega} / \delta_0$ as in the
case of symplectization $\hat Y$.
It is easy to check that $\widetilde{e}(p) = \widetilde{e}(\theta) \geq 1$
for any holomorphic building $p \in \widehat{\mathcal{M}}^{X, \leq (L^1_{\max}, L^2_{\max})}$
of type $\theta$.
Assume that domain curve representations
$(\mathcal{S}^{Y^-}, \mathcal{V}_{\theta, l}^{Y^-}, \mathcal{U}_{\theta, l}^{Y^-},
\mathcal{U}_{\theta, l}^{Y^-, \mathrm{D}})$ and
$(\mathcal{S}^{Y^+}, \mathcal{V}_{\theta, l}^{Y^+}, \mathcal{U}_{\theta, l}^{Y^+},
\mathcal{U}_{\theta, l}^{Y^+, \mathrm{D}})$
of $\widehat{\mathcal{M}}^{Y^-, \leq L^1_{\max}}_{\leq C}$ and
$\widehat{\mathcal{M}}^{Y^+, \leq L^2_{\max}}_{\leq C}$ are given respectively.
Then we can define a compatible domain curve representation
$(\mathcal{S}^X, \mathcal{V}_{\theta, l}^X, \allowbreak \mathcal{U}_{\theta, l}^X, \allowbreak \mathcal{U}_{\theta, l}^{X, \mathrm{D}})$
of $\widehat{\mathcal{M}}^{X, \leq (L^1_{\max}, L^2_{\max})}_{\leq C}$ similarly.
To distinguish the negative floors, the $0$-th floor and the positive floors
in the space of domain curves, we add new marked points
$z^{B^-}$, $z^{B^0}$ and $z^{B^+}$ similarly to $z^A$ so that
\begin{itemize}
\item
if a irreducible component contains a marked point in $z^{B^0}$, then
it is contained in the $0$-th floor, and
\item
otherwise, it contains a marked point in either $z^{B^-}$ or $z^{B^+}$ and
in the former case, it is contained in the negative floors, and
in the latter case, it is contained in the positive floors.
\end{itemize}
(The $0$-th floor may contain marked points in $z^{B^-}$ or $z^{B^+}$.)
Hence in this case, we construct $\mathcal{V}^X_{\theta, l}$ and $\mathcal{U}^X_{\theta, l}$
as the subspaces of the space
$\widehat{\mathcal{M}}^X_{(\mathcal{S}_X, \mathcal{S}_{Y^\pm}), A, B^-, B^0, B^+}$
consisting of holomorphic buildings with marked point $z$, $z^A$, $z^{B^-}$, $z^{B^0}$
and $z^{B^+}$.
To formulate the compatibility condition with the domain curve representations
of $(\mathcal{S}^{Y^\pm}, \mathcal{V}_{\theta, l}^{Y^\pm}, \mathcal{U}_{\theta, l}^{Y^\pm},
\mathcal{U}_{\theta, l}^{Y^\pm, \mathrm{D}})$, first we add marked points $z^{B^\pm}$ to
the curves in $\mathcal{U}_{\theta, l}^{Y^\pm}$ and $\mathcal{U}_{\theta, l}^{Y^\pm, \mathrm{D}}$ which satisfy
the conditions similar to the marked points $z^A$.
Then the compatibility conditions are formulated as follows instead of Condition
\ref{decomposition into parts U DM}, \ref{decomposition into parts U} and
\ref{decomposition into parts V}.
\begin{enumerate}
\item[\ref{decomposition into parts U DM}$\:\!\!^X$]
For any $\theta = (g, k, E_{\hat \omega})$,
$\hat p \in \mathcal{U}_{\theta, l}^{X, \mathrm{D}}$ and subset $\mathcal{N}$ of its nodal
points, replace each nodal point in $\mathcal{N}$ with a pair of marked points
(we regard the new marked points as points in the set $z$), and
let $\hat p'_i$ $(1 \leq i \leq N)$ be its connected components or an arbitrary
decomposition into unions of its connected components.
Let $g'_i$ and $k'_i$ be the genus and the number of marked points $z$ of
each $\hat p'_i$ respectively.
Then there exist some $E_{\hat \omega}^i \geq 0$ such that
$E_{\hat \omega} = \sum_i E_{\hat \omega}^i$ and the following hold for
$\theta'_i = (g'_i, k'_i, E_{\hat \omega}^i)$.
\begin{itemize}
\item
$\hat p'_i \in \mathcal{U}_{\theta'_i, l(\hat p'_i)}^{X, \mathrm{D}}$ if $\hat p'_i$
contains a marked point in $z^{B^0}$.
\item
$\hat p'_i \in \mathcal{U}_{\theta'_i, l(\hat p'_i)}^{Y^\pm, \mathrm{D}}$ if $\hat p'_i$ does not contain
any marked points in $z^{B^0}$ and it contains a marked point in $z^{B^\pm}$.
\end{itemize}
\item[\ref{decomposition into parts U}$\:\!\!^X$]
$\mathcal{U}_{\theta, l}^X$ satisfies the following conditions.
\begin{itemize}
\item
For any $p \in \mathcal{U}_{\theta, l}^X$ and any decomposition $p_i$ ($1 \leq k$)
into unions of its connected components, let $p'_i$ be the holomorphic buildings
obtained by collapsing trivial floors.
Then $p'_i \in \mathcal{U}_{\theta(p'_i), l(p'_i)}^X$ for all $i$.
\item
For any $p \in \mathcal{U}_{\theta, l}^X$ and any gap between non-positive floors,
let $p_1$ and $p_2$ be the holomorphic buildings obtained by separating $p$ at this gap.
($p_1$ is the part in the negative floors.)
Then $p'_1 \in \mathcal{U}_{\theta(p'_1), l(p'_1)}^{Y^-}$ and $p'_2 \in \mathcal{U}_{\theta(p'_2), l(p'_2)}^X$.
We also assume the similar condition for the gap between non-negative floors.
\item
For any $p \in \mathcal{U}_{\theta, l}^X$ and any subset of its nodal points,
the holomorphic building $p'$ obtained by replacing these nodal points to pairs of
marked points is contained in $\mathcal{U}_{\theta(p'), l(p')}^X$.
\end{itemize}
\item[\ref{decomposition into parts V}$\:\!\!^X$]
For each $p \in \widehat{\mathcal{M}}^{X, \leq L_{\max}}_{(\mathcal{S}_X, \mathcal{S}_{Y^\pm}),
A, B^-, B^0, B^+, \theta, l}$,
replace all nodal points and joint circles of $p$ to pairs of marked points and
pairs of limit circles respectively
(we regard the new marked points as points in the set $z$), and
let $p'_i$ $(1 \leq i \leq k)$ be their non-trivial connected components.
Then $p \in \mathcal{V}_{\theta, l}^X$ if and only if the following hold:
\begin{itemize}
\item
$p'_i \in \mathcal{V}_{\theta(p'_i), l(p'_i)}^X$ if $p'_i$ contains a marked point in $z^{B^0}$.
\item
$p'_i \in \mathcal{V}_{\theta(p'_i), l(p'_i)}^{Y^\pm}$ if $p'_i$ does not contain a marked point
in $z^{B^0}$ and it contains a marked point in $z^{B^\pm}$.
\end{itemize}
\end{enumerate}
The definition of compatible Kuranishi data for
$\widehat{\mathcal{M}}^{X, \leq (L^1_{\max}, L^2_{\max})}_{\leq C}$
are also similar, and we can construct them by the same argument.
Then the pre-Kuranishi structure of each
$\widehat{\mathcal{M}}^{X, \leq (L^1_{\max}, L^2_{\max})}_\theta$ is defined by these data
as in the case of symplectization.
\subsection{Smoothness of pre-Kuranishi structure in the case of $X$}
Recall that in Section \ref{smoothness},
to obtain a smooth pre-Kuranishi structure of the space of holomorphic buildings
for a contact manifold, we had to use a strong differential structure of
the parameter space of the deformation of a domain curve.
Such a strong differential structure is determined by a fixed pair of large constants
$\alpha$ and $\beta$, and to construct a pre-Kuranishi structure of
the space of holomorphic buildings of higher energy,
we need to choose larger constants in general.
Hence for a cobordism $(X, \omega)$ from $(Y^-, \lambda^-)$ to $(Y^-, \lambda^-)$,
we need to consider the case where
we use different constants $\alpha^\pm$ and $\beta^\pm$ for
the smooth pre-Kuranishi structure of $\widehat{\mathcal{M}}(Y^\pm, \lambda^\pm, J^\pm)$.
The difference of $\beta^\pm$ does not have a difficulty.
We can use the coordinates defined by
$\rho_\mu^{L_\mu} = \hat \rho_\mu^{\beta^-}$ for a joint circle $S_\mu^1$
between non-positive floors and
$\rho_\mu^{L_\mu} = \hat \rho_\mu^{\beta^+}$ for a joint circle $S_\mu^1$
between non-negative floors.
However, for nodal points in the $0$-th floor, there is not such a $\pm$-decomposition.
Hence we need to use a gradation of smooth structures.
We fix a smooth function $\alpha : X \to \mathbb{R}_{>0}$ such that
$\alpha|_{(-\infty, -T] \times Y^-} = \alpha^-$ and $\alpha|_{[T, \infty) \times Y^+}
= \alpha^+$ for some $T \geq 0$.
Roughly speaking, for a nodal point $q_\nu$ of a holomorphic building
$(\Sigma_0, z, u_0) \in \widehat{\mathcal{M}}(X, \omega, J)$, we use the coordinate
defined by $\rho_\nu = \hat \rho_\nu^{\alpha(u_0(q_\nu))}$.
In this section, we explain the precise definition of the smooth structure of
Kuranishi neighborhoods of $\widehat{\mathcal{M}}(X, \omega, J)$,
and prove the smoothness of an embedding between two Kuranishi neighborhoods
or an essential submersion from that of a disconnected holomorphic building
to products of those of its connected components.
Let $(V, E, s, \psi, G)$ be a Kuranishi neighborhood of a point
$(\Sigma_0, z, u_0) \in \widehat{\mathcal{M}}(X, \omega, J)$.
We assume that the height of $(\Sigma_0, z, u_0)$ is $(k_-, k_+)$.
Recall that $V$ is a subset of $\hat V = \mathring{X} \times B_\epsilon(0)$
defined by $V = \{(a, b, x) \in \hat V; s^0(a, b, x) = 0\}$,
where $B_\epsilon(0)$ is a ball in the kernel of a linear operator,
and $s^0 : \hat V \to \mathbb{R}^{k_-} \oplus \mathbb{R}^{k_+} \oplus \bigoplus_{z^{++}_l} \mathbb{R}^2$
is a function on $\hat V$ defined similarly to (\ref{s^0}).
Let $\{\mu\}$ and $\{\nu\}$ be the indices of joint circles and nodal points
of $\Sigma_0$ respectively.
For each $i \in \{-k_-, \dots, -1\}$, let $M_i \subset \{\mu\}$ be the index set of
the joint circles between $i$-th floor and $(i+1)$-th floor,
and for $i \in \{1, \dots, k_+\}$, let $M_i \subset \{\mu\}$ be the index set of
the joint circles between $(i-1)$-th floor and $i$-th floor.
For each pair of subsets $\Pi \subset \{-k_-, \dots, -1\} \cup \{1, \dots, k_+\}$
and $\Pi' \subset \{\nu\}$,
we define $\mathring{X}_{\Pi, \Pi'} \subset \mathring{X}$ by
\begin{align*}
\mathring{X}_{\Pi, \Pi'}
= \{ (a,b) \in \mathring{X};\, & \rho_\mu \neq 0 \text{ for all } \mu \in M_i
\text{ if and only if } i \in \Pi\\
& \zeta_\nu \neq 0 \text{ if and only if } \nu \in \Pi'\},
\end{align*}
\begin{defi}
For any $0 < \epsilon < 1$ and
$\tilde \delta_0 = (\tilde \delta_{0, i})_{i \in \{-k_-, \dots, -1\} \cup \{1, \dots, k_+\}}$,
we say a continuous function $f$ on $\hat V = \mathring{X} \times B_\epsilon(0)$
is $(\epsilon, \tilde \delta_0)$-admissible if for any
$\Pi \subset \{-k_-, \dots, -1\} \cup \{1, \dots, k_+\}$ and $\Pi' \subset \{\nu\}$,
the restriction of $f$ to $\mathring{X}_{\Pi, \Pi'} \times B_\epsilon(0) \subset
\hat V$ is smooth and its differentials satisfy the following estimates
similar to those of $\phi$ given in Corollary \ref{asymptotic estimates of phi}:
For any $l \geq 1$ and any multi-index $(k_x, k_j, k_b, \allowbreak (k_{\mu_i})_{i \in \Pi}, \allowbreak
(l_\mu)_\mu, \allowbreak
(k_\nu)_{\nu \in \Pi'}, \allowbreak (l_\nu)_{\nu \in \Pi'})$,
there exists some constant $C > 0$ such that
\[
|\partial_x^{k_x} \partial_j^{k_j} \partial_b^{k_b} \partial_{(\rho_{\mu_i})}^{(k_{\mu_i})}
\partial_{(\varphi_\mu)}^{(l_\mu)} \partial_{(\rho_\nu)}^{(k_\nu)}
\partial_{(\varphi_\nu)}^{(l_\nu)} f(a, b, x)|
\leq C \prod_{\substack{i \\ k_{\mu_i} \neq 0}}
\rho_\mu^{L_{\mu_i} \tilde \delta_{0, i}/2 - k_{\mu_i}}\!\!
\prod_{\substack{\nu \\ (k_\nu, l_\nu) \neq (0,0)}} \rho_\nu^{\epsilon - k_\nu}
\]
for all $(a, b, x) \in \mathring{X}_{\Pi, \Pi'} \times B_\epsilon(0)$.
We say a continuous function $f$ on $V \subset \hat V$ is
$(\epsilon, \tilde \delta_0)$-admissible if the composition of $f$ and the natural
projection $\hat V \to V$ is $(\epsilon, \tilde \delta_0)$-admissible.
See Remark \ref{natural projection from hat V to V} for the natural projection.
\end{defi}
Corollary \ref{asymptotic estimates of phi} implies that
$\phi : \hat V \to C^l(\Sigma_0 \setminus N_0,
(\mathbb{R}_{-k_-} \sqcup \dots \sqcup \mathbb{R}_{-1}) \times Y^-
\sqcup X \sqcup
(\mathbb{R}_1 \sqcup \dots \sqcup \mathbb{R}_{k_+}) \times Y^+) \times E^0$ is
$(\epsilon, \tilde \delta_0)$-admissible for any $0 < \epsilon < 1$ and
$0 < \tilde \delta_{0, i} < \min\{ \kappa_\mu / L_\mu; \mu \in M_i\}$.
For each $\nu$ such that the $\nu$-th nodal point of $(\Sigma_0, z, u_0)$
is contained in the $0$-th floor,
there exists an $(\epsilon, \tilde \delta_0)$-admissible
function $\alpha_\nu : \hat V \to \mathbb{R}_{> 0}$ such that
$\alpha_\nu(a, b, x) = \alpha(u_{a, b, x}(q_\nu))$ for any $(a, b, x) \in \hat V$ such that
$\rho_\nu = 0$, where $q_\nu$ is the $\nu$-th nodal point of $\widetilde{P}_a$
and $u_{a, b, x} = \Phi_{a, b}(\xi_{a, b, x})$ is the map for $(a, b, x)$.
For example, the composition of the projection $\hat V \to
\{(a, b, x) \in \hat V; \rho_\nu = 0\}$ and the map $\alpha_\nu(a, b, x)
= \alpha(u_{a, b, x}(q_\nu))$ on $\{(a, b, x) \in \hat V; \rho_\nu = 0\}$
satisfies this condition since
the map $u_{a, b, x}(q_\nu) : \{(a, b, x) \in \hat V; \rho_\nu = 0\} \to X$
is $(\epsilon, \tilde \delta_0)$-admissible as well as $\phi$.
We fix such an $(\epsilon, \tilde \delta_0)$-admissible
function $\alpha_\nu : \hat V \to \mathbb{R}_{> 0}$.
For each $\nu$ such that the $\nu$-th nodal point of $(\Sigma_0, z, u_0)$ is
contained in a negative floor or a positive floor, we define $\alpha_\nu$
by $\alpha_\nu = \alpha^-$ or $\alpha_\nu = \alpha^+$ respectively.
For each $\mu$, we define $\beta_\mu$ by $\beta_\mu = L_\mu^{-1} \beta^-$
if $\mu$ is a joint circle between non-positive floors and
$\beta_\mu = L_\mu^{-1} \beta^+$ if $\mu$ is a joint circle between non-negative
floors.
We define a smooth structure of $\hat V = \mathring{X} \times B_\epsilon(0)$
by the coordinate
{\belowdisplayskip= 0pt
\[
\hat V \subset \mathcal{J}_0 \times D^{l_0} \times \widetilde{D}^{l_1} \times B_\epsilon(0)
\to \mathcal{J}_0 \times D^{l_0} \times ([0, 1] \times S^1)^{l_1} \times B_\epsilon(0)
\]}
{\abovedisplayskip= 0pt
\begin{multline}
(j, (\zeta_\nu = \rho_\nu^2 e^{2\sqrt{-1} \varphi_\nu})_\nu,
(\zeta_\mu = \rho_\mu^{2\pi} e^{2\pi \sqrt{-1} \varphi_\mu})_\mu, x)\\
\mapsto (\hat \jmath, (\hat \zeta_\nu = \hat \rho_\nu^2
e^{2\sqrt{-1} \hat \varphi_\nu})_\nu, (\hat \rho_\mu, \hat \varphi_\mu)_\mu, \hat x) \label{admissible coordinate change}
\end{multline}
}
given by $\rho_\nu = \hat \rho_\nu^{\alpha_\nu}$,
$\rho_\mu = \hat \rho_\mu^{\beta_\mu}$
and $(\hat \jmath, \hat \varphi_\nu, \hat \varphi_\mu, \hat x)
= (j, \varphi_\nu, \varphi_\mu, x)$.
First we prove the smoothness of the map
\[
\phi : \hat V \to
C^l(\Sigma_0 \setminus N_0, (\mathbb{R}_{-k_-} \sqcup \dots \sqcup \mathbb{R}_{-1}) \times Y^-
\sqcup X \sqcup (\mathbb{R}_1 \sqcup \dots \sqcup \mathbb{R}_{k_+}) \times Y^+) \times E^0,
\]
which implies the smoothness of the evaluation maps at the marked points.
This follows from the following lemma.
\begin{lem}\label{admissibility implies smoothness}
For any $(\epsilon, \tilde \delta_0)$-admissible function $f$ on $\hat V$,
\begin{align*}
&\Bigl|\partial_{\hat x}^{k_x} \partial_{\hat \jmath}^{k_j} \partial_{\hat b}^{k_b}
\partial_{(\hat \rho_{\mu_i})}^{(k_{\mu_i})}
\partial_{(\hat \varphi_\mu)}^{(l_\mu)} \partial_{(\hat \rho_\nu)}^{(k_\nu)}
\Bigl(\prod_\nu \frac{1}{\hat \rho_\nu^{l_\nu}}\Bigr)\partial_{(\hat \varphi_\nu)}^{(l_\nu)}
f \Bigr|\\
&\lesssim \prod_{\substack{i \\ k_{\mu_i} \neq 0}}
(\hat \rho_{\mu_i})^{\beta \tilde \delta_{0, i} / 2 - k_{\mu_i}}
\prod_{\substack{\nu \\ (k_\nu, l_\nu) \neq (0,0)}}
(\hat \rho_\nu)^{\epsilon \alpha - (k_\nu + l_\nu)}
(- \log \hat \rho_\nu)^N,
\end{align*}
where $N = |k_x| + |k_j| + |k_b| + |(k_{\mu_i})| + |(l_\mu)| + |(k_\nu)| + |(l_\nu)|$.
\end{lem}
\begin{proof}
It is easy to check that the claim follows from the following estimates of
the differentials of the coordinate change and
the $(\epsilon, \tilde \delta_0)$-admissibility of $f$ and $\alpha$:
\begin{align}
&\Bigl|\partial_x^{k_x} \partial_j^{k_j} \partial_b^{k_b} \partial_{(\rho_{\mu_i})}^{(k_{\mu_i})}
\partial_{(\varphi_\mu)}^{(l_\mu)} \partial_{(\rho_\nu)}^{(k_\nu)}
\partial_{(\varphi_\nu)}^{(l_\nu)}
\Bigl(\frac{\partial \rho_{\nu_1}}{\partial \hat \rho_{\nu_0}}
- \delta^{\nu_0, \nu_1} \alpha_{\nu_1} \hat \rho_{\nu_1}^{\alpha_{\nu_1} - 1}
\Bigr)\Bigr| \notag\\
&\lesssim \rho_{\nu_1} (- \log \hat \rho_{\nu_1}) \prod_{\substack{i \\ k_{\mu_i} \neq 0}}
\rho_{\mu_i}^{L_{\mu_i} \tilde \delta_{0, i} / 2 - k_{\mu_i}}
\prod_{\substack{\nu \\ (k_\nu, l_\nu) \neq (0,0) \\\text{or } \nu = \nu_0}}
\rho_\nu^{\epsilon - k_\nu - \delta^{\nu, \nu_0} \alpha_{\nu_0}^{-1}},
\label{nu nu}
\end{align}
\begin{align}
&\Bigl|\partial_x^{k_x} \partial_j^{k_j} \partial_b^{k_b} \partial_{(\rho_{\mu_i})}^{(k_{\mu_i})}
\partial_{(\varphi_\mu)}^{(l_\mu)} \partial_{(\rho_\nu)}^{(k_\nu)}
\partial_{(\varphi_\nu)}^{(l_\nu)}
\frac{\partial \rho_{\nu_1}}{\partial \hat \rho_{\mu_{i_0}}}\Bigr| \notag\\
&\lesssim \rho_{\nu_1} (- \log \hat \rho_{\nu_1})
\prod_{\substack{i \\ k_{\mu_i} \neq 0\\ \text{or } i = i_0}}
\rho_{\mu_i}^{L_{\mu_i} \tilde \delta_{0, i} / 2 - k_{\mu_i}
- \delta^{i, i_0} \beta_{\mu_{i_0}}^{-1}}
\prod_{\substack{\nu \\ (k_\nu, l_\nu) \neq (0,0)}}
\rho_\nu^{\epsilon - k_\nu},
\label{nu mu}
\end{align}
and
\begin{align}
&|\partial_x^{k_x} \partial_j^{k_j} \partial_b^{k_b} \partial_{(\rho_{\mu_i})}^{(k_{\mu_i})}
\partial_{(\varphi_\mu)}^{(l_\mu)} \partial_{(\rho_\nu)}^{(k_\nu)}
\partial_{(\varphi_\nu)}^{(l_\nu)}g^{\nu_1}| \notag\\
&\lesssim \rho_{\nu_1} (- \log \hat \rho_{\nu_1})
\prod_{\substack{i \\ k_{\mu_i} \neq 0}}
\rho_{\mu_i}^{L_{\mu_i} \tilde \delta_{0, i} / 2 - k_{\mu_i}}
\prod_{\substack{\nu \\ (k_\nu, l_\nu) \neq (0,0)}}
\rho_\nu^{\epsilon - k_\nu}
\label{nu the others}
\end{align}
for
\[
g^{\nu_1} = \frac{\partial \rho_{\nu_1}}{\partial \hat \varphi_{\nu}},
\frac{\partial \rho_{\nu_1}}{\partial \hat \varphi_{\mu}},
\frac{\partial \rho_{\nu_1}}{\partial \hat x},
\frac{\partial \rho_{\nu_1}}{\partial \hat \jmath},
\frac{\partial \rho_{\nu_1}}{\partial \hat b_{\mu}},
\]
where $\delta^{\nu, \nu'}$ and $\delta^{i, i'}$ are the Kronecker deltas.
We sketch the proof of (\ref{nu nu}), (\ref{nu mu}) and (\ref{nu the others}).
Let $A$ be a square-matrix-valued function on
\[
\{(a, b, x) \in \hat V; \rho_\nu \neq 0, \rho_\mu \neq 0 \text{ for all }
\nu \text{ and } \mu\} \times B_\epsilon(0)
\]
defined by
\begin{align*}
&\lsuperscript{(\rho_\nu (-\log \hat \rho_\nu) \partial_{\rho_\nu},\
\partial_{\varphi_\nu},\
\rho_{\mu_i} \partial_{\rho_{\mu_i}},\
\partial_{\varphi_\mu},\
\partial_x,\ \partial_j,\ \partial_b)}{t}\\
&= A \cdot
\lsuperscript{
(\alpha_\nu^{-1} \hat \rho_\nu (-\log \hat \rho_\nu) \partial_{\hat \rho_\nu},\
\partial_{\hat \varphi_\nu},\
\beta_{\mu_i}^{-1} \hat \rho_{\mu_i} \partial_{\hat \rho_{\mu_i}},\
\partial_{\hat \varphi_\mu},\
\partial_{\hat x},\ \partial_{\hat \jmath},\ \partial_{\hat b})}{t}
\end{align*}
We can easily check the following estimates of the columns of $(A-1)$
corresponding to the vectors $\rho_\nu (-\log \hat \rho_\nu) \partial_{\rho_\nu}$.
It is also easy to check that the other columns of $(A-1)$ are zero.
In the inequalities below,
$(A-1)_{\rho_{\nu_0}, \rho_{\nu_1}}$ is the entry corresponding to
$\rho_{\nu_0} (-\log \hat \rho_{\nu_0}) \partial_{\rho_{\nu_0}}$
and $\rho_{\nu_1} (-\log \hat \rho_{\nu_1}) \partial_{\rho_{\nu_1}}$.
The other entries $(A-1)_{\rho_{\nu_{i_0}}, \rho_{\nu_1}}$ are similar.
In (\ref{A the others}), $\ast$ denotes the other rows:
$\ast = \varphi_{\nu}, \varphi_{\mu}, x, j, b$.
\begin{align}
&\bigl|
\partial_x^{k_x} \partial_j^{k_j} \partial_b^{k_b} \partial_{(\rho_{\mu_i})}^{(k_{\mu_i})}
\partial_{(\varphi_\mu)}^{(l_\mu)} \partial_{(\rho_\nu)}^{(k_\nu)}
\partial_{(\varphi_\nu)}^{(l_\nu)} (A-1)_{\rho_{\nu_0}, \rho_{\nu_1}}
\bigr| \notag\\
&\lesssim (-\log \hat \rho_{\nu_0}) \prod_{\substack{i \\ k_{\mu_i} \neq 0}}
\rho_{\mu_i}^{L_{\mu_i} \tilde \delta_{0, i} / 2 - k_{\mu_i}}
\prod_{\substack{\nu \\ (k_\nu, l_\nu) \neq (0,0) \\\text{or } \nu = \nu_0}}
\rho_\nu^{\epsilon - k_\nu},
\label{A nu}
\end{align}
\begin{align}
&\bigl|
\partial_x^{k_x} \partial_j^{k_j} \partial_b^{k_b} \partial_{(\rho_{\mu_i})}^{(k_{\mu_i})}
\partial_{(\varphi_\mu)}^{(l_\mu)} \partial_{(\rho_\nu)}^{(k_\nu)}
\partial_{(\varphi_\nu)}^{(l_\nu)} (A-1)_{\rho_{\mu_{i_0}}, \rho_{\nu_1}}
\bigr| \notag\\
&\lesssim \prod_{\substack{i \\ k_{\mu_i} \neq 0 \\\text{or } i = i_0}}
\rho_{\mu_i}^{L_{\mu_i} \tilde \delta_{0, i} / 2 - k_{\mu_i}}
\prod_{\substack{\nu \\ (k_\nu, l_\nu) \neq (0,0)}}
\rho_\nu^{\epsilon - k_\nu},
\label{A mu}
\end{align}
and
\begin{align}
&\bigl|
\partial_x^{k_x} \partial_j^{k_j} \partial_b^{k_b} \partial_{(\rho_{\mu_i})}^{(k_{\mu_i})}
\partial_{(\varphi_\mu)}^{(l_\mu)} \partial_{(\rho_\nu)}^{(k_\nu)}
\partial_{(\varphi_\nu)}^{(l_\nu)} (A-1)_{\ast, \rho_{\nu_1}}
\bigr| \notag\\
&\lesssim \prod_{\substack{i \\ k_{\mu_i} \neq 0}}
\rho_{\mu_i}^{L_{\mu_i} \tilde \delta_{0, i} / 2 - k_{\mu_i}}
\prod_{\substack{\nu \\ (k_\nu, l_\nu) \neq (0,0)}}
\rho_\nu^{\epsilon - k_\nu}.
\label{A the others}
\end{align}
These estimates follow from the $(\epsilon, \tilde \delta_0)$-admissibility of
$\alpha_{\nu}$ and the following equations:
\[
(A-1)_{\rho_{\nu_0}, \rho_{\nu_1}}
= (-\log \hat \rho_{\nu_0}) \rho_{\nu_0}
\frac{\partial \alpha_{\nu_1}}{\partial \rho_{\nu_0}},
\]
\[
(A-1)_{\rho_{\mu_{i_0}}, \rho_{\nu_1}}
= \rho_{\mu_{i_0}} \frac{\partial \alpha_{\nu_1}}{\partial \rho_{\mu_{i_0}}}
\]
and
\[
(A-1)_{\ast, \rho_{\nu_1}}
= \partial_\ast \alpha_{\nu_1},
\]
where $\partial_\ast = \partial_{\varphi_{\nu}}, \partial_{\varphi_{\mu}},
\partial_x, \partial_j, \partial_b$.
(\ref{A nu}), (\ref{A mu}) and (\ref{A the others}) imply that
the same inequalities hold for $A^{-1} - 1$ as well as $A-1$.
This is because the derivatives of $A^{-1}$ are polynomials of
$A^{-1} = 1 + (1-A) + (1-A)^2 + \cdots$ and the derivatives of $(A-1)$.
These inequalities are equivalent to (\ref{nu nu}), (\ref{nu mu}) and
(\ref{nu the others}).
\end{proof}
Next we prove the smoothness of the embedding between two Kuranishi neighborhoods.
For this proof, we directly use the admissibility of $\phi$
rather than its smoothness.
The definition of the embedding itself is the same as the case of
$\widehat{\mathcal{M}}(Y, \lambda, J)$.
We assume the similar condition to Section \ref{embed} and use the same notation.
Let $(a^1, b^1, u^1, h^1) \to (a^2, b^2, u^2, h^2)$ be the embedding of
$V^0_1 \subset V_1$ into $V_2$.
Let $N_{q_0} \subset \{\nu^1\}$ be the set of indices of nodal points of $\Sigma_1$
which remain to be nodal points in $\Sigma_0$, that is, $\rho_{\nu^1} = 0$ at $a_0^1$.
For each $\nu^1 \in N_{q_0}$, let $\iota(\nu^1)$ be the index of the corresponding
nodal point of $\Sigma_2$.
Similarly, let $M_{q_0} \subset \{\mu^1\}$ be the set of indices of joint circles of
$\Sigma_1$ which remain to be joint circles in $\Sigma_0$,
and let $\iota(\mu^1)$ be the index of the corresponding joint circle of $\Sigma_2$
for each $\mu^1 \in M_{q_0}$.
We assume that the maps $\phi$ and $\alpha_\nu$ for the Kuranishi neighborhood
$(V_1, E_1, s_1, \psi_1, G_1)$ are $(\epsilon, \tilde \delta_0)$-admissible,
those for $(V_2, E_2, s_2, \psi_2, G_2)$ are $(\epsilon, \tilde \delta'_0)$-admissible,
and $\tilde \delta_{0, i} \leq \tilde \delta'_{0, j}$ if
the joint circles of $\Sigma_1$ which belong to $M^1_i$
remain to be joint circles in $\Sigma_0$ and they correspond to those which
belong to $M^2_j$.
First we check the following:
\begin{itemize}
\item[($\clubsuit$)]
$\zeta^2_{\iota(\nu^1)} / \zeta^1_{\nu^1}$ for $\nu^1 \in N_{q_0}$
and $\rho^2_{\iota(\mu^1)} / \rho^1_{\mu^1}$ for $\mu^1 \in M_{q_0}$
are $(\epsilon, \tilde \delta_0)$-admissible and bounded away from zero
(i.e. the continuous extensions do not take zero on $V^0_1$).
$\zeta^2_{\nu^2}$ for $\nu^2 \notin \iota(N_{q_0})$,
$\rho^2_{\mu^2}$ for $\mu^2 \notin \iota(M_{q_0})$ and
$(\varphi^2_\mu, j^2, b^2_\mu, u^2, h^2)$ are $(\epsilon, \tilde \delta_0)$-admissible.
\end{itemize}
The $(\epsilon, \tilde \delta_0)$-admissibility of $h^2$ is clear.
$\mathcal{Z}_2^+ \in \Sigma_1 \setminus N_1$ is $(\epsilon, \tilde \delta_0)$-admissible
because of the $(\epsilon, \tilde \delta_0)$-admissibility of $u^1 \in
C^{l_1}(\Sigma_1 \setminus N_0, (\mathbb{R}_{-k_-} \sqcup \dots \sqcup \mathbb{R}_{-1}) \times Y^-
\sqcup X \sqcup (\mathbb{R}_1 \sqcup \dots \sqcup \mathbb{R}_{k_+}) \times Y^+)$.
Hence $\hat a^2 \in \hat U_2$ is also $(\epsilon, \tilde \delta_0)$-admissible.
Therefore in the definition of $\theta$,
$\sigma \circ u^1 \circ (\pi_1|_{(\widetilde{P}_1)_{a^1}})^{-1} \circ
\Theta|_{(\hat P_2)|_{\hat a^2}} (\hat R^2_i(\hat a^2)) \in \mathbb{R}_{i'}$ is
$(\epsilon, \tilde \delta_0)$-admissible.
($i'$ is the floor of $\Sigma_1$ which corresponds to the $i$-th floor of $\Sigma_2$
in $\Sigma_0$.)
Together with the $(\epsilon, \tilde \delta_0)$-admissibility of $u^1$,
it implies that $\mathcal{Z}_2^{++} \in \Sigma_1 \setminus N_1$ is also
$(\epsilon, \tilde \delta_0)$-admissible.
The $(\epsilon, \tilde \delta_0)$-admissibility of $\mathcal{Z}_2^+$ and
$\mathcal{Z}_2^{++}$ implies the $(\epsilon, \tilde \delta_0)$-admissibility of
$a^2 \in \widetilde{U}_2$.
Furthermore, it implies that
$\zeta^2_{\iota(\nu^1)} / \zeta^1_{\nu^1}$ for $\nu^1 \in N_{q_0}$
and $\rho^2_{\iota(\mu^1)} / \rho^1_{\mu^1}$ for $\mu^1 \in M_{q_0}$
are $(\epsilon, \tilde \delta_0)$-admissible and bounded away from zero.
The $(\epsilon, \tilde \delta_0)$-admissibility of $a^2$ and $u^1$ implies
$u^2 \in
C^{l_2}(\Sigma_2 \setminus N_2, (\mathbb{R}_{-k_-} \sqcup \dots \sqcup \mathbb{R}_{-1}) \times Y^-
\sqcup X \sqcup (\mathbb{R}_1 \sqcup \dots \sqcup \mathbb{R}_{k_+}) \times Y^+)$ is
$(\epsilon, \tilde \delta_0)$-admissible.
Finally, $b^2_\mu$ are also $(\epsilon, \tilde \delta_0)$-admissible because
the function $f_\mu$ in (\ref{b for any kappa}) is $(\epsilon, \tilde \delta_0)$-admissible.
It is easy to check that ($\clubsuit$) and the $(\epsilon, \tilde \delta'_0)$-admissibility
of $\alpha^2_{\nu^2}$ on $V_2$ imply that $\alpha^2_{\nu^2}$ are
$(\epsilon, \tilde \delta_0)$-admissible as functions on $V^0_1$.
Note that for any $\nu^1 \in N_{q_0}$, $\alpha^2_{\iota(\nu^1)} = \alpha^1_{\nu^1}$
on $\{(a^1, b^1, u^1, h^1) \in V^0_1; \rho^1_{\nu^1} = 0\}$ by definition.
Therefore their $(\epsilon, \tilde \delta'_0)$-admissibility implies that
for any multi-index $(k_{x^1}, k_{j^1}, k_{b^1}, k_{\mu^1_i}, \allowbreak l_{\mu^1},
\allowbreak k_{\nu^1}, \allowbreak l_{\nu^1})$
such that $(k_{\nu^1_0}, l_{\nu^1_0}) = (0, 0)$,
\begin{align}
&\bigl|\partial_{x^1}^{k_{x^1}} \partial_{j^1}^{k_{j^1}} \partial_{b^1}^{k_{b^1}}
\partial_{(\rho_{\mu^1_i})}^{(k_{\mu^1_i})}
\partial_{(\varphi_{\mu^1})}^{(l_{\mu^1})} \partial_{(\rho_{\nu^1})}^{(k_{\nu^1})}
\partial_{(\varphi_{\nu^1})}^{(l_{\nu^1})}
(\alpha^2_{\iota(\nu^1_0)} - \alpha^1_{\nu^1_0})\bigr| \notag\\
&\lesssim (\rho^1_{\nu^1_0})^\epsilon \cdot \prod_{\substack{i \\ k_{\mu_i} \neq 0}}
\rho_{\mu_i}^{L_{\mu_i} \tilde \delta_{0, i} / 2 - k_{\mu_i}}
\prod_{\substack{\nu \\ (k_\nu, l_\nu) \neq (0,0)}}
\rho_\nu^{\epsilon - k_\nu}.
\label{difference of alpha}
\end{align}
Now we prove the smoothness of the embedding.
For any $\nu^1 \in N_{q_0}$,
($\clubsuit$), (\ref{difference of alpha}) and the $(\epsilon, \tilde \delta_0)$-admissibility
of $\alpha^1_{\nu^1}$ and $\alpha^2_{\iota(\nu^1)}$ imply that
\[
\hat \rho^2_{\iota(\nu^1)} / \hat \rho^1_{\nu^1}
= (\rho^1_{\nu^1})^{(\alpha^2_{\iota(\nu^1)})^{-1} - (\alpha^1_{\nu^1})^{-1}}
\cdot \bigl(\rho^2_{\iota(\nu^1)} / \rho^1_{\nu^1}\bigr)^{(\alpha^2_{\iota(\nu^1)})^{-1}}
\]
is $(\epsilon, \tilde \delta_0)$-admissible.
Assume that $\mu^1_i \in M^1_i$ and that
$\mu^2_\ast = \iota(\mu^1_i) \in M^2_{i'}$.
Then ($\clubsuit$) implies
\[
\hat \rho^2_{\mu^2_{i'}} / \hat \rho^1_{\mu^1_i}
= (\rho^2_{\iota(\mu^1_i)} / \rho^1_{\mu^1_i})^{(\beta_{\mu^1_i})^{-1}}
\cdot e^{(b^2_{\mu^2_{i'}} - b^2_{\mu_\ast}) \beta}
\]
is also $(\epsilon, \tilde \delta_0)$-admissible.
Therefore, $(a^2, b^2, u^2, h^2) \in V_2$ is an $(\epsilon, \tilde \delta_0)$-admissible
function of $(a^1, b^1, u^1, h^1) \in V^0_1$ if the differential structure of $V_2$
is defined by $(\alpha^2_{\nu^2}, \beta^\pm)$.
Hence Lemma \ref{admissibility implies smoothness} implies the smoothness of
the embedding.
We can similarly prove the smoothness of the essential submersion
from a Kuranishi neighborhood of a disconnected holomorphic building
to the product of those of its connected components.
\subsection{Fiber products and multisections}\label{fiber prod for X}
Let $K_{Y^\pm} \hookrightarrow \overline{P}_{Y^\pm}$ be triangulations, and let $K^0_{Y^\pm}$ be
finite sets of smooth cycles in $Y^\pm$.
Assume that a finite sequence $K^0_X = (x)$ of smooth cycles
with closed supports in $X$ is given which satisfies the following conditions.
For each cycle $x \in K^0_X$, $\mathop{\mathrm{supp}}\nolimits x \cap (-\infty, 0] \times Y^-$ is empty set or
there exists some cycle $y \in K^0_{Y^-}$ such that
$x |_{(-\infty, 0] \times Y^-} = (-\infty, 0] \times y$.
Similarly, for each cycle $x \in K^0_X$, $\mathop{\mathrm{supp}}\nolimits x \cap [0, \infty) \times Y^+$ is
empty set or there exists some cycle $y \in K^0_{Y^+}$ such that
$x |_{[0, \infty) \times Y^+} = [0, \infty) \times y$.
Further we assume that these relations give bijections
\begin{align*}
\mu_- &: \{x \in K^0_X; \mathop{\mathrm{supp}}\nolimits x \cap (-\infty, 0] \times Y^- \neq \emptyset\}
\to K^0_{Y^-} \text{ and}\\
\mu_+ &: \{x \in K^0_X; \mathop{\mathrm{supp}}\nolimits x \cap [0, \infty) \times Y^+ \neq \emptyset\}
\to K^0_{Y^+}.
\end{align*}
First we explain about the construction of the multisections of the fiber products.
Assume that the multisection of
$(\widehat{\mathcal{M}}_{Y^\pm}^\diamond, \mathring{K}_{Y^\pm}^2, K_{Y^\pm}, K_{Y^\pm}^0)$
is given for each $Y^\pm$.
We define a space
$\widehat{\mathcal{M}}_X^\diamond$
as follows.
Its point
\[
((\Sigma^\alpha, z^\alpha, u^\alpha)_{\alpha \in A^- \sqcup A^0 \sqcup A^+},
M^{\mathrm{rel}})
\]
consists of holomorphic buildings
$(\Sigma^\alpha, z^\alpha, u^\alpha)_{\alpha \in A^-}$ for $Y^-$,
$(\Sigma^\alpha, z^\alpha, u^\alpha)_{\alpha \in A^0}$ for $X$,
$(\Sigma^\alpha, z^\alpha, u^\alpha)_{\alpha \in A^+}$ for $Y^+$, and
a set $M^{\mathrm{rel}} = \{(S^1_{+\infty_l}, S^1_{-\infty_l})\}$ of pairs of
limit circles which satisfy the following conditions:
\begin{itemize}
\item
Any two pairs in $M^{\mathrm{rel}}$ do not share the same limit circle.
\item
For each pair $\alpha_1, \alpha_2 \in A = A^- \sqcup A^0 \sqcup A^+$,
let $M^{\alpha_1, \alpha_2} \subset M^{\mathrm{rel}}$ be the subset of pairs
$(S^1_{+\infty_l}, S^1_{-\infty_l})$ such that $S^1_{+\infty_l}$ is a $+\infty$-limit circle
of $\Sigma^{\alpha_1}$ and $S^1_{-\infty_l}$ is a $-\infty$-limit circle of
$\Sigma^{\alpha_2}$.
Then there does not exists any sequence
$\alpha_0, \alpha_1, \dots, \alpha_k = \alpha_0 \in A$ such that
$M^{\alpha_i, \alpha_{i+1}} \neq \emptyset$ for all $i = 0, 1, \dots, k-1$.
\item
For subsets $A_1, A_2 \subset A$, define
$M^{A_1, A_2} = \bigcup_{\alpha_1 \in A_1, \alpha_2 \in A_2} M^{\alpha_1, \alpha_2}$.
Then $M^{\mathrm{rel}}$ is the union of
$M^{\mathrm{rel}, \leq 0} = M^{A^-, A^- \sqcup A^0}$
and $M^{\mathrm{rel}, \geq 0} = M^{A^0 \sqcup A^+, A^+}$.
\end{itemize}
We regard $\widehat{\mathcal{M}}^\diamond_{Y^-}$ and $\widehat{\mathcal{M}}^\diamond_{Y^+}$
as subspaces of $\widehat{\mathcal{M}}^\diamond_X$ consisting of points
such that $A^0 = A^+ = \emptyset$ and $A^- = A^0 = \emptyset$ respectively.
We say a point
$((\Sigma^\alpha, z^\alpha, u^\alpha)_{\alpha \in A^- \sqcup A^0 \sqcup A^+},
M^{\mathrm{rel}}) \in \widehat{\mathcal{M}}^\diamond_X$ is disconnected if
there exists a decomposition $A^- \sqcup A^0 \sqcup A^+ = A_1 \sqcup A_2$ such that
$M^{A_1, A_2} = M^{A_2, A_1} = \emptyset$.
Otherwise we say it is connected.
We denote the space of connected points of $\widehat{\mathcal{M}}^\diamond_X$ by
$(\widehat{\mathcal{M}}^\diamond_X)^0$.
Decomposition into connected components defines a map $\widehat{\mathcal{M}}^\diamond_X
\to \bigcup_N (\prod^N (\widehat{\mathcal{M}}^\diamond_X)^0) / \mathfrak{S}_N$.
Let
\begin{align*}
\Upsilon : \widehat{\mathcal{M}}^\diamond_X &\to
\prod (\overline{P}_{Y^-} \times \overline{P}_{Y^-}) / \mathfrak{S}
\times \prod (\overline{P}_{Y^+} \times \overline{P}_{Y^+}) / \mathfrak{S} \\
&\quad \quad
\times \prod \overline{P}_{Y^-} / \mathfrak{S}
\times \prod \overline{P}_{Y^+} / \mathfrak{S}
\times \prod (Y^- \cup X \cup Y^+) / \mathfrak{S}.
\end{align*}
be the continuous map which maps a point
$((\Sigma^\alpha, z^\alpha, u^\alpha)_{\alpha \in A^- \sqcup A^0 \sqcup A^+},
M^{\mathrm{rel}})$ to the following point.
Its $\prod (\overline{P}_{Y^-} \times \overline{P}_{Y^-}) / \mathfrak{S}$-factor is
\[
(\pi_{Y^-} \circ u|_{S^1_{+\infty_l}}, \pi_{Y^-} \circ u|_{S^1_{+\infty_l}})
_{(S^1_{+\infty_l}, S^1_{-\infty_l}) \in M^{\mathrm{rel}, \leq 0}},
\]
its $\prod (\overline{P}_{Y^+} \times \overline{P}_{Y^+}) / \mathfrak{S}$-factor is
\[
(\pi_{Y^+} \circ u|_{S^1_{+\infty_l}}, \pi_{Y^+} \circ u|_{S^1_{+\infty_l}})
_{(S^1_{+\infty_l}, S^1_{-\infty_l}) \in M^{\mathrm{rel}, \geq 0}},
\]
its $\prod \overline{P}_{Y^-} / \mathfrak{S}
\times \prod \overline{P}_{Y^+} / \mathfrak{S}$-factor is
$(\pi_{Y^\pm} \circ u|_{S^1_{\pm\infty}})_{S^1_{\pm\infty} \notin M^{\mathrm{rel}}}$,
and its $\prod (Y^- \cup X \cup Y^+) / \mathfrak{S}$-factor is
$(u(z))_{z \in \bigcup_\alpha z^\alpha}$, where if $z \in z^\alpha$ is contained in
a negative (or positive) floor of $(\Sigma^\alpha, z^\alpha, u^\alpha)$,
then we read $u(z)$ as $\pi_{Y^-} \circ u(z)$
(or $\pi_{Y^-} \circ u(z)$ respectively).
$\Upsilon$ is realized as a strong smooth map.
Define the fiber product
\[
(\widehat{\mathcal{M}}^\diamond_X, (\mathring{K}_{Y^-}^2, \mathring{K}_{Y^+}^2),
(K_{Y^-}, K_{Y^+}), (K^0_{Y^-}, K^0_X, K^0_{Y^+})) \subset
\widehat{\mathcal{M}}^\diamond_X
\]
by
\begin{align*}
&(\widehat{\mathcal{M}}^\diamond_X, (\mathring{K}_{Y^-}^2, \mathring{K}_{Y^+}^2),
(K_{Y^-}, K_{Y^+}), (K^0_{Y^-}, K^0_X, K^0_{Y^+})) \\
&= \Upsilon^{-1}\Bigl(\prod \mathring{K}_{Y^-}^2 / \mathfrak{S} \times
\prod \mathring{K}_{Y^+}^2 / \mathfrak{S} \times
\prod K_{Y^-} / \mathfrak{S} \times \prod K_{Y^+} / \mathfrak{S} \\
&\hphantom{= \Upsilon^{-1}\Bigl(}
\times \prod (K^0_{Y^-} \cup K^0_X \cup K^0_{Y^+}) / \mathfrak{S}\Bigr).
\end{align*}
We abbreviate the above fiber product
as $(\widehat{\mathcal{M}}^\diamond_X, \mathring{K}^2_X, K_X, K^0_X)$.
(This is simply an abbreviation, and $\mathring{K}^2_X$ or $K_X$
are not defined.)
Define a multi-valued partial submersion
$\Xi : \widehat{\mathcal{M}}^\diamond_X \to \widehat{\mathcal{M}}^\diamond_X$ by
\begin{align*}
&\Xi(((\Sigma^\alpha, z^\alpha, u^\alpha)_{\alpha \in A^- \sqcup A^0 \sqcup A^+},
M^{\mathrm{rel}})) \\
&= \{((\Sigma^\alpha, z^\alpha, u^\alpha)_{\alpha \in A^- \sqcup A^0 \sqcup A^+},
\mathring{M}^{\mathrm{rel}}); \mathring{M}^{\mathrm{rel}} \subsetneq M^{\mathrm{rel}}\},
\end{align*}
and let $\Xi^\circ$ be the restriction of
$\Xi \subset \widehat{\mathcal{M}}^\diamond_X \times \widehat{\mathcal{M}}^\diamond_X$
to the set of points
\[
(((\Sigma^\alpha, z^\alpha, u^\alpha)_{\alpha \in A^- \sqcup A^0 \sqcup A^+},
M^{\mathrm{rel}}),
((\Sigma^\alpha, z^\alpha, u^\alpha)_{\alpha \in A^- \sqcup A^0 \sqcup A^+},
\mathring{M}^{\mathrm{rel}}))
\]
such that
$((\Sigma^\alpha, z^\alpha, u^\alpha)_{\alpha \in A^- \sqcup A^0 \sqcup A^+},
M^{\mathrm{rel}})
\in (\widehat{\mathcal{M}}^\diamond_X, \mathring{K}^2_X, K_X, K^0_X)$
and $(\pi_Y \circ u|_{S^1_{+\infty_l}}, \pi_Y \circ u|_{S^1_{-\infty_l}})
\in \rho_\ast K_{Y^\pm}$
for all $(S^1_{+\infty_l}, S^1_{-\infty_l}) \in M^{\mathrm{rel}} \setminus
\mathring{M}^{\mathrm{rel}}$.
Then $\Xi^\circ$ is a multi-valued partial essential submersion
from $(\widehat{\mathcal{M}}^\diamond_X, \mathring{K}^2_X, K_X, K^0_X)$ to itself.
We also define a multi-valued partial essential submersion
\[
\Lambda : (\partial \widehat{\mathcal{M}}^\diamond_X, \mathring{K}^2_X, K_X, K^0_X)
\to (\widehat{\mathcal{M}}^\diamond_X, \mathring{K}^2_X, K_X, K^0_X)
\]
similarly to the case of $Y$.
Similarly to the case of symplectization, for each point
\[
p = ((\Sigma^\alpha, z^\alpha, u^\alpha)
_{\alpha \in A^- \sqcup A^0 \sqcup A^+}, M^{\mathrm{rel}})
\in (\widehat{\mathcal{M}}^\diamond_X, \mathring{K}^2_X, K_X, K^0_X),
\]
we define
$\widetilde{e}(p) = \widetilde{e}_{\delta_0}(p)
= \sum_\alpha \widehat{e}_{\delta_0}(\theta_\alpha)
+ \frac{1}{2} \# M^{\mathrm{rel}}$, where each
$\theta_\alpha$ is the type of $(\Sigma^\alpha, z^\alpha, u^\alpha)$.
Then the maps $\Xi^\circ$ and $\Lambda$ decrease $\widetilde{e}$.
Hence if we decompose $(\widehat{\mathcal{M}}^\diamond_X, \mathring{K}^2_X, K_X, K^0_X)$ by
$\widetilde{e}$, then
$\Xi^\circ$ and $\Lambda$ constitute a compatible system of
multi-valued partial submersions.
We can construct the perturbed multisections of
$(\widehat{\mathcal{M}}^\diamond_X, \mathring{K}^2_X, K_X, K^0_X)$
which satisfy the following conditions:
\begin{itemize}
\item
The restrictions of the grouped multisection of
$(\widehat{\mathcal{M}}^\diamond_X, \mathring{K}^2_X, K_X, K^0_X)$ to
$(\widehat{\mathcal{M}}_{Y^\pm}^\diamond, \mathring{K}_{Y^\pm}^2, K_{Y^\pm}, K_{Y^\pm}^0)$
coincide with the given grouped multisections.
\item
Let $((\widehat{\mathcal{M}}^\diamond_X)^0, \mathring{K}^2_X, K_X, K^0_X) \subset
(\widehat{\mathcal{M}}^\diamond_X, \mathring{K}^2_X, K_X, K^0_X)$
be the subset of connected points.
Its grouped multisection induces that of
\[
\bigcup_N (\prod^N (\widehat{\mathcal{M}}^\diamond_X)^0, \mathring{K}^2_X, K_X, K^0_X)
/ \mathfrak{S}_N.
\]
Then the grouped multisection of
$(\widehat{\mathcal{M}}^\diamond_X, \mathring{K}^2_X, K_X, K^0_X)$
coincides with its pull back by the submersion
\[
(\widehat{\mathcal{M}}^\diamond_X, \mathring{K}^2_X, K_X, K^0_X) \to
\bigcup_N (\prod^N (\widehat{\mathcal{M}}^\diamond_X)^0, \mathring{K}^2_X, K_X, K^0_X)
/ \mathfrak{S}_N
\]
defined by decomposition into connected components.
\item
The grouped multisections of
$(\widehat{\mathcal{M}}^\diamond_X, \mathring{K}^2_X, K_X, K^0_X)$
are compatible with respect to the compatible system of
multi-valued partial essential submersions defined by $\Xi^\circ$ and $\Lambda$.
\end{itemize}
Next we define the fiber products we use for the construction of the algebra.
As in Section \ref{fiber prod and orientation},
let $((\hat \epsilon^{i, j}_l), (\hat c^i_l), (x^i_l), (\eta^i_l))$ be sequences of simplices
with local coefficients such that
\begin{itemize}
\item
$\hat \epsilon^{i, j}_l = \theta^{\lsuperscript{D}{t}}_{\epsilon^{i, j}_l} \epsilon^{i, j}_l
\theta^D_{\epsilon^{i, j}_l}$ ($-m_- \leq i < j \leq 0$)
are products of simplices $\epsilon^{i, j}_l$ in $\mathring{K}^2_{Y^-}$not contained in
$\overline{P}_{Y^-}^{^t\text{bad}} \times \overline{P}_{Y^-} \cup
\overline{P}_{Y^-} \times \overline{P}_{Y^-}^{\text{bad}}$ and orientations
$\theta^{\lsuperscript{D}{t}}_{\epsilon^{i, j}_l}$ of $p_1^\ast S_{Y^-}^{\lsuperscript{D}{t}}$
and $\theta^D_{\epsilon^{i, j}_l}$ of $p_2^\ast S_{Y^-}^{\lsuperscript{D}{t}}$
on $\mathop{\mathrm{Int}}\nolimits \epsilon^{i, j}_l$,
\item
$\hat \epsilon^{i, j}_l = \theta^{\lsuperscript{D}{t}}_{\epsilon^{i, j}_l} \epsilon^{i, j}_l
\theta^D_{\epsilon^{i, j}_l}$ ($0 \leq i < j \leq m_+$)
are products of simplices $\epsilon^{i, j}_l$ in $\mathring{K}^2_{Y^+}$ not contained in
$\overline{P}_{Y^+}^{^t\text{bad}} \times \overline{P}_{Y^+} \cup
\overline{P}_{Y^+} \times \overline{P}_{Y^+}^{\text{bad}}$ and orientations
$\theta^{\lsuperscript{D}{t}}_{\epsilon^{i, j}_l}$ of $p_1^\ast S_{Y^+}^{\lsuperscript{D}{t}}$
and $\theta^D_{\epsilon^{i, j}_l}$ of $p_2^\ast S_{Y^+}^{\lsuperscript{D}{t}}$
on $\mathop{\mathrm{Int}}\nolimits \epsilon^{i, j}_l$,
\item
$\hat c^i_l = c^i_l \theta^D_{c^i_l}$ ($-m_- \leq i \leq 0$) are products of simplices
$c^i_l$ in $K_{Y^-}$ not contained in $\overline{P}_{Y^-}^\text{bad}$ and
orientations $\theta^D_{c^i_l}$ of $\S_{Y^-}^D$ on $\mathop{\mathrm{Int}}\nolimits c^i_l$,
\item
$\hat \eta^i_l = \theta^{\lsuperscript{D}{t}}_{\eta^i_l} \eta^i_l$ are products of simplices
$\eta^i_l$ in $K_{Y^+}$ not contained in $\overline{P}_{Y^+}^{^t\text{bad}}$ and
orientations $\theta^{\lsuperscript{D}{t}}_{\eta^i_l}$ of $\S_{Y^+}^{\lsuperscript{D}{t}}$
on $\mathop{\mathrm{Int}}\nolimits \eta^i_l$,
\item
$x^i_l$ ($-m_- \leq i < 0$) are cycles in $K^0_{Y^-}$,
\item
$x^0_l$ are cycles in $K^0_X$, and
\item
$x^i_l$ ($0 < i \leq m_+$) are cycles in $K^0_{Y^+}$.
\end{itemize}
Take lifts $\tilde \epsilon^{i, j}_l$, $\tilde c^i_l$ and $\tilde \eta^i_l$, and define
$\breve \epsilon^{i, j}_l = \theta^{\lsuperscript{D}{t}}_{\epsilon^{i, j}_l} \tilde \epsilon^{i, j}_l
\theta^D_{\epsilon^{i, j}_l}$, $\breve c^i_l = \tilde c^i_l \theta^D_{c^i_l}$ and
$\breve \eta^i_l = \theta^{\lsuperscript{D}{t}}_{\eta^i_l} \tilde \eta^i_l$ as in Section
\ref{fiber prod and orientation}.
For such a sequence, the pre-Kuranishi space
$\overline{\mathcal{M}}^{(m_-, X, m_+)}
_{((\breve \epsilon^{i, j}_l), (\breve c^i_l), (x^i_l), (\breve \eta^i_l))}$
is defined similarly.
Its grouped multisection is defined by the pull back of that of
$(\widehat{\mathcal{M}}^\diamond_X, \mathring{K}^2_X, K_X, K^0_X)$ by the natural map
\[
\overline{\mathcal{M}}^{(m_-, X, m_+)}
_{((\breve \epsilon^{i, j}_l), (\breve c^i_l), (x^i_l), (\breve \eta^i_l))} \to
(\widehat{\mathcal{M}}^\diamond_X, \mathring{K}^2_X, K_X, K^0_X).
\]
The definition of its orientation is almost the same with the case of $\hat Y$.
The only difference is that we define the orientation of
$\mathcal{W}^0 = TX^0 \times \mathcal{C}^0 / \mathbb{R}^{k_- + k_+} \oplus
\bigoplus_{z^{++}_{0, \beta}} \mathbb{R}^2$
by
\[
(-1)^{k_+} (\mathbb{R}^{k_- + k_+} \oplus \bigoplus_{z^{++}_{0, \beta}} \mathbb{R}^2) \oplus \mathcal{W}^0
= TX^0 \times \mathcal{C}^0
\]
if the range of the holomorphic building corresponding to the center of the Kuranishi
neighborhood is $(\overline{\mathbb{R}}_{-k_-} \cup \dots \cup \overline{\mathbb{R}}_{-1}) \times Y^-
\cup \overline{X} \cup (\overline{\mathbb{R}}_1 \cup \dots \cup \overline{\mathbb{R}}_{k_+}) \times Y^+$.
It is easy to check that this is well defined
and independent of the choice of the lifts of $\eta^i_j$, $c^i_j$ and $\epsilon^i_j$
under the natural isomorphism.
Hence we may denote the above pre-Kuranishi space by
$\overline{\mathcal{M}}^{(m_-, X, m_+)}_{((\hat \epsilon^{i, j}_l), (\hat c^i_l), (x^i_l), (\hat \eta^i_l))}$.
Similarly to equation (\ref{boundary of MM}),
it is easy to see that for any $((\hat c_l), (x_l), (\hat \eta_l))$ and
$(\hat \epsilon^{i, j}_l)$,
\begin{align}
0 &= \sum_{\star_{m_-, m_+}} (-1)^\ast
[\partial \overline{\mathcal{M}}^{(m_-, X, m_+)}_{((\hat \epsilon^{i, j}_l), (\hat c^i_l), (x^i_l),
(\hat \eta^i_l))}]^0 \notag\\
&= \sum_{\star_{m_-, m_+}} (-1)^{\ast + m_- + m_+}
[\overline{\mathcal{M}}^{(m_-, X, m_+)}_{\partial ((\hat \epsilon^{i, j}_l), (\hat c^i_l), (x^i_l),
(\hat \eta^i_l))}]^0 \notag\\
&\quad + \sum_{\substack{-m_- \leq i_0 \leq 0 \\ \star_{m_- +1, m_+}}}
(-1)^{\ast + m_- + 1 + i_0} [\overline{\mathcal{M}}^{(m_- +1, X, m_+)}
_{((e^{\Delta_\ast [\overline{P}_{Y^-}]})^{i_0 -1, i_0} \cup
(\tau^-_{i_0}\hat \epsilon^{i, j}_l), (\hat c^i_l),
(x^i_l), (\hat \eta^i_l))}]^0 \notag\\
&\quad + \sum_{\substack{0 \leq i_0 \leq m_+ \\ \star_{m_-, m_+ +1}}}
(-1)^{\ast + m_- + i_0} [\overline{\mathcal{M}}^{(m_-, X, m_+ +1)}
_{((e^{\Delta_\ast [\overline{P}_{Y^-}]})^{i_0, i_0 +1} \cup (\tau^+_{i_0} \hat \epsilon^{i, j}_l),
(\hat c^i_l), (x^i_l), (\hat \eta^i_l))}]^0, \label{boundary of MMX}
\end{align}
where the sum $\star_{m_-, m_+}$ is taken over all decompositions
\[
\{\hat c_l\} = \coprod_{-m_- \leq i \leq 0} \{\hat c^i_l\},
\quad \{\hat \eta_l\} = \coprod_{0 \leq i \leq m_+} \{\hat \eta^i_l\}
\]
as sets
and all decompositions
\[
\{x_l\} = \coprod_{-m_- \leq i \leq m_+} \{x^i_l\}
\]
such that $x^i_l \in K_{Y^-}^0$ for $-m_- \leq i <0$ and $x^i_l \in K_{Y^+}^0$
for $0 < i \leq m_+$.
(We identify $x \in K_X^0$ with $\mu_-(x) \in K_{Y^-}^0$ and $\mu_+(x) \in K_{Y^+}^0$
in the above decomposition.)
The sign $\ast$ is the weighted sign of the permutation
\[
\begin{pmatrix}
(\hat c^1_l)_l & \cdots & (\hat c^m_l)_l
& (x^1_l)_l & \cdots & (x^m_l)_l & (\hat \eta^1_l)_l & \cdots & (\hat \eta^m_l)_l\\
&(\hat c_l)_l&&&(x_l)_l&&&(\hat \eta_l)_l&
\end{pmatrix}.
\]
The definition of $\tau^\pm_{i_0} \hat \epsilon_l^{i, j}$ are similar to
$\tau_{i_0} \hat \epsilon^{i, j}_l$ in Section \ref{fiber prod and orientation}.
See the next section for the precise definition.
Let $((\hat c_l), (x_l), (\alpha_l))$ be a triple of
\begin{itemize}
\item
a sequence of chains $\hat c_l$ in
$C_\ast(\overline{P}_{Y^-}, \overline{P}_{Y^-}^{\text{bad}}; \S_{Y^-}^D \otimes \mathbb{Q})$
\item
a sequence of cycles $x_l$ in $K_X^0$, and
\item
a sequence of cochains $\alpha_l$ with compact supports in
$C^\ast(\overline{P}_{Y^+}, \overline{P}_{Y^+}^{\text{bad}}; \S_{Y^+}^D \otimes \mathbb{Q})$,
\end{itemize}
For such a triple $((\hat c_l), (x_l), (\alpha_l))$, we define a pre-Kuranishi space
(or a linear combination of pre-Kuranishi spaces)
$\overline{\mathcal{M}}^X((\hat c_l), (x_l), (\alpha_l))$ by
\begin{align*}
&\overline{\mathcal{M}}^X((\hat c_l), (x_l), (\alpha_l))\\
& = \sum_{m_-, m_+ \geq 0}\sum_{\star_{m_-, m_+}}(-1)^\ast \overline{\mathcal{M}}^{(m_-, X, m_+)}
_{((\widetilde{G}^+_{m_+}, \widetilde{G}^-_{-m_-}), (\hat c^i_l), (x^i_l),
([\overline{P}_{Y^+}] \cap \alpha^i_l))}
\end{align*}
where
$\widetilde{G}^\pm = \widetilde{G}^\pm_0 + \widetilde{G}^\pm_{\pm1} +
\widetilde{G}^\pm_{\pm2} + \cdots = \Theta^\pm(e^{\otimes G^\pm})$ are appropriate
linear combinations of
\begin{multline*}
((\kappa\rho\Delta_\ast [\overline{P}_{Y^\pm}])^{i, j}, \dots,
(\kappa\rho\Delta_\ast [\overline{P}_{Y^\pm}])^{i, j},
\epsilon_{\overline{P}_{Y^\pm}}^{i, j}, \dots, \epsilon_{\overline{P}_{Y^\pm}}^{i, j},\\
(\Delta_\ast [\overline{P}_{Y^\pm}])^{i, j}, \dots, (\Delta_\ast [\overline{P}_{Y^\pm}])^{i, j})
\end{multline*}
defined in the next section.
(Pay attention to the order of $(\widetilde{G}^+_{m_+}, \widetilde{G}^-_{-m_-})$.
This is equivalent to $(-1)^{m_- m_+}(\widetilde{G}^-_{-m_-}, \widetilde{G}^+_{m_+})$.)
The sum $\star_{m_-, m_+}$ is taken over all decompositions
\[
\{\hat c_l\} = \coprod_{-m_- \leq i \leq 0} \{\hat c^i_l\},
\quad \{\alpha_l\} = \coprod_{0 \leq i \leq m_+} \{\alpha^i_l\}
\]
as sets
and all decompositions
\[
\{x_l\} = \coprod_{-m_- \leq i \leq m_+} \{x^i_l\}
\]
such that $x^i_l \in K_{Y^-}^0$ for $-m_- \leq i <0$ and $x^i_l \in K_{Y^+}^0$
for $0 < i \leq m_+$.
The sign $\ast$ is the weighted sign of the permutation
\[
\begin{pmatrix}
(\hat c^1_l)_l & \cdots & (\hat c^m_l)_l
& (x^1_l)_l & \cdots & (x^m_l)_l & (\alpha^1_l)_l & \cdots & (\alpha^m_l)_l\\
&(\hat c_l)_l&&&(x_l)_l&&&(\alpha_l)_l&
\end{pmatrix}.
\]
We note $\widetilde{G}^\pm_0 = 1$.
Hence the main term is
\[
\overline{\mathcal{M}}^X_{((\hat c_l), (x_l), ([\overline{P}_{Y^+}] \cap \alpha_l))}.
\]
The following equation holds true.
\begin{align}
0 &= [\partial\overline{\mathcal{M}}^X((\hat c_l), (x_l), (\alpha_l))]^0\notag\\
&= [\overline{\mathcal{M}}^X(\partial((\hat c_l), (x_l), (\alpha_l)))]^0\notag\\
&\quad - \sum_{\star_-} (-1)^{\ast_-} \frac{1}{k !}
[\overline{\mathcal{M}}^{Y^-}((\hat c^-_l), (x^-_l), (\hat d_1^\ast, \hat d_2^\ast, \dots,
\hat d_k^\ast))]^0\notag\\
&\hphantom{\quad - \sum_{\star_-} (-1)^{\ast_-} \frac{1}{k !}}
\times [\overline{\mathcal{M}}^X((\hat d_k, \hat d_{k - 1}, \dots, \hat d_1) \cup (\hat c^0_l),
(x^0_l), (\alpha_l))]^0\notag\\
&\quad + \sum_{\star_+} (-1)^{\ast_+} \frac{1}{k !}
[\overline{\mathcal{M}}^X((\hat c_l), (x^0_l), (\alpha^0_l) \cup (\hat d_1^\ast, \hat d_2^\ast,
\dots, \hat d_k^\ast))]^0\notag\\
&\hphantom{\quad + \sum_{\star_+} (-1)^{\ast_+} \frac{1}{k !}}
\times [\overline{\mathcal{M}}^{Y^+}((\hat d_k, \hat d_{k - 1}, \dots, \hat d_1), (x^+_l), (\alpha^+_l))]^0
\label{boundary formula for X}
\end{align}
where the sum $\star_-$ is taken over $k \geq 0$, all simplices $d_l$ of $K_{Y^-}$
not contained in $\overline{P}_{Y^-}^{\text{bad}}$,
and all decompositions
\[
\{\hat c_l\} = \{\hat c^-_l\} \sqcup \{\hat c^0_l\}, \quad
\{x_l\} = \{x^-_l\} \sqcup \{x^0_l\}
\]
such that $x^-_l \in K_{Y^-}^0$.
The sign $\ast_-$ is the weighted sign of the permutation
\[
\begin{pmatrix}
(\hat c^-_l) \ (x^-_l) \ (\hat c^0_l) \ (x^0_l)\\
(\hat c_l) \quad (x_l)
\end{pmatrix}.
\]
The sum $\star_+$ is taken over $k \geq 0$, all simplices $d_l$ of $K_{Y^+}$ not contained
in $\overline{P}_{Y^+}^{\text{bad}}$, and all decompositions
\[
\{s_l\} = \{x^0_l\} \sqcup \{x^+_l\}, \quad \{\alpha_l\} = \{\alpha^0_l\} \sqcup \{\alpha^+_l\}
\]
such that $x^+_l \in K_{Y^+}^0$.
$\ast_+$ is the weighted sign of the permutation
\[
\begin{pmatrix}
(x^0_l) \ (\alpha^0_l) \ (x^+_l) \ (\alpha^+_l)\\
(x_l) \quad (\alpha_l)
\end{pmatrix}.
\]
To construct the algebra, we need to use the space of irreducible sequences of
holomorphic buildings.
Let $f^\pm_a$ be monomials of the form
\begin{multline*}
((\rho_\ast [\overline{P}_{Y^\pm}])^{i, j}, \dots,
(\rho_\ast [\overline{P}_{Y^\pm}])^{i, j},
\epsilon_{\overline{P}_{Y^\pm}}^{i, j}, \dots, \epsilon_{\overline{P}_{Y^\pm}}^{i, j},\\
(\Delta_\ast [\overline{P}_{Y^\pm}])^{i, j}, \dots, (\Delta_\ast [\overline{P}_{Y^\pm}])^{i, j}
)_{\substack{0 \leq i < j \leq m_{f_a^\pm}\\ (\text{or} -m_{f_a^\pm} \leq i < j \leq 0)}}
\end{multline*}
such that $m_\pm = \sum m_{f_a^\pm}$.
Then we define the space of irreducible sequences of holomorphic buildings
\begin{align*}
&(\overline{\mathcal{M}}^{(m_-, X, m_+)}
)_{(f^+_1 \otimes \dots \otimes f^+_{n^+}, f^-_1 \otimes \dots \otimes f^-_{n^-}),
(\breve c^i_l), (x^i_l), (\breve \eta^i_l))}^0\\
&\subset \overline{\mathcal{M}}^{(m_-, X, m_+)}
_{(\Theta^+(f^+_1 \otimes \dots \otimes f^+_{n^+}),
\Theta^-(f^-_1 \otimes \dots \otimes f^-_{n^-})), (\breve c^i_l), (x^i_l), (\breve \eta^i_l))}
\end{align*}
as follows. ($\Theta^\pm$ is defined by (\ref{Theta^pm}) in the next section.)
First we consider the case of $(n_-, n_+) \neq (0, 0)$.
A sequence of holomorphic buildings
\[
(\Sigma_i, s_i, u_i, \phi_i)_{-m_- \leq i \leq m_+} \in
\overline{\mathcal{M}}^{(m_-, X, m_+)}_{(\Theta^+(f^+_1 \otimes \dots \otimes f^+_{n^+}),
\Theta^-(f^-_1 \otimes \dots \otimes f^-_{n^-})), (\breve c^i_l), (x^i_l), (\breve \eta^i_l))}
\]
is contained in the above space if
\begin{itemize}
\item
each connected component of $\Sigma_0$ ($\Sigma_0$ is the $\overline{\mathcal{M}}^X$-factor
of $(\Sigma_i)_{-m_- \leq i \leq m_+}$. It is not necessarily of height one.)
concerns at least one monomial $f^\pm_i$,
that is, it contains at least one limit circle corresponding to
a variable in $f^\pm_i$,
and
\item
for any decomposition $\{f^-_1, \dots f^-_{n^-}, f^+_1, \dots, f^+_{n^+}\} = A \sqcup B$,
there exists a connected component of $\Sigma_0$ which concerns both of
some $f \in A$ and some $g \in B$.
\end{itemize}
If $(n_-, n_+) = (0, 0)$, then a holomorphic building $(\Sigma, z, u, \phi) \in
\overline{\mathcal{M}}^X_{((\breve c^i_l), (x^i_l), (\breve \eta^i_l))}$
is irreducible if it is connected.
First we note that all irreducible sequences of holomorphic buildings corresponding to
the zeros of the multisection of the $0$-dimensional component of the above
Kuranishi space have genera $\geq 0$ if each $f_a^\pm$ is contained
in $\mathring{\mathcal{B}}_{m_{f_a^\pm}}^\pm$
(this is also defined in the next section),
that is, if the number of variables in each $f_a^\pm$ each of which defines a relation of
the periodic orbit on one $+\infty$-limit circle of $\Sigma_i$ ($i \neq 0$) and
the periodic orbit on one $-\infty$-limit circle of $\Sigma_j$ ($j \neq 0$) is
$\geq m_{f_a^\pm} -1$.
This is because each factor $\Sigma_i$ except the $\overline{\mathcal{M}}^X$-factor is
connected by the dimensional reason.
We also note that for any sequence of holomorphic buildings
$(\Sigma_i, z_i, u_i, \phi_i)_i$ in
\[
\overline{\mathcal{M}}^{(m_-, X, m_+)}_{(\Theta^+(f^+_1 \otimes \dots \otimes f^+_{n^+}),
\Theta^-(f^-_1 \otimes f^-_2 \otimes \dots \otimes f^-_{n^-})), (\breve c^i_l), (x^i_l),
(\breve \eta^i_l))},
\]
we can decompose the set $\{f^\pm_a\}$ into sets $A_j$ such that
for any $j \neq j'$, there does not exist a connected component of $\Sigma_0$
which concerns both of some $f \in A_j$ and some $g \in A_j$,
and each $A_j$ cannot be decomposed further.
Hence each sequence of holomorphic buildings corresponding to a zero of the multisection
of the $0$-dimensional component can be decomposed into
ireducible sequences of holomorphic buildings contained
in the factors corresponding to $A_j$ and the connected holomorphic
buildings with height one.
For each connected holomorphic buildings with height one,
we add an empty set to $\{A_j\}$, and call $\{A_j\}$ as the irreducible decomposition
of $\{f^\pm_a\}$ corresponding to $(\Sigma_i, z_i, u_i, \phi_i)_i$.
For each triple $((\hat c_l), (x_l), (\alpha_l))$, we define a pre-Kuranishi space
\begin{align*}
&(\overline{\mathcal{M}}^X)^0((\hat c_l), (x_l), (\alpha_l))\\
& = \sum_{m_-, m_+ \geq 0}\sum_{\star_{m_-, m_+}}(-1)^\ast
(\overline{\mathcal{M}}^{(m_-, X, m_+)})^0_{((\widehat{G}^+_{m_+}, \widehat{G}^-_{-m_-}),
(\hat c^i_l), (x^i_l), ([\overline{P}_{Y^+}] \cap \alpha^i_l))}
\end{align*}
where
$e^{\otimes G^\pm} = \widehat{G}^\pm_0 + \widehat{G}^\pm_1 + \widehat{G}^\pm_2
+ \dots \in (\bigoplus_{m = 0}^\infty \bigotimes_{\sum l_i = m}(\mathcal{B}_{l_i}^+)^{l_i})^\wedge$.
Note that since $G^\pm$ is contained in
$(\bigoplus_{l = 1}^\infty (\mathring{\mathcal{B}}_l^+)^l)^\wedge$, the genera of the zero of the
multisection of the zero-dimensional component of
$(\overline{\mathcal{M}}^X)^0((\hat c_l), (x_l), (\alpha_l))$ are $\geq 0$.
The irreducible decomposition implies the following equation.
\begin{equation}
[\overline{\mathcal{M}}^X((\hat c_l), (x_l), (\alpha_l))]^0
= \sum (-1)^\ast \frac{1}{k !} \prod_{i = 1}^k
[(\overline{\mathcal{M}}^X)^0((\hat c^i_l), (x^i_l), (\alpha^i_l))]^0,
\label{irreducible decomposition}
\end{equation}
where the sum is taken over all $k \geq 0$ and all decompositions
\[
\{\hat c_l\} = \coprod_{i = 1}^k \{\hat c^i_l\}, \quad \{x_l\} = \coprod_{i = 1}^k \{x^i_l\}, \quad
\{\alpha_l\} = \coprod_{i = 1}^k \{\alpha^i_l\}
\]
as sets.
The sign $\ast$ is the weighted sign of the permutation
\[
\begin{pmatrix}
(c^1_l)\ (x^1_l)\ (\alpha^1_l) \dots (c^k_l)\ (x^k_l)\ (\alpha^k_l)\\
(c_l) \quad (x_l) \quad (\alpha_l)
\end{pmatrix}
\]
If $((\hat c_l), (x_l), (\alpha_l)) = (\emptyset, \emptyset, \emptyset)$, then the term
$\prod_{i = 1}^0 [(\overline{\mathcal{M}}^X)^0((\hat c^i_l), (x^i_l), (\alpha^i_l))]^0$ corresponding to
$k = 0$ on the right hand side of equation (\ref{irreducible decomposition}) is
defined by $1$, and otherwise it is defined by zero.
It corresponds to the number of the empty curve.
Equation (\ref{irreducible decomposition}) is proved as follows.
We write $\overline{\mathcal{M}}^X((\hat c_l), (x_l), (\alpha_l))$ as
\begin{align*}
&\overline{\mathcal{M}}^X((\hat c_l), (x_l), (\alpha_l)) \\
&= \sum_{N^-, N^+ \geq 0} (-1)^\ast \overline{\mathcal{M}}^{(\ast, X, \ast)}
_{((\Theta^+((G^+)^{\otimes N^+}), \Theta^-((G^-)^{\otimes N^-})), (\hat c^i_l), (x^i_l),
([\overline{P}_{Y^+}] \cap \alpha^i_l))},
\end{align*}
where we omit $m_\pm$ because they differ according to the variables in
$\Theta^\pm((G^+)^{\otimes N^\pm})$.
For each point $(\Sigma_i, z_i, u_i, \phi_i)_i$ in
\[
\overline{\mathcal{M}}^{(\ast, X, \ast)}
_{((\Theta^+((G^+)^{\otimes N^+}), \Theta^-((G^-)^{\otimes N^-})), (\hat c^i_l), (x^i_l),
([\overline{P}_{Y^+}] \cap \alpha^i_l))},
\]
we decompose the sequence of holomorphic buildings $(\Sigma_i, z_i, u_i, \phi_i)_i$
into irreducible sequences $(\Sigma^j_i, z^j_i, u^j_i, \phi^j_i)^j_i$ ($j = 1, \dots, k$).
Then it corresponds to a point in
$\prod^k (\overline{\mathcal{M}}^{(\ast, X, \ast)})^0_{((e^{\otimes G^+}, e^{\otimes G^-}),
\ast, \ast, \ast)} / \mathfrak{S}_k$,
and the group $\mathfrak{S}_k$ of permutation corresponds to the coefficient
$1 / k!$ in the right hand side of Equation (\ref{irreducible decomposition}).
Next we count the number of points $(\Sigma_i, z_i, u_i, \phi_i)_i$ in
\[
\overline{\mathcal{M}}^{(\ast, X, \ast)}
_{((\Theta^+((G^+)^{\otimes N^+}), \Theta^-((G^-)^{\otimes N^-})), \ast, \ast, \ast)}
\]
corresponding to a given
irreducible sequences $(\Sigma^j_i, z^j_i, u^j_i, \phi^j_i)^j_i$ ($j = 1, \dots, k$).
Define $n^\pm_j \geq 0$ by the condition that
for each $j$, $(\Sigma^j_i, z^j_i, u^j_i, \phi^j_i)_i$ is contained in
$(\overline{\mathcal{M}}^{(\ast, X, \ast)})^0_{(((G^+)^{\otimes n^+_j}, (G^-)^{\otimes n^-_j}),
\ast, \ast, \ast)}$.
Consider the irreducible decomposition $\{A_j\}_{j=1,\dots, k}$ of
$\{G^+_{(1)}, \dots, G^+_{(N^+)}, G^-_{(1)}, \dots, G^-_{(N^-)}\}$
($G^\pm_{(i)} = G^\pm$) corresponding to the point $(\Sigma_i, z_i, u_i, \phi_i)_i$,
and let $A_j = A^+_j \sqcup A^-_j$ be the decomposition into the sets
$A^\pm_j$ consisting of $G^\pm$.
Then $\# A^\pm_j = n^\pm_j$.
Conversely, for any decomposition $\{A_j\}_{j=1,\dots, k}$ of
$\{G^+_{(1)}, \dots, G^+_{(N^+)}, G^-_{(1)}, \dots, G^-_{(N^-)}\}$ such that
$\# A^\pm_j = n^\pm_j$, there exists a unique point in $\overline{\mathcal{M}}^{(\ast, X, \ast)}
_{((\Theta^+((G^+)^{\otimes N^+}), \Theta^-((G^-)^{\otimes N^-})), \ast, \ast, \ast)}$
corresponding to $(\Sigma^j_i, z^j_i, u^j_i, \phi^j_i)^j_i$ ($j = 1, \dots, k$) and
the decomposition $\{A_j\}_{j=1,\dots, k}$.
The number of such decompositions of the set is
\[
\frac{N^+ !}{n^+_1 ! n^+_2 ! \dots n^+_k !} \cdot \frac{N^- !}{n^-_1 ! n^-_2 ! \dots n^-_k !},
\]
and it coincides with the ratios of the product of the coefficients
$1 / N^\pm !$ of $(G^\pm)^{\otimes N^\pm}$ in $e^{\otimes G^\pm}$
on the left hand side of Equation (\ref{irreducible decomposition})
to the product of the coefficients
$1 / n^\pm_j !$ of $(G^\pm)^{\otimes n^\pm_j}$ in $e^{\otimes G^\pm}$
on the right hand side.
Hence Equation (\ref{irreducible decomposition}) holds true.
\begin{rem}
As in Remark \ref{connected or not}, we do not know whether or not we can choose
$G^\pm$ so that all irreducible sequences of holomorphic buildings
in the zero-dimensional
component of $(\overline{\mathcal{M}}^X)^0((\hat c_l), (x_l), (\alpha_l))$ are connected.
However, for the construction of the algebra in Section \ref{algebra for X},
it is enough to observe their genera are $\geq 0$.
\end{rem}
\subsection{Construction of the correction terms}\label{correction terms for X}
In this section, we construct $(G_{\pm m}^\pm)_{m \geq 1}$ used for the definition of
the correction terms of $\overline{\mathcal{M}}^X((\hat c_l), (x_l), (\alpha_l))$.
As in the case of the construction of $(F_m)_{m \geq 2}$, we consider algebras modeled
on the splitting of holomorphic buildings.
We construct $(G_m^+)_{m \geq 1}$ and $(G_{-m}^-)_{m \geq 1}$ independently.
First we construct $(G_m^+)_{m \geq 1}$.
For $m \geq 1$, let $B^+_m = \bigoplus_{n = 0}^{\frac{m(m + 1)}{2}} (B^+_m)^n$ be the
$\mathbb{Z}$-graded super-commutative algebra with coefficient $\mathbb{R}$ generated by variables
$\rho_{(e_i, e_j)}$, $\Delta_{(e_i, e_j)}$ and $\epsilon_{(e_i, e_j)}$ ($0 \leq i < j \leq m$).
The $\mathbb{Z}$-grading is defined by $\dim \rho_{(e_i, e_j)} = \dim \Delta_{(e_i, e_j)} = 0$ and
$\dim \epsilon_{(e_i, e_j)} = 1$.
For $m = 0$, we define $B^+_0 = \mathbb{R}$.
For each $m \geq 1$, the differential $\partial' : B^+_m \to B^+_m$ is defined by
$\partial' \epsilon_{(a, b)} = (-1)^m (\rho_{(a, b)} - \Delta_{(a, b)})$ and $\partial' \rho_{(a, b)}
= \partial' \Delta_{(a, b)} = 0$.
For $m = 0$, we define $\partial' = 0 : B_0^+ \to B_0^+$.
Homomorphisms $\tau^+_i : B^+_m \to B^+_{m + 1}$ ($0 \leq i \leq m$, $m \geq 1$)
are defined by
$\tau^+_i(x_{(a, b)}) = x_{(\hat \tau_i(a), \hat \tau_i(b))}$,
where $x$ is $\rho$, $\epsilon$ or $\Delta$, and each $\hat \tau^+_i$ is defined by
\[
\hat \tau^+_i (e_j) = \begin{cases}
e_j &j < i\\
e_i + e_{i + 1} &j = i\\
e_{j + 1} & j > i
\end{cases}.
\]
For $m = 0$, we define $\tau^+_0 = \mathrm{id}_{\mathbb{R}}$.
For $i > m$, we define $\tau^+_i = 0 : B^+_m \to B^+_{m + 1}$.
We define homomorphisms $\Diamond^+ : B^+_m \otimes A_{m'} \to B^+_{m+ m'}$
($m \geq 0$, $m' \geq 1$) by
\[
\Diamond^+ (f \otimes g) = (-1)^{1 + m m'} f \cdot
\exp(\rho_{(\sum_{0 \leq i \leq m} e_i, \sum_{m + 1 \leq j \leq m + m'} e_j)}) \cdot
g^{+ m}.
\]
We define homomorphisms $\Theta^+ : \bigotimes_{i = 1}^n B^+_{m_i} \to B^+_{\sum m_i}$
by
\begin{equation}
\Theta^+ (f_1 \otimes f_2 \otimes \dots \otimes f_n)
= f_1^{+\sum_{i = 2}^n m_i} \cdot f_2^{+\sum_{i = 3}^n m_i} \cdots f_n,
\label{Theta^pm}
\end{equation}
where $f^{+k}$ is defined by
\[
e^{+k}_j = \begin{cases}
e_0 &j = 0\\
e_{j + k} & j \neq 0
\end{cases}.
\]
For $n = 0$, we define $\Theta^+ = \mathrm{id}_\mathbb{R} : \mathbb{R} \to \mathbb{R}$.
Define a linear subspace $\Ddot B^+_m \subset B^+_m$ as follows.
For each $1 \leq i \leq m-1$ ($i \neq 0$) and each monomial
\[
f = x^{(1)}_{(a_1, b_1)} x^{(2)}_{(a_2, b_2)} \dots x^{(n)}_{(a_n, b_n)},
\]
such that $(a_j, b_j) \neq (e_i, e_{i+1})$, we define a monomial
\[
f^{(e_i, e_{i + 1})} = x^{(1)}_{(a'_1, b'_1)} x^{(2)}_{(a'_2, b'_2)} \dots x^{(n)}_{(a'_n, b'_n)}
\]
by permuting $e_i$ and $e_{i + 1}$ in $\{a_j, b_j\}$.
Then $\Ddot B^+_m \subset B^+_m$ is the subspace spanned by
$f + f^{(e_i, e_{i + 1})}$ for all such pair $i$ and $f$.
Define $\mathcal{B}^+_m = B^+_m / \Ddot B^+_m$.
This is not an algebra but the following maps are well defined.
\begin{align*}
\partial' &: \mathcal{B}^+_m \to \mathcal{B}^+_m & (m \geq 0)\\
\sum_{i \geq 1} (-1)^i e^{\Delta_{(e_i, e_{i + 1})}} \tau^+_i &: \mathcal{B}_m \to \mathcal{B}_{m + 1} &
(m \geq 0)\\
e^{\Delta_{(e_0, e_1)}} \tau^+_0 &: \mathcal{B}_m \to \mathcal{B}_{m + 1} & (m \geq 0)\\
\Diamond^+ &: \mathcal{B}^+_m \otimes \mathcal{A}_{m'} \to \mathcal{B}^+_{m + m'} & (m \geq 0, m' \geq 1)\\
\Theta^+ &: \bigotimes_{i = 1}^n \mathcal{B}^+_{m_i} \to \mathcal{B}^+_{\sum m_i} & (n \geq 0, m_i \geq 0)
\end{align*}
Further we define $\mathring{\mathcal{B}}^+_m \subset \mathcal{B}^+_m$ as follows.
First we define a new degree $\deg'$ by
\[
\deg' x_{(e_i, e_j)} = \begin{cases}
0 & i = 0\\
1 & i \geq 1
\end{cases}.
\]
Let $\mathring{B}^+_m \subset B^+_m$ be the ideal generated by monomials with
$\deg' \geq m - 1$ and define $\mathring{\mathcal{B}}^+_m = \mathring{B}^+_m
/ (\Ddot B^+_m \cap \mathring{B}^+_m)$.
It is easy to see that the homology of $((\mathring{\mathcal{B}}^+_m)^\ast, \partial')$ is zero
at $\ast \neq 0$.
($\ast$ is the dimension.)
Let $F^+ \in \mathcal{A}$ be a zero obtained in Section \ref{algebra for correction}.
We prove that there exists some $G^+ = G^+_1 +G^+_2 + \dots \in
(\bigoplus_{l = 1}^\infty (\mathring{\mathcal{B}}^+_l)^l)^\wedge$ such that
\begin{multline}
\partial' (\Theta^+(e^{\otimes G^+})) + \sum_i (-1)^i e^{\Delta{(e_i, e_{i + 1})}} \tau^+_i
\Theta^+(e^{\otimes G^+})\\
+ \Diamond^+ (\Theta^+(e^{\otimes G^+}) \otimes F^+) = 0,
\label{+G eq}
\end{multline}
where $e^{\otimes G^+} = 1 + G^+ + \frac{1}{2!} G^+ \otimes G^+
+ \frac{1}{3!} G^+ \otimes G^+ \otimes G^+ + \cdots$.
We inductively construct $G^+_{\leq m}= G^+_1 + G^+_2 + \cdots + G^+_m \in
\bigoplus_{l = 1}^m (\mathring{\mathcal{B}}_l^+)^l$ such that
\begin{multline}
\partial' (\Theta^+(e^{\otimes G^+_{\leq m}})) + \sum_i (-1)^i e^{\Delta{(e_i, e_{i + 1})}} \tau^+_i
\Theta^+(e^{\otimes G^+_{\leq m-1}})\\
+ \Diamond^+ (\Theta^+(e^{\otimes G^+_{\leq m-1}}) \otimes
F^+) \equiv 0 \label{G^+_m eq}
\end{multline}
in $(\bigoplus_{l = 1}^\infty (\mathcal{B}^+_l)^{l-1})^\wedge /
(\bigoplus_{l = m + 1}^\infty (\mathcal{B}^+_l)^{l - 1})^\wedge$.
First we define $G^+_{\leq 1} = G^+_1 \in (\mathring{\mathcal{B}}_1^+)^1$ by
\begin{align*}
G^+_1 = - \sum_{k = 1}^\infty \frac{1}{k !} (
&\underbrace{\epsilon_{(e_0, e_1)} \Delta_{(e_0, e_1)} \dots \Delta_{(e_0, e_1)}}_k\\
&+ \underbrace{\rho_{(e_0, e_1)} \epsilon_{(e_0, e_1)} \Delta_{(e_0, e_1)} \dots
\Delta_{(e_0, e_1)}}_k\\
&+ \dots + \underbrace{\rho_{(e_0, e_1)} \dots \rho_{(e_0, e_1)} \epsilon_{(e_0, e_1)}}_k).
\end{align*}
Then it is easy to check that this satisfies equation (\ref{G^+_m eq}).
Next assuming we have constructed $G^+_{\leq m - 1}$,
we prove there exists a required $G^+_{\leq m}$ ($m \geq 2$).
It is enough to show that
\begin{multline}
\partial' (\Theta^+(e^{\otimes G^+_{\leq m-1}})) + \sum_i (-1)^i e^{\Delta{(e_i, e_{i + 1})}} \tau^+_i
\Theta^+(e^{\otimes G^+_{\leq m-1}})\\
+ \Diamond^+ (\Theta^+(e^{\otimes G^+_{\leq m-1}}) \otimes
F^+) \equiv 0 \label{B mathring}
\end{multline}
in $(\bigoplus_{l = 1}^\infty (\mathcal{B}^+_l)^{l-1})^\wedge /((\bigoplus_{l = m + 1}^\infty
(\mathcal{B}^+_l)^{l - 1})^\wedge \oplus \bigoplus_{l = 1}^\infty (\mathring{\mathcal{B}}^+_l)^{l-1})$
and
\begin{equation}
\partial' \Bigl( \sum_i (-1)^i e^{\Delta{(e_i, e_{i + 1})}} \tau^+_i
\Theta^+(e^{\otimes G^+_{\leq m-1}}) + \Diamond^+ (\Theta^+(e^{\otimes G^+_{\leq m-1}})
\otimes F^+) \Bigr) \equiv 0 \label{B closed}
\end{equation}
in $(\bigoplus_{l = 2}^\infty (\mathcal{B}^+_l)^{l-2})^\wedge /
(\bigoplus_{l = m + 1}^\infty (\mathcal{B}^+_l)^{l-2})^\wedge$.
First we prove equation (\ref{B mathring}).
For the proof, we use the following maps
$\mathring{\tau}^+_0$ and $\mathring{\Diamond}^+$.
The linear map $\mathring{\tau}^+_0 : \bigotimes_{i = 1}^n B^+_{m_i} \to B^+_{\sum m_i}$ is
defined as follows.
Let $f_i \in B_{m_i}^+$ ($1 \leq i \leq n$) be monomials, and consider each term of
\[
\tau^+_0 \Theta^+(f_1 \otimes f_2 \otimes \dots \otimes f_n).
\]
In each term, some of $e_0$'s appearing in $f_i$ are changed to $e_1$ since
$\hat \tau_0^+$ maps $e_0$ to $e_0 + e_1$.
$\mathring{\tau}^+_0(f_1 \otimes f_2 \otimes \dots f_n)$ is defined by the sum of
the terms appearing in $\tau^+_0 \Theta^+(f_1 \otimes f_2 \otimes \dots \otimes f_n)$
such that each $f_i$ has at least one $e_0$ which is changed to $e_1$.
Then it induces a linear map $\mathring{\tau}^+_0 : \bigotimes_{i = 1}^n \mathcal{B}^+_{m_i}
\to \mathcal{B}^+_{\sum m_i}$.
For $n = 0$, we defined $\mathring{\tau}^+_0 = \mathrm{id}_\mathbb{R} : \mathbb{R} \to \mathbb{R}$.
For example, if $m_1 = 2$ and $m_2 = 1$, then
\begin{align*}
&\mathring{\tau}^+_0 (\Delta_{(e_0, e_1)} \epsilon_{(e_0, e_2)} \epsilon_{(e_1, e_2)} \otimes
\epsilon_{(e_0, e_1)})\\
&= (\Delta_{(e_0, e_3)} \epsilon_{(e_0, e_4)} + \Delta_{(e_0, e_3)} \epsilon_{(e_1, e_4)}
+ \Delta_{(e_1, e_3)} \epsilon_{(e_0, e_4)}) \epsilon_{(e_3, e_4)} \epsilon_{(e_0, e_2)}.
\end{align*}
The linear map $\mathring{\Diamond}^+ : (\bigotimes_{i = 1}^n B^+_{m_i}) \otimes A_{m'} \to
B^+_{\sum m_i + m'}$ is defined as follows.
Put $m = \sum_{i = 1}^n m_i$ and let
\[
\mathring{\exp}(\rho_{(\sum_{0 \leq i \leq m} e_i, \sum_{m + 1 \leq j \leq m + m'} e_j)})
\]
be the sum of all terms in
$\exp(\rho_{(\sum_{0 \leq i \leq m} e_i, \sum_{m + 1 \leq j \leq m + m'} e_j)})$
in which at least one $e_k$ appears for each $1 \leq l \leq n$ such that
$\sum_{1 \leq a < l} m_a + 1 \leq k \leq \sum_{1 \leq a \leq l} m_a$.
For example, if $n = 2$ and $m_1 = m_2 = m' = 2$, then
\begin{align*}
&\mathring{\exp}(\rho_{(\sum_{0 \leq i \leq 4} e_i, \sum_{j = 5, 6} e_j)})\\
& = \exp(\rho_{(\sum_{0 \leq i \leq 4} e_i, \sum_{j = 5, 6} e_j)})
- \exp(\rho_{(\sum_{i = 0, 3, 4} e_i, \sum_{j = 5, 6} e_j)})\\
& \quad - \exp(\rho_{(\sum_{i = 0, 1, 2} e_i, \sum_{j = 5, 6} e_j)})
+ \exp(\rho_{(e_0, \sum_{j = 5, 6} e_j)}).
\end{align*}
$\mathring{\Diamond}^+$ is defined by
\begin{align*}
\mathring{\Diamond}^+(f_1 \otimes f_2 \otimes \dots \otimes f_n \otimes g)
&= (-1)^{1 + (\sum m_i) m'} \Theta^+(f_1 \otimes f_2 \otimes \dots \otimes f_n)\\
&\quad \cdot \mathring{\exp}(\rho_{(\sum_{0 \leq i \leq m} e_i, \sum_{m + 1 \leq j \leq m + m'} e_j)})
g^{+ \sum m_i}.
\end{align*}
It also induces a linear map
$\mathring{\Diamond}^+ : (\bigotimes_{i = 1}^n \mathcal{B}^+_{m_i}) \otimes \mathcal{A}_{m'} \to
\mathcal{B}^+_{\sum m_i + m'}$.
We can easily check the following equations for any
$G \in (\bigoplus_{l = 1}^\infty (\mathring{\mathcal{B}}^+_l)^l)^\wedge$.
\begin{equation}
\partial' \Theta^+\biggl(\frac{1}{k !} G^{\otimes k}\biggr) = \Theta^+\biggl(\frac{1}{(k-1) !}
G^{\otimes (k-1)} \otimes \partial' G\biggr)\label{Geq 1}
\end{equation}
\begin{align}
&\sum_{i \geq 1} (-1)^i e^{\Delta_{(e_i, e_{i+1})}} \tau^+_i \Theta^+
\biggl(\frac{1}{k !} G^{\otimes k}\biggr) \notag\\
&\hspace{60pt}
= \Theta^+\biggl(\frac{1}{(k-1) !} G^{\otimes (k-1)} \otimes
\sum_{i \geq 1} (-1)^i e^{\Delta_{(e_i, e_{i+1})}} \tau^+_i G\biggr)\label{Geq 2}
\end{align}
\begin{align}
&e^{\Delta_{(e_0, e_1)}} \tau^+_0 \Theta^+\biggl(\frac{1}{k !} G^{\otimes k}\biggr) \notag\\
&\hspace{60pt}
= \sum_{l = 0}^k \Theta^+\biggl(\frac{1}{(k-l) ! l !}G^{\otimes (k-l)} \otimes
(e^{\Delta_{(e_0, e_1)}} \mathring{\tau}^+_0(G^{\otimes l}))\biggr)\label{Geq 3}
\end{align}
\begin{align}
&\Diamond^+\biggl(\Theta^+\biggl(\frac{1}{k !} G^{\otimes k}\biggr) \otimes F^+\biggr)
\notag\\
&\hspace{70pt}
=
\sum_{l = 0}^k \Theta^+\biggl(\frac{1}{(k-l) ! l !}G^{\otimes (k-l)} \otimes
\mathring{\Diamond}^+ (G^{\otimes l} \otimes F^+)\biggr)\label{Geq 4}
\end{align}
Furthermore, it is easy to see that
\begin{equation}
\Theta^+(f_1 \otimes \dots \otimes f_k \otimes
\Theta^+(f_{k + 1} \otimes \dots \otimes f_n))
= \Theta^+(f_1 \otimes \dots f_n).\label{Theta eq}
\end{equation}
The assumption of the induction implies
\begin{align*}
R^{(m-1)} =& \
\partial' (\Theta^+(e^{\otimes G^+_{\leq m-1}}))
+ \sum_i (-1)^i e^{\Delta{(e_i, e_{i + 1})}}
\tau^+_i \Theta^+(e^{\otimes G^+_{\leq m-1}})\\
&+ \Diamond^+ (\Theta^+(e^{\otimes G^+_{\leq m-1}}) \otimes F^+)
\end{align*}
is zero in $(\bigoplus_{l = 1}^\infty (\mathcal{B}^+_l)^{l-1})^\wedge /
(\bigoplus_{l = m}^\infty (\mathcal{B}^+_l)^{l-1})^\wedge$.
Hence
\[
\frac{(-1)^l}{l !} \Theta^+((G^+_{\leq m-1})^{\otimes l} \otimes R^{(m-1)})
\equiv 0
\]
in $(\bigoplus_{l = 1}^\infty (\mathcal{B}^+_l)^{l-1})^\wedge /
(\bigoplus_{l = m + 1}^\infty (\mathcal{B}^+_l)^{l-1})^\wedge$ for all $l \geq 1$.
Therefore, for the proof of (\ref{B mathring}), it is enough to prove that
\begin{equation}
\sum_{l \geq 0} \frac{(-1)^l}{l !} \Theta^+((G^+_{\leq m-1})^{\otimes l} \otimes R^{(m-1)})
\equiv 0 \label{sum eq}
\end{equation}
in $(\bigoplus_{l = 1}^\infty (\mathcal{B}^+_l)^{l-1})^\wedge /(
(\bigoplus_{l = m + 1}^\infty (\mathcal{B}^+_l)^{l-1})^\wedge \oplus
\bigoplus_{l=1}^m (\mathring{\mathcal{B}}^+_l)^{l-1})$.
Equations (\ref{Geq 1}) to (\ref{Theta eq}) imply that the left hand side of
(\ref{sum eq}) is equal to
the sum of the following terms:
\begin{equation}
\sum_{l \geq 0} \frac{(-1)^l}{l !} \Theta^+((G^+_{\leq m-1})^{\otimes l} \otimes
\partial' (\Theta^+(e^{\otimes G^+_{\leq m-1}})))
= \partial' G^+_{\leq m-1} \label{term 1}
\end{equation}
\begin{align}
&\sum_{l \geq 0} \frac{(-1)^l}{l !} \Theta^+\Bigl((G^+_{\leq m-1})^{\otimes l} \otimes
\sum_{i \geq 1} (-1)^i e^{\Delta{(e_i, e_{i + 1})}}
\tau^+_i \Theta^+(e^{\otimes G^+_{\leq m-1}})\Bigr) \notag\\
&= \sum_{i \geq 1} (-1)^i e^{\Delta{(e_i, e_{i + 1})}}
\tau^+_i G^+_{\leq m-1} \label{term 2}
\end{align}
\begin{align}
&\sum_{l \geq 0} \frac{(-1)^l}{l !} \Theta^+((G^+_{\leq m-1})^{\otimes l} \otimes
e^{\Delta{(e_0, e_1)}} \tau^+_0 \Theta^+(e^{\otimes G^+_{\leq m-1}})) \notag\\
&= e^{\Delta{(e_0, e_1)}} \mathring{\tau}^+_0 \Theta^+(e^{\otimes G^+_{\leq m-1}})
\label{term 3}
\end{align}
\begin{align}
&\sum_{l \geq 0} \frac{(-1)^l}{l !} \Theta^+((G^+_{\leq m-1})^{\otimes l} \otimes
\Diamond^+ (\Theta^+(e^{\otimes G^+_{\leq m-1}}) \otimes F^+)) \notag\\
&= \mathring{\Diamond}^+ (\Theta^+(e^{\otimes G^+_{\leq m-1}}) \otimes F^+)
\label{term 4}
\end{align}
Terms (\ref{term 1}), (\ref{term 3}), (\ref{term 4}) and
$(e^{\Delta_{(e_i, e_{i + 1})}} - 1) \tau^+_i G^+_{\leq m-1}$ ($i \geq 1$)
are contained in
$(\bigoplus_l (\mathring{\mathcal{B}}^+_l)^{l-1})^\wedge$,
and $\tau^+_i G^+_{\leq m-1} \equiv 0$ in
$\bigoplus_{l = 1}^\infty (\mathcal{B}_l^+)^{l-1}$ for $i > 0$.
(In general, $\tau_i^+ f$ is contained in $\ddot B_{m + 1}^+$ for any $f \in B_m^+$
and $i > 0$.)
These prove equation (\ref{sum eq}).
Therefore we can construct $G^+_{\leq m}$ inductively.
As with equation (\ref{A closed}) in Section \ref{algebra for correction},
equation (\ref{B closed}) is proved as follows.
Put $\widetilde{G}^+ = \Theta^+(e^{\otimes G^+_{\leq m-1}})$.
The left hand side of (\ref{B closed}) is
\begin{align*}
&\sum_i (-1)^{i+1} e^{\Delta_{(e_i, e_{i + 1})}} \tau^+_i \partial' \widetilde{G}^+
+ \Diamond^+ \Bigl(\partial' \widetilde{G}^+ \otimes \sum_j (-1)^j F^+_j \Bigr)
+ \Diamond^+(\widetilde{G}^+ \otimes \partial' F^+)\\
&= \sum_i (-1)^i e^{\Delta_{(e_i, e_{i + 1})}} \tau^+_i \Bigl(\sum_j (-1)^j
e^{\Delta_{(e_j, e_{j + 1})}}
\tau^+_j \widetilde{G}^+ + \Diamond^+(\widetilde{G}^+ \otimes F^+)\Bigr)\\
&\quad - \Diamond^+\Bigl(\Bigl(\sum_i (-1)^i e^{\Delta_{(e_i, e_{i + 1})}}
\tau^+_i \widetilde{G}^+ + \Diamond^+(\widetilde{G}^+ \otimes F^+)\Bigr)
\otimes \sum_j (-1)^j F^+_j\Bigr)\\
&\quad - \Diamond^+\Bigl(\widetilde{G}^+ \otimes \Bigl(\sum_i (-1)^i
e^{\Delta_{(e_i, e_{i + 1})}} \tau_i F^+ + \Box(F^+ \otimes F^+)\Bigr)\Bigr)
\end{align*}
and this is zero because
{\belowdisplayskip= 0pt
\[
\Bigl(\sum_i (-1)^i e^{\Delta_{(e_i, e_{i + 1})}} \tau^+_i \Bigr) \circ \Bigl(\sum_j (-1)^j
e^{\Delta_{(e_j, e_{j + 1})}} \tau^+_j \Bigr) = 0,
\]
}
\begin{align*}
\sum_i (-1)^i e^{\Delta_{(e_i, e_{i + 1})}} \tau^+_i \Diamond^+(f \otimes g)
- \Diamond^+\Bigl(\sum_i (-1)^i e^{\Delta_{(e_i, e_{i + 1})}} \tau^+_i f \otimes
(-1)^{\deg g} g\Bigr)\\
- \Diamond^+\Bigl(f \otimes \sum_i (-1)^i e^{\Delta_{(e_i, e_{i + 1})}} \tau^+_i g \Bigr)
= 0,
\end{align*}
\[
\Diamond^+(f \otimes \Box(g \otimes h)) + \Diamond^+(\Diamond^+(f \otimes g)
\otimes (-1)^{\deg h} h) = 0.
\]
Next, we construct $(G_{-m}^-)_{m \geq 1}$.
For $m \geq 1$, let $B^-_{-m}$ be the $\mathbb{Z}$-graded super-commutative algebra
with coefficient $\mathbb{R}$ generated by variables
$\rho_{(e_i, e_j)}$, $\Delta_{(e_i, e_j)}$ and $\epsilon_{(e_i, e_j)}$ ($-m \leq i < j \leq 0$).
The $\mathbb{Z}$-grading is defined by $\dim \rho_{(e_i, e_j)} = \dim \Delta_{(e_i, e_j)} = 0$ and
$\dim \epsilon_{(e_i, e_j)} = 1$.
For $m = 0$, we define $B^-_0 = \mathbb{R}$.
For each $m \geq 1$, the differential $\partial' : B^-_{-m} \to B^-_{-m}$ is defined by
$\partial' \epsilon_{(a, b)} = (-1)^m (\rho_{(a, b)} - \Delta_{(a, b)})$ and $\partial' \rho_{(a, b)}
= \partial' \Delta_{(a, b)} = 0$.
Homomorphisms $\tau^-_i : B^-_{-m} \to B^-_{- m - 1}$ ($-m \leq i \leq 0$, $m \geq 1$)
are defined by
$\tau^-_i(x_{(a, b)}) = x_{(\hat \tau_i(a), \hat \tau_i(b))}$,
where $\hat \tau^-_i$ is defined by
\[
\hat \tau^-_i (e_j) = \begin{cases}
e_{j - 1} &j < i\\
e_{i - 1} + e_i &j = i\\
e_j & j > i
\end{cases}.
\]
For $m = 0$, we define $\tau^-_0 = \mathrm{id}_{\mathbb{R}}$.
For $i < -m$, we define $\tau^-_i = 0 : B^-_{-m} \to B^-_{-m - 1}$.
We define $\tilde\tau^-_i = (-1)^{m + 1 + i} \tau^-_i : B^-_{-m} \to B^-_{-m - 1}$.
We define homomorphisms $\Diamond^- : A_m \otimes B^-_{-m'} \to B^-_{- m - m'}$
($m \geq 1$, $m' \geq 0$) by
\[
\Diamond^- (f \otimes g) = (-1)^{(m - 1)m'} f^{-m'} \cdot
\exp(\rho_{(\sum_{-m -m' \leq i \leq -m' - 1} e_i, \sum_{-m' \leq j \leq 0} e_j)}) \cdot g.
\]
We define homomorphisms $\Theta^- : \bigotimes_{i = 1}^n B^-_{-m_i} \to B^-_{\sum -m_i}$
by
\[
\Theta^- (f_1 \otimes f_2 \otimes \dots \otimes f_n)
= f_1 \cdot f_2^{-m_1} \cdots f_n^{-\sum_{i=1}^{n - 1} m_i},
\]
where $f^{-k}$ is defined by
\[
e^{-k}_j = \begin{cases}
e_0 &j = 0\\
e_{j - k} & j \neq 0
\end{cases}.
\]
Define $\mathcal{B}^-_m$ and $\mathring{\mathcal{B}}^-_m \subset \mathcal{B}^-_m$ similarly.
In this case, $\deg'$ is defined by
\[
\deg' x_{(e_i, e_j)} = \begin{cases}
0 & j = 0\\
1 & j \leq -1
\end{cases}.
\]
Let $F^- \in \mathcal{A}$ be a zero in Section \ref{algebra for correction}.
(We do not need to assume $F^- = F^+$.)
As in the case of $G^+$, we can construct
$G^- = G^-_{-1} + G^-_{-2} + \dots \in (\bigoplus_{l = 1} (\mathcal{B}^-_{-l})^l)^\wedge$ such that
\begin{align}
&\partial' (\Theta^-(e^{\otimes G^-})) + \sum_i e^{\Delta{(e_{i - 1}, e_i)}} \tilde\tau^-_i
\Theta^-(e^{\otimes G^-}) \notag\\
&\hspace{120pt}
+ \Diamond^- (F^- \otimes \Theta^-(e^{\otimes G^-})) = 0. \label{-G eq}
\end{align}
Note that
\begin{align*}
G^-_1 = \sum_{k=1}^\infty \frac{1}{k !}(
&\underbrace{\epsilon_{(e_{-1}, e_0)} \Delta_{(e_{-1}, e_0)} \dots \Delta_{(e_{-1}, e_0)}}_k\\
&+ \underbrace{\rho_{(e_{-1}, e_0)} \epsilon_{(e_{-1}, e_0)} \Delta_{(e_{-1}, e_0)} \dots
\Delta_{(e_{-1}, e_0)}}_k\\
&+ \dots + \underbrace{\rho_{(e_{-1}, e_0)} \dots
\rho_{(e_{-1}, e_0)} \epsilon_{(e_{-1}, e_0)}}_k).
\end{align*}
Equation (\ref{boundary formula for X}) is satisfied for the solutions
$G^+$ of (\ref{+G eq}) and $G^-$ of (\ref{-G eq}) because (\ref{boundary of MMX})
implies
\begin{align*}
&\sum_{\star_{m_-, m_+}} (-1)^\ast \partial' \overline{\mathcal{M}}^{(m_-, X, m_+)}
_{((\widetilde{G}^+_{m_+}, \widetilde{G}^-_{-m_-}), (\hat c^i_l), (x^i_l), (\hat \eta^i_l))}\\
&= \sum_{\star_{m_-, m_+}} (-1)^\ast \overline{\mathcal{M}}^{(m_-, X, m_+)}
_{((\widetilde{G}^+_{m_+}, \widetilde{G}^-_{-m_-}), \partial((\hat c^i_l), (x^i_l),
(\hat \eta^i_l)))}\\
& \quad + \sum_{\star_{m_-, m_+}} (-1)^{\ast + m_-} \overline{\mathcal{M}}^{(m_-, X, m_+)}
_{((\partial' \widetilde{G}^+_{m_+}, \widetilde{G}^-_{-m_-}), (\hat c^i_l), (x^i_l),
(\hat \eta^i_l))}\\
& \quad + \sum_{\star_{m_-, m_+ +1}}
(-1)^{\ast + m_-} \overline{\mathcal{M}}^{(m_-, X, m_+ +1)}_{((\sum_{i = 0}^{m_+}
(-1)^i e^{\Delta_{(e_i, e_{i+1})}}\widetilde{G}^+_{m_+},
\widetilde{G}^-_{-m_-}),
(\hat c^i_l), (x^i_l), (\hat \eta^i_l))}\\
& \quad + \sum_{\star_{m_-, m_+}} (-1)^\ast \overline{\mathcal{M}}^{(m_-, X, m_+)}
_{((\widetilde{G}^+_{m_+}, \partial' \widetilde{G}^-_{-m_-}), (\hat c^i_l), (x^i_l),
(\hat \eta^i_l))}\\
& \quad + \sum_{\star_{m_- + 1, m_+}}
(-1)^\ast \overline{\mathcal{M}}^{(m_- +1, X, m_+)}_{((\widetilde{G}^+_{m_+},
\sum_{i = -m_-}^0 e^{\Delta_{(e_{i-1}, e_i)}} \tilde \tau^-_i \widetilde{G}^-_{-m_-}),
(\hat c^i_l), (x^i_l), (\hat \eta^i_l))}
\end{align*}
and the following equations hold true.
For any $m$, $(m_-, m_+)$ and $((\hat c^i_l)_{i = -m - m_-}^{m_+},\allowbreak
(x^i_l)_{i = -m - m_-}^{m_+},\allowbreak (\hat \eta^i_l)_{i = - m_-}^{m_+})$,
\begin{align*}
&\sum \frac{1}{k!} \Bigl[(\overline{\mathcal{M}}_{Y^-})^m_{(F^-_m, (\hat c^{i -m_-}_l)_{i = -m}^{- 1},
(x^{i - m_-}_l)_{i = -m}^{- 1},
([\overline{P}_{Y^-}] \cap \hat d_1^\ast, \dots, [\overline{P}_{Y^-}] \cap
\hat d_k^\ast))}\Bigr]^0\\
& \hphantom{\sum \frac{1}{k!}} \cdot
\bigl[\overline{\mathcal{M}}^{(m_-, X, m_+)}_{((\widetilde{G}^+_{m_+}, \widetilde{G}^-_{-m_-}),
(\hat d_k, \dots, \hat d_1) \cup (\hat c^i_l), (x^i_l), (\hat \eta^i_l))}\bigr]^0\\
& = \bigl[\overline{\mathcal{M}}^{(m_- + m, X, m_+)}_{((\widetilde{G}^+_{m_+}, \Diamond^-(F^-_m
\otimes \widetilde{G}^-_{-m_-})), (\hat c^i_l), (x^i_l), (\hat \eta^i_l))}\bigr]^0
\end{align*}
where the sum is taken over all $k \geq 0$ and all simplices $d_l$ of $K_{Y^-}$
not contained in $\overline{P}_{Y^-}^{\text{bad}}$, and
for any $m$, $(m_-, m_+)$ and $((\hat c^i_l)_{i = -m_-}^{m_+},\allowbreak (x^i_l)_{i = -m_-}^{m_+ + m},
\allowbreak (\hat \eta^i_l)_{i = -m_-}^{m_+ + m})$,
\begin{align*}
&\sum \frac{1}{k!} \bigl[\overline{\mathcal{M}}^{(m_-, X, m_+)}_{((\widetilde{G}^+_{m_+},
\widetilde{G}^-_{-m_-}), (\hat c^i_l), (x^i_l), (\hat \eta^i_l) \cup
([\overline{P}_{Y^-}] \cap \hat d_1^\ast, \dots, [\overline{P}_{Y^-}] \cap
\hat d_k^\ast))}\bigr]^0\\
& \hphantom{\sum \frac{1}{k!}} \cdot\Bigl[(\overline{\mathcal{M}}_{Y^+})^m
_{(F^+_m, (\hat d_k, \dots, \hat d_1),
(x^{i + m_+}_l)_{i = 1}^m, (\hat \eta^{i+ m_+}_l)_{i = 1}^m)}\Bigr]^0\\
&= (-1)^{1 + m_-}\bigl[\overline{\mathcal{M}}^{(m_-, X, m_+ + m)}
_{((\Diamond^+(\widetilde{G}^+_{m_+} \otimes F^+_m), \widetilde{G}^-_{-m_-}),
(\hat c^i_l), (x^i_l), (\hat \eta^i_l))}\bigr]^0
\end{align*}
where the sum is taken over all $k \geq 0$ and all simplices $d_l$ of $K_{Y^+}$
not contained in $\overline{P}_{Y^+}^{\text{bad}}$
\subsection{Construction of the algebras}\label{algebra for X}
In this section, we construct the algebra for $X$.
It gives a kind of chain map between the algebras for $Y^-$ and $Y^+$
in the sense of SFT.
We follow the argument of \cite{EGH00}.
First we consider the case of general SFT.
We define a super-commutative algebra $\mathcal{D}_X = \mathcal{D}_{(X, \omega, Y^\pm, \lambda^\pm,
K_{Y^\pm}, \overline{K}_X^0)}$ as follows.
Its elements are formal series
\[
\sum_{(\hat c_i^\ast), (\hat c'_i), e}
f_{(\hat c_i^\ast), (\hat c'_i), e}(t, \hbar) q^-_{\hat c_1^\ast}
\dots q^-_{\hat c_{k_q}^\ast} p^+_{\hat c'_1} \dots p^+_{\hat c'_{k_p}} T^e,
\]
where $f_{(\hat c_i^\ast), (\hat c'_i), e}(t, \hbar) \in \mathbb{R}[[t, \hbar]]$ is a formal series
of the variables $t_x$ ($x \in K_X^0$) and $\hbar$,
and the infinite sum is taken over all sequences $((\hat c_i), (\hat c'_i))$ consisting of
simplices $\hat c_i$ of $K_{Y^-}$ not contained in $\overline{P}^{\text{bad}}_{Y^-}$ and
simplices $(\hat c'_i)$ of $K_{Y^+}$ not contained in $\overline{P}^{\text{bad}}_{Y^+}$,
and $e \in \tilde \omega H_2(\overline{X}, \partial \overline{X}; \mathbb{Z})$
($\cong H_2(\overline{X}, \partial \overline{X}; \mathbb{Z}) / \mathop{\mathrm{Ker}}\nolimits \tilde \omega$)
with the following Novikov condition:
for any $C \geq 0$, the number of the non-zero terms with
$\sum_j e(p^+_{\hat c'_j}) \geq - C$
and $e + \sum_j e(p^+_{\hat c'_j}) \geq - C$ is finite.
The product is defined so that all variables are super-commutative,
where $\mathbb{Z} / 2$-degree is similar to the case of $\mathcal{W}_Y$ except
$|t_x| = \mathop{\mathrm{codim}}\nolimits_X x$ and $|T^e| = 0$.
We also define a submodule $\mathcal{D}_X^{\leq \kappa} \subset \mathcal{D}_X$ for each
$\kappa \geq 0$ by the condition
$\sum_i e(q^-_{\hat c_i^\ast}) + e + e(p^+_{\hat c'_i}) \leq \kappa$.
To define differentials on quotients of $\mathcal{D}_X$, we use a bigger
super-commutative algebra
$\mathcal{D}\D_X = \mathcal{D}\D_{(X, \omega, Y^\pm, \lambda^\pm, K_{Y^\pm}, \overline{K}^0_X,
\overline{K}^0_{Y^\pm}, \mu^\pm)}$.
Its elements are formal series
\[
\sum_{(\hat c_i^\ast), (\hat c'_i), e}
f_{(\hat c_i^\ast), (\hat c'_i), e}(t, \hbar) q^-_{\hat c_1^\ast}
\dots q^-_{\hat c_{k_q}^\ast} p^+_{\hat c'_1} \dots p^+_{\hat c'_{k_p}} T^e,
\]
where in this case, $f_{(\hat c_i^\ast), (\hat c'_i), e}(t, \hbar) \in
\mathbb{R}[[\hbar]] [\hbar^{-1}] [[t]]$, namely,
the coefficient of each monomial of the $t$-variables in
$f_{(\hat c_i^\ast), (\hat c'_i), e}(t, \hbar)$ is arrowed to have a pole of finite degree
at $\hbar = 0$.
(The degrees do not need to be bounded.)
For each $\kappa \geq 0$, we define a submodule $\mathcal{D}\D_X^{\leq \kappa} \subset
\mathcal{D}\D_X$ by the condition
$\sum_i e(q^-_{\hat c_i^\ast}) + e + e(p^+_{\hat c'_i}) \leq \kappa$.
For each positive constant $\delta > 0$,
we also define a submodule $\mathcal{D}\D_X^{\leq \kappa, \delta} \subset
\mathcal{D}\D_X^{\leq \kappa}$ by the condition
\begin{equation}
\widetilde{g}_\delta := g + \frac{1}{2}(k_t + k_q + k_p)
- \frac{\sum_i e(q^-_{\hat c_i^\ast}) + e + e(p^+_{\hat c'_i})}{\delta}
\geq - \frac{\kappa}{\delta}. \label{tilde gC}
\end{equation}
Note that $\mathcal{D}_X^{\leq \kappa} \subset \mathcal{D}\D_X^{\leq \kappa, \delta}$
and $\mathcal{D}\D_X^{\leq \kappa, \delta} \subset \mathcal{D}\D_X^{\leq \kappa, \delta'}$
for $\delta \geq \delta'$.
Define submodules $\widetilde{J}^{\leq \kappa, \delta}_{C_0, C_1, C_2}
= \widetilde{J}^{\leq \kappa, \delta}_{X, C_0, C_1, C_2} \subset
\mathcal{D}\D_X^{\leq \kappa, \delta}$ by
\begin{align*}
\widetilde{J}^{\leq \kappa, \delta}_{C_0, C_1, C_2} &= \Bigl\{\sum
a_{(x_i), (\hat c_i^\ast), (\hat c'_i), g, e} t_{x_1} \dots t_{x_{k_t}} q^-_{\hat c_1^\ast}
\dots q^-_{\hat c_{k_q}^\ast} p^+_{\hat c'_1} \dots p^+_{\hat c'_{k_p}} \hbar^g T^e
\in \mathcal{D}\D_X^{\leq \kappa, \delta};\\
&\quad \quad a_{(x_i), (\hat c_i^\ast), (\hat c'_i), g, e} = 0 \text{ for all }
((x_i)_{i = 1}^{k_t}, (\hat c_i^\ast),_{i = 1}^{k_q} (\hat c'_i)_{i = 1}^{k_p}, g, e)
\text{ such that}\\
&\quad \quad
k_t \leq C_0,\ \widetilde{g}_\delta \leq C_1,\ \sum e(p^+_{\hat c'_i}) \geq - C_2
\text{ and } e + \sum e(p^+_{\hat c'_i}) \geq - C_2 \Bigr\}.
\end{align*}
Note that these are ideals if $\kappa = 0$.
Note also that
$\widetilde{J}^{\leq \kappa, \delta}_{C_0, C_1 + \kappa((\delta')^{-1} - \delta^{-1}), C_2}
\subset \widetilde{J}^{\leq \kappa, \delta'}_{C_0, C_1, C_2}$ for $\delta \geq \delta'$,
which implies that we have a natural map
\[
\mathcal{D}\D_X^{\leq \kappa, \delta}
/ \widetilde{J}^{\leq \kappa, \delta}_{C_0, C_1 + \kappa((\delta')^{-1} - \delta^{-1}), C_2}
\to \mathcal{D}\D_X^{\leq \kappa, \delta'}
/ \widetilde{J}^{\leq \kappa, \delta'}_{C_0, C_1, C_2}.
\]
We also define submodules $J^{\leq \kappa, \delta}_{C_0, C_1, C_2} \subset
\mathcal{D}_X^{\leq \kappa}$ by $J^{\leq \kappa, \delta}_{C_0, C_1, C_2}
= \widetilde{J}^{\leq \kappa, \delta}_{C_0, C_1, C_2} \cap \mathcal{D}_X^{\leq \kappa}$.
Let $(\hbar^{-1} \mathcal{D}_X^{\leq 0})^{\star, \delta} \subset \hbar^{-1} \mathcal{D}_X^{\leq 0}$
be a submodule defined by the following conditions:
\begin{itemize}
\item
$\widetilde{g}_\delta$ is nonnegative.
(Hence $(\hbar^{-1} \mathcal{D}_X^{\leq 0})^{\star, \delta} \subset \mathcal{D}\D_X^{\leq 0, \delta}$.)
\item
The constant term is zero.
\end{itemize}
We also define submodules $J^{\star, \delta}_{C_0, C_1, C_2}
= J^{\star, \delta}_{X, C_0, C_1, C_2} \subset
(\hbar^{-1} \mathcal{D}_X^{\leq 0})^{\star, \delta}$ by
\begin{align*}
&J^{\star, \delta}_{C_0, C_1, C_2} \\
&= \Bigl\{\sum
a_{(x_i), (\hat c_i^\ast), (\hat c'_i), g, e} t_{x_1} \dots t_{x_{k_t}} q^-_{\hat c_1^\ast}
\dots q^-_{\hat c_{k_q}^\ast} p^+_{\hat c'_1} \dots p^+_{\hat c'_{k_p}} \hbar^g T^e
\in (\hbar^{-1} \mathcal{D}_X^{\leq 0})^{\star, \delta};\\
&\quad \quad a_{(x_i), (\hat c_i^\ast), (\hat c'_i), g, e} = 0 \text{ for all }
((x_i)_{i = 1}^{k_t}, (\hat c_i^\ast),_{i = 1}^{k_q} (\hat c'_i)_{i = 1}^{k_p}, g, e)
\text{ such that}\\
&\quad \quad
k_t \leq C_0,\ \widetilde{g}_\delta \leq C_1,\ \sum e(p^+_{\hat c'_i}) \geq - C_2
\text{ and } e + \sum e(p^+_{\hat c'_i}) \geq - C_2 \Bigr\}.
\end{align*}
We say that $\delta > 0$ is admissible for $C_2$ if
\begin{itemize}
\item
$\delta \leq L^\pm_{\min}$, where $L^\pm_{\min}$ is the minimal period of
periodic orbits in $(Y^\pm, \lambda^\pm)$, and
\item
$\delta \leq E_{\hat \omega}(u)$
for any non-constant holomorphic building $(\Sigma, u)$ for $X$
of genus $0$ and height $1$ such that the number of the limit circle is $\leq 1$
and the period of the periodic orbit on the circle is $\leq C_2$ (if it exists).
\end{itemize}
If we fix a triple $(\overline{C}_0, \overline{C}_1, \overline{C}_2)$ and an admissible
constant $\delta$ for $\overline{C}_2$, then, choosing a compatible family of
perturbations $\mathcal{B}_X$ of the multisections of finite number of pre-Kuranishi spaces
(these also need to be compatible with $\mathcal{B}_{Y^\pm}$)
and using their virtual fundamental chains, we can define the generating functions
\begin{align*}
\mathcal{F} &= \hbar^{-1} \sum_{g \geq 0} \mathcal{F}_g \hbar^g
\in (\hbar^{-1} \mathcal{D}_X^{\leq 0})^{\star, \delta}
/ J^{\star, \delta}_{\overline{C}_0, \overline{C}_1, \overline{C}_2}\\
\widetilde{\mathcal{F}} &= \hbar^{-1} \sum_{g \in \mathbb{Z}} \widetilde{\mathcal{F}}_g \hbar^g
\in \mathcal{D}\D_X^{\leq 0, \delta}
/ \widetilde{J}^{\leq 0, \delta}_{\overline{C}_0, \overline{C}_1, \overline{C}_2}
\end{align*}
by
\[
\mathcal{F}_g = \sum_{k_q, k_t, k_p \geq 0, e} \frac{1}{k_q ! k_t ! k_p !} [ (\overline{\mathcal{M}}^X_{g, e})^0(
\underbrace{\mathbf{q}, \dots, \mathbf{q}}_{k_q},
\underbrace{\mathbf{t}, \dots, \mathbf{t}}_{k_t},
\underbrace{\mathbf{p}, \dots, \mathbf{p}}_{k_p})]^0 T^{-e}
\]
and
\[
\widetilde{\mathcal{F}}_g = \sum_{k_q, k_t, k_p \geq 0, e}
\frac{1}{k_q ! k_t ! k_p !} [ (\overline{\mathcal{M}}^X_{g, e})(
\underbrace{\mathbf{q}, \dots, \mathbf{q}}_{k_q},
\underbrace{\mathbf{t}, \dots, \mathbf{t}}_{k_t},
\underbrace{\mathbf{p}, \dots, \mathbf{p}}_{k_p})]^0 T^{-e},
\]
where $\mathbf{q} = \sum_c q_{\hat c^\ast} \hat c$, $\mathbf{t} = \sum_x t_x x$ and
$\mathbf{p} = \sum_c p_{\hat c} \hat c^\ast$ are formal series.
Sometimes we explicitly indicate the dependence of $\mathcal{F}$ to various data as
\[
\mathcal{F} = \mathcal{F}_{(X, \omega, Y^\pm, \lambda^\pm, K_{Y^\pm}, K_X^0, K_{Y^\pm}^0,
\mu^\pm, K_{Y^\pm}^2, J, \mathcal{B}_X)}.
\]
$\widetilde{\mathcal{F}}$ indeed satisfies the condition of
$\mathcal{D}\D_X^{\leq 0, \delta}
/ \widetilde{J}^{\leq 0, \delta}_{\overline{C}_0, \overline{C}_1, \overline{C}_2}$, that is,
$\widetilde{g}_\delta \geq 0$ for all terms such that
$k_t \leq \overline{C}_0$, $\sum e(p^+_{\hat c'_i}) \geq - \overline{C}_2$
and $e + \sum e(p^+_{\hat c'_i}) \geq - \overline{C}_2$.
It is enough to see that every holomorphic building $(\Sigma, z, u, \phi)$ such that
$\sum_j L_{\gamma_{+\infty_j}} \leq \overline{C}_2$
and $e + \sum_j L_{\gamma_{+\infty_j}} \leq \overline{C}_2$ satisfies
\begin{equation}
\widetilde{g}_\delta = g + \frac{1}{2}(k_t + k_q + k_p)
+ \frac{E_{\hat \omega}(u)}
{\delta} \geq 1,
\label{tilde gC ineq}
\end{equation}
where $g$ is its genus, $k_t$, $k_q$ and $k_p$ are the numbers of its marked points,
$-\infty$-limit circles, and $+\infty$-limit circles respectively.
($L_{\gamma_{\pm\infty_i}}$ are the periods of the periodic orbits on its limit circles.)
First note that $\widetilde{g}_\delta - 1$ is additive with respect to
disjoint union of holomorphic buildings, and that if a holomorphic building
$(\Sigma', z', u', \phi')$ for $Y^-$ or $Y^+$ is glued to a holomorphic
building for $X$, then $\widetilde{g}$ is changed by
more than or equal to the corresponding $\widetilde{g}$ of $(\Sigma', z', u', \phi')$
since $\delta \leq L^\pm_{\min}$.
Therefore, it is enough to show inequality (\ref{tilde gC ineq}) for a connected
holomorphic building of height one.
Assume contrary, that is, assume that there exists a holomorphic building
$(\Sigma, z, u, \phi)$ of height one such that $\tilde g < 1$.
Since $\tilde g < 1$ implies $g = 0$ and $k_t \leq 1$, $u$ is not a constant map.
Note that the period of the periodic orbits on its $-\infty$-limit circle is
$\leq e + \sum_j L_{\gamma_{+\infty_j}} \leq \overline{C}_2$ (if it exists) by
(\ref{E hat omega estimate}), and
$\tilde g < 1$ implies that the number of the limit circles ($= k_q + k_p$) is $\leq 1$.
Therefore, the assumption of $\delta$ implies that $\delta \leq E_{\hat \omega}(u)$,
which contradicts to the assumption $\tilde g < 1$.
Hence $\tilde F$ satisfies the condition $\widetilde{g}_\delta \geq 0$.
$\mathcal{F}$ also satisfies the condition $\widetilde{g}_\delta \geq 0$.
Furthermore, the degree of $\mathcal{F}$ is even because of the dimension of pre-Kuranishi
spaces, and $\mathcal{F}$ does not contain constant term because there does not exist
any holomorphic buildings of genus $g = 1$ without marked points or limit circles
whose $E_{\hat \omega}$-energy is zero.
It is easy to check that for any
$\mathcal{G} \in (\hbar^{-1} \mathcal{D}_X^{\leq 0})^{\star, \delta}
/ J^{\star, \delta}_{\overline{C}_0, \overline{C}_1, \overline{C}_2}$ of even degree
and any formal series $P(x) \in \mathbb{R}[[x]]$,
$P(\mathcal{G}) \in \mathcal{D}\D_X^{\leq 0, \delta}
/ \widetilde{J}^{\leq 0, \delta}_{\overline{C}_0, \overline{C}_1, \overline{C}_2}$ is well
defined.
Equation (\ref{irreducible decomposition}) implies that $\widetilde{\mathcal{F}} = e^{\mathcal{F}}$
in $\mathcal{D}\D_X^{\leq 0, \delta}
/ \widetilde{J}^{\leq 0, \delta}_{\overline{C}_0, \overline{C}_1, \overline{C}_2}$.
$\mathcal{D}\D_X$ has a structure of a left $D$-module
over $\mathcal{W}_{Y^-}$ as follows.
For each variable $p_{\hat c}$ ($c \in K^-$),
we define a differential operator on $\mathcal{D}\D_X$ by
\[
\overrightarrow{p_{\hat c}}
= \hbar \overrightarrow{\frac{\partial}{\partial q_{\hat c^\ast}}}.
\]
Then each
\[
f = \sum f_{(\hat c_i^\ast), (\hat c'_i)}(t, \hbar) q_{\hat c_1^\ast} q_{\hat c_2^\ast} \dots
q_{\hat c_k^\ast} p_{\hat c'_1} p_{\hat c'_2} \dots p_{\hat c'_l} \in \mathcal{W}_{Y^-}
\]
acts on $\mathcal{D}\D_X$ as a differential operator
\[
\overrightarrow{f}
= \sum f_{(\hat c_i^\ast), (\hat c'_i)}(\tilde t, \hbar)
q^-_{\hat c_1^\ast} q^-_{\hat c_2^\ast}
\dots q^-_{\hat c_k^\ast} \overrightarrow{p_{\hat c'_1}} \overrightarrow{p_{\hat c'_2}}
\dots \overrightarrow{p_{\hat c'_l}},
\]
where we replace each variable $t_x$ ($x \in K_{Y^-}^0$) with
$\tilde t_x = t_{(\mu_-)^{-1}(x)}$.
($\mu_-$ is the bijection defined in Section \ref{fiber prod for X}.)
Similarly, $\mathcal{D}\D_X$ has a structure of a right $D$-module
over $\mathcal{W}_{Y^+}$.
In this case, each variable $q_{\hat c^\ast}$ ($c \in K^+$)
defines a differential operator
\[
\overleftarrow{q_{\hat c^\ast}} = \hbar \overleftarrow{\frac{\partial}{\partial p_{\hat c}}}
\]
from right,
and each
\[
f = \sum f_{(\hat c_i^\ast), (\hat c'_i)}(t, \hbar) q_{\hat c_1^\ast} q_{\hat c_2^\ast} \dots
q_{\hat c_k^\ast} p_{\hat c'_1} p_{\hat c'_2} \dots p_{\hat c'_l} \in \mathcal{W}_{Y^+}
\]
acts on $\mathcal{D}\D_X$ as a differential operator
\[
\overleftarrow{f} = \sum f_{(\hat c_i^\ast), (\hat c'_i)}(\tilde t, \hbar)
\overleftarrow{q_{\hat c_1^\ast}}
\overleftarrow{q_{\hat c_2^\ast}} \dots \overleftarrow{q_{\hat c_k^\ast}}
p^+_{\hat c'_1} p^+_{\hat c'_2} \dots p^+_{\hat c'_l},
\]
where we replace each variable $t_x$ ($x \in K_{Y^+}^0$) with
$\tilde t_x = t_{(\mu_+)^{-1}(x)}$.
These $D$-module structures
\begin{align*}
\mathcal{W}_{Y^-} \times \mathcal{D}\D_X &\to \mathcal{D}\D_X,\\
\mathcal{D}\D_X \times \mathcal{W}_{Y^+} &\to \mathcal{D}\D_X
\end{align*}
induce the following maps:
{\belowdisplayskip=0pt
\begin{multline*}
\mathcal{W}_{Y^-}^{\leq \kappa_1}
/ I^{\leq \kappa_1}_{C_0, C_1 + \kappa_2 \delta^{-1}
+ \kappa_1 (\delta^{-1} - L_{\min}^{-1}), C_2 + \kappa_2} \times
\mathcal{D}\D_X^{\leq \kappa_2, \delta}
/ \widetilde{J}^{\leq \kappa_2, \delta}_{C_0, C_1 + \kappa_1 \delta^{-1}, C_2}\\
\to \mathcal{D}\D_X^{\leq \kappa_1 + \kappa_2, \delta}
/ \widetilde{J}^{\leq \kappa_1 + \kappa_2, \delta}_{C_0, C_1, C_2},
\end{multline*}
}
{\abovedisplayskip=0pt
\begin{multline*}
\mathcal{D}\D_X^{\leq \kappa_1, \delta}
/ \widetilde{J}^{\leq \kappa_1, \delta}_{C_0, C_1 + \kappa_2 \delta^{-1}, C_2 + \kappa_2}
\times \mathcal{W}_{Y^+}^{\leq \kappa_2} / I^{\leq \kappa_2}_{C_0, C_1 + \kappa_1 \delta^{-1}
+ \kappa_2 (\delta^{-1} - L_{\min}^{-1}), C_2}\\
\to \mathcal{D}\D_X^{\leq \kappa_1 + \kappa_2, \delta}
/ \widetilde{J}^{\leq \kappa_1 + \kappa_2, \delta}_{C_0, C_1, C_2}.
\end{multline*}
}
Assume that a generating function $\mathcal{H}_{Y^\pm} \in (\hbar^{-1} \mathcal{W}_Y^{\leq 0})^+
/ (\hbar^{-1} \mathcal{W}_Y^{\leq 0})^+_{\overline{C}_0, \overline{C}_1, \overline{C}_2}$
are defined and that $\overline{C}_0 \geq C_0$, $\overline{C}_1 \geq C_1
+ \kappa \delta^{-1}$ and $\overline{C}_2 \geq C_2 + \kappa$.
Then they define a linear map $\widehat{D}_X : \mathcal{D}\D_X^{\leq \kappa, \delta}
/ \widetilde{J}^{\kappa, \delta}_{C_0, C_1, C_2}
\to \mathcal{D}\D_X^{\leq \kappa, \delta}
/ \widetilde{J}^{\kappa, \delta}_{C_0, C_1, C_2}$ by
\[
\widehat{D}_X f = \delta f - \overrightarrow{\mathcal{H}_{Y^-}} f
+ (-1)^{|f|} f \overleftarrow{\mathcal{H}_{Y^+}}.
\]
Equations (\ref{main eq}) for $\mathcal{H}_{Y^\pm}$ imply that
$\widehat{D}_X$ is a differential of $\mathcal{D}\D_X^{\leq \kappa, \delta}
/ \widetilde{J}^{\kappa, \delta}_{C_0, C_1, C_2}$.
Equation (\ref{boundary formula for X}) implies $\widetilde{\mathcal{F}} = e^{\mathcal{F}}$ satisfies
\begin{equation}
\widehat{D}_X e^\mathcal{F} = 0 \label{main eq for X}
\end{equation}
in $\mathcal{D}\D_X^{\leq 0, \delta}
/ \widetilde{J}^{\leq 0, \delta}_{\overline{C}_0, \overline{C}_1, \overline{C}_2}$.
Define maps
\begin{multline*}
\mathcal{W}_{Y^-}^{\leq \kappa_1}
/ I^{\leq \kappa_1}_{C_0, C_1 + \kappa_2 \delta^{-1}
+ \kappa_1(\delta^{-1} - L_{\min}^{-1}), C_2 + \kappa_2} \times
\mathcal{D}_X^{\leq \kappa_2, \delta}
/ J^{\leq \kappa_2, \delta}_{C_0, C_1 + \kappa_1 \delta^{-1}, C_2 + \kappa_1}\\
\to \mathcal{D}_X^{\leq \kappa_1 + \kappa_2, \delta}
/ J^{\leq \kappa_1 + \kappa_2, \delta}_{C_0, C_1, C_2}
\end{multline*}
by
\[
(f, g) \mapsto f \underset{\mathcal{F}}{\overrightarrow{\ast}} g
= e^{- \mathcal{F}} \overrightarrow{f} (e^{\mathcal{F}} g)
\]
for $\kappa_1, \kappa_2, C_0, C_1, C_2$ such that
$\overline{C}_0 \geq C_0$, $\overline{C}_1 \geq C_1
+ (\kappa_1 + \kappa_2) \delta^{-1}$, $\overline{C}_2 \geq C_2 + \kappa_1 + \kappa_2$.
This family of maps defines a left module-like structure, that is,
the associativity law is satisfied if it is well defined.
Similarly, we define maps
\begin{multline*}
\mathcal{D}_X^{\leq \kappa_1, \delta}
/ J^{\leq \kappa_1, \delta}_{C_0, C_1 + \kappa_2 \delta^{-1}, C_2 + \kappa_2} \times
\mathcal{W}_{Y^+}^{\leq \kappa_2}
/ I^{\leq \kappa_2}_{C_0, C_1 + \kappa_1 \delta^{-1}
+ \kappa_2(\delta^{-1} - L_{\min}^{-1}), C_2 + \kappa_1}
\\
\to \mathcal{D}_X^{\leq \kappa_1 + \kappa_2, \delta}
/ J^{\leq \kappa_1 + \kappa_2, \delta}_{C_0, C_1, C_2}
\end{multline*}
by
\[
(g, f) \mapsto g \underset{\mathcal{F}}{\overleftarrow{\ast}} f
= (g e^{\mathcal{F}}) \overleftarrow{f} e^{-\mathcal{F}}
\]
for $\kappa_1, \kappa_2, C_0, C_1, C_2$ such that
$\overline{C}_0 \geq C_0$, $\overline{C}_1 \geq C_1
+ (\kappa_1 + \kappa_2) \delta^{-1}$, $\overline{C}_2 \geq C_2 + \kappa_1 + \kappa_2$.
This family of maps defines a right module-like structure.
Note that these module-like structures are a bimodule structure, that is,
\[
(f \underset{\mathcal{F}}{\overrightarrow{\ast}} g) \underset{\mathcal{F}}{\overleftarrow{\ast}} h
= f \underset{\mathcal{F}}{\overrightarrow{\ast}} (g \underset{\mathcal{F}}{\overleftarrow{\ast}} h)
\]
for all $f \in \mathcal{W}_{Y^-}$, $g \in \mathcal{D}_X$ and $h \in \mathcal{W}_{Y^+}$.
Define a linear map $D_{\mathcal{F}} : \mathcal{D}_X^{\leq \kappa} / J^{\leq \kappa, \delta}_{C_0, C_1, C_2}
\to \mathcal{D}_X^{\leq \kappa} / J^{\leq \kappa, \delta}_{C_0, C_1, C_2}$ by
\begin{align*}
D_{\mathcal{F}} f &= e^{-\mathcal{F}} [\widehat{D}_X, f] (e^{\mathcal{F}}) \\
&= e^{-\mathcal{F}}\widehat{D}_X(f e^{\mathcal{F}}). &(\text{by } (\ref{main eq for X}))
\end{align*}
Then it satisfies the following:
\begin{itemize}
\item
$D_{\mathcal{F}}$ is a differential, that is, $D_{\mathcal{F}}^2 = 0$.
\item
For any $f \in \mathcal{W}_{Y^-}^{\leq \kappa_1} / I^{\leq \kappa_1}_{C_0, C'_1, C_2 + \kappa_2}$
and $g \in \mathcal{D}_X^{\leq \kappa_2} / J^{\leq \kappa_2, \delta}_{C_0, C''_1, C_2}$,
\begin{equation}
D_{\mathcal{F}}(f \underset{\mathcal{F}}{\overrightarrow{\ast}} g)
= (D_{Y^-}f) \underset{\mathcal{F}}{\overrightarrow{\ast}} g
+ (-1)^{|f|} f \underset{\mathcal{F}}{\overrightarrow{\ast}} D_{\mathcal{F}} (g)
\label{D_F left Leibnitz}
\end{equation}
in $\mathcal{D}_X^{\leq \kappa_1 + \kappa_2} / J^{\leq \kappa_1 + \kappa_2, \delta}
_{C_0, C_1, C_2}$, where $C'_1 = C_1 + \kappa_1(\delta^{-1} - L_{\min}^{-1})
+ \kappa_2 \delta^{-1}$ and $C''_1 = C_1 + \kappa_1 \delta^{-1}$.
\item
For any $g \in \mathcal{D}_X^{\leq \kappa_2} / J^{\leq \kappa_2, \delta}_{C_0, C''_1,
C_2 + \kappa_1}$ and
$f \in \mathcal{W}_{Y^+}^{\leq \kappa_1} / I^{\leq \kappa_1}_{C_0, C'_1, C_2}$,
\begin{equation}
D_{\mathcal{F}}(g \underset{\mathcal{F}}{\overleftarrow{\ast}} f)
= D_{\mathcal{F}}(g) \underset{\mathcal{F}}{\overleftarrow{\ast}} f
+ (-1)^{|g|} g \underset{\mathcal{F}}{\overleftarrow{\ast}} (D_{Y^+} f)
\label{D_F right Leibnitz}
\end{equation}
in $\mathcal{D}_X^{\leq \kappa_1 + \kappa_2} / J^{\leq \kappa_1 + \kappa_2, \delta}
_{C_0, C_1, C_2}$.
\end{itemize}
They imply that the family of cohomology groups
$H^\ast(\mathcal{D}_X^{\leq \kappa} / J^{\leq \kappa, \delta}_{C_0, C_1, C_2}, D_{\mathcal{F}})$ has a
$(H^\ast(\mathcal{W}_{Y^-}^{\leq \kappa} / I^{\leq \kappa}_{C_0, C_1, C_2}, D_{Y^-}),
H^\ast(\mathcal{W}_{Y^+}^{\leq \kappa} / I^{\leq \kappa}_{C_0, C_1, C_2}, D_{Y^+}))$-bimodule-like
structure.
Sometimes we denote the linear maps $D_{\mathcal{F}}$ for
the generating function
$\mathcal{F} = \mathcal{F}_{(X, \omega, Y^\pm, \lambda^\pm, K_{Y^\pm}, K^0_X,
K^0_{Y^\pm}, \mu^\pm, K^2_{Y^\pm}, J, \mathcal{B}_X)}$ by
\[
D_{(X, \omega, Y^\pm, \lambda^\pm, K_{Y^\pm}, K^0_X,
K^0_{Y^\pm}, \mu^\pm, K^2_{Y^\pm}, J, \mathcal{B}_X)}.
\]
By definition, $D_{\mathcal{F}} 1 = 0$.
Therefore the linear maps
\begin{align*}
i_\mathcal{F}^-(f) &= f \underset{\mathcal{F}}{\overrightarrow{\ast}} 1 = e^{-\mathcal{F}} \overrightarrow{f} e^{\mathcal{F}}
: \mathcal{W}_{Y^-}^{\leq \kappa}
/ I^{\leq \kappa}_{C_0, C_1 + \kappa(\delta^{-1} - L_{\min}^{-1}), C_2}
\to \mathcal{D}_X^{\leq \kappa} / J^{\leq \kappa, \delta}_{C_0, C_1, C_2},\\
i_\mathcal{F}^+(f) &= 1 \underset{\mathcal{F}}{\overleftarrow{\ast}} f = e^{\mathcal{F}} \overleftarrow{f} e^{-\mathcal{F}}
: \mathcal{W}_{Y^+}^{\leq \kappa}
/ I^{\leq \kappa}_{C_0, C_1 + \kappa(\delta^{-1} - L_{\min}^{-1}), C_2}
\to \mathcal{D}_X^{\leq \kappa} / J^{\leq \kappa, \delta}_{C_0, C_1, C_2}
\end{align*}
induce homomorphisms
\[
i_\mathcal{F}^\pm : H^\ast(\mathcal{W}_{Y^\pm}^{\leq \kappa}
/ I^{\leq \kappa}_{C_0, C_1 + \kappa(\delta^{-1} - L_{\min}^{-1}), C_2}, D_{Y^\pm})
\to H^\ast(\mathcal{D}_X^{\leq \kappa} / J^{\leq \kappa, \delta}_{C_0, C_1, C_2}, \mathcal{D}_\mathcal{F}).
\]
This pair of homomorphisms $i_\mathcal{F}^\pm$ is the chain map in the sense of general SFT.
Next we consider rational SFT.
Define $\mathcal{L}_X = \mathcal{L}_{(X, \omega, Y^\pm, \lambda^\pm,
K_{Y^\pm}, \overline{K}_X^0)} = \mathcal{D}_X|_{\hbar = 0}$ as a quotient super-commutative
algebra of $\mathcal{D}_X$.
We also use a bigger super-commutative algebra $\widehat{\mathcal{L}}_X$.
Its elements are formal series
\[
\sum
f_{(\hat c_i), (\hat c'_i), (\hat c''_i), (\hat c'''_i), e}(t)
q_{\hat c_1^\ast}^- \dots q_{\hat c_{k^-_q}^\ast}^-
q_{(\hat c'_1)^\ast}^+ \dots q_{(\hat c'_{k^+_q})^\ast}^+
p_{\hat c''_1}^- \dots p_{\hat c''_{k^-_p}}^-
p_{\hat c'''_1}^+ \dots p_{\hat c'''_{k^+_p}}^+ T^e,
\]
where each $f_{(\hat c_i), (\hat c'_i), (\hat c''_i), (\hat c'''_i), e}(t) \in \mathbb{R}[[t]]$ is a formal
series of the variables $t_x$ ($x \in K^0_X$) and
the infinite sum is taken over all sequences
$((\hat c_i), (\hat c'_i), (\hat c''_i), (\hat c'''_i), e)$
consisting of the simplices $(\hat c_i)$ of $K_{Y^-}$ not contained in
$\overline{P}^{\text{bad}}_{Y^-}$,
$(\hat c'_i)$ of $K_{Y^+}$ not contained in $\overline{P}^{\text{bad}}_{Y^+}$,
$(\hat c''_i)$ of $K_{Y^-}$ not contained in $\overline{P}^{\text{bad}}_{Y^-}$,
$(\hat c'''_i)$ of $K_{Y^+}$ not contained in $\overline{P}^{\text{bad}}_{Y^+}$ and
$e \in \tilde \omega H_2(\overline{X}, \partial \overline{X})$.
We impose the following Novikov condition on the infinite sum:
for any $C > 0$, the number of the non-zero terms with
$\sum_j e(p^-_{\hat c''_j}) + \sum_j e(p^+_{\hat c'''_j}) \geq -C$
and $e + \sum_j e(p^-_{\hat c''_j}) + \sum_j e(p^+_{\hat c'''_j}) \geq -C$ is finite.
The Poisson structure of $\widehat{\mathcal{L}}_X$ is defined by
\begin{align*}
\{f, g\} &= \sum_{c \in K_{Y^-}} \biggl(\frac{\overleftarrow{\partial} f}{\partial p_{\hat c}^-}
\frac{\overrightarrow{\partial} g}{\partial q_{\hat c^\ast}^-}
- (-1)^{|f| |g|} \frac{\overleftarrow{\partial} g}{\partial p_{\hat c}^-}
\frac{\overrightarrow{\partial} f}{\partial q_{\hat c^\ast}^-}\biggr)\\
&\quad - \sum_{c \in K_{Y^+}} \biggl(\frac{\overleftarrow{\partial} f}{\partial p_{\hat c}^+}
\frac{\overrightarrow{\partial} g}{\partial q_{\hat c^\ast}^+}
- (-1)^{|f| |g|} \frac{\overleftarrow{\partial} g}{\partial p_{\hat c}^+}
\frac{\overrightarrow{\partial} f}{\partial q_{\hat c^\ast}^+}\biggr).
\end{align*}
We regard $\mathcal{P}_{Y^-}$ and $\mathcal{P}_{Y^+}$ as subspaces of
$\widehat{\mathcal{L}}_X$ by
$q_{\hat c^\ast} \mapsto q^-_{\hat c^\ast}$, $p_{\hat c} \mapsto p^-_{\hat c}$
and $q_{\hat c^\ast} \mapsto q^+_{\hat c^\ast}$, $p_{\hat c} \mapsto p^+_{\hat c}$
respectively.
Then the inclusions $\mathcal{P}_{Y^-} \hookrightarrow \widehat{\mathcal{L}}_X$ and
$\mathcal{P}_{Y^+} \hookrightarrow \widehat{\mathcal{L}}_X$
are a Poisson map and an anti-Poisson map respectively.
For each even element $g \in \mathcal{L}_X$,
define a map
\[
f \mapsto f|_g : \widehat{\mathcal{L}}_X \to \mathcal{L}_X
\]
by the evaluation map given by $p_{\hat c}^-
= \frac{\overrightarrow{\partial} g}{\partial q_{\hat c^\ast}^-}$ and
$q_{\hat c^\ast}^+ = \frac{\overleftarrow{\partial} g}{\partial p_{\hat c}^+}$.
For each $\kappa \geq 0$, we define submodules
$\mathcal{L}_X^{\leq \kappa} \subset \mathcal{L}_X$
and $\widehat{\mathcal{L}}_X^{\leq \kappa} \subset \widehat{\mathcal{L}}_X$
by the conditions
$\sum_i e(q_{\hat c_i^\ast}^-) + e + \sum_i e(p_{\hat c'_i}^+) \leq \kappa$
and $\sum_i e(q_{\hat c_i^\ast}^-) + \sum_i e(q_{(\hat c'_i)^\ast}^+) + e
+ \sum_i e(p_{\hat c''_i}^-) + \sum_i e(p_{\hat c''_i}^+) \leq \kappa$ respectively.
Define submodules
$J^{\leq \kappa}_{C_0, C_2} \subset \mathcal{L}_X^{\leq \kappa}$
and $\widetilde{J}^{\leq \kappa}_{C_0, C_2} \subset
\widehat{\mathcal{L}}_X^{\leq \kappa}$ by
\begin{align*}
J^{\leq \kappa}_{C_0, C_2} &= \Bigl\{\sum
a_{(x_i), (\hat c_i^\ast), (\hat c'_i), e} t_{x_1} \dots t_{x_{k_t}} q^-_{\hat c_1^\ast}
\dots q^-_{\hat c_{k_q}^\ast} p^+_{\hat c'_1} \dots p^+_{\hat c'_{k_p}} T^e
\in \mathcal{L}_X^{\leq \kappa};\\
&\quad \quad a_{(x_i), (\hat c_i^\ast), (\hat c'_i), e} = 0 \text{ for all }
((x_i)_{i = 1}^{k_t}, (\hat c_i^\ast),_{i = 1}^{k_q} (\hat c'_i)_{i = 1}^{k_p}, e)
\text{ such that}\\
&\quad \quad
k_t \leq C_0,\ \sum e(p^+_{\hat c'_i}) \geq - C_2
\text{ and } e + \sum e(p^+_{\hat c'_i}) \geq - C_2 \Bigr\}
\end{align*}
and
\begin{align*}
\widetilde{J}^{\leq \kappa}_{C_0, C_2} &= \Bigl\{\sum
a_{(x_i), (\hat c_i), (\hat c'_i), (\hat c''_i), (\hat c'''_i), e} t_{x_1} \dots t_{x_{k_t}}
q^-_{\hat c_1^\ast} \dots q^-_{\hat c_{k_q}^\ast}
q^+_{(\hat c'_1)^\ast} \dots q^+_{(\hat c'_{k_q})^\ast} \\
&\quad \hphantom{\Bigl\{\sum
a_{(x_i), (\hat c_i), (\hat c'_i), (\hat c''_i), (\hat c'''_i), e} t_{x_1} \dots t_{x_{k_t}}}
p^-_{\hat c''_1} \dots p^-_{\hat c''_{k_p}}
p^+_{\hat c'''_1} \dots p^+_{\hat c'''_{k_p}} T^e
\in \widehat{\mathcal{L}}_X^{\leq \kappa};\\
&\quad \quad a_{\alpha} = 0 \text{ for all }
\alpha = ((x_i)_{i = 1}^{k_t}, (\hat c_i)_{i = 1}^{k^-_q}, (\hat c'_i)_{i = 1}^{k^+_q},
(\hat c''_i)_{i = 1}^{k^-_p}, (\hat c'''_i)_{i = 1}^{k^+_p}, e)\\
&\quad \quad \text{such that }
k_t \leq C_0,\ \sum e(p^+_{\hat c''_i}) + \sum e(p^+_{\hat c'''_i}) \geq - C_2
\text{ and}\\
&\quad \quad \hphantom{\text{such that }
k_t \leq C_0,\ }
e + \sum e(p^+_{\hat c''_i}) + \sum e(p^+_{\hat c'''_i}) \geq - C_2 \Bigr\}.
\end{align*}
First we note that $\mathbf{h} = \mathcal{H}_{Y^-, 0} - \mathcal{H}_{Y^+, 0}
\in \mathcal{L}_X^{\leq 0} / J^{\leq 0}_{\overline{C}_0, \overline{C}_2}$ satisfies
\begin{equation}
\delta \mathbf{h} - \frac{1}{2} \{\mathbf{h}, \mathbf{h}\} = 0.
\label{eq for h}
\end{equation}
For each triple $(\kappa, C_0, C_2)$ such that $\overline{C}_0 \geq C_0$ and
$\overline{C}_2 \geq C_2 + \kappa$,
we define a linear map
$\widehat{d}_X : \widehat{\mathcal{L}}^{\leq \kappa}_X
/ \widetilde{J}^{\leq \kappa}_{C_0, C_2}
\to \widehat{\mathcal{L}}^{\leq \kappa}_X
/ \widetilde{J}^{\leq \kappa}_{C_0, C_2}$ by
\[
\widehat{d}_X f = \delta f - \{\mathbf{h}, f\}.
\]
Then (\ref{eq for h}) implies that $\widehat{d}_X^2 = 0$.
$\widehat{d}_X$ also satisfies
\begin{align}
\widehat{d}_X(fg) = (\widehat{d}_X f) g + (-1)^{|f|} f \widehat{d}_X g
\label{hat d_X Leibnitz}\\
\widehat{d}_X\{f, g\} = \{\widehat{d}_X f, g\} + (-1)^{|f|} \{f, \widehat{d}_X g\}
\label{hat d_X bracket}
\end{align}
if the multiplications or Poisson brackets are well defined.
We use the genus zero part $\mathcal{F}_0 \in \mathcal{L}_X^{\leq 0}
/ J^{\leq 0}_{\overline{C}_0, \overline{C}_2}$ of the generating function.
Equation (\ref{main eq for X}) implies that
\begin{equation}
\delta \mathcal{F}_0 - \mathbf{h}|_{\mathcal{F}_0} = 0
\label{main eq for X rational}
\end{equation}
in $\mathcal{L}_X^{\leq 0} / J^{\leq 0}_{\overline{C}_0, \overline{C}_2}$.
For each triple $(\kappa, C_0, C_2)$ such that $\overline{C}_0 \geq C_0$ and
$\overline{C}_2 \geq C_2 + \kappa$,
define linear maps
$d_{\mathcal{F}_0} : \mathcal{L}^{\leq \kappa}_X / J^{\leq \kappa}_{C_0, C_2}
\to \mathcal{L}^{\leq \kappa}_X / J^{\leq \kappa}_{C_0, C_2}$
and $i_{\mathcal{F}_0}^\pm : \mathcal{P}^{\leq \kappa}_{Y^\pm} / I^{\leq \kappa}_{C_0, C_2}
\to \mathcal{L}^{\leq \kappa}_X / J^{\leq \kappa}_{C_0, C_2}$ by
\begin{align*}
d_{\mathcal{F}_0} f &= (\widehat{d}_X f)|_{\mathcal{F}_0} \\
&= \delta f - \{\mathbf{h}, f\}|_{\mathcal{F}_0}
\ (= (D_{\mathcal{F}} f)|_{\hbar = 0})
\end{align*}
and
\[
i_{\mathcal{F}_0}^\pm (f) = f|_{\mathcal{F}_0}.
\]
We claim that (\ref{main eq for X rational}) and (\ref{eq for h}) imply
that $d_{\mathcal{F}_0}$ is a differential (i.e. $d_{\mathcal{F}_0}^2 = 0$) and $i_{\mathcal{F}_0}^\pm$
are chain maps.
For its proof,
it is convenient to introduce a linear map
\[
f \mapsto \widetilde{f} : \widehat{\mathcal{L}}_X \to \widehat{\mathcal{L}}_X
\]
defined by
\begin{align*}
\widetilde{f} &= (k^-_p + k^+_q - 1) f \\
&= \bigl(\sum p^-_{\hat c} \overrightarrow{\partial}_{p^-_{\hat c}}
+ \sum q^+_{\hat c^\ast} \overrightarrow{\partial}_{q^+_{\hat c^\ast}} - 1\bigr) f
\end{align*}
for each monomial
\[
f = t_{x_1} \dots t_{x_{k_t}} q_{\hat c_1^\ast}^- \dots q_{\hat c_{k^-_q}^\ast}^-
q_{(\hat c'_1)^\ast}^+ \dots q_{(\hat c'_{k^+_q})^\ast}^+
p_{\hat c''_1}^- \dots p_{\hat c''_{k^-_p}}^-
p_{\hat c'''_1}^+ \dots p_{\hat c'''_{k^+_p}}^+ T^e.
\]
\begin{lem}
\begin{enumerate}[label=\normalfont (\roman*)]
\item
For any $f, g \in \widehat{\mathcal{L}}_X$,
\begin{equation}
\widetilde{\{f, g\}} = \{ \widetilde{f}, g \} + \{f, \widetilde{g}\}.
\label{tilde{}}
\end{equation}
\item
For all $g \in \mathcal{L}_X^{\text{even}}$ and
$f \in \widehat{\mathcal{L}}_X$,
\begin{equation}
\{f, g\}|_g = \widetilde{f}|_g + f|_g
\label{{}ev}
\end{equation}
and
\begin{equation}
\delta(f|_g) = (\delta f)|_g - \{\delta g, f\}|_g.
\label{delta ev}
\end{equation}
\item
For all $g \in \mathcal{L}_X^{\text{even}}$ and $f, h \in \widehat{\mathcal{L}}_X$,
\begin{equation}
\{h, f|_g\}|_g = \{h, \{f, g\}\}|_g - \{h, \widetilde{f}\}|_g.
\label{{}evev}
\end{equation}
In particular,
\begin{equation}
\{h|_g, f\}|_g + \{h, f|_g\}|_g = \{h, f\}|_g.
\label{evev}
\end{equation}
\end{enumerate}
\end{lem}
\begin{proof}
(\ref{tilde{}}) and (\ref{{}ev}) are easy.
(\ref{delta ev}) is proved as follows.
First note that if we regard each side as an operator $A$ for $f$ then it satisfies
\[
A(f_1 f_2) = A(f_1) f_2|_g + (-1)^{|f_1|} f_1|_g A(f_2).
\]
Hence we may assume that $f$ is some variable $q^-$, $q^+$, $p^-$ or $p^+$.
If $f$ is $q^-$ or $p^+$, then it satisfies (\ref{delta ev}) since
$\delta(f|_g) = (\delta f)|_g = \delta f$ and $\{\delta g, f\}|_g = 0$.
Next we consider the case of $f = q^+_{\hat c^\ast}$.
Define $a_{c', c} \in \mathbb{Q}$ by $\partial \hat c' = \sum_c a_{c', c} \hat c$.
Then (\ref{delta ev}) is equivalent to
\[
\delta\biggl(\frac{\overleftarrow{\partial} g}{\partial p_{\hat c}^+}\biggr)
- \frac{\overleftarrow{\partial} (\delta g)}{\partial p_{\hat c}^+}
= (-1)^{|g|} \sum_{c'} a_{c', c} \frac{\overleftarrow{\partial} g}{\partial p_{\hat c'}^+}.
\]
($(-1)^{|g|} = 1$ for $g \in \mathcal{L}_X^{\text{even}}$.)
We prove that this equation holds for all $g \in \widehat{\mathcal{L}}_X$ as follows.
If we regard each side as an operator $B$ for $g$ then it satisfies
\[
B(g_1 g_2) = (-1)^{|\hat c| |g_2|} B(g_1) g_2 + (-1)^{|g_1|} g_1 B(g_2).
\]
Hence it is enough to prove the equation for the case where $g$ is
some variable $q^-$, $q^+$, $p^-$ or $p^+$ and it can be easily checked.
Finally, if $f = p^-_{\hat c}$ then (\ref{delta ev}) is equivalent to
\[
\delta\biggl(\frac{\overrightarrow{\partial} g}{\partial q_{\hat c^\ast}^-}\biggr)
- (-1)^{|\hat c|} \frac{\overrightarrow{\partial} (\delta g)}{\partial q_{\hat c^\ast}^-}
= (-1)^{1 + |\hat c|} \sum_{c'} a_{c, c'}
\frac{\overrightarrow{\partial} g}{\partial q_{(\hat c')^\ast}^-},
\]
and it can be proved similarly.
(\ref{{}evev}) is proved as follows.
If we regard each side as an operator $C$ for $f$ then it satisfies
\[
C(f_1 f_2) = C(f_1) f_2|_g + (-1)^{|h| |f_1|} f_1|_g C(f_2).
\]
Hence it is enough to prove (\ref{{}evev}) for the case where $f$ is
some variable $q^-$, $q^+$, $p^-$ or $p^+$ and it can be easily checked.
(\ref{evev}) is a corollary of (\ref{{}evev}).
\end{proof}
Now we prove the following proposition.
The fourth claim is used to define a Poisson structure of rational SFT cohomology of
$(Y, \lambda)$ in Section \ref{independence}.
\begin{prop}\label{properties of d_{F_0}}
\begin{enumerate}[label=\normalfont (\roman*)]
\item
$d_{\mathcal{F}_0}$ is a differential, that is, $d_{\mathcal{F}_0}^2 = 0$.
\item
$f \mapsto f|_{\mathcal{F}_0}
: (\widehat{\mathcal{L}}_X^{\leq \kappa} / \widetilde{J}^{\leq \kappa}_{C_0, C_2},
\widehat{d}_X)
\to (\mathcal{L}_X^{\leq \kappa} / J^{\leq \kappa}_{C_0, C_2}, d_{\mathcal{F}_0})$ is a chain map,
that is, $d_{\mathcal{F}_0}(f|_{\mathcal{F}_0}) = (\widehat{d}_X f)|_{\mathcal{F}_0}$ for all
$f \in \widehat{\mathcal{L}}_X^{\leq \kappa} / \widetilde{J}^{\leq \kappa}_{C_0, C_2}$.
In particular, $i_{\mathcal{F}_0}^\pm$ are chain maps, that is, $d_{\mathcal{F}_0} \circ i_{\mathcal{F}_0}^\pm
= i_{\mathcal{F}_0}^\pm \circ d_{Y^\pm}$.
\item
For any $f \in \mathcal{P}^{\leq \kappa_1}_{Y^\pm}
/ I^{\leq \kappa_1}_{C_0, C_2 + \kappa_2}$
and $g \in \mathcal{L}^{\leq \kappa_2}_X / J^{\leq \kappa_2}_{C_0, C_2}$,
\[
d_{\mathcal{F}_0}(i_{\mathcal{F}_0}^\pm(f) g)
= i_{\mathcal{F}_0}^\pm(d_{Y^\pm} f) g + (-1)^{|f|} i_{\mathcal{F}_0}^\pm(f) d_{\mathcal{F}_0} g
\]
in $\mathcal{L}^{\leq \kappa_1 + \kappa_2}_X / J^{\leq \kappa_1 + \kappa_2}_{C_0, C_2}$.
\item
Assume that $f \in \widehat{\mathcal{L}}_X^{\leq \kappa_1}
/ \widetilde{J}^{\leq \kappa_1}_{C_0, C_2 + \kappa_2}$,
$g \in \widehat{\mathcal{L}}_X^{\leq \kappa_2}
/ \widetilde{J}^{\leq \kappa_2}_{C_0, C_2 + \kappa_1}$,
$a \in \mathcal{L}_X^{\leq \kappa_1} / J^{\leq \kappa_1}_{C_0, C_2 + \kappa_2}$
and $b \in \mathcal{L}_X^{\leq \kappa_2} / J^{\leq \kappa_2}_{C_0, C_2 + \kappa_1}$
satisfy $f|_{\mathcal{F}_0} = d_{\mathcal{F}_0} a$ and $g|_{\mathcal{F}_0} = d_{\mathcal{F}_0} b$.
Then
\begin{align}
&\{f, g\}|_{\mathcal{F}_0}
+ (-1)^{|f|} \bigl(\{\widehat{d}_X f, b\}|_{\mathcal{F}_0}
- \{a, \widehat{d}_X g\}|_{\mathcal{F}_0}\bigr)\notag\\
&= d_{\mathcal{F}_0} \bigl( \{a, g\}|_{\mathcal{F}_0} + (-1)^{|f|} \{f, b\}|_{\mathcal{F}_0}
+ \{a, \{\mathbf{h}, b\}\}|_{\mathcal{F}_0} \bigr) \label{eq for rational independence}
\end{align}
in $\mathcal{L}_X^{\leq \kappa_1 + \kappa_2}
/ J^{\leq \kappa_1 + \kappa_2}_{C_0, C_2}$.
In particular, if in addition $\widehat{d}_X f = 0$ and $\widehat{d}_X g = 0$, then
$\{f, g\}|_{\mathcal{F}_0}$ is exact.
\end{enumerate}
\end{prop}
\begin{proof}
First note that (\ref{main eq for X rational}), (\ref{delta ev}) and (\ref{evev}) imply that
for any $f \in \widehat{\mathcal{L}}_X^{\leq \kappa}
/ \widetilde{J}^{\leq \kappa}_{C_0, C_2}$,
\begin{align}
d_{\mathcal{F}_0}(f|_{\mathcal{F}_0})
&= (\delta f)|_{\mathcal{F}_0} - \{ \delta \mathcal{F}_0, f\}|_{\mathcal{F}_0}
-\{\mathbf{h}, f|_{\mathcal{F}_0}\}|_{\mathcal{F}_0} \notag\\
&= (\delta f)|_{\mathcal{F}_0} - \{ \mathbf{h}|_{\mathcal{F}_0}, f\}|_{\mathcal{F}_0}
-\{\mathbf{h}, f|_{\mathcal{F}_0}\}|_{\mathcal{F}_0} \notag\\
&= (\delta f)|_{\mathcal{F}_0} - \{\mathbf{h}, f\}|_{\mathcal{F}_0} \notag\\
&= (\widehat{d}_X f)|_{\mathcal{F}_0}.
\label{d ev}
\end{align}
(i) is because (\ref{d ev}) implies
\[
d_{\mathcal{F}_0}^2 f = d_{\mathcal{F}_0} ((\widehat{d}_X f)|_{\mathcal{F}_0}) = (\widehat{d}_X^2 f)|_{\mathcal{F}_0} = 0.
\]
(ii) is due to (\ref{d ev}).
(iii) is because for any $f \in \widehat{\mathcal{L}}_X^{\leq \kappa_1}
/ \widetilde{J}^{\leq \kappa_1}_{C_0, C_2 + \kappa_1}$
and $g \in \mathcal{L}^{\leq \kappa_2}_X / J^{\leq \kappa_2}_{C_0, C_2}$,
\begin{align*}
d_{\mathcal{F}_0}(f|_{\mathcal{F}_0} \cdot g)
&= d_{\mathcal{F}_0}((f g)|_{\mathcal{F}_0}) & (\text{since } g = g|_{\mathcal{F}_0})\\
&= (\widehat{d}_X (fg))|_{\mathcal{F}_0} & (\text{by } (\ref{d ev}))\\
&= (\widehat{d}_X f)|_{\mathcal{F}_0} \cdot g + (-1)^{|f|} f|_{\mathcal{F}_0} \cdot d_{\mathcal{F}_0} g
& (\text{by } (\ref{hat d_X Leibnitz}))
\end{align*}
in $\mathcal{L}^{\leq \kappa_1 + \kappa_2}_X / J^{\leq \kappa_1 + \kappa_2}_{C_0, C_2}$.
(iv) is because
\begin{align*}
d_{\mathcal{F}_0}(\{a, g\}|_{\mathcal{F}_0})
&= (\widehat{d}_X \{a, g\})|_{\mathcal{F}_0} \\
&= \{\widehat{d}_X a, g\}|_{\mathcal{F}_0} + (-1)^{|a|}\{a, \widehat{d}_X g\}|_{\mathcal{F}_0} \\
&= \{(\widehat{d}_X a)|_{\mathcal{F}_0}, g\}|_{\mathcal{F}_0}
+ \{\widehat{d}_X a, g|_{\mathcal{F}_0}\}|_{\mathcal{F}_0}
+ (-1)^{|a|}\{a, \widehat{d}_X g\}|_{\mathcal{F}_0} \\
&= \{f|_{\mathcal{F}_0}, g\}|_{\mathcal{F}_0}
+ \{\widehat{d}_X a, (\widehat{d}_X b)|_{\mathcal{F}_0}\}|_{\mathcal{F}_0}
+ (-1)^{|a|}\{a, \widehat{d}_X g\}|_{\mathcal{F}_0},
\end{align*}
\begin{align*}
(-1)^{|f|} d_{\mathcal{F}_0}(\{f, b\}|_{\mathcal{F}_0})
&= (-1)^{|f|} (\widehat{d}_X \{f, b\})|_{\mathcal{F}_0} \\
&= \{f, \widehat{d}_X b\}|_{\mathcal{F}_0} + (-1)^{|f|} \{\widehat{d}_X f, b\}|_{\mathcal{F}_0} \\
&= \{f, (\widehat{d}_X b)|_{\mathcal{F}_0}\}|_{\mathcal{F}_0} + \{f|_{\mathcal{F}_0}, \widehat{d}_X b\}|_{\mathcal{F}_0}
+ (-1)^{|f|} \{\widehat{d}_X f, b\}|_{\mathcal{F}_0} \\
&= \{f, g|_{\mathcal{F}_0}\}|_{\mathcal{F}_0} + \{(\widehat{d}_X a)|_{\mathcal{F}_0}, \widehat{d}_X b\}|_{\mathcal{F}_0}
+ (-1)^{|f|} \{\widehat{d}_X f, b\}|_{\mathcal{F}_0}
\end{align*}
and
\begin{align*}
d_{\mathcal{F}_0}(\{a, \{\mathbf{h}, b\}\}|_{\mathcal{F}_0})
&= - d_{\mathcal{F}_0}(\{a, \widehat{d}_X b\}|_{\mathcal{F}_0}) \\
&= - (\widehat{d}_X \{a, \widehat{d}_X b\})|_{\mathcal{F}_0} \\
&= - \{\widehat{d}_X a, \widehat{d}_X b\}|_{\mathcal{F}_0} \\
&= - \{(\widehat{d}_X a)|_{\mathcal{F}_0}, \widehat{d}_X b\}|_{\mathcal{F}_0}
- \{\widehat{d}_X a, (\widehat{d}_X b)|_{\mathcal{F}_0}\}|_{\mathcal{F}_0}.
\end{align*}
\end{proof}
Finally we consider the case of contact homology.
Assume that $(X, \omega)$ is an exact cobordism, that is,
$\omega = d \theta$ for some 1-form $\theta$ on $X$ such that
\[
\theta|_{(-\infty, 0] \times Y^-} = e^\sigma \lambda^-\quad \text{and} \quad
\theta|_{[0, \infty) \times Y^+} = e^\sigma \lambda^+.
\]
Further we assume that the domains of $\mu_\pm$ are the whole of $K_X^0$ and
$\mu_\pm : K_X^0 \to K_{Y^\pm}^0$ are bijections.
Define
\[
\widehat{\mathcal{F}}_0 = \sum_{c}
\frac{\overleftarrow{\partial} \mathcal{F}_0}{\partial p^+_{\hat c}} \biggr|_{p^+ = 0}
\cdot p^+_{\hat c} \in \mathcal{L}^{\leq 0}_X / J^{\leq 0}_{\overline{C}_0, \overline{C}_2}.
\]
Exactness of $(X, \omega)$ implies
\[
\frac{\overrightarrow{\partial}}{\partial q_{\hat c^\ast}^-} (\mathcal{F}_0|_{p = 0}) = 0.
\]
Hence equation (\ref{main eq for X rational}) implies
\begin{equation}
\delta \widehat{\mathcal{F}}_0 - \widehat{\mathbf{h}}|_{\widehat{\mathcal{F}}_0} = 0
\label{main eq for X contact}
\end{equation}
in $\mathcal{L}^{\leq 0}_X / J^{\leq 0}_{\overline{C}_0, \overline{C}_2}$,
where $\widehat{\mathbf{h}}
= \widehat{\mathcal{H}}_{Y^-, 0} - \widehat{\mathcal{H}}_{Y^+, 0}$.
For each pair $(\kappa, C_0)$ such that $\overline{C}_0 \geq C_0$ and
$\overline{C}_2 \geq \kappa$, we define a homomorphism
$\Psi_{\widehat{\mathcal{F}}_0} :
\mathcal{A}_{Y^+}^{\leq \kappa} / I^{\leq \kappa}_{C_0}
\to \mathcal{A}_{Y^-}^{\leq \kappa} / I^{\leq \kappa}_{C_0}$
by the evaluation
\[
\Psi_{\widehat{\mathcal{F}}_0}(f) = f|_{\widehat{\mathcal{F}}_0}
= f\Big|_{q^+_{\hat c^\ast}
= \bigl(\frac{\overleftarrow{\partial} \mathcal{F}_0}{\partial p^+_{\hat c}}\bigr|_{p^+ = 0}\bigr)}.
\]
Then (\ref{main eq for X contact}) implies that this is a chain map, that is,
$\partial_{Y^-} \circ \Psi_{\widehat{\mathcal{F}}_0}
= \Psi_{\widehat{\mathcal{F}}_0} \circ \partial_{Y^+}$,
where we identify each $t_x$ ($x \in K_{Y^+}^0$) with
$t_{\mu_- \circ \mu_+^{-1}(x)}$.
Therefore it induces a homomorphism $(\Psi_{\widehat{\mathcal{F}}_0})_\ast :
H^\ast (\mathcal{A}_{Y^+}^{\leq \kappa} / I^{\leq \kappa}_{C_0}, \partial_{Y^+})
\to H^\ast(\mathcal{A}_{Y^-}^{\leq \kappa} / I^{\leq \kappa}_{C_0}, \partial_{Y^-})$.
\subsection{Algebras with further energy conditions}
\label{algebras with further energy conditions}
Assume that $Z$ contains contact manifolds $(Y_i, \lambda_i)$ ($1 \leq i \leq m$) and
that for each $i = 1, 2, \dots, m$, there is a pair of symplectic cobordisms
$Z_i^-$ and $Z_i^+$ such that $Z = Z_i^- \cup_{Y_i} Z_i^+$.
We assume that the pull back of the symplectic form $\omega$ to $Y_i$ is $d\lambda_i$.
Then we can construct the algebras which respect these decompositions as follows.
(We need these algebras for the definition of the composition of generating
functions in Section \ref{composition of generating functions}.)
Let $((-\epsilon, \epsilon) \times Y_i, d(e^\sigma \lambda_i)) \hookrightarrow (Z, \omega)$
be a neighborhood of each $Y_i$ and define a closed two form $\widetilde{\omega}_{Y_i}$
on $X$ by
$\widetilde{\omega}_{Y_i} = \omega$ on $Z_i^+$,
$\widetilde{\omega}_{Y_i} = d(\varphi \lambda^+)$ on $[0, \infty) \times Y^+$,
$\widetilde{\omega}_{Y_i} = d(\varphi \lambda_i)$ on $(-\epsilon, 0] \times Y_i$,
and $\widetilde{\omega}_{Y_i} = 0$
on $(-\infty, 0] \times Y^- \cup (Z_i^- \setminus (-\epsilon, 0] \times Y_i)$,
where $\varphi : \mathbb{R} \to \mathbb{R}_{\geq 0}$ is a smooth function with compact support
such that $\varphi(0) = 1$ and $\varphi|_{(-\infty, -\epsilon]} \equiv 0$.
For a holomorphic building $(\Sigma, z, u) \in \widehat{\mathcal{M}}(X, \omega, J)$,
define $e = \int u^\ast \widetilde{\omega}$ and
$e_{Y_i} = \int u^\ast \widetilde{\omega}_{Y_i}$.
Then these satisfy
\[
e + \sum_{+\infty\text{-limit circles}} L_{\gamma_{+\infty_j}} \geq
\sum_{-\infty\text{-limit circles}} L_{\gamma_{-\infty_j}}
\]
and
\[
e_{Y_i} + \sum_{+\infty\text{-limit circles}} L_{\gamma_{+\infty_j}} \geq 0.
\]
The former is due to (\ref{E hat omega estimate}), and the latter is because
\begin{align*}
&e_{Y_i} + \sum_{+\infty\text{-limit circles}} L_{\gamma_{+\infty_j}}\\
& = \int_{u^{-1}((-\epsilon, 0] \times Y^-)} u^\ast d(\varphi \lambda_i)
+ \int_{u^{-1}(Z^+_i)} u^\ast \omega
+ \int_{u^{-1}(([0, \infty) \cup \mathbb{R}_1 \cup \dots \cup \mathbb{R}_{k_+}) \times Y^+)}
u^\ast d\lambda^+\\
&\geq 0,
\end{align*}
where
\begin{align*}
&\int_{u^{-1}((-\epsilon, 0] \times Y^-)} u^\ast d(\varphi \lambda_i)\\
&= \int_{u^{-1}((-\epsilon, 0] \times Y^-)} u^\ast (d\varphi \wedge \lambda_i)
+ \int_{u^{-1}((-\epsilon, 0] \times Y^-)} u^\ast (\varphi d \lambda_i)\\
&\geq 0
\end{align*}
since we may assume $\partial_\sigma \varphi \geq 0$ on $(-\epsilon, 0]$.
Define a super-commutative algebra $\mathcal{D}_{X, (Y_i)}$ as follows.
Its elements are formal series
\[
\sum_{(\hat c_i^\ast), (\hat c'_i), A}
f_{(\hat c_i^\ast), (\hat c'_i), A}(t, \hbar) t_{x_1} \dots t_{x_{k_t}} q^-_{\hat c_1^\ast}
\dots q^-_{\hat c_{k_q}^\ast} p^+_{\hat c'_1} \dots p^+_{\hat c'_{k_p}} T^A,
\]
where
$f_{(\hat c_i^\ast), (\hat c'_i), A}(t, \hbar) \in \mathbb{R}[[t, \hbar]]$ are
formal series of the variables $t_x$ ($x \in K^0_X$) and $\hbar$, and
the infinite sum is taken over all sequences $((\hat c_i), (\hat c'_i))$
as in the usual case and
$A \in H_2(\overline{X}, \partial \overline{X}; \mathbb{Z}) \big/
(\mathop{\mathrm{Ker}}\nolimits e \cap \bigcap_i \mathop{\mathrm{Ker}}\nolimits e_{Y_i})$,
where $e(A) = \widetilde{\omega}A$ and $e_{Y_i}(A) = \widetilde{\omega}_{Y_i} A$.
We impose the following Novikov condition on the elements of $\mathcal{D}_{X, (Y_i)}$:
for any $C > 0$, the number of the non-zero terms with
$\sum_j e(p^+_{\hat c'_j}) \geq - C$,
$e(A) + \sum_j e(p^+_{\hat c'_j}) \geq - C$ and
$e_{Y_i}(A) + \sum_j e(p^+_{\hat c'_j}) \geq - C$ is finite.
We also define a bigger super-commutative algebra $\mathcal{D}\D_{X, (Y_i)}$.
Its elements are formal series
\[
\sum_{(\hat c_i^\ast), (\hat c'_i), A}
f_{(\hat c_i^\ast), (\hat c'_i), A}(t, \hbar) t_{x_1} \dots t_{x_{k_t}} q^-_{\hat c_1^\ast}
\dots q^-_{\hat c_{k_q}^\ast} p^+_{\hat c'_1} \dots p^+_{\hat c'_{k_p}} T^A,
\]
similar to those of $\mathcal{D}_{X, (Y_i)}$ except that
$f_{(\hat c_i^\ast), (\hat c'_i), A}(t, \hbar) \in \mathbb{R}[[\hbar]][\hbar^{-1}][[t]]$.
The submodules $\mathcal{D}_{X, (Y_i)}^{\leq \kappa} \subset \mathcal{D}_{X, (Y_i)}$
and $\mathcal{D}\D_{X, (Y_i)}^{\leq \kappa} \subset \mathcal{D}\D_{X, (Y_i)}$ are defined by
the conditions
$\sum_i e(q^-_{\hat c_i^\ast}) + e(A) + \sum_j e(p^+_{\hat c'_j}) \leq \kappa$
and $e_{Y_i}(A) + \sum_j e(p^+_{\hat c'_j}) \leq \kappa$.
For each positive constant $\delta > 0$, we define a submodule
$\mathcal{D}_{X, (Y_i)}^{\leq \kappa, \delta} \subset \mathcal{D}_{X, (Y_i)}^{\leq \kappa}$ by the condition
$\widetilde{g}_\delta \geq - \kappa / \delta$.
The submodules $\widetilde{J}^{\leq \kappa, \delta}_{C_0, C_1, C_2} \subset
\mathcal{D}\D_{X, (Y_i)}^{\leq \kappa, \delta}$ are defined by
\begin{align*}
&\widetilde{J}^{\leq \kappa, \delta}_{C_0, C_1, C_2} \\
&= \Bigl\{\sum
a_{(x_i), (\hat c_i^\ast), (\hat c'_i), g, e} t_{x_1} \dots t_{x_{k_t}} q^-_{\hat c_1^\ast}
\dots q^-_{\hat c_{k_q}^\ast} p^+_{\hat c'_1} \dots p^+_{\hat c'_{k_p}} \hbar^g T^A
\in \mathcal{D}\D_{X, (Y_i)}^{\leq \kappa, \delta};\\
&\quad \quad a_{(x_i), (\hat c_i^\ast), (\hat c'_i), g, A} = 0 \text{ for all }
((x_i)_{i = 1}^{k_t}, (\hat c_i^\ast),_{i = 1}^{k_q} (\hat c'_i)_{i = 1}^{k_p}, g, A)
\text{ such that}\\
&\quad \quad
k_t \leq C_0,\ \widetilde{g}_\delta \leq C_1,\ \sum e(p^+_{\hat c'_i}) \geq - C_2,\
e(A) + \sum e(p^+_{\hat c'_i}) \geq - C_2 \\
& \quad \quad \text{and }
e_{Y_j}(A) + \sum e(p^+_{\hat c'_i}) \geq - C_2 \text{ for all } j
\Bigr\},
\end{align*}
and submodules $J^{\leq \kappa, \delta}_{C_0, C_1, C_2} \subset
\mathcal{D}_{X, (Y_i)}^{\leq \kappa}$ are defined by
$J^{\leq \kappa, \delta}_{C_0, C_1, C_2} = \mathcal{D}_{X, (Y_i)} \cap
\widetilde{J}^{\leq \kappa, \delta}_{C_0, C_1, C_2}$.
The same argument are valid for $\mathcal{D}_{X, (Y_i)}$ and $\mathcal{D}\D_{X, (Y_i)}$.
Namely, a compatible finite family of virtual fundamental chains defines
generating functions $\mathcal{F} \in (\hbar^{-1} \mathcal{D}_{X, (Y_i)}^{\leq 0})^{\star, \delta}
/ J^{\star, \delta}_{\overline{C}_0, \overline{C}_1, \overline{C}_2}$ and
$\widetilde{\mathcal{F}} \in \mathcal{D}\D_{X, (Y_i)}^{\leq 0, \delta}
/ \widetilde{J}^{\leq 0, \delta}_{\overline{C}_0, \overline{C}_1, \overline{C}_2}$,
and they define differentials
$D_\mathcal{F} : \mathcal{D}_{X, (Y_i)}^{\leq \kappa} /J^{\leq \kappa, \delta}_{C_0, C_1, C_2}
\to \mathcal{D}_{X, (Y_i)}^{\leq \kappa} /J^{\leq \kappa, \delta}_{C_0, C_1, C_2}$.
The rational version is similarly defined.
Let $\mathcal{L}_{X, (Y_i)} = \mathcal{D}_{X, (Y_i)}|_{\hbar = 0}$ be the quotient super-commutative
algebra.
Elements of the Poisson space $\widehat{\mathcal{L}}_{X, (Y_i)}$ are formal series
\[
\sum
f_{\alpha}(t)
q_{\hat c_1^\ast}^- \dots q_{\hat c_{k^-_q}^\ast}^-
q_{(\hat c'_1)^\ast}^+ \dots q_{(\hat c'_{k^+_q})^\ast}^+
p_{\hat c''_1}^- \dots p_{\hat c''_{k^-_p}}^-
p_{\hat c'''_1}^+ \dots p_{\hat c'''_{k^+_p}}^+ T^A,
\]
as in the usual $\widehat{\mathcal{L}}_X$ with Novikov condition, that is,
for any $C > 0$, the number of the non-zero terms with
$\sum_j e(p^-_{\hat c''_j}) + \sum_j e(p^+_{\hat c'''_j}) \geq - C$,
$e(A) + \sum_j e(p^-_{\hat c''_j}) + \sum_j e(p^+_{\hat c'''_j}) \geq - C$
and $e_{Y_i}(A) + \sum_j e(p_{\hat c''_j}^-) + \sum_j e(p_{\hat c'''_j}^+) \geq - C$
is finite.
Then we can define a differential $d_{\mathcal{F}_0} : \mathcal{L}_{X, (Y_i)}^{\leq \kappa}
/ J^{\leq \kappa}_{C_0, C_2} \to \mathcal{L}_{X, (Y_i)}^{\leq \kappa}
/ J^{\leq \kappa}_{C_0, C_2}$ as in the usual case.
We note that we do not need to consider the case of contact homology since
in this case, we consider only exact cobordisms.
\section{The case of homotopy}
\label{Homotopy}
In this section, we prove that two generating functions for $(X, \omega)$ and $K_X^0$
defined by using different almost complex structures and perturbations are homotopic
in the sense of \cite{EGH00}.
Furthermore, we prove that its homotopy type does not change if we change
the symplectic form $\omega$ on $Z$ by an exact form, and $K_X^0$ by
boundaries.
\subsection{Fiber products and their orientations}
\label{fiber product and orientation for homotopy}
Let $(X^\tau, \omega^\tau)_{\tau \in I = [0, 1]}$ be a family of symplectic manifolds
with cylindrical ends such that
the manifold $X^\tau = X = (-\infty, 0] \times Y^- \cup Z \cup [0, \infty) \times Y^+$ is
independent of $\tau$ and the symplectic forms have the form
$\omega^\tau = \omega^0 + d \theta^\tau$ for some
one-forms $\theta^\tau$ whose supports are contained in $Z$.
Let $J^\tau$ be a family of $\omega^\tau$-compatible almost complex structures
whose restriction to $(-\infty, 0] \times Y^-$ and $[0, \infty) \times Y^+$ are
independent of $\tau$ and obtained by some complex structures of $\xi^\pm$.
For each $i = 0, 1$, let $K_{X^i}^0$ be a finite set of smooth cycles with closed
support in $X$ with bijections
\begin{align*}
\mu^i_- &: \{x \in K_{X^i}^0; \mathop{\mathrm{supp}}\nolimits x \cap (-\infty, 0] \times Y^- \neq \emptyset\}
\to K_{Y^-}^0\\
\mu^i_+ &: \{x \in K_{X^i}^0; \mathop{\mathrm{supp}}\nolimits x \cap [0, \infty) \times Y^+ \neq \emptyset\}
\to K_{Y^+}^0
\end{align*}
such that $x|_{(-\infty, 0] \times Y^-} = (-\infty, 0] \times \mu^i_-(x)$
and $x_{[0, \infty) \times Y^+} = [0, \infty) \times \mu^i_+(x)$.
Assume that a finite set $K_{X^I}^0 = \{(x^\tau)_{\tau \in I}\}$ of
$C^\infty(I, \mathbb{R})$-linear combinations of smooth cycles with closed supports in $X$
is given which satisfies the following conditions:
\begin{itemize}
\item
$\{x^0\} = K_{X^0}^0$ and $\{x^1\} = K_{X^1}^0$.
Furthermore, we assume that $x^\tau$ is constant near $\tau = 0,1$.
\item
$\frac{d}{d\tau} x^\tau$ are boundaries of some $C^\infty(I, \mathbb{R})$-linear combinations of
smooth chains $(y^\tau)_{\tau \in I}$ in $X$ whose supports are contained in $Z$
for each $(x^\tau)_{\tau \in I}$.
In particular, $x^\tau$ is independent of $\tau$ on the complement of $Z$.
\item
There exist bijections
\begin{align*}
\mu_- &: \{(x^\tau)_{\tau \in I} \in K_{X^I}^0; \mathop{\mathrm{supp}}\nolimits x^\tau \cap (-\infty, 0] \times Y^-
\neq \emptyset\} \to K_{Y^-}^0\\
\mu_+ &: \{(x^\tau)_{\tau \in I} \in K_{X^I}^0; \mathop{\mathrm{supp}}\nolimits x^\tau \cap [0, \infty) \times Y^+
\neq \emptyset\} \to K_{Y^+}^0
\end{align*}
such that
$x^\tau|_{(-\infty, 0] \times Y^-} = (-\infty, 0] \times \mu_-((x^\tau)_{\tau \in I})$,
$x^\tau_{[0, \infty) \times Y^+} = [0, \infty) \times \mu_+((x^\tau)_{\tau \in I})$ and
$\mu_\pm((x^\tau)_{\tau \in I}) = \mu^i_\pm(x^i)$ for $i = 0, 1$.
\end{itemize}
As with $\widehat{\mathcal{M}}(X, \omega, J)$ in Section \ref{Kuranishi for X},
we can construct a pre-Kuranishi structure of
\[
\widehat{\mathcal{M}}_{X^I} = \bigcup_{\tau \in I} \widehat{\mathcal{M}}(X^\tau, \omega^\tau, J^\tau).
\]
There exists a natural strong smooth map from $\widehat{\mathcal{M}}_{X^I}$ to
$I$ which maps $\widehat{\mathcal{M}}_{X^\tau} = \widehat{\mathcal{M}}(X^\tau, \omega^\tau, J^\tau)$
to $\tau \in I$, and the associated maps from the Kuranishi neighborhoods to
$I$ are submersive.
Assume that grouped multisections of the fiber products
$(\widehat{\mathcal{M}}_{Y^\pm}^\diamond, \mathring{K}_{Y^\pm}^2, \allowbreak K_{Y^\pm},
\allowbreak K_{Y^\pm}^0)$ and
$(\widehat{\mathcal{M}}^\diamond_{X^i}, \mathring{K}^2_{X^i}, K_{X^i}, K^0_{X^i})$ for $i = 0,1$
are given and that they satisfy the compatibility conditions.
We define $\widehat{\mathcal{M}}_{X^I}^\diamond$ by
$\widehat{\mathcal{M}}_{X^I}^\diamond = \bigcup_{\tau \in I} \widehat{\mathcal{M}}_{X^\tau}^\diamond$.
Note that this is not locally the product of $\widehat{\mathcal{M}}_{X^I}$ but
its fiber product over $I$.
Its element is written as $(\tau,
(\Sigma^\alpha, z^\alpha, u^\alpha)_{\alpha \in A^- \sqcup A^0 \sqcup A^+},
M^{\mathrm{rel}})$, and
all of $(\Sigma^\alpha, z^\alpha, u^\alpha)$ are holomorphic buildings
for the same $X^\tau$.
Note that $\widehat{\mathcal{M}}_{X^I}$ contains $I \times \widehat{\mathcal{M}}_{Y^\pm}$.
Similarly to the case of $X$ in Section \ref{Kuranishi for X},
we define the fiber product
\[
(\widehat{\mathcal{M}}^\diamond_{X^I}, (\mathring{K}_{Y^-}^2, \mathring{K}_{Y^+}^2),
(K_{Y^-}, K_{Y^+}), (K^0_{Y^-}, K^0_{X^I}, \partial_\tau K^0_{X^I}, K^0_{Y^+})) \subset
\widehat{\mathcal{M}}^\diamond_{X^I}.
\]
In this case, we need to explain the meaning of fiber product with
$K^0_{X^I}$ or $\partial_\tau K^0_{X^I}$.
Let $\boldsymbol{K}^0_{X^I}$ be the finite set of smooth chains
which appear in $x^\tau$ or $y^\tau$.
Assume that $y^\tau = 0$ on $[0, \epsilon] \cup [1 - \epsilon, 1]$
for all $(x^\tau)_{\tau \in I} \in K^0_{X^I}$ for some $\epsilon > 0$.
Then the meaning of fiber product with $K^0_{X^I}$ and $\partial_\tau K^0_{X^I}$
is that we assume the transversality condition of the grouped multisection
with respect to $K^0_{X^0}$, $\boldsymbol{K}^0_{X^I}$ and $K^0_{X^1}$
on $[0, \epsilon]$, $[\epsilon, 1 - \epsilon]$ and $[1 - \epsilon, 1]$ respectively.
We abbreviate the above fiber product as
$(\widehat{\mathcal{M}}^\diamond_{X^I}, \mathring{K}^2_{X^I}, K_{X^I}, K^0_{X^I})$.
We define multi-valued partial essential submersions
$\Xi^\circ$ and $\Lambda$ from
$(\widehat{\mathcal{M}}^\diamond_{X^I}, \allowbreak \mathring{K}^2_{X^I}, K_{X^I}, K^0_{X^I})$
to itself similarly.
We need to construct the grouped multisections of
$(\widehat{\mathcal{M}}^\diamond_{X^I}, \mathring{K}^2_{X^I}, K_{X^I}, K^0_{X^I})$
which satisfy the similar compatibility conditions.
Notice that for a disconnected holomorphic building of $\widehat{\mathcal{M}}_{X^I}$,
the perturbed multisection induced by the product of the perturbed multisections
transverse to the zero sections for the connected components is not always
transverse to the zero section.
This is because we need to use the same factor $I$ for all connected components.
In other words, it is not a product but a fiber product with respect to $I$.
To overcome this problem, we use a continuous family of grouped multisections.
We can construct the continuous families of grouped multisections of
$(\widehat{\mathcal{M}}^\diamond_{X^I}, \mathring{K}^2_{X^I}, K_{X^I}, K^0_{X^I})$
which satisfy the following conditions:
\begin{itemize}
\item
The restrictions of the natural map
$(\widehat{\mathcal{M}}^\diamond_{X^I}, \mathring{K}^2_{X^I}, K_{X^I}, K^0_{X^I}) \to I$
to the fiber products of the zero sets of the perturbed multisection
are essentially submersive, that is,
even if we restrict to the fiber product of the zero set with the simplices, it is
submersive.
\item
The restrictions of the grouped multisection of
$(\widehat{\mathcal{M}}^\diamond_{X^I}, \mathring{K}^2_{X^I}, K_{X^I}, K^0_{X^I})$ to
$I \times
(\widehat{\mathcal{M}}_{Y^\pm}^\diamond, \mathring{K}_{Y^\pm}^2, K_{Y^\pm}, K_{Y^\pm}^0)$
coincide with the pull backs of the grouped multisections of
$(\widehat{\mathcal{M}}_{Y^\pm}^\diamond, \mathring{K}_{Y^\pm}^2, K_{Y^\pm}, K_{Y^\pm}^0)$
by the projection.
\item
The restrictions of the grouped multisection of
$(\widehat{\mathcal{M}}^\diamond_{X^I}, \mathring{K}^2_{X^I}, K_{X^I}, K^0_{X^I})$ to
$(\widehat{\mathcal{M}}^\diamond_{X^i}, \mathring{K}^2_{X^i}, K_{X^i}, K^0_{X^i})$
for $i = 0,1$ coincide with the given grouped multisections.
\item
Let $((\widehat{\mathcal{M}}^\diamond_{X^I})^0, \mathring{K}^2_{X^I}, K_{X^I}, K^0_{X^I}) \subset
(\widehat{\mathcal{M}}^\diamond_{X^I}, \mathring{K}^2_{X^I}, K_{X^I}, K^0_{X^I})$
be the subset of connected points.
Its continuous family of grouped multisections induces that of
\[
\bigcup_N (\prod^N (\widehat{\mathcal{M}}^\diamond_{X^I})^0, \mathring{K}^2_{X^I}, K_{X^I},
K^0_{X^I})_I / \mathfrak{S}_N,
\]
where each $(\prod^N (\widehat{\mathcal{M}}^\diamond_{X^I})^0, \mathring{K}^2_{X^I}, K_{X^I},
K^0_{X^I})_I$ is the fiber product over $I$.
This is because of the first condition.
Then the grouped multisection of
$(\widehat{\mathcal{M}}^\diamond_{X^I}, \mathring{K}^2_{X^I}, K_{X^I}, K^0_{X^I})$
coincides with its pull back by the submersion
\[
(\widehat{\mathcal{M}}^\diamond_{X^I}, \mathring{K}^2_{X^I}, K_{X^I}, K^0_{X^I}) \to
\bigcup_N (\prod^N (\widehat{\mathcal{M}}^\diamond_{X^I})^0, \mathring{K}^2_{X^I}, K_{X^I},
K^0_{X^I})_I / \mathfrak{S}_N
\]
defined by decomposition into connected components.
\item
The grouped multisections of
$(\widehat{\mathcal{M}}^\diamond_{X^I}, \mathring{K}^2_{X^I}, K_{X^I}, K^0_{X^I})$
are compatible with respect to the compatible system of
multi-valued partial essential submersions defined by $\Xi^\circ$ and $\Lambda$.
\end{itemize}
The grouped multisection of each
$\overline{\mathcal{M}}^{(m_-, X^I, m_+)}_{((\hat \epsilon^{i, j}_l), (\hat c^i_l), (x^i_l), (\hat \eta^i_l))}$
is defined by the pull back of that of
$(\widehat{\mathcal{M}}^\diamond_{X^I}, \mathring{K}^2_{X^I}, K_{X^I}, K^0_{X^I})$
by the natural essential submersion.
The definition of the orientation of
$\overline{\mathcal{M}}^{(m_-, X^I, m_+)}_{((\hat \epsilon^{i, j}_l), (\hat c^i_l), (x^i_l), (\hat \eta^i_l))}$
is almost the same with the case of $X$.
The only difference is that it is defined by
\[
(TI \oplus \mathcal{W}^{-m_-} \oplus \dots \oplus \mathcal{W}^{m_+})_{\star}
\]
instead of (\ref{ori}).
Assume that two pairs of solutions $(G^\pm)^0$, $(G^\pm)^1$ of (\ref{+G eq})
and (\ref{-G eq}) in Section \ref{correction terms for X} are given.
Then we can construct a smooth family of the solutions $(G^\pm)^\tau$ ($\tau \in I$)
of the equations which coincide with the given solution at $\tau = 0, 1$.
For a triple of sequences $((\hat c_l), (x_l), (\alpha_l))$, we define a pre-Kuranishi space
(or a $C^\infty(I, \mathbb{R})$-linear combination of pre-Kuranishi spaces)
$\overline{\mathcal{M}}^{X^I}((\hat c_l), (x_l), (\alpha_l))$ by
\begin{align*}
&\overline{\mathcal{M}}^{X^I}((\hat c_l), (x_l), (\alpha_l))\\
& = \sum_{m_-, m_+ \geq 0}\sum_{\star_{m_-, m_+}}(-1)^\ast \,
\overline{\mathcal{M}}^{(m_-, X^I, m_+)}
_{(((\widetilde{G}^+_{m_+})^I, (\widetilde{G}^-_{-m_-})^I), (\hat c^i_l), (x^i_l),
([\overline{P}_{Y^+}] \cap \alpha^i_l))}
\end{align*}
where
$(\widetilde{G}^\pm)^\tau = (\widetilde{G}^\pm_0)^\tau
+ (\widetilde{G}^\pm_{\pm1})^\tau +
(\widetilde{G}^\pm_{\pm2})^\tau + \cdots = \Theta^\pm(e^{\otimes (G^\pm)^\tau})$.
The sum and the sign $\ast$ are the same as those of the non-parametrized case.
Let
\begin{align*}
\bigl[\overline{\mathcal{M}}^{X^I \!, e}_g((\hat c_l), (x_l), (\alpha_l))\bigr]
&= (f_{0, g}^e)^\tau((\hat c_l), (x_l), (\alpha_l))
\oplus (f_{1, g}^e)^\tau((\hat c_l), (x_l), (\alpha_l)) d\tau\\
\bigl[(\overline{\mathcal{M}}^{X^I\!, e}_g)^0((\hat c_l), (x_l), (\alpha_l))\bigr]
&= (h_{0, g}^e)^\tau((\hat c_l), (x_l), (\alpha_l))
\oplus (h_{1, g}^e)^\tau((\hat c_l), (x_l), (\alpha_l)) d\tau
\end{align*}
be the counterparts of virtual fundamental chains, where $f_{j, g}^e((\hat c_l), (x_l), (\alpha_l))$
and $h_{j, g}^e((\hat c_l), (x_l), (\alpha_l))$ ($j = 0, 1$) are smooth functions of
$\tau \in I = [0, 1]$.
Let $(H^\pm)^\tau = (H^\pm)^\tau_2 + (H^\pm)^\tau_3 + \cdots$ be
an appropriate $C^\infty(I, \mathbb{R})$-linear combination of
\begin{multline*}
((\rho_\ast [\overline{P}_{Y^\pm}])^{i, j}, \dots, (\rho_\ast [\overline{P}_{Y^\pm}])^{i, j},
\epsilon_{\overline{P}_{Y^\pm}}^{i, j}, \dots, \epsilon_{\overline{P}_{Y^\pm}}^{i, j},\\
(\Delta_\ast [\overline{P}_{Y^\pm}])^{i, j}, \dots, (\Delta_\ast [\overline{P}_{Y^\pm}])^{i, j})
\end{multline*}
defined in the next section, and define
$(\mathring{f}_{0, g}^e)^\tau((\hat c_l), (x_l), (\alpha_l))$
by the $\Omega^0(I)$ part of the virtual fundamental chain of the $(g, e)$-part of
\[
\sum_{\substack{m_- \geq 0\\ m_+ \geq 0}}\sum_{\star_{m_-, m_+}}
(-1)^\ast\bigl(\overline{\mathcal{M}}^{(m_-, X^I, m_+)}
_{\kappa_1} + \overline{\mathcal{M}}^{(m_-, X^I, m_+)}
_{\kappa_2}\bigr),
\]
where
\[
\kappa_1 = (\Theta^+(e^{\otimes (G^+)^\tau})_{m_+},
\Theta^-(e^{\otimes (G^-)^\tau} \otimes (H^-)^\tau)_{-m_-}, (\hat c^i_l), (x^i_l), ([\overline{P}_{Y^+}] \cap \alpha^i_l))
\]
and
\[
\kappa_2 = (\Theta^+(e^{\otimes (G^+)^\tau} \otimes (H^+)^\tau)_{m_+},
\Theta^-((-1)^{m_-}e^{\otimes (G^-)^\tau})_{-m_-},
(\hat c^i_l), (x^i_l), ([\overline{P}_{Y^+}] \cap \alpha^i_l)).
\]
We also define $(\mathring{h}_{0, g}^e)^\tau((\hat c_l), (x_l), (\alpha_l))$ by
the $\Omega^0(I)$ part of the virtual fundamental chain of its irreducible part.
Define $\hat f_{1, g}^e((\hat c_l), (x_l), (\alpha_l))$ and
$\hat h_{1, g}^e((\hat c_l), (x_l), (\alpha_l))$ by
\begin{align*}
&\hat f_{1, g}^e((\hat c_l), (x_l), (\alpha_l))\\
&= - f_{1, g}^e((\hat c_l), (x_l), (\alpha_l)) + \mathring{f}_{0, g}^e((\hat c_l), (x_l), (\alpha_l))\\
&\quad + \sum_j (-1)^{\sum |\hat c_l| + \sum_{i < j} |x_i|}
f_{0, g}^e((\hat c_l), (x_1, x_2, \dots, y_j, \dots, x_{k_t}), (\alpha_l))
\end{align*}
and
\begin{align*}
&\hat h_{1, g}^e((\hat c_l), (x_l), (\alpha_l))\\
&= - h_{1, g}^e((\hat c_l), (x_l), (\alpha_l))
+ \mathring{h}_{0, g}^e((\hat c_l), (x_l), (\alpha_l))\\
&\quad + \sum_j (-1)^{\sum |\hat c_l| + \sum_{i < j} |x_i|}
h_{0, g}^e((\hat c_l), (x_1, x_2, \dots, y_j, \dots, x_{k_t}), (\alpha_l)).
\end{align*}
The second terms $\mathring{f}_{0, g}^e((\hat c_l), (x_l), (\alpha_l))$ and
$\mathring{h}_{0, g}^e((\hat c_l), (x_l), (\alpha_l))$ correspond to the differential of
$(G^\pm)^\tau$, and the third terms corresponds to the differential of $x^\tau$.
There terms are added to make equation (\ref{boundary formula for X^I}) below
hold true.
Then $f_{0, g}^e$, $h_{0, g}^e$, $\hat f_{1, g}^e$ and $\hat h_{1, g}^e$ satisfy
the following equations.
\begin{equation}
f_{0, g}^e((\hat c_l), (x_l), (\alpha_l)) = \sum_{\star_0} (-1)^{\ast_0} \frac{1}{k !} \prod_{i = 1}^k
h_{0, g_i}^{e_i}((\hat c^i_l), (x^i_l), (\alpha^i_l)) \label{f_0 equation}
\end{equation}
\begin{equation}
\hat f_{1, g}^e((\hat c_l), (x_l), (\alpha_l)) = \sum_{\star_1} (-1)^{\ast_1}
f_{0, g_0}^{e_0}((\hat c^0_l), (x^0_l), (\alpha^0_l))
\, \hat h_{1, g_1}^{e_1}((\hat c^1_l), (x^1_l), (\alpha^1_l))
\label{f_1 equation}
\end{equation}
\begin{align}
&df_{0, g}^e((\hat c_l), (x_l), (\alpha_l)) \notag\\
&= \hat f_{1, g}^e(\partial((\hat c_l), (x_l), (\alpha_l))) d\tau \notag\\
&\quad - \sum_{\star'_-} (-1)^{\ast'_-} \frac{1}{k !}
[\overline{\mathcal{M}}^{Y^-}_{g_{-}}((\hat c^{-}_l), (x^{-}_l), (\hat d_1^\ast, \hat d_2^\ast, \dots,
\hat d_k^\ast))]^0 \notag\\
&\hphantom{\quad + \sum_{\star_-} (-1)^{\ast_-} \frac{1}{k !}}
\cdot \hat f_{1, g_0}^{e_0}((\hat d_k, \hat d_{k - 1}, \dots, \hat d_1) \cup (\hat c^0_l),
(x^0_l), (\alpha_l)) d\tau \notag\\
&\quad - \sum_{\star'_+} (-1)^{\ast'_+} \frac{1}{k !}
\hat f_{1, g_0}^e((\hat c_l), (x^0_l), (\alpha^0_l) \cup (\hat d_1^\ast, \hat d_2^\ast, \dots,
\hat d_k^\ast)) \notag\\
&\hphantom{\quad + \sum_{\star_+} (-1)^{\ast_+} \frac{1}{k !}}
\cdot [\overline{\mathcal{M}}^{Y^+}_{g_+}((\hat d_k, \hat d_{k - 1}, \dots, \hat d_1), (x^+_l), (\alpha^+_l))]^0 d\tau
\label{boundary formula for X^I}
\end{align}
\begin{align}
0 &= f_0(\partial((\hat c_l), (x_l), (\alpha_l)))\notag\\
&\quad - \sum_{\star_-} (-1)^{\ast_-} \frac{1}{k !}
[\overline{\mathcal{M}}^{Y^-}((\hat c^-_l), (x^-_l), (\hat d_1^\ast, \hat d_2^\ast, \dots, \hat d_k^\ast))]^0\notag\\
&\hphantom{\quad + \sum_{\star_+} (-1)^{\ast_+} \frac{1}{k !}}
\times f_0((\hat d_k, \hat d_{k - 1}, \dots, \hat d_1) \cup (\hat c^0_l), (x^0_l), (\alpha_l))\notag\\
&\quad + \sum_{\star_+} (-1)^{\ast_+} \frac{1}{k !}
f_0((\hat c_l), (x^0_l), (\alpha^0_l) \cup (\hat d_1^\ast, \hat d_2^\ast, \dots, \hat d_k^\ast))\notag\\
&\hphantom{\quad + \sum_{\star_+} (-1)^{\ast_+} \frac{1}{k !}}
\times [\overline{\mathcal{M}}^{Y^+}((\hat d_k, \hat d_{k - 1}, \dots, \hat d_1), (x^+_l), (\alpha^+_l))]^0
\label{boundary formula for each tau}
\end{align}
The sum $\star_0$ is taken over all $k \geq 0$, all decompositions
$g - 1 = \sum_{i = 1}^k (g_i - 1)$,
$e = \sum_{i = 1}^k e_i$ and all decompositions
\[
\{\hat c_l\} = \coprod_{i = 1}^k \{\hat c^i_l\}, \quad
\{x_l\} = \coprod_{i = 1}^k \{x^i_l\}, \quad
\{\alpha_l\} = \coprod_{i = 1}^k \{\alpha^i_l\}
\]
as sets.
The sign $\ast_0$ is the weighted sign of the permutation
\[
\begin{pmatrix}
(c^1_l) \ (x^1_l) \ (\alpha^1_l) \dots (c^k_l) \ (x^i_l) \ (\alpha^i_l)\\
(c_l) \quad (x_l) \quad (\alpha_l)
\end{pmatrix}.
\]
The sum $\star_1$ is taken over all decompositions
$g - 1= (g_0 - 1) + (g_1 - 1)$, $e = e_0 + e_1$
and all decompositions
\[
\{c_l\} = \{c^0_l\} \sqcup \{c^1_l\}, \quad \{x_l\} = \{x^0_l\} \sqcup \{x^1_l\}, \quad
\{\alpha_l\} = \{\alpha^0_l\} \sqcup \{\alpha^1_l\}
\]
as sets, and the sign $\ast_1$ is the weighted sign of the permutation
\[
\begin{pmatrix}
(c^0_l) \ (x^0_l) \ (\alpha^0_l) \ (c^1_l) \ (x^1_l) \ (\alpha^1_l)\\
(c_l) \ (x_l) \ (\alpha_l)
\end{pmatrix}.
\]
The sum $\star'_-$ is taken over $k \geq 0$, all simplices $d_l$ of $K_{Y^-}$ not contained
in $\overline{P}_{Y^-}^{\text{bad}}$,
all decompositions
\[
\{\hat c_l\} = \{\hat c^-_l\} \sqcup \{\hat c^0_l\}, \quad
\{x_l\} = \{x^-_l\} \sqcup \{x^0_l\}
\]
such that $x^-_l \in K^0_{Y^-}$, and all pairs
$(g_-, g_0)$ such that $g = g_- + g_0 + k -1$.
The sign $\ast'_-$ is the weighted sign of the permutation
\[
\begin{pmatrix}
(\hat c^-_l) \ (x^-_l) \ (\hat c^0_l) \ (x^0_l)\\
(\hat c_l) \quad (x_l)
\end{pmatrix}.
\]
The sum $\star'_+$ is taken over $k \geq 0$, all simplices $d_l$ of $K_{Y^+}$ not contained
in $\overline{P}_{Y^+}^{\text{bad}}$, and all decompositions
\[
\{x_l\} = \{x^0_l\} \sqcup \{x^+_l\}, \quad \{\alpha_l\} = \{\alpha^0_l\} \sqcup \{\alpha^+_l\}
\]
such that $x^+_l \in K^0_{Y^+}$, and all pairs
$(g_0, g_+)$ such that $g = g_0 + g_+ + k - 1$.
The sign $\ast'_+$ is the weighted sign of the permutation
\[
\begin{pmatrix}
(x^0_l) \ (\alpha^0_l) \ (x^+_l) \ (\alpha^+_l)\\
(x_l) \quad (\alpha_l)
\end{pmatrix}.
\]
Equation (\ref{boundary formula for each tau}) is a counterpart of equation
(\ref{boundary formula for X}), and the meaning of the sums and the signs are
the same.
As with equation (\ref{irreducible decomposition}),
(\ref{f_0 equation}) and (\ref{f_1 equation}) are due to the irreducible decomposition.
(\ref{boundary formula for X^I}) is proved in the next section.
(\ref{boundary formula for each tau}) is same as (\ref{boundary formula for X}).
\subsection{Construction of $H^\tau$}
In this section, we construct smooth families
$(H^+)^\tau = (H^+_2)^\tau + (H^+_3)^\tau + \dots \in (\bigoplus_{m \geq 2}
(\mathring{\mathcal{B}}_m^+)^{m+1})^\wedge$
and $(H^-)^\tau = (H^-_{-2})^\tau + (H^-_{-3})^\tau + \dots \in (\bigoplus_{m \geq 2}
(\mathring{\mathcal{B}}_{-m}^-)^{m+1})^\wedge$ such that
\begin{multline}
\partial \Theta^+(e^{\otimes G^+} \otimes H^+)
+ \sum_{i \geq 0} (-1)^i e^{\Delta_{(e_i, e_{i+1})}} \tau^+_i
\Theta^+(e^{\otimes G^+} \otimes H^+)\\
- \Diamond^+ \Bigl(\Theta^+(e^{\otimes G^+} \otimes H^+) \otimes
\sum_{j \geq 1} (-1)^j F^+_j\Bigr)
- \Theta^+\Bigl(e^{\otimes G^+} \otimes \frac{d}{d\tau}G^+\Bigr) = 0, \label{H^+ eq}
\end{multline}
\begin{multline}
\partial \Theta^-(e^{\otimes G^-} \otimes H^-)
+ \sum_{i \leq 0} e^{\Delta_{(e_{i-1}, e_i)}} \tilde \tau^-_i
\Theta^-(e^{\otimes G^-} \otimes H^-)\\
+ \Diamond^-(F^- \otimes \Theta^-(e^{\otimes G^-} \otimes H^-))
- \Theta^-(e^{\otimes G^-} \otimes \frac{d}{d\tau}G^-) = 0, \label{H^- eq}
\end{multline}
and prove (\ref{boundary formula for X^I}) for these $(H^+)^\tau$ and $(H^-)^\tau$.
First we construct $(H^+)^\tau$.
We inductively construct smooth families $(H^+)^\tau_{\leq m} = (H^+_2)^\tau
+ (H^+_3)^\tau + \dots + (H^+_m)^\tau \in
\bigoplus_{l = 2}^m (\mathring{\mathcal{B}}_l^+)^{l+1}$ such that
\begin{multline}
\partial \Theta^+(e^{\otimes G^+} \otimes H^+_{\leq m})
+ \sum_{i \geq 0} (-1)^i e^{\Delta_{(e_i, e_{i+1})}} \tau^+_i
\Theta^+(e^{\otimes G^+} \otimes H^+_{\leq m})\\
- \Diamond^+ \Bigl(\Theta^+(e^{\otimes G^+} \otimes H^+_{\leq m}) \otimes
\sum_{j \geq 1} (-1)^j F^+_j\Bigr)
- \Theta^+\Bigl(e^{\otimes G^+} \otimes \frac{d}{d\tau}G^+\Bigr) \equiv 0
\end{multline}
in $(\bigoplus_{l = 1}^\infty (\mathcal{B}_l^+)^l)^\wedge
/(\bigoplus_{l = m+1}^\infty (\mathcal{B}_l^+)^l)^\wedge$
First note that $\frac{d}{d\tau}G^+ \in (\bigoplus_{l \geq 2}
(\mathring{\mathcal{B}}_l^+)^l)^\wedge$
since $G^+_1$ is independent of $\tau$.
Hence we do not need $H_1^+$-part.
Assume we have already constructed $H^+_{\leq m-1} = H^+_2 + H^+_3 + \dots
+ H^+_{m-1} \in (\bigoplus_{l = 2}^{m-1} (\mathring{\mathcal{B}}_l^+)^{l+1})^\wedge$.
Then it is enough to prove that
\begin{multline}
\partial \Theta^+(e^{\otimes G^+} \otimes H^+_{\leq m-1})
+ \sum_{i \geq 0} (-1)^i e^{\Delta_{(e_i, e_{i+1})}} \tau^+_i
\Theta^+(e^{\otimes G^+} \otimes H^+_{\leq m-1})\\
- \Diamond^+ \Bigl(\Theta^+(e^{\otimes G^+} \otimes H^+_{\leq m-1}) \otimes
\sum_{j \geq 1} (-1)^j F^+_j\Bigr)
- \Theta^+ \Bigl(e^{\otimes G^+} \otimes \frac{d}{d\tau}G^+\Bigr) \equiv 0
\label{H tau mathring}
\end{multline}
in $(\bigoplus_{l = 1}^\infty (\mathcal{B}_l^+)^l)^\wedge
/((\bigoplus_{l = m+1}^\infty (\mathcal{B}_l^+)^l)^\wedge \oplus
\bigoplus_{l = 1}^\infty (\mathring{\mathcal{B}}_l^+)^l) $ and
\begin{align}
&\partial\Bigl(\sum_{i \geq 0} (-1)^i e^{\Delta_{(e_i, e_{i+1})}} \tau^+_i
\Theta^+(e^{\otimes G^+} \otimes H^+_{\leq m-1})\notag\\
&\quad - \Diamond^+\Bigl(\Theta^+(e^{\otimes G^+} \otimes H^+_{\leq m-1}) \otimes
\sum_{j \geq 1} (-1)^j F^+_j\Bigr)
- \Theta^+\Bigl(e^{\otimes G^+} \otimes \frac{d}{d\tau}G^+\Bigr)\Bigr)\notag\\
& \equiv 0
\label{H tau closed}
\end{align}
in $(\bigoplus_{l = 1}^\infty (\mathcal{B}_l^+)^{l-1})^\wedge
/(\bigoplus_{l = m+1}^\infty (\mathcal{B}_l^+)^{l-1})^\wedge$
The latter equation is proved by an argument similar to those for (\ref{A closed}) or
(\ref{B closed}).
The former can be proved in a similar way to equation (\ref{B mathring}) by using
the following equations.
\begin{align}
&\partial \Theta^+\biggl(\frac{1}{k !} G^{\otimes k} \otimes H \biggr)\notag\\
&= \Theta^+\biggl(\frac{1}{k !} G^{\otimes k} \otimes \partial H \biggr)
+ \Theta^+\biggl(\frac{1}{(k-1) !} G^{\otimes (k-1)} \otimes \partial G \otimes
\sum_j (-1)^j H_j \biggr)
\end{align}
\begin{align}
&\sum_{i \geq 1} (-1)^i e^{\Delta_{(e_i, e_{i+1})}} \tau^+_i \Theta^+
\biggl(\frac{1}{k !} G^{\otimes k} \otimes H \biggr)\notag\\
&= \Theta^+\biggl(\frac{1}{k !} G^{\otimes k} \otimes
\sum_{i \geq 1} (-1)^i e^{\Delta_{(e_i, e_{i+1})}} \tau^+_i H \biggr)\notag\\
&\quad + \Theta^+\biggl(\frac{1}{(k-1) !} G^{\otimes (k-1)} \otimes
\sum_{i \geq 1} (-1)^i e^{\Delta_{(e_i, e_{i+1})}} \tau^+_i G \otimes \sum_j (-1)^j H_j\biggr)
\end{align}
\begin{align}
&e^{\Delta_{(e_0, e_1)}} \tau^+_0 \Theta^+\biggl(\frac{1}{k !} G^{\otimes k} \otimes
H \biggr)\notag\\
&= \sum_{l = 0}^k \Theta^+\biggl(\frac{1}{(k-l) ! l !}G^{\otimes (k-l)} \otimes
(e^{\Delta_{(e_0, e_1)}} \mathring{\tau}^+_0(G^{\otimes l} \otimes H))\biggr)\notag\\
&\quad + \sum_{l = 0}^k \Theta^+\biggl(\frac{1}{(k-l) ! l !}G^{\otimes (k-l)} \otimes
(e^{\Delta_{(e_0, e_1)}} \mathring{\tau}^+_0(G^{\otimes l})) \otimes \sum_j (-1)^j H_j\biggr)
\end{align}
\begin{align}
&\Diamond^+\biggl(\Theta^+\biggl(\frac{1}{k !} G^{\otimes k} \otimes H \biggr)
\otimes \sum_j (-1)^j F^+_j\biggr)\notag\\
&= \sum_{l = 0}^k \Theta^+\biggl(\frac{1}{(k-l) ! l !}G^{\otimes (k-l)} \otimes
\mathring{\Diamond}^+ (G^{\otimes l} \otimes H \otimes
\sum_j (-1)^j F^+_j)\biggr)\notag\\
&\quad - \sum_{l = 0}^k \Theta^+\biggl(\frac{1}{(k-l) ! l !}G^{\otimes (k-l)} \otimes
\mathring{\Diamond}^+ (G^{\otimes l} \otimes F^+) \otimes \sum_j (-1)^j H_j\biggr)
\end{align}
Similarly, we can inductively construct a smooth family
$(H^-)^\tau_{\geq -m} = (H^-_{-2})^\tau + (H^-_{-3})^\tau + \dots
+ (H^-_{-m}) \in (\bigoplus_{l = 2}^m
(\mathring{\mathcal{B}}_{-l}^-)^{l+1})^\wedge$ such that
\begin{multline}
\partial \Theta^-(e^{\otimes G^-} \otimes H^-_{\geq -m})
+ \sum_{i \leq 0} e^{\Delta_{(e_{i-1}, e_i)}} \tilde \tau^-_i
\Theta^-(e^{\otimes G^-} \otimes H^-_{\geq -m})\\
+ \Diamond^-(F^- \otimes \Theta^-(e^{\otimes G^-} \otimes H^-_{\geq -m}))
- \Theta^-(e^{\otimes G^-} \otimes \frac{d}{d\tau}G^-) \equiv 0
\end{multline}
in $(\bigoplus_{l = 1}^\infty (\mathcal{B}_{-l}^-)^l)^\wedge
/(\bigoplus_{l = m+1}^\infty (\mathcal{B}_{-l}^-)^l)^\wedge$,
and we obtain a required solution $(H^-)^\tau \in (\bigoplus_{m \geq 2}
(\mathring{\mathcal{B}}_{-m}^-)^{m+1})^\wedge$.
Now we prove (\ref{boundary formula for X^I}) for these $H^\pm$.
In what follows, we omit the subscripts $g$ or $e$ for the simplification of
notation.
We abbreviate
\[
\sum (-1)^\ast ((\hat c^i_l), (x^i_l), ([\overline{P}_{Y^+}] \cap \alpha^i_l))
\]
as $((\hat c_l), (x_l), (\alpha_l))$,
where $(-1)^\ast$ is the weighted sign of the permutation corresponding to
$((\hat c^i_l), (x^i_l), (\alpha^i_l))$, and the sum is taken over all decomposition of
$(\hat c_l)$, $(x_l)$ and $(\alpha_l)$.
The sums below are taken over all $(m_-, m_+)$
(and all $m$, $k$, all sequences of simplices
$(\hat d_l)_{l = 1}^k$ of $K_{Y^\pm}$ not contained in $\overline{P}_{Y^\pm}^{\text{bad}}$,
and all decomposition $((\hat c_l), (x_l), (\alpha_l)) = ((\hat c'_l), (x'_l), (\alpha'_l)) \sqcup
((\hat c''_l), (x''_l), (\alpha''_l))$
if they appear).
We abbreviate $\Theta^-(e^{\otimes G^-} \otimes H^-)$ and
$\Theta^+(e^{\otimes G^+} \otimes H^+)$ to $\widetilde{H}^-$ and $\widetilde{H}^+$
respectively.
It is easy to check that
\begin{align}
&d f_0((\hat c_l), (x_l), (\alpha_l)) \notag\\
& = \sum d\bigl[\overline{\mathcal{M}}^{(m_-, X^I, m_+)}
_{((\widetilde{G}^+_{m_+}, \widetilde{G}^-_{-m_-}), (\hat c_l), (x_l),
(\alpha_l))}\bigr]^0 \notag\\
&= \sum \bigl[\overline{\mathcal{M}}^{(m_-, X^I, m_+)}
_{(\partial_\tau(\widetilde{G}^+_{m_+}, \widetilde{G}^-_{-m_-}), (\hat c_l), (x_l),
(\alpha_l))}
\bigr]^0 \notag\\
& \quad + \sum \bigl[\overline{\mathcal{M}}^{(m_-, X^I, m_+)}
_{((\widetilde{G}^+_{m_+}, \widetilde{G}^-_{-m_-}), (\hat c_l), \partial_\tau (x_l),
(\alpha_l))}\bigr]^0 \notag\\
& \quad - \sum \bigl[\overline{\mathcal{M}}^{(m_-, X^I, m_+)}
_{((\widetilde{G}^+_{m_+}, \widetilde{G}^-_{-m_-}), \partial((\hat c_l), (x_l),
(\alpha_l)))}\bigr]^1 \notag\\
& \quad - \sum (-1)^{m_-} \bigl[\overline{\mathcal{M}}^{(m_-, X^I, m_+)}
_{((\partial' \widetilde{G}^+_{m_+}, \widetilde{G}^-_{-m_-}), (\hat c_l), (x_l),
(\alpha_l))}\bigr]^1 \notag\\
& \quad - \sum \bigl[\overline{\mathcal{M}}^{(m_-, X^I, m_+)}
_{((\widetilde{G}^+_{m_+}, \partial'\widetilde{G}^-_{-m_-}), (\hat c_l), (x_l),
(\alpha_l))}\bigr]^1 \notag\\
& \quad -\sum (-1)^{m_-} \bigl[\overline{\mathcal{M}}^{(m_-, X^I, m_+ + 1)}
_{((\sum_i (-1)^i e^{\Delta_\ast [\overline{P}_{Y^+}]^{i, i+1}} \tau^+_i \widetilde{G}^+_{m_+},
\widetilde{G}^-_{-m_-}), (\hat c_l), (x_l),
(\alpha_l))}\bigr]^1 \notag\\
& \quad - \sum \bigl[\overline{\mathcal{M}}^{(m_- +1, X^I, m_+)}
_{((\widetilde{G}^+_{m_+}, \sum_i e^{\Delta_\ast [\overline{P}_{Y^-}]^{i-1, i}}
\tilde \tau^-_i \widetilde{G}^-_{-m_-}), (\hat c_l), (x_l),
(\alpha_l))}\bigr]^1,
\label{df_0}
\end{align}
(\ref{+G eq}) implies
\begin{align}
&- \sum (-1)^{m_-}\bigl(\bigl[\overline{\mathcal{M}}^{(m_-, X^I, m_+)}
_{((\partial' \widetilde{G}^+_{m_+}, \widetilde{G}^-_{-m_-}), (\hat c_l), (x_l),
(\alpha_l))}\bigr]^1 \notag\\
& \hphantom{- \sum (-1)^{m_-}}
+ \bigl[\overline{\mathcal{M}}^{(m_-, X^I, m_+)}
_{((\sum_i (-1)^i e^{\Delta_\ast [\overline{P}_{Y^+}]^{i, i+1}} \tau^+_i \widetilde{G}^+_{m_+},
\widetilde{G}^-_{-m_-}), (\hat c_l), (x_l),
(\alpha_l))}\bigr]^1\bigr) \notag\\
&= \sum (-1)^{m_-} \bigl(\bigl[\overline{\mathcal{M}}^{(m_-, X^I, m_+ + m)}
_{((\Diamond^+(\widetilde{G}^+_{m_+} \otimes F^+_m), \widetilde{G}^-_{-m_-}),
(\hat c_l), (x_l), (\alpha_l))}\bigr]^1 \bigr) \notag\\
&= \sum \frac{1}{k!}
\bigl[\overline{\mathcal{M}}^{(m_-, X^I, m_+)}
_{((\widetilde{G}^+_{m_+}, \widetilde{G}^-_{-m_-}), (\hat c'_l), (x'_l),
(\alpha'_l) \cup
(\hat d_l^\ast)_{l= 1}^k)}\bigr]^1 \notag\\
& \quad \ \hphantom{\sum \frac{1}{k!}} \cdot
\bigl[(\overline{\mathcal{M}}_{Y^+})^m
_{(F^+_m, (\hat d_l)_{l=k}^1, (x''_l),
(\alpha''_l))} \bigr]^0,
\label{G^+ 2}
\end{align}
and (\ref{-G eq}) implies
\begin{align}
& - \sum \bigl(\bigl[\overline{\mathcal{M}}^{(m_-, X^I, m_+)}
_{((\widetilde{G}^+_{m_+}, \partial' \widetilde{G}^-_{-m_-}), (\hat c_l), (x_l),
(\alpha_l))}\bigr]^1 \notag\\
& \hphantom{- \sum \bigl(}
+ \bigl[\overline{\mathcal{M}}^{(m_-, X^I, m_+)}
_{((\widetilde{G}^+_{m_+}, \sum_i e^{\Delta_\ast [\overline{P}_{Y^-}]^{i-1, i}} \tilde \tau^-_i
\widetilde{G}^-_{-m_-}), (\hat c_l), (x_l),
(\alpha_l))}\bigr]^1 \bigr) \notag\\
&= \sum \bigl[\overline{\mathcal{M}}^{(m_- + m, X^I, m_+)}
_{((\widetilde{G}^+_{m_+}, \Diamond^-(F^-_m \otimes \widetilde{G}^-_{-m_-})),
(\hat c_l), (x_l),
(\alpha_l))}\bigr]^1 \notag\\
&= \sum \frac{1}{k!} \bigl[(\overline{\mathcal{M}}_{Y^-})^m
_{(F^-_m, (\hat c'_l), (x'_l),
(\hat d_l^\ast)_{l=1}^k)} \bigr]^0 \notag\\
&\quad \ \hphantom{\sum \frac{1}{k!}} \cdot
\bigl[\overline{\mathcal{M}}^{(m_-, X^I, m_+)}
_{((\widetilde{G}^+_{m_+}, \widetilde{G}^-_{-m_-}), (\hat d_l)_{l=k}^1
\cup (\hat c''_l), (x''_l),
(\alpha_l))}\bigr]^1.
\label{G^- 2}
\end{align}
(\ref{H^- eq}) implies
\begin{align}
&\sum \bigl[\overline{\mathcal{M}}^{(m_-, X^I, m_+)}
_{((\widetilde{G}^+_{m_+}, \partial_\tau \widetilde{G}^-_{-m_-}), (\hat c_l), (x_l),
(\alpha_l))}\bigr]^0 \notag\\
& = \sum \bigl[\overline{\mathcal{M}}^{(m_-, X^I, m_+)}
_{((\widetilde{G}^+_{m_+}, \partial' \widetilde{H}^-_{-m_-}),
(\hat c_l), (x_l),
(\alpha_l))}\bigr]^0 \notag\\
& \quad + \sum \bigl[\overline{\mathcal{M}}^{(m_-, X^I, m_+)}
_{((\widetilde{G}^+_{m_+},
\sum_i e^{\Delta_\ast [\overline{P}_{Y^-}]^{i-1, i}} \tilde \tau^-_i
\widetilde{H}^-_{-m_-}), (\hat c_l), (x_l),
(\alpha_l))}\bigr]^0 \notag\\
& \quad + \sum \bigl[\overline{\mathcal{M}}^{(m_-, X^I, m_+)}
_{((\widetilde{G}^+_{m_+},
\Diamond^-(F^-_m \otimes \widetilde{H}^-_{-m_-})),
(\hat c_l), (x_l),
(\alpha_l))}\bigr]^0.
\label{partial tau - first}
\end{align}
(\ref{boundary of MMX}) implies
\begin{align}
0 &= \sum \bigl[\partial \overline{\mathcal{M}}^{(m_-, X^\tau, m_+)}
_{((\widetilde{G}^+_{m_+}, \widetilde{H}^-_{-m_-}),
(\hat c_l), (x_l),
(\alpha_l))}\bigr]^0 \notag\\
&= \sum (-1)^{m_-} \bigl( \bigl[\overline{\mathcal{M}}^{(m_-, X^I, m_+)}
_{((\partial' \widetilde{G}^+_{m_+}, \widetilde{H}^-_{-m_-}),
(\hat c_l), (x_l),
(\alpha_l))}\bigr]^0 \notag\\
&\quad \hphantom{\sum (-1)^{m_-} \bigl(}
+ \bigl[\overline{\mathcal{M}}^{(m_-, X^I, m_+ +1)}
_{((\sum_i (-1)^i e^{\Delta_\ast [\overline{P}_{Y^+}]^{i, i+1}} \tau^+_i \widetilde{G}^+_{m_+},
\widetilde{H}^-_{-m_-}), (\hat c_l), (x_l),
(\alpha_l))}\bigr]^0
\bigr) \notag\\
&\quad + \sum \bigl(\bigl[\overline{\mathcal{M}}^{(m_-, X^I, m_+)}
_{((\widetilde{G}^+_{m_+}, \partial' \widetilde{H}^-_{-m_-}),
(\hat c_l), (x_l),
(\alpha_l))}\bigr]^0 \notag\\
& \quad \hphantom{+ \sum \bigl(}
+ \bigl[\overline{\mathcal{M}}^{(m_- + 1, X^I, m_+)}
_{((\widetilde{G}^+_{m_+},
\sum_i e^{\Delta_\ast [\overline{P}_{Y^-}]^{i-1, i}} \tilde \tau^-_i
\widetilde{H}^-_{-m_-}), (\hat c_l), (x_l),
(\alpha_l))}\bigr]^0
\bigr) \notag\\
&\quad - \sum \bigl[\overline{\mathcal{M}}^{(m_-, X^I, m_+)}
_{((\widetilde{G}^+_{m_+}, \widetilde{H}^-_{-m_-}),
\partial((\hat c_l), (x_l),
(\alpha_l)))}\bigr]^0.
\end{align}
(\ref{+G eq}) implies
\begin{align}
&\sum (-1)^{m_-} \bigl( \bigl[\overline{\mathcal{M}}^{(m_-, X^I, m_+)}
_{((\partial' \widetilde{G}^+_{m_+}, \widetilde{H}^-_{-m_-}),
(\hat c_l), (x_l),
(\alpha_l))}\bigr]^0 \notag\\
&\hphantom{\sum (-1)^{m_-} \bigl(}
+ \bigl[\overline{\mathcal{M}}^{(m_-, X^I, m_+ +1)}
_{((\sum_i (-1)^i e^{\Delta_\ast [\overline{P}_{Y^+}]^{i, i+1}} \tau^+_i \widetilde{G}^+_{m_+},
\widetilde{H}^-_{-m_-}), (\hat c_l), (x_l),
(\alpha_l))}\bigr]^0
\bigr) \notag\\
&= -\sum (-1)^{m_-} \bigl[\overline{\mathcal{M}}^{(m_-, X^I, m_+ + m)}
_{((\Diamond^+(\widetilde{G}^+_{m_+} \otimes F^+_m), \widetilde{H}^-_{-m_-}), (\hat c_l), (x_l),
(\alpha_l))}\bigr]^0
\end{align}
It is easy to check the following equations.
\begin{align}
&\sum (-1)^{m_-} \bigl[\overline{\mathcal{M}}^{(m_-, X^I, m_+ + m)}
_{((\Diamond^+(\widetilde{G}^+_{m_+} \otimes F^+_m),
\widetilde{H}^-_{-m_-}),
(\hat c_l), (x_l),
(\alpha_l))}\bigr]^0 \notag\\
&= - \sum \frac{1}{k!} \bigl[\overline{\mathcal{M}}^{(m_-, X^I, m_+)}
_{((\widetilde{G}^+_{m_+}, \widetilde{H}^-_{-m_-}),
(\hat c_l), (x'_l),
(\alpha'_l) \cup (
\hat d_l^\ast)_{l=1}^k)}\bigr]^0 \notag\\
& \quad \hphantom{- \sum \frac{1}{k!}} \cdot
\bigl[(\overline{\mathcal{M}}_{Y^+})^m_{(F^+_m, (\hat d_l)_{l=k}^1, (x''_l), (\alpha''_l))} \bigr]^0,
\end{align}
\begin{align}
&\sum \bigl[\overline{\mathcal{M}}^{(m_- + m, X^I, m_+)}
_{((\widetilde{G}^+_{m_+},
\Diamond^-(F^-_m \otimes \widetilde{H}^-_{-m_-})),
(\hat c_l), (x_l),
(\alpha_l))}\bigr]^0 \notag\\
&= - \sum \frac{1}{k!}
\bigl[(\overline{\mathcal{M}}_{Y^-})^m
_{(F^-_m, (\hat c'_l), (x'_l), (\hat d_l^\ast)_{l=1}^k)} \bigr]^0 \notag\\
& \quad \hphantom{- \sum \frac{1}{k!}} \cdot
\bigl[\overline{\mathcal{M}}^{(m_-, X^I, m_+)}
_{((\widetilde{G}^+_{m_+}, \widetilde{H}^-_{-m_-}), (\hat d_l)_{l=k}^1,
(\hat c''_l), (x''_l),
(\alpha_l))}\bigr]^0.
\label{partial tau - end}
\end{align}
(\ref{partial tau - first}) to (\ref{partial tau - end}) imply
\begin{align}
&\sum \bigl[\overline{\mathcal{M}}^{(m_-, X^I, m_+)}
_{((\widetilde{G}^+_{m_+}, \partial_\tau \widetilde{G}^-_{-m_-}), (\hat c_l), (x_l),
(\alpha_l))}\bigr]^0 \notag\\
&= \sum (-1)^{m_-} \bigl[\overline{\mathcal{M}}^{(m_-, X^I, m_+ + m)}
_{((\Diamond^+(\widetilde{G}^+_{m_+} \otimes F^+_m), \widetilde{H}^-_{-m_-}), (\hat c_l), (x_l),
(\alpha_l))}\bigr]^0 \notag\\
&\quad + \sum \bigl[\overline{\mathcal{M}}^{(m_-, X^I, m_+)}
_{((\widetilde{G}^+_{m_+}, \widetilde{H}^-_{-m_-}),
\partial((\hat c_l), (x_l), (\alpha_l)))}\bigr]^0 \notag\\
&\quad + \sum \bigl[\overline{\mathcal{M}}^{(m_-, X^I, m_+)}
_{((\widetilde{G}^+_{m_+},
\Diamond^-(F^-_m \otimes \widetilde{H}^-_{-m_-})), (\hat c_l), (x_l),
(\alpha_l))}\bigr]^0 \notag\\
&= \sum \bigl[\overline{\mathcal{M}}^{(m_-, X^I, m_+)}
_{((\widetilde{G}^+_{m_+}, \widetilde{H}^-_{-m_-}),
\partial((\hat c_l), (x_l), (\alpha_l)))}\bigr]^0 \notag\\
&\quad - \sum \frac{1}{k!} \bigl[\overline{\mathcal{M}}^{(m_-, X^I, m_+)}
_{((\widetilde{G}^+_{m_+}, \widetilde{H}^-_{-m_-}),
(\hat c_l), (x'_l), (\alpha'_l) \cup (\hat d_l^\ast)_{l=1}^k)}\bigr]^0 \notag\\
& \quad \hphantom{- \sum \frac{1}{k!}} \cdot
\bigl[(\overline{\mathcal{M}}_{Y^+})^m_{(F^+_m, (\hat d_l)_{l=k}^1, (x''_l),
(\alpha''_l))} \bigr]^0 \notag\\
&\quad - \sum \frac{1}{k!}
\bigl[(\overline{\mathcal{M}}_{Y^-})^m_{(F^-_m, (\hat c'_l), (x'_l),
(\hat d_l^\ast)_{l=1}^k)} \bigr]^0 \notag\\
& \quad \hphantom{- \sum \frac{1}{k!}} \cdot
\bigl[\overline{\mathcal{M}}^{(m_-, X^I, m_+)}
_{((\widetilde{G}^+_{m_+}, \widetilde{H}^-_{-m_-}), (\hat d_l)_{l=k}^1,
(\hat c''_l), (x''_l), (\alpha_l))}\bigr]^0.
\label{partial tau -}
\end{align}
Similarly,
\begin{align}
&\sum \bigl[\overline{\mathcal{M}}^{(m_-, X^I, m_+)}
_{((\partial_\tau \widetilde{G}^+_{m_+}, \widetilde{G}^-_{-m_-}), (\hat c_l), (x_l),
(\alpha_l))}\bigr]^0 \notag\\
&= \sum (-1)^{m_-} \bigl[\overline{\mathcal{M}}^{(m_-, X^I, m_+)}
_{((\widetilde{H}^+_{m_+}, \widetilde{G}^-_{-m_-}),
\partial((\hat c_l), (x_l),
(\alpha_l)))}\bigr]^0 \notag\\
&\quad -\sum (-1)^{m_-} \frac{1}{k!}
\bigl[\overline{\mathcal{M}}^{(m_-, X^I, m_+)}
_{((\widetilde{H}^+_{m_+}, \widetilde{G}^-_{-m_-}),
(\hat c_l), (x'_l),
(\alpha'_l) \cup
(\hat d_l^\ast)_{l=1}^k)}\bigr]^0 \notag\\
& \quad \hphantom{-\sum (-1)^{m_-} \frac{1}{k!}}
\cdot [\bigl[(\overline{\mathcal{M}}_{Y^+})^m_{(F^+_m, (\hat d_l)_{l=k}^1, (x''_l),
(\alpha''_l))} \bigr]^0 \notag\\
&\quad - \sum (-1)^{m_-} \frac{1}{k!}
\bigl[(\overline{\mathcal{M}}_{Y^-})^m_{(F^-_m, (\hat c'_l), (x'_l),
(\hat d_l^\ast)_{l=1}^k)} \bigr]^0
\notag\\
& \quad \hphantom{- \sum (-1)^{m_-} \frac{1}{k!}}
\cdot \bigl[\overline{\mathcal{M}}^{(m_-, X^I, m_+)}
_{((\widetilde{H}^+_{m_+}, \widetilde{G}^-_{-m_-}),
(\hat d_l)_{l=k}^1,
(\hat c''_l), (x''_l),
(\alpha_l))}\bigr]^0.
\label{partial tau +}
\end{align}
(\ref{boundary of MMX}) implies
\begin{align}
&\sum \bigl[\partial \overline{\mathcal{M}}^{(m_-, X^\tau, m_+)}
_{((\widetilde{G}^+_{m_+}, \widetilde{G}^-_{-m_-}), (\hat c_l), (x_1, \dots, y_r, \dots, x_{k_t}),
(\alpha_l))}\bigr]^0 \notag\\
&= \sum \bigl[\overline{\mathcal{M}}^{(m_-, X^I, m_+)}
_{((\widetilde{G}^+_{m_+}, \widetilde{G}^-_{-m_-}), \partial((\hat c_l),
(x_1, \dots, y_r, \dots, x_{k_t}),
(\alpha_l)))}\bigr]^0 \notag\\
&\quad + \sum (-1)^{m_-} \bigl(\bigl[\overline{\mathcal{M}}^{(m_-, X^I, m_+)}
_{((\partial' \widetilde{G}^+_{m_+}, \widetilde{G}^-_{-m_-}), (\hat c_l),
(x_1, \dots, y_r, \dots, x_{k_t}),
(\alpha_l))}\bigr]^0 \notag\\
& \quad \hphantom{+ \sum (-1)^{m_-} \bigl(}
+ \bigl[\overline{\mathcal{M}}^{(m_-, X^I, m_+)}
_{((\sum_i (-1)^i e^{\Delta_\ast [\overline{P}]^{i, i+1}} \tau^+_i \widetilde{G}^+_{m_+},
\widetilde{G}^-_{-m_-}), (\hat c_l), (x_1, \dots, y_r, \dots, x_{k_t}),
(\alpha_l))}\bigr]^0 \bigr) \notag\\
& \quad + \sum \bigl(\bigl[\overline{\mathcal{M}}^{(m_-, X^I, m_+)}
_{((\widetilde{G}^+_{m_+}, \partial' \widetilde{G}^-_{-m_-}), (\hat c_l),
(x_1, \dots, y_r, \dots, x_{k_t}),
(\alpha_l))}\bigr]^0 \notag\\
& \quad \hphantom{+ \sum \bigl(}
+ \bigl[\overline{\mathcal{M}}^{(m_-, X^I, m_+)}
_{((\widetilde{G}^+_{m_+}, \sum_i e^{\Delta_\ast [\overline{P}_{Y^-}]^{i-1, i}} \tilde \tau^-_i
\widetilde{G}^-_{-m_-}), (\hat c_l), (x_1, \dots, y_r, \dots, x_{k_t}),
(\alpha_l))}\bigr]^0 \bigr) \notag\\
&= \sum \bigl[\overline{\mathcal{M}}^{(m_-, X^I, m_+)}
_{((\widetilde{G}^+_{m_+}, \widetilde{G}^-_{-m_-}), (\hat c_l),
\partial (x_1, \dots, y_r, \dots, x_{k_t}),
(\alpha_l))}\bigr]^0 \notag\\
& \quad - \sum (-1)^{m_-}\bigl[\overline{\mathcal{M}}^{(m_-, X^I, m_+ + m)}
_{((\Diamond^+(\widetilde{G}^+_{m_+} \otimes F^+_m), \widetilde{G}^-_{-m_-}), (\hat c_l),
(x_1, \dots, y_r, \dots, x_{k_t}),
(\alpha_l))}\bigr]^0 \notag\\
& \quad - \sum \bigl[\overline{\mathcal{M}}^{(m_-, X^I, m_+)}
_{((\widetilde{G}^+_{m_+}, \Diamond^-(F^-_m \otimes \widetilde{G}^-_{-m_-})), (\hat c_l),
(x_1, \dots, y_r, \dots, x_{k_t}),
(\alpha_l))}\bigr]^0.
\end{align}
Hence
\begin{align}
&\sum (-1)^{\sum |c_l| + \sum_{l < r}|x_l|}\bigl[\overline{\mathcal{M}}^{(m_-, X^I, m_+)}
_{((\widetilde{G}^+_{m_+}, \widetilde{G}^-_{-m_-}), \partial((\hat c_l),
(x_1, \dots, y_r, \dots, x_{k_t}),
(\alpha_l)))}\bigr]^0 \notag\\
&= - \sum (-1)^{\sum |c_l| + \sum_{l < r'}|x'_l|}\bigl[\overline{\mathcal{M}}^{(m_-, X^I, m_+)}
_{((\widetilde{G}^+_{m_+}, \widetilde{G}^-_{-m_-}), (\hat c_l),
(x'_1, \dots, y_{r'}, \dots, x'_{k'_t}),
(\alpha'_l) \cup
(\hat d_l^\ast)_{l=1}^k)}\bigr]^0 \notag\\
& \quad \hphantom{- \sum (-1)^{\sum |c_l| + \sum_{l < r'}|x'_l|}}
\cdot \bigl[(\overline{\mathcal{M}}_{Y^+})^m_{(F^+_m, (\hat d_l)_{l_=k}^1, (x''_l),
(\alpha''_l))} \bigr]^0 \notag\\
&\quad - \sum \bigl[(\overline{\mathcal{M}}_{Y^-})^m_{(F^-_m, (\hat c'_l), (x'_l),
(\hat d_l^\ast)_{l=1}^k)} \bigr]^0 \notag\\
& \quad \hphantom{- \sum}
\cdot (-1)^{\sum |c''_l| + \sum_{l < r''}|x''_l|}\bigl[\overline{\mathcal{M}}^{(m_-, X^I, m_+)}
_{((\widetilde{G}^+_{m_+}, \widetilde{G}^-_{-m_-}), (\hat c''_l),
(x''_1, \dots, y_{r''}, \dots, x''_{k''_t}), (\alpha_l))}\bigr]^0
\label{partial tau x}
\end{align}
(\ref{df_0}), (\ref{G^+ 2}), (\ref{G^- 2}), (\ref{partial tau -}), (\ref{partial tau +})
and (\ref{partial tau x})
imply (\ref{boundary formula for X^I}).
\subsection{Construction of homotopies}\label{algebra for homotopy}
We define families of generating functions
$\mathcal{F}^\tau, \mathcal{K}^\tau \in (\hbar^{-1} \mathcal{D}_X^{\leq 0})^{\star, \delta}
/ J^{\star, \delta}_{\overline{C}_0, \overline{C}_1, \overline{C}_2}$
and $\widetilde{\mathcal{F}}^\tau, \widetilde{\mathcal{K}}^\tau \in \mathcal{D}\D_X^{\leq 0, \delta}
/ \widetilde{J}^{\leq 0, \delta}_{\overline{C}_0, \overline{C}_1, \overline{C}_2}$
by
\begin{align*}
\mathcal{F}^\tau &= \hbar^{-1} \sum \frac{1}{k_q ! k_t ! k_p !} (h_{0, g}^e)^\tau
(\underbrace{\mathbf{q}, \dots, \mathbf{q}}_{k_q},
\underbrace{\mathbf{t}, \dots, \mathbf{t}}_{k_t},
\underbrace{\mathbf{p}, \dots, \mathbf{p}}_{k_p}) \hbar^g T^e\\
\widetilde{\mathcal{F}}^\tau &= \hbar^{-1} \sum \frac{1}{k_q ! k_t ! k_p !} (f_{0, g}^e)^\tau
(\underbrace{\mathbf{q}, \dots, \mathbf{q}}_{k_q},
\underbrace{\mathbf{t}, \dots, \mathbf{t}}_{k_t},
\underbrace{\mathbf{p}, \dots, \mathbf{p}}_{k_p}) \hbar^g T^e\\
\mathcal{K}^\tau &= \hbar^{-1} \sum \frac{1}{k_q ! k_t ! k_p !} (\hat h_{1, g}^e)^\tau
(\underbrace{\mathbf{q}, \dots, \mathbf{q}}_{k_q},
\underbrace{\mathbf{t}, \dots, \mathbf{t}}_{k_t},
\underbrace{\mathbf{p}, \dots, \mathbf{p}}_{k_p}) \hbar^g T^e\\
\widetilde{\mathcal{K}}^\tau &= \hbar^{-1} \sum \frac{1}{k_q ! k_t ! k_p !} (\hat f_{1, g}^e)^\tau
(\underbrace{\mathbf{q}, \dots, \mathbf{q}}_{k_q},
\underbrace{\mathbf{t}, \dots, \mathbf{t}}_{k_t},
\underbrace{\mathbf{p}, \dots, \mathbf{p}}_{k_p}) \hbar^g T^e
\end{align*}
Then (\ref{f_0 equation}), (\ref{f_1 equation}),
(\ref{boundary formula for X^I}) and (\ref{boundary formula for each tau})
imply the following equations.
\begin{gather*}
\widetilde{\mathcal{F}}^\tau = e^{\mathcal{F}^\tau}\\
\widetilde{\mathcal{K}}^\tau = \mathcal{K}^\tau \widetilde{\mathcal{F}}^\tau\\
\frac{d}{d\tau} \widetilde{\mathcal{F}}^\tau = \delta \widetilde{\mathcal{K}}^\tau
- \overrightarrow{\mathcal{H}^-} \widetilde{\mathcal{K}}^\tau
- \widetilde{\mathcal{K}}^\tau \overleftarrow{\mathcal{H}^+}\\
\widehat{D}_X(\widetilde{\mathcal{F}}^\tau) = 0
\end{gather*}
Therefore, the following equation holds true in $\mathcal{D}\D_X^{\leq 0, \delta}
/ \widetilde{J}^{\leq 0, \delta}_{\overline{C}_0, \overline{C}_1, \overline{C}_2}$.
\begin{align}
\frac{d}{d\tau} (e^{\mathcal{F}^\tau}) &= \widehat{D}_X (\mathcal{K}^\tau e^{\mathcal{F}^\tau}) \notag\\
&= [\widehat{D}_X, \mathcal{K}^\tau] (e^{\mathcal{F}^\tau}) \label{main eq for X^I}
\end{align}
Namely, the family of functions $\mathcal{F}^\tau$ is a homotopy in the sense
of \cite{EGH00}.
\begin{defi}
One-parameter family of functions $F^\tau \in (\hbar^{-1} \mathcal{D}_X^{\leq 0})^{\star, \delta}
/ J^{\star, \delta}_{\overline{C}_0, \overline{C}_1, \overline{C}_2}$ ($\tau \in [0, 1]$)
of even degree is said to be a homotopy
if (\ref{main eq for X}) holds for all $\mathcal{F} = \mathcal{F}^\tau$ and
there exists a family of functions $\mathcal{K}^\tau \in (\hbar^{-1} \mathcal{D}_X^{\leq 0})^{\star, \delta}
/ J^{\star, \delta}_{\overline{C}_0, \overline{C}_1, \overline{C}_2}$ of odd degree
which makes equation (\ref{main eq for X^I}) holds for all $\tau \in [0, 1]$.
\end{defi}
\begin{rem}
If (\ref{main eq for X}) holds for some $\mathcal{F}^\tau$ and (\ref{main eq for X^I}) is satisfied
for all $\tau \in [0, 1]$, then (\ref{main eq for X}) holds for all $\mathcal{F}^\tau$.
\end{rem}
\begin{rem}
(\ref{main eq for X^I}) is equivalent to
\begin{equation}
\frac{d}{d\tau} \mathcal{F}^\tau = D_{\mathcal{F}^\tau}(\mathcal{K}^\tau)
\label{main eq for X F}
\end{equation}
in $(\hbar^{-1} \mathcal{D}_X^{\leq 0})^{\star, \delta}
/ J^{\star, \delta}_{\overline{C}_0, \overline{C}_1, \overline{C}_2}$.
\end{rem}
First we consider the case of general SFT.
As in \cite{EGH00}, we define flows by linear differential equations.
For each four-tuple $(\kappa, C_0, C_1, C_2)$ such that
$\overline{C}_0 \geq C_0$, $\overline{C}_1 \geq C_1
+ \kappa \delta^{-1}$ and $\overline{C}_2 \geq C_2 + \kappa$,
we define a flow $\Phi^\tau : \mathcal{D}\D_X^{\leq \kappa, \delta}
/ \widetilde{J}^{\kappa, \delta}_{C_0, C_1, C_2}
\to \mathcal{D}\D_X^{\leq \kappa, \delta}
/ \widetilde{J}^{\kappa, \delta}_{C_0, C_1, C_2}$ by
\[
\frac{d}{d\tau} \Phi^\tau(f) = [\widehat{D}_X, \mathcal{K}^\tau] \Phi^\tau(f), \quad \Phi^0 = \mathrm{id},
\]
and define $T^\tau : \mathcal{D}_X^{\leq \kappa} / J^{\leq \kappa, \delta}_{C_0, C_1, C_2}
\to \mathcal{D}_X^{\leq \kappa} / J^{\leq \kappa, \delta}_{C_0, C_1, C_2}$ by
\[
T^\tau(f) = e^{-\mathcal{F}^\tau} \Phi^\tau (e^{\mathcal{F}^0} f).
\]
$\Phi^\tau$ is well defined because it is defined by a linear differential equation
on a finite dimensional vector space.
$T^\tau$ is well defined because it is also defined by
\[
\frac{d}{d\tau} T^\tau(f)
= [[D_{\mathcal{F}^\tau}, \mathcal{K}^\tau], T^\tau(f)] (1),
\quad T^0 = \mathrm{id}.
\]
Some of the following were proved in \cite{EGH00}, and some are straightforward,
but we prove all of them for the convenience of the reader.
\begin{lem}\label{homotopy and T}
\begin{enumerate}[label = \normalfont (\roman*)]\
\item
$T^\tau$ is a chain map from
$(\mathcal{D}_X^{\leq \kappa} / J^{\leq \kappa, \delta}_{C_0, C_1, C_2}, D_{\mathcal{F}^0})$ to
$(\mathcal{D}_X^{\leq \kappa} / J^{\leq \kappa, \delta}_{C_0, C_1, C_2}, D_{\mathcal{F}^\tau})$
for each $\tau$, that is,
$D_{\mathcal{F}^\tau} \circ T^\tau = T^\tau \circ D_{\mathcal{F}^0}$.
(This is equivalent to $\widehat{D}_X \circ \Phi^\tau = \Phi^\tau \circ \widehat{D}_X$.)
Furthermore, up to chain homotopy, it is determined by $(\mathcal{F}^{\tau'})_{\tau' \in [0, \tau]}$
and independent of the choice of the family $(\mathcal{K}^{\tau'})_{\tau' \in [0, \tau]}$
which satisfies equation (\ref{main eq for X^I}).
\item
If a smooth family of generating functions $\mathcal{F}^{\tau, \sigma}
\in (\hbar^{-1} \mathcal{D}_X^{\leq 0})^{\star, \delta}
/ J^{\star, \delta}_{\overline{C}_0, \overline{C}_1, \overline{C}_2}$
$((\tau, \sigma) \in [0, 1] \times [0, 1])$
satisfies $\mathcal{F}^{0, \sigma} \equiv \mathcal{F}^{0, 0}$ and the one-parameter family
$(\mathcal{F}^{\tau, \sigma})_{\tau \in [0, 1]}$ is a homotopy for each $\sigma \in [0, 1]$,
then the one-parameter family $(\mathcal{F}^{\tau, \sigma})_{\sigma \in [0, 1]}$ is also
a homotopy for each $\tau \in [0, 1]$.
\item
Further assume that the above family of generating functions satisfies
$\mathcal{F}^{1, \sigma} = \mathcal{F}^{0, 0}$ and $\mathcal{F}^{\tau, 0} \equiv \mathcal{F}^{0, 0}$.
Let $T^\tau : \mathcal{D}_X^{\leq \kappa} / J^{\leq \kappa, \delta}_{C_0, C_1, C_2}
\to \mathcal{D}_X^{\leq \kappa} / J^{\leq \kappa, \delta}_{C_0, C_1, C_2}$
be the flow defined by the one-parameter homotopy $(\mathcal{F}^{\tau, 1})_{\tau \in [0, 1]}$.
Then $T^1$ is equal to the identity map up to chain homotopy.
In other words, if a loop homotopy $(\mathcal{F}^{\tau, 1})_{\tau \in S^1}$ is contractible
in the space of loop homotopies with the base point $\mathcal{F}^{0, 1}$,
then the chain map $T^1$ is the identity map up to chain homotopy.
Hence for a general one-parameter homotopy $(\mathcal{F}^\tau)_{\tau \in [0, 1]}$,
the end $T^1$ of the family of the chain maps $(T^\tau)_{\tau \in [0, 1]}$
is determined up to chain homotopy by the homotopy type of the homotopy
$(\mathcal{F}^\tau)_{\tau \in [0, 1]}$ relative to the end points.
\item
There exists a family of linear maps
$A^{\pm, \tau} : \mathcal{W}^{\leq \kappa}_{Y^\pm}
/ I^{\leq \kappa}_{C_0, C_1 + \kappa(\delta^{-1} - L^{-1}_{\min}), C_2}
\to \mathcal{D}^{\leq \kappa}_X / J^{\leq \kappa, \delta}_{C_0, C_1, C_2}$
such that
\begin{equation}
i_{\mathcal{F}^0}^\pm - (T^\tau)^{-1} \circ i_{\mathcal{F}^\tau}^\pm = D_{\mathcal{F}^0} \circ A^{\pm, \tau}
+ A^{\pm, \tau} \circ D_{Y^\pm},
\label{A chain homotopy}
\end{equation}
that is, the following diagrams are commutative up to chain homotopy.
\[
\begin{tikzcd}
(\mathcal{D}_X^{\leq \kappa} / J^{\leq \kappa, \delta}_{C_0, C_1, C_2}
, D_{\mathcal{F}^0})\ar{r}{T^\tau}
& (\mathcal{D}_X^{\leq \kappa} / J^{\leq \kappa, \delta}_{C_0, C_1, C_2}, D_{\mathcal{F}^\tau})\\
(\mathcal{W}_{Y^\pm}^{\leq \kappa}
/ I^{\leq \kappa}_{C_0, C_1 + \kappa(\delta^{-1} - L^{-1}_{\min}), C_2}, D_{Y^\pm})
\ar{u}{i_{\mathcal{F}^0}^\pm} \ar{ur}[swap]{i_{\mathcal{F}^\tau}^\pm}&
\end{tikzcd}
\]
\end{enumerate}
\end{lem}
\begin{rem}
In the following proof, we need to take care of the degree with respect to $\hbar$.
Multiplication of $\mathcal{F}$ or $\mathcal{K}$ may decrease the degree at most by one,
but super-commutator $[ , ]$ increase the degree at least by one.
Hence in order to see that the linear maps defined below are well-defined,
we need to check that the number of super-commutators are greater than or equal to
the number of multiplications of $\mathcal{F}$ or $\mathcal{K}$.
\end{rem}
\begin{proof}
(i)
To prove that $T^\tau$ is a chain map,
it is enough to see that $(\Phi^\tau)^{-1} \widehat{D}_X \Phi^\tau
: \mathcal{D}\D_X^{\leq \kappa, \delta} / J^{\leq \kappa, \delta}_{C_0, C_1, C_2}
\to \mathcal{D}\D_X^{\leq \kappa, \delta} / J^{\leq \kappa, \delta}_{C_0, C_1, C_2}$ is
independent of $\tau \in [0, 1]$. This can be proved by
\begin{align*}
\frac{d}{d\tau} (\Phi^\tau)^{-1} \widehat{D}_X \Phi^\tau(f)
&= - (\Phi^\tau)^{-1} [\widehat{D}_X, \mathcal{K}^\tau] \widehat{D}_X \Phi^\tau(f)
+ (\Phi^\tau)^{-1} \widehat{D}_X [\widehat{D}_X, \mathcal{K}^\tau] \Phi^\tau(f)\\
&= 0.
\end{align*}
The latter claim is proved as follows.
If $\mathcal{F}^\tau$ and $\mathring{\mathcal{K}}^\tau$ also satisfy equation (\ref{main eq for X^I}),
then $\mathcal{G}^\tau = \mathring{\mathcal{K}}^\tau - \mathcal{K}^\tau \in \mathop{\mathrm{Ker}}\nolimits D_{\mathcal{F}^\tau}$.
Let $\mathring{T}^\tau$ be the flow defined by $\mathcal{F}^\tau$ and $\mathring{\mathcal{K}}^\tau$.
Then
\begin{align*}
\frac{d}{d\tau} (\mathring{T}^\tau)^{-1} T^\tau (f)
&= - (\mathring{T}^\tau)^{-1} [[D_{\mathcal{F}^\tau}, \mathring{\mathcal{K}}^\tau], T^\tau(f)] (1)
+ (\mathring{T}^\tau)^{-1} [[D_{\mathcal{F}^\tau}, \mathcal{K}^\tau], T^\tau(f)] (1)\\
&= - (\mathring{T}^\tau)^{-1} [D_{\mathcal{F}^\tau}, \mathcal{G}^\tau] T^\tau(f)\\
&= - [D_{\mathcal{F}^0}, (\mathring{T}^\tau)^{-1} \mathcal{G}^\tau T^\tau] f.
\end{align*}
Therefore $(\mathring{T}^\tau)^{-1} T^\tau
: (\mathcal{D}_X^{\leq \kappa} / J^{\leq \kappa, \delta}_{C_0, C_1, C_2}, D_{\mathcal{F}^0})
\to (\mathcal{D}_X^{\leq \kappa} / J^{\leq \kappa, \delta}_{C_0, C_1, C_2}, D_{\mathcal{F}^0})$
are chain homotopic to the identity map for all $\tau$,
which implies the claim.
(ii)
Let $T^{\tau, \sigma} : (\hbar^{-1} \mathcal{D}_X^{\leq 0})^{\star, \delta}
/ J^{\star, \delta}_{\overline{C}_0, \overline{C}_1, \overline{C}_2}
\to (\hbar^{-1} \mathcal{D}_X^{\leq 0})^{\star, \delta}
/ J^{\star, \delta}_{\overline{C}_0, \overline{C}_1, \overline{C}_2}$ be the flow
defined similarly for each pair of one-parameter families
$(\mathcal{F}^{\tau, \sigma})_{\tau \in [0, 1]}$
and $(\mathcal{K}^{\tau, \sigma})_{\tau \in [0, 1]}$ satisfying (\ref{main eq for X F}).
Namely, they are defined by
\[
\partial_\tau T^{\tau, \sigma}(f)
= [[D_{\mathcal{F}^{\tau, \sigma}}, \mathcal{K}^{\tau, \sigma}], T^{\tau, \sigma}(f)] (1),
\quad T^{0, \sigma} = \mathrm{id}.
\]
Similarly to (i), each $T^{\tau, \sigma}$ is a chain map from
$((\hbar^{-1} \mathcal{D}_X^{\leq 0})^{\star, \delta}
/ J^{\star, \delta}_{\overline{C}_0, \overline{C}_1, \overline{C}_2}, D_{\mathcal{F}^{0, 0}})$
to $((\hbar^{-1} \mathcal{D}_X^{\leq 0})^{\star, \delta}
/ J^{\star, \delta}_{\overline{C}_0, \overline{C}_1, \overline{C}_2}, D_{\mathcal{F}^{\tau, \sigma}})$.
Hence it is enough to show that there exists a family of functions
$\mathring{\mathcal{K}}^{\tau, \sigma} \in (\hbar^{-1} \mathcal{D}_X^{\leq 0})^{\star, \delta}
/ J^{\star, \delta}_{\overline{C}_0, \overline{C}_1, \overline{C}_2}$ of odd degree
satisfying the following equations.
\begin{equation}
(T^{\tau, \sigma})^{-1} \partial_\sigma \mathcal{F}^{\tau, \sigma}
= D_{\mathcal{F}^{0, 0}}\bigl((T^{\tau, \sigma})^{-1} \mathring{\mathcal{K}}^{\tau, \sigma}\bigr)
\label{circle K condition}
\end{equation}
This is proved by the following calculations.
\begin{align}
&\partial_\tau
\bigl((T^{\tau, \sigma})^{-1} \partial_\sigma \mathcal{F}^{\tau, \sigma} \bigr) \notag\\
&= (T^{\tau, \sigma})^{-1}
\partial_\sigma \partial_\tau \mathcal{F}^{\tau, \sigma}
- (T^{\tau, \sigma})^{-1} \partial_\tau T^{\tau, \sigma} (T^{\tau, \sigma})^{-1}
\partial_\sigma \mathcal{F}^{\tau, \sigma} \notag\\
&= (T^{\tau, \sigma})^{-1}
\partial_\sigma (D_{\mathcal{F}^{\tau, \sigma}}\mathcal{K}^{\tau, \sigma})
- (T^{\tau, \sigma})^{-1} [[D_{\mathcal{F}^{\tau, \sigma}}, \mathcal{K}^{\tau, \sigma}],
\partial_\sigma \mathcal{F}^{\tau, \sigma}](1) \notag\\
& =(T^{\tau, \sigma})^{-1}
D_{\mathcal{F}^{\tau, \sigma}} \partial_\sigma \mathcal{K}^{\tau, \sigma}
+ (T^{\tau, \sigma})^{-1}
[D_{\mathcal{F}^{\tau, \sigma}}, \partial_\sigma\mathcal{F}^{\tau, \sigma}](\mathcal{K}^{\tau, \sigma}) \notag\\
&\quad - (T^{\tau, \sigma})^{-1}
[[D_{\mathcal{F}^{\tau, \sigma}}, \partial_\sigma\mathcal{F}^{\tau, \sigma}], \mathcal{K}^{\tau, \sigma}](1)
- (T^{\tau, \sigma})^{-1}
[D_{\mathcal{F}^{\tau, \sigma}}, [\mathcal{K}^{\tau, \sigma}, \partial_\sigma\mathcal{F}^{\tau, \sigma}]](1) \notag\\
&= D_{\mathcal{F}^{0, 0}}((T^{\tau, \sigma})^{-1} \partial_\sigma\mathcal{K}^{\tau, \sigma})
- (T^{\tau, \sigma})^{-1}
\mathcal{K}^{\tau, \sigma} D_{\mathcal{F}^{\tau, \sigma}}(\partial_\sigma\mathcal{F}^{\tau, \sigma}) \notag\\
&\quad - (T^{\tau, \sigma})^{-1}
[D_{\mathcal{F}^{\tau, \sigma}}, [\mathcal{K}^{\tau, \sigma}, \partial_\sigma\mathcal{F}^{\tau, \sigma}]](1) \notag\\
&= D_{\mathcal{F}^{0, 0}}((T^{\tau, \sigma})^{-1} \partial_\sigma\mathcal{K}^{\tau, \sigma})
\label{diff of sigma homotopy}
\end{align}
In the last equality, we have used the following facts:
\begin{itemize}
\item
$D_{\mathcal{F}^{\tau, \sigma}}(\partial_\sigma\mathcal{F}^{\tau, \sigma}) = 0$
because
\begin{align*}
D_{\mathcal{F}^{\tau, \sigma}}(\partial_\sigma\mathcal{F}^{\tau, \sigma})
&= e^{-\mathcal{F}^{\tau, \sigma}}
\widehat{D}_X(\partial_\sigma\mathcal{F}^{\tau, \sigma} e^{\mathcal{F}^{\tau, \sigma}})\\
&= e^{-\mathcal{F}^{\tau, \sigma}} \partial_\sigma \widehat{D}_X(e^{\mathcal{F}^{\tau, \sigma}})\\
&= 0.
\end{align*}
\item
$[\mathcal{K}^{\tau, \sigma}, \partial_\sigma\mathcal{F}^{\tau, \sigma}] = 0$ because multiplication
in $\mathcal{D}\D_X$ is super-commutative.
\end{itemize}
(\ref{diff of sigma homotopy}) implies that
\begin{equation}
\mathring{\mathcal{K}}^{\tau, \sigma}
:= T^{\tau, \sigma} \int_0^\tau
(T^{{\tau'}, \sigma})^{-1} \partial_\sigma\mathcal{K}^{{\tau'}, \sigma} d\tau'
\in (\hbar^{-1} \mathcal{D}_X^{\leq 0})^{\star, \delta}
/ J^{\star, \delta}_{\overline{C}_0, \overline{C}_1, \overline{C}_2}
\label{circle K}
\end{equation}
satisfies equation (\ref{circle K condition}).
(iii)
Let $\mathcal{K}^{\tau, \sigma}, \mathring{\mathcal{K}}^{\tau, \sigma}
\in (\hbar^{-1} \mathcal{D}_X^{\leq 0})^{\star, \delta}
/ J^{\star, \delta}_{\overline{C}_0, \overline{C}_1, \overline{C}_2}$ be families of functions
satisfying
\begin{align}
\partial_\tau\mathcal{F}^{\tau, \sigma} &= D_{\mathcal{F}^{\tau, \sigma}}(\mathcal{K}^{\tau, \sigma}),\\
\partial_\sigma\mathcal{F}^{\tau, \sigma} &= D_{\mathcal{F}^{\tau, \sigma}}(\mathring{\mathcal{K}}^{\tau, \sigma}).
\end{align}
Let $T^{\tau, \sigma}, \mathring{T}^{\tau, \sigma}
: \mathcal{D}_X^{\leq \kappa} / J^{\leq \kappa, \delta}_{C_0, C_1, C_2}
\to \mathcal{D}_X^{\leq \kappa} / J^{\leq \kappa, \delta}_{C_0, C_1, C_2}$ be flows defined by
\begin{align}
\partial_\tau T^{\tau, \sigma}(f)
&= [[D_{\mathcal{F}^{\tau, \sigma}}, \mathcal{K}^{\tau, \sigma}], T^{\tau, \sigma}(f)] (1),
\quad T^{0, \sigma} = \mathrm{id},\\
\partial_\sigma \mathring{T}^{\tau, \sigma}(f)
&= [[D_{\mathcal{F}^{\tau, \sigma}}, \mathring{\mathcal{K}}^{\tau, \sigma}],
\mathring{T}^{\tau, \sigma}(f)] (1),
\quad \mathring{T}^{\tau, 0} = \mathrm{id}.
\end{align}
Since $\mathring{T}^{1, \sigma} = \mathrm{id}$, it is enough to prove that the chain maps
$(\mathring{T}^{\tau, \sigma})^{-1} T^{\tau, \sigma}$ from
$(\mathcal{D}_X^{\leq \kappa} / J^{\leq \kappa, \delta}_{C_0, C_1, C_2}, D_{\mathcal{F}^{0, 0}})$ to itself are
equal to the identity map up to chain homotopy for all $\sigma, \tau \in [0, 1]$.
The latter claim of (i) implies that we may assume that
$\mathcal{K}^{\tau, 0} = 0$ (since $\mathcal{F}^{\tau, 0} = \mathcal{F}^{0, 0}$)
and that the family $\mathring{\mathcal{K}}^{\tau, \sigma}$ is defined by (\ref{circle K}).
(\ref{circle K}) implies that the following equation holds in
$(\hbar^{-1} \mathcal{D}_X^{\leq 0})^{\star, \delta}
/ J^{\star, \delta}_{\overline{C}_0, \overline{C}_1, \overline{C}_2}$.
\begin{equation}
\partial_\sigma\mathcal{K}^{\tau, \sigma} - \partial_\tau\mathring{\mathcal{K}}^{\tau, \sigma}
+ [[D_{\mathcal{F}^\sigma_\tau}, \mathcal{K}^{\tau, \sigma}], \mathring{\mathcal{K}}^{\tau, \sigma}](1) = 0
\label{exactness of K}
\end{equation}
Define linear maps $S^{\tau, \sigma}, U^{\tau, \sigma}
: \mathcal{D}_X^{\leq \kappa} / J^{\leq \kappa, \delta}_{C_0, C_1, C_2}
\to \mathcal{D}_X^{\leq \kappa} / J^{\leq \kappa, \delta}_{C_0, C_1, C_2}$
($\tau, \sigma \in [0, 1]$) by
\begin{equation}
S^{\tau, \sigma}(f)
= (T^{\tau, \sigma})^{-1} \bigl(-\partial_\tau \mathring{T}^{\tau, \sigma}(f) +
[[D_{\mathcal{F}^{\tau, \sigma}}, \mathcal{K}^{\tau, \sigma}], \mathring{T}^{\tau, \sigma}(f)](1)\bigr)
\end{equation}
\begin{equation}
U^{\tau, \sigma}(f) = (\mathring{T}^{\tau, \sigma})^{-1}
[[[D_{\mathcal{F}^{\tau, \sigma}}, \mathcal{K}^{\tau, \sigma}], \mathring{K}^{\tau, \sigma}],
T^{\tau, \sigma} (f)](1)
\end{equation}
Then the following hold true.
\begin{enumerate}[label=\normalfont(\alph*)]
\item
Each $S^{\tau, \sigma}$ is a chain map from
$(\mathcal{D}_X^{\leq \kappa} / J^{\leq \kappa, \delta}_{C_0, C_1, C_2}, D_{\mathcal{F}^{0, 0}})$ to itself.
\item
$\partial_\sigma (\mathring{T}^{\tau, \sigma})^{-1} T^{\tau, \sigma}(f) \in
\mathcal{D}_X^{\leq \kappa} / J^{\leq \kappa, \delta}_{C_0, C_1, C_2}$
satisfies the following differential equation in variable $\tau$
for any $f \in \mathcal{D}_X^{\leq \kappa} / J^{\leq \kappa, \delta}_{C_0, C_1, C_2}$
and $\sigma \in [0, 1]$:
\begin{align}
&\partial_\tau\partial_\sigma (\mathring{T}^{\tau, \sigma})^{-1} T^{\tau, \sigma}(f)
= S^{\tau, \sigma} \partial_\sigma
\bigl((\mathring{T}^{\tau, \sigma})^{-1} T^{\tau, \sigma}(f)\bigr)\notag\\
&\hphantom{\partial_\tau\partial_\sigma (\mathring{T}^{\tau, \sigma})^{-1} T^{\tau, \sigma}(f) =}
+ D_{\mathcal{F}^{0, 0}} U(f) + U(D_{\mathcal{F}^{0, 0}} f), \label{eq of circle T T}\\
&\partial_\sigma (\mathring{T}^{\tau, \sigma})^{-1} T^{\tau, \sigma}(f)\bigr|_{\tau = 0} = 0.
\label{initial circle T T}
\end{align}
\end{enumerate}
First we prove the claim assuming the above two.
(a) implies that we can regard (\ref{eq of circle T T}) and (\ref{initial circle T T})
as equations of one-parameter families
$\partial_\sigma((\mathring{T}^{\tau, \sigma})^{-1} T^{\tau, \sigma}(\cdot))$
in the quotient space of chain maps from
$(\mathcal{D}_X^{\leq \kappa} / J^{\leq \kappa, \delta}_{C_0, C_1, C_2}, D_{\mathcal{F}^{0, 0}})$ to itself
modulo null homotopies.
Then they become a linear differential equation with the trivial initial condition,
which implies that $\partial_\sigma \bigl((\mathring{T}^{\tau, \sigma})^{-1}
T^{\tau, \sigma}(\cdot)\bigr)$ is a family of null homotopies from
$(\mathcal{D}_X^{\leq \kappa} / J^{\leq \kappa, \delta}_{C_0, C_1, C_2}, D_{\mathcal{F}^{0, 0}})$ to itself.
Hence their integrations $(\mathring{T}^{\tau, \sigma})^{-1} T^{\tau, \sigma}(\cdot)$
are chain homotopic to $(\mathring{T}^{\tau, 0})^{-1} T^{\tau, 0}(\cdot) = \mathrm{id}$.
Now we prove the above two claims.
First we check (a).
By direct calculations, we see
\begin{align*}
D_{\mathcal{F}^{0, 0}} S^{\tau, \sigma}(f)
&= (T^{\tau, \sigma})^{-1} D_{\mathcal{F}^{\tau, \sigma}}
\bigl(-\partial_\tau \mathring{T}^{\tau, \sigma}(f)
+ [[D_{\mathcal{F}^{\tau, \sigma}}, \mathcal{K}^{\tau, \sigma}], \mathring{T}^{\tau, \sigma}(f)](1)\bigr) \\
&= (T^{\tau, \sigma})^{-1}
\bigl(-\partial_\tau(D_{\mathcal{F}^{\tau, \sigma}} \mathring{T}^{\tau, \sigma}(f))
+ [D_{\mathcal{F}^{\tau, \sigma}}, D_{\mathcal{F}^{\tau, \sigma}}(\mathcal{K}^{\tau, \sigma})]
(\mathring{T}^{\tau, \sigma}(f)) \\
&\hphantom{= (T^{\tau, \sigma})^{-1}
\bigl(}
+ D_{\mathcal{F}^{\tau, \sigma}}
([[D_{\mathcal{F}^{\tau, \sigma}}, \mathcal{K}^{\tau, \sigma}], \mathring{T}^{\tau, \sigma}(f)](1))
\bigr)
\\
&= (T^{\tau, \sigma})^{-1}
\bigl(-\partial_\tau(D_{\mathcal{F}^{\tau, \sigma}} \mathring{T}^{\tau, \sigma}(f))
- [D_{\mathcal{F}^{\tau, \sigma}}(\mathring{T}^{\tau, \sigma}(f)), D_{\mathcal{F}^{\tau, \sigma}}]
(\mathcal{K}^{\tau, \sigma})
\bigr)
\end{align*}
and
\begin{align*}
S^{\tau, \sigma} D_{\mathcal{F}^{0, 0}}(f)
&= (T^{\tau, \sigma})^{-1} \bigl(-\partial_\tau \mathring{T}^{\tau, \sigma}(D_{\mathcal{F}^{0, 0}}f) +
[[D_{\mathcal{F}^{\tau, \sigma}}, \mathcal{K}^{\tau, \sigma}],
D_{\mathcal{F}^{\tau, \sigma}}(\mathring{T}^{\tau, \sigma}(f))](1)\bigr) \\
&= (T^{\tau, \sigma})^{-1}
\bigl(-\partial_\tau(\mathring{T}^{\tau, \sigma} D_{\mathcal{F}^{0, 0}}(f))
- [D_{\mathcal{F}^{\tau, \sigma}}(\mathring{T}^{\tau, \sigma}(f)), D_{\mathcal{F}^{\tau, \sigma}}]
(\mathcal{K}^{\tau, \sigma})
\bigr)
\end{align*}
Hence $D_{\mathcal{F}^{0, 0}} S^{\tau, \sigma} = S^{\tau, \sigma} D_{\mathcal{F}^{0, 0}}$.
Next we prove (b).
This is also proved by direct calculation.
The key is equation (\ref{exactness of K}).
First we separate
\begin{align}
&\partial_\tau\partial_\sigma((\mathring{T}^{\tau, \sigma})^{-1}T^{\tau, \sigma}(f))
\notag\\
&= (\partial_\tau\partial_\sigma (\mathring{T}^{\tau, \sigma})^{-1}) T^{\tau, \sigma}(f)
+ (\mathring{T}^{\tau, \sigma})^{-1} \partial_\tau\partial_\sigma T^{\tau, \sigma}(f)
\notag\\
&\quad
+ (\partial_\sigma(\mathring{T}^{\tau, \sigma})^{-1}) \partial_\tau T^{\tau, \sigma}(f)
+ (\partial_\tau(\mathring{T}^{\tau, \sigma})^{-1}) \partial_\sigma T^{\tau, \sigma}(f)
\end{align}
into four parts and calculate each of them as follows.
\begin{align}
&(\partial_\tau\partial_\sigma (\mathring{T}^{\tau, \sigma})^{-1}) T^{\tau, \sigma}(f)
\notag\\
&= - \partial_\tau\bigl((\mathring{T}^{\tau, \sigma})^{-1}
[[D_{\mathcal{F}^{\tau, \sigma}}, \mathring{\mathcal{K}}^{\tau, \sigma}], \cdot](1)\bigr)(T^{\tau, \sigma}(f))
\notag\\
&= - (\mathring{T}^{\tau, \sigma})^{-1}
[[D_{\mathcal{F}^{\tau, \sigma}}, \partial_\tau \mathring{\mathcal{K}}^{\tau, \sigma}], T^{\tau, \sigma}(f)](1)
\notag \\
& \quad - (\mathring{T}^{\tau, \sigma})^{-1}
[[[D_{\mathcal{F}^{\tau, \sigma}}, D_{\mathcal{F}^{\tau, \sigma}}(\mathcal{K}^{\tau, \sigma})], \mathring{\mathcal{K}}^{\tau, \sigma}],
T^{\tau, \sigma}(f)](1) \notag\\
&\quad
+ (\mathring{T}^{\tau, \sigma})^{-1} \partial_\tau\mathring{T}^{\tau, \sigma}
(\mathring{T}^{\tau, \sigma})^{-1} [[D_{\mathcal{F}^{\tau, \sigma}}, \mathring{\mathcal{K}}^{\tau, \sigma}],
T^{\tau, \sigma}(f)](1)
\label{00 first}
\end{align}
\begin{align}
&(\mathring{T}^{\tau, \sigma})^{-1} \partial_\tau\partial_\sigma T^{\tau, \sigma}(f)
\notag\\
&= (\mathring{T}^{\tau, \sigma})^{-1}[[D_{\mathcal{F}^{\tau, \sigma}}, \mathcal{K}^{\tau, \sigma}],
\partial_\sigma T^{\tau, \sigma}(f)](1) \notag\\
& \quad
+ (\mathring{T}^{\tau, \sigma})^{-1}
[[D_{\mathcal{F}^{\tau, \sigma}}, \partial_\sigma \mathcal{K}^{\tau, \sigma}],
T^{\tau, \sigma}(f)](1) \notag\\
&\quad
+ (\mathring{T}^{\tau, \sigma})^{-1}
[[[D_{\mathcal{F}^{\tau, \sigma}}, D_{\mathcal{F}^{\tau, \sigma}}(\mathring{\mathcal{K}}^{\tau, \sigma})],
\mathcal{K}^{\tau, \sigma}],
T^{\tau, \sigma}(f)](1)
\label{00 second}
\end{align}
\begin{align}
&(\partial_\sigma(\mathring{T}^{\tau, \sigma})^{-1}) \partial_\tau T^{\tau, \sigma}(f)
\notag\\
&= - (\mathring{T}^{\tau, \sigma})^{-1} [[D_{\mathcal{F}^{\tau, \sigma}},
\mathring{\mathcal{K}}^{\tau, \sigma}],
[[D_{\mathcal{F}^{\tau, \sigma}}, \mathcal{K}^{\tau, \sigma}], T^{\tau, \sigma}(f)](1)](1)
\label{00 third}
\end{align}
\begin{align}
(\partial_\tau(\mathring{T}^{\tau, \sigma})^{-1}) \partial_\sigma T^{\tau, \sigma}(f)
= - (\mathring{T}^{\tau, \sigma})^{-1} \partial_\tau \mathring{T}^{\tau, \sigma}
(\mathring{T}^{\tau, \sigma})^{-1} \partial_\sigma T^{\tau, \sigma}(f)
\label{00 fourth}
\end{align}
We also calculate the following two.
\begin{align}
&- S^{\tau, \sigma} \partial_\sigma
\bigl((\mathring{T}^{\tau, \sigma})^{-1} T^{\tau, \sigma}(f)\bigr) \notag\\
&= (\mathring{T}^{\tau, \sigma})^{-1} \partial_\tau \mathring{T}^{\tau, \sigma}
(\mathring{T}^{\tau, \sigma})^{-1} \partial_\sigma T^{\tau, \sigma}(f) \notag\\
&\quad
- (\mathring{T}^{\tau, \sigma})^{-1} \partial_\tau \mathring{T}^{\tau, \sigma}
(\mathring{T}^{\tau, \sigma})^{-1} [[D_{\mathcal{F}^{\tau, \sigma}}, \mathring{K}^{\tau, \sigma}],
T^{\tau, \sigma}(f)](1) \notag\\
& \quad
- (\mathring{T}^{\tau, \sigma})^{-1} [[D_{\mathcal{F}^{\tau, \sigma}}, \mathcal{K}^{\tau, \sigma}],
\partial_\sigma T^{\tau, \sigma}(f)](1) \notag\\
&\quad
+ (\mathring{T}^{\tau, \sigma})^{-1} [[D_{\mathcal{F}^{\tau, \sigma}}, \mathcal{K}^{\tau, \sigma}],
[[D_{\mathcal{F}^{\tau, \sigma}}, \mathring{\mathcal{K}}^{\tau, \sigma}], T^{\tau, \sigma}(f)](1)](1)
\label{00 fifth}
\end{align}
\begin{align}
&-(D_{\mathcal{F}^{0, 0}} U(f) + U(D_{\mathcal{F}^{0, 0}} f)) \notag\\
&= -(\mathring{T}^{\tau, \sigma})^{-1}D_{\mathcal{F}^{\tau, \sigma}}(
[[[D_{\mathcal{F}^{\tau, \sigma}}, \mathcal{K}^{\tau, \sigma}], \mathring{K}^{\tau, \sigma}],
T^{\tau, \sigma} (f)](1)) \notag\\
& \quad
- (\mathring{T}^{\tau, \sigma})^{-1}
[[[D_{\mathcal{F}^{\tau, \sigma}}, \mathcal{K}^{\tau, \sigma}], \mathring{K}^{\tau, \sigma}],
D_{\mathcal{F}^{\tau, \sigma}}(T^{\tau, \sigma} (f))](1)
\notag\\
&= - (\mathring{T}^{\tau, \sigma})^{-1} [D_{\mathcal{F}^{\tau, \sigma}}, [[D_{\mathcal{F}^{\tau, \sigma}},
\mathcal{K}^{\tau, \sigma}], \mathring{\mathcal{K}}^{\tau, \sigma}]] T^{\tau, \sigma}(f) \notag\\
&\quad
+ (\mathring{T}^{\tau, \sigma})^{-1}
[D_{\mathcal{F}^{\tau, \sigma}}, [[D_{\mathcal{F}^{\tau, \sigma}}, \mathcal{K}^{\tau, \sigma}],
\mathring{\mathcal{K}}^{\tau, \sigma}](1)] T^{\tau, \sigma}(f) \notag\\
&= - (\mathring{T}^{\tau, \sigma})^{-1} [[D_{\mathcal{F}^{\tau, \sigma}}, \mathcal{K}^{\tau, \sigma}],
[D_{\mathcal{F}^{\tau, \sigma}}, \mathring{\mathcal{K}}^{\tau, \sigma}]] T^{\tau, \sigma}(f) \notag\\
&\quad
+ (\mathring{T}^{\tau, \sigma})^{-1}
[D_{\mathcal{F}^{\tau, \sigma}}, [[D_{\mathcal{F}^{\tau, \sigma}}, \mathcal{K}^{\tau, \sigma}],
\mathring{\mathcal{K}}^{\tau, \sigma}](1)] T^{\tau, \sigma}(f)
\label{00 sixth}
\end{align}
We need to show that the sum of (\ref{00 first}) to (\ref{00 sixth}) is zero.
The sum of the third term of (\ref{00 second}) and the second term of (\ref{00 first}) is
\begin{align}
&(\mathring{T}^{\tau, \sigma})^{-1}
[[[D_{\mathcal{F}^{\tau, \sigma}}, D_{\mathcal{F}^{\tau, \sigma}}(\mathring{\mathcal{K}}^{\tau, \sigma})],
\mathcal{K}^{\tau, \sigma}],
T^{\tau, \sigma}(f)](1) \notag\\
&- (\mathring{T}^{\tau, \sigma})^{-1}
[[[D_{\mathcal{F}^{\tau, \sigma}}, D_{\mathcal{F}^{\tau, \sigma}}(\mathcal{K}^{\tau, \sigma})],
\mathring{\mathcal{K}}^{\tau, \sigma}],
T^{\tau, \sigma}(f)](1) \notag\\
&= (\mathring{T}^{\tau, \sigma})^{-1}
([[D_{\mathcal{F}^{\tau, \sigma}}, D_{\mathcal{F}^{\tau, \sigma}}(\mathring{\mathcal{K}}^{\tau, \sigma})],
\mathcal{K}^{\tau, \sigma}] \notag\\
& \hphantom{= (\mathring{T}^{\tau, \sigma})^{-1}
(}
- [[D_{\mathcal{F}^{\tau, \sigma}}, D_{\mathcal{F}^{\tau, \sigma}}(\mathcal{K}^{\tau, \sigma})],
\mathring{\mathcal{K}}^{\tau, \sigma}]) T^{\tau, \sigma}(f) \notag\\
&\quad - (\mathring{T}^{\tau, \sigma})^{-1}
\bigl(T^{\tau, \sigma}(f)
([[D_{\mathcal{F}^{\tau, \sigma}}, D_{\mathcal{F}^{\tau, \sigma}}(\mathring{\mathcal{K}}^{\tau, \sigma})],
\mathcal{K}^{\tau, \sigma}] \notag\\
&\quad \hphantom{- (\mathring{T}^{\tau, \sigma})^{-1}
\bigl(T^{\tau, \sigma}(f)
(}
- [[D_{\mathcal{F}^{\tau, \sigma}}, D_{\mathcal{F}^{\tau, \sigma}}(\mathcal{K}^{\tau, \sigma})],
\mathring{\mathcal{K}}^{\tau, \sigma}])(1) \bigr) \notag\\
&= (\mathring{T}^{\tau, \sigma})^{-1}
(- [D_{\mathcal{F}^{\tau, \sigma}}(\mathring{\mathcal{K}}^{\tau, \sigma}), [D_{\mathcal{F}^{\tau, \sigma}},
\mathcal{K}^{\tau, \sigma}]] \notag\\
&\quad \hphantom{(\mathring{T}^{\tau, \sigma})^{-1}
(}
+ [D_{\mathcal{F}^{\tau, \sigma}}(\mathcal{K}^{\tau, \sigma}), [D_{\mathcal{F}^{\tau, \sigma}},
\mathring{\mathcal{K}}^{\tau, \sigma}]]) T^{\tau, \sigma}(f) \notag\\
&\quad - (\mathring{T}^{\tau, \sigma})^{-1} T^{\tau, \sigma}(f)
([D_{\mathcal{F}^{\tau, \sigma}}, D_{\mathcal{F}^{\tau, \sigma}}(\mathring{\mathcal{K}}^{\tau, \sigma})]
(\mathcal{K}^{\tau, \sigma}) \notag\\
&\quad \hphantom{- (\mathring{T}^{\tau, \sigma})^{-1} T^{\tau, \sigma}(f)
(}
- [D_{\mathcal{F}^{\tau, \sigma}}, D_{\mathcal{F}^{\tau, \sigma}}(\mathcal{K}^{\tau, \sigma})]
(\mathring{\mathcal{K}}^{\tau, \sigma})) \notag\\
&= (\mathring{T}^{\tau, \sigma})^{-1}
([D_{\mathcal{F}^{\tau, \sigma}}(\mathcal{K}^{\tau, \sigma}), [D_{\mathcal{F}^{\tau, \sigma}},
\mathring{\mathcal{K}}^{\tau, \sigma}]] \notag\\
&\quad \hphantom{(\mathring{T}^{\tau, \sigma})^{-1}
(}
- [D_{\mathcal{F}^{\tau, \sigma}}(\mathring{\mathcal{K}}^{\tau, \sigma}), [D_{\mathcal{F}^{\tau, \sigma}},
\mathcal{K}^{\tau, \sigma}]] ) T^{\tau, \sigma}(f) \notag\\
&\quad - (\mathring{T}^{\tau, \sigma})^{-1} T^{\tau, \sigma}(f)
D_{\mathcal{F}^{\tau, \sigma}}[[D_{\mathcal{F}^{\tau, \sigma}},
\mathcal{K}^{\tau, \sigma}], \mathring{\mathcal{K}}^{\tau, \sigma}](1)
\end{align}
The sum of the fourth term of (\ref{00 fifth}) and (\ref{00 third}) is
\begin{align}
&(\mathring{T}^{\tau, \sigma})^{-1} [[D_{\mathcal{F}^{\tau, \sigma}}, \mathcal{K}^{\tau, \sigma}],
[[D_{\mathcal{F}^{\tau, \sigma}}, \mathring{\mathcal{K}}^{\tau, \sigma}], T^{\tau, \sigma}(f)](1)](1) \notag\\
&- (\mathring{T}^{\tau, \sigma})^{-1} [[D_{\mathcal{F}^{\tau, \sigma}}, \mathring{\mathcal{K}}^{\tau, \sigma}],
[[D_{\mathcal{F}^{\tau, \sigma}}, \mathcal{K}^{\tau, \sigma}], T^{\tau, \sigma}(f)](1)](1) \notag\\
&= (\mathring{T}^{\tau, \sigma})^{-1} [[D_{\mathcal{F}^{\tau, \sigma}}, \mathcal{K}^{\tau, \sigma}],
[D_{\mathcal{F}^{\tau, \sigma}}, \mathring{\mathcal{K}}^{\tau, \sigma}](T^{\tau, \sigma}(f))](1) \notag\\
&\quad
- (\mathring{T}^{\tau, \sigma})^{-1} [[D_{\mathcal{F}^{\tau, \sigma}}, \mathcal{K}^{\tau, \sigma}],
D_{\mathcal{F}^{\tau, \sigma}}(\mathring{\mathcal{K}}^{\tau, \sigma}) T^{\tau, \sigma}(f)](1) \notag\\
&\quad
- (\mathring{T}^{\tau, \sigma})^{-1} [[D_{\mathcal{F}^{\tau, \sigma}}, \mathring{\mathcal{K}}^{\tau, \sigma}],
[D_{\mathcal{F}^{\tau, \sigma}}, \mathcal{K}^{\tau, \sigma}](T^{\tau, \sigma}(f))](1) \notag\\
&\quad
+ (\mathring{T}^{\tau, \sigma})^{-1} [[D_{\mathcal{F}^{\tau, \sigma}}, \mathring{\mathcal{K}}^{\tau, \sigma}],
D_{\mathcal{F}^{\tau, \sigma}}(\mathcal{K}^{\tau, \sigma}) T^{\tau, \sigma}(f)](1) \notag\\
&= (\mathring{T}^{\tau, \sigma})^{-1} [[D_{\mathcal{F}^{\tau, \sigma}}, \mathcal{K}^{\tau, \sigma}],
[D_{\mathcal{F}^{\tau, \sigma}}, \mathring{\mathcal{K}}^{\tau, \sigma}]] T^{\tau, \sigma}(f) \notag\\
&\quad - (\mathring{T}^{\tau, \sigma})^{-1}
\bigl([D_{\mathcal{F}^{\tau, \sigma}}(\mathcal{K}^{\tau, \sigma}), [D_{\mathcal{F}^{\tau, \sigma}},
\mathring{\mathcal{K}}^{\tau, \sigma}]] \notag\\
&\quad - [D_{\mathcal{F}^{\tau, \sigma}}(\mathring{\mathcal{K}}^{\tau, \sigma}), [D_{\mathcal{F}^{\tau, \sigma}},
\mathcal{K}^{\tau, \sigma}]] \bigr) T^{\tau, \sigma}(f)
\end{align}
The sum of the second term of (\ref{00 second}) and the first term of (\ref{00 first}) is
\begin{align}
& (\mathring{T}^{\tau, \sigma})^{-1}
[[D_{\mathcal{F}^{\tau, \sigma}}, \partial_\sigma \mathcal{K}^{\tau, \sigma}],
T^{\tau, \sigma}(f)](1) \notag\\
& \quad
- (\mathring{T}^{\tau, \sigma})^{-1}
[[D_{\mathcal{F}^{\tau, \sigma}}, \partial_\tau \mathring{\mathcal{K}}^{\tau, \sigma}], T^{\tau, \sigma}(f)](1)
\notag\\
&= (\mathring{T}^{\tau, \sigma})^{-1}
[[D_{\mathcal{F}^{\tau, \sigma}}, \partial_\sigma \mathcal{K}^{\tau, \sigma}
- \partial_\tau \mathring{\mathcal{K}}^{\tau, \sigma}],
T^{\tau, \sigma}(f)](1)
\notag\\
&= - (\mathring{T}^{\tau, \sigma})^{-1}
[[D_{\mathcal{F}^{\tau, \sigma}}, [[D_{\mathcal{F}^{\tau, \sigma}}, \mathcal{K}^{\tau, \sigma}],
\mathring{\mathcal{K}}^{\tau, \sigma}](1)], T^{\tau, \sigma}(f)](1)
\quad (\text{by (\ref{exactness of K})})
\notag\\
&= - (\mathring{T}^{\tau, \sigma})^{-1}
[D_{\mathcal{F}^{\tau, \sigma}}, [[D_{\mathcal{F}^{\tau, \sigma}}, \mathcal{K}^{\tau, \sigma}],
\mathring{\mathcal{K}}^{\tau, \sigma}](1)]T^{\tau, \sigma}(f) \notag\\
&\quad
+ (\mathring{T}^{\tau, \sigma})^{-1} T^{\tau, \sigma}(f)
D_{\mathcal{F}^{\tau, \sigma}}([[D_{\mathcal{F}^{\tau, \sigma}}, \mathcal{K}^{\tau, \sigma}],
\mathring{\mathcal{K}}^{\tau, \sigma}](1))
\end{align}
Therefore the sum of (\ref{00 first}) to (\ref{00 sixth}) is zero.
(iv)
We prove the existence of $A^{-, \tau}$.
Since
\begin{align*}
&\frac{d}{d \tau} (\mathring{T}^\tau)^{-1} \circ i_{\mathcal{F}^\tau}^-(f) \\
&= (\mathring{T}^\tau)^{-1} [f \underset{\mathcal{F}^\tau}{\overrightarrow{\ast}},
D_{\mathcal{F}^\tau}(\mathcal{K}^\tau)](1)
- (\mathring{T}^\tau)^{-1}
[[D_{\mathcal{F}^\tau}, \mathcal{K}^\tau], i_{\mathcal{F}^\tau}^-(f)](1) \\
&= - (\mathring{T}^\tau)^{-1} [[D_{\mathcal{F}^\tau}, \mathcal{K}^\tau],
f \underset{\mathcal{F}^\tau}{\overrightarrow{\ast}}](1) \\
&= - (T^\tau)^{-1}
[D_{\mathcal{F}^\tau}, [\mathcal{K}^\tau, f \underset{\mathcal{F}^\tau}{\overrightarrow{\ast}}]](1)
- (T^\tau)^{-1}
[\mathcal{K}^\tau, [D_{\mathcal{F}^\tau}, f \underset{\mathcal{F}^\tau}{\overrightarrow{\ast}}]](1) \\
&= - (T^\tau)^{-1} D_{\mathcal{F}^\tau}
\bigl([\mathcal{K}^\tau, f \underset{\mathcal{F}^\tau}{\overrightarrow{\ast}}](1)\bigr)
- (T^\tau)^{-1}
[\mathcal{K}^\tau, (D_{Y^-} f) \underset{\mathcal{F}^\tau}{\overrightarrow{\ast}}](1),
\end{align*}
(\ref{A chain homotopy}) is satisfied for
$A^{-, \tau} : \mathcal{W}^{\leq \kappa}_{Y^-}
/ I^{\leq \kappa}_{C_0, C_1 + \kappa(\delta^{-1} - L^{-1}_{\min}), C_2}
\to \mathcal{D}^{\leq \kappa}_X / J^{\leq \kappa, \delta}_{C_0, C_1, C_2}$ defined by
\[
A^{-, \tau}(h) = - \int_0^\tau (T^{\tau'})^{-1}
[\mathcal{K}^{\tau'}, h \underset{\mathcal{F}^{\tau'}}{\overrightarrow{\ast}}](1) d\tau'.
\]
We can similarly construct a family of chain homotopies
$A^{+, \tau}$.
\end{proof}
Next we consider the case of rational SFT.
Equation (\ref{main eq for X^I}) implies that the two families of
functions $\mathcal{F}^\tau_0, \mathcal{K}^\tau_0 \in
\mathcal{L}_X^{\leq 0} / J^{\leq 0}_{\overline{C}_0, \overline{C}_2}$ satisfy
\[
\frac{d}{d\tau} \mathcal{F}^\tau_0 = \delta \mathcal{K}^\tau_0
- \{\mathbf{h}, \mathcal{K}^\tau_0\}|_{\mathcal{F}^\tau_0}
\ (= d_{\mathcal{F}^\tau_0} \mathcal{K}^\tau_0)
\]
in $\mathcal{L}_X^{\leq 0}
/ J^{\leq 0}_{\overline{C}_0, \overline{C}_2}$, where
$\mathbf{h} = \mathcal{H}_{Y^-, 0} - \mathcal{H}_{Y^+, 0}$.
Namely, the family of functions $\mathcal{F}^\tau_0$ is a homotopy in the following sense.
\begin{defi}
One-parameter family of functions $F^\tau_0 \in \mathcal{L}_X^{\leq 0}
/ J^{\leq 0}_{\overline{C}_0, \overline{C}_2}$ ($\tau \in [0, 1]$)
of even degree is said to be a homotopy
if (\ref{main eq for X rational}) holds for all $\mathcal{F}_0 = \mathcal{F}^\tau_0$ and
there exists a family of functions $\mathcal{K}^\tau_0 \in \mathcal{L}_X^{\leq 0}
/ J^{\leq 0}_{\overline{C}_0, \overline{C}_2}$ of odd degree
which makes the following equation hold for all $\tau \in [0, 1]$.
\begin{equation}
\frac{d}{d\tau} \mathcal{F}^\tau_0 = d_{\mathcal{F}^\tau_0} \mathcal{K}^\tau_0 \label{main eq for X^I rational}
\end{equation}
\end{defi}
For each triple $(\kappa, C_0, C_2)$ such that $\overline{C}_0 \geq C_0$ and
$\overline{C}_2 \geq C_2 + \kappa$,
define a flow $T_0^\tau : \mathcal{L}^{\leq \kappa}_X / J^{\leq \kappa}_{C_0, C_2}
\to \mathcal{L}^{\leq \kappa}_X / J^{\leq \kappa}_{C_0, C_2}$ by
\[
\frac{d}{d\tau} T^\tau_0(f) = -\{\{\mathbf{h}, \mathcal{K}^\tau_0\},
T^\tau_0(f)\}|_{\mathcal{F}^\tau_0}.
\]
(It is related to the flow $T^\tau$ by $T^\tau_0(f) = T^\tau(f)|_{\hbar = 0}$.)
Then the following hold true as in the case of general SFT.
\begin{lem}
\begin{enumerate}[label = \normalfont (\roman*)]\
\item
$T^\tau_0$ is a chain map from
$(\mathcal{L}^{\leq \kappa}_X / J^{\leq \kappa}_{C_0, C_2}, d_{\mathcal{F}^0_0})$ to
$(\mathcal{L}^{\leq \kappa}_X / J^{\leq \kappa}_{C_0, C_2}, d_{\mathcal{F}^\tau_0})$
for each $\tau$.
Furthermore, up to chain homotopy, it is determined by
$(\mathcal{F}^{\tau'}_0)_{\tau' \in [0, \tau]}$
and independent of the choice of the family $(\mathcal{K}^{\tau'}_0)_{\tau' \in [0, \tau]}$
which satisfies equation $(\ref{main eq for X^I rational})$.
\item
If a smooth family of generating functions $\mathcal{F}^{\tau, \sigma}_0
\in \mathcal{L}_X^{\leq 0} / J^{\leq 0}_{\overline{C}_0, \overline{C}_2}$
$((\tau, \sigma) \in [0, 1] \times [0, 1])$
satisfies $\mathcal{F}^{0, \sigma}_0 \equiv \mathcal{F}^{0, 0}_0$ and the one-parameter family
$(\mathcal{F}^{\tau, \sigma}_0)_{\tau \in [0, 1]}$ is a homotopy for each $\sigma \in [0, 1]$,
then the one-parameter family $(\mathcal{F}^{\tau, \sigma}_0)_{\sigma \in [0, 1]}$ is also
a homotopy for each $\tau \in [0, 1]$.
\item
Further assume that the above family of generating functions satisfies
$\mathcal{F}^{1, \sigma}_0 = \mathcal{F}^{0, 0}_0$ and $\mathcal{F}^{\tau, 0}_0 \equiv \mathcal{F}^{0, 0}_0$.
Let $T^\tau_0 : \mathcal{L}^{\leq \kappa}_X / J^{\leq \kappa}_{C_0, C_2}
\to \mathcal{L}^{\leq \kappa}_X / J^{\leq \kappa}_{C_0, C_2}$
be the flow defined by the one-parameter homotopy $(\mathcal{F}^{\tau, 1}_0)_{\tau \in [0, 1]}$.
Then $T^1_0$ is equal to the identity map up to chain homotopy.
In other words, if a loop homotopy $(\mathcal{F}^{\tau, 1}_0)_{\tau \in S^1}$ is contractible
in the space of loop homotopies with the base point $\mathcal{F}^{0, 1}_0$,
then the chain map $T^1_0$ is the identity map up to chain homotopy.
Hence for a general one-parameter homotopy $(\mathcal{F}^\tau_0)_{\tau \in [0, 1]}$,
the end $T^1_0$ of the family of the chain maps $(T^\tau_0)_{\tau \in [0, 1]}$
is determined up to chain homotopy by the homotopy type of the homotopy
$(\mathcal{F}^\tau_0)_{\tau \in [0, 1]}$ relative to the end points.
\item
There exists a family of linear maps
$A^{\pm, \tau}_0 : \mathcal{P}_Y^{\leq \kappa} / I^{\leq \kappa}_{C_0, C_2}
\to \mathcal{L}^{\leq \kappa}_X / J^{\leq \kappa}_{C_0, C_2}$
such that
\begin{equation}
i_{\mathcal{F}^0_0}^\pm - (T^\tau_0)^{-1} \circ i_{\mathcal{F}^\tau_0}^\pm = d_{\mathcal{F}^0_0} \circ A^{\pm, \tau}_0
+ A^{\pm, \tau}_0 \circ d_{Y^\pm},
\label{A chain homotopy rational}
\end{equation}
that is, the following diagrams are commutative up to chain homotopy.
\[
\begin{tikzcd}
(\mathcal{L}^{\leq \kappa}_X / J^{\leq \kappa}_{C_0, C_2}, d_{\mathcal{F}^0_0}) \ar{rr}{T^\tau_0}
& & (\mathcal{L}^{\leq \kappa}_X / J^{\leq \kappa}_{C_0, C_2}, d_{\mathcal{F}^\tau_0})\\
&(\mathcal{P}_Y^{\leq \kappa} / I^{\leq \kappa}_{C_0, C_2}, d_{Y^\pm})
\ar{ul}{i_{\mathcal{F}^0_0}^\pm} \ar{ur}[swap]{i_{\mathcal{F}^\tau_0}^\pm}&
\end{tikzcd}
\]
\end{enumerate}
\end{lem}
Finally, we consider the case of contact homology.
Define
\[
\widehat{\mathcal{K}}^\tau_0 :=
\sum \frac{\overleftarrow{\partial} \mathcal{K}^\tau_0}{\partial p_{\hat c}^+} \bigg|_{p^+=0} \cdot
p^+_{\hat c}
\in \mathcal{L}_X^{\leq 0} / J^{\leq 0}_{\overline{C}_0, \overline{C}_2}.
\]
Then $\widehat{\mathcal{F}}^\tau_0$ and $\widehat{\mathcal{K}}^\tau_0$ satisfy
\begin{gather*}
\frac{d}{d\tau} \widehat{\mathcal{F}}^\tau_0 =
\delta \widehat{\mathcal{K}}^\tau_0
- \{\widehat{\mathbf{h}},
\widehat{\mathcal{K}}^\tau_0\} |_{\widehat{\mathcal{F}}^\tau_0},\\
\delta \widehat{\mathcal{F}}^\tau_0
= \widehat{\mathbf{h}}
|_{\widehat{\mathcal{F}}^\tau_0},
\end{gather*}
where $\widehat{\mathbf{h}}
= \widehat{\mathcal{H}}_{Y^-, 0} - \widehat{\mathcal{H}}_{Y^+, 0}$.
For pairs $(\kappa, C_0)$ such that $\overline{C}_0 \geq C_0$ and
$\overline{C}_2 \geq \kappa$, define linear maps
$\Delta^\tau : \mathcal{A}^{\leq \kappa}_{Y^+} / I^{\leq \kappa}_{C_0}
\to \mathcal{A}^{\leq \kappa}_{Y^-} / I^{\leq \kappa}_{C_0}$ by
\[
\Delta^\tau(f) = - \int_0^\tau \{\widehat{\mathcal{K}}^s_0, f\}|
_{\widehat{\mathcal{F}}^s_0} ds.
\]
Then the above equations imply that these are chain homotopies from $\Psi^\tau
= \Psi_{\widehat{\mathcal{F}}^\tau_0}$
to $\Psi^0$, that is,
\[
\Psi^\tau - \Psi^0 = \partial_{Y^-} \circ \Delta^\tau + \Delta^\tau \circ \partial_{Y^+}.
\]
In fact, $\widehat{\mathcal{F}}^\tau_0$ and $\widehat{\mathcal{K}}^\tau_0$
give a DGA homotopy in the sense that the following maps satisfy
the conditions of DGA homomorphism.
\begin{align}
\Psi : (\mathcal{A}^{\leq \kappa}_{Y^+} / I^{\leq \kappa}_{C_0}, \partial_{Y^+})
&\to (\Omega^\ast(I), d) \otimes
(\mathcal{A}^{\leq \kappa}_{Y^-} / I^{\leq \kappa}_{C_0}, \partial_{Y^-}) \label{DGA homotopy?}\\
f &\mapsto f|_{\widehat{\mathcal{F}}^\tau_0}
- d \tau \otimes \{\widehat{\mathcal{K}}^\tau_0, f\}|_{\widehat{\mathcal{F}}^\tau_0}
\notag
\end{align}
More precisely, $\Psi : \mathcal{A}^{\leq \kappa}_{Y^+} / I^{\leq \kappa}_{C_0} \to
\Omega^\ast_{C^N}(I) \otimes \mathcal{A}^{\leq \kappa}_{Y^-} / I^{\leq \kappa}_{C_0}$
is a linear map which satisfies
\[
(d \otimes 1 + (-1)^\ast \otimes \partial_{Y^-}) \Psi(f)
= \Psi(\partial_{Y^+} f)
\]
for $f \in \mathcal{A}^{\leq \kappa}_{Y^+} / I^{\leq \kappa}_{C_0}$, and
\[
\Psi(fg) = \Psi (f) \Psi(g)
\]
in $\Omega^\ast_{C^N}(I) \otimes \mathcal{A}^{\leq \kappa_1 + \kappa_2}_{Y^-}
/ I^{\leq \kappa_1 + \kappa_2}_{C_0}$
for all $f \in \mathcal{A}^{\leq \kappa_1}_{Y^+} / I^{\leq \kappa_1}_{C_0}$ and
$g \in \mathcal{A}^{\leq \kappa_2}_{Y^+} / I^{\leq \kappa_2}_{C_0}$
if $\overline{C}_0 \geq C_0$ and $\overline{C}_2 \geq \kappa_1 + \kappa_2$.
($\widehat{\mathcal{F}}^\tau_0$ is not of class $C^\infty$, but
$(\Omega^\ast(I), d)$ is a DGA of differential forms of class $C^\infty$.
Hence (\ref{DGA homotopy?}) is not strictly a DGA homomorphism.)
\section{Composition}
\label{composition}
Let $X^- = (-\infty, 0] \times Y^- \cup Z^- \cup [0, \infty) \times Y^0$ and
$X^+ = (-\infty, 0] \times Y^0 \cup Z^+ \cup [0, \infty) \times Y^+$ be two
symplectic manifolds with cylindrical ends.
We regard them as symplectic cobordisms.
Then their composition $X = X^- \# X^+$ is defined by
\[
X = (-\infty, 0] \times Y^- \cup Z^- \cup Z^+ \cup [0, \infty) \times Y^+.
\]
Let $K_{X}^0$ be the set of cycles consisting of
\begin{itemize}
\item
cycles $x$ in $K_{X^-}^0$ such that $\mathop{\mathrm{supp}}\nolimits x \cap [0, \infty) \times Y^0
= \emptyset$,
\item
cycles $x$ in $K_{X^+}^0$ such that $\mathop{\mathrm{supp}}\nolimits x \cap (-\infty, 0] \times Y^0
= \emptyset$ and
\item
the cycles $x = x^- \# x^+$ obtained by the sums of the restrictions of
cycles $x^-$ in $K_{X^-}^0$ to $(-\infty, 0] \times Y^- \cup Z^-$
and the restrictions of cycles $x^+$ in $K_{X^+}^0$ to
$Z^+ \cup [0, \infty) \times Y^+$
corresponding to the same cycles $y$ in $K_{Y^0}^0$.
\end{itemize}
In this section, we prove that the composition of symplectic cobordisms corresponds to
the composition of the algebras.
First in Section \ref{composition of generating functions}, we recall the composition
of generating functions, and in Section \ref{composition of cobordisms},
we prove that the generating function of $X$ is homotopic to the composition of the
generating functions of $X^-$ and $X^+$.
In Section \ref{correction terms for composition}, we construct the correction terms
needed for Section \ref{composition of cobordisms}.
\subsection{Composition of generating functions}
\label{composition of generating functions}
In this section, we recall the definition of the composition of generating functions of
$X^-$ and $X^+$ and its linearizations defined in \cite{EGH00}.
First we consider the case of general SFT.
The composition map $\star : \mathcal{D}\D_{X^-} \otimes \mathcal{D}\D_{X^+}
\to \mathcal{D}\D_{X, Y^0}$ is defined by
\[
f \star g = (\overrightarrow{f} g)|_{q^0_{\hat c^\ast} = 0 \text{ for } c \in K_{Y^0}},
\]
where $\overrightarrow{f}$ is the differential operator obtained from $f$ by replacing
the variables $p^0_{\hat c}$ ($c \in K_{Y^0}$) with
$\hbar \overrightarrow{\frac{\partial}{\partial q^0_{\hat c^\ast}}}$,
and we replace the variables $t_{x^-}$ in $f$ and $t_{x^+}$ in $g$ with $t_{x^- \# t_x+}$.
(We denote the two variables corresponding to each simplex $c$ of $K_{Y^0}$
by $p^0_{\hat c}$ and $q^0_{\hat c^\ast}$.)
In the above definition, we regard $A \in \tilde \omega_{X^-}
H_2(\overline{X^-}, \partial \overline{X^-}; \mathbb{Z}) \cong
H_2(\overline{X^-}, \partial \overline{X^-}; \mathbb{Z}) / \mathop{\mathrm{Ker}}\nolimits \tilde \omega_{X^-}$
of the variables $T^A$ appearing in $f$
and $B \in \tilde \omega_{X^+}
H_2(\overline{X^+}, \partial \overline{X^+}; \mathbb{Z}) \cong
H_2(\overline{X^+}, \partial \overline{X^+}; \mathbb{Z}) / \mathop{\mathrm{Ker}}\nolimits \tilde \omega_{X^+}$ of the
variable $T^B$ in $g$
as elements of $H_2(\overline{X}, \partial{X}; \mathbb{Z}) \big/ (\mathop{\mathrm{Ker}}\nolimits e \cap \mathop{\mathrm{Ker}}\nolimits e_{Y^0})$
by the isomorphism
\begin{align*}
&H_2(\overline{X}, \partial \overline{X}) \big/ (\mathop{\mathrm{Ker}}\nolimits e \cap \mathop{\mathrm{Ker}}\nolimits e_{Y^0})\\
& \cong H_2(\overline{X}, Y^0 \cup \partial \overline{X}) \big/
(\mathop{\mathrm{Ker}}\nolimits e \cap \mathop{\mathrm{Ker}}\nolimits e_{Y^0})\\
& \cong H_2(Z^-, \partial Z^-) / \mathop{\mathrm{Ker}}\nolimits \tilde \omega_{X^-} \oplus
H_2(Z^+, \partial Z^+) / \mathop{\mathrm{Ker}}\nolimits \tilde \omega_{X^+}\\
& \cong H_2(\overline{X^-}, \partial \overline{X^-}) / \mathop{\mathrm{Ker}}\nolimits \tilde \omega_{X^-} \oplus
H_2(\overline{X^+}, \partial \overline{X^+}) / \mathop{\mathrm{Ker}}\nolimits \tilde \omega_{X^+}.
\end{align*}
(In the above equation, we use $\tilde \omega_{X^+} = e_{Y^0}$ and $\tilde \omega_{X^-}
+ \tilde \omega_{X^+} = e$.)
Note that the above composition map induces maps
\begin{multline*}
\star : \mathcal{D}\D_{X^-}^{\leq \kappa_1, \delta}
/ \widetilde{J}^{\leq \kappa_1, \delta}_{C_0, C_1 + \kappa_2 \delta^{-1}, C_2 + \kappa_2}
\otimes \mathcal{D}\D_{X^+}^{\leq \kappa_2, \delta}
/ \widetilde{J}^{\leq \kappa_2, \delta}_{C_0, C_1 + \kappa_1 \delta^{-1}, C_2}\\
\to \mathcal{D}\D_X^{\leq \kappa_1 + \kappa_2, \delta}
/ \widetilde{J}^{\leq \kappa_1 + \kappa_2, \delta}_{C_0, C_1, C_2}.
\end{multline*}
The composition
\[
\mathcal{F}^- \Diamond \mathcal{F}^+ \in (\hbar^{-1} \mathcal{D}_X^{\leq 0})^{\star, \delta}
/ J^{\star, \delta}_{\overline{C}_0, \overline{C}_1, \overline{C}_2}
\]
of generating functions
$\mathcal{F}^\pm \in (\hbar^{-1} \mathcal{D}_{X^\pm}^{\leq 0})^{\star, \delta}
/ J^{\star, \delta}_{\overline{C}_0, \overline{C}_1, \overline{C}_2}$ of $X^\pm$
are defined by
\[
e^{\mathcal{F}^- \Diamond \mathcal{F}^+} = e^{\mathcal{F}^-} \star e^{\mathcal{F}^+}
\]
in $\mathcal{D}\D_X^{\leq 0, \delta}
/ \widetilde{J}^{\leq 0, \delta}_{\overline{C}_0, \overline{C}_1, \overline{C}_2}$.
Then equations (\ref{main eq for X}) for $\mathcal{F}^\pm$ imply
\[
\widehat{D}_X (e^{\mathcal{F}^- \Diamond \mathcal{F}^+}) = 0.
\]
in $\mathcal{D}\D_X^{\leq 0, \delta}
/ \widetilde{J}^{\leq 0, \delta}_{\overline{C}_0, \overline{C}_1, \overline{C}_2}$.
In fact, any
$f^- \in \mathcal{D}\D_{X^-}^{\leq \kappa_1, \delta}
/ \widetilde{J}^{\leq \kappa_1, \delta}_{C_0, C_1 + \kappa_2 \delta^{-1}, C_2 + \kappa_2}$
and $f^+ \in \mathcal{D}\D_{X^+}^{\leq \kappa_2, \delta}
/ \widetilde{J}^{\leq \kappa_2, \delta}_{C_0, C_1 + \kappa_1 \delta^{-1}, C_2}$
satisfy
\begin{equation}
\widehat{D}_{X} (f^- \star f^+)
= (\widehat{D}_{X^-} f^-) \star f^+ + (-1)^{|f^-|} f^- \star (\widehat{D}_{X^+} f^+)
\label{star eq}
\end{equation}
in $\mathcal{D}\D_X^{\leq \kappa_1 + \kappa_2, \delta}
/ \widetilde{J}^{\leq \kappa_1 + \kappa_2, \delta}_{C_0, C_1, C_2}$.
More generally,
if $X^\pm$ contains contact manifolds $(Y^\pm_i, \lambda^\pm_i)$ as in
Section \ref{algebras with further energy conditions}, then
for $\mathcal{F}^\pm \in (\hbar^{-1} \mathcal{D}_{X^\pm, (Y^\pm_i)}^{\leq 0})^{\star, \delta}
/ J^{\star, \delta}_{\overline{C}_0, \overline{C}_1, \overline{C}_2}$,
we can define the composition
$\mathcal{F}^- \Diamond \mathcal{F}^+ \in (\hbar^{-1} \mathcal{D}_{X, (Y^-_i, Y^0, Y^+_i)}^{\leq 0})^{\star, \delta}
/ J^{\star, \delta}_{\overline{C}_0, \overline{C}_1, \overline{C}_2}$.
Define linear maps
$T_{\mathcal{F}^-}(\cdot \Diamond \mathcal{F}^+)
: \mathcal{D}^{\leq \kappa}_{X^-} / J^{\leq \kappa, \delta}_{C_0, C_1, C_2} \to
\mathcal{D}^{\leq \kappa}_{X, Y^0} / J^{\leq \kappa, \delta}_{C_0, C_1, C_2}$
and $T_{\mathcal{F}^+}(\mathcal{F}^- \Diamond \cdot) : \mathcal{D}^{\leq \kappa}_{X^+}
/ J^{\leq \kappa, \delta}_{C_0, C_1, C_2} \to
\mathcal{D}^{\leq \kappa}_{X, Y^0} / J^{\leq \kappa, \delta}_{C_0, C_1, C_2}$ by
\begin{align*}
T_{\mathcal{F}^-}( \cdot \Diamond \mathcal{F}^+)(f)
&= e^{- \mathcal{F}^- \Diamond \mathcal{F}^+}((f e^{\mathcal{F}^-}) \star e^{\mathcal{F}^+})\\
T_{\mathcal{F}^+}(\mathcal{F}^- \Diamond \cdot)(f)
&= e^{- \mathcal{F}^- \Diamond \mathcal{F}^+}(e^{\mathcal{F}^-} \star (f e^{\mathcal{F}^+})).
\end{align*}
(These are the linearizations of the composition map.)
We also define a map
\begin{align*}
&T^2_{\mathcal{F}^-, \mathcal{F}^+} : \mathcal{D}^{\leq \kappa_1}_{X^-}
/ J^{\leq \kappa_1, \delta}_{C_0, C_1 + \kappa_2 \delta^{-1}, C_2 + \kappa_2}
\otimes \mathcal{D}^{\leq \kappa_2}_{X^+}
/ J^{\leq \kappa_2, \delta}_{C_0, C_1 + \kappa_1 \delta^{-1}, C_2} \\
&\hspace{170pt}
\to \mathcal{D}^{\leq \kappa_1 + \kappa_2}_{X, Y^0}
/ J^{\leq \kappa_1 + \kappa_2, \delta}_{C_0, C_1, C_2}
\end{align*}
by
\[
T^2_{\mathcal{F}^-, \mathcal{F}^+}(f \otimes g) = e^{- \mathcal{F}^- \Diamond \mathcal{F}^+} ((f e^{\mathcal{F}^-}) \star (g e^{\mathcal{F}^+})).
\]
Note that $T_{\mathcal{F}^-}(\cdot \Diamond \mathcal{F}^+)(f) = T^2_{\mathcal{F}^-, \mathcal{F}^+}(f \otimes 1)$
and $T_{\mathcal{F}^+}(\mathcal{F}^- \Diamond \cdot)(f) = T^2_{\mathcal{F}^-, \mathcal{F}^+}(1 \otimes f)$.
Some of the following properties of these maps were proved in \cite{EGH00}.
\begin{lem}\label{linearized T}
The linearizations of the composition map satisfy the following.
\begin{enumerate}[label=\normalfont(\roman*)]
\item
They are chain maps, that is,
\begin{align*}
T_{\mathcal{F}^-}(\cdot \Diamond \mathcal{F}^+) \circ D_{\mathcal{F}^-}
&= D_{\mathcal{F}^- \Diamond \mathcal{F}^+} \circ T_{\mathcal{F}^-}(\cdot \Diamond \mathcal{F}^+),\\
T_{\mathcal{F}^+}(\mathcal{F}^- \Diamond \cdot) \circ D_{\mathcal{F}^+}
&= D_{\mathcal{F}^- \Diamond \mathcal{F}^+} \circ T_{\mathcal{F}^+}(\mathcal{F}^- \Diamond \cdot).
\end{align*}
More generally,
\[
T^2_{\mathcal{F}^-, \mathcal{F}^+} \circ (D_{\mathcal{F}^-} \otimes 1 + (-1)^\ast \otimes D_{\mathcal{F}^+})
= D_{\mathcal{F}^- \Diamond \mathcal{F}^+} \circ T^2_{\mathcal{F}^-, \mathcal{F}^+}.
\]
\item
They satisfy the following compatibility conditions with $i_{\mathcal{F}^\pm}^\pm$ and
$i_{\mathcal{F}^- \Diamond \mathcal{F}^+}^\pm$.
{\belowdisplayskip=0pt
\begin{multline*}
T_{\mathcal{F}^-}(\cdot \Diamond \mathcal{F}^+) \circ i_{\mathcal{F}^-}^- = i_{\mathcal{F}^- \Diamond \mathcal{F}^+}^-\\
: \mathcal{W}_{Y^-}^{\leq \kappa}
/ I^{\leq \kappa}_{C_0, C_1 + \kappa(\delta^{-1} - L_{\min}^{-1}), C_2}
\to \mathcal{D}_{X, Y^0}^{\leq \kappa, \delta} / J^{\leq \kappa, \delta}_{C_0, C_1, C_2},
\end{multline*}
}{\abovedisplayskip=0pt
\begin{multline*}
T_{\mathcal{F}^+}(\mathcal{F}^- \Diamond \cdot) \circ i_{\mathcal{F}^+}^+ = i_{\mathcal{F}^- \Diamond \mathcal{F}^+}^+\\
: \mathcal{W}_{Y^+}^{\leq \kappa}
/ I^{\leq \kappa}_{C_0, C_1 + \kappa(\delta^{-1} - L_{\min}^{-1}), C_2}
\to \mathcal{D}_{X, Y^0}^{\leq \kappa, \delta} / J^{\leq \kappa, \delta}_{C_0, C_1, C_2}.
\end{multline*}
}
More generally, they are compatible with the multiplication as follows.
For any $f \in \mathcal{W}_{Y^-}^{\leq \kappa_1} / I^{\leq \kappa}_{C_0, C'_1, C_2 + \kappa_1}$
and $g \in \mathcal{D}_{X^-}^{\leq \kappa_2} / J^{\leq \kappa_2, \delta}_{C_0, C_1
+ \kappa_1 \delta^{-1}, C_2}$,
\[
T_{\mathcal{F}^-}(\cdot \Diamond \mathcal{F}^+) (f \underset{\mathcal{F}^-}{\overrightarrow{\ast}} g)
= f \underset{\mathcal{F}^- \Diamond \mathcal{F}^+}{\overrightarrow{\ast}}
(T_{\mathcal{F}^-}(\cdot \Diamond \mathcal{F}^+) g)
\]
in $\mathcal{D}_{X, Y^0}^{\leq \kappa_1 + \kappa_2, \delta}
/ J^{\leq \kappa_1 + \kappa_2, \delta}_{C_0, C_1, C_2}$,
where $C'_1 = C_1 + \kappa_1 (\delta^{-1} - L_{\min}^{-1}) + \kappa_2 \delta^{-1}$.
For any $g \in \mathcal{D}_{X^+}^{\leq \kappa_2} / J^{\leq \kappa_2, \delta}_{C_0, C_1
+ \kappa_1 \delta^{-1}, C_2 + \kappa_1}$ and
$f \in \mathcal{W}_{Y^+}^{\leq \kappa_1} / I^{\leq \kappa_1}_{C_0, C'_1, C_2}$,
\[
T_{\mathcal{F}^+}(\mathcal{F}^- \Diamond \cdot) (g \underset{\mathcal{F}^+}{\overleftarrow{\ast}} f)
= (T_{\mathcal{F}^+}(\mathcal{F}^- \Diamond \cdot) g)
\underset{\mathcal{F}^- \Diamond \mathcal{F}^+}{\overleftarrow{\ast}} f
\]
in $\mathcal{D}_{X, Y^0}^{\leq \kappa_1 + \kappa_2, \delta}
/ J^{\leq \kappa_1 + \kappa_2, \delta}_{C_0, C_1, C_2}$.
\item
They satisfy the following compatibility condition with $i^{\pm}_{\mathcal{F}^\mp}$.
\begin{align*}
T_{\mathcal{F}^-}(\cdot \Diamond \mathcal{F}^+) \circ i^+_{\mathcal{F}^-}
&= T_{\mathcal{F}^+}(\mathcal{F}^- \Diamond \cdot) \circ i^-_{\mathcal{F}^+}\\
&: \mathcal{W}_{Y^0}^{\leq \kappa}
/ I^{\leq \kappa}_{C_0, C_1 + \kappa(\delta^{-1} - L_{\min}^{-1}), C_2}
\to \mathcal{D}_{X, Y^0}^{\leq \kappa, \delta} / J^{\leq \kappa, \delta}_{C_0, C_1, C_2}
\end{align*}
More generally, they are compatible with the multiplication:
\[
T^2_{\mathcal{F}^-, \mathcal{F}^+}((g \underset{\mathcal{F}^-}{\overleftarrow{\ast}} f) \otimes h)
= T^2_{\mathcal{F}^-, \mathcal{F}^+}(g \otimes (f \underset{\mathcal{F}^+}{\overrightarrow{\ast}} h))
\]
in $\mathcal{D}^{\leq \kappa_1 + \kappa_2 + \kappa_3}_{X, Y^0}
/ J^{\leq \kappa_1 + \kappa_2 + \kappa_3, \delta}_{C_0, C_1, C_2}$
for any
\begin{align*}
f &\in \mathcal{W}_{Y^0}^{\leq \kappa_1}
/ I^{\leq \kappa_1}_{C_0, C_1 + \kappa_1 (\delta^{-1} - L_{\min}^{-1})
+ (\kappa_2 + \kappa_3) \delta^{-1},
C_2 + \kappa_3}, \\
g &\in \mathcal{D}^{\leq \kappa_2}_{X^-} / J^{\leq \kappa_2, \delta}_{C_0, C_1
+ (\kappa_1 + \kappa_3) \delta^{-1}, C_2 + \kappa_1 + \kappa_3}, \\
h &\in \mathcal{D}^{\leq \kappa_3}_{X^+} / J^{\leq \kappa_3, \delta}_{C_0, C_1
+ (\kappa_1 + \kappa_2) \delta^{-1}, C_2}.
\end{align*}
\item
Let $X^i$ ($i = 1, 2, 3$) be symplectic cobordisms from $Y^{i-1}$ to $Y^i$,
and let $\mathcal{F}^i$ be a generating function for each $X^i$.
Then
\begin{align*}
T_{\mathcal{F}^1 \Diamond \mathcal{F}^2}(\cdot \Diamond \mathcal{F}^3) \circ T_{\mathcal{F}^1}(\cdot \Diamond \mathcal{F}^2)
&= T_{\mathcal{F}^1}(\cdot \Diamond (\mathcal{F}^2 \Diamond \mathcal{F}^3)),\\
T_{\mathcal{F}^2 \Diamond \mathcal{F}^3}(\mathcal{F}^1 \Diamond \cdot) \circ T_{\mathcal{F}^3}(\mathcal{F}^2 \Diamond \cdot)
&= T_{\mathcal{F}^3}((\mathcal{F}^1 \Diamond \mathcal{F}^2) \Diamond \cdot).
\end{align*}
More generally,
\[
T^2_{\mathcal{F}^1 \Diamond \mathcal{F}^2, \mathcal{F}^3} \circ (T^2_{\mathcal{F}^1, \mathcal{F}^2} \otimes 1)
= T^2_{\mathcal{F}^1, \mathcal{F}^2 \Diamond \mathcal{F}^3} \circ (1 \otimes T^2_{\mathcal{F}^2, \mathcal{F}^3}).
\]
\item
Let $(\mathcal{F}^{\pm, \tau}, \mathcal{K}^{\pm, \tau})$ be homotopies of generating functions for
$X^\pm$.
Then
\[
(\mathcal{F} ^\tau = \mathcal{F}^{-, \tau} \Diamond \mathcal{F}^{+, \tau},\ \mathcal{K}^\tau
= T^2_{\mathcal{F}^{-, \tau}, \mathcal{F}^{+, \tau}}
(\mathcal{K}^{-, \tau} \otimes 1 + 1 \otimes \mathcal{K}^{+, \tau}))
\]
is a homotopy of generating functions of $X$.
Furthermore, there exist families of linear maps $A^{\pm, \tau}
: \mathcal{D}_{X^\pm}^{\leq \kappa} / J^{\leq \kappa, \delta}_{C_0, C_1, C_2}
\to \mathcal{D}_{X, Y^0}^{\leq \kappa} / J^{\leq \kappa, \delta}_{C_0, C_1, C_2}$
such that
\begin{multline*}
(T^\tau)^{-1} \circ T_{\mathcal{F}^{-, \tau}}(\cdot \Diamond \mathcal{F}^{+, \tau}) \circ T^{-, \tau}
- T_{\mathcal{F}^{-, 0}}(\cdot \Diamond \mathcal{F}^{+, 0})\\
= D_{\mathcal{F}^{-, 0} \Diamond \mathcal{F}^{+, 0}} \circ A^{-, \tau} + A^{-, \tau} \circ D_{\mathcal{F}^{-, 0}},
\end{multline*}
\begin{multline*}
(T^\tau)^{-1} \circ T_{\mathcal{F}^{+, \tau}}(\mathcal{F}^{-, \tau} \Diamond \cdot) \circ T^{+, \tau}
- T_{\mathcal{F}^{+, 0}}(\mathcal{F}^{-, 0} \Diamond \cdot)\\
= D_{\mathcal{F}^{-, 0} \Diamond \mathcal{F}^{+, 0}} \circ A^{+, \tau} + A^{+, \tau} \circ D_{\mathcal{F}^{+, 0}},
\end{multline*}
where $T^{\pm, \tau}$ and $T^\tau$ are the flows for the homotopies
$(\mathcal{F}^{\pm, \tau}, \mathcal{K}^{\pm, \tau})$ and $(\mathcal{F}^\tau, \mathcal{K}^\tau)$ respectively.
Namely, the following diagrams are commutative up to chain homotopy.
\[
\begin{tikzcd}[column sep= huge]
(\mathcal{D}_{X^-}^{\leq \kappa} / J^{\leq \kappa, \delta}_{C_0, C_1, C_2}, D_{\mathcal{F}^{-, 0}})
\ar{r}{T_{\mathcal{F}^{-, 0}}(\cdot \Diamond \mathcal{F}^{+, 0})}
\ar{d}{T^{-, \tau}}
& (\mathcal{D}_{X, Y^0}^{\leq \kappa} / J^{\leq \kappa, \delta}_{C_0, C_1, C_2},
D_{\mathcal{F}^{-, 0} \Diamond \mathcal{F}^{+, 0}}) \ar{d}{T^{\tau}}\\
(\mathcal{D}_{X^-}^{\leq \kappa} / J^{\leq \kappa, \delta}_{C_0, C_1, C_2}, D_{\mathcal{F}^{-, \tau}})
\ar{r}{T_{\mathcal{F}^{-, \tau}}(\cdot \Diamond \mathcal{F}^{+, \tau})}
& (\mathcal{D}_{X, Y^0}^{\leq \kappa} / J^{\leq \kappa, \delta}_{C_0, C_1, C_2},
D_{\mathcal{F}^{-, \tau} \Diamond \mathcal{F}^{+, \tau}})
\end{tikzcd}
\]
\[
\begin{tikzcd}[column sep= huge]
(\mathcal{D}_{X^+}^{\leq \kappa} / J^{\leq \kappa, \delta}_{C_0, C_1, C_2}, D_{\mathcal{F}^{+, 0}})
\ar{r}{T_{\mathcal{F}^{+, 0}}(\mathcal{F}^{-, 0} \Diamond \cdot)}
\ar{d}{T^{+, \tau}}
& (\mathcal{D}_{X, Y^0}^{\leq \kappa} / J^{\leq \kappa, \delta}_{C_0, C_1, C_2},
D_{\mathcal{F}^{-, 0} \Diamond \mathcal{F}^{+, 0}}) \ar{d}{T^{\tau}}\\
(\mathcal{D}_{X^+}^{\leq \kappa} / J^{\leq \kappa, \delta}_{C_0, C_1, C_2}, D_{\mathcal{F}^{+, \tau}})
\ar{r}{T_{\mathcal{F}^{+, \tau}}(\mathcal{F}^{-, \tau} \Diamond \cdot)}
& (\mathcal{D}_{X, Y^0}^{\leq \kappa} / J^{\leq \kappa, \delta}_{C_0, C_1, C_2},
D_{\mathcal{F}^{-, \tau} \Diamond \mathcal{F}^{+, \tau}})
\end{tikzcd}
\]
More generally, there exists a family of linear maps
\begin{multline*}
A^\tau
: \mathcal{D}_{X^-}^{\leq \kappa_1}
/ J^{\leq \kappa_1, \delta}_{C_0, C_1 + \kappa_2 \delta^{-1}, C_2 + \kappa_2}
\otimes \mathcal{D}_{X^+}^{\leq \kappa_2}
/ J^{\leq \kappa_2, \delta}_{C_0, C_1 + \kappa_1 \delta^{-1}, C_2}\\
\to \mathcal{D}_{X, Y^0}^{\leq \kappa_1 + \kappa_2, \delta}
/ J^{\leq \kappa_1 + \kappa_2, \delta}_{C_0, C_1, C_2}
\end{multline*}
such that
\begin{multline*}
(T^\tau)^{-1} \circ T^2_{\mathcal{F}^{-, \tau}, \mathcal{F}^{+, \tau}} \circ (T^{-, \tau} \otimes T^{+, \tau})
- T^2_{\mathcal{F}^{-, 0}, \mathcal{F}^{+, 0}} \\
= D_{\mathcal{F}^{-, 0} \Diamond \mathcal{F}^{+, 0}} \circ A^\tau
+ A^\tau \circ (D_{\mathcal{F}^{-, 0}} \otimes 1 + (-1)^\ast \otimes D_{\mathcal{F}^{+, 0}}).
\end{multline*}
\end{enumerate}
\end{lem}
\begin{proof}
(i) is due to (\ref{star eq}).
(ii), (iii) and (iv) are straightforward.
(v) is proved as follows.
Using (\ref{star eq}), we can easily check that $(\mathcal{F}^\tau, \mathcal{K}^\tau)$ is a homotopy.
We construct $A^\tau$.
For any $f \in \mathcal{D}_{X^-}^{\leq \kappa_1}
/ J^{\leq \kappa_1, \delta}_{C_0, C_1 - \kappa_2 \delta^{-1}, C_2 + \kappa_2}$
and $g \in \mathcal{D}_{X^+}^{\leq \kappa_2}
/ J^{\leq \kappa_2, \delta}_{C_0, C_1 - \kappa_1 \delta^{-1}, C_2}$,
\begin{align*}
&\frac{d}{d \tau} (T^\tau)^{-1} \circ T^2_{\mathcal{F}^{-, \tau}, \mathcal{F}^{+, \tau}} \circ
(T^{-, \tau} \otimes T^{+, \tau})(f \otimes g)\\
&= (T^\tau)^{-1} \circ T^2_{\mathcal{F}^{-, \tau}, \mathcal{F}^{+, \tau}}
([[D_{\mathcal{F}^{-, \tau}}, \mathcal{K}^{-, \tau}], T^{-, \tau}(f)](1) \otimes T^{+, \tau}(g))\\
& \quad + (T^\tau)^{-1} \circ T^2_{\mathcal{F}^{-, \tau}, \mathcal{F}^{+, \tau}}
(T^{-, \tau}(f) \otimes [[D_{\mathcal{F}^{+, \tau}}, \mathcal{K}^{+, \tau}], T^{+, \tau}(g)](1))\\
&\quad + (T^\tau)^{-1} T^2_{\mathcal{F}^{-, \tau}, \mathcal{F}^{+, \tau}}
(D_{\mathcal{F}^{-, \tau}}(\mathcal{K}^{-, \tau}) T^{-, \tau}(f) \otimes T^{+, \tau}(g)) \\
&\quad + (T^\tau)^{-1} T^2_{\mathcal{F}^{-, \tau}, \mathcal{F}^{+, \tau}}
(T^{-, \tau}(f) \otimes D_{\mathcal{F}^{+, \tau}}(\mathcal{K}^{+, \tau}) T^{+, \tau}(g))\\
&\quad - (T^\tau)^{-1} D_{\mathcal{F}^\tau}(\mathcal{K}^\tau)
T^2_{\mathcal{F}^{-, \tau}, \mathcal{F}^{+, \tau}}(T^{-, \tau}(f) \otimes T^{+, \tau}(g))\\
&\quad - (T^\tau)^{-1} [[D_{\mathcal{F}^\tau}, \mathcal{K}^\tau],
T^2_{\mathcal{F}^{-, \tau}, \mathcal{F}^{+, \tau}}(T^\tau(f) \otimes T^{+, \tau}(g))](1)\\
&= (T^\tau)^{-1} T^2_{\mathcal{F}^{-, \tau}, \mathcal{F}^{+, \tau}}
([D_{\mathcal{F}^{-, \tau}}, \mathcal{K}^{-, \tau}] T^{-, \tau}(f) \otimes T^{+, \tau}(g)) \\
&\quad + (T^\tau)^{-1} T^2_{\mathcal{F}^{-, \tau}, \mathcal{F}^{+, \tau}}
(T^{-, \tau}(f) \otimes [D_{\mathcal{F}^{+, \tau}}, \mathcal{K}^{+, \tau}] T^{+, \tau}(g)) \\
&\quad - (T^\tau)^{-1}
[D_{\mathcal{F}^\tau}, \mathcal{K}^\tau](T^2_{\mathcal{F}^{-, \tau}, \mathcal{F}^{+, \tau}}
(T^{-, \tau}(f) \otimes T^{+, \tau}(g))\\
&= D_{\mathcal{F}^0} S^\tau(f) + S^\tau(D_{\mathcal{F}^{-, 0}}f \otimes g
+ (-1)^{|f|} f \otimes D_{\mathcal{F}^{+, 0}}g)\\
\end{align*}
where
\begin{align*}
S^\tau(f \otimes g)
&= (T^\tau)^{-1}\bigl( T^2_{\mathcal{F}^-, \mathcal{F}^+}
(\mathcal{K}^{-, \tau} T^{-, \tau}(f) \otimes T^{+, \tau}(g))\\
&\quad \hphantom{(T^\tau)^{-1}\bigl(}
+ (-1)^{|f|} T^2_{\mathcal{F}^-, \mathcal{F}^+}(T^{-, \tau}(f) \otimes \mathcal{K}^{+, \tau} T^{+, \tau}(g))\\
&\quad \hphantom{(T^\tau)^{-1}\bigl(}
- \mathcal{K}^\tau T^2_{\mathcal{F}^-, \mathcal{F}^+}(T^{-, \tau}(f) \otimes T^{+, \tau}(g)) \bigr).
\end{align*}
Therefore,
\[
A^\tau(f \otimes g) = \int_0^\tau S^{\tau'}(f \otimes g) d \tau'
\]
is a required family of linear maps.
\end{proof}
Next we consider rational SFT.
The composition $\mathcal{F}^-_0 \sharp \mathcal{F}^+_0 \in \mathcal{L}^{\leq 0}_{X, Y^0}
/ J^{\leq 0}_{\overline{C}_0, \overline{C}_2}$ of
generating functions $\mathcal{F}^\pm \in \mathcal{L}^{\leq 0}_{X^\pm}
/ J^{\leq 0}_{\overline{C}_0, \overline{C}_2}$ is defined by
\begin{align*}
\mathcal{F}^-_0 \sharp \mathcal{F}^+_0 &= ((\hbar^{-1} \mathcal{F}^-_0 \Diamond \hbar^{-1} \mathcal{F}^+)
\cdot \hbar)|_{\hbar = 0}\\
&= (\mathcal{F}^- \Diamond \mathcal{F}^+)_0.
\end{align*}
We define linear maps
\begin{align*}
T_{\mathcal{F}^-_0}(\cdot \sharp \mathcal{F}^+_0) &: \mathcal{L}^{\leq \kappa}_{X^-}
/ J^{\leq \kappa}_{C_0, C_2}
\to \mathcal{L}^{\leq \kappa}_{X, Y^0} / J^{\leq \kappa}_{C_0, C_2}, \\
T_{\mathcal{F}^+_0}(\mathcal{F}^-_0 \sharp \cdot) &: \mathcal{L}^{\leq \kappa}_{X^+}
/ J^{\leq \kappa}_{C_0, C_2}
\to \mathcal{L}^{\leq \kappa}_{X, Y^0} / J^{\leq \kappa}_{C_0, C_2}, \\
(T_0)^2_{\mathcal{F}^-_0, \mathcal{F}^+_0} &: \mathcal{L}^{\leq \kappa_1}_{X^-}
/ J^{\leq \kappa_1}_{C_0, C_2 + \kappa_2} \otimes \mathcal{L}^{\leq \kappa_2}_{X^+}
/ J^{\leq \kappa_2}_{C_0, C_2} \to
\mathcal{L}^{\leq \kappa_1 + \kappa_2}_{X, Y^0}
/ J^{\leq \kappa_1 + \kappa_2}_{C_0, C_2}
\end{align*}
by
\begin{align*}
T_{\mathcal{F}^-_0}(\cdot \sharp \mathcal{F}^+_0)(f)
&= T_{\hbar^{-1}\mathcal{F}^-_0}(\cdot \Diamond \hbar^{-1}\mathcal{F}^+_0)(f)|_{\hbar = 0},\\
T_{\mathcal{F}^+_0}(\mathcal{F}^-_0 \sharp \cdot)(f)
&= T_{\hbar^{-1}\mathcal{F}^+_0}(\hbar^{-1}\mathcal{F}^-_0 \Diamond \cdot)(f)|_{\hbar = 0},\\
(T_0)^2_{\mathcal{F}^-_0, \mathcal{F}^+_0}(f \otimes g)
&= T^2_{\hbar^{-1} \mathcal{F}^-_0, \hbar^{-1} \mathcal{F}^+_0}(f \otimes g)|_{\hbar = 0}.
\end{align*}
Then they satisfy the counterpart of Lemma \ref{linearized T}.
Finally we consider the case of contact homology.
Note that
\[
\widehat{\mathcal{F}^-_0 \sharp \mathcal{F}^+_0} =
\sum_{c}
\frac{\overleftarrow{\partial} (\mathcal{F}^-_0 \sharp \mathcal{F}^+_0)}
{\partial p^+_{\hat c}} \biggr|_{p^+ = 0} \cdot p^+_{\hat c}
= \widehat{\mathcal{F}}^+_0\Bigr|_{q^0_{\hat c^\ast} =
\frac{\overleftarrow{\partial} \widehat{\mathcal{F}}^-_0}{\partial p^0_{\hat c}}}.
\]
This implies that the composition $\Psi_{\widehat{\mathcal{F}}^-_0} \circ
\Psi_{\widehat{\mathcal{F}}^+_0} : \mathcal{A}_{Y^+}^{\leq \kappa} / I^{\leq \kappa}_{C_0} \to
\mathcal{A}_{Y^-}^{\leq \kappa} / I^{\leq \kappa}_{C_0}$ coincides with the chain map
defined by $\widehat{\mathcal{F}^-_0 \sharp \mathcal{F}^+_0}$.
\subsection{Composition of cobordisms}\label{composition of cobordisms}
In this section, we construct a homotopy between the generating function of $X$
and the composition of the generating functions of $X^-$ and $X^+$.
For each $0 \leq T < \infty$, a new manifold $X^T$ is defined by
\[
X^T = (-\infty, 0] \times Y^- \cup Z^- \cup ([0, T]_{0^-} \cup [-T, 0]_{0^+}) \times Y^0
\cup Z^+ \cup [0, \infty) \times Y^+,
\]
where we identify $T \in [0, T]_{0^-}$ with $-T \in [-T, 0]_{0^+}$.
First we define a holomorphic building for $X^{[0, \infty]}$.
\begin{defi}
A holomorphic building $(T, \Sigma, z, u, \phi)$ for $X^{[0, \infty]}$
consists of the following:
\begin{itemize}
\item
$0 \leq T \leq \infty$
\item
A marked curve $(\Sigma, z)$ which is obtained from some union of
marked semistable curves $(\check \Sigma, z \cup (\pm \infty_i))$
with a floor structure.
In this case, floor takes values in $\{-k_-, \dots, -1, 0, 1, \dots, k_+\}$
if $0 \leq T < \infty$,
and $\{-k_-, \dots, -1,\allowbreak 0^-,\allowbreak 0_1, \dots, 0_l, 0^+, 1, \dots, k_+\}$ ($l \geq 0$)
if $T = \infty$.
\item
If $T < \infty$, then $u$ is a continuous map
$u : \Sigma \to (\overline{\mathbb{R}}_{-k_-} \cup \dots \cup \overline{\mathbb{R}}_{-1}) \times Y^-
\cup X^T \cup (\overline{\mathbb{R}}_1 \cup \dots \cup \overline{\mathbb{R}}_{k_+}) \times Y^+$,
and if $T = \infty$, then $u$ is a continuous map
$u : \Sigma \to (\overline{\mathbb{R}}_{-k_-} \cup \dots \cup \overline{\mathbb{R}}_{-1}) \times Y^-
\cup X^- \cup (\overline{\mathbb{R}}_{0_1} \cup \dots \cup \overline{\mathbb{R}}_{0_l})
\times Y^0 \cup X^+ \cup (\overline{\mathbb{R}}_{1} \cup \dots \cup
\overline{\mathbb{R}}_{k_+}) \times Y^+$.
\item
$\phi_{\pm\infty_i} : S^1 \to S^1_{\pm\infty_i}$ is a family of coordinates of limit circles.
\end{itemize}
We assume the following conditions:
If $T < \infty$, then $(\Sigma, z, u, \phi)$ is a holomorphic building for $X^T$.
In this case, the energies $E_\lambda(u)$ and $E_{\hat \omega}(u)$ are defined by
\begin{align*}
E_\lambda(u) &= \max \biggl\{
\sup_{I \subset \mathbb{R}_{-k_-} \cup \dots \cup \mathbb{R}_{-1} \cup (-\infty, 0]}
\frac{1}{|I|} \int_{u^{-1}(I \times Y^-)} u^\ast (d\sigma \wedge \lambda^-),\\
&\hphantom{= \max \biggl\{}
\sup_{I \subset [0, T] \cup [-T, 0]} \frac{1}{|I|} \int_{u^{-1}(I \times Y^0)}
u^\ast (d\sigma \wedge \lambda^0),\\
&\hphantom{= \max \biggl\{}
\sup_{I \subset [0, \infty) \cup \mathbb{R}_1 \cup \dots \cup \mathbb{R}_{k_+}}
\frac{1}{|I|} \int_{u^{-1}(I \times Y^+)} u^\ast (d\sigma \wedge \lambda^+) \biggr\},\\
E_{\hat \omega}(u) &= \int_{u^{-1}(X^T)} u^\ast \hat \omega^T
+ \int_{u^{-1}((\overline{\mathbb{R}}_{-k_-} \cup \dots \cup \overline{\mathbb{R}}_{-1}) \times Y^-)} u^\ast d\lambda^-\\
&\quad + \int_{u^{-1}((\overline{\mathbb{R}}_1 \cup \dots \cup \overline{\mathbb{R}}_{k_+}) \times Y^+)} u^\ast d\lambda^+,
\end{align*}
where $\hat \omega^T$ is defined by
$\hat \omega^T|_{Z^\pm} = \omega^\pm$,
$\hat \omega^T|_{(-\infty, 0] \times Y^-} = d\lambda^-$,
$\hat \omega^T|_{([0, T] \cup [-T, 0]) \times Y^0} = d\lambda^0$, and
$\hat \omega^T |_{[0, \infty) \times Y^+} = d\lambda^+$.
If $T = \infty$, then we assume that $(\Sigma, z, u, \phi)$ satisfies the following conditions:
\begin{itemize}
\item
If $i(\alpha) <0^-$ then
$u(\Sigma_\alpha \setminus \coprod S^1) \subset \mathbb{R}_{i(\alpha)} \times Y^-$, and
$u|_{\Sigma_\alpha \setminus \coprod S^1} : \Sigma_\alpha \setminus \coprod S^1 \to \mathbb{R}_{i(\alpha)} \times Y^-$ is $J$-holomorphic.
\item
If $i(\alpha) = 0^-$ then
$u(\Sigma_\alpha \setminus \coprod S^1) \subset X^-$, and
$u|_{\Sigma_\alpha \setminus \coprod S^1} : \Sigma_\alpha \setminus \coprod S^1 \to X^-$
is $J$-holomorphic.
\item
If $0_1 \leq i(\alpha) \leq 0_l$ then
$u(\Sigma_\alpha \setminus \coprod S^1) \subset \mathbb{R}_{i(\alpha)} \times Y^0$, and
$u|_{\Sigma_\alpha \setminus \coprod S^1} : \Sigma_\alpha \setminus \coprod S^1 \to \mathbb{R}_{i(\alpha)} \times Y^0$ is $J$-holomorphic.
\item
If $i(\alpha) = 0^+$ then
$u(\Sigma_\alpha \setminus \coprod S^1) \subset X^+$, and
$u|_{\Sigma_\alpha \setminus \coprod S^1} : \Sigma_\alpha \setminus \coprod S^1 \to X^+$
is $J$-holomorphic.
\item
If $i(\alpha) >0^+$ then
$u(\Sigma_\alpha \setminus \coprod S^1) \subset \mathbb{R}_{i(\alpha)} \times Y^+$, and
$u|_{\Sigma_\alpha \setminus \coprod S^1} : \Sigma_\alpha \setminus \coprod S^1 \to \mathbb{R}_{i(\alpha)} \times Y^+$ is $J$-holomorphic.
\item
The energies $E_\lambda(u) <\infty$ and $E_{\hat \omega}(u) <\infty$ are finite
which are defined by
\begin{align*}
E_\lambda(u) &= \max \biggl\{
\sup_{I \subset \mathbb{R}_{-k_-} \cup \dots \cup \mathbb{R}_{-1} \cup (-\infty, 0]}
\frac{1}{|I|} \int_{u^{-1}(I \times Y^-)} u^\ast (d\sigma \wedge \lambda^-),\\
&\hphantom{= \max \biggl\{}
\sup_{I \subset [0, \infty)_{0^-} \cup \mathbb{R}_{0_1} \cup \dots \cup \mathbb{R}_{0_l} \cup (-\infty, 0]_{0^+}}
\frac{1}{|I|} \int_{u^{-1}(I \times Y^0)}
u^\ast (d\sigma \wedge \lambda^0),\\
&\hphantom{= \max \biggl\{}
\sup_{I \subset [0, \infty) \cup \mathbb{R}_1 \cup \dots \cup \mathbb{R}_{k_+}}
\frac{1}{|I|} \int_{u^{-1}(I \times Y^+)} u^\ast (d\sigma \wedge \lambda^+) \biggr\}, \\
E_{\hat \omega}(u) &= \int_{u^{-1}(X^-)} u^\ast \hat \omega^-
+ \int_{u^{-1}(X^+)} u^\ast \hat \omega^+\\
&\quad + \int_{u^{-1}((\overline{\mathbb{R}}_{-k_-} \cup \dots \cup \overline{\mathbb{R}}_{-1}) \times Y^-)} u^\ast d\lambda^-
+ \int_{u^{-1}((\overline{\mathbb{R}}_{0_1} \cup \dots \cup \overline{\mathbb{R}}_{0_l}) \times Y^0)}
u^\ast d\lambda^0\\
&\quad + \int_{u^{-1}((\overline{\mathbb{R}}_1 \cup \dots \cup \overline{\mathbb{R}}_{k_+}) \times Y^+)} u^\ast d\lambda^+.
\end{align*}
\item
$u$ is positively asymptotic to a periodic orbit $\gamma_{+\infty_i} = \pi_Y \circ u \circ \phi_{+\infty_i} \in P_{Y^+}$ at each $S^1_{+\infty_i}$,
and negatively asymptotic to a periodic orbit $\gamma_{-\infty_i} = \pi_Y \circ u \circ \phi_{-\infty_i} \in P_{Y^-}$ at each $S^1_{-\infty_i}$.
At every joint circle, $u$ is positively asymptotic to a periodic orbit
on the side of lower floor and negatively asymptotic to the same periodic orbit
on the side of higher floor.
\item
For each component $\hat\Sigma_\alpha$, if $u|_{\Sigma_\alpha}$ is a constant map,
then $2g_\alpha + m_\alpha \geq 3$.
\item
For each $i \neq 0^\pm$, the $i$-th floor $u^{-1}(\overline{\mathbb{R}}_i \times Y^\pm) \subset
\Sigma$ (or $u^{-1}(\overline{\mathbb{R}}_i \times Y^0) \subset \Sigma$) contains
nontrivial components.
\end{itemize}
\end{defi}
We denote the space of holomorphic buildings for $X^{[0, \infty]}$ by
$\overline{\mathcal{M}}_{X^{[0, \infty]}}$.
We define $\widehat{\mathcal{M}}_{X^{[0, \infty]}}$ similarly.
Kuranishi neighborhoods of $\widehat{\mathcal{M}}_{X^{[0, \infty]}}$ are defined in a similar way to
those of $\widehat{\mathcal{M}}_X$.
Let $\beta_0 > 0$ be the constant used for the definition of the strong differential
structure of the space of deformation of the domain curves in the case of $Y^0$
in order to construct smooth Kuranishi neighborhoods.
(Recall that in Section \ref{smoothness}, we use the coordinate change
$\rho_\mu = \hat \rho_\mu^{L_\mu^{-1} \beta}$ to define the strong differential
structure. We denote this $\beta$ in the case of $Y^0$ by $\beta_0$.)
We define the differential structure of $[0, \infty]$ by
the coordinate $\varphi : [0, \infty] \to [0, 1]$ given by
$\varphi(T) = \exp(-2T / \beta_0)$.
Then it is easy to check that the natural map
$\widehat{\mathcal{M}}_{X^{[0, \infty]}} \to [0, \infty]$ is a strong smooth map.
The associated maps from the Kuranishi neighborhoods to $[0, \infty]$ is
not submersive in general, but if we regard $[0, \infty]$ as
a $1$-dimensional simplex, then these maps are essentially submersive.
(Locally, at the corner corresponding to the splitting of $Y^0$-floors,
these maps looks like
$(t_1, \dots, t_k) \to t_1 \cdots t_k : [0, 1)^k \to [0, 1)$.)
First we define a space $\widehat{\mathcal{M}}^\diamond_{X^{[0, \infty]}}$.
Its point $(T, (\Sigma^\alpha, z^\alpha, u^\alpha)_{\alpha \in A_Y^- \sqcup A_X^0
\sqcup A_Y^+}, M^{\mathrm{rel}})$ consists of $0 \leq T \leq \infty$,
holomorphic buildings
$(\Sigma^\alpha, z^\alpha, u^\alpha)_{\alpha \in A_Y^-}$ for $Y^-$,
$(\Sigma^\alpha, \allowbreak z^\alpha, \allowbreak u^\alpha)_{\alpha \in A_X^0}$ for $X^T$
(that is, $(T, \Sigma^\alpha, z^\alpha, u^\alpha)_{\alpha \in A_X^0}$ are
holomorphic buildings for $X^{[0, \infty]}$),
$(\Sigma^\alpha, z^\alpha, u^\alpha)_{\alpha \in A_Y^+}$ for $Y^+$, and
a set $M^{\mathrm{rel}} = \{(S^1_{+\infty_l}, S^1_{-\infty_l})\}$ of pairs of
limit circles which satisfy the following conditions:
\begin{itemize}
\item
Any two pairs in $M^{\mathrm{rel}}$ do not share the same limit circle.
\item
For each pair $\alpha_1, \alpha_2 \in A = A_Y^- \sqcup A_X^0 \sqcup A_Y^+$,
let $M^{\alpha_1, \alpha_2} \subset M^{\mathrm{rel}}$ be the subset of pairs
$(S^1_{+\infty_l}, S^1_{-\infty_l})$ such that $S^1_{+\infty_l}$ is a $+\infty$-limit circle
of $\Sigma^{\alpha_1}$ and $S^1_{-\infty_l}$ is a $-\infty$-limit circle of
$\Sigma^{\alpha_2}$.
Then there does not exists any sequence
$\alpha_0, \alpha_1, \dots, \alpha_k = \alpha_0 \in A$ such that
$M^{\alpha_i, \alpha_{i+1}} \neq \emptyset$ for all $i = 0, 1, \dots, k-1$.
\item
For subsets $A_1, A_2 \subset A$, define
$M^{(A_1, A_2)} = \bigcup_{\alpha_1 \in A_1, \alpha_2 \in A_2} M^{\alpha_1, \alpha_2}$.
Then $M^{\mathrm{rel}}$ is the union of
$M^{\mathrm{rel}, \leq 0} = M^{(A_Y^-, A_Y^- \sqcup A_X^0)}$
and $M^{\mathrm{rel}, \geq 0} = M^{(A_X^0 \sqcup A_Y^+, A_Y^+)}$.
\end{itemize}
This is not a usual pre-Kuranishi structure but is a fiber product with
the diagonal in the product of $[0, \infty]$ by the essential submersion.
We also define a space $\widehat{\mathcal{M}}^\diamond_{X^-, X^+}$ as follows.
Its point
\[
((\Sigma^\alpha, z^\alpha, u^\alpha)_{\alpha \in A_Y^- \sqcup A_X^-
\sqcup A_Y^0 \sqcup A_X^+ \sqcup A_Y^+}, M^{\mathrm{rel}})
\]
consists of holomorphic buildings
$(\Sigma^\alpha, z^\alpha, u^\alpha)_{\alpha \in A_Y^-}$ for $Y^-$,
$(\Sigma^\alpha, z^\alpha, u^\alpha)_{\alpha \in A_X^-}$ for $X^-$,
$(\Sigma^\alpha, z^\alpha, u^\alpha)_{\alpha \in A_Y^0}$ for $Y^0$,
$(\Sigma^\alpha, z^\alpha, u^\alpha)_{\alpha \in A_X^+}$ for $X^+$,
$(\Sigma^\alpha, z^\alpha, u^\alpha)_{\alpha \in A_Y^+}$ for $Y^+$, and
a set $M^{\mathrm{rel}} = \{(S^1_{+\infty_l}, S^1_{-\infty_l})\}$ of pairs of
limit circles which satisfy the following conditions:
\begin{itemize}
\item
Any two pairs in $M^{\mathrm{rel}}$ do not share the same limit circle.
\item
For each pair $\alpha_1, \alpha_2 \in A = A_Y^- \sqcup A_X^- \sqcup A_Y^0
\sqcup A_X^+ \sqcup A_Y^+$,
let $M^{\alpha_1, \alpha_2} \subset M^{\mathrm{rel}}$ be the subset of pairs
$(S^1_{+\infty_l}, S^1_{-\infty_l})$ such that $S^1_{+\infty_l}$ is a $+\infty$-limit circle
of $\Sigma^{\alpha_1}$ and $S^1_{-\infty_l}$ is a $-\infty$-limit circle of
$\Sigma^{\alpha_2}$.
Then there does not exists any sequence
$\alpha_0, \alpha_1, \dots, \alpha_k = \alpha_0 \in A$ such that
$M^{\alpha_i, \alpha_{i+1}} \neq \emptyset$ for all $i = 0, 1, \dots, k-1$.
\item
For subsets $A_1, A_2 \subset A$, define
$M^{(A_1, A_2)} = \bigcup_{\alpha_1 \in A_1, \alpha_2 \in A_2} M^{\alpha_1, \alpha_2}$.
Then $M^{\mathrm{rel}}$ is the union of
$M^{\mathrm{rel}, -} = M^{(A_Y^-, A_Y^- \sqcup A_X^-)}$,
$M^{\mathrm{rel}, 0} = M^{(A_X^- \sqcup A_Y^0, A_Y^0 \sqcup A_X^+)}$
and $M^{\mathrm{rel}, +} = M^{(A_X^+ \sqcup A_Y^+, A_Y^+)}$.
\end{itemize}
The definition of the connected points of $\widehat{\mathcal{M}}^\diamond_{X^{[0, \infty]}}$ and
$\widehat{\mathcal{M}}^\diamond_{X^-, X^+}$ is similarly to the case of $X$ and $Y$.
$\widehat{\mathcal{M}}^\diamond_{X^{[0, \infty]}}$ contains
$[0, \infty] \times \widehat{\mathcal{M}}^\diamond_{Y^\pm}$, and
$\widehat{\mathcal{M}}^\diamond_{X^-, X^+}$
contains $\widehat{\mathcal{M}}^\diamond_{Y^\pm}$, $\widehat{\mathcal{M}}^\diamond_{Y^0}$ and
$\widehat{\mathcal{M}}^\diamond_{X^\pm}$.
We define
\begin{align*}
&(\widehat{\mathcal{M}}^\diamond_{X^{[0, \infty]}}, (\mathring{K}_{Y^-}^2,
\mathring{K}_{Y^0}^2, \mathring{K}_{Y^+}^2),
(K_{Y^-}, K_{Y^+}), \\
& \hspace{50pt}
(K^0_{Y^-}, K^0_{X^-}, K^0_{Y^0}, K^0_{X^+}, K^0_{Y^+})) \subset
\widehat{\mathcal{M}}^\diamond_{X^{[0, \infty]}}
\end{align*}
and
\begin{align*}
&(\widehat{\mathcal{M}}^\diamond_{X^-, X^+}, (\mathring{K}_{Y^-}^2,
\mathring{K}_{Y^0}^2, \mathring{K}_{Y^+}^2),
(K_{Y^-}, K_{Y^0}, K_{Y^+}), \\
& \hspace{50pt}
(K^0_{Y^-}, K^0_{X^-}, K^0_{Y^0}, K^0_{X^+}, K^0_{Y^+})) \subset
\widehat{\mathcal{M}}^\diamond_{X^-, X^+}
\end{align*}
similarly.
We abbreviate the above two spaces by
$(\widehat{\mathcal{M}}^\diamond_{X^{[0, \infty]}}, \mathring{K}_{X^{[0, \infty]}}, \allowbreak
K_{X^{[0, \infty]}}, \allowbreak K^0_{X^{[0, \infty]}})$ and
$(\widehat{\mathcal{M}}^\diamond_{X^-, X^+}, \mathring{K}_{X^-, X^+},
K_{X^-, X^+}, K^0_{X^-, X^+})$ respectively.
Let $(\widehat{\mathcal{M}}^\diamond_{X^\infty}, \mathring{K}_{X^\infty},
K_{X^\infty}, K^0_{X^\infty})$ be the fiber product of
$(\widehat{\mathcal{M}}^\diamond_{X^{[0, \infty]}}, \mathring{K}_{X^{[0, \infty]}}, \allowbreak
K_{X^{[0, \infty]}}, \allowbreak K^0_{X^{[0, \infty]}})$ with $\infty \in [0, \infty]$.
Then both of $(\widehat{\mathcal{M}}^\diamond_{X^{[0, \infty]}}, \mathring{K}_{X^{[0, \infty]}}, \allowbreak
K_{X^{[0, \infty]}}, \allowbreak K^0_{X^{[0, \infty]}})$ and
$(\widehat{\mathcal{M}}^\diamond_{X^-, X^+}, \mathring{K}_{X^-, X^+},
K_{X^-, X^+}, K^0_{X^-, X^+})$ contains
$(\widehat{\mathcal{M}}^\diamond_{X^\infty}, \mathring{K}_{X^\infty}, \allowbreak
K_{X^\infty}, \allowbreak K^0_{X^\infty})$.
Similarly to the usual case, we can define multi-valued partial essential submersions
$\Xi^\circ$ and $\Lambda$ for both of
$(\widehat{\mathcal{M}}^\diamond_{X^{[0, \infty]}}, \mathring{K}_{X^{[0, \infty]}}, \allowbreak
K_{X^{[0, \infty]}}, \allowbreak K^0_{X^{[0, \infty]}})$ and
$(\widehat{\mathcal{M}}^\diamond_{X^-, X^+}, \allowbreak \mathring{K}_{X^-, X^+}, \allowbreak
K_{X^-, X^+}, \allowbreak K^0_{X^-, X^+})$.
We can construct a continuous family of grouped multisections of
$(\widehat{\mathcal{M}}^\diamond_{X^{[0, \infty]}}, \allowbreak \mathring{K}_{X^{[0, \infty]}}, \allowbreak
K_{X^{[0, \infty]}}, \allowbreak K^0_{X^{[0, \infty]}})$ and
a usual grouped multisection of
$(\widehat{\mathcal{M}}^\diamond_{X^-, X^+}, \allowbreak \mathring{K}_{X^-, X^+}, \allowbreak
K_{X^-, X^+}, \allowbreak K^0_{X^-, X^+})$
which satisfy the following conditions.
\begin{itemize}
\item
On a neighborhood of $T \in \{0\} \cup \{\infty\} \subset [0, \infty]$,
the continuous family of grouped multisections of
$(\widehat{\mathcal{M}}^\diamond_{X^{[0, \infty]}}, \allowbreak \mathring{K}_{X^{[0, \infty]}}, \allowbreak
K_{X^{[0, \infty]}}, \allowbreak K^0_{X^{[0, \infty]}})$ is a usual grouped multisection.
\item
The restrictions of the natural map
$(\widehat{\mathcal{M}}^\diamond_{X^{[0, \infty]}}, \allowbreak \mathring{K}_{X^{[0, \infty]}}, \allowbreak
K_{X^{[0, \infty]}}, \allowbreak K^0_{X^{[0, \infty]}}) \to [0, \infty]$
to the fiber products of the zero sets of the perturbed multisection
are essentially submersive.
\item
The restrictions of the continuous family of
grouped multisections of
$(\widehat{\mathcal{M}}^\diamond_{X^{[0, \infty]}}, \allowbreak \mathring{K}_{X^{[0, \infty]}}, \allowbreak
K_{X^{[0, \infty]}}, \allowbreak K^0_{X^{[0, \infty]}})$ to
$[0, \infty] \times
(\widehat{\mathcal{M}}_{Y^\pm}^\diamond, \allowbreak \mathring{K}_{Y^\pm}^2, \allowbreak
K_{Y^\pm}, K_{Y^\pm}^0)$
coincide with the pull backs of the grouped multisections of
$(\widehat{\mathcal{M}}_{Y^\pm}^\diamond, \mathring{K}_{Y^\pm}^2, K_{Y^\pm}, K_{Y^\pm}^0)$
by the projection.
\item
The restrictions of the grouped multisection of
$(\widehat{\mathcal{M}}^\diamond_{X^-, X^+}, \allowbreak \mathring{K}_{X^-, X^+}, \allowbreak
K_{X^-, X^+}, \allowbreak K^0_{X^-, X^+})$ to
$(\widehat{\mathcal{M}}_{Y^\pm}^\diamond, \allowbreak \mathring{K}_{Y^\pm}^2, \allowbreak
K_{Y^\pm}, K_{Y^\pm}^0)$,
$(\widehat{\mathcal{M}}_{Y^0}^\diamond, \allowbreak \mathring{K}_{Y^0}^2, \allowbreak
K_{Y^0}, K_{Y^0}^0)$ and
$(\widehat{\mathcal{M}}^\diamond_{X^\pm}, \allowbreak \mathring{K}^2_{X^\pm}, \allowbreak
K_{X^\pm}, K^0_{X^\pm})$
coincide with the given grouped multisections.
\item
The restrictions of the
grouped multisections of
$(\widehat{\mathcal{M}}^\diamond_{X^{[0, \infty]}}, \allowbreak \mathring{K}_{X^{[0, \infty]}}, \allowbreak
K_{X^{[0, \infty]}}, \allowbreak K^0_{X^{[0, \infty]}})$ and
$(\widehat{\mathcal{M}}^\diamond_{X^-, X^+}, \allowbreak \mathring{K}_{X^-, X^+}, \allowbreak
K_{X^-, X^+}, \allowbreak K^0_{X^-, X^+})$ to
$(\widehat{\mathcal{M}}^\diamond_{X^\infty}, \allowbreak \mathring{K}_{X^\infty}, \allowbreak
K_{X^\infty}, \allowbreak K^0_{X^\infty})$ coincide.
\item
Let
\begin{align*}
&((\widehat{\mathcal{M}}^\diamond_{X^{[0, \infty]}})^0, \mathring{K}^2_{X^{[0, \infty]}},
K_{X^{[0, \infty]}}, K^0_{X^{[0, \infty]}}) \\
&\quad \subset
(\widehat{\mathcal{M}}^\diamond_{X^{[0, \infty]}}, \mathring{K}^2_{X^{[0, \infty]}}, \allowbreak
K_{X^{[0, \infty]}}, \allowbreak K^0_{X^{[0, \infty]}})
\end{align*}
be the subset of connected points.
Its continuous family of grouped multisections induces that of
\[
\bigcup_N (\prod^N (\widehat{\mathcal{M}}^\diamond_{X^{[0, \infty]}})^0,
\mathring{K}^2_{X^{[0, \infty]}}, K_{X^{[0, \infty]}},
K^0_{X^{[0, \infty]}})_{[0, \infty]} / \mathfrak{S}_N,
\]
where each $(\prod^N (\widehat{\mathcal{M}}^\diamond_{X^{[0, \infty]}})^0,
\mathring{K}^2_{X^{[0, \infty]}}, K_{X^{[0, \infty]}},
K^0_{X^{[0, \infty]}})_{[0, \infty]}$ is the fiber product over ${[0, \infty]}$.
Then the grouped multisection of
$(\widehat{\mathcal{M}}^\diamond_{X^{[0, \infty]}}, \mathring{K}^2_{X^{[0, \infty]}},
\allowbreak K_{X^{[0, \infty]}}, \allowbreak K^0_{X^{[0, \infty]}})$
coincides with its pull back by the essential submersion
\begin{align*}
&(\widehat{\mathcal{M}}^\diamond_{X^{[0, \infty]}}, \mathring{K}^2_{X^{[0, \infty]}},
K_{X^{[0, \infty]}}, K^0_{X^{[0, \infty]}}) \\
&\to
\bigcup_N (\prod^N (\widehat{\mathcal{M}}^\diamond_{X^{[0, \infty]}})^0,
\mathring{K}^2_{X^{[0, \infty]}}, K_{X^{[0, \infty]}},
K^0_{X^{[0, \infty]}})_{[0, \infty]} / \mathfrak{S}_N
\end{align*}
defined by decomposition into connected components.
\item
Let
\begin{align*}
&((\widehat{\mathcal{M}}^\diamond_{X^-, X^+})^0, \mathring{K}^2_{X^-, X^+},
K_{X^-, X^+}, K^0_{X^-, X^+}) \\
&\quad \subset
(\widehat{\mathcal{M}}^\diamond_{X^-, X^+}, \mathring{K}^2_{X^-, X^+},
\allowbreak K_{X^-, X^+}, \allowbreak K^0_{X^-, X^+})
\end{align*}
be the subset of connected points.
Its grouped multisection induces that of
\[
\bigcup_N (\prod^N (\widehat{\mathcal{M}}^\diamond_{X^-, X^+})^0,
\mathring{K}^2_{X^-, X^+}, K_{X^-, X^+},
K^0_{X^-, X^+}) / \mathfrak{S}_N.
\]
Then the grouped multisection of
$(\widehat{\mathcal{M}}^\diamond_{X^-, X^+}, \mathring{K}^2_{X^-, X^+},
K_{X^-, X^+}, K^0_{X^-, X^+})$
coincides with its pull back by the submersion
\begin{align*}
&(\widehat{\mathcal{M}}^\diamond_{X^-, X^+}, \mathring{K}^2_{X^-, X^+},
K_{X^-, X^+}, K^0_{X^-, X^+}) \\
&\to
\bigcup_N (\prod^N (\widehat{\mathcal{M}}^\diamond_{X^-, X^+})^0,
\mathring{K}^2_{X^-, X^+}, K_{X^-, X^+},
K^0_{X^-, X^+}) / \mathfrak{S}_N
\end{align*}
defined by decomposition into connected components.
\item
The (continuous families of) grouped multisections of
$(\widehat{\mathcal{M}}^\diamond_{X^{[0, \infty]}}, \allowbreak \mathring{K}_{X^{[0, \infty]}}, \allowbreak
K_{X^{[0, \infty]}}, \allowbreak K^0_{X^{[0, \infty]}})$ and
$(\widehat{\mathcal{M}}^\diamond_{X^-, X^+}, \allowbreak \mathring{K}_{X^-, X^+}, \allowbreak
K_{X^-, X^+}, \allowbreak K^0_{X^-, X^+})$ are compatible with respect to
the compatible systems of
multi-valued partial essential submersions defined by $\Xi^\circ$ and $\Lambda$.
\end{itemize}
We define pre-Kuranishi spaces
$\overline{\mathcal{M}}^{(m_-, X^{[0, \infty]}, m_+)}
_{(\hat \epsilon^{i, j}_l, \hat c^i_l, x^i_l, \hat \eta^i_l)}$ and
$\overline{\mathcal{M}}^{(m_-, X^-, m, X^+, m_+)}
_{(\hat \epsilon^{i, j}_l, \hat c^i_l, x^i_l, \hat \eta^i_l)}$ similarly
to $\overline{\mathcal{M}}^{(m_-, X^I, m_+)}_{((\hat \epsilon^{i, j}_l), (\hat c^i_l), (x^i_l),
(\hat \eta^i_l))}$ and
$\overline{\mathcal{M}}^{(m_-, X, m_+)}_{((\hat \epsilon^{i, j}_l), (\hat c^i_l), (x^i_l), (\hat \eta^i_l))}$,
and define their grouped multisections (or a continuous family of grouped multisections
for the former)
by the pull back by the natural maps to
$(\widehat{\mathcal{M}}^\diamond_{X^{[0, \infty]}}, \mathring{K}_{X^{[0, \infty]}}, \allowbreak
K_{X^{[0, \infty]}}, \allowbreak K^0_{X^{[0, \infty]}})$ and
$(\widehat{\mathcal{M}}^\diamond_{X^-, X^+}, \allowbreak \mathring{K}_{X^-, X^+}, \allowbreak
K_{X^-, X^+}, \allowbreak K^0_{X^-, X^+})$ respectively.
For a triple $((\hat c_l), (x_l), (\alpha_l))$, we define a pre-Kuranishi space
(or a linear combination of pre-Kuranishi spaces)
$\overline{\mathcal{M}}^{X^{[0, \infty]}}((\hat c_l), (x_l), (\alpha_l))$ by
\[
\overline{\mathcal{M}}^{X^{[0, \infty]}}((\hat c_l), (x_l), (\alpha_l))
= \sum_{\star} (-1)^\ast
\overline{\mathcal{M}}^{(m_-, X^{[0, \infty]}, m_+)}
_{(\Theta^+(e^{\otimes G_{X^+}^+}), \Theta^-(e^{\otimes G_{X^-}^-}),
(\hat c^i_l), (x^i_l), ([\overline{P}] \cap \alpha^i_l))},
\]
where
($G_{X^\pm}^\pm$ are the solutions of (\ref{+G eq}) and (\ref{-G eq})
used for the definition of the generating functions of $X^\pm$)
and the sum $\star$ is taken over all decompositions
\[
\{\hat c_l\} = \coprod_{-m_- \leq i \leq 0} \{\hat c^i_l\}, \quad
\{x_l\} = \coprod_{-m_- \leq i \leq m_+} \{x^i_l\}, \quad
\{\alpha_l\} = \coprod_{0 \leq i \leq m_+} \{\alpha^i_l\}
\]
as sets, and the sign $\ast$ is the weighted sign of the permutation
\[
\begin{pmatrix}
(\hat c^{-m_-}_l) \ (x^{-m_-}_l) \dots (x^{m_+}_l) \ (\alpha^{m_+}_l)\\
(\hat c_l) \quad (x_l) \quad (\alpha_l)
\end{pmatrix}.
\]
Similarly, we define its subspace of irreducible sequences of holomorphic buildings by
\[
\bigl(\overline{\mathcal{M}}^{X^{[0, \infty]}}\bigr)^0((\hat c_l), (x_l), (\alpha_l))
= \sum_{\star} (-1)^\ast
\Bigl(\overline{\mathcal{M}}^{(m_-, X^{[0, \infty]}, m_+)}_{((e^{\otimes G_{X^+}^+}),
(e^{\otimes G_{X^-}^-}),
(\hat c^i_l), (x^i_l), ([\overline{P}] \cap \alpha^i_l))}\Bigr)^0.
\]
Let
\[
\bigl[\overline{\mathcal{M}}^{X^{[0, \infty]}\!, e}_g((\hat c_l), (x_l), (\alpha_l))\bigr]
= (f_{0, g}^e)^\tau((\hat c_l), (x_l), (\alpha_l))
\oplus (f_{1, g}^e)^\tau((\hat c_l), (x_l), (\alpha_l)) d\tau
\]
and
\[
\bigl[(\overline{\mathcal{M}}^{X^{[0, \infty]}\!, e}_g)^0((\hat c_l), (x_l), (\alpha_l))\bigr]
= (h_{0, g}^e)^\tau((\hat c_l), (x_l), (\alpha_l))
\oplus (h_{1, g}^e)^\tau((\hat c_l), (x_l), (\alpha_l)) d\tau
\]
be the counterparts of the virtual fundamental chains, where
$(f_{j, g}^e)((\hat c_l), (x_l), (\alpha_l))$ and $(h_{j, g}^e)((\hat c_l), (x_l), (\alpha_l))$
are smooth functions of $\tau \in [0, \infty] \cong [0, 1]$.
Note that since we assume that the continuous family of grouped multisections
is a usual grouped multisection on a neighborhood of
$\tau \in \{0\} \cup \{\infty\} \subset [0, \infty]$, the zero set of the
perturbed multisections of the $1$-dimensional parts of the above
pre-Kuranishi spaces do not intersect with the corner of codimension $\geq 2$.
In particular, on the zero set, the strong continuous map to
$[0, \infty] \cong [0, 1]$ is submersive.
Hence on a neighborhood of $\tau \in \{0\} \cup \{\infty\} \subset [0, \infty]$,
$(f_{0, g}^e)^\tau$ and $(h_{0, g}^e)^\tau$ are constant as functions of $\tau$,
and $(f_{1, g}^e)^\tau$ and $(h_{1, g}^e)^\tau$ are zero.
Then $f_{0, g}^e$, $h_{0, g}^e$, $\hat f_{1, g}^e = f_{1, g}^e$ and
$\hat h_{1, g}^e = h_{1, g}^e$ satisfy (\ref{f_0 equation}), (\ref{f_1 equation}),
and (\ref{boundary formula for X^I}).
Furthermore, $(f_{0, g}^e)^{\tau = \infty}$ coincides with the $(g, e)$ part of
\[
\sum_{\star} (-1)^\ast
\overline{\mathcal{M}}^{(m_-, X^-, 0, X^+, m_+)}
_{(\Theta^+(e^{\otimes G_{X^+}^+}),
(e^{(\Delta_\ast[\overline{P}])^{0^-, 0^+}}), \Theta^-(e^{\otimes G_{X^-}^-}),
(\hat c^i_l), (x^i_l), ([\overline{P}] \cap \alpha^i_l))}.
\]
If $(Y^0, \lambda^0)$ satisfies Morse condition (i.e. if $P$ is a union of circles),
then the above pre-Kuranishi space is enough for the construction of
a homotopy from the generating function of $X^0$ to the composition
of the generating functions of $X^-$ and $X^+$.
However, in general, we need another parametrized pre-Kuranishi space.
Let $G^\theta$ ($\theta \in [0, 1]$) be an appropriate $C^\infty(I, \mathbb{R})$-linear
combination of
\[
((\rho_\ast [\overline{P}])^{i, j}, \dots,
(\rho_\ast [\overline{P}])^{i, j},
\epsilon_{\overline{P}}^{i, j}, \dots, \epsilon_{\overline{P}}^{i, j},
(\Delta_\ast [\overline{P}])^{i, j}, \dots, (\Delta_\ast [\overline{P}])^{i, j})_{(i, j)}
\]
defined in the next section.
For each family $((\hat c_l), (x_l), (\alpha_l))$, we define
a $C^\infty(I, \mathbb{R})$-linear combination of pre-Kuranishi spaces
$\overline{\mathcal{M}}^{X^\infty, \theta \in [0, 1]}((\hat c_l), (x_l), (\alpha_l))$ by
\begin{align*}
&\overline{\mathcal{M}}^{X^\infty, \theta \in [0, 1]}((\hat c_l), (x_l), (\alpha_l)) \\
&= \sum_{\star} (-1)^\ast
\overline{\mathcal{M}}^{(m_-, X^-, m, X^+, m_+)}
_{(\Theta^+(e^{\otimes G_{X^-}^+}), \Theta(e^{\otimes G^\theta}),
\Theta^-(e^{\otimes G_{X^+}^-}), (\hat c^i_l), (x^i_l), ([\overline{P}] \cap \alpha^i_l))}.
\end{align*}
Similarly, we define a $C^\infty(I, \mathbb{R})$-linear combination of Kuranishi spaces
of irreducible sequences of holomorphic buildings
$(\overline{\mathcal{M}}^{X^\infty, \theta \in [0, 1]})^0((\hat c_l), (x_l), (\alpha_l))$ by
\begin{align*}
&(\overline{\mathcal{M}}^{X^\infty, \theta \in [0, 1]})^0((\hat c_l), (x_l), (\alpha_l))\\
&= \sum_{\star} (-1)^\ast
\Bigl(\overline{\mathcal{M}}^{(m_-, X^-, m, X^+, m_+)}_{((e^{\otimes G_{X^-}^+}), (e^{\otimes G^\theta}),
(e^{\otimes G_{X^+}^-}), (\hat c^i_l), (x^i_l), ([\overline{P}] \cap \alpha^i_l))}\Bigr)^0
\end{align*}
where in this case, the irreducibility is defined as follows.
First we consider the case of $(n_-, n, n_+) \neq (0, 0, 0)$.
A sequence of holomorphic buildings
\[
(\Sigma_i, z_i, u_i, \phi_i)_{i \in
\{-m_-, \dots, -1, 0^-, 1, \dots, m, 0^+, 1, \dots, m_+\}}
\]
in
\[
\overline{\mathcal{M}}^{(m_-, X^-, m, X^+, m_+)}
_{(\Theta^+(f_1^+ \otimes \dots \otimes f_{n_+}^+),
\Theta(f_1 \otimes \dots \otimes f_n),
\Theta^-(f_1^- \otimes \dots \otimes f_{n_-}^-),
(\hat c^i_l), (x^i_l), ([\overline{P}] \cap \alpha^i_l))}
\]
is said to be irreducible if
\begin{itemize}
\item
each connected component of $\Sigma_{0^-}$ and $\Sigma_{0^+}$ concerns at least one
monomial in $\{f_i^\pm, f_i\}$, and
\item
for any decomposition $\{f_i^\pm, f_i\} = A \sqcup B$, there exists some connected
component of $\Sigma_{0^-}$ or $\Sigma_{0^+}$ which concerns both of some $f \in A$
and some $g \in B$.
\end{itemize}
If $(n_-, n, n_+) = (0, 0, 0)$, then a point
$((\Sigma_{0^-}, z_{0^-}, u_{0^-}, \phi_{0^-}), (\Sigma_{0^+}, z_{0^+}, u_{0^+}, \phi_{0^+}))$
is irreducible if one of $(\Sigma_{0^\pm}, z_{0^\pm}, u_{0^\pm}, \phi_{0^\pm})$ is
connected and the other is the empty curve.
Let
\begin{align*}
\bigl[\overline{\mathcal{M}}^{X^\infty, \theta \in [0, 1], e}_g((\hat c_l), (x_l), (\alpha_l))\bigr]
= (\mathring{f}_{0, g}^e)^\theta((\hat c_l), (x_l), (\alpha_l))
\end{align*}
and
\begin{align*}
\bigl[(\overline{\mathcal{M}}^{X^\infty, \theta \in [0, 1], e}_g)^0((\hat c_l), (x_l), (\alpha_l))\bigr]
= (\mathring{h}_{0, g}^e)^\theta((\hat c_l), (x_l), (\alpha_l))
\end{align*}
be the virtual fundamental chains.
Equation (\ref{G^0 eq}) in the next section implies that
$(\mathring{f}_{0, g}^e)^{\theta = 0}$ coincides with $(f_{0, g}^e)^{\tau = \infty}$.
Let $H^\theta = H^\theta_1 + H^\theta_2 + \dots$ be an appropriate
$C^\infty(I, \mathbb{R})$-linear combination of
\[
((\rho_\ast [\overline{P}])^{i, j}, \dots,
(\rho_\ast [\overline{P}])^{i, j},
\epsilon_{\overline{P}}^{i, j}, \dots, \epsilon_{\overline{P}}^{i, j},
(\Delta_\ast [\overline{P}])^{i, j}, \dots, (\Delta_\ast [\overline{P}])^{i, j})_{(i, j)}
\]
defined in the next section, and define
$(\Ddot f_{0, g}^e)^\theta((\hat c_l), (x_l), (\alpha_l))$ by the virtual fundamental chain of
\[
\sum_{\star} (-1)^\ast
\overline{\mathcal{M}}^{(m_-, X^-, m, X^+, m_+)}
_{(\Theta^+(e^{\otimes G_{X^+}^+}),
\Theta(e^{\otimes G^\theta} \otimes H^\theta),
\sum_{m_-}(-1)^{m_-}\Theta^-(e^{\otimes G_{X^-}^-})_{-m_-}, (\hat c^i_l), (x^i_l),
([\overline{P}] \cap \alpha^i_l))}.
\]
We also define $(\Ddot h_{0, g}^e)^\theta((\hat c_l), (x_l), (\alpha_l))$ by
the virtual fundamental chains of its irreducible part.
Then $f_{0, g}^e = \mathring{f}_{0, g}^e$, $h_{0, g}^e = \mathring{h}_{0, g}^e$,
$\hat f_{1, g}^e = \Ddot f_{0, g}^e$ and $\hat h_{1, g}^e = \Ddot h_{0, g}^e$ also satisfy
(\ref{f_0 equation}), (\ref{f_1 equation}), and (\ref{boundary formula for X^I}).
Define the following families of generating functions.
\begin{align*}
\mathcal{F}^\tau &= \hbar^{-1} \sum \frac{1}{k_q ! k_t ! k_p !} (h_{0, g}^e)^\tau
(\underbrace{\mathbf{q}, \dots, \mathbf{q}}_{k_q},
\underbrace{\mathbf{t}, \dots, \mathbf{t}}_{k_t},
\underbrace{\mathbf{p}, \dots, \mathbf{p}}_{k_p}) \hbar^g T^e\\
\widetilde{\mathcal{F}}^\tau &= \hbar^{-1} \sum \frac{1}{k_q ! k_t ! k_p !} (f_{0, g}^e)^\tau
(\underbrace{\mathbf{q}, \dots, \mathbf{q}}_{k_q},
\underbrace{\mathbf{t}, \dots, \mathbf{t}}_{k_t},
\underbrace{\mathbf{p}, \dots, \mathbf{p}}_{k_p}) \hbar^g T^e\\
\mathcal{K}^\tau_g &= \hbar^{-1} \sum \frac{1}{k_q ! k_t ! k_p !} (h_{1, g}^e)^\tau
(\underbrace{\mathbf{q}, \dots, \mathbf{q}}_{k_q},
\underbrace{\mathbf{t}, \dots, \mathbf{t}}_{k_t},
\underbrace{\mathbf{p}, \dots, \mathbf{p}}_{k_p}) \hbar^g T^e\\
\widetilde{\mathcal{K}}^\tau_g &= \hbar^{-1} \sum \frac{1}{k_q ! k_t ! k_p !} (f_{1, g}^e)^\tau
(\underbrace{\mathbf{q}, \dots, \mathbf{q}}_{k_q},
\underbrace{\mathbf{t}, \dots, \mathbf{t}}_{k_t},
\underbrace{\mathbf{p}, \dots, \mathbf{p}}_{k_p}) \hbar^g T^e\\
\mathcal{F}^\theta &= \hbar^{-1} \sum \frac{1}{k_q ! k_t ! k_p !} (\mathring{h}_{0, g}^e)^\theta
(\underbrace{\mathbf{q}, \dots, \mathbf{q}}_{k_q},
\underbrace{\mathbf{t}, \dots, \mathbf{t}}_{k_t},
\underbrace{\mathbf{p}, \dots, \mathbf{p}}_{k_p}) \hbar^g T^e\\
\widetilde{\mathcal{F}}^\theta &= \hbar^{-1} \sum \frac{1}{k_q ! k_t ! k_p !}
(\mathring{f}_{0, g}^e)^\theta
(\underbrace{\mathbf{q}, \dots, \mathbf{q}}_{k_q},
\underbrace{\mathbf{t}, \dots, \mathbf{t}}_{k_t},
\underbrace{\mathbf{p}, \dots, \mathbf{p}}_{k_p}) \hbar^g T^e\\
\mathcal{K}^\theta_g &= \hbar^{-1} \sum \frac{1}{k_q ! k_t ! k_p !} (\Ddot h_{0, g}^e)^\theta
(\underbrace{\mathbf{q}, \dots, \mathbf{q}}_{k_q},
\underbrace{\mathbf{t}, \dots, \mathbf{t}}_{k_t},
\underbrace{\mathbf{p}, \dots, \mathbf{p}}_{k_p}) \hbar^g T^e\\
\widetilde{\mathcal{K}}^\theta_g &= \hbar^{-1} \sum \frac{1}{k_q ! k_t ! k_p !}
(\Ddot f_{0, g}^e)^\theta
(\underbrace{\mathbf{q}, \dots, \mathbf{q}}_{k_q},
\underbrace{\mathbf{t}, \dots, \mathbf{t}}_{k_t},
\underbrace{\mathbf{p}, \dots, \mathbf{p}}_{k_p}) \hbar^g T^e
\end{align*}
Then it is easy to see that the concatenation of the homotopies $\mathcal{F}^\tau$ and
$\mathcal{F}^\theta$ defined by the above generating functions
gives a homotopy from the generating function of $X^0$ to the composition of
the generating functions of $X^-$ and $X^+$.
\subsection{Construction of the correction terms}\label{correction terms for composition}
\subsubsection{Constuction of $G^\theta$}
For $m \geq 1$, let $C_m = \bigoplus_{n = 0}^{\frac{m(m+1)}{2}} C_m^n$ be the $\mathbb{Z}$-graded
super-commutative algebra with coefficient $\mathbb{R}$ generated by variables
$\rho_{(e_i, e_j)}$, $\Delta_{(e_i, e_j)}$ and $\epsilon_{(e_i, e_j)}$ ($0 \leq i < j \leq m$).
The $\mathbb{Z}$-grading is defined by $\dim \rho_{(e_i, e_j)} = \dim \Delta_{(e_i, e_j)} = 0$ and
$\dim \epsilon_{(e_i, e_j)} = 1$.
For each $m \geq 1$, the differential $\partial' : C_m^n \to C_m^{n-1}$ is defined by
$\partial' \epsilon_{(a, b)} = (-1)^{m-1} (\rho_{(a, b)} - \Delta_{(a, b)})$ and
$\partial' \rho_{(a, b)} = \partial' \Delta_{(a, b)} = 0$.
Homomorphisms $\tau_i : C_m \to C_{m + 1}$ ($0 \leq i \leq m$) are defined by
$\tau_i (x_{(a, b)}) = x_{(\hat \tau_i(a), \hat \tau_i(b))}$,
where each $\hat \tau_i$ is defined by
\[
\hat \tau_i(e_j) = \begin{cases}
e_j & j < i\\
e_i + e_{i + 1} & j = i\\
e_{j + 1} & j > i
\end{cases}.
\]
Define homomorphism $\Theta : \bigotimes_{i = 1}^n C_{m_i} \to C_{1 + \sum_{i=1}^n
(m_i - 1)}$ by
\[
\Theta(f_1 \otimes f_2 \otimes \dots f_n) = f_1^{+\sum_{i = 2}^n (m_i - 1)} f_2^{+\sum_{i = 3}^n
(m_i - 1)} \dots f_n,
\]
where each $f_a^{+\sum_{i = a + 1}^n (m_i - 1)}$ is defined by
\[
e_j^{+\sum_{i = a + 1}^n (m_i - 1)} = \begin{cases}
e_0 & j = 0\\
e_{j + \sum_{i = a + 1}^n (m_i - 1)} & j \neq 0, m_a\\
e_{1 + \sum_{i=1}^n (m_i - 1)} & j = m_a
\end{cases}.
\]
We also define $\boxminus : B_m^+ \otimes B_{m'}^- \to C_{m + m' + 1}$ by
\[
\boxminus(f \otimes g) = (-1)^{m m'} f \cdot \exp(\rho_{(\sum_{0 \leq i \leq m} e_i,
\sum_{m + 1 \leq j \leq m + m' + 1} e_j)}) \cdot g^{+(m + m' + 1)}.
\]
We define a linear subspace $\Ddot C_m \subset C_m$ as follows.
For each $1 \leq i \leq m - 2$ and each monomial
\[
f = x^{(1)}_{(a_1, b_1)} x^{(2)}_{(a_2, b_2)} \dots x^{(n)}_{(a_n, b_n)}
\]
such that $(a_j, b_j) \neq (i, i + 1)$, we define a monomial
\[
f^{(e_i, e_{i + 1})} = x^{(1)}_{(a'_1, b'_1)} x^{(2)}_{(a'_2, b'_2)} \dots x^{(n)}_{(a'_n, b'_n)}
\]
by permuting $i$ and $i + 1$ of $\{a_j, b_j\}$.
Then $\Ddot C_m \subset C_m$ is the subspace spanned by $f + f^{(e_i, e_{i + 1})}$ for all
such pair $i$ and $f$.
Define $\mathcal{C}_m = C_m / \Ddot C_m$.
Then the following maps are well defined.
\begin{align*}
\partial' &: \mathcal{C}_m \to \mathcal{C}_m \\
\sum_{0 < i < \max} (-1)^i e^{\Delta_{(e_i, e_{i + 1})}}
\tau_i &: \mathcal{C}_m \to \mathcal{C}_{m + 1} \quad\quad ({\max} = m) \\
e^{\Delta_{(e_0, e_1)}} \tau_0 &: \mathcal{C}_m \to \mathcal{C}_{m + 1} \\
(-1)^{\max} e^{\Delta_{(e_{\max}, e_{\max + 1})}} \tau_{\max} &: \mathcal{C}_m \to \mathcal{C}_{m + 1}\\
\Theta &: \otimes_{i = 1}^n \mathcal{C}_{m_i} \to \mathcal{C}_{1 + \sum_i (m_i - 1)}\\
\boxminus &: \mathcal{B}_m^+ \otimes \mathcal{B}_{m'}^- \to \mathcal{C}_{m + m' + 1}
\end{align*}
Further we define $\mathring{\mathcal{C}}_m \subset \mathcal{C}_m$ as follows.
We define a new degree $\deg'$ by
\[
\deg' x_{(e_i, e_j)} = \begin{cases}
0 & \text{if } i = 0 \text{ or } j = m\\
1 & \text{otherwise}
\end{cases}
\]
For $m \geq 2$, let $\mathring{C}_m \subset C_m$ be the subspace spanned by monomials
with $\deg' \geq m - 2$ which do not contain variables $\rho_{(e_0, e_m)}$,
$\Delta_{(e_0, e_m)}$ or $\epsilon_{(e_0, e_m)}$.
Define $\mathring{\mathcal{C}}_m = \mathring{C}_m
/ (\Ddot{C}_m \cap \mathring{C}_m) \subset \mathcal{C}_m$.
In this section, we prove that there exists a smooth family
$G^\theta = G^\theta_1 + G^\theta_2 + \dots
\in (\bigoplus_{m = 1}^\infty \mathring{C}_m^{m-1})^\wedge$ ($\theta \in [0, 1]$)
which satisfies the following equations.
\begin{gather}
\partial' (\Theta(e^{\otimes G^\theta})) + \sum_{i \geq 0} e^{\Delta_{(e_i, e_{i+1})}} \tau_i
\Theta(e^{\otimes G^\theta}) = 0\\
G^\theta_1 = (1 - \theta) \Delta_{(e_0, e_1)} + \theta \rho_{(e_0, e_1)}\\
G^0 = G^0_1= \Delta_{(e_0, e_1)} \label{G^0 eq}\\
\boxminus(e^{\otimes G^+_{X^-}} \otimes e^{\otimes G^-_{X^+}}) = \Theta(e^{\otimes G^1})
\label{G^1 eq}
\end{gather}
In the previous section, we replace $\rho_{(e_i, e_j)}$, $\Delta_{(e_i, e_j)}$ and
$\epsilon_{(e_i, e_j)}$ in $G^\theta$ with $(\rho_\ast [\overline{P_{Y^0}}])^{0_i, 0_j}$,
$(\Delta_\ast [\overline{P_{Y^0}}])^{0_i, 0_j}$ and $(\epsilon_{\overline{P_{Y^0}}})^{0_i, 0_j}$
respectively, where $0_0$ and $0_{\max}$ should be read as $0^-$ and $0^+$ respectively.
First we note that the last two equations define $G^0$ and $G^1$.
We inductively construct $G^\theta_{\leq m} = G^\theta_1 + \dots + G^\theta_m \in
\bigoplus_{l = 1}^m \mathring{\mathcal{C}}_l^{l-1}
$ such that
\begin{equation}
\partial' (\Theta(e^{\otimes G^\theta_{\leq m}})) + \sum_{i \geq 0} e^{\Delta_{(e_i, e_{i+1})}} \tau_i
\Theta(e^{\otimes G^\theta_{\leq m-1}}) \equiv 0 \label{G theta eq}
\end{equation}
in $\bigoplus_{l=2}^\infty \mathcal{C}_l^{l-2} / \bigoplus_{l=m+1}^\infty \mathcal{C}_l^{l-2}$.
First we define $G^\theta_2 \in \mathring{\mathcal{C}}_2^1$ by
\begin{align*}
G^\theta_2 &= e^{(1-\theta)(\Delta_{(e_0, e_1)} + \Delta_{(e_1, e_2)})}\\
&\quad \cdot \biggl(-\sum_{k \geq 1} \frac{\theta^k}{k !} e^{\theta\rho_{(e_1, e_2)}}
(\underbrace{\epsilon_{(e_0, e_1)} \Delta_{(e_0, e_1)} \dots \Delta_{(e_0, e_1)}}_k\\
&\hphantom{\quad \cdot \biggl(-\sum_{k \geq 1} \frac{\theta^k}{k !} e^{\theta\rho_{(e_1, e_2)}}(}
+ \underbrace{\rho_{(e_0, e_1)} \epsilon_{(e_0, e_1)} \Delta_{(e_0, e_1)} \dots
\Delta_{(e_0, e_1)}}_k\\
&\hphantom{\quad \cdot \biggl(-\sum_{k \geq 1} \frac{\theta^k}{k !} e^{\theta\rho_{(e_1, e_2)}}(}
+ \dots + \underbrace{\rho_{(e_0, e_1)} \dots \rho_{(e_0, e_1)} \epsilon_{(e_0, e_1)}}_k)\\
&\quad \quad + \sum_{k \geq 1} \frac{\theta^k}{k !} e^{\theta\rho_{(e_0, e_1)}}
(\underbrace{\epsilon_{(e_1, e_2)} \Delta_{(e_1, e_2)} \dots \Delta_{(e_1, e_2)}}_k\\
&\hphantom{\quad \quad + \sum_{k \geq 1} \frac{\theta^k}{k !} e^{\theta\rho_{(e_0, e_1)}}(}
+ \underbrace{\rho_{(e_1, e_2)} \epsilon_{(e_1, e_2)} \Delta_{(e_1, e_2)} \dots
\Delta_{(e_1, e_2)}}_k\\
&\hphantom{\quad \quad + \sum_{k \geq 1} \frac{\theta^k}{k !} e^{\theta\rho_{(e_0, e_1)}}(}
+ \dots + \underbrace{\rho_{(e_1, e_2)} \dots \rho_{(e_1, e_2)} \epsilon_{(e_1, e_2)}}_k)\biggr).
\end{align*}
Then it is easy to see that this satisfies equation (\ref{G theta eq}) for $m = 2$.
Next assuming we have constructed $G^\theta_{\leq m-1}$, we prove there exists
a required family $G^\theta_m$. It is enough to show that
\begin{equation}
\Theta\Bigl(\Bigl(\partial' \Theta(e^{\otimes G^\theta_{\leq m-1}}) + \sum (-1)^i
e^{\Delta_{(e_i, e_{i+1})}} \tau_i \Theta(e^{\otimes G^\theta_{\leq m-1}})\Bigr)
\otimes e^{-G^\theta_1}\Bigr) \equiv 0 \label{G theta mathring}
\end{equation}
in $\bigoplus_{l=2}^\infty \mathcal{C}_l^{l-2} / (\bigoplus_{l = m+1}^\infty \mathcal{C}_l^{l-2}
\oplus \bigoplus_{l = 2}^\infty \mathring{\mathcal{C}}_l^{l-2})$ and
\begin{equation}
\partial'\Bigl(\sum_{0 \leq i \leq \max} (-1)^i e^{(e_i, e_{i+1})} \tau_i
\Theta (e^{\otimes G^\theta_{\leq m-1}})\Bigr) \equiv 0 \label{G theta closed}
\end{equation}
in $\bigoplus_{l = 3}^\infty \mathcal{C}_l^{l-3} / \bigoplus_{l = m+1}^\infty \mathcal{C}_l^{l-3}$.
The latter is proved by an argument similar to that for equation (\ref{B mathring}).
We can prove the former similarly to equation (\ref{B closed}) using the following
equations.
\begin{align}
&\partial' \Theta\Bigl(\frac{1}{k !}(G^\theta_{\geq m-1} - G^\theta_1)^{\otimes k}\Bigr)
\notag\\
&= \Theta\Bigl(\frac{1}{(k-1) !} (G^\theta_{\geq m-1} - G^\theta_1)^{\otimes (k-1)} \otimes
\partial' (G^\theta_{\geq m-1} - G^\theta_1)\Bigr)
\end{align}
\begin{align}
&\sum_{0 < i < \max} (-1)^i e^{\Delta_{(e_i, e_{i+1})}} \tau_i
\Theta\Bigl(\frac{1}{k !} (G^\theta_{\geq m-1} - G^\theta_1)^{\otimes k}\Bigr)\notag\\
&= \Theta\Bigl(\frac{1}{(k-1) !} (G^\theta_{\geq m-1} - G^\theta_1)^{\otimes (k-1)}
\notag\\
&\hphantom{= \Theta\Bigl(}
\otimes \sum_{0 < i < \max} (-1)^i e^{\Delta_{(e_i, e_{i+1})}}
\tau_i (G^\theta_{\geq m-1} - G^\theta_1)
\Bigr)
\end{align}
\begin{align}
&e^{\Delta_{(e_0, e_1)}} \tau_0
\Theta\Bigl(\frac{1}{k !}(G^\theta_{\geq m-1})^{\otimes k}\Bigr)
\notag\\
&= \sum_{l_1 + l_2 + l_3 = k} \Theta\Bigl(\frac{1}{l_1 ! l_2 ! l_3 !}
(G^\theta_{\geq m-1} - G^\theta_1)^{\otimes l_1} \otimes
\mathring{\tau}_0((G^\theta_{\geq m-1})^{\otimes l_2}) \notag \\
&\hphantom{= \sum_{l_1 + l_2 + l_3 = k} \Theta\Bigl(}
\otimes
(G^\theta_1)^{\otimes l_3} \Bigr)
\end{align}
\begin{align}
&(-1)^{\max} e^{\Delta_{(e_{\max}, e_{\max +1})}} \tau_{\max}
\Theta\Bigl(\frac{1}{k !}(G^\theta_{\geq m-1})^{\otimes k}\Bigr)\notag\\
&= \sum_{l_1 + l_2 + l_3 = k} \Theta\Bigl(\frac{1}{l_1 ! l_2 ! l_3 !}
(G^\theta_{\geq m-1} - G^\theta_1)^{\otimes l_1} \otimes
(-1)^{\max}\mathring{\tau}_{\max}((G^\theta_{\geq m-1})^{\otimes l_2})
\notag\\
&\hphantom{= \sum_{l_1 + l_2 + l_3 = k} \Theta\Bigl(}
\otimes (G^\theta_1)^{\otimes l_3}\Bigr)
\end{align}
In the above equations, $\mathring{\tau}_0$ and $\mathring{\tau}_{\max}$ are defined
in a similar way to $\mathring{\tau}^+_0$ in Section \ref{correction terms for X}.
Therefore we can inductively construct a required family
$G^\theta_{\leq m} = G^\theta_1 + \dots + G^\theta_m \in
\bigoplus_{l = 1}^m \mathring{\mathcal{C}}_l^{l-1} \cong \bigoplus_{l = 1}^\infty
\mathring{\mathcal{C}}_l^{l-1} / \bigoplus_{l = m+1}^\infty \mathring{\mathcal{C}}_l^{l-1}$.
\subsubsection{Construction of $H^\theta$}
Next we construct a smooth family
$H^\theta = H^\theta_1 + H^\theta_2 + \dots
\in (\bigoplus_{m=1}^\infty \mathring{\mathcal{C}}_m^m)^\wedge$ which satisfies
the following equation.
\[
\partial' \Theta(e^{\otimes G} \otimes H)
+ \sum_{i \geq 0} (-1)^i e^{\Delta_{(e_i, e_j)}} \tau_i \Theta(e^{\otimes G} \otimes H)
+ \Theta\Bigl(e^{\otimes G} \otimes \frac{d}{d\theta} G\Bigr) = 0.
\]
We inductively construct $H^\theta_{\leq m} = H^\theta_1 + \dots + H^\theta_m \in
\bigoplus_{l = 1}^m \mathring{\mathcal{C}}_l^l$ such that
\begin{multline}
\partial' \Theta(e^{\otimes G} \otimes H_{\leq m})
+ \sum_{i \geq 0} (-1)^i e^{\Delta_{(e_i, e_j)}} \tau_i
\Theta(e^{\otimes G} \otimes H_{\leq m}) \\
+ \Theta\Bigl(e^{\otimes G} \otimes \frac{d}{d\theta} G\Bigr) \equiv 0 \label{H eq}
\end{multline}
in $(\bigoplus_{l=1}^\infty \mathcal{C}_l^{l-1})^\wedge /
(\bigoplus_{l=m+1}^\infty \mathcal{C}_l^{l-1})^\wedge$
Since $\frac{d}{d\theta} G_1^\theta = \rho_{(e_0, e_1)} - \Delta_{(e_0, e_1)}$,
$H_1^\theta = - \epsilon_{(e_0, e_1)}$ satisfies equation (\ref{H eq}) for $m = 1$.
Assuming we have already constructed $H_{\leq m-1}$,
we prove that there exists a required family $H_m^\theta$.
It is enough to show that
\begin{multline*}
\Bigl(\partial' \Theta(e^{\otimes G} \otimes H_{\leq m-1})
+ \sum_{i \geq 0} (-1)^i e^{\Delta_{(e_i, e_j)}} \tau_i
\Theta(e^{\otimes G} \otimes H_{\leq m-1})\\
+ \Theta\Bigl(e^{\otimes G} \otimes \frac{d}{d\theta} G\Bigr)\Bigr) \otimes
e^{-\otimes G_1} \equiv 0
\end{multline*}
in $(\bigoplus_{l=1}^\infty \mathcal{C}_l^{l-1})^\wedge /
((\bigoplus_{l=m+1}^\infty \mathcal{C}_l^{l-1})^\wedge
\oplus \bigoplus_{l = 1}^\infty \mathring{\mathcal{C}}_l^{l-1})$
and
\[
\partial'\Bigl(\sum_{i \geq 0} (-1)^i e^{\Delta_{(e_i, e_j)}}
\tau_i \Theta(e^{\otimes G} \otimes H_{\leq m-1})
+ \Theta\Bigl(e^{\otimes G} \otimes \frac{d}{d\theta} G\Bigr)\Bigr) \equiv 0
\]
in $(\bigoplus_{l=2}^\infty \mathcal{C}_l^{l-2})^\wedge /
(\bigoplus_{l=m+1}^\infty \mathcal{C}_l^{l-2})^\wedge$.
The former can be proved by a similar argument to
those for (\ref{H tau mathring}) or (\ref{G theta mathring}),
and the latter can be proved similarly to (\ref{A closed}), (\ref{B closed}),
(\ref{H tau closed}) or (\ref{G theta closed}).
Therefore, we can inductively construct a required family $H^\theta \in
(\bigoplus_{m=1}^\infty \mathring{\mathcal{C}}_m^m)^\wedge$.
\section{Independence}\label{independence}
Let $(Y, \xi)$ be a contact manifold and let $\overline{K}_Y^0 \subset H_\ast(Y, \mathbb{Z})$
be a finite subset (or a finite sequence).
We have seen that if we fix
a contact form $\lambda$, a triangulation $K_Y$ of $\overline{P}_Y$,
a Euclidean cell complex $K^2_Y$, a representative $K^0_Y$ of $\overline{K}_Y^0$,
and a complex structure $J$ of $\ker \lambda$,
construct a family of pre-Kuranishi spaces and choose a compatible family of
perturbed multisections, then
we obtain chain complexes
\begin{gather*}
(\mathcal{W}^{\leq \kappa}_{(Y, \lambda, K_Y, \overline{K}_Y^0)} / I^{\leq \kappa}_{C_0, C_1, C_2},
D_{(Y, \lambda, K_Y, K_Y^0, K_Y^2, J, \mathcal{B})}), \\
(\mathcal{P}^{\leq \kappa}_{(Y, \lambda, K_Y, \overline{K}_Y^0)}
/ I^{\leq \kappa}_{C_0, C_2},
d_{(Y, \lambda, K_Y, K_Y^0, K_Y^2, J, \mathcal{B})}),\\
(\mathcal{A}^{\leq \kappa}_{(Y, \lambda, K_Y, \overline{K}_Y^0)} / I^{\leq \kappa}_{C_0},
\partial_{(Y, \lambda, K_Y, K_Y^0, K_Y^2, J, \mathcal{B})}),
\end{gather*}
where $\mathcal{B}$ denotes the other choices for the construction of
the pre-Kuranishi structure and the perturbed multisections.
The aim of this section is to construct SFT cohomologies of a contact manifold
by the limits of the cohomologies of the above chain complexes and
to prove that they are invariants of $(Y, \xi, \overline{K}_Y^0)$.
We also construct SFT cohomologies of a symplectic cobordism as limits.
First we note that for any constant $a > 0$,
chain complexes for $(Y, a \lambda)$ can be constructed by using
the same $(K_Y, K^2_Y, K^0_Y, J, \mathcal{B})$ as those of $(Y, \lambda)$.
Then the chain complex
$(\mathcal{W}^{\leq a \kappa}_{(Y, a \lambda, K_Y, K_Y^0)} / I^{\leq a \kappa}_{C_0, C_1, a C_2},
D_{(Y, \lambda, K_Y, K_Y^0, K_Y^2, J, \mathcal{B})})$ is naturally isomorphic to
$(\mathcal{W}^{\leq \kappa}_{(Y, \lambda, K_Y, K_Y^0)} / I^{\leq \kappa}_{C_0, C_1, C_2},
D_{(Y, \lambda, K_Y, K_Y^0, K_Y^2, J, \mathcal{B})})$.
The cases of the other two chain complexes are similar.
Let $\mathcal{C}_{Y^\pm}
= (Y^\pm (= Y), \lambda^\pm, K_{Y^\pm}, K_{Y^\pm}^0, K_{Y^\pm}^2, J^\pm, \mathcal{B}_{Y^\pm})$
be two choices to define the above chain complexes.
A concordance $\mathcal{C}_X
= (X, \omega, Y^\pm, \lambda^\pm, K_{Y^\pm},\allowbreak K_X^0,\allowbreak K^0_{Y^\pm},\allowbreak
\mu^\pm, K^2_{Y^\pm}, J, \mathcal{B}_X)$
from $\mathcal{C}_{Y^-}$ to $\mathcal{C}_{Y^+}$
consists of
\begin{itemize}
\item
a cobordism $(X, \omega)$ from $(Y^-, \lambda^-)$ to $(Y^+, \lambda^+)$ of the form
$X = (-\infty, 0] \times Y^- \cup [0, T_0] \times Y \cup [0, \infty) \times Y^+$
for $T_0 \geq 0$ and $\omega|_{[0, T_0] \times Y} = d(f \lambda^-)$ for some
smooth function $f : [0, T_0] \times Y \to \mathbb{R}_{>0}$ such that
$f \lambda^-|_{\{0\} \times Y} = \lambda^-$ and
$f \lambda^-|_{\{T_0\} \times Y} = \lambda^+$,
\item
a sequence $K_X^0$ of smooth cycles in $X$ with closed support and
bijections $\mu^\pm : K_X^0 \to K_{Y^\pm}^0$ such that
for some $T \geq 0$,
$x|_{(-\infty, -T] \times Y^-} = (-\infty, -T] \times \mu^-(x)$
and $x|_{[T, \infty) \times Y^+} = [T, \infty) \times \mu^+(x)$,
\item
an $\omega$-compatible almost complex structure $J$ of $X$ whose restrictions to
$(-\infty, -T] \times Y^-$ and $[T, \infty) \times Y^+$ coincide with those induced
by $J^-$ and $J^+$ respectively for some $T \geq 0$ and
\item
a pre-Kuranishi structure of $\widehat{\mathcal{M}}(X, \omega, J)$ and a family of multisections
of its fiber products compatible with $\mathcal{B}_{Y^\pm}$, which is denoted by $\mathcal{B}_X$.
\end{itemize}
We note that for the algebra of SFT of $X$,
$\delta = \min(L_{Y^-, \min}, L_{Y^+, \min})$ is admissible for any $C_2 \geq 0$.
(We can define the generating function $\mathcal{F}$ for $X$ as an element of
$(\hbar^{-1} \mathcal{D}_X^{\leq 0})^{\star, \delta}
/ J^{\star, \delta}_{\overline{C}_0, \overline{C}_1, \overline{C}_2}$.)
We say a concordance
$\mathcal{C}_X$ is trivial if
$(Y^-, \lambda^-, K_{Y^-}, K_{Y^-}^0, K_{Y^-}^2, J^-, \mathcal{B}_{Y^-})
= (Y^+, a \lambda^+, K_{Y^+}, K_{Y^+}^0, K_{Y^+}^2, J^+, \mathcal{B}_{Y^+})$ for some
$a > 0$.
A short concordance $\mathcal{C}_X$ is a concordance such that
$(Y^-, \lambda^-) = (Y^+, \lambda^+)$ and $T_0 = 0$, that is,
$X = (-\infty, 0] \times Y^- \cup [0, \infty) \times Y^+$.
First we prove the following.
\begin{lem}\label{trivial generating function}
For a trivial concordance $\mathcal{C}_X$,
the generating function $\mathcal{F}$
is homotopic to the trivial generating function
\[
\mathcal{F}^{\text{\it tri}} = \hbar^{-1} \sum_c q^-_{\hat c^\ast} p^+_{\hat c},
\]
where the sum is taken over all simplices in $K_{Y^+}$ not contained in
$\overline{P}_{Y^+}^{\text{bad}}$.
\end{lem}
First we consider the case of a trivial short concordance.
We denote the same
$(Y^\pm, \lambda^\pm, K_{Y^\pm}, K_{Y^\pm}^0, K_{Y^\pm}^2, J^\pm, \mathcal{B}_{Y^\pm})$
by $(Y, \lambda, K_Y, K_Y^0, K_Y^2, J, \mathcal{B}_Y)$,
and regard the symplectization $(X, \omega) = (Y \times \mathbb{R}, d(e^\sigma \lambda))$
as a trivial short concordance.
For each pair $(\hat c = c \theta^D_c, \hat \eta = \theta^{^t\! D}_\eta \eta)$,
let $\overline{\mathcal{M}}^{E_{\hat \omega}=0}_{g = 0, \# \pm\infty = 1}(\hat c, \hat \eta)
\subset (\overline{\mathcal{M}}^X)^0_{(\hat c, \emptyset, \hat \eta)}$
be the component which consists of connected holomorphic buildings of genera $g = 0$
with one limit circle for each end and
without marked points whose $E_{\hat \omega}$-energies are zero.
(Namely, these are trivial cylinders in the $0$-th floor.)
\begin{lem}
The chain map
$\varphi : C_\ast(\overline{P}_Y, \overline{P}_Y^{\text{\normalfont bad}};
\S^D \otimes \mathbb{Q})
\to C_\ast(\overline{P}_Y, \overline{P}_Y^{\text{\normalfont bad}}; \S^D \otimes \mathbb{Q})$
defined by
\[
\varphi(\hat c) = \sum_{c'} \bigl[\overline{\mathcal{M}}^{E_{\hat \omega}=0}_{g = 0, \# \pm\infty = 1}(\hat c,
[\overline{P}] \cap (\hat c')^\ast)\bigr]^0 \hat c'
\]
is chain homotopic to the identity, where
the sum is taken over all simplices $c'$ in $K_Y^0$ not contained in
$\overline{P}_Y^{\text{\normalfont bad}}$.
\end{lem}
\begin{proof}
It is easy to check that $\varphi$ is indeed a chain map.
Therefore it is enough to show that
\begin{equation}
\bigl[\overline{\mathcal{M}}^{E_{\hat \omega}=0}_{g = 0, \# \pm\infty = 1}(x, [\overline{P}] \cap \alpha)\bigr]^0
= \langle x, \alpha \rangle \label{equation of intersection theory}
\end{equation}
for any cycle
$x \in C_\ast(\overline{P}_Y, \overline{P}_Y^{\text{bad}}; \S^D \otimes \mathbb{Q})$
and any cocycle $\alpha \in C^\ast(\overline{P}_Y, \overline{P}_Y^{\text{bad}};
\S^D \otimes \mathbb{Q})$.
Recall that for the fundamental chain
\[
[\overline{P}] = \sum_\zeta \frac{1}{m_\zeta} \zeta \theta^{\overline{P}}_\zeta
\in C_{\dim P - 1}(\overline{P}, \overline{P}^{\text{no}}; \S^{\overline{P}} \otimes \mathbb{Q}),
\]
$\rho_\ast [\overline{P}] \in C_{\dim P -1} (\overline{P} \times \overline{P},
\overline{P}^{^t\text{bad}} \times \overline{P} \cup
\overline{P} \times \overline{P}^{\text{bad}};
p_1^\ast \S^{\lsuperscript{D}{t}} \otimes p_2^\ast \S^D \otimes \mathbb{Q})$ is defined by
\[
\rho_\ast [\overline{P}] = \sum_\zeta \frac{1}{m_\zeta} \theta^{\lsuperscript{D}{t}}_\zeta
(\rho_\ast \zeta) \theta^D_\zeta,
\]
where
\[
\rho_\ast \zeta = \sum_{0 \leq p \leq n} \partial_{p+1} \dots \partial_n \zeta
\times \partial_0 \dots \partial_{p+1} \zeta.
\]
Note that by definition,
the left hand side of Equation (\ref{equation of intersection theory}) coincides with
\begin{equation}
\Bigl\langle \sum_{\zeta, p}
\Bigl[\overline{\mathcal{M}}^{E_{\hat \omega}=0}_{g = 0, \# \pm\infty = 1}(x,
\theta^{\lsuperscript{D}{t}}_\zeta \partial_{p+1} \dots \partial_n \zeta)\Bigr]^0
\partial_0 \dots \partial_{p+1} \zeta \theta^D_\zeta, \alpha \Bigr\rangle.
\label{expansion of cap product}
\end{equation}
We rewrite $\sum_{\zeta, p} [\overline{\mathcal{M}}^{E_{\hat \omega}=0}_{g = 0, \# \pm\infty = 1}(x,
\theta^{\lsuperscript{D}{t}}_\zeta \partial_{p+1} \dots \partial_n \zeta)]^0
\partial_0 \dots \partial_{p+1} \zeta \theta^D_\zeta$ as
the virtual fundamental chain of the fiber product of
$\overline{\mathcal{M}}^{E_{\hat \omega}=0}_{g = 0, \# \pm\infty = 1}(x, \cdot)$ with
$\rho_\ast [\overline{P}]$,
and prove that this is homologous to the virtual fundamental chain of
the fiber product with $\Delta_\ast [\overline{P}]$.
More precisely, we construct these fiber products as follows.
For simplices $c \subset \overline{P}$ and
$\eta \subset \overline{P} \times \overline{P}$,
we define $\widehat{\mathcal{M}}^{E_{\hat \omega}=0}_{g = 0, \# \pm\infty = 1}(c, \eta)$
by the inverse image of $c \times \Delta_{\overline{P}} \subset
\overline{P} \times (\overline{P} \times \overline{P})$ by the map
\[
(\mathrm{ev}_{-\infty} \times \mathrm{ev}_{+\infty}) \times \pi_1
: \widehat{\mathcal{M}}^{E_{\hat \omega}=0}_{g = 0, \# \pm\infty = 1} \times \eta
\to (\overline{P} \times \overline{P}) \times \overline{P},
\]
where $\pi_1 : \overline{P} \times \overline{P} \to \overline{P}$ is the first projection.
Similarly, for simplices with local coefficients $\hat c = c \theta^D_c$ and
$\hat \eta = \theta^{^t\! D}_\eta \eta \theta^D_\eta$,
we define $\overline{\mathcal{M}}^{E_{\hat \omega}=0}_{g = 0, \# \pm\infty = 1}(\hat c, \hat \eta)$
by choosing lifts $\tilde c \subset P$ and
$\tilde \eta \subset \overline{P} \times \overline{P}$ of $c$ and $\eta$ respectively.
Its orientation is defined by using $\theta^D_c$ and $\theta^{^t\! D}_\eta$.
For each
\[
\widehat{\mathcal{M}}^{E_{\hat \omega}=0}_{g = 0, \# \pm\infty = 1}
(c, \partial_{p+1} \dots \partial_n \zeta \times \partial_0 \dots \partial_{p+1} \zeta)
\subset \widehat{\mathcal{M}}^{E_{\hat \omega}=0}_{g = 0, \# \pm\infty = 1}
(x, \rho_\ast [\overline{P}]),
\]
we use the perturbed multisection
defined by the pull back by the submersion to
$\widehat{\mathcal{M}}^{E_{\hat \omega}=0}_{g = 0, \# \pm\infty = 1}
(c, \partial_{p+1} \dots \partial_n \zeta)$.
Then (\ref{expansion of cap product}) coincides with
\begin{equation}
\bigl\langle (\pi_2)_\ast
\bigl(\overline{\mathcal{M}}^{E_{\hat \omega}=0}_{g = 0}(x, \rho_\ast [\overline{P}])\bigr),
\alpha\bigr\rangle,
\label{another expression of cap product}
\end{equation}
where $\pi_2$ is the strong smooth map defined by
the second projection $\overline{P} \times \overline{P} \to \overline{P}$.
In (\ref{another expression of cap product}), we can replace $\alpha$ with
a closed form $\tilde \alpha$ (with local coefficient) which represents
$\alpha \in H^\ast(\overline{P}_Y, \overline{P}_Y^{\text{bad}}; \S^D \otimes \mathbb{Q})$
and rewrite (\ref{another expression of cap product}) as
\begin{equation}
\int_{\overline{\mathcal{M}}^{E_{\hat \omega}=0}_{g = 0}(x, \rho_\ast [\overline{P}])}
\pi_2^\ast \tilde \alpha.
\label{int on rho ast}
\end{equation}
Since $\overline{\mathcal{M}}^{E_{\hat \omega}=0}_{g = 0}(x, \rho_\ast [\overline{P}])$ and
$\overline{\mathcal{M}}^{E_{\hat \omega}=0}_{g = 0}(x, \Delta_\ast [\overline{P}])$ are
cobordant by $\overline{\mathcal{M}}^{E_{\hat \omega}=0}_{g = 0}(x, \epsilon_\ast [\overline{P}])$,
(\ref{int on rho ast}) coincides with
\begin{equation}
\int_{\overline{\mathcal{M}}^{E_{\hat \omega}=0}_{g = 0}(x, \Delta_\ast [\overline{P}])}
\pi_2^\ast \tilde \alpha.
\label{int on Delta ast}
\end{equation}
For a simplex $c \subset \overline{P}$,
let $\widehat{\mathcal{M}}^{E_{\hat \omega}=0}_{g = 0}(c, \cdot)$ be the space
defined by the fiber product with $c$ on the $-\infty$-side. (For $+\infty$-side,
we do not take fiber product.)
Note that there exists a submersion from
$\widehat{\mathcal{M}}^{E_{\hat \omega}=0}_{g = 0}(c, \cdot)$ to
$\widehat{\mathcal{M}}^{E_{\hat \omega}=0}_{g = 0}(c, \Delta_\ast [\overline{P}])$.
In fact, the only difference is that for the construction of a perturbed multisection
of the latter, we need to make the zero set transverse to all simplices in $\overline{P}$.
Define $\overset{\,\!_-\,\!_\wedge}{\mathcal{M}}\,\!_{g = 0}^{E_{\hat \omega}=0}(x, \cdot)$
by the space of holomorphic buildings with $S^1$-coordinates
only on $-\infty$-limit circle.
Then (\ref{int on Delta ast}) coincides with
\begin{equation}
\int_{\overset{\,\!_-\,\!_\wedge}{\mathcal{M}}\,\!_{g = 0}^{E_{\hat \omega}=0}(x, \cdot)}
\pi_{+\infty}^\ast \tilde \alpha.
\label{int on cdot}
\end{equation}
Since we do not need perturbation for
$\overset{\,\!_-\,\!_\wedge}{\mathcal{M}}\,\!_{g = 0}^{E_{\hat \omega}=0}(x, \cdot)$,
(\ref{int on cdot}) coincides with $\langle x, \alpha \rangle$.
\end{proof}
\begin{proof}[Proof of Lemma \ref{trivial generating function}]
First we prove the case of trivial short concordance.
For each $A \geq 0$, define an ideal $\mathcal{I}^\delta_A \subset \mathcal{D}\D_X^{\leq 0, \delta}$ by
\begin{align*}
\mathcal{I}^\delta_A = \{&\sum
a_{(x_i), (\hat c_i^\ast), (\hat c'_i), g} t_{x_1} \dots t_{x_{k_t}} q^-_{\hat c_1^\ast}
\dots q^-_{\hat c_{k_q}^\ast} p^+_{\hat c'_1} \dots p^+_{\hat c'_{k_p}} \hbar^g
\in \mathcal{D}\D_X^{\leq 0, \delta};\\
&a_{(x_i), (\hat c_i^\ast), (\hat c'_i), g} = 0 \text{ if }
\widetilde{g}_\delta \leq A\},
\end{align*}
and define $\mathcal{I}_A^{\star, \delta}
= \mathcal{I}^\delta_A \cap (\hbar^{-1} \mathcal{D}_X^{\leq 0})^{\star, \delta}$.
Then the generating function $\mathcal{F}$ satisfies
\[
\mathcal{F} \equiv \hbar^{-1} \sum_{\hat c, \hat c'} [\overline{\mathcal{M}}^{E_{\hat \omega}=0}_{g = 0, \# \pm\infty = 1}
(\hat c, [\overline{P}] \cap (\hat c')^\ast)]^0 q^-_{\hat c^\ast} p^+_{\hat c'}
\]
in $(\hbar^{-1} \mathcal{D}_X^{\leq 0})^{\star, \delta}
/ (J^{\star, \delta}_{\overline{C}_0, \overline{C}_1, \overline{C}_2} + \mathcal{I}_0^{\star, \delta})$
Let
\begin{align*}
C_\ast(\overline{P}_Y, \overline{P}_Y^{\text{bad}}; \S^D \otimes \mathbb{Q})
&\to C_\ast(\overline{P}_Y, \overline{P}_Y^{\text{bad}};\allowbreak \S^D \otimes \mathbb{Q})\\
\hat c &\mapsto \sum_{\hat c'} a_{\hat c, \hat c'} \hat c'
\end{align*}
be the chain homotopy from $\varphi$ to $\mathrm{id}$ given in the above lemma,
that is, the family $a_{\hat c, \hat c'}$ satisfies
\[
\hat c - \sum_{c'} [\overline{\mathcal{M}}^{E_{\hat \omega}=0}_{g = 0, \# \pm\infty = 1}(\hat c,
[\overline{P}] \cap (\hat c')^\ast)]^0 \hat c'
= \sum_{\hat c'} a_{\hat c, \hat c'} \partial \hat c'
+ \sum_{\hat c'} a_{\partial \hat c, \hat c'} \hat c'
\]
for any $\hat c$.
Define
\[
\mathcal{K} = \hbar^{-1} \sum_{c, c'} a_{\hat c, \hat c'} q_{\hat c^\ast} p_{\hat c'}
\in (\hbar^{-1} \mathcal{D}_X^{\leq 0})^{\star, \delta}
/ J^{\star, \delta}_{\overline{C}_0, \overline{C}_1, \overline{C}_2}.
\]
Then $e^{[\widehat{D}_X, \tau \mathcal{K}]} e^{\mathcal{F}}$ ($\tau \in [0, 1]$) is a homotopy from
$\mathcal{F}$ to a generating function $\mathcal{F}^1$ which satisfies
\begin{equation}
\mathcal{F}^1 \equiv \hbar^{-1} \sum_{\hat c} q^-_{\hat c^\ast} p^+_{\hat c} \label{F trivial}
\end{equation}
in $(\hbar^{-1} \mathcal{D}_X^{\leq 0})^{\star, \delta}
/ (J^{\star, \delta}_{\overline{C}_0, \overline{C}_1, \overline{C}_2} + \mathcal{I}_0^{\star, \delta})$.
Hence we may assume $\mathcal{F}$ also satisfies the above equation
in $(\hbar^{-1} \mathcal{D}_X^{\leq 0})^{\star, \delta}
/ (J^{\star, \delta}_{\overline{C}_0, \overline{C}_1, \overline{C}_2} + \mathcal{I}_0^{\star, \delta})$.
We claim that there exists $\mathcal{G} \in (\hbar^{-1} \mathcal{D}_X^{\leq 0})^{\star, \delta}
/ J^{\star, \delta}_{\overline{C}_0, \overline{C}_1, \overline{C}_2}$ such that
\[
e^{\mathcal{F}} \star e^{\mathcal{G}} = e^{\mathcal{F}^{\text{\it tri}}}
\]
in $\mathcal{D}\D_X^{\leq 0, \delta}
/ \widetilde{J}^{\leq 0, \delta}_{\overline{C}_0, \overline{C}_1, \overline{C}_2}$, that is,
$\mathcal{F} \Diamond \mathcal{G} = \mathcal{F}^{\text{\it tri}}$ in
$(\hbar^{-1} \mathcal{D}_X^{\leq 0})^{\star, \delta}
/ J^{\star, \delta}_{\overline{C}_0, \overline{C}_1, \overline{C}_2}$.
This can be proved as follows.
Let $0 = A_0 < A_1 < A_2 < \dots $ be all constants $A$ such that
$\bigcap_{\epsilon > 0} (\widetilde{J}^{\leq 0, \delta}_{\overline{C}_0, \overline{C}_1,
\overline{C}_2} + \mathcal{I}^\delta_{A - \epsilon}) \supsetneq
\widetilde{J}^{\leq 0, \delta}_{\overline{C}_0, \overline{C}_1, \overline{C}_2} + \mathcal{I}^\delta_A$.
Since $\mathcal{F}$ satisfies equation (\ref{F trivial}),
it is easy to construct $\mathcal{G}_{\leq m} = \mathcal{G}_0 + \mathcal{G}_1 + \dots + \mathcal{G}_m \in
(\hbar^{-1} \mathcal{D}_X^{\leq 0})^{\star, \delta}
/ (J^{\star, \delta}_{\overline{C}_0, \overline{C}_1, \overline{C}_2}
+ \mathcal{I}^{\star, \delta}_{A_m})$ inductively such that
$\mathcal{G}_{\leq m} \equiv \mathcal{G}_{\leq m-1}$ in
$(\hbar^{-1} \mathcal{D}_X^{\leq 0})^{\star, \delta}
/ (J^{\star, \delta}_{\overline{C}_0, \overline{C}_1, \overline{C}_2}
+ \mathcal{I}^{\star, \delta}_{A_{m-1}})$ and
\[
e^{\mathcal{F}} \star e^{\mathcal{G}_{\leq m}} \equiv e^{\mathcal{F}^{\text{\it tri}}}
\]
in $\mathcal{D}\D_X^{\leq 0, \delta}
/ (\widetilde{J}^{\leq 0, \delta}_{\overline{C}_0, \overline{C}_1, \overline{C}_2}
+ \mathcal{I}^\delta_{A_m})$.
Therefore we can construct a required $\mathcal{G}$.
Since the composition of $X$ and $X$ is isomorphic to $X$,
$\mathcal{F} \Diamond \mathcal{F}$ is homotopic to $\mathcal{F}$.
Hence $\mathcal{F} \Diamond \mathcal{F} \Diamond \mathcal{G}$ is homotopic to $\mathcal{F} \Diamond \mathcal{G}$.
Therefore, any generating function $\mathcal{F}$ ($= \mathcal{F} \Diamond \mathcal{F}^{\text{\it tri}}$) of $X$
is homotopic to $\mathcal{F}^{\text{\it tri}}$.
(All generating functions
$\mathcal{F}$, $\mathcal{F} \Diamond \mathcal{F}$, $\mathcal{F} \Diamond \mathcal{F} \Diamond \mathcal{G}$ and $\mathcal{F} \Diamond \mathcal{G}$
are elements of $(\hbar^{-1} \mathcal{D}_X^{\leq 0})^{\star, \delta}
/ J^{\star, \delta}_{\overline{C}_0, \overline{C}_1, \overline{C}_2}$.)
Finally we consider the case of general trivial concordance.
Since $\omega|_{[0, T_0] \times Y} = d(f \lambda^-)$ for some
smooth function $f : [0, T_0] \times Y \to \mathbb{R}_{>0}$ such that
$f \lambda^-|_{\{0\} \times Y} = \lambda^-$ and
$f \lambda^-|_{\{T_0\} \times Y} = \lambda^+$,
$(X, \omega)$ is isomorphic to the trivial short concordance
$((-\infty, 0] \cup [0, \infty)) \times Y^+$ of $Y^+$ by
\begin{align*}
[0, T_0] \times Y &\hookrightarrow (-\infty, 0] \times Y^+ \\
(\sigma, y) &\mapsto (\log f(\sigma), y)
\end{align*}
and
\begin{align*}
(-\infty, 0] \times Y^- &\hookrightarrow (-\infty, 0] \times Y^+ \\
(\sigma, y) &\mapsto (\sigma + \log a, y).
\end{align*}
We can construct the generating function for $(X, \omega)$ by
the same data as those for the trivial short concordance of $Y^+$.
Then it is easy to check that this generating function is also homotopic to
the trivial generating function.
\end{proof}
Let $(Y^\pm, \lambda^\pm)$ be two arbitrary contact manifolds and
$(X, \omega)$ be an arbitrary cobordism from $(Y^-, \lambda^-)$ to $(Y^+, \lambda)$.
We assume that the generating functions for $Y^-$ and $Y^+$ are defined by
$\mathcal{C}_{Y^-} = (Y^-, \lambda^-, K_{Y^-}, K_{Y^-}^0, K_{Y^-}^2, J^-,
\mathcal{B}_{Y^-})$ and $\mathcal{C}_{Y^+}
= (Y^+, \lambda^+, K_{Y^+}, K_{Y^+}^0, K_{Y^+}^2, J^+, \mathcal{B}_{Y^+})$ respectively,
and that the generating function $\mathcal{F}_X \in (\hbar^{-1} \mathcal{D}_X^{\leq 0})^{\star, \delta}
/ J^{\star, \delta}_{\overline{C}_0, \overline{C}_1, \overline{C}_2}$ for $X$
are defined by the data
$\mathcal{C}_X = (X, \omega, Y^\pm, \lambda^\pm, K_{Y^\pm},\allowbreak
K_X^0,\allowbreak K^0_{Y^\pm},\allowbreak \mu^\pm, K^2_{Y^\pm}, J, \mathcal{B}_X)$ compatible with
$\mathcal{C}_{Y^-}$ and $\mathcal{C}_{Y^+}$.
The argument in Section \ref{Homotopy} implies that the homotopy type of
$\mathcal{F}_X$ does not depend on
the choice of $\mathcal{C}_X$ if we fix $\mathcal{C}_{Y^\pm}$.
We denote the cohomology
$H^\ast(\mathcal{D}_X^{\leq \kappa} / J^{\leq \kappa, \delta}_{C_0, C_1, C_2}, D_{\mathcal{F}})$
for $\mathcal{C}_X$ by
$H^\ast(\mathcal{D}_{\mathcal{C}_X}^{\leq \kappa} / J^{\leq \kappa, \delta}_{C_0, C_1, C_2},
D_{\mathcal{C}_X})$.
Then this implies that cohomologies
$H^\ast(\mathcal{D}_{\mathcal{C}_X}^{\leq \kappa} / J^{\leq \kappa, \delta}_{C_0, C_1, C_2},
D_{\mathcal{C}_X})$
for $\mathcal{C}_X$ compatible with a fixed pair
$(\mathcal{C}_{Y^-}, \mathcal{C}_{Y^+})$ (and with the same $\mu^\pm$)
are naturally isomorphic.
Namely, for every pair $(\mathcal{C}_X, \mathcal{C}'_X)$,
there exists a unique isomorphism
\[
T_{\mathcal{C}'_X, \mathcal{C}_X}
: H^\ast(\mathcal{D}_{\mathcal{C}_X}^{\leq \kappa} / J^{\leq \kappa, \delta}_{C_0, C_1, C_2},
D_{\mathcal{C}_X})
\to H^\ast(\mathcal{D}_{\mathcal{C}'_X}^{\leq \kappa} / J^{\leq \kappa, \delta}_{C_0, C_1, C_2},
D_{\mathcal{C}'_X}),
\]
and these isomorphisms satisfy
$T_{\mathcal{C}_X, \mathcal{C}_X} = \mathrm{id}$ and
$T_{\mathcal{C}''_X, \mathcal{C}'_X} \circ T_{\mathcal{C}'_X, \mathcal{C}_X}
= T_{\mathcal{C}''_X, \mathcal{C}_X}$.
Similarly, cohomologies $H^\ast(\mathcal{L}_{\mathcal{C}_X}^{\leq \kappa}
/ J^{\leq \kappa}_{C_0, C_2}, d_{\mathcal{C}_X})
= H^\ast(\mathcal{L}_{X}^{\leq \kappa}
/ J^{\leq \kappa}_{C_0, C_2}, d_{\mathcal{F}_0})$
for $\mathcal{C}_X$ compatible with a fixed pair
$(\mathcal{C}_{Y^-}, \mathcal{C}_{Y^+})$ (and with the same $\mu^\pm$)
are naturally isomorphic.
Next we compare two SFT cohomologies of $X$ compatible with different pairs
$\mathcal{C}_{Y^\pm}$ for $(Y^\pm, \xi^\pm)$.
First we treat the case where we do not change the contact forms $\lambda^\pm$.
(To treat the general case, we cannot fix a filtration and need to take the limit
with respect to the filtration.)
\begin{lem}\label{short concordance for X}
Let $\mathcal{C}_X$ be a cobordism from $\mathcal{C}_{Y^-}$ to $\mathcal{C}_{Y^+}$,
and let
\begin{multline*}
\mathcal{C}_{X_1}
= (X_1, \omega_1, (Y^+, Y^+), (\lambda^+, \lambda^+), (K_{Y^+}, K_{Y^+_1}), K_X^0,
(K^0_{Y^+}, K^0_{Y^+_1}),\\
\mu^\pm, (K^2_{Y^+}, K^2_{Y^+_1}), J, \mathcal{B}_X)
\end{multline*}
be a short concordance from
$\mathcal{C}_{Y^+}$
to $\mathcal{C}_{Y^+_1}
= (Y^+, \lambda^+, K_{Y^+_1}, K_{Y^+_1}^0, K_{Y^+_1}^2, J^+_1, \mathcal{B}_{Y^+_1})$.
Then
\[
T_{\mathcal{F}_X}(\cdot \Diamond \mathcal{F}_{X_1})
: (\mathcal{D}^{\leq \kappa}_X / J^{\leq \kappa, \delta}_{C_0, C_1, C_2}, D_{\mathcal{F}_X})
\to (\mathcal{D}^{\leq \kappa}_{X \# X_1} / J^{\leq \kappa, \delta}_{C_0, C_1, C_2},
D_{\mathcal{F}_X \Diamond \mathcal{F}_{X_1}})
\]
and
\[
T_{(\mathcal{F}_X)_0}(\cdot \sharp (\mathcal{F}_{X_1})_0)
: (\mathcal{L}^{\leq \kappa}_X / J^{\leq \kappa}_{C_0, C_2}, d_{(\mathcal{F}_X)_0})
\to (\mathcal{L}^{\leq \kappa}_{X \# X_1} / J^{\leq \kappa}_{C_0, C_2},
d_{(\mathcal{F}_X) \sharp (\mathcal{F}_{X_1})_0})
\]
are chain homotopy equivalences.
\end{lem}
\begin{proof}
First we consider the case of general SFT.
Let
\begin{multline*}
\mathcal{C}_{X_2}
= (X_2, \omega_2, (Y^+, Y^+), (\lambda^+, \lambda^+), (K_{Y^+_1}, K_{Y^+}), K_X^0,
(K^0_{Y^+_1}, K^0_{Y^+}),\\
\mu^\pm, (K^2_{Y^+_1}, K^2_{Y^+}), J, \mathcal{B}_X)
\end{multline*}
be a short concordance from
$\mathcal{C}_{Y^+_1}$
to $\mathcal{C}_{Y^+}$.
Since $X_1 \# X_2$ is a trivial short concordance, its generating function
$\mathcal{F}_{X_1} \Diamond \mathcal{F}_{X_2}$ is homotopic to the trivial generating function
$\mathcal{F}^{\text{\it tri}}$.
Hence Lemma \ref{linearized T} (v) implies that
\begin{align*}
T_{\mathcal{F}_X}(\cdot \Diamond (\mathcal{F}_{X_1} \Diamond \mathcal{F}_{X_2}))
&: (\mathcal{D}^{\leq \kappa}_X / J^{\leq \kappa, \delta}_{C_0, C_1, C_2}, D_{\mathcal{F}_X}) \\
&\quad \to (\mathcal{D}^{\leq \kappa}_{X \# X_1 \# X_2} / J^{\leq \kappa, \delta}_{C_0, C_1, C_2},
D_{\mathcal{F}_X \Diamond \mathcal{F}_{X_1} \Diamond \mathcal{F}_{X_2}})
\end{align*}
is a chain homotopy equivalence.
By Lemma \ref{linearized T} (iv), this map coincides with the composition
$T_{\mathcal{F}_X \Diamond \mathcal{F}_{X_1}}(\cdot \Diamond \mathcal{F}_{X_2}) \circ
T_{\mathcal{F}_X}(\cdot \Diamond \mathcal{F}_{X_1})$.
Hence $T_{\mathcal{F}_X}(\cdot \Diamond \mathcal{F}_{X_1})$ has a left homotopy inverse and
$T_{\mathcal{F}_X \Diamond \mathcal{F}_{X_1}}(\cdot \Diamond \mathcal{F}_{X_2})$ has a right homotopy inverse.
Since we can apply the above argument for a cobordism $X \# X_1$ and
a short concordance $\mathcal{C}_{X_2}$,
$T_{\mathcal{F}_X \Diamond \mathcal{F}_{X_1}}(\cdot \Diamond \mathcal{F}_{X_2})$ has a right inverse.
Hence $T_{\mathcal{F}_X}(\cdot \Diamond \mathcal{F}_{X_1})$ is a chain homotopy equivalence.
The case of rational SFT is similar.
\end{proof}
The above lemma and the counterpart of a short concordance from
$\mathcal{C}_{Y^-_1}$ to $\mathcal{C}_{Y^-}$ imply that homologies
$H^\ast(\mathcal{D}_{\mathcal{C}_X}^{\leq \kappa} / J^{\leq \kappa, \delta}_{C_0, C_1, C_2},
D_{\mathcal{C}_X})$ and
$H^\ast(\mathcal{L}_{\mathcal{C}_X}^{\leq \kappa}
/ J^{\leq \kappa}_{C_0, C_2}, d_{\mathcal{C}_X})$
for $\mathcal{C}_X$ compatible with a fixed pair
$((Y^-, \lambda^-, \overline{K}^0_{Y^-}),
(Y^+, \lambda^+, \overline{K}^0_{Y^+}))$ (and with the same $\mu^\pm$)
are naturally isomorphic respectively.
(The naturality is due to Lemma \ref{linearized T} (iv), (v) (or its rational version) and
Lemma \ref{trivial generating function}.)
Therefore for any cobordism $(X, \omega)$ between two strict contact manifolds
$(Y^\pm, \lambda^\pm)$ and any $(\overline{K}^0_X, \overline{K}^0_{Y^\pm},
\mu^\pm)$,
we can define the limits
\begin{align}
&H^\ast_{\mathrm{SFT}}(X, \omega, Y^\pm, \lambda^\pm, \overline{K}^0_X,
\overline{K}^0_{Y^\pm}, \mu^\pm) \notag\\
&= \varprojlim_{C_2} \varinjlim_{\kappa, \delta} \varprojlim_{C_0, C_1}
H^\ast(\mathcal{D}_{\mathcal{C}_X}^{\leq \kappa} / J^{\leq \kappa, \delta}_{C_0, C_1, C_2},
D_{\mathcal{C}_X}).
\label{limit general X}
\end{align}
and
\begin{equation}
H^\ast_{\mathrm{RSFT}}(X, \omega, Y^\pm, \lambda^\pm, \overline{K}^0_X,
\overline{K}^0_{Y^\pm}, \mu^\pm)
= \varprojlim_{C_2} \varinjlim_{\kappa} \varprojlim_{C_0}
H^\ast(\mathcal{L}_{\mathcal{C}_X}^{\leq \kappa} / J^{\leq \kappa}_{C_0, C_2},
d_{\mathcal{C}_X}).
\label{limit rational X}
\end{equation}
We sometimes abbreviate these limits as $H^\ast(\mathcal{D}_X, D_X)$ and
$H^\ast(\mathcal{L}_X, d_X)$ respectively.
We will prove that these cohomology groups do not depend on the choice of
the contact forms of $(Y^\pm, \xi^\pm)$ later.
It is easy to check that for a pair of composable cobordism $(X, \omega)$ and
$(X', \omega')$, the limit of the linearizations of the composition maps define maps
\[
T_{\mathcal{F}_X}(\cdot \Diamond \mathcal{F}_{X'}) : H^\ast(\mathcal{D}_X, D_X)
\to H^\ast(\mathcal{D}_{X \# X'}, D_{X \# X'}),
\]
\[
T_{\mathcal{F}_{X'}}(\mathcal{F}_X \Diamond \cdot) : H^\ast(\mathcal{D}_{X'}, D_{X'})
\to H^\ast(\mathcal{D}_{X \# X'}, D_{X \# X'}),
\]
\[
T_{(\mathcal{F}_X)_0}(\cdot \sharp (\mathcal{F}_{X'})_0) : H^\ast(\mathcal{L}_X, d_X)
\to H^\ast(\mathcal{L}_{X \# X'}, d_{X \# X'}),
\]
and
\[
T_{(\mathcal{F}_{X'})_0}((\mathcal{F}_X)_0 \sharp \cdot) : H^\ast(\mathcal{L}_{X'}, d_{X'})
\to H^\ast(\mathcal{L}_{X \# X'}, d_{X \# X'}).
\]
Next we consider the SFT cohomologies of a contact manifold.
First we compare two cohomology groups defined by the same contact form with
different other data.
\begin{lem}\label{short concordance for Y}
For a short concordance $\mathcal{C}_X$ from $\mathcal{C}_{Y^-}$
to $\mathcal{C}_{Y^+}$, the linear maps
\[
i_{\mathcal{F}_X}^\pm : (\mathcal{W}^{\leq \kappa}_{Y^\pm} / I^{\leq \kappa}_{C_0, C_1, C_2}, D_{Y^\pm})
\to (\mathcal{D}^{\leq \kappa, L_{\min}}_X / J^{\leq \kappa, L_{\min}}_{C_0, C_1, C_2}, D_{\mathcal{F}_X})
\]
and
\[
i_{(\mathcal{F}_X)_0}^\pm
: (\mathcal{P}^{\leq \kappa}_{Y^\pm} / I^{\leq \kappa}_{C_0, C_2}, d_{Y^\pm})
\to (\mathcal{L}^{\leq \kappa}_X / J^{\leq \kappa}_{C_0, C_2}, d_{(\mathcal{F}_X)_0})
\]
are chain homotopy equivalences,
and the compositions of the induced maps
\[
A = i_{\mathcal{F}_X}^- \circ (i_{\mathcal{F}_X}^+)^{-1}
: H^\ast(\mathcal{W}^{\leq \kappa}_{Y^+} / I^{\leq \kappa}_{C_0, C_1, C_2}, D_{Y^+})
\to H^\ast(\mathcal{W}^{\leq \kappa}_{Y^-} / I^{\leq \kappa}_{C_0, C_1, C_2}, D_{Y^-})
\]
and
\[
A^0 = i_{(\mathcal{F}_X)_0}^- \circ (i_{(\mathcal{F}_X)_0}^+)^{-1}
: H^\ast(\mathcal{P}^{\leq \kappa}_{Y^+} / I^{\leq \kappa}_{C_0, C_2}, d_{Y^+})
\to H^\ast(\mathcal{P}^{\leq \kappa}_{Y^-} / I^{\leq \kappa}_{C_0, C_2}, d_{Y^-})
\]
do not depend on the short concordance $\mathcal{C}_X$.
\end{lem}
\begin{proof}
We prove the case of general SFT. The case of rational SFT is similar.
First we consider the case of a trivial short concordance.
Note that for the trivial generating function $\mathcal{F}^{\text{\it tri}}$,
$i_{\mathcal{F}^{\text{\it tri}}}^\pm$ coincide with the identity map under the natural
identification $\mathcal{W}^{\leq \kappa}_{Y^\pm} / I^{\leq \kappa}_{C_0, C_1, C_2}
\cong \mathcal{D}^{\leq \kappa, L_{\min}}_X / J^{\leq \kappa, L_{\min}}_{C_0, C_1, C_2}$
which maps $q_{\hat c^\ast}$ and $p_{\hat c}$ to $q^-_{\hat c^\ast}$ and
$p^+_{\hat c}$ respectively.
Since the generating function $\mathcal{F}_X$ is homotopic to the trivial generating function,
Lemma \ref{homotopy and T} (iv) implies that
$i_{\mathcal{F}_X}^\pm$ is chain homotopic to the composition of $i_{\mathcal{F}^{\text{\it tri}}}^\pm$
and the isomorphism defined by the homotopy.
Hence $i_{\mathcal{F}_X}^\pm$ are also chain homotopy equivalence.
Next we consider the general case.
Let $\mathcal{C}_{X'}$ be a short concordance from $\mathcal{C}_{Y^+}$ to
$\mathcal{C}_{Y^-}$.
Then $T_{\mathcal{F}_X}(\cdot \Diamond \mathcal{F}_{X'}) \circ i_{\mathcal{F}_X}^-
= i_{\mathcal{F}_X \Diamond \mathcal{F}_{X'}}^-
: (\mathcal{W}^{\leq \kappa}_{Y^-} / I^{\leq \kappa}_{C_0, C_1, C_2}, D_{Y^\pm})
\to (\mathcal{D}^{\leq \kappa}_{X \# X'} / J^{\leq \kappa, L_{\min}}_{C_0, C_1, C_2},
D_{\mathcal{F}_X \Diamond \mathcal{F}_{X'}})$
are chain homotopy equivalence since $\mathcal{F}_X \Diamond \mathcal{F}_{X'}$ is homotopic to the
generating function of a trivial short concordance.
Since $T_{\mathcal{F}_X}(\cdot \Diamond \mathcal{F}_{X'})$ is also a chain homotopy equivalence by
Lemma \ref{short concordance for X}, so is $i_{\mathcal{F}_X}^-$.
Similarly, $i_{\mathcal{F}_X}^+$ is also a chain homotopy equivalence.
Finally we check the independence of $A = i_{\mathcal{F}_X}^- \circ (i_{\mathcal{F}_X}^+)^{-1}$.
For any two short concordances $\mathcal{C}_X$ and $\mathcal{C}'_X$,
there exists a smooth family of exact cobordisms
$(X^\tau, \omega^\tau)_{\tau \in I}$ such that $(X^0, \omega^0) = (X, \omega)$
and $(X^1, \omega^1) = (X', \omega')$, and
we can construct a homotopy from $\mathcal{F}_X$ to $\mathcal{F}_{X'}$.
This implies that there exists an isomorphism
$T : (\mathcal{D}_X / J_{C_0, C_1}^\delta, D_\mathcal{F}) \to
(\mathcal{D}_{X'} / J_{C_0, C_1}^\delta, D_{\mathcal{F}'})$
such that $i_{\mathcal{F}_{X'}}^\pm$ coincides with $T \circ i_{\mathcal{F}_X}^\pm$ up to chain homotopy.
Hence $A$ does not depend on the choice of the short concordance.
\end{proof}
We denote the isomorphisms $A$ and $A^0$ in the above lemma by
$A_{\mathcal{C}_{Y^-}, \mathcal{C}_{Y^+}}$ and
$A_{\mathcal{C}_{Y^-}, \mathcal{C}_{Y^+}}^0$ respectively.
The above lemma implies that if we fix $(Y, \lambda, \overline{K}_Y^0)$,
the cohomologies
\[
H^\ast(\mathcal{W}_{(Y, \lambda, K_Y, \overline{K}_Y^0)}^{\leq \kappa}
/ I^{\leq \kappa}_{C_0, C_1, C_2}, D_{(Y, \lambda, K_Y, K_Y^0, K_Y^2, J, \mathcal{B})})
\]
and
\[
H^\ast(\mathcal{P}_{(Y, \lambda, K_Y, \overline{K}_Y^0)}^{\leq \kappa}
/ I^{\leq \kappa}_{C_0, C_2}, d_{(Y, \lambda, K_Y, K_Y^0, K_Y^2, J, \mathcal{B})})
\]
defined by various data $\mathcal{C}_Y = (Y, \lambda, K_Y, K_Y^0, K_Y^2, J, \mathcal{B})$
of the same $(Y, \lambda, \overline{K}_Y^0)$ are naturally isomorphic respectively, and
the isomorphisms are given by the above $A_{\mathcal{C}_{Y^-}, \mathcal{C}_{Y^+}}$
and $A_{\mathcal{C}_{Y^-}, \mathcal{C}_{Y^+}}^0$.
The naturality of isomorphisms $A_{\mathcal{C}_{Y^-}, \mathcal{C}_{Y^+}}$
is proved as follows.
$A_{\mathcal{C}_Y, \mathcal{C}_Y} = \mathrm{id}$ is due to Lemma \ref{trivial generating function}.
$A_{\mathcal{C}_{Y''}, \mathcal{C}_{Y'}} \circ A_{\mathcal{C}_{Y'}, \mathcal{C}_Y}
= A_{\mathcal{C}_{Y''}, \mathcal{C}_Y}$ is because the following diagram is commutative
by Lemma \ref{linearized T},
where $\mathcal{C}_{X^-}$ is a short concordance from
$\mathcal{C}_{Y''}$ to $\mathcal{C}_{Y'}$ with a generating function $\mathcal{F}^-$
and $\mathcal{C}_{X^+}$ is a short concordance from
$\mathcal{C}_{Y'}$ to $\mathcal{C}_Y$ with a generating function $\mathcal{F}^+$,
and we abbreviate $H^\ast(\mathcal{W}_Y^{\leq \kappa} / I^{\leq \kappa}_{C_0, C_1, C_2}, D_Y)$
or $H^\ast(\mathcal{D}_{\mathcal{C}_{X}}^{\leq \kappa} / J^{\leq \kappa, L_{\min}}_{C_0, C_1, C_2})$
by $H^\ast(\mathcal{C}_Y)$ or $H^\ast(\mathcal{C}_X)$ respectively.
\[
\begin{tikzcd}
H^\ast(\mathcal{C}_{Y''})
\ar{r}{i^-_{\mathcal{F}^-}}
\ar{ddrr}[swap]{i^-_{\mathcal{F}^- \Diamond \mathcal{F}^+}}
&H^\ast(\mathcal{C}_{X^-})
\ar{ddr}{\hspace{-5pt}T_{\mathcal{F}^-}(\cdot \Diamond \mathcal{F}^+)}
&\ar{l}[swap]{i^+_{\mathcal{F}^-}}
H^\ast(\mathcal{C}_{Y'})
\ar{r}{i^-_{\mathcal{F}^+}}
&H^\ast(\mathcal{C}_{X^+})
\ar{ddl}[swap, near end]{T_{\mathcal{F}^+}(\mathcal{F}^- \Diamond \cdot) \hspace{-5pt}}
&\ar{l}[swap]{i^+_{\mathcal{F}^+}}
H^\ast(\mathcal{C}_Y)
\ar{ddll}{i^+_{\mathcal{F}^- \Diamond \mathcal{F}^+}}\\
\\
&&H^\ast(\mathcal{C}_{X^- \# X^+})&&
\end{tikzcd}
\]
Therefore we can define the limits
\begin{align}
&H^\ast_{\mathrm{SFT}}(Y, \lambda, \overline{K}^0_Y) \notag\\
&= \varprojlim_{C_2} \varinjlim_{\kappa} \varprojlim_{C_0, C_1}
H^\ast(\mathcal{W}_{(Y, \lambda, K_Y, \overline{K}_Y^0)}^{\leq \kappa}
/ I^{\leq \kappa}_{C_0, C_1, C_2}, D_{(Y, \lambda, K_Y, K_Y^0, K_Y^2, J, \mathcal{B})}).
\label{limit general Y}
\end{align}
and
\begin{align}
&H^\ast_{\mathrm{RSFT}}(Y, \lambda, \overline{K}^0_Y) \notag\\
&= \varprojlim_{C_2} \varinjlim_{\kappa} \varprojlim_{C_0}
H^\ast(\mathcal{P}_{(Y, \lambda, K_Y, \overline{K}_Y^0)}^{\leq \kappa}
/ I^{\leq \kappa}_{C_0, C_2}, d_{(Y, \lambda, K_Y, K_Y^0, K_Y^2, J, \mathcal{B})}).
\label{limit rational Y}
\end{align}
We sometimes abbreviate these limits as $H^\ast(\mathcal{W}_Y, D_Y)$ and
$H^\ast(\mathcal{P}_Y, d_Y)$ respectively.
For any cobordism $(X, \omega)$ from $(Y^-, \lambda^-)$ to
$(Y^+, \lambda^+)$, we can define
\begin{equation}
i_{X}^\pm : H^\ast_{\mathrm{SFT}}(Y^\pm, \lambda^\pm, \overline{K}_{Y^\pm}^0)
\to H^\ast_{\mathrm{SFT}}(X, \omega, Y^\pm, \lambda^\pm, \overline{K}^0_X,
\overline{K}^0_{Y^\pm}, \mu^\pm)
\label{well def of i_X^pm}
\end{equation}
and
\begin{equation}
i_{X, 0}^\pm : H^\ast_{\mathrm{RSFT}}(Y^\pm, \lambda^\pm, \overline{K}^0_{Y^\pm})
\to H^\ast_{\mathrm{RSFT}}(X, \omega, Y^\pm, \lambda^\pm, \overline{K}^0_X,
\overline{K}^0_{Y^\pm}, \mu^\pm)
\label{well def of i_{X, 0}^pm}
\end{equation}
by the limits of $i_{\mathcal{F}}^\pm$ and $i_{\mathcal{F}_0}^\pm$ respectively.
For example, the well-definedness of $i_X^+$ is due to the following fact:
Let $\mathcal{C}_{Y^\pm}$ and $\mathcal{C}'_{Y^\pm}$ be two data for
$(Y^\pm, \lambda^\pm)$,
$\mathcal{C}_{X}$ be a cobordism from $\mathcal{C}_{Y^-}$ to $\mathcal{C}_{Y^+}$,
and $\mathcal{C}_{X'}$ be a cobordism from $\mathcal{C}'_{Y^-}$ to $\mathcal{C}'_{Y^+}$.
Assume that both of $\mathcal{C}_{X}$ and $\mathcal{C}_{X'}$ are data of
the same cobordism $(X, \omega)$.
Then the following diagram is commutative by Lemma \ref{linearized T},
where $\mathcal{C}_{X_0}$ is a short concordance from $\mathcal{C}_{Y^+}$ to
$\mathcal{C}'_{Y^+}$, $\mathcal{C}_{X_1}$ is a short concordance from
$\mathcal{C}_{Y^-}$ to $\mathcal{C}'_{Y^-}$, and
$T : H^\ast(\mathcal{C}_{X \# X_0}) \to H^\ast(\mathcal{C}_{X_1 \# X'})$
is the isomorphism for a homotopy from $\mathcal{F}_{X} \Diamond \mathcal{F}_{X_0}$ to
$\mathcal{F}_{X_1} \Diamond \mathcal{F}_{X'}$.
The left column is the natural isomorphism for the SFT cohomology for
$(Y^+, \lambda^+)$,
and the right column is the natural isomorphism for the SFT cohomology for
$(X, \omega)$.
Therefore the compatibility of these isomorphisms and the maps $i^+_{\mathcal{F}_X}$,
$i^+_{\mathcal{F}_{X'}}$ implies the well-definedness of $i^+_{X}$.
\[
\begin{tikzcd}[column sep = huge]
H^\ast(\mathcal{C}_{Y^+})
\ar{rr}{i^+_{\mathcal{F}_X}}
\ar{d}[swap]{i^-_{\mathcal{F}_{X_0}}}
&&H^\ast(\mathcal{C}_{X})
\ar{d}{T_{\mathcal{F}_X}(\cdot \Diamond \mathcal{F}_{X_0})}\\
H^\ast(\mathcal{C}_{X_0})
\ar{rr}{T_{\mathcal{F}_{X_0}}(\mathcal{F}_X \Diamond \cdot)}
&&H^\ast(\mathcal{C}_{X \# X_0})
\ar{d}{T}\\
H^\ast(\mathcal{C}'_{Y^+})
\ar{u}{i^+_{\mathcal{F}_{X_0}}}
\ar{urr}{i^+_{\mathcal{F}_{X} \Diamond \mathcal{F}_{X_0}}\hspace{-10pt}}
\ar{rr}[swap]{\hspace{30pt} i^+_{\mathcal{F}_{X_1} \Diamond \mathcal{F}_{X'}}}
\ar{drr}[swap]{i^+_{\mathcal{F}_{X'}}}
&&H^\ast(\mathcal{C}_{X_1 \# X'})\\
&&H^\ast(\mathcal{C}_{X'})
\ar{u}[swap]{T_{\mathcal{F}_{X'}}(\mathcal{F}_{X_1} \Diamond \cdot)}
\end{tikzcd}
\]
\begin{prop}\label{algebraic structures of limits}
The limits of SFT cohomologies satisfies the following.
\begin{enumerate}[label= \normalfont (\roman*)]
\item
$H^\ast_{\mathrm{SFT}}(Y, \lambda, \overline{K}^0)$ inherits a structure of algebra.
\item
$H^\ast_{\mathrm{RSFT}}(Y, \lambda, \overline{K}^0)$ inherits a structure
of Poisson algebra.
\item
$H^\ast_{\mathrm{SFT}}(X, \omega, Y^\pm, \lambda^\pm, \overline{K}^0_X,
\overline{K}^0_{Y^\pm}, \mu^\pm)$ has a structure of
$H^\ast_{\mathrm{SFT}}(Y^\pm, \lambda^\pm, \overline{K}_{Y^\pm}^0)$-bimodule.
\item
$H^\ast_{\mathrm{RSFT}}(X, \omega, Y^\pm, \lambda^\pm, \overline{K}^0_X,
\overline{K}^0_{Y^\pm}, \mu^\pm)$ has a structure of
$H^\ast_{\mathrm{RSFT}}(Y^\pm, \lambda^\pm, \overline{K}_{Y^\pm}^0)$-bimodule.
\end{enumerate}
\end{prop}
\begin{rem}
$H^\ast_{\mathrm{RSFT}}(X, \omega, Y^\pm, \lambda^\pm, \overline{K}^0_X,
\overline{K}^0_{Y^\pm}, \mu^\pm)$ does not have a structure of Poisson module over
$H^\ast_{\mathrm{RSFT}}(Y^\pm, \lambda^\pm, \overline{K}_{Y^\pm}^0)$.
\end{rem}
\begin{proof}
First we consider (i) and (iii).
(\ref{D_Y diff alg}) implies that the multiplication of $\mathcal{W}_Y$ induces maps
\begin{align}
&H^\ast(\mathcal{W}_Y^{\leq \kappa_1} / I^{\leq \kappa_1}_{C_0, C_1 + \kappa_2 L_{\min}^{-1},
C_2 + \kappa_2}, D_Y) \times
H^\ast(\mathcal{W}_Y^{\leq \kappa_2} / I^{\leq \kappa_2}_{C_0, C_1 + \kappa_1 L_{\min}^{-1}, C_2},
D_Y) \notag\\
&\hphantom{H^\ast(\mathcal{W}_Y^{\leq \kappa_1} / I^{\leq \kappa_1}_{C_0, C_1 + \kappa_2 L_{\min}^{-1},
C_2 + \kappa_2}, D_Y)}
\to H^\ast(\mathcal{W}_Y^{\leq \kappa_1 + \kappa_2}
/ I^{\leq \kappa_1 + \kappa_2}_{C_0, C_1, C_2}, D_Y).
\label{multiplication of homology of W_Y with filtration}
\end{align}
Similarly, (\ref{D_F left Leibnitz}) and (\ref{D_F right Leibnitz}) imply that for
any cobordism $(X, \omega)$ from $(Y^-, \lambda^-)$ to $(Y^+, \lambda^+)$, the
$\mathcal{W}_{Y^\pm}$-bimodule structure of $\mathcal{D}_X$ induces maps
\begin{multline*}
H^\ast(\mathcal{W}_{Y^-}^{\leq \kappa_1} / I^{\leq \kappa_1}_{C_0, C'_1, C_2 + \kappa_2}, D_Y)
\times H^\ast(\mathcal{D}_X^{\leq \kappa_2} / J^{\leq \kappa_2, \delta}_{C_0, C_1
+ \kappa_1 \delta^{-1}, C_2}, D_{\mathcal{F}_X})\\
\to H^\ast(\mathcal{D}_X^{\leq \kappa_1 + \kappa_2} / J^{\leq \kappa_1 + \kappa_2, \delta}
_{C_0, C_1, C_2}, D_{\mathcal{F}_X}),
\end{multline*}
where $C'_1 = C_1 + \kappa_1 (\delta^{-1} - L_{\min}^{-1}) + \kappa_2 L_{\min}^{-1}$,
and
\begin{multline*}
H^\ast(\mathcal{D}_X^{\leq \kappa_1} / J^{\leq \kappa_1, \delta}_{C_0, C_1
+ \kappa_2 \delta^{-1}, C_2 + \kappa_1}, D_{\mathcal{F}_X}) \times
H^\ast(\mathcal{W}_{Y^+}^{\leq \kappa_2} / I^{\leq \kappa_2}_{C_0, C''_1, C_2}, D_{Y^+})\\
\to H^\ast(\mathcal{D}_X^{\leq \kappa_1 +\kappa_2} / J^{\leq \kappa_1 + \kappa_2, \delta}
_{C_0, C_1, C_2}, D_{\mathcal{F}_X}),
\end{multline*}
where $C''_1 = C_1 + \kappa_1 (\delta^{-1} - L_{\min}^{-1})$.
These multiplications satisfy the associativity condition.
Therefore the map $A = i_X^- \circ (i_X^+)^{-1}$ in Lemma
\ref{short concordance for Y} preserves the multiplication.
Namely, for any
$f \in H^\ast(\mathcal{W}^{\leq \kappa_1}_{Y^+}
/ I^{\leq \kappa}_{C_0, C_1 + \kappa_2 L_{\min}^{-1}, C_2 + \kappa_2}, D_{Y^+})$ and
$g \in H^\ast(\mathcal{W}^{\leq \kappa_2}_{Y^+}
/ I^{\leq \kappa}_{C_0, C_1 + \kappa_1 L_{\min}^{-1}, C_2}, D_{Y^+})$,
\[
i_{\mathcal{F}}^+(fg)
= 1 \underset{\mathcal{F}}{\overleftarrow{\ast}} f \underset{\mathcal{F}}{\overleftarrow{\ast}} g
= A(f) \underset{\mathcal{F}}{\overrightarrow{\ast}} A(g) \underset{\mathcal{F}}{\overrightarrow{\ast}} 1
= i_{\mathcal{F}}^-(A(f) A(g))
\]
in $H^\ast(\mathcal{D}^{\leq \kappa_1 + \kappa_2, L_{\min}}_X
/ J^{\leq \kappa_1 + \kappa_2, L_{\min}}_{C_0, C_1, C_2}, D_{\mathcal{F}_X})$.
Hence (\ref{multiplication of homology of W_Y with filtration}) depends only on
the triple $(Y, \lambda, \overline{K}^0)$, and it does not depend on the other choices
of $\mathcal{C}_Y$.
(\ref{multiplication of homology of W_Y with filtration}) induces the multiplication of
the limit $H^\ast_{\mathrm{SFT}}(Y, \lambda, \overline{K}^0)$ as follows.
First (\ref{multiplication of homology of W_Y with filtration}) induces
\begin{multline*}
\varinjlim_{\kappa_1} \varprojlim_{C_0, C_1}
H^\ast(\mathcal{W}_Y^{\leq \kappa_1} / I^{\leq \kappa_1}_{C_0, C_1, C_2 + \kappa_2}, D_Y)
\times \varprojlim_{C_0, C_1}
H^\ast(\mathcal{W}_Y^{\leq \kappa_2} / I^{\leq \kappa_2}_{C_0, C_1, C_2}, D_Y) \\
\to \varinjlim_{\kappa} \varprojlim_{C_0, C_1}
H^\ast(\mathcal{W}_Y^{\leq \kappa} / I^{\leq \kappa}_{C_0, C_1, C_2}, D_Y),
\end{multline*}
and this induces
\begin{multline*}
\varprojlim_{C'_2} \varinjlim_{\kappa_1} \varprojlim_{C_0, C_1}
H^\ast(\mathcal{W}_Y^{\leq \kappa_1} / I^{\leq \kappa_1}_{C_0, C_1, C'_2}, D_Y)
\times \varprojlim_{C_0, C_1}
H^\ast(\mathcal{W}_Y^{\leq \kappa_2} / I^{\leq \kappa_2}_{C_0, C_1, C_2}, D_Y) \\
\to \varinjlim_{\kappa} \varprojlim_{C_0, C_1}
H^\ast(\mathcal{W}_Y^{\leq \kappa} / I^{\leq \kappa}_{C_0, C_1, C_2}, D_Y).
\end{multline*}
Then this induces
\begin{align*}
\varprojlim_{C'_2} \varinjlim_{\kappa_1} \varprojlim_{C_0, C_1}
H^\ast(\mathcal{W}_Y^{\leq \kappa_1} / I^{\leq \kappa_1}_{C_0, C_1, C'_2}, D_Y)
\times \varinjlim_{\kappa_2} \varprojlim_{C_0, C_1}
H^\ast(\mathcal{W}_Y^{\leq \kappa_2} / I^{\leq \kappa_2}_{C_0, C_1, C_2}, D_Y) \ \\
\to \varinjlim_{\kappa} \varprojlim_{C_0, C_1}
H^\ast(\mathcal{W}_Y^{\leq \kappa} / I^{\leq \kappa}_{C_0, C_1, C_2}, D_Y),
\end{align*}
and finally this induces the multiplication of the limit.
(iii) also follows from the above argument and a similar argument to the proof of
well-definedness of (\ref{well def of i_X^pm}).
($i_{\mathcal{F}_X}^\pm$ are special case of multiplication.)
Next we consider (ii) and (iv).
A similar argument implies
$H^\ast_{\mathrm{RSFT}}(Y, \lambda, \overline{K}^0)$ inherits a structure of algebra
and that
$H^\ast_{\mathrm{RSFT}}(X, \omega, Y^\pm, \lambda^\pm, \overline{K}^0_X,
\overline{K}^0_{Y^\pm}, \mu^\pm)$ has a structure of
$H^\ast_{\mathrm{RSFT}}(Y^\pm, \lambda^\pm, \overline{K}_{Y^\pm}^0)$-bimodule.
We need to prove that $H^\ast_{\mathrm{RSFT}}(Y, \lambda, \overline{K}^0)$ inherits
a Poisson structure.
First we prove that the map $A^0$ in Lemma \ref{short concordance for Y}
preserves the Poisson structure.
Namely, we prove that for any $f \in H^\ast(\mathcal{P}^{\leq \kappa_1}_{Y^+}
/ I^{\leq \kappa_1}_{C_0, C_2 + \kappa_2}, d_{Y^+})$ and
$g \in H^\ast(\mathcal{P}^{\leq \kappa_2}_{Y^+}
/ I^{\leq \kappa_2}_{C_0, C_2 + \kappa_1}, d_{Y^+})$,
\begin{equation}
A^0(\{f, g\}) = \{A^0(f), A^0(g)\} \label{A^0 Poisson}
\end{equation}
in $H^\ast(\mathcal{P}^{\leq \kappa_1 + \kappa_2}_{Y^-}
/ I^{\leq \kappa_1 + \kappa_2}_{C_0, C_2}, d_{Y^-})$.
We denote the subspace of cycles of a chain complex $(C^\ast, d)$ by $Z(C^\ast, d)$.
Assume that $f^\pm \in Z(\mathcal{P}^{\leq \kappa_1}_{Y^\pm}
/ I^{\leq \kappa_1}_{C_0, C_2 + \kappa_2}, d_{Y^\pm})$ and
$g^\pm \in Z(\mathcal{P}^{\leq \kappa_2}_{Y^\pm}
/ I^{\leq \kappa_2}_{C_0, C_2 + \kappa_1}, d_{Y^\pm})$ satisfy
\begin{align*}
i_{(\mathcal{F}_X)_0}^- f^- - i_{(\mathcal{F}_X)_0}^+ f^+ &= d_{\mathcal{F}_0} a, \\
i_{(\mathcal{F}_X)_0}^- g^- - i_{(\mathcal{F}_X)_0}^+ g^+ &= d_{\mathcal{F}_0} b
\end{align*}
for some $a, b \in \mathcal{L}^{\leq \kappa_1 + \kappa_2}_X
/ J^{\leq \kappa_1 + \kappa_2}_{C_0, C_2}$.
Note that $\{f^-, g^+\} = \{f^+, g^-\} = 0$.
Then
\begin{align*}
i_{(\mathcal{F}_X)_0}^-(\{f^-, g^-\}) - i_{(\mathcal{F}_X)_0}^+(\{f^+, g^+\})
&= (\{f^-, g^-\} - \{f^+, g^+\})|_{\mathcal{F}_0} \\
&= \{f^- - f^+, g^- - g^+\}|_{\mathcal{F}_0}
\end{align*}
is exact in $(\mathcal{L}^{\leq \kappa_1 + \kappa_2}_X
/ J^{\leq \kappa_1 + \kappa_2}_{C_0, C_2}, d_{(\mathcal{F}_X)_0})$
by Proposition \ref{properties of d_{F_0}} (iv) since $\widehat{d}_X(f^- - f^+) = 0$
and $\widehat{d}_X(g^- - g^+) = 0$.
This proves equation (\ref{A^0 Poisson}).
Therefore $A^0$ in Lemma \ref{short concordance for Y} preserves
the Poisson structure.
Recall that for $\kappa^\circ \leq \kappa$ and $C^\circ \leq C$,
$(\mathcal{P}_Y^{\leq \kappa^\circ} + I^{\leq \kappa}_{C_0, C^\circ_2})
/ I^{\leq \kappa}_{C_0, C_2}$ is the fiber product of
$\mathcal{P}_Y^{\leq \kappa^\circ} / I^{\leq \kappa^\circ}_{C_0, C^\circ_2}$
and $\mathcal{P}_Y^{\leq \kappa} / I^{\leq \kappa}_{C_0, C_2}$ over
$\mathcal{P}_Y^{\leq \kappa} / I^{\leq \kappa}_{C_0, C^\circ_2}$,
and the Poisson bracket of $\mathcal{P}_Y$
induces (\ref{Poisson bracket for fiber product}).
Its homology
$H^\ast((\mathcal{P}_Y^{\leq \kappa^\circ} + I^{\leq \kappa}_{C_0, C^\circ_2})
/ I^{\leq \kappa}_{C_0, C_2}, d_Y)$ is also well-defined.
Since $H^\ast$ preserves fiber product structure, it is the fiber product of
$H^\ast(\mathcal{P}_Y^{\leq \kappa^\circ} / I^{\leq \kappa^\circ}_{C_0, C^\circ_2}, d_Y)$
and $H^\ast(\mathcal{P}_Y^{\leq \kappa} / I^{\leq \kappa}_{C_0, C_2}, d_Y)$ over
$H^\ast(\mathcal{P}_Y^{\leq \kappa} / I^{\leq \kappa}_{C_0, C^\circ_2}, \allowbreak d_Y)$.
Furthermore, since fiber product commutes with limits,
$H^\ast_{\mathrm{RSFT}}(Y, \lambda, \overline{K}^0_Y)$ is isomorphic to
\[
\varprojlim_{C_2^\circ} \varinjlim_{\kappa^\circ}
\varprojlim_{C_2} \varinjlim_{\kappa} \varprojlim_{C_0}
H^\ast((\mathcal{P}_Y^{\leq \kappa^\circ} + I^{\leq \kappa}_{C_0, C^\circ_2})
/ I^{\leq \kappa}_{C_0, C_2}, d_Y).
\]
First, (\ref{Poisson bracket for fiber product}) induces the map
\begin{align*}
H^\ast((\mathcal{P}_Y^{\leq \kappa^\circ_1} + I^{\leq \kappa_1}_{C_0, C_2})
/ I^{\leq \kappa_1}_{C_0, C'_2}, d_Y) \times
H^\ast((\mathcal{P}_Y^{\leq \kappa^\circ_2} + I^{\leq \kappa_2}_{C_0, C_2})
/ I^{\leq \kappa_1}_{C_0, C''_2}, d_Y) \\
\to H^\ast(\mathcal{P}_Y^{\leq \kappa_1 + \kappa_2}
/ I^{\leq \kappa_1 + \kappa_2}_{C_0, C_2}, d_Y)
\end{align*}
for $C'_2 \geq C_2 + \kappa^\circ_2$ and $C''_2 \geq C_2 + \kappa^\circ_1$,
and then this induces the map
\begin{align*}
&\varprojlim_{C'_2} \varinjlim_{\kappa_1} \varprojlim_{C_0}
H^\ast((\mathcal{P}_Y^{\leq \kappa^\circ_1} + I^{\leq \kappa_1}_{C_0, C_2})
/ I^{\leq \kappa_1}_{C_0, C'_2}, d_Y) \\
&\quad
\times
\varprojlim_{C''_2} \varinjlim_{\kappa_2} \varprojlim_{C_0}
H^\ast((\mathcal{P}_Y^{\leq \kappa^\circ_2} + I^{\leq \kappa_2}_{C_0, C_2})
/ I^{\leq \kappa_1}_{C_0, C''_2}, d_Y) \\
&\to \varinjlim_{\kappa} \varprojlim_{C_0}
H^\ast(\mathcal{P}_Y^{\leq \kappa}
/ I^{\leq \kappa}_{C_0, C_2}, d_Y).
\end{align*}
Finally, this induces the map
\begin{align*}
&\varprojlim_{C_2} \varinjlim_{\kappa_1^\circ}
\varprojlim_{C'_2} \varinjlim_{\kappa_1} \varprojlim_{C_0}
H^\ast((\mathcal{P}_Y^{\leq \kappa^\circ_1} + I^{\leq \kappa_1}_{C_0, C_2})
/ I^{\leq \kappa_1}_{C_0, C'_2}, d_Y) \\
&\quad
\times \varprojlim_{C_2} \varinjlim_{\kappa_2^\circ}
\varprojlim_{C''_2} \varinjlim_{\kappa_2} \varprojlim_{C_0}
H^\ast((\mathcal{P}_Y^{\leq \kappa^\circ_2} + I^{\leq \kappa_2}_{C_0, C_2})
/ I^{\leq \kappa_1}_{C_0, C''_2}, d_Y) \\
&\to \varprojlim_{C_2} \varinjlim_{\kappa} \varprojlim_{C_0}
H^\ast(\mathcal{P}_Y^{\leq \kappa}
/ I^{\leq \kappa}_{C_0, C_2}, d_Y),
\end{align*}
which is the Poisson bracket of $H^\ast_{\mathrm{RSFT}}(Y, \lambda, \overline{K}^0_Y)$.
\end{proof}
Next we show that SFT cohomologies of cobordisms $(X, \omega)$ from $(Y^-, \xi^-)$
to $(Y^+, \xi^+)$ does not depend on the choice of the contact structure of
$(Y^\pm, \xi^\pm)$.
\begin{prop}\label{general concordance for X}
Let $(X, \omega)$ be a cobordism from $(Y^-, \lambda^-)$ to $(Y^+, \lambda^+)$,
and let $(X_1, \omega_1)$ be a (general) concordance from
$(Y^+, \lambda^+)$ to $(Y^+, \lambda_1^+)$.
($\lambda^+$ and $\lambda_1^+$ are contact forms for the same contact structure
$\xi^+$.)
Then
\[
T_{\mathcal{F}_X}(\cdot \Diamond \mathcal{F}_{X_1})
: H^\ast(\mathcal{D}_X, D_X) \to H^\ast(\mathcal{D}_{X \# X_1}, D_{X \# X_1})
\]
and
\[
T_{(\mathcal{F}_X)_0}(\cdot \sharp (\mathcal{F}_{X_1})_0)
: H^\ast(\mathcal{L}_X, d_X) \to H^\ast(\mathcal{L}_{X \# X_1}, d_{X \# X_1})
\]
are isomorphisms.
\end{prop}
\begin{proof}
We consider the case of general SFT. The case of rational SFT is similar.
By the argument similar to Lemma \ref{short concordance for X},
it is enough to prove for the case where $(X_1, \omega_1)$ is a trivial concordance.
In this case, Lemma \ref{trivial generating function} implies that
$T_{\mathcal{F}_X}(\cdot \Diamond \mathcal{F}_{X_1}) : H^\ast(\mathcal{D}_X, D_X) \to H^\ast(\mathcal{D}_X, D_X)$
is the limit of the inclusion-quotient maps similar to those given by the filtration.
Hence this limit is the identity map.
Therefore $T_{\mathcal{F}_X}(\cdot \Diamond \mathcal{F}_{X_1})$ is an isomorphism.
\end{proof}
This Proposition implies that SFT cohomologies
\[
H^\ast_{\mathrm{SFT}}(X, \omega, Y^\pm, \lambda^\pm, \overline{K}^0_X,
\overline{K}^0_{Y^\pm}, \mu^\pm)
\]
and
\[
H^\ast_{\mathrm{RSFT}}(X, \omega, Y^\pm, \lambda^\pm, \overline{K}^0_X,
\overline{K}^0_{Y^\pm}, \mu^\pm)
\]
defined by (\ref{limit general X}) and (\ref{limit rational X}) for cobordisms
$(X, \omega)$ from $(Y^-, \xi^-)$ to $(Y^+, \lambda^+)$ with different contact forms
are naturally isomorphic respectively.
We denote these isomorphic cohomology groups by
\[
H^\ast_{\mathrm{SFT}}(X, \omega, Y^\pm, \xi^\pm, \overline{K}^0_X,
\overline{K}^0_{Y^\pm}, \mu^\pm)
\]
and
\[
H^\ast_{\mathrm{RSFT}}(X, \omega, Y^\pm, \xi^\pm, \overline{K}^0_X,
\overline{K}^0_{Y^\pm}, \mu^\pm)
\]
respectively.
Finally we show that SFT cohomologies of $(Y, \xi)$ do not depend on the choice of
the contact structure $\lambda$.
\begin{prop}\label{general concordance for Y}
For any concordance $(X, \omega)$ from $(Y^-, \lambda^-)$ to $(Y^+, \lambda^+)$,
the homomorphisms
\[
i_X^\pm : H^\ast (\mathcal{W}_{Y^\pm} , D_{Y^\pm}) \to H^\ast(\mathcal{D}_X, D_X)
\]
and
\[
i_{X, 0}^\pm : H^\ast (\mathcal{P}_{Y^\pm} , d_{Y^\pm}) \to H^\ast(\mathcal{L}_X, d_X)
\]
are isomorphisms of modules, and the composition
\[
A = i_X^- \circ (i_X^+)^{-1} : H^\ast(\mathcal{W}_{Y^+}, D_{Y^+})
\to H^\ast(\mathcal{W}_{Y^-}, D_{Y^-})
\]
is an isomorphism of algebras, and
the composition
\[
A^0 = i_{X, 0}^- \circ (i_{X, 0}^+)^{-1} : H^\ast(\mathcal{P}_{Y^+}, d_{Y^+})
\to H^\ast(\mathcal{P}_{Y^-}, d_{Y^-})
\]
is an isomorphism of Poisson algebras.
Furthermore, $A$ and $A^0$ do not depend on the concordance $(X, \omega)$.
\end{prop}
\begin{proof}
The proof of the first claim is similar to that of Lemma \ref{short concordance for Y}.
First we consider the case of trivial concordance.
As in the proof of Proposition \ref{general concordance for X},
in this case, Lemma \ref{trivial generating function} implies that
$i_X^\pm : H^\ast (\mathcal{W}_{Y^\pm} , D_{Y^\pm}) \to H^\ast(\mathcal{D}_X, D_X)$ is the limit of
the inclusion-quotient maps similar to those given by the filtration.
Hence the limit is an isomorphism.
In general case, there exists a concordance $(X', \omega')$ from
$(Y^+, \lambda^+)$ to $(Y^-, \lambda^-)$ such that $X \# X'$ is a trivial concordance.
Then $T_{\mathcal{F}_X}(\cdot \Diamond \mathcal{F}_{X'}) \circ i_X^- = i_{X \# X'}^-
: H^\ast (\mathcal{W}_{Y^-} , D_{Y^-}) \to H^\ast(\mathcal{D}_{X \# X'}, D_{X \# X'})$ is an isomorphism.
Since $T_{\mathcal{F}_X}(\cdot \Diamond \mathcal{F}_{X'})$ is also an isomorphism by Proposition
\ref{general concordance for X}, this implies that $i_X^-$ is an isomorphism.
The cases of $i_X^+$ or $i_{X, 0}^\pm$ are similar.
By an argument similar to that of Proposition \ref{algebraic structures of limits},
we can prove that $A = i_X^- \circ (i_X^+)^{-1}$ is an algebra homomorphism,
and $A^0 = i_{X, 0}^- \circ (i_{X, 0}^+)^{-1}$ is an isomorphism of Poisson algebras.
The independence of $A = i_X^- \circ (i_X^+)^{-1}$ and
$A^0 = i_{X, 0}^- \circ (i_{X, 0}^+)^{-1}$ are similar to
Lemma \ref{short concordance for Y}.
\end{proof}
This proposition implies that SFT cohomologies
$H^\ast(\mathcal{W}_{(Y, \lambda, \overline{K}_Y^0)}, D_{(Y, \lambda, \overline{K}_Y^0)})$
and
$H^\ast(\mathcal{P}_{(Y, \lambda, \overline{K}_Y^0)}, d_{(Y, \lambda, \overline{K}_Y^0)})$
defined by (\ref{limit general Y}) and (\ref{limit rational Y}) for different strict
contact manifolds $(Y, \lambda)$ of the same contact structure $\xi$ are
naturally isomorphic respectively.
Hence we denote these cohomology groups by
$H^\ast_{\mathrm{SFT}}(Y, \xi, \overline{K}^0)$ and
$H^\ast_{\mathrm{RSFT}}(Y, \xi, \overline{K}^0)$ respectively.
It is easy to check that
$H^\ast_{\mathrm{SFT}}(X, \omega, Y^\pm, \xi^\pm, \overline{K}^0_X,
\overline{K}^0_{Y^\pm}, \mu^\pm)$ has a structure of
$H^\ast_{\mathrm{SFT}}(Y^\pm, \xi^\pm, \overline{K}_{Y^\pm}^0)$-bimodule,
and
$H^\ast_{\mathrm{RSFT}}(X, \omega, Y^\pm, \xi^\pm,\allowbreak \overline{K}^0_X,
\overline{K}^0_{Y^\pm}, \mu^\pm)$ has a structure of
$H^\ast_{\mathrm{RSFT}}(Y^\pm, \xi^\pm, \overline{K}_{Y^\pm}^0)$-bimodule.
Finally we consider the case of contact homology.
This case is more standard.
\begin{prop}\label{short concordance for contact homology}
For any short concordance $\mathcal{C}_X$ from $\mathcal{C}_{Y^-}$
to $\mathcal{C}_{Y^+}$, the homomorphism
\[
\Psi_{(\widehat{\mathcal{F}}_X)_0} :
H^\ast (\mathcal{A}_{Y^+}^{\leq \kappa} / I^{\leq \kappa}_{C_0}, \partial_{Y^+}) \to
H^\ast (\mathcal{A}_{Y^-}^{\leq \kappa} / I^{\leq \kappa}_{C_0}, \partial_{Y^-})
\]
is an isomorphism.
Furthermore, it does not depend on the short concordance $\mathcal{C}_X$.
\end{prop}
\begin{proof}
If $\mathcal{C}_X$ is a trivial short concordance, then the claim follows from
Lemma \ref{trivial generating function}.
For a general short concordance $\mathcal{C}_X$, let $\mathcal{C}_{X'}$ be
a short concordance from $\mathcal{C}_{Y^+}$ to $\mathcal{C}_{Y^-}$.
Then since the compositions of these two short concordance are
trivial short concordances, $\Psi_{(\widehat{\mathcal{F}}_{X'})_0} \circ
\Psi_{(\widehat{\mathcal{F}}_X)_0}$ and $\Psi_{(\widehat{\mathcal{F}}_X)_0} \circ
\Psi_{(\widehat{\mathcal{F}}_{X'})_0}$ are isomorphisms.
Hence $\Psi_{(\widehat{\mathcal{F}}_X)_0}$ is also an isomorphism.
\end{proof}
Therefore we can define the limit
\[
H^\ast_{\mathrm{CH}}(Y, \lambda, \overline{K}^0)
= \varinjlim_{\kappa} \varprojlim_{C_0}
H^\ast(\mathcal{A}_{(Y, \lambda, K_Y, \overline{K}_Y^0)}^{\leq \kappa}
/ I^{\leq \kappa}_{C_0}, \partial_{(Y, \lambda, K_Y, K_Y^0, K_Y^2, J, \mathcal{B})}).
\]
We sometimes abbreviate this limit as $H^\ast(\mathcal{A}_Y, \partial_Y)$.
For any exact cobordism $(X, \omega)$ from $(Y^-, \lambda^-)$ and
$(Y^+, \lambda^+)$, we can define
\[
\Psi_X : H^\ast_{\mathrm{CH}}(Y^+, \lambda^+, \overline{K}^0_{Y^+})
\to H^\ast_{\mathrm{CH}}(Y^-, \lambda^-, \overline{K}^0_{Y^-})
\]
by the limit of $\Psi_{(\widehat{\mathcal{F}}_X)_0}$.
We can easily prove the following.
\begin{prop}
For any concordance $(X, \omega)$ from $(Y^-, \lambda^-)$ to $(Y^+, \lambda^+)$,
the homomorphism
\[
\Psi_X : H^\ast_{\mathrm{CH}}(Y^+, \lambda^+, \overline{K}^0_{Y^+})
\to H^\ast_{\mathrm{CH}}(Y^-, \lambda^-, \overline{K}^0_{Y^-})
\]
is an isomorphism of algebras.
Furthermore, it does not depend on the concordance $(X, \omega)$.
\end{prop}
We denote the isomorphism class of contact homology by
$H^\ast_{\mathrm{CH}}(Y, \xi, \overline{K}^0)$.
\section{SFT of a contact manifold with the $S^1$-action induced by the Reeb flow}
\label{S^1 action}
The arguments in \cite{EGH00} or \cite{Bo02} are easily adapted to our construction of
SFT.
In this section, we demonstrate how to calculate the SFT cohomology of pre-quantization
spaces, or more generally, contact manifolds with the locally free $S^1$-action
generated by the Reeb vector field.
Let $(Y, \lambda)$ be a closed contact manifold and assume that there exists a constant
$L > 0$ such that $\varphi^\lambda_L = \mathrm{id}$.
Then $S^1 = \mathbb{R} / L\mathbb{Z}$ acts on $Y$ by $t \cdot y = \varphi^\lambda_t(y)$.
We consider the SFT of such a contact manifold.
We may assume $L = 1$.
First we consider the case where every cycle in $K^0$ is invariant by this action.
In this case, we can calculated the SFT cohomology by the following proposition.
\begin{prop}\label{H = 0}
All periodic orbits are good, and the local systems $\S^D$ and $\S^{\overline{P}}$ are
trivial on $\overline{P}$.
Furthermore, we can construct the virtual fundamental chains which make
$\mathcal{H} = 0$.
\end{prop}
Theorem \ref{H vanishes} is a corollary of this proposition.
Let $J$ be an $S^1$-invariant $d\lambda$-compatible complex structure of
$\xi = \mathop{\mathrm{Ker}}\nolimits \lambda$.
First we prove the claim about the local systems $\S^D$ and $\S^{\overline{P}}$.
For each $l \geq 1$, let $Y^{l^{-1} \mathbb{Z} / \mathbb{Z}} = \mathrm{ev}_0 P_{l^{-1}} \subset Y$ be the
fixed manifold of the subgroup $l^{-1} \mathbb{Z} / \mathbb{Z} \subset S^1$.
Then $l^{-1} \mathbb{Z} / \mathbb{Z}$ acts on each fiber of $\xi|_{Y^{l^{-1} \mathbb{Z} / \mathbb{Z}}}$.
Since this is a unitary action, we can decompose this complex vector bundle by
the eigenvalues:
\[
\xi|_{Y^{l^{-1} \mathbb{Z} / \mathbb{Z}}} = W_0 \oplus W_1 \oplus \dots \oplus W_{l-1},
\]
where $(\varphi^\lambda_{l^{-1}})_\ast$ acts on each $W_k$ by $e^{2\pi \sqrt{-1} k / l}$.
Then for each point $y \in Y^{l^{-1} \mathbb{Z} / \mathbb{Z}}$, we can define a unitary trivialization of
$\xi$ on the periodic orbit $\gamma(t) = \varphi^\lambda_{l^{-1}t}(y)$ by
\[
(\varphi^\lambda_{l^{-1}t})_\ast \circ
\bigl(\bigoplus_k e^{-2\pi \sqrt{-1} kt / l} 1_{W_k}\bigr):\\
\xi_{\gamma(0)} = W_0 \oplus W_1 \oplus \dots
\oplus W_{l-1} \stackrel{\cong}{\to} \xi_{\gamma(t)}
\]
if we fix a unitary basis of each $W_k$.
Under this trivialization, $(\varphi^\lambda_{l^{-1}t})_\ast$ are given by the diagonal
matrices
\[
\bigoplus_k e^{2\pi \sqrt{-1} kt / l} 1_{W_k}.
\]
Hence the linear operator $\mathring{D}^+_\gamma$ is complex linear.
In particular, its kernel has the complex orientation.
Therefore $\S^D$ is a trivial local system on $P_{l^{-1}}$.
Similarly, $\S^D$ is trivial on $P_{k / l}$ for each $k / l$
since $\mathrm{ev}_0 P_{k / l} = \mathrm{ev}_0 P_{1 / l}$ if $k$ and $l$ are coprime.
Hence there are no bad orbits and the induced local system on
$\overline{P}$ is also trivial.
Similarly, $\overline{P}$ does not contain any non-orientable points,
and $\S^{\overline{P}}$ is trivial on $\overline{P}$.
Next we construct required virtual fundamental chains.
$\widehat{\mathcal{M}} = \widehat{\mathcal{M}}(Y, \lambda, J)$ has a locally free $S^1$-action
defined by $t \cdot (\Sigma, z, u)
= (\Sigma, z, (1 \times \varphi^\lambda_t) \circ u)$.
We will construct a pre-Kuranishi structure of the quotient space
$\widehat{\mathcal{M}}^\bullet = \widehat{\mathcal{M}} / S^1$ which induces a pre-Kuranishi structure of
$\widehat{\mathcal{M}}$.
Since the evaluation maps to $\overline{P}$ or $Y / S^1$ are well-defined on
$\widehat{\mathcal{M}}^\bullet$, we can define its fiber products
$((\widehat{\mathcal{M}}^\bullet)^\diamond, \mathring{K}^2, K, K^0 / S^1)$
and multi-valued partial submersions $\Xi^\circ$ and $\Lambda$ similarly.
We can construct a grouped multisection of
$((\widehat{\mathcal{M}}^\bullet)^\diamond, \mathring{K}^2, K, K^0 / S^1)$ which satisfies
the similar compatibility conditions.
Then we define the grouped multisection of
$(\widehat{\mathcal{M}}^\diamond, \mathring{K}^2, K,K^0)$ by its pull back.
Since the $S^1$-action is locally free, it makes the virtual fundamental chains of
the zero-dimensional fiber products used for the definition of $\mathcal{H}$ vanish.
Therefore, it is enough to define a required pre-Kuranishi structure of
$\widehat{\mathcal{M}}^\bullet$.
First we explain the construction of a Kuranishi neighborhood of a point $\bar p_0
= (\Sigma_0, z, u_0) \in \widehat{\mathcal{M}}^\bullet$.
Define a finite group $G^+_0$ by
\[
G^+_0 = \{(g, t) \in \mathop{\mathrm{Aut}}\nolimits (\Sigma_0) \times S^1; g (\{z_i\}) = \{z_i\},
u_0 \circ g = (1 \times \varphi^\lambda_t) \circ u_0\}.
\]
We also define a group $G^{++}_0 \subset \mathop{\mathrm{Aut}}\nolimits (\Sigma_0) \times S^1$ by
$G^{++}_0 = G^+_0 \cdot S^1$.
We assume that the following data
$(\bar p_0^+, S, (\mathcal{O}_a, \mathcal{N}_a, E^0_a, \lambda_a)_{a \in A})$
are given instead of the data in the usual case.
$\bar p_0^+ = (\Sigma_0, z, z^+, u_0) \in \widehat{\mathcal{M}}^\bullet$ is, as in the usual case,
a curve obtained by adding marked points on the nontrivial components of $\Sigma_0$.
We assume that all unstable components of $(\Sigma_0, z, z^+)$ are trivial cylinders
of $\bar p_0$ and $G^+_0$ preserves $z^+ = \{z^+_i\}$ as a set.
$S \subset Y$ is a finite union of $S^1$-invariant codimension-two submanifolds
such that $\pi_Y \circ u_0$ intersects with $S$ at $z^+$ transversely.
We can take such an $S^1$-invariant submanifold for the following reason.
Choosing appropriate additional marked points $z^+$, we assume that
the differential $d^\xi u_0$ does not vanish at $z^+$.
Let $l^{-1} \mathbb{Z} / \mathbb{Z}$ be the stabilizer of the point $y = u_0(z_i^+)$.
Then an $S^1$-equivariant tubular neighborhood of the orbit $S^1 \cdot y$ is
isomorphic to $\mathbb{R} / \mathbb{Z} \times_{l^{-1} \mathbb{Z} / \mathbb{Z}} \xi_y$.
Since the $l^{-1} \mathbb{Z} / \mathbb{Z}$-action on $\xi_y$ is unitary and commutative,
$\xi_y$ can be decomposed into irreducible representations of complex dimension one.
Therefore there exists an $l^{-1} \mathbb{Z} / \mathbb{Z}$-invariant subspace $\xi^0_y \subset \xi_y$ of
complex codimension one such that $\mathop{\mathrm{Im}}\nolimits d^\xi u_0 (z_i^+) \pitchfork \xi^0_y$.
Then $\pi_Y \circ u_0$ intersects with the $S^1$-invariant submanifold
$S = \mathbb{R} / \mathbb{Z} \times_{l^{-1} \mathbb{Z} / \mathbb{Z}} \xi^0_y$ transversely at $z_i^+$.
For the construction of the global pre-Kuranishi structure, we used an infinite family of
disjoint submanifolds $\{S^x\}_{x \in \mathbb{R}^2}$.
(See the proof of Lemma \ref{existence of a domain curve representation}.)
It was constructed as constant sections of the trivial tubular neighborhood of $S$.
To construct such a family of $S^1$-invariant submanifolds,
it is enough to make the $l^{-1} \mathbb{Z} / \mathbb{Z}$-action on $\xi_y / \xi^0_y \cong
\mathop{\mathrm{Im}}\nolimits d^\xi u_0 (z_i^+)$ trivial.
In particular, it is enough to choose $z^+$ so that the stabilizer $l^{-1} \mathbb{Z} / \mathbb{Z}$ of
each $\pi_Y \circ u_0(z^+_i)$ is locally minimal in the image of $\pi_Y \circ u_0$.
Let $(\hat P \to \hat X, Z, Z^+, Z_{\pm\infty})$ be the local universal family of
the stabilization $(\hat \Sigma, z, z^+, \pm\infty)$ of the blow down curve of
$(\Sigma, z, z^+)$
We need an additional vector space $E^0$ and a linear map $\lambda$.
If we can take a $G^+_0$-equivariant linear map
$\lambda : E^0 \to C^\infty(\hat P \times Y, {\textstyle\bigwedge}^{0, 1} V^\ast \hat P
\otimes (\mathbb{R} \partial_\sigma \oplus TY))$
which is $S^1$-invariant, that is,
\[
\lambda(h)(z, \varphi^\lambda_t(y)) = (1 \otimes (\varphi^\lambda_t)_\ast)
\lambda(h)(z, y)
\]
for all $t \in S^1$, and which makes the linear operator $D_{p_0}^+$ defined in
Section \ref{construction of nbds} surjective, then it is easy to construct
a Kuranishi neighborhood of $\bar p_0 \in \widehat{\mathcal{M}}^\bullet$
which is independent of the choice of the representative $p_0 \in \widehat{\mathcal{M}}$.
However, since the $S^1$-action on $Y$ is not necessarily free, we cannot construct
such a $G^+_0$-equivariant linear map in general.
Instead, we take the following data
$(\mathcal{O}_a, \mathcal{N}_a, E^0_a, \lambda_a, I_a)_{a \in A}$:
\begin{itemize}
\item
$A$ is a finite index set.
\item
For each $a \in A$,
$\mathcal{O}_a \subset Y$ is an $S^1$-orbit,
$\mathcal{N}_a \subset Y$ is its $S^1$-invariant tubular neighborhood,
and $\pi_{\mathcal{N}_a} : \mathcal{N}_a \to \mathcal{O}_a$ is its $S^1$-equivariant
projection.
\item
Let $\pi_{\widetilde{\mathcal{O}}_a} : \widetilde{\mathcal{O}}_a \to \mathcal{O}_a$ be
the covering space of $\mathcal{O}_a$ such that the $S^1$-action lifts to
$\widetilde{\mathcal{O}}_a$ as a free (and transitive) action.
Then $\pi_a : E^0_a \to \widetilde{\mathcal{O}}_a$ is a finite dimensional
$G^{++}_0$-vector bundle.
(The action of $G^{++}_0 \subset \mathop{\mathrm{Aut}}\nolimits (\Sigma_0) \times S^1$ on
$\widetilde{\mathcal{O}}_a$ is defined by the projection $G^{++}_0 \to S^1$.)
\item
Define $\widetilde{\mathcal{N}}_a = \mathcal{N}_a \times_{\mathcal{O}_a}
\widetilde{\mathcal{O}}_a$ and let
$\widetilde{\pi}_{\mathcal{N}_a} : \widetilde{\mathcal{N}}_a \to
\widetilde{\mathcal{O}}_a$ be the projection.
Define $\pi_{\hat P \times \widetilde{\mathcal{N}}_a} : \hat P \times
\widetilde{\mathcal{N}}_a \to \widetilde{\mathcal{O}}_a$ by
$\pi_{\hat P \times \widetilde{\mathcal{N}}_a}(z, y)
= \widetilde{\pi}_{\mathcal{N}_a}(y)$.
Then
$\lambda_a : \pi_{\hat P \times \widetilde{\mathcal{N}}_a}^\ast E^0_a \to
{\textstyle\bigwedge}^{0,1} V^\ast \hat P \otimes_{\mathbb{C}} (\mathbb{R} \partial_\sigma \oplus TY)
|_{\hat P \times \widetilde{\mathcal{N}}_a}$
is a $G^{++}_0$-equivariant bundle map with compact support
$\mathop{\mathrm{supp}}\nolimits \lambda_a \subset \hat P \times \widetilde{\mathcal{N}}_a$.
\item
$I_a \subset S^1$ is a union of finite number of intervals
which is invariant by the $G^+_0$-action.
\end{itemize}
We impose the following conditions on them:
\begin{enumerate}[label=(\arabic*)]
\item
The projection of $\mathop{\mathrm{supp}}\nolimits \lambda_a \subset \hat P \times \widetilde{\mathcal{N}}_a$ to
$\hat P$ does not intersect with the nodal points of $\hat P$ or $Z_{\pm\infty}$.
\item
\label{condition of I}
There exists a simply connected neighborhood $\mathcal{I}_a \subset S^1$ of $0$ and
a finite subgroup $\Gamma \subset S^1$ such that
$I_a = \mathcal{I}_a + \Gamma$,
$\mathcal{I}_a = - \mathcal{I}_a$ and
$(\mathcal{I}_a + \mathcal{I}_a) \cap \Gamma = \{0\}$.
(Namely, the intervals in $I_a$ have the same length, and
the intervals in the complement $S^1 \setminus I_a$ also have the same length.
Furthermore, the former is smaller than the latter.)
\item
\label{support lambda small}
Let $p_0 = (\Sigma_0, z, u_0) \in \widehat{\mathcal{M}}$ be a representative of
$\bar p_0 \in \widehat{\mathcal{M}}^\bullet$.
Then there exists a point $x_a \in \widetilde{\mathcal{O}}_a$ such that
\[
\mathop{\mathrm{supp}}\nolimits \lambda_a \cap (1 \times \pi_{\widetilde{\mathcal{N}}_a})^{-1}
\mathop{\mathrm{graph}}\nolimits (\pi_Y \circ u_0) \subset \pi_{\hat P \times \widetilde{\mathcal{N}}_a}^{-1}
(I_a \cdot x_a).
\]
\item
\label{transverse condition of lambda}
Let $E^0_{a, x_a}$ be the vector space of locally $S^1$-invariant sections of
$E^0_a|_{I_a \cdot x_a}$.
(A locally $S^1$-invariant section is a section which is $S^1$-invariant on each connected
component of $I_a \cdot x_a$.
Namely, if we trivialize $E^0_a|_{I_a \cdot x_a}$ by the $S^1$-action,
then it is a locally constant section.)
Note that the $G^+_0$-action on $E^0_a$ induces a $G^+_0$-action on $E^0_{a, x_a}$.
Define $\widetilde{\mathcal{N}}_{a, x_a} = \widetilde{\pi}_{\mathcal{N}_a}^{-1}(I_a \cdot x_a)
\subset \widetilde{\mathcal{N}}_a$ and $\mathcal{N}_{a, x_a}
= \pi_{\widetilde{\mathcal{N}}_a}(\widetilde{\mathcal{N}}_{a, x_a}) \subset \mathcal{N}_a$,
where $\pi_{\widetilde{\mathcal{N}}_a} : \widetilde{\mathcal{N}}_a \to \mathcal{N}_a$ is
the projection.
Define a $G^+_0$-equivariant linear map
\[
\lambda_{a, x_a} : E^0_{a, x_a} \to C^\infty(\hat P \times \mathcal{N}_{a, x_a},
{\textstyle\bigwedge}^{0,1} V^\ast \hat P \otimes_{\mathbb{C}} (\mathbb{R} \partial_\sigma \oplus TY)).
\]
by
\[
\lambda_{a, x_a}(h)(z, y) = \sum_{\tilde y \in \widetilde{\mathcal{N}}_a,
\pi_{\widetilde{\mathcal{N}}_a}(\tilde y) = y} \lambda_a(h(z, \tilde y)).
\]
Let $E^0$ and $\lambda$ be the direct sums of $E^0_{a, x_a}$ and $\lambda_{a, x_a}$
over $a \in A$ respectively.
Then the liner map
\begin{align*}
&D_{p_0}^+ : \widetilde{W}_\delta^{1, p}(\Sigma_0, u_0^\ast T \hat Y) \oplus E^0\\
&\to
L_\delta^p(\Sigma_0, {\textstyle\bigwedge}^{0, 1}T^\ast \Sigma_0 \otimes u_0^\ast T \hat Y)
\oplus
\bigoplus_{\text{limit circles}} \mathop{\mathrm{Ker}}\nolimits A_{\gamma_{\pm\infty_i}} / (\mathbb{R} \partial_\sigma
\oplus \mathbb{R} R_\lambda)\\
&\quad \oplus \bigoplus_{z_i} T_{\pi_Y \circ u_0(z_i)} Y\\
&(\xi, h) \mapsto (D_{p_0} \xi(z) + \lambda(h)(z, \pi_Y \circ u_0(z)),
\sum_j \langle\xi|_{S^1_{\pm\infty_i}}, \eta_j^{\pm\infty_i}\rangle
\eta_j^{\pm\infty_i}, \pi_Y \circ \xi (z_i))
\end{align*}
is surjective,
where $D_{p_0}$ is the linearization of the equation of $J$-holomorphic maps,
and $\{\eta_j^{\pm\infty_i}\}_j$ is an orthonormal basis of
the orthogonal complement of $\mathbb{R} \partial_\sigma \oplus \mathbb{R} R_\lambda$ in
$\mathop{\mathrm{Ker}}\nolimits A_{\gamma_{\pm\infty_i}}$ for each $\pm\infty_i$.
\end{enumerate}
We can construct such data $(\mathcal{O}_a, \mathcal{N}_a, E^0_a, \lambda_a)_{a \in A}$
as follows.
First we explain the construction of $E^0_a$
for each $S^1$-orbit $\mathcal{O}_a \subset Y$.
Define a map $\pi_{\widetilde{\mathcal{O}}^+_a} : \widetilde{\mathcal{O}}^+_a
= G^{++}_0 \times_{S^1} \widetilde{\mathcal{O}}_a \to \widetilde{\mathcal{O}}_a$ by
$\pi_{\widetilde{\mathcal{O}}^+_a}(g, t, x) = t \cdot x$.
Let $\widehat{E}^0_a \to \widetilde{\mathcal{O}}^+_a$ be the pull back of
$(\mathbb{R} \partial_\sigma \oplus TY)|_{\mathcal{O}_a}$ by $\pi_{\widetilde{\mathcal{O}}_a} \circ
\pi_{\widetilde{\mathcal{O}}^+_a} : \widetilde{\mathcal{O}}^+_a \to \mathcal{O}_a$,
and define a $G^{++}_0$-vector bundle $\pi_a : E^0_a \to \widetilde{\mathcal{O}}_a$ by
$E^0_a|_x = \bigoplus_{y \in \pi_{\widetilde{\mathcal{O}}^+_a}^{-1}(x)} \widehat{E}^0_a|_y$.
Define maps $\pi_{\widetilde{\mathcal{N}}^+_a} : \widetilde{\mathcal{N}}^+_a
= G^{++}_0 \times_{S^1} \widetilde{\mathcal{N}}_a \to \widetilde{\mathcal{N}}_a$
and $\pi_{\hat P \times \widetilde{\mathcal{N}}^+_a} : \hat P \times
\widetilde{\mathcal{N}}^+_a \to \widetilde{\mathcal{O}}^+_a$ by
$\pi_{\widetilde{\mathcal{N}}^+_a}(g, t, x) = t \cdot x$ and
$\pi_{\hat P \times \widetilde{\mathcal{N}}^+_a}(z, g, t, x)
= (g, t, \widetilde{\pi}_{\mathcal{N}_a}(x))$ respectively.
We note that
\[
(\pi_{\hat P \times \widetilde{\mathcal{N}}_a}^\ast E^0_a)|_\gamma
= \bigoplus_{\delta \in (1 \times \pi_{\widetilde{\mathcal{N}}^+_a})^{-1}(\gamma)}
(\pi_{\hat P \times \widetilde{\mathcal{N}}^+_a}^\ast \widehat{E}^0_a)|_\delta.
\]
We construct $\lambda_a$ as follows.
Take a $G^{++}_0$-invariant section $\rho_a$ of the pull back of
${\textstyle\bigwedge}^{0,1} V^\ast \hat P$ to $\hat P \times \widetilde{\mathcal{N}}^+_a$
such that the projection of its support to $\hat P$ is contained in a small neighborhood of
some $G^+_0$-orbit.
Since $\pi_{\hat P \times \widetilde{\mathcal{N}}^+_a}^\ast \widehat{E}^0_a$ is a pull back
of $(\mathbb{R} \partial_\sigma \oplus TY)|_{\mathcal{O}_a}$,
$\rho_a$ defines a linear map $\pi_{\hat P \times \widetilde{\mathcal{N}}^+_a}^\ast
\widehat{E}^0_a \to {\textstyle\bigwedge}^{0,1} V^\ast \hat P \otimes_{\mathbb{C}}
(\mathbb{R} \partial_\sigma \oplus TY)$, which defines the $G^{++}_0$-linear map
$\lambda_a : \pi_{\hat P \times \widetilde{\mathcal{N}}_a}^\ast E^0_a \to
{\textstyle\bigwedge}^{0,1} V^\ast \hat P \otimes_{\mathbb{C}} (\mathbb{R} \partial_\sigma \oplus TY)$.
If the support of $\rho_a$ is sufficiently small, then there exists a union of intervals
$I_a \subset S^1$ which satisfies Condition \ref{condition of I} and
\ref{support lambda small}.
Since the $G^{++}_0$-action on $\hat P \times \widetilde{\mathcal{N}}^+_a$ is free,
if we choose appropriate $\mathcal{O}_a$ and $\rho_a$
($a \in A$), then Condition \ref{transverse condition of lambda} also holds true.
Using the above data, we construct the Kuranishi neighborhood of
$\bar p_0 \in \widehat{\mathcal{M}}^\bullet$ as follows.
As in the usual case, we fix a temporally data $(z^{++}, S', \hat R_i)$,
where in this case, we assume that they are $G^+_0$-invariant.
In addition, we take a $G^+_0$-invariant family of sections
$\hat R_{S^1} = (\hat R_{S^1, l})$ of $\hat P \to \hat X$ and
a codimension-one submanifold $S_{S^1} \subset Y$ transverse to the Reeb vector field
such that $\pi_Y \circ u_0(\widetilde{R}_{S^1, l}(0)) \in S_{S^1}$ for all $l$,
where $\widetilde{R}_{S^1, l}$ is the section of $\widetilde{P} \to \widetilde{X}$
induced by $\hat R_{S^1, l}$.
Define a function $p_{S_{S^1}}$ on a small neighborhood of $S_{S^1}$
by $y \in \varphi^\lambda_{p_{S_{S^1}}(y)}(S_{S^1})$ and $|p_{S_{S^1}}(y)| \ll 1$.
These data are used to kill the $S^1$-action.
As in the usual case, we define a smooth manifold $\hat V = X \times B_\epsilon(0)$
and define a smooth map $s^0 : \hat V \to \mathbb{R}^k \oplus \bigoplus_{z_\beta^{++}} \mathbb{R}^2$.
In addition, we define a smooth map $s^1 : \hat V \to \mathbb{R}$ by
\[
s^1(a, b, x) = \frac{1}{m_i} \sum_{l=1}^{m_i} p_{S_{S^1}} \circ \pi_Y \circ
\Phi_{a, b}(\xi_x)(\widetilde{R}_{S~1, l}(a)),
\]
and define $\mathring{V} = \{s^0 = 0, s^1 = 0\} \subset \hat V$.
As in the usual case, we define a smooth map
$s : \mathring{V} \to E := E^0_{p_0} \oplus \bigoplus_{z_\alpha^+} \mathbb{R}^2$.
It is easy to see that the natural map
$\bar \psi : s^{-1}(0) / G^+_0 \to \widehat{\mathcal{M}}^\bullet$
is a homeomorphism onto a neighborhood of $\bar p_0 \in \widehat{\mathcal{M}}^\bullet$.
Hence $(\mathring{V}, E, s, \bar \psi, G^+_0)$
define the Kuranishi neighborhood of $\bar p_0 \in \widehat{\mathcal{M}}^\bullet$.
We note that the Kuranishi neighborhood is independent of the choice of
$x_a \in \widetilde{\mathcal{O}}_a$ because of Condition \ref{condition of I}.
The definition of the embedding of Kuranishi neighborhoods are similar to the usual
one explained in Section \ref{embed}.
A global pre-Kuranishi structure of $\widehat{\mathcal{M}}^\bullet$ is defined similarly,
and it induces a pre-Kuranishi structure of $\widehat{\mathcal{M}}$.
As we noted, we can define the fiber products
$((\widehat{\mathcal{M}}^\bullet)^\diamond, \mathring{K}^2, K, K^0 / S^1)$
and its multi-valued partial submersions $\Xi^\circ$ and $\Lambda$ similarly.
We can construct its grouped multisection satisfying the conditions similar
to those for $(\widehat{\mathcal{M}}^\diamond, \mathring{K}^2, K,K^0 / S^1)$.
Then its pull back defines the grouped multisection of
$(\widehat{\mathcal{M}}^\diamond, \mathring{K}^2, K,K^0)$.
As we have explained, the virtual fundamental chains defined by these grouped
multisections are the required ones.
Therefore Proposition \ref{H = 0} holds true.
Next we consider the case where $K^0$ contains cycles which are not invariant by
the $S^1$-action.
We assume that the $S^1$-action is free, that is, we only consider the case of
a pre-quantization space of some closed symplectic manifold.
We show that some terms of the rational part $\mathcal{H}_0$ of the generating function
are calculated by the Gromov-Witten invariants of the closed symplectic manifold.
The following argument is an adaptation of that given in \cite{EGH00} and \cite{Bo02}.
Let $(M, \omega)$ be a closed symplectic manifold of dimension $2(n-1)$ with an
integral cohomology class $[\omega] \in H^\ast (M; \mathbb{Z})$.
Let $\pi_M : Y \to M$ be a principal $U(1)$-bundle with first Chern class
$c_1(Y) = [\omega]$, and $\alpha$ be a connection form such that $\pi_M^\ast \omega
= -\frac{1}{2\pi\sqrt{-1}} d\alpha$.
Then $\lambda = -\frac{1}{2\pi\sqrt{-1}} \alpha$ is a contact form of $Y$ such that
$d\lambda = \pi_M^\ast \omega$.
Note that the Reeb flow of the pre-quantization space $(Y, \lambda)$ is opposite
to the usual $U(1)$-action on $Y$.
Since $\overline{P} = \bigcup_{k = 1}^\infty \overline{P}_k$ and $\overline{P}_k \cong M$,
a smooth triangulation of $M$ defines a triangulation $K$ of $\overline{P}$.
Let $J$ be an $\omega$-compatible almost complex structure on $M$.
It induces a complex structure of $\xi = \mathop{\mathrm{Ker}}\nolimits \lambda \cong \pi_M^\ast TM$, which
we also denote by $J$.
Then as an almost complex manifold,
$\hat Y = \mathbb{R} \times Y$ is isomorphic to
\[
Y \underset{U(1)}{\times} (\mathbb{C} \setminus 0)
= Y \underset{U(1)}{\times} (\mathbb{C} P^1 \setminus \{0, \infty\})
\]
by $(\sigma, y) \mapsto [y, e^{-2\pi \sigma}]$,
where the almost complex structure of $\mathcal{L} = Y \times_{U(1)} \mathbb{C}$ is defined by
$T_{[y, z]} \mathcal{L} \cong \xi_y \oplus T_z \mathbb{C}$.
The almost complex structure of $Y \times_{U(1)} \mathbb{C} P^1$ is similar.
Holomorphic buildings for $(Y, \lambda, J)$ and stable maps in $(M, J)$ are related
as follows.
For a holomorphic building $(\Sigma, z, u) \in \widehat{\mathcal{M}}(Y, \lambda, J)$ of height $k$,
a $J$-holomorphic map
\[
\check u : \check \Sigma \to Y \underset{U(1)}{\times} (
\underbrace{\mathbb{C} P^1 \lrsubscripts{\cup}{0}{\infty} \mathbb{C} P^1 \lrsubscripts{\cup}{0}{\infty}
\dots \lrsubscripts{\cup}{0}{\infty} \mathbb{C} P^1}_k)
\]
is defined by $\check u|_{\Sigma \setminus \coprod S^1}
= u|_{\Sigma \setminus \coprod S^1}$
(and removal of singularity), where
$(\check \Sigma, z, \pm\infty)$ is the blow down curve of $(\Sigma, z)$,
$\coprod S^1 \subset \Sigma$ is the union of imaginary cirlces in $\Sigma$
and we regard $\Sigma \setminus \coprod S^1$ as a subset of $\check \Sigma$.
Let $\hat \pi_M : Y \times_{U(1)} (
\mathbb{C} P^1 \lrsubscripts{\cup}{0}{\infty}
\dots \lrsubscripts{\cup}{0}{\infty} \mathbb{C} P^1) \to M$ be the projection.
Then $\bar u = \hat \pi_M \circ \check u : \check \Sigma \to M$ is a
$J$-holomorphic map, and the restriction of $\check u$ to the $i$-th floor component
$\check \Sigma_i \subset \check \Sigma$ can be regarded as a meromorphic section of
$\bar u^\ast \mathcal{L}$ on $\check \Sigma_i$.
Then each zero of $\check u$ with degree $k$ corresponds to a $+\infty$-limit circle of
$\Sigma$, and the asymptotic periodic orbit of $u$ on this circle has multiplicity $k$.
Similarly, each pole of $\check u$ with degree $k$ corresponds to a $-\infty$-limit circle
of $\Sigma$, and the asymptotic periodic orbit of $u$ on this circle has multiplicity $k$.
Let $(\hat \Sigma, z, \pm\infty)$ be the curve obtained by collapsing the irreducible
component of $(\check \Sigma, z, \pm\infty)$ corresponding to the trivial cylinders of
$(\Sigma, z, u)$.
Note that $\bar u$ induces a stable map $(\hat \Sigma, z \cup \{\pm\infty_i\}, \hat u)$
of $(M, J)$ since $\bar u$ is constant on each irreducible component of $\check \Sigma$
corresponding to a trivial cylinder of $(\Sigma, z, u)$.
Then the $E_{\hat \omega}$-energy of $(\Sigma, z, u)$ is
\begin{equation}
E_{\hat \omega}(u) = E(\hat u) := \int_{\hat \Sigma} \hat u^\ast \omega
= \sum_{+\infty_i} k_{\gamma_{+\infty_i}} - \sum_{-\infty_i} k_{\gamma_{-\infty_i}},
\label{energy of stable map}
\end{equation}
where $k_{\gamma_{\pm\infty_i}}$ is the multiplicity of $\gamma_{\pm\infty_i}$,
which is equivalent to the degree of the corresponding zero or pole of $\hat u$.
Conversely, let $(\check \Sigma, z \cup \{\pm\infty_i\})$ be a semistable curve
of genus $g = 0$ with a floor structure and
$\bar u : \check \Sigma \to M$ be a $J$-holomorphic map.
We assume that an integer $k_{\pm\infty_i} \geq 1$ is attached to
each marked point $\pm\infty_i$, and an integer $k_\mu \geq 1$ to
each nodal point $p_\mu$ which joints two components with different floors.
We assume that these integers satisfy the energy condition
for each component of $\check \Sigma$.
Namely, we assume that the sum of $k_{+\infty_i}$ and $k_\mu$ corresponding to
the zeros on the component is larger than the sum of $k_{-\infty}$ and $k_\mu$
corresponding to the poles on the component.
Then there exists a $J$-holomorphic map
\[
\check u : \check \Sigma \to Y \underset{U(1)}{\times} (
\mathbb{C} P^1 \lrsubscripts{\cup}{0}{\infty} \mathbb{C} P^1 \lrsubscripts{\cup}{0}{\infty}
\dots \lrsubscripts{\cup}{0}{\infty} \mathbb{C} P^1)
\]
which is obtained by patching meromorphic sections of $\bar u^\ast \mathcal{L}$ on
$\check \Sigma_i$
such that each $+\infty_i$ is a zero of degree $k_{+\infty_i}$,
each $-\infty_i$ is a pole of degree $k_{-\infty_i}$, and each nodal point $p_\mu$ is
a pole on the component of higher floor and a zero on the component of the lower floor
of degree $k_\mu$.
Furthermore, $\check u$ is unique modulo $\mathbb{C}^\ast$-valued holomorphic functions
on $\coprod_i \check \Sigma_i$.
(The uniqueness is true for $g \geq 1$ but the existence is not always true for
$g \geq 1$.)
Let $(\Sigma, z)$ be the curve obtained by the oriented blow up of $(\check \Sigma, z)$
at $\pm\infty_i$ and $p_\mu$ with appropriate $\varphi_\mu \in S^1$.
Then $\check u$ defines a holomorphic building $(\Sigma, z, u) \in \widehat{\mathcal{M}}$.
(There are $k_\mu$ choices of $\varphi_\mu \in S^1$ for each $\mu$.)
Assume that all cycles in $K^0$ except one cycle $y$ are $S^1$-invariant.
We show that if we use an appropriate virtual fundamental cycles then
$\mathcal{H}_0 \in \mathcal{W}_Y|_{g = 0} / (t_y^2)$ is calculated by
the rational Gromov-Witten invariants of $(M, \omega)$.
First we recall the definition of Gromov-Witten invariants.
Since the pre-Kuranishi spaces used for its definition do not have boundary of codimension
one, usually we do not need any compatibility conditions of the virtual fundamental
chains of them for construction.
However, in order to use the induced grouped multisection of the pre-Kuranishi spaces
for the definition of SFT cohomology of $(Y, \lambda)$,
we need some compatibility conditions.
We need the following data $(\hat p_0^+, \hat S, \hat E^0, \hat \lambda)$
to define a Kuranishi neighborhood of $\hat p_0 = (\hat \Sigma_0, z, \hat u_0)
\in \overline{\mathcal{M}}(M, J)$:
\begin{itemize}
\item
$\hat p_0^+ = (\hat \Sigma_0, z \cup z^+, \hat u_0)$ is a curve obtained
by adding marked points to make $(\hat \Sigma_0, z \cup z^+)$ stable.
We assume that $G_0 = \mathop{\mathrm{Aut}}\nolimits(\hat \Sigma_0, z, \hat u_0)$ preserves $z^+$ as a set.
\item
$\hat S \subset M$ is a finite union of codimension-two submanifolds such that
$u_0$ intersects with $\hat S$ at $z^+$ transversely.
\item
Let $(\hat P \to \hat X, Z \cup Z^+)$ be the local universal family of
$(\hat \Sigma_0, z \cup z^+)$.
Then $\hat E^0$ is a finite dimensional $G_0$-vector space
and $\hat \lambda : \hat E^0 \to C^\infty(\hat P \times M;
{\textstyle\bigwedge}^{0, 1} V^\ast \hat P \otimes TM)$
is a $G_0$-equivariant linear map which satisfies following conditions:
\begin{itemize}
\item
For each $h \in \hat E^0$, the projection of the support of $\hat \lambda(h)$ to
$\hat P$ does not intersect with the nodal points or marked points $Z$.
(It may intersect with $Z^+$.)
\item
The linear map
\begin{align*}
\widehat{D}_{\hat p_0}^+ : \widetilde{W}^{1, p}(\hat \Sigma_0, \hat u_0^\ast TM)
\oplus \hat E^0
&\to L_\delta^p(\hat \Sigma_0, {\textstyle\bigwedge}^{0, 1}T^\ast \hat \Sigma_0 \otimes
\hat u_0^\ast T M)\\
&\quad \oplus \bigoplus_{z_i} T_{\hat u_0(z_i)} M\\
(\xi, h) &\mapsto (\widehat{D}_{\hat p_0} \xi + \hat \lambda(h), \xi (z_i))
\end{align*}
is surjective,
where $\widehat{D}_{\hat p_0}$ is a linearization of the equation of
the $J$-holomorphic maps, that is,
\[
\widehat{D}_{\hat p_0} \xi = \nabla \xi + J(\hat u_0) \nabla \xi j
+ \nabla_\xi J(u_0) d\hat u_0 j.
\]
\end{itemize}
\end{itemize}
Using the above data, we can construct a Kuranishi neighborhood of
$\hat p_0 \in \overline{\mathcal{M}}(M, J)$ similarly.
A global pre-Kuranishi structure of $\overline{\mathcal{M}}(M, J)$ is also constructed similarly.
Define its fiber products $\overline{\mathcal{M}}(M, J)^m
_{(\epsilon^{i, j}_l, c^i_l, \mathring{x}^i_l, \eta^i_l)}$
for all sequences $(\epsilon^{i, j}_l, c^i_l, \mathring{x}^i_l, \eta^i_l)$ consisting of
$\epsilon^{i, j}_l \in K^2$, $c^i_l \in K$,
$\mathring{x}^i_l \in \{ x / S^1; x \in K^0 \setminus \{y\}\} \cup \{\pi_M(y)\}$ and
$\eta^i_l \in K$,
where we regard each $x / S^1$ ($x \in K^0 \setminus \{y\}\}$)
as a cycle of dimension $\dim x - 1$ defined by the map $x / S^1 \to \pi_M(x)$,
and $\pi_M(y)$ as a cycle of dimension $\dim y$ defined by the map $y \to \pi_M(y)$.
We also define the fiber product $(\overline{\mathcal{M}}(M, J)^\diamond, \mathring{K}^2, K,
K^0 / S^1)$
similarly to $(\widehat{\mathcal{M}}^\diamond, \mathring{K}^2, K,K^0)$, and construct its grouped
multisection satisfying the compatibility conditions.
Then using the induced grouped multisection of the fiber products $\overline{\mathcal{M}}(M, J)^m
_{(\epsilon^{i, j}_l, c^i_l, \mathring{x}^i_l, \eta^i_l)}$, we can construct
the Gromov-Witten invariant of $(M, \omega)$.
Now we consider the pre-Kuranishi structure of $\widehat{\mathcal{M}}_{g = 0}(Y, \lambda, J)$.
The Kuranishi neighborhood of a point $p_0 \in \widehat{\mathcal{M}}_{g = 0}(Y, \lambda, J)$
is defined by the data $(p_0^+, S, E^0, \lambda)$ obtained from the data
$(\hat p_0^+, \hat S, \hat E^0, \hat \lambda)$
for the stable curve $\hat p_0 = (\hat \Sigma, z, \hat u)$,
where
$p_0^+ = (\Sigma, z \cup z^+, u) \in \widehat{\mathcal{M}}(Y, \lambda, J)$ is a curve obtained
by adding the marked points $z^+$ to $(\Sigma, z)$ corresponding to the
additional marked points of $\hat p_0^+$, $S$ and $E^0$ are defined by
$S = \pi_M^{-1}(\hat S)$ and $E^0 = \hat E^0$, and
$\lambda : E^0 \to C^\infty(\hat P \times Y; {\textstyle\bigwedge}^{0, 1} V^\ast \hat P
\otimes (\mathbb{R} \partial_\sigma \oplus TY))$ is the map defined by the pull back of $\hat \lambda$ and
the isomorphism
\[
T_{(\sigma, y)} \hat Y = \xi_y \oplus (\mathbb{R} \partial_\sigma \oplus \mathbb{R} R_\lambda(y))
\cong (\pi_M^\ast TM)_y \oplus \mathbb{C}.
\]
Then the linear operator $D_{p_0}^+$ is not necessarily surjective,
but if we replace all vector spaces
$T_{\pi_Y \circ u_0(z_i)} Y$ except one in the range of $D_{p_0}^+$ with
$T_{\pi_M \circ \pi_Y \circ u_0(z_i)} M$, then it becomes surjective.
Hence we can define the generating function $\mathcal{H}_0$ modulo $(t_y^2)$ using
the grouped multisections of the fiber products of $\widehat{\mathcal{M}}$ induced by those of
the corresponding fiber products of $\overline{\mathcal{M}}(M, j)$.
Then it is easy to see that
\begin{align*}
&[\overline{\mathcal{M}}^Y((\hat c_l), (y, x_l), (\alpha_l))]^0\\
&= (-1)^{\sum |\hat c_l|} \prod k_{\hat c_l} \cdot
[\overline{\mathcal{M}}(M, J)_{((\hat c_l), (\pi_M(y), (x_l / S^1)), ([M] \cap \alpha_l))}]^0,
\end{align*}
where $x_l$ are cycles in $M$, each $k_{\hat c_l}$ is the multiplicity of the
periodic orbits in $c_l$.
Note that in the left hand side of the above equation, the correction terms vanish
because they correspond to linear combinations of fiber products of several pre-Kuranishi
spaces, and for each fiber product, at least one factor has a locally free $S^1$-action.
(See \cite{EGH00} or \cite{Bo02} for more sophisticated expression of the above
equation.)
\section{Notation of differential}\label{diff notation}
We use the following notation in Section \ref{smoothness}.
\begin{defi}
Let $X$, $Y$ be real Banach spaces
(or finite dimensional vector spaces).
A continuous map $A : X \to Y$ is said to be differentiable at $x\in X$
if there exists a bounded operator
$DA_x :X\to Y$
such that for any $\epsilon >0$ there exists some constant $\delta>0$
such that
$||A(x+v) - A(x) + DA_x \cdot v ||_Y \leq \epsilon ||v||_X$
for any $||v||_X\leq \delta$.
We call $DA_x$ the differential of $A$ at $x\in X$.
$A$ is said to be of class $C^1$ if it is differentiable at every point of $X$ and
$DA : X\to L(X,Y)$ is continuous.
$A$ is said to be of class $C^k$ if it is of class $C^1$ and $DA$ is of class $C^{k-1}$.
Define $D^kA=D(D^{k-1}A) : X \to L(X,L(X,\dots, L(X,Y)\dots ))$
inductively.
Hence
\begin{align*}
&D^kA_x\cdot v^k\cdot v^{k-1} \cdot \dots \cdot v^1\\
&= \frac{\partial^k}
{\partial t^1\partial t^2 \dots \partial t^k}A(x+t^1v^1+t^2v^2+\dots
+t^kv^k)\Bigr|_{t^1=t^2=\dots=t^k=0} \in Y
\end{align*}
for any $v^1,v^2,\dots,v^k\in X$.
\end{defi}
|
2,877,628,088,979 | arxiv | \section{Introduction}\label{S:intro}
Characterizing quantum entanglement~\cite{Schrodinger1935},
\cite{Bruss2002} is an important open problem in quantum
information theory~\cite{Nielsen2000}. The nonclassical
correlations associated with entanglement have been of immense
interest since the very inception of quantum mechanics
\cite{Einstein1935}, \cite{Bell1964}. Quantum information science
has identified entanglement as a potential resource. The ability
of quantum computers to solve classically hard problems
efficiently, the increased security of quantum cryptographic
protocols, the enhanced capacity of quantum channels---all these
are attributed to entanglement~\cite{Nielsen2000}. The presence of
entanglement has been related to quantum phase transitions and the
behavior of condensed systems~\cite{Ghosh2003},
\cite{Osborne2002}, \cite{Osterloh2002}. Entanglement has also
allowed the understanding of techniques such as
density-matrix-renormalization group in a new
light~\cite{Vidal2003}. A significant part of recent research in
theoretical quantum information science has centered around
understanding and characterizing entanglement. In spite of this,
entanglement remains a poorly understood feature of quantum
systems.
Although many tests have been devised which attempt to decide whether
a general quantum state is separable or not, this problem is known to
be NP-Hard~\cite{Gurvits2003}. Quantifying entanglement involves
devising functions acting on quantum states that, in some reasonable
way, order entangled states according to the degree of nonclassical
correlation possessed by them. Measures of entanglement can be
broadly divided into two classes depending on whether an efficient
way of computing them for arbitrary states exists or not. Tests for
separability can also be classified in a similar
fashion~\cite{Bruss2002}. {\em Computationally operational} measures of
entanglement are easy to calculate for any state, while there is no
known procedure for efficiently calculating {\em computationally nonoperational}
measures for an arbitrary state. From here on we abbreviate the descriptions
computationally operational and computationally nonoperational to
simply operational and nonoperational, respectively. Several
physically significant measures of entanglement are of the
nonoperational variety. This makes it important to place bounds on
the values of such measures. In this paper, we investigate the
problem of placing lower bounds on nonoperational measures of
entanglement for a quantum state assuming that we know the values of
one or more operational measures for that state.
The outline of this paper is as follows. In Sec.~\ref{general} we start
with examples of both operational and nonoperational measures of
entanglement. We then discuss the general scheme of placing bounds on
nonoperational measures using operational ones as constraints. In
Sec.~\ref{sec:Phimap} we start with the separability criterion due to
Breuer~\cite{Breuer2006} and then show that a new, operational entanglement
measure, called the $\Phi$-negativity, can be extracted from it. In
Sec.~\ref{singly} we use the $\Phi$-negativity to bound three
nonoperational measures of entanglement for $4\times N$ systems, namely,
the entanglement of formation, the tangle, and the concurrence. We compare
our results to the bounds based on another operational measure, the
negativity. In the process, we present a different way of deriving the
results in~\cite{Chen2005}. In Sec.~\ref{doubly} we obtain bounds on the
three nonoperational measures using both the negativity and
$\Phi$-negativity simultaneously as constraints. We also discuss how our
new bounds relate to previously known bounds in this section. Our
conclusions and future prospects are summarized in Sec.~\ref{conclusion}.
\section{General considerations}\label{general}
\subsection{Operational and nonoperational measures of entanglement}
A commonly used measure of entanglement for a pure-state $\ket{\Psi}$ of two
systems $A$ and $B$ is the entropy of the reduced density operator $\rho_A$ (or
$\rho_B$),
\begin{equation}
\label{vonneumann}
S(\rho_A)=- {\rm{Tr }}(\rho_A \log\rho_A)=S(\rho_B)=- {\rm{Tr }}(\rho_B \log\rho_B).
\end{equation}
We write this entropy either as a function $h(\Psi)$ of the state $\ket{\Psi}$
or as a function $H(\vec{\mu})$ of the vector of Schmidt coefficients of
$\ket{\Psi}$. It is a physically motivated quantity, in that it gives the rate
at which copies of a pure state can be converted, by using only local operations
and classical communication (LOCC), into copies of maximally entangled states
and vice versa~\cite{Bennett1996}. This measure can be elevated so that it
applies to bipartite mixed states also by taking the so-called convex-roof extension of
Eq.~(\ref{vonneumann}). This extended quantity is the entanglement of formation (EOF),
and it is defined as
\begin{equation}
\label{eof}
h(\rho)\equiv \min_{\{p_j, \ket{\Psi_j}\}}\bigg\{\sum_j p_j
h(\Psi_j)\biggr|\rho=\sum_jp_j \ket{\Psi_j}\bra{\Psi_j} \bigg\}.
\end{equation}
The EOF provides an upper bound on the rate at which maximally
entangled states can be distilled from $\rho$ and a lower bound on
the rate at which maximally entangled states must be supplied to
create copies of $\rho$~\cite{Hayden2001}. Exact expressions for
the EOF of several classes of states are known. One of the
earliest, and simplest, was for an arbitrary state of two qubits
\cite{Wootters1998}. The EOF in that case, was presented in terms
of the concurrence, a subsidiary quantity. The concurrence itself
has since been identified as an entanglement monotone and extended
to higher-dimensional systems~\cite{Rungta2001},\cite{Rungta2003}.
The EOF and the concurrence are examples of a more general framework of defining
entanglement measures. Suppose we have an entanglement measure $g$ defined only
on pure states $\ket{\Psi}$, which is a concave function $G$ of Schmidt
coefficients $\vec{\mu}$ of the marginal density operator of $\ket{\Psi}$. That
is, suppose $g$ has the form $g(\Psi)= G(\vec{\mu})$ on pure states. This can
be extended to a measure on mixed states via the convex-roof extension,
\begin{equation}
\label{monotone}
g(\rho)=\min_{\{p_j, \ket{\Psi_j}\}}\bigg\{\sum_j p_j
g(\Psi_j)\biggr|\rho=\sum_j p_j\ket{\Psi_j}\bra{\Psi_j} \bigg\}.
\end{equation}
It has been proven~\cite{Vidal2000} that any $g(\rho)$ constructed in this
way is, on average, nonincreasing under LOCCs. An entanglement measure with this property is known
as an entanglement monotone. Besides the EOF and concurrence, other
examples of entanglement monotones include the tangle, relative entropy,
entanglement of distillation, etc. Each has its use in particular physical
contexts. All the entanglement measures just mentioned have one feature
in common: they are nonoperational. The bottleneck in evaluating most of
these measures for mixed states is the minimization over all pure-state
decompositions. As a consequence, placing lower bounds on these measures
of entanglement for arbitrary states becomes important.
An alternate approach to detecting and quantifying entanglement is
based on the application of positive (but not completely positive)
maps on density operators
\cite{Stinespring1955},\cite{Stormer1963}, \cite{Choi1972},
\cite{Choi1974}, \cite{Choi1975}, \cite{Horodecki1996},
\cite{Terhal2001}, \cite{Rudolph2000}, \cite{Rudolph2002},
\cite{Chen2003}. In particular, a quantum state is separable if
and only if it remains positive semidefinite under the action of
{\em any}\/ positive map. Given a positive map, we can construct
an entanglement measure based on the spectrum of the density
operators under the action of the map \cite{Plenio2007},
\cite{Vidal2002}. Such measures are typically much easier to
calculate for general quantum states than the ones discussed
earlier because they do not involve the convex-roof construction.
Measures of entanglement based on positive maps are therefore
operational in nature. The negativity~\cite{Zyczkowski1998},
\cite{Vidal2002} is an example of an entanglement measure of this
sort, derived from the transpose map~\cite{Peres1996},
\cite{Horodecki1996}.
We can use the operational measures of entanglement as constraints
to obtain bounds on nonoperational, convex-roof-extended ones. The
complexity of the minimization in Eq.~(\ref{monotone}) is reduced
by solving it over a constrained set, instead of over all
pure-state decompositions. This was done in~\cite{Chen2005a},
\cite{Chen2005} for the EOF and the concurrence by minimizing over
states with a given value of negativity. We turn now to
describing the general procedure for constructing bounds based on
the use of one or more operational entanglement measures as
constraints.
\subsection{Multiply-constrained bounds on nonoperational measures of
entanglement}\label{subsec:multiply}
Let $f_1,\cdots,f_K$ be $K$ operational measures used to characterize the
entanglement in a bipartite system. Assume that they have values ${\bf n} \equiv
(n_1,\ldots,n_K)$ for a state $\rho$. Their action on pure states can be
expressed as functions of the Schmidt coefficients, i.e.,
\begin{equation}
\label{eq:constraints} f_i(\Psi) =F_i(\vec {\mu}),\;\; i =
{1,\cdots,K}.
\end{equation}
We are interested in a lower bound on the value of an independent,
nonoperational measure $g$, which is a monotone defined on mixed
states via the convex-roof construction. Let us assume that for the
state $\rho$, the optimal pure-state decomposition with respect to
$g$ is $\rho = \sum_j p_j |\Psi^j \rangle \langle \Psi^j|$. Then
\begin{equation}
\label{eq:doublyA2}
g(\rho) = \sum_j p_j \, g\big( \Psi^j \big) = \sum_j p_j \, G \big(\vec{\mu}^j
\big).
\end{equation}
Now define the function
\begin{equation}
\label{eq:doublyA3}
\widetilde{G}(m_1,\ldots,m_K) \equiv \widetilde{G}({\bf m})=\min_{\vec
{\mu}}\big\{G(\vec {\mu}) \big| F_1(\vec{\mu})=m_1,\ldots, F_K(\vec{\mu})=m_K
\big\}.
\end{equation}
Let ${\mathcal G}({\bf m}) = {\mbox{co}} \bigl[ \widetilde{G} ({\bf m}) \bigr]$
be the convex hull of $\widetilde{G}({\bf m})$, i.e., the largest convex
function of $K$ variables $(m_1, \ldots , m_K)$ that is bounded from above by
$\widetilde{G}({\bf m})$. Using Eq.~(\ref{eq:doublyA3}) and the convexity of
${\mathcal G}$, we can write
\begin{equation}
\label{eq:doublyA5}
g(\rho) \geq \sum_j p_j \, {\mathcal G}\big( {\bf n}^j \big) \geq {\mathcal G}
\biggl( \sum_j p_j {\bf n}^j \biggr).
\end{equation}
If ${\mathcal G}$ is a monotonically nondecreasing function of all its
arguments and if the operational measures $F_i$ are convex functions so that
$\sum_j p_j n^j_i \geq n_i$, we obtain
\begin{equation}
\label{eq:doublyA6}
g(\rho) \geq {\mathcal G} ({\bf n}).
\end{equation}
If the conditions for the validity of the inequality~(\ref{eq:doublyA6}) are
met, then we obtain a lower bound on $g(\rho)$ by knowing the operational
measures ${\bf n}$ for $\rho$.
Regrettably, the first assumption leading to inequality
(\ref{eq:doublyA6}) is not always valid: the function
$\mathcal{G}({\bf n})$ is not guaranteed to be monotonic. If it is
not, then we have to impose monotonicity by introducing a new
monotonically nondecreasing function $\widetilde G_{\uparrow}({\bf
n})$, constructed from $\widetilde{G}({\bf n})$. In the examples we
consider in Sec.~\ref{doubly}, $\widetilde{G}({\bf n})$ turns out to
be monotonic, so we do not have to construct the new function
$\widetilde G_{\uparrow}({\bf n})$. For the sake of completeness, the
general construction of $\widetilde G_{\uparrow}({\bf n})$ is
presented in Appendix~{\ref{A:construction}}.
We can now redefine ${\mathcal G}({\bf n})$ as the convex hull of
$\widetilde G_{\uparrow}({\bf n})$, rather than simply the convex
hull of $\widetilde G({\bf n})$. It is not immediately obvious that
the convex hull of a monotonically nondecreasing function is also
monotonically nondecreasing. The proof that this is so is given in
Appendix~\ref{AppProof}.
The only requirement on the operational entanglement
measures $F_i$ for using them as constraints is that they are convex
functions on the set of states. Furthermore, even if we do not have
the functions $F_i$ themselves, but have instead functions
$\hat{F}_i$ that bound $F_i$ from above for pure states, then the
functions $\hat{F}_i$ can be used as constraints in the
definition~(\ref{eq:doublyA3}) of $\widetilde G$, in place of the
functions $F_i$. The arguments leading to
inequality~(\ref{eq:doublyA6}) go through exactly as before, i.e.,
$g(\rho)\geq\sum_j p_j{\mathcal G}(\hat{\bf n}^j)\geq{\mathcal G}
\Bigl(\sum_j p_j \hat{\bf n}^j\Bigr)\geq\mathcal{G}\Bigl(\sum_j
p_j{\bf n}^j\Bigr)\geq{\mathcal G}({\bf n})$, the only difference
being that there is an additional step, the second-to-last one, where
we use $\hat{F}_i (\vec{\mu}) = \hat {n}_i \geq n_i = F_i$ to
conclude that $\sum_j p_j \hat n^j_i \geq \sum_j p_j n_j$. The
danger in using upper bounds instead of the actual values of the
functions $F_i$ is that the final bound on $g(\rho)$ might turn out
to be less useful or even meaningless. In the example we consider in
Sec.~\ref{doubly}, however, one of the constraints we use is an upper
bound on an operational entanglement measure, rather than the measure
itself, yet the bound we get turns out to be stronger than previous
bounds.
Since our bound is intended for arbitrary states, there is one more
subtlety to address, and that is the domain of the functions
$G,\widetilde{G}$, and $\mathcal{G}$. The
operational measures ${\bf n}$ map the state $\rho$ to a point in a
$K$-dimensional hypercube in the space of the $K$ independent
constraints $n_k$. Pure states correspond to a simply connected
subset in this hypercube, which we call the pure-state region. The
pure-state region is the domain of the functions $G$ and $\widetilde{G}$. This
domain is not always convex, and so $\mathcal{G}({\bf n})$ is defined on the
convex hull of the pure-state region, which is generally bigger than the
pure-state region, though only a subset of the full hypercube available to a
general state.
Finally, we have to extend ${\mathcal G}({\bf n})$ to the entire hypercube of
states. Note that for inequalities (\ref{eq:doublyA5}) and (\ref{eq:doublyA6})
to hold, ${\mathcal G}({\bf n})$ must be a monotonically nondecreasing function
in the entire hypercube while it has to be convex only on the convex hull
of the pure-state region. So, in extending ${\mathcal G}({\bf n})$ outside the
hull, we only have to take into account the monotonicity requirement
(\ref{eq:doublyA6}). To construct such an extension of ${\mathcal G}({\bf n})$,
start from a point on the boundary of the hull and begin traversing out along
{\em decreasing} directions parallel to the axes of the hypercube. Outside the
hull, and till reaching the boundaries of the hypercube, the extension is
defined as the constant function with value equal to that at the point on the
boundary of the hull. To generate the complete extension, this simple procedure
is repeated for every point on all the boundaries of the hull. This procedure
is also demonstrated in detail in Sec.~\ref{doubly} for the example we consider.
In this paper, we carry out the general program just described
with two particular constraints ($K=2$). One of them is the
negativity \cite{Vidal2002}. For the second, we develop a new
entanglement measure, called $\Phi$-negativity, based on a
recently presented separability criterion \cite{Breuer2006}
(see~\cite{Breuer2006a} for another measure based on the same
criterion). Like the negativity, it is easily computable for any
$\rho$ and there are no convex-roof constructions involved in the
computation. The $\Phi$-negativity, unlike the negativity, is not
a simple function of the Schmidt coefficients for pure states. We
find a simple function of the Schmidt coefficients that is an
upper bound on the $\Phi$-negativity and, as described above, we
use this function instead as the constraint to simplify our
computations. We use both the (upper bound on) $\Phi$-negativity
and the negativity simultaneously as constraints to place new
bounds on the EOF, tangle, and concurrence of $4\times N$ systems.
Ours is the first instance of a doubly-constrained bound on
entanglement measures for a family of states. It puts bounds that
are tighter than those obtained in~\cite{Chen2005a},
\cite{Chen2005}. Multiply constrained bounds based on entanglement
witnesses that can be applied to individual quantum states have
been obtained using a different approach in \cite{Guhne2006},
\cite{Eisert2006}.
Although all of the results in this paper are obtained using the
negativity and $\Phi$-negativity, a third constraint based on the
realignment criterion~\cite{Rudolph2000}, \cite{Rudolph2002},
\cite{Chen2003} can be added to improve the bounds for certain
classes of states. On pure states, the negativity and the
realignment criterion lead to the same constraint. This means that
in deriving both the singly and doubly constrained bounds we could
have modified the negativity to take advantage of this, as was
done in~\cite{Chen2005a}, \cite{Breuer2006a}. Furthermore, the
addition of the realignment criterion adds very little complexity
to the procedure described below.
Before concluding this section, we review the notation used in this
paper. We use lower case Latin letters, say $g$, to denote
entanglement measures. The corresponding upper case character, $G$,
denotes the same entanglement measure defined on pure states,
expressed as a function of the Schmidt coefficients. The same letter
with a tilde on top, $\widetilde{G}$, stands for the minimum of $G$
subject to constraints. Calligraphic letters like ${\mathcal G}$
denote the bound on $g$ obtained by taking the convex hull of
$\widetilde{G}$. If we have to impose monotonicity on $\widetilde{G}$
as an intermediate step, we define a new
function~$\widetilde{G}_{\uparrow}$.
\section{\texorpdfstring{$\Phi$-}{Phi-}Map}\label{sec:Phimap}
Recently, a new separability criterion has been proposed based on
a positive nondecomposable map \cite{Breuer2006}. It is a
combination of the Peres criterion and the reduction
criterion~\cite{Cerf1999}, \cite{Horodecki1999} for detecting
entangled states. In this section we construct a new entanglement
measure from this map and calculate it for pure states.
\subsection{Separability criterion}
Let us consider a finite-dimensional Hilbert space $\mathbb{C}^D$. It
can be regarded as the space of a spin-$j$ particle with $D=2j+1$. A
natural basis for this space is the ``angular-momentum basis''
$\ket{j,m}$, where $m=-j, -j +1, \dots, j-1, j$. The separability
criterion to be presented involves the time-reversal operator
$\vartheta$ whose action on an operator $\sigma$ acting on
$\mathbb{C}^D$ is given as
\begin{equation}
\vartheta \sigma = V \sigma^T V^{\dag},
\end{equation}
where the superscript $T$ stands for transposition in the
angular-momentum basis and $V$ is a unitary operator defined as
\begin{equation}
\label{eq:vmat}
\bra{j,m}V\ket{j,m'}= (-1)^{j-m}\delta_{m,-m'}.
\end{equation}
This map was initially introduced by Breuer to study the entanglement of
$4 \times 4 \; \mathrm{SU}(2)$ invariant states; in that case, the
$\vartheta$ map, together with the Peres criterion, was found to be a
necessary and sufficient separability condition~\cite{Breuer2005}. In even
dimensions, an additional property holds: $V^T=-V$, i.e., $V$ is
skew-symmetric in addition to being unitary.
The condition for positivity under the partial time-reversal map $(I
\otimes \vartheta)\rho \geq 0$ is unitarily equivalent to the Peres PPT
criterion $(I \otimes T)\rho \geq 0$. This means that partial time
reversal can be used as an entanglement detection criterion. Breuer
\cite{Breuer2006} defines a positive map
\begin{equation}
\label{phimap}
\Phi(\rho) = {\rm{Tr }}(\rho)I -\rho - V\rho^T V^{\dag}\;,
\end{equation}
which conjoins the time reversal map with the so-called reduction
criterion~\cite{Cerf1999}. The map $\Phi$ then defines for any joint density
operator $\rho_{AB}$ a necessary condition for separability as
\begin{equation}
\label{totalphi} (I\otimes\Phi)\rho_{AB} = {\rm{Tr }}_{B}(\rho_{AB})\otimes
I_B-\rho_{AB} - (I_A\otimes V)\rho_{AB}^{T_B}(I_A\otimes
V^{\dag})\geq 0.
\end{equation}
Any state that violates the above condition must be entangled.
Consider the space $\mathcal{H}_A\otimes
\mathcal{H}_B=\mathbb{C}^D \otimes \mathbb{C}^D$. It can be
regarded, without loss of generality, as the Hilbert space of two
spin-$j$ particles with $j=(D-1)/2$. The total spin of the system,
denoted by $J$ ranges over the values $J=0,1,\dots,2j=D-1$. Let
$P_J$ be the projector onto the ($2J+1$)-dimensional spin-$J$
manifold. It can then be shown that $\Phi$ is a nondecomposable
positive, but not completely positive map~\cite{Breuer2006},
\cite{Breuer2006a} in all even dimensions $D$ greater than or
equal to 4. The proof of positivity cannot be extended to odd
dimensions as it exploits the skew-symmetric nature of the unitary
operator $V$. In addition, the hermitian operator
\begin{equation}
W \equiv (I\otimes \Phi)P_0
\end{equation}
is an optimal entanglement witness~\cite{Lewenstein2000},
\cite{Breuer2006}, in that the set of PPT states detected by $W$
is not contained in the set detected by any other \emph{single}
witness. There, of course, exist families of PPT states that $W$
fails to detect. The optimal nature of $W$ provides motivation for
contructing a measure of entanglement based on the $\Phi$-map.
\subsection{Entanglement measures from maps}\label{sec2B}
Our endeavor here is to define a quantitative operational
measure of entanglement based on the $\Phi$ map. We call
this quantity the $\Phi$-negativity, denote it by $n_{\Phi}$, and
define it for a general mixed state as
\begin{equation}
\label{nphi}
n_{\Phi}(\rho)=
\frac{D(D-1)}{4}\left[\frac{||(I\otimes\Phi)\rho||}{D-2}-1\right],
\end{equation}
where $D=\min(\dim(\mathcal{H}_A),\dim(\mathcal{H}_B))$, and the
trace norm of an operator is defined as
$||O||= {\rm{Tr }}(\sqrt{OO^{\dag}})$. For a separable state $\sigma$,
$(I\otimes\Phi)\sigma$ has no negative eigenvalues, so
$||(I\otimes\Phi)\sigma||= {\rm{Tr }}{[(I\otimes\Phi) \sigma]} = (D-2)
{\rm{Tr }}(\sigma)=D-2.$ Hence the $\Phi$-negativity is zero on separable
states. This calculation also shows that $\Phi$ is not trace
preserving, and this motivates the factor of $D-2$ in the denominator
of Eq.~(\ref{nphi}). The $\Phi$-negativity is a shifted and scaled
version of the sum of the negative eigenvalues of a state under the
action of the map~(\ref{totalphi}). Since this sum can be
expressed in terms of the trace norm of an operator,
$||(I\otimes\Phi)\rho_{AB} ||$, it is a convex function of $\rho$ as
required in the general scheme described in Sec.~\ref{general}. By
defining the $\Phi$-negativity in terms of a map, we make sure that
it is an operational measure that involves no convex-roof extensions.
Similar measures of entanglement based on other positive, but not
completely positive maps have previously been proposed and
investigated~\cite{Plenio2007}. The negativity~\cite{Vidal2002}, which is based on
the Peres partial transpose criterion, is defined as
\begin{equation}
n_T(\rho) = \frac{||\rho^{T_A}||-1}{2},
\end{equation}
where $\rho$ is a joint density operator, $T_A$ is the partial
transposition with respect to system $A$. A positive value of $n_T$
indicates an entangled state. The Peres negativity, in addition to
being a measure of entanglement, is also an entanglement monotone,
since it is nonincreasing on average under LOCC
operations~\cite{Vidal2002}. The $\Phi$-negativity is not an
entanglement monotone, but it is a convex function of $\rho$.
The $\Phi$-negativity is a new operational measure of entanglement
for any quantum state. To use the $\Phi$-negativity as a constraint
in bounding nonoperational measures we need expressions for
$n_{\Phi}$ for pure states. We start from the Schmidt decomposition
of any pure state,
\begin{equation}
\label{schmidt}
\ket{\Psi_{AB}}=\sum_{i=1}^{D} \sqrt{\mu_i}\ket{a_i,b_i}
\end{equation}
for $\ket{\Psi_{AB}}\in \mathbb{C}^D\otimes\mathbb{C}^N$ and $D\leq
N$. The $\mu_i$ are the Schmidt coefficients, satisfying $\mu_i \geq
0 \;\; \forall\;i$ and $\sum_{i=1}^D \mu_{i}=1$.
Before we apply the map $(I \otimes \Phi)$ to this state we note that
the matrix $V$ appearing in the definition of the $\Phi$-map has the
form given in Eq.~(\ref{eq:vmat}) only in the angular-momentum basis
for system $B$, and the required transposition is also carried out in
this basis. Relabeling the angular-momentum eigenvectors
$\{\ket{j,m} \}$ as $\{ \ket{l} \}$ with $l=1,\ldots,D=2j+1$, we
transform $|\Psi_{AB} \rangle$ to the angular-momentum basis for
subsystem $B$ and obtain
\begin{equation}
\rho_{AB} =\ket{\Psi_{AB}}\bra{\Psi_{AB}}= \sum_{i,j,l,m=1}^D\sqrt{\mu_i \mu_j}
\langle l | b_i \rangle \langle b_j | m \rangle |a_i, l \rangle \langle a_j,
m|,
\end{equation}
and
\begin{eqnarray}
\label{phionrho1}
(I\otimes \Phi)\rho_{AB} & = & \sum_{i=1}^D \mu_i\ket{a_i}\bra{a_i} \otimes
\sum_{l=1}^D \ket{l}\bra{l} \nonumber \\
&& -\sum_{i,j,l,m=1}^D\sqrt{\mu_i \mu_j}
\langle l | b_i \rangle \langle b_j | m \rangle |a_i, l \rangle \langle a_j,
m| \nonumber \\
&& -\sum_{i,j,l,m=1}^D \sqrt{\mu_i \mu_j}
(-1)^{l+m} \langle b_i | l \rangle \langle m | b_j \rangle
\ket{a_i,D+1-m}\bra{a_j, D+1-l}.
\end{eqnarray}
The trace norm $||(I \otimes \Phi)\rho_{AB}||$ and the entanglement
measure $n_{\Phi}$ defined using the trace norm are rather
complicated functions of the Schmidt coefficients $\mu_i$ and the
matrix elements $\langle l| b_i\rangle$ of the unitary matrix that
transforms between the Schmidt basis of subsystem $B$ the angular
momentum basis used in Eq.~(\ref{eq:vmat}). Computing the numerical
value of $n_{\Phi}$ for any state is relatively easy, but the
analytic expression for the entanglement measure is quite unwieldy.
All we really need to generate a constraint from the $\Phi$-map,
which can be used to place lower bounds on nonoperational measures of
entanglement, is an upper bound on $n_{\Phi}$. Such a bound is
obtained by considering the special case in which the Schmidt basis
of subsystem $B$ is the same as the basis of the angular-momentum
eigenstates $\{ \ket{l} \}$, i.e., $\langle l | b_i \rangle =
\delta_{li}$. We then have
\begin{eqnarray}
\label{phionrho}
(I\otimes \Phi)\rho_{AB} & = & \sum_{i=1}^D \mu_i \ket{a_i}\bra{a_i} \otimes
\sum_{j=1}^D \ket{b_j}\bra{b_j} \nonumber \\
&& -\sum_{i,j=1}^D \sqrt{\mu_i\mu_j}\Bigl[\ket{a_i b_i}\bra{a_j
b_j}+(-1)^{i+j}\ket{a_i}\bra{a_j}\otimes\ket{b_{D-j+1}}\bra{b_{D-i+1}}\Bigr].
\end{eqnarray}
For the first nontrivial case, $D=4$, which we will be using
extensively, explicit diagonalization of the above operator is
possible. As shown in Appendix~\ref{appA}, it has six nonzero
eigenvalues, one of which is negative. The trace norm can then be
evaluated as the sum of the absolute values of the eigenvalues. Thus,
\begin{equation}
\label{tnorm}
||(I \otimes \Phi)\rho_{AB}||= {\rm{Tr }}\Bigl(\sqrt{[(I \otimes
\Phi)\rho_{AB}]^2}\Bigr) = 2[1+\sqrt{(\mu_1+\mu_4)(\mu_2+\mu_3)}],
\end{equation}
where we use the fact that the $\Phi$-map is hermiticity preserving.
Therefore, for all $4\times N$ pure states that have the Schmidt
basis for subsystem $B$ the same as the angular-momentum basis,
\begin{equation}
\label{nphi4}
n_{\Phi}=3\sqrt{(\mu_1+\mu_4)(\mu_2+\mu_3)} \equiv \hat{n}_{\Phi}.
\end{equation}
In Eqs.~(\ref{tnorm}) and (\ref{nphi4}) and in all our subsequent
discussion of the function $\hat n_\Phi$, the Schmidt coefficients
are ordered from largest to smallest, i.e.,
$\mu_1\ge\mu_2\ge\mu_3\ge\mu_4$.
The function $\hat{n}_{\Phi}$ is a simple function of the Schmidt
coefficients for any $4 \times N$ pure state and as shown numerically
by the results displayed in Fig.~\ref{bound}, the true
$\Phi$-negativity, $n_{\Phi}$, calculated with respect to a fixed
angular-momentum basis, is always bounded from above by
$\hat{n}_{\phi}$.
\begin{figure}[!ht]
\resizebox{8.1cm}{5.5cm}{\includegraphics{fig_bound.png}}
\caption{A histogram of the difference $\hat{n}_{\Phi} - n_{\Phi}$
for five million $4 \times 4$ pure states picked randomly from the
Haar measure. The bin size in the histogram is $0.001$. The
difference is always found to be positive. These results carry over
to $4 \times N$ states when $N>4$ ($N$ even), since the difference
only depends on the Schmidt coefficients. We have tried to prove
that the difference is nonnegative without success and thus rely on
this numerical demonstration instead.}
\label{bound}
\end{figure}
In the rest of this paper we use $\hat{n}_{\Phi}$ instead of
$n_{\Phi}$ as the constraint for bounding nonoperational measures
because of its simple algebraic form. When we refer to constraints
based on the $\Phi$-negativity we are referring to fixing the value
of $\hat{n}_{\Phi}$. Expressions for $\hat{n}_{\Phi}$ for pure states
in higher dimensions are discussed in Appendix~\ref{appA}.
In the next section, we will use $\hat{n}_{\Phi}$ to put lower
bounds on the EOF, tangle, and concurrence for $4 \times N$ mixed
states. We then compare our results to such bounds that have
already been derived based on the Peres negativity $n_T$. For $D
\times N$ pure states, the Peres negativity is given
by~\cite{Zyczkowski1998},\cite{Vidal2002}
\begin{equation}
n_T = \frac{\left(\sum_{i=1}^D\sqrt{\mu_i} \right)^2-1}{2}.
\end{equation}
\section{Singly Constrained Bounds}\label{singly}
\subsection{Entanglement of formation}\label{singly:eof}
A lower bound ${\mathcal H}\big( \hat{n}_{\Phi} \big)$ on the EOF, constrained
by pure states having a certain value for $\hat{n}_{\Phi}$,
can be obtained using the steps described in Sec.~\ref{general}.
All the subsequent results presented in this section and the
next are for $4\times N$ states $\rho$, with $N\ge4$.
Firstly, we have to find
\begin{equation}
\widetilde{H}\big( \hat{n}_{\Phi}
\big)=\min_{\vec{\mu}}\left\{H(\vec{\mu})\Bigl|3\sqrt{(\mu_1 + \mu_4)(\mu_2 +
\mu_3)}=\hat{n}_{\Phi} \right\},
\end{equation}
and then its convex hull,
\begin{equation}
{\mathcal H }(\rho) = \mathrm{co}\big[\widetilde{H}\big( \hat{n}_{\Phi}
\big)\big],
\end{equation}
provided $\widetilde{H} \big( \hat{n}_{\Phi} \big)$ is a monotonically
increasing
function of $\hat{n}_{\Phi}$. Defining $\mu_1+\mu_4=\alpha$ and
$\mu_2+\mu_3=\beta$, we can write the normalization and $\hat{n}_{\Phi}$
constraints as
\begin{equation}
\label{cons}
\alpha + \beta = 1\qquad\mbox{and}\qquad\alpha\beta =
\frac{\hat{n}_{\Phi}^2}{9},
\end{equation}
which give
\begin{equation}
\label{phialpha}
\alpha=\frac{1\pm\sqrt{1-4\hat{n}_{\Phi}^2/9}}{2}
\quad\mbox{and}\quad
\beta=\frac{1\mp\sqrt{1-4\hat{n}_{\Phi}^2/9}}{2}
\end{equation}
Minimizing
\begin{equation}
H(\vec\mu)=-\mu_1\log\mu_1-\mu_4\log\mu_4
-\mu_2\log\mu_2-\mu_3\log\mu_3=
H_2(\alpha)+\alpha H_2(\mu_1/\alpha)+\beta H_2(\mu_2/\beta),
\end{equation}
where $H_2(\cdot)$ is the binary entropy function, is trivial,
because we simply make the last two terms zero by choosing
$\mu_1=\alpha$ and $\mu_2=\beta$ [and choosing the upper sign in
Eq.~(\ref{phialpha}) so as to be consistent with the assumed ordering
of the Schmidt coefficients]. Then the minimum entropy is
\begin{equation}
\widetilde{H}\big( \hat{n}_{\Phi} \big) = H_2(\alpha).
\end{equation}
\begin{figure}[!ht]
\resizebox{8.1cm}{6cm}{\includegraphics{fig_rphi.png}}
\resizebox{8.1cm}{6cm}{\includegraphics{fig_rt.png}}
\caption{On the left is the bound on the EOF based on a constrained
$\hat{n}_{\Phi}$, Eq.~(\ref{phibound}). The plot on the right is the
bound on the EOF based on a constrained negativity,
Eq.~(\ref{Tbound}).}
\label{Rtphifig}
\end{figure}
That $\widetilde{H}\big( \hat{n}_{\Phi} \big)$ is a convex, monotonically
increasing function of $n_\Phi$ can be shown by considering its first and
second derivatives. Its convex roof is the function itself, i.e.,
\begin{equation}
{\mathcal H}\big( \hat{n}_{\Phi} \big)
=\mathrm{co}\big[\widetilde{H}\big(
\hat{n}_{\Phi} \big)\big]=\widetilde{H}\big( \hat{n}_{\Phi} \big),
\end{equation}
and the bound can thus be extended to mixed states, giving
\begin{equation}
\label{phibound}
h(\rho) \geq H_2\!\left(\frac{1+\sqrt{1-4{n}_{\Phi}^2/9}}{2}\right),
\end{equation}
with $n_\Phi$ being the $\Phi$-negativity of $\rho$.
The first step in bounding the EOF with only a single constraint on the
negativity is to determine the function
\begin{equation}
\label{negmin}
\widetilde{H}\big( n_T \big)
=\min_{\vec{\mu}}\left\{H(\vec{\mu})\left|\frac{\left(\sum_{j=1}^4
\sqrt{\mu_j}\right)^2-1}{2} = n_T \right.\right\}.
\end{equation}
This was solved in~\cite{Terhal2000},\cite{Chen2005a} for $2$ or
$3$ Schmidt coefficients and recently shown to be valid for any
number of Schmidt coefficients~\cite{Fei2006}. In particular, for
four Schmidt coefficients, the case of interest here, we obtain
\begin{equation}
\widetilde{H}\big( n_T \big) = H_2(\gamma) + (1-\gamma)\log_2 3,
\end{equation}
with
\begin{equation}
\label{eq:gamma}
\gamma=\frac{\left(\sqrt{2n_T+1} + \sqrt{3(3-2n_T)}\right)^2}{16}.
\end{equation}
\begin{figure}[!ht]
\resizebox{8 cm}{8
cm}{\includegraphics{fig_singleboundcompareF.png}}
\caption{In Region 1, the singly-constrained $n_{\Phi}$ bound is better
than the singly-constrained $n_T$ bound. In Region 2, the opposite is
true.}
\label{singlecompareF}
\end{figure}
Unlike $\widetilde{H}\big( \hat{n}_{\Phi} \big)$, $\widetilde{H}\big( n_T \big)$
is not convex over the entire range of $n_T$. It is, however, a
monotonically increasing function of $n_T$. The actual bound on the EOF is
thus the convex-roof extension of this function,
$\mathrm{co}[\widetilde{H}\big( n_T \big)]$, which is given
as~\cite{Chen2005a}
\begin{equation}
\label{Tbound}
\begin{array}{l}
h(\rho) \geq {\mathcal H}\big( n_T \big) \equiv
\mathrm{co}\big[\widetilde{H}\big( n_T \big)\big]=
\left\{ \begin{array}{ll}
H_{2}(\gamma)+(1-\gamma)\log _{2} 3, & n_T \in [0,1],\\[2mm]
\big(n_T -\frac{3}{2}\big)\log _2 3+ 2, & n_T \in [1,\frac{3}{2}].
\end{array} \right. \end{array}
\end{equation}
Both the singly constrained bounds are plotted in Fig.~\ref{Rtphifig}. It
might seem that the bound based on the $\hat{n}_{\Phi}$ constraint is
always poorer than that in Eq.~(\ref{Tbound}), but this is not the case.
There is a region in the $\hat{n}_{\Phi}$-$n_T$ plane where the bound of
Eq.~(\ref{phibound}) is better than that of Eq.~(\ref{Tbound}). This is
depicted in Fig.~\ref{singlecompareF}.
\subsection{Tangle and concurrence}\label{singly:tangle}
The procedure in the previous section can be undertaken for the
tangle $t(\rho)$ and the concurrence $c(\rho)$ \cite{Rungta2001},
\cite{Rungta2003}. To place bounds on the tangle, we start by
finding
\begin{equation}
\widetilde{T}\big( \hat{n}_{\Phi}
\big)=\min_{\vec{\mu}}\left\{2\left(1-|\vec{\mu}|^2\right) \Biggl|3\sqrt{(\mu_1
+ \mu_4)(\mu_2 + \mu_3)}=\hat{n}_{\Phi} \right\},
\end{equation}
which gives a bound for pure states. Then, just as for the EOF, the bound
on the tangle for mixed states is given by the convex hull of
$\widetilde{T}\big( \hat{n}_{\Phi} \big)$,
\begin{equation}
t (\rho) \geq {\mathcal T}\big( \hat{n}_{\Phi} \big) \equiv
\mathrm{co}\big[\widetilde{T}\big( \hat{n}_{\Phi} \big)\big],
\end{equation}
provided $\widetilde{T}\big(\hat{n}_{\Phi} \big)$ is a monotonically
nondecreasing
function of $\hat{n}_{\Phi}$.
Using the normalization and $\hat{n}_{\Phi}$ constraints of
Eq.~(\ref{phialpha}), we have
\begin{equation}
2\left(1-|\vec{\mu}|^2\right)=2\left(1-\sum_{i=1}^4\mu_i^2\right)=
4\sum_{i<j}\mu_i\mu_j
=4\left(\frac{\hat{n}_{\Phi}^2}{9}+\mu_1\mu_4+\mu_2\mu_3\right).
\end{equation}
Just as for the EOF, the minimization is trivial, the minimum
occurring for the upper sign in Eq.~(\ref{phialpha}), with $\mu_4=0$
($\mu_1=\alpha$) and $\mu_3=0$ ($\mu_2=\beta$), thus giving
\begin{equation}
\widetilde{T}\big( \hat{n}_{\Phi} \big)=\frac{4}{9}\hat{n}_{\Phi}^2.
\end{equation}
Since this is both monotonically increasing and convex in
$\hat{n}_{\Phi}$, the same bound holds for mixed states, but in terms
of the actual negativity $n_\Phi$, i.e.,
\begin{equation}
\label{concnphibound}
t(\rho) \geq {\mathcal T}\big({n}_{\Phi} \big) =
\frac{4}{9}{n}_{\Phi}^2.
\end{equation}
The lower bound on the tangle, subject to a constraint on the negativity,
is found by starting from
\begin{equation}
\widetilde{T} \big( n_T
\big)=\min_{\vec{\mu}}\left\{2\left(1-|\vec{\mu}|^2\right)\Biggl|\frac{
\left(\sum_{j=1}^4 \sqrt{\mu_j}\right)^2-1}{2}=n_T \right\}.
\end{equation}
This is a relatively involved minimization, but it is exactly the same as
the minimization problem that arises in evaluating a bound on the tangle
for isotropic states, so we can adapt the result of~\cite{Rungta2003} to
give
\begin{equation}
\label{rungtacavesbound}
\widetilde{T} \big( n_T \big) =
\frac{1}{12}\left(9+4n_T^2+\sqrt{3\left(3+4n_T-4n_T^2\right)}(2n_T-3)
\right).
\end{equation}
This quantity is monotonically increasing, but is not convex over the
complete range of $n_T$. The convex hull ${\mathcal T}\big( n_T \big)
\equiv \mathrm{co}\big[\widetilde{T} \big( n_T \big)\big]$ is required to
extend the bound to mixed states. Again using the results
of~\cite{Rungta2003}, we obtain
\begin{equation}
\label{concntbound}
\begin{array}{l}
t(\rho) \geq {\mathcal T}\big( n_T \big) = \left\{
\begin{array}{ll}
\frac{1}{12}\left(9+4n_T^2+\sqrt{3\left( 3+4n_T-4n_T^2\right)}(2n_T-3) \right),
& n_T \in [0,1],\\[2mm]
\frac{4}{3}n_T-\frac{1}{2}, & n_T \in [1,\frac{3}{2}]. \end{array}
\right. \end{array}
\end{equation}
We can derive from Eq.~(\ref{concnphibound}) an expression for the
lower bound on the concurrence of $4 \times N$ states with a given
value of ${n}_{\Phi}$:
\begin{equation}
c(\rho) \geq {\mathcal C}\big( n_{\Phi} \big) = \widetilde{C} \big(
n_{\Phi}
\big) = \sqrt{\widetilde{T} \big( n_{\Phi} \big)} = \frac{2}{3}
n_{\Phi}.
\end{equation}
An expression for the minimum of the concurrence, subject to the
negativity constraint, can be obtained from Eq.~(\ref{rungtacavesbound}).
The resulting function is everywhere concave, and thus its convex hull is
a straight line joining the end points. This line is
\begin{equation}
\label{chenbound}
c(\rho) \geq \mathcal{C} \big( n_T \big) = \sqrt{\frac{2}{3}}n_{T}.
\end{equation}
The bounds on both the tangle and the concurrence are plotted in Fig
\ref{Ctphifig}. As was true for the EOF, the $n_\Phi$ bound is better than the
$n_T$ bound in some parts of the ${n}_{\Phi}$-$n_T$ plane. This is shown in
Fig.~\ref{singlecompareT}.
\begin{figure}[!ht]
\resizebox{8.1cm}{6cm}{\includegraphics{fig_tphi.png}}
\resizebox{8.1cm}{6cm}{\includegraphics{fig_tt.png}}
\caption{The plot on the left shows the bounds on the tangle and the
concurrence based on the $n_{\Phi}$ constraint. The solid line is
the bound on the tangle and the dashed line is the bound on the
concurrence. On the right is a plot of the analogous bounds based on the
$n_T$ constraint.}
\label{Ctphifig}
\end{figure}
\begin{figure}[!ht]
\resizebox{8 cm}{8 cm}{\includegraphics{fig_singleboundcompareT.png}}
\caption{Region 1 is where the $n_{\Phi}$ constraint is better than the
$n_T$ constraint for bounding the tangle and concurrence. Region 2 is where the
converse is true.}
\label{singlecompareT}
\end{figure}
Recently, a lower bound on the concurrence has been derived based on the
negativity constraint~\cite{Chen2005}, using techniques different from those
employed here. That lower bound is exactly the one in Eq.~(\ref{chenbound}). We
have thus provided an independent derivation of the bound presented
in~\cite{Chen2005}. In addition, we can use the procedure from \cite{Chen2005} to
derive a lower bound on the tangle based on the $\hat{n}_{\Phi}$ constraint.
Then we obtain \begin{equation}
\frac{\widetilde{T} \big( \hat{n}_{\Phi} \big)}{4} -
\frac{\hat{n}_{\Phi}^2}{9}=
\mu_1\mu_4+\mu_2\mu_3\geq 0,
\end{equation}
which for general mixed states, leads exactly to the bound in
Eq.~(\ref{concnphibound}).
\section{Doubly Constrained Bounds} \label{doubly}
In this section we place new lower bounds on the EOF, tangle, and
concurrence for $4 \times N$ density operators by using $n_T$ and
$\hat{n}_{\Phi}$ simultaneously as constraints.
\subsection{Pure states of \texorpdfstring{$4 \times N$}{4 by N} systems}
\label{doubly:pure}
For a $4 \times N$ pure state, described by the Schmidt coefficients
$\mu_i$, $i=1,\ldots,4$, we have three constraint equations,
\begin{eqnarray}
\label{eq:doublyB1}
\frac{1}{2} \left[ \left( \sqrt{\mu_1} + \sqrt{\mu_2} + \sqrt{\mu_3} +
\sqrt{\mu_4}\,\right)^2 -1 \right] & = & n_T, \nonumber \\
3 \sqrt{(\mu_1 + \mu_4)(\mu_2 + \mu_3)} &=& \hat{n}_{\Phi}, \nonumber \\
\mu_1 + \mu_2 + \mu_3 + \mu_4 &=&1,
\end{eqnarray}
in addition to the inequality constraints
\begin{equation}
1\ge\mu_1\ge\mu_2\ge\mu_3\ge\mu_4\ge0.
\end{equation}
Both $\hat{n}_{\Phi}$ and $n_T$ take on values between $0$ and $3/2$,
so all $4 \times N$ states, pure or mixed, are mapped to a square of
side $3/2$ in the $\hat{n}_{\Phi}$-$n_T$ plane. Not all points in the
square correspond to pure states. If we solve the three equations in
(\ref{eq:doublyB1}) simultaneously and express $\mu_1$, $\mu_2$ and
$\mu_3$ in terms of $n_T$, $\hat{n}_{\Phi}$ and $\mu_4$ (see
Appendix~\ref{appB}), we find that for some allowed values of
$\hat{n}_{\Phi}$ and $n_T$, there is no allowed value of $\mu_4$ for
which the other three Schmidt coefficients are real numbers between
$0$ and $1$ in even one of the solution branches of
(\ref{eq:doublyB1}).
To find the region occupied by pure states in the $\hat{n}_{\Phi}$-$n_T$ plane,
let us use the pure-state expressions for $n_T$ and $\hat{n}_{\Phi}$ in
Eq.~(\ref{eq:doublyB1}) to find the largest and smallest values that $n_T$
can take on for a fixed value of $\hat{n}_{\Phi}$. We proceed as in the
minimization of $H(\vec\mu)$ in Sec.~\ref{singly}. Defining
$\alpha=\mu_1+\mu_4$ and $\beta=\mu_2+\mu_3$, the normalization and
$\hat{n}_{\Phi}$ constraints can be solved to give $\alpha$ and $\beta$ as in
Eq.~(\ref{phialpha}). The negativity takes the form
\begin{equation}
\sqrt{2n_T+1}=\sqrt{\mu_1}+\sqrt{\alpha-\mu_1}+\sqrt{\mu_2}+\sqrt{\beta-\mu_2}\;.
\end{equation}
It is trivial to see that the maximum of $n_T$ occurs when
$\mu_1=\mu_4=\alpha/2$ and $\mu_2=\mu_3=\beta/2$. This maximum cannot
be achieved, however, because we must respect the ordering $\mu_1
\geq \mu_2 \geq \mu_3 \geq \mu_4$ that we assumed in our definition
of $\hat{n}_{\Phi}$. We should always choose $\mu_2=\mu_3=\beta/2$,
but the best we can then do with $\mu_1$ and $\mu_4$ is to choose
$\mu_4=\beta/2$, $\mu_1=\alpha-\beta/2$ when $\alpha\geq\beta$ [upper
sign in Eq.~(\ref{phialpha})] or $\mu_1=\beta/2$,
$\mu_4=\alpha-\beta/2$ when $\beta\geq\alpha$ [lower sign in
Eq.~(\ref{phialpha})]. The requirement that $\mu_1\le\alpha$ implies
that the latter case can only be used when $\hat n_\Phi\ge\sqrt2$. In
both cases, the the maximum value of $n_T$ for fixed $\hat{n}_{\Phi}$
has the form
\begin{equation}
\label{eq:doublyB5}
n_T = \frac{1}{2}\left[\left(\sqrt{\alpha - \beta/2} + 3\sqrt{\beta/2}\right)^2 - 1\right].
\end{equation}
It turns out that the upper sign in Eq.~(\ref{phialpha}) always gives
a larger value for $n_T$. Using the upper sign, we find that the
maximum of $n_T$ for fixed values of $\hat{n}_{\Phi}$ is given by
\begin{equation}
\label{eq:doublyB5b}
n_T = \frac{3}{4}\!\left( 1- \sqrt{1 - \frac{4}{9} \hat{n}_{\Phi}^2} +
\sqrt{\frac{4}{3} \hat{n}_{\Phi}^2 + 2 \sqrt{1-\frac{4}{9} \hat{n}_{\Phi}^2} -2}
\; \right).
\end{equation}
The minimum value of $n_T$ occurs on the boundary of allowed Schmidt
coefficients, i.e., when $\mu_1=\alpha$ and $\mu_2=\beta$, with the
upper sign in Eq.~(\ref{phialpha}). Thus the minimum value of $n_T$
for a fixed value of $\hat{n}_{\Phi}$ is given by
\begin{equation}
\label{eq:doublyB8}
n_T =
\frac{1}{2}\left[(\sqrt{\alpha} + \sqrt{\beta}\,)^2 - 1\right]=
\frac{1}{3} \hat{n}_{\Phi}.
\end{equation}
From Eqs.~(\ref{eq:doublyB5}) and (\ref{eq:doublyB8}) we find that the
pure states of a $4\times N$ system lie in the region shown in Fig
\ref{fig:pure}. Notice that for this case of two constraints, the
pure-state region is not convex.
\begin{figure}[!ht]
\resizebox{8 cm}{8 cm}{\includegraphics{fig_pure.png}}
\caption{The pure-state region in the $\hat{n}_{\Phi}$-$n_T$ plane for $4
\times N$
systems.}
\label{fig:pure}
\end{figure}
\subsection{Entanglement of formation} \label{doubly:eof}
The EOF for pure bipartite states is a concave function of the marginal
density operator obtained by tracing over one of the subsystems. This
means that it is a concave function of the Schmidt coefficients
$\vec{\mu}$. Searching for a minimum is not the most natural thing one can
do with a concave function, yet this is what we are instructed to do by
the procedure for bounding the EOF outlined in Sec.~\ref{subsec:multiply}.
Starting from the EOF $H(\vec{\mu})$ for pure bipartite $4 \times N$
states, our objective is to find a convex, monotonic function ${\mathcal
H}({\bf n})$ as outlined in the Sec.~\ref{general}. This function will be
our lower bound on the EOF for all states.
The first step is to find the function
\begin{equation}
\label{eq:doublyC1}
\widetilde{H}({\bf n}) = \widetilde{H}\big(\hat{n}_{\Phi}, n_T \big) \equiv
\min_{\vec{\mu}} \left\{ H(\vec{\mu}) \left| 3 \sqrt{(\mu_1+
\mu_4)(\mu_2+\mu_3)} = \hat{n}_{\Phi}, \; \frac{ \left(\sum_j \sqrt{\mu_j}
\right)^2
-1 }{2} = n_T \right. \right\},
\end{equation}
which is defined on the pure-state region. The method of Lagrange multipliers
is not suitable for finding the minimum in Eq.~(\ref{eq:doublyC1}) because the
problem is over-constrained. The equations that we obtain using Lagrange
multipliers have a consistent solution only if $\hat{n}_{\Phi}$ and $n_T$ are
related as in Eq.~(\ref{eq:doublyB5b}) and therefore lie on the upper boundary
of the pure-state region. This does not mean that there is no minimum for
$H(\vec{\mu})$, but rather that the minimum lies on a boundary of the allowed
values of $\vec{\mu}$.
We already know $\widetilde{H}\big(\hat{n}_{\Phi},\,n_T \big)$ on
the boundaries of the pure-state region. The boundary with three
of the Schmidt coefficients being zero is the origin in the
$\hat{n}_{\Phi}$-$n_T$ plane where
$\widetilde{H}\big(\hat{n}_{\Phi},\,n_T \big)=H(\vec{\mu})=0$. The
boundary with two of the Schmidt coefficients zero lies on the
line $n_T=\hat{n}_{\Phi}/3$. To find the value of
$\widetilde{H}\big(\hat{n}_{\Phi},\, n_T \big)$ along this
boundary, note that the minimum of the EOF subject to just the
$\hat{n}_{\Phi}$ constraint occurs for $\vec{\mu}_{\Phi}=(\alpha,
1-\alpha,0,0)$, where $\alpha$ is given in Eq.~(\ref{phialpha}).
Substituting $\vec{\mu}_{\Phi}$ into $n_T$ we get $n_T=
\sqrt{\alpha (1-\alpha)}=\hat{n}_{\Phi}/3$. This means that along
the line $ n_T = \hat{n}_{\Phi}/3$, the $n_T$ constraint is
automatically satisfied if the $\hat{n}_{\Phi}$ constraint is
satisfied. Thus along the lower boundary of the pure-state region,
we have $\widetilde{H}\big(\hat{n}_{\Phi},\, n_T \big) =
\widetilde{H}\big( \hat{n}_{\Phi} \big)$. Similarly, along the
upper boundary of the pure-state region, the $\hat{n}_{\Phi}$
constraint comes for free. This is because the minimum of the EOF
subject to the $n_T$ constraint occurs when the Schmidt
coefficients are given by $\vec{\mu}_T = (\gamma, \gamma',
\gamma', \gamma')$ with $\gamma$ given by Eq.~(\ref{eq:gamma}) and
$\gamma' = (1-\gamma)/3$. The doubly-constrained problem reduces
to the singly-constrained problem when
$\hat{n}_{\Phi}=\sqrt{2(2\gamma+1) (1-\gamma)}$. Relabelling
$\gamma$ as $\alpha - \beta/2$ and $\gamma'$ as $\beta/2$ we see
that the $\hat{n}_{\Phi}$ constraint is automatically satisfied
along the upper boundary of the pure-state region if the $n_T$
constraint is satisfied. Hence along the upper boundary of the
pure-state region, we have $\widetilde{H} \big(\hat{n}_{\Phi},\,
n_T \big)=\widetilde{H}\big( n_T \big)$.
These considerations mean that for the entanglement of formation, the
monotone boundaries that we define in Appendix~\ref{A:construction}
coincide with the boundaries of the pure-state region, making it
unnecessary to construct the monotonically nondecreasing function
$\widetilde H_{\uparrow}\big(\hat n_\Phi,n_T\big)$, since $\widetilde
H\big(\hat n_\Phi,n_T\big)$ is itself monotonically nondecreasing.
The minimum of $H(\vec{\mu})$ in the remaining part of the pure-state
region can be found using the straightforward numerical procedure
described below. We start from the two distinct sets of solutions
$\vec{\mu}^{(1)}$ and $\vec{\mu}^{(2)}$ of the three constraint
equations (see Appendix~\ref{appB}). We go to the boundary where one
of the Schmidt coefficients is zero by setting $\mu_4=0$ in the
solutions. Now compute $H\big(\vec{\mu}^{(1)} \big)$ and
$H\big(\vec{\mu}^{(2)} \big)$ corresponding to the two solutions in
the regions in the $\hat{n}_{\Phi}$-$n_T$ plane where each of the
solutions is valid. The solutions are not valid in the whole
pure-state region because the three Schmidt coefficients have to be
real, nonnegative numbers less than one. All points in the pure-state
region cannot be covered if we set $\mu_4=0$. This is easily seen by
noticing that the point $\hat{n}_{\Phi}=n_T=3/2$ corresponds to the
fully entangled $4 \times N$ state and for this state all four
Schmidt coefficients have the value $1/4$. The fully entangled state
and other states close to it cannot be reached using the procedure
described above if we stay on the boundary defined by $\mu_4=0$. So
we start increasing the value of $\mu_4$ in small steps until it
reaches $1/4$. The parts of the 2-constraint region that are covered
by different choices of $\mu_4$ are shown in
Fig.~\ref{fig:m4regions}.
\begin{figure}[!ht]
\resizebox{8 cm}{8 cm}{\includegraphics{fig_m4_regions.png}}
\caption{The part of the 2-constraint region in which a value for
$\widetilde{H}({\bf n})$ can be computed is shown for four values of
$\mu_4=0$, 0.02, 0.1, and 0.25. The two lines are the boundaries of
the pure-state region.}
\label{fig:m4regions}
\end{figure}
This numerical procedure gives us ranges of values of $\mu_4$ over which
$H\big(\vec{\mu}^{(1)} \big)$ and/or $H\big(\vec{\mu}^{(2)} \big)$ can be
calculated at each point in the pure-state region. For the value of
$\widetilde{H}({\bf n})$ at each point, we pick the minimum over the
allowed range of values for $\mu_4$ at that point.
\begin{figure}[!ht]
\resizebox{16 cm}{8 cm}{\includegraphics{fig_eof_2constraint.png}}
\caption{(Color online) Plots of $\widetilde{H}({\bf n})$, the minimum of
the entropy of formation, $H(\vec{\mu})$, in the pure-state region. On
the left side is a 3-dimensional plot of $\widetilde{H}({\bf n})$ and on
the right is a contour plot of the same function.}
\label{fig:eof2constraint}
\end{figure}
The function $\widetilde{H}({\bf n})$ in the pure-state region is
shown in Fig.~\ref{fig:eof2constraint}. It is, as required, a
monotonically increasing function of both $\hat{n}_{\Phi}$ and $n_{T}$. Along
the upper boundary of the pure-state region, the numerically
computed value of $\widetilde{H}({\bf n})$ matches the value of
$\widetilde{H}\big(n_T\big)$ from Eq.~(\ref{negmin}). In addition to this,
from the contour plot of $\widetilde{H}({\bf n})$ in
Fig.~\ref{fig:eof2constraint}, we see that along the upper boundary, the
function has zero slope along the $\hat{n}_{\Phi}$ direction.
The function $\widetilde{H}({\bf n})$ is not convex, which
can be seen by computing the Hessian at every point in the pure-state
region. If the function were convex, both eigenvalues of the Hessian would
be positive at all points. It turns out that one of the eigenvalues of the
Hessian is negative in a region in the upper right corner of the
$\hat{n}_{\Phi}$-$n_T$ plane, close to the maximally entangled state.
Since $\widetilde{H}({\bf n})$ is not convex, we have to compute its
convex hull,
\begin{equation}
\label{eq:doublyC7}
{\mathcal H}({\bf n}) = {\mbox{co}}\left[ \widetilde{H}({\bf n})
\right],
\end{equation}
to obtain the bound on the EOF in the pure-state region. The convex hull
of $\widetilde{H}({\bf n})$ can be computed numerically, and it
turns out that the difference between ${\mathcal H}({\bf n})$ and
$\widetilde{H}({\bf n})$ is quite small ($\sim 10^{-3}$), the
two differing differ only in a small region in the upper right corner of
the pure-state region. As shown in Appendix~\ref{AppProof}, taking the
convex hull preserves monotonicity.
\begin{figure}[!ht]
\resizebox{16 cm}{8 cm}{\includegraphics{fig_eof_bound.png}}
\caption{(Color online) The doubly-constrained bound ${\mathcal H}({\bf n})$ on
the EOF of all $4 \times N$ states. On the right side is a contour plot of the
same function.}
\label{fig:eofbound}
\end{figure}
To obtain a bound on the EOF of {\em all}\/ $4 \times N$ states, we have
to extend ${\mathcal H}({\bf n})$ out of the pure-state region to the rest
of the $\hat{n}_{\Phi}$-$n_T$ plane. The extension has to respect the
monotonicity of ${\mathcal H}({\bf n})$ so that the string of inequalities
Eq.~(\ref{eq:doublyA6}) holds. This can be achieved by extending
${\mathcal H}({\bf n})$ using surfaces that match the function at the
lower and upper boundaries of the pure-state region. To preserve
monotonicity, the surface added on to the region below the lower boundary
of the set of pure states has zero slope along the $n_T$ direction, and the
surface added on to the region above the upper boundary of the set of pure
states has zero slope along the $\hat{n}_{\Phi}$ direction. The resulting
doubly-constrained bound ${\mathcal H}({\bf n})$ on the EOF is shown in
Fig.~\ref{fig:eofbound}. We see from the figure that the extension to the
whole $n_{\Phi}$-$n_T$ plane produces a smooth and seamless surface.
One final point worth mentioning involves the use of our bound for general
mixed states. To do so, one must calculate $n_\Phi$ for the mixed state, and
this calculation depends on the choice of an angular-momentum basis for system
$B$ in order to define the $\Phi$-map. The bound itself thus depends on this
choice of basis, and the best bound would generally be found for the basis
choice that gives the largest value of $n_\Phi$. For pure states, for example,
the results in Fig. \ref{bound} show that the best choice of basis is the
Schmidt basis for system $B$.
The isotropic states, which lie along the diagonal in the
$\hat{n}_{\Phi}$-$n_T$ plane, are special in that they saturate the
singly-constrained bound ${\mathcal H}\big(n_T \big)$ from
Eq.~(\ref{Tbound}). These states thus furnish a good consistency test
of our doubly-constrained bound because our bound must match the
singly-constrained bound when applied to isotropic states. A
comparison of the two bounds for isotropic states is given in
Fig.~\ref{fig:eofcompare}.
\begin{figure}[!ht]
\resizebox{7 cm}{7
cm}{\includegraphics{fig_eofcompare.png}} \caption{The
thick black line is the doubly-constrained bound on the EOF for
isotropic states. The dashed white line, lying on top of the black
line, is the singly-constrained bound $\mathcal{H}\big(n_T\big)$
from Eq.~(\ref{Tbound}).}
\label{fig:eofcompare}
\end{figure}
We can make a second comparison between the singly and doubly
constrained bounds using Fig.~\ref{fig:eofcompare}. From the way we
constructed ${\mathcal H}({\bf n})$, we know that its value on the
diagonal in the $\hat{n}_{\Phi}$-$n_T$ plane is the same as its value
on the upper boundary of the pure-state region. We also know that the
upper boundary is where the singly-constrained bound and the
doubly-constrained bound are the same. From
Fig.~\ref{fig:eofcompare}, we see that the convex hull
$\mathcal{H}\big(n_T\big)$ of the function $\widetilde{H}\big( n_T
\big)$ of one variable matches the convex hull ${\mathcal H}(\bf n)$
of the function $\widetilde{H}({\bf n})$ of two variables on the
upper pure-state boundary. These consistency checks give us
increased confidence in the accuracy of our results.
\subsection{Tangle and concurrence}\label{doubly:tangle}
Doubly-constrained bounds can be placed on the tangle and the
concurrence of $4 \times N$ states by extending the procedure used
for the EOF. For the tangle, we start by finding the function,
\begin{equation}
\label{eq:doublyD1}
\widetilde{T}({\bf n}) = \min_{\vec{\mu}} \left\{ 2\left( 1- \left| \vec{\mu}
\right|^2 \right) \left| 3\sqrt{(\mu_1 + \mu_4)(\mu_2 + \mu_3)} =
\hat{n}_{\Phi},
\frac{\left( \sum_j \sqrt{\mu_j} \right)^2-1}{2} = n_T \right. \right\},
\end{equation}
in the pure-state region. For the concurrence, we want the function
$\widetilde C({\bf n})= \sqrt{\widetilde T({\bf n})}$, since for pure
states the concurrence is the square root of the tangle.
\begin{figure}[!ht]
\resizebox{16 cm}{8 cm}{\includegraphics{fig_tangle_bound.png}}
\caption{(Color online) The doubly-constrained bound ${\mathcal T}({\bf n})$ on
the tangle of $4 \times N$ states. On the right side is a contour plot of the
same function.}
\label{fig:tanglebound}
\end{figure}
\begin{figure}[!htb]
\resizebox{16 cm}{8 cm}{\includegraphics{fig_conc_bound.png}}
\caption{(Color online) The doubly-constrained bound ${\mathcal C}({\bf n})$ on
the concurrence of $4 \times N$ states. On the right side is a
contour plot of the same function.}
\label{fig:concbound}
\end{figure}
For all three of the entanglement monotones, EOF, tangle and
concurrence, the monotone boundaries we define in
Appendix~\ref{A:construction} coincide with the boundaries of the
pure-state region. This is because the singly-constrained bounds for
all three measures correspond to the same sets of Schmidt
coefficients, $\vec{\mu}_T=(\gamma,\gamma',\gamma', \gamma')$ and
$\vec{\mu}_{\Phi}=(\alpha, 1-\alpha, 0, 0)$, and we have already seen
for the EOF that these Schmidt coefficients define the boundaries of
the pure-state region. This makes it unnecessary for these
entanglement monotones to construct the monotonically nondecreasing
function discussed in Appendix~\ref{A:construction}. In general, for
two different measures of entanglement and two constraints, the
singly-constrained bounds for the two measures need not correspond to
the same Schmidt coefficients.
Once we have $\widetilde{T}({\bf n})$, the convex hull of this function
extended to the whole $\hat{n}_{\Phi}$-$n_T$ plane is the doubly-constrained
bound on the tangle, ${\mathcal T}({\bf n})$. A three-dimensional plot and a
contour plot of ${\mathcal T}({\bf n})$ are shown in Fig.~\ref{fig:tanglebound}.
The bound on the concurrence is the convex hull of the surface
obtained from $\widetilde C({\bf n})$. The resulting bound on the
concurrence, ${\mathcal C}({\bf n})$, is shown in
Fig.~\ref{fig:concbound}.
\section{Conclusion}\label{conclusion}
We focused on two aspects of the problem of quantifying entanglement in
this paper. The first was a comparison between the bounds on different
measures of entanglement obtained by using $n_T$ and $\hat{n}_{\Phi}$
independently as constraints. The second was the construction of
doubly-constrained bounds on the three measures of entanglement that we
considered.
Starting from the $\Phi$-map~\cite{Breuer2006}, we found that we
can define an entanglement measure, which we call the
$\Phi$-negativity. The $\Phi$-negativity of arbitrary quantum
states can be calculated in a straightforward manner, just like
their negativity. We also found that we can obtain a much simpler
function $\hat{n}_{\Phi}$ of the Schmidt coefficients of pure
states that is an upper bound on their $\Phi$-negativity. Previous
work~\cite{Chen2005a},\cite{Chen2005} has shown that the
negativity can be used as a constraint to place bounds on the EOF,
the tangle, and the concurrence of bipartite states. We obtained a
different set of bounds on these three measures of entanglement
for $4 \times N$ mixed states by using $\hat{n}_{\Phi}$ instead as
the constraint. The scheme for placing lower bounds on
nonoperational measures of entanglement is general enough to allow
us to use $\hat{n}_{\Phi}$ instead of $n_{\Phi}$ as the
constraint. We were then able to compare the two sets of bounds on
the measures of entanglement coming from using either one of the
two operational entanglement measures as a single constraint.
We found that the $\hat{n}_{\Phi}$-$n_T$ plane for pure states can
be divided into two regions depending on which constraint led to
the better bound on a given measure of entanglement. This prompted
us to consider whether we can construct a single, composite bound
for each measure of entanglement, applicable to the entire
$\hat{n}_{\Phi}$-$n_T$ plane, by using both constraints
simultaneously. It turned out that for $4 \times N$ systems this
is a tractable problem, and we obtained doubly-constrained lower
bounds for the first time for the EOF, the tangle, and the
concurrence. We showed how the bounds on the different measures of
entanglement obtained for pure states can be extended to include
all states. We found that the requirement of monotonicity on the
bound defined on pure states dictates how to extend the bound to
all states.
|
2,877,628,088,980 | arxiv | \section{Conclusions}
The role mining process, usually, returns a role infrastructure on the basis of the relationships among users and permissions contained in the $\upa-m$ matrix. However, the definition of a role-set really reflecting the internal functionalities of the examined organization remains a challenging task.
The need for managing different kind of constraints in role engineering has recently been the focus of many works in literature \cite{kumar10, HinSur11, MaLiLuWa11}. The definition and the management of constraints in role mining are very important aspects in role mining, since they allow the role engineer to control the automatic process and introduce some rules that can have impact on the retrieved structure.
In this paper, we have proposed a heuristic capable of returning a complete role-set satisfying constraints on the maximum number of permissions included in each role. The comparisons made show how the results in terms of accuracy, distance, size, and computation time improve on a previously presented algorithm \cite{kumar10}.
Our simple algorithm is easily extensible to consider other kinds of cardinality constraints, such as maximum number of users assigned to a role or mutually exclusive permissions or roles \cite{MaLiLuWa11}.
Furthermore, it is possible to investigate on the definition of other kinds of constraints regarding the role hierarchy and the semantic associated to each role \cite{semantic08}, and try to adapt the proposed algorithm in order to return a role set satisfying the newly defined constraints.
\label{conclusions}
\section{Introduction}
Complex organizations need to establish access control policies in order
to manage access to restricted resources. A simple way to accomplish this
task is to collect set of permissions in roles and then assign roles
according to the responsibilities and qualifications of each employee.
The Role Based Access Control (RBAC) is a well known paradigm to define and
organize roles and permissions in an efficient way. Introduced in the
early '90 years \cite{coyne,nist},such a paradigm has been
investigated for long time and has become recently used in different
commercial systems to manage identities and accounts \cite{ibmtivoli}.
The goal of RBAC is to collect set of permissions in roles and define a complete
and efficient set of roles that can be assigned to users in order
to access restricted resources.
The advantage is that access control can
be centralized and decoupled from users and the costs and the overhead of
the security management can be reduced.
The correct definition of the set of roles which satisfies the needs of the
organization is the most difficult and costly task to be performed during
the implementation of a RBAC system. Such an activity is often referred to
as {\it role engineering} and includes
the correct identification of roles from the current structural
organization of the enterprise.
Mainly this task, i.e. the extraction of a complete and
efficient set of roles, can be performed using two approaches:
\emph{top-down} or \emph{bottom-up} role
engineering. In the first case, roles are defined after that the
functionalities of the organization have been well defined and studied, and
elicitation activities have been performed.
The top down approach usually is labor intensive and
involves a large amount of work and time done by humans especially in large
enterprises with a large number of business processes, as reported in some case study are
available in the literature \cite{Roeckle00}.
On the other hand, the bottom-up process, often denoted also as \emph{role mining}
starts from the analysis of the current set of permissions assigned to
users, and tries to aggregate them in order to extract and define roles.
Obviously, hybrid approaches can exist
in which both directions, top-down and bottom-up, can be used in subsequent
steps of the analysis in order to refine the returned set of roles.
Bottom-up approach to role mining has been more investigated, since many
techniques borrowed from data mining can be applied in an automatic way to
the existing configuration of user-permission assignments.
A RBAC system can be easily constructed in this way and a starting set of roles can
be fastly generated. The problem with such an approach, is that the quality and the
number of returned roles often are not so good, since no semantics is taken into consideration when
the role mining process is started. In many situations the returned set of roles might
not match any functional organization within the analyzed enterprise and the
existing business processes might not be adequately represented. An
accurate analysis of the returned results is needed to better tune the
retrieved representation of roles to the current organizational requirements of
the enterprise.
A formal definition of the Role
Mining Problem (RMP) and some of its variants has been given and deeply analyzed in
\cite{sacmat07}. There, the NP-completeness of the (decisional)
RMP problem has been proved, and the formulation of RMP as a graph covering
problem has been done in \cite{biclique,zhang07}.
The problem of imposing constraints on the set of roles resulting after the
role mining process has been considered in the past. Statically or
dynamically mutually exclusive roles constraints have been included in RBAC
models \cite{Sandhu96} and in the NASI/NIST standards \cite{nist}. According to these
constraints, for examples, no user can be assigned contemporary a given set
of mutually exclusive roles, or no user can activate simultaneously a set of
roles in the same session. Such constraints are often used as mechanisms
to achieve separation of duty principles, preventing one user from having a
large number of roles to complete a task, or imposing restrictions on the
minimum number of users needed to perform a critical activity \cite{Li2007}.
Recently, a simple classification of the constraints on the cardinality of
the number of roles, users and permissions for a given RBAC system has been
proposed \cite{MaLiLuWa11}. The first heuristic taking into account a
cardinality constraint on the number of permissions contained in a role has
been proposed by \cite{kumar10}. In its work, however, the proposed results
have been compared only on other heuristics which were not able to consider
constrained roles.
In this work we propose a novel heuristic for mining RBAC roles under
cardinality constraint. The algorithm is based on a previous
proposal \cite{sac10}, where an initial candidate role set was constructed by
considering one role for each user on the basis of the current assignment
of permissions. The role set is then refined and updated by eliminating the
roles obtained as union of other roles already included in the set and
ensuring that the cardinality constraint is respected. Finally, an optimization of
the role set is performed by running a lattice reduction procedure,
previously described in \cite{biclique}.
The resulting procedure is very efficient in terms of computation time and
quality of returned role set.
To this aim we present
the results obtained by running our heuristics on different datasets, some
available over the network, some artificially created. The results are
compared with our implementation of the algorithm presented in \cite{kumar10}
and analyzed in terms of the metrics presented in \cite{MolloyLLMWL09}.
The remainder of this paper is organized as follows. In the next section we
discuss related works. Section \ref{np} contains the preliminary concepts needed to
define the constrained role mining problem and the discussion on its
complexity. In section \ref{ksma} we introduce our heuristics and compare the solution with related
work in section \ref{experiments}. Finally Section \ref{conclusions} presents our conclusions and
ongoing work.
\section{Related Works}
\label{related}
Role engineering has been firstly introduced by Coyne et al in \cite{coyne} where the definition of a top down process for the definition of roles
has been discussed.
Along the same research line, several other works have been presented \cite{Roeckle00},
but recently, the focus of role engineering has turned to consider more automated techniques, based on the bottom up approach, where data mining techniques are applied for the definition of roles \cite{Kuh03}. Role mining algorithms have been presented based on set covering \cite{ChCr09}, graph theory \cite{biclique, zhang07}, subset enumeration \cite{ccs06}, database tiling \cite{sacmat07}.
The theoretical aspects of the RMP have been considered in \cite{VaidyaAG10,sacmat07,ChCr09}, where the complexity of the problem has been analyzed and its equivalence to other known optimization problem showed. Another interrelated problem, i.e. dealing with the semantic meaning of roles, has been addressed in \cite{semantic08}.
Cardinality constraints on the number of permissions included in a role have been firstly considered in \cite{kumar10}, and a heuristic algorithm called Constrained Role Miner (CRM) has been proposed. The CRM algorithm takes in input the \up\ matrix and returns a set of roles, each one satisfying the given cardinality constraint. CRM is based on the idea of clustering users having the same set of permissions and selecting, as candidate roles, the roles composed of the set of permissions satisfying the constraint and having the highest number of associated users.
In \cite{kumar10}, the performances of the algorithm are evaluated on real world datasets considering different metrics (such as the number of returned roles, the sum of the size of the user assignment and permission assignment matrices and so on), with respect to other previously proposed algorithms. However the comparison is performed without considering constraints, since the other algorithms return a complete set of roles but have not the capability of managing constraints. In section \ref{experiments} we evaluate our proposal against the result obtained after our implementation of the CRM algorithm, considering both real world and synthetic datasets.
A different kind of cardinality constraints, considering the number of user-role assignments, have been considered in \cite{HinSur11}. Such constraints can be useful when the number of persons that can be assigned to a given role (e.g. the number of directors, managers, etc) in a given organization is known or can be fixed. In the paper, three algorithms have been proposed based on a graph approach, where the role mining problem is mapped to the problem of finding minimum biclique cover of the edges of a bipartite graph. The three heuristics are obtained by modifying the basic greedy algorithm proposed in \cite{biclique}, and experimental results on real world datasets are reported considering some constraints on the number of users that can be assigned to any role.
Finally cardinality constraints have also been considered in \cite{MaLiLuWa11} where a representation of the constraints in terms of association rules is proposed: permissions are regarded as attributes, and each user-permission assignment as a transaction. To generate mutually exclusive permissions constraints, i.e. set of permissions that cannot be assigned to the same role, an algorithm is proposed, based on known techniques for mining association rules in databases, and its performance evaluated on synthetically generated datasets
|
2,877,628,088,981 | arxiv | \section{Black hole masses from the disk emission}
Determining the masses of the accreting black holes (BHs)
in ultraluminous X-ray sources (ULXs) is a key
unsolved problem in X-ray astrophysics. In the absence
of direct kinematic measurements of their mass functions,
indirect methods based on X-ray spectral modelling have
sometimes been used, by analogy with Galactic BH X-ray
binaries, whose spectra can be approximated,
in the ``canonical'' $0.3$--$10$ keV range,
with a thermal component
plus a power-law. The thermal component is consistent
with disk-blackbody emission from an optically
thick disk, while the power-law is generally attributed
to inverse-Compton scattering of softer disk photons off
high-energy electrons. The peak temperature and luminosity
of the thermal component are good indicators
of the size of the X-ray-emitting inner-disk region,
which in turn is related to the BH mass.
In the simplest and most commonly used disk-blackbody
approximation \cite{mak86},
\begin{eqnarray}
L_0 &\approx& 4 \pi \sigma T_0^4 R_{0}^2 \\
L_0 &=& \eta \dot{M} c^2 \\
\sigma T_0^4 &\approx& \frac{3GM\dot{M}}{8 \pi R_{0}^3},
\end{eqnarray}
where $R_{0}$ is a characteristic size $\approx R_{\rm in}$
(inner-disk radius) and $T_0$ is a characteristic temperature
$\approx T_{\rm max}$, and where we have ignored a factor
related to the no-torque condition at $R_{\rm in}$ and a hardening
factor. (Those terms can be taken into account later as a numerical
factor $g$ \cite{fab04}.)
$L_0$ is the integrated disk luminosity, under the assumption
that the disk dissipates most of the gravitational energy
of the inflow. The radiative efficiency
$\eta \approx 0.1$--$0.3$ for a source in a high/soft state
(dominated by a bright disk).
From (1), (2), and (3):
\begin{equation}
M \approx \frac{c^2 g^2 \eta L_0^{1/2} T_0^{-2}}{3G(\sigma \pi)^{1/2}}\,
\approx 10.2 \, \left(\frac{g}{1.35}\right)^2 \, \left(\frac{\eta}{0.2}\right)\,
\left(\frac{L_0}{5 \times 10^{38}
{\rm{~erg~s}}^{-1}}\right)^{1/2}\, \left(\frac{T_0}{1
{\rm{~keV}}}\right)^{-2} \, M_{\odot},
\end{equation}
which also implies $L_0 \sim M^2 T_0^{4}$. The effective
hardening factor $g \approx 1.35$ \cite{fab04}. The
observable quantities $L_0$ and $T_0$ may come from
X-ray spectral fitting, and the efficiency is known
within a factor of $2$ based on standard accretion models.
Despite the various approximations, (4) works well
(within a factor of $2$) when applied to the masses of Galactic BHs.
It is also observationally verified that the X-ray luminosity
of each source varies over months or years
following the relation $L_{\rm X} \sim T_0^{4}$ at constant $M$
\cite{mil04}, at least until it approaches the Eddington limit,
$L_{\rm Edd} \approx 1.3 \times 10^{38} (M/M_{\odot})$ erg s$^{-1}$.
\section{Soft X-ray emission in ULXs}
ULX X-ray spectra can also be modelled
with a power-law plus a thermal component, much cooler
than that observed from Galactic BHs --- typically, with
temperatures $kT_0 \sim 0.12$--$0.20$ keV for the brightest
sources. If (4) is applied, with total
$0.3$--$10$ keV luminosities $\approx 1$--$2 \times 10^{40}$
erg s$^{-1}$, the inferred masses are $\sim 1000 M_{\odot}$.
This argument has been used in support
of the intermediate-mass BH interpretation of ULXs
\cite{mil04}; however, it is based on questionable assumptions.
Firstly, there is little direct evidence
that the ``soft excess'' in ULXs is due to disk emission,
partly because of our limited observing band. Analogous
soft-excesses in AGN, and particularly in narrow-line
Seyfert 1 galaxies (NLS1) can be explained as blurred,
ionized absorption (mostly in the $\sim 0.5$--$2$ keV
band) and reprocessing of the primary
power-law-like spectrum in a fast outflow \cite{gie04}\cite{che06}.
We showed \cite{gon06} that a similar interpretation
can be applied to ULXs, whose X-ray spectra are very
similar to those of NLS1. We do not speculate here
why the primary spectrum is non-thermal. As a possible
analogy, the power-law component becomes dominant over
the disk component in Galactic BHs in the very high state,
at luminosities $\lower.5ex\hbox{$\buildrel>\over\sim$} L_{\rm Edd}$.
Secondly, (4) holds only if the disk is emitting
most of the energy liberated by accretion, as in (2).
In ULXs, this is not the case. Even if the deviation
from the power-law spectrum at soft energies is indeed from
disk emission, the X-ray spectrum is still dominated
by the power-law component, and the thermal component
would represent only $\sim 10\%$ of the $0.3$--$10$ keV emission
\cite{sto06}. Thus, (2) and (3) no longer apply in this form.
Physically, this suggests that most of the accretion power
is not radiated by the disk, but is efficiently transferred
in other forms (mechanical, thermal or magnetic energy)
to an upscattering medium, and then partly radiated
with a non-thermal (power-law-like) spectrum.
In summary, we have {\it two alternative scenarios to describe
the first-order deviations from a power-law spectrum}
in the soft X-ray band: either outflow reprocessing (mostly
ionized absorption), or a modified disk that is
only radiating a small fraction of the energy released
by accretion; or a combination of both.
Both are consistent with the X-ray
observations. We discuss the ionized outflow
scenario elsewhere in these Proceedings \cite{gon06b};
see also \cite{gon06}.
Here we outline the main implications of the disk hypothesis.
\section{The chilled disk scenario}
The key physical question for such a scenario is why
the disk is so cold for its luminosity. Even when we take
into account only the luminosity in the fitted thermal
component, ULX disks would radiate up to $\approx$ a few $10^{39}$
erg s$^{-1}$ with a much lower peak temperature
than that observed from Galactic BHs in their high state.
A standard, truncated disk, replaced in the inner region
by a radiatively-inefficient flow, would produce a power-law
dominated spectrum with a cold disk component, as is the case
for stellar-mass BHs in the low/hard state. However, if ULXs
are in the low/hard state at luminosities $> 10^{40}$
erg s$^{-1}$, their BH masses should be $\lower.5ex\hbox{$\buildrel>\over\sim$} 10^4 M_{\odot}$.
The formation of such massive remnants is difficult to explain
with existing models of stellar or star cluster evolution.
Therefore, we consider an alternative scenario based
on the following phenomenological arguments:
\begin{enumerate}
\item the mass inflow rate
is sufficiently high that the disk extends all the way to the innermost
stable circular orbit;
\item hence, the conversion
of gravitational energy to accretion power is efficient:
the total power is $\lower.5ex\hbox{$\buildrel>\over\sim$} 0.1 \dot{M}c^2$;
\item the outer region, at radii $R \ge R_{\rm c}$, is a standard
optically-thick disk ($T \sim R^{-3/4}$)
radiating its accretion power with a multicolour diskbb spectrum;
\item in the inner region, at $R_{\rm in} \le R \le R_{\rm c}$,
only a small fraction of the released power is directly radiated
by the disk. The rest is efficiently transferred to a corona
or jet or magnetized outflow, acting as an upscattering medium.
\end{enumerate}
One possible way to model this process is by assuming
that a constant fraction of power is extracted from the disk
at all radii. The alternative possibility is that the outer disk
region, at radii $R \ge R_{\rm c}$, is a standard
optically-thick disk ($T \sim R^{-3/4}$),
while the non-thermal component (jet, magnetized outflow
or corona) becomes dominant in the inner region,
at $R_{\rm in} \le R \le R_{\rm c}$.
In any case, the inner disk is {\it cooler
than a standard disk}, because it radiates only a flux
\begin{equation}
\sigma T_{\rm eff}(R)^4 \approx \alpha \, \frac{3GM\dot{M}}{8 \pi R^3} - F_{\rm nt}(R)
< \frac{3GM\dot{M}}{8 \pi R^3},
\end{equation}
where $(1-\alpha) > 0$ is the constant fraction of power
extracted from the disk at all radii,
and $F_{\rm nt}(R)$ is the additional flux removed
from the innermost region, inside the transition radius
$R_{\rm c}$ \citep{SoK06}.
If/where $F_{\rm nt}(R) =0$, $T_{\rm eff} \sim R^{-3/4}$,
and the emitted spectrum is still a disk blackbody.
If $F_{\rm nt}(R) = 0$ at all radii, the mass estimate (4) holds true,
independent of $\alpha$. As the accretion rate is varied,
the disk follows the same $L_0 \sim M^2 T_0^{4}$ track,
athough cooler and less luminous than in the $\alpha = 1$ case,
for a given accretion rate.
However, we argue that
the extraction of power from the disk is more likely
to occur in the inner region, while the disk may be undisturbed
at large radii.
It follows from (5) that the temperature profile
flattens out in the inner region, where $F_{\rm nt}$ dominates.
In particular, if $T(R)$ increases more slowly that $R^{-1/2}$ for
$R \rightarrow R_{\rm in}$, the maximum contribution to the disk emission
occurs at $R \approx R_{\rm c}$, $T \approx T(R_{\rm c})$.
Those values will also be proportional (accounting
for the hardening factor) to the fitted coulor
peak temperature and radii derived from an observed spectrum.
For simplicity, here we assume that
$T = T(R_{\rm c}) =$ constant
inside $R_{\rm c}$; our conclusions do not depend on the exact
choice of a temperature law. Moreover, only a fraction
$\beta < 1$ of the disk emission from the inner region
may be directly visible; if all the disk photons from that region
are comptonized, then $\beta \approx 0$.
We can now re-write (1), (2), (3) as:
\begin{eqnarray}
L_0 &\approx& 4\pi R_{\rm c}^2 \sigma T_{\rm eff}(R_{\rm c})^4
+ 2\pi \beta (R_{\rm c}^2 - R_{\rm in}^2)\sigma T_{\rm eff}(R_{\rm c})^4
\approx 4\pi R_{0}^2 \sigma T_0^4\\
L_0 &=& \alpha f \eta \dot{M} c^2 \\
\sigma T_{\rm eff}(R_{\rm c})^4 &\approx &
\alpha \frac{3GM\dot{M}}{8\pi R_{\rm c}^3}.
\end{eqnarray}
For simplicity, we have assumed here that $\beta \ll 1$,
so that there is effectively no observational
difference between a disk truncated at $R_{\rm c}$
and replaced by an efficient comptonizing medium
or outflow inside that radius,
and a disk extending to the innermost stable orbit but
with a flat temperature distribution. Physically, there is
of course a significant difference, because the mechanisms
responsible for extracting accretion power and transferring it
to a jet or a corona may require the presence of an accretion
disk in the inner region, even though we cannot see it directly.
See for example the model of \cite{kun04},
where power is removed
from the disk by the vertical component of the magnetic torque.
In (7), we have defined $\alpha f$ as the fraction
of accretion power radiated by the disk,
with $0 < \alpha \lower.5ex\hbox{$\buildrel<\over\sim$} 1$ and $0 < f \lower.5ex\hbox{$\buildrel<\over\sim$} 1$.
A fraction $(1-\alpha)$ of the accretion power is extracted
uniformly over the whole disk surface, and a further fraction
$\alpha (1-f)$ is extracted from the disk inside the transition
radius, as discussed above.
It is difficult to determine $f$ (or, equivalently,
estimate the flattening of the temperature profile
in the inner disk) observationally. We can
estimate $f \sim 0.1$, based on the fitted ratio of soft thermal
emission over total X-ray luminosity in ULXs.
If anything, that ratio is an overestimate of $f$,
because non-thermal processes are less radiatively efficient than
thermal disk emission, and some power is still likely
to escape in non-radiative forms.
The other two observable parameters, as before,
are the peak colour temperature $T_0$ and the luminosity $L_0$
of the thermal component.
We can now solve (6), (7), (8) for $M$, $\dot{M}$ and $R_{\rm c}$
as a function of the parameters $f$, $\alpha$, $g$,
and of the observable quantities $T_0$ and $L_0$ (or $R_0$):
\begin{eqnarray}
M &\approx& 90.7 \left(\frac{\eta}{0.2}\right)
\left(\frac{f}{0.1}\right) \left(\frac{g}{1.35}\right)^2
\left(\frac{L_0}{2 \times 10^{39}
{\rm{~erg~s}}^{-1}}\right)^{1/2} \nonumber\\
&\times& \left(\frac{T_0}{0.15~{\rm keV}}\right)^{-2} \, M_{\odot} \\
\dot{M} &\approx& 1.8 \times 10^{-6} \, \left(\frac{0.2}{\eta}\right)
\left(\frac{0.1}{f}\right)\, \left(\frac{1}{\alpha}\right)\,
\left(\frac{L_0}{2 \times 10^{39}
{\rm{~erg~s}}^{-1}}\right) \ \ M_{\odot} \ {\rm yr}^{-1}\\
R_{\rm c} &\approx& g^2 R_0
\approx \frac{3}{2f\eta} \, \frac{GM}{c^2}
\ \approx \ 75 \, \left(\frac{0.2}{\eta}\right)
\, \left(\frac{0.1}{f}\right)\, \frac{GM}{c^2}.
\end{eqnarray}
We note again that $M$ depends on $f$, but not on $\alpha$.
We conclude that the fitted spectral features of ULXs
(X-ray luminosity, temperature and ratio of thermal/non-thermal
contribution) suggest masses $\sim 50$--$100 M_{\odot}$, at the extreme
end of, but still consistent with models of stellar evolution.
The emitted luminosity is a few times $L_{\rm Edd}$, but
the disk radiative contribution alone is $\lower.5ex\hbox{$\buildrel<\over\sim$} L_{\rm Edd}$.
The rest is generated outside the disk by non-thermal processes.
The largest contribution to the thermal disk emission comes
from $R \approx R_{\rm c} \approx 100$ gravitational radii;
inside this region, non-thermal processes dominate. In principle,
this can be tested by studying the short-term variability timescale
of the thermal and non-thermal components.
We speculate that an efficient disk-corona-outflow coupling
with vertical transport of energy can be achieved through
large-scale magnetic fields, when the azimuthal-vertical
component of the magnetic stress is properly taken into
account \cite{kun04}. See \cite{SoK06} for a more detailed
discussion.
\begin{theacknowledgments}
RS acknowledges support from an OIF
Marie Curie Fellowship, through University College London.
ACG acknowledges support from the
{\it Funda\c{c}\~ao para a Ci\^encia e a Tecnologia (FCT)},
under grant BPD/11641/2002.
ZK acknowledges a University of Sydney Bridging Support
research grant.
\end{theacknowledgments}
\bibliographystyle{aipprocl}
|
2,877,628,088,982 | arxiv | \section{Introduction}
Turbulent flows appear in problems of interest to many fields of engineering sciences such as aeronautics, mechanical, chemical engineering and in oceanographic, meteorological and astrophysical sciences, besides others. Improved understanding of turbulence evolution would lead to important advances in these fields.
In academic and industrial applications, most investigations into turbulent flow problems use turbulence models. Turbulence models are simplified relations that express quantities that are difficult to compute in terms of simpler flow parameters. They relate higher-order unknown correlations to lower-order quantities. These unknown correlations represent the actions of viscous dissipation, pressure-velocity interactions, etc. For example pressure strain correlation is a non-local phenomenon and is difficult to compute. Using models for pressure strain correlation, it is expressed as a function of Reynolds stresses, dissipation and mean velocity gradients which are local quantities. This enables us to estimate the pressure strain correlation and its effects on flow evolution in a simpler manner that is computationally inexpensive. Turbulence models are an essential component of all computational fluid dynamics software and are used in almost all simulations into real life fluid flows of engineering importance.
A majority of industrial applications use simple two-equation turbulence models like the $k-\epsilon$ and $k-\omega$ models. However recent emphasis in the scientific community has markedly shifted to Reynolds stress models (\cite{hanjalic2011,durbin2017,klifi2013,mishra3,jakirlic2015,manceau2015,eisfeld2016,schwarzkopf2016,moosaie2016,lee2016,mishra6,sun2017}). Reynolds stress models have the potential to provide better predictions than turbulent viscosity based models at a computational expense significantly lower than DNS studies. They may be able to model the directional effects of the Reynolds stresses and additional complex interactions in turbulent flows (\cite{johansson1994}). They have the ability to accurately model the return to isotropy of decaying turbulence and the behavior of turbulence in the rapid distortion limit (\cite{pope2000}).
Reynolds Stress Models are based on the Reynolds Stress Transport Equation that describes the evolution of individual components of the Reynolds stress tensor. This is in contrast to two-equation modeling approach where evolution equations for scalars like the turbulent kinetic energy $k$ and dissipation $\epsilon$ are solved and the eddy viscosity hypothesis is used to approximate the Reynolds stresses. The Reynolds Stress Transport Equations describe the production, dissipation and redistribution each of the components of the Reynolds stress tensor. Different physical mechanisms in this evolution are represented by the separate terms in this equation. The general form of the Reynolds Stress Transport Equation is given by \cite{pope2000}
\begin{equation}
\begin{split}
&\partial_{t} \overline{u_iu_j}+U_k \frac{\partial \overline{u_iu_j}}{\partial x_k}=P_{ij}-\frac{\partial T_{ijk}}{\partial x_k}-\eta_{ij}+\phi_{ij},\\
&\mbox{where},\\
& P_{ij}=-\overline{u_ku_j}\frac{\partial U_i}{\partial x_k}-\overline{u_iu_k}\frac{\partial U_j}{\partial x_k},\\
&\ T_{kij}=\overline{u_iu_ju_k}-\nu \frac{\partial \overline{u_iu_j}}{\partial{x_k}}+\delta_{jk}\overline{ u_i \frac{p}{\rho}}+\delta_{ik}\overline{ u_j \frac{p}{\rho}},\\
&\eta_{ij}=-2\nu\overline{\frac{\partial u_i}{\partial x_k}\frac{\partial u_j}{\partial x_k}} \\
&\phi_{ij}= \overline{\frac{p}{\rho}(\frac{\partial u_i}{\partial x_j}+\frac{\partial u_j}{\partial x_i})}\\
\end{split}
\end{equation}
The turbulence production process is represented by $P_{ij}$ and is an inner product between the Reynolds stress tensor and the mean velocity gradient. In physical terms this mechanism models the action of the mean velocity gradients working against the Reynolds stresses and is a transfer of kinetic energy from the mean flow to the fluctuating velocity field. The production mechanism acts as a source of energy for the turbulent flow. $\eta_{ij}$ represents the dissipation process and is the product of the fluctuating velocity gradients and the fluctuating rate of strain. Physically it models the fluctuating velocity gradients working against the deviatoric fluctuating stresses transforming turbulence kinetic energy into internal energy. The dissipation mechanism acts as a sink of energy for the turbulent flow. The turbulent transport process is represented by $T_{ijk}$ and represents the transfer of turbulent kinetic energy between different locations in the flow domain. This has contributions from viscous diffusion, pressure transport and turbulent convection. Finally $\phi_{ij}$ represents the pressure strain correlation and redistributes turbulent kinetic energy among the components of the Reynolds stresses. Of these terms, production is the only process that is closed at the single point level. The other terms require models for their closure. The accuracy of the Reynolds stress modeling approach depends on the quality of the models developed for these turbulence processes.
Of the terms that require models for their closure the pressure strain correlation term is generally considered to be the most important \cite{lumley1975,sc}. There are three reasons behind this. Firstly the pressure strain correlation term is active in all turbulent flows. For instance in homogeneous turbulence, turbulence transport is absent due to spatial homogeneity. Similarly in decaying turbulence, turbulence production is zero due to the absence of mean velocity gradients. In the rapidly distorted turbulent flows, the dissipation mechanism is negligible as its time scale is much larger than the applied distortion. However in all these flows, the pressure strain correlation is present and actively transforming the evolution of the turbulent flow.
The second reason is due to the action of pressure strain correlation being very important in the evolution of turbulent flows. In important flow regimes like elliptic streamline flow the flow instability is initiated by pressure action \cite{kerswell2002}. In strained mean flows like plane strain, axisymmetric strained mean flows pressure action stabilizes the flow instability \cite{mishra4}. The pressure strain correlation term determines if turbulence grows or decays in many turbulent flows. Hence its accurate modeling is highly important.
The final reason is due to the complexity of the pressure strain correlation mechanism and the challenges in its modeling. The central ideas for pressure strain correlation modeling were introduced by \cite{chou}. The first model for the pressure strain correlation was formulated by \cite{rotta1951}. Since their foundational investigations, many researchers have developed more advanced and complex closure models for the pressure strain correlation term \cite{lrr, ssg,johansson1994,kr1995}. However all the available pressure strain correlation models have notable shortcomings. These shortcomings exist in the accuracy of their predictions and the realizability of their predictions. For example in rotation dominated flows the predictions of available models is incorrect both quantitatively and qualitatively. While Direct Numerical Simulations show that turbulence should be growing, available models predict that turbulence decays for these flows. Such rotation dominated flows include many flows of aerospace engineering like the trailing vortex, flap edge vortices and leading edge vortex flows. Another type of flows where the available pressure strain correlation
models are unsatisfactory is non-equilibrium turbulent flows. Non equilibrium turbulent flows include highly strained flows for example flows with shock turbulence interactions. The shortcomings of available models in accurately predicting such important classes of engineering flows is limits engineering investigations in such flows. In addition to accuracy in predictions the available pressure strain correlation models are unable to provide realizable predictions. The realizability condition \cite{realizability2, realizability1} tries to guarantee that the predictions of the turbulence model are not unphysical and correspond to flows that can exist in real life. In mathematical terms, the realizability condition requires turbulence models to predict a positive semi-definite Reynolds stress tensor. Unrealizable turbulence models may lead to issues in numerical convergence and numerical instability. Many investigators have found that the available pressure strain correlation models can guarantee realizable predictions only for low to moderate levels of Reynolds stress anisotropy \cite{mishra3}.
The rest of this paper is organized as follows. In Section II we outline the mathematical details of pressure strain correlation modeling. In Section III we discuss the action of the slow pressure strain correlation term, introduce established models for this term and compare their performance for different flows. In Section IV we discuss the action of the rapid pressure strain correlation term, introduce established rapid pressure strain correlation models and compare their performance for different flows. In the first four sections we focus on outlining developments in the modeling of the pressure strain correlation for incompressible flows. In Section V we outline and discuss important challenges and hurdles for making pressure strain correlation models an indispensable tool in the engineering design process. This paper concludes with a summary and concluding remarks in Section VI.
\section{Mathematical and modeling details}
In incompressible flows fluctuating pressure is governed by a Poisson equation:
\begin{equation}
\frac{1}{\rho}{\nabla}^2 p=-2\frac{\partial{U}_j}{\partial{x}_i}\frac{\partial{u}_i}{\partial{x}_j}-\frac{\partial^2}{\partial x_i \partial x_j} ( u_iu_j-\overline{u_iu_j})
\end{equation}
Poisson equation is an elliptic partial differential equation. The Laplacian $\nabla^2$ is an elliptic operator and because of this the Poisson equation has no real characteristic directions. This elliptic nature of the operator leads to the non-local nature of the solution for fluctuating pressure. This elliptic nature of the governing equation indicates that the pressure at a single point in the flow is affected by changes to the flow at all points in the flow domain. This non-local character is inherited by the pressure strain correlation as well.
In literature this fluctuating pressure is decomposed into two components, rapid and slow pressure.
\begin{equation}
p=p^S+p^R
\end{equation}
Rapid pressure corresponds to the linear part of the source term in the Poisson equation. This term is directly and instantaneously affected by any changes in the mean velocity gradient and is referred to as rapid pressure. It is governed by
\begin{equation}
\frac{1}{\rho}{\nabla}^2 p^R=-2\frac{\partial{U}_j}{\partial{x}_i}\frac{\partial{u}_i}{\partial{x}_j}
\end{equation}
Slow pressure corresponds to the nonlinear part of the source term in the Poisson equation. This term is not directly affected by changes in the mean velocity gradient and is referred to as slow pressure. It is governed by
\begin{equation}
\frac{1}{\rho}{\nabla}^2 p^S=-\frac{\partial^2 }{\partial x_i \partial x_j} ( u_iu_j-\overline{u_iu_j})
\end{equation}
The most general form of the slow pressure strain correlation has the form \cite{ssmodel}:
\begin{equation}
\phi^{(S)}_{ij}=\beta_1 b_{ij} + \beta_2 (b_{ik}b_{kj}- \frac{1}{3}II_b \delta_{ij})
\end{equation}
$\beta_1$ and $\beta_2$ can be the functions of second and third invariants of Reynolds stress anisotropy or can be a function of turbulent Reynolds number. $b_{ij}=\frac{\overline{u_iu_j}}{2k}-\frac{\delta_{ij}}{3}$ is the Reynolds stress anisotropy tensor, $II_b$ and $III_b$ are the second and third invariants of the Reynolds stress anisotropy respectively.
The most general form of the rapid pressure strain correlation is:
\begin{equation}
\begin{split}
&\phi_{ij}^{(R)}=S_{pq}[Q_1\delta_{ip}\delta_{jq}+Q_2(b_{ip}\delta{jq}+b_{jp}\delta{iq-2/3b_{pq}}\delta{ij})+Q_3b_{pq}b_{ij}+Q_4(b_{iq}b_{jp}-1/3b_{pk}b_{kq}\delta_{ij})\\
&+Q_5b_{pl}b_{lq}b_{ij}+(Q_5b_{pq}+Q_6b_{pl}b_{lq})(b_{ik}b_{kj}-1/3II_{b}\delta_{ij}]\\
&+\Omega_{pq}[Q_7(b_{ip}\delta{jq}+b_{jp}\delta_{iq})+Q_8b_{pk}(b_{jk}\delta{iq}+b_{ik}\delta{jq}+Q_9b_{pk}(b_{jk}b_{ik}+b_{ik}b_{jq})]
\end{split}
\end{equation}
where, $S_{ij}$ is the mean rate of strain, $W_{ij}$ is the mean rate of rotation and $K$ is the turbulent kinetic energy. $II_b = b_{ij}b_{ji}$ is the second invariant of the Reynolds stress anisotropy tensor. Different models differ in the choice of the values of the model coefficients. Choosing specific sets of coefficients to be non-zero determines the order of the model with respect to the Reynolds stress tensor. This is outlined in Table 1.
\begin{table}\scriptsize
\begin{center}
\def~{\hphantom{0}}
\begin{tabular}{c c c}
\textbf{} \textbf{RPSC Model}&\ \textbf{Expression Order}&\ \textbf{Non-zero Coefficients}\\[3pt]
LRR \cite{lrr} \ \ &\ \ \ \ linear in Reynolds stresses &\ \ \ \ $Q_1, Q_2, Q_7$\\ \\
SSG \cite{ssg} \ \ &\ \ \ \ quadratic in Reynolds stresses &\ \ \ \ $Q_1, Q_2, Q_3, Q_7$\\ \\
Johansson \& Hallback \cite{johansson1994} \ \ &\ \ \ \ fourth order in Reynolds stresses &\ \ \ $Q_1-Q_9$ \\ \\
\end{tabular}
\\
\caption{Rapid pressure strain correlation models compared with respect to their non-zero coefficients and order of expression}
\label{tab:comp}
\end{center}
\end{table}
As is seen in the general model expressions given in Equations (6) and (7) both the slow and the rapid components of the pressure strain correlation are modeled using tensors like the Reynolds stress anisotropy ($b_{ij}$), mean rate of strain ($S_{ij}$), mean rate of rotation ($\Omega_{ij}$), etc. These tensors in the modeling basis are local tensors but the pressure strain correlation has non-local dynamics. Because of this inconsistency most pressure strain correlation models have serious shortcomings in the accuracy of their predictions and in maintaining realizability of their predictions.
In the next sections we discuss the slow and rapid pressure strain correlation models respectively. We find a similar trend in their development where the first few models are simpler and attempt to replicate some basic details of the pressure strain correlation. The succeeding models attempt to address shortcomings by incorporating more complex expressions, such as higher degrees of nonlinearity with respect to the Reynolds stress anisotropy terms. The final models attempt to add additional tensors to the modeling basis that can admit missing non-local information to the model.
\section{Slow pressure strain correlation models}
Numerous experimental and numerical investigations into decaying turbulent flows \cite{lumley1977,warhaft1978,le1985} have observed that along with a decay in the turbulent kinetic energy the anisotropy of the Reynolds stresses reduces towards an isotropic state. This is also observed in experimental investigations \cite{choi1984,choi2001,hallback1995} that initially anisotropic Reynolds stresses relax towards an isotropic state in the absence of external mean velocity gradients. This is known as the return to isotropy phenomenon of turbulence. In turbulence modeling the slow pressure strain correlation is chiefly responsible for the return to isotropy of turbulence. Slow pressure strain correlation models aim to replicate the details of this return to isotropy phenomenon.
\subsection{Rotta Model:}
The first model for the slow pressure strain correlation was proposed by \cite{rotta1951}. The form of this model is given by
\begin{equation}
\phi^{(S)}_{ij}=-2C_{R}\epsilon b_{ij}
\end{equation}
The evolution equation for the Reynolds stress anisotropy for the Rotta model can be written as \cite{pope2000}
\begin{equation}
\frac{db_{ij}}{dt}=(-C_{R}-1)\frac {\epsilon} {k} b_{ij}
\end{equation}
The slow pressure strain correlation model of Rotta \cite{rotta1951} is linear in the Reynolds stresses. While it captures the return to anisotropy it is unable to capture the nonlinear nature of this return to isotropy process. For example on the Lumley triangle the paths predicted by the Rotta model are straight lines. Experimental data clearly shows that the return to isotropy is via curved trajectories \cite{chung1995}. The dependence of the rate of return to isotropy on the invariants of the Reynolds stress anisotropies is also not accounted for in the Rotta model.
\subsection{Lumley Model:}
The nonlinear effects were incorporated in the model of \cite{lumley1979}, in which the nonlinearities were introduced in the model through the functions of Reynolds stress anisotropy or the invariants of the Reynolds stress anisotropy. The coefficients of the model were taken as the function of turbulence Reynolds number
\begin{equation}
\begin{split}
\beta_1=2.0+\frac{F}{9}\exp\frac{-7.7}{\sqrt{Re_t}}\Bigg(\frac{72}{\sqrt{Re_t}}+80.1 ln[1+62.4(-II_b+2.3III_b)]\Bigg),and\\
\beta_2=0
\end{split}
\end{equation}
where, $q^2=2K$, $II_b$ and $III_b$ are the second and third invariants of the Reynolds stress anisotropy tensor. $F$ is the determinant of the normalized Reynolds stress tensor.
\subsection{Sarkar and Speziale Model:}
The model of Sarkar and Speziale \cite{ssmodel} is a quadratic model, the coefficients of the model are constants.
\begin{equation}
\beta_1=3.4 \quad and \quad \beta_2=3(\beta_1-2)
\end{equation}
The transport equations for the Reynolds stress anisotropy were considered as follows:
\begin{equation}
\begin{split}
\frac{dII_b}{d\tau}=1.4II_b+8.4III_b\\
\frac{dIII_b}{d\tau}=-4.2III_b+2.1II_b^2
\end{split}
\end{equation}
This simple quadratic model with only one independent constant is able to account for most of the nonlinear character of the return to isotropy phenomenon. It has been widely adopted in engineering simulations of turbulent flows.
\subsection{Slow pressure strain correlation models with extended tensor bases:}
All the slow pressure strain correlation models discussed in this section assume that the characteristic length scale of turbulence is the same in all directions. This is markedly true in flows where the geometry of the flow domain or body forces lead to a co-ordinate direction in the flow being decidedly preferred. For example axisymmetric expansion and axisymmetric contraction mean flows. In many anisotropic turbulent flows, the characteristic length scale is observed to be varying in different directions \cite{panda2017}. At the most basic level, we must try to include this anisotropy in the length scale in the modeling basis for the pressure strain correlation. \cite{panda2017} proposed a modeled formulation of length scale anisotropy tensor by assuming a linear relationship of length scale anisotropy with dissipation and Reynolds stress anisotropy tensors. The length scale anisotropy \cite{warrior2014} has the form
\begin{equation}
L_{ij}/l=0.75(c_1^*b_{ij}+c_2^*e_{ij})
\end{equation}
The model for the slow pressure strain correlation has the form:
\begin{equation}
\begin{split}
\phi_{ij}=c_1b_{ij}+c_2e_{ij}+c_3l_{ij}+c_4(b_{ik}b_{kj}-1/3b_{mn}b_{mn}\delta_{ij})\\
+c_5(b_{ik}e_{kj}-1/3b_{mn}e_{mn}\delta{ij})+c_6(e_{ik}e_{kj}-1/3e_{mn}e_{mn}\delta{ij})+\\
c_7(b_{ik}l_{kj}-1/3b_{mn}l_{mn}\delta{ij})+c_8(e_{ik}l_{kj}-1/3e_{mn}l_{mn}\delta{ij})+\\
c_9(l_{ik}l_{kj}-1/3l_{mn}l_{mn}\delta{ij})
\end{split}
\end{equation}
The anisotropy states in the turbulent flows can be characterized by two variables $\xi$ and $\eta$ given by
\begin{equation}
\begin{split}
\xi=(III_b/2)^{1/3}, \quad \eta=(-II_b/3)^{1/2},
\end{split}
\end{equation}
In a turbulent flow , $\xi$ and $\eta$ can be determined at any point and time from the Reynolds stresses. The $\xi$-$\eta$ phase space is bounded by two straight line segments denoting axisymmetric turbulence and from above by a curved line representing two-dimensional turbulence. This representation is referred to as the Lumley triangle \cite{pope2000} and is a simple manner to visualize the anisotropy of the Reynolds stress tensor. All realizable states of the Reynolds stress tensor lie inside the Lumley triangle.
We compare the predictions of the slow pressure strain correlation models introduced in this section and contrast them to experimental data. Here we focus on three models specifically: the slow pressure strain correlation models of \cite{rotta1951}, \cite{ssmodel} and \cite{panda2017}. The model of \cite{rotta1951} is chosen as due to its simplicity it is widely used in turbulence simulations. The model of \cite{ssmodel} attempts to address the deficiencies of the linear \cite{rotta1951} model by adding nonlinear terms in the Reynolds stress anisotropy. This is one methodology to address the limitations in the models and is thus included. The model of \cite{panda2017} attempts to address the deficiencies of slow pressure strain correlation models by adding additional tensors to the modeling basis. This represents another approach to address the limitations in the models and is thus included in the comparisons.
\begin{figure}
\centering
\includegraphics[height=8cm]{Lepenven_neg_IIIb.jpg}
\caption{Lumley triangle trajectories of the model predictions against the plane contraction experiment of \cite{le1985} ($III_b<0$) }
\end{figure}
\begin{figure}
\centering
\includegraphics[height=8cm]{choi_lumley.jpg}
\caption{Lumley triangle trajectories of model predictions against the plane distortion experiment of \cite{choi1984}}
\end{figure}
\begin{figure}
\centering
\includegraphics[height=12cm]{Warhaft.jpg}
\caption{Lumley triangle trajectories of model predictions against the experiment of \cite{warhaft1978}}
\end{figure}
Figure 1 represents the evolution of trajectories for the plane contraction experiment of \cite{le1985}, for this case the initial value of third invariant of Reynolds stress anisotropy is negative. The experimental data shows mild curvature in the phase plane. The nonlinear quadratic model of \cite{ssmodel} has predictions that better fit the experimental results, in comparison to \cite{rotta1951} and \cite{panda2017} models. The trajectories on the Lumley triangle in figure 2 are strongly curved indicating the nonlinear effects in the return to isotropy behavior. It is noticed in figure 2 that model of \cite{ssmodel} has predictions that are better in comparison other two models. The phase space comparison for the experiments of \cite{warhaft1978} is presented in figure 3. The curvature in the experimental results is very small. Both models of \cite{ssmodel} and \cite{rotta1951} have predictions that are very similar same. This indicates that the nonlinear effects were not dominant in the flows.
\begin{figure}
\centering
\subfloat[$II_b$]{\includegraphics[height=8cm]{UBEROI_IIb.jpg}}
\subfloat[$III_b$]{\includegraphics[height=8cm]{UBEROI_IIIb.jpg}}
\caption{Comparisons of model predictions against the experiment of \cite{uberoi1963}}
\end{figure}
Figure 4 represents the temporal evolution of second ($II_b$) and third ($III_b$) invariants for the experimental results of \cite{uberoi1963}. The \cite{panda2017} model has predictions that are better in comparison to the model of \cite{ssmodel}.
From the comparison against experimental data the predictions of the model of \cite{ssmodel} show best agreement with data across different experimental studies.
\section{Rapid pressure strain correlation models}
The rapid component of the pressure strain correlation accounts for the linear interactions between the fluctuating velocity field and the mean velocity gradient. This rapid pressure strain correlation has behavior that is very dependent on the mean velocity field. As an illustrative example we can demarcate the behavior of the rapid pressure strain correlation in two regimes of planar flows: hyperbolic streamline flow and elliptic streamline flow. In elliptic streamline flows the elliptical flow instability is initiated by the rapid pressure strain correlation \cite{kerswell2002}. In hyperbolic streamline flows the rapid pressure strain correlation stabilizes the flow instability \cite{mishra4}. The effect of the rapid pressure strain correlation is highly dependent on the mean gradient and substantially varies between different flows.
Besides giving accurate predictions of the flow evolution and ensuring realizable Reynolds stresses there are additional properties required of the ideal RPSC model. These include
\begin{enumerate}
\item The RPSC model ($\phi_{ij}^{(R)}$) should have a model expression linear in the Reynolds stresses \cite{reynolds1976,pope2000}.
\item The RPSC model should have a model expression linear in the mean velocity gradient \cite{johansson1994, pope2000}.
\item The RPSC model should obey the Crow constraint (from isotropic initial conditions) \cite{crow1968}.
\end{enumerate}
An ideal model is expected to conform to these properties. However no available model is able to meet all these properties and still produce accurate predictions. While there are many available models for the rapid pressure strain correlation, we discuss three established popular models. These include the models by Launder, Reece and Rodi\cite{lrr} (termed the LRR model), Speziale, Sarkar and Gatski \cite{ssg} (termed the SSG model) and Johansson and Hallback \cite{johansson1994} (termed the Johansson-Hallback model).
\subsection{LRR Model:}
The model proposed by Launder, Reece and Rodi\cite{lrr} has the form
\begin{equation}
\begin{split}
\phi_{ij}^{(R)}=C_1K S_{ij}+C_2K(b_{ik} S_{jk}+b_{jk} S_{ik}-2/3b_{mn} S_{mn}\delta_{ij})+C_3K(b_{ik} W_{jk}+b_{jk} W_{ik})
\end{split}
\end{equation}
The closure coefficients are given as $C_1=0.8$, $C_2=1.75$ and $C_3=1.31$.
The model proposed by \cite{lrr} conforms to all the properties for a RPSC model. It is linear in the Reynolds stresses and the mean velocity gradient. It also conforms to the Crow constraint. However it is not able to show accurate predictions for complex flows for example flows dominated by rotational effects. It is also not able to maintain realizability of the Reynolds stress or their evolution for moderate to high levels of anisotropy in the flow \cite{mishra3}.
The LRR model has been widely adopted in turbulence simulations and is available in most computational fluid dynamics software. Many variants of this model have also been developed. For example the model of \cite{jones1988} retains the model form of the LRR model but changes the coefficient values to improve performance in turbulent flows with high rates of shear.
\subsection{SSG Model:}
The model proposed by Speziale, Sarkar and Gatski \cite{ssg} has the form
\begin{equation}
\begin{split}
\phi_{ij}^{(R)}=(C_1-C_1^*II^{0.5})K S_{ij}+C_2K(b_{ik} S_{jk}+b_{jk} S_{ik}-2/3b_{mn} S_{mn}\delta_{ij})+C_3K(b_{ik} W_{jk}+b_{jk} W_{ik})
\end{split}
\end{equation}
The closure coefficients are given as $C_{1}=0.8$, $C_1^{*}=1.3$, $C_2=1.25$ and $C_3=0.4$.
The model expression does not conform to all the properties for RPSC models stated above and is quadratic in the Reynolds stresses. However it is able to show much improved accuracy in predictions and better realizability behavior than other linear models. For turbulent flows in non-inertial frames of reference this model is much better than other RPSC models.
\subsection {Johansson-Hallback Model:}
Johansson and Hallback\cite{johansson1994} derived the most general expression for the RPSC model. This is given by
\begin{equation}
\begin{split}
&\phi_{ij}^{(R)}=S_{pq}[Q_1\delta_{ip}\delta_{jq}+Q_2(b_{ip}\delta{jq}+b_{jp}\delta{iq-2/3b_{pq}}\delta{ij})+Q_3b_{pq}b_{ij}+Q_4(b_{iq}b_{jp}\\
&-1/3b_{pk}b_{kq}\delta_{ij})+Q_5b_{pl}b_{lq}b_{ij}+(Q_5b_{pq}+Q_6b_{pl}b_{lq})(b_{ik}b_{kj}-1/3II_{b}\delta_{ij}]\\
&+\Omega_{pq}[Q_7(b_{ip}\delta{jq}+b_{jp}\delta_{iq})+Q_8b_{pk}(b_{jk}\delta{iq}+b_{ik}\delta{jq}+Q_9b_{pk}(b_{jk}b_{ik}+b_{ik}b_{jq})]
\end{split}
\end{equation}
Here $Q_i$ are scalar functions of the invariants of the Reynolds stress anisotropy and the mean velocity gradient. These can in turn be expressed in terms of scalars $B_{\alpha}$ as
\begin{equation}
\begin{split}
&Q_1=4/5-2/5(4B_2+15B_3)II_\alpha-2/5B_5III_\alpha-1/220(19B_6-120B_7)II_b^2,\\
&Q_2=-12B_1-1/2B_5II_b-1/2(B_6-8B_7)III_b,\\
&Q_3=-8B_2+36B_3+1/22(7B_6-72B_7)II_b,\\
&Q_4=96B_2-36B_3-1/22(7B_6-72B_7)II_b,\\
&Q_5=B_5,\\
&Q_6=B_6,\\
&Q_7=-4/3-28/3B_1+1/6(2B_4-B_5)II_b-1/18(3B_6-56B_7)III_b,\\
&Q_8=-16B_2+28B_3+1/22(3B_6-56B_7)II_b,\\
&Q_9=B_4
\end{split}
\end{equation}
Based on the choices for the $B_{\alpha}$ \cite{johansson1994,sjogren1998,sjogren2000,hallback1995} outlined models that were second-, third- and fourth-order with respect to the Reynolds stresses. All these models conform to the strong realizability condition. The fourth-order model shows high agreement with data from experiments for strain-dominated mean flows. The performance of this model for rotation-dominated mean flows is still lacking.
In figures 5, 6 and 7, we show the performance of these models in the rapid distortion limit by comparing them to the results of RDT simulations. For purely strained flows such as the plane strain mean flow shown in figure 5 the models have good agreement with the trends observed in the RDT simulation. The fourth-order model of \cite{johansson1994} shows better performance as compared to the models of \cite{lrr} and \cite{ssg}.
This trend is observed again in figure 6 for a planar strained mean flow at the rapid distortion limit. All the models show acceptable agreement with the RDT simulation for the Reynolds stress anisotropy. The agreement of the fourth-order model of \cite{johansson1994} for the evolution of the turbulent kinetic energy is very accurate.
In figure 7 we compare the predictions of these models for an elliptic streamline flow at the rapid distortion limit. The figure shows that none of the models give satisfactory predictions for elliptic streamline flows. For example in figure 7 (b) the RDT simulations suggest that the turbulent kinetic energy of the flow is growing exponentially. All the RPSC models predict otherwise. For elliptic streamline flows RPSC models are inexact.
\begin{figure}
\centering
\subfloat[$b_{11}$]{\includegraphics[height=8cm]{Review1aPSb11.jpg}}
\subfloat[$log(k)$]{\includegraphics[height=8cm]{Review1bPStke.jpg}}
\caption{Rapid pressures train correlation model predictions for a plane strain mean flow at the rapid distortion limit}
\end{figure}
\begin{figure}
\centering
\subfloat[$b_{12}$]{\includegraphics[height=8cm]{Review2aHSb12.jpg}}
\subfloat[$log(k)$]{\includegraphics[height=8cm]{Review2bHStke.jpg}}
\caption{Rapid pressures train correlation model predictions for a planar strained mean flow at the rapid distortion limit}
\end{figure}
\begin{figure}
\centering
\subfloat[$b_{22}$]{\includegraphics[height=8cm]{Review3aEFb22.jpg}}
\subfloat[$log(k)$]{\includegraphics[height=8cm]{Review3bEFtke.jpg}}
\caption{Rapid pressures train correlation model predictions for an elliptic streamline mean flow at the rapid distortion limit}
\end{figure}
\subsection {Rapid pressure strain correlation models with extended bases}
One of the primary challenges in RPSC modeling is to replicate the non-local dynamics of pressure while using local tensors such as the Reynolds stresses and mean velocity gradients. The models of \cite{lrr, ssg,johansson1994} attempt to do this but have unsatisfactory performance in rotation dominated flows, etc. Some investigators have tried to formulate RPSC models by appending additional tensors to the modeling basis. We discuss a few such notable models here and analyze one of these in detail.
Kassinos and Reynolds\cite{kr1995} attempted to formulate a RPSC model using additional tensors in the modeling basis including as stropholysis, circulicity, etc. This was justified by differentiating between the componentiality and the
dimensionality of turbulent flow field. Using single point (or local) tensors such as the Reynolds stresses informs the model about the componentiality of the turbulent flow field but not about the dimensionality of the turbulent flow field. \cite{kr1995} define the structure dimensionality tensor $D_{ij}=M_{kkij}$, where the Reynolds stress is given by $R_{ij}=M_{ijkk}$. Addition of this tensor to the modeling basis would bring in important information and improve predictions. The final model in \cite{kr1995} did not show much improved predictions for rotation dominated flows. The model also had realizability issues \cite{mishra3}. The structure dimensionality tensor $D_{ij}$ is non-local and is not available in most engineering simulations. This made the usage of this model more problematic.
Cambon et al\cite{cambon1992} posited that using just the deviatoric component of the Reynolds stresses was unable to describe the turbulent flow field completely especially in the presence of mean rotation. They decomposed the Reynolds stress anisotropy tensor into two components: directional and polarization anisotropy ($b_{ij}=b^e_{ij}+b^z_{ij}$). Transport equations for these two components separately were developed. This model was able to show some improvements in rotation dominated flows. The transport equations for the decomposed anisotropy components were not unique and the closure coefficients were tuned to give agreement with the experiments used in the investigation. Above all in a real life engineering problem there is no clear manner on how to decompose the Reynolds stress anisotropy as information about the decomposition is non-local.
\cite{mishra6} developed an illustrative model where the model closure coefficients were functions of the mean velocity gradient invariants. In previous investigations \cite{mishra1,mishra2} have illustrated the details of the intercomponent energy transfer caused by the rapid pressure strain correlation. Using spectral analysis, \cite{mishra1,mishra2} establish a most likely evolution based on the statistics of the turbulent velocity that models should aim to reproduce. They have shown that including the mean velocity gradient in the modeling basis would lead to the addition of missing physics and improved model predictions. This is in agreement with \cite{lee1989} where it is shown that adding the mean strain rate information would improve the predictions of the pressure strain correlation model. In \cite{mishra3} a new approach to realizability is developed. Using this realizability approach it was shown that addition of the mean velocity gradient information would lead to better realizability behavior. \cite{mishra6,mishrathesis} have shown that including the mean gradient information by making the model coefficients functions of the mean velocity gradient would lead to a simple model structure, better realizability behavior and improved accuracy of predictions.
Instead of adding non-local tensors to the modeling basis the model of \cite{mishra6} uses the mean velocity gradient invariants to add missing information to the model expression. This model may be considered as compliant to use in real life engineering problems as it does not require the estimation of non-local tensors. The model expression is given by
\begin{equation}
\begin{split}
\phi_{ij}^{(R)}=4/5K\overline S_{ij}+6A_5\beta S_{pq}K(b_{ip}\delta {jq} +b_{jp}\delta {iq} + 2/3b_{pq}\delta {ij})+\\
2/3(4+7A_5(\beta))W_{pq}(b_{ip}\delta{jq}+b_{jp}\delta{iq})
\end{split}
\end{equation}
\begin{equation}
\begin{split}
A_5(\beta)=0.22\beta -0.44,\beta\in [0,0.5] \quad and\\
A_5(\beta)=-0.83\beta^2-0.44,\beta\in [0.5,1]
\end{split}
\end{equation}
where ,$\beta$ is the ellipticity parameter and is defined as
\begin{equation}
\begin{split}
\beta=\frac{W_{mn}W_{mn}}{W_{mn}W_{mn}+s_{mn}s_{mn}}
\end{split}
\end{equation}
In figures 8 and 9 we compare the predictions of this model to the DNS investigation of \cite{bns} for different elliptic flows. The predictions of the LRR model are included for contrast. Blaisdell and Shariff \cite{bns} have simulated homogeneous turbulence subjected to elliptic mean flows:
\begin{equation}
\frac{\partial U_i}{\partial x_j}=
\begin{bmatrix}
0 & 0 & -\gamma-e \\
0 & 0 & 0 \\
\gamma-e & 0 & 0
\end{bmatrix}
\end{equation}
where $e=\sqrt{\frac{1-\beta}{2}}$ and $\gamma=\sqrt{\frac{\beta}{2}}$. For $e>\gamma$ the mean flow has elliptic streamlines of aspect ratio $E=\sqrt{(\gamma+e)(\gamma-e)}$. We use this data from two simulations with mean flows having aspect ratios $E=2$ and $1.5$. The turbulent velocity field is initially isotropic and the initial $\frac{\eta}{Sk} = 0.167$.
In figure 8 the case with $E=1.5$ is shown. In figure 8 (a) The model of \cite{mishra6} shows better agreement with DNS data than the predictions of the LRR model. In figure 8 (b) the LRR model does predict turbulent kinetic energy growth but at a rate much smaller than DNS. The rate of turbulent kinetic energy growth predicted by the model of \cite{mishra6} is in agreement with the DNS data.
In figure 9 the case with $E=2$ is shown. (In this case the effect of rotation on flow evolution has increased over $E=1.5$). In figure 9 (b) while the DNS simulations predict the turbulent kinetic energy to be growing the model of LRR predicts decay. The rate of turbulent kinetic energy growth predicted by the model of \cite{mishra6} is in agreement with the DNS data.
\begin{figure}
\centering
\subfloat[$b_{22}$]{\includegraphics[height=8cm]{Review4aEFb13.jpg}}
\subfloat[$log(k)$]{\includegraphics[height=8cm]{Review4bEFtke.jpg}}
\caption{Rapid pressures train correlation model predictions for an elliptic streamline mean flow at the rapid distortion limit}
\end{figure}
\begin{figure}
\centering
\subfloat[$b_{22}$]{\includegraphics[height=8cm]{Review5aEFb13.jpg}}
\subfloat[$log(k)$]{\includegraphics[height=8cm]{Review5bEFtke.jpg}}
\caption{Rapid pressures train correlation model predictions for an elliptic streamline mean flow at the rapid distortion limit}
\end{figure}
\section{Modeling challenges and directions}
In this section of the review we outline and discuss some of the important present challenges and future research directions in pressure strain correlation modeling. It is our intention to make researchers cognizant of the details of these issues hampering the development and application of pressure strain correlation models in particular and the Reynolds stress modeling approach in general.
\subsection{Quantification of errors, uncertainties and variability in predictions}
Owing to the simplifications made during the formulation of turbulence models they suffer from inaccuracies in their predictions. This uncertainty in the predictions can be epistemic or aleatoric. Epistemic uncertainties in turbulence modeling occur owing to the empirical nature of the closure modeling expressions, the inaccuracies in the closure modeling coefficients, etc \cite{uq3}. In the recent past uncertainty quantification for turbulence models has become a very active field where investigators are trying to estimate the discrepancy between model predictions and the true evolution of turbulence \cite{uq2, uq2}. Investigators have studied the nature of and the variability in the closure coefficients \cite{edeling}. The uncertainty arising due to the limitation of the closure expression have been investigated in detail by other researchers \cite{dow, smith}.
A vast majority of the study of uncertainty in turbulence models has been restricted to simpler 2-equation models like the $k-\epsilon$ and $k-\omega$ models. There has been very limited work done to quantify the uncertainties in and the variability of Reynolds stress modeling approaches. Reynolds stress models are as susceptible to such uncertainties and errors as zero-, 1- and 2-equation turbulence models. \cite{mishrauq2,mishrauq1} have studied the uncertainty in Reynolds stress model predictions arising due to the use of the Reynolds stress tensor as the only descriptor of the turbulent flow field. Using the Reynolds stress tensor as the only characteristic of the fluctuating velocity field ignores the history of the flow and assumes that all flows with the same Reynolds stresses will evolve identically under similar conditions. \cite{mishrauq2} have shown that this is not a satisfactory assumption and flows with the same Reynolds stress tensor can show evolution that is completely different. In light of these results it is essential to carry out careful studies analyzing the uncertainties in pressure strain correlation models.
\subsection{Turbulent flows with compressibility effects}
In this article we have focused on the pressure strain correlation modeling for incompressible turbulent flows. Many flows of engineering interest have significant compressibility effects. These include high speed turbulent flows found in super-sonic and hyper-sonic design problems, turbulent flows in diffusers, etc. These flows are profoundly different from their incompressible counterparts \cite{cambon1993}. Compressible turbulent flows
have dilatational components of the velocity field leading to shocks, density variations and similar effects absent in incompressible flows \cite{sc}. Variability in the values of the transport coefficients that are functions of temperature and pressure are significant for compressible turbulent flows \cite{a1999}. The turbulent velocity field in compressible flows is highly dependent on the Mach number where the anisotropy of the turbulent flow increases with higher Mach numbers \cite{pantano}
It is accepted in the modeling community that majority of the compressibility effects are applied through the pressure strain correlation \cite{sarkar1}. The key hurdle in extending Reynolds stress modeling approach to compressible flows is the absence of an accurate model for the pressure strain correlation for these flows \cite{sarkar2}. While many investigators have studied pressure strain correlation modeling for incompressible flows there is much lesser work done for compressible flows. A majority of the pressure strain correlation models developed for compressible turbulent flows are extensions of available incompressible pressure strain correlation models with the addition of a blending function dependent on the turbulent Mach number \cite{c1,c2,c3,c4,c5,c6}. In numerical investigations it has been found that these models may give satisfactory results in very weakly compressible flows but in flows with significant compressibility their predictions are unsatisfactory \cite{gomez}.
This approach of making small modifications to incompressible turbulent flow models for the pressure strain correlation has both advantages and disadvantages. As an advantage it is able to transition from weakly compressible to incompressible flows consistently. However it tends to ignore the significant changes in the nature of pressure between incompressible and compressible flows. In incompressible turbulent flows pressure acts as a Lagrange multiplier and ensures that the flow velocity remains divergence free. However in compressible flows pressure is a thermodynamic variable governed by the energy equation, the equation of state and calorific equation. At high levels of compressibility pressure has wave like behavior and this causes complex interactions with the
velocity field. This results in a significant change in the nature and evolution of the turbulent flow in compressible flows that is substantially different form the incompressible flows. The development of an accurate model for the pressure strain correlation that can capture the effects of compressibility in high speed flows and still exhibit consistency with incompressible physics in the limit of low speed flows is a key hurdle in turbulence modeling.
\subsection{Improved modeling of the rate of dissipation tensor}
While models for the pressure strain correlation may be formulated in isolation their testing uses them with the rate of dissipation in simulating the flow. The coupled interaction of the pressure strain correlation model with the rate of dissipation model obfuscates the exact source of prediction errors. An important shortcoming for the Reynolds Stress Modeling approach is the approximate nature of the rate of dissipation equation. While the model equations for the evolution of the Reynolds stress anisotropy components are exact and based on the Reynolds stress transport equation the evolution equation for the rate of dissipation is empirically derived \cite{pope2000}. This model expression is
\begin{equation}
\frac{D\epsilon}{Dt}=\frac{\partial }{\partial x_i} (\nu +\frac{\nu_t}{\sigma_k})\frac{\partial \epsilon}{\partial x_i} +C_{\epsilon 1} \frac{P\epsilon}{k} - C_{\epsilon 2} \frac{\epsilon^2}{k}
\end{equation}
The first term on the right hand side represents the diffusive transport of $\epsilon$. The second and third terms on the right side represent the generation of $\epsilon$ due to vortex stretching and the destruction of $\epsilon$ by viscous action. The standard values for the closure coefficients are given by $\sigma_{\epsilon}=1.3$, $C_{\epsilon 1}=1.44$ and $C_{\epsilon 1}=1.92$, based on the constants determined by \cite{launder1974}. The value of the $C_{\epsilon 2}$ coefficient is calibrated to be in agreement with the power law decay observed in decaying turbulence. Here the decay exponent corresponds to the power law decay observed as $k(t)=k(t_0)(t/t_0)^{-n}$ and $\epsilon(t)=\epsilon(t_0)(t/t_0)^{-n-1}$. In terms of the decay exponent $n$ this is given by
\begin{equation}
n=\frac{1}{C_{\epsilon 2} -1}
\end{equation}
\begin{figure}
\begin{centering}
\includegraphics[width=0.7\textwidth]{foo.pdf}
\caption{Contrasting the relationship between $C_{\epsilon 2}$ and $n$ based on experimental studies (solid black line) and the values used in Reynolds Stress Modeling investigations. \label{fig:ce2}}
\end{centering}
\end{figure}
Most experimental investigations have found the decay exponent to lie in the range of $1.15-1.45$. This obligates the value of $C_{\epsilon 2}$ to approximately lie in the range $1.69-1.87$. However the values used in different models often lies well outside this bound. Based on \cite{batchelor1948}, \cite{hanjalic1972} chose $C_{\epsilon 2}=2.0$ to make the turbulent kinetic energy vary inversely with distance from the origin. Both the investigations of \cite{lrr} and \cite{ssmodel} changed it to $C_{\epsilon 2}=1.9$ so as to get faster rate of decay for their model simulations. \cite{ssg} chose to adopt the value of $C_{\epsilon 2}=1.92$ for better calibration of their model. Since then, different modeling investigations have used different values for the coefficient varying from $1.90$ to $2.0$. All these chosen values lie outside the range prescribed by experimental investigations and are often varying from one numerical investigation to another.
The values of the $C_{\epsilon 1}$ is chosen to match the steady state parameters in homogeneous turbulent shear flow. The form is given by $\frac{P}{\epsilon}=\frac{C_{\epsilon 2} -1}{C_{\epsilon 1} -1}$. It can be seen from this relationship that the choice of the value of the coefficient $C_{\epsilon 2}$ also in turn affects the value of the $C_{\epsilon 1}$ coefficient. Any errors in the values of $C_{\epsilon 2}$ will have a cascading effect and will affect the accuracy of the entire model. As can be seen in this discussion there is significant inconsistency in the modeling of the rate of dissipation term. The errors due to the rate of dissipation term affect the evaluation of the pressure strain correlation models. Improved models for the rate of dissipation term may be of great assistance in developing and testing improved pressure strain correlation models.
\section{Concluding remarks}
In this article we provide a thorough review of pressure strain correlation modeling for turbulent flows. Starting from the Reynolds stress transport equations the numerical and mathematical foundations of pressure strain correlation modeling are established. The key challenges in this modeling effort arising due to the non-local nature of the behavior of the pressure strain correlation are established.
Established slow pressure strain correlation models are introduced. Their predictions are compared and contrasted against experimental data from a range of experiments.
Popular rapid pressure strain correlation were introduced. Their predictions were contrasted against rapid distortion theory based simulations. It was shown that most rapid pressure strain correlation models have satisfactory behavior in strain dominated turbulent flows but unsatisfactory predictions in rotation dominated flows. Alternative models that add to the modeling basis were introduced and their predictions for elliptic streamline flows were shown. We outlined and discussed important challenges and hurdles for making pressure strain correlation models an indispensable tool in the engineering design process.
\newpage
\bibliographystyle{asmems4}
|
2,877,628,088,983 | arxiv | \section{Introduction}
We investigate the injective types and the algebraically injective
types in univalent mathematics, both in the absence and in the
presence of propositional resizing axioms. These notions of
injectivity are about the extension problem
\begin{diagram}
X & & \rInto^j & & Y \\
& \rdTo_f & & \ldEto & \\
& & D. & &
\end{diagram}
The injectivity of a type \m{D:\mathcal{U}} is defined by the surjectivity
of the restriction map \m{(-) \mathrel{\circ} j} along any embedding~\m{j}:
\M{ \Pi(X,Y : \mathcal{U})\, \Pi (j : X \hookrightarrow Y)\, \Pi(f : X \to D)\, \exists (g :
Y \to D)\, g \mathrel{\circ} j = f,
}
so that we get an \emph{unspecified} extension~\m{g} of~\m{f}
along~\m{j}.
The algebraic injectivity of \m{D} is defined by a given section
\m{(-) \mid j} of the restriction map \m{(-) \mathrel{\circ} j}, following Bourke's
terminology~\cite{bourke:2017}. By \m{\Sigma{-}\Pi}-distributivity,
this amounts to
\M{
\Pi(X,Y : \mathcal{U})\,
\Pi (j : X \hookrightarrow Y)\, \Pi(f : X \to D)\, \Sigma (f \mid j : Y \to D),
f \mid j \mathrel{\circ} j = f,
}
so that we get a \emph{designated} extension~\m{f \mid j} of~\m{f}
along~\m{j}. Formally, in this definition, \m{f \mid j} can be
regarded as a variable, but we instead think of the symbol ``\m{\mid}'' as
a binary operator.
For the sake of generality, we work without assuming or rejecting the
principle of excluded middle, and hence without assuming the axiom of
choice either. Moreover, we show that the principle of excluded middle
holds if and only if all pointed types are algebraically injective,
and, assuming resizing, if and only if all inhabited types are
injective, so that there is nothing interesting to say about
(algebraic) injectivity in its presence. That pointness and
inhabitedness are needed is seen by considering the embedding \m{\mathbb{0}
\hookrightarrow \mathbb{1}}.
Under propositional resizing principles~\cite{hottbook}
(Definitions~\ref{resizing} and~\ref{omega:resizing} below), the main
results are easy to state:
\begin{enumerate}
\item Injectivity is equivalent to the propositional truncation of
algebraic injectivity.
(This can be seen as a form of choice that
just holds, as it moves a propositional truncation inside a
\m{\Pi}-type to outside the \m{\Pi}-type, and may be related to
\cite{kenney:2011}.)
\item The algebraically injective types are precisely the retracts of
exponential powers of universes. Here by an exponential power of a type \m{B} we mean a type of the form \m{A \to B}, also written \m{B^A}.
In particular,
\begin{enumerate}
\item The algebraically injective sets are precisely the
retracts of powersets.
\item The algebraically injective \m{(n+1)}-types are precisely
retracts of exponential powers of the universes of \m{n}-types.
\end{enumerate}
Another consequence is that any universe is embedded as a retract of any
larger universe.
\item The algebraically injective types are also precisely the
underlying objects of the algebras of the partial-map
classifier.
\end{enumerate}
In the absence of propositional resizing, we have similar results
that have subtler statements that need to keep track of universe
levels rather explicitly.
Most constructions developed in this paper are in the absence of
propositional resizing. We apply them, with the aid of a notion of
algebraic flabbiness, which is related to the partial-map classifier,
to derive the results that rely on resizing mentioned above.
\paragraph{Acknowledgements.} Mike Shulman has acted as a sounding
board over the years, with many helpful remarks, including in
particular the suggestion of the terminology \emph{algebraic
injectivity} from~\cite{bourke:2017} for the notion we consider
here.
\section{Underlying formal system} \label{foundations}
Our handling of universes has a model in \m{\infty}-toposes following
Shulman~\cite{2019arXiv190407004S}. It differs from that of the HoTT
book~\cite{hottbook}, and Coq~\cite{coq}, in that we don't assume
cumulativity, and it agrees with that of Agda~\cite{agda}.
\subsection{Our univalent type theory}
Our underlying formal system can be considered to be a subsystem of that
used in UniMath~\cite{unimath}.
\begin{enumerate}
\item We work within an intensional Martin-L\"of type theory with
types \m{\mathbb{0}} (empty type), \m{\mathbb{1}} (one-element type with
\m{\operatorname{\star}:\mathbb{1}}), \m{\mathbb{N}} (natural numbers), and type formers \m{+}
(binary sum), \m{\Pi} (product), \m{\Sigma} (sum) and \m{\operatorname{Id}}
(identity type), and a hierarchy of type universes ranged over by
\m{\mathcal{U},\mathcal{V},\mathcal{W},\mathcal{T}}, closed under them in a suitable sense discussed
below.
We take these as required closure properties of our formal system,
rather than as an inductive definition.
\item We assume a universe \m{\mathcal{U}_0}, and for each universe \m{\mathcal{U}} we
assume a successor universe \m{\mathcal{U}^+} with \m{\mathcal{U} : \mathcal{U}^+}, and for any
two universes \m{\mathcal{U},\mathcal{V}} a least upper bound \m{\mathcal{U} \sqcup \mathcal{V}}. We
stipulate that we have \m{\mathcal{U}_0 \sqcup \mathcal{U} = \mathcal{U}} and \m{\mathcal{U} \sqcup \mathcal{U}^+
= \mathcal{U}^+} definitionally, and that the operation \m{(-)\sqcup(-)} is
definitionally idempotent, commutative, and associative, and that
the successor operation \m{(-)^+} distributes over \m{(-)\sqcup(-)}
definitionally.
\item We don't assume that the universes are cumulative on the nose,
in the sense that from \m{X : \mathcal{U}} we would be able to deduce that
\m{X : \mathcal{U} \sqcup \mathcal{V}} for any \m{\mathcal{V}}, but we also don't assume that
they are not. However, from the assumptions formulated below, it
follows that for any two universes \m{\mathcal{U},\mathcal{V}} there is a map
\m{\operatorname{lift}_{\mathcal{U},\mathcal{V}} : \mathcal{U} \to \mathcal{U} \sqcup \mathcal{V}}, for instance \m{X \mapsto X
+ \mathbb{0}_\mathcal{V}}, which is an embedding with \m{\operatorname{lift} X \simeq X} if
univalence holds (we cannot write the identity type \m{\operatorname{lift} X = X},
as the left- and right-hand sides live in the different types \m{\mathcal{U}}
and \m{\mathcal{U} \sqcup \mathcal{V}}, which are not (definitionally) the same in
general).
\item We stipulate that we have copies \m{\mathbb{0}_\mathcal{U}} and \m{\mathbb{1}_\mathcal{V}} of the
empty and singleton types in each universe \m{\mathcal{U}} (with the subscripts
often elided).
\item We stipulate that if \m{X : \mathcal{U}} and \m{Y : \mathcal{V}}, then \m{X+Y : \mathcal{U} \sqcup \mathcal{V}}.
\item We stipulate that if \m{X : \mathcal{U}} and \m{A : X \to \mathcal{V}} then
\m{\Pi_X A : \mathcal{U} \sqcup \mathcal{V}}. We abbreviate this product type as \m{\Pi A}
when \m{X} can be inferred from \m{A}, and sometimes we write it
verbosely as \m{\Pi (x:X), A \, x}.
In particular, for types \m{X : \mathcal{U}} and \m{Y : \mathcal{V}}, we have the function
type \m{X \to Y : \mathcal{U} \sqcup \mathcal{V}}.
\item The same type stipulations as for \m{\Pi}, and the same
grammatical conventions apply to the sum type former \m{\Sigma}.
In particular, for types \m{X : \mathcal{U}} and \m{Y : \mathcal{V}}, we have the cartesian product \m{X \times Y : \mathcal{U} \sqcup \mathcal{V}}.
\item We assume the \m{\eta} rules for \m{\Pi} and \m{\Sigma}, namely
that \m{f = \lambda x, f \, x} holds definitionally for any \m{f} in
a \m{\Pi}-type and that \m{z=(\operatorname{pr}_1 z , \operatorname{pr}_2 z)} holds definitionally
for any \m{z} in a \m{\Sigma} type, where \m{\operatorname{pr}_1} and \m{\operatorname{pr}_2} are
the projections.
\item For a type \m{X} and points \m{x,y:X}, the identity type \m{\operatorname{Id}_{X} x \, y} is abbreviated as \m{\operatorname{Id} x \, y} and often written \m{x =_X y} or simply \m{x = y}.
The elements of the identity type \m{x=y} are called identifications
or paths from \m{x} to~\m{y}.
\item When making definitions, definitional equality is written ``$\overset{\text{def}}{=}$''. When it is invoked, it is written e.g.\ ``\m{x = y} definitionally''. This is consistent with the fact that any definitional equality \m{x = y} gives rise to an element of the identity type \m{x = y} and should therefore be unambiguous.
\item When we say that something is the case by construction, this means we
are expanding definitional equalities.
\item We tacitly assume univalence~\cite{hottbook}, which gives
function extensionality (pointwise equal functions are equal) and
propositional extensionality (logically equivalent subsingletons are
equal).
\item We work with the existence of propositional, or subsingleton, truncations as an
assumption, also tacit. The HoTT book~\cite{hottbook}, instead,
defines type formation \df{rules} for propositional truncation as a
syntactical construct of the formal system. Here we take
propositional truncation as an axiom for any pair of
universes \m{\mathcal{U},\mathcal{V}}: \textcolor{darkblue}{
\begin{align*}
\Pi (X:\mathcal{U})\, \Sigma & (\trunc{X} : \mathcal{U}), \\
& \mathrel{\phantom{\times}} \text{\m{\trunc{X}} is a proposition} \times (X \to \trunc{X}) \\
& \times \bracket{\Pi (P : \mathcal{V}), \text{\m{P} is a proposition} \to (X \to P) \to \trunc{X} \to P}.
\end{align*}}
We write \m{\mid x \mid} for the insertion of \m{x:X} into the type
\m{\trunc{X}} by the assumed function \m{X \to \trunc{X}}. We also
denote by \m{\bar{f}} the function \m{\trunc{X} \to P} obtained by
the given ``elimination rule'' \m{(X \to P) \to \trunc{X} \to P}
applied to a function \m{f:X \to P}. The universe \m{\mathcal{U}} is that of
types we truncate, and \m{\mathcal{V}} is the universe where the propositions
we eliminate into live. Because the existence of propositional
truncations is an assumption rather than a type formation rule, its
so-called ``computation'' rule \M{\bar{f} \mid x \mid = f x} doesn't
hold definitionally, of course, but is established as a derived
identification, by the definition of proposition.
\end{enumerate}
\subsection{Terminology and notation}
\label{existence:terminology}
We assume that the readers are already familiar with the notions of
univalent mathematics, e.g.\ from the HoTT book~\cite{hottbook}. The
purpose of this section is to establish terminology and notation only,
particularly regarding our modes of expression that diverge from the HoTT
book.
\begin{enumerate}
\item A type \m{X} is a singleton, or contractible, if there is a
designated \m{c:X} with \m{x = c} for all \m{x:X}:
\M{
\text{\m{X} is a singleton} \overset{\text{def}}{=} \Sigma (c : X), \Pi (x:X), x = c.
}
\item A proposition, or subsingleton, or truth value, is a type with
at most one element, meaning that any two of its elements are
identified:
\M{
\text{\m{X} is a proposition} \overset{\text{def}}{=} \Pi(x,y:X), x=y.
}
\item
By an unspecified element of a type \m{X} we mean a (specified)
element of its propositional truncation~\m{\trunc{X}}.
We say that a type is inhabited if it has an unspecified element.
If the type \m{X} codifies a mathematical statement, we say that
\m{X} holds in an unspecified way to mean the assertion
\m{\trunc{X}}. For example, if we say that the type \m{A} is a
retract of the type \m{B} in an unspecified way, what we mean is that
\m{\trunc{\text{\m{A} is a retract of \m{B}}}}.
\item Phrases such as ``there exists'', ``there is'', ``there is some'', ``for some'' etc.\ indicate a propositionally truncated \m{\Sigma}, and symbolically we write \M{(\exists (x:X), A \, x) \overset{\text{def}}{=} \trunc{\Sigma (x:X), A \, x}.}
For emphasis, we may say that there is an unspecified \m{x:X} with \m{A\,x}.
When the meaning of existence is intended to be (untruncated)
\m{\Sigma}, we use phrases such as ``there is a designated'', ``there
is a specified'', ``there is a distinguished'', ``there is a given'', ``there is a chosen'', ``for some chosen'', ``we can find'' etc.
The statement that there is a unique \m{x:X} with \m{A \, x} amounts to
the assertion that the type \m{\Sigma (x:X), A \, x} is a singleton:
\M{
(\exists! (x:X), A \, x) \overset{\text{def}}{=} \text{the type \m{\Sigma (x:X), A \, x} is a singleton}.
}
That is, there is a unique pair \m{(x,a)} with \m{x:X} and \m{a : A\,
x}. This doesn't need to be explicitly propositionally truncated,
because singleton types are automatically propositions.
The statement that there is at most one \m{x:X} with \m{A \, x}
amounts to the assertion that the type \m{\Sigma (x:X), A \, x} is a
subsingleton (so we have at most one pair \m{(x,a)} with \m{x:X} and
\m{a : A\, x}).
\item We often express a type of the form \m{\Sigma(x:X), A \, x} by
phrases such as ``the type of \m{x:X} with \m{A \, x}''.
For example, if we define the fiber of a point \m{y:Y} under a
function \m{f : X \to Y} to be the type \m{f^{-1}(y)} of points \m{x:X}
that are mapped by \m{f} to a point identified with \m{y}, it
should be clear from the above conventions that we mean
\M{
f^{-1}(y) \overset{\text{def}}{=} \Sigma (x : X), f x = y.
}
Also, with the above terminological conventions, saying that the
fibers of \m{f} are singletons (that is, that \m{f} is an equivalence)
amounts to the same thing as saying that for every \m{y:Y} there is a
unique \m{x:X} with \m{f(x)=y}.
Similarly, we say that such an \m{f} is an embedding if for every
\m{y:Y} there is at most one \m{x:X} with \m{f(x)=y}. In passing, we
remark that, in general, this is stronger than \m{f} being
left-cancellable, but coincides with left-cancellability if the type
\m{Y} is a set (its identity types are all subsingletons).
\item We sometimes use the mathematically more familiar ``maps to''
notation~\m{\mapsto} instead of type-theoretical lambda notation
\m{\lambda} for defining nameless functions.
\item Contrarily to an existing convention among some practitioners,
we will not reserve the word \df{is} for mathematical statements
that are subsingleton types. For example, we say that a type is
algebraically injective to mean that it comes equipped with suitable
data, or that a type \m{X} is a retract of a type \m{Y} to mean that
there are designated functions \m{s : X \to Y} and \m{r : Y \to X},
and a designated pointwise identification \m{r \mathrel{\circ} s \sim \operatorname{id}}.
\item Similarly, we don't reserve the words \df{theorem}, \df{lemma},
\df{corollary} and \df{proof} for constructions of elements of
subsingleton types, and all our constructions are indicated by the
word proof, including the construction of data or structure.
Because \df{proposition} is a semantical rather than syntactical
notion in univalent mathematics, we often have situations when we
know that a type is a proposition only much later in the
mathematical development. An example of this is univalence. To know
that this is a proposition, we first need to state and prove many
lemmas, and even if these lemmas are propositions themselves, we
will not know this at the time they are stated and proved. For
instance, knowing that the notion of being an equivalence is a
proposition requires function extensionality, which follows from
univalence. Then this is used to prove that univalence is a
proposition.
\end{enumerate}
\subsection{Formal development}
A computer-aided formal development of the material of this paper has
been performed in Agda~\cite{agda}, occasionally preceded by pencil
and paper scribbles, but mostly directly in the computer with the aid
of Agda's interactive features. This paper is an unformalization of
that development. We emphasize that not only numbered statements in
this paper have formal counterparts, but also the comments in passing,
and that the formal version has more information than what we
choose to report here.
We have two versions. One of them~\cite{injective:blackboard} is in
\df{blackboard style}, with the ideas in the order they have come to
our mind over the years, in a fairly disorganized way, and with local
assumptions of univalence, function extensionality, propositional
extensionality and propositional truncation. The other
one~\cite{injective:article} is in \df{article style}, with univalence
and existence of propositional truncations as global assumptions, and
functional and propositional extensionality derived from
univalence. This second version follows closely this paper (or rather
this paper follows closely that version), organized in a way more
suitable for dissemination, repeating the blackboard definitions, in a
definitionally equal way, and reproducing the proofs and constructions
that we consider to be relevant while invoking the blackboard for the
routine, unenlightening ones. The blackboard version also has
additional information that we have chosen not to include in the
article version of the Agda development or this paper.
An advantage of the availability of a formal version is that, whatever
steps we have omitted here because we considered them to be
obvious or routine, can be found there, in case of doubt.
\section{Injectivity with universe levels}
As discussed in the introduction, in the absence of propositional
resizing we are forced to keep track of universe levels rather
explicitly.
\begin{definition}
We say that a type \m{D} in a universe \m{\mathcal{W}} is \df{\m{\mathcal{U},\mathcal{V}}-injective}
to mean
\M{ \Pi(X : \mathcal{U})\, \Pi(Y : \mathcal{V})\, \Pi (j : X \hookrightarrow Y)\, \Pi(f : X \to D),\, \exists (g :
Y \to D), g \mathrel{\circ} j \sim f,
}
and that it is \df{algebraically \m{\mathcal{U},\mathcal{V}}-injective} to mean
\M{
\Pi(X : \mathcal{U}) \,\Pi(Y : \mathcal{V})\,
\Pi (j : X \hookrightarrow Y) \, \Pi(f : X \to D) ,\, \Sigma (f \mid j : Y \to D),
f \mid j \mathrel{\circ} j \sim f.
}
\end{definition}
\noindent Notice that, because we have function extensionality, pointwise
equality~\m{\sim} of functions is equivalent to equality, and
hence equal to equality by univalence. But it is more convenient for
the purposes of this paper to work with pointwise equality in these
definitions.
\section{The algebraic injectivity of universes}
Let \m{\mathcal{U},\mathcal{V},\mathcal{W}} be universes, \m{X:\mathcal{U}} and \m{Y : \mathcal{V}} be types, and
\m{f : X \to \mathcal{W}} and \m{j : X \to Y} be given functions, where \m{j} is not necessarily an embedding.
We define functions \m{f \scaleobj{0.7}{\,\downarrow\,} j} and \m{f \scaleobj{0.7}{\,\uparrow\,} j} of type \m{Y \to \mathcal{U} \sqcup \mathcal{V} \sqcup \mathcal{W}}
by
\textcolor{darkblue}{\begin{eqnarray*}
(f \scaleobj{0.7}{\,\downarrow\,} j) \, y & \overset{\text{def}}{=} & \Sigma (w : j^{-1}(y)), f(\operatorname{pr}_1 w), \\
(f \scaleobj{0.7}{\,\uparrow\,} j) \, y & \overset{\text{def}}{=} & \Pi (w : j^{-1}(y)), f(\operatorname{pr}_1 w).
\end{eqnarray*}}
\begin{lemma}
If \m{j} is an embedding, then both \m{f \scaleobj{0.7}{\,\downarrow\,} j} and \m{f \scaleobj{0.7}{\,\uparrow\,} j} are extensions of \m{f} along~\m{j} up to equivalence, in the sense that \M{(f \scaleobj{0.7}{\,\downarrow\,} j \mathrel{\circ} j) \, x \simeq f x \simeq (f \scaleobj{0.7}{\,\uparrow\,} j \mathrel{\circ} j) \, x,}
and hence extensions up to equality if \m{\mathcal{W}} is taken to be \m{\mathcal{U} \sqcup \mathcal{V}}, by univalence.
\end{lemma}
\noindent Notice that if \m{\mathcal{W}} is kept arbitrary, then univalence cannot be applied because equality is defined only for elements of the same type.
\begin{proof}
Because a sum indexed by a subsingleton is equivalent to any of its
summands, and similarly a product indexed by a subsingleton is equivalent to
any of its factors, and because a map is an embedding precisely when
its fibers are all subsingletons.
\end{proof}
\noindent We record this corollary:
\begin{lemma} \label{ref:16:1}
The universe \m{\mathcal{U} \sqcup \mathcal{V}} is algebraically \m{\mathcal{U},\mathcal{V}}-injective, in at least two ways.
\end{lemma}
\noindent And in particular, e.g.\ \m{\mathcal{U}} is \m{\mathcal{U},\mathcal{U}}-injective, but of course
\m{\mathcal{U}} doesn't live in \m{\mathcal{U}} and doesn't even have a copy in
\m{\mathcal{U}}. For the following, we say that \m{y : Y} is not in the image
of \m{j} to mean that \m{j \, x \ne y} for all \m{x:X}.
\begin{proposition}
For \m{y:Y} not in the image of \m{j}, we have
\m{(f \scaleobj{0.7}{\,\downarrow\,} j) \, y \simeq \mathbb{0}} and
\m{(f \scaleobj{0.7}{\,\uparrow\,} j) \, y \simeq \mathbb{1}}.
\end{proposition}
\noindent With excluded middle, this would give that the two extensions have
the same sum and product as the non-extended map, respectively, but
excluded middle is not needed, as it is not hard to see:
\begin{remark} We have canonical equivalences
\m{\Sigma f \simeq \Sigma (f \scaleobj{0.7}{\,\downarrow\,} j)} and
\m{\Pi f \simeq \Pi (f \scaleobj{0.7}{\,\uparrow\,} j)}.
\end{remark}
Notice that the functions \m{f}, \m{f \scaleobj{0.7}{\,\downarrow\,} j} and \m{f \scaleobj{0.7}{\,\uparrow\,} j},
being universe valued, are type families, and hence the notations
\m{\Sigma f}, \m{\Sigma(f \scaleobj{0.7}{\,\downarrow\,} j)}, \m{\Pi f} and \m{\Pi(f \scaleobj{0.7}{\,\uparrow\,} j)}
are just particular cases of the notations for the sum and product of
a type family.
The two extensions are left and right Kan extensions in the following
sense, without the need to assume that \m{j} is an embedding. First, a
map \m{f:X \to \mathcal{U}}, when \m{X} is viewed as an \m{\infty}-groupoid and
hence an \m{\infty}-category, and when \m{\mathcal{U}} is viewed as the
\m{\infty}-generalization of the category of sets, can be considered
as a sort of \m{\infty}-presheaf, because its functoriality is
automatic: If we define
\M{f [ p ] \overset{\text{def}}{=} \operatorname{transport} f p}
of type \m{f\, x \to f\, y} for \m{p : \operatorname{Id} \, x \, y}, then for
\m{q : \operatorname{Id} \, y \, z} we have
\M{ f [
\operatorname{refl}_x ] = \operatorname{id}_{f \, x}, \qquad\qquad f [p \operatorname{\bullet} q] = f [q] \mathrel{\circ} f
[p].
}
Then we can consider the type of transformations between such
\m{\infty}-presheaves \m{f : X \to \mathcal{W}} and \m{f' : X \to \mathcal{W}'} defined by
\M{
f \mathrel{\,\,\preceq\,\,} f' \overset{\text{def}}{=} \Pi (x : X), f \, x \to f' x,
}
which are automatically natural in the sense that for all \m{\tau: f \mathrel{\,\,\preceq\,\,} f'} and \m{p : \operatorname{Id} \, x \, y},
\M{
\tau_y \mathrel{\circ} f [ p ] = f' [p] \mathrel{\circ} \tau_x.
}
It is easy to check that we have the following canonical transformations:
\begin{remark}
\m{f \scaleobj{0.7}{\,\downarrow\,} j \mathrel{\,\,\preceq\,\,} f \scaleobj{0.7}{\,\uparrow\,} j} if \m{j} is an embedding.
\end{remark}
It is also easy to see that, without assuming \m{j} to be an embedding,
\begin{enumerate}
\item \m{f \mathrel{\,\,\preceq\,\,} f \scaleobj{0.7}{\,\downarrow\,} j \mathrel{\circ} j},
\item \m{f \scaleobj{0.7}{\,\uparrow\,} j \mathrel{\circ} j \mathrel{\,\,\preceq\,\,} f}.
\end{enumerate}
These are particular cases of the following constructions, which are
evident and canonical, even if they may be a bit
laborious:
\begin{remark} For any \m{g : Y \to \mathcal{T}}, we have canonical equivalences
\begin{enumerate}
\item \m{(f \scaleobj{0.7}{\,\downarrow\,} j \mathrel{\,\,\preceq\,\,} g) \simeq (f \mathrel{\,\,\preceq\,\,} g \mathrel{\circ} j),} \quad i.e.\ \m{f \scaleobj{0.7}{\,\downarrow\,} j} is a left Kan extension,
\item \m{(g \mathrel{\,\,\preceq\,\,} f \scaleobj{0.7}{\,\uparrow\,} j) \simeq (g \mathrel{\circ} j \mathrel{\,\,\preceq\,\,} f),} \quad i.e.\ \m{f \scaleobj{0.7}{\,\uparrow\,} j} is a right Kan extension.
\end{enumerate}
\end{remark}
We also have that the left and right Kan extension operators along an
embedding are themselves embeddings, as we now show.
\begin{theorem}
For any types \m{X,Y:\mathcal{U}} and any embedding \m{j : X \to Y}, left Kan extension along \m{j} is an embedding of the function type \m{X \to \mathcal{U}} into the function type \m{Y \to \mathcal{U}}.
\end{theorem}
\begin{proof}
Define \m{s : (X \to \mathcal{U}) \to (Y \to \mathcal{U})} and \m{r : (Y \to \mathcal{U}) \to (X \to \mathcal{U})} by
%
\M{
\begin{array}{lll}
s \, f & \overset{\text{def}}{=} & f \scaleobj{0.7}{\,\downarrow\,} j, \\
r \, g & \overset{\text{def}}{=} & g \mathrel{\circ} j.
\end{array}
}
%
By function extensionality, we have that \m{r (s \, f) = f}, because
\m{s} is a pointwise-extension operator as \m{j} is an embedding,
and by construction we have that \m{s (r \, g) = (g \mathrel{\circ} j) \scaleobj{0.7}{\,\downarrow\,}
j}. Now define \m{\kappa : \Pi (g : Y \to \mathcal{U}), s(r \,g) \mathrel{\,\,\preceq\,\,}
g} by
%
\M{
\kappa \, g \, y \, ((x , p) , C) \overset{\text{def}}{=} \operatorname{transport} \, g \, p \, C
}
for all \m{g : Y \to \mathcal{U}}, \m{y : Y}, \m{x : X}, \m{p : j \, x = y} and \m{C : g(j \, x)}, so that
\m{\operatorname{transport} \, g \, p \, C} has type \m{g \, y },
and consider the type
%
\M{
M \overset{\text{def}}{=} \Sigma (g : Y \to \mathcal{U})\,\Pi(y:Y), \text{the map \m{\kappa \, g \, y : s (r \, g) \, y \to g \, y} is an equivalence.}
}
%
Because the notion of being an equivalence is a proposition and
because products of propositions are propositions, the first projection
\M{\operatorname{pr}_1 : M \to (Y \to \mathcal{U})} is an embedding. To complete the proof,
we show that there is an equivalence \m{\phi : (X \to \mathcal{U}) \to M}
whose composition with this projection is \m{s}, so that \m{s},
being the composition of two embeddings, is itself an embedding. We
construct \m{\phi} and its inverse \m{\gamma} by
%
\M{
\begin{array}{lll}
\phi \, f & \overset{\text{def}}{=} & (s f , \varepsilon \, f), \\
\gamma \, (g , e) & \overset{\text{def}}{=} & r \, g,
\end{array}
}
%
where \m{\varepsilon \, f} is a proof that the map \m{\kappa \, (s
f) \, y} is an equivalence for every \m{y : Y}, to be constructed
shortly. Before we know this construction, we can see that \m{\gamma
(\phi \, f) = r (s \, f) = f} so that \m{\gamma \mathrel{\circ} \phi \sim
\operatorname{id}}, and that \m{\phi (\gamma (g , e)) = (s(r g) , \varepsilon (r
g))}. To check that the pairs \m{(s(r g) , \varepsilon (r g))}
and \m{(g , e)} are equal and hence \m{\phi \mathrel{\circ} \gamma \sim \operatorname{id}},
it suffices to check the equality of the first components, because
the second components live in subsingleton types. But \m{e \, y}
says that \m{s (r \, g) \, y \simeq g \, y} for any \m{y:Y}, and
hence by univalence and function extensionality, \m{s (r \, g) = g}.
Thus the functions \m{\phi} and \m{\gamma} are mutually
inverse. Now, \m{\operatorname{pr}_1 \mathrel{\circ} \phi = s} definitionally using the
$\eta$-rule for \m{\Pi}, so that indeed \m{s} is the composition of
two embeddings, as we wanted to show.
It remains to show that the map \m{\kappa \, (s f) \, y : s (f \, y) \to s(r(s \,
f)) \, y} is indeed an equivalence. The domain and
codomain of this function amount, by construction, to respectively
%
\M{
\begin{array}{lll}
A & \overset{\text{def}}{=} & \Sigma (t : j^{-1}(y)), \Sigma (w : j^{-1}(j (\operatorname{pr}_1 t))), f (\operatorname{pr}_1 w)\\
B & \overset{\text{def}}{=} & \Sigma (w : j^{-1}(y)), f(\operatorname{pr}_1 w).
\end{array}
}
We construct an inverse \m{\delta : B \to A} by
%
\M{
\delta \, ((x , p),C) \overset{\text{def}}{=} ((x , p) , (x , \operatorname{refl}_{j \, x}) , C).
}
%
It is routine to check that the functions \m{\kappa \, (s f) \, y}
and \m{\delta} are mutually inverse, which concludes the proof.
\end{proof}
The proof of the theorem below follows the same pattern as the
previous one with some portions ``dualized'' in some sense, and so we
are slightly more economic with its formulation this time.
\begin{theorem}
For any types \m{X,Y:\mathcal{U}} and any embedding \m{j : X \to Y}, the right Kan extension operation along \m{j} is an embedding of the function type \m{X \to \mathcal{U}} into the function type \m{Y \to \mathcal{U}}.
\end{theorem}
\begin{proof}
Define \m{s : (X \to \mathcal{U}) \to (Y \to \mathcal{U})} and \m{r : (Y \to \mathcal{U}) \to (X \to \mathcal{U})} by
%
\M{
\begin{array}{lll}
s \, f & \overset{\text{def}}{=} & f \scaleobj{0.7}{\,\uparrow\,} j, \\
r \, g & \overset{\text{def}}{=} & g \mathrel{\circ} j.
\end{array}
}
%
By function extensionality, we have that \m{r (s \, f) = f},
and, by construction, \m{s (r \, g) = (g \mathrel{\circ} j) \scaleobj{0.7}{\,\uparrow\,}
j}. Now define \m{\kappa : \Pi (g : Y \to \mathcal{U}), g \mathrel{\,\,\preceq\,\,} s(r \,g)
} by
%
\M{
\kappa \, g \, y \, C (x , p) \overset{\text{def}}{=} \operatorname{transport} \, g \, p^{-1} \, C
}
for all \m{g : Y \to \mathcal{U}}, \m{y : Y}, \m{C : g \, y}, \m{x : X}, \m{p : j \, x = y}, so that
\m{\operatorname{transport} \, g \, p^{-1} \, C} has type \m{g (j \, x) },
and consider the type
%
\M{
M \overset{\text{def}}{=} \Sigma (g : Y \to \mathcal{U})\,\Pi(y:Y), \text{the map \m{\kappa \, g \, y : g \, y \to s (r \, g) \,y} is an equivalence.}
}
%
Then the first projection \m{\operatorname{pr}_1 : M \to (Y \to \mathcal{U})} is an embedding. To complete the proof,
we show that there is an equivalence \m{\phi : (X \to \mathcal{U}) \to M}
whose composition with this projection is \m{s}, so that it follows that \m{s}
is an embedding. We
construct \m{\phi} and its inverse \m{\gamma} by
%
\M{
\begin{array}{lll}
\phi \, f & \overset{\text{def}}{=} & (s f , \varepsilon \, f), \\
\gamma \, (g , e) & \overset{\text{def}}{=} & r \, g,
\end{array}
}
%
where \m{\varepsilon \, f} is a proof that the map \m{\kappa \, (s
f) \, y} is an equivalence for every \m{y : Y}, so that \m{\phi}
and \m{\gamma} are mutually inverse by the argument of the previous
proof.
To prove that the map \m{\kappa \, (s f) \, y : s(r(s \,
f)) \, y \to s (f \, y)} is an equivalence, notice that its domain and
codomain amount, by construction, to respectively
%
\M{
\begin{array}{lll}
A & \overset{\text{def}}{=} & \Pi (w : j^{-1}(y)), f(\operatorname{pr}_1 w), \\
B & \overset{\text{def}}{=} & \Pi (t : j^{-1}(y)), \Pi (w : j^{-1}(j (\operatorname{pr}_1 t))), f (\operatorname{pr}_1 w).
\end{array}
}
We construct an inverse \m{\delta : B \to A} by
%
\M{
\delta \, C \, (x , p) \overset{\text{def}}{=} C (x , p) (x , \operatorname{refl}_{j \, x}).
}
%
It is routine to check that the functions \m{\kappa \, (s f) \, y}
and \m{\delta} are mutually inverse, which concludes the proof.
\end{proof}
The left and right Kan extensions trivially satisfy \m{f \scaleobj{0.7}{\,\downarrow\,} \operatorname{id} \sim f}
and \m{f \scaleobj{0.7}{\,\uparrow\,} \operatorname{id} \sim f} because the identity map is an embedding, by
the extension property, and so are contravariantly functorial in view
of the following.
\begin{remark} \label{iterated}
For types \m{X : \mathcal{U}}, \m{Y : \mathcal{V}} and \m{Z : \mathcal{W}}, and functions \m{j : X \to Y}, \m{k : Y \to Z} and \m{f : X \to \mathcal{U} \sqcup \mathcal{V} \sqcup \mathcal{W}}, we have canonical identifications
\M{
\begin{array}{lll}
f \scaleobj{0.7}{\,\downarrow\,} (k \mathrel{\circ} j) & \sim & (f \scaleobj{0.7}{\,\downarrow\,} j) \scaleobj{0.7}{\,\downarrow\,} k, \\
f \scaleobj{0.7}{\,\uparrow\,} (k \mathrel{\circ} j) & \sim & (f \scaleobj{0.7}{\,\uparrow\,} j) \scaleobj{0.7}{\,\uparrow\,} k.
\end{array}
}
%
\end{remark}
\begin{proof}
This is a direct consequence of the canonical equivalences
%
\M{
\begin{array}{lll}
(\Sigma (t : \Sigma B) , C \, t) \simeq (\Sigma (a : A)\,
\Sigma (b : B \, a), C(a,b)) \\
(\Pi (t : \Sigma B) , C \, t) \simeq (\Pi (a : A)\,
\Pi (b : B \, a), C(a,b))
\end{array}
}
%
for arbitrary universes \m{\mathcal{U},\mathcal{V},\mathcal{W}} and \m{A:\mathcal{U}}, \m{B: A \to \mathcal{V}}, and \m{C : \Sigma \, B \to \mathcal{W}}.
\end{proof}
The above and the following are applied in work on
compact ordinals (reported in our repository~\cite{TypeTopology}).
\begin{remark}
For types \m{X : \mathcal{U}} and \m{Y : \mathcal{V}}, and functions \m{j : X \to Y}, \m{f : X \to \mathcal{W}}
and \m{f' : X \to \mathcal{W}'}, if the type \m{f \, x} is a retract of \m{f'
\, x} for any
\m{x:X}, then the type \m{(f \scaleobj{0.7}{\,\uparrow\,} j) \, y} is a
retract of \m{(f' \scaleobj{0.7}{\,\uparrow\,} j) \, y} for any \m{y : Y}.
\end{remark}
\noindent The construction is routine, and presumably can be
performed for left Kan extensions too, but we haven't paused to check
this.
\section{Constructions with algebraically injective types}
Algebraic injectives are closed under retracts:
\begin{lemma}
If a type \m{D} in a universe \m{\mathcal{W}} is algebraically
\m{\mathcal{U},\mathcal{V}}-injective, then so is any retract \m{D' : \mathcal{W}'} of \m{D} in
any universe \m{\mathcal{W}'}.
\end{lemma}
\noindent In particular, any type equivalent to an algebraically injective type
is itself algebraically injective, without the need to invoke univalence.
\begin{proof}
\M{\begin{diagram}[p=0.4em]
X & & \rTo^j & & Y \\
& \rdTo^f\rdTo(2,4)_{s \mathrel{\circ} f} & & \ldEto^{f \mid j}\ldTo(2,4)_{(s \mathrel{\circ} f) \mid j} & \\
& & D' & & \\
& & \dTo^s \uTo_r & & \\
& & D.
\end{diagram}}
\noindent For a given section-retraction pair \m{(s,r)}, the construction of the
extension operator for \m{D'} from that of \m{D} is given by \m{f \mid
j \overset{\text{def}}{=} r \mathrel{\circ} ((s \mathrel{\circ} f) \mid j)}.
\end{proof}
\begin{lemma}
The product of any family \m{D_a} of algebraically
\m{\mathcal{U},\mathcal{V}}-injective types in a universe \m{\mathcal{W}}, with indices \m{a} in a type
\m{A} of any universe \m{\mathcal{T}}, is itself algebraically \m{\mathcal{U},\mathcal{V}}-injective.
\end{lemma}
\noindent In particular, if a type \m{D} in a universe \m{\mathcal{W}} is algebraically
\m{\mathcal{U},\mathcal{V}}-injective, then so is any exponential power \m{A \to D : \mathcal{T} \sqcup \mathcal{W}} for
any type \m{A} in any universe \m{\mathcal{T}}.
\begin{proof}
We construct the extension operator \m{(-)\mid(-)} of the product
\m{\Pi D : \mathcal{T} \sqcup \mathcal{W}} in a pointwise fashion from the extension
operators \m{(-)\mid_a(-)} of the algebraically injective types
\m{D_a}: For \m{f : X \to \Pi D}, we let \m{f \mid
j : Y \to \Pi D} be
\M{
(f \mid j) \, y \overset{\text{def}}{=} a \mapsto ((x \mapsto f \, x \, a) \mid_a j) \, y.
}
\end{proof}
\begin{lemma}
Every algebraically \m{\mathcal{U},\mathcal{V}}-injective type \m{D:\mathcal{W}} is a retract
of any type \m{Y:\mathcal{V}} in which it is embedded into.
\end{lemma}
\begin{proof}
\M{\begin{diagram}
D & & \rInto^j & & Y \\
& \rdTo_\operatorname{id} & & \ldEto_{r \overset{\text{def}}{=} \operatorname{id} \mid j} & \\
& & D. & &
\end{diagram}}
\noindent
We just extend the identity function along the embedding to get the desired retraction~\m{r}.
\end{proof}
The following is a sort of \m{\infty}-Yoneda embedding:
\begin{lemma}
The identity type former \m{\operatorname{Id}_X} of any type \m{X:\mathcal{U}} is an embedding of the type~\m{X} into the type~\m{X \to \mathcal{U}}.
\end{lemma}
\begin{proof}
To show that the \m{\operatorname{Id}}-fiber of a given \m{A : X \to \mathcal{U}} is a
subsingleton, it suffices to show that if is pointed then it is
a singleton. So let \m{(x,p):\Sigma (x : X), \operatorname{Id} x = A} be a point
of the fiber. Applying \m{\Sigma}, seen as a map of type \m{(X \to
\mathcal{U}) \to \mathcal{U}}, to the identification~\m{p : \operatorname{Id} \, x = A}, we get an
identification
%
\M {
\operatorname{ap} \, \Sigma \, p : \Sigma (\operatorname{Id} x) = \Sigma A,
}
%
and hence, being equal to the singleton type \m{\Sigma (\operatorname{Id} x)},
the type \m{\Sigma A} is itself a singleton. Hence we have
%
\M{\begin{array}{llll}
A \, x & \simeq & \operatorname{Id} x \mathrel{\,\,\preceq\,\,} A & \text{By the Yoneda Lemma~\cite{rijke:msc},} \\
& = & \Pi (y : X), \operatorname{Id} \, x \, y \to A \, y & \text{by definition of \m{\mathrel{\,\,\preceq\,\,}},} \\
& \simeq & \Pi (y : X), \operatorname{Id} \, x \, y \simeq A \, y & \text{because \m{\Sigma A} is a singleton (Yoneda corollary),} \\
& \simeq & \Pi (y : X), \operatorname{Id} \, x \, y = A \, y & \text{by univalence,} \\
& \simeq & \operatorname{Id} \, x = A & \text{by function extensionality.}
\end{array}
}
%
So by a second application of univalence we get \m{A \, x = (\operatorname{Id} \, x
= A)}. Hence, applying \m{\Sigma} on both sides, we get
\m{\Sigma A = (\Sigma (x : X), \operatorname{Id} \, x = A)}. Therefore, because
the type \m{\Sigma A} is a singleton, so is the fiber \m{\Sigma (x
: X), \operatorname{Id} \, x = A} of~\m{A}.
\end{proof}
\begin{lemma} \label{ref:16:3}
If a type \m{D} in a universe \m{\mathcal{U}} is algebraically \m{\mathcal{U},\mathcal{U}^+}-injective, then \m{D} is a retract of the exponential power \m{D \to \mathcal{U}} of \m{\mathcal{U}}.
\end{lemma}
\begin{proof}
\M{\begin{diagram}
D & & \rInto^\operatorname{Id} & & (D \to \mathcal{U}) \\
& \rdTo_\operatorname{id} & & \ldEto_{r \overset{\text{def}}{=} \operatorname{id} \mid \operatorname{Id}} & \\
& & D. & &
\end{diagram}}
\noindent This is obtained by combining the previous two
constructions, using the fact that \m{D \to \mathcal{U}} lives in the successor
universe \m{\mathcal{U}^+}.
\end{proof}
\section{Algebraic flabbiness and resizing constructions}
We now discuss resizing constructions that don't assume resizing
axioms. The above results, when combined together in the obvious way,
almost give directly that the algebraically injective types are
precisely the retracts of exponential powers of universes, but there
is a universe mismatch. Keeping track of the universes to avoid the
mismatch, what we get instead is a resizing construction without the
need for resizing axioms:
\begin{lemma}
Algebraically \m{\mathcal{U},\mathcal{U}^+}-injective types \m{D:\mathcal{U}} are
algebraically \m{\mathcal{U},\mathcal{U}}-injective too.
\end{lemma}
\begin{proof}
By the above constructions, we
first get that \m{D}, being algebraically \m{\mathcal{U},\mathcal{U}^+}-injective, is a
retract of \m{D \to \mathcal{U}}. But then \m{\mathcal{U}} is algebraically
\m{\mathcal{U},\mathcal{U}}-injective, and, being a power of \m{\mathcal{U}}, so is \m{D \to \mathcal{U}}.
Finally, being a retract of \m{D \to \mathcal{U}}, we have that \m{D} is
algebraically \m{\mathcal{U},\mathcal{U}}-injective.
\end{proof}
This is resizing down and so is not surprising.
Of course, such a construction can be performed directly by considering
an embedding \m{\mathcal{U} \to \mathcal{U}^+}, but the idea is to generalize it to
obtain further resizing-for-free constructions, and, later,
resizing-for-a-price constructions. We achieve this by considering a
notion of flabbiness as data, rather than as property as in the
1-topos literature (see e.g.\
Blechschmidt~\cite{Blechschmidt:2018}). The notion of flabbiness
considered in topos theory is defined with truncated \m{\Sigma}, that
is, the existential quantifier \m{\exists} with values in the
subobject classifier \m{\Omega}. We refer to the notion defined with
untruncated \m{\Sigma} as algebraic flabbiness.
\begin{definition}
We say that a type \m{D : \mathcal{W}} is \df{algebraically \m{\mathcal{U}}-flabby} if
\M{
\Pi (P : \mathcal{U}), \text{if \m{P} is a subsingleton then \m{\Pi(f : P \to D)\, \Sigma (d : D)\, \Pi(p : P), d = f \, p}.}
}
\end{definition}
\noindent This terminology is more than a mere analogy with
algebraic injectivity: notice that flabbiness and algebraic flabbiness
amount to simply injectivity and algebraic injectivity with respect to
the class of embeddings \m{P \to \mathbb{1}} with \m{P} ranging over
subsingletons:
\begin{diagram}
P & & \rInto & & \mathbb{1} \\
& \rdTo_f & & \ldEto & \\
& & D. & &
\end{diagram}
Notice also that an algebraically flabby type \m{D} is pointed, by
considering the case when \m{f} is the unique map \m{\mathbb{0} \to D}.
\begin{lemma} \label{for27:1}
If a type \m{D} in the universe \m{\mathcal{W}} is algebraically
\m{\mathcal{U},\mathcal{V}}-injective, then it is algebraically \m{\mathcal{U}}-flabby.
\end{lemma}
\begin{proof}
Given a subsingleton \m{P:\mathcal{U}} and a map \m{f : P \to D}, we can take its extension \m{f \mid \operatorname{!}: \mathbb{1} \to D} along the unique map \m{!:P \to \mathbb{1}}, because it is an embedding, and then we let \m{d \overset{\text{def}}{=} (f \mid \operatorname{!})\, \operatorname{\star}}, and the extension property gives \m{d = f \, p} for any \m{p:P}.
\end{proof}
The interesting thing about this is that the
universe~\m{\mathcal{V}} is forgotten, and then we can put any other universe
below \m{\mathcal{U}} back, as follows.
\begin{lemma} \label{for27:2}
If a type \m{D} in the universe \m{\mathcal{W}} is algebraically \m{\mathcal{U} \sqcup
\mathcal{V}}-flabby, then it is also algebraically \m{\mathcal{U},\mathcal{V}}-injective.
\end{lemma}
\begin{proof}
Given an embedding \m{j : X \to Y} of types \m{X:\mathcal{U}} and \m{\mathcal{V}}, a map \m{f : X \to D} and a
point \m{y:Y}, in order to construct \m{(f \mid j) \, y} we consider
the map \m{f_y : j^{-1}(y) \to D} defined by \m{(x,p) \mapsto
f\,x}. Because the fiber \m{j^{-1}(y) : \mathcal{U} \sqcup \mathcal{V}} is a subsingleton as \m{j} is an
embedding, we can apply algebraic flabbiness to get \m{d_y : D} with
\m{d_y = f_y (x,p)} for all \m{(x,p):j^{-1}(y)}. By the construction of
\m{f_y} and the definition of fiber, this amounts to saying that for
any \m{x : X} and \m{p : j \, x = y}, we have \m{d_y = f \,
x}. Therefore we can take
\M{(f \mid j) \, y \overset{\text{def}}{=} d_y,}
because we then have
\M{(f \mid j) (j \, x) = d_{j \, x} = f_{j \, x} (x , \operatorname{refl}_{j \, x}) = f \, x}
for any \m{x:X}, as required.
\end{proof}
\noindent We then get the following resizing construction by composing the above
two conversions between algebraic flabbiness and injectivity:
\begin{lemma}
If a type \m{D} in the universe \m{\mathcal{W}} is algebraically \m{(\mathcal{U} \sqcup
\mathcal{T}),\mathcal{V}}-injective, then it is also algebraically \m{\mathcal{U},\mathcal{T}}-injective.
\end{lemma}
\noindent In particular, algebraic \m{\mathcal{U},\mathcal{V}}-injectivity gives
algebraic \m{\mathcal{U},\mathcal{U}}- and \m{\mathcal{U}_0,\mathcal{U}}-injectivity. So this is no
longer necessarily resizing down, by taking \m{\mathcal{V}} to be
e.g.\ the first universe~\m{\mathcal{U}_0}.
\section{Injectivity of subuniverses}
We now apply algebraic flabbiness to show that any subuniverse closed
under subsingletons and under sums, or alternatively under products,
is also algebraically injective.
\begin{definition}
By a \df{subuniverse} of \m{\mathcal{U}} we mean a projection \m{\Sigma \, A
\to \mathcal{U}} with \m{A : \mathcal{U} \to \mathcal{T}} subsingleton-valued and the
universe \m{\mathcal{T}} arbitrary. By a customary abuse of language, we also
sometimes refer to the domain of the projection as the
subuniverse. Closure under subsingletons means that \m{A\,P} holds
for any subsingleton \m{P:\mathcal{U}}. Closure under sums amounts to
saying that if \m{X:\mathcal{U}} satisfies \m{A} and every \m{Y \, x}
satisfies \m{A} for a family \m{Y : X \to \mathcal{U}}, then so does
\m{\Sigma \, Y}. Closure under products is defined in the same way
with \m{\Pi} in place of \m{\Sigma}.
\end{definition}
\noindent Notice that \m{A} being subsingleton-valued is
precisely what is needed for the projection to be an embedding, and
that all embeddings are of this form up to equivalence (more
precisely, every embedding of any two types is the composition of an
equivalence into a sum type followed by the first projection).
\begin{lemma}
Any subuniverse of \m{\mathcal{U}} which is closed
under subsingletons and sums, or alternatively under subsingletons and
products, is algebraically \m{\mathcal{U}}-flabby and hence
algebraically \m{\mathcal{U},\mathcal{U}}-injective.
\end{lemma}
\begin{proof}
Let \m{\Sigma\,A} be a subuniverse of \m{\mathcal{U}}, let \m{P:\mathcal{U}} be a
subsingleton and \m{f : P \to \Sigma \, A} be given. Then define
\begin{quote}
(1)~\m{ X \overset{\text{def}}{=} \Sigma (\operatorname{pr}_1 \mathrel{\circ} f)} \qquad or \qquad (2)~\m{X \overset{\text{def}}{=} \Pi (\operatorname{pr}_1 \mathrel{\circ} f)}
\end{quote}
according to whether we have closure under sums or products. Because
\m{P}, being a subsingleton satisfies \m{A} and because the values
of the map \m{\operatorname{pr}_1 \mathrel{\circ} f : P \to \mathcal{U}} satisfy \m{A} by definition
of subuniverse, we have \m{a : A\, X} by the sum or product closure
property, and \m{d \overset{\text{def}}{=} (X,a)} has type \m{\Sigma \,A}. To
conclude the proof, we need to show that \m{d = f\,p} for any
\m{p:P}. Because the second component \m{a} lives in a subsingleton
by definition of subuniverse, it suffices to show that the first
components are equal, that is, that \m{X = \operatorname{pr}_1 (f p)}. But this
follows by univalence, because a sum indexed by a subsingleton is
equivalent to any of summands, and a product indexed by a
subsingleton is equivalent to any of its factors.
\end{proof}
We index \m{n}-types from \m{n=-2} as in the HoTT book, where the
\m{-2}-types are the singletons. We have the following as a corollary.
\begin{theorem}
The subuniverse of \m{n}-types in a universe \m{\mathcal{U}} is algebraically
\m{\mathcal{U}}-flabby, in at least two ways, and hence algebraically
\m{\mathcal{U},\mathcal{U}}-injective.
\end{theorem}
\begin{proof}
We have a subuniverse because the notion of being an \m{n}-type is a
proposition. For \m{n=-2}, the subuniverse of singletons is itself a
singleton, and hence trivially injective. For \m{n>-2}, the
\m{n}-types are known to be closed under subsingletons and both sums
and products.
\end{proof}
\noindent In particular:
\begin{enumerate}
\item The type \m{\Omega_\mathcal{U}} of subsingletons in a universe \m{\mathcal{U}} is
algebraically \m{\mathcal{U},\mathcal{U}}-injective.
(Another way to see that \m{\Omega_\mathcal{U}} is algebraically injective is
that it is a retract of the universe by propositional
truncation. The same would be the case for \m{n}-types if we were
assuming \m{n}-truncations, which we are not.)
\item Powersets, being exponential powers of \m{\Omega_\mathcal{U}}, are
algebraically \m{\mathcal{U},\mathcal{U}}-injective.
\end{enumerate}
An anonymous referee suggested the following additional examples: (i)
The subuniverse of subfinite types, i.e., subtypes of types for which
there is an uunspecified equivalence with \m{\operatorname{Fin}(n)}
for some~\m{n}. This subuniverse is closed under both \m{\Pi} and
\m{\Sigma}. (ii) Reflective subuniverses, as they are closed under
\m{\Pi}. (iii) Any universe \m{\mathcal{U}} seen as a subuniverse of \m{\mathcal{U}
\sqcup \mathcal{V}}.
\section{Algebraic flabbiness with resizing axioms}
Returning to size issues, we now apply algebraic flabbiness to show
that propositional resizing gives unrestricted algebraic injective
resizing.
\begin{definition} \label{resizing}
The propositional resizing principle, from \m{\mathcal{U}} to \m{\mathcal{V}}, that we
consider here says that every proposition in the universe \m{\mathcal{U}} has
an equivalent copy in the universe~\m{\mathcal{V}}. By propositional resizing without
qualification, we mean propositional resizing between any of the
universes involved in the discussion.
\end{definition}
This is consistent because
it is implied by excluded middle, but, as far as we are aware, there
is no known computational interpretation of this axiom. A model in
which excluded middle fails but propositional resizing holds is given
by Shulman~\cite{MR3340541}.
We begin with the following construction, which says that algebraic
flabbiness is universe independent in the presence of propositional
resizing:
\begin{lemma}
If propositional resizing holds, then the algebraic \m{\mathcal{V}}-flabbiness
of a type in any universe gives its algebraic \m{\mathcal{U}}-flabbiness.
\end{lemma}
\begin{proof}
Let \m{D:\mathcal{W}} be a type in any universe \m{\mathcal{W}}, let \m{P : \mathcal{U}} be a
proposition and \m{f : P \to D}. By resizing, we have an equivalence
\m{\beta : Q \to P} for a suitable proposition \m{Q:\mathcal{V}}. Then the
algebraic \m{\mathcal{V}}-flabbiness of \m{D} gives a point \m{d:D} with \m{d
= (f \mathrel{\circ} \beta) \, q} for all \m{q : Q}, and hence with \m{d = f
\, p} for all \m{p : P}, because we have \m{p=\beta \, q} for \m{q
= \alpha \, p} where \m{\alpha} is a quasi-inverse of \m{\beta},
which establishes the algebraic \m{\mathcal{U}}-flabbiness of~\m{D}.
\end{proof}
And from this it follows that algebraic injectivity is also universe
independent in the presence of propositional resizing: we convert
back-and-forth between algebraic injectivity and algebraic flabbiness.
\begin{lemma} \label{universe:independence}
If propositional resizing holds, then for any type \m{D} in any universe
\m{\mathcal{W}}, the algebraic \m{\mathcal{U},\mathcal{V}}-injectivity of \m{D} gives its
algebraic \m{\mathcal{U}',\mathcal{V}'}-injectivity.
\end{lemma}
\begin{proof}
We first get the \m{\mathcal{U}}-flabbiness of \m{D} by~\ref{for27:1}, and then its
\m{\mathcal{U}' \sqcup \mathcal{V}'}-flabbiness by~\ref{universe:independence}, and finally its algebraic
\m{\mathcal{U}',\mathcal{V}'}-injectivity by~\ref{for27:2}.
\end{proof}
As an application of this and of the algebraic injectivity of
universes, we get that any universe is a retract of any larger
universe. We remark that for types that are not sets, sections are
not automatically embeddings~\cite{MR3548859}. But we can choose the
retraction so that the section is an embedding in our situation.
\begin{lemma} \label{canonical}
We have an embedding of any universe \m{\mathcal{U}} into any larger universe \m{\mathcal{U} \sqcup \mathcal{V}}.
\end{lemma}
\begin{proof}
For example, we have the embedding given by \m{X \mapsto X +
\mathbb{0}_\mathcal{V}}. We don't consider an argument that this is indeed an
embedding to be entirely
routine without a significant amount of experience in univalent
mathematics, even if this may seem obvious. Nevertheless, it is certainly
safe to leave it as a challenge to the reader, and a proof can be
found in~\cite{injective:article} in case of doubt.
\end{proof}
\begin{theorem}
If propositional resizing holds, then any universe \m{\mathcal{U}} is a
retract of any larger universe \m{\mathcal{U} \sqcup \mathcal{V}} with a section that
is an embedding.
\end{theorem}
\begin{proof}
The universe \m{\mathcal{U}} is algebraically \m{\mathcal{U},\mathcal{U}}-injective
by~\ref{ref:16:1}, and hence it is algebraically \m{\mathcal{U}^+,(\mathcal{U} \sqcup
\mathcal{V})^+}-injective by~\ref{universe:independence}, which has the
right universe assignments to apply the construction~\ref{ref:16:3}
that gives a retraction from an embedding of an injective type into
a larger type, in this case the embedding of the universe \m{\mathcal{U}}
into the larger universe \m{\mathcal{U} \sqcup \mathcal{V}} constructed
in~\ref{canonical}.
\end{proof}
As mentioned above, we almost have that the algebraically injective
types are precisely the retracts of exponential powers of universes,
up to a universe mismatch. This mismatch is side-stepped by
propositional resizing. The following is one of the main results of
this paper:
\begin{theorem} \df{(First characterization of algebraic
injectives.)} If propositional resizing holds, then a type \m{D}
in a universe \m{\mathcal{U}} is algebraically \m{\mathcal{U},\mathcal{U}}-injective if and
only if \m{D} is a retract of an exponential power of \m{\mathcal{U}} with
exponent in \m{\mathcal{U}}.
\end{theorem}
\noindent We emphasize that this is a logical equivalence ``if and
only if'' rather than an \m{\infty}-groupoid equivalence
``\m{\simeq}''. More precisely, the theorem gives two constructions in
opposite directions. So this characterizes the types that \df{can} be
equipped with algebraic-injective structure.
\begin{proof}
\m{(\Rightarrow)}: Because \m{D} is algebraically
\m{\mathcal{U},\mathcal{U}}-injective, it is algebraically \m{\mathcal{U},\mathcal{U}^+}-injective by
resizing, and hence it is a retract of \m{D \to \mathcal{U}} because it is
embedded into it by the identity type former, by taking the
extension of the identity function along this embedding.
\m{(\Leftarrow)}: If \m{D} is a retract of \m{X \to \mathcal{U}} for some
given \m{X:\mathcal{U}}, then, because \m{X \to \mathcal{U}}, being an exponential
power of the algebraically \m{\mathcal{U} ,\mathcal{U}}-injective type \m{\mathcal{U}}, is
algebraically \m{\mathcal{U},\mathcal{U}}-injective, and hence so is \m{D} because it
is a retract of this power.
\end{proof}
We also have that any algebraically injective \m{(n+1)}-type is a retract
of an exponential power of the universe of \m{n}-types. We establish something
more general first.
\begin{lemma}
Under propositional resizing, for any subuniverse \m{\Sigma \, A} of
a universe \m{\mathcal{U}} closed under subsingletons, we have that any
algebraically \m{\mathcal{U},\mathcal{U}}-injective type \m{X:\mathcal{U}} whose identity types
\m{x=_X x'} all satisfy the property \m{A} is a retract of
the type \m{X \to \Sigma \, A}.
\end{lemma}
\begin{proof}
Because the first projection \m{j : \Sigma \, A \to \mathcal{U}} is an
embedding by the assumption, so is the map \m{k \overset{\text{def}}{=} j \mathrel{\circ} (-) :
(X \to \Sigma A) \to (X \to \mathcal{U})} by a general property of
embeddings. Now consider the map \m{l : X \to (X \to \Sigma \, A)}
defined by \m{x \mapsto (x' \mapsto (x=x', p \, x \, x'))}, where
\m{p \, x \, x' : A(x=x')} is given by the assumption. We have that
\m{k \mathrel{\circ} l = \operatorname{Id}_X} by construction. Hence \m{l} is an embedding
because \m{l} and \m{\operatorname{Id}_X} are, where we are using the general fact
that if \m{g \mathrel{\circ} f} and \m{g} are embeddings then so is the
factor~\m{f}. But \m{X}, being algebraically \m{\mathcal{U},\mathcal{U}}-injective by
assumption, is algebraically \m{\mathcal{U},(\mathcal{U}^+ \sqcup \mathcal{T})}-injective by
resizing, and hence so is the exponential power \m{X \to \Sigma \,
A}, and therefore we get the desired retraction by extending its
identity map along~\m{l}.
\end{proof}
Using this, we get the following as an immediate consequence.
\begin{theorem} \df{(Characterization of algebraic injective \m{(n+1)}-types.)} If propositional resizing holds, then an
\m{(n+1)}-type \m{D} in \m{\mathcal{U}} is algebraically \m{\mathcal{U},\mathcal{U}}-injective
if and only if \m{D} is a retract of an exponential power of the
universe of \m{n}-types in \m{\mathcal{U}}, with exponent in \m{\mathcal{U}}.
\end{theorem}
\begin{corollary}
The algebraically injective sets in \m{\mathcal{U}} are the retracts of
powersets of (arbitrary) types in \m{\mathcal{U}}, assuming propositional resizing.
\end{corollary}
\noindent Notice that the powerset of any type is a set, because
\m{\Omega_\mathcal{U}} is a set and because sets (and more generally
\m{n}-types) form an exponential ideal.
\section{Injectivity in terms of algebraic injectivity in the absence of resizing}
We now compare injectivity with algebraic injectivity. The following observation follows from the fact that retractions are surjections:
\begin{lemma}
If a type \m{D} in a universe \m{\mathcal{W}} is algebraically
\m{\mathcal{U},\mathcal{V}}-injective, then it is \m{\mathcal{U},\mathcal{V}}-injective
\end{lemma}
\noindent The following observation follows from the fact that propositions are
closed under products.
\begin{lemma}
Injectivity is a proposition.
\end{lemma}
\noindent But of course algebraic injectivity is not. From this we immediately
get the following by the universal property of propositional
truncation:
\begin{lemma}
For any type \m{D} in a universe \m{\mathcal{W}}, the truncation of the
algebraic \m{\mathcal{U},\mathcal{V}}-injectivity of \m{D} gives its
\m{\mathcal{U},\mathcal{V}}-injectivity.
\end{lemma}
In order to relate injectivity to the propositional truncation of
algebraic injectivity in the other direction, we first establish some
facts about injectivity that we already proved for algebraic
injectivity. These facts cannot be obtained by reduction (in
particular products of injectives are not necessarily injective, in
the absence of choice, but exponential powers are).
\begin{lemma} \label{embedding-||retract||}
Any \m{\mathcal{W},\mathcal{V}}-injective type \m{D} in a universe \m{\mathcal{W}} is a retract
of any type in \m{\mathcal{V}} it is embedded into, in an unspecified way.
\end{lemma}
\begin{proof}
Given \m{Y:\mathcal{V}} with an embedding \m{j : D \to Y}, by the
\m{\mathcal{W},\mathcal{V}}-injectivity of \m{D} there is an \df{unspecified} \m{r : Y
\to D} with \m{r \mathrel{\circ} j \sim \operatorname{id}}. Now, if there is a
\df{specified} \m{r : Y \to D} with \m{r \mathrel{\circ} j \sim \operatorname{id}} then
there is a specified retraction. Therefore, by the functoriality of
propositional truncation on objects applied to the previous
statement, there is an unspecified retraction.
\end{proof}
\begin{lemma}
If a type \m{D' : \mathcal{U}'} is a retract of a type \m{D : \mathcal{U}} then the \m{\mathcal{W},\mathcal{T}}-injectivity of
\m{D} implies that of \m{D'}.
\end{lemma}
\begin{proof}
Let \m{r : D \to D'} and \m{s : D' \to D} be the given section
retraction pair, and, to show that \m{D'} is \m{\mathcal{W},\mathcal{T}}-injective,
let an embedding \m{j : X \to Y} and a function \m{f : X \to D'} be
given. By the injectivity of \m{D}, we have some unspecified
extension \m{f' : Y \to D} of \m{s \mathrel{\circ} f : X \to D}. If such a
designated extension is given, then we get the designated extension
\m{r \mathrel{\circ} f'} of \m{f}. By the functoriality of propositional
truncation on objects and the previous two statements, we get the
required, unspecified extension.
\end{proof}
The universe assignments in the following are probably not very
friendly, but we are aiming for maximum generality.
\begin{lemma}
If a type \m{D : \mathcal{W}} is \m{(\mathcal{U} \sqcup \mathcal{T}),(\mathcal{V} \sqcup \mathcal{T})}-injective,
then the exponential power \m{A \to D} is \m{\mathcal{U},\mathcal{V}}-injective for any \m{A:\mathcal{T}}.
\end{lemma}
\begin{proof}
For a given embedding \m{j : X \to Y} and a given map \m{f : X \to
(A \to D)}, take the exponential transpose \m{g : X \times A \to
D} of \m{f}, then extend it along the embedding \m{j \times \operatorname{id} :
X \times A \to Y \times A} to get \m{g' : Y \times A \to D} and
then back-transpose to get \m{f' : Y \to (A \to D)}, and check that
this construction of \m{f'} does give an extension of \m{f} along
\m{j}. For this, we need to know that if \m{j} is an embedding then
so is \m{j \times \operatorname{id}}, but this is not hard to check. The result
then follows by the functoriality-on-objects of the propositional
truncation.
\end{proof}
\begin{lemma}
If a type \m{D:\mathcal{U}} is \m{\mathcal{U},\mathcal{U}^+} injective, then it is a retract of
\m{D \to \mathcal{U}} in an unspecified way.
\end{lemma}
\begin{proof}
This is an immediate consequence of~\ref{embedding-||retract||} and the fact that the
identity type former \m{\operatorname{Id}_X : X \to (X \to \mathcal{U})} is an embedding.
\end{proof}
With this we get an almost converse to the fact that truncated
algebraic injectivity implies injectivity: the universe levels are
different in the converse:
\begin{lemma}
If a type \m{D:\mathcal{U}} is \m{\mathcal{U},\mathcal{U}^+}-injective, then it is algebraically \m{\mathcal{U},\mathcal{U}^+}-injective in an unspecified way.
\end{lemma}
So, in summary, regarding the relationship between injectivity and
truncated algebraic injectivity, so far we know that
\begin{quote}
if \m{D} is algebraically \m{\mathcal{U},\mathcal{V}}-injective in an unspecified way
then it is \m{\mathcal{U},\mathcal{V}}-injective,
\end{quote}
and, not quite conversely,
\begin{quote}
if \m{D} is \m{\mathcal{U},\mathcal{U}^+}-injective then it is algebraically \m{\mathcal{U},\mathcal{U}}-injective in an unspecified way.
\end{quote}
Therefore, using propositional resizing, we get the following
characterization of a particular case of injectivity in terms of
algebraic injectivity.
\begin{proposition} \label{worse} \df{(Injectivity in terms of algebraic injectivity.)}
If propositional resizing holds, then a type \m{D : \mathcal{U}} is \m{\mathcal{U},\mathcal{U}^+}-injective if and only if it is algebraically \m{\mathcal{U},\mathcal{U}^+}-injective in an unspecified way.
\end{proposition}
\noindent We would like to do better than this. For that purpose, we consider
the partial-map classifier in conjunction with flabbiness and resizing.
\section{Algebraic flabbiness via the partial-map classifier}
We begin with a generalization~\cite{MR3695545} of a familiar construction
in \m{1}-topos theory~\cite{MR1173017}.
\begin{definition}
The lifting \m{\mathcal{L}_{\mathcal{T}} \, X : \mathcal{T}^+ \sqcup \mathcal{U}} of a type \m{X:\mathcal{U}}
with respect to a universe \m{\mathcal{T}}
is defined by
\M{ \mathcal{L}_{\mathcal{T}}\, X \overset{\text{def}}{=} \Sigma (P : \mathcal{T}), (P \to X) \times
\text{\m{P} is a subsingleton}. }
\end{definition}
When the universes \m{\mathcal{T}} and \m{\mathcal{U}} are the same and the last
component of the triple is omitted, we have the familiar
canonical correspondence
\M{
(X \to \mathcal{T}) \simeq (\Sigma (P : \mathcal{T}), P \to X)
}
that maps \m{A : X \to \mathcal{T}} to \m{P \overset{\text{def}}{=} \Sigma \, A} and the
projection \m{\Sigma \, A \to X}. If the universe~\m{\mathcal{U}} is not
necessarily the same as \m{\mathcal{T}}, then the equivalence becomes
\M{
(\Sigma (A : X \to \mathcal{T} \sqcup \mathcal{U}), \Sigma(T : \mathcal{T}), T \simeq \Sigma \, A) \simeq (\Sigma (P : \mathcal{T}), P \to X).
}
This says that although the total space \m{\Sigma \, A} doesn't
live in the universe \m{\mathcal{T}}, it must have a copy in \m{\mathcal{T}}.
What the third component of the triple does is to restrict the above
equivalences to the subtype of those \m{A} whose total spaces \m{\Sigma
\, A} are subsingletons. If we define the type of partial maps by
\M{(X \rightharpoonup Y) \overset{\text{def}}{=} \Sigma (A : \mathcal{T}), (A \hookrightarrow X) \times (A \to Y),
}
where \m{A \hookrightarrow X} is the type of embeddings, then for any \m{X,Y :
\mathcal{T}}, we have an equivalence
\M{
(X \rightharpoonup Y)
\simeq (X \to \mathcal{L}_{\mathcal{T}} \, Y),
}
so that \m{\mathcal{L}_{\mathcal{T}}} is the partial-map classifier for the universe
\m{\mathcal{T}}. When the universe~\m{\mathcal{U}} is not necessarily the same
as~\m{\mathcal{T}}, the lifting classifies partial maps in~\m{\mathcal{U}} whose
embeddings have fibers with copies in~\m{\mathcal{T}}.
This is a sort of an \m{\infty}-monad ``across
universes''~\cite{TypeTopology}, and modulo providing coherence data,
which we haven't done at the time of writing, but which is not needed
for our purposes. We could call this a ``wild monad'', but we will
refer to it as simply a monad with this warning.
In order to discuss the lifting in more detail, we first characterize
its equality types. We denote the projections from \m{\mathcal{L}_{\mathcal{T}}
\, X} by
\M{
\begin{array}{llll}
\delta (P , \phi , i) & \overset{\text{def}}{=} & P & \text{(domain of definition),} \\
\upsilon (P , \phi , i) & \overset{\text{def}}{=} & \phi & \text{(value function),} \\
\sigma(P , \phi , i) & \overset{\text{def}}{=} & i & \text{(subsingleton-hood of the domain of definition).}
\end{array}
}
For \m{l , m : \mathcal{L}_{\mathcal{T}} \, X}, define
\M{
(l \backsimeq m) \overset{\text{def}}{=} \Sigma (e : \delta \, l \simeq \delta \, m), \upsilon \, l = \upsilon \, m \mathrel{\circ} e,
}
as indicated in the commuting triangle
\M{\begin{diagram}[p=0.4em]
\delta l & & \rTo^e & & \delta m \\
& \rdTo_{v l} & & \ldTo_{v m} & \\
& & X & &
\end{diagram}}
\begin{lemma}
The canonical transformation \m{(l = m) \to (l \backsimeq m)} that
sends \m{\operatorname{refl}_l} to the identity equivalence paired with \m{\operatorname{refl}_{\upsilon \,
l}} is an equivalence.
\end{lemma}
The unit \m{\eta : X
\to \mathcal{L}_\mathcal{T} X} is given by \M{\eta_X \, x = (\mathbb{1}, (p \mapsto x), i)}
where \m{i} is a proof that \m{\mathbb{1}} is a proposition.
\begin{lemma}
The unit \m{\eta_X : X\to\mathcal{L}_\mathcal{T} X} is an embedding.
\end{lemma}
\begin{proof}
This is easily proved using the above characterization of equality.
\end{proof}
\begin{lemma}
The unit satisfies the unit equations for a monad.
\end{lemma}
\begin{proof}
Using the above characterization of equality, the left and right
unit laws amount to the fact that the type \m{\mathbb{1}} is the left
and right unit for the operation \m{(-)\times(-)} on types.
\end{proof}
\noindent Next, \m{\mathcal{L}_\mathcal{T}} is functorial by mapping a function
\m{f : X \to Y} to the function \m{\mathcal{L}_\mathcal{T} f : \mathcal{L}_\mathcal{T} X \to \mathcal{L}_\mathcal{T} Y}
defined by
\M{
\mathcal{L}_\mathcal{T} f (P , \phi , i) = (P , f \mathrel{\circ} \phi , i).
}
This commutes with identities and composition definitionally.
We define the multiplication \m{\mu_X : \mathcal{L}_{\mathcal{T}} (\mathcal{L}_{\mathcal{T}}\, X) \to \mathcal{L}_{\mathcal{T}}\, X} by
\M{
\begin{array}{lll}
\delta (\mu (P , \phi , i)) & \overset{\text{def}}{=} & \Sigma (p : P), \delta (\phi \, p), \\
\upsilon (\mu (P , \phi , i)) & \overset{\text{def}}{=} & (p , q) \mapsto \upsilon (\phi \, p) \, q , \\
\sigma (\mu (P , \phi , i)) & \overset{\text{def}}{=} & \text{because subsingletons are closed under sums.} \\
\end{array}
}
\begin{lemma}
The multiplication satisfies the associativity equation for a monad.
\end{lemma}
\begin{proof}
Using the above characterization of equality, we see that this
amounts to the associativity of \m{\Sigma}, which says that for
\m{P:\mathcal{T}}, \m{Q: X \to \mathcal{T}}, \m{R : \Sigma \, Q \to \mathcal{T}} we have
\m{(\Sigma (t : \Sigma \, Q), R \, t) \simeq (\Sigma (p : P)\,
\Sigma (q : Q \, p), R(p,q))}.
\end{proof}
\noindent The naturality conditions for the unit and multiplication are
even easier to check, and we omit the verification. We now turn to
algebras. We omit the direct verification of the following.
\begin{lemma} Let \m{X:\mathcal{U}} be any type.
\begin{enumerate}
\item A function \m{\alpha : \mathcal{L}_\mathcal{T} X \to X}, that is, a functor
algebra, amounts to a family of functions \m{\bigsqcup_P : (P \to
X) \to X} with \m{P : \mathcal{T}} ranging over subsingletons.
\medskip We will write \m{\bigsqcup_P \phi} as \m{\bigsqcup_{p : P} \, \phi \, p}.
\item The unit law for monad algebras amounts to, for any \m{x:X},
%
\M{
\bigsqcup_{p : \mathbb{1}} x = x,
}
%
which is equivalent to, for all subsingletons \m{P}, functions \m{\phi : P \to X} and points \m{p_0 : P},
%
\M{
\bigsqcup_{p : P} \phi \, p = \phi \, p_0.
}
%
\medskip Therefore a functor algebra satisfying the unit law
amounts to the same thing as algebraic flabbiness data. In other
words, the algebraically \m{\mathcal{T}}-flabby types are the algebras of
the pointed functor \m{(\mathcal{L}_\mathcal{T},\eta)}. In particular,
monad algebras are algebraically flabby.
\item The associativity law for monad algebras amounts to, for any subsingleton \m{P :
\mathcal{T}} and family \m{Q : P \to \mathcal{T}} of subsingletons, and any \m{\phi : \Sigma \, Q \to X},
%
\M{
\bigsqcup_{t : \Sigma Q} \phi \, t = \bigsqcup_{p : P} \bigsqcup_{q : Q \, p} \phi (p ,q).
}
%
\end{enumerate}
\end{lemma}
\noindent So the associativity law for algebras plays no role in
flabbiness. But of course we can have algebraic flabbiness data that
is associative, such as not only the free algebra \m{\mathcal{L}_\mathcal{T} X}, but
also the following two examples that connect to the opening
development of this paper on the injectivity of universes, in
particular the construction~\ref{iterated}:
\begin{lemma} The universe \m{\mathcal{T}} is a monad algebra of
\m{\mathcal{L}_\mathcal{T}} in at least two ways, with \m{\bigsqcup = \Sigma} and
\m{\bigsqcup = \Pi}.
\end{lemma}
We now apply these ideas to injectivity.
\begin{lemma}
Any algebraically \m{\mathcal{T},\mathcal{T}^+}-injective type \m{D:\mathcal{T}} is a retract of \m{\mathcal{L}_\mathcal{T} D}.
\end{lemma}
\begin{proof}
Because the unit is an embedding, and so we can extend the identity of~\m{D} along it.
\end{proof}
\begin{theorem} \df{(Second characterization of algebraic injectives.)}
With propositional resizing, a type \m{D:\mathcal{T}} is algebraically
\m{\mathcal{T},\mathcal{T}}-injective if and only if it is a retract of a monad
algebra of \m{\mathcal{L}_\mathcal{T}}.
\end{theorem}
\begin{proof}
\m{(\Rightarrow)}: Because \m{D} is algebraically
\m{\mathcal{T},\mathcal{T}}-injective, it is algebraically \m{\mathcal{T},\mathcal{T}^+}-injective by
resizing, and hence it is a retract of \m{\mathcal{L}_\mathcal{T} D}.
\m{(\Leftarrow)}: Algebraic injectivity is closed under retracts.
\end{proof}
\begin{definition} \label{omega:resizing}
Now, instead of propositional resizing, we consider the propositional
impredicativity of the universe \m{\mathcal{U}}, which says that the type
\m{\Omega_\mathcal{U}} of propositions in \m{\mathcal{U}}, which lives in the next
universe \m{\mathcal{U}^+}, has an equivalent copy in \m{\mathcal{U}}. We refer to this
kind of impredicativity as \m{\Omega}-resizing.
\end{definition}
It is not hard to see
that propositional resizing implies \m{\Omega}-resizing for all
universes other than the first one~\cite{TypeTopology}, and so all
the assumption of \m{\Omega}-resizing does is to account
for the first universe too.
\begin{lemma}
Under \m{\Omega}-resizing, for any type \m{X:\mathcal{T}}, the type
\m{\mathcal{L}_{\mathcal{T}} X : \mathcal{T}^+} has an equivalent copy in the universe \m{\mathcal{T}}.
\end{lemma}
\begin{proof}
We can take \m{\Sigma (p : \Omega'), \operatorname{pr}_1(\rho \, p) \to X} where \m{\rho : \Omega' \to \Omega_\mathcal{T}} is the given equivalence.
\end{proof}
We apply this lifting machinery to get the following, which doesn't
mention lifting in its formulation.
\begin{theorem} \label{better}
(Characterization of injectivity in terms of algebraic injectivity.)
In the presence of \m{\Omega}-resizing, the
\m{\mathcal{T},\mathcal{T}}-injectivity of a type \m{D} in a universe \m{\mathcal{T}} is
equivalent to the propositional truncation of
its algebraic \m{\mathcal{T},\mathcal{T}}-injectivity.
\end{theorem}
\begin{proof}
We already know that the truncation of algebraic injectivity
(trivially) gives injectivity. For the other direction, let $L$ be
a resized copy of \m{\mathcal{L}_\mathcal{T} D} in the universe \m{\mathcal{T}}. Composing
the unit with the equivalence given by resizing, we get an embedding
\m{D \to L}, because embeddings are closed under composition and
equivalences are embeddings. Hence \m{D} is a retract of \m{L} in
an unspecified way by the injectivity of~\m{D}, by extending its
identity. But \m{L}, being equivalent to a free algebra, is
algebraically injective, and hence, being a retract of \m{L} in an
unspecified way, \m{D} is algebraically injective in an unspecified
way, because retracts of algebraically injectives are algebraically
injective, by the functoriality of truncation on objects.
\end{proof}
As an immediate consequence, by reduction to the above results about algebraic
injectivity, we have the following corollary.
\begin{theorem}
Under \m{\Omega}-resizing and propositional resizing, if a type
\m{D} in a universe \m{\mathcal{T}} is \m{\mathcal{T},\mathcal{T}}-injective , then it is also
\m{\mathcal{U},\mathcal{V}}-injective for any universes \m{\mathcal{U}} and \m{\mathcal{V}}.
\end{theorem}
\begin{proof}
The type \m{D} is algebraically \m{\mathcal{T},\mathcal{T}}-injective in an
unspecified way, and so by functoriality of truncation on objects
and algebraic injective resizing, it is algebraically
\m{\mathcal{U},\mathcal{V}}-injective in an unspecified way, and hence it is
\m{\mathcal{U},\mathcal{V}}-injective.
\end{proof}
At the time of writing, we are not able to establish the converse. In
particular, we don't have the analogue of~\ref{universe:independence}.
\section{The equivalence of excluded middle with the (algebraic) injectivity of all pointed types}
Algebraic flabbiness can also be applied to show that all pointed
types are (algebraically) injective if and only if excluded middle
holds, where for injectivity resizing is needed as an assumption, but
for algebraic injectivity it is not.
The decidability of a type \m{X} is defined to be the assertion \m{X +
(X \to \mathbb{0})}, which says that we can exhibit a point of \m{X} or
else tell that \m{X} is empty.
The principle of excluded middle in univalent mathematics, for the
universe \m{\mathcal{U}}, is taken to mean that all subsingleton types in
\m{\mathcal{U}} are decidable:
\M{
\operatorname{EM}_\mathcal{U} \overset{\text{def}}{=} \Pi (P : \mathcal{U}), \text{\m{P} is a subsingleton \m{\to P + (P \to \mathbb{0})}.}
}
As discussed in the introduction, we are not assuming or rejecting
this principle, which is independent of the other axioms. Notice
that, in the presence of function extensionality, this principle is
a subsingleton, because products of subsingletons are subsingletons and
because \m{P + (P \to \mathbb{0})} is a subsingleton for any
subsingleton \m{P}. So in the following we get data out of a proposition.
\begin{lemma}
If excluded middle holds in the universe \m{\mathcal{U}}, then every pointed
type \m{D} in any universe \m{\mathcal{W}} is algebraically \m{\mathcal{U}}-flabby.
\end{lemma}
\begin{proof}
Let \m{d} be the given point of \m{D} and \m{f : P \to D} be a function with
subsingleton domain. If we have a point \m{p : P}, then we can take
\m{f \, p} as the flabbiness witness. Otherwise, if \m{P \to \mathbb{0}},
we can take \m{d} as the flabbiness witness.
\end{proof}
\noindent
For the converse, we use the following.
\begin{lemma}
If the type \m{P + (P \to \mathbb{0}) + \mathbb{1}} is algebraically
\m{\mathcal{W}}-flabby for a given subsingleton \m{P} in a universe \m{\mathcal{W}},
then \m{P} is decidable.
\end{lemma}
\begin{proof}
Denote by \m{D} the type \m{P + (P \to \mathbb{0}) + \mathbb{1}} and let \m{f :
P + (P \to \mathbb{0}) \to D} be the inclusion. Because \m{P + (P \to
\mathbb{0})} is a subsingleton, the algebraic flabbiness of \m{D} gives
\m{d : D} with \m{d = f \, z} for all \m{z : P + (P \to \mathbb{0})}.
Now, by definition of binary sum, \m{d} must be in one of the three
components of the sum that defines~\m{D}. If it were in the third
component, namely \m{\mathbb{1}}, then \m{P} couldn't hold, because if it
did we would have \m{p:P} and hence, omitting the inclusions into
sums, and considering \m{z=p}, we would have, \m{d = f p = p},
because \m{f} is the inclusion, which is not in the \m{\mathbb{1}}
component. But also \m{P \to \mathbb{0}} couldn't hold, because if it did
we would have \m{\phi:P \to \mathbb{0}} and hence, again omitting the
inclusion, and considering \m{z=\phi}, we would have \m{d = f \,
\phi = \phi}, which again is not in the \m{\mathbb{1}} component. But
it is impossible for both \m{P} and \m{P \to \mathbb{0}} to fail, because
this would mean that we would have functions \m{P \to \mathbb{0}} (the
failure of \m{P}) and \m{(P \to \mathbb{0}) \to \mathbb{0}} (the failure of
\m{P \to \mathbb{0}}), and so we could apply the second function to the
first to get a point of the empty type, which is not
available. Therefore \m{d} can't be in the third component, and so
it must be in the first or the second, which means that \m{P} is
decidable.
\end{proof}
\noindent From this we immediately conclude the following:
\begin{lemma}
If all pointed types in a universe \m{\mathcal{W}} are algebraically \m{\mathcal{W}}-flabby, then excluded middle holds in~\m{\mathcal{W}}.
\end{lemma}
\noindent
And then we have the same situation for algebraically injective types,
by reduction to algebraic flabbiness:
\begin{lemma}
If excluded middle holds in the universe \m{\mathcal{U} \sqcup \mathcal{V}}, then any
pointed type \m{D} in any universe \m{\mathcal{W}} is algebraically
\m{\mathcal{U},\mathcal{V}}-injective.
\end{lemma}
\noindent Putting this together with some universe specializations, we
have the following construction.
\begin{theorem}
All pointed types in a universe \m{\mathcal{U}} are algebraically \m{\mathcal{U},\mathcal{U}}-injective if and only if excluded middle holds in~\m{\mathcal{U}}.
\end{theorem}
\noindent
And we have a similar situation with injective types.
\begin{lemma}
If excluded middle holds, then every inhabited type of any universe is
injective with respect to any two universes.
\end{lemma}
\begin{proof}
Because excluded middle gives algebraic injectivity, which in turn gives
injectivity.
\end{proof}
\noindent Without resizing, we have the following.
\begin{lemma}
If every inhabited type \m{D:\mathcal{W}} is \m{\mathcal{W},\mathcal{W}^+}-injective, then
excluded middle holds in the universe \m{\mathcal{W}}.
\end{lemma}
\begin{proof}
Given a proposition \m{P}, we have that the type \m{D \overset{\text{def}}{=} P + (P
\to \mathbb{0}) + \mathbb{1}_{\mathcal{W}}} is injective by the assumption. Hence it
is algebraically injective in an unspecified way by
Proposition~\ref{worse}. And so it is algebraically flabby in an
unspecified way. By the lemma, \m{P} is decidable in an unspecified
way, but then it is decidable because the decidability of a
proposition is a proposition.
\end{proof}
\noindent With resizing we can do better:
\begin{lemma}
Under \m{\Omega}-resizing, if every inhabited type in a universe \m{\mathcal{U}} is
\m{\mathcal{U},\mathcal{U}}-injective, then excluded middle holds in \m{\mathcal{U}}.
\end{lemma}
\begin{proof}
Given a proposition \m{P}, we have that the type \m{D \overset{\text{def}}{=} P + (P
\to \mathbb{0}) + \mathbb{1}_{\mathcal{U}}} is injective by the assumption. Hence it
is injective in an unspecified way by
Theorem~\ref{better}. And so it is algebraically flabby in an
unspecified way. By the lemma, \m{P} is decidable in an unspecified
way, and hence decidable.
\end{proof}
\begin{theorem}
Under \m{\Omega}-resizing, all inhabited types in a universe \m{\mathcal{U}} are
\m{\mathcal{U},\mathcal{U}}-injective if and only if excluded middles
holds in~\m{\mathcal{U}}.
\end{theorem}
\noindent It would be interesting to get rid of the resizing assumption, which,
as we have seen, is not needed for the equivalence of the algebraic
injectivity of all pointed types with excluded middle.
\bibliographystyle{plain}
|
2,877,628,088,984 | arxiv | \section{Introduction}
Recently a new class of cosmological models based on the string field
theory (SFT)~\cite{review-sft} and the $p$-adic string theory emerges
and attracts a lot of attention \cite{IA1}--\cite{GK}. It is known that
the SFT and the $p$-adic string theory are UV-complete ones. Thus, one
can expect that resulting (effective) models should be free of
pathologies. These models exhibit one general non-standard property,
namely, their actions have terms with infinitely many derivatives, i.e.
nonlocal terms. The higher derivative terms usually produce phantom
fields \cite{Ostrogradski:1850,PaisU} (see also~\cite{AV-NEC}). Models
that includes phantoms violate the null energy condition (NEC), and,
therefore, are unstable. Models with higher derivative terms produce
also well-known problems with quantum instability~\cite{AV-NEC}.
To obtain a stable model with the NEC violation (the state parameter
$w_{\mathrm{DE}}<-1$) one should construct this model as an effective
model, connected with the fundamental theory, which is stable and
admits quantization. With the lack of quantum gravity, we can just
trust string theory or deal with an effective theory admitting the UV
completion.
The purpose of this paper is to study $f(R)$ gravity models with a
nonlocal scalar field. We consider a general form of nonlocal action
for the scalar field with a quadratic potential, keeping the main
ingredient, the analytic function $\mathcal{F}(\Box_g)$, which in fact produces
the nonlocality, almost unrestricted.
\section{Nonlocal gravitation models}
The SFT inspired nonlocal gravitation models~\cite{IA1} are introduced
as a sum of the SFT action of the tachyon field $\phi$ plus the gravity
part of the action. One cannot deduce this form of the action from the
SFT. In this paper we study the $f(R)$ gravity, which is a
straightforward modification of the general relativity. We consider the
following action:
\begin{equation}
S_f=\int d^4x \sqrt{-g}\left(\frac{f(L^2R)}{16\pi
G_NL^2}+\frac{1}{\alpha^{\prime}g_o^2}\left(\frac{1}{2}\phi\,\mathcal{F}\left(\alpha^{\prime}\Box_g\right)\phi
-V(\phi) \right)-\Lambda\right),
\end{equation}
where $f(L^2R)$ is an arbitrary differentiable function. We use the
signature $(-,+,+,+)$, $g_{\mu\nu}$ is the metric tensor, $G_N$ is the
Newtonian constant. The potential $V(\phi)$ is a quadratic polynomial
$V(\phi)=C_2\phi^2+C_1\phi+C_0$, where $C_2$, $C_1$, and $C_0$ are
arbitrary real constants.
The function $\mathcal{F}$ is assumed to be analytic at all finite points of
the complex plane, in other words, to be an entire function. The
function $\mathcal{F}$ can be represented by the convergent series expansion:
$\mathcal{F}(\Box_g)=\sum\limits_{n=0}^{\infty}f_n\Box_g^{\;n}$. The
Weierstrass factorization theorem asserts that the function $\mathcal{F}$ can
be represented by a product involving its zeroes $J_k$:
\begin{equation}
\mathcal{F}(J)=J^me^{Y(J)}\prod_{k=1}^\infty\left(1-\frac{J}{J_k}\right)e^{\frac{J}{J_k}+\frac{J^2}{2J_k^2}
+\dots+\frac{1}{p_k}\left(\frac{J}{J_k}\right)^{p_k}},
\end{equation}
where $m$ is an order of the root $J=0$ ($m$ can be equal to zero),
$Y(J)$ is an entire function, natural numbers $p_n$ are chosen such
that the series
$\sum\limits_{n=1}^\infty\left(\frac{J}{J_n}\right)^{p_n+1}$
is an absolutely and uniformly convergent one.
Scalar fields $\phi$ (associated with the open string tachyon) is
dimensionless, while $[\alpha^{\prime}]=\mbox{length}^2$, $[L]=\mbox{length}$ and
$[g_o]=\mbox{length}$. Let us introduce dimensionless coordinates
$\bar{x}_\mu=x_\mu/\sqrt{\alpha'}$, the dimensionless Newtonian
constant $\bar{G}_N=G_N/\alpha'$, the dimensionless parameter $\bar
L=L/\sqrt{\alpha'}$, and the dimensionless open string coupling
constant $\bar g_o=g_o/\sqrt{\alpha^{\prime}}$. The dimensionless cosmological
constant $\bar\Lambda=\Lambda{\alpha^{\prime}}^2$, $\bar{R}$ is the curvature
scalar in the coordinates $\bar{x}_\mu$:
\begin{equation}
S_f=\int d^4 \bar{x} \sqrt{-g}\left(\frac{f(\bar{L}^2 \bar{R})}{16\pi
\bar{G}_N\bar{L}^2}+\frac{1}{\bar{g}_o^2}\left(\frac{1}{2}\phi\,\mathcal{F}\left(\bar{\Box}_g\right)\phi
-V(\phi) \right)-\bar{\Lambda}\right). \label{action_model2}
\end{equation}
In the following formulae we omit bars, but use only dimensionless
coordinates and parameters.
It is well-known~\cite{Mukhanov1} that at $f'(R)>0$ any $f(R)$ gravity
models in the metric variational approach are equivalent to the
Einstein gravity with a scalar field\footnote{There are two types of
$f(R)$ gravity: the metric variational approach and the Palatini
formalism. In the first case the equations of motion are obtained by
variation with respect to metric. Connections are the function of
metric in this formalism. In the Palatini formalism one should vary the
action independently with respect to metric and the connections.}. In
the metric variational approach the equations of gravity are as
follows:
\begin{equation}
\label{fr_equ} G_{\mu\nu}\equiv f'(R)R_{\mu\nu}-
\frac{f(R)}{2}g_{\mu\nu}-D_\mu
\partial_\nu f'(R)+g_{\mu\nu}\Box_g f'(R)=8\pi
G_N T_{\mu\nu}, \quad
\mathcal{F}(\Box_g)\phi=\frac{dV}{d\phi},
\end{equation}
where the energy--momentum (stress) tensor $T_{\mu\nu}$ is:
\begin{equation}
\label{TEV}
T_{\mu\nu}\equiv{}-\frac{2}{\sqrt{-g}}\frac{\delta{S}}{\delta
g^{\mu\nu}}
=\frac{1}{g_o^2}\Bigl(E_{\mu\nu}+E_{\nu\mu}-g_{\mu\nu}\left(g^{\rho\sigma}
E_{\rho\sigma}+W\right)\Bigr),
\end{equation}
\begin{equation}
E_{\mu\nu}\equiv\frac{1}{2}\sum_{n=1}^\infty
f_n\sum_{l=0}^{n-1}\partial_\mu\Box_g^l\phi\partial_\nu\Box_g^{n-1-l}\phi,\quad
W\equiv\frac{1}{2}\sum_{n=2}^\infty
f_n\sum_{l=1}^{n-1}\Box_g^l\phi\Box_g^{n-l}\phi-\frac{f_0}{2}\phi^2+C_1\phi.
\end{equation}
\section{Localization of nonlocal gravitational actions}
The Ostrogradski representation has been proposed for polynomial
$\mathcal{F}(\Box)$ in the Minkowski space-time~\cite{Ostrogradski:1850,PaisU}.
Our goal is to generalize this result on gravitational models with an
arbitrary analytic function $\mathcal{F}(\Box)$ with simple and double roots.
We also generalize the Ostrogradski representation on the models with a
linear potential. The nonlocal cosmological models with quadratic
potentials have been studied
in~\cite{Koshelev07,AJV0701,AJV0711,MN,KV,Vernov2010,VernovSQS}.
Let us start with the case $C_1=0$. We consider a function $\mathcal{F}(J)$,
which has simple roots $J_i$ and double roots $\tilde{J}_k$, and the
function
\begin{equation}
\label{phi0}
\phi_0=\sum\limits_{i=1}^{N_1}\phi_i+\sum\limits_{k=1}^{N_2}\tilde\phi_k,
\end{equation}
where
\begin{equation}
(\Box_g-J_i)\phi_i=0 \quad\mbox{and}\quad (\Box_g-\tilde{J}_k)^2\tilde\phi_k=0\quad\Leftrightarrow\quad
(\Box_g-\tilde{J_k})\tilde\phi_k=\varphi_k,\quad
(\Box_g-\tilde{J_k})\varphi_k=0.
\label{equphi}
\end{equation}
Without loss of generality we assume that for any $i_1$ and $i_2\neq
i_1$ conditions $J_{i_1}\neq J_{i_2}$ and
${\tilde{J}}_{i_1}\neq{\tilde{J}}_{i_2}$ are satisfied.
The energy--momentum tensor, which corresponds to $\phi_0$, has the
following form:
\begin{equation}
T_{\mu\nu}\left(\phi_0\right)=
T_{\mu\nu}\left(\sum\limits_{i=1}^{N_1}\phi_i+\sum\limits_{k=1}^{N_2}\tilde\phi_k\right)=
\sum\limits_{i=1}^{N_1}T_{\mu\nu}(\phi_i)+\sum\limits_{k=1}^{N_2}T_{\mu\nu}(\tilde\phi_k),
\label{Tmunugen}
\end{equation}
where all $T_{\mu\nu}$ are given by (\ref{TEV}) and
\begin{equation}
E_{\mu\nu}(\phi_i)=\frac{{
\mathcal{F}'(J_i)}}{2}\partial_{\mu}\phi_i\partial_{\nu}\phi_i,\quad
E_{\mu\nu}(\tilde\phi_k)= \frac{{
\mathcal{F}''(\tilde{J}_k)}}{4}\left(\partial_\mu\tilde\phi_k\partial_\nu\varphi_k
+\partial_\nu\tilde\phi_k\partial_\mu\varphi_k\right)+
\frac{\mathcal{F}'''(\tilde{J}_k)}{12}\partial_\mu\varphi_k\partial_\nu\varphi_k,
\end{equation}
\begin{equation}
\label{Vdr} W(\phi_i)=\frac{J_i \mathcal{F}'(J_i)}{2}\phi_i^2,\quad W(\tilde{\phi}_k)=\frac{\tilde{J}_k
\mathcal{F}''(\tilde{J}_k)}{2}\tilde\phi_k\varphi_k+ \left(\frac{{\tilde{J}_k
\mathcal{F}'''(\tilde{J}_k)}}{12}+\frac{{
\mathcal{F}''(\tilde{J}_k)}}{4}\right)\varphi_k^2,
\end{equation}
where a prime denotes a derivative with respect to $J$: $\mathcal{F}'\equiv
\frac{d\mathcal{F}}{dJ}$, \ $\mathcal{F}''\equiv \frac{d^2\mathcal{F}}{dJ^2}$ and $\mathcal{F}'''\equiv
\frac{d^3 \mathcal{F}}{dJ^3}$.
Considering the following local action
\begin{equation}
S_{loc}=\int d^4x\sqrt{-g}\left(\frac{f(R)}{16\pi
G_N}-\Lambda\right)+\sum_{i=1}^{N_1}S_i+\sum_{k=1}^{N_2}\tilde{S}_k,
\label{Sloc}
\end{equation}
where
\begin{equation}
S_i=\!{}-\frac{1}{g_o^2}\int d^4x\sqrt{-g}
\frac{\mathcal{F}'(J_i)}{2}\left(g^{\mu\nu}\partial_\mu\phi_i\partial_\nu\phi_i
+J_i\phi_i^2\right),
\end{equation}
\begin{equation}
\begin{array}{l}
\!\displaystyle\tilde{S}_k=\!\displaystyle\! {}-\frac{1}{g_o^2}\int
d^4x\sqrt{-g}\left(g^{\mu\nu}\left(\frac{{
\mathcal{F}''(\tilde{J}_k)}}{4}\left(\partial_\mu
\tilde{\phi}_k\partial_\nu\varphi_k+\partial_\nu
\tilde{\phi}_k\partial_\mu\varphi_k\right)+{}\right.\right.\\[2.7mm]
\displaystyle + \left.\frac{
\mathcal{F}'''(\tilde{J}_k)}{12}\partial_\mu\varphi_k\partial_\nu\varphi_k\right)+
\left. \frac{\tilde{J}_k \mathcal{F}''(\tilde{J}_k)}{2}\tilde\phi_k\varphi_k
+\left(\frac{{\tilde{J}_k \mathcal{F}'''(\tilde{J}_k)}}{12}+\frac{{
\mathcal{F}''(\tilde{J}_k)}}{4}\right)\varphi_k^2\right), \label{Slocdr}
\end{array}
\end{equation}
we can see that solutions of the Einstein equations and equations in
$\phi_k$, $\tilde{\phi}_k$ and $\varphi_k$, obtained from this action,
solve the initial nonlocal equations (\ref{fr_equ}). Thus, we
obtain that special solutions to nonlocal equations can be found as
solutions to system of local (differential) equations. If $ \mathcal{F}(J)$ has
an infinity number of roots then one nonlocal model corresponds to
infinity number of different local models and the initial nonlocal
action (\ref{action_model2}) generates infinity number of local actions
(\ref{Sloc}).
We should prove that the way of localization is self-consistent. To
construct local action (\ref{Sloc}) we assume that equations
(\ref{equphi}) are satisfied. Therefore, the method of localization is
correct only if these equations can be obtained from the local action
$S_{loc}$. The straightforward calculations show that the way of
localization is self-consistent because:
\begin{equation}
\frac{\delta{S_{loc}}}{\delta \phi_i}=0 \, \Leftrightarrow \,
\Box_g\phi_i=J_i\phi_i; \, \frac{\delta{S_{loc}}}{\delta
\tilde{\phi}_k}=0 \, \Leftrightarrow \,
\Box_g\varphi_k=\tilde{J}_k\varphi_k; \,
\frac{\delta{S_{loc}}}{\delta \varphi_k}=0 \, \Leftrightarrow \,
\Box_g\tilde{\phi}_k=\tilde{J}_k\tilde{\phi}_k+\varphi_k.
\end{equation}
In spite of the above-mention equations we obtain from $S_{loc}$ the
equations:
\begin{equation}
G_{\mu\nu}=8\pi G_N\left(T_{\mu\nu}(\phi_0)-\Lambda g_{\mu\nu}\right),
\end{equation}
where $\phi_0$ is given by (\ref{phi0}) and $T_{\mu\nu}(\phi_0)$ can be
calculated by (\ref{Tmunugen}). So, we get such systems of
differential equations that any solutions of these systems are
particular solutions of the initial nonlocal equations (\ref{fr_equ}).
Let us consider functions $\mathcal{F}(J)$ with two and only two simple roots.
If $\mathcal{F}(J)$ has two real simple roots, then $\mathcal{F}'(J)>0$ at one root and
$\mathcal{F}'(J)<0$ at another root, so we get a quintom
model~\cite{Quinmodrev1}, in other words, local model with one standard
scalar field and one phantom scalar field. In the case of two complex
conjugated simple roots $J_j$ and $J_j^*$ one gets the following
action:
\begin{equation}
S_c=\!\int\!\! d^4x\frac{\sqrt{-g}}{2g_o^2}\left(
\mathcal{F}'(J_j)\left(g^{\mu\nu}\partial_\mu\phi_j\partial_\nu\phi_j
+J_j\phi_j^2\right)+{\mathcal{F}'}^*(J_j)\left(g^{\mu\nu}\partial_\mu\phi^*_j\partial_\nu\phi^*_j
+J^*_j{\phi_i^*}^2\right)\right).
\end{equation}
We introduce real fields $\xi$ and $\eta$ such that $\phi_j=\xi+i\eta$,
\ $\phi_j^*=\xi-i\eta$, denote $d_r\equiv\Re e(\mathcal{F}'(J))$, \
$d_i\equiv\Im m(\mathcal{F}'(J))$, and obtain:
\begin{equation}
S_c=\int d^4x\frac{\sqrt{-g}}{2g_o^2}\Bigl(d_r
g^{\mu\nu}\left(\partial_\mu\xi\partial_\nu\xi-
\partial_\mu\eta\partial_\nu\eta\right)+
d_ig^{\mu\nu}(\partial_\mu\xi\partial_\nu\eta-\partial_\mu\eta\partial_\nu\xi)+V_1\Bigr),
\end{equation}
where $V_1$ is a potential term. In the case $d_i=0$ we get a quintom
model, in opposite case the kinetic term in $S_c$ has a nondiagonal
form. To diagonalize the kinetic term we make the transformation:
$\chi=\upsilon+\tilde{C}\sigma$, $\eta={}-\tilde{C}\upsilon+\sigma$,
where $\tilde{C}\equiv\left(d_r+\sqrt{d_r^2+d_i^2}\right)/d_i$, and get
a quintom model:
\begin{equation}
S_c=\int
d^4x\frac{\sqrt{-g}}{2g_o^2}\left(\frac{2\left(d_r^2+d_i^2\right)}{d_i^2}\left(d_r+\sqrt{d_r^2+d_i^2}\right)
\left(\partial_\mu\upsilon\partial_\nu\upsilon
-\partial_\mu\sigma\partial_\nu\sigma\right)+V_1\right).
\end{equation}
In the case of a real double root $\tilde{J}_k$ we express
$\tilde{\phi}_k$ and $\varphi_k$ in terms of new fields $\xi_k$ and
$\chi_k$:
\begin{eqnarray}
\tilde{\phi}_k&=&\frac{1}{2\mathcal{F}''(\tilde{J}_k)}\left(\left(\mathcal{F}''(\tilde{J}_k)-\frac{2}{3}\mathcal{F}'''(\tilde{J}_k)\right)
\xi_k-\left(\mathcal{F}''(\tilde{J}_k)+\frac{2}{3}\mathcal{F}'''(\tilde{J}_k)\right)\chi_k\right),
\quad \varphi_k=\xi_k+\chi_k,\nonumber
\end{eqnarray}
we obtain the corresponding $\tilde{S}_k$ in the following form:
\begin{eqnarray}
\tilde{S}_k&=&\frac{{}-1}{2g_o^2}\!\int\!
d^4\!x\sqrt{-g}\left(g^{\mu\nu}\frac{\mathcal{F}''(\tilde{J}_k)}{4}(\partial_\mu
\xi_k\partial_\nu\xi_k-\partial_\nu
\chi_k\partial_\mu\chi_k)+\left[\frac{{\tilde{J}_k\mathcal{F}'''(\tilde{J}_k)}}{12}+\frac{{\mathcal{F}''(\tilde{J}_k)}}{4}\right]
(\xi_k+\chi_k)^2+\right.\nonumber\\
&+&\left.\frac{\tilde{J}_k}{4}\left[(\mathcal{F}''(\tilde{J}_k)-\frac{2}{3}\mathcal{F}'''(\tilde{J}_k))
\xi_k-(\mathcal{F}''(\tilde{J}_k)+\frac{2}{3}\mathcal{F}'''(\tilde{J}_k))\chi_k\right](\xi_k+\chi_k)\right).\nonumber
\end{eqnarray}
It is easy to see that each $\tilde{S}_k$ includes one phantom scalar
field and one standard scalar field. So, in the case of one double root
we obtain a quintom model. In the Minkowski space appearance of phantom
fields in models, when $\mathcal{F}(J)$ has a double root, has been obtained
in~\cite{PaisU}. So, we come to conclusion that both two simple roots
and one double root of $\mathcal{F}(J)$ generate quintom models.
The model with action (\ref{action_model2}) in the case $C_1\neq 0$ has
been considered in detail in~\cite{VernovSQS}. Here we present only the
obtained algorithm of localization for an arbitrary quadratic potential
$V(\phi)=C_2\phi^2+C_1\phi+C_0$:
\begin{itemize}
\item Change values of $f_0$ and $\Lambda$ such that the potential
takes the form $V(\phi)=C_1\phi$.
\item Find roots of the function $ \mathcal{F}(J)$ and calculate orders of
them. Select an finite number of simple and double roots.
\item Construct the corresponding local action. In the case $C_1=0$
one should use formula (\ref{Sloc}). In the case $C_1\neq 0$ and
$f_0\neq 0$ one should use (\ref{Sloc}) with the replacement of the
scalar field $\phi$ by $\chi$ and the corresponding modification of
the cosmological constant. In the case $C_1\neq 0$ and $f_0=0$ the
local action is the sum of (\ref{Sloc}) and either
\begin{equation}
S_{\psi}={}-\frac{1}{2g_o^2}\int\! d^4x\sqrt{-g}\left(
f_1g^{\mu\nu}\partial_\mu\psi\partial_\nu\psi+2C_1\psi+\frac{f_2C_1^2}{f_1^2}\right),
\end{equation}
in the case of simple root $J=0$, or
\begin{eqnarray}
S_{\tilde{\psi}}&=&{}-\!\int\! d^4x\frac{\sqrt{-g}}{2g_o^2}\left[
g^{\mu\nu}\left(f_2(\partial_\mu\tilde{\psi}\partial_\nu\tau
+\partial_\nu\tilde{\psi}\partial_\mu\tau)+f_3\partial_\mu\tau\partial_\nu\tau\right)
+f_2\tau^2+2C_1\tilde{\psi}+\frac{f_3C_1}{2f_2}\tau\right]\nonumber
\end{eqnarray}
in the case of double root $J=0$. Note that in the case $C_1\neq 0$
and $f_0=0$ the local action (\ref{Sloc}) has no term, which
corresponds to the root $J=0$.
\item Vary the obtained local action and get a system of the Einstein
equations and equations of motion. The obtained system is a finite
order system of differential equations, \textit{i.e.} we get a local
system. Seek solutions of the obtained local system.
\end{itemize}
\section{Conclusion}
The main result of this paper is the generalization of the algorithm of
localization on the $f(R)$ gravity models with a nonlocal scalar
field. The algorithm of localization is proposed for an arbitrary
analytic function $\mathcal{F}(\Box_g)$, which has both simple and double
roots. We have proved that the same functions solve the initial
nonlocal Einstein equations and the obtained local Einstein equations.
We have found the corresponding local actions and proved the
self-consistence of our approach. In the case of two simple roots as
well as in the case of one double root we get a quintom
model~\cite{Quinmodrev1}. The algorithm of localization does not depend
on metric, so it can be used to find solutions for any metric.
The author wishes to express his thanks to I.~Ya.~Aref'eva for useful
and stimulating discussions. The research
has been supported in part by RFBR grant 08-01-00798, grant of Russian
Ministry of Education and Science NSh-4142.2010.2 and by Federal
Agency for Science and Innovation under state contract
02.740.11.0244.
|
2,877,628,088,985 | arxiv | \section{Introduction}
The goal of this paper is to introduce Ricci Yang-Mills solitons on nilpotent Lie groups. In this setting, Ricci Yang-Mills solitons are weaker than Ricci soliton metrics in a sense to be made precise below. We provide some examples of manifolds known not to admit Ricci solitons that do admit Ricci Yang-Mills solitons.
To study this problem, we rephrase our questions in the language of moment maps for a particular representation of $GL_n\mathbb R$. A similar implementation of Geometric Invariant Theory has been carried out by Lauret, Eberlein, Jablonski, et.~al, in the study of Ricci solitons on nilpotent Lie groups (see, for example, \cite{La}, \cite{Eber07}, \cite{Jab:Thesis}). Moreover, our approach to the study of Ricci Yang-Mills solitons answers a technical question asked by Eberlein concerning moment maps in the 2-step nilpotent setting (cf. Question \ref{question: distinguished pts of GLqR action}).\\
The Ricci flow is a differential equation on the space of Riemannian metrics on $M$, $\mathfrak{Met}$. In this space, the only fixed points of this equation are the Ricci-flat metrics, whereas general Einstein metrics are fixed points of the volume normalized Ricci flow. However, if one works in the space of $\mathfrak{Met}$/$\mathfrak{Diff}$, where $\mathfrak{Diff}$ is the group of diffeomorphisms on $M$, then one allows for a new family of fixed points, namely the metrics that flow by scaling and diffeomorphism; i.e. $g(t)=\sigma(t)\phi(t)^*g_0$, where $\phi(t):M\to M$ is a one parameter family of diffeomorphisms. These are the Ricci soliton metrics. One can show that Ricci soliton metrics satisfy the following equation:
\begin{equation}
\label{soliton}
Rc+\mathcal{L}_Xg+\frac{\epsilon}{2}g=0,
\end{equation}
where $X$ is the vector field generating the diffeomorphisms, and $\epsilon=-1, 0, 1$ corresponds to shrinking, steady, and expanding solitons, respectively. If $X$ is the gradient of some function, i.e. $X=\nabla f$, then a solution to Eq.~\ref{soliton} is said to be a gradient Ricci soliton.
Ricci soliton structures on Lie groups were first discovered by Baird and Danielo [BD] and independently by Lott [Lo]. Baird and Danielo discovered the first known examples of nongradient soliton structures on the Lie groups Nil and Sol [BD]. They studied semiconformal maps from 3-manifolds to Riemann surfaces and described completely the soliton structures on all the 3-dimensional geometries. More generally, one knows quite a bit of information about left-invariant Ricci solitons on Lie groups. If $G$ is a semi-simple group, then any Ricci soliton metric must be Einstein, and all compact semi-simple groups admit Einstein metrics. Within the set of solvable groups, it is known that there exist solvable $G$ which admit non-Einstein Ricci soliton metrics. There also exist solvable groups that do admit Einstein metrics. Further restricting to nilpotent Lie groups, it is known that there are no Einstein metrics on $G$, but there can be Ricci soliton metrics on $G$ (see \cite{La} for more details).
However, there are some spaces that are known not to admit even Ricci solitons. Consider a Lie group $G$ with left-invariant metric. Recently there have been many new families of nilpotent Lie groups constructed which cannot admit left-invariant Ricci soliton metrics \cite{Jab:Moduli}. To better understand these spaces, it would be useful to have an even weaker notion of `best' metric. It is our intention to show that a suitable notion could be that of a Ricci Yang-Mills soliton.
The Ricci Yang-Mills flow was defined independently in \cite{St} and \cite{Yo}. These equations are motivated by the study of Ricci flow on principal bundles and can be written as a modified Ricci flow coupled to the Yang-Mills heat flow. Long-time behavior of the flow has been studied in \cite{St1}, and stability properties have been considered in \cite{Yo1}.
Gradient Ricci Yang-Mills solitons have been studied in \cite{St}, \cite{St1}, and \cite{Yo}. In the case of a $U(1)$-bundle over a compact surface, these were classified in \cite{St1}. Ricci Yang-Mills solitons have also been studied in the context of dynamical systems. In \cite{Jane}, it was discovered that the Ricci Yang-Mills flow is an ideal candidate for studying magnetic flows. There is work in progress to determine whether Ricci Yang-Mills solitons have new dynamical properties.\\
This note is organized as follows. In $\S 2$, we define both the Ricci Yang-Mills flow and the Ricci Yang-Mills soliton equations. We provide the framework for the Ricci Yang-Mills soliton equations on 2-step nilpotent Lie groups in $\S 3$. In this section we translate the notion of Ricci Yang-Mills solitons into the notion of distinguished points from Geometric Invariant Theory. Finally in $\S 4$, we provide several examples (using algebraic techniques) of Lie groups that do not admit Ricci solitons but that do admit Ricci Yang-Mills solitons.\\
\textit{Acknowledgements.} This note is a component of a larger project to understand the Ricci Yang-Mills flow and its special solutions. The authors would like to thank Dan Jane for many enlightening conversations and Pat Eberlein for spotting a critical typo.
\section{Ricci Yang-Mills solitons}
The Ricci Yang-Mills flow is a natural coupling of the Ricci flow and the Yang-Mills heat flow. Let $\pi:P\to M$ be a principal bundle with structure group $G$. Also, let $g$ be a metric on $M$, $k$ an $\mathfrak{Ad}$-invariant metric on $\mathfrak{g}$, and $\omega$ the connection 1-form on $P$. We will consider so-called \emph{bundle metrics} on $P$ of the form
\[h=\pi^\ast g+k\omega,
\]
where $k \omega$ acts on vector fields by $k\omega(Y,Z)=k(\omega(Y),\omega(Z))$.
Writing the Ricci flow equations for a metric of this form with the additional hypothesis that the size of the fiber remains fixed, one can define the Ricci Yang-Mills flow to be
\begin{equation}
\label{rym total}
\frac{\partial h}{\partial t}=-2(Rc-Rc^V),
\end{equation}
where $Rc^V$ is the projection of the Ricci tensor onto its vertical component (cf.~\cite{St} and the proof of Theorem \ref{thm: RYM is invariant under bundle auts}). If $G$ is abelian, using the definition of $h$ and the structure of $Rc(h)$, one can show that this equation is equivalent to the following system of equations:
\begin{subequations}
\label{rym}
\begin{align}
\frac{\partial g}{\partial t}&=-2Rc(g)+\tilde\Omega^2\\
\frac{\partial \tilde \omega}{\partial t}&=-\delta \tilde \Omega.
\end{align}
\end{subequations}
Here $\tilde \omega$ and $\tilde\Omega$ are the pullbacks under a local section of the connection 1-form and the bundle curvature, respectively. Recall that when $G$ is abelian, $\tilde \Omega$ is a well-defined Lie algebra-valued 2-form on the base. In coordinates, $\tilde \Omega^2_{ij}=g^{kl}k^{\alpha \beta}\tilde \Omega_{\alpha ki} \tilde{\Omega}_{\beta lj}$, where the greek indices are the Lie algebra indices and the Roman indices correspond to quantities measured with respect to $g$. Existence and uniqueness of solutions to the Ricci Yang-Mills flow have been studied in \cite{St} and \cite{Yo}.\\
We would like to define Ricci Yang-Mills solitons in a way that is analogous to Ricci solitons. Namely, due to the diffeomorphism invariance of the Ricci flow, one can define Ricci solitons to be fixed points of the Ricci flow in the space $\mathfrak{Met}/\mathfrak{Diff}$. However, since a principal bundle is a manifold endowed with a group action of $G$, in defining Ricci Yang-Mills solitons, we are interested in diffeomorphisms that preserve the full structure of the principal bundle.
\begin{defin}
An automorphism of a principal bundle $\pi:P\to M$ is a diffeomorphism $F:P\to P$ such that $F(pg)=F(p)g$ for all $g\in G, p\in P$. We denote this set by $\mathfrak{Diff}_G$.
\end{defin}
Let $F\in \mathfrak{Diff}_G$ be an automorphism of the principal bundle $P$. Then $F$ descends to a diffeomorphism $f$ on $M$ satisfying $f\circ \pi = \pi \circ F$, and if $\omega$ is a connection on $P$, then $F^*\omega$ is also a connection.
\begin{thm}\label{thm: RYM is invariant under bundle auts}
The Ricci Yang-Mills flow is invariant under automorphisms of $P$.
\end{thm}
\begin{proof}
Using the properties above, one immediately sees that $F^*h$ is a bundle metric for $F\in \mathfrak{Diff}_G$. More precisely,
$$F^*h = \pi^* (f^*g) + k F^*\omega.$$
For each $z\in \mathfrak{g}$, there is a canonical vector field on $P$ defined by $Z_p = \restrictto{\frac{d}{dt}}{t=0} p\cdot exp(tz)$. As $F$ preserves the $G$ action and $\omega (\restrictto{\frac{d}{dt}}{t=0} p\cdot exp(tz) ) = z$, we see that $F_* (Z_p) = Z_{F(p)}$ and $\omega (Z_p) = \omega (Z_{F(p)})=(F^*\omega)(Z_p) = z$. Observe that if $\{z_i\}$ is an orthonormal basis of $\mathfrak{g}$ then the induced vector fields $\{Z_i\}$ form an orthonormal frame of the vertical space relative to both metrics, $h$ and $F^* h$.
We can write the Ricci Yang-Mills flow as $\frac{\partial h}{\partial t}=-2(Rc-Rc^V)$, where $Rc^V$ is the projection of the Ricci tensor onto its vertical component. Specifically, if $U$ is a vector field on $P$, then we can define the projection onto its vertical component to be $p_h(U)=\sum_i h(Z_i,U) Z_i$, where $\{Z_i\}$ is an orthonormal basis (relative to $h$) of the vertical space as above. In this notation, we have $Rc^V(h)(U,V)=Rc(p_h(U),p_h(V))$.
Let $F:P\to P$ be a bundle automorphism. Since $F$ is a diffeomorphism, clearly $Rc(F^\ast h)=F^\ast Rc(h)$. It remains only to check that $Rc^V(F^\ast h)=F^\ast Rc^V(h)$. First we show that $p_{F^\ast h} = p_h \circ F_* $. By definition,
\begin{eqnarray*}
p_{F^\ast h}(U)&=& \sum_i (F^* h)(Z_i,U) Z_i\\
&=& \sum_i (f^* g) (\pi_* Z_i, \pi_* U) + k(
\omega(F_*U),\omega(F_*Z_i))Z_i\\
&=& \sum_i k( \omega(F_*U),\omega(Z_i) )Z_i\\
&=& \sum_i h(F_* U, Z_i) Z_i\\
&=& p_h ( F_* U)
\end{eqnarray*}
Here we have used the fact that $\{Z_i\}$ will be orthonormal in both metrics $h$ and $F^*h$. Thus $p_{F^\ast h} = p_h \circ F_\ast $. Using this fact and the diffeomorphism invariance of $Rc$, one sees that in fact $Rc^V(F^\ast h)=F^\ast Rc^V(h)$. Thus the Ricci Yang-Mills flow is invariant under bundle automorphisms of $P$.
\end{proof}
\begin{cor} The Ricci Yang-Mills flow preserves the set of left-invariant metrics on a Lie group $N$.
\end{cor}
\begin{proof}
Left multiplication $L_g(p)=gp$ is a bundle automorphism since left and right multiplication commute. Thus the result follows from above.
\end{proof}
Since the set of left-invariant metrics is preserved under this evolution, we can interpret the Ricci Yang-Mills flow as an evolution of the metric on a single tangent space; more precisely, we will evolve an inner product on the Lie algebra. This is the standard approach to studying geometric evolutions on Lie groups with left-invariant metrics.
\subsection{Self-similar solutions to Ricci Yang-Mills equations.}
Analogous to the case of Ricci flow, we define Ricci Yang-Mills solitons to be generalized fixed points of Eq.~\ref{rym}.
\begin{defin}
A solution $(g_t, \tilde\omega_t)$ to the Ricci Yang-Mills equations is a self similar solution if there exists a scaling $\sigma(t)$ and a family of diffeomorphisms $\psi_t$ $ \in \mathfrak{Diff}(M)$ such that $g_t = \sigma(t) \cdot \psi_t^* g$ and $\tilde\Omega_t = \psi_t^* \tilde\Omega$ (with $\sigma(0) = 1$ and $\psi_0$ the identity). Let $X \in \Gamma(M, TM)$ generate $\psi$ near $t = 0$. As in the case of Ricci flow, one can show that the notion of self-similar solutions is equivalent to $(g,\tilde\omega)$ satisfying
\begin{subequations}
\label{RYM soliton}
\begin{align}
(\tilde\Omega^2-2Rc)(g) &= \sigma'(0) g + \mathcal{L}_X g,\\
\Delta_d \tilde\Omega &= \mathcal{L}_X \tilde\Omega,
\end{align}
\end{subequations}
where $\Delta_d$ is the Hodge Laplacian. We will call solutions satisfying Eq.~\ref{RYM soliton} \emph{(weak) Ricci Yang-Mills solitons}.
\end{defin}
\textit{Notation:} Let $\lambda = \sigma '(0)$. If $\lambda>0$ we say the Ricci Yang-Mills soliton is an expander, if $\lambda=0$ the Ricci Yang-Mills soliton is called steady, and $\lambda<0$ is called a shrinker.
\begin{defin}\label{defin: strong RYM soliton} In contrast to weak Ricci Yang-Mills solitons, we say that a Ricci Yang-Mills soliton is a strong Ricci Yang-Mills soliton when there exists a family $\tilde \psi_t \in \mathfrak{Diff}_G$ which induces $\psi_t$; that is, such that $\pi \circ \psi_t = \varphi_t \circ \pi$.
\end{defin}
\noindent
\begin{remark}
The notion of being a weak Ricci Yang-Mills soliton is intrinsic to the base manifold; as the group $G$ is abelian, we may consider $g$ and $\tilde\Omega$ as objects living on $M$. From the perspective of the base, one doesn't see diffeomorphisms of the total space. It is not known, even in the case of $U(1)$-bundles over surfaces, whether weak solitons are always strong. Considering not all diffeomorphisms on $M$ are induced by bundle automorphisms of $P\to M$, we expect there to exist Ricci Yang-Mills solitons which are precisely weak. This will be the object of future study.
\end{remark}
In the setting of nilpotent Lie groups, our principal bundles are $N \to N/Z$ where $Z$ is the center. Here every diffeomorphism lifts to a diffeomorphism of the total space and hence all Ricci Yang-Mills solitons will be strong in this paper, see Section \ref{section: RYM sol of Lie type}.\\
\noindent
\textit{Remark.}
An Einstein Yang-Mills metric as defined in \cite{St} and \cite{Yo} is one such that the metric on the base is Einstein and the connection is Yang-Mills; i.e. $\delta \tilde\Omega=0$. On a 2-step nilpotent Lie group, every metric is a metric of this type (cf. Lemmas \ref{lemma: base is flat} and \ref{lemma: Delta Omega = 0}).
Ricci Yang-Mills solitons are not direct generalizations of Einstein Yang-Mills metrics (as defined above) in the same way that Ricci solitons are generalizations of Einstein metrics. Recall that Ricci solitons are fixed points of the volume-normalized Ricci flow, which differs from the Ricci flow only by a change of scale in space and time. Einstein Yang-Mills metrics (as defined above) are fixed points of a certain volume normalized Ricci Yang-Mills flow; however, due to the lack of scale invariance of this equation, the volume normalized flow does not differ only by a change of scale (see \cite{St} or \cite{Yo}).
An alternate definition of Einstein Yang-Mills, which is more natural from the view point of special solutions to the Ricci Yang-Mills flow, would be a Ricci Yang-Mills soliton generated by a trivial vector field; that is, $L_X=0$. This is consistent with the idea of an Einstein metric from the perspective of Ricci solitons and Ricci flow. To avoid confusion with previous definitions, we make the following definition
\begin{defin} A \emph{trivial Ricci Yang-Mills soliton} is one that is generated by the trivial vector field $X=0$.\end{defin}
Below we will construct many examples of trivial Ricci Yang-Mills solitons; these correspond to so-called minimal points of a particular representation (see Corollary \ref{cor: closed SL q C orbit implies RYM soliton}).
\section{Ricci Yang-Mills solitons on nilmanifolds}
A natural test case in the search for Ricci Yang-Mills solitons on principal bundles is the case of a torus bundle over a torus. These compact manifolds are precisely the locally homogeneous manifolds which are modeled on 2-step nilpotent Lie groups (see \cite{Pal61}). More precisely, these spaces are quotients of nilpotent Lie groups by cocompact lattices. As in the case of Ricci flow, to understand the dynamics of this geometric evolution on a compact manifold, we study the evolution on the simply connected cover, a nilpotent Lie group with a left-invariant metric. This cover is also a principal bundle and the covering map is a morphism of bundles. We recall some basic facts for the convenience of the reader.
\begin{defin}Let $\mathfrak{N}$ be a finite dimensional Lie algebra, and for $i\geq 1$, let $\mathfrak{N}^i=[\mathfrak{N},\mathfrak{N}^{i-1}]$, where $\mathfrak{N}^0=\mathfrak{N}$. Then $\mathfrak{N}$ is said to be nilpotent if $\mathfrak{N}^i=\{0\}$ for some $i$. A nilpotent Lie algebra is $k$-step if $\mathfrak{N}^k=\{0\}$ but $\mathfrak{N}^{k-1}\neq \{0\}$. A Lie group is said to be ($k$-step) nilpotent if its Lie algebra is ($k$-step) nilpotent.
\end{defin}
\begin{defin}\label{defin: type p,q} A 2-step nilpotent Lie group $N$ or Lie algebra $\mathfrak N$ is said to be of type $(p,q)$ if dim $[\mathfrak N,\mathfrak N] = p$ and codim $[\mathfrak N,\mathfrak N] =q$.
\end{defin}
Observe that $p$ above satisfies $1\leq p \leq \frac{1}{2}q(q-1) = \dim \mathfrak{so}(q)$. Stratifying the space of 2-step nilpotent Lie algebras into types $(p,q)$ is very convenient in terms of phrasing generic results.
\begin{defin} Let $\{X_i\}$ be a basis of the Lie algebra $\mathfrak N$. The structure constants relative to the basis $\{X_i\}$ are the coefficients $\{c_{ij}^k\}$ defined by $[X_i,X_j]=\sum_k c_{ij}^k X_k$. \end{defin}
\begin{defin} Let $N$ be a Lie group with Lie algebra $\mathfrak N$. There exists a map $exp_N : \mathfrak N \to N$ called the Lie group exponential. When there is no confusion, we write $exp$ for $exp_N$.
\end{defin}
Recall that when $N$ is a simply connected nilpotent Lie group, the map $exp_N$ is a diffeomorphism from $\mathfrak N$ to $N$.
\begin{thm}[Campbell-Baker-Hausdorff formula]\label{thm: CBH formula} For $X,Y \in \mathfrak N$, we have
$$exp(X) \ exp(Y) = exp( X+Y + \frac{1}{2} [X,Y] + \ higher \ order \ terms )$$
where `higher order terms' means combinations of 2 or more brackets involving $X$ and $Y$.
\end{thm}
This formula explicitly relates the Lie product of the group and the Lie bracket of the algebra. The higher order terms can be written explicitly, but we will only use this formula in the case that one of $X$ or $Y$ is in the center; in this case we have $exp(X)exp(Y)=exp(X+Y)$ (see Section \ref{section: RYM sol of Lie type}).\\
Every nilpotent Lie group $N$ can be viewed as a non-trivial principal bundle. The total space will be $P=N$, and the group $G$ will be the center of $N$ acting on the right. We describe this in detail in the 2-step case. The details in the general case are similar.
\subsection{2-step nilmanifolds}
We would like to consider the Ricci Yang-Mills soliton equations on simply-connected 2-step nilpotent Lie groups. Let $N$ be a 2-step nilpotent Lie group with Lie algebra $\mathfrak N$. We endow $N$ with a left-invariant metric $h$; this is equivalent to endowing the Lie algebra $\mathfrak N$ with an inner product. Let $Z=[N,N]$ be the commutator subgroup, $\mathfrak Z = Lie \ Z$, and $\mathcal H = \mathfrak Z ^ \perp$ the orthogonal complement relative to the given metric on $\mathfrak N$. We point out that $Z$ is a central subgroup since $N$ is 2-step nilpotent. One could choose to work with either the full center of $N$ or the commutator $[N,N]$ in what follows.
Let $\{X_1,\dots, X_q\} \cup \{Z_1,\dots, Z_p\}$ be an orthonormal basis of $\mathfrak N = \mathcal H \oplus \mathfrak Z$. Here $q=\dim \mathcal H$, $p=\dim \mathfrak Z$, and $n = q+p = \dim N$; this 2-step nilpotent algebra is of type $(p,q)$. By left-translating, we can treat this basis of $\mathfrak N$ as a left-invariant frame on $N$. Relative to this basis we have the (Lie algebra) structure coefficients defined via
$$ [ X_i, X_j] = \sum_k c_{ij}^k Z_k$$.
Notice that $\pi:N\to N/Z$ is naturally a principal $G$-bundle where $G = \mathbb R^p \simeq Z$. The action of $G\simeq Z$ will be given by first injecting $Z$ into $N$ and then multiplying on the right. To distinguish between $\mathfrak Z$ abstractly versus embedded in $\mathfrak N$, we will use lower case letters to denote elements of $\mathfrak g$ and upper case letters to denote elements of $\mathfrak Z\subset \mathfrak N$; that is, given $z\in \mathfrak g$, $z\to Z \in \mathfrak Z \subset \mathfrak N$. A choice of a horizontal subspace $\mathcal{H}$ yields a connection 1-form $\omega$ which vanishes on $\mathcal H$ and takes values in the Lie algebra $\mathfrak{g}$. More precisely, we define our connection so that $\omega(Z)=z$ and $\omega(X)=0$ for $Z\in \mathfrak Z$, $X \in \mathcal H$. Thus far, we have defined our connection on $\mathfrak N = T_e N$. We extend the definition of the connection to the rest of $N$ by imposing $\omega$ be left-invariant under $N$.
Recall that the Ricci Yang-Mills soliton equations are
\begin{subequations}
\label{rym 1}
\begin{align}
-2Rc_g+\tilde\Omega^2&=\mathcal{L}_Xg+\lambda g\\
\Delta_d\tilde\Omega&=\mathcal{L}_X\tilde\Omega.
\end{align}
\end{subequations}
Here $g$ is the induced metric on $M=N/Z$. As $Z$ is a normal subgroup of $N$, $N/Z$ is a Lie group and the metric $g$ is left $N/Z$-invariant.\\
\begin{lemma}\label{lemma: base is flat} In the 2-step nilpotent setting, Eq.~\ref{rym 1}a becomes $\tilde\Omega^2=\mathcal{L}_Xg+\lambda g$.
\end{lemma}
\begin{proof}
As the base is an abelian Lie group $N/Z$ with left-invariant metric, it is flat and hence $Rc_g=0$.
\end{proof}
\begin{lemma}\label{lemma: Omega vs structure matrices} Let $X_i, X_j$ be horizontal vectors in the basis of $\mathfrak N$ above, then
$\Omega (X_i,X_j) = \sum_\alpha -c_{ij}^\alpha z_\alpha$.
\end{lemma}
\begin{proof}
By definition of $\Omega$,
\begin{displaymath}
\Omega(X_i, X_j) = d \omega (X_i, X_j) = X_i( \omega (X_j)) - X_j( \omega(X_i)) - \omega ([X_i, X_j])=-\omega(\sum_k c_{ij}^kZ_k)=\sum_k -c_{ij}^kz_k.
\end{displaymath}
Here we have used that fact that $\omega$ is left-invariant and $X_i$ is a left-invariant vector field on $N$. Thus, $\omega(X_i)$ is a constant function on $N$ and $X_j(\omega(X_i))=0$.
\end{proof}
\begin{lemma}\label{lemma: Delta Omega = 0} The connection $\tilde \omega$ is Yang-Mills; that is,
$\Delta_d \tilde{\Omega}=0$.
\end{lemma}
\begin{proof}
Recall that $\tilde\Omega = d\tilde\omega$ and so $\delta \tilde\Omega = 0$ if and only if $\Delta_d \tilde\Omega = 0$. Let $U \subset M$ be an open set, and let $s:U\to N$ be a local section. We can define $\tilde{X}_i=\pi_\ast X_i$ to be a left-$N/Z$-invariant vector field on the base. We have that $[X_i,X_j]=c_{ij}^kZ_k$. Thus
\begin{eqnarray*}
\tilde{\Omega}(\tilde{X}_i,\tilde{X}_j)&=&s^\ast \Omega(\tilde{X}_i,\tilde{X}_j)\\
&=&\Omega(s_\ast\tilde{X}_i,s_\ast\tilde{X}_j)\\
&=&\Omega(X_i,X_j).
\end{eqnarray*}
So $\tilde{\Omega}=\sum_{ijk}-c_{ij}^k z_k\tilde \sigma^i \wedge \tilde \sigma^j$, where $\{\tilde \sigma^i\}$ are dual to $\{\tilde X_i \}$. To compute $\Delta_d\tilde{\Omega}$, we only need to compute $d\star d\star (\tilde{\sigma}^i\wedge \tilde{\sigma}^j)$, as the structure constants come out. We compute that $d\star(\tilde{\sigma}^i\wedge \tilde{\sigma}^j)=(-1)^{i+j-1}d(\tilde{\sigma}^1\wedge\cdots \wedge \hat{i}\hat{j} \wedge \cdots \wedge \tilde{\sigma}^q)$, and
\[
d\tilde{\sigma}^k(\tilde{X}_i,\tilde{X}_j)=\tilde{X}_i(\tilde{\sigma}^k(\tilde{X}_j)-\tilde{X}_j(\tilde{\sigma}^k(\tilde{X}_i)-\tilde{\sigma}^k([\tilde{X}_i,\tilde{X}_j])=-\tilde{\sigma}^k([\tilde{X}_i,\tilde{X}_j]).
\]
Again, we have used left-invariance to make two of the middle terms vanish in the above equation. Lastly, $[\tilde{X}_i,\tilde{X}_j]=0$ as $\mathfrak N / \mathfrak Z$ is abelian.
Thus $\Delta_d\tilde{\Omega}=0$.
\end{proof}
\begin{lemma} In the 2-step nilpotent setting, Eq.~\ref{rym 1}b becomes $\mathcal{L}_X\tilde{\Omega}=0$.
\end{lemma}
\begin{prop}\label{prop: RYM from deriv on N/Z}
Let $D$ be a derivation of $\mathfrak N/ \mathfrak Z$ and $exp(tD)$ the associated 1-parameter family of automorphisms of $N/Z$. Using this family of diffeomorphisms, the Ricci Yang-Mills soliton equations on 2-step nilpotent Lie groups become
\begin{subequations}
\label{rym w/deriv}
\begin{align}
\tilde \Omega^2(v,w)&=g((D+D^t)v,w)+\lambda g(v,w)\\
0&=\tilde \Omega(Dv,w)+\tilde \Omega(v,Dw),
\end{align}
\end{subequations}
where $v, w$ are vector fields on $M = N/Z$.
\end{prop}
\begin{proof} We postpone presenting the details of this proof. See Appendix A for information regarding one-parameter families of automorphisms.
\end{proof}
\begin{observ} Every linear map of an abelian Lie algebra is an automorphism of the Lie algebra.
\end{observ}
\noindent Thus, $\mathfrak N$ being a 2-step nilpotent Lie algebra means $\mathfrak N/\mathfrak Z$ is abelian and we can use any linear map ${ D: \mathfrak N / \mathfrak Z \to \mathfrak N / \mathfrak Z}$ in Eq.~\ref{rym w/deriv}.
\begin{defin}\label{defin: RYM soliton of Lie type} We say that a left-invariant Ricci Yang-Mills soliton is of \emph{Lie type} if it comes from an automorphism of the base $N/Z$ as in Proposition \ref{prop: RYM from deriv on N/Z}. We say that such a metric is of \emph{symmetric Lie type} if the derivation is symmetric with respect to the inner product on $\mathfrak N/ \mathfrak Z$.
\end{defin}
In the sequel, we build many examples of such Ricci Yang-Mills solitons. Presently, we have no examples of left-invariant Ricci Yang-Mills solitons which are not of symmetric Lie type. This question will be investigated in future work.
\subsection{2-step nilpotent from the perspective of structure matrices}
In this section we approach this problem of finding Ricci Yang-Mills solitons using so-called `structure matrices'. Studying 2-step nilpotent Lie groups from this point of view is quite natural and has been used by Eberlein, Jablonski, and others to obtain results about Ricci solitons. Using this approach we prove the existence of a large class of Ricci Yang-Mills solitons using Geometric Invariant Theory (cf. Theorem \ref{thm: RYM soliton vs distinguished pt}) and construct examples of nilmanifolds that do not admit left-invariant Ricci solitons but do admit left-invariant Ricci Yang-Mills solitons (see Section \ref{section: examples}).\\
The geometry of $N$ can be completely encoded by a tuple of structure matrices $(C^1,\dots, C^p)$ which is an element of $\mathfrak{so}(q)^p$. We establish this perspective below. Our main references are \cite{Eber07} and \cite{Jab:Thesis}.
Recall that a nilpotent Lie group $N$ with left-invariant metric $\langle,\rangle $ is equivalent to a nilpotent Lie algebra $\mathfrak N$ with inner product, also denoted $\langle,\rangle $. Let $\{X_1,\dots,X_q\} \cup \{Z_1,\dots ,Z_p\}$ be an orthonormal basis of $\mathfrak N$ where the $Z_k$ form an orthonormal basis of $\mathfrak Z$. Relative to this basis we may compute the structure constants $\{c_{ij}^k\}$ defined via $[X_i,X_j] = \sum_k c_{ij}^k Z_k$. Thus we may associate to our basis a $p$-tuple of matrices $(C^1,\dots,C^p)$ where $(C^k)_{ij}=c_{ij}^k$. Notice that different $p$-tuples of matrices can be associated to a given $\mathfrak N$. We describe below how these different tuples of matrices are related to each other.
Conversely, given a tuple $C=(C^1,\dots,C^p)$, we can naturally associate to it a metric 2-step nilpotent Lie algebra. This construction is dual to the construction of $p$-tuples above. We will require the $C^k$ to be linearly independent in $\mathfrak{so}(q)$ so that the commutator of the constructed nilpotent algebra will have dimension $p$.
Let $\{e_1,\dots,e_q,e_{q+1},\dots,e_{q+p} \}$ be the standard basis of $\mathbb R^q\oplus \mathbb R^p$. Endow $\mathbb R^q\oplus \mathbb R^p$ with the standard inner product so that this basis is orthonormal. Define the Lie bracket as
$$[e_i,e_j] = \sum_k C^k_{ij}e_{q+k}$$
for $1\leq i,j \leq q$, and let all other brackets be trivial. This Lie algebra is clearly 2-step nilpotent with commutator equal to $\mathbb R^p = span\langle e_{q+1}, \dots , e_{q+p}\rangle $. We denote the metric 2-step nilpotent Lie algebra associated to $C$ by $\mathfrak N_C$ and the 2-step nilpotent Lie group with left-invariant metric associated to $\mathfrak N_C$ by $N_C$.
Using tuples of matrices, we may study 2-step nilpotent Lie algebras not just individually, but as elements of the much larger space $\mathfrak{so}(q)^p$. As stated above, elements of $\mathfrak{so}(q)^p$ whose coordinates are linearly independent correspond to 2-step (metric) nilpotent Lie algebras of type $(p,q)$ (cf. Definition \ref{defin: type p,q}).
\subsection{Geometric Invariant Theory and 2-step nilpotent Lie algebras}
On the vector space $\mathfrak{so}(q)^p$ there is a natural action of $GL_q\mathbb R \times GL_p\mathbb R$ which is a linear representation. Given $g\in GL_q\mathbb R$ and $C=(C^1,\dots, C^p)$, we define
$$ g\cdot C = (gC^1g^t, \dots, gC^pg^t)$$
which acts on each coordinate individually. It is clear that $g\cdot C^k = gC^kg^t \in \mathfrak{so}(q)$ given that $C^k\in \mathfrak{so}(q)$. The action of $GL_p\mathbb R$ takes linear combinations of the coordinates of $C=(C^1,\dots ,C^p)$. Given $h\in GL_p\mathbb R$ we define
$$h\cdot C = D=(D^1,\dots, D^p) \quad \mbox{ with } \quad D^k = \sum_l h_{lk}C^l$$
One can compute that these actions commute and thus we have an action of $GL_q\mathbb R \times GL_p\mathbb R$ on $\mathfrak{so}(q)^p$.
Using this action, we can easily describe when two different structure matrices produce the same nilpotent Lie group. For proofs of the next two theorems we refer the reader to \cite{Eber07}.
\begin{thm} Let $C, D \in \mathfrak{so}(q)^p$ correspond to 2-step nilpotent Lie algebras $N_C, N_D$, respectively. Then $N_C$ and $N_D$ are isomorphic Lie groups if and only if $D \in GL_q\mathbb R \times GL_p \mathbb R \cdot C$, the orbit of $C$ under the group action of $GL_q\mathbb R \times GL_p\mathbb R$.
\end{thm}
Here we were only concerned with the underlying Lie group structure of $N_C$ and $N_D$. The next theorem considers the metric structures as well.
\begin{thm} Let $C, D \in \mathfrak{so}(q)^p$ correspond to 2-step nilpotent Lie algebras $N_C, N_D$, respectively, with left-invariant metrics. Then $N_C$ and $N_D$ are isometric as Riemannian manifolds if and only if $D \in O(q)\times O(p) \cdot C$, the orbit of $C$ under the compact subgroup $O(q)\times O(p) \subset GL_q\mathbb R \times GL_p\mathbb R$.
\end{thm}
This representation of $GL_q\mathbb R\times GL_p\mathbb R $ on $\mathfrak{so}(q)^p$ has even more structure from the view point of Geometric Invariant Theory. Once translated, these extra structures on the representation space have strong consequences on the Riemannian geometry of associated nilpotent Lie groups. We present a brief discussion below and refer the reader to \cite{Jab:DetectingOrbits} for a more thorough treatment.\\
Associated to the representation of $GL_q\mathbb R\times GL_p\mathbb R$ on $\mathfrak{so}(q)^p$, we have a Lie algebra representation of $\mathfrak{gl}_q\mathbb R \times \mathfrak{gl}_p\mathbb R$ on $\mathfrak{so}(q)^p$. This is obtained in the usual way via differentiation; that is, given $(X,Y)\in \mathfrak{gl}_q\mathbb R \times \mathfrak{gl}_p\mathbb R$ and $C\in \mathfrak{so}(q)^p$ we have
$$ (X,Y)\cdot C = X\cdot C + Y\cdot C,$$
where $X\cdot C = (XC^1+C^1X^t, \dots , XC^p + C^pX^t)$ and $Y\cdot C = D$ with $D^k = \sum_l Y_{lk} C^l$.
The space $\mathfrak{so}(q)$ has the inner product $\langle C,D\rangle = tr(CD^t) = - tr (CD)$. This inner product extends to $\mathfrak{so}(q)^p$ by making the coordinates of the tuple orthogonal; that is, consider $C=(C^1,\dots,C^p)$ and $D=(D^1,\dots,D^p)$ then
$$\langle C,D\rangle = \langle (C^1,\dots,C^p),(D^1,\dots,D^p)\rangle = \sum_\alpha \langle C^\alpha,D^\alpha\rangle = \sum_\alpha -tr(C^\alpha D^\alpha).$$
We define two polynomials, $m_1$ and $m_2$, which are associated to our representation of $GL_q\mathbb R \times GL_p\mathbb R$. Here $m_1$ is the moment map for the action of $GL(q,\mathbb{R})$ on $\mathfrak{so}(q)^p$, and $m_2$ is the moment map for the action of $GL(p,\mathbb{R})$ on $\mathfrak{so}(q)^p$. Notice that the polynomial $m_1$ will be valued in $symm_q$, the symmetric $q\times q$ matrices, while the polynomial $m_2$ will be valued in $symm_p$. For $C\in \mathfrak{so}(q)^p$, we define
\begin{eqnarray*} m_1(C) &=& -2 \sum_\alpha (C^\alpha)^2 \\
m_2(C)_{ij} &=& \langle C^i,C^j\rangle .
\end{eqnarray*}
Adding these together one has the moment map $m=m_1+m_2$ for the action of $GL_q\mathbb R \times GL_p\mathbb R$. This is valued in $symm_q \oplus symm_p$.
We are interested in three different group actions on $\mathfrak{so}(q)^p$; namely, the actions of the full group $GL_q\mathbb R\times GL_p\mathbb R$ and its subgroups $GL_q\mathbb R$, the first factor, and $SL_q\mathbb R \subset GL_q\mathbb R$. In the following definition, $G$ will denote one of these three groups, and $m_G$ will denote the moment map corresponding to $G$.
\begin{defin}\label{defin: distinguished points} We call a point $C\in \mathfrak{so}(q)^p$ $G$-distinguished if $m_G(C)\cdot C = r C$ for some $r\in \mathbb R$. We call a point $G$-minimal if $m_G(C)\cdot C =0$. Minimal points are obviously a special kind of distinguished point.
\end{defin}
Here $m_G(C) \in \mathfrak{gl}_q\mathbb R \times \mathfrak{gl}_p\mathbb R$, since $G$ is a subgroup of $GL_q\mathbb R \times GL_p\mathbb R$, and it acts via the Lie algebra action of $\mathfrak{gl}_q\mathbb R \times \mathfrak{gl}_p\mathbb R$. Distinguished and minimal points can be defined more generally for any representation of a reductive group on a vector space \cite{Jab:DistinguishedOrbits}.
\begin{prop} Let $C$ be a distinguished point as above, then $r\geq 0$.
\end{prop}
This is a consequence of the facts that $\nabla ||m_G||^2 (C) = m_G(C)\cdot C$ and that a function is non-decreasing along its gradient flow; for more details see \cite{Jab:DistinguishedOrbits}. In this setting distinguished points play a very interesting role (cf. Theorems \ref{thm: Ricci solion vs. distinguished point}, \ref{thm: geod flow inv vs. distinguished pt}, and \ref{thm: RYM soliton vs distinguished pt}).
\begin{thm}\label{thm: Ricci solion vs. distinguished point} Let $C\in \mathfrak{so}(q)^p$ correspond to a 2-step nilpotent Lie group $N_C$ with left-invariant metric. Then $N_C$ is a nilsoliton if and only if $C$ is a distinguished point of the $GL_q\mathbb R\times GL_p\mathbb R$ action.
\end{thm}
By nilsoliton we mean a nilpotent Lie group with left-invariant Ricci soliton metric. This was originally proven for all nilpotent Lie groups (not just 2-step) by Jorge Lauret \cite{La}. In the 2-step nilpotent setting, Eberlein \cite{Eber07} proved this in the frame work of structure matrices; this approach has proven very fruitful for constructing examples.
\begin{thm}\label{thm: geod flow inv vs. distinguished pt} Let $C\in \mathfrak{so}(q)^p$ corresponding to a 2-step nilpotent Lie group $N_C$ with left-invariant metric. Then the metric on $N_C$ is so-called geodesically flow invariant if and only if $C$ is a minimal point of the $SL_q \mathbb R$ action.
\end{thm}
We have not defined the notion of a metric being `geodesically flow invariant' and refer the reader to \cite{Eber07} for more details. This theorem is presented so that the reader may place Theorem \ref{thm: RYM soliton vs distinguished pt} in a more general context with Theorems \ref{thm: Ricci solion vs. distinguished point} and \ref{thm: geod flow inv vs. distinguished pt}.
\begin{question}\label{question: distinguished pts of GLqR action} Is there good geometric meaning to Lie groups $N_C$ which correspond to a point $C$ which is a distinguished point of the $GL_q\mathbb R$-action?
\end{question}
This question was asked to us by Pat Eberlein and has been answered in Theorem \ref{thm: RYM soliton vs distinguished pt}.
\subsection{Ricci Yang-Mills equations and Geometric Invariant Theory}
As Ricci solitons are naturally associated to distinguished points (cf. Definition \ref{defin: distinguished points} and Theorem \ref{thm: Ricci solion vs. distinguished point}), one is naturally lead to investigate if there is a similar connection between Ricci Yang-Mills solitons and Geometric Invariant Theory. There is a strong (and similar) relationship in the Ricci Yang-Mills setting. We will study the Ricci Yang-Mills soliton equations from the perspective of structure matrices. We begin by translating Eq.~\ref{rym w/deriv} into a system of equations on tuples of matrices.
In the following proposition, $C\in \mathfrak{so}(q)^p$ corresponds to the metric Lie group $N_C$, and $D$ is an element of $\mathfrak{gl}_q\mathbb R$. Additionally, we present the Ricci Yang-Mills equations here as $(1,1)$ tensors as opposed to $(2,0)$ tensors. From the perspective of structure matrices, it is more natural to present the equations in the following form.
\begin{prop}
The Ricci Yang-Mills soliton equations of Lie type (Eqns. \ref{rym w/deriv} a \& b) on 2-step nilpotent Lie groups can be written as
\begin{subequations}
\label{rym 2}
\begin{align}
m_1(C) = -2\sum_\alpha (C^\alpha)^2 &= 2\lambda Id + 2 (D+D^t)\\
0 &= D^t \cdot C
\end{align}
\end{subequations}
where $D^t \cdot C$ denotes the action of $\mathfrak{gl}_q$ on $\mathfrak{so}(q)^p$; that is, $D^t \cdot C = (D^t \cdot C^1, \dots , D^t \cdot C^p)$ and $D^t \cdot C^i = D^t C^i+C^iD$.
\end{prop}
\begin{proof}
We begin by raising an index on $\tilde \Omega^2$ and show that this is $\frac{1}{2}m_1(C)$. Recall that \[\tilde \Omega^2_{ij} = \sum_{\alpha \beta k l} g^{kl}k^{\alpha \beta} \tilde \Omega_{\alpha k i} \tilde \Omega_{\beta l j}.\]
We will use the orthonormal basis $\{X_i\} \cup \{Z_\alpha\}$ of $\mathfrak N$ that was used to calculate our structure matrix. Thus we have
\begin{eqnarray*}
\tilde \Omega^2_{ij} = \sum_{\alpha \beta k l} g^{kl}k^{\alpha \beta} \tilde \Omega_{\alpha k i} \tilde \Omega_{\beta l j} &=& \sum_{\alpha l} \tilde \Omega_{\alpha l i} \tilde \Omega_{\alpha l j}\\
&=& \sum_{\alpha l} (-C^\alpha)_{ l i} (-C^\alpha)_{ l j} = - \sum_{\alpha} (C^\alpha)^2_{ij} = \frac{1}{2} m_1(C)_{ij}
\end{eqnarray*}
Here we have used Lemma \ref{lemma: Omega vs structure matrices} to compare $\tilde \Omega ^2$ and $C$. The right-hand side of Eq.~\ref{rym w/deriv}a is easily converted to a $(1,1)$-tensor to obtain the claimed result.
For the second equation, recall that $\tilde \Omega = - \sum z_k c_{ij}^k \tilde \sigma_i \wedge \tilde \sigma_j$ and $c_{ij}^k = \langle C^k X_i,X_j\rangle $. Thus
\[\tilde \Omega(v,w) = - \sum_k z_k \langle C^kv,w\rangle ,\]
where we are identifying $\mathcal H \simeq \mathfrak N/ \mathfrak Z$ isometrically via $\pi_*$. Therefore
\[\tilde \Omega(Dv,w) + \tilde \Omega(v,Dw) = - \sum_k z_k \langle C^kDv,w\rangle + \langle C^kv,Dw\rangle = - \sum_k z_k \langle (C^k D + D^t C^k) v,w \rangle \]
as required.
\end{proof}
\begin{thm}
\label{thm: RYM soliton vs distinguished pt}
Let $N_C$ be the metric 2-step nilpotent Lie group corresponding to a tuple $C\in \mathfrak{so}(q)^p$. Then the metric on $N$ is a left-invariant Ricci Yang-Mills soliton of symmetric Lie type (cf. Definition \ref{defin: RYM soliton of Lie type}) if and only if $C$ is a distinguished point of the action of $GL_q\mathbb R$ on $\mathfrak{so}(q)^p$ (cf. Definition \ref{defin: distinguished points}).
\end{thm}
This theorem should be compared to its analogue for Ricci solitons (Theorem \ref{thm: Ricci solion vs. distinguished point}). An interesting and easily proved consequence of the above theorem is the following.
\begin{cor} Ricci Yang-Mills solitons of symmetric Lie type are expanders; i.e. have $\lambda >0$.
\end{cor}
\begin{proof}[Proof of theorem and corollary] By definition, $C$ being a distinguished point of the action of $GL_q$ on $\mathfrak{so}(q)^p$ is equivalent to $m_1(C)\cdot C = a(C) C$ where $a(C) > 0$. This holds if and only if $m_1(C)= \frac{1}{2} a(C) + B$ where $B\in Stab_C$, that is, $B\cdot C = 0$. Since $m_1(C)$ is always a symmetric matrix, $B$ is a symmetric matrix.
Using Eq.~\ref{rym 2}, we see that if $D$ is symmetric, we have our equivalence using $a=2\lambda$ and $D=-4B$. Lastly, $a>0$ implies $\lambda >0$.
\end{proof}
\begin{remark}
Notice that $\lambda >0$ agrees with the sign convention of \cite{La08}. There for nilpotent Lie groups, he defines the Ricci soliton equation to be $Rc=cI+D$ and shows that $c<0$. Our first soliton equation has that $Rc^H=-\lambda I+D$, where $Rc^H$ is the horizontal component of the Ricci tensor. \end{remark}
At this point we are able to use general theorems from Geometric Invariant Theory to prove results about the existence of Ricci Yang-Mills solitons.
\begin{cor}\label{cor: closed SL q C orbit implies RYM soliton} If the orbit $SL_q\mathbb R \cdot C$ is closed in $\mathfrak{so}(q)^p$ then the corresponding Lie group admits a (trivial) Ricci Yang-Mills soliton. Moreover, such metrics are precisely the geodesically flow-invariant metrics (cf. Theorem \ref{thm: geod flow inv vs. distinguished pt}).
\end{cor}
This corollary follows from the fact that if $SL_q\mathbb R \cdot C$ is closed, then there exists a minimal point on the orbit (assume it is $C$) satisfying $m_1(C) = r\ Id$. See \cite{Jab:Thesis} for more details. Once $m_1(C)$ has this form, it is clear that $C$ will be a (trivial) Ricci Yang-Mills soliton. This corollary provides us with a very general procedure for building examples of (trivial) Ricci Yang-Mills solitons.
\begin{remark} Consider 2-step nilpotent Lie groups of type $(p,q)$. If $ p \leq \frac{1}{2}q(q-1) -2$ then almost every $SL_q\mathbb R$-orbit is closed \cite{Jab:Thesis}. Hence, almost every 2-step nilpotent Lie group admits a Ricci Yang-Mills soliton when $p \leq \frac{1}{2}q(q-1) -2$. \end{remark}
In the nilpotent setting, we can make precise the sense in which Ricci Yang-Mills solitons are weaker than Ricci solitons. For this observation, we need the following theorem from \cite{Eber07}.
\begin{thm}\label{thm: Ricci soliton and geod flow invr} Let $N_C$ be a 2-step nilpotent Lie group of type $(p,q)$ with left-invariant metric corresponding to $C \in \mathfrak{so}(q)^p$. The metric nilpotent group $N_C$ is both a Ricci soliton and `geodesic flow invariant' if and only if $m_1(C)=r\ Id_q$ and $m_2(C)=s\ Id_p$ for some $r,s\in \mathbb R$.
\end{thm}
\begin{cor} If $N_C$ admits a geodesic flow invariant Ricci soliton, then such a metric is also a Ricci Yang-Mills soliton.
\end{cor}
Notice that, in general, a manifold that admits a Ricci soliton will not necessarily admit a Ricci Yang-Mills soliton, as Ricci Yang-Mills solitons are only defined on manifolds that are also principal bundles.
\subsection{Ricci Yang-Mills solitons on nilpotent Lie groups are always strong}\label{section: RYM sol of Lie type}
In this section we show that Ricci Yang-Mills solitons on nilpotent Lie groups are strong Ricci Yang-Mills solitons (cf. Definition \ref{defin: strong RYM soliton}). This is true for nilpotent Lie groups of arbitrary steps, not just 2-step nilpotent.
Let $N$ be a simply-connected nilpotent Lie group with central subgroup $Z$. The usual projection \linebreak $\pi : N \to N/Z$ is a principal bundle with structure group $G\simeq Z$. Here $Z$ is connected and so $N/Z$ is also simply-connected. At the identity element $e\in N$, $\pi_*:\mathfrak N \to \mathfrak N/\mathfrak Z$ has $Ker \ \pi_* = \mathfrak Z$, and the restriction $\pi_*|_\mathcal H : \mathcal H = \mathfrak Z^\perp \to \mathfrak N / \mathfrak Z$ is a linear isometry.
\begin{lemma} The projection $\pi$ is a Lie group homomorphism and thus
$$\pi \circ exp_N = exp_{N/Z} \circ \pi_*. $$
Moreover, since $exp$ is a diffeomorphism we also have
$$log_{N/Z} \circ \pi = \pi_* \circ log_N, $$
where $log$ is the inverse of $exp$.
\end{lemma}
Let $\varphi \in \mathfrak{Diff}(N/Z)$ be a diffeomorphism on the base $N/Z$. Then we define $\psi \in \mathfrak{Diff}_Z N$ by
$$\psi \circ exp_N ( X+ Z)= exp_N ( \pi_*|_\mathcal H ^{-1} \circ log_{N/Z} \circ \varphi \circ exp_{N/Z} \circ \pi_* (X) + Z ),$$
where $X\in \mathcal H$ and $Z\in \mathfrak Z$. Notice that $\psi$ is well-defined as $N$ being simply-connected implies $exp :\mathfrak N \to N$ is a diffeomorphism. By the Campbell-Baker-Hausdorff formula (cf. Theorem \ref{thm: CBH formula}) we see that $exp(Y)exp(Z) = exp(Y+Z)$ for any $Y\in \mathfrak N$ and $Z\in \mathfrak Z$, and hence this map is a bundle automorphism. Using the above lemma, it is straight-forward to show that $\psi$ induces $\varphi$, that is, $\pi \circ \psi = \varphi \circ \pi$. Thus we have the following proposition.
\begin{prop} The map $\psi$ induces $\varphi$ and hence Ricci Yang-Mills solitons on nilpotent Lie group are strong Ricci Yang-Mills solitons.
\end{prop}
\noindent
Notice that for a general principal bundle $\pi : P \to M$, one would not expect a diffeomorphism on $M$ to lift to a bundle automorphism on $P$. The above proposition works for our nilpotent groups since the quotient is a homomorphism of Lie groups.
\section{Examples}\label{section: examples}
We now provide examples of nilpotent Lie groups that do not admit Ricci solitons but that do admit Ricci Yang-Mills solitons. From the perspective of Geometric Invariant Theory, that such examples exist is not a surprise. It should be rare but not too uncommon because there should be plenty of points $C\in \mathfrak{so}(q)^p$ whose $SL_q\mathbb R\times SL_p\mathbb R$-orbit is not closed but whose $SL_q\mathbb R$-orbit is closed (cf. Corollary \ref{cor: closed SL q C orbit implies RYM soliton} and Theorem \ref{thm: Ricci soliton and geod flow invr}).
\begin{example}
\end{example}
The first example can be produced from the work of Cynthia Will. In \cite{Wi}, Will constructs a curve of (pairwise) non-isomorphic nilpotent Lie groups which do not admit Ricci solitons. We use the algebra corresponding to $t=1$ in her curve $\overline \mu_t$. This algebra has structure matrices
$$C^1 = \left[ \begin{BMAT}{cc.cc.cc}{cc.cc.cc}
& a^2&&&&\\
-a^2&&&&&\\
&&&&& 1\\
&&&&-1&\\
&&& 1&&\\
&&-1&&&
\end{BMAT} \right] ,
C^2 = \left[ \begin{BMAT}{cc.cc.cc}{cc.cc.cc}
&&&&& a\\
&&&&-a&\\
&&&\ 0&&\\
&&\ 0&&&\\
& a&&&&\\
-a&&&&&
\end{BMAT} \right] ,
C^3 = \left[ \begin{BMAT}{cc.cc.cc}{cc.cc.cc}
&&& a&&\\
&&-a&&&\\
& a&&&&\\
-a&&&&&\\
&&&&&\ 0\\
&&&&\ 0&
\end{BMAT} \right]$$
The algebra presented above is isomorphic to Will's example but has different structure matrices. The above is $g\cdot C= gCg^t$ where $g=diag\{a,a,1,1,1,1\}$ for Will's set of structure matrices $C$.
A simple computation shows that
\[m_1(C)=-2\left[ \begin{BMAT}{cc.cc.cc}{cc.cc.cc}
-a^4-2a^2&&&&&\\
&-a^4-2a^2&&&&\\
&&-1-a^2&&&\\
&&&-1-a^2&&\\
&&&&-1-a^2&\\
&&&&&-1-a^2
\end{BMAT} \right]
\]
By Theorem~\ref{thm: RYM soliton vs distinguished pt}, the above will be an Ricci Yang-Mills soliton if we can show $C$ is a distinguished point; that is, if $m_1(C)\cdot C = rC$ for some $r\in \mathbb R$. This is possible for $a^2 = \frac{-1+\sqrt 5}{2}>0$, and we have the desired result. Notice that in this case, our soliton is generated by the trivial vector field (i.e., $D=0$ in Eqn. \ref{rym 2}), since $m_1(C)$ is a multiple of the identity. We will provide examples below which are not generated by trivial vector fields.
\begin{example}
\end{example}
The second family of examples uses the manifolds constructed in \cite{Jab:Moduli}. In particular, one can construct continuous familes of algebras of type $(p,q)$ for $2\leq p \leq 6$ that are shown to not admit Ricci soliton metrics. To do so, we must first describe a process called concatenation.
Consider $A=(A_1, \dots , A_{p})\in \mathfrak{so}(q_1)^p$ and $B=(B_1,\dots ,B_{p})\in \mathfrak{so}(q_2)^{p}$ which are structure matrices associated to nilpotent Lie algebras $N_A$ and $N_B$ of types $(p,q_i)$, respectively. Then we can build a new nilpotent Lie algebra $N_C$ corresponding to the structure matrix $C\in \mathfrak{so}(q)^p$, where $q=q_1+q_2$ and $$C_i=\begin{pmatrix}A_i \\ & B_i \end{pmatrix}.$$
We call this process \textit{concatenation} and denote it by $C= A+_cB$. As $A$ and $B$ have linearly independent components, the same is true for $C$ and hence $C$ corresponds to a nilalgebra of type $(p,q)$. Additionally, we will abuse notation and concatenate $A\in \mathfrak{so}(q_1)^{p_1}$ and $B\in \mathfrak{so}(q_2)^{p_2}$ where $p_1 < p_2$. This is an element of $\mathfrak{so}(q_1+q_2) ^{p_2}$ defined as
$$ (A_1,\dots , A_{p_1}, \underbrace{0,\dots,0}_{p_2-p_1} ) +_c (B_1,\dots, B_{p_2} ).$$
We are interested in concatenating the following structure matrices.
Denote by $J$ the $2\times 2$ matrix $\begin{bmatrix} 0 & 1\\ -1 &0\end{bmatrix}$. Define $A_1 \in \mathfrak{so}(2k)$ to be the concatenation $A_1 = J\underbrace{+_c\dots +_c}_k J$. This is just a block diagonal matrix with all blocks being copies of $J$. Define $B_1,B_2,\dots, B_6 \in \mathfrak{so}(4)$ as
$$B_1= \left[ \begin{BMAT}{cc.cc}{cc.cc} 0 &1 &&\\ -1&0 &&\\ && 0&1\\ && -1&0\end{BMAT} \right], \
B_2 = \left[ \begin{BMAT}{cc.cc}{cc.cc} && 0&\ 1 \ \\ && \ 1\ &0\\ 0&-1&&\\ -1&0&&\end{BMAT} \right], \
B_3 = \left[ \begin{BMAT}{cc.cc}{cc.cc} && \ 1 \ &0 \\ &&0& \ 1\ \\ -1&0&&\\ 0&-1&&\end{BMAT} \right],
$$
$$B_4= \left[ \begin{BMAT}{cc.cc}{cc.cc} 0 &1 &&\\ -1&0 &&\\ && 0&-1\\ && 1&0\end{BMAT} \right], \
B_5 = \left[ \begin{BMAT}{cc.cc}{cc.cc} && 0&\ 1 \ \\ && -1\ &0\\ 0&\ 1&&\\ -1&0&&\end{BMAT} \right],\
B_6 = \left[ \begin{BMAT}{cc.cc}{cc.cc} && \ 1 \ &0 \\ &&0& -1\ \\ -1&0&&\\ 0&\ 1&&\end{BMAT} \right].
$$
Now define $C=(C_1, \ldots, C_j)=a_1A_1+_c(b_1B_1,c_1B_2)+_c\cdots +_c(b_{n-1}B_1,c_{n-1}B_2)+_c(d_1B_1, \ldots, d_jB_j)$ for $b_i, c_i, d_i\in \mathbb{R}$. As concatenations have such a simple presentation, it is easy to compute the value of $m_1$ at such an element. For details see \cite{Jab:Moduli}.
$$ m_1(C) = -2\sum_i C_i^2 = \left[\begin{BMAT}{cccccc}{cccccc}
2a_1^2 Id_{2k}& &&&&\\
&2(b_1^2+c_1^2)Id_4 &&&&\\
&& \ddots &&&\\
&&& 2(b_{n-1}^2+c_{n-1}^2)Id_4&&\\
&&&& 2(d_1^2 +\dots + d_j^2) Id_4 &\\
&&& & &
\addpath{(0,5,.)rudrdlu}
\addpath{(3,3,.)rldruddur}
\end{BMAT}\right]$$
Then by Theorem~\ref{thm: RYM soliton vs distinguished pt}, an algebra of this type will admit a Ricci Yang-Mills soliton as long as
\[a_1^2=b_1^2+c_1^2=\cdots=b_{n-1}^2+c_{n-1}^2=d_1^2+\cdots+d_j^2.
\]
Thus we have a $n-1$-parameter family of non-isomorphic algebras (by letting the $b_i$ vary) that admit Ricci Yang-Mills solitons but that do not admit Ricci solitons.
Notice that in this example, since $m_1(w)$ is a multiple of the identity, $D\equiv 0$. \\
\begin{example}
\end{example}
These examples of Ricci Yang-Mills solitons that are generated by non-trivial vector fields will be of types $(3,9), \ldots, (6,9)$. Again, it is shown in \cite{Jab:Moduli} that these algebras do not admit Ricci soliton metrics. Let $C$ be the concatenation
$$a_1 \left[ \J \right] +_c \lambda \left( \left[ \begin{BMAT}{c.c}{c.c}\J & \\&0 \end{BMAT}\right], \left[ \begin{BMAT}{c.c}{c.c}0&\\ &\J \end{BMAT} \right] \right) +_c (b_1 \ B_1,\dots,b_j \ B_j).$$
In this case, $m_1(C)\neq rId$, so if the manifold admits a Ricci Yang-Mills soliton, it will be nontrivial.
Specifically, we compute $m_1(C)$ to be
\[m_1(C)=-2\left[ \begin{BMAT}{c.ccc.c}{c.ccc.c}
2a_1^2 Id_2&&&&\\
&2\lambda^2&&&\\
&&4\lambda^2&&\\
&&&2\lambda^2&\\
&&&&2(b_1^2+\cdots+b_j^2) Id_4
\end{BMAT} \right].
\]
Then a Ricci Yang-Mills soliton is admitted if $4a_1^2=6\lambda^2=4(b_1^2+\cdots+b_j^2)=r$. Using the notation of Theorem~\ref{thm: RYM soliton vs distinguished pt}, we see that
\[B=\left[ \begin{BMAT}{c.ccc.c}{c.ccc.c}
0&&&&\\
&-\lambda^2&&&\\
&&\lambda^2&&\\
&&&-\lambda^2&\\
&&&&0
\end{BMAT} \right].
\]
A simple computation confirms that $B$ is a stabilizer of $C$. Thus we obtain a $j-1$-parameter family of non-trivial Ricci Yang-Mills soliton metrics on this algebra. Most algebras should admit many non-isometric Ricci Yang-Mills solitons.
|
2,877,628,088,986 | arxiv | \section{Introduction}
In a recent paper \cite{BaroneHidalgo}, we have investigated the role of couplings between quantum fields and external stationary currents (time-independent sources) concentrated along parallel branes with arbitrary co-dimensions. To do that, we have calculated the vacuum energies for a variety of models of quantum fields that interact with external stationary currents concentrated along parallel $D$-branes. As particular cases, for bosonic fields, we have considered external currents which could describe charge distributions and stationary dipole distributions along the branes.
Is is worthy mentioning that systems of quantum fields interacting with external potentials concentrated along branes have been treated in the literature; see, for instance, \cite{bordag,milton,S} and references cited therein. But, the coupling of quantum fields to external currents concentrated along branes is not a well-explored subject.
In order to fulfill a question left aside in reference \cite{BaroneHidalgo}, in this paper we make a deeper discussion on the description of multipole density distributions along branes with arbitrary dimensions by the use of external currents concentrated at specific regions of space. The results for charges and dipole distributions, which can be taken as generalizations of the ones exposed in reference \cite{BaroneHidalgo}, are presented, for completeness, in this paper and also in order to make clearer some discussions and correct a flaw in reference \cite{BaroneHidalgo}. The results obtained here for currents describing four-pole distributions and $N$-pole distributions are novel.
Along the paper, we shall consider models in $d+D+1$ dimensions described by a quantum field coupled to an external current concentrated along a $D$-brane and another one concentrated at a given point of space. The latter current represents a point-like test-charge which is used to investigate the force field produced by the former one.
We shall also use the same notation as in reference \cite{BaroneHidalgo}, where the coordinate $(D+d+1)$-vector is given by
\begin{equation}
\label{def4vetor}
x=(x^{0},x^{1},...,x^{d},x^{d+1},...,x^{d+D})\ ,
\end{equation}
and its perpendicular and parallel parts to the $D$-brane are, respectively,
\begin{eqnarray}
\label{defxperpx|}
{\bf x}_{\perp}&=&(x^{1},...,x^{d})\ ,\nonumber\\
{\bf x}_{\|}&=&(x^{d+1},...,x^{d+D})\ .
\end{eqnarray}
We shall also use similar notations for the momenta $k$, as well as for any other vector considered in this paper.
This work is outlined as follows: in section (\ref{escalar}), we consider models for the scalar field, with and without mass, coupled to currents describing distributions of charges, dipoles, quadrupoles and $N$-poles. Section (\ref{eletromagnetico}) is devoted to extending the previous results for the electromagnetic case. In section (\ref{conclusao}), we draw some final remarks and conclusions.
\section{Scalar Field}
\setcounter{equation}{0}
\label{escalar}
All over this section, we consider models for the massive scalar field, $\phi$, in $d+D+1$ dimensions, always interacting with an external current $J$ different for each model. The lagrangians of all models investigated have the same structure
\begin{equation}
\label{Lescalar}
{\cal L}_{scalar}=\frac{1}{2}(\partial_{\mu}\phi)(\partial^{\mu}\phi)-\frac{1}{2}m^{2}\phi^{2}+J\phi\ ,
\end{equation}
with the corresponding generating functional of the Green's functions
\begin{equation}
\label{rfv}
{\cal Z}=\exp\Biggl(-\frac{i}{2}\int\int\ d^{d+D+1}x\ d^{d+D+1}y\ \ J(x)\Delta_{F}(x,y)J(y)\Biggr)\ ,
\end{equation}
where $\Delta_{F}(x,y)$ is the Green's function
\begin{equation}
\label{defDeltaF}
\Delta_{F}(x,y)=\lim_{\varepsilon\to0}\int{\frac{d^{d+D+1}k}{(2\pi)^{d+D+1}}\ \frac{\exp\Bigl[ik(x-y)\Bigr]}{k^{2}-m^{2}+i\varepsilon}}\ .
\end{equation}
It is worthy mentioning that the current $J$ in the functional (\ref{rfv}) is not an auxiliary parameter introduced in order to perform perturbative calculations, as usualy considered in the literature. The current $J$ has here a physical meaning and shall not be set equal to zero.
In the limit $T\to\infty$, the generating functional of any quantum system, whose lagrangian density does not depend explicitly on the time coordinate (what is the case we are considering), can be written in the form \cite{Zee,Itzykson,Peskin}
\begin{equation}
\label{FG}
{\cal Z}=\exp(-iET)\ ,
\end{equation}
where $E$ is the lowest energy of the system and $T=\int_{-\infty}^{\infty}dx^{0}$.
Comparing Eq's (\ref{rfv}) and (\ref{FG}), we have
\begin{equation}
\label{rfv1}
E=\lim_{T\rightarrow 0}\frac{1}{2T}\int\int\ d^{d+D+1}x\ d^{d+D+1}y\ \ J(x)\Delta_{F}(x,y)J(y)\ .
\end{equation}
From now on, let us take the current $J$ to be composed by a fuction concentrated along a $D$-dimensional brane along with another function concentrated at a given point of space.
The first model we study is a generalization of the one exposed in \cite{BaroneHidalgo} and it is considered in this work for completeness and in order to make clearer the method employed in this whole paper in a simple example. The current corresponding to the first model is taken to be given by
\begin{equation}
\label{corrente1}
J_{I}({\bf x})=\sigma\delta^{d}({\bf x}_{\perp}-{\bf A})+\sigma_{0}\delta^{d+D}({\bf x}-{\bf a})\ ,
\end{equation}
where ${\bf A}=(A^{1},A^{2},...,A^{d})$ and ${\bf a}=(a^{1},a^{2},...,a^{d+D})$. The first term at the right-hand side of (\ref{corrente1}) is a distribution concentrated along a $D$-dimensional brane denoted by the vector ${\bf A}$, the last term at the right-hand side is a distribution concentrated at the point ${\bf a}$.
Substituting (\ref{defDeltaF}) and (\ref{corrente1}) into (\ref{rfv1}), discarding terms due to the brane self-interaction and the test-charge self-interaction, performing, in the following order, the integrals $dx^{0}dk^{0}dy^{0}d^{D}{\bf x}_{\|}d^{D}{\bf y}_{\|}d^{D}{\bf k}_{\|}d^{d}{\bf x}_{\perp}d^{d}{\bf y}_{\perp}$ and making a trivial change of variables, we arrive at
\begin{equation}
\label{qwe1}
E_{I}=-\sigma\sigma_{0}\int\frac{d^{d}{\bf k}_{\perp}}{(2\pi)^{d}}\frac{\exp[-i{\bf k}_{\perp}\cdot({\bf a}_{\perp}-{\bf A})]}{{\bf k}_{\perp}^{2}+m^{2}}\ ,
\end{equation}
where we have used that $T=\int dx^{0}$.
The analytic extension for the integral in (\ref{qwe1}) is calculated in reference \cite{BaroneHidalgo}, where we consider, separately, the situations with and without mass. For $m=0$ and $d\not=2$, we have
\begin{eqnarray}
\label{qwe2}
E_{I}(m&=&0,d\not=2)\nonumber \\
&=&-\frac{\sigma\sigma_{0}}{(2\pi)^{d/2}}2^{(d/2)-2}\Gamma\Bigl((d/2)-1\Bigr)|{\bf a}_{\perp}-{\bf A}|^{2-d}\ ,\cr
&\ &
\end{eqnarray}
where $\Gamma$ is the gamma function.
For a massive field, we have
\begin{eqnarray}
\label{qwe3}
&&E_{I}(m,d)=-\frac{\sigma\sigma_{0}}{(2\pi)^{d/2}}m^{d-2}\nonumber \\
&&\Bigl(m|{\bf a}_{\perp}-{\bf A}|\Bigr)^{1-(d/2)}K_{(d/2)-1}(m|{\bf a}_{\perp}-{\bf A}|),\
\end{eqnarray}
where $K_{\mu}(x)$ designates the $K$-Bessel function \cite{Arfken}. It is worth mentioning that expression (\ref{qwe3}) is valid for any $d>0$, even for $d=2$.
Assuming that $d\not=2$ and taking the limit $m\to 0$ in Eq. (\ref{qwe3}), we obtain the result (\ref{qwe2}), what can be done with the aid of the expression $K_{\nu}(z)\stackrel{z\to0}{\longrightarrow}\Gamma(\nu)2^{\nu-1}/z^{\nu}\ ,\ \nu\not=0$ . The energy for the situation where $m=0$ and $d=2$ is obtained with the help of (\ref{qwe3}) and the expression $K_{0}(z)\stackrel{z\to 0}{\longrightarrow}-\ln(z/2)-\gamma$ \cite{Arfken}, with $\gamma$ designating the Euler constant, as follows
\begin{eqnarray}
\label{qwe4}
E_{I}(m=0,d=2)&=&-\frac{\sigma\sigma_{0}}{2\pi}\lim_{m\to 0}\Biggl[K_{0}(m|{\bf a}_{\perp}-{\bf A}|)\Biggr]\cr\cr
&\cong&-\frac{\sigma\sigma_{0}}{2\pi}\lim_{m\to 0}\Biggl[-\ln\Biggl(\frac{m|{\bf a}_{\perp}-{\bf A}|}{2}\Biggr)-\gamma\Biggr]\cr\cr
&\cong&-\frac{\sigma\sigma_{0}}{2\pi}\lim_{m\to 0}\Biggl[-\ln\Biggl(\frac{m|{\bf a}_{\perp}-{\bf A}|}{2}\Biggr)-\gamma\cr\cr
&\ &+\ln(ma_{0})-\ln(ma_{0})\Biggr]\cr\cr
&\cong&\frac{\sigma\sigma_{0}}{2\pi}\ln\Biggl(\frac{|{\bf a}_{\perp}-{\bf A}|}{a_{0}}\Biggr)\cr\cr
&\ &+\frac{\sigma\sigma_{0}}{2\pi}\Bigl[\gamma-\ln(2)+\lim_{m\to 0}\ln(ma_{0})\Bigr]\cr\cr
&\rightarrow& \frac{\sigma\sigma_{0}}{2\pi}\ln\Biggl(\frac{|{\bf a}_{\perp}-{\bf A}|}{a_{0}}\Biggr)\ ,
\end{eqnarray}
where, in the fourth line, we have added and subtracted the term $\ln(ma_{0})$ inside the brackets, introducing an arbitrary length-dimensional finite constant $a_{0}$.
In the last line of the above expression, we have discarded all terms which do not depend on the distance $|{\bf a}_{\perp}-{\bf A}|$, even the divergent ones, once they do not contribute to the force between the test charge and the brane.
The presence of the arbitrary constant $a_{0}$ in the energy (\ref{qwe4}) does not produce any physical result, once the force between the test charge and the brane do not depend on the distance $|{\bf a}_{\perp}-{\bf A}|$. In fact, one could add the constant term $(\sigma\sigma_{0})/(2\pi)\ln(a_{0})$ to the energy (\ref{qwe4}), what leads to $E_{I}(m=0,d=2)=(\sigma\sigma_{0})/(2\pi)\ln(|{\bf a}_{\perp}-{\bf A}|)$. The constant $a_{0}$ was introduced for convenience, in order to make the argument of the logarithm dimensionless.
Let us take the restriction $D+d=3$, which corresponds to adopting a space-time with $3+1$ dimensions. In this case, we have, for $d=1$, $d=2$ and $d=3$, respectively, a uniform distribution of charges along a plane, a straight line and a point. The energy corresponding to the masless case and $d=2$ is given by (\ref{qwe4}). For the masless case, with $d=1$ and $d=3$, the results for the energy (\ref{qwe2}) read, respectively,
\begin{eqnarray}
\label{qwe5}
E_{I}(m=0,d=1)&=&\frac{\sigma\sigma_{0}}{2}|{\bf a}_{\perp}-{\bf A}|\cr\cr
E_{I}(m=0,d=3)&=&\frac{\sigma\sigma_{0}}{4\pi}|{\bf a}-{\bf A}|^{-1}\ ,
\end{eqnarray}
where, in the last line, we have suppressed the sub-index $\perp$ for the vector ${\bf a}$, once its parallel part does not exist whenever $D=0$, what happens once $D+d=3$ and $d=3$.
The results (\ref{qwe4}) and (\ref{qwe5}) agree with the ones obtained for the electromagnetic field in classical electrodynamics up to an overall sign, always present in comparing the scalar and electromagnetic fields. The second Eq. (\ref{qwe5}) is the coulombian interaction between two point charges $\sigma_{0}$ and $\sigma$ placed at positions ${\bf a}$ and ${\bf A}$, respectively, obtained in references \cite{Zee,BaroneHidalgo,Itzykson}.
The second current we study is given by
\begin{equation}
\label{corrente2}
J_{II}({\bf x})=\sigma V^{\mu}\Bigl(\partial_{\mu}\delta^{d}({\bf x}_{\perp}-{\bf A})\Bigr)+\sigma_{0}\delta^{d+D}({\bf x}-{\bf a})\ ,
\end{equation}
where $V^{\mu}$ is a four-vector taken to be constant and uniform in the reference frame we are performing the calculations, and also, with vanishing time and perpendicular components in this frame, that is, $V^{0}=0$ and ${\bf V}_{\|}=0$. The partial derivative in the above equation is with respect to the ${\bf x}$ coordinates.
Substituting (\ref{corrente2}) in (\ref{rfv1}), discarding terms due to self interactions, as before, performing a change of integration variables and an integration by parts we have
\begin{eqnarray}
\label{qwe6}
E_{II}&=&-\frac{\sigma\sigma_{0}}{T}\int\int d^{d+D+1}x\ d^{d+D+1}y\ \delta^{d}({\bf x}_{\perp}-{\bf A})\cr\cr
&\ &\times\ \delta^{d+D}({\bf y}-{\bf a})({\bf V}\cdot{\bf\nabla}_{\perp})\Delta_{F}(x,y)\ ,
\end{eqnarray}
where we have defined the operator ${\bf\nabla}_{\perp}=(\partial/\partial x^{1},\partial/\partial x^{2},...,\partial/\partial x^{d})$.
Using the Fourier representatin (\ref{defDeltaF}) in (\ref{qwe6}), operating with $({\bf V}\cdot{\bf \nabla}_{\perp})$, performing, in the following order, the integrals $dx^{0}dk^{0}dy^{0}d^{D}{\bf x}_{\|}d^{D}{\bf y}_{\|}d^{D}{\bf k}_{\|}d^{d}{\bf x}_{\perp}d^{d}{\bf y}_{\perp}$ and using the fact that $T=\int dx^{0}$ we have
\begin{eqnarray}
\label{azxc1}
E_{II}&=&-\sigma\sigma_{0}\int\frac{d^{d}{\bf k}_{\perp}}{(2\pi)^{d}}\frac{{\bf V}_{\perp}\cdot i{\bf k}_{\perp}}{{\bf k}_{\perp}^2+m^{2}}\exp\Bigl[i{\bf k}_{\perp}\cdot({\bf a}_{\perp}-{\bf A})\Bigr]\cr\cr
&=&-\sigma\sigma_{0}\bigl({\bf V}_{\perp}\cdot{\bf \nabla}_{a\perp}\bigr)\int\frac{d^{d}{\bf k}_{\perp}}{(2\pi)^{d}}\frac{1}{{\bf k}_{\perp}^2+m^{2}}\cr\cr
&\ &\exp\Bigl[i{\bf k}_{\perp}\cdot({\bf a}_{\perp}-{\bf A})\Bigr] ,
\end{eqnarray}
where we have defined the differential operator ${\bf\nabla}_{a\perp}=(\partial/\partial a^{1},...,\partial/\partial a^{d})$.
The integral which appears in (\ref{azxc1}) is the same one present in (\ref{qwe1}) up to the sign of the exponential argument. This different sign is irrelevant for the result \footnote{With a change of variables this sign can be inverted} . As already said, this integral is calculated in reference \cite{BaroneHidalgo} for $m=0$ and $m\not=0$, separately.
By using (\ref{qwe1}) and (\ref{qwe2}), the energy (\ref{azxc1}) for the masless case reads
\begin{eqnarray}
\label{azxc2}
E_{II}(m=0,d)&=&-\sigma\sigma_{0}\bigl({\bf V}_{\perp}\cdot{\bf \nabla}_{a\perp}\bigr)\frac{1}{(2\pi)^{d/2}}2^{(d/2)-2}\cr\cr
&\ &\Gamma\Bigl((d/2)-1\Bigr)|{\bf a}_{\perp}-{\bf A}|^{2-d}\cr\cr
&=&-\frac{\sigma_{0}}{(2\pi)^{d/2}}2^{(d/2)-1}\ \Gamma\bigl(d/2\bigr)\cr\cr
&\ &|{\bf a}_{\perp}-{\bf A}|^{1-d}\ (-\sigma{\bf V})\cdot\frac{{\bf a}_{\perp}-{\bf A}}{|{\bf a}_{\perp}-{\bf A}|} .
\end{eqnarray}
As we shall see, result (\ref{azxc2}) is the interaction energy between a point charge $\sigma_{0}$ at the position ${\bf a}$ and a dipole distribution along a $D$-brane placed at ${\bf A}$ with dipole density $-\sigma{\bf V}$.
Comparison of Eq's (\ref{qwe1}) and (\ref{qwe3}) allows us to write down the energy (\ref{azxc1}), for the massive field, in the form
\begin{eqnarray}
\label{azxc3}
E_{II}(m,d)&=&-\frac{\sigma\sigma_{0}}{(2\pi)^{d/2}}m^{d-2}\bigl({\bf V}_{\perp}\cdot{\bf \nabla}_{a\perp}\bigr)\cr\cr
&\ &\Bigl[\Bigl(m|{\bf a}_{\perp}-{\bf A}|\Bigr)^{1-(d/2)}K_{(d/2)-1}\Bigl(m|{\bf a}_{\perp}-{\bf A}|\Bigr)\Bigr]\cr\cr
&=&\frac{\sigma_{0}}{(2\pi)^{d/2}}m^{d/2}|{\bf a}_{\perp}-{\bf A}|^{-d/2}\cr\cr
&\ &K_{d/2}\Bigl(m|{\bf a}_{\perp}-{\bf A}|\Bigr)\Bigl((-\sigma{\bf V})\cdot({\bf a}_{\perp}-{\bf A})\Bigl) ,
\end{eqnarray}
where, in the last line, we used the fact that
\begin{equation}
\frac{d}{dx}\Bigl(x^{1-(d/2)}K_{(d/2)-1}(x)\Bigr)=-x^{1-(d/2)}K_{d/2}(x)\ .
\end{equation}
Taking the limit of vanishing mass in the result (\ref{azxc3}), we obtain the energy (\ref{azxc2}), even for the case $d=2$.
Now, we restrict to the case $D+d=3$, which means that we are in a $3+1$ space-time. In this case, taking $d=1$, $d=2$ and $d=3$ means that the brane is reduced to a plane, a line and a point, respectively, and the corresponding energies read
\begin{eqnarray}
\label{azxc4}
E_{II}(m,d=1)&=&\frac{\sigma\sigma_{0}}{2}|{\bf a}_{\perp}-{\bf A}|^{-1}\cr\cr
&\ &\exp{(m|{\bf a}_{\perp}-{\bf A}|)}{\bf V}\cdot({\bf a}_{\perp}-{\bf A})\ ,\cr\cr
E_{II}(m,d=2)&=&\frac{\sigma\sigma_{0}}{2\pi}m|{\bf a}_{\perp}-{\bf A}|^{-1}\cr\cr
&\ &K_{1}(m|{\bf a}_{\perp}-{\bf A}|){\bf V}\cdot({\bf a}_{\perp}-{\bf A})\ ,\cr\cr
E_{II}(m,d=3)&=&\frac{\sigma\sigma_{0}}{4\pi}m|{\bf a}-{\bf A}|^{-2}\exp{(-m|{\bf a}-{\bf A}|)}\cr\cr
&\ &\Biggl(1+\frac{1}{m|{\bf a}-{\bf A}|}\Biggr){\bf V}\cdot({\bf a}-{\bf A})\ ,
\end{eqnarray}
where, in the last line, we have discarded the sub-index $\perp$, once $D=0$.
For a vanishing mass, the last equation (\ref{azxc4}) reads
\begin{equation}
\label{dipolospontuais}
E_{II}(m=0,d=3)=-\frac{\sigma_{0}\sigma}{4\pi}\frac{(-\sigma{\bf V})\cdot({\bf a}-{\bf A})}{|{\bf a}-{\bf A}|^{-2}}\ .
\end{equation}
which is the interaction energy between a test scalar charge $\sigma$ placed at the point ${\bf a}$ and a scalar dipole $-\sigma{\bf V}$ lying in the position ${\bf A}$. It is important to notice that, in comparing with the electromagnetic field, we have an overall minus sign.
From the above computations, we can interpret the first term in the current (\ref{corrente2}) as a uniform distribution of stationary dipoles with dipole momentum density given by $-\sigma{\bf V}$ along a $D$-dimensional brane placed at ${\bf A}$.
In the paper \cite{BaroneHidalgo}, we have considered a scalar current composed by an arbitrary number $N$ of terms similar to the first one present in (\ref{corrente2}). Each term was taken to be concentrated along a different brane and all of them were taken to be parallel to one another. We have also taken a different four-vector $V^{\mu}_{(i)}$ ($i=1..N$) for each term. In order to analyze the meaning of the considered current, we have considered two point-like branes and a $3+1$ space-time. Once the interaction energy we have obtained contains only terms proportional to the products ${\bf V}_{i}\cdot{\bf V}_{j}$, ($i=1,2$), as shown in equation (43) of reference \cite{BaroneHidalgo}, that is
\begin{equation}
{\cal E}=\frac{-1}{4\pi a^{3}}\Bigl[\Bigr(\sigma_{1}{\bf V}_{(1)}\cdot\sigma_{2}{\bf V}_{(2)}\Bigr)-3\Bigl(\sigma_{1}{\bf V}_{(1)}\cdot{\hat a}\Bigr)\!\!\!\Bigl(\sigma_{2}{\bf V}_{(2)}\cdot{\hat a}\Bigr)\Bigr]\ ,
\end{equation}
we have interpreted each term in the current as a dipole with the wrong sign. In this paper, we correct this point and the true interpretation is the one exposed after Eq. (\ref{dipolospontuais}).
The third and last model we consider for the scalar field is determined by the current
\begin{equation}
\label{corrente3}
J_{III}({\bf x})=\sigma V^{\mu\nu}\Bigl(\partial_{\mu}\partial_{\nu}\delta^{d}({\bf x}_{\perp}-{\bf A})\Bigr)+\sigma_{0}\delta^{d+D}({\bf x}-{\bf a})\ ,
\end{equation}
where $V^{\mu\nu}$ is a symmetric tensor with rank-2 taken to be constant and uniform in the refrence frame we are performing the calculations. From (\ref{corrente3}), it can be noticed that we can take $V^{0\mu}=V^{i\mu}=0$, $i=d+1,...,D$ with no loss of generality.
Substituting the current (\ref{corrente3}) in equation (\ref{rfv1}), performing two integrations by parts, integrating in the variables $dx^{0}dk^{0}dy^{0}d^{D}{\bf x}_{\|}d^{D}{\bf y}_{\|}d^{D}{\bf k}_{\|}d^{d}{\bf x}_{\perp}d^{d}{\bf y}_{\perp}$ and using the definitions of $T$ and the operator ${\bf\nabla}_{a\perp}$, both employed in equation (\ref{azxc1}), we have
\begin{eqnarray}
\label{azxc5}
&&E_{III}(m,d)=\cr\cr
&&\!\!\!\!\!\!\!\!\!\! =-\sigma\sigma_{0}\sum_{i,j=1}^{d}\int\frac{d^{d}{\bf k}_{\perp}}{(2\pi)^{d}}\frac{(i{\bf k}_{\perp})^{i}\ V^{ij}\ (i{\bf k}_{\perp})^{j}}{{\bf k}_{\perp}^{2}+m^{2}}\cr\cr
&& \exp{\bigl[i{\bf k}_{\perp}\cdot({\bf a}_{\perp}-{\bf A})\bigr]}\cr\cr
&&\!\!\!\!\!\!\!\!\!\! =-\sigma\sigma_{0}\sum_{i,j=1}^{d}V^{ij}{\bf\nabla}_{a\perp}^{i}{\bf\nabla}_{a\perp}^{j}\int\frac{d^{d}{\bf k}_{\perp}}{(2\pi)^{d}}\frac{1}{{\bf k}_{\perp}^{2}+m^{2}}\cr\cr
&&\exp{\bigl[i{\bf k}_{\perp}\cdot({\bf a}_{\perp}-{\bf A})\bigr]} .
\end{eqnarray}
As before, we first consider the masless case and, next, the massive field. For this model, this two-step analysis is important in order to identify a freedom in the choice of $V^{\mu\nu}$.
Once we do not take the test charge in the brane, that is, ${\bf a}_{\perp}\not={\bf A}$, and using Fourier representation for the Dirac delta function, we can write
\begin{eqnarray}
\label{azxc6}
&&\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! {\bf\nabla}_{a\perp}^{2}\int\frac{d^{d}{\bf k}_{\perp}}{(2\pi)^{d}}\frac{1}{{\bf k}_{\perp}^{2}}\exp{\bigl[i{\bf k}_{\perp}\cdot({\bf a}_{\perp}-{\bf A})\bigr]}=\nonumber \\
&&\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! -\int\frac{d^{d}{\bf k}_{\perp}}{(2\pi)^{d}}\exp{\bigl[i{\bf k}_{\perp}\cdot({\bf a}_{\perp}-{\bf A})\bigr]}=\delta^{d}({\bf a}_{\perp}-{\bf A})=0\ .
\end{eqnarray}
For the masless case, the energy (\ref{azxc5}) reads
\begin{eqnarray}
\label{azxc7}
&&E_{III}(m=0,d)=\cr\cr
&&-\sigma\sigma_{0}\sum_{i,j=1}^{d}V^{ij}{\bf\nabla}_{a\perp}^{i}{\bf\nabla}_{a\perp}^{j}\int\frac{d^{d}{\bf k}_{\perp}}{(2\pi)^{d}}\frac{1}{{\bf k}_{\perp}^{2}}\exp{\bigl[i{\bf k}_{\perp}\cdot({\bf a}_{\perp}-{\bf A})\bigr]}\cr\cr
&&+\sigma\sigma_{0}\frac{1}{d}(tr V)\left[{\bf\nabla}_{a\perp}^{2}\int\frac{d^{d}{\bf k}_{\perp}}{(2\pi)^{d}}\frac{1}{{\bf k}_{\perp}^{2}}\exp{\bigl[i{\bf k}_{\perp}\cdot({\bf a}_{\perp}-{\bf A})\bigr]}\right]\ ,\cr
&&\
\end{eqnarray}
where we have introduced a vanishing term (the second one in the right hand side), as stated in expression (\ref{azxc6}). In the above equation, $tr V$ stands for the trace of tensor $V$.
Defining the traceless tensor
\begin{equation}
\label{defD}
D^{ij}=V^{ij}-\frac{tr V}{d}\delta^{ij}\ ,
\end{equation}
the energy (\ref{azxc7}) can be rewritten in the form
\begin{eqnarray}
\label{aqwe1}
&&E_{III}(m=0,d)=\nonumber \\
&&-\sigma\sigma_{0}\sum_{i,j=1}^{d}D^{ij}{\bf\nabla}_{a\perp}^{i}{\bf\nabla}_{a\perp}^{j}\int\frac{d^{d}{\bf k}_{\perp}}{(2\pi)^{d}}\frac{1}{{\bf k}_{\perp}^{2}}\exp{\bigl[i{\bf k}_{\perp}\cdot({\bf a}_{\perp}-{\bf A})\bigr]}\ .\cr
&&
\end{eqnarray}
If we compare Eq's (\ref{qwe1}) and (\ref{qwe2}), we can obtain the integral which appears in the above equation and write the energy (\ref{aqwe1}) in the form
\begin{eqnarray}
\label{aqwe2}
E_{III}(m=0,d)=
-\frac{\sigma\sigma_{0}}{(2\pi)^{d/2}}2^{(d/2)-2}\Gamma\bigl((d/2)-1\bigr)\cr\cr
\sum_{i,j=1}^{d}D^{ij}{\bf\nabla}_{a\perp}^{i}{\bf\nabla}_{a\perp}^{j}|{\bf a}_{\perp}-{\bf A}|^{2-d}\cr\cr
=-\frac{\sigma_{0}}{\pi^{d/2}}\Gamma\bigl((d/2)+1\bigr)\frac{1}{|{\bf a}_{\perp}-{\bf A}|^{d}}\cr\cr
\sum_{i,j=1}^{d}\frac{({\bf a}_{\perp}-{\bf A})^{i}}{|{\bf a}_{\perp}-{\bf A}|}(\sigma D^{ij})\frac{({\bf a}_{\perp}-{\bf A})^{j}}{|{\bf a}_{\perp}-{\bf A}|}\ .
\end{eqnarray}
The result (\ref{aqwe2}) is the interaction energy between a point charge and a four-pole distribution along a $D$-brane with four-pole momentum desnsity given by $\sigma D^{ij}$. This point can be made clearer if we consider a $3+1$ space-time, which corresponds to take $D+d=3$. With this restriction $d$ can assume the values $1$, $2$ and $3$, leading to the energies
\begin{widetext}
\begin{eqnarray}
\label{aqwe3}
E_{III}(m=0,d=1)&=&-\frac{\sigma_{0}}{2}\frac{1}{|{\bf a}_{\perp}-{\bf A}|}\frac{({\bf a}_{\perp}-{\bf A})^{i}}{|{\bf a}_{\perp}-{\bf A}|}(\sigma D^{ij})\frac{({\bf a}_{\perp}-{\bf A})^{j}}{|{\bf a}_{\perp}-{\bf A}|}\ ,\cr\cr
E_{III}(m=0,d=2)&=&-\frac{\sigma_{0}}{\pi}\frac{1}{|{\bf a}_{\perp}-{\bf A}|^{2}}\frac{({\bf a}_{\perp}-{\bf A})^{i}}{|{\bf a}_{\perp}-{\bf A}|}(\sigma D^{ij})\frac{({\bf a}_{\perp}-{\bf A})^{j}}{|{\bf a}_{\perp}-{\bf A}|}\ ,\cr\cr
E_{III}(m=0,d=3)&=&-\frac{3\sigma_{0}}{4\pi}\frac{1}{|{\bf a}-{\bf A}|^{3}}\frac{({\bf a}-{\bf A})^{i}}{|{\bf a}-{\bf A}|}(\sigma D^{ij})\frac{({\bf a}-{\bf A})^{j}}{|{\bf a}-{\bf A}|}\ .
\end{eqnarray}
\end{widetext}
The last equation ({\ref{aqwe3}) is the interaction energy between a point charge placed at ${\bf a}$ and a point-like four-pole placed at ${\bf A}$ with four pole momentum given by $\sigma D^{ij}$.
It is interesting to notice that the trace of the tensor $V$ in (\ref{corrente3}) is irrelevant for the energy (\ref{aqwe2}). This situation is different when we consider the field with mass, as it shall be shown.
When $m\not=0$, we can compare Eq's (\ref{qwe1}) and (\ref{qwe3}) in order to obtain the integral of Eq. (\ref{azxc5}), which takes the form
\begin{eqnarray}
\label{aqwe4}
&&E_{III}(m,d)=-\frac{\sigma\sigma_{0}}{(2\pi)^{d/2}}m^{d-2}\sum_{i,j=1}^{d}V^{ij}{\bf\nabla}_{a\perp}^{i}{\bf\nabla}_{a\perp}^{j}\cr\cr
&&\Bigl[(m|{\bf a}_{\perp}-{\bf A}|)^{1-(d/2)}K_{(d/2)-1}(m|{\bf a}_{\perp}-{\bf A}|)\Bigr]\cr\cr
&&=\frac{\sigma_{0}}{(2\pi)^{d/2}}m^{d/2}|{\bf a}_{\perp}-{\bf A}|^{-d/2}\cr\cr
&&\sum_{i,j=1}^{d}(\sigma V^{ij})\Biggl[\delta^{ij}K_{d/2}(m|{\bf a}_{\perp}-{\bf A}|)\cr\cr
&&-m|{\bf a}_{\perp}-{\bf A}|\frac{({\bf a}_{\perp}-{\bf A})^{i}}{|{\bf a}_{\perp}-{\bf A}|}\frac{({\bf a}_{\perp}-{\bf A})^{j}}{|{\bf a}_{\perp}-{\bf A}|}K_{(d/2)+1}(m|{\bf a}_{\perp}-{\bf A}|)\Biggr]\ .\cr
&&\
\end{eqnarray}
Expression (\ref{aqwe4}) exhibits a dependence on the trace $tr(V)$, which cannot be removed as before, in the masless case. So, the four-pole tensor $V$ cannot be defined as being traceless for the massive field.
In order to compare the four-pole energies for the field with and without mass and write a single expression for both cases, let us proceed similarly to what we have done in the masless case and add a vanishing term to the energy (\ref{aqwe4}) given by
\begin{eqnarray}
\label{defDeltaE}
&&\!\!\!\!\!\!\! \Delta E_{III}(m,d)=\cr\cr
&&\!\!\!\!\!\!\! =\sigma\sigma_{0}\frac{tr V}{d}\Bigl({\bf\nabla}_{a\perp}^{2}-m^{2}\Bigr)\int\frac{d^{d}{\bf k}_{\perp}}{(2\pi)^{d}}\frac{1}{{\bf k}_{\perp}^{2}+m^{2}}\cr\cr
&&\ \ \ \ \ \ \ \exp{\bigl[i{\bf k}_{\perp}\cdot({\bf a}_{\perp}-{\bf A})\bigr]}\cr\cr
&&\!\!\!\!\! =-\frac{\sigma\sigma_{0}}{(2\pi)^{d/2}}m^{d/2}|{\bf a}_{\perp}-{\bf A}|^{-d/2}\cr\cr
&&\!\!\!\!\!\!\! \sum_{i,j=1}^{d}\frac{tr V}{d}\delta^{ij}\Biggl[\delta^{ij}K_{d/2}(m|{\bf a}_{\perp}-{\bf A}|)\cr\cr
&&\!\!\!\!\!\!\! -m|{\bf a}_{\perp}-{\bf A}|\frac{({\bf a}_{\perp}-{\bf A})^{i}}{|{\bf a}_{\perp}-{\bf A}|}\frac{({\bf a}_{\perp}-{\bf A})^{j}}{|{\bf a}_{\perp}-{\bf A}|}K_{(d/2)+1}(m|{\bf a}_{\perp}-{\bf A}|)\Biggr]\cr\cr
&&\!\!\!\!\!\!\! -\frac{\sigma\sigma_{0}}{(2\pi)^{d/2}}m^{d} \frac{tr V}{d}\ (m|{\bf a}_{\perp}-{\bf A}|)^{1-(d/2)}K_{(d/2)-1}(m|{\bf a}_{\perp}-{\bf A}|) .\cr
&\ &\!\!\!\!\!
\end{eqnarray}
Using the Fourier representation for the Dirac delta function and the fact that ${\bf a}_{\perp}\not={\bf A}$, one can show that the right-hand side of the first line of Eq. (\ref{defDeltaE}) is equal to zero, so $\Delta E_{III}(m,d)=0$. Combining equations (\ref{aqwe4}) and (\ref{defDeltaE}), we get a new expression for the four-pole energy
\begin{eqnarray}
\label{tgb1}
&&\!\!\!\!\!\!\!\!\!\!\!\! E_{III}(m,d)\rightarrow E_{III}(m,d)+\Delta E_{III}(m,d)\cr\cr
&&\!\!\!\!\!\!\!\!\!\!\!\! =\frac{\sigma\sigma_{0}}{(2\pi)^{d/2}}m^{d/2}|{\bf a}_{\perp}-{\bf A}|^{-d/2}\cr\cr
&&\!\!\!\!\!\!\!\!\!\!\!\! \sum_{i,j=1}^{d}\Biggl[V^{ij}-\frac{tr V}{d}\delta^{ij}\Biggr] \Biggl[\delta^{ij}K_{d/2}(m|{\bf a}_{\perp}-{\bf A}|)\cr\cr
&&\!\!\!\!\!\!\!\!\!\!\!\! -m|{\bf a}_{\perp}-{\bf A}|\frac{({\bf a}_{\perp}-{\bf A})^{i}}{|{\bf a}_{\perp}-{\bf A}|}\frac{({\bf a}_{\perp}-{\bf A})^{j}}{|{\bf a}_{\perp}-{\bf A}|}K_{(d/2)+1}(m|{\bf a}_{\perp}-{\bf A}|)\Biggr]\cr\cr
&&\!\!\!\!\!\!\!\!\!\!\!\! -\frac{\sigma\sigma_{0}}{(2\pi)^{d/2}}m^{d} \frac{tr V}{d}\ (m|{\bf a}_{\perp}-{\bf A}|)^{1-(d/2)}K_{(d/2)-1}(m|{\bf a}_{\perp}-{\bf A}|)\ .\cr
&&\
\end{eqnarray}
By using the definition of the traceless tensor (\ref{defD}}), Eq. (\ref{tgb1}) can be rewritten in the form
\begin{eqnarray}
\label{aqwe5}
E_{III}(m,d)=
-\frac{\sigma_{0}}{(2\pi)^{d/2}}m^{1+(d/2)}|{\bf a}_{\perp}-{\bf A}|^{1-d/2}\cr\cr
\Biggl[\frac{tr(\sigma V)}{d}K_{(d/2)-1}(m|{\bf a}_{\perp}-{\bf A}|)\cr\cr
+\sum_{i,j=1}^{d}\!\!\!\frac{({\bf a}_{\perp}-{\bf A})^{i}}{|{\bf a}_{\perp}-{\bf A}|}(\sigma D^{ij})\frac{({\bf a}_{\perp}-{\bf A})^{j}}{|{\bf a}_{\perp}-{\bf A}|}K_{(d/2)+1}(m|{\bf a}_{\perp}-{\bf A}|)\Biggr] .\cr
\ \!\!
\end{eqnarray}
Result (\ref{aqwe5}) is equivalent to (\ref{aqwe4}) and gives the interaction energy, intermediated by the massive scalar field, between a point charge placed at ${\bf a}$ and an uniform distribution of four-poles lying along a brane, placed at ${\bf A}$, and with four pole density $\sigma V$. In the limit $m\to0$, Eq. (\ref{aqwe5}) reduces to (\ref{aqwe2}). In (\ref{aqwe5}) we have, explicitly, separated the contribution to the energy due to the trace of the tensor $V$ and a contribution which does not come from the trace of $V$.
For $d=3$, and considering a space-time with $3+1$ dimensions, we have a system composed by a point-like four-pole and a test charge, which has the corresponding interaction energy
\begin{eqnarray}
E_{III}(m,d=3)=-\frac{\sigma_{0}}{4\pi}m^{2}\frac{1}{|{\bf a}-{\bf A}|}\exp{(-m|{\bf a}-{\bf A}|)}\cr\cr
\Biggl[\frac{tr(\sigma V)}{3}+\frac{({\bf a}-{\bf A})^{i}}{|{\bf a}-{\bf A}|}(\sigma D^{ij})\frac{({\bf a}-{\bf A})^{j}}{|{\bf a}-{\bf A}|}\cr\cr
\times\Biggl(1+\frac{3}{m|{\bf a}-{\bf A}|}+\frac{3}{m^{2}|{\bf a}-{\bf A}|^{2}}\Biggr)\Biggr]\ .
\end{eqnarray}
To conclude this section, we consider the stationary current distribution in $3+1$ dimensions given by
\begin{equation}
\label{ade1}
J_{n}({\bf x})=\sigma^{\mu_1\mu_2\mu_3...\mu_n}
[\partial_{\mu_1\mu_2\mu_3...\mu_n}^n\delta^{3}({\bf x}-{\bf A})]
+\sigma_{0}\delta^{3}({\bf x}-{\bf a}),
\end{equation}
that is, a point charge located at ${\bf a}$ and the derivative of arbitrary order of a Dirac's delta function concentrated at the point ${\bf A}$.
The quantity $\sigma^{\mu_1\mu_2\mu_3...\mu_n}$ is a completely symmetric tensor.
For the sake of simplicity, we consider only the four-dimensional case and point-like Dirac's delta functions.
Replacing the current (\ref{ade1}) in (\ref{rfv1}) and proceeding as before, we obtain
\begin{equation}
\label{ade2}
E_{n}=-\frac{\sigma_0}{2}
\sigma^{\mu_1\mu_2\mu_3...\mu_n}
\partial_{({\bf a})\mu_1\mu_2\mu_3...\mu_n}^n\left(\frac{e^{-m|{\bf a}-{\bf A}|}}{4\pi |{\bf a}-{\bf A}|}\right)\ ,
\end{equation}
where $\partial_{({\bf a})\mu_1\mu_2\mu_3...\mu_n}^n$ means the derivative with respect to the ${\bf a}$ coordinates.
Considering the limit $m\to 0$ and defining the vector ${\bf r}={\bf a}-{\bf A}$ we get
\begin{equation}
\label{ade3}
E_n=-\frac{\sigma_0}{8\pi}\sigma^{\mu_1\mu_2\mu_3...\mu_n}
\partial_{\mu_{1}\mu_{2}\mu_{3}...\mu_{n}}^n\frac{1}{r}.
\end{equation}
It can be easily verified that the expression above gives the interaction between a point charge and an $N$-pole. For instance, whenever $n=3$, we have the interaction between a point charge and a four-pole; for $n=4$, the interaction between a point charge and an octupole, and so on.
\section{Maxwell Field}
\setcounter{equation}{0}
\label{eletromagnetico}
In this section, we extend the results obteined for the scalar field, in the previous section, for the electromagnecic field. We always take models described by the lagrangian density
\begin{equation}
\label{Leletromagnetico}
{\cal L}=-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}-\frac{1}{2\alpha}(\partial_{\mu}A^{\mu})^{2}+J^{\mu}A_{\mu}\ ,
\end{equation}
where $A^{\mu}$ is the electromagnetic field, $F_{\mu\nu}=\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}$ is the field strength and $J^{\mu}$ is a stationary external current, different for each model we consider. In the above equation $\alpha$ is a gauge parameter.
Following similar steps which lead to equation (\ref{rfv1}), the vacuum energy corresponding to the lagrangian (\ref{Leletromagnetico}) can be writen in the form
\begin{equation}
\label{arfv1}
E=\frac{1}{2T}\int\int d^{d+D+1}x d^{d+D+1}x\ J^{\mu}(x)\ \Delta_{\mu\nu}(x,y)\ J^{\nu}(y)\ ,
\end{equation}
where $\Delta_{\mu\nu}(x,y)$ is the photon propagator
\begin{eqnarray}
\label{defDeltamunu}
&&\!\!\!\!\! \Delta_{\mu\nu}(x,y)=\nonumber \\
&&\!\!\!\!\! -\int\frac{d^{d+D+1}k}{(2\pi)^{d+D+1}}\frac{1}{k^{2}}\Biggl[\eta_{\mu\nu}-(1-\alpha)\frac{k_{\mu}k_{\nu}}{k^{2}}\Biggl]\exp{[-ik(x-y)]}\ .\cr
&&\
\end{eqnarray}
The first model we study is given by the current
\begin{equation}
\label{corrente4}
J_{IV}^{\mu}=\sigma W^{\mu} \delta^{d}({\bf x}_{\perp}-{\bf A})+\sigma_{0} \eta^{\mu 0} \delta^{d+D}({\bf x}-{\bf a})\ ,
\end{equation}
which is a generalization of the one specified by the current (\ref{corrente1}). The quantity $\ W^{\mu}$ is a $(d+D+1)$-vector taken to be constant and uniform in the reference frame we are performing the calculations. In order to ensure gauge invariance for the last term in the right-hand side of the lagrangian (\ref{Leletromagnetico}), the $(d+D+1)$-vector $W^{\mu}$ must satisfy the condition ${\bf W}_{\perp}=0$.
Inserting (\ref{corrente4}) into (\ref{arfv1}), using the Fourier representation (\ref{defDeltamunu}), performing, in the following order, the integrals $dx^{0}dk^{0}dy^{0}d^{D}{\bf x}_{\|}d^{D}{\bf y}_{\|}d^{D}{\bf k}_{\|}d^{d}{\bf x}_{\perp}d^{d}{\bf y}_{\perp}$ and taking into account that ${\bf W}_{\perp}=0$ we have
\begin{equation}
E_{IV}(d)=\sigma_{0}(\sigma W_{0})\int\frac{d^{d}{\bf k}_{\perp}}{(2\pi)^{d}}\frac{1}{{\bf k}_{\perp}^{2}}\exp{[i{\bf k}_{\perp}\cdot({\bf a}_{\perp}-{\bf A})]}\ .
\end{equation}
In reference \cite{BaroneHidalgo}, the above integral is calculated for any $d\not=2$,
\begin{equation}
\label{xsw1}
E_{IV}(d)=\frac{\sigma_{0}(\sigma W_{0})}{(2\pi)^{d/2}}\Gamma\bigl((d/2)-1\bigr)2^{(d/2)-2}|{\bf a}_{\perp}-{\bf A}|^{2-d}\ ,\ d\not=2\ .
\end{equation}
If we take $W^{\mu}\sim\eta^{\mu0}$, the result (\ref{xsw1}) becomes the interaction energy between a point-like test charge placed at ${\bf a}$ and a uniform charge distribution along a $D$-brane with charge density $\sigma$ and placed at ${\bf A}$.
For $d=2$, we insert a mass parameter in the propagator (\ref{defDeltamunu}) as a regulator parameter in order to identify infrared divergences. This procedure leads to the expression
\begin{eqnarray}
E_{IV}(d=2)&=&\lim_{m\to 0}\sigma_{0}(\sigma W_{0})\int\frac{d^{d}{\bf k}_{\perp}}{(2\pi)^{d}}\frac{1}{{\bf k}_{\perp}^{2}+m^{2}}\nonumber \\
&&\exp{[i{\bf k}_{\perp}\cdot({\bf a}_{\perp}-{\bf A})]}\cr\cr
&=&\lim_{m\to 0}\frac{\sigma_{0}(\sigma W_{0})}{2\pi}K_{0}(m|{\bf a}_{\perp}-{\bf A}|)\cr\cr
&\to&-\frac{\sigma_{0}(\sigma W_{0})}{2\pi}\ln\Biggl(\frac{|{\bf a}_{\perp}-{\bf A}|}{a_{0}}\Biggr)\ ,
\end{eqnarray}
where we have proceeded similarly to what we have done in Eq. (\ref{qwe4}).
The second model we study for the Maxwell field is given by the current
\begin{equation}
\label{corrente5}
J_{V}^{\mu}=\sigma W^{\mu} V^{\alpha}\partial_{\alpha}\bigl(\delta^{d}({\bf x}_{\perp}-{\bf A})\bigr)+\sigma_{0} \eta^{\mu 0} \delta^{d+D}({\bf x}-{\bf a})\ ,
\end{equation}
where $W^{\mu}$ and $V^{\mu}$ are four-vectors defined in the same way as in Eq's (\ref{corrente4}) and (\ref{corrente2}), respectively.
Inserting the current (\ref{corrente5}) in the energy (\ref{arfv1}), using definition (\ref{defDeltamunu}) and performing similar steps which lead to the result (\ref{azxc2}) from (\ref{rfv1}) we have
\begin{eqnarray}
\label{EV}
E_{V}(d)&=&\frac{\sigma_{0}}{(2\pi)^{d/2}}\Gamma(d/2)2^{(d/2)-1}\cr\cr
&\ &|{\bf a}_{\perp}-{\bf A}|^{-d}(-\sigma W^{0}{\bf V})\cdot({\bf a}_{\perp}-{\bf A})\ .
\end{eqnarray}
If we take $W^{\mu}\sim\eta^{\mu 0}$, Eq. (\ref{EV}) can be interpreted as the interaction energy between a point-like charge and a uniform distribution of electric dipoles along a $D$-brane, with dipole density given by $-\sigma V^{\alpha}$. In order to make this fact clearer let us take $d=3$, $W^{\mu}=\eta^{\mu0}$ and restrict ourselves to a $3+1$ spacetime. In this case, we have a point-like dipole and the energy (\ref{EV}) reads \cite{Jackson,Landau}
\begin{equation}
E_{V}(d=3)=\frac{\sigma_{0}}{4\pi}\frac{(-\sigma{\bf V})\cdot({\bf a}-{\bf A})}{|{\bf a}-{\bf A}|^{3}}\ .
\end{equation}
The third and last model we consider for the Maxwell field is described by the current
\begin{equation}
\label{corrente6}
J_{VI}^{\mu}=\sigma W^{\mu}V^{\alpha\beta}\partial_{\alpha}\partial_{\beta}\bigl(\delta^{d}({\bf x}_{\perp}-{\bf A})\bigr)+\sigma_{0}\eta^{\mu\nu}\delta^{d+D}({\bf x}-{\bf a})\ ,
\end{equation}
where $V^{\alpha\beta}$ is a tensor with the same features of the one used in the current (\ref{corrente3}) and $W^{\mu}$ is the same $(d+D+1)$-vector used in (\ref{corrente4}).
Inserting the current (\ref{corrente6}) in the expression (\ref{arfv1}) and performing similar steps which lead to the result (\ref{aqwe2}), we have, for the electromagnetic field, the energy
\begin{eqnarray}
\label{EVI}
E_{VI}(d)&=&\frac{\sigma_{0}}{\pi^{d/2}}\Gamma\bigl((d/2)+1\bigr)|{\bf a}_{\perp}-{\bf A}|^{-d-2}\cr\cr
&\ &\sum_{i,j=1}^{d}({\bf a}_{\perp}-{\bf A})^{i}(\sigma W^{0}D^{ij})({\bf a}_{\perp}-{\bf A})^{j} ,
\end{eqnarray}
where we have used definition (\ref{defD}).
For $W^{\mu}\sim\eta^{\mu0}$, we can interpret Eq. (\ref{EVI}) as the interaction energy between a point-charge and an uniform four-pole distribution along a $D$-brane, with four-pole density given by $\sigma D^{ij}$. In $3+1$ dimensions with $W^{\mu}=\eta^{\mu0}$ the energy (\ref{EVI}) reads
\begin{equation}
E_{VI}(d=3)=\frac{3\sigma_{0}}{4\pi}|{\bf a}_{\perp}-{\bf A}|^{-5}\sum_{i,j=1}^{d}({\bf a}-{\bf A})^{i}(\sigma D^{ij})({\bf a}-{\bf A})^{j}\ ,
\end{equation}
which is the interaction energy between a point-like four-pole $(\sigma D^{ij})$ placed at ${\bf A}$ and a test charge $\sigma_{0}$ placed at ${\bf a}$ \cite{Jackson,Landau}.
To conclude this section, we stress that we could obtain, for the electromagnetic field, a result similar to the one presented in equation (\ref{ade3}) for $n$-pole distributions.
\section{Conclusions and Final Remarks}
\label{conclusao}
In this paper, we have carried out an investigation on the role of external currents concentrated at specific regions of space ($D$-dimensional branes) and coupled to bosonic fields, specifically, the scalar and electromagnetic ones.
We have considered a $D+d+1$ dimensional space-time and three kinds of currents for each field. All currents are composed by two parts where the second one describe the presence of a stationary point-like test-charge used to investigate the force field produced by the former one. From the results obtained for masless fields, we could notice that the first term of each current describe the presence of stationary charges, dipoles or four-poles distributions along $D$-dimensional branes.
As for the results for dipole-distributions (the second and fifth models studied), we corrected a flaw in reference \cite{BaroneHidalgo} on the interpretation of the dipole-distribution sign described by the currents.
We have shown that, for masless fields, the four-pole tensor density corresponding to a uniform four-pole distribution along a $D$-brane can be defined as being traceless in any dimension, once its trace does not contribute to the interaction energy between a test charge and the corresponding four-pole distribution.
For massive fields, we have shown that the trace of the four-pole tensor density always contributes to the interaction energy between a point-like test charge and the corresponding four-pole distribution. This fact can be seen from Eq. (\ref{aqwe5}), where we have the interaction energy between a point-like test-charge and a four-pole distribution mediated by the massive scalar field. In this result, we have separated the contributions to the energy due to the traceless part of the four-pole density tensor, $\sigma D$, and the contribution due strictly to the trace of this tensor, $\sigma\ tr D$.
Finally, we have obtained that, if we consider a massless scalar field in four-dimensional space-time in interaction with a stationary current composed by a point-like Dirac's delta function and the $n$-th derivative of a point-like Dirac's delta function, the result is the interaction energy between a point charge and a point-like $N$-pole. We stress that an identical result can be obtained also in the electromagnetic case.
\
{\bf Acknowledgements}
The authors would like to thank C. Farina, J.A. Helay\"el-Neto and N.F Svaiter for discussions and suggestions, J.A. Helay\"el-Neto and F.E. Barone for reading the manuscript and FAPEMIG for invaluable financial support.
|
2,877,628,088,987 | arxiv | \section{Introduction}
Interstellar dust plays an important role in galaxies: it helps to balance gas heating and cooling and the surface of dust grains provides a favourable place for chemical reactions to occur.
Dust contributes only a small fraction of the mass of the interstellar medium (ISM), but in normal star-forming galaxies it can re-radiate up to $\sim30\%$ of the stellar light in the infrared \citep[e.g.][]{Clements1996}.
The two main places where dust is formed is in the ejecta of core-collapse supernovae and in the envelopes of asymptotic giant branch (AGB) stars \citep{Galliano2018b}. These two production mechanisms alone however can not account for the amount of dust observed in high redshift galaxies \citep{Bertoldi2003, Priddey2003, Rowlands2014, Watson2015, Michalowski2015}. Grain growth is another mechanism that can increase the dust content of a galaxy, but it is not well understood how much this process can contribute to the total dust production \citep{Barlow1978c, Ferrara2016, Ceccarelli2018}. In order to resolve this tension, we need first to improve our understanding of all the mechanisms of dust production and growth. Second, it is the necessary to have tools to accurately measure the dust content of distant galaxies and have a good understanding of the uncertainties on these measurements; this is the question this paper tackles.
Dust masses are measured by fitting the spectral energy distribution (SED) of galaxies in the far-infrared/sub-millimeter spectral range.
The standard model used is a modified black-body function (MBB), which depends on the dust mass, temperature ($T$) and emissivity index $\beta$.
An anti-correlation between temperature and $\beta$ has been observed in galactic sources and luminous infrared galaxies \citep{Dupac2003, Desert2008, Yang2007}. However, it has been shown that noise in the data can introduce an artificial anti-correlation between $T$ and $\beta$ \citep[e.g.][]{Shetty2009a, Shetty2009b}. An incorrect estimate of $T$ and $\beta$ would consequently bias the measurement of the dust mass.
A way to overcome this problem and break the $T-\beta$ degeneracy is to use a hierarchical Bayesian approach
\citep{Kelly2012, Juvela2013, Veneziani2013, Galliano2018}.
The hierarchical approach uses the information from the parameter distribution of the entire sample of galaxies to better constrain temperature and $\beta$ for each single galaxy.
The hierarchical method has the advantage that it does not require knowing the prior distribution of the parameters before the fitting, but can infer the parameters describing the prior directly during the fitting procedure, after assuming the shape of the distribution. The limitation of this is that the prior is only valid for the sample of galaxies under consideration, i.e. the prior depends on the population that one is considering.
The \Herschel\ Space Observatory\footnote{\textit{Herschel} is an ESA space observatory with science instruments provided by European-led Principal Investigator consortia and with important participation from NASA.} \citep{Pilbratt2010} has been key for the study of dust in nearby galaxies, providing photometric observations in the wavelength range 100-500\micron, that allowed to characterize the shape of their far-infrared SED.
The \Herschel\ Reference Survey \citep[HRS, ][]{Boselli2010b} is a guaranteed time program that measured the far-infrared SED of $\sim$ 300 nearby galaxies.
Using HRS galaxies, \cite{Cortese2014} show that their far-infrared and submm colors are inconsistent with a single modified black-body model with the same emissivity index $\beta$ for all galaxies.
Dust continuum observations can also be used to infer the molecular gas mass of a galaxy.
It has been shown that the dust continuum luminosity of galaxies correlates with the CO luminosity \citep{Hildebrand1983, Magdis2012, Eales2012, Scoville2014, Groves2015} and this relation can be used to infer the molecular gas mass of a galaxy by applying a molecular gas-to-dust ratio.
This method can be extremely useful for faint or high-redshift galaxies, since the dust emission is brighter and therefore easier to observe than the CO line emission.
This method can therefore be beneficial for measuring the molecular gas content of large samples of galaxies.
The JINGLE (JCMT dust and gas In Nearby Galaxies Legacy Exploration), survey is a large program on the James Clerk Maxwell Telescope (JCMT) which aims to characterize the dust and molecular gas in nearby galaxies and study the relation between the two \citep{Saintonge2018}.
JINGLE combines dust observations from the SCUBA-2 camera on the JCMT (and from \Herschel), with the cold gas measurements obtained with the JCMT RxA instrument.
With both measurements of the dust and cold gas properties for a statistical sample of nearby galaxies, we can study the variations in the dust-to-gas mass ratio as a function of galaxy and dust properties.
One of the objectives of the survey is to benchmark dust scaling relations with other galaxy properties such as stellar mass, metallicity, and star-formation rate.
These relations can be used to estimate the dust temperature and dust emissivity index in galaxies for which there are not enough photometric data available to measure them directly through SED fitting. This can be useful especially for high redshift galaxies.
An excess of emission at wavelengths $\geq500$\micron\ with respect to the modified black-body model has been observed in numerous dwarf galaxies \citep[e.g.][]{Galametz2011, Remy-Ruyer2013, Remy-Ruyer2015}, in late-type galaxies \citep{Dumke2004, Bendo2006, Galametz2009}, in the Magellanic Clouds \citep{Israel2010, Bot2010b}, and in M33 \citep{Hermelo2016, Relano2018}. The origin of this `submm' excess is still an open question.
The SCUBA-2 observations at 850\micron\ can help to place better constraints on the submm slope and investigate the presence of this excess in the JINGLE sample.
In this paper we take advantage of the large and homogeneous JINGLE sample and apply a hierarchical Bayesian approach to reduce the $T-\beta$ degeneracy and obtain more accurate measurements of the dust parameters using MBB models.
The hierarchical approach is crucial to disentangle dust temperature $T$ and emissivity index $\beta$ and allows us for the first time to study the independent relations of these two dust quantities with other galaxy global properties.
This paper is organised as follows. In Section \ref{sec:sample} we present the sample and the data used in this paper. Then we describe the classical and hierarchical Bayesian SED fitting methods and compare the two methods using simulated SEDs (Section \ref{sec:method}). Section \ref{sec:results} illustrates the results of the SED fitting of the JINGLE sample, the $T$-$\beta$ relation, and comparison of different modified black-body models.
In Section \ref{sec:dust_scal_relation} we derive scaling relations between dust quantities and other global galaxy properties. Finally in Section \ref{sec:conclusions} we summarize the main results and our conclusions. Readers who are less interested in the statistical methods and tests of the fitting methods may wish to skip ahead to Section \ref{sec:results}.
\section{Sample and data}
\label{sec:sample}
\subsection{JINGLE sample}
The \Ntot\ galaxies in the JINGLE sample have stellar masses in the range $\log M_*/M_\odot=9-11.3$ and are in the redshift range $0.01 < z < 0.05$.
The targets were selected from the \textit{H}-ATLAS survey \citep{Eales2010, Maddox2018} with the requirement to have a detection $\geq 3\sigma$ in the 250\micron\ and 350\micron\ SPIRE bands.
Additionally, they have been selected to have a flat logarithmic stellar mass distribution.
Due to these requirements, they are mainly main-sequence star-forming galaxies with $-1.5 <\ \log \text{SFR}/[$\Msun\ yr$^{-1}] <\ 1.5$ (see Figure~\ref{fig:SFR_Mstar_JINGLE_HRS}). A detailed description of the selection criteria is provided in \cite{Saintonge2018}.
Most of the JINGLE objects are late-type galaxies, with only seven classified as early-type galaxies \citep{Saintonge2018}.
Properties of the JINGLE galaxies used in this work (such as SFR, metallicity, distances,...) are taken from the JINGLE catalog \citep{Saintonge2018}. In particular, we use the star-formation rates and stellar masses measured with {\tt MAGPHYS} \citep{daCunha2008}. In this paper we refer to JINGLE galaxies using their corresponding JINGLE ID, as described in the JINGLE catalog \citep{Saintonge2018}.
\subsection{HRS sample}
To extend our analysis to a larger range in galaxy properties, we include in our analysis also galaxies from the \Herschel\ Reference Survey \citep[HRS, ][]{Boselli2010b}.
The HRS is a volume-limited sample (15 Mpc $\leq D \leq$ 25 Mpc) of 323 galaxies, with flux limits in the $K$-band to minimize selection effects due to dust and young high-mass stars.
A large fraction of HRS galaxies lie in clusters, with 47\% of the HRS galaxies listed in the Virgo Cluster Catalogue alone. They have stellar masses in the range $\log M_*/M_\odot= 8.4-11.3$.
Galaxies from the HRS have been observed in the five \Herschel\ bands (at 100\micron, 160\micron, 250\micron, 350\micron, and 500\micron), but do not have observations at 850\micron. In our analysis we use the SFR and stellar masses measured with {{\tt MAGPHYS}} by \cite{DeVis2017}, to be consistent with the JINGLE measurements.
Figure \ref{fig:SFR_Mstar_JINGLE_HRS} shows the JINGLE and HRS galaxies on the SFR-$M_*$ plane.
With respect to the JINGLE galaxies, the HRS sample includes galaxies which are less massive ($\log M_* < 9$) and with lower SFR ($-2 < \log$ (SFR/[\Msun\ yr$^{-1}]) < 0.6$, mean $\log$ (SFR/[\Msun\ yr$^{-1}] )= -0.71$) compared to JINGLE, which has a mean $\log (\text{SFR}$/[\Msun\ yr$^{-1}]) = 0.04$. HRS galaxies are also less dusty than JINGLE targets (De Looze et al., in prep.), since contrary to JINGLE they have not been selected based on detection in the infrared bands.
The HRS sample includes also a large number of early-type galaxies \citep[62/323, ][]{Smith2012}, which are not well represented in the JINGLE sample (7/\Ntot).
Therefore by including this sample in our analysis, we can test whether the dust scaling relations that we find with the JINGLE sample hold also for other types of galaxies.
Additionally, increasing the dynamical range of galaxy properties will help to constrain better the dust scaling relations.
\begin{figure
\centering
\subfigure{\includegraphics[width=0.45\textwidth]
{Figures/SFR_Mstar/SFR_Mstar_JINGLE_HRS_SDSS_z_cut_MagPhys_low_SFR}}
\caption{Distribution of the JINGLE and HRS sample in the SFR-$M_*$ plane. The position of the star formation main sequence \citep{Saintonge2016} is shown as a dashed line, the 0.4 dex dispersion is shown by dotted lines. The grey contours show the distribution of SDSS galaxies at redshift $z < 0.05$. }
\label{fig:SFR_Mstar_JINGLE_HRS}
\end{figure}
\subsection{Data}
\subsubsection{JINGLE}
Our data set consists of photometric points at 22\micron\ (WISE), 60\micron\ (IRAS), 100\micron, 160\micron\ (\Herschel /PACS), 250\micron, 350\micron, 500\micron\ (\Herschel /SPIRE), and 850\micron\ (SCUBA-2).
A detailed description of the JINGLE photometric data set is given in \cite{Smith2019} and De Looze et al. (in prep.). Here we summarize the most important points.
The fluxes of the WISE, \Herschel, and SCUBA-2 bands have been extracted from matched apertures based on the SPIRE 250\micron\ band. The flux extraction is described in detail by \cite{Smith2019}. One galaxy (JINGLE 62) has been removed from the sample since it is not detected in the 250\micron\ band and therefore it is not listed in the release version of the \textit{H}-ATLAS DR2 catalogue \citep{Maddox2018}. Thus the sample analysed in this work consists of \Ntot\ galaxies.
We consider upper limits for fluxes with peak signal-to-noise ratio (S/N) < 3.
Since the CO(3-2) 345.79 GHz line emits in the 850\micron\ band, we corrected the SCUBA-2 flux by subtracting the estimated contribution of the CO(3-2) line \citep[for details see][]{Smith2019}.
After subtracting the CO(3-2) emission, some of the fluxes become negative, due to the uncertainties in the 850\micron\ fluxes and in the CO(3-2) predictions. These fluxes are consistent with zero within the uncertainties and are considered as upper limits.
In our sample, there are 66 galaxies with peak S/N<3 and additionally 4 galaxies have negative 850\micron\ flux, even though their peak S/N> 3 before subtraction of the CO(3-2) contribution.
For all these cases, we use conservative upper limits equal to five times the flux uncertainty in that band.
The IRAS 60\micron\ fluxes are derived using the Scan Processing and Integration Tool (SCANPI\footnote{http://irsa.ipac.caltech.edu/applications/Scanpi/}), following the strategy of \cite{Sanders2003}.
In our sample, 69/\Ntot\ galaxies have 5$\sigma$ upper limits for the 60\micron\ flux and 22/\Ntot\ do not have IRAS 60\micron\ observations.
\subsubsection{HRS}
For the HRS sample, we have flux measurements in the \Herschel /PACS \citep{Cortese2014} and \Herschel /SPIRE bands \citep{Ciesla2012}, from 100\micron\ to 500\micron. We note that, contrary to JINGLE, this sample does not have observations at 850\micron, therefore the long-wavelength slope of the SED can be constrained only by the 500\micron\ point.
In the case of non-detections, we consider upper limits equal to five times the flux uncertainties as we do for the JINGLE sample.
We exclude from the sample 39 galaxies which are not detected in all of the \Herschel\ bands, and therefore do not have constraints on their dust properties. We also exclude four galaxies which do not have SFR and stellar mass measurements from \cite{DeVis2017}. They were excluded from the sample because their SEDs show signs of contamination from dust heated by an active galactic nucleus or a hot X-ray halo or from synchrotron radiation emission \citep{Eales2017}.
The final sample consists of 41 early-type and 239 late-type galaxies, for a total of 280 galaxies.
\section{Method}
\label{sec:method}
\subsection{Models}
\label{sec:models}
To describe the far-infrared and sub-millimeter spectral energy distribution (SED) we adopt the three models employed by \cite{Gordon2014} for the SED fit of the Magellanic Clouds: single modified black-body (SMBB), broken emissivity law modified black-body (BMBB), and two modified black-bodies (TMBB). We describe below the analytic functions and the parameters used for the three models:
\begin{itemize}
\item \textbf{SMBB:} The single modified black-body model describes the dust emission $F_{\lambda}$ (in units of W m$^{-2}$ Hz$^{-1}$ sr$^{-1}$) at each wavelength $\lambda$ in the following way \citep{Hildebrand1983}:
\begin{equation}
F_{\lambda} = \frac{M_{\text{dust}}}{D^{2}} \kappa_{\lambda} B_{\lambda}(T)
\end{equation}
where $M_{\text{dust}}$ is the dust mass in the galaxy and $D$ is the distance of the galaxy. $B_{\lambda}$($T$) is the Planck function for the emission of a black-body with a dust temperature $T$ given by:
\begin{equation}
B_{\lambda}(T) = \frac{2hc^2}{\lambda^{5}} \frac{1}{\exp\left( \frac{hc}{k_{B}T\lambda} \right)-1} .
\end{equation}
The dust mass absorption coefficient $\kappa$ describes which dust mass gives rise to an observed luminosity. The value of $\kappa$ depends on the physical properties of the dust, such as the mass density of the constituent materials, the efficiency with which they emit, the grain surface-to-volume ratio, and the grain size distribution \citep{Koehler2015, Ysard2018}.
The SMBB applies a dust emissivity power law to characterise the behaviour of $\kappa$ as a function of wavelength:
\begin{equation}
\kappa_{\lambda}= \kappa_{0}\left( \frac{\lambda_{0}}{\lambda}\right)^{\beta}
\end{equation}
where $\kappa_{0}$ is the reference dust mass absorption coefficient. Laboratory studies found that the absorption coefficient depends also on the dust temperature and dust emissivity index $\beta$, with higher $\kappa$ values observed for higher temperatures and lower $\beta$ values \citep{Coupeaud2011}. For simplicity, here we assume a constant value $\kappa_0$= $\kappa(500\mu\text{m} ) = 0.051\ \text{m}^2 \ \text{kg}^{-1}$ from \cite{Clark2016}.
This model has three free parameters ($M_{dust}$, $T$, and $\beta$), and assumes that the dust emission can be described by a dust component with a single temperature. At wavelengths shorter than 100 $\mu$m, a second warmer dust component can contribute to the FIR emission \citep[e.g.][]{Relano2018}. Therefore for this model, we use only the flux bands with wavelengths $\geq 100 \mu$m.
Additionally, we use the 60$\mu$m point as an upper limit, in order to better constrain the dust temperature.\\
\item \textbf{BMBB:} When fitting the FIR SED with a SMBB model, some galaxies show an excess in the flux at wavelengths $\geq $ 500\micron, called `sub-millimeter' excess
\citep{Lisenfeld2002, Galliano2003, Dumke2004, Bendo2006, Galametz2009,Israel2010, Bot2010b, Hermelo2016}.
The broken emissivity law modified black-body (BMBB) model assumes that the submm excess is due to variations in the wavelength dependence of the dust emissivity law.
These variations are parametrized by a broken power law:
\begin{equation}
\kappa_{\lambda}=
\Bigg \{\begin{array}{ll}
\kappa_0\left(\frac{\lambda_0}{\lambda}\right)^{\beta_1} & \text{if } \lambda < \lambda_b \\
\kappa_0\left(\frac{\lambda_0}{\lambda_b} \right)^{\beta_1} \left(\frac{\lambda_b}{\lambda}\right)^{\beta_2}& \text{if } \lambda > \lambda_b \\
\end{array}
\end{equation}
where $\lambda_b$ is the wavelength of the break.
This model has five free parameters: $M_{\text{dust}}$, $T$, $\beta_1$, $\beta_2$, and $\lambda_b$.
Also for this model, we use only the flux bands with wavelengths $\geq 100$\micron. In order to have good constraints on the fitting parameters, it is crucial to have a detection of the 850\micron\ flux. If the SCUBA-2 point is not detected, an upper limit is not enough to constrain the parameters of this model. Without the 850\micron\ flux point, the 500\micron\ flux point is the only one that can be used to determine $\beta_2$ and $\lambda_b$, leading to large uncertainties on their values.\\
\item \textbf{TMBB:} The two modified black-body model assumes that the FIR SED is emitted by two dust populations with different temperatures. The dust emission is parametrized by two modified black-bodies: one for the cold dust (indicatively $T < 40$ K) and one for the warm dust (indicatively $T > 40$ K):
\begin{equation}
F_{\lambda} = F_{\lambda}^{\text{SMBB}_{cold}}+F_{\lambda}^{\text{SMBB}_{warm}}
\end{equation}
where the two SMBB components are defined as above.
In order to reduce the number of free parameters, we fix the $\beta$ value of the warm component to 1.5 \citep{Coupeaud2011, Boselli2012}, while we leave the $\beta$ value of the cold component as a free parameter.
So in this model we have five free parameters: $M_{\text{cold}}$, $T_{\text{cold}}$, $\beta_{\text{cold}}$, $M_{\text{warm}}$, and $T_{\text{warm}}$. For the fitting, we use the fluxes in all available bands from 22 to 850\micron.
\end{itemize}
All these models assume that dust grains are optically thin. According to dust models, this assumption holds for wavelengths $\geq 100$\micron, while at shorter wavelengths it is possible that dust is optically thick \citep{Draine2007}. \cite{Casey2012} modelled the SED of 65 luminous infrared galaxies from the GOALS survey \citep{Armus2009} and found that even if the dust is optically thick, the difference in the SED shape at 22\micron\ would be small. \cite{Utomo2019} studied the dust emission at resolved scales in four nearby galaxies (Small and Large Magellanic Clouds, M31, and M33) and found that most of the dust emitting at wavelengths longer than 100\micron\ is optically thin. They observe that at wavelengths $\sim$20\micron\ some regions of the galaxies become optically thick, but on global galaxy scales we do not expect these regions to dominate the emission.
We apply the SMBB model to both the JINGLE and HRS sample, while we apply the BMBB and TMBB models only to the JINGLE sample. We make this decision because for the HRS sample we do not have the 850\micron\ flux point, and therefore we do not have enough flux points for models with a large number of free parameters.
Additionally, for the BMBB model it is very important to have the 850\micron\ point to constrain the emissivity index $\beta_2$ after the break.
Fig. \ref{fig:example_SED_models} shows an example of the SED fitting of one galaxy from the JINGLE sample using the three models.
\begin{figure*
\centering
\subfigure{\includegraphics[width=0.32\textwidth]
{Figures/SED_models/JINGLE147_SMBB_non_hier_SED_fit}}
\subfigure{\includegraphics[width=0.32\textwidth]
{Figures/SED_models/JINGLE147_BMBB_non_hier_SED_fit}}
\subfigure{\includegraphics[width=0.32\textwidth]
{Figures/SED_models/JINGLE147_TMBB_non_hier_SED_fit}}
\caption{Example of FIR SED of one galaxy from the JINGLE sample, fitted with the non-hierarchical approach using the three models: single modified black-body (SMBB, left panel), broken emissivity law modified black-body (BMBB, middle panel) and two modified black-bodies (TMBB, right panel). The shaded regions show the lower and upper 1-$\sigma$ uncertainties on the SED models, defined by taking the maximum and minimum flux values of the models with likelihood values in the highest 68th percentile.}
\label{fig:example_SED_models}
\end{figure*}
\subsection{Introduction to the Bayesian SED fitting method}
\label{sec:Bayesian_method}
In this section we briefly describe the Bayesian approach
used for the SED fitting \citep[we follow the same notation as in][]{Galliano2018}.
Readers who are less interested in the statistical methods may wish to go directly to the results presented in Section \ref{sec:results}.
The observed SED of a galaxy ($F^{obs}$) can be described in the following way:
\begin{equation}
F^{obs}(\lambda_j) = F^{mod}(\lambda_j, \vec{\theta})+\epsilon(\lambda_j)\cdot F^{err}(\lambda_j)
\end{equation}
where $F^{obs}(\lambda_j)$ is the flux observed at the wavelength $\lambda_j$ and $F^{mod}(\lambda_j,\vec{\theta})$ is the flux described by our model with parameters $\vec{\theta}$. The last term describes the deviation of the observed flux from the model due to random noise: $F^{err}(\lambda_j)$ is the amplitude of the noise and $\epsilon(\lambda_j)$ is a random variable with mean $< \epsilon> = 0$ and standard deviation $\sigma(\epsilon) =1$.
We can reverse the previous formula to express $\epsilon(\lambda_j)$ as a function of the other quantities:
\begin{equation}
\epsilon(\lambda_j) = \frac{F^{obs}(\lambda_j) - F^{mod}(\lambda_j, \vec{\theta})}{F^{err}(\lambda_j)}.
\label{eq:eps}
\end{equation}
The goal is to find the best parameters to fit the data by minimising the offset between the model and the data. From a Bayesian point of view, this is equivalent to maximising the likelihood of the model, given the data.
The probability of the data given the model parameters $\vec{\theta}$ can be expressed as:
\begin{equation}
p(\vec{F}^{obs}|\vec{\theta}) = \prod_{j=1}^m p(\epsilon(\lambda_j, \vec{\theta}))
\end{equation}
where $\vec{F}^{obs} = \left(F^{obs}(\lambda_1), . . . , F^{obs}(\lambda_m)\right)$ is the vector containing the flux emission at each waveband $j=1,...,m$.
We are interested in the probability of the model parameters, knowing the observations. Thus we can use the Bayes' theorem to write the expression:
\begin{equation}
p(\vec{\theta}|\vec{F}^{obs}) = \frac{p(\vec{F}^{obs}|\vec{\theta}) \cdot p(\vec{\theta})}{p(\vec{F}^{obs})} \propto p(\vec{F}^{obs}|\vec{\theta}) \cdot p(\vec{\theta}),
\label{eq:ptheta_given_data}
\end{equation}
where $p(\vec{\theta})$ is the `prior' distribution, and $p(\vec{\theta}|\vec{F}_{obs})$ is the `posterior' distribution. The denominator $p(\vec{F}^{obs})$ can be neglected since it is constant for a given set of observed fluxes. By sampling the posterior distribution in the parameter space we can construct the posterior probability density function (PDF).
Examples of posterior probability density functions (PDF) are shown in the appendix (Fig. \ref{fig:post_triangle_plot}). The figure shows the PDFs obtained from the SED fit of one galaxy using the SMBB, BMBB, and TMBB models.
\subsection{Hierarchical Bayesian method}
\label{sec:Hier_method}
The difference between the classical and hierarchical Bayesian method is that in the former the prior distribution is an assumption and in the latter it is defined by the data sample \citep[e.g.][]{Gelman2004, Galliano2018}. Hierarchical methods require therefore a population of objects, which are used to define the prior distributions. In the case of SED fitting, the sample can be formed by multiple spatially resolved regions of the same galaxy or by a sample of galaxies with similar properties.
The entire sample is then fitted simultaneously, in order to extract both the information about the prior distribution of the sample and the posterior distribution of the single elements of the sample.
\cite{Kelly2012} showed that the hierarchical method can be used to reduce the degeneracy between $T$ and $\beta$. This approach has subsequently been used in other studies to reduce the $T$-$\beta$ degeneracy \citep{Juvela2013, Veneziani2013,Galliano2018}. The key assumption behind the hierarchical approach is that the dust parameters of our sample of galaxies follow a common distribution. In our case we assume that they follow a Student's \textit{t}-distribution. Thanks to this assumption, we are able to better constrain model parameters, especially for galaxies with low S/N, where a large range of combinations of $T$ and $\beta$ provide reasonably good fits to the data. In those cases, the prior helps to constrain the range of possible $T$ and $\beta$. The key point of the hierarchical approach is that we do not need to specify the mean and standard deviation of the prior distribution before doing the fit, but they can be inferred by the data.
The new parameters describing the prior distribution of the parameters $\vec{\theta}$ are called \textit{hyper-parameters}. The commonly used hyper-parameters are:
\begin{itemize}
\item $\vec{\mu}$: the average of the parameter vector $\vec{\theta}$;
\item $\Sigma$: the covariance matrix describing the standard deviation and correlation of $\vec{\theta}$.
\end{itemize}
Using this formalism, the posterior distribution of the parameters given the data $p(\vec{\theta}|\vec{F}_{obs})$ for the $i$-th galaxy in the sample becomes:
\begin{equation}
p(\vec{\theta_i}|\vec{F_i}^{obs}, \vec{\mu}, \Sigma) \propto p(\vec{F_i}^{obs}|\vec{\theta_i}) \cdot p(\vec{\theta_i}|\vec{\mu}, \Sigma).
\label{eq:ptheta_given_data_hier_one_galaxy}
\end{equation}
This is the hierarchical equivalent of eq. (\ref{eq:ptheta_given_data}).
The posterior distribution of the parameters and hyper-parameters for the entire sample of $n$ galaxies is:
\begin{multline}
p(\vec{\theta_1}, ..., \vec{\theta_n}, \vec{\mu}, \Sigma| \vec{F_1}^{obs},...,\vec{F_n}^{obs})
\propto \prod_{i=1}^n p(\vec{\theta_i}|\vec{F_i}^{obs}, \vec{\mu}, \Sigma) \cdot p(\vec{\mu}) \cdot p(\Sigma) \\
\propto \prod_{i=1}^n p(\vec{F_i}^{obs}|\vec{\theta_i}) \cdot p(\vec{\theta_i}|\vec{\mu}, \Sigma)
\cdot p(\vec{\mu}) \cdot p(\Sigma),
\end{multline}
\label{eq:ptheta_given_data_hier}
where $p(\vec{\mu})$ and $p(\Sigma)$ are the prior distributions of the hyper-parameters. When compared to the classical Bayesian method, the hierarchical method is able to recover the distribution of parameters with better precision, especially if the noise in the data is high \citep{Kelly2012, Galliano2018}.
In that case, the hierarchical approach uses the information about the parameter distribution obtained from the rest of the sample to better constrain the parameters for the particular objects where the quality of the data is low.
The hierarchical method will not necessarily perform better in measuring the parameter of a single object, but it will be less biased when measuring the distribution of parameters of the entire population.
\subsection{Noise distribution}
\label{sec:noise_distr}
In this section we describe the functions used to model the noise distribution for both the non-hierarchical and hierarchical approaches.
The noise is usually modelled with a normal distribution or a Student's \textit{t}-distribution . The Student's \textit{t}-distribution\ has a higher probability in the tails with respect to the normal distribution, allowing for more outliers. Its shape is described by the number of degrees of freedom $f$: as $f$ decreases, more probability will be in the tails of the distribution.
The normal distribution is a special case of the \textit{t}-distribution with the number of the degrees of freedom that goes to infinity, $f \rightarrow \infty$.
The probability density of a normal distribution is defined as:
\begin{equation}
\text{Normal}(y|\mu, \sigma) = \frac{1}{\sqrt{2 \pi} \sigma}\exp \left(-\frac{1}{2}\left(\frac{y-\mu}{\sigma} \right)^2 \right),
\end{equation}
where $\mu$ is the mean and $\sigma$ is the standard deviation.
The multivariate normal distribution is the generalization of the one-dimensional normal distribution to a higher dimension $m$:
\begin{multline}
\text{MultiNormal}(\vec{y}|\vec{\mu}, \Sigma) = \\
\frac{1}{(2 \pi)^{m/2}} \frac{1}{\sqrt{|\Sigma|}}\exp \left(-\frac{1}{2} (\vec{y}-\vec{\mu})^T \Sigma^{-1} (\vec{y}-\vec{\mu}) \right),
\end{multline}
where $m$ is the dimension of the vector $\vec{y}$, $\Sigma$ is the $m\times m$ covariance matrix, and $(\vec{y}-\vec{\mu})^T$ indicates the transpose of the vector $(\vec{y}-\vec{\mu})$.
The Student's \textit{t}-distribution\ is defined as:
\begin{equation}
\text{Student}(y|\mu, \sigma, f) = \frac{\Gamma((f+1)/2)}{\Gamma (f/2)}
\frac{1}{\sqrt{f \pi} \sigma} \left( 1+\frac{1}{f}\left( \frac{y-\mu}{\sigma} \right)^2 \right) ^{-\frac{f+1}{2}},
\end{equation}
where $f$ is the number of degrees of freedom. The multivariate Student's \textit{t}-distribution\ is the generalization of the one-dimensional distribution to a higher dimension $m$:
\begin{multline}
\text{MultiStudent}(\vec{y}|\vec{\mu}, \Sigma, f) = \\
\frac{\Gamma((f+m)/2)}{\Gamma (f/2)}
\frac{1}{(f \pi)^{m/2}} \frac{1}{\sqrt{|\Sigma|}}
\left( 1+\frac{1}{f} (\vec{y}-\vec{\mu})^T \Sigma^{-1} (\vec{y}-\vec{\mu}) \right)^{-\frac{f+m}{2}},
\end{multline}
where $m$ is the dimension of the vector $\vec{y}$.
We expect to observe a flux excess at 850\micron\ for some galaxies, given the fact that the submm excess has been reported in numerous studies \citep[e.g.][]{Galametz2011, Remy-Ruyer2013, Remy-Ruyer2015, Hermelo2016}. Since the 850\micron\ fluxes have usually larger uncertainties than the other points, if we use a Student's \textit{t}-distribution, the SMBB model will assume that every change in slope at 850\micron\ is due to the error being underestimated, rather than to a physical effect. The model will then `ignore' the 850\micron\ point, and produce a fit considering only the \Herschel\ points. Since we believe that there is information in the longer wavelength points, we therefore decide to use a normal distribution for the error.
In Section \ref{sec:normal_vs_student} of the appendix we compare the results obtained using the Student and normal distribution.
In both the non-hierarchical and hierarchical case, we model the noise as:
\begin{equation}
p(\vec{F}^{obs}|\vec{F}^{mod}(\vec{\theta}), C) = \text{MultiNormal}(\vec{F}^{obs}|\vec{F}^{mod}(\vec{\theta}), C),
\end{equation}
where $C$ is the covariance matrix, which describes the uncertainties associated with the flux densities in the different wavebands (see Section \ref{sec:cov_matrix} for the definition of the covariance matrix).
\subsection{Prior distributions}
In this section we describe the prior distributions assumed for the hierarchical and non-hierarchical method.\\
\\
\textbf{Non-hierarchical:} For the prior distribution of the parameters $\vec{\theta}$, we assume uniformly distributed (``flat") priors, i.e. $p(\theta)=1$, in the ranges described in Table \ref{tab:para_ranges}.\\
\\
\textbf{Hierarchical:}
For the definition of the prior distributions in the hierarchical framework, we follow \cite{Kelly2012}, \cite{Galliano2018} and the {\tt Stan}\ manual \citep{Stan_manual}.
\begin{itemize}
\item \textbf{\it parameters}: for the definition of the prior distributions of the parameters given the hyper-parameters, we follow \cite{Kelly2012} and \cite{Galliano2018}. We assume a multivariate Student's \textit{t} distribution with $f=8$ degrees of freedom:
\begin{equation}
p(\vec{\theta_i}|\vec{\mu}, \Sigma) = \text{MultiStudent} (\vec{\theta_i}|\vec{\mu}, \Sigma, f=8).
\end{equation}
We also tried to vary the number of degrees of freedom and did not see any differences in the results. Assuming a Student's \textit{t}-distribution\ allows one to have more galaxies with dust parameters which are `outliers' from the mean of the sample. In this way, we make sure that our assumption that the galaxies belong to the same population is not too stringent.
We note that the parameters $\vec{\theta_i}$ are not constrained within a certain range but they are allowed to take any value. Their distribution is described by the prior distribution and we set some constraints on the allowed range of the hyper-priors (mean and standard deviation) that determine the shape of the priors (see next point).
\\
\item \textbf{\it hyper-parameters}: For the mean $\vec{\mu}$ of the parameters,
we assume a uniform prior with a large parameter range.
In this way we ensure that the prior is proper (i.e $\int p(\theta) d\theta < \infty$), and at the same time we maintain the prior vague enough to not constrain the results \citep{Tak2018, Gelman2007}.
The prior ranges for $\vec{\mu}$ are shown in Table \ref{tab:mu_prior_range}.
We note that we set the prior range of $\mu(T_{warm})$ to be > 50 K, because we want the distribution of warm temperatures to be well separated from the distribution of cold temperatures.
For the covariance matrix $\Sigma$, we use the \textit{separation strategy} from \cite{Barnard2000}. This formalism ensures that the prior distributions of the correlations between parameters are uniform over the range $[-1,1]$, meaning that all values of the correlations are equally likely. The separation strategy breaks down the covariance matrix in:
\begin{equation}
\Sigma=SRS \ ,
\end{equation}
where $S$ is a diagonal matrix with the values of the standard deviation, and $R$ is the correlation matrix. Both $S$ and $R$ have dimension $q\times q$, where $q$ is the number of free parameters in the model.
The prior distribution of the hyper-parameters is then:
\begin{equation}
p(\vec{\mu}) \cdot p(\Sigma) \propto p(\vec{\mu}) \cdot p(S) \cdot p(R)\ .
\end{equation}
For the priors on the $S$ and $R$ we follow the recommendations given by the {\tt Stan}\ manual \citep{Stan_manual}.
For the priors on the diagonal elements of $S$, we use a weakly informative prior, parametrized by a half-Cauchy distribution with a small scale $\sigma$ = 2.5 \citep{Stan_manual}
\begin{equation}
p(S_{k,k}) = \text{Cauchy}(0, \sigma) =
\frac{1}{\pi \sigma} \frac{1}{1+\left( \frac{S_{k,k}}{\sigma} \right)^2},
\end{equation}
where $S_{k,k}$ > 0, for $k=1,..,q$.
For the priors on the correlation matrix $R$, we use a LKJ correlation distribution with shape $\nu = 2$:
\begin{equation}
p(R) = \text{LKJ\ Corr}(R, \nu) \propto \det(R)^{\nu-1}
\end{equation}
(see \cite{Lewandowski2009} for definitions).
The basic idea of the LKJ correlation distribution is that as $\nu$ increases, the prior increasingly concentrates around the identity matrix.\\
\end{itemize}
\begin{table}
\centering
\caption{Prior parameter ranges assumed for the Bayesian non-hierarchical SED modelling using the SMBB function.}
\begin{tabular}{|lc|}
\hline
Parameter & Range \\ \hline \hline
$\log M_{dust}/M_{\odot}$ & (5, 9) \\
$T$ [K] & (5, 50) \\
$\beta$ & ($0.1$, 3) \\
\hline
\end{tabular}
\label{tab:para_ranges}
\end{table}
\begin{table}
\centering
\caption{Ranges of the priors on the hyper-parameter $\vec{\mu}$ (sample mean) for the Bayesian hierarchical SED modelling using the SMBB, BMBB and TMBB functions. }
\begin{tabular}{|lc|}
\hline
\textbf{Hyper-parameter} & Range \\ \hline \hline
\multicolumn{2}{c}{SMBB} \\ \hline
$\mu(\log M_{dust}/M_{\odot})$ & (6, 9) \\
$\mu(T)$ [K] & (15, 50) \\
$\mu(\beta)$ & (0.5, 3) \\
\hline\hline
\multicolumn{2}{c}{BMBB} \\ \hline
$\mu(\log M_{dust}/M_{\odot})$ & (5, 9) \\
$\mu(T)$ [K] & (5, 50) \\
$\mu(\beta_{1})$ & (0, 5) \\
$\mu(\beta_{2})$ & (0, 5) \\
$\mu(\lambda_b)$ [\micron ] & (420, 500) \\
\hline \hline
\multicolumn{2}{c}{TMBB} \\ \hline
$\mu(\log M_\text{cold}/M_{\odot})$ & (6, 10) \\
$\mu(T_\text{cold})$ [K] & (5, 40)\\
$\mu(\beta_\text{cold})$ & (0.5, 5) \\
$\mu(\log M_\text{warm}/M_{\odot})$ & (2, 7) \\
$\mu(T_\text{warm})$ [K] & (50, 90)\\
\hline
\end{tabular}
\label{tab:mu_prior_range}
\end{table}
\subsection{Covariance matrix, beam and filter corrections}
\label{sec:cov_matrix}
In order to perform an accurate fit, it is important to take into account correctly the uncertainties associated with each flux measurement as well as the correlation between these uncertainties.
The covariance matrix $C$ describes the uncertainties associated with the flux densities in the different wave bands, and includes both calibration and measurement uncertainties. Calibration uncertainties can be correlated between bands observed with the same instrument.
For the definition of the covariance matrix, we follow \cite{Gordon2014}. The calibration covariance matrix is defined as:
\begin{equation}
C_{j,k}^{cal} = [A_{cor,j,k}+ A_{uncor, j,k}] = [\sigma_{cor,j,k}^2+ \delta_{j,k}\sigma_{uncor, j,k}^2]
\end{equation}
where $A_{cor}$ is the matrix of the noise correlated between bands, $A_{uncor}$ is the diagonal matrix of repeatability that is uncorrelated between bands. $\sigma_{cor, j,k}$ and $\sigma_{uncor, j,k}$ are the percentage of correlated and uncorrelated uncertainties, respectively, between the $j$-th and $k$-th band, and $\delta_{j,k}$ is one for $j=k$ and zero otherwise.
The calibration uncertainty values that we use are reported in Table \ref{tab:corr_noise}, given in percentage of the flux.
The total covariance matrix $C$ is a combination of the calibration and measurement uncertainties:
\begin{equation}
C_{j,k} = C_{j,k}^{cal} \cdot F_{j} \cdot F_{k} + F_{j}^{err} \cdot F_{k}^{err}
\end{equation}
where $F_{j}$ and $F_{k}$ are the fluxes in the $j$-th and $k$-th waveband, and $F_{j}^{err}$ and $F_{k}^{err}$ are the corresponding measurement uncertainties.
The colour and beam corrections applied to our data are described in detail in De Looze et al. (in prep.).\\
\\
\begin{table*}
\centering
\caption{Percentage of correlated and uncorrelated uncertainties for the different instruments.}
\begin{tabular}{lcccc}
\hline
Instrument & Waveband & Correlated & Uncorrelated & Reference \\
& [\micron] & uncertainty & uncertainty & \\
\hline \hline
WISE & 22 & - & 5.7 $\%$ & \cite{Jarrett2011}\\
IRAS & 60 & - & 20 $\%$ & \cite{Sanders2003, Miville-Deschenes2005} \\
PACS & 100, 160 & 5 $\%$ & 2 $\%$ & \cite{Balog2014}, \cite{Decin2007}\\
SPIRE & 250, 350, 500 & 4 $\%$ & 1.5 $\%$ & \cite{Bendo2013} \\
SCUBA & 850 & - & 10 $\%$ & \cite{Smith2019}\\
\end{tabular}
\label{tab:corr_noise}
\end{table*}
\\
\textbf{Non-hierarchical}: The filter corrections are applied to the model SED by convolving the model flux points with the appropriate filter response curve in each band.
The \textit{Herschel}/SPIRE fluxes were corrected also for the effective beam area, which depends on the shape of the spectrum due to the absolute SPIRE calibration in units of flux density per beam. The SED shape is described by the dust temperature $T$ and the emissivity index $\beta$. At each step of the Markov chain Monte Carlo (MCMC) algorithm, the \Herschel /SPIRE fluxes are corrected according to the two model parameters, before comparing them to the fluxes of the SED model.
For the BMBB model, we applied the beam and color corrections using $\beta_1$ or $\beta_2$ depending on the wavelength position of the break $\lambda_{b}$. For the TMBB model, we calculate which of the two components (warm or cold) contribute the most to the flux in every band. Then we calculate the corrections using the temperature $T$ and $\beta$ values of the dominant component in each band.\\
\\
\textbf{Hierarchical}:
The beam and filter corrections make it more difficult for the code to converge, since in every MCMC step the fluxes are slightly modified. This is more problematic for the hierarchical approach, because it has a larger number of free parameters. Therefore, in order to achieve convergence in a reasonable amount of time, we apply a slightly different approach to implement the beam and filter corrections in the hierarchical case. We first do the hierarchical fit without beam and filter corrections. Then we apply the beam and filter corrections on the fluxes based on the values of $T$ and $\beta$ measured from the fit with no corrections, and finally we repeat the hierarchical fit using the `corrected' fluxes. The beam and filter corrections are generally small compared to the flux uncertainties, therefore this approximation of the corrections does not affect the results significantly.
\subsection{Implementation of the SED fitting}
\label{sec:implementation}
\textbf{Non-hierarchical method:}
For the implementation of the classical Bayesian SED fitting method, we employ the affine-invariant ensemble sampler for Markov Chain Monte Carlo \citep[MCMC, ][]{Metropolis1953} code {\tt emcee} \citep{Goodman2010,Foreman-Mackey2013}.
The MCMC algorithm is designed to sample the posterior distribution of the unknown parameters, i.e. the probability of the parameters given the data. The values of the parameters with the corresponding uncertainties can then be inferred from the posterior distribution. We consider as results the median values of the marginalized posterior probability distributions, and we estimate the uncertainties from the values corresponding to the 16th and 84th percentiles.
To monitor the convergence we look at the effective sample size ($N_{eff}$), which is defined as the number of iterations divided by the integrated autocorrelation time $
N_{eff} = N_{iter}/\tau_{int}$. The autocorrelation time $\tau_{int}$ measures the number of steps after which the drawings are truly independent \citep{Foreman-Mackey2013}. It is recommended to have at least $N_{eff}> 10$, to ensure that the sequence has converged \citep{Gelman2004}. \\
\\
\textbf{Hierarchical method:}
For the implementation of the hierarchical Bayesian fitting we use {\tt Stan}\ \citep[][http://mc-stan.org/]{Carpenter2017}, a software for Bayesian inference which employs the No-U-Turn sampler (NUTS), a variant of Hamiltonian Monte Carlo sampler. The Hamiltonian Monte Carlo (HMC) sampling \citep{Duane1987, Neal1994, Neal2011} is a form of MCMC sampling which uses the gradient of the logarithmic probability function to accelerate the parameter exploration and the convergence to the stationary distribution \citep{Stan_manual}.
The HMC algorithm is more efficient than other MCMC algorithms (as for example the Metropolis-Hastings algorithm) in sampling the parameter space and in finding the region of high likelihood, because it samples the probability distribution with fewer samples. Therefore it is particularly well suited for problems with high dimension, as is the case for hierarchical models.
For example, for the hierarchical fit of 100 galaxies using the SMBB model, which has three free parameters, the dimension is of the order $\sim 300$.
Another advantage of {\tt Stan}\ is that it can sample simultaneously the posterior distribution of parameters and hyper-parameters. {\tt Stan}\ allows to define the model by specifying the probability distribution of each parameter (or hyper-parameter) independently, without the need of computing the full posterior distribution.
For the practical implementation, we used {\tt PyStan}\footnote{http://pystan.readthedocs.io/en/latest/ \\ http://mc-stan.org}, which is the Python interface to {\tt Stan}\ \citep{pystan}.
The recommended method for monitoring the convergence of the MCMC chains in {\tt Stan}\ is computing the potential scale reduction statistics $\hat{R}$ \citep{Gelman1992}, which gives an estimate of the factor by which the scale of the posterior distribution may be reduced as the number of iterations goes to infinity. If $\hat{R}$ is large, it means that increasing the number of iterations is likely to improve the inference. If $\hat{R}\sim 1$, then we can be confident that the number of iterations that we are using is large enough. Thus we set the requirement that for our runs $\hat{R}< 1.15$. We also check that the effective sample size $N_{eff}$ is always larger than 10.
\subsection{Validation of the method with simulations of mock SEDs}
We test our fitting methods using simulated FIR SEDs. For the mock SEDs, we know the input parameter values, thus we can assess how well our fitting procedure is able to recover them.
The simulation code takes as input parameters the dust mass ($\log M_{dust}$), temperature $T$, and emissivity index $\beta$, and it uses these parameters to generate an SED assuming a single modified black-body (SMBB) model. Then it extracts the flux density in the selected wavebands and it adds random noise at each flux point. We assume the noise to be Gaussian distributed around zero, with amplitude equal to the noise level.
We assume a different noise level in every band. For the wavebands (100, 160, 250, 350, 500, 850) \micron, we use the following noise levels, given as percentages of the flux: (20, 10, 5, 10, 20, 25)\%, respectively. We estimate these values by taking the mean of the error fraction in each band from our data.
The goal of the test is to assess how well the non-hierarchical Bayesian approach can measure the values of temperature and $\beta$.
We simulate 100 SEDs with the same input parameters ($\log M_{dust}=8 $ \Msun, $T= 30 $ K, $ \beta =1.5$), adding to every SED random noise in every band as explained above.
Figure \ref{fig:T-beta_single_input} shows the results in the $T$-$\beta$ plane. As we can see from the figure, an artificial anti-correlation is generated only from the effect of adding noise to the fluxes. This suggests that the non-hierarchical Bayesian approach will always measure a $T$-$\beta$ anti-correlation, even if it is not present in the data. Thus, in order to asses if the $T$-$\beta$ anti-correlation is indeed present in our sample, we need a more sophisticated fitting method.
We run the same simulation, but this time we use the hierarchical code to fit the SEDs.
The results are in better agreement with the input value, and do not show any artificial correlation or anti-correlation between $T$ and $\beta$.
The non-hierarchical method measures a large range of temperatures ($T=22-42$~K) and $\beta$ values ($\beta=0.8-2.3$).
The hierarchical method measures smaller ranges of $T=27-30$~K and $\beta=1.50-1.55$, which are closer to the input values.
Consequently, also the dust masses are better measured with the hierarchical method. The dust masses measured with the non-hierarchical method are in the range $\log M_{dust}/M_{\odot}=7.87-8.23$, with typical uncertainties of $\sim0.13$~dex, while the ones measured with the hierarchical method are in the range $\log M_{dust}/M_{\odot}=8.06-8.09$, with typical uncertainties 0.02~dex.
\begin{figure*}
\centering
\subfigure{\includegraphics[width=0.48\textwidth]
{Figures/Mock_SED/Comp_sim_T-beta_sim_single_nonhier_noise_per_band_for_paper_v2.pdf}}
\subfigure{\includegraphics[width=0.48\textwidth]{Figures/Mock_SED/Comp_sim_T-beta_sim_single_hier_noise_per_band_for_paper_v2.pdf}}
\caption{Results of temperature and $\beta$ from the fit of 100 simulated SMBB SEDs with the same input parameters ($\log M_{dust}/M_\odot=8$, $T=30$ K, and $\beta = 1.5$) and 10\% added noise. The output values are derived with the non-hierarchical (left panel) and hierarchical (right panel) SED fitting method. In red is shown the input value and in blue are the measured values.
}
\label{fig:T-beta_single_input}
\end{figure*}
We also test whether the codes can recover a positive or negative $T$-$\beta$ correlation. In both cases, the hierarchical method perform equally or better than the non-hierarchical code. Details of these simulations can be found in appendix~\ref{sec:simulation}.
\section{Results}
\label{sec:results}
\subsection{JINGLE sample: non-hierarchical vs. hierarchical results}
In the previous section we have demonstrated, using simulated SEDs, that the hierarchical method works better than the non-hierarchical approach. Here we apply both methods to the \Ntot\ galaxies of the JINGLE sample and we show the advantages of using the hierarchical method.
We start by using the simplest model, the single modified black-body (SMBB).
Figure \ref{fig:comp_hier_nonhier} shows the comparison of the dust masses, dust temperatures and $\beta$ derived with the two approaches.
In general, dust masses agree quite well between the two methods (median difference = 0.07 dex). The dust masses derived using the hierarchical method are slightly smaller, and this is probably due to the variations in dust temperatures. For a given constant flux, higher dust temperatures correspond to lower dust masses.
In the range $15-25$~K the dust temperatures from the hierarchical approach are indeed slightly higher.
At high temperatures, the differences between the two methods are larger and the non-hierarchical method measures much higher temperatures ($T>\ 30$ K) than the hierarchical method.
This is because as the dust temperature increases, the peak of the SED moves to shorter wavelengths. If the SED peaks at wavelengths shorter than 100\micron, it is not sampled by the flux bands considered in the fit, since for the SMBB we are considering the 60\micron\ point as an upper limit. Therefore it is more difficult to constrain the temperature. If we were to include flux points at shorter wavelengths we would need to consider a second MBB component with a warmer temperature, because the assumption of a single temperature MBB does not hold over such a large wavelength range. Instead, in the hierarchical framework, the code uses the information from the temperature distribution of the galaxy population to constrain $T$, and it will consider more likely for the galaxy to have a temperature close to the population mean temperature than an extreme value. Therefore the hierarchical method can better constrain the dust temperature.
The range of temperatures is smaller in the hierarchical case ($T = 17-30$ K), than in the non-hierarchical case ($T = 15-48$ K). The same is true for the range of $\beta$: in the hierarchical case $\beta= 0.6-2.2$, while in the non-hierarchical case $\beta= 0.0-2.5$. In the hierarchical approach, we assume that the population follows a common distribution, thus the fitting is less likely to return extreme values of $\beta$. However, the hierarchical code can accommodate some outliers, since we do not define a priori the standard deviation of the prior distribution. Thus if the data require it, the standard deviation can be large, allowing for more `extreme' values of $\beta$.
But if the extreme objects have large noise on the flux values, then the hierarchical method considers more likely that they are not `true outliers', but that their extreme SED shape is only due to the noise in the data points.
If we believe that the hierarchical approach gives more accurate results for the cases with high noise level, we conclude that the extreme values found with the non-hierarchical approach are likely not reliable, but only due to the noise in the data.
The results of the hierarchical fit using the SMBB model are given in Table \ref{tab:results_SMBB}.
\begin{figure*}
\centering
\subfigure{\includegraphics[width=1.0\textwidth]
{Figures/Hier_nonhier/Comp_SMBB_nonhier-SMBB_hier_3plots_res}}
\caption{Comparison of dust properties of the JINGLE sample obtained through the fit of a single modified black-body (SMBB) using the non-hierarchical and hierarchical approaches. The lower panels show the difference between the hierarchical and non-hierarchical fit in each of the derived properties.
}
\label{fig:comp_hier_nonhier}
\end{figure*}
\subsection{$T$-$\beta$ relation in the JINGLE sample}
We use the results of the SED fitting using the SMBB model to investigate whether there is a relation between dust temperature and $\beta$ in our sample of galaxies. An anti-correlation between $T$ and $\beta$ has been observed in many studies \citep[e.g.][]{Dupac2003, Desert2008}, but it has been demonstrated that it can be attributed to the degeneracy between the two parameters and the effect of noise on the data \citep{Shetty2009a, Shetty2009b}.
Figure \ref{fig:T-beta} shows the results from the non-hierarchical and hierarchical approach applied to our sample of \Ntot\ galaxies.
The results from the non-hierarchical method show a significant anti-correlation between $T$ and $\beta$. The Pearson correlation coefficient is $R_{pear}= -0.79$ (p-value $= 1.19\cdot 10^{-41}$). The results from the hierarchical method shows a weaker anti-correlation ($R_{pear}= -0.52$, p-value $= 9.79\cdot 10^{-15}$).
This shows that the choice of the method used is really important and can deeply influence the results.
This result confirms previous findings \citep{Shetty2009a,Shetty2009b,Kelly2012, Veneziani2013, Juvela2013} that the observed $T-\beta$ anti-correlation is mainly driven by the fact that they are degenerate parameters, and by the noise on the data.
There is still an anti-correlation between $T$ and $\beta$ even using the hierarchical approach ($R_{pear}= -0.52$). This could mean that there is indeed a physical relation between these two quantities. However, it is also possible that the hierarchical method is not able to remove completely the $T-\beta$ degeneracy, leaving a residual anti-correlation. With our current data we are not able to distinguish whether the observed relation is a physical effect or whether it is due to a residual degeneracy.
\begin{figure*}
\centering
\subfigure{\includegraphics[width=0.45\textwidth]
{Figures/T-beta/Comp_SMBB_T_beta-nonhier_v12_scuba_errorbar_v5}}
\subfigure{\includegraphics[width=0.45\textwidth]
{Figures/T-beta/Comp_SMBB_T_beta-hier_multivar_2UL_192_gal_filter_scuba_errorbar_v5}}
\caption{Relation between the dust temperature and dust emissivity index ($T$-$\beta$ relation) for the JINGLE sample derived with non-hierarchical (left panel) and hierarchical (right panel) Bayesian methods. In both cases, we fit the SED using a single modified black-body (SMBB) model and we include the 850\micron\ flux point in the fit.
}
\label{fig:T-beta}
\end{figure*}
We also compare the results obtained with and without including the 850\micron\ flux point in the fit using the hierarchical approach (see Figure \ref{fig:comp_with_without_scuba}).
In general, the emissivity indices $\beta$ measured with the 850\micron\ flux point are equal or lower than the ones measured without the 850\micron\ point. This means that without the SCUBA-2 flux, the fits of the \Herschel\ points alone have steeper slopes. This suggests that there is indeed a `submm' excess visible at 850\micron, at least in some galaxies. This is visible especially for low values of $\beta < 1$. We note that not all galaxies show this behaviour: for some galaxies the $\beta$ values measured with and without SCUBA-2 flux are in good agreement, or they show a small deficit at 850\micron.
Consequently, the dust temperatures show the opposite trend: they are in general larger when the 850\micron\ point is included in the fit, because they have to compensate for the lower $\beta$ values.
The mass measurements are only slightly affected by the presence of the SCUBA-2 flux point (median difference: 0.002 dex). The largest difference in the dust masses measured with and without the SCUBA-2 flux point is 0.07 dex.
The fact that the dust masses do not show a larger variation depends on the fact that we assumed a constant absorption coefficient $\kappa_0$. Laboratory studies show that $\kappa$ changes with dust temperature $T$ and $\beta$ \citep{Coupeaud2011, Demyk2017a, Demyk2017b}.
Therefore, by keeping $\kappa$ constant we erase the difference in dust masses that would arise from the different temperature and $\beta$ values. A certain value of $\kappa_0$ will give an accurate dust mass only if the $\beta$ value used for the fit is the same that was used to measure $\kappa_0$ \citep{Bianchi2013}.
However, a recent laboratory study by \cite{Demyk2017a} shows that variations in $\kappa_0$ are more prominent for high temperatures ($T > 30$~K) than for low temperatures. For the temperature range considered in this study ($10-30$~K) they do not observe variations in $\kappa_0$.
A possible approach to account for variations in $\kappa_0$ would be to change the value of $\kappa_0$ according to the value of $T$ and $\beta$ used for the fitting in an iterative way.
We plan to investigate this in the future.
\begin{figure*}
\centering
\subfigure{\includegraphics[width=0.9\textwidth]
{Figures/Results_hier/SMBB/Comp_SMBB_hier-SMBB_hier_no_scuba_3plots_res.pdf}}
\caption{Comparison of the dust masses, temperatures and emissivity indices obtained through the fit of a single modified black-body (SMBB) using the hierarchical approach, with and without the SCUBA-2 flux point at 850\micron. The lower panels show the difference between fit with and without the SCUBA-2 flux in each of the derived properties.
}
\label{fig:comp_with_without_scuba}
\end{figure*}
\subsection{Comparison of models: SMBB, BMBB, TMBB}
In many cases, the SMBB model is not enough to fit the FIR/submm SED accurately. Especially at long wavelengths, the SED often shows a change in the slope. Therefore we consider also two other models: the broken emissivity law modified black-body (BMBB) and two modified black-bodies TMBB models, described in Section \ref{sec:models}. In this section we compare the results obtained applying these models to the SED fit of the JINGLE sample. The results of the hierarchical fit using the BMBB and TMBB models are given in Tables \ref{tab:results_BMBB} and \ref{tab:results_TMBB}.
\subsubsection{BMBB}
The broken emissivity law modified black-body model \citep[BMBB,][]{Gordon2014} allows for a variation in the wavelength dependence of the dust emissivity law, to account for a submm excess. This is parametrized by using two emissivity indices for shorter and longer wavelengths.
The break wavelength is a free parameter in our model.
For the JINGLE sample we find values in the range $480-488$\micron. The emissivity index at wavelengths shorter than $\lambda_{break}$ ($\beta_1$) is in the range $0.6-2.2$. The range of the second emissivity index at wavelengths $> \lambda_{break}$ ($\beta_2$) is larger ($0.1-3.3$).
We compared the results obtained using the BMBB model with the results from the SMBB model (Fig.\ref{fig:comp_SMBB_BMBB_TTMBB}).
The dust masses measured with the BMBB model are in agreement with the ones measured with the SMBB model, with a maximum difference of 0.1~dex.
The BMBB model measures generally slightly lower temperatures than the SMBB model (median difference of 1 K). In the case of a shallower slope of the submm SED, the SMBB model fits it by using a lower value of $\beta$ and a higher $T$. The BMBB can correct using a smaller value of $\beta_2$, without affecting the temperature measurement. Thus $T$ does not depend anymore on the longer wavelength points and can have a lower value. We compare also the emissivity index $\beta$ from the SMBB model, with the parameter $\beta_1$ which describes the slope of the BMBB model before the break. $\beta_1$ tends to be larger than $\beta$ from the SMBB for low values of $\beta$. This is due to the fact that any excess at longer wavelength can be modelled by a second index $\beta_2$, while in the case of the SMBB the excess needs to be taken into account by $\beta$.
The results from the BMBB model are more similar to the SMBB fit without the 850\micron\ point. This is due to the fact that the BMBB model fits the fluxes at longer wavelengths (500\micron\ and 850\micron\ point) using a second emissivity index $\beta_2$, thus the measurements of $T$ and $\beta_1$ are not sensitive to the flux measurement at 500\micron\ and 850\micron. Figure \ref{fig:outliers_comp_SMBB_BEMBB} shows an example of the SMBB and BMBB fit of one galaxy for which the difference in temperature is more evident (JINGLE 1).
This model is especially useful to quantify the possible sub-mm excess, given by the difference between the two emissivity indices $\beta_1$ and $\beta_2$. Further discussion on the submm excess is presented in Section \ref{sec:submm_excess}.
\begin{figure*}
\centering
\subfigure{\includegraphics[width=0.9\textwidth]
{Figures/Results_hier/BMBB/Comp_SMBB-BMBB_multivar_2UL_192_gal_filter_corr_3plots_res.pdf}}
\subfigure{\includegraphics[width=0.9\textwidth]
{Figures/Results_hier/TMBB/Comp_SMBB-TMBB_multivar_2UL_192_gal_filter_corr_v2_1000iter_3plots_res.pdf}}
\caption{ \textit{Upper panels}: Comparison of the cold dust masses, temperatures and emissivity index obtained through the fit of a single modified black-body (SMBB) and a broken emissivity power law MBB model (BMBB). For the BMBB model, the $\beta$ value shown in the plot is $\beta_1$, i.e. the emissivity index at wavelength < $\lambda_{break}$. The lower sub-panels show the difference between the two models in each of the derived properties. \textit{Bottom panels:} Comparison of the results from the SMBB and two modified black-bodies (TMBB) model. For the TMBB model, the values shown in the plot are the parameters of the cold component ($\log M_{cold}$, $T_{cold}$, $\beta_{cold}$).}
\label{fig:comp_SMBB_BMBB_TTMBB}
\end{figure*}
\begin{figure*}
\centering
\subfigure{\includegraphics[width=0.44\textwidth]
{Figures/SED_models/JINGLE/SMBB/JINGLE1_SMBB_hier_SED_fit.pdf}}
\subfigure{\includegraphics[width=0.44\textwidth]
{Figures/SED_models/JINGLE/BMBB/JINGLE1_BMBB_hier_SED_fit.pdf}}
\caption{SMBB and BMBB fit for the galaxy JINGLE 1, where there is a clear difference in the dust temperature measured with the two different models.}
\label{fig:outliers_comp_SMBB_BEMBB}
\end{figure*}
\subsubsection{TMBB}
The bottom panels of Figure \ref{fig:comp_SMBB_BMBB_TTMBB} show the comparison of the SMBB and the two MBB model (TMBB). The dust masses are in good agreement, with the cold dust masses derived from the TMBB being slightly higher (median offset: 0.03 dex).
The dust temperatures of the cold component obtained with the TMBB model tend to be lower than the ones measured from the SMBB model by about 3\% (or 0.8 K). This is expected, since the warm component is contributing to the fit of the 100\micron\ flux, allowing the cold component to shift to longer wavelengths, corresponding to colder temperatures. Consequently, the $\beta_{cold}$ values from the TMBB are also slightly higher (median offset: 0.05).
The outlier is JINGLE 33 (Fig.~\ref{fig:outliers_comp_SMBB_TMBB}). This galaxy has a high 60\micron\ flux, compared to the 100\micron\ flux, which results in the warm dust component (with $T_{warm}=52.3$~K) reproducing most of the emission, and skewing the cold dust component to a lower temperature ($T_{cold}=17.2$~K) and a higher dust mass.
The warm dust component does not contribute much to the entire dust mass. Warm dust masses are in the range $10^{3.4}-10^{6.6}$ M$_\odot$, which correspond to only 0.01-4.4\% of the total dust mass of the galaxies.
Nevertheless, it is important to take into account this component because, as we have shown, it can affect the measurement of the temperature and emissivity $\beta$ of the cold component. The temperatures of the warm component are in the range $66-76$~K, with the exception of JINGLE 33 which has a lower temperature (52.3~K).
If we compare the total dust masses ($M_{dust, tot} = M_{cold}+M_{warm}$) from the TMBB with the cold dust masses $M_{cold}$ from the SMBB, the latter are smaller by 10\% ($\sim0.08$~dex) on average.
Other studies found that fitting the SED using the TMBB model will result in higher cold dust masses. For example \cite{Gordon2014} found that the dust masses of the Small and Large Magellanic Clouds are 6-15 times larger when estimated using a TMBB model instead of the SMBB model.
\cite{Clark2015} found that the warm dust mass can contribute up to 38\% of the total dust mass of galaxies in the \textit{Herschel}-ATLAS survey.
The disagreement with our findings is probably due the fact that these studies do not include the 22\micron\ flux point in their fit. Consequently, their warm component is shifted to longer wavelength and has lower temperature than ours, thus contributing more to the total dust mass. The cold dust temperature of the TMBB will also be smaller than in the SMBB case, thus resulting in higher cold dust masses.
\begin{figure*}
\centering
\subfigure{\includegraphics[width=0.44\textwidth]{Figures/SED_models/JINGLE/SMBB/JINGLE33_SMBB_hier_SED_fit}}
\subfigure{\includegraphics[width=0.44\textwidth]
{Figures/SED_models/JINGLE/TMBB/JINGLE33_TMBB_hier_SED_fit}}
\caption{SMBB and TMBB fit for the galaxy JINGLE 33, which shows a clear difference in the cold dust mass measured with the two different methods. The warm component has a large contribution to the total dust emission in this galaxy.}
\label{fig:outliers_comp_SMBB_TMBB}
\end{figure*}
\subsection{Model comparison with information criterion}
In order to decide which of the models provides a better fit to the data, we applied a criterion based on the comparison of the likelihoods. We consider the Bayesian Information Criterion (BIC) \citep{Schwarz1978} which takes into account not only the likelihood of the fit, but also the number of free parameters of the models.
The latter point is important, since increasing the number of free parameters would generally lead to better fits.
The Bayesian Information Criterion (BIC) \citep{Schwarz1978} is defined as:
\begin{equation}
BIC = -2\cdot \ln(L) + q\cdot \ln(m)\,
\end{equation}
where $L$ is the likelihood (i.e. the probability of the data given the parameter $p(\vec{F}|\vec{\theta})$),
$q$ is the number of free parameters of the model, and $m$ is the number of data points (wavebands).
The model with the lowest BIC value is the preferred model according to this criterion. To calculate the likelihood $L_i$ for the $i$-th galaxy we consider the product of the likelihood $p(F_{i,j}^{obs}|\vec{\theta_i}, F_{i,j}^{err}, \delta_j)$ in all wavebands $j=1,...,m$.
\begin{equation}
L_i = \prod_{j=1}^m p(F_{i,j}^{obs}|\vec{\theta_i}, F_{i,j}^{err}, \delta_j)
\end{equation}
Figure \ref{fig:comp_info_crit} shows the BIC values for the BMBB and TMBB models compared to the SMBB model. For most of the galaxies (180/\Ntot, 94\%), the TMBB model is preferred. This is probably due to the fact that the additional warm component can help to improve the fit at 100\micron, without affecting the fit of the points at longer wavelengths.
For seven galaxies the preferred model is the BMBB model (JINGLE ID: 35, 56, 77, 101, 118, 133, and 147). In all these galaxies there is a clear submm excess at 850\micron.
The BIC criterion does not identify all galaxies for which the 850\micron\ flux is enhanced with respect to the SMBB model, but selects the ones for which the discrepancy can not be attributed to flux uncertainties or uncertainties in the model.
There are five galaxies which are best modelled with the SMBB model (JINGLE ID 83, 110, 142, 159, and 186).
The TMBB model is not able to fit well the 60\micron\ and 100\micron\ flux points of these galaxies. For JINGLE 83 and JINGLE 159 the 60\micron\ flux is too low and is not well fitted by the TMBB model. For JINGLE 110 the 60\micron\ flux is instead too high compared to the 100\micron\ flux.
For JINGLE 186, the uncertainty on the 60\micron\ flux is very small, and therefore even a small deviation from the perfect fit of that data point results in a low likelihood. In JINGLE 142, the 500$\micron$ point is enhanced with respect to the 350\micron\ flux point and the 850\micron\ upper limit. In general neither the SMBB and TMBB models are able to produce a good fit for this galaxy.
The SED fits with the BMBB and TMBB models for all galaxies are shown in Figure \ref{fig:three_SED_models}.
We conclude that the TMBB model produces the best fit of the FIR SED for most of the galaxies. Additionally, the comparison of the BIC of the SMBB and BMBB model can be used to identify galaxies which show a strong submm excess or deficit.
\begin{figure}
\centering
\subfigure{\includegraphics[width=0.44\textwidth]
{Figures/BIC/Comp_info_crit_SMBB-BMBB-TMBB_BIC_v5.pdf}}
\caption{Comparison of the negative Bayesian Information Criterion (-BIC) for the fit using the three models: SMBB, BMBB, TMBB. The model with the largest value of -BIC is the preferred model. If the difference between the BICs is smaller than two (shown by the dotted lines) there is little evidence to prefer one model over an other.
}
\label{fig:comp_info_crit}
\end{figure}
\section{Relation between dust properties and galaxy properties}
\label{sec:dust_scal_relation}
In this section, we investigate how dust properties correlate with global galaxy properties.
We use the results obtained using the SMBB model, even though the TMBB model is preferred according to the Bayesian information criterion. We decide to use the SMBB model because one of the goals of this analysis is to provide prescriptions to estimate $T$ and $\beta$ from other galaxy quantities. These prescriptions can be useful in those cases where only a few photometric data points are available and in such cases it is preferred to use the model with the smallest number of free parameter (i.e. the SMBB model). Additionally, as we have shown in the previous section, the differences in $T$ and $\beta$ derived from the SMBB and the TMBB models are not very large and they are mainly systematic shifts, that can be accounted for.
We include in this analysis also the galaxies from the \Herschel\ Reference Survey \citep[HRS,][]{Boselli2010b}, which allow us to extend the parameter range to lower SFR and specific SFR, since a large fraction of the HRS sample are galaxies which lie below the star-formation main-sequence (see Fig.~\ref{fig:SFR_Mstar_JINGLE_HRS}).
In this case, the total sample of galaxies consists of two populations: star-forming galaxies (main-sequence galaxies) and passive galaxies (below main-sequence). Therefore the basic assumption for the use of the hierarchical method that all galaxies belong to the same population does not hold any more. We therefore divide the `total' sample (JINGLE+HRS) into two sub-samples according to their position in the SFR-$M_*$ plane and fit each separately.
In this way, the assumption that the galaxies in one sub-sample belong to the same population is still valid.
We define the two sub-samples as follows:
\begin{itemize}
\item \textit{main-sequence galaxies/ star-forming sample}: galaxies belonging to the SF main-sequence or laying above it.
This sample consists of all galaxies which fall above the lower limit of the SF main-sequence, defined as 0.4 dex below the SF main-sequence from \cite{Saintonge2016}.
\item \textit{below main-sequence sample/passive sample}: galaxies laying below the SF main sequence. These are the galaxies which lie more than 0.4 dex below the SF main-sequence defined by \cite{Saintonge2016}.
\end{itemize}
The star-forming sample consists of 313 galaxies (177 from JINGLE and 136 from HRS) and the passive sample of 159 galaxies (15 from JINGLE and 144 from HRS). We did a test fitting galaxies belonging to the two sub-samples together. This test confirms that it is necessary to separate the sample in two, to avoid to force the two sub-samples to move toward a common mean, introducing systematic biases in the results.
Figure \ref{fig:SFR_Mstar_temp_beta} shows the galaxies on the SFR-$M_*$ plane, color-coded by dust temperature $T$ and emissivity index $\beta$. The dust temperature increases when moving from the bottom-right corner (high $M_*$, low SFR) to the upper-left corner (low $M_*$, high SFR). The emissivity $\beta$ instead tends to increase with $M_*$. From this figure we can already see that $T$ and $\beta$ are related to different galaxy properties, with $T$ varying depending on the SSFR and $\beta$ on the stellar mass.
\begin{figure*}
\centering
\subfigure{\includegraphics[width=0.44\textwidth]
{Figures/Dust_vs_galaxy_prop/SFR_Mstar_JINGLE_HRS_T}}
\subfigure{\includegraphics[width=0.44\textwidth]
{Figures/Dust_vs_galaxy_prop/SFR_Mstar_JINGLE_HRS_beta}}
\caption{Distribution of the JINGLE and HRS sample in the SFR-$M_*$ plane, color coded by dust temperature (left) and emissivity index $\beta$ (right). Dust temperatures and $\beta$ are measured using the SMBB model and the hierarchical SED fitting approach. The position of the star formation main sequence \citep{Saintonge2016} is shown as a dashed lines, the 0.4 dex dispersion is shown by dotted lines.}
\label{fig:SFR_Mstar_temp_beta}
\end{figure*}
We quantify the strength of these relations by calculating the correlation coefficients between $T$, $\beta$ and the following quantities:
stellar mass, stellar mass surface density ($\mu_*=M_*/(2\pi R_{50}^2)$, where $R_{50}$ is the optical half-light radius in the $i$ band from SDSS), metallicity \citep[12+log(O/H), using the O3N2 calibration of][]{Pettini2004}, HI mass fraction ($M_\text{HI}/M_*$), star-formation rate (SFR), specific SFR (SSFR), SFR surface density ($\Sigma_\text{SFR}$), and SFR divided by dust mass. We consider all quantities in log space.
We calculate the Pearson correlation coefficient $R$ and perform a linear fit when the absolute value of the correlation coefficient is higher than 0.4, both for the total sample and for the JINGLE and HRS samples separately. We did the fit also for the two samples separately to see whether there are differences in the correlations derived using JINGLE or HRS.
We apply a correction to account for the fact that the stellar mass distribution of our sample does not exactly represent the stellar mass distribution in the local Universe, using the method developed for the xCOLD GASS survey \citep{Saintonge2017}.
We compare the mass distribution of our sample, in bins of 0.1 dex in $\log M_{*}$, to the expected mass distribution of a volume-limited sample based on the stellar mass function from \cite{Baldry2012}. For each mass bin, we calculate the ratio between the normalized number of galaxies in our sample and in the mass distribution from \cite{Baldry2012}. We apply this ratio as a statistical weight when we fit the dust scaling relations.
The correlation coefficients and parameters of the linear fits are summarized in Table \ref{tab:corr_coeff}.
\begin{table*}
\centering
\caption{The table shows the Pearson correlation coefficient $R$ between dust properties (dust emissivity index $\beta$ and dust temperature $T$) and global galaxy properties. If |$R$|> 0.4 we provide the best fit relation (slope and intercept) between the selected galaxy property ($p$) and $T$ (or $\beta$).}
\label{tab:corr_coeff}
\begin{tabular}{|lccccccc|}
\hline
Properties $p$ & \multicolumn{3}{c}{correlation with $\beta$} & \multicolumn{3}{c}{correlation with $T$} \\
& $R$ & slope & intercept & $R$ & slope & intercept \\
\hline
\hline
$\log M_*$ & $0.58$ & $0.23\pm0.02$ & $-0.60\pm0.22$ &$-0.29$ & & \\
$\log \mu_*$ & $0.62$ & $0.30\pm0.03$ & $-0.84\pm0.27$ &$-0.19$ & & \\
12+log(O/H) & $0.58$ & $0.95\pm0.13$ & $-6.64\pm1.16$ &$-0.19$ & & \\
$\log M_{HI}/M_*$ & $-0.65$ & $-0.25\pm0.04$ & $1.56\pm0.02$ &$0.41$ & $0.38\pm0.23$ & $23.07\pm0.15$ \\
log SFR & $0.20$ & & &$0.21$ & & \\
log SSFR & $-0.40$ & & &$0.54$ & $1.83\pm0.19$ & $41.02\pm1.90$ \\
log $\Sigma_\text{SFR}$ & $0.13$ & & &$0.49$ & $2.49\pm0.23$ & $26.74\pm0.38$ \\
log SFR/$M_{dust}$ & $-0.15$ & & &$0.73$ & $3.40\pm0.29$ & $49.52\pm2.32$ \\
\hline
\end{tabular}
\end{table*}
\begin{figure*}
\centering
\subfigure{\includegraphics[width=0.9\textwidth]
{Figures/Dust_vs_galaxy_prop/Comp_prop_beta_T_4_Q_line_fit_MHI_fraction_2samples_2Q_for_paper_weight_rvalue_w.pdf}}
\subfigure{\includegraphics[width=0.9\textwidth]
{Figures/Dust_vs_galaxy_prop/Comp_prop_beta_T_4_Q_line_fit_SFR_2samples_2Q_for_paper_weight_rvalue_w.pdf}}
\caption{Dust scaling relations: correlation of dust temperature $T$ and effective $\beta$ with other global galaxy properties: stellar mass ($M_*$), stellar mass surface density ($\mu_*=M_*/(2\pi R_{50}^2)$, where $R_{50}^2$ is the optical half-light radius in the $i$ band from SDSS in kpc), metallicity \citep[12+log(O/H), O3N2 calibration of][]{Pettini2004}, HI mass fraction ($M_{HI}/M_*$), star-formation rate (SFR), specific SFR (SSFR), SFR surface density ($\Sigma_\text{SFR}$), and SFR over dust mass (SFR/M$_\text{dust}$).
Dust temperatures and $\beta$ are measured using the SMBB model and the hierarchical SED fitting approach.
The JINGLE sample is shown in blue and the HRS sample in magenta. Galaxies of the `main-sequence' sample are shown with circles and galaxies of the `below main-sequence' sample are shown with triangles. In every panel we show the Pearson correlation coefficient $R$. For the cases where $R>0.4$, the plot shows the linear fit to the JINGLE sample (in blue), to the HRS sample (in magenta), and to the two samples together (in black).}
\label{fig:comp_T_beta_prop}
\end{figure*}
We find that the emissivity $\beta$ shows a positive correlation with log $M_*$ (Pearson correlation coefficient $R=0.58$), log $\mu_*$ ($R=0.62$), and metallicity ($R=0.58$).
Since these galaxy properties are all correlated with each other, it is not surprising that they all correlate with $\beta$.
These trends were already observed by \cite{Cortese2014} in the HRS sample. They also observed negative correlations of these quantities with dust temperature $T$, due to the fact that they used a non-hierarchical method for the fitting and therefore they could not break the degeneracy between $T$ and $\beta$.
Thus, they were not able to distinguish whether the fundamental physical correlations were driven by the temperature or by the emissivity index. In our analysis, these three quantities do not show a strong anti-correlation with temperature ($-0.29\leq R\leq -0.19$).
We note that for the JINGLE galaxies the metallicities are measured from the SDSS fibre spectra and therefore represent only the metallicities in the central 3 arcsec of the galaxies. For the HRS sample, metallicities are measured from long-slit integrated optical spectra \citep{Boselli2013, Hughes2013}, and thus represent better the global metallicities of the galaxies. Indeed we find that the correlation between $\beta$ and metallicity is higher ($R=0.67$ if we consider only the HRS sample.
We also find an anti-correlation between $\beta$ and the HI mass fraction ($R=-0.65$), that was already observed in \cite{Cortese2014}. In this case, the HI mass fraction shows a weaker correlation with dust temperature ($R= 0.41$). The HI mass fraction is known to correlate with the inverse of the stellar mass surface density and with SSFR \citep{Catinella2013}. Thus it is expected to see an anti-correlation with $\beta$ and a positive correlation with $T$, due to the correlation of SSFR with $T$.
The dust temperature correlates with log SSFR ($R=0.54$), log $\Sigma_{SFR}$ ($R=0.49$), and log SFR/$M_{dust}$ ($R=0.73$).
These correlations have already been observed by \cite{Clemens2013} and \cite{Cortese2014}.
As stated in \cite{Clemens2013}, the fact the cold dust temperature correlates with SFR surface density but not with stellar mass surface density suggests that the cold dust is heated more by ongoing star-formation or by young stars. Also \cite{Kirkpatrick2014} observed a correlation between cold dust temperature and SFR normalized by the 500\micron\ luminosity, that is a proxy for the dust mass, on spatially resolved scales in galaxies from the KINGFISH sample \citep{Kennicutt2011}. According to their work, this correlation suggests that the number of photons from young stars relative to the amount of dust has an important heating effect on the diffuse cold dust component. Moreover, \cite{Galametz2012} studied a sub-sample of galaxies from the KINGFISH sample and observed that the higher dust temperatures coincide with the center of star-forming regions, showing a connection between dust temperature and star-formation.
The temperature of the dust is regulated by the radiation from star-formation, weighted by the amount of dust present in the galaxy.
The relation between $T$ and SSFR shows more scatter at low SSFR. This may be related in part to the fact that SFR measurements are less accurate for low SSFR \citep[log~SSFR< -10.6, ][]{Hunt2019}.
Also it is likely that the contribution of the older stellar population to the dust heating is higher in low SSFR galaxies, since the star-formation is weak and the contribution from old stars can be more significant.
\subsection{Primary correlation analysis}
In this section we investigate which are the primary parameters driving the correlation with dust properties. This analysis has two goals: 1) to provide prescriptions to estimate the temperature $T$ and the emissivity index $\beta$ of the dust from other galaxy properties, 2) to understand which are the physical quantities that influence and set $T$ and $\beta$ in a galaxy.
We perform a Bayesian inference analysis to find the best combination of parameters that can be used to estimate the dust properties.
We consider the galaxy parameters which, alone or combined, show some correlation with $\beta$ and $T$: stellar mass $M_*$, star-formation rate, dust mass, metallicity, and surface area ($A=2\pi R_{50}^2$, where $R_{50}^2$ is the optical half-light radius in the $i$ band from SDSS in kpc). The surface area is used to calculate for example the SFR and stellar mass `surface density'.
We fit first-order polynomial models with a different number of parameters, exploring all possible combinations of parameters. The number of possible combination of $k$ parameters selected from a total sample of $n$ parameters is $C_{n,k} =\frac{n!}{k!(n-k)!}$. We use a first-order polynomial model in log space:
\begin{equation}
Q_{model}(x_1, ..., x_k) = \sum_{j=1}^{k} a_j \log (x_j) +b,
\end{equation}
where $k$ is the number of galaxy properties $x_j$ considered, and $Q_{model}$ is the value of the dust quantity ($T$ or $\beta$) approximated by the model.
We use a Bayesian inference method to determine the optimal number of parameters needed to fit the data and the best fitting relations. We model the probability of observing our data, given the model and the uncertainties, as a normal distribution:
\begin{multline}
p(Q_i |Q_{model,i}(x_{1,i}, ..., x_{k,i}), Q_{err,i}) = \\ w_i \cdot \text{Normal}(Q_{model,i}, Q_{err,i}),
\end{multline}
for each galaxy $i$ in our sample, where $w_i$ is the weight correcting for the flat $M_*$ distribution (see Sec.~\ref{sec:dust_scal_relation}).
We consider only the uncertainties on the dust quantity $Q_i$, but not on the galaxy properties $x_{j,i}$. We make this choice because we want to minimise the difference between $Q_{i}$ and $Q_{model,i}$, given the quantities $x_{j,i}$.
We perform a MCMC fit using {\tt Stan}\ to find the best fitting parameters and measure the likelihood of the different models. Then we apply the Bayesian Information Criterion (BIC) to find the optimal number of parameters and the best model.
We consider first the models to estimate $\beta$. According to the BIC, the preferred model has five parameters: stellar mass, surface area, metallicity, star-formation rate, and HI mass. The best fit relation is given by:
\begin{multline}
\beta_{model} = 0.26_{-0.03}^{+0.03}\cdot \log M_{*}
- 0.27_{-0.03}^{+0.03}\cdot \log Area \\
+ 0.60_{-0.09}^{+0.09}\cdot \left[12+\log(O/H)\right]
+ 0.18_{-0.03}^{+0.03}\cdot log SFR \\
- 0.23_{-0.03}^{+0.03}\cdot log M_{HI}
-3.54_{-0.84}^{+0.82}.
\end{multline}
This model includes five parameters, several of which are known to be correlated, therefore it is difficult to know which one is more fundamentally related to $\beta$.
To assess this, we measure the increase in $R^2$ that each parameter produces when it is added to a model that contains already all other parameters. This change represents the amount of variance that can be explained by each parameter and that is not explained by the other variables. We measure $R^2$ ($0<R^2<1$) as the squared Pearson correlation coefficient between the dust parameter ($\beta$ or $T$) and the `modelled' parameter ($\beta_{model}$, $T_{model}$), i.e. the parameter estimated by the linear combination of galaxy properties.
Table \ref{tab:beta_R2} shows the results.
From the analysis of the increase of the $R^2$, we can see that the most fundamental parameter determining $\beta$ is the stellar mass (increase in $R^2$: $\Delta R^2=11.2\%$). The second one is the surface area ($\Delta R^2= 8.0\%$). Since they have opposite coefficients in the fit with almost the same magnitude ($0.26\pm0.03$ for $M_*$ and $-0.27\pm0.03$ for the surface area), this can be interpreted as the stellar mass surface density correlating with $\beta$. If we consider stellar mass and surface area combined as a single parameter in the analysis, the increase in $R^2$ due to stellar mass surface density is $\Delta R^2= 17.9\%$. The following parameter in order of importance is the metallicity ($\Delta R^2= 7.1\%$). SFR and HI mass cause a smaller increase in $R^2$ ($\Delta R^2=5.0\%$ and $5.7\%$ respectively), and the dust mass has a negligible contribution ($\Delta R^2 = 0.5\%$).
\cite{Smith2012b} studied the variation of $\beta$ in M31 (Andromeda). They found that $\beta$ decreases with galactocentric radius. Since also the stellar mass surface density, $\mu_*$, in M31 decreases with radius \citep{Tamm2012}, their result is consistent with a correlation between $\beta$ and $\mu_*$.
\cite{Koehler2015} found that the emissivity index of grains evolve from lower to higher $\beta$ values when transitioning from diffuse to denser inter-stellar medium (ISM) due to grain coagulations. If the stellar mass density is related to the density of the ISM, this could explain the relation between $\beta$ and the stellar mass surface density.
As we have seen in the previous section, $\beta$ correlates also with metallicity and with the inverse of the HI mass fraction. This indicates a relation between $\beta$ and the state of evolution of a galaxy: more evolved galaxies tend to have higher metallicity and lower HI fraction.
A possible interpretation of the variation of $\beta$ with metallicity and HI mass fraction is related to the structure and composition of dust grains. Crystalline or carbonaceous dust is characterized by a lower $\beta$ with respect to amorphous or silicate dust \citep{Desert1990,Jones2013}.
We expect less evolved (metal-poor) galaxies undergoing an elevated period of star formation activity to produce a lot of dust in stars \citep{Zhukovska2014}, and this dust has a more crystalline structure at the beginning \citep{Waters1996, Waelkens1996, deVries2010} and tends to become more `amorphous' with time \citep[e.g.][]{Demyk2001}. Therefore more evolved galaxies can be expected to have more amorphous dust and higher $\beta$. Additionally, silicate dust is thought to survive for a longer time compared to carbon dust \citep[e.g.][]{Jones2011}. Thus we expect dust in a more evolved galaxy to have a larger fraction of silicate grains that are associated with higher values of $\beta$.
Another possible explanation for the relation between $\beta$ and metallicity is the observation that the abundance of carbon stars, which produce carbon dust, decreases at high metallicities \citep{Boyer2019}. Thus we can expect high-metallicity galaxies to have less carbonaceous dust and consequently a higher $\beta$.
Another possibility is that the low $\beta$ values are due to temperature mixing. In our analysis we are not measuring directly the emissivity of dust grains but we are measuring an `effective $\beta$', which includes both the actual emissivity of the dust and the effect of temperature mixing \citep[e.g.][]{Hunt2015}.
It has been shown that variations of the dust temperatures along the line-of-sight can broaden the SED and mimic the effect of a low $\beta$ value \citep{Shetty2009a}.
\citet{Remy-Ruyer2015} find the SED of low-metallicity dwarf galaxies to be broader than the one of higher metallicity galaxies, consistent with our finding of lower $\beta$ in low-metallicity galaxies.
They explain this effect with the fact that dwarf galaxies have a clumpier ISM that produces a wider distribution of dust temperatures.
Since the preferred relation to approximate $\beta$ needs a large number of parameters, we also provide the
best relation with two parameters (stellar mass and surface area) and with three parameters (stellar mass, surface area, and metallicity), that are more practical to use:
\begin{multline}
\beta_{model} = 0.42_{-0.02}^{+0.02}\cdot \log M_{*}
- 0.37_{-0.03}^{+0.03}\cdot \log Area
-1.97_{-0.18}^{+0.18}.
\end{multline}
\begin{multline}
\beta_{model} = 0.28_{-0.03}^{+0.03}\cdot \log M_{*}
- 0.38_{-0.03}^{+0.03}\cdot \log Area \\
+ 0.80_{-0.09}^{+0.09}\cdot \left[12+\log \text{(O/H)}\right]
-7.48_{-0.67}^{+0.64}.
\end{multline}
A summary with the best relations for every number $k$ of parameters can be found in Table \ref{tab:corr_coeff_best}.
We perform a similar analysis to investigate which combination of parameters gives the better approximation of the dust temperature $T$.
According to the BIC, the preferred model has three parameters: SFR, dust mass, and metallicity (BIC$=848.8$). Also the two-parameter model with SFR and dust mass has a similar BIC (BIC$=849.6$), meaning that adding the metallicity parameter has only a small effect on improving the correlation.
This confirms our previous finding that dust temperature correlates strongly with SFR per unit dust mass.
The $R^2$ analysis gives the same result: the most important parameter is clearly the SFR ($\Delta R^2= 87.9\%$), with a secondary dependence on the dust mass ($\Delta R^2= 16.6\%$). The other four parameters have a very small effect ($\Delta R^2< 3\%$).
This relation is however of limited practical interest since it requires prior knowledge of the dust mass.
Therefore we consider also the two-parameter model with the best BIC that do not include $\log M_{dust}$ as a parameter. The two-parameter model uses SFR and stellar mass ($R=0.50$)
\begin{equation}
T_{model} = 2.50_{-0.22}^{+0.22}\cdot \log SFR -2.14_{-0.19}^{+0.20}\cdot \log M_{*} +44.24_{-2.02}^{+1.93}.
\end{equation}
Tables for $T$ and $\beta$ with all the relations with two or three parameters are in the appendix (Tables \ref{tab:corr_beta} and \ref{tab:corr_T}).
\begin{table}
\centering
\caption{Increase in $R^2$ when the parameter is added to a model that already contains the other parameters.}
\begin{tabular}{|lcc|}
\hline
& $\beta$ & $T$ \\
Parameter & increase in $\Delta R^2$ ($\%$) & increase in $\Delta R^2$ ($\%$) \\
\hline \hline
$\log M_*$ & 11.2 & 0.5\\
log SFR & 5.0 & 80.0 \\
log Area & 8.0 & 2.4\\
12+log(O/H) & 7.1 & 1.5 \\
$ \log M_{dust}$ & 0.5 & 13.6\\
$ \log M_{HI}$ & 5.7 & 0.5 \\
\hline
\end{tabular}
\label{tab:beta_R2}
\end{table}
\begin{table*}
\centering
\caption{Results of the correlation analysis to derive an expression to approximate the emissivity $\beta$ and the dust temperature using global galaxy properties. The table shows the coefficients $a_j$ of the best polynomial expression ($Q_{model}(x_1, ..., x_k) = \sum_{j=1}^{k} a_j \log (x_j) +b$) to estimate $\beta$ and $T$ using a different number of parameters $k$. The table also shows the Baysian Information Criterion (BIC) and the Pearson correlation coefficient $R$ between the dust parameter ($\beta$ or $T$) and the `modelled' parameter ($\beta_{model}$, $T_{model}$), i.e. the parameter estimated by the linear combination of galaxy properties.}
\label{tab:corr_coeff_best}
\begin{tabular}{|lccccccccc|}
\hline
\multicolumn{9}{c}{\bf emissivity index $\beta$}\\
Parameters & $\log M_*$ & log SFR & log Area & 12+log(O/H) & $\log M_{dust}$ & $ \log M_{HI}$ & intercept & BIC & $R$\\
& [M$_\odot$] &[M$_\odot$ yr$^{-1}$] & [kpc$^2$] & & [M$_\odot$] &[M$_\odot$] & & & \\
\hline
$k$ = 1 & & & & 0.98 $\pm$ 0.06 & & & -6.77 $\pm$ 0.59 & 170.56 & 0.61 \\
$k$ = 2 &0.42 $\pm$ 0.02 & & -0.37 $\pm$ 0.03 & & & & -1.97 $\pm$ 0.18 & 53.19 & 0.64 \\
$k$ = 3 &0.28 $\pm$ 0.03 & & -0.38 $\pm$ 0.03 & 0.80 $\pm$ 0.09 & & & -7.48 $\pm$ 0.64 & -14.37 & 0.70 \\
$k$ = 4 &0.33 $\pm$ 0.03 & & -0.29 $\pm$ 0.03 & 0.69 $\pm$ 0.10 & & -0.13 $\pm$ 0.03 & -5.92 $\pm$ 0.69 & -27.87 & 0.71 \\
$k$ = 5 &0.26 $\pm$ 0.03 & 0.18 $\pm$ 0.03 & -0.27 $\pm$ 0.03 & 0.60 $\pm$ 0.09 & & -0.23 $\pm$ 0.03 & -3.54 $\pm$ 0.82 & -54.40 & 0.73 \\
$k$ = 6 &0.31 $\pm$ 0.04 & 0.23 $\pm$ 0.04 & -0.25 $\pm$ 0.04 & 0.66 $\pm$ 0.10 & -0.13 $\pm$ 0.08 & -0.20 $\pm$ 0.04 & -3.84 $\pm$ 0.89 & -51.19 & 0.73 \\
\hline \hline
\multicolumn{10}{c}{\bf Temperature} \\
Parameters & $\log M_*$ & log SFR & log Area & 12+log(O/H) & $\log M_{dust}$ & $ \log M_{HI}$ & intercept & BIC & $R$\\
& [M$_\odot$] &[M$_\odot$ yr$^{-1}$] & [kpc$^2$] & & [M$_\odot$] &[M$_\odot$] & & & \\
\hline
$k$ = 1 & & 0.65 $\pm$ 0.13 & & & & & 22.93 $\pm$ 0.08 & 1024.78 & 0.15 \\
$k$ = 2 & & 4.19 $\pm$ 0.29 & & & -3.73 $\pm$ 0.30 & & 51.88 $\pm$ 2.20 & 849.60 & 0.68 \\
$k$ = 3 & & 4.06 $\pm$ 0.29 & & -1.85 $\pm$ 0.75 & -3.31 $\pm$ 0.31 & & 64.7 $\pm$ 5.44 & 848.76 & 0.68 \\
$k$ = 4 & & 3.93 $\pm$ 0.31 & -0.66 $\pm$ 0.28 & -2.24 $\pm$ 0.79 & -2.71 $\pm$ 0.41 & & 64.13 $\pm$ 5.67 & 849.08 & 0.69 \\
$k$ = 5 &0.36 $\pm$ 0.35 & 3.99 $\pm$ 0.32 & -0.63 $\pm$ 0.29 & -2.36 $\pm$ 0.76 & -3.08 $\pm$ 0.57 & & 64.86 $\pm$ 5.82 & 853.77 & 0.69 \\
$k$ = 6 &0.29 $\pm$ 0.39 & 4.01 $\pm$ 0.33 & -0.58 $\pm$ 0.30 & -2.59 $\pm$ 0.81 & -2.86 $\pm$ 0.64 & -0.23 $\pm$ 0.29 & 67.87 $\pm$ 7.28 & 859.06 & 0.70 \\
\hline
\end{tabular}
\end{table*}
\subsection{Submm excess}
\label{sec:submm_excess}
In this section we discuss the behaviour of the SED at long wavelengths ($\lambda > 500\micron$). In particular, we are interested in galaxies which show a so-called `submm excess'.
An excess at submm wavelength has been observed in dwarf galaxies \citep{Lisenfeld2002, Galliano2003}, in late-type galaxies \citep{Dumke2004, Bendo2006, Galametz2009}, and in the Magellanic Clouds \citep{Israel2010, Bot2010b}. The most significant excesses can not be explained by contribution from synchrotron, free-free or molecular line emission \citep[e.g.][]{Galliano2003}. Different explanations proposed to explain this phenomenon are for example the presence of a very cold dust component, a temperature-dependent emissivity \citep{Meny2007}, and the presence of rotating or magnetic grains \citep{Draine2012}.
We identify the galaxies with an excess at 850\micron\ with respect to the SMBB model, taking into account uncertainties on the SCUBA-2 fluxes and on the SMBB model:
\begin{equation}
F^{obs} -F^{model} > F^{obs}_{err} +F^{model}_{err}.
\end{equation}
There are 27/\Ntot (14\%) galaxies which satisfy this criterion. If we adopt a more stringent criterion, requiring the galaxy to have an excess above 2$\sigma$ (i.e. $(F^{obs} -F^{model}) > 2\cdot F^{obs}_{err}$), we find that 24 galaxies (12\%) satisfy this criterion. From a normal distribution, we would expect to find only 2.5\% of the galaxies with an excess above 2$\sigma$, thus we think that it is a statistically significant result.
The galaxies with submm excess do not appear to be in a particular region of the SFR-$M_*$ plane (see Fig. \ref{fig:SFR_Mstar_submm_excess}).
There also some galaxies which show a deficit at 850\micron.
A weak point of this analysis is that the submm excess is determined only by a single point, the 850\micron\ SCUBA-2 flux. Therefore the presence of an excess can also be due to a number of factors including measurement errors, uncertainties on the apertures, contamination by other sources, and uncertainties on the CO(3-2) contribution. In order to better characterise and quantify the submm excess, additional flux points at longer wavelengths are needed. We plan to investigate this in the future.
We have an accepted proposal to observe 18 JINGLE targets at 1mm and 2mm with NIKA-2 on the IRAM-30m telescope.
With two additional flux points we will be able to characterize better the submm excess and to test different models proposed to account for it.
\begin{figure}
\centering
\subfigure{\includegraphics[width=0.44\textwidth]
{Figures/submm_excess/SFR_Mstar_JINGLE_submm_ex_submm_ex_galaxies.pdf}}
\caption{JINGLE galaxies with a submm excess are shown by red stars symbols, while the JINGLE sample is shown in light blue. The position of the star formation main sequence \citep{Saintonge2016} is shown as a dashed lines, the 0.4 dex dispersion is shown by dotted lines.}
\label{fig:SFR_Mstar_submm_excess}
\end{figure}
\section{Conclusions}
\label{sec:conclusions}
In this paper we analyse a sample of \Ntot\ star-forming galaxies from the JINGLE survey. We also include in the analysis 323 galaxies from the Herschel Reference Survey (HRS) to expand our analysis to galaxies with lower specific star-formation rate.
We fit their far-infrared/submm SED with modified black-body (MBB) models using a hierarchical Bayesian approach that allows to reduce the degeneracy between parameters, especially between dust temperature and emissivity index $\beta$. We consider three models: single modified black-body (SMBB), two modified black-bodies (TMBB), and MBB with a broken emissivity law (BMBB).
The main results of our study are:
\begin{itemize}
\item \textit{Dust masses}: the choice of the model (SMBB, BMBB or TMBB) has only a small effect on the dust mass estimates. The cold dust masses measured with the TMBB are larger than the ones measured by the SMBB by only 0.04 dex on average, and the dust masses measured with the BMBB model agree very well with the SMBB results.\\
\item $T$-$\beta$ relation: the use of the hierarchical Bayesian approach to fit the FIR SED is crucial to infer the intrinsic relation between dust temperature and dust emissivity index $\beta$.
In the JINGLE sample, the anti-correlation between $T$ and $\beta$ is reduced when we use the hierarchical approach ($R=-0.52$) with respect to the non-hierarchical result ($R=-0.79$). Using the hierarchical approach, both $T$ and $\beta$ span smaller ranges ($17\ \text{K} < T < 30$ K, $0.6 < \beta < 2.2$) with fewer outliers.
\\
\item \textit{Dust scaling relations:} the hierarchical approach is able to reduce the degeneracy between $T$ and $\beta$ and to separate their relations with other galaxy properties. We find that the dust emissivity index $\beta$ correlates with stellar mass surface density, metallicity and anti-correlates with HI mass fraction ($M_{HI}/M_*$). The strongest relation is with stellar mass surface density.
The dust temperature correlates with HI mass fraction, SSFR, SFR surface density and SFR per unit dust mass. The strongest relation is with SFR per unit dust mass.
These relations can be used to estimate the dust temperature or emissivity index in galaxies where insufficient data prevents determining them directly through SED fitting.\\
\item \textit{Submm excess}: we observe an excess at 850\micron\ with respect to the flux predicted from the SMBB fit in 26/\Ntot\ (14\%) galaxies, but we do not find these galaxies to lie in a particular region in the stellar mass-SFR plane.
Additional flux points at longer wavelengths are needed to better characterize the submm excess and to investigate its origin.
\end{itemize}
The dust scaling relations derived in this work based on low-redshift galaxies show that dust properties correlate with global galaxy properties. After calibrating these relations with data at higher redshift, they could be applied to the study of high-redshift galaxies.
Thanks to ALMA it is now possible to detect dust emission in galaxies at redshifts as high as $z>7$ \citep[e.g.][]{Watson2015, Laporte2017}, but the measurement of dust masses in these objects is difficult due to the scarcity of photometric points.
The possibility to use scaling relations to predict what dust properties to apply in the SED modelling will increase the precision of the dust mass measurements in the early Universe, and consequently will help our understanding of dust evolution over cosmic time.
\section*{Acknowledgements}
We thank Boris Leistedt, Fr{\'e}d{\'e}ric Galliano, Luca Cortese, and Lorne Whiteway for useful discussions.
A.S.~acknowledges support from the Royal Society through the award of a University Research Fellowship.
I.D.L.~gratefully acknowledges the support of the Research
Foundation Flanders (FWO).
C.D.W. acknowledges support from the Natural Science and Engineering Research Council of Canada and the Canada Research Chairs program.
E.B.~acknowledges support from the UK Science and Technology Facilities Council [grant number ST/M001008/1].
M.J.M.~acknowledges the support of the National Science Centre, Poland, through the SONATA BIS grant 2018/30/E/ST9/00208.
The James Clerk Maxwell Telescope is operated by the East Asian Observatory on behalf of The National Astronomical Observatory of Japan; Academia Sinica Institute of Astronomy and Astrophysics; the Korea Astronomy and Space Science Institute; the Operation, Maintenance and Upgrading Fund for Astronomical Telescopes and Facility Instruments, budgeted from the Ministry of Finance (MOF) of China and administrated by the Chinese Academy of Sciences (CAS), as well as the National Key R$\&$D Program of China (No. 2017YFA0402700). Additional funding support is provided by the Science and Technology Facilities Council of the United Kingdom and participating universities in the United Kingdom and Canada. The authors wish to recognize and acknowledge the very significant cultural role and reverence that the summit of Mauna Kea has always had within the indigenous Hawaiian community. We are most fortunate to have the opportunity to conduct observations from this mountain.
This research has made use of data from HRS project. HRS is a Herschel Key Programme utilising Guaranteed Time from the SPIRE instrument team, ESAC scientists and a mission scientist.
The HRS data was accessed through the Herschel Database in Marseille (HeDaM - http://hedam.lam.fr) operated by CeSAM and hosted by the Laboratoire d'Astrophysique de Marseille.
This research made use of Astropy, a community-developed core Python package for Astronomy \citep{astropy}, {\tt Matplotlib} \citep{Hunter2007} and {\tt NumPy} \citep{VanDerWalt2011}. This research used the {\tt TOPCAT} tool for catalogue cross-matching \citep{Taylor2005}.
This research used the {\tt Stan}\ interface for Python {\tt PyStan} \citep{pystan}.
This research used the {\tt CORNER} Python package \citep{corner}.
\bibliographystyle{mn2e}
|
2,877,628,088,988 | arxiv | \subsection{RQ1. To what extent do SE participant studies report the diversity of participants?}
The first step to understanding the awareness of diversity in SE research is to examine how often SE studies report the diversity of their participants.
This analysis will help us understand how well-established are the practices of collecting, analyzing and reporting demographic data about participants.
Hence, to answer RQ1, we performed the qualitative content analysis described in Section~\ref{sec:creatingDivCategories}.
We also break our results by type of participant study (e.g., survey, task-based) and for this purpose we rely on how researchers describe their own study.
\textbf{We found that 98.01\% of the examined participant studies report the diversity of participants.} Out of our sample of 105 studies that we analyzed in our research, 103 studies used descriptive words and phrases when characterizing participants. Only 2 studies out of 105 did not report any participant diversity.
To make better sense of the results, we break our analysis per type of participant study in Table~\ref{tab:study-categories}.
First, we note that SE papers use participant studies on 11 different study modalities, the most common being: surveys (51 studies), task-based (33 studies), interviews (22 studies), and validation (19 studies).
Second, most types of studies tend to report an average of 3 different categories when describing the participants, with Lab studies reporting the highest average (4.6 categories) across the samples we investigated.
Some studies, such as tasked-based and lab studies, reported a wide array of diversity categories when describing participants, including 8 different types of descriptors.
Our results indicate that the practice of reporting participant diversity is well-established in participant studies in Software Engineering.
Researchers tend to select between 2-4 categories of descriptors when describing participants.
Only in rare cases (2 out of 105 studies) do researchers omit to describe the participants included in their study.
\begin{table}[]
\centering
\caption{The frequency in which studies report diversity category (Reported column) per type of participant study.}
\label{tab:study-categories}
\input{tables/categories-participant-studies}
\end{table}
\subsection{RQ2. What diversity categories are reported in SE participant studies?}
\begin{table*}[ht]
\centering
\caption{Diversity categories identified in participant study description of 105 studies from ICSE 2019, 2020, and 2021.}
\label{tab:diversity-categories-with-definition}
\input{tables/diversity-categories-with-definition}
\end{table*}
After determining that most SE participant studies do report aspects of participant diversity, we want to identify what kinds of diversity are reported. Such analysis will help us better understand what characteristics of participant diversity are emphasized by SE researchers and the aspects that might be overlooked and require more attention.
To answer RQ2, we created a coding scheme where we identified descriptors about participants in SE studies. We then grouped similar descriptors to form diversity categories as mentioned in section \ref{sec:creatingDivCategories}.
\textbf{We found 12 different diversity categories reported in SE participant studies.}
Table~\ref{tab:diversity-categories-with-definition} shows the diversity categories we identify in the content analysis, as well as their definition and frequency (Freq) in our sample. The frequencies do not sum up to 100\% as, on average, studies report multiple categories of diversities (as shown in Table~\ref{tab:study-categories}).
Overall, two types of diversity remained dominant in the participant diversity reported by our sample of studies, the \cat{Experience} and the \cat{Main Occupation} of participants.
This shows that SE studies are often concerned with showcasing a diverse sample of participants regarding professional backgrounds.
The \cat{Experience} of participants was reported in 89\% of the studies.
We group in this category different kinds of professional experiences, from years of industrial experience to experience in a specific technology, such as programming languages.
As Zhang and Yang report in their study~\cite{Zhang&Yang:2019}:
\begin{quotebox}
``Eleven participants had two to \terms{five years of Java experience}, while the other five were \terms{novice programmers with one-year Java experience}, showing a good mix of different levels of Java programming experience.''
\end{quotebox}
\cat{Main Occupation} was reported in 77\% of the studies, showing a well-established practice to collect and report participation occupations.
This category includes professional occupations (e.g., developer, quality analyst) and education-related occupations (e.g., Ph.D. student, MSc. student).
For example, as Ju \textit{et al.}~\cite{Ju&Sajnani:2021} describes:
\begin{quotebox}
``This process yielded 397 \terms{developers} and 1167 \terms{developer managers} for interviews and 1629 \terms{developers} and 754 \terms{developer managers} for surveys''
\end{quotebox}
Three other categories were reported to describe participants in a comparable frequency: \cat{Gender/Sex} (39\%), \cat{Education} (37\%), and \cat{Location} (33\%).
Reported in at least a third of the studies, these categories help qualify the diversity of participants in terms of gender, scholarly level and geographic location.
For example, Dias \textit{et al.} report the location diversity of their participants as follows:
\begin{quotebox}
``Our interviewees are \terms{located in different countries}, such as Brazil, Canada, Czech Republic, USA, Germany, and Portugal.''~\cite{Dias&Meirelles:2021}
\end{quotebox}
\cat{Age} (18\%) and \cat{Language} (8\%) of participants are less commonly reported in our sample of SE studies.
The age of participants is a characteristic that may be indirectly captured by participants' years of experience.
However, it is remarkable that the language spoken by participants is seldom reported, given the international audience of SE papers and the importance of effective communication in any type of study with participants.
For example, Xia \textit{et al.} reported on the efforts of translating the survey content to ensure effective communication with participants:
\begin{quotebox}
``To support respondents from China, we translated our survey to \terms{Chinese} before publishing the survey.''~\cite{Xia&Wan:2019}
\end{quotebox}
Finally, some categories were reported only in a handful of SE participant studies.
These categories include \cat{Nationality} (5\%), \cat{Race} (4\%), \cat{Physiological} (2\%), \cat{Psychological} (2\%) and \cat{Socioeconomic status} (1\%).
These categories describe specific racial and social aspects of participants and seem to only be collected and reported in specific cases.
For example, Krueger~\textit{et al.}~\cite{Krueger&Huang:2020} ask participants to complete psychological measurement surveys, such as the Positive and Negative Affect Scale (PANAS, emotional health).
Peitek~\textit{et al.}\cite{Peitek&Apel:2021} reported that all participants in their study had a normal or corrected-to-normal vision and were right-handed.
Once we analyze the frequency of categories based on two major themes of categories, professional and social category (Themes column of Table~\ref{tab:diversity-categories-with-definition}), it becomes clear that \textbf{SE studies emphasize professional diversity over social diversity when selecting and reporting participants.}
All professional categories are reported in at least a third of all studies, while only social categories of \cat{Gender} and \cat{Location} reach similar levels of frequency.
Social diversity categories, also known as personal or identity categories, are considered particularly important for EDI efforts towards social equity \cite{GovCanada:2021}, however, our results show that SE studies do not often report this type of diversity.
\subsection{RQ3.What is the function of diversity in SE participant studies?}
In the previous RQ we found that SE studies commonly report characteristics of participants in their studies, with emphasis on professional background.
In this RQ, we aim to assess the function of participant diversity in SE research.
To that end, we follow the methodology described in Section~\ref{sec:divCategoriesInSections}, where we classify the function of participant diversity in four categories: 1) describing participants, 2) analyzing the impact of diversity in the study results, 3) reflecting upon participant diversity in the study conclusion, and 4) assessing the limitation of diversity in the participant study.
\begin{figure}[]
\centering
\includegraphics[scale=0.67]{figures/categories-per-section.pdf}
\caption{How participant diversity is used in the sample of 105 SE studies. We report the frequency of studies that use participant diversity for \textbf{describing} participants, \textbf{analyzing} results, \textbf{reflecting} the study outcomes, and assessing the study \textbf{limitations}. }
\label{fig:diversity-categories-per-section}
\end{figure}
\begin{table}[]
\centering
\caption{
The function of participant diversity in the 105 studies, broken down by diversity category.}
\label{tab:percent-div-cats-per-section}
\input{tables/percent-diversity-per-section}
\end{table}
Figure~\ref{fig:diversity-categories-per-section} shows the frequency of every function of participant diversity in the sample of 105 studies.
We determine that 98\% of studies describe participant diversity by considering 103 out of 105 studies that report diversity.
Naturally, this result is expected as the primary function of participant descriptors is to describe the participants.
Interestingly, however, participant diversity is less frequently referred to when researchers analyze their study results (50\%), reflect upon the results (35\%), and assess the limitations of their study (44\%).
This leads us to conclude that \textbf{in most SE participant studies, diversity is reported as a means to describe participants but is less frequently used for further analysis or reflection in the research}.
Table~\ref{tab:percent-div-cats-per-section} displays a breakdown of the participant diversity function, across diversity categories.
To make it easier to identify patterns across the diversity categories, we highlight the highest frequency (dark blue) and the second-highest frequency (light blue) in each category.
It can be seen that \cat{Experience} is the diversity category that most consistently presents the four functions of describing, analyzing, reflecting and assessing limitations in SE studies.
While also a dominant professional category (as discussed in RQ2), \cat{Main Occupation} is mostly used to describe participants (76\%) and is only used for reflection or assessing limitations in 10\% of the studies in our sample. This leads us to conclude that \textbf{SE researchers use the \cat{Experience} of participants consistently in the four functions of diversity we analyzed}.
At least a quarter of the studies in our sample discusses the experience of participants when analyzing the results, outcomes, and limitations of their study.
In contrast, \textbf{all other diversity categories are used mostly to describe participants, with no further reflection on its possible impact in the research.}
Some interesting patterns emerge when we look at the second most frequent function (light blue) per diversity category in Table~\ref{tab:percent-div-cats-per-section}.
We identified \cat{Gender/sex}, \cat{Location}, and \cat{Language} as social diversity categories which researchers use more frequently to assess the study limitations.
Often, these categories are used to discuss the limitations of the study's generalizability (external validity).
The following quote from Alsuhaibani \textit{et al.}~\cite{Alsuhaibani&Newman:2021} is an example of the use of participation diversity to discuss the study's limitations:
\begin{quotebox}
``With regards to external validity, we did not directly collect geolocation data. However, we did collect IP addresses, which gave us country information. From this we determined that participants came from 72 different countries, mainly from Europe and North America. Thus, we feel that the results are generalizable to a broad population.''~ \cite{Alsuhaibani&Newman:2021}
\end{quotebox}
\cat{Race} and \cat{Socioeconomic status} which are two eminent social diversity categories, were only reported in 4\% and 1\% of all studies in our sample, respectively. These two social diversity categories played a role when describing the participants but were not used in further analysis or reflection in any SE study we sampled. This finding leads us to the conclusion that although some SE researchers collect and report data about participants' race and socioeconomic status, they do not use this data for further analysis or reflection in their research. This is not necessarily problematic. For example, considering to which extent the sample of research participants represents the demography is not necessarily linked to investigating social differences. However, especially in the case that participants are far from representing the demography, a further reflection on the impact of this fact on the results of the research might be important.
To conclude, diversity plays four major functions: describing, analyzing, reflecting upon and assessing the limitations of participants in SE studies. Our results indicate that in most studies, SE researchers report diversity when describing participants. However, participant diversity is used less frequently by researchers when discussing the study results, reflecting on the study outcomes, or when assessing the study's limitations.
\section*{General Abstract}
Incorporating diversity considerations in research, development, and innovation has become an increasingly important topic. It is a well-known fact that diverse teams produce better outcomes, whereas the lack of diversity might result in biased and discriminatory technologies. Therefore, the inclusion of diverse stakeholders is considered paramount for the creation of an ethical and social-responsible future. With this study, we aim to contribute to the conversation on how EDI (equity, diversity, inclusion) can be implemented in Software Engineering (SE) research. In our study, we focus on SE research that includes research participants since this is an evident opportunity to consider diversity, and we investigate to which extent and with what purpose SE researchers consider and report diversity in their research papers. Our results demonstrate that only a few studies do not consider diversity at all, however, the examined studies differ greatly in the range of the consideration and reporting of diversity. From these outcomes, we draw the conclusion of differences in the diversity awareness among SE researchers. Finally, we propose a model of diversity awareness for participant studies as a tool to support SE researchers in reflecting on diversity and incorporating it systematically in their research.
\section{Introduction}
\label{sec:introduction}
In recent years, the consideration of the diversity of people who are involved in research and innovation has become an increasingly important topic. Diversity of developers and researchers is regarded – and has already been proven - beneficial for numerous reasons: diverse teams produce better outcomes \cite{Menezes&Prikladnicki:2018, Pieterse&Kourie:2006,Patrick&Kumar:2012}, research and development that systematically includes diversity are of benefit for a broader population whereas the lack of diversity might result in biased and discriminatory technologies \cite{Schiebinger&Klinge:2011,Buolamwini&Gebru:2018,Tannerbaum&Ellis:2019}, the inclusion of diverse stakeholders is paramount for the creation of an ethical and social-responsible future \cite{Sarewitz:2005} \cite{VanOudheusden&Shelley-Egan:2021}, and, last but not least, excluding large parts of society from research and the development of future technologies violates human rights \cite{Oberleitner:2021,Al-Nashif:2021,PARL:2010}. One size does not fit all, and this applies to research and development, too.
To mitigate the negative effects of non-diverse research and to stimulate researchers to incorporate diversity in their research, national and international research funding programs are developing policies and guidelines for equity, diversity, and inclusion (EDI) \cite{NSERC:2022}, for responsible research and innovation (RRI) \cite{EU:2013} to bind the research funding to increasingly important conditions: the sufficient consideration of EDI.
In our study, we focus on research that includes research participants as we regard this as an evident opportunity for SE researchers to include diversity considerations in their research. Therefore, our research questions are as follows:
\begin{itemize}
\item RQ1. To what extent do SE participant studies report the diversity of participants?
\item RQ2. What diversity categories are reported in SE participant studies?
\item RQ3. What is the function of diversity in SE participant studies?
\end{itemize}
To capture all variants of diversity categories, we choose an inductive approach, which means: we do not predefine the categories to examine the content according to these categories. Instead, we reconstruct the diversity categories through open coding of the content. Therefore, for the purpose of this study, we choose a broader definition of diversity categories: A diversity category is a category based on which individuals (in our case, research participants) are distinguished and grouped. This open approach allows for identifying categories that might be relevant for research but are not included in the commonly discussed diversity categories such as gender, race, and ethnicity, amongst others \cite{Nkomo&Stewart:2006}.
Our results demonstrate that only a few studies do not consider diversity at all, however, the examined studies differ greatly in the range of the consideration and reporting of diversity. From these outcomes, we draw the conclusion of differences in diversity awareness among SE researchers. We propose a model of diversity awareness for participant studies as a tool to support SE researchers in reflecting on diversity and incorporating it systematically in their research.
Finally, we publish our classification coding scheme and the dataset used to conduct this study to encourage further studies on diversity awareness and facilitate replicability\footnote{\url{https://doi.org/10.5281/zenodo.7587076}}.
This paper is organized as follows.
Section~\ref{sec:relatedWork} describes the related literature.
We detail our study methodology in Section~\ref{sec:methodology} and present the results of our three research questions in Section~\ref{sec:results}.
Upon reflecting on the results, we discuss a model of diversity awareness and guidelines for the community in Section~\ref{sec:discussion}.
Finally, we discuss the limitations of our study in Section~\ref{sec:threats} and our final remarks in Section~\ref{sec:conclusion}.
\section{Related Work}
\label{sec:relatedWork}
Time and time again, diversity, and the importance of diversity, has been a topic of discussion for many researchers in various fields. This has led to different definitions of diversity emerging in different contexts \cite{Tamtik:2022, Zanoni&Janssens:2004, Nagappan&Zimmermann:2013}. In recent years, diversity, especially gender diversity, has received increasing attention in the field of SE \cite{Catolino&Palomba:2019, Burnett&Peters:2016, Padala&Mendez:2022, Bosu&Sultana:2019}.
\noindent\textbf{Lack of Diversity in SE.}
The issue of lack of diversity, also referred to as the diversity crisis, in SE has been investigated by many SE research papers as can be seen in the IEEE special issue article by Albusays \textit{et al.} \cite{Albusays&Bjorn:2021}. The importance of diversity and inclusion in the field of SE has been clearly stated in the article with one of the questions being ``What are the relevant diversity parameters we should consider when exploring software development practices and technology'' \cite{Albusays&Bjorn:2021}. Tamtik~\cite{Tamtik:2022} notes that only understanding diversity in terms of gender, language, and socio-economic categories limits the understanding of diversity. Similarly, Rodriguez and Nadri~\textit{et al.} conclude that more perceived diversity aspects need to be considered in SE research \cite{Rodriguez&Nadri:2021}. One of the goals of our research is to address these challenges in the context of SE research, by (i) revealing how SE researchers are addressing diversity in their studies and reporting about it in their publications, and (ii) by providing SE researchers with guidelines they can use when conducting participant studies.
Other work by Menezes and Prikladnicki~\cite{Menezes&Prikladnicki:2018} investigated various publications about diversity in SE by performing a literature review. They also investigated the impact of diversity on processes in software development by conducting semi-structured interviews. They concluded that many challenges still need to be overcome to make SE work environments more diverse. Our work complements the work by Menzes and Prikladnicki since we investigate to what extent SE research considers diversity, the types of categories used and the function of diversity.
\noindent\textbf{Human Values in SE Research.}
Perera~\textit{et al.} examine to what extent human values are considered in SE research instead of diversity \cite{Perera&Hussain:2020}. They use a similar methodology to ours, where they analyze research papers from top-tier SE conferences and journals to assess their consideration of human values. The found that very few SE publications consider human values.
Storey~\textit{et al.} conducted a similar study where they investigate the consideration of human aspects in SE research by analyzing SE research papers \cite{Storey&Ernst:2020}. They found most SE studies focus on technical aspects of SE even though these studies claim to impact human stakeholders. Storey~\textit{et al.} concluded that there is a need for SE studies to consider more human aspects and recommended a framework they created that can be used to consider more human and social aspects in SE research.
Although the work by Perera~\textit{et al.} \cite{Perera&Hussain:2020} and Storey \textit{et al.}~\cite{Storey&Ernst:2020} used a similar approach to ours, these two studies investigate the considerations of human aspects/values in SE research whereas we study the consideration of diversity in SE research, more specifically participant studies, while providing guidelines for SE researchers conducting participant studies.
The work closest to ours is the work by Lenarduzzi \textit{et al.}~\cite{Lenarduzzi&Dieste:2021} who investigated participant studies in SE, where they examined the current participant selection guidelines and practices in empirical SE research. They analyze existing guidelines for participant selection in SE and present the participant selection strategies being currently used in their results. Our work complements their study since we study actual research papers involving participants and create guidelines
for SE researchers, whereas their study only examined guidelines (and not actual research papers).
\section{Methodology}
\label{sec:methodology}
The goal of our research is to examine to which extent diversity is considered in SE research and to provide guidelines to SE researchers that they can use when conducting participant studies. As a methodological approach, we apply qualitative and quantitative content analysis \cite{krippendorff&Bock:2009} \cite{krippendorff:2018} where we used open coding and axial coding methods \cite{Saldana:2009} to examine software engineering research papers which include participant studies. Due to the novel nature of this study, we start by performing a pilot study to assess the viability of this study and develop an appropriate framework.
Fig.~\ref{fig:method-overview} shows an overview of our methodology. As the first step, we selected a venue to collect data for our study. Then, we conducted a pilot study to assess the viability of our study and establish a coding scheme. After the coding scheme was developed, we extended the data collection process to include more studies with participants. To collect our sample data, we filtered through abstracts of research papers from our selected venues to identify studies with participants. Once we gathered our data, the first author read each paper and classified each study according to our coding scheme. We leveraged a mixed methods approach that uses qualitative and quantitative methods to reach our results.
\begin{figure*}
\centering
\includegraphics[width=\textwidth]{figures/methodOverviewUpdated}
\caption{Overview of methodology as explained in Section \ref{sec:methodology}.}
\label{fig:method-overview}
\end{figure*}
\subsection{Research Questions}
To assess to what extent SE research papers with participants display diversity awareness, we first investigate if SE research papers report on diversity in their studies:
\\
\noindent \textbf{RQ1. To what extent do SE participant studies report the diversity of participants?}\\
\noindent After examining whether diversity is even reported, we set out to identify the kinds of diversity reported:\\
\noindent \textbf{RQ2. What diversity categories are reported in SE participant studies?}\\
Finally, after assessing what kind of diversity categories can be identified in SE research with participants, we investigate if the reported diversity is actually analysed/leveraged throughout the research paper:\\
\noindent \textbf{RQ3. What is the function of diversity in SE participant studies?} \\
\subsection{Selecting the Target Venue}
To take a more systematic approach, we collected the sample for this research from the International Conference of Software Engineering proceedings (ICSE). We choose ICSE as our venue because it is considered the flagship software engineering conference \cite{Damian&Zeller:2022} and the proceedings of ICSE are easily accessible online \cite{ICSE2020Proceedings}. We collected our sample from ICSE 2019, 2020 and 2021 technical tracks to enrich the sample of studies and ensure our findings cover multiple editions of the conference.
In total, the technical track of ICSE 2019, 2020, and 2021 contained 379 papers.
Sampling three venues were also in line with software engineering meta-studies, such as the study of Storey \textit{et al.}~\cite{Storey&Ernst:2020}, which used two venues, ICSE 2017 technical track and ESME 2017 paper, as their sample.
For completeness, we list below all of the inclusion criteria used for the data set of our study:
\begin{itemize}
\item Technical research track papers from ICSE 2019, 2020 and 2021.
\item Studies that contained participants such as surveys, control groups, action research, grounded theory, focus groups, interviews, field studies, lab studies, validation studies, task-based studies and judgement studies.
\item Studies with one or more participants
\end{itemize}
\subsection{Selecting Papers with Participant Studies}
\label{sec:selectingParticipantStudies}
Software engineering research employs a variety of research methods. As we are interested in studies with participants, we need to identify papers that have employed a particular set of methods, e.g., surveys, interviews, field studies, etc. To this aim, we read the abstracts of all 376 papers in the technical research track of ICSE, from 2019 to 2021.
We choose to read the abstracts of each paper in order to find research papers containing participant studies. Abstracts often convey the main methodology of a research therefore if a participant study was conducted as part of a research, it would likely be mentioned in the abstract. This is particularly the case with participant studies, as the inclusion of interviews and surveys with practitioners is well-valued by the SE community. Analyzing paper abstracts is commonly used in related literature when conducting similar content analyses~\cite{Shaw:2003,Perera&Hussain:2020}. Given our initial sample contains 379 papers, reading the abstracts was a suitable and efficient method for filtering papers that are candidates for containing participant studies.
To select papers with participant studies, we identified words and phrases that indicated that the research involves research participants. Some examples of these words and phrases are given in Table.~\ref{tab:participant-studies-keywords}.
\begin{table}[]
\centering
\caption{The table shows some of the words and phrases we recognized in abstracts of papers containing participant studies.}
\label{tab:participant-studies-keywords}
\input{tables/participant-studies-keywords}
\end{table}
Sometimes, the abstract of a paper does not explicitly describe the entire methodology adopted by the studies. In the case of only implicit indication of research involving participants, we read the entire paper for verification. For example, in the following abstract, the indication of a study with research participants is not clear.
\begin{quotebox}
\textbf{``Using screencasts of developers' real work}, we demonstrate the usefulness of our technique in a practical application for action-aware extraction of key-code frames in developers' work.'' \cite{Zhao&Xing:2019}
\end{quotebox}
The highlighted text here indicates a possibility that the ``developers'' mentioned in the abstract are participants in this study. In such a case, we go beyond reading the abstract to verifying the contents of the paper to see if participants were actually involved in the study.
We did not consider studies where participants were not directly or actively involved in the research. For example, Wu \textit{et al.} \cite{Wu&Deng:2021} analyzed app user reviews in their research. Since the authors use already available reviews from the users and do not contact the app users directly to gather data from them, we do not consider this paper to have a participant study.
\begin{table}[H]
\centering
\caption{The table shows the number of abstracts we read to select our sample, the number of research papers in our sample and the number of studies we found per venue.}
\label{tab:summary-of-sample}
\input{tables/summary-of-sample}
\end{table}
Of a total of 376 papers from the three venues of ICSE 2019, 2020, and 2021 technical tracks, we identified 79 publications that met our selection criteria to be included in our study. A summary of this data is presented in Table~\ref{tab:summary-of-sample}. Out of our sample of 79 research papers with participants, some of the papers included more than one participant study. For example, Miller \textit{et al.} \cite{Miller&Rodeghero:2021} conducted two different surveys with different sets of participants.
Since the two surveys that were described in the paper had different sets of participants, we coded the two different surveys as two different participant studies, noting that they are part of the same research paper. Similarly, several other research papers consisted of two or more studies with participants. Therefore, even though we selected 79 research papers with participant studies as our sample data, we found a total of 105 studies with participants in those research papers. Thus, our main sample consists of a total of 105 participant studies.
\subsection{Developing the Diversity Coding Scheme}
\label{sec:developingCodingScheme}
To the best of our knowledge, there are no other studies that attempt to characterize diversity awareness of SE research. Hence, we cannot reuse an established framework to conduct our study. We start the study by developing an appropriate framework for encoding different diversity categories by conducting a pilot study using the ICSE 2020 technical research track.
The ICSE 2020 technical research track contains a total of 129 papers. We read the abstracts of the 129 papers and identified 25 research papers with participant studies.
Once we selected our sample of 25 research papers, the first author read each research paper and identified references creating diversity categories and developing a coding scheme. By following an inductive approach, codes were developed while reading the papers. We used an open coding method where the codes emerged directly from the content of the paper \cite{Saldana:2009}. In the next step, we used the axial coding method where codes were grouped into diversity categories \cite{Saldana:2009}. Tamtik \cite{Tamtik:2022} and Storey \textit{et al.} \cite{Storey&Ernst:2020} used a similar approach in their study. The coding scheme grew as new diversity categories were identified in the sample data.
Throughout the pilot study, the research team met weekly between February 2021 and June 2021 to discuss each paper in our sample. We discussed, defined and refined the diversity categories that emerged from each paper.
The initial coding scheme developed through our pilot study which was agreed upon by the research team. While the pilot study was fundamental for developing the initial coding scheme, the coding scheme was further refined and new codes were added to the coding scheme as we went through the rest of our sample. All new codes added, as the coding book evolved, were discussed and agreed upon by the research team.
\subsection{Qualitative Analysis of Papers to Create Diversity Categories}
\label{sec:creatingDivCategories}
The developed coding scheme mentioned in section~\ref{sec:developingCodingScheme} was used to code the sample of 105 studies. The first author read each paper and identified different words and phrases that describe and distinguish participants in the studies. For example, in the following quote from Krueger \textit{et al.}:
\begin{quotebox}
``When participants elected to participate in the study, we collected basic demographic data (\textbf{sex}, \textbf{gender}, \textbf{age}, \textbf{cumulative GPA}, and \textbf{years of experience}) and \textbf{socioeconomic status} (SES) data.'' \cite{Krueger&Huang:2020}
\end{quotebox}
the words highlighted in bold are descriptive words since they are used to describe the participants. We marked such descriptive words in each research paper.
We grouped codes to create the diversity categories. For example the words ``sex'', ''gender'', ``gender fluid'', ``male'', ``female'', ``women'', ``sexual orientation'' were all grouped into the diversity category labeled ``gender/sex''. The research team met each week to discuss any potential ambiguities that arouse in forming the diversity categories and find agreement.
These diversity categories were then added to our coding scheme. Whenever we discovered a descriptive word or phrase that could not previously be included in our coding scheme, we either added the new code to an existing category or added a new category to the coding scheme. Thus, our coding scheme included the diversity categories identified in our sample papers. It is important to note here that the diversity categories in our coding scheme are not all the diversity categories that exist. The ones in our scheme are just the ones we created based on the words and phrases we identified in our sample data. We wanted to inductively create diversity categories to add to our coding scheme rather than using preexisting frameworks for diversity because we did not want to limit our view of diversity to the already existing diversity category definitions. This approach is commonly applied when using the open coding method \cite{Saldana:2009}.
\subsection{Identify the Function of Diversity Categories in Software Engineering Research}
\label{sec:divCategoriesInSections}
The goal of RQ3 is to understand how diversity is embedded in SE research.
We want to identify the function that participant diversity serves in the selected SE studies.
We consider four different functions of diversity in participant studies: 1) describing participants, 2) analyzing the impact of diversity in the study results, 3) reflecting upon participant diversity in the study conclusion, and 4) assessing the limitation of diversity in the participant study.
Given our sample contains 105 studies covering a multitude of different study types and goals, we evaluate the function of diversity in a study using a two-step approach.
First, we identify the section in which diversity is discussed, as the study section also serves a clear function in the study report.
For example, we illustrate how a hypothetical ICSE paper would have its sections classified by our method in Fig.~\ref{fig:ICSE-format}.
Second, we carefully analyze the context in which diversity is inserted to confirm its function in the study.
In detail, we proceed as follows to identify the four functions of diversity:
\begin{itemize}
\item \textbf{Describing} the diversity of participants.
This information is usually reported in the methodology sections of a research paper to describe the diverse sample of participants, e.g., in the ``experimental design'', ``study design'', and ``research design'' sections.
In some studies, the diversity information of participants is not reported in the methodology section but the user evaluation part of their study. Hence the importance to consider the context in which diversity is described to identify its function in SE participant studies.
\item \textbf{Analysing} the diversity of participants in study results.
We evaluated the results reported in a research paper to identify if authors analyze the impact of participant diversity when discussing the results of their research. This information is usually reported in the ``results'' section of the paper but can also be found in ``experimental results'', ``findings'', etc.
\item \textbf{Reflecting} upon the diversity of participants when concluding the study.
We identified if authors reflect on participant diversity and use this data to draw outcomes or conclusions in a paper.
This information is typically discussed in the sections named ``discussion'', ``conclusions'' and ``future work''.
\item Assessing the \textbf{limitations} of participant diversity in the study.
To identify if authors assess the limitations of participant diversity or the lack of collection of diversity data about participants, we evaluate the ``threats to validity'' or ``limitations'' section of research papers. It is important to identify if authors assess the limitations of their participant diversity, as it reflects how aware authors are about diversity or lack thereof.
\end{itemize}
\begin{figure}[]
\frame{\includegraphics[scale=0.4]{figures/ICSE-format.pdf}}
\caption{This image shows a generic layout format of a typical ICSE paper. The sections circled in red are the sections that typically correspond to the four functions of diversity, namely, describing, analyzing, reflecting and assessing limitations in a study.}
\label{fig:ICSE-format}
\end{figure}
\section{Results}
\label{sec:results}
The goal of our overall research is to assess the diversity awareness in software engineering research with participants. To do so, we first want to know if researchers report any kind of diversity data about participants (RQ1); what kind of diversity they report (RQ2); and the function of diversity in the software research (RQ3). The aforementioned research questions help us assess the awareness of diversity in software engineering participant research.
\input{RQ1}
\input{RQ2}
\input{RQ3}
\input{discussion}
\section{Threats to Validity}
\label{sec:threats}
In this section, we recognize the threats to validity of our research and discuss methods we applied in our research design to mitigate these threats.
\noindent\textbf{Internal Validity:}
A threat to the internal validity of our research is the selection of our sample of participant studies. This process is described in Section~\ref{sec:selectingParticipantStudies}. The first author read all the abstracts from our selected venue to identify participant studies for our sample.
There is a chance that participant studies were not included because the authors have not described their methodology in their abstract. To mitigate this threat of subjectivity, we took two steps. Firstly, whenever ambiguity arose, the first author read the entire paper to confirm (or not) its inclusion in the study. Secondly, if after verifying the contents of the paper there was still ambiguity, the research team discussed if the study would be considered part of the sample or removed. Given our sample consists of 105 participant studies, it is unlikely that any missing study would have drastically changed our overall results.
The classification of diversity categories in the coding scheme for RQ1 and RQ2 could also pose a threat to the internal validity of our findings. This is because this classification was done manually and can be regarded as subjective. However, to mitigate this threat, we took the following steps, described in Section~\ref{sec:developingCodingScheme}. First, we conducted a pilot study and met regularly for the duration of this study in order to discuss the classification of each diversity category in our pilot sample. Second, if ever there was ambiguity during coding, we discussed it among the authors to reach a consensus.
\noindent\textbf{External Validity:}
A threat to external validity is that our selection sample was limited to the ICSE technical tracks of the 2019, 2020, and 2021 editions.
ICSE is highly regarded as a top-tier conference in the SE research community and even though choosing three years of sample papers is within industry standards (e.g.,~\cite{Storey&Ernst:2020}), it is possible that our results might differ if we had selected a different venue for our sample. This might limit the generalizability of our results. However, we believe that our contributions of creating a coding scheme to identify diversity categories, the model for diversity awareness, and guidelines for diversity awareness would still remain valid.
\noindent\textbf{Construct Validity:}
Our coding scheme poses a threat to our construct validity. We defined four different functions of diversity, namely describing, analyzing, reflecting and assessing limitations of participant diversity. These functions were based on our interpretation of what we thought were relevant functions for this research. However, we acknowledge that there could be other characterizations of different functions of diversity in participant studies. Our classification of diversity categories also poses a similar threat where the diversity categories we created could be labelled differently. However, due to the lack of classification schemes in SE that suited our research, we felt the need to create our own classification scheme for diversity categories. Our approach to identifying diversity categories in SE participant studies is one of the contributions of our research that we hope to see in future works.
\input{conclusion}
\section{Discussion}
\label{sec:discussion}
\subsection{Diversity Awareness Model for SE Participant Studies}
\begin{table*}
\centering
\caption{Diversity Awareness Model for SE participant studies.}
\label{tab:diversity-awareness-model}
\input{tables/diversity-awareness-model-v2.tex}
\end{table*}
While studying the extent to which SE participant studies report on the diversity of participants, we identified five types of studies which were distinguishable in terms of considering diversity in the overall research and publication approach. Based on these five types of studies, we propose a model for diversity awareness as shown in Table~\ref{tab:diversity-awareness-model}.
The model contains five levels of diversity awareness and it can be helpful in assessing the level of diversity awareness of authors of a participant study and supporting SE researchers in reflecting on diversity.
In Table~\ref{tab:diversity-awareness-model}, we describe short fictional examples for each level.
We refrain from providing real examples for all levels, as our goal is not to single out past studies but to discuss paths for improving future ones.
Good examples, however, can help illustrate and guide SE researchers.
As a good example of a study with high-level diversity awareness, we report on the work of Dias \textit{et al.}~\cite{Dias&Meirelles:2021}.
The authors of this study demonstrate a high level of diversity awareness (level 4), as they consciously ``choose'' to include diversity and report in detail about their choice and the reasons for it.
\begin{quotebox}
``To foster diversity, when inviting the participants, we prioritize women and non-US based maintainers. We took this decision to avoid having too many “Silicon Valley” participants, as they are over-represented amongst OSS maintainers.'' \cite{Dias&Meirelles:2021}
\end{quotebox}
A few studies in our sample even reported the diversity of the researchers involved. The following quote from Gerosa \textit{et al.}~\cite{Gerosa&Wiese:2021} shows how the authors report the diversity of the researchers. This indicates diversity awareness that goes beyond considering the diversity of research participants.
\begin{quotebox}
``we formed an international and diverse team of researchers, who are originally from South America (4), Europe (3), and Asia (1) and were working, at the time of this study, in North America (5), Europe (1), South America (1), and Australia (1). Seven researchers work in academia with extensive experience with OSS, and one researcher is a practitioner working in an OSS company.''~\cite{Gerosa&Wiese:2021}
\end{quotebox}
\subsection{Diversity Awareness Guidelines}
It is difficult to determine the level of diversity awareness quantitatively. For example, reporting several diversity categories or stating that the gender diversity of participants was not considered because the research is not about gender does not necessarily mean a high level of diversity awareness.
\textbf{It is the combination of reporting, analyzing, addressing and reflecting on the considerations of diversity that can demonstrate the diversity awareness of authors.} Here, we propose questions that can guide researchers when self-assessing their diversity awareness that they apply on their research.
\begin{itemize}
\item Do you consider diversity in your research?
\item Which diversity categories did you consider when selecting research participants?
\item How do you make sure that your research participants are diverse (e.g., a demographic representation of society)?
\item What recruitment efforts did you undertake to reach a demographic representation?
\item Which groups/diversity categories are over represented?
\item Which professional diversity categories do you consider (e.g., work experience, ...)?
\item Which social diversity categories do you consider (e.g., gender, language, nationality, age, ...)?
\item Which diversity categories do you NOT consider? Why do you not consider them?
\item Which diversity categories might be relevant in the context of your research?
\item Are you reporting your diversity considerations in your research publication?
\item Are the diversity categories considered relevant to your research?
\end{itemize}
\subsection{Impact on Society}
We believe this research to have a significant impact on the design of future SE research, in particular research that includes research participants. Our work both highlights the importance of considering and reporting the diversity of participants as well as provides guiding questions to successfully integrate considerations of diversity and inclusion into the research design. Furthermore, our proposed model of diversity awareness helps researchers to self-assess and review their research to identify lacks which, otherwise, they would have missed. For research that includes participant studies, defining, selecting and recruiting research participants is an evident opportunity to consider and implement diversity and inclusion into research. The better diversity is considered and implemented in the research design the broader will be the acceptance and the benefit of the research and development for society.
\section{Conclusion and Future Work}
\label{sec:conclusion}
In this paper, we examine the diversity considerations in SE participant studies.
We apply content analysis to investigate participant studies from three ICSE technical tracks, from 2019 to 2021.
Our investigation focused on understanding 1) the extent to which participants are described, 2) what diversity categories are more prominent in SE research, and 3) the function participant diversity serves in SE studies.
On one hand, our findings shed light on some positive remarks for the SE research community.
Reporting participant diversity is a well-established practice, with studies reporting on multiple characteristics of participants.
On the other hand, our results also point to some gaps/challenges that may need further addressing.
Studies emphasize participants' professional backgrounds over their social backgrounds, which may prevent important reflections that are needed in a research community.
Furthermore, participant diversity is often only reported initially in the studies, to describe participants, but is seldom analyzed or reflected upon, when researchers discuss their study results, outcomes and limitations.
Our study sheds light on the strengths and gaps of diversity awareness in SE participant studies and provides a model that future studies can use to improve diversity awareness. This research is as a starting point for discussions about diversity awareness in SE participant research.
There is a large potential for future work to assess the impact of the lack of diversity consideration in SE participant research, and it's effect on representation and inclusion. Our model can also be applied to other engineering fields to uncover differences of reporting participant diversity across various engineering disciplines.
\subsection{RQ3.What is the function of diversity in SE participant studies?}
In the previous RQ we found that SE studies commonly report characteristics of participants in their studies, with emphasis on professional background.
In this RQ, we aim to assess the function of participant diversity in SE research.
To that end, we follow the methodology described in Section~\ref{sec:divCategoriesInSections}, where we classify the function of participant diversity in four categories: 1) describing participants, 2) analyzing the impact of diversity in the study results, 3) reflecting upon participant diversity in the study conclusion, and 4) assessing the limitation of diversity in the participant study.
\begin{figure}[]
\centering
\includegraphics[scale=0.67]{figures/categories-per-section.pdf}
\caption{How participant diversity is used in the sample of 105 SE studies. We report the frequency of studies that use participant diversity for \textbf{describing} participants, \textbf{analyzing} results, \textbf{reflecting} the study outcomes, and assessing the study \textbf{limitations}. }
\label{fig:diversity-categories-per-section}
\end{figure}
\begin{table}[]
\centering
\caption{
The function of participant diversity in the 105 studies, broken down by diversity category.}
\label{tab:percent-div-cats-per-section}
\input{tables/percent-diversity-per-section}
\end{table}
Figure~\ref{fig:diversity-categories-per-section} shows the frequency of every function of participant diversity in the sample of 105 studies.
We determine that 98\% of studies describe participant diversity by considering 103 out of 105 studies that report diversity.
Naturally, this result is expected as the primary function of participant descriptors is to describe the participants.
Interestingly, however, participant diversity is less frequently referred to when researchers analyze their study results (50\%), reflect upon the results (35\%), and assess the limitations of their study (44\%).
This leads us to conclude that \textbf{in most SE participant studies, diversity is reported as a means to describe participants but is less frequently used for further analysis or reflection in the research}.
Table~\ref{tab:percent-div-cats-per-section} displays a breakdown of the participant diversity function, across diversity categories.
To make it easier to identify patterns across the diversity categories, we highlight the highest frequency (dark blue) and the second-highest frequency (light blue) in each category.
It can be seen that \cat{Experience} is the diversity category that most consistently presents the four functions of describing, analyzing, reflecting and assessing limitations in SE studies.
While also a dominant professional category (as discussed in RQ2), \cat{Main Occupation} is mostly used to describe participants (76\%) and is only used for reflection or assessing limitations in 10\% of the studies in our sample. This leads us to conclude that \textbf{SE researchers use the \cat{Experience} of participants consistently in the four functions of diversity we analyzed}.
At least a quarter of the studies in our sample discusses the experience of participants when analyzing the results, outcomes, and limitations of their study.
In contrast, \textbf{all other diversity categories are used mostly to describe participants, with no further reflection on its possible impact in the research.}
Some interesting patterns emerge when we look at the second most frequent function (light blue) per diversity category in Table~\ref{tab:percent-div-cats-per-section}.
We identified \cat{Gender/sex}, \cat{Location}, and \cat{Language} as social diversity categories which researchers use more frequently to assess the study limitations.
Often, these categories are used to discuss the limitations of the study's generalizability (external validity).
The following quote from Alsuhaibani \textit{et al.}~\cite{Alsuhaibani&Newman:2021} is an example of the use of participation diversity to discuss the study's limitations:
\begin{quotebox}
``With regards to external validity, we did not directly collect geolocation data. However, we did collect IP addresses, which gave us country information. From this we determined that participants came from 72 different countries, mainly from Europe and North America. Thus, we feel that the results are generalizable to a broad population.''~ \cite{Alsuhaibani&Newman:2021}
\end{quotebox}
\cat{Race} and \cat{Socioeconomic status} which are two eminent social diversity categories, were only reported in 4\% and 1\% of all studies in our sample, respectively. These two social diversity categories played a role when describing the participants but were not used in further analysis or reflection in any SE study we sampled. This finding leads us to the conclusion that although some SE researchers collect and report data about participants' race and socioeconomic status, they do not use this data for further analysis or reflection in their research. This is not necessarily problematic. For example, considering to which extent the sample of research participants represents the demography is not necessarily linked to investigating social differences. However, especially in the case that participants are far from representing the demography, a further reflection on the impact of this fact on the results of the research might be important.
To conclude, diversity plays four major functions: describing, analyzing, reflecting upon and assessing the limitations of participants in SE studies. Our results indicate that in most studies, SE researchers report diversity when describing participants. However, participant diversity is used less frequently by researchers when discussing the study results, reflecting on the study outcomes, or when assessing the study's limitations.
\subsection{RQ2. What diversity categories are reported in SE participant studies?}
\begin{table*}[ht]
\centering
\caption{Diversity categories identified in participant study description of 105 studies from ICSE 2019, 2020, and 2021.}
\label{tab:diversity-categories-with-definition}
\input{tables/diversity-categories-with-definition}
\end{table*}
After determining that most SE participant studies do report aspects of participant diversity, we want to identify what kinds of diversity are reported. Such analysis will help us better understand what characteristics of participant diversity are emphasized by SE researchers and the aspects that might be overlooked and require more attention.
To answer RQ2, we created a coding scheme where we identified descriptors about participants in SE studies. We then grouped similar descriptors to form diversity categories as mentioned in section \ref{sec:creatingDivCategories}.
\textbf{We found 12 different diversity categories reported in SE participant studies.}
Table~\ref{tab:diversity-categories-with-definition} shows the diversity categories we identify in the content analysis, as well as their definition and frequency (Freq) in our sample. The frequencies do not sum up to 100\% as, on average, studies report multiple categories of diversities (as shown in Table~\ref{tab:study-categories}).
Overall, two types of diversity remained dominant in the participant diversity reported by our sample of studies, the \cat{Experience} and the \cat{Main Occupation} of participants.
This shows that SE studies are often concerned with showcasing a diverse sample of participants regarding professional backgrounds.
The \cat{Experience} of participants was reported in 89\% of the studies.
We group in this category different kinds of professional experiences, from years of industrial experience to experience in a specific technology, such as programming languages.
As Zhang and Yang report in their study~\cite{Zhang&Yang:2019}:
\begin{quotebox}
``Eleven participants had two to \terms{five years of Java experience}, while the other five were \terms{novice programmers with one-year Java experience}, showing a good mix of different levels of Java programming experience.''
\end{quotebox}
\cat{Main Occupation} was reported in 77\% of the studies, showing a well-established practice to collect and report participation occupations.
This category includes professional occupations (e.g., developer, quality analyst) and education-related occupations (e.g., Ph.D. student, MSc. student).
For example, as Ju \textit{et al.}~\cite{Ju&Sajnani:2021} describes:
\begin{quotebox}
``This process yielded 397 \terms{developers} and 1167 \terms{developer managers} for interviews and 1629 \terms{developers} and 754 \terms{developer managers} for surveys''
\end{quotebox}
Three other categories were reported to describe participants in a comparable frequency: \cat{Gender/Sex} (39\%), \cat{Education} (37\%), and \cat{Location} (33\%).
Reported in at least a third of the studies, these categories help qualify the diversity of participants in terms of gender, scholarly level and geographic location.
For example, Dias \textit{et al.} report the location diversity of their participants as follows:
\begin{quotebox}
``Our interviewees are \terms{located in different countries}, such as Brazil, Canada, Czech Republic, USA, Germany, and Portugal.''~\cite{Dias&Meirelles:2021}
\end{quotebox}
\cat{Age} (18\%) and \cat{Language} (8\%) of participants are less commonly reported in our sample of SE studies.
The age of participants is a characteristic that may be indirectly captured by participants' years of experience.
However, it is remarkable that the language spoken by participants is seldom reported, given the international audience of SE papers and the importance of effective communication in any type of study with participants.
For example, Xia \textit{et al.} reported on the efforts of translating the survey content to ensure effective communication with participants:
\begin{quotebox}
``To support respondents from China, we translated our survey to \terms{Chinese} before publishing the survey.''~\cite{Xia&Wan:2019}
\end{quotebox}
Finally, some categories were reported only in a handful of SE participant studies.
These categories include \cat{Nationality} (5\%), \cat{Race} (4\%), \cat{Physiological} (2\%), \cat{Psychological} (2\%) and \cat{Socioeconomic status} (1\%).
These categories describe specific racial and social aspects of participants and seem to only be collected and reported in specific cases.
For example, Krueger~\textit{et al.}~\cite{Krueger&Huang:2020} ask participants to complete psychological measurement surveys, such as the Positive and Negative Affect Scale (PANAS, emotional health).
Peitek~\textit{et al.}\cite{Peitek&Apel:2021} reported that all participants in their study had a normal or corrected-to-normal vision and were right-handed.
Once we analyze the frequency of categories based on two major themes of categories, professional and social category (Themes column of Table~\ref{tab:diversity-categories-with-definition}), it becomes clear that \textbf{SE studies emphasize professional diversity over social diversity when selecting and reporting participants.}
All professional categories are reported in at least a third of all studies, while only social categories of \cat{Gender} and \cat{Location} reach similar levels of frequency.
Social diversity categories, also known as personal or identity categories, are considered particularly important for EDI efforts towards social equity \cite{GovCanada:2021}, however, our results show that SE studies do not often report this type of diversity.
\subsection{RQ1. To what extent do SE participant studies report the diversity of participants?}
The first step to understanding the awareness of diversity in SE research is to examine how often SE studies report the diversity of their participants.
This analysis will help us understand how well-established are the practices of collecting, analyzing and reporting demographic data about participants.
Hence, to answer RQ1, we performed the qualitative content analysis described in Section~\ref{sec:creatingDivCategories}.
We also break our results by type of participant study (e.g., survey, task-based) and for this purpose we rely on how researchers describe their own study.
\textbf{We found that 98.01\% of the examined participant studies report the diversity of participants.} Out of our sample of 105 studies that we analyzed in our research, 103 studies used descriptive words and phrases when characterizing participants. Only 2 studies out of 105 did not report any participant diversity.
To make better sense of the results, we break our analysis per type of participant study in Table~\ref{tab:study-categories}.
First, we note that SE papers use participant studies on 11 different study modalities, the most common being: surveys (51 studies), task-based (33 studies), interviews (22 studies), and validation (19 studies).
Second, most types of studies tend to report an average of 3 different categories when describing the participants, with Lab studies reporting the highest average (4.6 categories) across the samples we investigated.
Some studies, such as tasked-based and lab studies, reported a wide array of diversity categories when describing participants, including 8 different types of descriptors.
Our results indicate that the practice of reporting participant diversity is well-established in participant studies in Software Engineering.
Researchers tend to select between 2-4 categories of descriptors when describing participants.
Only in rare cases (2 out of 105 studies) do researchers omit to describe the participants included in their study.
\begin{table}[]
\centering
\caption{The frequency in which studies report diversity category (Reported column) per type of participant study.}
\label{tab:study-categories}
\input{tables/categories-participant-studies}
\end{table}
\section{Conclusion and Future Work}
\label{sec:conclusion}
In this paper, we examine the diversity considerations in SE participant studies.
We apply content analysis to investigate participant studies from three ICSE technical tracks, from 2019 to 2021.
Our investigation focused on understanding 1) the extent to which participants are described, 2) what diversity categories are more prominent in SE research, and 3) the function participant diversity serves in SE studies.
On one hand, our findings shed light on some positive remarks for the SE research community.
Reporting participant diversity is a well-established practice, with studies reporting on multiple characteristics of participants.
On the other hand, our results also point to some gaps/challenges that may need further addressing.
Studies emphasize participants' professional backgrounds over their social backgrounds, which may prevent important reflections that are needed in a research community.
Furthermore, participant diversity is often only reported initially in the studies, to describe participants, but is seldom analyzed or reflected upon, when researchers discuss their study results, outcomes and limitations.
Our study sheds light on the strengths and gaps of diversity awareness in SE participant studies and provides a model that future studies can use to improve diversity awareness. This research is as a starting point for discussions about diversity awareness in SE participant research.
There is a large potential for future work to assess the impact of the lack of diversity consideration in SE participant research, and it's effect on representation and inclusion. Our model can also be applied to other engineering fields to uncover differences of reporting participant diversity across various engineering disciplines.
\section*{General Abstract}
Incorporating diversity considerations in research, development, and innovation has become an increasingly important topic. It is a well-known fact that diverse teams produce better outcomes, whereas the lack of diversity might result in biased and discriminatory technologies. Therefore, the inclusion of diverse stakeholders is considered paramount for the creation of an ethical and social-responsible future. With this study, we aim to contribute to the conversation on how EDI (equity, diversity, inclusion) can be implemented in Software Engineering (SE) research. In our study, we focus on SE research that includes research participants since this is an evident opportunity to consider diversity, and we investigate to which extent and with what purpose SE researchers consider and report diversity in their research papers. Our results demonstrate that only a few studies do not consider diversity at all, however, the examined studies differ greatly in the range of the consideration and reporting of diversity. From these outcomes, we draw the conclusion of differences in the diversity awareness among SE researchers. Finally, we propose a model of diversity awareness for participant studies as a tool to support SE researchers in reflecting on diversity and incorporating it systematically in their research.
\section{Introduction}
\label{sec:introduction}
In recent years, the consideration of the diversity of people who are involved in research and innovation has become an increasingly important topic. Diversity of developers and researchers is regarded – and has already been proven - beneficial for numerous reasons: diverse teams produce better outcomes \cite{Menezes&Prikladnicki:2018, Pieterse&Kourie:2006,Patrick&Kumar:2012}, research and development that systematically includes diversity are of benefit for a broader population whereas the lack of diversity might result in biased and discriminatory technologies \cite{Schiebinger&Klinge:2011,Buolamwini&Gebru:2018,Tannerbaum&Ellis:2019}, the inclusion of diverse stakeholders is paramount for the creation of an ethical and social-responsible future \cite{Sarewitz:2005} \cite{VanOudheusden&Shelley-Egan:2021}, and, last but not least, excluding large parts of society from research and the development of future technologies violates human rights \cite{Oberleitner:2021,Al-Nashif:2021,PARL:2010}. One size does not fit all, and this applies to research and development, too.
To mitigate the negative effects of non-diverse research and to stimulate researchers to incorporate diversity in their research, national and international research funding programs are developing policies and guidelines for equity, diversity, and inclusion (EDI) \cite{NSERC:2022}, for responsible research and innovation (RRI) \cite{EU:2013} to bind the research funding to increasingly important conditions: the sufficient consideration of EDI.
In our study, we focus on research that includes research participants as we regard this as an evident opportunity for SE researchers to include diversity considerations in their research. Therefore, our research questions are as follows:
\begin{itemize}
\item RQ1. To what extent do SE participant studies report the diversity of participants?
\item RQ2. What diversity categories are reported in SE participant studies?
\item RQ3. What is the function of diversity in SE participant studies?
\end{itemize}
To capture all variants of diversity categories, we choose an inductive approach, which means: we do not predefine the categories to examine the content according to these categories. Instead, we reconstruct the diversity categories through open coding of the content. Therefore, for the purpose of this study, we choose a broader definition of diversity categories: A diversity category is a category based on which individuals (in our case, research participants) are distinguished and grouped. This open approach allows for identifying categories that might be relevant for research but are not included in the commonly discussed diversity categories such as gender, race, and ethnicity, amongst others \cite{Nkomo&Stewart:2006}.
Our results demonstrate that only a few studies do not consider diversity at all, however, the examined studies differ greatly in the range of the consideration and reporting of diversity. From these outcomes, we draw the conclusion of differences in diversity awareness among SE researchers. We propose a model of diversity awareness for participant studies as a tool to support SE researchers in reflecting on diversity and incorporating it systematically in their research.
Finally, we publish our classification coding scheme and the dataset used to conduct this study to encourage further studies on diversity awareness and facilitate replicability\footnote{\url{https://doi.org/10.5281/zenodo.7587076}}.
This paper is organized as follows.
Section~\ref{sec:relatedWork} describes the related literature.
We detail our study methodology in Section~\ref{sec:methodology} and present the results of our three research questions in Section~\ref{sec:results}.
Upon reflecting on the results, we discuss a model of diversity awareness and guidelines for the community in Section~\ref{sec:discussion}.
Finally, we discuss the limitations of our study in Section~\ref{sec:threats} and our final remarks in Section~\ref{sec:conclusion}.
\section{Related Work}
\label{sec:relatedWork}
Time and time again, diversity, and the importance of diversity, has been a topic of discussion for many researchers in various fields. This has led to different definitions of diversity emerging in different contexts \cite{Tamtik:2022, Zanoni&Janssens:2004, Nagappan&Zimmermann:2013}. In recent years, diversity, especially gender diversity, has received increasing attention in the field of SE \cite{Catolino&Palomba:2019, Burnett&Peters:2016, Padala&Mendez:2022, Bosu&Sultana:2019}.
\noindent\textbf{Lack of Diversity in SE.}
The issue of lack of diversity, also referred to as the diversity crisis, in SE has been investigated by many SE research papers as can be seen in the IEEE special issue article by Albusays \textit{et al.} \cite{Albusays&Bjorn:2021}. The importance of diversity and inclusion in the field of SE has been clearly stated in the article with one of the questions being ``What are the relevant diversity parameters we should consider when exploring software development practices and technology'' \cite{Albusays&Bjorn:2021}. Tamtik~\cite{Tamtik:2022} notes that only understanding diversity in terms of gender, language, and socio-economic categories limits the understanding of diversity. Similarly, Rodriguez and Nadri~\textit{et al.} conclude that more perceived diversity aspects need to be considered in SE research \cite{Rodriguez&Nadri:2021}. One of the goals of our research is to address these challenges in the context of SE research, by (i) revealing how SE researchers are addressing diversity in their studies and reporting about it in their publications, and (ii) by providing SE researchers with guidelines they can use when conducting participant studies.
Other work by Menezes and Prikladnicki~\cite{Menezes&Prikladnicki:2018} investigated various publications about diversity in SE by performing a literature review. They also investigated the impact of diversity on processes in software development by conducting semi-structured interviews. They concluded that many challenges still need to be overcome to make SE work environments more diverse. Our work complements the work by Menzes and Prikladnicki since we investigate to what extent SE research considers diversity, the types of categories used and the function of diversity.
\noindent\textbf{Human Values in SE Research.}
Perera~\textit{et al.} examine to what extent human values are considered in SE research instead of diversity \cite{Perera&Hussain:2020}. They use a similar methodology to ours, where they analyze research papers from top-tier SE conferences and journals to assess their consideration of human values. The found that very few SE publications consider human values.
Storey~\textit{et al.} conducted a similar study where they investigate the consideration of human aspects in SE research by analyzing SE research papers \cite{Storey&Ernst:2020}. They found most SE studies focus on technical aspects of SE even though these studies claim to impact human stakeholders. Storey~\textit{et al.} concluded that there is a need for SE studies to consider more human aspects and recommended a framework they created that can be used to consider more human and social aspects in SE research.
Although the work by Perera~\textit{et al.} \cite{Perera&Hussain:2020} and Storey \textit{et al.}~\cite{Storey&Ernst:2020} used a similar approach to ours, these two studies investigate the considerations of human aspects/values in SE research whereas we study the consideration of diversity in SE research, more specifically participant studies, while providing guidelines for SE researchers conducting participant studies.
The work closest to ours is the work by Lenarduzzi \textit{et al.}~\cite{Lenarduzzi&Dieste:2021} who investigated participant studies in SE, where they examined the current participant selection guidelines and practices in empirical SE research. They analyze existing guidelines for participant selection in SE and present the participant selection strategies being currently used in their results. Our work complements their study since we study actual research papers involving participants and create guidelines
for SE researchers, whereas their study only examined guidelines (and not actual research papers).
\section{Methodology}
\label{sec:methodology}
The goal of our research is to examine to which extent diversity is considered in SE research and to provide guidelines to SE researchers that they can use when conducting participant studies. As a methodological approach, we apply qualitative and quantitative content analysis \cite{krippendorff&Bock:2009} \cite{krippendorff:2018} where we used open coding and axial coding methods \cite{Saldana:2009} to examine software engineering research papers which include participant studies. Due to the novel nature of this study, we start by performing a pilot study to assess the viability of this study and develop an appropriate framework.
Fig.~\ref{fig:method-overview} shows an overview of our methodology. As the first step, we selected a venue to collect data for our study. Then, we conducted a pilot study to assess the viability of our study and establish a coding scheme. After the coding scheme was developed, we extended the data collection process to include more studies with participants. To collect our sample data, we filtered through abstracts of research papers from our selected venues to identify studies with participants. Once we gathered our data, the first author read each paper and classified each study according to our coding scheme. We leveraged a mixed methods approach that uses qualitative and quantitative methods to reach our results.
\begin{figure*}
\centering
\includegraphics[width=\textwidth]{figures/methodOverviewUpdated}
\caption{Overview of methodology as explained in Section \ref{sec:methodology}.}
\label{fig:method-overview}
\end{figure*}
\subsection{Research Questions}
To assess to what extent SE research papers with participants display diversity awareness, we first investigate if SE research papers report on diversity in their studies:
\\
\noindent \textbf{RQ1. To what extent do SE participant studies report the diversity of participants?}\\
\noindent After examining whether diversity is even reported, we set out to identify the kinds of diversity reported:\\
\noindent \textbf{RQ2. What diversity categories are reported in SE participant studies?}\\
Finally, after assessing what kind of diversity categories can be identified in SE research with participants, we investigate if the reported diversity is actually analysed/leveraged throughout the research paper:\\
\noindent \textbf{RQ3. What is the function of diversity in SE participant studies?} \\
\subsection{Selecting the Target Venue}
To take a more systematic approach, we collected the sample for this research from the International Conference of Software Engineering proceedings (ICSE). We choose ICSE as our venue because it is considered the flagship software engineering conference \cite{Damian&Zeller:2022} and the proceedings of ICSE are easily accessible online \cite{ICSE2020Proceedings}. We collected our sample from ICSE 2019, 2020 and 2021 technical tracks to enrich the sample of studies and ensure our findings cover multiple editions of the conference.
In total, the technical track of ICSE 2019, 2020, and 2021 contained 379 papers.
Sampling three venues were also in line with software engineering meta-studies, such as the study of Storey \textit{et al.}~\cite{Storey&Ernst:2020}, which used two venues, ICSE 2017 technical track and ESME 2017 paper, as their sample.
For completeness, we list below all of the inclusion criteria used for the data set of our study:
\begin{itemize}
\item Technical research track papers from ICSE 2019, 2020 and 2021.
\item Studies that contained participants such as surveys, control groups, action research, grounded theory, focus groups, interviews, field studies, lab studies, validation studies, task-based studies and judgement studies.
\item Studies with one or more participants
\end{itemize}
\subsection{Selecting Papers with Participant Studies}
\label{sec:selectingParticipantStudies}
Software engineering research employs a variety of research methods. As we are interested in studies with participants, we need to identify papers that have employed a particular set of methods, e.g., surveys, interviews, field studies, etc. To this aim, we read the abstracts of all 376 papers in the technical research track of ICSE, from 2019 to 2021.
We choose to read the abstracts of each paper in order to find research papers containing participant studies. Abstracts often convey the main methodology of a research therefore if a participant study was conducted as part of a research, it would likely be mentioned in the abstract. This is particularly the case with participant studies, as the inclusion of interviews and surveys with practitioners is well-valued by the SE community. Analyzing paper abstracts is commonly used in related literature when conducting similar content analyses~\cite{Shaw:2003,Perera&Hussain:2020}. Given our initial sample contains 379 papers, reading the abstracts was a suitable and efficient method for filtering papers that are candidates for containing participant studies.
To select papers with participant studies, we identified words and phrases that indicated that the research involves research participants. Some examples of these words and phrases are given in Table.~\ref{tab:participant-studies-keywords}.
\begin{table}[]
\centering
\caption{The table shows some of the words and phrases we recognized in abstracts of papers containing participant studies.}
\label{tab:participant-studies-keywords}
\input{tables/participant-studies-keywords}
\end{table}
Sometimes, the abstract of a paper does not explicitly describe the entire methodology adopted by the studies. In the case of only implicit indication of research involving participants, we read the entire paper for verification. For example, in the following abstract, the indication of a study with research participants is not clear.
\begin{quotebox}
\textbf{``Using screencasts of developers' real work}, we demonstrate the usefulness of our technique in a practical application for action-aware extraction of key-code frames in developers' work.'' \cite{Zhao&Xing:2019}
\end{quotebox}
The highlighted text here indicates a possibility that the ``developers'' mentioned in the abstract are participants in this study. In such a case, we go beyond reading the abstract to verifying the contents of the paper to see if participants were actually involved in the study.
We did not consider studies where participants were not directly or actively involved in the research. For example, Wu \textit{et al.} \cite{Wu&Deng:2021} analyzed app user reviews in their research. Since the authors use already available reviews from the users and do not contact the app users directly to gather data from them, we do not consider this paper to have a participant study.
\begin{table}[H]
\centering
\caption{The table shows the number of abstracts we read to select our sample, the number of research papers in our sample and the number of studies we found per venue.}
\label{tab:summary-of-sample}
\input{tables/summary-of-sample}
\end{table}
Of a total of 376 papers from the three venues of ICSE 2019, 2020, and 2021 technical tracks, we identified 79 publications that met our selection criteria to be included in our study. A summary of this data is presented in Table~\ref{tab:summary-of-sample}. Out of our sample of 79 research papers with participants, some of the papers included more than one participant study. For example, Miller \textit{et al.} \cite{Miller&Rodeghero:2021} conducted two different surveys with different sets of participants.
Since the two surveys that were described in the paper had different sets of participants, we coded the two different surveys as two different participant studies, noting that they are part of the same research paper. Similarly, several other research papers consisted of two or more studies with participants. Therefore, even though we selected 79 research papers with participant studies as our sample data, we found a total of 105 studies with participants in those research papers. Thus, our main sample consists of a total of 105 participant studies.
\subsection{Developing the Diversity Coding Scheme}
\label{sec:developingCodingScheme}
To the best of our knowledge, there are no other studies that attempt to characterize diversity awareness of SE research. Hence, we cannot reuse an established framework to conduct our study. We start the study by developing an appropriate framework for encoding different diversity categories by conducting a pilot study using the ICSE 2020 technical research track.
The ICSE 2020 technical research track contains a total of 129 papers. We read the abstracts of the 129 papers and identified 25 research papers with participant studies.
Once we selected our sample of 25 research papers, the first author read each research paper and identified references creating diversity categories and developing a coding scheme. By following an inductive approach, codes were developed while reading the papers. We used an open coding method where the codes emerged directly from the content of the paper \cite{Saldana:2009}. In the next step, we used the axial coding method where codes were grouped into diversity categories \cite{Saldana:2009}. Tamtik \cite{Tamtik:2022} and Storey \textit{et al.} \cite{Storey&Ernst:2020} used a similar approach in their study. The coding scheme grew as new diversity categories were identified in the sample data.
Throughout the pilot study, the research team met weekly between February 2021 and June 2021 to discuss each paper in our sample. We discussed, defined and refined the diversity categories that emerged from each paper.
The initial coding scheme developed through our pilot study which was agreed upon by the research team. While the pilot study was fundamental for developing the initial coding scheme, the coding scheme was further refined and new codes were added to the coding scheme as we went through the rest of our sample. All new codes added, as the coding book evolved, were discussed and agreed upon by the research team.
\subsection{Qualitative Analysis of Papers to Create Diversity Categories}
\label{sec:creatingDivCategories}
The developed coding scheme mentioned in section~\ref{sec:developingCodingScheme} was used to code the sample of 105 studies. The first author read each paper and identified different words and phrases that describe and distinguish participants in the studies. For example, in the following quote from Krueger \textit{et al.}:
\begin{quotebox}
``When participants elected to participate in the study, we collected basic demographic data (\textbf{sex}, \textbf{gender}, \textbf{age}, \textbf{cumulative GPA}, and \textbf{years of experience}) and \textbf{socioeconomic status} (SES) data.'' \cite{Krueger&Huang:2020}
\end{quotebox}
the words highlighted in bold are descriptive words since they are used to describe the participants. We marked such descriptive words in each research paper.
We grouped codes to create the diversity categories. For example the words ``sex'', ''gender'', ``gender fluid'', ``male'', ``female'', ``women'', ``sexual orientation'' were all grouped into the diversity category labeled ``gender/sex''. The research team met each week to discuss any potential ambiguities that arouse in forming the diversity categories and find agreement.
These diversity categories were then added to our coding scheme. Whenever we discovered a descriptive word or phrase that could not previously be included in our coding scheme, we either added the new code to an existing category or added a new category to the coding scheme. Thus, our coding scheme included the diversity categories identified in our sample papers. It is important to note here that the diversity categories in our coding scheme are not all the diversity categories that exist. The ones in our scheme are just the ones we created based on the words and phrases we identified in our sample data. We wanted to inductively create diversity categories to add to our coding scheme rather than using preexisting frameworks for diversity because we did not want to limit our view of diversity to the already existing diversity category definitions. This approach is commonly applied when using the open coding method \cite{Saldana:2009}.
\subsection{Identify the Function of Diversity Categories in Software Engineering Research}
\label{sec:divCategoriesInSections}
The goal of RQ3 is to understand how diversity is embedded in SE research.
We want to identify the function that participant diversity serves in the selected SE studies.
We consider four different functions of diversity in participant studies: 1) describing participants, 2) analyzing the impact of diversity in the study results, 3) reflecting upon participant diversity in the study conclusion, and 4) assessing the limitation of diversity in the participant study.
Given our sample contains 105 studies covering a multitude of different study types and goals, we evaluate the function of diversity in a study using a two-step approach.
First, we identify the section in which diversity is discussed, as the study section also serves a clear function in the study report.
For example, we illustrate how a hypothetical ICSE paper would have its sections classified by our method in Fig.~\ref{fig:ICSE-format}.
Second, we carefully analyze the context in which diversity is inserted to confirm its function in the study.
In detail, we proceed as follows to identify the four functions of diversity:
\begin{itemize}
\item \textbf{Describing} the diversity of participants.
This information is usually reported in the methodology sections of a research paper to describe the diverse sample of participants, e.g., in the ``experimental design'', ``study design'', and ``research design'' sections.
In some studies, the diversity information of participants is not reported in the methodology section but the user evaluation part of their study. Hence the importance to consider the context in which diversity is described to identify its function in SE participant studies.
\item \textbf{Analysing} the diversity of participants in study results.
We evaluated the results reported in a research paper to identify if authors analyze the impact of participant diversity when discussing the results of their research. This information is usually reported in the ``results'' section of the paper but can also be found in ``experimental results'', ``findings'', etc.
\item \textbf{Reflecting} upon the diversity of participants when concluding the study.
We identified if authors reflect on participant diversity and use this data to draw outcomes or conclusions in a paper.
This information is typically discussed in the sections named ``discussion'', ``conclusions'' and ``future work''.
\item Assessing the \textbf{limitations} of participant diversity in the study.
To identify if authors assess the limitations of participant diversity or the lack of collection of diversity data about participants, we evaluate the ``threats to validity'' or ``limitations'' section of research papers. It is important to identify if authors assess the limitations of their participant diversity, as it reflects how aware authors are about diversity or lack thereof.
\end{itemize}
\begin{figure}[]
\frame{\includegraphics[scale=0.4]{figures/ICSE-format.pdf}}
\caption{This image shows a generic layout format of a typical ICSE paper. The sections circled in red are the sections that typically correspond to the four functions of diversity, namely, describing, analyzing, reflecting and assessing limitations in a study.}
\label{fig:ICSE-format}
\end{figure}
\section{Results}
\label{sec:results}
The goal of our overall research is to assess the diversity awareness in software engineering research with participants. To do so, we first want to know if researchers report any kind of diversity data about participants (RQ1); what kind of diversity they report (RQ2); and the function of diversity in the software research (RQ3). The aforementioned research questions help us assess the awareness of diversity in software engineering participant research.
\input{RQ1}
\input{RQ2}
\input{RQ3}
\input{discussion}
\section{Threats to Validity}
\label{sec:threats}
In this section, we recognize the threats to validity of our research and discuss methods we applied in our research design to mitigate these threats.
\noindent\textbf{Internal Validity:}
A threat to the internal validity of our research is the selection of our sample of participant studies. This process is described in Section~\ref{sec:selectingParticipantStudies}. The first author read all the abstracts from our selected venue to identify participant studies for our sample.
There is a chance that participant studies were not included because the authors have not described their methodology in their abstract. To mitigate this threat of subjectivity, we took two steps. Firstly, whenever ambiguity arose, the first author read the entire paper to confirm (or not) its inclusion in the study. Secondly, if after verifying the contents of the paper there was still ambiguity, the research team discussed if the study would be considered part of the sample or removed. Given our sample consists of 105 participant studies, it is unlikely that any missing study would have drastically changed our overall results.
The classification of diversity categories in the coding scheme for RQ1 and RQ2 could also pose a threat to the internal validity of our findings. This is because this classification was done manually and can be regarded as subjective. However, to mitigate this threat, we took the following steps, described in Section~\ref{sec:developingCodingScheme}. First, we conducted a pilot study and met regularly for the duration of this study in order to discuss the classification of each diversity category in our pilot sample. Second, if ever there was ambiguity during coding, we discussed it among the authors to reach a consensus.
\noindent\textbf{External Validity:}
A threat to external validity is that our selection sample was limited to the ICSE technical tracks of the 2019, 2020, and 2021 editions.
ICSE is highly regarded as a top-tier conference in the SE research community and even though choosing three years of sample papers is within industry standards (e.g.,~\cite{Storey&Ernst:2020}), it is possible that our results might differ if we had selected a different venue for our sample. This might limit the generalizability of our results. However, we believe that our contributions of creating a coding scheme to identify diversity categories, the model for diversity awareness, and guidelines for diversity awareness would still remain valid.
\noindent\textbf{Construct Validity:}
Our coding scheme poses a threat to our construct validity. We defined four different functions of diversity, namely describing, analyzing, reflecting and assessing limitations of participant diversity. These functions were based on our interpretation of what we thought were relevant functions for this research. However, we acknowledge that there could be other characterizations of different functions of diversity in participant studies. Our classification of diversity categories also poses a similar threat where the diversity categories we created could be labelled differently. However, due to the lack of classification schemes in SE that suited our research, we felt the need to create our own classification scheme for diversity categories. Our approach to identifying diversity categories in SE participant studies is one of the contributions of our research that we hope to see in future works.
\input{conclusion}
\section{Discussion}
\label{sec:discussion}
\subsection{Diversity Awareness Model for SE Participant Studies}
\begin{table*}
\centering
\caption{Diversity Awareness Model for SE participant studies.}
\label{tab:diversity-awareness-model}
\input{tables/diversity-awareness-model-v2.tex}
\end{table*}
While studying the extent to which SE participant studies report on the diversity of participants, we identified five types of studies which were distinguishable in terms of considering diversity in the overall research and publication approach. Based on these five types of studies, we propose a model for diversity awareness as shown in Table~\ref{tab:diversity-awareness-model}.
The model contains five levels of diversity awareness and it can be helpful in assessing the level of diversity awareness of authors of a participant study and supporting SE researchers in reflecting on diversity.
In Table~\ref{tab:diversity-awareness-model}, we describe short fictional examples for each level.
We refrain from providing real examples for all levels, as our goal is not to single out past studies but to discuss paths for improving future ones.
Good examples, however, can help illustrate and guide SE researchers.
As a good example of a study with high-level diversity awareness, we report on the work of Dias \textit{et al.}~\cite{Dias&Meirelles:2021}.
The authors of this study demonstrate a high level of diversity awareness (level 4), as they consciously ``choose'' to include diversity and report in detail about their choice and the reasons for it.
\begin{quotebox}
``To foster diversity, when inviting the participants, we prioritize women and non-US based maintainers. We took this decision to avoid having too many “Silicon Valley” participants, as they are over-represented amongst OSS maintainers.'' \cite{Dias&Meirelles:2021}
\end{quotebox}
A few studies in our sample even reported the diversity of the researchers involved. The following quote from Gerosa \textit{et al.}~\cite{Gerosa&Wiese:2021} shows how the authors report the diversity of the researchers. This indicates diversity awareness that goes beyond considering the diversity of research participants.
\begin{quotebox}
``we formed an international and diverse team of researchers, who are originally from South America (4), Europe (3), and Asia (1) and were working, at the time of this study, in North America (5), Europe (1), South America (1), and Australia (1). Seven researchers work in academia with extensive experience with OSS, and one researcher is a practitioner working in an OSS company.''~\cite{Gerosa&Wiese:2021}
\end{quotebox}
\subsection{Diversity Awareness Guidelines}
It is difficult to determine the level of diversity awareness quantitatively. For example, reporting several diversity categories or stating that the gender diversity of participants was not considered because the research is not about gender does not necessarily mean a high level of diversity awareness.
\textbf{It is the combination of reporting, analyzing, addressing and reflecting on the considerations of diversity that can demonstrate the diversity awareness of authors.} Here, we propose questions that can guide researchers when self-assessing their diversity awareness that they apply on their research.
\begin{itemize}
\item Do you consider diversity in your research?
\item Which diversity categories did you consider when selecting research participants?
\item How do you make sure that your research participants are diverse (e.g., a demographic representation of society)?
\item What recruitment efforts did you undertake to reach a demographic representation?
\item Which groups/diversity categories are over represented?
\item Which professional diversity categories do you consider (e.g., work experience, ...)?
\item Which social diversity categories do you consider (e.g., gender, language, nationality, age, ...)?
\item Which diversity categories do you NOT consider? Why do you not consider them?
\item Which diversity categories might be relevant in the context of your research?
\item Are you reporting your diversity considerations in your research publication?
\item Are the diversity categories considered relevant to your research?
\end{itemize}
\subsection{Impact on Society}
We believe this research to have a significant impact on the design of future SE research, in particular research that includes research participants. Our work both highlights the importance of considering and reporting the diversity of participants as well as provides guiding questions to successfully integrate considerations of diversity and inclusion into the research design. Furthermore, our proposed model of diversity awareness helps researchers to self-assess and review their research to identify lacks which, otherwise, they would have missed. For research that includes participant studies, defining, selecting and recruiting research participants is an evident opportunity to consider and implement diversity and inclusion into research. The better diversity is considered and implemented in the research design the broader will be the acceptance and the benefit of the research and development for society.
|
2,877,628,088,989 | arxiv | \section{Introduction}
It is known that there are a few direct connections between physical quantities in $\mathcal{N}=4$ SYM and QCD. At the tree level, gluon amplitudes are equivalent in the two theories. At one-loop level, through the supersymmetric decomposition \cite{Bern:1994cg}, one-loop $\mathcal{N}=4$ amplitudes are also useful building blocks for QCD amplitudes.
An intriguing relation between $\mathcal{N}=4$ SYM and QCD also exists at high loop orders, known as the principle of maximal transcendentality \cite{Kotikov:2002ab, Kotikov:2004er}.
The \emph{``transcendentality"} here refers to the transcendentality degree which is a mathematical notion characterizing the ``complexity" of transcendental functions and numbers.
For example, the transcendentality degree of algebraic numbers or rational functions is zero, the $\log(x)$ function and $\pi$ have transcendentality degree $1$, and the more general classical polylogrithm $\mathrm{Li}_n(x)$ and the Riemann zeta value $\zeta_n$ has degree $n$, recalling their definition:
\begin{equation}
\mathrm{Li}_n(x) = \sum_{k=1}^\infty {x^k \over k^n} = \int_0^x { \mathrm{Li}_{n-1}(t) \over t} d t \,, \qquad \mathrm{Li}_1(x) = - \log(1-x) \,, \qquad \zeta_n = \mathrm{Li}_n(x=1) \,.
\end{equation}
The maximal transcendentality principle (MTP) conjectures that for certain physical quantities, the maximally transcendental parts, \emph{i.e.}~the parts with the highest transcendentality degree, are equal in ${\cal N}=4$ SYM and QCD (up to certain identification of the fermions in the two theories).
The MTP was originally proposed in \cite{Kotikov:2002ab, Kotikov:2004er}, suggesting that the anomalous dimensions of twist-two operators in $\mathcal{N}=4$ SYM can be obtained from the maximally transcendental part of the corresponding QCD results \cite{Moch:2004pa}.
A further observation beyond the anomalous dimensions was made for the form factors and Higgs amplitudes: the study of the two-loop three-point form factor of stress-tensor multiplet in $\mathcal{N}=4$ SYM in \cite{Brandhuber:2012vm} shows that it coincides with the maximally transcendental part of the two-loop Higgs plus three-gluon amplitudes in the heavy top-mass limit obtained in \cite{Gehrmann:2011aa}. This generalizes the scope of the MTP from anomalous dimensions to kinematics-dependent functions.
More correspondence was also observed to Higgs-plus-three-parton amplitudes with high dimensional operators \cite{Brandhuber:2014ica, Brandhuber:2017bkg, Jin:2018fak, Brandhuber:2018xzk, Brandhuber:2018kqb, Jin:2019ile, Jin:2019opr, Jin:2020pwh} and with external quark states \cite{Jin:2019ile, Banerjee:2017faz}.
There is also other evidence of the correspondence for Wilson lines \cite{Li:2014afw, Li:2016ctv}.
The MTP was used to obtain the planar four-loop collinear anomalous dimension in $\mathcal{N}=4$ SYM \cite{Dixon:2017nat}, which was confirmed by \cite{Agarwal:2021zft}.
It is also worth pointing out that the MTP was found to be true for the non-planar cusp anomalous dimensions at four loops \cite{Henn:2019swt,Huber:2019fxe}, suggesting that it should apply beyond the planar (\emph{i.e.}~large $N_c$) limit.
So far the maximal transcendentality principle is still a conjecture and is known in the above-mentioned cases through explicit computations. (There are also known counterexamples such as gluon amplitudes, which we will discuss in the Discuss section.)
In this paper, we make a concrete step toward the understanding of the MTP by proving it for the known cases of form factors.
We recall that an $n$-point form factor is defined as a matrix element between a local gauge-invariant operator and $n$ asymptotic on-shell states (see \cite{Yang:2019vag} for a recent review):
\begin{equation}
\label{eq:FF-def}
\mathcal{F}_{\mathcal{O}, n} = \int d^D x e^{-i q\cdot x} \langle \Phi(p_1) \ldots \Phi(p_n) | \mathcal{O}(x) |0 \rangle \,,
\end{equation}
where $p_i, i=1,..,n$ are on-shell momenta and $q=\sum_{i=1}^n p_i$ is the off-shell momentum associated to the local operator.
As mentioned, the form factors are also related to Higgs amplitudes in the Higgs effective theory where the top quark is integrated out \cite{Wilczek:1977zn, Shifman:1979eb, Dawson:1990zj, Djouadi:1991tka, Kniehl:1995tn}.
The central idea in our proof of the MTP for form factors is to apply a recently developed bootstrap method \cite{Guo:2021bym}.
In this method, one starts with a general ansatz of loop quantities in terms of a set of master integrals and then determines the integral coefficients via various physical constraints, such as infrared divergences and collinear limits.
A further key idea is to apply a small set of unitarity cuts that are universal for general gauge theories as constraints, which are enough to fix the possibly remaining degrees of freedom.
It turns out that for a large class of form factors, these universal physical constraints are sufficient to determine the maximally transcendental parts uniquely, irrespective of which gauge theory is under consideration.
We will discuss these in detail for the minimal form factors and the three-point form factor of ${\rm tr}(F^2)$ up to two loops.
To test the MTP beyond the known examples, we also consider the two-loop four-gluon form factor of the $\mathrm{tr}(F^3)=\mathrm{tr}(F_{\mu}^{~\nu} F_{\nu}^{~\rho} F_{\rho}^{~\mu})$ operator and obtain its maximally transcendental parts for the first time.
This is a more non-trivial four-point next-to-minimal form factor and has much richer structures.
The form factor depends on seven independent Lorentz invariants: six Mandelstam variables $s_{ij}=(p_i+p_j)^2$ with $1\leqslant i<j \leqslant4$ and one parity-odd variable ${\rm tr}_5$. The general ansatz contains 221 master integrals, and the coefficients depend on five spinor factors.
We find that the physical constraints from the infrared divergences and the collinear limits, together with the cancellation of spurious poles, can fix a significant part of the ansatz. The remaining degrees of freedom can be fixed by a simple type of quadruple unitarity cuts.
A nice fact we find is that the difference of the two-loop form factors in different gauge theories can only depend on two free parameters.
We obtain the maximally transcendental parts of the form factor in both $\mathcal{N}=4$ SYM and pure YM theories. Interestingly, the results are different in the two theories.
The reason is that there are extra maximally transcendental contributions in ${\cal N}=4$ SYM from the diagrams involving the fermion (\emph{i.e.}~gluino) loop.
In contrast, the scalar-loop diagrams in ${\cal N}=4$ SYM have no maximally transcendental contribution.
This crucial fact implies that the results of $\mathcal{N}=4$ SYM and QCD are still identical, once one converts the fundamental quarks in QCD to be adjoint fermions, which can be achieved by a proper change of the color factors in the QCD form factor.
Thus, the MTP still holds in this case.
As another intriguing observation, we find that the four-point form factor of the half-BPS $\mathrm{tr}(\phi^3)$ operator \cite{Guo:2021bym} is identical to a part of the $\mathrm{tr}(F^3)$ form factor that carries the same spinor factors. This seems to be not trivial since the two form factors have different spinor structures.
This paper is organized as follows.
In Section~\ref{sec:strategy}, we give a brief review of the general bootstrap strategy and then discuss the collinear limit for form factors in detail.
In Section~\ref{sec:miniFF}, we apply the bootstrap strategy to prove the MTP for two-loop minimal form factors, and some constraints on the lower transcendental parts are also considered.
In Section~\ref{sec:3ptFF} we consider the two-loop three-point form factors of $\mathrm{tr}(F^2)$ via the bootstrap method together with unitarity cuts.
In Section~\ref{sec:2loop4ptF3}, we compute the four-point form factors of $\mathrm{tr}(F^3)$ up to two loops and discuss the maximal transcendentality properties.
A summary and discussion are given in Section~\ref{sec:discussion}.
Several appendices provide the integral conventions, some explicit results, as well as some technical details.
Appendix~\ref{app:UT} provides the definition of pure UT master integrals used in the paper.
Appendix~\ref{app:catani} discusses the Catani IR subtraction formula and its relation to the BDS subtraction.
Appendix~\ref{app:letterandcollinear} gives the definition of symbol letters as well as their collinear limit behavior for the four-point two-loop form factors.
Appendix~\ref{app:HigherOrder} discusses a technical point about the constraints from the $\mathcal{O}(\epsilon)$ order of form factors.
Appendix~\ref{app:BuildingBlocks} provides the building blocks obtained from the bootstrap of the four-point form factor.
Appendix~\ref{app:fullFF} gives the results of the finite remainder function of the four-point form factor.
Appendix~\ref{app:A4noMT} discusses the one-loop four-gluon amplitude, which provides a counterexample of the maximally transcendental principle for the amplitude case.
\section{Bootstrap strategy based on master integrals}
\label{sec:strategy}
It is known that an $l$-loop amplitude or form factor can be expanded in a set of integral basis as
\begin{equation}
\label{eq:masterexpansion}
\mathcal{F}^{(l)} = \sum_i \alpha_{i} \, I_{i}^{(l)} \,,
\end{equation}
where $I_{i}^{(l)}$ can be chosen as a set of master integrals obtained via the integration-by-part reduction \cite{Chetyrkin:1981qh, Tkachov:1981wb}.
The number of IBP master integrals for a given quantity is also known to be finite \cite{Smirnov:2010hn}.
While the basis integrals are theory independent, the intrinsic physical information is contained in the coefficients $\alpha_i$, which will be the main target we investigate.
In the traditional Feynman diagram method, one usually starts with Feynman diagrams and then performs the integral reduction to get the coefficients. This typically requires complicated intermediate steps and the results are also often given in incomprehensible forms.
Besides, the physical properties (mentioned below) are not manifest in such a computation but only provide consistency checks for the final results.
The bootstrap strategy takes a very different route: the final form of the result such as \eqref{eq:masterexpansion} is taken as the starting ansatz, and the physical consistency conditions are used at the very beginning of the computation, namely, they are used as constraints to solve the coefficients in the ansatz.
In this way, the physical properties are manifest in each step, and this often leads to a result in a compact form.
In Section~\ref{sec:constraints} we first briefly discuss various physical properties that will be used as constraints in later computations, then we will provide some details about the collinear limit of form factors in Section~\ref{sec:collinearFF}.
\subsection{Physical constraints}
\label{sec:constraints}
The constraints are from the general properties of physical quantities,
including: (1) the loop quantity should reproduce the general infrared (IR) divergences, (2) it should satisfy the collinear factorization property, (3) the spurious poles must cancel in the full result, and (4) it should satisfy unitarity cuts or other possible constraints.
Below we discuss them in more detail.
\paragraph{IR divergences.}
Amplitudes and form factors with massless external states have IR divergences, which have universal structures and are related to the number and types of external massless particles.
In the planar limit, for example, IR divergences are captured by the two-point Sudakov form factors \cite{Mueller:1979ih,Collins:1980ih,Sen:1981sd,Magnea:1990zb},
which are determined by two kinematics-independent nubmers:
the cusp anomalous dimension $\gamma_\text{cusp}$ \cite{Korchemsky:1985xj, Korchemsky:1988si}
and the collinear anomalous dimension $\mathcal{G}_{\rm coll}$ (see \emph{e.g.}~\cite{Dixon:2017nat}).
For amplitudes or form factors with multiple external legs, IR divergences for general massless gauge theories can be conveniently taken into account by the Catani formula \cite{Catani:1998bh}. Since our main focus is on the maximally transcendental parts, it is convenient to use the Bern-Dixon-Smirnov (BDS) ansatz \cite{Bern:2005iz, Anastasiou:2003kj} which also captures the collinear behavior, as will be explained shortly below.
Some details of the Catani formula and its relation to the BDS form are given in Appendix~\ref{app:catani}.
\paragraph{Collinear limits.}
When two external legs are taken in the collinear limit, the form factors satisfy factorization formula as (see \emph{e.g.}~\cite{Kosower:1999xi}):
\begin{equation}
\label{eq:collinear-general}
\mathcal{F}^{(L)}_n(1,\ldots, a^{h_a}, b^{h_b}, \ldots, n) \xrightarrow{ p_a || p_b } \sum_\ell \sum_\sigma \mathbf{Sp}_{-\sigma}^{(\ell)}(a^{h_a},b^{h_b}) \, \mathcal{F}^{(L-\ell)}_{n-1}(1,\ldots, (a+b)^\sigma, \ldots, n) \,.
\end{equation}
For example, in the linear limit $p_a \, || \, p_b \, || \, P=p_a+p_b$
\begin{equation}
p_a \rightarrow z P , \qquad p_b \rightarrow (1-z)P \,,
\end{equation}
the one-loop splitting amplitude $\mathbf{Sp}^{(1)}$ can be given as \cite{Bern:1994zx, Kosower:1999rx, Bern:1999ry}
\begin{equation}
\label{eq:splittingAmp00}
\mathbf{Sp}^{(1)}(P\rightarrow a \, b; z) = \mathbf{Sp}^{(0)}(P \rightarrow a\,b; z)\ r_1^{[1], \text{MT}}(P^2, z) + \textrm{(lower transendental part)} \,,
\end{equation}
where the maximally transcendental part of the one-loop splitting function (denoted by the superscript `MT') is
\begin{equation}
\label{eq:splittingAmp}
r_1^{[1], \text{MT}}(P^2, z) = \frac{ e^{\epsilon \gamma_\text{E}}\Gamma (-\epsilon)^2 \Gamma (\epsilon +1)}{\Gamma (1-2 \epsilon )} (- P^2 )^{-\epsilon}\Big\{ 1-z^{-\epsilon}-\left({1-z}\right)^{-\epsilon} + \epsilon ^2 \big[ \log (z)\log (1-z) -\zeta_2 \big] + {\cal O}(\epsilon^3) \Big\} \,.
\end{equation}
We stress that \eqref{eq:splittingAmp} is universal for general gauge theories, and this formula will be used to bootstrap the one-loop three- and four-point form factors in Section~\ref{subsec:3ptff1loop} and Section~\ref{sec:solveFF4pt1loop}.
Beyond one-loop order, there is a convenient way to capture both the IR and collinear behavior
by using the BDS ansatz \cite{Bern:2005iz} for $\mathcal{N}=4$ SYM (or the maximally transcendental parts in general gauge theories, see more discussion in Appendix~\ref{app:catani}). The loop correction at two loops can be given as
\begin{equation}
\label{eq:BDSansatz}
\mathcal{I}^{(2)} = \frac{1}{2} \left( \mathcal{I}^{(1)}(\epsilon) \right)^2 + f^{(2)}(\epsilon) \mathcal{I}^{(1)}(2\epsilon) + \mathcal{R}^{(2)} + \mathcal{O}(\epsilon) \,,
\end{equation}
where
\begin{equation}
\label{eq:f2def}
f^{(2)}(\epsilon)=-2 \zeta_{2}-2 \zeta_{3} \epsilon-2 \zeta_{4} \epsilon^{2} \,.
\end{equation}
The original two-loop BDS ansatz is proposed with only the first two terms in \eqref{eq:BDSansatz} \cite{Bern:2005iz, Anastasiou:2003kj}:
\begin{equation}
\label{eq:BDSansatzOrigin}
\mathcal{I}^{(2), {\rm BDS}} = \frac{1}{2} \left( \mathcal{I}^{(1)}(\epsilon) \right)^2 + f^{(2)}(\epsilon) \mathcal{I}^{(1)}(2\epsilon) \,,
\end{equation}
which were constructed in a way that they capture all the IR divergences and also have correct collinear behavior of amplitudes. This original ansatz is correct for the four- and five-point amplitudes in ${\cal N}=4$ SYM, but for higher-point amplitudes an extra finite remainder function is needed \cite{Drummond:2008aq, Bern:2008ap}, denoted as $\mathcal{R}^{(2)}$ in \eqref{eq:BDSansatz}.
The same BDS-ansatz structure also generalizes to form factors \cite{Brandhuber:2012vm, Dixon:2020bbt}.
Since the remainder function is free from both IR and collinear singularities, it has the important property that the $n$-point remainder reduces trivially to $(n-1)$-point remainder in the collinear limit as
\begin{equation}
\label{eq:remainderCL}
\mathcal{R}_n^{(2)} \ \xlongrightarrow[\mbox{}]{\mbox{$p_i \parallel p_{i+1}$}} \ \mathcal{R}_{n-1}^{(2)} \,.
\end{equation}
This will provide useful constraints for the two-loop three- and four-point form factors in Section~\ref{sec:3ptFF} and Section~\ref{sec:2loop4ptF3}.
\paragraph{Spurious pole cancellation.}
For form factors with non-trivial spinor structures (such as the four-point form factors considered later), the coefficients of master integrals can contain spurious poles (\emph{i.e.}~unphysical poles).
The cancellation of spurious poles typically requires a combination of both the spinor factors and the master integrals, which can provide non-trivial constraints on the coefficients of master integrals.
The details of the spurious poles as well as applying their cancellation as constraints will be given in Section~\ref{sec:2loop4ptF3} for the discussion of four-point form factors.
\paragraph{Lightlike limit of $q$.}
The form factor as defined in \eqref{eq:FF-def} contains an off-shell momentum $q=\sum_i p_i$ which is carried by the operator. An interesting limit to consider is the lightlike limit $q^2 \rightarrow 0$.\footnote{This limit of $q^2\rightarrow0$ has also been considered for the three-point form factor of $\mathrm{tr}(F^2)$ in \cite{Lin:2021lqo}.}
Since the form factors are equivalent Higgs-plus-gluons amplitudes where $q^2 = m_H^2$ \cite{Brandhuber:2012vm, Gehrmann:2011aa}, this lightlike limit of $q$ can be understood as the massless limit $m_H \rightarrow0$ of the Higgs particle, therefore it is reasonable to expect that
the form factor should have a smooth limit.
This can provide useful constraints and will play a role in Section~\ref{sec:3ptFF}.
\paragraph{Unitarity cuts.}
When the above constraints are not enough to fix the full results, one can use another powerful tool -- the unitarity cut constraints \cite{Bern:1994zx, Bern:1994cg, Britto:2004nc}.
The unitarity cut method is a powerful method that in principle can determine the full result.
Here in our application together with the bootstrap strategy, the nice point is that after using the above physical constraints, only a small number of simple unitarity cuts are needed to fix the remaining free parameters.
Moreover, these cuts can often be chosen such that they are universal for general gauge theories, and this fact will play an important role in the proof of MTP.
At this point, it may be good to compare our strategy with the symbol-bootstrap method that has also been used for computing the finite remainder of amplitudes (see \emph{e.g.}~\cite{Dixon:2011pw, Dixon:2013eka,Dixon:2014iba,Golden:2014pua,Drummond:2014ffa, Caron-Huot:2016owq,Dixon:2016nkn, Drummond:2018caf, Caron-Huot:2019vjl,Dixon:2020cnr, Zhang:2019vnm, He:2020vob, Golden:2021ggj}) and form factors \cite{Brandhuber:2012vm, Dixon:2020bbt}.
Unlike the symbol bootstrap where one considers only the finite remainder functions,
the main difference for the bootstrap used here is that we start with an ansatz of the full form factor in terms of master integrals. This requires the knowledge of master integrals and in this sense, it contains more input information than the symbol bootstrap. On the other hand, the master integrals are theory-independent and can be used for general observables in general theories.
Importantly, physical constraints that are not available in symbol bootstrap can be applied here, such as the IR divergences and the unitarity-cut constraints.
As we will see, it is these new features that make it possible to prove the maximal transcendentality principle that relates form factors in different theories.
\subsection{Collinear limit of form factors}
\label{sec:collinearFF}
In this subsection, we provide some details about the collinear limit for form factors. This is mainly used in the computation of the four-point form factor in Section~\ref{sec:2loop4ptF3}. In that cases, the kinematic variables are a bit complicated and the collinear limit must be taken properly. In the meanwhile, the introduced momentum twistor variables will also help to understand the structure of the symbol letter variables of the four-point form factors.
We introduce the periodic Wilson line configuration in the momentum space \cite{Alday:2007hr, Maldacena:2010kp,Brandhuber:2010ad,Gao:2013dza}, as shown in Figure~\ref{fig:WL3pt_x} for the three-point form factor case.
One has
\begin{equation}
x_i - x_{i+1} = p_i = \lambda_i \widetilde\lambda_i \,, \qquad x_{\underline{i}} - x_i = x_i - x_{\bar{i}} = q \,.
\end{equation}
As in the case of scattering amplitudes, each dual coordinate $x_i$ corresponds to a line in the (dual) twistor space which is represented by two momentum twistor variables $Z_{i-1}, Z_i$ \cite{Hodges:2009hk, Mason:2009qx}. The momentum twistor variables can be defined as
\begin{equation}
\label{eq:Z-def}
Z_{i}^A = (\lambda_i^\alpha, \mu_i^\beta) \,, \qquad \mu_i^\beta = x_i^{\alpha\beta} \cdot \lambda_{i \alpha} = x_{i+1}^{\alpha\beta} \cdot \lambda_{i \alpha} \,.
\end{equation}
The periodic Wilson line configuration in momentum twistor space is shown in Figure~\ref{fig:WL3pt}.
\begin{figure}[tb]
\centering
\includegraphics[scale=0.5]{figs/WL3pt_x.eps}
\caption{Dual periodic Wilson line configuration for the three-point form factor.}
\label{fig:WL3pt_x}
\end{figure}
\begin{figure}[tb]
\centering
\includegraphics[scale=0.5]{figs/WL3pt.eps}
\caption{Momentum twistor space picture for the dual periodic Wilson line configuration for the three-point form factor. The red dashed line corresponds to the $x$-configuration in Figure~\ref{fig:WL3pt_x}.}
\label{fig:WL3pt}
\end{figure}
The momentum twistor variables and dual spacetime coordinates can be related using the following formula
\begin{equation}
\label{eq:REL}
\langle i | \, x_{i j}\, x_{j k} \, | k \rangle = \frac{\langle Z_i Z_{j-1} Z_j Z_k \rangle}{\langle j-1, j \rangle } \,,
\end{equation}
where $x_{ij} \equiv x_i - x_j$, $\langle i j \rangle \equiv \epsilon^{\alpha\beta} \lambda_{i \alpha} \lambda_{j \beta}$, and $\langle Z_i Z_j Z_k Z_l \rangle = \epsilon_{ABCD}Z_i^A Z_j^B Z_k^C Z_l^D$.
From \eqref{eq:REL}, one can obtain following useful relations
\begin{equation}
\label{eq:REL-x2}
x_{i j}^2 = {\langle Z_{i-1} Z_i Z_{j-1} Z_j \rangle \over \langle i-1, i \rangle \langle j-1, j \rangle} \,.
\end{equation}
In the following, we will use the abbrevation for the four-brackets $\langle Z_i Z_j Z_k Z_l \rangle = \langle i j k l \rangle$.
For the three-point case, using \eqref{eq:REL}-\eqref{eq:REL-x2}, one can rewrite the ratio variables in terms of twistor four-brackets and spinor products, such as
\begin{align}
& u = \frac{s_{12}}{q^2} = {x_{13}^2 \over x_{1 \bar{1}}^2} = {\langle \underline{3} 1 2 3 \rangle \over \langle \underline{3} 1 3 \bar{1} \rangle} {\langle 3 \bar{1} \rangle \over \langle 2 3 \rangle} \,, \qquad\qquad
1 - {1\over u} = {\langle \underline{3} \bar{1} 2 3 \rangle \over \langle \underline{3} 1 2 3 \rangle}
{\langle \underline{3} 1 \rangle \over \langle \underline{3} \bar{1} \rangle} \,, \\
& v = \frac{s_{23}}{q^2} = {x_{2\bar{1}}^2 \over x_{2 \bar{2}}^2} = {\langle 123 \bar{1} \rangle \over \langle 1 2 \bar{1} \bar{2} \rangle} {\langle \bar{1}\bar{2}\rangle \over \langle 3 \bar{1} \rangle} \,, \qquad\qquad 1 - {1\over v} = {\langle 1 \bar{2} 3\bar{1} \rangle \over \langle 1 2 3 \bar{1} \rangle}{\langle 12 \rangle \over \langle 1 \bar{2} \rangle} \,, \\
& w = \frac{s_{13}}{q^2} = {x_{3\bar{2}}^2 \over x_{3 \bar{3}}^2} = {\langle 2 3 \bar{1}\bar{2} \rangle \over \langle 2 3 \bar{2}\bar{3} \rangle} {\langle\bar{2}\bar{3} \rangle\over\langle \bar{1} \bar{2} \rangle}\,,\qquad\qquad 1- {1\over w} = {\langle 2 \bar{3} \bar{1} \bar{2} \rangle \over \langle 2 3 \bar{1} \bar{2} \rangle}{\langle 23 \rangle \over \langle 2 \bar{3} \rangle} \,.
\end{align}
These variables are enough to provide the symbol letters for the remainder functions of three-point form factors of ${\rm tr}(F^2)$ \cite{Brandhuber:2012vm, Dixon:2020bbt}. In this case, the collinear limit is relatively trivial, for example, in the limit of $p_1 \parallel p_3$, one has $s_{13}\rightarrow0$, $q^2 \rightarrow s_{12}+s_{23}$ and
\begin{equation}
u \rightarrow u, \qquad v \rightarrow 1-u, \qquad w \rightarrow 0 \,.
\end{equation}
\begin{figure}[tb]
\centering
\includegraphics[scale=0.5]{figs/WL4pt.eps}
\caption{Dual periodic Wilson line configuration for the four-point form factor in momentum twistor space.}
\label{fig:WL4pt}
\end{figure}
The kinematics of the four-point form factor is more complicated. The periodic Wilson line configuration in momentum twistor space is shown in Figure~\ref{fig:WL4pt}.
We define the ratio variables:
\begin{equation}\label{eq:defuij}
u_{ij} = \frac{s_{ij}}{q^2} \,, \qquad u_{ijk} = \frac{s_{ijk}}{q^2} \,,
\end{equation}
where $s_{ij \ldots k} = (p_i+p_j+\ldots+p_k)^2$.
They can be represented by momentum twistor as
\begin{align}
& u_{i,i+1} = \frac{x_{i,i+2}^2}{x_{\underline{i},i}^2} = \frac{ \langle i-1,i, i+1, i+2 \rangle \langle \underline{i}-1, \underline{i} \rangle }{ \langle \underline{i}-1, \underline{i}, i-1, i \rangle \langle i+1, i+2 \rangle } \, , \\
& u_{i,i+1,i+2} = \frac{x_{i,i+3}^2}{x_{\underline{i}, i}^2} = \frac{\langle i-1, i, i+2, i+3 \rangle \langle \underline{i}-1, \underline{i} \rangle }{\langle \underline{i}-1, \underline{i}, i-1, i \rangle \langle i+2, i+3 \rangle } \, . \nonumber
\end{align}
Note that there are only five independent ratio variables, which can be chosen as five of $u_{ij}$:
\begin{equation}
\{ u_{12}, \, u_{23}, \, u_{34}, \, u_{14}, \, u_{13}, \, u_{24}\} \,, \qquad u_{12}+u_{23}+u_{34}+u_{14}+u_{13}+u_{24} = 1 \,.
\end{equation}
There are also other variables which necessarily appear in the two-loop functions, which we define as
\begin{align}
x_{ijkl}^{\pm} = \frac{q^2+s_{ij}-s_{kl}\pm\sqrt{\Delta_{3,ijkl}}}{2s_{ij}}, \quad
y_{ijkl}^{\pm} = \frac{\mathrm{tr}_{\pm}(ijkl)}{2s_{ij}s_{il}}, \quad
z_{ijkl}^{\pm\pm} = 1+y_{ijkl}^{\pm}-x_{lijk}^{\pm} ,
\end{align}
where $\Delta_{3,ijkl} = {\rm Gram}(p_i+p_j, p_k+p_l)$ also appears in the one-loop three-massive triangle integral,
and $\mathrm{tr}_{\pm}(ijkl) = s_{ij} s_{kl}-s_{ik} s_{jl}+s_{il} s_{jk} \pm \mathrm{tr}_5$, in which the parity-odd kinematics
\begin{equation}\label{eq:tr5}
\mathrm{tr}_5 = 4i \epsilon_{\mu\nu\rho\sigma}p_1^{\mu}p_2^{\nu}p_3^{\rho}p_4^{\sigma} = \langle 1|2|3|4|1 ]-[1|2|3|4|1\rangle
\end{equation}
is related to the Gram determinant $\Delta_5 = {\rm Gram}(p_1, p_2, p_3, p_4)$ by $\Delta_5 = \mathrm{tr}_5^2$.
These variables are used to define the symbol letters in the remainder function where we are defined in Appendix~\ref{app:letters}.
Since the $y^\pm$ variables are related to the parity-odd variable $\mathrm{tr}_5$, we discuss them in detail below.
Using the relations
\begin{equation}
y_{ijkl}^+ = \frac{\mathrm{tr}_{+}(ijkl)}{2s_{ij}s_{il}} = \frac{\langle l | k | j ]}{\langle l | i | j ]} \,, \qquad
y_{ijkl}^- = \frac{\mathrm{tr}_{-}(ijkl)}{2 s_{ij} s_{il}} = \frac{\langle j | k | l ]}{\langle j | i | l ]} \, ,
\end{equation}
one has
\begin{align}
& y_{1234}^+ = \frac{\langle 1234 \rangle}{\langle \underline{4}123 \rangle} \frac{\langle 1 \underline{4} \rangle}{\langle 1 4 \rangle} \,, \qquad\qquad\qquad\qquad\quad\ y_{ijkl}^- = \frac{u_{jk} u_{kl}}{u_{ij} u_{il}} \big(y_{ijkl}^+ \big)^{-1} \,, \nonumber \\
& y_{1324}^+ = \frac{\langle 13 \rangle \langle 24 \rangle}{\langle 14 \rangle \langle 23 \rangle} \Big(\frac{u_{123}-u_{12}}{u_{23}}-1 \Big)^{-1} \,, \qquad y_{3124}^+ = y_{1324}^+ \big|_{p_1 \leftrightarrow p_3} \,, \nonumber \\
& y_{1342}^+ = \frac{\langle 13 \rangle \langle 24 \rangle}{\langle 12 \rangle \langle 34 \rangle} \Big(\frac{u_{134}-u_{14}}{u_{34}}-1 \Big)^{-1} \,, \qquad y_{3142}^+ = y_{1342}^+ \big|_{p_1 \leftrightarrow p_3} \,. \nonumber
\end{align}
Unlike the three-point case, the collinear limit for the four-point form factor should be carefully taken.
Consider the collinear limit $p_4 \parallel p_3$, one can parametrize $Z_4$ in the following way:
\begin{equation}
Z_4 = Z_3 + \delta {\langle \bar{1} \bar{2} 1 3 \rangle \over \langle \bar{1} \bar{2} 1 2 \rangle} Z_2 + \tau \delta {\langle \bar{2} 1 2 3 \rangle \over \langle \bar{1} \bar{2} 1 2 \rangle} \bar{Z}_1 + \eta {\langle \bar{1} 1 2 3 \rangle \over \langle \bar{1} \bar{2} 1 2 \rangle} \bar{Z}_2 \,,
\end{equation}
where the ratio of four brackets are introduced to balance the twistor weight. The collinear limit can be achieved by taking first $\eta \rightarrow 0$, followed by $\delta \rightarrow 0$. The parameter $\tau$ is finite which gives the momentum fraction shared by $p_4$ in the limit.
Such a parametrization for the collinear limit was introduced for the amplitudes case in \cite{CaronHuot:2011ky}.
The same limit applies simultaneously to $\underline{Z}_4, \bar{Z}_4$ due the periodicity condition.
And we can also define the collinear limit for spinors using \eqref{eq:Z-def} as
\begin{align}
& \lambda_4 = \lambda_3 + \delta {\langle \bar{1} \bar{2} 1 3 \rangle \over \langle \bar{1} \bar{2} 1 2 \rangle} \lambda_2 + \tau \delta {\langle \bar{2} 1 2 3 \rangle \over \langle \bar{1} \bar{2} 1 2 \rangle} \bar{\lambda}_1 + \eta {\langle \bar{1} 1 2 3 \rangle \over \langle \bar{1} \bar{2} 1 2 \rangle} \bar{\lambda}_2 \,.
\end{align}
The explicit collinear limit for all letter variables are given in Appendix~\ref{app:letterCL}.
Finally, let us mention an alternative way to consider the collinear limit by expressing all variables in terms of spinor variables $\{\lambda_i, \tilde\lambda_j\}$. For example, the collinear limit $p_i \parallel p_j$ can be taking by using following parameterization of spinor variables
\begin{align}
& \lambda_i \rightarrow \lambda_i \,, \qquad\qquad\quad\qquad \tilde{\lambda}_i \rightarrow (1-t) \tilde{\lambda}_i \,, \\
& \lambda_j \rightarrow \lambda_i + \delta \frac{\langle il \rangle}{\langle kl \rangle} \lambda_k \,, \qquad \tilde{\lambda}_j \rightarrow t \tilde{\lambda}_i + \tilde{\delta} \frac{\left[il\right]}{\left[kl\right]} \tilde{\lambda}_k \,, \nonumber
\end{align}
where $\{\lambda_k, \tilde{\lambda}_k\}$ and $\{\lambda_l, \tilde{\lambda}_l\}$ are the two pairs of reference spinors and the indices $\{i,j,k,l\}$ are not equal to each other. Then the collinear limit can be achieved by taking $\delta, \tilde{\delta} \rightarrow 0$, where there is no need to distinguish the order of limits here. The formula gives the limit behavior as $p_i \rightarrow (1-t) p_i$ and $p_j \rightarrow t p_i$, thus the momentum conservation is satisfied by $p_i+p_j = \lambda_i \tilde{\lambda}_i + \lambda_j \tilde{\lambda}_j \rightarrow \lambda_i \tilde{\lambda}_i + \mathcal{O}(\delta,\tilde{\delta})$.
\section{Two-loop minimal form factors}
\label{sec:miniFF}
In this section, we consider a special class of form factors, the so-called minimal form factors:
\begin{equation}
\mathcal{F}_{\mathcal{O}_n, \text{min}} = \int d^D x e^{-i q\cdot x} \langle p_1\, p_2 \ldots p_n | \mathcal{O}_n(x)|0 \rangle \,.
\end{equation}
There are two requirements in the definition of minimal form factors.
Consider the minimal form factor $\mathcal{F}_{\mathcal{O}_{n}, {\rm min}}$ of a length-$n$ operator $\mathcal{O}_{n} = \mathrm{tr}(\mathcal{W}_1 ... \mathcal{W}_n)$.
First, the number of external on-shell states should be equal to the the length of the operator.
Second, the minimal form factor at tree level is required to be non-zero, \emph{i.e.}~$\mathcal{F}_{\mathcal{O}_{n}, {\rm min}}^{(0)} \neq 0$, thus the external states should have same field configuration as $\{{\cal W}_i\}$.
For example, the form factor $\mathcal{F}_{\mathrm{tr}(F^3)}(1^{q},2^{\bar{q}},3^-)$ is not a minimal form factor, as it is zero at tree-level.
On the other hand,
the form factor of $\mathrm{tr}(\phi^n)$ and $n$ external on-shell scalar states, or the form factor of $\mathrm{tr}(F^n)$ with $n$ external gluon states, are minimal form factors.
Such form factors have been studied to two-loop order in $\mathcal{N}=4$ SYM and QCD \cite{Wilhelm:2014qua, Brandhuber:2014ica, Loebbert:2015ova, Brandhuber:2016fni, Loebbert:2016xkw, Brandhuber:2017bkg, Jin:2018fak, Brandhuber:2018xzk, Brandhuber:2018kqb, Jin:2019ile, Jin:2019opr, Jin:2020pwh,Lin:2020dyj,Jin:2022ivc}. These minimal form factors have played an important role for computing anomalous dimensions of high-length operators. The results of minimal form factors show that their maximally transcendental parts are given by same functions up to two-loop order, and lower transcendentality parts also present some universal structures \cite{Jin:2019ile, Jin:2019opr}.
The goal of the section is to prove these universal structures.
We will show that by imposing only the IR constraint, it is enough to fix the maximally transcendental part of minimal form factors up to two loops.
We also show that the IR divergences can be used to put strong constraints on the lower transcendental parts.
As a brief outline,
we will first consider the maximally transcendental parts up to two loops in Section~\ref{sec:MTminimal}.
Next, we will consider the constraints for the lower transcendental part in Section~\ref{sec:LTminimal}.
Finally, we apply the bootstrap results together with some unitarity-cut arguments to explain the universality of the results in different theories in Section~\ref{sec:univer-mimi}.
\paragraph{Setup.}
We define the $L$-loop $n$-point correction function $\mathcal{I}_{\text{min}, n}^{(L)}$ by factorizing out the tree-level minimal form factor as
\begin{equation}
\mathcal{F}_{\mathcal{O}_n, \text{min}}^{(L)} = \mathcal{F}_{\mathcal{O}_n, \text{min}}^{(0)} \mathcal{I}_{\text{min}, n}^{(L)} \,.
\end{equation}
The loop correction function can be expanded in set of master integrals
\begin{equation}
\mathcal{I}_{\text{min}, n}^{(L)} = \sum_i c_i(\epsilon) {I}_{i}^{(L)} \,,
\end{equation}
where $\{{I}_{i}^{(L)}\}$ is a set of master integrals and the coefficient $c_i(\epsilon)$ depends on $\epsilon = {(4-D)/2}$ and Mandelstam variables. An important remark follows: throughout this paper, we will always choose the set of master integrals, such that they have uniform transcendentality degree $2L$ at $L$ loops. All such masters which are used in this paper are collected in Appendix~\ref{app:UT}. They are often called pure uniformly transcendental (UT) integrals as defined in the canonical differential equations \cite{Henn:2013pwa}. Since the maximal transcendentality degree of $L$-loop form factors is $2L$, the coefficients $c_i(\epsilon)$ must be polynomials in $\epsilon$; here as in usual case, $\epsilon$ is assigned to carry transcendentality degree of $-1$.
For the convenience of later discussion, we reorganize the loop correction function in terms of different transcendentality degree as
\begin{equation}
\label{eq:minidegCalF}
\mathcal{I}_{\text{min}, n}^{(L)} = \sum_{a=2L}^{-\infty} \epsilon^{2L-a} \mathcal{I}_{\text{min}, n}^{(L), \text{deg-}a} \, ,
\end{equation}
where $\mathcal{I}_{\text{min}, n}^{(L), \text{deg-}a}$ corresponds to the correction to the transcendentality degree-$a$ part and is a linear combination of pure UT master integrals with coefficients free of $\epsilon$.
Explicitly, at one and two loops one has
\begin{align}
\mathcal{I}_{\text{min}, n}^{(1)} &= \mathcal{I}_{\text{min}, n}^{(1), \text{deg-2}} + \epsilon \, \mathcal{I}_{\text{min}, n}^{(1), \text{deg-1}} + (\textrm{lower degrees}) \,, \\
\mathcal{I}_{\text{min}, n}^{(2)} &= \mathcal{I}_{\text{min}, n}^{(2), \text{deg-4}} + \epsilon \, \mathcal{I}_{\text{min}, n}^{(2), \text{deg-3}} + \epsilon^2 \, \mathcal{I}_{\text{min}, n}^{(2), \text{deg-2}} + (\textrm{lower degrees}) \,,
\end{align}
where $\mathcal{I}_{\text{min}, n}^{(L), \text{deg-a}}$ is free from $\epsilon$.
Our goal is to compute $\mathcal{I}_{\text{min}, n}^{(L), \text{deg-a}}$ in terms of master integral expansions.
\subsection{Maximally transcendental part}
\label{sec:MTminimal}
The maximally transcendental parts are given by $\mathcal{I}_{\text{min}, n}^{(1), \text{deg-2}}$ at one loop and $\mathcal{I}_{\text{min}, n}^{(2), \text{deg-4}}$ at two loops. For simplicity, we will focus on the planar form factors with all particles in adjoint representation. We will discuss other representations such quarks in QCD in Section~\ref{sec:univer-mimi}.
\subsubsection{One-loop case}
\label{sec:MTmini1loop}
We start with the simple one-loop case as a warm-up.
For the minimal form factor at the one-loop level, the interactions involve at most two external legs at one time, thus the only type of master integrals are one-loop bubble integrals as defined in \eqref{eq:bubbleintegral}:\footnote{We consider gauge theories with all fields being massless in the paper, thus there are no tadpole integrals.}
\begin{equation}
I_{\text{Bub}}^{(1)}(i, j) = \frac{1}{\epsilon^2} - \frac{\log(-s_{ij})}{\epsilon} + \mathcal{O}(\epsilon^0) \,.
\end{equation}
The general ansatz of the planar form factor can be given by the summation of the bubble integrals as:
\begin{equation}
\mathcal{I}_{\text{min}, n}^{(1), \text{deg-2}} = \sum_{i=1}^n c_{(i,i+1)} I_{\text{Bub}}^{(1)}(i, i+1) \,,
\end{equation}
where the external legs are circular as $I_{\text{Bub}}^{(1)}(n, n+1) = I_{\text{Bub}}^{(1)}(n, 1)$, and the similar convention is also adopted later.
The maximally transcendental part of the one-loop minimal form factor has universal IR divergences (which are determined by the maximally transcendental part of one-loop Sudakov form factor, see \emph{e.g.}~\cite{Bern:2005iz} for a review):
\begin{equation}
\mathcal{I}_{\text{min}, n}^{(1), \text{deg-2}} \big|_\text{div.} = - \sum_{i=1}^n \bigg( \frac{1}{\epsilon^2} - \frac{\log(-s_{i,i+1})}{\epsilon} \bigg) \,.
\end{equation}
Matching the expression of bubble integrals, we find that
\begin{equation}
c_{(i,i+1)} = -1 \,.
\end{equation}
Therefore the maximally transcendental part of the one-loop minimal form factor is fixed uniquely as
\begin{equation}
\label{eq:miniMT1loop}
\mathcal{I}_{\text{min}, n}^{(1), \text{deg-2}} = -\sum_{i=1}^n I_{\text{Bub}}^{(1)}(i, i+1) \, .
\end{equation}
\subsubsection{Two-loop case}
\label{sec:MTmini2loop}
Next, we consider the maximally transcendental part of the two-loop minimal form factor.
In this case, the interactions involve at most three external legs at one time.
The two-loop master integrals for minimal form factors are shown in Figure~\ref{fig:master-mini-FF}. They are explicitly defined in Appendix~\ref{app:UT}.
\begin{figure}[t]
\centering
\subfloat[Sun]{\includegraphics[scale=0.5]{figs/Sun.eps}}
\subfloat[dBub]{\includegraphics[scale=0.4]{figs/dBub.eps}}
\subfloat[TBub0]{\includegraphics[scale=0.4]{figs/TBub0.eps}}
\subfloat[TT2]{\includegraphics[scale=0.3]{figs/TT20.eps}}
\subfloat[TBub2]{\includegraphics[scale=0.4]{figs/TBub2.eps}}
\subfloat[BubBox]{\includegraphics[scale=0.4]{figs/BubBox.eps}}
\subfloat[NTBox1]{\includegraphics[scale=0.4]{figs/TBoxNP1.eps}}
\caption{Master integrals for the two-loop minimal form factors. \label{fig:master-mini-FF}}
\end{figure}
It is convenient to introduce the two-loop density function which involves up to four external legs $\{i,j,k,l\}$, and its most general form can be given as an expansion of master integrals:
\begin{align}
\mathcal{I}_{2}^{(2)}(i, j) = & c_1 I_{\text{Sun}}^{(2)}(i, j) + c_2 I_{\text{dBub}}^{(2)}(i, j; i, j) + c_3 I_{\text{TBub0}}^{(2)}(i, j) + c_4 I_{\text{NTBox1}}^{(2)}(i, j) \,, \nonumber\\
\mathcal{I}_{3}^{(2)}(i, j, k) = & c_5 I_{\text{Sun}}^{(2)}(i, j, k) + c_6 I_{\text{dBub}}^{(2)}(i, j; i, j, k) + c_7 I_{\text{TT2}}^{(2)}(i, j, k) + c_8 I_{\text{TBub2}}^{(2)}(i, j, k) \nonumber \\
&
+ c_9 I_{\text{BubBox}}^{(2)}(i, j, k) \,, \nonumber \\
\mathcal{I}_{4}^{(2)}(i, j, k, l) = & c_{10} I_{\text{dBub}}^{(2)}(i, j; k, l) \,,
\label{eq:MinAnsatz}
\end{align}
where $I_{\text{NTBox1}}^{(2)}(i,j)$ in $\mathcal{I}_{2}^{(2)}(i,j)$ are need to notice that they contribute to the leading-color part of Sudakov form factor($n=2$) although which are the non-planar master integrals. Without loss of generality, we take them into account, and which should be left in the case of higher points($n \geqslant 3$).
Then the two-loop ansatz for the planar minimal form factor can be given by the summation of the above functions as
\begin{equation}
\mathcal{I}_{\text{min}, n}^{(2), \text{deg-4}} = \sum_{i=1}^n \bigg( \mathcal{I}_{2}^{(2)}(i, i+1) + \mathcal{I}_{3}^{(2)}(i, i+1, i+2) + \sum_{j=i+2}^{n+1} \mathcal{I}_{4}^{(2)}(i, i+1, j, j+1) \bigg) \,.
\end{equation}
Since we will use IR divergences to constrain the results, we can organize the two-loop results in terms of two types of building-blocks as
\begin{equation}
\mathcal{I}_{\text{min}, n}^{(2), \text{deg-4}} = \sum_{\alpha} x_{\alpha} G_{\text{min}, \alpha}^{(2)} + \sum_{\beta} y_{\beta} \tilde{G}_{\text{min}, \beta}^{(2)} \,,
\end{equation}
and we will ask the first part $G_{\text{min},\alpha}^{(2)}$ to provide the correct IR divergences, and require the second part $\tilde{G}_{\text{min},\beta}^{(2)}$ are IR finite. Note $G_{\text{min},\alpha}^{(2)}$ and $\tilde{G}_{\text{min},\beta}^{(2)}$ are linear combination of master integrals.
\paragraph{Constructing building-blocks $G_{\text{min}, \alpha}^{(2)}$.}
We construct the building-blocks $G_{\text{min},\alpha}^{(2)}$ such that they provide the correct infrared divergences.
As reviewed in Section~\ref{sec:constraints}, the IR divergence can be captured by the BDS-ansatz function $\mathcal{I}_{\text{min}, n}^{(2), \text{BDS}}$, which is determined by the one-loop correction as
\begin{equation}
\label{eq:BDS2loopmim}
\mathcal{I}_{\text{min}, n}^{(2), \text{BDS}, \text{deg-4}} = \frac{1}{2}\left( \mathcal{I}_{\text{min}, n}^{(1), \text{deg-2}}(\epsilon) \right)^2 + f^{(2)}(\epsilon) \mathcal{I}_{\text{min}, n}^{(1), \text{deg-2}}(2\epsilon) \,.
\end{equation}
Note that we keep only the maximally transcendental part of degree $4$.
As reviewed in Appendix~\ref{app:catani}, the BDS ansatz and the Catani IR formula have the same maximally transcendental IR divergences, therefore the discussion with BDS ansatz here will not only apply to $\mathcal{N}=4$ SYM but also to general gauge theories.
Using the one-loop result \eqref{eq:miniMT1loop}, one can expand \eqref{eq:BDS2loopmim} as
\begin{align}
\label{eq:nptInfraredStructure}
\mathcal{I}_{\text{min}, n}^{(2), \text{BDS}, \text{deg-4}}
= & \sum_{i = 1}^{n} \left( I_\text{Bub}^{(1)}(i, i+1)I_\text{Bub}^{(1)}(i+1, i+2) - f^{(2)}(\epsilon) I_\text{Bub}^{(1)}(i, i+1; 2\epsilon) \right) \nonumber \\
& + \frac{1}{2} \sum_{j \neq i-1, i, i+1} \left( I_\text{Bub}^{(1)}(i, i+1)^2 + I_\text{Bub}^{(1)}(i, i+1) I_\text{Bub}^{(1)}(j, j+1) \right) \,.
\end{align}
We can divide the BDS-ansatz function into three parts, and introduce three $G_{\text{min}, \alpha}^{(2)}$ functions to capture their divergences as
\begin{align}
G_{\text{min}, 1}^{(2)}(i, i+1, i+2) \big|_{\text{div.}} &= I_{\text{Bub}}^{(1)}(i, i+1) I_{\text{Bub}}^{(1)}(i+1, i+2) \big|_{\text{div.}} \, , \nonumber\\
G_{\text{min},2}^{(2)}(i, i+1) \big|_{\text{div.}} &= f^{(2)}(\epsilon) I_{\text{Bub}}^{(1)}(i, i+1; 2\epsilon) \big|_{\text{div.}} \, , \nonumber \\
G_{\text{min},3}^{(2)}(i, j) &= I_\text{dBub}^{(2)}(i, i+1; j, j+1) \,,
\label{eq:MinFFIR}
\end{align}
where $ I_\text{dBub}^{(2)}(i, i+1; j, j+1) = I_\text{Bub}^{(1)}(i, i+1) I_\text{Bub}^{(1)}(j, j+1)$ with $j \neq i-1$ or $i+1$.
The choice of $G_{\text{min}, 1}^{(2)}$ and $G_{\text{min}, 2}^{(2)}$ is not unique. To pick up a simple solution, we require that:
(1) $G_{\text{min}, 1}^{(2)}(i, j, k)$ have flip symmetry by exchanging external legs $p_i$ and $p_k$;
(2) $G_{\text{min},2}^{(2)}(i, j)$ only contain the master integrals with two external legs $p_i$ and $p_j$;
and (3) neither $G_{\text{min}, 1}^{(2)}(i, j, k)$ nor $G_{\text{min}, 2}^{(2)}(i, j)$ involve the non-planar master integrals $I_{\text{NTBox1}}^{(2)}$.
We can then obtain the following unique combinations
\begin{align}
G_{\text{min}, 1}^{(2)}(i, j, k) = & 4 I_\text{Sun}^{(2)}(i, j)+\frac{1}{2} I_{\text{TBub0}}^{(2)}(i, j) + \frac{1}{4} I_\text{dBub}^{(2)}(i, j; i, j) \nonumber \\
& +I_\text{BubBox}^{(2)}(i, j, k) - \frac{1}{2} I_{\text{TT2}}^{(2)}(i, j, k) + \left( p_i \leftrightarrow p_k \right) \,, \nonumber \\
G_{\text{min}, 2}^{(2)}(i, j) = & - \left( \frac{1}{2} I_{\text{TBub0}}^{(2)}(i, j)+I_\text{Sun}^{(2)}(i, j) \right) \,.
\end{align}
\paragraph{Constructing building-blocks $\tilde{G}_{\text{min}, \beta}^{(2)}$.}
Next, we consider all possible building-blocks $\tilde{G}_{\text{min}, \beta}^{(2)}$ which are infrared finite.
By considering all possible combinations of master integrals, it is not hard to find that there are two independent blocks of $\tilde{G}_{\text{min}, \beta}^{(2), \text{deg-4}}$ which can be given as
\begin{align}
\label{eq:tildeGmindeg4}
\tilde{G}_{\text{min}, 1}^{(2), \text{deg-4}}(i, j) = & \frac{1}{2} I_{\text{NTBox1}}^{(2)}(i, j) + I_{\text{dBub}}^{(2)}(i, j; i, j) + 11 I_{\text{Sun}}^{(2)}(i, j) + \frac{5}{2} I_{\text{TBub0}}^{(2)}(i, j) \nonumber \\
= & -\zeta_4 + \mathcal{O}(\epsilon) \,, \\
\tilde{G}_{\text{min}, 2}^{(2), \text{deg-4}}(i, j, k) = & I_{\text{dBub}}^{(2)}(i, j; i, j) +4 I_{\text{Sun}}^{(2)}(i, j) - (i \leftrightarrow k) = -12 \zeta_3 \log\left( \frac{s_{ij}}{s_{jk}} \right) + \mathcal{O}(\epsilon) \nonumber \,.
\end{align}
It is worthwhile noticing that all the master integrals occurred in \eqref{eq:tildeGmindeg4} are range-two integrals that involve only two external legs.
There are some considerations when we define the above two functions:
\begin{itemize}
\item[1)] $\tilde{G}_{\text{min}, 1}^{(2), \text{deg-4}}$ contains the non-planar integrals $I_{\text{NTBox1}}^{(2)}(i, j)$. For planar minimal form factor with $n \geqslant 3$, $I_{\text{NTBox1}}^{(2)}(i, j)$ will not contribute, thus the block $\tilde{G}_{\text{min}, 1}^{(2), \text{deg-4}}$ will not occur. In particular, the color factor of the non-planar topologies will vanish for the $n=3$ case with external particles in adjoint representation (such as three-gluon).
\item[2)] $\tilde{G}_{\text{min}, 2}^{(2), \text{deg-4}}(i, j, k)$ is anti-symmetric by exchanging the external momenta $p_i$ and $p_k$. Such building blocks will vanish if their coefficients are free from kinematics invariants and the form factor has the symmetry of flipping $p_i\leftrightarrow p_k$ or cyclicly permuting external momentum, the later will then be explained.
\end{itemize}
\paragraph{Full form factor.}
Now we can give the form factor results in terms of the above two types of building blocks.
For general $n \geqslant 3$ minimal form factors, the infrared structure \eqref{eq:nptInfraredStructure} require the following form
\begin{align}
\label{eq:Min3ptFF}
\mathcal{I}_{\text{min}, n}^{(2), \text{deg-4}} = & \sum_{j \neq i-1, i, i+1} \frac{1}{2} \left( G_{\text{min}, 3}^{(2)}(i, i) + G_{\text{min}, 3}^{(2)}(i, j) \right)
+ \sum_{i=1}^n \left( G_{\text{min}, 1}^{(2)}(i, i+1, i+2) - G_{\text{min}, 2}^{(2)}(i, i+1) \right) \nonumber \\
& + \sum_{i=1}^{n} \left( y_{1,i} \tilde{G}_{\text{min}, 1}^{(2), \text{deg-4}}(i, i+1) +y_{2,i} \tilde{G}_{\text{min}, 2}^{(2), \text{deg-4}}(i, i+1, i+2) \right) \,,
\end{align}
where $y_{\beta,i}$ are parameters which can be fixed to be $0$ for the following cases. For planar minimal form factors, as mentioned $\tilde{G}_{\text{min}, 1}^{(2), \text{deg-4}}$ cannot occur since they contain non-planar master integrals, thus $y_{1,i}$ are zero. The maximally transcendental part of the minimal form factor has constant coefficients, moreover the form factors of operators $\mathrm{tr}(\phi^n)$ and $\mathrm{tr}((F_{\mu\nu})^n)$ have the symmetry of cycling external particles, which require all $y_{2,i}$ must be the same, in such case the summation of $\tilde{G}_{\text{min}, 2}(i, i+1, i+2)$ will vanish, thus one has also $y_{2,i}=0$. Actually, with some simple argument of unitarity cuts, one can show that this is also true for general minimal form factors; we will discuss this more in Section~\ref{sec:univer-mimi}.
For the $n=2$ Sudakov form factor, the master integrals with three external particles do not contribute, thus only building-blocks $G_{\text{min}, 2}^{(2), \text{deg-4}}(1,2)$, $G_{\text{min}, 3}^{(2), \text{deg-4}}(1,2)$, and $\tilde{G}_{\text{min}, 1}^{(2), \text{deg-4}}(1, 2)$
can appear, the last one contains the non-planar master integrals $I_{\text{NTBox1}}^{(2)}(i, j)$ which are necessary. By matching the infrared structure \eqref{eq:nptInfraredStructure} at $n=2$:
\begin{equation}
\label{eq:2ptInfraredStructure}
\mathcal{I}_{\text{min}, 2}^{(2), \text{BDS}, \text{deg-4}} = 2 \left( I_\text{Bub}^{(1)}(i, i+1)^2 - f^{(2)}(\epsilon) I_\text{Bub}^{(1)}(i, i+1; 2\epsilon) \right) \,,
\end{equation}
we obtain
\begin{align}
\mathcal{I}_{\text{min}, 2}^{(2), \text{deg-4}} = & 2 G_{\text{min}, 3}^{(2)}(1, 2) - 2 G_{\text{min}, 2}^{(2)}(1, 2) + y_0 \tilde{G}_{\text{min}, 2}^{(2), \text{deg-4}}(1, 2) \\
= & 2 ( 1+y_0 ) I_\text{dBub}^{(2)}(1, 2; 1, 2) + \left( 1 +\frac{5y_0}{2} \right) I_{\text{TBub0}}^{(2)}(1, 2) \nonumber \\
& + (2+11y_0) I_\text{Sun}^{(2)}(1, 2) + \frac{y_0}{2} I_\text{NTBox1}^{(2)}(1, 2) \,, \nonumber
\end{align}
with the single unfixed parameter $y_0$, which in principle can be fixed by a unitarity cut. Later in Section~\ref{sec:3ptFF}, we will show the parameter $y_0$ can also be fixed by considering the collinear and the $q^2\rightarrow0$ limits of two-loop three-point form factor, which gives $y_0 = 2$.
\paragraph{Remainder functions.}
The above discussion is about the coefficients of the master integrals and provides the maximally transcendental part to all orders in $\epsilon$. Below we focus on the finite order and consider the funite remainder function $\mathcal{R}_{\text{min}, n}^{(2), \text{deg-4}}$, which can be compared with the known results in literature.
We first compute the building blocks that compose the remainder:
\begin{align}
G_{\text{min}, 1}^{(2)}(i, j,k) - I_{\text{Bub}}^{(1)}(i, j) I_{\text{Bub}}^{(1)}(j, k) = & T_4\left( \frac{s_{ij}}{s_{ijk}}, \frac{s_{jk}}{s_{ijk}}, \frac{s_{ik}}{s_{ijk}} \right) + \left( i \leftrightarrow k \right) + \mathcal{O}(\epsilon) \,, \nonumber \\
G_{\text{min}, 2}^{(2)}(i, j) - f^{(2)}(\epsilon) I_{\text{Bub}}^{(1)}(i, j; 2\epsilon) = & -3 \zeta_4 + \mathcal{O}(\epsilon) \,,
\end{align}
where $T_4(u,v,w)$ is defined as
\begin{align}
T_4(u, v, w) = & G(1-u,1-u,1,0,w) \\
& -\mathrm{Li}_4(u) -\mathrm{Li}_4(1-u)+\mathrm{Li}_4\left(\frac{u-1}{u}\right) -\mathrm{Li}_2\left(\frac{u-1}{u}\right) \mathrm{Li}_2\left(\frac{w}{1-u}\right) \nonumber \\
& -\log (u) \left(\mathrm{Li}_3\left(\frac{w}{1-u}\right)+\mathrm{Li}_3\left(-\frac{v}{w}\right)+\mathrm{Li}_3\left(\frac{w-1}{w}\right) \right) \nonumber \\
& -\left(\mathrm{Li}_3\left(\frac{u-1}{u}\right)-\mathrm{Li}_3(1-u)\right) \log \left(\frac{1-u}{v}\right) \nonumber \\
& +\mathrm{Li}_2(u) \left(\log (w) \log \left(\frac{1-u}{v}\right)+\frac{1}{2} \log ^2\left(\frac{1-u}{v}\right)\right) \nonumber \\
& -\log (u) \left( \frac{\log ^2(1-u) \log \left(\frac{v}{w}\right)}{2} -\frac{\log ^3(1-u)}{3} -\frac{\log ^3(w)}{3} \right) \nonumber \\
& -\log^2(u) \left( \frac{\log (u) \log (v)}{6} + \frac{\log ^2(w)}{8} + \frac{\log (1-u) \log (w)}{2} - \frac{\log ^2(u)}{24} \right) \nonumber \\
& - \zeta_2 \left(\frac{1}{2} \log ^2\left(\frac{1-u}{v}\right)+\log (u) \log \left(\frac{1-w}{w}\right)-\frac{1}{2} \log ^2(u)\right) +
\zeta_3 \log (u)-\zeta_4 \, . \nonumber
\end{align}
Therefore, we can define the density function $\mathcal{R}_{3}^{(2), \text{deg-4}}(i,j,k)$ as
\begin{equation}
\mathcal{R}_{3}^{(2), \text{deg-4}}(1,2,3) = T_4(u, v, w) + T_4(w, v, u) + 3 \zeta_4 \,.
\end{equation}
The result is consistent with the known results \cite{Brandhuber:2014ica, Lin:2020dyj}.
The planar minimal form factors with $\mathrm{tr}(\phi^n)$ are
\begin{equation}
\mathcal{R}_{\text{min}, n}^{(2), \text{deg-4}} = \sum_{i=1}^{n} \mathcal{R}_{3}^{(2), \text{deg-4}}(i,i+1,i+2) \, .
\end{equation}
Finally, we consider the form factor results for the special $n=2, 3$ cases. The $n=2$ Sudakov form factor is \cite{vanNeerven:1985ja}
\begin{equation}
\mathcal{R}_{\text{min}, 2}^{(2), \text{deg-4}} = 6 \zeta_4 - y_0 \zeta_4 = 4 \zeta_4 \,,
\end{equation}
where $y_0=2$ is used.
The $n=3$ minimal form factor of $\mathrm{tr}(\phi^3)$ has a compact form as
\begin{align}
\mathcal{R}_{\text{min}, 3}^{(2), \text{deg-4}} = & {\cal R}_{3}^{(2), \text{deg-4}}(1,2,3)+{\cal R}_{3}^{(2), \text{deg-4}}(2,3,1)+{\cal R}_{3}^{(2), \text{deg-4}}(3,1,2) \\
= & \frac{3}{4} \text{Li}_4\left(-\frac{u v}{w}\right)-\frac{3}{2} \log (w) \text{Li}_3\left(-\frac{u}{v}\right)-\frac{3}{2} \text{Li}_4(u) \nonumber \\
& + \frac{1}{32} \log ^2(u) \left( \log ^2(u)+\log ^2(v)+\log ^2(w)-4 \log (v) \log (w) \right) \nonumber \\
& +\frac{1}{8} \zeta_2 \left( 5 \log ^2(u) - 2 \log (v) \log (w) \right)+\frac{1}{2} \zeta_3 \log (u)+\frac{7}{16} \zeta_4 + \left( \text{full perm.}(u,v,w) \right) \,. \nonumber
\end{align}
\subsection{Lower transcendentality parts}
\label{sec:LTminimal}
The above discussion can be generalized to lower transcendentality parts.
We will show that a similar procedure will explain the universal building blocks of transcendentality degree-$3$ and degree-$2$ functions for QCD minimal form factors observed in \cite{Jin:2019opr}.
We will focus on the two-loop case. To apply the infrared structure, we consider again the BDS function $\mathcal{I}_{\text{min}, n}^{(2), \text{BDS}}$:\footnote{Here we use BDS ansatz for the $\mathcal{N}=4$ case. In the case of QCD, one can use Catani subtraction, as explained in Appendix~\ref{app:catani}.}
\begin{equation}
\mathcal{I}_{\text{min}, n}^{(2), \text{BDS}, \text{deg-a}} =\sum_{b=a-2}^{2} \frac{\mathcal{I}_{\text{min}, n}^{(1), b}(\epsilon) \mathcal{I}_{\text{min}, n}^{(1), \text{deg-(a-b)}}(\epsilon)}{2} + f^{(2)}(\epsilon) \mathcal{I}_{\text{min}, n}^{(1), \text{deg-a}}(2\epsilon) \,,
\end{equation}
where full $\mathcal{I}_{\text{min}, n}^{(2), \text{BDS}}$ is decomposed similarly as \eqref{eq:minidegCalF} as
\begin{equation}
\mathcal{I}_{\text{min}, n}^{(2), \text{BDS}} = \sum_{a=2L}^{-\infty} \epsilon^{2L-a} \mathcal{I}_{\text{min}, n}^{(2), \text{BDS}, \text{deg-a}} \,.
\end{equation}
We also assume the lower transcendentality part of one-loop form factor are known as
\begin{equation}
\mathcal{I}_{\text{min}, n}^{(1), \text{deg-a}} = \sum_{i = 1}^{n} c_{a,i}^{(1)} I_{\text{Bub}}^{(1)}(i, i + 1) + \frac{1}{\epsilon} Z_i^{(1)} \delta_{a,1} \,,
\end{equation}
where $Z_i^{(1)}$ represents the possible UV renormalization constant which contributes only in degree-$1$ part.
One can see that the BDS function $\mathcal{I}_{\text{min}, n}^{(2), \text{BDS}, \text{deg-a}}$ consists of bubbles integrals similar to \eqref{eq:nptInfraredStructure}. Therefore, the degree-$a$ part of the form factor can be expressed as
\begin{align}
\label{eq:nptLowerTransFF}
\mathcal{I}_{\text{min}, n}^{(2), \text{deg-a}} = & \sum_{j \neq i-1, i, i+1} \frac{1}{2} \left[ \left(c_{ai}^{(1)} \right)^2 G_{\text{min}, 3}^{(2)}(i, i) + c_{ai}^{(1)} c_{aj}^{(1)} G_{\text{min}, 3}^{(2)}(i, j) \right] \nonumber \\
& + \sum_{i=1}^n \left( c_{ai}^{(1)} c_{a,i+1}^{(1)} G_{\text{min}, 1}^{(2)}(i, i+1, i+2) + c_{ai}^{(1)} G_{\text{min}, 2}^{(2)}(i, i+1) \right)
+ (\textrm{UV-part}) \nonumber \\
& + \sum_{\beta} y_{\beta} \tilde{G}_{\text{min}, \beta}^{(2), \text{deg-a}} \,,
\end{align}
where the first two lines capture the full divergences and the UV-part contains terms depending on $Z_i^{(1)}$, we will not discuss the latter in detail here.
The remaining problem is to construct the concrete form of the building-blocks $\tilde{G}_{\text{min}, \beta}^{(2), \text{deg-a}}$, which can be determined by requiring that $\tilde{G}_{\text{min}, \beta}^{(2), \text{deg-a}}$ starting at the order of $\mathcal{O}(\epsilon^{a-2L})$ because of the factor $\epsilon^{2L-a}$.
Additionally, the first and second summations in LHS of \eqref{eq:nptLowerTransFF} will not contribute to the finite remainder. These terms in the complete form factor result will be multiplied by a factor $\epsilon^{2L-a}$ where $a<4$, thus they will only contribute to the $\mathcal{O}(\epsilon)$ part of the remainder.
Thus the lower transcendental part of the finite remainder will only depend on the $\tilde{G}_{\text{min}, \beta}^{(2), \text{deg-a}}$ part and the UV part, the latter is trivially determined by the one-loop result. In other words, only some special function blocks will appear in the remainder, we will give them as follows.
For transcendentality degree-$3$, there are
\begin{align}
\tilde{G}_{\text{min},1}^{(2), \text{deg-3}}(i,j) = & \tilde{G}_{\text{min},1}^{(2), \text{deg-4}}(i,j) = \mathcal{O}(\epsilon^0) \,, \nonumber\\
\tilde{G}_{\text{min},2}^{(2), \text{deg-3}}(i,j) = & I_{\text{dBub}}^{(2)}(i, j; i, j) +4 I_{\text{Sun}}^{(2)}(i,j) = \frac{6\zeta_3}{\epsilon} + \mathcal{O}(\epsilon^0) \,, \nonumber \\
\tilde{G}_{\text{min},3}^{(2), \text{deg-3}}(i,j,k) = & I_{\text{Sun}}^{(2)}(i,j) +\frac{1}{2} I_{\text{TBub}0}^{(2)}(i,j) - ( i \leftrightarrow k) = -\frac{ \zeta_2 }{\epsilon} \log \left( \frac{s_{ij}}{s_{jk}} \right) + \mathcal{O}(\epsilon^0) \,, \nonumber \\
\tilde{G}_{\text{min},4}^{(2), \text{deg-3}}(i,j,k) = & I_{\text{TBub}2}^{(2)}(i,j,k) + I_{\text{TT}2}^{(2)}(i,j,k) - I_{\text{BubBox}}^{(2)}(k,j,i) - I_{\text{Sun}}^{(2)}(i,j) \nonumber \\
= & \frac{1}{\epsilon} T_3\left( \frac{s_{ij}}{s_{ijk}}, \frac{s_{jk}}{s_{ijk}}, \frac{s_{ik}}{s_{ijk}} \right) + \mathcal{O}(\epsilon^0) \,,
\label{eq:buidlingblock1}
\end{align}
where $T_3(u, v, w)$ occurs in \cite{Jin:2019opr},
\begin{align}
T_3(u, v, w) = & \mathrm{Li}_3\left(1-v\right) - \mathrm{Li}_3\left(u\right) + \frac{1}{2} \log \left(\frac{1-v}{u}\right) \log ^2\left(v\right) - \zeta_2 \log \left(\frac{u v}{w}\right) + \frac{1}{6} \log ^3 \left(w\right) \nonumber \\
&+ \left( -\mathrm{Li}_3\left(-\frac{u}{w}\right) + \frac{1}{2}\mathrm{Li}_3\left(-\frac{u v}{w}\right) + \log \left(u\right) \mathrm{Li}_2\left(\frac{v}{1-u}\right) \right. \nonumber \\
& \left. + \frac{1}{2} \log \left(u\right) \log \left(\frac{w}{1-u}\right) \log \left(\frac{v}{1-u}\right) + (u \leftrightarrow v) \right) \,.
\label{eq:T3}
\end{align}
For transcendentality degree-$2$, there are
\begin{align}
\tilde{G}_{\text{min}, 1}^{(2), \text{deg-2}}(i, j) = & \tilde{G}_{\text{min},1}^{(2), \text{deg-4}}(i, j) = \mathcal{O}(\epsilon^{-1}) \,, \nonumber\\
\tilde{G}_{\text{min}, 2}^{(2), \text{deg-2}}(i, j) = & \tilde{G}_{\text{min}, 1}^{(2), \text{deg-3}}(i, j) = \mathcal{O}(\epsilon^{-1}) \,, \nonumber \\
\tilde{G}_{\text{min}, 3}^{(2), \text{deg-4}}(i, j) = & \tilde{G}_{\text{min}, 3}^{(2), \text{deg-3}}(i, j, k) = \mathcal{O}(\epsilon^{-1}) \,, \nonumber \\
\tilde{G}_{\text{min}, 4}^{(2), \text{deg-2}}(i,j) = & 2I_{\text{Sun}}^{(2)}(i,j) + I_{\text{TBub0}}^{(2)}(i,j) = \frac{\zeta_2}{\epsilon^2} + \mathcal{O}(\epsilon^{-1}) \, , \nonumber \\
\tilde{G}_{\text{min}, 5}^{(2), \text{deg-2}}(i, j, k) = & 2 I_{\text{Sun}}^{(2)}(i, j) + I_{\text{TBub2}}^{(2)}(k, j, i) = -\frac{ 1 }{\epsilon^2} T'_2 \left( \frac{s_{ij}}{s_{ijk}} \right) + \mathcal{O}(\epsilon^{-1}) \,, \nonumber \\
\tilde{G}_{\text{min}, 6}^{(2), \text{deg-2}}(i, j, k) = & I_{\text{TT2}}^{(2)}(i, j, k) = \frac{1}{\epsilon^2} T_2 \left( \frac{s_{ij}}{s_{ijk}}, \frac{s_{jk}}{s_{ijk}} \right) + \mathcal{O}(\epsilon^{-1}) \,.
\label{eq:buidlingblock2}
\end{align}
where $T'_2(u)$ and $T_2(u, v)$ occur in reference~\cite{Jin:2019opr},
\begin{align}
T'_2(u) = & \mathrm{Li}_2(1-u) + \frac{1}{2}\log^2(u) \, , \nonumber\\
T_2(u, v) = & \mathrm{Li}_2(1-u) + \mathrm{Li}_2(1-v) + \log(u) \log(v) -\zeta_2 \, .
\label{eq:T2}
\end{align}
We will not discuss the building blocks for transcendentality degree-$1$ and degree-$0$ because there are too many possibilities of building blocks for these parts; however, they consist only of $\log(-s_{ij})$ and rational functions that depend on kinematics.
\subsection{Universal transcendentality structures}
\label{sec:univer-mimi}
In previous subsections, we impose the constraints that form factors should have the correct universal IR divergences.
We will show that the remaining degrees of freedom can be classified by functions that are free from IR divergences, such as $\tilde{G}_{\text{min}, \beta}^{(2)}$ in \eqref{eq:tildeGmindeg4}.
In this subsection, we will further determine these remaining degrees of freedom by applying unitarity cuts.
We will show that the minimal form factors in $\mathcal{N}=4$ SYM and QCD theories will have the same maximally transcendental parts,
and moreover, they also contain universal lower transcendentality building blocks up to simple logarithm functions and Riemann zeta numbers.\footnote{Our discussion for the lower transcendental part is for the bare form factors. The UV renormalization will only modify the log and rational functions up to two-loop order.}
\paragraph{One-loop case.}
Let us consider first the one-loop case.
We show in Section~\ref{sec:MTmini1loop} that by matching the universal IR divergence, it is enough to fix the maximally transcendental part.
In this way, we arrive at the conclusion that maximally transcendental one-loop corrections $\mathcal{I}_{\text{min}, n}^{(1), \text{deg-2}}$ are universal in different theories and for different operators (up to a simple change of color factors, see discussion below).
We stress again that this is due to the fact that the maximally transcendent part of one-loop IR divergence is the same for general gauge theories.
\begin{figure}[t]
\centering
\includegraphics[scale=0.5]{figs/samecutmin1.eps}
\caption{Unitarity cut for one-loop minimal form factors in gauge theories.}
\label{fig:samecutsmin1loop}
\end{figure}
Now we will use a different argument based on unitarity cuts which will not only constrain the maximally transcendental but also lower transcendental parts.
Considering the double cut in Figure~\ref{fig:samecutsmin1loop}, which is enough to determine the full coefficient of $I_{\text{Bub}}^{(1)}(i,j)$. The cut integrand is
\begin{equation}
\label{eq:min1loopcut}
\mathcal{F}_{\mathcal{O}_{n}, n}^{(0)}(..,-l_1,-l_2,..) \mathcal{A}_{4}^{(0)}(l_2,l_1,i,j) \,,
\end{equation}
where the tree-level form factor $\mathcal{F}_{\mathcal{O}_{n}, n}^{(0)}(..,-l_1,-l_2,..)$ are non-zero only if the types of internal particles $l_1$ and $l_2$ are the same as external particles $i$ and $j$, namely, it must be a minimal form factor.
To be more concrete, we consider minimal form factors in $\mathcal{N}=4$ SYM and QCD with operators that contain only gluon and quark (or gluino) fields. Then it should be clear that the kinematic parts of the cut integrands \eqref{eq:min1loopcut} are the same for $\mathcal{N}=4$ SYM and QCD, and the difference only appears in the color factors that involve fermions: since the fermion is in adjoint representation in $\mathcal{N}=4$ SYM while being fundamental in QCD.
In this case, one can identify the two results if one converts the quark representation in QCD results from the fundamental to the adjoint.
For the one-loop correction function $\mathcal{I}_{\text{min}, n}^{(1)}$, this can be achieved by a simple replacement for the quadratic Casimir as $C_F \rightarrow C_A$.
\begin{figure}[t]
\centering
\subfloat[]{\includegraphics[scale=0.5]{figs/samecutmin2.eps}}
\subfloat[]{\includegraphics[scale=0.5]{figs/samecutmin3.eps}}
\caption{Unitarity cuts which are same for two-loop minimal form factors in gauge theories. }
\label{fig:samecutsmin}
\end{figure}
\paragraph{Two-loop case.}
In previous two subsections, we find that after the IR constraints, the remaining degrees of freedom for the two-loop minimal form factors are
related to functions $\tilde{G}_{\text{min}, \beta}^{(2), \text{deg-a}}$ in \eqref{eq:tildeGmindeg4}, \eqref{eq:buidlingblock1}, and \eqref{eq:buidlingblock2}.
For the maximally transcendental part, since the blocks $\tilde{G}_{\text{min}, 1}^{(2), \text{deg-4}}$ contain non-planar integrals, they will not contribute if we focus on the planar form factors.
We have also argued that if the form factor has cyclic symmetry, the terms $\tilde{G}_{\text{min}, 2}^{(2), \text{deg-4}}$ also vanish.
Below we would like to apply unitarity cuts to prove the universality of these functions in general gauge theories, including also the lower transcendental parts.
We first consider the maximally transcendental parts and focus on the planar form factor for simplicity.
To fix the coefficients of $\tilde{G}_{\text{min}, 2}^{(2), \text{deg-4}}$, one can consider the double-cut (a) in Figure~\ref{fig:samecutsmin} which can determine the coefficients of $I_{\text{dBub}}^{(2)}(i, j; i, j)$.\footnote{The other block $\tilde{G}_{\text{min}, 1}^{(2), \text{deg-4}}$ contains also $I_{\text{dBub}}^{(2)}(i, j; i, j)$. By doing the full-color double-cut Figure~\ref{fig:samecutsmin}(a), one can also fix the non-planar corrections. Our following arguments also apply to such cases.}
The cut integrand is
\begin{align}
& \mathcal{F}_{\mathcal{O}_{n}, n}^{(1)}(\ldots,-l_1,-l_2,\ldots) \mathcal{A}_{4}^{(0)}(l_2,l_1,i,j) \\
= & \mathcal{I}_{\text{min}, n}^{(1), \text{deg-2}}(\ldots,-l_1,-l_2,\ldots) \times \left[ \mathcal{F}_{\mathcal{O}_{n}, {\rm min}}^{(0)}(\ldots,-l_1,-l_2,\ldots) \mathcal{A}_{4}^{(0)}(l_2,l_1,i,j) \right] + (\textrm{lower trans.}) \,.\nonumber
\end{align}
We stress that only the first term in the second line contributes to maximally transcendental part, in which we have used the universality of one-loop results. In particular, the universal one-loop correction function $\mathcal{I}_{\text{min}, n}^{(1), \text{deg-2}}(\ldots,-l_1,-l_2,\ldots)$ contributes only if the internal particles $l_1$ and $l_2$ are the same types from external particles $i$ and $j$. In addition, the terms in the square brackets provide the same one-loop cut integrand \eqref{eq:min1loopcut}.
These show that the coefficients of double-bubble masters $I_{\text{dBub}}^{(2)}(i, j; i, j)$ are the same for general operators in general gauge theories.
Since we know that $\tilde{G}_{\text{min}, 2}^{(2), \text{deg-4}}$ blocks are zero for form factors with cyclic symmetry, they also vanish in form factors of general operators.
For two-loop minimal form factors in $\mathcal{N}=4$ SYM and QCD, their maximally transcendental parts are equivalent by a similar change of color factors for fermions, as discussed in the one-loop case.
Next we use unitarity-cut argument to derive constraints on the lower transcendental parts.
Consider the triple-cut (b) in Figure~\ref{fig:samecutsmin} with the cut integrand given as
\begin{equation}
\label{eq:min2looptriplecut}
\mathcal{F}_{\mathcal{O}_{n}, \text{min}}^{(0)}(\ldots, -l_1, -l_2, -l_3, \ldots) \mathcal{A}_{6}^{(0)}(l_3,l_2,l_1,i,j,k) \,,
\end{equation}
which are able to detect all range-3 master integrals $I_{\text{Sun}}^{(2)}(i,j,k)$, $I_{\text{BubBox}}^{(2)}(i,j,k)$, $I_{\text{TBub2}}^{(2)}(i,j,k)$, and $I_{\text{TT2}}^{(2)}(i,j,k)$. Since the tree form factor in the cut must be the minimal form factor (otherwise it is zero), the internal particles for the cut legs $\{l_1, l_2, l_3\}$ are the same types as external particles $\{i, j, k\}$. Similar to the one-loop case, this implies that this cut computation is the same also for $\mathcal{N}=4$ SYM and QCD,\footnote{Strictly speaking, the lower transcendentality part of this cut can be different in ${\cal N}=4$ SYM and QCD because of different regularization-scheme choices. But it can be shown that $T_3$, $T_2$, and $T'_2$ terms are not affected by choices of schemes.} if we change the color factors for fermions accordingly like $C_F \rightarrow C_A$. Thus the coefficients of all range-3 master integrals in a bare form factor will be the same for the two theories,
and the results only differ by the range-2 master integrals, which depend only on logarithm functions and Riemann zeta numbers. In particular, in the finite remainder, the function blocks $T_3(u,v,w)$, $T_2(u,v)$ and $T'_2(u)$ (\eqref{eq:T3} and \eqref{eq:T2}) for the lower transcendental parts must be the same for $\mathcal{N}=4$ SYM and QCD, as observed in \cite{Jin:2019ile, Jin:2019opr}.
\section{Two-loop three-point form factor of $\mathrm{tr}(F^2)$}
\label{sec:3ptFF}
In this section, we consider three-point form factor of $O_2 = \mathrm{tr}(F^2)$:
\begin{equation}
\mathcal{F}_{\mathcal{O}_2, 3} = \int d^D x e^{-i q\cdot x} \langle p_1 \, p_2 \, p_3 | \mathcal{O}_2(x)|0\rangle \,.
\end{equation}
This is a next-to-minimal form factor, and whose collinear limits will provide new important constraints. Moreover, this form factor is equivalent to the Higgs-plus-three-parton amplitudes in the heavy top mass limit by integrating out the heavy top quark \cite{Wilczek:1977zn, Shifman:1979eb, Dawson:1990zj, Djouadi:1991tka, Kniehl:1995tn, Chetyrkin:1997sg, Chetyrkin:1997un}. The full two-loop QCD corrections were obtained in \cite{Gehrmann:2011aa}, and the result in $\mathcal{N}=4$ SYM was obtained in \cite{Brandhuber:2012vm}.
It turns out that the maximally transcendental parts of the results in QCD and ${\cal N}=4$ SYM satisfy \cite{Brandhuber:2012vm, Jin:2019ile}:
\begin{equation}
\label{eq:MTP3pt}
\mathcal{F}^{(L), \mathcal{N}=4}_{\mathcal{L}\sim{\mathrm{tr}(F^2)}}(1,2,3) = \mathcal{F}^{(L), \text{QCD}}_{\mathrm{tr}(F^2), \text{M.T.}}(1^g,2^g,3^g)
= \mathcal{F}^{(L), \text{QCD}}_{\mathrm{tr}(F^2), \text{M.T.}}(1^q, 2^{\bar{q}}, 3^g) \Big|_{C_F \rightarrow C_A} ,
\end{equation}
for $L=1,2$.
One main goal of this section is to provide a proof for the relation \eqref{eq:MTP3pt}.
Here in ${\cal N}=4$ SYM theory, ${\cal L}$ is the chiral Lagrangian which contains ${\rm tr}(F^2)$ as a component, see \emph{e.g.}~\cite{Eden:2011yp, Brandhuber:2011tv}.\footnote{The chiral Lagrangian belongs also to the larger stress-tensor supermultiplet which is half-BPS. We mention that the same maximally transcendental function was also found in the two-loop three-point form factor of the non-BPS Konishi operator in $\mathcal{N}=4$ SYM \cite{Banerjee:2016kri}.} For our discussion of the maximally transcendental part, it is enough to focus on ${\rm tr}(F^2)$ since other components in the supermultiplet have only contribution of lower transcendentality.
Since the field strength operator can be decomposed as self-dual and anti-self-dual parts as
\begin{equation}
{\rm tr}(F_{\mu\nu}F^{\mu\nu}) = {1\over2} \big[ {\rm tr}(F_{\alpha\beta}F^{\alpha\beta}) + {\rm tr}({\bar F}_{\dot\alpha \dot\beta} {\bar F}^{\dot\alpha \dot\beta}) \big] \,,
\end{equation}
for simplicity (and without loss of generality), in the following discussion we will take the operator as the self-dual part ${\cal O}_2 = {\rm tr}(F_{\alpha\beta}F^{\alpha\beta})$. The minimal two-point tree-level form factors is ${\cal F}_{{\cal O}_2}^{(0)}(1^-,2^-) = \langle 12 \rangle^2$.
As a brief outline,
in Section~\ref{sec:ansatzN=4}, we will show that the constraints from IR together with collinear limits can fix the form factor in $\mathcal{N}=4$ up to the two-loop order. In Section~\ref{sec:MTP3pt}, we consider further the form factors in QCD, and together with the use of unitarity cuts for the $\mathcal{F}^{(l), \text{QCD}}_{\mathrm{tr}(F^2)}(1^q,2^{\bar{q}},3^g)$ case, then the relations \eqref{eq:MTP3pt} can be proven.
\subsection{Bootstrapping the $\mathcal{N}=4$ form factor}
\label{sec:ansatzN=4}
As a warm-up, we consider the form factors of stress-tensor supermultiplet in $\mathcal{N}=4$ super-Yang-Mills theory, which are uniformly transcendental with weight $2L$ at $L$ loops. The loop correction $\mathcal{I}_{\mathcal{O}_2, 3}^{(L)}$ can be defined by factorizing out the tree-level form factor from the loop-level as
\begin{equation}
\mathcal{F}_{\mathcal{O}_2, 3}^{(L)}(1,2,3) = \mathcal{F}_{\mathcal{O}_2, 3}^{(0)} \mathcal{I}_{\mathcal{O}_2, 3}^{(L)}(1,2,3) \, .
\end{equation}
and $\mathcal{I}_{\mathcal{O}_2, 3}^{(L)}$ are functions depending on three Mandelstam variables \{$s_{12}$, $s_{23}$, $s_{13}$\}.
\subsubsection{One-loop case}
\label{subsec:3ptff1loop}
At one loop, $\mathcal{I}_{\mathcal{O}_2, 3}^{(1)}$ can be expanded in terms of 7 master integrals as
\begin{align}
\label{eq:3pt1loopAnsatz}
\mathcal{I}_{\mathcal{O}_2, 3}^{(1)} = & c_1 I^{(1)}_{\text{Bub}}(1, 2) + c_2 I^{(1)}_{\text{Bub}}(2, 3) + c_3 I^{(1)}_{\text{Bub}}(1, 3) + c_4 I^{(1)}_{\text{Bub}}(1, 2, 3) \nonumber \\
& + c_5 I^{(1)}_{\text{Box}}(1, 2, 3) + c_6 I^{(1)}_{\text{Box}}(2, 3, 1) + c_7 I^{(1)}_{\text{Box}}(3, 1, 2) \, .
\end{align}
First, the infrared structure for one-loop takes an uniform form as
\begin{align}
\label{eq:3pt1loopIR}
\mathcal{I}_{\mathcal{O}_2, 3}^{(1)} \Big|_{\text{IR}} = -\frac{3}{\epsilon^2} +\frac{\log(-s_{12}) +\log(-s_{23}) +\log(-s_{13})}{\epsilon} \,.
\end{align}
By requiring that the infrared part of \eqref{eq:3pt1loopAnsatz} to match \eqref{eq:3pt1loopIR}, one finds the coefficients should satisfy
\begin{align}
& c_4 = -{3\over2} (c_1 + c_2 + c_3 + 3) \,, \qquad c_5 = -{1\over4}( c_1 + c_2 - c_3 + 1) \,, \nonumber\\
& c_6 = -{1\over4}( -c_1 + c_2 + c_3 + 1) \,, \qquad c_7 = -{1\over4}( c_1 - c_2 + c_3 + 1) \,.
\end{align}
Next we consider the constraints of the collinear limits.
In the linear limit $p_3 \, || \, p_1 \, || \, (p_1+p_3)=p_1'$, one has
\begin{equation}
s_{13} \rightarrow \delta q^2, \qquad s_{12} \rightarrow z q^2, \qquad s_{23} \rightarrow (1- z - \delta) q^2 \,,
\end{equation}
where $q^2 = (p_1+p_2+p_3)^2 = (p_1'+p_2)^2$, $0 \leqslant z \leqslant 1$ is a finite number and $\delta \ll 1$. As discussed in Section~\ref{sec:constraints}, using \eqref{eq:collinear-general}, the three-point form factor in the limit satisfies
\begin{equation}
\mathcal{F}^{(1)}_{\mathcal{O}_2, 3}(3, 1,2) \xrightarrow{ p_1 || p_3 } \sum_\sigma \Big[ \mathbf{Sp}_{-\sigma}^{(0)}(3^{h_3},1^{h_1}) \, \mathcal{F}_{\mathcal{O}_2, 2}^{(1)}(1'^\sigma,2) + \mathbf{Sp}_{-\sigma}^{(1)}(3^{h_3},1^{h_1}) \, \mathcal{F}^{(0)}_{\mathcal{O}_2, 2}(1'^\sigma,2) \Big] \,.
\end{equation}
Since only $\mathcal{F}_{\mathcal{O}_2, 2}(1'^-,2^-)$ is non-zero, after subtracting the tree factors, one obtains
\begin{align}
\label{eq:3pt1loopCL1}
\mathcal{I}_{\mathcal{O}_2, 3}^{(1)} \xrightarrow{ p_1 || p_3 } \, \mathcal{I}_{\mathcal{O}_2, 2}^{(1)}(p_1', p_2) + r_1^{[1], \text{MT}}(s_{13}, z) \,,
\end{align}
where $\mathcal{I}_{\mathcal{O}_2, 2}^{(1)}=- 2 I_{\text{Bub}}^{(1)}(p_1', p_2)$ is the known minimal two-point form factor, and $r_1^{[1], \text{MT}}$ is the splitting function given in \eqref{eq:splittingAmp}.
Explicitly, the form factor in the collinear limit is given by
\begin{align}
\label{eq:3pt1loopCL2}
\mathcal{I}_{\mathcal{O}_2, 3}^{(1)} \, \xrightarrow{ p_1 \parallel p_3 } & -\frac{3}{\epsilon^2} +
\frac{\log (\delta) + 3 \log (-q^2)+\log (1-z)+\log (z)}{\epsilon} \\
& - \log (\delta) \left( \log(-q^2) + \log (1-z) + \log (z) \right) - \frac{1}{2} \log ^2(\delta) \nonumber \\
& +\log (-q^2) (-\log (1-z)-\log (z)) - \frac{3}{2} \log ^2(-q^2) \nonumber\\
& +\frac{1}{2} \left(- \log ^2(1-z)- \log ^2(z)+2 \log (z) \log (1-z)+\zeta_2 \right) \,. \nonumber
\end{align}
By using the ansatz expression \eqref{eq:3pt1loopAnsatz} for $\mathcal{I}_{\mathcal{O}_2, 3}^{(1)}$, and requiring its collinear limit to match with \eqref{eq:3pt1loopCL2}, the coefficients in \eqref{eq:3pt1loopAnsatz} can be fixed to be
\begin{align}
c_2 = c_1 \,, \quad c_3 = 1 \,, \quad c_4 = -c_1-2 \,, \quad c_5 = -\frac{c_1}{2} \,, \quad c_6 = -\frac{1}{2} \,, \quad c_7 = -\frac{1}{2} \,,
\end{align}
leaving one parameter $c_1$ unfixed.
Note that we only use one collinear limit, the other collinear limits $p_3 \parallel p_2$ and $p_2 \parallel p_1$ can be further considered, which will give $c_1=1$.
To summarize, the infrared structure and the collinear limits are enough to fix the one-loop form factor uniquely as
\begin{equation}
\mathcal{I}_{{\cal O}_2, 3}^{(1)} = I_{\text{Bub}}^{(1)}(1,2) - I_{\text{Bub}}^{(1)}(1,2,3) - \frac{1}{2}I_{\text{Box}}^{(1)}(1,2,3) +(\text{cyclic perm.(1,2,3)}) \,.
\end{equation}
\subsubsection{Two-loop case}
At two loops, there are in general 89 master integrals, the integral topologies with maximal number of propagators, which are enough to cover all the master integrals we need, are shown in Figure~\ref{fig:maxTopology3pt}. Then ansatz for the maximally transcendental part can be generally written as
\begin{align}
\label{eq:3pt2loopAnsatz}
\mathcal{I}_{\mathcal{O}_2, 3}^{(2)} = & c_1 I_{\text{Sun}}^{(2)}(1,2) +c_2 I_{\text{Sun}}^{(2)}(1,2,3) +c_3 I_{\text{TBub0}}^{(2)}(1,2) +c_4 I_{\text{dBub}}^{(2)}(1,2;1,2) \\
& + c_5 I_{\text{dBub}}^{(2)}(1,2;1,2,3) + c_6 I_{\text{dBub}}^{(2)}(1,2,3;1,2,3) + c_{7} I_{\text{TBub1}}^{(2)}(1,2,3) \nonumber \\
& +c_{8} I_{\text{TBub2}}^{(2)}(1,2,3) + c_{9} I_{\text{TT0}}^{(2)}(1,2,3)+c_{10} I_{\text{TT1}}^{(2)}(1,2,3) + c_{11} I_{\text{TT1a}}^{(2)}(1,2,3) \nonumber \\
& + c_{12} I_{\text{TT2}}^{(2)}(1,2,3) + c_{13} I_{\text{TBox0}}^{(2)}(1,2,3) + c_{14} I_{\text{BoxBub}}^{(2)}(1,2,3)+c_{15} I_{\text{BubBox0}}^{(2)}(1,2,3) \nonumber \\
& +c_{16} I_{\text{BubBox}}^{(2)}(1,2,3) + c_{17} I_{\text{dBox1a}}^{(2)}(1,2,3)+c_{18} I_{\text{dBox1b}}^{(2)}(1,2,3) +c_{19} I_{\text{NTBox1}}^{(2)}(1,2) \nonumber \\
& +c_{20} I_{\text{NTBox2}}^{(2)}(1,2,3) + c_{21} I_{\text{NTBox3a}}^{(2)}(1,2,3) + c_{22} I_{\text{NTBox3b}}^{(2)}(1,2,3)+c_{23} I_{\text{NdBox1a}}^{(2)}(1,2,3) \nonumber \\
& +c_{24} I_{\text{NdBox1b}}^{(2)}(1,2,3) + c_{25} I_{\text{NdBox2a}}^{(2)}(1,2,3)+c_{26} I_{\text{NdBox2b}}^{(2)}(1,2,3) + \left( \text{full perm.(1,2,3)} \right) \nonumber \,,
\end{align}
where the parameters $c_i$ in ansatz can be solved by imposing constraints. We comment that some master integrals have symmetry of the external momentum, thus there are only 89 independent master integrals and parameters.
\begin{figure}[tb]
\centering
\includegraphics[scale=0.5]{figs/maxtopology3pt.eps}
\caption{Integral topologies of maximal number of propagators for the two-loop three-point form factor.}
\label{fig:maxTopology3pt}
\end{figure}
As we emphasized, all these master integrals are pure UT integrals with uniform transcendentality degree-$4$ and the explicit definitions are given in Appendix~\ref{app:UT}. Their expressions were obtained in \cite{Gehrmann:2000zt, Gehrmann:2001ck} as two-dimensional harmonic polylogarithms which depend on Mandelstam variables \{$s_{12}$, $s_{23}$, $s_{13}$\}, plugging them into our ansatz \eqref{eq:3pt2loopAnsatz}, we can obtain the form factor expression in powers of $\epsilon$, which starts from $\epsilon^{-4}$.
We can apply the constraints to solve for the coefficients of the ansatz in \eqref{eq:3pt2loopAnsatz}.
First, we consider the form factor $\mathcal{F}_{\mathcal{O}_2, 3}^{(2)}$ has the symmetry of cycling external momentum, $\mathcal{F}_{\mathcal{O}_2, 3}^{(2)}(p_1,p_2,p_3)=\mathcal{F}_{\mathcal{O}_2, 3}^{(2)}(p_2,p_3,p_1)$. The symmetry will reduce the number of parameters from 89 to 24.
To apply further physical constraints, we consider the BDS ansatz function
\begin{equation}
\mathcal{I}_{\mathcal{O}_2, 3, \text{BDS}}^{(2)} = \frac{1}{2} \left(\mathcal{I}_{\mathcal{O}_2, 3}^{(1)}(\epsilon)\right)^2+f^{(2)}(\epsilon) \mathcal{I}_{\mathcal{O}_2, 3}^{(1)}(2\epsilon) \,.
\end{equation}
As reviewed in Section~\ref{sec:constraints}, the BDS ansatz (using only one-loop data) provides the divergent part of $1/\epsilon^m, m=4,3,2,1$. By matching with our ansatz \eqref{eq:3pt2loopAnsatz}, the number of parameters is reduced to 13.
Furthermore, the finite remainder obtained by subtracting the BDS ansatz part has the collinear limits
\begin{equation}
\label{eq:3ptremainderCL}
\mathcal{R}_{\mathcal{O}_2, 3}^{(2)} = \mathcal{I}_{\mathcal{O}_2, 3}^{(2)}-\mathcal{I}_{\mathcal{O}_2, 3, \text{BDS}}^{(2)} \ \xlongrightarrow{\mbox{$p_i \parallel p_j$}} \mathcal{R}_{\mathcal{O}_2, 2}^{(2)} \, ,
\end{equation}
where $p_i$ and $p_j$ can be any pair of momentum, and the Sudakov form factor result $\mathcal{R}_{\mathcal{O}_2, 2}^{(2)} = (6 - y_0) \zeta_4$ is given in Section~\ref{sec:miniFF}, here we take $y_0$ as an input known parameter.
The limit can be calculated with the series of two-dimensional harmonic polylogarithms, or by evaluating the master integrals with a very small $s_{ij} = \delta q^2$ with $\delta \ll 1$ in high-precision. After applying all three collinear limits, there is only one parameter unfixed.
After this step, we can organize the form factor result in the following form
\begin{equation}
\mathcal{I}_{\mathcal{O}_2, 3}^{(2)} = G_1^{(2)} + y_0 G_2^{(2)} + c' \tilde{G}^{(2)} \,,
\end{equation}
where $c'$ is the unfixed parameter, and each function on the RHS can be expanded by master integrals as
\begin{align}
G_1^{(2)} = & -I_{\text{dBub}}^{(2)}(1,2;1,2,3) +\frac{2}{3} I_{\text{dBub}}^{(2)}(1,2,3;1,2,3) -3 I_{\text{BubBox}}^{(2)}(1,2,3) \\
& +\frac{1}{2} I_{\text{BubBox0}}^{(2)}(1,2,3) +\frac{1}{2} I_{\text{dBox1a}}^{(2)}(1,2,3) +\frac{1}{2} I_{\text{dBox1b}}^{(2)}(1,2,3) +\frac{1}{8} I_{\text{NdBox1a}}^{(2)}(1,2,3) \nonumber \\
& +\frac{1}{8} I_{\text{NTBox2}}^{(2)}(1,2,3) -\frac{1}{4} I_{\text{NTBox3b}}^{(2)}(1,2,3) -\frac{11}{6} I_{\text{Sun}}^{(2)}(1,2,3) -\frac{11}{4} I_{\text{Sun}}^{(2)}(1,2) \nonumber \\
& +\frac{7}{4} I_{\text{TBub2}}^{(2)}(1,2,3) +\frac{1}{2} I_{\text{TT0}}^{(2)}(1,2,3) -\frac{1}{4} I_{\text{TT1}}^{(2)}(1,2,3) +\frac{5}{4} I_{\text{TT2}}^{(2)}(1,2,3) + (\text{full permute}) \,, \nonumber \\
G_2^{(2)} = & \frac{3}{4} I_{\text{Sun}}^{(2)}(1,2) +\frac{13}{3} I_{\text{Sun}}^{(2)}(1,2,3) +\frac{3}{2} I_{\text{TBub1}}^{(2)}(1,2,3) -\frac{1}{4} I_{\text{TT1}}^{(2)}(1,2,3) \\
& -\frac{1}{2} I_{\text{TT1a}}^{(2)}(1,2,3) -\frac{5}{4} I_{\text{TT2}}(1,2,3) +\frac{1}{3} I_{\text{dBub}}^{(2)}(1,2,3;1,2,3) \nonumber \\
& -\frac{1}{4} I_{\text{NTBox3a}}^{(2)}(1,2,3) +\frac{3}{4} I_{\text{NTBox3b}}^{(2)}(1,2,3) + (\text{cyclic perm.(1,2,3)}) \,, \nonumber \\
\tilde{G}^{(2)} = & I_{\text{dBub}}^{(2)}(1,2;1,2) - I_{\text{dBub}}^{(2)}(1,2,3;1,2,3) +\frac{35}{4} I_{\text{Sun}}^{(2)}(1,2) -13 I_{\text{Sun}}^{(2)}(1,2,3) \\
& +\frac{5}{2} I_{\text{TBub0}}^{(2)}(1,2) -\frac{9}{2} I_{\text{TBub1}}^{(2)}(1,2,3) +\frac{3}{4} I_{\text{TT1}}^{(2)}(1,2,3) +\frac{3}{2} I_{\text{TT1a}}^{(2)}(1,2,3) +\frac{15}{4} I_{\text{TT2}}^{(2)}(1,2,3) \nonumber \\
& +\frac{1}{2} I_{\text{NTBox1}}^{(2)}(1,2) +\frac{3}{4} I_{\text{NTBox3a}}^{(2)}(1,2,3) -\frac{9}{4} I_{\text{NTBox3b}}^{(2)}(1,2,3) + (\text{cyclic perm.(1,2,3)}) \,, \nonumber
\end{align}
where the building-block $\tilde{G}^{(2)}$ is finite and vanishes in the collinear limits, thus it is unconstrained by infrared structure and collinear limit. The function $G_2^{(2)}$ is infrared finite and has collinear limit behavior as $G_2^{(2)} \rightarrow - \zeta_4 + \mathcal{O}(\epsilon)$ for any $p_i \parallel p_j$.
We note that the building-block $\tilde{G}^{(2)}$ contains the nonplanar master integrals $I_{\text{NTBox1}}^{(2)}(i,j)$, which cannot occur in two-loop three-point cases since its color factor is zero when the particles are adjoint representation. This gives $c' = 0$. Then the parameters in ansatz are all fixed.
We summarize the above process of applying constraints in Table~\ref{tab:solvingAnsatz3ptphi2}.
\begin{table}[t]
\centering
\vskip .1 cm
\begin{tabular}{| c | c |}
\hline
Constraints & Remaining parameters afterwards \\ \hline \hline
Starting ansatz & 89 \\ \hline
Symmetry(cycles) & 24 \\ \hline
IR & 11 \\ \hline
Collinear limit & 1 \\ \hline
Color factor structure & 0 \\ \hline
\end{tabular}
\caption{Solving ansatz via various constraints for two-loop three-point $\mathrm{tr}(\phi^2)$.
\label{tab:solvingAnsatz3ptphi2}}
\end{table}
Finally, we comment on the input parameter $y_0$, which is introduced by the Sudakov form factor.
Interestingly, supposing we do not know its value, which can be fixed by requiring that the form factor $\mathcal{F}_{\mathcal{O}_2, 3}^{(2)}$ has a smooth limit behavior when $q^2 \rightarrow 0$.
Let us explain how this works. It is not hard to see that for each of many master integrals, the $\epsilon$-expansion series are divergent when taking $q^2 \rightarrow 0$. For example, $I_{\text{Sun}}^{(2)}(1,2,3)$ contains the logarithm terms as
\begin{equation}
I_{\text{Sun}}^{(2)}(1,2,3) \propto (-q^2)^{-2\epsilon} = \sum_{k=0}^{\infty} \frac{\log^k(-q^2)}{k!} \epsilon^k \,.
\end{equation}
Some of harmonic polylogarithms contained in other master integrals also diverge logarithmically in the limit.
Therefore, the smooth limit behavior of $q^2 \rightarrow 0$ provides non-trivial constraints on the form factor: these logarithmic divergences from different master integrals should cancel with each other in the full form factor result.
One can study the above limit by taking the series expansion analytically, or an alternative way is to apply Cauchy's convergence test which equivalent to the smooth limit: for any positive number $\eta$, there should always exist $\delta>0$, such that the following inequality holds for $0<|x_1|<\delta$ and $0<|x_2|<\delta$:
\begin{equation}
\left| \left(\mathcal{I}_{\mathcal{O}_2, 3}^{(2)}\right) \Big|_{q^2 = x_1} - \left(\mathcal{I}_{\mathcal{O}_2, 3}^{(2)}\right) \Big|_{q^2 = x_2} \right| < \eta \,.
\end{equation}
There is easy to be implemented numerically, and in this way, one obtains constraints on the master coefficients. We mention that the limit of $q^2 = s_{12}+s_{23}+s_{13} \rightarrow 0$ implies that some $s_{ij}$ must be positive, so it is not possible to stay in the Euclidean region (with all $s_{ij}<0$) in the limit, and proper analytical continuation is required.
By applying this constraint, one can find that $y_0 = 2$, which is consistent with the known result.
\subsection{Understanding maximal transcendentality principle}
\label{sec:MTP3pt}
Now we consider the form factors in QCD. Unlike the $\mathcal{N}=4$ case where there is only one \emph{supersymmetric} form factor, in QCD one needs to distinguish three types of external particles, which are $(1^-, 2^-, 3^-)$, $(1^-, 2^-, 3^+)$ and $(1^q, 2^{\bar{q}}, 3^-)$ \cite{Gehrmann:2011aa}.
We will only focus on their maximally transcendental parts and prove the relations of maximal transcendentality \eqref{eq:MTP3pt}.
We first consider the bootstrap constraints and then apply also the unitarity-cut arguments.
\paragraph{Bootstrap constraints.}
The three types of external particles have different physical constraints:
\begin{itemize}
\item
The configuration $(1^-, 2^-, 3^-)$ has full $S_3$ permutational symmetries as the $\mathcal{N}=4$ SYM form factor, and thus the constraints one can use in this case are identical to the case of $\mathcal{N}=4$ SYM.
\item
The case $(1^-, 2^-, 3^+)$ in QCD has only the sub-symmetry of $S_3$ by exchanging $p_1$ and $p_2$. The collinear limits are also different, and one can only take $p_2 \parallel p_3$ or $p_1 \parallel p_3$. (Recall that we focus on the self-dual operator ${\cal O}_2 = {\rm tr}(F_{\alpha\beta}F^{\alpha\beta})$, and the collinear $p_1 \parallel p_2$ is zero in the limit since ${\cal F}_{{\cal O}_2}^{(0)}(1'^\pm,3^+) = 0$.)
\item
For the $(1^-, 2^-, 3^-)$ and $(1^-, 2^-, 3^+)$ cases, an analysis of color factor for Feynman diagrams shows that only the pure gluon configurations contribute the leading $N_c^2$ color parts, and they have no $N_c$-subleading corrections. In particular, the fifth topology in Figure~\ref{fig:maxTopology3pt} has zero color factors which means that the non-planar master integrals $I_{\text{NTBox1}}^{(2)}$, $I_{\text{NTBox2a}}^{(2)}$ and $I_{\text{NTBox2b}}^{(2)}$ will not contribute.
\item
The $(1^q, 2^{\bar{q}}, 3^-)$ case has least constraints: there is no symmetry property to use and the only collinear limit constraint is $p_1 \parallel p_2$.\footnote{The form factor ${\cal F}^{(0)}(1^q, 2^{\bar{q}}, 3^-) = {\langle 2 3\rangle^2 / \langle12\rangle}$ is non-zero by taking the limit $p_1 \parallel p_3$. However, to apply the factorization property \eqref{eq:collinear-general}, the collinear limit requires a physical pole, which does not exist in the above limit.}
\item
Finally, one can apply the condition that form factors should have a smooth limit when $q^2 \rightarrow 0$. This will be automatically satisfied for the first two configurations but provide non-trivial constraints for the $(1^q, 2^{\bar{q}}, 3^-)$ case.
\end{itemize}
Since the bootstrap procedures are similar to the $\mathcal{N}=4$ case in the previous subsection, we will not go into details but only
summarize the main steps in Table~\ref{tab:solvingAnsatz3ptphi2F2-1loop} and Table~\ref{tab:solvingAnsatz3ptphi2F2},
for one and two loops respectively.
\begin{table}[t]
\centering
\vskip .1 cm
\begin{tabular}{| p{5cm}<{\centering} | p{2.5cm}<{\centering} | p{2.5cm}<{\centering} | p{2.5cm}<{\centering} |}
\hline
External particles & $(1^-, 2^-, 3^-)$ & $(1^-, 2^-, 3^+)$ & $(1^q, 2^{\bar{q}}, 3^-)$ \\ \hline\hline
Constraints & \multicolumn{3}{c|}{Remaining parameters} \\ \hline
Starting ansatz & 7 & 7 & 7 \\ \hline
IR & 4 & 4 & 4 \\ \hline
Collinear limit & 0 & 0 & 1 \\ \hline
\end{tabular}
\caption{Bootstrap for one-loop form factors of $\mathrm{tr}(F^2)$.
\label{tab:solvingAnsatz3ptphi2F2-1loop}}
\end{table}
\begin{table}[t]
\centering
\vskip .1 cm
\begin{tabular}{| p{5cm}<{\centering} | p{2.5cm}<{\centering} | p{2.5cm}<{\centering} | p{2.5cm}<{\centering} |}
\hline
External particles & $(1^-, 2^-, 3^-)$ & $(1^-, 2^-, 3^+)$ & $(1^q, 2^{\bar{q}}, 3^-)$ \\ \hline\hline
Constraints & \multicolumn{3}{c|}{Remaining parameters} \\ \hline
Starting ansatz & 89 & 89 & 89 \\ \hline
Symmetry & 24 & 53 & 89 \\ \hline
IR & 11 & 21 & 48 \\ \hline
Collinear limit & 1 & 5 & 21 \\ \hline
Color factor & 0 & 2 & 21 \\ \hline
Smooth light-like limit of $q$
& 0 & 0 & 11 \\ \hline
\end{tabular}
\caption{Bootstrap for two-loop form factors of $\mathrm{tr}(F^2)$.
\label{tab:solvingAnsatz3ptphi2F2}}
\end{table}
We can see that all parameters in the cases $(1^-, 2^-, 3^-)$ and $(1^-, 2^-, 3^+)$ can be fixed with the constraints. We would like to stress that the bootstrap constraints are theory-independent, therefore the maximally transcendental parts of form factors $\mathcal{F}_{\mathrm{tr}(F^2), 3}^{(L)}(1^-,2^-,3^\pm)$ for $L=1,2$ should be the same for general gauge theories.
On the other hand, for the $(1^q, 2^{\bar{q}}, 3^-)$ case, there is one parameter left that is not fixed by IR and collinear constraints at one loop,
and there are 11 parameters left at two loops as shown in Table~\ref{tab:solvingAnsatz3ptphi2F2}.
Below we show that with the unitarity-cut method, its maximally transcendental part can be also proved to be universal.
\paragraph{Unitarity-cut for the $(1^q, 2^{\bar{q}}, 3^-)$ case.}
Below we would like to show that the QCD result is equivalent to the $\mathcal{N}=4$ result for $(1^q, 2^{\bar{q}}, 3^-)$ case by converting QCD quarks from the fundamental to the adjoint representation.
Note that due to supersymmetry, the $\mathcal{N}=4$ SYM result is the same for all possible choices of external states, this is then enough to prove \eqref{eq:MTP3pt}.
To determine this remaining parameter at one loop, one can use the unitarity cut shown in Figure~\ref{fig:cut1loop3point}. Actually, there is no need to perform this computation, since this cut is the same for $\mathcal{N}=4$ SYM and QCD if one changes the quadratic Casimir $C_F \rightarrow C_A$ in QCD. Thus we prove the relations in \eqref{eq:MTP3pt} at one loop.
\begin{figure}[t]
\centering
\includegraphics[scale=0.55]{figs/samecut1loop3point.eps}
\caption{A unitarity cut for the one-loop form factor ${\cal F}^{(1)}(1^{q}, 2^{\bar{q}}, 3^-)$.}
\label{fig:cut1loop3point}
\end{figure}
\begin{figure}[t]
\centering
\includegraphics[scale=0.5]{figs/colorF2fun.eps}
\caption{The QCD Feynman diagram that contributes to the fifth topology in Figure~\ref{fig:maxTopology3pt}.}
\label{fig:colorF2}
\end{figure}
The two-loop case is less trivial.
We first recall that the fifth topology in Figure~\ref{fig:maxTopology3pt} has zero color factors for adjoint particles, therefore, when we convert quarks to be adjoint, the non-planar master integrals $I_{\text{NTBox1}}^{(2)}$, $I_{\text{NTBox2a}}^{(2)}$ and $I_{\text{NTBox2b}}^{(2)}$ can not contribute.
The related Feynman diagram contribution in QCD is shown in Figure~\ref{fig:colorF2}. Its color factor can be computed as
\begin{equation}
t_F^2 (C_A-C_F) (C_A-2C_F) (T^{a_3})_{i_1}^{~\bar{j}_2} \,,
\end{equation}
which indeed vanishes when taking $C_F \rightarrow C_A$.
In this way, one can eliminate 6 parameters that are relate to these non-planar master integrals.
\begin{figure}[t]
\centering
\subfloat[]{\includegraphics[scale=0.55]{figs/samecut2loop3point1.eps}}
\subfloat[]{\includegraphics[scale=0.5]{figs/samecut2loop3point2.eps}}
\subfloat[]{\includegraphics[scale=0.5]{figs/samecut2loop3point3.eps}}
\caption{Unitarity cuts for the two-loop form factor $\mathcal{F}^{(2)}(1^{q}, 2^{\bar{q}}, 3^-)$.}
\label{fig:cut2loop3point-MT}
\end{figure}
For the remaining 5 parameters, we will apply the unitarity cuts in Figures~\ref{fig:cut2loop3point-MT} to fix the related five master integrals as follows:
\begin{align}
\textrm{cut-(a)}: & \quad I_{\text{dBub}}^{(2)}(1,2,3;1,2,3) \,, \nonumber\\
\textrm{cut-(b)}: & \quad I_{\text{TT1}}^{(2)}(3,1,2) \,, \quad I_{\text{dBox1a}}^{(2)}(2,3,1) \,, \nonumber\\
\textrm{cut-(c)}: & \quad I_{\text{TT1}}^{(2)}(1,2,3) \,, \quad I_{\text{dBox1a}}^{(2)}(3,1,2) \,. \nonumber
\end{align}
The cut-(a) is universal for general gauge theory since the cut legs can be only gluons.
On the other hand, the cut-(b) and (c) are more complicated, since the cut legs $\{l_1,l_2\}$ can have different particle configurations.
For example, the cut-(b) integrand in $\mathcal{N}=4$ SYM is:
\begin{align}
\label{eq:cutintF2A4A4}
\int \prod_{i=1}^2 d^4\eta_{l_i} \mathcal{F}_{\mathcal{O}_{2},2}^{(0), \text{MHV}}(-l_3^g,-l_4^g) \mathcal{A}_4^{(0), \text{MHV}}(1^\psi,l_2, l_1,l_3^g) \mathcal{A}_4^{(0), \text{MHV}} (2^{\bar{\psi}},3^g,l_4^g,-l_1,-l_2) \,,
\end{align}
where the fermionic integration corresponds to sum $l_i$ over all possible super-states in the $\mathcal{N}=4$ on-shell superfield \cite{Nair:1988bq}:
\begin{equation}
\Phi(l,\eta)=g_+(l) + \eta^A \, \bar\psi_A(l) + {\eta^A\eta^B \over 2!} \, \phi_{AB}(l) + { \epsilon_{ABCD} \eta^A\eta^B\eta^C \over 3!} \, \psi^D(l) + \eta^1\eta^2\eta^3\eta^4 \, g_-(l) \,.
\label{eq:onshellN=4superspace}
\end{equation}
More explicitly, \eqref{eq:cutintF2A4A4} can be expanded as
\begin{align}
& \mathcal{F}_{\mathcal{O}_{2}, 2}^{(0)}(-l_3^g, -l_4^g) \mathcal{A}_{4}^{(0)}(1^\psi, l_2^{{\bar\psi}/g}, l_1^{g/{\bar\psi}}, l_3^g) \mathcal{A}_{5}^{(0)}(2^{\bar\psi}, 3^g, l_4^g, -l_1^{g/\psi}, -l_2^{\psi/g}) \nonumber\\
& +\mathcal{F}_{\mathcal{O}_{2}, 2}^{(0)}(-l_3^g, -l_4^g) \mathcal{A}_{4}^{(0)}(1^\psi, l_2^{\psi/\phi}, l_1^{\phi/\psi}, l_3^g) \mathcal{A}_{5}^{(0)}(2^{\bar\psi}, 3^g, l_4^g, -l_1^{\phi/\bar\psi}, -l_2^{\bar\psi/\phi}) \,.
\label{eq:cutB3pt}
\end{align}
The configuration in the first line contains only gluon and gluino states, thus they will map to the QCD result by converting color factors of fermions accordingly.
The second line in \eqref{eq:cutB3pt}, however, involves the scalar particles which are special for the $\mathcal{N}=4$ SYM theory.
To have the maximally transcendental principle, it is crucial that the scalar configuration have not any leading transcendental contribution.
An explicit calculation shows that the configuration involving scalars indeed do not contribute to the maximally transcendental part.
The case of cut-(c) is similar.
Therefore, all the configurations that contribute to maximally transcendental parts are the same for $\mathcal{N}=4$ SYM and QCD, and if one changes the quadratic Casimir $C_F \rightarrow C_A$ in QCD, the QCD form factor result is the same as that of $\mathcal{N}=4$ SYM in the case $(1^q, 2^{\bar{q}}, 3^-)$.
\section{Two-loop four-point form factor of $\mathrm{tr}(F^3)$}
\label{sec:2loop4ptF3}
In this section, we consider further two-loop four-point form factor of length-three operators:
\begin{align}
\mathcal{F}_{\mathcal{O}_3,4} = \int d^D x e^{-i q\cdot x} \langle p_1 \, p_2 \, p_3 \, p_4 |\mathcal{O}_3(x)|0\rangle \,. \nonumber
\end{align}
The case of $\mathrm{tr}(\phi^3)$ form factor has been obtained by bootstrapping recently in \cite{Guo:2021bym}. In this paper we will compute a similar form factor which contains a length-three operator $\mathrm{tr}(F^3)$ and four external on-shell gluon states, defined concretely as
\begin{align}
\mathcal{F}_{\mathrm{tr}(F^3),4} := & \mathcal{F}_{\mathrm{tr}(F^3),4}(1^-, 2^-, 3^-, 4^+;q) \\
= & \int d^D x e^{-i q\cdot x} \langle g_-(p_1) g_-(p_2) g_-(p_3) g_+(p_4) | \mathrm{tr}(F^3)(x)|0\rangle \,. \nonumber
\end{align}
As in the previous three-point form factor, this form factor can be understood as the Higgs-plus-four-gluon scattering amplitudes in the Higgs EFT with a dimension-six operator.
Unlike $\mathrm{tr}(\phi^3)$, the operator $\mathrm{tr}(F^3)$ is non-BPS in $\mathcal{N}=4$ SYM, thus the form factor also receives contribution from lower transcendental parts in $\mathcal{N}=4$ SYM.
In this paper, we will focus on the maximal transcendental part.
We will first consider the form factors in $\mathcal{N}=4$ SYM and pure YM, and we apply bootstrap strategy to obtain the maximally transcendental parts up to two loops in Section~\ref{sec:ansatzTrF3}-\ref{sec:solvingAnsatz}. Then we will discuss the correspondence between QCD and $\mathcal{N}=4$ results in Section~\ref{sec:MTPtrF3}.
We finally discuss the connection between form factors of $\mathrm{tr}(F^3)$ and $\mathrm{tr}(\phi^3)$ in Section~\ref{sec:phi3andF3}.
\subsection{Ansatz of the form factor up to two loops}
\label{sec:ansatzTrF3}
We define the maximally transcendental part of the loop corrections $\mathcal{I}_{\mathrm{tr}(F^{3}), 4}^{(L), \text{M.T.}}$ as
\begin{equation}
\mathcal{F}_{\mathrm{tr}(F^{3}), 4}^{(L), \text{M.T.}} = \mathcal{F}_{\mathrm{tr}(F^{3}), 4}^{(0)} \, \mathcal{I}_{\mathrm{tr}(F^{3}), 4}^{(L), \text{M.T.}} \,,
\end{equation}
in which the tree-level result takes the simple form as
\begin{equation}
\label{eq:treeF3}
\mathcal{F}_{\mathrm{tr}(F^{3}), 4}^{(0)} = \frac{ \langle12\rangle \langle23\rangle \langle31\rangle^2 }{ \langle34\rangle \langle41\rangle } \, .
\end{equation}
Our goal is to compute the loop corrections in terms of master integral expansion:
\begin{equation}
\mathcal{I}_{\mathrm{tr}(F^{3}), 4}^{(L), \text{M.T.}} = \sum_i x_i I_i^{(L)} \,.
\end{equation}
Since we choose $I_i^{(L)}$ as pure UT integrals of transcendentality $2L$, the coefficients $x_i$ are independent of dimensional parameter $\epsilon$.
Unlike the discussion in previous sections, there is a major complication for the four-point form factor studied here, namely, the master coefficients will have non-trivial dependence on kinematics factors as $x_i = \sum_a c_{i,a} B_a$, where $B_a$ are kinematic factors. We will classify the possible $B_a$ factors.
First, since the tree factor has been factorized out, the $B_a$ factors must carry no helicity weight, in other words, they can only be functions of cross ratios of spinor products, or Mandelstam $s_{ij}$ variables.
Inspired by result of the four-point form factor of $\mathrm{tr}(\phi^3)$ \cite{Guo:2021bym}, we can assume that the factor $B_a$ can be expressed with only the cross ratios of angle brackets, which take the following form
\begin{equation}
\frac{\langle i j \rangle \langle k l \rangle}{\langle i k \rangle \langle j l \rangle} \,.
\end{equation}
The appropriate basis of cross ratios can be selected as
\begin{equation}
\label{eq:DefB1B2}
B_1 = \frac{\langle12\rangle \langle34\rangle}{\langle13\rangle \langle24\rangle} \,, \qquad B_2 = \frac{\langle14\rangle \langle23\rangle}{\langle13\rangle \langle24\rangle} \,,
\end{equation}
which appears in the result of $\mathrm{tr}(\phi^3)$ \cite{Guo:2021bym}.
Next, we require the $B_a$ factors contain only poles that appear in the numerator of the tree-level form factor \eqref{eq:treeF3}, and one has
\begin{equation}
\label{eq:PoleFromTree}
B_a \sim \left\{ \frac{1}{\langle 12 \rangle}\,, \frac{1}{\langle 23 \rangle}\,, \frac{1}{\langle 13 \rangle } \,, \frac{1}{\langle 13 \rangle^2 } \right\} \,.
\end{equation}
Besides $B_1$ and $B_2$, the set of $B_a$ factors which contain these poles can be given as
\begin{equation}
B_3 = B_1 B_2 = \frac{\langle12\rangle \langle34\rangle \langle14\rangle \langle23\rangle }{\langle13\rangle^2 \langle24\rangle^2} \, , \quad
B_4 = \frac{B_1}{B_2} = \frac{\langle12\rangle \langle34\rangle}{\langle14\rangle \langle23\rangle} \, , \quad
B_5 = \frac{B_2}{B_1} = \frac{\langle14\rangle \langle23\rangle}{\langle12\rangle \langle34\rangle} \,.
\end{equation}
We will assume that all master coefficients is a linear combination fo $B_a$, where $a =1, ..,5$.
We would like to comment that the poles \eqref{eq:PoleFromTree} will disappear when timing together with the tree form factors in the full form factor. On the other hand, since $B_a$ are cross ratios, new poles are introduced:
\begin{equation}
\label{eq:SpuriousPoleFromTree}
\textrm{spurious poles} : \quad \left\{ \frac{1}{\langle 34 \rangle} \,, \frac{1}{\langle 14 \rangle} \,, \frac{1}{\langle 24 \rangle^2} \right\} ,
\end{equation}
which can not be canceled by tree form factor.
These poles are spurious poles in the sense that they must cancel within the loop correction function $\mathcal{I}_{\mathrm{tr}(F^{3}), 4}^{(L), \text{M.T.}}$.
As we will see, this requirement will provide important constraints on the ansatz results.
It is convenient to reorganize the loop corrections as
\begin{equation}
\label{eq:ansatzFullF3}
\mathcal{I}_{\mathrm{tr}(F^{3}), 4}^{(L), \text{M.T.}} = \sum_{a=1}^5 B_a \mathcal{G}_a^{(L)} \, ,
\end{equation}
where $\mathcal{G}_a^{(L)}$ are the expansion of a set of pure UT master integrals with only numerical coefficients. In addition, they are not all independent because of the symmetry as follows
\begin{equation}
\label{eq:Gsymmetry}
\mathcal{G}_1^{(L)} = \mathcal{G}_2^{(L)} |_{(p_1\leftrightarrow p_3)} \, , \qquad \mathcal{G}_3^{(L)} = \mathcal{G}_3^{(L)} |_{(p_1\leftrightarrow p_3)} \, , \qquad \mathcal{G}_4^{(L)} = \mathcal{G}_5^{(L)} |_{(p_1\leftrightarrow p_3)} \,,
\end{equation}
which is determined by the symmetry of full form factor by ${p_1\leftrightarrow p_3}$ and the symmetry properties of $B_a$.
Therefore, one only needs to focus on $\mathcal{G}_1^{(L)}$, $\mathcal{G}_3^{(L)}$ and $\mathcal{G}_5^{(L)}$.
Below we discuss the ansatz in terms of master integrals in more detail.
For the one-loop form factor, there are 12 master integrals
\begin{equation}
\label{eq:ansatz1loop}
\mathcal{G}_a^{(1)} = c_{a, 1}^{(1)} I_{\text{Bub}}^{(1)}(1, 2) + c_{a, 2}^{(1)} I_{\text{Bub}}^{(1)}(1, 2, 3) + c_{a, 3}^{(1)} I_{\text{Box}}^{(1)}(1, 2, 3) + \text{cyclic perm.}(1,2,3,4) \,.
\end{equation}
Using the symmetry properties \eqref{eq:Gsymmetry}, one can find there are in total 32 = 12+12+8 (for $\mathcal{G}_1^{(1)}$, $\mathcal{G}_5^{(1)}$ and $\mathcal{G}_3^{(1)}$ respectively) independent parameters for the one-loop correction.
\begin{figure}[tb]
\centering
\includegraphics[scale=0.5]{figs/maxtopology.eps}
\caption{Integral topologies of maximal number of propagators for the planar two-loop form factor.}
\label{fig:maxTopology}
\end{figure}
For the two-loop form factor, the integral topologies with the maximal number of propagators (which are enough to cover all the master integrals we need) are shown in Figure~\ref{fig:maxTopology}.
Since the operator has length-three, the massive $q$-leg (denoted by blue color) should be connected to a four-vertex.
To obtain the full form factor, one needs to consider all possible insertions of the $q$-leg, since the operator is a color-singlet.
The most general ansatz contains 221 pure UT master integrals:
\begin{equation}
\label{eq:ansatz2loop}
\mathcal{G}_a^{(2)} = \sum_{i=1}^{221} c_{a,i}^{(2)} I_{i}^{(2)} \,.
\end{equation}
The definition of the involved master integrals are given in Appendix~\ref{app:UT}, and they have been computed in \cite{Abreu:2020jxa, Canko:2020ylt}, see also \cite{Chicherin:2021dyp, Abreu:2021smk}.
The main goal is to solve the coefficients $c_{a, i}^{(2)}$.
Using again the symmetry properties \eqref{eq:Gsymmetry},
one finds there are 221 free parameters from $\mathcal{G}_1^{(2)}$ and $\mathcal{G}_5^{(2)}$ respectively, and 118 from $\mathcal{G}_3^{(L)}$. In total, our two-loop ansatz contains 560 free parameters.
\subsection{Solving one-loop result}
\label{sec:solveFF4pt1loop}
We start with the one-loop ansatz given by \eqref{eq:ansatzFullF3} and \eqref{eq:ansatz1loop}. The bootstrap method can be applied using similar steps as in Section~\ref{subsec:3ptff1loop}.
First, the infrared structure is
\begin{equation}
\mathcal{I}_{\mathrm{tr}(F^3),4}^{(1)} \Big|_{\text{IR}} = \left( B_1+B_2 \right) \sum_{i=1}^4 \left( -\frac{1}{\epsilon^2} + \frac{\log(-s_{i,i+1})}{\epsilon} \right) \, .
\end{equation}
where $B_1+B_2=1$ is used. To match with the ansatz, one finds that
\begin{equation}
\mathcal{G}_1^{(1)} \Big|_\text{IR} = \sum_{i=1}^n \left( -\frac{1}{\epsilon^2}+\frac{\log(-s_{i,i+1})}{\epsilon} \right) \, , \qquad \mathcal{G}_3^{(1)} \Big|_\text{IR} = \mathcal{G}_5^{(1)} \Big|_\text{IR} = 0 \, .
\end{equation}
This can fix 21 degrees of freedom, leaving 11 free parameters.
Next, we consider the constraints of collinear limit $p_3 \parallel p_4 \parallel p_3' = p_3+p_4$ (the other collinear limit $p_1 \parallel p_4 \parallel p_1' = p_1+p_4$ is related by symmetry).
The Mandelstam variables in the limit can be taken as
\begin{equation}
\label{eq:colllinearofs}
s_{34} \rightarrow \delta q^2 \,, \quad s_{i3} \rightarrow z(1-\delta) s_{i3}'\,, \quad s_{i4} \rightarrow (1-z)(1-\delta) s_{i3}' \,,
\end{equation}
where $\delta\ll1$, $q^2 = (p_1+p_2+p_3+p_4)^2 = (p_1+p_2+p_3')^2$ and $s_{i3}' = (p_i+p_3')^2$ with $i = 1, 2$. Meanwhile the limits of $B_a$ factors are
\begin{equation}
\label{eq:colllinearofB}
\{B_1, B_3, B_4\} \rightarrow 0 \,, \qquad B_2 \rightarrow 1 \,, \qquad B_5 \rightarrow \infty \,,
\end{equation}
since $B_{1,3,4} \propto \langle 34 \rangle$, $B_5 \propto 1/\langle 34 \rangle$, and $B_2 \rightarrow1$ because $B_1 + B_2 = 1$. The parameterization \eqref{eq:colllinearofs} can be used directly for the one-loop master integrals since they are free from $\mathrm{tr}_5$.
Similar to the discussion of the three-point form factor in Section~\ref{subsec:3ptff1loop},
the four-point form factor in the collinear limit satisfies
\begin{equation}
\mathcal{I}^{(1), \text{M.T.}}_{\mathrm{tr}(F^3), 4} \xrightarrow{ p_3 || p_4 } \mathcal{I}_{\mathrm{tr}(F^3), 3}^{(1), \text{M.T.}}(1, 2, 3') +r_1^{[1], \text{MT}}(s_{34}, z) \,.
\end{equation}
where $\mathcal{I}_{\mathrm{tr}(F^3), 3}^{(1), \text{M.T.}}$ is the three-point minimal form factor, and $r_1^{[1], \text{MT}}$ is the splitting function given in \eqref{eq:splittingAmp}.
Together with \eqref{eq:ansatzFullF3} and \eqref{eq:colllinearofB}, one can find the following constraints on $\mathcal{G}_i$:
\begin{equation}
\label{eq:4pt1loopCL}
\mathcal{G}_2^{(1)} \rightarrow \mathcal{I}_{\mathrm{tr}(F^3), 3}^{(1), \text{M.T.}}(1, 2, 3') +r_1^{[1], \text{MT}}(s_{34}, z) \,, \qquad \mathcal{G}_5^{(1)} \rightarrow 0 \,.
\end{equation}
We point out that $\mathcal{G}_{a}^{(1)}$ can only have logarithmic divergence in the collinear limit, thus the terms $B_a \mathcal{G}_{a}^{(1)}$ for $a=1,3,4$ will vanish directly, leaving only constraints on $\mathcal{G}_2^{(1)}$ and $\mathcal{G}_5^{(1)}$.\footnote{Note that in principle $B_5 \mathcal{G}_5^{(1)}$ is allowed to have lower transcendental remnants in the collinear limit, which is irrelevant here since we focus only on maximally transcendental part; see also the discussion at the end of this subsection.}
By using the constraints \eqref{eq:4pt1loopCL}, one can solve for 8 parameters and the number of free parameters is reduced to 3.
Third, we consider the constraint from the cancellation of spurious poles \eqref{eq:SpuriousPoleFromTree}. The form factor should be finite, when $\langle 14 \rangle$, $\langle 24 \rangle$, and $\langle 34 \rangle$ approach 0, namely
\begin{equation}
\mathcal{F}_{\mathrm{tr}(F^3), 4}^{(1)} \ \xlongrightarrow{\mbox{$\langle ij \rangle \rightarrow 0$}} \text{finite} \,, \qquad \langle ij \rangle \in \left\{ \langle 14 \rangle, \langle 24 \rangle, \langle 34 \rangle \right\} \,.
\end{equation}
Only $B_5$ has spurious pole $\langle 34 \rangle$ and this provides a constraint on $\mathcal{G}_5^{(1)}$, which is equivalent to the constraint of spurious pole $\langle 14 \rangle$ on $\mathcal{G}_4^{(1)}$ through the symmetry of exchanging external momentum $p_1$ and $p_3$. And $B_1$, $B_2$ and $B_3$ all contain spurious pole $\langle 24 \rangle$, and one should consider them together to cancel the spurious pole. It is convenient to rearrange the sum of the three terms as
\begin{equation}
\label{eq:1loopnewform}
\sum_{a=1}^3 B_a \mathcal{G}_a^{(1)} = \frac{1}{2} \left( \mathcal{G}_1^{(1)} + \mathcal{G}_2^{(1)} + 2\mathcal{G}_3^{(1)} \right) + \frac{B_1-B_2}{2} \left( \mathcal{G}_1^{(1)} - \mathcal{G}_2^{(1)} \right) - \left( \frac{B_1-B_2}{2} \right)^2 \mathcal{G}_3^{(1)} \, ,
\end{equation}
and the spurious pole $\langle 24 \rangle$ only appears in $(B_1-B_2)/2$, which constrains the second and third terms.
To summarize, the spurious pole cancellation impose the constraints:
\begin{equation}
\label{eq:4pt1loopSP}
\mathcal{G}_5^{(1)} \, \xlongrightarrow{\mbox{$\langle 34 \rangle \rightarrow 0$}} 0 \,, \qquad \mathcal{G}_1^{(1)}-\mathcal{G}_2^{(1)} \xlongrightarrow{\mbox{$ \langle 24 \rangle \rightarrow 0$}} 0 \,, \qquad \ \mathcal{G}_3^{(1)} \xlongrightarrow{\mbox{$ \langle 24 \rangle \rightarrow 0$}} 0 \,,
\end{equation}
in the formula, we will not consider the cancellation in higher order of $\langle ij \rangle$, the reason will be discussed later.
We point out that the limit $\langle ij \rangle \rightarrow 0$ can be naviely treated as $s_{ij} \rightarrow 0$, because $\mathrm{tr}_5$ doesn't appear in the one-loop master integrals we used, whose limit behavior needs special treatment.
After this constraint, only 1 parameter left.
Now the result that contains a free parameter $x_0$ and satisfies the above constraints can be summarized as follows
\begin{align}
\label{eq:F3oneloop}
& \mathcal{G}_1^{(1)} = G^{(1)} - \tilde{G}^{(1)}(4,1,2) \,, \qquad \mathcal{G}_2^{(1)} = \mathcal{G}_1^{(1)} |_{\left( p_1 \leftrightarrow p_3 \right)} \,, \qquad \mathcal{G}_4^{(1)} = \mathcal{G}_5^{(1)} = 0 \,, \\
& \mathcal{G}_3^{(1)} = x_0 \left( \tilde{G}^{(1)}(4,1,2)+\tilde{G}^{(1)}(2,3,4) \right) \,, \nonumber
\end{align}
where the building-blocks $G^{(1)}$, $\tilde{G}^{(1)}(i,j,k)$ are
\begin{align}
& G^{(1)} = -I_{\text{Bub}}^{(1)}(1,2) - I_{\text{Bub}}^{(1)}(2,3) - I_{\text{Bub}}^{(1)}(3,4,1) - \frac{1}{2}I_{\text{Box}}^{(1)}(3,4,1) \, , \\
& \tilde{G}^{(1)}(i,j,k) = I_{\text{Bub}}^{(1)}(i,j) + I_{\text{Bub}}^{(1)}(j,k) - I_{\text{Bub}}^{(1)}(i,j,k) - \frac{1}{2}I_{\text{Box}}^{(1)}(i,j,k) \,. \nonumber
\end{align}
In particular, the building-block $\tilde{G}^{(1)}(i,j,k)$ is not only IR finite but is also trivial in the collinear limits as
\begin{equation}
\tilde{G}^{(1)}(i,j,k) \xrightarrow{ \langle ik \rangle \rightarrow 0 } \mathcal{O}(\langle ik \rangle) \,.
\end{equation}
Thus it is not constrained by the above limit.
Finally, to fix the remaining single parameter, we use the unitarity method. We note that $x_0$ appears in the coefficient $-(B_2 + x_0 B_3)/2$ of $I_{\text{Box}}^{(1)}(2,3,4)$ in \eqref{eq:F3oneloop}. By performing a simple quadruple-cut as Figure~\ref{fig:fourcut}:
\begin{equation}
\sum \mathcal{F}_{3}^{(0)} \mathcal{A}_{3}^{(0)} \mathcal{A}_{3}^{(0)} \mathcal{A}_{3}^{(0)} \longrightarrow - \mathcal{F}_4^{(0)} \frac{B_1}{2} I_{\text{Box}}^{(1)}(2,3,4) \big|_{\text{cut integrand}} \,,
\end{equation}
one obtains the box coefficient as $-(B_2-B_3)/2$ which fixes the parameter as $x_0 = -1$.
\begin{figure}[t]
\centering
\includegraphics[scale=0.5]{figs/fourcut.eps}
\caption{A simple quadruple cut to determine the coefficient of $I_{\text{Box}}^{(1)}(2,3,4)$.}
\label{fig:fourcut}
\end{figure}
This completes the construction of the one-loop form factor. We summarize the above steps in Table~\ref{tab:solvingAnsatz1loop4ptF3}.
We point out that in the above construct, the same constraints apply to general gauge theories. In particular, the unitarity cut in Figure~\ref{fig:fourcut} only involves gluon states, thus it gives the same coefficients for any gauge theory containing a Yang-Mills sector.
Therefore, the result we obtain applies to the form factor in general gauge theories, as long as one starts with the ansatz \eqref{eq:ansatzFullF3}.
As a cross-check, we have also performed an independent full unitarity-cut computation and found the same result as above using bootstrap.
\begin{table}[t]
\centering
\vskip .1 cm
\begin{tabular}{| c | c |}
\hline
Constraints & Parameters left \\ \hline \hline
Symmetry of $(p_1\leftrightarrow p_3)$ & 32 \\ \hline
IR & 11 \\ \hline
Collinear limit & 3 \\ \hline
Spurious pole & 1 \\ \hline
Unitarity & 0 \\ \hline
\end{tabular}
\caption{Solving ansatz via constraints for the one-loop four-point form factor of $\mathrm{tr}(F^3)$.
\label{tab:solvingAnsatz1loop4ptF3}
}
\end{table}
Finally, we make a comment about the term $(B_1-B_2)^2 \mathcal{G}_3^{(1)}/4$ in \eqref{eq:1loopnewform}.
Since the spurious pole in $(B_1-B_2)^2 \sim 1/\langle 24 \rangle^2$ is second-order, one may wonder if it could provide stronger constraint on $\mathcal{G}_3^{(1)}$ than \eqref{eq:4pt1loopSP}.
Indeed, if one considers the limit of $\mathcal{G}_3^{(1)}$ in the final result:
\begin{equation}
\label{eq:doublespuriouslimit}
\mathcal{G}_3^{(1)} \ \xlongrightarrow{\mbox{$ \langle 24 \rangle \rightarrow 0$}} s_{24}\left( {1\over s_{12}}{\log \left( \frac{s_{14}}{s_{12}+s_{14}} \right)} +(p_1 \leftrightarrow p_3, p_2 \leftrightarrow p_4)\right) + \mathcal{O}(s_{24}^2) \, ,
\end{equation}
one find it is only of $\langle 24 \rangle$ order,
which means that the term $(B_1-B_2)^2 \mathcal{G}_3^{(1)} \rightarrow \infty$ when $\langle 24 \rangle \rightarrow 0$.
This seems to contradict the spurious pole cancellation condition.
However, one can see from the RHS of \eqref{eq:doublespuriouslimit} that, the divergent term is of transcendental degree one, thus its cancellation should involve the lower transcendental part of the one-loop form factor.
This will indeed impose new constraints on the lower transcendental part of the form factor, but it is in no contradiction with our above computation on the leading transcendental part.\footnote{This could also happen for the collinear limit. For example, in the collinear limit $p_3 \parallel p_4$, $B_5$ is divergent, and $B_5 \mathcal{G}_5^{(L)}$ is in principle allowed to be non-zero but only contributes to lower transcendental parts; though we find that $B_5 \mathcal{G}_5^{(L)}$ vanish in the limit up to two loops.}
It is a general feature that certain limit of a transcendental function may generate functions of lower transcendentality, for example, $\mathrm{Li}_2(1-\delta)$ has series expansion for $\delta \ll 1$
\begin{equation}
\mathrm{Li}_2(1-\delta) = \zeta_2 + \sum_{n=1}^N \left( \frac{\delta^n }{n} \log (\delta) - \frac{\delta^n }{n^2} \right) + {\cal O}(\delta^{N+1}) \, ,
\end{equation}
where $\zeta_2$ has degree 2, but the other terms (truncated to certain order ${\cal O}(\delta^N)$) have lower transcendental degrees.
Such property can be used to impose constraints for lower transcendental parts.
\subsection{Solving two-loop result}
\label{sec:solvingAnsatz}
We consider two-loop case in this subsection. We start with the ansatz \eqref{eq:ansatzFullF3} and \eqref{eq:ansatz2loop}.
To simplify the discussion of each step, we will first apply the constraints at the level of \emph{symbol} and then at the level of full functions.
The \emph{symbol} $\mathcal{S}$ of a function $T^{(k)}$ of transcendentality $k$ is represented in a tensor product form as \cite{Goncharov:2010jf}
\begin{equation}
\mathcal{S} (T^{(k)}) = \sum_{i_1,\ldots, i_k} R_{i_1} \otimes \cdots \otimes R_{i_k} \,,
\end{equation}
where $R_i$ are rational functions of kinematic variables. (Rational function has transcendentality degree 0, and by definition, its symbol is zero.)
It can be understood as a mathematical tool to simplify transcendental functions into tensor products of function arguments, for simple examples:
$\mathcal{S}(\log (x)) = x, \mathcal{S}(\mathrm{Li}_2(x)) = -(1-x) \otimes x$.
For the problem we consider, the symbol of all two-loop masters have been given in \cite{Abreu:2020jxa}. After plugging them into our ansatz \eqref{eq:ansatz2loop}, we obtain an $\epsilon$-expansion expression of the form factor
\begin{equation}
\label{eq:ansatzSymbol}
\mathcal{S} \left(\mathcal{F}_{\mathrm{tr}(F^3), 4}^{(2), \text{M.T.}} \right) =
\sum_{k\geq0} \epsilon^{k-4} \sum_I \alpha_I(c) \otimes_{i=1}^k w_{I_i} \,,
\end{equation}
where are $w_I$ are rational function of Mandelstam variables and are classified as symbol \emph{letters}.
There are 46 independent letters and their definitions are given in Appendix~\ref{app:letters}.
Since the loop correction is uniformly transcendental, the tensor degree at given order in $\epsilon$-expansion is fixed, \emph{e.g.}~the finite order has degree $k=4$.
Moreover, the coefficient $\alpha_I(c)$ are linear combinations of $c_{a,i}$ in \eqref{eq:ansatz2loop}.
\subsubsection{Constraints of IR, collinear and spurious poles}
\label{sec:bootstrapFF4pt2loop}
To impose the constraint of IR divergences and collinear limit, as reviewed in Section~\ref{sec:constraints}, one can consider the BDS function
\begin{equation}
\label{eq:badBDS}
\tilde{\mathcal{I}}_{\mathrm{tr}(F^3), 4, \text{BDS}}^{(2), \text{M.T.}} = \frac{1}{2} \left(\mathcal{I}_{\mathrm{tr}(F^3), 4}^{(1), \text{M.T.}}(\epsilon)\right)^2+f(\epsilon) \mathcal{I}_{\mathrm{tr}(F^3), 4}^{(1), \text{M.T.}}(2\epsilon) \,.
\end{equation}
The one-loop result has been obtained in the previous subsection.
One complication is that the four-point form factor has multiple spinor factors $B_i$,
thus the one-loop square will introduce terms with new spinor factors such as $B_1^2$.
Here a nice solution to avoid this complication is to introduce another BDS function which is linear in $B_a$:
\begin{equation}
\label{eq:goodBDS}
\mathcal{I}_{\mathrm{tr}(F^3), 4, \text{BDS}}^{(2), \text{M.T.}} = \sum_{a=1}^2 \frac{1}{2} \mathcal{G}_a^{(1)} \left( B_a \mathcal{G}_a^{(1)} + B_3 \mathcal{G}_3^{(1)} \right) + f(\epsilon) \sum_{a=1}^2 B_a\mathcal{G}_a^{(1)}(2\epsilon) \, .
\end{equation}
Similar form was used for the four-point form factor of $\mathrm{tr}(\phi^3)$ \cite{Guo:2021bym}.
One can prove that the new BDS function \eqref{eq:goodBDS} has same infrared part and same collinear limit behavior as \eqref{eq:badBDS} by computing the difference between the two functions:
\begin{equation}
\tilde{\mathcal{I}}_{\mathrm{tr}(F^3), 4, \text{BDS}}^{(2), \text{M.T.}} - \mathcal{I}_{\mathrm{tr}(F^3), 4, \text{BDS}}^{(2), \text{M.T.}} = -B_3 \left( \mathcal{G}_1^{(1)} - \mathcal{G}_2^{(1)} - B_1 \mathcal{G}_3^{(1)} \right) \left( \mathcal{G}_1^{(1)} - \mathcal{G}_2^{(1)} + B_2 \mathcal{G}_3^{(1)} \right) \,.
\label{eq:diffBDS4pt}
\end{equation}
Since $\mathcal{G}_1^{(1)} \big|_{\text{div.}} = \mathcal{G}_2^{(1)} \big|_{\text{div.}}$ and $\mathcal{G}_3^{(1)} \big|_{\text{div.}} = 0$, the difference is IR finite. Moreover, the above formula will vanish in the collinear limits $p_1 \parallel p_4$ or $p_3 \parallel p_4$, because the factor $B_3$ vanishes in the limit.\footnote{$\mathcal{G}_{a}^{(1)}$ have only logarithmic divergence, thus the difference \eqref{eq:diffBDS4pt} still goes to zero in the collinear limits.}
\paragraph{Constraints at the symbol level.}
Using the BDS function \eqref{eq:goodBDS} and one-loop result, one obtains the two-loop divergent terms at orders $1/\epsilon^m, m=4,3,2,1$. By matching their symbol with our ansatz \eqref{eq:ansatzSymbol}:
\begin{equation}
\mathcal{S} \Big(\mathcal{I}_{\mathrm{tr}(F^3), 4, \text{BDS}}^{(2), \text{M.T.}} \Big) \Big|_\text{div.} = {\cal S} \Big( \mathcal{I}_{\mathrm{tr}(F^3), 4}^{(2), \text{M.T.}} \Big) \Big|_\text{div.} \,,
\end{equation}
we can solve for 353 parameters and the remaining degree of freedom is 207.
Next, the collinear limits for the finite remainder function $\mathcal{R}_{\mathrm{tr}(F^3), 4}^{(2), \text{M.T.}}$ should match the two-loop three-point remainder for the maximum transcendence part as
\begin{equation}
\label{eq:4ptremainderCL}
\mathcal{R}_{\mathrm{tr}(F^3), 4}^{(2), \text{M.T.}} =
\Big(\mathcal{I}_{\mathrm{tr}(F^3), 4}^{(2), \text{M.T.}} - \mathcal{I}_{\mathrm{tr}(F^3), 4, \text{BDS}}^{(2), \text{M.T.}} \Big)_\text{fin.}
\ \xlongrightarrow[\mbox{or {$p_4 \parallel p_1$}}]{\mbox{$p_4 \parallel p_3$}} \ \mathcal{R}_{\mathrm{tr}(F^3), 3}^{(2), \text{M.T.}} \, .
\end{equation}
The formula should hold at Symbol level, same as one-loop case, we have
\begin{equation}
\label{eq:4pt2loopCL}
\mathcal{S} \left(\mathcal{G}_2^{(2)} \right) \ \xlongrightarrow{\mbox{$p_4 \parallel p_3$}} \ \mathcal{S} \left({\cal R}_{\mathrm{tr}(F^3), 3}^{(2), \text{M.T.}}\right) \, , \qquad \mathcal{S} \left(\mathcal{G}_5^{(2)}\right) \ \xlongrightarrow{\mbox{$p_4 \parallel p_3$}} 0 \, .
\end{equation}
The collinear limits of the symbol letters are given explicitly in Appendix~\ref{app:letterCL}. After this step, the number of free parameters reduces to 119.
Furthermore, the spurious poles should be eliminated similar to the one-loop formula \eqref{eq:4pt1loopSP}, and this provides the following constraints on the two-loop correction functions:
\begin{equation}
\label{eq:4pt2loopSP}
\mathcal{G}_5^{(2)} \, \xlongrightarrow{\mbox{$\langle 34 \rangle \rightarrow 0$}} 0 \,, \qquad \mathcal{G}_1^{(2)}-\mathcal{G}_2^{(2)} \xlongrightarrow{\mbox{$ \langle 24 \rangle \rightarrow 0$}} 0 \,, \qquad \ \mathcal{G}_3^{(2)} \xlongrightarrow{\mbox{$ \langle 24 \rangle \rightarrow 0$}} 0 \,.
\end{equation}
As a technical point, we mention that the variable $\mathrm{tr}_5$, defined in \eqref{eq:tr5}, occurs in the two-loop master integrals. Its limit can be taken in the following way:
\begin{equation}
\mathrm{tr}_5 = s_{14}s_{23} - s_{12}s_{34} + s_{24} s_{13} - 2 \langle 24 \rangle \left[ 41 \right] \langle 13 \rangle \left[ 32 \right] \ \xlongrightarrow{\mbox{$ \langle 24 \rangle \rightarrow 0$}} s_{14} s_{23} - s_{12} s_{34} \,,
\end{equation}
where a proper definition on the LHS is used.
Since the spurious pole must cancel to all order in the $\epsilon$ expansion of the form factor, we also consider its cancellation at $\mathcal{O}(\epsilon)$ order. We find that the $\epsilon^1$-order also provide useful new constraints. They will constrain the coefficients of $I_{\text{TP}}^{(2)}$ and $I_{\text{BPb}}^{(2)}$ and $I_{\text{dBox2c}}^{(2)}$, which are all $\mu$-term master integrals. (Recall that the integral $I_{\text{dBox2c}}^{(2)}$ has contribution only starting from $\epsilon$-order.)
We also find that once the cancellation of spurious poles is satisfied at $\epsilon^1$-order, it will hold for any order of $\epsilon$; further details are given in Appendix~\ref{app:HigherOrder}.
The condition of spurious-pole cancellations can solve for 66 parameters, and the remaining freedom of degree is 53.
\paragraph{Constraints at the function level.}
Since the Symbol does not concern the terms that contain transcendental numbers such as $\pi, \zeta_n$, possible constraints may not be captured by using the symbol alone. Thus we need to consider the full functional form of the master integrals, which have been computed in \cite{Canko:2020ylt}.
Since we only need to fix the coefficients, it is convenient to do numerical computation with high enough precision. Details of performing numerics will be discussed in Appendix~\ref{app:fullFF}, and here we focus on the solution to the constraints.
By repeating the above steps at the function level, the remaining degrees of freedom can be reduced to
40 (by IR), 24 (by collinear limits), and 20 (by the cancellation of spurious poles).\footnote{We mention that the numerical collinear limit can be taken with the parameterization \eqref{eq:colllinearofs}.}
We summarize the constraints and corresponding fixed parameters in Table~\ref{tab:solvingAnsatz2loop4ptF3}.
We point out that two of 20 degrees of freedom only change results at $\mathcal{O}(\epsilon)$ order which will be explained in the next subsection.
All remaining parameters can be fixed by simple unitarity cuts as discussed later in Section~\ref{sec:unitarity4pt}.
\begin{table}[t]
\centering
\vskip .1 cm
\begin{tabular}{| l | c |}
\hline
Constraints & Parameters left \\ \hline \hline
Starting ansatz & 1105 \\ \hline
Symmetry of $(p_1\leftrightarrow p_3)$ & 560 \\ \hline
IR (Symbol) & 207 \\ \hline
Collinear limit (Symbol) & 119 \\ \hline
Spurious pole (Symbol) & 53 \\ \hline
IR (Function) & 40 \\ \hline
Collinear limit (Funcion) & 24 \\ \hline
Spurious pole (Funcion) & 20 \\ \hline
Simple unitarity cuts & 0 \\ \hline
\end{tabular}
\caption{Solving for parameters via constraints.
\label{tab:solvingAnsatz2loop4ptF3}
}
\end{table}
\subsubsection{Building blocks for the remaining parameters}
\label{sec:buildingblocksFF4pt}
Before applying further unitarity-cut constraints, it is instructive to first analyze the remaining degrees of freedom.
The terms depending on the remaining free parameters can be organized into three groups:
\begin{align}
\label{eq:BuildingBlock}
\left( B_1+B_2 \right) \tilde{G}_{1,\alpha}^{(2)} \,, \qquad & \alpha = 1, \ldots, 8 \,, \\
B_3 \tilde{G}_{2,\beta}^{(2)} \,, \qquad & \beta = 1, \ldots, 7\,, \nonumber \\
B_4 \tilde{G}_{3,\gamma}^{(2)} + \left( p_1\leftrightarrow p_3 \right) \,, \qquad & \gamma = 1, \ldots, 5\,,\nonumber
\end{align}
where $\tilde{G}_{1,\alpha}^{(2)}$ are
\begin{align}
& \tilde{G}_{1,1}^{(2)} = I_{\text{dBox2c}}^{(2)}(1,2,3,4) + I_{\text{dBox2c}}^{(2)}(3,2,1,4) \,, \qquad\quad\qquad\quad \tilde{G}_{1,2}^{(2)} = \tilde{G}_{1,1}^{(2)} |_{\left( p_2 \leftrightarrow p_4 \right)} \,, \nonumber \\
& \tilde{G}_{1,3} = I_{\text{BPb}}^{(2)}(1,2,3,4)- I_{\text{BPb}}^{(2)}(4,3,2,1) + \left(p_1 \leftrightarrow p_3 \right) \,, \qquad \ \tilde{G}_{1,4}^{(2)} = \tilde{G}_{1,3}^{(2)} |_{\left( p_2 \leftrightarrow p_4 \right)}\,, \nonumber \\
& \tilde{G}_{1,5} = I_{\text{TP}}^{(2)}(1,2,3,4) + I_{\text{TP}}^{(2)}(3,2,1,4) \, , \nonumber\\
& \tilde{G}_{1,6}^{(2)} = \tilde{G}_{1,5} |_{\left( p_i \rightarrow p_{i+1} \right)} \,, \qquad\quad
\tilde{G}_{1,7}^{(2)} = \tilde{G}_{1,5} |_{\left( p_i \rightarrow p_{i+2} \right)} \,, \qquad\quad
\tilde{G}_{1,8}^{(2)} = \tilde{G}_{1,5} |_{\left( p_i \rightarrow p_{i+3} \right)} \,,
\end{align}
and expressions for $\tilde{G}_{2,\beta}^{(2)}$ and $\tilde{G}_{3,\gamma}^{(2)}$ are a little lengthy and we give them in Appendix~\ref{app:BuildingBlocks}.
The eight functions $\tilde{G}_{1,\alpha}^{(2)}$ in the first group functions are a little special, because they are combinations of three special class of UT integrals:
\begin{equation}
\label{eq:3mutermintegrals}
I_{\text{TP}}^{(2)} \,, \qquad I_{\text{BPb}}^{(2)}\,, \qquad I_{\text{dBox2c}}^{(2)} \,,
\end{equation}
of which the numerators are proportional to $\mu_{ij} = l_i^{-2\epsilon} \cdot l_j^{-2\epsilon}$, see Appendix~\ref{app:UT}.
It will be necessary to apply $D$-dimensional unitarity cuts to determine their coefficients.
On the other hand, the other two groups $\tilde{G}_{2,\beta}^{(2)}$ and $\tilde{G}_{3,\gamma}^{(2)}$ are free of these $\mu$-term integrals.
We also point out that $\tilde{G}_{1,1}^{(2)}$ and $\tilde{G}_{1,2}^{(2)}$ contain only $I_{\text{dBox2c}}^{(2)}$ which is of $\mathcal{O}(\epsilon)$ order, thus they are irrelevant if one is only interested in getting the $\epsilon^0$ order results of the form factor.
All the above functions are free from infrared divergences, and they also satisfy the following collinear behavior:
\begin{equation}
\tilde{G}_{1,\alpha}^{(2)} \xlongrightarrow{\mbox{$ p_i \parallel p_j $}} 0 \,, \qquad \tilde{G}_{2,\beta}^{(2)} \xlongrightarrow{\mbox{$ \langle 24 \rangle \rightarrow 0 $}} 0 \,, \qquad \tilde{G}_{3,\gamma}^{(2)} \xlongrightarrow{\mbox{$ \langle 14 \rangle \rightarrow 0 $}} 0 \,,
\end{equation}
where $p_i \parallel p_j$ means the collinear limits of any pair of momentum.
We can understand why they are not constrained in the above procedure as follows:
\begin{itemize}
\item $\tilde{G}_{1,\alpha}^{(2)}$ free from all spurious poles because $B_1+B_2 = 1$.
\item $\tilde{G}_{2,\beta}^{(2)}$ free from the collinear limits $p_3 \parallel p_4$ and $p_1 \parallel p_4$ because they both give $B_3 \rightarrow 0$.
\item $\tilde{G}_{3,\gamma}^{(2)}$ vanish in the colliear limit $p_1 \parallel p_4$(meanwhile $B_4$ turns to infinity), which is covered by the requirement of spurious pole $\langle 14 \rangle \rightarrow 0$.
\end{itemize}
By analyzing the $\tilde{G}^{(2)}$ functions in \eqref{eq:BuildingBlock}, we find that the remaining 20 parameters can be fix by the coefficients of the master integrals
\begin{equation}
\label{eq:2loop4pointmaster}
I_\text{BubBox}^{(2)} \,, \quad I_\text{TBox0}^{(2)} \,, \quad I_\text{dBox1a}^{(2)} \,, \quad I_\text{BPb}^{(2)} \,, \quad I_\text{dBox2c}^{(2)} \,, \quad I_\text{TP}^{(2)} \,.
\end{equation}
Each above master integral has 8 different external-particle orders $(i,j,k,l)$, thus there are 48 coefficients but only 20 of them are independent.
We would like to stress that the constraints that we apply here, including the infrared structures, the collinear limits, and the spurious-pole cancellations, are the same for general gauge theories. Therefore, form factors of different gauge theories are different only by the value for the remaining 20 parameters.
\subsubsection{Pure-YM result via $D$-dimensional unitarity cut}
\label{sec:unitarity4pt}
In this subsection, we apply unitarity cuts to fix the remaining free parameters.
We will use the $D$-dimensional unitarity-IBP method to determine the form factor in pure YM theory.
Form factors in other theories will be discussed in the next subsection.
\begin{figure}[t]
\centering
\includegraphics[scale=0.6]{figs/simpletwocut_new.eps}
\caption{Unitarity cuts that can determine the remaining degrees of freedom.}
\label{fig:unitaritycut}
\end{figure}
The remaining degrees of freedom are the coefficients of the master integrals \eqref{eq:2loop4pointmaster}. To determine them, we find it is sufficient to consider only one type of unitarity cuts shown in Figure~\ref{fig:unitaritycut}.
Since \eqref{eq:2loop4pointmaster} involve $\mu$-term integrals $\{I_{\text{TP}}^{(2)}, I_{\text{BPb}}^{(2)}, I_{\text{dBox2c}}^{(2)}\}$ whose numerators depends on $\mu_{ij} = l_i^{-2\epsilon} \cdot l_j^{-2\epsilon}$. To determine their coefficients, it is not enough to consider four-dimensional cuts.
We will employ the $D$-dimensional unitarity cut based on the strategy of unitarity-IBP that has been applied to multi-loop computation of form factors (and Higgs amplitudes) in \cite{Jin:2018fak, Jin:2019ile, Jin:2019opr, Jin:2020pwh} and pure gluon amplitudes in \cite{Boels:2018nrr, Jin:2019nya}.\footnote{Similar strategy has also been used in the numerical unitarity approach \cite{Abreu:2017xsl, Abreu:2017hqn},
and idea of applying cuts to simplify IBP has been used in \emph{e.g.}~\cite{Kosower:2011ty, Larsen:2015ped, Ita:2015tya, Georgoudis:2016wff}.}
The strategy can be outlined as follows:
\begin{equation}
\mathcal{F}_4^{(2)}\Bigr|_{\text{cut}}
= \sum_{\rm hel.} {\cal F}_4^{(0)} {\cal A}^{(0)}_4 {\cal A}^{(0)}_4
\xrightarrow[\text{gauge-invariant basis}]{\text{projection at}} \sum_{\alpha} {f}^{\alpha}\mathbf{B}_{\alpha}
\xrightarrow{\ \text{cut-IBP}\ } \sum_{\text{cut permitted } I_i} c_i I^{(2)}_i \ .
\end{equation}
Below we explain each step in more detail.
First, the product of the tree amplitudes under $D$-dimensional cuts provides the cut integrands. We compute the tree form factor by Feynman rules and the perform the helicity sum by using
\begin{align}
\label{eq:polarsum}
\varepsilon^\mu(l_i) \circ \varepsilon^\nu(l_i) \equiv \sum_{\rm hel.}
\varepsilon^\mu(l_i) \varepsilon^\nu(l_i)
=\eta^{\mu\nu}-\frac{q^\mu_i l_i^\nu+q_i^\nu l_i^\mu}{q_i\cdot l_i},
\quad i=1,..,4 \,,
\end{align}
where $q_i^\mu$ are arbitrary light-like reference momenta.
In this way, we obtain the cut integrands which are tensor integrals containing inner products of polarization vectors and loop momenta.
Next, we perform integrand reduction and obtain scalar integrals by projecting the integrand to a set of gauge-invariant basis, as introduced in \emph{e.g.}~\cite{Boels:2018nrr}.
Following \cite{Boels:2018nrr}, the basis of $n$-gluon form factors and scattering amplitudes can be constructed using gauge invariant building blocks $\mathbf{A}_i^{jk}$ and $\mathbf{D}_{ij}$:
\begin{equation}
\mathbf{A}_i^{jk} = \delta^{\varepsilon_i p_i}_{p_j p_k} \,, \qquad \mathbf{D}_{ij} = \varepsilon^{\perp}_i \cdot \varepsilon^{\perp}_j=\frac{\delta^{\varepsilon_i p_1 p_2 p_3 p_4}_{\varepsilon_j p_1 p_2 p_3 p_4}}{\delta^{p_1 p_2 p_3 p_4}_{p_1 p_2 p_3 p_4}} \,,
\end{equation}
in which $\varepsilon_i^{\perp}$ is the component of $\varepsilon_i$ which is perpendicular to all $p_j$, and
\begin{equation}
\delta^{a_1 \cdots a_n}_{b_1 \cdots b_n} = \det \left( a_i\cdot b_j \right)_{n\times n} \, .
\end{equation}
For the four-gluon form factor under consideration, one can choose the following set of $\mathbf{A}$ and $\mathbf{D}$:
\begin{align}
& \mathbf{A}_1^{23} \,, \ \mathbf{A}_1^{24} \,, \ \mathbf{A}_2^{34} \,, \ \mathbf{A}_2^{31} \,, \ \mathbf{A}_3^{41} \,, \ \mathbf{A}_3^{42} \,, \ \mathbf{A}_4^{12} \,, \ \mathbf{A}_4^{13} \,, \ \nonumber \\
& \mathbf{D}_{12} \,, \ \mathbf{D}_{13} \,, \ \mathbf{D}_{14} \,, \ \mathbf{D}_{23} \,, \ \mathbf{D}_{24} \,, \ \mathbf{D}_{34} \,.
\end{align}
Each gauge invariant basis should contain all four polarization vectors $\varepsilon_i$ and also depend on them linearly. In total, there are 43 elements in the set of gauge invariant basis, including:
16 $(\mathbf{A})^4$-type, 24 $(\mathbf{A})^2\mathbf{D}$-type, and 3 $(\mathbf{D})^2$-type ones.
Since 43 is not a small number, it would be non-trivial to project the integrand on this set of basis.
Fortunately, there is an important simplification for our problem. The $\mathbf{D}$-terms vanish in four-dimension, so $(\mathbf{A})^2\mathbf{D}$ and $(\mathbf{D})^2$-type basis can be neglected in the HV scheme \cite{tHooft:1972tcz}, which is enough for our consideration of the maximally transcendental part.\footnote{Note that all internal momenta, as well as the helicity sum for the cut gluon states, are in $D$ dimensions.} Therefore we are left with a simpler gauge-invariant basis with only 16 $\mathbf{A}^4$-type elements, which will be denoted by $\mathbf{B}_{\alpha}$.
We would like to project the cut integrand in this set of basis as
\begin{equation}
\mathcal{F}_4^{(2)}\Bigr|_{\text{cut}} = \sum_{\alpha} {f}^{\alpha}(l_i, p_j)\mathbf{B}_{\alpha}\,,
\end{equation}
where only the $\mathbf{B}_{\beta}$ term contains polarization vectors.
This can computed by performing the following contraction with the gauge invariant basis as (see also \cite{Jin:2019opr})
\begin{equation}
\label{contract-gib}
f^\alpha= \left( \mathcal{F}_{\text{cut}} \circ \mathbf{B}_{\alpha} \right) \mathbf{G}^{\alpha\beta} \, ,
\end{equation}
in which $\mathbf{G}_{\alpha\beta} = \mathbf{B}_{\alpha} \circ \mathbf{B}_{\beta}$, and $\mathbf{G}^{\alpha\beta}$ is the inverse of $\mathbf{G}_{\alpha\beta}$.
The `$\circ$' product is defined as in \eqref{eq:polarsum}, and here the helicity sum is for the four external polarization vectors $\varepsilon_i$.
A technical challenge here is that the $16\times 16$ matrix $\mathbf{G}^{\alpha\beta}$ is still quite complicated, and it is the main obstacle to performing an analytical evaluation; therefore in this step, we have carried out the gauge-invariant basis contraction numerically.
The basis coefficients $f^\alpha$ are functions of Lorentz product of momenta, and thus it can be directly reduced using IBP reduction \cite{Chetyrkin:1981qh, Tkachov:1981wb}, with \emph{e.g.}~public codes \cite{Smirnov:2019qkx, Klappert:2020nbg}.
In this way, we obtain the coefficients $c_i^\alpha$ of master integrals $I_i$ that contain four cut propagators as shown in Figure~\ref{fig:unitaritycut}.
So far we have not specified the helicities of external gluons. The polarization vectors can be set to the $\pm$ helicities by the following rules for the basis:
\begin{equation}
\mathbf{A}_i^{jk}\bigr|_{\epsilon_i\rightarrow \epsilon_i^+}\rightarrow [i|j|k|i] \, , \qquad \mathbf{A}_i^{jk}\bigr|_{\epsilon_i\rightarrow \epsilon_i^-}\rightarrow \langle i|j|k|i\rangle \, .
\end{equation}
After being divided by the tree-level form factor $\mathcal{F}_4^{(0)}$, $\frac{\mathbf{B}_{\alpha}}{\mathcal{F}_4^{(0)}}\bigr|_{\text{helicity}}$ can be rewritten in terms of $s_{ij}$ and $\mathrm{tr}_5$. For example
\begin{equation}
{\mathbf{A}_1^{23} \mathbf{A}_2^{34} \mathbf{A}_3^{41} \mathbf{A}_4^{12} / \mathcal{F}_4^{(0)}} \bigr|_{(+---)} = -\frac{s_{12}s_{13}s_{14}}{2} \left( s_{14}s_{23}-s_{13}s_{24}+s_{12}s_{34} - \mathrm{tr}_5 \right).
\end{equation}
We can now reconstructed the analytical expression of the maximal transcendentality part of the master integral coefficients.
The coefficients are generally non-trivial rational functions of kinematics and dimensional regularization parameter $\epsilon$. For the four-gluon form factor at hand, there are a lot of kinematic variables, and in general, it can be difficult to reconstruct their complete form.
Since the focus of this paper is the maximal transcendentality part, the job is significantly simplified: using the simple ansatz \eqref{eq:ansatzFullF3}, the kinematic structure is known in advance, and there are 5 numerical parameters to determine which can be fixed by using only 5 numerical points.
We find the numerical results match perfectly with the ansatz form \eqref{eq:ansatzFullF3}, and the coefficients are also all small rational numbers. In this way, we fix the coefficient of all coefficients for the master integrals \eqref{eq:2loop4pointmaster}.
The final solutions of master coefficients are provided in the auxiliary files.
We mention that besides the master integrals \eqref{eq:2loop4pointmaster}, the cut will also determine the following other master integrals:
\begin{align}
& I_\text{BPa}^{(2)} \,, \ I_\text{BubBox2}^{(2)} \,, \ I_\text{dBox1b}^{(2)} \,, \ I_\text{dBox2a}^{(2)} \,, \ I_\text{dBox2b}^{(2)} \,, \ I_\text{TBub2}^{(2)} \,, \\
& I_\text{TT0}^{(2)} \,, \ I_\text{TT3a}^{(2)} \,, \ I_\text{TT3b}^{(2)}\,, \ I_\text{TBox1}^{(2)} \,, \ I_\text{TBox2a}^{(2)} \,, \ I_\text{TBox2b}^{(2)} \,. \nonumber
\end{align}
Their coefficients are related to the coefficients of other integrals by the previous bootstrap constraints. We find that the unitarity-cut results are fully consistent with the bootstrap computation. This provides a strong consistency check for our results.
\subsection{Equivalence between $\mathcal{N}=4$ SYM and QCD}
\label{sec:MTPtrF3}
In this subsection, we will first consider the difference of the form factors between different gauge theories. We find that for the maximally transcendental parts of the form factors, the difference between any gauge theory and the pure YM theory will only depend on two free parameters, where the key idea is to apply some universal unitarity cuts.
Next, we obtain the form factor result in $\mathcal{N}=4$ SYM by computing the difference with a simple four-dimensional unitarity cut. We will show that the difference comes from only the contribution that contains a fermion loop.
Finally, we compare the form factors between $\mathcal{N}=4$ SYM and QCD and show that the maximal transcendentality principle still holds.
\subsubsection{Difference between different theories}
As already mentioned at the end of Section~\ref{sec:bootstrapFF4pt2loop}, different gauge theories can only be different by the value for the remaining 20 parameters. Let us denote a general gauge theory that contains a YM sector as ``Theory-X''. We define the difference between the form factor results of Theory-X and pure YM theory as $\Delta^{(2), \text{Theory-X}}_{\text{M.T.}}$:
\begin{equation}
\Delta^{(2), \text{Theory-X}}_{\text{M.T.}} = \left( \mathcal{I}_{\mathrm{tr}(F^3),4}^{(2), \text{M.T.}} \right) \Big|_{\text{Theory-X}} - \left( \mathcal{I}_{\mathrm{tr}(F^3),4}^{(2), \text{M.T.}} \right) \Big|_{\text{pure YM}} \, .
\end{equation}
It should be clear that $\Delta^{(2), \text{Theory-X}}_{\text{M.T.}}$ can only come from the contribution of particles other than gluons.
Rather than naively expressing $\Delta^{(2)}_{\text{M.T.}}$ as a linear combination of 20 building-blocks listed in \eqref{eq:BuildingBlock},
one can show that $\Delta^{(2)}_{\text{M.T.}}$ has a much more compact form that has only two free parameters.
The idea is to consider some universal unitarity cuts that are theory-independent.
To be concrete, we consider four types of unitarity cuts shown in Figure~\ref{fig:samecuts2loop4point}. These cuts are special in the sense that they can only allow internal gluon configuration, therefore, the coefficients of the master integrals which can be detected by these cuts must be the same for any gauge theory that contains a Yang-Mills sector. The master integrals detected by these cuts are listed below:
\begin{align}
\textrm{cut-(a)}: \quad & I_\text{dBox2a}^{(2)},\ I_\text{dBox2b}^{(2)},\ I_\text{dBox2c}^{(2)},\ I_\text{BPa}^{(2)},\ I_\text{BPb}^{(2)}, \ I_\text{TP}^{(2)} \ \ \textrm{for all orderings of external particles.} \nonumber\\
\textrm{cut-(b)}: \quad& I_\text{BubBox0}^{(2)} \ \ \textrm{for all orderings of external particles.} \nonumber\\
\textrm{cut-(c)}: \quad& I_\text{dBub}^{(2)}(1,2;1,2,4),\ I_\text{TT0}^{(2)}(4,1,2),\ I_\text{TBox0}^{(2)}(2,3,4).\nonumber \\
\textrm{cut-(d)}: \quad& I_\text{TT1a}^{(2)}(4,1,2),\ I_\text{dBox1a}^{(2)}(2,3,4),\ I_\text{TBox0}^{(2)}(4,1,2).\nonumber
\end{align}
\begin{figure}[t]
\centering
\subfloat[]{\includegraphics[scale=0.45]{figs/samecut2loop4point1.eps}}
\subfloat[]{\includegraphics[scale=0.45]{figs/samecut2loop4point2.eps}}
\subfloat[]{\includegraphics[scale=0.45]{figs/samecut2loop4point3.eps}}
\subfloat[]{\includegraphics[scale=0.45]{figs/samecut2loop4point4.eps}}
\caption{The unitarity cuts which are same for general gauge theories.}
\label{fig:samecuts2loop4point}
\end{figure}
Since these master integrals are determined by the pure YM theory, they cannot occur in $\Delta^{(2)}_{\text{M.T.}}$. By inspecting their relation with the remaining building blocks in \eqref{eq:BuildingBlock}, we find that $\Delta^{(2)}_{\text{M.T.}}$ only depends on two functions of $\tilde{G}_{3,\gamma}$ and can be given as
\begin{equation}
\label{eq:difference}
\Delta^{(2)}_{\text{M.T.}} = B_4 \left( x_1 \tilde{G}_{3,1}^{(2)} + x_2 \tilde{G}_{3,3}^{(2)} \right) + (p_1 \leftrightarrow p_3) \,.
\end{equation}
The two free parameters $x_{1,2}$ can be fixed by the coefficients of $I_{\text{TBox0}}^{(2)}(1, 4, 3)$ and $I_{\text{dBox1a}}^{(2)}(1, 4, 3)$, which have coefficients $B_5( -\frac{3}{4} x_1 + x_2)$ and $\frac{1}{4} B_5 x_1$ respectively in the above formula.
Since they are free from $\mu$-terms, it is enough to apply a four-dimensional unitarity cut to fix them.
\subsubsection{Two-loop result of $\mathcal{N}=4$ SYM}
\begin{figure}[t]
\centering
\includegraphics[scale=0.5]{figs/simpletwocut_new2.eps}
\caption{The unitarity cut to determine $\Delta^{(2), \mathcal{N}=4}_{\text{M.T.}}$.}
\label{fig:unitaritycut2}
\end{figure}
Now we can consider the form factor result in $\mathcal{N}=4$ super Yang-Mills.
Since we have already obtained the pure YM result, it will be enough to compute $\Delta^{(2), \mathcal{N}=4}_{\text{M.T.}}$ given in \eqref{eq:difference}.
As we discussed before, we will consider a four-dimensional cut as in Figure~\ref{fig:unitaritycut} with the ordering of external particles chosen as $(ijkl)=(2341)$, which is given in Figure~\ref{fig:unitaritycut2}. This cut can fix the coefficients of $I_{\text{TBox0}}^{(2)}(1,4,3)$ and $I_{\text{dBox1a}}^{(2)}(1,4,3)$.
The cut integrand corresponds to the tree product of a four-point tree form factor and two four-point tree amplitudes:
\begin{align}
\int \prod_{i=1}^4 d\eta_{l_i}^4 \mathcal{F}_{\mathrm{tr}(F^3), 4}^{(0), \text{MHV}}(2^-, l_3, l_2, l_1) \mathcal{A}_4^{(0), \text{MHV}}(3^-, l_4, -l_2, -l_3) \mathcal{A}_4^{(0), \text{MHV}} (4^+, 1^-, -l_1, -l_4) \,.
\end{align}
To determine $\Delta^{(2), \mathcal{N}=4}_{\text{M.T.}}$, it is enough to consider configurations that are different from pure YM theory.
This means that we only need to consider the internal particles that contain scalars or fermions, which are
\begin{align}
& \mathcal{F}_{\mathrm{tr}(F^3), 4}^{(0)}(2^-, l_3^{\psi/\bar{\psi} /\phi}, l_2^{\bar{\psi}/\psi /\bar{\phi}}, l_1^{-}) \mathcal{A}_4^{(0)}(3^-, l_4^{+}, -l_2^{\psi/\bar{\psi} /\phi}, -l_3^{\bar{\psi}/\psi/\bar{\phi}}) \mathcal{A}_4^{(0)}(4^+,1^-,-l_1^{+},-l_4^{-}) \,, \\
& \mathcal{F}_{\mathrm{tr}(F^3), 4}^{(0)}(2^-, l_3^-, l_2^{\psi/\bar{\psi} /\phi}, l_1^{\bar{\psi}/\psi/\bar{\phi}}) \mathcal{A}_4^{(0)}(3^-, l_4^{\psi/\bar{\psi}/\phi}, -l_2^{\bar{\psi}/\psi/\bar{\phi}}, -l_3^+) \mathcal{A}_4^{(0)}(4^+,1^-,-l_1^{\psi/ \bar{\psi}/\phi},-l_4^{\bar{\psi}/\psi/\bar{\phi}})\,,\nonumber
\end{align}
After performing a unitarity-IBP computation similar to Section~\ref{sec:unitarity4pt}, we find that only the following configuration contributes to the maximally transcendental part:
\begin{align}
\label{eq:fermioncut}
\mathcal{F}_{\mathrm{tr}(F^3), 4}^{(0)}(2^-, l_3^-, l_2^{\psi/\bar{\psi}}, l_1^{\bar{\psi}/\psi}) \mathcal{A}_4^{(0)}(3^-, l_4^{\psi/\bar{\psi}}, -l_2^{\bar{\psi}/\psi}, -l_3^+) \mathcal{A}_4^{(0)}(4^+, 1^-, -l_1^{\psi/\bar{\psi}}, -l_4^{\bar{\psi}/\psi}) \,.
\end{align}
In particular, all configurations that involve scalars contribute only to lower transcendental parts.
This is a very important fact as we will discuss shortly: it implies that the maximal transcendentality principle still holds between ${\cal N}$=4 SYM and QCD.
\begin{table}[t]
\centering
\vskip .1 cm
\begin{tabular}{|m{2.5cm}<{\centering} |m{2.5cm}<{\centering}|m{3.5cm}<{\centering}| m{3.5cm}<{\centering} |}
\hline
Master & Topology & $\Delta^{(2), \mathcal{N}=4}_{\text{M.T.}}$ & pure YM \\ \hline
$I_{\text{TBub2}}^{(2)}(3, 4, 1)$ & \includegraphics[scale=0.3]{figs/TBub2_341.eps} & $\left( 0, 0, 0, 0, -\frac{9}{2} \right)$ & $\left( \frac{11}{8}, \frac{7}{4}, -\frac{3}{8}, 0, \frac{9}{2} \right)$ \\ \hline
$I_{\text{BubBox}}^{(2)}(1, 4, 3)$ & \includegraphics[scale=0.3]{figs/BubBox_143.eps} & $\left( 0, 0, 0, 0, 7 \right)$ & $\left( -3, -3, 1, 0, -7 \right)$ \\ \hline
$I_{\text{dBox1a}}^{(2)}(1, 4, 3)$ & \includegraphics[scale=0.3]{figs/dBox1a_143.eps} & $\left( 0, 0, 0, 0, -1 \right)$ & $\left( \frac{1}{2}, \frac{1}{2}, 0, 0, 1 \right)$ \\ \hline
$I_{\text{dBox1b}}^{(2)}(1, 4, 3)$ & \includegraphics[scale=0.3]{figs/dBox1b_143.eps} & $\left( 0, 0, 0, 0, -3 \right)$ & $\left( \frac{1}{2}, \frac{1}{2}, 0, 0, 3 \right)$ \\ \hline
$I_{\text{TT0}}^{(2)}(3, 1, 4)$ & \includegraphics[scale=0.3]{figs/TT0_341.eps} & $\left( 0, 0, 0, 0, -2 \right)$ & $\left( \frac{1}{2}, \frac{1}{2}, 0, 0, 2 \right)$ \\ \hline
$I_{\text{TBox0}}^{(2)}(1, 4, 3)$ & \includegraphics[scale=0.3]{figs/TBox0_143.eps} & $\left( 0, 0, 0, 0, 1 \right)$ & $\left( 0, 0, 0, 0, -1 \right)$ \\ \hline
\end{tabular}
\caption{The coefficients of the master integrals detected by the unitarity cut shown in Figure~\ref{fig:unitaritycut} in $\mathcal{N}=4$ super Yang-Mills and pure Yang-Mills, the vector $(c_1,c_2,c_3,c_4,c_5)$ means $\sum_{a} c_a B_a$.
\label{table:unitarityresult}
}
\end{table}
The maximally transcendental contributions detected by the cut of \eqref{eq:fermioncut} are collected in Table~\ref{table:unitarityresult}.
We also show the corresponding result in pure YM for comparison.
One note that the difference $\Delta^{(2), \mathcal{N}=4}_{\text{M.T.}}$ are related to $B_5$.
By comparing the coefficients of $I_{\text{TBox0}}^{(2)}(1,4,3)$ and $I_{\text{dBox1a}}^{(2)}(1,4,3)$ with that of $\Delta^{(2), \mathcal{N}=4}_{\text{M.T.}}$ in \eqref{eq:difference}, one can fix the two free parameters $x_i$. One has (note that $B_4$ and $B_5$ are related by symmetry)
\begin{equation}
B_5 \left( -\frac{3}{4} x_1 + x_2 \right) = -B_5 \,, \qquad \frac{1}{4} B_5 x_1 = B_5 \,, \nonumber
\end{equation}
which gives $x_1=4$ and $x_2 = 2$. Thus we obtain the difference
\begin{equation}
\Delta^{(2), \mathcal{N}=4}_{\text{M.T.}}= B_4 \Big( 4 \tilde{G}_{3,1}^{(2)} +2 \tilde{G}_{3,3}^{(2)} \Big) + \left( p_1 \leftrightarrow p_3 \right) \,.
\end{equation}
Together with the pure-YM result, we obtain the full maximally transcendental part for the form factor in $\mathcal{N} = 4$ SYM theory.
The results of the form factors are provided in the auxiliary files.
See also Appendix~\ref{app:fullFF} for some further discussion of the results.
Finally, we can understand the result of $\Delta^{(2), \mathcal{\cal N}=4}_{\text{M.T.}}$ from the Feynman diagram point of view.
The cut configuration \eqref{eq:fermioncut} means that the internal legs $\{ l_1, l_2, l_4\}$ are all fermions. Since the internal fermions must form a fermion loop, this implies that the diagrams that contribute to $\Delta^{(2), \mathcal{\cal N}=4}_{\text{M.T.}}$ must contain a fermion loop as shown in Figure~\ref{fig:fermionloop}.
This observation will be used in the next subsection.
\begin{figure}[t]
\centering
\includegraphics[scale=0.65]{figs/fermionloop.eps}
\caption{Diagrams that contributing to $\Delta^{(2), \mathcal{N}=4}_{\text{M.T.}}$ must involve a fermions loop, as indicated by the red color color.}
\label{fig:fermionloop}
\end{figure}
\subsubsection{Maximal transcendentality principle and color factors}
The previous computation shows that the difference $\Delta^{(2), {\cal N}=4}_{\text{M.T.}}$ comes from the configurations that involve only fermions but not scalars. Since the theory of QCD consists of the pure YM theory plus fermions, this means that the kinematic parts of the planar form factors between $\mathcal{N}=4$ SYM and QCD should be identical, while only the color factors can be different because of the different color representations of fermions in the two theories.\footnote{Note that both fermions are Dirac fermions in the two theories.}
Therefore, by properly converting the fermion representation from fundamental to adjoint, one will find the maximally transcendental part in QCD and $\mathcal{N}=4$ SYM are equivalent.
Below we discuss the color factors in more detail.
It is instructive to start with a one-loop Feynman diagram shown in Figure~\ref{fig:colorfactor}(a), in which there are $n$ external gluons and all propagators are fermions.
The corresponding color factors for the case of fundamental fermions in QCD and adjoint fermions ${\cal N}=4$ SYM are respectively
\begin{align}
\text{QCD (fundamental)} : & \qquad n_f \mathbf{C}_{\text{single-trace}} \,, \label{eq:qcdcolor1} \\
\text{${\cal N}=4$ (adjoint)} : & \qquad 4 \prod_{k = 1}^n f^{b_{k} a_k b_{k+1}} = 4 N_c \mathbf{C}_{\text{single-trace}} + (\text{double traces}) \,, \label{eq:neq4color1}
\end{align}
where
\begin{equation}\label{eq:Csingletrace}
\mathbf{C}_{\text{single-trace}} = \mathrm{tr}(T^{a_1} T^{a_2} \ldots T^{a_n}) + (-1)^n \mathrm{tr}(T^{a_n} T^{a_{n-1}} \ldots T^{a_1}) \,,
\end{equation}
and we have included $n_f$ in \eqref{eq:qcdcolor1} as the flavor number of quarks in QCD and similarly, the factor $4$ in \eqref{eq:neq4color1} is because there are four gluinos in the ${\cal N}=4$ supermultiplet.
We can see that the two color factors can be identified if we apply the following rule:
\begin{equation}
\label{eq:colorRule4pt}
n_f \rightarrow 4N_c \,, \qquad \textrm{and} \qquad N_c \rightarrow \infty \,,
\end{equation}
where the large $N_c$ limit is to keep only the single-trace terms.
\begin{figure}
\centering
\subfloat[]{\includegraphics[scale=0.45]{figs/oneloopcolor.eps}}
\hskip 2cm
\subfloat[]{\includegraphics[scale=0.45]{figs/twoloopcolor.eps}}
\caption{Color factors of the one- and two-loop cases.\label{fig:colorfactor}}
\end{figure}
Now we consider the two-loop $n$-gluon form factor.
For the form factor problem under consideration, a related two-loop diagram is shown in Figure~\ref{fig:colorfactor}(b).
The part on the LHS is a one-loop $(n-k+2)$-gluon amplitude with a fermion loop, which can be regarded as Figure~\ref{fig:colorfactor}(a), and we denote its color factor as $\mathbf{C}^{(1)}_{\cal A}$. The part on the RHS is a tree-level $(k+2)$-gluon form factor, and its color factor has only single-trace terms which we denote as $\mathbf{C}^{(0)}_{\cal F}$. The color factor of the two-loop diagram is given by the product as
\begin{equation}
\sum_{a_{l_1}, a_{l_2}} \mathbf{C}^{(1)}_{\cal A}(T^{a_1,} , \ldots, T^{a_{n-k}}, T^{a_{l_1}}, T^{a_{l_2}}) \mathbf{C}^{(0)}_{\cal F}(T^{a_{n-k+1}}, \ldots, T^{a_{n}}, T^{a_{l_2}}, T^{a_{l_1}}) \,.
\end{equation}
Note that if $l_1$ and $l_2$ are not adjacent in the fermion loop, the color factor will be sub-leading in the large $N_c$ expansion.
Keeping only the leading-$N_c$ color factors, one has
\begin{equation}
\begin{aligned}
\text{QCD} : & \qquad n_f N_c \mathbf{C}_{\text{single-trace}} \,,\\
\text{${\cal N}=4$} : & \qquad 4 N_c^2 \mathbf{C}_{\text{single-trace}} \,,
\end{aligned}
\end{equation}
where $\mathbf{C}_{\text{single-trace}}$ is given in \eqref{eq:Csingletrace}.
Therefore, the two-loop color factors are also identified with each other by using the rule \eqref{eq:colorRule4pt}.
The same argument holds for other two-loop diagrams that involve a fermion loop.
Thus we can conclude that
\begin{equation}
\Delta^{(2), {\rm QCD}}_{\text{M.T.}} \Big|_{n_f \rightarrow 4N_c, N_c \rightarrow \infty} = \Delta^{(2), \mathcal{N}=4}_{\text{M.T.}} \,.
\end{equation}
In other words, we find the following correspondence for the four-point form factor up to two-loop order in the limit of $N_c\rightarrow \infty$:
\begin{align}
& \mathcal{F}_{\mathrm{tr}(F^3)}^{(1),{\cal N}=4}(1^-,2^-,3^-,4^+) = \mathcal{F}_{\mathrm{tr}(F^3)}^{(1), \text{QCD}}(1^-,2^-,3^-,4^+) \,, \\
& \mathcal{F}_{\mathrm{tr}(F^3)}^{(2),{\cal N}=4}(1^-,2^-,3^-,4^+) = \mathcal{F}_{\mathrm{tr}(F^3)}^{(2), \text{QCD}}(1^-,2^-,3^-,4^+) \Big|_{n_f \rightarrow 4N_c} \,, \nonumber
\end{align}
which mean that the principle of maximally transcendental principle still holds for the four-point form factor of $\mathrm{tr}(F^3)$.
To the best of the authors' knowledge, this form factor provides the first example of the MTP where the fermion-loop diagrams contribute to the maximally transcendental part and thus the changing of color factors involves the $n_f$ factor.
\subsection{A relation to the form factor of $\mathrm{tr}(\phi^3)$}
\label{sec:phi3andF3}
There is another interesting correspondence between two different operators $\mathrm{tr}(F^3)$ and $\mathrm{tr}(\phi^3)$ in $\mathcal{N}=4$ super Yang-Mills theory. The four-point form factor $\mathcal{F}_{\mathrm{tr}(\phi^3), 4}^{(L)}$ is half-BPS and has only the maximally transcendentality part. The results up to two loops was obtained in \cite{Guo:2021bym} and take the following form:
\begin{equation}\label{eq:F4phi3structure}
\mathcal{F}_{\mathrm{tr}(\phi^3), 4}^{(L)}(1^\phi, 2^\phi, 3^\phi, 4^+) = \mathcal{F}_{\mathrm{tr}(\phi^3), 4}^{(0)}(1^\phi, 2^\phi, 3^\phi, 4^+) \left( B_1 \mathcal{G}_1^{(L)} + B_2 \mathcal{G}_2^{(L)} \right) \,,
\end{equation}
which has two spinor factors $B_1$ and $B_2$, while the form factor of $\mathrm{tr}(F^3)$ in \eqref{eq:ansatzFullF3} contains five $B_i$'s.
Interestingly, we find that the loop correction functions $\mathcal{G}_1^{(L)}$ and $\mathcal{G}_2^{(L)}$ in \eqref{eq:F4phi3structure} are precisely the same as that of $\mathcal{F}_{\mathrm{tr}(F^3), 4}^{(L), \text{M.T.}}$ in \eqref{eq:ansatzFullF3}.
In other words, we have the relation
\begin{equation}
\mathcal{I}_{\mathrm{tr}(F^3), 4}^{(L), \text{M.T.}}(1^-, 2^-, 3^-, 4^+) = \mathcal{I}_{\mathrm{tr}(\phi^3), 4}^{(L)}(1^\phi, 2^\phi, 3^\phi, 4^+) + \sum_{a = 3}^5 B_a \mathcal{G}_{a}^{(L)} \,,
\end{equation}
at least up to two loops.
To understand this relation, we briefly review the computation for the form factor of ${\mathrm{tr}(\phi^3)}$.
The tree-level four-point form factor of ${\mathrm{tr}(\phi^3)}$ is
\begin{equation}
\mathcal{F}_{\mathrm{tr}(\phi^3), 4}^{(0)}(1^\phi, 2^\phi, 3^\phi, 4^+) = \frac{\langle 13 \rangle}{\langle 34 \rangle \langle 14 \rangle} \,.
\end{equation}
The pole structure implies that only $B_1$ and $B_2$ should occur in the coefficients of the loop correction, as given in \eqref{eq:F4phi3structure}.
Via the bootstrap method, the form factor is first constrained by infrared structure, and then the constraints of the collinear limit at one and two loops respectively are
\begin{align}
& \mathcal{I}_{\mathrm{tr}(\phi^3), 4}^{(1)} \, \xlongrightarrow{\mbox{$ p_3 \parallel p_4 $}} \mathcal{I}_{\mathrm{tr}(\phi^3), 3}^{(1)} + r_1^{[1], \text{MT}}(s_{34}, z) \,, \\
& \mathcal{R}_{\mathrm{tr}(\phi^3), 4}^{(2)} \, \xlongrightarrow{\mbox{$ p_3 \parallel p_4 $}} \mathcal{R}_{\mathrm{tr}(\phi^3), 3}^{(2)} \,,
\end{align}
and the cancellation of the spurious pole $\langle 24 \rangle$ provides the constraint
\begin{equation}
\mathcal{G}_1^{(L)}-\mathcal{G}_2^{(L)} \, \xlongrightarrow{\mbox{$ \langle 24 \rangle \rightarrow 0$}} 0 \, .
\end{equation}
At one loop, the form factor is fixed uniquely by these constraints, and since the same constraints apply also to the one-loop form factor of ${\rm tr}(F^3)$, the one-loop correction functions the $\mathcal{G}_{1,2}^{(1)}$ must be the same for both form factors.
The case is more non-trivial at two loops.
At the two-loop level, there are still some remaining degrees of freedom that are free from the above constraints.
These can be organized as eight building blocks which are the same as $\tilde{G}_{1,\alpha}^{(2)}$ listed in \eqref{eq:BuildingBlock}.\footnote{In \cite{Guo:2021bym}, the number of remaining parameters are 10. Here we reduce the number to 8, which is equal to the number of $\tilde{G}_{1,\alpha}^{(2)}$. The difference is because here we apply further the constraints come from the $\mathcal{O}(\epsilon)$ order for the spurious pole $\langle 24 \rangle$ cancellation (see Appendix~\ref{app:HigherOrder}), which was not used in \cite{Guo:2021bym}.}
These building blocks can be fixed by using simple quadruple cuts.
Since the unitarity cuts are apparently quite different for the form factors of $\mathrm{tr}(F^3)$ and $\mathrm{tr}(\phi^3)$, it is a prior not obvious at all that the coefficients of $\tilde{G}_{1,\alpha}^{(2)}$ are the same for the two form factors.
Interestingly, the explicit computations show that they are identical for the two form factors.
We do not yet have a physical explanation for this, and it would be interesting to check if this is true for more general cases.
It is worth pointing out that $\tilde{G}_{1,\alpha}^{(2)}$ contain all the $\mu$-term master integrals and $D$-dimensional cuts are needed to determine them.
Since the unitarity computation for the half-BPS form factor of ${\rm tr}(\phi^3)$ is much simpler, this relation (if true generally) may be used to simplify the computation for the ${\rm tr}(F^3)$ form factor.
\section{Summary and discussion}
\label{sec:discussion}
In this paper, we study the principle of maximal transcendentality for a class of form factors using the bootstrap method.
For the minimal form factors up to two loops, we show that the IR divergences plus some symmetry arguments are sufficient to determine the maximally transcendental part.
Some non-trivial constraints on the lower transcendental parts are also discussed in a similar way.
For the two-loop three-gluon form factors of ${\rm tr}(F^2)$, we show that the IR divergences together with collinear limit constraints are able to fix the maximal transcendental part of form factors.
While the IR and collinear constraints may not fix all parameters, an important further insight is to apply some unitarity cuts that only depend on gluon states and thus are universal for general gauge theories.
As another important part of the paper, we obtain the maximal transcendental parts of the two-loop four-point planar form factor with $\mathrm{tr}(F^3)$ operator in both $\mathcal{N}$=4 SYM and pure YM theories.
The bootstrap computation using various physical constraints and certain simple $D$-dimensional unitarity cuts are given in detail.
We find that the form factor results are different between the $\mathcal{N}$=4 SYM and pure YM theories. The difference is due to the contribution from the gluino-loop diagrams in $\mathcal{N}=4$ SYM. Importantly, the scalar-loop diagrams in $\mathcal{N}=4$ SYM do not contribute to the maximally transcendental part.
This implies that the maximally transcendental part of the $\mathcal{N}=4$ SYM result is equivalent to the corresponding form factor result in QCD, up to a proper change of color factors associated with the quark loops.
Thus the maximal transcendentality principle still holds for this form factor.
It is instructive to compare with this four-point case of the simpler three-point form factor of ${\rm tr}(F^2)$. In the latter case, the maximally transcendental parts are all the same for the $\mathcal{N}=4$ SYM, QCD, and pure YM theories,
which can be understood as neither fermion- nor scalar-loops contribute to the maximal transcendentality part.
In contrast, the MTP property for the four-point form factors appears to be more non-trivial, due to the new contribution of fermion loops.
This seems to be the first such example for the maximal transcendentality principle and suggests the MTP may hold for more general form factors, or Higgs-plue-multi-gluon amplitudes.
Similar to the four-point form factor we consider, a key step to understanding or proving MTP would be to show that the scalar-loop diagrams would have no maximally transcendental contributions in more general cases.
We should also point out there are counterexamples of MTP for other observables. For example, for the gluon amplitudes, the MTP is violated even for the simple one-loop four-gluon amplitudes. In Appendix~\ref{app:A4noMT}, we discuss this in detail and show that while the MTP applies to $A_4^{(1)}(1^-,2^-,3^+,4^+)$, it is explicitly violated by $A_4^{(1)}(1^-,2^+,3^-,4^+)$.
Some counterexamples were found in the study of high energy limit of amplitudes \cite{DelDuca:2017peo}.
Through unitarity cuts, the amplitudes can enter as building blocks in high-loop form factors, and it would be very interesting to understand further the different properties between form factors and amplitudes.
\section*{Acknowledgements}
We thank Guanda Lin for the discussion.
This work is supported in part by the National Natural Science Foundation of China (Grants No.~12175291, 11935013, 11822508, 12047503)
and by the Key Research Program of the Chinese Academy of Sciences, Grant NO. XDPB15.
We also thank the support of the HPC Cluster of ITP-CAS.
|
2,877,628,088,990 | arxiv | \section{Introduction} \label{S_INTRO}
Since the first experimental demonstration of the microwave invisibility cloak~\cite{schurig}, there has been an explosion of theoretical and practical advances in the design and analysis of electromagnetic metamaterials~\cite{Fleury & another review article}.
In contrast, the much more challenging problem of creating invisibility cloaks and metamaterials for elastodynamics has been much less studied.
Notable recent advances in the theoretical analysis of cloaks for elastic waves have been made by Milton \emph{et al.}~\cite{MiltonNJP,MiltonPRS}, Norris \& Shuvalov~\cite{NorrisWaveMotion}, Brun \emph{et al.}~\cite{BrunAPL,BrunNJP}, Farhat \emph{et al.}~\cite{FarhatSebastienPlates}, Jones \emph{et al.}~\cite{JonesIntJSolsStruct}, Colquitt \emph{et al.}~\cite{ColquittPRS,ColquittJMPS}, Guennau \emph{et al.}~\cite{Guenneau} and Parnell \emph{et al.}~\cite{ParnellPRS,ParnellWaveMotion,NorrisPRS,NorrisAPL}.
These theoretical developments have been complemented with a series of experimental implementations of multi-scale mechanical cloaks performed by the group led by Wegener~{\cite{Wegener2012,UnfealabilityCloak,WegenerPNAS}}.
\begin{figure}[h]
\renewcommand{\figurename}{\footnotesize{Fig.}}
\begin{center}
\includegraphics[width=10.0 cm]{Abaqus.pdf}
\caption{\footnotesize Two-dimensional ABAQUS simulations performed in the case of a plate without a hole (upper), a plate with an uncloaked hole (middle), and a plate with a hole surrounded by the cloak (lower). For a frequency range between 120 Hz and 240 Hz, the simulations show that the cloak reduces significantly the scattered field. }
\label{CloakEffect}
\end{center}
\end{figure}
In particular, the work by Milton, Briane, \& Willis~\cite{MiltonNJP,MiltonPRS} identifies the effects of `negative inertia' in the elastic cloak and analyses the requisite strong anisotropy, not only in the elastic compliance, but also in the inertial properties of the cloak.
The fact that, in the framework of Milton \emph{et al.}~\cite{MiltonNJP,MiltonPRS}, the mass density of the material should behave as a tensorial quantity rather than simply as a scalar is an important and striking observation.
A novel approach of dynamic homogenisation, capable of encapsulating such striking effects, was introduced and systematically studied by Craster and co-authors \cite{Craster1, Craster2}. The asymptotic theory leads to an effective equation for the envelope function in the perturbation approximation relative to a standing wave associated with a periodic lattice or continuum structure.
Norris \& Shuvalov~\cite{NorrisWaveMotion} later generalised the framework of Milton, Briane, \& Willis and showed that, although one cannot create an invisibility cloak for elastodynamics without recourse to a non-classical generalised theory of elasticity, with an appropriate choice of Gauge one can choose between creating a micropolar elastic cloak or a cloak with tensorial density, for example.
The concept, design, and theoretical analysis of elastodynamic invisibility cloaks are much more challenging compared to cloaks for membrane, acoustic, optical, and anti-plane shear waves, all of which are governed by the transformed Helmholtz equation.
Although the theoretical framework for elastic cloaks is well defined, its experimental implementation has never been successfully achieved for dynamic vector problems of elasticity.
The pioneering work by Wegener and his group for Kirchhoff-Love plates~\cite{Wegener2012} is the only significant experimental contribution in this extremely challenging area.
In a different context, multi-scale resonators were discussed in \cite{Colombi} in relation to an approximate cloaking referred to as \lq Directional cloaking' for elastic plates containing voids.
The elastic Kirchhoff-Love plate provides an efficient and rigorous framework for the analysis of elastic waves in thin plates.
The propagation of flexural waves in a Kirchhoff-Love plate is governed by the biharmonic operator and waves in an homogeneous isotropic plate can be expressed as a linear combination of solutions of the Helmholtz equation (which we refer to as ``membrane waves'') and solutions of the modified Helmholtz equation.
The latter are evanescent fields but, nevertheless, may make a significant contribution through the boundary conditions and hence play a crucial role in the dispersion of flexural waves in structured plates~\cite{NatashaMovchan1,NatashaMovchan2,EvansPorter}, in addition to the design of flexural invisibility cloaks~\cite{JonesIntJSolsStruct}.
\begin{figure}
\renewcommand{\figurename}{\footnotesize{Fig.}}
\begin{center}
\includegraphics[width= 10 cm]{Experiments.pdf}
\caption{\footnotesize { Comparison between experiments and numerical simulations in the case of an applied displacement with a frequency of 120 Hz}} \label{Pict_Exp}
\end{center}
\end{figure}
The design of the cloak presented here is distinct from the cloaks presented by Wegener \emph{et al.}~\cite{Wegener2012}, in so far as the cloak designed and constructed here is matched with a multi-scale mechanical structure, as opposed to a homogeneous continuous ambient matrix.
Moreover, the cloak which we construct here is not singular, i.e. it does not require infinite wave speeds on the interior boundary, but is based on the regularised cloaking transformation and theoretical design presented by Colquitt \emph{et al.}~\cite{ColquittPRS} for membrane waves and later for flexural waves in plates~\cite{ColquittJMPS}.
The cloak design is shown in Fig.~\ref{Abaqus_geometry}A and the numerical and experimental implementations demonstrate that the invisibility cloak is efficient.
This is illustrated in Fig.~\ref{CloakEffect}, which shows a uniform vibrating plate without a hole, a plate with an uncloaked hole, and a plate with a hole surrounded by the cloak -- the latter reduces significantly the scattered field, as predicted by the analysis.
It is emphasised that here we consider the full elastodynamical problem of wave propagation in a discrete metamaterial lattice cloak embedded within a multi-scale ambient medium; this should be distinguished from the earlier work of Wegener \emph{et al.} on mechanical lattice cloaks~\cite{UnfealabilityCloak,WegenerPNAS} wherein the static cloaking problem is considered and no waves propagate within the system.
The structure of the paper is as follows. Section \ref{sqcloak} outlines the idea of the regularised cloaking transformation leading to the practical implementation of the multi-scale structured square cloak, which is surrounded by an ambient square lattice.
The computational and experimental implementations are discussed in Section \ref{computational_exp}.
In particular, the computational section \ref{computational} presents several ABAQUS simulations to verify the efficacy of the cloaking effect, as shown in Fig. \ref{CloakEffect}.
The experimental design and results are discussed in section \ref{experiments}.
Finally, we draw together important concluding remarks in section \ref{concluding}.
\section{The Hooke-Chladni-Faraday visualisation}
\label{HCF}
Before presenting the mathematical model of the cloak and the experimental implementation, we first discuss an elegant technique that we will employ to visualise the time-harmonic vibration of plates.
The method, known as the \emph{``Hooke-Chladni-Faraday''} technique, has been employed by many researchers in physics and mechanics for almost four centuries and continues to be used today.
Indeed, the recent paper~\cite{Chen_Luo} examines the visualisation of standing waves in dynamically reconfigurable liquid-based templates, which were used to assemble micro-scale materials into ordered structures with desired properties.
This novel approach at a micro-scale level employed the idea of Faraday waves.
A three-dimensional visualisation of acoustic standing waves was reported in a recent paper \cite{3DStandingWave} that demonstrated the elegant efficiency of the classical Hooke-Chladni-Faraday method, which continues to generate new ideas and exciting results.
In particular, the experiment reported in \cite{3DStandingWave}, shows levitating micro-particles along the nodal lines of a three-dimensional standing wave.
Well before the time of Michael Faraday, Robert Hooke and then Ernst Chladni discovered an ingenious method to visualise standing waves in elastodynamics.
This was especially effective for flexural resonances in elastic plates and membranes, as described in Chladni's book~\cite{Chladni}.
The technique consists of drawing a violin bow over a metallic or glass plate that is covered with flour.
Once the plate reaches a resonance the flour collects along the nodal lines of the resonant mode providing an elegant yet efficient visualisation of the standing wave present in the plate.
Following Chladni's experimental demonstration, the eigenvalue problem for the free vibrations of a square plate with free edges has been studied by many scientists, most notably, Lord Rayleigh~\cite{RayleighBook,Rayleigh1911} and Ritz~\cite{Ritz1908,Ritz1909} during the development of the now well known Rayleigh-Ritz method.
Although no closed form solution currently exists, Ritz~\cite{Ritz1909}, over a century ago, was able to construct remarkably accurate approximate solutions for the eigenfrequencies and nodal patterns.
These classical studies have led to a fascinating area of Cymatics, which analyses methods of making sound and vibration visible and has attracted attention of engineers, mathematicians, physicists and musicians around the world.
\begin{figure}[h]
\renewcommand{\figurename}{\footnotesize{Fig.}}
\begin{center}
\includegraphics[width=15 cm]{Mode_Sand.pdf}
\caption{\footnotesize Hooke-Chladni-Faraday visualization of four eigenmodes of a square elastic plate with a free boundary.
These illustrative experiments were produced in the \lq Instabilities Lab' of the University of Trento.
}
\label{Chladni_Figures}
\end{center}
\end{figure}
For the purpose of illustration, we include in Fig. \ref{Chladni_Figures} examples of the Chladni patterns for eigenmodes of a square elastic plate with a free boundary, which are accurate and fully consistent with analytical findings. The patterns depend on the boundary conditions and on any inhomogeneities that may occur in the plate, such as voids or inclusions. In particular, Chladni patterns were never constructed for a plate with a hole surrounded by a structured cloak.
In the present paper, we visualise Hooke-Chladni-Faraday patterns for flexural waves around an obstacle surrounded by a multi-scale structured cloak and elegantly illustrate the efficacy of the cloak. An ABAQUS simulation for the mechanical configuration, identical to the one used in the experiment, provides accurate numerical data on the displacement amplitudes and stress distribution.
We present the experimental visualisation to confirm the predicted wavefront profiles.
Compared to the classical settings for the above mentioned frequency response problem for a rectangular Kirchhoff plate, we go further and consider a structured plate with a cloaked hole.
Fig. \ref{Pict_Exp} includes Hooke-Chladni-Faraday patterns used for visualisation of the cloaking effect of flexural waves. These observations are new and demonstrate scattering patterns for three configurations, which include (a) a rectangular lattice-type plate, (b) a plate with a square hole, and (c) a plate with a structured cloak enclosing the hole.
As in the original experiments by Hooke, the powder collects along the nodal lines of the vibrating plate thus indicating the position of the wavefronts, i.e. the locus of points on the wave with the same phase and zero displacement.
The Hooke-Chladni-Faraday patterns allow us to conveniently visualise the propagation of waves in the structured plate and, in particular, we can observe a significant reduction of the scattering pattern for the case (c) (as in the bottom part of Fig. \ref{Pict_Exp}). A detailed discussion of the experiment is given in the main text of the paper.
\section{The square cloak}
\label{sqcloak}
In most of the papers addressing the design of cloaks for linear waves, a singular radially symmetric push-out transformation is employed (see, for example, \cite{Pendry, ParnellPRS}) which maps a point to a finite disc.
For theoretical and computational models of continuous media, such an approach is adequate.
However, for practical implementations it poses substantial difficulties.
In particular, not only does the transformation lead to infinite wave speeds on the interior boundary of the cloak, but also the required cloaking material is characterised by an unrealistic strong anisotropy.
Indeed a lattice structure, instead of a continuum, can be used to create an invisibility cloak.
A significant advantage of lattices is that they naturally accommodate high contrasts in their compliance leading to strong anisotropy.
However, a radial discrete cloak that fits inside a circular ring would not match any periodic lattice, and the presence of a geometrical mis-match on the interface boundary leads to a substantial mis-match in the interface boundary conditions.
For the design presented here, we choose to employ a square cloak following the theoretical framework recently established in~\cite{ColquittJMPS}.
The cloak is embedded in an ambient square lattice, which is subjected to out-of-plane flexural vibrations.
We consider a thin structured plate with a defect represented by a traction free void, which significantly influences the wave field, as illustrated in Fig.
\ref{CloakEffect}.
A structured cloak, as in \cite{ColquittPRS,ColquittJMPS}, is then installed around the void.
In the cloaked configuration, we observe a significant reduction in the scattered field and, in particular, the reduction in the shadow region behind the scatterer and the restoration of the incident field represented by a plane wave.
\subsection{The regularised cloaking transformation}
Following Colquitt et al. \cite{ColquittJMPS}, we choose a regularised near-cloak, which is obtained by four push-out transformations applied to trapezoidal regions, as illustrated in Fig. \ref{Deformation_Schematic}A.
The idea of regularisation is to map a domain with a small hole (e.g. a square of semi-width $\epsilon$ as in Fig. \ref{Deformation_Schematic}A) at the centre into another domain, whose exterior boundary is preserved while the interior boundary is expanded to the required finite size.
For the regularised problem, we set the Neumann boundary condition on the boundary of the hole, which means (in the case of an elastic plate) the free edge boundary condition.
As showed recently~\cite{JonesIntJSolsStruct}, the correct choice of boundary condition is of vital importance in order to achieve cloaking.
The mapping is defined in such a way that ${\bf x} = \vec{\mathcal{F}}^{(i)}({\bf X})$ for each trapezoidal region ($j=1,2,3,4$) shown in Fig.\ref{Deformation_Schematic}A. In particular, the map for the trapezoidal region (1), with bases perpendicular to the $X_1-$axis, is defined by the formula:
\begin{equation}
\vec{\mathcal{F}}^{(1)}({\bf X}) = \begin{bmatrix}
\alpha_1 X_1 + \alpha_2 \\
\alpha_1 X_2 + \alpha_2X_2/X_1
\end{bmatrix},\qquad
\label{FF}
\end{equation}
where
\begin{equation}
\alpha_1= \frac{w}{a+w-\epsilon}, \hspace{1cm} \alpha_2= \frac{(a+w)(a-\epsilon)}{a+w-\epsilon}.
\label{coeff}
\end{equation}
Here, $0<\epsilon\ll1$ is the regularisation parameter and uppercase letters denote the undeformed configuration, whilst lowercase letters denote the deformed configuration. Prior to the transformation, the interior boundary of the trapezoidal region (1) corresponds to $X_1=\epsilon$. Following the transformation, the interior boundary of this trapezoidal region is moved to $x_1=a$.
The outer boundary of the cloak is invariant with respect to the transformation, and for the region (1), it corresponds to $X_1 = a+w$.
The other three trapezoidal regions are transformed in the same way, subject to a rotation.
\begin{figure}
\renewcommand{\figurename}{\footnotesize{Fig.}}
\begin{center}
\includegraphics[width=15.0 cm]{Transform_square_lattice.pdf}
\caption{\footnotesize (A) The function $\vec{\mathcal{F}}^{(1)}$ maps the undeformed trapezoidal region (1) to the deformed trapezoidal configuration. (B) A discrete lattice structure where the curved ligaments are oriented following the principal directions of the stiffness matrix for the continuum cloak.}
\label{Deformation_Schematic}
\end{center}
\end{figure}
\subsection{Transformed equations after the cloaking transformation}
The time-harmonic flexural deformations of a homogeneous thin elastic plate are governed by the Kirchhoff plate equation, which before the transformation has the form
$$
\Big(\nabla_X^4 - \frac{\rho h}{D} \omega^2 \Big) w({\bf X}) =0.
$$
After the application of a non-singular transformation we have
\begin{equation}
\Big(\nabla \cdot J^{-1} {\bf F} {\bf F}^{T} \nabla J \nabla \cdot J^{-1} {\bf F} {\bf F}^{T} \nabla - \frac{\rho h}{J D} \omega^2 \Big) w({\bf x}) =0,
\label{transeq}
\end{equation}
where $ {\bf F} = \nabla_X {\bf x}$ is the non-degenerate Jacobi matrix of the geometrical transformation, and $J = \det {\bf F}.$
The above equation can be interpreted as the equation of an anisotropic pre-stressed plate, as discussed in \cite{ColquittJMPS}.
In the context of asymptotic approximations, the paper \cite{BrunNJP} discusses the configurations where the effects of pre-stress are small, and when an approximate factorisation of the transformed differential operator is possible.
Our goal is to construct an approximate cloak, which may be conveniently implemented experimentaly.
We observe that in the successful experiment for the continuous plate led by Wegener \cite{Wegener2012}, the authors assumed that the membrane waves are dominant and configured their radially symmetric approximate cloak accordingly.
Although such an assumption may appear to be inappropriate, it has been demonstrated that, within a certain frequency range, the membrane waves indeed are dominant and the approximate cloaking effect is apparent.
This approach is further reinforced for the square cloak by the observation made in \cite{ColquittJMPS} that the principal directions of stiffness for the membrane cloak and the flexural cloak are exactly the same.
Compared to implementations of invisibility cloaks for continua, such as in~\cite{Wegener2012}, the square lattice cloak developed in the present paper is relatively straightforward.
We now proceed to describe the implementation of our novel square lattice cloak.
\subsection{The lattice approximation of the cloak}
We consider an approximate cloak, realised using a discrete lattice structure with curved ligaments as derived in \cite{ColquittJMPS}.
These elastic ligaments are aligned with the the principal directions of the stiffness matrix for the continuum cloak, as
illustrated in Fig. \ref{Deformation_Schematic}B, and yield a structured flexural lattice system.
The thin elastic ligaments may be treated as beams of rectangular cross section characterised by the bending stiffnesses $D_1$ and $D_2$ chosen in accordance with the analytical formulae from \cite{ColquittJMPS}
\begin{equation}
D_1= \frac{x_1-\alpha_2}{x_1}, \hspace{1cm} D_2= \frac{x_1^4+\alpha_2^2 x_2^2}{x_1^3(x_1-\alpha_2)}.
\label{rigidcoeff}
\end{equation}
These flexural rigidities are visualised in Fig. \ref{Rigidity} for the right-hand quadrant.
For this quadrant, on the $X_1$ axis, $D_1$ represents the $X_1$ stiffness in the principal direction, whereas $D_2$ represents the $X_2$ stiffness in the other principal direction.
In every other point inside the cloak, $D_1$ and $D_2$ represent the stiffnesses in the principal directions of the locally orthotropic cloak constructed here.
We note that on the interior boundary of the cloak the tangential rigidity $D_2$ is much higher compared to the normal rigidity $D_1$
and emphasise that these stiffnesses are finite on this boundary of the cloak (see Fig. \ref{Rigidity}).
According to equation (\ref{transeq}) the mass density $\rho'$ inside the cloak is also non-uniform and obeys the formula
\begin{equation}
\rho' = \rho \frac{x_1-\alpha_2}{\alpha_1^2 x_1},
\label{density}
\end{equation}
where $\rho$ is the density outside the cloak and the coefficients $\alpha_1$ and $\alpha_2$ are defined in (\ref{coeff}).
Equations (\ref{rigidcoeff}) and (\ref{density}), together with the accompanying geometrical design of the cloak, provide the essential information for the experimental implementation discussed in the text below.
\begin{figure}
\renewcommand{\figurename}{\footnotesize{Fig.}}
\begin{center}
\includegraphics[width=13.0 cm]{Rigidity_Graphs.pdf}
\caption{\footnotesize The required stiffnesses $D_1$ and $D_2$ for the cloak ligaments reported as function of $x_1$ and $x_2$ .}
\label{Rigidity}
\end{center}
\end{figure}
\section{Numerical simulation and experimental verification}
\label{computational_exp}
We have performed both finite element (FE) computations, in ABAQUS, and also experiments to verify the efficacy of the square structured cloak in reducing the scattering of flexural waves from voids.
The structured cloak has been designed and implemented in SOLIDWORKS.
Each elastic ligament has a specified variable cross section that provides the required rigidities $D_1$ and $D_2$ as reported in Colquitt \emph{et al.} \cite{ColquittJMPS}.
The physical lattice cloak was created by milling holes into a polycarbonate plate;
both ABAQUS and the milling machine were programmed using the same SOLIDWORKS original code. In so doing, experiments and simulations have identical geometry, material parameters, constraints and applied out-of-plane vibrations.
Figure \ref{Abaqus_geometry} illustrates both the lattice geometries implemented in ABAQUS and the experiment.
\subsection{ABAQUS simulation}
\label{computational}
We have compared the wave field for three cases: the first for a homogeneous lattice in the absence of any void, the second in the presence of a void, and the third in the presence of a void surrounded by our specially designed invisibility cloak.
The simulations have been performed using a parametric python script for ABAQUS,
run by means of MATLAB.
We have computed the steady-state frequency response of each lattice using the Dynamic/Explicit package already implemented in ABAQUS.
Since the elastic lattices are constructed from thin elastic ligaments, we have performed the simulations employing 3D beam elements.
\begin{figure}
\renewcommand{\figurename}{\footnotesize{Fig.}}
\begin{center}
\includegraphics[width= 14 cm]{Cloak_design_milling.pdf}
\caption{\footnotesize {The comparison between the geometry of the real cloaked lattice drilled by a milling machine (A) and a frontal view (B) and a 3D view (C) of the geometry implemented in ABAQUS.
The zoomed area of the part (A) shows {the physical} realisation of the square cloak embedded in a square lattice. By observing the enlarged detail of the square cloak, the different cross sectional areas of the ligaments are clearly visible and chosen to match the required $D_1$ and $D_2$ principal flexural rigidities.
}} \label{Abaqus_geometry}
\end{center}
\end{figure}
The cloak design is an approximate one, in the sense that it works well for a certain range of frequencies.
Using the simulations we were be able to determine the range of frequencies over which the cloak is effective.
An out-of-plane vibration in the range between 100 Hz and 250 Hz has shown the predicted cloaking action. In particular, in Fig. \ref{CloakEffect} we show the cases of 120 and 240Hz, where the presence of the cloak leads the holed plate to behave in a similar way to an homogenous lattice without a void.
\subsection{The experimental implementation}
\label{experiments}
The model, designed theoretically and implemented in ABAQUS, has also been verified experimentally (the geometry is as in Fig. \ref{Abaqus_geometry}).
The experiment has been performed at the \lq Instabilities Lab' of the University of Trento.
In the current setup, we use the Hooke-Chladni-Faraday visualisation described earlier in Section \ref{HCF}.
Three structured flexural systems, namely a homogenous lattice, a lattice with a hole, and a lattice with a cloaked hole, have been produced by drilling polycarbonate plates (white 2099 Makrolon UV from Bayer) with an EGX-600 Engraving Machine (accuracy 0.01mm, by Roland).
The mechanical properties of the polycarbonate, namely elastic modulus (E), density ($\rho$) and Poisson's ratio ($\nu$), are respectively 2350 MPa, 1200 kg/m$^3$ and 0.35.
The lattice cloak as well as the geometrical and material parameters are chosen consistently, according to the same data set stored in the
SOLIDWORKS file, as illustrated in Fig. \ref{Abaqus_geometry}A, which shows the experimental sample.
Thin transparent film was used to cover the flexural lattice system to enable the use of powder for the Hooke-Chladni-Faraday visualisation.
The lattices, externally measuring 600mm by 400mm, have ligaments of constant cross-section (1.75 mm by 2.5mm) outside the cloak, whereas within the cloak the ligaments have variable width and height according to the analytical algorithm of Section \ref{sqcloak}.
The lattices, constrained by clamps on the two shorter sides and having the other two free, have been excited by using a TIRA Vibrations Test System TV51144 and BOSE ElectroForce 3300 Series II, connected to the left clamp of the lattices, as shown in the experimental arrangement Fig. \ref{Exp_Setup}.
The maximum amplitude (1 mm peak to peak) of the sinusoidal displacement has been imposed by the oscillating clamp connecting the Vibrations Test System to an NI CompactRIO system, interfaced with LabVIEW 2014 (National Instruments).
\subsection{The experimental results and comparison with the computational model}
The qualitative assessment of the effect of the cloak has been carried out using the Hooke-Chladni-Faraday technique that shows the positions of the nodal lines of the vibrating plate.
The boundary conditions are chosen consistently, both in the numerical simulation, and the physical experiment: one side of the plate is rigidly clamped, while the opposite side is attached to a moving clamp and excited by applying a time-harmonic displacement. The remaining two sides are traction free, i.e. the bending moments and the transverse forces are equal to zero at these boundaries.
\begin{figure}
\renewcommand{\figurename}{\footnotesize{Fig.}}
\begin{center}
\includegraphics[width= 14 cm]{Exp_setup.pdf}
\caption{\footnotesize {The vibration apparatus employed in the
experiments (A) the details of the constraints on both sides of the plate, namely a shaking clamp (B) and a rigid clamped (C).}} \label{Exp_Setup}
\end{center}
\end{figure}
Photos have been taken with a Sony NEX 5N digital camera (equipped with 3.5-5.6/18-55 lens, optical
steady shot manufactured by Sony Corporation) and with a Nikon D200 digital camera (equipped with a AF-S micro Nikkor lens 105 mm 1:2.8G ED).
Fig. \ref{Pict_Exp} shows the comparison between the experiment and the numerical simulations in the case of an applied displacement with a frequency of 120 Hz.
In particular, in the upper part of Fig. \ref{Pict_Exp} we see the unperturbed plate without a hole, with the one-dimensional frequency response pattern shown.
In the middle part of Fig. \ref{Pict_Exp} we show that plate containing an uncloaked square hole, with
a traction free boundary.
We observe similar shadow regions both, in the ABAQUS simulation, and the physical experiment with a real structured plate.
Finally, we demonstrate in the lower part of Fig. \ref{Pict_Exp} that the shadow region is significantly reduced when the cloaking region, depicted in Fig. \ref{Abaqus_geometry}A, is introduced around the hole, so that the wave pattern outside the hole becomes almost flat, as expected.
It is clear that there is good qualitative agreement between the experimental and numerical results.
In fact, in the case of a cloaked void, the nodal lines showing the incident field represented by a plane wave are almost straight, similar to the case of the homogeneous lattice. On the other hand, in the absence of the cloak, the pattern lines appear deeply influenced by the hole.
In this case we observe rounded nodal lines that differ significantly from the straight wavefronts observed for the homogenous lattice.
\section{Concluding remarks}
\label{concluding}
In this paper, we have presented a proof of concept design for a square invisibility cloak.
Having constructed the cloak, we proceeded to examined its effectiveness using both, in a computational ABAQUS model, as well as in real physical experiment.
This novel design, proposed in an earlier theoretical paper \cite{ColquittJMPS}, was appealing due its simplicity and elegance, which made an experimental implementation feasible.
The regularisation introduced into this design of cloak enables careful and precise implementation of boundary conditions on the interior boundary of the cloak in both the numerical simulations and the experiments.
The approximate cloak presented here, proves to be efficient within a predicted frequency range, but the results become frequency sensitive as the frequency of the incident wave increases.
This effect has been expected, and similar phenomena of the high frequency sensitivity were noted in \cite{Wegener2012}.
The range of applications of the proposed cloaking device is wide and it covers, in particular, earthquake resistant systems, as well as novel designs of foundations of civil engineering structures.
\vspace*{5mm} \noindent
{\sl Acknowledgments } The authors would like to thank Prof Davide Bigoni for stimulating discussions and valuable suggestions on the text of the paper, and for his strong support. D.M. and A.B.M. gratefully acknowledge financial support from the ERC Advanced Grant \lq Instabilities and nonlocal multiscale modelling of materials' FP7-PEOPLE-IDEAS-ERC-2013-AdG (2014-2019) and provision of experimental facilities via the \lq Instabilities Lab' of the University of Trento (http://www.ing.unitn.it/dims/ssmg/). N.V.M gratefully acknowledge financial support from European FP7 - INTERCER-2 project (PIAP-GA-2011-286110-INTERCER2).
D.J.C. thanks the EPSRC for support through research grant EP/J009636/1.
|
2,877,628,088,991 | arxiv | \section{Introduction}
Consider the following Hamiltonian:
\begin{align} \label{hmail}
H = \frac{1}{2} ( \pi_A^2 + (\vec{\nabla}A)^2 +m_A^2 A^2 +\pi_B^2 +(\vec{\nabla}B)^2 + m_B^2B^2)+ \frac{g}{4} (\vec{\nabla}A)^2 B^2 + \frac{1}{2} \pi_A^2 \left(\frac{\frac{g}{2}B^2}{1+\frac{g}{2}B^2}\right)
\end{align}
where $A$, $B$ are scalar fields and $g$ is a coupling constant. Is Lorentz invariance a symmetry of the system described by the above Hamiltonian? The answer is yes. The corresponding Lagrangian is given by:
\begin{align}\label{lag}
L = \frac{1}{2} ( (\partial_{\mu} A)^2 - m_A^2 A^2 +(\partial_{\mu} B)^2 -m_B^2B^2)-\frac{g}{4}(\partial_{\mu} A)^2 B^2
\end{align}
Which is manifestly Lorentz invariant. But this was not at all obvious from the Hamiltonian. Given a Hamiltonian how do we check for Lorentz invariance?
Generally, one has three avenues :
One, solve for the equations of motion and see if these are relativistic.
Two, calculate the Poisson brackets of the Noether charges of Lorentz symmetry with each other and check if this gives a representation of the Lorentz algebra. We do not elaborate on this method here, the reader is referred to \cite{greiner} for details. The non-trivial part of this method involves calculating the Poisson Bracket relations of the generators of boost and angular momenta with the Hamiltonian.
Finally, one may construct the Lagrangian and check if it is Lorentz invariant. If one is given a quantum Hamiltonian, then this last method would involve obtaining a path integral representation.
In this paper we will show that for a class of physical systems, it is possible to obtain the condition for Lorentz invariance directly from a Hamiltonian. As we will see, the condition states that certain vectors and tensors that may be constructed from the Hamiltonian are proportional to one another. We note that this method works when the representation of Lorentz symmetry is linear. There are cases where Lorentz symmetry is non-linearly realized, for instance gauge-fixed or reduced phase space treatments of gauge theories (see for instance \cite{Bernstein:1974rd}), but these are not considered here.
The systems for which the result holds must satisfy certain criteria. These are:
where the field momentum or the space derivative of one of the fields occurs in the Lagrangian they must
(i) Come in even powers . That is, any of the terms in the Lagrangian may contain a term like $(\partial_{x} A)^2 $ but a factor like $(\partial_{\mu} A) C^{\mu}$, where $C^{\mu}$ may be a (pseudo)vector, cannot occur in any of the terms.
(ii) Not be multiplied with the derivatives of one of the other fields. That is a term like $(\partial_{x} A)^2 B^2$ is allowed but $(\partial_{x} A)^2(\partial_{x} B)^2$ is not.
This work is particularly important in the context of those quantum theories where the other avenues may not be readily available. One such case was polymer quantized scalar field theory \cite{Ashtekar:2002vh}. The status of Lorentz invariance of this theory was open till a path integral formulation was found recently \cite{Kajuri:2014kva}, establishing that the theory is not Lorentz invariant. However our work makes the deduction of Lorentz non-invariance of this theory extremely simple, as we will show.
The plan of the paper is as follows. In the next section we start from the Lagrangian formulation and show how to derive the condition for Lorentz invariance for these Hamiltonians. First we restrict ourselves to an even more limited (but still non-trivial) class of Hamiltonians, for which the condition for Lorentz invariance takes a particularly simple form. Then after presenting some examples we sketch how to extend the derivation for all Hamiltonians satisfying (i) and (ii) above. The final section summarizes our findings.
\section{Encoding Relativistic Invariance in the Hamiltonian: The Restricted Case}
In this section we derive a sufficient condition for Lorentz invariance for physical systems which satisfy (i) and (ii) above. First we will present the derivation for systems which satisfy another criteria :
(iii) The derivative terms appearing in the Lagrangian must be quadratic. That is, any of the terms in the Lagrangian may contain the factor $(\partial_{x} A)^2 $ but a factor like $(\partial_{\mu} A)^4$ cannot occur in any of the terms.
Note that the system described by \eqref{lag} (equivalently \eqref{hmail}) satisfies these criteria.
For all such Hamiltonians, the condition for Lorentz invariance takes a particularly simple form. It goes as: construct for each field A, the following two column matrices -
\begin{align}
\label{def1}F_A^{\mu}&=(\frac{\partial H}{\partial \pi_A}, \vec{\nabla}A) \\
\label{def2}G_A^{\mu} &=( \pi_A, \frac{\partial H}{\partial \vec{\nabla}A})
\end{align}
The condition for Lorentz invariance is, for each $A$:
\begin{align}\label{condition}
F_A^{\mu} = k G_A^{\mu}
\end{align}
where $k$ is a scalar which may be the function of fields and its derivatives.
In the following subsection we derive \eqref{condition} starting from a Lagrangian formulation. This will be followed by some examples. We will complete this section by sketching the steps of the derivation for a system satisfying only (i) and (ii).
\subsection{Derivation from the Lagrangian}
In this section we will demonstrate how, for systems satisfying criteria (i) and (ii), the Lorentz invariance of a Lagrangian is encoded in the Hamiltonian through \eqref{condition}. We'll exhibit the proof for a system of scalar fields. The extension to higher spin fields is straightforward.
Let us consider a relativistic Lagrangian L which is a function of scalar fields $\phi_n$. We write this as:
$$L = L_1 +L_2 $$
where all terms containing derivatives in the field are put in $L_1$ and the rest of the terms are in $L_2$. For instance, in the Lagrangian \eqref{lag} we will have
\begin{align}
L_1 = \frac{1}{2} ( (\partial_{\mu} A)^2 +(\partial_{\mu} B)^2 )+\frac{g}{4}(\partial_{\mu} A)^2 B^2
L_1 = - m_A^2 A^2 -m_B^2B^2
\end{align}
Now since $L_2$ does not contain any terms containing field derivatives, it follows that $L_1$ and $L_2$ must be separately Lorentz invariant. Now for a Lagrangian satisfying the criteria (i) and (ii), $L_1$ may be written as:
\begin{align}\label{lag1}
L_1 = \sum_n \frac{1}{2} \frac{\partial L_1} {\partial (\partial_{\mu} \phi_n)} \partial_{\mu} \phi_n
\end{align}
Again, each term in $L_1$ must be Lorentz invariant in itself. But from the above, each term in $L_1$ can be written in the form
\begin{align}
(L_1)_n =\frac{1}{2} \eta_{\mu \nu} W_n^{\mu} Z_n^{\nu}
\end{align}
where
\begin{align}
W_n^{\mu} = \partial_{\mu} \phi_n\\
Z_n^{\nu} = \frac{\partial L_1} {\partial (\partial_{\mu} \phi_n)}
\end{align}
This is manifestly a Lorentz invariant quantity if $W_n^{\mu}$ and $Z_n{\nu}$ transform as four vectors under Lorentz transformations. But $W_n^{\mu}$ are a Lorentz four vectors by definition. It therefore follows that the condition of Lorentz invariance of the Lagrangian $L$ translates to the condition that the terms $Z_n{\nu}$ transform as four vectors under Lorentz transformations. Now for a given n $Z_n^{\mu}$ is formed by differentiating the Lagrangian with respect to spacetime derivatives of the field $\phi_n$. Therefore $Z_n^{\mu}$ must contain spacetime derivatives of the field itself. Therefore to transform similarly as $W_n^{\mu}$, it must be that
\begin{align}\label{condi}
W_n^{\mu} = kZ_n^{\mu}
\end{align}
where k is a scalar which may be a function of the fields and their derivatives. This relation therefore encodes the Lorentz invariance of the Lagrangian.
Now let us proceed to construct the Hamiltonian:
\begin{align}
\nonumber H &= \sum_n \frac{\partial L} {\partial (\partial_{0} \phi_n)} \partial_{0} \phi_n - L_1 -L_2 \\
&= \sum_n \left(\frac{\partial L_1} {\partial (\partial_{0} \phi_n)} \partial_{0} \phi_n -\frac{1}{2} \frac{\partial L_1} {\partial (\partial_{\mu} \phi_n)} \partial_{\mu} \phi_n \right)-L_2\\
&= -L_2 + \sum_n \frac{1}{2} \left(\frac{\partial L_1} {\partial (\partial_{0} \phi_n)} \partial_{0} \phi_n + \frac{\partial L_1} {\partial (\partial_{i} \phi_n)} \partial_{i} \phi_n \right)\\
&= -L_2 + \sum_n \frac{1}{2}\delta_{\mu \nu} W_n^{\mu} Z_n^{\nu}
\end{align}
As $\eta_{\mu \nu} W_n^{\mu} Z_n^{\nu}$ was invariant under Lorentz transformations, it follows that $\delta_{\mu \nu} W_n^{\mu} Z_n^{\nu}$ would be invariant under orthogonal transformations.
Let us convert these into functions of $\pi_n$ and $\phi_n$. We define
\begin{align}
&F_n^0 \equiv W_n^{0} = \partial_{0} \phi_n = \frac{\partial_{0} H}{\partial \pi_n} \\
&F_n^i \equiv W_n^{i} = \partial_{i} \phi_n \\
&G_n^0 \equiv Z_n^{0} = \frac{\partial L_1} {\partial (\partial_{0} \phi_n)} =\pi_n\\
&G_n^i \equiv Z_n^{i} = \frac{\partial L_1} {\partial (\partial_{i} \phi_n)} =\frac{\partial H} {\partial (\partial_{i} \phi_n)}
\end{align}
Where we have used one of the Hamilton's equations of motion in the first step.
Now if we decompose the Hamiltonian into $H_1$ and $H_2$ using the same logic as we used for the Lagrangian, we will have $H_2 = L_2$ and the Hamiltonian may be written as
\begin{align}
H = H_2 + \sum_n \frac{1}{2}\delta_{\mu \nu} F_n^{\mu} G_n^{\nu}
\end{align}
The condition \eqref{condi} for Lorentz invariance of the Lagrangian becomes, in terms of phase space functions, exactly the condition \eqref{condition} advertised before:
\begin{align}\label{condi2}
F_n^{\mu} = k G_n^{\mu}
\end{align}
Thus for the given class of systems, the relativistic invariance of the dynamics can be read off from the Hamiltonian by constructing the column matrices $F_n^{\mu}$ and $G_n^{\mu}$ and checking if \eqref{condi2} is satisfied.
\subsection{ A couple of examples}
As a first example let us consider the Hamiltonian of \eqref{hmail}. We know from \eqref{lag} that this is Lorentz invariant. Now let us check whether it satisfies our criteria.
For the A field the relevant vectors are
\begin{align}
F_n^{\mu}= (\frac{\partial H}{\partial \pi_A}, \vec{\nabla}A) = \left( \left( \frac{1+gB^2}{1+\frac{g}{2}B^2}\right) \pi_A, \vec{\nabla}A \right)\\
G_n^{\mu} =( \pi_A, \frac{\partial H}{\partial \vec{\nabla}A}) = \left(\pi_A, \left( 1+\frac{g}{2}B^2\right)\vec{\nabla}A \right)
\end{align}
It is easy to check that for this case
\begin{align}
F_n^{\mu} = k G_n^{\mu}
\end{align}
with
$$k = \left( 1+ \frac{g}{2}B^2 \right)^{-1}$$
Let us consider another example, this time from polymer quantization of scalar fields \cite{Ashtekar:2002vh}. That this system is not Lorentz invariant is known from its path integral formulation \cite{Kajuri:2014kva}. Here we have the following Hamiltonian:
\begin{align}
H = \frac{1}{2} \left( \pi^2 + (\nabla \phi)^2 \cos^2 (\mu \phi)\right)
\end{align}
where $\mu$ is a dimensionless quantity.
Let us consider the vectors for this case:
\begin{align}
F_n^{\mu} =( \pi, \nabla \phi)\\
G_n^{\mu} =(\pi, \nabla \phi)\cos^2 (\mu \phi))
\end{align}
Clearly these are not proportional to each other. Thus our method agrees with the known result.
\subsection{Encoding Relativistic Invariance in the Hamiltonian: more general case}
In this section we briefly sketch how to obtain the criteria for relativistic invariance for systems satisfying the criteria (i) and (ii) only.
The Lagrangian for such a system may be divided into $L_1$ and $L_2$ as before and $L_1$ may be expanded as:
\begin{align}
L_1 = \frac{1}{2} \frac{\partial L_1}{\partial(\partial_{\mu} \phi)} \partial_{\mu} \phi + \frac{1}{4} \frac{ \partial^2 L_1}{\partial(\partial_{\mu} \phi) \partial( \partial_{\nu}\phi)} \partial_{\nu}\phi \partial_{\mu} \phi +...
\end{align}
This may be written as
\begin{align}
L_1 = \frac{1}{2} \eta_{\mu \nu} A^{\mu} B^{\nu} + \frac{1}{4}\eta_{\alpha \beta} \eta_{\gamma \chi} C^{\alpha \gamma} B^{\beta} B^{\chi} +....
\end{align}
where
\begin{align}
A^{\mu}= \frac{\partial L_1}{\partial(\partial_{\mu} \phi)}\\
B^{\mu} = \partial_{\mu} \phi\\
C^{\mu \nu} = \frac{ \partial^2 L_1}{\partial(\partial_{\mu} \phi) \partial( \partial_{\nu}\phi)}
\end{align}
and so on.
Again this expression is manifestly Lorentz invariant, given that $A^{\mu} \propto B^{\mu},C^{\mu \nu}\propto B^{\mu}B^{\nu}...$ by the same logic as before. Now all these terms may be expressed in the Hamiltonian formulation in terms of H, $\pi, \vec{\nabla} \phi$ and derivatives of H to obtain the conditions in the Hamiltonian language, just as before.
\section{Summary and Outlook}
In this paper we have shown that, for a class of systems, there exists a simple way to check for Lorentz invariance directly from the Hamiltonian formulation. This class of systems were defined by conditions (i) and (ii) given above. For an even more restricted class, defined by conditions (i), (ii) \textit{and} (iii) given in section II, the condition reduces to a simple proportionality between two column matrices which may be constructed from the Hamiltonian.
This is particularly important in the context of quantum theories where other methods of checking for Lorentz invariance may be immediately available. One place where this is the case is polymer quantized field theory. Here the only check on Lorentz invariance so far has come from the path integral formulation \cite{Kajuri:2014kva}. We managed to reach the same conclusion in a much simpler way here.
\begin{acknowledgements}
We thank James Edwards for illuminating discussions and helpful comments and Gaurav Narain for helpful comments on the draft.
\end{acknowledgements}
|
2,877,628,088,992 | arxiv | \section{Introduction}
The dynamics of quantum matter is linked to several important phenomena in physics, such as thermalization or lack thereof~\cite{nandkishore2015many}, dynamical phase transitions~\cite{eckstein2009thermalization, heyl2018dynamical}, and universality in out-of-equilibrium dynamics~\cite{prufer2018observation, langen:ultracold_2015, polkovnikov:nonequilibrium_2011, lamacraft-moore:potential-insights_2012, altman2015non}. Understanding these phenomena is challenging, partly due to the lack of theoretical tools to accurately simulate them. There is an urgent need for such tools because recent experiments have made strides in measuring out-of-equilibrium dynamics; see, for example, Refs.~\cite{kim2010quantum, islam2013emergence, britton2012engineered, bohnet2016quantum, de2013nonequilibrium, de2016probing, zeiher2016many, mukherjee2016accessing, low2009universal, takei2016direct, guardado2018probing, nichols2019spin, lienhard2018observing, hazzard2014many, garttner2017measuring}. Several numerical methods, such as exact diagonalization~\cite{manmana2005time, rigol2008thermalization, prelovvsek2013ground, sandvik2010computational}, time-dependent density-matrix renormalization group~\cite{white2004real, daley2004time, vidal2004efficient, wolf2014solving, schuch2011classifying}, perturbative and Keldysh techniques~\cite{bray2002theory, calabrese2005ageing, henkel2008non, henkel2011non, kamenev2011field, tauber2014critical}, kinetic theories and phase-space methods~\cite{walls2007quantum, blakie2008dynamics, gardiner2004quantum, orioli2017nonequilibrium}, and numerical linked-cluster expansions~\cite{tang2013short, rigol2014quantum, white2017quantum, guardado2018probing, nichols2019spin, mallayya2018quantum}, have been used to calculate such dynamics. However, all these methods have limitations, ranging from being restricted to small or low-dimensional systems, to being accurate only for weakly interacting, close-to-equilibrium, or short-time situations.
In this paper we compare two popular and related semiclassical approximations for the dynamics of quantum matter, namely, the continuous truncated Wigner approximation (TWA) and discrete truncated Wigner approximations (DTWA)~\cite{wootters1987wigner, polkovnikov2003quantum, polkovnikov2010phase, schachenmayer2015many}, with each other and with exact analytical or numerical solutions. These approximations have been used frequently in recent years to simulate the dynamics of spin models~\cite{schachenmayer2015many, schachenmayer2015dynamics, pucci2016simulation, orioli2017nonequilibrium, orioli2018relaxation, babadi2015far, valtierra2017twa, czischek2018quenches, wurtz2018cluster}, which are some of the most ubiquitous dynamics probed in experiments~\cite{kim2010quantum, islam2013emergence, britton2012engineered, bohnet2016quantum, de2013nonequilibrium, de2016probing, zeiher2016many, mukherjee2016accessing, low2009universal, takei2016direct, guardado2018probing, nichols2019spin, lienhard2018observing, hazzard2014many, garttner2017measuring}. The approximations estimate the quantum expectation of observables as the average over classical trajectories of initial phase-space points which are sampled from the Wigner distribution associated with the initial state. They are simple to implement, and offer accuracy consistent with being semiclassical expansions~\cite{schachenmayer2015many, schachenmayer2015dynamics, pucci2016simulation}.
Earlier works~\cite{schachenmayer2015many, schachenmayer2015dynamics} have argued that DTWA is a superior approximation to calculate the dynamics of spin-spin correlations than TWA, based on specific examples considered. As an example of a case where DTWA is superior, Fig.~\ref{fig: correlation2}(a) shows the dynamics of correlations of neighboring spins in a one-dimensional (1D) Ising chain with no transverse field, obtained from the exact solution, DTWA, and TWA. (The initial conditions and Hamiltonians are described in the figure caption, while the DTWA and TWA calculations will be explained later.) For this case, DTWA exactly captures the dynamics of a specific component of spin correlations, while TWA is accurate for this component only at relatively short times.
However, we must exercise caution when claiming that one method is superior to another based on examples like the ones above, especially because there are nine components, $\langle\hat{S}^{\mu}_i\hat{S}^\nu_j\rangle-\langle\hat{S}^\mu_i\rangle\langle\hat{S}^\nu_j\rangle\ (\mu,\nu\in\{x,y,z\})$, of spin-spin correlations to assess. In contrast to Fig.~\ref{fig: correlation2}(a), Fig.~\ref{fig: correlation2}(b) shows that even for the same model, DTWA performs significantly worse and is qualitatively wrong when we look at a different component of the correlations and a different initial condition (described in the figure caption.) It is often not obvious which correlations, if any, are the most important, especially in dynamics far from equilibrium. Therefore, a more comprehensive comparison of the two Wigner approximations is necessary.
\begin{figure}[t]\centering
\includegraphics[width = 0.7\columnwidth]{fig1v2.pdf}
\caption{(Color online) Dynamics of one component of the spin-spin correlations for the 1D Ising model with no transverse field [whose Hamiltonian is Eq.~\eqref{eqn: HIsing}], obtained from the exact solution (black solid curve), DTWA (blue curve with circles), and TWA (red curve with squares): (a) $C_{ij}^{yz}$ for an initial state with all spins along $\mathbf{x}$, and (b) $C_{ij}^{xx}$ for an initial state with all spins $45^\circ$ between $\mathbf{x}$ and $\mathbf{z}$. $C^{\mu\nu}_{ij}$ is defined in Eq.~\eqref{eqn: C}. The black and blue curves overlap in (a).}
\label{fig: correlation2}
\end{figure}
The key finding in this paper is that both DTWA and TWA suppress spin correlations along one direction for a broad class of spin dynamics. We show strong numerical evidence for this, and then rigorously prove this for short times. We also find that the accuracy of DTWA versus TWA is more nuanced than simply one being better than the other. These insights are not readily apparent from looking at plots of the nine Cartesian components of spin-spin correlations. We are able to gain insight into the workings of TWA and DTWA and isolate the nuanced differences between them by utilizing the correlation matrix visualization (CMV) technique, which was recently introduced in Ref.~\cite{mukherjee2018geometric} building on geometrical visualization techniques in Refs.~\cite{kimura2003bloch, byrd2003characterization, jakobczyk2001geometry, tilma2002parametrization, rundle2017simple, bertlmann2008bloch, giraud2015tensor, jevtic2014quantum, dunkl2011numerical, kurzynski2016three, sorensen1984product, halstead1984multipole, donne1997pictorial, philp2005way, merkel2008quantum, dowling1994wigner, harland2012towards, gamel2016entangled, garon2015visualizing, leiner2017wigner}. Correlation matrix visualizations encode all the information contained in spin-spin correlations into three-dimensional shapes and allow us to compare all components of the spin-spin correlations on equal footing.
This article is organized as follows. In Sec.~\ref{sec: Wigner} we introduce TWA and DTWA. In Sec.~\ref{sec: CMV} we describe the tools and metrics that we use to analyze the results of TWA and DTWA. In Sec.~\ref{subsec: Ising} we compare spin-spin correlation dynamics for the exact solution, DTWA, and TWA applied to the Ising model with no transverse field. In Sec.~\ref{subsec: other} we compare spin-spin correlation dynamics calculated with these three methods for the nearest-neighbor 1D transverse Ising and XX models. In Sec.~\ref{sec: rigorous proof} we present a rigorous mathematical argument for one of the key findings in Sec.~\ref{sec: results}, that DTWA and TWA always suppress spin-spin correlations along one direction at short times. We distill the lessons of these comparisons and summarize in Sec.~\ref{sec: conclusions}.
\section{Wigner approximations}\label{sec: Wigner}
Wigner approximations approximate dynamics of quantum systems. The implementation of the technique has three steps, schematically illustrated in Fig.~\ref{fig: wigner}.
In the first step, we sample phase-space coordinates from the Wigner function associated with the initial density matrix $\hat{\rho}(0) = \ket{\psi(0)}\bra{\psi(0)}$. The Wigner function, denoted by $W(\mathbf{S})$, is a quasiprobability distribution that represents $\hat{\rho}(0)$ in an appropriate phase space, with phase points described by coordinates $\mathbf{S}$. The Wigner function $W(\mathbf{S})$ is defined via
\begin{equation}
\hat{\rho} = \int\! d\mathbf{S}~W(\mathbf{S}) \hat{A}(\mathbf{S}), \label{eqn: W}
\end{equation}
where $\hat{A}$ is called a phase-point operator and the integral runs over all of phase space. The phase-space coordinates that describe motional degrees of freedom are position and momentum. For spins, the coordinates can be the spin vector elements $(S^x, S^y, S^z)$. (For spins, the choice of phase space is not unique, and possible phase spaces are discussed in Secs.~\ref{subsec: TWA} and~\ref{subsec: DTWA}.) This step in the algorithm does not contain any approximation, as any observable in a quantum state can be obtained by averaging over phase-space points sampled from the Wigner distribution for that state.
In the second step, we evolve the sampled initial phase-space points in time according to classical equations for the spins. The equations of motion for the specific models we consider [Eqs.~\eqref{eqn: HIsing},~\eqref{eqn: HtransIsing}, and~\eqref{eqn: HXX}] are given in Eqs.~\eqref{eqn: IsingEOM},~\eqref{eqn: transIsingEOM}, and~\eqref{eqn: XXEOM}, respectively. We denote the classical trajectory of an initial point $\mathbf{S}$ by $\mathbf{S}_{\rm cl}(\mathbf{S},t)$.
In the third and final step, we calculate the expectation of an operator $\hat{O}$ at time $t$ by averaging over the trajectories of the phase points as
\begin{equation}\label{eqn: O}
\smallexpect{\hat{O}} = \int\! d\mathbf{S}~{\rm wl}(\hat{O},\mathbf{S}_{\rm cl}(\mathbf{S},t)) W(\mathbf{S}).
\end{equation}
Here, ${\rm wl}(\hat{O},\mathbf{S})$ is the Weyl symbol for $\hat{O}$ at the phase point $\mathbf{S}$. As examples, ${\rm wl}(\hat{S}^\mu_i,\mathbf{S}) = S^\mu_i$ and ${\rm wl}(\hat{S}^\mu_i\hat{S}^\nu_j+\hat{S}^\nu_j\hat{S}^\mu_i,\mathbf{S}) = S^\mu_iS^\nu_j+S^\nu_iS^\mu_j$. The procedure to obtain the Weyl symbol for other observables is more involved~\cite{polkovnikov2010phase}, but in this paper, we only need the examples listed here.
The essence of the Wigner approximations lies in the third step, where we estimate an observable at time $t$ from the classically evolved trajectories of the initial phase-space points. While this step might be intuitive, nevertheless the phase points at time $t$, which are evolved from the initial phase points, \textit{do not} sample the Wigner distribution of the quantum state at $t$. It is for this reason that, sometimes, Wigner approximations give results differing from the exact results. The main purpose of this paper is to explore different cases where the Wigner approximations give results differing from the exact results, extract generic trends regarding how they differ, and give a physical insight for these differences. We focus on spin models in this paper.
\begin{figure}[t]\centering
\includegraphics[width = 0.7\columnwidth]{wigner2.pdf}
\caption{(Color online) Illustration of Wigner approximations. The method consists of three steps: (a) Randomly sample points in phase space from the Wigner distribution for the initial state, (b) evolve the phase points classically through time, and (c) calculate the desired observable from the ensemble average of the observable at time $t$, evaluated from the time-evolved classical trajectories of the initial phase-space points.}
\label{fig: wigner}
\end{figure}
Different Wigner approximations differ in their choice of phase space. In this article we focus on two kinds of approximations with two different kinds of phase spaces: TWA samples from a finite continuous area of phase space and DTWA samples from a discrete set of phase points. We describe these schemes in Secs.~\ref{subsec: TWA} and~\ref{subsec: DTWA}, respectively.
\subsection{The TWA}\label{subsec: TWA}
In (continuous) TWA~\cite{polkovnikov2010phase}, the initial values of the spins are allowed to take any value in the continuous phase space spanning the points $(s^x,s^y,s^z)^{\otimes N}$, where $N$ is the number of spins. Reference~\cite{polkovnikov2010phase} derives the Wigner function for the state with all the spins pointing along the $\mathbf{z}$ direction to be
\begin{equation}
W(\mathbf{S}_{\rm tot}) \approx \frac{2}{\pi N}{\rm exp}\left(-\frac{(S^x_{\rm tot})^2+(S^y_{\rm tot})^2}{N/2}\right)\delta\left(S^z_{\rm tot}-N/2\right), \label{eqn: W_TWATot}
\end{equation}
where $S^\mu_{\rm tot} = \sum_i S^\mu_i$. Equation~\eqref{eqn: W_TWATot} is exact in the limit $N\to\infty$. Then the Wigner function for a single spin can be taken to be
\begin{equation}
W(\mathbf{S}_i) = \frac{2}{\pi}e^{-2(S^x_i)^2-2(S^y_i)^2}\delta\left(S^z_i-1/2\right). \label{eqn: W_TWA}
\end{equation}
This is one choice for the single-spin Wigner function that is consistent with Eq.~\eqref{eqn: W_TWATot}; other choices may be possible too. When the system has spins all uniformly pointing along a direction besides $\mathbf{z}$ at the initial time, we first initialize the spins along $\mathbf{z}$ by sampling from Eq.~\eqref{eqn: W_TWA} and then rotate all the spins. We always assume that the spins initially point in the $x$-$z$ plane.
\subsection{The DTWA}\label{subsec: DTWA}
In DTWA~\cite{schachenmayer2015many, schachenmayer2015dynamics}, the initial phase space is chosen to be a discrete set of points $\vec{\alpha}=(\vec{\alpha}_1,\vec{\alpha}_2,..\vec{\alpha}_N)$, where $\vec{\alpha}_i$ is the three-component spin vector for the $i$th spin. As a result, the continuous integral in Eq.~\eqref{eqn: O} is replaced by the sum
\begin{equation}
\expect{O}(t) = \sum_{\vec \alpha}~{\rm wl}(\hat{O},\vec{\alpha}_{\rm cl}(\vec\alpha,t))W_{\vec \alpha},
\end{equation}
where $\vec{\alpha}_{\rm cl}(\vec\alpha,t)$ is the classical trajectory of the initial phase point $\vec\alpha$.
The discrete locations where the initial points $\vec{\alpha}_i$ can lie are nonunique, and different works in the literature have made different choices. For example, Ref.~\cite{schachenmayer2015many} describes the case where the phase space for each spin consists of eight points given by
\begin{align}\label{eqn: S8_alpha}
&\mathbf{S}_1 = \frac{1}{2}(1,1,1),\nonumber\\
&\mathbf{S}_2 = \frac{1}{2}(-1,-1,1),\nonumber\\
&\mathbf{S}_3 = \frac{1}{2}(1,-1,-1),\nonumber\\
&\mathbf{S}_4 = \frac{1}{2}(-1,1,-1),\nonumber\\
&\mathbf{S}_{4+r} = -\mathbf{S}_r\ (1\leq r\leq4).
\end{align}
The phase-point operators are defined as $\hat{A}_{\vec{\alpha}_i} = \frac{1}{2}+\vec{\alpha}_i\cdot\hat{\vec{\sigma}}$, where $\hat{\vec{\sigma}}=(\hat{\sigma}^x,\hat{\sigma}^y,\hat{\sigma}^z)$ is the vector of Pauli matrices $\hat{\sigma}^\mu$ ($\mu=x,y,z$). The phase-point operator for $N$ spins is the product $\hat{A}_{\vec\alpha} = \Pi_i \hat{A}_{\vec{\alpha}_i}$. The Wigner function at $\vec{\alpha}$ is $W_{\vec \alpha} = \frac{1}{2^N}{\rm Tr}(\hat{\rho}\hat{A}_{\vec{\alpha}})$. We initialize the spins by sampling them from the probability distribution $|W_{\vec \alpha}|/\sum_\beta |W_{\vec\beta}|$, and when calculating the dynamics of an operator $\hat{O}$, we multiply its Weyl symbol ${\rm wl}(\hat{O},\vec\alpha)$ by the sign of $W_{\vec\alpha}$.
There is flexibility to choose other discrete sets of points in DTWA. Some of these choices are described in Ref.~\cite{pucci2016simulation}. The dynamics of spin systems sampled from different discrete phase spaces differ, as explored in detail in Ref.~\cite{pucci2016simulation}. While the phase spaces chosen in Ref.~\cite{pucci2016simulation} and other references work well for the models and initial conditions studied there, we find that those phase spaces yield significantly worse results for some of the models and conditions we consider in this paper. Therefore, we use only the phase space comprised of the phase points defined in Eq.~\eqref{eqn: S8_alpha}. For this phase space, the correlations in DTWA are accurate to linear order $O(t)$, although as we explain later, differences from the exact dynamics appear at longer times. We have not explored the question of finding the optimal phase space that will most accurately approximate the dynamics in our study.
\section{Geometric analysis of the spin correlations}\label{sec: CMV}
The connected correlations between a pair of spins $i$ and $j$ are
\begin{equation}\label{eqn: c}
c_{ij}^{\mu\nu} = \expect{\hat{S}_i^\mu \hat{S}_j^\nu}-\expect{\hat{S}_i^\mu}\expect{\hat{S}_j^\nu},\ \mu,\nu\in\{x,y,z,+,-\},
\end{equation}
and their symmetric part is given by
\begin{equation}
C_{ij}^{\mu\nu} = \frac{c_{ij}^{\mu\nu} + c_{ij}^{\nu\mu}}{2},\label{eqn: C}
\end{equation}
where $\hat{S}_j^\pm=\frac{\hat{S}_j^x\pm i\hat{S}_j^y}{2}$. The correlation matrix $C_{ij}$ is a $3\times3$ matrix with components $C^{\mu\nu}_{ij},\ \mu,\nu\in\{x,y,z\}$.
Reference~\cite{mukherjee2018geometric} introduced a geometric tool to visualize $C_{ij}$ using a three-dimensional contour called a CMV. We use this tool to analyze the results of the Wigner approximations. We define the CMV below, and refer the reader to Ref.~\cite{mukherjee2018geometric} for a detailed understanding of the CMV.
We define a function proportional to a homogeneous quadratic polynomial,
\begin{equation}
Q_{ij}(\mathbf{r}) = \frac{\mathbf{r}^T\cdot C_{ij}\cdot\mathbf{r}}{(1+r^2)^{3/2}},
\end{equation}
where $\mathbf{r}$ is a three-dimensional vector. The CMV is the locus of points $\mathbf{r}$ where $Q_{ij}(\mathbf{r})$ has a constant magnitude, $Q_{ij}(\mathbf{r}) = \pm P$. Each sign is assigned a different color. We shade points where $Q_{ij}(\mathbf{r})>0$ as red, and points where $Q_{ij}(\mathbf{r})<0$ as blue. Defining the correlation along the direction $\mathbf{n}$ as $C_{ij}^{nn}=\expect{(\hat{\vec{S_i}}\cdot \mathbf{n}) (\hat{\vec{S_j}}\cdot \mathbf{n})}-\langle\hat{\vec{S_i}}\cdot \mathbf{n}\rangle\langle\hat{\vec{S_j}}\cdot \mathbf{n}\rangle$, the points on the CMV along $\mathbf{n}$ can be obtained by solving the equation $|C_{ij}^{nn}/P| = (1+r^2)^{3/2}/r^2$. This equation has exactly two real solutions for $r$ in the limit that $|C_{ij}^{nn}/P|\gg1$, and these solutions are $r\simeq|C_{ij}^{nn}/P|$ and $r\simeq\sqrt{|P/C_{ij}^{nn}|}$. The size of the CMV along this direction is the difference between these solutions, which is roughly $|C_{ij}^{nn}/P|$. Based on this, we can interpret the size of the CMV along $\mathbf{n}$ as being proportional to $C_{ij}^{nn}$ and therefore the lobes of the CMV point along the eigenvectors of the matrix $C_{ij}$.
\begin{figure}[t]\centering
\includegraphics[width = 0.8\columnwidth]{CMV.pdf}
\caption{(Color online) Typical CMV shapes for four different cases of the correlation matrix written beside the CMV: (a) A dumbbell, (b) a clover, (c) a sphere, and (d) a wheel and axle.
}
\label{fig: CMV}
\end{figure}
We characterize spin-spin correlations via four main features of the CMV. These features are the CMV's size, shape, dimensionality, and orientation. The CMV's size roughly translates to the magnitude of the eigenvalues of $C_{ij}$. The CMV's shape is related to the ratio of the three eigenvalues to each other. The shape generally falls into one of a few categories, depicted in Fig.~\ref{fig: CMV}. When one of the eigenvalues is much larger than the other two, the CMV has the shape of a dumbbell, as in Fig.~\ref{fig: CMV}(a). When two eigenvalues are comparable, have opposite signs, and are larger than the third, the shape is a clover, as in Fig.~\ref{fig: CMV}(b). When all three eigenvalues are comparable, then the shape is a sphere or ellipsoid as in Fig.~\ref{fig: CMV}(c) if they have the same sign, and the shape resembles a wheel and axle as in Fig.~\ref{fig: CMV}(d) if one eigenvalue has a different sign. The CMV's dimensionality is contained in the description of its shape, but this feature is so important in our comparisons that we classify it separately. A dumbbell-shaped CMV is ``one dimensional,'' a clover-shaped one is ``two dimensional,'' and a sphere is ``three dimensional.'' The CMV's orientation tells us the directions of the eigenvectors of $C_{ij}$.
The features described above, despite being qualitative, nevertheless allow us to characterize the differences between Wigner approximations and the exact dynamics, as well as to identify the missing aspects of Wigner approximations. For example, we observe distinct and fairly simple trends such as that DTWA captures the revivals in the size of the CMVs more accurately than TWA (as already shown in Refs.~\cite{schachenmayer2015many, schachenmayer2015dynamics}.) Our most surprising finding is that both DTWA and TWA suppress correlations along one direction, thereby reducing the dimensionality of the CMV. On the other hand, the trends for the accuracy of TWA and DTWA are less apparent in the conventional way of plotting all components of the correlation matrix. Appendix~\ref{sec: component plots} shows the conventional componentwise analysis of correlations for the dynamics considered in the main text, so a curious reader can explore these themselves.
\section{Results}\label{sec: results}
In this section, we compare the dynamics of spin-spin correlations in DTWA, TWA, and the exact solution for various spin models. Specifically, in Sec.~\ref{subsec: Ising} we present the spin dynamics in the nearest-neighbor Ising model with no transverse field, in different dimensions, with different range of interactions, and from different initial states. Section~\ref{subsec: other} presents the spin dynamics in the 1D transverse field nearest-neighbor Ising model and the 1D nearest-neighbor XX model.
\subsection{Ising model}\label{subsec: Ising}
First, we consider the Ising model
\begin{equation}\label{eqn: HIsing}
\hat{H}_I = -\sum_{i\neq j} J_{ij} \hat{S}_i^z\hat{S}_j^z
\end{equation}
with arbitrary interactions $J_{ij}$.
The time-dependent equations for the quantum-mechanical spin operators are obtained from Heisenberg's equation $i\partial_t\hat{S}^\mu_i = [\hat{S}^\mu_i,\hat{H}]$, resulting in
\begin{align}\label{eqn: IsingEOM}
\dot{\hat{S}}_i^x &= \hat{S}_i^y\hat{B}_i^z,\nonumber\\
\dot{\hat{S}}_i^y &= -\hat{S}_i^x\hat{B}_i^z,\\
\dot{\hat{S}}_i^z &= 0,\nonumber
\end{align}
where $\hat{B}_i^\mu = \sum_{j\neq i} J_{ij}\hat{S}_j^\mu$. The same equations give the classical equations of motion for DTWA and TWA as well, with the quantum-mechanical operator $\hat{S}^\mu_i$ replaced by its classical counterpart $S^\mu_i$.
We initialize the system in the product state $\ket{\theta\theta\theta\textellipsis}$ with $\ket{\theta}=\cos\theta\ket{\uparrow}+\sin\theta\ket{\downarrow}$. We consider two different representative cases in the following sections: $\theta=\frac{\pi}{2}$ and $\theta=\frac{\pi}{4}$.
First, we will analytically solve this model. Equations~\eqref{eqn: IsingEOM} are integrable, and the solutions are
\begin{equation}\label{eqn: IsingEOMSolns}
\left(\begin{array}{c}\hat{S}^+_j(t)\\ \hat{S}^-_j(t)\\ \hat{S}^z_j(t)\end{array}\right) = \left(\begin{array}{ccc} e^{-i\hat{B}^z_jt}&0&0 \\ 0&e^{i\hat{B}^z_jt}&0 \\ 0&0&1\end{array}\right)
\left(\begin{array}{c}\hat{S}^+_j(0)\\ \hat{S}^-_j(0)\\ \hat{S}^z_j(0)\end{array}\right).
\end{equation}
The time dependence of $\hat{S}^x_j$ and $\hat{S}^y_j$ can be trivially obtained from $\hat{S}^\pm_j$. Note that $\hat{B}^z_j$ commutes with $\hat{H}_I$, and is therefore a constant. Using the relation that $\langle \hat{S}_i^\mu(0)\hat{S}_j^\nu(0)\rangle = \langle \hat{S}_i^\mu(0)\rangle\langle\hat{S}_j^\nu(0)\rangle$ for $i\neq j$ because the spins are initially independent, we obtain the solutions
\begin{align}\label{eqn: explicitSolns}
\langle \hat{S}_j^+(t)\rangle =& \left(\prod_{l\neq j} \langle e^{-iJ_{jl}t\hat{S}_l^z}\rangle\right) \langle \hat{S}_j^+(0)\rangle,\nonumber\\
\langle\hat{S}_j^+(t)\hat{S}_k^+(t)\rangle =& \left(\prod_{l\neq j,k} \langle e^{-i(J_{jl}+J_{kl})t\hat{S}_l^z}\rangle\right) \langle \hat{S}_j^+(0)e^{-iJ_{jk}t\hat{S}_j^z}\rangle \nonumber\\&\times \langle e^{-iJ_{jk}t\hat{S}_k^z}\hat{S}_k^+(0)\rangle, \nonumber\\
\langle\hat{S}_j^+(t)\hat{S}_k^-(t)\rangle =& \left(\prod_{l\neq j,k} \langle e^{-i(J_{jl}-J_{kl})t\hat{S}_l^z}\rangle\right) \langle \hat{S}_j^+(0)e^{iJ_{jk}t\hat{S}_j^z}\rangle \nonumber\\&\times \langle e^{-iJ_{jk}t\hat{S}_k^z}\hat{S}_k^-(0)\rangle, \nonumber\\
\langle\hat{S}_j^+(t)\hat{S}_k^z(t)\rangle =& \left(\prod_{l\neq j,k} \langle e^{-iJ_{jl}t\hat{S}_l^z}\rangle\right) \langle \hat{S}_j^+(0)\rangle \langle e^{-iJ_{jk}t\hat{S}_k^z}\hat{S}_k^z\rangle, \nonumber\\
\langle \hat{S}_j^-\rangle =& \langle\hat{S}_j^+\rangle^*,\nonumber\\
\langle\hat{S}_j^-(t)\hat{S}_k^-(t)\rangle =&\ \langle\hat{S}_j^+(t)\hat{S}_k^+(t)\rangle^*,\nonumber\\
\langle\hat{S}_j^-(t)\hat{S}_k^+(t)\rangle =&\ \langle\hat{S}_j^+(t)\hat{S}_k^-(t)\rangle^*,\nonumber\\
\langle\hat{S}_j^-(t)\hat{S}_k^z(t)\rangle =&\ \langle\hat{S}_j^+(t)\hat{S}_k^z(t)\rangle^*.
\end{align}
The Cartesian components of the magnetization and spin correlations can be obtained from
\begin{align}
&\langle \hat{S}_j^x\rangle = \langle\hat{S}_j^+\rangle + \langle\hat{S}_j^-\rangle,\nonumber\\
&\langle \hat{S}_j^y\rangle = -i(\langle\hat{S}_j^+\rangle - \langle\hat{S}_j^-\rangle),\nonumber\\
&C_{jk}^{xx} = C_{jk}^{++}+C_{jk}^{+-}+C_{jk}^{-+}+C_{jk}^{--},\nonumber\\
&C_{jk}^{xy} = -i(C_{jk}^{++}-C_{jk}^{+-}+C_{jk}^{-+}-C_{jk}^{--}),\nonumber\\
&C_{jk}^{yy} = -(C_{jk}^{++}-C_{jk}^{+-}-C_{jk}^{-+}+C_{jk}^{--}),\nonumber\\
&C_{jk}^{xz} = C_{jk}^{+z}+C_{jk}^{-z},\nonumber\\
&C_{jk}^{yz} = -i(C_{jk}^{+z}-C_{jk}^{-z}),\nonumber\\
&C_{jk}^{\mu\nu} = C_{jk}^{\nu\mu}.
\end{align}
All that remains is to evaluate the expectations in Eq.~\eqref{eqn: explicitSolns} in the exact solution, DTWA, and TWA. In DTWA and TWA, $\langle\hdots\rangle$ should be interpreted as average over the classical phase-space trajectories. Crucially, the explicit results for Eq.~\eqref{eqn: explicitSolns} in DTWA and TWA differ from the exact solution. This is because DTWA and TWA incorrectly estimate averages for products of spin operators on the same site at the initial time. It is worth noting that despite this crucial error, DTWA and TWA still qualitatively capture much of the dynamics of spin correlations, as we will see shortly. The mismatches with the exact solution have simple trends, which we explore in this section. The dynamics in DTWA can be much improved by going to higher order in the BBGKY hierarchy (which also integrates the Heisenberg equations for products of operators $\hat{S}_i^\mu\hat{S}_j^\nu$) and choosing a different phase space (see, e.g, Ref.~\cite{pucci2016simulation}).
We present explicit closed forms of Eq.~\eqref{eqn: explicitSolns} separately for the exact solution, DTWA and TWA in Eqs.~\eqref{eqn: explicitExactSolns},~\eqref{eqn: explicitDTWASolns}, and~\eqref{eqn: explicitTWASolns} in Appendix~\ref{sec: analytical_expns}. Closed forms for the spin correlations in the exact solution have also been calculated in Refs.~\cite{van2013relaxation, hazzard2014quantum}. To numerically evaluate Eqs.~\eqref{eqn: explicitExactSolns},~\eqref{eqn: explicitDTWASolns}, and~\eqref{eqn: explicitTWASolns} for an arbitrary $J_{ij}$ and $\theta$, we assume a chain with $11$ spins and periodic boundaries in the case of 1D models, and a $4\times4$ lattice with periodic boundary conditions for 2D models.
For the other models we consider in Sec.~\ref{subsec: other}, the solutions are more complicated although still integrable~\cite{calabrese2011quantum, calabrese2012quantum1, calabrese2012quantum2}, so we resort to numerically calculating the correlations. We again show that the mismatch between DTWA, TWA, and the exact solution has a simple trend. We also perturbatively calculate $C_{jk}^{\mu\nu}$ at short times in Sec.~\ref{sec: rigorous proof} for arbitrary spin models and rigorously prove our numerical observation.
\subsubsection{Nearest-neighbor 1D Ising model}
First, we study the case $\theta=\pi/2$ and nearest-neighbor interactions in a 1D chain, $J_{ij}=J\delta_{|i-j|=1}$. Figure~\ref{fig: Ising} shows the nearest-neighbor spin correlations for the exact dynamics, DTWA, and TWA.
We find that the shape and orientation of the CMVs are captured well by both TWA and DTWA, and the size is captured well at short times. All the CMVs have a clover shape [as in Fig.~\ref{fig: CMV}(b)]. All the CMVs have the right orientation: They all have large lobes along $\mathbf{y}+\mathbf{z}$ and $\mathbf{y}-\mathbf{z}$.
Despite the similarities listed above, there are two main differences between the exact solution, DTWA, and TWA. The first difference is the well-known inability of TWA to capture the periodic revivals present in the exact solution and DTWA. In fact, DTWA was invented mainly to capture these periodic revivals~\cite{schachenmayer2015many, schachenmayer2015dynamics}. The second difference these results reveal is that in DTWA and TWA, the CMVs are two dimensional, that is, the correlations vanish along the $x$ direction. This can be seen from looking at the components of the correlations in Eq.~\eqref{eqn: explicitTWASolns}. We will see that these differences are general features of spin model dynamics with product state initial conditions.
Our observations in Fig.~\ref{fig: Ising} about the inaccuracies of DTWA and TWA, especially the missing $C^{xx}_{ij}$ correlation, substantiate our argument that it is important to look at all components of the correlations while assessing these approximations. Plotting specific components, as in Fig.~\ref{fig: correlation2}(a), may be misleading about the performance of the approximations. For the model and initial condition considered here, the $C^{yz}$ component in Fig.~\ref{fig: correlation2}(a), which may be viewed as a slice of the CMVs in Fig.~\ref{fig: Ising} along $\frac{\mathbf{y}+\mathbf{z}}{\sqrt{2}}$ (because $C^{yy}_{ij}$ and $C^{zz}_{ij}$ are zero at all times), coincidentally happens to be a component which DTWA captures accurately. These coincidences may not occur for other models or initial conditions, as we will see in the following sections, because the direction misrepresented by the Wigner approximations is often not aligned along a Cartesian direction. All the nonzero Cartesian components of the correlations are plotted in Fig.~\ref{fig: Ising components}.
\begin{figure}[t]\centering
\includegraphics[width = \columnwidth]{1DIsingNNIntpi2.pdf}
\caption{(Color online) The CMVs for nearest-neighbor spin-spin correlations at different times in the nearest-neighbor 1D Ising model in the absence of a transverse field, for the exact solution (left), DTWA (middle), and TWA (right). At $t=0$, all the spins are aligned along $\mathbf{x}$, i.e., $\theta = \frac{\pi}{2}$. An animated movie showing this dynamics is included in the Supplemental Material~\cite{supplement}.}
\label{fig: Ising}
\end{figure}
\begin{figure}[t]\centering
\includegraphics[width = \columnwidth]{2DIsingNNIntpi2.pdf}
\caption{(Color online) The CMVs for nearest-neighbor spin-spin correlations at different times in the nearest-neighbor 2D Ising model in the absence of a transverse field, for the exact solution (left), DTWA (middle), and TWA (right). At $t=0$, all the spins are aligned along $\mathbf{x}$, i.e., $\theta = \frac{\pi}{2}$. An animated movie showing this dynamics is included in the Supplemental Material~\cite{supplement}.}
\label{fig: 2DIsing}
\end{figure}
\begin{figure}[t]\centering
\includegraphics[width = \columnwidth]{1DIsingR3Intpi2.pdf}
\caption{(Color online) The CMVs for nearest-neighbor spin-spin correlations at different times in the 1D Ising model in the absence of a transverse field and $\frac{1}{r^3}$ Ising interaction, for the exact solution (left), DTWA (middle), and TWA (right). At $t=0$, all the spins are aligned along $\mathbf{x}$, i.e., $\theta = \frac{\pi}{2}$. An animated movie showing this dynamics is included in the Supplemental Material~\cite{supplement}.}
\label{fig: Ising R3}
\end{figure}
\begin{figure}[t]\centering
\includegraphics[width = \columnwidth]{1DIsingAllIntpi2.pdf}
\caption{(Color online) The CMVs for spin-spin correlations at different times in the infinite-range Ising model in the absence of a transverse field, for the exact solution (left), DTWA (middle), and TWA (right). At $t=0$, all the spins are aligned along $\mathbf{x}$, i.e., $\theta = \frac{\pi}{2}$. An animated movie showing this dynamics is included in the Supplemental Material~\cite{supplement}.}
\label{fig: Ising AllInt}
\end{figure}
\begin{figure}[t]\centering
\includegraphics[width = \columnwidth]{1DIsingNNIntpi2NNN.pdf}
\caption{(Color online) The CMVs for next-nearest-neighbor spin-spin correlations at different times in the nearest-neighbor 1D Ising model in the absence of a transverse field, for the exact solution (left), DTWA (middle), and TWA (right). At $t=0$, all the spins are aligned along $\mathbf{x}$, i.e., $\theta = \frac{\pi}{2}$. An animated movie showing this dynamics is included in the Supplemental Material~\cite{supplement}.}
\label{fig: Ising NNN}
\end{figure}
\begin{figure}[t]\centering
\includegraphics[width = \columnwidth]{1DIsingNNIntpi4.pdf}
\caption{(Color online) The CMVs for nearest-neighbor spin-spin correlations at different times in the nearest-neighbor 1D Ising model in the absence of a transverse field, for the exact solution (left), DTWA (middle), and TWA (right). At $t=0$, all the spins are aligned halfway between $\mathbf{x}$ and $\mathbf{z}$, i.e., $\theta = \frac{\pi}{4}$. An animated movie showing this dynamics is included in the Supplemental Material~\cite{supplement}.}
\label{fig: Ising pi4}
\end{figure}
\begin{figure}[t]\centering
\includegraphics[width = \columnwidth]{1DTIM.pdf}
\caption{(Color online) The CMVs for nearest-neighbor spin-spin correlations in a nearest-neighbor 1D transverse Ising ($h=J/3$) system at different times, numerically calculated for the exact solution (left), DTWA (middle), and TWA (right). At $t=0$, all the spins are aligned along $\mathbf{x}$, i.e., $\theta = \frac{\pi}{2}$. An animated movie showing this dynamics is included in the Supplemental Material~\cite{supplement}.}
\label{fig: transIsing}
\end{figure}
\begin{figure}[t]\centering
\includegraphics[width = \columnwidth]{XX.pdf}
\caption{(Color online) The CMVs for nearest-neighbor spin-spin correlations in a 1D system with an $XX$ Hamiltonian, numerically calculated for the exact solution (left), DTWA (middle), and TWA (right), at different times. At $t=0$, all the spins are aligned along $\mathbf{x}$, i.e., $\theta = \frac{\pi}{2}$. An animated movie showing this dynamics is included in the Supplemental Material~\cite{supplement}.}
\label{fig: XX}
\end{figure}
\subsubsection{Dependence on dimension}\label{subsubsec: 2D}
It is a common expectation that semiclassical approximations perform better in higher dimensions, because the Wigner function does not spread much with time, due to small quantum fluctuations~\cite{polkovnikov2010phase}. To address this, we next study the case $\theta=\pi/2$ and nearest-neighbor interactions in a 2D lattice, $J_{ij}=J\delta_{|\vec{i}-\vec{j}|=1}$.
Figure~\ref{fig: 2DIsing} shows the nearest-neighbor spin correlations for the exact dynamics, DTWA, and TWA. We find that the comparison with the exact solution is similar to the 1D case : The shape and orientation of the CMVs are captured well by both TWA and DTWA, and the size is captured well at short times. Importantly though, the differences in the 1D Ising model also persist in the 2D model: The CMVs in DTWA and TWA are again two dimensional because the correlations completely vanish along $\mathbf{x}$, and the CMVs in TWA exponentially shrink with time. In fact, we rigorously prove in Appendix~\ref{sec: analytical_expns} that the CMV is two dimensional in DTWA and TWA in the nearest-neighbor Ising model in an arbitrary dimension and for any arbitrary initial state. Thus, although going to a higher dimension may improve some aspects of the performance of DTWA or TWA, it does not necessarily remedy the suppression of one correlation component. Further, we show in Sec.~\ref{sec: rigorous proof} that the correlations along the initial Bloch vector in TWA and DTWA are suppressed even for an arbitrary spin model in an arbitrary dimension. All the nonzero Cartesian components of the correlations for this model are plotted in Fig.~\ref{fig: 2DIsing components}.
\subsubsection{Dependence on range of interaction}\label{subsubsec: long-range}
It is also commonly expected that semiclassical approximations perform better for models with long-range interactions, again because the Wigner function does not spread much with time, due to small quantum fluctuations~\cite{polkovnikov2010phase}. To address this, we study two cases: first, Ising interactions decaying as $J_{ij}=\frac{J}{r_{ij}^3}$ in a 1D chain, which is typical in experiments with particles with a dipole moment, and second, infinite-range Ising interactions $J_{ij}=J$, as commonly realized in ion trap experiments. In both cases, we consider the initial state to have $\theta=\pi/2$. The infinite-range Ising model is well studied in the literature and leads to one-axis twisting of the total spin on the Bloch sphere~\cite{kitagawa1993squeezed, ma2011quantum}
Figure~\ref{fig: Ising R3} plots the nearest-neighbor spin correlations for the exact solution, DTWA and TWA in the $1/r^3$ Ising model. These CMVs also have clover shapes [as in Fig.~\ref{fig: CMV}(b)], and are still nearly two dimensional. The component $C^{xx}_{ij}$ is not zero in DTWA and TWA, but is much smaller than it is in the exact solution, as can be observed from the componentwise plots in Fig.~\ref{fig: Ising R3 components}. We will return to a general understanding of this suppression in Sec.~\ref{sec: rigorous proof}. The orientation of the CMVs is captured well by DTWA and TWA, and their size is captured well at short times.
Figure~\ref{fig: Ising AllInt} plots spin-spin correlations for the exact solution, DTWA and TWA in the infinite-range Ising model. Here the DTWA and TWA are capable of reproducing the dynamics at short times. The physical reason for this is that the correlations rapidly develop on a timescale $tJ\sim 1/\sqrt{N}$ (with $N$ being the number of spins), which is faster than the timescale for nearest-neighbor Ising models, essentially because more terms contribute to the dynamics. There is still a small suppression of correlations, but this suppression is much smaller than the magnitude of the correlations, because, as we show in Sec.~\ref{sec: rigorous proof}, the suppression grows on a much slower timescale $tJ\sim 1$. As a result, TWA and DTWA appear to accurately capture the initial rapid growth of correlations. The TWA and DTWA will lead to a noticeable suppression of correlations when $tJ\sim1$, as can be observed in the component wise plots in Fig.~\ref{fig: Ising All components}.
\subsubsection{Dependence on distance between spins}
In the models we study here, correlations in Wigner approximations generally get more accurate as the distance between the two spins increases. Here we calculate the correlations between next-nearest-neighbor spins in the nearest-neighbor 1D Ising model, with spins initialized to $\ket{\theta=\pi/2}$.
Figure~\ref{fig: Ising NNN} shows the next-nearest-neighbor spin correlations for the exact dynamics, DTWA, and TWA. In this case, DTWA agrees perfectly with the exact solution, and this can also be observed in the componentwise plots in Fig.~\ref{fig: Ising NNN components}. The CMVs in the exact solution and DTWA are one dimensional, while the CMVs in TWA are two dimensional, with a small $C^{xx}_{ij}$ component that is absent in the exact solution.
In all nearest-neighbor Ising models in an arbitrary dimension, and with no transverse field as considered throughout this section, all components of the correlations between spins with Manhattan distance greater than $2$ are zero in the exact solution, DTWA, and TWA. This can be easily verified from Eqs.~\eqref{eqn: explicitExactSolns},~\eqref{eqn: explicitDTWASolns}, and~\eqref{eqn: explicitTWASolns}. Correlations between faraway spins are generally not zero in long-range Ising models, and DTWA and TWA are expected to perform well in capturing the dynamics of these long-range correlations as the distance between spins increases. This will get clearer from our rigorous arguments for the dependence of the suppression with distance, which we will present in Sec.~\ref{sec: rigorous proof}.
\subsubsection{Dependence on initial states}
The accuracy and efficiency (i.e., number of samples required) of Wigner approximations depend strongly on the initial state. They become less accurate and significantly more numerically challenging for initial states different from $\theta=\pi/2$ and $\theta=0$. To demonstrate their accuracy, we calculate the nearest-neighbor correlations in the nearest-neighbor 1D Ising model (which is integrable) for $\ket{\theta=\pi/4}$.
Figure~\ref{fig: Ising pi4} shows the nearest-neighbor spin correlations in the exact solution, DTWA, and TWA. The CMVs in both Wigner approximations are again two dimensional at all times, as observed in all nearest-neighbor interaction cases above, and as rigorously proven in Sec.~\ref{sec: rigorous proof} for short times and Appendix~\ref{sec: analytical_expns} for all times. That is, correlations completely vanish along one direction. More interestingly, for this case, the suppressed direction rotates with time (for a closed-form expression of the direction of the vanishing correlations, see Appendix~\ref{sec: analytical_expns}). Aside from the two-dimensionality, the shape of the CMVs in the Wigner approximation reasonably agrees with the exact solution. Again, as expected, the CMVs in TWA exponentially shrinks in size, while the CMVs in DTWA and the exact solution undergo periodic oscillations at a period somewhat longer than the longest time presented in Fig.~\ref{fig: Ising pi4}. Further, there are also hints that the orientation of the CMVs in TWA is closer to the exact solution than the DTWA's is. This is to be expected from looking at Fig.~\ref{fig: correlation2}(b), for example, which showed that even the initial dynamics of $C^{xx}_{ij}$ in DTWA differed significantly from the exact solution and TWA. All the nonzero Cartesian components of the correlations are plotted in Fig.~\ref{fig: Ising pi4 components}.
The real advantage of visualizing the correlations as CMVs is demonstrated by the dynamics considered here: Plotting the CMVs clearly shows that DTWA and TWA completely miss correlations along one eigen direction, a fact which is obscured in the componentwise plots in Fig.~\ref{fig: Ising pi4 components} because the misrepresented direction is not aligned along a Cartesian direction.
For $\theta\notin \{0,\pi/2\}$, we note that DTWA presents a serious numerical obstacle in its implementation: There is a sign problem. The sign problem is notorious in quantum Monte Carlo algorithms, where it arises in fermionic systems as a result of negative wave functions due to anticommutations. The sign problem arises in DTWA because the Wigner function is negative at some of the phase-space points. In these cases, one way to sample the initial points $\mathbf{S}$ in phase space is with the weights $\frac{\left|W(\mathbf{S})\right|}{\int\! d\mathbf{S}~\left|W(\mathbf{S})\right|}$ and then multiply the Weyl symbol for the trajectory of $\mathbf{S}$ by the sign of $W(\mathbf{S})$.
When the sign problem occurs, a sample size scaling exponentially with $N$ is required to obtain a precise ensemble average (i.e with a small sampling error) for any observable in a system with $N$ spins~\footnote{The sampling error for the Bloch vector, i.e., the variance of the sample mean averaged over the classical trajectories, scales as $\alpha^N/N_s$, with $N_s$ the sample size, $N$ the number of spins, and $\alpha = \sum_{S_i}|P(S_i)|$ the sum of absolute values of Wigner functions at the initial phase points for a single spin. When $\theta\neq0,\pi/2$, $\alpha>1$, so the sample error increases exponentially with $N$ for fixed $N_s$.}.
While the results presented in this section were obtained from analytically integrating Eq.~\eqref{eqn: IsingEOM}, which is equivalent to implementing the Wigner approximations with an infinite sample size, a numerical implementation of the Wigner approximations would be computationally expensive. For example, the sampling error for $C^{yy}_{ij}$ at $t=0$ for $\theta=\pi/4$ and a sample size of $10^4$ is $0.019$. This error is comparable to the magnitude of $C^{yy}_{ij}$ during the dynamics and therefore we do not get much useful information about the correlation dynamics. The sampling error for $C^{yy}_{ij}$ reduces to $0.003$ for a larger sample size of $10^5$. This obstacle is not present for $\theta=\pi/2$, where the sampling error for $C^{yy}_{ij}$ for a sample size of $10^4$ is only $0.002$. Other components have similar errors for these sample sizes.
\footnotetext{OK}
The sign problem in DTWA can be ameliorated by rotating the phase space, such that the Wigner function is always positive at the initial phase points that are sampled. However, due to the different alignment between these points and the distinguished directions in the Hamiltonian (e.g., the $\mathbf{z}$ direction in the Ising model), the accuracy of the DTWA would need to be re-evaluated.
\subsubsection{Summary of Ising models}
Based on the integrable examples so far, we are able to observe simple trends regarding Wigner approximations: (a) For nearest-neighbor Ising models on a chain, square, or cubic lattice, the approximations completely miss correlations along one direction relative to the exact solution (this is true on any bipartite lattice, and is rigorously proven in Appendix~\ref{sec: analytical_expns}), (b) for longer-range Ising models, the approximations suppress correlations in the same direction as the nearest-neighbor case at short times, and, as expected, (c) correlations in TWA exponentially decay with time. There are also hints that the correlations are oriented incorrectly in DTWA for initial states different from $\ket{\theta=\pi/2}$. These trends were elegantly captured by plotting CMVs, while they are obscured in the componentwise correlation plots such as Fig.~\ref{fig: correlation2}(b) or Fig.~\ref{fig: Ising pi4 components}. The TWA and DTWA are more accurate in capturing correlations between spins that are far away from each other. The TWA and DTWA also perform better for models with long-range interactions, but their accuracy is limited to shorter times, as can be observed in the infinite-range interaction case. The TWA and DTWA have the same qualitative inaccuracies in nearest-neighbor models in higher dimensions as they do in one dimension.
Next we apply DTWA and TWA to the nearest-neighbor 1D transverse Ising model and the nearest-neighbor 1D XX model. We will find that the discrepancies between the Wigner approximations and the true dynamics have the same qualitative structure as observed in the zero-transverse-field Ising model.
\subsection{XX and transverse Ising models}\label{subsec: other}
For the nearest-neighbor 1D transverse Ising model given by
\begin{equation}\label{eqn: HtransIsing}
\hat{H}_{\rm T} = \hat{H}_I - h \sum_i \hat{S}^x_i,
\end{equation}
the time-dependent equations for the spins are
\begin{align}\label{eqn: transIsingEOM}
&\dot{\hat{S}}_i^x = \hat{S}_i^y \hat{B}_i^z,\nonumber\\
&\dot{\hat{S}}_i^y = - \hat{S}_i^x \hat{B}_i^z + h \hat{S}_i^z,\\
&\dot{\hat{S}}_i^z = - h \hat{S}_i^y.\nonumber
\end{align}
Equations~\eqref{eqn: transIsingEOM} are not analytically integrable. We numerically integrate them on a periodic chain with 11 spins.
Figure~\ref{fig: transIsing} depicts the CMVs obtained from a numerical implementation of exact diagonalization, DTWA, and TWA, when the system is initialized in $\theta=\pi/2$ and evolves under the model with $h=J/3$. The size, shape, and orientation of the CMVs in TWA and DTWA all approximately match with the exact solution, but as in the $h=0$ cases, the CMVs are somewhat two dimensional in both approximations. That is, the correlation along the direction perpendicular to the obvious clover shape is still much smaller in DTWA and TWA than it is in the exact solution. All the CMVs in these dynamics precess around the magnetic field. All the nonzero Cartesian components of the correlations are plotted in Fig.~\ref{fig: transIsing components}.
For the nearest-neighbor (NN) 1D XX model given by
\begin{equation}\label{eqn: HXX}
\hat{H}_{\rm XX} = - J\sum_{i} (\hat{S}^x_i \hat{S}^x_{i+1} + \hat{S}^y_i \hat{S}^y_{i+1}),
\end{equation}
the time-dependent equations for the spins are
\begin{align}\label{eqn: XXEOM}
&\dot{\hat{S}}_i^x = - \hat{S}_i^z \hat{B}_i^y,\nonumber\\
&\dot{\hat{S}}_i^y = \hat{S}_i^z \hat{B}_i^x,\\
&\dot{\hat{S}}_i^z = \hat{S}_i^x\hat{B}_i^y - \hat{S}_i^y\hat{B}_i^x.\nonumber
\end{align}
Equations.~\eqref{eqn: XXEOM} are not analytically solvable either. We numerically integrate them on a periodic chain with 11 spins.
Figure~\ref{fig: XX} depicts the CMVs obtained from a numerical implementation of exact diagonalization, DTWA, and TWA when the system is initialized in $\theta=\pi/2$. The size, shape, and orientation of the CMVs in TWA and DTWA all approximately match with the exact solution, but the CMVs are again two dimensional in both approximations at short times. Interestingly, at longer times, the direction along which the correlations are dominantly suppressed in DTWA and TWA seems to change somewhat independently of the CMVs' orientations: The CMVs are more squished along $\mathbf{x}$ for $tJ<2.1$ and they are more squished along $\mathbf{z}$ for $tJ>2.1$. All the nonzero Cartesian components of the correlations are plotted in Fig.~\ref{fig: XX components}.
\section{Why do DTWA and TWA suppress correlations?}\label{sec: rigorous proof}
We have observed a suppression of correlations in TWA and DTWA for the Ising, transverse Ising, and XX models. For the $h=0$ Ising models, where we explicitly calculated analytical expressions for the correlations, we attributed the suppression to DTWA and TWA incorrectly estimating averages for initial products of spin operators.
Here we present a general argument that shows that in any spin model for a generic initial product state $\ket{\theta\theta\textellipsis}$, the spin correlation along the initial spin direction $\mathbf{n} = \sin\theta\mathbf{x} + \cos\theta\mathbf{z}$ is always suppressed in DTWA and TWA, at $O(t^2)$. That is, we will show that
\begin{equation}\label{eqn: suppression}
\delta C^{nn}_{ij,\rm DTWA}(t) = |C^{nn}_{ij,\rm exact}(t)| - |C^{nn}_{ij,\rm DTWA}(t)| = At^2 + O(t^3)
\end{equation}
for $A>0$ and similarly for TWA, where $C_{ij}^{nn}$ is the correlation along the initial spin direction, defined as $C_{ij}^{nn} = \mathbf{n}\cdot C_{ij}\cdot\mathbf{n} = \sin^2\theta C_{ij}^{xx} + 2\sin\theta\cos\theta C_{ij}^{xz} + \cos^2\theta C_{ij}^{zz}$. [Note that there is no error to $O(t)$.]
Our argument makes use of the numerical observation that $C_{ij,\rm exact}^{nn}(t)>0$ and $C_{ij,\rm DTWA}^{nn}(t)\geq 0$ at short times. Therefore, to prove Eq.~\eqref{eqn: suppression}, it suffices to show that $C_{ij,\rm exact}^{nn}(t)> C_{ij,\rm DTWA}^{nn}(t)$ at $O(t^2)$.
We consider a general translationally invariant Hamiltonian with two-body interactions,
\begin{equation}
\hat{H} = -\sum_{i\mu} h^\mu \hat{S}_i^\mu - \sum_{i\neq j}\sum_\mu J_{ij}^\mu \hat{S}_i^\mu \hat{S}_j^\mu,
\end{equation}
and the initial product state $\ket{\theta\theta\textellipsis}$ as stated before. This covers all the cases we have considered in this paper.
The time-dependent equation for any spin is
\begin{equation}\label{eqn: arbitrary diffeq}
\dot{\hat{S}}_i^\mu = \epsilon^{\mu\nu\alpha} \hat{S}_i^\nu (h^\alpha + \hat{B}_i^\alpha),
\end{equation}
with $\epsilon$ being the Levi-Civit\`a tensor. We use the Einstein summation convention for the greek indices throughout this section. At short times, $\hat{S}_i^\mu(t)$ is [up to $O(t^2)$]
\begin{align}\label{eqn: Taylor}
\hat{S}_i^\mu(t) = & \hat{S}_i^\mu(0) + t \dot{\hat{S}}_i^\mu + \frac{t^2}{2}\ddot{\hat{S}}_i^\mu \nonumber\\
=& \hat{S}_i^\mu(0) + t \epsilon^{\mu\nu\alpha} \hat{S}_i^\nu(0) [h^\alpha + \hat{B}_i^\alpha(0)]\nonumber\\
&+ \frac{t^2}{2} \hat{S}_i^\lambda(0) \{ \epsilon^{\nu\lambda\beta}\epsilon^{\mu\nu\alpha} [h^\beta + \hat{B}_i^\beta(0)] [h^\alpha + \hat{B}_i^\alpha(0)] \nonumber\\
&+ \epsilon^{\mu\lambda\alpha} \epsilon^{\alpha\nu\beta} J_{ij}^\alpha \hat{S}_j^\nu(0) [h^\beta + \hat{B}_j^\beta(0)] \},
\end{align}
where $\ddot{\hat{S}}_i^\mu$ is obtained by differentiating Eq.~\eqref{eqn: arbitrary diffeq}.
We substitute Eq.~\eqref{eqn: Taylor} to calculate $C_{ij}^{\mu\nu}(t)$ in the exact solution, TWA, and DTWA up to $O(t^2)$. We define
\begin{align}
&\mathscr{S}^\mu = \expect{\hat{S}_i^\mu(0)},\nonumber\\
&\mathscr{C}^{\mu\nu}_2 = \expect{\hat{S}_i^\mu(0) \hat{S}_i^\nu(0)},\nonumber\\
&\mathscr{C}^{\mu\nu\lambda}_3 = \expect{\hat{S}_i^\mu(0) \hat{S}_i^\nu(0)\hat{S}_i^\lambda(0)}
\end{align}
and use the relations
\begin{align}\label{eqn: mathscrC}
& \mathscr{S}^\mu_{\rm exact} = \mathscr{S}^\mu_{\rm DTWA} = \mathscr{S}^\mu_{\rm TWA}, \nonumber\\
& \mathscr{C}^{\mu\nu}_{2,\rm exact} = \frac{1}{4}\delta_{\mu\nu} + \frac{i}{2}\mathscr{S}^\alpha \epsilon^{\mu\nu\alpha}, \nonumber\\
& \mathscr{C}^{\mu\nu}_{2,\rm TWA} = \mathscr{C}^{\mu\nu}_{2,\rm DTWA} = \frac{1}{4}\delta_{\mu\nu}.
\end{align}
We reemphasize that $\mathscr{C}_{2,\rm exact}$ is the quantum expectation of operators, while $\mathscr{C}_{2,\rm DTWA}$ and $\mathscr{C}_{2,\rm TWA}$ are averages over classical trajectories. Note that $\mathscr{C}_3$ can be written similarly to Eq.~\eqref{eqn: mathscrC}, but there are more cases to write, so we do not present them here.
It is straightforward to show that $C_{ij}^{\mu\nu}(t)$ in the exact solution, TWA, and DTWA are identical to each other at $O(1)$ and $O(t)$. Further, it can be verified, although somewhat tediously, that the difference between the exact solution and the Wigner methods arises at $O(t^2)$, and that the only terms that evaluate to different results are
\begin{align}
C_{ij}^{'\mu\nu}(t) &= t^2 \epsilon^{\mu\mu'\alpha} J_{ij}^\alpha J_{ij}^\beta [ \epsilon^{\nu\nu'\beta}\mathscr{C}^{\mu'\beta}_2\mathscr{C}^{\alpha\nu'}_2 \nonumber\\
& + \frac{1}{2}\epsilon^{\mu'\lambda\beta} \mathscr{S}^\lambda (\mathscr{C}^{\beta\alpha\nu}_3 + \mathscr{C}^{\nu\beta\alpha}_3) \nonumber\\
& +\frac{1}{2}\epsilon^{\alpha\lambda\beta} \mathscr{C}^{\mu'\beta}_2(\mathscr{C}^{\lambda\nu}_2 + \mathscr{C}^{\nu\lambda}_2) ].
\end{align}
The difference between TWA or DTWA and the exact solution can then be evaluated using Eq.~\eqref{eqn: mathscrC}, yielding
\begin{align}
\delta C^{xx}_{ij,\rm DTWA}(t) = &\frac{t^2}{4} \{ (\mathscr{S}^x)^2[(J_{ij}^y)^2+(J_{ij}^z)^2] - (\mathscr{S}^y)^2J^x_{ij}J^y_{ij} \nonumber\\ &- (\mathscr{S}^z)^2J^x_{ij}J^z_{ij} \}, \nonumber\\
\delta C^{xy}_{ij,\rm DTWA}(t) = &\frac{t^2}{4}\{ \mathscr{S}^x\mathscr{S}^yJ_{ij}^z [ J_{ij}^z - 2(\mathscr{S}^z)^2(J_{ij}^x+J_{ij}^y) ] \}.
\end{align}
The other components can be found by cyclic permutation, and $\delta C_{ij,\rm TWA}(t)$ can be similarly obtained from Eq.~\eqref{eqn: mathscrC}.
Specifically, setting $(\mathscr{S}^x,\mathscr{S}^y,\mathscr{S}^z)=\frac{1}{2}(\sin\theta,0,\cos\theta)$,
\begin{align}\label{eqn: error}
\delta C^{nn}_{ij,\rm DTWA}(t) = &\frac{t^2}{16}(J_{ij}^y)^2 + \frac{t^2}{16}( J_{ij}^x\cos^2\theta - J_{ij}^z\sin^2\theta)^2,\nonumber\\
\delta C^{nn}_{ij,\rm TWA}(t) = &\frac{t^2}{16}(J_{ij}^y)^2 + \frac{t^2}{16}( J_{ij}^x\cos^2\theta + J_{ij}^z\sin^2\theta)^2,
\end{align}
which are both nonnegative. This proves that TWA and DTWA always suppress correlations along the initial spin direction at short times, for arbitrary spin models. Our results in this section, which identify the error in TWA and DTWA [Eq.~\eqref{eqn: error}] and their source [Eq.~\eqref{eqn: mathscrC}], could potentially open avenues to modify the semiclassical equations to develop more accurate approximations.
\section{Conclusions}\label{sec: conclusions}
We have demonstrated that the accuracy of Wigner approximations is more nuanced than previously believed, and uncovered properties seemingly intrinsic to both TWA and DTWA, namely, that they incorrectly predict suppressed correlations along one direction. We presented a rigorous perturbative argument to explain the suppressed correlations at short times. The suppressed correlations are often difficult to catch in conventional componentwise plots due to the number and complexity of the correlations and often a misalignment of the suppressed correlation with any Cartesian directions. We also found hints that the orientation of the correlations at short times, at least when the spins do not initially point along a special direction of the Hamiltonian, is sometimes more accurate in TWA than in DTWA. We have systematically explored the performance of DTWA and TWA by changing various parameters, including the dimension of the model, the range of interactions, the distance between the correlated spins, and the initial state, as well as adding external fields to the model, and found that the major source of error in all cases is suppressed correlations along one direction. This observation persists even in cases where semiclassical approximations are expected to work well, such as higher dimensions and long-ranged interactions, as well as other nonintegrable models [such as the 2D transverse Ising model with short- and long-range interactions] that we have studied but not shown in this paper. We have condensed these observations into Table~\ref{table: conclusion}. Understanding the capabilities of TWA and DTWA that we have developed in this paper will better enable practitioners to choose the approximations that are most suited to capture the features they are interested in.
\begin{table}[t]\centering
\begin{tabular}{cccccc
\hline\hline
{\textbf Model} & Size & Revivals & Shape & 3D nature & Orientation\\
\hline
1D NN Ising & \checkmark & DTWA & \checkmark & $\times$ & \checkmark\\
\hline
2D NN Ising & \checkmark & DTWA & \checkmark & $\times$ & \checkmark\\
\hline
1D $\frac{1}{r^3}$ Ising & \checkmark & DTWA & \checkmark & $\times$ & \checkmark\\
\hline
infinite-range Ising & \checkmark & DTWA & \checkmark & $\times$ & \checkmark\\
\hline
NN Ising $\ket{\theta=\frac{\pi}{4}}$ & \checkmark & DTWA & \checkmark & $\times$ & TWA\\
\hline
NN Ising $C_{\langle\langle ij\rangle\rangle}$ & \checkmark & DTWA & \checkmark & & \checkmark\\
\hline
TIM & \checkmark & & \checkmark & $\times$ & $\checkmark$\\
\hline
XX & \checkmark & & \checkmark & $\times$ & \checkmark\\
\hline\hline
\end{tabular}
\caption{Summary of DTWA's and TWA's abilities in capturing different aspects of spin-spin correlation dynamics in a variety of spin models. We categorize their ability to correctly capture the overall size of CMVs at short times, revival of CMVs at longer times (if applicable), the rough shape up to any suppressed correlations, their 3D nature at short times (i.e., whether DTWA and TWA capture the three-dimensionality of CMVs present in the exact solution), and orientation of CMVs. Any text in the cells means that only the indicated method reasonably captures that category. The DTWA and TWA never have three dimensional CMVs at short times because one correlation component is suppressed in all the cases.}
\label{table: conclusion}
\end{table}
\section*{Acknowledgments}
This material was based upon work supported with funds from the Welch Foundation, Grant No. C-1872. K.R.A.H. thanks the Aspen Center for Physics, supported by the National Science Foundation Grant No. PHY-1066293, for its hospitality while part of this work was performed. We thank Rick Mukherjee and Anthony Mirasola for useful conversations.
B.S. and K.C.W. contributed equally to this work.
|
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\begin{document}
\begin{titlepage}
\rightline{\vbox{\halign{&#\hfil\cr
&ANL-HEP-PR-94-84\cr
&\today\cr}}}
\vspace{0.25in}
\begin{center}
{\large\bf
THE SPECTRUM OF THE $O(g^4)$ SCALE-INVARIANT LIPATOV KERNEL}
\medskip
Claudio Corian\`{o} and Alan R. White
\footnote{Work supported by the U.S. Department of
Energy, Division of High Energy Physics, Contract\newline W-31-109-ENG-38}
\\ \smallskip
High Energy Physics Division, Argonne National Laboratory, Argonne, IL
60439.\\ \end{center}
\begin{abstract}
The scale-invariant $O(g^4)$ Lipatov kernel has been determined by t-channel
unitarity. The forward kernel responsible for parton evolution is evaluated
and its eigenvalue spectrum determined. In addition to a logarithmic
modification of the $O(g^2)$ kernel a distinct new kinematic component
appears. This component is infra-red finite without regularization and
has the holomorphic factorization property necessary for conformal
invariance. It gives a reduction (of ~$\sim 65\alpha_s^2/\pi^2\sim 0.15$) in
the power growth of parton distributions at small-x.
\end{abstract}
\renewcommand{\thefootnote}{\arabic{footnote}} \end{titlepage}
\subhead{1. INTRODUCTION}
The BFKL pomeron\cite{lip} or, more simply, the Lipatov pomeron, has
recently attracted growing attention, both from the theoretical and the
experimental side. The BFKL equation resums leading logarithms in $1/x$.
When applied in the forward direction, at large $Q^2$, it
becomes an evolution equation for parton distributions. The Lipatov pomeron
solution of the equation predicts that a growth of the form
$$ \eqalign {F_2(x,Q^2) ~\sim ~ x^{1-\alpha_0} ~\sim ~x^{-{1 \over 2}}} ~,
\eqno(\refstepcounter{equation}\theequation)\label{F2}
$$
where $\alpha_0 - 1$ is the leading eigenvalue of the forward $O(g^2)$
Lipatov kernel, should be observed in the small-x behaviour of structure
functions. The BFKL pomeron is important in hard diffractive processes in
general, for example deep-inelastic diffraction\cite{bar}, and, perhaps,
in rapidity-gap jet production\cite{ahm}. BFKL resummation is also
anticipated to play a key role in all semi-hard QCD processes
\cite{marchesini1}, where there is a direct coupling of the hard scattering
process to the pomeron. It is one of the major results of the HERA
experimental program that a growth similar to that of (\ref{F2}) is
observed\cite{der}.
{}From both a theoretical and an experimental viewpoint, it is vital to
understand how the BFKL equation, and (\ref{F2}) in particular, is affected
by next-to-leading logarithm contributions. In recent papers \cite{ker,ca}
the scale invariant part of the $O(g^4)$ next-to-leading kernel has been
determined by reggeon diagram and t-channel unitarity techniques. In this
paper we summarise some newly derived properties of this kernel,
concentrating on the forward direction relevant for the evolution of parton
distributions.
The new kernel is initially expressed in terms of transverse momentum
integrals. We have evaluated these integrals explicitly in the forward
direction. The results for the connected part of the kernel can be presented
in terms of finite combinations of logarithms. We find that there
are two components. The first simply has the structure of the $O(g^2)$ kernel
but with additional logarithms of all the transverse momenta involved. This
component can also be obtained by squaring the $O(g^2)$ kernel. The infra-red
divergences it produces after integration are regulated by the disconnected
part of the kernel. Also, for this component the new eigenvalues are
trivially obtained by squaring the $O(g^2)$ eigenvalues.
The second component is a new kinematic form which appears for the first
time at $O(g^4)$. It has a number of important properties. Firstly not only
is it separately finite, but it has no singularities generating infra-red
divergences after integration. It therefore requires no regulation. A
completely new eigenvalue spectrum is produced, which we give an explicit
expression for. We find that the spectrum posesses the fundamental property
of holomorphic factorization, which is a necessary condition for conformal
symmetry of the kernel\cite{lk}.
Since the new component appears first at $O(g^4)$ and also has the same
conformal invariance property as the leading-order kernel, we anticipate that
scale-ambiguities in its absolute evaluation will appear only in
higher-orders. That is to say it makes as much sense to evaluate this new
component at a fixed value of $\alpha_s$ as it did to evaluate the
leading-order contribution with such a value. Consequently we can quote a
result for the modification of $\alpha_0$ by this contribution. There is a
reduction of just the right order of magnitude to give an improvement in the
phenomenology, while preserving a significant effect.
We are unable, as yet, to give a complete result for how (\ref{F2}) is
modified by our results. This is because we must first determine how the
scale-invariance of the $O(g^4)$ kernel is broken by the off-shell
renormalization scale so that, presumably, $g^2 /4\pi \to \alpha_s(Q^2)$. This
is non-trivial since we expect that all the transverse momenta in the diagrams
of the kernel will be involved in the scale-breaking. Fadin and Lipatov have
already calculated\cite{fad} the full reggeon trajectory function, that
gives the disconnected piece of the kernel, in the next-to-leading log
approximation - including renormalization effects. The diagram structure we
have anticipated is what is found, but there are additional scale-breaking
internal logarithm factors involved. As outlined in \cite{ca}, we hope to
determine the scale-breaking logarithms, that occur in the remainder of the
kernel, by an extension of the Ward identity plus infra-red finiteness
analysis that gives the scale-invariant kernel.
The contribution of ($t$-channel) four-particle nonsense states to the
connected part of the $O(g^4)$ kernel is given in \cite{ker} as a sum of
transverse momentum integrals
$$
\eqalign{(g^2N)^{-2} K^{(4n)}_{2,2}(k_1&,k_2,k_3,k_4)_c~=~K_2~+~K_3~+K_4~}.
\eqno(\refstepcounter{equation}\theequation)\label{sum}
$$
To be consistent with the diagrammatic notation used below, we
introduce a momentum conserving $\delta$-function - compared to the definition
given in \cite{ker} - and write
$$
\eqalign{K_i~=~(2\pi)^3~\delta^2(k_1+k_2 - k_3 -k_4)~\tilde{K_i} , }
\eqno(\refstepcounter{equation}\theequation)\label{del}
$$
with
$$
\eqalign{\tilde{K_2}~=~- \sum_{\scriptscriptstyle 1<->2}
\Biggl({k_1^2J_1(k_1^2)k_2^2k_3^2+k_1^2J_1(k_1^2)k_2^2k_4^2+
k_1^2k_3^2J_1(k_3^2)k_4^2+k_1^2k_3^2k_4^2J_1(k_4^2) \over
(k_1-k_3)^2} \Biggr),}
\eqno(\refstepcounter{equation}\theequation)
$$
$$
\eqalign{\tilde{K_3}~=~\sum_{\scriptscriptstyle 1<->2}~
J_1((k_1-k_3)^2)\Bigl(k_2^2k_3^2+k_1^2k_4^2\Bigr),}
\eqno(\refstepcounter{equation}\theequation)
$$
and
$$
\eqalign{\tilde{K_4}~=~\sum_{\scriptscriptstyle 1<->2}~
k_1^2k_2^2k_3^2k_4^2~I(k_1,k_2,k_3,k_4), }
\eqno(\refstepcounter{equation}\theequation)
$$
where
$$
\eqalign{J_1(k^2)~=~{1 \over (2\pi)^3}\int d^2q {1 \over q^2(k-q)^2}}
\eqno(\refstepcounter{equation}\theequation)
$$
and
$$
\eqalign{ I(k_1,k_2,k_3,k_4)~=~{1 \over (2\pi)^3}\int d^2p {1 \over
p^2(p+k_1)^2(p+k_4)^2(p+k_1-k_3)^2}.}
\eqno(\refstepcounter{equation}\theequation)\label{box}
$$
The Ward Identity constraint that the kernel should vanish when $k_i \to
0,~i=1,..,4$,
together with infra-red finiteness, determine the relative weights of $K_2,
K_3$ and $K_4$.
It will be convenient to introduce a diagrammatic notation for transverse
momentum integrals. We define
\epsffile{fig1.ps}
\noindent $~=~~(2\pi)^3\delta^2(\sum k_i~ - \sum k_i')(\sum k_i~)^2~~~~
{}~~~~~~~~=~~(1/2\pi)^{3n}\int d^2k_1...d^2k_n~ /~k_1^2...k_n^2 .$
\vspace{.1in}
\noindent As we indicated above, we will define all kernels (and parts of
kernels) to include a factor $(2\pi)^3\delta^2(\sum k_i -\sum k_i') $. They
are then dimensionless and formally scale-invariant. $K_2, K_3$ and $K_4$
can be represented as a sum of diagrams of the form shown in Figs.~1(a), 1(b)
and 1(c) respectively.
\epsffile{fig2.ps}
\begin{description}
\item[Fig.~1] (a), (b), (c) - connected diagrams for the $O(g^4)$
kernel;~~ (d), (e) - disconnected diagrams.
\end{description}
In \cite{ker}, the disconnected part of the kernel was assumed to include
diagrams of the form of Fig.~1(d) only. Although generated by four-particle
nonsense states, diagrams of the form of Fig.~1(e) were not included. This
was essentially because they can not be associated with higher-order
reggeization. In fact such diagrams should be included in $K^{(4n)}_{2,2}$.
The point being that there is a further contribution to the $O(g^4)$ kernel,
from iteration of the two-particle nonsense state, which cancels the
contribution of such diagrams. Iteration of the two-particle nonsense state
gives a contribution of the form $[K^{(2)}_{2.2}]^2$, which can be
represented diagrammatically as in Fig.~2.
\vspace{.2in}
\epsffile{fig3.ps}
\begin{description}
\item[Fig.~2] Iteration of the leading-order kernel via two-particle
nonsense states.
\end{description}
For $K^{(4n)}_{2,2}$ to be properly regulated after integration, diagrams of
the form of both Fig.~1(d) and Fig.~1(e) must be included and the result is
$$
\eqalign{(g^2N)^{-2} K^{(4n)}_{2,2}(k_1&,k_2,k_3,k_4)_c~=
{}~K_0~+~K_1~+~K_2~+~K_3~+K_4~,}
\eqno(\refstepcounter{equation}\theequation)\label{sum2}
$$
where $K_0$ contains the diagrams of the form of Fig.~1(e). If we use
(\ref{del}), $K_0$ is given by
$$
\eqalign{\tilde{K_0}~=~
\sum_{\scriptscriptstyle 1<->2}~
(2\pi)^3 J_1(k_1^2)J_1(k_2^2\Bigl(\delta^2(k_2-k_4)
+~\delta^2(k_2-k_3)\Bigr),}
\eqno(\refstepcounter{equation}\theequation)
$$
$K_1$ contains the diagrams of the form of Fig.~1(d) and
$$
\eqalign{\tilde{K_1}~=~-{2 \over 3}
\sum_{\scriptscriptstyle 1<->2}~
(2\pi)^3 k_1^2J_2(k_1^2)k_2^2\Bigl(k_3^2\delta^2(k_2-k_4)
+~k_4^2\delta^2(k_2-k_3)\Bigr),}
\eqno(\refstepcounter{equation}\theequation)
$$
with
$$
\eqalign{J_2(k^2)~=~{1 \over (2\pi)^3}\int d^2q {1 \over (k-q)^2}J_1(q^2).}
\eqno(\refstepcounter{equation}\theequation)
$$
That the $K_0$ contributions from $K^{(4n)}_{2,2}$ and $(K^{(2)}_{2,2})^2$
must cancel, determines that the full $O(g^4)$ kernel is given by
$$
\eqalign{ K^{(4)}_{2,2}~~=~~{1 \over 2^3}K^{(4n)}_{2,2} ~ -~(K^{(2)}_{2,2})^2}
\eqno(\refstepcounter{equation}\theequation)\label{full}
$$
This is the kernel that we wish to evaluate in the ``forward'' direction
$k_1=-k_2=k,~k_3=-k_4=k'$. Our result for $K^{(4)}_{2,2}(k,-k,k',-k')$ is
a much simpler expression than the full result given by (\ref{full}).
In writing down(\ref{full}) we have determined the overall sign by the
requirement that the contribution of the four-particle state should
be positive. The overall magnitude has been determined by noting that
the diagrams of the form of Fig.~1(e) contain only elements that appear in
the leading-order kernel and their contribution in $K^{(4n)}_{2,2}$ is
unambiguous. This implies that these diagrams should occur in
$K^{(4n)}_{2,2}$ (and therefore $\bigl(K^{(2)}_{2,2}\bigr)^2$) with an
absolute magnitude that is equal to that obtained by simple-minded iteration
of the leading-order kernel.
The major technical problem in determining $K^{(4)}_{2,2}(k,-k,k',-k')$ is the
evaluation of the box graph, i.e. $I(k,-k,k',-k')$ defined in (\ref{box}).
As we will show in detail in \cite{ca2}, if we regularize $I$ with a mass
term $m^2$ in each propagator it can be evaluated as a sum of logarithms
associated with each of the possible ``two-particle'' thresholds in the
external momenta. As $m^2 \to 0$, we obtain
\begin{eqnarray}
&& 2\pi^2 I[k,k']= A_{12} Log[{ k'}^2/m^2] +
A_{23} Log[k^2/m^2] + A_{34} Log[{k'}^2/m^2]\nonumber \\
&& + A_{13} Log[(k+k')^2/m^2] +
A_{14}Log[k^2/m^2] +A_{24} Log[(k-k')^2/m^2
\end{eqnarray}
where
$$
\eqalign{ &A_{12} = {k^2 - {k'}^2 \over k^2 (k+k')^2(k-k')^2}
{}~~~~~~~~~~~~~~~~~~~~~~A_{13}={1\over k^2 {k'}^2}\cr
&A_{14}={{k'}^2-k^2\over k'^2 (k+ k')^2 (k-k')^2}~~~~~~~~~~~~~~~~~~~~~
A_{23}={k'^2-k^2\over k'^2 (k+k')^2 (k-k')^2}\cr
&A_{24}={1\over k^2 {k'}^2}~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
A_{34}={k^2- {k'}^2\over k^2 (k+k')^2(k-k')^2}}
\eqno(\refstepcounter{equation}\theequation)
\label{f1}
$$
and so, as $m^2 \to 0$,
$$
\eqalign{
\tilde{K}_4 ~\to ~ {-~k^2 {k'}^2 \over (2\pi^2)}&\Biggl(
{ 2({k'}^2 -k^2) \over (k+k')^2 (k-k')^2}
Log\Biggl[{{k'}^2\over k^2} \Biggr] \cr
&+ {1\over (k-k')^2} Log\Biggl[{ (k-k')^2\over m^2}\Biggr]
{}~+~{1\over (k+k')^2} Log\Biggl[{ (k +k')^2\over m^2}\Biggr]
\Biggr)~ .}
\eqno(\refstepcounter{equation}\theequation)\label{bm2}
$$
$\tilde{K}_3$ simply gives a contribution of the same form as the last
two terms in (\ref{bm2}), i.e. as $m^2 \to 0$
\begin{eqnarray}
\tilde{K}_3~\to~{k^2 {k'}^2 \over (2\pi^2)} \left(
{1\over (k-k')^2} Log\left[{ (k-k')^2\over m^2}\right]
+ { 1\over (k+k')^2} Log\left[{ (k+k')^2\over m^2}\right]\right)
\end{eqnarray}
Similarly $\tilde{K}_2$ gives
$$
\eqalign{
\tilde{K_2} \to {-k^2 {k'}^2 \over (2\pi^2) } & \Biggl(
{1 \over (k-k')^2} (Log\Biggl[{ k^2\over m^2}\Biggr] +
Log\Biggl[{ {k'}^2\over m^2}\Biggr] ) \cr
+ & { 1\over (k+k')^2} (Log\Biggl[ { k^2\over m^2} \Biggr] +
Log\Biggl[{ {k'}^2 \over m^2} \Biggr] ) \Biggr).}
\eqno(\refstepcounter{equation}\theequation)
$$
The infra-red finiteness of $\tilde{K}^{(4n)}_c ~=~ \tilde{K}_2 +
\tilde{K}_3 +\tilde{K}_4$ is now apparent and we can write
$$
\eqalign{2\pi^2\tilde{K}^{(4n)}_c~&=~
\Biggl( {k^2{k'}^2 \over (k-k')^2}Log\left[{(k-k')^4 \over k^2{k'}^2 }\right]
{}~+~ {k^2{k'}^2 \over (k+k')^2} Log\left[{(k+k')^4 \over k^2{k'}^2 }\right]
\Biggr)\cr
&~~~~~-~~~~
\Biggl( {2 k^2{k'}^2 (k^2 - {k'}^2) \over (k-k')^2(k+k')^2}Log\left[{k^2 \over
{k'}^2}\right] \Biggr) \cr
&=~~~~ \Biggl( ~~{\cal K}_1~~\Biggr) ~~-~~\Biggl( ~~{\cal K}_2~~\Biggr)
. }
\eqno(\refstepcounter{equation}\theequation)\label{4nc}
$$
Note that only ${\cal K}_1$ gives infra-red divergences (at $k'= \pm k$) when
integrated over $k'$. These divergences are cancelled by $\tilde{K}_0$ and
$\tilde{K}_1$. We will not discuss this cancellation explicitly, but
implicitly include $\tilde{K}_0$ and $\tilde{K}_1$ in ${\cal K}_1$ for the
rest of our discussion.
Apart from the logarithmic factors, ${\cal K}_1$ has the same structure as
the forward (connected) $O(g^2)$ kernel. Indeed, if we evaluate
all of the diagrams generated by Fig.~2 that survive in the forward
direction, it is straightforward to show that
$$
\eqalign{ {\cal K}_1 ~=~ (2\pi)^2 \widetilde{(K^{(2)}_{2,2})^2}~, }
\eqno(\refstepcounter{equation}\theequation)
$$
implying from (\ref{full}) and (\ref{4nc}) that
$$
\eqalign{ \tilde{K}^{(4)}_{2,2}~~=~~- {1 \over 2^4\pi^2} (3{\cal K}_1 ~ +~{\cal
K}_2)~.}
\eqno(\refstepcounter{equation}\theequation)\label{full2}
$$
There are a number of reasons to believe that the contribution of
$(K^{(2)}_{2,2})^2$ to the $O(g^4)$ kernel should produce the
scale-dependence of the $O(g^2)$ kernel. Indeed one might be tempted to
directly interpret the logarithms in ${\cal K}_1$ as associated with coupling
constant renormalization in the leading-order kernel. While this can not
be simply correct (real ulta-violet renormalization has to be involved to
bring in the correct asymptotic freedom coefficients) there may well be some
sense in which this is the case. We clearly have to carry out a full
scale-breaking analysis to determine what this sense may be.
The interesting part of (\ref{full2}) is the ${\cal K}_2$ component. This
is finite at $k=\pm k'$, and so does not generate any divergences when
integrated. The symmetry properties also determine that this term can only
appear at the first logarithmic level (since the antisymmetry of
$Log[k^2/{k'}^2]$ compensates for the antisymmetry of $(k^2-{k'}^2)$ ). It is
therefore a completely new feature of the $O(g^4)$ kernel.
We now move on to the eigenvalues of $K^{(4)}_{2,2}$. We use as a complete
set of orthogonal eigenfunctions
$$
\eqalign{ \phi_{\mu,n}(k')~=~({k'}^2)^{\mu}~e^{i n \theta}}~~~~~~~~\mu=~{1
\over2}
+i\nu,~~ n=0,\pm1,\pm2,...
\eqno(\refstepcounter{equation}\theequation)
$$
(Our definition of the kernel requires that we keep a factor of ${k'}^{-2}$ in
the measure of the completeness relation for eigenfunctions relative to
\cite{lip}). The eigenvalues of $(K^{(2)}_{2,2})^2$ are trivially given by
the square of the $O(g^2)$ eigenvalues, and so the essential problem is to
determine the eigenvalues of ${\cal K}_2$. As a preliminary we first define
${\cal K}_2$ for non-integer dimensions.
Since each logarithm in ${\cal K}_2$ originates from an integral of the form
of $J_1$ we can replace it by a simple integral of the form
\begin{eqnarray}
{k^2 \over 2\pi}\int {d^D q\over q^2 (k-q)^2}~=~\eta[k^2]^{D/2~-1}~,
\,\,\,\,\,\,\,\eta={\Gamma[2-D/2]\Gamma[D/2-1]^2\over \Gamma[D-2]},
\label{j1}
\end{eqnarray}
where $\eta \to 2(D-2)^{-1}$ when $D \to 2$. This gives
\begin{eqnarray}
{\cal K}_2~=~2\eta~ {k^2 {k'}^2 (k^2-{k'}^2)\over
(k+k')^2 (k-k')^2}\left( (k^2)^{D/2~-1} -
({k'}^2)^{D/2~-1}\right).
\label{cbox}
\end{eqnarray}
We now write
\begin{eqnarray}
{\cal K}_2 \otimes \phi_{\mu,n} &=& {\cal K}_2^1 \otimes \phi_{\mu,n} -
{\cal K}_2^2 \otimes \phi_{\mu,n} \nonumber \\
&=& \lambda_1(\mu,n) \phi_{\mu,n} -
\lambda_2(\mu,n) \phi_{\mu,n} \nonumber \\
&=& \lambda(\mu,n) \phi_{\mu,n},
\end{eqnarray}
where
\begin{equation}
{\cal K}_2^1 \otimes \phi_{\mu,n}
=~2\eta \int {d^D k' \over ({k'}^2)^2} {(k^2)^{D/2} {k'}^2 (k^2-{k'}^2)
\phi_{\mu,n}(k')\over (k-k')^2(k+k')^2 ,}
\label{e1}
\end{equation}
and
\begin{equation}
{\cal K}_2^2 \otimes \phi_{\mu,n}
=~2\eta \int {d^D k' \over ({k'}^2)^2} { k^2 ({k'}^2)^{D/2} (k^2-{k'}^2)
\phi_{\mu,n}(k')\over (k-k')^2(k+k')^2 .}
\label{e2}
\end{equation}
We take the eigenfunction $\phi_{\mu,n}$ to be defined on a D-dimensional
angular space parameterized by $(\theta_1,\theta_2,...,\theta_{D-1})$
by assuming that $\theta\equiv \theta_{D-1}$. If we define
$cos\chi= k\cdot \hat{x}$ and $cos\theta ={k'}\cdot \hat{x}$, where $\hat{x}$
is an arbitrarily chosen unit vector, the only non-trivial angular integral
is
\begin{eqnarray}
I_{\chi}[n] &=& \int_{0}^{2 \pi} d\theta{ e^{i n \theta}
\over 1- z(k,k')sin^2\,\,(\theta- \chi)}
{}~~~~~~~~~z[{k,k'}]=-{4 k^2{k'}^2\over (k^2 -{k'}^2)^2} \nonumber \\
&=& \,2 \pi e^{i n \chi}
\left({k^2-{k'}^2\over k^2 + {k'}^2}\right)
\left[ \left( {k'\over k}\right)^n\Theta[k-k']
- \left({k\over k'}\right)^n \Theta[k'-k] \right].
\label{Ith}
\end{eqnarray}
if $n$ is an even integer ($\geq 0$). $I_{\chi}[n]$ vanishes if $n$ is an
odd integer and $I_{\chi}[-n] = I_{\chi}[|n|]$.
$I_{\chi}[n]$ is symmetric under the exchange of $k$ and $k'$, and also is
invariant under $k \to 1/k, k' \to 1/k'$. This last invariance is sufficient
to show from (\ref{e1}) and (\ref{e2}) that
$$
\eqalign{ \lambda(\mu~,n) ~=~\lambda(1 -\mu~,n)}
\eqno(\refstepcounter{equation}\theequation)\label{sym}
$$
Using (\ref{Ith}) we obtain from (\ref{e1}) and (\ref{e2}) that, as $D \to
2$,
\begin{eqnarray}
\lambda_1(\mu,n)~\to~2\eta
{\pi^{D/2}\over \Gamma[D/2]} \biggl(\beta\bigl(|n|/2 +D/2 +\mu - 1\bigr)
{}~-~\beta\bigl(|n|/2-D/2-\mu + 2\bigr) \biggr),
\end{eqnarray}
and
\begin{eqnarray}
\lambda_2(\mu,n)~\to~2\eta
{\pi^{D/2}\over \Gamma[D/2]} \biggl(\beta\bigl(|n|/2 +D +\mu - 2\bigr)
{}~-~ \beta\bigl(|n|/2-D -\mu + 3\bigr) \biggr),
\end{eqnarray}
where $\beta(x)$ is the incomplete beta function, i.e.
$$
\eqalign{
\beta(x)~&=~\int^1_0 dy~y^{x -1}[1+y]^{-1} \cr
&=~{1\over 2}\biggl(\psi\bigl({x+1\over 2}\bigr) -
\psi\bigl({x\over 2}\bigr)\biggr), \,\,\,\,\,\,
\psi (x)={d\over d\,\,x}log \Gamma[x]~.}
\eqno(\refstepcounter{equation}\theequation)
$$
$\lambda_1(\mu,n)$ and $\lambda_2(\mu,n)$ are separately singular at $D=2$,
but $\lambda(\mu,n)$ is finite, and writing $\Lambda(\nu,n)~\equiv~
\lambda( {1 \over 2} + i\nu~,n)$, we obtain
$$
\eqalign{ \Lambda(\nu,n) ~=~-~2\pi \biggl(\beta'\bigl({|n| + 1\over 2} +
i\nu\bigr)
{}~+~\beta'\bigl({|n| + 1 \over 2} -i\nu\bigr)\biggr). }
\eqno(\refstepcounter{equation}\theequation)\label{lam}
$$
We comment first on the general properties of (\ref{lam}). The symmetry
property (\ref{sym}) is clearly reflected in the presence of the two terms.
The two terms also give directly the property of holomorphic
factorization\cite{lk} necessary for conformal symmetry. That is
$\Lambda(\nu,n)$ is a sum of two terms, one depending on $(i\nu + 1/2 +
n/2)$ and the other on $( i\nu + 1/2 -n/2)$. These two combinations
determine respectively the eigenvalues of the holomorphic and
anti-holomorphic Casimir operators of linear conformal transformations.
Since
$$
\eqalign{
\beta'(x)~=~{1\over 4}\biggl(\psi'\bigl({x+1\over 2}\bigr) -
\psi'\bigl({x\over 2}\bigr)\biggr) , }
\eqno(\refstepcounter{equation}\theequation)\label{ps1}
$$
and
$$
\eqalign { \psi'(x)~=~\sum_{n=0}^{\infty} {1
\over (n+x)^2}, }
\eqno(\refstepcounter{equation}\theequation)\label{ps2}
$$
$\beta'(x)$ is a real analytic function and it follows from (\ref{lam}) that
the eigenvalues $\Lambda(\nu,n)$ are all real.
Note that since the eigenvalues of $K^{(2)}_{2,2}$ can be written as a sum of
holomorphic and antiholomorphic components it is clear that the eigenvalues
of $(K^{(2)}_{2,2})^2$ can not be. Therefore this part of the $O(g^4)$
kernel is not conformally invariant. This is one of the arguments, referred
to earlier, that this term is inter-related with the scale dependendence of
the $O(g^2)$ kernel.
Moving on to the modification of $\alpha_0$, we note that to obtain the
contribution to the eigenvalue of $\tilde{K}^{(4)}_{2,2}$ we multiply
$\Lambda(\nu,n)$ by $-1/~2^4\pi^2$. To compare with $\alpha_0-1$ we
have to multiply, in addition, by $N^2g^4/~(2\pi)^3$, where $N=3$ for QCD.
As we discussed above, since ${\cal K}_2$ represents a new kinematic form at
$O(g^4)$ we do not expect it to mix with renormalization effects and so it
should be legitimate to compare its contribution with $\alpha_0 -1$ by
setting $\alpha_s=g^2/4\pi$. It follows from the above that the leading
eigenvalue is $\Lambda(0,0)$, as it is for the $O(g^2)$ kernel. From
(\ref{lam})-(\ref{ps2}) we obtain the contribution to $\alpha_0 -1$ from the
${\cal K}_2$ term in (\ref{full2}) as
$$
\eqalign{ ~-~{ 9\alpha_s^2 \over 2\pi^3} \Lambda(0,0) ~
&= ~{18\alpha_s^2 \over \pi^2} \beta'(1/2)
= ~-~{9\alpha_s^2 \over 2\pi^2}\biggl(~\sum_{n=0}^{\infty} {1
\over (n+~1/4)^2} ~- ~\sum_{n=0}^{\infty} {1 \over (n+~3/4)^2}\biggr)\cr
&=~-~{9\alpha_s^2 \over 2\pi^2}\biggl(~16~+~{16 \over 25}~+~{16 \over 81}
{}~+~...~-~{16 \over 9}~-~{16 \over 49}~+~...\biggr)\cr
{}~&\sim ~-~{9\alpha_s^2 \over 2 \pi^2} \times 14.5
{}~\sim ~-~65~{\alpha_s^2 \over \pi^2} ~\sim ~-~0.15 }
\eqno(\refstepcounter{equation}\theequation)
$$
The corresponding contribution from the ${\cal K}_1$ term in (\ref{full2})
would be
$$
\eqalign{ - {3 \over 4 } \Biggl({12 \over \pi } Log[2] ~\alpha_s \Biggr)^2
{}~\sim~-~0.18}
\eqno(\refstepcounter{equation}\theequation)
$$
However, since we have no understanding how the logarithms in this term mix
with the renormalization of $\alpha_s$, this could well be essentially
accounted for by the choice of scale in the $O(\alpha_s)$ term. Therefore we
believe no attention should be paid to this last number.
\vspace{1in}
\centerline{\bf Acknowledgements}
We thank J. Bartels, R. K. Ellis, V. Fadin, R. Kirschner, L. Lipatov and
M. Wuesthoff for informative discussions and comments.
Finally, we dedicate this work to Roberto Baggio for the inspiration
provided. A mistake may be the final element leading to perfection, that
the midsummer dream of one of us did not come true notwithstanding.
\newpage
|
2,877,628,088,994 | arxiv | \section{Introduction}
\label{sec:intro}
Direct imaging of exoplanets requires significant light amplification and very high angular resolution. The reason is that exoplanets are very small, extremely dim, and very distant targets for observation. When we consider traditional astronomical instruments---telescopes and interferometers---for this purpose, we face the sobering reality of requiring prohibitively large apertures, prohibitively long baselines, or a combination of both. For instance, to capture a single-pixel image of an Earth-like exoplanet from a distance of 100 light-years, a diffraction-limited telescope with an aperture of $\sim 90$~km would be required. Optical interferometers with moderate-size telescopes and large baselines would require signal integration times of hundreds of thousands if not millions of years to achieve a reasonable signal-to-noise ratio (SNR). Clearly, these scenarios are impractical. These challenges lead us to examine other ways that have the potential to produce high-resolution, multipixel images of such distant, small, dim targets. This is our primary motivation for the ongoing study \cite{Turyshev:2017,Turyshev-Toth:2017,Turyshev-Toth:2019} of the solar gravitational lens (SGL).
The large heliocentric distances separating us from the beginning of the SGL focal region were previously hard to contemplate. Multiple recent developments in deep space exploration technologies (for review, see \cite{KISS:2015,Alkalai-etal:2017}), along with the fact that the Voyager 1 spacecraft was able to reach heliocentric distances beyond 140 astronomical units (AU) while still transmitting valuable data, place such distances within reach. This allows us to consider practical applications of the SGL as an ``optical instrument'' that could be used for multipixel imaging and spatially resolved spectroscopy of an exoplanet \cite{Turyshev-etal:2018}. Motivated by such a unique opportunity, we recently studied the optical properties of the SGL in various conditions. Specifically, we developed a wave-optical theory of the SGL \cite{Turyshev:2017,Turyshev-Toth:2017} and discussed its key features, including the SGL's light amplification and resolution. In \cite{Turyshev-Toth:2019} we studied the impact of the plasma in the solar corona on light propagation in the vicinity of the Sun and the extended solar atmosphere. As a result, most of the analytical tools that are needed to model the imaging of exoplanets have became available. With these analyses concluded, we found no major obstacles for imaging and spectroscopy applications of the SGL.
The next step was the development of realistic imaging scenarios and relevant simulations. However, we realized that a treatment of extended sources imaged by the SGL was still missing. Most of the published wave-optical analyses of the SGL assumed that the light comes from a source positioned at an infinite distance from the Sun. In reality, an exoplanet is a small, but {\em extended} object positioned at a large, but {\em finite} distance from us. In this paper we develop a wave-optical theory of the SGL for such sources.
Our paper is organized as follows:
Section~\ref{sec:em-waves-gr+pl} introduces the SGL and presents Maxwell's field equations for the electromagnetic (EM) field on the background of the solar gravitational monopole.
Section \ref{sec:EM-field-fin-dist} introduces the problem of finding the EM field from the source at a finite distance and establishes the general principles to finding the needed solution.
Section \ref{sec:go-em-outside} discusses the EM field in the geometric optics and weak interference regions.
Section \ref{sec:IF-region} is devoted to determining the EM field in the strong interference region.
Section \ref{sec:image_form} addresses the process of image formation with the SGL for an extended source.
Appendix~\ref{sec:rad_eq_wkb} discusses an approximate solution for the radial function that relies on the Wentzel--Kramers--Brillouin (WKB) approximation. A solution for this function is derived for the case when a plane EM wave originates at a large, but finite distance from the Sun.
\section{General properties of the Solar Gravitational Lens}
\label{sec:em-waves-gr+pl}
We consider the propagation of monochromatic light originating at a source that is positioned at a large but finite distance from the Sun, and received by a detector in the focal region of the SGL. Our objective is to investigate the effect of the finite distance from the source on image formation by the SGL.
\subsection{EM waves in a static gravitational field}
\label{sec:maxwell}
We focus on solving Maxwell's equations on the background set by the solar gravitational field. Following \cite{Turyshev:2017,Turyshev-Toth:2017,Turyshev-Toth:2018-grav-shadow}, we begin with the generally covariant form of Maxwell's equations:
{}
\begin{eqnarray}
\partial_lF_{ik}+\partial_iF_{kl}+\partial_kF_{li}=0, \qquad
\frac{1}{\sqrt{-g}}\partial_k\Big(\sqrt{-g}F^{ik}\Big)=-\frac{4\pi}{c}j^i,
\label{eq:max-eqs}
\end{eqnarray}
where $g_{mn}$ is the metric tensor and $g=\det g_{mn}$ is its determinant.\footnote{The notational conventions used in this paper are the same as in \cite{Landau-Lifshitz:1988,Turyshev-Toth:2017}: Latin indices ($i,j,k,...$) are spacetime indices that run from 0 to 3. Greek indices $\alpha,\beta,...$ are spatial indices that run from 1 to 3. In case of repeated indices in products, the Einstein summation rule applies: e.g., $a_mb^m=\sum_{m=0}^3a_mb^m$. Bold letters denote spatial (three-dimensional) vectors: e.g., ${\vec a} = (a_1, a_2, a_3), {\vec b} = (b_1, b_2, b_3)$.} To describe the SGL in the first post-Newtonian approximation, we use a static harmonic metric with the line element in spherical coordinates $(r,\theta,\phi)$ (see Fig.~\ref{fig:go}), given as:
\begin{eqnarray}
ds^2&=&u^{-2}c^2dt^2-u^2\big(dr^2+r^2(d\theta^2+\sin^2\theta d\phi^2)\big),
\label{eq:metric-gen}
\end{eqnarray}
where, to the accuracy sufficient to describe light propagation in the solar system, the quantity $u$ has the form $u=1+c^{-2}U+{\cal O}(c^{-4}),$ with $U$ being the Newtonian gravitational potential. As in \cite{Turyshev-Toth:2017}, we focus our discussion on the largest contribution to the gravitational scattering of light, which, in the case of the Sun, is due to the gravity field produced by a static monopole. In this case, the Newtonian potential may be given by $c^{-2}U({\vec r})={r_g}/{2r}+{\cal O}(r^{-3},c^{-4}),$ where $r_g=2GM_\odot/c^2$ is the Schwarzschild radius of the Sun. Therefore, the quantity $u$ in (\ref{eq:metric-gen}) has the form
{}
\begin{eqnarray}
u(r)&=&1+\frac{r_g}{2r}+{\cal O}(r^{-3},c^{-4}).
\label{eq:pot_w_1**}
\end{eqnarray}
In the case of a static, spherically symmetric gravitational field (\ref{eq:metric-gen})--(\ref{eq:pot_w_1**}), in the absence of sources or currents, $j^k\equiv (\rho,{\vec j})=0$, solving the field equations (\ref{eq:max-eqs}) is straightforward. We align the polar $z$-axis of the coordinate system along the wavevector $\vec k$ of the incident wave. The resulting complete solution of Maxwell's equations, following \cite{Born-Wolf:1999,Herlt-Stephani:1976}, was developed in \cite{Turyshev-Toth:2017,Turyshev-Toth:2019} with the components of the EM field ${\vec D}=u{\vec E}$ and ${\vec { B}}=u{\vec { H}}$:
{}
\begin{align}
\left( \begin{aligned}
{ D}_r& \\
{ B}_r& \\
\end{aligned} \right) =& \left( \begin{aligned}
\cos\phi \\
\sin\phi \\
\end{aligned} \right) \,e^{-i\omega t}\alpha(r, \theta), &
\left( \begin{aligned}
{ D}_\theta& \\
{ B}_\theta& \\
\end{aligned} \right) =& \left( \begin{aligned}
\cos\phi \\
\sin\phi \\
\end{aligned} \right) \,e^{-i\omega t}\beta(r, \theta), &
\left( \begin{aligned}
{ D}_\phi& \\
{ B}_\phi& \\
\end{aligned} \right) =& \left( \begin{aligned}
-\sin\phi \\
\cos\phi \\
\end{aligned} \right) \,e^{-i\omega t}\gamma(r, \theta),
\label{eq:DB-sol00p*}
\end{align}
with the quantities $\alpha, \beta$ and $\gamma$ computed from the following expressions:
{}
\begin{eqnarray}
\alpha(r, \theta)&=&
\frac{1}{u}\Big\{\frac{\partial^2 }{\partial r^2}
\Big[\frac{r\,{\hskip -1pt}\Pi}{u}\Big]+k^2 u^4\Big[\frac{r\,{\hskip -1pt}\Pi}{u}\Big]\Big\}+{\cal O}\Big(\big(\frac{1}{u}\big)''\Big),
\label{eq:alpha*}\\
\beta(r, \theta)&=&\frac{1}{u^2r}
\frac{\partial^2 \big(r\,{\hskip -1pt}\Pi\big)}{\partial r\partial \theta}+\frac{ik\big(r\,{\hskip -1pt}\Pi\big)}{r\sin\theta},
\label{eq:beta*}\\[0pt]
\gamma(r, \theta)&=&\frac{1}{u^2r\sin\theta}
\frac{\partial \big(r\,{\hskip -1pt}\Pi\big)}{\partial r}+\frac{ik}{r}
\frac{\partial\big(r\,{\hskip -1pt}\Pi\big)}{\partial \theta},
\label{eq:gamma*}
\end{eqnarray}
with $k=\omega/c$ being the wavenumber of the monochromatic EM wave and $\Pi (r, \theta)$ is the Debye potential given as \cite{Turyshev:2017,Turyshev-Toth:2017}
{}
\begin{eqnarray}
\Pi(r, \theta)&=& \frac{E_0}{2ik^2}\frac{u}{r}\sum_{\ell=kR^\star_\odot}^\infty i^{\ell-1}\frac{2\ell+1}{\ell(\ell+1)}e^{i\sigma_\ell}
H^+_\ell(kr_g, kr)P^{(1)}_\ell(\cos\theta)-\nonumber\\
&&\hskip 10 pt -\,
\frac{E_0}{2ik^2}\frac{u}{r}\sum_{\ell=1}^\infty i^{\ell-1}\frac{2\ell+1}{\ell(\ell+1)}e^{i\sigma_\ell}
H^-_\ell(kr_g, kr)P^{(1)}_\ell(\cos\theta) +{\cal O}(r_g^2),
\label{eq:Pi-s_a+0}
\end{eqnarray}
where $H^{+}_\ell$ and $H^{-}_\ell$ are the Coulomb--Hankel functions \cite{Abramovitz-Stegun:1965} representing outgoing and incident waves, correspondingly, $\sigma_\ell$ is known as the Coulomb phase shift (see \cite{Turyshev-Toth:2017} for details), and $P^{(1)}_\ell(\cos\theta) $ are the associated Legendre polynomials.
To derive the solution for the Debye potential $\Pi(r, \theta)$ given by (\ref{eq:Pi-s_a+0}), we used the fully absorbing boundary conditions that account for the physical size of the Sun (see details in \cite{Turyshev-Toth:2017,Turyshev-Toth:2018-grav-shadow,Turyshev-Toth:2019}). Specifically, we required that rays with impact parameters $b\le R_\odot^\star=R_\odot +r_g$ are completely absorbed by the Sun \cite{Turyshev-Toth:2017} and no reflection or coherent reemission occurs. Technically, such formulation relies on the semiclassical relationship between the partial momentum, $\ell$, and the impact parameter, $b$, that is given as as $\ell=k b$ (see relevant discussion in \cite{Turyshev-Toth:2019}.) Therefore, we require that no outgoing waves (i.e., $\propto H^{+}_\ell$) exist in the region behind the Sun for rays of light with impact parameter $b\leq R_\odot^\star$ or, equivalently, for $\ell \leq kR_\odot^\star$. This results in the Debye potential given by (\ref{eq:Pi-s_a+0}) that is valid for all distances outside the Sun $r>R^\star_\odot$ and all angles.
The expression (\ref{eq:Pi-s_a+0}) for the Debye potential is rather complex. It requires the tools of numerical analysis to fully explore its behavior and the resulting EM field \cite{Kerker-book:1969,vandeHulst-book-1981,Grandy-book-2005}. However, in most practical applications, we only need to know the field in the forward direction. Furthermore, our main interest is to study the largest effect of the solar gravitational field on light propagation, which corresponds to the smallest values of the impact parameter. In this situation, we may simplify the result (\ref{eq:Pi-s_a+0}) by taking into account the asymptotic behavior of the functions $H^{\pm}_\ell(kr_g,kr)$, while considering the resulting EM field at large heliocentric distances, such that $kr\gg\ell$, where $\ell$ is the order of the Coulomb function (see p.~631 of \cite{Morse-Feshbach:1953}). Given the fact that $r_g\ll R^*_\odot$, this approach may be used to describe the EM field immediately outside the solar photosphere all the way to the focal region.
\subsection{Optical properties of the SGL}
\label{sec:general}
The solution for the EM field given by Eqs.~(\ref{eq:DB-sol00p*})--(\ref{eq:Pi-s_a+0}) was analyzed extensively in \cite{Turyshev:2017,Turyshev-Toth:2017,Turyshev-Toth:2019}. The optical properties of the SGL are now well known \cite{Turyshev-Toth:2018,Turyshev-Toth:2018-plasma,Turyshev-Toth:2018-grav-shadow}. Below, we summarize the most important of these.
As was shown in \cite{Turyshev:2017,Turyshev-Toth:2017}, the SGL exists due to the effect of gravitation on the refractive properties of spacetime, focusing light. According to Einstein's general theory of relativity, the trajectory of a photon that travels near the Sun is deflected towards the Sun (its largest effect on light propagation described by the solar gravitational monopole) by the angle of $\theta_{\tt gr}=2r_g/b$, where $r_g=2GM_\odot/c^2$ is the Schwarzschild radius of the Sun and $b$ is the photon's solar impact parameter; see Fig.~\ref{fig:go}. Because solar gravity is weak, the actual deflection angle is very small, so that parallel rays of light passing by the Sun near the solar surface, $b=R_\odot$, focus at the large heliocentric distance of $R_\odot^2/2r_g=547.6~(b/R_\odot)^2$ AU. The SGL does not have a single focal point. Rays with larger impact parameters focus at greater distances from the Sun; thus, a focal half-line forms, as shown in Fig.~\ref{fig:regions}.
\begin{figure}
\includegraphics[scale=0.30]{p-source-inf-go}
\caption{\label{fig:go}Focusing of light by the SGL. The heliocentric coordinate system is such that the $z$-axis is along the incoming direction of the wave propagation, given by the wavevector $\vec k$. Spherical ($r,\theta$) and cylindrical ($\rho,z$) coordinates, used in the text, are shown. The azimuthal angle $\phi$ is suppressed. The trajectory of a light ray with impact parameter $b$ with respect to the Sun is deflected towards the Sun by the angle $\theta_{\tt gr}=2r_g/b$, causing it to intersect the $z$-axis at the heliocentric distance $z=b^2/2r_g$.
}
\end{figure}
\begin{figure}
\includegraphics[scale=0.27]{regions}
\caption{\label{fig:regions}The different optical regions of the SGL with respect to light from a source at infinity. Rays with a larger impact parameter intersect at a greater distance, forming a focal half-line (shown by the dashed line). The shaded area is the strong interference region in the immediate proximity of the optical axis as discussed in the text.}
\end{figure}
\begin{figure}
\includegraphics[scale=0.40]{p-source-inf-wo}
\caption{\label{fig:geom-wo}Wave-optical treatment of the diffraction of light on the solar gravitation field. In the strong interference region, rays with different impact parameters would have different optical path lengths which leads to formation of the interference pattern (shown conceptually, not to scale) on the image plane. }
\end{figure}
Based on the analysis presented in \cite{Turyshev-Toth:2017, Turyshev-Toth:2019}, we know that diffraction of light on the solar gravitational monopole together with the effect of light's interaction with the Sun results in the formation of four regions behind the Sun (Fig.~\ref{fig:regions}) with characteristically different EM field behavior, namely:
\begin{enumerate}[i)]
\item Rays with impact parameters $b\leq R^*_\odot$ are completely absorbed by the Sun, resulting in the shadow region directly behind the Sun. Because of the gravitational bending of light, the shadow has a hyperboloidal shape with its vertex at the heliocentric distance of $R_\odot^2/2r_g=547.6$~AU and the rim touching the Sun. As the edge of the solar disk is not optically smooth, no Arago spot forms along the shadow centerline. Thus, no discernible light from the distant source reaches this region \cite{Turyshev-Toth:2018-grav-shadow}.
\item Most of the area outside the shadow region is characterized by distances $r\geq R^*_\odot$ and the angles $\theta\gg \sqrt{2r_g/z}$. As the Sun blocks rays with impact parameter $b\leq R^*_\odot$, only one light ray passes through any given point in this region. This is the region of geometric optics. At any given point in this region, the EM field is represented by the incident wave \cite{Turyshev-Toth:2017}. The phase of the incident ray is well described by the geometric optics approximation, thus giving the name for this region. The structure of the EM field here was discussed in \cite{Turyshev-Toth:2017}.
\item As the light rays propagate towards the focal line, their deflected trajectories define a plane. The path of propagation ultimately begins to intersect trajectories in the same plane, passing by the opposite side of the solar monopole with impact parameter $r\geq R^*_\odot$. The smaller the angle $\theta$ at the intersection point, the smaller the optical path difference (OPD) between the rays. As $|\theta|\gg \sqrt{2r_g/z}$, from (\ref{eq:mu-point}) we see that the path difference is quite large, namely ${\rm OPD}=\sqrt{{2r_g z}}\,|\theta|\gg 2r_g\gg \lambda$. Although a weak interference pattern forms, the geometric optics approximation remains applicable here. This is what we call the region of the weak interference. At any given point in this region, the EM field can be represented by the incident and scattered waves (see \cite{Turyshev-Toth:2017} for details). An observer in this region would see two images of uneven brightness representing the same point source, situated on opposite sides of the Sun. As the point of intersection gets closer to the optical axis, the difference in brightness decreases.
\item For impact parameters $b> R^*_\odot$ and thus the distances beyond 547.6~AU, in the immediate vicinity of the optical axis, $0\leq |\theta| \simeq \sqrt{2r_g/z}$, we enter the region of strong interference, i.e., ``the focal beam of extreme intensity'' \cite{Herlt-Stephani:1976}. Because of azimuthal symmetry, rays with different azimuthal angles (defining different planes of propagation) intersect at or near the focal line. These intersecting rays have the smallest possible ${\rm OPD} \simeq \lambda$, thus creating a strong interference pattern (Fig.~\ref{fig:regions}). The components of the Poynting vector and the EM field intensity are oscillating with a spatial period of $\rho=\sqrt{z/2r_g}\lambda$. If positioned exactly on the optical axis (i.e., $\theta={\rm OPD}=0$), the observer would see a complete Einstein ring with the brightness of the source being greatly amplified. If positioned slightly away from the optical axis (i.e., $0<|\theta|\simeq \sqrt{2r_g/z}$ and, thus, $\lambda<{\rm OPD}\lesssim r_g$), two incomplete arcs with slightly different angular sizes and intensities (albeit both strongly amplified) would appear. This is the region where the SGL acquires its most impressive optical properties. To describe the image formation processes here, one needs a wave-optical treatment that was developed in \cite{Turyshev:2017,Turyshev-Toth:2017,Turyshev-Toth:2019}.
\end{enumerate}
The strong interference region is of the greatest practical importance. This is the region where the SGL offers major light amplification and high angular resolution, which are both needed for imaging of exoplanets. Using a wave-optical treatment of light diffraction on the solar gravitational monopole, we established \cite{Turyshev-Toth:2017} that for a point source at infinity and for an observer at the heliocentric distance $z$, the SGL's light amplification factor is given by
{}
\begin{eqnarray}
{\bar \mu}_z&=&
\mu_0 J^2_0\Big(\frac{2\pi}{\lambda}\sqrt{{2r_g z}}\,\theta
\Big),\qquad {\rm with}
\qquad
\mu_0=\frac{4\pi^2}{1-e^{-4\pi^2 r_g/\lambda}}
\frac{r_g}{\lambda}.
\label{eq:mu-point}
\end{eqnarray}
For any given $z$, light amplification reaches its maximum value when $\theta=\rho/z=0$. Away from the optical axis, destructive interference becomes significant as light rays with various optical path lengths converge on the same point. An observer positioned on the optical axis would register a major increase in brightness, characterized by the factor $\sim 4\pi^2 r_g/\lambda$ in (\ref{eq:mu-point}), which, for the wavelength of $\lambda=1~\mu$m, is $\sim 10^{11}$. Elsewhere in the image plane, rays with a non-vanishing difference of their optical paths form the interference pattern, as conceptually shown in Fig.~\ref{fig:geom-wo}. The average amplitude of the intensity of the pattern falls off approximately with the inverse of the distance from the optical axis, characterized by the radial coordinate $\rho\simeq \theta z$.
This level of light amplification is one of the major benefits of the SGL. The SGL's other main advantage is its angular resolution of $\sim$0.5 nanoarcseconds that is determined at the first null of the Bessel function in (\ref{eq:mu-point}) which occurs quite close to the optical axis:
$\rho_{\tt SGL0}\simeq 4.5~({\lambda}/{1~\mu{\rm m}})({z}/547.6~{\rm AU})^\frac{1}{2}~{\rm cm}$
\cite{Turyshev-Toth:2017}. These impressive values have motivated us to consider using the SGL for imaging distant, small and faint sources.
As we began the development of the appropriate simulation tools, we realized that the description above is based on studying the EM field received from a point source that is positioned at the infinite distance from the Sun. However, this assumption is not valid when we attempt to describe the imaging of extended and resolved sources analytically. It became necessary to develop analytical tools exactly for this purpose. This is discussed next.
\section{EM field from the source at a finite distance}
\label{sec:EM-field-fin-dist}
Our previous work on the wave-optical theory of the SGL relied on developments in atomic physics, dealing with nuclear scattering on a Coulomb potential. The relevant theoretical efforts from the twentieth century provided us with a rich set of mathematical tools and methods \cite{Messiah:1968,Schiff:1968,Morse-Feshbach:1953,Sharma-book:2006,Grandy-book-2005,Friedrich-book-2013} that can be used to study the scattering of light in the solar gravity field. These methods are directly applicable to problems in atomic physics, where the focus is on the asymptotic behavior of a scattered field.
In case of a source at a finite distance, however, these tools require additional development. We can still use the geometric optics approximation to describe light propagation in the region of geometric optics. The solution to the appropriate geodetic equation is well known and describes the trajectory and phase of a light ray along the path from the source at a finite distance to the observer position (see, for instance, discussion in Appendices B.1 and B.2 of \cite{Turyshev-Toth:2017}). However, to describe the optical processes in the weak and, especially, the strong interference regions, this approach is inadequate. We need to solve the time-independent Schr\"odinger equation with a Coulomb potential, as was done in \cite{Turyshev-Toth:2017}.
Technically, the solution must be sought using a form of the incident spherical wave for the source at a finite distance \cite{Born-Wolf:1999} and would involve a set of appropriately modified Coulomb functions.
No exact solution to this problem is known. Instead, we are forced to develop an approximation.
\subsection{The geometry of the problem}
\label{sec:opt-axis}
We consider a point source located at a distance $r_0$ from the Sun (Fig.~\ref{fig:point-3D}). The line connecting the point source and the center of the Sun defines the optical axis of the SGL, $\overline z$. Clearly, there are an infinite number of rays that are emitted by the point source in $4\pi$ steradians; many of these rays travel towards the Sun. Depending on the impact parameter, some these rays will either be absorbed by the Sun or else will travel beyond the Sun, eventually entering the geometric optics and the interference regions. In the spherically symmetric gravitational field (\ref{eq:metric-gen})--(\ref{eq:pot_w_1**}), the optical axis is the axis of axial symmetry. I.e., the geometry of the problem is invariant under a rotation around the $\overline z$-axis.
\begin{figure}
\includegraphics[scale=0.38]{point-3D}
\caption{\label{fig:point-3D}The three-dimensional geometry of the SGL, focusing light from a point source located at a finite distance. Two rays of light with wavevectors $\vec k_1$ and $\vec k_2$ are shown. The rays move in different planes, which intersect along the optical axis. Note that the $z$-axis is no longer uniquely defined. However, the optical axis $\overline z$ is unique and preserves the axial symmetry.}
\end{figure}
We introduce a heliocentric spherical coordinate system with the polar axis directed along the $\overline z$-axis. The coordinates of the point source in this coordinate system are $(r_0,0, 0)$. Next, we take a ray of light emitted by the point source in the direction towards the Sun with the solar impact parameter $b\geq R^*_\odot$. This wavevector $\vec k$ of this ray and the optical $\overline z$-axis define a plane. We set up a $z$-axis in this plane, parallel with $\vec{k}$. This axis corresponds to the $z$-axis that was used to describe the problem with the source being at infinity (shown in Fig.~\ref{fig:go} and discussed in Sec.~\ref{sec:em-waves-gr+pl}). In the spherical coordinate system corresponding to this $z$-axis, the coordinates of the point source are given by $(r_0,b/r_0, \phi_0)+{\cal O}(b^2/r_0^2)$.
At this point, we face a technical challenge. On the one hand, we have the solution (\ref{eq:DB-sol00p*})--(\ref{eq:Pi-s_a+0}), which was obtained under the assumption that the source is at infinity. This solution was obtained in the coordinate system corresponding to the $z$-axis that we just defined. However, to properly describe the problem, we need to find a solution for the ray propagating with the incident wavevector $\vec k$ given in a coordinate system corresponding to the optical $\overline z$-axis associated with a point source at a large, but finite distance. This problem requires one either solving the Maxwell equations in the $\overline z$-coordinate system with an appropriate form of the incident logarithm-modified plane wave (see \cite{Turyshev-Toth:2017} for details) or to transform the solution (\ref{eq:DB-sol00p*})--(\ref{eq:Pi-s_a+0}) from $z$-coordinates to $\overline z$-coordinates. No such solution is currently known.
To develop the needed solution for the EM field, we first note that because of the separation of variables that was used to solve the Maxwell equations (see \cite{Turyshev-Toth:2017} for details), the solution (\ref{eq:DB-sol00p*})--(\ref{eq:Pi-s_a+0}) isolates the dependence on the azimuthal angle $\phi$ from the dependence on the other two variables, $r$ and $\theta$, present in $\alpha(r,\theta)$, $\beta(r,\theta)$ and $\gamma(r,\theta)$, given by (\ref{eq:alpha*})--(\ref{eq:gamma*}). Thus, for any given angle $\phi=\phi_0$, the needed solution is already available in the form of (\ref{eq:alpha*})--(\ref{eq:Pi-s_a+0}), but the solution then needs to be rotated by the angle
\begin{equation}
\beta = \frac{b}{r_0}+{\cal O}(b^2/r_0^2),
\label{eq:beta}
\end{equation}
in the plane defined by $\phi=\phi_0$ and containing the optical axis. This rotation transforms the polar angle as
\begin{equation}
{\overline\theta}=\theta+\beta,
\label{eq:otheta}
\end{equation}
but leaves $r\to r+{\cal O}(b^2/r_0^2)$. The azimuthal angle, however, changes according to
\begin{equation}
\tan\overline\phi=\tan\phi+\frac{\sin(\phi-\phi_0)}{\tan\theta\cos^2\phi}\sin\beta+{\cal O}(b^2/r_0^2).
\label{eq:ophi}
\end{equation}
In other words, there is an additional rotation in the azimuthal plane, $\phi\to\overline\phi$. As it turns out, the actual magnitude of this rotation does not need to be computed, for the following reasons:
\begin{enumerate}[1)]
\item In regions other than the region of strong interference, at any point in space, light arrives in the form of at most two rays, both of which travel in the same plane. The intensity of light, therefore, does not depend on $\phi$.
\item In the strong interference region, we are near the optical axis, axial symmetry is restored, and dependence on the transformed azimuthal angle $\overline\phi$ vanishes, which is also apparent from the obvious degeneracy of Eq.~(\ref{eq:ophi}) when $\theta\to 0$. Therefore, although in this region, multiple rays of light traveling in different azimuthal planes are combined, the result remains independent of either $\phi$ or $\overline\phi$.
\end{enumerate}
These considerations allow us to greatly simplify the problem by considering light propagation in a plane only.
Another important concern regarding a point source at a finite distance is that light from that source no longer arrives in the form of a plane wave. However, when the source is at a great distance and we are studying a narrow beam of light, we may use the paraxial approximation, which allows us to continue using the formalism developed for incident plane waves instead of reformulating the problem with spherical waves. Using this approximation, we must rescale the field intensity of the plane wave, $E_0$, to account for its distance dependence from the source, namely $E_0\rightarrow E^{\tt s}_0/r_0$, where $E^{\tt s}_0$ is the field intensity of the corresponding spherical wave.
\subsection{Solving Maxwell's equations}
Considering the solution for the Debye potential given by Eq.~(\ref{eq:Pi-s_a+0}), we observe that it depends on the associated Legendre polynomials, $P^{(1)}_\ell(\cos\theta)$ and the Coulomb--Hankel functions, $H^{\pm}_\ell(kr_g,kr)$. Both of these quantities are given with respect to the $z$-axis, which, as discussed above, is not the most optimal axis to use in the case the source is at a finite distance. Not only is the $z$-axis not uniquely defined but it breaks axial symmetry.
Given the coordinate transformation (\ref{eq:otheta}), we see that the Legendre polynomials written with respect to the $\overline z$-axis may be given as $P_\ell(\cos\overline \theta)=P_\ell(\cos (\theta+\beta))=P_\ell(\cos \theta)+{\cal O}(\theta b/r_0)$, which is sufficient for our purposes.
\begin{figure}
\includegraphics[scale=0.40]{p-source-fin-int}
\caption{\label{fig:geom-opt}The SGL focusing light from a point source located at a finite distance. Two rays of light with the same impact parameter $b$, traveling in the same plane but on opposite sides of the Sun,
are shown. The incident rays are no longer parallel. The diagram is arranged such that the top incident ray appears horizontal, as in Fig.~\ref{fig:go}. The wavevector $\vec k$ is inclined with respect to the optical axis $\overline z$ by the angle $\beta=b/z_0$. Both rays intersect $\overline z$ at the distance $\overline z=(b^2/2r_g)(1+b^2/2r_g z_0)$.
}
\end{figure}
Next, we turn our attention to the solution for the radial function in (\ref{eq:Pi-s_a+0}) which is given by the Coulomb--Hankel functions $H^{\pm}_\ell(kr_g,kr)$. The asymptotic form of these functions has to account for the fact that the light source is at a finite distance, at spherical coordinates $(r_0, b/r_0, \phi_0)$. We follow the path of propagation of a light ray in the plane $\phi=\phi_0$. The corresponding asymptotic behavior of $H^{\pm}_\ell(kr_g,kr)$ was established in Appendix \ref{sec:rad_eq_wkb} for $kr\rightarrow\infty $ and $r\gg r_{\tt t}=\sqrt{\ell(\ell+1)}/k$ (see \cite{Turyshev-Toth:2017,Turyshev-Toth:2018-plasma}) in the form of (\ref{eq:R_solWKB+=_bar-imp}) and is given by
{}
\begin{eqnarray}
\lim_{kr\rightarrow\infty} H^{\pm}_\ell(kr_g,kr)&\sim&
\exp\Big[\pm ik\big(r+r_0+r_g\ln(4k^2rr_0)\big)+\frac{\ell(\ell+1)}{2k}\Big(\frac{1}{r}+\frac{1}{r_0}\Big)+\sigma_\ell-\frac{\pi \ell}{2}\Big)\Big] +{\cal O}\big((kr)^{-2}, r_g^2\big),
\label{eq:Fass*}
\end{eqnarray}
which includes the contribution from the centrifugal term, $\propto \ell(\ell+1)/2kr$, in the radial equation for the Debye potential (see Appendix \ref{sec:rad_eq_wkb} here or Appendix A in \cite{Turyshev-Toth:2018}). Including the centrifugal term lets us better describe the bending of the light ray's trajectory under the influence of solar gravity. In addition, Eq.~(\ref{eq:Fass*}) contains terms that describe the dependence of the total phase of the EM wave on the distance to the source, $r_0$.
Furthermore, as the optical axis preserves the axial symmetry, all the rays emitted towards the Sun from a point on that axis with coordinates $(r_0,0, \phi_0)$ and impact parameter $b>R^*_\odot$, will intersect the optical axis behind the Sun at one point, at the heliocentric distance of $\overline z=b^2/2r_g(1+b^2/2r_g r_0)$. As a result of this axial symmetry, the solution does not depend on the specific choice of $\phi_0$.
The presence of the $\propto \ell^2$ term in (\ref{eq:Fass*}) (and also in (\ref{eq:R_solWKB+=_bar-imp})) is important is it allows for a better description of the light ray's trajectory. Taking the semiclassical representation of the partial momenta via the impact parameter as $\ell=kb$, for $\ell\gg1$ we may present the Euclidean part of the phase of (\ref{eq:Fass*}) as
\begin{equation}
k(r+r_0)+\frac{\ell(\ell+1)}{2k}\Big(\frac{1}{r}+\frac{1}{r_0}\Big)=
k\big(r+\frac{b^2}{2r}\big)+k\big(r_0+\frac{b^2}{2r_0}\big)
= k\sqrt{r^2+b^2}+k\sqrt{r^2_0+b^2}+{\cal O}(b^4/r^4,b^4/r_0^4),
\end{equation}
which now correctly describes the Euclidean distance that the light travels from its emission at $r_0$ to the point of its detection, $\overline z$. The remaining part of the phase in (\ref{eq:Fass*}) that is given as $kr_g\ln(4k^2rr_0)$ is due to the lengthening of the light ray's path in the curved spacetime that is induced by the solar gravitational field.
Finally, as in the realistic exoplanet imaging situations, the ratio $\beta=b/r_0$ is small, any effect on the amplitude of the EM field from the relevant $\beta$-rotation of the EM field is negligible. As we shall see in Sec.~\ref{sec:go-em-outside}, most of the contribution from $\beta$ affects the phase of the EM wave, which is our primary interest.
\subsection{EM field in the shadow region}
\label{sec:shadow}
In the shadow behind the Sun (i.e., for impact parameters $b\leq R^\star_\odot$, see Fig.~\ref{fig:regions}), the EM field is represented by the Debye potential of the shadow, $\Pi_{\tt sh}$, which, from (\ref{eq:Pi-s_a+0}), is determined solely by the incoming wave, as prescribed by the fully absorbing boundary conditions:
{}
\begin{eqnarray}
\Pi_{\tt sh} (r, \theta)&=&-
\frac{E_0}{2ik^2}\frac{u}{r}\sum_{\ell=1}^\infty i^{\ell-1}\frac{2\ell+1}{\ell(\ell+1)}e^{i\sigma_\ell}H^-_\ell(kr_g, kr)P^{(1)}_\ell(\cos\theta).
\label{eq:Pi_sh}
\end{eqnarray}
As discussed in \cite{Turyshev-Toth:2017,Turyshev-Toth:2018-grav-shadow}, the potential (\ref{eq:Pi_sh}), to the required level of accuracy, produces no EM field in the area $-\pi/2\le \theta\le\pi/2$. In other words, there is no light in the shadow. Additionally, one can show that even outside the shadow region behind the Sun, the potential (\ref{eq:Pi_sh}) results in a very small EM field, negligible for our analysis \cite{Turyshev-Toth:2018,Turyshev-Toth:2018-grav-shadow}. As a result, we omit this term from the Debye potential (\ref{eq:Pi-s_a+0}) in our analysis when we discuss the EM field in the geometric optics and interference regions. Therefore, we focus on the EM field produced only by the outgoing part of the Debye potential (\ref{eq:Pi-s_a+0}), namely $\propto H^+_\ell(kr_g, kr)$.
\subsection{EM field outside the shadow}
\label{sec:EM-field}
In the region outside the solar shadow (i.e., for light rays with impact parameters $b>R_\odot^\star$), which includes the geometric optics and both interference regions (see Fig.~\ref{fig:regions}), the EM field is derived from the Debye potential (\ref{eq:Pi-s_a+0}). For this, we substitute (\ref{eq:Fass*}) in (\ref{eq:Pi-s_a+0}) and derive the Debye potential without the incident wave (discussed in Sec.~\ref{sec:shadow}):
{}
\begin{eqnarray}
\Pi(r, \theta)&=& -\frac{E_0}{k^2}\frac{u}{r}e^{ik(r+r_0+r_g\ln 4k^2rr_0)}\sum_{\ell=kR^\star_\odot}^\infty \ell^{-1}
e^{i\big(2\sigma_\ell+\frac{\ell^2}{2k}(\frac{1}{r}+\frac{1}{r_0})\big)}
P^{(1)}_\ell(\cos\theta) +{\cal O}(r_g^2, (kr)^{-3}),
\label{eq:Pi-0+}
\end{eqnarray}
where we recognize that for large $\ell\geq kR^\star_\odot$, we may replace $\ell+1\rightarrow \ell$ and $\ell+\textstyle\frac{1}{2}\rightarrow \ell$. Expression (\ref{eq:Pi-0+}) is the Debye potential that yields the EM field in the regions of geometric optics and the interference region.
Introducing, for convenience, the effective distance $\tilde r$ in the form
\begin{equation}
\frac{1}{\tilde r}=\frac{1}{r}+\frac{1}{r_0} \qquad \Rightarrow \qquad
{\tilde r}=\frac{rr_0}{r+r_0},
\label{eq:r-tilde}
\end{equation}
we derive the components of the EM field. For this, we use (\ref{eq:Pi-0+}) and (\ref{eq:r-tilde}) in the expressions (\ref{eq:alpha*})--(\ref{eq:gamma*}) to derive the factors $\alpha(r,\theta), \beta(r,\theta)$ and $\gamma(\theta)$, which, to the order of ${\cal O}\big(r^2_g,(kr)^{-3}\big)$, are computed as
{}
\begin{eqnarray}
\alpha(r,\theta) &=& -E_0\frac{ue^{ik(r+r_0+r_g\ln 4k^2rr_0)}}{k^2r^2}\sum_{\ell=kR^\star_\odot}^\infty \ell e^{i\big(2\sigma_\ell+\frac{\ell^2}{2k\tilde r}\big)}
\Big\{1-\frac{\ell^2}{4u^2k^2r^2}
\Big\}P^{(1)}_\ell(\cos\theta),
\label{eq:alpha*1*}\\
\beta(r,\theta) &=& E_0\frac{ue^{ik(r+r_0+r_g\ln 4k^2rr_0)}}{ikr}\sum_{\ell=kR^\star_\odot}^\infty\ell^{-1}e^{i\big(2\sigma_\ell+\frac{\ell^2}{2k\tilde r}\big)}
\Big\{\frac{\partial P^{(1)}_\ell(\cos\theta)}
{\partial \theta}\Big(1-\frac{\ell^2}{2u^2k^2r^2}
\Big)+\frac{P^{(1)}_\ell(\cos\theta)}{\sin\theta}
\Big\},
\label{eq:beta*1*}\\
\gamma(r,\theta) &=& E_0\frac{ue^{ik(r+r_0+r_g\ln 4k^2rr_0)}}{ikr}\sum_{\ell=kR^\star_\odot}^\infty
\ell^{-1}e^{i\big(2\sigma_\ell+\frac{\ell^2}{2k\tilde r}\big)}
\Big\{\frac{\partial P^{(1)}_\ell(\cos\theta)}
{\partial \theta}+\frac{P^{(1)}_\ell(\cos\theta)}{\sin\theta}\Big(1-
\frac{\ell^2}{2u^2k^2r^2}\Big) \Big\}.~~~~
\label{eq:gamma*1*}
\end{eqnarray}
Here we neglected small terms that behave as $\propto i/(u^2kr)$; terms $\propto ikr_g/\ell^2$ were also omitted because of the large partial momenta involved, $\ell\geq kR^\star_\odot$. Terms in both of these groups are negligably small when compared to the leading terms in each of these expressions above (a similar conclusion was reached in \cite{Turyshev-Toth:2018-plasma,Turyshev-Toth:2019}.)
Expressions (\ref{eq:alpha*1*})--(\ref{eq:gamma*1*}) represent an important result, allowing us to describe the EM field in the regions of interest for the SGL, namely the geometric optics region and the interference region.
\section{EM field in the geometric optics and weak interference regions}
\label{sec:go-em-outside}
We are interested in the area that can be reached by light rays with impact parameters $b>R^\star_\odot$ and located behind the Sun at heliocentric distances $r> R^\star_\odot$. This is the region of geometric optics for which
angles $\theta$ are rather large, satisfying the condition $|\theta|\gg \sqrt{2r_g/r}$ \cite{Turyshev-Toth:2017}. To establish the EM field (\ref{eq:DB-sol00p*}) in this region, we need to develop expressions (\ref{eq:alpha*1*})--(\ref{eq:gamma*1*}) evaluating them to the appropriate level of accuracy. We begin with the investigation of $\alpha(r,\theta)$ from (\ref{eq:alpha*1*}).
\subsection{Solution for the function $\alpha(r,\theta)$ and the radial components of the EM field}
\label{sec:radial-comp}
To evaluate the expression for $\alpha(r,\theta)$ in the region of geometric optics and, thus, for $\theta \gg \sqrt{2r_g/r}$, we use the asymptotic representation for $P^{(1)}_l(\cos\theta)$ \cite{Bateman-Erdelyi:1953,Korn-Korn:1968,Kerker-book:1969}, valid when $\ell\to\infty$:
{}
\begin{align}
P^{(1)}_\ell(\cos\theta) &=
\dfrac{-\ell}{\sqrt{2\pi \ell \sin\theta}}\Big(e^{i(\ell+\frac{1}{2})\theta+i\frac{\pi}{4}}+e^{-i(\ell+\frac{1}{2})\theta-i\frac{\pi}{4}}\Big)+{\cal O}(\ell^{-\textstyle\frac{3}{2}}), ~~~~~\textrm{for}~~~~~ 0<\theta<\pi.
\label{eq:P1l<}
\end{align}
With the approximation above, we may replace the sum in the (\ref{eq:alpha*1*})
with an integral yielding an expression for $\alpha(r, \theta)$ that to the order of $
{\cal O}\big(r^2_g,(kr)^{-3}\big)$ has the form:
{}
\begin{eqnarray}
\alpha(r, \theta)&=&\frac{E_0u}{k^2r^2}e^{ik(r+r_0+r_g\ln4k^2rr_0)}\hskip-4pt
\int_{\ell=kR_\odot^\star}^\infty \hskip 0pt
\frac{\ell\sqrt{\ell}d\ell}{\sqrt{2\pi \sin\theta}}\Big(1-\frac{\ell^2}{4u^2k^2r^2} \Big)
e^{i\big(2\sigma_\ell+\frac{\ell^2}{2k\tilde r}\big)}
\Big(e^{i(\ell\theta+\frac{\pi}{4})}+e^{-i(\ell\theta+\frac{\pi}{4})}\Big).
\label{eq:Pi_s_exp1}
\end{eqnarray}
We note that integrating over $\ell$ from $kR_\odot^\star$ to infinity is equivalent to integrating over the impact parameter $b$ ranging from grazing the Sun at $b=R_\odot^\star$ to infinity. We evaluate this integral by the method of stationary phase \cite{Turyshev-Toth:2017}, which is suitable for the evaluation of oscillatory integrals of the type
{}
\begin{equation}
I=\int A(\ell)e^{i\varphi(\ell)}d\ell, \qquad
\ell\in\mathbb{R},
\label{eq:stp-1}
\end{equation}
where the amplitude $A(\ell)$ is a slowly varying function of $\ell$, while $\varphi(\ell)$ is a rapidly varying function of $\ell$.
The integral (\ref{eq:stp-1}) may be replaced, to good approximation, with a sum over the points of stationary phase, $\ell_0\in\{\ell_{1,2,..}\}$, for which $d\varphi/d\ell=0$. Defining $\varphi''=d^2\varphi/d\ell^2$, we obtain the integral
{}
\begin{equation}
I\simeq\sum_{\ell_0\in\{\ell_{1,2,..}\}} A(\ell_0)\sqrt{\frac{2\pi}{\varphi''(\ell_0)}}e^{i\big(\varphi(\ell_0)+{\textstyle\frac{\pi}{4}}\big)}.
\label{eq:stp-2}
\end{equation}
Following \cite{Turyshev-Toth:2018-plasma}, we see that the relevant $\ell$-dependent part of the phase in (\ref{eq:Pi_s_exp1}) is of the form
{}
\begin{equation}
\varphi_{\pm}(\ell)=\pm\big(\ell\theta+\textstyle{\frac{\pi}{4}}\big)+2\sigma_\ell +\dfrac{\ell^2}{2k \tilde r}+{\cal O}\big(r_g^2, (kr)^{-3}\big).
\label{eq:S-l}
\end{equation}
Similarly to the approach used in \cite{Turyshev-Toth:2018-grav-shadow,Turyshev-Toth:2019}, we evaluate $\sigma_\ell$ for $\ell\gg kr_g$ as:
{}
\begin{eqnarray}
\sigma_\ell&=& -kr_g\ln \ell.
\label{eq:sig-l*}
\end{eqnarray}
The phase is stationary when $d\varphi^{[0]}_{\pm}/d\ell=0$, which, together with (\ref{eq:sig-l*}), implies
{}
\begin{equation}
\pm\theta-\frac{2kr_g}{\ell}+\frac{\ell}{k\tilde r}={\cal O}\big(r_g^2, (kr)^{-3}\big).
\label{eq:S-l-pri=}
\end{equation}
Relying on the semiclassical approximation that connects the partial momentum, $\ell$, to the impact parameter, $b$,
{}
\begin{equation}
\ell\simeq kb,
\label{eq:S-l-pri-p-g}
\end{equation}
for small angles $\theta$ (or, large heliocentric distances, $R_\odot/r<b/r\ll 1$), we see that the points of stationary phase that must satisfy (\ref{eq:S-l-pri=}) are given by (see \cite{Turyshev-Toth:2017} for details):
{}
\begin{equation}
\frac{b}{r} +\frac{b}{r_0} = \mp{ {\theta}}+\frac{2r_g}{b}+{\cal O}(\theta^3, r_g^2),
\label{eq:S-l-pri*}
\end{equation}
which describes hyperbolae representing the geodesic trajectories of light rays in the post-Newtonian gravitational field of a mass monopole \cite{Turyshev-Toth:2017}. For impact parameters $b\geq R_\odot^\star$, these trajectories are outside the Sun, crossing from the geometric optics region behind the Sun into the interference region. In essence, (\ref{eq:S-l-pri*}) is a classical thin lens equation that is familiar from geometric optics \cite{Sharma-book:2006,Nambu:2013b,Turyshev-Toth:2017}. Similarly to the approach demonstrated in \cite{Turyshev-Toth:2019}, the validity of this expression may be extended to higher powers of the small angle $\theta$, yielding complete trigonometric identities.
The result given by (\ref{eq:S-l-pri*}) differs from a similar expression given in \cite{Turyshev-Toth:2017,Turyshev-Toth:2019} for a source located at infinity. The finite distance to the source is captured by the term $\beta=b/r_0$, defined in Eq.~(\ref{eq:beta}); see also Fig.~\ref{fig:geom-opt}. In the coordinate system rotated by angle $\beta$ in the $\phi=\phi_0$ plane, we can easily see that light rays intersect the optical axis at slightly greater heliocentric distances.
For any given impact parameter, the focal point of the SGL, located at $r=b^2/2r_g$ for a source at infinity ($\beta=0$), is shifted $\overline r=b^2/(2r_g)(1+b^2/(2r_g z_0))$. The extra distance that a light ray needs to propagate before it intersects the optical axis is $\delta r=(b^2/2r_g)^2/z_0$, which, for nominal values of the parameters, is computed to be $\delta r=0.05 \,(b/R_\odot)^4(30\, {\rm pc}/z_0)$~AU. The extra distance is small but nonvanishing. The heliocentric distance to the focal point associated with a source at a finite distance, calculated as $\overline r$ with respect to the optical axis, is related to the heliocentric distance $r$ of the focal point associated with a source at infinity by
{}
\begin{equation}
\overline r=r(1+r/r_0)+{\cal O}(r^3/r^2_0).
\label{eq:r-bar}
\end{equation}
Equation~(\ref{eq:r-bar}) represents a rescaling of all relevant results by an extra a factor that depends on the distance to the source. This mapping between the distances provides an interpretation of the results that we obtain below.
\subsection{Incident and scattered waves}
We now continue to investigate (\ref{eq:S-l-pri*}). For small but finite angles, $|\theta|\gg \sqrt{2r_g/r}>0$, and large impact parameters, $b\geq R^\star_\odot\gg r_g$, Eq.~(\ref{eq:S-l-pri*}) yields two families of solutions for the points of stationary phase:
{}
\begin{equation}
b_{\tt in}= \mp \Big(\tilde r \theta+\frac{2r_g}{\theta}\Big)+{\cal O}(\theta^3,r_g^2), \qquad {\rm and} \qquad
b_{\tt s}= \pm \frac{2r_g}{\theta}+{\cal O}(\theta^3,r_g^2),
\label{eq:S-l-pri}
\end{equation}
where the family $b_{\tt in}$ represents the incident wave with light ray trajectories bent towards the Sun, obeying the eikonal approximation of geometric optics and the family $b_{\tt s}$ describes the scattered wave.
\begin{figure}
\includegraphics[scale=0.40]{regions-fin}
\caption{\label{fig:regions-fin}The incident and scattered rays in the case of finite distance to the source. Scattered rays with impact parameter $|b_{\tt s}|<R^*_\odot$ (shown by the third line from the bottom) are blocked by the Sun, resulting in the formation of the shadow and the geometric optics regions (as shown in Fig.~\ref{fig:regions}). Scattered rays with $|b_{\tt s}|>R^*_\odot$ (first and second lines from the bottom) pass by the Sun and, after crossing the optical axis at $\overline z$ or beyond, intersect incident rays, leading to the formation of the weak interference region. When the intersection occurs near or on the optical axis, the incident and scattered rays have similar optical path lengths, leading to the formation of the strong interference region.
}
\end{figure}
The solution (\ref{eq:S-l-pri}) offers a very nice representation of the gravitational scattering of light. Consider the first term in $b_{\tt in}$, given as $b_0= \tilde r \theta\geq R^*_\odot$. This term describes the propagation of light at the distance $b_0$ from the optical axis that is representative for an empty (or Euclidian) space-time. The second term in $b_{\tt in}$, is identical to $b_{\tt s}$ in magnitude, but has the opposite sign. This second term describes the scattering of light in the presence of the solar gravitational monopole. For a given impact parameter with respect to the optical axis, $b_0$, as the heliocentric distance $r$ increases, the magnitude of the angle $\theta$ decreases. As $\theta$ gets smaller, the scattering term becomes more significant, effectively deflecting the trajectory of the light ray. At this point we are still in the geometric optics region (see Figs.~\ref{fig:regions} and \ref{fig:regions-fin}) characterized by only one ray of light from the point source passing through any given point of space. For $\theta\geq 2r_g/R^*_\odot$, the impact parameter for the scattered wave is $|b_{\tt s}|\leq R^*_\odot$ and therefore, the scattered wave is blocked by the Sun, as prescribed by the fully absorbing boundary conditions \cite{Turyshev-Toth:2017}.
Closer to the optical axis, where $\theta<2r_g/R^*_\odot$, the impact parameter of the scattered ray is large enough: $|b_{\tt s}|=2r_g/\theta>R^*_\odot$, so that the scattered ray is no longer blocked by the Sun. As the solution for $b_{\tt s}$ has the sign opposite to that of $b_{\tt in}$, the scattered ray enters the region behind the Sun on the opposite side. After it crosses the optical axis, the scattered ray intersects the incident ray. As the two rays have a drastically different optical paths, no notable interference pattern emerges. An observer would see these two rays correspond to two images of the same source with uneven brightness on opposite sides of the Sun. This is characteristic of the region of weak interference, shown in Figs.~\ref{fig:regions} and \ref{fig:regions-fin}.
As a result, the two families of solutions (\ref{eq:S-l-pri}) represent the incident and scattered part of the same EM wave shown by Eq.~(21) of \cite{Turyshev-Toth:2017}. This provides support for the interpretation given in \cite{Deguchi-Watson:1987}, where these two solutions were interpreted as distinct rays of light on opposite sides of the Sun. (Note that the strong interference region is not covered by the approximation (\ref{eq:P1l<}). It is described in detail in Sec.~\ref{sec:IF-region}.) As discussed in \cite{Turyshev-Toth:2017,Turyshev-Toth:2018-grav-shadow,Turyshev-Toth:2019}, the presence of both of these families of light rays determines the physical properties of the EM field in three the regions relevant for the SGL, namely the shadow, the geometric optics region and both interference regions. In addition, the `$\pm$' or `$\mp$' signs represent light rays that propagate on opposite sides of the Sun, as a manifestation of the existing axial symmetry.
\subsection{Computing $\alpha_{\tt in}(r,\theta)$ and $\alpha_{\tt s}(r,\theta)$}
By extending the asymptotic expansion of $H^{+}_\ell(kr_g,kr)$ from (\ref{eq:Fass*}) to the order of ${\cal O}((kr)^{-(2n+1)})$ (i.e., using the WKB approximation as was done in Appendix \ref{sec:rad_eq_wkb}), the validity of the result (\ref{eq:S-l-pri}) may be extended to ${\cal O}(\theta^{2n+1})$. This fact was observed in \cite{Turyshev-Toth:2018-plasma} and used to improve the solution by including terms of higher order in $\theta$.
The first family of solutions of (\ref{eq:S-l-pri}), yielding $\ell_{\tt in}=kb_{\tt in}$, allows us to compute the phase for the points of stationary phase (\ref{eq:S-l}) for the EM waves moving towards the interference region (a similar calculation was done in \cite{Turyshev-Toth:2018-plasma}):
{}
\begin{eqnarray}
\varphi_{\pm}(\ell_{\tt in})&=& \pm\textstyle{\frac{\pi}{4}}-\textstyle{\frac{1}{2}}k\tilde r\theta^2-kr_g\ln k^2\tilde r^2\theta^2+{\cal O}(kr\theta^4,kr_g\theta^2).
\label{eq:S-l2p}
\end{eqnarray}
From (\ref{eq:S-l}) we compute $\varphi''(\ell)$ as
\begin{eqnarray}
\dfrac{d^2\varphi_{\pm}}{d\ell^2} &=& \dfrac{1}{k\tilde r}+\frac{2kr_g}{\ell^2} +{\cal O}\big((kr)^{-3}\big).
\label{eq:S-l2+}
\end{eqnarray}
After substituting $\ell_{\tt in}$ in (\ref{eq:S-l2+}), we derive $\varphi''(\ell_{\tt in})= {d^2\varphi_{\pm}}/{d\ell^2} \big|_{\ell=\ell_{\tt in}} $ which yields
\begin{eqnarray}
\sqrt{\frac{2\pi}{\varphi''(\ell_{\tt in})}}&=&\sqrt{2\pi k\tilde r}\Big\{1-\frac{r_g}{\tilde r\theta^2}
+{\cal O}(\theta^2,\frac{r_g}{r}\theta^2)\Big\}.
\label{eq:sf*}
\end{eqnarray}
Now, using (\ref{eq:sf*}), we have the amplitude of the integrand in (\ref{eq:Pi_s_exp1}), for $\ell_{\tt in}\gg1$, taking the form
{}
\begin{eqnarray}
\frac{\ell_{\tt in}\sqrt{\ell_{\tt in}}}{\sqrt{2\pi \sin\theta}}\Big(1-\frac{\ell^2_{\tt in}}{4u^2k^2r^2} \Big)
\sqrt{\frac{2\pi}{\varphi''(\ell_{\tt in})}}
&=&
(\mp1)^{\textstyle\frac{3}{2}} k^2 r\tilde ru^{-2}\sin(\theta+\beta)\Big(1+\frac{r_g}{r(1-\cos(\theta+\beta))}+{\cal O}(\theta^3, \beta^3,\frac{r_g}{r}\theta^2)\Big),\label{eq:S-l3p+*}
\end{eqnarray}
where the angle $\beta$ is from (\ref{eq:beta}) and we used the definition (\ref{eq:r-tilde}) for $\tilde r$ and the solution (\ref{eq:S-l-pri}) for the impact parameter $b_{\tt in}$, namely
{}
\begin{eqnarray}
\tilde r\theta&=&r\big(\theta-\frac{r\theta}{r_0}\big)+{\cal O}(r^3/r_0^2)=r\big(\theta+\frac{b}{r_0}\big)+{\cal O}(r^3/r_0^2)=r\big(\theta+\beta\big)+{\cal O}(r^3/r_0^2).
\label{eq:tildebeta}
\end{eqnarray}
As a result, the expression for $\alpha_{\tt in}(r,\theta)$ from (\ref{eq:Pi_s_exp1}) for the incident wave takes the form
{}
\begin{eqnarray}
\alpha_{\tt in}(r,\theta)&=&
E_0u^{-1}\frac{r_0}{(r+r_0)}\sin(\theta+\beta) \Big(1+\frac{r_g}{r\big(1-\cos(\theta+\beta)\big)}+{\cal O}(\theta^2,\frac{r_g}{r}\theta^2)\Big)e^{i \varphi_{\tt in}(r,\theta)},
\label{eq:Pi_s_exp4+1*}
\end{eqnarray}
where to derive the phase, $\varphi_{\tt in}(r,\theta)$, according to (\ref{eq:stp-2}), we combined the WKB phase from the exponent in front of the integral in (\ref{eq:Pi_s_exp1}) and the $\ell$-dependent contribution from (\ref{eq:S-l2p}):
$k(r+r_0+r_g\ln4k^2rr_0)+ \varphi_{\pm}(\ell_{\tt in})+\textstyle{\frac{\pi}{4}}
$, yielding
{}
\begin{eqnarray}
\varphi_{\tt in}(r,\theta)&=&
k\Big(r_0\cos\beta+r_g\ln 2kr_0+r\cos(\theta+\beta)-r_g\ln kr\big(1-\cos(\theta+\beta)\big)\Big).
\label{eq:phase_rr0}
\end{eqnarray}
We define $\vec k_\beta=-{\vec n}_0$ being the vector along the direction from the point source to the center of the Sun, i.e., along the optical axis. Then, the following identity holds $2kr_0= kr_0-k(\vec k_\beta\cdot\vec r_0)$. With this, the expression (\ref{eq:phase_rr0}) takes a familiar form:
{}
\begin{eqnarray}
\varphi_{\tt in}(r,\theta)&=&k\Big(
\vec k_\beta \cdot(\vec x-\vec x_0)-r_g\ln \frac{r-({\vec k}_\beta\cdot\vec x)}{r_0-(\vec k_\beta\cdot\vec x_0)}\Big),
\label{eq:phase_rr0-vec}
\end{eqnarray}
that describes the phase of an EM wave obtained with the geodesic equation (discussed in Appendix B of Ref.~\cite{Turyshev-Toth:2017}, with relevant results given there by (B22) and (B33), correspondingly.) The results (\ref{eq:Pi_s_exp4+1*})--(\ref{eq:phase_rr0}) represent good evidence that in the case of a finite distance to the source, the overall solution for the EM field is rotated by the angle $\beta$, aligning it with the optical axis specified by a point source with coordinates $({\vec b},r_0)$ and the center of the Sun. However, an obvious difference compared to a solution of the geodesic equation is the fact that (\ref{eq:phase_rr0-vec}) describes the evolution of the phase along the trajectory with a zero impact parameter.
Result (\ref{eq:phase_rr0-vec}) justifies our approach of modifying the existing solution for the EM field by applying the eikonal approximation, as discussed in Sec.~\ref{sec:opt-axis}. The use of the optical axis $\overline z$ restores the axial symmetry of the problem and the entire EM field representing the family of rays emitted towards the Sun with the same impact parameter $b$, may now be obtained by a simple rotation around the $\overline z$-axis by the angle $\phi_0$.
Now we consider the second family of solutions in (\ref{eq:S-l-pri}), given by $\ell_{\tt s}=kb_{\tt s}$ (similar derivations were made in \cite{Turyshev-Toth:2018-grav-shadow}), which allow us to compute the stationary phase as
{}
\begin{eqnarray}
\varphi_{\pm}(\ell_{\tt s})&=&
\pm\textstyle{\frac{\pi}{4}}-kr_g\ln 2k\tilde r+kr_g\ln k\tilde r \textstyle{\frac{1}{2}}\theta^2-2kr_g\ln \frac{kr_g}{e}+{\cal O}(kr_g \theta^2).
\label{eq:S-l27*+}
\end{eqnarray}
Using (\ref{eq:S-l2+}) and $\ell_{\tt s}=kb_{\tt s}$ from (\ref{eq:S-l-pri}), we compute the second derivative of the phase with respect to $\ell$:
{}
\begin{equation}
\varphi''_{\pm}(\ell_{\tt s})=\frac{\theta^2}{2kr_g}\Big(1+\frac{2r_g}{\tilde r\theta^2}\Big)+{\cal O}(\theta^3), \qquad {\rm thus, } \qquad
\sqrt{\frac{2\pi}{\varphi''(\ell_{\tt s})}}=\frac{\sqrt{4\pi kr_g}}{\theta}\Big(1-\frac{r_g}{\tilde r\theta^2}\Big).
\label{eq:S-l2202}
\end{equation}
At this point, we may evaluate the amplitude of the integrand in (\ref{eq:Pi_s_exp1}), which, for $\ell_{\tt s}\gg1$, is given as
{}
\begin{eqnarray}
\frac{\ell_{\tt s}\sqrt{\ell_{\tt s}}}{\sqrt{2\pi \sin\theta}} \Big(1-\frac{\ell^2_{\tt s}}{4u^2k^2r^2} \Big)\sqrt{\frac{2\pi}{\varphi''(\ell_{\tt s})}}&=&
(\mp1)^{\textstyle\frac{3}{2}}\Big( \frac{2kr_g}{\theta}\Big)^2\frac{1}{\theta}\Big(1-\frac{r_g}{\tilde r\theta^2}\Big).
\label{eq:S-l3p+*=}
\end{eqnarray}
As a result, the expression for $\alpha_{\tt s}(r,\theta)$ representing the scattered wave in (\ref{eq:Pi_s_exp1}) takes the form
{}
\begin{eqnarray}
\alpha_{\tt s}(r,\theta)&=&-E_0\Big(\frac{2r_g}{r}\Big)^2\frac{1}{\theta^3}
e^{ik\big(r_0+r_g\ln 2kr_0+r+r_g\ln kr(1-\cos(\theta+\beta))\big)+2i\sigma_0}
\sim {\cal O}(r_g^2),
\label{eq:alpha-2}
\end{eqnarray}
where the phase was computed by combining the WKB phase from (\ref{eq:Pi_s_exp1}) and the $\ell$-dependent contribution (\ref{eq:S-l27*+}), as we did to compute (\ref{eq:phase_rr0}). We conclude that to the order of ${\cal O}(r_g^2)$, there is no scattered wave in the radial direction, which is consistent with the results reported in \cite{Turyshev-Toth:2017}, extending those to include dependence on the source position.
The results (\ref{eq:Pi_s_exp4+1*}), (\ref{eq:alpha-2}) are the radial components of the EM wave corresponding to the two families of the impact parameters (\ref{eq:S-l-pri}). We use these solutions to discuss the EM field in the geometric optics region.
\subsection{Evaluating the function $\beta(r,\theta)$}
\label{sec:amp_func-beta}
To evaluate the magnitude of the function $\beta(r, \theta)$ in (\ref{eq:beta*1*}), we rely on the asymptotic behavior of $P^{(1)}_{l}(\cos\theta)/\sin\theta$ and $\partial P^{(1)}_{l}(\cos\theta)/\partial \theta$, which for fixed $\theta$ and $\ell\rightarrow\infty$ is given \cite{vandeHulst-book-1981,Turyshev-Toth:2019} as
{}
\begin{eqnarray}
\frac{P^{(1)}_\ell(\cos\theta)}{\sin\theta}
&=& \Big(\frac{2\ell}{\pi\sin^3\theta}\Big)^{\frac{1}{2}} \sin\Big((\ell+{\textstyle\frac{1}{2}})\theta-{\textstyle\frac{\pi}{4}}\Big)+{\cal O}(\ell^{-\textstyle\frac{3}{2}}),
\label{eq:pi-l*}\\
\frac{dP^{(1)}_\ell(\cos\theta)}{d\theta}
&=& \Big(\frac{2\ell^3}{\pi\sin\theta}\Big)^{\frac{1}{2}} \cos\Big((\ell+{\textstyle\frac{1}{2}})\theta-{\textstyle\frac{\pi}{4}}\Big)+{\cal O}(\ell^{-\textstyle\frac{1}{2}}).
\label{eq:tau-l*}
\end{eqnarray}
With these approximations, in the region of geometric optics the function $\beta(r,\theta)$ from (\ref{eq:beta*1*}) takes the form:
{}
\begin{eqnarray}
\beta(r,\theta)&=&E_0\frac{ue^{ik(r+r_0+r_g\ln 4k^2rr_0)}}{ikr}
\sum_{\ell=kR_\odot^\star}^\infty \ell^{-1}
e^{i\big(2\sigma_\ell+\frac{\ell^2}{2k\tilde r}\big)}
\times\nonumber\\
&&\times\,
\Big\{
\Big(\frac{2\ell^3}{\pi\sin\theta}\Big)^{\frac{1}{2}} \Big(1-\frac{\ell^2}{2k^2r^2}\Big)\cos\Big(\ell\theta-{\textstyle\frac{\pi}{4}}\Big)+\Big(\frac{2\ell}{\pi\sin^3\theta}\Big)^{\frac{1}{2}} \sin\Big(\ell\theta-{\textstyle\frac{\pi}{4}}\Big)
\Big\}.~~~~~~~
\label{eq:S1-v0s}
\end{eqnarray}
For large $\ell\gg1$, the first term in the curly brackets of (\ref{eq:S1-v0s}) dominates, so that this expression may be given as
{}
\begin{eqnarray}
\beta(r,\theta)&=&E_0\frac{ue^{ik(r+r_0+r_g\ln 4k^2rr_0)}}{ikr}
\sum_{\ell=kR_\odot^\star}^\infty
\Big(\frac{2\ell}{\pi\sin\theta}\Big)^{\frac{1}{2}} \Big(1-\frac{\ell^2}{2u^2k^2r^2}\Big)
e^{i\big(2\sigma_\ell+\frac{\ell^2}{2k\tilde r}\big)}
\cos\big(\ell\theta-{\textstyle\frac{\pi}{4}}\big).~~~~~~~
\label{eq:S1-v0s+}
\end{eqnarray}
To evaluate this expression, we again use the method of stationary phase. For this, representing (\ref{eq:S1-v0s+}) in the form of an integral over $\ell$, we have:
{}
\begin{eqnarray}
\beta(r,\theta)&=&-E_0\frac{ue^{ik(r+r_0+r_g\ln 4k^2rr_0)}}{kr}
\int_{\ell=kR_\odot^\star}^\infty \hskip-5pt \frac{\sqrt{\ell}d\ell}{\sqrt{2\pi\sin\theta}} \Big(1-\frac{\ell^2}{2u^2k^2r^2}\Big)
e^{i\big(2\sigma_\ell+\frac{\ell^2}{2k\tilde r}\big)}
\Big(e^{i(\ell\theta+{\textstyle\frac{\pi}{4}})}-e^{-i(\ell\theta+{\textstyle\frac{\pi}{4}})}\Big).~~~~~~~
\label{eq:S1-v0s+int*}
\end{eqnarray}
Expression (\ref{eq:S1-v0s+int*}) shows that the $\ell$-dependent part of the phase has a structure identical to that of (\ref{eq:S-l}). Therefore, the same solutions for the points of stationary phase apply. As a result, using (\ref{eq:S-l-pri}) and (\ref{eq:sf*}), from (\ref{eq:S1-v0s+int*}) and for the first family of solutions (\ref{eq:S-l-pri}) or $\ell_{\tt in}=kb_{\tt in}$, to the order of ${\cal O}(\theta^4)$, we have the amplitude in the stationary phase solution
{}
\begin{eqnarray}
\frac{\sqrt{\ell_{\tt in}}}{\sqrt{2\pi\sin\theta}}\Big(1-\frac{\ell^2_{\tt in}}{2u^2k^2r^2}\Big)\sqrt{\frac{2\pi}{\varphi''(\ell_{\tt in})}}&=&
\sqrt{\mp1}k\tilde ru^{-1}\Big(\cos(\theta+\beta)-\frac{r_g}{r}\Big).~~~
\label{eq:S-l3p}
\end{eqnarray}
As a result, similarly to (\ref{eq:Pi_s_exp4+1*}), the expression for the $\beta_{\tt in}(r,\theta)$ takes the form (with $\varphi_{\tt in}(r,\theta)$ is given by (\ref{eq:phase_rr0}))
{}
\begin{eqnarray}
\beta_{\tt in}(r,\theta)&=&
E_0u^{-1}\frac{r_0}{r+r_0}\Big(\cos(\theta+\beta)-\frac{r_g}{r}\Big) e^{ik\varphi_{\tt in}(r,\theta)}
+{\cal O}(\theta^4, \frac{r_g}{r}\theta^2).~~~~~
\label{eq:Pi_s_exp4+1pp}
\end{eqnarray}
Now we turn our attention to the second family of solutions in (\ref{eq:S-l-pri}) or for $\ell_{\tt s}=kb_{\tt s}$. Similarly to (\ref{eq:S-l3p+*=}), we have
{}
\begin{eqnarray}
\frac{\sqrt{\ell_{\tt s}}}{\sqrt{2\pi\sin\theta}} \Big(1-\frac{\ell^2_{\tt s}}{2u^2k^2r^2}\Big)\sqrt{\frac{2\pi}{\varphi''(\ell_{\tt s})}}&=&
\sqrt{\pm1} \frac{2kr_g}{\theta^2}+{\cal O}(\theta^2,r^2_g),~~~~
\label{eq:S-l3p+*=2}
\end{eqnarray}
which yields the following result for $\beta_{\tt s}(r,\theta)$:
{}
\begin{eqnarray}
\beta_{\tt s}(r,\theta)&=&
E_0\frac{r_g}{2r\sin^2\textstyle{\frac{1}{2}}(\theta+\beta)}
e^{ik\big(r_0+r_g\ln 2kr_0+r+r_g\ln kr(1-\cos(\theta+\beta))\big)+2i\sigma_0}
+{\cal O}(\theta^2, \frac{r_g}{r}\theta^2).~~~~~
\label{eq:beta-2}
\end{eqnarray}
\subsection{Evaluating the function $\gamma(r,\theta)$}
\label{sec:amp_func-der}
To determine the remaining components of the EM field (\ref{eq:DB-sol00p*}), we need to evaluate the behavior of the function $\gamma(r,\theta)$ from (\ref{eq:gamma*1*}). For that, we use the asymptotic behavior of $P^{(1)}_{l}(\cos\theta)/\sin\theta$ and $\partial P^{(1)}_{l}(\cos\theta)/\partial \theta$ from (\ref{eq:pi-l*}) and (\ref{eq:tau-l*}), and rely on the method of stationary phase.
Similarly to (\ref{eq:S1-v0s}), we drop the second term in the curly brackets in (\ref{eq:gamma*1*}). The resulting expression for $\gamma(r, \theta)$, for large partial momenta, $\ell\gg1$, is now determined from the following integral:
{}
\begin{eqnarray}
\gamma(r, \theta)&=& E_0\frac{ue^{ik(r+r_0+r_g\ln 4k^2rr_0)}}{kr} \int_{\ell=kR_\odot^\star}^\infty \hskip -5pt
\frac{\sqrt{\ell}d\ell}{\sqrt{2\pi\sin\theta}}e^{i\big(2\sigma_\ell+\frac{\ell^2}{2k\tilde r}\big)}
\Big(e^{i(\ell\theta+{\textstyle\frac{\pi}{4}})}-e^{-i(\ell\theta+{\textstyle\frac{\pi}{4}})}\Big).
\label{eq:gamma**1*}
\end{eqnarray}
Clearly, this expression yields the same points of stationary phase (\ref{eq:S-l}), thus, all the results obtained in Sec.~\ref{sec:radial-comp} are also relevant here. Therefore, the $\ell$-dependent amplitude of (\ref{eq:gamma**1*}), for the first family of solutions (\ref{eq:S-l-pri}), $\ell_{\tt in}=kb_{\tt in}$, is evaluated to be
{}
\begin{equation}
\frac{\sqrt{\ell_{\tt in}}}{\sqrt{2\pi\sin\theta}}\sqrt{\frac{2\pi}{\varphi''(\ell_{\tt in})}}=\pm\sqrt{\mp1}k\tilde r+{\cal O}(\theta^3, \frac{r_g}{r}\theta^2).
\label{eq:S-l3p+0*}
\end{equation}
With $\varphi_{\tt in}(r,\theta)$ from (\ref{eq:phase_rr0}), the function $ \gamma_{\tt in}(r,\theta)$ is given as
{}
\begin{eqnarray}
\gamma_{\tt in}(r,\theta)&=&
E_0u \frac{r_0}{r+r_0}e^{i\varphi_{\tt in}(r,\theta)}+{\cal O}(\theta^3,\frac{r_g}{r}\theta^2).~~~~~
\label{eq:Pi_s_exp4+1gg}
\end{eqnarray}
Finally, for the second family of solutions (\ref{eq:S-l-pri}), $\ell_{\tt s}=kb_{\tt s}$, the result for $\gamma_{\tt s}(r,\theta)$ is identical to that given by (\ref{eq:beta-2}).
\subsection{Solution for the EM field in the region of geometric optics}
\label{sec:EM-fieldsol}
To determine the components of the entire incident EM field in the region of geometric optics, we use the expressions that were obtained for the functions $\alpha_{\tt in}(r,\theta)$, $\beta_{\tt in}(r,\theta)$ and $\gamma_{\tt in}(r,\theta)$, which are given by (\ref{eq:Pi_s_exp4+1*}), (\ref{eq:Pi_s_exp4+1pp}) and (\ref{eq:Pi_s_exp4+1gg}), correspondingly, and substitute them in (\ref{eq:DB-sol00p*}).
Before we present the resulting solution for the EM field, we recognize that these functions $\alpha_{\tt in}(r,\theta)$, $\beta_{\tt in}(r,\theta)$ and $\gamma_{\tt in}(r,\theta)$ are given with respect to the optical axis (as evident from (\ref{eq:phase_rr0-vec})) that connects the point source with the center of the Sun. However, these functions are still constrained to the plane $\phi=\phi_0$. In order to regain the axial symmetry, we need to rotate the solution (\ref{eq:DB-sol00p*}) by the angle $\beta$ in the plane defined by $\phi_0$. As we mentioned earlier, because $\beta$ is very small, any contribution of such a rotation to the amplitude to the EM field is negligibly small. After performing the needed substitutions and implementing the rotation by $\beta$, we establish the solution for the incident wave produced by the Debye potential $\Pi_0$ from (\ref{eq:Pi-s_a+0}):
{}
\begin{eqnarray}
\left( \begin{aligned}
{ D}_r& \\
{ B}_r& \\
\end{aligned} \right)_{\tt \hskip -4pt in}&=& \frac{E^{\tt s}_0u^{-1}}{r+r_0}
\sin(\theta+\beta)\Big(1+\frac{r_g}{r(1-\cos(\theta+\beta))}\Big)
\left( \begin{aligned}
\cos\overline\phi\\
\sin\overline\phi\\
\end{aligned} \right)
e^{i \big(\varphi_{\tt in}(r,\theta)-\omega t\big)},
\label{eq:DB-t-pl=p10} \\
\left( \begin{aligned}
{ D}_\theta& \\
{ B}_\theta& \\
\end{aligned} \right)_{\tt \hskip -4pt in} &=&
\frac{E^{\tt s}_0u^{-1}}{r+r_0}
\Big(\cos(\theta+\beta)-\frac{r_g}{r}\Big)
\left( \begin{aligned}
\cos\overline\phi\\
\sin\overline\phi\\
\end{aligned} \right) e^{i \big(\varphi_{\tt in}(r,\theta)-\omega t\big)},
\label{eq:DB-th=p10} \\
\left( \begin{aligned}
{ D}_\phi& \\
{ B}_\phi& \\
\end{aligned} \right)_{\tt \hskip -4pt in} &=&
\frac{E^{\tt s}_0u}{r+r_0}
\left( \begin{aligned}
-\sin\overline\phi\\
\cos\overline\phi\\
\end{aligned} \right) \, e^{i \big(\varphi_{\tt in}(r,\theta)-\omega t\big)},
\label{eq:DB-t-pl=p20}
\end{eqnarray}
where the phase $\varphi_{\tt in}(r,\theta)$ is given by (\ref{eq:phase_rr0}) or, equivalently, by (\ref{eq:phase_rr0-vec}).
This is the solution for the EM field in the geometric optics region formed by the solar gravitational monopole.
\subsection{Solution for the EM field in the weak interference region}
\label{sec:EM-field-sol-wi}
We recall (see \cite{Turyshev-Toth:2017} for details) that in the case of gravitational scattering, there are two waves that characterize the overall scattering process in the region of weak interference: the incident wave given by (\ref{eq:DB-t-pl=p10})--(\ref{eq:DB-t-pl=p20}) and the scattered wave with $\alpha_{\tt s}(r,\theta)$ and $\beta_{\tt s}(r,\theta)=\gamma_{\tt s}(r,\theta)$ are given by (\ref{eq:alpha-2}), (\ref{eq:beta-2}), correspondingly, leading to the following form of the scattered wave with $(D_r, B_r)_{\tt \hskip 0pt s}= {\cal O}(r^2_g)$ and the rest of the components given as
{}
\begin{eqnarray}
\left( \begin{aligned}
{ D}_\theta& \\
{ B}_\theta& \\
\end{aligned} \right)_{\tt \hskip -4pt s} = \left( \begin{aligned}
{ B}_\phi& \\
-{ D}_\phi& \\
\end{aligned} \right)_{\tt \hskip -4pt s}= \frac{E^{\tt s}_0}{r+r_0}
\frac{r_g}{2r\sin^2\frac{1}{2}(\theta+\beta)}
\left( \begin{aligned}
\cos\overline\phi\\
\sin\overline\phi\\
\end{aligned} \right)
e^{i\big(k\big(r_0+r_g\ln 2kr_0+r+r_g\ln kr(1-\cos(\theta+\beta))\big)+2\sigma_0-\omega t \big)}.
\label{eq:scat-th-tot}
\end{eqnarray}
As a result, we established the EM field in the region of weak interference in the presence of the incident wave and the scattered wave, given by (\ref{eq:DB-t-pl=p10})--(\ref{eq:DB-t-pl=p20}) and by (\ref{eq:scat-th-tot}), correspondingly.
Note that the way we handled the sums in (\ref{eq:alpha*1*})--(\ref{eq:gamma*1*})---replacing them throughout this section with integrals over $\ell$ and then evaluating the integrals via the method of a stationary phase---amounts to an integral in the lens plane typically encountered in the models for weak gravitational lensing. In fact, results (\ref{eq:DB-t-pl=p10})--(\ref{eq:DB-t-pl=p20}) and (\ref{eq:scat-th-tot}) may now be used to compute the energy flux at the image plane, similarly to that done in Sec.~II.F of \cite{Turyshev-Toth:2017}. Although this is a rather simple step technically, we will not discuss it here as such a development is beyond the scope of the present paper.
This completes the derivation for the EM field in the geometric optics and weak interference regions formed by the solar gravitational monopole. We now turn out attention to the strong interference region, which is the region of greatest importance for imaging with the SGL.
\section{EM field in the strong interference region}
\label{sec:IF-region}
We are interested in the area behind the Sun, reachable by light rays with impact parameters $b\geq R_\odot^\star$. The focal region of the SGL begins where $r\geq b^2/2r_g$ and $0\leq \theta\simeq \sqrt{2r_g/r}$. The EM field in this region is derived from the Debye potential (\ref{eq:Pi-s_a+0}) and is given by the factors $\alpha(r,\theta)$, $\beta(r,\theta)$ and $\gamma(r,\theta)$ from (\ref{eq:alpha*1*})--(\ref{eq:gamma*1*}), which we now calculate.
\subsection{The function $\alpha(r,\theta)$ and the radial components of the EM field}
\label{sec:alpha-IF}
We again begin with the investigation of $\alpha(r,\theta)$ from (\ref{eq:alpha*1*}).
To evaluate this expression in the interference region where $0\leq \theta \simeq \sqrt{2r_g/r}$, we use the asymptotic representation for $P^{(1)}_l(\cos\theta)$ from \cite{Bateman-Erdelyi:1953,Korn-Korn:1968,Kerker-book:1969}, valid when $\ell\to\infty$:
{}
\begin{eqnarray}
P^{(1)}_\ell(\cos\theta)&=& \frac{\ell+{\textstyle\frac{1}{2}}}{\cos{\textstyle\frac{1}{2}}\theta}J_1\big((\ell+{\textstyle\frac{1}{2}})2\sin{\textstyle\frac{1}{2}}\theta\big).
\label{eq:pi-l0}
\end{eqnarray}
We use the approximation above and replace the sum in the resulting expression
(\ref{eq:alpha*1*}) with an integral to be evaluated with the method of stationary phase:
{}
\begin{eqnarray}
\alpha(r,\theta) &=& -E_0\frac{ue^{ik(r+r_0+r_g\ln 4k^2rr_0)}}{k^2r^2}\int_{\ell=kR^\star_\odot}^\infty \ell^2 d\ell e^{i\big(2\sigma_\ell+\frac{\ell^2}{2k\tilde r}\big)}\Big(1-\frac{\ell^2}{4u^2k^2r^2} \Big)J_1\big(\ell \theta\big)
+{\cal O}\big(r^2_g,(kr)^{-3},\theta^2\big).
\label{eq:alpha*IF*int}
\end{eqnarray}
We see that the $\ell$-dependent phase in this expression is given as
{}
\begin{eqnarray}
\varphi(\ell)&=& 2\sigma_\ell+\frac{\ell^2}{2k\tilde r}+{\cal O}((kr)^{-3})=-2kr_g\ln \ell+\frac{\ell^2}{2k\tilde r}+{\cal O}((kr)^{-3}).
\label{eq:IF-phase}
\end{eqnarray}
The phase is stationary when $d\varphi(\ell)/d\ell=0$, resulting in
{}
\begin{eqnarray}
-\frac{2kr_g}{\ell}+\frac{\ell}{k\tilde r}={\cal O}((kr)^{-3})\qquad \Rightarrow\qquad
\ell^2 =2 k^2r_g\tilde r +{\cal O}((kr)^{-1}) \qquad {\rm or} \qquad \ell_0=k\sqrt{2r_g\tilde r}.
\label{eq:IF-phase3}
\end{eqnarray}
This solution, $\ell_0$, represents the smallest partial momenta for the light trajectories to reach a particular heliocentric distance, $r$, on the optical axis (Fig.~\ref{fig:geom-opt}). It is consistent with the solution to the equation for geodesics (see Appendix~B in \cite{Turyshev-Toth:2017}) which yields the solution for the impact parameter of $b=\sqrt{2r_g\tilde r}$. Note that we choose $\ell$ to be positive.
We now return to evaluating (\ref{eq:alpha*IF*int}). The solution given by (\ref{eq:IF-phase3}) allows us to compute the stationary phase (\ref{eq:IF-phase}):
{}
\begin{eqnarray}
\varphi(\ell_0)&=& -kr_g\ln 2k\tilde r+\sigma_0 +{\textstyle\frac{\pi}{2}},
\label{eq:IF-phase00}
\end{eqnarray}
where $\sigma_0$ is the constant given as $\sigma_0=\arg \Gamma(1-ikr_g)$ \cite{Morse-Feshbach:1953}. For large values of $kr_g\rightarrow\infty$ this constant is evaluated as $\sigma_0=-kr_g\ln (kr_g/e)-\frac{\pi}{4}$ (see details in \cite{Turyshev-Toth:2017}).
Next, using (\ref{eq:IF-phase}), we compute the relevant
$\varphi''(\ell)={d^2\varphi_{}}/{d\ell^2} $ as
\begin{eqnarray}
\dfrac{d^2\varphi_{}}{d\ell^2} &=& \dfrac{1}{k\tilde r}+\frac{2kr_g}{\ell^2}
\qquad \Rightarrow \qquad
\sqrt{\frac{2\pi}{\varphi''(\ell_0)}}=\sqrt{\pi k\tilde r}.
\label{eq:IF-phase*6}
\end{eqnarray}
With (\ref{eq:IF-phase*6}), we now have the amplitude of the integrand in (\ref{eq:alpha*IF*int}), for $\ell$ from (\ref{eq:IF-phase3}), taking the form
{}
\begin{eqnarray}
\ell^2_0\Big(1-\frac{\ell^2_0}{4u^2k^2r^2}\Big)J_1\big(\ell \theta\big)\sqrt{\frac{2\pi}{\varphi''(\ell_0)}}&=&
k^22r_g\tilde r \sqrt{\pi k\tilde r}\Big(1-\frac{r_g\tilde r}{4r^2} +{\cal O}(r_g^2, (kr)^{-1})\Big)J_1\big(k\sqrt{2r_g\tilde r} \theta\big).
\label{eq:IF-phase*7}
\end{eqnarray}
As a result, the expression for $\alpha(r,\theta)$ from (\ref{eq:alpha*IF*int}) becomes
{}
\begin{eqnarray}
\alpha(r,\theta)&=&
-iE_0\sqrt{\frac{2r_g}{\tilde r} }\sqrt{2\pi kr_g}e^{i\sigma_0}\Big(\frac{r_0}{r+r_0}\big)^2J_1\big(k\sqrt{2r_g\tilde r} \theta\big)e^{ik\big(r+r_0+r_g\ln 2k(r+r_0)\big)}.
\label{eq:IF-phase*7*}
\end{eqnarray}
We can use the same approach to compute the remaining two factors $\beta(r,\theta)$ and $\gamma(r,\theta)$.
\subsection{The function $\beta(r,\theta)$ and the $\theta$-components of the EM field}
\label{sec:beta-IF}
Similarly to \cite{Turyshev-Toth:2017,Turyshev-Toth:2019}, to evaluate the magnitude of the function $\beta(r,\theta)$, we need to establish the asymptotic behavior of the Legendre polynomials $P^{(1)}_{l}(\cos\theta)$ in the relevant regime. The asymptotic formulae for the Legendre polynomials if $w=(\ell+{\textstyle\frac{1}{2}})\theta$ is fixed and $\ell\rightarrow \infty$ are \cite{vandeHulst-book-1981}:
{}
\begin{eqnarray}
\frac{P^{(1)}_\ell(\cos\theta)}{\sin\theta}&=& {\textstyle\frac{1}{2}}\ell(\ell+1)\Big(J_0(w)+J_2(w)\Big),
\label{eq:pi-l}
\qquad
\frac{dP^{(1)}_\ell(\cos\theta)}{d\theta}= {\textstyle\frac{1}{2}}\ell(\ell+1)\Big(J_0(w)-J_2(w)\Big).
\label{eq:tau-l}
\end{eqnarray}
Using (\ref{eq:tau-l}), we transform (\ref{eq:beta*1*}) by also replacing the sum in the resulting expression with an integral to derive $\beta$ as
{}
\begin{eqnarray}
\beta(r,\theta) &=& E_0\frac{ue^{ik(r+r_0+r_g\ln 4k^2rr_0)}}{ikr}\int_{\ell=kR^\star_\odot}^\infty \ell d\ell e^{i\big(2\sigma_\ell+\frac{\ell^2}{2k\tilde r}\big)}
\Big\{J_0\big(\ell\theta\big)- {\textstyle\frac{1}{2}}\Big(J_0\big(\ell\theta\big)-J_2\big(\ell\theta\big)\Big)\frac{\ell^2}{2u^2k^2r^2} \Big\}.
\label{eq:beta*IF*int}
\end{eqnarray}
As the $\ell$-dependent phase in (\ref{eq:beta*IF*int}) is the same as (\ref{eq:alpha*IF*int}), corresponding results from Sec.~\ref{sec:alpha-IF} are also applicable here. In fact, the same solutions for the points of stationary phase apply. As a result, using (\ref{eq:IF-phase3}), (\ref{eq:IF-phase*6}), from (\ref{eq:beta*IF*int}), we have
{}
\begin{eqnarray}
\ell_0\Big\{J_0\big(\ell_0\theta\big)- {\textstyle\frac{1}{2}}\Big(J_0\big(\ell_0\theta\big)-J_2\big(\ell_0\theta\big)\Big)
\frac{\ell^2_0}{2u^2k^2r^2} \Big\}\sqrt{\frac{2\pi}{\varphi''(\ell_0)}}=
k\tilde r\sqrt{2\pi kr_g}
\Big\{J_0\big(k\sqrt{2r_g\tilde r}\theta\big) +{\cal O}\big(\frac{r_g}{r}, r_g^2\big)\Big\}.~~~
\label{eq:beta*IF*=1}
\end{eqnarray}
As a result, the expression for $ \beta(r,\theta)$ in the interference region takes the form
{}
\begin{eqnarray}
\beta(r,\theta)&=& E_0\sqrt{2\pi kr_g}e^{i\sigma_0}\frac{r_0}{r+r_0}
J_0\Big(k\sqrt{2r_g\tilde r}\theta\Big)e^{ik(r+r_0+r_g\ln 2k(r+r_0))}\Big(1+{\cal O}\big(\frac{r_g}{r}, r_g^2\big)\Big).
\label{eq:beta*IF*=2}
\end{eqnarray}
\subsection{The function $\gamma(r,\theta)$ and the $\phi$-components of the EM field}
\label{sec:gamma-IF}
The $\phi$-components of the EM field is govern by the factor $\gamma(r,\theta)$ from (\ref{eq:gamma*1*}). Similarly to the discussion in Section \ref{sec:beta-IF}, we use (\ref{eq:pi-l*})--(\ref{eq:tau-l*}) to transform resulting expression for $\gamma(r,\theta)$ to an integral, also taking $\ell\gg1$:
{}
\begin{eqnarray}
\gamma(r,\theta) &=& E_0\frac{ue^{ik(r+r_0+r_g\ln 4k^2rr_0)}}{ikr}\int_{\ell=kR^\star_\odot}^\infty\ell d\ell e^{i\big(2\sigma_\ell+\frac{\ell^2}{2k\tilde r}\big)}
\Big\{J_0(\ell\theta)-{\textstyle\frac{1}{2}}\Big(J_0(\ell\theta)+J_2(\ell\theta)\Big)\frac{\ell^2}{2u^2k^2r^2}\Big\}.
\label{eq:gamma*IF*int}
\end{eqnarray}
As we noticed while deriving (\ref{eq:beta*IF*=1}), the term $\propto \ell^2/2u^2k^2r^2$ produces a contribution of ${\cal O}(r_g/r)$, which is negligible in the interference region. This term may also be neglected in the integrand of (\ref{eq:gamma*IF*int}). With this simplification, the resulting Eq.~(\ref{eq:gamma*IF*int}) is identical to that of (\ref{eq:beta*IF*int}). Therefore, we conclude that in the interference region $\gamma(r,\theta)=\beta(r,\theta)+{\cal O}({r_g}/{r}, r_g^2)$.
\subsection{The EM field in the interference region}
\label{sec:amp_func-IF}
Now we are ready to present the components of the EM field in the interference region. We do that by using the expressions that we obtained for the functions $\alpha(r,\theta)$, $\beta(r,\theta)=\gamma(r,\theta)+{\cal O}(r_g/r)$, which are given by (\ref{eq:IF-phase*7*}), (\ref{eq:beta*IF*=2}), correspondingly, and substitute them in (\ref{eq:DB-sol00p*}). As a result, we establish the solution for the EM field produced by the Debye potential $\Pi$ given by (\ref{eq:Pi-s_a+0}) for the EM wave in the interference region.
After implementing, as before, the rotation by $\beta$ in the plane defined by $\phi_0$, we obtain the solution
in the following form:
{}
\begin{eqnarray}
\left( \begin{aligned}
{ D}_r& \\
{ B}_r& \\
\end{aligned} \right) &=& -iE_0
\sqrt{\frac{2r_g}{\tilde r} }\sqrt{2\pi kr_g}e^{i\sigma_0}\Big(\frac{r_0}{r+r_0}\Big)^2J_1\Big(k\sqrt{2r_g\tilde r}
(\theta+\beta)
\Big)e^{i\big(k(r+r_0+r_g\ln2k(r+r_0))-\omega t\big)}\left( \begin{aligned}
\cos\overline\phi\\
\sin\overline\phi\\
\end{aligned} \right),~~~~~
\label{eq:IF-DBr0} \\
\left( \begin{aligned}
{ D}_\theta& \\
{ B}_\theta& \\
\end{aligned} \right) &=& \left( \begin{aligned}
{ B}_\phi& \\
-{ D}_\phi& \\
\end{aligned} \right)= E_0
\sqrt{2\pi kr_g}e^{i\sigma_0}\frac{r_0}{r+r_0}J_0\Big(k\sqrt{2r_g\tilde r}
(\theta+\beta)
\Big)e^{i\big(k(r+r_0+r_g\ln 2k(r+r_0))-\omega t\big)}\left( \begin{aligned}
\cos\overline\phi\\
\sin\overline\phi\\
\end{aligned} \right).
\label{eq:IF-DBth0}
\end{eqnarray}
The radial component of the EM field (\ref{eq:IF-DBr0}) is negligibly small compared to the other two components, which is consistent with the fact that while passing through solar gravity the EM wave preserves its transverse structure.
Eqs.~(\ref{eq:IF-DBr0})--(\ref{eq:IF-DBth0}) describe the EM field in the interference region of the SGL in the spherical coordinate system. To study this field on the image plane, we follow the approach demonstrated in \cite{Turyshev-Toth:2017}, where instead of spherical coordinates $(r,\theta,\phi)$, we introduced a cylindrical coordinate system $(\rho,\phi,z)$, more convenient for these purposes. In the region $r \gg r_g$, this can be done by defining $R=ur = r+r_g/2+{\cal O}(r_g^2)$ and introducing the coordinate transformations $ \rho=R\sin(\theta+\beta),$ $ z=R\cos(\theta+\beta)$, which, from (\ref{eq:metric-gen}), result in the following line element:
{}
\begin{eqnarray}
ds^2&=&u^{-2}c^2dt^2-u^2\big(dr^2+r^2(d\theta^2+\sin^2\theta d\phi^2)\big)=u^{-2}c^2dt^2-\big(d\rho^2+(\rho-\beta z)^2d\phi^2+u^2dz^2\big)+{\cal O}(r_g^2, \beta^2).
\label{eq:cyl_coord}
\end{eqnarray}
\begin{figure}
\includegraphics[scale=0.35]{p-source-fin-solo}
\caption{\label{fig:single}The SGL maps a point source with coordinates $(x',y')$ in the source plane to a point with coordinates $(x,y)=-(z/z_0)(x',y')$ in the image plane. The rotation of the PSF pattern, evident in Fig.~\ref{fig:geom-opt}, is not emphasized here.
}
\end{figure}
As a result, using (\ref{eq:IF-DBr0})--(\ref{eq:IF-DBth0}), for a high-frequency EM wave (i.e., neglecting terms $\propto(kr)^{-1}$) and for $r\gg r_g$, we derive the components of the EM field near the optical axis, which, together with (\ref{eq:tildebeta}) and up to terms of ${\cal O}(\rho^2/z^2, \beta)$, in the paraxial approximation (see discussion after (\ref{eq:DB-t-pl=p20})), take the form
{}
\begin{eqnarray}
\left( \begin{aligned}
{E}_\rho& \\
{H}_\rho& \\
\end{aligned} \right) = \left( \begin{aligned}
{H}_\phi& \\
-{E}_\phi& \\
\end{aligned} \right)&=&
\frac{ {E}^{\tt s}_0}{r+r_0}
\sqrt{2\pi kr_g}e^{i\sigma_0}
J_0\Big(k\sqrt{2r_g\overline r}
(\theta+\beta) \Big)
e^{i\big(k(r+r_0+r_g\ln 2k(r+r_0))-\omega t\big)}
\left( \begin{aligned}
\cos\overline\phi\\
\sin\overline\phi\\
\end{aligned} \right),
\label{eq:DB-sol-rho}
\end{eqnarray}
with the $z$-components of the EM wave behave as $({E}_z, {H}_z)\sim {\cal O}({\rho}/{z},\beta)$. The quantity $\overline r=r(1+r/r_0+{\cal O}(r^2/r_0^2))$ from (\ref{eq:r-bar}) denotes heliocentric distances along the line connecting the point source and the center of the Sun. Also, $r=\sqrt{z^2+\rho^2}=z+{\cal O}(\rho^2/z)$), $\theta=\rho/z+{\cal O}(\rho^2/z^2)$ and $\beta=b/z_0+{\cal O}(b^2/z_0^2)$. Note that these expressions were obtained using the approximations (\ref{eq:pi-l}) and are valid for forward scattering when $\theta+\beta\approx 0$, or when $0\leq \rho\leq r_g$.
Using (\ref{eq:DB-sol-rho}), we now compute the energy flux on the image plane in the interference region of the SGL (see Fig.~\ref{fig:single}). The relevant components of the time-averaged Poynting vector for the EM field in the image volume, as a result, may be given as \cite{Turyshev-Toth:2017}:
{}
\begin{eqnarray}
{\bar S}_z&=&\frac{c}{8\pi}
\Big(\frac{E^{\tt s}_0}{r+r_0}\Big)^2
\frac{2\pi kr_g}{1-e^{-2\pi k r_g}}
J^2_0\Big(k\sqrt{2r_g \overline r}
(\theta+\beta)\Big),
\label{eq:S_z*6z}
\end{eqnarray}
with ${\bar S}_\rho= {\bar S}_\phi=0$ for any practical purposes. Therefore, the non-vanishing component of the light amplification vector $ {\vec \mu}$, defined as ${\vec \mu}={\vec {\bar S}}/|{\vec{\bar S}}_0|$ (where $|{\bar {\vec S}}_0|=(c/8\pi)(E_0^{\tt s}/(r+r_0))^2$ is the time-averaged Poynting vector of a plane wave propagating in empty spacetime) takes the form:
{}
\begin{eqnarray}
{\bar \mu}_z&=&
\mu_0 J^2_0\Big(\frac{2\pi}{\lambda}\sqrt{2r_g \overline r}
(\theta+\beta)
\Big),\qquad {\rm with}
\qquad
\mu_0=\frac{4\pi^2}{1-e^{-4\pi^2 r_g/\lambda}}
\frac{r_g}{\lambda},
\label{eq:S_z*6z-mu}
\end{eqnarray}
where $\mu_0$ is the light amplification factor for imaging a point source with observer positioned on the optical axis that is set by position of that source and the origin of the heliocentric coordinate system. The amplification reaches its maximum value when the argument of the Bessel function vanishes which is happening when, for a given value of $\beta$ (that is set by the position of the point source), the angle $\theta$ takes opposite value, namely $\theta=-\beta$. Clearly, one recovers (\ref{eq:mu-point}) by taking in (\ref{eq:S_z*6z-mu}) the limit of $z_0\rightarrow \infty$ or, equivalently, by setting $\beta=0$.
The result given by Eq.~(\ref{eq:S_z*6z-mu}) can be used to study the image formation process in the case of an extended source.
\section{Image formation for an extended source with the SGL}
\label{sec:image_form}
To study how the SGL forms an image of an extended source, we model that extended source as a collection of point sources in the source plane. We integrate the point-spread function (PSF) for a single point source (\ref{eq:S_z*6z-mu}) over the extended source (Fig.~\ref{fig:multi}).
\subsection{Generalization to the case of an extended source}
\label{sec:SGL-gen-ext}
\begin{figure}
\includegraphics[scale=0.38]{p-source-ext}
\caption{\label{fig:multi}When the SGL maps an extended source (represented as multiple point sources) from the source to the image plane, the result is a set of overlapping PSF patterns. The center of each PSF represents the location in the image plane where an observer, looking back at the Sun, would see a complete Einstein ring attributed to that specific point source. Other point sources contribute light in the form of partial Einstein rings. $z_0$ and $\overline z$ denote heliocentric distances to the source plane and image plane, correspondingly. As in Fig.~\ref{fig:single}, rotation of the individual PSF patterns, evident in Fig.~\ref{fig:geom-opt}, is not emphasized here. }
\end{figure}
To discuss image formation in the case of an extended source, we introduce the vector of position of an imaging telescope in the coordinate system corresponding to the optical axis, which can be done as $\overline{\vec r}=\overline r( {\vec n}+{\vec k}_\beta)+{\cal O}(\beta^2)$. Following \cite{Turyshev-Toth:2017}, we recognize that for small angles $\theta$ and $\beta$ the argument of the Bessel function in (\ref{eq:S_z*6z}) to the order of ${\cal O}(\theta^2,\beta^2)$ has the form
{}
\begin{eqnarray}
k\sqrt{2r_g \overline r} \big(\theta+\beta\big)&=&
2\sqrt{kr_g(k\overline r-({\vec k}\cdot{\overline {\vec r}}))}= k\sqrt{2r_g\overline r}|{\vec k} \times({\vec k}\times({\vec n}+{\vec k}_\beta))|
=k\sqrt{2r_g\overline r}|{\vec n}_\perp+{\vec k}_{\beta\perp}|
=\nonumber\\
&=&k\sqrt{2r_g\overline r}
\sqrt{\Big(\frac{x}{\overline r}+\frac{b_x}{{ r}_0}\Big)^2+\Big(\frac{y}{{\overline r}}+\frac{b_y}{{ r}_0}\Big)^2}=k\sqrt{2r_g\overline r}\,\Big|\frac{\vec x}{\overline r}+\frac{\vec x'}{{ r}_0}\Big|,
\label{eq:arg}
\end{eqnarray}
where ${\vec n}_\perp=({x/{\overline r},y/{\overline r},0)}$, ${\vec k}_{\beta\perp}=(b_x/{ r_0},b_y/{ r_0},0)$ are the components of these two vectors perpendicular to $\vec k$.
We introduce the coordinate system where the $z$-axis lies on the principal optical axis, which is the line that connects the center of the exoplanet and the center of the Sun, before intersecting the image plane (see Fig.~\ref{fig:single}). It is convenient to express the results in terms of the distances along the principal optical axis, $z$ and $z_0$, rather than the distances along a particular optical axis $r$ and $r_0$. In addition, in this coordinate system, any point in the source and/or image planes is described by a deviation from that principal optical axis and corresponding angles, namely $(x',y',z_0) \leftrightarrow (\rho',\phi',z_0)$ for the source and $(x,y,z) \leftrightarrow (\rho,\phi,z)$ for the image, correspondingly.
Using the new coordinate system, the distances in (\ref{eq:arg}) are related to each other as $r=\sqrt{z^2+\rho^2}= z+{\cal O}(\rho^2/z)$, and $r_0=\sqrt{z_0^2+\rho'^2}= z_0+{\cal O}({\rho'^2}/{z_0})$, where we neglected terms that are second order in $\rho$ and $\rho'$, as ${\rho}/{z}\ll1$ and ${\rho'}/{z_0}\ll 1$. As a result, the distance $\overline r$ is expressed as $\overline r\rightarrow \overline z=z(1+z/z_0)$.
Now we can describe the imaging of an extended source. For that we can generalize (\ref{eq:S_z*6z-mu}) by using (\ref{eq:arg}), where, relying on the axial symmetry of the problem, we replace one point on the surface of the source $(b_y,b_y,r_0)$ with a generic point with coordinates $(x',y',z_0)$. This yields the PSF of the SGL, expressed as a function of the location $\vec{x}$ of a point source at a finite distance from the Sun, and a location $\vec{x'}$ in the image plane:
{}
\begin{eqnarray}
{\bar \mu}_z({\vec x},{\vec x}')&=&
\mu_0J^2_0\Big(\frac{2\pi}{\lambda}
\sqrt{\frac{2r_g}{\overline z}}
|{\vec x}+\frac{\overline z}{{ z}_0}{\vec x'}|\Big).
\label{eq:S_z*6z-mu2}
\end{eqnarray}
This is our main result. It allows one to study the image formation process, develop realistic imaging scenarios, and perform relevant simulations.
The result (\ref{eq:S_z*6z-mu2}) is different form the one that was obtained earlier \cite{Turyshev-Toth:2017} and shown by (\ref{eq:mu-point}). The new result explicitly accounts for the fact that the distance to the source is finite. In addition, this expression also explicitly depends on the coordinates ${\vec x}$ and ${\vec x'}$, not restricting them to a plane as was done in (\ref{eq:mu-point}) and (\ref{eq:S_z*6z-mu}).
\subsection{Photometric imaging of an extended source at a finite distance}
\label{sec:SGL-imaging-ext}
To produce an image of an astronomical source, the sources are assumed to be non-coherent. This allows us to use photometric imaging techniques. In this case, a telescope is used as a ``light bucket'', measuring the total brightness of the Einstein ring at various locations in the image plane, corresponding to different parts of the source (see Fig.~\ref{fig:single}).
For an extended luminous source with the surface brightness of $B(x',y')$, the power density, $I_0(x,y)$, received on the image plane at a distance of $\overline z+z_0$ from the source (see Fig.~\ref{fig:multi}) is computed by integrating the PSF (\ref{eq:S_z*6z-mu2}) over the surface of the extended source, which may be expressed as
\begin{eqnarray}
I_0(x,y)&=&\frac{\mu_0}{4\pi (\overline z+z_0)^2}\iint\displaylimits_{-\infty}^{+\infty}dx'dy'\, B(x',y')\,
J^2_0\Big(\frac{2\pi}{\lambda}\sqrt{\frac{2r_g}{\overline z}}
|{\vec x}+\frac{\overline z}{z_0}{\vec x}'|\Big),
\label{eq:power_dens}
\end{eqnarray}
where $B(x',y')$ is a function with compact support having non-zero values only within the source's dimensions.
Examining (\ref{eq:power_dens}), we see that monopole gravitational lens acts as a convex lens by focusing light, according to
{}
\begin{equation}
x=-\frac{\overline z}{z_0}x', \qquad y=-\frac{\overline z}{z_0}y'.
\end{equation}
This expression implies that the lens focuses light in the opposite quadrant in the image plane by also compressing the projected size of the source by a factor of ${z}/{z_0}\sim1.0\times 10^{-4}\,(z/650 ~{\rm AU}) (30~{\rm pc}/z_0)$. Thus, the diameter of the projection of an Earth-like exoplanet at those distances is reduced to $r_\oplus=R_\oplus (z/z_0)=1.34 (z/650 ~{\rm AU}) (30~{\rm pc}/z_0)$~km.
Given the image radius of $r_\oplus$, a telescope with aperture $d<2r_\oplus$ centered at a particular point $(x_0,y_0)$ on the image plane will receive the signal $P_d(x_0,y_0)=\iint dxdy \,I_0\big(x_0+x,y_0+y\big),$ where the integration is done within the telescope's aperture $|\vec x|\leq d/2$, and with $|{\vec x}_0+{\vec x}|\leq r_\oplus$, yielding the following result:
{}
\begin{eqnarray}
P_d(x_0,y_0)&=&\frac{\mu_0}{4\pi (\overline z+z_0)^2}\hskip -5pt\iint\displaylimits_{|{\vec x}|^2\leq (\frac{1}{2}d)^2}\hskip -5pt dxdy \iint\displaylimits_{-\infty}^{+\infty}dx'dy'\, B(x',y')\,
J^2_0\Big(\frac{2\pi}{\lambda}\sqrt{\frac{2r_g}{\overline z}}
|{\vec x}_0+{\vec x}+\frac{\overline z}{z_0}{\vec x}'|\Big).
\label{eq:power_rec2}
\end{eqnarray}
We note that although this result does not include the contribution from plasma in the solar corona \cite{Turyshev-Toth:2018-plasma,Turyshev-Toth:2019}, such a contribution may be easily incorporated if needed. This may be the case if we were to use the SGL for an application at microwave frequencies where the effect of solar plasma is significant. For optical and IR wavelengths the effect of the solar plasma on the optical properties of the SGL is negligible.
This is our main result that may be used to study the image formation for an extended source. This result opens the way for using the SGL for imaging of faint targets positioned at a large but finite distance from the Sun.
Expression (\ref{eq:power_rec2}) is rather complex and must be evaluated numerically. However, an interesting limiting case exists that may still be treated analytically: that of a point source at a finite distance. For an incoherent source, the power received by the telescope is given by (\ref{eq:power_rec2}). For a point source positioned at the optical axis, ${\vec x}'=0$ and the telescope at the center of the image, ${\vec x}_0=0$, integral (\ref{eq:power_rec2}) is easy to evaluate analytically. Assuming, a uniform source brightness, $B(x',y')=B_{\tt s}$, we integrate (\ref{eq:power_rec2}):
{}
\begin{eqnarray}
P^0_d(0,0)&=&
\frac{\pi R_\oplus^2d^2B_{\tt s}}{16(\overline z+z_0)^2}\mu_0\Big\{ J^2_0\Big(\pi\frac{d}{\lambda}\sqrt{\frac{2r_g}{\overline z}}\Big)+J^2_1\Big(\pi\frac{d}{\lambda}\sqrt{\frac{2r_g}{\overline z}}\Big)\Big\}\approx
\frac{\pi R_\oplus^2dB_{\tt s}}{4(\overline z+z_0)^2}\sqrt{2r_g \overline z},
\label{eq:mu_av}
\end{eqnarray}
where we used the approximations for the Bessel functions for large arguments \cite{Abramovitz-Stegun:1965}:
\begin{eqnarray}
J_0(x)\simeq \sqrt{\frac{2}{\pi x}}\cos(x-{\textstyle\frac{\pi}{4}})+{\cal O}\big(x^{-1}\big)
\qquad {\rm and} \qquad
J_1(x)\simeq \sqrt{\frac{2}{\pi x}}\sin(x-{\textstyle\frac{\pi}{4}})+{\cal O}\big(x^{-1}\big),
\label{eq:BF}
\end{eqnarray}
which is appropriate for typical parameters relevant to imaging with the SGL.
Expression (\ref{eq:mu_av}) may be used to estimate the signal received from a distant unresolved source. Some approximations exist to allow for a semi-analytical treatment of extended sources; however, their description is out of the scope of this paper and will be given elsewhere.
\subsection{Image formation by an optical telescope at the image plane}
\label{sec:image-form-Fourier}
A classical imaging technique relies on an imaging telescope with a large focal plane sensor array capable of capturing high-resolution images. This technique is well developed for coherent sources; however, astronomical sources may not be treated as such. Nevertheless, it is interesting to consider the image formation process with the SGL in the case when the extended source may be treated as coherent.
To produce images with the SGL, we represent an imaging telescope by a convex lens with focal distance $f$, and position the telescope at the image plane in the interference region \cite{Nambu:2013,Kanai-Nambu:2013,Nambu:2013b,Born-Wolf:1999}. The amplitude of the wave just in front of the lens from (\ref{eq:DB-sol-rho}) and (\ref{eq:S_z*6z-mu2}) is given as
{}
\begin{eqnarray}
{\cal A}({\vec x},{\vec x}')&=&
{E}_0 \sqrt{\mu_0}
J_0\Big(k
\sqrt{\frac{2r_g}{\overline z}} |{\vec x}+\frac{\overline z}{{ z}_0}{\vec x}'|\Big).
\label{eq:amp-w}
\end{eqnarray}
The presence of a convex lens with focal distance $f$ is equivalent to a Fourier transform of the wave (\ref{eq:amp-w}). Consider ${\cal A}({\vec x},{\vec x}')$ to be the EM field just in front of the convex lens with focal distance of $\overline z=f$. We position the detector at the focal distance of the convex lens. Using the Fresnel-Kirchhoff diffraction formula, the wave's amplitude on the detector's focal plane at a pixel location of ${\vec p}=(x_i,y_i)$ is given by
{}
\begin{eqnarray}
{\cal A}({\vec p},{\vec x}')=\frac{i}{\lambda}\iint \displaylimits_{|{\vec x}|^2\leq (d/2)^2} \hskip -7pt {\cal A}({\vec x},{\vec x}')e^{-i\frac{k}{2f}|{\vec x}|^2}\frac{e^{iks}}{s}d^2{\vec x},
\label{eq:amp-w-f0}
\end{eqnarray}
where $s$ is the optical path.
The function $e^{-i\frac{k}{2f}|{\vec x}|^2}=e^{-i\frac{k}{2f}(x^2+y^2)}$ represents the action of the convex lens which transforms incident plane waves to spherical waves focusing at the focal point. Assuming that the focal length is sufficiently larger than the radius of the lens, we may approximate the optical path as $s=\sqrt{(x-x_i)^2+(y-y_i)^2+f^2}\sim f+\big((x-x_i)^2+(y-y_i)^2\big)/2f$. This allows us to present (\ref{eq:amp-w-f0}) as
{}
\begin{eqnarray}
{\cal A}({\vec p},{\vec x}')&=&
- {E}_0 \sqrt{\mu_0}\frac{e^{ikf(1+{{\vec p}^2}/{2f^2})}}{i\lambda f}\iint\displaylimits_{|{\vec x}|^2\leq (\frac{1}{2}d)^2} d^2{\vec x}
J_0\Big(k
\sqrt{\frac{2r_g}{\overline z}} |{\vec x}+\frac{\overline z}{{ z}_0}{\vec x}'|\Big) e^{-i\frac{k}{f}({\vec x}\cdot{\vec p})}.
\label{eq:amp-w-f}
\end{eqnarray}
Integrating over the surface of the source, we have the amplitude of the total EM field on the imaging detector:
{}
\begin{eqnarray}
{\cal A}({\vec p})&=&
\iint \displaylimits_{|{\vec x}'|^2\leq R_\oplus^2}\hskip -5pt {\cal A}({\vec p},{\vec x}')d^2{\vec x}'.
\label{eq:amp-w-f3}
\end{eqnarray}
With the amplitude ${\cal A}({\vec p})$ given by (\ref{eq:amp-w-f3}), the EM field on the detector (denoted by subscript $\vec{p}$) is given as
{}
\begin{eqnarray}
\left( \begin{aligned}
{E}_\rho& \\
{H}_\rho& \\
\end{aligned} \right)_{\tt \hskip -3pt \vec p} = \left( \begin{aligned}
{H}_\phi& \\
-{E}_\phi& \\
\end{aligned} \right)_{\tt \hskip -3pt \vec p} &=&
\frac{{\cal A}({\vec p})}{\overline z +z_0}
e^{i\big(k(r+r_0+r_g\ln 2k(r+r_0))-\omega t\big)}
\left( \begin{aligned}
\cos\big(\phi +\phi_0\big)& \\
\sin\big(\phi +\phi_0\big)& \\
\end{aligned} \right).
\label{eq:DB-sol-rho2}
\end{eqnarray}
Using these results, we compute the Poynting vector for the EM field emitted by an extended source and received at a particular pixel ${\vec p}$ on the imaging detector. Given the form of EM field (\ref{eq:DB-sol-rho2}) and (\ref{eq:amp-w-f3}), the Poynting vector will have only one non-zero component, $S_z$. With overline and brackets denoting time averaging and ensemble averaging (over the oscillators on the source's surface), correspondingly, we compute $S_z$ as
{}
\begin{eqnarray}
S_z({\vec p})=\frac{c}{4\pi}\big<\overline{[{\rm Re}{\vec E}\times {\rm Re}{\vec H}]}_z\big>=\frac{c}{8\pi}\frac{1}{(\overline z+z_0)^2}
\big<\big({\rm Re}{\cal A}({\vec p})e^{i\Omega(t)}\big)^2\big>,
\label{eq:Pv}
\end{eqnarray}
where $\Omega(t)$ is the entire time-dependent phase given as $\Omega(t)=kf(1+{{\vec p}^2}/{2f^2})+k(r+r_0+r_g\ln 2k(r+r_0))-\omega t$.
The expectation value for $\big<\big({\rm Re}{\cal A}({\vec p})e^{i\Omega(t)}\big)^2\big>$ is given by the following expression:
{}
\begin{eqnarray}
\big<\big({\rm Re}{\cal A}({\vec p})e^{i\Omega(t)}\big)^2\big>&=&\frac{{\mu_0}}{(\lambda f)^2}\bigg<{\rm Re}\, \Big(e^{i\Omega(t)}
\hskip -5pt
\iint \displaylimits_{|{\vec x}'_a|^2\leq R_\oplus^2} \hskip -5pt
d^2{\vec x}'_a
{E}_0({\vec x}'_a) \hskip -5pt
\iint\displaylimits_{|{\vec x}|^2\leq (d/2)^2}
\hskip -5pt
d^2{\vec x}\,
J_0\Big(k
\sqrt{\frac{2r_g}{\overline r}} |{\vec x}+\frac{\overline r}{{\overline r}_0}{\vec x}_a'|\Big) e^{-i\frac{k}{f}({\vec x}\cdot{\vec p})}\Big) \times\nonumber\\
&&\hskip 25pt \times \, {\rm Re}\, \Big(e^{i\Omega(t)}\hskip -5pt
\iint \displaylimits_{|{\vec x}_b'|^2\leq R_\oplus^2} \hskip -5pt
d^2{\vec x}_b'
{E}_0({\vec x}_b') \hskip -5pt
\iint\displaylimits_{|{\vec x}|^2\leq (d/2)^2}
\hskip -5pt d^2{\vec x}\,
J_0\Big(k
\sqrt{\frac{2r_g}{\overline r}} |{\vec x}+\frac{\overline r}{{\overline r}_0}{\vec x}_b'|\Big) e^{-i\frac{k}{f}({\vec x}\cdot{\vec p})}\Big)\bigg>,
\label{eq:Pv2}
\end{eqnarray}
where we introduced ${E}_0({\vec x}_a')$ and ${E}_0({\vec x}_b')$ to account for the fact that the extended source may be incoherent.
Expression (\ref{eq:Pv2}) is rather complex to be evaluated analytically in the general case. However, it can be evaluated for a point source on the optical axis, ${\vec x}'=0$. In this case, the integral in (\ref{eq:Pv2}) is easy to compute analytically. As we deal with the point source, we may model ${E}_0({\vec x}_a')={E}_0({\vec x}_b')=E_0\delta ({\vec x}')$, which takes care of the outer double integrals over $d^2 {\vec x}'$ in (\ref{eq:Pv2}). With this, we only need to integrate one inner double integral over $d^2 {\vec x}$. To do this, we introduce the relevant lens and detector coordinates as $(x,y)=\rho(\cos\phi,\sin\phi)$, $(p_x,p_y)=\rho_i(\cos\phi_i,\sin\phi_i)$, correspondingly, and, using these new variables, we compute the integral $d^2 {\vec x}$ in (\ref{eq:Pv2}):
{}
\begin{eqnarray}
\int_0^{2\pi}\hskip-7pt d\phi \int_0^{d/2}\hskip-7pt\rho d\rho
J_0\Big(k
\sqrt{\frac{2r_g}{\overline z}} \rho\Big) e^{-i\frac{k}{f}\rho \rho_i \cos(\phi-\phi_i)}&=&2\pi\int_0^{d/2}\hskip-7pt\rho d\rho
J_0\Big(k
\sqrt{\frac{2r_g}{\overline z}} \rho\Big) J_0\Big(\frac{k \rho_i }{f}\rho\Big)=\nonumber\\
&&\hskip-150pt =\,
\frac{\pi kd}{k^2(\frac{2r_g}{\overline z}-\frac{\rho_i^2}{f^2})} \Big\{
\sqrt{\frac{2r_g}{\overline z}}J_0\Big({\textstyle\frac{1}{2}}kd \frac{\rho_i}{f}\Big)J_1\Big({\textstyle\frac{1}{2}}kd \sqrt{\frac{2r_g}{\overline z}}\Big)- \frac{\rho_i}{f}J_0\Big({\textstyle\frac{1}{2}}kd\sqrt{\frac{2r_g}{\overline z}}\Big)J_1\Big({\textstyle\frac{1}{2}}kd \frac{\rho_i}{f}\Big)\Big\}.
\label{eq:amp-w-f2d}
\end{eqnarray}
This result allows us to express the intensity of the EM signal at the detector.
After averaging over time, and using (\ref{eq:amp-w-f2d}) from (\ref{eq:Pv2}), we get the following components of the Poynting vector (NB: we arrange factors, e.g., of $d^2$ in order to present the Poynting vector in the form usually found in the literature \cite{Born-Wolf:1999}), to ${\cal O}({\overline z}/z_0)$:
{}
\begin{eqnarray}
S_z(\rho_i)=\frac{c}{8\pi}\frac{\mu_0E^{\tt s2}_0}{z_0^2}\Big(\frac{kd^2}{8 f}\Big)^2\bigg(
\frac{2}{{\textstyle\frac{1}{2}}kd(\frac{2r_g}{\overline z}-\frac{\rho_i^2}{f^2})} \Big\{
\sqrt{\frac{2r_g}{\overline z}}J_0\Big({\textstyle\frac{1}{2}}kd \frac{\rho_i}{f}\Big)J_1\Big({\textstyle\frac{1}{2}}kd \sqrt{\frac{2r_g}{\overline z}}\Big)- \frac{\rho_i}{f}J_0\Big({\textstyle\frac{1}{2}}kd\sqrt{\frac{2r_g}{\overline z}}\Big)J_1\Big({\textstyle\frac{1}{2}}kd \frac{\rho_i}{f}\Big)\Big\}\bigg)^2.~
\label{eq:amp-w-f2dp}
\end{eqnarray}
Taking a limit of $r_g \rightarrow 0$ in (\ref{eq:amp-w-f2dp}), we obtain
the Poynting vector showing the classic Airy pattern:
{}
\begin{eqnarray}
S^0_z(\rho_i)
\big|_{r_g\rightarrow 0}=\frac{c}{8\pi}\frac{E^{\tt s2}_0}{(\overline z+z_0)^2}\Big(\frac{kd^2}{8f}\Big)^2\bigg(\frac{2J_1\big( {\textstyle\frac{1}{2}}kd \frac{\rho_i}{f}\big)}{{\textstyle\frac{1}{2}}kd\frac{\rho_i}{f}}\bigg)^2.
\label{eq:amp-w-f2d2p}
\end{eqnarray}
One may show that expression (\ref{eq:amp-w-f2dp}) is always finite. In fact, even
when ${2r_g}/{\overline z}-{\rho_i^2}/{f^2}=0$, (\ref{eq:amp-w-f2dp}) remains finite and describes the Einstein ring as it seen is at the detector at the position given as
{}
\begin{eqnarray}
\rho_i=f\sqrt{\frac{2r_g}{\overline z}}.
\label{eq:amp-det}
\end{eqnarray}
For the image of the Einstein ring to be resolved on the detector, the image size has to occupy several pixels (Fig.~\ref{fig:ER-plot}). Assuming that the pixel size is $\delta_p=10\,\mu$m, and the image occupies $n_0=10$ pixels, so that $\rho_i=n_0\delta_p$, such an imaging system would require a lens with the focal length of
{}
\begin{eqnarray}
f=n_0\delta_p\Big({\frac{\overline r}{2r_g}}\Big)^\frac{1}{2}=12.83\,{\rm m}\,\Big(\frac{n_0}{10}\Big)\Big(\frac{\delta_p}{10\,\mu{\rm m}}\Big)\Big({\frac{\overline z}{650\,{\rm AU}}}\Big)^\frac{1}{2}.
\label{eq:amp-de2t}
\end{eqnarray}
We can also derive the power at the Einstein ring deposited on the detector. For this, in (\ref{eq:amp-w-f2dp}), we take a limit $\rho_i\rightarrow f\sqrt{{2r_g}/{\overline z}}$ and, relying on (\ref{eq:BF}), obtain
{}
\begin{eqnarray}
S_z(\rho_i^{\tt ER})=
\frac{c}{8\pi}\frac{E^{\tt s2}_0}{(\overline z+z_0)^2}\Big(\frac{kd^2}{8 f}\Big)^2
\,\frac{8\lambda {\overline z}}{\pi^2 d^2}.
\label{eq:amp-w-f2dps}
\end{eqnarray}
Comparing this expression with (\ref{eq:amp-w-f2d2p}), we see that the light at the Einstein ring is amplified with the amplification factor given by the following expression:
{}
\begin{eqnarray}
\mu_{\tt det}=\frac{8\lambda {\overline z}}{\pi^2 d^2}=7.88\times 10^7\,\Big(\frac{\lambda}{1\,\mu{\rm m}}\Big)\Big(\frac{{\overline z}}{650\,{\rm AU}}\Big)\Big(\frac{1\,{\rm m}}{d}\Big)^2.
\label{eq:amp-w-f2dps1}
\end{eqnarray}
\begin{figure}
\begin{center}
\includegraphics[width=0.25\linewidth]{ER-plot-25cm}
\includegraphics[width=0.25\linewidth]{ER-plot-1m}
\includegraphics[width=0.25\linewidth]{ER-plot-2m}
\end{center}
\caption{\label{fig:ER-plot}Image formation in the sensor plane of an optical telescope with a 10~m focal length and an aperture of 25~cm (left), 1~m (center), vs. 2~m (right). The images depict the Einstein ring of a point source on the optical axis, as seen by the telescope. The image produced by a 25~cm aperture is dominated by the diffraction pattern of the telescope. Larger telescope apertures, though still diffraction-limited, offer sufficient resolution, e.g., for the use of a coronagraph to block out light from the Sun inside the Einstein ring.}
\end{figure}
We note that (\ref{eq:amp-w-f2dp}) is also finite when $|\rho_i|=0$ with the corresponding value computed as
{}
\begin{eqnarray}
S_z(0)=
\frac{c}{8\pi}\frac{E^{\tt s 2}_0}{(\overline z+z_0)^2}\Big(\frac{kd^2}{8 f}\Big)^2
\mu_0\bigg(\frac{2J_1\Big({\textstyle\frac{1}{2}}kd \sqrt{\frac{2r_g}{\overline z}}\Big)}{{\textstyle\frac{1}{2}}kd \sqrt{\frac{2r_g}{\overline z}}}\bigg)^2.
\label{eq:amp-w-f2dp7}
\end{eqnarray}
From (\ref{eq:amp-w-f2dp7}), using (\ref{eq:BF}), for $r_g\not=0$, the amplification factor at the center of the detector is evaluated to be
{}
\begin{eqnarray}
\mu^0_{\tt det}=\mu_0\bigg(\frac{2J_1\Big({\textstyle\frac{1}{2}}kd \sqrt{\frac{2r_g}{\overline z}}\Big)}{{\textstyle\frac{1}{2}}kd \sqrt{\frac{2r_g}{\overline z}}}\bigg)^2=\frac{16\lambda^2 \overline z}{\pi^2 d^3}\sqrt{\frac{\overline z}{2r_g}}\sin^2\Big[\frac{\pi d}{\lambda} \sqrt{\frac{2r_g}{\overline z}}-{\frac{\pi}{4}}\Big]=2.02\times 10^7\,\Big(\frac{\lambda}{1\,\mu{\rm m}}\Big)^2\Big(\frac{{\overline z}}{650\,{\rm AU}}\Big)^{\frac{3}{2}}\Big(\frac{1\,{\rm m}}{d}\Big)^3.
\label{eq:amp-w-f2dps2}
\end{eqnarray}
For $r_g\rightarrow 0$, the amplification factor $\mu^0_{\tt det}$ reduces to $\mu^0_{\tt det}=1$ and the result (\ref{eq:amp-w-f2dp7}) is equivalent to (\ref{eq:amp-w-f2d2p}) for $\rho_i=0$.
\section{Discussion and Conclusions}
\label{sec:disc}
Our main motivation for this paper was to study image formation by the SGL in the case of an extended source located at a large, but finite distance from the Sun. This is the situation that we encounter when considering the use of the SGL for multipixel imaging and spatially-resolved spectroscopy of exoplanets. It is also relevant to many scenarios that involve imaging of extended sources with microlensing techniques. Yet surprisingly, no theoretical description of this scenario could be found in the literature that we surveyed, especially from the wave-optical perspective.
We began by recalling results of our investigations were we studied the propagation of plane EM waves on the background of a gravitational monopole. We relied on a Mie theory that we developed \cite{Turyshev-Toth:2017} to account for the refractive properties of the gravitational field in the vicinity of the Sun. The resulting EM field is described in full by the Debye potential (\ref{eq:Pi-s_a+0}), which accounts for the fully-absorbing boundary conditions introduced at the solar surface.
A key step in the new derivation is the development of the radial function (\ref{eq:Fass*}) that accounts for the phase shift acquired along the path from the source to the image plane. We obtained such a function by using the WKB approximation (\ref{eq:R_solWKB+=_bar-imp}), which is typically used to solve similar problems in nuclear scattering. Using the approximate solution to the radial function, we developed the corresponding Debye potential and show that no EM field exists directly behind the Sun in the shadow region.
Using the Debye potential (\ref{eq:Pi-s_a+0}) and the radial function (\ref{eq:Fass*}) in the region of geometric optics (see Sec.~\ref{sec:go-em-outside}) yields a solution for the EM wave propagating in this region. The EM field in the region of geometric optics can be described using incident and scattered waves, given by (\ref{eq:DB-t-pl=p10})--(\ref{eq:DB-t-pl=p20}) and (\ref{eq:scat-th-tot}), correspondingly. These solutions extend our previous results \cite{Turyshev-Toth:2017,Turyshev-Toth:2019} to the case of a source at finite distance. A source that is positioned at a finite distance from the Sun with coordinates $({\vec b},r_0)$ can be dealt with by simply introducing a rotation by the angle $\beta=b/r_0$, in the plane defined by $\vec{b}$, thus defining an optical axis of the SGL for this particular point source. This optical axis connects the point source and the center of the Sun, extending towards the image plane. Rays of light envelop the entire solar circumference and propagate toward this optical axis, ultimately reaching it in the interference region (see Sec.~\ref{sec:IF-region}). An observer at this location would see a perfect Einstein ring formed around the Sun. A deviation from the optical axis results in breaking the ring into arcs of uneven brightness.
The most practically interesting solution was obtained for the EM field in the interference region (\ref{eq:DB-sol-rho}). In the case of a source at finite distance, for any given impact parameter, the focal point of the SGL, which is nominally given as $b^2/(2r_g)$, is shifted further out from the Sun by the extra distance $(b^2/2r_g)^2/z_0$ that must be accounted for in any SGL mission design and simulations.
We extend our approach to imaging of extended sources (see Sec.~\ref{sec:image_form}). For this we represent the surface of an extended source as a collection of point sources, each selecting its own optical axis while imaged via the SGL. We show that for each point in the source plane, there exists an optical axis that points to a specific point on the image plane, where the intensity of light from that point source is maximal. As the SGL's point-spread function is much broader than the classical Airy pattern, there will be light deposited at widely separated points on the image plane, albeit at weaker levels. A telescope of modest size, positioned in the image plane would see an Einstein ring containing light from the entire extended body. At each location on the image plane, such a ring is composed of the complete Einstein ring produced by light from the source location exactly opposite to the image plane location with respect to the center of the Sun, and partial arcs from all other points in the source plane. These rings and arcs together determine the power density of the signal received by a telescope in the image plane and, ultimately, the image formed on the telescope's optical sensor.
Our treatment leads to an expression for the surface power density in the image plane (\ref{eq:power_dens}) that describes the total power received by an imaging telescope (\ref{eq:power_rec2}). The expressions obtained in this paper generalize previously known results to the case of a resolved extended source at a finite distance from the Sun.
The analysis can be extended to incorporate the optics of a telescope that would be positioned at the focal plane to collect light from the distant source. We presented an analytic derivation of the Einstein ring that appears due to a distant point source. This result is directly relevant to studying the required aperture of the telescope and the coronagraph that will be used to block out direct sunlight and much of the light from the solar corona, without obscuring light from the Einstein ring.
The next step is to analyze the signals received from realistic sources in our stellar neighborhood. This work is ongoing and results, when available, will be reported elsewhere.
\begin{acknowledgments}
This work in part was performed at the Jet Propulsion Laboratory, California Institute of Technology, under a contract with the National Aeronautics and Space Administration.
\end{acknowledgments}
|
2,877,628,088,995 | arxiv | \section{Introduction}
The ionosphere represents the region of the Earth's upper atmosphere where significant quantities of ions and electrons occur. Rocket launches can result in significant perturbations to this plasma, including chemical depletion of the \ch{O^+} dominated F-region by rocket exhaust gases. Such ionospheric depletions may impact high frequency radio communications or Global Navigation Satellite System (GNSS) signals. These depletions also provide a test to help validate numerical models of the ionosphere-thermosphere system.
A range of measurement techniques have been used to observe the depletion of the ionosphere by rocket exhaust since the dawn of the space age. The detection of an ionospheric hole was first reported by Booker \cite{booker1961} using ionosonde measurements following a Vanguard II missile launch in February 1959. Subsequently, such depletions have been studied using geostationary satellite signal Faraday rotation \cite{mendillo1975}, airglow emission \cite{mendillo1982}, incoherent scatter radar \cite{wand1984}, satellite Langmuir probe \cite{park2016}, and GNSS total electron content (TEC) \cite{furuya2008,mendillo2008,chou2018} measurements. Chou et al. \cite{chou2018} reported GPS observations of a large-scale ionospheric depletion resulting from the launch of FORMOSAT-5.
Analytical and numerical techniques have previously been applied to model these ionospheric depletions. Bernhardt, Park, and Banks \cite{bernhardt1975} derived an analytical expression for diffusion of rocket exhaust gases from the continuity equation and considered their injection into a numerical magnetic flux tube model. Subsequently, Bernhardt \cite{bernhardt1979} developed a numerical model for this process incorporating chemical reactions, diffusion of multiple species, and neutral winds. Mendillo \cite{mendillo1978} analytically derived expressions for the diffusion, chemical depletion, and recovery processes assuming the first two processes proceeded much faster than the third and neglecting changes in the ambient atmosphere with time.
In this work, a numerical model is adapted to simulate ionospheric depletions due to rocket launches and its output is compared to measurements for two launches. An overview of the launches considered is provided in section~\ref{sec_F9_launches}. The numerical model used and the ways in which it is adapted to simulate ionospheric depletions are described in section~\ref{sec_modelling}. Results of the numerical simulation are presented in section~\ref{sec_results} and compared with observational data in section~\ref{sec_observation}. These results are further discussed and possible extensions to this work identified in section~\ref{sec_discussion}.
\section{Falcon 9 launches} \label{sec_F9_launches}
This study investigated depletion of the ionosphere due to launches of the Jason-3 and FORMOSAT-5 satellites. These launches took place at 18:42:18 UT 17 January 2016 and at 18:51:00 UT 24 August 2017 respectively from Vandenberg Air Force Base (located at $34^{\circ}44' \, \mathrm{N}$, $120^{\circ}34' \, \mathrm{W}$). In both cases SpaceX Falcon 9 launch vehicles were used (v1.1 in the former case and v1.2/Full Thrust in the latter). The depletions were attributed to the second stage Merlin 1D Vacuum engines, which operated at F-region heights using liquid oxygen (LOX) and kerosene (RP-1) propellants. Specifications for the Falcon 9 second stage are listed in Table~\ref{tab_F9_data}. Complete combustion of LOX and RP-1 produces \ch{H_2O} and \ch{CO_2}, while exhaust from incomplete combustion also contains \ch{H_2} and \ch{CO}.
\begin{table}
\caption{Falcon 9 second stage data from versions of the SpaceX website (https://www.spacex.com/falcon9) archived on 29 November 2013 and 9 December 2015.}
\label{tab_F9_data}
\begin{tabular}{@{}lcc}
\hline
& v1.1 & v1.2 \\
\hline
Burn time, $\Delta T$ & $375 \, \mathrm{s}$ & $397 \, \mathrm{s}$ \\
Specific impulse, $I_{sp}$ & $340 \, \mathrm{s}$ & $348 \, \mathrm{s}$ \\
Thrust, $T$ & $801 \, \mathrm{kN}$ & $934 \, \mathrm{kN}$ \\
\hline
\end{tabular}
\end{table}
Speed and altitude (Fig.~\ref{fig_altitude}) were determined as functions of time based on launch videos released online by SpaceX, as in Chou et al. \cite{chou2018}. These data were used along with the final orbital inclinations to infer trajectories for both launches, which are plotted in Fig.~\ref{fig_trajectory}. The light FORMOSAT-5 payload enabled the launch vehicle to take an unusually steep trajectory through the F-region of the ionosphere to directly insert the satellite into an orbit at an altitude of $720 \, \mathrm{km}$. By contrast, the more conventional Jason-3 launch inserted the satellite into an initial orbit at $200 \, \mathrm{km}$, remaining below the predicted F-region peak.
\begin{figure}
\includegraphics[width=\linewidth]{Altitude_vs_time_after_launch_ionosphere.pdf}
\caption{Launch vehicle altitude as a function of time after launch (black lines) for Jason-3 (solid) and FORMOSAT-5 (dashed). The MSIS altitude profiles for electron number density, $n_e$, above the launch site are plotted alongside (grey lines), }
\label{fig_altitude}
\end{figure}
\begin{figure}
\includegraphics[width=\linewidth]{ground_track_times.pdf}
\caption{Inferred Jason-3 (solid line) and FORMOSAT-5 (dashed line) ground tracks.}
\label{fig_trajectory}
\end{figure}
\section{Modelling} \label{sec_modelling}
\subsection{GITM}
The Global Ionosphere-Thermosphere Model (GITM) is a general circulation model of the coupled charged and neutral components of the upper atmosphere \cite{ridley2006}. This model uses a finite difference method to solve a set of coupled fluid equations representing a set of ion and neutral species on a 3-dimensional spherical grid which can be stretched in latitude and altitude. Physical and chemical processes in the ionosphere and thermosphere are represented by realistic source terms in the continuity, momentum, and energy equations. Density is explicitly solved for individual major neutral and ion species. Simulations may encompass either the whole globe or a particular region, wherein boundary conditions specify velocities and density and temperature gradients.
A modified version of GITM was developed in this work which incorporates chemical processes, diffusion coefficients, and source terms associated with the rocket exhaust plume. GITM was run in the regional mode to ensure that resolution was sufficient to resolve the features of ionospheric depletions. Boundaries were set at $15^{\circ} \mathrm{N}$, $55^{\circ} \mathrm{N}$, $150^{\circ} \mathrm{W}$, and $90^{\circ} \mathrm{W}$ and an approximate resolution of $0.4^{\circ} \times 0.4^{\circ}$. The model was initialised with input from the empirical International Reference Ionosphere (IRI) and Mass-Spectrometer-Incoherent-Scatter (MSIS) models \cite{bilitza1990,picone2002}.
\subsection{Chemical processes}
In the F-region, \ch{O^+} ions undergo charge exchange reactions with rocket exhaust molecules which proceed ${\sim}2$ to ${\sim}3$ orders of magnitude faster than the corresponding reactions with the \ch{N_2} and \ch{O_2} neutral molecules present in the undisturbed thermosphere \cite{mendillo1981}. The most important reactions are:
\begin{equation}
\mathrm{O^{+} + H_{2}O \to H_{2}O^{+} + O,} \quad \kappa_1 = 2.42 \times 10^{-15} \, \mathrm{m^3.s^{-1}}, \label{H2O_CE}
\end{equation}
\begin{equation}
\mathrm{O^{+} + H_{2} \to OH^{+} + H,} \quad \kappa_2 = 1.62 \times 10^{-15} \, \mathrm{m^3.s^{-1}},
\end{equation}
\begin{equation}
\mathrm{O^{+} + CO_{2}} \begin{matrix}
\mathrm{\overset{0.5}{\rightarrow} O_{2}^{+} + CO} \\
\mathrm{\overset{0.5}{\rightarrow} CO_{2}^{+} + O}
\end{matrix} \mathrm{,} \quad \kappa_3 = 1.08 \times 10^{-15} \, \mathrm{m^3.s^{-1}}, \label{CO2_CE}
\end{equation}
where the reaction rate coefficients $\kappa_1$, $\kappa_2$, and $\kappa_3$ are derived from the measurements summarised by Anicich \cite{anicich2003}, weighted based on the uncertainties reported there. The branching ratios in eq.~\ref{CO2_CE} are chosen based on the results of Lindinger et al. \cite{lindinger1974}, who found that the dominant product ion changed from \ch{O_2^+} to \ch{CO_{2}^{+}} as temperature was increased from $300 \, \mathrm{K}$ to $900 \, \mathrm{K}$. Note that \ch{CO} is taken to be inert with respect to charge exchange with \ch{O^+} \cite{smith1978}.
The resulting molecular ions undergo dissociative recombination with electrons:
\begin{equation}
\mathrm{H_{2}O^{+} + e^{-}} \begin{array}{ll}
\mathrm{\overset{0.22}{\rightarrow} OH + H } \\
\mathrm{\overset{0.1}{\rightarrow} O + H_2 } \\
\mathrm{\overset{0.68}{\rightarrow} O + H + H } \label{H2O+_DR}
\end{array}
\mathrm{,} \quad \alpha_1 = 3.0 \times 10^{-13} \, \mathrm{m^3.s^{-1}} ,
\end{equation}
\begin{equation}
\mathrm{OH^{+} + e^{-} \to H + O,} \quad \alpha_2 = 1.0 \times 10^{-13} \, \mathrm{m^3.s^{-1}} ,
\end{equation}
\begin{equation}
\mathrm{O_{2}^{+} + e^{-} \to O + O,} \quad \alpha_3 = 2.0 \times 10^{-13} \, \mathrm{m^3.s^{-1}} , \label{O2+_DR}
\end{equation}
where the reaction rate coefficients $\alpha_1$, $\alpha_2$, and $\alpha_3$ are those provided by Mendillo \cite{mendillo1981}. The branching ratios in eq.~\ref{H2O+_DR} are taken from the work of Vejby-Christensen et al. \cite{vejbychristensen1997}.
The above reactions were implemented in the modified version of GITM. Reaction rate coefficients in eq.~\ref{H2O_CE}-\ref{O2+_DR} indicate that the dissociative recombination reactions occur ${\sim}2$ orders of magnitude faster than the charge exchange reactions and the former reactions are therefore treated as instantaneous for the purposes of this study. The enthalpies of reaction for the charge exchange and recombination processes were computed based on the ionisation threshold potentials and enthalpies of formation tabulated by Schunk \& Nagy \cite{schunk2009}. Thermal energy from exothermic reactions was partitioned between reaction products inversely proportional to their masses. Neutral reactants in eq.~\ref{H2O_CE}-\ref{CO2_CE} were implemented as major species. Additionally, those neutral products in eq.~\ref{CO2_CE}-\ref{O2+_DR} which were not previously considered in GITM were implemented as minor species. These changes are summarised in Table~\ref{tab_major_minor_species}.
\begin{table}
\caption{Changes to major and minor neutral species in GITM.}
\label{tab_major_minor_species}
\begin{tabular}{@{}lc}
Major (old) & \ch{O(^3P)}, \ch{O_2}, \ch{N_2}, \ch{N(^4S)}, \ch{NO}, and \ch{He} \\
Minor (old) & \ch{N(^2D)}, \ch{N(^2P)}, \ch{H}, \ch{CO_2}, and \ch{O(^1D)} \\
Major (new) & \ch{O(^3P)}, \ch{O_2}, \ch{N_2}, \ch{N(^4S)}, \ch{NO}, \ch{He}, \ch{H_2O}, \ch{H_2}, and \ch{CO_2} \\
Minor (new) & \ch{N(^2D)}, \ch{N(^2P)}, \ch{H}, \ch{O(^1D)}, \ch{CO}, and \ch{OH} \\
\end{tabular}
\end{table}
\subsection{Diffusion coefficients}
Where they were not previously implemented in GITM, binary gas diffusion coefficients were approximated using the empirical equation and data given by Fuller, Schettler, \& Giddings \cite{fuller1966} in the modified version. According to this equation
\begin{equation}
D_{ab} = \frac{10^{-3} T^{1.75} \left ( \frac{1}{M_a} + \frac{1}{M_b} \right )}{P \left [ \left ( \Sigma V_a \right )^{\frac{1}{3}} + \left ( \Sigma V_b \right )^{\frac{1}{3}} \right ]} ,
\end{equation}
where $D_{ab}$ is the binary diffusion coefficient, $P$ is total pressure (in $\mathrm{atm}$), $M_i$ is molecular weight (in $\mathrm{g.mol}^{-1}$), $T$ is temperature (in $\mathrm{K}$), and $\Sigma V_i$ is diffusion volume (based on tabulated data reflecting structural features of diffusing species $i$). The diffusion coefficient for species $a$ in the gas mixture was estimated using
\begin{equation}
D_{0a} = \frac{1-Y_a}{\sum_{i \ne a} Y_i / D_{ai}} ,
\end{equation}
where $Y_i$ is the molar fraction of species $i$.
\subsection{Rocket exhaust plume source term}
Initial diffusion of the rocket exhaust plume in the modified version of GITM was determined analytically using the theory of Bernhardt \cite{bernhardt1979}. This treatment neglected the self-continuum (fluid) and collisionless (free-streaming) initial stages of the exhaust plume expansion \cite{bernhardt1979b}. The plume length scale, $L$, and mean free path, $\lambda$, were assumed to be small compared to the dimensions of the problem. These quantities are plotted for the FORMOSAT-5 launch in Fig.~\ref{fig_length_scale}, based on the hard sphere approximation and gas dynamic theory of rocket plumes developed by Jarvinen et al. \cite{jarvinen1966} respectively. It is noted that the latter theory assumes continuum flow and a homogenous atmosphere, and therefore that $L \ll H$ and $L \ll \lambda$, where $H$ is the scale height of the background atmosphere. Thus, the rocket exhaust species were considered to expand diffusively from rest at the point where they were released in accordance with the approximation given by Bernhardt \cite{bernhardt1979};
\begin{multline}
\sigma_i \left ( x , y , z ; t \right ) = \frac{\dot{N}_{0i}}{\left ( 4 \pi D_{0i} \right )^{\frac{3}{2}}} \exp \left [ -z \left ( \frac{3}{4H} + \frac{1}{2H_i} \right ) \right. \\
\left. -\frac{H^2 \left ( 1 - \exp \left [ - z / \left ( 2 H \right ) \right ] \right )}{D_{0i}t} - \frac{\left ( x^2 + y^2 \right ) \exp \left [ -z/ \left ( 2 H \right ) \right ]}{4D_{0i}t} \right. \\
\left. -\left ( \frac{1}{H} -\frac{1}{H_i} \right )^2 \frac{D_{0i} t \exp \left [ z / \left ( 2H \right ) \right ]}{4} \right ] , \label{gas_diffusion_point}
\end{multline}
where $\sigma_i$ is the source term, $\dot{N}_{0i}$ is the molecular flow rate, and $D_{0i}$ is the diffusion coefficient for species $i$. Here we define $H = k T / \left ( m g \right )$ as the scale height of the background atmosphere and $H_i = k T / \left ( m_i g \right )$ as the scale height of species $i$, where $k$ is the Boltzmann constant, $T$ is the temperature, $m$ is the mean molecular mass of the background atmosphere, $m_i$ is the molecular mass of species $i$, and $g$ is gravitational acceleration. In the implementation of eq.~\ref{gas_diffusion_point} in GITM, the $x$, $y$, and $z$ coordinates are approximated by the zonal, meridional, and vertical displacements with respect to the point of release respectively. Thus, $x \approx r_0 \cos \theta_0 \left ( \phi - \phi_0 \right )$, $y \approx r_0 \left ( \theta - \theta_0 \right )$, and $z = r - r_0$, where $\theta_0$, $\phi_0$, and $r_0$ are the spherical coordinates of the point of release. The diffusion time, $t$, represents the difference between the time between the the exhaust being released and the corresponding addition of the source term in the numerical model. A sufficiently large value of $t$ was required to avoid failure of the GITM numerical solver, as excessive gradients in rocket exhaust species led to the occurence of negative densities. Chemical reactions during that time period were neglected in this treatment.
\begin{figure}
\includegraphics[width=\linewidth]{Altitude_vs_L_ionosphere_hole.pdf}
\caption{Length scales as functions of altitude for the FORMOSAT-5 launch. Plume length scale, $L$, (solid line), scale height, $H$, (dashed line), and mean free path, (dash-dotted line) are plotted.}
\label{fig_length_scale}
\end{figure}
Estimates of $\dot{N}_{0i}$ for each species were obtained based on the manufacturer data stated in Table~\ref{tab_F9_data}. Simulations were run for two different methods of determining the chemical composition of the exhaust; firstly assuming complete combustion and secondly using Chemical Equilibrium with Applications (CEA) software \cite{gordon1996}. In the latter case a combustion pressure of $6.895 \, \mathrm{MPa}$, combustion temperature of $3670 \, \mathrm{K}$, nozzle expansion ratio of $165$, and oxidiser to fuel ratio of $2.56$ were chosen. Conditions at the throat and exit were considered, representing the bounding cases for combustion. The resulting molecular flow rates are summarised in Table~\ref{tab_mfr_data}. It was assumed that the heating of neutral gasses in the thermosphere as the exhaust gasses as they came to rest was equal to the kinetic energy of the latter as they exited the rocket (in a reference frame stationary with respect to Earth), such that
\begin{equation}
\dot{Q} = \dot{N}_{0i} \left ( v - I_{sp} g_0 \right )^2 ,
\end{equation}
where $\dot{Q}$ is the heating rate, $v$ is the speed of the rocket, $I_{sp}$ is the specific impulse of the rocket, and $g_0$ is the reference gravitational acceleration. The source was added until $480 \, \mathrm{s}$ since launch. This preceded the actual termination of the second stage burn by approximately $60 \, \mathrm{s}$ in each case, though in the FORMOSAT-5 launch case the rocket was above the top of the simulation domain by time.
\begin{table*}
\begin{minipage}{130mm}
\caption{Molecular flow rates, $\dot{N}_{0i}$, for the Falcon 9 second stage.}
\label{tab_mfr_data}
\begin{tabular}{@{}lcccc}
\hline
& \ch{H_2O} & \ch{H_2} & \ch{CO_2} & \ch{CO} \\
\hline
Complete combustion, v1.1 & $2.47 \times 10^{27} \, \mathrm{s}^{-1}$ & $0 \, \mathrm{s}^{-1}$ & $2.28 \times 10^{27} \, \mathrm{s}^{-1}$ & $0 \, \mathrm{s}^{-1}$ \\
Complete combustion, v1.2 & $2.81 \times 10^{27} \, \mathrm{s}^{-1}$ & $0 \, \mathrm{s}^{-1}$ & $2.59 \times 10^{27} \, \mathrm{s}^{-1}$ & $0 \, \mathrm{s}^{-1}$ \\
CEA throat, v1.2 & $2.09 \times 10^{27} \, \mathrm{s}^{-1}$ & $5.07 \times 10^{26} \, \mathrm{s}^{-1}$ & $9.69 \times 10^{26} \, \mathrm{s}^{-1}$ & $1.94 \times 10^{27} \, \mathrm{s}^{-1}$ \\
CEA exit, v1.2 & $2.21 \times 10^{27} \, \mathrm{s}^{-1}$ & $1.02 \times 10^{27} \, \mathrm{s}^{-1}$ & $1.89 \times 10^{27} \, \mathrm{s}^{-1}$ & $1.43 \times 10^{27} \, \mathrm{s}^{-1}$ \\
\hline
\end{tabular}
\end{minipage}
\end{table*}
\section{Results} \label{sec_results}
Ionospheric depletions due to the Jason-3 and FORMOSAT-5 launches were modelled using GITM assuming complete combustion of the propellant. Evolution of the depletions over the $3$~hours following launch in each simulation are shown in the TEC maps in Fig.~\ref{fig_GITM_TEC_map_data}. The two launches coincided with periods of relatively low geomagnetic activity ($\mathrm{Kp} = 1-$ and $0+$ respectively) and solar activity ($\mathrm{F}10.7 = 101 \, \mathrm{ SFU}$ and $79 \, \mathrm{ SFU}$ respectively). Nevertheless, substantially higher background TEC values were computed for the Jason-3 launch ($15.5 \, \mathrm{ TECU}$ at Vandenberg Air Force Base) than for the FORMOSAT-5 launch ($11.8 \, \mathrm{ TECU}$).
\begin{figure*}
\centering
\subfloat[Jason-3]{
\includegraphics[width=\linewidth]{GITM_CC_TEC_J3.pdf}
}
\hspace{0mm}
\subfloat[FORMOSAT-5]{
\includegraphics[width=\linewidth]{GITM_CC_TEC_F5_For_Real.pdf}
}
\caption{TEC maps from GITM simulation of the ionospheric depletion following rocket launches assuming complete combustion.}
\label{fig_GITM_TEC_map_data}
\end{figure*}
In the Jason-3 launch case, the simulation indicated that an elongated ionospheric depletion formed rapidly, aligned with the launch vehicle trajectory. The difference in TEC between a simulation in which no rocket exhaust gases were added to the atmosphere and the simulation including these gases was taken to represent the ionospheric depletion and is shown in Fig.~\ref{fig_GPS_TEC_diff_J3_data}. The horizontal spatial extent of the depletion is represented by $A_d$, the area of the Earth's surface over which this difference exceeds $1 \, \mathrm{ TECU}$, which is plotted as a function of time in Fig.~\ref{fig_hole_area}. A maximum depletion of $5.26 \, \mathrm{ TECU}$ was recorded at 19:50 UT when $A_d = 1.97 \times 10^5 \, \mathrm{km}^2$. Over subsequent hours, the magnitude and spatial extent of the depletion decreased and it became more circular. Its centre moved northwestward, towards the launch site. The depletion in TEC was greater on the side of the depletion nearer to the launch site, as the slower speed of the launch vehicle earlier in its trajectory produced greater concentrations of exhaust gases while flight was approximately level from $300 \, \mathrm{s}$ after launch onwards.
\begin{figure}
\includegraphics[width=\linewidth]{GITM_TEC_Difference_J3.png}
\caption{TEC depletion map for 19:50 UT 24 August 2017, the time when the maximum TEC depletion was recorded, from GITM simulations of the launch of Jason-3 assuming complete combustion. Contours of $\Delta \mathrm{TEC} = 1 \mathrm{ TECU}$ are also plotted for 19:40 UT (solid line), 20:40 UT (dashed line), 21:40 UT (dotted line), and 22:40 UT (dash-dotted line).}
\label{fig_GPS_TEC_diff_J3_data}
\end{figure}
\begin{figure}
\includegraphics[width=\linewidth]{hole_area.pdf}
\caption{Evolution of the horizontal spatial extent of the ionosphere depletion, $A_d$, in simulations of the Jason-3 (dashed line) and FORMOSAT-5 (dash-dotted line) launches.}
\label{fig_hole_area}
\end{figure}
By contrast, in the FORMOSAT-5 launch case, a large, persistent, and nearly circular ionospheric depletion formed. The decrease in TEC in this case relative to that without addition of rocket exhaust gases is shown in Fig.~\ref{fig_GPS_TEC_diff_data}, with a maximum depletion of $11.5 \, \mathrm{ TECU}$ which was recorded at 21:40 UT. However, the maximum spatial extent occurred after further diffusion at 23:20 UT, when $A_d = 7.46 \times 10^5 \, \mathrm{km}^2$. Despite the release of similar quantities of rocket exhaust as in the previous case, much larger depletions occurred across a much larger geographical region in this instance. The depletion initially moved westward for several hours before coming to a halt and beginning to move back eastward. This movement was attributed to transport of the exhaust by neutral winds. A trail of reduced TEC resulted as the ionosphere, the upper regions of which are largely constrained to move along magnetic field lines, recovered. The ionospheric depletion continued to exist following sunset (which occurred at 02:38 UT August 25 at Vandenberg Air Force Base), though absolute depletion values decreased at night.
\begin{figure}
\includegraphics[width=\linewidth]{GITM_TEC_Difference.png}
\caption{TEC depletion map for 21:40 UT 24 August 2017, the time when the maximum TEC depletion was recorded, from GITM simulations of the launch of FORMOSAT-5 assuming complete combustion. Contours of $\Delta \mathrm{TEC} = 5 \mathrm{ TECU}$ are also plotted for 20:50 UT (solid line), 22:50 UT (dashed line), 00:50 UT (dotted line), and 02:50 UT (dash-dotted line).}
\label{fig_GPS_TEC_diff_data}
\end{figure}
The altitude distributions of changes in the ionosphere and thermosphere due to the rocket launches provide insight into the difference between the two cases above. These distributions are plotted as functions of time for the Jason-3 and FORMOSAT-5 launch cases in Fig.~\ref{fig_J3_altitude_profiles} and Fig.~\ref{fig_F5_altitude_profiles} respectively. We adopt the convention that positive $\Delta N_e$ represents a decrease in the number of electrons in the simulation domain due to the introduction of rocket exhaust, whereas positive $\Delta N_i$ represents an increase in the number of any other species $i$. In the Jason-3 case, the increases in \ch{CO_2} and \ch{H_2O} number were initially concentrated around $200 \, \mathrm{km}$ altitude. The centres of their non-hydrostatic altitude distributions fell and they broadened through diffusion. Overall, increases in \ch{CO_2} and \ch{H_2O} due to rocket exhaust decreased with time as a result of the charge exchange and dissociative recombination reactions in eq.~\ref{H2O_CE}-\ref{O2+_DR}. In the FORMOSAT-5 case, the initial increases in \ch{CO_2} and \ch{H_2O} number occurred across a broader range of altitudes. The centres of their altitude distributions fell over the course of several hours, with losses occurring at the lower boundary of the simulation. In both cases artificial changes in the distributions were found near the lower boundary.
\begin{figure}
\includegraphics[width=\linewidth]{altitude_profiles_J3_CC.png}
\caption{Distribution of differences in (a) $N_e$, (b) $N_{\mathrm{CO_2}}$, and (c) $N_{\mathrm{H_2O}}$ as a function of altitude and time due to the introduction of rocket exhaust in the GITM simulation of the Jason-3 launch.}
\label{fig_J3_altitude_profiles}
\end{figure}
\begin{figure}
\includegraphics[width=\linewidth]{altitude_profiles_F5_CC.png}
\caption{Distribution of differences in (a) $N_e$, (b) $N_{\mathrm{CO_2}}$, and (c) $N_{\mathrm{H_2O}}$ as a function of altitude and time due to the introduction of rocket exhaust in the GITM simulation of the FORMOSAT-5 launch.}
\label{fig_F5_altitude_profiles}
\end{figure}
The altitude distributions of increases in \ch{CO_2} and \ch{H_2O} number due to the rocket exhaust discussed above help explain the distribution of the reduction in the electron number. In the Jason-3 case the decrease in electron number was centred around $230 \, \mathrm{km}$ altitude, attaining its greatest value about $1$~hour after launch. These observations reflected the vertical diffusion of rocket exhaust upward into the \ch{O^+} dominated F-region with which it could react. Thereafter, the fall of the rocket exhaust gases and their consumption through charge exchange and dissociative recombination reactions resulted in a reduction in the electron loss rate, allowing the ionosphere to recover. The decrease in electron number in the FORMOSAT-5 case was distributed over a broad range of altitudes reflecting the distribution of increases in \ch{CO_2} and \ch{H_2O}. Even after the exhaust gasses had fallen to lower altitudes, the depletion remained at higher altitudes due to the combination of low electron production rates and the importance of diffusion from lower altitudes.
Total changes in the number of electrons and each of the rocket exhaust species considered in the simulation are plotted in Fig.~\ref{fig_Number_data}. As seen in Fig.~\ref{fig_J3_altitude_profiles} and Fig.~\ref{fig_F5_altitude_profiles}, reductions in electron number density are largely confined to altitudes greater than $200 \, \mathrm{km}$, therefore only particles above this altitude are considered. Differences in \ch{CO_2} and \ch{H_2O} declined rapidly in the Jason-3 launch case, shown in Fig~\ref{subfig_J3_CC_Number_data}, and more slowly in the Formosat-5 launch case, shown in Fig~\ref{subfig_F5_CC_Number_data}, reflecting the combined effects of diffusion and gravity. Charge exchange and dissociative recombination reactions led to small differences in \ch{CO} and \ch{H_2}, even though these were not present in the exhaust. The total number of electrons lost in this region due to the rocket exhaust was much higher in the FORMOSAT-5 launch case than the Jason-3 launch case (a factor of $\approx 6.8$ greater maximum $\Delta N_e$).
\begin{figure*}
\centering
\subfloat[Jason-3 complete combustion]{
\includegraphics[scale=.5]{altitude_profiles_combined_J3_CC_200km+bige.pdf}
\label{subfig_J3_CC_Number_data}
}
\hspace{0mm}
\subfloat[FORMOSAT-5 complete combustion]{
\includegraphics[scale=.5]{species_totals_combined_F5_CC_200km+bige.pdf}
\label{subfig_F5_CC_Number_data}
}
\hspace{0mm}
\subfloat[FORMOSAT-5 CEA throat]{
\includegraphics[scale=.5]{altitude_profiles_combined_F5_throat_200km+bige.pdf}
\label{subfig_F5_throat_Number_data}
}
\hspace{0mm}
\subfloat[FORMOSAT-5 CEA exit]{
\includegraphics[scale=.5]{altitude_profiles_combined_F5_exit_200km+bige.pdf}
\label{subfig_F5_exit_Number_data}
}
\caption{Differences in $N_e$ (solid line, multiplied by a factor of $10$), $N_{\mathrm{CO_2}}$ (dashed line), $N_{\mathrm{CO}}$ (dotted line), $N_{\mathrm{H_2O}}$ (dash-dotted line), and $N_{\mathrm{H_2}}$ (thin line)} above $200 \, \mathrm{ km}$ altitude as a function of time due to the introduction of rocket exhaust.
\label{fig_Number_data}
\end{figure*}
Simulations of the FORMOSAT-5 launch case were also conducted using the rocket exhaust composition computed by CEA at the throat and at the exit. The resulting changes in the numbers of different species are shown in Fig.~\ref{subfig_F5_throat_Number_data} and Fig~\ref{subfig_F5_exit_Number_data}. In each case the number of \ch{CO_2} and \ch{H_2O} molecules added by the rocket were significantly reduced, while the number of \ch{CO} and \ch{H_2} molecules (previously only present as products of reactions between the exhaust gasses and ionosphere) greatly increased. However, it was found that the overall decrease in the number of electrons was not sensitive to these changes in exhaust composition, with less than $6\%$ change in its greatest value.
\section{Observational data} \label{sec_observation}
TEC data maps based on GNSS observations following the FORMOSAT-5 launch show ionospheric depletions similar to those seen in the GITM simulation. Maps with $1^{\circ} \times 1^{\circ}$ resolution, which were derived from MAPGPS software \cite{rideout2006}, are plotted in Fig.~\ref{fig_MAPGPS_TEC_data}. While such measurements contain D-region and plasmasphere contributions to TEC that are excluded from the GITM simulation, these are typically small during the daytime \cite{yizengaw2008}. A region of reduced TEC was observed to form near the launch site and subsequently move westward, as seen in the simulation. Moreover, the depletion was circular and had a maximum spatial extent of approximately $10^{\circ}$ in latitude and $10^{\circ}$ in longitude. However, contrary to the simulation, there is little evidence of the depletion after 21:30 UT. Moreover, the GNSS data indicated a maximum TEC decrease of approximately $4 \, \mathrm{ TECU}$ compared with the approximately $10.8 \, \mathrm{ TECU}$ decrease found in the simulation.
\begin{figure*}
\centering
\subfloat[Jason-3]{
\includegraphics[width=\linewidth]{MAPGPS_data_J3.pdf}
\label{subfig_MAPGPS_J3_TEC_data}
}
\hspace{0mm}
\subfloat[FORMOSAT-5]{
\includegraphics[width=\linewidth]{MAPGPS_data.pdf}
\label{subfig_MAPGPS_F5_TEC_data}
}
\caption{TEC maps derived from ground-based GNSS observations showing the ionospheric depletion following rocket launches.}
\label{fig_MAPGPS_TEC_data}
\end{figure*}
The ionospheric depletion due to the FORMOSAT-5 launch was also evident in Swarm Langmuir probe data \cite{jin2019} for 24 August 2017, plotted in Fig.~\ref{fig_F5_SWARM_data}. Park \cite{park2016} performed a similar analysis on both Swarm and DMSP Langmuir probe data and argued that observed depletions in the absence of geomagnetic, seismic, or tropospheric disturbances were due to rocket exhaust rather than natural phenomena. Local minima in $n_e$ are encountered by Swarm-A and Swarm-C at $33^{\circ}11' 30'' \, \mathrm{N}$, $123^{\circ}12' 27'' \, \mathrm{W}$ and $33^{\circ}37' 03'' \, \mathrm{N}$, $121^{\circ}45' 39'' \, \mathrm{W}$ respectively. Such minima were not seen in data for 23 August or 25 August. In each case the minima corresponded to an approximate altitude of $442 \, \mathrm{km}$ and time 19:48:50 UT. As the signal in $n_e$ is similar for the two satellites, the midpoint between the aforementioned locations can be assumed to approximate the centre of the depletion. This would imply that the minimum travelled at a mean ground track velocity of $66 \, \mathrm{m.s}^{-1}$ at $231^{\circ} \, \mathrm{N}$ from the launch site. This location corresponds closely to the TEC minimum in the GITM simulation at 19:50 UT, estimated to lie at $33^{\circ}12' 04'' \, \mathrm{N}$, $122^{\circ}36' 27'' \, \mathrm{W}$ and implying a ground track velocity of $72 \, \mathrm{m.s}^{-1}$ at $228^{\circ} \, \mathrm{N}$.
\begin{figure}
\includegraphics[width=\linewidth]{SWARM_Langmuir_Data_F5_wGITM.pdf}
\caption{Electron (a) number density, (b) temperature, and (c) pressure measured by the Swarm-A (thick solid line) and Swarm-C (thick dashed line) satellites following the FORMOSAT-5 launch. Corresponding predictions from the GITM simulation assuming complete combustion at 19:50 UT are also shown (thin lines).}
\label{fig_F5_SWARM_data}
\end{figure}
The Swarm Langmuir probe observations may be compared with values interpolated from the GITM simulations at 19:50 UT, which are also plotted in Fig.~\ref{fig_F5_SWARM_data}. Ionospheric depletions are seen in both cases over a similar range of latitudes, approximately $33^{\circ} \pm 6^{\circ}$. However, the decrease in $n_e$ and increase in $T_e$ are far more pronounced in the GITM simulation than the Langmuir probe data. This difference implies that the quantity of rocket exhaust introduced into the upper thermosphere was overestimated in the numerical model. It is also noted that the undisturbed $n_e$ and $T_e$ obtained from the simulation are respectively much higher and lower than the Langmuir probe data. The former discrepancy appears to reflect the overestimation of the height of the F-region peak by GITM and the IRI model which initialises it, which can be seen in Fig.~\ref{fig_PA836_ionograms}.
Finally, the ionospheric depletion due to the FORMOSAT-5 launch was evident in vertical incidence soundings taken by the Point Arguello ionosonde. These measurements, plotted in Fig.~\ref{fig_PA836_ionograms_measured}, show rapid decreases in the maximum electron number density and altitude at which it occurred following the launch which are qualitatively similar to those found using the GITM model, plotted in Fig.~\ref{fig_PA836_ionograms_GITM}. Both the observational data and simulation indicated large depletions of the F-region with little impact on lower regions of the ionosphere. By 21:30 UT, the F-region had partly recovered in each case, and the electron number density maximum altitude had returned to near its value prior to launch. However, the ionosonde measurements show that the F-region peak prior to the launch was at lower altitude than in the IRI model which is used to intialise GITM. Furthermore, following the recovery of the F-region the ionosonde measurments show that this peak was lower than in the GITM simulation. Consequently, it is believed that GITM underestimated F-region electron production rates, which decrease with altitude. This is thought to have caused the model to underestimate the rate at which the ionosphere recovered from the depletion.
\begin{figure}
\centering
\subfloat[GITM simulation]{
\includegraphics[width=\linewidth]{GITM_PA836.png}
\label{fig_PA836_ionograms_GITM}
}
\hspace{0mm}
\subfloat[Ionosonde data]{
\includegraphics[width=\linewidth]{Ionosonde_PA836.png}
\label{fig_PA836_ionograms_measured}
}
\caption{Electron number density profile and peak altitude (red line) at Point Arguello ionosonde location ($33^{\circ}36' \, \mathrm{N}$, $120^{\circ}36' \, \mathrm{W}$) following the FORMOSAT-5 launch.}
\label{fig_PA836_ionograms}
\end{figure}
In contrast, the aforementioned features were not clearly observed in ground-based GNSS, satellite Langmuir probe, or ionosonde data for 17 January 2016 following the Jason-3 launch. This was unsurprising, given the much smaller and shorter-lived ionospheric depletion predicted by the GITM simulation. Moreover, ground-based GPS TEC and satellite Langmuir probe $n_e$ data following the Jason-5 launch had greater magnitude and greater spatio-temporal variation than those following the FORMOSAT-5 launch, making it more difficult to detect ionospheric depletion. Data from the Point Arguello ionosonde were sparse for the relevant period.
\section{Discussion} \label{sec_discussion}
\subsection{Comparison of launches}
Both the GITM simulations and observational data show that the FORMOSAT-5 launch resulted in a far larger and longer-lived depletion of the ionosphere than the Jason-3 launch, with several factors contributing to this difference. As noted above, the steeper trajectory in the former case resulted in a far greater proportion of exhaust gases being deposited at heights coinciding with the F-region and remaining in this region longer before falling to lower layers of the atmosphere. Additionally, the mass flow rate from the second stage in the latter case was $14 \%$ larger than in the former case. Moreover, UV flux was lesser in the FORMOSAT-5 launch case resulting in a lesser electron production rate. The F-region peak was higher in this case, as seen in Fig.~\ref{fig_altitude}, corresponding to a lower electron production rate.
Due to the different trajectories of the Jason-3 and FORMOSAT-5 launches, the approximation of the rocket exhaust gasses as expanding diffusively from a point source at rest would have affected the corresponding simulated results differently. Values of $L$ in the upper thermosphere in Fig.~\ref{fig_length_scale} are significant compared with the dimensions of the ionospheric depletions being considered. This comparison suggests that advection from the point of release is important in determining the spatial distribution of the rocket exhaust products. In the case of the FORMOSAT-5 launch, the near vertical trajectory would result in rocket exhaust gasses coming to rest at significantly lower altitudes than modelled here. The simulation would therefore in this case be expected to overestimate the overall ionospheric depletion, $\Delta N_e$, and consequently local reductions in TEC, as well as the longevity of these effects. In contrast, during the near horizontal phase of the trajectory for the Jason-3 launch, the exhaust gasses would have travelled significant horizontal distances. Thus, the simulation would be expected to overestimate the concentrations of rocket exhaust gasses and therefore local reductions in TEC.
\subsection{Comparison of simulation and observational data}
The size, shape, and initial motion of the ionospheric depletion in the first $2$ hours following the FORMOSAT-5 launch were modelled reasonably accurately by the GITM simulation. According to the previous analysis of Chou et al. \cite{chou2018} the maximum spatial extent was $6.36 \times 10^5 \, \mathrm{km^2}$, within $15\%$ of the simulated value. This suggests that the diffusion of the exhaust gasses through the thermosphere was modelled well by GITM. The horizontal movement of the ionospheric depletion was largely determined by that of the rocket exhaust gasses and therefore the horizontal winds in the thermosphere. Thus, the close agreement between simulation and observation regarding the location of the depletion during the Swarm satellite passes implies that horizontal winds were predicted accurately by GITM, which was initialised based on the empirical HWM07 model \cite{drob2008}.
In contrast, there were significant discrepancies between the simulations and observations regarding TEC and electron number density depletions and their longevity. These were likely due in large part to approximations made in the rocket exhaust source model. As noted above, the initial downward velocity of rocket exhaust molecules would result in their reaching lower altitudes before entering a diffusive expansion regime. This would reduce their impact on electron concentrations in the upper F-region, which takes longer to recover from depletions. Moreover, it results in the exhaust molecules falling below the F-region sooner and being depleted through charge exchange reactions more rapidly. The initial horizontal velocity of rocket exhaust molecules would disperse them over a wider area before entering the diffusive expansion regime. This would reduce the magnitude of TEC and electron number density depletions. The simulations described above neglected condensation of rocket exhaust, which reduces the number of molecules available for charge exchange reactions. In a previous study, condensation reduced the amount of \ch{H_2O} by $16.7 \%$ \cite{mendillo1981}.
Differences between the simulated and observed depletions also resulted from the ionosphere and thermosphere models used in this work. The IRI and GITM models provided reasonable estimates for the background TEC in both of the cases examined, as seen by comparison of the TEC maps Fig.~\ref{fig_GITM_TEC_map_data} and Fig.~\ref{fig_MAPGPS_TEC_data}. However, these models overestimated the height at which the F-region peak occurred without the depletion, as shown in Fig.~\ref{fig_PA836_ionograms}. As electron production rates decrease with altitude, this disagreement in turn contributes to the overestimate of the recovery time by the simulation. Furthermore, the boundary conditions of the regional GITM simulation do not incorporate general global circulation patterns. Thus, the accuracy of the simulation is expected to degrade significantly with time, particularly post sunset when day-to-night transport is neglected.
The GNSS TEC data were derived based on ionospheric pierce points at $450 \, \mathrm{km}$ altitude \cite{rideout2006}. However, the numerical modelling and ionosonde data shown in Fig.~\ref{fig_PA836_ionograms} indicate that the ionospheric plasma is predominantly concentrated at lower altitudes, particularly within the region affected by the depletion. Thus the depletion seen in Fig.~\ref{subfig_MAPGPS_F5_TEC_data} may be somewhat distorted. The oblique angles at which the GNSS measurements were taken together with the vertical distribution of the ionosphere limited resolution in latitude and longitude. This issue could have helped obscure the narrow ionospheric depletion the simulation suggested would occur following the Jason-3 launch.
Prior studies have indicated lifetimes of ionospheric depletions due to rocket launches ranging from $0.5$ to $6$ hours \cite{park2016}. This range is consistent with observational data for the FORMOSAT-5 launch case and the simulation of the Jason-3 launch case, but not the simulation of the FORMOSAT-5 launch case. However, large rocket launches have typically occurred over sea, making it difficult to take observations of ionospheric depletions throughout their evolution. Therefore, the possibility of rocket launches causing very long lived depletions by depositing exhaust gasses high in the ionosphere, as seen in the FORMOSAT-5 simulation, cannot be precluded.
\subsection{Future work}
Further simulations using the modified version of GITM could be used to investigate the sensitivity of ionospheric depletions to launch trajectory, the upper atmosphere environment, and physical and chemical parameters. The present work indicated the strong dependence of the ionospheric depletion on the altitude reached by the second stage and this should be further investigated. It would be possible to use the numerical model to consider the effects of increased solar and geomagnetic activity on the formation and recovery of ionospheric depletions. Variation with the local time and latitude of the launch might also be considered. Uncertainties in ionospheric depletion behaviour due to those in reaction rate and diffusion coefficients may be quantified through a sensitivity analysis.
An improved model of the rocket exhaust gas source could be obtained through the direct simulation Monte-Carlo (DSMC) method. This technique was previously used to model the transport of rocket exhaust in order in order to study ionospheric interactions of rocket exhaust by Bernhardt et al. \cite{bernhardt2012}. Rocket exhaust plume gasses push aside ionospheric plasma, resulting in a redistribution of $n_e$ on much shorter time-scales than the ionospheric depletions considered above. This is referred to as the snowplow effect and could be incorporated into such a DSMC simulation. The DSMC technique developed by Zhong et al. \cite{zhong2005} could also be used to determine the amount of exhaust lost through condensation.
It is envisaged that GITM could be further modified in future to incorporate other ionospheric effects of rocket launches. On shorter time-scales than considered above, small amplitude wave disturbances were observed propagating away from the rocket trajectory. These were generated by the expansion of the rocket exhaust gasses subsequent to their addition to the simulation (after $300 \, \mathrm{s}$ analytical expansion using eq.~\ref{gas_diffusion_point}), which did not include the sub-grid scale dynamics responsible for the waves which drive observed travelling ionospheric disturbances accompanying rocket launches. However, appropriate source terms could be incorporated into GITM to represent the excitation of the acoustic and atmospheric gravity waves during the passage of rockets through the thermosphere. The model described above may be used to study the potential for the rocket launches to trigger equatorial plasma bubbles. Using GITM to comprehensively model the state of the ionosphere-thermosphere system would enable the study of this phenomenon by coupling to an ionospheric code which self-consistently solves for electric fields, such as SAMI3 \cite{huba2008}.
\section{Conclusions}
A modified version of GITM has been developed to model ionospheric depletions due to rocket launches. The rocket exhaust gasses \ch{H_2O}, \ch{H_2}, and \ch{CO_2} were implemented as major species. A source term was added incorporating an initial diffusive expansion which was modelled analytically. Combined charge exchange and dissociative recombination reactions of these species were incorporated into the GITM chemical model.
The numerical model was applied to the Jason-3 and FORMOSAT-5 launches. Magnitude and longevity of the resulting depletions were found to depend on the altitudinal distribution of the rocket exhaust species which determined their residence times in the F-region. Thus, it was found that the steeper trajectory in the FORMOSAT-5 launch case produced larger and longer-lived depletions. Comparison of results for plume compositions determined based on complete combustion and CEA output indicated that the ionospheric depletion was not sensitive to this composition.
Results of the FORMOSAT-5 launch simulation were compared with GNSS, ionosonde, and satellite Langmuir probe measurements. The horizontal movement and expansion of the depletion following the FORMOSAT-5 launch in the numerical model were in reasonable agreement with GNSS TEC and Swarm Langmuir probe electron number density observations. However, the simulated depletions of TEC and electron number density and the time period over which they occurred were much greater than those observed. These disagreements were ascribed primarily to the neglect of the downward transport of rocket exhaust gasses prior to diffusive expansion and the overestimation of the F-region peak height by IRI and GITM.
\section{Acknowledgments}
This work was carried out with funding from the Royal Australian Air Force. Resources from the National Computational Infrastructure (NCI), supported by the Australian Government were used in this work. We thank Aaron Ridley for his invaluable guidance regarding the GITM code he has developed. Additionally, we gratefully acknowledge James Gilmore for providing data format conversion tools used in this work.
We acknowledge several organisations for supplying empirical ionospheric data. MAPGPS TEC map data were obtained through the Madrigal CEDAR Database, accessible via the World Wide Web at http://cedar.openmadrigal.org/index.html/. This service is provided by the Massachusetts Institute of Technology under support from US National Science Foundation grant AGS-1242204. Langmuir probe data from the ESA Swarm spacecraft were derived from the SWARM Data Access website, which can be found at https://swarm-diss.eo.esa.int/. NOAA ionosonde data were also used, which can be downloaded via FTP from ftp://ftp.ngdc.noaa.gov/ionosonde.
|
2,877,628,088,996 | arxiv | \section{Introduction}
In \cite{Merca15}, the second author considered a bisection of Euler's pentagonal number theorem
\begin{equation*}
(q;q)_\infty = \sum_{k=0}^\infty (-1)^{\lceil k/2 \rceil} q^{G_k}
\end{equation*}
based on the parity of the $k$-th generalized pentagonal number
$$G_{k} = \frac{1}{2} \left\lceil \frac{k}{2} \right\rceil \left\lceil \frac{3k+1}{2} \right\rceil,$$
and obtained the following result:
\begin{equation}\label{bisection1}
\sum_{k=0}^{\infty}\frac{1+(-1)^{G_k}}{2}(-1)^{\lceil k/2 \rceil}q^{G_k}=(q^2,q^{12},q^{14},q^{16},q^{18},q^{20},q^{30},q^{32};q^{32})_{\infty},
\end{equation}
where
$$(a_1,a_2\ldots,a_n;q)_{\infty}=(a_1;q)_{\infty}(a_2;q)_{\infty}\cdots(a_n;q)_{\infty}.$$
Because the infinite product
$$(a;q)_\infty = \prod_{k=0}^{\infty} (1-aq^k)$$
diverges when $a\neq 0$
and $|q|\geqslant 1$,
whenever $(a;q)_\infty$ appears in a formula, we shall assume that $|q| <1$.
The following identity for Euler's partition function $p(n)$ was obtained in \cite{Merca15} as a combinatorial interpretation of \eqref{bisection1}:
\begin{equation}
\sum_{k=0}^\infty \frac{1+(-1)^{G_k}}{2}(-1)^{\lceil k/2 \rceil} p(n-G_k) = L(n),
\end{equation}
where $L(n)$ is the number of partitions of $n$ into parts not congruent to
$0$, $2$, $12$, $14$, $16$, $18$, $20$ or $30 \mod{32}$. This identity is a bisection of Euler's well-known recurrence relation for the partition function $p(n)$:
\begin{equation}\label{ER}\sum_{k=0}^\infty (-1)^{\lceil k/2 \rceil} p(n-G_k) = \delta_{0,n},\end{equation}
where $\delta_{i,j}$ is the Kronecker delta function. For details on \eqref{ER} see Andrews's book \cite{Andrews76}.
In this paper, motivated by these results, we consider a bisection of another classical theta identity \cite[eq. 2.2.13]{Andrews76}
\begin{equation}\label{eq1}
\frac{(q^2;q^2)_\infty}{(-q;q^2)_\infty} = \sum_{k=0}^{\infty} (-q)^{k(k+1)/2}
\end{equation}
in order to derive new identities for Euler's partition function. These identities involve new partition functions which we define below.
For what follows, we denote by $g_r(\lambda)$ the smallest part of the partition $\lambda$ appearing less than $r$ times.
\begin{table}[t]
\centering
\begin{tabular}{c c c c c c c c}
\hline
$\lambda$ & 5 & 4+1 & 3+2 & 3+1+1 & 2+2+1 & 2+1+1+1 & 1+1+1+1+1 \\ [0.5ex]
$g_1(\lambda)$ & 1 & 2 & 1 & 2 & 3 & 3 & 2 \\ [0.5ex]
$g_2(\lambda)$ & 1 & 1 & 1 & 2 & 1 & 2 & 2 \\ [0.5ex]
$g_3(\lambda)$ & 1 & 1 & 1 & 1 & 1 & 2 & 2 \\ [0.5ex]
$g_4(\lambda)$ & 1 & 1 & 1 & 1 & 1 & 1 & 2 \\ [0.5ex]
$g_5(\lambda)$ & 1 & 1 & 1 & 1 & 1 & 1 & 2 \\ [0.5ex]
$g_6(\lambda)$ & 1 & 1 & 1 & 1 & 1 & 1 & 1 \\ [0.5ex]
\hline
\end{tabular}
\label{table:1}
\caption{The partition functions $g_r$ for $\lambda\vdash 5$}
\end{table}
The limit distribution of $g_r(\lambda)$ has been studied in \cite{Wagner}. In the literature, $g_1(\lambda)$ is referred to as the least gap of $\lambda$. By analogy, we refer to $g_r(\lambda)$ as the \textit{least $r$-gap} of $\lambda$. To make formulas more concise, we set $g_0(n)=\infty$.
We denote by $S_r(n)$ the sum of the least $r$-gaps in all partitions of $\lambda$, i.e., $$S_r(n)=\sum_{\lambda \vdash n}g_r(\lambda).$$ Thus, $S_1(n)$ is the sum of the least gaps in all partitions of $n$. By Table \ref{table:1}, we see, for example, that
$$S_1(5) = 1+2+1+2+3+3+2 = 14$$ and $$S_4(n)=1+1+1+1+1+1+2=8.$$
When $r \geqslant 2$, for each partition $\lambda$ we have $g_r(\lambda)\leqslant g_{r-1}(\lambda)$. Let $G_r(n)$ be the number of partitions $\lambda$ of $n$ satisfying $g_r(\lambda)<g_{r-1}(\lambda)$. It is clear that $G_1(n)=p(n)$ and $G_r(n)=0$ for $r\geqslant n+2$.
To our knowledge, the functions $S_r(n)$ and $G_r(n)$ have not been considered previously in the literature.
It is known \cite[A022567]{Sloane} that the sum of the least gaps in all partitions of $n$ can be expressed in terms of the Euler's partition function $p(n)$:
\begin{equation}\label{eq0}
\sum_{k=0}^\infty p(n-T_k) = S_1(n),
\end{equation}
where $T_n= n(n+1)/2$ is the $n$-th triangular number.
Upon reflection, one expects that there might be infinite families of such identities where \eqref{eq0} is the first entry.
As far as we know, the following identity has not been remarked before.
\begin{theorem}\label{T0}
For $n\geqslant 0$ and $r\geqslant 1$,
\begin{equation}\label{eqT0}
\sum_{k=0}^\infty p(n-rT_k) = S_r(n).
\end{equation}
\end{theorem}
In section \ref{sec2}, we provide a combinatorial proof of Theorem \ref{T0}.
Then, theta identity \eqref{eq1} and Theorem \ref{T0} allow us to find the generating function for $S_r(n)$ and to prove the following result that also involves the partitions of $n$ into parts not congruent to $0$, $r$ or $3r \bmod {4r}$. We denote by $U_r(n)$ the number of these partitions.
\begin{theorem}\label{T1}
For $n\geqslant 0$ and $r\geqslant 1$,
\begin{enumerate}
\item[(i)] $\displaystyle{ \sum_{k=0}^\infty \left(p(n-rT_{4k})+p(n-rT_{4k+3}) \right) = \frac{S_r(n)}{2}+\frac{U_r(n)}{2} ;}$
\item[(ii)] $\displaystyle{ \sum_{k=0}^\infty \left(p(n-rT_{4k+1})+p(n-rT_{4k+2}) \right) = \frac{S_r(n)}{2}-\frac{U_r(n)}{2} .}$
\end{enumerate}
\end{theorem}
By this theorem, we see that $S_r(n)$ and $U_r(n)$ have the same parity.
\begin{corollary}
For $n\geqslant 0$ and $r\geqslant 1$, the sum of the least $r$-gaps in all partitions of $n$ and the number of partitions of $n$ into parts not congruent to $0$, $r$ or $3r \bmod {4r}$ have the same parity.
\end{corollary}
In addition, we have the following identity.
\begin{corollary}\label{Cor1}
For $n\geqslant 0$ and $r\geqslant 1$,
$$\sum_{k=0}^\infty (-1)^{T_k} p(n-rT_k) = U_r(n).$$
\end{corollary}
Replacing $r$ by $1$ in Corollary \ref{Cor1}, we obtain another known identity (see \cite[the proof of Theorem 2.3]{BM}).
\begin{corollary}\label{C1}
For $n\geqslant 0$,
$$ \sum_{k=0}^\infty (-1)^{T_k} p(n-T_k) =
\begin{cases}
q\left(\frac{n}{2} \right), & \text{for $n$ even,}\\
0, & \text{for $n$ odd,}
\end{cases}
$$
where $q(n)$ is the number of partitions of $n$ into distinct parts.
\end{corollary}
It is shown in \cite[Corollary 4.7]{Merca16} that $q(n)$ is odd if and only if $n$ is a generalized pentagonal number. Thus, we deduce the following result related to the parity of $S_1(n)$.
\begin{corollary}\label{C6}
For $n\geqslant 0$, the sum of the least gaps in all partitions of $n$ is even except when $n$ is twice a generalized pentagonal number.
\end{corollary}
If $s\geqslant 3$ is the number of sides of a polygon, the $n$th $s$-polygonal number (or $s$-gonal number) is $$P(s,n)=\frac{n^2(s-2)-n(s-4)}{2}.$$ If we allow $n \in \mathbb Z$, we obtain generalized $s$-gonal numbers. Note that, for $n>0$, we have $P(3,-n)=P(3,n-1)$ and for all $n$ we have $P(4,-n)=P(4,n)$. For $s \geqslant 5$ and $n >0$, $P(s,-n)$ is not an ordinary $s$-gonal number. We remark that the $n$-th $s$-gonal number can be expressed in term of the triangular numbers $T_n$ as follows:
$$P(s,n)=(s-3)T_{n-1}+T_n.$$
Beside Theorem \ref{T0}, there is another infinite family of identities involving Euler's partition function $p(n)$ for which \eqref{eq0} is the special case $r=1$.
\begin{theorem}\label{T2}
For $n\geqslant 0$ and $r\geqslant 1$
\begin{equation}\label{pn}\sum_{k=0}^{\infty}p\left( n-P(r+2,-k)\right) =S_r(n)+G_r(n).\end{equation}
\end{theorem}
In this paper, we provide a purely combinatorial proof of this result and some applications involving partitions into even numbers of parts, partitions with nonnegative rank, and partitions with nonnegative crank.
\section{Combinatorial proof of Theorem \ref{T0}} \label{sec2}
Fix $r\geqslant 1$ and, for each $k\geqslant 0$ consider the fat staircase partition (written in exponential notation) $$\delta_r(k)=(1^r, 2^r, \ldots, (k-1)^r, k^{r}).$$ This is the staircase partition with largest part $k$ in which each part is repeated $r$ times. Its size is equal to $rT_k$.
As before, fix $r\geqslant 1$ and also fix $n \geqslant 0$. For each $k\geqslant 0$ we create an injection from the set of partitions of $n-rT_k$ into the set of partitions of $n$ $$\varphi_{r,n,k}:\{\mu \vdash n-rT_k\} \hookrightarrow \{\lambda \vdash n\}$$ where $\varphi_{r,n,k}(\mu)$ is the partition obtained from $\mu$ by inserting the parts of the staircase $\delta_r(k)$. Denote by $\mathcal A_{r,n,k}$ the image of $\{\mu \vdash n-rT_k\}$ under $\varphi_{r,n,k}$. Thus, $p(n-rT_k)=|\mathcal A_{r,n,k}|$ and $\mathcal A_{r,n,k}$ consists precisely of the partitions $\lambda$ of $n$ satisfying $g_r(\lambda)> k$.
Consider an arbitrary partition $\lambda$ of $n$ with $g_r(\lambda)=k$. Then $\lambda\in \mathcal A_{r,n,i}$, $i=0, 1, \ldots k-1$ and $\lambda \not \in \mathcal A_{r,n,j}$ with $j\geqslant k$. Therefore, each partition of $n$ with $g_r(\lambda)=k$ is counted by the left hand side of \eqref{eqT0} exactly $k$ times.
\section{Proof of Theorem \ref{T1}}
We rewrite the identity \eqref{eq1} as
\begin{equation}\label{eq2}
\sum_{k=0}^{\infty} (-q)^{T_k} = \frac{(q;q)_\infty}{(q^2;q^4)_\infty}.
\end{equation}
Applying bisection on \eqref{eq2}, we obtain:
\begin{equation}\label{eq3}
\frac{1}{2} \sum_{k=0}^\infty \left(q^{T_k} \pm (-q)^{T_k}\right) = \frac{1}{2} \frac{(-q;-q)_\infty \pm (q;q)_\infty}{(q^2;q^4)_\infty}.
\end{equation}
Multiplying both sides of \eqref{eq3} by the reciprocal of $(q;q)_\infty$, we give
\begin{align*}
\frac{1}{2(q;q)_\infty} \sum_{k=0}^\infty \left( q^{T_k} \pm (-q)^{T_k}\right)
& = \frac{(-q^2;q^2)_\infty}{2} \frac{(-q;-q)_\infty \pm (q;q)_\infty}{(q;q)_\infty} \nonumber \\
& = \frac{(-q^2;q^2)_\infty}{2} \left(\frac{(-q;-q)_\infty}{(q;q)_\infty} \pm 1\right)\nonumber \\
& = \frac{(-q;q)^2_\infty \pm (-q^2;q^2)_\infty}{2}.
\end{align*}
By this identity, with $q$ replaced by $q^r$, we obtain the relation
\begin{equation*}
\frac{1}{2(q^r;q^r)_\infty} \sum_{k=0}^\infty \left( q^{rT_k} \pm (-q^r)^{T_k}\right)=
\frac{(-q^r;q^r)^2_\infty \pm (-q^{2r};q^{2r})_\infty}{2},
\end{equation*}
that can be rewritten as
\begin{align}
&\frac{1}{2(q;q)_\infty} \sum_{k=0}^\infty \left( q^{rT_k} \pm (-q^r)^{T_k}\right) \nonumber\\
&\qquad = \frac{1}{2} \left( \frac{(-q^r;q^r)^2_\infty (q^r;q^r)_\infty}{(q;q)_\infty} \pm \frac{(-q^{2r};q^{2r})_\infty (q^r;q^r)_\infty}{(q;q)_\infty} \right) \nonumber\\
&\qquad = \frac{1}{2} \left( \frac{(-q^{r};q^{r})_\infty (q^{r};q^{r})_\infty}{(q;q)_\infty (q^r;q^{2r})_\infty} \pm \frac{ (q^r,q^{2r};q^{2r})_\infty}{(q;q)_\infty (q^{2r};q^{4r})_\infty} \right) \nonumber \\
&\qquad = \frac{1}{2} \left( \frac{(q^{2r};q^{2r})_\infty}{(q;q)_\infty (q^r;q^{2r})_\infty} \pm \frac{ (q^r,q^{3r},q^{4r};q^{4r})_\infty}{(q;q)_\infty} \right). \label{eq4}
\end{align}
Considering the generating function for $p(n)$, i.e.,
$$\sum_{n=0}^\infty p(n) q^n =\frac{1}{(q;q)_\infty}$$
and the theta identity \eqref{eq1}, by Theorem \ref{T0} we deduce that
$$\sum_{k=0}^\infty S_r(k) q^k = \frac{(q^{2r};q^{2r})_\infty}{(q;q)_\infty (q^r;q^{2r})_\infty}.$$
On the other hand, we have
$$ \sum_{k=0}^\infty U_r(k) q^k = \frac{ (q^r,q^{3r},q^{4r};q^{4r})_\infty}{(q;q)_\infty}.$$
Taking into account the well-known Cauchy multiplication of two power series,
we deduce our identities as combinatorial interpretations of \eqref{eq4}.
\section{Combinatorial proof of Theorem \ref{T2}}
The proof of Theorem \ref{T2} is analogous to the proof of Theorem \ref{T0}.
For fixed $r\geqslant 1$ and, for each $k\geqslant 0$we denote by ' $$\delta_r(k)=(1^r, 2^r, \ldots, (k-1)^r, k^{r-1})$$ the staircase partition in which the largest part is $k$ and is repeated $r-1$ times and all other parts are repeated $r$ times. Its size is equal to $P(r+2,-k)$.
As before, fix $r\geqslant 1$ and also fix $n \geqslant 0$. For each $k\geqslant 0$ we create an injection from the set of partitions of $n-P(r+2,-k)$ into the set of partitions of $n$ $$\varphi'_{r,n,k}:\{\mu \vdash n-P(r+2,-k)\} \hookrightarrow \{\lambda \vdash n\}$$ where $\varphi'_{r,n,k}(\mu)$ is the partition obtained from $\mu$ by inserting the parts of the staircase $\delta'_r(k)$. If $\mathcal A'_{r,n,k}$ denotes the image of $\{\mu \vdash n-P(r+2,-k)\}$ under $\varphi_{r,n,k}$, we have that $p(n-P(r+2,-k))=|\mathcal A'_{r,n,k}|$ and $\mathcal A'_{r,n,k}$ consists precisely of the partitions $\lambda$ of $n$ satisfying $g_r(\lambda)\geqslant k$ and $g_{r-1}(\lambda)> k$.
If $\lambda \vdash n$ has $g_r(\lambda)=k$, then $\lambda\in \mathcal A'_{r,n,i}$, $i=0, 1, \ldots k-1$. If $g_{r-1}(\lambda)=k$, then $\lambda \not \in \mathcal A'_{r,n,j}$ with $j\geqslant k$. If $g_{r-1}>k$, then $\lambda \in \mathcal A'_{r,n,k}$ but $\lambda \not \in \mathcal A'_{r,n,j}$ with $j> k$. Therefore, each partition of $n$ with $g_r(\lambda)=k$ is counted by the left hand side of \eqref{pn} exactly $k$ times if $g_r(\lambda)=g_{r-1}(\lambda)$ and exactly $k+1$ times if $g_r(\lambda)<g_{r-1}(\lambda)$.
\section{Applications of Theorem \ref{T2}}
In this section we consider some special cases of Theorem \ref{T2} in order to discover and prove new identities involving Euler's partition function $p(n)$.
\subsection{Partitions into even numbers of parts}
Now we consider the following classical theta identity \cite[eq. 2.2.12]{Andrews76}
\begin{equation}
\frac{(q;q)_\infty}{(-q;q)_\infty} =1+2 \sum_{k=1}^\infty (-1)^k q^{k^2}.
\end{equation}
Elementary techniques in the theory of partition \cite{Andrews76} allow us to derive a known combinatorial interpretation of this identity, namely
\begin{equation}\label{eq11}
p(n)+2\sum_{j=k}^n (-1)^k p(n-k^2) = p_e(n)-p_o(n),
\end{equation}
where $p_e(n)$ is the number of partitions of $n$ into even number of parts and
$p_o(n)$ is the number of partitions of $n$ into odd number of parts.
Moreover, it is known that
\begin{equation}\label{eq12}
p_e(n) = p(n)+\sum_{k=1}^n (-1)^k p(n-k^2)
\end{equation}
and
$$p_o(n) = -\sum_{k=1}^n (-1)^k p(n-k^2).$$
These relations can be considered a bisection of the identity \eqref{eq11}.
Combining identity \eqref{eq12} with the case $r=2$ of Theorem \ref{T2}, we derive the following result.
\begin{corollary}
For $n\geqslant 0$,
\begin{enumerate}
\item[(i)] $\displaystyle{ \sum_{k=0}^\infty p\left( n-(2k)^2\right) = \frac{S_2(n)+G_2(n)+p_e(n)}{2};}$
\item[(ii)] $\displaystyle{ \sum_{k=0}^\infty p\left( n-(2k+1)^2\right) = \frac{S_2(n)+G_2(n)-p_e(n)}{2}.}$
\end{enumerate}
\end{corollary}
\subsection{Partitions with nonnegative rank}
In 1944, Dyson \cite{Dyson} defined the rank of a partition as the difference between its largest part and the number of its parts.
Then he observed empirically that the partitions of $5n + 4$ (respectively $7n + 5$) form $5$ (respectively $7$) groups of equal size when sorted by their ranks modulo $5$ (respectively $7$). This interesting conjecture of Dyson was proved ten years later by Atkin and Swinnerton-Dyer \cite{Atkin}.
In this section, we denote by $R(n)$ the number of partitions of $n$ with nonnegative rank.
It is known \cite[A064174]{Sloane} that the number of partitions of $n$ with nonnegative rank can be expressed in terms of Euler's partition function as follows:
\begin{equation}
R(n)=\sum_{k=0}^n (-1)^k p\left( n-k(3k+1)/2\right).
\end{equation}
Considering the case $r=3$ of Theorem \ref{T2}, we obtain the following result.
\begin{corollary}
For $n\geqslant 0$,
\begin{enumerate}
\item[(i)] $\displaystyle{ \sum_{k=0}^\infty p\left( n-k(6k+1)\right) = \frac{S_3(n)+G_3(n)+R(n)}{2};}$
\item[(ii)] $\displaystyle{ \sum_{k=0}^\infty p\left( n-(2k+1)(3k+2)\right) = \frac{S_3(n)+G_3(n)-R(n)}{2}.}$
\end{enumerate}
\end{corollary}
\subsection{Partitions with nonnegative crank}
Dyson \cite{Dyson} conjectured the existence of a crank function for partitions that would provide a combinatorial proof of Ramanujan's congruence modulo $11$. Forty-four years later, Andrews and Garvan \cite{Andrews88} successfully found such a function which yields a combinatorial explanation of Ramanujan congruences modulo $5$, $7$, and $11$.
For a partition $\lambda$, let $l(\lambda)$ denote the largest part of $\lambda$, $\omega(\lambda)$ denote the number of $1$'s in $\lambda$, and $\mu(\lambda)$ denote the number of parts of $\lambda$ greater than $\omega(\lambda)$. The crank $c(\lambda)$ is defined by
$$
c(\lambda) =
\begin{cases}
l(\lambda), & \text{for $\omega(\lambda)=0$,}\\
\mu(\lambda)-\omega(\lambda), & \text{for $\omega(\lambda)>0$.}
\end{cases}
$$
In this section, we denote by $C(n)$ the number of partitions of $n$ with nonnegative crank.
We known \cite[A064428]{Sloane} that the number of partitions of $n$ with nonnegative crank can be expressed in terms of Euler's partition function $p(n)$:
\begin{equation}\label{crank}
C(n)=\sum_{k=0}^\infty (-1)^k p(n-T_k).
\end{equation}
We have the following result related to the parity of $C(n)$.
\begin{corollary}
For $n\geqslant 0$, the number of partitions of $n$ with nonnegative crank is even except when $n$ is twice a generalized pentagonal number.
\end{corollary}
\begin{proof}
Considering the case $r=1$ of Theorem \ref{T2} and the identity \eqref{crank}, we obtain
$$\sum_{k=0}^\infty p(n-T_{2k}) = \frac {C(n)+S_1(n)}{2}.$$
We see that the number of partitions of $n$ with nonnegative crank and the sum of the least gaps in all partitions of $n$ have the same parity. According to Corollary \ref{C6} the proof is finished.
\end{proof}
By the identity \eqref{crank} and Corollary \ref{Cor1}, we easily get two identities.
\begin{corollary}
For $n\geqslant 0$,
\begin{enumerate}
\item[(i)] $\displaystyle{ \sum_{k=0}^\infty (-1)^k p\left( n-T_{k+2\lfloor k/2 \rfloor}\right) = \frac{C(n)+U_1(n)}{2};}$
\item[(ii)] $\displaystyle{ \sum_{k=0}^\infty (-1)^k p\left( n-T_{k+2\lfloor k/2 \rfloor+2}\right) = \frac{C(n)-U_1(n)}{2}.}$
\end{enumerate}
\end{corollary}
|
2,877,628,088,997 | arxiv | \section{Introduction}
Despite the many successes achieved so far, the major challenge of time-dependent density functional theory (TDDFT) is to find good approximations to the Kohn-Sham potential, $\hat V^{KS}$, for a non-interacting system. This is a notoriously difficult problem and leads to failures of TDDFT in situations involving charge-transfer excitations~\cite{Dreuw00}, conical intersections~\cite{Tapavicza08} or photoionization~\cite{Petersilka99}. Naturally, this raises the following question: \emph{what is the complexity of generating of the necessary potentials?} We answer this question and show that access to a universal quantum computer is sufficient.
The present work, in addition to contributing to on-going research about the foundations of TDDFT,
is the latest application of quantum computational complexity theory to a growing list of problems in the physics and chemistry community~\cite{Whitfield13}. Our result emphasizes that the foundations of TDDFT are not devoid of computational considerations, even theoretically. Further, our work highlights the utility of reasoning using hypothetical quantum computers to classify the computational complexity of problems. The practical implications are that, within the interior of the domain of existence, it is efficient to compute the necessary potentials using a computer with access to an oracle capable of polynomial-time quantum computation.
Quantum computers are devices which use quantum systems themselves to store and process data. On the one hand, one of the selling points of quantum computation is to have efficient algorithms for calculations in quantum chemistry and quantum physics~\cite{Brown10,Kassal11,Yung12}. On the other hand, in the worst case, quantum computers are not expected to solve all NP (non-deterministic polynomial time) problems efficiently~\cite{Bennett97}. Therefore, it is an on-going investigation into when a quantum computer would be more useful than a classical computer. Our current result points towards evidence of computational differences between quantum computers and classical computers. In this way, we provide additional insights to one of the driving questions of information and communication processing in the past decades concerning practical application areas of quantum computing.
Our findings are in contrast to a previous result by Schuch and Verstraete~\cite{Schuch09},
which showed that, in the worst-case, polynomial approximation to the universal functional of ground state density functional theory (DFT) is likely to be impossible even with a quantum computer.
Remarkably, this discrepancy between the computational difficulty of TDDFT and ground state DFT is often reversed in practice where for common place systems encountered by physicists and chemists, TDDFT calculations are often more challenging than DFT calculations. Therefore,
our findings provide more reasons why quantum computers should be built.
The practical utility of our results can be understood in multiple ways. First, we have demonstrated a new theoretical understanding of TDDFT highlighting its relative simplicity as compared to ground state DFT computations. Second, we have introduced a $V$-representability parameter, which similar to the condition number of a matrix, diverges as the Kohn-Sham formalism becomes less applicable. Finally, for analysis purposes, it is often useful to know what the exact Kohn-Sham potential looks like in order to compare and contrast approximations to the exchange-correlation functionals. However, this has been limited to small dimensional or model systems and our results show that, with a quantum computer, one could perform such exploratory studies for larger systems.
\medskip
\section{Background}
\subsection{Time-dependent Kohn-Sham systems}
To introduce TDDFT and its Kohn-Sham formalism, it is instructive to view the Schr\"odinger equation as a map
\cite{Maitra10}
\begin{equation}
\{\hat V(t),\Psi(t_0)\}\mapsto\{n(t),\Psi(t)\}.
\label{eq:SEmap}
\end{equation}
The inputs to the map are an initial state of $N$ electrons, $\Psi{(t=t_0)}$, and a Hamiltonian, $\hat H(t)=\hat T+\hat W+\hat V{(t)}$ that contains a kinetic-energy term, $\hat T$, a two-body interaction term such as the Coulomb potential, $\hat W$, and a scalar time-dependent potential, $\hat V(t)$. The outputs of the map are the state at later time, $\Psi(t)$ and the one-particle probability density normalized to $N$ (referred to as the density),
\begin{eqnarray}
\langle \hat n(x) \rangle _ {\Psi(t)}&=&\bra{\Psi(t)}\hat n(x) \ket{\Psi(t)}\nonumber\\
&=&N\int |\Psi(x,x_2,...,x_N;t)|^2 dx_2...dx_N.
\end{eqnarray}
TDDFT is predicated on the use of the time-dependent density as the fundamental variable and all observables and properties are functionals of the density.
The crux of the theoretical foundations of TDDFT is an inverse map which has as inputs the density at all times and the initial state. It outputs the potential and the wave function at later times $t$,
\begin{equation}
\{\langle \hat n\rangle_{\Psi(t)},\Psi(t_0)\}\mapsto\{\hat V(t),\Psi(t)\}.
\label{eq:RGmap}
\end{equation}
This mapping exists via the Runge-Gross theorem \cite{Runge84} which shows that, apart from a gauge degree of freedom represented by spatially homogeneous variations, the potential is bijectively related to the density. However, the problem of time-dependent simulation has not been simplified; the dimension of the Hilbert space scales exponentially with the number of electrons due to the two-body interaction $\hat W$. As a result, the time-dependent Schr\"odinger equation quickly becomes intractable to solve with controlled precision on a classical computer.
Practical computational approaches to TDDFT rely on constructing the non-interacting time-dependent Kohn-Sham potential. If at time $t$ the density of a system described by potential and wave function, $\{\hat V(t),\Psi(t)\}$, is $\langle \hat n\rangle_{\Psi(t)}$, then the non-interacting Kohn-Sham system ($\hat W=0$) reproduces the same density but using a different potential, $\hat V^{KS}$. The key difficulty of TDDFT is obtaining the time-dependent Kohn-Sham potential.
Typically, the Kohn-Sham potential is broken into three parts: $\hat V^{KS}=\hat V+\hat V^H+\hat V^{xc}$. The first potential is the external potential given in the problem specification and the second is the Hartree potential $V^H(x,t)=\int n(x',t)|x-x'|^{-1} d^3x'$. The third is the exchange-correlation potential and requires an approximation to be specified wherein lies the difficulty of the Kohn-Sham scheme. In this article, we discuss how difficult approximating the full potential is but we make note that only the exchange-correlation is unknown. While we discuss the computation of the full Kohn-Sham potential from a given external potential and initial density, we will not construct an explicit functional for the exchange-correlation potential.
The route to obtaining the Kohn-Sham potentials we focus on is the evaluation of the map,
\begin{equation}
\{\langle \hat n \rangle_{\Psi(t)}, \Phi(t_0)\}\mapsto\{\hat V^{KS}(t),\Phi(t)\}.
\label{eq:KSmap}
\end{equation}
Here, the wave function of the Kohn-Sham system, $\Phi(t)=\mathcal A [\phi^{1}{(t)}\phi^{2}{(t)}...\phi^{N}{(t)}]$, is an anti-symmetric combination of single particle wave functions, $\phi^i(t)$, such that for all times $t$, the Kohn-Sham density, $n^{KS}{(t)}=\langle \hat n\rangle_{\Phi(t)}=\sum_{i=1}^N |\phi^i{(t)}|^2$, matches the interacting density $\langle\hat n\rangle_{\Psi(t)}$. If such a map exists, we call the system $V$-representable while implicitly referring to non-interacting $V^{KS}$-representablity.
As the map in \eqref{eq:KSmap} is foundational for TDDFT implementations based on the Kohn-Sham system, there are many articles~\cite{Leeuwen99,Baer08,Li08,Ruggenthaler11,Ruggenthaler12,Farzanehpour12} examining the existence of such a map. Instead of attempting to merely prove the existence of the Kohn-Sham potential, we will explore the limits on the efficient computation of this map and go beyond the scope of the previous works by addressing questions from the vantage of computational complexity.
The first approach to the Kohn-Sham inverse map found in \eqref{eq:KSmap}, was due to van Leeuwen~\cite{Leeuwen99} who constructed a Taylor expansion in $t$ of the Kohn-Sham potential to prove its existence. The construction relied on the continuity equation, $-\nabla\cdot\hat j=\partial_t \hat n$, and the Heisenberg equation of motion for the density operator to derive the local force balance equation at a given time $t$:
\begin{equation}
\partial_t^2\hat{n}-i[\hat W,\partial_t \hat n]=-\nabla\cdot(\hat n\nabla V)+\hat Q,
\label{eq:vL}
\end{equation}
where $\hat Q=i[\hat T,\partial_t \hat n]$ is the momentum-stress tensor.
In the past few years, several results have appeared extending van Leeuwen's construction~\cite{Baer08,Li08,Ruggenthaler11,Ruggenthaler12,Farzanehpour12}
to avoid technical problems (related to convergence and analyticity requirements). Here previous rigorous results by Farzanehpour and Tokatly~\cite{Farzanehpour12} on lattice TDDFT are directly applicable to our quantum computational setting.
\medskip
\subsection{The discrete force balance equation}
We summarize the details of the discretized local force-balance equation from~\cite{Farzanehpour12}. More detailed derivations are found in~\cite{Farzanehpour12} and as well as a more general derivation we provide in~\ref{appx:A}.
Consider a system discretized on a lattice of $M$ points forming a
Fock space. In second quantization, the creation $\hat a_i$ and annihilation $\hat a_j^\dag$ operators for arbitrary sites $i$ and $j$ must satisfy $ \hat a_i\hat a_j=-\hat a_j\hat a_i$ and $\hat a_i\hat a_j^\dag=\delta_{ij}-\hat a_j^\dag \hat a_i$. We define a discretized one-body operator as $\hat A=\sum_{n}^M\sum_{m}^MA_{mn}\hat a_m^\dag \hat a_n$ and designate $A$ as the coefficient matrix of the operator. The matrix elements are $A_{mn}=\bra{m}\hat A\ket{n}$ where $\ket{m}$ and $\ket{n}$ are the single electron sites corresponding to operators $\hat a_m$ and $\hat a_n$. Similar notation and definitions hold for the two-body operators.
The Hamiltonian, the density at site $j$, and the continuity equation are then given respectively by
\begin{eqnarray}
\hat H(t)&=& \sum_{ij}[T_{ij}+\delta_{ij}V_{i}(t)]\hat a_i^\dag \hat a_j+\sum_{ijkl} W_{ijkl}\hat a_i^\dag \hat a_j^\dag \hat a_k\hat a_l,\phantom{spac}\label{eq:H}\\
\hat n_j&=& \hat a_j^\dag \hat a_j, \\
\partial_t \hat n_j&=& -\sum_k \hat J_{jk}=-i\sum_k T_{kj}(\hat a_j^\dag \hat a_k-\hat a_k^\dag \hat a_j).
\end{eqnarray}
For the density of the Kohn-Sham system, $n^{KS}(t)=\langle \hat n\rangle_{\Phi(t)}$, to match the density of the interacting system, $n(t)=\langle \hat n\rangle_{\Psi(t)}$,
the discretized local force balance equation~\cite{Farzanehpour12} must be satisfied,
\begin{eqnarray}
S_j^{aim} &=&
\sum_k(V^{KS}_j-V^{KS}_k) T_{kj}\langle \hat a_j^\dag \hat a_k+\hat a_k^\dag \hat a_j\rangle_{\Phi(t)} \label{eq:1}\\
&=& \sum_k \left\langle -T_{kj}\hat \Gamma_{jk}+\delta_{jk}\sum_m T_{mj}\hat\Gamma_{jm}\right\rangle_{\Phi(t)}V^{KS}_k\phantom{space}\label{eq:2}\\
&=& \sum_k K_{jk}V^{KS}_k.\label{eq:K}
\end{eqnarray}
Here $\hat \Gamma_{ij}=\hat a_i^\dag \hat a_j +\hat a_j^\dag \hat a_i$ is
twice the real part of the one-body reduced density operator. A complete derivation of this equation is found the~\ref{appx:A}. The vector $S^{aim}$ is defined as $S^{aim}_j(\Psi,\Phi)=\partial_t^2\langle \hat{n}_j\rangle_{\Psi(t)}-\langle \hat Q^{KS}_j\rangle_{\Phi(t)}$. The force balance coefficient matrix, $K= \langle \hat K\rangle_{\Phi(t)}$, is defined through \eqref{eq:2} and \eqref{eq:K}.
Since the target density enters only through the second derivative appearing in $S^{aim}$, the initial state $\Phi(t_0)$ must reproduce the initial density, $\langle \hat n\rangle_{\Psi(t_0)}$, and the initial time-derivative of the density, $\partial_t \langle \hat n\rangle_{\Psi(t_0)}$.
The system is non-interacting $V$-representable so long as $K$ is invertible on the domain of spatial inhomogeneous potentials. Moreover, the Kohn-Sham potential is unique~\cite{Farzanehpour12}. Hence, the domain of $V$-representability is $\Omega=\left\{\Phi\;|\;\textrm{kern}\; K(\Phi)=\{V_{const}\}\right\}$. To ensure efficiency, we must further restrict attention to the interior of this domain where $K$ is sufficiently well-condition with respect to matrix inversion. The cost of the algorithm grows exponentially as one approaches this boundary but can in some cases be mitigated by increasing the number of lattice points.
\begin{figure}[t!]
\includegraphics[width =\columnwidth]{figure1.pdf}
\caption{In part \textbf{a}, the quantum computer takes as inputs the initial state and the time-dependent Hamiltonian and outputs the density at sufficiently many times. The output allows the numerical computation of the second derivative of the density at each time step which is then utilized by the classical computer to solve the discrete force balance equation \eqref{eq:K}. A consistent initial state at time $t=0$ must also be given which reproduces $n(0)$ and $\partial_t n(0)$. Note that while the wave function is obtained from the quantum computation, it cannot be processed for use in the classical part of the computation. The classical algorithm uses the density to obtain the Kohn-Sham potential at each subsequent time step through an iterated marching process as depicted in part \textbf{b}.
}
\label{fig:algorithm}
\end{figure}
\medskip
\section{Results overview}
\subsection{Quantum algorithm for the Kohn-Sham potentials}
We consider an algorithm to compute the density with error $\epsilon$ in the 1-norm to be efficient
when the temporal computational cost grows no more than polynomially in $1/\epsilon$, polynomially in $(\max_{0<s<t}\|H(s)\|)t$, polynomially in $M$, the number of sites, and polynomially in, $N$, the number of electrons. We will describe such an algorithm within the interior of the domain of $V$-representability.
To ensure that the algorithm is efficient, we must assume that the local kinetic energy and the local potential energy are both bounded by constant $E_L$ and that there is a fixed number, $\kappa$ such that $ \|K^{-1}\|_{\infty}=\max_i \sum_j |(K^{-1})_{ij}|\leq \kappa$. Note that, as we work in the Fock space, this condition does not preclude Coulombic interactions with nuclei so long as the site orbitals have finite spatial extent.
We will show that as long as $E_L\leq\sqrt{\log N}$, the algorithm remains efficient for fixed $\kappa$.
As is typical in numerical matrix analysis~\cite{Horn05,Golub13}, the inversion of a matrix become extremely sensitive to errors as the condition number, $C=\|K\|\;\|K^{-1}\|$, grows. The Lipschitz constant of the Kohn-Sham potential must also scale polynomially with the number of electrons.
The Lipschitz constant of the Kohn-Sham system could be different than that of the interacting system~\cite{Elliott12,Maitra10} and understanding of the relationship between these timescales requires a better understanding of the initial state $\Phi(t_0)$ dependence. What can be done, in practice, is to begin with an estimate of the maximum Lipschitz constant and if any two consecutive Kohn-Sham potentials violate this bound, restart with a larger Lipschitz constant.
Our efficient algorithm for computing the time-dependent potential, is depicted in Figure~\ref{fig:algorithm}. There are two stages. The first stage involves a quantum computer and its inputs are the initial many-body state $\Psi(t_0)$ and the external potential $V(t)$ on a given interval $[t_0,t_1]$. The quantum computer then evolves the initial state with the given external potential and obtains the time-evolved wave function at a series of discrete time-steps. The detailed analysis of the expectation estimation algorithm found in Ref.~\cite{Knill07} is used to bound errors in the measurement of the density and to estimate its second time derivative. In order to rigorously bound the error term, we assume that the fourth time derivative of the density is bounded by a constant, $c_4$.
The total cost of both stages of the algorithm is dominated by the cost of obtaining the wave function as this is the only step that depends directly on the number of electrons. Fortunately, quantum computers can perform time-dependent simulation efficiently~\cite{Wiebe10,Poulin11,Berry13}. The cost depends on the requested error in the wave function, $\delta_\psi$, and depends on the length of time propagated when time is measured relative to the norm of the Hamiltonian being simulated. The essential idea is to leverage the evolution of a controllable system (the quantum computer) with an imposed (simulation) Hamiltonian~\cite{Kassal11}. It should be highlighted that obtaining the density through experimental spectroscopic means is equivalent to the quantum computation provided the necessary criteria for efficiency and accuracy are satisfied.
The second stage involves only a classical computer, with the inputs being a consistent initial Kohn-Sham state $\Phi(t_0)$ and the interacting $\partial_t^2 \langle \hat n\rangle_{\Psi(t)}$ on the given interval $[t_0,t_1]$. The output is the Kohn-Sham potential at sufficiently many time steps to ensure the target accuracy is achieved.
The classical algorithm performs matrix inversion of a $M$ by $M$ matrix. The cost for the matrix inversion is $O(M^3)$ regardless of the other problem parameters (such as the number of electrons).
In our analysis detailed in the next section, we only consider errors from the quantum and classical aspects of our algorithm and we avoided some unnecessary complications by omitting detailed analysis of the classical problem of propagating the non-interacting Kohn-Sham system. Kohn-Sham propagation in the classical computer is well studied and can be done efficiently using various methods \cite{Castro04}. Further, we have also assumed that errors in the measured data are large enough that issues of machine precision do not enter. Thus, we have ignored the device dependent issue of machine precision in our analysis and refer to standard treatments~\cite{Horn05,Golub13} for the proper handling of this issue.
\medskip
\subsection{Overview of error bounds}
We demonstrate that our algorithm has the desired scaling by bounding the final error in the density. We follow an explicit-type marching process to obtain the solution at time $q\Delta t$ from the solution at $(q-1)\Delta t$. The full technique is elaborated in the next section.
As the classical matrix inversion algorithm at each time step is independent of the number of electrons and the quantum algorithm requires $\textrm{poly}(N,t_1-t_0,\delta_\psi^{-1},\epsilon^{-1})$ per time step (recall that $\delta_\psi$ is the allowed error in the wave function due to the quantum simulation algorithm), we can utilize error analysis for matrix inversion and an explicit marching process to get a final estimate of the classical and quantum costs for the desired precision $\epsilon$
\begin{eqnarray}
\textrm{cost }&Classical&=\textrm{poly}(L,t_1-t_0,\epsilon^{-1},M){e^{64 \kappa E_L^2}}\label{eq:13}\\
\textrm{cost }&Quantum&=\textrm{poly}(L,t_1-t_0,\epsilon^{-1},r,M,N)\; {e^{16\kappa E_L^2}}\label{eq:14}\phantom{spc}
\end{eqnarray}
The parameter $r$ is the number of repetitions of the quantum measurement required to obtain a suitably large confidence interval. We define the $V$-representability parameter as $R=\kappa E_L^2$ and if $R$ is bounded by a constant, then the algorithm is efficient.
The intractability of the algorithm with growing $R$ indicates the breakdown of $V$-representability.
Despite the exponential dependence of the algorithm on the representability parameter,
the domain of $V$-representability is known to encompass all time-analytic Kohn-Sham potentials in the continuum limit~\cite{Baer08,Li08,Ruggenthaler11,Ruggenthaler12}. Examining the exponential dependence, it is clear that increases in $\kappa$ can be offset by decreases in the local energy.
\section{Derivation of error bounds}
\subsection{Description of techniques used to bound cost}
Before diving into the details, let us give an overview of our techniques and what is to follow. In the first subsection, we look at the error in the wave function at time $t$. In each time step, the error is bounded from the errors in the previous steps. This leads to a recursion relation which we solve to get a bound for the total error at any time step. This error is propagated forward because we must solve $KV=S=Q+\partial_t^2 n$ for $V$ based on the data from the previous time step. The error in $\partial_t^2 n$ is due to the finite precision of the quantum computation and is independent of previous times. In the second subsection, the error in the density is then derived followed by a cost analysis in the final subsection.
We rescale time by factor $c$ such $t_1-t_0=1$ to get the final time step $z=1/\Delta t$. This rescaling is possible because there is no preferred units of time.
That said the rescaling of time cannot be done indefinitely for two reasons. First, the Lipschitz constant of both the real and the KS system must be rescaled by same factor of $c$. Since the cost
of the algorithm depends on the Lipschitz constant, increasingly long times will require more resources. Second, the quantum simulation algorithm does have an intrinsic time scale
set by the norm of the $H$ and its time derivatives~\cite{Wiebe10,Poulin11,Berry13}. Rescaling time by $c$ increases the norm of $H$ by the same factor; consequently, the difficulty of
the quantum simulation is invariant to trivial rescaling of the dynamics.
It is important to get estimates which do not directly depend on the number of sites. To do this, we assume that the lattice is locally connected under the hopping term such that there are at most $d$ elements per row of $T$ (since $T$ is symmetric, it is also $d$-col-sparse). This is equivalent to a bound for the local kinetic energy.
Throughout, we work with the matrix representations of the operators and the states.
The $L_p$ vector norms \cite{Horn05} with $p=1$, $2$, and $\infty$ are defined by $|x|_p=\left(\sum |x_i|^p \right)^{1/p}$.
The induced matrix norms are defined by $\|A\|_p=\max_{|x|_p=1}|Ax|_p$. Induced norms are important because they are compatible with the vector norm such that $|Mx|_p=\|M\|_p|x|_p$.
The vector 1-norm is appropriate for probability distributions and the vector 2-norm is appropriate for wave functions. The matrix 2-norm is also called the spectral norm and is equal to the maximum absolute value of an eigenvalue. For a diagonal matrix, $D$, the matrix 2-norm is the vector $\infty$-norm of $\textrm{diag}(D)$. Note that $|x|_p\geq|x|_{p'}$ for $p<p'$. Important, non-trivial characterizations of the infinity norms are $|x|_\infty=\max_i |x_i|$ and $\|A\|_\infty=\max_i \sum_j |A_{ij}|$.
\subsection{Error in the wave function via recursion relations}
We bound the error of the evolution operator from time $k\Delta t$ to $(k-1)\Delta t$, denoted $\|\Delta U({k,k-1)}\|_2$, in terms of the previous time step in order
to obtain a recursion relation. We first bound the errors in the potential
due to the time discretization and then those due to the computation errors using Lemma \ref{lem:unitary} found in~\ref{C}. The computation errors will depend on the error at the previous time step which will lead to the recursion relation sought after.
To bound the error in $\|\Delta U\|_2$ we must bound the error in the potential
$|\Delta V|_\infty\leq|\Delta V^{\Delta t}|_\infty+|\Delta V^{comp}|_\infty$. We define $ V^{\Delta t}(t)=V({t_k})$ with
$k$ such that $|t-t_k|\leq |t-t_m|$ for all $m$. Here, $\{V(t_k)\}$ is the discretized potential
with time step $|t_j-t_{j+1}|=\Delta t$. The error due to temporal discretization can be controlled
assuming a Lipschitz constant $L$ for the potential such that for all $t$ and $t'$,
$|V(t)-V({t'})|_\infty/|t-t'|\leq L$. Thus, for all $t$,
\begin{equation}
|\Delta V^{\Delta t}|_\infty=|V(t)-V^{\Delta t}(t)|_\infty\leq L\Delta t.
\end{equation}
The computational error
$|\Delta V^{comp}|_\infty$ is bounded using Lemma \ref{lem:linalg} in~\ref{C} with $\|K^{-1}\|_\infty\leq \kappa$ and the assumption $|V|_\infty\leq E_L$,
\begin{equation}
|\Delta V^{comp}|_\infty\leq \kappa \left( |\Delta Q|_\infty +|\Delta \partial_t^2 n|_\infty+\|\Delta K\|_\infty E_L \right)
\label{eq:comp}
\end{equation}
Now we need to bound the errors in $|\Delta Q|_\infty$ and $\|\Delta K\|_\infty$ in terms of the error $\delta^\Gamma_k=\max_{ij}|\Delta \Gamma_{ij}(k-1)|$ at time step $k-1$.
The error bound for $|\Delta Q|_\infty$ is obtained as
\begin{eqnarray}
|\Delta Q|_\infty&\leq& \max_i|([T,\Delta\Gamma]T)_{i}|\\
&\leq&\max_i \left|\sum_{pq}T_{ip}\Delta\Gamma_{pq}T_{qi}-\sum_{mn}\Delta\Gamma_{im}T_{mn}T_{ni}\right|\nonumber\\
&\leq&
2\delta_{k-1}^\Gamma d^2 \left(\max_{ij}|T_{ij}|\right)^2\nonumber\\
|\Delta Q|_\infty&\leq&2\delta_{k-1}^\Gamma E_L^2
\label{eq:Q}
\end{eqnarray}
The product $d\max |T_{ij}|$ is the maximum local kinetic energy and is, by assumption, bounded by $E_L$. Similarly,
\begin{eqnarray}
\|\Delta K\|_{\infty}&=&\max_i\sum_j|K_{ij}-\tilde{K}_{ij}|\\
&=&
\max_i\sum_j |T_{ij}\Delta\Gamma_{ij}-\delta_{ij}\sum_m T_{mj}\Delta\Gamma_{mj}|\nonumber\\
&\leq& \max_i\sum_j |T_{ij}\Delta\Gamma_{ij}|+\max_i\left|\sum_m T_{mi}\Delta\Gamma_{mi}\right|\nonumber\\
&\leq&\delta_{k-1}^\Gamma \max_i\sum_j |T_{ij}|+\delta_k^\Gamma\max_i\left|\sum_m T_{mi}\right|\nonumber\\
&\leq&2d\delta_{k-1}^\Gamma \left( \max_{ij}|T_{ij}| \right)\nonumber\\
\|\Delta K\|_{\infty}&\leq&2\delta_{k-1}^\Gamma E_L\label{eq:Kerr}
\end{eqnarray}
We convert from errors in the real part of the 1-RDM to errors in the wave function via
\begin{eqnarray}
\delta^{\Gamma_{ij}}&=& |\Delta\Gamma_{ij}|\nonumber\\
&\leq& |(\bra{\Phi}\Gamma_{ij})\ket{\Delta\Phi}|+|\bra{\Delta\Phi}(\Gamma_{ij}\ket{\Phi})|\label{eq:expn}\\
&\leq& 2|\Delta\Phi|_2\;|\Gamma_{ij}\ket{\Phi}|_2\leq 2 |\Delta\Phi|_2\;\|\Gamma_{ij}\|_2\nonumber\\
&\leq&4|\Delta\Phi|_2\label{eq:ev}
\end{eqnarray}
The inequality \eqref{eq:ev} follows because the maximum eigenvalue of $\langle a_i^\dag a_j\rangle_\psi$ for all $\psi$ is bounded by $1$ and $\Gamma_{ij}=2\textrm{ real}\langle a_i^\dag a_j\rangle_\psi$.
Taking the maximum over all $i$, $j$ we have
\begin{equation}
\delta_{k-1}^\Gamma=\max_{ij} (\delta^{\Gamma_{ij}}_{k-1}) \leq4\delta_{k-1}^\Phi\label{eq:G}
\end{equation}
Here $\delta_{k-1}^\Phi$ bounds the error in the two-norm $|\Delta \Phi|_2$ at time step $k-1$.
Putting together \eqref{eq:comp}, \eqref{eq:Q}, \eqref{eq:Kerr}, and \eqref{eq:G} gives
\begin{eqnarray}
|\Delta V^{comp}|_\infty\leq16\kappa E_L^2\delta_{k-1}^{\Phi}+\kappa|\Delta \partial_t^2n|_\infty
\label{eq:comp2}
\end{eqnarray}
To obtain the desired recursion relation, we note that at time step $k$ the error can be bounded via
\begin{eqnarray}
|\Phi(k)-\tilde{\Phi}(k)|_2&\leq& \|\Delta U({k,k-1})\|_2
+\delta_{k-1}^\Phi
\end{eqnarray}
obtained using an expansion similar to the one found in \eqref{eq:expn}.
Utilizing Lemma \ref{lem:unitary} (see~\ref{C}) and bound \eqref{eq:comp2}, we arrive at
\begin{eqnarray}
|\Phi(k)-\tilde{\Phi}(k)|_2
&\leq&\delta^\Phi_{k-1}+\Delta t|\Delta_{k,k-1}V|_\infty\nonumber\\
&\leq&\delta^\Phi_{k-1}+\Delta t(|\Delta V^{\Delta t}|_\infty+|\Delta V^{comp}|_\infty)\nonumber\\
&\leq&\delta^\Phi_{k-1}+\Delta t(L\Delta t+16\kappa E_L^2\delta_{k-1}^{\Phi}+\kappa|\Delta \partial_t^2n|_\infty)\nonumber\\
&\leq&(16\kappa E_L^2\Delta t+1)\delta_{k-1}^{\Phi}\nonumber\\
&&+\Delta t(L\Delta t+\kappa|\Delta \partial_t^2n|_\infty)\label{eq:prerecus}
\end{eqnarray}
To obtain a recursion relation we let the LHS of \eqref{eq:prerecus} define the new upper bound at time step $k$.
Recursion relations of the form $f_k=af_{k-1}+b$ have closed solution
$f_k=b(a^k-1)(a-1)^{-1}$. Thus, we have for the bound at time step $k$
\begin{eqnarray}
\delta_k^\Phi
&=&\frac{L\Delta t+\kappa|\Delta \partial_t^2n|_\infty}{16\kappa E_L^2} \left\{(16\kappa E_L^2\Delta t+1)^k-1\right\}
\end{eqnarray}
Now consider the final time step at $z=1/\Delta t$, and $e^x\geq(xz^{-1}+1)^z$ for $z<\infty$,
\begin{eqnarray}
\delta_z^\Phi&=&\frac{L\Delta t+\kappa|\Delta \partial_t^2n|_\infty}{16\kappa E_L^2} \left\{\left(\frac{16\kappa E_L^2}{z}+1\right)^z-1\right\}\\
&\leq&\left( \frac{1}{z}\frac{L}{16\kappa E_L^2}+\frac{|\Delta \partial_t^2n|_\infty}{16E_L^2}\right) \left\{e^{16\kappa E^2_L}-1\right\}\\
&\leq&\left( \frac{1}{z}\frac{L}{16\kappa E_L^2}+\frac{\sqrt{2c_4\delta_n}}{16E_L^2}\right) \left\{e^{16\kappa E^2_L}-1\right\}\label{eq:wferr}
\end{eqnarray}
We applied Lemma \ref{lem:ddot} from~\ref{C} to obtain the last line.
This bound is similar to the Euler formula for the global error but arises from the iterative
dependence of the potential on the previous error; not from any approximate solution to
an ordinary differential equation.
To ensure that the cost is polynomial in $M$ and $N$ for fixed $\kappa$, we must insist that $E_L\leq \sqrt{\log N}$. Consider the exponential factor
and assume that $E_L>1$. Then $\exp(16\kappa E^2_L)\leq \exp(16\kappa \log N)=N^{16\kappa}$ is a polynomial for fixed $\kappa$.
\subsection{Error bound on the density}
To finish the derivation, we utilize our bound for the wave function at the final time to get a bound on the error of the density at the final time. This will translate into conditions for the number of steps needed and the precision required for the density.
The error in the density is bounded by the error in the wave function through the following,
\begin{eqnarray*}
|\Delta n|_1&=& |\bra{\Phi}n\ket{\Phi}-\bra{\tilde \Phi}n\ket{\tilde\Phi}|_1\\
&=&|\bra{\Phi}n\ket{\Phi}-\bra{\Phi}n\ket{\tilde \Phi} +\bra{\Phi}n\ket{\tilde \Phi} -\bra{\tilde \Phi}n\ket{\tilde\Phi}|_1\\
&\leq&|\bra{\Phi}n\ket{\Delta\Phi} |_1+|\bra{\Delta\Phi}n\ket{\Phi}|_1
\end{eqnarray*}
Now consider the $i$-th element, $n_i=a_i^\dag a_i$, and the Cauchy-Schwarz $|\bra{x}y\rangle| \leq|x|_2\;|y|_2$,
\begin{eqnarray*}
\left|\left(\bra{\Phi}a_i^\dag a_i\right)\ket{\Delta\Phi}\right|&\leq& \left|\bra{\Phi}a_i^\dag a_i\right|_2\; \left|\Delta\Phi\right|_2
\leq\|a_i^\dag a_i\|_2\; |\Delta\Phi|_2\\
|\bra{\Phi}n_i\ket{\Delta\Phi} |_1 &\leq &|\Delta\Phi|_2
\end{eqnarray*}
Finally, from the definition of the 1-norm,
\begin{eqnarray}
|\Delta n(z)|_1&\leq& \sum_i \left(|\bra{\Delta\Phi(z)}n_i\ket{\tilde\Phi(z)}|+|\bra{\Phi(z)}n_i\ket{\Delta\Phi(z)}|\right)\nonumber\\&\leq&2M|\Delta\Phi(z)|_2\leq2M\delta_z^\Phi
\label{eq:err1}
\end{eqnarray}
For final error $\epsilon$ in the 1-norm of the density, we allow error $\epsilon/2$ due to the time step error and
$\epsilon/2$ error due to the density measurement. Following \eqref{eq:wferr} and \eqref{eq:err1}, we have for the number of time steps,
\begin{equation}
\left( \frac{ML}{4\epsilon\kappa E_L^2}\right) \left\{e^{16\kappa E^2_L}-1\right\} \leq z.
\label{eq:z}
\end{equation}
The bound for the measurement precision also follows as,
\begin{equation}
\left( \frac{\sqrt{2}Mc_4^{1/2}}{4\epsilon E_L^2}\right)^{2} \left\{e^{16\kappa E^2_L}-1\right\}^{2} \leq\delta_n^{-1}
\label{eq:n}
\end{equation}
\subsection{Cost analysis}
To obtain the cost for the quantum simulation and the subsequent measurement, we leverage detailed analysis of the expectation estimation algorithm \cite{Knill07}. To measure the density at time $t\in[t_0, t_1]$, a quantum simulation~\cite{Wiebe10,Poulin11,Berry13} of $\psi({t_0})\mapsto\psi(t)$ is performed at cost $q\leq \textrm{poly}(N,t_1-t_0,\delta_\psi^{-1})$ following an assumption that $H(t)$ is simulatable on a quantum computer which is usually the case for physical systems. In order to simplify the analysis, we assume that $\delta_\psi$ is such that $\delta_n+\delta_\psi\approx\delta_n$ is a reasonable approximation. Given the recent algorithm for logarithmically small errors \cite{Berry13}, this assumption is reasonable.
The expectation estimation algorithm (EEA) was analyzed in \cite{Knill07}. The algorithm EEA$(\psi,A,\delta,c)$ measures $\bra\psi A \ket \psi$ with precision $\delta$ and confidence $c$ such that Prob$(\tilde a-\delta\leq \bra\psi A \ket \psi\leq \tilde a+\delta)>c$ , that is, the probability that the measured value $\tilde a$ is within $\delta$ of $\bra\psi A\ket\psi$ is bounded from below by $c$. The idea is to use an approximate Taylor expansion: $$\bra\psi A \ket \psi\approx i\left( \bra\psi e^{-iAs}\ket\psi -1 \right)/s$$
The confidence interval is improved by repeating the protocol $r=|\log(1-c)|$ times. If the spectrum of $A$ is bounded by $1$, then the algorithm requires on the order $O(r/\delta^{3/2})$ copies of $\psi$ and $O(r/\delta^{3/2})$ uses of $\exp(-iAs)$ with $s=\sqrt{3\delta}/2$.
To perform the measurement of the density, we assume that the wave function is represented in first quantization~\cite{Kassal11} such that the necessary evolution operator is: $\exp(-i\hat n_j s)=\prod_k^N\exp(-i \ket{j}\bra{j}^{(k)}t)$. Here each Hamiltonian $\ket{j}\bra{j}^{(k)}$ acts on site $j$ of the $k$th electron simulation grid. Hence, each operation is local with disjoint support. Since there are $NM$ sites, this can be done efficiently. Comparing the costs, we will assume that the generation of the state dominates the cost.
Combining these facts, we arrive at the conclusion that the cost to measure the density to within $\delta_n$ precision is
\begin{eqnarray}
\textrm{cost }{Quantum}&=& \textrm{cost }StateGen+\textrm{cost }EEA\nonumber\\
&\approx& \textrm{cost }StateGen\nonumber\\
&=& O\left(rq\delta_n^{-3/2}\right)\phantom{spc}
\label{eq:q0}
\end{eqnarray}
Pairing this with \eqref{eq:z} and \eqref{eq:n}, we have an estimate for the number of quantum operations
\begin{eqnarray*}
\textrm{cost } Quantum &=& O\left(rqz\delta_n^{-3/2}\right)\\
&=&\textrm{poly}(L,\epsilon^{-1},r,M,N)\; {e^{64\kappa E_L^2}}
\end{eqnarray*}
The classical computational algorithm is an $[M\times M]$ matrix inversion at each time step costing
\begin{eqnarray*}
\textrm{cost } Classical &=& O(z M^3)\\
&=&O\left(M^3\left( \frac{ML}{4\epsilon\kappa E_L^2}\right) \left\{e^{16\kappa E^2_L}-1\right\}\right)\\
&=& \textrm{poly}(L,\epsilon^{-1},M)e^{16\kappa E^2_L}
\end{eqnarray*}
\medskip
\section{Quantum computation and the computational complexity of TDDFT}
Since the cost of both the quantum and classical algorithms scale as a polynomial of the input parameters, we can say that this is an efficient quantum algorithm for computing the time-dependent Kohn-Sham potential. Therefore, the computation of the Kohn-Sham potential is in the complexity class described by bounded error quantum computers running in polynomial time (BQP). This is the class of problems that can be solved efficiently on a quantum computer.
Quantum computers have long been considered as a tool for simulating quantum physics~\cite{Feynman82,Lloyd96,Brown10,Kassal11,Yung12}. The applications of quantum simulation fall into two broad categories: (1) dynamics~\cite{Zalka98,Lidar99,Kassal08} and (2) ground state properties~\cite{Somma02,Aspuru05,Whitfield11}. The first problem is in the spirit of the original proposal by Feynman~\cite{Feynman82} and is the focus of the current work.
Unfortunately, unlike classical simulations, the final wave function of a quantum simulation cannot be readily extracted due to the exponentially large size of the simulated Hilbert space. The retrieval of the full state would require quantum state tomography, which in the worst case, requires an exponential number of copies of the state and would take an exponentially large amount of space to even store the data classically. If, instead, the simulation results can be encoded into a minimal set of information and the simulation algorithm can be efficiently executed on a quantum computer, then the problem is in the complexity class BQP. Extraction of the density~\cite{Knill07} is the relevant example of such a quantity that can be obtained. Note that the density's time-evolution is dictated by wave function and hence the Schr\"odinger equation.
In summary, what we have proven is that computing the Kohn-Sham potential at bounded $\kappa E_L^2$ is in the complexity class BQP. To be precise, two technical comments are in order. First, we point out that we are really focused on promise problems since we require constraints on the inputs to be satisfied (i.e. $\kappa E_L^2<$constant). Second, computing the map \eqref{eq:KSmap} is not a decision problem and cannot technically be in the complexity class BQP. However, we can define the map to $b$ bits of precision by solving $M\log b$ accept-reject instances from the corresponding decision problem, which is in BQP. These concepts are further elaborated in \cite{Kitaev02,Watrous09,Whitfield13}.
While the quantum computer would allow most dynamical quantities to be extracted without resorting to the Kohn-Sham formalism, we have attempted to understand the difficulty of generating the Kohn-Sham potential. We only consider a polynomial time quantum computer as a tool for reasoning about the complexity of computing Kohn-Sham potentials. In essence, the Kohn-Sham potentials are a compressed classically tractable encoding of the quantum dynamics that allows the quantum simulation to be performed in polynomial time on a classical computer. This may have implications for the question of whether a classical witness can be used in place of quantum witness in the quantum Merlin Arthur game~\cite{Watrous09} (i.e. QMA$\stackrel{?}{=}$QCMA). A second useful by-product of our result is the introduction of the $V$-representability parameter which has general significance for practical computational settings.
\medskip
\section{Concluding remarks}
In this article, we introduced a $V$-representability parameter and have rigorously demonstrated two fundamental results concerning the computational complexity of time dependent density functional theory with bounded representability parameter. First, we showed that with a quantum computer, one need only provide the initial state and external potential on the interval $[t_0, t_1]$ in order to generate the time-dependent Kohn-Sham potentials. Second, we show that if one provides the density on the interval $[t_0, t_1]$, the Kohn-Sham potential can be obtained efficiently with a classical computer.
We point out that an alternative to our lattice approach may exist using tools from partial differential equations. Early results in this direction have been pioneered using an iterated map whose domain of convergence defines $V$-representability~\cite{Ruggenthaler11,Ruggenthaler12}. The convergence properties of the map have been studied in several one-dimensional numerical examples~\cite{Nielsen13, Ruggenthaler11,Ruggenthaler12}. Analytical understanding of the rate of convergence to the fixed point would complement the present work with an alternate formulation directly in real space.
While this paper focuses on the simulation of quantum dynamics, the complexity of the ground state problem is interesting in its own right \cite{Kitaev02,Watrous09,Whitfield13, Schuch09}. In this context, ground state DFT was formally shown \cite{Schuch09} to be difficult even with polynomial time quantum computation. Interestingly,
in that work, the Levy-minimization procedure \cite{Levy79} was utilized for the interacting system to avoid discussing the non-interacting ground state Kohn-Sham system and its existence. We have worked within the Kohn-Sham picture, but it may be interesting to construct a functional approach directly.
Future research involves improving the scaling with the condition number or showing that our observed exponential dependence on the representability parameter is optimal. Our work can likely be extended to bosonic and spin systems~\cite{Tempel12} since we have relied minimally on the fermionic properties of electrons. Finally, pre-conditioning the matrix $K$ can also help increase the domain of computationally feasible $V$-representability.
Our findings provide further illustration of how the fields
of quantum computing and quantum information can contribute to our understanding of
physical systems through the examination of quantum complexity theory.
\subsection*{Acknowledgements:} We appreciate helpful discussions with F. Verstraete and D. Nagaj. JDW thanks Vienna Center for Quantum Science and Technology for the VCQ Postdoctoral Fellowship and acknowledges support from the Ford Foundation. MHY acknowledges funding support from the National Basic Research Program of China Grant 2011CBA00300, 2011CBA00301, the National Natural Science Foundation of China Grant 61033001, 61061130540. MHY, DGT, and AAG acknowledge the National Science Foundation under grant CHE-1152291 as well as the Air Force Office of Scientific Research under grant FA9550-12-1-0046. AAG acknowledges generous support from the Corning Foundation.
\section*{Bibliography}
|
2,877,628,088,998 | arxiv | \section{Introduction}
\label{s1}
Intensity mapping with the neutral hydrogen (\ion{H}{I}~) 21-cm line is a progressing tool to probe the large scale structures of the post-reionization Universe \citep{BNS, BS01}. It can independently assess the expansion history of the Universe by measuring the Baryon Acoustic Oscillation (BAO) in the 21-cm power spectrum (PS) \citep{w08, Chang08, Seo_2010}. It is also possible to constrain various cosmological parameters using measurements of the 21-cm PS without reference to the BAO \citep{Bh09, Visbal_2009}. Further it is possible to quantify higher order statistics, such as bispectrum, to study non-Gaussianity \citep{BA5, Hazra2012}.
There has been several successful 21-cm intensity mapping experiments \citep{Pen2009a, chang10, masui2013, SW13, Anderson2018, Wolz2021} at low-redshifts $(z<1)$. Most of these experiments have used single dish telescopes, and they have detected the 21-cm signal by cross-correlating their measurements with existing galaxy redshift surveys. \citet{SW13} have detected the 21-cm intensity mapping signal in auto-correlation at $z=0.8$ using the Green Bank Telescope. Recently, CHIME\footnote{\url{https://chime-experiment.ca/en/}} \citep{chime22} has made the first interferometric 21-cm intensity mapping measurements in the redshift range $0.78<z<1.43$ using cross-correlations with luminous red galaxies, emission line galaxies, and quasars. Several other radio interferometers such as BINGO\footnote{\url{https://bingotelescope.org/}} \citep{Wuensche_2019}, the Tianlai project\footnote{\url{http://tianlai.bao.ac.cn/}} \citep{tian}, HIRAX\footnote{\url{https://hirax.ukzn.ac.za/}} \citep{Newburgh16} and MeerKAT\footnote{\url{https://www.sarao.ac.za/science/meerkat/}} \citep{Kennedy21} particularly focus on measuring the BAO to study the nature of Dark Energy. The Ooty Radio Telescope (ORT; \citealt{GS71}) is being upgraded to the Ooty Wide Field Array (OWFA\footnote{\url{http://rac.ncra.tifr.res.in/ort.html}}; \citealt{OWFA}) to measure the $21$-cm PS at $z\sim3.35$. Furthermore, the next-generation intensity mapping surveys, with the Square Kilometer Array (SKA\footnote{\url{https://www.skatelescope.org/}}; \citealt{SKA15}) hold the potential to provide a large cosmological window to the post-reionization era.
The Giant Metrewave Radio Telescope (GMRT\footnote{\url{http://www.gmrt.ncra.tifr.res.in/}}; \citealt{swarup91}) is sensitive to the post-reionization 21-cm intensity mapping signal from a broad redshift range ($z \le 6$; \citealt{Bharadwaj-Pandey-2003, bh_sri2004}). In an effort towards this \cite{ghosh1, ghosh2} have analyzed $610 \, {\rm MHz}$ GMRT data to place an upper limit $[\bar{x}_{\rm \ion{H}{I}~} \, b_{\rm \ion{H}{I}~}] < 2.9$ where $\bar{x}_{\rm \ion{H}{I}~}$ and $ b_{\rm \ion{H}{I}~}$ are the mean neutral fraction and bias parameter respectively at redshift $z=1.32$. We note that this corresponds to $[\Omega_{\rm \ion{H}{I}~} b_{\rm \ion{H}{I}~}] < 0.11$ where $\Omega_{\rm \ion{H}{I}~}$ is the comoving \ion{H}{I}~ mass density in units of the present critical density. In a recent work \cite{Ch21} (hereafter Ch21) have analyzed several $8 \, {\rm MHz} $ subsets drawn from $200 \, \rm{MHz}$ upgraded GMRT (uGMRT; \citealt{uGMRT}) data to estimate the 21-cm PS at multiple redshifts in the range $1.96<z<3.58$ and place the upper limits $[\Omega_{\rm \ion{H}{I}~} b_{\rm \ion{H}{I}~}] < 0.09, 0.11,0.12, \, {\rm and} \, 0.24$ at $z=1.96,2.19,2.62 \, {\rm and} \, 3.58$ respectively.
The Tapered Gridded Estimator (TGE; \citealt{samir14, samir16}) is a visibility based PS estimator which allows us to taper the sky response to suppress the contribution from bright sources located in the side-lobes and the periphery of the telescope's field of view. Further, the TGE works with the gridded visibilities which makes it computationally fast. The TGE also uses the measured visibility data to internally evaluate the noise bias and subtracts this to provide an unbiased estimate of the PS. \cite{Bh18} (hereafter, Paper I) have proposed the TGE to estimate $C_{\ell}(\Delta\nu)$ the multi-frequency angular power spectrum (MAPS; \citealt{KD07, Mondal19}) which characterizes the second order statistics of the sky signal jointly as a function of the angular multipole $\ell$ and frequency separation $\Delta\nu$. They use $C_{\ell}(\Delta\nu)$ to determine the cylindrical power spectrum of the 21-cm brightness temperature fluctuations $P(k_{\perp},k_{\parallel})$ which is related to $C_{\ell}(\Delta \nu)$ through a Fourier transform with respect to $\Delta \nu$. Using simulated visibility data, they show that this estimator can accurately recover the input model PS even when $80\%$ randomly chosen frequency channels are flagged. A salient feature of this estimator is that it only uses the available data, and it is not necessary to make any assumption regarding the data values in the missing frequency channels. In a recent paper \citet{Pal20} (hereafter Paper II) have demonstrated the capabilities of the TGE by using the TGE to estimate $P(k_{\perp},k_{\parallel})$ from a 150 MHz GMRT observational data where $47 \%$ of the frequency channels are flagged due to Radio Frequency Interference (RFI). They obtain a $ 2 \sigma$ upper limit of $(72.66)^2\,{\rm K}^{2}$ on the mean squared \ion{H}{I}~ 21-cm brightness temperature fluctuations at $k = 1.59\, {\rm Mpc}^{-1}$. We note that the two dimensional (2D) TGE for the angular power spectrum $C_{\ell}$ has been extensively used to study the foregrounds for cosmological 21-cm observations \citep{samir17a, samir20, Cha1, M20} and also magnetohydrodynamics turbulence in supernova remnants \citep{Preetha19, Preetha21}.
In this work we consider uGMRT Band 3 $(300-500\,{\rm MHz})$ data of the ELAIS N1 field. \cite{Cha2} have analysed this data and used the 2D TGE to study the angular and spectral variation of $C_{\ell}(\nu)$ for the diffuse galactic synchrotron emission. As mentioned earlier, \citetalias{Ch21} have analysed this data using a delay spectrum approach to estimate the PS of the 21-cm intensity mapping signal. The difficulty arises because the missing frequency channels (flagged due to RFI) introduce artefacts in delay space which corrupt the estimated PS. \citetalias{Ch21} have overcome this by using one dimensional complex CLEAN \citep{Parsons_2009} to compensate for the missing frequency channels. Considering the same data, the present work uses a bandwidth of $24.4\,{\rm MHz}$ centred at $432.8\,{\rm MHz}$. Here we have applied the TGE to estimate the MAPS and PS $P(k_{\perp},k_{\parallel})$. We study the capabilities of this estimator to (1) suppress the wide-field foregrounds, and (2) deal with the missing frequency channels. We also present results for the spherically binned power spectrum, and present an upper limit for $[\Omega_{\rm \ion{H}{I}~} b_{\rm \ion{H}{I}~}]$ at $z=2.28$.
The paper is arranged as follows. Section~\ref{s2} summarizes the observations and the preliminary processing of the data which has been used here. Section~\ref{s3} presents the methodology how TGE is used to estimate $C_{\ell}(\Delta \nu)$ and $P(k_{\perp}, k_{\parallel})$ from the observed visibility data, and in Section~\ref{sec:simulation} we have validated the methodology using simulations. Our results are presented in Section~\ref{s6}, and we present a summary and conclusion in Section~\ref{s7}.
Throughout this paper, unless mentioned otherwise, we have used a $\Lambda$CDM cosmology with the parameters $\Omega_{m} = 0.309$, $h = 0.67$, $n_s = 0.965$, and $\Omega_{b}h^2 = 0.0224$ which are in reasonable agreement with the present observations \citep{Planck18f}.
\section{Observations and data analysis}
\label{s2}
\begin{table}
\centering
\caption{Observation summary}
\label{t_1}
\begin{tabular}{|l|c|}
\hline
\hline
Working antennas & $28$\\
\hline
Central Frequency & $400$ MHz \\
\hline
Number of Channels & $8192$ \\
\hline
Channel width & $24.4$ kHz \\
\hline
Bandwidth & $200$ MHz \\
\hline
Total observation time & $25 $ h \\
\hline
Integration time & $2$ s \\
\hline
Target field $(\alpha,\delta)_{2000}$ & ($16^{h}10^{m}1^{s}$,\\
& $+54^{\circ}30^{'}36^{''}$) \\
\hline
Galactic coordinates $(l,b)$ & $86.95^{\circ},+44.48^{\circ}$ \\
\hline
\hline
\end{tabular}
\end{table}
The observations were carried out using the GMRT array configuration \citep{swarup91}. The recently upgraded version of GMRT (uGMRT) provides a frequency coverage of $120-1500\,\textrm{MHz}$ with $400~ \textrm{MHz}$ maximum instantaneous available bandwidth and an improved receiver system with higher $G/T_{sys}$ for high dynamic range imaging \citep{uGMRT}.
We have observed the field European Large-Area ISO Survey-North 1 (ELAIS-N1; $\alpha_{2000}=16^{h}10^{m}1^{s}, \delta_{2000}=54^{\circ}30^{'}36^{''}$) with the uGMRT at Band 3 ($300-500\,{\rm MHz}$) during May 2017 for 25 hours over four days. The primary calibrators 3C286 and 3C48 have been used to scale the overall flux of the observation. We have also used a nearby phase calibrator (J1549+506) to correct the antenna gains' temporal variation. The total bandwidth of the observation is 200 MHz with a frequency resolution of 24.4 kHz. We have taken the data with a high time resolution (2 s) to identify and remove the RFI. The observations were carried out mainly at night to minimize the RFI. The relevant observational parameters have been summarized in Table \ref{t_1}.
\begin{figure}
\centering
\includegraphics[width=\columnwidth]{Plots/uv1.pdf}
\caption{The upper panels show the uv-coverage for the May 6 data (left panel) and the combined nights data (right panel) considering baselines of length $U \leq 3000 \lambda$ at $\nu_{c}=432.8\,{\rm MHz}$. The corresponding baseline density (number of baselines per unit area of the uv plane) is shown as a function of $U$ in the lower panels.}
\label{fig:uv}
\end{figure}
The details of the initial data analysis are given in \cite{Cha1, Cha2}. Here, we briefly summarize the flagging and calibration steps adopted. For initial RFI flagging, we use the {\scriptsize AOFLAGGER} which detects any anomaly in time-frequency domain per baseline per polarization and discards the corrupt data \citep{Off10, Off12}. We discard $5\%$ of the total number of channels ($2.5\% $ on each side) due to a bad bandshape at the edge of the bandwidth. We take the direction-independent approach to calibrate the data using the Common Astronomy Software Applications ({\scriptsize CASA}; \citealt{casa07}). We start with an initial gain and bandpass calibration of the primary calibrators and remove residual RFIs from the calibrated data using the {\scriptsize RFLAG} routine of {\scriptsize CASA}. After doing this twice, we perform a final bandpass and delay calibration on the primary calibrators. We next rectify the temporal variation of the amplitude and the phase of the antenna gains using the secondary calibrator. Finally, we apply the gain solutions to the target field and split this out for imaging. The final off-source r.m.s. noise is $\sim 15\mu \textrm{Jy}\,\textrm{beam}^{-1}$ which is nearly $3.5$ times higher than the theoretically expected noise. Note that we have calculated the theoretical noise using the specifications of uGMRT for Band 3 \citep{uGMRT}, and considering the parameters of our observations. The excess noise in the image can thus be caused due to a difference in the system temperature depending on the direction of sky we have observed; this may also be caused due to residual deconvolution errors during imaging, residual calibration errors etc. Hence, we expect this excess noise to further contribute as excess power in the estimated power spectrum as residual foregrounds and systematics, as well as increase the error budget (the r.m.s. fluctuations) of the power spectrum estimated from the data \citep{JK20,JK22}. Post four rounds of self-calibration (only phase), we have identified and subtracted out the compact and discrete sources with flux densities $> 100\mu \textrm{Jy}$ within an area of $1.8 \, {\rm deg^2}$ using the task {\scriptsize UVSUB} in {\scriptsize CASA}. The residual visibility data is used for the subsequent analysis presented here.
The baseline distribution and various telescope parameters change with frequency across the $200 \, {\rm MHz}$ uGMRT bandwidth. For the subsequent analysis we have divided the total bandwidth into eight sub-bands (namely, sub-bands 0 to 7). The bandwidth of each sub-band $\sim 24.4 \, {\rm MHz}$ is less than $10 \%$ of the central frequency of the corresponding sub-band, which allows us to ignore the baseline migration and change in telescope parameters within the sub-band. This is a fair assumption considering Section \ref{sec:simulation} where we validate our estimator for the data considered here. We have checked the quality of the data in each of the sub-bands in terms of the percentage of flagged channels and the variance of the data. The May 6 observation from the sub-band 2 is found to have the least percentage of flagging and the smallest visibility r.m.s. Guided by this, we have entirely restricted the subsequent analysis of this paper to sub-band 2 which is centred at $\nu_{c}=432.8\,{\rm MHz}$ and contains 1000 channels with spectral resolution $\Delta\nu_{c}=24.4\,\,{\rm kHz}$. In addition to the individual nights data, we have also combined the four nights data using the {\scriptsize CONCAT} task of {\scriptsize CASA}. The flagging statistics and r.m.s. values of the individual nights and the combined nights data are given in Table \ref{tab:flagandrms}.
The upper panels of Figure~\ref{fig:uv} show the baseline distributions for the May 6 data (left panel) and the combined nights data (right panel) for a single channel at the central frequency. The lower panels show the baseline density, {\it i.e.}, the number of baselines per unit area of the uv-plane, as a function of baseline length $U=\lvert{\bf U}\rvert$.
\begin{table}
\centering
\caption{Considering the sub-band 2 with central frequency $\nu_{c}=432.8\,{\rm MHz}$, spectral resolution $\Delta\nu_{c}=24.4\,\,{\rm kHz}$ and bandwidth $B_{bw}=24.4\,\,{\rm MHz}$, we tabulate the flagging fraction and r.m.s. of the visibilities $\sigma_{N}$ for different nights of observation}
\label{tab:flagandrms}
\begin{tabular}{|c|c|c|}
\hline
\hline
Night of observation & flag (\%) & r.m.s. $\sigma_{N}$ (Jy)\\
\hline
\hline
May 5 & 70.97 & 0.431337 \\
\hline
May 6 & 13.07 & 0.394103 \\
\hline
May 7 & 42.75 & 0.473156 \\
\hline
May 27 & 71.29 & 0.445912 \\
\hline
All nights combined & 54.81 & 0.430112 \\
\hline
\hline
\end{tabular}
\end{table}
\section{Methodology}
\label{s3}
The multi-frequency angular power spectrum (MAPS) $C_{\ell}(\nu_a, \nu_b)$ jointly characterizes the statistical properties of the sky signal as a function of the angular multipoles and frequency. The brightness temperature fluctuations in the sky is decomposed in terms of the spherical harmonics $Y_{\ell}^{\rm m}(\hat{\bm{n}})$ as,
\begin{equation}
\delta T_{\rm b} (\hat{\bm{n}},\,\nu)=\sum_{\ell,m} a_{\ell {\rm m}} (\nu) \,
Y_{\ell}^{\rm m}(\hat{\bm{n}}) \,,
\label{eq:alm}
\end{equation}
and the MAPS is defined as
\begin{equation}
C_{\ell}(\nu_a, \nu_b) = \big\langle a_{\ell {\rm m}} (\nu_a)\, a^*_{\ell
{\rm m}} (\nu_b) \big\rangle\,
\label{eq:cl}
\end{equation}
where $\langle ... \rangle$ denotes an ensemble average over different statistically independent realizations of the random field $\delta T_{\rm b} (\hat{\bm{n}},\,\nu)$.
The details of the visibility based TGE for measuring the MAPS and the PS are given in \citetalias{Bh18} and \citetalias{Pal20}. Here, we briefly summarize the mathematical formalism for this estimator. Starting from the visibility data, $\mathcal{V}_{i}(\nu_{a})$ corresponding to the i-th baseline $\textbf{U}_{i}$ and frequency $\nu_{a}$, the TGE first convolves the measured $\mathcal{V}_{i}(\nu_{a})$ with $\tilde{w}(\u)$ which is the Fourier transform of a window function ${\cal W}(\theta)$ suitably chosen to taper the primary beam (PB) of the telescope far away from the phase center. We divide the uv plane in a rectangular grid and the convolved visibilities $\mathcal{V}_{cg}$ at the grid-point $\u_{g}$ is given by,
\begin{equation}
\mathcal{V}_{cg}(\nu_{a}) = \sum_{i}\tilde{w}(\u_g-\u_i) \, \mathcal{V}_i(\nu_{a}) \,F_i(\nu_a).
\label{eq:a1}
\end{equation}
Here the subscript `$a$' denotes an individual channel and $a=1,2....N_{c}$ where $N_{c}$ is the total number of channels that cover a bandwidth $B_{bw}$. The factor $F_i(\nu_a)$ is used to incorporate the flagging information. $F_i(\nu_a)$ is assigned a value `$0$' if the data at the baseline $\textbf{U}_{i}$ and frequency $\nu_{a}$ is flagged and $F_i(\nu_a)$ is `$1$' otherwise.
We use a Gaussian window function, ${\cal W}(\theta)=e^{-\theta^{2}/\theta^{2}_{w}}$, to taper the sky signal away from the phase center. The main lobe of the PB of any telescope with a circular aperture can be approximated as, $\mathcal{A}(\theta)=e^{-\theta^{2}/\theta^{2}_{0}}$ where $\theta_{0}\sim 0.6 \times \theta_{\rm FWHM}$, $\theta_{\rm FWHM}$ being the full width at half maxima of $\mathcal{A}(\theta)$\citep{BS01, samir14}. We choose $\theta_{w}=f\theta_{0}$, where `$f$' represents the tapering parameter and controls the degree to which the PB pattern is tapered. The convolution implemented in eq.~(\ref{eq:a1}) equivalently amounts to modulating $\mathcal{A}(\theta)$ with ${\cal W}(\theta)$ by a multiplication, where $f>1$ will provide very little tapering and $f<1$ can be used to highly suppress the PB away from the phase center.
We define the TGE for MAPS in \citetalias{Bh18} as,
\begin{align}
\hat{E}_g(\nu_a,\nu_b) &= M_g^{-1}(\nu_a,\nu_b) {\mathcal Re} \Big[\mathcal{V}_{cg}(\nu_a) \mathcal{V}_{cg}^{*}(\nu_b) \nonumber \\
& - \delta_{a,b} \, \sum_i F_i(\nu_a) \mid
\tilde{w}(\u_g-\u_i) \mid^2 | \mathcal{V}_i(\nu_a) |^2 \Big]
\label{eq:TGEI}
\end{align}
where ${\mathcal Re}[..]$ refers to real part of the expression within the brackets [..] and $M_g(\nu_a,\nu_b)$ is a normalization constant (discussed in detail later in this section). Along with the sky signal, each visibility $\mathcal{V}_{i}(\nu_{a})$ contains an additive noise component $\mathcal{N}_{i}(\nu_{a})$ that is assumed to be a Gaussian random variable with zero mean and variance $ 2 \sigma_N^2$. The noise in different baselines, frequency channels and timestamps are uncorrelated. Consequently, the noise contribution in MAPS is restricted only to the self-correlations of the visibilities
\begin{equation}
\langle \mathcal{N}_{i}(\nu_{a}) \mathcal{N}_{j}^{*}(\nu_{b}) \rangle = \delta_{i,j}\delta_{a,b} 2 \, \sigma_N^2 \,.
\end{equation}
The second term in the square brackets $[...]$ in eq.~(\ref{eq:TGEI}) subtracts out the contribution from the self correlation of a visibility {\it i.e.} same baseline, frequency channel and timestamp. This exactly cancels out the noise contribution in the first term, and we obtain an unbiased estimate of MAPS.\\
We have validated eq.~(\ref{eq:TGEI}) (hereafter referred to as TGE-I) in \citetalias{Bh18} using realistic $150\, {\rm MHz}$ GMRT simulations. We have shown there that in the absence of foregrounds TGE-I can recover an input model 21-cm PS with a very high accuracy even in the presence of noise and $80\%$ flagging in the visibility data. However, the data available from the past and current $21$-cm experiments are dominated by various foregrounds that overshadow the noise and the $21$-cm signal by a few orders of magnitude \citep{ghosh1,ghosh2}. We find (shown later) that $C_{\ell}(\nu_a,\nu_b)$ estimated by applying TGE-I to such foreground dominated data shows a discontinuity at $\nu_{a}=\nu_{b}$.
This discontinuity arises due to the self-correlation term which is subtracted out only for $\nu_a=\nu_b$ in TGE-I. We also find that this discontinuity introduces a negative bias in the estimated PS $P(k)$. To deal with this problem, we have slightly modified TGE-I in \citetalias{Pal20} to obtain
\begin{align}
\hat{E}_g(\nu_a,&\nu_b) = M_g^{-1}(\nu_a,\nu_b) {\mathcal Re} \Big[\mathcal{V}_{cg}(\nu_a) \mathcal{V}_{cg}^{*}(\nu_b) \nonumber \\
& - \sum_i F_i(\nu_a)F_i(\nu_b) \mid \tilde{w}(\u_g-\u_i) \mid^2 \mathcal{V}_i(\nu_a) \mathcal{V}_i^{*}(\nu_b) \Big]
\label{eq:a4}
\end{align}
where all the terms hold the same meaning as in TGE-I. The modified estimator in eq.~(\ref{eq:a4}) (hereafter referred to as TGE-II) differs from TGE-I in the second term within the square brackets. This term now subtracts out the self-correlation of a visibility
with itself {\it i.e.} same baseline and timestamp considering all possible combinations of frequencies $\nu_a$ and $\nu_b$.
This removes the discontinuity at $\nu_{a}=\nu_{b}$ in the estimated MAPS, and also avoids the negative bias in the estimated $P(k)$. We shall demonstrate this later in Section \ref{s4}, and we have validated TGE-II using simulations in Section~\ref{sec:simulation}.
We now consider the normalization factor $M_g(\nu_a,\nu_b)$. Here
we have used simulations to estimate the value of $M_g(\nu_a,\nu_b)$. We first simulate multiple realizations of a Gaussian random field having unit multi-frequency angular power spectrum (UMAPS; $C_{\ell}(\nu_a, \nu_b)=1$). We use this as the sky signal to simulate the corresponding visibilities $ [\mathcal{V}_i(\nu_a)]_{\rm UMAPS}$ at the baselines and frequency channels identical to the data. The flagging of the actual data $F_i(\nu_a)$ was applied to the simulated visibilities $ [\mathcal{V}_i(\nu_a)]_{\rm UMAPS}$ and these are then analyzed identically to the actual data to obtain
\begin{align}
{M}_g & (\nu_a,\nu_b) = {\mathcal Re} \Big[\mathcal{V}_{cg}(\nu_a) \mathcal{V}_{cg}^{*}(\nu_b) \nonumber \\
& - \sum_i F_i(\nu_a)F_i(\nu_b) \mid \tilde{w}(\u_g-\u_i) \mid^2 \mathcal{V}_i(\nu_a) \mathcal{V}_i^{*}(\nu_b) \Big]_{\rm {UMAPS}}.
\label{eq:a4a}
\end{align}
We average over multiple realizations of the simulated UMAPS to reduce the statistical uncertainties in the estimated values of $M_g(\nu_a,\nu_b)$. For the subsequent analysis, we have simulated $50$ realizations of UMAPS and used these to estimate $M_{g}$. Note that our estimator does not incorporate the migration of the baselines with frequency and considers the values of the baselines to be fixed at the reference frequency $\nu_{c}$.
The estimator in eq.~(\ref{eq:a4}) gives an unbiased estimate of the MAPS $\langle {\hat E}_g(\nu_a,\nu_b) \rangle =C_{\ell_g}(\nu_a,\nu_b)$ at the grid point $\u_g$, or equivalently at angular multipole $\ell_g=2\,\pi\,\mid \u_g \mid$.
To increase the signal-to-noise ratio, we further bin the entire $\ell$ range into $10\,\,\ell$ bins. The bin averaged Tapered Gridded Estimator is defined as,
\begin{equation}
{\hat E}_G[q](\nu_a,\nu_b) = \frac{\sum_g w_g {\hat E}_g(\nu_a,\nu_b)}
{\sum_g w_g } \,,
\label{eq:a6}
\end{equation}
where the sum is over all the grid points $\u_g$ in the $q$'th bin and the $w_g$'s are the corresponding weights. Here, we have used $w_g=M_g(\nu_a,\nu_b)$ which implies that the weight is proportional to the baseline density of the particular grid point. The ensemble average of ${\hat E}_G[q](\nu_a,\nu_b)$ gives an unbiased estimate of the bin averaged MAPS $\bar{C}_{\bar{\ell}_q} (\nu_a,\nu_b)$ at the effective angular multipole $\bar{\ell}_q = \frac{ \sum_g w_g \ell_g}{ \sum_g w_g}$. Throughout this work we have considered baselines within $U\le3000\lambda$ (equivalently, $\ell\le18850$) and divided this into $10\,\,\ell$ bins. The effective angular multipoles corresponding to these bins cover a range $535 \lesssim \bar{\ell}_q \lesssim 15850$. Note that $\bar{\ell}_q$ vary slightly with the value of $f$ and the values quoted in this paper have been estimated at$f=0.6$. In the subsequent discussion we have used the simplified notation $C_{\ell}(\nu_{a},\nu_{b})$ and $\ell$ to denote $\bar{C}_{\bar{\ell}_q}(\nu_a,\nu_b)$ and $\bar{\ell}_q$ respectively.
Considering a sufficiently small bandwidth of observation, the redshifted $21$-cm signal can be assumed to be statistically homogeneous (ergodic) along the line-of-sight (e.g. \citealt{Mondal19}). This allows us to express $C_{\ell}(\nu_a,\nu_b)$ in terms of ${C_{\ell}(\Delta\nu)}$ where $\Delta \nu = \mid \nu_b-\nu_a \mid$. This means that the statistical properties of the signal can now be entirely described as a function of the frequency separations $\Delta \nu$. Under the flat sky approximation, $P(k_{\perp}, k_{\parallel})$ the 3D power spectrum of the 21-cm brightness temperature fluctuations
is then given by the Fourier transform of ${C_{\ell}(\Delta\nu)}$ along the line-of-sight \citep{KD07},
\begin{equation}
P(k_{\perp},\,k_{\parallel})= r^2\,r^{\prime} \int_{-\infty}^{\infty} d (\Delta \nu) \,
e^{-i k_{\parallel} r^{\prime} \Delta \nu}\, C_{\ell}(\Delta \nu)
\label{eq:cl_Pk}
\end{equation}
where $k_{\parallel}$ and $k_{\perp}=\ell/r$ are the components of ${\mathbf k}$ respectively parallel and perpendicular to the line-of-sight, $r$ and $r^{\prime}=dr/d\nu$ are respectively the comoving distance and its derivative with respect to $\nu$, both evaluated at the reference frequency $\nu_{c}=432.8\,{\rm MHz}$. Here $r$ and $r^{\prime}$ are evaluated to have values $5703\,{\rm Mpc}$ and $9.85\,{\rm Mpc/MHz}$ respectively.
We use a maximum likelihood estimator to estimate the PS $\bar{P}(k_{\perp},k_{\parallel m})$ from the measured $C_{\ell}(n \, \Delta\nu_{c})$, where $n,\,m\, \epsilon\, [0,N_{E}-1]$ and $N_{E} = N_{c}/2$. Note that here we have used half of the available frequency separations $0\le\Delta\nu\le (N_{c}/2-1)\Delta\nu_{c}$ to avoid the poorly sampled higher frequency separations. In matrix notation,
\begin{equation}
C_{\ell}(n \, \Delta\nu_{c})= \sum_{m} \textbf{A}_{nm} \, \bar{P}(k_{\perp},k_{\parallel m}) + [\textrm{Noise}]_{n}
\label{eq:b2}
\end{equation}
where $\textbf{A}_{nm}$ are the components of the $N_{E} \times N_{E}$ Hermitian matrix $\textbf{A}$ containing the coefficients of the Fourier transform and $[\textrm{Noise}]_{n}$ is an additive noise associated with each estimated $C_{\ell}(n \, \Delta\nu_{c})$.
The maximum likelihood estimate of $\bar{P}(k_{\perp},k_{\parallel m})$ is given by,
\begin{align}
\bar{P}(k_{\perp},k_{\parallel m}) = \sum_n \{ [\textbf{A} ^{\dagger} \textbf{N}^{-1} \textbf{A}]^{-1} \textbf{A}^{\dagger} & \textbf{N}^{-1} \}_{mn} \nonumber\\ &\{\mathcal{W}_{\rm BN}(n\Delta\nu_{c}) C_{\ell}(n\Delta\nu_{c})\}
\label{eq:ML}
\end{align}
where $\textbf{N}$ is the noise covariance matrix and `$\dagger$' denotes the Hermitian conjugate. We have also introduced a Blackman-Nuttall (BN; \citealt{nut81}) window function $\mathcal{W}_{\rm BN}(n\Delta\nu_{c})$ along the $\Delta\nu$ to reduce any unwanted ripples in the estimated PS along $k_{\parallel}$ arising due to the finite bandwidth of observation.
We have estimated $\textbf{N}$ through `noise-only' simulations. As mentioned earlier, we have assumed that the noise in the visibilities are drawn from a Gaussian random distribution with zero mean and variance $2 \sigma^{2}_{N}$, and are uncorrelated at different baselines, frequencies and timestamps. We have simulated visibilities corresponding to the system noise only at baselines and frequencies identical to the actual data along with the flagging statistics. We use the value of $\sigma^{2}_{N}$ estimated from the data itself (Table \ref{tab:flagandrms}). We apply our estimator (eq.~\ref{eq:a4}) to estimate the MAPS corresponding to the simulated noise only visibilities. We generate multiple statistically independent noise-only visibility realizations to estimate the noise covariance matrix $\textbf{N}$ from the estimated MAPS. This method has been validated in \citetalias{Pal20}. The reader is referred to \citetalias{Pal20} for further details. Throughout the work, we have used $50$ noise realizations to estimate the noise covariance matrix $\textbf{N}$. Further, we have also estimated the PS $P(k_{\perp}, k_{\parallel})$ for each of these noise-only simulations, and we have determined the mean and variance $[\delta P_{N}]^{2}$ of these values. As expected, the mean is consistent with zero. We have used $[\delta P_{N}]$ to quantify the system noise contribution to the statistical fluctuations of the estimated PS of the actual data.
We use eq.~(\ref{eq:ML}) to estimate of the 3D PS $\bar{P}(k_{\perp},k_{\parallel m})$ from the measured $C_{\ell}(n\Delta\nu_{c})$. We have further binned $\bar{P}(k_{\perp},k_{\parallel m})$ along $k_{\parallel m}$ to obtain the bin averaged $P(k_{\perp}, k_{\parallel})$ which we present in the subsequent analysis. The estimated bin averaged cylindrical power spectra $P(k_{\perp}, k_{\parallel})$ span a $(k_{\perp}, k_{\parallel})$ range of $0.09\le k_{\perp} \le2.78\,\,{\rm Mpc}^{-1}$ and $0\le k_{\parallel} \le13.1$ Mpc$^{-1}$ respectively.
\subsection{A comparison between TGE-I and TGE-II}
\label{s4}
\begin{figure}
\begin{center}
\includegraphics[width=\columnwidth]{Plots/comp.pdf}
\caption{A comparison of TGE-I (blue solid lines) with TGE-II (orange dashed lines) applied on the combined nights data with $f=0.6$. The upper panels show $C_{\ell}(\Delta\nu)$ as a function of $\Delta\nu$ for two different $\ell$ values. The lower panels show slices of the estimated $P(k_{\perp}, k_{\parallel})$ as a function of $k_{\parallel}$ at a fixed value of $k_{\perp}$ corresponding to the $C_{\ell}(\Delta\nu)$ shown in the upper panels. The yellow shaded regions show $1\sigma$ errors $\delta P_{N}$ due to system noise.}
\label{fig:comp}
\end{center}
\end{figure}
In this sub-section, we briefly demonstrate the shortcoming of TGE-I (eq.~\ref{eq:TGEI}) which was originally defined in \citetalias{Bh18}, and we also show that these can be overcome by TGE-II (eq.~\ref{eq:a4}) which we have used here. We apply both estimators to the combined nights data for the tapering parameter $f=0.6$. The upper panels of Figure~\ref{fig:comp} show the estimated $C_{\ell}(\Delta\nu)$ as a function of $\Delta\nu$ for two $\ell$ values for both TGE-I and TGE-II. We have restricted the frequency range to $4\,{\rm MHz}$ in the figure to highlight the abrupt discontinuity observed at $\Delta\nu=0$ for $C_{\ell}(\Delta\nu)$ estimated with TGE-I. As discussed in Section \ref{s3}, this dip arises due to the self-correlation term which is only subtracted for $\nu_{a}=\nu_{b}$ to remove the noise bias. We see that the discontinuity is not present for TGE-II where we have subtracted out the self-correlation at all $\Delta\nu$. We also note that, as expected, the results from TGE-I and II both match for large $\Delta \nu$.
The lower panels of Figure~\ref{fig:comp} show slices of the PS $P(k_{\perp}, k_{\parallel})$ along $k_{\parallel}$ at a fixed $k_{\perp}$ estimated using eq.~(\ref{eq:ML}) from the $C_{\ell}(\Delta\nu)$ shown in the upper panels. The yellow shaded regions show the $1\sigma$ statistical fluctuations ($\delta P_{N}$) arising due to system noise. In all cases $P(k_{\perp}, k_{\parallel})$ has relatively large values at small $k_{\parallel}$ which correspond to modes within the foreground wedge. The values of $P(k_{\perp}, k_{\parallel})$ fall with increasing $k_{\parallel}$ up to $k_{\parallel}\sim4\,{\rm Mpc}^{-1}$ beyond which the results from the two estimators are quite different. For TGE-I (blue solid line), in both the panels we notice that $P(k_{\perp}, k_{\parallel})$ has negative values for $k_{\parallel} \gtrsim 4\,{\rm Mpc}^{-1}$ and the values fall to $\sim -0.8\,{\rm K}^{2}{\rm Mpc}^{3}$ at the largest $k_{\parallel}$-bins. In contrast, we find that this negative bias is absent in TGE-II (orange dashed lines) where the values of $P(k_{\perp}, k_{\parallel})$ oscillate around zero for $k_{\parallel}\gtrsim 4\,{\rm Mpc}^{-1}$. Further, we also see that these oscillations are roughly within the yellow shaded region, indicating that these are consistent with the fluctuations expected from the system noise in the data. The negative bias in TGE-I arises from the abrupt dip at $\Delta \nu=0$ seen in $C_{\ell}(\Delta\nu)$.
We have also noticed large negative values in $P(k_{\parallel})$ at a few grid points $\textbf{U}_{g}$ for TGE-II near the wedge boundary. This is mostly originating due to a combination of corrupted baselines (most possibly due to bandpass calibration errors) at a few grid points. At these grid points we also observe a small dip near $\Delta \nu=0$ in $C_{\ell}(\Delta\nu)$. At this stage, we have decided to flag these grid points, favouring less data rather than bad data. Typically about $\sim 17 \%$ of grid points are flagged. We see that the negative bias is not present for TGE-II after the flagging (Figure~\ref{fig:comp}), and we use the rest of the grid points for further analysis. We also drop the suffix ``-II'' and refer to this as TGE throughout the rest of the Paper.
\section{Simulation}
\label{sec:simulation}
\begin{figure}
\begin{center}
\includegraphics[width=\columnwidth]{Plots/cl_val.pdf}
\caption{The data points show the mean $C_{\ell}(\Delta\nu)$ with $2\,\sigma$ errors (shaded region) estimated from 16 realizations of the simulated sky signal. We have restricted $\Delta\nu$ to $\le 12.2\,\,{\rm MHz}$ in the plot, which we have used to estimate the PS. The solid lines show the analytical predictions corresponding to the input model $P^m(k)$. The dotted line shows $[C_{\ell}(\Delta \nu)]_T$ the cosmological 21-cm signal predicted at $\ell=1635$ for the $\Lambda$CDM model with $[\Omega_{\rm \ion{H}{I}~} b_{\rm \ion{H}{I}~}]=10^{-3}$. Note that the $\Delta \nu=0$ points are shifted slightly for plotting on a logarithmic scale.}
\label{cl}
\end{center}
\end{figure}
\begin{figure}
\begin{center}
\includegraphics[width=\columnwidth]{Plots/pk-val.pdf}
\caption{The upper panel shows the estimated spherically-binned power spectrum $P(k)$ (data points) and $2\,\sigma$ error-bars for the simulations along with the input model $P^{m}(k)$ (purple solid line). The bottom panel shows the fractional deviation $\delta=[P(k)-P^m(k)]/P^m(k)$ (data points) and the expected $2 \sigma$ statistical fluctuations for the same (blue shaded region). The (red) dotted lines demarcate the region where $\mid \delta \mid \le 0.05 $.}
\label{fig:pk_val}
\end{center}
\end{figure}
We have already validated TGE in \citetalias{Pal20} using $150\,{\rm MHz}$ GMRT simulations where $\sim47\%$ of the data were flagged. We found that TGE could recover the input PS with $<8\%$ fractional deviation over the entire $k$-range used for the analysis. In the present work, we have repeated a similar analysis for the sub-band 2 data, which we have analyzed here. The aim is to validate the estimator and quantify the accuracy to which TGE is expected to recover $P(k)$ for the data analyzed here.
The 21-cm brightness temperature fluctuations $\delta T_{\rm b} (\hat{\bm{n}},\,\nu)$ in the simulations are assumed to be a Gaussian random field corresponding to an input model
\begin{equation}
\label{eq:modelps}
P^{m}(\bm{k}) = A \left( \frac{k}{k_0}\right)^n {\rm mK^{2} \, Mpc^{3}}\,.
\end{equation}
where we have arbitrarily set $A=10$, $k_0 = 1 \, \mathrm{Mpc}^{-1}$, and used a power law index $n=-2$. The simulations closely follow the prescriptions of \cite{samir17} and \citetalias{Pal20}. We have carried out simulations on a $N^{3}=[1024]^3$ cubic grid with a grid spacing $\Delta L = 0.24$ Mpc which matches the spectral resolution $\Delta\nu_{c} = 24.4 \, {\rm kHz}$ of our data ($\Delta L = r^{\prime} \times \Delta\nu_{c}$). This results in an angular resolution of $\Delta\theta \sim 8.4^{''}$ ($\Delta L = r \Delta\theta$), and the angular extent of the simulation box ($N\Delta\theta$) covers $\sim2.5$ times the $\theta_{\rm FWHM}$ of GMRT at the frequency $\nu_c = 432.8 \, {\rm MHz}$. We have converted the simulated images into visibilities using the baseline distribution of the combined nights data. The simulations incorporate the frequency dependence of the PB and baseline migration.
We have applied the TGE (eq.~\ref{eq:a4}) on the simulated visibilities, and analyzed the simulated data identical to the actual data, to estimate the MAPS $C_{\ell}(\Delta\nu)$. We have used $N_r=16$ independent realizations of the simulation to estimate the mean $C_{\ell}(\Delta\nu)$ and the $2\,\sigma$ errors shown in Figure~\ref{cl} at three values of $\ell$ for $f=0.6$. We have also shown (solid lines) the analytical model predictions $C^m_{\ell}(\Delta\nu)$ calculated using \citep{KD07,ali14}
\begin{equation}
C_{\ell}(\Delta\nu) = \frac{1}{\pi r^2} \int_{0}^{\infty} d k_{\parallel} \cos(k_{\parallel}r^{\prime}\Delta\nu) P(\bm{k})\,.
\label{eq:CT}
\end{equation}
We see that the $C_{\ell}(\Delta\nu)$ estimated from the simulations closely matches the analytical prediction $C^m_{\ell}(\Delta\nu)$ which are mostly within the shaded region showing the $2\,\sigma$ uncertainty. The deviations between $C_{\ell}(\Delta\nu)$ and $C^m_{\ell}(\Delta\nu)$ are found to lie within $~\rlap{$<$}{\lower 1.0ex\hbox{$\sim$}}\, 10\%$ at $\Delta \nu =0$. These deviations are primarily due to uncertainties in the normalization factors $M_{g}(\nu_{a},\nu_{b})$ which have been estimated from $50$ UMAPS realizations. To test this we have checked that these deviations decrease if the number of UMAPS realizations is increased. We also note that the large $\Delta\nu$ are poorly sampled compared to the small $\Delta\nu$, and the cosmic variance increases as we go to larger frequency separations.
The dotted line in Figure~\ref{cl} shows $[C_{\ell}(\Delta \nu)]_T$ an estimate of the cosmological 21-cm signal expected in the observed data. It is assumed that the fluctuations of the \ion{H}{I}~ distribution trace the underlying matter distribution with a
linear bias $b_{\rm \ion{H}{I}~}$. This allows us to express $P_{T}(\bm{k})$ the predicted 21-cm brightness temperature power spectrum in terms of $P^s_m(\bm{k})$ the underlying matter power spectrum in redshift space. Here we have rewritten
eq.~(23) of \cite{BA5} as
\begin{equation}
P_{T}(\bm{k}) \, = \,[\Omega_{\rm \ion{H}{I}~} b_{\rm \ion{H}{I}~}]^{2}\, \bar{T}^{2}\, P^s_{m}(\bm{k})
\label{eq:ph1}
\end{equation}
with
\begin{equation}
\bar{T}(z) = 133 \,{\rm mK}\,(1+z)^{2}\,\bigg(\frac{h}{0.7}\bigg)\,\bigg(\frac{H_{0}}{H(z)}\bigg)
\label{eq:tbar}
\end{equation}
where the cosmological \ion{H}{I}~ mass density $\Omega_{\rm \ion{H}{I}~}$ is the comoving \ion{H}{I}~ mass density in units of the present critical density.
DLA observations (e.g. \citealt{Not, Zafar}) show that $\Omega_{\rm \ion{H}{I}~} \sim 10^{-3}$ across $1.5 < z < 5$, whereas various simulations (e.g. \citealt{Deb16}) indicate $1 \le b_{\rm \ion{H}{I}~} \le 2$ across $(2 \le z \le 3)$. For the estimates presented here we have used $[\Omega_{\rm \ion{H}{I}~} b_{\rm \ion{H}{I}~}]=10^{-3} $ and a fitting formula for $P_m(k)$ \citep{Eisenstein_1998}, ignoring the effect of redshift space distortion. We find that $[C_{\ell}(\Delta \nu)]_T$ has a peak value of $\approx 0.9 \times 10^{-6} \, {\rm mK}^2$ at $\Delta \nu=0$ for the value of $\ell$ $(=1635)$ shown here. The value of $[C_{\ell}(\Delta \nu)]_T$ decreases with increasing $\Delta \nu$ and it is $\approx 0$ for $\Delta \nu > 1 \, {\rm MHz}$. In fact, a similar behaviour is also seen for the model predictions $[C_{\ell}^m(\Delta \nu)]$ where we find that the value peaks at $\Delta \nu =0$ and decorrelates rapidly with increasing $\Delta \nu$ with a value $\approx 0$ at $\Delta \nu > 1 \, {\rm MHz}$. The peak value reduces with increasing $\ell$ for which the signal also decorrelates faster. These are generic features of the expected 21-cm signal \citep{Bharadwaj-Pandey-2003} irrespective of the details of the 21-cm PS.
We have implemented eq.~(\ref{eq:ML}) to estimate the PS of the simulated sky signal. Identical to the actual data, we have also used a BN window function along the frequency separation for these simulations. The simulations differ from the data in that the error-covariance $\textbf{N}$ is dominated by cosmic variance, whereas the data is system noise dominated. Here we have used the covariance of the simulated $C_{\ell}(\Delta\nu)$ to estimate the noise covariance matrix $\textbf{N}$. The upper panel of Figure~\ref{fig:pk_val} shows the estimated spherically-binned PS $P(k)$ and the associated $2\,\sigma$ errors along with the model PS $P^{m}(k)$. We see that $P(k)$ is in reasonably good agreement with $P^{m}(k)$ across the entire $k$ range considered here. The lower panel of Figure~\ref{fig:pk_val} shows the fractional deviation $\delta=[P(k)-P^{m}(k)]/P^{m}(k)$ and the expected $2\,\sigma$ statistical fluctuations for the same. We have $\mid \delta \mid ~\rlap{$<$}{\lower 1.0ex\hbox{$\sim$}}\,\, 5 \%$ in most of the $k$-bins shown here. We have somewhat larger deviation $(\mid \delta \mid \sim 10\%)$ at the smallest $k$-bin. The convolution with the window function (eq.~\ref{eq:a1}) is expected to become important at the small baselines \citep{samir14}, and this possibly contributes to enhance the deviations in the small $k$-bins. A part of the deviations could also arise from the low baseline density in some of the bins (Figure~\ref{fig:uv}). We see that the $\delta$ values are all consistent with the predicted $2 \sigma$ errors. In the analysis of the actual observed data, as presented later in this paper, we have identified some of the $(k_{\perp},k_{\parallel})$ modes as being foreground contaminated. These modes have been excluded for estimating the spherically-binned PS $P(k)$ of the actual data. In keeping with this, we have also excluded these modes for the simulations presented here. The entire validation presented here used {\it exactly} the same $(k_{\perp},k_{\parallel})$ modes as those that have been used for the actual data. In summary, we have validated the TGE and we find that it is able recover the input model PS to an accuracy better than $~\rlap{$<$}{\lower 1.0ex\hbox{$\sim$}} \,\, 10 \%$ across the entire $k$ range considered here, and $~\rlap{$<$}{\lower 1.0ex\hbox{$\sim$}}\,\,5\%$ across $0.54 \le k \le 7.58\,\,{\rm Mpc}^{-1}$. The results are not very different even if we include all the available $(k_{\perp},k_{\parallel})$ modes to estimate $P(k)$.
\section{Results}
\label{s6}
\subsection{The Estimated MAPS}
\label{maps}
We have used the TGE (eqs.~\ref{eq:a4} and \ref{eq:a6}) to estimate the MAPS $C_{\ell}(\Delta\nu)$ from the calibrated and compact source subtracted visibility data for the individual nights of observation as well as the combined data. We see that the May 6 data (Table ~\ref{tab:flagandrms}) has the least flagging as well as the smallest visibility r.m.s. Guided by this, we first consider the results for the May 6 data and subsequently use this as a reference for comparing the results for the other nights (not shown here) and the combined data. The two polarizations (LL and RR) were treated as independent measurements from the same baseline. We have repeated the analysis for three values of the tapering parameter $f=5.0,\,2.0,\,\textrm{and}\,0.6$. As mentioned earlier, the tapering increases with decreasing value of $f$, and $f=5.0$ can be considered equivalent to an untapered PB.
\begin{figure}
\begin{center}
\includegraphics[width=\columnwidth]{Plots/cl-f.pdf}
\caption{$C_{\ell}(\Delta\nu)$ as a function of $\Delta\nu$ for the May 6 observation are shown at $f=5.0,\, 2.0,\,{\rm and} \,0.6$ at two values of the angular multipole $\ell$. The left panels show the $C_{\ell}(\Delta\nu)$ for the entire $24.4\, {\rm MHz}$ bandwidth considered here. The right panels show the same but we restrict the frequency separation upto $5\, {\rm MHz}$. The black solid lines represent the estimated $C_{\ell}(\Delta\nu)$ using $f=0.6$ for the combined data.}
\label{fig:6may_f}
\end{center}
\end{figure}
In Figure~\ref{fig:6may_f} we have shown $C_{\ell}(\Delta\nu)$ as a function of the frequency separation $\Delta\nu$ at different values of $f$ for the May 6 data. The upper and lower panels correspond to two representative $\ell$-values, $\ell=1635$ and $4450$ respectively. The left and right panels show $C_{\ell}(\Delta\nu)$ over the entire $24.4\,{\rm MHz}$ bandwidth and $\Delta\nu\le 5\,{\rm MHz}$ respectively. We see that $C_{\ell}(\Delta\nu)$ exhibit an oscillatory pattern whose frequency increases with $\ell$. This increase in the frequency of oscillation is more evident in the right panels which show a small part of the $\Delta\nu$ range. These oscillatory patterns are consistent with the expected foreground behaviour \citep{ghosh1,ghosh2,ghosh3}. The contribution to $C_{\ell}(\Delta\nu)$ from a single point source is predicted (\citetalias{Pal20}) to be
\begin{equation}
C_\ell (\Delta\nu) \propto \cos{(\ell \theta \Delta \nu/\nu_c)} \,
\label{eq:oscillation}
\end{equation}
where $\theta $ is the sine of the angle between the source position and the phase center of the observation. As mentioned earlier, the compact and discrete sources within the main lobe of the PB have been modelled and subtracted out \citep{Cha2}. However, far-field residual sources remain that are difficult to model and clean out from the data. The oscillations in the estimated $C_{\ell}(\Delta\nu)$ is essentially a superposition of the oscillatory contributions from all the residual sources outside the main lobe of the PB. Note that the oscillation in $C_{\ell}(\Delta\nu)$ is fundamentally due to the chromatic nature of radio-interferometric measurements. The increase in the frequency of oscillation in $C_{\ell}(\Delta\nu)$ at larger baselines yields the `wedge' shape in the PS, which we shall present shortly. The extent of this `foreground wedge' is determined by the position ($\theta$) of the wide-field source, which can maximally reach the horizon limit $\theta \sim 1$. The oscillations seen here, or equivalently the foreground wedge, arises due to `baseline migration'
\citep{adatta10, Morales_2012, parsons12, vedantham12, Murray_2018}. TGE allows us to suppress the antenna response at large angular distances relative to the phase center, reducing the large angular-scale foreground contributions present in the data. This is illustrated in Figure~\ref{fig:6may_f} where we see that for both the $\ell$ values the overall amplitude of $C_{\ell}(\Delta\nu)$ goes down as the value of $f$ is reduced (or equivalently, the tapering is increased). Comparing with respect to $f=5.0$, we find that the amplitude of $C_{\ell}(0)$ drops by a factor $3-4$ for $f=0.6$ at the $\ell$ values shown here. Further, the amplitude of the oscillatory pattern also decreases considerably as the value of $f$ is reduced. The implication of this on the PS will be discussed in Section \ref{power} where we consider the PS for different values of $f$.
\begin{figure}
\begin{center}
\includegraphics[width=\columnwidth]{Plots/heatmap.pdf}
\caption{This shows $C_{\ell}(\Delta\nu)$ across the entire ($\ell, \Delta\nu$) range for May 6 at $f=5$ (left panel), May 6 at $f=0.6$ (middle panel) and four nights combined data at $f=0.6$ (right panel).}
\label{fig:MAPS3D}
\end{center}
\end{figure}
Figure~\ref{fig:MAPS3D} shows $C_{\ell}(\Delta\nu)$ across the entire $(\ell,\Delta\nu)$ range. A comparison of the results obtained from the May 6 data with $f=5.0$ (left panel) and $f=0.6$ (middle panel) demonstrates the effect of tapering. Considering the left panel, we see that the oscillations along $\Delta\nu$ are prominently visible in most of the $\ell$-bins. For larger $\ell$, the oscillations become so rapid that we cannot discern them in the figure. The overall amplitude and that of the oscillatory pattern are both visibly reduced in the middle panel. The rightmost panel shows $C_{\ell}(\Delta\nu)$ for the combined nights data for $f=0.6$. It is evident from the last two panels of the Figure~\ref{fig:MAPS3D} that combining different nights data yields a further smoothing of the oscillatory patterns. This is also illustrated in Figure~\ref{fig:6may_f} which shows $C_{\ell}(\Delta\nu)$ for the combined data in black-solid lines for $f=0.6$. We see, in both the Figures \ref{fig:6may_f} and \ref{fig:MAPS3D}, that the overall amplitude of $C_{\ell}(\Delta\nu)$ as well as the amplitude of the oscillations are smaller for the combined nights in comparison to the May 6 data. The reason is that the convolution (eq.~\ref{eq:a1}), which incorporates the tapering in TGE, is sensitive to the baseline distribution \citep{samir14}. The baseline densities for May 6 and the combined nights are shown in the lower panels of Figure~\ref{fig:uv}. We see that the baseline density increases by a factor $\sim 3.5$ for the combined nights data. The uv-coverage is also considerably less patchy in comparison to the May 6 data. We expect the tapering to be more effective for the denser and more uniform baseline coverage of the combined nights. We see that this expectation is borne out in the estimated $C_{\ell}(\Delta\nu)$.
In addition to the rapid oscillations in ${C_{\ell}(\Delta\nu)}$ (Figures~\ref{fig:6may_f} and \ref{fig:MAPS3D}) which arise from the residual compact source contribution due to baseline migration, ${C_{\ell}(\Delta\nu)}$ also exhibits a gradual de-correlation {\it i.e.} the value of ${C_{\ell}(\Delta\nu)}$ decreases as $\Delta \nu$ increases. We expect the intrinsic frequency spectrum of the compact sources to cause a smooth de-correlation of ${C_{\ell}(\Delta\nu)}$. However, it is interesting to note that the values of ${C_{\ell}(\Delta\nu)}$ do not fall monotonically with increasing $\Delta\nu$. We can observe this in the lower left panel of Figure~\ref{fig:6may_f} where the amplitude of the oscillations in ${C_{\ell}(\Delta\nu)}$ decreases till $\Delta\nu \sim 5 \, \rm{MHz}$, then increases again up to $\Delta\nu \sim 10 \, \rm{MHz}$, and then decreases again. This modulation, we believe, arises because of the PB pattern which changes with frequency across the frequency bandwidth considered here. The position of the null points of the PB changes considerably with frequency, and this possibly causes the slow modulation seen in ${C_{\ell}(\Delta\nu)}$.
\subsection{The Estimated PS}
\label{power}
\begin{figure*}
\begin{center}
\includegraphics[width=\textwidth]{Plots/PS_binned.pdf}
\caption{The first three panels from the left show the absolute values of the cylindrical power spectra $P(k_{\perp},k_{\parallel})$ for the May 6 data for different values of tapering. The rightmost panel shows the same for the combined nights data for $f=0.6$. In all the cases the black dashed lines denote $[k_{\parallel}]_{H}$.}
\label{fig:Pk_f}
\end{center}
\end{figure*}
We have applied the maximum likelihood method described in Section \ref{s3} on $C_{\ell}(\Delta\nu)$ to estimate the cylindrical power spectra $P(k_{\perp},k_{\parallel})$.
The different panels of Figure~\ref{fig:Pk_f} show the absolute values of the PS.
The first three panels, starting from the left, respectively correspond to
$f=5.0, \, 2.0, \, \rm{and} \, 0.6$ for the May 6 data while the right panel corresponds to $f=0.6$ for the combined nights. In all cases, the foregrounds are found to be largely confined within a wedge in the $(k_{\perp},k_{\parallel})$ plane. The wedge boundary (also called the `horizon limit') can be mapped to a straight line $[k_{\parallel}]_{H} = (r/r^{\prime}\nu_{c}) k_{\perp}$ in the $(k_{\perp},k_{\parallel})$ plane \citep{pober16}. The region $k_{\parallel} \le [k_{\parallel}]_{H}$ is referred to as the ``foreground wedge'', and the PS estimated in this region of the $(k_{\perp},k_{\parallel})$ plane is largely dominated by the foregrounds. The region $k_{\parallel} > [k_{\parallel}]_{H}$ is relatively foreground-free, and we refer to this as the ``21-cm window''. While the bulk of the foregrounds in Figure~\ref{fig:Pk_f} are localized within the foreground wedge, all the panels also show some foreground leakage outside the predicted wedge boundary. Various factors like the chromaticity of the sky signal and the PB, sparse sampling of baselines, calibration errors, etc., cause this foreground to leak into the 21-cm window \citep{bowman09, Thyagarajan_2016}, and it is often necessary to discard a part of the 21-cm window for estimating the PS of the 21-cm signal.
We now compare the PS values in the three left panels of Figure~\ref{fig:Pk_f} which respectively correspond to $f=5.0, 2.0$ and $0.6$ for the May~6 data. We see that the overall foreground contamination comes down as the value of $f$ is reduced {\it i.e.} the tapering is increased, and we have a larger side-lobe suppression. In addition to a decrease in amplitude within the foreground wedge, we also notice a reduction in the leakage outside the wedge. However, this effect is not uniform across the different $k_{\perp}$-bins. The convolution which incorporates the tapering is expected to be more effective when we have a denser and more uniform $uv$ coverage of the baselines. As discussed in Section $\ref{maps}$, we have a denser baseline distribution at the small baselines, which is reflected in the fact that the foreground suppression is more effective at the lower $k_{\perp}$-bins. The rightmost panel of Figure~\ref{fig:Pk_f} corresponds to $f=0.6$
for the combined nights data. Comparing the two rightmost panels, both of which correspond to $f=0.6$, we see that we have a smaller foreground contribution for the combined nights data compared to the May 6 data. This is expected because the combined nights data has a $\sim 3.5$ times larger baseline density in comparison to the May 6 data (Figure \ref{fig:uv}). We see that we have the lowest level of foreground contribution for the combined nights data with $f=0.6$, and we have focused on this for the subsequent analysis of this paper.
\begin{figure}
\begin{center}
\includegraphics[width=\columnwidth]{Plots/pkslice.pdf}
\caption{The absolute values of the cylindrical power spectra $P(k_{\perp},k_{\parallel})$ for the combined nights data are shown as a function of $k_{\parallel}$ for $f=5.0\,{\rm and}\,0.6$ at two representative values of $k_{\perp}$. The value at $k_{\parallel}=0$ has been slightly shifted for plotting on a log scale. The vertical red solid line denotes $[k_{\parallel}]_{H}$ at the respective $k_{\perp}$-bin. The green and light-blue shaded regions show the $1\sigma$ errors $[\delta P_{N}]$ due to the system noise estimated at $f=5.0\,{\rm and}\,0.6$ respectively.}
\label{fig:Pk_f_slice}
\end{center}
\end{figure}
We next consider the estimated values of $P(k_{\perp},k_{\parallel})$ in some detail. To study this, in Figure~\ref{fig:Pk_f_slice} we have shown the absolute values of $P(k_{\perp},k_{\parallel})$ as a function of $k_{\parallel}$ for two different fixed $k_{\perp}$ bins corresponding to $k_{\perp}=0.50\,{\rm Mpc}^{-1}\,$ (upper panel) and $ \,0.78\,{\rm Mpc}^{-1}$ (lower panel). The results are shown for the combined nights data, considering two values of tapering namely $f=5.0$ and $f=0.6$. In each panel, the vertical red solid line denotes $[k_{\parallel}]_{H}$ which is the predicted foreground wedge boundary for the particular $k_{\perp}$-bin. Further, the shaded region denotes the predicted $1\sigma$ errors $([\delta P_{N}])$ due to the system noise contribution considering $f=5.0$ (green) and $0.6$ (light-blue). In all cases we have the largest value of $P(k_{\perp},k_{\parallel})$ $(\sim 10^9 \, {\rm mK}^2 \, {\rm Mpc}^3)$ at $k_{\parallel}=0$. The value of $P(k_{\perp},k_{\parallel})$ falls $(\sim 10^6 \, - \, 10^7 \, {\rm mK}^2 \, {\rm Mpc}^3)$ till $k_{\parallel}$ approaches $[k_{\parallel}]_{H}$ where it flattens out, and then rises slightly in a few $k_{\parallel}$ bins just beyond $[k_{\parallel}]_{H}$. Further beyond this, the value of $\mid P(k_{\perp},k_{\parallel}) \mid $ falls with increasing $k_{\parallel}$. We find $\mid P(k_{\perp},k_{\parallel}) \mid \sim 10^4 \, - \, 10^5 \, {\rm mK}^2 \, {\rm Mpc}^3$ at the largest $k_{\parallel}$ bins where the power oscillates between positive and negative values which are comparable with the $1-\sigma$ error-bars. We interpret the estimated power in this region to be arising due to a combination of system noise and some residual foreground leakage. In both the panels we find that the values of $\mid P(k_{\perp},k_{\parallel}) \mid $ decrease when $f$ is reduced from $f=5.0$ to $f=0.6$.
Considering Figure~\ref{fig:Pk_f_slice}, as noted earlier, we find that there is a $k_{\parallel}$ range within the foreground wedge where $P(k_{\perp},k_{\parallel})$ has a relatively small value in comparison to the values at $k_{\parallel}=0$ and $k_{\parallel} \approx [k_{\parallel}]_{H}$. This is also seen in the various panels of Figure~\ref{fig:Pk_f} where we see that the values of $P(k_{\perp},k_{\parallel})$ fall as we move away from $k_{\parallel}=0$ and then increases again at $k_{\parallel} \approx [k_{\parallel}]_{H}$. We identify the rise in $P(k_{\perp},k_{\parallel})$ close to the horizon limit as a wide-field foreground effect known as the ``{\it pitchfork effect}'' \citep{thyag15_1,thyag15}.
The pitchfork effect has been previously reported in observations with the MWA \citep{thyag15}, PAPER \citep{kohn16} and LOFAR \citep{gehlot17} telescopes at lower frequencies $(\sim 150 \, {\rm MHz})$ which target the redshifted 21-cm signal from the Epoch of Reionization ($z>6$). The present work is possibly the first time this effect is being observed at higher frequencies which correspond to the post-reionization 21-cm signal.
We notice (Figure~\ref{fig:Pk_f}) that the magnitude of the pitchfork effect is reduced as the tapering parameter is reduced from $f=5$ to $f=0.6$.
\begin{figure}
\begin{center}
\includegraphics[width=\columnwidth]{Plots/pk-sph.pdf}
\caption{The cylindrical power spectra $\mid P(k_{\perp},k_{\parallel}) \mid $ for the combined nights data for $f=0.6$. Here the black-dashed line denotes $[k_{\parallel}]_{H}$. The region above the green solid line has been used for spherical binning.}
\label{fig:pk-sph}
\end{center}
\end{figure}
\begin{figure}
\begin{center}
\includegraphics[width=85mm]{Plots/Nstat.pdf}
\caption{The histogram of the variable $X=\frac{P(k_{\perp},\,k_{\parallel})}{\delta P_{N}(k_{\perp},\,k_{\parallel})}$. The black-solid curve shows the fit with t-distribution. The mean $1.21$ and standard deviation $6.09$ are obtained from $\mid X \mid \le 20$, demarcated by the vertical black-dashed lines.}
\label{pkbysigma}
\end{center}
\end{figure}
Figure~\ref{fig:pk-sph} shows the best results for the cylindrical PS $\mid P(k_{\perp},k_{\parallel}) \mid$ which has been used to estimate the spherically binned PS $P(k)$ where $k=\sqrt{k_{\perp}^2 + k_{\parallel}^2}$. This has been obtained from the combined nights data for $f=0.6$. The black dashed line shows $[k_{\parallel}]_H$ the predicted boundary of the foreground wedge. We notice that the foreground leakage extends beyond $[k_{\parallel}]_H$. We have selected the entire $k_{\perp}$ range, and $k_{\parallel}$ modes inside the ``21-cm window'', with a buffer region of $0.5-1.0\,\,{\rm Mpc}^{-1}$ outside the wedge boundary as shown in Figure~\ref{fig:pk-sph}. We have included all the modes beyond the green solid lines in the $(k_{\perp},k_{\parallel})$ plane for the spherical binning. Figure \ref{pkbysigma} shows the statistics of the estimated $P(k_{\perp},k_{\parallel})$ for the selected region through a variable `$X$', which is defined by the ratio of the PS to the statistical error due to the system noise at a $(k_{\perp},k_{\parallel})$,
\begin{equation}
X=\frac{P(k_{\perp},k_{\parallel})}{\delta P_{N}(k_{\perp},k_{\parallel})}.
\label{eq:stat}
\end{equation}
We see that the probability density function (P.D.F.) is mostly symmetric within $\mid X \mid\le\,\, 20$ (area demarcated by the vertical black-dashed lines) with mean $mean(X)=\mu=1.21$ and $\sqrt{var(X)}=\sigma_{Est}=6.09$. The negative values of $X$ are consistent with roughly 3 times $\sigma_{Est}$, and no negative values are observed outside $\mid X \mid\le\,\, 20$. This, along with the positive mean within this region, indicate that no negative bias can arise from the modes selected here for the spherical binning. Further, in the scenario where the PS in the selected region are solely dominated by the system noise which follows a Gaussian distribution with zero mean and variance $\sigma_{N}^{2}$, we expect the variable $X$ to follow a Gaussian distribution with $mean(X)=\mu$ and variance $var(X)=\sigma_{Est}^{2}$ as estimated from a sample of N-points, with $\mu$ being comparable to $\frac{\sigma_{Est}}{\sqrt{N}}$ (i.e. $\mu \lesssim \frac{\sigma_{Est}}{\sqrt{N}}$), and as $N \to \infty$, $\sigma_{Est}^{2}\to1$. The standard deviation $\sigma_{Est}>1$ suggests that the estimated r.m.s. statistical fluctuations $\delta P_{N}(k_{\perp},k_{\parallel})$ are underestimated by a factor $\sigma_{Est}$. We note that factors such as RFI, residual point source contributions, residual calibration errors etc., can additionally contribute to the error and may have caused the variance to exceed the predicted value estimated from the system noise only. We account for this underestimation by multiplying the $\delta P_{N}(k_{\perp},k_{\parallel})$ with $\sigma_{Est}$ henceforth to denote the actual error due to statistical fluctuations, $\delta P^{True}_{N}(k_{\perp},k_{\parallel})=\sigma_{Est}\times\delta P_{N}(k_{\perp},k_{\parallel})$. The black-solid curve in Figure \ref{pkbysigma} shows the P.D.F. fitted with a t-distribution. We find that the P.D.F. is well described by the t-distribution within $\mid X \mid~\rlap{$<$}{\lower 1.0ex\hbox{$\sim$}}\,\, 20$; however, the long positive tail observed at higher $X$ is not modelled well and is somewhat underestimated by the t-distribution function. From these observations, we interpret that the estimated $P(k_{\perp},k_{\parallel})$ contain contributions from statistical fluctuations as well as residual foreground emission in the region considered here. Note that we have considered $(k_{\perp},k_{\parallel})$ modes up to $\frac{P(k_{\perp},k_{\parallel})}{\delta P^{True}_{N}(k_{\perp},k_{\parallel})}\approx 30$ for the spherical binning.
\begin{figure}
\begin{center}
\includegraphics[width=\columnwidth]{Plots/deltaT.pdf}
\caption{The mean square brightness temperature fluctuations $\Delta^2(k)$ shown as a function of $k$ along with $2 \sigma$ error bars for the selected regions shown in Figure~\ref{fig:pk-sph}. The orange line shows the result at $z=2.19$ from \citetalias{Ch21} along with $2\,\sigma$ error bars as reported in the paper.}
\label{fig:sp-pk}
\end{center}
\end{figure}
We have estimated the spherically binned PS $P(k)$ using the cylindrical PS $P(k_{\perp},k_{\parallel})$ values in all the $(k_{\perp},k_{\parallel})$ modes which lie beyond the green solid line in Figure~\ref{fig:pk-sph}. The entire $k$ range has been divided into $8$ equally spaced logarithmic bins. The solid blue line in Figure~\ref{fig:sp-pk} shows $\mid \Delta^2(k) \mid $ where $\Delta^2(k) \equiv {k^{3}}P(k)/{2\pi^{2}} $ the estimated mean squared brightness temperature fluctuations along with the $2\,\sigma$ error bars. Here $\sigma={k^{3}}[\delta P^{True}_{N}(k)]/{2\pi^{2}}$, where $[\delta P^{True}_{N}(k)]$ is the r.m.s. estimated using the spherically binned PS from $50$ realisations of noise only simulations multiplied by $\sigma_{Est}=6.09$. The values of $ \Delta^2(k)$ and $\sigma$ are tabulated in Table~\ref{tab:ul} for reference. Considering the values of $\Delta^2(k)$, we find that this has the smallest value $\Delta^2(k)=(128.91)^2\,\,{\rm mK}^2$ for the first bin at $k=0.347\,\,{\rm Mpc}^{-1}$. The value of $\Delta^2(k)$ increases approximately as $\Delta^2(k)=(244.16)^2 \, {\rm mK}^2 \, (k/1 {\rm Mpc}^{-1})$ in the entire $k$ range $0.347 < k < 7.584 \,\textrm{Mpc}^{-1}$. The values of $\Delta^2(k)$ in all of the first five bins and the seventh $k$-bin are well in excess of $0 \, + \, 2 \, \sigma$, and we interpret the values estimated in these bins as arising from residual foregrounds and systematics. The remaining two $k$-bins are consistent with $0 \, + \, 2 \, \sigma$. We have used the estimated $\Delta^2(k)$ and $\sigma$ values to place $2 \sigma$ upper limits $\Delta_{UL}^{2}(k) = \Delta^{2}(k) + 2\sigma$ on the 21-cm brightness temperature fluctuations at different $k$ bins (Table~\ref{tab:ul}). We find the tightest constraint on the upper limits to be $(133.97)^2 \, {\rm mK}^{2}$ at the smallest bin $k = 0.347\,\textrm{Mpc}^{-1}$. We have used the estimated $\Delta_{UL}^{2}(k)$ to place corresponding $2 \sigma$ upper limits on $[\Omega_{\rm \ion{H}{I}~}b_{\rm \ion{H}{I}~}]_{UL}$ (eq.~\ref{eq:ph1}). The values corresponding to the different $k$-bins are tabulated in Table~\ref{tab:ul}. We obtain the tightest constraint of $[\Omega_{\rm \ion{H}{I}~}b_{\rm \ion{H}{I}~}]_{UL} \le 0.23$ from the smallest bin $k = 0.347 \, \textrm{Mpc}^{-1}$.
\begin{table}
\centering
\caption{Spherically binned mean square brightness temperature fluctuations $\Delta^2(k)$ and the corresponding statistical error predictions $\sigma$ for different $k$-bins. The $2\,\sigma$ upper limits $\Delta^{2}_{UL}(k)=\Delta^{2}(k)+2\,\sigma$ and corresponding $[\Omega_{\rm \ion{H}{I}~}b_{\rm \ion{H}{I}~}]_{UL}$ values are also provided.}
\begin{tabular}{ccccc}
\hline
\hline
$k$ & $\Delta^2(k)$ & $1\sigma$ & $\Delta_{UL}^{2}(k)$ & $[\Omega_{\rm \ion{H}{I}~}b_{\rm \ion{H}{I}~}]_{UL}$\\
Mpc$^{-1}$ & (mK)$^2$ & (mK)$^2$ & (mK)$^2$ &\\
\hline
$0.347$ & $(128.91)^2$ & $(25.79)^2$ & $(133.97)^2$ & $0.230$\\
$0.539$ & $(152.14)^2$ & $(43.92)^2$ & $(164.33)^2$ & $0.234$\\
$0.837$ & $(168.54)^2$ & $(62.99)^2$ & $(190.64)^2$ & $0.230$\\
$1.301$ & $(278.48)^2$ & $(100.37)^2$ & $(312.57)^2$ & $0.326$\\
$2.021$ & $(406.20)^2$ & $(159.58)^2$ & $(464.68)^2$ & $0.425$\\
$3.141$ & $(375.19)^2$ & $(271.49)^2$ & $(536.83)^2$ & $0.436$\\
$4.881$ & $(705.60)^2$ & $(449.38)^2$ & $(949.61)^2$ & $0.694$\\
$7.584$ & $(704.35)^2$ & $(701.07)^2$ & $(1216.18)^2$ & $0.807$\\
\hline
\label{tab:ul}
\end{tabular}
\end{table}
\citetalias{Ch21} have carried out a multi-redshift analysis of the same observational data after splitting it into several sub-bands of $8$ MHz each. Their $z=2.19$ sub-band is closest to our analysis, and we have also shown their results in Figure~\ref{fig:sp-pk} (orange line). We see that our present analysis extends to substantially smaller $k$ values $(0.347 < k < 7.584 \,\textrm{Mpc}^{-1})$ compared to \citetalias{Ch21} who have considered the $k$-range $1 < k < 10 \,\textrm{Mpc}^{-1}$. We find that in the common $k$-range $1 < k < 8 \,\textrm{Mpc}^{-1}$ our $\Delta^{2}(k)$ estimates are $\sim 7$ times larger than those of \citetalias{Ch21}. We also see that the $2\,\sigma$ error bars on $\Delta^{2}(k)$ are smaller in the present analysis as compared to \citetalias{Ch21} in the first five $k$-bins. The difference can be attributed to the larger frequency bandwidth $(24.4 \, {\rm MHz})$ used here. For the remaining three $k$-bins, the larger error bars may be attributed to lower sampling at longer baselines. Comparing the $2 \sigma$ upper limits, \citetalias{Ch21} have obtained $(61.49)^{2} \, {\rm mK}^{2}$ and $0.11$ respectively for $\Delta^{2}(k)$ and $[\Omega_{\rm \ion{H}{I}~}b_{\rm \ion{H}{I}~}] $ at $k=1 \, {\rm Mpc}^{-1}$ whereas the present analysis reports $(133.97)^{2} \, {\rm mK}^{2}$ and $0.23$ respectively at $k=0.347 \, {\rm Mpc}^{-1}$.
We take this opportunity to highlight a key difference between the analysis method of \citetalias{Ch21} and the one used here. Considering the visibilities measured at the individual baselines, \citetalias{Ch21} have carried out a Fourier transform along frequency to estimate the visibilities in delay space \citep{Morales04} which were then used \citep{parsons12} to estimate the PS. The difficulty arises because the missing frequency channels (flagged due to RFI) introduce artefacts in the Fourier transform which corrupt the estimated PS. \citetalias{Ch21} have overcome this by using one dimensional complex CLEAN \citep{Parsons_2009} to compensate for the missing frequency channels. In contrast, the method used here first correlates the visibility data across frequency channels to estimate $C_{\ell}(\Delta \nu)$. There are no missing frequency separations $\Delta \nu$
in the estimated $C_{\ell}(\Delta \nu)$ even though the visibility data has a substantial number of missing frequency channels. We have then used maximum likelihood to estimate the PS which is related to $C_{\ell}(\Delta \nu)$ through a Fourier transform with respect to $\Delta \nu$. This method uses only the available frequency channels to estimate the PS, and it is not necessary to compensate for the missing frequency channels. For the present analysis we have validated this using simulations (Section~\ref{sec:simulation}) where the flagging of the simulated data exactly matches that of the actual data $(55 \%)$. An earlier work \citep{Bh18} has used simulations to demonstrate that the present estimator is able to successfully recover the PS even when the data in $ 80 \%$ randomly chosen frequency channels are flagged.
\section{Summary and Conclusions}
\label{s7}
\ion{H}{I}~ 21-cm intensity mapping is a promising tool to probe the large-scale structures in the Universe across a wide redshift range. In this paper we employ the TGE to estimate the MAPS $C_{\ell}(\Delta\nu)$ and the PS $P(k_{\perp}, k_{\parallel})$ using data from four nights of uGMRT Band 3 observations of the ELAIS-N1 field. Our analysis is restricted to a $24.4\,{\rm MHz}$ sub-band which has a frequency resolution of $24.4 \,{\rm kHz}$ and is centered at $432.8\, {\rm MHz}$ which corresponds to 21-cm signal from a redshift $z=2.28$.
Compact and discrete sources with flux densities $> 100\mu \textrm{Jy}$ within an area of $1.8 \, {\rm deg^2}$ were identified and subtracted out. The residual visibility data was used for the analysis presented here. In addition to the individual nights data, we have also analysed the combined nights data.
The TGE (eq.~\ref{eq:a4}) uses the measured visibilities to directly estimate $C_{\ell}(\Delta\nu)$ which characterizes the second order statistics of the sky signal jointly as a function of the angular multipole $\ell$ and frequency separation $\Delta \nu$. The TGE has three inherent advantages namely (1) it works with the gridded visibility data which makes it computationally fast; (2) it allows us to taper the sky response which reduces the foreground contamination from bright sources located in the side-lobes and the periphery of the telescope's field of view; (3) it uses the data to internally estimate the noise bias and subtracts this to provide an unbiased estimate of $C_{\ell}(\Delta\nu)$. We have used maximum likelihood (eq.~\ref{eq:ML}) to determine $P(k_{\perp}, k_{\parallel})$ from the estimated $C_{\ell}(\Delta\nu)$, the two being related through a Fourier transform (eq.~\ref{eq:cl_Pk}).
We have validated the power-spectrum estimator using simulations (Section~\ref{sec:simulation}) which incorporate the flagging, frequency and baseline coverage of the actual data. As noted earlier, it is not necessary to compensate for the missing frequency channels. Our analysis demonstrates that the estimator can recover the input model power spectrum with high accuracy over the entire $k$ range used for the analysis presented in this paper (Figure \ref{fig:pk_val}).
The May 6 data has the least flagging and the minimum visibility r.m.s. (Table~\ref{tab:flagandrms}), and Figure~\ref{fig:6may_f} shows $C_{\ell}(\Delta\nu)$ for different values of the tapering parameter $f$ considering two values of $\ell$. Note that the tapering is more effective (sky response is narrower) as the value of $f$ is reduced. Figure~\ref{fig:MAPS3D} shows $C_{\ell}(\Delta\nu)$ for the full $(\ell,\Delta \nu)$ range considered here. Considering $f=5$, which may be loosely interpreted as no tapering, we find that $C_{\ell}(\Delta\nu)$ exhibits oscillations in $\Delta \nu$, the frequency of the oscillations increases with $\ell$. We interpret these oscillations as arising from residual compact sources located at large angles from the phase center. We find that the overall amplitude as well as the amplitude of the oscillations in $C_{\ell}(\Delta\nu)$ both decrease as $f$ is varied from $f=5$ to $f=0.6$. This demonstrates that the TGE is effective in tapering the sky response to suppress the contribution from sources in the outer region of the field of view. Both Figures~\ref{fig:6may_f} and \ref{fig:MAPS3D} also show the results for the combined nights data with $f=0.6$. Comparing this with May 6 with $f=0.6$, we find that the oscillations and the overall amplitude of $C_{\ell}(\Delta\nu)$ is even further reduced when we consider the combined nights data. This is a direct consequence of the higher baseline density (Figure \ref{fig:uv}) which makes the tapering more effective for the combined nights data in comparison to the May 6 data. This is due to the fact that tapering in the TGE is implemented through a convolution (eq.~\ref{eq:a1}) which is more effective for the higher baseline density.
The different panels of Figure~\ref{fig:Pk_f} shows $\mid P(k_{\perp}, k_{\parallel}) \mid $ for four different cases. In all cases, the large values of $\mid P(k_{\perp}, k_{\parallel}) \mid $ are mainly localized within the foreground wedge $k_{\parallel} \le [k_{\parallel}]_{H}$ (horizon), however we also find some foreground leakage beyond the predicted wedge boundary. Considering the May 6 data, we find that overall foreground amplitude and also the foreground leakage outside the wedge both come down as the value of $f$ is reduced from $f=5$ to $f=0.6$. There is an even further reduction when we consider the combined nights data with $f=0.6$. We see that the combined nights data gives better results in comparison to the May 6 data, and we have used the combined nights data for the subsequent results and discussion here. Considering $f=5.0$ and $f=0.6$, Figure~\ref{fig:Pk_f_slice} shows $\mid P(k_{\perp},k_{\parallel}) \mid $ as a function of $k_{\parallel}$ for two different fixed $k_{\perp}$ bins. We find the largest values of $P(k_{\perp},k_{\parallel})$ $(\sim 10^9 \, {\rm mK}^2 \, {\rm Mpc}^3)$ at $k_{\parallel}=0$. The value of $\mid P(k_{\perp},k_{\parallel}) \mid $ fall with increasing $k_{\parallel}$, and we find $\mid P(k_{\perp},k_{\parallel}) \mid \sim 10^4 \, - \, 10^5 \, {\rm mK}^2 \, {\rm Mpc}^3$ at the largest $k_{\parallel}$ bins where the power oscillates between positive and negative values which are comparable with the $1-\sigma$ error-bars computed from system noise only simulations. We interpret the estimated power in this region as a combination of system noise and some residual foreground leakage. We also find that the values of $\mid P(k_{\perp},k_{\parallel}) \mid $ decrease when $f$ is reduced from $f=5.0$ to $f=0.6$.
An interesting feature seen in the various panels of Figure~\ref{fig:Pk_f} and also in both the panels of Figure~\ref{fig:Pk_f_slice} is that the value of $P(k_{\perp},k_{\parallel})$ does not decrease monotonically with increasing $k_{\parallel} $. Rather, it initially decreases and then increases again just beyond the horizon $k_{\parallel} \approx [k_{\parallel}]_{H}$ after which it decreases again. This is more clearly visible at large $k_{\perp}$. We identify the rise in $P(k_{\perp},k_{\parallel})$ close to the horizon limit as the pitchfork effect which has been reported earlier \citep{thyag15,kohn16,gehlot17} in low frequency observations $(\sim 150 \, {\rm MHz})$ which target the EoR 21-cm signal. The present work is possibly the first time that this effect is being reported at higher frequencies which target the post-reionization 21-cm signal.
The solid green curve in Figure~\ref{fig:pk-sph} demarcates the region of $(k_{\perp},k_{\parallel})$ space which we have identified to be relatively foreground-free and has been used to estimate the spherically binned power spectrum $P(k)$. We use the variable $X$, defined in eq.~(\ref{eq:stat}), to study the statistics of the estimated PS in this region (Figure \ref{pkbysigma}). We find that for $\mid X\mid~\rlap{$<$}{\lower 1.0ex\hbox{$\sim$}}\,\,20$ the P.D.F. is roughly symmetric with a positive mean $\mu=1.21$. For $\mid X\mid~\rlap{$<$}{\lower 1.0ex\hbox{$\sim$}}\,\,20$, the P.D.F. is well described the t-distribution, beyond which $(X>20)$ the t-distribution function underestimates the P.D.F. This indicates that the PS consists of some noise contributions as well as residual foregrounds. Modes within $\mid X\mid~\rlap{$<$}{\lower 1.0ex\hbox{$\sim$}}\,\,20$ have a standard deviation $\sigma_{Est}=6.09$, which suggests that the r.m.s. fluctuations estimated using the noise simulations underestimate the true errors by a factor $\sigma_{Est}=6.09$. We rectify for this by considering the true errors as $\delta P^{True}_{N}(k_{\perp},k_{\parallel})=\sigma_{Est}\times\delta P_{N}(k_{\perp},k_{\parallel})$, which we carry forward for further analysis. Figure~\ref{fig:sp-pk} shows the mean square brightness temperature fluctuations $\Delta^2(k)$ along with $2 \sigma$ error bars considering $8$ bins across the range $0.347 \le k \le 7.584 \,\textrm{Mpc}^{-1}$. Table \ref{tab:ul} lists these values along with $\Delta_{UL}^{2}(k)$ the corresponding $2 \sigma$ upper limits. We find the tightest $2 \sigma$ upper limit of $\Delta_{UL}^{2}(k)\le(133.97)^2\,{\rm mK}^{2}$ at $k=0.347\,{\rm Mpc}^{-1}$ which translates to an upper limit $[\Omega_{\rm \ion{H}{I}~}b_{\rm \ion{H}{I}~}]_{UL}\le 0.23$. \citetalias{Ch21} reported $\Delta_{UL}^{2}(k)\le(61.49)^{2} \, {\rm mK}^{2}$ and $[\Omega_{\rm \ion{H}{I}~} b_{\rm \ion{H}{I}~}]_{UL} \le 0.11$ at $k=1 \, {\rm Mpc}^{-1}$. The upper limits presented here are still orders of magnitude larger than the expected signal corresponding to $\Omega_{\rm \ion{H}{I}~} \sim 10^{-3}$ and $b_{\rm \ion{H}{I}~} \sim 2 $.
\section*{Acknowledgements}
We thank the scientific editor and the anonymous referee for their comments which have helped us in improving this work. We thank the staff of GMRT for making this observation possible. GMRT is run by National Centre for Radio Astrophysics of the Tata Institute of Fundamental Research. AG would like to acknowledge IUCAA, Pune for providing support through the associateship programme. SB would like to acknowledge funding provided under the MATRICS grant SERB/F/9805/2019-2020 of the Science \& Engineering Research Board, a statutory body of Department of Science \& Technology (DST), Government of India. Part of this work has used the Supercomputing facility `PARAM Shakti' of IIT Kharagpur established under National Supercomputing Mission (NSM), Government of India and supported by Centre for Development of Advanced Computing (CDAC), Pune.
\section*{Data Availability}
The data from this study are available upon reasonable request to the corresponding author.
\bibliographystyle{mnras}
|
2,877,628,088,999 | arxiv | \section{Principles of cosmoparticle physics}
CosmoParticle Physics studies mutual relationship and fundamental physical grounds of Cosmology
and Particle Physics \cite{1}. It provides unified treatment of the basic laws of the Universe and
elementary particles, establishes mutual correspondence between them and probes the fundamental
nature of micro- and macro-worlds in the proper combination of its indirect physical, astrophysical
and cosmological effects. It offers the nontrivial way out of the wrong circle of problems, to which
fundamental physics comes in its one-dimensional development.
Cosmoparticle physics is now being formed into selfconsistent new science, following internal
basic principles in its future development. This development revives the tradition of Natural
philosophy of the universal knowledge, the tradition to consider the world in its universal
completeness and unity.
Cosmoparticle physics reproduces in the largest and smallest scales the general feature of the
fundamental physics: the mutual correspondence between microscopic and macroscopic
descriptions, say, between thermodynamics, atomic theory, hydrodynamics and kinetics, or between
the fundamental macroscopic and microscopic quantities, e.g., between the Avogadro number and
the mass of proton. However, at the level of fundamental cosmology and particle physics this
correspondence acquires the new quality of their unity.
That is why the first basic principle of cosmoparticle physics is the idea of a world system, treating
in the unique framework the foundations of macro- and micro-physics. The second principle
assumes, that the world system establishes strict quantitatively definite mutual correspondence
between fundamental cosmological, astrophysical and micro-physical laws, i.e. postulates the
quantitatively definite correspondence between the structures at macro- and micro- levels. Finally,
the third principle assumes, that the set of world system parameters does not exceed the number of
its macro- and micro-scopic signatures.
One may easily find, that the first principle simply postulates the existence of a world system,
whereas the two other principles specify its necessary properties. The crucial point in this approach
is multidimensional solution, offered by the cosmoparticle physics to the problems, both cosmology
and particle theory face on. It may be shown, that this approach naturally embeds all the widely
known existing trends in studying links between cosmology and particle physics, such as
astroparticle physics, theories of everything, particle astrophysics, cosmoarcheology.
Here we'd like to specify some new types of links, following with necessity from the basic
principles of cosmoparticle physics and lying outside these widely discussed trends.
\section{Unified models of cosmology and particle physics}
Intensive efforts to construct the finite Theory of Everything, undertaken last decade on the base of
Superstring models, have not lead, unfortunately, to extensive theoretical framework, putting
together the modern cosmology and particle physics into the detailed and quantitatively definite
picture. The point is that the space of classical string vacuum has a vary large degeneracy, and there
is no objective criteria that distinguishes a particular string vacuum among the numerous
possibilities. The mathematical complexity is multiplied by the enormous variety of possible
embeddings of the Standard model (SM) of particle interactions into the structure of superstring
models. Indeed, the guiding principle of superstring phenomenology is very simple: it is to
reproduce the SM within the effective low energy field theory of a string model. Since only general
features such as the gauge group, number of families, etc. are considered, it leads to numerous
possibilities for embedding the SM in superstring phenomenology. For example \cite{kaku}, within
the framework of perturbative heterotic superstring, the total rank of the gauge group (for $N=1$,
space-time supersymmetric models) can be as large as $22$. After the SM
$SU(3)_C\bigotimes SU(2)_W\bigotimes U(1)_Y $
symmetry with the rank $4$ is reproduced, the rank of the residual gauge symmetry can be still as
large as $18$. Taking into account that the number of models grows (roughly) as a factorial of the
rank of the residual gauge symmetry, it becomes clear that we need additional arguments to restrict
the amount of models. One of them is to use grand unification and to embed the SM symmetry
within a simple gauge group $G\supset SU(3)_C\bigotimes SU(2)_W\bigotimes U(1)_Y$. To break
the grand unified gauge group $G$ down to that of the SM an adjoint representation of Higgs fields
must be present in effective field theory among the light degrees of freedom. In perturbative
heterotic superstring such states in the massless spectrum are compatible with $N=1$
supersymmetry and chiral fermions only if the grand unified gauge group is realized via a current
algebra at level $k>1$ (see \cite{2}). This condition leads to reduction of the total rank of the gauge
group, and, therefore, restricts the number of possible models. However, for example, for a grand
unified gauge group $G=SO(10)$ with, $k=3$, the rank of the residual gauge symmetry can be still
as large as $7$. Thus even grand unification constraint allows unacceptable amount of SM
embedding. In the case of more sophisticated and extensive string models the ambiguity grows,
making virtually impossible to use the main advantage of the string theory -- to calculate all the
fundamental macro- and microphysical quantities from the first principles.
Moreover, however extensive String models are, they do not represent the most general embedding
for the particle physics and the physics of space-time. The following motivations illustrate some
idea on the possible form of such a general framework.
Events are basic elements of space-time in relativistic theory. The intervals between them maintain
the geometry of space-time. So it seems physically meaningful to treat material processes, causing
the events, together with the space-time, they take place in. But such mutual dependence formally
should correspond to specific structure of the world, in which unified treatment of internal degrees
of freedom (reduced to gauge symmetries) and space-time coordinates may not be completely
covered by the string theory. Some more general mathematical framework may be appropriate, e.g.
the invariant formulation of the apparatus of fiber bundle theory (see \cite{3} and Refs. wherein),
treating space-time and internal variables on equal footing and making it possible to fix the true
symmetry of fundamental interactions and geometry of space-time from exact solutions for the
functional integral. The realization of such program can lead to the true physically selfconsistent
theory of space-time, elementary particles and fundamental natural forces. As a step in this
direction, elaboration of unified models of cosmology and particle physics is important.
Such models treat physically selfconsistent complete cosmological scenarios. Physical
selfconsitency means, that the physical grounds for inflation, baryosynthesis and dark matter are
considered in the unified theoretical framework on the base of the unique particle model, and the
degree of completeness assumes the accuracy, with which the astronomical observational data are
reproduced in the considered cosmological scenario. The degree of completeness of the
cosmological model should depend on the properties of the physical model only.
The easiest way to construct cosmologically selfconsistent particle models is to extend the SM by
addition to its $SU(3)_C\bigotimes SU(2)_W\bigotimes U(1)_Y $
symmetry some other global or local gauge symmetries or by inclusion of the SM symmetry group
into more general gauge group. As a result, the extended gauge model contains new particles and
fields, related to new symmetries added to the standard model. In the most cases, the masses of new
particles and strength of new interactions, mediated by new fields, correspond to superhigh energy
scales, inaccessible to direct experimental test at accelerators. At best, experimental high-energy
physics can put lower limits on some parameters, related to these scales. The only possibility is to
elaborate a system of indirect physical, astrophysical and cosmological constraints on the free
parameters of the "hidden" sector of particle model, to fix them and to specify the cosmological
scenario, following from this choice.
The strategy of cosmoparticle physics approach to unified models of cosmology and particles can
be stipulated as follows:
\begin{enumerate}
\item Physically motivated choice for extended gauge particle model.
\item Test for its cosmological selfconsistency -- study of its possibility to reproduce cosmological
and astrophysical phenomena and effects
\item Determination of free parameters of the "hidden" sector of particle model or set of constraints
on them from the combination of indirect cosmological, astrophysical and experimental physical
restrictions.
\item Elaboration of complete quantitatively definite cosmological scenario.
\item Formulation of the system of indirect experimental physical and astronomical effects,
providing the detailed test of the physical model and cosmological scenario, based on it.
\item Estimation of completeness of this scenario.
\end{enumerate}
Cosmoparticle physics puts traditional methods of observational astronomy and experimental
physics into nontrivial multidimensional complex system of links, thus enriching substantially the
collaboration between physics and astronomy established by astroparticle physics.
\section{The system of links between astronomical observations and laboratory physics
experiments}
Links between particle physics and cosmology are generally viewed by astroparticle physics as
system of linear relations. So, statements \cite{4}, that electron neutrino mass is about $30eV$,
immediately lead to cosmological consequences, since Big Bang cosmology predicts primordial
neutrino background with the concentration, equal to $3/11$ of the one of relic photons. By
multiplying the neutrino mass on the concentration of cosmological neutrino background one
immediately found, that the massive neutrino density should dominate in the modern Universe and
that gravitational instability in the nonrelativistic gas of massive neutrinos should play the dominant
role in the formation of the large scale structure of the Universe. Primordial massive neutrinos were
identified with the hot dark matter in the halo being one of the three classes of elementary particle
dark matter (DM) candidates.
In general hot DM refers to low mass neutral particles that where still in thermal equilibrium after
the QCD phase transition. Hot DM particles have a cosmological number density roughly
comparable to that of microwave background photons, which implies an upper limit to their mass
of a few ten $eV$. Neutrinos are the standard example of hot DM, although other possibilities such
as Majorons are discussed in the literature. Majorons are the pseudo-Goldstone bosons connected
with the Majorana nature of the mass of neutrino. Majorana mass of neutrino corresponds to lepton
number violation. In this case lepton number violating processes such as nuclear neutrinoless
double beta decay can take place. If at least two types of neutrino are massive and neutrino states of
definite mass do not coincide with the states with definite lepton number, neutrino oscillations
should take place. In the matter resonant enhancement of neutrino oscillations can take place, what
may be the solution for Solar neutrino puzzle at very small values of the difference of neutrino mass
squares $\delta m^2\simeq10^{-6}eV$. The detailed analysis of all these crossdisciplinary links,
undertaken by astroparticle physics, could not however lead to any definite conclusion in view of
evident troubles of the simple model of massive electron neutrinos in its confrontation with the
observational and experimental data.
The successive experimental measurements of electron neutrino mass in studies of beta
spectrum of tritium lead to ambiguous results, not confirming the original claims on the value of
$\simeq 30eV$. The upper limit on the electron neutrino mass is roughly $10eV\div 15eV$, a more
precise limit cannot be given since unexplained effects have resulted in the negative value of
$m(\nu_e)^2$ in recent tritium beta decay experiments. The (90\% C.L.) upper limit on an effective
Majorana neutrino mass $0.65eV$ from Heidelberg-Moscow $^{76}Ge$ neutrinoless $2\beta$
decay experiments \cite{5}. The upper limits from accelerator experiments on the masses of the
other neutrinos are $m(\nu_{\mu})<0.17MeV$ and $m(\nu_{\tau})<24MeV$ (95\% C.L.). The
events that appear to represent $\bar\nu_{\mu}\to\bar\nu_e$ oscillations followed by
$\bar\nu_e+p\to n+e^+$, $n+p\to D+\gamma$, with coincident detection of $e^+$ and the
$2.2MeV$ neutron-capture $\gamma$ ray in the Liquid Scintillator Neutrino Detector (LSND)
experiment at Los Alamos suggest that
$\Delta m^2_{e\mu }=\mid m(\nu_{\mu})^2- m(\nu_{e})^2\mid >0$ \cite{6}. Comparison with
exclusion plots from other experiments implies a lover limit
$\Delta m^2_{e\mu }=\mid m(\nu_{\mu})^2- m(\nu_{e})^2\mid >0.2eV^2$, implying in turn a
lower limit $m_{\nu}\ge 0.45eV$, or $\Omega_{\nu}\ge0.02(0.5/h)^2$. More data and analysis are
needed from LSND's $\nu_{\mu}\to\nu_e$ channel before the initial hint \cite{7} that
$\Delta m^2_{\mu e}\approx 6eV^2$ can be confirmed. Recent Super-Kamiokande data following
the Kamiokande data \cite{8} show that the deficit of $E>1.3GeV$ atmospheric $\nu_{\mu}$
increases with zenith angle. These data suggested that $\nu_{\mu}\to\nu_{\tau}$ oscillations length
is comparable to the height of the atmosphere, implying that
$\Delta m^2_{\tau\mu}\simeq 10^{-3}eV^2$ -- which in turn implies that if either $\nu_{\mu}$
or $\nu_{\tau}$ have large enough mass ($\ge 1eV$) to be a hot dark matter particles, then they
must be nearly degenerate in mass, i.e., the hot dark matter mass is shared between these two
neutrino species. However, the deficit of atmospheric $\nu_{\mu}$ even at small zenith angles,
corresponding to paths much smaller than oscillation length, causes serious doubts in the
interpretation of Super-Kamiokande and Kamiokande data \cite{9}. At
$\Omega_{\nu}\simeq 1$ neutrino free streaming strongly suppresses adiabatic fluctuations at
scales smaller than galaxy superclusters ($\simeq 10^{15}M_{\odot}$). With the use of the COBE
upper limit, hot DM with adiabatic fluctuations would hardly lead to any structure formation at all.
The proper choice of a possible solution for this problem - transition to more complicated cases, hot
DM plus some sort of seeds, such as cosmic strings (see for example \cite{10}) or to other class of
dark matter candidates, corresponding to cold DM (CDM) scenario - has in fact no fundamental
grounds in the framework of astroparticle physics. Moreover the physical grounds for neutrino
instability or for CDM particles are not alternative to the ones for neutrino rest mass, and from the
physical viewpoint the general case should account for all these possibilities. Cold DM consists of
particles for which the scale of free streaming is very small and its existence leads to strong
dynamical effects at galaxy scale.
The development of CDM models and their troubles in the framework of astroparticle physics seem
to confirm the general wisdom on true complexity of the world system. The two sorts for cold DM
that are best motivated remain supersymmetric particles (WIMPs) and axions.
Supesymmetry underlies almost all new ideas in particle physics, including superstrings. There are
two key feature of supesymmetry that make it especially relevant to DM, $R$ -- parity and the
connection between supersymmetry breaking and the electroweak scale. The $R$ -- parity of any
particle is $R\equiv (-1)^{L+B+S}$, where $L$, $B$, and $S$ are its lepton number, baryon
number, and spin. In most version of supersymmetry, $R$ -- parity is exactly conserved. This has
the powerful consequence that the lightest $R$ -- odd particle -- often called the "lightest
supersymmetric partner" (LSP)- must be stable, for there is no lighter $R$ -- odd particle for it to
decay into. The LSP is thus natural candidate to be the dark matter. In the standard version of
supersymmetry, there is an answer to the deep puzzle why there should be such a large difference in
mass between the GUT scale $M_{GUT}\simeq 10^{16}GeV$ and the electroweak scale
$M_W=80GeV$. Since both gauge symmetries are supposed to be broken by Higgs bosons which
moreover must interact with each other, the natural expectation would be that
$M_{GUT}\simeq M_W$
or that $M_W$ is induced by radiative correction $M_W\sim\alpha M_{GUT}$. The
supersymmetric answer to this "gauge hierarchy" problem is that the masses of the weak boson
$W^{\pm}$ and all other light particles are zero until supersymmetry itself breaks. Thus, there is a
close relationship between the masses of the supersymmetric partner particles and the electroweak
scale. Since the abundance of the LSP is determined by its annihilation in the early Universe, and
the corresponding cross section involves exchanges of weak bosons or supersymmetric particles -- all
of which have electromagnetic-strength couplings and masses $\simeq M_W$ -- the cross section
will be $\sigma\simeq e^2s/M^4_W$ (where $s$ is the square of the center of mass energy) i.e.,
comparable to the that of typical weak interaction processes. This in turn has the remarkable
consequence that the modern density of LSPs can be close to the critical density, i.e.
$\Omega_{LSP}\simeq 1$. The LSP is in the most cases a spin -- $1/2$ Majorana particle called
"neutralino", which represents the linear combination of photino (supersymmetric partner of the
photon), zino (partner of the $Z^0$), Higgsinos (partners of the two Higgs bosons associated with
electroweak symmetry breaking in supersymmetric theory), and axinos (partner of the axion).
Neutralinos are Weakly Interacting Massive Particles (WIMPs) with the mass from tens to hundreds
GeV, and thus are natural candidates for the cold DM.
The prediction of invisible axion follows from another line of theoretical argumentation, related to
the solution of the strong CP violation problem in QCD. Searches for axion emission in $\mu$,
$K$ decays and nuclear decays put lower limit on the scale of axion physics. Constraints on stellar
energy losses due to axion emission put this limit even higher: up to $10^6GeV$ in the case of
archion and up to $10^8GeV$ for the bulk of other invisible axion models. In cosmology,
primordial coherent axion field oscillations were found to behave in respect to gravitational
instability as gas of very heavy particles, making invisible axion popular CDM candidate.
Experimental searches for cosmic and Solar axion fluxes are under way, based on the predicted
effect of axion-photon conversion in time--varying electromagnetic field.
In the framework of astroparticle physics it is not possible to find physical motivations which
candidate on CDM particle -- neutralino or axion -- is more preferable. From particle physics
viewpoint the both candidates are important, since both supersymmetry and invisible axion solution
are necessary to remove internal inconsistencies of the standard model: supersymmetry removes
quadratic divergence of Higgs boson mass in the electroweak theory and axion recovers from strong
CP violation in QCD. Astroparticle physics has no theoretical tools to find the proper combination
for the both hypothetical phenomena. Moreover, recent analysis of the observational data on the
large scale structure and of the anisotropy of thermal electromagnetic background find troubles in
simple CDM model and favors more sophisticated dark matter scenario, such as mixed cold+hot
dark matter (see for example \cite{11}). It appeals for necessity in special methods to deal with
multiparameter space of physical and cosmological parameters, which astroparticle physics does not
possess.
Together with the proper combination of studies of cosmological large scale structure, relic
radiation, nucleosynthesis, tests for inflational, baryosynthesis and dark matter models
cosmoparticle physics invokes such forms of crossdisciplinary studies as cosmoarcheology or
experimental physical cosmology.
\section{Cosmoparticle approach to the problem of fermion masses and mixing}
The problem of fermion families is one of key problems in the modern particle physics. It has
different aspects, questioning the origin of family replication, quark and lepton mass spectrum and
mixing pattern, CP violation in weak interactions, CP conservation in strong interactions,
suppression of flavor changing neutral currents (FCNC), pattern of neutrino masses and oscillations,
etc. Thus the particle model of fermion families should offer the solution to all these problems.
The standard model (SM) is successful in describing various experimental data (see for example
\cite{12}) and it can be considered as a minimal necessary element of any theory of flavor. In SM
the three families, sharing the same quantum numbers under the
$SU(3)_C\bigotimes SU(2)_W\bigotimes U(1)_Y $ gauge symmetry, are introduced as an anomaly
free set of chiral left-handed fermions $q_i=(u_i,d_i)$, $u^c_i$, $d^c_i$;
$l_i(\nu_i,e_i)$, $e^c_i$,
where $i=1,2,3$ is a family index. In SM the masses of fermions and $W^{\pm}$, $Z$ gauge
bosons have the common origin in the Higgs mechanism. Quarks and charged leptons get masses
through the Yukawa couplings to the Higgs doublet $\phi$:
\begin{equation}
\label{1}
L_{Yuk}=\lambda^u_{ij}q_iCu^c_j\tilde\phi +\lambda^d_{ij}q_iCd^c_j\phi +
\lambda^e_{ij}l_iCe^c_j\phi\qquad (\tilde\phi =i\tau_2\phi^*)
\end{equation}
So, the fermion masses are related to the weak scale
$\langle\phi\rangle =v=174GeV$. However, the Yukawa constants are arbitrary, namely
$\hat\lambda^{u,d,e}$ are in general complex $3\times 3$ matrices. To reproduce the masses of
quarks and leptons one has to put by hands $27$ values of these matrix elements. The SM contains
no renormalizable couplings that could generate the neutrino masses:
\begin{equation}
\label{2}
L_{\nu}=\frac{\lambda^{\nu}_{ij}}{M}(l_i\tilde\phi )C(l_j\tilde\phi ),\qquad
\lambda^{\nu}_{ij}=\lambda^{\nu}_{ji}
\end{equation}
where $M>>v$ is the regulator mass, which depends on the mechanism of neutrino mass ({2})
generation. The matrices of coupling constants and the corresponding fermion mass matrices
$\hat m^f=\hat\lambda^fv$ $(f=u,d,e)$ and $\hat m^{\nu}=\hat\lambda^{\nu}(\nu^2/M)$
can be reduced to the diagonal form by the unitary transformations $V_f$ and $V_{\nu}$. Hence,
quarks are mixed in the charged current interactions, and these mixings are determined by Cabibbo-
Kobayashi-Maskawa (CKM) matrix. The CKM matrix is parameterized by three mixing angles and
CP-violating phase. In the case of massive neutrinos, a similar mixing matrix emerges also in the
lepton sector.
The fermion family puzzle consists in the following phenomena:
the mass spectrum of quarks and charged leptons is spread over five orders of magnitude, from
MeVs to hundred GeVs;
the weak transitions dominantly occur inside the families, and are suppressed between different
families thereby the SM exhibits the natural suppression of the flavor changing neutral currents
(FCNC), both in the gauge boson and Higgs exchanges;
the Yukawa constants in {1} are generally complex, the observed CP- violating phenomena can be
explained by the CKM mechanism with sufficiently large CP-phase $\simeq 1$. However, at the
same time it induces the strong CP violation problem (see for example \cite{13}): the overall phase
of the Yukawa matrices gives effective contribution to the vacuum $\Theta$- term in QCD and thus
induces the P and CP violation in strong interactions. On the other hand, the measurements of
dipole electric moment of neutron impose the strong bound $\Theta <10^{-9}$;
the experimental data show some ambiguous indications for neutrino masses and mixing.
The fermion mass and mixing problem can be formulated as a problem of matrices of the Yukawa
couplings $\hat\lambda^f$, which remain arbitrary in the SM. There is no explanation, what is the
origin of the observed hierarchy between their eigenvalues, why $\hat\lambda^u$ and
$\hat\lambda^d$ are small, what is the origin of the complex structure needed for the CP- violation
in weak interactions, why the $\Theta$- term is vanishingly small in spite of the complex Yukawa
matrices. It is attractive to think that at some scale above the electroweak scale there exists a more
fundamental theory which could allow to calculate the Yukawa couplings, or at least to fix the
relationship between them.
The structure of mass matrix can be related with the spontaneously broken horizontal symmetry
between fermion families. Consider, for example, model with all quark and lepton states
transforming as triplets
$f_{\alpha}=(q,l,u^c,d^c,e^c)_{\alpha}$ of the horizontal $SU(3)_H$ symmetry \cite{14},
($\alpha =1,2,3$ is a family index). Such a horizontal symmetry does not allow quarks and leptons
to have renormalizable Yukawa couplings. Thus, the fermion mass generation is possible only after
the $SU(3)_H$ breaking, through the high order (non-renormalizable) operators (HOPs) involving
some "horizontal" Higgses inducing this breaking at the scale $V_H>>v$. This suggests that the
observed mass hierarchy may emerge due to the hierarchy in the $SU(3)_H$ breaking. Full
$SU(3)_H$ breaking is achieved by introducing the horizontal scalars: a sextet
$\chi_3^{\{\alpha\beta\}}$
and two other sextets or triplets
$\chi_{1,2}^{[\alpha\beta ]}\simeq\varepsilon^{\alpha\beta\gamma}\chi_{\gamma}$.
The pattern of their $3\times 3$ VEV matrix can be chosen so that the first sextet VEV is acquired
by $(3,3)$ component, and in sextets (or triplets) $\chi_2$ and $\chi_1$ the smaller VEVs
$V_{23}$ and $V_{12}$ are acquired by $(2,3)$ and $(1,2)$ (or first and third ) components.
VEVs follow the hierarchy $V_{33}>>V_{23}>>V_{12}$, which is stable relative to radiative
corrections. Thus in the context of the $SU(5)\otimes SU(3)_H$ theory with fermions in
representations $(\bar 5+10)_{\alpha}$, the relevant HOPs \cite{14} can be induced through the
renormalizable interactions, as a result of integrating out the effects of hypothetical superheavy
particles (see, for example, \cite{15,16}). In the other words, the quark and lepton masses can be
induced through their mixing with superheavy -- fermions, in a direct analogy to the see--saw
mechanism of neutrino mass generation. In this case the VEV pattern of Higgs multiplets $\chi$ is
reflected in the Yukawa matrices, and the fermion mass hierarchy follows the hierarchy of
$SU(3)_H$ symmetry breaking. There are two possible choices for the representation of $F$ --
fermions, and, respectively, one can generate two types of the pattern of Yukawa mass matrices
\cite{17,18}. The first case corresponds to a direct hierarchy pattern. In particular, the VEV pattern
leads directly to the Fritzch texture. Another possibility is the inverse hierarchy. In the latter case
the VEV pattern is inverted in the fermion mass structure (see more detail \cite{17,18,16}).
Thus, the horizontal $SU(3)_H$ symmetry is attractive since it unifies all families. For the solution
of the strong CP- problem one can introduce the Peccei-Quinn (PQ) type symmetries \cite{19},
which in additionally could further restrict the mass matrix structure. In particular, in the horizontal
$SU(3)_H$ symmetry models the PQ symmetry can be naturally related to the phase
transformation of the horizontal scalars $\chi$ \cite{17,18}.
Consider as an example the application of the approach of cosmoparticle physics (section 2) to the
problem of fermion flavours. This strategy can be stipulated as follows.
{\bf Step 1.} The class of physically motivated extensions of SM is considered, namely, the class
of gauge models with horizontal family symmetry.
{\bf Step 2.} The inevitable consequences of chosen class of models, which are able to reproduce
cosmological and astrophysical phenomena and effects are the following:
\begin{itemize}
\item the existence of the specific type of invisible axion (archion), which is simultaneously
Majoron and familon \cite{17,18};
\item the existence of horizontal scalars $\chi$ with superhigh energy scale of VEVs;
\item the existence of neutrino Majorana mass with the hierarchy of neutrino masses;
\item the nonconservation of lepton number $\Delta L=2$;
\item the instability of neutrino relative to decays on more light neutrino and archion;
\item the Dirac see-saw mechanism and singlet scalar $\eta$ , connected with it;
\end{itemize}
{\bf Step 3.} One introduces the main free parameter $V_H$ of the hidden sector of the considered
model, namely, the scale of horizontal $SU(3)_H$ symmetry breaking. The set of indirect
cosmological, astrophysical and experimental physical restrictions on the hidden sector is revealed
from following phenomena:
\begin{itemize}
\item from the analysis of data of nondiagonal transitions (for example $\mu\to ea$ and $K\to\pi a$
(where $a$ is archion)) \cite{20,21};
\item from the astrophysical estimations of stellar energy losses due to archion emission \cite{18};
\item from the analysis of archion emission influence the time scale and energetics of neutrino flux
from collapsing star \cite{18};
\item from the analysis of inhomogeneities generated by the large scale modulation of coherent
axion field oscillations \cite {22,23,24};
\item from the analysis of primordial black holes formation in the second order phase transitions
connected with three stages of horizontal $SU(3)_H$ -- symmetry breaking, which take place at the
inflationary stage \cite{25,24};
\item from the effect of nonthermal horizontal symmetry restoration at postinflational dust-like
stage \cite{24,26};
\end{itemize}
Taking together all limits imposed by the pointed phenomena it is possible to extract two narrow
windows for the value of the parameter $V_H$. They are the "low" energy branch $V_6$
\cite{23,27} and the "high" energy branch $V_{10}$ \cite{24}.
{\bf Step 4.} With the use of the above restrictions one can elaborate the physically motivated full
cosmological model, which is based on the chosen horizontal extension of SM. This model has been
called the model of "horizontal" unification (MHU) \cite{23,24}.
\begin{itemize}
\item MHU solves the problems of SM connected with family problem and strong CP violation
problem in QCD; it predicts qualitatively new type of invisible axion (archion) \cite{17,18,28}; it
predicts the neutrino masses and neutrino flavour nondiagonal transitions with emission of archion.
\item MHU predicts the following history of Universe:
\begin{itemize}
\item The early Universe starts from the inflational stage \cite{23,24}, driven by the inflaton field
$\eta$, being singlet relative to all gauge groups. The VEV of this field plays the role of the
universal energy scale in the Dirac see-saw mechanism of the generation of masses of charged
fermions \cite{16,17,23,24}. When the inflational stage is finished the inflaton field decays due to
interactions assumed by the Dirac see-saw mechanism \cite{23,24}. It leads to reheating of the
Universe and consequently to transition to the standard Friedman cosmology.
\item The reheating temperature is sufficiently high for generation of the observed baryon
asymmetry. The baryogenesis mechanism in the MHU combines the $(B+L)$ nonperturbative
electroweak nonconservation at high temperatures with $\Delta L=2$ nonequilibrium transitions,
induced by Majorana neutrino interaction \cite{23}. The mechanism can provide inhomogeneous
baryosynthesis and even to the existence of antimatter domains in baryon asymmetrical Universe
\cite{24}.
\item There are two possible scenarios of large scale structure (LSS) formation:
\begin{itemize}
\item Hierarchic decay scenario (HDS) \cite{23,21}, realized at the "low" energetic scale ($V_6$).
In the HDS the LSS formation takes place in the succession of stages of dominance of unstable
neutrino and their relativistic decay products.
\item Mixed stable dark matter, realized at "high" energetic scale ($V_{10}$) \cite{24}. The
formation of LSS in this case occurs at the conditions of dominance of coherent oscillations of
axion field and massive stable neutrino (see \cite{24} in more detail).
\end{itemize}
\end{itemize}
\end{itemize}
{\bf Step 5.} The system of the detailed indirect test of MHU and MHU-based cosmological
scenario can use the following signatures:
\begin{itemize}
\item MHU predicts flavour nondiagonal decays of leptons, mesons and hyperons (see \cite{21,27}
in more detail);
\item MHU predicts the level of oscillations $K\to\bar K$, $B\to\bar B$ \cite{21};
\item astronomical search for invisible axions (see for example \cite{29}) and their two-photon
decays;
\item experimental searches for solar axions (see for example \cite{30});
\item experimental searches for the force, violating the Equivalence Principle, which is connected
with the existence of invisible axion (see for example \cite{31}).
\end{itemize}
{\bf Step 6.} The estimation of completeness of obtained scenario is necessary to determine the
direction of the further extension of the considered approach. In the other words the elaborated
cosmological model should incorporate the cosmological consequence of some other extensions of
the SM such as GUT and SUSY. In particular, the estimation of completeness of MHU can be
obtained by the comparison of the predicted consequences of the MHU-based scenario of inflation,
baryosynthesis and LSS formation with the astronomical observations (see \cite{24} in more
details).
To conclude, the development of cosmology and particle physics and the nontrivial tests of their
foundations in combination of indirect evidences follow the laws of cosmoparticle physics, that will
unify on the basis of its principles the existing trends in studies of mutual relationship of elementary
particles and the Universe, widely represented in the present proceedings.
|
2,877,628,089,000 | arxiv | \section{Introduction}
The contacts between metals and semiconductors (M/S) play an important role in many practical electronic devices with far reaching applications. A controlled tuning of contact properties~\cite{Ivanco2000,Xi2014}, that is of technological relevance, relies on the fundamental understanding of the underlying physics and material properties.
Recent progress in planar GaAs Schottky barrier diodes, primarily developed for the X-ray detection applications, stimulated interest in the low work function M/S contacts, since the diodes possessing them revealed unexpectedly and favorably low leakage current~\cite{Bohacek2008,Dubecky2009,Dubecky2013}. Further improvement of these devices with a promising application potential in sensorics, $\gamma$/X-ray detection and medical imaging~\cite{Zatko2004}, relies on the properties of the interfaces between the reactive low work function metal layers and surface passivisation metals.
In one of the most striking cases~\cite{Dubecky2013}, the Mg overlayer on GaAs was passivated by the Au. Since the XPS analysis of the formed contact suggests a presence of a non-ideal disordered AuMg alloy~\cite{Dubecky2014} with the changing composition instead of distinct Au/Mg phases, the question remains to be answered: what is the work function of the alloy right at the alloy/GaAs interface? This value would be decisive for further understanding of the non-trivial transport phenomena observed~\cite{Dubecky2013}.
An AuMg alloy is resistant to oxygen, water and organic solvents~\cite{Oyamada2005} and it has been successfully used e.g. as a cathode material in organic light emitting diodes~\cite{Tang1987,Oyamada2005}. To best of our knowledge, its work function is known only from one experiment~\cite{Oyamada2005} and conclusive theoretical work on this subject is not
available. In order to fill this gap, in the present work we theoretically study \mbox{Au/Mg/AuMg} surface models with the primary goal to understand how the work function of these systems behaves in non-ideal scenarios. To this end, we consider a defective Mg decorated Au(100) surface, Au decorated Mg(001) surface, and various low-index ideal/disordered CsCl-like~\cite{Thomas1986} AuMg surfaces. The selected systems help to clarify how is the work function of the Au-Mg systems affected by the structure, defects and surface composition.
As we show below, within the set of the considered models, the work functions are Mg-dominated. I.e. they always lie below the average of Au and Mg work functions, and mostly close to the Mg one, in agreement with the experimental observations. {We find, that this trend is caused by the surface dipole formation due to the charge transfer from Mg to Au.} The reported results may find practical applications in Au/Mg/AuMg surface physics and in the technology of low-work function M/S contacts.
\section{Methods}
The calculations were performed using the plane-wave density functional (DFT) theory with the ab-initio Perdew-Wang (PW91) exchange correlation (xc) functional~\cite{PW91}, as implemented in the VASP code~\cite{VASP}. {Initial tests of the PBE xc functional~\cite{Perdew1996} are reported in addition.} The nuclei were represented by the projected augmented wave pseudopotentials~\cite{Blochl1994,Kresse1999} and electronic wave functions were expanded in the plane-wave basis set with the 400~eV energy cutoff. The forces acting on nuclei in optimizations were converged to 0.001 eV/\AA.
The conventional cell parameters of the bulk Au (fcc), Mg (hcp) and AuMg (CsCl structure) were optimized using a 10$\times$10$\times$10 k-point grid and used to construct 1$\times$1 or 2$\times$2 (to keep the even number of electrons per cell) surface five-layer (enough for our purposes~\cite{Fall1999}) symmetric slab models with at least 20\AA~vacuum region. Atomic positions were subsequently optimized using a 10x10x1 k-point grid, whereas the final single-point runs were obtained with the 14x14x1 {(or equivalent)} k-point grids.
The work-functions~($\Phi$) were obtained from the differences between the plane-averaged local potential~{(cf. e.g. Ref.~\citenum{Rusu2006})} energies at the vacuum level ($E_{\textrm{vac}}$) and the Fermi energy ($E_{\textrm{F}}$), i.e. $\Phi=E_\textrm{vac}-E_\textrm{F}$.
{The surface dipole properties are understood in terms of the relative changes of the surface dipole component perpendicular to the ideal optimized surfaces, $\Delta\mu_\bot$, calculated as a dipole correction ~\cite{Makov1995}. A positive $\Delta\mu_\bot$ implies lower $\Phi$ with respect to the reference surface.}
The AuMg surface energies $\gamma$ (in the limit of $T\rightarrow0$), that determine the stability, were estimated from thermodynamic considerations~\cite{Reuter2001,Reuter2008,Bendavid2013} assuming limiting cases of Au- ($\gamma^{\text{Au-rich}}$) and Mg-rich ($\gamma^{\text{Mg-rich}}$) atmospheres. The Au-rich limit is relevant to the experimental conditions of the GaAs/Mg/Au contact preparation that is of our interest~\cite{Dubecky2013}. We neglect $pV$ terms and zero-point vibrations~\cite{Reuter2001} and the Gibbs free energies are approximated by the total energies ($E$) from the zero-temperature DFT calculations~\cite{Bendavid2013}.
The chemical potential of Au (per atom) in the Au-rich limit is solely determined by the Au bulk which is the preferred phase for Au, i.e.
\begin{equation}
\mu_{\text{Au}}\approx E_{\text{tot}}^{\text{Au}}/N_{\text{Au}},
\end{equation}
where $E_{\text{tot}}^{\text{Au}}$ is the total Au bulk energy and $N_{\text{Au}}$ is the number of Au atoms per bulk simulation cell.
The chemical potential of Mg is subsequently fixed by the thermodynamic equilibrium condition,
\begin{equation}
\mu_{\text{Mg}}^{\text{Au-rich}}=\mu_{\text{AuMg}}-\mu_{\text{Au}},
\end{equation}
leading to the approximation
\begin{equation}
\mu_{\text{Mg}}^{\text{Au-rich}}\approx E_{\text{tot}}^{\text{AuMg}}/N_{\text{AuMg}}-E_{\text{tot}}^{\text{Au}}/N_{\text{Au}}.
\end{equation}
The surface energy of AuMg expressed in terms of the approximate chemical potentials defined above reads
\begin{equation}
\gamma^{\text{Au-rich}}\approx \frac{1}{2A}[E_{\text{tot}}^{\text{slab}}-N^{\text{slab}}_{\text{Au}}\mu_{\text{Au}}-N^{\text{slab}}_{\text{Mg}}\mu_{\text{Mg}}^{\text{Au-rich}}],
\end{equation}
where $A$ is the slab-model surface area, the factor $1/2$ accounts for the two surfaces in the simulation cell, $N^{\text{slab}}_{\text{Au}}$ and $N^{\text{slab}}_{\text{Mg}}$ are counts of the number of Au and Mg atoms in the relaxed slab, respectively.
In a similar way, one can define the following chemical potentials relevant for the Mg-rich case,
\begin{equation}
\mu_{\text{Mg}}\approx E_{\text{tot}}^{\text{Mg}}/N_{\text{Mg}},
\end{equation}
\begin{equation}
\mu_{\text{Au}}^{\text{Mg-rich}}\approx E_{\text{tot}}^{\text{AuMg}}/N_{\text{AuMg}}-E_{\text{tot}}^{\text{Mg}}/N_{\text{Mg}},
\end{equation}
and use them to obtain
\begin{equation}
\gamma^{\text{Mg-rich}}\approx \frac{1}{2A}[E_{\text{tot}}^{\text{slab}}-N^{\text{slab}}_{\text{Au}}\mu_{\text{Au}}^{\text{Mg-rich}}-N^{\text{slab}}_{\text{Mg}}\mu_{\text{Mg}}],
\end{equation}
a surface energy in the limit of Mg-rich atmosphere.
\section{Models}
The studied surface models (cf. Tab.~\ref{tabresA}, Tab.~\ref{tabresB} and Fig.~\ref{fig_disord}) include~\cite{suppInfo}:
i) {clean Au(100) (A1), Au(100) with surface missing-atom defects~(A2-A4), and Au(100) with a subsurface missing atom defect (A5).}
ii) Au(100) surface with one Au atom substituted by Mg (B1), Mg decorated Au(100) with Mg at the top, bridge and hollow positions (B2-B4), B1 with an additional Mg atom on top of Au, i.e. Mg$_2$/Au(100) (B5-6) and Mg(1ML)/Au(100), i.e. Au covered by a single Mg monolayer (B7).
iii) In the case of Mg, the studied models include a pure Mg(001) surface (M1), Mg(001) with a missing-atom defect (M2).
iv) Concerning the Au decorated Mg, only a single Au atom decorated Au/Mg(001) model (N1) where the Au atom resides at the hollow site is considered. The calculations starting from the top and bridge configurations converged to the same state as N1 and are therefore not reported.
v) The models of AuMg surface include ideal CsCl-like (100), (110), and (111) surfaces (G1-G5), Mg decorated AuMg(110) (G6), and two disordered $3\times3$ (110) surfaces (G7-8) generated by the Born-Oppenheimer {ab-initio} molecular dynamics performed at 2500~K~{\cite{Konopka2009}} and subsequent optimization of the two random snapshots to their respective nearest local minima.
\section{Results and Discussion}
{\subsection{Benchmarks}}
The bulk lattice parameters of Au and Mg and work functions of clean Au(100) and Mg(001) surfaces, {calculated using PW91 and PBE xc correlation functionals}, are reported in the Tab.~\ref{tabref}. The results {from both functionals } well agree to the experimental data and previous theoretical calculations~\cite{Prada2008}, confirming the reliability of {the used DFT} approach {(for an extended discussion regarding suitability of DFT functionals for modeling of Au, cf. e.g. Ref.~\citenum{Dubecky2012}). The data also reveal that the PBE functional performs slightly better with respect to the reported experiment, nevertheless, since both approaches generate similar results and the PW91 is known to perform well in various solid-state surface models containing Au/Mg~\cite{Giordano2005,Rusu2006,Prada2008}, we continue to use PW91 throughout the work.
\begin{table}[h!]
\scriptsize
\caption{The calculated lattice parameters ($a_0$, $c_0$) of Au (fcc), Mg (hcp), and the work functions ($\Phi$) of Au(100) and Mg(001) surfaces, compared to the experiment.}
\begin{tabular}{llcccccc}
\hline
\hline
& &$a_0$/\AA & $c_0$/\AA & $\Phi$/eV \\
\hline
Au/Au(100) & PW91 & 4.18 & - & 5.10 \\
& {PBE} & {4.16} & - & {5.11} \\
& experiment & 4.08$^a$ & - & 5.22$^b$ \\\\
Mg/Mg(001) & PW91 & 3.20 & 5.18 & 3.72 \\
& {PBE} & {3.20} & {5.19} & {3.72} \\
& experiment & 3.21$^c$ & 5.21$^c$ & 3.66$^b$ \\
\hline\hline
$^a$~Ref.~\citenum{Maeland1964}\\
$^b$~Ref.~\citenum{Michaelson1977}\\
$^c$~Ref.~\citenum{Walker1959}\\
\end{tabular}
\label{tabref}
\end{table}
{\subsection{Clean and Au/Mg Decorated Au(100) and Mg(001) Surfaces}}
The top-view illustrations~\cite{VESTA} of the considered Au(100) and Mg(001) surface models together with the calculated and experimental $\Phi$ (where available) {and $\Delta\mu_\bot$} are summarized in the Tab.~\ref{tabresA}. Apparently, the surface defects (missing atoms) lower $\Phi$ with respect to the ideal surface in both, Au (A2-4 vs. A1) and Mg (M2 vs. M1), respectively. The presence of a subsurface defect may, on the other hand, slightly increase $\Phi$ (A5). {These changes correlate with $\Delta\mu_\bot$ defined with respect to the clean Au (A1), as reported for A2 and A5 and clean Mg (N1), respectively.}
\begin{table}[ht!]
\scriptsize
\caption{The illustrations of the considered Au and Mg surface models, the related work functions ($\Phi$) {and surface dipole differences ($\Delta\mu_\bot$)}. Colors: yellow - Au, blue - Mg belonging to the contiguous surface layer, royal blue - Mg ad-atoms.
}
\newcolumntype{C}{>{\centering\arraybackslash} m{1.1cm} }
\begin{tabular}{Cm{3.3cm}Cc|C|c}
\hline
\hline
Label&Model & Top view & & $\Phi$/eV & {$\Delta\mu_\bot$/eV\AA{}} \\
\hline
&&&& \\
& Au {(experiment)} && & 5.22$^a$ \\
A1& Au(100) & \includegraphics[width=25pt,bb=0 0 344 344]{./Au100.jpg} & & 5.10 \\
A2& Au(100) w/ defect 1 & \includegraphics[width=25pt,bb=0 0 314 314]{./Au100def1.jpg} & & 5.06 & {0.03}\\
A3& Au(100) w/ defect 2 & \includegraphics[width=25pt,bb=0 0 344 344]{./Au100def2.jpg} & & 4.95 \\
A4& Au(100) w/ defect 3 & \includegraphics[width=25pt,bb=0 0 314 314]{./Au100def3.jpg} & & 4.75 \\
A5& Au(100) w/ bulk defect & \includegraphics[width=25pt,bb=0 0 344 344]{./Au100.jpg} & & 5.21 & {-0.02}\\
\hline
&&&&\\
B1& Au(100) Mg substituted & \includegraphics[width=25pt,bb=0 0 304 304]{./Au100Mgreplaced.jpg} & & 4.59 & {0.14}\\
B2& Au(100)/Mg top & \includegraphics[width=25pt,bb=0 0 356 356]{./Au100Mgatop.jpg} & & 3.57 & {0.47}\\
B3& Au(100)/Mg bridge & \includegraphics[width=25pt,bb=0 0 311 320]{./Au100Mgbridge.jpg} & & 3.46 & {0.50}\\
B4& Au(100)/Mg hollow & \includegraphics[width=25pt,bb=0 0 311 311]{./Au100Mghollow.jpg} & & 3.40 & {0.48}\\
B5& Au(100)/Mg$_2$ X & \includegraphics[width=25pt,bb=0 0 392 392]{./Au100Mg2case1.jpg} & & 3.42 & {0.48}\\
B6& Au(100)/Mg$_2$ XY & \includegraphics[width=25pt,bb=0 0 392 392]{./Au100Mg2case2.jpg} & & 3.37 & {0.48}\\
B7& Au(100)/Mg(1ML) & \includegraphics[width=25pt,bb=0 0 344 344]{./Au100Mg1ML.jpg} & & 3.87 & {0.38}\\
\hline
&&&&\\
& Mg {(experiment)} & & & 3.66$^a$\\
M1 & Mg(001) & \includegraphics[width=34pt,bb=0 0 794 541]{./Mgsurf.jpg} & & 3.72 \\
M2 & Mg(001) w/ defect & \includegraphics[width=34pt,bb=0 0 936 636]{./Mgsurf_def.jpg} & & 3.71 \\
\hline
&&&&\\
N1 & Mg(001)/Au hollow & \includegraphics[width=34pt,bb=0 0 786 535]{./Mgsurf_Au_hollow.jpg}& & 4.10 & {-0.12}\\
\hline
\hline
$^a$~Ref.~\citenum{Michaelson1977}\\
\end{tabular}
\label{tabresA}
\end{table}
In the case of Mg-substituted Au(100) surface (B1), where one of the surface Au atoms per cell is replaced by Mg, the surface $\Phi=4.59$~eV is significantly lowered (by 0.51~eV) with respect to the clean Au(100) ($\Phi=5.10$~eV, A1). The work functions drop further {by an additional $\sim$~1~eV,} to $\Phi=3.37-3.57$~eV, when Mg decorates the Au surface. In all the considered cases {where Mg decorates Au} (B2-B6), these values are even below the {theoretical value of} $\Phi$ of an ideal clean Mg(001) surface ($\Phi=3.72$~eV).
The work function of the Au surface covered by an ideal single Mg monolayer, i.e. Mg(1ML)/Au(100) (B7), amounts to 3.87~eV. {Here, the drop of $\Phi$ is not as pronounced, since $\Delta\mu_\bot$ induced by the adsorbed {\it ideal} monolayer of Mg is smaller than in the cases where Mg atom lies on the clean/substituted Au surface (B2-B6)}.
For completeness, we mention that the Au decorated Mg(001) reveals the $\Phi=4.10$~eV, i.e. higher by 0.44~eV with respect to the clean Mg, nevertheless still below the average of clean Au and Mg~($\Phi_{\text{avg}}$=4.41~eV). The data clearly indicate, that Mg strongly dominates the surface work functions of Au/Mg surfaces containing Mg.
\begin{figure}\label{figDOS1}
\centering
\includegraphics[width=240pt, bb= 0 0 1600 798]{./PDOS_B1.jpg}
\newline
\newline\\
\includegraphics[width=240pt, bb= 0 0 1600 798]{./PDOS_B4.jpg}
{\caption{Densities of states (DOS) for the B1 (top) and B4 (bottom) models of the Mg-modified Au surface (cf.~Tab.~\ref{tabresA}). The vacuum energy is aligned to zero and the vertical lines indicate the Fermi level. The charge density differences (right; decrease - red, increase - blue, isosurface 0.0015 a.u.) were obtained from the final Mg-modified Au(100) surface models and their pure unoptimized Au/Mg constituents in the identic cell.}}
\end{figure}
{In order to understand why the presence of Mg induces such a strong lowering of the Au surface work function, we further analyze the selected representative models, B1 and B4, in terms of the density of states~\cite{Giordano2005} (DOS) and electron density differences (Fig.~1). The Fermi levels in Fig.~1 in both cases, B1 and B4, indicate that the charge must flow from Mg to Au in order to maintain equilibrium. This effect is demonstrated in the electron density difference plots (Fig.~1), which reveal a qualitative difference between the models, even though the $s$ charge transfer from Mg due to the reaction with Au is very similar (0.18$e$ in B1 vs. 0.13$e$ in B4; estimates from incomplete projections within the Wigner-Seitz radii) and even slightly more pronounced in case with higher $\Phi$. In B1 where one of the surface atoms is substituted by Mg, the charge density is primarily rearranged within the surface plane, whereas in B4, the rearrangement takes place primarily along the surface
normal, which leads to a more enhanced $\Delta\mu_\bot$, thus lowering $\Phi$ more significantly. }
\begin{figure} \label{figcorr}
\centering
\includegraphics[width=175pt,bb=0 0 1150 793]{./correlation.jpg}
{ \caption{Dependence of the work functions $\Phi$ on the relative out-of-surface dipole component $\Delta\mu_\bot$ in the studied Mg-modified Au surfaces.}}
\end{figure}
{An effect of the surface dipole and its directionality is thus identified as a primary reason responsible for the significant lowering of $\Phi$ observed in all the Mg-modified Au surface models (B1-B7), which is further corroborated by the remarkably good correlation between $\Delta\mu_\bot$ and $\Phi$, as reported in Fig.~2.}
{\subsection{AuMg Alloy Surfaces}}
{The optimum lattice constant of bulk AuMg with CsCl structure was found to be $a=3.31$~\AA{}, in agreement with the previous theoretical and experimental values~\cite{Methfessel2000}.}
The results concerning the studied low-index CsCl-like AuMg surfaces, together with the related illustrations, are reported in the Tab.~\ref{tabresB}. The Mg-rich (G1) and Mg-top (G4) surfaces reveal the lowest $\Phi$, and the Au-rich and Au-top surfaces (G2 and G5) reveal the highest $\Phi$, as expected. The surface energies $\gamma${, that determine the surface stability at realistic conditions,} in both considered limits (Au-rich and Mg-rich {atmospheres}) reveal that the most stable surface is AuMg(110), containing Au and Mg in the same plane (G3, $\Phi=4.12$~eV).
{In the following, we further analyze the most thermodynamically stable AuMg(110) surface. First, we consider an additional Mg adatom on the G3 surface, that leads to the G6 model. The Mg adatom further lowers $\Phi$ by a significant amount of 0.53~eV, that is fully attributable to the change in the surface dipole component $\Delta\mu_\bot$ (Tab.~\ref{tabresB}).}
{Finally, an effect of disorder on the work function of AuMg(110) is considered. In order to produce non-ideal} structures, the AuMg(110) was annealed at \mbox{2500~K} and two randomly chosen snapshots were subsequently reoptimized (G7 and G8, Fig.~\ref{fig_disord}). A resulting disorder{ed structures reveal} lowering {of} $\Phi$ from the original 4.12~eV to 4.04 (G7) and 3.82~eV (G8), respectively.
In the case of G8, a more pronounced lowering of $\Phi$ is attributed to the presence of Mg dimer lying out-of-plane, a structural feature of this model { (qualitatively similar to the B5 and B6 models of Mg-modified Au)}. Therefore, a nonideality enables an easier electron withdrawal (lower $\Phi$) from the surface, similar to the Mg adatom cases (B3, B4-6 and G6), compared to the the ideal case (G3) or non-ideal case (G7) that is {more} flat {(or closer to the ideal clean surface)}.
\begin{figure}[t!]
\centering
\includegraphics[width=98pt,bb=0 0 1000 500]{./disorder_top1.jpg} (G7)
\includegraphics[width=98pt, bb= 0 0 1000 500]{./disorder_top2.jpg} (G8)
\caption{The disordered AuMg(110) 3$\times$3 surface models produced by molecular dynamics (at 2500~K) and subsequent optimization to the nearest local minimum on the potential energy surface.}
\label{fig_disord}
\end{figure}
\begin{table*}[ht!]
\scriptsize
\caption{The illustrations of the considered AuMg surface models, the related work functions ($\Phi$), surface energies, $\gamma^{\text{Au-rich}}$ and $\gamma^{\text{Mg-rich}}$ {and surface normal dipole component changes $\Delta\mu_\bot$} (for definitions, cf. Methods). Colors: yellow - Au, blue - Mg.
}
\newcolumntype{C}{>{\centering\arraybackslash} m{1.4cm} }
\newcolumntype{D}{>{\centering\arraybackslash} m{2.05cm} }
\begin{tabular}{Cm{3.3cm}Cc|C|Dc|c}
\hline
\hline
Label&Model & Top view & & $\Phi$/eV & $\gamma^{\text{Au-rich}}$/eV\AA{}$^{-2}$ & $\gamma^{\text{Mg-rich}}$/eV\AA{}$^{-2}$ & {$\Delta\mu_\bot$/eV\AA{}} \\
\hline
&&&&&&\\
& AuMg {(experiment)} & & & 3.70$^a$ & & \\
G1& AuMg(100) CsCl, Mg-rich & \includegraphics[width=27pt,bb=0 0 506 506]{./AuMg100_Mgrich.jpg} & & 3.37 & 1.23 & 0.08 \\
G2& AuMg(100) CsCl, Au-rich & \includegraphics[width=27pt,bb=0 0 538 538]{./AuMg100_Aurich.jpg} & & 4.83 & 1.08 & 0.15 \\
G3& AuMg(110) CsCl & \includegraphics[width=23pt,bb=0 0 346 272]{./AuMg110-CsCl.jpg} & & 4.12 & 0.73 & 0.06 & \\%{0.39$^c$}\\
G4& AuMg(111) CsCl, Mg-top & \includegraphics[width=30pt,bb=0 0 591 419]{./AuMg111_Mgtop.jpg} & & 3.58 & 1.02$^b$ & 0.09$^b$\\
G5& AuMg(111) CsCl, Au-top & \includegraphics[width=30pt,bb=0 0 651 462]{./AuMg111_Autop.jpg} & & 4.28 & 1.02$^b$ & 0.09$^b$\\
&&&&&&\\
\hline&&&&&&\\
G6& AuMg(110) Mg ad-atom & \includegraphics[width=22pt,bb=0 0 792 544]{./AuMg110-CsCl-adatom.jpg} & & 3.59 & - & - & {0.10}\\
&&&&&&\\
G7& AuMg(110) disordered & Fig.~\ref{fig_disord} & & 4.04 & - & - \\
&&&&&&\\
G8& AuMg(110) disordered & Fig.~\ref{fig_disord} & & 3.82 & - & -\\
&&&&&&\\
\hline\hline
$^a$~Ref.~\citenum{Oyamada2005}\\
$^b$~Average\\
\end{tabular}
\label{tabresB}
\end{table*}
Overall, {the considered AuMg surfaces} (except for the ideal pure Au-terminated surface) show a trend observed in Mg decorated Au, i.e. that the work functions are Mg dominated (lie below $\Phi_{\text{avg}}$) and typically approach {the work function of} pure Mg. Nonideality/disorder further {lowers} $\Phi$. Based on the reported results, {we theorize} that the work function of an amorphous AuMg is similar to the pure Mg, a conclusion in agreement with the experimental observations~\cite{Oyamada2005} and expectations~\cite{Dubecky2013}.
\section{Conclusions}
The work functions of the non-ideal Au, Mg and AuMg surfaces were calculated by the ab-initio plane-wave density functional theory. Irrespective of the details, the considered models containing Mg on the surface, {including} AuMg alloys, reveal strongly Mg-dominated work functions, i.e. significantly shifted toward the work function of pure Mg(001) surface. {This effect is dominantly caused by the enhancement of the surface dipole due to the charge transfer from Mg to Au and more pronounced if the charge transfer occurs along the surface normal direction.} A stable AuMg alloy possesses a low work function similar to the reactive Mg, while being remarkably stable against water and air~\cite{Oyamada2005}, and is thus well suited for technological applications including contact metallizations in devices based on metal/semiconductor interfaces.
\begin{acknowledgments}
The support from the Operational Programme Research and Development (OP R\&D) for Innovations - European Regional Development
Fund (ERDF, project CZ.1.05/2.1.00/03.0058), the OP Education for
Competitiveness - European Social Fund (projects CZ.1.07/2.3.00/30.0004 and CZ.1.07/2.3.00/20.0058), and by the Slovak Grant Agency for Science (VEGA 2/0167/13 and 2/0175/13), is gratefully acknowledged.
The calculations were in part performed at the Slovak HPC infrastructure (projects ITMS 26230120002 and 26210120002)
supported by the ERDF OP R\&D.
\end{acknowledgments}
|
2,877,628,089,001 | arxiv | \section{Introduction}
A wide range of imaging modalities use tomographic reconstruction
algorithms to form 2-D and 3-D images from their projection data. While
the classical Filtered Back Projection (FBP) algorithm, and its
variants, are widely used in practice \cite{pan2009commercial}, iterative
reconstruction algorithms hold great potential to enable high-quality
imaging from limited projection data (e.g., few views, limited-view, low-dose) and reduce exposure to radiation. Developments, over last several
decades, in this area often formulate the image reconstruction as an
(ill-posed) inverse problem where a regularized solution is found by an
iterative optimization algorithm. Several aspects of iterative
reconstruction algorithms \cite{gilbert1972iterative, andersen1984simultaneous, andersen1989algebraic, sidky2008image} overlap with active areas of research in
solving these optimization problems efficiently as well as image modeling
with regularization (e.g., sparsity-based, network-based) that enhance
the quality of recovered image from limited data.
An important issue in the performance of iterative reconstruction
algorithms is the discretization, and more generally the representation,
of images. The expansion of an image in a basis allows for derivation of
a {\em forward model}, $\mathbf{A}$, that relates the image coefficients $\mathbf c$ to the measurements $\mathbf y$ in the projection domain by a linear system: $\mathbf y = \mathbf{A} \mathbf c$. The entries of the forward model are computed from the contributions of
each basis function to each measurement in the projection domain.
Specifically, the contribution of a basis function is computed by (1)
integrating the basis function along an incident ray to form its
{\em footprint} and (2) integrating these footprints across a detector
cell in the projection domain (often called the {\em detector blur}).
Common choices for expansion are the pixel- and voxel-basis that provide
a piecewise-constant model for image representation. Kaiser-Bessel \cite{lewitt1990multidimensional}
functions have also been considered as a smooth basis for image
representation.
Given a finite set of measurements $\mathbf y \in \mathbb{R}^m$ and a choice for basis function, one can setup linear systems for images at various resolutions (i.e., discretizations) to be reconstructed from $\mathbf y$. For an image resolution with $N$ elements, characterized by coefficients $\mathbf c \in \mathbb{R}^N$, the forward model is constructed from the footprint of scaled basis functions. To reduce the degrees of freedom in the inverse problem, one wants to build the forward model at the coarsest possible resolution for image discretization; however, doing so limits our ability to resolve for the features of interest in the image. The ability of (a space spanned by translates of) a basis function to approximate images with the smallest resolution is often characterized by approximation order -- scales of a basis function with higher approximation order provide discretizations that approximate the underlying image more accurately, compared to the scales of a basis function with a lower approximation order. While the choice of pixel-basis provides a partition of unity and hence a first-order method for approximation, Kaiser-Bessel functions require filtering operations\cite{nilchian2015optimized} to achieve first-order approximation. A different class of basis functions, called box splines, have been proposed in \cite{entezari2012box} for image discretization in the context of tomographic reconstruction. The first-order box splines coincide with the pixel-basis (in 2-D) and voxel-basis in (3-D), but higher order box splines allow for higher orders of approximation.
Besides approximation order, the effectiveness of basis functions for tomographic reconstruction also depends on the accuracy and efficiency of calculating integrals involved in the footprint and detector blur. The pixel-basis and Kaiser-Bessel functions as well as box splines have closed-form Radon transforms and their footprints can be computed analytically. However, computation of detector blur is more challenging, and several approaches have been proposed for approximating the underlying integrals, such as strip-integral method\cite{lo1988strip}, natural pixel decomposition \cite{byonocore1981natural}, Fourier-based methods \cite{tabei1992backprojection}, distance-driven \cite{de2002distance} approximation, and separable footprints (SF)~\cite{long20103d}. More recently a look-up table-based integration (LTRI) approach was proposed~\cite{ha2018look} that provides speedups compared to the SF method at the cost of further errors in approximating the integrals.
In this paper, we demonstrate that the detector blur can be calculated with a convolution -- in the continuous domain -- using box spline methodology; we also demonstrate a practical approach for computing these projections in fan-beam geometry using box spline evaluation algorithms. The convolutional approach leads to efficient computations that are exact in parallel geometry and highly accurate in fan-beam geometry. While our method encompasses both flat and arc detectors, we will discuss flat detectors, since the extension to the arc geometry is easily obtained as a special case.
Specific contributions of this paper include:
\begin{itemize}
\item Derivation of fast forward and back-projection operators for exact computation of footprint integral in the fan-beam geometry.
\item Derivation of the accurate detector blur computation in both parallel and fan-beam geometry.
\end{itemize}
\section{X-ray optics Model}\label{Fan Beam Projection Geometry}
\subsection{Fan Beam Geometry}
To specify the fan-beam geometry in the general $2$-dimensional configuration,
let $\mathbf{u} \in S$ denote viewing direction and point $\mathbf{o} \in
\mathbb{R}^2$ as rotation center (or origin). A point $\mathbf{p} = D_{po}\mathbf{u}$ is the location of projector, where $D_{po} \in \mathbb{R}^+$ is the (unsigned) distance from projector to rotation center.
A hyperplane $\mathbf{u}^\perp$ orthogonal to the direction $\mathbf{u}$ denotes the detector plane and point $\mathbf{d} \in \mathbf{u}^\perp$ is the center of the detector plane. Let $\ProjT$ be a matrix whose columns span the hyperplane $\mathbf{u}^\perp$
, $\mathbf{x} \in \mathbb{R}^2$ be the spatial coordinates of the input function (image domain), and the parameterized form is
$\mathbf{x} =\mathbf{p} + t\mathbf{v}(s)$,
where ${s} \in \mathbb{R}$ is the coordinate on the 1-D detector plane (sinogram domain) and $\mathbf v(s)$ is the unit direction from $\mathbf p$ to $\ProjT s$, which can be calculated by
$\frac{\mathbf{p} - \ProjT s}{ \|\mathbf{p} - \ProjT s \| }$. The detector-rotation center distance is
$D_{so}$ and projector-detector distance is $D_{ps}$. The geometry is
illustrated in Fig. \ref{fig:fanBeamGeometry}.
\begin{figure}[ht]
\centering
{\includegraphics[width=0.8\columnwidth]{fanBeamGeom-mps}}
\caption{Fan-beam X-ray CT system, a discretized model}\label{fig:fanBeamGeometry}
\end{figure}
\subsection{Analytical Model}
In geometric tomography $f$ is often the indicator function of a convex polytope and in biomedical \cite{brahme2014comprehensive}, scientific imaging \cite{rebollo2013sparse}, and industrial applications \cite{natterer1986mathematics} $f$ is the relative radiodensity modeling material's attenuation index as the X-ray passes through the object (often described by linearization of the Lambert-Beer law \cite{long20103d}). When considering that the projector is ideal point source, the 2-dimensional fan-beam X-ray transform $\mathcal{P}$ maps $f(\mathbf x)$,
$\mathbf x \in \mathbb{R}^2$, into the set of its \emph{line integrals} to form the projection:
\begin{equation}\label{eq:xRayIntegration}
\mathcal{P}\{f\}(s,\mathbf{u}) = \int_{0}^{\infty} f(\mathbf{p} + t\mathbf{v}(s)) {\rm d}t.
\end{equation}
We denote above map as $\mathcal{P}_\mathbf{u}\{f\}$ for short.
In a simple model of forward projection, one can do point-sampling on the projected function $\mathcal{P}_\mathbf{u} \{f\}$,
whereas in more realistic modeling of the transform model the projections are integrated across a detector cell with a finite bin width ${\tau} \in \mathbb{R}$ such that equation
(\ref{eq:xRayIntegration}) can be modified to
\begin{equation}\label{eq:radonBoxBlur}
\begin{split}
\mathcal{P}_{\mathbf{u}, {\tau}}\{f\}({s}) = & \int_{{s} - \frac{\boldsymbol{\tau}}{2} }^{ {s} + \frac{\boldsymbol{\tau}}{2} } h_\tau(s)\int_0^\infty f(\mathbf{p} + t\mathbf{v}(s)){\rm d}t {\rm d}s
\end{split}
\end{equation}
where $h_\tau$ is the detector blur function over the support of bin width ${\tau}$. Since the detectors are usually uniformly placed on the projection
domain, the blur function is often modeled as shift-invariant function \cite{herman2009fundamentals, lu2009computable}. Because the detector sensitivity is often modeled as a constant function over the detector cell or with a drop-off at the cell boundary \cite{lu2009computable, garcia2009study}, the analytical model can be simplified\cite{yu2012finite}:
\begin{equation}\label{eq:anlyticalProjection}
\begin{split}
\mathcal{P}_{\mathbf{u}, {\tau}}\{f\}({s}) &= \int_{s-\frac{\tau}{2}}^{s+\frac{\tau}{2}}\int_{0}^{\infty} f(\mathbf{p} + t\mathbf{v}(s)){\rm d}t {\rm d}s\\
&= \int_{\Omega_s}\frac{f(\mathbf{x})}{\gamma(s,\tau)\|\mathbf{x-p}\|} {\rm d}\mathbf{x},
\end{split}
\end{equation}
where $\Omega_s$ is the detector source-bin triangle shown as the gray area in
Fig. \ref{fig:fanBeamGeometry} and $\gamma(s, \tau)$ is the fan-beam angle at coordinate $s$ with bin width $\tau$.
Due to the linearity of integration, X-ray transform pseudo-commutes with the
translation:
\begin{equation}\label{eq:translationProperty}
\begin{split}
\mathcal{P}_\mathbf{u}\{f(\cdot - \mathbf{x}_0)\}( s) &= (\mathcal{P}_\mathbf{u} f)(s - \mathbf{P(x}_0))\\
\mathcal{P}_{\mathbf{u},\tau}\{f(\cdot - \mathbf{x}_0)\}( s) &= (\mathcal{P}_{\mathbf{u},\tau} f)(s - \mathbf{P(x}_0)),
\end{split}
\end{equation}
where the operator $\mathbf{P}$ represents the \emph{perspective
projection} with perspective division. As the geometric projection of a point from image domain onto sinogram domain is not orthogonal in fan-beam geometry (e.g., $\mathbf{ps}_i$ is not perpendicular to detector plane in Fig. \ref{fig:fanBeamGeometry}.), the perspective distortion achieved
by perspective division is necessary to correctly reflect the relation between distance of a pixel to projector and its location on the projection plane.
\subsection{Discretized Model}
Discretization or a finite-dimensional approximation of a continuous-domain signal (image) $f$ is an important issue in signal processing. With an $N$-dimensional model for approximation (see Fig. \ref{fig:fanBeamGeometry}), an expansion in a basis set of the form
\begin{equation}\label{eq:modelofInputFunction}
f_N(\mathbf x) = \sum_{n=1}^Nc_n \varphi(\mathbf{x-k}_n),
\end{equation}
allows us to derive a discretized forward model in tomography. Here $\varphi$ is a basis function, and $c_n$ is the expansion coefficient corresponding to the $n^{\rm th}$ grid point $\mathbf k_n$. The combination of (\ref{eq:xRayIntegration}), (\ref{eq:radonBoxBlur}), and this expansion, together with the translation property (\ref{eq:translationProperty}), provides the discretized forward models:
\begin{equation}\label{eq:modelofXrayIntegration}
\begin{split}
\mathcal{P}_\mathbf{u}\{f_N\}( s) &= \sum_{n=1}^N c_n\mathcal{P}_\mathbf{u}\{\varphi\}( s - \mathbf{P(k}_n)) \\
\mathcal{P}_{\mathbf{u},\tau}\{f_N\}( s) &= \sum_{n=1}^N c_n\mathcal{P}_{\mathbf{u},\tau}\{\varphi\}( s - \mathbf{P(k}_n)).
\end{split}
\end{equation}
The line integral of the basis function, $\mathcal{P}_\mathbf{u}\{\varphi\}$, is called \emph{footprint} of the basis and $\mathcal{P}_{\mathbf{u},\tau}\{\varphi\}$ is \emph{detector blur}.
{Equations (\ref{eq:modelofXrayIntegration}) show that the exact X-ray transform of the discretized model can be modeled by linear combination of the integral of the basis function.
In 2-D, the commonest choice of basis functions for image representation is the indicator function of pixels (aka the pixel-basis). With pixel-basis and a constant function to model the detector blur, a geometric approach for the combined footprint and detector blur computations can be derived~\cite{yu2012finite} as:
\begin{equation}\label{eq:anlyticalProjectionDiscrete}
\mathcal{P}_{\mathbf{u},\tau}\{f_N\}( s) \approx \sum_{{n} =1}^{N} c_n \frac{A_n}{\gamma(s, \tau)\|\mathbf k_n- \mathbf p\|},
\end{equation}
where $A_{n}$ is the intersection area of the detector source-bin triangle with the pixel located at $n^{\rm th}$ grid $\mathbf{k}_n$.
This intersection area can be computed exactly by the Gauss's area formula:
\begin{equation}\label{eq:groundTruth}
A_{n} = \frac{1}{2!} \sum_{i=1}^{M-2} \Bigl| ( \mathbf{v}_i - \mathbf{v}_{i+1}) \times (\mathbf{v}_M - \mathbf{v}_{i+1}) \Bigr|,
\end{equation}
where $\mathbf{v}_i$ are the anticlockwise sorted vertices of a convex polygon obtained by intersection of detector source-bin triangle with a pixel, which can be found algorithmically (e.g., by applying Sutherland-Hodgman algorithm \cite{sutherland1974reentrant}), and $M(\geq 3)$ denotes the total number of intersections.
{Searching for the intersection vertices is computationally expensive and hard to be parallelized due to the sequential nature of polygon clipping and sorting algorithm. The LTRI method~\cite{ha2018look} provides a solution to speedup these computations by using a look-up table to store a finite set of precomputed intersection areas and interpolating them for each combination of viewing angle, pixel and detector bin. This interpolation naturally introduces an extra source of error that can be controlled, to some extent, by increasing the resolution of the look-up table, but the data transfer involved in accessing a large look up table slows down GPU computations. In addition, a constant approximation of $\|\mathbf{x - p}\|$ by $\|\mathbf{k-p}\|$ degrades the accuracy in approximating the analytical model (\ref{eq:anlyticalProjection}). Furthermore, this approximation is not flexible for higher-order cases because the intersection area needs to be generalized by with geometric integration when higher-order basis functions are used and this renders the volume computation infeasible.}
\subsection{Roadmap} Our method models the footprint, $\mathcal{P}_\mathbf{u}\{ \varphi\}$, in the continuous sinogram domain using box spline calculus (instead of approximating the optics integrals through intersection areas). This formulation allows us to leverage an exact closed-form formula in both parallel and fan-beam geometries. The detector blur, $\mathcal{P}_{\mathbf{u},\tau} \{\varphi\}$, calculated by integral of $\mathcal{P}_\mathbf{u}\{ \varphi\}$ within a detector bin, in our framework, also has an exact closed-form expression for parallel geometry. In fan-beam geometry, although this integral can be computed exactly, it has a prohibitively large computational cost. While we use that exact integration as the $\emph{reference projector}$, we also introduce the concept of \emph{effective blur} for efficient approximation of the detector blur. The highly efficient computation of detector blur, discussed in section \ref{sec:fanbeamXray}, has a closed-form solution in box spline methodology which leads to highly accurate and efficient computations that are the main results of this paper.
\section{Discretization in Box Spline Basis}
Box splines are piecewise polynomial functions that can be used as basis functions for approximation in discrete-continuous modeling. The pixel-basis coincides with the first order box spline in 2-D and higher order box splines can be considered as more general choices for discretization of images. However, since the pixel-basis is the most commonly-used choice in CT, we will view the pixel-basis using the terminology of box splines. While viewing pixel-basis as a box spline may appear as a complication, benefits of this formulation becomes apparent once we establish that footprint and detector blur integrals of the pixel-basis result in higher-order box splines. The resulting higher-order box splines allow us to efficiently and accurately model these operations that are essential to forward and back-projection.
\subsection{Box Spline Review}
Box splines generalize B-splines to multivariate setting where they include tensor-product B-splines as a special case, but are generally non-separable functions. A box spline is a smooth piecewise polynomial function, $M: \mathbb{R}^d \mapsto \mathbb{R}$, that is associated with $N$ vectors in $\mathbb{R}^d$~\cite{deboor1993box}. The simplest box spline (i.e., $N=d$) can be constructed in $\mathbb{R}^d$ as the indicator function of the $d$-dimensional hypercube that is the pixel- and voxel-basis function when d = 2 and 3. Box splines have a convolutional construction that is essential to our derivation of footprint and detector blur.
An {\em elementary} box spline, $M_{\boldsymbol{\xi}}$, associated with a vector
$\boldsymbol{\xi} \in \mathbb{R}^d$ can be thought of as the indicator function of the set $\{t\boldsymbol{\xi} | 0 \leq t < 1\}$, and is formally defined as a Dirac-line distribution (generalized function) by its directional ``moving-average'' action on a
function $f(\mathbf{x})$ in $\mathbb{R}^d$:
$(M_{\boldsymbol{\xi}} * f)(\mathbf{x}) = \int_0^1 f(\mathbf{x} - t \boldsymbol{\xi}){\rm d}t$. Given a set of $N \ge d$ directions, arranged in columns, as
$\boldsymbol{\Xi} := [\boldsymbol{\xi}_1, \boldsymbol{\xi}_2, \cdots,
\boldsymbol{\xi}_N]$, the associated box spline can be constructed by:
\begin{equation}
M_{\boldsymbol{\Xi}}(\mathbf{x}) = (M_{\boldsymbol{\xi}_1} *
\cdots * M_{\boldsymbol{\xi}_N})(\mathbf{x}),
\end{equation}
and this is illustrated in 2D (i.e., $d=2$) in Fig. \ref{fig:boxsplinesConvolution}.
\begin{figure}[ht]
\centering
{\includegraphics[width=1.0\columnwidth]{boxsplinesConvolution.eps}}
\caption{From left to right, the figures show that the Box Splines can be constructed by convolving an elementary Box Splines in specific direction.}\label{fig:boxsplinesConvolution}
\end{figure}
When the directions are orthogonal, $M_{\boldsymbol{\Xi}}$ is a tensor-product B-spline with the repeats of a direction elevating the degree of the B-spline.
\subsection{X-ray projection in Parallel Geometry}
Previous work~\cite{entezari2012box} has demonstrated that in \emph{parallel} geometry the projection (e.g., X-ray and Radon transforms) of a box spline $M_{\boldsymbol{\Xi}}$ is another box spline, $M_{\boldsymbol{Z}}$ (i.e., in the sinogram domain), where $\boldsymbol{Z}$ is the \emph{geometric} projection of the original directions in $\boldsymbol{\Xi}$.
Let $\mathcal{R}_\mathbf{u}$ denote the X-ray projection for a viewing
direction specified by vector $\mathbf{u}$, then we have~\cite{entezari2012box}:
\begin{equation}\label{eq:projectionBoxSplineParallel}
\mathcal{R}_\mathbf{u}\{M_{\boldsymbol{\Xi}}\}({s}) = M_{\boldsymbol{Z}}({s})=
(M_{\mathbf{R}_{\mathbf{u}^\perp}\boldsymbol{\xi}_1} * \cdots *
M_{\mathbf{R}_{\mathbf{u}^\perp} \boldsymbol{\xi}_N})({s}),
\end{equation}
where $\mathbf{R}_{\mathbf{u}^\perp}$ is the transformation matrix that geometrically projects the coordinates of image domain onto detector plane perpendicular to
$\mathbf u$. Using this notation, we have $\boldsymbol{Z} := \mathbf{R}_{\mathbf{u}^\perp} \boldsymbol{\Xi}$.
Fig. \ref{fig:boxSplineProjConv} shows a box spline in $\mathbb{R}^2$ specified by directions $\boldsymbol{\Xi} = [\boldsymbol{\xi}_1, \boldsymbol{\xi}_2]$,
which coincides with a pixel-basis, when projected to the sinogram domain is
a univariate box spline in $\mathbb{R}$ with 2 directions that is the convolution of 2 elementary box splines. In this
example, the matrix $\mathbf{R}_{\mathbf{u}^\perp} = [\sin(\theta),
-\cos{\theta}]$ specifies the direction of the projection plane, where
$\theta$ is the viewing angle, and $\boldsymbol{\xi}_1 =
\begin{bmatrix}
1\\
0
\end{bmatrix}
$, and $\boldsymbol{\xi}_2 =
\begin{bmatrix}
0\\
1
\end{bmatrix}
$.
\begin{figure}[ht]
\centering
{\includegraphics[width=0.9\columnwidth]{boxSplineProjConv}}
\caption{Projection of pixel basis as convolution. }\label{fig:boxSplineProjConv}
\end{figure}
\subsection{Detector Blur}
Since the detector sensitivity is often modeled
as a constant function, the blur function $h_{{\tau}}$ can be
modeled as a box spline in an elementary box spline so that
$h_{{\tau}} = M_{{\tau}}$.
With this model, we can derive\footnote{A preliminary version of this manuscript appeared in conference proceedings in ISBI~\cite{Zhang2019box}.} the result of detector blur in parallel-beam geometry.
\begin{theorem}\label{theory:detectorBlurParallel}
The parallel-beam projection of pixel-basis with detector blur is a 3-direction box spline.
\end{theorem}
\begin{proof}
Because detector blur is the convolution of blur function $h_\tau$ with footprint $\mathcal{R}_\mathbf{u}\{M_{\boldsymbol{\Xi}}\}$.
By applying the convolutional construction of box spline, the detector blur can be computed by a single evaluation of a box spline:
\begin{equation*}
\begin{split}
\mathcal{R}_{\mathbf{u}, {\tau}}\{M_{\boldsymbol{\Xi}}\} ({s})=
\mathcal{R}_{\mathbf{u}}\{M_{\boldsymbol{\Xi}}\} * h_{{\tau} }({s}) = M_{[\mathbf{R}_{\mathbf{u}^\perp}\boldsymbol{\Xi}, {\tau}]}({s}).
\end{split}
\end{equation*}
We also note that as the detector cells lie on the detector plane, it's naturally perpendicular to
the direction of projection such that $\mathbf{R}_{\mathbf{u}^\perp}\boldsymbol{\tau} = \tau$ and $[\mathbf{R}_{\mathbf{u}^\perp}\boldsymbol{\Xi}, {\tau}] = \mathbf{R}_{\mathbf{u}^\perp}[\boldsymbol{\Xi}, \boldsymbol{\tau}]$, where $\boldsymbol{\tau}$ is a vector in image domain parallel to detector plane. Hence, it indicates that the detector blur can be modeled in image domain as convolution of basis function with an elementary box spline (see Fig. \ref{fig:boxSplineConvEff}):
\begin{equation}\label{eq:boxspineProjectionBinBlurParallel}
\begin{split}
\mathcal{R}_{\mathbf{u}, {\tau}}\{M_{\boldsymbol{\Xi}}\}( s) = \mathcal{R}_{\mathbf{u}}\{M_{[\boldsymbol{\Xi, \tau}]}\}( s).
\end{split}
\end{equation}
\end{proof}
\begin{figure}[ht]
\centering
{\includegraphics[width=0.65\columnwidth]{boxSplineConvEff}}
\caption{Projection of box spline with detector blur}\label{fig:boxSplineConvEff}
\end{figure}
\section{Forward Projection with Box Splines in Fan-Beam}\label{sec:fanbeamXray}
In fan-beam geometry, although the concept of X-ray and Radon transform of box splines still holds, the divergent nature of rays does not allow a direct application of (\ref{eq:projectionBoxSplineParallel}). The rays, in fan-beam geometry, are not perpendicular to the detector plane, however we can construct an orthogonal coordinate system for each ray by a rotation of detector plane and evaluating the projection at the corresponding coordinate system of individual rays. This transformation allows us to extend the theory in parallel geometry to fan-beam setting. Unlike the parallel setting that projection results in a single box spline with a fixed direction set, $\mathbf{Z}$, for each viewing angle, we introduce a box spline for each viewing ray (i.e., projection point in the sinogram domain).
\begin{theorem}\label{threorem:fan_parall}
The fan-beam projection of a box spline onto sinogram domain is a box spline whose direction set depends on and varies with the sinogram coordinate.
\end{theorem}
\begin{proof}
The X-ray projection of a box spline in image domain is the line integral:
\begin{equation}\label{eq:boxintegral}
\begin{split}
\mathcal{P}_\mathbf{u} \{M_{\boldsymbol {\Xi} }\}({s}) &= \int_{l_s(t)}M_{\boldsymbol {\Xi}}({\bf x}){\rm d}l_s(t) \\
& = \int_{0}^{\infty}M_{\boldsymbol {\Xi}}(\mathbf p +
t( \frac{ \mathbf{p - P_{u^{\perp}}}s }{||\mathbf{p - P_{u^{\perp}}}s ||_2} ) ) {\rm d}t
\end{split}
\end{equation}
where $l_s(t)$ is the parametric equation of line. This line can also be reparameterized as:
\begin{equation}\label{eq:reparam_line}
l_s(t) = t \mathbf v(s) + \mathbf R_{\mathbf{v(s)^\perp}}^T s',
\end{equation}
where $\mathbf v(s)$ is the direction of the ray with unit length, $\mathbf{R}_{\mathbf{v}(s)^{\perp}}^T$ is a matrix whose columns span the hyperplane, $\mathbf{v}(s)^{\perp}$, perpendicular to ${\mathbf{v}(s)}$ and ${s}' = \mathbf{R}_{\mathbf{v}(s)^{\perp}}\mathbf{p}$ is the orthogonal projection of $\mathbf p$ on $\mathbf{v}(s)^{\perp}$. In this new parameterized form, the
matrix $[\mathbf R_{\mathbf v(s)^{\perp}}^T, \mathbf v(s)]$ forms a orthonormal frame rotated from image domain where the rotation angle varies with the sinogram coordinate, $s$.
With the reparameterized line equation (\ref{eq:reparam_line}) , the integral (\ref{eq:boxintegral}) is reformulated as:
\begin{equation}\label{eq:newboxintegral}
\begin{split}
\mathcal{P}_\mathbf{u} \{M_{\boldsymbol {\Xi} }\}({s}) =
\int_{0}^{\infty}M_{\boldsymbol {\Xi}}(t \mathbf v(s) + \mathbf R_{\mathbf{v(s)^\perp}}^T s') {\rm d}t,
\end{split}
\end{equation}
which is the exact form of the X-ray projection of box spline in parallel geometry in \cite{entezari2010box} and \cite{entezari2012box}.
Therefore the right-hand side of (\ref{eq:newboxintegral}) can be derived as:
\begin{equation}\label{eq:newboxintegralparallel}
\begin{split}
\int_{0}^{\infty}M_{\boldsymbol {\Xi}}(t \mathbf v(s) + \mathbf R_{\mathbf{v(s)^\perp}}^T s') dt &= \mathcal{R}_{\mathbf v(s)}\{M_{\boldsymbol {\Xi}}\}(s') \\
&=M_{\mathbf{R}_{ \mathbf{v}(s)^{\perp} } \boldsymbol{\Xi} }
(s')\\
&= M_{\boldsymbol Z (s) }({s'}).
\end{split}
\end{equation}
The equations (\ref{eq:newboxintegral}) and (\ref{eq:newboxintegralparallel}) indicate
\begin{equation*}\label{eq:fanbeamBoxSplineProj}
\mathcal{P}_\mathbf{u} \{M_{\boldsymbol {\Xi} }\}({s}) = M_{\boldsymbol Z(s) }({s'}).
\end{equation*}
Because $M_{\boldsymbol Z(s) }$ is a box spline associated with direction set $\mathbf{Z}(s)$ varying with sinogram coordinate, the Theorem \ref{threorem:fan_parall} is proved. The geometric explanation of this proof is shown in Fig. \ref{fig:pointWiseBoxspline}.
\begin{figure}[ht]
\centering
{\includegraphics[width=0.9\columnwidth]{pointWiseBoxspline-0}}
\caption{Fan-beam projection of box spline. The direction set of the projected box spline depends on coordinate $s$.}\label{fig:pointWiseBoxspline}
\end{figure}
\end{proof}
As a corollary we have:
\begin{corollary}
The fan-beam projection of the pixel-basis is a 2-direction box spline whose direction set varies with sinogram coordinates.
\end{corollary}
When the directions in $\boldsymbol \Xi = [\boldsymbol \xi_1, \boldsymbol \xi_2]$ are orthogonal,
$M_{\boldsymbol \Xi}$ describes the pixel-basis. In the sinogram domain, direct evaluation
of box spline is possible by using rectangular box function corresponding to segments for each non-zero direction $\zeta(s) \in \boldsymbol Z(s)$:
\begin{equation*}
\begin{split}
M_{ \zeta(s)}(s)
&= \frac{1}{| \zeta(s)|}{\rm box}(\frac{s}{|\zeta(s)|}) \\
&= \frac{u(s) - u(s - \zeta(s))}{|\zeta(s)|}\\
&= \Delta_{\zeta(s)}u(s),
\end{split}
\end{equation*}
where $u(s)$ is the step function, and $\Delta_{\zeta(s)}$ denotes forward-differencing with a step size $\zeta(s)$. When $\zeta(s) = 0$, the elementary box spline reduces to a delta function, $M_0(s) = \delta(s)$, that gets eliminated in the convolutions in $\boldsymbol Z(s)$. According to the convolutional property of box splines, the fan beam X-ray projection of a pixel-basis can be expanded as:
\begin{equation}\label{eq:boxsplineProjectFanbeam}
\begin{split}
\mathcal{P}_\mathbf{u} \{M_{ \boldsymbol{\Xi} } \} (s) &= M_{\boldsymbol Z(s)}(s') = M_{ \mathbf{R}_{\mathbf{v}(s) ^{\perp} }
\boldsymbol{\Xi} }(s') \\
&=M_{ [\zeta_1(s), \zeta_2(s)] }(s')\\
&=(\Delta_{\zeta_1(s)}u * \Delta_{\zeta_2(s)}u)(\mathbf{R}_
{\mathbf{v}(s)^{\perp} }\mathbf{p})\\
&=\Delta_{\zeta_1(s)} \Delta_{\zeta_2(s)}(\mathbf{R}_
{\mathbf{v}(s)^{\perp} }\mathbf{p})_+.
\end{split}
\end{equation}
Here the projected basis is a \emph{2-direction box spline} whose direction set $\boldsymbol Z(s)$ varies with the sinogram coordinate $s$. This box spline can be evaluated by forward differencing applied to the one-sided power function (i.e., ramp function here) defined as: $y_+ = \max(y, 0)$.
\subsection{Detector Blur}
For detector blur in \emph{fan-beam} geometry, because of the non-parallel structure of
rays, the convolution of elementary box spline in image domain does not hold. Nevertheless, we derive an effective blur ${\tau}'$, which intersects an area with
pixel very close to the area cut by the detector source-bin triangle (solid blue) as illustrated in Fig. \ref{fig:pointWiseBoxsplineBinWidth}.
To derive the ${\tau}'$, we use the fact that ${\tau}'$ is a \emph{perspective projection} of ${\tau}$ on the plane $\mathbf{v}(s)^{\perp}$. Therefore, let $\mathcal{B}_{\mathbf{v}(s)}^T = [\mathbf{R}_{\mathbf v(s)^\perp}^T, \mathbf v(s)]$ denote the matrix whose columns are the basis vectors
of the coordinate system built by $\mathbf{v}(s)^{\perp}$ and $\mathbf{v}(s)$, $\mathcal{B}_{P}$ is the perspective projection matrix. By utilizing the homogeneous
transformation, ${\tau}'$ is computed as:
\begin{equation}\label{eq:newDelta}
{\tau}' =
\frac{(\mathcal{B}_{P}\mathcal{B}_{\mathbf{v}(s)}\ProjT{(s+\frac{\tau}{2} }))_{x_1}}
{(\mathcal{B}_{P}\mathcal{B}_{\mathbf{v}(s)}\ProjT{(s+\frac{\tau}{2} }))_{x_2}} -
\frac{(\mathcal{B}_{P}\mathcal{B}_{\mathbf{v}(s)}\ProjT{(s-\frac{\tau}{2} }))_{x_1}}
{(\mathcal{B}_{P}\mathcal{B}_{\mathbf{v}(s)}\ProjT{(s-\frac{\tau}{2} }))_{x_2}} .
\end{equation}
The division involved in transform (\ref{eq:newDelta}) is
\emph{perspective division} that is requisite after perspective transform, and $\tau'$ is achieved by differing the two transformed coordinates. At last,
\begin{figure}[ht]
\centering
{\includegraphics[width=0.7\columnwidth]{pointWiseBoxsplineBinWidth}}
\caption{Projection of box spline with bin blur}\label{fig:pointWiseBoxsplineBinWidth}
\end{figure}
the combination of the effective blur (\ref{eq:newDelta}), equations
(\ref{eq:boxspineProjectionBinBlurParallel}) and
(\ref{eq:boxsplineProjectFanbeam}) from Theorem \ref{theory:detectorBlurParallel} and Theorem \ref{threorem:fan_parall}
yields the closed-form formula for X-ray transform of
box spline in fan-beam with detector blur:
\begin{equation}\label{eq:pointwiseBoxSplineBlur}
\begin{split}
&\mathcal{P}_{\mathbf{u}, {\tau}} \{M_{ \boldsymbol{\Xi} } \} ({s})
= M_{ \mathbf{R}_{\mathbf{v}({s}) ^{\perp} } \boldsymbol{\Xi} }({s}') \\
&= (\Delta_{\zeta_1({s})}u *
\Delta_{{\zeta}_2({s})}u * \Delta_{{\tau}'}u)
(\mathbf{R}_{\mathbf{v}({s})^{\perp} }\mathbf{p})\\
&=\frac{\Delta_{\zeta_1({s})} \Delta_{\zeta_2({s}) } \Delta_{{\tau}'}
(\mathbf{R}_{\mathbf{v}({s})^{\perp} }\mathbf{p})_+^2}{2!}.
\end{split}
\end{equation}
Equation (\ref{eq:pointwiseBoxSplineBlur}) is the analytical closed-form approximation
of X-ray transform of a box spline in fan-beam geometry.
In a common fan-beam setup, the size of pixels and detector bin width are usually very
small and projector-detector distance is much larger so that the approximation made by the effective blur is relatively highly accurate and this will be demonstrated in experiment section.
\section{Experiments and results}
The goal in the proposed convolutional non-separable footprint (CNSF) formalism is to provide an efficient and accurate method to model the forward and back-projection in fan-beam X-ray CT problem, as these operations are the fundamental computations in almost all the iterative reconstruction algorithms. To examine the accuracy and efficiency of our CNSF framework, we compare it with the
state-of-art algorithms designed for efficient computation of forward and back-projection, namely: the Separable Footprints (SF)~\cite{long20103d} and the Look-up Table-based Ray Integration (LTRI)~\cite{ha2018look}. For a \emph{reference projector} (Ref), as the function (\ref{eq:boxsplineProjectFanbeam}) is the exact formula for X-ray transform
of pixel-basis in fan-beam without detector blur, we use symbolic function to evaluate the integral in an interval specified by detector bin width with relative error tolerance to $0$ and absolute error tolerance to $10^{-12}$, where the computation is extremely expensive.
\subsection{Forward Projection for Single Pixel}
In order to demonstrate accuracy of approximation made by our method, we first show the ``microscopic'' view of
the projections of a single pixel achieved by different methods.
We simulate a 2-D fan-beam flat detector X-ray CT system with a single pixel centered at rotation center with size $(1 \text{ mm} \times 1\text{ mm})$. The detector bin width $\tau$ is $0.5$ mm, the projector to rotation center distance $D_{po} = 3$ mm, and detector to rotation center distance $D_{so} = 3$ mm.
In order to visualize the approximations made by different methods as continuous functions, we over-sample the projections
by setting the interval of detectors $\Delta_s$ to $0.01$ mm and the number of detectors $N_s$ to $601$.
\begin{figure*}[ht!]
\begin{minipage}[l]{0.5\columnwidth}
\centering
\includegraphics[width=4.5cm]
{projectionCompare_center_0_0_0_0_angle_0}
\subcaption{}\label{fig:AAA}
\end{minipage}
\begin{minipage}[l]{0.5\columnwidth}
\centering
\includegraphics[width=4.5cm]
{projectionCompare_center_0_0_0_0_angle_15}
\subcaption{}\label{fig:BBB}
\end{minipage}
\begin{minipage}[l]{0.5\columnwidth}
\centering
\includegraphics[width=4.5cm]
{projectionCompare_center_0_0_0_0_angle_35}
\subcaption{}\label{fig:CCC}
\end{minipage}
\begin{minipage}[l]{0.5\columnwidth}
\centering
\includegraphics[width=4.5cm]
{projectionCompare_center_0_0_0_0_angle_45}
\subcaption{}\label{fig:DDD}
\end{minipage}
\caption{From (a)-(d), the projection angles are $0^{\circ}$,
$15^{\circ}$, $35^{\circ}$ and $45^{\circ}$ respectively. }
\label{fig:experiment_1PixelProjBlur}
\end{figure*}
We use the MIRT toolbox \cite{irtFessler} and LTRI source code \cite{ltriHa} to generate the comparisons and we implement the CNSF
projector in CUDA. Fig. \ref{fig:experiment_1PixelProjBlur} shows the projection of pixel-basis for all four projectors with detector blur. Our proposed method in all these angles is capable to closely approximate the nonlinear functions. SF can only use trapezoid-shape functions to approximate the functions, but obviously in most situations, the analytical projection function has irregular shape. Although LTRI can also fit the curve, the accuracies are much less than our method in most locations in
sinogram domain.
The geometric setting in above experiment is to illustrate the performance of the approximations made by the compared algorithms in extremely detailed level, while
in order to show the comprehensive performance of the projection in more practical setting, we compare the
\emph{maximum errors }of forward projectors by defining:
\begin{equation}
e(\mathbf{u}) = \max_{s \in \mathbb{R} }|F(s, \mathbf{u}) - F_{ref}(s, \mathbf{u})|,
\end{equation}
where $F$ is the projection function approximated by any of SF, LTRI, and CNSF projectors, and $F_{ref}$ is generated by {reference} projector.
The pixel is centered at $(0,0)$ mm and $(100.5, 50.5)$ mm respectively with size $(1 \text{ mm}\times 1\text{ mm}) $,
$\tau = 0.5$ mm, $\Delta_s = 0.5$ mm, $D_{po} = 200$ mm, and $D_{so} = 200$ mm. Since the pixel centered at $(0, 0)$ mm is symmetric in all four
quadrants, we only evenly select $90$ angle over $90^{\circ}$ that is shown in
Fig. \ref{subfig:maxErrorA}, while in Fig. \ref{subfig:maxErrorB},
$360$ angles over $360^{\circ}$ are uniformly selected.
\begin{figure}[ht!]
\begin{subfigure}{.47\linewidth}
\includegraphics[scale=0.18]{maxError_center_0_0_0_0}
\caption{}\label{subfig:maxErrorA}
\end{subfigure}
\hskip1em
\begin{subfigure}{.47\linewidth}
\includegraphics[scale=0.18]{maxError_center_100_5_50_5}
\caption{}\label{subfig:maxErrorB}
\end{subfigure}
\caption{Maximum errors comparison among CNSF, SF and LTRI projectors for a pixel centered at different locations.}\label{fig:experiment_BoxplineProjBlur}
\end{figure}
When the pixel is located at origin, the errors presented in LTRI are slightly smaller than CNSF at the angles ranged from $0^{\circ}$ to $10^{\circ}$ and $78^{\circ}$ to $90^{\circ}$, whereas our method performs much better at other angles. If pixel offsets a large distance from origin, the accuracy will degrade as in \ref{subfig:maxErrorB}, but our method suffers the least impact from the asymmetric location
of pixel. Overall, the proposed method outperforms other methods in most angles of forward projections in respect of accuracy.
\subsection{Radon Transform of Image}
We compare the accuracies of forward projection for complete 2-D images with a size of $(128 \text{ mm} \times 128 \text{ mm})$ and a size of $(256 \text{ mm}\times 256\text{ mm})$. The
numbers of detectors $N_s$ are $409$ and $815$ respectively, and they are spaced by $\Delta_s = 1$ mm with bin width $\tau = 0.5$ mm.
Fig. \ref{fig:RadonTransformCompare} shows the absolute errors of forward projections from 360 angles of Shepp-Logan and Brain phantom, and the errors are scaled by $512$ and $1024$ respectively for visualization purposes. The projector-rotation center
and detector-rotation center distance pair, $(D_{po}, D_{so}$), is $(200,200)$ mm for Shepp-Logan and $(400, 400)$ mm for brain. The results substantiate the significant improvements in accuracy compared to LTRI and SF.
\begin{figure}[ht!]
\begin{tabular}{ccccr}
\includegraphics[width=0.21\linewidth]{./experiment_radon_groundTruth}&
\includegraphics[width=0.21\linewidth]{./radon_shepp_logan_ltri_ref}&
\includegraphics[width=0.21\linewidth]{./radon_shepp_logan_sf_ref}&
\includegraphics[width=0.21\linewidth]{./radon_shepp_logan_box_ref}\\
\includegraphics[width=0.21\linewidth]{./radon_brain_ref}&
\includegraphics[width=0.21\linewidth]{./radon_brain_ltri_ref}&
\includegraphics[width=0.21\linewidth]{./radon_brain_sf_ref}&
\includegraphics[width=0.21\linewidth]{./radon_brain_box_ref}\\
Ref<RI&SF&CNSF
\end{tabular}
\caption{Absolute error in the projection (sinogram) domain. First row shows the projection
of Shepp-Logan dataset with size of $128 \text{ mm} \times 128\text{ mm}$ while second row is brain dataset with
size of $256 \text{ mm}\times 256\text{ mm}$.} \label{fig:RadonTransformCompare}
\end{figure}
It is also worth noting the fact that when the projector and detector are located farther away each other, as the second row, the error becomes smaller, which also corroborates our statement in section \ref{sec:fanbeamXray}.
\subsection{Time performance of Forward and Back-projection}
The other important advantage of our proposed method is
its ability to achieve high performance with on-the-fly computations of the forward projection (FP) and back-projection (BP), eliminating the
needs to store the system matrix and/or a look-up table. This is accomplished
by efficient evaluation of the box spline on the right hand
side of (\ref{eq:pointwiseBoxSplineBlur}).
In this experiment, we simulate an flat detector fan-beam X-ray CT system with $360$ angles over $360^{\circ}$ and bin width $\tau = 1$ mm. The image resolutions used in the simulation are ($64 \text{ mm}\times 64\text{ mm} $), ($128 \text{ mm}\times 128 \text{ mm}$), ($256 \text{ mm}\times 256\text{ mm}$), ($512\text{ mm}\times 512\text{ mm}$), ($1024\text{ mm}\times 1024\text{ mm}$), and ($2048\text{ mm}\times 2048\text{ mm}$). The images are all-ones images (the intensity of each pixel in the image is 1) for equitable comparison. The detector bin is spaced by $\Delta_s = 1$ mm, and the numbers of the detector $N_s$ are $205$, $409$, $815$, $1627$, $3250$, and $6499$ corresponding to the different image resolutions.
In order to adapt different field-of-views and image sizes, several projector-rotation center and detector-rotation center distances are selected as $(D_{po}, D_{so})$ =
($100, 100$) mm, ($200, 200$) mm, ($400, 400$) mm, ($800, 800$) mm,
($1600, 1600$) mm, and ($3200, 3200$) mm.
Fig. \ref{fig:timeComparisonWithLTRI} shows our speedup factors over LTRI for the average projection of all angles. All experiments are performed on NVIDIA-GTX 1080 GPU with CUDA-10.0, Intel i7-6800K 6-core CPU with 3.40 GHz.
\begin{figure}[ht]
\centering
{\includegraphics[width=1\columnwidth]{./speedupOverLTRI}}
\caption{Run time comparisons with LTRI}
\label{fig:timeComparisonWithLTRI}
\end{figure}
Elimination of the necessity to access a look-up table leads
to high throughput in GPU implementations. Our method does not store any intermediate data for projections, unlike the precomputed table in LTRI. Therefore, there is not a lot of GPU
memory access, which is usually the bottleneck in GPU computing, in our implementation.
It is also noteworthy that the speedup in back-projection is always higher than forward projection in
all resolutions. The reason for this phenomenon is that in forward projection, each projection value
is weighted sum of several pixels, thus, in CUDA implementation, each kernel thread will
execute the atomic add instruction that degrades the efficiency of parallel computing.
This serialized behavior from atomic operation occurs much less in back-projection of our implementation leading to improved speedup.
Since there is no publicly available GPU version of SF method, we also implement the CPU version of proposed method with Intel Threading Building Blocks (TBB) library that can parallelize code and compare the run time with SF in CPU parallel computing.
\begin{figure}[ht]
\centering
{\includegraphics[width=1\columnwidth]{./speedupOverSF}}
\caption{Run time comparisons with SF}
\label{fig:timeComparisonWithSF}
\end{figure}
Fig. \ref{fig:timeComparisonWithSF} shows the speedup of average projection of
all angles over SF. Since the computational resources on CPU are much less than
GPU, the speedups are not so significant as ones achieved by GPU. However, our method can still achieve as least 2 times speedups over SF.
\subsection{Reconstruction Performance}
To evaluate the improvements of accuracy in our method in the actual
reconstruction problems, we perform experiments where the measurement was
generated by the \emph{reference projector} and reconstructed with different approximating projectors (i.e., SF, LTRI and CNSF).
The optimizer used in the experiments is
adaptive steepest descent POCS regularized by total-variation (TV) called
ASD-POCS \cite{sidky2008image} and all the
hyper-parameters are adjusted to achieve the highest reconstruction quality for each model.
In this experiment, we use Shepp-Logan and the Brain phantom with resolution ($128 \text{ mm}\times 128 \text{ mm}$) and ($256 \text{ mm}\times 256\text{ mm}$) respectively as the benchmark dataset. The simulated flat detector X-ray system is configured with $N_s = 409$, $D_{po} = 200$ mm, $D_{so} = 200$ mm and $N_s = 815$, $D_{po} = 400$ mm, $D_{so} = 400$ mm. The detectors are spaced by $\Delta_s = 1$ mm with bin width $\tau = 1$ mm and the projectors are uniformly spaced over $360^{\circ}$. Fig. \ref{fig:reconstructionSheppLogan} shows the reconstruction result of Shepp-Logan phantom. The (imperfect) reconstruction achieved by reference
projector in Fig. \ref{fig:reconstructionSheppLoganRef} illustrates the
practical reconstruction problem from limited-view projection. The approximating projector made by LTRI results in a less accurate reconstruction (the resolution of look-up table in LTRI is $10000 \times 180$). SF makes a slightly more accurate approximation and achieves a little higher quality over LTRI. Our method provides a reconstruction
that can achieve the same quality as the one provided by reference projector.
\begin{figure}[ht!]
\centering
\hskip1em
\begin{subfigure}{.37\linewidth}
\includegraphics[scale=0.4]{shepp_logan_reconst_ref}
\caption{Ref, SNR = 26.39dB}\label{fig:reconstructionSheppLoganRef}
\end{subfigure}
\hskip3em
\begin{subfigure}{.4\linewidth}
\includegraphics[scale=0.4]{shepp_logan_reconst_ltri}
\caption{LTRI, SNR = 25.63dB}
\end{subfigure}
\vskip2em
\hskip1em
\begin{subfigure}{.37\linewidth}
\includegraphics[scale=0.4]{shepp_logan_reconst_sf}
\caption{SF, SNR = 25.80dB}
\end{subfigure}
\hskip3em
\begin{subfigure}{.4\linewidth}
\includegraphics[scale=0.4]{shepp_logan_reconst_box}
\caption{CNSF, SNR = 26.39dB}
\end{subfigure}
\caption{Reconstruction of Shepp-Logan phantom from $16$ uniformly spaced projections using ASD-POCS. }\label{fig:reconstructionSheppLogan}
\end{figure}
Fig. \ref{fig:reconstructionBrain} shows the reconstruction of the brain phantom. In order to evaluate the impact of the accuracy of the
forward model in image reconstruction, we visualize the differences of
all reconstructions from the reconstruction provided by
reference model that is shown in \ref{fig:reconstructionBrainRef}. For
visualization purpose, we scale the errors by appropriate factors shown
in captions. The SNRs for these results are (Ref) $19.38$dB, (LTRI) $18.96$dB, (SF) $19.07$dB, (CNSF) $19.38$dB respectively.
\begin{figure}[ht!]
\centering
\begin{subfigure}{.45\linewidth}
\includegraphics[scale=0.32]{Brain_reconst_ref}
\caption{Ref}\label{fig:reconstructionBrainRef}
\end{subfigure}
\hskip2em
\begin{subfigure}{.45\linewidth}
\includegraphics[scale=0.33]{Brain_reconst_ref_ltri}
\caption{(LTRI - Ref)$\times 10^2$}
\end{subfigure}
\vskip2em
\begin{subfigure}{.45\linewidth}
\includegraphics[scale=0.33]{Brain_reconst_ref_sf}
\caption{(SF - Ref)$\times 10^2$}
\end{subfigure}
\hskip2em
\begin{subfigure}{.45\linewidth}
\includegraphics[scale=0.33]{Brain_reconst_ref_box}
\caption{(CNSF - Ref)$\times (4\times10^3)$}
\end{subfigure}
\caption{Reconstruction of brain phantom from $30$ uniformly spaced projections using ASD-POCS. }\label{fig:reconstructionBrain}
\end{figure}
This experiment shows significant improvements over LTRI and SF methods in image reconstruction.
\section{Conclusion}
Accurate and efficient modeling the CT system is essential to the iterative image reconstruction problem.
We presented a convolutional non-separable footprint framework for forward and back-projection in fan-beam X-ray tomographic reconstruction. We show the detailed derivation from parallel X-ray transform to fan-beam setting.
The experiments, in a 2-D setting, show significant improvements in the approximation error of our method compared to other state-of-the-art methods designed for this purpose. The increase of the accuracy in forward model also results in the improvement of quality in image reconstruction. In addition, several times of speedup over the GPU and CPU implementations of other methods also shows the efficiency of our method. We believe that the implementation of evaluation of the CNSF is not fully optimized. Our future research will focus on this
optimization and meanwhile we are developing an extension to 3-D for cone-beam geometry.
\bibliographystyle{IEEEtran}
|
2,877,628,089,002 | arxiv |
\section{Submission of conference papers to ICLR 2022}
\input{tex_files/0_abstract}
\input{tex_files/1_introduction}
\input{tex_files/2_related_work}
\input{tex_files/3_analysis}
\input{tex_files/4_method}
\input{tex_files/5_experiments}
\input{tex_files/6_conclusion}
\section{Introduction}
Real-time inference on resource-constrained and efficiency-demanding platforms has long been desired and extensively studied in the last decades, resulting in significant improvement on the trade-off between efficiency and accuracy~\citep{han2015deep,liu2018rethinking,mei2019atomnas,tanaka2020pruning,ma2020image,mishra2020survey,liang2021pruning,jin2021teachers,liu2021lottery}.
As a model compression technique, quantization is promising compared to other methods, such as network pruning~\citep{tanaka2020pruning,li2021npas,ma2020image,ma2021sanity,yuan2021mest} and slimming~\citep{liu2017learning,liu2018rethinking}, as it achieves a large compression ratio~\citep{krishnamoorthi2018quantizing,nagel2021white} and is computationally beneficial for integer-only hardware.
The latter one is especially important because many hardwares
(e.g., most brands of DSPs~\citep{dsp_hvx,hexagon}) only support integer or fixed-point arithmetic for accelerated implementation and cannot deploy models with floating-point operations.
{However, the drop in performance, such as classification accuracy, caused by quantization errors, restricts wide applications of such methods~\citep{zhu2016trained}.}
To address this challenge, many approaches have been proposed, which can be categorized into \textit{simulated} quantization, \textit{integer-only} quantization, and \textit{fixed-point} quantization~\citep{gholami2021survey}.
Fig.~\ref{fig:quant_comp} shows a comparison between these implementations.
For simulated quantization, previous works propose to use trainable clipping-levels~\citep{choi2018pact}, together with scaling techniques on activations~\citep{jin2020neural} and/or gradients~\citep{esser2019learned}, to facilitate training for the quantized models. However, some operations in these works, such as batch normalization (BN), are conducted with full-precision to stabilize training~\citep{jin2020neural,esser2019learned}, limiting the practical application of integer-only hardware. Meanwhile, integer-only quantization, where the model inference can be implemented with integer multiplication, addition, and bit shifting, has shown significant progress in recent studies~\citep{jacob2018quantization,yao2021hawq,kim2021bert}. Albeit floating-point operations are removed to enable models running on devices with limited support of operation types, INT32 multiplication is still required for these methods. On the other hand, fixed-point quantization, which also applies low-precision logic for arithmetic, does not require INT32 multiplication or integer division. For example, to replace multiplication by bit shifting, \citet{jain2019trained} utilize trainable power-of-2 scale factors to quantize the model.
\input{figs/fig_quant_comp}
In this work, we adopt fixed-point quantization. Our work differs from previous efforts~\citep{jain2019trained} in three major aspects.
{First, to determine the minimum error quantization threshold, we conduct statistical analysis on fixed-point numbers}.
Second, we unify parameterized clipping activation (PACT) and fixed-point arithmetic to achieve high performance and high efficiency.
Third, we discuss and propose quantization fine-tuning methods for different models. We dub our method as F8Net, as it consists in only \textbf{F}ixed-point \textbf{8}-bit multiplication employed for \textbf{Net}work quantization. We thoroughly study the problem with fixed-point numbers, where only INT8 multiplication is involved, without any INT32 multiplication, neither floating-point nor fixed-point types. Throughout this paper we focus on 8-bit quantization, the most widely supported case for different devices and is typically sufficient for efficiency and performance requirements. Our contribution can be elaborated as follows.
\begin{itemize}[leftmargin=1em]
\item We show 8-bit fixed-point number is able to represent a wide range of values with negligible relative error, once the format is properly chosen (see Fig.~\ref{fig:fix_point_stats_analysis} and Fig.~\ref{fig:fix_point_format_analysis}).
This critical characteristic enables fixed-point numbers a much stronger representative capability than integer values.
\item
We propose a method
to determine the fixed-point format, also known as fractional length, for weights and activations using their variance. This is achieved by analyzing the statistical behaviors of fixed-point values of different formats, especially those quantized from random variables with normal distribution of different variances. The analysis reveals the relationship between relative quantization error and variance, which further helps us build an approximated formula to determine the fractional length from the variance.
\item
We develop a novel training algorithm for fixed-point models by unifying fixed-point quantization and PACT~\citep{choi2018pact}.
Besides, we show the impact of fractional length sharing for residual blocks, which is also important to obtain good performance for quantized models.
\item We validate our approach for various models, including MobileNet V1/V2 and ResNet18/50 on ImageNet for image classification,
and demonstrate better performance than existing methods that resort to 32-bit multiplication. We also integrate the recent proposed fine-tuning method to train quantized models from pre-trained full-precision models with ours for further verification.
\end{itemize}
\section{Related Work}
Quantization is one of the most widely-used techniques for neural network compression~\citep{courbariaux2015binaryconnect,han2015deep,zhu2016trained,zhou2016dorefa,zhou2017incremental,mishra2017wrpn,park2017weighted,banner2018post}, with two types of training strategies: Post-Training Quantization directly quantizes a pre-trained full-precision model~\citep{he2018learning,nagel2019data,fang2020near,fang2020post,garg2021dynamic}; Quantization-Aware Training uses training data to optimize quantized models for better performance~\citep{gysel2018ristretto,esser2019learned,hubara2020improving,tailor2020degree}. In this work, we focus on the latter one, which is explored in several directions. One area uses uniform-precision quantization where the model shares the same precision~\rev{\citep{zhou2018explicit,wang2018two,choukroun2019low,gong2019differentiable,langroudi2019cheetah,jin2020adabits,bhalgat2020lsq+,chen2020statistical,yang2020searching,darvish2020pushing,oh2021automated}}. Another direction studies mixed-precision that determines bit-width for each layer through search algorithms, aiming at better accuracy-efficiency trade-off~\citep{dong2019hawq,wang2019haq,habi2020hmq,fu2020fractrain,fu2021cpt,yang2020fracbits,zhao2021distributionadaptive,zhao2021distribution,ma2021ompq}. There is also binarization network, which only applies 1-bit~\citep{rastegari2016xnor,hubara2016binarized,cai2017deep,bulat2020high,guo2021boolnet}.
Despite the fact that quantization helps reduce energy consumption and inference latency, it is usually accompanied by performance degradation. To alleviate this problem, several methods are proposed.
One type of effort focuses on simulated quantization. The strategy is to leave some operations, e.g., BN, in full-precision for the stabilized training of quantized models~\citep{choi2018pact,esser2019learned,jin2020neural}. Nevertheless, these methods limit the application of the quantized models on resource-demanding hardware, such as DSP, where full-precision arithmetic is not supported for accelerated computing~\citep{hexagon,dsp_hvx}.
To completely eliminate floating-point operations from the quantized model,
integer-only quantization techniques emulate the full-precision multiplication by 32-bit integer multiplication followed by bit shifting~\citep{jacob2018quantization,zhu2020towards,wu2020integer,yao2021hawq,kim2021bert}.
However, the calculation of INT32 multiplication in these works requires one more operation, which results in extra energy and higher latency~\citep{gholami2021survey}. In parallel, recent work~\citep{jain2019trained} proposes to restrict all scaling factors as power-of-2 values for all weights and activations, which belongs to fixed-point quantization methods~\citep{lin2016fixed,jain2019trained,kim2021zero,mitschke2019fixed,enderich2019learning,chen2017fxpnet,enderich2019fix,zhang2020fixed,goyal2021fixed}. This enables the model to only incorporate INT8 or even INT4 multiplications, followed by INT32 bit shifting.
However, there still a lack of a thorough study of the benefits of using fixed-point arithmetic.
Also, the power-of-2 scaling factors are directly determined from the training data without theoretical analysis and guidance.
In this work, we give an extensive analysis, especially on the potential and theoretical principle of using fixed-point values for quantized models, and demonstrate that with proper analysis and design, a model quantized with only INT8 multiplication involved is able to achieve comparable and even better performance to the integer-only methods implemented with INT32 multiplication.
\section{Analysis of Fixed-Point Representation}
In this section, we first introduce the fixed-point multiplication~\citep{smith1997scientist,tan2018digital} and analyze the distribution of weight from different layers in a well-trained full-precision model (Sec.~\ref{sec:fixed-point}).
We then investigate the statistical property of fixed-point numbers, and demonstrate the potential of approximating full-precision values by 8-bit fixed-point numbers with different formats (Sec.~\ref{sec:statis_fixed_point}).
After that, we study the relationship between standard deviation of random variables and the optimal fixed-point format with the smallest quantization error. Finally, we derive an approximated formula relating the standard deviation and fixed-point format, which is
verified empirically and
employed in our final algorithms (Sec.~\ref{sec:choosing_fixed_point}).
\subsection{Advantages of Fixed-Point Arithmetic}\label{sec:fixed-point}
\input{figs/fig_weight_and_format_vs_layer}
Fixed-point number is characterized by its format, which includes both the word length indicating the whole bit-width of the number and the fractional length (FL) characterizing the range and resolution of the represented values~\citep{smith1997scientist}. Fixed-point arithmetic---especially fixed-point multiplication---is widely utilized for applications in, e.g., digital signal processing~\citep{smith1997scientist,tan2018digital}.
Compared with integer or floating-point multiplication, fixed-point multiplication has two major characteristics:
First, multiplying two fixed-point numbers is more efficient than multiplying two floating-point numbers, especially on resource-constrained devices such as DSP.
Second, it is more powerful than its integer counterpart due to its versatility and the representative ability of fixed-point numbers (there can be tens of different implementations for fixed-point multiplication but only one for integer and floating-point ones~\citep{smith1997scientist}).
This efficiency and versatility make fixed-point quantization a more appealing solution than integer-only quantization.
Specifically, as shown in Fig.~\ref{fig:weight_range_vs_layer}, the scales of weights from different layers in a pre-trained full-precision model can vary in orders, ranging from less than $0.1$ to nearly $4$.
Direct quantization with only integers inevitably introduces considerable quantization error, unless more precision and more operations are involved, such as using INT32 multiplication together with bit shifting for scaling as shown in Fig.~\ref{fig:int_only}. On the other hand, employing fixed-point numbers has the potential to reduce quantization error without relying on high-precision multiplication, as weights and activations from different layers have the extra degree of using different formats during quantization. Indeed, as shown in Fig.~\ref{fig:format_vs_layer} for a well-trained MobileNet V2 with 8-bit fixed-point numbers, the fractional lengths for weights and activations vary from layer to layer. This raises the question of how to determine the formats for each layer. In the following, we study this for 8-bit fixed-point models.
\input{figs/fix_point_stats_analysis}
\subsection{Statistical Analysis for Fixed-Point Format}\label{sec:statis_fixed_point}
For a predefined bit-width, integer, which is a special case of fixed-point numbers with zero fractional length, has a predefined set of values that it can take, which severely constrains the potential of integer-only quantization.
On the other hand, fixed-point numbers, with an extra degree of freedom, i.e., the fractional length, are able to represent a much wider range of full-precision values by selecting the proper format, and thus they are more suitable for quantization. As an example, Fig.~\ref{fig:fix_point_stats_analysis} shows the relative quantization error with 8-bit fixed-point values using different formats for a set of random variables, which are sampled from normal distributions (both signed and unsigned, with the latter processed by ReLU before quantization) with zero-mean and different standard deviations $\sigma$ (more experimental details in Appx.~\ref{app:statistical}).
From the experiments, we make the following two observations.
\textbf{Observation 1}: \textit{Fixed-point numbers with different formats have different optimal representing regions\rev{, and the minimum relative error and optimal standard deviation (annotated as a star) varies for different fractional lengths (Fig.~\ref{fig:fix_point_stats_analysis}).}} This is because the format controls the value magnitude and the representation resolution (the least significant bit).
\textbf{Observation 2:} \rev{\textit{Larger fractional lengths are more robust to represent smaller numbers, while smaller fractional lengths are more suitable for larger ones}. For a given standard deviation, using small fractional length has the risk of underflow, while large fractional length might cause overflow issue. Specifically, integers (black curves in Fig.~\ref{fig:fix_point_format_analysis}) are much more prone to underflow issues and have large relative errors for small enough values to quantize.
}
\input{figs/fix_point_format_analysis}
\vspace{-2pt}
\subsection{Choosing Optimal Fixed-Point Format}\label{sec:choosing_fixed_point}
With the above observations, we are interested in answering two questions:
\textit{(1) Can we achieve a small fixed-point quantization error for a wide range of full-precision values by
always
using the optimal fractional length corresponding to the smallest relative error?}
To answer this, we first plot the smallest possible relative error amongst all the candidate fixed-point formats against the standard deviation. As shown in red lines from Fig.~\ref{fig:std_opt_fl_err_signed} and Fig.~\ref{fig:std_opt_fl_err_unsigned}, for zero-mean normal distribution, by always choosing the optimal fixed-point format, we are able to achieve a relative quantization error smaller than 1\% for standard deviation with a range of order of at least around $3$. For example, for signed quantization, the standard deviation can range from $0.1$ to around $40$ to achieve less than 1\% error, and for unsigned quantization, the standard deviation can range from $0.1$ to $100$.
The experiments verify our presumption that using fixed-point values with the optimal formats is able to achieve negligible quantization error.
\textit{(2) Can we have a simple way to determine the optimal fractional length?}
To answer this, we plot the optimal fractional length from the statistics of the full-precision values against the standard deviation, as shown in the blue lines in Fig.~\ref{fig:std_opt_fl_err_signed} and Fig.~\ref{fig:std_opt_fl_err_unsigned}.
We find that the threshold $\sigma$ value corresponding to the jumping point is almost equidistant on the log scale of the standard deviation. This is expected as the representing region of different formats are differed by a factor of 2's exponents. Plotting the threshold standard deviation (on a log-scale) against the corresponding optimal fractional length (Fig.~\ref{fig:sigma_threshold_signed} and Fig.~\ref{fig:sigma_threshold_unsigned}), we find their relationship is almost linear, leading to the following semi-empirical approximating formulas to determine the optimal fractional length $\mathrm{FL}^*$ from the standard deviation~\rev{(more discussion in Appendix~\ref{sec:dynamic_range_vs_std})}:
\begin{align}
\small
\mathrm{Signed:}\qquad\mathrm{FL}^*&\approx\lfloor\log_2\frac{40}{\sigma}\rfloor
&\mathrm{Unsigned:}\qquad\mathrm{FL}^*&\approx\lfloor\log_2\frac{70}{\sigma}\rfloor
\label{eq:opt_fl}
\end{align}
In the following, unless specifically stated, we use~\eqref{eq:opt_fl} to determine the fractional length for both weight and activation quantization.~\rev{Note that we only calculate the standard deviation during training.}
\section{Methods}
In this section, we discuss our proposed training technique for neural network quantization with fixed-point numbers, where the formats of weights and activations in each layer are determined based on~\eqref{eq:opt_fl} during training. We first analyze how to unify PACT and fixed-point quantization (Sec.~\ref{sec:method_relashionship}). Then we show how to quantize weights and activations, especially updating for BN running statistics and fractional lengths (Sec.~\ref{sec:method_double_forward}).
Finally, we discuss the necessity of relating scaling factors from two adjacent layers to calculate the effective weights for quantization, especially for residual blocks where some layers have several layers following them (Sec.~\ref{sec:method_relating}).
\subsection{Unifying PACT and Fixed-Point Quantization}\label{sec:method_relashionship}
To quantize a positive value $x$ with unsigned fixed-point number of format $(\ensuremath{\mathrm{WL}}, \ensuremath{\mathrm{FL}})$, where $\ensuremath{\mathrm{WL}}$ and $\ensuremath{\mathrm{FL}}$ denotes word length and fractional length for the fixed-point number, respectively, we have the quantization function $\mathrm{fix\_quant}$ as:
\begin{equation}
\small
\mathrm{fix\_quant}(x)=\frac{1}{2^\ensuremath{\mathrm{FL}}}\round\left(\clip\left(x \cdot 2^\ensuremath{\mathrm{FL}}, 0, 2^\ensuremath{\mathrm{WL}} - 1\right)\right),\label{eq:fix_quant}
\end{equation}
where $\clip$ is the clipping function, and $0\le\ensuremath{\mathrm{FL}}\le\ensuremath{\mathrm{WL}}$ for unsigned fixed-point numbers.
Note that fixed-point quantization has two limitations:
overflow, which is caused by clipping into its representing region, and underflow, which is introduced by the rounding function.
Both of these introduce approximation errors.
To minimize the error, we determine the optimal fractional length for each layer based on the analysis in Sec.~\ref{sec:choosing_fixed_point}.
To achieve a better way to quantize a model using fixed-point numbers, we take a look at one of the most successful quantization techniques, PACT~\citep{choi2018pact}, which clips on the full-precision value with a learned clipping-level $\alpha$ before quantization:
\begin{equation}
\small
\mathrm{PACT}(x) = \frac{\alpha}{M}\round\left(\frac{M}{\alpha}\clip\left(x, 0, \alpha\right)\right),\label{eq:pact}
\end{equation}
where $M$ is a pre-defined scale factor mapping the value from $[0, 1]$ to $[0, M]$. The formal similarity between~\eqref{eq:fix_quant} and~\eqref{eq:pact} inspires us to relate them with each other as (more details in the Appx.~\ref{sec:derivation_fix_point_PACT}):
\begin{equation}
\small
\mathrm{PACT}(x) = \frac{2^\ensuremath{\mathrm{FL}}\alpha}{2^\ensuremath{\mathrm{WL}}-1}\mathrm{fix\_quant}(\frac{2^\ensuremath{\mathrm{WL}}-1}{2^\ensuremath{\mathrm{FL}}\alpha}x),\label{eq:pact_as_fix_quant}
\end{equation}
where we have set $M=2^\ensuremath{\mathrm{WL}} - 1$, which is the typical setting. With this relationship, we can implement PACT and train the clipping-level $\alpha$ implicitly with fixed-point quantization.
\subsection{Updating BN and Fractional Length}\label{sec:method_double_forward}
\noindent\textbf{Double Forward for BN Fusion}.
To quantize the whole model with only 8-bit fixed-point multiplication involved, we need to tackle the scaling factor from BN layer, including both the weight and running variance. Specifically, we need to quantize the effective weight that fuses the weight of convolution layers with the weight and running variance from BN~\citep{jacob2018quantization,yao2021hawq}. This raises the question of how to determine the running statistics during training. To solve this problem, we apply forward computation twice. \textit{For the first forward}, we apply the convolution using quantized input yet full-precision weight of the convolution layer, and use the output to update the running statistics of BN. In this way, the effective weight to quantize is available. Note there is no backpropagation for this step. \textit{For the second forward}, we quantize the combined effective weight to get the final output of the two layers of convolution and BN and do the backpropagation.
\noindent\textbf{Updating Fractional Length}.
Different from existing work that directly trains the fractional length~\citep{jain2019trained}, we define the fractional length for weight on-the-fly during training by inferring from current value of weight, using~\eqref{eq:opt_fl}. For the fractional length of activation, we use a buffer to store and update the value with a momentum of $0.1$, similar to how to update BN running statistics. Once the fractional lengths are determined after training, we keep them fixed for inference.
\subsection{Relating Scaling Factors between Adjacent Layers}\label{sec:method_relating}
As shown in~\eqref{eq:pact_as_fix_quant}, there are still two extra factors during the quantization operation, which we denote as a fix scaling factor $\eta_\mathrm{fix}$:
\begin{equation}
\small
\eta_\mathrm{fix}=\frac{2^\ensuremath{\mathrm{FL}}\alpha}{2^\ensuremath{\mathrm{WL}}-1}.
\label{eq:fix_scaling}
\end{equation}
Now $\alpha$ is a trainable parameter with full-precision, which means the fix scaling factor is also in full-precision. To eliminate undesired extra computation, we absorb it into the above-mentioned effective weights for quantization (Sec.~\ref{sec:method_double_forward}). However, the fix scaling factor occurs twice, one for rescaling after quantization ($\eta_\mathrm{fix}$) and the other for scaling before quantization ($1/\eta_\mathrm{fix}$). To completely absorb it, we need to relate two adjacent layers. In fact, for a mapping that includes convolution, BN, and ReLU (more details are shown in Appx.~\ref{sec:two_layers_scaling}), we apply PACT quantization to relate the activation between two adjacent layers as
\begin{equation}
\small
q^{(l+1)}_i=\mathrm{fix\_quant}\left(\sum_{j=1}^{n^{(l)}}\textcolor{black}{\underbrace{\mystrut{4ex} \frac{\gamma^{(l)}_i}{\sigma^{(l)}_i}\frac{\eta_\mathrm{fix}^{(l)}}{\eta_\mathrm{fix}^{(l+1)}}W^{(l)}_{ij}}_{\text{Effective Weight}}}\textcolor{black}{q^{(l)}_j}+
\textcolor{black}{\underbrace{\mystrut{4ex}\frac{1}{\eta_\mathrm{fix}^{(l+1)}}\left(\beta^{(l)}_i-\frac{\gamma^{(l)}_i}{\sigma^{(l)}_i}\mu^{(l)}_i\right)}_{\text{Effective Bias}}}\right),\label{eq:relating}
\end{equation}
where $q$ is the fixed-point activation, $W$ the full-precision weight of the convolution layer, $i$ and $j$ the spatial indices, $n$ the total number of multiplication, and the superscript $(l)$ indicates the $l$-th block consisting of convolution and BN. $\gamma$, $\beta$, $\sigma$, $\mu$ are the learned weight, bias, running standard deviation, and running mean for the BN layer, respectively. Also, we set $\ensuremath{\mathrm{WL}}=8$ for all layers.
As can be seen from~\eqref{eq:relating}, to obtain the final effective weight for fixed-point quantization, for the $l$-th Conv-BN block, we need to access the fix scaling factor, or equivalently, the clipping-level $\alpha$ and the activation fractional length $\ensuremath{\mathrm{FL}}$, from its following $(l+1)$-th block(s). To achieve this, we apply two techniques.
\noindent\textbf{Pre-estimating Fractional Length}.
As mentioned above, we determine the activation fractional length from its standard deviation.
\rev{Also,~\eqref{eq:fix_scaling} indicates that the fix scaling factor relies on such fractional length for each layer. However, in~\eqref{eq:relating}, we need the fix scaling factor from the next layer to determine the effective weight under quantization, which we have not yet updated. Thus, when calculating the effective weights during training, we use the activation fractional length stored in the buffer, instead of the one for quantizing the input of the next layer.}
\noindent\textbf{Clipping-Level Sharing}.
As shown in Fig.~\ref{fig:res_connect}, for residual blocks, some layers have two following layers (which we also name as child layer).
\rev{Since we need the fix scaling factor from the child layer to calculate the effective weight for the parent (see~\eqref{eq:relating}), inconsistent fix scaling factors between all children layers will be a problem. To this end,} we define one layer as master and force all its siblings to share its clipping-level.
In fact, the best way is to share both the clipping-level and the fractional length~\rev{among siblings}, but we find sharing fractional length leads to considerable performance drop, especially for deep models such as MobileNet V2 and ResNet50.
This is because the fractional lengths play two roles here: one is for the fix scaling factor, and the other is for the representing region (or equivalently the clipping-level).
Using different fractional lengths effectively enables different clipping-levels (although only differ by a factor of power-of-2, see Appx.~\ref{sec:different_fraclen}), which can be beneficial because the activation scales might vary from layer to layer.
Moreover, breaking the constraint of sharing activation fractional length does not introduce much computational cost, as the value only differs in storing format, and typically the values are stored in 32-bit, i.e., the accumulation results are only quantized into 8-bit for multiplication.
Note that when computing the effective weight of the parent layer, we only use the master child's activation fractional length. For effective weight of each child layer and fixed-point quantization on its input, we use its own fractional length
\input{figs/fig_res_connect}
\section{Experiments}
\label{sec:experiment}
\input{tabs/tab_conventional}
In this section, we present our results for various models on ImageNet~\citep{deng2009imagenet} for classification task and compare the results with previous works that focus on quantization-aware training to verify the effectiveness of our method. We show the results for two sets of training. First, we discuss the conventional training method following~\citet{jin2020neural}. Second, we unify our method with one recent fine-tuning method that quantizes full-precision models with high accuracy~\citep{yao2021hawq}. More detailed experimental settings are described in Appx.~\ref{sec:experiment_detail}.
\noindent\textbf{Conventional training}.
We first apply our method using conventional training~\citep{choi2018pact,esser2019learned,jin2020neural,fu2021cpt}, where the quantized model is trained with the simplest setting as those for full-precision model (more details in Appx.~\ref{sec:experiment_detail}). To verify the effectiveness of our method, we perform experiments on several models including ResNet18 and MobileNet V1/V2. As shown in Table~\ref{tab:8bit_conventional}, our method achieves the state-of-the-art results for all models. Additionally, we obtain
comparable or even better performance than the full-precision counterparts.
Compared with previous works on simulated quantization~\citep{choi2018pact,park2018value,esser2019learned,jin2020neural,fu2021cpt} that requires full-precision rescaling after INT8 convolution, our approach is not only more efficient but also achieves better performance. On the other hand, compared with previous fixed-point quantization~\citep{jain2019trained}, our approach gives better results. This might partially due to that our method is based on a more systematic analysis, as explained above in Section~\ref{sec:choosing_fixed_point}.
To further understand the significance of our method, we plot the fractional lengths for weight and activation for each layer. Illustrated in Fig.~\ref{fig:format_vs_layer} for MobileNet V2, we find that the fractional lengths for both weight and activation vary from layer to layer. Specifically, for weight quantization, since some layers have relatively large value range of effective weight, especially some depthwise layers, small fractional length is necessary to avoid overflow issue. On the other hand, for layers with small weight scale, large fractional length has more advantages to overcome the underflow problem. The same conclusion also applies for the fractional length for activation. Indeed, for some early layers in front of depthwise convolution layer, the activation fractional length needs to be small, yet for the later-stages, larger fractional length is desired. This further verifies our finding that using different fractional lengths for layers with the same parent is critical for good performance, because layers at different depths might be siblings and requires different fractional lengths (see Fig.~\ref{fig:res_connect}).
\input{tabs/tab_finetune}
\noindent\textbf{Tiny fine-tuning on full-precision model}. Recent work~\citep{yao2021hawq} focus on investigating the potential of neural network quantization. To this end, they suggest to tiny fine-tune on a well-pretrained full-precision model with high accuracy. In this way, it might help to avoid misleading conclusion coming from improper comparison between weak full-precision models with strong quantized model. To further investigate the power of our method and compare it with these advanced techniques, we also apply our method and fine-tune on several full-precision models with high accuracy. Also, given the number of total fine-tuing steps is very small, we apply grid search to determine the optimal fractional lengths for this experiment. The results are listed in Table~\ref{tab:8bit_finetune}, and we can find that our method is able to achieve better performance than previous method~\citep{yao2021hawq}, without time- and energy-consuming high-precision multiplication (namely dyadic scaling shown in Fig.~\ref{fig:int_only}).
Our method reveals that the high-precision rescaling, no matter implemented in full-precision, or approximated or quantized with INT32 multiplication followed by bit-shifting (a.k.a. dyadic multiplication), is indeed unnecessary and is not the key part for quantized model to have good performance. This is not well-understood in previous literature. Specifically, we demonstrate that by properly choosing the formats for weight and activation in each layer, we are able to achieve comparable and even better performance with 8-bit fixed-point numbers, which can be implemented more efficiently on specific hardwares such as DSP that only supports integer operation.
\section{Conclusion}
\rev{Previous works on neural network quantization typically rely on 32-bit multiplication, either in full-precision or with INT32 multiplication followed by bit-shifting (termed dyadic multiplication).}
This raises the question of whether high-precision multiplication is critical to guarantee high-performance for quantized models, or whether it is possible to eliminate it to save cost.
In this work, we study the opportunities and challenges of quantizing neual networks with 8-bit only fixed-point multiplication, via thorough statistical analysis and novel algorithm design.
\rev{We validate our method on ResNet18/50 and MobileNet V1/V2 on ImageNet classification.}
With our method, we achieve the state-of-the-art performance without 32-bit multiplication, and the quantized model is able to achieve comparable or even better performance than their full-precision counterparts. Our method demonstrates that high-precision multiplication, implemented with either floating-point or dyadic scaling, is not necessary for model quantization to achieve good performance. One future direction is to perform an in-depth statistical analysis of fixed-point numbers with smaller word-lengths
for neural network quantization.
\section{Appendix}
\subsection{More Experimental Details}
\label{sec:experiment_detail}
\noindent\textbf{More Details for Conventional Training}. For conventional training method, we train the quantized model initialized with a pre-trained full-precision one. The training of full-precision and quantized models shares the same hyperparameters, including learning rate and its scheduler, weight decay, number of epochs, optimizer, and batch size. For ResNet18 and MobileNet V1, we use an initial learning rate of $0.05$, and for MobileNet V2, it is $0.1$. We find the value of learning rate, i.e., $0.1$ and $0.05$, does not have much impact on the final performance. Totally, $150$ epochs of training are conducted, with cosine learning rate scheduler without restart. The warmup strategy is adopted with linear increasing ($\mathrm{batch size}/256\times0.05$)~\citep{goyal2017accurate} during the first five epochs before cosine learning rate scheduler. The input image is randomly cropped to $224\times224$ and randomly flipped horizontally, and is kept as 8-bit unsigned fixed-point numbers with $\ensuremath{\mathrm{FL}}=8$ and without standardization.
For ResNet18 and MobileNet V1/V2, we use batch size of $2048$ and run the experiments on $8$ A100 GPUs. The parameters are updated with SGD optimizer and Nesterov momentum with a momentum weight of $0.9$ without damping. The original structure of MobileNet V2 uses ReLU6 as its activation. Since our unified PACT and the fixed-point quantization already has clipping operation, and can be equivalently formulated with ReLU6 by rescaling weight or activation, we eliminate ReLU6 in our implementation.
\noindent\textbf{Discussion for Weight Decay}. We set weight decay to $4\times10^{-5}$, and find the weight decay scheme is critical for good performance, especially for the quantized model. We analyze weight decay for different models as follows:
\begin{itemize}[leftmargin=1em]
\item For ResNet18, we apply weight decay on all layers, including convolution, fully-connected, and BN layers.
\item For MobileNet V1, previous methods only apply weight decay on conventional convolution and fully-connected layers, but not on depthwise convolution and BN~\citep{howard2017mobilenets}.
We find this leads to the overfitting problem, making some early convolution layers have large weights, which is not friendly for quantization.
We further observe that some channels of some depthwise convolution layers have all zero inputs, due to some channels of previous layer become all negative and ReLU is applied afterwards, making the running statistics of the corresponding channels in the following BN layer almost zero. This breaks the regularization effect of BN~\citep{luo2018towards}. Since each output channel only depends on one input channel for depthwise convolution layers, the weights connecting them become uncontrolled, and the effective weights become large, leading to an overfitting problem. Applying weight decay on the depthwise convolution and BN layers helps to alleviate this problem, and the resulting effective weights become small.
\item For MobileNet V2, we find overfitting plays the role of reducing the validation error (although the training error is lower), and applying weight decay on depthwise convolution or BN weights impairs the training procedure. The underlying reason might be related to the residual connecting structure of this model (note MobileNet V1 does not use residual connection).
\end{itemize}
In summary, we apply weight decay on all layers, including depthwise convolution and BN layers for ResNet18 and MobileNet V1, and do not apply weight decay on depthwise convolution and BN layers for MobileNet V2.
\noindent\textbf{More Details for Tiny Fine-tuning}. For tiny fine-tuning on full-precision models, we follow the same strategy proposed in~\citet{yao2021hawq}. Specifically, we use a constant learning rate of $10^{-4}$, with $500$ iterations of fine-tuning (or equivalently data ratio of around $0.05$ with batch size of $128$). Different from~\citep{yao2021hawq}, we find fixed BN is not helpful, and we allow it to update during the whole fine-tuning step. As mentioned in Sec.~\ref{sec:experiment}, we apply grid search to determine the fractional lengths for both weight and input, as the training cost is very small and applying grid search does not introduce too much effort or training time. Also, since the original full-precision model uses the normalized input, we also apply normalization on the images and quantize images with signed fixed-point numbers (and format determined with grid search) before being fed into the first convolution layer of the model.
\subsection{More Details for Statistical Analysis}
\label{app:statistical}
For the toy example in Fig.~\ref{fig:fix_point_stats_analysis}, we sample $10,000$ zero-mean Gaussian random variables with different standard deviations, and apply ReLU activation for the rectified Gaussian variables with unsigned quantization. The variables are then quantized with fixed-point quantization given in~\eqref{eq:fix_quant} and~\eqref{eq:fix_quant_signed}, respectively. We calculate the relative quantization error and plot against the standard deviation for each fixed-point format. Note that zero-mean is a reasonable simplifying assumption if we assume to neglect the impact of bias in BN for analysis purposes.
\subsection{derivation for fixed-point and PACT relation}
\label{sec:derivation_fix_point_PACT}
\input{tex_files/derive_PACT_fix_point}
\subsection{Double Side Quantization for Weight and MobileNet V2}
\label{sec:double_side_quant}
In~\eqref{eq:fix_quant}, we only give the formula for fixed-point quantization of unsigned case. For weight and activation from some layer without following ReLU nonlinearity (such as some layers in MobileNet V2), signed quantization is necessary, and the expression is similarly given as:
\begin{equation}
\mathrm{fix\_quant}(x)=\frac{1}{2^\ensuremath{\mathrm{FL}}}\round\left(\clip\left(x \cdot 2^\ensuremath{\mathrm{FL}}, - 2^{\ensuremath{\mathrm{WL}}-1} + 1, 2^{\ensuremath{\mathrm{WL}}-1} - 1\right)\right),\label{eq:fix_quant_signed}
\end{equation}
where $\clip$ is the clipping function, and $0\le\ensuremath{\mathrm{FL}}\le\ensuremath{\mathrm{WL}}-1$.
\subsection{Derivation of Effective Weight}
\label{sec:two_layers_scaling}
\input{tex_files/two_layer_scaling_factor}
\subsection{Private Fractional Lengths Enabling Different Clipping-Levels}
\label{sec:different_fraclen}
\input{tex_files/different_fraclen}
\rev{
\subsection{More Discussion of the Optimal Fractional Length}
\label{sec:dynamic_range_vs_std}
Here we give some further discussion of using standard deviation to determine the optimal fractional length. The main reason is that standard deviation is a more robust statistics than others, such as dynamic range, and is an easily-estimated parameter for Gaussian distributed weights and pre-activations.
Considering depth-wise convolution layers that contain much fewer weights and inputs, using robust statistics becomes essential as these layers might include weights or inputs with strange behavior, \emph{e.g.}, the pre-activation values of some channels become all negative with large magnitude. Therefore, the standard deviation is more suitable and robust than the dynamic range.
}
\rev{
\subsection{Fractional Length for ResNet50}
Here we provide more results of fractional lengths distribution in Fig.~\ref{fig:res50_fraclen} for the well-trained ResNet50 with 8-bit fixed-point numbers finetuned from the Baseline \#2 in Table~\ref{tab:res50_finetune}. As we can see, the optimal fractional lengths are layer-dependent and their distribution is highly different from those in MobileNet V2 (as shown in Fig.~\ref{fig:format_vs_layer}). Specifically, for MobileNet V2, some layers have vanishing weight fractional lengths and less than $4\%$ of all layers have an activation fractional length less than 4, while for ResNet50, more than $88\%$ of all layers have an activation fractional length that is less or equal to 4.}
\input{figs/fig_res50_format_vs_layer}
\input{tabs/tab_ablation_res50_6_to_8}
\rev{
\subsection{Analyzing Searching Space of Fractional Lengths}
In the main paper, we adopt the largest possible searching space for the fractional lengths of 8-bit fixed-point. As shown in Fig.~\ref{fig:format_vs_layer}, many layers have a fractional length less than $4$, either for input or weight. Here we study whether it is possible to use only fractional lengths between $6$ and $8$. To this end, we finetune on ResNet50b using the Baseline \#1. The results are listed in Table~\ref{tab:ablation_res50b_6_to_8}, from which we find that restricting the fractional lengths between $6$ to $8$ significantly impacts the performance of the final quantized model, as the top-1 accuracy drops from $77.6\%$ to $72.4\%$. }
\section{Acknowledgement}
This research is partially supported by.
|
2,877,628,089,003 | arxiv | \section{Introduction}
At the heart of quantum chemistry lies the accurate description of the electronic structure of molecular systems,
which is a very challenging task since the corresponding mathematical problem to be solved scales exponentially with the system size.
Wave function theory (WFT) aims at solving the Schrödinger equation for a general molecular system
and provides a systematic way of improving the accuracy of the computed properties following a two-fold path: \textbf{(i)} improving the quality of the wave function in a given basis to get as close as possible from the full-configuration interaction (FCI), \textbf{(ii)} improving the quality of the one-electron basis set used to project the Schrödinger equation.
The exact properties of the system would be obtain with the FCI wave function in a complete basis set (CBS).
There exists many different flavours of wave function ansätze which approximate the FCI wave function and energy,
and they all have an unfavorable computational scaling with the system size and most importantly, with the size of the basis set in common.
Therefore a major drawback of WFT is the slow convergence of the results with respect to the basis set size,
which mainly originates from the poor description of two-body density matrix near the electron-electron coalescence point (\textit{i.e.} $r_{12}\approx 0$).
A central idea shared by the theories aiming to improve the convergence of WFT with respect to the basis set is related to the so-called
electron-electron cusp-condition derived by Kato\cite{Kat-CPAM-57}:
one multiply the cusp-less wave function developed in an incomplete basis set by a correlation factor explicitly depending on the $r_{12}$ coordinate which restores the cusp-condition.
Beside the cusp, the most important role of the correlation factor is to lower the probability of finding two electrons near one another, which is often referred as digging the short-range part of the Coulomb hole.
There are mainly three classes of theories dealing with a correlation factor:
\textbf{(i)} F12 theory\cite{Ten-TCA-12,TenNog-WIREs-12,HatKloKohTew-CR-12, KonBisVal-CR-12, GruHirOhnTen-JCP-17, MaWer-WIREs-18} where one projects out the effect of the correlation factor from the incomplete basis set used to compute the cusp-less wave function,
\textbf{(ii)} variational Monte Carlo (VMC) methods\cite{TouAssUmr-book-vmc} where the full effect of the correlation factor is retained in the wave function and all parameters are variationally optimized and
\textbf{(iii)} the transcorrelated theory (TC) \cite{Hirschfelder-JCP-63,BoyHan-PRSLA-69,BoyHan-2-PRSLA-69} where the effect of the full correlation factor is incorporated through a non hermitian effective Hamiltonian.
All these three theories have been shown to strongly reduce the basis set convergence problem of WFT.
The main advantage of the TC theory is that it combines favourable aspects of both VMC and F12: \textbf{(i)} since only up to effective three-electron terms are needed (compared to the N$-$body terms of VMC), usual post-Hartree Fock methods can be designed to solve the TC Hamiltonian, \textbf{(ii)} no more than $\mathbb{R}^6$ integrals are needed, and \textbf{(iii)} compact wave function can be obtained because the full correlation factor is taken into account.
Despite these attractive features, the two main drawbacks of the TC theory are that \textbf{(i)} the non-hermitian nature of the TC operator which induces the loss of variationality, and \textbf{(ii)} that the three-body terms generate an $M^6$ tensor -- $M$ is the number of basis set functions -- which becomes rapidly prohibitive to store during calculations.
Nevertheless, because it originates from a similarity transformation, the exact eigenvalues are obtained when reaching the CBS limit, which suggests that the loss of variational property in TC is a signature of a too constrained form of the wave function.
Regarding the functional form of the cusp-less wave function and of the correlation factors,
the seminal work of Boys and Handy\cite{BoyHan-PRSLA-69,BoyHan-2-PRSLA-69} proposed to optimize both the orbitals of a single Slater determinant and a sophisticated correlation factor.
Then, Ten-No\cite{TenNo-CPL-00-a} proposed to significantly change of paradigm since he used the combination of a rather simple universal correlation factor whose shape was optimized for the range of valence electrons, and a rather sophisticated ansatz for the wave function (M{\o}ller-Plesset at second order in Refs. \onlinecite{TenNo-CPL-00-a,HinTanTen-JCP-01} and linearised coupled-cluster in Ref. \cite{HinTanTen-CPL-02}).
The works of Ten-No have shown a faster convergence of the TC theory (such as TC-MP2) towards the exact energies with respect to the basis set with
respect to their parent usual WFT theory (such as regular MP2).
Nevertheless, it should be mentioned that because the correlation factor was optimized for valence electrons,
the use of basis sets explicitly optimized for core electrons (\textit{e.g.} the cc-pCVXZ family) is mandatory
in order to maintain a sensible value for the energy in all-electron calculations.
More recently, Cohen \textit{et. al}\cite{CohLuoGutDobTewAla-JCP-19} applied the TC methodology with an elaborate correlation factor,
and proposed to use the full configuration interaction Monte Carlo (FCIQMC) method to obtain the exact ground state energy and the corresponding right eigenvector of the TC Hamiltonian in given basis set.
In their work\cite{CohLuoGutDobTewAla-JCP-19}, the authors used the Jastrow factors of Moskowitz \textit{et. al.}\cite{SchMos-JCP-90}
optimized in the context of VMC for the He-Ne neutral series, and which explicitly take into account electron-electron-nucleus (e-e-n) correlation effects.
This work has showed the beneficial impact of the e-e-n terms in yielding highly accurate total energies and ionisation potentials using the TC-FCIQMC method with modest basis sets. The TC-FCIQMC method has also been applied to the binding curve of the Be$_2$ system, yielding spectroscopic accuracy across the entire binding curve using only triple-zeta basis sets, demonstrating how the TC-FCIQMC method can be used in ab initio problems with a delicate balance between static and dynamical correlation \cite{GutCohLuoAla-JCP-21}. In other work from the Alavi group, the application of the TC-FCIQMC method to the 2D Hubbard model\cite{DobLuoAla-PRB-19} showed how transcorrelation can be beneficial in the treatment of {\em strongly correlated} systems, by compressing the right eigenvector of the ground state so that it becomes largely dominated by the Hartree-Fock determinant, in a regime where the ground-state eigenvector of the non-transcorrelated Hubbard Hamiltonian is strongly multi-configurational. Similar application of the Gutzwiller Ansatz\cite{Gutzwiller-PRL-63,BriRic-PRB-70} was recently reported by Reiher \textit{et. al.}\cite{BaiRei-JCP-20} using density matrix renormalisation group
and various methods based on the TC approach have been used to reduce the resource requirements for accurate
electronic structure calculations on state-of-the-art quantum computing hardware\cite{ValTak-PCCP-20, McArdle2020, Schleich2021, Kumar2022, Sokolov2022}.
Recently, one of the present authors introduced a single-parameter correlation factor\cite{Gin-JCP-21} inspired by range-separated density functional theory (RS--DFT).
The main idea developed in this work was to find a mapping between the leading order terms in $1/r_{12}$ of the effective scalar potential obtained in the TC equations and the non divergent long-range interaction $\text{erf}(\mu r_{12})/r_{12}$ used in RS--DFT. The correlation factor obtained with such a procedure has an explicit analytical form which depends on a single parameter $\mu$: the lower the $\mu$, the deeper is the correlation hole dug by the correlation factor, and in the $\mu \rightarrow \infty$ limit the effect of the correlation factor vanishes.
Preliminary tests on atomic and molecular two-electron systems have shown that this TC framework also improves the convergence of the energy, and that a good value of the parameter $\mu$ could be systematically obtained with nothing more than the knowledge of the Hartree-Fock (HF) density.
The advantage of this simple correlation factor is that the corresponding TC Hamiltonian has a rather simple analytical form for which the two- and three-body integrals can be very efficiently obtained using a mixed numerical and analytical scheme.
The aim of the present work is to study how this relatively simple correlation factor performs for systems with more than two electrons.
In order to be able to eliminate any source of errors within a basis set, we use the FCIQMC approach to obtain the exact right eigenvector in a given basis set.
We are then able to compare with the results obtained with the more sophisticated correlation factor used in the recent work of Cohen \textit{et. al.}.\cite{CohLuoGutDobTewAla-JCP-19}
The remainder of this article is organized as follows:
In Section~\ref{sec:theo}, we recap the main equations of the TC theory together with the explicit form of the TC Hamiltonian obtained
in Ref.~\onlinecite{Gin-JCP-21}.
Then, in Section~\ref{sec:results_BNe} we investigate the sensitivity of the present approach with the quality of description of core electrons and in Section~\ref{sec:results_5idx} we investigate a possible approximation to the numerous $N^6$ tensor of the three-body integrals inherent to the TC approach within the Ansatz considered here.
In Section~\ref{sec:results_full} we report the results for total energies of neutral and first cation species on the Li--Ne series as a function of increasing basis set size.
We also compare the quality of the ionization potentials (IPs) obtained with the present approach with the existing literature.
Eventually we conclude in Section~\ref{sec:conclusion}.
\section{Theory}
\label{sec:theo}
\subsection{General equations and concepts of TC theory}
The general form of the transcorrelated Hamiltonian for a symmetric correlation factor $\uu{1}{2}$ is given by
\begin{equation}
\label{ht_def_g}
\begin{aligned}
\hu &\equiv e^{-\hat{\tau}_u} \hat{H} e^{\hat{\tau}_u} \\
& = H + \big[ H,\hat{\tau}_u \big] + \frac{1}{2}\bigg[ \big[H,\hat{\tau}_u\big],\hat{\tau}_u\bigg],
\end{aligned}
\end{equation}
where $\hat{\tau}_u = \sum_{i<j}u(\br{i},\br{j})$ and $\hat{H} = -\sum_i \frac{1}{2} \nabla^2_i + v(\br{}_i) + \sum_{i<j} \frac{1}{r_{ij}}$.
Eq. \eqref{ht_def_g} leads to the following transcorrelated Hamiltonian
\begin{equation}
\begin{aligned}
\label{ht_def_g2}
\hu& = H - \sum_{i<j} \ku{i}{j} - \sum_{i<j<k} \lu{i}{j}{k}
\end{aligned}
\end{equation}
where the effective two- and three-body operators $\ku{1}{2}$ and $\lu{1}{2}{3}$ are defined as
\begin{equation}
\begin{aligned}
\ku{1}{2} = \frac{1}{2} \bigg( &\Delta_1 \uu{1}{2} + \Delta_2 \uu{1}{2} \\
+ &\big(\nabla_1 \uu{1}{2} \big) ^2 + \big(\nabla_2 \uu{1}{2} \big) ^2 \bigg) \\
+ &\nabla_1 \uu{1}{2} \cdot \nabla_1 + \nabla_2 \uu{1}{2}\cdot \nabla_2
\end{aligned}
\end{equation}
and
\begin{equation}
\begin{aligned}
\lu{1}{2}{3} = & \nabla_1 \uu{1}{2} \cdot \nabla_1 \uu{1}{3} \\
+ & \nabla_2 \uu{2}{1} \cdot \nabla_2 \uu{2}{3} \\
+ & \nabla_3 \uu{3}{1} \cdot \nabla_3 \uu{3}{2} .
\end{aligned}
\end{equation}
In practice, the TC Hamiltonian is projected into a basis set $\basis$
\begin{equation}
\label{eq:def_hub}
\begin{aligned}
&\hub = P^{\basis} \hu P^\basis,
\end{aligned}
\end{equation}
where $P^{\basis}$ is the projector onto a given basis set $\basis$.
Using real-valued orthonormal spatial molecular orbitals (MOs) $\{\phi_i(\br{})\}$, $\hub$ can be written in a second-quantized form as
\begin{equation}
\label{eq:def_hub_sec_q}
\begin{aligned}
&\hub = \sum_{i,j \in \basis} \,\, \sum_{\sigma = \uparrow,\downarrow} h_{ij} \adi{j,\sigma}\ai{i,\sigma}\\
& + \frac{1}{2}\sum_{i,j,k,l \in \basis} \,\, \sum_{\sigma,\lambda = \uparrow,\downarrow}
\big( V_{ij}^{kl} - \kijkl\big) \adi{k,\sigma} \adi{l,\lambda} \ai{j,\lambda} \ai{i,\sigma} \\
& - \frac{1}{6} \sum_{i,j,m,k,l,n \in \basis} \,\, \sum_{\sigma,\lambda,\kappa = \uparrow,\downarrow}
\lmuijmkln \adi{k,\sigma} \adi{l,\lambda} \adi{n,\kappa} \ai{m,\kappa} \ai{j,\lambda} \ai{i,\sigma}
\end{aligned}
\end{equation}
where $h_{ij}$ are the usual one-electron integrals, $V_{ij}^{kl}$ are the usual two-electron integrals,
$\kijkl$ are the two-electron integrals corresponding to the effective two-body operator $\ku{1}{2}$ operator
\begin{equation}
\kijkl = \int \text{d} \br{1} \text{d} \br{2} \phi_k(\br{1}) \phi_l(\br{2}) \ku{1}{2} \phi_i(\br{1}) \phi_j(\br{2}),
\end{equation}
and $\lmuijmkln$ are the three-electron integrals corresponding to the effective three-body operator $\lu{1}{2}{3}$
\begin{equation}
\begin{aligned}
\lmuijmkln = \int \text{d} \br{1} \text{d} \br{2} \text{d} & \br{3} \phi_k(\br{1}) \phi_l(\br{2}) \phi_n(\br{3}) \\ & \lu{1}{2}{3} \phi_i(\br{1}) \phi_j(\br{2}) \phi_m(\br{3}).
\end{aligned}
\end{equation}
The ground state eigenvalue and the associated right eigenvector fulfill the eigenvalue equation
\begin{equation}
\hub \phiu = \eob \phiu,
\end{equation}
and because of the properties of the similarity transformation the exact ground state energy $E_0$ is recovered in the CBS limit
\begin{equation}
\lim_{\basis \rightarrow \text{CBS}} \eob = E_0,
\end{equation}
for all correlation factors $u(\br{i},\br{j})$ chosen to obtain $\hub$.
If $u(\br{i},\br{j})$ is properly chosen one expects a fast convergence of $\eob$ towards $E_0$.
Nevertheless, because of the loss the of variational principle of $\eob$ due to the non hermitian character of $\hub$, this convergence is not guaranteed to be monotonic as in the usual WFT calculations, and $\eob$ can be bellow the exact ground state energy.
\begin{comment}
\subsection{Frozen-core TC Hamiltonian}
As usually done in WFT calculations, it is convenient to define a frozen core (FC) version of the TC Hamiltonian to avoid the burden of correlating core electrons which contribute very weakly to most of the chemically relevant energy differences.
Here we propose the following strategy for the FC version of the TC Hamiltonian when the MOs of a basis set $\basis$ is split into a "core" subset $\corset$ and an "active" subset $\actset$ (with of course $\corset \cup \actset = \emptyset $):
we take the normal-ordered part of the regular Hamiltonian on a Slater determinant with core orbitals doubly occupied,
and we use the usual TC Hamiltonian in $\actset$.
Therefore, the normal-ordering of the regular Hamiltonian produces a scalar quantity $E^{\text{core}}$ corresponding to the one- and two-electron energy from core electrons, and an effective one-body Fock operator coming from mean field bare coulomb interaction of the core electrons with the rest of the system.
Mathematically we can therefore write
\begin{equation}
\label{eq:ht_fc}
\begin{aligned}
&\hubfc = E^{\text{core}} + \sum_{i,j \in \actset} \,\, \sum_{\sigma = \uparrow,\downarrow} \big( h_{ij} + f_{ij}^{\text{core}} \big)\adi{j,\sigma}\ai{i,\sigma}\\
& + \frac{1}{2}\sum_{i,j,k,l \in \actset} \,\, \sum_{\sigma,\lambda = \uparrow,\downarrow}
\big( V_{ij}^{kl} - \kijkl\big) \adi{k,\sigma} \adi{l,\lambda} \ai{j,\lambda} \ai{i,\sigma} \\
& - \frac{1}{6} \sum_{i,j,m,k,l,n \in \actset} \,\, \sum_{\sigma,\lambda,\kappa = \uparrow,\downarrow}
\lmuijmkln \adi{k,\sigma} \adi{l,\lambda} \adi{n,\kappa} \ai{m,\kappa} \ai{j,\lambda} \ai{i,\sigma},
\end{aligned}
\end{equation}
where
\begin{equation}
E^{\text{core}} = \sum_{r \in \corset} \big( 2 h_{rr} + W_{rr}^{rr}\big) + 2 \sum_{r \in \corset} \sum_{s >r \in \corset} \big( 2 W_{rs}^{rs} - W_{rs}^{sr} \big),
\end{equation}
and
\begin{equation}
f_{ij}^{\text{core}} = \sum_{r \in \corset} \big( 2 W_{ir}^{jr} - W_{ir}^{rj} \big), \quad \{i,j\} \in \actset.
\end{equation}
\end{comment}
\subsection{One-parameter TC Hamiltonian: $\hmu$}
Recently, one of the present authors\cite{Gin-JCP-21} have introduced a one-parameter correlation factor $\umu{1}{2}$
based on a mapping between the $r_{12} \approx 0$ limit of the TC Hamiltonian and the range separated DFT effective Hamiltonian.
The explicit form of $\umu{1}{2}$ derived in Ref. \onlinecite{Gin-JCP-21} reads as
\begin{equation}
\label{eq:def_j}
\umu{1}{2} = \frac{1}{2}r_{12}\bigg( 1 - \text{erf}(\mu r_{12}) \bigg) - \frac{1}{2\sqrt{\pi}\mu}e^{-(r_{12}\mu)^2}.
\end{equation}
Because of the simple analytical expression of $\umu{1}{2}$, the corresponding TC Hamiltonian $\hmu$ defined as
\begin{equation}
\label{eq:def_h_mu}
\begin{aligned}
\hmu &\equiv e^{-\hat{\tau}_\mu} \hat{H} e^{\hat{\tau}_\mu} \\
& = H - \sum_{i<j} \kmu{i}{j} - \sum_{i<j<k} \lmu{i}{j}{k},
\end{aligned}
\end{equation}
with $\hat{\tau}_\mu = \sum_{i<j}\umu{i}{j}$, has a relatively simple analytical form
with the effective two- and three-body operators
\begin{equation}
\label{eq:k_final}
\begin{aligned}
& \kmu{i}{j}= \frac{1 - \text{erf}(\mu r_{12})}{r_{12}} - \frac{\mu}{\sqrt{\pi}} e^{-\big(\mu r_{12} \big)^2} \\
&+ \frac{\bigg(1 - \text{erf}(\mu r_{12}) \bigg)^2}{4} - \bigg( \text{erf}(\mu r_{12}) - 1\bigg) \deriv{}{r_{12}}{}
\end{aligned}
\end{equation}
and
\begin{equation}
\label{eq:l_final}
\begin{aligned}
\lmu{i}{j}{k} = & \frac{1 - \text{erf}(\mu r_{12})}{2 r_{12}} \br{12} \cdot \frac{1 - \text{erf}(\mu r_{13})}{2 r_{13}} \br{13} \\
+ & \frac{1 - \text{erf}(\mu r_{12})}{2 r_{12}} \br{21} \cdot \frac{1 - \text{erf}(\mu r_{23})}{2 r_{23}} \br{23} \\
+ & \frac{1 - \text{erf}(\mu r_{13})}{2 r_{13}} \br{31} \cdot \frac{1 - \text{erf}(\mu r_{32})}{2 r_{32}} \br{32},
\end{aligned}
\end{equation}
respectively.
The correlation factor $\umu{1}{2}$ exactly restores the cusp conditions and the effective Hamiltonian $\hmu$ obtained with the scalar two- and three-body effective interaction in Eqs.~\eqref{eq:k_final} and~\eqref{eq:l_final} is non divergent, yielding \enquote{cusp-less} eigenvectors as illustrated in Ref~\onlinecite{Gin-JCP-21}.
As apparent from the definitions of Eq~\eqref{eq:k_final} and Eq.~\eqref{eq:l_final}, the global shape of $\hmu$ depends on a unique parameter $\mu$, which can be seen either as the inverse of the typical range of the correlation effects, or the typical value of the effective interaction at $r_{12}=0$.
In the $\mu \rightarrow +\infty$ limit one obtains the usual Hamiltonian, while in $\mu \rightarrow 0 $ limit one obtains an attractive non hermitian Hamiltonian.
Similarly to Eq.~\eqref{eq:def_hub}, we define the projection onto a basis set $\basis$ of the TC Hamiltonian $\hmu$
\begin{equation}
\hmub \equiv P^\basis \hmu P^\basis,
\end{equation}
whose ground state eigenvalue and associated right-eigenvector satisfy
\begin{equation}
\hmub \phimu = \eomub \phimu.
\end{equation}
Because of its relatively simple form, the correlation factor $\umu{1}{2}$ has the advantage that it leads
to effective operators $\kmu{1}{2}$ and $\lmu{1}{2}{3}$ with a simple-enough analytical form for which integrals
can be computed efficiently using a mixed numerical-analytical scheme (see Ref \onlinecite{Gin-JCP-21} for explicit formulas).
This is in contrast to the ${\mathbb R}^6$ numerical integrals needed when using more sophisticated correlation factors.
\section{Results}
\label{sec:results}
\subsection{Computational details}
To obtain the ground state eigenvalue $\eob$ of a given TC Hamiltonian $\hub$, we use the recently developed similarity transformed full-configuration interaction quantum Monte Carlo (ST-FCIQMC) technique\cite{CohLuoGutDobTewAla-JCP-19, GutCohLuoAla-JCP-21} which extends
the original stochastic projection technique of FCIQMC\cite{BooThoAla-JCP-09, BooAla-JCP-10, BooCleThoAla-JCP-11, GhaLozAla-JCP-19, VitAlaKat-JCTC-20, Guther2020, Dobrautz2019, Dobrautz2021}
to a non-hermitian and three-body Hamiltonian.
The FCIQMC parameters were $10^6$ walkers, an initiator threshold of $n_{init} = 3$ and a semi-stochastic space of $N_D = 1000$.
Provided a $\hub$ and a given basis set $\basis$, the necessary one-, two- and three-body integrals are computed using restricted Hartree-Fock (RHF) MOs.
When the correlation factor is $\umu{1}{2}$, we label the results by $\tcfcimu$, whereas when using the correlation factor of Moskowitz \textit{et. al.}\cite{SchMos-JCP-90} we label the results by $\tcfcihint$.
Regarding the integrals involved in $\hmub$, the scalar two-body part is computed analytically and the non hermitian together with the three-body parts are computed using a mixed analytical-numerical scheme where the Becke's numerical grid\cite{Bec-JCP-88b} contains 30 radial points and a Lebedev angular grid of 50 grid points. Numerical tests have shown that these relatively small number of grid points ensures a sub $\mu\text{Ha}$ convergence of the total energies.
Regarding the value of $\mu$ chosen here, thorough this article we use the so-called RSC+LDA system-dependent value defined in Eq. (57) of Ref. \onlinecite{Gin-JCP-21} as such a strategy was found to be the most accurate on the study of two-electron systems in the latter work.
Estimates of the FCI in a given basis set $\basis$ and within a sub mH precision were obtained with the configuration interaction perturbatively selected iteratively\cite{malrieu_cipsi} (CIPSI) as implemented in the Quantum Package\cite{QP2}.
The estimated CBS all-electron results for atoms and cations are taken from Ref. \onlinecite{ChaGwaDavParFro-PRA-93}.
Except for the ST-FCI-QMC, all calculations where performed using the Quantum Package\cite{QP2}.
\subsection{Preliminary investigation on B, B$^+$, Ne and Ne$^+$ }
\label{sec:results_BNe}
Before performing the study on the whole Li--Ne series together with their first cations, we perform a detailed study on the neutral and first cations of the boron and neon atoms.
The main questions we address are \textbf{(i)} how to treat core electrons in all-electron calculations using $\hmu$ and \textbf{(ii)} to investigate a possible reduction of the computational cost involved in the three-body operator while maintaining the accuracy.
\subsubsection{Treatment of core electrons in all-electron calculations}
We begin our preliminary investigation by studying the treatment of core electrons with $\hmu$ in the case of the boron neutral atom.
We performed all electron $\tcfcimu$ calculations with the cc-pVXZ and cc-pCVXZ basis sets (X=D,T) to study the impact of functions suited for core-valence correlation, and we report the results in Table \ref{tab:no-core-mu}.
From Table \ref{tab:no-core-mu} we can observe that the all electron $\tcfcimu$ calculations without core-valence functions significantly underestimate the exact ground state energy of the boron atom by 36 mH and 26 mH in the cc-pVDZ and cc-pVTZ basis sets, respectively.
On the other hand such effect is strongly reduced when using core-valence functions as the underestimation of the ground state energy is of 2.2 mH and 1.8 mH with the cc-pCVDZ and cc-pCVTZ basis sets, respectively.
Regarding now the effect on the IP, it can be noticed that while the energy difference computed using a cc-pCVDZ is already
within a sub mH precision with respect to the CBS value, the results obtained without the core-valence functions are far from such an accuracy as the error is of 17 and 15 mH using the cc-pVDZ and cc-pVTZ, respectively.
Therefore, as already shown by Ten-No \textit{et. al.}\cite{HinTanTen-JCP-01},
core-valence correlation functions are mandatory when performing all electron calculations in the context of TC methods,
unless when the correlation factor includes explicit electron-electron-nucleus correlation
as for instance in the work of Cohen \textit{et. al.}\cite{CohLuoGutDobTewAla-JCP-19}.
\begin{table}
\label{table_b}
\caption{\label{tab:no-core-mu}Boron and boron cation total energies results (in a.u.) and ionization potentials (IP) from all-electron calculations for $\mu_{\text{RSC+LDA}} \approx 1.02$ for Boron and $\mu_{\text{RSC+LDA}} \approx 1.15$ for the boron cation with and without core-valence basis functions. }
\begin{tabular}{ccccc}
\toprule
Atom & & DZ & TZ & CBS\protect\cite{ChaGwaDavParFro-PRA-93, DavHagChaMeiFro-PRA-91} \\
\hline
\multirow{2}{*}{B} & with core-valence & -24.65613 & -24.65568 & \multirow{2}{*}{-24.65391} \\
& w/o core-valence & -24.69075 & -24.68063 & \\
\hline
\multirow{2}{*}{B$^+$} & with core-valence & -24.35147 & -24.34960 & \multirow{2}{*}{-24.34889} \\
& w/o core-valence & -24.37284 & -24.36495 & \\
\hline
\multirow{2}{*}{IP} & with core-valence & 0.30466 & 0.30608 & \multirow{2}{*}{0.30502} \\
& w/o core-valence & 0.31791 & 0.31568 & \\
\botrule
\end{tabular}
\end{table}
\subsubsection{The \enquote{5-idx} approximation on the three-body term}
\label{sec:results_5idx}
Another important computational aspect of the TC method are the numerous $N^6$ integrals to be computed for the three-body effective operator $\lu{1}{2}{3}$.
The problems regarding these terms are two-fold: \textbf{(i)} the computation of the intermediate quantities which can be quite demanding and \textbf{(ii)} the computation and storage of all the $N^6$ integrals.
In the context of $\hmub$, point \textbf{(i)} is not really a problem since all intermediate quantities are computed analytically and not numerically in contrast to more complex correlation factors\cite{CohLuoGutDobTewAla-JCP-19, GutCohLuoAla-JCP-21} (see Appendix of Ref.~\onlinecite{Gin-JCP-21}).
Therefore, the main computational bottleneck is the computation and storage of the $N^6$ integrals.
Nevertheless, one can notice that the most numerous terms in the $\lmuijmkln$ tensor are those corresponding to 6 different indices, which corresponds to pure triple excitations operators.
We propose here the \enquote{5-idx} approximation of the three-body term which consists in neglecting all integrals $\lmuijmkln$ with six different indices, which reduces to $N^5$ the number of integrals to compute and store for the treatment of $\lu{1}{2}{3}$.
We performed numerical calculation with the full treatment of the $\lu{1}{2}{3}$ operator and the \enquote{5-idx} approximation using the cc-pCVXZ (X=D,T,Q) for the neon atom and its first cation, and report the results in Table~\ref{tab:5idx}.
From Table~\ref{tab:5idx} we can observe that the results are almost insensitive to the 5-idx approximation as the differences between the energies are of about 10$^{-5}$H for both the neon and the first cation.
This result therefore indicates that the 5-idx approximation drastically reduces both the memory and CPU bottleneck of the TC calculations
while leaving the numerical results unchanged to a sub $m\text{Ha}$ precision.
This is nevertheless still more expensive than the normal-ordered approaches proposed by some of the present authors in Ref. \onlinecite{SchCohAla-JCP-21}, but has nevertheless the advantage not to depend on the one- and two-body density of some reference wave function.
\begin{table}
\small
\renewcommand{\arraystretch}{1.2}
\caption{\label{tab:5idx}Effect of neglecting the full 3-body terms on total energies (reported in a.u.) for all electron calculations in $\tcfcimu$ in a cc-pCVXZ basis sets (X=D,T,Q) for Neon and Ne$^+$ and the corresponding ionization potential (reported in m a.u.).
}
\begin{tabular}{ccccc}
\toprule
Method & Basis & Ne [H] & Ne$^+$ [H] & IP [mH] \\
\hline
Full & cc-pCVDZ & -128.96435(1)\phantom{0} & -128.16345(1)\phantom{0} & 800.90(2)\phantom{0} \\
5idx & cc-pCVDZ & -128.96437(2)\phantom{0} & -128.16344(2)\phantom{0} & 800.93(4)\phantom{0} \\
Full & cc-pCVTZ & -128.93230(1)\phantom{0} & -128.14083(8)\phantom{0} & 791.47(9)\phantom{0} \\
5idx & cc-pCVTZ & -128.93201(2)\phantom{0} & -128.14064(6)\phantom{0} & 791.38(8)\phantom{0} \\
Full & cc-pCVQZ & -128.93569(3)\phantom{0} & -128.14295(8)\phantom{0} & 792.7(1)\phantom{00}\\
5idx & cc-pCVQZ & -128.93545(2)\phantom{0} & -128.14264(1)\phantom{0} & 792.81(3)\phantom{0}\\
\hline
\botrule
\end{tabular}
\end{table}
\subsection{All electrons calculations the Li-Ne species and first cations}
\begin{table*}
\caption{\label{tab:full-core-a}Ground state all electron calculations for the Li--Ne species, together with their first cations and the corresponding ionization potential (IP) computed in the cc-pCVXZ (X=D,T,Q) family of basis set. SM-17 results stand for TC-FCIQMC with the correlation factor of Moskowitz \textit{et. al.}\cite{SchMos-JCP-90} to obtain the TC Hamiltonian.
Estimated non relativistic CBS results are obtained from Ref.~\onlinecite{ChaGwaDavParFro-PRA-93}. All results are reported in atomic units.}
\renewcommand{\arraystretch}{1.1}
\begin{tabular}{lccccc|cccccc}
\toprule
Atom &Method &\multicolumn{1}{c}{CVDZ}&\multicolumn{1}{c}{CVTZ}& \multicolumn{1}{c}{CVQZ}& \multicolumn{1}{c}{Est. CBS$^a$} & Atom &Method &\multicolumn{1}{c}{CVDZ}&\multicolumn{1}{c}{CVTZ}& \multicolumn{1}{c}{CVQZ}& \multicolumn{1}{c}{Est. CBS$^a$} \\
\hline
&$\tcfcimu$ & -7.47909 & -7.47857 & -7.47832 & & &$\tcfcimu$ & -54.59896 & -54.59382 & -54.59019 & \\
Li & SM-17 & -7.47748 & -7.47824 &\textendash& -7.47806 & N & SM-17 & -54.56695 & -54.58658 &\textendash& -54.58920 \\
& CIPSI & -7.46602 & -7.47424 & -7.47636 & & & CIPSI & -54.51765 & -54.56793 & -54.58197 & \\[3pt]
&$\tcfcimu$ & -7.27988 & -7.28003 & -7.27999 & & &$\tcfcimu$ & -54.05787 & -54.05721 & -54.05478 & \\
Li$^+$ & SM-17 & -7.27951 & -7.28016 &\textendash& -7.27991 & N$^+$ & SM-17 & -54.03875 & -54.05159 &\textendash& -54.05460 \\
& CIPSI & -7.26919 & -7.27655 & -7.27833 & & & CIPSI & -53.99612 & -54.03672 & -54.04850 & \\[3pt]
&$\tcfcimu$ & 0.19921 & 0.19853 & 0.19833 & & &$\tcfcimu$ & 0.54109 & 0.53661 & 0.53545 & \\
IP(Li) & SM-17 & 0.19797 & 0.19808 &\textendash& 0.19815 & IP(N) & SM-17 & 0.52820 & 0.53499 &\textendash& 0.53460 \\
& CIPSI & 0.19683 & 0.19769 & 0.19803 & & & CIPSI & 0.52153 & 0.53121 & 0.53347 & \\
\hline
&$\tcfcimu$ & -14.67037 & -14.66855 & -14.66777 & & &$\tcfcimu$ & -75.07412 & -75.06774 & -75.06729 & \\
Be & SM-17 & -14.66969 & -14.66863 &\textendash& -14.66736 & O & SM-17 & -75.02676 & -75.06082 &\textendash& -75.06730 \\
& CIPSI & -14.65182 & -14.66236 & -14.66556 & & & CIPSI & -74.95051 & -75.03122 & -75.05447 & \\[3pt]
&$\tcfcimu$ & -14.32565 & -14.32538 & -14.32504 & & &$\tcfcimu$ & -74.57748 & -74.57055 & -74.56829 & \\
Be$^+$ & SM-17 & -14.32570 & -14.3254 &\textendash & -14.32476 & O$^+$ & SM-17 & -74.54189 & -74.56201 &\textendash& -74.56680 \\
& CIPSI & -14.31102 & -14.32048 & -14.32317 & & & CIPSI & -74.47796 & -74.54098 & -74.55815 & \\[3pt]
&$\tcfcimu$ & 0.34472 & 0.34317 & 0.34272 & & &$\tcfcimu$ & 0.49664 & 0.49719 & 0.49900 & \\
IP(Be) & SM-17 & 0.34399 & 0.34323 &\textendash& 0.34258 & IP(O) & SM-17 & 0.48487 & 0.49881 &\textendash& 0.50050 \\
& CIPSI & 0.34080 & 0.34188 & 0.34239 & & & CIPSI & 0.47255 & 0.49024 & 0.49632 & \\
\hline
&$\tcfcimu$ & -24.65612 & -24.65568 & -24.65467 & & &$\tcfcimu$ & -99.74701 & -99.73164 & -99.73326 & \\
B & SM-17 & -24.65169 & -24.65459 &\textendash& -24.65391 & F & SM-17 & -99.67001 & -99.72284 &\textendash& -99.73390\\
& CIPSI & -24.62603 & -24.64485 & -24.65083 & & & CIPSI & -99.56965 & -99.68185 & -99.71509 & \\[3pt]
&$\tcfcimu$ & -24.35147 & -24.34960 & -24.34953 & & &$\tcfcimu$ & -99.10398 & -99.09299 & -99.09358 & \\
B$^+$ & SM-17 & -24.35028 & -24.34916 &\textendash& -24.34892 & F$^+$ & SM-17 & -99.04116 & -99.08228 &\textendash& -99.09280 \\
& CIPSI & -24.32985 & -24.34213 & -24.34664 & & & CIPSI & -98.95526 & -99.05157 & -99.07850 & \\[3pt]
&$\tcfcimu$ & 0.30466 & 0.30608 & 0.30513 & & &$\tcfcimu$ & 0.64303 & 0.63865 & 0.63968 & \\
IP(B) & SM-17 & 0.30141 & 0.30542 &\textendash& 0.30499 & IP(F) & SM-17 & 0.62885 & 0.64056 &\textendash& 0.64110 \\
& CIPSI & 0.29618 & 0.30272 & 0.30419 & & & CIPSI & 0.61439 & 0.63028 & 0.63659 & \\
\hline
&$\tcfcimu$ & -37.84888 & -37.84793 & -37.84617 & & &$\tcfcimu$ &-128.96435 &-128.93221 &-128.93569 & \\
C & SM-17 & -37.83537 & -37.84462 &\textendash& -37.84500 & Ne & SM-17 &-128.84774 &-128.91945 &\textendash& -128.93760 \\
& CIPSI & -37.79798 & -37.83003 & -37.83962 & & & CIPSI &-128.72254 &-128.86823 &-128.91235 & \\[3pt]
&$\tcfcimu$ & -37.43162 & -37.43214 & -37.43170 & & &$\tcfcimu$ &-128.16345 &-128.14082 &-128.14298 & \\
C$^+$ & SM-17 & -37.42606 & -37.43018 &\textendash& -37.43103 & Ne$^+$& SM-17 &-128.06691 &-128.12553 &\textendash & -128.14310 \\
& CIPSI & -37.39487 & -37.41932 & -37.42711 & & & CIPSI &-127.95437 &-128.08498 &-128.12259 & \\[3pt]
&$\tcfcimu$ & 0.41727 & 0.41579 & 0.41447 & & &$\tcfcimu$ & 0.80090 & 0.79139 & 0.79271 & \\
IP(C) & SM-17 & 0.40931 & 0.41444 &\textendash& 0.41397 & IP(Ne)& SM-17 & 0.78083 & 0.79392 &\textendash& 0.79450 \\
& CIPSI & 0.40345 & 0.41123 & 0.41303 & & & CIPSI & 0.76817 & 0.78325 & 0.78976 & \\
\botrule
\end{tabular}
\end{table*}
\begin{table}
\caption{\label{tab:mae}Mean absolute errors (MAE) in mH for the ionization potentials at the $\tcfcimu$ and CIPSI levels of theory for the Li--Ne series in the cc-pCVXZ basis sets.
The results labelled by SM-17 are the TC-FCIQMC results using of Ref.~\onlinecite{CohLuoGutDobTewAla-JCP-19} using a flexible correlation factor. }
\begin{tabular}{lccc}
\toprule
& CVDZ & CVTZ & CVQZ \\
\hline
CIPSI & 14.65& 5.24 & 2.08 \\
$\tcfcimu$ & 3.19 & 1.85 & 0.81 \\
SM-17 & 7.22 & 0.60 & - \\
\botrule
\end{tabular}
\end{table}
\begin{figure*}
\centering
\includegraphics[width=0.9\textwidth]{Fig1.pdf}
\caption{\label{fig:boron-neon}Boron (a), B$^+$ (b), Ne (d) and Ne$^+$ (e) absolute energy difference with respect to the estimated exact total energies\protect\cite{ChaGwaDavParFro-PRA-93} and ionization potential for boron (c) and neon(f) of the range separated TC ($\mu$-TC), and Cohen~\emph{et al.}\cite{CohLuoGutDobTewAla-JCP-19} (SM-17) and non-transcorrelated results (no TC) for fully correlated calculations with cc-pCVXz core-valence basis sets.
The gray areas indicates sub mH chemical accuracy.}
\end{figure*}
\label{sec:results_full}
We report in Tables~\ref{tab:full-core-a} the performance of all-electron ground state calculations on the Li--Ne series in the cc-pCVXZ basis sets (X=D,T,Q), together with their first cations at the $\tcfcimu$ and CIPSI levels of theory.
We also report in Tables~\ref{tab:full-core-a} the estimated non relativistic CBS results of Ref.~\onlinecite{ChaGwaDavParFro-PRA-93}, together with the so-called $\tcfcihint$ results which are
the TC-FCIQMC calculations in the cc-pCVXZ basis set family (X=D,T) using the same methodology of Ref. ~\onlinecite{CohLuoGutDobTewAla-JCP-19}
where the correlation factor of Moskowitz \textit{et. al.}\cite{SchMos-JCP-90} was used to obtain the TC Hamiltonian.
The $\tcfcihint$ correlation factor is very flexible as it contains explicit electron-nucleus, electron-electron and electron-electron-nucleus terms and have been optimized at the VMC level for each neutral species considered here.
The mean absolute errors (MAE) of the IPs are reported in Table~\ref{tab:mae}.
From Tables~\ref{tab:full-core-a} and \ref{tab:mae} one can observe that the convergence of both the total energies and the ionization potential using $\tcfcimu$ and $\tcfcihint$ is strongly improved with respect to usual WFT calculations, which is expected due to the presence of explicit correlation.
Several specifics aspects have to be pointed out from these Tables~\ref{tab:full-core-a} and \ref{tab:mae}. \\
\textbf{(i)} For $Z>5$, the total energies provided by $\tcfcimu$ are always closer to the exact ones than that of $\tcfcihint$. \\
\textbf{(ii)} With increasing nuclear charge, the discrepancy between the total energies at triple-zeta basis set level using $\tcfcimu$ and $\tcfcihint$ increases. This suggests that, the electron-electron-nucleus term of the $\tcfcihint$ takes into account only a part of the correlation effects arising from the core (\textit{i.e.} core-core and core-valence correlation effects). \\
\textbf{(iii)} Although the total energies obtained with $\tcfcimu$ in double-zeta basis sets can be way below the exact ground state energy (by about 26 mH and 20 mH in the case of the Ne and Ne$^+$ in the cc-pCVDZ basis set, respectively), the energy differences are of good quality (the accuracy of the ionization potential of Ne using $\tcfcimu$ in the cc-pCVDZ basis set is comparable to that of regular WFT in a cc-pCVQZ basis set). \\
\textbf{(iv)} While at the double-zeta level the error with respect to the exact IP is significantly smaller using the $\tcfcimu$ than the $\tcfcihint$ approach for all systems, the errors at the triple-zeta level are smaller with the $\tcfcihint$ by approximatively 1 kcal on average. Nevertheless, the error with respect to the exact IP obtained with $\tcfcimu$ at the quadruple-zeta level decrease below 0.001 a.u., showing a systematic convergence pattern.
\section{Conclusion}
\label{sec:conclusion}
In the present work, we further investigated a new strategy based on the TC method which was previously applied on two-electron systems only\cite{Gin-JCP-21}.
One of the focus of the paper is to test its validity one more realistic systems where many body effects arrise and therefore where the effective three-body terms of the TC Hamiltonian have to be included.
In order to avoid particle-hole truncation errors due to approximations of the right eigenvector in a given basis set,
the ground state energy of the TC Hamiltonian have been obtained using the recently proposed non-hermitian and three-body variant of the FCIQMC\cite{CohLuoGutDobTewAla-JCP-19}.
The main feature of the recently introduced\cite{Gin-JCP-21} TC correlation factor is that, beside producing a strictly non divergent TC Hamiltonian, it has a simple parametrization which depends only on a single parameter $\mu$.
Such a parameter $\mu$ determines the impact of the correlation factor through the depth and typical range of the correlation hole that it induces in the wave function.
Also, thanks to the simple analytical structure of the obtained TC Hamiltonian, all needed integrals can be very efficiently computed in a mixed numerical-analytical scheme.
The parameter $\mu$ is determined efficiently for each system,
according to the method described in Ref.~[\citen{Gin-JCP-21}],
and depends only on the density of the system under study,
which essentially results in a parameter-free correlation factor.
The main focus of this work is the study of the convergence of the TC eigenvalues and energy differences with respect to the quality of the basis set
and its ability to treat both core and valence electrons.
We performed calculations on the Li--Ne series in the cc-pCVXZ (X=D,T,Q), together with their first cations in order to investigate
the convergence towards the CBS limit of both total energies and IPs.
The main conclusion of this study is that, provided that the basis set contains core-valence functions, very accurate total energies can already be obtained from the triple-zeta quality basis sets.
Regarding the accuracy of the IPs computed here, while the MAE is significantly smaller in double-zeta quality basis sets using the single-parameter $\tcfcimu$ compared to the more elaborate SM-17 correlation factor,
the results at the triple-zeta level of theory are outperformed by the latter by about 1.2 mH on average.
Nevertheless, the MAE of both these methods are within chemical accuracy (below 1 mH) with CBS limit results at the quadruple zeta level.
In the context of TC calculations, this study shows that the results obtained with a simple one-parameter correlation factor such as $\umu{1}{2}$ are comparable with those obtained with much more sophisticated correlation factors including electron-electron-nucleus terms.
On the other hand, it should be pointed out that in the TC work of Cohen
\emph{et al.} the employed SM17 correlation factors were taken from the literature, and were not further optimized for use in the TC method.
In subsequent work, some of the present authors will investigate a better suited method
to optimize general Jastrow factors for the TC approach.
Among the perspective of this work, the use of a correlation factor $\umu{1}{2}$ with a $\mu$ varying in real space could be of interest as it could possibly mimic the electron-electron-nucleus correlation effects.
|
2,877,628,089,004 | arxiv | \section{Introduction}
This document provides instructions for submitting papers to the 55\textsuperscript{th} IEEE/ACM International Symposium on Microarchitecture\textsuperscript{\textregistered} (MICRO 2022). In an effort to respect the efforts of reviewers and in the interest of fairness to all prospective authors, we request that all submissions to MICRO 2022 follow the formatting and submission rules detailed below. Submissions that violate these instructions may not be reviewed, at the discretion of the program chairs, in order to maintain a review process that is fair to all potential authors.
This document is itself formatted using the MICRO 2022 submission format. The content of this document mirrors that of the submission instructions that appear on the conference website. All questions regarding paper formatting and submission should be directed to the program chairs.
\subsection{Format Highlights}
\begin{itemize}
\item Paper must be submitted in printable PDF format.
\item Text must be in a minimum 10pt font, see Table~\ref{table:formatting}.
\item Papers must be at most 11 pages, not including references.
\item No page limit for references.
\item Each reference must specify {\em all} authors (no {\em et al.}).
\item Author anonymity must be fully preserved, including in any referenced artifacts (e.g., GitHub repository).
\end{itemize}
\subsection{Paper Evaluation Objectives}
The committee will make every effort to judge each submitted paper on its own merits. There will be no target acceptance rate. We expect to accept a wide range of papers with appropriate expectations for evaluation---while papers that build on significant past work with strong evaluations are valuable, papers that open new areas with less rigorous evaluation are equally welcome and especially encouraged.
\section{Paper Preparation Instructions}
\subsection{Paper Formatting}
Papers must be submitted in printable PDF format and should contain a {\em maximum of 11 pages} of single-spaced two-column text, {\bf not including references}. You may include any number of pages for references, but see below for more instructions. If you are using \LaTeX~\cite{lamport94} to typeset your paper, then we suggest that you use the template here: \href{https://www.microarch.org/micro55/submit/micro55-latex-template.zip}{\LaTeX~Template}. This document was prepared with that template. Note that the template and sample paper may render slightly differently on different \LaTeX~engines, due to typesetting changes between versions. If you use a different software package to typeset your paper, then please adhere to the guidelines given in Table~\ref{table:formatting}.
\begin{scriptsize}
\begin{table}[h!]
\centering
\begin{tabular}{|l|l|}
\hline
\textbf{Field} & \textbf{Value}\\
\hline
\hline
File format & PDF \\
\hline
Page limit & 11 pages, {\bf not including}\\
& {\bf references}\\
\hline
Paper size & US Letter 8.5in $\times$ 11in\\
\hline
Top margin & 1in\\
\hline
Bottom margin & 1in\\
\hline
Left margin & 0.75in\\
\hline
Right margin & 0.75in\\
\hline
Body & 2-column, single-spaced\\
\hline
Space between columns & 0.25in\\
\hline
Line spacing (leading) & 11pt \\
\hline
Body font & 10pt, Times\\
\hline
Abstract font & 10pt, Times\\
\hline
Section heading font & 12pt, bold\\
\hline
Subsection heading font & 10pt, bold\\
\hline
Caption font & 9pt (minimum), bold\\
\hline
References & 8pt, no page limit, list \\
& all authors' names\\
\hline
\end{tabular}
\caption{Formatting guidelines for submission.}
\label{table:formatting}
\end{table}
\end{scriptsize}
{\em Please ensure that you include page numbers with your submission}. This makes it easier for the reviewers to refer to different parts of your paper when they provide comments. Please ensure that your submission has a banner at the top of the title page, similar to this document, which contains the submission number and the notice of confidentiality. If using the template, just replace XXX with your submission number.
\subsection{Content}
Reviewing will be {\em double blind} (no author list); therefore, please do not include any author names on any submitted documents except in the space provided on the submission form. You must also ensure that the metadata included in the PDF does not give away the authors. You must fully anonymize any links to artifacts (e.g., GitHub repository) and remove any links to artifacts that cannot be fully anonymized. {\bf Papers that violate the anonymization policy may be rejected without review} at the discretion of the program chairs.
If you are improving upon your prior work, refer to your prior work in the third person and include a full citation for the work in the bibliography. For example, if you are building on {\em your own} prior work in the papers \cite{nicepaper1,nicepaper2,nicepaper3}, you would say something like: "While the authors of \cite{nicepaper1,nicepaper2,nicepaper3} did X, Y, and Z, this paper additionally does W, and is therefore much better." Do NOT omit or anonymize references for blind review. There is one exception to this for your own prior work that appeared in IEEE CAL, arXiv, workshops without archived proceedings, etc.\, as discussed later in this document.
\noindent\textbf{Figures and Tables:} Ensure that the figures and tables are legible. Please also ensure that you refer to your figures in the main text. Many reviewers print the papers in gray-scale. Therefore, if you use colors for your figures, ensure that the different colors are highly distinguishable in gray-scale.
\noindent\textbf{References:} There is no length limit for references. {\em Each reference must explicitly list all authors of the paper. Papers not meeting this requirement will be rejected.} Authors of NSF proposals should be familiar with this requirement. Knowing all authors of related work will help find the best reviewers. Since there is no length limit for the number of pages used for references, there is no need to save space here.
\section{Paper Submission Instructions}
\subsection{Guidelines for Determining Authorship}
IEEE guidelines dictate that authorship should be based on a {\em substantial intellectual contribution}. It is assumed that all authors have had a significant role in the creation of an article that bears their names. In particular, the authorship credit must be reserved only for individuals who have met each of the following conditions:
\begin{enumerate}
\item Made a significant intellectual contribution to the theoretical development, system or experimental design, prototype development, and/or the analysis and interpretation of data associated with the work contained in the article;
\item Contributed to drafting the article or reviewing and/or revising it for intellectual content; and
\item Approved the final version of the article as accepted for publication, including references.
\end{enumerate}
A detailed description of the IEEE authorship guidelines and responsibilities is available \href{https://www.ieee.org/publications_standards/publications/rights/Section821.html}{here}. Per these guidelines, it is not acceptable to award {\em honorary } authorship or {\em gift} authorship. Please keep these guidelines in mind while determining the author list of your paper.
\subsection{Declaring Authors}
Declare all the authors of the paper upfront. Addition/removal of authors once the paper is accepted will have to be approved by the program chairs, since it potentially undermines the goal of eliminating conflicts for reviewer assignment.
\subsection{Areas and Topics}
Authors should indicate these areas on the submission form as well as specific topics covered by the paper for optimal reviewer match. If you are unsure whether your paper falls within the scope of MICRO, please check with the program chairs -- MICRO is a broad, multidisciplinary conference and encourages new topics.
\subsection{Revision of Previously-Reviewed\\ Manuscript}
If the manuscript has been previously reviewed and rejected and is now being submitted to MICRO, the authors have an option of providing a letter explaining how the paper has been revised for this current submission. We expect this revision information to improve both the submission and the review process. Authors choosing to provide such a letter have control over who has access to it by specifying one of the following options:
\begin{enumerate}
\item Shared with all reviewers of the paper
\item Shared with reviewers who declare that they reviewed a prior version and who request the revision information
\item Not shared with any PC member but available to the program chairs
\end{enumerate}
We encourage you to keep this letter concise and optionally append additional information, such as a version of the paper that highlights the differences or any other material of your choice.
\subsection{Declaring Conflicts of Interest}
Authors must register all their conflicts for their paper submission. Conflicts are needed to ensure appropriate assignment of reviewers. {\bf If a paper is found to have an undeclared conflict that causes a problem OR if a paper is found to declare false conflicts in order to abuse or ``game'' the review system, the paper may be rejected without review.} We use the following conflict of interest guidelines for determining the conflict period for MICRO 2022. Please declare a conflict of interest (COI) with the following people for any author of your paper:
\begin{enumerate}
\item Your Ph.D. advisor(s), post-doctoral advisor(s), Ph.D. students,
and post-doctoral advisees, forever.
\item Family relations by blood or marriage, or their equivalent,
forever (if they might be potential reviewers).
\item People with whom you have collaborated in the last FOUR years, including:
\begin{itemize}
\item co-authors of accepted/rejected/pending papers
\item co-PIs on accepted/rejected/pending grant proposals
\end{itemize}
\item Ongoing collaboration that has not yet resulted in a paper or proposal submission. Justification may be queried.
\item When there is a direct funding relationship between an author and the potential reviewer (e.g., the reviewer is a sponsor of an author's research on behalf of his/her company or vice versa) in the last FOUR years.
\item People (including students) who shared your primary institution(s) in the last FOUR years.
\item Other relationships, such as close personal friendship, that you think might tend
to affect your judgment or be seen as doing so by a reasonable person familiar
with the relationship.
\end{enumerate}
We would also like to emphasize that the following scenarios do {\em not} constitute a conflict:
\begin{enumerate}
\item Authors of previously-published, closely related work on that basis alone.
\item ``Service'' collaborations such as co-authoring a report for a professional organization, serving on a program committee, or co-presenting tutorials.
\item Co-authoring a paper that is a compendium of various projects with no true collaboration among the projects.
\item People who work on topics similar to or related to those in your papers.
\item Collaborators on large funded projects where there is no close collaboration and no joint benefit in the paper being accepted.
\end{enumerate}
We hope to draw most reviewers from the program committee, but others
from the community may also write reviews. {\bf Please declare all your conflicts (not just restricted to the PC).} When in doubt, please contact the program chairs.
\subsection{Concurrent Submissions and Workshops}
By submitting a manuscript to MICRO 2022, the authors guarantee that the manuscript has not been previously published or accepted for publication in a substantially similar form in any conference, journal, or the archived proceedings of a workshop (e.g., in the ACM/IEEE digital library) -- see exceptions below. The authors also guarantee that no paper that contains significant overlap with the contributions of the submitted paper will be under review for any other conference or journal or an archived proceedings of a workshop during the MICRO 2022 review period. Violation of any of these conditions will lead to rejection.
The only exceptions to the above rules are for the authors' own papers in (1) workshops without archived proceedings such as in the ACM/IEEE digital library (or where the authors chose not to have their paper appear in the archived proceedings), or (2) venues such as IEEE CAL or arXiv where there is an explicit policy that such publication does not preclude longer conference submissions. In all such cases, the submitted manuscript may ignore the above work to preserve author anonymity. This information must, however, be provided on the submission form -- the program chairs will make this information available to reviewers if it becomes necessary to ensure a fair review. As always, if you are in doubt, it is best to contact the program chairs.
Finally, the ACM/IEEE Plagiarism Policy (\href{http://www.acm.org/publications/policies/plagiarism_policy}{here} and \href{https://www.ieee.org/publications_standards/publications/rights/plagiarism.html}{here}) covers a range of ethical issues concerning the misrepresentation of other works or one's own work.
\section{Ethics}
\begin{enumerate}
\item Authors must abide by the ACM code of ethics and the IEEE code of ethics
\item Authors must not contact reviewers or PC members about any submission, including their own. This includes attempting to sway a reviewer, requesting information about any aspect of the reviewing process, and/or asking about the outcome of a submission. Similarly, authors are not allowed to ask another party to contact the reviewers on their behalf.
\item Authors must not disclose the content of reviews for their paper publicly (e.g., on social media) before the results are announced.
\item Authors must report any allegations of submission or reviewing misconduct to the program chairs. The only exception is if the complaint is about the program chairs; in this case, the Steering Committee should be contacted.
\end{enumerate}
\section*{ACKNOWLEDGMENTS}
This document is derived from previous conferences, in particular MICRO 2013, ASPLOS 2015, MICRO 2015, MICRO 2016, MICRO 2017, MICRO
2018, MICRO 2019, MICRO 2020 and MICRO 2021, as well as SIGARCH/TCCA's Recommended Best Practices for the Conference Reviewing Process.
\bibliographystyle{IEEEtranS}
\section{Introduction}
Superconducting quantum computing \cite{clarke2008superconducting, dicarlo2009demonstration,stassi2020scalable} is one of the major technologies of quantum computing that aims to solve problems that are intractable on a classical computer by leveraging mature silicon technology. Compared to other quantum computing hardware technologies, it has advantages in scalability, microwave operation control, and nanosecond-level gate operation \cite{fu2017experimental,fu2018microarchitecture,ding2020systematic}.
\begin{figure}[!t]
\centerline{\includegraphics[width=0.5\textwidth]{Figures/wholestruc.png}}
\caption{One complete cycle of validating new quantum ideas with proposed CQV as SR estimator, bypassing the real quantum machine. Original image from \cite{solenov2018potential}}
\label{WholeStruct}
\end{figure}
However, superconducting quantum computers suffer from a great variety of noise channels, which can be classified into retention errors or operational errors \cite{tannu2019not}.
Although quantum error correction (QEC) can be performed to fix the errors, this requires an enormous number of physical qubits that cannot be realized in current noisy intermediate-scale quantum (NISQ) era machines which contain at most a few hundred noisy qubits \cite{preskill2018quantum}.
Therefore, in NISQ, quantum computing performs noisy operations and accepts the fact that errors can happen on any physical qubit at any time during the execution. To combat the high error rate, correct output is captured by repeating the same computations for a sufficient number of trials, and the number of correct trials out of the total number of trials is called Success Rate (SR).
Figure \ref{WholeStruct} shows one complete validation round of a researcher testing a new idea, and the idea needs to be transferred to a compiled circuit before executing on a real machine.
Repeatedly waiting for results executed by noisy, crowded, and privileged quantum computers to validate designs is becoming an unavoidable bottleneck for researchers in many quantum computing fields, revealing the demand for an alternative way to easily and quickly get the final SR.
Surprisingly, compared with the popular research on improving the success rate, fewer studies are trying to understand how a given compiled circuit results in a certain experimental success rate after executing on a target quantum computer. The reason behind that is the lack of an accurate method to model the error and simulate the cumulative success rate based on that.
Knowing the error behavior is one of the unavoidable prerequisites for improving the success rate and knowing how much and where to reduce the errors.
Currently, there are two ways to predict the success rate: the estimated probability of success (ESP) \cite{tannu2019ensemble} and statistical fault injection \cite{Qiskit}. Both suffer from low accuracy and increasing prediction errors when algorithm size is increased. Inspired by the famous classical chip vulnerability study AVF \cite{walcott2007dynamic, duan2009versatile}, we have proposed and designed a quantum vulnerability factor (QVF) that locates error matter qubits and applies an accurate error modeling scheme to generate a cumulative quantum vulnerability (CQV). As shown in Fig. \ref{WholeStruct}, the CQV would simulate the error behaviors for the target quantum chip and estimate the SR.
To predict the SR, we first redesign the compiled circuits to enable cycle view, promising each physical qubit will be involved with at most one operation within a cycle, which isolates errors to cycles.
Next, based on the observation that some errors do not impact the output, we define such qubits to be in an un-architecturally correct execution required (un-ACE) state for the given cycle and exhausted all possible cases of un-ACE state.
Then, based on the ACE states, we calculated the QVF and applied it to help the calculation of CQV. In addition, by presenting the un-ACE states for every gate at the compiled time, researchers will have a clear and direct view of gate error behaviors straight from the compiled circuit.
Ultimately, the CQV is calculated by accumulating the gate success rate, which is calculated from calibrated gate errors including the crosstalk error, through all the ACE states within the QVF model. The error modeling scheme of CQV also presents a novel and accurate way to quantify the error impact across the CNOT gates and produce the 1-CQV in the end for predicting actual SR for a given system.
We validate our design on four of the state-of-the-art quantum machines with 27 and 65 qubits hosted by IBM written with the Qiskit open-source quantum platform \mbox{\cite{Qiskit}}. As a quantum system SR estimator, with the pre-calibrated system on a different day, the CQV outperforms the state-of-the-art SR estimator ESP by reducing on average 6 times in relative prediction error, with best cases at 20 times. We also verify that the CQV can achieve good accuracy for unseen benchmarks and unseen machines.
The contributions of our paper can be listed as follows:
\begin{itemize}
\item To our best knowledge, we are the first to bring the classical vulnerability study analysis to the quantum computing analyses, which changes the way of viewing errors and compiled circuits.
\item We define and propose a systematic methodology to demonstrate all possible different un-ACE state cases.
\item We design and build an accurate and simple error modeling scheme in which the first work quantifies error impact flowing across the CNOT gate.
\item The proposed CQV outperforms the current state-of-the-art success rate estimation model and achieves 6 times less relative prediction error on average.
\item When the size of the algorithm reaches and surpasses the quantum volume of a device, the CQV has a 10x improvement in relative prediction error rate compared with the state-of-the-art estimates.
\end{itemize}
\section{Background}
\subsection{Quantum Circuit}
The qubit is the quantum counterpart of the classical bit, capable of taking state $|0\rangle$, state $|1\rangle$, and all the other linear combinations of the $|0\rangle$ and $|1\rangle$ states. A $CNOT$ gate is the two-qubit gate that reverses the magnitude of the target qubit if the input state of the control qubit’s magnitude is at state one. During the CNOT operation, the phase information on the target qubit’s state will transfer through the $CNOT$ gate and apply to the control qubit’s state at the output, which is known as \textbf{phase-kickback}. The SWAP gate is used when performing state exchange of two qubits, which is equivalent to three $CNOT$ gates in series.
The processes of generating a quantum executable from a given algorithm and executing it on a quantum chip have been shown in Figure \ref{WholeStruct}. The logical circuits are generated assuming an all-to-all connection between qubits, which can be applied on a superconducting quantum computer, which supports only adjacent connections, after a series of compiling steps. As shown, the quantum compiler will be given information about the target quantum chip and compiler strategy, such as optimization levels, initial layout method, mapping method, etc. Based on that, the compiler will follow all the intermediate compiling steps to generate a compiled circuit for executing on the quantum computer. For more details on quantum computing, please refer to \cite{nielsen2002quantum}.
\subsection{NISQ Era Quantum Error}
Superconducting quantum computers are currently one of the most promising technologies for quantum computation, with advantages in scalability, operation control, operating on nanosecond scale, microwave control, etc. However, noise is the biggest challenge that impedes the growth of superconducting quantum computing toward solving large-size algorithms. The noise that a superconducting quantum computer suffers can be classified as operational errors and retention errors \cite{tannu2019not}.
\textbf{Operational Errors:}
When operating a superconducting quantum computer, the control pulses for gate operations may introduce errors into the target qubits' quantum state or the nearby qubits. For instance, on 4/21/2022 the ibmq\_montreal quantum chip had an average single-qubit operation error rate at $5.04\times10^{-4}$, average measurement error rate at $3.05\times10^{-2}$, and average CNOT error rate at $2.11\times10^{-2}$ \cite{Ibmquantum}, which varies over time. More complex forms of operational errors exist, such as the recently observed crosstalk errors \cite{murali2020software}.
\textbf{Retention Errors:}
The lifetime of a qubit is determined by its relaxation time (T1) and decoherence time (T2). The decoherence and relaxation time represent the qubit's average time to retain its energized and superpositioned states, respectively. These times act as an upper limit for the current quantum computing execution time.
\begin{figure}[!b]
\centerline{\includegraphics[width=0.5\textwidth,height=3.5cm]{Figures/predictdrawback.png}}
\caption{Current success rate estimator performance}
\label{currentESP}
\end{figure}
\textbf{Success Rate:}
While performing computation on current quantum computers, errors can occur at almost every qubit and every cycle with a frequency as high as once in every few hundred cycles. The solution to such noisy computation is Quantum Error Correction (QEC), correcting the noise and achieving fault-tolerant quantum computing. However, QEC cannot be applied in the current Noise-Intermediate Scale Quantum Computers (NISQ) machines, defined as machines with noisy qubits ranging from a few tens to a few hundred \cite{preskill2018quantum}. Quantum computers in the NISQ class do not have enough qubits to perform QEC. Therefore, the current quantum computers use repetition methods to increase the chance of getting correct output results against the noise as shown in Figure \ref{WholeStruct}. The current number of repetitions can range from a few hundred to a few thousand. We compute the Success Rate (SR) by dividing the number of correct outputs by the total number of executions, which is shown in equation \ref{SR}. The Failure Rate (FR) is the counterpart driven by one minus the SR.
\begin{equation}
Success\ Rate = \frac{Number\ of\ trials\ with\ correct\ output }{Number \ of\ N\ repeated\ trails\ }\
\label{SR}
\end{equation}
\begin{figure}[!t]
\centerline{\includegraphics[width=0.5\textwidth, height=1.2cm]{Figures/espwrong.png}}
\caption{ESP for different circuits}
\label{espissue}
\end{figure}
\section{Motivation}
\subsection{Limitations of Success Rate Metric}
Currently, quantum algorithms are evaluated mainly based on the success rate metric, or metrics derived from success rate, such as PST or MIBF\cite{tannu2019not}. However, the success rate has a few inherent drawbacks at both hardware and software levels, which prevents it to be used directly to study the error behavior.
The first problem is that the correct result must be known in advance, either through mathematical predictions or simulated executions on a classical computer\mbox{\cite{Qiskit}}. Superconducting quantum computers are currently one of the most promising technologies for quantum computation, with advantages in scalability, operation control, operating on nanosecond scale, microwave control, etc. However, noise is the biggest challenge that impedes the growth of superconducting quantum computing toward solving large-size algorithms.
\subsection{Limitations of Current Error Analyzing Methods}
Currently, compared with error modeling studies, research on improving the success rate by optimizing the compiler strategy has been receiving more focus, which is presented in detail in section \ref{sect:relatedwork}. Among the existing methods of predicting SR, the major two are statistical fault injection \cite{Ibmquantum} and estimated success probability (ESP) \cite{tannu2019ensemble}. Statistical Fault Injection utilizes the classical computer to simulate the quantum computation with errors injected to each basis gate with certain triggering probabilities and log the output\cite{Qiskit}. Such a method has an inherent problem of scale-up. The estimated success probability (ESP), shown in equation \ref{ESPequaiton}, predicts the correct output trial probability by multiplying the success rate of each gate, generated by one minus the error rate in equation \ref{errortosr}, for the full circuit. As shown in Fig. \ref{currentESP}, we have applied both methods to estimate the SR of the QFT algorithm with various algorithm sizes and quantum chips. The predicted results of both methods fail to capture the real SR with a range of 25\% to 60\% offset and their relative error rates range from 70\% to 470\%. Meanwhile, both models have decreased prediction accuracy when the algorithm size gets large, which fails to support scaled-up computing.
\begin{equation}
g_{i}^{s}= (1- g_{i}^{e}),\ m_{i}^{s}= (1- m_{i}^{e})
\label{errortosr}
\end{equation}
\begin{equation}
ESP = \displaystyle \prod_{i=1}^{N_{gates}} g_{i}^{s} * \displaystyle \prod_{i=1}^{N_{Measurement}} m_{i}^{s}
\label{ESPequaiton}
\end{equation}
\subsection{Some Error Matter and Some Do Not } \label{sect:someerrronotmatter}
By inspecting compiled circuits based on cycles, we can examine the ESP model to identify why it fails to predict the success rate accurately.
As shown in Fig. \ref{espissue}, the ESP model could make a correct prediction of the SR only for the second circuit. It will overestimate the error for cases similar to the first circuit since the error of the red-boxed $Z$ gate cannot impact the final result.
It will also underestimate the error for cases like the third circuit since the error that happened on the two red-boxed $H$ gates will influence both cumulative SRs of the two measurements. However, based on the gate error rate to gate success rate transformation presented in equation \ref{errortosr} from ESP, the impact of such error had been calculated only once. The error effect could be addressed as error propagation and is completely ignored in this model.
From such analysis, it is clear that some errors in the compiled circuit are not going to impact the output result no matter what the error states are, and some are making more impact.
\begin{figure}[t]
\centerline{\includegraphics[width=0.50\textwidth, height =3.5cm]{Figures/cycleview.png}}
\caption{Cycle view maps operation and virtual qubit onto quantum chip, with Cumulative SR of the qubit and gate success rate appear for each physical qubit at every cycles}
\label{cycleview}
\end{figure}
\section{Quantum Vulnerability Factor}
\subsection{Cycle View } \label{sect:Cycle View}
For understanding the error behavior while executing on a quantum chip, a clear view of where and when the error occurs and how much it affects the compiled circuit has to be established first.
Therefore, this paper quantifies the compiled circuit to a finer degree, the cycle level, analogous to the classical electrical circuit diagram, as defined in Figure \ref{cycleview}, to replace the previous analysis at the level of the entire compiled circuit. Based on the compiled circuit with cycle restrictions, a snapshot of the operating quantum chip at a given cycle can be linked with the corresponding cycle in the compiled circuit. By using the cycle view, at most one type of error will happen to the physical qubit during the cycle time since at most one gate will happen to any physical qubit within a cycle. If we insert identity gates into cycles where physical qubits are idle with corresponding identity operation error rates, the error and gate count of the cycle-view-enabled compiled circuit could be simplified as the production of cycle count and physical qubit count. In addition, each physical qubit at any cycle could display two numbers associated with it identifying its cumulative success rate and current gate success rate.
Such error isolation provides a direct view of computation by viewing complicated compiled circuits as clear quantum chip snapshots and assigning time attributes to all the errors.
\subsection{Quantum ACE State and QVF}
Inspired by the classical architecturally correct execution (ACE) and architectural vulnerability factor (AVF) \cite{walcott2007dynamic, duan2009versatile} and error matter example seen in the last section, we develop the Quantum Vulnerability factor (QVF) and Cumulative Quantum Vulnerability (CQV), to locate the overestimate errors and perform error modeling include the error flowing across a CNOT gate, respectfully. The CQV is present in Sec. \ref{sect:CQV}.
The QVF uses the ACE state to represent the cases where the error does not matter. \textbf{The definition of the un-ACE state}: If an error happens to a physical qubit at any given cycle, the infected quantum states will not affect the output bits. In contrast, if an error on a physical qubit at a given cycle might impact the output, the qubit is considered to be in the ACE state. To be conservative, we only change qubits from the ACE to un-ACE state when any incident error that can occur on it at a given cycle cannot change the output.
To determine the ACE state, instead of giving a wrong state to a target qubit and simulating the output state like the fault injection method used in Qiskit and other work \cite{resch2020day}, we focus on determining whether there exists a path between the error and any measurement gate. We have given a detailed description of determining all the cases of the un-ACE state in Sec. \ref{sect:determinun-ace}
After using the cycle view to present errors and identify all un-ACE states out of the circuit, all the left are ACE states. The factor of ACE states over the sum of ACE and un-ACE states for a given time can be used to quantify the vulnerability of the quantum chip, which we name Quantum Vulnerability Factor (QVF). \textbf{The definition of QVF}: For a given compiled circuit arranged in cycle view, the QVF is equal to the average failure rate, which uses the calibrated error rate of the target physical qubit and its operation, of all ACE states over total physical qubits for all the cycles. It can be rewritten as the failure rate of all ACE states across the quantum chip at any given cycle. Identifying more un-ACE states will result in achieving a lower QVF and a better approximation of the real error effect during execution. More un-ACE states mean fewer ACE states and therefore a less vulnerable circuit. Additionally, understanding the ACE states distribution across the chip can guide architects to make trade-offs between cost and reliability when designing future quantum computers. One of the core advantages of QVF is that it can be known at compile time. We summarize acronyms used throughout this work in Table \ref{table:acronyms}.
\begin{scriptsize}
\begin{table}[ht]
\centering
\caption{Table of acronyms}
\begin{tabular}{|l|l|}
\hline
\textbf{Term} & \textbf{Explanation}\\
\hline
\hline
NOISE & Error rate for Operations or States \\
\hline
ACE & Architecturally Correction Execution\\
\hline
Un-ACE & Unnecessary for Architecturally \\
& Correction Execution\\
\hline
QVF & Quantum Vulnerability Factor \\
\hline
CQV & Cumulative Quantum Vulnerability\\
\hline
ESP & Estimated Probability of Success\\
\hline
SR & Success Rate of correct result \\
\hline
FR & Failure rate: one minus Success Rate\\
\hline
\end{tabular}
\label{table:acronyms}
\end{table}
\end{scriptsize}
\section{Determining Un-ACE States} \label{sect:determinun-ace}
\subsection{Virtual Qubit Terminology}
When identifying un-ACE states, we find that the two different categories, logical and ancilla qubits form the set of \textbf{virtual qubits} which equals the number of physical qubits.
Logical qubits are initially defined by a logical circuit and can be divided into outputting logical qubits and assisting logical qubits based on whether the qubit is measured in the end.
The virtual qubits added to fill the gap between logical and physical qubits are named ancilla qubits, based on the wildly used concept in the quantum compiling community \cite{Ibmquantum}. Ancilla qubits are further classified as used ancilla qubits and unused ancilla qubits based on whether they participate in computation. In this way, discussing all the un-ACE cases belonging to each type of virtual qubit will cover all the un-ACE cases.
\subsection{Ancilla Qubits in the Un-ACE States}
The compiler performs expands the number of virtual qubits by adding ancilla qubits as placeholders. For such reason, the ancilla qubits can stay idle for the entire circuit or be used to perform swaps to transfer logical qubits when there are connection limitations among targeted physical qubit pairs. However, when necessary, the compiler can use ancilla qubits to perform operations and later pass the prepared quantum states to the logical qubits. Therefore, ancilla qubits tend to have a smaller chance to meet a two-qubit gate, except for three CNOT gates grouped into a swap gate. Therefore, errors that happen to them have a lower chance to propagate toward output bits. Below are the two cases of ancilla qubit usage and the corresponding ACE analysis.
\textbf{Case 1: Unused Ancilla Qubits.}
To determine whether an ancilla qubit can be classified in this case, we must examine its entire lifetime. First, the ancilla qubit should stay in the same physical qubit for the entire circuit, which means that it is not involved in any swap operations. Next, the ancilla qubit must not encounter any operations except for barriers and idle (identity) operations as these do not affect the qubit state. The ancilla qubit in question stays in the un-ACE state in all cycles if meeting these requirements. It is simple to verify as no paths are available to such virtual qubits to spread any error to other virtual qubits.
\begin{figure*}[!t]
\centerline{\includegraphics[width=1\textwidth,height = 3cm]{Figures/cases.png}}
\caption{Un-ACE state cases. 1 and 2): Non-entangled un-ACE states cases. 3-5): Entangled qubits determination cases }
\label{fig5}
\end{figure*}
\textbf{Case 2: Used Ancilla Qubits.}
Ancilla qubits involved in operations other than identity or barrier operations are classified as used ancilla qubits. We choose to consider the used ancilla qubits the same as assisting logical qubits for un-ACE qubit state determination. Since they are both used to prepare quantum states for outputting logical qubits, their primary job is to assist final output. Therefore, errors on the used ancilla qubits can impact the output, and detailed ACE determination steps are presented in section \ref{sect:acelogicqubit}.
\subsection{Logical Qubits in the Un-ACE States} \label{sect:acelogicqubit}
Here we introduce cases and methods to identify un-ACE states for the two categories of logical qubits. The first group are logical qubits that are measured and directly written to at least one of the output bits, which we name as \textbf{outputting logical qubits}. On the contrary, the second group contains the logical qubit that does not have a measurement gate, which is named as \textbf{assisting logical qubits}. Most logical qubits belong to the first group. They tend to stay in the ACE state for most of their cycles, except for the cycles before being initialized and after measurement, where the states are not being used or transferred. The second group is used to prepare the desired quantum state and transfer it to other logical qubits to assist the final output. Therefore, for the qubits in the second group, they become un-ACE after the last time they transfer states. The previously discussed used ancilla qubits overlap with this second group.
However, there are cases where assisting logical qubits can stay in the ACE state even after passing the state. Such cases happen when the virtual qubits are entangled. The entangled correlation between virtual qubits will maintain until either one of the entangled qubits is measured or the entanglement is terminated. Therefore, we will first introduce the process of determining the un-ACE state in non-entangled cases. Then we will present a similar procedure for qubits in entangled cases.
\subsubsection{Non-Entangled Logical Qubit in Un-ACE State}
In this subsection, the targeted logical qubit will stay non-entangled with any other qubits for the entire circuit time. Here we will introduce how to identify the un-ACE state for both outputting and assisting logical qubits. We choose to inspect the logical qubits’ cycles from end to start (right to left), starting from the last cycle of the circuit. This way, a segment of un-ACE qubit states can be determined in one search until the condition changes.
\textbf{Case 1: Measurement logical qubit.}
This case contains outputting logical qubits that will be written to at least one of the output bits after measurement. The current examples shown in the paper will only have one measurement gate at the end of the circuit for a virtual qubit, and each of the measurements will be saved to one classical register. All the classical registers together form an output bit string. However, measuring on a specific virtual qubit multiple times is allowed as some algorithms require an intermediate result to control a controlled gate. Therefore, only the result will be used as a metric to determine the un-ACE status. Thus, some of the intermediate measurements will not act as an indicator. Being conservative, logical qubits in this case will largely remain in an ACE state. This is because the qubit is being measured, and an error in any earlier cycle can change the measured result. Therefore, it stays in the ACE state during the cycles between its initialization and measurement. There is still an opportunity to define un-ACE cycles for these logical measurement qubits, specifically where the quantum state it contains will not be needed. For example, the cycles before initialization and after the measurement with no remaining operation gates, which can be seen in Fig. \ref{fig5}.2 labeled as none used after measurement.
\textbf{Case 2: Trashed state.}
If a logical qubit enters the trashed state, then that logical qubit enters the un-ACE state for all cycles until the end of the circuit. The prerequisite for being in the trashed state is that the logical qubit does not entangle with other qubits and stays idle without any computational operations for the rest of the circuit. Such a case can happen on assisting logical qubits and used ancilla qubits, with an example shown in Fig. \ref{fig5}.1-2. It is easy to prove the un-ACE property for the target qubit since any error that occurs in the trashed state cannot influence other qubits and its information is not measured, therefore errors will not affect the output.
\textbf{Case 3: First-level non-spread logical qubit.}
When a logical qubit does not write its state to the output bits and there is still some single-qubit gate operating on it after the last CNOT gate, then the logical qubit enters the un-ACE state after the last CNOT gate. A simple example is shown in Fig. \ref{fig5}.1, which leaves $Q_2$ to become an un-ACE qubit starting from cycle 2. This can be verified by quantum states after the CNOT gate is no longer needed and transferred to others.
\textbf{Case 4: Second-level non-spread logical qubit.}
After determining the first-level non-spread logical qubits and marking their cycles into un-ACE states, there is a unique situation where both operands of a CNOT gate are followed by an un-ACE state. Therefore, the CNOT operation for both logical qubits can also be in an un-ACE state. Then a second-level non-spread logical qubit can start to search the path for both logical qubits until reaching its next CNOT, a simple example shown in Fig \ref{fig5}.2. The second-level non-spread determination process can be performed multiple times to reach out to all the possible non-spread states with the upper limit for the number of searched cycles equal to the total cycle count.
\subsubsection{Entangled Logical Qubit}
When qubits become entangled, if one of the qubits is measured, then the other qubit will also instantaneously collapse to a predefined state based on the structure of the entanglement. Therefore, before the entanglement is terminated either by measurement or inserting specific gates, an error in one logical qubit does not require a CNOT gate to propagate. Consequently, when determining the un-ACE qubit states, entanglement and CNOT gate paths should be considered simultaneously.
\textbf{Entangled logical qubits.}
In the circuit shown above in Fig. \ref{fig5}.3.a-b, the two logical qubits form a Bell state, which is one way to create entanglement \cite{nielsen2002quantum}. Since $Q_1$ has a measurement that belongs to outputting logical qubit group, both $Q_1$ and $Q_2$ will stay in ACE states for the entire circuit. Errors that occur at any cycle could be transferred to the output. In Fig. \ref{fig5}.3.b, we demonstrate one way to create an entanglement qubit group with more than two qubits, which also remains ACE for all cycles.
\textbf{Logical qubit operating with entanglement qubits.}
As shown in Figs. \ref{fig5}.4.a-b and \ref{fig5}.5.a, we have demonstrated cases where an assisting logical qubit could have un-ACE states when involved in CNOT operations with entangled qubits. Those three cases plus the case in \ref{fig5}.3.b illustrate all the possible CNOT gate interactions between an assisting logical qubit and entangled qubit pairs involving at least one outputting logical qubit. We can entangle the third qubit only after successfully creating a Bell state and targeting the third qubit with a CNOT operation. The entanglement created from more than two qubits forms the GHZ states \cite{greenberger1989bell} which follows the same relation as discussed.
In those sub-figures that fail to entangle the third qubit, we can identify un-ACE states for the third logical qubit.
For Fig \ref{fig5}.4.a, even if $Q_3$ serves as the target for the CNOT operation, its phase state will pass to $Q_2$, known as the phase-kickback phenomenon \cite{cleve1998quantum}. That is the reason we cannot list $Q_3$’s cycles before the CNOT gate as the un-ACE state.
As illustrated in Fig. \ref{fig5}.5.b, assisting logical qubits can also be classified into the secondary non-spread state. However, a careful examination of the additional logical qubit’s entanglement status should be considered.
\section{Computing QVF and CQV} \label{sect:CQV}
When computing the AVF for classical computers, Little's Law and Performances Model \cite{mukherjee2005soft} are the primary candidates to make a low-cost estimation of the ACE bits for a given hardware structure based on un-ACE instructions. However, we cannot perform such analysis when computing QVF because we do not have instructions (such as no-ops or mask instructions) where we can determine specific un-ACE bits. From the last section's analysis, we have not found any distinct instruction from which we can identify the un-ACE qubit states.
For computing the QVF for a given circuit on a specific chip, information on the entire final circuit is required since any missing cycles could lead to possible error propagation paths. Moreover, even though the compiler can merge subcircuits to generate a new total circuit for solving sophisticated quantum algorithms, the QVF for the subcircuits may not be used to compute the total QVF. There are cases where a logical qubit identified as an un-ACE qubit state in one subcircuit might be used again when combined with other subcircuits, changing the ACE status of that qubit. These later-used states will conflict with the prior defined un-ACE states. Therefore, it is recommended to perform the QVF determination procedure every time after creating a new final circuit.
The equation for computing the QVF of a given chip running a specific benchmark is simply the average QVF for all the ACE states on the chip for a given cycle. Such QVF at cycle $C_t$ can be written as:
\begin{equation}
QVF\_C_t = \frac{\sum_{ACE \ States \ at \ C_t } \ Calibrated\ Error \ Rate }{Device \ physical\ qubit\ count\ }\
\end{equation}
Based on this, the QVF for a quantum chip shows the average QVF for the entire circuit, which can be written as:
\begin{equation}
QVF = \frac{\sum_{cycles} \sum_{ACE \ States } \ Calibrated \ Error \ Rate }{Number\ of\ physical\ qubits \times total\ cycles}\
\end{equation}
It is common for small algorithms to use many ancilla qubits that stay in the unused case as the algorithm does not require many qubits. For these cases, we propose the Used QVF (UQVF) which only considers the virtual qubits used.
Since QVF is designed to present the average FR of quantum execution at the cycle level, we also propose the \textbf{Cumulative Quantum Vulnerability} (CQV) to present the final circuit's vulnerability by predicting the FR for the compiled circuit. To be clear, the CQV will not predict the correct result but the possibility of not receiving a correct result when repeating execution. In contrast, $1-CQV$ represents the estimated success rate driven by using the CQV model. To compute the CQV, we build a Monte Carlo simulator based on the QVF model to predict the value of $1-CQV$.
The simulator iterates through the compiled circuit for each physical qubit in every cycle. If the physical qubit in question is in the ACE state, the cumulative success rate of the corresponding virtual qubit will be updated by multiplying the running success rate with the gate success rate calculated in equation \ref{errortosr}. We perform a crosstalk error calibration based on \cite{murali2020software} and update the gate success rate of the victim by multiplying it with the calibrated crosstalk gate success rate.
If the targeting operation is a CNOT gate, then the error from its paired qubit will also be affecting the target qubit. For all such cases in a given compiled circuit, we propose a weight, ranging from 0 to 1, to represent the portion of cumulative errors, one minus the corresponding accumulative success rate, which could flow across the CNOT gate.
Then the accumulating success rate of such testing qubit will multiply with both the CNOT gate success rate and the Success Rate from its paired virtual qubit determined by only using the weight portion of the cumulative error. A heuristic method to provide a proper weight selection for a given compiled circuit is discussed in detail in Section \ref{determineweight}. In the end, the value of $1-CQV$ is determined by multiplying the final cumulative success rates of all the outputting logical qubits.
\section{Determining Weight } \label{determineweight}
Now that we have designed the error model, we will analyze the weight, one of the major coefficients in the CQV calculation. We ultimately identify a method to choose a proper weight for CQV to provide an accurate prediction for the real success rate at compile time.
\subsection{Is the Weight Trivial?}
Before identifying the value of the weight, we first demonstrate the prediction performance when the weight is set equal to zero or one representing no error or full error crossover the CNOT gate, respectively.
The compiled circuits of the experiments are generated from different combinations of compiler settings for all four benchmarks at both algorithm sizes. The CQV calculation is performed using the calibration error and execution results from IBMQ\_Montreal on Apr. 1st 2022.
As shown in Fig. \ref{figure: weightexist}, though the $1-CQV\_0$ results are closer to the real success rate than ESP, there are still noticeable differences between $1-CQV\_0$ and the real SR which means some errors are not being well represented.
Meanwhile, $1-CQV\_1$ makes predictions close to zero all the time and loses track of the real SR, meaning the errors are being overestimated.
The experiment results show that, for those benchmarks, using zero or one as the weight will lead to inaccurate predictions.
Since assigning the weight as zero or one for those benchmarks is not accurate, we would want to know the relationship between weight and prediction accuracy. As shown in Fig. \ref{figure: weightopt}, by plotting the absolute prediction error with all the weights with 1\% granularity, we can identify which weight achieves the lowest prediction error compared to all the other weights.
We also calculated the correlation between the weight value and its corresponding 1-CQV prediction, and all three correlations are approximate to -1, which is consistent with the relationship in Fig. \ref{figure: weightopt}.
\begin{figure}[!t]
\centerline{\includegraphics[width=1.0\linewidth, height=3cm]{Figures/weightexist.png}}
\caption{Average Success Rate Prediction Comparison Between Weight as 0 and 1 for benchmarks on IBMQ\_Montreal }
\label{figure: weightexist}
\end{figure}
\subsection{Determine a Proper Weight} \label{sec:weightcapture}
After knowing that the best weight should be a number between zero and one, how can we approximate it at compile time to assist the CQV calculation in predicting the real success rate?
To answer that question, we propose a heuristic method that first calibrates the best weights among different compiled circuits on a target machine and then uses a heuristic function generated from the calibration data to locate a proper weight.
The first step of the calibration process is to execute many compiled circuits generated from target benchmarks with various compile strategies on the target quantum machine. The two prerequisites for choosing a target benchmark are that the benchmark has only one correct output and that the correct output can be determined to calculate the real SR.
After getting the real SRs, the second step of the calibration process will provide the full spectrum of weights for the CQV calculation and determine the best weight for each compiled circuit using the same method as in Fig. \ref{figure: weightopt}. As shown in Fig. \ref{figure: weightdepth}, we have plotted the best weight against the depth of the compiled circuit. The result is consistent with our expectation -- the best weight will be very arbitrary when the depth is short, but as the depth of the compiled circuit increases, the best weight begins to converge and shows an overall decreasing trend.
Here, we give a short analysis of the best weight distribution. By checking the experiment data, we find out that, the SR of these benchmarks will drop to less than 10\% when their depth is greater than 100 and less than one percent when the depth is greater than 150 cycles, accordingly.
Furthermore, if the depth is greater than 150 cycles, the best weight tends to approach zero.
A possible explanation behind this would be that, as the depth increases, the quantum states tend to mix more, which possibly decreases on average the effect of errors.
Another possibility might be that errors may cancel to some degree.
Since the vulnerability model we use treats any operation error on an ACE qubit as a sign that will lead to incorrect output. However, in the real operating environment, later errors could adjust the previous error in the dynamic noise system, causing the original wrong quantum state to return closer to the correct state. At the same time, since noise would cause almost every possible output result to appear for certain trials, certain correct trials may be caused by noise as well. These two low-probability cases will stand out as the SR of a benchmark decrease, especially when the SR stays between 10\% and 0.1\%. Ultimately, with a very deep circuit, the best weight will decrease even more to accommodate for such small probability cases due to fewer correct trials.
The second step of the heuristic method is to use a heuristic function to determine the proper weight for a given compiled circuit. The heuristic function used in this paper is the logarithmic average of all the best weights within the target depth section according to the data in Fig. \ref{figure: weightdepth}.
To predict a compiled circuit on a target machine, one can calculate the CQV based on the weight captured by either repeating our full heuristic process or using the heuristic function to choose weight based on our calibrated data in Fig. \ref{figure: weightdepth}. Starting from now, our heuristic method will be the default way to find the weight used in CQV calculations.
Please notice that our heuristic method of choosing the weight is designed to accommodate any hardware upgrades.
\begin{figure}[!t]
\centerline{\includegraphics[width=1.0\linewidth, height=3cm]{Figures/weightopt.png}}
\caption{Capture the best weight for benchmarks with 1\% steps }
\label{figure: weightopt}
\end{figure}
\begin{figure}[!t]
\centerline{\includegraphics[width=1.0\linewidth, height =3.5cm]{Figures/weightcalib.png}}
\caption{Comparing the Best Weight among all experiments }
\label{figure: weightdepth}
\end{figure}
\section{Implementation}
We use Qiskit \cite{Qiskit}, an open-source framework for quantum computing, to evaluate our QVF determination process. We augment Qiskit version 0.34.2, which performs the QVF and CQV calculations based on the executable final circuit called the transpiled circuit or compiled circuit. We implement four functions to perform the simplified QVF calculation workflow as shown in Fig. \ref{fig8}.a.
\begin{figure}[!t]
\centerline{\includegraphics[width=0.4\textwidth,height=9cm]{Figures/fig8.png}}
\caption{Complete QVF and CQV process and circuit booking example}
\label{fig8}
\end{figure}
\begin{scriptsize}
\begin{table}[!h]
\centering
\caption{Benchmarks and Quantum Chips Description}
\label{tab:algorihtms}
\begin{tabular}{|l|l|}
\hline
\textbf{Item} & \textbf{Description}\\
\hline
\hline
Bernstein-Vazirani (BV) & contain assisting logical \\
& qubit\\
\hline
Deutsch-Jozsa (DJ) & contain assisting logical \\
& qubit\\
\hline
Quantum Fourier Transform & no assisting logical qubit\\
(QFT) & \\
\hline
Quantum Phase Estimation & no assisting logical qubit\\
(QPE) & \\
\hline
\hline
ibmq\_montreal & 27-qubits with Hexagon \\
\hline
ibmq\_toronto & 27-qubits with Hexagon \\
\hline
ibmq\_mumbai & 27-qubits with Hexagon \\
\hline
ibmq\_brooklyn & 65-qubits with Hexagon \\
\hline
\end{tabular}
\end{table}
\end{scriptsize}
\textbf{Entanglement Finder.}
Determining the entanglement relationships among the qubits in the circuit is critical for performing an accurate un-ACE state analysis and thus the later QVF calculation, though detecting entanglement in a quantum system is one of the active research fields with no simple solutions \mbox{\cite{dimic2018single,saggio2019experimental}}. For our design, there are two ways to identify the entangled qubits and their entangled cycles. The first method requires the software engineer who builds the logical qubits to input the designated entangled qubits and their entanglement durations. The second method will do a brute-force search to locate all Bell states and GHZ states from the logical circuit. For now, Bell states and GHZ states are the only two entangled states our entanglement finder can identify.
\textbf{Circuit Booking.}
At this step, the circuit booking function will receive a list of entangled qubits with the information of their entangled virtual qubit names and the start cycles and end cycles of their entanglement. The final compiled circuit is also sent to the circuit booking function, which builds a table based on this final circuit. The rows are the physical qubits, and the columns are the cycle index. Each cell contains some variables of the circuit. An example of the output table generated by the circuit booking function is shown in Fig. \ref{fig8}b, which presents all the variables for the cell with indices below them. The first variable is the name of the virtual qubit the current physical qubit holds at this cycle. The next variable contains the entanglement information of the virtual qubits. If not empty, the second variable presents the name of the virtual qubit it entangled with. If entangled among more than 2 qubits, to reduce conflict, all the others in the entanglement will write the first entanglement qubit in the second index. The third variable holds S if it belongs to an intermediate process of a swap gate. The fourth index represents O if the virtual qubit is an outputting logical qubit. The last variable represents the ACE status of the virtual qubit at such a cycle.
\textbf{Identifying Un-ACE Cases.}
This function takes the table generated by the circuit booking function and determines the un-ACE status from the final circuit which changes the table's ACE status variable. The determination will run over all the physical qubits starting from the last cycle to the first cycle. The first iteration will determine the virtual qubit's ACE status variable based on the status of the outputting logical qubit variable and entanglement information variable in this cycle. The inspection will go over all the cases listed in section 4 and perform the search until it exhausts all the second-level non-spread cases.
\textbf{Calculating QVF and CQV.}
This function will calculate the QVF and CQV based on the table's ACE variable modified by the last function. The equations and processes for generating those numbers are listed in section \ref{sect:CQV}.
Table \ref{tab:algorihtms} is the description of the backend and benchmark used in our evaluation. We run our experiments on state-of-the-art quantum machines with 27 and 65 qubits. The description column lists the information of the quantum chip, with qubit size and its connecting shape.
\section{Results}
\subsection{CQV Accuracy }
\textbf{Algorithm size within Quantum Volume}
In this section, we will present the CQV prediction performance for BV, DJ, and QPE benchmarks on the three 27-qubit machines IBMQ\_Montreal, IBMQ\_Toronto, and IBMQ\_Mumbai with weights calculated from the Apr. 1st error calibrations.
As shown in Figure \ref{fig：predict}, we present the 1-CQV prediction accuracy compared with ESP for QPE with varying compiled circuits on IBMQ\_Montreal on Apr. 5th. The different configurations are guidelines for the compiler to generate the final compiled circuit based on the given logical circuit and target device. The configurations used in Figure \ref{fig：predict} are generated for all the three algorithm sizes (4, 5, and 6 qubits) of the QPE benchmark. The first integer of the configuration lists the size of the algorithm. The second integer ranging from 0 to 2 indicates the optimization level used in the configuration, which implements different optimization steps. The next variable illustrates the layout method used, which corresponds to different qubit allocation strategies. The last variable describes the routing method used, which directs qubit movement. By following different configurations, Qiskit will generate different compiled circuits based on the same QPE logical circuit with the same number of qubits, which is used to generate many different inputs for our experiments showing the accuracy of the quantum SR estimators. The blue bar indicates the real success rate after executing each configuration. The grey and blue bar indicate our success rate prediction driven from $1-CQV$ and the ESP prediction, respectively. The closer to the blue bar, the more accurate the prediction. It is clear that $1-CQV$ is much closer to the real SR compared to the ESP. After shown to the right of Fig. \ref{fig：predict}, the average absolute error rate for ESP over all the configurations compared to ground truth SR is 37\%, while $1-CQV$ achieves an average error of 4\%. Such an error rate difference means that $1-CQV$ achieves a 94\% error reduction compared to the ESP, which also means the CQV calculation is adaptable to the variation errors in different calibration periods and provides excellent predictions.
By inspecting Fig. \ref{fig：predict}, SR drops clearly when the algorithm size increases, which follows the nature of the benchmark by employing significantly more two-qubit gates. When increasing algorithm size, not only does the number of two-qubit gates increase, but the number of swapping operations also increases.
As shown in Fig. \ref{figure:singlemachine}, we present the 1-CQV prediction for all the benchmarks on the single machine IBMQ\_Montreal at Apr. 8th. In the figure, we include experiments with an SR higher than 0.1\%. The relative prediction error is a metric produced by dividing the absolute prediction error by the SR. From the figure, it is easy to find that the 1-CQV prediction outperforms the ESP by achieving an average of 6 times and best at 20 times less relative prediction error rate. As shown in Fig. \ref{allmachine}, 1-CQV presents a stable and accurate average relative prediction error across all machines and benchmarks with data collected on Apr. 9th.
When calculating CQV, our simulator takes the two-qubit gates into account and considers the influence of both qubits' accumulated success rates. From these observations, we can say that the CQV has better error modeling than the ESP noise model.
\begin{figure*}[!t]
\centerline{\includegraphics[width=1.0\linewidth, height=4cm]{Figures/singlemachineSinglebench.png}}
\caption{Predicted and real success rates of QPE with IBMQ$\_$Montreal Quantum machine 4/5/22}
\label{fig：predict}
\end{figure*}
\textbf{Algorithm size beyond Quantum Volume}
From the results shown in Fig. \ref{fig：predict}, \ref{figure:singlemachine}, \ref{allmachine}, it is surprising that the $1-CQV$ has an accurate and stable prediction of the real success rate even when it falls in the 10\% to 0.1\% range, which was ignored in previous research. The major reason for the low success rate is that the size of the compiled circuits equals or exceeds the desired quantum volume of the target quantum machine. The simple explanation of quantum volume could be the product of virtual qubits and circuit depth. Such observation leads to a conclusion that the $1-CQV$ prediction is accurate across the full spectrum of algorithm sizes and success rates.
After making a full spectrum comparison among all the backends and benchmarks, from the observations of the results, we can say that the CQV has better error modeling than the ESP noise model.
\begin{figure}[!t]
\centerline{\includegraphics[width=1.0\linewidth,height=3cm]{Figures/singleMachineAllBench.png}}
\caption{Average relative predict error for all the benchmarks on single Quantum machine IBMQ\_Montreal}
\label{figure:singlemachine}
\end{figure}
\begin{figure}[!t]
\centerline{\includegraphics[width=1.0\linewidth, height=3cm]{Figures/allbenallmachine.png}}
\caption{Average relative predict error across all benchmarks and backends}
\label{allmachine}
\end{figure}
As shown in Fig. \ref{figure:singlemachine}, we have demonstrated the CQV prediction is stable across different algorithms on single machines, and the result shows that CQV has an overall stable and less relative error rate and performs much better when the algorithm reaching and passing the quantum volume.
\subsection{Scalability Analysis of CQV }
We performed the scalability analysis of the CQV in this section by reporting the execution time. We chose to test QFT with input sizes varying from 5 to 120 using the fake backend of IMBQ\_Washington with 127-qubits. Instead of focusing on the predicted result which should be close to zero when getting too large, we are interested in the execution time of different prediction methods. As shown in Table \ref {tab: scaling}, it is clear that the execution time of CQV scales linearly with the CX gate count, which would provide reliable scalability to help future quantum research.
\begin{scriptsize}
\begin{table}[!h]
\centering
\caption{Quantum Estimator Scalability Comparison(Second) }
\label{tab: scaling}
\begin{tabular}{|l|l|l|l|l|}
\hline
\textbf{Algorithm} &\textbf{CX count}& \textbf{ESP}& \textbf{Qiskit}& \textbf{CQV}\\
\hline
QFT\_5 &59 & 0.00299& 1.31142 & 0.00700\\
\hline
QFT\_10 & 408 &0.02004 & 1.86264 &0.03100 \\
\hline
QFT\_15 & 1383 &0.06101& N/A & 0.15903 \\
\hline
QFT\_20 & 2657 &0.09802 & N/A & 0.22605\\
\hline
QFT\_50 & 26408 &0.71115 & N/A & 5.97225\\
\hline
QFT\_100 & 157428 & 1.83842 & N/A &7.76095 \\
\hline
QFT\_120 & 208260 & 3.10630 & N/A & 20.3832 \\
\hline
\hline
\end{tabular}
\end{table}
\end{scriptsize}
\subsection{Weight Sensitivity Study}
\begin{figure*}[!t]
\centerline{\includegraphics[width=1.0\linewidth,height=3cm]{Figures/weightsens.png}}
\caption{Weight stable Study for different weight settings for a week on IBMQ\_Montreal}
\label{figure: weightstable}
\end{figure*}
The goal of this subsection is to understand whether the prediction accuracy of CQV will be subject to the volatile nature of current NISQ quantum machines' error rates and whether the best weight predictions for a certain day will remain optimal for the same experiment in the future. We performed two repeated experiments for more than a week and presented the CQV with various weight settings in Fig. \ref{figure: weightstable}. Both experiments are chosen because the best weights of the experiments found on April 1st are not equal to the weights determined by the heuristic method. For CQV\_HW and CQV\_BW, the experiments on different dates used the same weight determined on the Apr. 1st, respectively. As shown in Fig. \ref{figure: weightstable}, starting from Apr. 3rd, the CQV\_HW presents less prediction error for seven out of eight times compared to the CQV\_BW using the best weight of Apr. 1st. For QPE\_4, the CQV\_HW is more accurate than CQV\_BW four out of seven times when predicting the SR. The CQV\_0.1 is representing the weight as 10\% which is a simple and quick way to perform prediction without the need to make the weight calibration in advance. It is clear that the heuristic weight is stable enough to provide accurate prediction against noise variations over days and the best weight for one compile configuration might change over days.
After inspecting those experimental results, we found out that the machine was later reporting partially disabled functionality, such as certain CX gates having a 100\% error rate. Those dates’ results were ignored and left gaps in the date axis.
\section{Case Study }
\subsection{Case 1: Unseen Benchmark}
In the previous section, we demonstrated that - at compile-time, CQV could make an accurate prediction for algorithms on machines that have been previously used to determine the weight.
In this section, we will explore the predictive power of CQV to estimate real SR for unseen algorithms and then unseen machines.
We choose the QFT algorithm as the unseen algorithm. The QFT has been widely used as a building block for other algorithms and is also used to form part of the QPE algorithm. To decrease the association between the QPE and QFT, we generate the QFT benchmarks inversely compared to its counterpart in QPE. The weight for each machine was chosen based on the heuristic method illustrated in Sec. \ref{sec:weightcapture} using the other three algorithms, accordingly. As shown in Table \ref{tab: unseenbench}, we have presented the detailed CQV prediction performance of QFT on all three machines. The CQV result is surprisingly accurate with an average of 3\% absolute prediction error across all machines and algorithm sizes. The algorithm also achieves at least 6 times less average relative prediction error for CQV compared to the ESP. The 0.41 average relative prediction error of 1-CQV across all the machines and algorithm size supports that the CQV could be used as a stable and accurate SR estimator for unseen algorithms.
\subsection{Case 2: Unseen Machine}
\begin{figure}[!t]
\centerline{\includegraphics[width=1.0\linewidth, height=2.5cm]{Figures/unseenmachine.png}}
\caption{Average abs. predict error of BV on unseen IBMQ\_Brooklyn}
\label{unseenmachine}
\end{figure}
\begin{table*}[]
\caption{CQV Prediction for Unseen Algorithm QFT}
\label{tab: unseenbench}
\resizebox{\textwidth}{!}{
\begin{tabular}{|lll|ll|ll|ll|}
\hline
\multicolumn{3}{|l|}{2022\_04\_12} & \multicolumn{2}{l|}{Avg. Prediciotn} & \multicolumn{2}{l|}{Avg. Abs. Prediciotn Error} & \multicolumn{2}{l|}{Avg. Relative Prediction Error} \\ \hline
\multicolumn{1}{|l|}{Machine} & \multicolumn{1}{l|}{Benchmark} & RealSR & \multicolumn{1}{l|}{ESP} & 1-CQV & \multicolumn{1}{l|}{ESP} & 1-CQV & \multicolumn{1}{l|}{ESP} & 1-CQV \\ \hline
\multicolumn{1}{|l|}{\multirow{3}{10em}{ibmq\_montreal}} & \multicolumn{1}{l|}{QFT\_4} & 0.340843 & \multicolumn{1}{l|}{0.643122} & 0.298255 & \multicolumn{1}{l|}{0.302279} & 0.04258817 & \multicolumn{1}{l|}{0.886858} & 0.124949565 \\ \cline{2-9}
\multicolumn{1}{|l|}{} & \multicolumn{1}{l|}{QFT\_5} & 0.129277 & \multicolumn{1}{l|}{0.522711} & 0.122285 & \multicolumn{1}{l|}{0.393434} & 0.00699177 & \multicolumn{1}{l|}{3.043343} & 0.054083654 \\ \cline{2-9}
\multicolumn{1}{|l|}{} & \multicolumn{1}{l|}{QFT\_6} & 0.028583 & \multicolumn{1}{l|}{0.289672} & 0.006614 & \multicolumn{1}{l|}{0.261089} & 0.0219689 & \multicolumn{1}{l|}{9.134565} & 0.768612629 \\ \hline
\multicolumn{1}{|l|}{\multirow{3}{10em}{ibmq\_mumbai}} & \multicolumn{1}{l|}{QFT\_4} & 0.173173 & \multicolumn{1}{l|}{0.585265} & 0.241069 & \multicolumn{1}{l|}{0.412092} & 0.06789655 & \multicolumn{1}{l|}{2.379662} & 0.392074505 \\ \cline{2-9}
\multicolumn{1}{|l|}{} & \multicolumn{1}{l|}{QFT\_5} & 0.056254 & \multicolumn{1}{l|}{0.437791} & 0.081249 & \multicolumn{1}{l|}{0.381537} & 0.02499528 & \multicolumn{1}{l|}{6.782389} & 0.444328395 \\ \cline{2-9}
\multicolumn{1}{|l|}{} & \multicolumn{1}{l|}{QFT\_6} & 0.032936 & \multicolumn{1}{l|}{0.272236} & 0.009243 & \multicolumn{1}{l|}{0.2393} & 0.02369366 & \multicolumn{1}{l|}{7.265511} & 0.719376703 \\ \hline
\multicolumn{1}{|l|}{\multirow{3}{10em}{ibmq\_toronto}} & \multicolumn{1}{l|}{QFT\_4} & 0.212094 & \multicolumn{1}{l|}{0.571954} & 0.254039 & \multicolumn{1}{l|}{0.359861} & 0.04194556 & \multicolumn{1}{l|}{1.696706} & 0.197769098 \\ \cline{2-9}
\multicolumn{1}{|l|}{} & \multicolumn{1}{l|}{QFT\_5} & 0.070116 & \multicolumn{1}{l|}{0.426818} & 0.084775 & \multicolumn{1}{l|}{0.356702} & 0.01465873 & \multicolumn{1}{l|}{5.087331} & 0.209064487 \\ \cline{2-9}
\multicolumn{1}{|l|}{} & \multicolumn{1}{l|}{QFT\_6} & 0.032585 & \multicolumn{1}{l|}{0.250512} & 0.006355 & \multicolumn{1}{l|}{0.217926} & 0.02622974 & \multicolumn{1}{l|}{6.687909} & 0.804960147 \\ \hline
\end{tabular}}
\end{table*}
One of the challenges the current quantum researchers face is lack of the access to physical machines, which impacts the validation of their design. Therefore, performing the full heuristic method to determine the corresponding weight could not be done on a machine without access. In such a scenario, one can use the weight calibrations from other machines to perform the CQV prediction with a seen benchmark on the unseen machine at compile time. We have performed such an experiment and validated the CQV by comparing it with the real SR. As shown in \ref{unseenmachine}, we used the machine IBMQ\_Brooklyn to represent the unseen machine with weight calibrated from IBMQ\_Montreal. Due to the limited QV of IBMQ\_Brooklyn, BV is the only benchmark that is completed without reporting errors. The 1-CQV achieves an average 10\% absolute prediction across all the algorithm sizes. Though such prediction error is two times higher than the corresponding prediction of BV on IBMQ\_Montreal, it still outperforms the ESP by three times. The IBMQ\_Montreal has 27 qubits and IBMQ\_Brooklyn has 65 qubits, which could be the reason for less accurate prediction. If 10\% average prediction accuracy is suitable for the validation, then CQV would help to make predictions on the unseen machines.
\section{Related Work} \label{sect:relatedwork}
The early research on quantum computing was focused on designing quantum instruction set architecture \cite{fu2019eqasm} and quantum computer microarchitecture \cite{fu2017experimental,fu2018microarchitecture, murali2020architecting, murali2019full} for solving the constraints to apply logical circuits to machines.
After that, the temporal and spatial noise variation problems of superconducting quantum computers have been discovered and studied in \cite{tannu2019not, murali2019noise}, which also provide mapping and allocation-enhanced compilations based on such variations.
Currently, the focus of quantum computing is on optimizing the success rate by applying different compiler strategies, such as mitigating the effect of errors by enhancing the quantum instructions \cite{shi2019optimized, gokhale2020optimized}, decreasing measurement errors \cite{tannu2019mitigating,gokhale2020optimization}, mitigating crosstalk errors \cite{murali2020software, ding2020systematic}, combining the mappings to reduce dissimilar mistakes \cite{tannu2019ensemble}, and compiling with specific constraints \cite{li2019tackling,ding2020systematic}.
Although fault paths within quantum circuits have been studied to trace error propagation\mbox{\cite{janardan2016analytical}}, such a method requires the Clifford-gate error rates which are not available in the current quantum computers. Quantum fault injection \cite{resch2020day} is also being studied but fails to scale up due to the exponential growth requirement on computation power.
Though ESP mode created in \cite{nishio2020extracting} is widely used as a lightweight success rate predictor, such method sufferers from low prediction accuracy and high relative error rate while increasing algorithm size.
To the best of our knowledge, there is no prior work that provides a systematic approach to determining all ACE cases and adding such information to each gate on whether the error will result in fault output. In addition, based on the QVF model, this work provides a CQV metric to provide accurate FR.
ACE analysis \cite{mukherjee2003systematic, mukherjee2005soft}, architectural vulnerability factor (AVF) prediction \cite{walcott2007dynamic, duan2009versatile}, and program
vulnerability factor (PVF) \cite{sridharan2009eliminating} estimation have been well studied for conventional computer systems. This work is inspired by the prior research in conventional computers but performs a different ACE analysis procedure for quantum computers during the compilation stage instead of at runtime.
\section{Conclusion}
In conclusion, inspired by AVF our paper proposes QVF to quantify the average ACE states of a chip at any given cycle, which can present the circuit and noise behavior. Based on the QVF, we propose the CQV metrics to predict the failure rate of a given compiled circuit. By validating our design on state-of-the-art quantum machines with well-known benchmarks, in both seen and partially unseen systems, we have achieved an average of 6 times less relative prediction error rate compared to the state-of-the-art SR estimator ESP, which can assist and speed up the quantum computing research at both the hardware and software levels.
\bibliographystyle{IEEEtranS}
|
2,877,628,089,005 | arxiv | \section{\label{sec:level1}Introduction}
There is a lot of evidence for a significant multinucleon ejection contribution to the inclusive neutrino charge current (CC)
cross section in the 1~GeV energy region \cite{ccqe}.
On the experimental side, several recent nuclear target CCQE (CC quasi-elastic) cross section measurements
reported a large value for the
axial mass ($M_A$), in disagreement both with older deuterium target measurements \cite{bodek_MA} and also with electroproduction
arguments \cite{pion_MA}.
A possible explanation for the discrepancy is that some events interpreted as CCQE are in fact due to a
different dynamical mechanism
that typically leads to multinucleon emission \cite{martini1}. In fact, in the case of the MiniBooNE (MB)
CCQE measurement, nucleons in the final state were not analyzed at all.
The MB collaboration reported a high-statistics 2-dimensional muon inclusive cross section likely to contain a large multinucleon
ejection contribution, and thus provide a challenge for theoretical models \cite{MB_MA}.
On the theoretical side there are several two-body current computations that support the idea of a
significant multinucleon contribution to CC
neutrino scattering.
First was the Marteau model, based on the earlier works of Ericson and Delorme \cite{marteau}.
The model is formulated in the framework of the non-relativistic Fermi Gas in the Local Density Approximation (LDA) approach.
It includes the QE and $\Delta$ excitation
elementary interactions, Random Phase Approximation effects
(RPA are in medium polarization corrections important in the low four-momentum transfer
region \cite{luis}),
and $\Delta$ self-energy in the nuclear matter \cite{oset_salcedo}.
The model was later upgraded by
Martini, Ericson, Chanfray and Marteau (MEChM model) and compared to the MB data \cite{martini1}.
It predicts a large np-nh contribution to the CC cross section that
can explain the MB CCQE anomalous $M_A$ measurement (n particles and
n holes; in the Fermi Gas picture ejection of n particles means that there are also n holes left in the Fermi sea).
With the inclusion of relativistic corrections the MB 2D differential
cross section can be reproduced \cite{martini2}.
Another approach to the multinucleon ejection was proposed by Nieves, Ruiz Simo and Vicente Vacas \cite{nieves1}.
The model uses techniques developed for the analysis of the
inclusive electron-nucleus cross section in the kinematical region contaning both QE and $\Delta$ excitation peaks
\cite{gil}. The model is relativistic and incorporates similar nuclear effects as the MEChM model.
In the paper \cite{nieves2} a comparison to the MB 2D CCQE data gave a best fit axial mass value $1.08 \pm 0.03$~GeV.
The third microscopic computation of the multinucleon ejection in electron and neutrino interactions was discussed in a series of papers
\cite{amaro}. There are many similarities but also some differences between the three approaches and a
detailed discussion of them may be found in \cite{comparison}.
Recently, an effective approach to describe multinucleon contribution to the neutrino inclusive cross section was also
proposed.
The Transverse Enhancement model (TEM) \cite{bodek} is based on analysis of electron-carbon scattering data and
parameterizes the multinucleon ejection contribution to the muon inclusive cross section in terms of a
modification of the magnetic electromagnetic form factor. The model predicts that the two-body current contribution
is less important at
larger neutrino energies which can reduce a tension between the MB and NOMAD \cite{nomad} axial mass measurements.
In all the approaches the calculated quantity is the contribution to the muon inclusive differential cross section corresponding to
multinucleon ejection.
This is sufficient if the aim is to
reproduce the MB 2D
CCQE data. However, any attempt to separate the dynamical mechanism leading to the CCQE-like events (defined here as those with no pions in the
final state)
must be based on a careful investigation of the hadrons in the final state. This introduces a new ingredient
to the discussion, Final State
Interactions (FSI) effects. There are always CCQE-like events which are not really CCQE due to FSI effects.
The major contribution comes from the
pion absorption but there are also events with several nucleons in the final state originating from a
primary CCQE interaction with an energetic
outgoing proton.
A further clarification is necessary: in this paper the term {\it CCQE} will always refer to primary interactions. The basic picture is that
of a nucleus as composed from quasi-free nucleons.
In many neutrino oscillation experiments neutrino energy is reconstructed based on the muon information only.
It is clear that
the two-body current contribution can introduce a significant bias and
should be considered in the evaluation of systematic errors.
It is important to develop models of the two-body current contribution which can be implemented in neutrino Monte Carlo (MC) event
generators \cite{MC},
and for that one needs predictions for nucleons in the final state.
The aim of this paper is to propose a model to provide this information. The nucleon ejection part of the model is quite universal and can be
used together with any model of the double differential muon inclusive cross section.
The paper is organized in the following way. In Sect. 2 the multinucleon ejection model is introduced. In order
to make numerical predictions a model of the
muon 2D differential cross section is necessary. Two such models are described: a microscopic
model inspired by the Marteau approach and the effective TE model. Both models have been implemented in the
NuWro MC event generator \cite{nuwro}. In Sect. 3 some predictions from the model are shown and the focus is on demostration of the
relevance of the FSI effects. Section 4 contains a discussion of possible ways to measure the two-body contribution experimentally and
a few final remarks can be found in Sect. 5.
\section{The model}
The multinucleon ejection model proposed in this paper is quite general. It
needs as an input a muon inclusive differential cross section model. Two models
used in the numerical computations will be described in \ref{inclu_models}.
\subsection{Nucleon ejection model\label{model}}
One basic assumption of the model is that the energy and momentum are transfered to two (or three) nucleons simultaneously.
The procedure to generate nucleon final states is as follows:
\begin{itemize}
\item Two (or three) nucleons are selected from a Fermi sphere of radius $220$~MeV
(we assume the interaction occurs on carbon).
\item Th four momentum of the hadronic system is calculated by adding the four momenta of selected nucleons and the energy and momentum transfered
by the interacting neutrino.
\item A Lorentz boost to the hadronic center of mass system is performed.
\item Two (or three) nucleons are selected isotropically in the hadronic center of mass system
\item The boost back to the laboratory frame is performed.
\end{itemize}
Each event is weighted by the muon inclusive differential cross section. The
energy balance is done based on the assumption that inital state nucleons are in the potential well of the depth $V=E_f + 8$~MeV ($E_f$ is
the Fermi energy). In the numerical computations:
\begin{itemize}
\item The Fermi energy is subtracted from each initial state nucleon.
\item For each nucleon in the final state (in the LAB frame) its energy is reduced by $8$~MeV, adjusting the momentum
so that it remains on-shell.
\end{itemize}
The above procedure allows for a smooth distribution of the nucleon momenta in the final state.
When the model is implemented in a MC event generator, after the initial interaction nucleons propagating
through a nucleus are subject to
rescatterings. The treatment of energy balance must be consistent with the way the cascade model is designed.
The algorithm introduces some correlations between the initial state nucleons:
not all the initial
configurations give rise to a hadronic system with center of mass invariant mass larger than $2M$ (or $3M$ for
three nucleon ejection, where $M$ is nucleon's mass). In the numerical code initial state nucleon configurations are selected until
acceptable nucleon momenta are found.
Tests were also done with fully correlated pairs of nucleons in the
deuteron-like configuration with opposite three-momenta in the initial state and the results remained virtually unchanged. The most important
assumption of the model is that of an isotropic distribution of final state nucleons in the hadronic center of mass frame.
In the Monte Carlo implementation it is assumed that
in the CC neutrino reactions $\sim 80$\% of the final state nucleons are protons and only $\sim 20$\% are neutrons. For the
CC antineutrino reactions
the isospin composition of the final state is opposite with $\sim 80$\% of neutrons and $\sim 20$\% of protons.
The authors of the MEChM model claim that in the case of CC
neutrino interactions there should be much more proton-neutron than neutron-neutron pairs in the initial state
and as a consequence
more proton-proton than proton-neutron pairs in the final state \cite{martini1}.
We note the above procedure to get nucleons in the final state is similar to the one considered earlier
in \cite{oset_cascade}.
\subsection{Muon cross section models}
\label{inclu_models}
In this subsection two muon inclusive cross section models are presented.
\subsubsection{Microscopic model}
The model is based on the Marteau approach \cite{marteau_thesis}. It was developed about 10 years ago and discussed at the NuInt02 workshop
\cite{sobczyk_nuint02}. The cross section is given as:
\begin{equation}
\frac{d^2\sigma}{dq d\omega} = \frac{G_F\cos^2\theta_C q}{32\pi E_\nu^2} L_{\mu\nu}H^{\mu\nu},
\label{section}
\end{equation}
where $E_\nu$ is the neutrino energy, $q^\mu=(\omega, \vec q)$ is the four-momentum transfer, $q=|\vec q|$, $L_{\mu\nu}$ is the leptonic tensor
\begin{equation}
L_{\mu\nu} = 8\left( k'_\mu k_\nu + k_\mu k'_\nu -g_{\mu\nu}k\cdot k' \pm i\epsilon_{\mu\nu\alpha\beta}k'^\alpha k^\beta \right),
\label{leptonic}
\end{equation}
$k$ and $k'$ are neutrino and charged lepton four-momenta, $\pm$ signs refer to the neutrino and antineutrino cases.
The hadronic current is:
\begin{equation}
H^{\mu\nu} = H^{\mu\nu}_{NN} + H^{\mu\nu}_{N\Delta} + H^{\mu\nu}_{\Delta N} + H^{\mu\nu}_{\Delta\Delta},
\end{equation}
where $H^{\mu\nu}_{XY}$, $X,Y\in (N,\Delta)$ are expressed in terms of expectation values of spin operators acting in the
nucleus Hilbert space, so called nuclear response functions. The model uses quasi elastic form factors to describe the
$\Delta$ excitation and consistency of the approach was investigated in \cite{marteau_delta}.
More details about the model can be found in \cite{sobczyk_nuint02, marteau}.
The model includes the RPA and $\Delta$ in-medium self-energy effects. Recently, the model was upgraded and it
now contains LDA effects. The elementary 2p-2h responses were added following the procedures described in \cite{marteau_thesis}.
The model in this paper -- like the MEChM model -- does not include RPA corrections to the np-nh
contribution (exceptions are terms describing
pionless $\Delta$ decays). The elementary response functions used in the microscopic model
are similar to those used in the MEChM paper, as shown in Fig. 3 in \cite{martini1}.
In Appendix A we collect basic formulae which allow for a comparison with other approaches.
\begin{figure}
\includegraphics[width = \columnwidth]{total_npnh.eps
\caption{\label{fig:after_rpa} (Color online) Predictions for the np-nh contributions to the CC scattering on carbon from two models
discussed in this paper. For comparison the QE cross section is also shown.}
\end{figure}
\begin{figure}
\includegraphics[width = \columnwidth]{diff_750.eps
\caption{\label{fig:diff_700} (Color online) Predictions for the two-body current contribution to the
CC carbon target differential cross section versus energy transfer from the two models
discussed in this paper.}
\end{figure}
In the microscopic model there is the possibility of two- and three- nucleon ejection.
The three nucleon ejection contribution comes from
pionless $\Delta$ decays i.e. from reactions $N\Delta \rightarrow NN$ and $NN\Delta \rightarrow NNN$ with $N$ standing for a nucleon
\cite{oset_salcedo}.
\subsubsection{Transverse Enhancement Model}
A new approach to describe CCQE scattering on nuclear targets is proposed in~\cite{bodek}. The model
is easy to implement in MC event generators.
It is sufficient to modify the vector magnetic form factor keeping all other ingredients of the CCQE model as in the free nucleon target case.
The authors of \cite{bodek} proposed for the carbon target a universal transverse enhancement function of $Q^2$.
For low $Q^2$ its form is determined by scaling arguments while for large $Q^2$ ($>0.5$~GeV$^2$) it is obtained as a
fit to
inclusive electron cross section data from the JUPITER experiment.
The prescription to include transverse enhancement contribution in the numerical computations amounts to the replacement:
\begin{equation}
G_M^{p,n}(Q^2)\rightarrow \tilde G_M^{p,n}(Q^2)=\sqrt{ 1 + AQ^2 \exp (-\frac{Q^2}{B}) } G_M^{p,n}(Q^2)
\end{equation}
where $G_M^{p,n}(Q^2)$ are electromagnetic form-factors, $A= 6$ GeV$^{-2}$ and $B=0.34$~GeV$^{2}$.
The most interesting feature of the TEM model is that it offers a possible explanation to the
apparent contradiction between low (MB) and high (NOMAD)
neutrino energy $M_A$ measurements: for energies up to $\sim 700$~MeV the TEM predicts the
cross section to be similar to CCQE with $M_A=1.3$~GeV.
For higher neutrino energies the TEM cross section becomes significantly smaller and at $E_\nu \sim 5$~GeV it corresponds to CCQE with
$M_A\sim 1.15$~GeV.
As the TEM prediction for the two-body current contribution one takes the difference between the cross sections calculated with the modified
and default magnetic form factors:
\begin{equation}
\frac{d^2\sigma^{TEM}}{dqd\omega} = \frac{d^2\sigma^{CCQE}}{dqd\omega} (\tilde G_M^{p,n}) - \frac{d^2\sigma^{CCQE}}{dqd\omega} (G_M^{p,n}).
\end{equation}
In the TEM only two-nucleon ejection takes place.
\subsection{Muon CC models comparison}
Fig. \ref{fig:after_rpa} shows the predictions for the np-nh contribution from both models.
They are quite similar in size and only for neutrino energies above $700$~MeV does
the microscopic model
predict a larger cross section. For comparison, the predictions for the QE cross section (one outgoing nucleon at the interaction point)
are shown as well.
In the presented models, the multinucleon ejection contribution amounts to about 25\% of the QE cross section.
Fig. \ref{fig:diff_700} shows the differential cross section for the multinucleon ejection contribution
as a function of the energy transfer. It should be stressed that so far no assumptions about the final state nucleons
were necessary. The model predictions are quite different. In the case of the microscopic model there is
a lot of structure
coming from various ingredients of the model: contributions from pionless $\Delta$ decays, elementary $NN$ and $N\Delta$ 2p-2h
responses. Similar structure can be seen also in Fig. 2 in \cite{martini1}.
There are two versions of the MEChM model which differ by elementary 2p-2h response functions.
Following the procedures described in \cite{martini1}, by making a fit to the data contained in \cite{aem}
new responses as functions of the variable
$x \equiv Q^2/2M\omega = (q^2-\omega^2)/2M\omega$ (where $M$ is the nucleon mass)
were obtained.
However, within the microscopic model of this paper the new responses lead to a too rapidly increasing np-nh cross section
as a function of neutrino energy and in the rest of this paper the original Marteau responses are used \cite{marteau_thesis}.
In the case of the TE model a sharp fall in the differential cross section at $\sim 460$~MeV comes from the way in which the model was implemented.
We followed the original paper and assumed the target nucleon to be at rest. Pauli blocking effects are introduced by means of a $Q^2$
dependent suppression function. An alternative implementation is to use the Fermi Gas model, but it is not obvious which approach
leads to better agreement with the MB 2D CCQE data \cite{sobczyk_tem}.
\begin{figure}
\includegraphics[width = \columnwidth]{npnh_vert_act_750.eps
\caption{\label{fig:kin_800} (Color online) Total kinetic energy of the final state protons coming from two-body current interactions
as modeled by the microscopic model
implemented in NuWro. Predictions from the model with and without nucleon rescatterings are compared. The neutrino energy is $750$~MeV. }
\end{figure}
\begin{figure}
\includegraphics[width = \columnwidth]{bodek_vert_act_750.eps
\caption{\label{fig:kin2_800} (Color online) The same as in Fig. \ref{fig:kin_800} but for the TE model. }
\end{figure}
\section{Results}
\begin{figure}
\includegraphics[width = \columnwidth]{npnh_maximal_proton_momentum_750.eps
\caption{\label{fig:max_800} (Color online) Momentum of most energetic final state protons coming from two-body current interactions
as modeled by the microscopic model
implemented in NuWro. Predictions from the model with and without nucleon rescatterings are compared. The neutrino energy is $750$~MeV.
}
\end{figure}
\begin{figure}
\includegraphics[width = \columnwidth]{bodek_maximal_proton_momentum_750.eps
\caption{\label{fig:max2_800} (Color online) The same as in Fig. \ref{fig:max_800} but for the TE model.
}
\end{figure}
\begin{figure}
\includegraphics[width = \columnwidth]{npnh_second_proton_momentum_750_cut100MeV.eps
\caption{\label{fig:second_800} (Color online) Momentum of second most
energetic final state protons coming from two-body current interactions
as modeled by the microscopic model
implemented in NuWro. Predictions from the model with and without nucleon rescatterings are compared. The
neutrino energy is $750$~MeV.
}
\end{figure}
\begin{figure}
\includegraphics[width = \columnwidth]{bodek_second_proton_momentum_750_cut100MeV.eps
\caption{\label{fig:second2_800} (Color online) The same as in Fig. \ref{fig:second_800} but for the TE model.
}
\end{figure}
\begin{figure}
\includegraphics[width = \columnwidth]{npnp_fsi.eps
\caption{\label{D2_bodek} Two dimensional distribution of proton momenta: highest energy with respect to the
second highest energy protons.
Muon neutrino energy is $750$~MeV, simulations are done for the microscopic model.
}
\end{figure}
\begin{figure}
\includegraphics[width = \columnwidth]{bodeck_fsi.eps
\caption{\label{D2_npnh}
The same as in Fig. \ref{D2_npnh} but for the TE model.
}
\end{figure}
Figures \ref{fig:kin_800} -- \ref{fig:second2_800} show predictions from both models
implemented in the NuWro MC event generator. All of them are for protons only
because in the experimental analysis neutrons are usually not detected. In Section III we focus on the impact of FSI effects
on the results.
\subsection{Total kinetic energy}
Figs \ref{fig:kin_800} and \ref{fig:kin2_800} show the kinetic energy of all the protons in the final state.
Due to rescatterings protons lose a fraction
of their kinetic energy. No protons in the final state is a possible outcome in the NuWro FSI model.
\subsection{Nucleon momenta}
Figs \ref{fig:max_800} -- \ref{fig:second2_800}
show the distributions of the momenta of highest energy and second highest energy protons in the final state.
Again, because of reinteractions protons becomes less energetic.
On average their momenta are
reduced by $\sim 100$~MeV/c.
An interesting feature of the distribution of the second most energetic protons is that the rescatterings make
the fraction of events in which there is at most one
proton in the final state smaller.
The distributions seen in Figs \ref{fig:second_800} and \ref{fig:second2_800} are important
because they allow for an estimate of the probability that there is a pair of reconstructed protons in the final state.
It is interesting that at the neutrino energy of only $750$~MeV the second nucleon can be quite energetic.
Figs \ref{D2_npnh} -- \ref{D2_bodek} show two-dimensional distributions: highest energy versus second highest energy protons for both models.
Because the range of energy transfers for the TEM is smaller, the phase space covered by pairs of protons
is also reduced.
\subsection{Energy reconstruction}
\begin{figure}
\includegraphics[width = \columnwidth]{energy_reconstr_750.eps
\caption{\label{fig:reconstr_800} (Color online) Reconstructed neutrino energy for two
body current interactions of $E_\nu = 750$~MeV muon neutrino.}
\end{figure}
Fig. \ref{fig:reconstr_800} shows the
distribution of reconstructed energy $E_{rec}^{QE}$ obtained within the microscopic model.
The true neutrino energy
is always $750$~MeV. $E_{rec}^{QE}$ is defined based on the information infered from the final muon only, assuming that the interaction
is CCQE and the target nucleon is at rest:
\begin{equation}
E_{rec}^{QE} = \frac{2ME_\mu-m^2}{2(M-E_\mu + \sqrt{E_\mu^2 -m^2}\cos\theta_\mu)}
\end{equation}
where $E_\mu$ is muon energy, $m$ is muon mass, and $\theta_\mu$ is an angle between neutrino and muon three momenta.
The presence of the two-body current contribution introduces an important bias and
it can be seen that on average the true neutrino energy is larger by $\sim 150-200$~MeV than $E_{rec}^{QE}$.
The TEM uses the QE kinematics and it is not suitable to study the bias in energy reconstruction.
\section{Discussion}
\subsection{A role of energy transfer}
The predictions from both models implemented in NuWro are quite similar in shape; the only major difference
is that in the microscopic model protons are on average more energetic.
This can be understood as a consequence of different shapes for the muon differential cross
sections in energy transfer. In fact, this energy (a fraction of) is seen as the kinetic energy of ejected protons.
From the point of view of the multinucleon ejection model,
the most important features of muon CC two-body current contribution to the differential cross section are:
(i) the integrated size, and (ii)
the distribution of events in the energy transfer.
\subsection{How to see two-body current events}
In the $1$~GeV energy region (e.g. as it is in the T2K experiment) two-body current events are almost all CCQE-like.
The probability to produce pions due to FSI effects
is negligible. It follows that in the experimental analysis it is important to
develop an effective veto on pions.
There are two promising observables which contain an information about the two-body current contribution.
The first one is pairs of protons in the final state
both with momenta above some threshold value. The second one is the integrated kinetic energy of charged hadrons in the final state.
In fact, if the typical neutrinos energies are about or below $1$~GeV,
most of the protons will most likely remain not reconstructed and they will only contribute to the vertex activity.
All the computations discussed below are done for the $750$~MeV muon neutrinos. They can give an idea about the size of effect
to be expected e.g. in the T2K experiment. The computations are done for the models as implemented in NuWro with
all the FSI effects being included.
\subsubsection{Proton pairs}
Tables \ref{tabelka_tem} and \ref{tabelka_npnh} show the predicted number of proton pairs with both
momenta above various threshold values. The signal is defined as exactly two protons
above: $300$, $400$ and $500$~MeV/c. It is assumed that in the samples of events there are no $\pi^0$ ($\pi^0$ is either
reconstructed or at least
one energetic photon resulting from its decay is detected).
As for $\pi^\pm$ it is assumed that they can be identified if their momenta are above either $0$ (i.e. all of them are
detected and the sample of events contains no $\pi^\pm$), or $200$~MeV/c.
In each situation the numbers of signal and background events are shown separately. The background events are those with exactly two protons
satisfying the selection criteria but originating from other dynamical mechanisms.
\begin{table}
\begin{tabular}{c||l|l|l|l|}
$\pi^\pm$ cut [$\frac{MeV}{c}$]$\downarrow$& proton cut [$\frac{MeV}{c}$] $\rightarrow$ & 300 & 400 & 500 \\
\hline\hline
0 & signal & 7185 & 4201 & 1805 \\
& background & 13774 & 7928 & 2311 \\
\hline
200 & signal & 7231 & 4201 & 1805 \\
& background & 16158 & 8577 & 2388 \\
\hline
\end{tabular}
\caption{Predicted number of proton pairs with both momenta above various threshold values and two threshold values of the $\pi^\pm$
momentum.
Simulations were done for $750$MeV muon neutrinos. The number of generated events is $2.5\cdot 10^5$.
The microscopic model in a NuWro implementation was used to
produce signal events.
}
\label{tabelka_npnh}
\end{table}
\begin{table}
\begin{tabular}{c||l|l|l|l|}
$\pi^\pm$ cut [$\frac{MeV}{c}$]$\downarrow$& proton cut [$\frac{MeV}{c}$] $\rightarrow$ & 300 & 400 & 500 \\
\hline\hline
0 & signal & 5457 & 2271 & 651 \\
& background & 13780 & 7961 & 2267 \\
\hline
200 & signal & 5465 & 2271 & 651 \\
& background & 16112 & 8691 & 2349 \\
\hline
\end{tabular}
\caption{
The same as in TABLE \ref{tabelka_tem} but for the NuWro implementation of the TE model.
}
\label{tabelka_tem}
\end{table}
The total number of generated CC events is $2.5\cdot 10^5$. In the case of microscopic model for multinucleon ejection
the sample of events contains: $59.4$\% QE and $15.2$\% two-body current events. Remaining are
mostly single pion production events. For the TEM the composition of the sample of events is similar.
The background for the two proton signal comes mainly from the real pion production and its subsequent
absorption. The NuWro implementation of the absorption contains only two-body mechanism, and the size of the background can be
overestimated
\cite{lads}.
There are noticable differences in the predictions for the number of the signal events from the two analyzed models,
and they show an interesting pattern.
In the case of pairs of protons both with momenta above $500$~MeV/c the difference is by a factor of 3 while for protons with
momenta above $400$~MeV/c and $300$~MeV/c the differences are only $\sim 80$\% and $\sim 30$\%.
For larger proton momenta
the pattern can be explained by the fact that the probability of a large energy transfer is in the case of microscopic model much bigger.
For lower proton momenta the most important becomes the fact that overall cross
sections for multinucleon ejection are in both models
quite similar.
The numbers of background events are slightly different in both cases. They were generated in two separate simulations and also
the cross sections for multinucleon ejection are slightly different.
The numbers shown in Tables \ref{tabelka_tem} and \ref{tabelka_npnh}
suggest that in order to separate the signal from the background it is enough to have the precision in the
background estimation on the level of $50$\%, which seems to be a realistic goal to achieve.
We conclude that counting of proton pairs can provide an experimental proof of existence of the multinucleon ejection contribution
and also a tool to discriminate between various models. However, it is necessary to have large samples of event because the probability to
have a signal pair of protons with both momenta above $500$~MeV/c is only $0.26-0.72$\%.
\subsubsection{Integrated energy of charged hadrons}
\begin{figure}
\includegraphics[width = \columnwidth]{npnh_log_energy_deposit_contributions_cut_200_750.eps
\caption{\label{fig:deposit} (Color online) The distribution of hadronic kinetic energy for the events with cuts as described in the text.
Simulations were done with $750$~MeV muon neutrinos using NuWro implementation of the microscopic model.
The number of generated events is $2.5\cdot 10^5$. Contributions from CCQE, RES+DIS and np-nh events are also shown separately.
}
\end{figure}
\begin{figure}
\includegraphics[width = \columnwidth]{bodek_log_energy_deposit_contributions_cut_200_750.eps
\caption{\label{fig:deposit2} (Color online) The same as in Fig. \ref{fig:deposit} but for the TE model.
}
\end{figure}
\begin{figure}
\includegraphics[width = \columnwidth]{npnh_log_ratio_energy_deposit_contributions_cut_200_750.eps
\caption{\label{fig:ratio} (Color online) The distribution of hadronic kinetic energy normalized by the muon energy (see Eq. \ref{hadrons})
for the events with cuts as described in the text. Simulations
were done with $750$~MeV muon neutrinos using NuWro implementation of the microscopic model.
The number of generated events is $2.5\cdot 10^5$. Contributions from CCQE, RES+DIS and np-nh events are also shown separately.}
\end{figure}
\begin{figure}
\includegraphics[width = \columnwidth]{bodek_log_ratio_energy_deposit_contributions_cut_200_750.eps
\caption{\label{fig:ratio2} (Color online) The same as in Fig. \ref{fig:ratio} but for the TE model.
}
\end{figure}
For the overall charged hadron kinetic energy two observables can be considered:
\begin{equation}
\sum_j T_j, \qquad{\rm and}\qquad \frac{\sum_j T_j}{E_\mu}.
\label{hadrons}
\end{equation}
where $j$ runs over charged hadrons in the final state and $E_\mu$ is the muon energy.
All the protons contribute to the observables defined above, independent on how large their momenta are.
Also in this case one must
introduce some assumptions about
the pions. In the plots shown in Figs \ref{fig:deposit} -- \ref{fig:ratio2} in the signal events there are no $\pi^0$ and no $\pi^\pm$
with momenta above $200$~MeV/c. In the figures contributions from CCQE, RES+DIS and two-body current mechanism are also shown
separately. In the neutrino MC nomenclature
RES refers to the resonance region and DIS to more inelastic events but for the purpose of this study they are combined together.
All three contributions are different in shape. This is clearly seen in the case of RES+DIS. The shape
of the multinucleon ejection contribution depend on the underlying muon cross section model and on the FSI effects.
It is important to remember that in the
distributions shown in Figs \ref{fig:deposit} and \ref{fig:ratio} also reconstructed protons are included.
In the actual experimental analysis the directly observable quantity is the reconstructed energy. However, using neutrino MC generator one
can make predictions for the shape of the distribution of the reconstructed energy and compare to the data. It can happen that both measured
and predicted shapes do not match and that the inclusion of the contribution from the two-body current mechanism
one gets much better data/MC agreement.
\section{Final remarks}
The aim of this paper was to discuss the phenomenological consequences of the two-body current contribution to
the neutrino CC muon inclusive cross section. In order to estimate the size of effect for the hadron observables
the multinucleon ejection model
was proposed which can be easily implemented in neutrino MC event generators. Using the model
some
observables which contain an information about two-body current dynamics were discussed.
It is important that the multinucleon ejection model proposed in the Section \ref{model} can be used together with any model
providing predictions for the two-body contribution to the CC inclusive neutrino-nucleus cross section.
Identification of the two-body current experimental signal is possible if based on a careful data/MC comparison. For that
it is essential to have a reliable model of FSI effects. It is also central to have a good description of the
nucleon final states arising after pion absorption.
One should also keep in mind that on the theoretical side there is a lot of uncertainty as to how large the two-body current
contribution is expected to be. A careful comparison of the predictions from the models described in the Introduction for
the MiniBooNE flux averaged contribution in the region $\cos\theta_\mu\in (0.8, 0.9)$ (see Fig. 6 in \cite{martini2} , Fig. 3 in
\cite{nieves2} and Fig. 4 in \cite{barbaro}) reveals that they differ by a factor of two.
All the microscopic models discussed in this paper are based on the Local Fermi Gas model and the arguments of the paper
\cite{carlson} indicate that in order to evaluate the two-body current contribution it is necessary to use a more realistic
model of the nucleus ground state. For the multinucleon ejection model this means that one should include
more correlations between initial state nucleons.
During the work on the upgraded version of this paper an article on the similar subject was
published by Lalakulich et al \cite{olga}.
|
2,877,628,089,006 | arxiv | \section{Introduction}
Throughout recorded history, human population has always been haunted by the emergence and re-emergence of infectious diseases. Several lives have been lost due to lack of knowledge on the dynamical behavior of epidemic outbreaks of contagious diseases and measures to confront them \cite{Dobson1996}, \cite{Oldstone1998}. For decades, scientists have toiled to understand the transmission characteristics of such diseases so as to devise control strategies to prevent further spread of infection. The field of science that studies such epidemic diseases and in particular, the factors that influence the incidence, distribution, and control of infectious diseases in human populations is called \emph{epidemiology}. In this regard, mathematical modeling has proven to serve useful in analyzing, predicting, evaluating, detecting, and implementing efficient control programs. Such analytical models accompanied by computer simulations serve as experimental tools for building, testing, and assessing theories and understanding the relationship between various parametric values involved.
Practical use of epidemic models depends on how closely they realize actual biological diseases in real world. To keep the models simple and tractable, many assumptions and relaxations are taken into consideration at each level of the process. However, even such simplified models often pose significant questions regarding the underlying mechanisms of infection spread and possible control approaches. Hence, adopting the apt epidemic model for prediction of real phenomenon is of great importance.
Models that are useful in the study of infectious diseases at the population scale can be broadly classified into two types: \emph{deterministic} and \emph{stochastic}. Early models that were developed to study specific diseases such as tuberculosis, measles, and rubella were deterministic in nature. In deterministic models, the large population is divided into smaller groups called \emph{compartments} (or \emph{classes}) where each group represents a specific stage of the epidemic. Such models, often formulated in terms of a system of differential equations (in continuous time) or difference equations (in discrete time), attempt to explain what happens on the average at the population scale. A solution of a deterministic model is a function of time or space and is generally uniquely dependent on the initial data. On the other hand, a stochastic model is formulated in terms of a stochastic process which, in turn, is a set of random variables, $X(t;\omega)\equiv X(t)$, defined as $\{X(t;\omega)|t\in T and~ \omega\in \Omega\}$ where $T$ and $\Omega$ represent time and a common sample space, respectively. The solution of a stochastic model is a probability distribution for each of the random variables. Such models capture the variability inherent due to demographic and environment variability and are useful under small population sizes. More specifically, they allow follow-up of each individual in the population on a chance basis \cite{Trottier2001}, \cite{Britton2010}. Discrete-time Markov chain (DTMC), continuous-time Markov chain (CTMC), and stochastic differential equation (SDE) models are three types of stochastic modeling processes which have been deeply covered in \cite{Allen2008}. Figure~\ref{fig1_classification} shows the different classes under which epidemic models have been studied in the literature. The connecting blue lines in the figure highlight the main scope of this report.
\begin{figure}[!t]
\centering
\includegraphics[width=5.0in]{Figures/Fig1_n.eps}
\caption{Classification of various classes of epidemic models.}
\label{fig1_classification}
\end{figure}
Needless to say, both deterministic and stochastic epidemic models have their own applications. Deterministic models are used to address questions such as: \textit{what fraction of individuals would be infected in an epidemic outbreak?}, \textit{what conditions should be satisfied to prevent and control an epidemic?}, \textit{what happens if individuals are mixed non-homogeneously?}, and so on \cite{Brauer2008}. While such models are preferable in studying a large population, stochastic epidemic models are useful for a small community and answer questions such as: \textit{how long is the disease likely to persist?}, \textit{what is the probability of a major outbreak?}, and the like \cite{Allen2008}. Hence, stochastic epidemic models are generalized forms of simple deterministic counterparts. However, unlike deterministic models, stochastic models can be laborious to set up and may need many simulation runs to yield useful predictions. They can become mathematically very complex and result in misperception of the dynamics \cite{Trottier2001}. To this end, we focus on some widely-used deterministic models which are relatively easier to conceive, set up, and implement using various computer softwares at disposal.
Deterministic epidemiology is believed to have started in the early twentieth century \cite{Hethcote2000}. In 1906, Hamer was the first to assume that the incidence (number of new cases per unit time) is proportional to the product of the number of susceptible and infective individuals in his model for measles epidemics \cite{Hamer1906}. The exponential growth in mathematical epidemiology was boosted by the acclaimed work of Kermack and McKendrick which was published in 1927 \cite{Kermack1927}. This paper laid out a foundation for modeling infections where all members of the population are assumed to be initially equally susceptible to the disease and confer complete immunity only after recovery. After decades of neglect, the Kermack-McKendrick model was brought back to prominence by Anderson \emph{et al.} in 1979 \cite{Anderson1979}. Since then several models have been developed addressing aspects such as passive and disease-acquired immunity, vaccination, quarantine, vertical transmission, disease vectors, age structure, social and sexual mixing groups, as well as chemotherapy \cite{Brauer2001, Chavez1989,Hethcote1994,Keeling2011}. Improved models have also been designed for diseases such as measles, chickenpox, smallpox, whooping cough, malaria, rabies, diphtheria, filariasis, herpes, syphilis, and HIV/AIDS \cite{Keeling2011}, \cite{Vynnycky2010}.
The main objective of this article is to help the reader gain an insight on the basics of deterministic compartmental modeling through implementation. In this work, we formulate some well-known models and derive their steady-state solutions. Since the models under study are non-linear in nature, we investigate their qualitative behavior near their corresponding equilibria using linearization method \cite{Khalil2001}. All models discussed in this paper have been implemented using \emph{Wolfram Mathematica} \cite{mathem}, the codes of which are freely available.
The remainder of this article is structured as follows. Section II provides a demographic classification of deterministic models along with notations and assumptions that will be used throughout the paper. In Section III, we present the classical epidemic model known as the \emph{susceptible}-\emph{infected}-\emph{recovered} (\emph{SIR}) model which forms the basis for the extended models that follow in Sections IV to VII, accompanied by their implementation results. In Section VIII, we present some additional factors that impact the behavior of epidemic models, followed by some conclusive remarks in Section IX.
\section{Demographic classification and notations}
In order to analyze the structure of epidemic models as well as the relation between their structure and the resulting dynamics, it is important to classify models as clear and simple as possible. In our study, we classify and study compartmental models based upon \emph{demography} or \emph{vital dynamics}. Demography relates to the study of characteristics of human populations such as birth, death, incidence of disease, and so on. Epidemic models with vital dynamics consider an open population with births and deaths while models without vital dynamics have a closed and fixed population with no demographic turnover.
For better realization, we assume that the \emph{law of mass action} holds for all models in this paper. This law states that if individuals in a population mix homogeneously, then the encounters between infected and susceptible individuals occur at a rate proportional to their respective numbers in the population \cite{Anderson1992}. In other words, the rate at which the susceptible population becomes infected is directly proportional to the product of the sizes of the two populations. Table~\ref{table_1_notation} summarizes the notations that will be used in model derivation henceforth. Note that $S$, $I$, $R$, and $E$ are used to represent the compartments in the epidemic model as well as the proportion of the corresponding compartments at any time instant $t$.
\begin{table}[!hb]
\renewcommand{\arraystretch}{1.2}
\caption{Notations used in model derivation}
\label{table_1_notation}
\centering
\begin{tabular}{|c||c|}
\hline
\bfseries Notation & \bfseries Definition\\
\hline\hline
$N$ & Total population size\\
$S$ or $S(t)$ & Number of \emph{susceptible} individuals at time $t$\\
$I$ or $I(t)$ & Number of \emph{infected} individuals at time $t$\\
$R$ or $R(t)$ & Number of \emph{recovered} (or \emph{removed}) individuals at time $t$\\
$E$ or $E(t)$ & Number of \emph{exposed} individuals at time $t$\\
$M$ or $M(t)$ & Number of \emph{passively immune infants} at time $t$\\
$\beta$ & Contact (or transmission) rate\\
$\gamma$ & Recovery rate\\
$1/\varepsilon$ & Average latent period\\
$\nu$ & Loss of immunity rate of recovered individuals\\
$b$ & Birth rate\\
$\mu$ & Death rate\\
$\mathcal{R}_0$ & Basic reproduction number (or ratio)\\
$e_i$ & Equilibrium point indexed at $i$\\
$X^*$ & Equilibrium value of class $X$; $X \in \{S,I,R,E\}$\\
$DFE$ & Disease-free Equilibrium\\
$EE$ & Endemic Equilibrium\\
$\lambda_i$ & Eigenvalue indexed at $i$\\
\hline
\end{tabular}
\end{table}
\section{The basic \emph{SIR} model}
In their first paper, Kermack and McKendrick created a model in which the population is divided into three compartments: \emph{susceptible} ($S$), \emph{infected} ($I$), and \emph{recovered} ($R$) \cite{Kermack1927} as illustrated in Figure~\ref{fig2_SIR}. They assumed that all individuals are mutually equally susceptible to the disease and that complete immunity is obtained merely after recovery from infection. Moreover, they also assumed that the duration of the disease is same as the duration of infection with constant transmission and recovery rates. Based upon the demographic classification, the epidemic and endemic $SIR$ models are studied below.
\begin{figure}[!t]
\centering
\includegraphics[width=2.0in]{Figures/Fig2_n.eps}
\caption{Basic $SIR$ model without vital dynamics.}
\label{fig2_SIR}
\end{figure}
\subsection{\emph{SIR} model without vital dynamics}
For a closed population of size $N$, we assume that the mixing of individuals is homogeneous and the law of mass action holds. Also, for large classes of communicable diseases, it is more realistic to consider a \emph{force of infection} that depends on the fraction of infected population with respect to the total constant population $N$, rather than the absolute number of infectious subjects. Based upon this assumption, the \emph{standard} disease incidence rate is defined as $\beta S I/N$ and the overall rate of recovery is given as $\gamma I$. In spite of the above simplifying assumptions, the resulting non-linear system does not admit a closed-form solution. Nevertheless, we shall see how significant results can be derived analytically. Figure~\ref{fig2_SIR} can be translated into the following set of differential equations:
\begin{eqnarray}
\label{eqn123_SIR_wvd}
\frac{dS}{dt}&{}={}&-\beta S \frac{I}{N},\\
\frac{dI}{dt}&{}={}&\beta S \frac{I}{N} -\gamma I,\\
\frac{dR}{dt}&{}={}&\gamma I.
\end{eqnarray}
Summing up (1), (2), and (3) yields zero which implies that the population is of constant size with $S+I+R=N$. Dividing (2) by (1) gives:
\begin{equation}
\label{eqn4_SIR_wvd}
\frac{dI}{dS}=\frac{\gamma N}{\beta S}-1.
\end{equation}
Assume that the population is susceptible up to time zero at which a relatively small number, $I(0)$, become infected. Thus, at $t=0$, $S(0)=N-I(0)$ and $R(0)=0$. As time approaches infinity, $\lim_{t \rightarrow \infty} I(t)=0$, $\lim_{t \rightarrow \infty} S(t)=S(\infty)$, and the number of individuals that have been infected is $S(0)-S(\infty)$. Integrating (4) leads to:
\begin{equation}
\label{eqn5_SIR_wvd}
I(\infty)-I(0)=\frac{\gamma N}{\beta} \ln \left(\frac{S(\infty)}{S(0)}\right)-S(\infty)+S(0)+c,
\end{equation}
where $c$ is constant and $S(\infty)$ is the proportion of susceptibles at the end of the epidemic. Since the initial infection is small, (5) further reduces to:
\begin{equation}
\label{eqn6_SIR_wvd}
\ln \frac{S(\infty)}{S(0)}=\frac{\beta}{\gamma} \left(\frac{S(\infty)}{N}-1\right)+c^{'},
\end{equation}
where $c^{'}$ denotes some constant. Defining $\mathcal{R}_0=\beta/\gamma$ as the \emph{basic reproduction ratio}, we see in Figure~\ref{fig3_SIR_wvd_Densities} that for $\mathcal{R}_0>1$, a small infection to the population would create an epidemic. $\mathcal{R}_0$ describes the total number of secondary infections produced when one infected individual is introduced into a disease-free population. The importance of the role of $\mathcal{R}_0$ can be seen by rewriting (2) as follows:
\begin{figure*}[!t]
\centering
\begin{subfigure}[b]{0.485\textwidth}
\centering
\includegraphics[width=\textwidth]{Figures/Fig3_1_n.eps}
\caption{$\mathcal{R}_0=0.7$, $\beta=0.7$, and $\gamma=1$.}
\label{fig3_1}
\end{subfigure}%
~
\begin{subfigure}[b]{0.485\textwidth}
\centering
\includegraphics[width=\textwidth]{Figures/Fig3_2_n.eps}
\caption{$\mathcal{R}_0=1$, $\beta=0.5$, and $\gamma=0.5$.}
\label{fig3_2}
\end{subfigure}
~
\begin{subfigure}[b]{0.485\textwidth}
\centering
\includegraphics[width=\textwidth]{Figures/Fig3_3_n.eps}
\caption{$\mathcal{R}_0=3$, $\beta=0.3$, and $\gamma=0.1$.}
\label{fig3_3}
\end{subfigure}
~
\begin{subfigure}[b]{0.485\textwidth}
\centering
\includegraphics[width=\textwidth]{Figures/Fig3_4_n.eps}
\caption{$\mathcal{R}_0=1$, $\beta=1.0$, and $\gamma=0.2$.}
\label{fig3_4}
\end{subfigure}
\caption[Density versus time for $SIR$ model without vital dynamics]{Density versus time for $SIR$ model without vital dynamics where $N=1$, $S(0)=0.99$, $I(0)=0.01$, and $R(0)=0$.}
\label{fig3_SIR_wvd_Densities}
\end{figure*}
\begin{equation}
\label{eqn7_SIR_wvd}
\frac{dI}{dt}=(\mathcal{R}_0 \frac{S}{N}-1) \gamma I.
\end{equation}
In order to avoid an epidemic, (7) should be non-positive. This is possible only if $\mathcal{R}_0 S(0) \leq N$. On the other hand, if $\mathcal{R}_0 S(0) > N$, then (7) is positive and thus, there will be an epidemic outbreak. Figure~\ref{fig3_SIR_wvd_Densities} illustrates the limiting values of the $S$, $I$, and $R$ compartments for different values of $\mathcal{R}_0$ where $N$ is normalized to 1.
\subsection{\emph{SIR} model with vital dynamics}
Inclusion of demographic dynamics may permit a disease to persist in a population in the long term. A disease is said to be \emph{endemic} if it remains in a population for over a decade or two. Due to the long time period involved, an endemic disease model must include births as a source of new susceptibles and natural deaths in each compartment. In our study of the endemic $SIR$ model, we consider constant birth and death rates. Using the notations in Table~\ref{table_1_notation}, the scheme in Figure~\ref{fig4_SIR} can be expressed mathematically as:
\begin{figure}[!t]
\centering
\includegraphics[width=2.3in]{Figures/Fig4_n.eps}
\caption{Basic $SIR$ model with vital dynamics.}
\label{fig4_SIR}
\end{figure}
\setlength{\arraycolsep}{0.0em}
\begin{eqnarray}
\label{eqn8910_SIR_vd}
\frac{dS}{dt}&{}={}&b N - \beta S \frac{I}{N} - \mu S,\\
\frac{dI}{dt}&{}={}& \beta S \frac{I}{N} -(\gamma+\mu) I,\\
\frac{dR}{dt}&{}={}&\gamma I - \mu R.
\end{eqnarray}
Assuming that $b$ equals $\mu$, we can easily see that the sum of the above three equations yields zero when $S+I+R=N$ holds in a non-varying population. Moreover, we observe that the average time of an infection is $1/(\gamma+\mu)$, and since the infectious individuals infect others at rate $\beta$, $\mathcal{R}_0$ is defined as $\beta/(\gamma+\mu)$.
\subsubsection{Existence of equilibria}
By setting the left-hand side of the (8)-(10) to zero and solving for $S$, $I$, and $R$, we obtain the following two steady states (or equilibrium points) \cite{Khalil2001}:
\begin{equation}
\label{eqn11_SIR_vd}
\begin{split}
e_1:(S^*,I^*,R^*)&=(N, 0, 0),\\
e_2:(S^*,I^*,R^*)&=\left(\frac{N}{\mathcal{R}_0}, N c_1 (\mathcal{R}_0-1), N c_2 (\mathcal{R}_0-1)\right),
\end{split}
\end{equation}
where $c_1=\mu/\beta$ and $c_2=\gamma/\beta$. Points $e_1$ and $e_2$ denote the \emph{disease-free equilibrium} ($DFE$) and \emph{endemic equilibrium} ($EE$) points, respectively. Figure~\ref{Fig5a} depicts the system in a disease-free steady state when $\mathcal{R}_0\leq 1$, whereas Figure~\ref{Fig5b} shows the occurrence of an endemic as the infected population reaches a limiting value of 0.225 when $\mathcal{R}_0 >1$. The stability of the system is driven by $\mathcal{R}_0$ as it can be observed by linearizing the system of equations at these points.
\begin{figure}[!t]
\centering
\begin{subfigure}[b]{0.485\textwidth}
\centering
\includegraphics[width=\textwidth]{Figures/Fig5_1_new.eps}
\caption{$\mathcal{R}_0=0.5$, $\beta= .5$, $\gamma=0.4$, and $\mu=0.6$.}
\label{Fig5a}
\end{subfigure}
~
\begin{subfigure}[b]{0.485\textwidth}
\centering
\includegraphics[width=\textwidth]{Figures/Fig5_2_n.eps}
\caption{$\mathcal{R}_0=1.6$, $\beta=0.8$, $\gamma=0.2$, and $\mu=0.3$.}
\label{Fig5b}
\end{subfigure}
\caption[Density versus time for $SIR$ model with vital dynamics]{Density versus time for $SIR$ model with vital dynamics where $N=1$, $S(0)=0.5$, $I(0)=0.4$, and $R(0)=0.1$.}
\label{fig5_SIR_vd_Densities}
\end{figure}
\subsubsection{Equilibria stability analysis}
The local stability of the model at these equilibrium points is analyzed via linearization. The Jacobian matrix for (8) and (9) is given as:
\renewcommand{\arraystretch}{2.0}
\begin{equation}
\label{eqn12_SIR_vd}
J=\begin{bmatrix}
-\mu-\beta \dfrac{I}{N} &\quad -\beta \dfrac{S}{N}\\
\beta \dfrac{I}{N} &\quad \beta \dfrac{S}{N}-(\gamma+\mu)
\end{bmatrix}.
\end{equation}
Evaluating the above matrix at $e_1$ and solving the characteristic equation, $det(J-\mathbf{\lambda I})=0$, where $\mathbf{I}$ is the identity matrix of size 2, results in the following pair of eigenvalues:
\begin{eqnarray}
\label{eqn13_SIR_vd}
(\lambda_1,\lambda_2)|_{e_1}&{}={}&(-\mu, \beta-\gamma-\mu).
\end{eqnarray}
In order to be a \emph{stable node}, both eigenvalues should be negative. Therefore, $e_1$ is stable when $\beta < \gamma+\mu$ (or equivalently $\mathcal{R}_0 < 1$) and unstable when $\beta > \gamma+\mu$. Similarly, the Jacobian matrix evaluated at $e_2$ is as below:
\renewcommand{\arraystretch}{2.0}
\begin{equation}
\label{eqn14_SIR}
J=\begin{bmatrix}
-\mu \mathcal{R}_0 &\quad -\gamma-\mu\\
\mu (\mathcal{R}_0-1) &\quad 0
\end{bmatrix}.
\end{equation}
One can easily observe that the determinant of (14) is positive as long as $\mathcal{R}_0>1$. Hence, the endemic equilibrium is stable only when $\mathcal{R}_0>1$.
\setlength{\textfloatsep}{20pt}
\begin{figure*}[!t]
\centering
\begin{subfigure}[b]{0.48\textwidth}
\centering
\includegraphics[width=\textwidth]{Figures/Fig6_1_n.eps}
\caption{Disease-free equilibrium $(e_1)$.}
\label{Fig6a}
\end{subfigure}
~
\begin{subfigure}[b]{0.48\textwidth}
\centering
\includegraphics[width=\textwidth]{Figures/Fig6_2_n.eps}
\caption{Endemic equilibrium $(e_2)$.}
\label{Fig6b}
\end{subfigure}
\caption[Phase portrait of $SIR$ model with vital dynamics]{Phase portrait of $SIR$ model with vital dynamics for (a) $\mathcal{R}_0 \leq 1$ where the system converges to $e_1=(1,0)$ and for (b) $\mathcal{R}_0 > 1$ where it converges to $e_2=(0.625,0.225)$.}
\label{fig6_SIR_portrait}
\end{figure*}
The \emph{S-I phase portrait} in Figure~\ref{fig6_SIR_portrait} shows how the model approaches the $DFE$ and $EE$ points with different initial values for $S(0)$ and $I(0)$. For the sake of simplicity, $N$ has been normalized to 1. As depicted in Figure~\ref{Fig6a}, for $\mathcal{R}_0 = 0.5$, the system eventually ends up at $(1,0)$, irrespective of the initial values of $S(0)$ and $R(0)$. On the other hand, an endemic occurs at $(0.625, 0.225)$ for $\mathcal{R}_0=1.6$ as in Figure~\ref{Fig6b}.
The phenomenon in which a parameter variation causes the stability of an equilibrium to change is known as \emph{bifurcation} \cite{Khalil2001}. In continuous systems, this corresponds to the real part of an eigenvalue of an equilibrium passing through zero. With the basic reproduction number as the \emph{bifurcation parameter}, Figure~\ref{fig7_SIR_Bifurcation} shows a \emph{transcritical} bifurcation where the equilibrium points persist through the bifurcation, but their stability properties change. Hence, we can conclude that:
\begin{equation}
\label{eqn15_SIR_vd}
\begin{split}
DFE&:\mathcal{R}_0 \leq 1 \Rightarrow \lim_{t \to \infty}(S(t), I(t), R(t)) = e_1,\\
EE&:\mathcal{R}_0 > 1 \Rightarrow \lim_{t \to \infty}(S(t), I(t), R(t)) = e_2.
\end{split}
\end{equation}
A few recent studies have revealed interesting bifurcation behaviors in $SIR$ models incorporated with factors such as varying immunity period, saturated treatment, and vaccination. We refer the interested reader to \cite{Onofrio2007,Jiang2009,Blyuss2010,Wang2012} and the references therein for more details.
\begin{figure}[!t]
\centering
\includegraphics[width=3.2in]{Figures/Fig7_n.eps}
\caption[Bifurcation diagram for the endemic $SIR$ model]{Bifurcation diagram for the endemic $SIR$ model where the equilibrium points change stability properties at $\mathcal{R}_0=1$.}
\label{fig7_SIR_Bifurcation}
\end{figure}
\section{The $SIS$ model}
For viral diseases, such as measles and chickenpox, where the recovered individuals, in general, gain immunity against the virus, the $SIR$ model is applicable. However, there exist certain bacterial diseases such as gonorrhoea and encephalitis that do not confer immunity. In such diseases, an infectious individual is allowed to recover from the infection and return unprotected to the susceptible class where he/she is prone to get infected again. Cases as such can be modeled using the \emph{susceptible-infected-susceptible} (\emph{SIS}) model as shown in Figure~\ref{fig8_SIS_vd}, where the model variables are defined in Table~\ref{table_1_notation}.
\begin{figure}[!t]
\centering
\includegraphics[width=1.3in]{Figures/Fig8_n.eps}
\caption{The $SIS$ model without vital dynamics.}
\label{fig8_SIS_vd}
\end{figure}
\subsection{SIS Model without vital dynamics}
In a fixed population, where there is no birth or death and individuals recover from the disease at the \emph{per capita} rate of $\gamma$, the simplest form of the model in Figure~\ref{fig8_SIS_vd} is given by:
\begin{eqnarray}
\label{eqn1617_SIS_wvd}
\frac{dS}{dt}&{}={}&\gamma I-\beta S \frac{I}{N},\\
\frac{dI}{dt}&{}={}& \beta S \frac{I}{N}-\gamma I.
\end{eqnarray}
By substituting $S=N-I$ in (17), the system above can be reduced to:
\begin{eqnarray}
\label{eqn18_SIS_wvd}
\frac{dI}{dt}&{}={}& (\beta-\gamma) I - \frac{\beta}{N} I^2.
\end{eqnarray}
Solving (\ref{eqn18_SIS_wvd}) analytically with $I(0)=I_0$ gives the solution for the complete system at time $t$ as follows:
\begin{eqnarray}
\label{eqn1920_SIS_wvd}
S(t)&{}={}&N-I(t),\\
I(t)&{}={}&\frac{(\beta - \gamma) N I_0}{(\beta - \gamma) N e^{-(\beta - \gamma) t}+ \beta I_0 \left[1-e^{-(\beta - \gamma) t}\right]} .
\end{eqnarray}
The behavior of the system in long-term can be inferred by looking at the possible values of $(\beta-\gamma)$ that make (20) feasible. If $(\beta-\gamma) > 0$, then $e^{-(\beta-\gamma)t} \to 0$ as $t \to \infty$. This can be written as:
\begin{eqnarray}
\lim_{t \rightarrow \infty} I(t)=\frac{(\beta-\gamma) N I_0}{\beta I_0}=\left(1-\frac{\gamma}{\beta}\right) N.
\end{eqnarray}
If $(\beta-\gamma)<0$, then $e^{-(\beta-\gamma)t} \to \infty$ as $t \to \infty$ and thus, $lim_{t \rightarrow \infty} I(t)=0$.
\subsubsection{Existence of equilibria}
There exists two equilibrium points for this model which can be obtained by setting $\frac{dI}{dt}=0$ in (18) and solving for $S$ and $I$. With $\mathcal{R}_0$ as $\beta/\gamma$, we get:
\begin{figure}[!t]
\centering
\begin{subfigure}[b]{0.485\textwidth}
\centering
\includegraphics[width=\textwidth]{Figures/Fig9_1_n.eps}
\caption{$\mathcal{R}_0=0.78$, $\beta=0.7$, and $\gamma=0.9$.}
\label{Fig9a}
\end{subfigure}
~
\begin{subfigure}[b]{0.485\textwidth}
\centering
\includegraphics[width=\textwidth]{Figures/Fig9_2_n.eps}
\caption{$\mathcal{R}_0=2.5$, $\beta=0.5$, and $\gamma=0.2$.}
\label{Fig9b}
\end{subfigure}
\caption[Density versus time for $SIS$ model without vital dynamics]{Density versus time for $SIS$ model without vital dynamics where $N=1$, $S(0)=0.9$, and $I(0)=0.1$.}
\label{fig9_Densities}
\end{figure}
\begin{equation}
\label{eqn22_SIR_vd}
\begin{split}
e_1:(S^*,I^*)&=(N, 0),\\
e_2:(S^*,I^*)&=\left(\frac{N}{\mathcal{R}_0}, \frac{N}{\mathcal{R}_0}(\mathcal{R}_0-1)\right),
\end{split}
\end{equation}
where $e_1$ and $e_2$ denote the $DFE$ and $EE$ points, respectively. In terms of the basic reproduction number, if $\mathcal{R}_0 \leq 1$, the pathogen dies out as illustrated in Figure~\ref{Fig9a} because the infection in one individual cannot replace itself. If $\mathcal{R}_0 > 1$, an existing infectious individual leads to more than one infection thus, spreading the pathogen in the population as seen in Figure~\ref{Fig9b}.
\subsubsection{Equilibria stability analysis}
The Jacobian matrix constructed from (16) and (17) is as follows:
\begin{equation}
\label{eqn14_SIR_vd}
J=\begin{bmatrix}
-\beta \dfrac{I}{N} &\quad \gamma - \beta \dfrac{S}{N}\\
\beta \dfrac{I}{N} &\quad \beta \dfrac{S}{N} - \gamma
\end{bmatrix}.
\end{equation}
Linear stability analysis for $e_1$ is done by solving the corresponding characteristic equation to obtain the following pair of eigenvalues:
\begin{eqnarray}
\label{eqn_SIS_wvd_DFE}
(\lambda_1,\lambda_2)|_{e_1}&{}={}&(0, \beta-\gamma).
\end{eqnarray}
The stability of $e_1$ depends on the value taken by $\lambda_2$. The equilibrium point is a stable $DFE$ if $\beta < \gamma$ (or equivalently $\mathcal{R}_0<1$) and unstable if $\beta > \gamma$ (or $\mathcal{R}_0>1$). Similarly, the eigenvalues of the characteristic equation for $e_2$ are:
\begin{eqnarray}
\label{eqn_SIS_wvd_EE}
(\lambda_1,\lambda_2)|_{e_2}&{}={}&(0, -\beta+\gamma).
\end{eqnarray}
In this case, the $EE$ point is stable if $\beta >\gamma$ and unstable if $\beta<\gamma$. Figure~\ref{fig10_SIS_wvd_vectorplot} depicts the vector plots for examples where $e_1$ and $e_2$ are unstable. In Figure~\ref{Fig10a}, the system converges to some state (highlighted in red) other than $(1,0)$ when $\mathcal{R}_0=2.5$. Likewise, for $\mathcal{R}_0<1$, the system converges to an invalid state as illustrated in Figure~\ref{Fig10b}. At $\beta=\gamma$ or equivalently, $\mathcal{R}_0=1$, a \emph{bifurcation} occurs as the two equilibria collide ($DFE$ equals $EE$) and exchange stability \cite{Hethcote2000}. The forward bifurcation occurring at this threshold condition is similar to as seen previously in Section III.
\begin{figure}[!t]
\centering
\begin{subfigure}[b]{0.475\textwidth}
\centering
\includegraphics[width=\textwidth]{Figures/Fig10_1_n.eps}
\caption{Point $(e_1)$ when $\mathcal{R}_0=2.5$, $\beta=0.5$, and $\gamma=0.2$.}
\label{Fig10a}
\end{subfigure}
~
\vspace{0.1in}
\begin{subfigure}[b]{0.475\textwidth}
\centering
\includegraphics[width=\textwidth]{Figures/Fig10_2_n.eps}
\caption{Point $(e_2)$ when $\mathcal{R}_0=0.778$, $\beta=0.7$, and $\gamma=0.9$.}
\label{Fig10b}
\end{subfigure}
\caption[Instability of the equilibrium points in $SIS$ model without vital dynamics]{Vector plots showing the instability of the equilibrium points in $SIS$ model without vital dynamics for (a) $\mathcal{R}_0 > 1$ and ~(b) $\mathcal{R}_0 < 1$.}
\label{fig10_SIS_wvd_vectorplot}
\end{figure}
\subsection{SIS Model with vital dynamics}
The $SIS$ model with varying population of constant size is as shown in Figure~\ref{fig11_SIS_vd}. The corresponding system of differential equations for such a model is given below, where $b$ and $\mu$ are assumed to be equal and $S + I = N$:
\begin{eqnarray}
\label{eqn2627_SIS_vd}
\frac{dS}{dt}&{}={}& b N + \gamma I -\beta S \frac{I}{N} - \mu S ,\\
\frac{dI}{dt}&{}={}& \beta S \frac{I}{N} - (\gamma + \mu) I.
\end{eqnarray}
\begin{figure}[!t]
\centering
\includegraphics[width=1.5in]{Figures/Fig11_n.eps}
\caption{The $SIS$ model with vital dynamics.}
\label{fig11_SIS_vd}
\end{figure}
\subsubsection{Existence of equilibria}
To find the equilibrium points of the system, we set (26) and (27) to zero and solve for $S$ and $I$. This results in $e_1$ as the $DFE$ point and $e_2$ as the $EE$ point as given below:
\begin{equation}
\label{eqn28_SIR_vd}
\begin{split}
e_1:(S^*,I^*)&=(N, 0),\\
e_2:(S^*,I^*)&=\left(\frac{N}{\mathcal{R}_0}, \frac{N}{\mathcal{R}_0}(\mathcal{R}_0-1)\right),
\end{split}
\end{equation}
where $\mathcal{R}_0$ is $\beta/(\gamma+\mu)$. Figure~\ref{fig12_Densities} shows the system behavior for different values of $\mathcal{R}_0$. In Figure~\ref{Fig12a}, since $\mathcal{R}_0$ is less than 1, the disease dies out and the system enters the disease-free steady state. The same happens at $\mathcal{R}_0 = 1$, where the two equilibria meet. For $\mathcal{R}_0$ greater than 1, the disease does not die out, but instead remains in the population as an endemic with a limiting value. This can be seen in Figure~\ref{Fig12b} where an endemic occurs at $(S(t), I(t))=(0.56, 0.44)$ for $\mathcal{R}_0=1.8$.
\begin{figure*}[!t]
\centering
\begin{subfigure}[b]{0.485\textwidth}
\centering
\includegraphics[width=\textwidth]{Figures/Fig12_1_n.eps}
\caption{$\mathcal{R}_0=0.6$, $\beta=0.3$, $\gamma=0.3$, and $\mu=0.2$.}
\label{Fig12a}
\end{subfigure}
~
\begin{subfigure}[b]{0.48\textwidth}
\centering
\includegraphics[width=\textwidth]{Figures/Fig12_2_n.eps}
\caption{$\mathcal{R}_0=1.8$, $\beta=0.9$, $\gamma=0.3$, and $\mu=0.2$.}
\label{Fig12b}
\end{subfigure}%
\caption[Density versus time for $SIS$ model with vital dynamics]{Density versus time for $SIS$ model with vital dynamics where $N=1$, $S(0)=0.9$ and $I(0)=0.1$.}
\label{fig12_Densities}
\end{figure*}
\subsubsection{Equilibria stability analysis}
The corresponding Jacobian matrix for this model is given as:
\begin{figure}[!t]
\centering
\begin{subfigure}[b]{0.47\textwidth}
\centering
\includegraphics[width=\textwidth]{Figures/Fig13_1_n.eps}
\caption{Stability of $e_1$ when $\mathcal{R}_0=0.6$, $\beta=0.3$, $\gamma=0.3$, and $\mu=0.2$.}
\label{Fig13a}
\end{subfigure}~
~
\vspace{0.1in}
\begin{subfigure}[b]{0.47\textwidth}
\centering
\includegraphics[width=\textwidth]{Figures/Fig13_2_n.eps}
\caption{Stability of $e_2$ when $\mathcal{R}_0=1.8$, $\beta=0.9$, $\gamma=0.9$, and $\mu=0.2$.}
\label{Fig13b}
\end{subfigure}
\caption[Stability of the equilibrium points in $SIS$ model with vital dynamics]{Vector plots showing the stability of the equilibrium points in $SIS$ model with vital dynamics for (a) $\mathcal{R}_0 \leq 1$ and (b) $\mathcal{R}_0 > 1$.}
\label{fig13_SIS_vd_vectorplot}
\end{figure}
\renewcommand{\arraystretch}{2.0}
\begin{equation}
\label{eqn29_SIS_vd}
J=\begin{bmatrix}
-\beta \dfrac{I}{N} - \mu &\quad \gamma - \beta \dfrac{S}{N}\\
\beta \dfrac{I}{N} &\quad \beta \dfrac{S}{N} - (\gamma+\mu)
\end{bmatrix}.
\end{equation}
The eigenvalues of $J$ in (\ref{eqn29_SIS_vd}) for $e_1$ and $e_2$ are deduced as follows:
\begin{eqnarray}
(\lambda_1,\lambda_2)|_{e_1}&{}={}&(-\mu,~\beta-(\gamma+\mu)),\\
(\lambda_1,\lambda_2)|_{e_2}&{}={}&(-\mu,-\beta+\gamma+\mu).
\end{eqnarray}
Linear stability analysis reveals that the disease-free equilibrium ($e_1$) is asymptotically stable if $\beta - (\gamma + \mu) \leq 0$ (or $\mathcal{R}_0 \leq 1$) and unstable otherwise \cite{Vargas2011}. Similarly, the endemic steady state is asymptotically stable if $\mathcal{R}_0>1$. Figure~\ref{fig13_SIS_vd_vectorplot} portraits the stability of $e_1$ and $e_2$ for different values of $\mathcal{R}_0$. In Figure~\ref{Fig13a}, with $\mathcal{R}_0=0.6$, the system converges to $e_1=(1,0)$ where it is stable. In the same manner, as in Figure~\ref{Fig13b}, the system eventually ends up at $e_2=(0.56,0.44)$ for $\mathcal{R}_0=1.8$, which is a valid endemic state. The forward bifurcation for the simple $SIS$ model with demographic turnover and $\mathcal{R}_0$ as the bifurcation parameter occurs at $\mathcal{R}_0=1$ as studied earlier. Nevertheless, such models in presence of additional factors reveal interesting behaviors. Some recent works on $SIS$ model that exhibit bifurcations consider factors such as non-constant contact rate having multiple stable equilibria \cite{Driessche2000}, non-linear birth rate \cite{Liu2009}, treatment \cite{Wang2009}, and time delay \cite{Das2009}.
\section{The $SIRS$ model}
The $SIRS$ model is an extension of the basic $SIR$ model in which individuals recover with immunity to the disease and become susceptible again after some time recovering. Influenza is a contagious viral disease that is usually studied using this model. In what follows, we investigate the model in both, absence and presence of demographic turnover.
\subsection{$SIRS$ Model without vital dynamics}
\begin{figure}[!t]
\centering
\includegraphics[width=2.0in]{Figures/Fig14_n.eps}
\caption{The $SIRS$ model without vital dynamics.}
\label{fig14_SIRS_wvd}
\end{figure}
The system of differential equations describing the $SIRS$ flow diagram in Fig.~\ref{fig14_SIRS_wvd} is as below:
\begin{eqnarray}
\label{eqn181920_SIRS_wvd}
\frac{dS}{dt}&{}={}&\nu R - \beta S \frac{I}{N},\\
\frac{dI}{dt}&{}={}& \beta S \frac{I}{N}-\gamma I,\\
\frac{dR}{dt}&{}={}& \gamma I - \nu R.
\end{eqnarray}
\subsubsection{Existence of equilibria}
With the three compartments summing up to $N$, we obtain the following two equilibrium points where $e_1$ and $e_2$ denote the $DFE$ and $EE$, respectively. With $c_1=\nu/(\gamma+\nu)$, $c_2=\gamma/(\gamma+\nu)$, and $\mathcal{R}_0$ defined as $\beta / \gamma$, we have:
\begin{equation}
\label{eqn35_SIRS_wvd}
\begin{split}
e_1:(S^*,I^*,R^*)&=(N, 0, 0),\\
e_2:(S^*,I^*,R^*)&=\left(\frac{N}{\mathcal{R}_0}, \frac{N}{\mathcal{R}_0} c_1 (\mathcal{R}_0-1), \frac{N}{\mathcal{R}_0} c_2(\mathcal{R}_0-1)\right).
\end{split}
\end{equation}
As illustrated in Fig.~\ref{fig15_Densities}, the infection dies out and reaches the disease-free steady state for $\mathcal{R}_0 \leq 1$ and stays as an endemic when $\mathcal{R}_0>1$.
\begin{figure}[t]
\centering
\begin{subfigure}[b]{0.485\textwidth}
\centering
\includegraphics[width=\textwidth]{Figures/Fig15_1_n.eps}
\caption{$\mathcal{R}_0=0.95$, $\beta=0.95$, $\gamma=1$, and $\nu=0.5$.}
\label{Fig15a}
\end{subfigure}~
~
\begin{subfigure}[b]{0.485\textwidth}
\centering
\includegraphics[width=\textwidth]{Figures/Fig15_2_n.eps}
\caption{$\mathcal{R}_0=3$, $\beta=0.6$, $\gamma=0.2$, and $\nu=0.85$.}
\label{Fig15b}
\end{subfigure}
\caption[Density versus time for $SIRS$ model without vital dynamics]{Density versus time for $SIRS$ model without vital dynamics where $N=1$, $S(0)=0.9$, $I(0)=0.1$, and $R(0)=0$.}
\label{fig15_Densities}
\end{figure}
\subsubsection{Equilibria stability analysis}
The corresponding Jacobian matrix of the system is obtained by substituting $R$ with $N-S-I$ in (32). Hence,
\begin{equation}
\label{eqn36_SIRS_wvd}
J=\begin{bmatrix}
-\nu - \beta \dfrac{I}{N} &~~ -\nu - \beta \dfrac{S}{N}\\
\beta \dfrac{I}{N} &~~ \beta \dfrac{S}{N} - \gamma
\end{bmatrix}.
\end{equation}
At $e_1$, the matrix $J$ yields the following two eigenvalues:
\begin{eqnarray}
(\lambda_1,\lambda_2)|_{e_1}=(-\nu,~\beta-\gamma).
\end{eqnarray}
Therefore, $e_1$ is a stable node if $\lambda_2 \leq 0$ or $\mathcal{R}_0 \leq 1$, and is a saddle point (unstable) if $\lambda_2 > 0$ or $\mathcal{R}_0>1$. However, analyzing the stability of $e_2$ requires more care due to the structure complexity of the corresponding eigenvalues given as:
\begin{equation}
\label{eqn38_SIRS_wvd}
\begin{split}
\lambda_{1,2}|_{e_2}&=\frac{-\nu (\beta+\nu)\pm \sqrt{\nu^2 (\beta + \nu)^2-4 \nu(\beta-\gamma)(\gamma+\nu)^2}}{2(\gamma+\nu)}.
\end{split}
\end{equation}
For the sake of clarity, let us denote $\nu(\beta+\nu)$ and $2(\gamma+\nu)$ as $c$ and $b$, respectively. Rewriting the above eigenvalues gives:
\begin{eqnarray}
\lambda_{1,2}|_{e_2}=\frac{-c\pm \sqrt{c^2 - \nu(\beta-\gamma)b^2}}{b}.
\end{eqnarray}
Also, let us represent $c^2-\nu (\beta-\gamma) b^2$ as $\delta$ and $\nu (\beta-\gamma) b^2$ as $\zeta$. In order to study the stability of $e_2$, we need to consider the following two cases:
\begin{figure*}[!t]
\centering
\begin{subfigure}[b]{0.47\textwidth}
\centering
\includegraphics[width=\textwidth]{Figures/Fig16_1_n.eps}
\caption{Saddle point with $\mathcal{R}_0=0.95$, $\beta=0.95$, $\gamma=1$, and $\mu=0.5$.}
\label{Fig16a}
\end{subfigure}~
~
\vspace{0.1in}
\begin{subfigure}[b]{0.47\textwidth}
\centering
\includegraphics[width=\textwidth]{Figures/Fig16_2_n.eps}
\caption{Stable node with $\mathcal{R}_0=3$, $\beta=0.6$, $\gamma=0.2$, and $\mu=0.85$.}
\label{Fig16b}
\end{subfigure}
~
\begin{subfigure}[b]{0.47\textwidth}
\centering
\includegraphics[width=\textwidth]{Figures/Fig16_3_n.eps}
\caption{Stable focus with $\mathcal{R}_0=2$, $\beta=0.7$, $\gamma=0.35$, and $\mu=0.25$.}
\label{Fig16c}
\end{subfigure}
\caption[Stability of the $EE$ point in $SIRS$ model without vital dynamics]{Vector plots showing the stability of the $EE$ point in $SIRS$ model without vital dynamics for (a) $\delta \geq 0$ and $\mathcal{R}_0 \leq 1$ ($Case~A$ (i)), (b) $\delta \geq 0$ and $\mathcal{R}_0 > 1$ ($Case~A$ (ii)), and (c) $\delta < 0$ and $\mathcal{R}_0 > 1$ ($Case~B$).}
\label{fig16_SIRS_portrait}
\end{figure*}
\begin{enumerate}[~$\bullet$]
\item \emph{Case A:} When $\delta \geq 0$, depending on the possible values that $(\beta-\gamma)$ can take, we have the following two conditions :
\begin{enumerate}[(i)]
\item If $\beta - \gamma \leq 0$ or $\mathcal{R}_0 \leq 1$, then $\zeta \leq 0$ which implies that the magnitude of $\delta$ is always greater than or equal to $c^2$. Under this condition, the eigenvalues will always be real with opposite signs. Hence, the equilibrium point $e_2$ would be a \emph{saddle point} as portrayed in Figure~\ref{Fig16a}, where the parametric values are the same as that in Figure~\ref{Fig15a}.
\item If $\beta - \gamma > 0$ or $\mathcal{R}_0 > 1$, then $\zeta > 0$ and thus, the magnitude of $\delta$ would always be lesser than $c^2$. In this case, $\lambda_1$ and $\lambda_2$ will always be real values with negative signs. As shown in Figure~\ref{Fig16b}, $e_2$ converges to a \emph{stable node} with the same parametric values as given in Figure~\ref{Fig15b}. Here, $\delta=0.019$ which is lesser than $c^2=1.519$ and thus, results in a stable node at $(0.333, 0.539)$.
\end{enumerate}
\item \emph{Case B:} When $\delta < 0$, $\beta - \gamma$ should always be greater than zero and thus, the eigenvalues would be complex conjugates. Since the real parts of the eigenvalues are negative, the equilibrium would be a \emph{stable focus} point where the following condition holds:
\setlength{\arraycolsep}{0.0em}
\begin{eqnarray}
\mathcal{R}_0 > \frac{c^2}{\gamma \nu b^2} + 1.
\label{eqn40_SIRS_wvd}
\end{eqnarray}
As an example for this case, with $\mathcal{R}_0=2$, $\beta=0.7$, $\gamma=0.35$, and $\nu=0.25$, the stable focus occurs at $(0.5, 0.208)$ as depicted in Figure~\ref{Fig16c}.
\end{enumerate}
\subsection{$SIRS$ model with vital dynamics}
\begin{figure}[!t]
\centering
\includegraphics[width=2.3in]{Figures/Fig17_n.eps}
\caption{The $SIRS$ model with vital dynamics.}
\label{fig17_SIRS_vd}
\end{figure}
As in Figure~\ref{fig17_SIRS_vd}, the $SIRS$ model with standard incidence can be simply expressed as the following set of differential equations \cite{Vargas2011}:
\begin{eqnarray}
\label{eqn414243_SIRS_vd}
\frac{dS}{dt}&{}={}&b N + \nu R - \beta S \frac{I}{N} - \mu S,\\
\frac{dI}{dt}&{}={}& \beta S \frac{I}{N} - (\gamma + \mu) I,\\
\frac{dR}{dt}&{}={}& \gamma I - (\nu + \mu) R.
\end{eqnarray}
It is worth mentioning that $1/\gamma$ and $1/\nu$ can be regarded as the \emph{mean infectious period} and the \emph{mean immune period}, respectively. With $\nu =0$, the model reduces to an $SIR$ model with no transition from class $R$ to class $S$ due to life-long immunity.
\subsubsection{Existence of equilibria}
The system has a $DFE$ point and a unique $EE$ point denoted by $e_1$ and $e_2$, respectively, as given below:
\begin{equation}
\label{eqn44_SIRS_vd}
\begin{split}
e_1:(S^*,I^*,R^*)&=(N, 0, 0),\\
e_2:(S^*,I^*,R^*)&=\left(\frac{N}{\mathcal{R}_0}, \frac{N}{\mathcal{R}_0} c_1 (\mathcal{R}_0-1), \frac{N}{\mathcal{R}_0} c_2(\mathcal{R}_0-1)\right),
\end{split}
\end{equation}
where $\mathcal{R}_0$ is given by $\beta/(\gamma+\mu)$, $c_1=(\nu+\mu)/(\gamma+\nu+\mu)$, $c_2=\gamma/(\gamma+\nu+\mu)$, and $b$ is assumed to be equal to $\mu$. It should be noted that the basic reproduction number does not depend on the loss of immunity rate ($\nu$). The system reaches $DFE$ steady state for $\mathcal{R}_0 =0.28$ as shown in Figure~\ref{Fig18a}. Here, the parametric values are taken to be $\beta=0.04$, $\mu=0.043$, $\gamma=0.1$, and $\nu=0.01$. On the other hand, Figure~\ref{Fig18b} illustrates an example for which $\mathcal{R}_0 > 1$. In this case, the system reaches the endemic state $(0.625,0.125,0.25)$ for $\mathcal{R}_0 = 1.6$.
\begin{figure}[!t]
\centering
\begin{subfigure}[b]{0.485\textwidth}
\centering
\includegraphics[width=\textwidth]{Figures/Fig18_1_n.eps}
\caption{$\mathcal{R}_0=0.28$, $\beta=0.04$, $\gamma=0.1$, $\mu=0.043$, and $\nu=0.01$.}
\label{Fig18a}
\end{subfigure}
~
\begin{subfigure}[b]{0.485\textwidth}
\centering
\includegraphics[width=\textwidth]{Figures/Fig18_2_n.eps}
\caption{$\mathcal{R}_0=1.6$, $\beta=0.8$, $\gamma=0.4$, $\mu=0.1$, and $\nu=0.1$.}
\label{Fig18b}
\end{subfigure}%
\caption[Density versus time for $SIRS$ model with vital dynamics]{Density versus time for $SIRS$ model with vital dynamics where $N=1$, $S(0)=0.6$, $I(0)=0.2$, and $R(0)=0.2$.}
\label{fig18_Densities}
\end{figure}
\subsubsection{Equilibria stability analysis}
Substituting $R=N-S-I$ in (41) gives the following Jacobian matrix which is obtained from (41) and (42):
\begin{equation}
\label{eqn45_SIRS_vd}
J=\begin{bmatrix}
- \nu - \beta \dfrac{I}{N} - \mu &~~ - \nu - \beta \dfrac{S}{N}\\
\beta \dfrac{I}{N} &~~ \beta \dfrac{S}{N}-(\gamma+\mu)
\end{bmatrix}.
\end{equation}
Evaluating (45) at $e_1$ and solving its corresponding characteristic equations yields the following pair of eigenvalues:
\begin{eqnarray}
(\lambda_1,\lambda_2)|_{e_1}&{}={}&(-(\nu+\mu),~\beta-(\gamma+\mu)).
\end{eqnarray}
We see that the disease-free equilibrium ($e_1$) is stable if $\lambda_2 \leq 0$, i.e. $\beta-(\gamma+\mu) \leq 0$ or $\mathcal{R}_0 \leq 1$, and unstable otherwise. Similarly, on finding the eigenvalues for $e_2$, we see that this unique endemic equilibrium point is stable for $\mathcal{R}_0 > 1$. The instability of the equilibrium points can be seen in Figure~\ref{fig19_SIRS_portrait}. For $\mathcal{R}_0 = 1.6$, the vector plot in Figure~\ref{Fig19a} shows how the system does not converge to $(S^*,I^*)=(1,0)$. In the same manner, Figure~\ref{Fig19b} illustrates the instability of $e_2$ at $(S^*,I^*)=(3.575,-0.892)$ when $\mathcal{R}_0$ is less than unity. With the basic reproduction ratio as the bifurcation parameter, the system yields a transcritical forward bifurcation at $\mathcal{R}_0=1$. More interesting behaviors have been reported when studied under factors such as stage structure \cite{Zhang2010} and non-linear incidence rates \cite{Alexander2006}, \cite{Hu2011}.
Hitherto, we have dealt with models comprising of $S$, $I$, and $R$ compartments. In these models, the infected individuals become infectious immediately. In the next two models, an \emph{exposed} compartment in which all the individuals have been infected but are not yet infectious, is introduced. Such models take into consideration the latent period of the disease, resulting in an additional compartment denoted by $E(t)$. The progression rate coefficient from compartment $E$ to $I$ is given as $\varepsilon$ such that $1/\varepsilon$ is the mean latent period. Several other models with latent period such as $SEIR$ and $SEIRS$ have also been reported in the literature. However, such models are beyond the scope of this article. Interested readers can refer to \cite{Keeling2011}, \cite{Vynnycky2010}, and \cite{Ma2009} for more on epidemic models beyond two dimensions.
\begin{figure}[!t]
\centering
\begin{subfigure}[b]{0.46\textwidth}
\centering
\includegraphics[width=\textwidth]{Figures/Fig19_1_n.eps}
\caption{Instability of $DFE$ point when $\mathcal{R}_0=1.6$, $\beta = 0.8$, $\gamma = 0.4$, $\mu = 0.1$, and $\nu = 0.1$.}
\label{Fig19a}
\end{subfigure}~
~
\vspace{0.1in}
\begin{subfigure}[b]{0.47\textwidth}
\centering
\includegraphics[width=\textwidth]{Figures/Fig19_2_n.eps}
\caption{Instability of $EE$ point when $\mathcal{R}_0=0.28$, $\beta = 0.04$, $\gamma=0.1$, $\mu = 0.043$, and $\nu = 0.01$.}
\label{Fig19b}
\end{subfigure}
\caption[Instability of the equilibrium points in $SIRS$ model with vital dynamics]{Vector plots showing the instability of $SIRS$ model with vital dynamics for (a) disease-free equilibrium when $\mathcal{R}_0 > 1$ and ~(b) endemic equilibrium when $\mathcal{R}_0 \leq 1$.}
\label{fig19_SIRS_portrait}
\end{figure}
\section{The $SEI$ model}
Unlike $SIR$ models, the \emph{susceptible-exposed-infected} ($SEI$) model assumes that a susceptible individual first undergoes a latent (or exposed) period before becoming infectious \cite{Anderson1981}. One example of this model is the transmission of severe acute respiratory syndrome (SARS) coronavirus.
\begin{figure}[!t]
\centering
\includegraphics[width=2.0in]{Figures/Fig20_n.eps}
\caption{The $SEI$ model without vital dynamics.}
\label{fig20_SEI_wvd}
\end{figure}
\subsection{$SEI$ model without vital dynamics}
For a fixed population of size $N$, the following differential equations describe the flow diagram in Fig.~\ref{fig20_SEI_wvd}:
\begin{eqnarray}
\label{eqn484950_SEI_wvd}
\frac{dS}{dt}&{}={}& - \beta S \frac{I}{N},\\
\frac{dE}{dt}&{}={}& \beta S \frac{I}{N} - \varepsilon E,\\
\frac{dI}{dt}&{}={}& \varepsilon E.
\end{eqnarray}
The system should be analyzed asymptotically as it does not have a closed-form solution. We shall see that the population converges into a single compartment due to the straight-forward nature of the system. In what follows, we investigate the system behavior in terms of $\beta$ and $\varepsilon$.
\begin{enumerate}[~$\bullet$]
\item \emph{Case A:} When $\beta\neq 0$ and $\varepsilon\neq 0$, depending on the initial values of $E(0)$ and $I(0)$, the system approaches two different equilibrium points as below:
\begin{enumerate}[(i)]
\item If $E(0)=0$ and $I(0)=0$, the system remains in the following disease-free equilibrium:
\begin{eqnarray}
(S(t),E(t),I(t))=(N,0,0).
\end{eqnarray}
This scenario is illustrated in Figure~\ref{Fig21a} where the complete population is susceptible at all times for $N=1$, $\beta=0.8$, $\varepsilon=0.5$, and $S(0)=1$.
\item If $E(0)\neq0$ or $I(0)\neq0$, then the system approaches the following equilibrium point in long-term:
\begin{eqnarray}
\lim_{t \to \infty}(S(t),E(t),I(t))=(0,0,N).
\end{eqnarray}
To prove this, consider the solution of (49) which is:
\begin{eqnarray}
I(t)=\varepsilon \int_0^t E(\tau) d\tau+I(0).
\end{eqnarray}
$I(t)$ is a monotonically increasing function when $E(t)>0$. Since all compartments are always non-negative, $E(t)$ is greater than 0 when $E(t)\neq0$. Additionally, on solving (47), we get:
\begin{eqnarray}
S(t)=S(0) \exp\left[-\frac{\beta}{N}\int_0^t I(\tau) d\tau\right].
\end{eqnarray}
When $I(t)>0$, we see that $S(t)$ is a monotonically decreasing function. Unless when $E(t)=0$ for all $t$, $I(t)$ is a monotonically increasing function, and $\lim_{t \to \infty} S(t)=0$ as $\lim_{t \to \infty} \int_0^t I(\tau) d\tau=\infty$. In terms of $E(t)$, from (48), we see that the solution $E(t)=c e^{-\varepsilon t}$ goes to 0 as $t$ approaches infinity. In summary, if the condition $E(0)\neq0$ or $I(0)\neq0$ is satisfied, then $\lim_{t \to \infty}S(t)=0$. Since $S=0$, $\lim_{t \to \infty}E(t)=\lim_{t \to \infty}c e^{-\varepsilon t}=0$. Consequently, $\lim_{t \to \infty}I(t)=N$. This case is depicted in Figure~\ref{Fig21b}.
\end{enumerate}
\item \emph{Case B:} When $\beta=0$ and $\varepsilon\neq0$, the reduced system yields the following solution:
\begin{equation}
\label{eqn55_SEI_wvd}
\begin{split}
S(t)&= S(0),\\
E(t)&= E(0)e^{-\varepsilon t},\\
I(t)&= \varepsilon \int_0^t E(\tau) d\tau + I(0).
\end{split}
\end{equation}
Since $\lim_{t \to \infty}E(t)=0$, the system results in the following equilibrium solution as shown in Figure~\ref{Fig21c}, where the state of the system changes from the initial condition $(S(0),E(0),I(0))=(0.75,0.24,0.01)$ to steady state $(0.75,0,0.25)$ for $\varepsilon=0.5$:
\begin{eqnarray}
\lim_{t \to \infty}(S(t),E(t),I(t))=(S(0),0,N-S(0)).
\end{eqnarray}
\begin{figure*}[!t]
\centering
\begin{subfigure}[b]{0.485\textwidth}
\centering
\includegraphics[width=\textwidth]{Figures/Fig21_1_n.eps}
\caption{$S(0)=1$, $E(0)=I(0)=0$, $\beta=0.8$, and $\varepsilon=0.5$.}
\label{Fig21a}
\end{subfigure
~
\begin{subfigure}[b]{0.485\textwidth}
\centering
\includegraphics[width=\textwidth]{Figures/Fig21_2_n.eps}
\caption{$S(0)=0.99$, $E(0)=0$, $I(0)=0.01$, $\beta=0.8$ and $\varepsilon=0.5$.}
\label{Fig21b}
\end{subfigure}%
~
\begin{subfigure}[b]{0.485\textwidth}
\centering
\includegraphics[width=\textwidth]{Figures/Fig21_3_n.eps}
\caption{$S(0)=0.75$, $E(0)=0.24$, $I(0)=0.01$, $\beta=0$, and $\varepsilon=0.5$.}
\label{Fig21c}
\end{subfigure}%
~
\begin{subfigure}[b]{0.485\textwidth}
\centering
\includegraphics[width=\textwidth]{Figures/Fig21_4_n.eps}
\caption{$S(0)=0.85$, $E(0)=0.05$, $I(0)=0.1$, $\beta=0$, and $\varepsilon=0$.}
\label{Fig21d}
\end{subfigure}%
~
\begin{subfigure}[b]{0.485\textwidth}
\centering
\includegraphics[width=\textwidth]{Figures/Fig21_5_n.eps}
\caption{$S(0)=0.85$, $E(0)=0.15$, $I(0)=0$, $\beta=0.8$, and $\varepsilon=0$.}
\label{Fig21e}
\end{subfigure}%
~
\begin{subfigure}[b]{0.485\textwidth}
\centering
\includegraphics[width=\textwidth]{Figures/Fig21_6_n.eps}
\caption{$S(0)=0.85$, $E(0)=0.1$, $I(0)=0.05$, $\beta=0.8$, and $\varepsilon=0$.}
\label{Fig21f}
\end{subfigure}%
\caption[Density versus time for $SEI$ model without vital dynamics]{Density versus time for $SEI$ model without vital dynamics where (a) $E(0)=I(0)=0$ $(Case~A(i))$, (b) $E(0) \neq 0$ or $I(0) \neq 0$ $(Case~ A(ii))$, (c) $\beta=0$ and $\varepsilon \neq 0$ $(Case~B)$, (d) $\beta=\varepsilon=0$ $(Case~C)$, (e) $I(0)=0$ $(Case~D(i))$, and (f) $I(0)\neq0$ $(Case~D(ii))$.}
\label{fig21_SEI_Densities}
\end{figure*}
\item \emph{Case C:} When $\beta=0$ and $\varepsilon=0$, the solution is given as a set of constants. As $t$ approaches infinity, the system remains in the following equilibrium point as exemplified in Figure~\ref{Fig21d}, where $S(0)=0.85$, $E(0)=0.05$, and $I(0)=0.1$:
\begin{eqnarray}
\lim_{t \to \infty}(S(t),E(t),I(t))=(S(0),E(0),I(0)).
\end{eqnarray}
\item \emph{Case D:} When $\beta \neq 0$ and $\varepsilon=0$, the solution of the reduced system of differential equations is given as below:
\begin{equation}
\label{eqn58_SEI_wvd}
\begin{split}
S(t)&= S(0),\\
E(t)&= E(0)e^{-\varepsilon t},\\
I(t)&= \varepsilon \int_0^t E(\tau) d\tau + I(0).
\end{split}
\end{equation}
As Figures~\ref{Fig21e} and \ref{Fig21f} reveal, the equilibrium point that the system reaches depends upon the value of $I(0)$. Hence,
\begin{enumerate}[(i)]
\item If $I(0)=0$, we see that $S(t)=S(0)$ for all $t$. Thus,
\begin{eqnarray}
\lim_{t \to \infty}(S(t),E(t),I(t))&=(S(0),N-S(0),0).~~
\end{eqnarray}
\item If $I(0)\neq 0$, $\lim_{t \to \infty}S(t)=0$ and we get:
\begin{eqnarray}
\lim_{t \to \infty}(S(t),E(t),I(t))&=(0,N-I(0),I(0)).~~
\end{eqnarray}
\end{enumerate}
\end{enumerate}
\subsection{$SEI$ model with vital dynamics}
\begin{figure}[!t]
\centering
\includegraphics[width=2.3in]{Figures/Fig22_n.eps}
\caption{$SEI$ model with vital dynamics.}
\label{fig22_SEI_vd}
\end{figure}
With $b=\mu$ in a birth-death population of total size $N$ , the model illustrated in Fig.~\ref{fig22_SEI_vd} can be written as follows:
\begin{eqnarray}
\label{eqn616263_SEI_vd}
\frac{dS}{dt}&{}={}& b N - \beta S \frac{I}{N} - \mu S,\\
\frac{dE}{dt}&{}={}& \beta S \frac{I}{N} - \varepsilon E - \mu E,\\
\frac{dI}{dt}&{}={}& \varepsilon E - \mu I.
\end{eqnarray}
\subsubsection{Existence of equilibria}
The two set of equilibrium points obtained by setting the left-hand side of (60)-(62) to zero and solving for $S$, $E$, and $I$ are:
\begin{equation}
\label{eqn64_SEI_vd}
\begin{split}
e_1:(S^*,E^*,I^*)&=(N, 0, 0),\\
e_2:(S^*,E^*,I^*)&=\left(\frac{N}{\mathcal{R}_0}, \frac{N}{\mathcal{R}_0} c_1(\mathcal{R}_0-1), \frac{N}{\mathcal{R}_0} c_2 (\mathcal{R}_0-1)\right),
\end{split}
\end{equation}
with $c_1$ and $c_2$ defined as $\mu/(\varepsilon+\mu)$ and $\varepsilon/(\varepsilon+\mu)$, respectively, and $\mathcal{R}_0=\beta \varepsilon/(\mu(\varepsilon+\mu))$. The $DFE$ and $EE$ steady-states are respectively, $e_1$ and $e_2$. With $\beta=0.25$, $\varepsilon=0.4$, and $\mu=0.2$, Figure~\ref{Fig23a} depicts the disease-free steady state of the system as $\mathcal{R}_0=0.833$. Likewise, Figure~\ref{Fig23b} shows how the system reaches the endemic equilibrium for $\mathcal{R}_0=1.7$ when $\beta=0.54$, $\varepsilon=0.5$, and $\mu=0.22$.
\begin{figure*}[!t]
\centering
\begin{subfigure}[b]{0.485\textwidth}
\centering
\includegraphics[width=\textwidth]{Figures/Fig23_1_n.eps}
\caption{$\mathcal{R}_0=0.833$ with $\beta=0.25$, $\varepsilon=0.4$, and $\mu=0.2$.}
\label{Fig23a}
\end{subfigure}~
~
\begin{subfigure}[b]{0.485\textwidth}
\centering
\includegraphics[width=\textwidth]{Figures/Fig23_2_n.eps}
\caption{$\mathcal{R}_0=1.7$ with $\beta=0.54$, $\varepsilon=0.5$, and $\mu=0.22$.}
\label{Fig23b}
\end{subfigure}%
\caption[Density versus time for $SEI$ model with vital dynamics]{Density versus time for $SEI$ model with vital dynamics where $N=1$, $S(0)=0.6$, $E(0)=0.3$, and $I(0)=0.1$.}
\label{fig23_SEI_vd_Density}
\end{figure*}
\subsubsection{Equilibria stability analysis}
Considering (60) and (62), the Jacobian matrix for the system is as given below:
\renewcommand{\arraystretch}{2.0}
\begin{equation}
\label{eqn65_SEI_vd}
J=\begin{bmatrix}
-\beta \dfrac{I}{N} - \mu &~~ -\beta \dfrac{S}{N}\\
-\varepsilon &~~ -(\varepsilon + \mu)
\end{bmatrix}.
\end{equation}
Evaluating the matrix at $e_1$ and solving the characteristic equation gives the following two eigenvalues:
\begin{eqnarray}
\lambda_{1,2}|_{e_1}=-\frac{1}{2}\left(\varepsilon+2\mu\pm\sqrt{\varepsilon(4\beta+\varepsilon)}\right).
\end{eqnarray}
For $e_1$ to be a stable node, both eigenvalues should be negative which means that $\lambda_2$ should be less than zero. Hence, $\varepsilon(4\beta+\varepsilon)$ should be less than $(\varepsilon+2\mu)^2$, which implies that $\mathcal{R}_0<1$. In other words, $e_1$ is a stable point if $\mathcal{R}_0<1$ and is a saddle point if the eigenvalues have opposite signs. Doing the same for $e_2$, we get:
\begin{eqnarray}
\lambda_{1,2}|_{e_2} = \frac{-\beta\varepsilon-(\varepsilon+\mu)^2}{2(\varepsilon+\mu)} \pm
\sqrt{\frac{(\beta\varepsilon)^2}{4(\varepsilon+\mu)^2} - \frac{\beta\varepsilon}{2} + \frac{(\varepsilon + \mu)(\varepsilon + 5\mu)}{4}}.
\end{eqnarray}
Similar to the above case, if $\mathcal{R}_0>1$, then both eigenvalues are negative and thus, $e_2$ would be a stable node. Otherwise, $e_2$ would be a saddle point. In the simplest case, with $\mathcal{R}_0$ as the bifurcation parameter, the system exhibits a forward transcritical bifurcation as it switches between the two equilibria. However, not much has been done in bifurcation analysis of such models in presence of other factors.
\section{The $SEIS$ model}
The \emph{susceptible}-\emph{exposed}-\emph{infected}-\emph{susceptible} ($SEIS$) model is an extension of the $SEI$ model such that in this model, the individual does not remain infected forever, but instead recovers and returns back to being susceptible again. Many sexually transmitted diseases (STD) and chlamydial infections are known to result in little or no acquired immunity following recovery \cite{Anderson1979}. In such cases, this model may serve as a suitable choice.
\begin{figure}[!t]
\centering
\includegraphics[width=2.0in]{Figures/Fig24_n.eps}
\caption{The $SEIS$ model without vital dynamics.}
\label{fig24_SEIS_wvd}
\end{figure}
\subsection{$SEIS$ model without vital dynamics}
The dynamical transfer of hosts depicted in Figure~\ref{fig24_SEIS_wvd} can be formulated as follows, where $N=S+E+I$:
\begin{eqnarray}
\label{eqn686970_SEIS_wvd}
\frac{dS}{dt}&{}={}& \gamma I - \beta S \frac{I}{N},\\
\frac{dE}{dt}&{}={}& \beta S \frac{I}{N} - \varepsilon E,\\
\frac{dI}{dt}&{}={}& \varepsilon E - \gamma I.
\end{eqnarray}
\subsubsection{Existence of equilibria}
On solving (67)-(69) for $S$, $E$, and $I$, we obtain $e_1$ and $e_2$ which represent the disease-free and endemic equilibrium points, respectively:
\begin{equation}
\label{eqn71_SEIS_wvd}
\begin{split}
e_1:(S^*,E^*,I^*)&=(N, 0, 0),\\
e_2:(S^*,E^*,I^*)&=\left(\frac{N}{\mathcal{R}_0}, \frac{N}{\mathcal{R}_0} c_1 (\mathcal{R}_0-1), \frac{N}{\mathcal{R}_0} c_2 (\mathcal{R}_0-1)\right),
\end{split}
\end{equation}
where $c_1=\gamma/(\varepsilon+\gamma)$, $c_2=\varepsilon/(\varepsilon+\gamma)$, and $\mathcal{R}_0=\beta/\gamma$. As shown in Figure~\ref{fig25_SEIS_wvd_Density}, the system converges to $e_1$ when $\mathcal{R}_0 \leq 1$ and to $e_2$ when $\mathcal{R}_0>1$.
\begin{figure}[!t]
\centering
\begin{subfigure}[b]{0.485\textwidth}
\centering
\includegraphics[width=\textwidth]{Figures/Fig25_1_n.eps}
\caption{$\mathcal{R}_0=0.4$, $\beta=0.16$, $\varepsilon=0.12$, and $\gamma=04$.}
\end{subfigure}~
~
\begin{subfigure}[b]{0.485\textwidth}
\centering
\includegraphics[width=\textwidth]{Figures/Fig25_2_n.eps}
\caption{$\mathcal{R}_0=1.5$, $\beta=0.6$, $\varepsilon=0.12$, and $\gamma=0.4$.}
\end{subfigure}%
\caption[Density versus time for $SEIS$ model without vital dynamics]{Density versus time for $SEIS$ model without vital dynamics where $N=1$, $S(0)=0.8$, $E(0)=0.18$, and $I(0)=0.02$.}
\label{fig25_SEIS_wvd_Density}
\end{figure}
\subsubsection{Equilibria stability analysis}
\begin{figure}[!t]
\centering
\begin{subfigure}[b]{0.46\textwidth}
\centering
\includegraphics[width=\textwidth]{Figures/Fig26_1_n.eps}
\caption{Stability of $e_1$ when $\mathcal{R}_0=0.4$, $\beta=0.16$, $\varepsilon=0.12$, and $\gamma=0.4$.}
\label{Fig26a}
\end{subfigure}~
~
\vspace{0.1in}
\begin{subfigure}[b]{0.46\textwidth}
\centering
\includegraphics[width=\textwidth]{Figures/Fig26_2_n.eps}
\caption{Stability of $e_2$ when $\mathcal{R}_0=1.5$, $\beta=0.6$, $\varepsilon=0.12$, and $\gamma=0.4$.}
\label{Fig26b}
\end{subfigure}
\caption[Stability of the equilibrium points in $SEIS$ model without vital dynamics]{Vector plots showing the stability of the equilibrium points in $SEIS$ model without vital dynamics for (a) $\mathcal{R}_0 \leq 1$ and ~(b) $\mathcal{R}_0 > 1$.}
\label{fig26_SEIS_vector_wvd}
\end{figure}
The asymptotic behavior of the model can be analyzed by studying the stability conditions of the system near its equilibrium points. The Jacobian matrix formed by using (67) and (69) is as below:
\renewcommand{\arraystretch}{2.0}
\begin{equation}
\label{eqn72_SEIS_wvd}
J=\begin{bmatrix}
-\beta \dfrac{I}{N} &~~ \gamma-\beta \dfrac{S}{N}\\
-\varepsilon &~~ -(\varepsilon + \gamma)
\end{bmatrix}.
\end{equation}
At $e_1$, the matrix $J$ yields the following eigenvalues:
\begin{eqnarray}
\lambda_{1,2}|_{e_1}=-\frac{1}{2}\left(\varepsilon+\gamma\pm\sqrt{(\gamma-\varepsilon)^2+4 \beta\varepsilon}\right).
\end{eqnarray}
Once again, in order for $e_1$ to be stable, both eigenvalues should be negative and since $\lambda_1$ is already negative, $\lambda_2$ should be less than zero. Hence, for $\lambda_2$ to be negative, $-\gamma-\varepsilon + \sqrt{(\gamma-\varepsilon)^2+4\beta\varepsilon}$ should be negative, which on further simplification implies that $\mathcal{R}_0<1$. Therefore, $e_1$ is a stable $DFE$ when $\mathcal{R}_0<1$ and is unstable when $\mathcal{R}_0>1$. Similarly, calculating the eigenvalues of $J$ evaluated at $e_2$ results in the following pair of eigenvalues:
\begin{eqnarray}
\lambda_{1,2}|_{e_2}&{}={}&\left(-(\gamma+\varepsilon),~\frac{\gamma(\gamma-\beta)} {(\gamma+\varepsilon)}\right),
\end{eqnarray}
Since $\lambda_1$ is always negative, the stability of $e_2$ depends on $\lambda_2$. For $\lambda_2<0$, we observe that $e_2$ is a stable $EE$. On the contrary, when $\lambda_2>0$ (or equivalently, $\mathcal{R}>1$), the equilibrium point is unstable. This is clearly shown in Figure~\ref{fig26_SEIS_vector_wvd} where the stability of the system at the equilibrium points depends on $\mathcal{R}_0$. In Figure~\ref{Fig26a}, the system reaches the stable state $(S^*,I^*)=(1,0)$ for $\mathcal{R}_0=0.4$, whereas in Figure~\ref{Fig26b}, it converges to $(0.667,0.077)$ which is a stable endemic. As observed in previous cases, the model results in a forward bifurcation at $\mathcal{R}_0=1$ when switching from one steady-state to another. Thus, the stability of $e_1$ is persistent at $\mathcal{R}_0=1$.
\subsection{$SEIS$ model with vital dynamics}
\begin{figure}[!t]
\centering
\includegraphics[width=2.3in]{Figures/Fig27_n.eps}
\caption{The $SEIS$ model with vital dynamics.}
\label{fig27_SEIS_vd}
\end{figure}
In this subsection, we consider the $SEIS$ model for a population of size $N$ with birth and death rates that are constant. The set of equations given below refer to such a scheme illustrated in Figure~\ref{fig27_SEIS_vd}:
\begin{eqnarray}
\label{eqn757677_SEIS_wvd}
\frac{dS}{dt}&{}={}& b N +\gamma I - \beta S \frac{I}{N} - \mu S,\\
\frac{dE}{dt}&{}={}& \beta S \frac{I}{N} - (\varepsilon+\mu) E,\\
\frac{dI}{dt}&{}={}& \varepsilon E - (\gamma+\mu) I.
\end{eqnarray}
\subsubsection{Existence of equilibria}
The following two equilibrium points are calculated by setting $b=\mu$, the time-derivatives in (74)-(76) to zero, and solving for $S$, $E$, and $I$:
\begin{equation}
\label{eqn78_SEIS_vd}
\begin{split}
e_1:(S^*,E^*,I^*)&=(N, 0, 0),\\
e_2:(S^*,E^*,I^*)&=\left(\frac{N}{\mathcal{R}_0}, \frac{N}{\mathcal{R}_0} c_1 (\mathcal{R}_0-1), \frac{N}{\mathcal{R}_0} c_2 (\mathcal{R}_0-1)\right),
\end{split}
\end{equation}
where $c_1$, $c_2$, and $\mathcal{R}_0$ are defined as $(\gamma+\mu)/(\gamma+\varepsilon+\mu)$, $\varepsilon/(\gamma+\varepsilon+\mu)$, and $\beta\varepsilon/((\varepsilon+\mu)(\gamma+\mu))$, respectively. For $\mathcal{R}_0 \leq 1$, the system approaches $e_1$ which is disease-free, while for $\mathcal{R}_0 > 1$, it ends up at $e_2$.
\subsubsection{Equilibria stability analysis}
The Jacobian matrix for this system is given as follows:
\renewcommand{\arraystretch}{2.0}
\begin{equation}
\label{eqn79_SEIS_vd}
J=\begin{bmatrix}
-\beta \dfrac{I}{N} - \mu &~~ \gamma - \beta \dfrac{S}{N}\\
-\varepsilon &~~ -(\mu+\varepsilon+\gamma)
\end{bmatrix}.
\end{equation}
By evaluating the matrix in (78) at $e_1$, we see that the equilibrium point is a stable disease-free equilibrium when both of the following eigenvalues are negative and is unstable if the eigenvalues have opposite signs. In terms of $\mathcal{R}_0$, $e_1$ is stable if $\mathcal{R}_0<1$ and unstable if $\mathcal{R}_0>1$:
\begin{eqnarray}
\lambda_{1,2}|_{e_1}=-\frac{1}{2}\left(\varepsilon+\gamma+2\mu\pm\sqrt{(\gamma-\varepsilon)^2+4 \beta\varepsilon}\right).
\end{eqnarray}
Doing the same for $e_2$ yields a more complex pair of eigenvalues. However, on simplifying the eigenvalues, one can easily conclude that $e_2$ is stable when $\mathcal{R}_0>1$ and unstable otherwise. At $\mathcal{R}_0=1$, the system changes from $e_1$ to $e_2$ resulting in a forward bifurcation. A few works that reveal interesting bifurcation behaviors in $SEIS$ model can be found in \cite{Hethcote1991,Cao2011,Li2011}.
\section{Other factors in modeling}
Occurrences of certain events in nature and society may influence the behavior of the epidemic models seen so far. To guarantee that the model of choice mimics its counterpart in real-world, such influential factors should be taken into consideration. Many of these factors result in models with additional compartments which make them more complicated for mathematical analysis. In this section, a concise description on some of the most prominent factors that are considered in epidemic modeling is provided.
\subsection{Latent Period}
The time period between exposure and the onset of infectiousness is defined as the latent period. This is slightly different from the definition of \emph{incubation period} which is the time interval between exposure and appearance of the first symptom of the disease in question. Thus, the latent period could be shorter or even longer than the incubation period. As seen before, $SEI$ and $SEIS$ are two examples of models with latent period. Models with higher dimensions such as $SEIR$, $SEIRS$, $MSEIR$, and $MSEIRS$ have also been reported in the literature \cite{Ma2008}. However, since these models cannot be reduced to planar differential equation systems due to their complexity, only a few complete analytic results have been obtained.
\subsection{Quarantine and Vaccination}
In absence of vaccination for an outbreak of a new disease, isolation of diagnosed infectives and quarantine of people who are suspected of having been infected are some of the few control measures available. Models such as $SIQS$ and $SIQR$ with a quarantined compartment, denoted by $Q(t)$, assume that all infectives go through the quarantined compartment before recovering or becoming susceptible again \cite{Hethcote2002}. However, vaccination, if available, is one of the most cost-effective methods of preventing disease spread in a population. We refer the reader to \cite{Keeling2011}, \cite{Vynnycky2010}, and \cite{Arino2003} for more on models with vaccination and vaccine efficacy.
\subsection{Time Delay}
Models with time delay deal with the fact that the dynamic behavior of disease transmission at time $t$ depends not only on the current state but also on the state of previous time \cite{Ma2009}, \cite{Arino2006}. Time delays are of two types namely, \emph{discrete} (or \emph{fixed}) delay and \emph{continuous} (or \emph{distributed}) delay. In the case of discrete time delay, the behavior of the model at time $t$ depends on the state at time $t-\tau$ as well, where $\tau$ is some fixed constant. As an example, with $\tau$ being the latent period for some disease, the number of infectives at time $t$ also depends on the number of infectives at time $t-\tau$. On the contrary, the behavior of a model with continuous delay at time $t$ depends on the states during the whole period prior to $t$ as well.
\subsection{Age Structure}
Since individuals in different age groups have different infection and mortality rates, age is considered to be an important characteristic in modeling infectious diseases. Mostly, cases of sexually transmitted diseases (STDs) such as AIDS occur in younger individuals as they tend to be more active within or between populations. Likewise, malaria is responsible for nearly half of the death of infants under the age of 5 due to their weak immune system. Hence, such facts highlight the importance of age structure in epidemic modeling. Age-structured models are broadly classified into three types namely, \emph{discrete} \cite{Yicang2004}, \emph{continuous} \cite{Li2008}, and \emph{age groups} or \emph{stages} \cite{Hethcote1997}.
\subsection{Multiple Groups}
In epidemiology, multi-group models describe the spread of infectious diseases in heterogeneous populations where each heterogeneous host population can be divided into several homogeneous groups in terms of geographic distributions, models of transmissions, and contact patterns. One of the pioneer models with multiple groups was investigated by Lajmanovich \emph{et al.} in \cite{Lajmanovich1976} for the transmission of gonorrhea. However, recent studies include differentiation of susceptibility to infection (DS) due to genetic variation of susceptible individuals, variation in infectiousness, and disease spread in competing populations \cite{Ma2009}.
\subsection{Migration}
A central assumption in the classical models seen so far is that the rate of new infections is proportional to the mass action term. In these models, we assumed that the infected and susceptible individuals mix homogeneously. An increasingly important issue in epidemiology is how to extend these classical formulations to adequately describe the spatial heterogeneity in the distribution of susceptible and infected people and in the parameters of the spread of the infection observed in both, experimental data and computer simulations. Diffusion or migration of individuals in space are simply of two types: \emph{migration among different patches} and \emph{continuous diffusion in space}. In the former type, migration of individuals between patches depend on the connectivity of the patches. Models with migration between two patches \cite{Hethcote1976} and \emph{n} patches \cite{Wang2004} have been reported in the literature. The latter type, on the other hand, takes into account the fact that the distribution of individuals and their interactions depend not only on the time \emph{t}, but also the location in a given space.
\subsection{Non-linear Forces of Infection}
Most of the classical epidemic models admit threshold dynamics, i.e. a $DFE$ is stable if $\mathcal{R}_0<1$ and an $EE$ is stable if $\mathcal{R}_0>1$. However, Capasso \emph{et. al} \cite{Capasso1997} showed that it is likely possible for a $DFE$ and $EE$ to be stable simultaneously. Futhermore, periodic oscillations have been observed in the incidence of various diseases including mumps, chickenpox, influenza, and the like. The question that arises is why are classical epidemic models unable to capture these periodic phenomena? The main reason is the nature of the force of infection. Classical models frequently use mass incidence and standard incidence which imply that the contact rate and infection probability per contact are constant in time. Nonetheless, it is more realistic (with added complexity) to consider the force of infection as a periodic function in time. As a simple example, consider the $SIR$ model with a periodic incidence function $F(I,t)$, and birth and death rates taken to be $\mu$ as given below \cite{Diallo2008}:
\begin{eqnarray}
\label{nonlinearSIR}
\frac{dS}{dt}&{}={}& \mu N - F(I, t)\frac{S}{N} - \mu S,\\
\frac{dI}{dt}&{}={}& F(I, t)\frac{S}{N} - (\gamma+\mu) I,\\
\frac{dR}{dt}&{}={}& \gamma I - \mu R.
\end{eqnarray}
In recent years, much attention has been given to the study and analysis of chaotic behavior in epidemic models with non-linear infection forces.
\section{Conclusion}
Mathematical modeling of communicable diseases has received considerable attention over the last fifty years. A wide range of studies on epidemic models has been reported in the literature. However, there lacks a comprehensive study on understanding the dynamics of simple deterministic models through implementation. Aiming at filling such a gap, this work introduced some widely-appreciated epidemic models and studied each in terms of mathematical formulation, near equilibrium point stability analysis, and threshold dynamics with the aid of \emph{Mathematica}. In addition, important factors that may be considered for better modeling were also presented. We believe that this article would serve as a good starting point for readers new to this research area and/or with little mathematical background.
\bibliographystyle{ieeetran}
|
2,877,628,089,007 | arxiv | \section{INTRODUCTION}
When the surface emission from a spinning neutron star is not uniform,
a periodic brightness oscillation is produced as the hot and cold
spots spin in and out of the line of sight of a distant observer. Such
brightness variations may be caused by the magnetic field topology on
the stellar surface of a pulsar, by the non-uniform thermonuclear
burning on the surface of an X-ray burster, or by the anisotropic
accretion of matter from a companion star. The amplitudes and shapes
of the resulting pulsations are determined not only by the brightness
contours on the stellar surface but also by the degree of strong-field
gravitational lensing that photons experience on their paths to the
distant observer (Pechenick et al. 1983; Strohmayer et al. 1997). For
this reason, pulse profile modeling is a powerful method for measuring
neutron-star masses and radii (see Strohmayer 2004). An advantage of
this method is that it does not require a measurement of the distance
to the neutron star. Two future X-ray missions, NASA's approved {\em
NICER\/} (Gendreau et al.\ 2012) and ESA's proposed {\em LOFT\/}
(Feroci et al.\ 2012), rely on pulse profile modeling to measure the
masses and radii of neutron stars in two classes of sources that show
surface brightness oscillations in the X-rays. {\em NICER\/} targets
the pulsed surface emission that has been detected from
rotation-powered millisecond pulsars, while {\em LOFT\/} is designed
to measure the pulse profiles of accretion-powered millisecond pulsars
and of thermonuclear bursters.
Earlier attempts to measure neutron-star properties from rotation-powered
(e.g., Pavlov \& Zavlin 1997; Bogdanov et al.\ 2007) and
accretion-powered millisecond pulsars (e.g., Poutanen \& Gierlinski
2003; Leahy et al.\ 2008) and bursters (Nath et al.\ 2002) resulted in
large, correlated uncertainties between the inferred masses and radii.
{\em NICER\/}'s design, which will allow accumulating a large number of counts
for each of its targets over very long integration times, and {\em LOFT}'s
large collecting area, which will lead to highly accurate pulse
profiles even during the course of a 10~s X-ray burst, will address
the issue of reducing the statistical uncertainties of the
measurements. However, even when the statistical errors are reduced,
significant correlations between the inferred parameters remain. This
has been recognized in earlier studies (e.g., Nath et al.\ 2002; Poutanen
\& Beloborodov 2006) and has been demonstrated more recently in a detailed
study of parameter estimation using mock {\em LOFT} data (Lo et al.\ 2013).
In this paper, we use simulated pulse profiles from spinning neutron
stars in order to identify the origin of the degeneracies in the
measurements of masses and radii that are obtained with this
technique. We use a Fourier series decomposition of the pulse profiles
at different photon energies to quantify the number of distinct
observables that can be measured from each profile. We show that,
because gravitational lensing suppresses the amplitudes of the high
harmonics, this number is rather small and is practically independent
of the number of phase bins used in the measurement. However, the
number of independent parameters that are required to uniquely
characterize each system is rather large. Therefore, the effective
number of degrees of freedom in comparing theoretical models to data
is very small or zero, causing the observed correlations between
parameters.
We further demonstrate that obtaining pulse profiles at different photon
energies significantly reduces the extent of these correlations. This
is because the modulation of the spectrum due to Doppler effects at
moderate spin frequencies introduces a photon-energy dependent
structure to the pulse profiles. Observing, therefore, pulse profiles
in multiple energy bands leads to measuring additional, uncorrelated
observables, thereby increasing the effective number of degrees of
freedom. The Fourier series decomposition approach that we present
here can be used in defining the optimal ranges of photon energies and
in formulating analysis strategies that maximize the effective number
of degrees of freedom. Moreover, it provides a useful
order-of-magnitude estimate for the number of photons that are
required to be accumulated and for the level of background that can be
accommodated in order for a specified precision to be reached in radius
measurements.
\section{CALCULATIONS OF PULSE PROFILES}
We use the ray tracing algorithm described in Psaltis \& \"Ozel (2013)
to calculate the brightness oscillations detected by an observer at
infinity that arise from a circular, uniform hot spot of angular
radius $\rho$ on the surface of a spinning neutron star. The observer
and the center of the spot are located at an inclination $i$ and
at a colatitude $\theta_{\rm s}$, respectively, with respect to the
stellar spin axis. We assume that the emission from the hot spot has
a blackbody spectrum and is isotropic in the local Lorentz frame on
the stellar surface. Note that, in general, the beaming of the
emerging radiation may not be isotropic, but will depend on the
particular type of system under consideration (see also end of \S3.1).
The targets of interest for missions such as {\em NICER\/} and {\em
LOFT\/} spin at $\sim 200-700$~Hz. The spacetime around such
neutron stars can be uniquely described by the Hartle-Thorne metric
(Hartle \& Thorne 1968). Calculations within this metric allow us to
accurately account for the effects of Doppler shifts and aberration,
of frame dragging, as well as of the oblateness of the stellar surface
and of its quadrupole moment\footnote{Simulations of pulse profiles
for neutron stars spinning at $\gtrsim 700$~Hz can only be performed
with numerical spacetimes, which depend on the details of the
equation of state (see Cadeau et al.\ 2007 and discussion in Psaltis
\& \"Ozel 2013).}. Morsink et al.\ (2007) and Psaltis \& \"Ozel
(2013) showed that all these effects need to be taken into account in
order for measurements of neutron-star masses and radii via pulse
profile modeling to reach the $\sim 5-10$\% accuracy required to
distinguish between equations of state (e.g., Lattimer \& Prakash
2001; \"Ozel \& Psaltis 2009).
In this setup, eight distinct parameters are required to fully specify
the geometry of the system and the spacetime of a neutron star
spinning at a known frequency $f$:\\
\hspace*{0.5cm}{\em (i)\/} the mass $M$ of the neutron star;\\
\hspace*{0.5cm}{\em (ii)\/} the equatorial radius $R_{\rm eq}$ of the neutron
star;\\
\hspace*{0.5cm}{\em (iii)\/} the ellipticity of its surface
$\epsilon_{\rm s}$;\\
\hspace*{0.5cm}{\em (iv)\/} its specific spin angular momentum $a\equiv
2\pi I f c/GM^2$, where $I$ is its moment of inertia;\\
\hspace*{0.5cm}{\em (v)\/} the quadrupole moment of its spacetime as measured
by the parameter $\eta$;\\
\hspace*{0.5cm}{\em (vi)\/} the observer inclination $i$;\\
\hspace*{0.5cm}{\em (vii)\/} the colatitude of the spot $\theta_{\rm s}$; and \\
\hspace*{0.5cm}{\em (viii)\/} the angular radius of the spot $\rho$.
If the emission originates from two localized hot spots, as in
the case of polar-cap emission from rotation-powered millisecond
pulsars (see, e.g., Bogdanov et al.\ 2007), then up to two additional
angles may be needed to specify the relative position of the two spots
on the stellar surface.
\begin{figure}[t]
\psfig{file=f1.eps,width=3.5in,clip=}
\caption{Pulse profiles (photon flux as a function of rotational
phase) generated by a circular hot spot on the surface of a neutron
star spinning at 1~Hz. The solid line corresponds to a neutron star
with $M=1.6 M_\odot$ and $R_{\rm eq}=10$~km and the crosses to a
neutron star with $M=1.8 M_\odot$ and $R_{\rm eq}=11$~km, where we
chose the masses and radii so that the two stars have comparable
$M/R_{\rm eq}$. The hot spot has a radius of $\rho=10^\circ$, is
located at a colatitude of $\theta_{\rm s}=40^\circ$, and is
observed from an inclination of $i=30^\circ$ with respect to the
stellar spin axis. In order to demonstrate that, at low spin
frequencies and for a wide range of geometries, the pulse profile is
nearly sinusoidal, we use open circles to show the pulse profile for
the 1.6~$M_\odot$ star, when we have suppressed all the harmonics
beyond the fundamental. The similarity between all three pulse
profiles demonstrates visually that pulse-profile modeling for
slowly spinning neutron stars suffers from a large degeneracy
between the inferred mass and radius.}
\label{fig:profiles}
\end{figure}
\begin{figure}[t]
\psfig{file=f2.eps,width=3.5in,clip=}
\caption{Contours of constant fractional rms amplitude of pulsations
generated by a hot spot on the surface of a neutron star spinning at
1~Hz, as a function of the assumed stellar mass and radius. The
remaining parameters of the calculation are the same as in
Figure~\ref{fig:profiles}. The contours lie along lines of constant
$M/R_{\rm eq}$. This is expected given that the fractional rms
amplitude depends on the amount of gravitational lensing experienced
by the photons and the latter, for a slowly spinning neutron star,
depends only on $M/R_{\rm eq}$.}
\label{fig:rms_1Hz}
\end{figure}
The observed pulse profiles are affected by each of these eight
parameters and could, in principle, contain adequate structure to
allow for uncorrelated measurements of all of them. However, as we
will show in the following section, gravitational light bending smears
the profiles and effectively erases some of the structure that encodes
the detailed properties of the neutron star and of the spacetime. As a
result, realistic pulse profiles do not contain enough information to
measure these eight parameters independently, even at the
signal-to-noise ratios expected when a large number of
photons is collected.
However, tight relations exist between several of the above
macroscopic quantities that depend very weakly on the equation of
state (e.g., Morsink et al.\ 2007; Yagi \& Yunes 2013; Baub\"ock et
al.\ 2013) and can be used to reduce the number of free parameters
that are necessary to model pulse profiles. In particular, hereafter,
we will use relations that connect the parameters $\epsilon_{\rm s}$,
$a$, and $\eta$ to $M$ and $R_{\rm eq}$ (Baub\"ock et
al.\ 2013). Finally, when the angular size of the spot is small
($\rho\lesssim 10^\circ$), the pulse profile does not depend on this
parameter (see, e.g., Bogdanov et al.\ 2007). Therefore, for systems
in which the surface emission is highly localized, as is expected to
be the case during the first fraction of a second of an X-ray burst
before the burning front has propagated to a significant distance away
from the ignition point (Strohmayer et al.\ 1997, 1998) and for
polar-cap heating in the case of rotation-powered pulsars (e.g.,
Bogdanov 2013), the spot size can be eliminated as a parameter. As a
result, the pulse profile is determined only by four parameters: $M$,
$R_{\rm eq}$, $i$, and $\theta_{\rm s}$. In the following section, we
show that these four parameters can be independently inferred from
realistic pulse profiles if we use neutron stars that spin at moderate
rates and utilize the photon-energy dependence of the profiles.
\begin{figure}[t]
\psfig{file=f3.eps,width=3.5in,clip=}
\caption{Pulse profiles generated by a circular hot spot on the
surface of a neutron star spinning at 1~Hz. In this case, the hot
spot has a radius $\rho=10^\circ$, is located at a colatitude
$\theta_{\rm s}=80^\circ$, and is observed from an inclination
$i=90^\circ$ with
respect to the spin axis, such that it is occulted
by the neutron-star surface for a fraction of the spin period. The
solid line shows a truncated sinusoid that best describes the result
of the ray-tracing calculation. Even though the occultation generates
a large number of measurable harmonics, the pulse profile can be
accurately described by only two numbers: its amplitude and the
duration of the occultation.}
\label{fig:profile_occult}
\end{figure}
\section{Measuring Neutron-Star Parameters from Pulse Profile Modeling}
\subsection{Slowly Spinning Neutron Stars}
The external spacetime of a slowly spinning neutron star depends only
on its compactness $GM/Rc^2$. Therefore, modeling pulse profiles
observed from such systems can only lead to a measurement of $M/R_{\rm
eq}$ and not of the two parameters independently. We illustrate this
degeneracy in Figure~\ref{fig:profiles}, where we show the pulse
profiles from two slowly spinning (1~Hz) neutron stars, with
substantially different masses and radii but with very similar
compactness. In Figure~\ref{fig:rms_1Hz}, we further demonstrate the
degenerate dependence of pulse profiles on $M/R_{\rm eq}$. In
particular, we plot contours of constant fractional root-mean-squared
(rms) amplitude on the mass-radius plane for a neutron star spinning
at 1~Hz, while keeping fixed the inclination of the observer to
$i=30^\circ$ and the colatitude of the hot spot to $\theta_{\rm
s}=40^\circ$. As expected, the contours are lines of constant
$M/R_{\rm eq}$. In the next section, we will discuss how these
contours change as a function of the neutron-star spin frequency.
\begin{figure}[t]
\psfig{file=f4.eps,width=3.5in,clip=}
\caption{The pulse profile and the phase dependence of a spectral
color for a neutron star spinning at 600~Hz. In this calculation,
the colatitude of the spot is $\theta_{\rm s}=40^\circ$ and the
inclination of the observer is $i=60^\circ$. The spectral
color is defined here as the ratio of the number of photons with
energies above the temperature of the blackbody emission to the
number of those below. The peak of the spectral color occurs close
to the phase at which the tangential velocity of the surface is
maximum. On the other hand, the peak of the radiation flux occurs
close to the phase at which the projected area of the hot spot is
maximum. For this reason, the former precedes the latter. The dashed
line shows the sinusoid that has the same amplitude as that of
the fundamental harmonic of the oscillations.}
\label{fig:color}
\end{figure}
\begin{figure}[t]
\psfig{file=f5.eps,width=3.5in,clip=}
\caption{Amplitudes of the fundamental and second harmonic for the
pulsations originating on the surface of a 1.6~$M_\odot$, 10~km
neutron star, as a function of its spin frequency, for two different
inclinations of the observer (30$^\circ$ and 60$^\circ$). The
remaining parameters of the
calculation are the same as in Figure~\ref{fig:profiles}. At slow
spins, the pulse profiles are highly sinusoidal whereas, at higher
spins, Doppler effects introduce asymmetries to the pulse profiles
and increase the amplitudes of the higher harmonics. Even at spin
frequencies as high as 600~Hz, the amplitude of the second harmonic
is $\simeq 10$ times lower than that of the fundamental for these
geometries. The lines show the approximate scaling of
equation~(\ref{eq:scaling}) for the amplitude of the harmonic, given
the calculated amplitude of the fundamental. The amplitude of the
fundamental increases slightly with spin frequency because of the
increase in the peak-to-peak excursion caused by Doppler boosts.}
\label{fig:rms_spin}
\end{figure}
Figure~\ref{fig:profiles} also demonstrates that the pulse profile is
highly sinusoidal by comparing the result of the ray-tracing
calculation to a pure sinusoid with the appropriate phase and
amplitude. This implies that, even if the signal to noise of an
observation allows splitting the observed pulse profile into a large
number of phase bins, the complete information content in the profile
is captured by a single quantity: the amplitude of the sinusoid.
In other words, if we decompose the pulse profile of a slowly spinning
neutron star into a Fourier series, only the amplitude of the
fundamental will be measurable.
When the geometry is such that the hot spot is occulted by the stellar
surface for a fraction of the spin phase, a large number of harmonics
will be present. However, these additional harmonics acquire large
amplitudes only because of the truncation of the otherwise sinusoidal
profile due to the occultation (Gibbs phenomenon). In this case, the
total information content in the pulse profile is represented by only
two quantities, e.g., the fractional rms amplitude of the oscillation and the
duration of the occultation (see Fig.~\ref{fig:profile_occult}).
\begin{figure*}[t]
\centerline{
\psfig{file=f6a.eps,width=3.5in,clip=}
\psfig{file=f6b.eps,width=3.5in,clip=}}
\caption{Contours of constant {\em (Left)\/} fractional rms amplitude
and {\em (Right)\/} ratio of the amplitude of the harmonic to that of
the fundamental for pulsations generated by a hot spot on the
surface of a neutron star spinning at 600~Hz, as a function of the
stellar mass and radius. The remaining parameters of the
calculation are the same as in Figure~\ref{fig:profiles}. For a
fixed neutron-star spin frequency, the stellar radius determines the
magnitude of the Doppler effects, which themselves determine
predominantly the harmonic content of the pulse profiles. For this
reason, the contours shown in the right panel are nearly vertical
and correspond to lines of nearly constant radius.}
\label{fig:rms_600Hz}
\end{figure*}
\begin{figure*}[t]
\centerline{
\psfig{file=f7a.eps,width=3.5in,clip=}
\psfig{file=f7b.eps,width=3.5in,clip=}}
\caption{Contours of constant {\em (Left)\/} fractional rms amplitude and {\em
(Right)\/} ratio of the amplitude of the harmonic to that of
the fundamental for pulsations generated by a hot spot on the
surface of a neutron star spinning at 600~Hz, as a function of the
colatitude of the spot and the inclination of the observer.
The remaining parameters of the calculation are the same as in
Figure~\ref{fig:profiles}. In both cases, the contours lie
primarily along curves on which the product $\sin i \sin\theta_{\rm s}$
is constant.}
\label{fig:rms_600Hz_theta}
\end{figure*}
When we compare the number of unique pieces of information encoded in
the pulse profile of a slowly spinning neutron star (i.e., two in a
geometry with occultation or one without) to the number of parameters
required to describe the system (i.e., three: $M/R_{\rm eq}$,
$i$, and $\theta_{\rm s}$), it becomes apparent that pulse
profile modeling in such a system can only result in highly correlated
measurements of its parameters. Additional structure in the pulse
profiles may also be present due to an anisotropic beaming of
radiation on the stellar surface (e.g., Poutanen \& Beloborodov 2006).
However, measuring the amplitudes of the higher harmonics in this case
will only provide information about the beaming of radiation and not
about the neutron-star properties.
\subsection{Moderately Spinning Neutron Stars}
When a neutron star is spinning at moderate rates ($\sim 300-800$~Hz),
the nearly relativistic velocity of its surface causes three phenomena
that introduce complexity to the pulse profiles: Doppler shifts of the
photon energies, aberration in their angular distribution, and time
delays between photons emitted at different spin phases. We show in
Figure~\ref{fig:color} the deviation of the pulse profile from a pure
sinusoid for a spot at 40$^\circ$ colatitude on a neutron star
spinning at 600~Hz, and observed from a 30$^\circ$ inclination. As found
in earlier studies (e.g., Braje et al.\ 2000), the pulse profile
becomes asymmetric and peaks at an earlier phase compared to the sinusoid.
We quantify the degree of structure in the pulse profiles by comparing
the amplitude of the second harmonic to that of the fundamental as a
function of the spin frequency in Figure~\ref{fig:rms_spin}. As
expected, the amplitude of the second harmonic increases significantly
with increasing spin frequency and is more than an order of magnitude
larger for a 600~Hz star compared to a slowly spinning one. The
harmonic amplitudes also depend strongly on the observer's inclination
and are much larger for an observer located closer to the rotational
equator. Indeed, the ratio of the harmonic amplitude to that of the
fundamental scales approximately as (see Poutanen \& Beloborodov 2006)
\begin{eqnarray}
\frac{C_2}{C_1}&\simeq&2\left(\frac{2\pi f R_{\rm eq}}{c}\right)
\sin i \sin\theta_{\rm s}\nonumber\\
&=&0.126\left(\frac{f}{300~\mbox{Hz}}\right)
\left(\frac{R_{\rm eq}}{10~\mbox{km}}\right)
\sin i \sin\theta_{\rm s}\;.
\label{eq:scaling}
\end{eqnarray}
This approximate scaling is shown in Figure~\ref{fig:rms_spin}, for
two different observer inclinations, and matches the results of the
numerical calculation. Note that the primary scaling is due to the
first-order Doppler effect while higher-order corrections (due to the
oblateness and the quadrupole moment of the neutron star) affect
primarily the numerical factor in this last equation. Using the
simpler Schwarzschild+Doppler approximation (e.g., Miller \& Lamb
1998; Poutanen \& Beloborodov 2006; Lo et al. 2013), as opposed to the
Hartle-Thorne metric we use here, therefore, leads only to a
systematic bias in the measurement and not to qualitatively different
uncertainties.
It is evident from equation~(\ref{eq:scaling}) that a marked
difference between slowly and moderately spinning neutron stars is
that in the latter case, the amplitude of the second harmonic shows a
strong dependence on the neutron star radius. The amplitudes of
higher harmonics can, therefore, be useful for breaking the degeneracy
between the stellar mass and radius that we discussed in \S3.1. This
is shown in Figure~\ref{fig:rms_600Hz}, in which contours of constant
fractional rms amplitude (left panel) and of the ratio of the amplitudes of
the second harmonic to the fundamental (right panel) are plotted on
the mass-radius parameter space. In this calculation, the neutron star
is spinning at 600~Hz, the inclination of the observer is
$i=30^\circ$, and the colatitude of the hot spot is
$\theta_{\rm s}=40^\circ$. (Note that this geometry is far from the
one that maximizes the ratio $C_2/C_1$.) The contours of constant
fractional rms amplitude (left panel) are primarily along lines of constant
compactness, as in the case of slowly spinning neutron stars. However,
they bend upwards at large radii because of the increase in the
peak-to-peak flux excursion caused by Doppler boosts. In contrast,
the contours of constant amplitude ratios (right panel) are primarily
vertical, since the harmonic amplitudes increase with stellar radius,
as shown in eq.~[\ref{eq:scaling}]). The weak mass dependence of these
contours arises from the effects of gravitational lensing and from the
redshift factors that need to be taken into account when computing the
velocity of the stellar surface in the local Lorentz frame.
Figure~\ref{fig:rms_spin} demonstrates that, even at the high end of
the observed spin frequencies, the ratio of the amplitude of the
second harmonic to that of the fundamental is quite small; naturally,
the amplitudes of the higher harmonics are even smaller (see, e.g.,
Poutanen \& Beloborodov 2006). Moreover, the right panel of
Figure~\ref{fig:rms_600Hz} shows that the ratio of the amplitude of
the harmonic to that of the fundamental needs to be measured to a
$\sim 10$\% fractional accuracy in order for the observations to
distinguish between neutron-star radii that differ by $\sim 1$~km. As
we will show in \S4, this is quite a severe requirement and makes it
unlikely that future observations will be able to extract more than
two measurable quantities from bolometric pulse
profiles.\footnote{This number can be increased by one if the
geometry of the system is such that the hot spot is occulted for a
fraction of the spin phase (see the discussion in \S3.1 for the case
of slowly spinning neutron stars).} Therefore, even in this case,
the independent pieces of information in realistic measurements still
falls short of the number of system parameters (i.e., four) that need
to be determined. In principle, the relative phase of the harmonic and
the fundamental provides additional information. However, in practice,
the phase is too poorly determined due to the weakness of the harmonic.
In Figure~\ref{fig:rms_600Hz_theta}, we take a different cut through
the four-dimensional parameter space and plot the dependence of the
fractional rms amplitude (left panel) and the ratio of the amplitude of the
harmonic to that of the fundamental (right panel) as a function of the
spot colatitude and the observer's inclination. In both panels, the
contours of constant amplitude and the contours of constant harmonic
ratio lie along curves on which the product $\sin i
\sin\theta_{\rm s}$ is nearly constant. This implies that, if there is
no occultation, the two measurable quantities cannot be used to infer
the two angles independently. However, given that constraining these
two angles independently is often not of interest, the above
combination $\sin i \sin\theta_s$ can be treated as a single
nuisance parameter, thus reducing the number of system parameters that
need to be measured to three. We will now show that the photon energy
dependence of the pulse profiles can provide the additional pieces of
information and break the remaining degeneracy between parameters.
\begin{figure}[t]
\psfig{file=f8.eps,width=3.5in,clip=}
\caption{The evolution of the radiation spectrum, as observed at
infinity at an inclination of $30^\circ$, generated by a hot spot
on the surface of a neutron star spinning at 600~Hz. The other
parameters of the calculation are the same as in Figure~\ref{fig:rms_spin}.
The various curves correspond to different spin pßhases, with zero
representing the phase at which the center of the hot spot and the
observer are on the same meridian. Comparing the ordering of the
curves at very low and very high photon energies reveals a strong
color oscillation during a spin cycle.}
\label{fig:spectrum}
\end{figure}
\begin{figure*}[t]
\centerline{
\psfig{file=f9a.eps,width=3.5in,clip=}
\psfig{file=f9b.eps,width=3.5in,clip=}}
\caption{Contours of constant {\em (Left)\/} fractional rms amplitude of a
spectral color oscillation and {\em (Right)\/} difference between
the phase of peak spectral color and of peak radiation flux, for
pulsations generated by a hot spot on the surface of a neutron star
spinning at 600~Hz, as a function of the stellar mass and
radius. The remaining parameters of the calculation are the same as
in Figure~\ref{fig:spectrum}. The color oscillations are introduced
primarily by Doppler effects and, therefore, the contours on the
left panel are nearly vertical, as in the right panel of
Figure~\ref{fig:rms_600Hz}. Understanding the shape of the contours in
the right panel of this figure is more subtle and is described in
detail in the text.}
\label{fig:color_rms}
\end{figure*}
\begin{figure*}[t]
\centerline{
\psfig{file=f10a.eps,width=3.5in,clip=}
\psfig{file=f10b.eps,width=3.5in,clip=}}
\caption{Contours of constant {\em (Left)\/} fractional rms amplitude of a
spectral color oscillation and {\em (Right)\/} difference between
the phase of peak spectral color minus that of peak radiation flux,
for pulsations generated by a hot spot on the surface of a neutron
star spinning at 600~Hz, as a function of the colatitude of the spot
and the inclination of the observer. The remaining parameters of the
calculation are the same as in Figure~\ref{fig:spectrum}. The
contours of the fractional rms amplitude of the spectral oscillation
approximately follow curves on which the product $\sin i
\sin\theta_{\rm s}$ is constant, as in the case of the first two
observables shown in Figure~\ref{fig:rms_600Hz_theta}. In contrast,
the phase difference results in a nearly orthogonal constraint and
provides the information needed to measure the two angles
independently.}
\label{fig:color_rms_theta}
\end{figure*}
\subsection{The Photon Energy Dependence of Pulse Profiles}
It is well understood that the Doppler effects that increase the
amplitudes of the higher harmonics in the pulse profile also introduce
a modulation to the radiation spectrum observed at infinity. This is
shown for a typical set of parameters in Figure~\ref{fig:spectrum} at
four different spin phases. As expected, the radiation spectrum is
softer in the range of spin phases for which the hot spot is receding
from the observer and harder when the hot spot is approaching.
We can quantify the degree of spectral modulation during a pulse phase
by defining a color as the ratio between the number of photons
observed in two energy bands separated at an energy equal to the
hot-spot temperature. In Figure~\ref{fig:color}, we plot the evolution
of such a color with pulse phase for a typical configuration, where we
use a photon energy equal to the hot-spot temperature (as measured at
the neutron-star surface) to separate the two energy bands. The
evolution of the color with pulse phase shows a nearly sinusoidal
modulation with a significant fractional rms amplitude and a peak
phase that is offset from that of the flux oscillation. As we will
show below, the amplitudes and the peak phases of the color
oscillations do not have the same dependence on the system parameters
as the amplitudes of the harmonics of the bolometric flux. For this
reason, they provide the two additional observables needed to break
the degeneracies discussed in \S3.2 and allow a measurement of all the
system parameters. This is true even when using the minimal spectral
information encoded in one spectral color. If the number of photons
accumulated during an observation allow separating the pulse profiles
into several energy bands, additional consistency relations between
model parameters can be obtained.
Figure~\ref{fig:color_rms} shows contours of constant fractional
rms amplitude of the color oscillations (left panel) and of the
phase difference between the phase of peak color minus the phase of
peak flux (right panel) on the mass-radius parameter space. If there
were no gravitational lensing and redshift effects, the amplitude of
the color oscillations would be strictly proportional to the radius of
the neutron star and would have a scaling similar to the one given in
equation~(\ref{eq:scaling}). However, gravitational effects introduce
a weak dependence on the neutron-star compactness such that the
contours of constant color amplitude shown in
Figure~\ref{fig:color_rms} are not parallel to the contours of
constant harmonic ratios shown in the right panel of
Figure~\ref{fig:rms_600Hz}.
The dependence of the contours of constant phase difference between
the color and flux oscillations on mass and radius shown in the right
panel of Figure~\ref{fig:color_rms} is more subtle. The peak phases of
the two oscillations are determined by the combination of the
evolution of the projected surface area of the hot spot on pulse phase
(which peaks at phase zero) and of the Doppler effects (which peak at
phase 0.75). The flux oscillation is determined predominantly by the
former effect, peaks close to zero phase, and the Doppler boosts
introduce small corrections, moving the peak toward earlier phases. On
the other hand, the color oscillation is determined predominantly by
the Doppler effects, peaks close to phase 0.75, and the surface area
projection introduces a small correction, moving the peak toward later
phases. For small values of the neutron-star compactness, for which
the gravitational lensing effects are weak, the relative shift between
the two peak phases is dominated by Doppler effects and, therefore,
the contours shown in Figure~\ref{fig:color_rms} become more
vertical. At high values of the neutron-star compactness, for which
the relative shift between the two peak phases is dominated by
gravitational lensing effects, the contours become nearly horizontal.
The contours of the four observables shown in Figures~\ref{fig:rms_600Hz} and
\ref{fig:color_rms} on the mass-radius parameter space do not have the same
dependence on the system properties. This leads to the conclusion
that measuring with sufficient accuracy the amplitudes of the lowest
two harmonics of the bolometric flux oscillation, as well as the
amplitude and relative phase of the spectral color oscillation is
adequate to uniquely determine all four parameters of each observed
system, as we will discuss in the next section.
We explore in Figure~\ref{fig:color_rms_theta} the dependence of the
fractional rms amplitude of the color oscillations (left panel) and
the phase difference between the flux and the color oscillations
(right panel) on the spot colatitude and the inclination of the
observer. The contours of constant fractional rms color amplitude lie
along curves on which the product $\sin i \sin\theta_{\rm s}$ is
nearly constant, as was the case with the two observables obtained
from the bolometric flux oscillations (see
Fig.~\ref{fig:rms_600Hz_theta}). This is expected given that the color
oscillations are also generated by Doppler effects and are determined
by the projection of the vector of the surface velocity along the
instantaneous line-of-sight between the observer and the hot spot. In
contrast, the phase difference between the flux and the color
oscillations shows a significantly different dependence on the two
angles. Therefore, if measuring the two angles independently is a goal
in and of itself, this last observable provides the additional piece
of information necessary to achieve it.
\begin{figure*}[t]
\centerline{
\psfig{file=f11a.eps,width=3.5in,clip=}
\psfig{file=f11b.eps,width=3.5in,clip=}}
\caption{Constraints on {\em (Left)\/} the neutron-star mass and
equatorial radius and {\em (Right)\/} the inclination of the
observer and colatitude of the hot spot, obtained from measuring
four properties of a pulse profile. The measured values of the
fractional rms amplitude of the pulse profile, the fractional rms
amplitude $C_2$
of the harmonic, the fractional rms amplitude of the color oscillation,
and the phase difference $\Delta\phi$ between the flux and color
oscillations were assumed to be equal to $0.17\pm 0.005$, $0.018\pm
0.001$, $0.058\pm 0.001$, and $-0.136\pm 0.002$, respectively. The
central values for each parameter correspond to the results of the
calculation for a 1.6~$M_\odot$, 10~km neutron star, observed from
an inclination of $30^\circ$, with a hot spot on its surface at a
colatitude of $40^\circ$. Each panel represents a cut through the
four-dimensional parameter space on which the parameters not shown
remain constant. The distinct dependence of the four observables on
the system parameters allows a unique recovery of the assumed
neutron-star mass and radius.}
\label{fig:measurements}
\end{figure*}
\section{Prospects for Measuring Neutron-Star Properties from
Pulse Profile Modeling}
In \S2, we showed that the pulse profile observed from a neutron star
spinning at a moderate rate can be accurately described by four
parameters: the mass of the neutron star, its equatorial radius, the
inclination of the observer, and the colatitude of the hot spot on the
stellar surface. In \S3, we demonstrated that observations of the
photon energy dependent pulse profiles result in at least four
measurable quantities that have a distinct dependence on the model
parameters: the amplitude of the bolometric flux oscillation, the
amplitude of its second harmonic, the amplitude of the spectral color
oscillation, and the phase difference between the bolometric flux and
the color oscillations.
The first three of these four observables depend on the same
combination $\sin i \sin\theta_s$ of the two geometric
parameters. Therefore, if the main goal of the pulse profile modeling
is to measure the masses and radii of neutron stars, this combination
can be treated as a single parameter. In this case, the first three
observables lead to a unique determination of the system
parameters. Nonetheless, if all four observables can be measured with
sufficient accuracy, then the geometry of the system can also be
uniquely determined.
In \S 3, we also quantitatively explored the dependence of the four
observables on the neutron star mass and radius. Three of the four
observables acquire detectable amplitudes because of the relativistic
Doppler shifts on the rapidly spinning neutron-star surface. As a
result, the accuracy of the measurements will depend on the accuracy
at which the amplitudes of the harmonics and color oscillations can be
measured. Using, for example, equation~(\ref{eq:scaling}), we can
relate an uncertainty $\Delta C_2$ for the measurement of the amplitude
of the harmonic to an uncertainty for the inferred radius $\Delta R_{\rm eq}$
as
\begin{equation}
\frac{\Delta R_{\rm eq}}{R_{\rm eq}}=
\left[2C_1\left(\frac{2\pi f R_{\rm eq}}{c}\right)
\sin i \sin\theta_{\rm s}\right]^{-1}\Delta C_2 \;.
\label{eq:deltaR}
\end{equation}
As we show in the appendix, the uncertainty with which the amplitude
of a given harmonic can be measured depends on the total number of
source counts $S$ accumulated during the observations and on the total
number of background counts $B$ accumulated at the same time as
\begin{equation}
\Delta C_n=\frac{\sqrt{S+B}}{S}\;.
\label{eq:ampl_error}
\end{equation}
Combining the last two equations, we obtain
\begin{equation}
\frac{\Delta R_{\rm eq}}{R_{\rm eq}}=\left[
\left(\frac{4\pi f R_{\rm eq}}{c}\right)
\sin i \sin\theta_{\rm s}\right]^{-1} \left(\frac{\sqrt{S+B}}{C_1 S}\right)
\;.
\label{eq:final_unc}
\end{equation}
This equation provides the analytic understanding for the figure of
merit ${\cal R}$ introduced by Lo et al.\ (2013), which is closely
related to the quantity in the rightmost parentheses shown above.
When the number of source photons dominates that of the background,
i.e., when $S\gg B$, then the uncertainty in the measurement of the
neutron-star radius scales as
\begin{eqnarray}
\frac{\Delta R_{\rm eq}}{R_{\rm eq}}
&\simeq&0.055
\left(\frac{C_1}{0.3}\right)^{-1}
\left(\frac{f}{600~\mbox{Hz}}\right)^{-1}
\left(\frac{R_{\rm eq}}{10~\mbox{km}}\right)^{-1}\nonumber\\
&&\qquad
\left(\frac{\sin i}{0.5}\right)^{-1}
\left(\frac{\sin\theta_{\rm s}}{0.5}\right)^{-1}
\left(\frac{S}{10^6~\mbox{cts}}\right)^{-1/2}\;.
\end{eqnarray}
This relation suggests that achieving a 5\% accuracy in the
measurement of a neutron star radius from pulse profile modeling
requires accumulating of the order $10^6$ source counts.
Figure~\ref{fig:measurements} shows the constraints on the model
parameters that can be obtained from measuring the four observable
quantities discussed above with a precision that is characteristic of
an observation with one million source counts. In particular, we
assumed that all fractional amplitudes were measured with an accuracy
of $10^{-3}$ and the phase difference between flux and color
oscillations was measured with an accuracy of $5\times 10^{-4}$ (see
the Appendix). For this example, the simulated lightcurve was generated
for a $1.6 M_\odot$, 10~km neutron star spinning at 600~Hz, observed
at an inclination of $30^\circ$, with a small uniform hot spot on its
surface at a colatitude of $40^\circ$. The figure demonstrates that
the distinct dependence of the four observables on the system
parameters allows a unique recovery of the assumed neutron-star mass
and radius. Moreover, it also shows that the assumed uncertainties
lead to a measurement of the neutron star mass and radius that is
sufficient to distinguish between different equations of state.
\subsection{Importance of the Background Model}
We emphasize that three of the four observables
discussed above are fractional rms amplitudes of Fourier
harmonics. Measuring these fractional amplitudes requires obtaining
both the pulsed and unpulsed (``DC'') components of the pulse profiles. If an
additional background that does not originate on the neutron-star
surface is present in the observed energy band, this component needs
to be separately measured and subtracted. The alternative, i.e.,
measuring the properties of the additional background from the pulse
profiles themselves, introduces severe degeneracies between the
inferred model parameters, as shown in Lo et al.\ (2013). This is easy
to understand within the framework of counting system parameters and
observables that we followed here. Indeed, if we choose to perform
pulse profile analysis in two energy bands with unknown backgrounds,
then we introduce two additional parameters to our model, increasing
the total number to six. However, the number of observables that can
be inferred accurately from the pulse profiles remains equal to
four. This difference between the number of model parameters and
observables results in substantial degeneracies between the model
parameters of interest.
The approach we developed in this paper allows us to also investigate the
accuracy at which the number of background counts needs to be known
{\em a priori\/} in order for a desired accuracy in the mass and radius
measurement to be achieved. If we denote by $a_n$ the absolute amplitude
of the $n-$th Fourier component in a profile and by $N\equiv S+B$ the
total number of counts accumulated, then the fractional source amplitude
of the same Fourier component is simply
\begin{equation}
C_n=\frac{a_n}{S}=\frac{a_n}{N-B}\;.
\end{equation}
At least two effects, in principle, contribute to the uncertainty in
the measurement of the fractional source amplitude: the uncertainty in the
measurement of the absolute amplitude $\Delta a$ and the uncertainty
in the {\em a priori\/} knowledge of the background $\Delta B$.
Incorporating both sources of error, we obtain
\begin{eqnarray}
\Delta C_n^2&=&\frac{\Delta a_n^2}{(N-B)^2}+
\left[\frac{a_n}{(N-B)^2}\right]^2\Delta B^2\nonumber\\
&=&\left(\frac{\Delta a_n}{S}\right)^2+
\left(\frac{a_n}{S^2}\right)^2\Delta B^2\;.
\end{eqnarray}
The first term in the right-hand side of this equation is the Poisson
error in the measurement of the fractional amplitude and is given by
equation~(\ref{eq:ampl_error}). Assuming that the uncertainty in the
measurement of the background is Poisson dominated, $\Delta B=\sqrt{B}$.
Inserting these two expressions in the last equation, we obtain
\begin{equation}
\Delta C_n^2=\left(\frac{\sqrt{S+B}}{S}\right)^2+
C_n^2\frac{B}{S^2}
\end{equation}
or simply
\begin{equation}
\Delta C_n=\frac{\sqrt{S+B(1+C_n^2)}}{S}\;.
\label{eq:back_ampl}
\end{equation}
Equation~(\ref{eq:back_ampl}) shows that the background contributes in
two ways in the uncertainty of the measured fractional amplitude of
the source: the overall counts in the background increase the level of
the Poisson noise in the power spectrum and hence degrade the
measurement of the absolute amplitude of the pulsations. At the same
time, the uncertainty in the subtraction of the background counts
affects the inference of the fractional amplitude of the pulsations.
Because $C_n^2\ll 1$, the latter effect is always subdominant compared
to the former. We can, therefore, neglect it and simply use
equation~(\ref{eq:final_unc}) to infer the expected uncertainty in the
measurement of neutron-star radii when the observations have a
significant background.
\section{Conclusions}
In this paper, we investigated how pulse profiles generated by hot
spots on moderately spinning neutron stars can be used to infer the
stellar mass and radius. We showed that bolometric pulse profiles do
not contain sufficient information to break parameter degeneracies to
uniquely measure these quantities. However, a measurement of the
spectral color oscillations provides additional constraints that allow
a separate determination of $M$ and $R_{\rm eq}$. Extracting pulse
profiles even in just two different energy bands is sufficient to
derive this information. However, achieving $5\%$ precision in
neutron-star radius requires accumulating $\gtrsim 10^6$~counts in the
pulse profile measurements. This can be accomplished by long exposure
times (as in the case of {\em NICER}) or by a large collecting area
(as in the case of {\em LOFT}).
The requirements we presented here for making measurements of the
neutron star radius with a given precision are robust to the detailed
energy coverage and response of a particular instrument. Given that
our study has utilized idealized light curves, the details of, e.g.,
required number of counts, can be refined for a particular detector or
choice of energy bandpass. However, we note that because only two
energy channels are required to make the spectral color oscillation
measurement, even a modest energy resolution is sufficient. On the
other hand, a broad energy coverage is advantageous because the color
oscillations are more pronounced when measured over a wider energy
range. Ideally, the energy bandpass should include at least the
blackbody peak or the exponential tail above the peak.
X-ray spectral color oscillations have been previously reported in
several thermonuclear burst oscillations from two neutron stars
(Strohmayer et al. 1999; Strohmayer 2000). These color oscillations
were all measured during the burst decay phase and were found to be in
phase with the flux oscillations, contrary to the expectations discussed
above. This suggests that the color oscillations in the burst tails are not
generated by Doppler effects and are not appropriately modeled by the hot spot
model described above. This also corroborates other lines of arguments that
the oscillations in the burst tail are generated by a different mechanism
(e.g., surface modes) than those in the burst rise (see also Watts 2012 and
references therein).
In the case of oscillations observed during the rise phases of X-ray
bursts, which are the prime targets for {\em LOFT\/}, the predominant
background arises from the X-ray emission from the accretion flow. In
the case of surface emission from rotation powered pulsars, which are
the main targets for {\em NICER\/}, the non-thermal emission from the
neutron-star magnetosphere is the main source of the background. In
both cases, the energy spectrum of the pulsations is very different
from the energy spectrum of the background. If the shape of the energy
spectrum of the background is known, e.g., from theoretical models and
prior observations, then its overall normalization can be measured at
an energy band where the pulsed surface emission is negligible, i.e.,
at hard X-rays. Such an approach will lead to an independent
measurement of the background in the energy bands of interest and will
not adversely affect measuring neutron-star masses and radii from
pulse profile modeling.
\acknowledgements We thank the {\em NICER\/} and {\em LOFT\/} science
teams for many discussions on ray tracing in neutron-star spacetimes.
This work was supported in part by NSF grant AST-1108753, NSF CAREER
award AST-0746549, and {\em Chandra\/} Theory grant
TM2-13002X. F.\"O. gratefully acknowledges support from the Radcliffe
Institute for Advanced Study at Harvard University. D.P.\ and
F.\"O.\ thank the Institute for Theory and Computation at Harvard
University for their hospitality during the time that this work was
completed.
|
2,877,628,089,008 | arxiv |
\section{Introduction}
\subsection{Background}
A packing puzzle is a solitary game where a player tries to find a way to cover a given shape using polyominoes, where a polyomino is a set of squares joined together by their edges. The computational complexity of packing puzzles was studied by Demaine and Demaine \cite{DD07} who showed that tiling a shape or region using polyominoes is NP-complete.
In this work we study tilings of regions in the square lattice with L-shaped trominoes (a polyomino of three cells) called an \emph{L-Tromino} or simply tromino in this work. A cell in $\mathbb{Z}^2$ is a subset $[a,a+1]\times [b,b+1]$ and a region is any finite union of connected cells. At our disposal we have an infinite amount of trominoes and would like to know if a given region can be covered or tiled with trominoes.
The problem of tiling with trominoes was first studied by Conway and Lagarias \cite{CL90} who presented an algebraic necessary condition for a region in order to have a tiling. Moore and Robson \cite{MR01} showed that deciding if a region can be covered with trominoes is NP-complete. Later Horiyama \emph{et al.} \cite{HIN17} presented another proof of NP-completeness by constructing an one-one reduction which implies that counting the number of tilings with trominoes is \#P-complete. Counting the number of tilings with L-trominoes was also studied by Chin \emph{et al.} \cite{CGH07} using generating functions.
\subsection{Contributions}
In this work we aim at identifying instances of the tiling problem with trominoes that either have efficient algorithms
or it remains NP-complete. As a further generalization of the problem, we also consider regions with ``defects'' or holes,
that is, we want to know if there is a tiling with trominoes without covering the defects. First we study the
Aztec rectangle (and hence, also an Aztec diamond) \cite{EKLP92,MPS17} and show that any Aztec rectangle of side lengths $a,b$ can be covered with trominoes if and only if $a(b+1)+b(a+1)\equiv 0 \pmod 3$ (Theorem \ref{th-1}), which implies the existence of a polynomial time algorithm for
finding a tiling in an Aztec rectangle, and hence, an Aztec diamond. Then we showed that for the cases when
$a(b+1)+b(a+1)\equiv 0 \pmod 3$ does not hold, if an Aztec Rectangle has exactly one defect, then it can be covered with trominoes
(Theorem \ref{the:az-defect}). In general, however, deciding the tiling of an Aztec diamond with an unknown number of defects is
NP-complete (Theorem \ref{the:az-defect-hard}).
In the second part of this paper we study a restricted case of the tiling problem where we only have $180^\circ$ rotations
of the trominoes available. Here we show that the problem remains NP-complete (Theorem \ref{the:180tromino}) by slightly
modifying the one-one reduction from the 1-in-3 Graph Orientation Problem of Horiyama \emph{et al.} \cite{HIN17}, whereas
any Aztec rectangle has no tiling at all (Theorem \ref{the:Aztec-180}). Nevertheless, we show that if a region does not contain
any of the so-called ``forbidden polyominoes'' identified in this work, then that region has an efficient algorithm
for deciding a tiling (Theorem \ref{the:forbidden}). This latter result is proved by constructing a graph representation
of the region, called an intersection graph, and identifying independent sets of certain size. If the intersection graph
has a claw, then that claw will correspond to a forbidden polyomino; if the graph is claw-free, however, we can use
well-known efficient algorithms for finding independent sets, and hence, a tiling for the region.
Finally we close this paper in Section \ref{sec:li} where we study a relation between L-Trominoes and I-Trominoes. We introduce a technique for decomposing a region in simple parts that yields an efficient algorithm for finding L-Tromino covers. This tiling technique is a modification of the proof of Theorem \ref{the:Aztec-180} for tiling the Horiyama \emph{et al.} \cite{HIN17} gadgets with I-Trominoes to tiling general regions with L-Trominoes.
\section{Preliminaries}\label{sec:preliminaries}
In this work we will use $\mathbb{Z}$ to denote the set of integers and $[a,b]$ to denote the discrete interval $\{a,a+1,\dots, b\}$.
A region $R$ is a finite union of connected cells, where connected means that any two cells in $R$ share one common edge. If a cell is the set of points $[a,a+1]\times [b,b+1]$, we label such cell by $(a,b)$ which we refer to as the \emph{cell's coordinate}. Two cells are adjacent if the Manhattan distance, i.e., the $L_1$-norm, of their coordinates is 1; thus, two cells in diagonal to each other are not adjacent.
A \emph{tromino} is a polyomino of 3 cells. In general there are two types of trominoes, the L-tromino and the I-tromino. An L-tromino is a polyomino of 3 cells with an L shape. An I-tromino is a polyomino of 3 straight cells with the form of an I. In this work we will mostly be dealing with L-Trominos and we will refer to them simply as trominoes; I-trominoes will appear later but we will make sure to clarify to which type of tromino we are referring to.
A \emph{defect} is a cell that is ``marked'' in the sense that no tromino can be placed on top of that cell. A \emph{cover} or \emph{tiling} of a region $R$ is a set of trominoes covering all cells of $R$ that are not defects without overlapping and each tromino is packed inside $R$. The \emph{size} of a cover is the number of tiles in it.
\begin{definition}
$\mathrm{T{\footnotesize ROMINO}}$ is the following problem:
\begin{tabular}{ll}
INPUT &: a region $R$ with defects.\\
OUTPUT &: ``yes'' if $R$ has a cover and ``no'' otherwise.
\end{tabular}
\end{definition}
Moore and Robson \cite{MR01} proved that $\mathrm{T{\footnotesize ROMINO}}$ is NP-complete and Horiyama \emph{et al.} \cite{HIN17} proved that $\#\mathrm{T{\footnotesize ROMINO}}$, the counting version of $\mathrm{T{\footnotesize ROMINO}}$, is \#P-complete.
In this work we will also consider tilings where only trominoes with $180^\circ$ rotations are used. More precisely, given a region $R$ we want to find a cover where all trominoes are \emph{right-oriented} as in Fig.\ref{fig:180tromino}(a) or \emph{left-oriented} as in Figure \ref{fig:180tromino}(b). We will refer to trominoes where only their $180^\circ$ rotations are considered as \emph{180-trominoes}. A \emph{180-cover} of $R$ is a cover with 180-trominoes.
\begin{figure}[t]
\centering
\subfloat[Right-oriented]{
\begin{tikzpicture}[scale=0.8]
\draw (1,1) rectangle (2,2);
\draw (0,0) rectangle (1,1);
\draw (1,0) rectangle (2,1);
\end{tikzpicture}
\quad
\begin{tikzpicture}[scale=0.8]
\draw (1,1) rectangle (2,2);
\draw (0,0) rectangle (1,1);
\draw (0,1) rectangle (1,2);
\end{tikzpicture}
}
\qquad\qquad
\subfloat[Left-oriented oriented]{
\begin{tikzpicture}[scale=0.8]
\draw (1,1) rectangle (2,2);
\draw (0,1) rectangle (1,2);
\draw (1,0) rectangle (2,1);
\end{tikzpicture}
\quad
\begin{tikzpicture}[scale=0.8]
\draw (0,0) rectangle (1,1);
\draw (0,1) rectangle (1,2);
\draw (1,0) rectangle (2,1);
\end{tikzpicture}
}
\caption{The $\mathrm{180\mbox{-}T{\footnotesize ROMINO}}$ problem either takes trominoes from the left figure or the right figure.}
\label{fig:180tromino}
\end{figure}
\begin{definition}
$\mathrm{180\mbox{-}T{\footnotesize ROMINO}}$ is the following problem:
\begin{tabular}{ll}
INPUT &: a region $R$ with defects.\\
OUTPUT &: ``yes'' if $R$ has a 180-cover and ``no'' otherwise.
\end{tabular}
\end{definition}
\section{Tiling of the Aztec Rectangle}\label{sec:Aztec}
The \emph{Aztec Diamond of order $n$}, denoted $\operatorname{AD}(n)$, is the union of lattice squares $[a,a+1]\times [b,b+1]$, with $a,b\in \mathbb{Z}$, that lie completely inside the square $\{(x,y)\> | \> |x|+|y|\leq n+1\}$ \cite{EKLP92}. Figure \ref{fig:azn} shows the first four Aztec diamonds. Tilings of the Aztec diamond with dominoes was initially studied by Elkies \emph{et al.} \cite{EKLP92} and later by several other people.
The concept of an Aztec diamond can be very easily extended to that of an \emph{Aztec rectangle}. We denote by $\mathcal{AR}_{a,b}$ the Aztec rectangle
which has $a$ unit squares on the southwestern side and $b$ unit squares on the northwestern side; in the case when $a=b=n$ we get an Aztec diamond of order $n$. When dealing with Aztec rectangle, with no loss of generality, we always assume that $a<b$.
As an example Fig.\ref{fig:ar} shows $\mathcal{AR}_{4,10}$. Domino tilings of Aztec rectangles have been studied by various mathematicians starting with Mills \emph{et. al.} \cite{MRR83}.
In the following subsections we study tilings of the Aztec rectangle using trominoes with and without defects, and then specialize them to Aztec diamonds.
\begin{figure}[t]
\centering
\subfloat[$\operatorname{AD}(1)$]{
\includegraphics[scale=0.022]{./figs/az1_with_lines}
}
\subfloat[$\operatorname{AD}(2)$]{
\includegraphics[scale=0.033]{./figs/az2_with_lines}
}
\subfloat[$\operatorname{AD}(3)$]{
\includegraphics[scale=0.030]{./figs/az3_with_lines}
}
\subfloat[$\operatorname{AD}(4)$]{
\includegraphics[scale=0.023]{./figs/az4_with_lines}
}
\caption{Aztec diamonds of order 1, 2, 3 and 4.}
\label{fig:azn}
\end{figure}
\begin{figure}[t]
\centering
\includegraphics[scale=0.5]{./figs/rectangle-4-10}
\caption{Aztec rectangle $\mathcal{AR}_{4,10}$.}
\label{fig:ar}
\end{figure}
\subsection{Tilings with No Defects}\label{sec:az-defects}
For any Aztec rectangle $\mathcal{AR}_{a,b}$ with no defects, we can completely understand when there is a tiling. The following theorem gives a characterization.
\begin{theorem}\label{th-1}
$\mathcal{AR}_{a,b}$ has a cover if and only if $a(b+1)+b(a+1)\equiv 0 \pmod 3$.
\end{theorem}
\noindent As a corollary, we get the following for the Aztec diamond.
\begin{corollary}\label{the:az-tile}
$\operatorname{AD}(n)$ has a cover if and only if $n(n+1)\equiv 0 \pmod 3$.
\end{corollary}
\begin{figure}[t]
\centering
\subfloat[1-stair]{
\includegraphics[scale=0.1]{./figs/1stair}
}
\quad\quad\quad\quad
\subfloat[2-stair]{
\includegraphics[scale=0.07]{./figs/2stair}
}
\quad\quad\quad\quad
\subfloat[$k$-stair]{
\includegraphics[scale=0.06]{./figs/kstair}
}
\caption{A stair also includes all $90^\circ$ rotations.}
\label{fig:stair}
\end{figure}
To prove Theorem \ref{th-1}, first we present tilings of particular cases of the Aztec rectangle in Lemmas \ref{lem-1} and \ref{lem-2}. The following lemma is trivial.
\begin{lemma}\label{squares}
An Aztec rectangle, $\mathcal{AR}_{a,b}$ contains $a(b+1)+b(a+1)$ unit squares. Further, specializing $a=b=n$ we get that an Aztec diamond of order $n$ contains $2n(n+1)$ unit squares.
\end{lemma}
Define a \emph{stair} as a polyomino made-up only of trominoes with their $180^\circ$ rotations connected as in Fig.\ref{fig:stair}(a). The same stair can be rotated $90^\circ$ to obtain another stair. A \emph{$k$-stair} is a co-joined set of $k$ stairs, where a stair is joined to another stair by matching their extremes; for example, in Fig.\ref{fig:stair}(b) we can see two stairs where the lowest extreme of the upper stair is matched with the upper extreme of the lower stair. This idea is easily extended to a set of $k$ stairs thus giving a $k$-stair as in Fig.\ref{fig:stair}(c). A $k$-stair can also be rotated $90^\circ$ to obtain another $k$-stair. The \emph{height of a $k$-stair} is the number of steps in it. It is easy to see that the height of a $k$-stair is $3k+2$. In addition, a single tromino would be a $0$-stair.
\begin{figure}[t]
\centering
\subfloat[Tiling with a single stair.]{
\includegraphics[scale=0.5]{./figs/lemma-2}
}
\quad
\subfloat[Tiling with a double stair.]{
\includegraphics[scale=0.45]{./figs/2-5-ar-lemma}
}
\caption{Tilings of Lemmas \ref{lem-1} and \ref{lem-2}.}
\label{fig:borders}
\end{figure}
\begin{lemma}\label{lem-1}
If $3\mid a,b$ and $\mathcal{AR}_{a,b}$ has a cover, then $\mathcal{AR}_{a+2,b+2}$ has a cover.
\end{lemma}
\begin{proof}
If $a,b$ are multiples of 3, then an $a/3$-stair and an $b/3$-stair can be used to tile around $\mathcal{AR}_{a,b}$ along the shorter and longer sides
respectively, using the pattern of Fig.\ref{fig:borders}(a). This tiling increments the order of the Aztec rectangle
by 2, thus obtaining a tiling for $\mathcal{AR}_{a+2,b+2}$.\hfill$\Box$ \bs
\end{proof}
\begin{lemma}\label{lem-2}
If $3\mid a+1,b+1$ and $\mathcal{AR}_{a,b}$ has a cover, then $\mathcal{AR}_{a+4,b+4}$ has a cover.
\end{lemma}
\begin{proof}
To find a tiling for $\mathcal{AR}_{a+4,b+4}$ we use four copies of $\operatorname{AD}(2)$ added to the four corners of $\mathcal{AR}_{a,b}$. Then, to complete the tiling, we use two
$(a-2)/3$ and $(b-2)/3$-stairs one on top of each other along the shorter and longer sides respectively, to complete the border. The entire construction follows the pattern of Fig.\ref{fig:borders}(b). This tiling increments the order of the Aztec rectangle by 4, thus obtaining a tiling for $\mathcal{AR}_{a+4,b+4}$.\hfill$\Box$ \bs
\end{proof}
\noindent The above two Lemmas gives as easy corollaries the following.
\begin{corollary}\label{lem:plus2}
If $\operatorname{AD}(n)$ has a cover and $n$ is a multiple of 3, then $\operatorname{AD}(n+2)$ has a cover.
\end{corollary}
\begin{corollary}\label{lem:plus4}
If $\operatorname{AD}(n-1)$ has a cover and $n$ is a multiple of 3, then $\operatorname{AD}(n+3)$ has a cover.
\end{corollary}
Now, let us prove Theorem \ref{th-1}.
\begin{proof}[Proof of Theorem \ref{th-1}]
The values for which $a(b+1)+b(a+1)\equiv 0 \pmod 3$ holds are $a,b=3k$ and $a,b=3k-1$ for some $k\in \mathbb{Z}$.
Thus, the statement is equivalent to saying that for all positive integers $k$ there is a tiling of $\mathcal{AR}_{a,b}$ where $3\mid a,b$ or $3\mid a+1,b+1$ and that there are no tilings for $\mathcal{AR}_{a,b}$ when $3\mid a+2,b+2$.
We show the second part now, which is easy since if we have $\mathcal{AR}_{a,b}$ with $a,b$ of the form $3k+2$, then the number of lattice
squares inside $\mathcal{AR}_{a,b}$ is not divisible by $3$ and hence we cannot tile this region with trominoes.
We come to the first cases now. Using Lemmas \ref{lem-1} and \ref{lem-2}, this part is clear if we can show the base induction case to be true.
The base case of Lemma \ref{lem-1} is shown in Fig.\ref{fig:base-1}(a), which is $\mathcal{AR}_{3,6}$. Once we have a tiling of $\mathcal{AR}_{3,6}$, we
can use Lemma \ref{lem-1} to create a tiling of an Aztec rectangle whose sides are increased by $2$. We can also increase
$\mathcal{AR}_{3,6}$ by using the additional pieces shown in Fig.\ref{fig:base-1}(b,c) using them in combinations with any case of Aztec rectangle
satisfying the properties of Lemma \ref{lem-1} to increase either the longer or the shorter sides, and if all three additional pieces are
used then we can increase both sides of $\mathcal{AR}_{a,b}$.
\begin{figure}[t]
\centering
\subfloat[Base induction case.]{
\includegraphics[scale=0.5]{./figs/3-6-ar-tiled}
}
\quad
\subfloat[Length additional pieces.]{
\includegraphics[scale=0.45]{./figs/add-1-2-lemma1}
}
\quad
\subfloat[Breadth additional piece.]{
\includegraphics[scale=0.45]{./figs/add-3-lemma1}
}
\caption{Base case of Lemma \ref{lem-1}.}
\label{fig:base-1}
\end{figure}
Similarly, the base case of Lemma \ref{lem-2} is shown in Fig.\ref{fig:base-2}(a), which is $\mathcal{AR}_{2,5}$. Once we have a tiling of $\mathcal{AR}_{2,5}$, we
can use Lemma \ref{lem-2} to create a tiling of an Aztec rectangle whose sides are increased by $4$. We can also increase
$\mathcal{AR}_{a,b}$ by using the additional pieces shown in Fig.\ref{fig:base-2}(b,c,d) using them in combinations with any case of Aztec rectangle
satisfying the properties of Lemma \ref{lem-2} to increase either the longer or the shorter sides, and if all three additional pieces are
used then we can increase both sides of $\mathcal{AR}_{a,b}$.\hfill$\Box$ \bs
\end{proof}
\begin{figure}[t]
\centering
\subfloat[Base induction case.]{
\includegraphics[scale=0.5]{./figs/2-5-ar-tiled}
}
\quad
\subfloat[Length additional piece.]{
\includegraphics[scale=0.45]{./figs/piece-add-1-lemma2}
}
\quad
\subfloat[Length additional piece.]{
\includegraphics[scale=0.45]{./figs/piece-add-2-lemma2}
}
\quad
\subfloat[Breadth additional piece.]{
\includegraphics[scale=0.45]{./figs/piece-add-3-lemma2}
}
\caption{Base case of Lemma \ref{lem-2}.}
\label{fig:base-2}
\end{figure}
An $O(b^2)$ time algorithm is immediately obtained from the proof of Theorem \ref{th-1}, and also for the Aztec Diamond (see App. \ref{alg:ar}).
\subsection{Tiling with Defects}
From Theorem \ref{th-1} we know that for any positive integers $a,b$, the Aztec rectangles with no defects $\mathcal{AR}_{a,b}$ such that $3$ divides
$a,b$ or $3$ divides $a+1,b+1$ have a cover but if $3$ divides $a+2,b+2$, then $\mathcal{AR}_{a,b}$ does not have a tiling. We show that if such an Aztec rectangle has exactly one defect, then it can be covered with trominoes.
\begin{theorem}\label{the:ar-defect}
$\mathcal{AR}_{a,b}$ with $a,b$ of the form $3k-2$ with one defect has a cover.
\end{theorem}
\begin{proof}
To tile $\mathcal{AR}_{a,b}$ with one defect we use a construct which we call a \emph{fringe} appearing in Fig.\ref{fig:az-defect}(a). It is easy to check that if a fringe has exactly one defect, then it can be covered with trominoes.
To construct a tiling for $\mathcal{AR}_{a,b}$ with one defect we place a fringe in a way that includes the defect and the left and right ends of the fringe touches the boundaries of the Aztec rectangle as in Fig.\ref{fig:az-defect}(b). Then we use the tiling pattern of Fig.\ref{fig:az-defect}(b) where we put stairs above and below the fringe.\hfill$\Box$ \bs
\end{proof}
\begin{figure}[t]
\centering
\subfloat[Fringe]{
\includegraphics[scale=0.1]{./figs/fringe}
}
\quad\quad\quad\quad
\subfloat[Tiling pattern]{
\includegraphics[scale=0.5]{./figs/4-7-tiled}
}
\caption{Tiling of $\mathcal{AR}_{a,b}$ with one defect. A \emph{fringe} can be composed of any number of order 1 Aztec diamonds $\operatorname{AD}(1)$ joined by their upper right and lower left cells. An \emph{reversed fringe} is obtained by joining order 1 Aztec diamonds by their upper left and lower right cells.}
\label{fig:az-defect}
\end{figure}
\noindent As an easy corollary, we obtain the corresponding result for Aztec diamonds.
\begin{corollary}\label{the:az-defect}
For any positive integer $k$, the Aztec Diamond $\operatorname{AD}(3k-2)$ with one defect has a cover.
\end{corollary}
We can consider many different classes of defects, and it is observed that some of these classes have easy tilings, as an example,
we have in Fig.\ref{fig:four-d}(a) an Aztec rectangle with four defects on its corners. A tiling of this region is shown in
Fig.\ref{fig:four-d}(b). In the combinatorics literature, tilings of regions with defects of several kinds for Aztec rectangle have been studied (see
\cite{MPS17} for the most general class of boundary defects).
\begin{figure}[t]
\centering
\subfloat[$\mathcal{AR}_{a,b}$ with four defects]{
\includegraphics[scale=0.5]{./figs/4-7-ar-new-hole}
}
\quad\quad\quad\quad
\subfloat[Tiling pattern]{
\includegraphics[scale=0.5]{./figs/4-7-ar-new-hole-tiled}
}
\caption{Tiling of $\mathcal{AR}_{a,b}$ with four defects.}
\label{fig:four-d}
\end{figure}
\begin{remark}
Similar defects can be studied for Aztec Diamonds as well. In fact, we can delete all cells in a fringe and obtain a tiling.
\end{remark}
The proof of Theorem \ref{the:ar-defect} gives an optimal $O(b^2)$ time algorithm for finding a cover for $\mathcal{AR}_{a,b}$ with one defect. In general, however, it is computationally hard to determine if $\mathcal{AR}_{a,b}$ with an unknown number of defects has a cover.
\begin{theorem}\label{the:az-defect-hard}
It is NP-complete to decide whether $\mathcal{AR}_{a,b}$ with an unbounded number of defects has a cover.
\end{theorem}
\begin{proof}[Proof Sketch]
The reduction is from tiling an arbitrary region $R$ with defects. The idea is to embed $R$ into $\mathcal{AR}_{a,b}$ for some sufficiently large $n$ and insert defects in $\mathcal{AR}_{a,b}$ in a way that surrounds $R$.\hfill$\Box$ \bs
\end{proof}
\section{Tiling with 180-Trominos}\label{sec:180tromino}
In this section we study tilings of arbitrary regions using only 180-trominoes. With no loss of generality, we will only consider right-oriented 180-trominoes.
\subsection{Hardness}\label{sub:hardness}
It is easy to see that even when restricted to 180-trominoes, deciding the existence of a tiling of an arbitrary region is still hard.
\begin{theorem}\label{the:180tromino}
$\mathrm{180\mbox{-}T{\footnotesize ROMINO}}$ is NP-complete.
\end{theorem}
\begin{proof}[Proof Sketch]
The proof uses the same gadgets for the reduction for I-Trominoes from the \emph{1-in-3 Graph Orientation Problem} of Horiyama \emph{et al.} \cite{HIN17}. Take any gadget of Horiyama \emph{et al.} \cite{HIN17} and partition each cell into 4 new cells. Thus, each I-tromino is transformed in a new $2\times 6$ or $6\times 2$ region (depending on the orientation of the I-tromino) which can be covered with four 180-trominoes as in Fig \ref{fig:itromino-180ltromino}. If a gadget is covered with I-trominoes, then the same gadget, after partitioning each cell into four new cells, can also be covered with 180-trominoes. To see the other direction of this implication, we exhaustively examined all possible ways to cover each 4-cell-divided gadget with L-trominoes, and observed that each gadget with its original cells can also be covered with I-trominoes (see App.\ref{sec:gadgets}).\hfill$\Box$ \bs
\end{proof}
\begin{figure}[t]
\centering
\includegraphics[scale=0.028]{./figs/itrominos-360rotated}
\caption{I-Tromino to L-Tromino transformation using 180-Trominoes.}
\label{fig:itromino-180ltromino}
\end{figure}
Theorem \ref{the:180tromino} also implies that the \emph{Triangular Trihex Tiling Problem} of Conway and Lagarias \cite{CL90} is NP-complete.
It is natural to think along these lines about tiling the Aztec rectangle (and hence, Aztec diamond) with 180-trominoes. However, we show that it is impossible.
\begin{theorem}\label{the:Aztec-180}
$\mathcal{AR}_{a,b}$ does not have a 180-cover.
\end{theorem}
\begin{proof}
Consider the southwestern side of any Aztec rectangle as in Fig.\ref{fig:az-180tromino} and pick any one of the marked cells, say the cell at coordinate
$(c,d)$. There are only two ways to cover that cell with a right-oriented tromino. With one tromino we can cover the cells
with coordinates $(c,d),(c,d+1)$ and $(c+1,d+1)$, whereas with the other tromino we can cover the cells $(c,d), (c+1,d)$ and
$(c+1,d+1)$. In either case the cells at $(c,d)$ and $(c+1,d+1)$ are always covered, and depending on which tromino is
chosen either the cell at $(c,d+1)$ or $(c+1,d)$ is covered. Therefore, if we cover the entire bottom-left side of an Aztec
rectangle, there will always be a cell at $(c,d+1)$ or $(c+1,d)$ that cannot be covered. Note that any reversed fringe
that is on top of the bottom-left side of any Aztec rectangle can be covered with 180-trominoes if it has one defect.
\end{proof}
\begin{corollary}
$\operatorname{AD}(n)$ does not have a 180-cover.
\end{corollary}
\begin{figure}[t]
\centering
\includegraphics[scale=0.55]{./figs/rectangle-180}
\caption{Covering of an Aztec rectangle with right-oriented trominoes.}
\label{fig:az-180tromino}
\end{figure}
\subsection{Efficient Tilings}\label{sub:efficient}
In this section we identify a sufficient condition for a region to have an efficient algorithm that decides the existence of a 180-cover.
\begin{figure}[t]
\centering
\includegraphics[scale=0.8]{./figs/forbidden_grosor3}
\caption{Forbidden polyominoes. All $180^\circ$ rotations, reflections and shear transformations are also forbidden polyominoes.}\label{fig:forbidden}
\end{figure}
\begin{theorem}\label{the:forbidden}
If a region $R$ does not contain any of the forbidden polyominoes of Fig.\ref{fig:forbidden} as a subregion, then there exists a polynomial-time algorithm that decides whether $R$ has a 180-cover.
\end{theorem}
For the remaining of this section we present a proof of Theorem \ref{the:forbidden}. Remember that, with no loss of generality, we only consider right-oriented trominoes. Given a region $R$ we construct a graph $G_R$, which we call the \emph{region graph} of $R$, as follows. For each cell $(a,b)$ that is not a defect there is a vertex $v_{ab}$. There is an edge for each pair of adjacent vertices and for each pair $v_{ab}$ and $v_{(a+1)(b+1)}$. Note that this reduction is one-to-one. We present an example in Fig.\ref{fig:graphs}.
\begin{figure}[t]
\centering
\subfloat[Region $R$]{
\centering
\includegraphics[scale=0.1]{./figs/region}
}
\quad\quad\quad\quad\quad\quad
\subfloat[Region graph $G_R$]{
\centering
\includegraphics[scale=0.02]{./figs/region-graph}
}
\caption{Example of a region graph construction.}
\label{fig:graphs}
\end{figure}
From the region graph $G_R$ we construct a new graph $I_R$ which we call an \emph{intersection graph} and is constructed as follows. For each triangle in $G_R$ there is a vertex $t$ and there is an edge between vertices $t_i$ and $t_j$ if the corresponding triangles share a vertex in $G_R$; for example, the intersection graph for Fig.\ref{fig:graphs} is a triangle, because all triangles in the region graph share at least one vertex.
\begin{lemma}\label{lem:ind}
For any region $R$ with a fixed number of defects, the maximum number of 180-trominoes that fit in $R$ equals the size of a maximum independent set in $I_R$.
\end{lemma}
\begin{proof}
Let $k$ be the maximum number of tiles that fit in $R$ and let $S$ be a maximum independent set in the intersection graph $I_R$. We claim that $|S|=k$.
Each triangle in the region graph $G_R$ correspond to a position where a 180-tile can fit. If $k$ is the maximum number tiles that can fit in $R$, then there exists $k$ triangles in $G_R$, denoted $T$, that do not share any common vertex. Each triangle in $T$ corresponds to a vertex in $I_R$ and since none of the triangles in $T$ share a common vertex, $T$ defines an independent set in $I_R$ and $k\leq |S|$.
To prove that $|S|=k$ suppose by contradiction that $T$ is not a maximum independent set of $I_R$, that is, $k<|S|$. Since $S$ is an independent set in $I_R$, there are $|S|$ triangles in $G_R$ that do not share a common vertex. Thus, we can fit $|S|$ 180-trominos in $R$, which is a contradiction because $k<|S|$.\hfill$\Box$ \bs
\end{proof}
The idea for a proof of Theorem \ref{the:forbidden} is to construct a polynomial time algorithm that decides the existence of a 180-cover by deciding if a maximum independent set in $I_R$ equals the number of cells of $R$ divided by 3, which agrees with the number of trominoes covering $R$. Deciding the existence of a maximum independent set of a given size is a well-known NP-complete problem, nevertheless, it is known from the works of Minty \cite{Min80}, Sbihi \cite{Sbi80} and Nakamura and Tamura \cite{NT01} that for claw-free graphs\footnote{A graph is \emph{claw-free} if it does not have $K_{1,3}$ (a claw) as an induced subgraph.} finding independent sets can be done in polynomial time. Hence, if $I_R$ is claw-free, then we can use a polynomial time algorithm for finding independent sets to decide the existence of a 180-cover. If $I_R$ has a claw, however, each claw will give one of the forbidden polyominoes.
In Lemma \ref{lem:reduction} below we show that $\mathrm{180\mbox{-}T{\footnotesize ROMINO}}$ is polynomial time reducible to deciding independent sets, which allow us to construct algorithms for $\mathrm{180\mbox{-}T{\footnotesize ROMINO}}$ using known algorithms for deciding independent sets. Then in Lemma \ref{lem:claw} we show that if $I_R$ has a claw, then that claw corresponds to a forbidden polyomino in the region $R$.
\begin{lemma}\label{lem:reduction}
There is a many-one polynomial-time reduction from $\mathrm{180\mbox{-}T{\footnotesize ROMINO}}$ to the problem of deciding existence of an independent set of a given size.
\end{lemma}
\begin{proof}
First the reduction constructs the region graph $G_R$ and the intersection graph $I_R$. If the size of the largest independent set equals the number of cells of $R$ divided by 3, then output ``yes'' because $R$ has a 180-cover; otherwise output ``no'' because $R$ does not have a 180-cover.
Suppose $R$ has a 180-cover. If $n$ is the number of cells in $R$, then the number of tiles in the 180-cover is $n/3$. By Lemma \ref{lem:ind}, the largest independent set in $I_R$ equals $n/3$.
Now suppose $R$ does not have a 180-cover. If $n$ is the number of cells in $R$, then $n/3$ is not equal the maximum number of tiles that can fit in $R$. Thus, by Lemma \ref{lem:ind}, it holds that $n/3$ is not equal the size of the largest independent set in $I_R$.\hfill$\Box$ \bs
\end{proof}
\begin{lemma}\label{lem:claw}
If $I_R$ has a claw, then $R$ has at least one forbidden polyomino.
\end{lemma}
\begin{proof}[Proof Sketch]
For any claw in $I_R$ there is a vertex of degree 3 and three vertices of degree 1, and each vertex in $I_R$ corresponds to a triangle in the region graph $G_R$. We refer to the triangle that corresponds to the degree 3 vertex as the \emph{central triangle} and each degree 1 triangles is called an \emph{adjacent triangle}. Thus, to obtain all forbidden polyominoes, we look at all posible ways to connect (by the vertices) each adjacent triangle to the central triangle in such a way that
each adjacent triangle only connects to the central triangle in a single vertex and it is not connected to any other adjacent triangle; otherwise, if an adjacent triangle connects with two vertices of the central triangle or any two adjacent vertices connects with one another, then the induced graph does not corresponds to a claw. By exhaustively enumerating all possibilities, we can extract all polyominoes that correspond to claws in $I_R$. Then we partition this set of polyominoes in five equivalent classes, where two polyominoes are in the same class if and only if one can be obtained from the other by a $180^\circ$ rotation, a reflection or shear transformation (App.\ref{app:forbidden}).\hfill$\Box$ \bs
\end{proof}
Lemmas \ref{lem:reduction} and \ref{lem:claw} complete the proof of Theorem \ref{the:forbidden}.
\section{I-Trominoes vs L-Trominoes}\label{sec:li}
In Section \ref{sec:180tromino} we saw that any gadget of Horiyama \emph{et al.} \cite{HIN17} can be covered with I-trominoes if and only if the same gadget, after partitioning each cell into four new cells, can be covered with L-trominoes. In general, if $R$ is any region and $R^\boxplus$ is the region $R$ where each cell is partitioned into four cells, we have that if $R$ can be covered with I-trominoes, then $R^\boxplus$ can be covered with L-trominoes. We do not know, however, if the other way of this implication holds in the general case. The following theorem partly answers this open problem.
\begin{theorem}\label{the:detachable}
Let $R$ be a connected region of size $n$. The region $R^\boxplus$ has an L-Tromino cover if and only if 3 divides $n$.
\end{theorem}
\begin{proof}[Proof Sketch]
It is clear that if $R^\boxplus$ has an L-Tromino cover, then 3 divides $n$. Now suppose that 3 divides $n$. Say that a connected region $R$ with $n$ vertices is \emph{detachable} if there exists a way to separate $R$ in two connected subregions of sizes $n_1$ and $n_2$ such that 3 divides $n_1$ and 3 divides $n_2$. We can show that if $R$ is not detachable, then $R^\boxplus$ can always be covered with L-Trominoes.
In order to construct a tiling for $R^\boxplus$ we first decompose $R$ by recursively detaching it in connected subregions until all subregions obtained this way are not detachable. Since each subregion is not detachable, we construct an L-Tromino cover for each subregion and then join them to obtain a cover for $R^\boxplus$. We omit details due to the lack of space (see App.\ref{sec:detachable}).\hfill$\Box$ \bs
\end{proof}
The proof of Theorem \ref{the:detachable} gives an efficient algorithm to find covers for any $R^\boxplus$.
\section{Concluding Remarks and Open Problems}\label{sec:conclusions}
In this work we studied the computational hardness of tiling arbitrary regions with L-trominoes. We showed restrictions to the problem that keeps it computationally intractable and identified concrete instances where an efficient tiling exists.
We conclude this paper with some open problems that we consider challenging and that we believe will fuel future research in the subject.
\begin{enumerate}
\item \emph{Hardness of tiling the Aztec rectangle with a given number of defects.} In Section \ref{sec:Aztec} we saw that an Aztec rectangle with 0 or 1 defects can be covered with L-trominoes in polynomial time, whereas in general the problem is NP-complete when the Aztec rectangle has an unknown number of defects; with $2+3k$, for every $k$, an Aztec rectangle cannot be covered because the number of cells is not divisible by 3. It is open if there exists a polynomial time algorithm for deciding a tiling for an Aztec rectangle with a given number of defects.
\item \emph{Tiling of orthogonally-convex regions.} In this work we showed several instances where a tiling can be found in polynomial time. In general, it is open if an orthogonally-convex region with no defects can be covered in polynomial time or if it is NP-complete to decide if a tiling exists.
\end{enumerate}
\bibliographystyle{splncs03}
|
2,877,628,089,009 | arxiv | \section{Introduction}
Brownian motion is a physical phenomenon which was historically
modeled by the Wiener process, both directly identifying the particle
trajectories with the realizations of the process and indirectly assuming that
the particle velocity evolves according to a Ornstein-Uhlenbeck process.
Although in the last century the Wiener process has been used
to describe a variety of phenomena in finance, biology, engineering, electronics
and so on, its name remains strictly tied to the description of the motion
of random particles.
If one tries to extend its use to the description of the random motion of
relativistic particles, one clashes against one of its more characteristic
properties: trajectories are not differentiable, which means infinite speed
while relativity only allows luminal or subluminal velocities.
This fact doesn't imply that it is useless, on the contrary the
relativistic Brownian motion can be still modeled via a variety of modified Langevin
equations which produce trajectories with a speed which is never superluminal.
We just quote [1-19]
which are a few of the studies which followed this strategy in the last fifty
years.
In a recent research \cite{Serva:2020} we considered the extreme case in which
a particle moves at the speed of light, the aim was to produce a probabilistic
tool which is related to the Brownian motion of light-speed particles in the
same way as the Wiener process is related to the Brownian motion of classical
particles.
We do not derive here the erratic motion of a light-speed particle by some
limit procedure which involves collisions with other particles or obstacles as in
\cite{Scalas:2015}, but we directly provide the mathematical framework.
Although it appears very difficult to imagine a physical device where a mass-less
particle is trapped and scattered as a photon in a box of mirrors, this is, indeed,
closer to reality of what one could think.
Almost one century ago, Albert Einstein theoretically conceived a box in which a
single photon could be trapped in order to measure the relationship between mass
and energy.
Recently a team of physicists have created this box, a device that snares a photon
up to half a second \cite{Gleyzes:2007}.
One more reason for searching a Wiener description of relativistic random particles
is the possibility to extend the analogy between the Feynman integral
and the Wiener integral (Feynman-Kac formula) to the relativistic quantum domain.
The Schr\"odinger equation is solved by the Feynman integral
while the heat equation, which is connected to the first by analytic
continuation, is solved by the Wiener integral.
The relativistic versions of the Schr\"odinger equation are
the Klein-Gordon (zero spin particles) and Dirac (spin one half) equations.
Both are hyperbolic equations (Dirac equation in its second order formulation).
Analytic continuation gives rise to elliptic equations,
the point is: which process is associated to the elliptic equations?
The first answers were given in
\cite{Ichinose:1982, Ichinose:1984, Jacobson:1984, Ichinose:1986, Ichinose:1987}
and later implemented in \cite{De Angelis:1990, De Angelis:1991} where
the Wiener process was still the main ingredient, but a four dimensional
one with both position and time following trajectories which are the realizations
of a Wiener process with the proper time as index.
The proper time is then eliminated by a procedure based on hitting times.
Nevertheless, the resulting process is unphysical since the speed is
not bounded and the whole construction only results in a tool for obtaining
a probabilistic solution of some elliptic equations.
If one forces the approach to the realm of physics one has to abandon
Markov property and the single particle picture \cite{Serva:1988}.
There is a third way to approach the relativistic problem with
a process which is physical and allows to construct the solution
of the quantum hyperbolic equations.
In 1956 the Polish physicist and mathematician Mark Kac
considered a (1+1)-dimensional process (one space dimension + time)
where the particle travels at speed of light
(left or right) and randomly inverts its velocity and he
proved that the associated probability density
satisfies the telegrapher equation \cite{Kac:1956}.
About thirty years after the Kac pioneering work,
Gaveau {\it et al.} noticed that the telegrapher equation could be easily
associated both to the Dirac equation in 1+1 dimensions (first order formulation)
and to the Klein-Gordon equation also in 1+1 dimensions
(second order formulation).
Using this equivalence they were able to give a probabilistic solution
(by the backward Kolmogorov equation) to these fundamental
quantum equations \cite{Gaveau:1984}.
This result was later refined and extended in \cite{Blanchard:1986, Combe:1987}.
The weak point was that both Kac and these later constructions only
worked for particles in 1+1 dimensions .
Indeed, the process considered in \cite{Kac:1956,Gaveau:1984,Blanchard:1986,Combe:1987}
is part of a larger class, in fact, by Lorentz boosts new processes can be obtained
with particles moving at the speed of light (a simple consequence of the fact that
a light-speed particle in an inertial frame is also light-speed in any other inertial
frame).
The processes of this larger class have in general an unbalanced
probability rate of velocity inversion {\it i.e,}
the inversions from right to left occur with a different probability
rate of those from left to right, as a consequence,
the particle may have a non vanishing average velocity.
The class of these one-dimensional light-speed processes was further extended
by considering inversion rates which not only depend on the sign of the velocity
but also on position and time.
This extension gave the possibility to reformulate the quantum mechanics of a
relativistic particle in terms of stochastic processes \cite{Serva:1986}
in the spirit of Nelson's stochastic mechanics \cite{Nelson:1967}.
Again, this construction was limited to 1+1 dimensions.
All the processes mentioned above can be related to the Feynman checkerboard,
which was also proposed as a mathematical tool for Fermions path integrals
(see, for example, \cite{Kauffman:1996}).
The main difference is that in the (1+1)-dimensional Kac historical model and
in the related ones (included the processes in this work) both time and space
are continuous {\it ab initio}, while in the Feynman checkerboard they are
both discrete.
In case, continuity can be reached by a proper limit \cite{Molfetta:2012}.
Both types of approach have been used to define path integrals for Dirac equation,
but only the checkerboard can be placed at the origin of quantum walks
\cite{Molfetta:2012,Molfetta:2013,Jay:2018}.
In this paper we consider a family of processes which generalizes the Kac
approach to the (3+1)-dimensional case (three space dimensions + time).
The goal is to construct a Brownian motion
which is the most similar to the Wiener process among all those processes
which do not conflict with relativity.
We assume that the particle only moves at the speed of light $c$ which implies
that velocity can be represented by a point on the surface of a sphere of
radius $c$.
We also assume that in the 'rest frame', the velocity performs a isotropic
Wiener process on that
surface (which corresponds to anisotropic Wiener processes in general frames).
In this way the speed is always $c$ which is the largest among those compatible
with relativity, but velocity direction changes.
It should be remarked
the trajectories of the velocity are almost everywhere continuous
but they are not differentiable, on the contrary the trajectories of the
positions are continuous and differentiable.
In the 'rest frame' the particle is ballistic at short times (position
changes proportionally to time), and
ordinarly diffusive ($E[x^2] \sim t$) at large times for which
the average velocity vanishes.
Then, one has to consider all the processes generated by Lorentz boosts.
The instantaneous velocity of these processes must be also luminal,
because a luminal particle is luminal in any inertial frame.
Therefore, the velocity still remains on the surface of the sphere,
nevertheless, its diffusion is anisotropic
and the average velocity is unvanishing at large times.
The construction of this family of processes, which transform
one into the other by Lorentz boost, needs Ito calculus
which leeds to the core equation (\ref{steq'}) which
represents the entire family.
The paper is simply organized:
in section 2 we introduce the process in the 'rest frame'.
The velocity process on the sphere is formulated in a new and more
economic way which allow a simpler use of Ito calculus.
In section 3 we characterize the entire family of processes
generated by Lorentz boosts.
Nevertheless, the very long application of Ito calculus to reach this goal
is postponed in an Appendix that eventually the reader can skip.
Averages are computed in section 4 where we also
highlight and describe the short-term ballistic behavior
versus the long-term diffusive behavior of the particles in the 3D physical
space.
Summary and outlook can be finally found in section 5.
\section{Stochastic equations for the process in the 'rest frame'}
The particle velocity performs
a isotropic Wiener process on the surface of a sphere of radius $c$,
In this way, while the velocity direction changes, the speed always
equals $c$, which is the largest among those compatible with relativity.
The equations governing this process (Ito notation) are:
\begin{equation}
\begin{aligned}
& d{\bf x}(t) = {\bf c}(t) dt , \\
& d{\bf c}(t) = -\omega^2 {\bf c}(t)\, dt + \omega c \, d {\bf w}(t)
\end{aligned}
\label{steq}
\end{equation}
where, according to Ito, $d{\bf c}(t) = {\bf c}(t+dt) - {\bf c}(t)$
and $d{\bf w}(t) ={\bf w}(t+dt)-{\bf w}(t)$
is a two component standard Wiener increment on the plane perpendicular to
$\bf c$$(t)$ such that $E[\, |d{\bf w}(t)|^2]$ = $2dt$.
The second of the above equations, given that $d{\bf w}(t)$
is a two component Wiener increment tangent to the surface,
describes a isotropic Wiener process on that surface of a sphere.
This process was studied for the first time at least 70 years
ago \cite{Yosida:1949,Yosida:1952}.
It is straightforward to verify that $|{\bf c} (t)|$ = $c$
at any time $t \ge 0$. In fact, according to Ito, one trivially obtains
$d{\bf c}^2(t)=0$ (in next section this equality is explicitly proven for
the general family of processes generated by Lorentz boosts).
Therefore,
equation (\ref{steq}) describes a particle which has constant speed and whose
velocity changes direction following continuous but not differentiable
trajectories.
The present model can be seen as a 3 space dimensions version
of the Kac process. In fact, a constant speed process in one
space dimension can be only constructed by considering jumps between the
two possible velocities.
In three space dimensions, constant speed means that velocity is represented by
a point on the surface of a sphere (with the speed as radius). A process over this
surface can be realized by allowing jumps between points
(velocity jumps from one value to another) or by continuous Wiener trajectories
on the surface (velocity changes direction in a continuous way).
There are only these two choices for a Markovian
generalization of the Kac process to three space dimensions,
we followed the second.
Nevertheless, the present model can be also considered as a peculiar
Ornstein-Uhlenbeck process constructed in such a way that speed remains constant.
In the following pages we will omit the time as an explicit
argument when it is not strictly necessary.
For example, we will simply write ${\bf c}$, $d{\bf c}$, ${\bf w}$
and $d{\bf w}$ for ${\bf c}(t)$, $d{\bf c}(t)$, ${\bf w}(t)$
and $d{\bf w}(t)$.
The increment $d{\bf w}$ has to be a two
component Wiener differential perpendicular to $\bf c$
(which means tangent to the surface),
nevertheless, the recipe for its construction is not univocal.
In the early seventies Strook and then Ito \cite{Stroock:1971,Ito:1975}
constructed the increment $d\bf w$ in the second of the equations (\ref{steq})
as
\begin{equation}
d{\bf w} =
\left(\mathbb{ I}- \mathbf{n}\mathbf{n}^\mathsf{T} \right) \,d{\bf W}
\label{sigma}
\end{equation}
where ${\bf n}(t)= {\bf c}(t)/{c}$ is a time dependent unitary vector,
${\bf W}$ is a standard three dimensional Wiener process,
$\mathbb{ I} $ is the $3 \times 3$ identity matrix and the row vector
$\mathbf{n}^\mathsf{T}$ is the transposed of the column vector $\bf n$.
One gets $d{\bf c} = -\omega^2 {\, \bf c}\, dt + \sigma \,d{\bf W}$
where
$\sigma=
\omega c \left( \mathbb{I}-\mathbf{n}\mathbf{n}^\mathsf{T} \right) $
is a $3 \times 3$ matrix.
About ten years later a simpler choice was considered \cite{Price:1983,Berg:1985}:
\begin{equation}
d{\bf w} =
{\, \bf n} \times \,d{\bf W},
\label{sigma'}
\end{equation}
which leads to $ d{\bf c} =-\omega^2 {\, \bf c}\, dt + \hat \sigma \,d{\bf W} $
where
$\hat \sigma= \omega c [{\bf n}]$ with $[{\bf n}]$ being
the $3 \times 3$ skew matrix representation of the vector $\bf n$.
It is easy to check that
$\sigma \, \sigma^\mathsf{T} = \hat \sigma \, {\hat \sigma}^{\mathsf{T}}
= \omega^2 c^2 \left(\mathbb{ I}-\mathbf{n}\mathbf{n}^\mathsf{T}\right) $
which implies that
the Forward Kolmogorov Equation is the same for choices (\ref{sigma})
and (\ref{sigma'}).
See \cite{Brillinger:1997,Krishna:2000} for properties and applications.
Both implementations of the two-dimensional increment $d{\bf w}$ are made by a
three dimensional Wiener process $\bf W$$(t)$, which is somehow
redundant for the construction of a two-dimensional increment.
We propose here to use in place of the standard three-components Wiener process
$\bf W$ = $(W_1, \, W_2, \, W_3)$, a standard two-components
Wiener processes $w_2, w_3$ (we write $w_2,\, w_3$ in place of
$W_2, \, W_3$ to avoid confusion).
Our choice is
\begin{equation}
d{\bf w}=
{\bf n_2}\, d w_2 + {\bf n_3}\, dw_3
\label{m}
\end{equation}
where ${\bf n_2}(t)$ and ${\bf n_3}(t)$ are two unitary vectors
perpendicular each other and also perpendicular to ${\bf n}(t)$.
The Wiener increments are independent which
implies $E[dw_1(t) \, dw_2(t)] = 0$ and they are standard
which means $E[(d{\bf w}(t))^2]$ = $2dt$.
It must be clear that the orientation of ${\bf n_2}(t)$ and ${\bf n_3}(t)$
can be arbitrarily chosen on the plane perpendicular to ${\bf n}(t)$.
In the following we make a choice which is motivated by the fact that
it is the simplest for our goal, which is to construct, by Ito calculus,
the general family of processes generated by Lorentz boosts.
In \cite{Serva:2020} we made a totally different choice which allowed us
to separate the spacial variables from the velocity variables
in order to write down a Forward Kolmogorov Equation
directely in a 3D configuration space.
Given a constant vector ${\bf v}$, we chose here
\begin{equation}
{\bf n_2}= \frac{{\bf v} \times {\bf n}}{| {\bf v} \times {\bf n}|},
\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,
{\bf n_3}= {\bf n} \times {\bf n_2},
\label{nn}
\end{equation}
so that $\bf v$, $\bf n$=$\bf c$/c and $\bf n_3$ are on the same plane
and $\bf n_2$ is perpendicular to it.
To fix the ideas one can put the north pole in the $\bf v$ direction
with respect to the center of the sphere,
so that ${\bf n_3}$ is tangent to a meridian and points to north,
while ${\bf n_2}$ is tangent to a parallel and points to est.
In this way it is simple to pass to spherical coordinates.
At the poles (where $\bf n$ equals $\pm$ $ \bf v$/$| \bf v|$)
the unitary vectors ${\bf n_2}$ and ${\bf n_3}$
can be arbitrarily chosen perpendicularly to $\bf v$.
Equations (\ref{sigma}), (\ref{sigma'}) and (\ref{m})
are three totally equivalent way to
construct the same Wiener increment tangent to the surface.
According to our representation (\ref{m}), the second equation in (\ref{steq})
rewrites as:
\begin{equation}
d{\bf c} = -\omega^2 {\bf c}\, dt + \omega c \,
({\bf n_2}\, d w_2 + {\bf n_3}\, dw_3),
\label{steq2}
\end{equation}
the advantage being that we use only a two component Wiener process in
place of a three component one, moreover
this stochastic equation is straightforwardly associated
to the velocity spherical Laplacian in the Kolmogorov Equations
when it is expressed in terms of longitude and latitude.
We stress again that our specific choice (\ref{nn}) is only dictated by
convenience for later calculations in this paper, any other choice which
keeps $\bf n_2$ and $\bf n_3$ perpendicular to $\bf n$ and perpendicular
each other is equally admissible.
\section{Lorentz boosts and stochastic equations in a generic inertial frame}
In the 'rest frame' the velocity $\bf c$$(t)$ of the
particle evolves according to equation (\ref{steq2}) where ${\bf n_2}$
and ${\bf n_3}$ are defined by (\ref{nn}).
Then, assume that this 'rest frame' moves at constant velocity $\bf u$
(without rotating) with respect to a second inertial frame.
Since the choice of $\bf v$ in (\ref{nn}) is arbitrary, we can leave it
to coincide with $\bf u$. In the next we will only use $\bf v$
to indicate both the velocity in (\ref{nn}) and the velocity of the 'rest frame'.
According to special relativity, the velocity $\bf c'$$(t)$ of the particle in
the second frame is
\begin{equation}
{\bf c'}(t)= \frac{1}{1+\frac{{\bf v}\cdot{\bf c}(t)}{c^2}}
\left[\alpha{\bf c}(t)+{\bf v} + (1-\alpha)\frac{{\bf v}
\cdot {\bf c}(t)}{v^2}{\bf v} \right]
\label{c'}
\end{equation}
where $\bf v$, $c$ and
$\alpha =\left(1-\frac{v^2}{c^2} \right)^{\frac{1}{2}}$ are constant.
Notice that in the above equation the argument of ${\bf c'}$ is still $t$
and not the time $t'$ of the second frame.
By special relativity the velocity in this second inertial frame is also
luminal (${\bf |c'|}=c$) (at the end of this section we will show
that indeed $({\bf c'})^2= {\bf c}^2 =c^2$).
If one also takes into account that the time increment $dt'$ in
the second frame satisfies
\begin{equation}
\frac{dt}{dt'}= \frac{1}{\alpha}
\left( 1-\frac{{\bf v} \cdot {\bf c'}}{c^2} \right),
\label{tt'}
\end{equation}
one should be able to write from (\ref{c'})
and (\ref{tt'}) a stochastic equation for ${\bf c'} (t')$
analogous to (\ref{steq2}) and (\ref{nn}).
Notice that in the new equation the time will be $t'$ and the increment
$d{\bf c'} = {\bf c'}(t'+dt') - {\bf c'}(t')$ must be expressed
in terms of ${\bf c'}(t')$, $dt'$ and of the increments
$dw'_2 = w_2(t'+dt') - w_2(t')$ and $dw'_3 = w_3(t'+dt') - w_3(t')$.
In the second frame the particle will still
instantaneously move at the speed of light but, contrarily to the case of the process
in the 'rest frame', its average velocity will not vanish at large times.
We define $\delta {\bf c'} = {\bf c'}(t+dt) - {\bf c'}(t)$
(notice the difference with $d{\bf c'} = {\bf c'}(t'+dt') - {\bf c'}(t')$),
then a long and tedious application of Ito calculus (see the Appendix) leads to
\begin{equation}
\delta {\bf c'}= - \frac{\omega^2}{\alpha^2}
\left[ 1-\frac{{\bf v} \cdot {\bf c'}}{c^2} \right]^2 {\bf c'} dt
+ \frac{\omega \, c}{\alpha}
\left( 1-\frac{{\bf v} \cdot {\bf c'}}{c^2} \right)
({\bf n'_2}\, d w_2 + {\bf n'_3}\, dw_3)
\label{ito}
\end{equation}
where $\bf n'_2$ and $\bf n'_3$ are the two unitary vectors perpendicular
to $\bf c'$ defined as in (\ref{nn}) (with $\bf n$, $\bf n_2$ and $\bf n_3$
replaced by $\bf n'$ = $\bf c'$$/c$, $\bf n'_2$ and $\bf n'_3$).
Then, taking into account (\ref{tt'})
and remembering that $dw_3 /dw'_3 = dw_2/dw'_2 =(dt/dt')^\frac{1}{2}$ one gets
\begin{equation}
d{\bf c'}= - \frac{\omega^2}{\alpha^3}
\left[ 1-\frac{{\bf v} \cdot {\bf c'}}{c^2} \right]^3 {\bf c'} dt'
+ \omega \, c\, \left[ \frac{1}{\alpha}
\left( 1-\frac{{\bf v} \cdot {\bf c'}}{c^2} \right)
\right]^{\frac{3}{2}} d {\bf w'}
\label{steq'}
\end{equation}
where $d {\bf w'}= {\bf n'_2}\, d w'_2 + {\bf n'_3}\, dw'_3$ is a two component
increment perpendicular to ${\bf c'}$ in complete analogy with the
process in the 'rest frame'. Notice that by (\ref{c'}) the
three vectors $\bf v$, $\bf c$ and $\bf c'$ are on the same plane
so that $\bf n_3$, $\bf n'_3$ also are
on the same plane. As a consequence $\bf n_2$ and $\bf n'_2$ are both
perpendicular to that plane so that $\bf n_2$ = $\bf n'_2$.
This is the core equation, each process of the family is labeled by
the index ${\bf v}$, the case ${\bf v}=0$ corresponds to the
process in the 'rest frame'.
One can easily prove from (\ref{steq2}) that the speed of the
particle remains constantly luminal {\it i.e,} $ d|{\bf c'}(t)| =0$,
in fact by Ito calculus
\begin{equation}
d|{\bf c'}|^2 = - 2 \, \frac{\omega^2}{\alpha^3}
\left[ 1-\frac{{\bf v} \cdot {\bf c'}}{c^2} \right]^3 |{\bf c'}|^2 dt'
+ 2 \, c^2 \,\frac{\omega^2}{\alpha^3}
\left[ 1-\frac{{\bf v} \cdot {\bf c'}}{c^2} \right]^3 dt'
+2 \frac{c \, \omega^3}{\alpha^3} \,
\left[ 1-\frac{{\bf v} \cdot {\bf c'}}{c^2} \right]^3
{\bf c'} \cdot d {\bf w'}
\end{equation}
where the second term at the right comes from the second order
contribution to Ito differential.
Since ${\bf c'}$ and $d {\bf w'}$ are perpendicular the equation
reduces to
\begin{equation}
d|{\bf c'}|^2 = - 2 \, \frac{\omega^2}{\alpha^3}
\left[ 1-\frac{{\bf v} \cdot {\bf c'}}{c^2} \right]^3
\left(|{\bf c'}|^2 - c^2 \right) dt' =0
\end{equation}
where the last equality holds if the initial velocity is luminal
{\it i.e,} $|{\bf c}(0)|=c$.
The fact that the process remains luminal is not
astonishing since a particle moving at the speed of light
also moves at the speed of light in any other inertial frame.
Therefore, the equation (\ref{steq'}) defines a family of
light-speed processes (indexed by ${\bf v}$)
which transform one in the other by Lorentz boost.
We already mentioned that in a generic inertial frame (indexed by ${\bf v}$)
the particle has
a long term unvanishing average velocity. This is simple consequence
of the fact that the rest frame moves at velocity ${\bf v}$ with respect to the
generic frame and of the fact that the long term average velocity vanishes
for the process in the 'rest frame' (we will prove this in the next section).
The reason of this long term average lies in the fact that
the diffusion of ${\bf c'}$
slows down the more ${\bf v} \cdot {\bf c'}$ is large
as it can be inferred from equation (\ref{steq'}),
This means that the particle spends more time with values of
${\bf c'}$ aligned with ${\bf v}$ and less time when it is
anti-aligned.
This is exactly the same one has with the (1+1)-dimensional
Kac process associated to the telegrapher equation.
In that simpler case a non vanishing average velocity
is determined by the fact that unbalanced rates of inversions
lead to a longer permanence of the velocity in one of the two directions.
\section{Short-term ballistic behavior versus long-term diffusive behavior}
In this section we only consider the process in the 'rest frame',
all results concerning averages
can be eventually Lorentz transformed for the processes in a generic
inertial frame.
The stochastic equation (\ref{steq}) can be recast in an integral
equation:
\begin{equation}
\begin{aligned}
& {\bf x}(t) = {\bf x}(0) +\int_0^t {\bf c}(s) ds, \\
& {\bf c}(t) = e^{-\omega^2 t} \left[ {\bf c}(0) +
\omega \, c \int_0^t e^{\omega^2 s} d {\bf w}(s) \right].
\end{aligned}
\label{sol}
\end{equation}
This is not a solution because $d {\bf w}(s)$, according to (\ref{m})
and (\ref{nn}), depends on ${\bf c}(t)$.
Let us mention that the proof that the particle velocity remains constantly
luminal {\it i.e,} $ |{\bf c}(t)| =c$ can be eventually also obtained by the
second integral equation in (\ref{sol}).
Starting from the second integral equation in
(\ref{sol}) one can easily find out that
the following averages hold for $t \ge s \ge 0 $:
\begin{equation}
\begin{aligned}
& E[{\bf c}(t)] = e^{-\omega^2 t} {\bf c}(0), \\
& E[{\bf c}(t) \cdot {\bf c}(s)] = c^2 \, e^{-\omega^2 (t+s)}
+ 2\omega^2 c^2 e^{-\omega^2 (t+s)}\int_0^s e^{2\omega^2 u} du
=c^2 \, e^{-\omega^2 (t-s)}.
\end{aligned}
\end{equation}
Notices that the first of the equalities above says that the average velocity
$E[{\bf c}(t)]$ vanishes for large $t$ however, for a generic inertial
frame, $E[{\bf c'}(t')]$ doesn't vanish for large $t'$.
This is simple consequence
of the fact that the rest frame moves at velocity ${\bf v}$ with respect to the
generic frame.
Using these averages and the first of the integral equations in
(\ref{sol}), one also obtains
\begin{equation}
\begin{aligned}
& E[{\bf x}(t)]
= {\bf x}(0)+ \frac{1 \!- \! e^{-\omega^2 t}}{\omega^2} \, {\bf c}(0), \\
& E[ \left({\bf x}(t)-{\bf x}(0)\right)^2 ] =
2 c^2 \int_0^t \int_0^s e^{-\omega^2 (s-u)} du \, ds=
\frac{2 c^2}{\omega^2} \, t -
\frac{2c^2}{\omega^4} \left( 1\!- \! e^{-\omega^2 t} \right).
\end{aligned}
\label{xx}
\end{equation}
The above averages imply, for large times, a diffusive behavior with
coefficient $\frac{ c^2}{\omega^2}$, in this limit
one has in fact
\begin{equation}
E[ \left({\bf x}(t)-{\bf x}(0)\right)^2 ]
\sim \frac{2 c^2}{\omega^2} \, t,
\label{long}
\end{equation}
on the contrary, for short times one has
\begin{equation}
E[{\bf x}(t)-{\bf x}(0)] \sim {\bf c}(0) t, \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,
E[ \left({\bf x}(t)-{\bf x}(0)\right)^2 ] \sim c^2 t^2,
\label{short}
\end{equation}
which means ballistic behavior at the speed of light.
The short-term ballistic behavior it is not completely unexpected. In fact,
for a small time $\Delta t \ll 1/\omega^2$ the velocity of a particle remains
almost constant. This can be understood from the second equation in
(\ref{steq}) which, having defined
$\Delta{\bf c} ={\bf c}(\Delta t)-{\bf c}(0)$, implies $|\Delta{\bf c}|
\approx|\omega^2 {\bf c}\,\Delta t - \omega c \, {\bf w}(\Delta t)|
\le \omega^2 c \Delta t + \omega c |{\bf w}(\Delta t)|$.
Since $|{\bf w}(\Delta t)|$ is of the order of $ \sqrt{\Delta t} $ and given
that $\Delta t \ll 1/\omega^2$,
one finally obtains $|\Delta{\bf c}| /c \ll 1$.
This short-term ballistic
behavior was already observed and described in depth in \cite{Debbasch:2012}
for the Relativistic Ornstein-Uhlenbeck Process \cite{Debbasch:1997}.
In case, it could be also coarsely derived from equation (2) in
\cite{Debbasch:2012} by adapting the line of reasoning we followed above.
\section{Summary and outlook}
In conclusion we have found that equation (\ref{steq'}) describes a family of
light-speed processes which transform one in the other by Lorentz boost.
Their main characteristics can be resumed as follows:
\begin{itemize}
\item
the family of processes that we propose generalizes to 3+1 dimensions
the 1956 idea of Mark Kac in the sense that particles only move at
the speed of light.
Although the Kac process can be generalized to 3+1 dimensions in a different
way, for example considering a velocity which performs jumps in place of having
continuous trajectories, we think the process presented here,
having as a constitutive ingredient the Wiener process, is the most natural choice
for this generalization. Moreover, since the speed is always the maximum possible
given the relativistic constraint,
it posseses the trajectories which better mimics the (infinite speed) Wiener
trajectories;
\item
the long term average velocity vanishes in the 'rest frame',
but it does not in a generic frame.
This is is a consequence of the fact that
according to (\ref{steq'}) the diffusion of the velocity
slows down the more ${\bf v} \cdot {\bf c'}$ is large.
In turn, this means that the particle spends more time with values of
${\bf c'}$ aligned with ${\bf v}$ and less time when it is
anti-aligned.
This is exactly the same situation one has with the (1+1)-dimensional
Kac process associated to the telegrapher equation since the probability rate
of inversion of velocity can be different for left/right and right/left
inversions \cite{Serva:1986};
\item
for large times the behavior of the position is diffusive with coefficient
$\frac{ c^2}{\omega^2}$, one has in fact $E[ \left({\bf x}(t)-{\bf x}(0)\right)^2]
\sim \frac{2 c^2}{\omega^2} \, t$. On the contrary, for short times
$E[{\bf x}(t)-{\bf x}(0)] \sim c t$ and
$E[ \left({\bf x}(t)-{\bf x}(0)\right)^2 ] \sim c^2 t^2$
which means ballistic behavior at the speed of light.
The short-term ballistic behavior holds for $t$ smaller than $1/\omega^2$.
\end{itemize}
This process is the natural candidate for modeling the Brownian motion
of mass-less particles, nevertheless, its use should not be limited to this case.
The situation is similar in the non-relativistic realm; thought
a particle with infinite speed is unphysical, the Wiener process
is largely used to model its erratic movement.
Another point that deserves investigation and which contributed to prompt
this work is the possible connection of its Backward Kolmgorov Equation
with relativistic equations as Klein-Gordon and Dirac.
The goal would be to find a generalization of the Gaveau {\it et al.}
approach to the (3+1)-dimensional case. This topic is presently under study.
\section*{Appendix: Ito calculus}
In this appendix we apply Ito calculus, in order to obtain equation (\ref{ito})
from equation (\ref{c'}).
Since we defined $\delta {\bf c'} = {\bf c'}(t+dt) - {\bf c'}(t)$,
then from (\ref{c'}) we immediately get
\begin{equation}
\delta{\bf c'}=
\frac{1}{1+\frac{{\bf v}\cdot({\bf c}+d{\bf c})}{c^2}}
\left[\alpha({\bf c}+d{\bf c})+{\bf v} + (1-\alpha)\frac{{\bf v}
\cdot ({\bf c}+d{\bf c})}{v^2}{\bf v} \right]-
\frac{1}{1+\frac{{\bf v}\cdot{\bf c}}{c^2}}
\left[\alpha{\bf c}+{\bf v} + (1-\alpha)\frac{{\bf v}
\cdot {\bf c}}{v^2}{\bf v} \right]
\end{equation}
where $ {\bf c} + d{\bf c} = {\bf c}(t+dt) $ and ${\bf c} = {\bf c}(t)$.
This is still not a Ito increment, but it is the trivial
application of the definition $\delta {\bf c'} = {\bf c'}(t+dt) - {\bf c'}(t)$.
This equation can be exactly rewritten as
\begin{equation}
\delta{\bf c'}=
\frac{d \epsilon}{1+d \epsilon} \left[ (1-\alpha)\frac{c^2}{v^2}{\bf v}-
\frac{1}{1+\frac{{\bf v}\cdot{\bf c}}{c^2}}
\left(\alpha{\bf c}+{\bf v} + (1-\alpha)\frac{{\bf v}
\cdot {\bf c}}{v^2}{\bf v} \right)\right] +
\frac{1}{1+d \epsilon} \frac{\alpha{d\bf c}}{1+\frac{{\bf v}\cdot{\bf c}}{c^2}}
\label{dc'}
\end{equation}
where $d{\bf c}$ is given by (\ref{steq2}) and where
\begin{equation}
d \epsilon=\frac{1}{1+\frac{{\bf v}\cdot{\bf c}}{c2}} \,
\frac{ {\bf v}\cdot d{\bf c}}{c^2} = -\frac{\omega^2 }{c^2}
\frac{ {\bf v}\cdot {\bf c}}{1+\frac{{\bf v}\cdot{\bf c}}{c2}}
\, dt + \frac{\omega }{c} \,
\frac{ {\bf v}\cdot {\bf n_3}}{1+\frac{{\bf v}\cdot{\bf c}}{c2}} \, dw_3.
\label{eps}
\end{equation}
The next step is to calculate the right side of the equation (\ref{dc'})
in terms of the new
variables ${\bf c'}$, ${\bf n_2'}$ and ${\bf n_3'}$. First of all,
using (\ref{c'}) we immediately rewrite the equation (\ref{dc'}) as
\begin{equation}
\delta {\bf c'} = \frac{1}{1+d \epsilon}
\left[d \epsilon \, \left( (1-\alpha)\frac{c^2}{v^2}{\bf v} - {\bf c'} \right)
+ \frac{ 1-\frac{{\bf v}\cdot{\bf c'}}{c^2} }{\alpha}\, d{\bf c}\, \right],
\label{2dc'}
\end{equation}
but we also need to calculate $d \epsilon$ and $ d{\bf c}$ in
terms of the new coordinates. In order to reach this goal we need to recall that
\begin{equation}
{\bf c}= \frac{1}{1-\frac{{\bf v}\cdot{\bf c'}}{c^2}}
\left[\alpha{\bf c'}-{\bf v} + (1-\alpha)\frac{{\bf v}
\cdot {\bf c'}}{v^2}{\bf v} \right] ,
\,\,\,\,\,\,\, \to \,\,\,\,\,\,\,
{\bf v} \cdot {\bf c}= \frac{1}{1-\frac{{\bf v}\cdot{\bf c'}}{c^2}}
\left[ {\bf v}\cdot {\bf c'} - v^2 \right] \, ,
\label{c}
\end{equation}
moreover
\begin{equation}
c \, {\bf n_3}= \frac{1}{1-\frac{{\bf v}\cdot{\bf c'}}{c^2}}
\left[\alpha c {\bf n_3'}-{\bf v} \times {\bf n_2'} +
(1-\alpha)\frac{{\bf v}
\cdot {\bf c'}}{v^2} \, {\bf v} \times {\bf n_2'}\right],
\,\,\,\,\,\,\, \to \,\,\,\,\,\,\,
{\bf v} \cdot {\bf n_3}= \alpha \frac{{\bf v}
\cdot {\bf n_3'}}{1-\frac{{\bf v}\cdot{\bf c'}}{c^2}}.
\label{n3}
\end{equation}
From these two last equations one easily realize
that the three vectors $\bf c$, $\bf v$ and $\bf c'$ are co-planar.
As well, ${\bf n_3}$ and ${\bf n_3'}$ lie on the same plane.
Moreover, ${\bf n_2'}={\bf n_2}$ is perpendicular to that plane.
We get
\begin{equation}
d \epsilon = -\frac{\omega^2 }{c^2}
\frac{ {\bf v}\cdot {\bf c'}-v^2}{\alpha^2}
\, dt + \frac{\omega }{c} \,
\frac{ {\bf v}\cdot {\bf n_3}'}{\alpha} \, dw_3,
\label{eps'}
\end{equation}
where $d {\bf c}$ is given by (\ref{steq2}) with
$\bf n_2$=$\bf n_2'$ and $\bf c$ and $\bf n_3$ given respectively by
the first equation in (\ref{c}) and the first equation in (\ref{n3}).
We are now ready compute the Ito differential {\it i.e,} we are ready
to rewrite the differential $\delta {\bf c'}$ keeping only terms of order $dt$.
To obtain this result we have first of all to expand (\ref{2dc'})
to the second order with respect to the differentials:
\begin{equation}
\delta {\bf c'} \approx
\left[d \epsilon \, \left( (1-\alpha)\frac{c^2}{v^2}{\bf v} - {\bf c'} \right)
+ \frac{ 1-\frac{{\bf v}\cdot{\bf c'}}{c^2} }{\alpha}\, d{\bf c}\,\, \right] -
\left[(d \epsilon)^2 \, \left( (1-\alpha)\frac{c^2}{v^2}{\bf v} -
{\bf c'} \right) + d \epsilon d{\bf c}] \,
\frac{ 1-\frac{{\bf v}\cdot{\bf c'}}{c^2} }{\alpha}\, \right],
\label{3dc'}
\end{equation}
then, we have replace the second order differentials
$(d \epsilon)^2$ and $d \epsilon \, d{\bf c}$
by the terms proportional to $dt$ of their averages:
\begin{equation}
(d \epsilon)^2 \approx
\left(\frac{\omega }{c} \right)^2 \,
\frac{({\bf v} \cdot {\bf n_3'})^2}{\alpha^2}\, dt \, ,
\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,
d \epsilon d{\bf c} \approx
\frac{\omega^2 }{\alpha} \,
({\bf v} \cdot {\bf n_3'}) {\bf n_3}\, dt
\end{equation}
where $\bf n_3$ must be expressed in terms of the new variables
by the first equation in (\ref{n3}).
Let us rewrite equation (\ref{3dc'}) as
\begin{equation}
\delta {\bf c'}= \delta \mathcal{A}+ \delta \mathcal{B}+ \delta \mathcal{C}
\label{4dc'}
\end{equation}
where $\delta \mathcal{A}$ is the term proportional to $dt$ which comes from
the first order differentials (the deterministic part of the first
term between square parenthesis in (\ref{3dc'})),
$\delta \mathcal{B}$ is the term proportional to $dt$ which
comes from the second order differential (the second
term between square parenthesis in (\ref{3dc'})) and
$\delta \mathcal{C}$ is the random term (the random part of the first
term between square parenthesis in (\ref{3dc'})).
After having expressed all the old variables in terms of the new ones
(except $\bf n_3$), we have
\begin{equation}
\begin{aligned}
& \delta \mathcal{A}= -\left[\frac{\omega^2 }{c^2}
\frac{ {\bf v}\cdot {\bf c'}-v^2}{\alpha^2}
\left( (1-\alpha)\frac{c^2}{v^2}{\bf v} - {\bf c'} \right)
+ \frac{\omega^2}{\alpha}\,
\left(\alpha{\bf c'}-{\bf v} + (1-\alpha)\frac{{\bf v}
\cdot {\bf c'}}{v^2}{\bf v} \right) \,\, \right]dt , \\
& \delta \mathcal{B}= -\frac{\omega ^2}{\alpha^2} ({\bf v} \cdot {\bf n_3'})
\left[\frac{{\bf v} \cdot {\bf n_3'}}{c^2}\, \,
\left( (1-\alpha)\frac{c^2}{v^2}{\bf v} -
{\bf c'} \right) +
\left( 1-\frac{{\bf v}\cdot{\bf c'}}{c^2}\right)\, {\bf n_3}\right] dt , \\
& \delta \mathcal{C}= \left[ \frac{\omega }{c} \,
\frac{ {\bf v}\cdot {\bf n_3}'}{\alpha} \,
\left( (1-\alpha)\frac{c^2}{v^2}{\bf v} - {\bf c'} \right)\, dw_3
+ \frac{ 1-\frac{{\bf v}\cdot{\bf c'}}{c^2} }{\alpha}\,
\omega c \left( {\bf n'_2} \, dw_2 + {\bf n_3} \,dw_3\right) \right]
\end{aligned}
\label{5dc'}
\end{equation}
with $\bf n_3$ given by (\ref{n3}) in terms of the new variables.
After some rearrangement of terms we get:
\begin{equation}
\delta \mathcal{A} =
-\left[\frac{\omega^2 }{c^2}
\frac{ {\bf v}\cdot {\bf c'}}{\alpha^2}
- \frac{\omega^2}{\alpha^2}\, \right]{\bf v} dt
+\left[\frac{\omega^2 }{c^2}
\frac{ {\bf v}\cdot {\bf c'}-c^2}{\alpha^2}
\right]{\bf c'}dt
=\frac{\omega^2 }{\alpha^2} \left[ 1-\frac{{\bf v}\cdot{\bf c'}}{c^2}
\right]({\bf v}-{\bf c'})dt.
\end{equation}
This differential lies in the plane of $\bf c'$ and $\bf n_3'$
and can be decomposed along these two vectors:
\begin{equation}
\delta \mathcal{A}
=-\frac{\omega^2 }{\alpha^2} \left[ 1-\frac{{\bf v}\cdot{\bf c'}}{c^2}
\right]^2 {\bf c'}dt+\frac{\omega^2 }{\alpha^2}
\left[ 1-\frac{{\bf v}\cdot{\bf c'}}{c^2}
\right]({\bf v} \cdot{\bf n_3'}){\bf n_3'}dt.
\end{equation}
Analogously, the term $\delta \mathcal{B}$, after decomposition along
$\bf c'$ and $\bf n_3'$, can be rewritten as
\begin{equation}
\begin{aligned}
\delta \mathcal{B} = -\frac{\omega ^2}{\alpha^2} ({\bf v} \cdot {\bf n_3'}) &
\left( 1-\frac{{\bf v}\cdot{\bf c'}}{c^2}\right)\,
\left[\frac{{\bf v} \cdot {\bf n_3'}}
{ 1-\frac{{\bf v}\cdot{\bf c'}}{c^2}}\, \,
\left( (1-\alpha)\frac{{\bf v} \cdot {\bf c'}}{v^2} - 1 \right) +
{\bf n_3} \cdot {\bf c'}\right] \frac{{\bf c'}}{c^2} dt \\
& -\frac{\omega ^2}{\alpha^2} ({\bf v} \cdot {\bf n_3'})
\left( 1-\frac{{\bf v}\cdot{\bf c'}}{c^2}\right)\,
\left[\frac{{\bf v} \cdot {\bf n_3'}}
{\left( 1-\frac{{\bf v}\cdot{\bf c'}}{c^2}\right)\,}\, \,
\left( (1-\alpha)\frac{{\bf v} \cdot {\bf n_3'}}{v^2} \right) +
{\bf n_3} \cdot {\bf n_3'}\right] {\bf n_3'} dt .
\end{aligned}
\label{dd}
\end{equation}
Since ${\bf c'} \cdot {\bf n_3}$=$-{\bf c} \cdot {\bf n_3'}$, from (\ref{c})
by scalar product with ${\bf n_3'}$, one gets
\begin{equation}
{\bf c'} \cdot {\bf n_3}= \frac{1}{ 1-\frac{{\bf v}\cdot{\bf c'}}{c^2}}
\left[1 - (1-\alpha)\frac{{\bf v} \cdot {\bf c'}}{v^2 c^2}\right]
{\bf v} \cdot {\bf n_3'} ,
\label{vw}
\end{equation}
therefore the first term at the right side
of equality (\ref{dd}) vanishes, moreover from (\ref{c})
and by the definitions of $\bf n_3$ and $\bf n_3'$ one has that
\begin{equation}
{\bf n_3} \cdot {\bf n_3'}= \frac{{\bf c} \cdot {\bf c}'}{c^2}=
\frac{1}{1-\frac{{\bf v}\cdot{\bf c'}}{c^2}}
\left[\alpha - \frac{{\bf v}\cdot{\bf c'}}{ c^2} +
(1-\alpha)\frac{({\bf v}\cdot {\bf c'})^2}{v^2}\right]
\label{cc'} ,
\end{equation}
which, can be substituted in the second term of (\ref{dd}) in order
to obtain
\begin{equation}
\delta \mathcal{B} =
-\frac{\omega ^2}{\alpha^2} ({\bf v} \cdot {\bf n_3'})
\left( 1-\frac{{\bf v}\cdot{\bf c'}}{c^2}\right)\,{\bf n_3'} dt .
\end{equation}
A scalar product of the third of (\ref{5dc'}) with ${\bf c'}$ gives
\begin{equation}
{\bf c'} \cdot \delta \mathcal{C}=
\frac{\omega c}{\alpha} \left[ \,
{\bf v}\cdot {\bf n_3}' \,
\left( (1-\alpha)\frac{{\bf v} \cdot{\bf c'}}{v^2} - 1\right)\,
+ \left( 1-\frac{{\bf v}\cdot{\bf c'}}{c^2} \right)\,
\left( {\bf c'} \cdot {\bf n_3} \right) \right] dw_3 =0
\end{equation}
where the equality is obtained using (\ref{vw}).
Construction is coherent since the Wiener increment of ${\bf c'}$ has no
component parallel to ${\bf c'}$ itself.
Therefore, by decomposition along
$\bf n_2'$ = $\bf n_2'$ and $\bf n_3'$ we have
\begin{equation}
\delta \mathcal{C}=
\frac{\omega c}{\alpha}\left[ \,
1 - \frac{{\bf v}\cdot{\bf c'}}{c^2} \right] {\bf n'_2} \, dw_2
+
\frac{\omega c}{\alpha}\left[ \,
(1-\alpha)\frac{({\bf v}\cdot {\bf n_3}')^2}{v^2}
+ \left(1-\frac{{\bf v}\cdot{\bf c'}}{c^2} \right)
\frac{{\bf c'} \cdot {\bf c}}{c^2} \right] {\bf n_3'} \, dw_3 ,
\end{equation}
which given (\ref{cc'}) can be rewritten as
\begin{equation}
\delta \mathcal{C}= \frac{\omega \, c } {\alpha} \,
\left(1-\frac{{\bf v}\cdot{\bf c'} }{c^2} \right) \,
( {\bf n'_2} \, dw_2 + {\bf n'_3} \,dw_3).
\label{bw}
\end{equation}
Finally, by $\delta {\bf c'}= \delta \mathcal{A}+\delta \mathcal{B}
+\delta \mathcal{C}$
we finally obtain
\begin{equation}
\delta {\bf c'}= - \frac{\omega^2}{\alpha^2}
\left[ 1-\frac{{\bf v} \cdot {\bf c'}}{c^2} \right]^2 {\bf c'} dt
+ \frac{\omega \, c}{\alpha}
\left( 1-\frac{{\bf v} \cdot {\bf c'}}{c^2} \right)
({\bf n'_2}\, d w_2 + {\bf n'_3}\, dw_3),
\label{ito2}
\end{equation}
which is the equation (\ref{ito}) that we use in section 3 and which allows
to find out the equation (\ref{steq'}) which characterizes the general class
of the light-speed processes.
|
2,877,628,089,010 | arxiv | \section{Introduction}
Answer set programming (ASP) is a declarative formalism for knowledge representation and reasoning based on stable model semantics \cite{DBLP:journals/ngc/GelfondL91,DBLP:journals/cacm/BrewkaET11}, for which robust and efficient implementations are available~\cite{DBLP:conf/iclp/GebserKKOSW16}.
State-of-the-art ASP systems are usually based on the ``ground+solve'' approach~\cite{DBLP:journals/aim/KaufmannLPS16}, in which a \textit{grounder} module transforms the input program (containing variables) in an equivalent variable-free one, whose stable models are subsequently computed by the \textit{solver} module.
ASP implementations adopting this traditional approach are known to be effective for solving complex problems arising from academic and industrial applications, including: product configuration~\cite{DBLP:conf/scm/KojoMS03}, decision support systems for space shuttle flight controllers~\cite{DBLP:conf/asp/NogueiraBGWB01}, explanation of biomedical queries~\cite{DBLP:journals/tplp/ErdemO15}, construction of phylogenetic supertrees~\cite{DBLP:journals/tplp/KoponenOJS15}, data-integration~\cite{DBLP:journals/tplp/MannaRT15}, reconfiguration systems~\cite{DBLP:conf/cpaior/AschingerDFGJRT11}, and more.
Nonetheless, there are some classes of programs
(cf.\ \cite{DBLP:journals/ai/CalimeriGMR16})
whose evaluation is not feasible with the ``ground+solve'' approach just because the grounding phase induces a combinatorial blow-up.
An issue that is usually referred to as the \textit{grounding bottleneck} of ASP.
The grounding bottleneck has been subject of several studies in recent years, and various alternative approaches to overcome it have been proposed.
Some of these are based on syntactic extensions that enable the combination of ASP solvers with solvers for external theories~\cite{DBLP:journals/tplp/OstrowskiS12,DBLP:journals/corr/BalducciniL17,DBLP:journals/tplp/BalducciniL13,DBLP:journals/tplp/AzizCS13,DBLP:journals/jair/CatDBS15,DBLP:conf/iclp/SusmanL16,DBLP:conf/birthday/EiterRS16};
whereas, the most prominent approach working on plain ASP is \textit{lazy grounding}, which was implemented by \textsc{asperix}~\cite{DBLP:conf/lpnmr/LefevreN09a}, \textsc{gasp}~\cite{DBLP:journals/fuin/PaluDPR09}, and \textsc{omiga}~\cite{DBLP:conf/jelia/Dao-TranEFWW12}.
Roughly, the idea of lazy grounding is to instantiate rules only when it is required during the search for a stable model~\cite{DBLP:journals/ai/LiuPST10}. %
In this way, it is possible to prevent the grounding of rules that are unnecessary for the computation.
Albeit lazy grounding techniques obtained promising preliminary results,
they cannot yet reach the performance of state of the art systems in many
benchmarks~\cite{DBLP:journals/ai/CalimeriGMR16,DBLP:conf/lpnmr/LefevreN09a}.
One of the reasons is probably that fully-fledged lazy grounding techniques could not be easily integrated within solvers based on the very efficient Conflice-Driven Clause Learning (CDCL) algorithm~\cite{DBLP:journals/tc/Marques-SilvaS99,DBLP:journals/aim/KaufmannLPS16,Weinzierl2017}.
Nonetheless, in many applications, the grounding bottleneck is merely caused by rules of a specific kind, namely constraints.
For example, the following constraint has been identified
as the bottleneck in programs solving a problem of natural language understanding:
\begin{equation*}
\leftarrow eq(X,Y), \ eq(Y, Z),\ \ensuremath{\raise.17ex\hbox{\ensuremath{\scriptstyle\mathtt{\sim}}}}\xspace eq(X, Z)
\end{equation*}
Its grounding, which features a cubic number of instances with respect to the extension of predicate $eq$ in the worst case, is often not feasible for real world instances~\cite{DBLP:journals/fuin/Schuller16}.
In this paper, we focus on the above practically-relevant case of problematic constraints.
In particular, we systematically compare alternative strategies that avoid the instantiation of some constraints by extending a CDCL-based ASP solver.
In a nutshell, the input program is simplified by omitting problematic constraints and it is grounded; then, the resulting ground program is provided as input to a solver that is extended to emulate the presence of missing constraints.
Among the strategies for extending the solver, we considered \textit{lazy instantiation of constraints} and \textit{custom propagators}.
In the first strategy, the solver searches for a stable model $S$ of the simplified program. Then, $S$ is returned as a solution if it satisfies also the omitted constraints, otherwise the violated instances of these constraints are lazily instantiated, and the search continues (Sec.~\ref{sec:lazy}).
In the second strategy, the solver is extended (in possibly alternative ways) by custom \textit{propagators}, which emulate the presence of missing constraints during the search (Sec.~\ref{sec:propagators}).
The above-mentioned strategies can be implemented by using the API of existing CDCL-based ASP solvers~\cite{DBLP:conf/iclp/GebserKKOSW16,DBLP:conf/aiia/DodaroRS16}.
An empirical evaluation conducted on real and synthetic benchmarks (Sec.~\ref{sec:experiments}) confirms that the usage of lazy instantiation and custom propagators is effective when the grounding bottleneck is due to some constraint. The analysis of the results highlights strengths and weaknesses of the different strategies.
Moreover, it shows there is not always a clear winner for a given problem, and the choice depends also on the characteristics the instances to solve.
This observation suggested to investigate the applicability of algorithm selection techniques.
The results are positive, in the sense that already a basic portfolio is faster than the best approach.
\section{Answer Set Programming (ASP)}
An ASP program $\Pi$ is a finite set of rules of the form:
\begin{equation}\label{eq:rule}
a_1 \lor \ldots \lor a_n \lar b_1, \ldots, b_j, \ensuremath{\raise.17ex\hbox{\ensuremath{\scriptstyle\mathtt{\sim}}}}\xspace b_{j+1}, \ldots, \ensuremath{\raise.17ex\hbox{\ensuremath{\scriptstyle\mathtt{\sim}}}}\xspace b_m
\end{equation}
where $a_1,\ldots,a_n,b_1,\ldots,b_m$ are atoms and $n\geq 0,$
$m\geq j\geq 0$.
In particular, an \emph{atom} is an expression of the form $p(t_1, \ldots, t_k)$, where $p$ is a predicate symbol and $t_1, \ldots, t_k$ are \emph{terms}.
Terms are alphanumeric strings, and are distinguished in variables and constants.
According to the Prolog's convention, only variables start with an uppercase letter.
A \emph{literal} is an atom $a_i$ (positive) or its negation $\ensuremath{\raise.17ex\hbox{\ensuremath{\scriptstyle\mathtt{\sim}}}}\xspace a_i$ (negative), where $\ensuremath{\raise.17ex\hbox{\ensuremath{\scriptstyle\mathtt{\sim}}}}\xspace$ denotes the \emph{negation as failure}.
Given a rule $r$ of the form (\ref{eq:rule}), the disjunction $a_1 \lor \ldots \lor a_n$ is the {\em head} of $r$, while $b_1,\ldots,b_j, \ensuremath{\raise.17ex\hbox{\ensuremath{\scriptstyle\mathtt{\sim}}}}\xspace b_{j+1}, \ldots, \ensuremath{\raise.17ex\hbox{\ensuremath{\scriptstyle\mathtt{\sim}}}}\xspace b_m$ is the {\em body} of $r$, of which $b_1,\ldots,b_j$ is the {\em positive body}, and $\ensuremath{\raise.17ex\hbox{\ensuremath{\scriptstyle\mathtt{\sim}}}}\xspace b_{j+1}, \ldots, \ensuremath{\raise.17ex\hbox{\ensuremath{\scriptstyle\mathtt{\sim}}}}\xspace b_m$ is the {\em negative body} of $r$.
A rule $r$ of the form (\ref{eq:rule}) is called a \textit{fact} if $m=0$ and a \textit{constraint} if $n=0$.
An object (atom, rule, etc.) is called {\em ground} or {\em
propositional}, if it contains no variables.
Rules and programs are \textit{positive} if they contain no negative literals, and \textit{general} otherwise. %
Given a program $\Pi$, let the \emph{Herbrand Universe} \HU{\Pi} be the
set of all constants appearing in $\Pi$ and the \emph{Herbrand Base}
\HB{\Pi} be the set of all possible ground atoms which can be constructed
from the predicate symbols appearing in $\Pi$ with the constants of \HU{\Pi}.
Given a rule $r$, \GP{r} denotes the
set of rules obtained by applying all possible substitutions $\sigma$
from the variables in $r$ to elements of \HU{\Pi}. Similarly, given a
program $\Pi$, the {\em ground instantiation} \GP{\Pi} of $\Pi$
is the set \( \bigcup_{r \in \Pi} \GP{r} \).
For every program $\Pi$, its stable models are defined using its ground
instantiation \GP{\Pi} in two steps: First stable models of positive
programs are defined, then a reduction of general programs to positive
ones is given, which is used to define stable models of general
programs.
A set $L$ of ground literals is said to be {\em consistent} if, for every
literal $\ell \in L$, its negated literal $\ensuremath{\raise.17ex\hbox{\ensuremath{\scriptstyle\mathtt{\sim}}}}\xspace \ell$ is not contained in
$L$.
Given a set of ground literals $L$, $\posOf{L} \subseteq L$ denotes
the set of positive literals in $L$.
An interpretation $I$ for $\Pi$ is a consistent set of ground literals
over atoms in $\HB{\Pi}$. %
A ground literal $\ell$ is {\em true} w.r.t.\ $I$ if $\ell\in I$;
$\ell$ is {\em false} w.r.t.\ $I$ if its negated literal is in $I$;
$\ell$ is {\em undefined} w.r.t.\ $I$ if it is neither true nor false w.r.t.\ $I$.
A constraint $c$ is said to be \textit{violated} by an interpretation $I$ if all literals in the body of $c$ are true.
An interpretation $I$ is {\em total} if, for each atom $a$ in $\HB{\Pi}$,
either $a$ or $\ensuremath{\raise.17ex\hbox{\ensuremath{\scriptstyle\mathtt{\sim}}}}\xspace a$ is in $I$ (i.e., no atom in $\HB{\Pi}$ is undefined w.r.t.\ $I$).
Otherwise, it is \textit{partial}.
A total interpretation $M$ is a {\em model} for $\Pi$
if, for every $r \in \GP{\Pi}$, at
least one literal in the head of $r$ is true w.r.t.\ $M$ whenever all literals in the
body of $r$ are true w.r.t.\ $M$.
A model $X$ is a {\em stable model}
for a positive program $\Pi$ if any other model $Y$ of $\Pi$ is such that $\posOf{X} \subseteq \posOf{Y}$.
The {\em reduct} or {\em Gelfond-Lifschitz transform}
of a general ground program $\Pi$ w.r.t.\ an interpretation $X$ is the positive
ground program $\Pi^X$, obtained from $\Pi$ by (i) deleting all rules
$r \in \Pi$ whose negative body is false w.r.t.\ X and (ii)
deleting the negative body from the remaining rules.
A stable model of %
$\Pi$ is a model $X$ of $\Pi$ such
that $X$ is a stable model of $\GP{\Pi}^X$.
We denote by $SM(\Pi)$ the set of all stable models of $\Pi$,
and call $\Pi$ \textit{coherent} if $SM(\Pi) \neq \emptyset$, \textit{incoherent} otherwise.
\begin{example}\label{ex:grounding}
Consider the following program $\Pi_1$:
\begin{equation*}
\begin{array}{lll}
r_1: a(1) \leftarrow \ensuremath{\raise.17ex\hbox{\ensuremath{\scriptstyle\mathtt{\sim}}}}\xspace b(1) &\qquad
r_2: b(1) \leftarrow \ensuremath{\raise.17ex\hbox{\ensuremath{\scriptstyle\mathtt{\sim}}}}\xspace a(1) & \qquad
r_3: \leftarrow a(X), \ b(X) \\
r_4: c(1) \leftarrow \ensuremath{\raise.17ex\hbox{\ensuremath{\scriptstyle\mathtt{\sim}}}}\xspace d(1) & \qquad
r_5: d(1) \leftarrow \ensuremath{\raise.17ex\hbox{\ensuremath{\scriptstyle\mathtt{\sim}}}}\xspace c(1) & \qquad
r_6: \leftarrow a(X), \ \ensuremath{\raise.17ex\hbox{\ensuremath{\scriptstyle\mathtt{\sim}}}}\xspace b(X)
\end{array}
\end{equation*}
The ground instantiation $\GP{\Pi_1}$ of the program $\Pi_1$ is the following program:
\begin{equation*}
\begin{array}{lll}
g_1: a(1) \leftarrow \ensuremath{\raise.17ex\hbox{\ensuremath{\scriptstyle\mathtt{\sim}}}}\xspace b(1) &\qquad
g_2: b(1) \leftarrow \ensuremath{\raise.17ex\hbox{\ensuremath{\scriptstyle\mathtt{\sim}}}}\xspace a(1) & \qquad
g_3: \leftarrow a(1), \ b(1) \\
g_4: c(1) \leftarrow \ensuremath{\raise.17ex\hbox{\ensuremath{\scriptstyle\mathtt{\sim}}}}\xspace d(1) & \qquad
g_5: d(1) \leftarrow \ensuremath{\raise.17ex\hbox{\ensuremath{\scriptstyle\mathtt{\sim}}}}\xspace c(1) & \qquad
g_6: \leftarrow a(1), \ \ensuremath{\raise.17ex\hbox{\ensuremath{\scriptstyle\mathtt{\sim}}}}\xspace b(1)
\end{array}
\end{equation*}
Note that %
$M=\{\ensuremath{\raise.17ex\hbox{\ensuremath{\scriptstyle\mathtt{\sim}}}}\xspace a(1), \ b(1), \ c(1), \ \ensuremath{\raise.17ex\hbox{\ensuremath{\scriptstyle\mathtt{\sim}}}}\xspace d(1)\}$ is a model of $\GP{\Pi_1}$. Since $\GP{\Pi_1}^M$ comprises only the facts $b(1)$ and $c(1)$, and constraint $g_3$, $M$ is a stable model of $\Pi$.
%
$\hfill\lhd$
\end{example}
\paragraph{Support.}
Given a model $M$ for a ground program $\Pi$,
we say that a ground atom $a \in M$ is {\em supported}
with respect to $M$ if there exists a \emph{supporting} rule $r\in \Pi$
such that $a$ is the only true atom w.r.t. $M$ in the head of $r$, and all literals in the
body of $r$ are true w.r.t.\ $M$.
If $M$ is a stable model of a program $\Pi$, then all atoms in $M$ are supported.
\section{Solving Strategies}
\label{sec:solving}
\subsection{Classical Evaluation}\label{sec:solving:classical}
The standard solving approach for ASP is
instantiation followed by a procedure similar to CDCL for SAT
with extensions specific to ASP~\cite{DBLP:journals/aim/KaufmannLPS16}.
The basic algorithm
$\mi{ComputeStableModel}(\Pi)$
for finding a stable model of program $\Pi$
is shown in Algorithm~\ref{alg:mg}.
The Function~\ref{fn:propagatestd}
combines unit propagation (as in SAT) with
some additional ASP-specific propagations, which ensure
the model is stable (cf. \cite{DBLP:journals/aim/KaufmannLPS16}).
Given a partial interpretation $I$ consisting of literals,
and a set of rules $\Pi$,
\emph{unit propagation}
infers a literal $\ell$ to be true
if there is a rule $r \in \Pi$ such that
$r$ can be satisfied only by $I \cup \{\ell\}$.
Given the nogood representation
$C(r) = \{ \ensuremath{\raise.17ex\hbox{\ensuremath{\scriptstyle\mathtt{\sim}}}}\xspace a_1, \ldots, \ensuremath{\raise.17ex\hbox{\ensuremath{\scriptstyle\mathtt{\sim}}}}\xspace a_n, b_1, \ldots, b_j, \ensuremath{\raise.17ex\hbox{\ensuremath{\scriptstyle\mathtt{\sim}}}}\xspace b_{j+1}, \ldots, \ensuremath{\raise.17ex\hbox{\ensuremath{\scriptstyle\mathtt{\sim}}}}\xspace b_m \}$
of a rule $r$,
then the negation of a literal $\ell \in C(r)$
is unit propagated w.r.t.\ $I$ and rule $r$
iff $C(r) \setminus \{ \ell \} \subseteq I$.
To ensure that models are supported,
unit propagation is performed on the
Clark completion %
of $\Pi$ %
or alternatively a \emph{support propagator} is used~\cite{DBLP:conf/ijcai/AlvianoD16}.
\begin{example}\label{ex:prog}
Consider the ground program $\Pi_1$ from Example~\ref{ex:grounding}.
$\mi{ComputeStableModel}(\Pi_1)$
starts with $I = \emptyset$ and
does not propagate anything in line~\ref{ln:alg:propagate}.
$I$ is partial and consistent, so the algorithm continues
in line~\ref{ln:alg:partial}.
Assume no restart and no deletion is performed,
and assume $\mi{ChooseLiteral}$ returns $\{a(1)\}$, i.e., $I=\{a(1)\}$.
Next, $\mi{Propagate}(I)$ is called,
which yields $I = \{a(1),\ b(1),\ \ensuremath{\raise.17ex\hbox{\ensuremath{\scriptstyle\mathtt{\sim}}}}\xspace b(1)\}$:
$\ensuremath{\raise.17ex\hbox{\ensuremath{\scriptstyle\mathtt{\sim}}}}\xspace b(1)$ comes from unit propagation on $g_3$
and $b(1)$ from unit propagation on $g_6$.
Thus, $I$ is inconsistent and $I$ is analyzed to compute a reason explaining the conflict, i.e., $\mi{CreateConstraint}(I) = \{g_7\}$ with $g_7: \lar a(1)$.
Intuitively, the truth of $a(1)$ leads to an inconsistent interpretation, thus $a(1)$ must be false.
Then, the consistency of $I$ is restored (line~\ref{ln:alg:restore}), i.e., $I = \emptyset$, and $g_7$ is added to $\Pi_1$.
The algorithm again restarts at line~\ref{ln:alg:propagate}
with $I = \emptyset$
and propagates $I = \{\ensuremath{\raise.17ex\hbox{\ensuremath{\scriptstyle\mathtt{\sim}}}}\xspace a(1),\ b(1)\}$,
where $\ensuremath{\raise.17ex\hbox{\ensuremath{\scriptstyle\mathtt{\sim}}}}\xspace a(1)$ comes from unit propagation on $g_7$,
and $b$ from unit propagation on $g_2$.
$I$ is partial and consistent,
therefore lines~\ref{ln:alg:partial} and~\ref{ln:alg:choice} are executed.
Assume again that no restart and no constraint deletion
happens,
and that $\mi{ChooseLiteral}(I) = \{ c(1) \}$.
Therefore, the algorithm continues in line~\ref{ln:alg:propagate}
with $I = \{ \ensuremath{\raise.17ex\hbox{\ensuremath{\scriptstyle\mathtt{\sim}}}}\xspace a(1), \ b(1), \ c(1) \}$.
Propagation yields $I = \{ \ensuremath{\raise.17ex\hbox{\ensuremath{\scriptstyle\mathtt{\sim}}}}\xspace a(1),\ b(1),\ c(1),\ \ensuremath{\raise.17ex\hbox{\ensuremath{\scriptstyle\mathtt{\sim}}}}\xspace d(1) \}$
because $\ensuremath{\raise.17ex\hbox{\ensuremath{\scriptstyle\mathtt{\sim}}}}\xspace d(1)$ is support-propagated w.r.t.\ $g_4$ and $I$ (or unit-propagated w.r.t.\ the completion of $g_4$ and $I$).
$I$ is total and consistent, therefore the algorithm returns $I$ as the first stable model.
$\hfill\lhd$
\end{example}
For the performance of this search procedure,
several details are crucial:
learning effective constraints from inconsistencies as well as heuristics for restarting, constraint deletion,
and for choosing literals.
\newcommand{\mathcal{P}}{\mathcal{P}}
\begin{algorithm}[t]
\SetKwInOut{Input}{Input}
\SetKwInOut{Output}{Output}
\Input{A ground program $\mathcal{P}$}
\Output{A stable model for $\mathcal{P}$ or $\bot$}
\Begin{
%
$I := \emptyset$\;
$I := $ \textit{Propagate($I$)}\; \label{ln:alg:propagate}
\uIf{$I$ is inconsistent}
{
$r$ := \textit{CreateConstraint($I$) \label{ln:alg:learning}\;
$I := $ \textit{RestoreConsistency}($I$)\; \label{ln:alg:restore}
\lIf{$I$ is consistent\label{ln:alg:addconstraint}}{$\mathcal{P} := \mathcal{P} \ \cup \ \{r\}$}\;}
\lElse{\Return $\bot$\;}
}
\lElseIf{$I$ total}
{
{\Return $I$\;}
}
\Else{
$I := $ \textit{RestartIfNeeded($I$)}; \qquad $\mathcal{P} := $ \textit{DeleteConstraintsIfNeeded($\mathcal{P}$)}\; \label{ln:alg:partial}
$I := I \ \cup $ \textit{ChooseLiteral($I$)}\; \label{ln:alg:choice}
}
\textbf{goto}~\ref{ln:alg:propagate}\;
}
\caption{ComputeStableModel}\label{alg:mg}
\end{algorithm}
\begin{function}[t]
$\mathcal{I} = I$\;
\lFor{$\ell \in \mathcal{I}$} %
{
$\mathcal{I}$ := $\mathcal{I} \ \cup \ Propagation(\mathcal{I}, \ \ell)$ \; %
}
\Return $\mathcal{I}$\;
\caption{Propagate($I$)}\label{fn:propagatestd}
\end{function}
\subsection{Lazy Constraints}\label{sec:lazy}
\begin{algorithm}[t]
\SetKwInOut{Input}{Input}
\SetKwInOut{Output}{Output}
\Input{A nonground program $\Pi$, a set of nonground constraints $C \subseteq \Pi$}
\Output{A stable model for $\Pi$ or $\bot$}
\Begin{
$\mathcal{P}$ := $\GP{\Pi \setminus C}$\;
$I$ := $ComputeStableModel(\mathcal{P})$\; \label{ln:lazy:search}
\lIf{$I$ == $\bot$}{ \Return $\bot$ \label{ln:lazy:incoherent}\;}
$\mathcal{C} = \{c \mid c \in \GP{C}, \ c $ is violated$\}$\label{ln:lazy:violating}\;
\lIf {$\mathcal{C}$ == $\emptyset$}{\Return $I$\;}
$\mathcal{P}$ := $\mathcal{P} \cup \mathcal{C}$ \label{ln:lazy:constraint}\;
\textbf{goto}~\ref{ln:lazy:search}\;
}
\caption{LazyConstraintInstantiation}\label{alg:lazy}
\end{algorithm}
The algorithm presented in this section is reported as Algorithm~\ref{alg:lazy}.
The algorithm takes as input a program $\Pi$ and a set of constraints $C \subseteq \Pi$.
Then, the constraints in $C$ are removed from $\Pi$, obtaining the program $\mathcal{P}$.
A stable model of $\GP{\mathcal{P}}$ is searched (line~\ref{ln:lazy:search}).
Two cases are possible:
$(i)$ $\mathcal{P}$ is incoherent (line~\ref{ln:lazy:incoherent}). Thus, the original program $\Pi$ is also incoherent and the algorithm terminates returning $\bot$.
$(ii)$ $\mathcal{P}$ is coherent. Thus, a stable model, say $I$, is computed. In this case, a set of constraints $\mathcal{C} \in \GP{C}$ that are violated under the stable model $I$ are extracted (line \ref{ln:lazy:violating}) and added to $\mathcal{P}$ (line~\ref{ln:lazy:constraint}).
The process is repeated until either a stable model of $\mathcal{P}$ violating no constraints in $\GP{C}$ is found or $\mathcal{P}$ is incoherent.
Importantly, $\GP{C}$ is never represented explicitly in the implementation of line~5.
\begin{example}\label{ex:lazy}
Again consider program $\Pi_1$
from Example~\ref{ex:grounding}
and the set of constraints $C= \{r_3, \ r_6\}$.
The algorithm computes a stable model, say $I_1=\{a(1), \ \ensuremath{\raise.17ex\hbox{\ensuremath{\scriptstyle\mathtt{\sim}}}}\xspace b(1), \ c(1), \ \ensuremath{\raise.17ex\hbox{\ensuremath{\scriptstyle\mathtt{\sim}}}}\xspace d(1)\}$, of $\mathcal{P}_1 = \GP{\Pi_1\ \setminus \ C}$.
Thus, the ground instantiation $g_6$ of $r_6$ is violated under $I_1$ and therefore $g_6$ is added to $\mathcal{P}$.
Then, a stable model of $\mathcal{P}$ is computed, say $I_2=\{\ensuremath{\raise.17ex\hbox{\ensuremath{\scriptstyle\mathtt{\sim}}}}\xspace a(1), \ b(1), \ c(1), \ \ensuremath{\raise.17ex\hbox{\ensuremath{\scriptstyle\mathtt{\sim}}}}\xspace d(1)\}$.
At this point, $I_2$ violates no constraint in $\GP{C}$. Thus, the algorithm terminates returning $I_2$.
Note that all instantiations of constraint $r_3$ will be never violated since rules $r_1$ and $r_2$ enforce that exactly one of $a(1)$ and $b(1)$ can be true in a stable model. Thus, $r_3$ will never be instantiated by the algorithm.$\hfill \lhd$
\end{example}
An important feature of Algorithm~\ref{alg:lazy} is that it requires no modifications to the search procedure implemented by the underlying ASP solver.
\subsection{Constraints via Propagators}\label{sec:propagators}
In this section, constraints are replaced using the concept of \textit{propagator}, which can set truth values of atoms during the solving process,
based on truth values of other atoms.
An example of a propagator is the unit propagation, detailed in Section~\ref{sec:solving:classical}.
In contrast to the lazy instantiation of constraints that aims at adding violated constraints when a stable model candidate is found, propagators usually are used to evaluate the constraints during the computation of the stable model.
Given a program $\Pi$, traditional solvers usually apply propagators on the whole set of rules and constraints in $\GP{\Pi}$.
An alternative strategy is to consider a variant of the program, say $\mathcal{P} = \Pi \setminus C$, where $C$ is a set of constraints.
The solver is then executed on $\GP{\mathcal{P}}$ and a propagator is used to guarantee the coherence of partial interpretations with the constraints in $\GP{C}$.
Constraints in $C$ are not instantiated in practice but their inferences are simulated by an ad-hoc procedure
implemented for that purpose.
This approach requires a modification of the Propagation function in \ref{fn:propagatestd},
such that Propagation considers the additional set $C$ of constraints,
verifies which constraints would result in a propagation on the partial interpretation,
and propagate truth values due to inferences on $C$ in addition to unit propagation.
\begin{example}
Again consider program $\Pi_1$ from Example~\ref{ex:grounding} and the set of constraints $C= \{r_3, \ r_6\}$.
The idea is to execute Algorithm~\ref{alg:mg}
on $\GP{\mathcal{P}_1}$, where $\mathcal{P}_1 = \Pi_1 \setminus C$.
$\mi{ComputeStableModel}(\mathcal{P}_1)$
starts with $I = \emptyset$ and
does not propagate anything in line~\ref{ln:alg:propagate}.
$I$ is partial and consistent, so the algorithm continues
in line~\ref{ln:alg:partial}.
Assume no restart and no deletion is performed,
and assume $\mi{ChooseLiteral}$ returns $\{a(1)\}$, i.e., $I=\{a(1)\}$.
Next, $\mi{Propagate}(I, \ C)$ is called.
In this case, the propagation yields $I = \{a(1), \ b(1), \ \ensuremath{\raise.17ex\hbox{\ensuremath{\scriptstyle\mathtt{\sim}}}}\xspace b(1)\}$, where $\ensuremath{\raise.17ex\hbox{\ensuremath{\scriptstyle\mathtt{\sim}}}}\xspace b(1)$ comes from unit propagation on $g_1$, while $b(1)$ comes from unit propagation on the ground instantiation $g_6$ of the rule $r_6$.
Thus, $I$ is inconsistent and $I$ is analyzed to compute a reason that explains the conflict, i.e., $\mi{CreateConstraint}(I) = \{g_7\}$ with $g_7: \lar a(1)$.
Then, the algorithm continues as shown in Example~\ref{ex:prog}.
Note that, from this point of the computation, the ground instantiations of constraints $r_3$ and $r_6$ will never be violated again, since $g_7$ assure that $a(1)$ will be false in all partial interpretations under consideration. $\hfill \lhd$
\end{example}
We classify constraint propagators according to the priority given to them. In particular, they are considered \textit{eager} if propagation on non-ground constraints is executed as soon as possible, i.e., during unit propagation of already grounded constraints; moreover, they are called \textit{postponed} (or \textit{post}) if propagation on constraints is executed after all other (unit, support, etc.) propagations.
\section{Implementation and Experimental Analysis}\label{sec:experiments}
\subsection{Implementation}
\label{sec:implementation}
The lazy instantiation of constraints and the propagators have been implemented on top of the ASP solvers \wasp~\cite{DBLP:conf/lpnmr/AlvianoDLR15} and \clingo~\cite{DBLP:conf/iclp/GebserKKOSW16}.
The Python interface of \wasp~\cite{DBLP:conf/aiia/DodaroRS16} follows a synchronous message passing protocol implemented by means of method calls.
Basically, a Python program implements a predetermined set of methods that are later on called by \wasp whenever specific points of the computation are reached.
The methods may return some values that are then interpreted by \wasp.
For instance, when a literal is true the method \textit{onLiteralTrue} of the propagator is called, whose output is a list of literals to infer as true as a consequence (see~\cite{DBLP:conf/aiia/DodaroRS16} for further details).
\clingo~5~\cite{DBLP:conf/iclp/GebserKKOSW16}
provides a Python interface where a propagator class
with an interface similar to \wasp can be registered.
Two important differences exist between \wasp and \clingo.
Firstly \clingo provides only a post-propagator interface
and no possibility for realizing an eager propagator
(that runs before unit propagation is finished).
Secondly, \wasp first collects nogoods added in Python
and then internally applies them and handles conflicts,
while \clingo requires an explicit propagation call
after each added nogood.
If propagation returns a conflict
then no further nogoods can be added in \clingo,
even if further nogoods were detected.
After consulting the \clingo authors,
we implemented a queue for nogoods
and add them in subsequent propagations
if there is a conflict.
This yields higher performance than abandoning these nogoods.
\subsection{Description of Benchmarks}\label{sec:benchdescription}
In order to empirically compare the various strategies for avoiding the instantiation of constraints, we investigated several benchmarks of different nature, namely Stable Marriage, Packing, and Natural Language Understanding.
All benchmarks contain one or few constraints whose grounding can be problematic.
\paragraph{Stable Marriage.} The \emph{Stable Marriage} problem can be described as follows:
given $n$ men and $m$ women, where each person has a preference order over the opposite sex, marry them so that the marriage is stable.
In this case, the marriage is said to be stable if there is no couple $(M, W)$ for which both partners would rather be married with each other than their current partner.
We considered the encoding used for the fourth ASP Competition.
For the lazy instantiation and for the ad-hoc propagators the following constraint has been removed from the encoding:
\begin{equation*}
\begin{array}{l}
\leftarrow match(M,W1), \ manAssignsScore(M,W,Smw), \ W1 \ne W, \\
\quad \quad manAssignsScore(M,W1,Smw1), \ Smw > Smw1, \ match(M1,W),\\
\quad \quad womanAssignsScore(W,M,Swm), \ womanAssignsScore(W,M1,Swm1), \ Swm \ge Swm1.
\end{array}
\end{equation*}
Intuitively, this constraint guarantees that the stability condition is not violated.
\paragraph{Packing.} The \emph{Packing Problem} is related to a class of problems in which one has to pack objects together in a given container. We consider the variant of the problem submitted to the ASP Competition 2011. In that case, the problem was the packing of squares of possibly different sizes in a rectangular space and without the possibility of performing rotations.
The encoding follows the typical guess-and-check structure, where positions of squares are guessed and some constraints check whether the guessed solution is a stable model.
We identified 2 expensive sets of constraints.
The first set comprises the following two constraints:
\begin{align*}
\leftarrow pos(I,X,Y), pos(I,X_1,Y_1), X_1 \neq X & \qquad
\leftarrow pos(I,X,Y), pos(I,X_1,Y_1), Y_1 \neq Y
\end{align*}
which enforce that a square is not assigned to different positions.
The second set comprises constraints forbidding the overlap of squares.
One of these constraints is reported in the following:
\begin{align*}
\leftarrow pos(I_1,X_1,Y_1), \ square(I_1,D_1), \ pos(I_2,X_2,Y_2), \ square(I_2,D_2), \phantom{we are free to stay herei}\\
I1 \neq I2, \ W1 = X1+D1, \
H1 = Y1+D1, \ X2 \geq X1,\ X2 < W1, \ Y2 \geq Y1, \ Y2 < H1.
\end{align*}
Other constraints are similar thus they are not reported.
\paragraph{Natural Language Understanding (NLU).} The \emph{NLU} benchmark is an application of ASP
in the area of Natural Language Understanding,
in particular the computation of optimal solutions
for First Order Horn Abduction problems
under the following cost functions:
cardinality minimality,
cohesion%
, and weighted abduction%
.
This problem and these objective functions have been described by
Sch\"uller~\shortcite{DBLP:journals/fuin/Schuller16}.
In this problem,
we aim to find a set of explanatory atoms
that makes a set of goal atoms true with respect to a
First Order Horn background theory.
We here consider the acyclic version of the problem
where backward reasoning over axioms
is guaranteed to introduce a finite set of
new terms.
A specific challenge in this problem is
that input terms and terms invented via backward chaining
can be equivalent to other terms,
i.e., the unique names assumption is partially not true.
Equivalence of terms must be handled explicitly in ASP,
which is done by guessing an equivalence relation.
This makes the instantiation of most instances
infeasible, as the number of invented terms becomes large,
due to the grounding blow-up caused by the following constraint:
\begin{align*}
&\leftarrow \mi{eq(A,B)},\, \mi{eq(B,C)},\, \ensuremath{\raise.17ex\hbox{\ensuremath{\scriptstyle\mathtt{\sim}}}}\xspace \mi{eq(A,C)}.
\end{align*}
\subsection{Hardware and Software Settings}
The experiments were run on a Intel Xeon CPU X3430 2.4 GHz.
Time and memory were limited to 600 seconds and 4 GB, respectively.
In the following, \wasp-\textsc{lazy} refers to \wasp implementing lazy instantiation of constraints, while \wasp-\textsc{eager} and \wasp-\textsc{post} refer to \wasp implementing eager and postponed propagators, respectively.
All versions of \wasp use \gringo version 5.1.0 as grounder, whose grounding time is included in the execution time of \wasp.
Moreover, \clingo~\textsc{lazy} and \clingo~\textsc{post} refer to \clingo implementing lazy and postponed propagators, respectively. For the NLU benchmark, we always use unsat-core optimization.
\subsection{Discussion of Results}
\begin{table}[b!]
\caption{Stable Marriage: Number of solved instances and average running time (in seconds).} \label{tab:stable}
\centering
\footnotesize
\setlength{\tabcolsep}{0.2em}
\begin{tabular}{rrrrrrrrrrrrrrrr}
\toprule
\textbf{Pref. (\%)} & \multicolumn{2}{c}{\textbf{\wasp}} & \multicolumn{2}{c}{\textbf{\wasp-\textsc{lazy}}} & \multicolumn{2}{c}{\textbf{\wasp-\textsc{eager}}} & \multicolumn{2}{c}{\textbf{\wasp-\textsc{post}}} & \multicolumn{2}{c}{\textbf{\clingo}} & \multicolumn{2}{c}{\textbf{\clingo-\textsc{lazy}}} & \multicolumn{2}{c}{\textbf{\clingo-\textsc{post}}} &\\
& \textbf{sol.} & \textbf{avg t}
& \textbf{sol.} & \textbf{avg t} & \textbf{sol.} & \textbf{avg t} & \textbf{sol.} & \textbf{avg t} & \textbf{sol.} & \textbf{avg t} & \textbf{sol.} & \textbf{avg t} & \textbf{sol.} & \textbf{avg t}\\
0 & 10 & 4.1 & 10 & 4.7 & 10 & 4.6 & 10 & 4.7 & 10 & 10.6 & 10 & 4.2 & 10 & 4.2\\
5 & 9 & 16.2 & 10 & 4.7 & 10 & 4.3 & 10 & 4.9 & 10 & 23.0 & 10 & 4.6 & 10 & 4.4\\
10 & 10 & 19.2 & 10 & 4.7 & 10 & 4.3 & 10 & 4.6 & 10 & 34.6 & 10 & 6.4 & 10 & 8.2\\
15 & 9 & 24.3 & 10 & 4.7 & 10 & 4.4 & 10 & 4.8 & 10 & 42.9 & 10 & 9.6 & 10 & 17.5\\
20 & 8 & 35.2 & 10 & 4.8 & 10 & 4.6 & 10 & 5.2& 10 & 48.9 & 10 & 16.5 & 10 & 24.7\\
25 & 10 & 34.8 & 10 & 4.8 & 10 & 5.4 & 10 & 6.0& 10 & 53.9 & 10 & 22.2 & 10 & 42.8\\
30 & 6 & 97.0 & 10 & 5.0 & 10 & 7.7 & 10 & 7.6& 10 & 59.5 & 10 & 32.2 & 10 & 92.1\\
35 & 10 & 42.1 & 10 & 5.0 & 10 & 8.2 & 10 & 10.0& 10 & 65.8 & 10 & 62.4 & 10 & 115.9\\
40 & 9 & 51.3 & 10 & 5.2 & 10 & 7.6 & 10 & 9.2& 10 & 68.4 & 10 & 81.8 & 10 & 117.5\\
45 & 10 & 113.4 & 10 & 5.4 & 10 & 10.8 & 10 & 12.0& 10 & 71.0 & 10 & 97.7 & 10 & 140.8\\
50 & 6 & 74.6 & 10 & 5.1 & 10 & 22.4 & 10 & 20.3& 10 & 72.0 & 10 & 153.6 & 10 & 143.4\\
55 & 9 & 44.5 & 8 & 5.9 & 10 & 39.4 & 10 & 23.6& 10 & 72.9 & 10 & 193.8 & 10 & 166.5\\
60 & 9 & 70.9 & 10 & 7.7 & 10 & 23.8 & 10 & 25.0& 10 & 74.6 & 10 & 241.1 & 10 & 181.6\\
65 & 7 & 99.3 & 10 & 11.4 & 10 & 64.7 & 10 & 54.2& 10 & 74.7 & 10 & 295.6 & 10 & 209.8\\
70 & 9 & 89.3 & 5 & 25.5 & 10 & 121.8 & 10 & 101.8& 10 & 75.0 & 10 & 361.1 & 10 & 235.3\\
75 & 8 & 77.0 & 0 & - & 10 & 184.0 & 10 & 146.7& 10 & 75.1 & 6 & 472.1 & 10 & 311.0\\
80 & 7 & 85.5 & 0 & - & 10 & 248.6 & 8 & 274.7& 10 & 76.3 & 0 & - & 10 & 434.3\\
85 & 4 & 259.5 & 0 & - & 10 & 232.3 & 1 & 337.2& 10 & 82.3 & 0 & - & 7 & 569.7\\
90 & 9 & 79.2 & 0 & - & 5 & 449.4 & 0 & -& 10 & 251.1 & 0 & - & 1 & 577.7\\
95 & 10 & 46.3 & 0 & - & 0 & - & 0 & -& 6 & 273.6 & 0 & - & 3 & 580.8\\
100 & 8 & 67.6 & 1 & 81.2 & 10 & 133.3 & 10 & 153.6& 10 & 74.1 & 6 & 493.3 & 10 & 323.9\\
\bottomrule
\end{tabular}
\end{table}
\paragraph{Stable Marriage.}
Concerning Stable Marriage, we executed the 30 instances selected for the Fourth ASP Competition.
\clingo and \wasp executed on the full encoding are able to solve 29 out of the 30 instances with an average running time of 50 and 29 seconds, respectively.
On the same instances, ad-hoc propagators cannot reach the same performance.
Indeed, \wasp-\lazy and \wasp-\post perform the worst solving 0 and 5 instances, respectively, whereas \wasp-\eager is much better with 17 solved instances.
The same performance is obtained by \clingo-\lazy and \clingo-\post which can solve 0 and 17 instances in the allotted time, respectively.
The poor performance of the lazy instantiation can be explained by looking at the specific nature of the instances.
Indeed, each instance contains a randomly generated set of preferences of men for women (resp. women for men).
By looking at the instances we observed that each man (resp. woman)
has a clear, often total, preference order over each woman (resp. man).
This specific case represents a limitation for employing the lazy instantiation.
Indeed, \wasp and \clingo executed on the encoding without the stability constraint perform naive choices until a stable model candidate is found.
Then, each candidate contains several violations of the stability condition and many constraints are added.
However, those constraints are not helpful since they only invalidate the current stable model candidate.
In general, for instances where the program without the stability condition is under-constrained many stable model candidates need to be invalidated before an actual solution is found (intuitively, given a program $\Pi$ and a set of constraints $C \subseteq \Pi$,
$|SM(\Pi \setminus C)| \gg |SM(\Pi)|$).
In order to further analyze this behavior empirically, we have conducted an additional experiment on the same problem.
In particular, we randomly generated instances where each man (resp. woman) gives the same preference to each woman (resp. man), so basically the stability condition is never violated.
Then, we consider a percentage $k$ of preferences, i.e., each man (resp. woman) gives the same preference to all the women (resp. men) but to $k$\% of them a lower preference is given.
In this way, instances with small values of $k$ should be easily solved by lazy instantiation, whereas instances with high values of $k$ should be hard.
For each considered percentage $k$, we executed 10 randomly generated instances.
Results are reported in Table~\ref{tab:stable}, where the number of solved instances and the average running time are shown for each tested approach.
Concerning \wasp, as observed before, for instances where the value of $k$ is small (up to 50\%) the lazy approach can solve all the instances with an average running time of about 5 seconds.
On the other hand, for high values of $k$ the advantages of the lazy approach disappear, as observed for the competition instances.
Interestingly, the eager propagator obtained the best performance overall.
For the tested instances, it seems to benefit of a smaller program and generation of the inferences does not slow down the performance as observed for competition instances.
Concerning \clingo, the lazy approach is the best performing one for instances where the value of $k$ is up to 35\%.
As shown for \wasp, the performance of the lazy approach are worse for high values of $k$.
\paragraph{Packing.}
\begin{figure}[t]
\figrule
\begin{tikzpicture}[scale=0.9]
\pgfkeys{%
/pgf/number format/set thousands separator = {}}
\begin{axis}[
scale only axis
, font=\scriptsize
, x label style = {at={(axis description cs:0.5,0.04)}}
, y label style = {at={(axis description cs:0.05,0.5)}}
, xlabel={Number of instances}
, ylabel={Execution time (s)}
, xmin=0, xmax=50
, ymin=0, ymax=610
, legend style={at={(0.88,0.96)},anchor=north, draw=none,fill=none}
, legend columns=1
, width=0.7\textwidth
%
%
%
, height=0.3\textwidth
, ytick={0,200,400,600}
, xtick={0,10,20,30,40,50}
, major tick length=2pt
]
\addplot [mark size=2.5pt, color=blue, mark=o] [unbounded coords=jump] table[col sep=semicolon, y index=3] {./packing.csv};
\addlegendentry{\wasp-\textsc{lazy}}
\addplot [mark size=2.5pt, color=blue, mark=square] [unbounded coords=jump] table[col sep=semicolon, y index=1] {./packing.csv};
\addlegendentry{\wasp-\textsc{eager}}
\addplot [mark size=2pt, color=blue, mark=diamond] [unbounded coords=jump] table[col sep=semicolon, y index=2] {./packing.csv};
\addlegendentry{\wasp-\textsc{post}}
\addplot [mark size=2pt, color=black, mark=*] [unbounded coords=jump] table[col sep=semicolon, y index=5] {./packing.csv};
\addlegendentry{\clingo-\textsc{lazy}}
\addplot [mark size=2pt, color=black, mark=diamond*] [unbounded coords=jump] table[col sep=semicolon, y index=4] {./packing.csv};
\addlegendentry{\clingo-\textsc{post}}
\end{axis}
\end{tikzpicture}
%
\caption{Packing: Comparison of \textsc{lazy} and \textsc{propagators} approaches on 50 instances.}\label{fig:packing}
\figrule
\end{figure}
Concerning Packing problem, we considered all 50 instances submitted to the Third ASP Competition.
Interestingly, when all constraints are considered none of the instances can be instantiated within the time limit.
Thus, \clingo and \wasp do not even start the computation of a stable model.
The grounding time substantially decreases when the two sets of expensive constraints described in Section~\ref{sec:benchdescription} are removed from the encoding.
Indeed, in this case, the grounding time on the tested instances is 5 seconds on average, with a peak of 16 seconds.
Results of the lazy constraint instantiation and of constraint propagators on the resulting program are reported in the cactus plot of Figure~\ref{fig:packing}.
The graph highlights that \wasp-\textsc{eager}, \wasp-\textsc{post}, and \clingo-\textsc{post} basically obtained the same performance.
Indeed, the first two solve all the tested instances with an average running time of~22 and~23 seconds, respectively, while \clingo-\textsc{post} solves 49 out of 50 instances with an average running time of 25 seconds.
Both \wasp-\textsc{post} and \clingo-\textsc{post} outperform their lazy counterparts.
Indeed, \wasp-\textsc{lazy} solves~10 instances, with an average running time of~226 seconds, while \clingo-\textsc{lazy} solves~5 instances, with an average running time of~301 seconds.
As already observed on the Stable Marriage instances, lazy instantiation cannot compete with constraint propagators.
In this experiment, we observed that \wasp and \clingo perform naive choices on the encoding without the expensive constraints, thus each candidate stable model contains several violations of constraints, leading to inefficient search in harder instances.
\paragraph{Natural Language Understanding (NLU).}
\begin{table}[b!]
\caption{NLU Benchmark: Number of solved instances and average running time (in seconds).} \label{tab:nlu}
\footnotesize
\centering
\setlength{\tabcolsep}{0.2em}
\begin{tabular}{lrrrrrrrrrrrrrr}
\toprule
\textbf{Obj. Func.} & \multicolumn{2}{c}{\textbf{\wasp}} & \multicolumn{2}{c}{\textbf{\wasp-\textsc{lazy}}} & \multicolumn{2}{c}{\textbf{\wasp-\textsc{eager}}} & \multicolumn{2}{c}{\textbf{\wasp-\textsc{post}}} & \multicolumn{2}{c}{\textbf{\clingo}} & \multicolumn{2}{c}{\textbf{\clingo-\textsc{lazy}}} & \multicolumn{2}{c}{\textbf{\clingo-\textsc{post}}}\\
& \textbf{sol.} & \textbf{avg t} & \textbf{sol.} & \textbf{avg t} & \textbf{sol.} & \textbf{avg t} & \textbf{sol.} & \textbf{avg t} & \textbf{sol.} & \textbf{avg t} & \textbf{sol.} & \textbf{avg t} & \textbf{sol.} & \textbf{avg t} \\
Card. & 43 & 39.7 & 50 & 2.3 & 50 & 4.3 & 50 & 3.3 & 41& 30.7 & 50 & 4.5 & 50 & 1.5\\
Coh. & 43 & 40.1 & 50 & 18.5 & 50 & 8.8 & 50 & 6.3 & 41 & 30.7 & 49 & 24.6 & 49 & 15.8\\
W. Abd. & 43 & 49.3 & 50 & 26.6 & 49 & 66.1 & 50 & 62.6 & 41 & 33.9 & 48 & 31.9 & 50 & 24.0\\
\bottomrule
\end{tabular}
\end{table}
Concerning NLU, we considered all 50 instances and all three objective functions used in \cite{DBLP:journals/fuin/Schuller16}.
Results are reported in Table~\ref{tab:nlu}.
As a general observation, all the tested instances are solved by \wasp-\textsc{lazy} and \wasp-\textsc{post}, no matter the objective function.
Moreover, \wasp-\textsc{lazy} is on average faster than all other alternatives for both the objective functions cardinality and weighted abduction.
The good performance of lazy instantiation is related to the small number of failing stable model checks performed.
Indeed, only 2, 16, and 64 invalidations are on average required for cardinality, coherence, and weighted abduction, respectively.
The number of propagation calls is much higher for
\wasp-\textsc{eager} than for \wasp-\textsc{post}
(approximately \wasp-\textsc{eager} performs 3 times more propagation calls than \wasp-\textsc{post}).
However, the number of propagated literals that are not
immediately rolled back because of a conflict
is very similar,
hence it is clear that \wasp-\textsc{eager} performs
a lot of unnecessary propagations in this benchmark
and \wasp-\textsc{post} should be preferred.
Note that this is not generally the case for other benchmarks.
Concerning \clingo,
45, 248, and 321, stable model candidates are invalidated with \clingo-\lazy, respectively,
and a similar amount (26, 589, and 700, respectively) with \clingo-\post.
This shows that \clingo tends to produces more stable models
that violate lazy constraints.
These violations are detected earlier with \clingo-\post,
therefore it outperforms \clingo-\lazy in all objectives.
None of the \clingo propagators is able to solve all instances with all objectives,
whereas \wasp-\post solves all of them within 600~s.
In particular for objective functions cardinality and coherence,
\wasp is always slightly faster and uses slightly more memory than \clingo.
For weighted abduction, \clingo-\post is most efficient with \wasp-\lazy in second place.
Nevertheless, using \clingo or \wasp with a \lazy or \post propagator will always be an advantage
over using the pure ASP encoding where the constraints are instantiated prior to solving.
Hence the choice of the method for instantiating constraints
is more important than the choice of the solver.
\paragraph{Discussion.}\label{par:discussion}
We empirically investigated whether lazy instantiation or propagators can be a valid option for enhancing the traditional ``ground+solve'' approach.
When the full grounding is infeasible, then both lazy instantiation and propagators can overcome this limitation, even though they exhibit different behaviors depending on the features of the problem and of the instances.
This is particularly evident in Packing, where no instance can be grounded within the time limit.
Since propagators are activated during the search, while lazy instantiation intervenes only when a total interpretation is computed, propagators are preferable when the problematic constraint is important to lead the search toward a solution (as overlap constraints in Packing).
On the other hand, a high number of unnecessary propagations can make propagators inefficient and even slower than the lazy approach. In these cases, we observed that post propagators are better than eager propagators as remarked by the results on the objective function `weighted abduction' in the NLU benchmark.
The experiment on Stable Marriage highlights that lazy instantiation is effective when few constraints are instantiated during the search.
This is the case when: (i) it is very likely that a stable model of the simplified (i.e., without problematic constraints) input program also satisfies the lazy constraints; or (ii) the solver heuristics is such that one of the first candidate total interpretations also satisfies the lazy constraints.
This is also confirmed in the NLU benchmark where the instances often have the above characteristics, and the propagator is better only when the constraints generated by the lazy approach do not fit the working memory.
Moreover, from case (ii), we conjecture that the lazy approach can be effective in combination with domain-specific heuristics~\cite{DBLP:conf/aaai/GebserKROSW13,DBLP:journals/tplp/DodaroGLMRS16}.
Finally, we conducted an additional experiment, where we do not oppose our approaches with the ground+solve one as in the previous cases, but it only aims at comparing the lazy propagation versus propagators in a controlled setting.
In particular, we considered a synthetic benchmark based on the well-known 3-SAT problem that is interesting for our study since it allows us to control both the hardness of the instances and the probability that an interpretation satisfies the constraint.
Indeed, we generated the instances uniformly at random in a range centered on the phase transition~\cite{DBLP:series/faia/Achlioptas09}. %
We used a straightforward ASP encoding where we guess an interpretation and we check by a single (non-ground) constraint whether this satisfies all clauses.
The results are summarized in Figure \ref{fig:threesat} where we present two representative runs on formulas with 220 and 280 Boolean variables, respectively.
Since eager and post propagators behave very similarly we only show comparisons between eager propagator and a lazy instantiation.
Expectedly, execution times follow the easy-hard-easy pattern~\cite{DBLP:series/faia/Achlioptas09}, centered on the phase transition, while varying the ratio $R$ of clauses over variables. Initially, the problem is very easy and both approaches are equally fast. Then there is an interval in which the lazy approach is preferable, and finally the eager approach becomes definitely better than the lazy.
Note that, on formulas with 220 variables (see Figure~\ref{fig:3-sat-220}) the lazy approach is preferable also on the hardest instances, instead with 280 variables (see Figure~\ref{fig:3-sat-280}) the eager approach becomes more convenient before the phase transition.
To explain this phenomenon we observe that the lazy approach can be exemplified by assuming that the solver freely guesses a model and then the lazy instantiator checks it, until every clause is satisfied by an assignment or no model can be found.
The probability that a random model satisfies all clauses is $(\frac{7}{8})^k$ where $k$ is the number of clauses, thus fewer tries are needed on average to converge to a solution if the formula has fewer clauses.
This intuitively explains why, as the number of variables increases, the eager approach becomes more convenient at smaller and smaller values of $R$.
It is worth pointing out that this simplified model does not fully capture the behavior of lazy instantiation that is more efficient in practice, since the implementation learns from previous failures (by instantiating violated constraints).
\begin{figure}
\figrule
\begin{subfigure}{.5\textwidth}
%
\begin{tikzpicture}
\pgfkeys{%
/pgf/number format/set thousands separator = {}}
\begin{axis}[
scale only axis
, font=\scriptsize
, x label style = {at={(axis description cs:0.5,0.04)}}
, y label style = {at={(axis description cs:0.05,0.5)}}
, xlabel={Number of clauses / number of variables}
, ylabel={Execution time (s)}
, xmin=2.5, xmax=5.5
%
, ymin=0.04, ymax=40
, log basis y=10, ytickten={-1,0,1}
, ymode=log
%
, legend style={at={(0.25,0.85)},anchor=north, draw=none,fill=none, font=\tiny}
, legend columns=1
, width=0.65\textwidth
, height=0.3\textwidth
, xtick={3,3.5,4,4.5,5}
, major tick length=2pt
]
\addplot [mark size=1.5pt, color=blue, mark=x] [unbounded coords=jump] table[col sep=semicolon, y index=2] {./3-sat-220.csv};
\addlegendentry{\wasp-\textsc{lazy}}
\addplot [mark size=1.5pt, color=red, mark=o] [unbounded coords=jump] table[col sep=semicolon, y index=1] {./3-sat-220.csv};
\addlegendentry{\wasp-\textsc{eager}}
%
\draw[dotted] (axis cs: 4.26, 0.04) -- (axis cs: 4.26, 40);
\end{axis}
\begin{axis}[
scale only axis
, axis y line*=right
, font=\scriptsize
, x label style = {at={(axis description cs:0.5,0.04)}}
, y label style = {at={(axis description cs:0.05,0.5)}}
, xlabel={Number of clauses / number of variables}
, ylabel={Frequency of UNSAT}
, ylabel near ticks, yticklabel pos=right
, xmin=2.5, xmax=5.5
, ymin=-0.05, ymax=1.05
%
, legend style={at={(0.25,0.98)},anchor=north, draw=none,fill=none, font=\tiny}
, legend columns=1
, width=0.65\textwidth
%
%
, ytick={0,0.5,1}
, height=0.3\textwidth
, xtick={3,3.5,4,4.5,5}
, major tick length=2pt
]
\addplot [dashed, mark size=1pt, color=black] [unbounded coords=jump] table[col sep=semicolon, y index=1] {./phaseTransition.csv};
\addlegendentry{\textsc{unsat freq.}}
\end{axis}
\end{tikzpicture}
%
%
%
\caption{Results with 220 variables}\label{fig:3-sat-220}
\end{subfigure}%
\begin{subfigure}{.5\textwidth}
%
\begin{tikzpicture}
\pgfkeys{%
/pgf/number format/set thousands separator = {}}
\begin{axis}[
scale only axis
, axis y line*=left
, font=\scriptsize
, x label style = {at={(axis description cs:0.5,0.04)}}
, y label style = {at={(axis description cs:0.05,0.5)}}
, xlabel={Number of clauses / number of variables}
, ylabel={Execution time (s)}
, xmin=2.5, xmax=5.5
%
, ymin=0.04, ymax=4000
, ymode=log
%
, legend style={at={(0.25,0.85)},anchor=north, draw=none,fill=none, font=\tiny}
, legend columns=1
, width=0.65\textwidth
, height=0.3\textwidth
, xtick={3,3.5,4,4.5,5}
, major tick length=2pt
]
\addplot [mark size=1.5pt, color=blue, mark=x] [unbounded coords=jump] table[col sep=semicolon, y index=2] {./3-sat-280.csv};
\addlegendentry{\wasp-\textsc{lazy}}
\addplot [mark size=1.5pt, color=red, mark=o] [unbounded coords=jump] table[col sep=semicolon, y index=1] {./3-sat-280.csv};
\addlegendentry{\wasp-\textsc{eager}}
%
\draw[dotted] (axis cs: 4.28, 0.04) -- (axis cs: 4.28, 4000);
\end{axis}
\begin{axis}[
scale only axis
, axis y line*=right
, font=\scriptsize
, x label style = {at={(axis description cs:0.5,0.04)}}
, y label style = {at={(axis description cs:0.05,0.5)}}
, xlabel={Number of clauses / number of variables}
, ylabel={Frequency of UNSAT}
, ylabel near ticks, yticklabel pos=right
, xmin=2.5, xmax=5.5
, ymin=-0.05, ymax=1.05
%
, legend style={at={(0.25,0.98)},anchor=north, draw=none,fill=none, font=\tiny}
, legend columns=1
, width=0.65\textwidth
%
%
, ytick={0,0.5,1}
, height=0.3\textwidth
, xtick={3,3.5,4,4.5,5}
, major tick length=2pt
]
\addplot [dashed, mark size=1pt, color=black] [unbounded coords=jump] table[col sep=semicolon, y index=2] {./phaseTransition.csv};
\addlegendentry{\textsc{unsat freq.}}
\end{axis}
\end{tikzpicture}
%
\caption{Results with 280 variables}\label{fig:3-sat-280}
%
\end{subfigure}
\caption{3-SAT experiments. Red and blue lines correspond to eager propagators and lazy instantiation respectively. The dashed black line represents the percentage of UNSAT instances, while the vertical dotted line evidences the phase transition point (frequency is about $0.5$ at $R=4.26$).}%
\label{fig:threesat}
\figrule
\end{figure}
\portfoliotable{b!}
\subsection{On the applicability of techniques for automatic algorithm selection}\label{sec:portfolio}
The analysis conducted up to now shows that there is not always a clear winner among the strategies for realizing constraints,
since the best solving method depends on characteristics of the encoding and the instance at hand.
In similar scenarios, portfolio approaches
which automatically choose one out of a set of possible methods have proven to be very effective in increasing system performance, since they combine the strengths of the available methods.
Therefore, we investigated whether algorithm selection techniques can improve performance in our context.%
We apply basic algorithm selection based on classification
with machine learning:
we extract some natural features from each instance,
and train a C4.5 \cite{Quinlan1993c45} classifier
to predict the best solving method
(i.e., the one that required least amount of time) among all the available ones (including the plain solver).
We limit our analysis to Stable Marriage and NLU,
because in these domains none of the available methods is clearly superior.
As features for stable marriage we used the number of persons and the percentage or preferences, %
for NLU we used the number of facts and the number of distinct constants and (instance-specific) predicates.
We create portfolios for both \wasp-based and \clingo-based implementations.
Table \ref{tab:portfolio} shows the results
of our evaluation using 10-fold cross-validation
(i.e., we split the set of instances into 10 partitions
and use each partition as test set while training on the remaining partitions).
For each problem we report (weighted average) precision, recall, and f-measure of the prediction, as well as the average performance gain of the portfolio (i.e., by gain we mean the difference in percentage between the sum of the execution times measured for the portfolio and for its best method).
We observe that the classifier is able to choose the best algorithm in many cases, and the choice is almost ideal in NLU (f-measure of 0.9 for \wasp and 0.84 for \clasp). The portfolios are always faster (in terms of execution times) than the corresponding best method for the respective problem.
The performance gain peaks to 38\% for the \wasp-based, and is less pronounced for the \clingo-based (peak at 13.6\%).
This is expected since \clingo features a basic solver that is more competitive with propagator-based solutions in these domains.
Summarizing, these results confirm that already the application of basic portfolio techniques is a viable option for improving the performance when propagators are available.
\section{Related Work}
The grounding bottleneck in ASP has been subject of various studies.
The most prominent grounding-less approach that works on plain ASP is \textit{lazy grounding}, which was implemented by \textsc{asperix}~\cite{DBLP:conf/lpnmr/LefevreN09a}, \textsc{gasp}~\cite{DBLP:journals/fuin/PaluDPR09}, and \textsc{omiga}~\cite{DBLP:conf/jelia/Dao-TranEFWW12}.
Differently from our approach that is focused on constraints,
these solvers perform lazy instantiation for \emph{all} the rules of a program,
and do not perform (conflict) clause learning.
Weinzierl~\shortcite{Weinzierl2017} recently investigated learning of non-ground clauses.
Lazy instantiation of constraints was topic of
several works on \emph{integrating ASP with other formalisms}.
These include CASP~\cite{DBLP:conf/asp/BaseliceBG05,DBLP:journals/tplp/OstrowskiS12,DBLP:journals/corr/BalducciniL17},
ASPMT~\cite{DBLP:conf/iclp/SusmanL16}, BFASP~\cite{DBLP:journals/tplp/AzizCS13}, and
HEX \cite{DBLP:journals/tplp/EiterFIKRS16}.
Differently from our approach, these approaches are based on syntactic extensions
that enable the combination of ASP solvers with solvers for external theories.
HEX facilitates the integration
of generic computation oracles as literals in ASP rule bodies,
and allows these computations not only to return true or false,
but also to inject constraints into the search.
This gave rise to the `on-demand constraint' usage pattern
of external atoms \cite{DBLP:conf/birthday/EiterRS16}
which roughly corresponds with the \lazy propagators in this work.
HEX also permits a declarative specification of
properties of external computations \cite{DBLP:journals/tplp/Redl16},
e.g., antimonotonicity with respect to some part of the model.
Such specifications automatically generate additional
lazy constraints. %
Integration of ASP with continuous motion planning in robotics,
based on HEX, was investigated in \cite{DBLP:journals/aicom/ErdemPS16}:
adding motion constraints in a \post propagator
was found to be
significantly faster than checking only complete stable model candidates (\lazy).
For integrating \emph{CModels with BProlog}
\cite{DBLP:journals/tplp/BalducciniL13}
it was shown that using BProlog similar as a \post propagator
(clearbox)
performs better than using it as a \lazy propagator
(black-box).
De~Cat et al.~\shortcite{DBLP:journals/jair/CatDBS15}
provide a theory and implementation
for \emph{lazy model expansion}
within the FO(ID) formalism
which is based on \emph{justifications}
that prevent instantiation of certain constraints
under assumptions.
These assumptions are relative to a model candidate
and can be revised from encountered conflicts,
leading to a partially lazy instantiation
of these constraints.
We finally observe that lazy constraints can be seen as a simplified form of lazy clause generation that was originally introduced in Constraint Programming~\cite{DBLP:conf/cp/FeydyS09}.
\section{Conclusion}\label{sec:conclusion}
In this paper, we compared several solutions for addressing the problem of the grounding bottleneck focusing on the practically-relevant case of problematic constraints without resorting to any language extension.
The considered approach can be seen as a natural extension of the ``ground+solve'' paradigm, adopted by state of the art ASP systems, where some constraints are replaced either by lazy instantiators or propagators. The solutions fit CDCL-based solving strategies, and can be implemented using APIs provided by state of the art solvers.
Experiments conducted on both real-world and synthetic benchmarks clearly outline that all the approaches can solve instances that are out of reach of state of the art solvers because of the grounding blowup.
Lazy instantiation is the easiest to implement, and it is the best choice when the problematic constraints have a high probability to be satisfied. Otherwise, eager and post propagators perform better, with the latter being slightly more efficient when the constraint is activated more often during propagation.
Our empirical analysis shows that there is not always a clear winner for a given problem, thus we investigated the applicability of algorithm selection techniques.
We observed that a basic portfolio can improve on the best strategy also on these cases.
As far as future work is concerned, we will study what are the conditions under which an entire subprogram (and not just some constraints) can be replaced by a propagator. Another line of research might be to investigate the impact of applying rule decomposition techniques before handling the constraints~\cite{DBLP:journals/tplp/BichlerMW16}.
\section*{Acknowledgments}
The paper has been partially supported by the Italian Ministry for Economic Development (MISE) under project ``PIUCultura -- Paradigmi Innovativi per l'Utilizzo della Cultura'' (n. F/020016/01-02/X27),
under project ``Smarter Solutions in the Big Data World (S2BDW)'' (n. F/050389/01-03/X32) funded
within the call ``HORIZON2020'' PON I\&C 2014-2020,
and by the Scientific and Technological Research Council of Turkey
(TUBITAK) Grant 114E777.
\ifinlinerefs
\input{references.sty}
\else
\bibliographystyle{acmtrans}
|
2,877,628,089,011 | arxiv | \section{Introduction}
In quantum materials, long-range order and macroscopic functionality emerge from microscopic interactions between structural, electronic, orbital, and magnetic degrees of freedom. Quite often, these degrees of freedom are deeply intertwined, leading to competing, coexisting, or even cooperative orders \cite{Tokura:2000ck,Dagotto:2005ip,Keimer:2017gq}. It is imperative to experimentally disentangle these subtle interactions to unlock the full potential of the myriad of phenomena embedded in quantum matter. Directed property control using static-tuning knobs such as strain, electrostatic gating, or magnetic fields provides a promising route to explore complex energy landscapes with the possibility of selecting a particular ordered quantum phase \cite{Tokura:2017bh}. In the context of these materials, a fundamental transition to study is the insulator-to-metal transition (IMT), that is ubiquitous yet often not well understood\cite{Imada:1998er}.
The main challenge in understanding the IMT in many systems is the cooperative coupling between the lattice structure, electronic order, and the magnetic configuration that occurs in a wide variety of transition metal oxides (e.g. manganites\cite{Tokura:2006ff}, cobaltites\cite{Sundaram:2009bx}, ferrites\cite{Blasco:2018ep}, vanadates\cite{Yan:2019ft}, nickelates\cite{Catalano:2018kc}). This arises due to the highly connected nature of the lattice\cite{Woodward:1997th,Howard:1998uf,Howard:2004co,Carpenter:2009cf} that results in an interplay between changes in electronic and magnetic order with many structural degrees of freedom such as distortions, octahedral rotations and cation displacements. Recently, some data has been assembled and analyzed to explore correlations between structural, electronic, and magnetic order\cite{Balachandran:2013cg,Wagner:2016dx,Wagner:2018kz}, which is insightful but not always conclusive. This leads to a challenging problem that is difficult to disentangle especially in the case where multiple order parameters change under the same physical conditions.
By moving beyond equilibrium probes, an alternate approach is to utilize ultrafast stimulation with optical pulses to explore the energy landscape and follow how phases evolve with time during conversion between quantum states. This route can make it possible to disentangle how microscopic competing degrees of freedom lead to the emergence of long range order \cite{Averitt:2002dq,Basov:2011ht,Zhang:2014dq}, with the ultimate goal of light directed property control \cite{Basov:2017ix}. Indeed, ultrafast techniques now span the electromagnetic spectrum enabling multi-modal studies of complexity in solids. Ultrafast X-ray techniques have risen to prominence making it possible to directly and simultaneously probe different degrees of freedom at their fundamental timescales \cite{Lindenberg:2000ib,Cavalleri:2005fe,Ichikawa:2011js,Lee:2012ga,Caviglia:2013to,deJong:2013he,Park:2013jt,Beaud:2014bp,Forst:2015fv,Langner:2015ef,Lourembam:2015kb,Zhu:2016cp,ThielemannKuhn:2017cj}. To that end, ultrafast x-ray measurements can monitor the evolution of structural, electronic, and magnetic changes during the course of a photoinduced IMT. Here we will show how to achieve temporal discrimination together with the selectivity of these new X-ray probes to track specific time-scales for each degree of freedom, to gain a deeper understanding of systems with strongly coupled order parameters.
\begin{figure}[h]
\centering
\includegraphics[width=.35\textwidth]{AFMwCOFig1-eps-converted-to
\caption{\label{Fig1} {\bf Coupled order parameters} (A) Schematic of interconnected orders and how they are connected in NdNiO$_3$ . (B) NiO$_6$ octahedra sites connected with the (green) long bond (LB) and (purple) short bond (SB) charge order associated with the Ni-O breathing mode distortion. (C) E$^\prime$-type antiferromagnetic (AFM) unit cell with alternating planes of ferromagnetically aligned $S=1$ LB site (green) and $S=0$ SB sites (purple) shown in red in an $\cdots\uparrow\cdot\ 0\ \cdot\downarrow\cdot\ 0\ \cdots$ pattern.
}
\end{figure}
In this article, we illustrate a paradigm using ultrafast soft X-ray scattering and absorption capabilities at the Linac Coherent Light Source (LCLS) to probe the electronic and magnetic degrees of freedom directly in order to disentangle multiple interactions in a correlated oxide. NdNiO$_3$ has combined antiferromagnetic (AFM) and charge order (CO) that collapse simultaneously at the IMT. The magnetic scattering is sensitive to the long-range spin order associated with the AFM-E$^\prime$ wavevector, while XAS is sensitive to the local Ni coordination connected to the breathing mode distortion of the charge ordered (CO) state and changes in screening due to the IMT \cite{Freeland:2015iw,Green:2016dk}, which is supported by optical pump - THz probe measurements. Thus, following short pulse excitation, it is possible to independently track the magnetic and electronic dynamics on ultrafast timescales. As described in detail below, for above gap excitation, there is a rapid collapse of the magnetic state ($<$ 175 fs) followed by a slower relaxation of the breathing mode CO and IMT response ($\sim$ 450 fs). From comparison between experiment and theory, we can develop a clear picture of the pathway where an intersite charge transfer (ICT) that triggers a collapse in the magnetic state followed by a slower IMT triggered by a displacive excitation of a coherent Nd soft phonon mode, which was observed via modulation of the Nd magnetic order (see Fig.\ \ref{Fig1}(a)). We will refer here to the leading distortion modes\cite{Balachandran:2013cg,Wagner:2018kz} that are relevant for the monoclinic to orthorhombic (M-O) symmetry change that is always observed at the IMT in the structural refinements at equilibrium in RNiO$_3$ compounds to connect these with our ultrafast dynamics results as well as with theory. We depict the IMT transition pathway starting from ultrafast melting of AFM order, followed by lattice responses, including octahedral rotation changes ($R^{+}_4$ and $M^{+}_3$ distortions), and antipolar A-site cations displacements ($X^{+}_5$) that couple to the breathing mode ($R^{+}_1$) and ultimately dictate the response time of IMT transition. Our results can accordingly provide a direct visualization of the IMT, which was not accessible under steady state equilibrium interrogation.
\begin{figure}
\centering
\includegraphics[width=.45\textwidth]{SampleConfig_v5-eps-converted-to
\caption{\label{Fig1b} {\bf Experiment geometry, and x-ray resonance spectra} (A) Illustration of sample geometry and experiment. (B) Resonant X-ray absorption and magnetic scattering spectra at the Ni L$_3$ edge measured simultaneously. The vertical line marks the energy for the measurement of the time delay scans in\ Fig.\ \ref{Fig2}.}
\end{figure}
\section{Experiment Description}
The perovskite nickelate NdNiO$_3$ is a prototypical system with coupled order parameters, exhibiting concomitant charge and magnetic order associated with an IMT (Refs.\ \onlinecite{Freeland:2015iw,Middey:2016jc,Catalano:2018kc} and references therein). Charge order is associated with an orthorhombic to monoclinic structural transition involving two NiO$_6$ sites in the monoclinic phase, which are referred to as short-bond (SB) and long-bond (LB) (see Fig.\ \ref{Fig1}A ). The magnetic order is E$^\prime$-type antiferromagnetic (AFM) with a 4$\times$4$\times$4 pseudocubic unit cell (2$\times$1$\times$2 monoclinic unit cell) with large planes of ferromagnetically aligned LB ($S=1$) and small SB ($S=0$) Ni sites arranged in an $\cdots\uparrow\cdot\ 0\ \cdot\downarrow\cdot\ 0\ \cdots$
pattern as shown in Fig.\ \ref{Fig1}B \cite{GarciaMunoz:1992dj,RodriguezCarvajal:1998dy, Scagnoli:2006ja}. Although the nominal ionic ground-state is Ni$^{3+}$ in a low spin 3d$^7$ configuration ($t_{2g}^6e_g^1$), theory strongly supports a state that is $3d^8\underline{L}$ where $\underline{L}$ denotes a ligand hole on the oxygen site. In this scenario, the charge ordered phase corresponds to alternating $3d^8$\,LB and $3d^8\underline{L}^2$\,SB sites \cite{Mizokawa:2000wq,Mazin:2007jx,Lee:2011eg,Park:2012hg,Johnston:2014ca,Subedi:2015en,Varignon:2017is,Haule:2017ft}. For NdNiO$_3$ \ the question arises of how the cooperation between magnetism and charge order drives the IMT? Theory suggests that the magnetic order is the main driver and supports the CO phase \cite{Park:2012hg,Varignon:2017is,Mercy:2017il}, but this has been difficult to show directly on the experimental side and motivates our time domain studies.
The (001) oriented films were grown using pulsed laser deposition on NdGaO$_3$ substrates (tensile strain of 1.4 $\%$) with a thickness of 50 nm \cite{Liu:2013ep,Middey:2016jc}. Experiments were performed at the SXR end-station where the X-rays and laser pulses arrive colinearly with the polarization in the scattering plane (see Fig.\ \ref{Fig1b}A)\cite{Schlotter:2012ey}. The sample was cooled below the IMT transition temperature (T$_{IMT} \approx$ 150\,K) to 70\,K and aligned to measure the off-specular ($(1/4,1/4,1/4)$ pseudocubic Bragg peak corresponding to the AFM-E$^\prime$ order\cite{Scagnoli:2006ja,Scagnoli:2008iu} as well as the bulk sensitive X-ray absorption spectroscopy (XAS) in fluorescence yield (see Fig.\ \ref{Fig1b}B), where the vertical line shows the energy chosen for the time-resolved measurements. Since the main change in the XAS across the IMT is seen at the dip near 852.5 eV (see temperature dependent XAS in the supplement Fig.\ S2), this energy was chosen to measure both channels without changing the photon energy. The pulse duration for this experiment was $\sim$100\,fs for the X-ray pulses and $\sim$150\,fs for the laser, which gives a total time resolution of $\sim$175\,fs. Excitation pulses at 1.55\,eV were utilized in addition to mid-IR pulses generated using optical parametric generation and difference frequency generation ($\approx$ 83 - 135 meV) to enable above and below gap pumping (optical gap E$_{g} \approx $ 100meV) \cite{Dhaka:2015id}. Additionally, The insulator-to-metal transition dynamics of the NdNiO$_3$ thin film was explored with optical pump-THz probe spectroscopy to track the formation of the metallic phase. To achieve optimal temporal resolution, the experiment was implemented in the following way: (i) both excitation (optical) and probe (THz) beams were collinear and incident on the sample at normal incidence to eliminate temporal broadening of rising dynamics introduced by oblique incidence of excitation beam. The 1.55\,eV reflectivity ($\Delta$R/R) dynamics were measured with pulses having a duration of $\sim$20\,fs.
\begin{figure}[t]
\centering
\includegraphics[width=.45\textwidth]{DelayScans_Fig_v6-eps-converted-to
\caption{\label{Fig2} {\bf Magnetic and electronic dynamics} (A) Delay scans for the magnetic scattering and XAS data following 800 nm excitation. The scale for the magnetic scattering is 1 for static AFM order and zero for complete loss of long-range order. XAS is a relative change consistent with the magnitude of XAS change seen with crossing IMT (see Fig. S2). (B) THz transmission overlayed with XAS. (C) Nd magnetic scattering intensity at the same wave-vector as Ni. The thin vertical line is at time-zero while the second line is after 1 period of the coherent phonon.}
\end{figure}
\section{Results}
\subsection{Ultrafast spectroscopy and scattering}
The ultrafast electronic and magnetic dynamics are presented in Fig.\ \ref{Fig2}A, which shows the pump-probe delay scans from magnetic scattering and XAS measurements following 1.55 eV excitation.
At an excitation level of $\sim$0.01 electrons per nickel site (0.5 mJ/cm$^2$), the magnetic scattering is completely quenched within 175 fs. In dramatic contrast, the time-resolved XAS scan shows a longer transition time of 446 fs, which demonstrates that that the magnetic and electronic contributions to the IMT have different timescales. For all of the data, the transition times were determined by fitting to an error function together with a slower exponential recovery. Note that the quoted transition times are defined by the twice the Gaussian width (width defined by difference at 0.85 and 0.15 of maximum value) and fitting errors were all $<$ 10 fs. Clearly, the magnetic order collapse is at the limit of our temporal resolution while the XAS dynamics are substantially longer. This indicates a rapid and total collapse of the magnetic order prior to changes in the CO (breathing distortion) and IMT embodied in the XAS response. To track the IMT directly, we utilized optical-pump THz-probe measurements to directly follow the formation of the metallic state. As shown in Fig.\ \ref{Fig2}B, the THz response shows a transition to the metallic phase that correlates directly with the XAS signal. The transition time to the metallic phase from the THz transmission is 425 $\pm$ 15 fsec, consistent with the XAS timescale. From this we can confirm that the ultrafast IMT is occurring at a slower timescale than the collapse of the long-range magnetic order.
In addition, we performed ultrafast degenerate 800 nm pump-probe measurements of the optical response with 20 fs time resolution. The comparison of the magnetic scattering response with the $\Delta$R/R (see Figs. S3 and S4) shows that the $\Delta$R/R response (transition time 228 $\pm$ 3 fs), is somewhat slower than the time-resolution-limited magnetic response. However, both our calculations (supplemental Fig. S8) and Ref.\ \onlinecite{Ruppen:2017du} demonstrate that the near infrared reflectivity dynamics (800 nm R/R) are influenced by changes in both electronic and magnetic order. The transition in the optical data does not show any evidence for two separate timescales so without a comprehensive model of how the optical spectra evolve with time, it is hard to extract numbers beyond observing that the data supports more than one timescale in the system. In the supplement (see Fig.\ S5), we present the data for pump wavelength dependence suggesting that the fast time-scale is associated with above-gap excitation and does not depend on the energy in excess of the NdNiO$_3$ bandgap. Nevertheless, we argue that the magnetic sensitivity contribution of the optical reflectivity captures in turn a somewhat faster transition time than what we observe in the purely electronic channels, including XAS and THz signals.
We also tracked the induced order on the Nd site as an additional probe of the changes in the magnetic order where the long-range Nd order arises from a very weak magnetic coupling to Ni via the Nd-O-Ni bonding. By tuning to the Nd M$_4$ resonance ($\sim$ 1000 eV) at the same wave-vector as the Ni ordering\cite{Scagnoli:2008iu}, we explored how the Nd ordering responds to the change in Ni ordering. In addition to a fast initial drop at the same rate as the collapse of Ni magnetic order (see Fig.\ \ref{Fig2}C), there is a strong oscillatory component with a period of $\sim$ 450 fs that is rapidly damped within a few ps, due to a coherent phonon rather than a Nd magnon. The oscillation period is consistent with the soft phonon associated with the Nd site\cite{Zaghrioui:2001kk}, which we also observed with optical reflectivity (Figure S6). The ringing in the Nd magnetic scattering is related to changes in the Nd-O-Ni exchange path and the time-scale of the damped oscillation is consistent with time-resolved measurements of the collapse of the CO state\cite{Esposito:2018kb}. Note that the fit to Nd data was restricted to the early time region (between Nd intensity of 0.6 and 1) and results in the same width of initial drop in order as the Ni case, which implies the Nd senses the Ni long-range order collapses in the first 200 fs. This connection and the coherent phonon mode will be explored in more detail in the discussion section.
\subsection{Theory}
To obtain additional insight into how magnetic order couples to the lattice, we used density functional theory (DFT) to examine the electronic structure and energetics of various magnetic states as a function of static lattice distortions. The energetics of the AFM-E$^\prime$ monoclinic structure is compared against the ferromagnetic (FM) solution, which in this context serves as a proxy of the paramagnetic (PM) state, which is difficult to approximate at the DFT level. However, note that the FM solution that is provided solely by DFT has orthorhombic symmetry and thus captures well the essential structural characteristic of the structural M-O transition observed during IMT. To include the influence of correlations, a plus Hubbard $U$ = 2 eV correction was used on the Ni d orbitals, which accurately captures the details of the AFM insulating phase \cite{Varignon:2017is}. Specifically, we investigate the evolution of the magnetic and electronic properties as a function of the breathing mode distortion (Fig.~\ref{Fig3}A). We focus on this particular distortion as it is the primary distortion active at the $Pnma \rightarrow P2_1/n$ transition; in addition, the DFT studies indicate that Jahn-Teller and rumpling distortions are not operative in determining the magnetic order (see Figure S4).
\begin{figure}[t]
\centering
\vspace{-3em}
\includegraphics[width=.5\textwidth]{Nickelates_DFT_Results_Fig_v5-eps-converted-to
\caption{\label{Fig3}{ {\bf Calculated electronic and magnetic structure} (A) Dependence of AFM-E$^\prime$ and FM states as a function of breathing mode distortion\cite{Wagner:2018kz}. The two step path discussed in the text is illustrated by points A,B, and C. (B) Orbital dependent Ni density-of-states corresponding to points A, B, and C. indicated in panel A with the Fermi level set to 0\,eV. }}
\end{figure}
As a function of the cooperative breathing mode distortion, Fig.~\ref{Fig3}A plots two energy curves corresponding to insulating AFM-E$^\prime$ and ferromagnetic (FM) order\cite{Wagner:2018kz}. The minimum for the AFM case agrees with the experimental value of the breathing mode distortion and the resulting magnetic order has $S\sim$1 on the LB site and $S=0$ on the SB site. The AFM phase is insulating at all magnitudes of breathing distortion, but the FM phase is only insulating at values of breathing distortion greater than point C (additionally, see Figs. S7 and S9). Although a non-collinear spin structure also satisfies the symmetry for E$^\prime$-type order \cite{GarciaMunoz:1992dj,RodriguezCarvajal:1998dy, Scagnoli:2006ja,Scagnoli:2008iu}, our results together with other recent theoretical results strongly suggest that the collinear phase is the stable ground magnetic state \cite{Varignon:2017is,Haule:2017ft}. The energy of the FM state is always higher than the AFM state except close to zero breathing mode distortion, corresponding to the high temperature orthorhombic symmetry of the paramagnetic metallic phase. Further, we note that the gap between AFM and FM at the AFM minimum is $\sim$kT$_N$. To understand the evolution of these different degrees of freedom upon optical excitation, we consider a two step process. Step 1 (see Fig.\ \ref{Fig3}A), corresponds to direct excitation from points A to B at a timescale faster than the lattice can respond and alter the cooperative breathing mode distortion (i.e., a shift along the horizontal axis). As shown by the density of states (DOS) in Fig.\ \ref{Fig3}B, at this point on the FM curve, the system is still insulating. Since point B is not at the minimum of the FM energy curve, the system will then move towards smaller breathing distortion and the minimum located at point C, where the system becomes metallic (see insets of Fig.\ \ref{Fig3}B).
\section{Discussion}
Now we can link these results in order to understand the fundamental question of what controls the time-scales for phase transformation in a strongly coupled phase with multiple order parameters. This insight can allow us to build a more complete picture of how light interacts with these degrees of freedom. First, we discuss the nature of the optical excitation process. In the case of highly-covalent nickelates, our calculations of the optical spectra (see supplement) and those with DMFT theory in Ref. \cite{Ruppen:2017du} are consistent with an intersite charge transfer (ICT) from the $3d^8$ LB sites to $3d^8\underline{L}^2$ SB sites involving excitations between the Ni $3d$ and O $2p$ states. This ICT dynamically changes the charge distribution around the lattice sites. From our analysis of the calculated density of states above, two key components change between AFM (A) and FM (B) configurations. First, the local moment on the SB site, which is $0\,\mu_B$ for the AFM phase converts to $\sim0.5\,\mu_B$ in the FM state. This arises from a local rearrangement of the spin-dependent $e_g$ orbital occupancy on the SB site. Secondly, the difference in Born effective charges between the LB and SB sites increases (see details in supplement Table S3). This is associated with the redistribution of oxygen holes in the lattice due to the change in magnetic state, similar to the process that would be triggered via light induced ICT. Such an excitation has pathways for directly coupling to magnon excitation and allows the light to couple to the magnetic degree of freedom\cite{Hellsvik:2016cs}.
Understanding the excitation pathway allows one to address the question of what controls the fundamental timescales for collapse of these different degrees of freedom. For other complex oxide systems, the timescales for the charge transfer excitation and rearrangement of orbital occupancies has been shown to occur on sub $\sim$100 fs timescales \cite{Okamoto:2011da,Singla:2013kk,Beaud:2014bp}. As such, we consider ICT an operative pathway by which changes in the electronic configuration modify the magnetic order via optical modification of the exchange and the time-scale for the collapse of magnetism is then tied to the details of the spin-wave spectrum\cite{Bossini:2016iv}. This is due to the momentum conserving optical excitation of bimagnons as observed in other transition metal oxides\cite{Hellsvik:2016cs}. Indeed, recent resonant inelastic X-ray scattering measurements measured the magnons in NdNiO$_3$ for the first time and show that the the zone-boundary magnons in NdNiO$_3$ have $\sim$50 meV energy($\sim$85 fs period)\cite{Lu:2018cf}, which is consistent with our inference of a sub-100 fs timescale that leads to the vertical transition from points A to B in Fig.\ \ref{Fig3}A. Since the AFM order supports the charge order for NdNiO$_3$ , the collapse of magnetic order triggers a collapse of the CO phase. Nevertheless, since our experiments demonstrate a slowed response for CO with respect to the melting of the magnetic order, we discuss the origin of this observation in the following.
From bulk NdNiO$_3$ it is known that the paramagnetic phase can exist in the presence of CO so a change in lattice symmetry is not required for the collapse of magnetic order. However, the metallic state is only present in orthorhombic symmetry. On the structural side, the collapse of the CO is key to the transition to the higher symmetry metallic phase, but the breathing mode is coupled to a low energy Nd mode that is directly associated with the $P2_1/n \rightarrow Pnma$ structural phase transition\cite{Balachandran:2013cg}. In support of this point, we find a linear relationship between the $X^{+}_5$ distortion that is characteristic for A-site cation distortion and the breathing mode ($R^{+}_1$) distortion as observed from equilibrium state refinements\cite{Balachandran:2013cg,Wagner:2018kz} (see Fig. S10). Since $X^{+}_5$ is also coupled to the $R^{+}_4$ and $M^{+}_3$ octahedral rotation distortions involved in M-O symmetry change at IMT, we expect that a collective structural response is needed under a dynamical structural transformation. Owing to the coupling of these two modes, the timescale will be dictated by the slowest mode (i.e. phonon bottleneck) and is consistent with the timescale seen for the electronic changes, as was seen recently for the case of layered nickelates \cite{Coslovich:2017kg} and was already known for VO$_2$\cite{Cavalleri:2004eh}. This expected time-scale is also consistent with recent ultrafast measurements of the collapse of the CO state\cite{Esposito:2018kb} and connects to the clear coherent phonon oscillation seen via the Nd magnetic scattering (see Fig.\ \ref{Fig2}C).
\begin{figure}[t]
\centering
\includegraphics[width=.45\textwidth]{NdscatteringFig_v3-eps-converted-to
\caption{\label{ndfig} {\bf Coherent lattice contributions to Nd order dynamics} (A) An illustration of the Nd-O soft mode associated with a coordinated Nd-O bond changes and NiO$_6$ rotation. (B) Nd magnetic scattering as a function of fluence in direct comparison to the Ni magnetic response. (C) Difference between the fit and data to extract the time period of the coherent phonon (black lines).}
\end{figure}
To better understand this coherent phonon mode and how it connects to the induced Nd magnetic order\cite{Matsuda:1990fg,Scagnoli:2008iu}, we utilized theoretical calculations (see supplement) and insight from calculations of Nd-TM coupling for the case of NdFeO$_3$\cite{Chen:2012bf} as no such calculations exist for NdNiO$_3$ . Figure\ \ref{ndfig}A, shows a portion of the unit cell with the Nd atom and the shortest exchange pathway to the high-moment Ni$_{LB}$ site. Calculations show that this $\sim$80 cm$^{-1}$ A$_g$ symmetry mode involves motion of the Nd atom that drives a tilting/rotation of the NiO$_6$ octahedra without any motion of the Ni atom. Since the exchange, $J_{Nd-Ni}$, is directly connected with the Nd-O bond length, this provides a direct connection between the magnetic exchange and the lattice vibrations. If the bond length changes, $J_{Nd-Ni}$ varies and the induced Nd order varies correspondingly. To look at this mode in more detail, we show two fluences for the measurement of the Nd induced magnetic order. Both show a clear coherent mode and the fact that it has a longer lifetime at low fluence associates this with an excitation in the insulating phase (see Fig.\ \ref{ndfig}B). To quantify this mode more directly, we will break the Nd data into two parts: a fast initial decay followed by a slow recovery and the coherent oscillation.
In the first 200 fs, the Nd magnetic collapse has the same timescale as the Ni magnetic data implying a connection between the two processes. Usually the parasitic Nd order is considered as a paramagnetic moment in the large local field due to proximity to the ordered Ni atoms, as was evidenced in resonant soft X-ray scattering at the Nd M-edge\cite{Scagnoli:2008iu}. To first order, one would accordingly expect a slow paramagnetic relaxation and not an ultrafast response. However, recent work has shown that in ErFeO$_3$ the Fe magnons hybridize partially with Er spin fluctuations\cite{Li:2018ha}, providing a partial pathway for the Ni magnetic excitations to couple to the Nd. This seems consistent with the observation that the fast drop in Nd order is roughly proportional to what is seen for the case of Ni. However the the connection to Ni does not explain the coherent oscillation, which was not seen for Ni and has a longer period than expected for Ni magnons ($\sim$85 fs period). To quantify the oscillation, we take the difference between the data and a simple fit shown in Fig.\ \ref{ndfig}B and fit a cosine to this difference, we determined the oscillation periods for both fluences to be 378 $\pm$ 10 fs. This period is very consistent with the observed phonon modes\cite{Zaghrioui:2001kk} and our calculations (see supplement) associated with motion of the Nd atom. This phonon causes a coordinated increase/decrease of the shortest Nd-O-Ni bonds highlighted in figure Fig.\ \ref{ndfig}A. Since the magnetic exchange is strongly affected by the change in bond-length, this results in a modulation of the Nd order that has the period associated with the Nd-O phonon.
It is well known that the equilibrium atomic positions for all atoms in the unit cell are strongly affected by the IMT, as is shown in Fig.\ \ref{Fig3}A. An abrupt change in the equilibrium atomic positions arising at the first order transition provides a mechanism for the displacive excitation of phonons. Deeper insight will require developing a model linking the bond-length changes to the magnetic coupling between Nd and Ni. However, the displacive mechanism suggests that as soon as the magnetism is quenched, the lattice knows it must move to a new set of equilibrium positions. This together with the different timescales for magnetism and the IMT are consistent with a picture that for NdNiO$_3$ the antiferromagnetic order is the primary order that supports the CO phase. Such a conclusion has been suggested theoretically\cite{Park:2012hg,Varignon:2017is,Mercy:2017il}, but has been too difficult to prove experimentally given the concomitant nature of the electronic and magnetic phase transitions. Here using time-domain techniques we were able to extract more insight from the fundamental timescales and show clearly that magnetism is the driver of the IMT by showing the initial magnetic collapse is what triggers the loss of CO and subsequent IMT.
Now we can expand the context and discuss this in connection to a much wider class of materials with entangled order parameters. It is useful to consider the present results in the context of other ultrafast experiments on NdNiO$_3$ and more generally in the context of photoinduced IMT dynamics in other materials. In comparison to vanadates such as VO$_{2}$ and V$_{2}$O$_{3}$, the IMT in NdNiO$_3$ has a fluence threshold that is nearly an order of magnitude smaller \cite{Cavalleri:2004eh,Wall:2013dv}. This is, in part, related to the large (relative) latent heat in the vanadates, necessitating a larger absorbed energy density to obtain the high temperature metallic structure. V$_{2}$O$_{3}$ is similar to NdNiO$_{3}$, with a high temperature paramagnetic metallic state that transitions to a low temperature antiferromagnetic insulating phase (T$_{IMT}$ $\approx 150K$). Despite these similarities, the dynamics of the photoinduced IMT in V$_{2}$O$_{3}$ is more similar to VO$_{2}$. Specifically, there is no evidence of magnetism playing an important role in triggering the IMT in V$_{2}$O$_{3}$. This further highlights the unique cooperativity of magnetism and the lattice in triggering the IMT in NdNiO$_3$ following photoexcitation.
\section{Conclusion}
In summary, using X-rays to disentangle the electronic and magnetic degrees of freedom has provided a concise picture of how light interacts with strongly correlated matter. By tracking the distinct timescales one can gain not only a clearer picture of how charge, magnetic, orbital, and lattice orders evolve, but also which are key to driving the transition and which are triggered as a response to the changing fundamental order parameter. Using this approach, not only can we see that magnetic order is likely the fundamental order parameter for NdNiO$_{3}$, but also provides a paradigm to unravel entangled order parameters in many of the complex materials of contemporary interest.
\section{Acknowledgements}
\begin{acknowledgements} J.W.F. wants to acknowledge the help with science and data analysis from N. Laanait. V.S., J.Z., R.D.A, J.C., H.W., J.M.R, and J.W.F were supported by the Department of Energy grant DE-SC0012375 for work with ultrafast X-Ray and optical experiments and analysis of the data with theoretical support. D.P.\ was supported by the Army Research Office (Grant No.\ W911NF-15-1-0017). J.C. was also supported by DOD-ARO under Grant No. 0402-17291 and by the Gordon and Betty Moore Foundation's EPiQS Initiative through Grant No. GBMF4534. Work at the Advanced Photon Source, Argonne was supported by the U.S. Department of Energy, Office of Science under Grant No. DEAC02-06CH11357. Use of the Linac Coherent Light Source (LCLS), SLAC National Accelerator Laboratory, which is a DOE Office of Science User Facility, under Contract No. DE-AC02-76SF00515. Electronic structure calculations were performed using computational resources provided by the DOD-HPCMP. J.W.F. would like to acknowledge many insightful conversations with A.J. Millis and D. Khomskii.
\end{acknowledgements}
|
2,877,628,089,012 | arxiv | \section{Introduction}
The trend of population aging is currently an undisputed fact, observed throughout the globe, in developed and developing countries alike \cite{nations2015world}. By 2050 over 20\% of the population of the USA is projected to be 65+ \cite{ortman2014aging}, other developed countries, such as Japan are also expected to become aging populations by that time. Although this trend has been deemed a negative trajectory for population development \cite{harper2014economic}, and a giant challenge for social security frameworks, and in turn a major burden for the state, research shows that this trend may bear out other results. For instance, Spijker et al. argument that although demographic data are true in reality number of older adults requiring assistance in UK and other countries actually been falling in recent years \cite{spijker2013population}. This point of view is supported by Sanderson et al. \cite{sanderson2015faster} who point out that metrical age qualifying to being older adult will increase in coming years.
In parallel to the mentioned trend, an increase in smartphone and on-line tool usage among older adults can also be observed \cite{zickuhr2012older}. This trend of increasing usage of mobile ICT can be observed worldwide, and already has inspired many researchers and designers to introduce new and creative solutions tailored towards the needs of the older adults \cite{massimi2007using}. When considered in relation to the findings suggesting that in future aging societies, older adults may be more affluent, healthy and tech savvy than their current counterparts. This increase in use of mobile technology, opens up new possibilities for addressing the issue of population aging, and creating solutions tailored to a new growing customer base. With these new possibilities however, also come new challenges. The demographics increase in the share of mobile solutions users leads to new, and previously not addressed issues such as those related to the older adults safety.
Research shows that older adult are less likely to be tech savvy\cite{grimes2010older}, and often as a result, are more prone to suffer from threats related to their safety. Moreover, older adults tend to be more aware of the risks they face on-line \cite{adams2005psychological}. Due to the fact that on-line solutions such as social media, communication and financial services or other mobile based apps increasingly blur the line between the on-line and real life situation, it is understandable that, when addressing the safety issues of older adults, one must consider both aspects of safety. Discussing this topic is our main aim in this paper.
For the sake of this paper, we made an explicit distinction between two aspects of older adults safety; their on-line security, and real life safety. The former, as the name suggest, refers to risk the older adult may face when interacting with ICT (ie. identity theft, scamming, phishing, spread of malware and spam), the latter on the other hand deals with risks connected with interactions made in the real world, such as theft, assault, burglary, traffic accidents on so on. Although these two aspects were deeply reviewed by researchers (with work by \cite{grimes2010older}, and \cite{forjuoh2016172} serving as good examples), there has been no attempt, to the best of our knowledge, the analyze the interaction between the two aspects, let alone to analyze how developers can address them in the design process.
Therefore we decided, based also on previous endeavours and activities with older adults in our LivingLab, to use a case study of an intergenerational design team during a Hackathon event conducted at the Polish-Japanese Academy of Information Technology in Warsaw, Poland (PJAIT). In this study we addressed the issue of the connection between on-line security and real life safety in the design process by using participatory design approach. By describing the design process and group dynamics, as well as the final product, we wish to open further discussion, and provide some deeper insight into a topic that, although important in lieu of the aforementioned trend in stronger connection between the on-line and off-line, was not yet fully addressed by the research community. In this paper we provide more comprehensive approach to the security of older adults based on the concept described in case study section which we previously briefly outlined in our conference report \cite{balcerzak2017press}.
The rest of the paper is organized as follows. In the \textbf{Related work} section, we provide a comprehensive description of findings in regards to on-line security and real life safety of older adults, as well as security design in software engineering, as well as participatory design, and older adults motivation for taking part in design processes and similar tasks. In the section \textbf{LivingLab insight} we describe a broader context of their empirical studies and research activities with older adults in PJAIT LivingLab relevant to the above mentioned topics. The \textbf{Case study} section present in detail the design session conducted during the Hackathon, the specific features of the application created by the intergenerational team, as well as the design process itself. The \textbf{Conclusions and Future Work} section details how the observations made during the design session interact with the findings related to the topics of on-line security and real life safety.
\section{Related works}\label{related_work}
\subsection{Older adults and on-line security}
According to studies done by \cite{grimes2010older} older adults are, on average less knowledgeable and aware of on-line risks than younger adults. Also, as shown by \cite{adams2005psychological}, older adults show limited trust to on-line technology, however this effect diminishes with usage of ICT. Another dimension crucial with on-line security is trust. Studies done by \cite{mccloskey2006importance} as well as \cite{grimes2010older} show that trust among older adults is not affected by age, and relies more on experience that comes with the use of ICT. Our previous study on location-based game with mobile ICT technology \cite{kopec2017location} which we present briefly in next section also showed the connection between direct ICT usage and self-confidence of older adults in context of mobile technologies. Therefore, the design of applications that address the issue of trust is important. The topic of safety, also plays a role in research related to applying smart house technologies as tools for enabling older adult independence\cite{peek2017can}.
A topic also related to on-line safety is the issue using data and text mining techniques for the detection f scamming and the credibility of on-line material\cite{Wawer:2014:PWC:2567948.2579000}\cite{Jankowski-Lorek:2014:PCW:2639968.2640074}.
\subsection{Older adults and real life safety}
For the sake of this paper, when reviewing the real-life hazards faced by older adults, the authors will focus on a specific aspects of safety, which in the literature is refereed to as neighborhood safety. This refers to risk related to social interaction in the nearest environment of the older adult. As the comprehensive literature review conducted by \cite{forjuoh2016172} suggests, five main themes constitute the overall framework of neighborhood safety. These are: general neighborhood safety; crime-related safety; traffic-related safety; fall-related safety; and proxies for safety (e.g., vandalism, graffiti).
This literature review provides also a deep and comprehensive review of what main themes are the object of focus among researchers interested in real life hazards and their perception by older adults. It is noteworthy, that in the studies used in the review, main focus was put on health issues and their relation with the feeling of safety. This is a theme which appears in works such as those done by \cite{won2016neighborhood}, where the authors show that physical health is a correlate with the older adults sense of safety. Another interesting field of study is related to older adults emotion in relation with surrounding environment. We conducted a preliminary study on mapping senior citizens' emotions with urban space \cite{nielek2017emotions}, which is also connected with our previous work, namely the mentioned mobile game \cite{kopec2017location} which was related to the topics of well-being and happiness of older adults. This issue is a vital subject of many other interesting research, i.e a longitudinal study on a sample of over 10,000 older adults that pointed out that perception of one's psychological well being affects the older adults feeling of safety \cite{choi2017perceived}.
To the best of the authors knowledge, little focus was placed on interaction between thus defined levels and perceptions of neighborhood safety and use of mobile devices and apps. Therefore, besides our previous endeavours mentioned above, we decided also to pursue this topic in exploratory in-depth interviews with older adults form our LivingLab, which are elaborated in the next section.
\subsection{User participation and interaction}
Fundamental idea relevant for this case is connected with a concept of user-center design, as a part of general idea of human focused approach related to another important idea: participatory design, sometimes called co-design. While there are different origins for the two latter terms they actually refer to the same idea of bottom-up approach which is widely used besides software engineering from architecture and landscape design to healthcare industry.\cite{szebeko2010co} All those concepts put human in the center of designing process, however, there is a small, but significant distinction between user-center design and participatory design: the former refers the process of designing FOR users, while the latter WITH users.\cite{sanders2002user,sanders2008co} From the point of view of this study especially important are concepts that are related to user competences and empowerment provided e.g. by Ladner \cite{ladner2015design}.
Another case is the work done by \cite{xie2012connecting}, describing a cooperation between seniors and preschool children in a design task. An interesting observation made in this research was the fact that both groups needed equally time together, and time in separation in order to function properly. The broader context for these topics is covered by the contact theory, a widely recognized psychological concept coined by Allport \cite{allport1979nature} and developed for many years by other, e.g. Pettigrew.\cite{pettigrew1998intergroup} According to the theory the problem of intergroup stereotypes can be faced by intergroup contact. However, there is several condition for optimal intergroup contact, but many studies proved that intergroup contact is worthwhile approach since it typically reduces intergroup prejudice.\cite{pettigrew2006meta} We had also explored the topic of intergroup interaction in our various studies including mentioned intergenerational location-based game and the hackathon based on previous works and tools i.e. by Rosencranz \cite{rosencranz1969factor}.
\subsection{Volunteering of older adults}
The effects of volunteering has on older adults is a developing field of study within various disciplines. The consensus that can be reached throughout various studies, is that participation in volunteer activity has many positive effects on the elderly. \cite{lum2005effects} claim that older adults who frequently volunteer in various activities, tend to have improved physical and mental health, compared to those who do not participate in volunteering. The work of \cite{morrow2003effects} extends this correlation to well-being (with volunteering being correlated with higher levels of well-being) this is also reinforced by \cite{greenfield2004formal}and \cite{hao2008productive}. This is crucial since studies also show that in some regards, elders are more likely to be engaged in volunteer activity \cite{morrow2010volunteering}.
Motivation for volunteering among older adults is also important. In research done by \cite{itoko2014involving} involving the comparison of older and younger adults when participating in a crowd sourcing task of proof reading texts in Japanese, it was shown that older adults where successfully motivated by the use of gamification techniques within the task, which means according to the most widespread definition of gamification by Deterding \textit{the use of game design elements in non-game contexts}.\cite{deterding2011game} However, a similar task, conducted by \cite{brewerwould} on a group of American seniors showed a slightly different pattern, where older adults were bored by the task, and did not comprehend its' significance. Nevertheless gamification is strictly connected with motivation and therefore inevitably leads to the psychological context, e.g. Zichermann claims that \textit{gamification is 75 percent of psychology and 25 percent of technology}. \cite{zichermann2011gamification} At this point it is worth mentioning that according to the latest reviews more and more solutions are based on solid psychological theories and frameworks. \cite{mora2015literature} One of the most important theoretical approach in the field of motivation is self-determination theory (SDT) developed by Ryan and Deci. \cite{ryan2000self} It is based on subtheories formerly developed by the authors of SDT: cognitive evaluation theory (CET) related to the intristic motivation and organismic integration theory (OIT) related to extrinsic motivation. The theory was proven to be effective to the elderly as well, e.g. by Vallerand .\cite{vallerand1989motivation} We had explored the topic of older adults volunteering and motivation in our research i.e. Wikipedia content co-creation. \cite{nielek2017turned}
\section{LivingLab PJAIT insights} \label{livinglab}
In this section we provide some insights from our experience with older adults within Polish-Japanese Academy of Information Technology LivingLab. Further details of our LivingLab, it's origin and design as well as older adults activities are provided in separate description \cite{kopec2017livinglab}.
\subsection{About LivingLab PJAIT}
The term \textit{Living Lab} was coined by William Mitchell from MIT \cite{niitamo2006state} and was used to refer to the real environment, like a home or urban space, where routines and everyday life interactions of users and new technology can be observed and recorded to foster the process of designing new useful and acceptable products and services. The idea of LivingLab is therefore inherently coupled with broad concept human-centered approach described in previous section.\\
LivingLab at the Polish-Japanese Academy of Information Technology is a long-term framework project, whose goals are related to social inclusion and active engagement of the elderly in social life by facilitating the development of ICT literacy among them as well as creating an active community of stakeholders who are both the beneficiaries and enablers of research into their problems.
This framework project has been established in a long-term partnership with the City of Warsaw.\\
Throughout recent years we have organized a number of activities for older adults focused on various research areas relevant to the topics of this article as we mentioned in related work section. In particular during an intergenerational location based game older adults performed various everyday mobile ICT tasks, such as connecting to Wi-Fi, browsing the information, scanning QR codes or taking panoramic photos with the assistance of their teammates. On the other hand, the younger participants benefited from the background knowledge about the city and its history of the elderly. Thus a positive bi-directional intergenerational interaction was observed alongside positive older adults self-awareness of the technology in context of physical activities and well-being. In other LivingLab activities, including on-line courses and crowdsourcing tasks as well as real life workshops and activities i.e. devoted to both application and content co-creation we also noticed that security issues are important but a bit vague area for older adults. Based on the literature review we decided to conduct a more in-depth qualitative research described in next subsection.
\subsection{Older adults and security}
Based on literature and observations from LivingLab activities, workshops and consultations we decided to obtain a more in-depth insight in the topic, including both relevant perspectives: on-line security and real life safety.
We used individual in-depth interviews in order to extract additional security insights. In total, we conducted four such interviews with older adults from our LivingLab aged 65+, two less technology advanced (female participants P1 and P2) and a pair of more advanced older adults (female P3 and male P4). To obtain more in-depth information from them we decided to conduct individual semi-structured interviews related to several topics: Internet and mobile application usage, endeavours towards on-line security and real life safety alongside with perception of potential interaction between those two realms.
Based on interviews we figured three different strategies toward security: caution, separation and awareness. While awareness is related to the more technologically advanced older adults, the two former strategies are interesting since they are represented by older adults with lower ICT literacy. Generally speaking the caution strategy represented by P1 was based on carefulness in both real life and on-line activities while separation strategy represented by P2 was based on strong conviction on non-transition between virtual and real realm. In particular P2 claimed that \textit{these two worlds should not be comined}.
First we asked participants about their Internet and mobile experience. There are three major areas of their on-line interests:
\begin{itemize}
\item doing everyday tasks, like paying the bills
\item keeping in touch with family and friends
\item source of information.
\end{itemize}
The latter was the most extensive category and included various topics from health and medical issues (i.e. drugs and food ingredients, dietary information), through transportation to cultural and political news. In this context P2 claimed that \textit{The world has gone so far that it is difficult to live without the Internet these days}.
In order to obtain insight about on-line security we asked the participants about securing themselves in virtual space. In this context participants were aware of anti-virus protection as well trust issues, identity theft and identity verification (the need of verification the identity of on-line entities i.e. shops and companies). However, here we observed the major difference in separate-world approach. P2 explicitly stated, that there is no direct transition between the virtual and real realms. This was directly connected with careless websurfing. Moreover P2 wasn't afraid of her identity theft based on the claim \textit{I am no one special, I am an ordinary Smith}.
We also asked about the safeness endeavor made by the participant in their everyday life. Besides personal physical safety measures like traffic safety, observation of the surroundings and other persons they pointed out two major areas connected with virtual space: finance and health. However they cannot establish the connection between the two realms. In particular, besides techniques of safe cash carrying (P1: \textit{one can deposit the cash in various piece of garment}) they stated that in general they use credit cards instead of cash and they deposit money in bank. In reference to health, besides physical activity they pointed out healthy diet and food and medical ingredients verification.
As we can see some factors from virtual space activities can be directly mapped and connected with the real world. However, it was difficult to the participants to find the connection by themselves. More technologically advanced older adults (P3 and P4), were more aware of two realms i.e. bank account protection and identity theft alongside with interface as a vital concept between virtual and real world. On the other hand older adults with lower ICT literacy did not found themselves the connection between real life safety and increasing on-line security and vice versa. Surprisingly even though they provided a number of proper examples and behaviors from both realms they failed to find themselves spontaneously the connection. This leads to the conclusion that inevitably there is a room for designers to employ a participatory design approach in order to obtain insights from older adults not only to better understand their need but also to foster the process of idea development in order to find better connection between on-line and real life habits and safety. Because usually there is a generation gap between software development teams and end-users in case of older adults application we also decided to obtain a deeper insight into the dynamics of such collaboration, described in next section, based on our experience in the field of intergenerational interaction.
\section{Case Study}\label{design}
\subsection{Case study context}
The case study, involving the intergenerational developer team, was observed during a DEVmuster Hackathon organized in march 2016 in The Polish-Japanese Academy for Information Technology in Warsaw Poland, during which older adults and students of the academy had an opportunity to cooperate in designing application that would address the needs of older adults. The team consisted of 4 males, all in their twenties, who were students of the academy, and two older adults, a male, and a female, who were participants of the PJAIT LivingLab, a framework presented in previous section, which involves volunteers wishing to take part in various project aimed at activizing older adults with the use of ICT.
The team formed during the first hours of the event, during which ideas for the potential app were discussed. Upon reviewing the opinions of the older adults, the students decided to change their original idea of an application, in to an app designed for exchange of favor between volunteers and older adults. The name 'F1' was chosen, based on the function key for calling the help menu.
\subsection{Platform architecture}
The F1 platform was designed as client-server model and requires access to the Internet for the proper functioning. It might be a serious limitation for less developed and less populated countries but as the platform is intended to be deployed in Poland we decided to sacrifice versatility for simplicity.
Access to the system is possible either through a web site or a dedicated mobile application. Although both ways provide the same functionality there are also substantial differences. Mobile application was designed to be most convenient for people offering support. Web site is more focused on posting requests for help. The reason for this differentiation is that mobile application will be more frequently used by younger volunteers and web browsers are better suited for older adults. In the case described in this paper senior participants were more familiar with traditional desktop or mobile computers than with smartphones, mainly due to their professional background and prior LivingLab activities i.e. computer workshop organized by the City of Warsaw. Moreover computer web browsers can be more suitable for people with certain disabilities -- e.g. visually impaired or with limited hand dexterity. On the other hand mobile devices are becoming more and more widespread and the adoption of touch interface by the elderly can be faster and more effective than traditional computer interfaces (e.g. observed in our previous research studies). Thus in final product the application mode (web site or mobile app) is intended to be freely interchangeable at any time by the user.
\subsection{Functionality}
As presented in figure \ref{F_main}
The web based application contains all the key information and functions important for the user searching for the help of a volunteer. The screen informs the user about his or her previous favor requests, as well as the location of other users in the area. There is also an S.O.S button which can be used in case of emergency.
\begin{figure}[p]
\centering
\includegraphics[scale=0.35]{web_glowny.jpg}
\caption{Main screen of web-based application focused mostly on people requiring assistance.}
\label{F_main}
\end{figure}
The mobile application view, used by the potential volunteer is shown in figure\ref{fig:f1}. When using the mobile view, the user can view a map of the nearest area where favor requests are marked. When the user selects a favor a brief description of the favor and the requesting user is provided.
\begin{figure}[!tbp]
\centering
\subfloat[Welcome and login screen.]{\includegraphics[width=0.3\textwidth]{tel_logowanie.jpg}\label{fig:f3}}
\hfill
\subfloat[Screen presents requests for assists in neighborhood on the map.]{\includegraphics[width=0.3\textwidth]{tel_mapa01.jpg}\label{fig:f2}}
\hfill
\subfloat[Detailed information about request for assist.]{\includegraphics[width=0.3\textwidth]{tel_zgloszenie02.jpg}\label{fig:f1}}
\caption{Mobile application dedicated for those who offer assistance.}
\end{figure}
\begin{figure}[!ht]
\centering
\includegraphics[scale=0.15]{web_przyjecie_04_edited.png}
\caption{Pop-up window displaying the confirmation details for selected volunteer}
\label{F_popup}
\end{figure}
\subsection{Security related features}
Another direct benefit from applying participatory design approach refers to security issues. During the pre-design phase, older adults voiced their concerns related to user security, taking into account threats for both parties of the process. The main areas in which they stated older adults require aid when it comes to security were related with minimizing the risks of coming in contact with fake profiles, or malicious users, as well as dealing with problems of potential identity theft. The older adults involved in the project also stressed that the platform should be able to aid the older users when dealing with emergencies when swift help is needed. These issues were addressed by applying the following solutions into the design of the platform.
\subsubsection{Trusted profiles}
Sign up is free for all and requires only a valid email account. Lowering the barrier makes system more user-friendly but also prone to malicious users. Discussions with prospect users during design phase reveal that the elderly are afraid of letting unknown people visit their apartments. To address this problem a voluntary procedure for confirming profiles was implemented. User profile might be confirmed by external organizations that are trustworthy: e.g. schools or local NGOs. Confirmation of the verified status is visible for everyone next to profile picture -- see fig. \ref{F_popup}.
\subsubsection{Challenge-response authentication}
Next to the threat of fake or malicious profiles mentioned in previous section yet another problem was identified by participatory approach. Even if identity of volunteer is confirmed on the platform still we need a way to confirm it in real world when volunteer is knocking to the door of senior's apartment. This is the situation when digital system should face analogue world and bottom-up approach proved to be helpful once again. Therefore, a standard challenge-response authentication has been adapted and implemented. The platform generates two keywords for both users. Elderly should ask about the right password before letting someone in. Passwords are randomly selected from a subset of polish words to make them easy to remember and dictation by entry phone.
\subsubsection{Reputation score}
Limiting the amount of frauds is crucial for assuring wide acceptance of the platform but it is not enough. Next to deliberate and planed frauds we can also see a bad quality service. Therefore, the platform contains a reputation management system. Every agreed and conducted service can be evaluated on Likert-type scale. To make scale easier to understand for users first two grades are red, neutral score is gray and the two positive levels are green. Sum of all evaluation for give user are displayed next to the picture -- see fig. \ref{F_popup}.
\subsubsection{Emergency button}
In real life exhaustive list of risks and threats is impossible to complete. Therefore, an emergency button has been added to the F1 platform.
\section{Conclusions and future work}\label{conclusion}
As it was presented in the \textbf{Related Work} section and in our study, the issues of on-line security and real life safety faced by older adults are quite diverse. The span a wide set of threats ranging from health issues, to crime related issues, and problems connected with the spread of malware and spam, and even though older adults can point out a variety of those issues, usually it is difficult for them to spontaneously find the connection between the two realms. However, in our case study we showed that participatory design approach can benefit both younger tech-minded developers and older adults as end-users.
The observations made during the presented case study show, that the key for creating an on-line solution, that would not only aid them in their daily lives, but also but be resilient to safety and security issues mentioned in the literature, is not necessarily to address all potential dangerous scenarios. From the description of the app created through a participatory design within an intergenerational team, where older adults, were not only final users, but also active team members, one can see that the focal point for addressing issues of safety, both on-line and in real life, is modeling of trust within the user base of the application. Use of such elements like a reputation system, two step password verification between users who decided to exchange favors, as well as external confirmation of users provides a wide array of possibilities for encouraging the development of trust between older adults and younger volunteers. This corroborates the general ramifications of the intergroup contact theory, however it extends its scope beyond intergroup stereotyping into the field of limiting the feeling of insecurity among the group of older adults.
It is also important to notice that the participatory approach utilized by the team, allowed to overcome the aforementioned issue of understanding the link between real life safety and on-line security. In light of the results from the the interviews conducted within the framework of the LivingLab, where older adults had difficulties with connecting the two aspects of safety, the outcome of the team design process yields great promise.
The case study presented here, of course, serves mostly as a jumping-off point for further considerations in a topic that, although important, is not, in the authors opinion, amply researched. The findings made in this paper show, that by use of a participatory design framework, it is possible not only to address general issues of on-line security, but also, to create new methods of thinking about user safety outside of the paradigm of software engineering. With the increasing role of mobile technologies in real life situation, these initial exploratory findings offer interesting option for improving the initial design process of mobile applications.
In their future work, we plan to further explore this interesting emerging field of study. With the observation made in this paper being mostly exploratory in nature, it seems fitting to conduct a set of more methodologically rigid tests with the aim of verifying, what form of participatory design can further improve the process of addressing the threat on-line and real life threats faced by end users of an application.
\section{Acknowledgments.}
This project has received funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement No 690962.
\bibliographystyle{abbrv}
|
2,877,628,089,013 | arxiv | \section{Introduction}
The notion of \emph{centrality} aims at capturing the importance of a node in a network. This concept arises and finds application in many fields; for example, it selects the nodes in a network that have more chances to lead to cascade effects if hit by a shock (\cite{Ballester06}), or it identifies the nodes that have more influence in the opinion formation and diffusion in a social network (\cite{Kempe15}), in order to possibly perform optimal targeting interventions (\cite{Galeotti09}, \cite{Galeotti17}).
In the literature different definitions of centrality can be found, such as the \emph{degree} centrality or the \emph{eigenvalue} centrality (see for references \cite{latora17}, Section 2.3); in this paper we focus on the so-called \emph{Bonacich} centrality measure, introduced in a seminal paper by the American sociologist \cite{PB:87}.
Formally, the Bonacich centrality $ \pi_i $ of a node $ i $ in a directed unweighted network is defined as
\begin{equation}\label{centrality}
\pi_i=\beta\sum\limits_{j\in N_i^{-}}\frac{\pi_j}{d_j}+(1-\beta)\eta_i\, ,
\end{equation}
where $N_i^{-}$ is the in-neighborhood of node $i$ in the network, $d_j$ is the out-degree of node $ j $, $\eta_i$ can be interpreted as the a-priori centrality of $ i $ (possibly the same for all nodes), and $\beta\in (0,1)$ is some fixed parameter. Notice that by (\ref{centrality}), the centrality of node $ i $ depends on the centrality of the nodes $ j $ linking at $ i $ (discounted by the number of their out-links) and on its intrinsic centrality. The centrality of a node is then somewhat inherited by the nodes connected to it: a node is important in the measure that important nodes have a link to it.
The Bonacich centrality have found wide applications in many contexts, as in social networks (e.g.\ representing citations among scientists), in describing Nash equilibria in networked quadratic games (\cite{Ballester06}), in production networks among firms (\cite{Acemoglu12}), and in opinion dynamics models as the Friedkin-Johnsen model (\cite{NEF-ECJ:90}).
A famous instance of the Bonacich centrality is the so-called \emph{PageRank centrality} for web pages, introduced by \cite{SB-LP:98}, which is at the core of modern search engines like Google. Any search query on the web leads indeed to a set of possible related web pages that are sorted and presented according to their centrality ranking by the engine.
Due to the relevance of the PageRank centrality for the external visibility of a web page, the problem of understanding how this measure can be efficiently computed and how it can be modified by perturbing the network has recently become very popular; see for example \cite{Ishii.Tempo:2014}, \cite{Como.Fagnani:2015}. The effect on the centrality caused by adding or deleting links in the network is not obvious from the recursive definition (\ref{centrality}). It is not difficult to see that the addition of a link $ (i,j) $ always increases the centrality of the node $ j $; less clear is how it affects the centrality of node $ i $ or, possibly, of all the other nodes in the network. In a context like that of web pages, where each node can decide only where to point its out-links and the aim is to gain visibility (that is, to increase its centrality in the network), the question of how such choice modifies its centrality and what is the rewiring that can possibly optimize it, turns out to be a natural relevant question.
A first analysis in this sense can be found in \cite{Avrachenkov06} and \cite{dekerchove08}, while \cite{Jungers10} explore computational time issues of these problems.
In this paper, we take this point of view by assuming that nodes are left free to choose their out-links and we cast the problem into a game-theoretic setting where rewards of nodes are exactly their centralities. We investigate the shapes that the network assumes when maximazing the centrality is the only driving force: we study the Nash equilibria of our game, i.e. configurations of the network in which every node is playing its optimal action, and the behavior of the best response dynamics, i.e. a discrete dynamics in which, at every time step, a random player plays an optimal action
(see Section 3 for formal definitions).
We can see our problem as an instance of a \emph{network formation game}, where the actions of the players (the nodes of the network) are the ones defining the underlying network structure; we refer the reader to \cite{Jackson05} for a survey on network formation games and their applications in economy and sociology.
More in detail, we study the problem under the assumption that all nodes are allowed to place the same number $m$ of out-links. We obtain a complete classification of the Nash equilibria in the case $m=1$, and a fairly complete classification of Nash equilibria in the case $ m=2 $. Namely, we provide necessary conditions for a configuration to be a Nash equilibrium and a complete classification of strict Nash equilibria and Nash equilibria to which converges the best response dynamics (see Section 3 for formal definitions). The main message that comes from this analysis is that the centrality maximization performed by each node tends to create undirected and disconnected or loosely connected networks: the components are $2$-cliques for $m=1$, rings and a special \emph{Butterfly} graph for $m=2$.
While completing this research, we discovered that a similar game-theoretic formulation was considered in \cite{scarsini}, Section 7, where authors prove the existence of Nash equilibria for a generalized version of our game. While \cite{scarsini} just prove the existence of Nash equilibria and show few examples, in this work we provide an almost complete characterization of Nash equilibria, which is independent and, we believe, cannot be derived from their results.
The paper is structured as follows. In Section \ref{model} we present the game theoretical setting; Section \ref{pre} recalls classical results and definitions of game theory, while Section \ref{main} describes the main results of the paper. All technical results and proofs are in Section \ref{sec:proofs}. Section \ref{conclusions} concludes with summary and some open problems.
\section{The model}\label{model}
In this section, we formally define the centrality maximization game and we state the problems we want to address.
Consider a directed graph $\mathcal G=(\mathcal V,\mathcal E)$ where $\mathcal V=\{1,\dots ,n\}$ is the set of nodes and $\mathcal E\subseteq \mathcal V\times\mathcal V$ is the set of (directed) edges. We denote by $(i,j) \in \mathcal E$ a directed edge from node $i$ to node $j$. We assume throughout the paper that $\mathcal G$ does not contain self-loops. In- and out- neighborhoods of a node $i$ are indicated, respectively, by $N_i^-$ and $N_i$. Their cardinalities $d_i^-=|N_i^-|$ and $d_i= |N_i|$ are, respectively, the in- and the out-degree of node $i$. Under the assumption that $d_i>0$ for every $i\in\mathcal V$, we equip $\mathcal G$ with the normalized weight matrix $R$ whose entries $R_{ij}$ are defined as
\[ R_{ij}=\frac{1}{d_i}\mathbbm{1}_{\lbrace(i,j)\in\mathcal E\rbrace},\]
where $ \mathbbm{1} $ is the characteristic function. The entry $R_{ij}$ represents the weight attributed to the link $(i,j)$.
The Bonacich centrality $ \pi=(\pi_1,\dots ,\pi_n) $ of $\mathcal G$ in Eq.\ (\ref{centrality}) can be more compactly written as
\begin{equation}\label{Bonacich} \pi=(1-\beta)(I-\beta R^\top)^{-1}\eta\end{equation}
where $ I $ is the identity matrix, $\beta\in (0,1)$, $\eta\in\mathbb{R}^n$ is a fixed probability vector\footnote{$ v $ is
a \emph{probability} vector if $ \sum_iv_i=1 $ and $ v_i\geq 0 $ for all $ i $.} and $ R^\top $ denotes the transpose of the matrix $ R $. A direct check shows that $\pi$ is a probability vector.
Expanding (\ref{Bonacich}) in a power series, we can write the Bonacich centrality of node $i$ as
\begin{equation}\label{Bonacich2}\pi_i\!=\!(1-\beta)\!\left[\eta_i+\beta \sum_j\eta_jR_{ji}+\beta^2\sum_{j,l}\eta_jR_{jl}R_{li}+\cdots \right].\end{equation}
Interpreting $\eta$ as a vector assigning an a-priori centrality (not depending on the graph) to each node (possibly the uniform one $\eta_i=n^{-1}$ for all $ i $), formula $(\ref{Bonacich2})$ says that the Bonacich centrality of a node in the graph $\mathcal G$ is the discounted sum of its own centrality $\eta_i$ and of the centrality of the other nodes discounted by the weight of the paths connecting to $i$ through the constant $\beta$. Notice that the constant $(1-\beta)$ appears just to normalize $\pi$ to a probability vector.
In our setting, we start with the set of nodes $\mathcal V=\{1,\dots ,n\}$ and we suppose that each node $i$ is a player that assigns $m$ directed edges from $i$ to $m$ other distinct elements in $\mathcal V$. This construction results in a graph $\mathcal G$ and the Bonacich centrality of node $i$ in $ \mathcal G $ represents its utility. This can be thought as a classical game where
\begin{itemize}
\item $\mathcal V$ is the set of players;
\item given $i\in\mathcal V$, the corresponding set of actions $\mathcal A_i$ is the family of all subsets of $\mathcal V\setminus\{i\}$ of cardinality $m$;
\item let $ \mathcal{A}=\prod_i\mathcal A_i $ and $ x=(x_1,\dots ,x_n)\in \mathcal{A}$ a strategy profile (or \emph{configuration}).
We define the graph $\mathcal G(x)=(\mathcal V, \mathcal E(x))$ where
$\mathcal E(x)=\{(i,j)\;|\; i\in\mathcal V,\; j\in x_i\}$.
Notice that by construction $\mathcal G(x) $ has constant out-degree equal to $ m $.
We denote by $R(x)$ the normalized weight matrix of $\mathcal G(x)$\footnote{That is, $ R_{ij}(x)=m^{-1} $ if $ (i,j)\in\mathcal E(x) $, $ R_{ij}(x)=0 $ otherwise.}. Given $\beta\in (0,1)$ and $\eta\in\mathbb{R}^n$ a probability vector such that $ \eta_i>0 $ for all $ i $, we
define the utility vector $ u(x)=(u_1(x),\dots ,u_n(x)) $ as the Bonacich centrality of $\mathcal G(x)$:
$$u(x)=(1-\beta)(I-\beta R(x)^\top)^{-1}\eta.$$
\end{itemize}
The game we have introduced is denoted by $ \Gamma(\mathcal V,\beta,\eta, m) $ to recall all the parameters entering in the construction.
The main goal of this paper is to analyze the structure of Nash equilibria for the game $ \Gamma(\mathcal V,\beta,\eta, m) $ and to investigate the asymptotic behavior of its best response dynamics, which is defined in the next section.
The game is homogeneous in the sense that we give every node the chance to place the same number $m$ of out-links in the network. A natural generalization of this problem would be to consider a different number $ m_i $ of out-links for each node $ i $; we leave this to future work.
\section{Preliminaries}\label{pre}
In this section we recall some fundamental definitions and classical results in game theory that will be used in the next sections.
Given $x\in\mathcal A$ and $i\in\mathcal V$, we adopt the usual convention to indicate with $x_{-i}\in\mathcal A_{-i}=\prod_{k\neq i}\mathcal A_k$ the vector $x$ restricted to the components in $\mathcal V\setminus\{i\}$ and to use the notation $x=(x_i, x_{-i})$.
\begin{defn}\label{defn:bestResp}
Let $ i\in \mathcal V $ and $x_{-i}\in\mathcal A_{-i}$. We define the \emph{best response set} $\mathcal{B}_i(x_{-i})$ of node $ i $ given the strategy $ x_{-i} $ as
$$\mathcal{B}_i(x_{-i})= \text{argmax}_{x_i\in\mathcal{A}_i}u_i(x_i, x_{-i}).$$
\end{defn}
The best response set represents the set of actions of player $ i $ that maximize his utility $ u_i $, given the strategy $ x_{-i} $ played by all the other players.
We now recall the definition of (strict) Nash Equilibria and best response dynamics.
\begin{defn}\label{defn:nashEquilibrium}
Let $x\in \mathcal{A}$ be a strategy profile. If for all $i\in\mathcal V$, $x_i\in \mathcal{B}_i(x_{-i})$, then $x$ a \textit{Nash equilibrium}.
If for all $i\in\mathcal V$, $\mathcal{B}_i(x_{-i})=\lbrace x_i \rbrace$, then $x$ a \emph{strict Nash equilibrium}. We denote by $\mathcal N$ and $\mathcal N^{\text{st}}$ the set of, respectively, Nash equilibria and strict Nash equilibria.
\end{defn}
\begin{defn}\label{defn:bestRespDynamics}
The \emph{(asynchronous) best response dynamics} is a discrete time dynamics $ Y_t $ on the state space $\mathcal A$ in which at every time $t\in\mathbb{N}$, a player $i$ is chosen uniformly at random and he revises his action by picking an element $y$ in $\mathcal B_i\bigl((Y_{t-1})_{-i}\bigr)$ uniformly at random.
\end{defn}
A classical result of \cite{Monderer} states that if a game is ordinal potential\footnote{A game is ordinal potential if there exists a function $ \Psi:\mathcal{A}\to \mathbb{R} $ s.t.\ $u_i(x_i,x_{-i})<u_i(x'_i,x_{-i}) \Leftrightarrow \Psi(x_i,x_{-i})<\Psi(x'_i,x_{-i}) $.}, then its best response dynamics converges in finite time with probability one to (a subset of) Nash equilibria, independently on the initial condition.
\cite{scarsini} (Proposition 7.5 and Section 7.2) proved that our game is ordinal potential, which let us formulate the following result:
\begin{prop}\label{prop:Cominetti}
The best response dynamics on $ \Gamma(\mathcal V,\beta,\eta, m) $ always converges in finite time with probability one to a set $ \mathcal N^* \subseteq \mathcal N$ of Nash equilibria.
\end{prop}
Typically $\mathcal N^*$ is a proper subset of $\mathcal N$. Moreover, as strict Nash equilibria are absorbing points of the best response dynamics, it holds that $\mathcal N^{\text{st}}\subseteq \mathcal N^*$; however, in general they are not equal.
If we consider the transition graph on the configuration set $\mathcal A$ induced by the best response dynamics $ Y_t $,
the set $\mathcal N^*$ can be described as its smallest trapping set (no edge leading out of $\mathcal N^*$) that is globally reachable (from every configuration in $\mathcal A$ there is a path leading inside $\mathcal N^*$).
Nash equilibria in $\mathcal N^*$ play a crucial role in games as they are those the best response dynamics will eventually converge to, while Nash equilibria in $\mathcal N\setminus\mathcal N^*$ will only show up in the transient behavior.
Our aim is to investigate the structure of these three sets $\mathcal N^{\text{st}}\subseteq \mathcal N^*\subseteq \mathcal N$ for the game $ \Gamma(\mathcal V,\beta,\eta, m) $ that we have introduced in the previous section.
\section{Main results}\label{main}
In this paper we focus on the case when $m=1$ and $m=2$, namely when nodes are allowed to set, respectively, one or two out-links towards other nodes. Through a characterization of the best response set $\mathcal{B}_i(x_{-i})$, we are capable of giving a full description of the three sets $\mathcal N^{\text{st}}$, $\mathcal N^*$ and $\mathcal N$ of Nash equilibria for $ m\!=\!1 $, and a full description of $\mathcal N^{\text{st}}$ and $\mathcal N^*$ for $m\!=\!2$, together with a necessary condition for $ \mathcal N $. The case $ m\!=\!2 $ presents a much more complex behavior and, for certain aspects, as complex as the general case.
\subsection{The case of out-degree $m=1$}
In order to describe our results, it is convenient to introduce a particular family of graphs
\begin{defn}
We call a \emph{$2$-clique} the complete directed graph (without self-loops) with two nodes and we indicate it by $C_2$; we call a \textit{singleton} a node with zero in-degree. Given $l, r\in \mathbb{N}$, we define $C_2^{l,r}$ as the family of directed graph obtained by taking the disjoint union of $l$ copies of $C_2$ plus $r$ extra singletons, each of them having exactly one out-link towards a node in any of the $2$-cliques.
\end{defn}
Notice that $C_2^{l,r}$ has exactly $n=2l+r$ nodes and all nodes have out-degree equal to one. Figure \ref{fig:C1k} is an example of graph of type $C_2^{l,r}$ for $ l=3 $ and $ r=6 $.
The following theorem is our first main result for the case $m=1$.
\begin{figure}
\begin{center}
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\end{center}
\caption{An example of a graph of type $ C_2^{3,6} $.}
\label{fig:C1k}
\end{figure}
\begin{figure}
\begin{center}
(a)
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$ \qquad \quad$
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\end{center}
\normalsize
\caption{(a) A graph of type $ C_2^{n/2,0} $ with $n=8$; (b) A graph of type $ C_2^{(n-1)/2,1} $ with $ n=7 $.}
\label{fig:C1n-1/2}
\end{figure}
\begin{thm}\label{thm:nash_m=1} For any choice of $\beta$ and $\eta$, the game $ \Gamma(\mathcal V,\beta,\eta, 1) $ has the following properties:
\begin{enumerate}
\item the set of Nash equilibria $ \mathcal N $ coincides with all the configurations $x\in\mathcal A$ for which $\mathcal G(x)$ is of type $C_2^{l,r}$ with $2l+r=n$;
\item the set of strict Nash equilibria $\mathcal N^{st}$ is empty when $n$ is odd and it coincides with all the configurations $x\in\mathcal A$ for which $\mathcal G(x)$ is of type $C_2^{n/2,0}$ when $n$ is even.
\end{enumerate}
\end{thm}
Figure \ref{fig:C1n-1/2}(a) represents a strict Nash equilibrium for $ \Gamma(\mathcal V,\beta,\eta, 1) $ with $n\!=\!8$, while Fig.\ \ref{fig:C1n-1/2}(b) shows a nonstrict Nash equilibrium for $n\!=\!7$. The following corollary completely captures the asymptotic behavior of the best response dynamics of $ \Gamma(\mathcal V,\beta,\eta, 1) $; in particular it shows that the Nash equilibrium of Fig.\ \ref{fig:C1n-1/2}(b) belongs to $ \mathcal{N}^* $.
\begin{cor}\label{cor:trapping_sets}
Consider the best response dynamics for the game $ \Gamma(\mathcal V,\beta, \eta,1) $. For any choice of $\beta$ and $\eta$, it holds that:
\begin{itemize}
\item if $ n$ is even, the limit set $\mathcal N^*$ coincides with $\mathcal N^{\text{st}}$, namely it consists of those $x\in\mathcal A$ for which $\mathcal G(x)$ is of type $C_2^{n/2,0}$;
\item if $ n$ is odd, the limit set $\mathcal N^*$ coincides with those $x\in\mathcal A$ for which $\mathcal G(x)$ is of type $C_2^{(n-1)/2,1}$.
\end{itemize}
\end{cor}
Notice that when $n=2k$, the best response dynamics will eventually be absorbed in any of the $|\mathcal N^*|=n!2^{-k}(k!)^{-1}$ strict Nash equilibria with probability one. On the other hand, when $n=2k+1$ the best response dynamics will eventually reach the (unique) trapping set consisting of $|\mathcal N^*|=(n-1)n!2^{-k}(k!)^{-1}$ configurations of type $C_2^{(n-1)/2,1}$. In this case, it can be shown that the best response dynamics will keep fluctuating ergodically in the set $\mathcal N^*$ with uniform equilibrium probability.
\subsection{The case of out-degree $m=2$}
We call \emph{ring} graph an undirected graph whose vertices are arranged in a ring so that each vertex has exactly two neighbors (see for example Fig.\ \ref{fig:C2_nash}(a), where each connected component is a ring graph). The \emph{length} of a ring graph is the number of its vertices.
From now on we say that an edge $ (i,j) $ in $ \mathcal G $ is \emph{undirected} if also $ (j,i) $ is an edge of $ \mathcal G $, otherwise we call it \emph{directed}. We say that a graph is \emph{undirected} if all its edges are undirected. In figures, we represent directed edges with arrows and undirected edges with simple lines.
The first main result of this section is a complete characterization of the set of strict Nash equilibria.
\begin{thm}\label{thm:strictNash_m2}
For any choice of $\beta$ and $\eta$, the set of strict Nash equilibria $ \mathcal N^{st} $ of the game $ \Gamma(\mathcal V,\beta,\eta, 2) $ consists of all the configurations $x\in\mathcal A$ for which $\mathcal G(x)$ is the union of ring graphs.
\end{thm}
A consequence of this fact is that for any $n\geq 3$ there always exists a strict Nash equilibrium, as the ring graph of length $ n $ is always one of these.
Figure \ref{fig:C2_nash}(a) provides an example of strict Nash equilibrium with $ n=9 $.
\begin{figure}
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(a)
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(q_6) edge node {} (q_9)
(q_4) edge node {} (q_5)
;
\end{tikzpicture}
$ \qquad $(b)
\begin{tikzpicture}[roundnode/.style={circle,fill=black!30, inner sep=2pt, minimum size=3mm},node distance=0.7cm,line width=0.2mm,scale=0.6]
\node[roundnode] (q_1) at (0,0) {};
\node[roundnode] (q_2) at (0,-2) {};
\node[roundnode,circle,draw,fill=white] (q_3) at (2,-1) {};
\node[roundnode] (q_4) at (4,0) {};
\node[roundnode] (q_5) at (4,-2) {};
\path[->] (q_5) edge node {} (q_3)
(q_1) edge node {} (q_3)
;
\path[-]
(q_3) edge node {} (q_2)
(q_1) edge node {} (q_2)
(q_4) edge node {} (q_3)
(q_4) edge node {} (q_5)
;
\end{tikzpicture}
\end{center}
\caption{(a) Example of strict Nash equilibrium for the game $ \Gamma(\mathcal V,\beta,\eta, 2) $ with $ n=9 $. (b) The Butterfly graph. White nodes do not have unique best response.}
\label{fig:C2_nash}
\end{figure}
We now investigate the structure of all Nash equilibria.
Given a Nash equilibrium $x\in \mathcal A$, let $\lbrace \mathcal G_\lambda(x)\rbrace_{\lambda=1,\dots , \Lambda}$ be the decomposition of $\mathcal G(x)$ in terms of its strongly connected components. The \emph{condensation graph} of $\mathcal G(x)$ is defined as the graph $\mathcal H(x)$ whose nodes are the components $\lbrace \mathcal G_\lambda(x)\rbrace_{\lambda}$ and where there is an edge from $\mathcal G_{\lambda_1}(x)$ to $ \mathcal G_{\lambda_2}(x)$ if there exists an edge in $\mathcal G(x)$ from a node in $\mathcal G_{\lambda_1}(x)$ to a node in $\mathcal G_{\lambda_2}(x)$. The condensation graph $\mathcal H(x)$ is directed and acyclic. The following theorem describes the topology of $\mathcal H(x)$ when $ x\in\mathcal N $, thus characterizing the structure of the Nash equilibria of the game $ \Gamma(\mathcal V,\beta,\eta, 2) $. We remind that a vertex is called a \emph{sink} if it has zero out-degree and it is called a \emph{source} if it has zero in-degree.
\begin{thm}\label{thm:condensation_graph}
Let $x\in\mathcal A$ be a Nash equilibrium for the game $ \Gamma(\mathcal V,\beta,\eta, 2) $ and $\mathcal H(x)$ be its condensation graph on the components $\lbrace \mathcal G_\lambda(x)\rbrace_{\lambda}$. For any choice of $\beta$ and $\eta$, the following facts hold:
\begin{enumerate}
\item every component $\mathcal G_\lambda(x)$ is either a sink or a source in $\mathcal H(x)$ (or both if isolated);
\item every source component is either a single vertex (singleton) or a $2$-clique;
\item every sink component is either a ring or the Butterfly graph in Fig. \ref{fig:C2_nash}(b).
\end{enumerate}
\end{thm}
Notice that the Butterfly graph is a nonstrict Nash equilibrium as the best response of the node in the center is not unique, i.e.\ it can change action while maintaining the same utility. Figure \ref{fig:C3_nonStrictNash3} provides other two examples of nonstrict Nash equilibria: in both structures we can identify either a singleton or a 2-clique linking to rings; the nodes in white have not unique best response.
\begin{figure}
\begin{center}
(a)
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\node[roundnode] (q_20) at (3.75,-0.325){};
\node[roundnode] (q_30) at (3.75,0.325){};
\node[roundnode,circle,draw,fill=white, inner sep=2pt, minimum size=3mm ] (q_4) at (1.4,0.5){};
\node[roundnode,circle,draw,fill=white, inner sep=2pt, minimum size=3mm ](q_5) at (2.25,0.5){};
\path[-]
(q_3) edge node {} (q_2)
(q_1) edge node {} (q_3)
(q_1) edge node {} (q_2)
;
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(q_10) edge node {} (q_20)
(q_20) edge node {} (q_30)
(q_10) edge node {} (q_30)
(q_4) edge node {} (q_5)
;
\path[->]
(q_4) edge node {} (q_2)
(q_5) edge node {} (q_10)
;
\end{tikzpicture}
(b)
\begin{tikzpicture}[roundnode/.style={circle,fill=black!30, inner sep=1pt, minimum size=3mm},node distance=1cm,line width=0.2mm, ]
\node[roundnode] (q_1) at (0,-0.325){};
\node[roundnode] (q_2) at (0.75,0){};
\node[roundnode] (q_3) at (0,0.325){};
\node[roundnode][roundnode,circle,draw,fill=white, inner sep=2pt, minimum size=3mm ] (q_4) at (1.5,0.6){};
\node[roundnode] (q_10) at (2.25,0){};
\node[roundnode] (q_20) at (3,-0.325){};
\node[roundnode] (q_30) at (3,0.325){};
\path[-]
(q_3) edge node {} (q_2)
(q_1) edge node {} (q_2)
(q_3) edge node {} (q_1)
;
\path[-]
(q_10) edge node {} (q_20)
(q_20) edge node {} (q_30)
(q_30) edge node {} (q_10)
;
\path[->]
(q_4) edge [] node {} (q_10)
(q_4) edge [] node {} (q_2)
;
\end{tikzpicture}
\end{center}
\caption{Examples of nonstrict Nash equilibria for $ \Gamma(\mathcal V,\beta,\eta, 2) $. White nodes do not have unique best response.}
\label{fig:C3_nonStrictNash3}
\end{figure}
\begin{rem}
\label{remark_m2}
Not all the configurations $x\in \mathcal A$ that satisfy conditions (1), (2) and (3) of Theorem \ref{thm:condensation_graph} are Nash equilibria. Indeed, by direct computation it is easy to see that the following examples are not Nash equilibria:
\begin{enumerate}
\item a singleton linking to two adjacent nodes in a ring of length greater or equal than four (see Fig.\ \ref{fig:singleton_ring}(a));
\item a 2-clique linking to a single node in a ring of length greater or equal than four (see Fig.\ \ref{fig:singleton_ring}(b)).
\end{enumerate}
\end{rem}
\begin{figure}
\begin{center}
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\node[roundnode] (q_4) at (esa.corner 4) {n-2};
\node[roundnode] (q_5) at (esa.corner 5) {n-3};
\node[roundnode] (q_6) at (esa.corner 6) {$ \dots $};
\node[roundnode] (q_7) [above left=of q_3] {j};
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(q_7) edge node {} (q_2)
;
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(q_2) edge node {} (q_1)
(q_2) edge node {} (q_3)
(q_3) edge node {} (q_2)
(q_3) edge node {} (q_4)
(q_4) edge node {} (q_3)
(q_4) edge node {} (q_5)
(q_5) edge node {} (q_4)
(q_5) edge [dotted] node {} (q_6)
(q_1) edge [dotted] node {} (q_6)
;
\node at (-3.8,-1.5) {(a)};
\end{tikzpicture}
$ \,\,\, $
\begin{tikzpicture}[roundnode/.style={circle,fill=black!30, inner sep=0.5pt, minimum size=6mm},node distance=1cm,line width=0.2mm,scale=0.5]
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\node[roundnode,fill=black] (q_3) at (esa.corner 3) {\textcolor{white}{s}};
\node[roundnode] (q_4) at (esa.corner 4) {n-3};
\node[roundnode] (q_5) at (esa.corner 5) {n-4};
\node[roundnode] (q_6) at (esa.corner 6) {$ \dots $};
\node[roundnode] (q_7) [left=of q_3] {j};
\node[roundnode] (q_8) [above left=of q_3] {k};
\path[->] (q_7) edge node {} (q_3)
(q_8) edge node {} (q_3)
;
\path[-] (q_1) edge node {} (q_2)
(q_7) edge node {} (q_8)
(q_2) edge node {} (q_1)
(q_2) edge node {} (q_3)
(q_3) edge node {} (q_2)
(q_3) edge node {} (q_4)
(q_4) edge node {} (q_3)
(q_4) edge node {} (q_5)
(q_5) edge node {} (q_4)
(q_5) edge [dotted] node {} (q_6)
(q_1) edge [dotted] node {} (q_6)
;
\node at (-4.5,-1.5) {(b)};
\end{tikzpicture}
\end{center}
\caption{(a) Singleton linking to two adjacent nodes in a ring. (b) 2-clique linking to a single node in a ring. Black nodes are not in best response.}
\label{fig:singleton_ring}
\end{figure}
We are now ready to characterize the limit set $ \mathcal N^* \subseteq \mathcal N$ for the game $ \Gamma(\mathcal V,\beta, \eta,2) $, i.e.\ the absorbing points of its best response dynamics.
\begin{cor}\label{cor:trapping_setsM2}
Consider the game $ \Gamma(\mathcal V,\beta, \eta,2) $ and let $ i$ s.t.\ $ i=n\! \mod 3 $. Then for any choice of $\beta$ and $\eta$, it holds that:
\begin{itemize}
\item if $ i= 0,1 $, the limit set $\mathcal N^*$ coincides with $\mathcal N^{\text{st}}$
\item if $ i= 2$, the limit set $\mathcal N^*$ coincides with $\mathcal N^{\text{st}} \cup \mathcal G^3_b $, where $\mathcal G^3_b$ is the set of all graphs that are unions of rings of length three and a Butterfly graph or unions of rings of length three and a 2-clique linking to any nodes in the rings (see e.g.\ Fig.\ref{fig:C3_nonStrictNash3}(a), Fig.\ref{fig:trasitionButterfly}(b), (c)).
\end{itemize}
\end{cor}
Figure \ref{fig:dynamics} shows the convergence of the best response dynamics starting from the same initial configuration to two different equilibria, namely a strict Nash equilibrium (union of rings) and a nonstrict Nash equilibrium in $ \mathcal G_b^3 $ (union of rings of length three and a Butterfly graph). The simulations have been done using suitable \textsc{Matlab}\xspace routines.
\begin{figure}
\centering
\includegraphics[width=0.25\textwidth]{initial.png}
\\
(a)\includegraphics[width=0.2\textwidth]{strictNash.png}$\quad $(b)\includegraphics[width=0.2\textwidth]{nonStrict.png}
\caption{Convergence of the best response dynamics starting from the same initial configuration to (a) a strict Nash equibrium (b) a nonstrict Nash equilibrium.}
\label{fig:dynamics}
\end{figure}
\section{Proofs of the results}\label{sec:proofs}
The proofs of our results are based on a probabilistic interpretation of the game in terms of Markov chains. We first recall some preliminary notions on Markov chains and we apply them to our game. Then in Subsections \ref{subsec:onelink_proofs} and \ref{subsec:twolink_proofs} we prove the results presented in the previous section respectively for the case $ m=1 $ and $ m=2 $.
A (discrete-time) Markov chain $ X_t $ on a finite state space $ \mathcal V=\lbrace 1,\dots ,n\rbrace $ and with transition matrix $ P\in\mathbb{R}^{n\times n} $, $ P $ stochastic\footnote{A matrix $ P $ is stochastic if each row is a probability vector.}, is a sequence of random variables $ X_1,X_2,\dots $ with values in $ \mathcal V$ such that $ \mathbb{P}(X_{t+1}=i|X_1=j_1,\dots,X_t=j_t)=\mathbb{P}(X_{t+1}=i|X_{t}=j_t)=P_{j_ti} $. Given $s\in \mathcal V $, we define $ T_s:=\inf\lbrace t\geq 0: X_t=s \rbrace $ the \textit{hitting} time on $ s $ and $ T^+_s:=\inf\lbrace t\geq 1: X_t=s \rbrace $ the \textit{return} time to $ s $. Given $i,s\in \mathcal V $, we define $\tau_{i}^s:=\mathbb{E}_i[T_s] $ the \textit{expected} hitting time on $ s $ of the Markov chain $ X_t $ with initial state $ i $. It is known that if $ P$ is an irreducible matrix, then the Markov chain admits a unique invariant distribution, that is a probability vector $ \pi $ s.t.\ $ \pi=P^{\top}\pi $. The invariant distribution $ \pi $ can be written in terms of hitting times:
\begin{prop}\label{prop:hitting_times}
Let $ X_t $ be a Markov chain with finite state space $ \mathcal V $ and irreducible transition matrix $ P $, and let $ \pi $ be its (unique) invariant distribution. Then it holds that
\begin{equation}\label{eq:pi}
\pi_s=\left( 1+\sum_{i\in \mathcal V}P_{si}\tau_i^s\right)^{-1},
\end{equation}
where the expected hitting times $\tau_i^s $, $ i\!\in\! \mathcal V $, are the only family of values satisfying the following system:
\begin{equation}\label{eq:system_tau}
\begin{cases}
\tau_i^s=0 &\text{if }i=s,\\
\tau_i^s=1+\sum_{j\in \mathcal V}P_{ij}\tau_j^s &\text{if }i\neq s.
\end{cases}
\end{equation}
\end{prop}
\begin{pf}
Equation (\ref{eq:pi}) comes from the fact that $ \pi_s=(\mathbb{E}_s[T^+_s])^{-1}$ and $ \mathbb{E}_s[T^+_s]=1+\sum_{i\in \mathcal V}P_{si}\tau_i^s $, which are both standard results on Markov chains, as well as (\ref{eq:system_tau}). See for example \cite{norris_1997}.$ \qed $
\end{pf}
Manipulating (\ref{Bonacich}) and using the fact that $\mathbf{1}^\top\pi=1$ with $ \mathbf{1} $ the all-ones vector, we can see that the Bonacich centrality $\pi$ satisfies the relation
$$\pi=(\beta R^\top+(1-\beta) \eta\mathbf{1}^\top)\pi.$$ Since $P=\beta R+(1-\beta)\mathbf{1}\eta^\top$ is an irreducible stochastic matrix, it means that $\pi$ is the (unique) invariant distribution of the Markov chain having $ P $ as transition matrix.
We now use this characterization in the context of our game. Given a configuration $x\in\mathcal A$, we write
\begin{equation}\label{eq:P(x)}
P(x)=\beta R(x)+(1-\beta)\mathbf{1}\eta^\top
\end{equation}
and we denote by $\tau_i^s(x)$ the hitting time on $ s $ of the Markov chain having $P(x)$ as transition matrix and starting from $ i $. When the configuration $ x $ is clear from the context, sometimes we write $\tau_i^s$ instead of $\tau_i^s(x)$ to ease the notation.
The utility vector $u(x)$ can be written in terms of the formula (\ref{eq:pi}) as
$u_s(x_s,x_{-s})=\left(1+\sum_{i\in \mathcal V}P_{si}(x)\tau_i^s(x)\right)^{-1}$.
Since
the terms $P_{si}(x)$ only depend on $ x_s $ (the out-links from $s$), while the hitting times $\tau_i^s(x)$ only depend on $x_{-s}$, with slight abuse of notation we rewrite the utility function as
\begin{equation}\label{eq:best-times}
u_s(x_s,x_{-s})=\left(1+\sum_{i\in \mathcal V}P_{si}(x_{s})\tau_i^s(x_{-s})\right)^{-1}.
\end{equation}
A consequence of (\ref{eq:best-times}) is an explicit formula describing the best response set, as shown by the following remark.
\begin{rem}
\label{cor:best_response}
Consider the game $ \Gamma(\mathcal V,\beta,\eta, m) $, a node $ s\in \mathcal V $ and $ x_{-s}\in \mathcal{A}_{-s} $. Then the best response set $ \mathcal{B}_s(x_{-s}) $ of player $ s $ when all the other players are playing the actions $ x_{-s} $ can be written as:
\begin{equation}\label{eq:best_response}
\mathcal{B}_s(x_{-s})=\underset{x_s\in\mathcal{A}_s}{ \text{argmin}}\sum_{i\in \mathcal V}R_{si}(x_s)\tau_i^s(x_{-s}).
\end{equation}
\end{rem}
In the following, given $ x\in \mathcal A $ we denote by $ N_s^{-}(x) $ the in-neighborhood of the vertex $ s $ in the graph $ \mathcal{G}(x) $, that is $ i\in N_s^{-}(x)$ if and only if $ s\in x_i $ (or equivalently, if and only if $ R_{is}(x)>0 $). Notice that $ N_s^{-}(x) $ depends just on $ x_{-s} $ so with a slight abuse of notation we can write $ N_s^{-}(x_{-s}) $.
\subsection{The case of out-degree $ m=1 $}\label{subsec:onelink_proofs}
In order to prove Theorem \ref{thm:nash_m=1}, we need to better characterize the best response actions of a player. The first important observation is the following:
\begin{rem}\label{rem:best_response_m1}
If $ m=1 $, then for any $ s\in\mathcal V $ and $ x_s\in\mathcal{A}_s $ it holds that $ R_{sx_s}(x_s)=1 $ and $ R_{si}(x_s)=0 $ for all $ i\neq x_s $. Therefore (\ref{eq:best_response}) takes the form: \[ \mathcal{B}_s(x_{-s})=\text{argmin}_{i\in \mathcal V\setminus\lbrace s\rbrace}\tau_i^s(x_{-s}).\]
\end{rem}
The following proposition shows that the best response action of a player in the game $ \Gamma(\mathcal V,\beta,\eta, 1) $ takes always place in his in-neighborhood, as long as it is nonempty.
\begin{prop}\label{prop:best_response}
Consider the game $ \Gamma(\mathcal V,\beta,\eta, 1) $ and let $ s\in \mathcal V $ and $ x_{-s}\in \mathcal{A}_{-s} $. It holds that:
\begin{enumerate}
\item If $ N_s^{-}(x_{-s})\neq \emptyset $, then $\mathcal{B}_s(x_{-s})= N_s^{-}(x_{-s}) $;
\item If $ N_s^{-}(x_{-s})= \emptyset $, then $ \mathcal{B}_s(x_{-s})=\mathcal V \setminus\lbrace s\rbrace $
\end{enumerate}
\end{prop}
\begin{pf}
$ (1) $ Suppose that $ N_s^{-}(x_{-s})\neq \emptyset $ and let $ i,j,k\neq s $ s.t.\ $ i,j\in N_s^{-}(x_{-s})$ and $ k\notin N_s^{-}(x_{-s})$. We show that $ \tau_i^s=\tau_j^s $ and $ \tau_i^s<\tau_k^s $; by Remark \ref{rem:best_response_m1}, this implies that $\mathcal{B}_s(x_{-s})= N_s^{-}(x_{-s}) $. By Proposition \ref{prop:hitting_times}, it holds that
\begin{align*}
\tau_i^s&=1+(1-\beta)\sum_{v\in \mathcal V}\eta_v\tau_v^s,\,\,\,\,\, \tau_j^s=1+(1-\beta)\sum_{v\in \mathcal V}\eta_v\tau_v^s,\\
\tau_k^s&=1+(1-\beta)\sum_{v\in \mathcal V}\eta_v\tau_v^s +\beta \tau_{h}^s
\end{align*}
where $x_k=\lbrace h\rbrace $. Since $ \tau_{h}^s>0 $, it follows that $ \tau_i^s=\tau_j^s $ and $ \tau_i^s<\tau_k^s $.\\
$ (2) $ Suppose that $ N_s^{-}(x_{-s})= \emptyset $ and let $ j\neq s $. This implies that at every discrete time $ t $, the probability to arrive at node $ s $ from $ j $ is equal to $(1-\beta)\eta_s \left( 1- (1-\beta)\eta_s\right)^{t-1} $. Therefore it holds that
\begin{equation}\label{eq:tau_emptyinneigh}
\tau_j^s= (1-\beta)\eta_s\sum_{t=1}^{\infty}t\left( 1-(1-\beta)\eta_s \right)^{t-1},
\end{equation}
which does not depend on $ j$. We just proved that $ \tau_j^s=\tau_i^s $ for every $ i,j\neq s $, so we conclude by Remark \ref{rem:best_response_m1}. $ \quad\qed $
\end{pf}
We are now ready to prove Theorem \ref{thm:nash_m=1}.
\begin{comment}
\begin{figure}
\begin{center}
(a)
\begin{tikzpicture}[roundnode/.style={circle,fill=black!30, inner sep=1pt, minimum size=3mm},node distance=1cm,line width=0.2mm, ]
\node[roundnode] (q_1) at (0,0){$s$};
\node[roundnode] (q_2) at (1,0){};
\node[roundnode] (q_3) at (2,0){};
\node[roundnode] (q_4) at (1.5,-1){};
\node[roundnode] (q_5) at (1.5,1){};
\node[roundnode] (q_6) at (2,2){};
\node[roundnode] (q_7) at (1,2){};
\path[->]
(q_1) edge [] node {} (q_2)
(q_7) edge [] node {} (q_1)
(q_6) edge [] node {} (q_5)
(q_5) edge [] node {} (q_1)
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\draw [dashed] (0.7,0.7) rectangle (1.7,2.3);
\node at (0,1.7) {$ \mathcal B_s(x_{-s}) $};
\end{tikzpicture}
$ \qquad$ (b)
\begin{tikzpicture}[roundnode/.style={circle,fill=black!30, inner sep=1pt, minimum size=3mm},node distance=1cm,line width=0.2mm, ]
\node[roundnode] (q_1) at (0,0){$s$};
\node[roundnode] (q_2) at (1,0){};
\node[roundnode] (q_3) at (2,0){};
\node[roundnode] (q_4) at (1.5,-1){};
\node[roundnode] (q_5) at (1.5,1){};
\node[roundnode] (q_6) at (2,2){};
\node[roundnode] (q_7) at (1,2){};
\path[->]
(q_1) edge [] node {} (q_2)
(q_7) edge [] node {} (q_5)
(q_6) edge [] node {} (q_5)
(q_5) edge [] node {} (q_4)
(q_2) edge [] node {} (q_4)
(q_3) edge [] node {} (q_2)
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\draw [dashed] (0.7,-1.3) rectangle (2.3,2.3);
\node at (0,1.5) {$ \mathcal B_s(x_{-s}) $};
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\end{center}
\caption{The best response set $ \mathcal B_s(x_{-s}) $ of $ s $ in the game $ \Gamma(\mathcal V,\beta,\eta, 1) $ when its in-neighborhood is (a) not empty (b) empty.}
\label{fig:bestresponse_m1}
\end{figure}
\end{comment}
\textbf{Proof of Theorem \ref{thm:nash_m=1}.}
$ (1) $ A configuration $ x\in \mathcal{A} $ is a Nash equilibrium iff for all $ s\in\mathcal V $, it holds that $ x_s\in\mathcal{B}_s(x_{-s}) $. By Proposition \ref{prop:best_response}, this happens iff for all $ s\in \mathcal V $ s.t. $ N^{-}_s(x_{-s})\neq \emptyset $, we have that $x_s\in N^{-}_s(x_{-s}) $, thus forming the 2-clique $\lbrace s,x_s\rbrace $ in $ \mathcal{G}(x) $. Therefore $ x\in \mathcal{A} $ is a Nash equilibrium iff $ \mathcal{G}(x) $ is of type $ C_2^{l,r} $ where $ r $ is the number of vertices $ v$ such that $ N^{-}_v(x_{-v})= \emptyset $.\\
$ (2) $ A configuration $ x\in \mathcal{A} $ is a strict Nash equilibrium iff for all $ s\in\mathcal V $, it holds that $ \lbrace x_s\rbrace = \mathcal{B}_s(x_{-s}) $; by Proposition \ref{prop:best_response} this holds iff for all $ s\in\mathcal V $, $ N^{-}_s(x_{-s})=\lbrace x_s \rbrace$. Therefore for all $ s\in\mathcal V $, $ \lbrace s,x_s\rbrace $ must be a 2-clique in $ \mathcal{G}(x) $, and this is possible iff $ n $ is even and $ \mathcal{G}(x) $ is of type $ C_2^{n/2,0} $. $ \qed $
\textbf{Proof of Corollary \ref{cor:trapping_sets}.}
In view of Proposition \ref{prop:Cominetti} and Theorem \ref{thm:nash_m=1}, we just need to show that any configuration of type $ C_2^{l,r} $ will eventually converge in a best response dynamics to a configuration of type $ C_2^{n/2,0} $ when $ n $ is even and to a configuration of type $ C_2^{(n-1)/2,1} $ when $ n $ is odd. Suppose that the node $ v\in C_2^{l,r}$ is selected in the best response dynamics; we have the following cases:
(i) $ v $ belongs to a 2-clique and has in-degree equal to one: in this case its best response is unique so it does not change action;
(ii) $ v $ belongs to a 2-clique $ \lbrace v,w\rbrace $ and has in-degree $ >1 $: in this case by item (1) of Proposition \ref{prop:best_response}, it can change action (maintaining the same utility) by linking to some other vertex $ v_1 $ in $ N^{-}_v $. We have then two subcases: (iia) $ w $ has in-degree equal to one in $ C_2^{l,r} $ so when $ v $ changes its action, we still end up in a configuration of type $ C_2^{l,r} $; (iib) $ w $ has in-degree equal $ >1 $ in $ C_2^{l,r} $; in this case, once $ w $ is selected it \emph{has} to change action by linking back to some $ w_1\in N^{-}_w $, $ w_1\neq v $; we hence end up in a configuration of type $ C_2^{l+1,r-2} $.
Suppose now $ v $ is one of the $ r $ vertices with zero in-degree: by item (2) of Proposition \ref{prop:best_response}, $ v $ can change action (maintaining the same utility) by linking to any other vertex $ w $ in $C_2^{l,r} $. We have two cases:
(iii) $ w $ is a 2-clique; then we still end up in a configuration of type $ C_2^{l,r} $;
(iv) $ w $ is another vertex with zero in-degree. In this case, since now $ |N^{-}_w|>0 $, once $ w $ is selected it \emph{has} to change action by linking back to $ v $; we hence end up in a configuration of type $ C_2^{l+1,r-2} $.
We have just proved that in a best response dynamics, starting from a configuration of type $ C_2^{l,r} $ with positive probability we increase the number of two-cliques (and we can never reduce it). This implies that we will eventually converge to configurations with the maximal number of two cliques, that is $ C_2^{n/2,0} $ for $ n $ even, and $ C_2^{(n-1)/2,1} $ for $ n $ odd.
$ \qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qed $
\subsection{The case of out-degree $ m=2 $}\label{subsec:twolink_proofs}
As in the case of $ m=1 $, we want to better characterize the best response set of a player. The following two lemmas will be useful for proving the subsequent Proposition \ref{prop:bestresponse_m2}, in which we show that the best response actions of a node are always towards nodes that are at most at in-distance two from it.
\begin{lem}\label{lem:upperbound_tau}
Consider the game $ \Gamma(\mathcal V,\beta,\eta, 2) $, and let $ x\in \mathcal{A} $ and $ s\in \mathcal V $. It holds that:
\begin{enumerate}
\item for every $ i\neq s $, $ \tau_i^s(x)\leq \eta_s^{-1}(1-\beta)^{-1} $;
\item if there exists $ i\neq s $ such that $\tau_i^s(x)= \eta_s^{-1}(1-\beta)^{-1} $, then $ N^{-}_s(x)=\emptyset $.
\end{enumerate}
\end{lem}
\begin{pf}
$ (1) $ Let $ A $ be a matrix such that for all $ i\in \mathcal V $, $ A_{ii}= \beta+ (1-\beta)\eta_i $ and for all $ j\neq i$, $ A_{ij}= (1-\beta)\eta_j $ . If we denote by $ \hat{\tau}_i^s $ the expected hitting time of the Markov chain $ \hat{X}_t $ with transition matrix $ A $ and initial state $ s $, by solving the system (\ref{eq:system_tau}) it is easy to see that for all $ i,k\neq s $ it holds that $ \hat{\tau}_i^s=\hat{\tau}_k^s $. This in turn implies that for every $ i\neq s $, $ \hat{\tau}_i^s= \eta_s^{-1}(1-\beta)^{-1} $.
In $ \hat{X}_t $ the probability to jump from any node $ i $ to $ s $ is always equal to $ (1-\beta)\eta_s $, while in the Markov chain $ X_t $ associated to our game (with transition matrix as in (\ref{eq:P(x)})) the probability to jump from any node $ i$ to $ s $ is always greater or equal than $ (1-\beta)\eta_s $. It follows that $ \tau_i^s\leq \hat{\tau}_i^s $, so we conclude.\\
$ (2) $ Let $ i\neq s $ such that $\tau_i^s= \eta_s^{-1}(1-\beta)^{-1} $. We first show that for every $ j\neq s $, $\tau_j^s= \eta_s^{-1}(1-\beta)^{-1} $. Indeed, suppose by contrary that there exists $ j\neq s $ such that $\tau_j^s< \eta_s^{-1}(1-\beta)^{-1} $. If $ a,b\in \mathcal V $ are the vertices such that $ x_i=\lbrace a,b\rbrace $, then by system (\ref{eq:system_tau}) it holds that \[ \tau_i^s=1+(1-\beta)\sum_{v\in \mathcal V}\eta_v\tau_v^s+\frac{\beta}{2}(\tau_a^s+\tau_b^s).\] In view of item $ (1) $, this implies that $ \tau_i^s< \eta_s^{-1}(1-\beta)^{-1} $, which is a contradiction; therefore $\tau_j^s= \eta_s^{-1}(1-\beta)^{-1} $. Suppose now by contradiction that $N^{-}_s(x)\neq \emptyset $ and let $ k\in N^{-}_s(x)$ and $ a\in\mathcal V $ such that $ x_k=\lbrace a,s\rbrace $. By system (\ref{eq:system_tau}) it holds that
\begin{equation}\label{eq:lemma}
\tau_k^s=1+(1-\beta)\sum_{v\in \mathcal V}\eta_v\tau_v^s+\frac{\beta}{2}\tau_a^s.
\end{equation}
As $\tau_v^s= \eta_s^{-1}(1-\beta)^{-1} $ for every $ v\neq s $ and $\tau_s^s=0 $, equation (\ref{eq:lemma}) implies that $ \beta=0 $, which is a contradiction and so we conclude. $\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad \qed $
\end{pf}
The next lemma provides a different upper bound on the return times $ \tau_{i}^s(x) $ when $|N_s^-(x)| \geq 1$.
We denote by $ N_s^{-2}(x) $ the set $ N_s^{-}(x)\cup \lbrace N_t^{-}(x): t\in N_s^{-}(x) \rbrace$, that is the in-neighborhood of $ s $ in $ \mathcal{G}(x) $ at distance at most two. Notice that also $ N_s^{-2}(x) $ depends just on $ x_{-s} $ so we can write as well $ N_s^{-2}(x_{-s}) $.
\begin{lem}\label{lem:upperbound2_tau}
Consider the game $ \Gamma(\mathcal V,\beta,\eta, 2) $, and let $ x\in \mathcal{A} $ and $ s \in \mathcal V $ such that $|N_s^-(x)| \geq 1$. Let $k \in N_s^-(x)$ and set $ T_1 = ( 1-\frac{\beta}{2}) (1-\beta)^{-1}(\eta_s + \frac{\beta}{2}\eta_{k})^{-1} $ and $ T_2 = (1-\beta)^{-1}(\eta_s + \frac{\beta}{2} \eta_k)^{-1} $. Then it holds that:
\begin{enumerate}
\item $ \tau_{k}^s(x) \leq T_1$ and for all $i \neq k$, $ \tau_{i}^s(x)\leq T_2 $;
\item if $\tau_{k}^s(x) = T_1$ and for all $ i\neq k,s$, $\tau_{i}^s(x) = T_2 $, then $|N^{-2}_s(x)|= 1$.
\end{enumerate}
\end{lem}
\begin{pf}
$ (1) $
Let $ \tau_{\text{max}}^s= \max_{j \in \mathcal V} \;\tau_j^s$.
By system (\ref{eq:system_tau}) it holds that $\tau_{\text{max}}^s \leq 1 + (1-\beta)\eta_{k}\tau_{k}^s +(1-\beta)(1-\eta_s - \eta_{k})\tau_{\text{max}}^s + \beta \tau_{\text{max}}^s$,
which implies that
\begin{equation}
\label{eq:lemmaUpperBound2_eq2}
\tau_{\text{max}}^s \leq (1-\beta)^{-1}(\eta_{k} + \eta_s)^{-1} + \eta_{k}(\eta_{k}+\eta_s)^{-1}\tau_k^s.
\end{equation}
At the same time, by system (\ref{eq:system_tau}) it holds that $ \tau_{k}^s \leq 1 + (1-\beta)(1-\eta_s - \eta_{k}) \tau_{\text{max}}^s + (1-\beta)\eta_{k}\tau_{k}^s + (\beta/2) \tau_{\text{max}}^s$,
which implies that
\begin{equation}
\label{eq:lemmaUpperBound2_eq3}
\tau_{k}^s \leq \frac{1+\left[(1-\beta)(1-\eta_{k}-\eta_s)+\frac {\beta}{2}\right]\tau_{\text{max}}^s}{1 -(1-\beta)\eta_{k}}.
\end{equation}
By substituting inequality (\ref{eq:lemmaUpperBound2_eq2}) in (\ref{eq:lemmaUpperBound2_eq3}), the following upper bound is obtained:
\[
\tau_{k}^s\leq T_1 = \left(1-\frac{\beta}{2}\right)(1-\beta)^{-1}\left(\eta_s + \frac{\beta}{2}\eta_{k}\right)^{-1},
\]
while by substituting inequality (\ref{eq:lemmaUpperBound2_eq3}) in (\ref{eq:lemmaUpperBound2_eq2}) we obtain:
\[
\tau_{\text{max}}^s\leq T_2 = (1-\beta)^{-1}\left(\eta_s + \frac{\beta}{2} \eta_k\right)^{-1}.
\]
(2) Suppose that there exists $ j\neq k $ such that $j\in N_s^-(x) \cup N_k^-(x) $; we show that this leads to a contradiction.
There are three cases: either $ x_j=\lbrace s,k \rbrace $, or there exists $ b\neq k,s $ such that $ x_j=\lbrace s,b\rbrace $ or $ x_j=\lbrace k,b\rbrace $. By system (\ref{eq:system_tau}), $\tau_j^s$ satisfies:
\begin{equation}
\label{eq:Lemma11_item2_1}
\tau_j^s \leq 1 + (1-\beta)\sum_{i \in \mathcal V}\eta_i \tau_i^s + \frac{\beta}{2}\left(\tau_{k}^s + \tau_b^s\right).
\end{equation}
By substituting the values of the $ \tau_{i}^s $'s in the hypothesis and by observing that $ T_1<T_2 $, equation (\ref{eq:Lemma11_item2_1}) leads to:
\begin{align*}
T_2 &\leq 1 + (1-\beta)\left( T_2 + \eta_k T_1 -(\eta_k + \eta_s)T_2\right) + \frac{\beta}{2}\left(T_1 + T_2\right) \\
&< 1 + (1-\beta)\left( T_2 + \eta_k T_2 -(\eta_k + \eta_s)T_2\right) + \frac{\beta}{2}\left(2 T_2\right) \\
&< \left[ (1-\beta) \frac{\beta}{2} \eta_k + 1\right] (1-\beta)^{-1}\left(\eta_s + \frac{\beta}{2} \eta_k\right)^{-1} < T_2,
\end{align*}
which is a contradiction. This means that the set $N_s^-(x) \cup N_k^-(x)$ has to be equal to $\{k\}$ and so $|N_s^{-2}(x)|= 1$. $ \qed $
\end{pf}
The following proposition characterizes the best response set of a player in the game $ \Gamma(\mathcal V,\beta,\eta, 2) $ and it will play a key role in both the proofs of Theorem \ref{thm:strictNash_m2} and \ref{thm:condensation_graph}.
From now on, fixed $ s\in\mathcal V $ and $ x\in \mathcal A $, we label the elements of $ \mathcal V $ in such a way that $ \mathcal V=\lbrace s, v_1,\dots ,v_{n-1}\rbrace $ and
\begin{equation}\label{eq:tau_v}
0=\tau_s^s(x)<\tau_{v_1}^s(x)\leq \tau_{v_2}^s(x)\leq \dots \leq \tau_{v_{n-1}}^s(x).
\end{equation}
\begin{prop}\label{prop:bestresponse_m2}
Consider the game $ \Gamma(\mathcal V,\beta,\eta, 2) $, and let $ x\in\mathcal A $ and $ s\in \mathcal V $. It holds that:
\begin{enumerate}
\item if $ N_s^{-2}(x)\!=\! \emptyset $, then $ \mathcal{B}_s(x_{-s})\!=\!\bigl\lbrace \lbrace v,w\rbrace\! : v,w\!\in\! \mathcal V \setminus\lbrace s\rbrace, v\neq w \bigr\rbrace $;
\item if $ |N_s^{-2}(x)|\!=\!1 $, then $ \mathcal{B}_s(x_{-s})\!=\!\bigl\lbrace \lbrace r,v\rbrace \!: v\in \mathcal V \setminus\lbrace s,r\rbrace \bigr\rbrace $, where $ \lbrace r\rbrace=N_s^{-2}(x)=N_s^{-}(x) $;
\item if $ |N_s^{-2}(x)|\!\geq\! 2 $, then $ \mathcal{B}_s(x_{-s})\subseteq
\bigl\lbrace \lbrace v,w\rbrace \!: v,w\in N^{-}_s(x), v\!\neq\! w\bigr\rbrace \cup \bigl\lbrace \lbrace v,w\rbrace \!: v\!\in\! N^{-}_s(x) \text{ and } w\!\in\! N^{-}_v(x)\bigr\rbrace$.
\end{enumerate}
\end{prop}
\begin{pf}
$ (1) $ If $N_s^{-2}(x)= \emptyset $, then $ \tau_j^s $ can still be expressed as in (\ref{eq:tau_emptyinneigh}), so we conclude.\\
$ (2) $ We remind that we label the elements of $ \mathcal V $ in such a way that (\ref{eq:tau_v}) holds.
We first show that $v_1\in N_s^{-}(x) $. By contradiction, suppose that $v_1\notin N_s^{-}(x) $; then $ x_{v_1}=\lbrace a,b\rbrace $ for some $ a,b \neq s $. It holds that
\[
\tau_{v_1}^s=1+(1-\beta)\sum_{v\in \mathcal V}\eta_v\tau_v^s+\frac{\beta}{2}(\tau_a^s+\tau_b^s)\geq 1+\tau_{v_1}^s-\eta_s(1-\beta)\tau_{v_1}^s,
\]
which implies that $ \tau_{v_1}^s\geq \eta_s^{-1}(1-\beta)^{-1} $. By Lemma \ref{lem:upperbound_tau}, it follows that $ \tau_{v_1}^s=\eta_s^{-1}(1-\beta)^{-1} $ and $N_s^{-}(x)= \emptyset $, which is a contradiction. Therefore, if $ N_s^{-2}(x)=N_s^{-}(x)=\lbrace r\rbrace $, it holds that $ r=v_1 $ and so $ r\in x_s $ for any $ x_s\in \mathcal B_s(x_{-s}) $. We now show that $ \tau_j^s=\tau_k^s $ for every $ j,k\neq r,s $, which implies that $ \mathcal{B}_s(x_{-s})=\bigl\lbrace \lbrace r,v\rbrace : v\in \mathcal V \setminus\lbrace s,r\rbrace \bigr\rbrace $.
By hypothesis, for every $j\neq s,r $, the probability to jump from $ j $ to $ s $ is equal to $ (1-\beta)\eta_s $ and the probability to jump from $ j $ to $ r $ is equal to $ (1-\beta)\eta_r$. It follows that the probability to arrive in $ s $ from $ j $ in exactly $ t $ steps without passing through $ r $ is equal to $(1-\beta)\eta_s(1-(1-\beta)(\eta_s+\eta_r))^{t-1} $ and the probability to arrive in $ r $ from $ j $ in exactly $ t $ steps without passing through $ s $ is equal to $(1-\beta)\eta_r(1-(1-\beta)(\eta_s+\eta_r))^{t-1} $. Consequently,
\[
\tau_j^s=\sum_{t=1}^{\infty}(1-\beta)(t\eta_s+\eta_r(t+\tau_r^s))\bigl( 1-(1-\beta)(\eta_s+\eta_r)\bigr)^{t-1},
\]
which does not depend on $ j $.\\
$ (3) $ Suppose that $ |N_s^{-2}(x)|\geq 2 $. We already proved that $ v_1\in N^{-}_s(x) $; we need to prove that either $ v_2\in N^{-}_s(x)$ or $ v_2\in N^{-}_{v_1}(x) $. Suppose by contradiction that this is not the case and let $ a,b\neq s,v_1 $ such that $ x_{v_2}=\lbrace a,b\rbrace $. By applying system (\ref{eq:system_tau}) to express $ \tau_{v_2}^s $ and by using the fact that for all $ j\geq 2$, $ \tau_{v_j}^s\geq \tau_{v_2}^s $, it holds that:
\begin{equation}
\label{eq:PropUpperBound2_eq1}
\tau_{v_2}^s \geq \frac{1}{(1-\beta)(\eta_{v_1} + \eta_s)} + \frac{\eta_{v_1}}{\eta_{v_1}+\eta_s}\tau_{v_1}^s.
\end{equation}
Moreover, by applying system (\ref{eq:system_tau}) to express $ \tau_{v_1}^s $ and by using again the fact that for all $ j\geq 2$, $ \tau_{v_j}^s\geq \tau_{v_2}^s $, it holds that:
\begin{equation}
\label{eq:PropUpperBound2_eq2}
\tau_{v_1}^s \geq \frac{1+\left[(1-\beta)(1-\eta_{v_1}-\eta_s)+\frac {\beta}{2}\right]\tau_{v_2}^s}{1 -(1-\beta)\eta_{v_1}}.
\end{equation}
By substituting inequality (\ref{eq:PropUpperBound2_eq1}) in (\ref{eq:PropUpperBound2_eq2}) and inequality (\ref{eq:PropUpperBound2_eq2}) in (\ref{eq:PropUpperBound2_eq1}) we obtain respectively:
\[
\tau_{v_1}^s\geq T_1 \quad \text{and}\quad \tau_{v_2}^s\geq T_2,
\]
where $ T_1 $ and $ T_2 $ are defined in Lemma \ref{lem:upperbound2_tau}. Therefore, by (\ref{eq:tau_v}) and item (1) of Lemma \ref{lem:upperbound2_tau}, it holds that $\tau_{v_1}^s = T_1$, and for all $ j \geq 2 $, $\tau_{v_j}^s = T_2$. By applying item (2) of the same lemma it follows that $|N_s^{-2}(x)| = 1$, which contradicts the hypothesis. $\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\quad \qed $
\end{pf}
\begin{rem}
\label{rem:viciniAdist1-viciniAdist2}
Suppose that $|N_s^{-2}(x)|\!\geq\! 2 $ for some $ x\in\mathcal A $ and $ s\in \mathcal V $ and let $ x_s=\lbrace i,j\rbrace\in \mathcal{B}_s(x_{-s})$.
Item (3) of Proposition \ref{prop:bestresponse_m2} implies that, if $ j\notin N^{-}_s(x) $, then $ i\in N^{-}_s(x) $ and $j\in N^{-}_i(x)$. In other words, if
$ j $ is not an in-neighbor of $ s $, then $ (j,i) $ and $ (i,s) $ must be edges of $ \mathcal G(x) $, together with the edges $ (s,i) $ and $ (s,j) $ as $ s $ is playing $ \lbrace i,j\rbrace $.
\end{rem}
Figure \ref{fig:bestresponse_m2} graphically synthesizes Proposition \ref{prop:bestresponse_m2}.
Notice that in view of Proposition \ref{prop:bestresponse_m2}, the best response of a node $ s $ can be unique only in the case $ |N_s^{-2}(x)|\geq 2 $.
\begin{figure}
\begin{center}
\begin{tikzpicture}[roundnode/.style={circle,fill=black!30, inner sep=1pt, minimum size=3mm},node distance=1cm,line width=0.2mm, ]
\node[roundnode] (q_1) at (0,0){$s$};
\node[roundnode] (q_2) at (1,0){};
\node[roundnode] (q_3) at (2,0){};
\node[roundnode] (q_4) at (1.5,-1){};
\node[roundnode] (q_5) at (1.5,1){};
\node[roundnode] (q_6) at (2,2){};
\node[roundnode] (q_7) at (1,2){};
\path[-]
(q_7) edge [] node {} (q_6)
(q_4) edge [] node {} (q_5)
;
\path[->]
(q_1) edge [] node {} (q_2)
(q_1) edge [] node {} (q_4)
(q_7) edge [] node {} (q_5)
(q_6) edge [] node {} (q_5)
(q_2) edge [] node {} (q_4)
(q_2) edge [] node {} (q_5)
(q_3) edge [] node {} (q_2)
(q_3) edge [] node {} (q_5)
(q_4) edge [] node {} (q_3)
;
\draw [dashed] (0.7,-1.3) rectangle (2.3,2.3);
\node at (1.5,-1.7) {$ \mathcal B_s(x_{-s}) $};
\node at (0,-1.3) {(a)};
\end{tikzpicture}
\begin{tikzpicture}[roundnode/.style={circle,fill=black!30, inner sep=1pt, minimum size=3mm},node distance=1cm,line width=0.2mm, ]
\node[roundnode] (q_1) at (0,0){$s$};
\node[roundnode] (q_2) at (1,0){};
\node[roundnode] (q_3) at (2,0){};
\node[roundnode] (q_4) at (1.5,-1){};
\node[roundnode] (q_5) at (1.5,1){};
\node[roundnode] (q_6) at (2,2){};
\node[roundnode] (q_7) at (1,2){$ r $};
\path[-]
(q_2) edge [] node {} (q_4)
(q_6) edge [] node {} (q_3)
(q_2) edge [] node {} (q_3)
(q_1) edge [] node {} (q_7)
;
\path[->]
(q_1) edge [] node {} (q_4)
(q_7) edge [] node {} (q_5)
(q_6) edge [] node {} (q_5)
(q_5) edge [] node {} (q_2)
(q_5) edge [] node {} (q_3)
(q_4) edge [] node {} (q_3)
;
\draw [dashed] (0.7,1.7) rectangle (1.3,2.3);
\draw [dashed] (0.6,-1.3)--(0.6,1.5)--(1.5,1.5)--(1.5,2.4)--(2.3,2.4)--(2.3,-1.3)--(0.6,-1.3);
\node at (1.5,-1.7) {$ \mathcal B_s(x_{-s}) $};
\node at (0,-1.3) {(b)};
\end{tikzpicture}
\begin{tikzpicture}[roundnode/.style={circle,fill=black!30, inner sep=1pt, minimum size=3mm},node distance=1cm,line width=0.2mm, ]
\node[roundnode] (q_1) at (0,0){$s$};
\node[roundnode] (q_2) at (1,0){};
\node[roundnode] (q_3) at (2.3,0){};
\node[roundnode] (q_4) at (1.5,-1){};
\node[roundnode] (q_5) at (1.5,1){};
\node[roundnode] (q_6) at (2.3,2){};
\node[roundnode] (q_7) at (1,2){};
\path[-]
(q_1) edge [] node {} (q_2)
(q_5) edge [] node {} (q_3)
(q_3) edge [] node {} (q_6)
(q_5) edge [] node {} (q_7)
;
\path[->]
(q_1) edge [] node {} (q_4)
(q_7) edge [] node {} (q_1)
(q_6) edge [] node {} (q_5)
(q_2) edge [] node {} (q_3)
(q_4) edge [] node {} (q_3)
(q_4) edge [] node {} (q_2)
;
\draw [dashed] (0.6,-1.3) rectangle (1.8,2.2);
\node at (1,-1.7) {$ \mathcal B_s(x_{-s}) \subseteq $};
\node at (-0.3,-1.3) {(c)};
\end{tikzpicture}
\end{center}
\caption{The best response set $ \mathcal B_s(x_{-s}) $ of player $ s $ in the game $ \Gamma(\mathcal V,\beta,\eta, 2) $ when $ N^-_s(x) $ is (a) empty (b) of cardinality one (c) of cardinality greater than one. In case (b), node $s$ has to link to its unique in-neighbor $r$ and then it can place its second link anywhere else.}
\label{fig:bestresponse_m2}
\end{figure}
\textbf{Proof of Theorem \ref{thm:strictNash_m2}.}
We first prove that a ring graph on $ n $ vertices is a strict Nash equilibrium for $ \Gamma(\mathcal V,\beta, \eta,2) $. If $ n=3 $ the proof is trivial. Suppose that $ n>3 $ and
consider the ring graph as in Fig.\ \ref{fig:cycle_graph}(a); we want to show that the node $ s $ is in its (unique) best response, that is we want to show that $ \tau_1^s, \tau_{n-1}^s< \tau_v^s $ for all $ v\neq 1, n-1 $. By the symmetry of the graph, $ \tau_1^s=\tau_{n-1}^s$ and $ \tau_2^s=\tau_{n-2}^s$. In view of Remark \ref{rem:viciniAdist1-viciniAdist2}, it then suffices to show that $ \tau_1^s<\tau_2^s $. By system (\ref{eq:system_tau}), we have that $\tau_{2}^s-\tau_{1}^s= (\beta/2) (\tau_1^s-\tau_2^s)+(\beta/2)\tau_3^s$, which implies that $ \tau_2^s>\tau_1^s $ since $ \tau_3^s>0 $. \\
We now show that if $ x^* $ is a strict Nash equilibrium for $ \Gamma(\mathcal V,\beta, \eta,2) $, then $\mathcal{G}(x^*)$ is undirected, which implies that $\mathcal{G}(x^*)$ is the union of ring graphs since by construction each vertex of $\mathcal{G}(x^*)$ has out-degree equal to $ 2 $.
Assume by contradiction that there exists a strict Nash equilibrium $ x\in \mathcal A $ and two nodes $ s,j\in \mathcal V $ such that $ (s,j)\in \mathcal E(x)$ but $ (j,s)\notin \mathcal E(x) $. Since $ x $ is a strict Nash equilibrium, all the nodes are in their best response and $ |\mathcal{B}_v(x)|=1 $ for all $ v\in \mathcal V $. By Proposition \ref{prop:bestresponse_m2} we know that $ j\in N^{-2}_s(x) $: since $(j,s)\notin \mathcal E(x) $, it means that there exists $ i\neq j,s $ such that $(j,i),(i,s),(s,i)\in \mathcal E(x)$ by Remark \ref{rem:viciniAdist1-viciniAdist2} (see also Fig.\ \ref{fig:proof}(a)). This also implies that $ i\in N^{-2}_j(x) $.
If $ i\in N^{-}_j(x) $, by system (\ref{eq:system_tau}) it holds that $ \tau_i^j-\tau_s^j=(\beta/2)(\tau_s^j-\tau_i^j)$ and so $\tau_i^j=\tau_s^j $. Therefore we have that either $ (j,s)\in \mathcal E(x) $ or $ |\mathcal{B}_j(x)|>1 $, both cases leading to a contradiction. We now examine the case $ i\in N^{-2}_j(x)\setminus N^{-}_j(x) $: by Remark \ref{rem:viciniAdist1-viciniAdist2} there exists $ k\neq i,j $ such that $(i,k),(k,j),(j,k)\in \mathcal E(x) $ (see Fig.\ \ref{fig:proof}(b)). Proposition \ref{prop:bestresponse_m2} also implies that $ k\in N^{-2}_i(x) $. If $ k\in N^{-}_i(x) $, we are in the situation represented in Fig.\ \ref{fig:proof}(c); by using system (\ref{eq:system_tau}), it is easy to see that $ \tau_s^j=\tau_k^j $. This implies that either $ k=s $ (in which case $ (j,s)\in \mathcal E(x) $) or $ |\mathcal{B}_j(x)|>1 $, so we always arrive to a contradiction. Finally, we need to consider the case $ k\in N^{-2}_i(x)\setminus N^{-}_i(x) $: since the actions of $ i $ are determined as in Fig.\ \ref{fig:proof}(b), it must hold that $ (k,s)\in \mathcal E(x) $, as represented in Fig.\ \ref{fig:proof}(d). By using again system (\ref{eq:system_tau}) to express $ \tau_i^j $ and $ \tau_s^j $, we get that $(1+\beta/2)( \tau_i^j -\tau_s^j)=(\beta/2)\tau_k^j>0 $ and so $ \tau_i^j >\tau_s^j $. This implies that $ j $ is not in its best response, thus leading to a contradiction. $ \qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qed $
\begin{figure}
\begin{center}
(a)
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\node[roundnode] (q_6) at (esa.corner 6) {$ \dots $};
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(q_2) edge node {} (q_3)
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(q_3) edge node {} (q_4)
(q_4) edge node {} (q_3)
(q_4) edge node {} (q_5)
(q_5) edge node {} (q_4)
(q_5) edge [dotted] node {} (q_6)
(q_1) edge [dotted] node {} (q_6)
;
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$ \qquad $(b)
\begin{tikzpicture}[roundnode/.style={circle,fill=black!30, inner sep=2pt, minimum size=6mm},node distance=0.7cm,line width=0.2mm]
\node[roundnode] (q_1) at (1.5,0) {j};
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\path[-] (q_1) edge node {} (q_3)
(q_3) edge node {} (q_2)
;
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\end{center}
\caption{(a) A ring graph on $ n $ nodes. (b) The directed graph $T_{(s,j),i}$. }
\label{fig:cycle_graph}
\end{figure}
\begin{figure}
\begin{center}
(a)
\begin{tikzpicture}[roundnode/.style={circle,fill=black!30, inner sep=1pt, minimum size=6mm},node distance=0.8cm,line width=0.2mm]
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;
\path[->]
(q_2) edge node {} (q_3)
(q_3) edge node {} (q_1)
;
\end{tikzpicture}
$ \qquad $
(b)
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\node[roundnode] (q_1) {i};
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;
\path[->]
(q_2) edge node {} (q_3)
(q_3) edge node {} (q_1)
(q_1) edge node {} (q_4)
;
\end{tikzpicture}\\
$ $\\
(c)
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(q_3) edge node {} (q_4)
(q_4) edge node {} (q_1)
;
\path[->]
(q_2) edge node {} (q_3)
(q_3) edge node {} (q_1)
;
\end{tikzpicture}
$ \qquad $
(d)
\begin{tikzpicture}[roundnode/.style={circle,fill=black!30, inner sep=1pt, minimum size=6mm},node distance=0.8cm,line width=0.2mm]
\node[roundnode] (q_1) {i};
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\node[roundnode] (q_3) [right=of q_2] {j};
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\path[-] (q_1) edge node {} (q_2)
(q_3) edge node {} (q_4)
;
\path[->]
(q_2) edge node {} (q_3)
(q_3) edge node {} (q_1)
(q_4) edge node {} (q_2)
(q_1) edge node {} (q_4)
;
\end{tikzpicture}
\end{center}
\caption{Explanatory graphs for the proof of Theorem \ref{thm:strictNash_m2}.}
\label{fig:proof}
\end{figure}
Before proving Theorem \ref{thm:condensation_graph}, we first need the following definition and Lemma \ref{lem:triangle}.
\begin{defn}\label{defn:triangle}
We denote by $T_{(s,j),i}$ the directed graph on the vertices $ \lbrace i,j,s\rbrace $ having one directed edge $ (s,j) $ and all the other edges undirected (see Fig.\ \ref{fig:cycle_graph}(b)). We will sometimes refer to $T_{(s,j),i}$ as a \emph{triangle}.
\end{defn}
\begin{lem}\label{lem:triangle}
Let $ x\in \mathcal A $ be a Nash equilibrium for the game $ \Gamma(\mathcal V,\beta, \eta,2) $, $\mathcal H(x)$ be the condensation graph of $ \mathcal G(x) $ and let $\mathcal G_\lambda(x)=(\mathcal V_\lambda,\mathcal E_\lambda)$ be a sink in $\mathcal H(x)$. If there exists $(s,j)\in\mathcal E_{\lambda}$ that is directed, then $\mathcal G_\lambda(x)$ contains a structure of type $T_{(s,j),i}$.
\end{lem}
\begin{pf} Notice that since the out-degree of each node in $\mathcal G_\lambda(x)$ is equal to two, this graph must contain at least three nodes and $ |N_s^{-2}(x)|\geq 2$. It follows that $j\in N_s^{-2}(x)$ and so by Remark \ref{rem:viciniAdist1-viciniAdist2}, there exists $i\in\mathcal V_{\lambda}$ such that $(j,i), (i,s), (s,i)\in\mathcal E_{\lambda}$ (see Fig.\ \ref{fig:proof}(a)). We are left to prove that $(i,j)\in\mathcal E_{\lambda}$. If this was not the case, then by Remark \ref{rem:viciniAdist1-viciniAdist2} there would exist $ k\in\mathcal V_\lambda $ such that $(i,k), (k,j), (j,k)\in\mathcal E_{\lambda}$, i.e.\ the graph in Fig.\ \ref{fig:proof}(b) would be a subgraph of $\mathcal G_\lambda(x)$.
In this configuration, the only way $ i$ could be at equilibrium is that $ (k,i)\in \mathcal E_\lambda $, as otherwise $\lbrace s,j\rbrace$ would give it a strictly better utility than $\lbrace s,k\rbrace$. We would then be in the configuration of Fig.\ \ref{fig:proof}(c); but in this case $ j $ is not at equilibrium, as $\lbrace s,k\rbrace$ gives it a strictly better utility than $\lbrace i,k\rbrace$. This completes the proof. $\qquad\qquad\qquad\qquad \qed $
\end{pf}
We are now ready to prove Theorem \ref{thm:condensation_graph}.
\textbf{Proof of Theorem \ref{thm:condensation_graph}.}
Consider any component $\mathcal G_\lambda(x)=(\mathcal V_\lambda,\mathcal E_\lambda)$ that is not a sink in $\mathcal H(x)$. Necessarily, there must exist $i\in \mathcal V_\lambda$ such that $N_i(x)\not\subseteq \mathcal V_{\lambda}$. In particular, this implies that
$ |N_i^{-2}(x)|\leq 1$ by Proposition \ref{prop:bestresponse_m2}. If $ |N_i^{-2}(x)|=0$, it means that $\mathcal V_\lambda=\{i\}$ is a singleton. If $ |N_i^{-2}(x)|=1$, then necessarily $\mathcal V_\lambda=\{i,j\}$ for some $j\neq i$ and so $\mathcal G_\lambda(x)$ is the $2$-clique on $\{i,j\}$. Notice that in both cases, there cannot be any other component $\mathcal G_{\lambda'}(x)$ linking to $\mathcal G_{\lambda}(x)$ in the condensation graph, as otherwise the condition $ |N_i^{-2}(x)|\leq 1$ would be violated. This proves items $(1)$ and $(2)$. \\
We now study the structure of the sink components. Suppose that the component $\mathcal G_\lambda(x)=(\mathcal V_\lambda,\mathcal E_\lambda)$ is not a ring graph and thus not undirected; then there must exist at least two directed edges in $\mathcal E_{\lambda}$. Let $(s,j)$ be one of these directed edges and let $T_{(s,j),i}$ be the corresponding triangle (see Definition \ref{defn:triangle} and Lemma \ref{lem:triangle}). We now discuss how any other triangle $T_{(r,k), t}$ in $\mathcal G_\lambda(x)$ can possibly intersect with $T_{(s,j),i}$. Notice that, since the out-degree of all nodes in $\mathcal G_\lambda(x)$ is $2$, the two triangles cannot intersect in the nodes of out-degree equal to two in the corresponding triangles, namely
$\{i,s\}\cap\{r,k,t\}=\emptyset$ and $\{r,t\}\cap\{i,j,s\}=\emptyset$. Therefore the only possibility is that they have just one node in common, namely $j=k$; this corresponds to the Butterfly graph (see Fig. \ref{fig:C2_nash}(b)). Since in the Butterfly graph every node has out-degree equal to $2$, it necessarily coincides with the connected component $\mathcal G_\lambda(x)$. If instead $T_{(s,j),i}$ does not intersect any other triangle, there must exist a sequence of distinct nodes $j=j_1, j_2,\dots , j_l=r$, with $l\geq 2$, such that $\{j_a,j_{a+1}\}$ are $2$-cliques in $\mathcal G_\lambda(x)$ for $a=1,\dots ,l-1$ and such that there exists a triangle $T_{(r,k), t}$ in $\mathcal G_\lambda(x)$ for some $ k,t $. Since there cannot be any incoming directed edge in $r$ by hypothesis, we deduce that $N_r^{-2}(x)=\{j_{l-1}, j_{l-2}\}$ if $l\geq 3$ and $N_r^{-2}(x)=\{i,j,s\}$ if $l=2$. This last case is impossible since it would result that $k\in \{i,j,s\}$, contrarily to what we had assumed. In the case when $l\geq 3$, we obtain that $k=j_{l-2}$ that leads to the graph depicted in Fig. \ref{fig:long}.
A direct computation shows that nodes $j$ and $k$ are however not at equilibrium in this configuration. This completes the proof. $\qquad\qquad\qquad\qquad\qquad\qquad\quad \qed $
\begin{figure}
\begin{center}
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\node[roundnode] (q_2) at (0,-2) {i};
\node[roundnode,fill=black] (q_3) at (1.5,-1) {\textcolor{white}{j}};
\node[roundnode] (q_4) at (3,-1) {};
\node[roundnode] (q_5) at (5,-1) {};
\node[roundnode,fill=black] (q_6) at (6.5,-1) {\textcolor{white}{k}};
\node[roundnode] (q_7) at (8,0) {r};
\node[roundnode] (q_8) at (8,-2) {t};
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(q_1) edge node {} (q_3)
;
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(q_3) edge node {} (q_2)
(q_1) edge node {} (q_2)
(q_4) edge node {} (q_3)
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(q_8) edge node {} (q_7)
;
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\end{center}
\caption{Explanatory graph for the proof of Theorem \ref{thm:condensation_graph}. Black nodes are not in best response.}
\label{fig:long}
\end{figure}
We are left to prove Remark \ref{remark_m2} and Corollary \ref{cor:trapping_setsM2}.
\textbf{Proof of Remark \ref{remark_m2}.}
(1) Consider Fig.\ \ref{fig:singleton_ring}(a); we show that node $ s $ is not playing an action in its best response set. Indeed by system (\ref{eq:system_tau}), it holds that $\tau_j^{s}- \tau_{1}^{s} = \frac{\beta}{2} (\tau_{1}^{s}- \tau_{2}^{s})$. In the proof of Theorem \ref{thm:strictNash_m2} we showed that if the ring has length greater or equal than four, then $\tau_1^{s} < \tau_{2}^{s}$ and therefore $\tau_j^{s}< \tau_{1}^{s}$. It follows that node $s$ is not playing a best response action and so such configuration is not a Nash equilibrium.\\
(2) Consider Fig.\ \ref{fig:singleton_ring}(b); we show that node $ s $ is not playing an action in its best response set.
By symmetry $\tau_j^s = \tau_{k}^{s}$, so it sufficies to show that $ \tau_1^s< \tau_j^s $. By system (\ref{eq:system_tau}), $(1+\beta/2)(\tau_{1}^{s}- \tau_{j}^{s}) = (\beta/2) (\tau_{2}^{s}- \tau_{1}^{s})$; in the proof of Theorem \ref{thm:strictNash_m2} we showed that $\tau_{2}^{s} > \tau_{1}^{s}$, so we conclude.$\qquad\qquad \qed $
\textbf{Proof of Corollary \ref{cor:trapping_setsM2}.}
We know that $\mathcal N^{st} \subseteq \mathcal N^*$.
Let $ x\in \mathcal N^* \setminus \mathcal N^{st}$ and let $ \mathcal G(x) $ be its associated graph; $ \mathcal G(x) $ must have a directed link.
The first key observation is that the transition states of the Butterfly graph are the ones shown in Fig.\ \ref{fig:trasitionButterfly}, which are all Nash equilibria. Hence, every time the graph $\mathcal G(x)$ contains a Butterfly graph, there is a nonzero probability that the best response dynamics will assume the configurations (b) or (c) in Fig.\ \ref{fig:trasitionButterfly}, i.e.\ a configuration with a 2-clique linking to a ring of length three.
The second key observation is that $\mathcal G(x)$ can have at most one singleton or one 2-clique. In fact, since by Proposition \ref{prop:bestresponse_m2} both singletons and nodes in a 2-clique are always playing a best response action independently on the node they are linking to, there is a nonzero probability that they will direct their links to another singleton or node in a 2-clique, which will not be playing a best response action anymore.
Therefore $\mathcal G(x)$ has either a singleton, a 2-clique or a Butterfly graph, as the Butterfly graph transforms with nonzero probability into a 2-clique linking to a ring. We are left with the following cases:
\begin{description}
\item[-] $\mathcal G(x)$ is a collection of rings and a singleton. It follows from Remark \ref{remark_m2} that $\mathcal G(x)$ cannot have rings with more than three nodes. If all the rings have length three, there is a nonzero probability that the singleton $ s$ will link to two adjacent nodes $ j$ and $i $ of a ring $ \lbrace i,j,k\rbrace $. In this case it is easy to verify that $ \tau_j^i =\tau_s^i$, and so there is a nonzero probability to end up in a configuration as in Fig.\ \ref{fig:proof}(c), which has been proved not to be a Nash equilibrium. It follows that $\mathcal G(x)$ cannot contain singletons.
\item[-] $\mathcal G(x)$ is a collection of rings and a 2-clique. By Remark \ref{remark_m2}, $\mathcal G(x)$ cannot have rings with more than three nodes, so all the rings have length three. It follows that the 2-clique can either form configurations (b) or (c) in Fig.\ \ref{fig:trasitionButterfly} or configuration (a) in Fig.\ \ref{fig:C3_nonStrictNash3}, which are all Nash equilibria. Hence $\mathcal G(x) \in \mathcal G_b^3$.
\item[-] $\mathcal G(x)$ is a collection of rings and a Butterfly graph. As shown in Fig.\ \ref{fig:trasitionButterfly}, there is a nonzero probability to end up in the previous case, which implies that all the rings have length three. Consequently, $\mathcal G(x) \in \mathcal G_b^3$.
\end{description}
We just proved that $\mathcal N^* \setminus \mathcal N^{st} \subseteq \mathcal G_b^3 $. At the same time, every $ x\in \mathcal G_b^3 $ is a Nash equilibrium, so $ \mathcal G_b^3 \subseteq \mathcal N^* \setminus \mathcal N^{st}$. Hence $\mathcal N^*= \mathcal N^{st} \cup \mathcal G_b^3$, noticing that $\mathcal G_b^3$ is not empty if and only if $(n\!\mod 3) =1$.
$ \qquad\qquad\qquad\qquad\qquad\qquad\qed $
\begin{figure}
\centering
(a)
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(q_1) edge node {} (q_3)
;
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(q_3) edge node {} (q_2)
(q_1) edge node {} (q_2)
(q_4) edge node {} (q_3)
(q_4) edge node {} (q_5)
;
\end{tikzpicture}
(b)
\begin{tikzpicture}[roundnode/.style={circle,fill=black!30, inner sep=2pt, minimum size=3mm},node distance=0.3cm,line width=0.2mm,scale=0.5]
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\path[->] (q_5) edge node {} (q_3)
(q_4) edge node {} (q_3)
;
\path[-]
(q_3) edge node {} (q_2)
(q_1) edge node {} (q_2)
(q_1) edge node {} (q_3)
(q_4) edge node {} (q_5)
;
\end{tikzpicture}
(c)
\begin{tikzpicture}[roundnode/.style={circle,fill=black!30, inner sep=2pt, minimum size=3mm},node distance=0.3cm,line width=0.2mm,scale=0.5]
\node[roundnode] (q_1) at (0,0) {};
\node[roundnode] (q_2) at (0,-2) {};
\node[roundnode] (q_3) at (-2,-1) {};
\node[roundnode] (q_4) at (2,0) {};
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(q_4) edge node {} (q_1)
;
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(q_1) edge node {} (q_3)
(q_3) edge node {} (q_2)
(q_1) edge node {} (q_2)
(q_4) edge node {} (q_5)
;
\end{tikzpicture}
\caption{Transitions of the Butterfly graph.}
\label{fig:trasitionButterfly}
\end{figure}
\section{Conclusion}\label{conclusions}
In this paper we proposed a game in which every node of a network aims at maximizing its Bonacich centrality by choosing where to direct its out-links, whose number is fixed to be equal to $ m $. We have completely characterized the sets $ \mathcal N^{\text{st}} $, $ \mathcal N^* $ and $ \mathcal N $ of Nash equilibria when $ m=1 $ and the sets $ \mathcal N^{\text{st}} $ and $ \mathcal N^* $ when $ m=2 $, case in which we have also provided necessary conditions for a configuration $ x $ to be in $ \mathcal N $. Our results show that the centrality maximization performed by each node tends to create disconnected and undirected networks, partially due to the locality property of the best response actions. In particular, both for $ m=1 $ and $ m=2 $ all the $ m $-regular undirected networks result to be (strict) Nash equilibria. A natural follow-up of our work would be the analysis of Nash equilibria of the game for a general $ m $, possibly in an heterogeneous setting where $ m $ is different for each node. Preliminary numerical experiments show that this tendency to create disconnected networks show up also for bigger $ m $, and that the problem becomes much more complex. In particular, it seems that the set of Nash equilibria depends also on the parameter $ \beta $ and that not all $ m $-regular undirected networks are Nash equilibria.
|
2,877,628,089,014 | arxiv | \section{Introduction}
The reliance on automated face recognition systems to authenticate a person's identity is becoming commonplace. Whether we are attempting to unlock our smartphones or cross a country border, success (or failure) is increasingly defined by a machine's ability to recognise the link between our faces and our identities. Despite the numerous security benefits associated with this form of authentication, there are growing privacy concerns over how our sensitive face data is being handled by the increasing number of applications that are demanding this personal information for identity management.
Modern face recognition systems are based on deep learning architectures, whereby a compact face representation (commonly referred to as an \textit{embedding}) is learned from several example images of a person's face. Recently, it has been shown that a face embedding can be inverted to recover an approximation of the original face image \cite{z16, c17, m19}, and that certain soft biometric attributes (e.g., sex, race, age, hair colour) can be extracted from the representative face embeddings \cite{f20, t20}. These findings indicate that face embeddings contain a wealth of personally identifiable information, which, if leaked from the system(s) in which they are employed, would represent a threat to the privacy of face recognition system users.
To ensure the public's trust in face recognition technologies, it is imperative that we establish effective means of protecting the employed face embeddings. This is especially important in light of the recent EU General Data Protection Regulation (GDPR)\footnote{\url{https://bit.ly/3nMM1Qz}}, which imposes a legal obligation to exercise caution in handling biometric data to protect individuals' digital identities. Considering the urgency of this matter in view of the widespread use of face recognition in practice, work investigating the protection of face embeddings is surprisingly limited.
This paper aims to contribute towards preserving the privacy of face recognition system users, by proposing a method for converting our sensitive face embeddings to more secure representations. In particular, we propose a new method, called \textit{PolyProtect}, which maps face embeddings to protected templates with the help of multivariate polynomials, whose parameters are defined separately for each user of the underlying face recognition system. Considering the proliferating use of face recognition in mobile devices, such as smartphones, we present a comprehensive evaluation of PolyProtect in a cooperative-user mobile face verification scenario, on two open-source face recognition systems. Our results indicate that PolyProtect shows promise as an effective face embedding protection scheme in practice.
The remainder of the paper is structured as follows. Section 2 outlines the main approaches to face embedding protection in the literature; Section 3 proposes our new protection method, PolyProtect; Section 4 analyses PolyProtect in terms of the three most common evaluation criteria; and Section 5 presents conclusions and avenues for future work.
\section{Face embedding protection methods}
It is generally agreed upon that a face embedding protection method should possess the following three properties:
\begin{enumerate}
\item \textbf{Recognition accuracy:} The incorporation of the protection method into a face recognition system should not result in a (significant) degradation of the system's recognition accuracy.
\item \textbf{Irreversibility:} It should be impossible (or computationally infeasible) to recover the original face embedding (or face image) from its protected version.
\item \textbf{Unlinkability:} It should be possible to generate multiple, sufficiently different protected templates from the same subject's face embeddings, such that the templates cannot be linked to the same identity. This would allow for the renewal (replacement) of compromised templates and the use of the same face identity across multiple applications, without the risk of cross-matching the protected templates.
\end{enumerate}
In the literature, there are two main types of approaches towards face embedding protection, as illustrated in Fig. 1. The first type of approach consists of learning face embeddings from face images using a neural network optimised for this task, then mapping the learned embeddings to protected templates using a \textit{separate, handcrafted} protection algorithm. The second type of approach involves training a neural network to \textit{learn} a suitable protection algorithm, to transform a face image to its protected template.
\begin{figure}[!ht]
\centering
\includegraphics[width=\columnwidth]{fig_1.png}%
\caption{The two main types of approaches to face embedding protection.}
\end{figure}
Examples of the \textit{first} type of approach to face embedding protection, include: \cite{p15, a20, a19, s19, r21, p19, b18}. These methods use some sort of algorithmically defined transformation to convert the face embeddings (learned from the input face images) to more secure representations. The proposed transformations include one-way cryptographic \cite{p15} or Winner Takes All \cite{a20} hashing, convolution of the embedding with a random kernel \cite{a19}, use of the Fuzzy Commitment \cite{s19} or Fuzzy Vault \cite{r21} scheme, fusion of a subject's face embedding with a different subject's face embedding using keys extracted from the two sets of features \cite{p19}, and homomorphic encryption \cite{b18}. The main issue with these approaches is that they have not been comprehensively evaluated in terms of their ability to simultaneously satisfy \textit{all three} properties of face embedding protection methods. More specifically, although an evaluation of the recognition accuracy is presented for all the methods, the irreversibility and unlinkability analyses often lack depth. In particular, the irreversibility tends to be either: (i) assumed (e.g., based on the secrecy of certain transformation parameters or on the reputed irreversibility of the employed transforms) but not empirically justified in the evaluation context, or (ii) estimated from a purely theoretical point of view that does not reflect the method's robustness to an inversion attack in practice (where certain theoretical assumptions are unlikely to hold). Similarly, the renewability of protected templates is usually simply assumed by virtue of the randomness of external parameters, but the unlinkability property is seldom experimentally validated. So, we do not have a complete picture of each method's strengths and weaknesses.
Examples of the \textit{second} type of approach to face embedding protection, include: \cite{p16, c18, k18, c19, t19, m20, p21, l21}. These methods \textit{learn} the protected template from the input face image (with face embeddings being extracted, in some format, during the process). In \cite{p16, c18, k18, c19}, a random code is \textit{pre-defined} for each subject during enrollment, then the neural network is trained to map different samples of the same subject's face to their (same) corresponding code. A cryptographic hash of the random code represents the protected face template. The main limitation of these approaches is that, since the protected templates are pre-defined, we would need to re-train the neural network for the enrollment of each new user or the re-enrollment of existing users whose protected templates have been compromised. To get around this problem, the methods in \cite{t19, m20, p21, l21} train a neural network to learn its \textit{own representation} of a protected template, instead of training it to learn a mapping to a pre-defined (pre-hash) code. In \cite{t19}, the neural network is trained to map face images to intermediate binary codes then correct errors in these binary codes (followed by cryptographic hashing of the error-corrected codes). Since the network is trained to learn the same code for every presentation of the same subject's face, renewal of compromised protected templates would be impossible. To enable template renewability, \cite{m20, p21, l21} train their neural networks to incorporate external, user-specific randomness into the process of learning the protected templates. These methods appear promising in their ability to generate renewable and unlinkable protected templates, but there is no analysis providing insight into the expected scalability of the renewability effort before neural network re-training may need to be invoked. Furthermore, the irreversibility of most of the methods in the \textit{type 2} protection approach has been evaluated in terms of the \textit{final protected template}, based on certain assumptions (e.g., the secrecy of user-specific parameters or the one-way property of cryptographic hash functions); however, there has been no consideration of how irreversibility may be affected by potential information leakage in different layers of the trained neural network, assuming a fully-informed attacker with access to the network and its learned parameters. So, we have insufficient evidence of the methods' irreversibility in practice, particularly for a fully-informed attacker.
Considering both types of approaches towards face embedding protection (Fig. 1), the \textit{second} type appears to be gaining traction recently. This is because the promise of designing a potentially complex protection algorithm without the need to explicitly define it, is attractive. Unfortunately, this approach has several limitations. Firstly, the protection method is specific to the neural network within which it has been trained, so it cannot be readily adopted for the protection of face embeddings generated by other face recognition models. Secondly, template renewability appears infeasible or impractical for all but methods such as \cite{m20, p21, l21}; however, more empirical evidence is needed to justify the scalability of these methods without invoking network re-training. Thirdly, when we train a neural network to \textit{learn} a protection algorithm for the input face images, there is often uncertainty about what exactly the network is learning at each stage of the process, which makes it difficult to perform a comprehensive evaluation of the irreversibility of the resulting protected templates. This is because, if we assume the most challenging threat scenario where an adversary has access to the trained model (i.e., network architecture and all learned parameters), then the irreversibility analysis should consider how this knowledge could be used to extract additional information about the original face embedding or image from different layers of the neural network. Thus far, none of the face embedding protection methods in the literature has presented such a thorough irreversibility analysis, so we do not have a fair picture of how irreversible the protected face templates would be in practice.
For the aforementioned reasons, our work focuses on the \textit{first} type of approach, where a handcrafted protection algorithm is applied to learned face embeddings. This approach allows for: (1) a flexible protection method that can be applied to face embeddings generated using different face recognition models; (2) the enrollment of new users or re-enrollment of compromised users without the need to re-train any neural network; and (3) a more precise definition of the protection algorithm (without the uncertainty in neural-network-based learning), and thus a better understanding of appropriate evaluation techniques. Although a few face embedding protection methods in this category have already been proposed (e.g., \cite{p15, a20, a19, s19, r21, p19, b18}), they are incomplete in their evaluations (as explained earlier), making it difficult to draw concrete conclusions on their expected robustness in practice.
In light of this discussion, the main contribution of this paper is a new protection method, PolyProtect, which transforms face embeddings to their protected counterparts via user-specific multivariate polynomials. The following two sections describe PolyProtect and present an evaluation of its suitability as a face embedding protection method. It should be emphasized that our focus was on evaluating PolyProtect from a \textit{practical} point of view, particularly when analysing its irreversibility, which is usually neglected in favour of theoretical approaches in the literature. The presented results are, therefore, more tangible and realistic than the idealistic outcomes of purely theoretical evaluations, which makes it easier to grasp PolyProtect's practical value. Although theoretical evaluations can be valuable in certain cases, we would nevertheless encourage other researchers to also consider adopting practical methodologies when evaluating their proposed protection methods. This would help to provide a clearer picture of the methods' robustness in practice, thereby allowing for more direct method comparisons in specific application contexts.
\section{PolyProtect}
This section proposes PolyProtect, a new method for protecting face embeddings in neural-network-based face recognition systems. Let $V = [v_1, v_2, ..., v_n]$ denote an \textit{n}-dimensional, real-number face embedding. The aim of PolyProtect is to map $V$ to another real-number feature vector, $P = [p_1, p_2, ..., p_k]$ (where $k < n$), which is the protected version of $V$. This is achieved by mapping sets of $m$ (where $m << n$) consecutive elements from $V$ to single elements in $P$ via multivariate polynomials defined by a set of $m$ user-specific (i.e., distinct for each user of the face recognition system), ordered, unique, non-zero integer coefficients, $C = [c_1, c_2, ..., c_m]$, and exponents, $E = [e_1, e_2, ..., e_m]$.
The first $m$ consecutive elements of $V$ (i.e., $v_1, v_2, ..., v_m$) are mapped to the first element in $P$ (i.e., $p_1$) via Eq. (1):
\begin{equation}
p_1 = c_{1}v_{1}^{e_1} + c_{2}v_{2}^{e_2} + ... + c_{m}v_{m}^{e_m}
\end{equation}
The elements of $V$ used to generate $p_2$ depend on the desired amount of \textit{overlap} between successive sets of elements. The \textit{minimum} overlap is 0, in which case the elements of $V$ in each set would be unique. The \textit{maximum} overlap is $m - 1$, in which case successive element sets would share $m - 1$ elements. Eqs. (2) and (3) define the mapping from $V$ to $p_2$ for overlaps of 0 and $m - 1$, respectively:
\begin{equation}
p_2 = c_{1}v_{m+1}^{e_1} + c_{2}v_{m+2}^{e_2} + ... + c_{m}v_{m+m}^{e_m}
\end{equation}
\begin{equation}
p_2 = c_{1}v_{2}^{e_1} + c_{2}v_{3}^{e_2} + ... + c_{m}v_{m+1}^{e_m}
\end{equation}
The remaining elements in $P$ (i.e., $p_3, ..., p_k$) are generated in a similar manner, until all the elements in $V$ have been used up. If the last set of elements from $V$ is incomplete because the dimensionality of $V$ is not divisible by the required number of element sets (defined by $m$ and the amount of overlap), $V$ is padded by a sufficient number of zeros to complete the last set. Fig. 2 illustrates the mapping from $V$ to $P$ for overlaps of 0 to 4, when $V$ consists of 128 elements and $m = 5$.
\begin{figure*}
\centering
\includegraphics[width=\textwidth]{fig_2.png}%
\caption{Mapping 128-dimensional $V$ to $P$ via PolyProtect, using $C = [c_1, c_2, ..., c_5]$ and $E = [e_1, e_2, ..., e_5]$, for different amounts of overlap.}
\end{figure*}
From Fig. 2, it is evident that the dimensionality of $P$ is influenced by the amount of overlap used in the $V \rightarrow P$ mapping. Using an overlap of 0 results in the \textit{smallest} $P$ (consisting of 26 elements), and using an overlap of 4 results in the \textit{largest} $P$ (consisting of 124 elements). So, it is reasonable to conclude that, the greater the amount of overlap, the more information from $V$ will be contained in $P$. This will be shown to have an effect on the recognition accuracy and irreversibility properties of PolyProtect, which are evaluated in Section 4.
The main idea behind PolyProtect was to design a protection algorithm that introduces user-specific, tuneable non-linearities to a face embedding, such that the resulting protected template would be irreversible even if all the parameters of the mapping are known. The use of multiple multivariate polynomials with user-specific coefficients and exponents thus seemed like a natural choice. The requirement that the mapping from a face embedding to its protected template be user-specific, was motivated by a desire to satisfy the unlinkability property (see Section 1). Furthermore, we wished to be able to easily tune the protection algorithm to control the trade-off between the recognition accuracy and irreversibility properties, which is achievable in PolyProtect primarily by varying the amount of overlap. Since PolyProtect relies on \textit{user-specific} parameters ($C$ and $E$), its envisioned operating scenario is in cooperative-user face verification applications (such as proving one's identity in order to unlock a personal smartphone).
To the best of our knowledge, PolyProtect represents a novel approach towards face embedding protection. Although the reader may be tempted to liken PolyProtect to the Fuzzy Vault scheme \cite{js06} due to the use of polynomials in both methods, there are actually a number of fundamental differences between the two approaches. Firstly, the Fuzzy Vault scheme operates on \textit{unordered} sets of elements, whereas PolyProtect relies on an \textit{ordered} feature vector (e.g., an embedding). Secondly, the polynomial used in the Fuzzy Vault scheme is \textit{univariate} (i.e., each element in the biometric template serves as the input to the polynomial in turn), whereas PolyProtect's polynomials are \textit{multivariate} (i.e., groups of elements from the biometric template are simultaneously passed to the polynomial). Thirdly, the Fuzzy Vault scheme relies on the addition of random data points (\textit{chaff points}) to hide the polynomial outputs, whereas this is \textit{not} a feature of PolyProtect. Fourthly, the verification operation in the Fuzzy Vault scheme requires the reconstruction of the secret polynomial to extract its coefficients, which serve as the user's secret \textit{key}. The verification of PolyProtected templates does \textit{not} require polynomial reconstruction nor the extraction of a secret key; instead, PolyProtected templates are compared directly using a distance/similarity metric. So, we may conclude that PolyProtect and the Fuzzy Vault scheme represent fundamentally different approaches to biometric template protection.
Although its construction suggests the suitability of PolyProtect for protecting \textit{any} real-number biometric feature vector, the focus of this paper is on its applicability to \textit{face embeddings} alone. Similarly, although PolyProtect is envisioned for adoption in cooperative-user verification scenarios in general, the evaluation presented in this paper will target only a \textit{mobile application context}. This is because: (i) facial recognition in mobile devices (e.g., smartphones) is one of the most (if not \textit{the} most) common face verification applications in practice, and (ii) this type of facial recognition represents the least constrained (and thus most challenging) cooperative-user face verification scenario. The suitability of PolyProtect to serve as an effective face embedding protection scheme in this scenario in practice, is evaluated in Section 4.
\section{Analysis of PolyProtect for face\\ embeddings in mobile face verification}
This section evaluates the suitability of PolyProtect for securing face embeddings in a cooperative-user mobile face verification scenario. Section 4.1 describes our experimental set-up. Then, Sections 4.2 to 4.4, respectively, evaluate PolyProtect in terms of the three properties outlined in Section 1: recognition accuracy, irreversibility, and unlinkability.
\subsection{Experimental set-up}
To evaluate the suitability of PolyProtect for protecting face embeddings, we first needed to establish a \textit{baseline} deep-neural-network-based face recognition system, into which PolyProtect could be incorporated. We adopted \textit{two} open-source systems for this purpose, both implemented within the \texttt{bob.bio.face\_ongoing} PyPI package\footnote{\url{https://bit.ly/2XLsYLQ}}: \textit{facenet} and \textit{idiap\_msceleb\_inception\_v2\_centerloss\_rgb}, henceforth referred to as Facenet and Idiap, respectively. The main difference between the two systems lies in how they generate face embeddings from face images, which is defined by their adopted pre-trained deep neural network models: Facenet uses the open-source FaceNet model 20170512-110547 from David Sandberg\footnote{\url{https://bit.ly/39oYNMV}}, and Idiap uses a CNN model based on the Inception-ResNet-v2 architecture and trained on the MS-Celeb-1M dataset\footnote{\url{https://bit.ly/3Alwivn}}. Both the Facenet and Idiap systems work with 128-dimensional face embeddings, which were the inputs to our PolyProtect algorithm.
As noted earlier, we focus on face \textit{verification} rather than \textit{identification}. Fig. 3 illustrates the incorporation of PolyProtect into the enrollment and verification stages of our two baseline systems, which are differentiated by their feature extractors (i.e., deep neural network models trained to extract 128-dimensional face embeddings from face images).
\begin{figure*}[!h]
\centering
\includegraphics[width=\textwidth]{fig_3.png}%
\caption{Enrollment (green arrows) and verification (red arrows) in the PolyProtected Facenet and Idiap face recognition systems. $F^R$ and $F^Q$ are the reference and query input face images, $V^R$ and $V^Q$ are the corresponding 128-dimensional face embeddings, and $P^R$ and $P^Q$ are the PolyProtected templates. The PolyProtect mapping is defined by the user-specific coefficients, $C$, and exponents, $E$.}
\end{figure*}
Fig. 3 indicates that, during enrollment, the reference face embedding, $V^R$, is protected using PolyProtect (defined by the user-specific parameters $C$ and $E$), then the resulting protected template, $P^R$, is stored in the system's database. During verification, the query face embedding, $V^Q$, is likewise protected to generate $P^Q$. Then, $P^Q$ and $P^R$ are compared using the cosine distance metric, and the resulting score is processed to determine whether or not the underlying faces match. The cosine distance metric was chosen because the same metric was used to compare the unprotected face embeddings in our baseline systems (i.e., $V^R$ and $V^Q$), which makes it easier to evaluate the effect of PolyProtect on the systems' recognition accuracy. Technically, however, the score output by the Comparator in Fig. 3 is a \textit{similarity} score, because the cosine distances are multiplied by -1 (to turn them into similarity scores). So, the score range is [-2.0, 0.0], where -2.0 would indicate that $P^R$ and $P^Q$ (or $V^R$ and $V^Q$) are as different as possible, while 0.0 would imply that they are the same.
Note that the reason for selecting the Facenet and Idiap face recognition systems was that they were reported\footnote{\url{https://bit.ly/39jfCIT}} to have the best recognition accuracy when evaluated in the verification scenario on the face dataset that we deemed the most suitable for evaluating PolyProtect: Mobio \cite{m12}, which consists of bi-modal (audio and video) data captured from 152 people using two mobile devices (a phone and a laptop) in a cooperative-user scenario. We chose to evaluate PolyProtect on this dataset for four main reasons. Firstly, recall that the aim of this paper is to evaluate the suitability of PolyProtect for protecting face embeddings in a cooperative-user mobile face verification context. This is because the prevalence of mobile devices in our society, as well as current market trends, suggests that this is likely to be one of the (if not \textit{the}) biggest uses for face verification in practice (e.g., unlocking your smartphone with your face), so this represents PolyProtect's most likely application scenario. Since the Mobio database most closely imitates this target scenario, it was a very fitting choice for the PolyProtect evaluations presented in this paper. Secondly, the face videos in Mobio were captured in uncontrolled environments, so the face images extracted from the video frames are realistic and natural in terms of illumination, head poses, and facial expressions. Thirdly, the samples were acquired over 2 years across 5 countries, and 12 sessions in total were captured per person (database subject). This, together with the uncontrolled acquisition environment, makes the database challenging in the amount of session variability it exhibits. Fourthly, the dataset is publicly available\footnote{\url{https://bit.ly/39n3pDi}}, which allows for the reproducibility of our experiments.
To evaluate the PolyProtected systems, we first needed to establish three parameters: the value of $m$, and the ranges of $C$ and $E$. Recall that $m$ specifies the number of elements from the face embedding, $V$, used to generate each element in the protected template, $P$. We chose $m = 5$, meaning that each element in $P$ was generated using 5 consecutive elements from $V$, as illustrated for different overlaps in Fig. 2. Our thinking behind setting $m = 5$ was inspired by the Abel-Ruffini theorem, which states that there is no closed-form algebraic expression for solving polynomials of degree 5 or higher with arbitrary coefficients \cite{g18}. While this does not imply that it is impossible to find the roots of such polynomials, what it means is that a general expression does not exist (unlike, for example, for 2-degree polynomials). So, an attacker trying to reverse the $V \rightarrow P$ mapping to recover $V$ could not rely on an analytical approach using an existing, well-defined formula. While they could attempt to use a root-finding algorithm to find a numerical approximation for $V$, such methods are generally sensitive to initial guesses and are, therefore, prone to converging to a false solution. (More details on the feasibility of recovering $V$ from $P$ are provided in the irreversibility analysis in Section 4.3.) The reason we did not set $m > 5$ was because this would require using exponents larger than 5 in the PolyProtect mapping. Since the face embeddings consist of quite small floating point values, large powers would effectively obliterate certain embedding elements during the PolyProtect mapping. In the same vein of thought, the exponents, $E$, in our PolyProtected systems were randomly-permuted, unique integers in the range [1, 5].
The choice of a suitable range for the coefficients, $C$, was not as evident. We experimented with several $C$ ranges to generate the PolyProtected templates, but there appeared to be no significant differences in the resulting recognition accuracy of our PolyProtected face verification systems. We suspect that this is due to the use of the cosine distance metric in the comparison of PolyProtected templates. Since this metric calculates the difference between the \textit{directions} of the vectors being compared, their magnitudes are less important, so the effects of using a larger or smaller $C$ range are presumably diluted as a result. We expect that employing a magnitude-sensitive metric (e.g., Euclidean distance) would result in larger differences in the recognition accuracy of the PolyProtected systems when different $C$ ranges are employed; however, at this stage we have chosen to adopt the cosine distance metric for consistency with the comparison of unprotected face embeddings in the baseline systems, and we leave the investigation of alternative metrics to future work. For the evaluations presented in this paper, we used an arbitrarily selected $C$ range of [-50, 50], so all sets of $C$s consisted of 5 randomly-selected, unique, non-zero integers in this range.
We are now ready to present our evaluation of PolyProtect, when the protection method is incorporated into the Facenet and Idiap face verification systems, and when the analysis is performed on the Mobio dataset. The evaluation is based on the three properties of protection methods outlined in Section 1: recognition accuracy, irreversibility, and unlinkability. Sections 4.2 to 4.4, respectively, present the corresponding analysis. As mentioned in Section 2, our focus was on evaluating PolyProtect from a \textit{practical} point of view (as opposed to theoretical), to present a clearer picture of the method's practical value.
\subsection{Recognition accuracy}
This section investigates how the recognition accuracy of the PolyProtected face verification systems compares to that of the corresponding baseline (unprotected) systems. The aim of this analysis was to determine whether the incorporation of PolyProtect into a deep-neural-network-based face verification system would degrade its recognition accuracy.
To conduct this analysis, the recognition accuracy of each baseline system (Facenet and Idiap) was first evaluated on the Mobio dataset, by running the \textit{mobile0-male} verification protocol\footnote{\url{https://bit.ly/3nP9HnF}} used to generate the reported baseline results\footnote{\url{https://bit.ly/39jfCIT}}. The same protocol was then applied to the corresponding PolyProtected face verification systems.
The recognition accuracy of our PolyProtected face verification systems was
evaluated in two scenarios: (i) Normal (best-case), and (ii) Stolen Coefficients and Exponents (worst-case). The PolyProtected system should operate in the Normal scenario most of the time. Here, we assume that each enrolled user dutifully employs their own $C$ and $E$ parameters in the generation of their PolyProtected face templates, as envisioned by the design of the PolyProtect scheme. In the Stolen Coefficients and Exponents scenario, a user attempts to pass off as a different user by stealing the target's $C$ and $E$ parameters, and applying them to their own face embedding to generate their PolyProtected template. While it is reasonable to assume that the latter scenario should be uncommon in practice (provided that users' $C$ and $E$ parameters are stored securely), it is still important to consider this worst-case scenario (as is sometimes done in the literature) when analysing the expected recognition accuracy of PolyProtect in practice.
The Normal (N) scenario was simulated by randomly generating a set of \textit{different} $C$ and $E$ parameters for \textit{each subject}, then applying those parameters to map each of the subject's reference and query embeddings to their corresponding PolyProtected templates. So, for reference and query face embeddings from the \textit{same} subject, the \textit{same} $C$ and $E$ parameters were used to generate their corresponding PolyProtected templates, while reference and query face embeddings from \textit{different} subjects were protected using \textit{different} $C$ and $E$ parameters (since these parameters are subject/user-specific). Next, comparison scores were computed between all required pairs of reference and query PolyProtected templates (as defined by the aforementioned verification protocol), resulting in a set of \textit{genuine} scores (when the templates being compared originate from the \textit{same} subject) and a set of \textit{impostor} scores (when the templates originate from \textit{different} subjects).
To simulate the Stolen Coefficients and Exponents (SCE) scenario, we assumed the most extreme scenario where each subject's $C$ and $E$ parameters are stolen and used by the other subjects. So, for all query face embeddings that were meant to be compared to a particular reference embedding (according to the adopted verification protocol), the \textit{same} $C$ and $E$ parameters (i.e., those belonging to the reference identity) were used to generate \textit{all} the corresponding query PolyProtected templates. Consequently, although the genuine scores were calculated in the same way for both the N and SCE scenarios, the impostor scores were calculated using \textit{subject}-specific parameters in the N scenario and \textit{reference}-specific parameters in the SCE scenario.
This process was repeated for 10 trials, where for each trial a new set of $C$ and $E$ parameters was chosen for each subject. (This may be interpreted as simulating 10 different applications in which the same subjects are enrolled.) Then, the genuine and impostor scores across the 10 trials were concatenated (separately), and the recognition accuracy was calculated on the concatenated scores. These same sets of $C$ and $E$ parameters (10 per subject) were applied to each PolyProtected system (Facenet and Idiap), as well as for each \textit{overlap} parameter defining the PolyProtect mapping. Fig. 4 depicts the resulting ROC plots, generated on the \textit{evaluation} set of the Mobio database, in the N and SCE scenarios. Each plot compares the verification accuracy of the corresponding baseline system against the verification accuracy of the PolyProtected system for different amounts of overlap used in the PolyProtect mapping.
\begin{figure}[!h]
\centering
\includegraphics[width=0.49\columnwidth]{fig_4_facenet_N.png}
\hfill
\includegraphics[width=0.49\columnwidth]{fig_4_idiap_N.png}
\vfill
\includegraphics[width=0.49\columnwidth]{fig_4_facenet_SCE.png}
\hfill
\includegraphics[width=0.49\columnwidth]{fig_4_idiap_SCE.png}
\caption{ROC plots comparing the baseline and PolyProtected face verification systems, in the N and SCE scenarios. The vertical dashed line in each plot corresponds to the match threshold at a False Match Rate (FMR) of $10^{-3}$ or 0.1\%, which is a commonly used criterion.}
\end{figure}
Additionally, Table 1 summarises the True Match Rates (TMR = 1 - FNMR) from Fig. 4, at match thresholds corresponding to FMR = \{0.01\%, 0.1\%, 1\%\} (i.e., FMR = \{$10^{-4}$, $10^{-3}$, $10^{-2}$\}). We present only the TMRs for PolyProtect's N scenario, since this is its intended operational scenario.
\begin{table}[!h]
\renewcommand{\arraystretch}{1.3}
\caption{TMR at match thresholds corresponding to different FMR values, for the baseline and PolyProtected (N scenario) systems.}
\centering
\begin{tabular}{|c|c|c|c|c|c|}
\hline
\multicolumn{3}{|c}{\multirow{2}{*}{\textbf{System}}} & \multicolumn{3}{|c|}{\textbf{TMR @ FMR = }} \\
\cline{4-6}
\multicolumn{3}{|c|}{} & \textbf{0.01\%} & \textbf{0.1\%} & \textbf{1\%} \\
\hline
\multirow{6}{*}{Facenet} & \multicolumn{2}{c|}{\textit{Baseline}} & \textit{98.87\%} & \textit{99.62\%} & \textit{99.87\%} \\
\cline{2-6}
& \multirow{5}{*}{\shortstack{PolyProtected:\\Overlap =}} & 0 & 89.39\% & 96.70\% & 99.45\% \\
\cline{3-6}
& & 1 & 95.69\% & 98.88\% & 99.79\% \\
\cline{3-6}
& & 2 & 98.47\% & 99.59\% & 99.85\% \\
\cline{3-6}
& & 3 & 99.05\% & 99.78\% & 99.87\% \\
\cline{3-6}
& & 4 & 99.59\% & 99.87\% & 99.87\% \\
\hline
\multirow{6}{*}{Idiap} & \multicolumn{2}{c|}{\textit{Baseline}} & \textit{98.80\%} & \textit{99.60\%} & \textit{99.85\%} \\
\cline{2-6}
& \multirow{5}{*}{\shortstack{PolyProtected:\\Overlap =}} & 0 & 91.54\% & 96.98\% & 99.69\% \\
\cline{3-6}
& & 1 & 93.15\% & 97.82\% & 99.74\% \\
\cline{3-6}
& & 2 & 93.86\% & 98.76\% & 99.88\% \\
\cline{3-6}
& & 3 & 98.86\% & 99.78\% & 99.93\% \\
\cline{3-6}
& & 4 & 99.66\% & 99.85\% & 99.95\% \\
\hline
\end{tabular}
\end{table}
There are several important observations from Fig. 4 and Table 1. Firstly, in both the N and SCE scenarios, the recognition accuracy of the PolyProtected systems generally improves as the amount of overlap increases. This makes sense, because using a larger overlap in the PolyProtect mapping results in the generation of a higher-dimensional PolyProtected template (see Fig. 2), which contains more information from the original face embedding. So, the larger the overlap, the better the expected recognition accuracy.
Secondly, for the N scenario, at the most commonly used match threshold at FMR = $10^{-3}$ (or 0.1\%), marked by the vertical dashed line in Fig. 4, the recognition accuracy of both the Facenet and Idiap PolyProtected systems is slightly \textit{better} than that of the corresponding baseline, when an overlap of 3 or 4 is used in the PolyProtect mapping (see Table 1 for the exact values). This is despite the fact that these PolyProtected systems work with templates of lower dimensionality (63 and 124 for overlaps of 3 and 4, respectively) than that of the face embeddings used in the baseline systems (128). So, this improvement in the recognition accuracy of the aforementioned PolyProtected systems is most probably due to the use of user-specific $C$ and $E$ parameters in the generation of the PolyProtected templates, which increases the separation between different users in the protected feature domain. Although the recognition accuracy of the PolyProtected systems using overlaps of 0-2 is a little worse than that of the baseline for both Facenet and Idiap at the same match threshold, even the worst recognition accuracy (i.e., at overlap = 0) is still very good, resulting in a TMR of almost 97\%. Furthermore, if a higher FMR can be tolerated, then the difference in the recognition accuracy of all PolyProtected systems and the corresponding baseline becomes fairly insignificant, reaching approximate equivalence at an FMR of $10^{-2}$ (1\%). Finally, both Fig. 4 and Table 1 suggest that setting the match threshold to the stricter value corresponding to an FMR of $10^{-4}$ (0.01\%) results in a fairly significant degradation in the recognition accuracy of PolyProtected systems using an overlap of 0 (compared to the corresponding baseline). Although this drop in the recognition accuracy may be considered acceptable in some application scenarios, in practice the overlap parameter should be \textit{tuned} according to the desired ``recognition accuracy versus irreversibility'' trade-off (explored in the irreversibility analysis in Section 4.3). Having said that, we should emphasize that the aim of this paper is to evaluate PolyProtect in a cooperative-user mobile face verification context, which is the most likely operational scenario for PolyProtect in practice. In this case, a match threshold at an FMR of 0.01\% (or lower) is unlikely to be necessary, since we would not expect anywhere near 10,000 impostor verification attempts. In fact, a match threshold at an FMR of 0.1\% (corresponding to an impostor accept rate of 1 in 1,000) should be more than adequate for this application scenario, and applications where impostor access attempts are even less likely may adopt a more lenient match threshold (e.g., at an FMR of 1\%, which corresponds to 1 in 100 impostors being accepted).
Thirdly, Fig. 4 indicates that, in the SCE scenario, the recognition accuracy of the PolyProtected systems is worse, in general, than the baselines. This may be attributed to the fact that, in this scenario, we are essentially performing a dimensionality reduction in the mapping from each subject's face embedding to their PolyProtected template, without the benefits of the additional user-specific information as in the N scenario. Consequently, the amount of discriminative information in the protected templates may be expected to be less than that in the unprotected face embeddings, resulting in lower recognition accuracy. Nevertheless, even in this worst-case scenario, the recognition accuracy of both the Facenet and Idiap PolyProtected systems using an overlap of 4 is almost equivalent to that of the corresponding baselines at FMR = $10^{-3}$. Even the PolyProtected systems using an overlap of 3 appear to perform very well, with a TMR (1 - FNMR) of around 0.98 (98\%) for Facenet and 0.97 (97\%) for Idiap. Furthermore, similarly to the observation made for the N scenario, a higher FMR tolerance in the employed systems would ensure that all PolyProtected systems (i.e., for all overlaps) achieve high recognition accuracy even in the SCE scenario; for example, at an FMR of $10^{-2}$ (1\%) in Fig. 4, the TMR for all PolyProtected systems is over 0.95 (95\%), and the performance of PolyProtected systems using overlaps of 3 and 4 achieves equivalence with the baselines.
In summary, our analysis indicates that the recognition accuracy of a PolyProtected system depends on the amount of overlap used in the PolyProtect mapping, with the accuracy improving as the overlap increases. When the system operates in the envisioned (N) scenario, the recognition accuracy may be expected to be higher than, approximately equivalent to, or not significantly worse than, that of the baseline system (using unprotected face embeddings), depending on the chosen match threshold. In the worst-case (SCE) scenario where all users' parameters are stolen, the recognition accuracy of a PolyProtected system may be worse than that of the baseline; however, our results indicate that the performance can be improved by tuning the amount of overlap used in the PolyProtect mapping as well as the FMR tolerance of the underlying system. So, although this worst-case scenario is highly unlikely to occur in practice\footnote{It is unlikely that each user would steal \textit{all} the other users' PolyProtect parameters, especially at the same time. So, the N scenario results are more indicative of the expected recognition accuracy.}, even in this case the recognition accuracy should be acceptable, ensuring that the system does not suffer much in the time it takes to replace the compromised PolyProtected template(s). We may thus reasonably conclude that PolyProtect is capable of satisfying the \textit{recognition accuracy} property of a face embedding protection scheme.
It would be useful to present quantified comparisons of the recognition accuracy of PolyProtect to that of other face embedding protection schemes in the literature, but it is currently not possible to do this fairly. For the comparison to be as fair as possible, the following conditions should be satisfied: (1) the methods should be evaluated on the \textit{same dataset}, and to ensure that the comparison remains relevant to the context of the PolyProtect evaluation presented in this paper, this dataset should be Mobio; (2) the raw embeddings should be generated using the \textit{same feature extraction} process, and we should be able to \textit{de-couple} the feature extraction and protection steps, to ensure that the comparison targets \textit{solely} the protection algorithms; (3) we should have access to the methods' implementation code, to allow for better insight into whether or not a fair comparison would be possible. Condition (1) is not satisfied, because none of the existing face embedding protection methods have been evaluated on the Mobio dataset. Since the aim of this paper is to evaluate PolyProtect's suitability for adoption in a mobile face verification scenario, and as no other face embedding protection method has been evaluated on a database that is equally/more suitable (compared to Mobio) for representing this target scenario, it is not possible to present a fair numerical comparison against the recognition accuracy results reported for the existing protection methods. Conditions (2) and (3) are satisfied only by \cite{b18}, which employs fully homomorphic encryption to protect 128-dimensional embeddings generated using the same open-source FaceNet model adopted for our Facenet baseline. A known property of homomorphic encryption methods is their ability to ensure approximately zero loss in the recognition accuracy; indeed, \cite{b18} showed that this is achievable regardless of the evaluation dataset, depending on the precision of the face embedding quantisation scheme and the adopted match threshold. So, without an explicit comparison on the Mobio dataset, we may reasonably conclude that PolyProtect appears comparable to \cite{b18} in its ability to be tuned to approximately maintain the recognition accuracy of the baseline (unprotected) system, when the protected system operates as intended (N scenario for PolyProtect, and the use of user-specific encryption/decryption keys for \cite{b18}).
Having made this comparison, we also note that the use of PolyProtect may actually cause an \textit{increase} in the resulting recognition accuracy (as discussed in our analysis of Fig. 4 and Table 1), while this would not be possible using the homomorphic encryption method in \cite{b18}. On the other hand, although \cite{b18} does not present the equivalent of PolyProtect's SCE scenario (i.e., stolen encryption/decryption keys), we may expect that the use of \textit{same} encryption/decryption keys to enroll \textit{different} subjects in \cite{b18} would produce approximately the same comparison scores as for the \textit{user-specific} key scenario. Despite this potential advantage of \cite{b18} over PolyProtect in the worst-case (albeit practically unlikely) scenario, the main pitfall of \cite{b18} is that the face embeddings are secure only insofar as the decryption keys remain secret. PolyProtected templates, on the other hand, do not rely solely on the secrecy of the user-specific parameters to protect the original face embeddings. An analysis of the irreversibility of PolyProtect is presented in Section 4.3.
\subsection{Irreversibility}
A face embedding protection method is considered irreversible (non-invertible) if it is impossible (or computationally infeasible) to recover the original (unprotected) face embedding from its protected template. This section investigates the irreversibility of PolyProtect. Section 4.3.1 defines our adopted threat model, on which the subsequent irreversibility analysis will be based. We then consider the difficulty of recovering the original face embedding from one (Section 4.3.2) or more (Section 4.3.3) PolyProtected templates from the same subject.
\subsubsection{Threat model}
To analyse PolyProtect's irreversibility, we must first define our \textit{threat model}, as specified in ISO/IEC 30316 (the international standard on performance testing of biometric template protection schemes)\footnote{\url{https://bit.ly/3hLRnYM}}. The threat model characterises the type of attacker on which we wish to base our irreversibility analysis. The most difficult threat model in ISO/IEC 30316 is referred to as a \textit{full disclosure model}, which assumes that the attacker knows everything there is to know about the protection method (e.g., algorithms, secrets, etc.). Since this type of attacker would represent the worst-case scenario in practice, we decided to base our analysis of the irreversibility of PolyProtect on the \textit{full disclosure model}.
In the context of PolyProtect, we define the full disclosure threat model as assuming that the attacker has access to the following information: knowledge of the PolyProtect algorithm, including the number of embedding elements ($m$) used to generate each PolyProtected element, the amount of \textit{overlap} used in the PolyProtect mapping, as well as the user-specific $C$ and $E$ parameters that define PolyProtect's polynomials; one or more PolyProtected templates, $P$, corresponding to a particular face embedding, $V$; and knowledge of a face embedding element distribution, which is representative of the face embeddings used to create the PolyProtected templates to which the attacker has access. The attacker's goal, therefore, is to use all this information to attempt to recover a subject's original face embedding, $V$, from one or more of their PolyProtected templates, $P$.
Recall that, in our experimental set-up (see Section 4.1), a face embedding consists of 128 elements, i.e., $V = [v_1, v_2, ..., v_{128}]$. In the PolyProtect irreversibility analysis, the attacker's aim is to recover these 128 elements from the corresponding PolyProtected template, $P = [p_1, p_2, ..., p_k]$ (where $k$ depends on the amount of overlap used in the PolyProtect mapping, as illustrated in Fig. 2). Since our full disclosure threat model assumes that the attacker has some knowledge of the distribution of these 128 embedding elements, the first step in analysing the irreversibility of PolyProtect was to estimate this distribution. Note that, as defined by the Mobio database protocol we adopted, the face embeddings used in our experiments were divided into two sets: \textit{development} and \textit{evaluation}. Each set was further split into \textit{reference} embeddings (used for enrollment) and \textit{query} embeddings (used for verification). Since the intention of the adopted database protocol is to report results on the evaluation set, we may consider the evaluation set's reference face embeddings as the embeddings that are used to generate the \textit{enrolled} PolyProtected templates. It is, therefore, precisely these embeddings that the attacker in our irreversibility analysis is trying to recover. This means that the attacker should \textit{not} have access to these embeddings; however, they may be assumed to have access to the development set, since these embeddings are not used for enrollment\footnote{In general, the purpose of the development set is to establish certain evaluation parameters, such as the match threshold.}. The attacker could, therefore, use the \textit{development} set's reference face embeddings to estimate the distribution of the 128 elements in the \textit{evaluation} set's reference embeddings, which they are attempting to recover.
Our irreversibility analysis thus began with an estimation of the probability distribution for each of the 128 elements in a face embedding, using the reference embeddings from Mobio's \textit{development} dataset, separately for our Facenet and Idiap baseline systems. This means that each embedding element was considered separately across all the reference embeddings, to estimate its corresponding probability distribution. So, for each of our two baseline systems, we ended up with 128 different probability distributions, one for each of the 128 embedding elements.
Note that these probability distributions reveal a lot of information about the underlying embeddings (and thus the face images from the Mobio database), so we are unable to publish the distributions in this paper, because access to the Mobio dataset is granted based on an end-user license agreement. Since our PolyProtect code is publicly available, the interested reader may generate these results for themselves upon gaining lawful access to Mobio.
The probability distributions were next used to estimate the irreversibility of PolyProtect when the fully-informed attacker has access to one (Section 4.3.2) or more (Section 4.3.3) PolyProtected templates from the same person.
\subsubsection{Access to single PolyProtected template}
This section considers the feasibility of recovering the original face embedding, $V$, from its PolyProtected template, $P$. A fully-informed attacker could attempt this $P \rightarrow V$ inversion in one of two ways: (i) analytically, or (ii) numerically.
An analytical approach towards inverting $P$ would involve re-arranging the system of multivariate polynomial equations used in the $V \rightarrow P$ mapping (see Section 3), to obtain an explicit solution for each variable, $v_i$ ($i = 1, 2, ..., n$), in $V = [v_1, v_2, ..., v_n]$. Note that a \textit{variable} corresponds to an element in the original face embedding, so in the case of our experimental set-up, which employed 128-dimensional face embeddings, there are 128 variables, i.e., $V = [v_1, v_2, ..., v_{128}]$. This means that our $V \rightarrow P$ mapping is a mapping from a 128-dimensional space to a lower-dimensional space, where the dimensionality of the PolyProtected template, $k$, is determined by the amount of overlap used (see Fig. 2). Consequently, the inverse mapping, $P \rightarrow V$, would be a mapping from the $k$-dimensional space to the 128-dimensional space. These forward and inverse mappings are summarised in Table 2.
\begin{table}[!h]
\renewcommand{\arraystretch}{1.3}
\caption{$V \rightarrow P$ and $P \rightarrow V$ mappings for different overlap amounts.}
\centering
\begin{tabular}{|c|c|c|}
\hline
\textbf{Overlap} & $\mathbf{V \rightarrow P}$ & $\mathbf{P \rightarrow V}$ \\
\hline
0 & $\mathbb{R}^{128} \rightarrow \mathbb{R}^{26}$ & $\mathbb{R}^{26} \rightarrow \mathbb{R}^{128}$ \\
\hline
1 & $\mathbb{R}^{128} \rightarrow \mathbb{R}^{32}$ & $\mathbb{R}^{32} \rightarrow \mathbb{R}^{128}$ \\
\hline
2 & $\mathbb{R}^{128} \rightarrow \mathbb{R}^{42}$ & $\mathbb{R}^{42} \rightarrow \mathbb{R}^{128}$ \\
\hline
3 & $\mathbb{R}^{128} \rightarrow \mathbb{R}^{63}$ & $\mathbb{R}^{63} \rightarrow \mathbb{R}^{128}$ \\
\hline
4 & $\mathbb{R}^{128} \rightarrow \mathbb{R}^{124}$ & $\mathbb{R}^{124} \rightarrow \mathbb{R}^{128}$ \\
\hline
\end{tabular}
\end{table}
Considering Table 2 alongside the description of PolyProtect in Section 3, we can interpret the presented mappings as follows. When an overlap of 0 is used, the $V \rightarrow P$ mapping is defined by 26 equations in 128 variables (unknowns), resulting in a 26-dimensional PolyProtected template, $P$. Based on the description of PolyProtect in Section 3, we know that each of these 26 equations consists of a unique set of 5 variables. So, it is reasonable to assume that the 26 equations are linearly independent, meaning that the inverse mapping ($P \rightarrow V$) cannot be uniquely defined, due to the $128 - 26 = 102$ degrees of freedom.
Similar observations can be made for overlaps 1 to 4; however, in this case, each equation does not consist of an entirely unique set of variables, since there is \textit{some} overlap between the different sets of embedding elements used to generate each PolyProtected element, meaning that there are fewer degrees of freedom in the inverse mapping. Nevertheless, since every equation consists of at least one variable that is not used by any of the other equations, it is still reasonable to assume that all the equations, for all overlaps, are linearly independent. Consequently, we may conclude that the inverse mapping, $P \rightarrow V$, is defined by an \textit{underdetermined} system of equations and, therefore, technically does not exist. This is because there are (theoretically) infinitely many solutions for the elements in $V$ that could produce $P$, so there is \textit{no unique solution}.
Furthermore, as discussed in Section 4.1, the Abel-Ruffini theorem states that there is no closed-form algebraic expression for solving polynomials of degree 5 or higher. Since we used $m = 5$ in our PolyProtect implementation, each equation in the $V \rightarrow P$ mapping is a multivariable 5-degree polynomial. So, we may deduce that an analytical solution for $P \rightarrow V$ does \textit{not} exist, implying that PolyProtected templates are \textit{theoretically irreversible}.
Although PolyProtect is irreversible in theory, in practice there will not be an infinite number of solutions for $V$. This is because the values of $V$ will be limited in some way (e.g., by the range and precision of possible values) depending on the implementation of the underlying face recognition system. Consequently, the number of valid solutions will be constrained in practice. Furthermore, the mathematical impossibility of deriving an \textit{exact} solution to the problem of recovering $V$ from $P$ may present a smaller obstacle in practice, where an attacker might be satisfied with an \textit{approximation} of $V$. So, although the inverse mapping, $P \rightarrow V$, cannot be explicitly defined, a real-world attacker may attempt to use a numerical solver to converge to an approximate solution for $V$ from a set of initial guesses.
To estimate the feasibility of such an attempt in practice, we simulated this irreversibility attack using an open-source numerical solver: Python's \textit{scipy.optimize.root} function with the \textit{lm} method. This method adopts the Levenberg-Marquardt algorithm, which approximates a solution to a non-linear system of equations using a damped least-squares approach. This algorithm is reported to be very reliable for solving non-linear, medium-sized (i.e., a few hundred variables) optimization problems in practice\footnote{\url{https://bit.ly/3tSGKbo}}, and can be applied to underdetermined systems of equations\footnote{\url{https://bit.ly/3nQcvkc}}. Since the recovery of $V$ from $P$ is represented as an underdetermined 128-variable system of non-linear equations, we deemed this method very suitable for our purposes.
The first step in attempting to recover an approximation of the original face embedding, $V$, from its PolyProtected template, $P$, using the aforementioned numerical solver, was to set up a system of $k$ equations (where $k$ corresponds to the dimensionality of $P$, as per Table 2) for each overlap amount. For example, the system of 26 equations for overlap = 0 (see Fig. 2) was set up for the numerical solver as follows:
\begin{align*}
c_{1}v_{1}^{e_1} + c_{2}v_{2}^{e_2} + c_{3}v_{3}^{e_3} + c_{4}v_{4}^{e_4} + c_{5}v_{5}^{e_5} - p_{1} &= 0 \\
\\
c_{1}v_{6}^{e_1} + c_{2}v_{7}^{e_2} + c_{3}v_{8}^{e_3} + c_{4}v_{9}^{e_4} + c_{5}v_{10}^{e_5} - p_{2} &= 0 \\
& \vdots \\
c_{1}v_{126}^{e_1} + c_{2}v_{127}^{e_2} + c_{3}v_{128}^{e_3} + c_{4}0^{e_4} + c_{5}0^{e_5} - p_{26} &= 0
\end{align*}
Note the use of zeros in place of $v_{129}$ and $v_{130}$ in the last (26\textsuperscript{th}) equation, which is due to the padding required to make the 128 dimensions of our face embeddings divisible by 5 (our choice for $m$, as explained in Section 4.1). The systems of equations for overlaps 1 to 4 were set up in a similar way, except that each system consisted of a different number of equations (as per Table 2). For example, for overlap = 4, the system consisted of 124 equations, as follows:
\begin{align*}
c_{1}v_{1}^{e_1} + c_{2}v_{2}^{e_2} + c_{3}v_{3}^{e_3} + c_{4}v_{4}^{e_4} + c_{5}v_{5}^{e_5} - p_{1} &= 0 \\
\\
c_{1}v_{2}^{e_1} + c_{2}v_{3}^{e_2} + c_{3}v_{4}^{e_3} + c_{4}v_{5}^{e_4} + c_{5}v_{6}^{e_5} - p_{2} &= 0 \\
& \vdots \\
c_{1}v_{124}^{e_1} + c_{2}v_{125}^{e_2} + c_{3}v_{126}^{e_3} + c_{4}v_{127}^{e_4} + c_{5}v_{128}^{e_5} - p_{124} &= 0
\end{align*}
Since it was possible to fully formulate each of the 124 equations using the 128 embedding elements ($v_{1}, v_{2}, ..., v_{128}$), no padding of the face embeddings was necessary in this case. Please refer to Fig. 2 to visualise the systems of equations for all overlap amounts.
Next, the numerical solver was used to approximate a solution for $V = [v_1, v_2, ..., v_{128}]$ from every $P$ that was generated for the \textit{evaluation} set of reference face embeddings in Section 4.2 (i.e., a total of 380 $P$ templates for each overlap amount). Since the solver requires a set of initial guesses for $v_1, v_2, ..., v_{128}$, we randomly generated 100 guesses for each of the 128 embedding elements. The guesses were drawn from the probability distributions established for the corresponding embedding elements on the \textit{development} set of reference embeddings (as explained in Section 4.3.1).
The numerical solver then started from these initial guesses to estimate a solution for each of the 128 elements in $V$, from the corresponding $P$. As soon as the solver indicated that a solution for $V$ had been found (by setting the Boolean flag \texttt{success} to 1), the process was stopped. Since the error tolerance for the solution was set to a very small (default) value ($1.49012e-08$), an attacker attempting this inversion attack in practice should have no reason to doubt that the solution returned at this point in the process is a good approximation to $V$. So, not all 100 initial guesses necessarily needed to be tried. Once all 380 $P$ templates had passed through the solver, we calculated the \textit{solution rate} as the proportion of all $P$s for which a solution was found.
If a solution, $V^{*}$, for a particular $P$ was found, we then calculated the comparison score between this approximation of the corresponding face embedding and the true embedding, $V$, in terms of the negative cosine distance (as for the recognition accuracy in Section 4.2). The idea was to determine whether $V^{*}$ was a \textit{close enough} approximation to $V$, such that $V^{*}$ could be used to launch a replay attack in our baseline face recognition systems, which store the \textit{unprotected} $V$ as a reference face embedding. Since the closeness of a match between two face embeddings in practice would always be dependent on a match threshold (i.e., the match will never be perfect), it makes sense to judge the success of an inversion attack in the same way.
We used the aforementioned approach to calculate the \textit{match rate} for each set of $P$ templates for which a solution for $V$ was found by the numerical solver. The match rate refers to the proportion of $P$s for which the comparison score between $V^{*}$ and $V$ was greater than or equal to a pre-defined match threshold. The match rate was computed at three thresholds established on the baseline systems' \textit{development} set\footnote{Simulates the real-life scenario where match thresholds are tuned offline, on a different set of subjects to that encountered in the systems' deployment scenario (represented by the \textit{evaluation} set of embeddings).} of face embeddings: at FMR = 0.01\% (fairly strict), at FMR = 0.1\% (commonly used), and at FMR = 1\% (fairly lenient), to represent the progression from a higher-security to a lower-security application scenario.
Finally, we calculated the \textit{inversion success rate} = \textit{solution rate} $\times$ \textit{match rate}, to estimate the overall success rate for an attacker attempting the $P \rightarrow V$ inversion using the numerical solver. Table 3 presents the resulting inversion success rates for our Facenet and Idiap PolyProtected systems.
\begin{table}[!h]
\renewcommand{\arraystretch}{1.3}
\caption{Inversion success rates at different match thresholds.}
\centering
\begin{tabular}{|c|c||c|c|}
\hline
\textbf{Threshold} & \textbf{Overlap} & \textbf{Facenet} & \textbf{Idiap} \\
\hline
\multirow{5}{*}{@ FMR = 0.01\%} & 0 & 0.00 & 0.00 \\
& 1 & 0.00 & 0.00 \\
& 2 & 0.00 & 0.00 \\
& 3 & 0.05 & 0.02 \\
& 4 & 0.95 & 0.96 \\
\hline
\multirow{5}{*}{@ FMR = 0.1\%} & 0 & 0.00 & 0.00 \\
& 1 & 0.00 & 0.00 \\
& 2 & 0.01 & 0.01 \\
& 3 & 0.15 & 0.15 \\
& 4 & 0.95 & 0.96 \\
\hline
\multirow{5}{*}{@ FMR = 1\%} & 0 & 0.00 & 0.00 \\
& 1 & 0.01 & 0.01 \\
& 2 & 0.08 & 0.05 \\
& 3 & 0.52 & 0.49 \\
& 4 & 0.95 & 0.97 \\
\hline
\end{tabular}
\end{table}
Table 3 shows two important trends. Firstly, the inversion success rate is, in general, \textit{lower} when the baseline systems operate at a \textit{stricter} match threshold (at a lower FMR). This is because a stricter threshold would impose a tougher standard on what is meant by a \textit{good enough} approximation of $V$. So, the stricter the threshold in practice, the less likely the inversion attack would be to succeed, in the sense that it would become more difficult to match the recovered approximation of $V$ (i.e., $V^{*}$) to the unprotected $V$ used in the baseline face recognition system. This trend is most evident for an overlap of 3, for which the inversion success rate for both the Facenet and Idiap systems is close to 0 at the strictest match threshold (at FMR = 0.01\%), equal to 0.15 at the commonly used threshold (at FMR = 0.1\%), and around 0.5 at the most lenient threshold (at FMR = 1\%).
The second important trend from Table 3 is that the inversion success rate tends to \textit{increase} as the amount of overlap used in the PolyProtect mapping \textit{increases}. This makes sense, because a larger overlap results in a higher-dimensional PolyProtected template (see Table 2 and Fig. 2), which contains a greater amount of information from the original face embedding. Consequently, the system of equations in the $P \rightarrow V$ mapping for larger overlaps is more constrained than for smaller overlaps, meaning that, if a solution for $V$ is found, it is more likely to be closer to the true solution. For example, we see that for both the Facenet and Idiap systems, the inversion success rate for PolyProtected templates generated using overlaps of 0-2 is close to 0 at all three match thresholds, meaning that the templates can be considered \textit{practically irreversible} at these operating points. Although the inversion success rate for an overlap of 3 was found to be higher, the results indicate that the attacker would still \textit{fail} to invert the PolyProtected template 95\% and 98\% of the time for the Facenet and Idiap systems, respectively, when the baseline match threshold is set at FMR = 0.01\%, 85\% of the time for both systems when the threshold is set at FMR = 0.1\%, and about 50\% of the time when the threshold is set at FMR = 1\%. This is still significantly better than using the original (unprotected) face embeddings, especially considering that this evaluation was based on the assumption of a \textit{fully-informed attacker}, which should be extremely rare in practice.
Overall, our analysis suggests that using overlaps of 0, 1, or 2 would be the safest option, and that an overlap of 3 might be acceptable depending on the unprotected system's operating match threshold. An overlap of 4, on the other hand, appears to be an unwise choice, because the inversion success rate was found to be close to 1 for all three thresholds in Table 3. Having said this, recall that our analysis is based on the toughest threat model, whereby we assume that our simulated attacker knows \textit{everything} there is to know about the PolyProtected systems, including representative probability distributions of the original face embedding elements, as well as all the parameters employed in the PolyProtect mapping. This should \textit{not} be the case in practice, meaning that a real-life inversion attack should be significantly less likely to succeed. For example, in practice the user-specific parameters, $C$ and $E$, should \textit{not} be stored in the open, but rather secured separately (e.g., via encryption, on a separate token, etc.). Also, it should be difficult for the attacker to obtain representative embedding distributions from which to draw the initial guesses for the numerical solver. These factors combined would make it much more difficult for the solver to converge to a close approximation of the original embedding, meaning that, in a real-life inversion attack, we would expect the inversion success rate to be close to 0 most of the time.
Considering once again our worst-case evaluation scenario based on a fully-informed attacker, the two main trends observed in Table 3 imply that the inversion success rate of PolyProtected templates in practice will depend on the amount of overlap used in the PolyProtect mapping and the operating threshold of the baseline face recognition system in which the unprotected face embeddings are employed. If we were to form a general recommendation on what overlap amount to use for the PolyProtect mapping in practice (for the considered evaluation scenario), we should consider Table 3 alongside the trends shown by the ROC plots in Fig. 4 (and with the help of Table 1). Based on this comparison, we may conclude that, in general, an overlap of 2 would most likely provide the best trade-off between irreversibility and recognition accuracy. Since the corresponding PolyProtected templates were shown to be practically irreversible even for the worst-case (albeit unlikely) scenario of a fully-informed attacker, we may reasonably conclude that PolyProtect would be suitable for securing face embeddings in practice. Table 3 and Fig. 4 (with Table 1) further suggest that overlaps of 0 and 1 may be more suitable for lower-security PolyProtected applications, which operate at more lenient thresholds (i.e., corresponding to higher FMRs), while an overlap of 3 could offer the best accuracy versus irreversibility trade-off in higher-security PolyProtected applications (where a lower FMR is desired). Remember, however, that the \textit{smaller} the overlap, the more \textit{irreversible}, and thus more \textit{privacy-preserving}, the PolyProtected templates would be.
A meaningful, quantifiable comparison between the irreversibility of PolyProtect and that of the other face embedding protection schemes in the literature is currently extremely difficult. This is mainly due to inconsistencies in the adopted evaluation methodologies (there is no standardised approach, since the evaluation tends to be method-specific) and the assumed threat model (which is usually not even explicitly defined). Furthermore, it is often not evident how we should fairly select comparable parameters (e.g., user-specific transformation keys) across different protection methods, since such parameters (if they exist) do not always take the same form or have the same meaning. For such reasons, comparing the irreversibility of different face embedding protection methods in the literature may result in ambiguous conclusions on which method is better in this regard. So, irreversibility comparisons are usually avoided.
On a final note, we would like to make a case for \textit{practical} irreversibility evaluation methodologies. Recall that the irreversibility analysis presented earlier for PolyProtect was based on defining a \textit{successful inversion} as the ability to use the approximation of $V$ recovered from its $P$ (i.e., $V^{*}$) to impersonate the same identity in an unprotected face recognition system that employs $V$ itself. This seems like a reasonable approach for estimating PolyProtect's irreversibility \textit{in practice}, because it gives us direct insight into what exactly a successful inversion attack might mean in a real-life scenario. In our view, this is more useful (from a practical point of view) than a theoretical evaluation, since such evaluations tend to be based on unrealistic assumptions (like the assumption of the existence of an infinite number of solutions for $V$, as discussed earlier in the context of an \textit{analytical} irreversibility analysis for PolyProtect). We thus hope to encourage a greater focus on practical evaluation methodologies when analysing the irreversibility of face embedding protection methods in the literature, particularly in terms of inversion attacks for which readily-available tools can be used (such as Python's numerical solver in our analysis). This would help to produce more tangible results, which may make it easier to understand and compare the expected irreversibility of different protection methods in practice. Furthermore, since we are at the stage where the protection of face embeddings is becoming an urgent requirement in real-life applications, a practically-oriented irreversibility analysis would surely be appreciated by deployers of face recognition systems.
\subsubsection{Access to multiple PolyProtected templates}
Section 4.3.2 analysed the irreversibility of PolyProtect when an attacker has access to only one PolyProtected template, $P$, corresponding to a certain face embedding, $V$. In this section, we consider the scenario where the attacker has access to \textit{multiple} PolyProtected templates from the same $V$, which they attempt to combine to recover an approximation of $V$. This is referred to as a Record Multiplicity Attack (ARM) in the literature. This type of attack could occur in the scenario where the same face embedding is used to generate different PolyProtected templates (using different $C$ and $E$ parameters), then each PolyProtected template is either enrolled in a different application or used to replace a compromised PolyProtected template in the same application. Although such an attack should be extremely difficult to launch in practice\footnote{The attacker would need to know which applications to target and how to hack them to steal the subject's PolyProtected templates.}, it must still be considered when analysing PolyProtect's irreversibility.
To analyse the susceptibility of PolyProtect to ARM, we proceeded as follows. We assumed that different PolyProtected templates from the same subject are generated using the \textit{same} original face embedding, $V$. In practice, we should never use \textit{exactly the same} face embedding to generate different PolyProtected templates for the same subject. This is because the original face embedding should never be stored in the clear, meaning that each enrollment should request \textit{a new sample} of the enrollee's face. Consequently, each PolyProtected template belonging to the same subject should be generated using a different instance of $V$. In our ARM analysis, however, we consider the worst-case scenario where the \textit{same} $V$ is used to generate all of a subject's $P$ templates. This should make it easier to combine the information from these PolyProtected templates to recover the original face embedding, which means that this represents the best-case scenario for the attacker.
The face embeddings and corresponding PolyProtected templates used in this analysis were the same as those used for the irreversibility analysis in Section 4.3.2. This means that each $V$ was associated with 10 different $P$s (each generated using different $C$ and $E$ parameters). Our ARM analysis thus consisted of attempting to recover an approximation of $V$ using 1 to 10 of its corresponding $P$s. This was simulated using the numerical solver approach explained in Section 4.3.2. This time, however, we were trying to solve systems of $k \times p$ equations (where $k$ is the dimensionality of the $P$ templates, and $p$ is the number of $P$s that the attacker is assumed to have access to), instead of only $k$ equations as in Section 4.3.2.
So, the first step was to set up a system of $k \times p$ equations for each overlap amount\footnote{The $k$ values for overlaps 0 to 4 are indicated in Table 2.}, and for each value of $p$ in the range [1, 10]. For example, for overlap = 2, to simulate an attacker with access to the minimum of only \textit{one} 42-dimensional $P$, we set up a system of 42 equations in 128 unknowns (embedding elements), same as for the irreversibility analysis in Section 4.3.2. On the other hand, to simulate an attacker with access to the maximum of \textit{ten} 42-dimensional $P$s generated from the same $V$, we set up a system of 420 equations in 128 unknowns (i.e., one set of 42 equations used in the $V \rightarrow P$ mapping for each of the 10 PolyProtected templates). Note that, unlike the 42-equation system, the 420-equation system would not be \textit{underdetermined} but \textit{overdetermined}, meaning that the additional constraints to the solution space may allow the attacker to have a greater chance of recovering a good approximation of $V$ from ten $P$s than from one $P$. So, we would expect to see an increase in PolyProtect's susceptibility to ARM as the number of $P$s considered in the attack increases. However, an overdetermined system of equations may introduce inconsistencies to the solution space, so this expected trend cannot be considered certain in practice.
Similarly to the analysis in Section 4.3.2, we evaluated the \textit{inversion success rate} = \textit{solution rate} $\times$ \textit{match rate}. This was calculated separately for each $p$ in the range [1, 10] (i.e., 1 to 10 PolyProtected templates per $V$), and for each overlap amount. Note that, since the aim of this analysis was to determine whether using multiple PolyProtected templates would make it easier to recover the original face embedding than using only one PolyProtected template, the choice of match threshold was not so important, particularly because the irreversibility analysis in Section 4.3.2 already demonstrated the effects of using different match thresholds. So, Fig. 5 illustrates the inversion success rate for the Facenet and Idiap systems as the number of PolyProtected templates increases from 1 to 10, based only on the most commonly used match threshold at FMR = 0.1\% from Section 4.3.2.
\begin{figure}[!h]
\centering
\includegraphics[width=0.49\columnwidth]{fig_5_facenet.png}
\hfill
\includegraphics[width=0.49\columnwidth]{fig_5_idiap.png}
\caption{Inversion success rate when an attacker is assumed to have access to multiple PolyProtected templates from the same face embedding. The results are based on the match threshold calculated at FMR = 0.1\% on the \textit{development} set of baseline face embeddings.}
\end{figure}
From Fig. 5, it is evident that, in general, the inversion success rate increases as the number of PolyProtected templates used in the attack increases. This makes sense, because access to a larger number of PolyProtected templates from the same embedding provides more information about that embedding, thereby allowing the numerical solver to approximate a more accurate solution to the $P \rightarrow V$ system of equations. As stated earlier, however, an overdetermined system of equations may result in an inconsistent solution set, thereby actually confusing the solver as to which solution is the correct one. This may explain the sudden drop in the inversion success rate in various curves in Fig. 5 (e.g., between 5 and 6 PolyProtected templates for overlap = 1 in both plots). So, a larger number of templates may not always translate to a higher inversion success rate in practice, even if this does appear to be the general trend.
Another general trend from Fig. 5 is that, the greater the amount of overlap used in the PolyProtect mapping, the fewer PolyProtected templates would be required to launch a Record Multiplicity Attack. This is in line with the irreversibility analysis in Section 4.3.2, which showed that the inversion success rate for a single $P$ increases as the amount of overlap increases. Since PolyProtected templates generated using a greater overlap contain more information about the original face embedding, it makes sense that the attacker would require more PolyProtected templates generated using a smaller overlap to achieve the same inversion success rate as for fewer PolyProtected templates generated using a larger overlap. So, we may conclude that, the smaller the amount of overlap used in the PolyProtect mapping, the less susceptible PolyProtect would be to a Record Multiplicity Attack in practice.
Having made these observations, we must emphasize that our ARM analysis assumed that all PolyProtected templates to which an attacker has access, have been generated from exactly the same face embedding, $V$. This should \textit{not} be the case in practice, even if the PolyProtected templates were acquired from the same system during re-enrollment of the same (compromised) user. Additionally, multiple PolyProtected templates belonging to the same subject enrolled in different applications may be generated by different systems (e.g., Facenet and Idiap), using different amounts of \textit{overlap} in the $V \rightarrow P$ mapping. So, our ARM analysis represents the best-case scenario for our simulated attacker, but in reality the susceptibility of PolyProtect to this type of attack should be much lower, especially as the attacker is unlikely to be \textit{fully informed}.
Nevertheless, Fig. 5 indicates that even in this best-case scenario for the attacker, a high inversion success rate (e.g., $\approx$ 0.9 or above) for a commonly used match threshold is unlikely to be attainable for PolyProtected templates generated using overlaps of 0 or 1, even if the attacker had access to 10 templates from the same face embedding. Although such a high inversion success rate may be achievable for PolyProtected templates generated using an overlap of 2, the attacker would need to acquire a large number of templates from the same face embedding (i.e., 9 or 10), which may be considered practically infeasible\footnote{Hacking the database of even one system should be difficult, let alone 9 or 10 different systems, or the same system this many times.}. On the other hand, Fig. 5 suggests that a smaller number of PolyProtected templates would be needed when those templates have been generated using a larger overlap amount (i.e., 4-5 templates for an overlap of 3, or 1-2 templates for an overlap of 4). Although it is reasonable to assume that it would be difficult to acquire even 4-5 different PolyProtected templates from the same person in practice, 1-2 templates would be more feasible. So, as concluded for the single-template irreversibility analysis in Section 4.3.2, an overlap of 4 is not recommended in practice due to the high risk of inversion of the PolyProtected template(s) by a fully-informed attacker. In fact, in Section 4.3.2 we observed that an overlap of 2 generally seems to offer the best trade-off between irreversibility and recognition accuracy, while overlaps of 0 and 1 may be more suitable for lower-security applications, and an overlap of 3 could offer the best accuracy/irreversibility trade-off in higher-security applications. We make the same recommendation in light of our ARM analysis, while emphasizing once again that, if the priority is \textit{user privacy}, a smaller overlap should be employed in the PolyProtect mapping, as this would ensure the generation of \textit{more irreversible} PolyProtected templates.
Finally, as for the irreversibility analysis in Section 4.3.2, it is currently impossible to present a meaningful, quantifiable comparison between PolyProtect's ARM susceptibility to that of the other face embedding protection schemes in the literature. This time, however, such a comparison is not only limited by the evaluation inconsistencies explained in Section 4.3.2, but also by the fact that this type of analysis is lacking for most of the proposed protection methods. For example, of the methods mentioned in Section 2, ARM susceptibility was only evaluated in \cite{m20}. In \cite{m20}, the authors state that the ARM attacking complexity is lower bounded by the effort of exhaustively searching for the user-specific key, thereby concluding that ARM does not reduce the complexity of inverting the protected templates. No analysis is presented to validate this claim from a practical point of view, however, especially considering the possibility of the user-specific key being leaked when it is decoded during an authentication attempt. So, it is difficult to draw a fair comparison between the susceptibility to ARM of the method in \cite{m20} and PolyProtect, in a practical context.
\subsection{Unlinkability}
Assume that a certain PolyProtected face embedding, $P$, is enrolled in a face recognition system. This section investigates whether, in the event that $P$ is compromised (e.g., stolen from the system's database), we could \textit{renew} it, i.e., \textit{cancel} it and generate a replacement PolyProtected template, $P'$, by using different $C$ and $E$ parameters in the $V \rightarrow P$ mapping. The two templates, $P'$ and $P$, should be sufficiently different to ensure that they are \textit{unlinkable} (i.e., cannot be linked to the same identity). We also consider, therefore, the possibility of generating multiple \textit{diverse} PolyProtected templates from the same subject's face, for the purpose of enrolling this person in multiple applications without the risk of cross-matching their identity.
To conduct this evaluation in as realistic a setting as possible, we assumed that different PolyProtected templates belonging to the same subject would have been generated using different instances of that subject's face embedding. This is because the face embedding used to generate a particular PolyProtected template should be discarded during enrollment (i.e., only the PolyProtected template should be stored in the recognition system's database), so each new enrollment would require a new image of the subject's face, from which a new embedding would be generated. For example, let $V_1$ denote a subject's first face embedding, $P_1$ represent the corresponding PolyProtected template, and $C_1$ and $E_1$ denote the coefficients and exponents used to generate $P_1$, respectively. Now, assume that $P_1$ is compromised in some way, meaning that we must remove it from the database and replace it with a new PolyProtected template from the same subject's face. To achieve this, we ask the person to present a new sample of their face, from which the representative face embedding, $V_2$, is extracted. To protect $V_2$ via PolyProtect, we then generate new parameters, $C_2$ and $E_2$, which are used to create the new protected template, $P_2$. Alternatively, $P_1$ and $P_2$ could be used to enroll the same subject in two different applications. The following analysis considers whether $P_2$ is likely to be sufficiently different from $P_1$, such that they can effectively be seen as distinct, \textit{unlinkable} identities.
PolyProtect's unlinkability property was evaluated using the framework proposed in \cite{gb18}. We chose to adopt this framework, because it considers unlinkability from a \textit{practical} angle, which has been our focus in evaluating PolyProtect in this paper. Specifically, the method in \cite{gb18} measures unlinkability in the context of the \textit{mated} and \textit{non-mated} score distributions, which represent the comparison scores between different protected templates from the \textit{same} subject and between different protected templates from \textit{different} subjects, respectively. The unlinkability is measured in terms of two metrics: $D_{\leftrightarrow}(s)$, a local score-wise measure of the degree of linkability based on the likelihood ratio between mated and non-mated scores, and $D_{\leftrightarrow}^{\mathit{sys}}$, a global measure of the overall linkability of the underlying recognition system.
To evaluate the unlinkability of PolyProtected templates using the approach from \cite{gb18}, the first step was to select a number of different face embeddings from each subject in our adopted Mobio dataset, to simulate the enrollment of the same subject in multiple face recognition applications (or re-enrollment in the same application in the event that their protected template has been compromised). Then, the idea was to apply a \textit{different} set of $C$ and $E$ parameters to each face embedding, to generate its corresponding PolyProtected template, $P$. Based on the recommendation\footnote{At least 5 different protected templates per subject should be used. The authors used 10 in their experiments.} in \cite{gb18}, we randomly selected 10 different face embeddings per subject, resulting in 10 different PolyProtected templates per person. Then, each PolyProtected template was compared to every other PolyProtected template from the \textit{same subject} to generate a set of \textit{mated} comparison scores, and to all PolyProtected templates from \textit{every other subject} to generate a set of \textit{non-mated} comparison scores. This process was repeated for 10 trials, where in each trial a new set of 10 face embeddings was randomly selected for each subject in the dataset. The resulting 10 sets of mated and non-mated comparison scores were then concatenated (separately), and the concatenated scores were used to evaluate the unlinkability of the PolyProtected templates. For reference, we also calculated the unlinkability of the corresponding \textit{unprotected} embeddings in the same way.
Fig. 6 shows the unlinkability plots\footnote{Produced using open-source code at: \url{https://bit.ly/3tYv3jw}} for our Facenet and Idiap PolyProtected and baseline (unprotected) systems, on Mobio's \textit{development} subset. Due to space restrictions, for the PolyProtected systems we show only the plots for overlap = 2, since this was the generally recommended value in Section 4.3, and Table 4 summarises the global $D_{\leftrightarrow}^{\mathit{sys}}$ measures for all overlaps.
\begin{figure}[!h]
\centering
\includegraphics[width=0.48\columnwidth]{fig_6_facenet_baseline.png}
\hfill
\includegraphics[width=0.48\columnwidth]{fig_6_idiap_baseline.png}
\vfill
\includegraphics[width=0.48\columnwidth]{fig_6_facenet_polyprotect.png}
\hfill
\includegraphics[width=0.48\columnwidth]{fig_6_idiap_polyprotect.png}
\caption{Unlinkability plots for baseline and PolyProtected (overlap = 2) Facenet and Idiap systems, on Mobio's \textit{development} set. $D_{\leftrightarrow}(s)$ measures the score-wise degree of linkability, and the vertical dotted lines mark the score range within which linkability $\approx$ 0. $D_{\leftrightarrow}^{\mathit{sys}}$ (in the titles) indicates the overall system linkability, over the entire score range.}
\end{figure}
\begin{table}[!h]
\renewcommand{\arraystretch}{1.3}
\caption{$D_{\leftrightarrow}^{\mathit{sys}}$ for PolyProtected systems on Mobio's development set.}
\centering
\begin{tabular}{|l||c|c|c|c|c|}
\hline
\textbf{Overlap} & 0 & 1 & 2 & 3 & 4 \\
\hline
\textbf{Facenet} $\mathbf{D_{\leftrightarrow}^{\mathit{sys}}}$ & 0.14 & 0.15 & 0.15 & 0.15 & 0.16 \\
\hline
\textbf{Idiap} $\mathbf{D_{\leftrightarrow}^{\mathit{sys}}}$ & 0.09 & 0.10 & 0.10 & 0.11 & 0.12 \\
\hline
\end{tabular}
\end{table}
Note that we used the \textit{development} database subset instead of the \textit{evaluation} subset. This is because we wished to check whether \textit{full unlinkability} was attainable if the only requirement for the 10 different PolyProtected templates from the same subject was that their randomly generated $C$ and $E$ parameters were \textit{different}, but the extent of the differences between the resulting PolyProtected templates was not checked. In other words, our aim was to use the development subset as a sounding board for checking whether the aforementioned process would ensure the incorporation of sufficient \textit{diversity} into different PolyProtected templates from the same subject; if so, then we would apply the same procedure on the \textit{evaluation} Mobio subset, otherwise the approach would need to be reconsidered.
There are several important observations from Fig. 6 and Table 4. Firstly, note that $D_{\leftrightarrow}^{\mathit{sys}}$ measures the overall system linkability, where a value of 0 would indicate that the system is fully \textit{unlinkable}, whereas a value of 1 would indicate that the system is fully \textit{linkable}. We observe that $D_{\leftrightarrow}^{\mathit{sys}}$ for our baseline systems, which use \textit{unprotected} face embeddings, is closer to 1, indicating that unprotected face embeddings from the same subject (e.g., used across different applications) are almost fully \textit{linkable}. On the contrary, the $D_{\leftrightarrow}^{\mathit{sys}}$ values for our PolyProtected systems are closer to 0, suggesting that different PolyProtected templates generated from the same subject's face embeddings are almost fully \textit{unlinkable}.
Another observation is that $D_{\leftrightarrow}^{\mathit{sys}}$ increases slightly as the amount of overlap used in the PolyProtect mapping increases. This may be attributed to the fact that a greater overlap produces a higher-dimensional PolyProtected template, which contains more information from the original face embedding, so it may be a little easier to link higher-dimensional PolyProtected templates from the same identity. Having said this, the difference in $D_{\leftrightarrow}^{\mathit{sys}}$ between the different overlaps in Table 4 does not appear significant.
Fig. 6 and Table 4 also show lower $D_{\leftrightarrow}^{\mathit{sys}}$ values for the PolyProtected Idiap system compared to the Facenet system, even though $D_{\leftrightarrow}^{\mathit{sys}}$ for the two baseline systems is the same. This is probably due to differences in the original embedding distributions, which are emphasized as a result of the PolyProtect mapping. Although the differences in the $D_{\leftrightarrow}^{\mathit{sys}}$ values across the two PolyProtected systems seem minor, this finding nevertheless indicates that the unlinkability of PolyProtected templates depends in part on the underlying face features, which makes sense in practice.
Another important observation\footnote{The same observation was made for overlaps of 0, 1, 3, and 4.} from Fig. 6 relates to the local score-wise measure of linkability, $D_{\leftrightarrow}(s)$. In both PolyProtected system plots, the value of $D_{\leftrightarrow}(s)$ within the score range marked by the vertical dotted lines is approximately 0. This indicates that, if we were to compare two PolyProtected templates and the resulting score was within that range, it would be almost impossible to link the two templates to the same identity; in other words, the systems are fully \textit{unlinkable} within that score range. Outside this score range, however, $D_{\leftrightarrow}(s)$ in both PolyProtected system plots gradually increases to 1. This implies that, if we were to compare two PolyProtected templates and the resulting score was found to lie towards the extremes of the score range, we could be almost certain that the two templates belong to the same identity; in other words, the systems tend towards full \textit{linkability} outside the score range marked by the vertical dotted lines.
We believe that this phenomenon of full linkability at the extremes of the PolyProtected systems' score range (indicated by the two bumps in the \textit{mated} score distributions) is mainly due to the relationship between the \textit{signs} of the corresponding elements among each pair of PolyProtected templates that is compared to generate those scores. If the corresponding elements of the two PolyProtected templates all have the same signs (i.e., either both positive or both negative), then the resulting score will be close to 0.0. This is because the comparison score is a measure of the angle between the two template vectors, and the angle approaches 0 when the corresponding vector elements are similar in magnitude (since they come from the same subject's face) \textit{and} have the same signs (which is determined both by the signs of the original elements and the effects on the signs by the PolyProtect mapping). Alternatively, if the corresponding PolyProtected elements have opposite signs, the score will be close to -2.0. This effect is more prominent for PolyProtected templates from the \textit{same} subject's face embeddings, because these embeddings are much more similar than embeddings from different subjects' faces. Consequently, these similarities may be exaggerated to produce PolyProtected templates that are either as similar as possible (score $\approx$ 0.0) or as different as possible (score $\approx$ -2.0), depending on the choice of the $C$ and $E$ parameters used in the PolyProtect mappings.
Based on the aforementioned observations, we conclude that our PolyProtected systems achieved a promising overall degree of unlinkability (indicated by the low $D_{\leftrightarrow}^{\mathit{sys}}$ values), but the systems were only fully unlinkable within a certain part of the score range. So, we investigated the possibility of achieving full unlinkability across the \textit{entire} score range, by selecting the $C$ and $E$ PolyProtect parameters in a stricter way, such that the \textit{mated} scores would be forced (as much as practically possible) to lie within the aforementioned range. In other words, we tested the possibility of removing the mated score distribution bumps at the extreme ends of the score range, through smarter PolyProtect parameter selection. To achieve this, we proceeded as follows.
We established the full unlinkability score range on our \textit{development} dataset (as shown in Fig. 6), then applied that score range to the \textit{evaluation} dataset. Note that Mobio's development and evaluation datasets consist of \textit{different subjects}, so we can think of this approach as establishing the score range prior to system deployment on a dataset that simulates the application scenario, but when we do not know who will use the systems in practice. Since the full unlinkability score range was found to vary across the Facenet and Idiap face recognition systems and across the different overlaps, we used separate, system-specific and overlap-specific score ranges to select the $C$ and $E$ PolyProtect parameters that would be employed in generating the PolyProtected templates on Mobio's \textit{evaluation} dataset.
The same process that was applied on the \textit{development} subset to randomly select the face embeddings used in the unlinkability analysis (described earlier), was applied to the \textit{evaluation} dataset, and the same number of face embeddings and thus PolyProtected templates (i.e., 10 per subject, for each of 10 trials) was considered. The only difference was in the way that the $C$ and $E$ parameters were selected. For the \textit{development} dataset, we simply required the parameters to be \textit{different} for different PolyProtected templates for the same subject's face embeddings. For the \textit{evaluation} dataset, however, the parameter selection process was more strict.
Let $\{V_1, V_2, ..., V_{10}\}$ represent a set of 10 face embeddings randomly selected for a particular subject from Mobio's \textit{evaluation} dataset. For the unlinkability analysis, we needed to generate a PolyProtected template for each of these 10 embeddings, meaning that we had to generate 10 sets of $C$ and $E$ parameters. This was accomplished as follows. To produce $P_1$, which is the PolyProtected template of $V_1$, we simply generated the corresponding $C_1$ and $E_1$ randomly. Then, to produce $P_2$ from $V_2$, we also began by generating the corresponding $C_2$ and $E_2$ parameters randomly. At this point, ideally we would have liked to compare $P_2$ to $P_1$ and check if the resulting score was within the score range established for the corresponding face recognition system on Mobio's \textit{development} dataset; if this was the case, then $C_2$ and $E_2$ would be considered acceptable, otherwise a new set of parameters would be randomly generated until the aforementioned condition was satisfied. Unfortunately, however, such an approach would assume that the deployers of the face recognition system knew in advance all the PolyProtected templates that a particular person would use to enroll into different applications, which should not be the case in practice (but may be possible for template replacements within the \textit{same} system). What we \textit{can} assume, however, is that the \textit{parameters} used to generate those PolyProtected templates are known, because this is the only way to ensure that the same parameters are not used across different applications or for replacement PolyProtected templates within the same application.
So, we adapted the process of selecting $C_2$ and $E_2$, such that, instead of being compared to $P_1$, the $P_2$ produced using these parameters was compared to the PolyProtected template produced from applying $C_1$ and $E_1$ to the \textit{same} face embedding, $V_2$. If the comparison score between the two PolyProtected templates was within the required score range, then $C_2$ and $E_2$ were accepted; otherwise, a new set of parameters was randomly generated until the aforementioned condition was satisfied. Once $P_2$ was successfully produced, we moved on to the selection of parameters $C_3$ and $E_3$, which would be used to produce $P_3$ from $V_3$. Similarly to the process used for $P_2$, we began by randomly generating $C_3$ and $E_3$ to produce $P_3$. This time, $P_3$ was compared to \textit{two} other PolyProtected templates: one resulting from applying $C_1$ and $E_1$ to $V_3$, and the other resulting from applying $C_2$ and $E_2$ to $V_3$. If \textit{both} comparison scores were within the required score range, $P_3$ was considered successful; otherwise, $C_3$ and $E_3$ kept being randomly generated until the score condition was satisfied. This process was continued until all 10 PolyProtected templates were successfully generated, where $P_{10}$ was compared to PolyProtected templates produced using $V_{10}$ and all 9 sets of previously-generated $C$ and $E$ parameters.
The idea behind this strict process of selecting the $C$ and $E$ parameters was to ensure that \textit{different} PolyProtected templates generated from the \textit{same} face embedding would be \textit{unlinkable}. If we could achieve this, then it would be reasonable to assume that the selected parameters would also ensure unlinkability between PolyProtected templates generated from \textit{different face embeddings} belonging to the same subject (which should be the case if the face embeddings are quite similar). Note that this parameter selection process was conducted separately for each system (Facenet and Idiap) and each overlap in the set $\{0, 1, 2, 3, 4\}$, using the corresponding score ranges established on the \textit{development} dataset. Fig. 7 shows the unlinkability plots resulting from applying this \textit{strict} $C$ and $E$ parameter selection process on Mobio's \textit{evaluation} dataset. Due to space restrictions, Fig. 7 presents the unlinkability plots only for overlap = 2 for each of our two PolyProtected systems, and Table 5 summarises the $D_{\leftrightarrow}^{\mathit{sys}}$ values for all overlaps.
\begin{figure}[!h]
\centering
\includegraphics[width=0.48\columnwidth]{fig_7_facenet_polyprotect.png}
\hfill
\includegraphics[width=0.48\columnwidth]{fig_7_idiap_polyprotect.png}
\caption{Unlinkability plots for PolyProtected (overlap = 2) Facenet and Idiap systems, on Mobio's \textit{evaluation} set using strict $C$ and $E$ selection.}
\end{figure}
\begin{table}[!h]
\renewcommand{\arraystretch}{1.3}
\caption{$D_{\leftrightarrow}^{\mathit{sys}}$ for strict $C$ and $E$ selection on Mobio's evaluation set.}
\centering
\begin{tabular}{|l||c|c|c|c|c|}
\hline
\textbf{Overlap} & 0 & 1 & 2 & 3 & 4 \\
\hline
\textbf{Facenet} $\mathbf{D_{\leftrightarrow}^{\mathit{sys}}}$ & 0.03 & 0.04 & 0.03 & 0.04 & 0.03 \\
\hline
\textbf{Idiap} $\mathbf{D_{\leftrightarrow}^{\mathit{sys}}}$ & 0.03 & 0.02 & 0.02 & 0.01 & 0.01 \\
\hline
\end{tabular}
\end{table}
From Fig. 7 and Table 5, it is clear that $D_{\leftrightarrow}^{\mathit{sys}}$ for all PolyProtected systems is effectively 0. This suggests that, by employing a stricter parameter selection process for the PolyProtect mapping, it is possible to achieve almost full unlinkability for PolyProtected templates generated using all overlap amounts, on both the Facenet and Idiap systems. Furthermore, unlike in Fig. 6, $D_{\leftrightarrow}(s)$ in Fig. 7 does not approach anywhere near a value of 1 across the entire score range\footnote{The same trend was observed for all overlaps.}. A few scores are shown to have a non-zero $D_{\leftrightarrow}(s)$ measure, suggesting that there is a small probability of establishing a link between the PolyProtected templates being compared, if such a score is obtained; however, the low $D_{\leftrightarrow}(s)$ values even for these scores implies a very small degree of linkability, which is reflected in the near-zero $D_{\leftrightarrow}^{\mathit{sys}}$ measures. So, we may reasonably conclude that effectively full unlinkability of PolyProtected face embeddings is attainable in practice, provided that the $C$ and $E$ parameter generation is conducted in a manner smarter than naive random selection.
Note that, during our strict parameter selection process, we actually implemented a score range \textit{tolerance} if suitable $C$ and $E$ parameters could not be found within 100 tries. The tolerance started at 0 and was increased by 0.01 for every 100 failed parameter generation attempts. This was done to speed up the experiments, and we did not check for repetition across the 100 tries. So, we expect that more optimal unlinkability results could be obtained with a more stringent implementation, perhaps using a lower/zero tolerance. Nevertheless, the aim was to show that the unlinkability of our PolyProtected systems can be improved by adopting a smarter-than-random parameter selection method. The results in Fig. 7 and Table 5 prove that this is realisable in practice.
Of the example face embedding protection methods mentioned in Section 2, an unlinkability analysis was presented in \cite{m20, p21, l21}. These evaluations were also conducted using the framework proposed in \cite{gb18}, and the best $D_{\leftrightarrow}^{\mathit{sys}}$ values are approximately comparable to our results in Table 5. Since the full experimental procedure (e.g., the number of protected templates considered per subject) adopted for the unlinkability analysis of all these methods is not always clear, we cannot guarantee that our unlinkability comparison is perfectly fair. We may reasonably conclude, however, that in our respective evaluation scenarios, both PolyProtect and the protection methods in \cite{m20, p21, l21} were shown to satisfy the \textit{unlinkability} criterion to a high degree, using the same general evaluation framework.
\section{Conclusions and Future Work}
This paper proposed PolyProtect, a method for protecting face embeddings in neural-network-based face recognition systems. PolyProtect maps a face embedding to a lower-dimensional representation, via multivariate polynomials defined by user-specific coefficients and exponents. The recognition accuracy, irreversibility, and unlinkability of PolyProtect were evaluated on two open-source face recognition systems in a cooperative-user mobile face verification context, which represents PolyProtect's most likely application scenario in practice.
We showed that PolyProtect is capable of preserving or improving the baseline (unprotected) systems' recognition accuracy under normal operating conditions, and that the accuracy can be tuned by varying the amount of overlap used in the PolyProtect mapping. Acceptable accuracy is thus attainable even in the worst-case (albeit unlikely) scenario where all user-specific parameters are stolen.
Our irreversibility analysis, for a fully-informed attacker, simulated the feasibility of recovering an approximation of the face embedding from its PolyProtected template(s), using a numerical solver. The inversion success rate was calculated in terms of the comparison score between the returned solution and the true face embedding, which was assumed to be enrolled in the baseline (unprotected) face recognition system. At a commonly-used match threshold, PolyProtected templates generated using overlaps of 0-2 were found to be practically irreversible, those generated using an overlap of 3 partially reversible, and those generated using the maximum overlap of 4 almost fully reversible. As expected, the inversion success rate was demonstrated to decrease when the baseline face recognition system adopts a stricter match threshold, and increase when a more lenient threshold is used, though the difference was mainly noticeable for an overlap of 3. Access to multiple PolyProtected templates from the same face embedding was shown to increase the chances of a successful template inversion in the best-case scenario for the attacker, but in general a high success rate was found to be unattainable unless the maximum overlap was used in the PolyProtect mapping (not recommended) or the number of acquired templates was impractically large. We thus recommended carefully selecting the overlap parameter in practice, according to the desired irreversibility versus recognition accuracy trade-off.
Finally, an analysis of PolyProtect's unlinkability property showed that it is possible to achieve effectively full unlinkability between multiple PolyProtected templates from the same subject's face embeddings, particularly if the user-specific parameters employed in the PolyProtect mappings are selected in a smarter-than-random fashion. This suggests that it would be possible to generate sufficiently diverse PolyProtected templates from the same subject's face, such that: (i) compromised templates could be renewed (i.e., cancelled and safely replaced by new ones), and (ii) different templates could be enrolled in different applications without the risk of cross-matching.
Our focus in evaluating PolyProtect was on using \textit{practical} evaluation methodologies, to present insight into the method's robustness as a real-life face embedding protection scheme. The results indicate that the method is capable of satisfying the recognition accuracy, irreversibility, and unlinkability criteria, even under the toughest threat model that assumes a fully-informed attacker with complete knowledge of the system and all its parameters. We may thus reasonably conclude that PolyProtect shows promise in practice, which is important considering the urgent requirement for robust face embedding protection methods in real-life face recognition applications.
Current plans for future work mainly include extending our evaluation to analyse the generalisability of PolyProtect across different types of face recognition systems (e.g., for embeddings extracted using different pre-trained face recognition models) and in diverse application scenarios (i.e., on datasets representing other face recognition contexts, in addition to cooperative-user mobile verification). In the same vein of thought, we intend to evaluate the suitability of PolyProtect for protecting embeddings extracted from \textit{other} biometric modalities (besides the face). Furthermore, we plan to work on expanding the unlinkability analysis to estimate the number of different PolyProtected templates that can be generated from the same biometric identity, consider additional types of practical attacks that can complement our existing irreversibility analysis, and broaden the recognition accuracy evaluation by investigating alternative (e.g., magnitude-specific) metrics for measuring the similarity between two PolyProtected templates.
\ifCLASSOPTIONcompsoc
\section*{Acknowledgments}
\else
\section*{Acknowledgment}
\fi
This material is based upon work supported by the Center for Identification Technology Research (CITeR) under Grant No. 20F-01I, and CITeR affiliates IDEMIA and SICPA.
\bibliographystyle{IEEEtran}
|
2,877,628,089,015 | arxiv | \section{Introduction}
The determination of accurate current star formation rates (SFRs) is of
fundamental importance for the understanding and investigation
of the processes relevant for star formation and galaxy evolution. A commonly
used tracer to calculate the current SFR of galaxies is the
measurement of the H$\alpha$ emission line which is linked to
the presence of short-lived massive stars.
The standard way to construct relations between the total H$\alpha$ luminosity
and the total SFR of a galaxy is to apply on galaxy-wide scales
an invariant stellar initial mass function (IMF)
which is determined in star clusters
\citep[eg.][]{kennicutt1983a,gallagher1984a,kennicutt1994a,kennicutt1998b,kennicutt1998a}.
This method provides a linear SFR-$L_\mathrm{H\alpha}$ relation
as long as the assumed
galaxy-wide IMF is treated to be constant and independent of the SFR.
A basic result of SFR studies is that $SFR/M_\mathrm{gas}$
decreases with decreasing total neutral gas mass, $M_\mathrm{gas}$.
Low-gas-mass (dwarf irregular) galaxies turn their gas into stars
over much longer time scales than high-gas-mass (large disk) galaxies
\citep[eg.][]{skillman2003a,karachentsev2007a,kaisin2008a}.
That dwarf galaxies have lower star-formation efficiencies,
defined as $\tau_\mathrm{gas}^{-1}=SFR/M_\mathrm{gas}$,
than massive galaxies has been a generally accepted fact at the base of
most theoretical work on galaxy evolution.
But in the last years it has been shown that the application of the IMF in
star clusters to galaxy-wide scales is doubtful
\citep{weidner2003a,weidner2005a,weidner2006a,hoversten2008a,meurer2009a,lee2009a}.
Instead, the galaxy-wide IMF has to be calculated by adding all
IMFs of all young star clusters leading to an integrated galactic
stellar initial mass function (IGIMF). The fundamental property of the
IGIMF-theory is that for massive stars
it steepens with decreasing total SFR although the shape of the underlying
IMF in each star cluster is universal and constant. This is a direct
consequence of (i) the star formation process taking place in individual
embedded star clusters on the scale of a pc,
many of which dissolve when emerging from their molecular
cloud cores, rather than being uniformly distributed over the galaxy and
of (ii)
low-star-formation regions being dominated by low-mass star clusters which
are void of massive stars \citep*{weidner2003a,weidner2005a,weidner2006a,weidner2009a}.
The main aim of this paper is not to present a high precission data
analysis of galaxies but to demonstrate the change of fundamental galaxy
properties when switching from the constant IMF assumption
to the IGIMF theory.
We recap the basics of the
IGIMF theory and update its observational support
in Section~\ref{sec_igimf}.
The required data of the sample of star forming galaxies are tabulated in
Section~\ref{sec_data}. We then calculate the SFRs of galaxies
(Section~\ref{sec_sfr}), gas depletion time scales (Section~\ref{sec_gas_depl}),
and stellar mass build-up times (Section~\ref{sec_buildup_times})
using an IGIMF based SFR-$L_\mathrm{H\alpha}$ conversion and show
the dramatic differences that arise form using a constant-IMF based
SFR-$L_\mathrm{H\alpha}$ relation.
\section{IGIMF: theory and observation}
\label{sec_igimf}
The IMF within each star cluster seems to be universal
\citep{elmegreen1997a,elmegreen1999a,kroupa2001a,kroupa2002a,loeckmann2009a}.
The canonical form
of the IMF, $\xi(m)=dN/dm$, where $dN$ is the number of stars in the mass
interval $[m,m+dm]$, is a two-part power law,
$\xi(m)\propto m^{-\alpha_\mathrm{i}}$, in the stellar regime
with $\alpha_1 = 1.3$ for $0.1 \le m/\mathrm{M_\odot}<0.5$,
$\alpha_2 = 2.35$ for $0.5 \le m/\mathrm{M_\odot}< m_\mathrm{max}$,
where $m_\mathrm{max}$ is the maximal stellar mass
in a just-born star cluster with embedded stellar mass $M_\mathrm{ecl}$.
It can also be formulated as a log-normal distribution at low masses
and a power-law extension at high masses \citep{miller1979a,chabrier2003a}.
The maximum stellar mass, $m_\mathrm{max}$, up to which the IMF of
a star cluster is populated, is a function of the total stellar mass,
$M_\mathrm{ecl}$, \citep{weidner2006a}. This relation has been recently
confirmed by an updated much larger sample of young embedded star clusters
\citep{weidner2009a}.
The masses of young embedded star clusters are distributed according to the
embedded cluster mass function (ECMF) \citep[eg.][]{lada2003a}. The
ECMF is populated up to the most massive young star cluster,
$M_\mathrm{ecl,max}(SFR)$. This most massive young star cluster is found
to scale with the total SFR of the host galaxy \citep*{weidner2004b}.
As stars form in star clusters the galaxy-wide IMF has to be calculated
by adding all IMFs of all young star clusters. The embedded star cluster
population is dominated by low-mass star clusters and has only a few
high-mass star clusters. But due to the $m_\mathrm{max}$-$M_\mathrm{ecl}$
relation massive stars are predominately formed in high-mass clusters,
whereas low-mass stars are formed in all star clusters.
Thus the resulting integrated galactic stellar initial mass function
is steeper than the underlying universal canonical IMF in each star cluster.
With decreasing SFR the upper mass limit of the ECMF decreases and
the number ratio of high-mass to low-mass young star clusters decreases.
Consequently the galaxy-wide number ratio of high-mass to low-mass
young stars decreases, too. As a consequence the IGIMF steepens with decreasing
SFR, which we refer to as the IGIMF-effect.
Among all quantities which are relevant for the IGIMF theory
the slope of the ECMF is the least accurately constrained one. Thus,
two different ECMF-slopes, according to the observational and theoretical
range of possible slopes, are used to fully explore the possible range
of IGIMF effects: the standard IGIMF with a Salpeter
single power-law ECMF slope of
$\beta=2.35$, and the minimum-1 IGIMF with a power-law ECMF slope of
$\beta_2=2.00$ for star cluster more massive than 50~$M_\odot$ and
$\beta_1=1.00$ for star cluster masses between 5 and 50~$M_\odot$.
It has been recently suggested
that the mass
function of young star clusters might be better described by a
Schechter-function \citep{bastian2008a,larsen2009a}.
\citet{bastian2008a} obtains a Schechter function
with a low mass power-law slope of $\beta=2$ below
$M_\star =$ 1--5 $\times 10^{6}$~M$_\odot$. It follows from the SFR-most massive star cluster relation \citep{weidner2004b} that only galaxies with
SFR$\gtrsim$30~M$_\odot$~yr$^{-1}$ populate the ECMF beyond the mass regime described by the power law part of the Schechter function. All galaxies in our sample are below this threshould. Thus the power law description of the ECMF in the IGIMF model includes already a possible global Schechter function of the ECMF for these galaxies.
For a deeper introduction the reader is referred to section~2 of
\citet{pflamm-altenburg2007d} and references therein.
As a consequence, all galaxy-wide properties which depend predominantly
on the presence of high-mass stars scale non-linearly with the
total SFR, such as the total H$\alpha$ luminosity,
the oxygen yield, and the alpha-element [$\alpha$/Fe] ratios.
The observed mass-metallicty relation of galaxies is a direct
outcome of the IGIMF-theory \citep{koeppen2007a} and the
decreasing [$\alpha$/Fe] ratio with decreasing velocity dispersion
of galaxies can be easily explained in the IGIMF context \citep{recchi2009a}.
In both cases no extreme fine-tuning and parameter adjustment is required
as is the case if a constant IMF on galaxy scales is assumed.
The IGIMF-theory connects the galaxy-wide IMF with the current SFR and thus
refers to whole galaxies. Within galaxies star formation is described by
the corresponding surface star formation rate density,
$\Sigma_\mathrm{SFR} =dM/dt dx dy$, which defines the newly formed stellar
mass, $dM$, per time interval, $dt$, and per area, $dx dy$. In order to
construct a local relation between the local star formation rate surface
density,
$\Sigma_{SFR}$, and the produced local H$\alpha$ luminosity surface density,
$\Sigma_\mathrm{H\alpha}=dL_\mathrm{Halpha}/dt dx dy$, which defines the
produced H$\alpha$ luminosity, $dL_\mathrm{H\alpha}$, per time interval, $dt$,
and per area, $dx dy$, the embedded cluster mass function and the integrated
galactic stellar initial mass function are replaced in the IGIMF theory
by their corresponding surface densities. These are
the local embedded cluster mass
function, LECMF,
and the local integrated galactic stellar initial mass function, LECMF.
In the outer regions of disk galaxies the star formation rate surface density
is lower and the LECMF is populated to lower upper masses
than in the inner regions. Like the IGIMF effect for whole galaxies, a
local IGIMF effect emerges. Thus the H$\alpha$ luminosity surface
density decreases faster with increasing radial distance than the star
formation rate surface
denstiy, which naturally explains the radial H$\alpha$ cutoff in disk galaxies
\citep{pflamm-altenburg2008a}.
How the H$\alpha$ based SFRs
of galaxies and their related properties change when
the classical linear $L_\mathrm{H\alpha}$-SFR relation is replaced
by the non-linear IGIMF based conversion, presented in
\citet{pflamm-altenburg2007d}, is explored in this paper.
The classical relation between the total H$\alpha$ luminosity of a galaxy
and the underlying SFR is based on the assumption that the galaxy-wide IMF is identical to the IMF observed in star clusters and does not vary with SFR. Thus, the
classical SFR scales linearly with the total H$\alpha$ luminosity. The relation between the total SFR and the total H$\alpha$ luminosity in the IGIMF theory is nearly linear in the high H$\alpha$ luminosity range, i.e. for L$_*$ and more luminous galaxies.
But the IGIMF-based SFR-$L_\mathrm{Halpha}$ relation deviates
increasingly from linearity for H$\alpha$ faint galaxies, i.e. for galaxies with an H$\alpha$ luminosity comparable to the SMC ($\approx 5\times10^{39}$ erg s$^{-1}$) or less. Thus, the classical method underestimates the SFR of H$\alpha$ faint galaxies compared to the IGIMF prediction. This underestimation is
illutrated in Fig.~\ref{fig_sfr_lha_comp}. The SFR for both IGIMF models (standard and minimal) are calculted using the fith-order polynomal fit
published in secion~2.3 and tab.~2 in \citet{pflamm-altenburg2007d}. The classical SFR is calculated with the widly used linear relation by \citet{kennicutt1994a},
\begin{equation}
\frac{\mathrm{SFR}}{\mathrm{M_\odot\;yr^{-1}}} =
\frac{L_\mathrm{H\alpha}}{1.26\times 10^{41}\;\mathrm{erg\;s^{-1}}}\;.
\end{equation}
For example, a galaxy with an H$\alpha$ luminosity of $1.26\times 10^{38}$~erg~s$^{-1}$
has a classical Kennicutt-SFR of $10^{-3}$~M$_\odot$~yr$^{-1}$. The IGIMF SFR is 5 times larger for the minimal model and 10 times larger in the standard model.
\begin{figure}
\plotone{f1}
\caption{\label{fig_sfr_lha_comp} Ratio of the SFRs calculated from
the observed
H$\alpha$ luminosity with the classical constant-IMF based conversion from \citet{kennicutt1994a} and the IGIMF based conversion from \citet{pflamm-altenburg2007d}. The IGIMF-based SFRs are calculted using the fith-order polynomal fit
published in secion~2.3 and tab.~2 in \citet{pflamm-altenburg2007d}.}
\end{figure}
Additionally the IGIMF-effect is expected
to have different strenghts for H$\alpha$ and FUV radiation, because
the H$\alpha$ luminosity depends only on the presence of the
ionising radiation of short-lived
high-mass stars, whereas long-lived B~stars also contribute to
the FUV flux. It has been predicted that the H$\alpha$/FUV flux ratio
decreases with decreasing SFR \citep{pflamm-altenburg2009a}.
Recently \citet{meurer2009a} have shown
that the H$\alpha$/FUV flux ratio of star
forming galaxies decreases with decreasing average H$\alpha$ surface
luminosity density and conclude that a varying galaxy-wide IMF slope
in dependence of the SFR might be the most likely explanation.
A different study by \citet{lee2009a} of local volume galaxies
found a decreasing H$\alpha$/FUV flux ratio with decreasing total
H$\alpha$ luminosity which is in remarkable quantitative
agreement with the prediction
by the IGIMF theory by \citet{pflamm-altenburg2009a} down to the
least-massive dwarf galaxies.
A more detailed listing and discussion of the observational
support of the IGIMF theory is given in section~1 of
\citet*{pflamm-altenburg2009a}.
It should be emphasised here that one may argue that anything can
be explained by a suitably chosen variable IMF. But how the IGIMF varies is
not adjusted for the problem to be solved or explained. Instead,
the nature of clustered
star formation defines how the slope of the IGIMF scales with the
total SFR. This means the
slope of the IGIMF is defined by the SFR of the galaxy. The relation
between the IGIMF-slope and the current SFR is constant for all applications.
There is no freedom to match the galaxy-wide IMF slope
for a considered problem.
Although the slope of the IGIMF varies the IMF in star clusters is universal
and constant and not varying, i.e. the star formation physics is untouched.
\section{Data}
\label{sec_data}
In order to calculate current SFRs, gas depletion times and
stellar mass buildup time scales of star forming galaxies in the
IGIMF-context, the H$\alpha$ luminosity, $L_\mathrm{H\alpha}$,
the total neutral gas mass, $M_\mathrm{gas}$, and the blue band absolute magnitude,
$M_\mathrm{B}$, of 200 local volume galaxies (Tab.~\ref{tab_data})
are compiled from different samples of star forming galaxies:
the Canes Venatici I
group of galaxies \citep{kaisin2008a}, the M81 galaxy group
\citep{karachentsev2007a}, the dwarf irregular galaxies of the Sculptor
group \citep{skillman2003a}, a sample of isolated dwarf irregular galaxies
\citep{vanzee2001a}, local group dwarf irregular galaxies \citep{mateo1998a}
and the local group disk galaxies, M31, M33 \citep{walterbos1994a,dame1993a,karachentsev2004a,verley2007a,corbelli1997a,kennicutt1995a,hindman1967,westerlund1997a}. In most cases the H$\alpha$
luminosity was not given. Instead, the published SFR, based on the observed
H$\alpha$ luminosity, has been transformed into the
H$\alpha$ luminosity using the respective SFR-$L_\mathrm{H\alpha}$ relation
as described in the publication where the SFR is extracted from.
The published SFR, based on a linear SFR-$L_\mathrm{H\alpha}$ relation
\citep[eg.][]{kennicutt1994a},
is tabulated in Table~\ref{tab_data} (Col.~5). The observed total
H$\alpha$-luminosity (Col.~2) is converted into an IGIMF-SFR
using the relations given in \citet{pflamm-altenburg2007d} for
the standard-IGIMF model (Col.~6) and the minimal1-IGIMF model (Col.~7)
in order to cover the range of IGIMFs allowed by the empirical data
(Sec.~\ref{sec_igimf}).
The Milky Way is included in this study. The HI mass is taken from \citet{van_den_bergh1999a}. Due to the lack of H$\alpha$ data for the Milky Way we take the SFR from \citet{diehl2006a} which is based on the meassurement of the Al-26 and Fe-60 gamma ray flux. As this SFR determination is based on a Scalo slope of $\alpha=2.7$ in the high-mass regime of the IMF, the IGIMF-effect is already taken unknowingly into account.
The total neutral gas mass (Col.~3) is the total HI mass taken from the references
multiplied by 1.32, following \cite{skillman2003a}, to account for primordial helium.
The absolute B-band magnitudes are listed in Col.~4.
Other HI--total-gas-mass conversion factors have been applied in the literature, eg. 1.34 \citep{knuth1986a,legrand2001a} or 1.36 \citep{kennicutt2007a}.
The conversion factor may therefore be uncertain by 3~per cent, corresponding to a logarithmic uncertainty of the total neutral gas mass of $\Delta \log_\mathrm{HI} = 0.013$. For the purpose of this study this uncertainty can be
neglected and we use a factor of 1.32,
following \citet{skillman2003a}, throughout this study.
Finally it should be noted that the galaxies of our sample stem from
different observations. Values from 21~cm and optical observations are
compared with each other. As HI exists to much larger galactocentric radii than
H$\alpha$ radiation is produced, the areas covered by the
observations are diffently large. Because our study compares total, integrated
values with each other the diffently large areas do not impose a problem
here. The observed H$\alpha$ luminosities might suffer from a bias
as the different observations might have been performed with differently defined
surface brightness limits. Given that the H$\alpha$ surface brightess
strongly declines with increasing galactocentric radius
in star forming galaxies, the main contribution to the total
H$\alpha$ luminosity comes from the inner regions of the galaxies which should
be covered by each observation. The very small possible bias for the total
H$\alpha$ luminosities can be neglected here, because the main aim of our study
is to demonstrate the differences which arise when the classical method for
calculating SFRs from total H$\alpha$ luminosities based on a constant
galaxy-wide IMF is replaced by the IGIMF-based conversion resulting in new
SFRs being up to a factor 100 larger.
In order to detect possible differences between the different data sources
all galaxies which belong to the same sub-sample
are plotted by the same
symbol. These sub-samples are the Canes Venatici cloud of galaxies
(CVnI), the M81 group, the Sculptor group of dwarf irregular galaxies,
a sample of isolated dwarf irregular galaxies,
the local group of star forming galaxies, and the Milky Way.
It can be seen in the following plots that the changes of the
SFRs with decreasing galaxy mass are the same for all sub-samples.
A systematic bias between the various sub-samples is thus not evident.
Allthough these sub-samples cover different regions in the local volume,
some overlap exists with the sub-sample of \emph{isolated dwarf irregular
galaxies} by \citet{vanzee2001a}. Nine galaxies have been identified to be
listed twice in Table~\ref{tab_data} (see Table~\ref{tab_gal_twice}).
These galaxies appear twice in the following plot as independent data.
\include{tab1}
\include{tab2}
\section{Calculating SFRs}
\label{sec_sfr}
The SFRs, which are taken from the literature and based on
a classical linear conversion of the H$\alpha$ luminosity into a SFR
\citep[eg.][]{kennicutt1994a},
are plotted in dependence of the total galaxy neutral gas mass in
Fig.~\ref{fig_sfr_m_gas_imf}. With decreasing galaxy neutral gas mass
the relation between the traditionally calculated SFR and the galaxy neutral gas
mass becomes increasingly steeper. Here we find that
for massive galaxies of our sample with neutral gas masses
$\ge 5\times10^7\;M_\odot$ the SFR scales slightly
non-linear with neutral gas mass,
\begin{equation}
\label{eq_sfr_m_gas_imf_high}
\frac{SFR}{M_\odot\;\mathrm{yr}^{-1}} =
3.91\times 10^{-13}\;\left(\frac{M_\mathrm{gas}}{M_\odot}\right)^{1.26}\;,
\end{equation}
and scales
almost quadratically for less massive galaxies
($M_\mathrm{gas} \le 5\times 10^7\;M_\odot$),
\begin{equation}
\label{eq_sfr_m_gas_imf_low}
\frac{SFR}{M_\odot\;\mathrm{yr}^{-1}} =
7.96\times 10^{-18}\;\left(\frac{M_\mathrm{gas}}{M_\odot}\right)^{1.87}\;.
\end{equation}
This behaviour is already well known
\citep[e.g.][]{karachentsev2007a,kaisin2008a}.
The non-linear relation between the total H$\alpha$ luminosity and the
underlying SFR calculated with the IGIMF-theory diverges significantly from
these classical linear relations based on a constant galaxy-wide IMF below
SFRs typical for SMC-type galaxies. For comparison, the SFR-$M_\mathrm{gas}$
relation in the classical picture becomes increasingly steeper for
galaxy masses less than SMC-type galaxies. It can be expected
that the IGIMF theory revises the H$\alpha$-SFRs of galaxies with a small
neutral gas content such that their lower star formation efficiencies
will be increased.
The SFR-$M_\mathrm{gas}$ relation of this galaxy sample as resulting from
the IGIMF theory is shown in Fig.~\ref{fig_sfr_m_gas_std-igimf}
for the standard model and in Fig.~\ref{fig_sfr_m_gas_min-igimf}
for the minimum model.
It can be clearly seen that for both IGIMF models the turn-down
evident in the classical SFR-$M_\mathrm{gas}$ relation
(Fig.~\ref{fig_sfr_m_gas_imf})
disappears completely.
For the standard IGIMF a bivariate regression gives a linear scaling
of the total SFR and the galaxy neutral gas mass,
\begin{equation}
\label{eq_sfr_m_gas_std_igimf}
\frac{SFR}{M_\odot\;\mathrm{yr}^{-1}} =
3.97\times 10^{-10}\;\left(\frac{M_\mathrm{gas}}{M_\odot}\right)^{0.99}\;.
\end{equation}
For the minimum IGIMF the relation between the SFR and the galaxy
neutral gas mass is
\begin{equation}
\label{eq_sfr_m_gas_min_igimf}
\frac{SFR}{M_\odot\;\mathrm{yr}^{-1}} =
1.62\times 10^{-9}\;\left(\frac{M_\mathrm{gas}}{M_\odot}\right)^{0.87}\;.
\end{equation}
Note that the main parameter in the IGIMF theory is the slope of the
ECMF. Both IGIMF models, the standard model with an ECMF slope of $\beta=2.35$
and the minimum model with an ECMF slope of $\beta=2$, cover the full range
of possible ECMFs. Therefore, the change that emerges
in the SFR-$M_\mathrm{gas}$ relation when changing from the classical
invariant IMF description to the IGIMF theory
is a direct result of the nature of clustered star formation.
In conclusion, the IGIMF theory revises the SFRs of dwarf galaxies
significantly, such that the true SFRs may be larger than hitherto thought
by two or three orders of magnitude. The reason why this was not
evident until now comes about because in dwarf galaxies star formation
predominantly occurs in low-mass clusters which do not contain massive
stars.
\begin{figure}
\plotone{f2}
\caption{\label{fig_sfr_m_gas_imf}The original SFRs taken from the
literature are plotted versus the host galaxy total neutral gas mass. These
SFRs are calculated from the total H$\alpha$ luminosity using a classical
linear SFR-$L_\mathrm{H\alpha}$ relation. Galaxies with a total neutral
gas mass
$\gtrsim 10^8 M_\odot$ show a SFR$\propto M_\mathrm{gas}^{1.26}$ relation
(eq.~\ref{eq_sfr_m_gas_imf_high}),
whereas the SFRs of less massive galaxies scale
with $M_\mathrm{gas}^{1.87}$ (eq.~\ref{eq_sfr_m_gas_imf_low}),
both bivariate fits being shown
as thick-dotted lines. For comparison, the thin dotted line shows
the bivariate regression
from Fig.~\ref{fig_sfr_m_gas_std-igimf} of the standard IGIMF model
with slope 0.99.}
\end{figure}
\begin{figure}
\plotone{f3}
\caption{\label{fig_sfr_m_gas_std-igimf}The calculated SFRs based on the
standard IGIMF versus the total galaxy neutral gas mass. The solid line shows
the bivariate regression for the standard IGIMF
(eq.~\ref{eq_sfr_m_gas_std_igimf}). The symbols are the same as in Fig.~\ref{fig_sfr_m_gas_imf}.
}
\end{figure}
\begin{figure}
\plotone{f4}
\caption{\label{fig_sfr_m_gas_min-igimf}
Same as Fig.~\ref{fig_sfr_m_gas_std-igimf} but using the
SFR-$L_\mathrm{H\alpha}$ relation based on the minimum-IGIMF.
The solid line shows the bivariate regression for this minimum IGIMF
(eq.~\ref{eq_sfr_m_gas_min_igimf}). For comparison,
the dotted line shows the fit of
the standard model (eq.~\ref{eq_sfr_m_gas_std_igimf}).
The symbols are the same as in Fig.~\ref{fig_sfr_m_gas_imf}.}
\end{figure}
\section{Gas depletion time scales}
\label{sec_gas_depl}
The gas depletion time scale, $\tau_\mathrm{gas}$, is a measure
for how long
a galaxy with current total neutral gas mass, $M_\mathrm{gas}$,
can sustain its current SFR without being refueled by accretion
of fresh extra-galactic material. It is defined by
\begin{equation}
\tau_\mathrm{gas} = \frac{M_\mathrm{gas}}{SFR}\;.
\end{equation}
The corresponding gas depletion time scales, based on a constant
galaxy-wide IMF, are plotted in Fig.~\ref{fig_t_gas_depl__m_gas_imf}.
The SFR-$M_\mathrm{gas}$ relations for the IMF case
(eq.~\ref{eq_sfr_m_gas_imf_high} and \ref{eq_sfr_m_gas_imf_low}) can
now be converted into $\tau_\mathrm{gas}$-SFR relations, which are plotted
as dashed lines in Fig.~\ref{fig_t_gas_depl__m_gas_imf}.
The gas depletion time scale increases slightly with decreasing total
galaxy neutral gas mass for galaxies more massive than $5\times 10^7$~$M_\odot$
and it increases strongly for less-massive galaxies with masses less than
$5\times 10^7$~$M_\odot$
, in agreement with other studies
\citep*{skillman2003a,bothwell2009a}. This is commonly interpreted
as star forming dwarf galaxies having much lower star formation
efficiencies, $\tau_\mathrm{gas}^{-1}$, than large disk galaxies.
\begin{figure}
\plotone{f5}
\caption{\label{fig_t_gas_depl__m_gas_imf} Gas depletion time scales
in dependence of the total galaxy neutral gas mass for the case of
a constant galaxy-wide IMF. The thin dotted line marks the age
of the universe, 13.7~Gyr. The symbols are the same as in Fig.~\ref{fig_sfr_m_gas_imf}.
}
\end{figure}
In the previous section it has been shown that the SFRs of dwarf galaxies
are much higher if the constant galaxy-wide IMF is replaced by the
IGIMF, and consequently the gas depletion time scales are now expected
to be shorter.
The resulting gas depletion times scales for the
standard IGIMF are plotted in Fig.~\ref{fig_t_gas_depl__m_gas_std-igimf}
and for the minimum IGIMF in Fig.~\ref{fig_t_gas_depl__m_gas_min-igimf}.
In both cases the gas depletion time scales of dwarf galaxies are comparable
to the gas depletion time scales of large disk galaxies.
For the standard IGIMF the SFR scales linearly
with the total galaxy neutral gas mass (eq.~\ref{eq_sfr_m_gas_std_igimf}). The resulting
gas depletion times scale with the total galaxy neutral gas mass as
\begin{equation}
\label{eq_tau_gas_std_igimf}
\tau_\mathrm{gas} = \frac{M_\mathrm{gas}}{SFR} =
2.52\;\mathrm{Gyr}\,\left(\frac{M_\mathrm{gas}}{M_\odot}\right)^{0.01}\;.
\end{equation}
The mean gas depletion time scale of a galaxy with a
neutral gas mass of $10^6~M_\odot$ is 2.89~Gyr and for a galaxy with a neutral gas
mass of $10^{10}~M_\odot$ it is 3.17~Gyr. Thus,
the gas depletion time scales are constant for all galaxies, being
approximately 3~Gyr. It follows that low-mass dwarf irregular
galaxies have the same star formation efficiency as massive
disk galaxies, in disagreement with the currently widely accepted notion
according to which dwarf galaxies are inefficient in making stars.
The gas depletion time scales for the standard IGIMF are not only
shorter than the IMF gas depletion time scales, but they also have
a smaller scatter. Fig.~\ref{fig_t_gas_depl_histo_std-igimf}
shows the normalised histogram of the gas depletion time scales
for the IMF case (dashed histogram) and for the standard IGIMF case
(solid histogram). The scatter of the standard IGIMF gas depletion time
scales can be well described by a log-normal distribution,
\begin{equation}
\label{eq_log_norm}
\frac{dN_\mathrm{Gal}}{d log \tau_\mathrm{gas}}=
\frac{1}{\sqrt{2\pi\sigma^2}}\;
e^{-\frac{\left(\log\tau_\mathrm{gas}-\mu\right)^2}{2\sigma^2}}\;,
\end{equation}
with $\mu=9.50$ (3.19~Gyr), $\sigma=0.36$. If and to what degree
conclusions can be made for the physical conditions for star formation
on the basis of the reduced scatter in the IGIMF gas depletion time scales
is unclear at the moment. A volume limited sample in further studies
is required in order to avoid any kind of bias.
In the case of the minimum IGIMF the resultant
$\tau_\mathrm{gas}$-$M_\mathrm{gas}$ relation follows from
eq.~\ref{eq_sfr_m_gas_min_igimf},
\begin{equation}
\label{eq_tau_gas_min_igimf}
\tau_\mathrm{gas} = \frac{M_\mathrm{gas}}{SFR} =
0.62\;\mathrm{Gyr}\,\left(\frac{M_\mathrm{gas}}{M_\odot}\right)^{0.13}\;.
\end{equation}
This leads to a gas-depletion time scale of 3.74~Gyr for a $10^6$~$M_\odot$
galaxy and 12.37~Gyr for a $10^{10}$~$M_\odot$ galaxy. This means that in
the case of the minimum IGIMF dwarf irregular galaxies consume their
gas faster than large disk galaxies and would therefore have
higher star-formation efficiencies in contradiction to the
currently widely accepted picture.
Thus, independent of the IGIMF model details dwarf galaxies
do not have lower star
formation efficiencies than large disk galaxies.
\begin{figure}
\plotone{f6}
\caption{\label{fig_t_gas_depl__m_gas_std-igimf} Same as
Fig.~\ref{fig_t_gas_depl__m_gas_imf} but for the
case of the standard IGIMF. The solid line shows
eq.~\ref{eq_tau_gas_std_igimf}, while the thin
dotted line marks the age of the universe as in
Fig.~\ref{fig_t_gas_depl__m_gas_imf}. Note the reduced scatter
in comparison to Fig.~\ref{fig_t_gas_depl__m_gas_imf}.
The symbols are the same as in Fig.~\ref{fig_sfr_m_gas_imf}.
}
\end{figure}
\begin{figure}
\plotone{f7}
\caption{\label{fig_t_gas_depl__m_gas_min-igimf}
Same as
Fig.~\ref{fig_t_gas_depl__m_gas_imf} but for the
case of the minimum IGIMF. The solid line shows
eq.~\ref{eq_tau_gas_min_igimf}.
The symbols are the same as in Fig.~\ref{fig_sfr_m_gas_imf}.
}
\end{figure}
\begin{figure}
\plotone{f8}
\caption{\label{fig_t_gas_depl_histo_std-igimf} The normalised
histogram of gas depletion
time scales for the constant IMF (Fig.~\ref{fig_t_gas_depl__m_gas_imf})
and the standard IGIMF (Fig.~\ref{fig_t_gas_depl__m_gas_std-igimf})
overlayed with the log-normal distribution of eq.~\ref{eq_log_norm}.
Note that the IGIMF-based gas depletion time scales are significantly
shorter on average, have a symmetric distribution,
and have a significantly smaller dispersion than if
the galaxy-wide IMF is assumed to be invariant.}
\end{figure}
\section{Blue stellar mass buildup times}
\label{sec_buildup_times}
In the previous section the current SFR and available total neutral gas mass
of galaxies have been analysed to calculate their current
gas depletion time scales. This allows an estimation of their future
star formation activity. Here we now compare the current SFR, and the
already assembled total stellar mass, $M_*$, of galaxies to analyse their past
star formation by calculating the corresponding stellar mass buildup
time scale, $\tau_*$, which is defined by
\begin{equation}
\label{eq_tau_star}
\tau_* = \frac{M_*}{SFR}\;.
\end{equation}
The total stellar mass is here calculated from the blue-band magnitude,
$M_\mathrm{B}$, (Tab.~\ref{tab_data}, Col.~4). In order to account
for possible different metallicities and star formation histories (SFHs),
we use the stellar-mass-to-light-ratio--colour relations published
by \citet{bell2001a}.
These relations are obtained from synthetic spectra based on different
galaxy evolution models with a constant galaxy-wide IMF. \citet{bell2001a}
adopt a large variety of star formation histories as indicated in
Tab.~\ref{tab_M_L}.
\include{tab3}
It might be argued that the application of these relations is wrong because the
IGIMF will affect the stellar-mass-to-light-ratio--colour relations.
In \citet{pflamm-altenburg2009a} it has been shown that for small SFRs
the IGIMF effect
for the FUV-flux is less than one dex if the IGIMF effect for
H$\alpha$ is two dex.
The B-band and longer wavelength bands are expected to show
an even smaller IGIMF effect than the FUV-flux.
The dominating influence on the stellar
mass buildup time scale is by far the IGIMF-effect for the H$\alpha$
based SFRs (eq.~\ref{eq_tau_star}).
Thus, the \citet{bell2001a} stellar-mass-to-light-ratio--colour
relations are applicable for an order-of-magnitude estimate of the
stellar mass buildup times. However, precision galaxy evolution modeling would
require revised mass-to-light ratios, which we will adress in future projects.
The two stellar-mass-to-light-ratio--colour relations
\begin{equation}
\log\frac{M_\mathrm{*}}{L_\mathrm{B}} = a_\mathrm{B,B-V}
+ b_\mathrm{B,B-V}\left(M_\mathrm{B}-M_\mathrm{V}\right)\;,
\end{equation}
and
\begin{equation}
\log\frac{M_\mathrm{*}}{L_\mathrm{B}} = a_\mathrm{B,V-H}
+ b_\mathrm{B,V-H}\left(M_\mathrm{V}-M_\mathrm{H}\right)\;,
\end{equation}
tabulated in \citet{bell2001a} can be used to construct
a stellar-mass-to-light-ratio--colour relation between the B-band magnitude
and the \hbox{B-H}-colour,
\begin{equation}
\label{eq_B_B-H}
\log\frac{M_\mathrm{*}}{L_\mathrm{B}} = a_\mathrm{B,B-H}
+ b_\mathrm{B,B-H}\left(M_\mathrm{B}-M_\mathrm{H}\right)\;,
\end{equation}
where the two coefficients are given by
\begin{equation}
\label{eq_aa_b}
a_\mathrm{B,B-H}=
\frac{b_\mathrm{B,V-H}\;a_\mathrm{B,B-V}+b_\mathrm{B,B-V}\;a_\mathrm{B,V-H}}
{b_\mathrm{B,V-H}+b_\mathrm{B,B-V}}\;,
\end{equation}
and
\begin{equation}
\label{eq_a_b}
b_\mathrm{B,B-H}=
\frac{b_\mathrm{B,V-H}\;b_\mathrm{B,B-V}}{b_\mathrm{B,V-H}+b_\mathrm{B,B-V}}\;,
\end{equation}
The required colour is obtained using the tabulated blue-band magnitude,
and the H-band magnitude, $M_\mathrm{H}$. To assign an H-band magnitude
to each galaxy we use the observed extremely tight correlation between the
B- and H-band magnitude of galaxies \citep[eq.~9]{kirby2008a} ranging from
$M_\mathrm{B}=-8$ to $M_\mathrm{B}=-22$. The resulting empirical
relation between
the B-H colour and the B-band is then given by
\begin{equation}
\label{eq_B_H}
M_\mathrm{B}-M_\mathrm{H} = -0.14\;M_\mathrm{B}+0.74\;.
\end{equation}
In \citet{bell2001a} the stellar-mass-to-light-ratio--colour relations
are given for seven different galaxy evolution models, which lead to slightly
different values for $a_\mathrm{B,B-V}$, $a_\mathrm{B,V-H}$,
$b_\mathrm{B,B-V}$, and $b_\mathrm{B,V-H}$. The resulting
coefficients, $a_\mathrm{B,B-H}$ and $b_\mathrm{B,B-H}$,
are listed in Tab.~\ref{tab_M_L}. In the following analysis we use their
mean values for the
$\log M_\mathrm{*}/L_\mathrm{B}$--($M_\mathrm{B}-M_\mathrm{H}$) relation,
\begin{equation}
\label{eq_a_b_values}
a_\mathrm{B,B-H} = -1.633\;\;\;\mathrm{and}\;\;\; b_\mathrm{B,B-H}=0.602\;.
\end{equation}
We now have for each of our galaxies $M_\mathrm{B}$ (Tab.~\ref{tab_data}, Col~4),
$M_\mathrm{H}$ (from eq.~\ref{eq_B_H}) and thus $M_\mathrm{*}/L_\mathrm{B}$
(from eq.~\ref{eq_B_B-H}).
Each of the $M_\mathrm{B}$ values can therewith be converted to the total
stellar mass $M_\mathrm{*}$ allowing the computation of $\tau_\mathrm{*}$
for each galaxy.
For a few galaxies of our sample absolute H-band magnitudes are listed
in \citet{kirby2008a}. For these galaxies we use the observed absolute H-band
magnitude, for all the others we use the calculated B-H colour according to the
procedure described above. Table~\ref{tab_H_obs_cal}
lists those galaxies which have observed
absolute H-band magnitudes by \citet{kirby2008a}.
\include{tab4}
We now estimate the error of the calculated stellar mass
due to different galaxy evolution models and theoretical stellar-mass-to-light
ratios. Consider one galaxy with observed blue-band luminosity, $L_\mathrm{B}$,
and current star formation rate, SFR. Two different theoretical
stellar-mass-to-light ratios, $\Upsilon_1$ and $\Upsilon_2$, lead to two
different stellar masses, $M_{*,1}$ and $M_{*,2}$. The logarithmic difference of the
two corresponding stellar mass
buildup times, $\tau_{*,1}$ and $\tau_{*,2}$, is
\begin{eqnarray}\nonumber&&
\log\tau_{*,2}-\log\tau_{*1} = \log\frac{\tau_{*,2}}{\tau_{*,1}}=
\log\frac{M_{*,2}\; SFR}{SFR\;M_{*,1}}\\
\nonumber&&
=\log\frac{M_{*,2}\; SFR\;L_\mathrm{B}}{L_\mathrm{B}\;SFR\;M_{*,1}}=
\log\frac{\Upsilon_{2}}{\Upsilon_1}\\
&&
=\log\Upsilon_{2}-\log\Upsilon_1\;.
\end{eqnarray}
Note that $L_\mathrm{B}$ and $SFR$ can be crossed out, because they refer
to the same galaxy.
In Fig.~\ref{fig_M_L_error} the stellar-mass-to-light ratio is plotted
as a function of the total blue-band magnitude for the different galaxy
evolution models listed in Tab.~\ref{tab_M_L}. For the least massive
galaxies the model variation in the stellar-mass-to light ratio is
about $\pm 0.05$ dex around the mean value relation and becomes smaller for
more massive galaxies. Therefore, for the purpose of this analysis
the uncertainty in the calculation of the current stellar mass due
to galaxy evolution can be neglected.
\begin{figure}
\plotone{f9}
\caption{\label{fig_M_L_error} The stellar-mass-to-light ratio as a
function of the blue-band magnitude resulting from combining
eq.~\ref{eq_B_H} and \ref{eq_B_B-H} with Tab.~\ref{tab_M_L}.
Each dotted line represents one individual galaxy evolution model
and the solid line shows the mean value relation. Note that the increase
of $M_\mathrm{*}/L_\mathrm{B}$ with $M_\mathrm{B}$ is not a result
of the IGIMF effect but occurs as a result of using the empirical
relation (eq.~\ref{eq_B_H}) in the constant-IMF modelling of
\citet{bell2001a}.
}
\end{figure}
For the case of a constant galaxy-wide IMF the resulting stellar-mass
buildup times as a function of the total stellar mass are
plotted in Fig.~\ref{fig_t_star_m_star_imf}. The average value of the
stellar-mass buildup time is constant for galaxies more massive than
$10^8~M_\odot$ and increases by more than one order of magnitude for
less massive ones. This implies that the SFR in massive galaxies has
decreased much more slowly over cosmic time than for the least
massive galaxies. This
is in contradiction to the finding of downsizing, according to which
massive disk galaxies have on average older stellar populations than
dwarf galaxies.
\begin{figure}
\plotone{f10}
\caption{\label{fig_t_star_m_star_imf}
The stellar mass buildup time scale in units of the Hubble time,
$\tau_\mathrm{h}$~=~13.7~Gyr, in dependence of the total stellar mass
derived from the blue luminosity for a constant galaxy-wide IMF. Note
that the buildup times for dwarf star forming galaxies are longer than
the ones for large disk galaxies. This implies that the SFRs of dwarf
galaxies must have decreased faster over cosmic time than the SFRs
of large disk galaxies. This is in contradiction to the finding of
downsizing according to which the SFRs of large disk galaxies must have decreased
faster over cosmic time than the SFRs of dwarf galaxies.
The symbols are the same as in Fig.~\ref{fig_sfr_m_gas_imf}.
}
\end{figure}
When using H$\alpha$ as a SFR tracer,
in the IGIMF context the SFRs are much higher than the constant IMF-based
SFRs for low SFR galaxies, i.e. less massive galaxies. The corresponding
stellar-mass buildup times therefore become shorter. Indeed, for both
IGIMF models, standard (Fig.~\ref{fig_t_star_m_star_std_igimf})
and minimum (Fig.~\ref{fig_t_star_m_star_min_igimf}),
the stellar-mass buildup times
decrease monotonically with decreasing galaxy stellar mass.
This suggests that massive
disk galaxies have been forming stars at about a constant rate for a Hubble
time, while dwarf galaxies have either turned on more recently or have
a slightly increasing SFH, in full agreement with downsizing.
\begin{figure}
\plotone{f11}
\caption{\label{fig_t_star_m_star_std_igimf}
The same as Fig.~\ref{fig_t_star_m_star_imf} but for the standard IGIMF.
Note that the stellar mass build-up times, which result from SFR
calculations in the IGIMF context, decrease with decreasing galaxy mass.
This suggest that the SFR may have been increasing slighly with time
for dwarf galaxies or that they were forming stars over a more recent epoch
while for massive disk galaxies the SFHs may be constant,
which is in agreement with downsizing.
The symbols are the same as in Fig.~\ref{fig_sfr_m_gas_imf}.
}
\end{figure}
\begin{figure}
\plotone{f12}
\caption{\label{fig_t_star_m_star_min_igimf}
The same as Fig.~\ref{fig_t_star_m_star_std_igimf} but for the minimum IGIMF.
The symbols are the same as in Fig.~\ref{fig_sfr_m_gas_imf}.
}
\end{figure}
\section{Conclusion}
We have applied the revised SFR-$L_\mathrm{H\alpha}$
relation, which takes the nature of clustered star formation into account,
to a significantly larger sample of star forming galaxies of the local
volume than was available in \citet{pflamm-altenburg2007d}.
The SFRs of galaxies and the corresponding gas depletion and
stellar-mass buildup time scales are calculated.
Comparing these new values with the
classical ones which are derived from SFRs based on an assumed constant
galaxy-wide IMF, the changes are dramatic:
The SFRs of galaxies scale linearly with the total galaxy neutral gas mass and the
corresponding gas depletion time scales are independent of the galaxy
neutral gas mass. This implies that dwarf galaxies have the same star formation
efficiencies as large disk galaxies. Furthermore, the stellar-mass
buildup times are only compatible with downsizing in the IGIMF context.
They are inconsistent with downsizing in the classical theory which
assumes an invariant galaxy-wide IMF.
These results follow from the
SFR$_\mathrm{H\alpha}$/SFR$_\mathrm{FUV}$ data compiled by \citet{lee2009a},
independently of the well developed IGIMF theory. The IGIMF theory merely
explains the \citet{lee2009a} results within an empirically well established
star-formation framework. The success of the IGIMF theory,
developed by \citet{weidner2003a} and \citet{weidner2005a,weidner2006a}, is
that the \citet{lee2009a} results have been predicted
\citep{pflamm-altenburg2007d} rather than being
adjusted afterwards. The power of the IGIMF theory lies in that it
is readily computable and deterministic.
We emphasise that the galaxy properties derived in the IGIMF theory
are qualitatively
independent of the main parameter of the IGIMF, the slope of the ECMF.
For the full range of
possible slopes of the ECMF in the IGIMF theory
SFRs of dwarf galaxies are significantly higher than derived in the classical
paradigm assuming a constant galaxy-wide IMF independent
of the global SFR.
The revision of the SFRs of dwarf galaxies suggested here
now poses a major challenge for our theoretical
understanding of star formation in galaxies which had been developed with the
aim of explaining the low star formation efficiencies of dwarf galaxies.
It follows that galaxy and cosmic evolution models need a substantial revision
requiring further studies.
\bibliographystyle{aa}
|
2,877,628,089,016 | arxiv | \section{Usage Scenario}~\label{sec:usabilityScenario}
In this section, we describe an end-to-end example illustrating how our tool can be applied in practice by a requirements engineer. Let \textit{KoopaApp} be a new system under development. KoopaApp is a gym fitness app for maintaining users' workout information and other health-related data. KoopaApp accesses personal information such as the location from other apps on the users' smartphone.
During the pandemic, such applications often raised concerns about privacy. For example, several health applications were analyzed for privacy-related issues in the RE literature~\cite{Bano:21,Fazzini:22}.
A requirements engineer (\textit{Daisy}) is in charge of specifying the KoopaApp requirements, including compliance requirements. As an example, we focus only on a subset of compliance requirements related to privacy and data protection.
Daisy (as is often the case in most software projects) is not very familiar with the privacy regulations, yet she knows well the functionalities and characteristics of the KoopaApp. During the elicitation of requirements, Daisy identifies a set of functionalities that make use of personal data and are thus subject to compliance. Some of these functionalities are related to the security of collected personal data. We assume here that Daisy or her team are aware of the relevant legal documents for their project. Suggesting relevant documents is beyond the scope of COREQQA. Accounting to possible security threats, Daisy poses a question (``What is the procedure for handling a personal data breach?'') using the COREQQA tool on the GDPR~\cite{GDPR}. COREQQA in turn provides the output shown in Figure~\ref{fig:example}.
From the output of COREQQA, Daisy is able to formulate the following compliance requirements (prefixed with the ID $CR$) under the label \textit{Users Data Breach}.\\
\textbf{Notify Users about the Data Breach.} \\
$CR_1.$ If a data breach is identified on the KoopaApp server, the KoopaApp-NotifyService shall inform the affected users. \\
$CR_2.$ The KoopaApp-NotifyService shall email the affected users on the registered email address and store the `user informed' response on the server. \\
$CR_3.$ The KoopaApp-NotifyService shall notify the affected users on the app and store the `read' response on the server.\\
\textbf{Notify the CIO about the Data Breach.} \\
$CR_4.$ If a data breach is identified on the KoopaApp server, the KoopaApp-NotifyService shall send an email to the Chief Information Officer notifying the breach, within 72 hours of its occurrence.
The four compliance requirements fulfill the regulations provided in Figure~\ref{fig:example}.
\section{Evaluation}~\label{sec:evaluation}
COREQQA has been evaluated on four legal documents, wherein the question-answer pairs were identified by two experts -- one expert in legal informatics and the other in requirements engineering~\cite{Abualhaija:2022}. In the following, we describe the four documents:
\sectopic{$\bullet$ GDPR} or General Data Protection Regulation (EU) 2016/679 is the European privacy law that harmonises the data protection, privacy and personal data transfer requirements~\cite{GDPR}. The experts identified 36 question-answer pairs from the entire document. The document was partitioned in 301 context spans by the ``Text Preprocessing'' step of Figure~\ref{fig:tool}.
\sectopic{$\bullet$ Directive (EU) 2019/770} is the European directive for regulating the supply of digital content or digital services, and laying down rules for contracts between any trader and consumer of digital content or service~\cite{EU_770}. The experts identified 33 question-answer pairs in this document, and the document was split into 120 context spans.
\sectopic{$\bullet$ Directive (EU) 2019/771} is the European directive concerning the sale of goods~\cite{EU_771}. Directive (EU) 2019/771 complements Directive (EU) 2019/770, as it formalises the contracts on the sale of goods that contain digital elements that require digital content or service. For example, the regulations related to the contracts of the smartphone are covered by Directive (EU) 2019/771, whereas the regulations for operating systems or apps on the smartphone might be covered by Directive (EU) 2019/770. The experts identified 19 question-answer pairs in Directive (EU) 2019/771, and the document text was split into 102 context spans.
\sectopic{$\bullet$ Luxembourg Law of 25 March 2020} is an amendment to numerous existing finance and banking related laws in Luxembourg. The amendment was intended to set due diligence measures for a central electronic data retrieval system related to bank accounts and safe-deposit boxes in Luxembourg, and was a step towards tackling money laundering~\cite{law_2020}. The experts identified 19 question-answer pairs in this legal document. The document text was split into 23 context spans.
To identify the most accurate similarity metric for context span retrieval, we compared BCE (Section~\ref{subsec:contextRetrieval}) with TF-IDF similarity~\cite{Aizawa:03} -- a metric commonly used in the NLP domain. BCE was significantly more accurate than TF-IDF in our experiments. Overall, from the 107 questions over the four documents, BCE retrieves the correct context span for 100 questions (for the top-$5$ spans).
In addition to RoBERTa, we further evaluated three QA models, namely BERT~\cite{Devlin:18}, ALBERT~\cite{Lan:19-albert}, ELECTRA~\cite{Clark:20-electra} for answer extraction (Section~\ref{subsec:answerExtraction}). RoBERTa was deemed the most accurate as it correctly extracted the answers for 97 questions.
We also analyzed the questions where COREQQA did not highly rank the correct context span (within top-5) or RoBERTa model did not extract the correct answer. Our analysis showed that generic questions, such as the ones formulated for defining or elaborating on a legal concept, were not correctly answered. This is because the legal document (or even a given context span) would usually contain several instances of such legal concept, thus misleading both the context span retrieval and answer extraction steps of COREQQA. We also realized that complicated questions (e.g., with composite conditions) were difficult to answer for COREQQA.
Last but not least, COREQQA answers questions within reasonable execution time. Thus, in short, our evaluation indicates that COREQQA produces accurate results and is fit for use by requirements engineers in practice. For answering one question from a legal document including an average of 620 sentences, COREQQA requires a total of $\approx$34 seconds.
\section{Conclusion}
We presented COREQQA -- a tool for assisting requirements engineers in better understanding compliance requirements through question-answering based on regulatory or legal documents. COREQQA is developed using a manually created dataset that combines a joint effort of a requirements engineer and a legal expert over four diverse legal documents. The tool is based on recent large-scale language models that are pre-trained for question-answering. Specifically, the tool applies the Sentence BERT cross encoder for retrieving the most relevant text passages from a legal document for a given question. The tool further employs the RoBERTa question-answering model for highlighting the likely answers to the question in the retrieved text passages.
In future, we plan to conduct a user study to assess how useful COREQQA is in practice.
\section{Introduction}~\label{sec:introduction}
With the growing reliance on personal data and confidential information, software systems are increasingly subject to compliance against regulations to enforce necessary safeguards for information protection and human safety~\cite{Janssen:2020,Leidner:21}. Failing to comply with relevant regulations can lead to legal, fiscal or reputational implications for an organization. Regulatory compliance is regarded as an essential yet challenging task by the Requirements Engineering (RE) community~\cite{Otto:07,Berenbach:09,Sleimi:18}.
\begin{figure}
\centering
\includegraphics[width=0.5\textwidth]{Figures/example_Tool.pdf}
\vspace*{-1em}
\caption{{Example of COREQQA's QA assistance on GDPR~\cite{GDPR}.}}
\label{fig:example}
\vspace*{-1em}
\end{figure}
In this paper, we propose the tool COREQQA (\textbf{Co}mpliance \textbf{Req}uirements Understanding using \textbf{Q}uestion \textbf{A}nswering). COREQQA is motivated by actual practical needs, considering that requirements engineers are being increasingly involved in software compliance against relevant regulations, e.g., all software systems in Europe must comply with GDPR (Regulation (EU) 2016/679) -- the European regulation on data protection, privacy and personal data transfer~\cite{GDPR}. Manually handling compliance requirements is tedious and error-prone since requirements engineers have to read through entire legal documents. Such documents are usually hefty, contain complicated natural language (NL) structures, frequently refer to external regulations, and are not easy to peruse without legal expertise~\cite{Sleimi:19,Abualhaija:2022}.
An automated Question Answering (QA) tool such as COREQQA helps requirements engineers efficiently navigate through the compliance-related content of legal documents.
QA is the task of automatically finding the answer to a question posed in NL from a given text passage. In our work, we refer to a single text passage as a \textit{context span}.
Instead of reviewing long, complex legal documents, COREQQA enables requirements engineers to ask a question about a compliance-related topic, and then returns a list of relevant context spans in which the answer is likely to be found. This way, COREQQA pinpoints the requirements engineers to the portions of the legal document where they need to invest their efforts and time.
We illustrate in Figure~\ref{fig:example}, the QA assistance provided by COREQQA to a requirements engineer, who is interested in understanding the regulations related to personal data breach.
The example question is specifically related to the process for handling personal data breaches. The answer to this question is mined in the GDPR text.
The legal obligations with regard to handling data breaches can have a significant impact on the software development process, e.g., sending notifications to different responsible agents within legally-binding time constraints.
COREQQA assists the requirements engineer in retrieving relevant information to define the respective compliance requirements for handling data breaches for the software system under development.
As we elaborate in Section~\ref{sec:architecture}, COREQQA returns as output the top-$N$\footnote{$N$ is a configurable parameter and is set $N=3$ for the example question in the figure} relevant context spans from a given legal document and the potential answers highlighted in the context spans.
COREQQA builds on large-scale natural language processing (NLP) language models for solving the QA task (also widely known as \emph{machine reading comprehension (MRC) task}~\cite{Jurafsky:20}). QA models for MRC generally assume that for each question, the relevant context span (containing the answer) is known a priori.
Developing a practical QA tool with this restriction is infeasible, as requirements engineers have no means of knowing the exact context span with the correct answer in advance. Therefore, in COREQAA we first find top-$N$ relevant context spans which likely contain the answer. To do so, we compute the semantic similarity between each context span in the legal document and the input question. Then, we demarcate the answer to the input question using the QA models.
We further observe that information relevant to answering the question could be found in multiple non-contiguous context spans (i.e., different sections in the same legal document). For example, the top-3 context spans selected in Figure~\ref{fig:example} are all directly relevant for answering the question. The first two spans (retrieved from different sections of the document)
explain the process of communicating breach details to the data subject, while the third span
specifies how to communicate breach details to the supervisory authorities. Thus, by retrieving multiple relevant spans and further highlighting the likely answers, COREQQA enables the requirements engineer to specify a complete and precise set of compliance requirements.
We leave configuring the $N$ parameter to the requirements engineer. While selecting higher values of $N$ entails more time and effort for reviewing the retrieved context spans, we believe that using COREQQA is still much more cost-effective in practice than manually traversing the entire legal document for the answer.
In the remainder of this tool demonstration paper, we elaborate the architecture of the tool, the dataset that we generated for developing COREQQA as well as an end-to-end usage scenario.
\section{Tool Architecture}~\label{sec:architecture}
The end-to-end architecture of COREQQA is illustrated in Figure~\ref{fig:tool}.
COREQQA aims at answering a given question posed by a requirements engineer in NL on some legal document. Below, we elaborate the main steps of the tool marked as 1 -- 3 in Figure~\ref{fig:tool}. We implemented COREQQA in Python 3.8.
\begin{figure*}
\centering
\includegraphics[width=\textwidth]{Figures/tool.pdf}
\caption{Overview of the end-to-end architecture of COREQQA.}
\label{fig:tool}
\end{figure*}
\subsection{Text Preprocessing}~\label{subsec:preprocessing}
In the first step, COREQQA parses the legal document and applies a simple NLP pipeline which is composed of tokenization and sentence splitting.
The tool then applies a set of regular expressions for normalizing the text (e.g., removing periods from the ending of acronyms, ``Art.'' becomes ``Art''). The motivation for normalizing the text is to improve the accuracy of sentence splitting. We operationalize the NLP pipeline using NLTK library~\cite{Bird:09,NLTK}, and the \textit{re module} in Python for regular expressions\footnote{\url{https://docs/python.org/3.8/library/re/}}.
In this step, we further partition the legal document into context spans. Due to technical constraints of underlying QA models, the maximum size of each context span is 512 tokens. To maintain coherence, we split the document into paragraphs first, and then check their size. Each paragraph fitting this size limitation is regarded as one context span. Otherwise, we split the paragraph into half, and check the size again. This process is iteratively performed until size limitations are met.
The output of this step is the list of context spans representing the input legal document.
\subsection{Relevant Context Retrieval}~\label{subsec:contextRetrieval}
In the second step, COREQQA computes the semantic similarity between the input question and each context span generated from the previous step.
We implement this step using the BERT cross-encoder (BCE) model available in the Sentence-Transformer 2.1.0~\cite{Reimers:19} provided by Hugging Face\footnote{\url{https://huggingface.co/}}.
BCE takes as input two text fragments, and returns as output a score between 0 and 1 indicating how semantically similar the two fragments are, with 1 being identical. To assess the relevance of the context span, we first compute BCE between the question and each sentence in the context span and then assign to the context span the maximum score achieved by any sentence. The intuition behind this computation strategy is that only a portion of the context span is expected to contain the likely answer to the input question.
Once we compute a score per context span, we rank the spans in descending order. We do this using the sort function from \textit{pandas} in Python\footnote{\url{https://pandas.pydata.org}}.
The result of this step is a list of top-$N$ relevant context spans to the input question. We keep the value \textit{N} as a parameter that can be initialized by the requirements engineer.
The value of $N$ depends on the practical context in which COREQQA is applied. Selecting large values (e.g., $\geq 10$) entails that the requirements engineer will review many
context spans to gain a better understanding of the compliance requirements associated with the question. Selecting small $N$ values (e.g., $\leq 3$) entails less context spans to be reviewed, but also a higher risk for the right answer not to be found in any of the top-$N$ relevant spans.
In Figure~\ref{fig:example}, we show an example of top-3 context spans retrieved in this step. The default value of the configurable parameter $N$ is set to 5 in COREQQA, based on previous experiments~\cite{Abualhaija:2022}.
\subsection{Answer Extraction}~\label{subsec:answerExtraction}
In the last step, we pass on the top-$N$ context spans deemed relevant in the previous step together with the input question to a pre-trained QA model. In our work, we apply the Transformers library to extract answers for the given question using the RoBERTa QA model~({\emph{roberta-base-squad2-distilled}}). RoBERTa then extracts from each context span a potential answer for the input question.
In Figure~\ref{fig:example}, we highlight the extracted answers in green.
We note that the engineer has access to the context spans already in the previous step. Thus, this step is not essential for providing assistance to the requirements engineer in understanding the questions related to compliance requirements. However, highlighting the answer in the context span improves readability and leads to a more efficient reviewing process. In practice, when the engineer selects a larger number $N$ (say 10), it is then advantageous to demarcate the answer automatically to help the engineer quickly navigate through the context spans.
\subsection{Final Output Representation}
For presenting the final output of the tool to the requirements engineer, we export for each question the top-$N$ context spans and the highlighted answers within these context spans as a Microsoft Word document. The document also shows for each context span a confidence score (range 0--1) of the answer highlighted in the span. This confidence score is automatically assigned to the extracted answer by the QA model.
We use the python-docx library v0.8.11~(\url{https://python-docx.readthedocs.io}) for exporting the output and visualizing highlighted answers in the relevant top-$N$ context spans.
|
2,877,628,089,017 | arxiv | \section{History}
People from all over the world have been playing games for centuries with an early form of dice being frequently discovered in Assyrian and Simerian archeological digs. A man named Girolamo Cardano $(1501-1576)$ is sometimes considered the first to give a working definition for probability with his works on this not being published until close to a century after his passing, around $1676$. By this time though, others had begun writing about the mathematics behind games including Galileo specifically around $1613$ and $1623$, who especially focused on those involving dice. Galileo had noted in \textit{Concerning an Investigation of Dice} the amount of sums when given two or more dice and how many ways this could occur in addition. Antoine Gombaud, also known as The Chevalier de M\'{e}r\'{e}, $(1607-1684)$ became caught up in gambling in which he began to question explanations to certain aspects that are naturally mathematical. Specifically, he questioned rolling at least one six on a die in four rolls with an even money bet in which he had gained a considerable sum of money while, over time, he had noticed that he lost more money than he expected while betting on one or more double sixes in twenty-four rolls of two dice and wondered why. In $1654$, he asked for help from Blaise Pascal, the well-known mathematician, in order to understand. Sparking Pascal's interest, Pascal corresponded with Pierre de Fermat to answer de M\'{e}r\'{e}'s question in which probability was formally stated.
Using the Multiplication Principle, which is the product of the number of ways each task can be performed, we can explain de M\'{e}r\'{e}'s dice problem. To start with the game of obtaining at least one six in four rolls, there are $6^4=1296$ possible outcomes. Now, we need to account for how many of these actually contain at least one six. This is done by counting how many of these outcomes do \emph{not} contain a six. So, if one of the dice has to be a six, then we have five outcomes per one roll which gives $5^4=625$ outcomes that do not have a six. So, taking the first minus the second number of outcomes we have $1296-625=671$ with at least one six. Then the proportion of outcomes for at least one six with the four rolls is
$$ \frac{671}{1296}=.5178.$$ We can see from this that we should expect to win more often than not.
With the second game of one or more double sixes in 24 rolls with two dice, however, the sample space is comprised of $6^2=36$ separate outcomes. Within this sample space, there is only one pair of double sixes which means $35$ of these outcomes we don't want. So, the proportion of these outcomes that do not contain double sixes is
$$ \left(\frac{35}{36}\right)^{24}=.5086. $$
Since this is greater than one half, we know that we should expect to lose more often. Therefore, we can see the solution to de M\'{e}r\'{e}'s queries. This is all how probability started, leading to everything we know today about it.
\section{Introduction to Zombie Dice}
The game Zombie Dice was created by Steve Jackson Games, Inc. In the game of Zombie Dice, the player plays as if he is the zombie, eating brains and getting shotgunned. For each die, there are three different face types. First are the brains which means eating the victim's brain, and they are set aside for that player's round. Shotguns are the zombie, or player, being ``shot''. These dice also are set aside. The last face type are footprints. This is where the victim escaped, so these dice are kept in front to roll again if the player so chooses.
There are thirteen dice total with three red, four yellow, and six green. Each specific color of dice have separate amounts of brains and shotguns, but each have two footprints. The red dice have one brain and three shotguns, the yellow have two brains and two shotguns, and the green have three brains and one shotgun.
The player starts by shaking the cup of thirteen dice and picking three random dice. If the player rolls shotguns or brains, they are placed to the side until the end of their turn. The player's turn is over when at least three shotguns are rolled in total for that round in which he receives no points for that round. The player can also choose to stop and score his points instead of rolling again, in which he gains one point per one brain accumulated. When a player chooses to continue, he then takes enough random dice from the cup to total three, which includes any footprints from the previous roll. For instance, say the player had one footprint from the previous roll, then only two dice are taken from the cup for a total of three dice to roll. If there are not any dice left in the cup, or not enough to total three for the next roll, the player notes how many brains he has gained and keeps the shotguns out while returning the rest of the dice to the cup and continues as normal. The object of the game is to gain thirteen or more points total. Once a player has obtained this, the rest of the players in that round finish. Whoever has the most brains at the end of the round wins. A tiebreaker round is played if a tie occurs.
The question we analyzed was, given we currently have $X$ brains, should we keep rolling or score our points? We began by creating two functions in order to find the probability of getting three or more shotguns given our current number of shotguns and the probability of getting zero, one, two, or three brains on the next roll which gave the following probability distribution functions
\begin{multicols}{2}
\begin{center}\begin{tabular}{c|c} $Y$ & $P(Y)$ \\\hline 0 & 0.347449 \\ 1 & 0.444833 \\ 2 & 0.183372 \\ 3 & 0.024346 \end{tabular} \end{center}
\columnbreak
\begin{center}\begin{tabular}{c|c} $X$ & $P(X)$ \\\hline 0 & 0.245144 \\ 1 & 0.444056 \\ 2 & 0.261072 \\ 3 & 0.049728 \end{tabular} \end{center}
\end{multicols} \noindent where $Y$ is the number of shotguns and $X$ is the number of brains. We then created a model which led to a recursive program in order to obtain the expected values for all future rolls. With this, we created a strategy along with generalizations to a simpler strategy that's easily memorizable. In order to reach the goal of optimal strategy to help answer the question though, we need a reminder and the definition for expected value. \emph{Expected value} can be used to see how much one should expect to receive over repeated trials, or in the long run. To give a more formal definition, we need to know about random variables. With an experiment having a sample space $S$, a function $X$ that assigns only a single real number to each element of $S$, $X(s)=x$, is called a \emph{random variable}. Now to define expected value, we let $X$ be a random variable and $e_1, e_2, \ldots, e_k$ be the elements of the sample space with the corresponding probabilities $p_1, p_2, \ldots, p_k$. Then the expected value $(EV)$ of the random variable is
$$EV(X)=p_1\cdot X(e_1)+p_2\cdot X(e_2)+\ldots+p_k\cdot X(e_k).$$ Now we can look at conditional expectation. First, we need to let $X$ be a discrete random variable and $E$ to be an event where $P(E)>0$. Then the \emph{conditional expectation} of the random variable $X$ given $E$ has occurred is defined as
$$EV(X|E)=\sum_{x\in X}xP(X=x|E).$$
Our answer to the question posed above is that the player will continue to roll if
$$- X \cdot\ P(B|S=Y) + EV \cdot\ (1-P(B|S=Y)) > 0$$
where
$$\begin{aligned} X &= \text{current number of brains} \\ Y &= \text{current number of shotguns} \\ P(B|S=Y) &= \text{probability of busting given we have $Y$ shotguns} \\ EV &= \text{expected number of brains on next rolls.}\end{aligned}$$
\section{The Model}
In order to construct one recursive function to answer when the player should keep rolling or stop and score points, we use the multinomial distribution along with summations. The multinomial coefficient counts the number of ways to choose from $n$ objects a collection of $k$ types of objects where there are $x_1$ of one type, $x_2$ of another, and so on up to $x_k$ of the final kind. We require that $x_1+x_2+\ldots+x_k=n$ so each $n$ object will appear as one of the types. The multinomial coefficient is given by
$$\ds{n\choose x_{1},x_{2},...,x_{k}}=\frac{n!}{x_1!\cdot x_2!\cdots x_k!}.$$
The multinomial distribution uses the multinomial coefficient and has $n$ trials which can result in any number $k$ of different results. In our case, the different results are brains, footprints, and shotguns. The trials are identical so the probability of outcome $R_{i}$ stays the same for each repetition. If the probability of outcome $R_{i}$ is given as $p_{i}$ then the probability that the experiment yields an $x_{i}$ number of outcome $R_{i}$ is given by:
$$\text{P}(R_{1}=x_{1},R_{2}=x_{2},...,R_{k}=x_{k})=\ds{n\choose x_{1},x_{2},...,x_{k}}\cdot p_{1}^{x_{1}}\cdot p_{2}^{x_{2}}\cdot\cdot\cdot p_{k}^{x_{k}}$$
where we require that
$$\begin{aligned} x_{1}+x_{2}+...+x_{k}&=n \text{ and} \\
p_{1}+p_{2}+...+p_{k}&=1.\end{aligned}$$
With this definition we can now write our probabilities much easier. For example, we can define
$$\text{P(Green Shotguns)}=P_{\text{green}}^{(0,0,g)}=\underbrace{\ds{g\choose 0,0,g}}_{\text{all green shotguns}}\cdot \underbrace{\left(\frac{1}{2}\right)^0}_{\text{brains}}\cdot \underbrace{\left(\frac{1}{3}\right)^0}_{\text{footprints}}\cdot \underbrace{\left(\frac{1}{6}\right)^g}_{\text{shotguns}}.$$
This will now allow us to write the probability of getting three shotguns of some combination between the green, yellow, or red dice as
$$\text{P(3 Shotguns With }\{g,y,r\})=P_{\text{green}}^{(0,0,g)}\cdot P_{\text{yellow}}^{(0,0,y)}\cdot P_{\text{red}}^{(0,0,r)}.$$
The probability of obtaining a certain combination of dice from the cup, the coefficient on these probabilities, is is computed as
$$\text{P}(g\text{ Green, }y\text{ Yellow, }r\text{ Red Dice})=\frac{\overbrace{\ds{6\choose g}}^{\text{green dice}}\cdot \overbrace{\ds{4\choose y}}^{\text{yellow dice}}\cdot \overbrace{\ds{3\choose r}}^{\text{red dice}}}{\underbrace{\ds{13\choose 3}}_{\text{all dice combinations}}}.$$
In general, we can define the following which is the probability of getting $x_{b}$ brains, $x_{f}$ footprints, and $x_{s}$ shotguns for the green dice where we require that $x_{b}+x_{f}+x_{s}=n$ as
$$P_{\text{green}}^{(x_{b},x_{f},x_{s})}=\underbrace{\ds{n\choose x_{b},x_{f},x_{s}}}_{\text{orderings}}\cdot \underbrace{\left(\frac{1}{2}\right)^{x_{b}}}_{\text{brains}}\cdot \underbrace{\left(\frac{1}{3}\right)^{x_{f}}}_{\text{footprints}}\cdot \underbrace{\left(\frac{1}{6}\right)^{x_{s}}}_{\text{shotguns}}.$$
The formulas for the yellow and red dice are similar with the same restriction and can be written respectively as
$$P_{\text{yellow}}^{(x_{b},x_{f},x_{s})}=\ds{n\choose x_{b},x_{f},x_{s}}\cdot \left(\frac{1}{3}\right)^{x_{b}}\cdot \left(\frac{1}{3}\right)^{x_{f}}\cdot \left(\frac{1}{3}\right)^{x_{s}}$$
and
$$P_{\text{red}}^{(x_{b},x_{f},x_{s})}=\ds{n\choose x_{b},x_{f},x_{s}}\cdot \left(\frac{1}{6}\right)^{x_{b}}\cdot \left(\frac{1}{3}\right)^{x_{f}}\cdot \left(\frac{1}{2}\right)^{x_{s}}.$$
Note that the formula for the yellow dice can be simplified as
$$P_{\text{yellow}}^{(x_{b},x_{f},x_{s})}=\ds{n\choose x_{b},x_{f},x_{s}}\cdot \left(\frac{1}{3}\right)^{n}$$
since the probabilities for brains, footprints, and shotguns are the same.
The expected value of continuing is computed as
$$\text{Expected Value of Continuing}=\underbrace{-b\cdot PE}_{\text{round ends}}+\underbrace{EB\cdot (1-PE)}_{\text{round continues}}>0$$
meaning that, with this inequality, we can play the game as though we are betting with the current amount of brains being the wager and continue until the expected value becomes zero or negative.
As an example, using all of the above pieces together and defining $S$ as the set of all triples of the form $\{g,y,r\}$ where $0 \leq g,y,r \leq 3$ and $g+y+r=3$, we can compute the probability of gaining $3$ shotguns on the first round with the use of summations as
$$ \sum_{\{g,y,r\}\in S} \frac{\ds{6\choose g}\cdot \ds{4\choose y}\cdot \ds{3\choose r}}{\ds{13\choose 3}}\cdot p_{\text{green}}^{(0,0,g)} \cdot p_{\text{yellow}}^{(0,0,y)} \cdot p_{\text{red}}^{(0,0,r)}=\frac{94}{3,861}\approx0.02435$$
meaning that the chance of your turn ending is about $2.4\%$ on the first roll.
In order to generalize this formula even more, we can define the following $\mathbf{C}\{g,y,r\}$ where $g_c$ green, $y_c$ yellow, and $r_c$ red dice are left in the cup and $g_f$ green, $y_f$ yellow, and $r_f$ red footprints roll from the previous roll (the boldface reminds us that the functions depends on $g_c$, $y_c$, $r_c$, $g_f$, $y_f$, and $r_f$). It gives the probability of getting the combination $\{g,y,r\}$ from the cup. So,
$$C_{g_f,y_f,r_f}^{g_c,y_c,r_c}\{g,y,r\}=\mathbf{C}\{g,y,r\}=\frac{\ds{g_c\choose g-g_f}\cdot \ds{y_c\choose y-y_f}\cdot \ds{r_c\choose r-r_f}}{\ds{g_c+y_c+r_c\choose 3-(g_f+y_f+r_f)}}$$
where $\mathbf{C}\{g,y,r\}=0$ for impossible situations such as when $g_f=2$ with the combination $\{1,1,1\}$ since this would mean the player would reroll two green dice and therefore could not have a red \textit{and} a yellow die to roll as well.
This allows us to write the following which gives the probability of getting $3$ shotguns on the next roll as
$$P(\text{3 Shotguns})=\sum_{\{g,y,r\}\in S} \mathbf{C}\{g,y,r\}\cdot p_{\text{green}}^{(0,0,g)} \cdot p_{\text{yellow}}^{(0,0,y)} \cdot p_{\text{red}}^{(0,0,r)}.$$
We can tackle the expected amount of brains in a similar manner using the same notation as above. The expected number of brains on the next roll is then
$$\text{Expected Brains}=0\cdot \mathbf{B}(0)+1 \cdot \mathbf{B}(1)+2\cdot \mathbf{B}(2)+3\cdot \mathbf{B}(3)$$
where $\mathbf{B}(x)$ is the probability of obtaining $x$ brains on the next roll (which depends on the number and color of footprints and dice left in the cup).
Using the summation from the shotgun example above, we gain the following brain summation given the current values of $g_c$ and $g_f$ and the corresponding red and yellow dice in the cup and footprints. We write
$$\mathbf{B}(3)= \sum_{\{g,y,r\}\in S} \mathbf{C}\{g,y,r\}\cdot p_{\text{green}}^{(g,0,0)} \cdot p_{\text{yellow}}^{(y,0,0)} \cdot p_{\text{red}}^{(r,0,0)}.$$
There is a difference between the notation $\{g,y,r\}$ and $(i,j,k)$ though, in which the first represents a specific selection of dice where the player has $g$ green dice, $y$ yellow dice, and $r$ red dice to roll and the latter represents a specific color of dice getting $i$ brains, $j$ footprints, and $k$ shotguns. Each element in $S$, which is the set of all triples of the form $\{g,y,r\}$ where $0 \leq g,y,r \leq 3$ and $g+y+r=3$, gives a certain set of $(i,j,k)$ possibilities for each color separately. To make this easier, we denote $T(\{g,y,r\};x)$ to be the set of possibilities given values for $g, y, r, \text{and } x$. So the most general form for $\mathbf{B}(x)$ for other values of $x$ is then
$$\mathbf{B}(x)= \sum_{\{g,y,r\}\in S} \mathbf{C}\{g,y,r\}\sum_{T(\{g,y,r\};x)} p_{\text{green}}^{(g_i,g_j,g_k)} \cdot p_{\text{yellow}}^{(y_i,y_j,y_k)} \cdot p_{\text{red}}^{(r_i,r_j,r_k)}.$$
Then, we can construct a formula with summation notation to find the expected number of brains on the next roll given the dice that is left in the cup and the footprints to reroll as
$$\text{Expected Brains}=EB_{g_f,y_f,r_f}^{g_c,y_c,r_c}=\mathbf{EB}=\sum_{i=0}^3 i\cdot \mathbf{B}(i).$$
With this notation we can also denote the probability of getting $x$ shotguns given the regular parameters by $S_{g_f,y_f,r_f}^{g_c,y_c,r_c}=\mathbf{S}(x)$ which gives
$$\mathbf{S}(x)=\sum_{\{g,y,r\}\in S}\mathbf{C}\{g,y,r\}\sum_{T(\{g,y,r\};x)}p_{\text{green}}^{(g_k,g_j,g_i)} \cdot p_{\text{yellow}}^{(y_k,y_j,y_i)} \cdot p_{\text{red}}^{(r_k,r_j,r_i)}.$$
This is almost the same as that of the brains but with the difference of the use of $(g_k,g_j,g_i)$ where the $g_i$ and $g_k$ roles are reversed in $p_{\text{green}}^{(g_k,g_j,g_i)}$ since the order of $(g_k,g_j,g_i)$ corresponds to $g_k$ brains, $g_j$ footprints, and $g_i$ shotguns which is the reverse of the elements of $T$ when we deal with shotguns. The same goes for the corresponding red and yellow dice functions.
The probability that the round ends depends upon the number of current shotguns, so we need to address the probability of the round ending on the next roll due to shotguns. Since the player can only roll three shotguns maximum, the probability is as follows
$$\text{P(Round Ends)}=PE^{g_c,y_c,r_c}_{g_f,y_f,r_f}(s)=\mathbf{PE}(s)=\sum^{3}_{i=3-s}\mathbf{S}(i)$$
where we have $s$ current shotguns and the usual variables. If we have one current shotgun, then we need to add together the probabilities of attaining two or three shotguns on the next roll which is written as
$$\mathbf{PE}(1)=\mathbf{S}(2)+\mathbf{S}(3).$$
Now, we can rewrite the expected value of continuing with this new notation as
$$\text{Expected Value of Continuing}=\underbrace{-b\cdot\mathbf{PE}(s)}_{\text{round ends}}+\underbrace{\mathbf{EB}\cdot(1-\mathbf{PE}(s))}_{\text{round continues}}>0.$$
However, we need to correct this since the function $\mathbf{EB}$ gives brains that are not allowed. For example, if we have \{1,1,1\} as our element of $S$ that we are iterating over and we have two current shotguns, then the $T(\{1,1,1\};2)$ element has $(1,0,0)$ for the green dice, $(1,0,0)$ for the yellow dice, and $(0,0,1)$ for the red dice giving us two extra brains and ending the round since we also gained a third shotgun. Thus, some of the elements of $T$ need to be ignored for certain $S$ elements. Also, if the player chooses to reroll and does not obtain three shotguns, then he has the opportunity to roll again and get more brains. So, we need to consider the expected number of brains on all future rolls and not just the next roll. This is done recursively, easily through using computers.
Solving the Expected Value of Continuing equation for $b$, we can give a value of $b$ that is similar to a decision point for a fixed value of $s$ and the values for the quantity of dice remaining in the cup and current footprints. Thus, we get
$$\frac{\mathbf{EB}\cdot(1-\mathbf{PE}(s))}{\mathbf{PE}(s)}>b.$$
This means that the player should continue rolling as long as the number of brains the player has currently, $b$, is less than the value on the left-hand side after one roll. Each time the player rolls, the left-hand side needs to be recomputed since the expected value of brains and the probability of the round ending changes with each roll.
A program was written by Dr. David Taylor in which the limitations above were removed in order to obtain an optimal strategy for playing Zombie Dice. This program gives the decision point $b$ for each possible combination of the variables for the amount of dice in the cup, footprints, and current number of shotguns. For example, with the following set up
\begin{center}
\begin{tabular}{c|c|c|c|c}
Dice in Cup & Current FPs & Decision & Decision & Decision
\\ R Y G & R Y G & (SG 0) & (SG 1) & (SG 2)
\\\hline 2 3 1 & 1 0 1 & 78.338580 & 4.043669 & 0.180008
\end{tabular}
\end{center}
the player should continue rolling if he has less than or equal to $78$ brains with no current shotguns, or if he has less than or equal to $4$ brains with one current shotgun. If the player has two current shotguns, then he should stop rolling if he has one or more brains and he should only keep rolling if he has no brains because the left-hand side would have to be less than $0.180008$ to keep rolling. This would be the strategy when the player has $2$ red, $3$ yellow, and $1$ green dice in the cup as well as $1$ red, $0$ yellow, and $1$ green footprints to reroll. With every combination of colors and amounts of footprints and dice in the cup, the decision point changes and especially depends on the number of current shotguns.
\section{Strategy}
Using the table and strategy created through recursion as explained above, the player can be a formidable opponent throughout the entire game, especially during the beginning and middle. Near the end of the game however (which is when someone in the current round who is before the player in turn order reaches $13$ or more brains, or when an opponent who comes after the player gets $10$ or more brains) the player will want to reach $13$ or more brains no matter the cost in order to have a chance at winning. Since the expected number of brains with $3$ red, $4$ yellow, and $6$ green dice in the cup, no footprints, and no current shotguns (the first roll of a player's turn), is $3.315559$, then we know that if a player has $10$ dice from previous rounds and has yet to go, then you can expect them to attain at least $3$ more brains on their first roll for a total of $13$ to end the round without you having the chance to beat them and win. This is why a player would want to try and obtain as many points if someone before them has already gained $10$ points or more, which is deemed as a strategy for the end of the game.
There are many generalizations that can be made with this immense data set that is given by the written computer code which gives all possible combinations ($4867$ total) of the six usual parameters (the amount of red, yellow, and green dice left in the cup; and the amount of red, yellow, and green footprints) plus the number of current shotguns. Below is the table of the simple strategy sorted by the current number of shotguns which is slightly less optimal than using the enormous data set, but it is vastly easier.
\begin{center}
\begin{tabular}{|c|c|} \hline Current Shotguns & Rule \\ \hline\hline 0 & Always keep rolling. \\\hline
\multirow{5}{*}{1} & If we must roll 3 red dice, stop at 1 brain. \\
& If ($r_f = 2$ and $y_f=1$) or $y_c>g_c$, stop at 2 brains. \\
& If ($r_f = 2$ and $g_f=1$) or $g_c>y_c$, stop at 3 brains. \\
& If we must roll 3 green dice, roll again. \\
& If $g_f = 2$, roll again. \\\hline
\multirow{2}{*}{2} & With $g_f = 3$, stop at 2 brains. \\
& Otherwise, stop at 1 brain. \\\hline
\end{tabular}
\end{center}
\section{Further Questions}
Throughout this paper, we have thoroughly explored the game of Zombie Dice and how one should optimally play. There may be other ways to present general strategies besides the way described above. We are certain that our strategy, especially the results from the recursive program, will beat the AI code for the iPad application when compared to see which plays more optimally. We would like to thank Steve Jackson Games for sharing the iPad decision-making strategy with us (under an NDA) so that we will be able to do this comparison.
\end{document} |
2,877,628,089,018 | arxiv | \section{Introduction}
\subsection{Background and Related Works}
\par Next generation multiple access (NGMA) aims to support massive connectivity scenarios for many envisioned Internet of Things (IoT) applications, e.g. manufacturing, transportation, agriculture, and medicine, to be efficiently and flexibly connected \cite{wu2020wc,liu2018spm,Liu2018tsp,shyianov2021jsac,NGMA,B5G}. A generic scenario for IoT connectivity involves a massive number of machine-type connections \cite{liu2018spm}, known as the massive machine-type communication (mMTC), which is one of the main use cases of the fifth-generation (5G) and beyond \cite{B5G}. Compared to human-type communications (HTC), MTC has two distinct features: sporadic traffic and short packet communication. The sporadic access means that for any given time, for the sake of long battery life and energy saving, only a small fraction within a huge number of devices are active for data transmission \cite{Liu2018tsp}. While the small data payloads result in a fall of spectrum efficiency and an increase of latency when applied with traditional multiple access protocols. In this regard, a number of scalable random access schemes have been investigated in the content of massive connectivity. Conceptually, there are two lines of work for that end, namely, individual and common codebook-based approaches, respectively.
\par In the spirit of the individual codebook based approach, each device is equipped with a unique codebook for the identification conducted by the base station (BS) \cite{Liu2018tsp,gjy2021}. Since the BS is interested in the IDs of devices, this framework can be referred to as sourced random access (SRA). A typical transmission process includes two phases in SRA, namely, training and data transmission, respectively. In the training phase, the BS conducts the task of device activity detection (DAD) and channel estimation (CE) based on the pre-allocated unique pilot sequences transmitted by active devices, while the data transmission is performed in the following phase. The task of DAD-CE can be modeled as the compressed sensing (CS) problem and there have been quantities of algorithmic solutions for this issue. Among these CS recovery techniques, the approximate message passing (AMP) algorithm, which is first proposed in \cite{donoho2009message} for the single measurement vector (SMV) based problem, exhibits great performance in terms of accuracy and complexity. Since the publication of \cite{donoho2009message}, there have been vast variants based on the AMP algorithm, such as multiple measurement vector (MMV) AMP \cite{Liu2018tsp}, generalized MMV (GMMV) AMP \cite{ke2020tps}, orthogonal AMP (OAMP) \cite{ma2017oamp}, generalized AMP (GAMP) \cite{GAMP} and many other works. Another line of work is based on the belief propagation (BP) algorithm \cite{BP}, which models the problem as a graph and iteratively calculates the messages among different nodes \cite{zhang2020iot,hiroki2020arxiv,yuan2020twc,huang2019tsp}. Thanks to the consistency of the message updating rules, the messages can always be jointly updated within iterations to obtain improved performance. However, the premise of these algorithms is that each device should have a unique pilot sequence. As we jump out of these algorithmic solutions and go back to the essence of the individual codebook, we find this is inapplicable to assign each device a unique pilot in the mMTC scenario with a huge amount of potential devices. Meanwhile, it is a waste of resources especially with sporadic traffic. To mitigate this issue, the study on the framework of common codebooks is rapidly developed.
\par As opposed to the case of individual codebooks, active devices choose codewords from a common codebook for their messages. In this regard, the task of BS is only to produce the transmitted messages regardless of the corresponding IDs, thus leading to the so-called unsourced random access (URA). The main difference between URA with other grant-based and grant-free random access protocols is that the BS does not perform device identification but only decodes the list of active device messages up to permutations \cite{B5G}. Conceptually, there are two distinct advantages for this novel framework: $\left.i\right)$ the BS is capable of accommodating massive devices, since all devices share the common codebook instead of being assigned unique preamble sequences as in traditional schemes. $\left.ii\right)$ No device identification information is needed to be embedded in the transmitted sequence. As such, the overhead will be reduced, contributing to improved efficiency. The study on URA is first reported in \cite{yury2017isit}, in which Polyanskiy considered the scenario of a massive number of infrequent and uncoordinated devices and discussed a random coding existence bound in the Gaussian multiple-access channel (GMAC). Since the publication of \cite{yury2017isit}, there have been many works devoted to approaching that bound \cite{or2017isit,Amalladinne2020tit,Fengler2019isit,sparcs2021tit,polar2020icc,polar2020vtc,Vem2019gc,Vem2019tcom}. A low complexity coding scheme is investigated in \cite{or2017isit}. Based on the combination of compute-and-forward and coding for a binary adder channel, it exhibits an improved performance compared with the traditional schemes, such as slotted-ALOHA and treating interference as noise (TIN). However, the size of the codebook increases exponentially with the blocklength, resulting in difficulty in achieving that underlying bound. To mitigate this issue, a slotted transmission framework is proposed in \cite{Amalladinne2020tit}, referred to as the coded compressed sensing (CCS) scheme. This cascaded system includes inner and outer codes. The outer tree encoder first splits the data into several slots with parity bits added. The CS coding is then employed within each slot to pick codewords from the codebook. The blocklength, as well as the size of the codebook in each slot, are greatly reduced, thus leading to a relaxation of the computational burden. On the receiver side, the inner CS-based decoder is first utilized to recover the slot-wise transmitted messages within each slot. The outer tree decoder then stitches the decoded messages across different slots according to the prearranged parity. This structure is inherited by the later works in \cite{Fengler2019isit,sparcs2021tit}, where the authors exploit the concept of sparse regression code (SPARC) \cite{sparcs2012} to conduct the inner encoding and decode it with the AMP algorithm. Some other coding schemes for URA are also reported in \cite{polar2020icc,polar2020vtc,Vem2019gc,Vem2019tcom}. The polar coding schemes based on TIN and successive interference cancellation (SIC) are investigated in \cite{polar2020icc,polar2020vtc}. In \cite{Vem2019gc,Vem2019tcom}, the author consider a hierarchical framework, where the device's data is split into two parts. A portion of the data is encoded by a common CS codebook and recovered by the CS techniques, while the rest is encoded by the low-density parity check code (LDPC) \cite{LDPC} with the key parameters conveyed in the former part. Besides the above works considering the GMAC, there are also works in the fading channel \cite{fading2019acssc,fading2019isit,fading2020tcom}.
\par Besides the above works considering a single receive antenna at the BS, the study of massive URA in multiple-input multiple-output (MIMO) systems has also drawn increasing attention. The BS equipped with multiple antennas provides extra dimensions for the received signal, thus offering more opportunity for signal detection and access of a massive number of devices. A coordinated-wise descend activity detection (AD) algorithm is proposed in \cite{Haghighatshoar2018isit,Fengler2021tit}, which finds the large-scale fading coefficients (LSFCs) of all the devices (coordinates) iteratively. The authors adopt a concatenated coding scheme with the aforementioned tree code as the outer code, and a non-Bayesian method (i.e., maximum likelihood (ML) detection) is leveraged as the inner decoder based on the covariance of the received signal, referred to as the covariance-based ML (CB-ML) algorithm. However, the performance of the CB-ML algorithm degrades dramatically when the number of antennas is less than that of the active devices. There are also some improvements to this algorithm in terms of complexity or accuracy \cite{Shyianov2021jsac,fast}. The slotted transmission framework proposed in \cite{Shyianov2021jsac} eliminates the need for concatenated coding. That is, no parity bit is added within each slot and the slot-wise messages can be stitched together by clustering the estimated channels. However, the channels in \cite{Shyianov2021jsac} are assumed to be identically distributed over all the slots, which is difficult to hold in practice. Besides, the collision in codewords (more than one device chooses the same codeword) will lead to a superimposition of the estimated channels, resulting in a failure of the stitching process. Currently, the design of an efficient collision resolution scheme in URA remains missing. In \cite{fast}, a more efficient coordinate selection policy is developed based on the multi-armed bandit approaches, leading to a faster convergence than the CB-ML algorithm. Nevertheless, these slotted transmission frameworks, such as CB-ML and its variants, all map a piece of data within each slot to a long transmitted codeword based on the CS coding, resulting in low efficiency. In this regard, the algorithmic design for the MIMO massive URA with high efficiency and accuracy remains missing.
\subsection{Challenges}
\par In this paper, we investigate a two-phase transmission scheme for MIMO massive URA, where the device's transmitted data is split into two phases with CS and LDPC coded, respectively. In this framework, no parity bits are needed to be embedded in the CS phase, and the data is encoded linearly in the LDPC phase instead of being mapped to a long codeword. As such, it exhibits higher efficiency and lower latency than the aforementioned slotted transmission scheme. The existing schemes considering this framework all focus on the single-antenna case. For instance, Polar coding \cite{polar2020vtc} and LDPC coding schemes \cite{Vem2019gc,Vem2019tcom} consider the GMAC with perfect channel state information (CSI) at the BS. In this regard, the above channel coding schemes can obtain satisfying performance. However, such an assumption cannot be established under the MIMO case. Besides, because of the massive connectivity and sporadic traffic of devices, the positions of active devices in the estimated equations are not determined. Consequently, conventional linear receivers, such as zero-forcing (ZF), are not applicable. Moreover, compared with the fading channel models \cite{fading2019acssc,fading2019isit,fading2020tcom} where the channel of each device is a scalar, in MIMO, the channel is a vector, and elements among antennas share the same activity. Therefore, elements in the channel vector should be estimated jointly rather than separately, of which the distribution is formulated as a multi-dimensional Bernoulli-Gaussian distribution in this paper. In this regard, the conventional estimators, such as the minimum mean square error (MMSE) estimation or the maximum \emph{a posteriori} (MAP) estimation, are hard to carry out straightforwardly. Specifically, it is computationally intractable to obtain a precise posterior distribution, since it involves marginalizing a joint distribution of the activity and channel with high dimensions.
\par Another challenge for the proposed scheme in the MIMO case is activity detection. We note that whether in the GMAC \cite{polar2020vtc,Vem2019gc,Vem2019tcom} or fading channel \cite{fading2019acssc,fading2019isit,fading2020tcom} in the single-antenna case, the device activity can be directly detected. For instance, since the device's channel is a scalar in the fading channel, it can be simply determined to be active when the estimated channel is non-zero or larger than a given threshold. However, it will be much involved in MIMO because there are multiple observations of the activity in the estimated channel vector. The existing works, such as \cite{ke2020tps}, make a hard decision based on the energy of the channel and \cite{Shyianov2021jsac} considers the LLR at each antenna separately and sums them to obtain the final decision. However, the threshold for the decision is hard to obtain in practice and the distribution of the channel is not utilized in \cite{ke2020tps}. Although \cite{Shyianov2021jsac} considers the channel distribution and gives a closed-form expression for activity detection, the nature that channels at each antenna share the same activity is not considered.
\par Besides, the LDPC code is efficient but sensitive to the CE results. Therefore, the overall performance is limited by the accuracy of CE. Moreover, with the presence of collision in URA, if more than one device chooses the same codeword in the CS phase, the corresponding channels will be superimposed and thus the subsequent LDPC decoding process will fail.
\subsection{Contributions}
\par To cope with the arising issues, based on the message passing (MP) algorithm, we propose the Joint DAD-CE algorithm to conduct the task of joint activity detection and channel estimate in the CS phase, and MIMO-LDPC-SIC Decoding algorithm for data decoding embedded in the LDPC phase. Moreover, to further improve the performance, we propose the Joint DAD-CE-DD algorithm to jointly update the messages in these two phases by utilizing the belief of each other. Finally, we propose a collision resolution protocol to address the collision in URA. The key and distinguishing features of the proposed algorithms are listed below.
\begin{itemize}
\item Based on the principle of the BP algorithm, we develop a low-complexity iterative MP algorithm to decode the two parts of data. For the CS phase, we investigate the Joint DAD-CE algorithm to recover part of the data embedded in the device activities, the interleaving patterns, and channel coefficients, which are the key parameters for the remaining data. Specifically, we derive a close-form expression for DAD by utilizing the joint distribution of the channel among antennas. Combined with SIC and in the spirit of interleave-division multiple access (IDMA) \cite{ping2006twc}, we elaborate on the MIMO-LDPC-SIC Decoding algorithm to decode the remaining data embedded in the LDPC phase. In addition to complexity reduction, the proposed algorithm exhibits a substantial performance enhancement in terms of accuracy and efficiency compared to the state-of-the-art CB-ML algorithm.
\item Thanks to the consistency of the MP algorithm, we propose the Joint DAD-CE-DD algorithm to further improve the performance. The proposed algorithm suggests a paradigm connecting the two parts. That is, messages in the decoding process of CS and LDPC parts can be jointly updated by utilizing the belief of each other, thus leading to improved performance. We employ the correctly decoded codewords as soft pilots to conduct CE jointly with the codewords in the CS phase, which contributes to improved accuracy of the estimated channel. Combined with the SIC method, the accuracy of the residual signal can be improved, leading to the enhanced decoding performance.
\item Under the current framework based on the common codebook, a collision happens if there are more than one device having the same preamble, which leads to a superimposition of the estimated channels. Accordingly, the superimposed channel will cause a failure in the LDPC decoding process. To this end, we propose a collision resolution protocol based on the energy detection (ED) and sliding window protocol (SWP). Succinctly, the ED is performed on the estimated channel of each device to find out the superimposition. Then, the BS broadcasts the indexes of the superimposed channels to all devices. Afterwards, the devices in collision slide the window in the data sequence and the CS coding is again performed for retransmission.
\end{itemize}
\par The rest of the paper is organized as follows. In Section \ref{sec-2}, we introduce the system model for MIMO massive URA . In Section \ref{sec-3}, we implement the two-phase encoding scheme and the collision resolution protocol is developed in Section \ref{sec-4}. Then, we elaborate on the low-complexity iterative decoding algorithm based on BP and explain the joint update algorithm in Section \ref{sec-5}. After verifying the numerical results and analyzing the complexity in \ref{simulation}, we conclude the paper in Section \ref{sec-7}.
\subsection{Notations}
\par Unless otherwise noted, lower- and upper-case bold fonts, $\bm{x}$ and $\bm{X}$, are used to denote
vectors and matrices, respectively; the $(m,n)$-th entry of $\bm{X}$ is denoted as ${X}_{m,n}$; $\bm{X}_{i,:}$ denotes the $i$-th row of $\bm{X}$; $\{\cdot\}^*$ denotes the conjugate of a number; $\{\cdot\}^T$ and $\{\cdot\}^H$ denote transpose and conjugate transpose of a matrix, respectively; $\mathbb{E}\{\cdot\}$ and $\text{Var}\left\lbrace\cdot \right\rbrace $ denote the statistical expectation and variance, respectively; $\bm{X}\sim\mathcal{CN}(\bm{x};\bm{\mu},\bm{\Sigma})$ means that the random vector $\bm{x}$ follows a complex Gaussian distribution with mean $\bm{\mu}$ and auto-covariance matrix $\bm{\Sigma}$; $\{\cdot\}!$ denotes the factorial; $\left[1:M \right] $ denotes the set of integers between $1$ and $M$; $ \mathcal{L} \backslash l$ denotes the entries in set $\left\lbrace 1,2,\cdots,L\right\rbrace $ except $l$; $\text{Re}(\cdot)$ and $\text{Im}(\cdot)$ denote the real and imaginary parts of a complex number, respectively; $\lceil \cdot \rceil$ denotes ceiling; $i$ is the imaginary unit (i.e., $i^2=-1$).
\section{System Model} \label{sec-2}
\par Consider the uplink of a single-cell cellular network consisting of $K_{tot}$ single-antenna devices, which are being served by a base station (BS) equipped with $M$ antennas. This paper assumes sporadic device activity, i.e, a small number, $K_a \ll K_{tot}$ of devices are active within a coherence time. Each device has $B$ bits of information to be coded and transmitted into a block-fading channel with $L$ channel uses. Let $\bm{v}_k \in \left\lbrace 0,1\right\rbrace ^B$ denote device $k$'s binary message and $f(\cdot): \left\lbrace 0,1\right\rbrace ^B \rightarrow \mathbb{C}^L$ is some one-to-one encoding function. Typically, in URA scenario, the implementation of $f(\cdot)$ is to select the corresponding codeword $\bm{a}_k$ from a shared codebook $\bm{A}=\left[ \bm{a}_1,\bm{a}_2,\cdots,\bm{a}_{2^{B}} \right] \in \mathbb{C}^{L \times2^B}$ according to $\bm{v}_k$ \cite{Haghighatshoar2018isit,Fengler2019isit,Fengler2021tit}. The corresponding received signal can be written as
\begin{equation}
\bm{Y}= \sum\nolimits_{k \in \mathcal{K}_{tot}}{\phi_k f(\bm{v}_k) \tilde{\bm{h}}_k^T} +\bm{Z}, \label{equ-1}
\end{equation}
where $\phi_k$ is the device activity indicator, which is modeled as a Bernoulli random variable and defined as follows:
\begin{equation}
\phi_{k}=\left\{\begin{array}{l}
1, \text { if device } k \text { is active, } \\
0, \text { otherwise }
\end{array} \quad \forall k \in \mathcal{K}_{tot}.\right. \label{equ-2}
\end{equation}
$\tilde{\bm{h}}_k \in \mathbb{C}^{M \times 1}$ is the channel vector of device $k$ and $\bm{Z}\in \mathbb{C}^{L\times M}$ is the additive white Gaussian noise (AWGN) matrix distributed as $\mathcal{CN}(0,\sigma^2\bm{I}_M)$. In line with the state-of-the-art setting \cite{Fengler2021tit}, we assume for simplicity and fair comparison that $\bm{h}_k$ are i.i.d., i.e., independent of each other and uncorrelated among antennas. Specifically, the Rayleigh fading model is considered in this paper, where $\tilde{\bm{h}}_k=\sqrt{\beta_{k}}\bm{g}_{k}$ and $\bm{g_{k}}\sim \mathcal{CN}(0,\bm{I}_M)$ denotes the Rayleigh fading component, and $\beta_{k}$ denotes LSFC. We would like to mention that more realistic channel models in MIMO or massive MIMO have been discussed and well investigated, such as the spatially correlated fading channel \cite{ke2020tps,add-corre,gao2015tsp,you2015twc,xxy2021iccc}. Due to the limited angular spread, the channel in the virtual angular domain exhibits a sparsity among antennas. As such, the data of devices is not directly superimposed but staggered on different antennas, and the multi-user interference can be further reduced, contributing to improved performance. Nevertheless, for the sake of consistency with the benchmarks \cite{Liu2018tsp,Fengler2021tit} and isolating the fundamental aspects of the problem without additional model complication, we embark throughout this paper on the study of i.i.d. Rayleigh fading channel. The study on the spatially correlated channel is remarkable and left for future work.
\par The BS's task is to produce a list of transmitted messages $\mathcal{L}(\bm{Y})$ without identifying from whom they are sent, thereby leading to the so-called URA. The performance of a URA system is evaluated by the probability of missed detection and false alarm, denoted by $p_{md}$ and $p_{fa}$, respectively, which are given by:
\begin{align}
\label{equ-3} p_{md} &= \frac{1}{{{K_a}}}\sum\nolimits_{k \in {\mathcal{K}_a}} {P\left( {{\bm{v}_{{k}}} \notin \mathcal{L}} \right)} \\
\label{equ-4} p_{fa} &= \frac{{\left| {\mathcal{L}\backslash \left\{ {{\bm{v}_{{k}}}:k \in {\mathcal{K}_a}} \right\}} \right|}}{{\left| \mathcal{L} \right|}}.
\end{align}
In this system, the code rate $R_c={B}\slash{L}$ and the spectral efficiency $\mu=\frac{B\cdot K_a}{L \cdot M}$. Let $P$ denote the power (per symbol) of each device, then the energy-per-bit $E_b/N_0$ is defined by
\begin{equation}
E_b/N_0 = \frac{LP}{2B}.
\end{equation}
\section{Encoder} \label{sec-3}
\par As aforementioned, the existing slotted transmission scheme exhibits low efficiency by the CS coding. To mitigate this issue, a two-phases coding scheme is proposed in \cite{Vem2019tcom,Vem2019gc}, which considers the $T$-user Gaussian multiple access (GMAC) channel and also URA scenario. We extend this work to the MIMO system and refer to it as the CS-LDPC coding scheme. Similarly, the hierarchical form of the encoding process is considered in this paper. The $B$ bits of information are first divided into two parts, namely, $B_p$ and $B_c$ bits with $B_p+B_c=B$. Typically, one would want $B_p \ll B_c$. The former $B_p$ information bits are coded by a CS-based encoder to pick codeword from the common codebook. Based on the codebook, the BS is tasked to recover part of the messages, the number of active devices, channel coefficients as well as interleaving patterns for the latter part. The remaining $B_c$ bits are coded with LDPC codes. For clarity, we denote the former and latter encoding processes as CS and LDPC phases, respectively. Correspondingly, the total $L$ channel uses are split into two segments of lengths $L_p$ and $L_c$, respectively, with $L_p+L_c=L$. Since only a small fraction of data is CS coded and the rest is LDPC coded, the efficiency in our scheme is higher than those purely CS-coded schemes \cite{Shyianov2021jsac,Amalladinne2020tit,Fengler2019isit,Fengler2021tit}. The key features of this encoding process are summarized in Fig. \ref{pic-1}. We elaborate on these two encoding phases below.
\begin{figure}[htpb]
\centerline{\includegraphics[width=0.4\textwidth]{pic-10}}
\caption{The encoding process in the CS and LDPC phases.}
\label{pic-1}
\end{figure}
\subsection{CS Phase} \label{sec3-1}
\par The URA fashion is considered in this phase. Let $\mathbf{A}\in \mathbb{C}^{L_p \times 2^{B_p}}$ denote the common codebook shared by all the devices. That is, the columns of $\mathbf{A}=\left[ \bm{a}_1,\bm{a}_2,\cdots,\bm{a}_{2^{B_p}} \right] $ form a set of codewords with power constraint $\left\| {{\mathbf{a}_{i}}} \right\|_2^2 = L_p$, from which each active device chooses in order to encode its ${B_p}$ bits of information. With a slight abuse of notation, let $\bm{v}_k^p$ denote the first $B_p$ bits of device $k$'s binary message, i.e., $\bm{v}_k^p \triangleq \bm{v}_k(1:B_p) \in \left\lbrace 0,1\right\rbrace ^{B_p}$. To apply the encoding scheme, we convert $\bm{v}_k^p$ into the $\ell_1$-norm binary vector $\tilde{\bm{v}}_k^p \in \left\lbrace 0,1\right\rbrace ^{2^{B_p}}$, in which a single one is placed at the location $i_k$. The value $i_k$ of binary sequence $\bm{v}_k^p$ is obtained by regarding it as an integer expressed in radix-$2$ (plus one), which we write as $i_k=\left[ \bm{v}_k^p \right]_2 \in \left[ 1: 2^{B_p} \right] $. Then, the coded sequence is obtained by taking the transpose of the $i_k$-th column of $\bm{A}$, which we denote by $ \bm{a}_{i_k}^T$. This facilitates the CS architecture, which maps the information to an elementary vector $\tilde{\bm{v}}_k^p$, according to which the corresponding codeword is selected from a fixed codebook. The corresponding received signal can be written as
\begin{equation}
\bm{Y}= \sum\nolimits_{k \in \mathcal{K}_{tot}}{\phi_k \bm{a}_{i_k}\tilde{\bm{h}}_k^T} +\bm{Z}, \label{equ-5}
\end{equation}
where $\phi_k$ is the device activity indicator, as defined in (\ref{equ-2}). The matrix form of (\ref{equ-5}) is given by
\begin{equation}
\bm{Y}=\bm{A}\bm{\Gamma}\tilde{\bm{H}}+\bm{Z}, \label{equ-6}
\end{equation}
where $\mathbf{A}\in \mathbb{C}^{L \times 2^{B_p}}$ is the common codebook shared by all devices. $\mathbf{\Gamma} \in {\left\{ {0,1} \right\}^{{2^{B_p}} \times {K_{tot}}}}$ denotes the binary selection matrix. For each active device $k\in \mathcal{K}_{a}$, the corresponding column $\bm{\Gamma}_{:,k}$ is all-zero but a single one in position $i_k$, while for all $\mathcal{K}_{tot} \backslash \mathcal{K}_a$ the corresponding column $\bm{\Gamma}_{:,k}$ is all zeros. $\tilde{\bm{H}}=\left[ \tilde{\bm{h}}_1,\cdots,\tilde{\bm{h}}_{K_{tot}}\right]^T \in \mathbb{C}^{K_{tot}\times M}$ corresponds to the MIMO channel coefficient matrix. We note that the number of total devices $K_{tot}$ plays no role in URA. In order to get rid of this variable, with slight abuse of notation we define the modified activity indicator and selection matrix as $\phi_{r}=\sum\nolimits_{k\in \mathcal{K}_a}{\gamma_{r,k}}$ and $\bm{\Phi}=\text{diag}\left\lbrace \phi_1, \cdots, \phi_{2^{B_p}}\right\rbrace$, respectively, where $\gamma_{r,k}$ is the $(r,k)$-th entry of $\bm{\Gamma}$. Correspondingly, the modified channel is ${\bm{H}}=\left[\bm{h}_1,\cdots, \bm{h}_{2^{B_p}} \right]$ with $\bm{h}_r=\sum\nolimits_{k \in {\mathcal{K}_a}}{\gamma_{r,k}\tilde{\bm{h}}_k}$. Hence, \eqref{equ-6} can be written as
\begin{equation}
\bm{Y}=\bm{A}\bm{\Phi}\bm{H}+\bm{Z}. \label{equ-6-add}
\end{equation}
Let $\bm{X}=\bm{\Phi}\bm{H}=\left[ \bm{x}_1,\cdots,\bm{x}_{2^{B_p}}\right]^T $. The goal for the BS in the CS phase is to detect the non-zero positions of the selection matrix $\bm{\Phi}$ and the corresponding channel vectors by recovering $\bm{X}$ based on the noisy observation $\bm{Y}$. Since $\bm{X}$ is row sparse, i.e., many $\bm{x}_n$ are zero, such reconstruction problem can be modeled as the CS problem \cite{Liu2018tsp}. Once $\bm{\Phi}$ is recovered, the message indicators of active devices $\left\lbrace i_k, \forall k\in \mathcal{K}_a \right\rbrace $ are also recovered. Moreover, $i_k$ acts as the parameter of the LDPC code, since it determines the interleaving pattern of the data in the LDPC phase. Considering the fact that the $B_p$ bits of message carries key parameters for the latter phase, we name it the preamble. We note that it is different from the preamble in the traditional grant-free scenario, which is a pure pilot with no data embedded.
\subsection{LDPC Phase}
\par Likewise, let $\bm{v}_k^c \triangleq \bm{v}_k(B_p+1:B) \in \left\lbrace 0,1\right\rbrace ^{B_c}$ denote the remaining $B_c$ information bits of device $k$. $\bm{v}_k^c$ is first encoded into an LDPC code $\bm{b}_k^c \in \left\lbrace 0,1 \right\rbrace^{\tilde{L}_c}$, which is determined by the LDPC parity check matrix $\bm{C}$ with size $\left(\tilde{L}_c-B_c \right)\times \tilde{L}_c$. We note that in the decoding process, if $\hat{\bm{v}_k^c}$ is a valid codeword, then $\text{mod} \left( \bm{C}\hat{\bm{v}}_k^c,2\right) =0$. $\bm{b}_k^c$ is then modulated to $\bm{s}_k^c$. We adopt a sparse spreading scheme introduced in \cite{Vem2019gc}. That is, $\bm{s}_k^c$ is zero-padded into a length $L_c$ vector $\tilde{\bm{s}}_k^c$
\begin{equation}
\tilde{\bm{s}}_k^c = \left[\bm{s}_k^c, ~0, ~\cdots, ~0 \right]. \label{equ-7}
\end{equation}
We then employ the index representation $i_k$ to permute the ordering of $\tilde{\bm{s}}_k^c$. This is implemented by a random interleaver $\pi_{i_k}$ with interleaving pattern $i_k$. As mentioned in \cite{Vem2019tcom}, the purpose for permuting the codewords is to decorrelate the random access interference from other devices. This is similar to the IDMA scheme since the interleaving patterns for most of the devices are different because of the distinctive indices $i_k$. Hence, $\bm{v}_k^c$ is finally encoded to $\pi_{i_k}\left( \tilde{\bm{s}}_k^c\right)$. Appending it to the coded message in the CS phase yields the final codeword $\bm{x}_k$:
\begin{equation}
\bm{x}_k = \left[ \bm{a}_{i_k}^T, \pi_{i_k}\left( \tilde{\bm{s}}_k^c\right) \right]^T. \label{equ-8}
\end{equation}
\par The received signal including both the CS and LDPC phases is given by
\begin{equation}
\bm{Y}=\sum\nolimits_{k \in \mathcal{K}_{a}}{\bm{x}_k \bm{h}_{i_k}^T} + \bm{Z}, \label{equ-9}
\end{equation}
where $\bm{h}_{i_k}$ is assumed to follow independent quasi-static flat fading within the above two phases in this paper. The goal for the BS is to recover $\left\lbrace \bm{v}_k, \forall k\in \mathcal{K}_a \right\rbrace $ based on the received signal $\bm{Y}$ and the channel $\bm{h}_{i_k}$ estimated in the CS phase. We emphasize again that the BS only produces the transmitted messages without distinguishing the corresponding devices.
\par This hierarchical encoding process appears to be similar to the work of \cite{Vem2019tcom}, where CS and channel coding techniques are utilized to encode the messages. However, unlike in \cite{Vem2019tcom} considering the GMAC system, our approach is in the MIMO channel. Hence, in addition to recovering the parameters of LDPC codes conveyed by the codebook $\bm{A}$ as in \cite{Vem2019tcom}, the BS is also tasked to estimate the channel coefficients. Besides, as we will see shortly in Section \ref{sec-5-3}, a belief propagation decoder draws a connection between the CS and LDPC phases. That is, messages in the CE as well as DD processes can be jointly updated by utilizing the belief of each other. Whereas the two phases in \cite{Vem2019tcom} are two independent modules and work sequentially.
\section{Collision Resolution Protocol} \label{sec-4}
\par It is possible that two or more devices have the same preamble message, $\bm{v}^p$, which, although, may occur in a small probability. In this case, the collided devices will have the same interleaving pattern in the LDPC phase, which goes against the principles of the IDMA scheme. Moreover, they will choose the same codeword as their coded messages in the CS phase, which results in the received signal in the CS phase being
\begin{equation}
\bm{Y}_{colli} = \sum\nolimits_{i_k\in\mathcal{C}}{\bm{a}_{i_k}\sum\nolimits_{j \in \mathcal{K}_{i_k}}{\tilde{\bm{h}}_j^T}} + \bm{Z}, \label{equ-10}
\end{equation}
where $\mathcal{C}$ and $\mathcal{K}_{i_k}$ denote the set of collided indexes and collided devices corresponding to the message index $i_k$, respectively. In this regard, the BS can only recover the superimposed channels of these devices, instead of their own, which will lead to the failure of the LDPC decoding process. Although collision may occur in a small probability, it can occur. However, the design of an efficient collision resolution scheme in URA remains missing.
\begin{figure*}[t]
\centering
\subfloat[The BS broadcasts collided index representations $\left\lbrace i_k | \epsilon_{i_k}>\eta \right\rbrace $ to all the devices. Based on this, active devices can figure out whether they have been in a collision. Deivce $i$ to $j$ are in collision in this case.]{\includegraphics[width=0.38\linewidth]{pic-1} \label{pic2-1}}
\hfil
\subfloat[The collided devices slide the window with length $B_p$ bits forward within the total $B$ bits to get new sequences. The sliding length is $B_0$, satisfying $0<B_0<B_p$.]{\includegraphics[width=0.38\linewidth]{pic-2}\label{pic2-2}}
\caption{The diagram of the collision resolution protocol.}
\end{figure*}
\par In this paper, we develop a collision resolution protocol based on the ED and SWP. We note that in real scenarios, the near-far effects can be well solved with the existing power control schemes \cite{ Chandrasekhar2009twc,Patel1994jsac,Turin1984jsac}. In this regard, a flat fading channel is considered in this paper. That is, the LSFCs $\beta_k$ for all the devices are assumed to be identical, as also considered in \cite{Haghighatshoar2018isit,Fengler2021tit}. As mentioned above, if a collision happens, the recovered channels of the collided devices will be a superposition of their own, that is
\begin{equation}
\hat{\bm{h}}_{i_k} = \sum\nolimits_{j \in \mathcal{K}_{i_k}}{\tilde{\bm{h}}_j} + \bm{z}, \quad i_k\in \mathcal{C}, \label{equ-11}
\end{equation}
where $\bm{z} \sim \mathcal{CN}(0,\sigma^2\bm{I}_M)$ denotes the Gaussian noise. And $\hat{\bm{h}}_{i_k}$ is distributed as $\hat{\bm{h}}_{i_k} \sim \mathcal{CN}(0,\left( |\mathcal{K}_{i_k}|+\sigma^2)\bm{I}_M\right) $, which has a higher power than those without collision. Therefore, an effective way to detect collision is to perform ED on the estimated channel by the BS
\begin{equation}
\epsilon_{i_k} = \mathbb{E}\left[ \hat{\bm{h}}_{i_k}^H \hat{\bm{h}}_{i_k} \right]. \label{equ-12}
\end{equation}
If $\epsilon_{i_k}$ is greater than a given threshold $\eta$, then it is utilized as evidence that there are at least two devices that have the same preamble and thus they choose the same CS codeword $\bm{a}_{i_k}$. Since devices themselves do not know whether they have been in a collision, the BS needs to feed this information back. To this end, the BS will broadcast the collided index representations $\left\lbrace i_k | \epsilon_{i_k}>\eta \right\rbrace $ to all the devices to help them get the judgment. Fig. \ref{pic2-1} showcases that device $i$ to $j$ realized that a conflict occurred after receiving the indexes broadcast by the BS. We note that in the above process, the additional information required at the BS is only the threshold and it can be easily preset in practice.
\par As illustrated in Fig. \ref{pic2-2}, in order to get a new non-conflicting index representation, the collided devices will slide the window with length $B_p$ bits forward within the total $B$ bits to get new sequences, denoted by ${\bm{v}_k^p}'$. We denote $B_0$ as the sliding length which satisfies $0<B_0<B_p$. The reason behind $B_0<B_p$ is that there should be a common part, labeled in green in Fig. \ref{pic2-2}, between the sequences before and after sliding the window, so as to splice back the sequences between different windows.
\begin{algorithm}
\caption{Collision Resolution Protocol}
\label{alo-1}
\begin{algorithmic}[1]
\STATE{{\bf Input}: estimated channel $\hat{\bm{H}}$, maximum iteration $t_{max}$, maximum and minimum threshold $\eta$, $\gamma$ }\\
\STATE{{\bf Output}: recovered channel $\tilde{\bm{H}}$}
\STATE{{\bf Initialize}: iteration count $t=0$, $\tilde{\bm{H}}=\left[~ \right] $}\\
\REPEAT
\STATE{Energy detection:
\begin{equation}
\epsilon_{i_k} = \mathbb{E}\left[ \hat{\bm{H}}_{i_k,:} \hat{\bm{H}}_{i_k,:}^H \right] ,i_k \in \left[ 1:2^{B_p}\right] \notag
\end{equation}
}\\
\STATE{Feedback: the BS broadcasts $\left\lbrace i_k, \epsilon_{i_k}>\eta \right\rbrace $}
\STATE{Combine: $\tilde{\bm{H}} \leftarrow \left[ \tilde{\bm{H}}; \hat{\bm{H}}_{i_k,:}\right],\left\lbrace i_k\vert \gamma<\epsilon_{i_k}<\eta \right\rbrace $}
\STATE{Window sliding and retransmission}
\STATE{Channel estimation: $\hat{\bm{H}}$}
\STATE{$t \leftarrow t+1$}\\
\UNTIL{$t=t_{max}$ or $\{\epsilon_{i_k}<\eta, ~\forall i_k \in \left[ 1:2^{B_p} \right] \}$}
\RETURN{$\tilde{\bm{H}}$}
\end{algorithmic}
\end{algorithm}
\par After obtaining ${\bm{v}_k^p}'$, the CS-based encoder is again performed to encode ${\bm{v}_k^p}'$ to $\bm{a}_{{i_k}'}$ with ${i_k}'=\left[ {\bm{v}_k^p}' \right]_2 \in \left[ 1: 2^{B_p} \right] $. The encoding process is the same as that in \ref{sec3-1}. Channels of the collided devices are expected to be recovered separately after the retransmission. The ED will be again performed on the recovered channels. If the collision still exists, the window sliding process will be executed again until the maximum number of retransmission is reached or no collision exists. The above collision resolution protocol is summarized in Algorithm \ref{alo-1}. We give the analysis for this collision resolution protocol in Appendix \ref{append-1}, which illustrates that as the window sliding progresses, the number of collided devices will decrease and tend to zero.
\section{Decoder} \label{sec-5}
\par The decoding process can be distilled into two key operations: the recovery of the preamble as well as the key parameters for the LDPC code, and the LDPC decoding process combined with SIC \cite{tianya2020gc}. Both are carried out with the MP algorithm. We emphasize again that the beliefs of these two parts can be leveraged to jointly update the messages of the decoding process, which is not considered in \cite{Vem2019gc,Vem2019tcom}.
\subsection{Joint DAD-CE Algorithm} \label{sec5-1}
\par The recovery of the preamble as well as the channels in the CS phase is equivalent to the joint DAD-CE problem, which can be modeled as a CS problem. According to the formulation in (\ref{equ-6-add}), the recovery of the sparse matrix $\bm{X}$ can be addressed by the CS-based methods, such as the AMP algorithm and its variants \cite{donoho2009message,Liu2018tsp,ke2020tps,ma2017oamp,GAMP}. Besides, the MP-based approaches \cite{zhang2020iot,hiroki2020arxiv} also work well on the above issues. The Bernoulli and Rician messages are jointly updated in \cite{zhang2020iot}, which considers the fading channel and grant-free scenario. This scenario is also considered in \cite{hiroki2020arxiv}, which takes advantage of estimated data symbols as soft pilot sequences to perform joint channel and data estimation. In this subsection, we consider the MIMO system and derive the update rules of messages based on the BP algorithm.
\par We derive the update rules of the activity indicator $\phi_k$ and channel vector $\bm{h}_k$ in (\ref{equ-5}), which are modeled as Bernoulli and Gaussian messages, respectively. The Gaussian messages can be characterized by the estimation of mean-value $\bm{u}_k$ and auto-covariance matrix $\bm{\Sigma}_k$. That is, $\bm{u}_k$ and $\bm{\Sigma}_k$ are the estimation and estimating deviation of $\bm{h}_k$ in (\ref{equ-5}), respectively. Besides, the Bernoulli messages for the activity indicator can be represented by $p_k$, which is the probability of $\phi_k$ taking the value one. These messages are updated iteratively between the observation and variable nodes, which can be characterized by the factor graph in Fig. \ref{pic-3}, where the received signal $y_{l,m}$ represents the observation node, denoted by SN, and channel $h_{k,m}$ and the activity pattern $\phi_k$ are the variable nodes, denoted by VN. The edges in the factor graph represent the connections among nodes. In the BP algorithm, messages are passed along these edges. We elaborate on the update rules of these messages below.
\begin{figure}[htpb]
\centerline{\includegraphics[width=0.48\textwidth]{pic-3-new}}
\caption{Factor graph of the joint DAD-CE algorithm. }
\label{pic-3}
\end{figure}
\subsubsection{Message Update at Observation Nodes}
\par We denote $p_{i \rightarrow l}^{VN}(t)$ as the Bernoulli message for the activity of device $i$, which is passed from VN $i$ to SN $l$ in the $t$-th iteration. Accordingly, $\mu_{im \rightarrow lm}^{VN}(t)$ and $\bm{\Sigma}_{i \rightarrow l}^{VN}(t)$ denote the Gaussian messages passed from VN $im$ to SN $lm$, which represent the estimation and the estimating deviation of the channel $\bm{h}_i$, respectively. The index $m \in \left[ 1:M \right] $ denotes the $m$-th antenna and also the $m$-th value of $\bm{h}_i$. Since the update rules of the messages are the same with respect to different iterations, the index of iteration is omitted in the following derivation. For clarity, we assume there is no collision in the CS phase for the following derivation. As such, $i_k$, the original subscript of $\bm{h}$ is replaced by $k$, since there is a one-to-one mapping between these two terms. We emphasize that the collision is considered in our implementation and addressed by the proposed resolution protocol in Algorithm \ref{alo-1}.
To give the message update rules at the SN $y_{l,m}$ in Fig. \ref{pic-3-add}, we first rewrite \eqref{equ-5} as
\begin{equation}\label{equ-13}
\begin{aligned}
y_{lm}&= \sum\nolimits_{i=1}^{K}{A_{li}\phi_ih_{im}} + n_{lm} \\
&= A_{lk}\phi_kh_{km} + \underbrace{\sum\nolimits_{i\in \mathcal{K} \backslash k}{A_{li}\phi_ih_{im}} + n_{lm}}_{z_{lkm}},
\end{aligned}
\end{equation}
\begin{figure}[htpb]
\centerline{\includegraphics[width=0.45\textwidth]{pic-9}}
\caption{Message update rules at VNs and SNs. The output message on each edge is obtained by collecting the messages from the other edges connected with the same node. }
\label{pic-3-add}
\end{figure}
where $l$ is in $L_p$ and $ \mathcal{K} \backslash k$ denotes the entries in set $\left\lbrace 1,2,\cdots,K\right\rbrace $ except $k$. The term $\sum\nolimits_{i\in K \backslash k}{A_{li}\phi_ih_{im}} + n_{lm}$ is modeled as an equivalent Gaussian noise with $\bm{z}_{lk}\sim \mathcal{CN}(\bm{\mu}_{z_{lk}},\bm{\Sigma}_{z_{lk}})$, where $\bm{\mu}_{z_{lk}}=[\mu_{z_{lk1}},\cdots,\mu_{z_{lkm}},\cdots,\mu_{z_{lkM}}]$ and $\mu_{z_{lkm}}$ is given by
\begin{equation}
\mu_{z_{lkm}}= \sum\nolimits_{i \in \mathcal{K} \backslash k} {A_{li} \cdot p_{i \rightarrow l}^{VN} \cdot \mu_{im \rightarrow lm}^{VN}}.
\label{equ-14}
\end{equation}
For $\bm{\Sigma}_{z_{lk}}$, the auto-covariance of $\bm{z}_{lk}$, since $\phi_i$ is the same for $M$ antennas, resulting in the correlation among antennas, we should not consider the variance of $\bm{z}_{lkm}$ at each antenna $m$ separately. Instead, the covariance of $z_{lk}$ is considered in this paper. Rewrite (\ref{equ-13}) in a vector form:
\begin{equation}
\begin{aligned}
\mathbf{y}_l^T &= \left[ {\begin{array}{*{20}{c}}
y_{l1}\\
y_{l2}\\
\vdots \\
y_{lM}
\end{array}} \right] = A_{lk}\phi_k \cdot \left[ {\begin{array}{*{20}{c}}
h_{k1}\\
h_{k2}\\
\vdots \\
h_{kM}
\end{array}} \right] \\
&+\underbrace{\sum\nolimits_{i \in \mathcal{K} \backslash k}{A_{li}\phi_i\cdot \left[ {\begin{array}{*{20}{c}}
h_{i1}\\
h_{i2}\\
\vdots \\
h_{iM}
\end{array}} \right]} + \left[ {\begin{array}{*{20}{c}}
n_{l1}\\
n_{l2}\\
\vdots \\
n_{lM}
\end{array}} \right]}_{\mathbf{z}_{lk}}.
\end{aligned} \label{equ-15}
\end{equation}
The $(m,n)$-th $(m\neq n)$ entry for $\bm{\Sigma_{z_{lk}}}$ satisfies
\begin{equation}
\begin{aligned}
(\Sigma_{z_{lk}})_{(m,n)} & = \sum\nolimits_{i \in \mathcal{K} \backslash k}{{\left| {{A_{li}}} \right|}^2\cdot p_{i \to l}^{VN}\cdot\Big\{ (\Sigma_{i \to l}^{VN})_{(m,n)} } \\
&{+ ~q_{i \to l}^{VN} \cdot \mu_{im \to lm}^{VN}\cdot(\mu_{in \to ln }^{VN})^* \Big\}}, ~m \neq n,
\end{aligned}
\label{equ-16}
\end{equation}
where $q_{i \to l}^{VN} = 1 - p_{i \to l}^{VN}$ denotes the probability that the Bernoulli variable $\phi_k$ equals to zero. If $m=n$, we have
\begin{equation}
\begin{aligned}
(\Sigma_{z_{lk}})_{(m,m)} &= \sum\nolimits_{i \in \mathcal{K} \backslash k}{{\left| {{A_{li}}} \right|}^2\cdot p_{i \to l}^{VN}\cdot\Big\{(\Sigma_{i \to l}^{VN})_{(m,m)}}\\
&{\left.\quad \quad \quad +~q_{i \to l}^{VN} \cdot \left| \mu_{im \to lm}^{VN}\right| ^2 \right\rbrace} + \sigma_n^2.
\end{aligned}
\label{equ-17}
\end{equation}
\par Details about the derivation of $\bm{\Sigma_{z_{lk}}}$ are given in Appendix \ref{append-2-1}. After obtaining the mean and covariance of $\bm{z}_{lk}$, we can get the Gaussian messages $\mu_{lm \rightarrow km}^{SN}$ and $\bm{\Sigma}_{l \rightarrow k}^{SN}$ passed from SN $lm$ to VN $km$ as below
\begin{align}
\begin{split}
\mu_{lm \to km}^{SN} &= \mathbb{E} \left[ h_{km} \vert y_{lm}, \mu_{z_{lkm}}, \bm{\Sigma}_{z_{lk}}, \phi_k=1 \right] \\
&= (y_{lm}-\mu_{z_{lkm}}) \slash A_{lk} \label{equ-18}
\end{split}\\
\begin{split}
\bm{\Sigma}_{l \to k}^{SN}&= \text{Var} \left[ h_{km} \vert y_{lm}, \mu_{z_{lkm}}, \bm{\Sigma}_{z_{lk}}, \phi_k=1 \right] \\
&= \bm{\Sigma}_{z_{lk}} \slash \left| A_{lk} \right| ^2,\label{equ-19}
\end{split}
\end{align}
where $\mathbb{E}\left[ a|b\right] $ and $\text{Var}\left[ a|b\right] $ denote the expectation and variance of $a$ conditioned on $b$, respectively.
\par For the Bernoulli message $p_{l \to k}^{SN}$ passed for SN $l$ to VN $k$, we have
\begin{equation}
\begin{aligned}
p_{l \to k}^{SN} &= \left[ 1+ \frac{P(\mathbf{y}_l \vert \phi_k=0, \bm{\mu}_{z_{lk}},\bm{\Sigma}_{z_{lk}})}{P(\mathbf{y}_l \vert \phi_k=1, \bm{\mu}_{z_{lk}},\bm{\Sigma}_{z_{lk}})}\right] ^{-1} \\
&= \left[ 1+ \frac{P(\bm{y}_l=\bm{z}_{lk} \vert \bm{\mu}_{z_{lk}}, \bm{\Sigma}_{z_{lk}})}{P(\bm{y}_l= A_{lk}\cdot\bm{h}_k+ \bm{z}_{lk} \vert \bm{\mu}_{z_{lk}}, \bm{\Sigma}_{z_{lk}})} \right] ^{-1} \\
&= \left[ 1+ \frac{f(\bm{y}_l\vert \bm{\mu}_{z_{lk}}, \bm{\Sigma}_{z_{lk}})}{f(\bm{y}_l\vert \bm{\mu}_{z_{lk}}^{'}, \bm{\Sigma}_{z_{lk}}^{'})} \right] ^{-1} \\
&= \frac{f(\bm{y}_l\vert \bm{\mu}_{z_{lk}}^{'}, \bm{\Sigma}_{z_{lk}}^{'})}{f(\bm{y}_l\vert \bm{\mu}_{z_{lk}}, \bm{\Sigma}_{z_{lk}})+f(\bm{y}_l\vert \bm{\mu}_{z_{lk}}^{'}, \bm{\Sigma}_{z_{lk}}^{'})},
\end{aligned} \label{equ-20}
\end{equation}
where
\begin{align}
\bm{\mu}_{z_{lk}}^{'} &= A_{lk} \cdot \bm{\mu}_{k \to l}^{VN} + \bm{\mu}_{z_{lk}} \\
\bm{\Sigma}_{z_{lk}}^{'} &= \left| A_{lk} \right| ^2 \cdot \bm{\Sigma}_{k \to l}^{VN} + \bm{\Sigma}_{z_{lk}},
\end{align}
which denote the mean-value and covariance of $\bm{y}_l$ when $\phi_k=1$, respectively. And $f(\bm{x} \vert \bm{\mu}, \Sigma)$ denotes the probability density function (pdf) of the multi-dimensional complex Gaussian distribution $\mathcal{CN}_M(\bm{x} \vert \bm{\mu}, \Sigma)$, that is
\begin{equation}
f(\bm{x} \vert \bm{\mu}, \bm{\Sigma})= \frac{1}{\pi^M \!\cdot\! \det\left( \bm{\Sigma}\right) }\!\cdot \exp \left[ - (\bm{x}-\bm{\mu})^H \bm{\Sigma}^{-1} (\bm{x}-\bm{\mu})\right]. \label{equ-23}
\end{equation}
Moreover, the Bernoulli message can be simplified by the use of log-likelihood ratio (LLR) to reduce the complexity as well as to avoid the computation overflow. Hence, the LLR of the message in (\ref{equ-20}) can be represented as
\begin{equation}
\begin{aligned}
l_{l \to k}^{SN} & \triangleq \text{ln}\frac{P(\phi_{k}=1)}{P(\phi_{k}=0)} = \ln \frac{p_{l \to k}^{SN}}{1-p_{l \to k}^{SN}} \\
& \overset{(a)}{=} \ln \frac{\det\left( \bm{\Sigma}_{z_{lk}}\right) }{\det\left( \bm{\Sigma}_{z_{lk}}^{'}\right) } \!+\! (\bm{y}_l-\bm{\mu}_{z_{lk}})^H\cdot \bm{\Sigma}_{z_{lk}}^{-1} \cdot (\bm{y}_l-\bm{\mu}_{z_{lk}}) \\
& \quad \quad \quad - (\bm{y}_l-\bm{\mu}_{z_{lk}}^{'})^H\cdot (\bm{\Sigma}_{z_{lk}}^{'})^{-1} \cdot (\bm{y}_l-\bm{\mu}_{z_{lk}}^{'}),
\end{aligned} \label{equ-24}
\end{equation}
where $\overset{(a)}{=}$ is derived by the substitution of (\ref{equ-20}) and (\ref{equ-23}).
\subsubsection{Message Update at Variable Nodes}
\par Likewise, the Gaussian and Bernoulli messages at VNs are updated by collecting the incoming messages from SNs. To ensure convergence, messages from the VN's own are not included in the calculation \cite{BP}. Typically, for the Gaussian messages, since $\bm{h}$ follows the Gaussian distribution, the update rule at the VN is to multiply the pdfs observed at each SN to obtain a new one. We note that the prior Gaussian distribution of $\bm{h}$ is also included in the multiplication. The new pdf still follows the Gaussian distribution, of which the mean and covariance are the updated messages at the VN. The pdf of $\bm{h}_k$ passed from VN $k$ to SN $l$ is given by
\begin{equation}
\begin{aligned}
&f(\bm{h} \vert \bm{\mu}_{k \to l}^{VN}, \bm{\Sigma}_{k \to l}^{VN}) \\
& \quad \quad \propto \prod\nolimits_{i \in \mathcal{L} \backslash l} {f(\bm{h} \vert \bm{\mu}_{i \to k}^{SN}, \bm{\Sigma}_{i \to k}^{SN})} \cdot f(\bm{h} \vert \bm{\mu}_k^{pri}, \bm{\Sigma}_k^{pri}). \label{equ-25}
\end{aligned}
\end{equation}
Accordingly, for the Gaussian pdf $f(\bm{h} \vert \bm{\mu}_{k \to l}^{VN}, \bm{\Sigma}_{k \to l}^{VN})$, the mean and covariance are give by
\begin{align}
\begin{split}
\bm{\mu}_{k \to l}^{VN} &= \mathbb{E}\left[ \bm{h}_k \vert \bm{\mu}_{i \to k}^{SN}, \bm{\Sigma}_{i \to k}^{SN}, i\in \mathcal{L} \backslash l \right] \\
&= \bm{\Sigma}_{k \to l}^{VN} \cdot \Big[ (\bm{\Sigma}_{k}^{pri})^{-1}\cdot \bm{\mu}_k^{pri} \\
& \quad \quad \quad \quad + \sum\nolimits_{i \in \mathcal{L} \backslash l} {(\bm{\Sigma}_{i \to k}^{SN})^{-1} \cdot \bm{\mu}_{i \to k}^{SN}} \Big] \label{equ-26}
\end{split}\\
\begin{split}
\bm{\Sigma}_{k \to l}^{VN} &= \operatorname{Var}\left[ \bm{h}_k \vert \bm{\mu}_{i \to k}^{SN}, \bm{\Sigma}_{i \to k}^{SN}, i\in \mathcal{L} \backslash l \right] \\
&= \left[ \sum\nolimits_{i \in \mathcal{L} \backslash l}{(\bm{\Sigma}_{i \to k}^{SN})^{-1}} + (\bm{\Sigma}_k^{pri})^{-1} \right] ^{-1}, \label{equ-27}
\end{split}
\end{align}
where $\bm{\mu}_k^{pri}= \bm{0}_{M \times 1}$ and $ \bm{\Sigma}_k^{pri}= \bm{I}_M$ are the prior mean and covariance of $\bm{h}_k$, $ \mathcal{L} \backslash l$ denotes the entries in set $\left\lbrace 1,2,\cdots,L\right\rbrace $ except $l$. We give the derivations of (\ref{equ-26}) and (\ref{equ-27}) in Appendix \ref{append-2-2}.
\par The derivation of the Bernoulli messages is the same as above, which is updated by collecting the messages observed at SNs. For $p_{k \to l}^{VN}$ passed for VN $k$ to SN $l$, it is obtained by multiplying the probability of $\phi_k=1$ passed from all the SNs to VN $k$ and then normalizing. Likewise, for convergence, the message passed from SN $l$ to VN $k$ is not included. We emphasize that the prior activation probability of each device $p_a$ is also considered. As such, $p_{k \to l}^{VN}$ is given by
\begin{equation}
\begin{aligned}
p_{k \to l}^{VN} &= P\left( \phi_k=1 \vert \left\lbrace p_{i\to k}^{SN}, i\in \mathcal{L} \backslash l \right\rbrace, p_a \right) \\
&= \frac {p_{a}\cdot \prod _{i\in \mathcal {L}\backslash l}p^{SN}_{i\to k}}{p_{a} \cdot \prod _{i\in \mathcal {L} \backslash l}p^{SN}_{i\to k}+(1-p_{a})\cdot \prod _{i\in \mathcal {L}\backslash l}\left ({1-p^{SN}_{i\to k}}\right)}.
\end{aligned}
\end{equation}
Likewise, for complexity reduction, we also employ the LLR to represent this message in iterations, of which the relationship with the activation probability is
\begin{equation}
\begin{aligned}
l_{k \to l}^{VN} &= \ln \frac {p^{VN}_{k\to l}}{1-p^{VN}_{k\to l}}=l_{0}+\sum _{i\in \mathcal {L}\backslash l}l^{SN}_{i\to k}\\
p^{VN}_{k\to l} &=\frac {1}{1+\exp \left ({{-l^{VN}_{k\to l}}}\right)}, \label{equ-29}
\end{aligned}
\end{equation}
where $l_0= \ln \frac{p_a}{1-p_a}$ is the prior LLR of the probability for the device being active.
\subsubsection{DAD Decision and CE Output}
\par Since the messages above are iteratively updated between SNs and VNs, after reaching the maximum number of iterations, the Bernoulli and Gaussian messages will have an output at VNs. For the Gaussian messages, similar to the above update rules, the output is obtained by combining all the incoming messages from SNs, i.e,
\begin{align}
\begin{split}
\bm{\mu}_{k}^{dec} &= \bm{\Sigma}_{k}^{dec} \cdot \left[ (\bm{\Sigma}_{k}^{pri})^{-1}\cdot \bm{\mu}_k^{pri} \right. \\
& \quad \quad \quad \quad \left.+ \sum\nolimits_{i \in \mathcal{L}} {(\bm{\Sigma}_{i \to k}^{SN})^{-1} \cdot \bm{\mu}_{i \to k}^{SN}} \right] \label{equ-30}
\end{split}\\
\begin{split}
\bm{\Sigma}_{k}^{dec} &= \left[ \sum\nolimits_{i \in \mathcal{L}}{(\bm{\Sigma}_{i \to k}^{SN})^{-1}} + (\bm{\Sigma}_k^{pri})^{-1} \right] ^{-1}, \label{equ-31}
\end{split}
\end{align}
which denote the output estimation and estimating deviation of $\bm{h}_k$, respectively. For the Bernoulli messages, the LLR of the DAD decision is
\begin{equation}
l^{\text {dec}}_{k}=l_{0}+\sum\nolimits_{l\in \mathcal {L}}l^{SN}_{l\to k}+l^{ce}_{k}. \label{equ-32}
\end{equation}
Device $k$'s activity is detected as $\hat{\phi}_k=1$ if $ l^{\text {dec}}_{k}>0$ and vice versa. The term $l_k^{ce}$ in (\ref{equ-32}) is to improve the DAD accuracy by exploiting the CE result \cite{zhang2020iot}, which is derived as follows. The estimated channel $\bm{h}_k$ can be modeled as $\hat{\bm{h}}_k= \bm{h}_k + \bm{\epsilon}_k$, where $\bm{\epsilon}_k$ is the complex Gaussian noise distributed as $\bm{\epsilon}_k \sim \mathcal{CN}(0,\bm{\Sigma}_{k}^{dec})$. Accordingly, the distribution of $\hat{\bm{h}}_k$ with respect to value of $\phi_k$ is
\begin{equation}
\hat{\bm{h}}_k \sim \left\{\begin{array}{l c}
\mathcal{CN}(\bm{\mu}_k^{pri},\bm{\Sigma}_{k}^{pri}+\bm{\Sigma}_{k}^{dec}), &\phi_k=1 \\
\mathcal{CN}(\bm{0},\bm{\Sigma}_{k}^{dec}), &\phi_k=0
\end{array} \quad \forall k \in \mathcal{K}_{tot}.\right. \label{equ-33}
\end{equation}
Therefore, this information can be leveraged to give an extra belief to the DAD decision. Similar to (\ref{equ-20}), $l^{ce}_{k}$ is given by
\begin{equation}
\begin{aligned}
l^{ce}_{k} =&\ln \frac {P\left ({\hat {\bm{h}}_{k} =\bm{\mu}_{k}^{dec} \vert \phi _{k}=1, \bm{\mu}_{k}^{pri},\bm{\Sigma}_{k}^{dec},\bm{\Sigma}_{k}^{pri} }\right)}{P\left ({\hat {\bm{h}}_{k} =\bm{\mu}_{k}^{dec} \vert \phi _{k}=0,\bm{\Sigma}_{k}^{dec} }\right)} \\
=&\ln \frac {f\left ({\bm{\mu}_{k}^{dec} \vert \bm{\mu}_{k}^{pri},\bm{\Sigma}_{k}^{pri}+\bm{\Sigma}_{k}^{dec}}\right)}{f\left ({\bm{\mu}_{k}^{dec} \vert \bm{0},\bm{\Sigma}_{k}^{dec}}\right)} \\
=&\ln \frac {\det\left( \bm{\Sigma}_{k}^{dec}\right) }{\det\left( \bm{\Sigma}_{k}^{pri}\!+\!\bm{\Sigma}_{k}^{dec}\right) } + \left(\bm{\mu}_k^{dec}\right)^H\cdot \left(\bm{\Sigma}_{k}^{dec} \right) ^{-1} \cdot \bm{\mu}_k^{dec} \\
&- \left( \bm{\mu}_k^{dec} \!-\!\bm{\mu}_k^{pri}\right) ^H\ \!\!\!\cdot \! \left(\bm{\Sigma}_{k}^{pri} \!+\! \bm{\Sigma}_{k}^{dec} \right) ^{-1} \!\! \!\cdot \! \left( \bm{\mu}_k^{dec}\!-\!\bm{\mu}_k^{pri}\right) \!. \label{equ-34}
\end{aligned}
\end{equation}
As aforementioned, we derive the LLR expression by utilizing the joint distribution of the channel among antennas. Finally, we obtain the estimated channel of device $k$ as
\begin{equation}
\bm{\hat{h}}_k = \hat{\phi}_k \cdot \bm{\mu}_{k}^{dec}.
\end{equation}
\begin{figure}[htpb]
\centerline{\includegraphics[width=0.4\textwidth]{pic-11-new}}
\caption{NMSE of CE by the Joint DAD-CE algorithm. $B_p=9$, $K_a=30$, $M=30$, $L_p=200$.}
\label{pic-5-add}
\end{figure}
\par The above joint DAD-CE algorithm is summarized in Algorithm \ref{alo-2}, where $N_{iter}$ denotes the maximum number of iterations. We note that $\bm{\Sigma}^{VN}$ and $\bm{\Sigma}^{SN}$ both go to diagonal matrices over the iteration in our numerical verification. Hence, the corresponding matrix inverse operations can be simplified to the divisions to reduce the complexity with little performance loss. As shown in Fig. \ref{pic-5-add}, Simplified denotes the approximation by treating $\bm{\Sigma}^{VN}$ and $\bm{\Sigma}^{SN}$ as diagonal matrices and Original means there is no approximation in Algorithm \ref{alo-2}. We use normalized mean square error (NMSE) for the evaluation of the CE performance. Fig. \ref{pic-5-add} illustrates that this approximation introduces little performance loss, which, although, has greatly reduced the complexity as aforementioned.
\begin{algorithm}
\caption{Joint DAD-CE Algorithm}
\label{alo-2}
\begin{algorithmic}[1]
\STATE {{\bf Input}: $\bm{Y_p}$, $\bm{A}$, $\bm{\mu}^{pri}$, $\bm{\Sigma}^{pri}$, $\sigma_n^2$, $p_a$}\\
\STATE{{\bf Initialize}: $\bm{\mu}^{VN}=\bm{\mu}^{pri}, \bm{\Sigma}^{VN}=\bm{\Sigma}^{pri}$}
\REPEAT
\STATE {SN update: $\bm{\mu}^{SN}$, $\bm{\Sigma}^{SN}$ by \eqref{equ-18}-\eqref{equ-19}}\\
\STATE {SN update: $l^{SN}$ by \eqref{equ-24}}\\
\STATE {VN update: $\bm{\mu}^{VN}$, $\bm{\Sigma}^{VN}$ by \eqref{equ-26}-\eqref{equ-27}}\\
\STATE {VN update: $l^{VN}$ by \eqref{equ-29}}\\
\UNTIL {$N_{iter}$ reached}
\STATE {CE output: $\bm{\mu}^{dec}$, $\bm{\Sigma}^{dec}$ by \eqref{equ-30}-\eqref{equ-31}}\\
\STATE {DAD Decision: $l^{dec}$ by \eqref{equ-32}}\\
\RETURN {$\left\lbrace\hat{\bm{h}}_k, \hat{\phi}_k, \forall k \in \mathcal{K}_{tot} \right\rbrace $}
\end{algorithmic}
\end{algorithm}
\begin{figure*}[t]
\centerline{\includegraphics[width=1.0\textwidth]{pic-4}}
\caption{Factor graph for LDPC decoding. }
\label{pic-4}
\end{figure*}
\subsection{MIMO-LDPC-SIC Decoder}
\par After obtaining the key parameters, such as interleaving patterns and channels by the joint DAD-CE algorithm, the LDPC decoding problem can be addressed by the standard BP algorithm \cite{BP,LDPC-MIMO}. Likewise, the DD process is performed by updating messages iteratively at different nodes. Differently, since the LDPC is a forward error correction code, besides the observation and variable nodes, the check nodes (CNs) are considered in the factor graph to provide an extra belief, as shown in {Fig. \ref{pic-4}}. We rewrite the received signal in the LDPC phase as
\begin{equation}
{\mathbf{Y}_c} = {\mathbf{Y}_{{L_p} + 1:{L},:}} = \sum\limits_{k \in {\mathcal{K}_a}} {{\pi _{{i_k}}}\left( {{\tilde{\mathbf{s}}_k}} \right){\mathbf{h}_{{i_k}}^T} + {\mathbf{Z}_{_{{L_p} + 1:{L},:}}}},
\label{equ-35}
\end{equation}
where $\mathbf{Y}_c \in {C^{{L_c} \times M}}$ is the last $L_c$ rows of $\mathbf{Y}$. The LDPC decoder is tasked to recover the last $B_c$ bits of information based on the received signal $\mathbf{Y_c}$, estimated interleaving patterns and channels using the low-complexity iterative BP algorithm. Owing to the two-phase encoding scheme, these key parameters can be recovered in the decoding of the CS phase. We emphasize that once the active indicators $\phi_k$ are recovered, the positions of $\left\lbrace \phi_k=1, k\in \mathcal{K}_{tot}\right\rbrace $ in the selection matrix $\bm{\Phi}$, i.e., $\left\lbrace i_k, \forall k\in \mathcal{K}_a \right\rbrace $, are determined. Thereby, the interleaving patterns of active devices are also recovered. As shown in \eqref{equ-35}, the zero-padded sequences $\tilde{\mathbf{s}}_k$ are subject to different permutations, and each is determined by the interleaving pattern $i_k$. Therefore, the effect of these permutations needs to be considered when the messages are being sent to and from the VNs, i.e., interleaving and de-interleaving in Fig. \ref{pic-4}, respectively.
\par The connections among nodes in Fig. \ref{pic-4} appear to be more involved than those of Fig. \ref{pic-3}. In the upper part of Fig. \ref{pic-4}, the CNs (blue color) and VNs (green color) as well as the edges connecting them constitute the Tanner graph in LDPC. The subscript $N=\tilde{L}_c-B_c$ denotes the number of CNs in the LDPC code, which corresponds to the number of rows of the LDPC check matrix. $K$ denotes the number of active devices estimated in the CS phase. Other subscripts are consistent with the aforementioned. The edges between CNs and VNs are described by the LDPC check matrix, which cannot be marked explicitly in the graph. For example, in the check matrix of device $k$, if the entry $c_{i,j}=1$, there will be an edge between SN $c_{k,i}$ and VN $s_{k,j}$.
\begin{figure}[htpb]
\centerline{\includegraphics[width=0.3\textwidth]{pic-7}}
\caption{The set of VNs connected with an SN is decided by the interleaving patterns. For instance, in the diagram, the set of VNs related with $y_{1,3}$ is $\left\lbrace s_{1,5}, s_{3,4}, s_{5,2} \right\rbrace $. Correspondingly, the set of SNs connected to these VNs is $\left\lbrace y_{m,3}, m=1,2,\cdots,M \right\rbrace $.}
\label{pic-7}
\end{figure}
\par The lower part of Fig. \ref{pic-4} refers to the graph for MIMO detection, of which the edges between VNs and SNs (yellow color) are simply determined by \eqref{equ-35} though looking complicated. For example, the VN $s_{k,l_1} \left( l_1\in \left[1:\tilde{L}_c \right]\right) $ is connected to the $l_2$-th SN from all antennas (i.e., $y_{m,l_2},m=1,2,...,M,l_2\in \left[1:L_c \right] $). We note that $l_1$ is not necessarily equal to $l_2$ in the presence of zero-padding and interleaving. Correspondingly, the VNs connected to SN $y_{m,l_2}$ depend on whose data is interleaved to the $l_2$-th channel use. For instance, as illustrated in Fig. \ref{pic-7}, the set of VNs connected to SN $y_{1,3}$ is $\left\lbrace s_{1,5}, s_{3,4}, s_{5,2} \right\rbrace $. That is, after zero-padding and interleaving, the fifth, fourth, and second bits of devices $1$, $3$, and $5$ are mapped to the third channel use, respectively. Before conducting the MP algorithm, we define the types of messages as follows.
\begin{itemize}
\item ${R_{k,n \to l_1}}$: Messages passed from CN $c_{k,n}$ to VN $s_{k,l_1}$.
\item ${Q_{k,l_1 \to n}}$: Messages passed from VN $s_{k,l_1}$ to CN $c_{k,n}$.
\item ${P_{k,l_1 \to m,l_2}}$: Messages passed from VN $s_{k,l_1}$ to SN $y_{m,l_2}$.
\item ${\Lambda _{m,l_2 \to k,l_1}}$: Messages passed from SN $y_{m,l_2}$ to VN $s_{k,l_1}$.
\end{itemize}
\par The massages $R$ and $Q$ refer to the parity check constraints in the LDPC code, while $P$ and $\Lambda$ are related to the received signals in the MIMO system.
\begin{figure}[htpb]
\centerline{\includegraphics[width=0.5\textwidth]{pic-5}}
\caption{Update rules for messages $R$ and $Q$ at CNs and VNs on the Tanner graph for device $k$. The dashed and solid lines represent input and output messages, respectively.}
\label{pic-5}
\end{figure}
\par We first give the MP rules for the LDPC decoding with BPSK modulation, which is known as the sum-product algorithm. As illustrated in Fig. \ref{pic-5}, the messages $R$ and $Q$ are iteratively updated between CNs and VNs. Likewise, for the reduction of complexity, we give the updating rules in the LLR form.
\begin{align}
{Q_{k,l_1 \to n}} &= \sum\limits_{j \in M} {\pi_{i_j}^{-1}\left( \Lambda _{j,l_2 \to k,l_1}\right) } \! + \!\! \sum\limits_{j \in {\mathcal{N}_c}\left( {k,l_1} \right) \backslash n} {\!\!\!{R_{k,j \to l_1}}} \label{equ-36}\\
{R_{k,n \to l_1}} &= 2{\tanh ^{ - 1}}\left( {\prod\limits_{j \in {\mathcal{N}_v}\left( {k,n} \right) \backslash l_1} {\!\!\!\tanh \left( {\frac{{{Q_{k,j \to n}}}}{2}} \right)} } \right), \label{equ-37}
\end{align}
where ${\mathcal{N}_c}\left( {k,l_1} \right) \backslash n$ denotes the set of CNs connected to $s_{k,l_1}$ except $c_{k,n}$, i.e., $\left\lbrace c_{k,n},c_{k,n_1},\cdots, c_{k,n_i}\right\rbrace $ in Fig. \ref{pic-5}. Likewise, ${\mathcal{N}_v}\left( {k,n} \right) \backslash l_1$ denotes the set of VNs connected to $c_{k,n}$ except $s_{k,l_1}$, i.e., $\left\lbrace s_{k,l_1^1},\cdots,s_{k,l_1^i}\right\rbrace $ in Fig. \ref{pic-5}.
\begin{figure}[htpb]
\centerline{\includegraphics[width=0.5\textwidth]{pic-6-v3}}
\caption{Update rules for messages $\Lambda$ and $P$ at VNs and SNs on the factor graph. The dashed and solid lines represent input and output messages, respectively.}
\label{pic-6}
\end{figure}
\par The message ${\Lambda _{m,l_2 \to k,l_1}}$ in \eqref{equ-36} is the LLR with the probability of VN $s_{k,l_1}$ taking different values observed at SN $y_{m,l_2}$. For BPSK modulated system, it is given by
\begin{equation}
\begin{aligned}
{\Lambda _{m,l_2 \to k,l_1}} &= \log \frac{{P\left( {{y_{m,l_2}}|\mathbf{H},{s_{k,l_1}} = +1} \right)}}{{P\left( {{y_{m,l_2}}|\mathbf{H},{s_{k,l_1}} = - 1} \right)}} \\
&= \frac{2}{{\sigma _{{z_{k,l_2}}}^2}}\text{Re}\left( {h_{m,k}^ * \left( {{y_{m,l_2}} - {\mu _{{z_{k,l_2}}}}} \right)} \right),
\label{equ-38}
\end{aligned}
\end{equation}
where $\bm{H}$ is the channel matrix, which can be estimated in the CS phase. We note that similar to the joint DAD-CE algorithm, \eqref{equ-38} is also obtained by treating the interference from other devices as noise. We rewrite \eqref{equ-35} as
\begin{equation}
\begin{aligned}
{y_{m,l_2}} &= \sum\limits_{\substack{j \in \mathcal{K}(m,l_2),l \in \mathcal{L}(m,l_2)}} {{h_{m,j}}{\pi_{i_j}\left( \tilde{s}_{j,l}\right) } + {n_{m,l_2}}}\\
&={h_{m,k}}\pi_{i_k}\left( \tilde{s}_{k,l_1}\right) \!+\! \underbrace{\sum\limits_{\substack{j \in \mathcal{K}(m,l_2) \backslash k\\l \in \mathcal{L}(m,l_2)\backslash l_1}} {{h_{m,j}}\pi_{i_j}\left( \tilde{s}_{j,l}\right) \!+\! {n_{m,l_2}}}}_{z_{k,l_2}}, \label{equ-40}
\end{aligned}
\end{equation}
where $n_{m,l_2}$ is the Gaussian noise with zero mean and variance ${\sigma _n^2}$, and $\mathcal{K}(m,l_2),\mathcal{L}(m,l_2)$ denote the set of devices as well as the corresponding bits related with SN $y_{m,l_2}$, respectively. Accordingly, the set of related VNs is $\left\lbrace s_{j,l} \vert j \in \mathcal{K}(m,l_2), l \in \mathcal{L}(m,l_2) \right\rbrace $ and we note that the subscripts $j$ and $l$ are one-to-one mappings. For instance, Fig. \ref{pic-7} showcases the VNs related with SN $y_{1,3}$. In this regard, $\mathcal{K}(1,3) = \left\lbrace 1,3,5 \right\rbrace $ and $\mathcal{L}(1,3)= \left\lbrace5,4,2 \right\rbrace $ and the corresponding VNs are $\left\lbrace s_{1,5}, s_{3,4}, s_{5,2} \right\rbrace $. The Gaussian noise $n_{m,l_2}$ and the interference from other devices are all treated as noise denoted by $z_{k,l_2}$, which is a Gaussian variable with mean ${\mu _{{z_{k,l_2}}}}$ and variance ${\sigma _{{z_{k,l_2}}}^2}$ given by
\begin{align}
{\mu _{{z_{k,l_2}}}} &= \sum\limits_{\substack{j \in \mathcal{K}(m,l_2) \backslash k\\l \in \mathcal{L}(m,l_2)\backslash l_1}} {{h_{m,j}} \cdot \mathbb{E}\left[ \pi_{i_j}\left( \tilde{s}_{j,l}\right) \right] } \label{equ-41} \\
\sigma _{{z_{k,l_2}}}^2 &= \sum\limits_{\substack{j \in \mathcal{K}(m,l_2) \backslash k\\l \in \mathcal{L}(m,l_2)\backslash l_1}} {{{\left| {{h_{m,j}}} \right|}^2}\cdot \text{Var}\left[ \left(\pi_{i_j}\left( \tilde{s}_{j,l}\right) \right)\right] }+\sigma_n^2. \label{equ-42}
\end{align}
And for BPSK modulation, the mean and variance of $\pi_{i_j}\left( \tilde{s}_{j,l}\right) $ is given by
\begin{align}
\mathbb{E}\left[ \pi_{i_j}\left( \tilde{s}_{j,l}\right) \right] &=2 \cdot \pi_{i_j}\left({P_{j,l \to m,l_2}}\right) - 1 \\
\text{Var}\left[ \left(\pi_{i_j}\left( \tilde{s}_{j,l}\right) \right)\right] &= 4\cdot\pi_{i_j}\left( \left( {1 - {P_{j,l \to m,l_2}}} \right) \cdot {P_{j,l \to m,l_2}}\right),
\end{align}
where ${P_{j,l \to m,l_2}}$ denotes the probability of VN $s_{j,l}=1$, and is initialized to $0.5$. We note that it needs to be interleaved before the calculation. As shown in Fig. \ref{pic-6}, ${P_{k,l_1 \to m,l_2}}$ is updated by collecting the incoming messages from CNs related with VN $s_{k,l_1}$ and those from all SNs except $y_{m,l_2}$. We give the update rule as below.
\begin{equation}
{P_{k,l_1 \to m,l_2}} \!=\! \frac{{\exp \!\left( \Lambda + R\right)}}{{1 + \exp \left( \Lambda + R\right)}}, \label{equ-43}
\end{equation}
where
\begin{equation}
\Lambda =\!\!\! \sum\limits_{j \in M \backslash m} {\pi_{i_j}^{-1}\left({\Lambda _{j,l_2 \to k,l_1}}\right) }, ~~R=\!\!\!\sum\limits_{j \in {\mathcal{N}_c}\left( {k,l_1} \right)} {{R_{k,j \to l_1}}}. \label{equ-add-1}
\end{equation}
Similar to \eqref{equ-36}, the message $\Lambda _{j,l_2 \to k,l_1}$ needs to be de-interleaved in the update of $P$. The LLR of VN $s_{k,l_1}$ at the end of an iteration is given by
\begin{equation}
{L_{k,l_1}} = \sum\limits_{j \in M} {\pi_{i_j}^{-1}\left( {\Lambda _{j,l_2 \to k,l_1}}\right) + \sum\limits_{j \in {\mathcal{N}_c}\left( {k,l_1} \right)} {{R_{k,j \to l_1}}} }. \label{equ-46}
\end{equation}
\par The information bit $\hat{v}_{k,l_1}^c$ is decoded as one if $L_{k,l_1}>0$ and zero otherwise. Since LDPC codes are described by the parity matrix $\bm{C}$, the iteratively decoding process is continued till $\text{mod} \left( \bm{C}\hat{\bm{v}}_k^c,2\right) =0$ or the maximum number of iterations $N_{iter}$ is reached.
\par To further improve the spectrum efficiency, the QPSK modulation is also considered in this coding system, of which the constellation set is $\mathbb{S}=\left\lbrace \pm 1/ \sqrt{2},\pm 1/ \sqrt{2} i\right\rbrace $. Briefly, QPSK modulated signals can be split into two orthogonal BPSK ones. As such, we can implement the above MP algorithm on these two signals separately. Additionally, the real and imaginary parts of the messages $\Lambda$ and $P$ need to be considered separately, so does the parity of subscript $l_1$ in $Q_{k,l_1 \to n}$ and $R_{k,n \to l_1}$. It is worth noting that the range of $l_1$ in $\Lambda$ and $P$ is half of that in $Q$ and $P$, i.e., $\left[1:\tilde{L}_c/2 \right]$, since there are both messages on the real and imaginary parts. In what follows we give the updated rules of these messages.
\par Similar to \eqref{equ-38}, $\Lambda$ is the LLR of the probability that $s_{k,l_1}$ takes different values in $\mathbb{S}$. However, it is no longer a real number. Instead, it is given by
\begin{equation}
\Lambda _{m,l_2 \to k,l_1} = \frac{2\sqrt{2}}{{\sigma _{{z_{k,l_2}}}^2}}\cdot {h_{m,k}^ *\cdot \left( {{y_{m,l_2}} - {\mu _{{z_{k,l_2}}}}} \right)}, \label{equ-47}
\end{equation}
where the mean and variance of $z_{k,l_2}$ are given in \eqref{equ-41} and \eqref{equ-42}, respectively. And for QPSK modulation, the mean and variance of $\pi_{i_j}\left( \tilde{s}_{j,l}\right) $ is given by
\begin{align}
\mathbb{E}\left[ \pi_{i_j}\left( \tilde{s}_{j,l}\right) \right] &= 1/\sqrt{2} \cdot \left\lbrace 2\cdot \pi_{i_j} \left( {P_{j,l \to m,l_2}^r}\right) - 1 \right. \notag \\
& \quad \quad \quad ~\left. +\left[ 2\cdot \pi_{i_j} \left( {P_{j,l \to m,l_2}^i}\right) -1 \right] \cdot i\right\rbrace \label{equ-48}\\
\text{Var}\left[ \left(\pi_{i_j}\left( \tilde{s}_{j,l}\right) \right)\right] &= 2\cdot\pi_{i_j}\left[ {P_{j,l \to m,l_2}^r} - \left({P_{j,l \to m,l_2}^r} \right)^2 \right. \notag \\
&\quad \quad \quad ~ \left. + {P_{j,l \to m,l_2}^i} - \left({P_{j,l \to m,l_2}^i} \right)^2 \right] , \label{equ-49}
\end{align}
where $ {P_{j,l \to m,l_2}^r}$ and $ {P_{j,l \to m,l_2}^i}$ are the real and imaginary parts of $P_{j,l \to m,l_2}$, respectively. Similar to \eqref{equ-43} and \eqref{equ-add-1}, $ {P_{k,l_1\to m,l_2}^r}$ and $ {P_{k,l_1\to m,l_2}^i}$ are given by
\begin{align}
{P_{k,\lceil \frac{l_1}{2} \rceil \to m,l_2}^r} &= \frac{\exp \left[ \text{Re}\left( \Lambda \right) + R \right]}{{1 + \exp \left[\text{Re}\left( \Lambda \right) + R \right]}}, ~l_1~\text{is odd}, \label{equ-51} \\
{P_{k,\frac{l_1}{2} \to m,l_2}^i} &= \frac{\exp \left[ \text{Im}\left( \Lambda \right) +R \right]}{{1 + \exp \left[\text{Im}\left( \Lambda \right) +R \right]}}, ~l_1~\text{is even}. \label{equ-52}
\end{align}
As such, the range of subscript $l_1$ in $P$ is half of that in $R$ as aforementioned. $\Lambda$ and $R$ are defined in \eqref{equ-add-1}. The update rule for message $R$ is the same as \eqref{equ-37} and that for $Q$ is given by
\begin{equation}
{Q_{k,l_1 \to n}} = \left\{\begin{array}{l}
\sum\limits_{j \in M} {\pi_{i_j}^{-1}\left(\text{Re}\left( \Lambda _{j,l_2 \to k,\lceil \frac{l_1}{2} \rceil}\right) \right) } \! \\
\quad \quad \quad + \!\! \sum\limits_{j \in {\mathcal{N}_c}\left( {k,l_1} \right) \backslash n} {\!\!\!{R_{k,j \to l_1}}}, ~l_1~\text{is odd}\\
\sum\limits_{j \in M} {\pi_{i_j}^{-1}\left(\text{Im}\left( \Lambda _{j,l_2 \to k,\frac{l_1}{2}}\right) \right) } \! \\
\quad \quad \quad + \!\! \sum\limits_{j \in {\mathcal{N}_c}\left( {k,l_1} \right) \backslash n} {\!\!\!{R_{k,j \to l_1}}}, ~l_1~\text{is even}\\
\end{array} \right. \label{equ-53}
\end{equation}
The LLR of VN $s_{k,l_1}$ at the end of an iteration is given by
\begin{equation}
L_{k,l_1} = {Q_{k,l_1 \to n}} + R_{k,n \to l_1}, ~ \forall n\in\mathcal{N}_c(k,l_1). \label{equ-54}
\end{equation}
The decision rule and termination condition are the same as those in the BPSK system mentioned earlier. Finally, we obtain the decoded messages.
\par Note that the estimated number of active devices $K$ is not guaranteed to be equal to $K_a$. Therefore, not all the decoded messages satisfy the parity check. We denote $\widehat{\mathcal{V}}=\left\{\hat{\bm{v}}_k^c, k \in \widehat{\mathcal{K}}\right\}$ and $\widehat{\mathcal{K}}$ as the set of successfully decoded messages and the corresponding devices, respectively. And we have $\left| {\widehat {\mathcal{K}}} \right| \le {K_a}$. To further improve the performance, we combine the MIMO-LDPC decoder with the SIC method and we denote it as the MIMO-LDPC-SIC algorithm, which works as follows.
\par Let $\widehat {\mathbf{H}} \in {C^{K \times M}}$ and $\mathcal{K}$ denote the channel matrix and the set of active devices estimated in the CS phase. Let ${\hat {\mathcal{V}}_0}$ and ${\hat {\mathcal{K}}_0}$ respectively denote the sets of correctly decoded messages (i.e., those that satisfy the check) and the corresponding devices obtained by the decoder, which are initialized to empty sets. With $\bm{Y}_c$, interleaving patterns $\left\lbrace \pi_{i_k}, k \in \mathcal{K}\right\rbrace $ and $\hat{\bm{H}}$, the decoder outputs the set of successfully decoded messages ${\widehat {\mathcal{V}}} $ and the corresponding devices ${\widehat {\mathcal{K}}}$. Then we have ${\widehat {\mathcal{V}}_0} \leftarrow {\widehat {\mathcal{V}}_0} \cup \widehat {\mathcal{V}}$, ${\widehat {\mathcal{K}}_0} \leftarrow {\widehat {\mathcal{K}}_0} \cup \widehat {\mathcal{K}}$ and ${\mathbf{H}} = {\widehat {\mathbf{H}}_{k,:}}$ for $k \in {\mathcal{K}} \backslash {\widehat {\mathcal{K}}_0}$. The residual signal is updated by
\begin{equation}
\mathbf{Y} = {\mathbf{Y}_c} - \sum\nolimits_{k \in {{\hat {\mathcal{K}}}_0}} {{\pi _{{i_k}}}\left( {{\tilde{\mathbf{s}}_k}} \right){\widehat{\mathbf{H}}_{k,:}} },
\label{equ-55}
\end{equation}
where $\tilde{\mathbf{s}}_k \in \mathbb{C}^{L_c\times 1}$ is the $k$-th codeword in $\widehat {\mathcal{V}}_0$ after modulation and zero-padding. The updated $Y$ and $\bm{H}$ as well as the interleaving patterns are sent to the MIMO-LDPC decoder for the next round of decoding. This iterative process ends when ${\hat {\mathcal{K}}} = \emptyset $ or ${\mathcal{K}} \backslash {\widehat {\mathcal{K}}_0} =\emptyset$. The overall decoding algorithm is summarized in Algorithm \ref{alo-3}.
\begin{algorithm}
\caption{MIMO-LDPC-SIC Decoding Algorithm}
\label{alo-3}
\begin{algorithmic}[1]
\STATE {{\bf Input}: $\bm{Y_c}$, $\hat{\bm{H}}$, $\mathcal{L}= \left\{ {{i_k},k \in \mathcal{K}} \right\}$, $\sigma_n^2$}\\
\STATE {{\bf Initialize}:\\ $\bm{Y} \!=\!\bm{Y_c}$, $\bm{H}\!=\!\hat{\bm{H}}$, ${\widehat {\mathcal{V}}_0} \!\leftarrow\! \emptyset $, ${\widehat {\mathcal{K}}_0} \!\leftarrow\! \emptyset $, ${R_{k,n \to l_1}}=0$,\\ BPSK: ${P_{k,l_1 \to m,l_2}}=0.5$, QPSK: ${P_{k,l_1 \to m,l_2}}=0.5 + 0.5i$.}\\
\REPEAT
\REPEAT
\STATE{$\Lambda$ update: $\Lambda _{m,l_2 \to k,l_1}$ by \eqref{equ-38} or \eqref{equ-47}.}\\
\STATE{$Q$ update: $Q_{k,l_1 \to n}$ by \eqref{equ-36} or \eqref{equ-53}.}\\
\STATE{$R$ update: $R_{k,n \to l_1}$ by \eqref{equ-37}.}\\
\STATE{$P$ update: $P_{k,l_1 \to m,l_2}$ by \eqref{equ-43} or \eqref{equ-51}-\eqref{equ-52}.}\\
\STATE{$L$ update and hard decision: $ L_{k,l_1}$ by \eqref{equ-46} or \eqref{equ-54}.}\\
\UNTIL{$N_{iter}$ reached or $\text{mod}(\bm{C}\hat{\bm{v}}^c,2)=0$.}
\STATE{{\bf Output}: ${\widehat {\mathcal{V}}} $, ${\widehat {\mathcal{K}}}$}\\
\STATE{${\widehat {\mathcal{V}}_0} \leftarrow {\widehat {\mathcal{V}}_0} \cup \widehat {\mathcal{V}}$, ${\widehat {\mathcal{K}}_0} \leftarrow {\widehat {\mathcal{K}}_0} \cup \widehat {\mathcal{K}}$.}\\
\STATE{${\mathbf{H}} = {\widehat {\mathbf{H}}_{k,:}}$ for $k \in {\mathcal{K}} \backslash {\widehat {\mathcal{K}}_0}$.}\\
\STATE{$ \mathbf{Y} = {\mathbf{Y}_c} - \sum\nolimits_{k \in {{\hat {\mathcal{K}}}_0}} {{\pi _{{i_k}}}\left( {{\tilde{\mathbf{s}}_k}} \right){\widehat{\mathbf{H}}_{k,:}} }$.}\\
\UNTIL{${\widehat {\mathcal{K}}} = \emptyset $ or ${\mathcal{K}} \backslash {\widehat {\mathcal{K}}_0} =\emptyset$.}\\
\STATE{{\bf Return:} ${\widehat {\mathcal{V}}_0}$, ${\widehat {\mathcal{K}}_0}$}
\end{algorithmic}
\end{algorithm}
\par We note that the stitching of the messages in the CS and LDPC phases is easy. In URA, devices are not identified, as such, the IDs can not be employed to distinguish the messages. We have known that the interleaving patterns and channels $\left\lbrace \pi_{i_k}, \hat{\bm{h}}_k, k \in \widehat{\mathcal{K}}_0\right\rbrace $ obtained in the CS phase acting as key parameters participate in the decoding process in the LDPC phase. For each decoded message in $\widehat{\mathcal{V}}_0$, it is decoded with a specific interleaving pattern $\pi_{i_k}$ as well as the channel $\hat{\bm{h}}_k$. As aforementioned, $\pi_{i_k}$ is uniquely determined by the message index representation $i_k$, which directly corresponds to device $k$' preamble. Therefore, $\pi_{i_k}$ and $\hat{\bm{h}}_k$ establish a connection of devices' messages in the two phases. Briefly, if $\hat{\bm{v}}^c$ is successfully decoded with the participation of $\pi_{i_k}$ and $\hat{\bm{h}}_k$, it is exactly the latter $B_c$ bits of message of device $k$. As such, the stitching of the messages in two phases will not be a problem.
\subsection{Joint Update} \label{sec-5-3}
\par The above CS and LDPC decoders can recover the $B$ bits of information with their work in tandem. Besides, thanks to the consistency of the above MP algorithm, the BP-based decoder can draw a connection of the decoding process in the CS and LDPC phases. That is, messages in the CE as well as the MIMO-LDPC decoding processes can be jointly updated by utilizing the belief of each other, thus leading to improved performance. This joint update algorithm is denoted as joint DAD-CE-DD algorithm and elaborated as follows.
\par For the successfully decoded devices ${\widehat {\mathcal{K}}_0}$, the corresponding messages $\left\lbrace {\pi _{{i_k}}}\left( {{\tilde{\mathbf{s}}_k}} \right), k \in \widehat {\mathcal{K}}_0 \right\rbrace $ can be leveraged as soft pilot sequences joint with their codewords $\left\lbrace \bm{a}_{i_k}, k \in \widehat {\mathcal{K}}_0 \right\rbrace $ in the CS phase to carry out a second CE. This longer pilot sequence will lead to a better CE performance, which has been confirmed in our simulation in Fig. \ref{pic-11}. We note that the subsequent CE is conducted via the above joint DAD-CE algorithm with devices' activity fixed. In this regard, the Gaussian messages $\left\lbrace \mu_{lm \to km}^{SN}, \bm{\Sigma}_{l \to k}^{SN}, \mu_{lm \to km}^{VN}, \bm{\Sigma}_{l \to k}^{VN}\right\rbrace $ in \eqref{equ-18}-\eqref{equ-19} and \eqref{equ-26}-\eqref{equ-27} are iteratively updated with a longer observation sequence $\left\lbrace \bm{y}_l, l\in \left[ 1:L_p+L_c\right] \right\rbrace$. Besides, it is worth noting that in the CE output in \eqref{equ-30}-\eqref{equ-31}, the prior mean and covariance $\mu_{k}^{pri}, \bm{\Sigma}_{k}^{pri}$ are no longer zero and $\bm{I}_M$, respectively. Instead, they are the output estimation $\mu_{k}^{dec}$ and estimating deviation $\bm{\Sigma}_{k}^{dec}$ of $\bm{h}_k$ in the first CE, respectively.
\begin{algorithm}
\caption{Joint DAD-CE-DD Algorithm}
\label{alo-4}
\begin{algorithmic}[1]
\STATE {{\bf Input}: $\bm{Y_p}$, $\bm{Y_c}$, $\bm{A}$, $\bm{\mu}^{pri}$, $\bm{\Sigma}^{pri}$, $\sigma_n^2$, $p_a$}\\
\STATE {$ \triangleright $ CS Phase: \\\quad Joint DAD-CE Algorithm}\\
\STATE{\quad $ \triangleright $ Collision Resolution Protocol}\\
\STATE{\quad Output: $\bm{\mu}^{dec}$, $\bm{\Sigma}^{dec}$, $\mathcal{L}= \left\{ {{i_k},k \in \mathcal{K}} \right\}$}\\
\STATE {$ \triangleright $ Joint DAD-CE-DD Algorithm:}\\
\STATE{\quad Initialize: $\widetilde{\bm{Y}}_r= \bm{Y}_c$, $\widetilde{\mathcal{V}}=\emptyset$}, $\widetilde{\mathcal{K}}=\emptyset$\\
\STATE {\quad \bf repeat}
\STATE {\quad \quad $ \triangleright $ LDPC Phase:}
\STATE{\quad \quad \quad Input: $\widetilde{\bm{Y}}_r$, $\bm{\mu}^{dec}$, $\mathcal{L}$, $\sigma_n^2$}\\
\STATE{\quad \quad \quad MIMO-LDPC-SIC Decoding Algorithm}\\
\STATE{\quad \quad \quad Output: ${\widehat {\mathcal{V}}_0}$, ${\widehat {\mathcal{K}}_0}$}\\
\STATE{\quad \quad{${\widetilde {\mathcal{V}}} \leftarrow {\widetilde {\mathcal{V}}} \cup \widehat {\mathcal{V}}_0$, ${\widetilde {\mathcal{K}}} \leftarrow {\widetilde {\mathcal{K}}} \cup \widehat {\mathcal{K}}_0$}}
\STATE{\quad \quad $\widetilde{\bm{Y}}_c = \sum\nolimits_{k \in {{\widehat {\mathcal{K}}}_0}} {{\pi _{{i_k}}}\left( {{\tilde{\mathbf{s}}_k}} \right){\bm{\mu}^{dec}_k }}+ \bm{Z}$}\\
\STATE{ \quad \quad $ \triangleright $ CS Phase:}\\
\STATE{\quad \quad \quad Input: $\bm{Y}_p$, $\bm{A}$, $\widetilde{\bm{Y}}_c$, ${\widehat {\mathcal{V}}_0}$, $\bm{\mu}^{dec}$, $\bm{\Sigma}^{dec}$, $\sigma_n^2$}\\
\STATE{\quad \quad \quad CE (Activity fixed)}\\
\STATE{\quad \quad \quad $ \triangleright $ Collision Resolution Protocol }\\
\STATE{\quad \quad \quad Output: $\widetilde{\bm{\mu}}^{dec}$, $\widetilde{\bm{\Sigma}}^{dec}$ }\\
\STATE{\quad \quad $\widetilde{\bm{Y}}_r = {\mathbf{Y}_c} - \sum\nolimits_{k \in {{\widetilde {\mathcal{K}}}}} {{\pi _{{i_k}}}\left( {{\tilde{\mathbf{s}}_k}} \right){\widetilde{\bm{\mu}}^{dec}_k} }$}
\STATE {{\quad \bf until} ~ ${\widehat {\mathcal{K}}}_0 = \emptyset $ or ${\mathcal{K}} \backslash {\widetilde {\mathcal{K}}} =\emptyset$}\\
\STATE{{\bf Return:} $\widetilde{\mathcal{V}}$, $\widetilde{\mathcal{K}}$, $\mathcal{L}= \left\{ {{i_k},k \in \widetilde{\mathcal{K}}} \right\}$, $\widetilde{\bm{\mu}}^{dec}$}
\end{algorithmic}
\end{algorithm}
\emph{Remark 1:} We remark that messages in the MP algorithm exhibit the property of consistency and unity. As such, messages among different parts can always be jointly processed and updated. For instance, recalling the Joint DAD-CE algorithm, where the Bernoulli messages are updated by utilizing the Gaussian messages. Likewise, in the MIMO-LDPC-SIC Decoding algorithm, the MP algorithm can be applied to MIMO detection or LDPC decoding. In the proposed algorithm, we combine these two parts and update the underlying messages jointly, i.e., the LLR message of the symbol in MIMO detection can be involved in the LDPC decoding process, and vice versa. Moreover, the property is again exploited in the Joint DAD-CE-DD algorithm. By leveraging the belief of the estimated channel and the correctness of the LDPC codewords, the CE can be again performed aided with the correctly decoded codewords in the LDPC phase, thus connecting these two phases. Throughout the paper, we take into consideration the idea of the joint update for the messages in MP-based algorithms.
\par One might argue that for the correctly decoded messages, improving the accuracy of the corresponding channels will not bring substantial performance improvements. Nevertheless, according to the SIC method in \eqref{equ-55}, by improving the channel accuracy of those devices whose messages are successfully decoded, the residual signal of the incorrectly decoded messages can be obtained more accurately. We denote the residual signal with improved accuracy as $\widetilde{\bm{Y}}_r$. Consequently, $\widetilde{\bm{Y}}_r$ will improve the performance for those messages that have not been successfully decoded yet. And this is the reason why the channel needs to be estimated twice or more times. We give a brief diagram for this algorithm in Fig. \ref{pic-8}, where $\widetilde{\bm{Y}}_c$ denotes the noisy signal reconstructed from the correctly decoded messages.
\begin{figure}[htpb]
\centerline{\includegraphics[width=0.5\textwidth]{pic-8}}
\caption{The diagram of the joint DAD-CE-DD algorithm.}
\label{pic-8}
\end{figure}
\par We note this data-aided CE algorithm appears to be similar to the work in \cite{hiroki2020arxiv}, which employs the data as soft pilots to conduct CE jointly with the real pilot sequences. However, the convergence and correctness of the estimated data cannot be verified, which may result in the propagation of errors and failure of the joint data and channel estimation. Whereas the convergence of the proposed algorithm can be guaranteed. The proposed Joint DAD-CE-DD algorithm consists of two modules, namely, the Joint DAD-CE algorithm for activity detection and channel estimation, and the MIMO-LDPC-SIC Decoding algorithm for data decoding. Messages are passed between these two modules. For arbitrary channel estimation results as input, the decoding process will be executed by the MIMO-LDPC-SIC Decoding algorithm. Once a codeword is successfully decoded (i.e., satisfies the check), it will not be changed in subsequent iterations. Therefore, the number of correctly decoded codewords is monotonically non-decreasing at each iteration. Correspondingly, the error rate $P_e$ is monotonically non-increasing, and also, it is bounded below by zero. According to the well-known monotone convergence theorem \cite{converge}, the proposed algorithm is guaranteed to converge. Besides, with the correctly decoded codewords aided to estimate the channel, an improved CE accuracy can be guaranteed and it contributes to a higher probability of successfully decoding for those who have not yet. This iterative algorithm ends when no more messages are correctly decoded or all the messages are decoded successfully, which is summarized in Algorithm \ref{alo-4}.
\subsection{Complexity}
\par In this section, we compare the complexity of the proposed algorithm with the existing alternative in terms of the real-number multiplication (or division), addition (or subtraction), and other operations (eg., exp, log, tan, $\text{tan}^{-1}$), as shown in Table \ref{tab-2}. As a divide-and-conquer strategy, there are inner and outer decoders in the CB-ML algorithm, referred to as ML and tree decoder, respectively. The complexity of our algorithm is mainly incurred by the Joint DAD-CE algorithm and MIMO-LDPC decoder. In Table \ref{tab-2}, the parameters $\alpha$ and $c$ satisfy that $B_p=\alpha \log_2K_a, L_p=c\log_2K_a$, respectively. We note that the complexity of the tree decoder is defined by the total number of parity check constraints that need to be verified \cite{Amalladinne2020tit}, and this is obtained in the regime that $B_p$ and $K_a$ tend to infinity. Nevertheless, in practice, it increases exponentially with the number of slots $S$. Besides, the tree code in \cite{Fengler2021tit} verifies the estimated codewords in each segment successively, and the check and estimate are performed separately. While during the LDPC decoding process in our algorithm, the check and estimate of the codeword are performed jointly and simultaneously, both by calculating the soft messages. Once the codeword is estimated, the corresponding check is completed. As for the inner code, the complexity of the ML decoder is nearly the same order as that of our algorithm. However, as a consequence of the coordinate descent algorithm, there are $2^{B_p}$ cycles in the ML decoding per iteration, which can only be computed successively. On the contrary, as a remarkable property of the MP algorithm, all computations in our algorithm can be decomposed into many local ones, which can be performed in parallel over the factor graph. Hence, our algorithm has lower time complexity. In general, the proposed algorithm exhibits a complexity linear with $K, M, L$, which is lower than the existing CB-ML algorithm.
\begin{table*}[!t]
\caption{COMPLEXITY COMPARISON PER ITERATION }
\renewcommand{\arraystretch}{1.7}
\setlength{\tabcolsep}{4.2mm}{
\begin{tabular}{l|l|l|l|l}
\hline
\multicolumn{2}{l|}{\bf Algorithms} & {\bf Real Multi. / Div.} &{ \bf Real Add. / Sub. }& {\bf Others } \\
\hline
\multirow{2}{*}{\tabincell{c}{CB-ML}} & ML decoder& $\mathcal{O}(KL_p^2)$ & $\mathcal{O}(KL_p^2)$ & $\backslash$\\
\cline{2-5}
& Tree decoder & \multicolumn{2}{l|}{$\mathcal{O}\left(K_a^{\alpha/c}\text{log}_2K_a\right), B_p, K_a \rightarrow \infty $} & $\backslash$ \\
\hline
\multirow{2}{*}{\tabincell{c}{Joint DAD-CE-DD}} &Joint DAD-CE & $\mathcal{O}(KML_p)$ & $\mathcal{O}(KML_p)$ & exp / log: $\mathcal{O}(KL_p)$\\
\cline{2-5}
& MIMO-LDPC& $\mathcal{O}( \widehat{K}M\widetilde{L}_c) $ & $\mathcal{O}( \widehat{K}M\widetilde{L}_c) $ & exp: $\mathcal{O}( \widehat{K}M\widetilde{L}_c) $, tanh / $\text{tanh}^{-1}$: $\mathcal{O}(\widehat{K}NN_v)$ \\
\hline
\multicolumn{5}{l}{\tabincell{l}{Note: $K=2^{B_p}$, $\widehat{K}, N, N_v$ denote the estimated number of active devices, the rows and row weights of the parity check matrix, respectively.}}
\end{tabular}}
\label{tab-2}
\end{table*}
\section{Numerical Results} \label{simulation}
\subsection{Parameter Settings}
\par In this section, we assess the overall performance of the proposed framework with the metric defined in \eqref{equ-3}-\eqref{equ-4}. The CB-ML proposed by Fengler et. al. in \cite{Fengler2021tit} serves as the baseline in this paper. For the sake of fair comparison to the benchmarks, and isolating the fundamental aspects of the problem without additional model complication, we consider the flat path loss model in the simulation, i.e., the channel is i.i.d. Rayleigh fading model and the LSFC is fixed to $\beta_k=1$ in all schemes. We note again that this can be achieved by the well-studied power control schemes \cite{Chandrasekhar2009twc,Patel1994jsac,Turin1984jsac} in practice. As such, the carrier frequency is not specified in the simulation, since it can be arbitrary and does not affect the performance of the proposed algorithm, when considering flat path loss model. The user distribution is considered to be uniformly distributed in the cell. The noise variance is set to $\sigma_n^2=1$ and is known to all schemes. The maximum scheduling times for the collision resolution protocol is $t_{max}=3$. For the joint DAD-CE algorithm, MIMO-LDPC-SIC decoder, and Joint DAD-CE-DD algorithm, the maximum number of iterations is set to $N_{iter}=20$, $30$, and $20$, respectively. We fix the message length to $B=96$ with $B_p=12$ and $B_c=84$. The length of the CS codeword is $L_p=100$ in both schemes. In the channel coding part, we employ the $(3,6)$-regular LDPC code \cite{LDPC} with the rate $0.5$. In our scheme, the length of channel use varies with the number of zeros padded in the sequence. However, it is discretely valued in the CB-ML scheme and changes with the number of slots denoted as $S$. Since the length of the information is fixed, we can change $S$ by adjusting the parity check allocation, which is given in Table \ref{tab-1} according to the principle in \cite{Amalladinne2020tit}. We note that the slot length $J=12$ aligns with $B_p$.
\begin{table}[h]
\centering
\caption{The parity check allocation for different number of slots.}
\setlength{\tabcolsep}{8mm}{
\begin{tabular}{l l}
\toprule
$S$ & Parity Check Allocation \\
\midrule
12 & (12,0), (3,9), (9,3), $\cdots$, (9,3), (0,12)\\
\specialrule{0em}{0.5pt}{0.5pt}
13 & (12,0), (4,8), (8,4), $\cdots$, (0,12)\\
\specialrule{0em}{0.5pt}{0.5pt}
14 & (12,0), (7,5), (7,5), $\cdots$, (7,5), (0,12)\\
\specialrule{0em}{0.5pt}{0.5pt}
15 & (12,0), (7,5), (7,5), $\cdots$, (7,5), (0,12), (0,12)\\
\specialrule{0em}{0.5pt}{0.5pt}
16 & (12,0), (6,6), (6,6), $\cdots$, (6,6), (0,12)\\
\specialrule{0em}{0.5pt}{0.5pt}
17 & (12,0), (6,6), (6,6), $\cdots$, (6,6), (0,12), (0,12)\\
\bottomrule
\end{tabular}}
\label{tab-1}
\end{table}
\par In Table \ref{tab-1}, (3,9) means that the first three are data bits and the last nine are parity bits and so on. For more details, we refer the reader to \cite{Fengler2021tit} for CB-ML algorithm and \cite{Amalladinne2020tit} for the tree coding scheme.
\subsection{Results}
\begin{figure}[htpb]
\centerline{\includegraphics[width=0.46\textwidth]{pic-simu-1}}
\caption{Performance of the proposed URA schemes as a function a the code rate. $E_b/N_0=10$ dB, $M=30$, $K_a=50$.}
\label{pic-9}
\end{figure}
\begin{figure*}[htpb]
\centerline{\includegraphics[width=1.0\textwidth]{pic-simu-4}}
\caption{Performance of the proposed URA schemes as a function of $E_b/N_0$, the number of antennas $M$ and active devices $K_a$, respectively. Parameter settings: $E_b/N_0=10$ dB, $M=30$, $K_a=50$, $L=1600$.}
\label{pic-12}
\end{figure*}
\par We first evaluate the error rate performance of the proposed schemes compared with the CB-ML scheme as a function of the code rate. The relationship between the code rate $R_c$ and channel use $L$ is $R_c=B/L$. In Fig. \ref{pic-9}, we fix the energy-per-bit $E_b/N_0=10$ dB, the number of antennas and active devices $M=30$ and $K_a=50$, respectively. In our schemes, No-colli-avoid refers to the scheme that there is no collision avoidance or the joint update. That is, the CS and LDPC phases work sequentially and potential collision may exist in the CS phase. For the other three schemes, the collision resolution protocol has been implemented and No-SIC denotes that the SIC method is not utilized in the LDPC phase nor do the two phases work jointly. Joint-BPSK and Joint-QPSK refer to the joint update algorithm with the SIC method in BPSK and QPSK modulations, respectively. As illustrated in Fig. \ref{pic-9}, there is a substantial performance enhancement compared to the CB-ML algorithm. The main reason is that the employed LDPC code has a higher code rate than the tree code proposed in \cite{Amalladinne2020tit}. As such, the proposed algorithm can work well in a relatively high rate region while the CB-ML algorithm can not. Once we set a high rate, there will be a substantial performance enhancement. However, this improvement decreases with the decrease of rate. For the target error rate $P_e=0.1$, the required code rate of the proposed Joint-BPSK is increased by $1.45$ times that of CB-ML, while the Joint-QPSK increases even more. For instance, the proposed Joint-QPSK and Joint-BPSK outperform CB-ML with a nearly $0.8$ dB gap and a $1.5$ dB gap at $R_c=0.06$, respectively. Additionally, we note that Joint-QPSK exhibits an overall $0.7$ dB performance gain compared with Joint-BPSK in terms of the error rate. This is because only half of the channel use is needed for the QPSK than BPSK modulation and thus more zeros can be padded into the sequence. Consequently, the interference of devices in the QPSK system is further reduced, resulting in improved performance. Altogether, the gain of the collision resolution protocol increases with the decrease of the code rate. However, there is no much gain for the work of joint update. This is because the gain mainly comes from the reduction of interference of devices, which has been reduced to a relatively low level by the zero-padding. As such, the joint update cannot provide more gains. Nevertheless, as we will see shortly, there is a certain gain by the joint update under a higher interference scenario.
\par For the sake of complexity reduction, we employ BPSK modulation instead of QPSK in the subsequent simulations since both of them have already outperformed CB-ML. In Fig. \ref{pic-12}, we compare the error rate performance of the proposed algorithms with respect to $E_b/N_0$, the number of antennas $M$ and active devices $K_a$, respectively. The number of channel uses is fixed to $L=1600$. Correspondingly, the data is split into $16$ slots in the CB-ML algorithm, of which the parity check allocation is given in Table. \ref{tab-1}. The other parameters are set as $E_b/N_0=10$ dB, $M=30$ and $K_a=50$. As illustrated in Fig. \ref{pic-12}, the state-of-the-art method CB-ML suffers from high error floors, which stems from the poor parity check constraints. In contrast, with collision resolution, the proposed Joint and No-SIC schemes exhibit water-falling curves in terms of the error rate with respect to $E_b/N_0$ and $K_a$, while they gradually stabilize with respect to $M$. This is because the interference of devices cannot be reduced to infinitesimal by increasing $M$. As such, error still exits even for a large $M$.
\begin{figure}[htpb]
\centerline{\includegraphics[width=0.44\textwidth]{pic-simu-8}}
\caption{Performance of the proposed Joint-DAD-CE-DD algorithm as a function of $M$ and channel use. Parameter settings: $E_b\slash N_0 = 8$ dB, $K_a = 70$, $L_p=100$.}
\label{pic-13}
\end{figure}
\par Moreover, we evaluate the performance of the proposed Joint-DAD-CE-DD algorithm under a large-scale antenna with respect to different channel uses. As showcased in Fig. \ref{pic-13}, the performance of the proposed algorithm is almost linear with the antennas, but with different slopes under different channel uses. We note that CB-ML cannot work at $L=1600$, while the Joint-DAD-CE-DD algorithm can achieve $P_e < 10^{-4}$ when $M>140$. Besides, the proposed algorithm only needs half of the channel uses, i.e., $L=800$ to outperform CB-ML when $M\geq90$. To sum up, the proposed algorithms outperform CB-ML in terms of error rate and spectral efficiency, with an explicit gain with respect to various variables.
\begin{figure}[h]
\centerline{\includegraphics[width=0.43\textwidth]{pic-simu-2}}
\caption{Error rate of the proposed URA schemes as a function of $E_b/N_0$. $M=30$, $K_a=40$, $L=268$, $R_c=0.36$.}
\label{pic-10}
\end{figure}
\par In order to provide more insights about the performance gain of different methods, we compare the error rate and CE performance among the methods in terms of $E_b/N_0$. In this regard, we fix the channel use to $L=268$ with BPSK modulation and no zero is padded, resulting in the increase of interference among devices. Besides, the code rate $R_c=0.36$, which is relatively high and the CB-ML cannot work under this circumstance. In Fig. \ref{pic-10}, No-Joint refers to the scheme with the SIC method but no joint update. Firstly, it is obvious that the collision resolution-based schemes all outperform the No-colli-avoid scheme with the increase of $E_b/N_0$. Then, the performance gain of the SIC method in the LDPC phase is about $0.3$ dB or even more when $E_b/N_0$ increases according to the comparison of No-SIC and No-Joint. Finally, by the comparison of No-Joint and Joint, there is also about $0.3$ dB gain coming from the work of joint update and it decreases with the increase of $E_b/N_0$. At a higher $E_b/N_0$, such as $20$ dB, there is nearly no performance gain by the joint update. This demonstrates that under a higher interference level of devices, the joint update algorithm can provide a certain gain, which gradually decreases with the increase of $E_b/N_0$.
\begin{figure}[htpb]
\centerline{\includegraphics[width=0.44\textwidth]{pic-simu-3}}
\caption{NMSE of CE with the proposed URA schemes as a function of $E_b/N_0$. $M=30$, $K_a=40$, $L=268$, $R_c=0.36$.}
\label{pic-11}
\end{figure}
\par The NMSE is employed to evaluate the CE performance. In Fig. \ref{pic-11}, the AMP algorithm investigated in \cite{Liu2018tsp} is utilized as a baseline for comparison. When $E_b/N_0$ ranges from $13$ to $15$ dB, the performance of Joint and No-Joint is of slight difference and both are slightly worse than AMP. However, the Joint scheme outperforms AMP with an ultra-linear speed when $E_b/N_0$ exceeds $15$ dB. As aforementioned, this gain exactly comes from the Joint-DAD-CE-DD algorithm, where the real pilot sequences as well as correctly decoded messages jointly conduct the task of CE. Figs. \ref{pic-10} and \ref{pic-11} demonstrate that the joint update indeed contributes to the improved accuracy of both the DD and CE. In general, the proposed algorithms provide substantial performance enhancements compared with CB-ML in terms of efficiency and accuracy.
\section{Conclusion} \label{sec-7}
\par In this paper, we have investigated a joint DAD, CE, and DD algorithm for MIMO massive URA. Different from the state-of-the-art slotted transmission scheme, the data in the proposed framework has been split into only two parts. A portion of the data is coded by CS and the rest is LDPC coded. Based on the principle of the BP, the iterative MP algorithm has been utilized to decode these two parts of data. Moreover, by exploiting the concept of the belief within each part, a joint decoding framework has been proposed to further improve the performance. Additionally, based on the ED and SWP, a collision resolution protocol has been developed to resolve the codeword collision issue in the URA system. In addition to the complexity reduction, the proposed algorithm has exhibited a substantial performance enhancement compared to the state-of-the-art in terms of efficiency and accuracy. The possible avenues for future work are various. More realistic channel models can be taken into consideration for MIMO URA. By utilizing the sparsity in the virtual angular domain of the spatially correlated channel, the multi-user interference can be further reduced. In addition, the proposed algorithm can be extended to handle more practical scenarios in the presence of asynchronization and frequency offset. Moreover, although this paper only studies LDPC codes, other codes, such as Turbo codes and Polar codes, could be applied if iterative soft decodings with superior performance exist in a desired block length.
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2,877,628,089,019 | arxiv | \section{Introduction}
Perturbation theories of gravity have been very broadly used in both theoretical and experimental explorations during the past few decades. In the theoretical research, linear order perturbation theories have been frequently applied in analyzing the stability issues of different kinds of cosmological solutions and black holes \cite{Wald73,Eath76}, and perturbation theories on FRW cosmological background play an important role in explaining the small inhomogeneities of the cosmological microwave background which we have observed today \cite{Mukhanov92,Ma95,Noh04}. Combined with the inflation hypothesis, cosmological perturbation theories enable us to gain a possible deep insight into the mysterious early universe \cite{Lyth99}.
So far most of the perturbation analysis were done in the context of general relativity (GR). From the viewpoints of the solar system experiments, GR is currently the best candidate theory of gravity. Nevertheless, for a variety of reasons \cite{Wesson80}, alternative theories of gravity never cease to exist even from the day GR was proposed and attracted more and more attention in the last twenty years ever since the anomalous galaxy rotation curves and cosmological acceleration were observed \cite{Zwicky37,Riess98}. There are usually two ways to explain these two phenomenons. One is to introduce some unknown ``dark matter" and ``dark energy" components which interact with other fields only through gravity \cite{Navarro96,Copeland06}. The other is to modify the gravity theory. While to add a cosmological constant into GR could explain the cosmological acceleration, the extremely tiny value of this constant suffers from the ``fine tuning problem" \cite{Zlatev99}. To avoid this problem and also due to some other considerations, many people turned to look for modified gravity theories \cite{Nojiri07}. Among all the reasonable attempts, scalar-tensor theories (STT) of gravity are the ones that receive most attention. On one hand, STT provide the great possibility to solve both the anomalous rotation curve and cosmological acceleration problems \cite{Catena04,Elizalde04}. On the other hand, STT can include a lot of modified gravity models as its special sectors, such as $f(R)$ theory, Brans-Dicke theory, induced gravity, etc \cite{Hwang97,Fujii03}. What is worth mentioning is that, when studying the cosmological perturbations during the slow-roll inflation period of STT, people noticed that the physics there behaves much different from that of GR \cite{Hwang96,Weenink10}. Some predictions made by $f(R)=R+\alpha R^2$ gravity and certain STT theories can predict the scalar spectral index and the tensor-to-scalar ratio \emph{completely} consistent with the Planck data \cite{Ade13}. A question that cannot be answered in the context of STT is how matter couples to gravity, but is usually treated as an additional freedom of the model under consideration. The two most prominent choices for STT are the Einstein and the Jordan frame, which are related by a conformal transformation, and which will be also chosen in this paper. A systematic approach how alternative actions for gravitational theories can be derived once the information how matter couples to gravity is known, can for instance be found in \cite{Giesel2012}.
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GR can be understood as a gauge theory with the gauge group being the diffeomorphisms $Diff(M)$ of the spacetime manifold. Likewise to other gauge theories physically relevant quantities, called observables, are those that are gauge invariant, and in the context of GR this corresponds to diffeomorphism invariant quantities. In the canonical formalism, that is used in this paper, this carries over to the requirement that observables, also denoted as Dirac observables in this context, are tensors on phase space that do (weakly) Poisson commute with all constraints of the system, and particularly for GR in the ADM formalism this means, with the Hamiltonian and spatial diffeomorphism constraints. The elementary variables in the ADM formalism are the ADM 3-metric, given by the pull back of the spacetime metric onto the spatial hypersurfaces, and its conjugate momenta. The canonical Hamiltonian density consists of the sum of the spatial diffeomorphism and Hamiltonian constraints, weighted by the so called shift vector and lapse function respectively, which parametrize possible foliations into spatial hypersurfaces of the four dimensional spacetime manifold. These configuration and momentum variables are no observables, but their Poisson brackets with the canonical Hamiltonian, and hence with the constraints, together with the constraints themselves yield the analogue of Einstein's equations in the canonical framework. As a consequence, when we discuss observables in the context of GR, the natural question arises, how the evolution of those observables can be described. Certainly, it cannot be generated by the canonical Hamiltonian, since this would allow only trivial evolution contradicting what we observe, because we do observe the evolution of physical observables in our everyday life and in experiments. Thus, what we are looking for is a so called physical Hamiltonian having the property that it generates the evolution of the observables with respect to physical time in the canonical setup of GR. The initially missing physical time and along with this the missing physical Hamiltonian is a common feature of any diffeomorphism-invariant theory and is often called the "problem of time" for short. Many attempts have been made to clarify this conceptual issue in the history, see \cite{Ashtekar13} for some examples. One possible way to circumvent the problem of time was introduced by Rovelli \cite{Rovelli91} and mathematically further developed by Dittrich \cite{Dittrich04} in the context of so called relational observables. The main idea of relational observables is to introduce so called reference fields with respect to which the observables and the evolution of the remaining degrees of freedom are formulated. These reference fields could for instance be additional matter fields but also purely gravitational degrees of freedom. In a seminal paper by Brown and Kucha{\v r} \cite{Brown95} they considered to use pressureless dust particles as such reference fields, which were introduced as additional matter fields. The dust particles can be understood as a free falling observer that is dynamically coupled to the system and with respect to which observables for GR are constructed. The dust fields therefore serve as a physical reference system and provide an interpretation of physical spatial and time coordinates. In \cite{Brown95} observables were constructed explicitly only with respect to the spatial diffeomorphism constraint and for the Hamiltonian constraint a corresponding Schr\"odinger-like equation was discussed within the quantum theory. A combination of the methods of Brown and Kucha{\v r} and the relational observable framework was used in \cite{Giesel10} to derive a manifestly gauge invariant Hamiltonian formulation of GR. The corresponding physical Hamiltonian associated with the dust observer then generates evolution with respect to this particular chosen physical time. As an application in \cite{Giesel10} the manifestly gauge invariant Hamiltonian formalism was used in order to develop a formalism for general relativistic perturbation theory. The obtained results were then in \cite{Giesel10s} applied to the context of cosmological perturbation theory for which a comparison with the results of the standard formalism, in the following denoted as standard cosmological perturbation theory (SCPT), was analyzed for linear order perturbation theory. It turns out that they are consistent with each other only up to small corrections. The latter occur due to the reason that in the manifestly gauge invariant formalism the observer is dynamically coupled to the system and thus has an influence on the system, whereas this is not the case in SCPT.
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The main difference between the manifestly gauge invariant and the standard formalism lies in the way how gauge invariant quantities and thus observables are constructed. In the manifestly gauge invariant formalism first one constructs observables and the associated physical Hamiltonian, which itself is an observable. Then one derives the equations of motion for all observables and afterwards considers the perturbation of these evolution equations. By construction any quantity and their corresponding perturbations, which are involved in these evolution equations are themselves observables and hence gauge invariant. Following SCPT the strategy is different since perturbations of gauge variant quantities, the metric and and their equations of motion, are considered. Gauge invariant objects, and hence observables, are constructed afterwards and are required to be only gauge invariant up to the corresponding order of perturbation theory one is interested in. The manifestly gauge invariant formalism might be of advantage when higher order perturbation theory is considered because in SCPT one has to start from scratch again when gauge invariant quantities are constructed \cite{Nakamura05}, while using the manifestly gauge invariant framework any object and thus also higher order perturbations are already gauge invariant.
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In this paper we extend this manifestly gauge invariant formalism to the case of STT.
This paper is organized as follows. In section two, after introducing the Brown-Kucha{\v r} formalism, relational observables and the notion of a physical Hamiltonian, we construct the physical Hamiltonian in Jordan frame of STT and derive the second order evolution equations of the canonical variables. In section three the latter will be used to derive the evolution equations for linear perturbations in the context of a general relativistic spacetime. In section four we apply these equations to flat FRW background and get the cosmological perturbation equations. In the first part of section five we formulate the gauge invariant Hamiltonian formalism in Einstein frame of STT and prove it is conformally equivalent to that of Jordan frame. In the second part we extend our results and study the cosmological perturbations in a different reference system. Our results are then compared with the standard cosmological perturbation theory of STT. In the last section we summarize the above results and draw some conclusions. In the appendix the Lagrangian and Hamiltonian analysis of the action of STT in Einstein frame is presented.
\section{Manifestly Gauge Invariant Hamiltonian Formalism of STT}
For the convenience of readers, in the first part of this section, we will first give a brief review of the idea proposed by Brown and Kucha{\v r}. Then we introduce how to combine the Brown-Kucha{\v r}'s dust formalism with the idea of relational observables to build the physical reference system and the gauge invariant Hamiltonian which generates the evolution of every observable in this physical reference system. We refer to \cite{Giesel10,Giesel07} for the details. In the second part we will apply these ideas to STT of gravity. First we derive the physical Hamiltonian in the Jordan frame of STT and then use it to derive the evolution equations of the 3-metric and gravitational scalar field.
\subsection{The Brown -- Kucha{\v r} Lagrangian and Relational Observables}
\label{BKFormalism}
In this subsection we will briefly review the idea of Brown and Kuch{\v ar} to use dust as reference system for general relativity and discuss how this idea can be embedded in the framework of relational observables. The generalization from general relativity to STT is not difficult and will be discussed in section \ref{EvEqnJordan}.
\subsubsection{Brown -- Kucha{\v r} Lagrangian and Deparametrization of General Relativity}
In \cite{Brown95} Brown and Kucha{\v r} followed the idea that matter fields can be chosen as a physical reference systems. For the reason that they were mainly interested in GR they considered the Einstein-Hilbert action and the following additional matter action
\begin{equation}
\label{SBKaction}
S_{dust}=-\frac{1}{2}\int\limits_{M}d^4x\sqrt{|\det(g)|}\rho\left(g^{\mu\nu}U_{\mu}U_{\nu}+1\right)
\end{equation}
which, as we will see below, can be interpreted as an action for pressureless dust.
The action $S_{dust}$ is not taken as a functional of the one form $U$ but the latter is expressed in terms of the scalar fields $T,S^j,W_j$ defined through $U=-dT+W_jdS^j$ where we use the notation that Latin letters run from $1$ to $3$ and Greek letters from $0$ to $4$. Hence, the action above is a functional of the fields $\rho, T, S^j, W_j$ and $g_{\mu\nu}$ and hence in addition to GR we have introduced eight more degrees of freedom. As we will see below the system GR+dust has second class constraints and once these are reduced the additional number of degrees of freedom is reduced to four. As discussed in detail in \cite{Brown95} the Euler-Lagrange equations associated with this action show that the vector field $U^\mu=g^{\mu\nu}U_\nu$ satisfies the differential equation of a geodesic in affine parametrization. The fields $W_j$ and $S^j$ are constant along the geodesics and the field $T$ defines proper time along each geodesic. Furthermore, the energy-momentum tensor associated with $S_{dust}$ has the form of the energy-momentum tensor of a pressureless perfect fluid. In order to use the dust matter as a dynamically coupled observer in the canonical framework we have to discuss the Hamiltonian formulation of (\ref{SBKaction}), which is discussed in detail in \cite{Giesel10}. We assume that $M$ is globally hyperbolic and thus we can perform a 3+1-split of $M\simeq \mathbb{R}\times \chi$ into time and space, with $\chi$ being a spatial manifold of arbitrary topology. For this purpose we introduce a family of embeddings $X_t : \chi\to M$ $x \mapsto X_t(x):=X(t,x)$, also called a foliation of $M$, where $\chi_t:=X_t(\chi)$ are called the leaves of the foliation, where we denoted the coordinates on $\chi$ by $x^a$ with $a=1,2,3$. Given the family of embeddings $X_t$ we can construct a family of tangent vectors $X^\mu_{t,a}$ and a co-normal $n_{\mu}$ for each leave. Using the metric we can also work with the future orientated normal $n^\mu$ and can use it to decompose the variation of the embeddings with respect to the parameter $t$ into a tangential and normal part to the leaves $\chi_t$ given by $\partial_t X^\mu_t=nn^\mu+n^a X^{\mu}_{t,a}$ where this decomposition is parametrized by the so called lapse function $n$ and shift vector $n^a$. The spatial three metric $q_{ab}$ intrinsic to $\chi$ can be constructed by pulling back $g_{\mu\nu}$ using the tangent vectors yielding $q_{ab}=g_{\mu\nu}X^\mu_{,a}X^{\nu}_{,b}$. In order to derive the Hamiltonian formulation via the Legendre transformation we introduce conjugate momenta $p^{ab},p,p_a,P,P_j,I,I^j$ associated with the configuration variables $q_{ab},n,n^a,T,S^j,\rho,W_j$. Note that in case we would like to couple additional standard model matter to gravity we would need to introduce additional phase space variables for this matter, which we will not display explicitly here, but we will mention below how such additional matter degrees of freedom enter the model. The theory possesses the following primary constraints
\begin{equation}
z:=p=0,\quad z_a:=p_a=0,\quad Z:=I=0,\quad Z^j:=I^j=0,\quad Z_j:=P_j-PW_j=0.
\end{equation}
Following the Dirac procedure for constrained systems the stability analysis of the primary constraints with respect to the primary Hamiltonian yields the following secondary constraints
\begin{eqnarray}
c^{tot} &=& c+c^{dust},\;c^{dust}=\frac{1}{2}\left[\frac{P^2}{\rho\sqrt{\det(q)}}
+\rho\sqrt{\det(q)}(1+q^{ab} U_a U_b)\right],
\nonumber\\
c^{tot}_a &=& c_a+c_a^{dust},\;c_a^{dust}=P[T_{,a}-W_j S^j_{,a}],
\nonumber\\
\tilde{c} &=& \frac{n}{2}\left[-\frac{P^2}{\rho^2\sqrt{\det(q)}}
+\sqrt{\det(q)}(1+q^{ab} U_a U_b)\right],
\end{eqnarray}
here $c$ denotes the gravitational contribution as well as the contribution from other possible standard model matter to the usual Hamiltonian constraint and $c_a$ denotes the contributions from gravity as well as other possible standard model matter to the usual spatial diffeomorphism constraint. Furthermore we have $U_a=-T_{,a}+W_jS^j_{,a}$. A further application of Dirac's constraint algorithm shows that no tertiary constraints occur and thus the set of constraints has completely been determined and is given by $\{c^{tot},c_a^{tot},\tilde{c},z,z_a,Z,Z^j,Z_j\}$. The next step is to classify them into first and second class constraints. The constraints $\{\tilde{c},Z,Z^j,Z_j\}$ form second class constraints. We will solve them explicitly below using the associated Dirac bracket. Furthermore we will also solve the constraints $z,z_a$ by considering the lapse function $n$ and the shift vector $n^a$ as Lagrange multipliers. Now we extend $c^{tot}$ and $c_a^{tot}$ to ${\cal C}^{tot}$ and ${\cal C}_a^{tot}$ by adding terms proportional to the constraints $z,z_a,Z,Z^j,Z_j$ in the case of $c_a^{tot}$ and terms proportional to $Z,Z_j,Z^j$ in the case of $c^{tot}$. These terms are exactly chosen such that ${\cal C}^{tot}$ and ${\cal C}_a^{tot}$ are first class constraints, see \cite{Giesel10} for more details. Since $z_a,z$ are first class constraints as well we end up with the following set of first class constraints $\{{\cal C}^{tot},{\cal C}_a^{tot},z,z_a\}$. Now solving the second class constraints strongly leads to
\begin{equation}
\label{Sol2nd}
W_j:=\frac{P_j}{P},\quad \rho^2:=\frac{P^2}{\sqrt{q}}\left(q^{ab}U_aU_b+1\right),\quad I:=0,\quad I^j:=0.
\end{equation}
For the reason that $Z,Z^j$ and $Z_j$ do only depend on the dust variables the associated Dirac bracket reduces to the Poisson bracket when applied to the geometrical degrees of freedom $q_{ab}, p^{ab}$ and other possible standard model degrees of freedom. When we introduce the Dirac bracket and also solve the constraints $z,z_a$ by considering the lapse function and shift vector as Lagrange multipliers, as usually done the ADM framework, we work on a reduced phase space where ${\cal C}^{tot}=c^{tot}$ and ${\cal C}_a^{tot}=c_a^{tot}$. Inserting the explicit solutions of the second class constraints from (\ref{Sol2nd}) we end up with the following first class constraints
\begin{eqnarray}
c^{tot}&=&c+c^{dust},\quad\quad c^{dust}=-\sqrt{P^2+q^{ab}c_a^{dust}c_b^{dust}}, \\
c_a^{tot}&=&c_a+c_a^{dust},\quad\quad c_a^{dust}=PT_{,a}+P_jS^j_{,a}.
\end{eqnarray}
Note that in principle we have two possible choices for the sign of $\rho$ here but the one chosen by us $\rho<0$ yielding also to $P<0$ is the part of the phase space that involves also flat space solutions since for them $c>0$ is necessary, for a more detailed discussion about this aspect see \cite{Giesel10}.
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The Hamiltonian and spatial diffeomorphism constraints satisfy a complicated Poisson algebra also called the hypersurface deformation algebra for the reason that it can also be derived from purely geometrical considerations in the context of deformation of hypersurfaces \cite{Hojman1976}. One of the motivations in \cite{Brown95} to introduce the dust as a reference systems was that using the dust one can write down an equivalent set of first class constraints that has the property that the corresponding constraint algebra becomes Abelian and the final Hamiltonian constraint can be written in deparametrized form, as we will discuss below. The important observation by Brown and Kucha{\v r}, also denoted as the Brown-Kucha{\v r}-mechanism in \cite{Giesel10,Giesel10s}, was that on the constraint surface of the spatial diffeomorphism constraint $c^{tot}_a=0$, we have $c_a=-c_a^{dust}$ and thus we can rewrite the dust contribution $c^{dust}$ as
\begin{equation}
c^{dust}=-\sqrt{P^2+q^{ab}c_ac_b}.
\end{equation}
As a consequence the only dependence in $c^{tot}$ on the dust variables is via the dust momentum $P$. Thus on the constraint surface of the Hamiltonian constraint we can solve $c^{tot}$ for $P$ and obtain an equivalent form of the Hamiltonian constraint given by
\begin{equation}
\label{equivHam}
\tilde{c}^{tot}=P+h,\quad\quad h:=\sqrt{c^2-q^{ab}c_ac_b}.
\end{equation}
The fact that $h$ no longer depends on the dust variables is what is called deparametrization and a consequence of this is that we will end up with a time independent physical Hamiltonian as discussed in the next subsection. In order to obtain an Abelian constraint algebra we also solve the spatial diffeomorphism constraint for the momenta $P_j$. For this purpose we have to assume that the matrix $S^j_{,a}$ is everywhere non-degenerate, an assumption similar to the classical restriction $\det(q)>0$, meaning that the inverse matrix exist, which we denote by $S^a_j$. Then on the constraint surface we have $P_j+S^a_j(c_a+PT_{,a})=P_j+S^a_j(c_a-hT_{,a})$ thus we can write down the following equivalent form of the spatial diffeomorphism constraint
\begin{equation}
\label{equivDiffeo}
\tilde{c}_j^{tot}=P_j+h_j,\quad\quad h_j:=S^a_j(c_a-hT_{,a}).
\end{equation}
We realize in contrast to the Hamiltonian constraint $\tilde{c}^{tot}$ the spatial diffeomorphism constraint $\tilde{c}^{tot}_j$ does not deparametrize. However this is no problem at all because the construction of observables is not restricted to the deparametrized case and can therefore be equally well applied to the spatial diffeomorphism constraint. It might only be technically a little bit more involved. It is only at the level of the physical Hamiltonian, for which however only the form of $\tilde{c}^{tot}$ turns out to be important, where deparametrization yields to simplifications in the sense that the final physical Hamiltonian will be time independent.
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Now considering the constraints $\tilde{c}^{tot}$ and $\tilde{c}^{tot}_j$ one can indeed show that they satisfy an Abelian constraint algebra \cite{Brown95}. This follows also immediately from the following abstract argument \cite{Henneaux1992}: The equivalent constraints $\tilde{c}^{tot}$ and $\tilde{c}^{tot}_j$ are still first class. Therefore their Poisson brackets are again linear combinations of constraints. However since all constraints of the system are linear in the momenta $P,P_j$ their Poisson brackets are independent of $P,P_j$. Consequently, we can evaluate the linear combination of constraints that appear in the Poisson bracket computation in particular at $P=-h$ and $P_j=-h_j$. From the Abelian constraint algebra and the explicit form of $h$ and $h_j$ in (\ref{equivHam}) and (\ref{equivDiffeo}) respectively we can conclude that $h(x)$ are mutually Poisson commuting while $h(x)$ does not Poisson commute with $h_j(x)$ nor do the $h_j(x)$ mutually commute. In the next subsection we will introduce the relational framework and use the latter to construct observables in the context of GR.
\subsection{Relational Framework for Constructing Observables}
The main idea of the relational framework is to introduce so called reference fields, often also denoted as clocks, that will then be used to construct observables with respect to the constraints of the system under consideration. Let us assume we have a system with a set of constraints $\{C_I\}$ labeled by an index $I$, which is up to now arbitrary. The aim is to introduce for each constraint $C_I$ a corresponding reference field $T^I$ such that the constraint and the reference field build, at least weakly, a conjugate pair, that is $\{C_I,T^J\}\approx \delta^J_J$ where $\approx$ means equality up to terms that vanish on the constraint surface. Now since for a given set of constraints finding those reference fields might not be a simple task, one uses the freedom that one can always modify the set of constraints as long as the modified set defines the same constraint surface. Suppose we choose a set of reference field $\{T^I\}$, one for each $C_I$ with the property that $\{C_I,T^J\}=:M^{I}_J$ with $M$ being an invertible matrix, then we can define the equivalent set of constraints $\{C'_I\}$ defined through
\begin{equation}
C'_I:=\sum\limits_{J}(M^{-1})_I^JC_J .
\end{equation}
One can easily show that for $\{C'_I\}$ we have $\{C'_I,T^J\}\approx \delta^J_J$. Given these new set of constraints $\{C'_I\}$ we can use the reference fields $\{T^I\}$ to construct observables for a general phase space functions. This will be a particular combination of the original phase space function under considerations and the reference fields. To discuss this construction more in detail we consider the Hamiltonian vector field associated with $C'_I$, which is denoted by $X_I$. As can be shown and will be crucial in the following constructions the $X_I$ mutually weakly commute. Let us introduce a set of up to now arbitrary real numbers $\{\beta_I\}$, again one for each constraint $C'_I$, and consider the following sum of Hamiltonian vector fields
\begin{equation}
X_\beta:=\sum\limits_{I}\beta^IX_I~.
\end{equation}
Now we consider a function $f$ on phase space and define a map $f\to \alpha_\beta(f)$ on the set of smooth functions on phase space given by
\begin{equation}
\alpha_\beta(f):=\exp(X_\beta)\cdot f=\sum\limits_{n=1}^\infty \frac{1}{n!}X^n_\beta\cdot f~,
\end{equation}
here $X^n_\beta\cdot f=\{C_\beta,f\}_{(n)}$ where $\{.,.\}_{(n)}$ denotes the iterative Poisson bracket defined through $\{C_\beta,f\}_{(0)}=f$ and $\{C_\beta,f\}_{(n)}=\{C_\beta,\{C_\beta,f\}_{(n-1)}\}$. $\alpha_\beta$ is a Poisson automorphism on the algebra of functions on phase space associated with the Hamiltonian vector field $X_\beta$ of $C_\beta=\beta^IC'_I$.
We will use the map $\alpha_\beta$ as well as the set of reference fields to construct an observable associated with a phase space function $f$. A weak Dirac observable has to weakly Poisson commute with all constraints $\{C_I\}$. Now the idea of the relational observables is that although the phase space function $f$ as well as the reference fields $T^I$ have non-vanishing Poisson brackets with the constraints a particular combination of the two involving the map $\alpha_\beta$ has vanishing Poisson brackets with all constraints. We want to construct a map that returns the value of $f$ at those values where the reference fields $T^I$ take the values $\tau^I$. In order to do so let us choose another set of real numbers $\{\tau^I\}$. We are interested in those values of the gauge parameters $\beta^I$ for which $\alpha_\beta(T^I)=\tau_I$. If we apply $\alpha_\beta$ onto the reference fields we obtain $\alpha_\beta(T^I)\approx T^I+\beta^I$, which can easily be solved for $\beta_I$ yielding $\beta_I=\tau^I-T^I$. We will denote this equation for short as $\beta=\tau-T$ suppressing the indices. Using this we can construct the following map for the phase space function $f$
\begin{equation}
O_f(\tau):=\left[\alpha_\beta(f)\right]_{\beta=\tau-T}~.
\end{equation}
The notation with the square brackets means that only after one has computed the action of $X_\beta$ with $\beta$ treated as a constant on phase space then one sets $\beta=\tau-T$ which becomes then phase space dependent. As has been proven in \cite{Dittrich04,Thiemann2004} $O_f(\tau)$ is indeed a weak Dirac observable, that is for all $I$ we have
\begin{equation}
\{O_f(\tau),C_I\}\approx 0.
\end{equation}
We realize that we can also understand the map $O_f$ as a map that returns the value of $f$ in the gauge $\beta=\tau-T$.
As also shown in \cite{Henneaux1992,Thiemann2004} the multi parameter family of maps $O^{\tau}: f\to O_f(\tau)$ is a homomorphism from the commutative algebra of functions on phase space to the commutative algebra of weak Dirac observables, both with pointwise multiplication,
\begin{equation}
O_f(\tau)+O_g(\tau)=O_{f+g}(\tau),\quad\quad
O_f(\tau)O_g(\tau)\approx O_{fg}(\tau).
\end{equation}
This will be a particularly useful property when the explicit construction of the observables is considered for the following reason: Let us denote the coordinates on phase space by $(q^A,p_A)$ where the index $A$ is chosen such that all relevant phase space degrees of freedom are considered. Now for a phase space function $f=f(q^A,p_A)$ have we have
\begin{equation}
O_f(\tau)=f(O_{q_A},O_{p^A})(\tau)
\end{equation}
which has the important consequence that it is sufficient to construct observables for the elementary phase space variables, something we will use below.
Moreover, multi parameter family of maps $O^{\tau}: f\to O_f(\tau)$ is a Poisson homomorphism with respect to the Dirac bracket $\{.,.\}^*$ associated with the system of second class constraints $C_I,T^I$ \cite{Henneaux1992,Thiemann2004}
, this means
\begin{eqnarray}
\{O_f(\tau),O_{g}(\tau)\}\approx \{O_f(\tau),O_g(\tau)\}^*
\approx O_{\{f,g\}^*}(\tau)
\end{eqnarray}
where the Dirac bracket is defined as
\begin{equation}
\{f,g\}^*=\{f,g\}-\{f,C_I\}(M^{-1})^I_J\{T^J,g\}+\{g,C_I\}(M^{-1})^I_J\{T^J,f\}.
\end{equation}
In the following we want to discuss the special case of constraints that are in deparametrized form and understand how this simplifies the construction of the observables $O_f(\tau)$. In the case of deparametrization we can always find canonical coordinates that consists of two sets $(T^I,P_I)$ and $(q^a,P_a)$ such that all constraints $C_I$ of the system can be written in the following form
\begin{equation}
C_I=P_I+h_I(q^a,p_a),
\end{equation}
and thus do not depend on the configuration variables $T^I$. In practice this is a very special case and most constrained systems, if at all, can only be written in partially deparametrized form, in which only part of the constraints deparametrize. However, for the following discussion let us assume that we consider a fully deparametrized system. Now following the steps of the construction of observables from the discussion above first we observe that
\begin{equation}
\{C_I,T^J\}=\delta_I^J~.
\end{equation}
Using the notation above this means the equivalent constraints $C'_I$ are identical to $C_I$ and thus the task of inverting a in general complicated matrix $M_I^J$ is no longer necessary. Furthermore as already discussed above if all constraints are linearly in the momenta $P_I$ then the associated constraint algebra is Abelian. For the reason that here also non of the $h_I$ depends on the reference fields $T_I$ we immediately get $\{h_I,h_J\}=0$ from this we can follow $\{h_I,C_J\}=0$ showing that each $h_I$ is already a Dirac observable. Moreover from the Abelian constraint algebra it follows that also the associated Hamiltonian vector fields commute and in this case here not only on the constraint surface but on the entire phase space. As a consequence all weak equalities that we used above can be replaced by strong equalities here.
\\
\\
First let us discuss the construction of the observables for the elementary variables $(q^a,p_a)$. Since $q^a$ and $p_a$ both commute with all momenta $P_J$ we can consider the Hamiltonian vector field associated with the $h_I$'s instead of defining $X_\beta$ via $C'_I$. Moreover for the reason that also $q^a$ and $p_a$ commute with all reference fields $T^I$ we can already, when applying $X_\beta$ to $f$, replace $\beta$ by the corresponding gauge $\tau^I-T^I$ yielding the following form for the observables for a function $f$ that depends only on $(q^a,p_a)$
\begin{equation}
\label{OfDep}
O_f(\tau)=\sum\limits_{n=0}^\infty\frac{1}{n!}X^n_\tau\cdot f
\end{equation}
where $X_\tau$ is the Hamiltonian vector field of the function
\begin{equation}
H_{\tau}=(\tau^I-T^I)H_I
\end{equation}
where $H_I:=O_{h_I}(\tau)$ denotes the observables associated with $h_I$. Because $h_I=h_I(q^a,p_a)$ is a function of $q^a$ and $p_a$ only, once the observables for the elementary variables $O_{q^a}(\tau)=:Q^a(\tau)$ and $O_{p_a}(\tau)=:P_a(\tau)$ are constructed we obtain $H_I$ as $H_I=O_{h_I}(\tau)=h_I(Q^a,P_a)(\tau)$ using the homomorphism property of the observable map. In the particular case of deparametrization we have $H_I=h_I$ because $h_I$ is already a Dirac observable as discussed above. Now if we restrict to functions that do only depend on $q^a$ and $p_a$ the Dirac bracket reduces to the Poisson bracket because those $f$ commute with all reference fields $T^I$. In particular for the algebra of the observables $Q^a(\tau)$ and $P_a(\tau)$ we obtain
\begin{equation}
\{P_a(\tau),Q^b(\tau)\}=\{O_{p_a}(\tau),O_{q^b}(\tau)\}=O_{\{p_a,q^b\}}(\tau)=O_{\delta_a^b}(\tau)=\delta_a^b~,
\end{equation}
showing that the reduced phase space has a very simple symplectic structure in terms of the coordinates $Q^a,P_a$, an important property if the quantization of such systems is considered. Having finished the discussion about the non-reference field degrees of freedom let us discuss now the case of the remaining reference field degrees of freedom. The observable associated to the reference fields $T^I$ is given by
\begin{equation}
O_{T^I}(\tau)=\left[\alpha_\beta(T^I)\right]_{\beta=\tau^I-T^I}=\tau^I
\end{equation}
and therefore is just a constant function on phase space. Since all momenta $P_I$ Poisson commute with all constraints they are already Dirac observables. In addition they can also be expressed as function of the observables $Q^a(\tau)$ and $P_a(\tau)$, because on the constraint surface we have
\begin{equation}
P_I=O_{P_I}(\tau)=-O_{h_I}(\tau)=-h_I(Q^a(\tau),P_a(\tau))=-H_I~.
\end{equation}
Hence, what we finally be interested in is the reduced phase space with elementary variables $Q^a(\tau)$ and $P_a(\tau)$.
\\
\\
Let us again consider an observable $O_f(\tau)$ associated with a function that depends only on $q^a$ and $p_a$. How can we formulate the evolution of such observables? Certainly this cannot be generated by the constraints since by construction $O_f(\tau)$ Poisson commutes with all constraints. However, $O_f(\tau)$ gives us the value of $f$ when the reference fields $T^I$ take the values $\tau^I$. As it will be the case for GR and also for STT one of the chosen reference fields will be associated with physical time and let us without loss of generality denote this reference field by $T^0$ and the values that it takes by $\tau^0$. Then time evolution for $O_f(\tau)$ can be described by the derivative of $O_f(\tau)$ with respect to $\tau^0$ since this encodes how $O_f(\tau)$ changes with time $\tau^0$. However, considering the form of $O_f(\tau)$ in (\ref{OfDep}) we can explicitly compute this derivative and as shown in \cite{Thiemann2004} one obtains
\begin{equation}
\frac{\partial O_f(\tau)}{\partial\tau_0}=\{H_0,O_f(\tau)\}
\end{equation}
where $H_0$ is the observable associated with $h_0$ that occurs in the constraint $C_0:=P_0+h_0$ associated with the reference field $T^0$ that we interpret as a reference field for time. In the following we will call $H_0$ the physical Hamiltonian because in contrast to the constraint $C_0$, that is generating gauge transformations, $H_0$ does not vanish on the constraint surface and can therefore be understood as a true Hamiltonian, which generates evolution with respect to physical time $\tau^0$. Note that because $h_0$ does not depend on $T^0$ (and also not on any other reference field) the final physical Hamiltonian $H_0$ is time independent.
\subsubsection{Observables for GR Using the Brown-Kucha{\v r} Dust}
In this subsection we will discuss how the Brown-Kucha{\v r} dust can be used to construct relational observables. In the case of GR and STT we have four times infinitely many constraints because we have one Hamiltonian and three spatial diffeomorphism constraints per spacetime point. Following the discussion of the last subsection we therefore need to choose 4 times infinitely many $T^I$ making four scalar fields a natural choice for reference fields. These will become exactly the four additional degrees of freedom $(T,S^j)$ which we added to the system by considering the Brown-Kucha{\v r} Lagrangian. The remaining degrees of freedom $(q_{ab},p^{ab})$ and possible other standard model degrees of freedom will be referred to as non-dust degrees of freedom. In the following in order to keep the discussion more simple we will only consider the system of gravity and dust.
In order to define the Hamiltonian vector field $X_\beta$ in this case, we introduce arbitrary functions $\beta^0, \beta^j$ on $\chi$ and define using the constraints in (\ref{equivHam}) and (\ref{equivDiffeo})
\begin{equation}
c_\beta^{tot}:=\int\limits_{\chi}d^3x \beta^\mu(x)\tilde{c}_\mu^{tot}(x)
\end{equation}
where $\beta^\mu=(\beta^0,\beta^k)$ and we have defined $\tilde{c}_0^{tot}=\tilde{c}^{tot}$. We denote the Hamiltonian vector field of $c_\beta^{tot}$ by
$X_\beta$ and using it we can define the map $\alpha_\beta$ given by
\begin{equation}
\alpha_\beta(f):=\exp(X_\beta)\cdot f=\sum\limits_{n=0}^\infty \frac{1}{n!}X_\beta^n\cdot f~.
\end{equation}
We use the notation $T^\mu(x)=(T(x),S^j(x))$ then applying $\alpha_\beta$ onto $T^\mu$ yields $\alpha_\beta(T^\mu(x))=T^\mu(x)+\beta^\mu(x)$. Let us denote the values that the reference fields $T^\mu(x)$ can take by $\tau^\mu(x)$ with $\tau^0(x)=\tau(x)$ and $\tau^j(x)=\sigma^j(x)$ where up to know $\tau^\mu$ are arbitrary functions on $\chi$. Now solving $\alpha_\beta(T^\mu(x))=\tau^\mu(x)$ for $\beta^\mu(x)$ leads to $\beta^{\mu}(x)=\tau^\mu(x)-T^\mu(x)$, which we again write as $\beta=\tau-T$. The observable associated to $f$ reads
\begin{equation}
O_f(\tau)=\left[\alpha_\beta(f)\right]_{\beta=\tau-T}~.
\end{equation}
Looking at the explicit form of $\tilde{c}^{tot}$ in (\ref{equivHam}) we realize that $\tilde{c}^{tot}(x)$ commutes with $S^j(y)$ and because of this we can construct the observables in two steps. First we reduce with respect to the spatial diffeomorphism constraints $\tilde{c}^{tot}_j$ and afterwards we construct the complete observables that also Poisson commute with the Hamiltonian constraint. Hence, we can rewrite $O_f(\tau)$ as
\begin{equation}
\label{CompObs}
O_f(\tau,\sigma^j)=\left[\alpha_{\beta^0}(\left[\alpha_{\beta^j}(f)\right]_{\beta^j=\sigma^j-S^j})\right]_{\beta^0=\tau-T} ~.
\end{equation}
Let us first discuss the inner part, that is how spatially diffeomorphism invariant objects are constructed. The reference fields $S^j$ will be used for this and therefore here the remaining degrees of freedom are $(q_{ab},p^{ab},T,P)$ for which observables need to be constructed. As discussed in detail in \cite{Giesel10} for the choice of a constant function $\sigma^j(x)=\sigma^j$ the observable with respect to $\tilde{c}_j^{tot}$ associated with any scalar function $f$ build from the variables $(q_{ab},p^{ab},T,P)$ can be expressed as
\begin{equation}
\label{tildef}
\tilde{f}(\sigma^j)=\left[\alpha_{\beta^j}(f)\right]_{\beta^j=\sigma^j-S^j}=f(x)\Big|_{S^j(x)=\sigma^j}
\end{equation}
where we denote the partially reduced function as $\tilde{f}$. The interpretation of the formula above is the following: Whatever the value $x$ is at which the function $f$ is evaluated $\tilde{f}(\sigma^j)$ returns $f$ evaluated at the point $x_\sigma$ at which $S^j(x)=\sigma^j$. We call the range of $S^j$ the dust space and denote it by ${\cal S}$. Since by our assumption $S^j_{,a}$ is everywhere invertible it defines a diffeomorphism $S^j:\chi\to {\cal S}$ and hence the value of $x_\sigma$ is unique. Our strategy is therefore to use $S^j$ to construct scalars $f$ on $\chi$ for $(q_{ab},p^{ab},T,P)$ and then apply the formula in (\ref{tildef}) yielding the partially reduced quantities $(\tilde{q}_{ab},\tilde{p}^{ab},\tilde{T},\tilde{P})$, explicitly we get
\begin{equation}
\label{TildeNonDust}
\tilde{T}=T\quad \tilde{P}=\frac{1}{J}P\quad
\tilde{q}_{jk}=q_{ab}S^a_jS^b_k
\quad
\tilde{p}^{jk}=\frac{1}{J}p^{ab}S^j_{,a}S^k_{,b}
\end{equation}
where $J:=\det\left(\frac{\partial S}{\partial x}\right)$ was used to obtain the correct density weight. Note while these are scalars on $\chi$ there are tensors on the dust space ${\cal S}$ with the same density weight that they have on $\chi$, see also \cite{Giesel10} for more details. We realize that the evaluation of the functions in (\ref{TildeNonDust}) at $x_\sigma$ is nothing else than the pull back of the corresponding fields to ${\cal S}$ under the inverse of the diffeomorphism $S^j:\chi\to {\cal S}$.
\\
Our remaining task is to compute the complete observables in (\ref{CompObs}) that also Poisson commutes with the Hamiltonian constraint. For this purpose the reference field $\tilde{T}$ will be used and hence we need to construct observables for $\tilde{q}_{jk}$ and $\tilde{p}^{jk}$ that we will denote by $Q_{jk}$ and $P^{jk}$ respectively. The constraint $\tilde{c}^{tot}$ in (\ref{equivHam}) is in deparametrized form and thus wen can apply the simplified construction discussed in the last subsection. Let us look at the smeared version of the constraint given by
\begin{equation}
c^{tot}_\tau:=\int\limits_{\chi} d^3x (\tau-T)(x)\tilde{c}^{tot}(x)
\end{equation}
where we used $\beta^0(x)=(\tau-T)(x)$. Now in order to construct the observables $Q_{jk}$ and $P^{jk}$ we need to ensure that $c_\tau^{tot}$ is already an observable with respect to the spatial diffeomophism constraint $\tilde{c}_j^{tot}$. For the choice of constant $\tau$ the constraint $c^{tot}_\tau$ is an integral over a density of weight one and we can equivalently express it as an integral over the dust space ${\cal S}$ given by
\begin{equation}
c^{tot}_\tau=\int\limits_{\cal S}d^3\sigma (\tau-\tilde{T})(\sigma)(\tilde{P}+\tilde{h})(\sigma)
\end{equation}
with
\begin{equation}
\tilde{h}(\sigma)=\sqrt{\tilde{c}^2(\sigma)-\tilde{q}^{jk}(\sigma)\tilde{c}_j(\sigma)\tilde{c}_k(\sigma)}
\end{equation}
where $\tilde{c}(\sigma)$ and $\tilde{c}_j(\sigma)$ are the observables of $c$ and $c_j$ in (\ref{equivHam}) and (\ref{equivDiffeo}) respectively with respect to the spatially diffeomorphism constraint. Now since $c^{tot}_\tau$ deparametrizes we do not need to consider the Hamiltonian vector field of $c^{tot}_\tau$ but we can work with the Hamiltonian vector field $X_\tau$ of
\begin{equation}
H_\tau:=\int\limits_{\cal S}d^3\sigma (\tau-\tilde{T})\tilde{h}(\sigma)~.
\end{equation}
The observables for a function $f$ that depends only on $\tilde{q}_{jk}$ and $\tilde{p}^{jk}$ (and possible other standard model matter degrees of freedom) is then given by
\begin{equation}
O_f(\tau,\sigma)=\sum\limits_{n=0}^\infty\frac{1}{n!}X^n_\tau\cdot f=\sum\limits_{n=0}^\infty\frac{1}{n!}\{H_\tau,f\}_{(n)}
\end{equation}
where $\{.,.\}_{(n)}$ again denotes the iterated Poisson bracket. Considering the discussion about the physical Hamiltonian above we observe that in case of the dust as reference fields the physical time is given by $\tau$ and the corresponding evolution of the observables is given by
\begin{equation}
\frac{d O_f(\tau,\sigma)}{d\tau}=\{{\bf H},O_f(\tau,\sigma)\}
\end{equation}
with the physical Hamiltonian
\begin{equation}
{\bf H}=\int\limits_{\cal S}d^3\sigma H(\sigma)~.
\end{equation}
Here $H(\sigma)$ is the (complete) observable associated to $\tilde{h}(\sigma)$, that is
\begin{equation}
H(\sigma):=O_{\tilde{h}}(\tau,\sigma)=\tilde{h}(\sigma)
\end{equation}
and is therefore independent of $\tau$ and hence physical time as expected in the deparametrized case. In the following we will continue to use the notation $Q_{jk}$ and $P^{jk}$ for the observables of $q_{ab}$ and $p^{ab}$. Furthermore we will denote the observables associated with $c$ and $c_j$ also by capital letters $C(\tau,\sigma)$ and $C_j(\tau,\sigma)$ respectively. Using this notation we can also rewrite the physical Hamiltonian density as
\begin{equation}
H(\sigma )=\sqrt{C^2(\tau ,\sigma )-Q^{ij}(\tau ,\sigma )C_i(\tau ,\sigma )C_j(\tau ,\sigma)}.
\end{equation}
\subsection{Evolution Equations in Jordan Frame}
\label{EvEqnJordan}
As is well known, in the classical formulation of STT, there are different choices for basic variables, corresponding to different frames. These frames are related to each other by conformal transformations. There are many debates on which frame should be regarded as the physical one (the word ``physical" has nothing to do with the gauge invariance here) and the question carries over to the discussion what kind of coupling should be chosen. Two prominent choices are either the Jordan frame, where a scalar field is non-minimally coupled to the metric, or the Einstein frame, where a scalar field is minimally coupled as in GR. If one considers the Einstein frame as the relevant one, the corresponding STT would make no difference from GR with a scalar field, which was already presented in \cite{Giesel10}. Hence, to analyze the non-minimally coupled case, in this paper, we consider the Jordan frame and this choice has the consequence that also the evolution equations of the linear perturbations will be formulated using the Jordan frame.
We consider the following action
\begin{eqnarray}
S_{STT}=S_{Jordan}+S_{dust},\label{action}\nonumber
\end{eqnarray}
where the STT action in Jordan frame reads
\begin{eqnarray}
S\, _{Jordan}=\int d^4x\sqrt{|\det (g)|}\left[F(\phi)R^{(4)}-K(\phi)g^{\mu\nu}(\triangledown_\mu\phi)\triangledown_\nu\phi-V(\phi)\right].\label{actionjordan}
\end{eqnarray}
Here for simplicity we set $16\pi G=1$, $F (\phi)$ and $K (\phi)$
are positive coupling functions, and $V(\phi)$ is the potential of the gravitational scalar field $\phi$.
As mentioned in section 2.1, the dust action reads
\begin{eqnarray}
S_{dust}=-\frac{1}{2}\int d^4x\sqrt{|\det(g)|}\rho \left[g^{\mu\nu}U_{\mu}U_{\nu}+1\right]
\end{eqnarray}
with $U_{\mu}=-T_{,\mu}+W_jS^j_{,\mu}$. The Hamiltonian analysis of STT in the generalized Brans-Dicke form is given in Refs. \cite{Zhang11} in Jordan frame. Using the results there and following the procedures in the last subsection, the physical Hamiltonian density reads
\begin{eqnarray}
H(\sigma )=\sqrt{C^2(\tau ,\sigma )-Q^{ij}(\tau ,\sigma )C_i(\tau ,\sigma )C_j(\tau ,\sigma)},\label{phyham}
\end{eqnarray}
with
\begin{eqnarray}
C_j(\tau,\sigma )&=&\left[-2Q_{jl}D_kP^{kl}+\Pi D_j\Phi\right](\tau,\sigma) ,\label{Cj}\\
C(\tau, \sigma)&=&\Bigg[\frac{1}{\sqrt{\det Q}}\Big(\frac{\Big(Q_{i m}Q_{j n}-\frac{1}{2}Q_{i j}Q_{mn}\Big)P^{ij}P^{mn}}{F(\Phi)}+\frac{\Big(F'(\Phi )Q_{ij}P^{ij}-F(\Phi)\Pi\Big)^2}{2F(\Phi)\big[3(F'(\Phi))^2+2F(\Phi)K(\Phi)\big]}\Big)\nonumber\\
&&
+\sqrt{\det Q}\Big(-F(\Phi )R^{(3)}+K(\Phi)Q^{ij}\big(D_i\Phi \big)\big(D_j\Phi\big)+2Q^{ij}D_i D_j F(\Phi )+V(\Phi)\Big)\Bigg](\tau ,\sigma ),\label{C}\nonumber\\
\end{eqnarray}
and the elementary Poisson bracket read:
\begin{eqnarray}
\{P^{ij}(\sigma),Q_{kl}(\tilde{\sigma})\}=\delta ^i_{(k}\delta^j_{l)}\delta(\sigma,\tilde{\sigma}),\qquad\{\Pi(\sigma),\Phi(\tilde{\sigma})\}=\delta (\sigma,\tilde{\sigma}),
\end{eqnarray}
where a prime $'$ in Eq. (\ref{C}) means the derivative with respect to $\Phi$, i.e. $F'(\Phi)=\frac{F(\Phi)}{d\Phi}, F''(\Phi)=\frac{d^2F(\Phi)}{d\Phi^2}$. Here we denote all the objects after gauge completion with capital letters. All quantities are now evaluated on the dust manifold. Note that Eq. (\ref{C}) is valid in the case of $3(F'(\Phi))^2+2F(\Phi)K(\Phi)\neq0$, which corresponds to the case $\omega\neq -\frac{3}{2}$ in Refs. \cite{Zhang11}, on which we focus in the paper.
Given a function $f$ on phase space, that does not depend on the dust variables, we can construct its associated observable $O_f$ by applying the procedure described in section \ref{BKFormalism}.
We get the Hamiltonian equations of motion for $O_f$, being first order differential equations with respect to the physical time $\tau$, by simply calculating its Poisson bracket with the physical Hamiltonian denoted by $\textbf{H}$ and given by $\textbf{H}:=\int d\sigma H(\sigma)$:
\begin{eqnarray}
\dot{O}_f\equiv\frac{dO_f}{d\tau}=\{\textbf{H},O_f\}.
\end{eqnarray}
In our case we are interested in the equations of motion for $O_{q_{ab}}=:Q_{jk}, O_{p^{ab}}=:P^{jk}, O_{\phi}=:\Phi, O_{\pi}=:\Pi$ and using the physical Hamiltonian of the dust model we end up with the following Hamiltonian equations:
\begin{eqnarray}
\dot{\Phi}&=&-\frac{N}{\sqrt{\det Q}}\frac{F'(\Phi)Q_{i j}P^{i j}-F(\Phi)\Pi}{3(F'(\Phi))^2+2F(\Phi)K(\Phi)}+\mathcal{L}_{\overset{\rightharpoonup }{N}}\Phi ,\label{phifirst}\\
\dot{\Pi}&=&-\frac{N}{\sqrt{\det Q}}\bigg[\frac{\Big(F'(\Phi)Q_{ij}P^{ij}-F(\Phi)\Pi\Big)^2}{2\big(F(\Phi)\big)^2\big[3(F'(\Phi))^2+2F(\Phi)K(\Phi )\big]^2}\cdot\Big[-F(\Phi)\big(3(F'(\Phi))^2+2F(\Phi)K(\Phi)\big)\Big]'\nonumber\\
&&\qquad\qquad\quad
+\frac{F'(\Phi)Q_{ij}P^{ij}-F(\Phi)\Pi}{F(\Phi)\big[3(F'(\Phi))^2+2F(\Phi)K(\Phi)\big]}\Big(F''(\Phi)Q_{ij}P^{ij}-F'(\Phi)\Pi\Big) \nonumber\\
&&\qquad\qquad\quad
-\frac{F'(\Phi)}{(F(\Phi))^2}\Big(Q_{im}Q_{jn}-\frac{1}{2}Q_{ij}Q_{mn}\Big)P^{ij}P^{mn}\bigg]\nonumber\\
&&
-\sqrt{\det Q}\left[-NF'(\Phi)R^{(3)}+N K'(\Phi)Q^{ij}\big(D_i\Phi\big)D_j\Phi+2F'(\Phi)Q^{ij}D_i D_jN+NV'(\Phi)\right]\nonumber\\
&&
+2\partial_j\left[N\sqrt{\det Q}K(\Phi )Q^{jk}\Phi _{,k}\right]+\mathcal{L}_{\overset{\rightharpoonup}{N}}\Pi,\label{pifirst}\\
\dot{Q}_{j k}&=&\frac{N}{\sqrt{\det Q}F(\Phi)}\Big(2G_{j k m n}P^{m n}+\frac{(F'(\Phi))^2Q_{m n}P^{mn}-F'(\Phi)F(\Phi)\Pi}{3(F'(\Phi))^2+2F(\Phi )K(\Phi)}Q_{jk}\Big)+\big(\mathcal{L}_{\overset{\rightharpoonup }{N}}Q\big)_{j k} ,\label{qfirst}\\
\dot{P}^{j k}&=&\frac{N}{\sqrt{\det Q}}\left[-\frac{Q_{mn}\Big(2P^{jm}P^{kn}-P^{jk}P^{mn}\Big)}{F(\Phi)}-\frac{(F'(\Phi))^2Q_{mn}P^{m n}-F'(\Phi)F(\Phi)\Pi}{F(\Phi)\big[3(F'(\Phi))^2+2F(\Phi)K(\Phi)\big]}P^{jk}\right]\nonumber\\
&&
+\sqrt{\det Q}\bigg[N\Big(-K(\Phi)Q^{jk}\Phi^{,m}\Phi_{,m}-V(\Phi)Q^{jk}-Q^{jk}D^iD_i F(\Phi)+K(\Phi)\Phi^{,j}\Phi^{,k}\Big)\nonumber\\
&&\qquad\qquad\quad
+\left[G^{-1}\right]^{jkmn}\Big(D_m D_n(NF(\Phi))-NF(\Phi)R_{mn}\Big)-2(D^{(j}N)D^{k)}F(\Phi)\nonumber\\
&&\qquad\qquad\quad
+Q^{jk}\big(D^iN\big)D_i F(\Phi)\bigg]\nonumber\\
&&
+\frac{N}{2}Q^{j k}C(\tau,\sigma)-\frac{1}{2}H(\tau,\sigma )Q^{j m}Q^{kn}N_m N_n+\big(\mathcal{L}_{\overset{\rightharpoonup }{N}}P\big)^{j k},\label{pfirst}
\end{eqnarray}
where $N:=\frac{C}{H}$, $N_j:=-\frac{C_j}{H}$, $\Phi_{,k}:=\frac{\partial \Phi}{\partial x^k}$, $ G_{jkmn}:=\frac12 (Q_{jm}Q_{nk}+Q_{jn}Q_{mk}-Q_{jk}Q_{mn})$ and its inverse $ [G^{-1}]^{jkmn}:=\frac12(Q^{jm}Q^{nk}+Q^{jn}Q^{mk}-2Q^{jk}Q^{mn})$ satisfying $G_{jkmn}G^{mnrs}=\delta^{r}_{(j}\delta^{s}_{k)}$. All the indices are lowered and raised by the three-metric and its inverse. Note that both $C(\tau,\sigma)$ and $H(\tau,\sigma)$ are non-vanishing because they are no longer constraints by introducing the dust fields as dynamically coupled observers. It is only the total Hamiltonian and diffeomorphism constraints, which involve the dust as well as the gravitational and scalar field contributions, that is still vanishing.
Moreover, as shown in \cite{Giesel10}, both $H$ and $N_j$ are conserved quantities because their Poisson brackets with the physical Hamiltonian equal zero. Now it is easy to see from Eqs. (\ref{phifirst}) and (\ref{qfirst}) that
\begin{eqnarray}
\Pi &=&\frac{1}{F(\Phi)}\Big(F'(\Phi)Q_{ij}P^{ij}+\frac{\sqrt{\det Q}}{N}\big(\dot{\Phi}-\mathcal{L}_{\overset{\rightharpoonup}{N}}\Phi\big)\Big[3 (F'(\Phi))^2+2F(\Phi)K(\Phi)\Big]\Big), \label{Pi}\\
P^{j k}&=&\frac{\sqrt{\det Q}F(\Phi )}{2N}\left[G^{-1}\right]^{jkmn}\Big(\dot{Q}_{mn}-(\mathcal{L}_{\overset{\rightharpoonup }{N}}Q)_{mn}+\frac{F'(\Phi )}{F(\Phi)}\big(\dot{\Phi}-\mathcal{L}_{\overset{\rightharpoonup }{N}}\Phi\big)Q_{mn}\Big).\label{Pjk}
\end{eqnarray}
Taking the second order time derivative of Eqs. (\ref{phifirst}) and (\ref{qfirst}) and substituting Eqs. (\ref{pifirst}), (\ref{pfirst}), (\ref{Pi}), (\ref{Pjk}) into them, we get the following equations:
\begin{eqnarray}
\overset{\cdot\cdot}{\Phi}&=&\left[\frac{\dot{N}}{N}-\frac{\Big(\sqrt{\det Q}\Big)^{\cdot}}{\sqrt{\det Q}}+\frac{N}{\sqrt{\det Q}}\Big(\mathcal{L}_{\overset{\rightharpoonup }{N}}(\frac{\sqrt{\det Q}}{N})\Big)\right]\big(\dot{\Phi}
-\mathcal{L}_{\overset{\rightharpoonup }{N}}\Phi \big)+2\big(\mathcal{L}_{\overset{\rightharpoonup }{N}}\dot{\Phi}\big)+\big(\mathcal{L}_{\dot{\overset{\rightharpoonup}{N}}}\Phi\big)-\big(\mathcal{L}_{\overset{\rightharpoonup }{N}}(\mathcal{L}_{\overset{\rightharpoonup}{N}}\Phi)\big) \nonumber\\
&&
-\frac{3F'(\Phi)F''(\Phi)+F(\Phi)K'(\Phi)}{3(F'(\Phi))^2+2F(\Phi)K(\Phi)}\big(\dot{\Phi}-\mathcal{L}_{\overset{\rightharpoonup}{N}}\Phi\big)^2
-\frac32\frac{N^2}{\sqrt{\det Q}}\frac{F'(\Phi)}{3(F'(\Phi))^2+2F(\Phi)K(\Phi)}C(\tau, \sigma)\nonumber\\
&&
+\frac{2F(\Phi)}{3(F'(\Phi))^2+2F(\Phi)K(\Phi)}\Bigg[N^2\bigg[K(\Phi)D^iD_i\Phi+K(\Phi)[Q^{jk}]_{,j}\Phi _{,k}-\frac{1}{2}V'(\Phi)\bigg]\nonumber\\
&&\qquad\qquad\qquad\qquad\qquad\qquad
+K(\Phi)Q^{jk}\Phi_{,k}\Big[\frac{N}{\sqrt{\det Q}}\big(N\sqrt{\det Q}\big)_{,j}\Big]\Bigg]\nonumber\\
&&
+\frac{F'(\Phi)}{3(F'(\Phi))^2+2F(\Phi)K(\Phi)}\Bigg[\frac{F(\Phi)}{4}\left[G^{-1}\right]^{rstu}\Big(\dot{Q}_{rs}
-(\mathcal{L}_{\overset{\rightharpoonup }{N}}Q)_{rs}\Big)\Big(\dot{Q}_{tu}-(\mathcal{L}_{\overset{\rightharpoonup }{N}}Q)_{tu}\Big)\nonumber\\
&&\qquad\qquad\qquad\qquad\qquad\qquad\quad
-F'(\Phi)Q^{mn}\Big(\dot{Q}_{mn}-(\mathcal{L}_{\overset{\rightharpoonup }{N}}Q)_{mn}\Big)\big(\dot{\Phi}-\mathcal{L}_{\overset{\rightharpoonup }{N}}\Phi\big)\nonumber\\
&&\qquad\qquad\qquad\qquad\qquad\qquad\quad
+\Big(2N^2K(\Phi)+\frac{F(\Phi)}{F'(\Phi)}N^2K'(\Phi)\Big)\Phi^{,i}\Phi_{,i}+3N^2V(\Phi)\nonumber\\
&&\qquad\qquad\qquad\qquad\qquad\qquad\quad
+5N^2D^iD_i F(\Phi)+3N(D^iN)D_i F(\Phi)-N^2F(\Phi)R^{(3)}\nonumber\\
&&\qquad\qquad\qquad\qquad\qquad\qquad\quad
+\frac{1}{2}\frac{N}{\sqrt{\det Q}}H(\sigma)Q^{mn}N_m N_n\Bigg],\label{phisecond}\\
\overset{\cdot\cdot}{Q}_{jk}&=&\left[\frac{\dot{N}}{N}-\frac{\Big(\sqrt{\det Q}\Big)^{\cdot}}{\sqrt{\det Q}}+\frac{N}{\sqrt{\det Q}}\Big(\mathcal{L}_{\overset{\rightharpoonup}{N}}(\frac{\sqrt{\det Q}}{N})\Big)\right]\Big(\dot{Q}_{jk}-(\mathcal{L}_{\overset{\rightharpoonup }{N}}Q)_{jk}\Big)\nonumber\\
&&
+Q^{m n}\Big(\dot{Q}_{mj}-(\mathcal{L}_{\overset{\rightharpoonup }{N}}Q)_{mj}\Big)\Big(\dot{Q}_{nk}-(\mathcal{L}_{\overset{\rightharpoonup}{N}}Q)_{nk }\Big)+N^2\Big(\frac{2K(\Phi)}{F(\Phi)}\Phi _{,j}\Phi _{,k}-2R_{jk}\Big)\nonumber\\
&&
+2ND_j D_k N+2\big(\mathcal{L}_{\overset{\rightharpoonup }{N}}\dot{Q}\big)_{jk}+\big(\mathcal{L}_{\dot{\overset{\rightharpoonup}{N}}}Q \big)_{jk}-\big(\mathcal{L}_{\overset{\rightharpoonup}{N}}(\mathcal{L}_{\overset{\rightharpoonup}{N}}Q)\big)_{jk}
-\frac{F'(\Phi)}{F(\Phi)}\Big(\dot{Q}_{jk}-(\mathcal{L}_{\overset{\rightharpoonup }{N}}Q)_{jk}\Big)\big(\dot{\Phi}
-\mathcal{L}_{\overset{\rightharpoonup }{N}}\Phi\big)\nonumber\\
&&
+\frac{2N^2}{F(\Phi)}D_j D_k F(\Phi)-\frac{NH(\sigma)}{\sqrt{\det Q}F(\Phi)}G_{jkmn}N^mN^n+\frac{Q_{j k}}{F(\Phi)}\Big[N(D^iN)D_i F(\Phi)+N^2 D^iD_i F(\Phi)+N^2V(\Phi)\Big]\nonumber\\
&&
+\frac{-2K(\Phi)F''(\Phi)+K'(\Phi)F'(\Phi)}{3(F'(\Phi))^2+2F(\Phi)K(\Phi)}
\big(\dot{\Phi}-\mathcal{L}_{\overset{\rightharpoonup }{N}}\Phi \big)^2Q_{jk}-\frac{N^2}{\sqrt{\det Q}}\frac{K(\Phi)Q_{jk}}{3(F'(\Phi))^2+2F(\Phi)K(\Phi)}C(\tau, \sigma)\nonumber\\
&&
-\frac{2F'(\Phi)Q_{jk}}{3(F'(\Phi))^2+2F(\Phi)K(\Phi)}\Bigg[N^2\bigg(K(\Phi)D^iD_i\Phi+K(\Phi)[Q^{jk}]_{,j}\Phi _{,k}-\frac{1}{2}V'(\Phi)\bigg)\nonumber\\
&&\qquad\qquad\qquad\qquad\qquad\qquad\quad
+K(\Phi)Q^{jk}\Phi_{,k}\Big[\frac{N}{\sqrt{\det Q}}\big(N\sqrt{\det Q}\big)_{,j}\Big]\Bigg]\nonumber\\
&&
-\frac{(F'(\Phi))^2Q_{jk}}{F(\Phi)\big[3(F'(\Phi))^2+2F(\Phi)K(\Phi)\big]}\Bigg[\frac{F(\Phi)}{4}\left[G^{-1}\right]^{rstu}\Big(\dot{Q}_{r s}-(\mathcal{L}_{\overset{\rightharpoonup}{N}}Q)_{rs}\Big)\Big(\dot{Q}_{tu}-(\mathcal{L}_{\overset{\rightharpoonup }{N}}Q)_{tu}\Big)\nonumber\\
&&\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad
-F'(\Phi)Q^{mn}\Big(\dot{Q}_{mn}-(\mathcal{L}_{\overset{\rightharpoonup}{N}}Q)_{mn}\Big)\Big(\dot{\Phi}
-\mathcal{L}_{\overset{\rightharpoonup}{N}}\Phi \Big)\nonumber\\
&&\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad
+\Big(2N^2K(\Phi)+\frac{F(\Phi)}{F'(\Phi)}N^2K'(\Phi)\Big)\Phi ^{,i}\Phi_{,i}+3N^2V(\Phi)\nonumber\\
&&\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad
+5N^2D^iD_i F(\Phi)+3N(D^iN)D_i F(\Phi)-N^2F(\Phi)R^{(3)}\nonumber\\
&&\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad
+\frac{1}{2}\frac{N}{\sqrt{\det Q}}H(\sigma )Q^{mn}N_m N_n\Bigg],\label{qsecond}
\end{eqnarray}
where $C(\tau, \sigma)$ is now expressed as a function depending on the configuration variables and their first order time derivatives given by
\begin{eqnarray}
C&=&\frac{\sqrt{\det Q}F(\Phi)}{4N^2}\left[G^{-1}\right]^{rstu}\Big(\dot{Q}_{rs}-(\mathcal{L}_{\overset{\rightharpoonup }{N}}Q)_{r s}\Big)\Big(\dot{Q}_{t u}-(\mathcal{L}_{\overset{\rightharpoonup }{N}}Q)_{tu}\Big)-\sqrt{\det Q}F(\Phi)R^{(3)}+\sqrt{\det Q}K(\Phi )Q^{jk}\Phi _{, j}\Phi _{,k}\nonumber\\
&&
-\frac{\sqrt{\det Q}}{N^2}F'(\Phi )Q^{rs}\Big(\dot{Q}_{rs}-(\mathcal{L}_{\overset{\rightharpoonup }{N}}Q)_{rs}\Big)\big(\dot{\Phi }-\mathcal{L}_{\overset{\rightharpoonup }{N}}\Phi\big)+\frac{\sqrt{\det Q}}{N^2}K(\Phi)\big(\dot{\Phi}-\mathcal{L}_{\overset{\rightharpoonup }{N}}\Phi \big)^2+2\sqrt{\det Q}Q^{jk}D_j D_k F(\Phi)\nonumber\\&&+\sqrt{\det Q}V(\Phi).\label{Hamiltonianconstr}
\end{eqnarray}
So far we have derived the evolution equations for the gauge invariant 3-metric $Q_{jk}$ and the scalar field $\Phi$. Notice that if we choose in the equation of motions shown in (\ref{phisecond}) and (\ref{qsecond}) $F(\Phi)=1$ and $K(\Phi)=\frac{1}{2}$, all terms containing $F'(\Phi)$, $F''(\Phi)$, $K'(\Phi)$ naturally vanish and the remaining terms reduce to the evolution equations of the observables for a physical system consisting of dust, gravity and a minimally coupled scalar field. In this special case the evolution equations in (\ref{phisecond}) and (\ref{qsecond}) agree with the evolution equations for the observables shown in equations (4.23) and (4.24) in \cite{Giesel10}\footnote{Note that the definition of the potential $V(\phi)$ used in \cite{Giesel10} differs by a factor of 2 and this needs to be considered in order to have an exact agreement between the equations.}.
\section{Linear Perturbations of STT on a General Background}
In this section we will use the gauge invariant variables $(\Phi, Q_{jk})$ and their equations of motions, obtained in last section, to derive the evolution equations for their corresponding linearly perturbed variables. For this purpose, we first split the configuration variables into a background and a perturbed part
\begin{eqnarray}
\Phi=\overline{\Phi}+\delta \Phi, \qquad Q_{jk}=\overline{Q}_{jk}+\delta Q_{jk},
\end{eqnarray}
where $\overline{\Phi}$ and $\overline{Q}_{jk}$ denote the background variables satisfying the evolution equations shown in (\ref{phisecond}) and (\ref{qsecond}) and are thus solutions of the classical scalar-tensor equations. From now on we denote all occurring background variables with a bar. Introducing the dust as dynamically coupled observers has the consequence, that the shift vector $N_j$ and the lapse function $N$ are no longer arbitrary but become fixed as particular functions on the reduced phase space, spanned by $(Q_{jk},P^{jk},\Phi,\Pi)$. Therefore $N, N_j$ are not treated as independent variables and the same is true for the associated perturbations. Since we consider the perturbations of the second order evolution equations for $Q_{jk}$ and $\Phi$, the lapse function $N$ and the shift vector $N_j$ can be understood as functions of $Q_{jk},\Phi$ and their associated velocities.
The explicit form of their perturbations reads
\begin{eqnarray}
N_j=\overline{N}_j+\delta N_j=-\frac{\overline{C}_j}{\overline{H}}+\delta N_j,\qquad N=\overline{N}+\delta N=\frac{\overline{C}}{\overline{H}}-\frac{\overline{N}^j\overline{N}^k}{2\overline{N}}\delta Q_{jk}+\frac{\overline{N}^j}{\overline{N}}\delta N_j.
\end{eqnarray}
Note that in principle $\delta N_j$ can be expressed in terms of $\delta Q_{jk}$ and $\delta \Phi$, but we keep $\delta N_j$ here because as proved in \cite{Giesel10} both $\delta N_j$ and $\delta H$ are conserved quantities, while a conserved quantity is fixed at an initial time, it remains a constant during the evolution. Keeping $\delta N_j$ here helps to formulate the final evolutions equations for the linear perturbations in more compact form. Likewise we also keep the explicit expression $\delta H$ in the following for the same reason.
Since the evolution equations (\ref{phisecond}) and (\ref{qsecond}) are already very complicated, it is expected that the evolution equations for $\delta \Phi$ and $\delta Q_{jk}$ will be even more complicated. In order to express them in a concise form, we notice that there are three important common terms in Eqs. (\ref{phisecond}) and (\ref{qsecond}), namely $C(\tau,\sigma)$ and all terms inside the last two square brackets. We introduce the following abbreviations, which agree for both equations:
\begin{eqnarray}
X&\equiv&N^2\Big[K(\Phi)D^iD_i\Phi+K(\Phi)[Q^{jk}]_{,j}\Phi_{,k}-\frac{1}{2}V'(\Phi)\Big]
+K(\Phi)Q^{jk}\Phi_{,k}\Big[\frac{N}{\sqrt{\det Q}}\Big(N\sqrt{\det Q}\Big)_{,j}\Big],\label{X}\\
Y&\equiv&\frac{F(\Phi)}{4}\left[G^{-1}\right]^{rstu}\Big(\dot{Q}_{r s}-(\mathcal{L}_{\overset{\rightharpoonup}{N}}Q)_{rs}\Big)\Big(\dot{Q}_{tu}-(\mathcal{L}_{\overset{\rightharpoonup }{N}}Q)_{tu}\Big)
-F'(\Phi)Q^{mn}\Big(\dot{Q}_{mn}-(\mathcal{L}_{\overset{\rightharpoonup}{N}}Q)_{mn}\Big)\Big(\dot{\Phi}
-\mathcal{L}_{\overset{\rightharpoonup}{N}}\Phi \Big)\nonumber\\
&&
+\Big(2N^2K(\Phi)+\frac{F(\Phi)}{F'(\Phi)}N^2K'(\Phi)\Big)\Phi^{,i}\Phi_{,i}+3N^2V(\Phi)
+5N^2D^iD_i F(\Phi)+3N(D^iN)D_i F(\Phi)\nonumber\\
&&
-N^2F(\Phi)R^{(3)}+\frac{1}{2}\frac{N}{\sqrt{\det Q}}H(\sigma )Q^{mn}N_m N_n.\label{Y}
\end{eqnarray}
The next step is to derive the equations of motions for the linear perturbations. For this purpose we use the following identities:
\begin{eqnarray}
\lefteqn{\delta\left[\left[G^{-1}\right]^{rstu}\Big(\dot{Q}_{rs}-(\mathcal{L}_{\overset{\rightharpoonup }{N}}Q)_{rs}\Big)\Big(\dot{Q}_{t u}-(\mathcal{L}_{\overset{\rightharpoonup }{N}}{Q})_{tu}\Big)\right]}\nonumber\\
&=&2\Big(\dot{\overline{Q}}_{rs}-(\mathcal{L}_{\overset{\rightharpoonup }{\overline{N}}}\overline{Q})_{rs}\Big)\Bigg[\left[\overline{G}^{-1}\right]^{rsjk}(\frac{\partial}{\partial\tau}-\mathcal{L}_{\overset{\rightharpoonup }{\overline{N}}})-\left[\overline{G}^{-1}\right]^{turj}\overline{Q}^{sk}\Big(\dot{\overline{Q}}_{tu}-(\mathcal{L}_{\overset{\rightharpoonup }{\overline{N}}}\overline{Q})_{tu}\Big)\nonumber\\
&&\qquad\qquad\qquad\qquad\quad
+[\overline{G}^{-1}]^{rsmn}\Big[\overline{Q}^{jt}\overline{N}^k\big[\overline{Q}_{mn}\big]_{,t}+2\overline{Q}_{tn}\frac{\partial}{\partial x^m}\Big(\overline{Q}^{jt}\overline{N}^k\Big)\Big]\Bigg]\delta{Q}_{jk}\nonumber\\
&&
-2\Big(\dot{\overline{Q}}_{rs}-(\mathcal{L}_{\overset{\rightharpoonup }{\overline{N}}}\overline{Q})_{rs}\Big)\left[\overline{G}^{-1}\right]^{rsjn}\left[\overline{Q}^{mt}\left[\overline{Q}_{jn}\right]_{,t}
+2\overline{Q}_{tj}\frac{\partial}{\partial x^n}\Big(\overline{Q}^{mt}\Big)\right]\delta{N}_m,\\
\lefteqn{\delta \Big[Q^{mn}\Big(\dot{Q}_{mn}-(\mathcal{L}_{\overset{\rightharpoonup }{N}}Q)_{mn}\Big)(\dot{\Phi}-\mathcal{L}_{\overset{\rightharpoonup}{N}}\Phi)\Big]}\nonumber\\
&=&\Bigg[-\overline{Q}^{jm}\overline{Q}^{kn}\Big(\dot{\overline{Q}}_{mn}-(\mathcal{L}_{\overset{\rightharpoonup}{N}}\overline{Q})_{m n}\Big)\big(\dot{\overline{\Phi}}-\mathcal{L}_{\overset{\rightharpoonup}{\overline{N}}}\overline{\Phi}\big)+\overline{Q}^{jk}
\big(\dot{\overline{\Phi}}-\mathcal{L}_{\overset{\rightharpoonup}{\overline{N}}}\overline{\Phi}\big)(\frac{\partial}{\partial\tau }-\mathcal{L}_{\overset{\rightharpoonup}{\overline{N}}})\nonumber\\
&&
-\overline{Q}^{mn}\big(\dot{\overline{\Phi}}-\mathcal{L}_{\overset{\rightharpoonup}{\overline{N}}}\overline{\Phi}\big)\left[-\overline{Q}^{j r}\overline{N}^k[\overline{Q}_{mn}]_{,r}-\overline{Q}_{nr}\frac{\partial}{\partial x^m}(\overline{Q}^{jr}\overline{N}^k)
-\overline{Q}_{mr}\frac{\partial }{\partial x^n}(\overline{Q}^{jr}\overline{N}^k)\right]\nonumber\\
&&
+\overline{Q}^{mn}\big[\overline{Q}^{jr}\overline{\Phi}_{,r}\overline{N}^k\big]\Big(\dot{\overline{Q}}_{mn}-(\mathcal{L}_{\overset{\rightharpoonup }{N}}\overline{Q})_{mn}\Big)\Bigg]\delta Q_{jk}\nonumber\\
&&
+\Bigg[-\overline{Q}^{jk}(\dot{\overline{\Phi}}-\mathcal{L}_{\overset{\rightharpoonup}{\overline{N}}}\overline{\Phi})\left[\overline{Q}^{m n}[\overline{Q}_{jk}]_{,n}+\overline{Q}_{kn}\frac{\partial}{\partial x^j}(\overline{Q}^{mn})+\overline{Q}_{jn}\frac{\partial}{\partial x^k}(\overline{Q}^{mn})\right]\nonumber\\
&&\quad
-\overline{Q}^{jk}[\overline{Q}^{m n}\overline{\Phi}_{,n}]\Big(\dot{\overline{Q}}_{jk}-(\mathcal{L}_{\overset{\rightharpoonup }{\overline{N}}}\overline{Q})_{jk}\Big)\Bigg]\delta N_m\nonumber\\
&&
+\bigg[\overline{Q}^{mn}\Big(\dot{\overline{Q}}_{mn}-(\mathcal{L}_{\overset{\rightharpoonup }{\overline{N}}}\overline{Q})_{m n}\Big)(\frac{\partial }{\partial\tau}-\mathcal{L}_{\overset{\rightharpoonup}{\overline{N}}})\bigg]\delta\Phi,\\
\lefteqn{\delta\left[D_j D_k F(\Phi)\right]=\left[-\Big(\overline{D}_n F(\overline{\Phi})\Big)\overline{D}_{(k}\overline{Q}^{m n}\right]\delta Q_{j)m}+\left[\frac{1}{2}\big(\overline{D}_n F(\overline{\Phi})\big)\overline{D}_m\overline{Q}^{m n}\right]\delta Q_{j k}+\left[\overline{D}_j\overline{D}_k F'(\overline{\Phi})\right]\delta\Phi,}\\
\lefteqn{\delta\left[NQ^{jk}(D_j N)D_k F(\Phi)\right]=\left[-\frac{1}{2}\overline{Q}^{m n}\Big(\overline{D}_n F(\overline{\Phi})\Big)\overline{D}_m\big(\overline{N}^j\overline{N}^k\big)-\frac{1}{2}\overline{Q}^{jm}\overline{Q}^{n k}\big(\overline{D}_m\overline{N}^2\big)\Big(\overline{D}_n F(\overline{\Phi})\Big)\right]\delta Q_{jk}}\nonumber\\
&&\qquad\qquad\qquad\qquad\quad
+\left[\overline{Q}^{jk}\Big(\overline{D}_k F(\overline{\Phi})\Big)\overline{D}_j\overline{N}^m\right]\delta N_m
+\left[\frac{1}{2}\big(\overline{D}_j\overline{N}^2\big)\overline{Q}^{jk}\overline{D}_k F'(\overline{\Phi})\right]\delta\Phi,\\
\lefteqn{\delta\Big(-\frac{N}{\sqrt{\det {Q}}}H G_{jkmn}N^m N^n\Big)=\Big[\frac{1}{2}\frac{\overline{N}}{\sqrt{\det {\overline{Q}}}}(\overline{Q}^{m n}+\frac{\overline{N}^m\overline{N}^n}{\overline{N}^2})\overline{H} \overline{G}_{jkrs}\overline{N}^r \overline{N}^s+\frac{1}{2}\frac{\overline{N}}{\sqrt{\det {\overline{Q}}}}\overline{H}\overline{N}^m\overline{N}^n \overline{Q}_{jk}}\nonumber\\
&&\qquad\qquad\qquad\qquad\qquad\qquad\quad
+2\frac{\overline{N}}{\sqrt{\det {\overline{Q}}}}\overline{H} \overline{G}_{jkrs} \overline{N}^n \overline{N}^s \overline{Q}^{rm}\Big]\delta Q_{mn}
+\Big[-2\frac{\overline{N}}{\sqrt{\det {\overline{Q}}}}\overline{H} \overline{N}^r \overline{N}^t \overline{Q}^{s m} \overline{G} _{rts(j}\Big]\delta Q_{k)m}\nonumber\\
&&\qquad\qquad\qquad\qquad\qquad\qquad
+\Big[-\frac{1}{2}\frac{\overline{N}}{\sqrt{\det {\overline{Q}}}}\overline{H} \overline{N}^m \overline{N}^n \overline{Q}_{mn}\Big]\delta Q_{jk}\nonumber\\
&&\qquad\qquad\qquad\qquad\qquad\qquad
+\Big[-\frac{\overline{N}}{\sqrt{\det {\overline{Q}}}}\frac{\overline{N}^m}{\overline{N}^2}\overline{H} \overline{G}_{jkrs}\overline{N}^r \overline{N}^s-2\frac{\overline{N}}{\sqrt{\det {\overline{Q}}}}\overline{H} \overline{G}_{jkrs}\overline{N}^s \overline{Q}^{rm}\Big]\delta N_{m}\nonumber\\
&&\qquad\qquad\qquad\qquad\qquad\qquad
+\Big[-\frac{\overline{N}}{\sqrt{\det {\overline{Q}}}}\overline{G}_{jkrs}\overline{N}^r \overline{N}^s\Big]\delta H,
\end{eqnarray}
where the derivatives inside the square bracket act on all terms including the perturbed variables to its right side, while derivatives surrounded by the round bracket like $(\overline{D}_i\overline{D}_j...)$ act only on the elements inside it. With these identities, the perturbation of the expression $X$ in (\ref{X}) can be expressed as
\begin{eqnarray}
\delta X&=&\Bigg[\overline{N}^2\big[(\overline{D}^i\overline{D}_i\overline{\Phi})K'(\overline{\Phi})+ K(\overline{\Phi})\overline{D}^i\overline{D}_i+ (\overline{Q}^{jk})_{,j}\overline{\Phi}_{,k}K'(\overline{\Phi})
+K(\overline{\Phi})(\overline{Q}^{mn})_{,n}\frac{\partial}{\partial x^m}-\frac12V''(\overline{\Phi})\big]\nonumber\\
&&\quad
+K'(\overline{\Phi})\overline{Q}^{jk}\overline{\Phi}_{,k}\Big[\frac{\overline{N}}{\sqrt{\det\overline{Q}}}\big(\overline{N}\sqrt{\det \overline{Q}}\big)_{,j}\Big]
+K(\overline{\Phi})\Big[\frac{\overline{N}}{\sqrt{\det \overline{Q}}}\big(\overline{N}\sqrt{\det \overline{Q}}\big)_{,j}\Big]\overline{Q}^{j k}\frac{\partial}{\partial x^k}\Bigg]\delta \Phi\nonumber\\
&&
+\Bigg[-\overline{N}^j\overline{N}^k\Big[K(\overline{\Phi})\overline{D}^i\overline{D}_i\overline{\Phi} +K(\overline{\Phi})(\overline{Q}^{m n})_{, m}\overline{\Phi} _{,n}-\frac{1}{2}V'(\overline{\Phi})\Big]
+\overline{N}^2K(\overline{\Phi})\Big[-\frac{\partial}{\partial x^n}(\overline{Q}^{jm}\overline{Q}^{kn}\overline{\Phi}_{,m})\Big]\nonumber\\
&&\quad
+K(\overline{\Phi})\overline{Q}^{mn}\overline{\Phi}_{,n}\Big[-\frac12\frac{\overline{N}}{\sqrt{\det {\overline{Q}}}}(\overline{Q}^{j k}+\frac{\overline{N}^j\overline{N}^k}{\overline{N}^2})[\overline{N}\sqrt{\det {\overline{Q}}}]_{,m}
+\frac12\frac{\overline{N}}{\sqrt{\det {\overline{Q}}}}\frac{\partial}{\partial x^m}\big(\overline{N}\sqrt{\det {\overline{Q}}}(\overline{Q}^{j k}-\frac{\overline{N}^j\overline{N}^k}{\overline{N}^2})\big)\Big]\nonumber\\
&&\quad
+K(\overline{\Phi})\Big[\frac{\overline{N}}{\sqrt{\det {\overline{Q}}}}(\overline{N}\sqrt{\det {\overline{Q}}})_{,n}\Big]\Big[-\overline{Q}^{jm}\overline{Q}^{kn}\overline{\Phi}_{,m}\Big]\Bigg]\delta Q_{jk}\nonumber\\
&&
+\Bigg[2\overline{N}^m\Big(K(\overline{\Phi})\overline{D}^i\overline{D}_i\overline{\Phi} +K(\overline{\Phi})\left[\overline{Q}^{m n}\right]_{, m}\overline{\Phi} _{,n}-\frac{1}{2}V'(\overline{\Phi})\Big)
+K(\overline{\Phi})\overline{Q}^{jk}\overline{\Phi}_{,k}\Big[\frac{\overline{N}}{\sqrt{\det\overline{ Q}}}\frac{\overline{N}^m}{\overline{N}^2}\big(\overline{N}\sqrt{\det \overline{Q}}\big)_{,j}\nonumber\\
&&\quad
+\frac{\overline{N}}{\sqrt{\det \overline{Q}}}\frac{\partial}{\partial x^j}\big(\frac{\overline{N}^m}{\overline{N}^2}\overline{N}\sqrt{\det{\overline{Q}}}\big)\Big]\Bigg]\delta N_m ,\label{deltaX}
\end{eqnarray}
and the perturbation of the expression $Y$ in (\ref{Y}) yields
\begin{eqnarray}
\delta Y&=&\Bigg[\frac{F'(\overline{\Phi})}{4}[\overline{G}^{-1}]^{r s t u}\Big(\dot{\overline{Q}}_{r s}-(\mathcal{L}_{\overset{\rightharpoonup }{\overline{N}}}\overline{Q})_{r s}\Big)\Big(\dot{\overline{Q}}_{t u}-(\mathcal{L}_{\overset{\rightharpoonup }{\overline{N}}}{\overline{Q}})_{t u}\Big)
-F(\overline{\Phi})''(\Phi)\overline{Q}^{m n}\Big(\dot{\overline{Q}}_{m n}-(\mathcal{L}_{\overset{\rightharpoonup }{\overline{N}}}\overline{Q})_{m n}\Big)\Big(\dot{\overline{\Phi}}-\mathcal{L}_{\overset{\rightharpoonup }{\overline{N}}}\overline{\Phi }\Big)\nonumber\\
&&\quad
-F'(\overline{\Phi})\overline{Q}^{m n}\Big(\dot{\overline{Q}}_{m n}-(\mathcal{L}_{\overset{\rightharpoonup }{\overline{N}}}\overline{Q})_{m n}\Big)(\frac{\partial}{\partial \tau}-\mathcal{L}_{\overset{\rightharpoonup} {\overline{N}}})+\overline{N}^2\overline{Q}^{mn}\overline{\Phi}_{,m}\overline{\Phi}_{,n}\big(2\overline{N}^2K(\overline{\Phi})
+\overline{N}^2\frac{F(\overline{\Phi})}{F'(\overline{\Phi})}K'(\overline{\Phi})\big)'
\nonumber\\
&&\quad
+2\big(2K(\overline{\Phi})+\frac{F(\overline{\Phi})}{F'(\overline{\Phi})}K'(\overline{\Phi})\big)
\overline{Q}^{mn}\overline{\Phi}_{,m}\frac{\partial}{\partial x^n}+3\overline{N}^2V'(\overline{\Phi})
+5\overline{N}^2\overline{D}^i\overline{D}_i F'(\overline{\Phi})+3\overline{N}(\overline{D}^i\overline{N})\overline{D}_i F'(\overline{\Phi})\nonumber\\
&&\quad
-\overline{N}^2\overline{R}^{(3)}F'(\overline{\Phi})\Bigg]\delta\Phi\nonumber\\
&&
+\Bigg[\frac{F(\overline{\Phi})}{2}\Big(\dot{\overline{Q}}_{r s}-(\mathcal{L}_{\overset{\rightharpoonup }{\overline{N}}}\overline{Q})_{r s}\Big)
\bigg[[\overline{G}^{-1}]^{rsmn}(\frac{\partial}{\partial \tau} - \mathcal{L}_{\overset{\rightharpoonup} {\overline{N}}})
-[\overline{G}^{-1}]^{turm}\overline{Q}^{sn}\Big(\dot{\overline{Q}}_{tu}-\mathcal{L}_{\overset{\rightharpoonup} {\overline{N}}}\overline{Q}_{tu}\Big)\nonumber\\
&&\qquad
+[\overline{G}^{-1}]^{rsvw}\Big(\overline{Q}^{mt}\overline{N}^n[\overline{Q}_{vw}]_{,t}+2\overline{Q}_{tw}\frac{\partial}{\partial x^v}(\overline{Q}^{mt}\overline{N}^n)\Big)\bigg]\nonumber\\
&&\quad
-F'(\overline{\Phi})\bigg[-\overline{Q}^{v m}\overline{Q}^{nw}\Big(\dot{\overline{Q}}_{vw}-(\mathcal{L}_{\overset{\rightharpoonup }{\overline{N}}}\overline{Q})_{vw}\Big)\big(\dot{\overline{\Phi}}-\mathcal{L}_{\overset{\rightharpoonup }{\overline{N}}}\overline{\Phi}\big)
+\overline{Q}^{mn}\big(\dot{\overline{\Phi}}-\mathcal{L}_{\overset{\rightharpoonup}{\overline{N}}}\overline{\Phi}\big)(\frac{\partial}{\partial\tau}- \mathcal{L}_{\overset{\rightharpoonup}{\overline{N}}})\nonumber\\
&&\qquad
+ \overline{Q}^{vw}\Big(\dot{\overline{Q}}_{vw}-(\mathcal{L}_{\overset{\rightharpoonup }{\overline{N}}}\overline{Q})_{vw}\Big)[\overline{Q}^{mr}\overline{\Phi}_{,r}N^n]\nonumber\\
&&\qquad
- \overline{Q}^{vw}\big(\dot{\overline{\Phi}}-\mathcal{L}_{\overset{\rightharpoonup }{\overline{N}}}\overline{\Phi}\big)\Big[-\overline{Q}^{mr}\overline{N}^n(\overline{Q}_{vw})_{,r}-\overline{Q}_{wr}\frac{\partial}{\partial x^v}(\overline{Q}^{mr}\overline{N}^n)-\overline{Q}_{vr}\frac{\partial}{\partial x^w}(\overline{Q}^{mr}\overline{N}^n)\Big]\bigg]\nonumber\\
&&\quad
-\overline{N}^m\overline{N}^n\big(2K(\overline{\Phi})+\frac{F(\overline{\Phi})}{F'(\overline{\Phi})}K'(\overline{\Phi})\big)\overline{\Phi}^{,i}
\overline{\Phi}_{,i}-\overline{N}^2\big(2K(\overline{\Phi})+\frac{F(\overline{\Phi})}{F'(\overline{\Phi})}K'(\overline{\Phi})\big)\overline{\Phi}^{,m}
\overline{\Phi}^{,n}\nonumber\\
&&\quad
-3V(\overline{\Phi})\overline{N}^m\overline{N}^n+5\Big[-\overline{N}^m\overline{N}^n\overline{Q}^{vw}(\overline{D}_v \overline{D}_w F(\overline{\Phi}))+\frac12\overline{N}^2\overline{Q}^{mn}(\overline{D}_wF(\overline{\Phi}))\overline{D}_v\overline{Q}^{vw}
\nonumber\\
&&\qquad
-\overline{N}^2\overline{Q}^{mr}(\overline{D}_wF(\overline{\Phi}))\overline{D}_r\overline{Q}^{nw}-\overline{N}^2\overline{Q}^{mv}
\overline{Q}^{nw}(\overline{D}_v\overline{D}_wF(\overline{\Phi}))\Big]\nonumber\\
&&\quad
+3\Big[-\frac12\overline{Q}^{vw}(\overline{D}_vF(\overline{\Phi}))\overline{D}_w(\overline{N}^m\overline{N}^n)-\frac12\overline{Q}^{mv}\overline{Q}^{nw}
(\overline{D}_v\overline{N}^2)(\overline{D}_wF(\overline{\Phi}))\Big]\nonumber\\
&&\quad
+F(\overline{\Phi})\overline{R}^{(3)}\overline{N}^m\overline{N}^n-\overline{N}^2F(\overline{\Phi})\Big[[\overline{G}^{-1}]^{mnvw}\overline{D}_v \overline{D}_w-\overline{R}^{mn}\Big]\nonumber\\
&&\quad
+\frac12\overline{H}(\sigma)\overline{Q}^{vw}\overline{N}_v \overline{N}_w\Big[-\frac12\frac{\overline{N}}{\sqrt{\det {\overline{Q}}}}(\overline{Q}^{mn}+\frac{\overline{N}^m\overline{N}^n}{\overline{N}^2})\Big]-\frac12\frac{\overline{N}}{\sqrt{\det {\overline{Q}}}}\overline{H}(\sigma)\overline{N}^m\overline{N}^n\Bigg]\delta Q_{mn}\nonumber\\
&&
+\Bigg[-\frac{F(\overline{\Phi})}{2}\Big(\dot{\overline{Q}}_{r s}-(\mathcal{L}_{\overset{\rightharpoonup }{\overline{N}}}\overline{Q})_{r s}\Big)[\overline{G}^{-1}]^{rsvn}\Big(\overline{Q}^{mt}[\overline{Q}_{vn}]_{,t}
+2\overline{Q}_{tv}\frac{\partial}{\partial x^n}(\overline{Q}^{mt})\Big)\nonumber\\
&&\quad
-F'(\overline{\Phi})\bigg[-\overline{Q}^{v w}\big(\dot{\overline{\Phi}}-\mathcal{L}_{\overset{\rightharpoonup }{\overline{N}}}\overline{\Phi}\big)\big[\overline{Q}^{mr}(\overline{Q}_{vw})_{,r}+\overline{Q}_{w r}\frac{\partial}{\partial x^v}(\overline{Q}^{mr})+\overline{Q}_{vr}\frac{\partial}{\partial x^w}(\overline{Q}^{mr})\big]\nonumber\\
&&\qquad
-\overline{Q}^{vw}\Big(\dot{\overline{Q}}_{vw}-(\mathcal{L}_{\overset{\rightharpoonup }{\overline{N}}}\overline{Q})_{vw}\Big)[\overline{Q}^{mn}\overline{\Phi}_{,n}]\bigg]\nonumber\\
&&\quad
+2\overline{N}^m\big(2K(\overline{\Phi})+\frac{F(\overline{\Phi})}{F'(\overline{\Phi})}K'(\overline{\Phi})\big)\overline{\Phi}^{,i}\overline{\Phi}_{,i}
+6V(\overline{\Phi})\overline{N}^m\nonumber\\
&&\quad
+10\big[\overline{N}^m\overline{Q}^{vw}(\overline{D}_v\overline{D}_wF(\overline{\Phi}))\big]+3\big[\overline{Q}^{vw}(\overline{D}_wF(\overline{\Phi}))
\overline{D}_v\overline{N}^m\big]
-2F(\overline{\Phi})\overline{R}^{(3)}\overline{N}^m\nonumber\\
&&\quad
+\frac{1}{2}\frac{\overline{N}}{\sqrt{\det \overline{Q}}}\overline{H}(\sigma )\overline{Q}^{rs}\overline{N}_r \overline{N}_s(\frac{\overline{N}^m}{\overline{N}^2})+\frac{\overline{N}}{\sqrt{\det \overline{Q}}}\overline{H}(\sigma )\overline{Q}^{mn}\overline{N}_n\Bigg]\delta N_m\nonumber\\
&&
+\Bigg[\frac12\frac{\overline{N}}{\sqrt{\det \overline{Q}}}\overline{Q}^{mn}\overline{N}_m\overline{N}_n\Bigg]\delta H.\label{deltaY}
\end{eqnarray}
Furthermore, we will need the perturbation of $C(\tau, \sigma)$ that is given by
\begin{eqnarray}
\delta C&=&\Bigg[\frac{1}{4}\frac{\sqrt{\det \overline{Q}}}{\overline{N}^2} F(\overline{\Phi})' \left[\overline{G}^{-1}\right]^{r s t u}\Big(\dot{\overline{Q}}_{r s}-(\mathcal{L}_{\overset{\rightharpoonup }{\overline{N}}}\overline{Q})_{r s}\Big)\Big(\dot{\overline{Q}}_{t u}-(\mathcal{L}_{\overset{\rightharpoonup }{\overline{N}}}{\overline{Q}})_{t u}\Big)-\sqrt{\det \overline{Q}} \overline{R}^{(3)}F'(\overline{\Phi})\nonumber\\
&&\quad
+\sqrt{\det \overline{Q}} K'(\overline{\Phi})\overline{Q}^{jk}\overline{\Phi}_{,j}\overline{\Phi}_{,k}+2 \sqrt{\det \overline{Q}} K(\overline{\Phi}) \overline{Q}^{jk} \overline{\Phi}_{,j} \frac{\partial}{\partial x^k}- F'(\overline{\Phi}) \overline{Q}^{m n}\Big(\dot{\overline{Q}}_{m n}-(\mathcal{L}_{\overset{\rightharpoonup }{\overline{N}}}\overline{Q})_{m n}\Big)(\frac{\partial}{\partial \tau} - \mathcal{L}_{\overset{\rightharpoonup}{\overline{N}}})\nonumber\\
&&\quad
-F''(\overline{\Phi})\overline{Q}^{m n}\Big(\dot{\overline{Q}}_{m n}-(\mathcal{L}_{\overset{\rightharpoonup}{\overline{N}}}\overline{Q})_{m n}\Big)\big(\dot{\overline{\Phi}}-\mathcal{L}_{\overset{\rightharpoonup }{\overline{N}}}\overline{\Phi}\big)+ \frac{\sqrt{\det \overline{Q}}}{\overline{N}^2}K'(\overline{\Phi})\big(\dot{\overline{\Phi}}-\mathcal{L}_{\overset{\rightharpoonup }{\overline{N}}}\overline{\Phi}\big)^2
\nonumber\\
&&\quad
+2\frac{\sqrt{\det \overline{Q}}}{\overline{N}^2} K(\overline{\Phi})\big(\dot{\overline{\Phi}}-\mathcal{L}_{\overset{\rightharpoonup }{\overline{N}}}\overline{\Phi}\big)(\frac{\partial}{\partial \tau} - \mathcal{L}_{\overset{\rightharpoonup} {\overline{N}}})+2 \sqrt{\det\overline{Q}}\overline{Q}^{jk} \overline{D}_{j} \overline{D}_{k} F'(\overline{\Phi})+ \sqrt{\det \overline{Q}} V'(\overline{\Phi})\Bigg]\delta\Phi\nonumber\\
&&
+\Bigg[\frac12 \overline{Q}^{jk}\overline{C}+\frac{\overline{N}^j\overline{N}^k}{\overline{N}^2}\bigg(\frac{\sqrt{\det \overline{Q}}F(\overline{\Phi} )}{4\overline{N}^2}\left[\overline{G}^{-1}\right]^{rstu}\Big(\dot{\overline{Q}}_{rs}-(\mathcal{L}_{\overset{\rightharpoonup}{\overline{N}}}\overline{Q})_{r s}\Big)\Big(\dot{\overline{Q}}_{tu}-(\mathcal{L}_{\overset{\rightharpoonup }{\overline{N}}}\overline{Q})_{tu}\Big)\nonumber\\
&&\quad
-\frac{\sqrt{\det \overline{Q}}}{\overline{N}^2}F'(\overline{\Phi})\overline{Q}^{rs}\Big(\dot{\overline{Q}}_{r s}-(\mathcal{L}_{\overset{\rightharpoonup}{\overline{N}}}\overline{Q})_{r s}\Big)\big(\dot{\overline{\Phi}}-\mathcal{L}_{\overset{\rightharpoonup }{\overline{N}}}\overline{\Phi}\big)+\frac{\sqrt{\det\overline{Q}}}{\overline{N}^2}K(\overline{\Phi} )(\dot{\overline{\Phi}}-\mathcal{L}_{\overset{\rightharpoonup }{\overline{N}}}\overline{\Phi})^2\bigg)\nonumber\\
&&\quad
+\frac{\sqrt{\det \overline{Q}}}{\overline{N}^2}\frac{F(\overline{\Phi})}{2}\Big(\dot{\overline{Q}}_{rs}-(\mathcal{L}_{\overset{\rightharpoonup }{\overline{N}}}\overline{Q})_{r s}\Big)
\Big[[\overline{G}^{-1}]^{rsjk}(\frac{\partial}{\partial\tau}-\mathcal{L}_{\overset{\rightharpoonup}{\overline{N}}})
-[\overline{G}^{-1}]^{turj}\overline{Q}^{sk}(\dot{\overline{Q}}_{tu}-\mathcal{L}_{\overset{\rightharpoonup}{\overline{N}}}\overline{Q}_{tu})\nonumber\\
&&\qquad
+[\overline{G}^{-1}]^{rsmn}\Big(\overline{Q}^{jt}\overline{N}^k[\overline{Q}_{mn}]_{,t}+2\overline{Q}_{tn}\frac{\partial}{\partial x^m}(\overline{Q}^{jt}\overline{N}^k)\Big)\Big]-\sqrt{\det \overline{Q}}F(\overline{\Phi})\big [[\overline{G}^{-1}]^{jkmn}\overline{D}_m \overline{D}_n-\overline{R}^{jk}]\nonumber\\
&&\quad
-\frac{\sqrt{\det\overline{Q}}}{\overline{N}^2}K(\overline{\Phi})\overline{\Phi}^{,j}\overline{\Phi}^{,k}-F'(\overline{\Phi})\Big[-\overline{Q}^{m j}\overline{Q}^{k n}\Big(\dot{\overline{Q}}_{m n}-(\mathcal{L}_{\overset{\rightharpoonup}{\overline{N}}}\overline{Q})_{mn}\Big)\big(\dot{\overline{\Phi}}-\mathcal{L}_{\overset{\rightharpoonup }{\overline{N}}}\overline{\Phi}\big)\nonumber\\
&&\qquad
+\overline{Q}^{jk}\big(\dot{\overline{\Phi}}-\mathcal{L}_{\overset{\rightharpoonup }{\overline{N}}}\overline{\Phi}\big)(\frac{\partial}{\partial\tau}-\mathcal{L}_{\overset{\rightharpoonup} {\overline{N}}})+\overline{Q}^{m n}\Big(\dot{\overline{Q}}_{mn}-(\mathcal{L}_{\overset{\rightharpoonup }{\overline{N}}}\overline{Q})_{m n}\Big)[\overline{Q}^{jr}\overline{\Phi}_{,r}\overline{N}^k]\nonumber\\
&&\qquad
- \overline{Q}^{m n}\big(\dot{\overline{\Phi}}-\mathcal{L}_{\overset{\rightharpoonup }{\overline{N}}}\overline{\Phi}\big)\big[-\overline{Q}^{jr}\overline{N}^k(\overline{Q}_{mn})_{,r}-\overline{Q}_{nr}\frac{\partial}{\partial x^m}(\overline{Q}^{jr}\overline{N}^k)-\overline{Q}_{mr}\frac{\partial}{\partial x^n}(\overline{Q}^{jr}\overline{N}^k)\big]\Big]\nonumber\\
&&\quad
+2\frac{\sqrt{\det \overline{Q}}}{\overline{N}^2}K(\overline{\Phi})\big(\dot{\overline{\Phi}}-\mathcal{L}_{\overset{\rightharpoonup }{\overline{N}}}\overline{\Phi}\big)[\overline{Q}^{jm}\overline{\Phi}_{,m}\overline{N}^k]
+2\Big[\frac{1}{2}\sqrt{\det\overline{Q}}\overline{Q}^{j k}\Big(\overline{D}_nF(\overline{\Phi})\Big)\overline{D}_m\overline{Q}^{m n} \nonumber\\
&&\qquad
-\sqrt{\det \overline{Q}}\overline{Q}^{j m}\Big(\overline{D}_nF(\overline{\Phi})\Big)\overline{D}_{m}\overline{Q}^{k n} -\overline{Q}^{m j}\overline{Q}^{k n}\sqrt{\det \overline{Q}}\Big(\overline{D}_m\overline{D}_nF(\overline{\Phi})\Big)\Big]\Bigg]\delta Q_{jk}\nonumber\\
&&
+\Bigg[-2\frac{\overline{N}^j}{\overline{N}^2}\bigg(\frac{\sqrt{\det \overline{Q}}F(\overline{\Phi} )}{4\overline{N}^2}\left[\overline{G}^{-1}\right]^{r s t u}\Big(\dot{\overline{Q}}_{r s}-(\mathcal{L}_{\overset{\rightharpoonup }{\overline{N}}}\overline{Q})_{r s}\Big)\Big(\dot{\overline{Q}}_{t u}-(\mathcal{L}_{\overset{\rightharpoonup }{\overline{N}}}\overline{Q})_{t u}\Big)\nonumber\\
&&\qquad
-\frac{\sqrt{\det \overline{Q}}}{\overline{N}^2}\overline{F}'(\overline{\Phi})\overline{Q}^{rs}\Big(\dot{\overline{Q}}_{rs}-(\mathcal{L}_{\overset{\rightharpoonup}{\overline{N}}}\overline{Q})_{rs}\Big)
\big(\dot{\overline{\Phi}}-\mathcal{L}_{\overset{\rightharpoonup}{\overline{N}}}\overline{\Phi}\big)+\frac{\sqrt{\det\overline{Q}}}{\overline{N}^2}K(\overline{\Phi})(\dot{\overline{\Phi} }-\mathcal{L}_{\overset{\rightharpoonup}{\overline{N}}}\overline{\Phi})^2\bigg)\nonumber\\
&&\quad
-\frac{\sqrt{\det \overline{Q}}}{\overline{N}^2}\frac{F(\overline{\Phi})}{2}\Big(\dot{\overline{Q}}_{r s}-(\mathcal{L}_{\overset{\rightharpoonup }{\overline{N}}}\overline{Q})_{r s}\Big)[\overline{G}^{-1}]^{rsjn}\big[\overline{Q}^{mt}[\overline{Q}_{jn}]_{,t}+2\overline{Q}_{tj}\frac{\partial}{\partial x^n}(\overline{Q}^{mt})\big]\nonumber\\
&&\quad
-\frac{\sqrt{\det \overline{Q}}}{\overline{N}^2}F'(\overline{\Phi})\Big[-\overline{Q}^{jk}\big(\dot{\overline{\Phi}}-\mathcal{L}_{\overset{\rightharpoonup }{\overline{N}}}\overline{\Phi}\big)\big[\overline{Q}^{mn}[\overline{Q}_{jk}]_{,n}+\overline{Q}_{kn}\frac{\partial}{\partial x^j}(\overline{Q}^{mn})+\overline{Q}_{j n}\frac{\partial}{\partial x^k}(\overline{Q}^{mn})\big]\nonumber\\
&&\qquad
-\overline{Q}^{jk}\Big(\dot{\overline{Q}}_{jk}-(\mathcal{L}_{\overset{\rightharpoonup}{\overline{N}}}\overline{Q})_{jk}\Big)[\overline{Q}^{mn}\overline{\Phi}_{,n}]\Big]
+2\frac{\sqrt{\det \overline{Q}}}{\overline{N}^2}K(\overline{\Phi})\big(\dot{\overline{\Phi}}-\mathcal{L}_{\overset{\rightharpoonup }{\overline{N}}}\overline{\Phi}\big)[-\overline{Q}^{mn}\overline{\Phi}_{,n}]\Bigg]\delta N_m.\label{deltaC}
\end{eqnarray}
With the above formulas, we have all intermediate results available in order to derive the evolution equations for the linear perturbations $\delta \Phi$ and $\delta Q_{jk}$. We will start to discuss the case for $\delta \Phi$. First we write the evolution equation for $\delta\Phi$ in the following form
\begin{eqnarray}
\delta \ddot{\Phi}=\textbf{A}\delta\Phi+\textbf{B}^{jk}\delta Q_{jk}+\textbf{C}^m\delta N_m+\textbf{D}\delta H.\nonumber
\end{eqnarray}
This is always possible with appropriate choices for the coefficients $\textbf{A}$, $\textbf{B}^{jk}$, $\textbf{C}^m$ and $\textbf{D}$. Using the explicit form of the coefficients $\textbf{A}$, $\textbf{B}^{jk}$, $\textbf{C}^m$, $\textbf{D}$ in the above equation we obtain
\begin{eqnarray}
\delta \ddot{\Phi}&=&\Bigg[\bigg[\frac{\dot{\overline{N}}}{\overline{N}}-\frac{\Big(\sqrt{\det \overline{Q}}\Big)^{\cdot}}{\sqrt{\det \overline{Q}}}+\frac{\overline{N}}{\sqrt{\det \overline{Q}}}\Big(\mathcal{L}_{\overset{\rightharpoonup }{\overline{N}}}(\frac{\sqrt{\det \overline{Q}} }{\overline{N}})\Big)\bigg](\frac{\partial}{\partial \tau}
-\mathcal{L}_{\overset{\rightharpoonup}{\overline{N}}})+\Big[\mathcal{L}_{\overset{\rightharpoonup} {\overline{N}}}(\frac{\partial}{\partial \tau}
-\mathcal{L}_{\overset{\rightharpoonup}{\overline{N}}})+\frac{\partial}{\partial \tau}
\mathcal{L}_{\overset{\rightharpoonup}{\overline{N}}}\Big]\nonumber\\
&&\quad
-\bigg(\frac{3F'(\overline{\Phi})F''(\overline{\Phi})+F(\overline{\Phi})K'(\overline{\Phi})}
{3(F'(\overline{\Phi}))^2+2F(\overline{\Phi})K(\overline{\Phi})}\bigg)'\big(\dot{\overline{\Phi}}-\mathcal{L}_{\overset{\rightharpoonup}{N}}
\overline{\Phi}\big)^2
-2\bigg(\frac{3F'(\overline{\Phi})F''(\overline{\Phi})+F(\overline{\Phi})K'(\overline{\Phi})}{3(F'(\overline{\Phi}))^2
+2F(\overline{\Phi})K(\overline{\Phi})}\bigg)\nonumber\\
&&\quad
\cdot\big(\dot{\overline{\Phi}}-\mathcal{L}_{\overset{\rightharpoonup}{\overline{N}}}\overline{\Phi}\big)
(\frac{\partial}{\partial \tau}- \mathcal{L}_{\overset{\rightharpoonup}{\overline{N}}})-\frac32\frac{\overline{N}^2}{\sqrt{\det \overline{Q}}}\bigg(\frac{F'(\overline{\Phi})}{3(F'(\overline{\Phi}))^2+2F(\overline{\Phi})K(\overline{\Phi})}\bigg)'\overline{C}(\tau, \sigma)\nonumber\\
&&\quad
+\bigg(\frac{2F(\overline{\Phi})}{3(F'(\overline{\Phi}))^2+2F(\overline{\Phi})K(\overline{\Phi})}\bigg)'\, \overline{X}+\bigg(\frac{F'(\overline{\Phi})}{3(F'(\overline{\Phi}))^2+2F(\overline{\Phi})K(\overline{\Phi})}\bigg)'\, \overline{Y}\Bigg]\delta \Phi\nonumber\\
&&
+\Bigg[\big(\dot{\overline{\Phi}}-\mathcal{L}_{\overset{\rightharpoonup }{\overline{N}}}\overline{\Phi}\big)\Big[-\frac12(\frac{\partial}{\partial \tau}-\mathcal{L}_{\overset{\rightharpoonup} {\overline{N}}})(\overline{Q}^{jk}+\frac{\overline{N}^j\overline{N}^k}{\overline{N}^2})-\frac{\overline{N}}{\sqrt{\det {\overline{Q}}}}\frac{\partial}{\partial x^m}(\frac{\sqrt{\det {\overline{Q}}}}{\overline{N}}\overline{Q}^{jm}\overline{N}^k)\Big]\nonumber\\
&&\quad
+(\overline{Q}^{jm}\overline{\Phi}_{,m}\overline{N}^k)\bigg[\frac{\dot{\overline{N}}}{\overline{N}}-\frac{\Big(\sqrt{\det\overline{Q}}\Big)^{\cdot }}{\sqrt{\det\overline{Q}}}+\frac{\overline{N}}{\sqrt{\det \overline{Q}}}\Big(\mathcal{L}_{\overset{\rightharpoonup }{\overline{N}}}(\frac{ \sqrt{\det \overline{Q}}}{\overline{N}})\Big)\bigg]\nonumber\\
&&\quad
+\Big[-\overline{Q}^{jm}\overline{Q}^{kn}\overline{N}_m\big(\dot{\overline{\Phi}}-\mathcal{L}_{\overset{\rightharpoonup }{\overline{N}}}\overline{\Phi}\big)_{,n}-(\frac{\partial}{\partial \tau}-\mathcal{L}_{\overset{\rightharpoonup} {\overline{N}}})\Big(\overline{Q}^{jm}\overline{Q}^{kn}\overline{N}_m\overline{\Phi}_{,n}\Big)\Big]\nonumber\\
&&\quad
-2\bigg(\frac{3F'(\overline{\Phi})F''(\overline{\Phi})+F(\overline{\Phi})K'(\overline{\Phi})}
{3(F'(\overline{\Phi}))^2+2F(\overline{\Phi})K(\overline{\Phi})}\bigg)\big(\dot{\overline{\Phi}}-\mathcal{L}_{\overset{\rightharpoonup }{\overline{N}}}\overline{\Phi}\big)[\overline{Q}^{jr}\overline{\Phi}_{,r}\overline{N}^k]\nonumber\\
&&\quad
+\frac32\bigg(\frac{F'(\overline{\Phi})}{3(F'(\overline{\Phi}))^2+2F(\overline{\Phi})K(\overline{\Phi})}\bigg)\bigg[\frac{1}{\sqrt{\det \overline{Q}}}\Big(\overline{N}^j\overline{N}^k+\frac{1}{2}\overline{N}^2\overline{Q}^{jk }\Big)\bigg]\overline{C}(\tau, \sigma)\Bigg]\delta Q_{jk}\nonumber\\
&&
+\Bigg[\big(\dot{\overline{\Phi}}-\mathcal{L}_{\overset{\rightharpoonup}{\overline{N}}}
\overline{\Phi}\big)\Big[(\frac{\partial}{\partial \tau}-\mathcal{L}_{\overset{\rightharpoonup} {\overline{N}}})\frac{\overline{N}^m}{\overline{N}^2}+\frac{\overline{N}}{\sqrt{\det {\overline{Q}}}}\frac{\partial}{\partial x^k}(\frac{\sqrt{\det {\overline{Q}}}}{\overline{N}}\overline{Q}^{mk})\Big]\nonumber\\
&&\quad
+\left[\frac{\dot{\overline{N}}}{\overline{N}}-\frac{\Big(\sqrt{\det \overline{Q}}\Big)^{\cdot}}{\sqrt{\det \overline{Q}}}+\frac{\overline{N}}{\sqrt{\det \overline{Q}}}\Big(\mathcal{L}_{\overset{\rightharpoonup }{\overline{N}}}(\frac{ \sqrt{\det \overline{Q}} }{\overline{N}})\Big)\right][-\overline{Q}^{mk}\overline{\Phi}_{,k}]\nonumber\\
&&\quad
+\bigg[\overline{Q}^{mk}\big(\dot{\overline{\Phi}}-\mathcal{L}_{\overset{\rightharpoonup }{\overline{N}}}\overline{\Phi}\big)_{,k}+(\frac{\partial}{\partial \tau} - \mathcal{L}_{\overset{\rightharpoonup} {\overline{N}}})(\overline{Q}^{mk}\overline{\Phi}_{,k})\bigg]\nonumber\\
&&\quad
+2\bigg(\frac{3F'(\overline{\Phi})F''(\overline{\Phi})+F(\overline{\Phi})K'(\overline{\Phi})}
{3(F'(\overline{\Phi}))^2+2F(\overline{\Phi})K(\overline{\Phi})}\bigg)\big(\dot{\overline{\Phi}}-\mathcal{L}_{\overset{\rightharpoonup }{\overline{N}}}\overline{\Phi}\big)[\overline{Q}^{mn}\overline{\Phi}_{,n}]\nonumber\\
&&\quad
-\frac{3\overline{N}^m}{\sqrt{\det \overline{Q}}}\frac{F'(\overline{\Phi})}{3(F'(\overline{\Phi}))^2+2F(\overline{\Phi})K(\overline{\Phi})}\overline{C}(\tau, \sigma)\Bigg]\delta N_m\nonumber\\
&&
-\frac32\frac{\overline{N}^2}{\sqrt{\det \overline{Q}}}\bigg(\frac{F'(\overline{\Phi})}{3(F'(\overline{\Phi}))^2+2F(\overline{\Phi})K(\overline{\Phi})}\bigg)\delta C+\bigg(\frac{2F(\overline{\Phi})}{3(F'(\overline{\Phi}))^2+2F(\overline{\Phi})K(\overline{\Phi})}\bigg)\delta X\nonumber\\
&&\
+\bigg(\frac{F'(\overline{\Phi})}{3(F'(\overline{\Phi}))^2+2F(\overline{\Phi})K(\overline{\Phi})}\bigg)\delta Y,\label{deltaphi}
\end{eqnarray}
Notice that we did not write down the explicit expressions of $\delta C$, $\delta X$ and $\delta Y$ here in order to show the final equation of motion in more compact form. Now we consider the evolution equations for the linear perturbation $\delta Q_{jk}$. Analogue to the case of $\delta\Phi$ we can also formulate the final evolution equations in the following form:
\begin{eqnarray}
\delta Q_{jk}=\textbf{U}_{jk}\delta\Phi+\textbf{V}\delta Q_{jk}+\textbf{W}_{jk}^{mn}\delta Q_{mn}+\textbf{X}^{m}_{(j}\delta Q_{k)m}+\textbf{Y}_{jk}^{m}\delta N_{m}+\textbf{Z}\delta H,\nonumber
\end{eqnarray}
with appropriate choices for the coefficients $\textbf{U}_{jk}$, $\textbf{V}$, $\textbf{W}_{jk}^{mn}$, $\textbf{X}^{m}_{(j}$, $\textbf{Y}_{jk}^{m}$ and $\textbf{Z}$. Using the explicit form of these coefficients the final evolution equation is given by
\begin{eqnarray}
\delta \ddot{Q}_{jk}&=&\Bigg[2\overline{N}^2\overline{\Phi}_{,j}
\overline{\Phi}_{,k}\Big(\frac{K(\overline{\Phi})}{F(\overline{\Phi})}\Big)'+4\overline{N}^2\frac{K(\overline{\Phi})}{F(\overline{\Phi})}
\overline{\Phi}_{,j}\frac{\partial}{\partial x^k}-\Big(\dot{\overline{Q}}_{jk}-(\mathcal{L}_{\overset{\rightharpoonup }{\overline{N}}}\overline{Q})_{jk}\Big)\big(\dot{\overline{\Phi}}-\mathcal{L}_{\overset{\rightharpoonup }{\overline{N}}}\overline{\Phi}\big)\Big(\frac{F'(\overline{\Phi})}{F(\overline{\Phi})}\Big)'\nonumber\\
&&\quad
-\frac{F'(\overline{\Phi})}{F(\overline{\Phi})}
\Big(\dot{\overline{Q}}_{jk}-(\mathcal{L}_{\overset{\rightharpoonup }{\overline{N}}}\overline{Q})_{jk}\Big)(\frac{\partial}{\partial \tau} - \mathcal{L}_{\overset{\rightharpoonup} {\overline{N}}})-2\overline{N}^2\frac{F'(\overline{\Phi})}{(F(\overline{\Phi}))^2}\big(\overline{D}_j \overline{D}_k F(\overline{\Phi})\big)+\frac{2\overline{N}^2}{F(\overline{\Phi})}\overline{D}_j \overline{D}_k F'(\overline{\Phi})\nonumber\\
&&\quad
+\frac{\overline{N}F'(\overline{\Phi})}{\sqrt{\det \overline{Q}}F^2(\overline{\Phi})}\overline{H}(\sigma)\overline{G}_{jkmn}\overline{N}^m\overline{N}^n
-\overline{Q}_{jk}\frac{F'(\overline{\Phi})}{(F(\overline{\Phi}))^2}\Big[\overline{N}(\overline{D}^i\overline{N})(\overline{D}_i F(\overline{\Phi}))+\overline{N}^2(\overline{D}^i\overline{D}_i F(\overline{\Phi}))\nonumber\\
&&\qquad
+\overline{N}^2V(\overline{\Phi})\Big]+\frac{\overline{Q}_{jk}}{F(\overline{\Phi})}\Big[\overline{N}(\overline{D}^i\overline{N})\overline{D}_i F'(\overline{\Phi})+\overline{N}^2 \overline{D}^i\overline{D}_i F'(\overline{\Phi})+\overline{N}^2V'(\overline{\Phi})\Big]\nonumber\\
&&\quad
+\bigg(\frac{-2K(\overline{\Phi})F''(\overline{\Phi})+K'(\overline{\Phi})F'(\overline{\Phi})}{3(F'(\overline{\Phi}))^2
+2F(\overline{\Phi})K(\overline{\Phi})}\bigg)'
\big(\dot{\overline{\Phi}}-\mathcal{L}_{\overset{\rightharpoonup}{\overline{N}}}\overline{\Phi} \big)^2\overline{Q}_{jk}+2\overline{Q}_{jk}\bigg(\frac{-2K(\overline{\Phi})F''(\overline{\Phi})
+K'(\overline{\Phi})F'(\overline{\Phi})}{3(F'(\overline{\Phi}))^2+2F(\overline{\Phi})K(\overline{\Phi})}\bigg)\nonumber\\
&&\quad
\cdot\big(\dot{\overline{\Phi}}-\mathcal{L}_{\overset{\rightharpoonup }{\overline{N}}}\overline{\Phi} \big)(\frac{\partial}{\partial \tau} - \mathcal{L}_{\overset{\rightharpoonup} {\overline{N}}})
-\frac{\overline{N}^2}{\sqrt{\det \overline{Q}}}\overline{Q}_{jk}\bigg(\frac{K(\overline{\Phi})}{3(F'(\overline{\Phi}))^2
+2F(\overline{\Phi})K(\overline{\Phi})}\bigg)'\overline{C}(\tau, \sigma)\nonumber\\
&&\quad
-2\overline{Q}_{jk}\bigg(\frac{F'(\overline{\Phi})}{3(F'(\overline{\Phi}))^2+2F(\overline{\Phi})K(\overline{\Phi})}\bigg)'\, \overline{X}-\overline{Q}_{jk}\bigg(\frac{(F'(\overline{\Phi}))^2}{F(\overline{\Phi})\big[3(F'(\overline{\Phi}))^2
+2F(\overline{\Phi})K(\overline{\Phi})\big]}\bigg)'\, \overline{Y}\Bigg]\delta \Phi\nonumber\\
&&
+\Bigg[\bigg[\frac{\dot{\overline{N}}}{\overline{N}}-\frac{\Big(\sqrt{\det\overline{ Q}}\Big)^{\cdot}}{\sqrt{\det \overline{Q}}}+\frac{\overline{N}}{\sqrt{\det \overline{Q}}}\Big(\mathcal{L}_{\overset{\rightharpoonup }{\overline{N}}}(\frac{ \sqrt{\det \overline{Q}} }{\overline{N}})\Big)\bigg](\frac{\partial}{\partial\tau}-\mathcal{L}_{\overset{\rightharpoonup} {\overline{N}}})
-2\overline{N}^2[-\frac12\overline{D}_m \overline{D}_n\overline{Q}^{mn}]\nonumber\\
&&\quad
+\big[\overline{N}(\overline{D}_n\overline{N})\overline{D}_m\overline{Q}^{mn}\big]+\big[\mathcal{L}_{\overset{\rightharpoonup }{\overline{N}}}(\frac{\partial}{\partial \tau}-\mathcal{L}_{\overset{\rightharpoonup} {\overline{N}}})+\frac{\partial}{\partial \tau} \mathcal{L}_{\overset{\rightharpoonup} {\overline{N}}}\big]-\frac{F'(\overline{\Phi})}{F(\overline{\Phi})}\big(\dot{\overline{\Phi}}-\mathcal{L}_{\overset{\rightharpoonup }{\overline{N}}}\overline{\Phi}\big)(\frac{\partial}{\partial \tau}-\mathcal{L}_{\overset{\rightharpoonup}{\overline{N}}})\nonumber\\
&&\quad
+\frac{2\overline{N}^2}{F(\overline{\Phi})}\Big[\frac12(\overline{D}_n F(\overline{\Phi}))\overline{D}_m\overline{Q}^{mn}\Big]
+\Big[-\frac{1}{2}\frac{\overline{N}}{F(\overline{\Phi})\sqrt{\det {\overline{Q}}} }\overline{H} \overline{N}^m \overline{N}^n \overline{Q}_{mn}\Big]\nonumber\\
&&\quad
+\frac{1}{F(\overline{\Phi})}\Big[\overline{N}(\overline{D}^i\overline{N})(\overline{D}_iF(\overline{\Phi}))+\overline{N}^2 (\overline{D}^i\overline{D}_iF(\overline{\Phi}))+\overline{N}^2V(\overline{\Phi})\Big]\nonumber\\
&&\quad
+\bigg(\frac{-2K(\overline{\Phi})F''(\overline{\Phi})+K'(\overline{\Phi})F'(\overline{\Phi})}{3(F'(\overline{\Phi}))^2
+2F(\overline{\Phi})K(\overline{\Phi})}\bigg)
\big(\dot{\overline{\Phi}}-\mathcal{L}_{\overset{\rightharpoonup }{\overline{N}}}\overline{\Phi}\big)^2
-\frac{\overline{N}^2}{\sqrt{\det \overline{Q}}}\bigg(\frac{K(\overline{\Phi})}{3(F'(\overline{\Phi}))^2
+2F(\overline{\Phi})K(\overline{\Phi})}\bigg)\overline{C}(\tau, \sigma)\nonumber\\
&&\quad
-2\bigg(\frac{F'(\overline{\Phi})}{3(F'(\overline{\Phi}))^2+2F(\overline{\Phi})K(\overline{\Phi})}\bigg)\, \overline{X}-\bigg(\frac{(F'(\overline{\Phi}))^2}{F(\overline{\Phi})\big[3(F'(\overline{\Phi}))^2+2F(\overline{\Phi})K(\overline{\Phi})\big]}\bigg)\, \overline{Y}\Bigg]\delta Q_{jk}\nonumber\\
&&
+\Bigg[\Big(\dot{\overline{Q}}_{jk}-(\mathcal{L}_{\overset{\rightharpoonup }{\overline{N}}}\overline{Q})_{jk}\Big)\Big[-\frac12(\frac{\partial}{\partial \tau}-\mathcal{L}_{\overset{\rightharpoonup} {\overline{N}}})(\overline{Q}^{mn}+\frac{\overline{N}^m\overline{N}^n}{\overline{N}^2})-\frac{\overline{N}}{\sqrt{\det {\overline{Q}}}}\frac{\partial}{\partial x^r}(\frac{\sqrt{\det {\overline{Q}}}}{\overline{N}}\overline{Q}^{mr}\overline{N}^n)\Big]\nonumber\\
&&\quad
+\bigg[\frac{\dot{\overline{N}}}{\overline{N}}-\frac{\Big(\sqrt{\det \overline{Q}}\Big)^{\cdot }}{\sqrt{\det \overline{Q}}}+\frac{\overline{N}}{\sqrt{\det \overline{Q}}}\Big(\mathcal{L}_{\overset{\rightharpoonup }{\overline{N}}}(\frac{ \sqrt{\det \overline{Q}} }{\overline{N}})\Big)\bigg]\bigg(-\overline{Q}^{mr}\overline{N}^n(\overline{Q}_{jk})_{,r}-\overline{Q}_{k r}\frac{\partial}{\partial x^j}(\overline{Q}^{mr}\overline{N}^n)\nonumber\\
&&\qquad
-\overline{Q}_{jr}\frac{\partial}{\partial x^k}(\overline{Q}^{mr}\overline{N}^n)\bigg)+\bigg[-\overline{Q}^{mr}\overline{Q}^{ns}\Big(\dot{\overline{Q}}_{rj}-(\mathcal{L}_{\overset{\rightharpoonup }{\overline{N}}}\overline{Q})_{r j}\Big)\Big(\dot{\overline{Q}}_{sk}-(\mathcal{L}_{\overset{\rightharpoonup }{\overline{N}}}{\overline{Q}})_{sk}\Big)\nonumber\\
&&\qquad
+(-2\overline{Q}^{tu})\Big(\dot{\overline{Q}}_{t(k}-(\mathcal{L}_{\overset{\rightharpoonup }{\overline{N}}}\overline{Q})_{t(k}\Big)\Big(-\overline{Q}^{mr}\overline{N}^n(\overline{Q}_{j)u})_{,r}-\overline{Q}_{ur}\frac{\partial}{\partial x^{j)}}(\overline{Q}^{mr}\overline{N}^n)-\overline{Q}_{j)r}\frac{\partial}{\partial x^u}(\overline{Q}^{mr}\overline{N}^n)\Big)\bigg]\nonumber\\
&&\quad
-\overline{N}^m\overline{N}^n
\Big(\frac{2K(\overline{\Phi})}{F(\overline{\Phi})}\overline{\Phi}_{,j}\overline{\Phi}_{,k}-2\overline{R}_{jk}\Big)-2\overline{N}^2
[-\frac12\overline{D}_j\overline{D}_k\overline{Q}^{mn}]
+\Big[-2\frac{\overline{N}^m\overline{N}^n}{\overline{N}^2}\overline{N}(\overline{D}_j\overline{D}_k\overline{N})\nonumber\\
&&\qquad
-\overline{N}^2
\overline{D}_j\overline{D}_k(\frac{\overline{N}^m\overline{N}^n}{\overline{N}^2})\Big]
+\bigg[-\overline{Q}^{mr}\overline{N}^n\big[\dot{\overline{Q}}_{jk}-(\mathcal{L}_{\overset{\rightharpoonup }{\overline{N}}}\overline{Q})_{jk}\big]_{,r}-\Big(\dot{\overline{Q}}_{kr}-(\mathcal{L}_{\overset{\rightharpoonup }{\overline{N}}}\overline{Q})_{kr}\Big)\frac{\partial}{\partial x^j}(\overline{Q}^{mr}\overline{N}^n)\nonumber\\
&&\qquad
-\Big(\dot{\overline{Q}}_{jr}-(\mathcal{L}_{\overset{\rightharpoonup }{\overline{N}}}\overline{Q})_{jr}\Big)\frac{\partial}{\partial x^k}(\overline{Q}^{mr}\overline{N}^n)+(\frac{\partial}{\partial\tau}-\mathcal{L}_{\overset{\rightharpoonup} {\overline{N}}})\big[-\overline{Q}^{mr}\overline{N}^n(\overline{Q}_{jk})_{,r}-\overline{Q}_{k r}\frac{\partial}{\partial x^j}(\overline{Q}^{mr}\overline{N}^n)\nonumber\\
&&\qquad
-\overline{Q}_{jr}\frac{\partial}{\partial x^k}(\overline{Q}^{mr}\overline{N}^n)\big]\bigg]-\frac{F'(\overline{\Phi})}{F(\overline{\Phi})}\big(\dot{\overline{Q}}_{jk}
-(\mathcal{L}_{\overset{\rightharpoonup }{\overline{N}}}\overline{Q})_{jk}\big)[\overline{Q}^{mr}\overline{\Phi}_{,r}\overline{N}^n]-\frac{F'(\overline{\Phi})}{F(\overline{\Phi})
}\big(\dot{\overline{\Phi}}-\mathcal{L}_{\overset{\rightharpoonup }{\overline{N}}}\overline{\Phi}\big)\nonumber\\
&&\qquad
\cdot\Big[\overline{Q}^{mr}\overline{N}^n[\overline{Q}_{jk}]_{,r}+\overline{Q}_{rk}\frac{\partial}{\partial x^j}(\overline{Q}^{mr}\overline{N}^n)+\overline{Q}_{rj}\frac{\partial}{\partial x^k}(\overline{Q}^{mr}\overline{N}^n)\Big]-\frac{2}{F(\overline{\Phi})}\overline{N}^m\overline{N}^n(\overline{D}_j\overline{D}_kF(\overline{\Phi}))\nonumber\\
&&\quad
+\Big[\frac{1}{2}\frac{\overline{N}}{F(\overline{\Phi})\sqrt{\det {\overline{Q}}}}(\overline{Q}^{m n}+\frac{\overline{N}^m\overline{N}^n}{\overline{N}^2})\overline{H} \overline{G}_{jkrs}\overline{N}^r \overline{N}^s+\frac{1}{2}\frac{\overline{N}}{F(\overline{\Phi})\sqrt{\det {\overline{Q}}}}\overline{H} \overline{N}^m\overline{N}^n \overline{Q}_{jk}\nonumber\\
&&\qquad
+2\frac{\overline{N}}{F(\overline{\Phi})\sqrt{\det {\overline{Q}}}}\overline{H} \overline{G}_{jkrs} \overline{N}^n \overline{N}^s \overline{Q}^{rm}\Big]+\frac{\overline{Q}_{jk}}{F(\overline{\Phi})}\Big[-\frac12\overline{Q}^{rs}
(\overline{D}_sF(\overline{\Phi}))\overline{D}_r(\overline{N}^m\overline{N}^n)\nonumber\\
&&\qquad
-\frac12\overline{Q}^{mr}\overline{Q}^{ns}(\overline{D}_r\overline{N}^2)(\overline{D}_sF(\overline{\Phi}))-\overline{N}^m\overline{N}^n\overline{Q}^{rs}
(\overline{D}_r\overline{D}_sF(\overline{\Phi}))+\frac12\overline{N}^2\overline{Q}^{mn}(\overline{D}_sF(\overline{\Phi}))\overline{D}_r\overline{Q}^{rs}\nonumber\\
&&\qquad
-\overline{N}^2\overline{Q}^{nr}
(\overline{D}_sF(\overline{\Phi}))\overline{D}_r\overline{Q}^{sm}-\overline{Q}^{mr}\overline{Q}^{sn}\overline{N}^2(\overline{D}_r\overline{D}_s
F(\overline{\Phi}))-\overline{N}^m\overline{N}^nV(\overline{\Phi})\Big]\nonumber\\
&&\quad
+2\overline{Q}_{jk}\bigg(\frac{-2K(\overline{\Phi})F''(\overline{\Phi})+K'(\overline{\Phi})F'(\overline{\Phi})}{3(F'(\overline{\Phi}))^2
+2F(\overline{\Phi})K(\overline{\Phi})}\bigg)
\big(\dot{\overline{\Phi}}-\mathcal{L}_{\overset{\rightharpoonup}{\overline{N}}}\overline{\Phi}\big)[\overline{Q}^{mr}\overline{\Phi}_{,r}\overline{N}^n]\nonumber\\
&&\quad
+\frac{K(\overline{\Phi})\overline{Q}_{jk}}{3(F'(\overline{\Phi}))^2+2F(\overline{\Phi})K(\overline{\Phi})}\Big[\frac{1}{\sqrt{\det \overline{Q}}}\Big(\overline{N}^m\overline{N}^n+\frac{1}{2}\overline{N}^2\overline{Q}^{mn }\Big)\Big]\overline{C}(\tau, \sigma)\Bigg]\delta Q_{mn}\nonumber\\
&&
+\Bigg[\Big[2\overline{Q}^{mn}\Big(\dot{\overline{Q}}_{n(k}-(\mathcal{L}_{\overset{\rightharpoonup }{\overline{N}}}\overline{Q})_{n(k}\Big)(\frac{\partial}{\partial \tau} - \mathcal{L}_{\overset{\rightharpoonup} {\overline{N}}})\Big]-2\overline{N}^2\Big[\overline{D}_n\overline{D}_{(k}\overline{Q}^{mn}\Big]-2\overline{N}(D_n\overline{N})D_{(k}\overline{Q}^{mn}\nonumber\\
&&\quad
+\frac{2\overline{N}^2}
{F(\overline{\Phi})}\Big[-(\overline{D}_nF(\overline{\Phi}))\overline{D}_{(k}\overline{Q}^{mn}\Big]+\Big[-\frac{2\overline{N}}{F(\overline{\Phi})\sqrt{\det {\overline{Q}}}}\overline{H} \overline{N}^r \overline{N}^t \overline{Q}^{s m} \overline{G} _{r t s (k}\Big]\Bigg]\delta Q_{j)m}\nonumber\\
&&
+\Bigg[\Big(\dot{\overline{Q}}_{jk}-(\mathcal{L}_{\overset{\rightharpoonup }{\overline{N}}}\overline{Q})_{jk}\Big)\Big[(\frac{\partial}{\partial \tau} - \mathcal{L}_{\overset{\rightharpoonup} {\overline{N}}})\frac{\overline{N}^m}{\overline{N}^2}\Big]+\Big(\dot{\overline{Q}}_{jk}-(\mathcal{L}_{\overset{\rightharpoonup }{\overline{N}}}\overline{Q})_{jk}\Big)\Big[\frac{\overline{N}}{\sqrt{\det \overline{Q}}}\frac{\partial}{\partial x^n}(\frac{\sqrt{\det \overline{Q}}}{\overline{N}}\overline{Q}^{mn})\Big]\nonumber\\
&&\quad
+\bigg[\frac{\dot{\overline{N}}}{\overline{N}}-\frac{\Big(\sqrt{\det\overline{ Q}}\Big)^{\cdot }}{\sqrt{\det\overline{ Q}}}+\frac{\overline{N}}{\sqrt{\det \overline{Q}}}\Big(\mathcal{L}_{\overset{\rightharpoonup }{\overline{N}}}(\frac{ \sqrt{\det \overline{Q}} }{\overline{N}})\Big)\bigg]\Big[-\overline{Q}^{mr}(\overline{Q}_{jk})_{,r}-\overline{Q}_{k r}\frac{\partial}{\partial x^j}(\overline{Q}^{mr})-\overline{Q}_{jr}\frac{\partial}{\partial x^k}(\overline{Q}^{mr})\Big]\nonumber\\
&&\quad
+\bigg[\Big(-2\overline{Q}^{tu}\Big(\dot{\overline{Q}}_{t(k}-(\mathcal{L}_{\overset{\rightharpoonup }{\overline{N}}}\overline{Q})_{t(k}\Big)\Big)\Big(\overline{Q}^{mn}(\overline{Q}_{j)u})_{,n}+\overline{Q}_{un}\frac{\partial}{\partial x^{j)}}(\overline{Q}^{mn})+\overline{Q}_{j)n}\frac{\partial}{\partial x^u}(\overline{Q}^{mn})\Big)\bigg]\nonumber\\
&&\quad
+2\overline{N}^m\Big(\frac{2K(\overline{\Phi})}{F(\overline{\Phi})}\overline{\Phi } _{, j}\overline{\Phi} _{, k}-2\overline{R}_{j k}\Big)+\Big[\frac{\overline{N}^m}{\overline{N}^2}\Big(4\overline{N}(\overline{D}_j\overline{D}_k\overline{N})\Big)+2\overline{N}^2
\overline{D}_j\overline{D}_k(\frac{\overline{N}^m}{\overline{N}^2})\Big]\nonumber\\
&&\quad
+\bigg[\overline{Q}^{mn}\big[\dot{\overline{Q}}_{jk}-(\mathcal{L}_{\overset{\rightharpoonup }{\overline{N}}}\overline{Q})_{jk}\big]_{,n}+\big(\dot{\overline{Q}}_{kn}-(\mathcal{L}_{\overset{\rightharpoonup }{\overline{N}}}\overline{Q})_{kn}\big)\frac{\partial}{\partial x^j}(\overline{Q}^{mn})+\big(\dot{\overline{Q}}_{jn}-(\mathcal{L}_{\overset{\rightharpoonup }{\overline{N}}}\overline{Q})_{jn}\big)\frac{\partial}{\partial x^k}(\overline{Q}^{mn})\nonumber\\
&&\qquad
+(\frac{\partial}{\partial \tau} - \mathcal{L}_{\overset{\rightharpoonup} {\overline{N}}})\Big(\overline{Q}^{mn}[\overline{Q}_{jk}]_{,n}+\overline{Q}_{kn}\frac{\partial}{\partial x^j}(\overline{Q}^{mn})+\overline{Q}_{jn}\frac{\partial}{\partial x^k}(\overline{Q}^{mn})\Big)\bigg]\nonumber\\
&&\quad
+\frac{F'(\overline{\Phi})}{F(\overline{\Phi})}\big(\dot{\overline{\Phi}}-\mathcal{L}_{\overset{\rightharpoonup }{\overline{N}}}\overline{\Phi}\big)\big[\overline{Q}^{mn}[\overline{Q}_{jk}]_{,n}+\overline{Q}_{kn}\frac{\partial}{\partial x^j}(\overline{Q}^{mn})+\overline{Q}_{j n}\frac{\partial}{\partial x^k}(\overline{Q}^{mn})\big]\nonumber\\
&&\quad
+\frac{F'(\overline{\Phi})}{F(\overline{\Phi})}\Big(\dot{\overline{Q}}_{jk}-(\mathcal{L}_{\overset{\rightharpoonup }{\overline{N}}}\overline{Q})_{jk}\Big)(\overline{Q}^{mr}\overline{\Phi}_{,r})+\frac{4\overline{N}^m}{F(\overline{\Phi})}
(\overline{D}_j\overline{D}_kF(\overline{\Phi}))\nonumber\\
&&\quad
+\Big[-\frac{\overline{N}}{F(\overline{\Phi})\sqrt{\det {\overline{Q}}}}\frac{\overline{N}^m}{\overline{N}^2}\overline{H}(\sigma) \overline{G}_{jkrs}\overline{N}^r \overline{N}^s
-2\frac{\overline{N}}{F(\overline{\Phi})\sqrt{\det {\overline{Q}}}}\overline{H}(\sigma) \overline{G}_{jkrs}\overline{N}^s \overline{Q}^{rm}\Big]\nonumber\\
&&\quad
+\frac{\overline{Q}_{jk}}{F(\overline{\Phi})}\Big[2\overline{N}^m\overline{Q}^{rs}(\overline{D}_r\overline{D}_sF(\overline{\Phi}))
+\overline{Q}^{rs}(\overline{D}_sF(\overline{\Phi}))\overline{D}_r\overline{N}^m+2\overline{N}^mV(\overline{\Phi})\Big]\nonumber\\
&&\quad
-2\overline{Q}_{jk}\bigg(\frac{-2K(\overline{\Phi})F''(\overline{\Phi})+K'(\overline{\Phi})F'(\overline{\Phi})}{3(F'(\overline{\Phi}))^2
+2F(\overline{\Phi})K(\overline{\Phi})}\bigg)
\big(\dot{\overline{\Phi}}-\mathcal{L}_{\overset{\rightharpoonup }{\overline{N}}}\overline{\Phi}\big)\big[\overline{Q}^{mn}\overline{\Phi}_{,n}\big]\nonumber\\
&&\quad
-2\frac{\overline{N}^m}{\sqrt{\det \overline{Q}}}\frac{K(\overline{\Phi})\overline{Q}_{jk}}{3(F'(\overline{\Phi}))^2+2F(\overline{\Phi})K(\overline{\Phi})}\overline{C}(\tau, \sigma)\Bigg]\delta N_m\nonumber\\
&&
+\Bigg[-\frac{\overline{N}}{F(\overline{\Phi})\sqrt{\det \overline{Q}}}\overline{N}^n\overline{N}^m\overline{G}_{jkmn}\Bigg]\delta H-\frac{\overline{N}^2}{\sqrt{\det \overline{Q}}}\frac{K(\overline{\Phi})\overline{Q}_{jk}}{3(F'(\overline{\Phi}))^2+2F(\overline{\Phi})K(\overline{\Phi})}\delta C\nonumber\\
&&
-\frac{2F'(\overline{\Phi})\overline{Q}_{jk}}{3(F'(\overline{\Phi}))^2+2F(\overline{\Phi})K(\overline{\Phi})}\delta X
-\frac{(F'(\overline{\Phi}))^2\overline{Q}_{jk}}{F(\overline{\Phi})\big[3(F'(\overline{\Phi}))^2+2F(\overline{\Phi})K(\overline{\Phi})\big]}\delta Y.\label{deltaQ}
\end{eqnarray}
The long and complicated equations (\ref{deltaphi}) and (\ref{deltaQ}) describe the evolution of the linearly perturbed scalar and
metric fields when the background metric are solutions of the STT equations of motion without any further symmetry requirements. Therefore these equations describe the evolution of the linear perturbations for general, and not only cosmological, spacetimes in the context of STT theories. The evolution equations in (\ref{deltaphi}) and (\ref{deltaQ}) are the generalization of the evolution equations (6.13) and (6.31) in \cite{Giesel10} to the framework of STT theories.
However, for the purpose of practical use, in the next section we will choose a specific background, namely a cosmological one, and study the detailed behavior of the perturbation equations. This also allows to compare the results obtained in this manifestly gauge invariant framework with already existing results in the literature obtained in the standard framework of cosmological perturbation theory.
\section{Cosmological Perturbations on FRW Background }
In this section we specialize the general relativistic equations from the last section to homogenous and isotropic FRW backgrounds and compare them with the already existing results in the literature. For simplicity, we only focus on the spatially flat case.
The equations are greatly simplified because the spatial derivatives of background variables in Eqs. (\ref{deltaphi}) and (\ref{deltaQ}) vanish, and we have $\overline{N}_j=\frac{\overline{C}_j}{\overline{H}}=0$ and $\overline{N}=\frac{\overline{C}}{\overline{H}}=\sqrt{1+\overline{Q}^{jk}\overline{N}_j\overline{N}_k}=1$. We denote $\overline{Q}_{jk}\equiv A^2\delta_{jk}$ where $A$ represents the gauge invariant scale factor. Then from Eqs. (\ref{phisecond}) and (\ref{qsecond}) we obtain the following evolution equations for the background variables
\begin{eqnarray}
\ddot{\overline{\Phi}}&=&-3\frac{\dot{A}}{A}\dot{\overline{\Phi}}-\Big(\frac{F'(\overline{\Phi})K(\overline{\Phi})+F(\overline{\Phi})K'(\overline{\Phi})
+3F'(\overline{\Phi})F''(\overline{\Phi})}{3(F'(\overline{\Phi}))^2+2F(\overline{\Phi})K(\overline{\Phi})}\Big)(\dot{\overline{\Phi}})^2
+\frac{2F'(\overline{\Phi})V(\overline{\Phi})-F(\overline{\Phi})V'(\overline{\Phi})}{3(F'(\overline{\Phi}))^2+2F(\overline{\Phi})K(\overline{\Phi})}\nonumber\\
&&
-\frac{F'(\overline{\Phi})}{2\big(3(F'(\overline{\Phi}))^2+2F(\overline{\Phi})K(\overline{\Phi})\big)}\frac{\epsilon}{A^3},\label{ddotphi}\\
\frac{\ddot{A}}{A}&=&\frac{F'(\overline{\Phi})}{F(\overline{\Phi})}\frac{\dot{A}}{A}\dot{\overline{\Phi}}
-\frac{\Big(K(\overline{\Phi})F'(\overline{\Phi})^2
+\frac43F(\overline{\Phi})K^2(\overline{\Phi})+2F(\overline{\Phi})K(\overline{\Phi})F''(\overline{\Phi})
-F'(\overline{\Phi})F(\overline{\Phi})K'(\overline{\Phi})\Big)}{2F(\overline{\Phi})\big(3(F'(\overline{\Phi}))^2
+2F(\overline{\Phi})K(\overline{\Phi})\big)}(\dot{\overline{\Phi}})^2\nonumber\\
&&
+\Big(\frac16-\frac{(F'(\overline{\Phi}))^2}{3(F'(\overline{\Phi}))^2+2F(\overline{\Phi})K(\overline{\Phi})}\Big)
\frac{V(\overline{\Phi})}{F(\overline{\Phi})}
+\frac{F'(\overline{\Phi}))V'(\overline{\Phi})}{2\big(3(F'(\overline{\Phi}))^2
+2F(\overline{\Phi})K(\overline{\Phi})\big)}\nonumber\\
&&
+\Big(\frac{1}{12}\frac{1}{F(\overline{\Phi})}
+\frac14\frac{\big(F'(\overline{\Phi})\big)^2}{F(\overline{\Phi})\big(3(F'(\overline{\Phi}))^2
+2F(\overline{\Phi})K(\overline{\Phi})\big)}\Big)\frac{\bar{\epsilon}}{A^3},\label{ddotA}
\end{eqnarray}
where
\begin{eqnarray}
\overline{\epsilon}=\overline{H}=A^3\bigg[-6F(\overline{\Phi})\Big(\frac{\dot{A}}{A}\Big)^2-6\frac{\dot{A}}{A}F'(\overline{\Phi})\,{\dot{\overline{\Phi}}}
+K(\overline{\Phi})\,(\dot{\overline{\Phi}})^2+V(\overline{\Phi})\bigg].\label{Hamiltdensity}
\end{eqnarray}
Notice that $\overline{\epsilon}$ is a constant, because as mentioned above the physical Hamiltonian density $\overline{H}$ is conserved during the evolution. From Eqs. (\ref{ddotA}) and (\ref{Hamiltdensity}) we read the gauge invariant Friedmann and Raychaudhuri equations as:
\begin{eqnarray}
(\frac{\dot{A}}{A})^2=\frac{1}{6}(\rho^{e}_{\Phi}+\tilde{\rho}),\label{Friedmann}\\
2\frac{\ddot{A}}{A}-(\frac{\dot{A}}{A})^2=-\frac{1}{2}(P^{e}_{\Phi}+\tilde{P}),\label{Raychaudhuri}
\end{eqnarray}
where $\rho^{e}$ and $P^{e}$ denote the effective energy density and pressure of the scalar field, while $\tilde{\rho}$ and $\tilde{P}$ represent the correction terms caused by the non-vanishing physical Hamiltonian density. These quantities are defined as
\begin{eqnarray}
\rho^{e}_{\Phi}&:=&-\frac{1}{F(\overline{\Phi})}[6\frac{\dot{A}}{A}F'(\overline{\Phi})\dot{\overline{\Phi}}-K(\overline{\Phi})\cdot(\dot{\overline{\Phi}})^2
-V(\overline{\Phi})],\label{rhoe}\nonumber\\
P^{e}_{\Phi}&:=&-2\frac{F'(\overline{\Phi})}{F(\overline{\Phi})}\frac{\dot{A}}{A}\dot{\overline{\Phi}}
-4\Big(\frac14-\frac{(F'(\overline{\Phi}))^2}{3(F'(\overline{\Phi}))^2+2F(\overline{\Phi})K(\overline{\Phi})}\Big)
\frac{V(\overline{\Phi})}{F(\overline{\Phi})}
-\frac{2F'(\overline{\Phi}))V'(\overline{\Phi})}{\big(3(F'(\overline{\Phi}))^2
+2F(\overline{\Phi})K(\overline{\Phi})\big)}\nonumber\\
&&
+2\frac{\Big(\frac12K(\overline{\Phi})F'(\overline{\Phi})^2
+F(\overline{\Phi})K^2(\overline{\Phi})+2F(\overline{\Phi})K(\overline{\Phi})F''(\overline{\Phi})
-F'(\overline{\Phi})F(\overline{\Phi})K'(\overline{\Phi})\Big)}{F(\overline{\Phi})\big(3(F'(\overline{\Phi}))^2
+2F(\overline{\Phi})K(\overline{\Phi})\big)}(\dot{\overline{\Phi}})^2,\label{Pe}
\nonumber\\
\tilde{\rho}&:=&-\frac{\overline{\epsilon}}{F(\overline{\Phi})A^3},\nonumber\\
\tilde{P}&:=&-\frac{\big(F'(\overline{\Phi})\big)^2}{F(\overline{\Phi})\big(3(F'(\overline{\Phi}))^2
+2F(\overline{\Phi})K(\overline{\Phi})\big)}\frac{\overline{\epsilon}}{A^3}.\nonumber
\end{eqnarray}
Given the evolution equations for the background variables we can consider their linear perturbations around an FRW background. The evolution equations for the linear perturbations have the following form
\begin{eqnarray}
\delta \ddot{\Phi}&=&\Bigg[\bigg[-\frac{6(F'(\overline{\Phi}))^2
+6F(\overline{\Phi})K(\overline{\Phi})}{3(F'(\overline{\Phi}))^2
+2F(\overline{\Phi})K(\overline{\Phi})}\frac{\dot{A}}{A}
-2\Big(\frac{\frac32F'(\overline{\Phi})K(\overline{\Phi})+F(\overline{\Phi})K'(\overline{\Phi})
+3F'(\overline{\Phi})F''(\overline{\Phi})}{3(F'(\overline{\Phi}))^2
+2F(\overline{\Phi})K(\overline{\Phi})}\Big)(\dot{\overline{\Phi}})\bigg]\frac{\partial}{\partial\tau}\nonumber\\
&&\quad
+\frac{2F(\overline{\Phi})K(\overline{\Phi})+2(F'(\overline{\Phi}))^2}{3(F'(\overline{\Phi}))^2
+2F(\overline{\Phi})K(\overline{\Phi})}\frac{1}{A^2}\delta^{jk}\overline{D}_j \overline{D}_k+\Big(\frac{3F(\overline{\Phi})F'(\overline{\Phi})}{3(F'(\overline{\Phi}))^2
+2F(\overline{\Phi})K(\overline{\Phi})}\Big)'\Big(\frac{\dot{A}}{A}\Big)^2
\nonumber\\
&&\quad
-\Big(\frac{6(F'(\overline{\Phi}))^2
+6F(\overline{\Phi})K(\overline{\Phi})}{3(F'(\overline{\Phi}))^2
+2F(\overline{\Phi})K(\overline{\Phi})}\Big)'\frac{\dot{A}}{A}\dot{\overline{\Phi}}
-\Big(\frac{\frac32F'(\overline{\Phi})K(\overline{\Phi})
+F(\overline{\Phi})K'(\overline{\Phi})
+3F'(\overline{\Phi})F''(\overline{\Phi})}{3(F'(\overline{\Phi}))^2
+2F(\overline{\Phi})K(\overline{\Phi})}\Big)'(\dot{\overline{\Phi}})^2\nonumber\\
&&\quad
+\Big(\frac{\frac32F'(\overline{\Phi})V(\overline{\Phi})
-F(\overline{\Phi})V'(\overline{\Phi})}{3(F'(\overline{\Phi}))^2
+2F(\overline{\Phi})K(\overline{\Phi})}\Big)'\Bigg]\delta \Phi\nonumber\\
&&
+\Bigg[\bigg[-\frac{(F'\overline{(\Phi)})^2+F(\overline{\Phi})K(\overline{\Phi})}{3(F'(\overline{\Phi}))^2
+2F(\overline{\Phi})K(\overline{\Phi})}\frac{1}{A^2}\dot{\overline{\Phi}}
+\frac{F(\overline{\Phi})F'(\overline{\Phi})}{3(F'(\overline{\Phi}))^2
+2F(\overline{\Phi})K(\overline{\Phi})}\frac{\dot{A}}{A^3}\bigg]\frac{\partial}{\partial\tau}\nonumber\\
&&\quad
+\frac12\cdot\frac{F(\overline{\Phi})F'(\overline{\Phi})}{3(F'(\overline{\Phi}))^2
+2F(\overline{\Phi})K(\overline{\Phi})}(\overline{D}^j\overline{D}^k
-\frac{1}{A^2}\delta^{jk}\overline{D}^i\overline{D}_i)\nonumber\\
&&\quad
+\bigg[\frac{2(F'(\overline{\Phi}))^2+2F(\overline{\Phi})K(\overline{\Phi})}{3(F'(\overline{\Phi}))^2
+2F(\overline{\Phi})K(\overline{\Phi})}\frac{\dot{A}}{A^3}\dot{\overline{\Phi}}
-\frac{2F(\overline{\Phi})F'(\overline{\Phi})}{3(F'(\overline{\Phi}))^2
+2F(\overline{\Phi})K(\overline{\Phi})}\frac{\dot{A}^2}{A^4}\bigg]\delta^{jk}\Bigg]\delta Q_{jk}\nonumber\\
&&
+\Bigg[\bigg[\frac{2(F'(\overline{\Phi}))^2+2F(\overline{\Phi})K(\overline{\Phi})}{3(F'(\overline{\Phi}))^2
+2F(\overline{\Phi})K(\overline{\Phi})}\frac{1}{A^2}\dot{\overline{\Phi}}
-\frac{2F(\overline{\Phi})F'(\overline{\Phi})}{3(F'(\overline{\Phi}))^2
+2F(\overline{\Phi})K(\overline{\Phi})}\frac{\dot{A}}{A^3}\bigg]\frac{\partial}{\partial x^j}\Bigg]\delta N_j,\label{ddotphicos}
\end{eqnarray}
\begin{eqnarray}
\delta\ddot{Q}_{jk}&=&\Bigg[\bigg[\Big(\frac{2F'(\overline{\Phi})K(\overline{\Phi})}{3(F'(\overline{\Phi}))^2
+2F(\overline{\Phi})K(\overline{\Phi})}\Big)A\dot{A}-2\Big(\frac{2F''(\overline{\Phi})K(\overline{\Phi})+(K(\overline{\Phi}))^2
-F'(\overline{\Phi})K'(\overline{\Phi})}{3(F'(\overline{\Phi}))^2
+2F(\overline{\Phi})K(\overline{\Phi})}\Big)A^2\dot{\overline{\Phi}}\bigg]\delta_{jk}\frac{\partial}{\partial\tau}\nonumber\\
&&\quad
-\Big(\frac{2(F'(\overline{\Phi}))^3+2F(\overline{\Phi})F'(\overline{\Phi})K(\overline{\Phi})}
{F(\overline{\Phi})(3(F'(\overline{\Phi}))^2
+2F(\overline{\Phi})K(\overline{\Phi}))}\Big)A^2\delta_{jk}\overline{D}^i\overline{D}_i
+2\frac{F'(\overline{\Phi})}{F(\overline{\Phi})}\overline{D}_j\overline{D}_k\nonumber\\
&&\quad
+\bigg[\Big(\frac{2F(\overline{\Phi})K(\overline{\Phi})}
{3(F'(\overline{\Phi}))^2
+2F(\overline{\Phi})K(\overline{\Phi})}\Big)'(\dot{A})^2
+\Big(\frac{2F'(\overline{\Phi})K(\overline{\Phi})}{3(F'(\overline{\Phi}))^2
+2F(\overline{\Phi})K(\overline{\Phi})}\Big)' A\dot{A}\dot{\overline{\Phi}}\nonumber\\
&&\qquad
-\Big(\frac{2F''(\overline{\Phi})K(\overline{\Phi})+(K(\overline{\Phi}))^2
-F'(\overline{\Phi})K'(\overline{\Phi})}{3(F'(\overline{\Phi}))^2
+2F(\overline{\Phi})K(\overline{\Phi})}\Big)'A^2(\dot{\overline{\Phi}})^2
+\Big(\frac{K(\overline{\Phi})V(\overline{\Phi})
+F'(\overline{\Phi})V'(\overline{\Phi})}{3(F'(\overline{\Phi}))^2
+2F(\overline{\Phi})K(\overline{\Phi})}\Big)'A^2\bigg]\delta_{jk}\Bigg]\delta \Phi\nonumber\\
&&
+\Bigg[\bigg[-3\frac{\dot{A}}{A}-\frac{F'(\overline{\Phi})}
{F(\overline{\Phi})}\dot{\overline{\Phi}}\bigg]\frac{\partial}{\partial\tau}+\overline{D}^i\overline{D}_i
+\Big(\frac{6F(\overline{\Phi})K(\overline{\Phi})+6(F'(\overline{\Phi}))^2}{3(F'(\overline{\Phi}))^2
+2F(\overline{\Phi})K(\overline{\Phi})}\Big)(\frac{\dot{A}}{A})^2
+\Big(\frac{6F'(\overline{\Phi})K(\overline{\Phi})+6\frac{(F'(\overline{\Phi}))^3}{F(\overline{\Phi})}}{3(F'(\overline{\Phi}))^2
+2F(\overline{\Phi})K(\overline{\Phi})}\Big)\frac{\dot{A}}{A}\dot{\overline{\Phi}}\nonumber\\
&&\quad
-\Big(\frac{2F''(\overline{\Phi})K(\overline{\Phi})+(K(\overline{\Phi}))^2
-F'(\overline{\Phi})K'(\overline{\Phi})}{3(F'(\overline{\Phi}))^2
+2F(\overline{\Phi})K(\overline{\Phi})}\Big)(\dot{\overline{\Phi}})^2
+\frac{K(\overline{\Phi})V(\overline{\Phi})+F'(\overline{\Phi})V'(\overline{\Phi})}{3(F'(\overline{\Phi}))^2
+2F(\overline{\Phi})K(\overline{\Phi})}
\bigg]\delta Q_{jk}\nonumber\\
&&
+\Bigg[\Big(-\frac{F'(\overline{\Phi})^2}{3(F'(\overline{\Phi}))^2
+2F(\overline{\Phi})K(\overline{\Phi})}\frac{\dot{A}}{A}
+\frac{F(\overline{\Phi})K(\overline{\Phi})
+(F'(\overline{\Phi}))^2}{3(F'(\overline{\Phi}))^2
+2F(\overline{\Phi})K(\overline{\Phi})}\frac{F'(\overline{\Phi})}
{F(\overline{\Phi})}\dot{\Phi}\Big)\delta^{mn}\delta_{jk}
\frac{\partial}{\partial\tau}\nonumber\\
&&\quad
+\Big(2\frac{F'(\overline{\Phi})^2}{3(F'(\overline{\Phi}))^2+2F(\overline{\Phi})K(\overline{\Phi})}(\frac{\dot{A}}{A})^2
-2\frac{F(\overline{\Phi})K(\overline{\Phi})+(F'(\overline{\Phi}))^2}{3(F'(\overline{\Phi}))^2
+2F(\overline{\Phi})K(\overline{\Phi})}\frac{F'(\overline{\Phi})}{F(\overline{\Phi})}
\frac{\dot{A}}{A}\dot{\overline{\Phi}}\Big)\delta^{mn}\delta_{jk}\nonumber\\
&&\quad
+\frac{F(\overline{\Phi})K(\overline{\Phi})+(F'(\overline{\Phi}))^2}{3(F'(\overline{\Phi}))^2
+2F(\overline{\Phi})K(\overline{\Phi})}A^2\delta_{jk}(\overline{D}^m\overline{D}^n
-\frac{1}{A^2}\delta^{mn}\overline{D}^i\overline{D}_i)
-4(\frac{\dot{A}}{A})^2\delta^{m}_{(j}\delta^{n}_{k)}+\overline{Q}^{mn}\overline{D}_j\overline{D}_k\Bigg]\delta Q_{mn}\nonumber\\
&&
+\Bigg[4\frac{\dot{A}}{A}\delta^{m}_{(k}\frac{\partial}{\partial\tau}-2\overline{D}^m\overline{D}_{(k}\Bigg]\delta Q_{j)m}
+\Bigg[2\frac{F'(\overline{\Phi})}{F(\overline{\Phi})}\dot{\overline{\Phi}}\Big(\delta^{m}_{(j}\frac{\partial}{\partial x^{k)}}-2\delta_{jk}\delta^{mw}\frac{\partial}{\partial x^w}\Big)
+2\frac{\dot{A}}{A}\delta^{m}_{(j}\frac{\partial}{\partial x^{k)}}+2\delta^{m}_{(j}\frac{\partial}{\partial x^{k)}}\frac{\partial}{\partial\tau}\Bigg]\delta N_m.\label{ddotqcos}\nonumber\\
\end{eqnarray}
To study the detailed behavior of the perturbation equations (\ref{ddotphicos}) and (\ref{ddotqcos}) we will work in conformal time $x^0=\eta$ for which $\overline{g}_{00}=-A^2$. However due to the dust observers we always have $g_{\tau\tau}=-N^2+Q^{jk}N_jN_k=-1$ where $\tau$ denotes the physical time, meaning that we automatically work with the proper time of the dust. Using the relation $dx^0=\frac{d\tau}{A}$ we obtain
\begin{equation}
g_{\tau\tau}=\frac{g_{00}}{A^2}, \quad g_{\tau j}=\frac{g_{0j}}{A}=N_j,\quad
g_{jk}=Q_{jk}.
\end{equation}
Following the procedure in standard cosmological perturbation theory (SCPT), we decompose the spacetime metric $g_{\mu\nu}=\overline{g}_{\mu\nu}+\delta g_{\mu\nu}$ into tensor, vector and scalar modes, parametrized by ten functions $\phi,B,E,\psi,S_j,F_j,h_{jk}$ where the vector modes $S_j$ and $F_j$ are transversal with respect to the spatial Euclidian metric and the tensor mode $h_{jk}$ is transversal and traceless,
\begin{eqnarray}
g_{00}=(-1+2\phi)A^2,\quad
g_{0j}=(B_{,j}+S_j)A^2,\quad
g_{jk}=\delta_{jk}+A^2[2\psi\delta_{jk}+2E_{,jk}+2F_{(j,k)}+h_{jk}].\label{decompmetric}
\end{eqnarray}
Now from $g_{\tau\tau}=-1$, we immediately get $\phi=0$ and therefore with the dust as dynamical observers we will always work in partly synchronous gauge\footnote{Here partly refers to the fact that in SCPT the gauge $\phi=B=0$ is denoted by synchronous gauge, while longitudinal gauge means $B=E=0$.}. Furthermore since $N_j$ can be expressed in terms of the other phase space variables, the functions $B$ and $S_j$ are not independent variables in our formalism. We remark that each mode in equations (\ref{decompmetric}) is already gauge invariant in our formalism, which is different from the case of SCPT. At the level of linear perturbation theory the individual modes decouple and we can study their evolution separately. Let us denote the derivative with respect to conformal time $\frac{d}{d\eta}$ by an upper prime ${}^{'}$. Substituting the expressions (\ref{decompmetric}) into (\ref{ddotqcos}) we obtain for the tensor modes
\begin{eqnarray}
2\mathcal{H}h^{'}_{jk}+\frac{F'(\overline{\Phi})}{F(\overline{\Phi})}\dot{\overline{\Phi}}Ah^{'}_{jk}+h^{''}_{jk}-\triangle h_{jk}=0,\label{eomtensor}
\end{eqnarray}
where $\mathcal{H}:=\frac{A^{'}}{A}$, $\triangle:=\delta^{ij}\overline{D}_i\overline{D}_j$. For the vector modes we consider the decomposition of the shift vector from (\ref{decompmetric}) given by
\begin{eqnarray}
\delta N_{j}=A[B_{,j}+S_{j}].\label{decompshift}
\end{eqnarray}
Since $\dot{\delta N_i}=0$ and in the case of the vector modes we also have $B=0$, we obtain
\begin{eqnarray}
\frac{dA}{d\eta}S_{j}+A\frac{dS_{j}}{d\eta}=0&\Longrightarrow& {\cal H}S_j+{S}^{'}_j=0.\label{eoms}
\end{eqnarray}
From (\ref{ddotqcos}) we get for the vector mode the following equation
\begin{eqnarray}
2F^{''}_{(j,k)}+4\mathcal{H}F^{'}_{(j,k)}+2\frac{F'(\overline{\Phi})}{F(\overline{\Phi})}\dot{\overline{\Phi}}AF^{'}_{(j,k)}
=2\mathcal{H}S_{(j,k)}+2\frac{F'(\overline{\Phi})}{F(\overline{\Phi})}\dot{\overline{\Phi}}AS_{(j,k)}.\label{vector}
\end{eqnarray}
Denoting $V_{j}=S_j-F^{'}_j$ and using Eq. (\ref{eoms}), Eq. (\ref{vector}) simplifies to
\begin{eqnarray}
2\mathcal{H}V_{(j,k)}+\frac{F'(\overline{\Phi})}{F(\overline{\Phi})}\dot{\overline{\Phi}}A V_{(j,k)}+V^{'}_{(j,k)}=0.\label{eomvector}
\end{eqnarray}
For the scalar mode equation (\ref{ddotqcos}) leads to
\begin{eqnarray}
&&\quad\Big(\mathcal{H}+\big(\ln\sqrt{F(\overline{\Phi})}\big)^{'}\Big)\Big(4\psi
+2\delta\big(\ln\sqrt{F(\overline{\Phi}})\big)\Big)^{'}\delta_{jk}
+2\Big(\psi+\delta\big(\ln\sqrt{F(\Phi})\big)\Big)^{''}\delta_{jk}\nonumber\\
&&\quad
+4\Big(\mathcal{H}+\big(\ln\sqrt{F(\overline{\Phi}})\big)^{'}\Big)\big(E^{'}_{,jk}\big)+2E^{''}_{,jk}\nonumber\\
&=&2\Big(\mathcal{H}+2\big(\ln\sqrt{F(\overline{\Phi})}\big)^{'}\Big)B_{,jk}+2\psi_{,jk}+4\Big(\delta \ln\sqrt{ F(\Phi)}\Big)_{,jk}-\delta_{jk}\bigg[\frac12\big(\overline{\Xi}^{'}\delta\overline{\Xi}^{'}
-A^2(\frac{V(\overline{\Xi})}{F(\overline{\Xi})})'_{\Xi}\delta \Xi\big)\bigg],\label{eomscalar}
\end{eqnarray}
where we used the field redefinition
\begin{eqnarray}
\frac{d\Xi(\Phi)}{d\Phi}=\sqrt{3(\frac{F'(\Phi)}{F(\Phi)})^2+2\frac{K(\Phi)}{F(\Phi)}}.\label{defxi}
\end{eqnarray}
Eq. (\ref{eomscalar}) is of the form $f_{,jk}+g\delta_{jk}=0$ for appropriate definitions of the functions $f$ and $g$. Applying the trace to this equation we have $3g+\Delta f=0$. Now operating with $\partial_j\partial_k$ on $f_{,jk}+g\delta_{jk}$ yields $\Delta(g+\Delta f)=0$. Since due to our boundary conditions there are no zero modes of the Laplacian, we can conclude $f=g=0$ and therefore the $(.)_{jk}$ part and $\delta_{jk}$ part of Eq. (\ref{eomscalar}) are independent of each other. Considering the separated contribution we get
\begin{eqnarray}
&&\bigg[4\big[\mathcal{H}+(\ln\sqrt{F(\overline{\Phi})})^{'}\big]E^{'}+2E^{''}-2\Big(\big[\mathcal{H}+(\ln\sqrt{F(\overline{\Phi})})^{'}\big]
+\big(\ln\sqrt{F(\overline{\Phi})}\big)^{'}\Big)B-2\big[\psi+\delta\ln\sqrt{F(\Phi)}\big]\nonumber\\
&&\quad
-2\delta \ln\sqrt{F(\Phi)}\bigg]_{,jk}=0,\label{eomsc1}\\
&&\bigg[4\big[\mathcal{H}+(\ln\sqrt{F(\overline{\Phi})})^{'}\big]\big[\psi+\delta\ln\sqrt{F(\Phi)}\big]^{'}
-2\big[\mathcal{H}+(\ln\sqrt{F(\overline{\Phi})})^{'}\big]
\big(\delta\ln\sqrt{F(\Phi})\big)^{'}
+2\big[\psi+\delta\ln\sqrt{F(\Phi)}\big]^{''}\bigg]\delta_{jk}\nonumber\\
&=&-\bigg[\frac12\big(\overline{\Xi}^{'}\delta\Xi^{'}
-A^2(\frac{V(\overline{\Xi})}{F(\overline{\Xi})})'_{\Xi}\delta \Xi\big)\bigg]\delta_{jk}.\label{eomsc2}
\end{eqnarray}
In SCPT the system does not involve any physical observers \cite{Mukhanov92}, the gauge invariant variables of STT are defined as:
\begin{eqnarray}
\Theta_A &=&-\delta\ln\sqrt{F(\Phi)}-\big[\mathcal{H}+(\ln\sqrt{F(\overline{\Phi})})^{'}\big](B-E^{'})-(B-E^{'})^{'},\nonumber\\
\Theta_B &=&\big[\psi+\delta\ln\sqrt{F(\Phi)}\big]
+\big[\mathcal{H}+(\ln\sqrt{F(\overline{\Phi})})^{'}\big](B-E^{'}),\nonumber\\
\delta Z&=&\delta\Xi+\overline{\Xi}^{'}(B-E^{'}).\label{newvariables}
\end{eqnarray}
Notice that in Ref. \cite{Mukhanov92}, there is an additional term proportional to $\delta N$ in the definition of $\Theta_A$. However in our model $\delta N$ is not an independent variable and we have
\begin{eqnarray}
\delta N=-\frac{\overline{N}^i\overline{N}^j}{2\overline{N}}\delta Q_{jk}+\frac{\overline{N}^j}{\overline{N}}\delta{N_j}.
\end{eqnarray}
Now for an FRW background we have $\overline{N}=1$ and $\overline{N}_j=0$, and thus $\delta N=0$, explaining why no contribution to $\delta N$ occurs in our equations. In the case of the scalar mode we have $\delta N_j=AB_{,j}$ and consequently
\begin{eqnarray}
\frac{dA}{d\eta}B_{,j}+A\frac{dB_{,j}}{d\eta}=0&\Longrightarrow& {\cal H}B_{,j}+{B}^{'}_{,j}=0,\label{eomb}
\end{eqnarray}
where the last relation follows from the fact that the Laplacian has no zero modes.
Using Eq. (\ref{eomb}), Eqs. (\ref{eomsc1}) and (\ref{eomsc2}) are simplified to
\begin{eqnarray}
&&2[\Theta_A-\Theta_B]_{,jk}=0,\label{equaltheta}\\
&&\Theta^{''}_B+\big[\mathcal{H}+(\ln\sqrt{F(\overline{\Phi})})^{'}\big](\Theta_A+2\Theta_B)^{'}
+\Big(2\big[\mathcal{H}+(\ln\sqrt{F(\overline{\Phi})})^{'}\big]^{'}+\big[\mathcal{H}+(\ln\sqrt{F(\overline{\Phi})})^{'}\big]^2\Big)\Theta_A\nonumber\\
&=&
-\frac14\big[(\overline{\Xi}^{'})^2\Theta_A+\overline{\Xi}^{'}\delta Z^{'}
-F(\overline{\Xi})A^2(\frac{V(\overline{\Xi})}{F^2(\overline{\Xi})})'_{\Xi}\delta Z\big],\label{eomscalar1}
\end{eqnarray}
where $(\frac{V(\overline{\Xi})}{F^2(\overline{\Xi})})'_{\Xi}:=d(\frac{V(\overline{\Xi})}{F^2(\overline{\Xi})})/d\Xi$. Note that in \cite{Giesel10s} a slightly different notation was used but the notations can be related to each other by specializing to $F(\Phi)=1$ and identifying the variables $\Phi$ and $\Psi$ in \cite{Giesel10s} with $\Theta_A=\Phi$ and $\Theta_B=\Psi$.
\\
As the next step we derive the equation of motion for the perturbation $\delta \Phi$. Using the field redefinition (\ref{defxi}), the equation of motion for the scalar field perturbations in (\ref{ddotphicos}) can be written in a more compact form as
\begin{eqnarray}
\delta \ddot{\Xi}&=&\Big[\dot{\overline{\Xi}}^2\big(\ln\sqrt{F(\overline{\Xi})}\big)''_{\Xi\Xi}-F(\overline{\Xi})
(\frac{V(\overline{\Xi})}{F^2(\Xi)})''_{\Xi\Xi}
-(\frac{V(\overline{\Xi})}{F^2(\overline{\Xi})})'_{\Xi}F'(\overline{\Xi})+3\big(\dot{(\ln\sqrt{F(\overline{\Xi})})}+\frac{\dot{A}}{A}\big)(\ln F(\overline{\Xi}))'_{\Xi}\dot{\overline{\Xi}}\nonumber\\
&&
-\frac32(\dot{\overline{\Xi}})^2\frac{F''(\overline{\Xi})}{F(\overline{\Xi})}
-3\frac{\dot{A}}{A}\dot{\overline{\Xi}}(\ln F(\overline{\Xi}))'_{\Xi}+\frac{1}{A^2}\delta^{jk}\overline{D}_j \overline{D}_k\Big]\delta \Xi+\Big[-4(\ln\sqrt{F(\Xi)})'_{\Xi}\dot{\overline{\Xi}}-3\frac{\dot{A}}{A}\Big]\delta \dot{\Xi}\nonumber\\
&&
+\frac{1}{A^2}(\frac{\dot{A}}{A})\dot{\overline{\Xi}}\delta Q_{jj}-\frac12\frac{1}{A^2}\dot{\overline{\Xi}}\delta\dot{Q}_{jj}+\frac{1}{A^2}\dot{\overline{\Xi}}\delta N_{j,j}-\delta \big(\frac12(\frac{1}{F(\Xi)})'_{\Xi}\frac{C}{\sqrt{\det Q}}\big).
\end{eqnarray}
Using the definition (\ref{newvariables}), it simplifies to
\begin{eqnarray}
&&\delta Z^{''}+2\big[\mathcal{H}+(\ln\sqrt{F(\overline{\Xi})})^{'}\big]\delta Z-\Delta \delta Z +(\frac{V(\overline{\Xi})}{F^2(\overline{\Xi})})'_{\Xi\Xi}F(\overline{\Xi})A^2\delta Z+\Xi^{'}\Theta^{'}_A-2(\frac{V(\overline{\Xi})}{F^2(\overline{\Xi})})'_{\Xi}F(\overline{\Xi})A^2\Theta^{'}_A+3\overline{\Xi}^{'}\Theta^{'}_B\nonumber\\
&=&\big(\frac12(\frac{1}{F(\overline{\Xi})})'_{\Xi}\frac{\overline{\epsilon}}{A}(B-E^{'})\big)^{'}
+\frac12(\frac{1}{F(\overline{\Xi})})'_{\Xi}\frac{\overline{\epsilon}}{A}(B-E^{'})^{'}-A^2\delta \big(\frac12(\frac{1}{F(\Xi)})'_{\Xi}\frac{\epsilon}{\sqrt{\det Q}}\big).\label{eomperscalar}
\end{eqnarray}
Note that compared to SCPT in the manifestly gauge invariant formalism we start with STT plus dust and thus increase the number of degrees of freedom by four. These four additional degrees of freedom are used to construct gauge invariant quantities associated with the spatial 3-metric $q_{jk}$ and the scalar field $\phi$, which we denoted by $Q_{jk}$ and $\Phi$ respectively. In order to compare our framework with the results of SCPT we expressed $\delta Q_{jk}$ in terms of the invariants $\Theta_A,\Theta_B,V_j$ and $h_{jk}$, usually used in SCPT, and also introduced the field redefinition in Eq. (\ref{defxi}) to finally work with $\delta\Xi$ and $\delta Z$ respectively. Now if we properly identify $\Theta_A,\Theta_B,V_j$, $h_{jk}$ and $\delta Z$ with the analogue quantities in the SCPT formalism (see \cite{Giesel10s} for a detailed discussion), we realize that the evolution equations for the seven perturbed invariants $\Theta_A,\Theta_B,V_j$,$h_{jk}$ and $\delta Z$ take a form almost identical to those equations that one obtains in the Lagrangian formalism using SCPT. The almost refers to the fact that in the manifestly gauge invariant formalism the dust as dynamically coupled observers has the effect that the physical Hamiltonian, which generates the evolution of the gauge invariant quantities $\delta Q_{jk}$ and $\delta\Phi$, is no longer a constraint as the case for SCPT. Instead it becomes a constant of motion. This constant of motion reflects the influence of the dust and hence the dynamically coupled observers on the system. In case of Eq. (\ref{eomperscalar}) these are exactly the terms on the right hand side of this equation, which are corrections caused by the non-vanishing Hamiltonian density $\bar{\epsilon}$ compared to the corresponding equation in SCPT. These corrections also encode the fact that in the manifestly gauge invariant model we have seven physical degrees of freedom whereas for STT in SCPT the theory possesses only three physical degrees of freedom. This can be seen from the fact that in the case of SCPT the seven invariants $\Theta_A,\Theta_B,V_j$,$h_{jk}$ and $\delta Z$ are still subject to constraints which reduce their independent number of physical degrees of freedom down to three. In contrast for the manifestly gauge invariant formalism the constraints have already been reduced by constructing the observables $Q_{jk}$ and $\Phi$. In particular in the limit of vanishing influence of the dust, meaning that its energy and momentum density vanishes, the conservation equations turn into constraint equations again and the corrections that occur in the evolution equations for the linear perturbations vanish. Then we obtain an exact match with the results obtained in SCPT. However the whole manifestly gauge invariant formalism is based on the idea that the influence of the dynamically coupled observers cannot totally be neglected and thus small corrections would always be needed to taken into account. As discussed in \cite{Giesel10s} for the dust observer in the late universe these modifications decay as compared to the usual terms.
\\
\\
As mentioned above the constraints in the SCPT become conservation equations in the manifestly gauge invariant framework due to the dynamically coupled observers since here only the total constraints consisting of STT contribution and the dust contribution vanish. First, we have two conserved quantities associated with the energy and momentum conservation respectively:
\begin{eqnarray}
{\epsilon}_j(\sigma)=-{C}_j(\sigma), \qquad {\epsilon}(\sigma)={H}(\sigma).
\end{eqnarray}
In the following we will derive the equations for the perturbed conserved charges $\delta\epsilon_j$ and $\delta\epsilon$ around an FRW background defined through $\delta\epsilon_j=\epsilon_j-\overline{\epsilon}_j$ and $\delta\epsilon=\epsilon-\overline{\epsilon}$ where for FRW $\overline{\epsilon}_j=0$ and $\overline{\epsilon}=\overline{H}$. We obtain for the linear perturbations
\begin{eqnarray}
\delta \epsilon_j&=&F(\overline{\Phi})A(\delta \dot{Q}_{jk,k}-\delta \dot{Q}_{kk,j})-2F(\overline{\Phi})\dot{A}(\delta Q_{jk,k}-\delta Q_{kk,j})-A^3F'(\overline{\Phi})(\delta \dot{\Phi})_{,j}-F(\overline{\Phi})A[\Delta\delta N_j-\delta N_{k,kj}]\nonumber\\
&&
-\big[2\dot{A}A^2F'(\overline{\Phi})+A^3F''(\overline{\Phi})\dot{\overline{\Phi}}-6A^2\dot{A}F'(\overline{\Phi})
+2A^3\dot{\overline{\Phi}}K(\overline{\Phi})\big](\delta \Phi)_{,j},\nonumber\\\label{perturbmomentum}
\end{eqnarray}
and
\begin{eqnarray}
\delta \epsilon &=& \bigg[-6F'(\overline{\Phi})A^3(\frac{\dot{A}}{A})^2
-6F''(\overline{\Phi})A^3\frac{\dot{A}}{A}\dot{\overline{\Phi}}
-6A^3\frac{\dot{A}}{A}F'(\overline{\Phi})\frac{\partial}{\partial \tau}
+A^3K'(\overline{\Phi})(\overline{\Phi})^2+2A^3K(\overline{\Phi})
\overline{\Phi}\frac{\partial}{\partial \tau}\nonumber\\
&&\quad
+2A^3F'(\overline{\Phi})\overline{D}^i\overline{D}_i
+A^3\frac{V'(\overline{\Phi})}{F(\overline{\Phi})}\bigg]\delta \Phi+\bigg[\frac12\frac{\overline{\epsilon}}{A^2F(\overline{\Phi})} \delta^{jk}
-2\delta^{jk}F(\overline{\Phi})\dot{A}\frac{\partial}{\partial \tau}+4F(\overline{\Phi})\frac{(\dot{A})^2}{A}\delta^{jk}\nonumber\\
&&\quad
-A^3F(\overline{\Phi})[G^{-1}]^{jkmn}\overline{D}_m\overline{D}_n-F'(\overline{\Phi})[-\frac{\dot{A}}{A^3}\delta^{jk}\dot{\overline{\Phi}}
+\frac{1}{A^2}\delta^{jk}\dot{\overline{\Phi}}\frac{\partial}{\partial \tau}]\bigg]\delta Q_{jk}\nonumber\\
&&
+\bigg[4F(\overline{\Phi})\dot{A}\delta^{mn}\frac{\partial}{\partial x^n}
+2F'(\overline{\Phi})A\dot{\overline{\Phi}}\delta^{mj}\frac{\partial}{\partial x^j}\bigg]\delta N_m.\label{perturbhamiltonian}
\end{eqnarray}
As the next step we decompose the vector equation (\ref{perturbmomentum}) into the longitudinal part
$\delta\epsilon^{\|}_j=\Delta^{-1}\delta\epsilon_{k,kj}$, where $\Delta^{-1}$ is the Green's function of the Laplacian $\Delta$, and the transversal part $\delta\epsilon^{\bot}=\delta\epsilon_{j}-\delta\epsilon^{\|}_j$. Inserting Eqs. (\ref{decompmetric}) and (\ref{decompshift}) into (\ref{perturbmomentum}),
the longitudinal part of Eq. (\ref{perturbmomentum}) gives
\begin{eqnarray}
(\psi +\delta \sqrt{F(\Phi)})^{'}_{,j}-\frac34(\frac{F'(\overline{\Phi})}{F(\overline{\Phi})})^2(\delta\Phi)_{,j}
=-\frac{1}{4}(\frac{\delta \epsilon^{\|}_j}{F(\overline{\Phi})A^2}+\overline{\Xi}^{'}\delta\Xi_{,j}),\label{constvector}
\end{eqnarray}
and the transversal part gives
\begin{eqnarray}
\Delta V_j=-\frac{\delta\epsilon^{\bot}_j}{F(\overline{\Phi})A^2}.\label{constvector1}
\end{eqnarray}
Written in terms of variables $\Theta_A$, $\Theta_B$ and $\delta Z$, Eq. (\ref{constvector}) becomes
\begin{eqnarray}
\Big[\Theta^{'}_B+\big[\mathcal{H}+(\ln\sqrt{F(\overline{\Phi})})^{'}\big]\Theta_A+\frac{1}{4}\overline{\Xi}^{'}\delta Z\Big]_{,j}=-\frac{1}{F(\overline{\Phi})A}\Big[\frac{\delta\epsilon^{\|}_j}{A}-\overline{\epsilon}(B-E^{'}_{,j})\Big].\label{constvector2}
\end{eqnarray}
Finally, Eq. (\ref{perturbhamiltonian}) gives
\begin{eqnarray}
&&\Delta \Theta_B-3\big[\mathcal{H}+(\ln\sqrt{F(\overline{\Phi})})^{'}\big]\Theta^{'}_B
+\frac14\big[-A^2\frac{V(\overline{\Xi})}{F(\overline{\Xi})}\Theta_A+\overline{\Xi}^{'}\delta Z^{'}-F(\overline{\Xi})A^2(\frac{V(\overline{\Xi})}{F^2(\overline{\Xi})})'_{\Xi}\delta Z\big]\nonumber\\
=&&
\frac{1}{4F(\overline{\Phi})A}\bigg[\delta\epsilon-\overline{\epsilon}\Big(\delta\ln\sqrt{F(\Phi)}+3(\psi +\delta \sqrt{F(\Phi)})+\Delta E-2(B-E^{'})^{'}+\big[\mathcal{H}+(\ln\sqrt{F(\overline{\Phi})})^{'}\big](B-E^{'})\Big)\bigg].\nonumber\\\label{constscalarmode}
\end{eqnarray}
Up to now we have derived the equations of motion for the background variables in (\ref{ddotphi}), (\ref{Friedmann}) and (\ref{Raychaudhuri}) and the perturbed variables in (\ref{eomtensor}), (\ref{eomvector}), (\ref{equaltheta}), (\ref{eomscalar1}), (\ref{eomperscalar}) and (\ref{constvector1}), (\ref{constvector2}), (\ref{constscalarmode}) in manifestly gauge invariant formalism in the case of STT.
Again we find if we choose $F(\Phi)=1$ and $K(\Phi)=\frac{1}{2}$ the STT + dust reduces to GR + scalar field + dust in \cite{Giesel10s}, which allows us to test the correctness of our equations here.
\section{Gauge Invariant Hamiltonian Formulation in Einstein Frame}
It is well known that in SCPT of STT the Jordan frame is related to the Einstein frame by a conformal transformation and can therefore be transformed into each other by appropriate field redefinitions. However, it is not clear whether this kind of equivalence will still hold in the manifestly gauge invariant formalism since such a transformation need to be formulated at the level of the reduced phase space here. In the first part of this section we will prove this equivalence. In the second part, using this equivalence, we extend our results to slightly different reference systems.
\subsection{Conformal Equivalence}
Under the conformal transformation $\tilde{g}_{ab}=F(\phi)g_{ab}$ and the field redefinition $\frac{d\xi}{d\phi}= \sqrt{3(\frac{F'(\phi)}{F(\phi)})^2+2\frac{K(\phi)}{F(\phi)}}$,
the action (\ref{actionjordan}) in Jordan frame is transformed into the following action in Einstein frame,
\begin{eqnarray}
S\, _{(\xi ,\tilde{g},dust)}=\int d^4x\sqrt{|\det (\tilde{g})|}\left[\tilde{R}^{(4)}-\frac12\tilde{g}^{\mu\nu}(\tilde{\triangledown} _\mu\xi) \tilde{\triangledown} _\nu\xi -\frac{V(\xi)}{F^2(\xi)} )\right]-\frac{1}{2}\int d^4x\sqrt{\det (\tilde{g})}\frac{\rho}{F(\xi)} \left[\tilde{g}^{\mu \nu }U_{\mu }U_{\nu }+\frac{1}{F(\xi)}\right],\nonumber\\\label{actionEinstein}
\end{eqnarray}
where $\tilde{g}_{ab}$ is now taken as the basic variable,
$\tilde{R}^{(4)}$ stands for the scalar curvature of the Einstein frame metric $\tilde{g}_{ab}$, $\tilde{g}^{ab}$ is the inverse of $\tilde{g}_{ab}$, $\tilde{\triangledown}$ is the covariant derivative compatible with $\tilde{g}_{ab}$, $V(\xi):=V(\phi(\xi))$ and $F(\xi):=F(\phi(\xi))$. Notice that the definition for the dust variables $\rho$ and $U_{\mu}$ are those in Jordan frame. This means that the reference system remains unchanged.
Hamiltonian analysis of action (\ref{actionEinstein}) is performed in appendix A, where we get the following physical Hamiltonian,
\begin{eqnarray}
\tilde{\textbf{H}}(\tau)=\int_{\mathcal{S}}d^3\sigma \tilde{H}(\tau,\sigma),\qquad \tilde{H}(\tau, \sigma)=\sqrt{F(\Xi)}\sqrt{\tilde{C}^2-\tilde{Q}^{ij}\tilde{C}_i\tilde{C}_j}(\tau, \sigma),
\end{eqnarray}
with
\begin{eqnarray}
\tilde{C}_j(\tau,\sigma )&=&\left[-2\tilde{Q}_{jl}\tilde{D}_k\tilde{P}^{kl}+\tilde{\Pi} \tilde{D}_j\Xi\right](\tau ,\sigma ),\label{CjEinstein}\\
\tilde{C}(\tau,\sigma)&=&\Bigg[\frac{1}{\sqrt{\det \tilde{Q}}}\big(\tilde{Q}_{i m}\tilde{Q}_{j n}-\frac{1}{2}\tilde{Q}_{i j}\tilde{Q}_{m n}\big)\tilde{P}^{i j}\tilde{P}^{m n}-\sqrt{\det \tilde{Q}}\tilde{R}^{(3)}\nonumber\\
&&
+\frac12\frac{\tilde{\Pi}^2}{\sqrt{\det \tilde{Q}}}+\frac12\sqrt{\det\tilde{Q}}\big(\tilde{Q}^{i j}(\tilde{D}_i\Phi)(\tilde{D}_j\Phi)+\frac{V(\Xi)}{F^2(\Xi)}\big)\Bigg](\tau ,\sigma).\label{CEinstein}
\end{eqnarray}
The elementary Poisson brackets read
\begin{eqnarray}
\{\tilde{P}^{jk}(\sigma),\tilde{Q}_{mn}(\sigma')\}=\delta ^j_{(m}\delta^k_{n)}\delta (\sigma,\sigma'),\qquad\{\tilde{\Pi}(\sigma),\Xi(\sigma' )\}=\delta (\sigma,\sigma').
\end{eqnarray}
The evolution of a general observable is given by
\begin{eqnarray}
\frac{d O}{d\tau}&=&\{\tilde{\textbf{H}}, O\}\nonumber\\
&=&\int_{\mathcal{S}}d^3\sigma
\frac{F(\sigma)}{\tilde{H}(\sigma)}\bigg(\tilde{C}(\sigma)\{\tilde{C}(\sigma),O\}-\tilde{Q}^{jk}(\sigma)\tilde{C}_j(\sigma)\{\tilde{C}_k (\sigma), O\}+\frac12\tilde{Q}^{jm}(\sigma)\tilde{Q}^{kn}(\sigma)\tilde{C}_j(\sigma)\tilde{C}_k(\sigma)\{\tilde{Q}_{jk}(\sigma), O\}\bigg)\nonumber\\
&&\qquad
+\frac{\tilde{H}(\sigma)}{2F(\sigma)}\{F(\sigma),O\}\nonumber\\
&=&\int_{\mathcal{S}}d^3\sigma \bigg(\tilde{N}(\sigma)\{\tilde{C}(\sigma),O\}+\tilde{N}^j\{\tilde{C}_j(\sigma), O\}+\frac{\tilde{H}(\sigma)}{2F(\sigma)}\tilde{N}^i(\sigma)\tilde{N}^j(\sigma)\{\tilde{Q}_{ij}(\sigma),O\}
+\frac{\tilde{H}(\sigma)}{2F(\sigma)}\{F(\sigma),O\}\bigg),
\end{eqnarray}
where the dynamical shift vector and lapse function are defined as
\begin{eqnarray}
\tilde{N}_j\equiv-\frac{F\tilde{C}_j}{\tilde{H}}, \qquad \tilde{N}\equiv\frac{F\tilde{C}}{\tilde{H}}=\sqrt{F+\tilde{Q}^{ij}\tilde{N}_i\tilde{N}_j}.\label{eomofO}
\end{eqnarray}
By calculating their Poisson brackets with $\tilde{\textbf{H}}$, we easily verify that $\tilde{\epsilon}_j:=-\tilde{C}_j(\sigma)$ and $\tilde{\epsilon}:=\tilde{H}(\sigma)$ still compose a pair of conserved quantities. Hence $\frac{\tilde{N}_j}{F(\Xi)}:=-\frac{\tilde{C}_j(\sigma)}{\tilde{H}(\sigma)}$ also remains a constant during the evolution.
Simply following the procedures in section two, we get the second order equations of motion for $\Xi$ and $\tilde{Q}_{jk}$ as
\begin{eqnarray}
\ddot{\Xi}&=&\Big[\frac{\dot{\tilde{N}}}{\tilde{N}}-\frac{(\sqrt{\det \tilde{Q}})^{\cdot}}{\sqrt{\det \tilde{Q}}}+\frac{\tilde{N}}{\sqrt{\det \tilde{Q}}}\Big(\mathcal{L}_{\overset{\rightharpoonup }{\tilde{N}}}\frac{\sqrt{\det \tilde{Q}}}{\tilde{N}}\Big)\Big](\dot{\Xi}-\mathcal{L}_{\overset{\rightharpoonup }{\tilde{N}}}\Xi)+\tilde{Q}^{jk}\Xi_{,k}\Big[\frac{\tilde{N}}{\sqrt{\det \tilde{Q}}}\Big([\tilde{N}\sqrt{\det\tilde{Q}}]_{,j}-\frac{\tilde{H}F'(\Xi)}{2F(\Xi)}\Big)\Big]\nonumber\\
&&\quad
+\tilde{N}^2\Big[\tilde{D}^i\tilde{D}_i\Xi+[\tilde{Q}^{jk}]_{,j}\Xi_{,k}-\frac12 (\frac{V(\Xi)}{(F(\Xi))^2})\Big]+2\big(\mathcal{L}_{\overset{\rightharpoonup }{\tilde{N}}}\dot{\Xi}\big)+\big(\mathcal{L}_{\dot{\overset{\rightharpoonup}{N}}}\Xi\big)-\big(\mathcal{L}_{\overset{\rightharpoonup }{\tilde{N}}}(\mathcal{L}_{\overset{\rightharpoonup}{\tilde{N}}}\Xi)\big),\label{ddotxi}\\
\ddot{\tilde{Q}}_{jk}&=&\left[\frac{\dot{\tilde{N}}}{\tilde{N}}-\frac{\Big(\sqrt{\det \tilde{Q}}\Big)^{\cdot}}{\sqrt{\det \tilde{Q}}}+\frac{\tilde{N}}{\sqrt{\det \tilde{Q}}}\Big(\mathcal{L}_{\overset{\rightharpoonup}{\tilde{N}}}(\frac{\sqrt{\det \tilde{Q}}}{\tilde{N}})\Big)\right]\Big(\dot{\tilde{Q}}_{jk}-(\mathcal{L}_{\overset{\rightharpoonup }{\tilde{N}}}\tilde{Q})_{jk}\Big)\nonumber\\
&&
+\tilde{Q}^{m n}\Big(\dot{\tilde{Q}}_{mj}-(\mathcal{L}_{\overset{\rightharpoonup }{\tilde{N}}}\tilde{Q})_{mj}\Big)\Big(\dot{\tilde{Q}}_{nk}-(\mathcal{L}_{\overset{\rightharpoonup}{\tilde{N}}}\tilde{Q})_{nk }\Big)+\tilde{N}^2\Big(\Xi _{,j}\Xi _{,k}-2\tilde{R}_{jk}\Big)\nonumber\\
&&
+2\tilde{N} \tilde{D}_j \tilde{D}_k\tilde{N}+2\big(\mathcal{L}_{\overset{\rightharpoonup }{\tilde{N}}}\dot{\tilde{Q}}\big)_{jk}+\big(\mathcal{L}_{\dot{\overset{\rightharpoonup}{N}}}\tilde{Q} \big)_{jk}-\big(\mathcal{L}_{\overset{\rightharpoonup}{\tilde{N}}}(\mathcal{L}_{\overset{\rightharpoonup}{\tilde{N}}}\tilde{Q})\big)_{jk}\nonumber\\
&&
-\frac{\tilde{N}\tilde{H}(\sigma)}{\sqrt{\det \tilde{Q}}F(\Xi)}\tilde{G}_{jkmn}\tilde{N}^m\tilde{N}^n+\tilde{Q}_{j k}\Big[-\frac{\tilde{N}^2}{2\sqrt{\det \tilde{Q}}}\tilde{C}+\tilde{N}^2\frac{V(\Xi)}{(F(\Xi))^2}\Big].\label{ddottildeq}
\end{eqnarray}
It is not difficult to check that Eqs. (\ref{ddotxi}) and (\ref{ddottildeq}) exactly reproduce Eqs. (\ref{phisecond}) and (\ref{qsecond}) after substituting $\tilde{Q}_{jk}\rightarrow F(\Phi)Q_{jk}$, $\frac{d\Xi(\Phi)}{d\Phi}\rightarrow\sqrt{3(\frac{F'(\Phi)}{F(\Phi)})^2+2\frac{K(\Phi)}{F(\Phi)}}$. Thus the evolution equations in Einstein frame can be related to those in Jordan frame through the conformal transformation and field redefinition. Since the equivalence holds for general variables, obviously it also holds for the linear perturbed variables under the assumption that the transformations between the two frames are non-singular everywhere. Thus we conclude that the Jordan and Einstein frames are still equivalent to each other in the gauge invariant formulation.
\subsection{Generalizing to Different Reference Systems}
In above sections we chose the dust particles as the observers, whose equations of motion satisfy $g^{\mu\nu}U_{\mu}U_{\nu}=-1$. An interesting question is whether one can choose different observers, for example the ones satisfying $g^{\mu\nu}U_{\mu}U_{\nu}=-X(\phi)$ with $X(\phi)$ an arbitrary (positive) function. To answer this question, we first generalize the action (\ref{actionEinstein}) to
\begin{eqnarray}
S\, _{(\xi ,\tilde{g},dust)}=\int d^4x\sqrt{|\det (\tilde{g})|}\left[\tilde{R}^{(4)}-\frac12\tilde{g}^{\mu\nu}(\tilde{\triangledown}_{\mu}\xi) \tilde{\triangledown}_{\nu}\xi -\frac{V(\xi)}{F^2(\xi)} )\right]-\frac{1}{2}\int d^4x\sqrt{|\det(\tilde{g})|}\frac{\rho}{J(\xi)}\left[\tilde{g}^{\mu \nu}U_{\mu }U_{\nu }+\frac{1}{L(\xi)}\right],\nonumber\\\label{modactEinstein}
\end{eqnarray}
where $J(\xi)$ and $L(\xi)$ are arbitrary positive functions. Since it has been shown in the last subsection that the two frames are equivalent, we choose Einstein frame in which the equations look simpler. The physical Hamiltonian reads (see Appendix A)
\begin{eqnarray}
\tilde{\textbf{H}}(\tau)=\int_{\mathcal{S}}d^3\sigma \tilde{H}(\tau,\sigma),\qquad \tilde{H}(\tau, \sigma)=\sqrt{L(\Xi)}\sqrt{\tilde{C}^2-\tilde{Q}^{ij}\tilde{C}_i\tilde{C}_j}(\tau, \sigma),
\end{eqnarray}
where expressions for $\tilde{C}_j$ and $\tilde{C}$ are the same as Eqs. (\ref{CjEinstein}) and (\ref{CEinstein}). The dynamical shift vector and lapse function are defined by
\begin{eqnarray}
\tilde{N}_j\equiv-\frac{L\tilde{C}_j}{\tilde{H}}, \qquad \tilde{N}\equiv\frac{L\tilde{C}}{\tilde{H}}=\sqrt{L+\tilde{Q}^{ij}\tilde{N}_i\tilde{N}_j}.\label{eomofO}
\end{eqnarray}
In FRW background, we have $\tilde{\overline{N}}_j=0$ and $\tilde{\overline{N}}=\sqrt{L(\overline{\Xi})}$. The equations of motion of background variables read
\begin{eqnarray}
\ddot{\overline{\Xi}}&=&\dot{(\ln\sqrt{L(\Xi)})}\dot{\overline{\Xi}}-3(\frac{\dot{\tilde{A}}}{\tilde{A}})\dot{\overline{\Xi}}
-L(\overline{\Xi})(\frac{V(\overline{\Xi})}{F^2(\overline{\Xi})})'_{\Xi}
-\frac{L(\overline{\Xi})'_{\Xi}}{2\sqrt{L(\overline{\Xi})}}\frac{\tilde{\bar{\epsilon}}}{\tilde{A}^3},\label{eomxil0}\\
3(\frac{\dot{\tilde{A}}}{\tilde{A}})^2&=&\frac{1}{4}\big(\dot{\overline{\Xi}}^2+\frac{L(\overline{\Xi})V(\overline{\Xi})}{F^2(\overline{\Xi})}\big)
-\frac{\tilde{\bar{\epsilon}}}{2\sqrt{L(\overline{\Xi})}\tilde{A}^3},\label{constrfrwl0}\\
3(\frac{\ddot{\tilde{A}}}{\tilde{A}})&=&-\frac{1}{4}\big[\frac{1}{2}(\dot{\overline{\Xi}}+\frac{L(\overline{\Xi})V(\overline{\Xi})}{F^2(\overline{\Xi})}
+\frac{3}{2}(\dot{\overline{\Xi}}^2-\frac{L(\overline{\Xi})V(\overline{\Xi})}{F^2(\overline{\Xi})}))\big]+\frac{\tilde{\bar{\epsilon}}}
{4\sqrt{L(\overline{\Xi})}\tilde{A}^3},
\label{friedmannl0}
\end{eqnarray}
where we denote $\tilde{Q}_{jk}\equiv\tilde{A}^2\delta_{jk}$ and $\tilde{\bar{\epsilon}}\equiv\tilde{H}$. The reader can easily check that the above equations reproduce the results in Eqs. (\ref{ddotphi}), (\ref{Friedmann}) and (\ref{Raychaudhuri}) by setting $L(\overline{\Xi})=F(\overline{\Xi})$ and replacing $\tilde{A}$ with $\sqrt{F(\overline{\Xi})}A$. For the linear perturbation equations, using the following decomposition
\begin{eqnarray}
\delta \tilde{N}=\frac{L'(\Xi)}{2\sqrt{L(\overline{\Xi})}}\delta \Xi,\qquad
\delta \tilde{N}_j=\sqrt{L(\overline{\Xi})}\tilde{A}(S_i+B_{,i}), \qquad \delta \tilde{Q}_{jk}:=\tilde{A}^2[2\tilde{\psi}\delta_{jk}+2E_{,jk}+2F_{(j,k)}+h_{jk}],
\end{eqnarray}
the evolution equations of the tensor and vector modes in the conformal time frame $\frac{d}{d\eta}=\frac{\tilde{A}}{\tilde{\overline{N}}}\frac{d}{d\tau}$ are
\begin{eqnarray}
2\mathcal{\tilde{H}}h^{'}_{jk}+h^{''}_{jk}-\Delta h_{jk}=0,\\
\Delta V_j=-\frac{\delta\tilde{\epsilon}^{\bot}_j}{\tilde{A}^2},\qquad 2\mathcal{\tilde{H}}V_j+V^{'}_{j,k}=0,
\end{eqnarray}
where $\mathcal{\tilde{H}}:=\frac{\tilde{A}^{'}}{\tilde{A}}$. The scalar mode contribution gives
\begin{eqnarray}
&&\Big[\tilde{\Theta}^{'}_B+\mathcal{\tilde{H}}\tilde{\Theta}_A+\frac{1}{4}\overline{\Xi}'\delta Z\Big]_{,j}=-\frac{1}{\tilde{A}}\Big[\frac{\delta \tilde{\epsilon}^{\|}_j}{\tilde{A}}-\frac{\tilde{\overline{\epsilon}}}{\sqrt{L(\overline{\Xi})}}(B-E'_{,j})\Big],\\
&&\Delta \tilde{\Theta}_B-3\mathcal{\tilde{H}}\tilde{\Theta}^{'}_B
+\frac14\big[-\tilde{A}^2\frac{V(\overline{\Xi})}{F^2(\overline{\Xi})}\tilde{\Theta}_A+\overline{\Xi}^{'}\delta Z^{'}-\tilde{A}^2(\frac{V(\overline{\Xi})}{F^2(\overline{\Xi})})^{'}_{\Xi}\delta Z\big]\nonumber\\
&=&\frac{1}{4\sqrt{L(\overline{\Xi})}\tilde{A}}\bigg[\delta \tilde{\epsilon}-\tilde{\overline{\epsilon}}\big(\delta\ln\sqrt{F(\Phi)}+3\tilde{\psi}+\Delta E-2(B-E^{'})^{'}+\mathcal{\tilde{H}}(B-E^{'})\big)\bigg],\\
&&\tilde{\Theta}^{''}_B+\mathcal{\tilde{H}}(\tilde{\Theta}_A+2\tilde{\Theta}_B)+(2\mathcal{\tilde{H}}^{'}+\mathcal{\tilde{H}}^2)\tilde{\Theta}_A
=-\frac14\big[(\overline{\Xi}^{'})^2\tilde{\Theta}_A+\overline{\Xi}^{'}\delta Z^{'}-\tilde{A}^2(\frac{V(\overline{\Xi})}{F^2(\overline{\Xi})})'_{\Xi}\delta Z\big],
\end{eqnarray}
where $\tilde{\Theta}_A:=-\delta \ln\sqrt{ L(\Xi)}-\mathcal{\tilde{H}}(B-E^{'})-(B-E^{'})^{'}$, $\tilde{\Theta}_B:=\tilde{\psi}+\mathcal{\tilde{H}}(B-E^{'})$, $\delta Z:=\delta\Xi+\overline{\Xi}^{'}(B-E^{'})$. It is easy to see that the above definitions are the same as those in (\ref{newvariables}) when $L(\Xi)=F(\Phi(\Xi))$. The evolution of the perturbed gravitational scalar field reads
\begin{eqnarray}
&&\delta Z^{''}+2\mathcal{\tilde{H}}\delta Z-\Delta \delta Z +(\frac{V(\overline{\Xi})}{F^2(\overline{\Xi})})'_{\Xi\Xi}\tilde{A}^2\delta Z+\overline{\Xi}^{'}\tilde{\Theta}^{'}_A-2(\frac{V(\overline{\Xi})}{F^2(\overline{\Xi})})'_{\Xi}\tilde{A}^2\tilde{\Theta}^{'}_A
+3\overline{\Xi}^{'}\tilde{\Theta}^{'}_B\nonumber\\
&=&\big(-\frac{L(\overline{\Xi})'_\Xi}{2L^{\frac{3}{2}}(\overline{\Xi})}\frac{\tilde{\epsilon}}{\tilde{A}}(B-E^{'})\big)^{'}
-\frac{L(\overline{\Xi})'_\Xi}{2L^{\frac{3}{2}}(\overline{\Xi})}\frac{\tilde{\epsilon}}{\tilde{A}}(B-E^{'})^{'}
-\frac{\tilde{A}^2}{L(\overline{\Xi})}\delta \big(\frac{L(\Xi)'_\Xi}{2\sqrt{L(\Xi)}}\frac{\tilde{\epsilon}}{\sqrt{\det \tilde{Q}}}\big).
\end{eqnarray}
Again we can easily match these equations with the ones in Jordan frame. We find that the different choice with $\tilde{g}^{\mu\nu}U_{\mu}U_{\nu}=-\frac{1}{L(\Xi)}$ leads to the change of dynamical lapse function $\tilde{N}$ and the correction terms.
Now we compare our manifestly gauge invariant cosmological perturbation theory (MGICPT) with the standard cosmological perturbation theory (SCPT). For the convenience of readers, we list all the equations in table 1. We denote the non gauge invariant background variables in SCPT with lowercase letters and the gauge invariant linearly perturbed variables with a hat. Since the Jordan and Einstein frames are equivalent in our manifestly gauge invariant Hamiltonian formalism as well as in the standard formalism, we write all equations in Einstein frame where they take a concise form. In Einstein frame $\tilde{A}:=\sqrt{F(\overline{\Xi})}A$, $\mathcal{\tilde{H}}:=\frac{d\tilde{A}/d\eta}{\tilde{A}}=\frac{1}{\sqrt{F(\overline{\Xi})}}\frac{d\tilde{A}}{d\tau}$, $\tilde{a}:=\sqrt{F(\overline{\xi})}a$, $\tilde{h}:=\frac{d\tilde{a}/d\eta}{\tilde{a}}=\frac{1}{\sqrt{F(\overline{\xi})}}\frac{d\tilde{a}}{dt}$, while the definitions for $\Theta_A$, $\Theta_B$ and $\delta Z$ are given in Eq. (\ref{newvariables}). From the comparison we see that these equations match each other precisely provided that the Hamiltonian density $\overline{\epsilon}$ goes to zero.
\begin{table}
\caption{\large\textbf{Comparison between MGICPT and SCPT of STT in Einstein frame}}
\begin{tabular}{|c|l|l|l|}\hline
\backslashbox{\textbf{E.O.M.}}{}
& Manifestly Gauge Invariant Formalism & Standard Formalism\\\hline
\multirow{2}*{} &
$\overline{\Xi}^{''}=-2\mathcal{\tilde{H}}\overline{\Xi}^{'}-\tilde{A}^2
\big(\frac{V(\overline{\Xi})}{F^2(\overline{\Xi})}\big)'_\Xi
-\frac{F(\overline{\Xi})'_\Xi}{2F^{\frac32}(\overline{\Xi})}\frac{\bar{\epsilon}}{\tilde{A}},$
&$\overline{\xi}^{''}=-2\tilde{h}\overline{\xi}^{'}-\tilde{a}^2
\big(\frac{V(\overline{\xi})}{F^2(\overline{\xi})}\big)'_\xi,
$\\
\textbf{Background}
&$-6\mathcal{\tilde{H}}^2+\frac12(\overline{\Xi}^{'})^2+\tilde{A}^2\frac{V(\overline{\Xi})}{F^2(\overline{\Xi})}
=\frac{\bar{\epsilon}}{F^{\frac12}(\overline{\Xi})\tilde{A}},$
&
$-6\tilde{h}^2+\frac12(\overline{\xi}^{'})^2+\tilde{a}^2\frac{V(\overline{\xi})}{F^2(\overline{\xi})}
=0,$
\\
\textbf{Variables}
&
$2\mathcal{\tilde{H}}^{'}+4\mathcal{\tilde{H}}^2
-\tilde{A}^2\frac{V(\overline{\Xi})}{F^2(\overline{\Xi})}=\frac{-\bar{\epsilon}}{2F^{\frac12}(\overline{\Xi})\tilde{A}},$
&
$2\tilde{h}^{'}+4\tilde{h}^2-\tilde{a}^2\frac{V(\overline{\xi})}{F^2(\overline{\xi})}=0,$\\\hline
\textbf{Tensor-Mode}&$2\mathcal{\tilde{H}}h^{'}_{jk}+h^{''}_{jk}-\Delta h_{jk}=0,$
&$2\tilde{h}\hat{h}^{'}_{ab}+\hat{h}^{''}_{ab}-\Delta \hat{h}_{ab}=0,$\\\hline
\multirow{2}*{\textbf{Vector-Mode}}
&$\Delta V_j=-\frac{\delta\epsilon^{\bot}_j}{\tilde{A}^2},$
&$\Delta \hat{V}_a=0,$\\
&$2\mathcal{\tilde{H}}\hat{V}_j+\hat{V}^{'}_{j,k}=0,$
&$2\tilde{h}\hat{V}_a+\hat{V}^{'}_{a,b}=0,$\\\hline
\multirow{6}*{\textbf{Scalar-Mode}}
&\quad$\Big[\Theta^{'}_B+\mathcal{\tilde{H}}\Theta_A+\frac{1}{4}\overline{\Xi}^{'}\delta Z\Big]_{,j}$
&$\hat{\Theta}^{'}_B+\tilde{h}\hat{\Theta}_A+\frac{1}{4}\overline{\xi}'\delta \hat{Z}=0, $\\
&$=-\frac{1}{\tilde{A}}\Big[\frac{\delta \epsilon^{\|}_j}{\tilde{A}}-F^{-\frac{1}{2}}(\overline{\Xi})\overline{\epsilon}(B-E^{'}_{,j})\Big],$
&\\
&\quad$\Delta \Theta_B-3\mathcal{\tilde{H}}\Theta^{'}_B+\frac14\big[-\tilde{A}^2\frac{V(\overline{\Xi})}{F^2(\overline{\Xi})}\Theta_A$
&\quad$\Delta \hat{\Theta}_B-3\tilde{h}\hat{\Theta}^{'}_B+\frac14\big[-\tilde{a}^2\frac{V(\overline{\xi})}{F^2(\overline{\xi})}\hat{\Theta}_A$\\
&$ +\overline{\Xi}^{'}\delta Z^{'}-\tilde{A}^2(\frac{V(\overline{\Xi})}{F^2(\overline{\Xi})})'_{\Xi}\delta Z\big]$
&$ +\overline{\xi}^{'}\delta \hat{Z}^{'}-\tilde{a}^2(\frac{V(\overline{\xi})}{F^2(\overline{\xi})})'_{\xi}\delta \hat{Z}\big]$\\
&$=\frac{1}{4F^{\frac12}(\overline{\Xi})\tilde{A}}\bigg[\delta \epsilon-\overline{\epsilon}(-\delta\ln\sqrt{F(\Xi)}$
&$=0,$\\
&$+3\tilde{\psi}+\Delta E)+2\bar{\epsilon}(B-E^{'})^{'}-\mathcal{\tilde{H}}(B-E^{'})\bigg],$
&\\
&\quad$\Theta^{''}_B+\mathcal{\tilde{H}}(\Theta_A+2\Theta_B)^{'}+(2\mathcal{\tilde{H}}^{'}+\mathcal{\tilde{H}}^2)\Theta_A$
&\quad$\hat{\Theta}^{''}_B+\tilde{h}(\hat{\Theta}_A+2\hat{\Theta}_B)^{'}+(2\tilde{h}^{'}+\tilde{h}^2)\hat{\Theta}_A$
\\
&$=-\frac14\big[(\overline{\Xi}^{'})^2\Theta_A+\overline{\Xi}^{'}\delta Z^{'}
-\tilde{A}^2(\frac{V(\overline{\Xi})}{F^2(\overline{\Xi})})'_{\Xi}\delta Z\big],$
&$=-\frac14\big[(\overline{\xi}^{'})^2\hat{\Theta}_A+\overline{\xi}^{'}\delta \hat{Z}^{'}
-\tilde{a}^2(\frac{V(\overline{\xi})}{F^2(\overline{\xi})})'_{\xi}\delta \hat{Z}\big],$\\
&
&\\\hline
\multirow{2}*{\textbf{Perturbed}}
&$\quad\delta Z^{''}+2\mathcal{\tilde{H}}\delta Z-\Delta \delta Z +(\frac{V(\overline{\Xi})}{F^2(\overline{\Xi})})'_{\Xi\Xi}\tilde{A}^2\delta Z$
&$\quad\delta \hat{Z}^{''}+2\tilde{h}\delta \hat{Z}-\Delta \delta \hat{Z} +(\frac{V(\overline{\xi})}{F^2(\overline{\xi})})'_{\xi\xi}\tilde{a}^2\delta \hat{Z}$\\
\textbf{Gravitational}
&\quad$+\Xi^{'}\Theta^{'}_A-2(\frac{V(\overline{\Xi})}{F^2(\overline{\Xi})})'_{\Xi}\tilde{A}^2\Theta^{'}_A+3\overline{\Xi}^{'}\Theta^{'}_B$
&$\quad+\xi^{'}\hat{\Theta}^{'}_A-2(\frac{V(\overline{\xi})}{F^2(\overline{\xi})})'_{\xi}\tilde{a}^2\hat{\Theta}^{'}_A+3\xi^{'}\hat{\Theta}^{'}_B$\\
\textbf{Scalar Field}
&$=\big(\frac12(\frac{1}{F(\overline{\Xi})})'_{\Xi}\frac{\sqrt{F}\overline{\epsilon}}{\tilde{A}}(B-E^{'})\big)^{'}
+\frac12(\frac{1}{F(\overline{\Xi})})'_{\Xi}\frac{\sqrt{F}\overline{\epsilon}}{\tilde{A}}(B-E^{'})^{'}$
&$=0.$\\
&$-\frac{\tilde{A}^2}{F(\Xi)}\delta \big(\frac12(\frac{1}{F(\Xi)})'_{\Xi}\frac{\overline{\epsilon}}{\sqrt{\det Q}}\big).$
&\\
\hline
\end{tabular}
\end{table}
\newpage
\section{Concluding Remarks}
In this paper we extended the manifestly gauge invariant Hamiltonian formalism introduced in \cite{Giesel10} to the case of STT. Our results are summarized as follows:
First, we derived the Hamiltonian equations of motion in the manifestly gauge invariant framework using the Brown-Kuch{\v r}-dust as reference fields in the Jordan frame. Afterwards we used these equations to derive the evolution equations for the linear perturbations of the 3-metric and the gravitational scalar field on a general relativistic spacetime background. These are the generalization of the results obtained in \cite{Giesel10} to the case of STT. Then we applied our general result to the special case of a flat FRW background. This allowed to compare the results obtained in our formalism to the results from standard cosmological perturbation theory (SCPT). Likewise to the analysis in \cite{Giesel10} where gravity and a minimally coupled scalar field were considered, also in the case of the STT we have shown that our results and the one from SCPT exactly agree in the limit when the energy momentum and its perturbation of the dust vanishes. However, taking the idea of the manifestly gauge invariant framework seriously this limit is rather artificial since the dynamically coupled observer will always have an imprint on the system and here these are exactly the computed corrections compared to the results of SCPT.
Second, we analyzed the question whether the Jordan frame and the Einstein frame are still equivalent to each other in the manifestly gauge invariant Hamiltonian formalism by showing their equations of motions can be related to each other through conformal transformations. Third, we generalized our results from the original reference system to a slightly different reference system in the Einstein frame and we showed that these equations are consistent with the ones in the Jordan frame.
\\
\\
There are several possible directions to extend our results. First, since we have derived the linear perturbation equations upon a general background, the direct extension is to choose a background different from the flat FRW case and study the cosmological perturbations in STT. For instance, the homogeneous but anisotropic Bianchi models are interesting candidates. Second, since STT can include a large variety of modified gravity models, we can choose some specific functions of $F(\Phi)$, $K(\Phi)$ and $V(\Phi)$ to study the detailed physical predictions in the inflation period and compare them with the predictions of SCPT. Third, our formalism has the advantage over the procedure used in the context of standard cosmological perturbations theory that we can directly get the gauge invariant perturbed variables from the full metric to \emph{any} order without worrying about the difficulty in constructing the higher order gauge invariant variables as in the standard way, thus it will be very convenient to use our formalism to study the non-linear effects such as the non-Gaussianity in the primordial perturbations.
\section*{ACKNOWLEDGMENTS}
Y.H. thanks the Chinese Scholarship Council (CSC) for financial support. K.G. thanks the Emerging Field Project ``Quantum Geometry" of the FAU Erlangen-N\"urnberg for financial support. This work is also supported in part by the NSFC (Grant Nos. 11235003 and 11475023) and the Research Fund for the Doctoral Program of Higher Education of China.
|
2,877,628,089,020 | arxiv | \section{Introduction}
Wormholes are solutions of the equations of gravitation which imply a nontrivial topology of spacetime, connecting two regions of the universe --or of two different universes-- by a traversable throat. For compact configurations, the throat is a minimal area surface \cite{hovis}; this feature is determined by the fact that the geodesics open up at the throat (for an alternative definition in terms of trapped surfaces see Ref. \cite{hayward} and references therein). Following the leading work by Morris and Thorne \cite{mo} in which their main aspects were introduced, traversable Lorentzian wormholes have received considerable attention \cite{visser}. If traversable wormholes exist, they could imply some unusual consequences, as for example the possibility of time travel \cite{mo-fro}. However, difficulties with their mechanical stability and the requirement of exotic matter (matter not fulfilling the energy conditions), which seems unavoidable within the framework of General Relativity, constitute the main objections against the actual existence of these objects. In particular, these two central aspects have been thoroughly analyzed for wormholes of the thin-shell class \cite{povi,visser}, that is, geometries which are mathematically constructed by cutting and pasting two manifolds, so that a matter layer is located at the throat (see Ref. \cite{ernesto} and references therein).
About two decades ago, the geometrical aspects of topological defects like cosmic strings began to be the object of a detailed study, mainly because they could have played an important role in structure formation in the early universe; it was also pointed out that they could manifest by gravitational lensing effects (see Ref. \cite{vilenkin}). From a different point of view, one-dimensional (open or closed) strings are the objects more seriously considered by present theoretical developments to be the fundamental building blocks of nature. As a consequence, the interest in the gravitational effects of both fundamental and cosmic strings has been recently renewed (for example, see Refs. \cite{strings}). It is thus natural that cylindrically symmetric wormhole geometries, as those associated to cosmic strings are, have been considered in the last years.
Recent works in which cylindrical wormholes have been studied include those by Cl\'ement \cite{cle1,cle2}, Aros and Zamorano \cite{arza}, Kuhfittig \cite{ku}, Bronnikov and Lemos \cite{brle} and our papers \cite{ei1,ei2,mc}. The thin-shell wormhole configurations associated to local and global cosmic strings analyzed in Refs. \cite{ei1,ei2,mc} turned to be unstable under velocity perturbations conserving the cylindrical symmetry. Moreover, it was pointed out \cite{ei2} that this feature seemed to be independent of the particular geometry considered, as long as the symmetry and the form of the equations of state of the static configurations were preserved. Mechanical instability would constitute an additional restriction to the possibility of finding these objects in the present day universe, besides the usual problem with exotic matter. However, regarding the issue of exotic matter, there is a subtle point in the case of non compact wormholes --as cylindrically symmetric ones are--. It was noted in Ref. \cite{brle} that for such configurations there are two admissible definitions of the throat, and that one of them --that the geodesics {\it restricted to a plane normal to the symmetry axis} open up-- can be compatible with a positive energy density.
In this work we extend our previous analysis of non-rotating cylindrical thin-shell wormholes. We first construct them by cut and paste of two equal regions of the most general static cylindrically symmetric geometry. Then we consider small velocity perturbations, under the hypothesis that the form of the equations of state of matter on the shell are preserved in the subsequent evolution. Our results demonstrate the conjecture about the stability of these objects introduced in Ref. \cite{ei2}. Second, we discuss the implications of relaxing the flare-out condition, as proposed in \cite{brle}. We find that within this new approach, for certain values of the parameters characterizing the configurations, cylindrical thin-shell wormholes could be supported by matter of positive energy density. The analysis is exemplified with wormholes constructed starting from Einstein--Maxwell cylindrical geometries. Throughout the paper we adopt units such $G=c=1$.
\section{Wormhole construction and characterization}\label{cylind}
In coordinates $X^{\alpha}=(t,r,\varphi ,z)$, the general static metric with cylindrical symmetry can be written in the form
\begin{equation}
ds^2 = -A(r)dt^2 +B(r)dr^2 +C(r)d\varphi ^2+D(r)dz^2,
\label{e1}
\end{equation}
where $A$, $B$, $C$ and $D$ are positive functions of $r$. From this geometry, we take two copies $ \mathcal{M}^{\pm} = \{ x / r \geq a \}$ of the region with $r \geq a$ and join them at the hypersurface
$ \Sigma \equiv \Sigma^{\pm} = \{ x / r - a = 0 \}$ to make a manifold $\mathcal{M}=\mathcal{M}^{+} \cup \mathcal{M}^{-}$, which is geodesically complete. If the geometry opens up at the shell (flare-out condition), this construction creates a cylindrically symmetric thin-shell wormhole with two regions connected by a throat at $\Sigma $. For the analysis of this traversable wormhole, we use the standard Darmois-Israel formalism \cite{daris} (for a modern review see Ref. \cite{mus}).
The throat of the wormhole is a synchronous timelike hypersurface, in which we adopt coordinates $\xi ^i=(\tau , \varphi,z )$, with $\tau $ the proper time on the shell. For the subsequent analysis of the stability of the static configurations, we let the radius of the throat be a function of $\tau $, i.e. $a = a ( \tau )$. In this case, the shell is defined by $\Sigma : \mathcal{H} ( r, \tau ) = r - a ( \tau ) = 0$. The extrinsic curvature (second fundamental forms) associated with the two sides of the shell is then given by
\begin{equation}
K_{ij}^{\pm} = - n_{\gamma}^{\pm} \left. \left( \frac{\partial^2
X^{\gamma}}{\partial \xi^i \partial \xi^j} +\Gamma_{\alpha\beta}^{\gamma}
\frac{\partial X^{\alpha}}{\partial \xi^i} \frac{\partial
X^{\beta}}{\partial \xi^j} \right) \right|_{\Sigma}, \label{e2}
\end{equation}
where $n_{\gamma}^{\pm}$ are the unit normals ($n^{\gamma} n_{\gamma} = 1$) to
$\Sigma$ in $\mathcal{M}$:
\begin{equation}
n_{\gamma}^{\pm} = \pm \left| g^{\alpha \beta} \frac{\partial
\mathcal{H}}{\partial X^{\alpha}} \frac{\partial \mathcal{H}}{\partial
X^{\beta}} \right|^{- 1 / 2} \frac{\partial \mathcal{H}}{\partial
X^{\gamma}} . \label{e3}
\end{equation}
It is convenient to use the orthonormal basis $\{ e_{\hat{\tau}} = \sqrt{1/A(r)}e_{t}$, $e_{\hat{\varphi}}
= \sqrt{1/C(r)}e_{\varphi}$, $e_{\hat{z}} =\sqrt{1/D(r)}e_{z}\}$,
for which $g_{_{\hat{\imath} \hat{\jmath}}} = \eta_{_{\hat{\imath}
\hat{\jmath}}}=diag(-1,1,1)$, so we obtain
\begin{equation}
K_{\hat{\tau} \hat{\tau}}^{\pm} = \mp \frac{2B(a)^2 \ddot{a}+B'(a)
+2B'(a)B(a)\dot{a}^2}{2B(a) \sqrt{B(a)} \sqrt{1+B(a)\dot{a}^2}},
\label{e4}
\end{equation}
\begin{equation}
K_{\hat{\varphi} \hat{\varphi}}^{\pm} = \pm \frac{C'(a)\sqrt{1+B(a)
\dot{a}^2}}{2C(a)\sqrt{B(a)}},
\label{e5}
\end{equation}
and
\begin{equation}
K_{\hat{z} \hat{z}}^{\pm} = \pm \frac{D'(a)\sqrt{1+B(a)\dot{a}^2}}
{2D(a) \sqrt{B(a)}},
\label{e6}
\end{equation}
where the dot and the prime represent $d / d \tau$ and $d/dr$, respectively. The Einstein equations on the shell, also called the Lanczos equations, can be written in the form:
\begin{equation}
-[K_{\hat{\imath} \hat{\jmath}}]+[K]g_{\hat{\imath} \hat{\jmath}}=
8\pi S_{\hat{\imath} \hat{\jmath}},
\label{e7}
\end{equation}
where $[K_{_{\hat{\imath} \hat{\jmath}}}]\equiv K_{_{\hat{\imath}
\hat{\jmath}}}^+ - K_{_{\hat{\imath} \hat{\jmath}}}^-$,
$[K]=g^{\hat{\imath} \hat{\jmath}}[K_{\hat{\imath} \hat{\jmath}}]$ is the
trace of $[K_{\hat{\imath} \hat{\jmath}}]$ and
$S_{_{\hat{\imath} \hat{\jmath}}} = \text{\textrm{diag}} ( \sigma,
p_{\varphi }, p_{z} )$ is the surface stress-energy tensor, with $\sigma$ the surface energy density and $p_{\varphi , z}$ the surface pressures. By replacing Eqs. (\ref{e4}), (\ref{e5}) and (\ref{e6}) in Eq. (\ref{e7}) we have
\begin{equation}
\sigma = - \frac{\sqrt{1 + B(a) \dot{a}^2}}{8 \pi \sqrt{B(a)}} \left[
\frac{C'(a)}{C(a)} + \frac{D'(a)}{D(a)} \right],
\label{e8}
\end{equation}
\begin{equation}
p_{\varphi} = \frac{1}{8 \pi \sqrt{B(a)} \sqrt{1 + B(a)\dot{a}^2}}
\left\{ 2 B(a) \ddot{a} + B(a) \left[ \frac{D'(a)}{D(a)} + \frac{2 B'(a)}
{B(a)} \right] \dot{a}^2 + \frac{D'(a)}{D(a)} + \frac{B'(a)}{B(a)} \right\},
\label{e9}
\end{equation}
\begin{equation}
p_{z} = \frac{1}{8 \pi \sqrt{B(a)} \sqrt{1 + B(a) \dot{a}^2}}
\left\{ 2 B(a) \ddot{a} + B(a) \left[ \frac{C'(a)}{C(a)} + \frac{2 B'(a)}
{B(a)} \right] \dot{a}^2 + \frac{C'(a)}{C(a)} + \frac{B'(a)}{B(a)} \right\} .
\label{e10}
\end{equation}
It is easy to see that $p_{\varphi }$, $p_{z}$ and $\sigma$
satisfy the equation
\begin{equation}
p_z-p_{\varphi }=\frac{C(a)D'(a)-C'(a)D(a)}
{C(a)D'(a)+C'(a)D(a)}\sigma .
\label{e11}
\end{equation}
The static equations are obtained by putting $\dot{a}=0$ and $\ddot{a}=0$ in Eqs.
(\ref{e8}), (\ref{e9}) and (\ref{e10}):
\begin{equation}
\sigma = - \frac{1}{8 \pi \sqrt{B(a)}} \left[
\frac{C'(a)}{C(a)} + \frac{D'(a)}{D(a)} \right],
\label{e12}
\end{equation}
\begin{equation}
p_{\varphi} = \frac{1}{8 \pi \sqrt{B(a)} }
\left[\frac{D'(a)}{D(a)}+ \frac{B'(a)}{B(a)} \right],
\label{e13}
\end{equation}
\begin{equation}
p_{z} = \frac{1}{8 \pi \sqrt{B(a)}}
\left[\frac{C'(a)}{C(a)} + \frac{B'(a)}{B(a)} \right] .
\label{e14}
\end{equation}
By using Eq. (\ref{e12}), the Eqs. (\ref{e13}) and (\ref{e14}) can be recast in the form
\begin{equation}
p_{\varphi}=-\frac{C(a)[B(a)D'(a)+B'(a)D(a)]}{B(a)[C(a)D'(a)+C'(a)D(a)]} \sigma,
\label{e15}
\end{equation}
\begin{equation}
p_{z}=-\frac{D(a)[B(a)C'(a)+B'(a)C(a)]}{B(a)[C(a)D'(a)+C'(a)D(a)]} \sigma .
\label{e16}
\end{equation}
Then, the functions $B$, $C$ and $D$ determine the equations of state
$p_{\varphi}(\sigma)$ and $p_{z}(\sigma)$ of the exotic matter on the shell.
If $\left( CD \right)'(a)=C'(a)D(a)+C(a)D'(a)>0$, from Eq. (\ref{e12}) it follows that the surface energy density is negative, indicating the presence of \textit{exotic} matter at the throat. The usual definition of the wormhole throat --suitable for compact configurations-- states that it is a {\it minimal area surface}. In the cylindrical case one can define the area function ${\cal A}(r)=\sqrt{C(r)D(r)}$, which should increase at both sides of the throat; then ${\cal A}'(a)>0$, so that $\left( CD \right)'(a)>0$. We shall call it the {\it areal} flare-out condition, for which negative energy density at the throat is clearly unavoidable. However, Bronnikov and Lemos \cite{brle} recently pointed out that, for infinite cylindrical configurations, a less restrictive definition of the wormhole throat seems to be the most natural. This new definition requires that the circular radius function ${\cal R}(r)=\sqrt{C(r)}$ has a minimum at the throat\footnote{Wormholes have a non trivial topology, which constitutes a global property of spacetime. For static cylindrically symmetric geometries, the global properties are determined by the behavior of the circular radius function \cite{brle}.}, so this function should increase at both sides of it (the geodesics within a plane normal to the symmetry axis open up at the throat); thus we shall call this one the {\it radial} flare-out condition. This definition implies $C'(a)>0$, leaving free the sign of $(CD)'(a)$; this feature is of central relevance for the physical viability of the wormhole, because allows for the possibility of a {\it positive} energy density\footnote{In our paper \cite{ei1} the less restrictive flare-out condition was adopted, while in \cite{ei2} and \cite{mc} we worked with the other definition.}.
\section{Stability analysis}
In what follows, we assume that the equations of state for the dynamic case have the same form as in the static one, i.e. that they do not depend on the derivatives of $a(\tau)$, so $p_{\varphi}(\sigma)$ and $p_{z}(\sigma)$ are given by Eqs. (\ref{e15}) and (\ref{e16}). This assumption is justified by the fact that we are interested in small velocity perturbations starting from a static solution, so that the evolution of the shell matter can be considered as a succession of static states \cite{ei2,mc}. Then, replacing Eqs. (\ref{e8}) and (\ref{e9}) in Eq. (\ref{e15}) (or Eqs. (\ref{e8}) and (\ref{e10}) in Eq. (\ref{e16})), a simple second order differential equation for $a(\tau )$ is obtained:
\begin{equation}
2B(a)\ddot{a}+B'(a)\dot{a}^{2}=0.
\label{e17}
\end{equation}
It is easy to verify that
\begin{equation}
\dot{a}(\tau )=\dot{a}(\tau _{0})\sqrt{\frac{B(a(\tau _{0}))}{B(a(\tau))}},
\label{e18}
\end{equation}
satisfies Eq. (\ref{e17}), where $\tau _{0}$ is an arbitrary fixed time.
Then, Eq. (\ref{e18}) can be rewritten in the form
\begin{equation}
\sqrt{B(a)}da=\dot{a}(\tau _{0})\sqrt{B(a(\tau _{0}))}d\tau ,
\label{e19}
\end{equation}
which by integrating both sides gives
\begin{equation}
\int^{a(\tau )}_{a(\tau _{0})}\sqrt{B(a)}da=\dot{a}(\tau _{0})
\sqrt{B(a(\tau _{0}))}(\tau - \tau _{0}).
\label{e20}
\end{equation}
So we have that the time evolution of the radius of the throat $a(\tau )$ is formally obtained by calculating the integral and inverting Eq. (\ref{e20}). From Eq. (\ref{e18}) one concludes that the sign of the velocity of the shell after being perturbed is completely determined by the sign of the initial velocity. Depending on the fact that the metric function $B$ is an increasing or a decreasing function of the throat radius, for a positive initial velocity the absolute value of the velocity will respectively decrease or increase, while for a negative initial velocity the opposite result is obtained. No oscillatory behavior can take place: the shell can only undergo a monotonous evolution. Because we have started from the most general static cylindrically symmetric geometry, we then find that the conjecture introduced in Ref. \cite{ei2} is true: under the assumption that the symmetry is preserved and that the equations of state of matter on the shell corresponding to static configurations are kept valid after a perturbation, cylindrical thin-shell wormholes are unstable.
\section{Examples: Einstein--Maxwell spacetimes}
There exist three kinds of static cylindrically symmetric geometries associated to Maxwell electromagnetism coupled to Einstein gravity \cite{ks}: with the definition $G(r)=k_1r^m+k_2r^{-m}$, we have the metric associated to an axial current (angular magnetic field)
\begin{equation}
ds^2=r^{2m^2}G^2(r)\left(-dt^2+dr^2\right)+r^2G^2(r)d\varphi^2+G^{-2}(r)dz^2,
\end{equation}
the metric corresponding to an angular current (magnetic field in the direction of the symmetry axis)
\begin{equation}
ds^2=r^{2m^2}G^2(r)\left(-dt^2+dr^2\right)+G^{-2}(r)d\varphi^2+r^2G^2(r)dz^2,
\end{equation}
and the metric associated to an axial charge (radial electric field):
\begin{equation}
ds^2=-G^{-2}(r)dt^2+r^{2m^2}G^2(r)\left( dr^2+dz^2\right)+r^2G^2(r)d\varphi^2.\label{E}
\end{equation}
The constants $k_1$, $k_2$ and $m$ are real numbers that must fulfill $k_1k_2 > 0$ in the magnetic cases and $k_1k_2 < 0$ in the electric case, so the metric and the electromagnetic potential are real \cite{ks}. The coordinates adopted above have been suitably adimensionalized in order to avoid constants which do not play an important physical role.
\begin{figure}[t!]
\begin{center}
\vspace{0cm}
\includegraphics[width=15cm]{figure1.eps}
\vspace{0cm}
\end{center}
\caption{Wormholes of throat radius $a$ with a radial electric field: the gray zones correspond to the values of the parameters $m$, $k_1$ and $k_2$ for which the radial flare-out condition is satisfied.}
\label{fig1}
\end{figure}
\begin{figure}[t!]
\begin{center}
\vspace{0cm}
\includegraphics[width=15cm]{figure2.eps}
\vspace{0cm}
\end{center}
\caption{Wormholes of throat radius $a$ with a radial electric field: the gray zones correspond to the values of the parameters $m$, $k_1$ and $k_2$ for which the radial flare-out condition is satisfied and the energy density at the throat is positive.}
\label{fig2}
\end{figure}
In terms of the functions of the general static metric (\ref{e1}), in the first magnetic case we have $A(r)=B(r)=G^2(r)r^{2m^2}$, $C(r)=r^2G^2(r)$, $D(r)=G^{-2}(r)$, in the second magnetic case $A(r)=B(r)=G^2(r)r^{2m^2}$, $C(r)=G^{-2}(r)$, $D(r)=r^2G^2(r)$, and in the electric case $A(r)=G^{-2}(r)$, $B(r)=D(r)=r^{2m^2}G^2(r)$, $C(r)=r^2G^2(r)$. Thus in both magnetic cases we obtain the same behaviour for the product $CD$, whose radial derivative, after the mathematical construction of a wormhole of throat radius $a$, determines the sign of the surface energy density $\sigma$: $C(r)D(r)=r^2$, so that $\left( CD\right)'(a)=2a >0$ and $\sigma$ can only be negative. In the case with a radial electric field, instead, we have $C(r)D(r)=r^{2m^2+2}G^4(r)$, which not necessarily is an increasing function of $r$. Thus we can have $\left( CD \right)' (a)<0$ and, if this is compatible with $ C'(a)>0$, working with the radial flare-out condition a cylindrical thin-shell wormhole would be possible with $\sigma >0$. For the metric (\ref{E}) associated to the electric field we have
\begin{eqnarray}
C'(a)& = & 2aG(a)\left[ G(a)+aG'(a)\right] \nonumber\\
& = & 2a\left( k_1a^m+k_2a^{-m}\right) \left[ (1+m)k_1 a^m+(1-m)k_2a^{-m}\right],\label{geo}
\end{eqnarray}
\begin{eqnarray}
\left( CD\right) '(a)& = & 2a^{2m^2+1}G^3(a)\left[(m^2+1)G(a)+2aG'(a)\right]\nonumber\\
& = & 2a^{2m^2+1}\left( k_1a^m+k_2a^{-m}\right)^3\left[ (1+m)^2 k_1a^m+(1-m)^2k_2a^{-m}\right].\label{ener}
\end{eqnarray}
If values of the parameters $k_1$, $k_2$ and $m$ exist such that the expression (\ref{geo}) is positive definite and the expression (\ref{ener}) is negative, then there exist cylindrical thin-shell wormholes of throat radius $a$ which are supported by matter of positive energy density. Let us see for which values of the parameters we obtain $\sigma > 0$. Eq. (\ref{ener}) can be written in the form
\begin{equation}
\left( CD\right) '(a) = 2a^{2m^2+1}G^2(a)\left\{2G(a)\left[ G(a)+ aG'(a)\right]+(m^2-1)G^2(a)\right\}.
\end{equation}
The first term within the curly brackets is always positive if $C'(a)>0$; then a necessary (but not sufficient) condition for $\left( CD\right) '(a)<0$ is that $|m|<1$. The values of the parameters $m$, $k_1$ and $k_2$ for which the radial flare-out condition is satisfied are plotted in Fig. \ref{fig1}, and the values for which the energy density is positive are shown in Fig. \ref{fig2}, in both cases for different values of the wormhole throat radius $a$.
Now let us analyze the energy conditions: non exotic matter satisfies the weak energy condition (WEC) that states $\rho \geq 0$, $\rho+P_\mu\geq 0$ ($\mu=r,\varphi,z$), or at least the null energy condition (NEC) $\rho+P_\mu\geq 0$. In our thin-shell construction the energy conditions are satisfied outside the shell. On the throat, instead of the energy density $\rho$ we have the surface energy density $\sigma$, and instead of the pressures $P_\mu$ we have the surface pressures $p_{\varphi}$, $p_z$ and $p_r$. It is straightforward to see that $\sigma + p_z=0$, and that
\begin{equation}
\sigma+p_{\varphi} = \frac{m^2-1}{4\pi a^{m^2+1}|k_1a^m+k_2a^{-m}|}.\label{spf}\end{equation}
So we find that $\sigma + p_{\varphi}\geq 0$ requires $|m|\geq 1$, while $\sigma >0$ requires, as noted above, that $|m|<1$. On the other hand, $\sigma+p_r=\sigma$ because the radial pressure is non singular on the shell. Then it is not possible to satisfy the energy conditions, though the energy density can be positive. If the parameters are such that $\sigma >0$, then an observer at rest at the throat, or an observer moving radially from one side of the throat to the other side would find only positive energy matter.\footnote{But an observer in a sufficiently fast circular orbit around the throat would find matter of negative energy, because of the violation of the condition $\sigma + p_{\varphi}\geq 0$ which results from $|m|<1$.}
\section{Summary}\label{discu}
We have constructed cylindrical thin shell wormholes of a general class, symmetric with respect to the throat. Under the assumption that the equations of state on the shell which defines the throat have the same form as in the static case when it is perturbed preserving the symmetry, we have shown that these wormholes are always mechanically unstable. We have therefore demonstrated our conjecture presented in \cite{ei2}: this is a general property and does not depend on the particular form of the metric adopted for the construction. Two possible definitions of the flare-out condition were considered. The areal flare-out condition is more restrictive because it always leads to a negative energy density on the throat. Instead, in this work we have shown that the radial one is less restrictive, allowing for positive energy density, depending on the particular form of the original metric from which the construction was performed. Examples of wormholes with electromagnetic fields were presented. In the two cases with a magnetic field, the energy density is always negative for both definitions of the flare-out condition. For a radial electric field, with the adoption of the radial flare-out condition we have found values of the parameters such that the energy density at the throat is positive. In the case of the radial flare-out condition, the existence of thin-shell wormholes satisfying the energy conditions, in principle not discarded by our general analysis, is still an open question.
\section*{Acknowledgments}
This work has been supported by Universidad de Buenos Aires and CONICET.
|
2,877,628,089,021 | arxiv | \section{Introduction}
Given a universe of elements, such as the vertices or edges of a graph, and a property on them, such as being a clique or a tree, a listing problem asks to return all subsets of the universe which satisfy the given property.
Listing structures, within graphs or other types of data, is a basic problem in computer science, and it is at the core of data analysis. While many problems can be solved by optimization approaches for the best solution, e.g., by finding the shortest path, or the largest clique, others require finding several solutions to the input problem: in community detection, for example, finding just one ``best'' community only gives us local information regarding some part of the data, so we may want to find several communities to make sense of the input.
Furthermore, many real-world scenarios may not have a clear objective function for the best solution: We may define an algorithm to optimize some desired property, but the optimal solution found may be lacking further properties that emerge during listing or simply not be practical.
We may want instead to quickly list several solutions, suitable according to some metrics, then analyze them a posteriori to find the desired one.
In these scenarios, listing only the solutions that are \textit{maximal} under inclusion is a common-sense requirement whenever it can be applied,\footnote{In other problems, we may want \textit{minimal} solutions instead, although this is usually an equivalent concept, as it corresponds to the complement of a solution being maximal.} as maximal solutions subsume the information contained in all others, and can be exponentially fewer: For example, a graph may have up to $2^n$ cliques, but only $3^{n/3}$ maximal ones~\cite{moon1965cliques}. For brevity, we call \textit{maximal listing problem} a listing problem where only the inclusion-maximal solutions should be output.
From a theoretical point of view, listing provides many challenging problems, especially when maximality is required. When dealing with listing algorithms, we are often interested in their complexity with respect to both $n$, the input size, and $\nsol$, the size of the output. Algorithms whose complexity can be bounded by a polynomial of these two factors are called \textit{output polynomial} or \textit{polynomial total time}~\cite{JOHNSON1988119}.
Interestingly, the hardness of listing problems does not seem to be correlated with that of optimization: there are several \textsc{np-h}ard\xspace maximum optimization problems whose corresponding maximal listing problem admits an output-polynomial solution (see, e.g.,~\cite{tsukiyama1977new,avis1996reverse}); on the other hand, there are problems for which one maximal (or maximum) solution can be identified in polynomial time, but an output-polynomial algorithm for listing maximal solutions would imply \textsc{p=np}~\cite{lawler1980generating}.
A long-standing question in the area is to find a characterization of which listing problems allow for output-polynomial solutions and which do not.
Furthermore, within output-polynomial algorithms stricter complexity classes exist, such as \textit{incremental polynomial time}, where the time to output the $i$-th solution is polynomial in $n$ and $i$, and \textit{polynomial delay}, where the time elapsed for outputting the next solution is upper bounded by a polynomial in $n$.
The latter class is of particular interest in practical scenarios, as it guarantees that solutions are output at a regular pace.
In this paper we add a few points to the latter class, by showing that there exist polynomial delay algorithms for some subgraph listing problems. More formally, we prove Theorem~\ref{thm:main}.
\begin{theorem}\label{thm:main}
The following problems allow polynomial delay listing algorithms by proximity search:
\begin{center}
\begin{small}
\begin{tabular}{cc}
\textsc{problem} & \textsc{delay}\\
maximal induced bipartite sg. & $O(n( m + n\iack(n)))$ \\
maximal connected induced bipartite sg. & $O(mn)$\\
maximal bipartite edge-induced sg. & $O(m^3)$ \smallskip\\
maximal induced $k$-degenerate sg. & $O(mn^{k+2})$\\
maximal edge-induced $k$-degenerate sg. & $O(m^3n^{k-1})$ \smallskip\\
maximal induced chordal sg. & $O(m^2n)$ \\
maximal connected induced chordal sg. & $O(m^2n)$ \\
maximal edge-induced chordal sg. & $O(m^4n)$ \smallskip\\
maximal induced proper interval sg. & $O(m^2n^3)$ \\
maximal connected induced proper interval sg. & $O(mn^3)$\smallskip\\
maximal connected obstacle-free convex hulls & $O(n^4)$ \smallskip\\
maximal induced trees & $O(m^2)$ \smallskip\\
maximal connected induced directed acyclic sg. & $O(mn^2)$ \\
maximal connected edge-induced directed acyclic sg. & $O(m^3)$\\
\end{tabular}
\noindent
Where ``sg.'' stands for subgraphs, $n$ and $m$ are the number of vertices and edges, $\iack(\cdot)$ is the functional inverse of the Ackermann function~\cite{Tarjan75UF}.
All the algorithms use $O(\nsol n)$ space, where $\nsol$ is the number of solutions.\\
\end{small}
\end{center}
\end{theorem}
To the best of our knowledge, no output-polynomial result was previously known for these problems. For completeness, we consider both \textit{induced} subgraphs (i.e., sets of vertices) and \textit{edge-induced} subgraphs (i.e., sets of edges), as well as the \textit{connected} case where solutions are required to be connected, as the structure of such variants can differ significantly.
Furthermore, we abstract a general technique that can be used to obtain similar results on other problems.
We do so by defining a graph whose vertices are the maximal solutions to the listing problem, and with directed edges between pairs of solutions, which we call \textit{solution graph}. The listing problem is solved by traversing the solution graph, and proving that all solutions are found this way.
The concept of solution graph is common to existing approaches, and general techniques already exist for building them, e.g.,~\cite{Cohen20081147}. However, the solution graph built with known approaches such as~\cite{Cohen20081147} may have too many edges, resulting in a traversal with exponential delay.
The key concept given in this paper is a technique to build a solution graph with fewer edges, while proving that all solutions are still found by its traversal.
An interesting property of this approach is that the resulting algorithms are remarkably simple to implement, while the complexity lies in proving their correctness.
We call this technique \textit{proximity search} since at its core lies a problem-specific notion of proximity. This notion acts as a sort of compass on the solution graph built by our algorithm, as given any two solutions $S$ and $S^*$, we will show that we always traverse an edge from $S$ to another solution $S'$ that has higher proximity to $S^*$; as $S^*$ has the highest proximity to itself, this implies that a traversal of the solution graph from any solution finds all others.
While others, such as~\cite{Cohen20081147,schwikowski2002enumerating}, already used the principle of reachability in the solution graph, we aim to define a looser set of necessary condition in order to guarantee this reachability, allowing more freedom in the design of algorithms, while at the same time formalize a technique called \textit{canonical reconstruction} that is effective in decomposing the structure of several problems to fit these rules.
The combination of these two parts creates algorithms that overcome the exponential burden imposed by the so-called \textit{input-restricted problem}, a reduced instance of the original problem that dominates the cost per solution of such approaches whose cost may be inherently exponential.
While the space required for a traversal of the graph is inherently proportional to the number of solutions, i.e., can be exponential in $n$, some output-polynomial techniques such as reverse search are able to work in polynomial space by inducing a tree-like structure on the solution graph, provided that the problem at hand is hereditary (i.e.~its property holds for the induced subgraphs) and the input-restricted problem is solvable efficiently.
By adding suitable constraints to the problems considered, we show a technique that combines proximity search with a recent generalization of reverse search to non-hereditary problems~\cite{conte2019framework}, obtaining algorithms with both polynomial-delay and polynomial space for some instances of proximity search. In particular, we prove that:
\begin{theorem}\label{thm:pspace}
The following problems allow polynomial delay listing and polynomial space algorithms by proximity search, with the following bounds:
\smallskip
\begin{small}
\begin{center}
\begin{tabular}{ccc}
\textsc{problem} & \textsc{delay} & \textsc{space} \\
maximal induced bipartite sg. & $O(n^2( m + n\iack(n)))$ & $O(m)$ \\
maximal connected induced bipartite sg. & $O(mn^2)$ & $O(m)$ \smallskip\\
maximal obstacle-free convex hulls & $O(n^4)$ & $O(n)$ \smallskip\\
maximal induced trees & $O(m^2n^2)$ & $O(m)$\\
maximal induced forests & $O(m^2n^2)$ & $O(m)$ \smallskip\\
\end{tabular}
\end{center}
\noindent
Where notation is as in Theorem~\ref{thm:main}.
\end{small}
\end{theorem}
\subsection{Related Work}
The listing problems considered in this paper model solutions as sets of elements (e.g., sets of vertices or edges of a graph), and consist in listing sets of elements with some required property, e.g., inducing a bipartite subgraph, or a tree.
We observe that the output is a family of sets, we can associate properties with the corresponding set systems: for example, a property is hereditary when each subset of a solution is a solution, which corresponds to the well-known independence systems~\cite{lawler1980generating}.
In this context, a simple yet powerful technique is recursively partitioning the search space into all solutions containing a certain element, and all that do not.
This technique, usually called binary partition or simply backtracking, proves efficient when listing all solutions ~\cite{Ruskey03combinatorialgeneration}, and can be used to design algorithms that are fast in practice,\footnote{E.g., implementations of the Bron-Kerbosh~\cite{tomita2006worst} algorithm tend to be faster than those of output-polynomial algorithms~\cite{DBLP:conf/icalp/ConteGMV16} for listing maximal cliques.} or that can bound the number of solutions in the worst-case~\cite{Fomin:2008:CBV:1435375.1435384}.
On the other hand, this strategy rarely gives output-polynomial algorithms when dealing with maximal solutions, as we may spend time exploring a solution subspace that contains many solutions but no maximal one.
To obtain output-polynomial algorithms for maximal solutions, many algorithms rely on the following idea: given a maximal solution $S$, and some element $x\not\in S$, the hardness of listing solutions \textit{maximal within} $S\cup \{x\}$ is linked to the hardness of listing them in a general instance. One of the earliest mentions of the idea can be found in the seminal paper by Lawler et al.~\cite{lawler1980generating}, that generalizes ideas from Paull et al.~\cite{paull1959minimizing} and Tsukiyama et al.~\cite{tsukiyama1977new}, and has been formally defined as \textit{input-restricted problem} by Cohen et al.~\cite{Cohen20081147}.
The intuition is that the solutions obtained this way, using a maximal solution $S$ and an element not in $S$, can be used to generate new maximal solutions of the original problem. We can thus traverse an implicit directed graph, which we will call \textit{solution graph}, where the vertices are the maximal solutions and the out-neighbors are obtained by means of the input-restricted problem.
In particular,~\cite{lawler1980generating} showed how solving this problem could yield an output-polynomial and polynomial space listing algorithm for properties corresponding to \textit{independence systems}, assuming the input-restricted problem has a bounded number of solutions. ~\cite{Cohen20081147} showed that the strategy could be extended to the more challenging \textit{connected-hereditary} graph properties (i.e., where \textit{connected} subsets of solutions are solutions) using exponential space, and recently,~\cite{conte2019framework} showed that the same result can be obtained in polynomial space for \textit{commutable} set systems (which include connected-hereditary properties).
A clear limitation of this approach is that, in order to obtain polynomial-delay algorithms, the input-restricted problem needs to be solved in polynomial time. This is possible for some problems (e.g., cliques and independent sets), but impossible for others, simply because their input-restricted problems may have exponentially many solutions. Figure~\ref{fig:bip:restr} shows an example for maximal bipartite subgraphs.
\begin{figure*}
\centering
\includegraphics[width=.9\textwidth]{img/bad-restr.pdf}
\caption{Instances of input-restricted problem for maximal bipartite subgraphs.
%
On the left: the black dots define a maximal bipartite induced subgraph; adding the vertex $v$ creates a graph with exponentially many maximal induced bipartite subgraphs, as we can obtain one by removing a vertex from each connected pair in the bottom in any combination.
%
On the right: the black edges define a maximal bipartite subgraph, and the addition of edge $e$ creates a graph with exponentially many maximal edge-induced bipartite subgraphs: every vertex on the bottom is incident to two edges; removing exactly one for each vertex yields a maximal edge-induced bipartite subgraph.}
\label{fig:bip:restr}
\end{figure*}
The literature contains many more results concerning the enumeration of maximal/minimal solutions, e.g.,~\cite{avis1996reverse,koch1996algorithm,schwikowski2002enumerating,Golovach2018,GELY20091447,CARMELI2020}, and in particular regarding challenging problems such as the well-known minimal hypergraph transversals/dominating sets problem~\cite{kante2014enumeration,golovach2015incremental,elbassioni2009output}.
However, to the best of our knowledge, the only two effective general techniques for listing maximal solutions in an output-sensitive fashion are the extension problem (binary partition, flashlight search), and the input-restricted problem: proximity search can be a valuable tool when the previous two fail.
We motivate this by showing the first polynomial delay algorithms for several maximal listing problems whose associated input-restricted problem is not solvable in polynomial time.
As mentioned above, a preliminary version of this paper containing some of the exponential-space algorithms has appeared in~\cite{Conte2019proximity}. Since its publication, some preprints~\cite{brosse2020efficient,kurita2020efficient,Cao:arXiv:2020} have appeared that apply the technique to obtain new output-polynomial algorithms.
In particular,~\cite{Cao:arXiv:2020} solves the enumeration of Maximal Induced Interval Subgraphs by proposing some variations to proximity search~\cite{Conte2019proximity}.
\subsection{Overview}
The main contribution of the paper is presenting \textit{proximity search}, a general technique that can be used to solve several enumeration problems in polynomial delay, and \textit{canonical reconstruction}, a way to design a proximity search algorithm by exploiting orderings of solutions of the problem at hand.
exponentially many solutions.
By using this technique we show polynomial delay algorithms for several maximal listing problems such as maximal bipartite subgraphs and the others mentioned in Theorem~\ref{thm:main}. Other than providing efficient algorithms, we remark that the technique may help gain further insight on which classes of problems allow output-polynomial listing algorithms and which do not.
The paper is organized as follows:
First, we introduce some basic concepts and notation in Section~\ref{sec:prelim}.
We then explain the proximity search technique, and formally define a class of problems, called \textit{proximity searchable}, which allow for a polynomial delay algorithm by its application.
Generality comes sometimes at the expense of efficiency but allows for a more intuitive understanding of the concepts at hand. For this reason, we divide the explanation in two parts: the first one, in Section~\ref{sec:outline}, formalizes the constraints required for a proximity search algorithm. The second, Section~\ref{sec:reconstruction}, introduces a technique which we call \textit{canonical reconstruction} for implementing proximity search. While canonical reconstruction is not the only way to obtain a proximity search algorithm, we observed that is often a powerful and elegant way to model the problem at hand.
Following, Sections~\ref{sec:bip}-\ref{sec:list:last}, shows how to prove that the problems in Theorem~\ref{thm:main} are proximity searchable and thus allow polynomial-delay algorithms.
As a drawback of the above algorithms is an exponential space requirement, we then propose a technique to address this issue, when suitable conditions are met: define a parent-child relation between solutions, in the style of reverse-search, as detailed in Section~\ref{sec:pspace-expl}, and give the algorithms in Section~\ref{sec:pspace-algs}. The resulting bounds are shown in Theorem~\ref{thm:pspace}. While this technique does not apply to all problems in Theorem~\ref{thm:main}, when it does it allows us to obtain polynomial-delay and polynomial-space algorithms for several problems whose input-restricted problem cannot be solved in polynomial time, including non-hereditary ones.
\section{Preliminaries}\label{sec:prelim}
Most of the enumeration problems addressed in this paper consider a simple undirected graph $G$, whose vertex set is denoted as $V(G)$ and edge set as $E(G)$, or simply $G=(V,E)$ when it is clear from the context. The neighborhood of a vertex $v$ is denoted as $N(v)$. For brevity, we refer to $|V(G)|$ as the number $n$ of vertices, to $|E(G)|$ as the number $m$ of edges, and to the maximum degree of a vertex in $G$ as $\Delta = \max_{v \in V} |N(v)|$. Furthermore, we assume the vertices to be labeled arbitrarily in increasing order $v_1, \ldots, v_n$, and say that $v_i$ is
smaller than $v_j$ if $i<j$.
We say that a neighbor of $v_i$ is a \textit{forward} neighbor if it comes later than $v_i$ in the order, and a \textit{backward} neighbor otherwise.
For a set of vertices $A\subseteq V(G)$, $E[A]$ denotes the edges of $G$ whose endpoints are both in $A$, and $G[A]$ the graph $(A, E[A])$, i.e., the subgraph induced in $G$ by $A$. Similarly, for a set $B$ of edges, $V[B]$ denotes the vertices incident to an edge in $B$ and $G[B] = (V[B], B)$. As common in the literature, we call \textit{induced subgraphs} those of the former kind, defined by a set of vertices, and \textit{edge-induced subgraphs} (or simply subgraphs) those of the latter, defined by a set of edges.
When dealing with subgraphs defined by a set of vertices (resp. edges) $A$, we will sometimes use $A$ to refer to both the vertex set (resp. edge set) and the subgraph $G[A]$ it induces, when this causes no ambiguity. We will also use $\texttt{cc}\xspace_v(A)$ to refer to the \textit{connected component} of $G[A]$ which includes the vertex $v$. For further notation, we refer to the standard terminology in~\cite{Diestel2005}.
For a set of vertices $A\subseteq V(G)$ which corresponds to a solution of the problem at hand, we say that $A$ is \emph{maximal} if there is no $A' \subseteq V(G)$ such that $A' \supset A$ and $A'$ is also a solution.
While not strictly necessary for the proximity search technique, in the following we will often rely on a simple ``maximalization'' function, named $\textsc{complete}\xspace(A)$: this function takes a (not necessarily maximal) solution $A$ and ``completes'' it, returning some maximal solution $A'\supseteq A$. We will refer to the computational cost of this function as $\mathcal{C}_t$.
Note that it is always possible to devise a polynomial-time computable $\textsc{complete}\xspace(\cdot)$ function for hereditary and connected-hereditary properties where solutions can be recognized in polynomial time, by simply trying to add vertices until no longer possible~\cite{Cohen20081147}.
For simplicity, we disregard the presence of isolated vertices in the complexity analysis of the algorithms provided: these are trivially handled for the problems considered in this paper (either they can all be added ``in bulk'' to every solution, as for bipartite subgraphs, or each constitutes a maximal solution by itself, as for connected bipartite subgraphs), can be removed with an $O(n)$ time preprocessing; this means we are able to perform operations like a visit of the graph in $O(m)$ time rather than $O(m+n)$ time.
\section{Proximity search outline}\label{sec:outline}
Proximity search is based on traversing an implicit solution graph, where the vertices are all the solutions to be listed and each directed arc goes from a solution to another using a neighboring function. Several solution graphs are possible, depending on how the neighboring function is defined. Apart from the fact that the resulting solution graphs are not necessarily strongly connected and some care should be taken to list all the solutions, the main hurdle is that the degree of the solution graphs can be exponential (as the number of solutions can be exponentially large in the input size), thus preventing to achieve polynomial delay when running a simple traversal. Proximity search circumvents these issues by designing a suitable neighboring function, denoted $\textsc{neighbors}\xspace(\cdot)$, that guarantees that the resulting solution graph it implicitly defines is strongly connected and of polynomial degree. Both these properties cannot be guaranteed
with the current state of the art for a number of problems discussed later.
We devote this section to formalize the general structure of proximity search, and the class of problems to which the technique can be applied. Also, we introduce the notion of proximity, symbolized by $\ensuremath{\mathrel{\stackrel{\makebox[0pt]{\ensuremath{_\rightsquigarrow}}}{\cap}}}\xspace$, to act as a sort of oracle for navigating the solution graph.
For reference in what we discuss next, we give the pseudo-code of the generic traversal of a solution graph based on the $\textsc{neighbors}\xspace(\cdot)$ function, as shown in Algorithm~\ref{alg:general}. As noted earlier, the algorithms obtained by specializing this generic traversal are remarkably simple: In a depth-first search traversal where the set \ensuremath{\mathcal{S}}\xspace keeps track of just the last visited solution, we only need to implement the $\textsc{neighbors}\xspace(\cdot)$ function.
On the other hand, the complexity is mostly hidden behind proving their completeness: Notably, the very notion of proximity $\ensuremath{\mathrel{\stackrel{\makebox[0pt]{\ensuremath{_\rightsquigarrow}}}{\cap}}}\xspace$ is only used in the proofs, and never actually appears in Algorithm~\ref{alg:general}.
\begin{algorithm2e}[ht]
\DontPrintSemicolon
\SetKwInOut{Input}{input}
\SetKwInOut{Output}{output}
\SetKwInOut{Global}{global}
\Input{Graph $G = (V,E)$ and listing problem $\mathcal{P}$}
\Output{All (maximal) solutions of $\mathcal{P}$ in $G$}
\Global{Set $\ensuremath{\mathcal{S}}\xspace$ of solutions found, initially empty}
\BlankLine
$S \gets$ an arbitrary solution of $\mathcal{P}$ \label{ln:g:first}\tcc{ (e.g. $\textsc{complete}\xspace(\emptyset)$)}
Call $\textsc{spawn}\xspace(S)$\;
\BlankLine
\SetKwProg{myproc}{Function}{}{}
\myproc{$\textsc{spawn}\xspace(S)$}{
Add $S$ to $\ensuremath{\mathcal{S}}\xspace$\label{ln:g:add}\;
\tcc{\small Output $S$ if recursion depth is even}
\ForEach{$S'\in \textsc{neighbors}\xspace(S)$\label{ln:g:neighs}}{
\lIf{$S' \not \in\ensuremath{\mathcal{S}}\xspace$}{$\textsc{spawn}\xspace(S')$\label{ln:g:rec}}
}
\tcc{\small Output $S$ if recursion depth is odd}
}
\caption{Traversal of the solution graph by proximity search.}\label{alg:general}
\end{algorithm2e}
In order to start the algorithm, we need one arbitrary maximal solution $S$. We remark that identifying one maximal (not maximum) solution is typically trivial, and can be achieved for example
by running $\textsc{complete}\xspace(\emptyset)$
when the $\textsc{complete}\xspace(\cdot)$ function is computable in polynomial time.
We formally define the class of problems which allow for a polynomial delay algorithm using this structure as \textit{proximity searchable}.
\begin{definition}[Proximity searchable]\label{def:searchable}
Let $\mathcal{P}$ be a listing problem over a universe $\mathcal{U}$ with set of solutions $\mathcal{S}\subseteq 2^\mathcal{U}$, where each solution is a subset of the universe. $\mathcal{P}$ is \textit{proximity searchable} if there exists a proximity function $\ensuremath{\mathrel{\stackrel{\makebox[0pt]{\ensuremath{_\rightsquigarrow}}}{\cap}}}\xspace : \mathcal{S}\times \mathcal{S}\rightarrow 2^\mathcal{U}$ and a neighboring function $\textsc{neighbors}\xspace(\cdot) : \mathcal{S} \rightarrow 2^{\mathcal{S}}$,
such that the following holds:
\begin{enumerate}
\item \label{item:prox1} One solution of $\mathcal{P}$ can be identified in time polynomial in $|\mathcal{U}|$.
\item \label{item:prox2} $\textsc{neighbors}\xspace(\cdot)$ is computable in time polynomial in $|\mathcal{U}|$.
\item \label{item:prox3} Given any two distinct solutions $S,S^*\in \mathcal{S}$, there exists $S' \in \textsc{neighbors}\xspace(S)$ such that $|S'\ensuremath{\mathrel{\stackrel{\makebox[0pt]{\ensuremath{_\rightsquigarrow}}}{\cap}}}\xspace S^*| > |S\ensuremath{\mathrel{\stackrel{\makebox[0pt]{\ensuremath{_\rightsquigarrow}}}{\cap}}}\xspace S^*|$.
\end{enumerate}
The above conditions imply the following one, which is reported for the sake of clarity.
\begin{enumerate}
\setcounter{enumi}{3}
\item \label{item:prox4} For any fixed $S^*$, $|S\ensuremath{\mathrel{\stackrel{\makebox[0pt]{\ensuremath{_\rightsquigarrow}}}{\cap}}}\xspace S^*|$ is maximized for (and only for) $S=S^*$.
\end{enumerate}
\end{definition}
If a problem is proximity searchable, then it is straightforward to see that we obtain a polynomial delay algorithm for it by using the corresponding $\textsc{neighbors}\xspace(\cdot)$ function in Algorithm~\ref{alg:general}. Let us formally prove it.
\begin{theorem}
\label{thm:strong}
All proximity searchable listing problems have a polynomial delay listing algorithm.
\end{theorem}
\begin{proof}
We first show that if a $\textsc{neighbors}\xspace(\cdot)$ function satisfies Definition~\ref{def:searchable}, the implicit solution graph it induces is strongly connected. Given any two distinct solutions $S, S^* \in \mathcal{S}$, we know by Definition~\ref{def:searchable}.(\ref{item:prox3}) that there exists $S'\in\textsc{neighbors}\xspace(S)$ such that $|S'\ensuremath{\mathrel{\stackrel{\makebox[0pt]{\ensuremath{_\rightsquigarrow}}}{\cap}}}\xspace S^*| > |S\ensuremath{\mathrel{\stackrel{\makebox[0pt]{\ensuremath{_\rightsquigarrow}}}{\cap}}}\xspace S^*|$. By induction on $S$ and $S'$, it follows that we will eventually reach a solution $S$ that globally maximizes $|\cdot \ensuremath{\mathrel{\stackrel{\makebox[0pt]{\ensuremath{_\rightsquigarrow}}}{\cap}}}\xspace S^*|$, which by Definition~\ref{def:searchable}.(\ref{item:prox4}) is precisely $S^*$.
Based on the above properties, we next show that Algorithm~\ref{alg:general} outputs (all and only) the solutions of any proximity searchable problem with no duplication.
Firstly, Algorithm~\ref{alg:general} returns only maximal solutions, as it only outputs the initial maximal solution found on line~\ref{ln:g:first}, found polynomial time by Definition~\ref{def:searchable}.(\ref{item:prox1}), and the output of calls to $\textsc{neighbors}\xspace(S)$ which contain maximal solutions.
We say that a solution is \textit{visited} when $\textsc{spawn}\xspace(S)$ is called. In Algorithm~\ref{alg:general} all solutions added to $\ensuremath{\mathcal{S}}\xspace$ are visited at most once, thanks to the membership test in the set $\ensuremath{\mathcal{S}}\xspace$; this guarantees that the same solution is never output twice. As the graph defined by $\textsc{neighbors}\xspace(S)$ is strongly connected, the traversal done by Algorithm~\ref{alg:general} starting from the solution found on line~\ref{ln:g:first} must find all solutions.
To complete the proof, we show that Algorithm~\ref{alg:general} runs in polynomial delay.
Firstly, $\textsc{neighbors}\xspace(S)$ requires polynomial time by Definition~\ref{def:searchable}.(\ref{item:prox2}) and thus can only return a polynomial number of solutions; this means the out-degree of every node in the implicit solution graph is polynomial and we can iterate over it in polynomial time.
As a new recursive call is performed only when a new solution is found, the amortized cost per solution is bound by the cost of a recursive call, i.e., the cost of lines~\ref{ln:g:add}--\ref{ln:g:rec}. As the cost of $\textsc{neighbors}\xspace(\cdot)$ is polynomial, and $\ensuremath{\mathcal{S}}\xspace$ can be easily maintained in polynomial time (in Appendix~\ref{sec:sol} we show this latter cost to be negligible for all algorithms presented here), it follows that the amortized cost per solution is polynomial. In order to get polynomial delay, we can employ the \textit{alternative output}~\cite{Uno2003} method, that can be applied to any recursive algorithm that outputs a solution in each recursive call: by performing output in pre-order when the recursion depth is even, and post-order when it is odd, the delay will be bounded by that of a constant number of recursive calls, i.e., polynomial.
\end{proof}
The following observations are in order:
\begin{itemize}
\item $\ensuremath{\mathrel{\stackrel{\makebox[0pt]{\ensuremath{_\rightsquigarrow}}}{\cap}}}\xspace$ is not actually used in Algorithm~\ref{alg:general}, and does \textit{not} need to be computed.
\item Proximity search can be applied to all listing domains where solutions are modeled by set systems, not just graphs.
\item Proximity search is mainly intended for maximal listing problems, however, it is not strictly limited to it.
\item Maximal listing problems in which the input-restricted problem is computable in polynomial time (as well as the $\textsc{complete}\xspace(\cdot)$ function) are proximity searchable.\footnote{In essence, we obtain as a special case the same solution graph as known algorithms based on the input-restricted problem~\cite{tsukiyama1977new,lawler1980generating,Cohen20081147}: For a solution $S$ we compute $\textsc{neighbors}\xspace(S)$ by sequentially taking all elements $v\in \mathcal{U}\setminus S$, solving the input-restricted problem for $S \cup \{v\}$, and applying $\textsc{complete}\xspace(\cdot)$ on the results.}
\item The polynomiality constraint on $\textsc{neighbors}\xspace(\cdot)$ can be relaxed: it can be trivially seen how computing $\textsc{neighbors}\xspace(\cdot)$ in Incremental Polynomial Time (resp. Polynomial Total Time) yields and Incremental Polynomial Time (resp. Polynomial Total Time) algorithm.
\item The cost per solution and delay of the algorithm is the complexity of the $\textsc{neighbors}\xspace(\cdot)$ function (we show in Appendix~\ref{sec:sol} how maintaining $\ensuremath{\mathcal{S}}\xspace$ is negligible).
\end{itemize}
In the rest of the paper, we show how to suitably model several problems to obtain new polynomial-delay algorithms for problems that, to the best of our knowledge, could not be previously solved in polynomial delay.
We show these algorithms by providing suitable $\ensuremath{\mathrel{\stackrel{\makebox[0pt]{\ensuremath{_\rightsquigarrow}}}{\cap}}}\xspace$ and $\textsc{neighbors}\xspace(\cdot)$ functions, proving that they satisfy Definition~\ref{def:searchable}, which automatically give us a polynomial delay listing algorithm by Algorithm~\ref{alg:general}.
We will use a common notation: $S$ is an arbitrary solution, and $S^*$ the ``target'' solution.
\begin{figure*}[t]
\centering
\includegraphics[width=\textwidth]{img/ex-bip.pdf}
\put (-505,145){\small(a)}
\put (-320,145){\small(b)}
\put (-140,145){\small(c)}
\put (-505,60){\small(d)}
\put (-320,60){\small(e)}
\put (-140,60){\small(f)}
%
%
\caption{\textbf{a}: a graph. \textbf{b},\textbf{c}: two maximal connected induced bipartite subgraphs of (a). \textbf{d},\textbf{e}: two maximal induced bipartite subgraphs of (a). \textbf{f}: a maximal edge-induced bipartite subgraph of (a).}
\label{fig:ex:bip}
\end{figure*}
\section{Proximity search by canonical reconstruction}\label{sec:reconstruction}
We make concrete use of the abstract notion of proximity search and introduce a technique, which we call \textit{canonical reconstruction}.
While it is kept separate from the previous section for cohesiveness, we find this
technique to often be the right way to look at maximal subgraph listing problems.
Since we deal with graphs, the universe $\mathcal{U}$ is the vertex set, unless explicitly specified.
To accompany the explanation, we detail its implementation in the case of Maximal Connected Induced Bipartite Subgraphs in Section~\ref{sec:bip}.
The technique is based on the definitions of \textit{canonical order} and \textit{canonical extender} for solutions, which depend entirely on the problem at hand, and it is intuitively a way to harness its structure.
\begin{example*}
For a Maximal Connected Induced Bipartite Subgraph $S$, we will use as canonical order a BFS-order of $G[S]$ starting from its vertex of smallest id, where ties are broken by vertex id:
For the subgraph in Figure~\ref{fig:ex:bip} (b) this order is\linebreak $2,3,5,8,11,7,10$, and for the one in (c) it is $2,3,8,12,11$.
\end{example*}
\paragraph{Canonical order and proximity}
Simply assume that each solution $S$ is given an ordering $s_1,\ldots, s_{|S|}$ of its elements which will satisfy some problem-specific conditions. We require that any \textit{prefix} $s_1,\ldots, s_i$ of this order corresponds to a (non-maximal) solution $\{ s_1,\ldots, s_i\}$. In the rest of the paper, we will refer as prefix of the order to both the sequence $s_1,\ldots, s_i$ and the corresponding set of elements $\{ s_1,\ldots, s_i\}$.
Note that the ordering is \emph{not} required to be efficiently computable, as the proximity search algorithm never actually computes it: it is only used in the correctness proof of the neighboring function. Moreover, the ordering is adaptive to each solution, so the same elements can be ranked differently in distinct solutions.
Given the order, we define the proximity function $\ensuremath{\mathrel{\stackrel{\makebox[0pt]{\ensuremath{_\rightsquigarrow}}}{\cap}}}\xspace$ as follows.
\begin{definition}[proximity]\label{def:proximity}
Given two solutions $S$ and $S^*$, let $ s^*_1,\ldots s^*_{|S^*|}$ be the canonical order of $S^*$: the proximity $S \ensuremath{\mathrel{\stackrel{\makebox[0pt]{\ensuremath{_\rightsquigarrow}}}{\cap}}}\xspace S^*$ between $S$ and $S^*$ is the longest \emph{prefix} $ s^*_1,\ldots,s^*_i$ of the canonical order of $S^*$ whose elements are all contained in $S$.
\end{definition}
It should also be noted that the operation is \textit{not} symmetric, i.e., we may have $S \ensuremath{\mathrel{\stackrel{\makebox[0pt]{\ensuremath{_\rightsquigarrow}}}{\cap}}}\xspace S^* \ne S^* \ensuremath{\mathrel{\stackrel{\makebox[0pt]{\ensuremath{_\rightsquigarrow}}}{\cap}}}\xspace S$.
\begin{example*}
Let $S$ be the subgraph shown in Figure~\ref{fig:ex:bip} (b) and $S^*$ the one shown in (c). Considering the canonical orders mentioned above, we can see that $S \ensuremath{\mathrel{\stackrel{\makebox[0pt]{\ensuremath{_\rightsquigarrow}}}{\cap}}}\xspace S^* = \{2,3,8\}$, while $S^* \ensuremath{\mathrel{\stackrel{\makebox[0pt]{\ensuremath{_\rightsquigarrow}}}{\cap}}}\xspace S = \{2,3\}$.
\end{example*}
\paragraph{Canonical extender}
The goal of a proximity search algorithm is to exploit Definition~\ref{def:searchable}.(\ref{item:prox3}): given $S$, for any $S^*$, find some $S'$ such that $|S'\ensuremath{\mathrel{\stackrel{\makebox[0pt]{\ensuremath{_\rightsquigarrow}}}{\cap}}}\xspace S^*|>|S\ensuremath{\mathrel{\stackrel{\makebox[0pt]{\ensuremath{_\rightsquigarrow}}}{\cap}}}\xspace S^*|$. Using Definition~\ref{def:proximity}, $S\ensuremath{\mathrel{\stackrel{\makebox[0pt]{\ensuremath{_\rightsquigarrow}}}{\cap}}}\xspace S^*$ is a prefix $s^*_1,\ldots,s^*_i$ of the canonical order of $S^*$, so we want to find any solution $S'$ that contains a longer prefix, i.e., $s^*_1,\ldots,s^*_{i+1}$ (possibly ordered differently and interspersed in the canonical order of $S'$). Since we must at least add the vertex $s^*_{i+1}$, we call $s^*_{i+1}$ the \textit{canonical extender} of $S,S^*$. Armed with this notion, we want to proceed conceptually as follows for a given solution $S$.
\begin{enumerate}
\item Guess which node $v \not \in S$ is the canonical extender $s^*_{i+1}$ (try all possibilities, $n$ at most).
\item Guess a \textit{removable set} $X\subseteq S$ from $S\cup\{v\}$, i.e., such that $S\setminus X\cup\{v\}$ is a solution and $X \cap \{s^*_1,\ldots,s^*_{i}\} = \emptyset$.
\item Obtain $S'$ as the outcome of $\textsc{complete}\xspace(S\setminus X\cup\{v\})$.
\end{enumerate}
In essence, we want to add $s^*_{i+1}$ to $S$, then turn the result back into a solution by removing some elements, but without affecting the proximity $s^*_1,\ldots,s^*_{i}$.
Recalling that prefixes of a canonical order are required to be (non-maximal) solutions, indeed $s^*_1,\ldots,s^*_{i+1}$ is a solution; hence, a removable set $X$ always exists (e.g., $X=S\setminus \{s^*_1,\ldots,s^*_{i}\}$).
The key point is that we want to satisfy the proximity requirement for all $S^*$ (that can be exponentially many) using only a \textit{polynomial} number of removable sets $X$.
While there is no general rule for this, and indeed, solving this for some problems would imply \textsc{p}=\textsc{np}, we will observe in this paper how it is possible to do so in some cases where a canonical order can efficiently decompose the underlying structure of the solution.
\paragraph{Canonical reconstruction}
Now we have all the ingredients to formalize below the required structure for adopting our strategy.
\begin{definition}(Proximity search by canonical reconstruction)
\label{def:crecon}
Given a maximal listing problem $\mathcal{P}$, in which each maximal solution $S$ is associated with a canonical ordering $s_1,\ldots, s_{|S|}$, we say that $\mathcal{P}$ admits a \emph{canonical reconstruction} if the following holds.
\begin{enumerate}
\item \label{item:canonical1} Any prefix $s_1,\ldots, s_i$ of the canonical order of any maximal solution $S$ is a (non-maximal) solution of $~\mathcal{P}$.
\item \label{item:canonical2} Given a maximal solution $S$ and any $v\not\in S$, there is set $\mathcal{X}\subseteq 2^S$ of \emph{removables}, such that
\begin{itemize}
\item $\mathcal{X} = \{ X_1, X_2,~\ldots \}$ can be computed in polynomial time.
\item $S\setminus X_i\cup\{v\}$ is a solution of $\mathcal{P}$ for any $X_i\in \mathcal{X}$.
\item For any $S^*$ such that $v$ is the canonical extender of $S,S^*$, there is at least one $X_i\in \mathcal{X}$ such that $(S\ensuremath{\mathrel{\stackrel{\makebox[0pt]{\ensuremath{_\rightsquigarrow}}}{\cap}}}\xspace S^*) \cap X_i = \emptyset$.
\footnote{Indeed, $(S\ensuremath{\mathrel{\stackrel{\makebox[0pt]{\ensuremath{_\rightsquigarrow}}}{\cap}}}\xspace S^*) \cap X_i = \emptyset$, i.e., $X_i$ does not intersect the proximity, implies that $S\setminus X_i\cup\{v\}$ contains $(S\ensuremath{\mathrel{\stackrel{\makebox[0pt]{\ensuremath{_\rightsquigarrow}}}{\cap}}}\xspace S^*) \cup \{v\}$, which extends the proximity with $v$.}
\end{itemize}
\item There is a polynomial-time computable function $\textsc{complete}\xspace(A)$ which takes a solution $A$ of $~\mathcal{P}$ and returns a maximal solution $A'\supseteq A$ of $~\mathcal{P}$.
\end{enumerate}
We further define the \emph{canonical reconstruction function}, as
\[
\textsc{neighbors}\xspace(S,v) = \bigcup_{X_i\in\mathcal{X}}\textsc{complete}\xspace((S\setminus X_i)\cup\{v\})
\]
and this corresponds to the solutions $S' \in \textsc{neighbors}\xspace(S)$ for which $v$ is the canonical extender of $S,S'$.
Hence, $\textsc{neighbors}\xspace(S)$ is obtained as $\bigcup_{v \in V(G)} \textsc{neighbors}\xspace(S,v)$.\footnote{For completeness, we define $\textsc{neighbors}\xspace(S,v) = \{S\}$ for $v\in S$, as $S\cup \{v\} = S$ is already a solution of $\mathcal{P}$.}
\end{definition}
We observe how the removables and the neighboring function can be derived from one another, so the algorithm can be defined by providing either one: one can focus on defining removables from $S\cup\{v\}$ that do not intersect the proximity, or equivalently solutions contained in $S\cup\{v\}$ that fully contain the proximity.
We also recall that a polynomial-time computable function $\textsc{complete}\xspace(\cdot)$ trivially exists for all hereditary and connected-hereditary properties that can be recognized in polynomial time (these include all the example problems shown in this paper).
\begin{figure*}
\centering
\includegraphics[width=\textwidth]{img/bip-run.pdf}
\put(-470,50){\begin{scriptsize}$S = \{2,3,5,7,8,10,11\}$, with $S_0 = \{2,8,11\}$, $S_1 = \{3,5,7,10\}$, and $v = 12$\end{scriptsize}}
\put(-470,33){\begin{scriptsize}$X_0 = N(v)\cap S_0 = \{8,11\}$,\quad $X_1 = N(v)\cap S_1 = \{5,7,10\}$\end{scriptsize} }
\put(-470,16){\begin{scriptsize}$S \setminus X_1 \cup \{v\} = \{2,3,8,11,12\}$ (top)\end{scriptsize}}
\put(-470,0){\begin{scriptsize}$S \setminus X_0 \cup \{v\} = \{2,3,5,7,10,12\}$, with vertex $9$ added by $\textsc{complete}\xspace(\cdot)$ (bottom)\end{scriptsize} }
\caption{The steps taken by the neighboring function $\textsc{neighbors}\xspace(S,v)$, for a possible solution $S$ of the graph in Figure~\ref{fig:ex:bip} and $v=12$. The bottom two lines correspond to the neighboring solutions produced by the function $\textsc{neighbors}\xspace(S,12)$.}
\label{fig:bip-running}
\end{figure*}
Finally, we show how canonical reconstruction immediately implies the maximal listing problem at hand is proximity searchable, using the $\textsc{neighbors}\xspace(\cdot)$ function defined above:
\begin{theorem}
\label{thm:canonical-reconstructio}
All maximal listing problems that allow a canonical reconstruction are proximity searchable.
\end{theorem}
\begin{proof}
Let us show that a listing problem $\mathcal{P}$ that satisfies Definition~\ref{def:crecon} satisfies the four conditions of Definition~\ref{def:searchable}.
Condition~(\ref{item:prox1}) is trivially satisfied, say, using $\textsc{complete}\xspace(\emptyset)$.
As for condition~(\ref{item:prox2}), recall that
\[
\textsc{neighbors}\xspace(S) =
\bigcup_{v \in V(G)} \bigcup_{X_i\in\mathcal{X}}(\textsc{complete}\xspace(S\setminus X_i\cup\{v\})).
\]
Considering that both $|V(G)|$ and $|\mathcal{X}|$ are polynomial, and $\textsc{complete}\xspace(\cdot)$ takes polynomial time, it follows that computing $\textsc{neighbors}\xspace(S)$ takes polynomial time.
For condition~(\ref{item:prox3}), consider the canonical extender $v$ for $S,S^*$: By Definition~\ref{def:crecon} there is $X_i\in \mathcal{X}$ such that $X_i\cap (S\ensuremath{\mathrel{\stackrel{\makebox[0pt]{\ensuremath{_\rightsquigarrow}}}{\cap}}}\xspace S^*) = \emptyset$; it holds that $S' = \textsc{complete}\xspace(S\setminus X_i\cup\{v\}) \in \textsc{neighbors}\xspace(S,v) \subseteq \textsc{neighbors}\xspace(S)$, and $(S\ensuremath{\mathrel{\stackrel{\makebox[0pt]{\ensuremath{_\rightsquigarrow}}}{\cap}}}\xspace S^*) \cup \{v\}\subseteq S'$, thus $|S'\ensuremath{\mathrel{\stackrel{\makebox[0pt]{\ensuremath{_\rightsquigarrow}}}{\cap}}}\xspace S^*|>|S\ensuremath{\mathrel{\stackrel{\makebox[0pt]{\ensuremath{_\rightsquigarrow}}}{\cap}}}\xspace S^*|$ because of $v$.
Finally, condition~(\ref{item:prox4}) is satisfied by looking at the definition of proximity in Definition~\ref{def:proximity}: fixed $S^*$, the proximity $S\ensuremath{\mathrel{\stackrel{\makebox[0pt]{\ensuremath{_\rightsquigarrow}}}{\cap}}}\xspace S^*$ is maximized if $S \supseteq S^*$, as we have $S\ensuremath{\mathrel{\stackrel{\makebox[0pt]{\ensuremath{_\rightsquigarrow}}}{\cap}}}\xspace S^* = S^* = S$; however, as this is a maximal listing problem, all solutions are inclusion-wise maximal, meaning that $S \supseteq S^*$ is only true for $S=S^*$.
\end{proof}
As a final remark, we note a somewhat surprising feature of this technique: while in general connected-hereditary properties (e.g., Maximal Connected Induced Bipartite Subgraphs) are more challenging to deal with than hereditary ones (e.g., Maximal Induced Bipartite Subgraphs), in the case of proximity search there is typically no difference, and in some instances, we even use the connected case as a starting point for the non-connected one (see, e.g., Section~\ref{sec:binon}).
\section{Maximal Bipartite Subgraphs}\label{sec:bip}
We now illustrate how to apply proximity search to maximal bipartite subgraph enumeration, giving the full details for the example in the previous section.
A graph $G$ is bipartite if its vertices can be partitioned into two sets $V_0,V_1$, such that $V_0\cap V_1 = \emptyset$, $V_0 \cup V_1 = V(G)$, and both $G[V_0]$ and $G[V_1]$ are edge-less graphs. Equivalently, $G$ is bipartite if it has no cycle of odd length.
Maximal bipartite subgraphs have also been studied as \textit{minimal odd cycle transversals}~\cite{kratsch2014compression}, as one is the complement of the other.
The problem of listing \textit{all} bipartite (and induced bipartite) subgraphs has been efficiently solved in~\cite{wasa2018bipartite}. However, to the best of our knowledge, neither the techniques in~\cite{wasa2018bipartite} nor other known ones extend to efficiently listing \textit{maximal} bipartite subgraphs, which poses a challenge. Consider the instance of input-restricted problem shown in Figure~\ref{fig:bip:restr} (left). We can exploit the fact that a subgraph of a bipartite graph is itself bipartite, meaning that the property is hereditary.
Hence, we could take the current solution $S$ (which are the endpoints of the bold edges) and a vertex $v \not \in S$, to then try to list all the maximal solutions contained in the induced subgraph $G[S \cup \{v\}]$; however, $G[S \cup \{v\}]$ has exponentially many solutions, meaning we cannot solve the input-restricted problem in polynomial time and thus we cannot get polynomial delay with the techniques from~\cite{lawler1980generating,Cohen20081147,conte2019framework}.
The best we could hope for is solving the input-restricted problem in polynomial delay or incremental polynomial time, which would yield an incremental polynomial time algorithm for the general problem~\cite{Cohen20081147}. Figure~\ref{fig:bip:restr} (right) shows an analogous situation for edge-induced subgraphs.
We thus turn to proximity search. First, let us introduce some preliminary notions: We denote an induced bipartite \textit{subgraph} of $G$ as a pair of vertex sets $\langle B_0,B_1 \rangle$, with $B_0\cap B_1 = \emptyset$ and $B_0 \cup B_1 \subseteq V(G)$, such that $G[B_0]$ and $G[B_1]$ are edge-less graphs. By convention, $B_0$ is the side of the bipartition containing the vertex of smallest label among those in the subgraph. In case $G[B_0 \cup B_1]$ has multiple connected components, this applies to all components. This way, any bipartite subgraph (connected or not) always has a \emph{unique} representation $\langle B_0,B_1 \rangle$.
We will sometimes use simply $B$ to refer to the subgraph $G[B_0 \cup B_1]$ induced by $\langle B_0,B_1 \rangle$. When performing $\textsc{complete}\xspace(B)$ (defined at the end of Section~\ref{sec:prelim}) and $B$ is not connected, this may move some vertices from $B_0$ to $B_1$ and vice versa due to different components becoming connected; even when $B$ is connected, if a vertex with smaller label than all others in $B$ is added to $B_1$, then $B_0$ and $B_1$ are immediately swapped to preserve the invariant of the smallest vertex being in $B_0$.
We define the intersection between two bipartite subgraphs $B$ and $B'$ as the set of all shared vertices, i.e.: $B\cap B' = (B_0 \cup B_1) \cap (B'_0\cup B'_1)$.
We consider the case of connected induced bipartite subgraphs in Section~\ref{sec:bicon}. We will later briefly show how this structure can be adapted to cover the non-connected and non-induced cases with small changes in Sections~\ref{sec:binon} and~\ref{sec:ebi}. Their complexity will be discussed in Section~\ref{sec:bitime}.
\subsection{Listing Maximal Connected Induced Bipartite Subgraphs}
\label{sec:bicon}
Let $B = \langle B_0,B_1 \rangle$ be a maximal induced bipartite subgraph of $G$, and $v$ a vertex not in $B$, i.e., in $v \in V(G)\setminus B$.
Looking at Definition~\ref{def:crecon}, we immediately observe that a polynomial-time computable function $\textsc{complete}\xspace(\cdot)$ exists since the problem is connected-hereditary.
Then, we need to define a suitable canonical order, and prove the existence of the corresponding removables.
Consider a BFS order of $G[B]$ starting from its vertex of smallest label, say $b_1$. In this order, a vertex $u$ precedes a vertex $v$ if the distance of $u$ from $b_1$ is smaller than that of $v$ or, in case the distance is equal, $u$'s label is smaller than $v$'s.
\begin{definition}[canonical order for connected induced bipartite subgraphs]\label{def:bip-con-canon}
The \emph{canonical} order of a \emph{connected induced bipartite subgraph} $B$ is the sequence $b_1, \ldots, b_{|B|}$ given by a BFS order of $G[B]$ rooted at the vertex $b_1$ of smallest label, where ties are broken by placing the vertex of smallest label first.
\end{definition}
For the subgraph in Figure~\ref{fig:ex:bip} (b) the canonical order is $2,3,5,8,11,7,10$, and for the one in (c) it is $2,3,8,12,11$. The definition of proximity is then automatically given by Definition~\ref{def:proximity}.
Last ingredient for Definition~\ref{def:crecon} is the set $\mathcal{X}\subseteq 2^B$, that contains just \emph{two} removables.
In order to get a bipartite graph, it is possible to make two removables as follows:
\begin{itemize}
\item $X_0 = N(v) \cap B_0 $,
\item $X_1 = N(v) \cap B_1 $.
\end{itemize}
That is, remove all the neighbors of $v$ in one of the two sides $B_i$: clearly, $v$ can be included in $B_i$ as it is now only adjacent to vertices of $B_{1-i}$.
While this works for the Maximal Induced Bipartite Subgraphs problem, we have the further constraint of \textit{connectivity}, so we must also discard every vertex that is not in the same connected component as $v$. The removables become as follows:
\begin{itemize}
\item $X_0 = B \setminus \texttt{cc}\xspace_v( \{v\} \cup (B_0 \setminus N(v)) \cup B_{1} )$,
\item $X_1 = B \setminus \texttt{cc}\xspace_v( \{v\} \cup (B_1 \setminus N(v)) \cup B_{0} )$.
\end{itemize}
That is, we remove all vertices not in the same connected component as $v$, after introducing $v$ and removing all its neighbors in either $B_0$ or $B_1$.
We can use these to create the neighboring function to be plugged in Algorithm~\ref{alg:general}, following Definition~\ref{def:crecon}:
\begin{definition}[neighboring function for maximal connected induced bipartite subgraphs]\label{def:neigh-bip-con}
$$\textsc{neighbors}\xspace(B,v) = \{ \textsc{complete}\xspace( \texttt{cc}\xspace_{v} ( \{v\} \cup (B_i \setminus N(v)) \cup B_{1-i} ) ) \mid i =0,1 \}$$
\end{definition}
A graphical example of this procedure is given in Figure~\ref{fig:bip-running} (for simplicity, we adopt an example where the subgraphs are connected after removing $v$'s neighbors, so the removables are equivalent to the simpler ones of the non-connected version).
\begin{lemma}\label{lem:bip-con-scap}
The problem of listing all Maximal Connected Induced Bipartite Subgraphs admits a canonical reconstruction.
\end{lemma}
\begin{proof}
For the canonical order given in Definition~\ref{def:bip-con-canon}, any prefix induces a graph that is connected because of the BFS order, and bipartite because bipartite subgraphs are hereditary, so condition~(\ref{item:canonical1}) of Definition~\ref{def:crecon} is satisfied.
As for condition~(\ref{item:canonical2}), it is evident from the definition of removables (alternatively, of the neighboring function) that they can be computed in polynomial time, and that they produce connected bipartite subgraphs.
We only need to show that the third item holds: given $B$, $B^*$ and their canonical extender $\ensuremath{\dot v}\xspace$, we have $B\ensuremath{\mathrel{\stackrel{\makebox[0pt]{\ensuremath{_\rightsquigarrow}}}{\cap}}}\xspace B^*\cap X_i = \emptyset$ for either $i=0$ or $i=1$; this will imply that $|B' \ensuremath{\mathrel{\stackrel{\makebox[0pt]{\ensuremath{_\rightsquigarrow}}}{\cap}}}\xspace B^*| > |B \ensuremath{\mathrel{\stackrel{\makebox[0pt]{\ensuremath{_\rightsquigarrow}}}{\cap}}}\xspace B^*|$ for $B' = \textsc{complete}\xspace( \texttt{cc}\xspace_{\ensuremath{\dot v}\xspace} ( \{\ensuremath{\dot v}\xspace\} \cup (B_i \setminus N(\ensuremath{\dot v}\xspace)) \cup B_{1-i} ) )$, so the proximity is successfully increased.
If $B\ensuremath{\mathrel{\stackrel{\makebox[0pt]{\ensuremath{_\rightsquigarrow}}}{\cap}}}\xspace B^* = \emptyset$ the claim is trivially true, as we can consider $b^*_1$ as canonical extender.
Otherwise, let $Z = B \ensuremath{\mathrel{\stackrel{\makebox[0pt]{\ensuremath{_\rightsquigarrow}}}{\cap}}}\xspace B^* = \{b^*_1, \ldots, b^*_h\}$, and we have $\ensuremath{\dot v}\xspace = b^*_{h+1}$. By Definition~\ref{def:bip-con-canon}, $Z$ is a connected induced bipartite subgraph, meaning that it allows a unique bipartition $Z_0,Z_1$ (with $Z_0$ being the set containing the vertex of smallest label in $Z$, that is, $b^*_1$).
Since $b^*_1$ is the vertex of smallest label in $B^*$, it will be in $B^*_0$, so it follows that $Z_0 \subseteq B^*_0$ and $Z_1 \subseteq B^*_1$.
Let $j$ be the value in $\{0,1\}$ such that $\ensuremath{\dot v}\xspace\in B^*_j$, and observe that $N(\ensuremath{\dot v}\xspace)\cap Z_j \subseteq N(\ensuremath{\dot v}\xspace)\cap B^*_j = \emptyset$.
Furthermore, we know that $b^*_1\in B^*_0$ and $b^*_1\in B$, but we do not know whether $b^*_1\in B_0$ or $b^*_1\in B_0$; however, there exists a value $i$ in $\{0,1\}$ such that either $b^*_1 \in (B_i\cap B^*_j)$ or $b^*_1 \in (B_{1-i}\cap B^*_{1-j})$. Observe that $Z_j \subseteq B^*_j \cap B_i$ and $Z_{1-j} \subseteq B^*_{1-j} \cap B_{1-i}$.
Finally, let $B' = \texttt{cc}\xspace_{\ensuremath{\dot v}\xspace} ( \{\ensuremath{\dot v}\xspace\} \cup (B_i \setminus N(\ensuremath{\dot v}\xspace)) \cup B_{1-i} )$, and consequently we have $X_i = B \setminus B'$.
$Z$ is fully contained in $B'$: the only vertices removed from $B$ by $X_i$ are (i) those in $N(\ensuremath{\dot v}\xspace)\cap B_i$, but $N(\ensuremath{\dot v}\xspace)\cap B_i \cap Z \subseteq N(\ensuremath{\dot v}\xspace)\cap B_i \cap Z_j \subseteq N(\ensuremath{\dot v}\xspace)\cap Z_j = \emptyset$, and (ii) the vertices not in the connected component of $\ensuremath{\dot v}\xspace$ in $G[\{\ensuremath{\dot v}\xspace\} \cup (B_i\setminus N(\ensuremath{\dot v}\xspace)) \cup B_{1-i}]$, but no such vertex can be in $Z$ as $Z\cup \{\ensuremath{\dot v}\xspace\}$ is a prefix of the canonical order of $B^*$, so it induces a connected subgraph.
We thus have that $Z\cup \{\ensuremath{\dot v}\xspace\} \subseteq B'$, meaning that $X_i \cap Z = \emptyset$, which proves the claim.
Constructively, we can finally observe how the maximal solution $B''=\textsc{complete}\xspace(B')$ is the one produces by the algorithm which increases the proximity to $B^*$, as we have $\{b^*_1, \ldots, b^*_h, b^*_{h+1}\} \subseteq B''\ensuremath{\mathrel{\stackrel{\makebox[0pt]{\ensuremath{_\rightsquigarrow}}}{\cap}}}\xspace B^*$ and thus $|B'' \ensuremath{\mathrel{\stackrel{\makebox[0pt]{\ensuremath{_\rightsquigarrow}}}{\cap}}}\xspace B^*| \ge |B \ensuremath{\mathrel{\stackrel{\makebox[0pt]{\ensuremath{_\rightsquigarrow}}}{\cap}}}\xspace B^*|+1$.
\end{proof}
From this, we immediately obtain the correctness of the algorithm.
\begin{theorem}\label{thm:bip-con}
A proximity search algorithm (Algorithm~\ref{alg:general}), using the\linebreak $\textsc{neighbors}\xspace(\cdot)$ function from Definition~\ref{def:neigh-bip-con} outputs all Maximal Connected Induced Bipartite Subgraphs of a graph $G$ without duplication with $O(nm)$ delay.
\end{theorem}
\begin{proof}
The correctness follows from Theorem~\ref{thm:canonical-reconstructio}.
The delay is dominated by the cost of the $\textsc{neighbors}\xspace(B)$ function, i.e., calling $O(n)$ times $\textsc{neighbors}\xspace(B,v)$. The cost of the latter is $O(m)$ time to compute $\texttt{cc}\xspace_v(\cdot)$, and $O(m)$ time to compute the $\textsc{complete}\xspace(\cdot)$ function by Lemma~\ref{lem:bip:comp} (delayed to Section~\ref{sec:bitime} for compactness). The statement follows.
\end{proof}
\subsection{Listing Maximal Induced Bipartite Subgraphs}
\label{sec:binon}
We can extend our solution to the non-connected case by building one connected component at a time. We obtain the canonical order by Definition~\ref{def:canonBFSnon}, that is, a BFS order of each component:
\begin{definition}[canonical order for induced bipartite subgraphs]\label{def:bip-non-canon}
The \emph{canonical} order of an \emph{induced bipartite subgraph} $B$ is the sequence $b_1, \ldots, b_{|B|}$ obtained by first ordering the connected components of $G[B]$ by incremental order of smallest-id vertex, then ordering each component by a BFS order (given in Definition~\ref{def:bip-con-canon}) rooted in its smallest-id vertex.
\end{definition}
In essence, this corresponds to ordering each connected component as in the connected case (Definition~\ref{def:neigh-bip-con}), and placing earlier components whose smallest-id vertex is smaller.
Looking again at Figure~\ref{fig:ex:bip}, and letting $B$ be the subgraph shown in (d) and $B^*$ as that shown in (e), the canonical order of $B$ is $\langle 1,2,7,8,11,10\rangle $, that of $B^*$ is $\langle 1,2,7,9,12,10\rangle $. By the definition of proximity for canonical reconstruction, we also obtain $B\ensuremath{\mathrel{\stackrel{\makebox[0pt]{\ensuremath{_\rightsquigarrow}}}{\cap}}}\xspace B^* = \{1,2,7\}$.
The removables become simpler for this case, as we can simply remove $N(v)\cap B_i$ for $i=0,1$.
As a result, the neighboring function is essentially the same as the connected case (Definition~\ref{def:neigh-bip-con}), with minor changes as we do not require the connectivity:
\begin{definition}[neighboring function for maximal induced bipartite subgraphs]\label{def:neigh-bip-non}
$$\textsc{neighbors}\xspace(B,v) = \{ \textsc{complete}\xspace(\{v\} \cup (B_i \setminus N(v)) \cup B_{1-i} ) \mid i=0,1\}$$
\end{definition}
We can then proceed to prove correctness and complexity of this case:
\begin{theorem}\label{thm:nbip}
A proximity search algorithm (Algorithm~\ref{alg:general}), using the\linebreak $\textsc{neighbors}\xspace(\cdot)$ function from Definition~\ref{def:neigh-bip-non} outputs all maximal induced bipartite subgraphs of a graph $G$ without duplication with $O(n(m+n\iack(n)))$ delay.
\end{theorem}
\begin{proof}
Consider the solutions $B$ and $B^*$. Let $b^*_1,\ldots,b^*_{|B^*|}$ be the canonical ordering of $B^*$ by Definition~\ref{def:neigh-bip-non}, $B\ensuremath{\mathrel{\stackrel{\makebox[0pt]{\ensuremath{_\rightsquigarrow}}}{\cap}}}\xspace B^* = b^*_1,\ldots,b^*_{i}$, and $u=b^*_{i+1}$ the canonical extender for $B,B^*$. Let $C$ be the connected component of $B^*$ containing $u$. Since all the neighbors of $u$ in $B^*$ must be in its same connected component $C^{B^ *}_x$, and the neighbouring function (Definition~\ref{def:neigh-bip-non}) only removes neighbors of $u$ from $B$, the function may not remove from $B$ any vertex of $B^*$ that is \textit{not} in $C$. As for vertices in $C$, $B\ensuremath{\mathrel{\stackrel{\makebox[0pt]{\ensuremath{_\rightsquigarrow}}}{\cap}}}\xspace B^*$ contains a (possibly empty) prefix of its BFS order, which is itself a connected bipartite subgraph in canonical order. By the correctness of Lemma~\ref{lem:bip-con-scap}, for either $B' = \textsc{complete}\xspace( \texttt{cc}\xspace_{v} ( \{v\} \cup (B_0 \setminus N(v)) \cup B_{1} ) ) $ or $B' = \textsc{complete}\xspace( \texttt{cc}\xspace_{v} ( \{v\} \cup (B_1 \setminus N(v)) \cup B_{0} ) )$, this prefix is expanded with $u$, giving us $B'\ensuremath{\mathrel{\stackrel{\makebox[0pt]{\ensuremath{_\rightsquigarrow}}}{\cap}}}\xspace B^* \supseteq (B\ensuremath{\mathrel{\stackrel{\makebox[0pt]{\ensuremath{_\rightsquigarrow}}}{\cap}}}\xspace B^*)\cup\{u\}$ and proving correctness.
As for the delay, we can see that the cost of $\textsc{neighbors}\xspace(B)$ is bounded as for the connected case by $O(n)$ times the cost of $\textsc{neighbors}\xspace(B,v)$, which is in turn bounded by $O(m + n\iack(n))$ by Lemma~\ref{lem:bip:comp}, proving the statement.
\end{proof}
\subsection{Maximal Edge Bipartite Subgraphs}
\label{sec:ebi}
Finally, we show how to adapt the above algorithm to Maximal \textit{Edge} Bipartite Subgraphs, where edge-induced subgraphs are denoted by a set of edges, rather than vertices. In the following, given two sets of vertices $A$ and $B$, let $E(A,B)$ be the set of edges with one endpoint in $A$ and the other in $B$. We observe that the \textit{Maximal} Edge Bipartite Subgraphs of a connected graph are always connected, otherwise some edge could be added to joint components without creating cycles; by the same logic they span all vertices, and may thus be represented by simply a bipartition $\langle B_0,B_1 \rangle$ of $V(G)$, where the bipartite subgraph corresponds to the edges in $E(B_0,B_1)$. For readability, we use the shorthand $E_B \equiv E(B_0,B_{1})$ to refer to the edges of the bipartite subgraph $B$.
We also observe that the problem is hereditary and allows for a polynomial time computable $\textsc{complete}\xspace(\cdot)$ function.
We define the canonical order of a solution $B$ by taking the canonical order $ b_1, \ldots, b_{|B|} $ of the \textit{vertices} of $G[B]$ according to Definition~\ref{def:bip-con-canon},\footnote{Note that the vertices of $G[B]$ are all of $V(G)$, but to compute the canonical order we need to consider only the edges in the bipartite subgraph $G[B]$.} then taking the edges of $B$ in increasing order of their \textit{latter} vertex in the vertex order, and breaking ties by increasing order of the earlier endpoint. This essentially corresponds to ``building'' $B$ in a similar fashion as in the induced version, but adding one edge at a time incident to the newly selected vertex. The removables for an edge $e = \{a,b\}$, where $a < b$, are as follows.
\begin{itemize}
\item $X_0 = (E_B \setminus N_E(a))\cup \{e\} )$,
\item $X_1 = (E_B \setminus N_E(b))\cup \{e\} )$.
\end{itemize}
The principle behind the neighboring function is different but inspired by the induced case: rather than taking a vertex out of the solution and trying to add it to $B_0$ or $B_1$, we take an edge $e = \{a,b\}$ with both endpoints in the same $B_i$, and try to move the two vertices $a$ and $b$ to opposite sides of the bipartition.
This can be achieved by including the edge $e$ in the solution, and then, to preserve the subgraph being bipartite, removing either $N_E(a)$ or $N_E(b)$ from it. Finally we apply the $\textsc{complete}\xspace(\cdot)$ function to obtain a solution that is maximal.
More formally, recalling $E_B \equiv E(B_0,B_{1})$, we define $\textsc{neighbors}\xspace(B)$ as
$$\bigcup\limits_{e=\{a,b\}\in E(G)\setminus E_B}\{
\textsc{complete}\xspace((E_B \setminus N_E(a))\cup \{e\}),\; \textsc{complete}\xspace((E_B \setminus N_E(b))\cup \{e\})
\}$$
Consider two solution $B$ and $B^*$, with $e_1, \ldots, e_{|E(B^*_0,B^*_1)|}$ being the canonical order of $B^*$. Furthermore, let $B\ensuremath{\mathrel{\stackrel{\makebox[0pt]{\ensuremath{_\rightsquigarrow}}}{\cap}}}\xspace B^* = \{e_1, \ldots, e_h\}$ and $\ensuremath{\dot e}\xspace = e_{h+1} = \{a,b\}$ the canonical extender, i.e., the first edge in the ordering of $B^*$ which is not in $B$.
By the definition of the canonical ordering, we have that $\{e_1, \ldots, e_h\}$ is a connected bipartite subgraph, meaning that it allows a unique bipartition $B' = B'_0, B'_1$ of its incident vertices. As $\{e_1, \ldots, e_h\}\cup \{\ensuremath{\dot e}\xspace\}$ is also a connected bipartite subgraph, for some $j \in \{0,1\}$ we must have both $N(a)\cap B'_j = \emptyset$ and $N(b)\cap B'_{1-j} = \emptyset$.
Since $B'$ is included in $B$, we must have either (i) $B'_j\subseteq B_i$ and $B'_{1-j}\subseteq B_{1-i}$ or (ii) $B'_{1-j}\subseteq B_i$ and $B'_{j}\subseteq B_{1-i}$.
Recall now that both $a$ and $b$ are assumed wlog to be in $B_i$, meaning that $N(a)\cap B_i = N(b)\cap B_i = \emptyset$.
In the (i) case, we have $N(b) \cap B_{1-i} \cap B'_{1-j} = \emptyset$, so removing $N_E(b)$ from $B$ may not remove any edge of $B'$.
Thus
$\textsc{complete}\xspace((E_B \setminus N_E(b))\cup \{\ensuremath{\dot e}\xspace\})$, which belongs to $\textsc{neighbors}\xspace(B)$,
will contain $(B\ensuremath{\mathrel{\stackrel{\makebox[0pt]{\ensuremath{_\rightsquigarrow}}}{\cap}}}\xspace B^*)\cup\{e\}$.
In the (ii) case, we have $N(a) \cap B_{1-i} \cap B'_{1-j} = \emptyset$, removing $N_E(a)$ may not remove any edge of $B'$.
Thus
$\textsc{complete}\xspace((E_B \setminus N_E(a))\cup \{\ensuremath{\dot e}\xspace\})$, which also belongs to $\textsc{neighbors}\xspace(B)$,
will contain $(B\ensuremath{\mathrel{\stackrel{\makebox[0pt]{\ensuremath{_\rightsquigarrow}}}{\cap}}}\xspace B^*)\cup\{e\}$.
This means that in both cases, $\textsc{neighbors}\xspace(B)$ will yield a solution $S'$ that contains $(B\ensuremath{\mathrel{\stackrel{\makebox[0pt]{\ensuremath{_\rightsquigarrow}}}{\cap}}}\xspace B^*)\cup\{e\}$, i.e., such that $|S'\ensuremath{\mathrel{\stackrel{\makebox[0pt]{\ensuremath{_\rightsquigarrow}}}{\cap}}}\xspace S^*| > |S \ensuremath{\mathrel{\stackrel{\makebox[0pt]{\ensuremath{_\rightsquigarrow}}}{\cap}}}\xspace S^*|$.
As the complexity is bounded by $O(m)$ calls to the $\textsc{neighbors}\xspace(B_i,B_{1-i},e)$ function, whose cost is again bounded by that of $\textsc{complete}\xspace(\cdot)$, that is $O(m^2)$ time (By Lemma~\ref{lem:bip:comp}), the following theorem holds:
\begin{theorem}
Maximal (edge-induced) Bipartite Subgraphs can be listed in $O(m^3)$ time delay.
\end{theorem}
\subsection{Complexity}
\label{sec:bitime}
In order to complete the analysis, let us look at the cost $\mathcal{C}_t$ for the three variants considered:
\begin{lemma}\label{lem:bip:comp}
$\mathcal{C}_t$ is $O(m)$ for Maximal Connected Induced Bipartite Subgraphs, $O(m+n\iack(n))$ for Maximal Induced Bipartite Subgraphs, and $O(m^2)$ for Maximal Edge-induced Bipartite Subgraphs,
where $n$ and $m$ are the number of vertices and edges, and $\iack(\cdot)$ is the functional inverse of the Ackermann function~\cite{Tarjan75UF}.
\end{lemma}
\begin{proof}
$\mathcal{C}_t$ is a bound for computing the $\textsc{complete}\xspace(S)$ function as well as a canonical order. As the latter is computed by a BFS, it takes $O(m)$ time in all three cases, let us then focus on $\textsc{complete}\xspace(S)$:
Firstly, observe that using $O(n)$ space and standard data structures, we can mark to which bipartition each vertex of $S$ belongs to (using $O(m)$ time to compute the initial bipartition of $S$), and which vertices have been already tested for addition, in $O(1)$ time per vertex. If a vertex fails the test to be added, it will not be possible to add it later on, so the total cost of $\textsc{complete}\xspace(S)$ comes from selecting which vertices to test, and testing each of these vertices once.
For the connected case, we must test only vertices adjacent to $S$ (in no particular order): we can find these initial ``candidates'' in $O(\sum_{x\in S}|N(x)|) = O(m)$ time, marking each vertex as tested the first time so it is not tested again. Whenever trying to add a vertex $v$ to $S$, we must pay $O(|N(v)|)$ time to check that all its neighbors belong to the same bipartition of $S$, in which case $v$ belongs to the other one.
If $v$ is not addible, we immediately discard it. If instead we add it to $S$, we mark it with the correct bipartition, and update the list of candidate with its neighbors in $O(|N(v)|)$ time. As each vertex is only tested once and only added once, the total cost of $\mathcal{C}_t$ is bounded by $O(m)$.
For the non-connected case, we further keep track of connected components via union-find~\cite{Tarjan75UF} (actually, for each connected component we will keep track of its two partitions). To test a vertex $v$ we must just check that it does not connect to two vertices in different partitions $C_0$ and $C_1$ of the same connected component $C$ of $X$: this can be done in $O(|N(v)|)$. Updating the union-find can be done in total $O(n \iack(n))$, where $\iack(\cdot)$ is the functional inverse of the Ackermann function~\cite{Tarjan75UF}.\footnote{As $\iack(n)$ grows extremely slowly, we remark that $\iack(n)$ is in essence $O(1)$ on real, finite, graphs.}
Once we tested a vertex, if this was not addible, it will never become addible, thus we only need to test each vertex once. The cumulative cost for testing will be the sum of the degrees of all tested vertices, that is bounded by $O(m)$.
The total time is thus $O(m + n \iack(n))$.
Finally, for Maximal Edge-induced Bipartite Subgraphs, we need to test each edge for addition just once as the property is hereditary. For each test we can simply check if the resulting graph is bipartite, which takes $O(m)$ time, for a total cost of $O(m^2)$.
\end{proof}
\section{Maximal k-Degenerate Subgraphs}\label{sec:list:first}
We here consider the enumeration of maximal $k$-degenerate subgraphs, giving an algorithm that has polynomial delay when $k$ is bounded.
A graph $G$ is $k$-degenerate if it allows an elimination order where each vertex has degree at most $k$ when deleted.
Equivalently, it is $k$-degenerate if no subgraph of $G$ is a $(k+1)$-core, that is a graph where each vertex has degree greater or equal to $k+1$.
The \textit{degeneracy} $d$ of $G$ is the smallest $k$ for which $G$ is $k$-degenerate.
A \textit{degeneracy ordering} of $G$ is an order of its vertices in which each vertex $v$ has at most $d$ neighbors occurring later than $v$, where $d$ is the degeneracy of $G$. It is well known that a degeneracy ordering can be found in $O(m)$ time by iteratively removing the vertex of smallest degree~\cite{DBLP:journals/corr/cs-DS-0310049}. To remove ambiguity, when multiple vertices have the same degree we can remove the one with smallest label.
The degeneracy is a well-known sparsity measure~\cite{EppsteinLS13}; its definition generalizes that of independent sets ($0$-degenerate graphs) and trees and forests (connected and non-connected $1$-degenerate graphs). Furthermore, degeneracy is linked to planarity as all planar graphs are $5$-degenerate, while outerplanar graphs are $2$-degenerate~\cite{lick1970k}.
We are interested in listing all maximal $k$-degenerate subgraphs of a graph $G$. An output-polynomial algorithm is known for maximal \textit{induced} $k$-degenerate subgraphs if $G$ is chordal~\cite{DBLP:conf/cocoon/ConteKOUW17}, but no output-polynomial results are known for general graphs.
\subsection{Maximal Induced k-Degenerate Subgraphs}\label{sec:kdegen:ind}
A subgraph of a $k$-degen\-erate graph is $k$-degenerate so the property is hereditary, and degeneracy can be computed in linear time so we can implement the $\textsc{complete}\xspace(\cdot)$ function in polynomial time.
Given a maximal induced $k$-degenerate subgraph $S$, we define its \textit{canonical order} as the \textit{reverse} of its degeneracy ordering, i.e., an ordering $s_1,\ldots, s_{|S|}$, such that $s_{|S|},\ldots, s_1$ is the degeneracy ordering of $S$. In the case of non connected subgraphs, this is adapted by considering the connected components one at a time in lexicographical order. Then, the proximity is defined by Definition~\ref{def:proximity}.
In the resulting ordering we have $|N(s_i)\cap \{s_1,\ldots,s_{i-1}\}| \le k$, i.e., the neighbors of $s_i$ in $S$ that precede $s_i$ in the canonical order are at most $k$. This is the key property that gives us the intuition for the algorithm: the removables correspond to all neighbors of the canonical extender \textit{except} a set of size at most $k$.
The neighboring function is obtained as follows.
\begin{definition}[Neighboring function for Maximal Induced $k$-Degenerate Subgraphs]\label{def:kdeg:neigh}
$$\textsc{neighbors}\xspace(S) = \bigcup\limits_{v\in V(G)} \textsc{neighbors}\xspace(S,v)$$
Where
$\textsc{neighbors}\xspace(S,v) = \{\textsc{complete}\xspace(\{v\} \cup S \setminus (N(v) \setminus K) : K\subseteq (S\cap N(v)) \text{ and } |K|\le k\}$
\end{definition}
Less formally, when computing $\textsc{neighbors}\xspace(S,v)$, we try to add $v$ to $S$ as canonical extender. Since $S$ is maximal, this violates the degeneracy constraint, so we remove all neighbors of $v$ except at most $k$ (the \textit{removable} set being $N(v)\setminus K$). The resulting subgraph $D = \{v\} \cup S \setminus (N(v) \setminus K)$ is $k$-degenerate: as $D\setminus \{v\}$ is $k$-degenerate as it is a subgraph of $S$, and any degeneracy ordering of $D\setminus\{v\}$ becomes a $k$-degenerate ordering for $D$ if we prepend $v$ in the beginning, because $v$ has at most $k$ neighbors in $D$. This means $N(v)\setminus K$ is a suitable removable according to Definition~\ref{def:crecon}.
We now show how these choices for $K$ satisfy Definition~\ref{def:crecon}: we iteratively try for $K$ all possible subsets of $S\cap N(v)$ of size at most $k$. These combinations, i.e., the number of removables, are $O(\sum_{i\in\{1, \ldots, k\}}\binom{|N(v)|}{i}) = O(n^k)$, which is polynomial when $k$ is bounded.
Let us now look at a target solution $S^*$ such that $v$ is the canonical extender for $S,S^*$, and let $s^*_1,\ldots,s^*_{|S^*|}$ be the canonical order of $S^*$: if $v = s_i$ in this order, it follows that $S\ensuremath{\mathrel{\stackrel{\makebox[0pt]{\ensuremath{_\rightsquigarrow}}}{\cap}}}\xspace S^* = \{s^*_1,\ldots,s^*_{i-1}\}$, and $|N(v)\cap \{s^*_1,\ldots,s^*_{i-1}\}| \le k$.
We also have that $\{s^*_1,\ldots,s^*_{i-1}\}\subseteq S$, so $N(v)\cap \{s^*_1,\ldots,s^*_{i-1}\} \subseteq N(v)\cap S$: since we try as $K$ all possible subsets of $N(v)\cap S$ of size at most $k$, we will eventually have $K = N(v)\cap \{s^*_1,\ldots,s^*_{i-1}\}$.
At this point the neighboring function will yield $S' = \textsc{complete}\xspace(\{v\} \cup S \setminus (N(v) \setminus K) = \textsc{complete}\xspace(\{v\} \cup S \setminus (N(v) \setminus (S\ensuremath{\mathrel{\stackrel{\makebox[0pt]{\ensuremath{_\rightsquigarrow}}}{\cap}}}\xspace S^*))$. In other words, we only remove some neighbors of $v$ from $S$, but all the neighbors that are part of $S\ensuremath{\mathrel{\stackrel{\makebox[0pt]{\ensuremath{_\rightsquigarrow}}}{\cap}}}\xspace S^*$ are not removed, thus $S' \supseteq \{v\} \cup (S\ensuremath{\mathrel{\stackrel{\makebox[0pt]{\ensuremath{_\rightsquigarrow}}}{\cap}}}\xspace S^*)$, meaning $|S'\ensuremath{\mathrel{\stackrel{\makebox[0pt]{\ensuremath{_\rightsquigarrow}}}{\cap}}}\xspace S^*| > |S\ensuremath{\mathrel{\stackrel{\makebox[0pt]{\ensuremath{_\rightsquigarrow}}}{\cap}}}\xspace S^*|$.
As for the running time, let us consider the cost $\mathcal{C}_t$ of a $\textsc{complete}\xspace(X)$ call. $k$-degenerate graphs are hereditary, i.e., if a vertex is not addible it will not become addible later, so we need to test each $v\in V(G)\setminus X$ for addition at most once. As testing the degeneracy takes $O(m)$ time, $\mathcal{C}_t = O(mn)$ time.
Consider now $\textsc{neighbors}\xspace(S,v)$: firstly, we enumerate each possible $K \subseteq N(v)\cap S$, which takes $O(\sum_{i\in\{1,\ldots,k\}}\binom{|N(v)|}{i}) = O(|N(v)|^k)$ time. For each, we run $\textsc{complete}\xspace(\{v\} \cup (S\setminus N(v)) \cup K)$, which takes $O(mn)$ time. The total cost is $O(n^{k+1} m)$ time.
The problem is thus proximity searchable, and the delay of the listing algorithm is the cost of $\textsc{neighbors}\xspace(S)$, i.e., running $O(n)$ times $\textsc{neighbors}\xspace(S,v)$ (maintaining $\ensuremath{\mathcal{S}}\xspace$ is negligible). More formally:
\begin{theorem}\label{thm:kdegen}
Maximal Induced $k$-degenerate Subgraphs are proximity searchable when $k$ is constant, and can be enumerated in $O(mn^{k+2})$ time delay.
\end{theorem}
We now observe that $1$-degenerate subgraphs are exactly forests, and the connected ones are trees; setting $k=1$ we immediately obtain polynomial-delay algorithms for listing Maximal Induced Forests that could be easily adapted to Maximal Induced Trees. However, an ad-hoc analysis, delayed to Section~\ref{sec:indtrees}, shows we can obtain algorithms with better delay, and even reduce the space usage to polynomial for these problems.
\subsection{Maximal Edge-induced \textit{k}-Degenerate Subgraphs}
We now consider Maximal Edge-induced \textit{k}-Degenerate Subgraphs, i.e., maximal sets of edges $E\subseteq E(G)$ that correspond to a $k$-degenerate subgraph of $G$.
An algorithm for this case can be obtained by exploiting the structure of the induced one.
In the following, let $N_E(v)$ be the \textit{edge neighborhood} of $v$, i.e., the set of edges of $G$ incident to the vertex $v$.
Note that edge-induced $k$-degenerate subgraphs are also hereditary, and so $\textsc{complete}\xspace(\cdot)$ takes polynomial time.
Let $S$ be an edge-induced $k$-degenerate subgraph, and let $v_1, \ldots , v_l$ be the canonical order of the vertices of $G[S]$ (i.e., the graph containing only edges in $S$ and vertices incident to them), as in Section~\ref{sec:kdegen:ind}.
The canonical ordering of $S$ is obtained by selecting the edges of $B$ by increasing order w.r.t. their \textit{later} endpoint in the vertex order, breaking ties by order of the other (earlier) endpoint.
This corresponds to selecting the vertices $v_1, \ldots , v_l$ in order, and for each adding the edges towards the preceding vertices one by one. Whenever all the edges from $v_i$ to the preceding vertices have been added, we can observe that the graph corresponds to that induced in $G[S]$ by the vertices $\{v_1, \ldots , v_i\}$. By the canonical order of the vertices defined in Section~\ref{sec:kdegen:ind}, this means $v_i$ has at most $k$ neighbors in $\{v_1, \ldots , v_{i-1}\}$.
Again, the proximity $\ensuremath{\mathrel{\stackrel{\makebox[0pt]{\ensuremath{_\rightsquigarrow}}}{\cap}}}\xspace$ is given by Definition~\ref{def:proximity}.
We can now define the neighboring function:
\begin{definition}[Neighboring function for Maximal Edge-induced $k$-Degenerate\linebreak Subgraphs]\label{def:e:kdeg:neigh}
Let $S$ be a maximal edge-induced $k$-degenerate subgraph, and $e = \{a,b\}$ an edge not in $S$.
We define:
$$\textsc{neighbors}\xspace(S) = \bigcup\limits_{e=\{a,b\} \in E\setminus S} \textsc{neighbors}\xspace(S,a,b) \cup \textsc{neighbors}\xspace(S,b,a)$$
Where
$\textsc{neighbors}\xspace(S,a,b) = \{\textsc{complete}\xspace(\{e\} \cup (S\setminus N_E(a)) \cup K) : K\subseteq (S\cap N_E(a)) \text{ and }$\linebreak $ |K|\le k-1\}$
\end{definition}
In other words, we add an edge $e=\{a,b\}$ to $S$, then force $a$ (or, respectively, $b$) to have degree at most $k$, by removing all other edges incident to it except at most $k-1$, as well as adding $e$.
The resulting graph is $k$-degenerate as $a$ (respectively $b$) has degree $k$, and the residual graph is a subgraph of $S$, which is $k$-degenerate, so it is possible to compute a degeneracy ordering.
Consider now two solutions $S$, $S^*$, with $S \ensuremath{\mathrel{\stackrel{\makebox[0pt]{\ensuremath{_\rightsquigarrow}}}{\cap}}}\xspace S^* = \{e_1, \ldots , e_h\}$, and let $\ensuremath{\dot e}\xspace = \{x,y\}$ be the earliest edge in the canonical order of $S^*$ that is not in $S$, i.e, $e_{h+1}$.
Assume wlog that $x$ comes before $y$ in the canonical (vertex) ordering of $S^*$. In this ordering, $y$ has at most $k$ neighbors preceding it, i.e, $|\{e_1, \ldots , e_h\} \cap N_E(y)| \le k$. Furthermore, by the same definition, all edges incident to $y$ that precede $\ensuremath{\dot e}\xspace$ in the ordering must be between $y$ and another vertex which comes earlier than $x$, and thus than $y$, in the ordering, thus they may be at most $k-1$ ($k$, including $\ensuremath{\dot e}\xspace$ itself, from $y$ to $x$). Let $K'$ be the set of these edges (not including $\ensuremath{\dot e}\xspace$).
When computing $\textsc{neighbors}\xspace(S,y,x)$, we consider all subsets of edges in $S$ incident to $y$ of size at most $k-1$. By what stated above, at some point we will consider exactly $K'$. In this case, we will obtain $S' = \textsc{complete}\xspace(\{\ensuremath{\dot e}\xspace\} \cup (S\setminus N_E(y)) \cup K')$. This must contain all edges in $\{e_1, \ldots , e_h\}$, as we only removed edges neighboring $y$, but all those in $\{e_1, \ldots , e_h\}$ were in $K'$. Thus we have $\{e_1, \ldots , e_h\} \cup \ensuremath{\dot e}\xspace = \{e_1, \ldots , e_h, e_{h+1}\} \subseteq S'$, which implies $|S'\ensuremath{\mathrel{\stackrel{\makebox[0pt]{\ensuremath{_\rightsquigarrow}}}{\cap}}}\xspace S^*| > |S\ensuremath{\mathrel{\stackrel{\makebox[0pt]{\ensuremath{_\rightsquigarrow}}}{\cap}}}\xspace S^*|$.
The case in which $x$ comes after $y$ in the ordering is similarly satisfied by $\textsc{neighbors}\xspace(S,x,y)$.
Finally, we only need to show that $\textsc{neighbors}\xspace(S)$ takes polynomial time to compute: indeed this is $O(m)$ times the cost of $\textsc{neighbors}\xspace(S,y,x)$, which in turn has the cost of computing $\textsc{complete}\xspace(\cdot)$ once for each possible considered set $K$. These latter are $O(\binom{N_E(y)}{k-1})$, and the $\textsc{complete}\xspace(\cdot)$ can be easily implemented in $O(m^2)$ (as above, testing degeneracy takes $O(m)$ time and each edge needs to be considered at most once for addition since the problem is hereditary), for a total cost that is polynomial when $k$ is constant. We can thus state the following:
\begin{theorem}\label{thm:e:kdegen}
Maximal Edge-induced $k$-degenerate Subgraphs are proximity searchable when $k$ is constant, and can be enumerated with delay $O(\binom{n}{k-1}m^3)$.
\end{theorem}
\section{Maximal Chordal Subgraphs}\label{sec:chordal}
\subsection{Maximal Induced Chordal Subgraphs}
A graph $G$ is chordal if every cycle in $G$ of length greater than 3 has a chord, i.e., an edge between two non-consecutive vertices in the cycle.
Chordal graphs have been widely studied, and it is known that several problems which are challenging on general graphs become easier on chordal graphs (see, e.g.,~\cite{Chandran2001,Okamoto2005,blair1993introduction}). While the problem of finding a largest chordal subgraph has been studied~\cite{bliznets2016largest}, to the best of our knowledge there are no known enumeration results.
We here aim at listing Maximal Induced Chordal Subgraphs of $G$.
The problem is hereditary, and chordality can be tested in $O(m)$ time~\cite{rose1976algorithmic}, thus $\textsc{complete}\xspace(\cdot)$ takes $O(mn)$ time.
A (sub)graph is chordal iff it allows a \textit{perfect elimination ordering} $\{v_1, \ldots, v_n\}$ of its vertices, i.e., such that for all $i$, $N(v_i) \cap \{v_{i+1},\ldots,v_{n}\}$ is a clique~\cite{DBLP:conf/cocoon/ConteKOUW17}.
We can obtain this by recursively removing simplicial vertices, i.e., vertices whose neighborhood in the residual graph is a clique.\footnote{To remove ambiguity, we can remove the lexicographically smallest when multiple simplicial vertices are present.}
As the neighbors of a simplicial vertex form a clique, we observe that removing a simplicial vertex cannot disconnect the residual graph.
It is also known that a chordal graph has $O(n)$ maximal cliques, and a vertex $v$ participates in $O(|N(v)|)$ maximal cliques~\cite{DBLP:conf/cocoon/ConteKOUW17}.
We use this to define the canonical order, which is then combined with Definition~\ref{def:proximity} to obtain the proximity function~$\ensuremath{\mathrel{\stackrel{\makebox[0pt]{\ensuremath{_\rightsquigarrow}}}{\cap}}}\xspace$.
\begin{definition}[Canonical Order for Maximal (Connected) Induced Chordal Subgraphs]\label{canon:chordal}
The canonical order $\{s_1, \ldots, s_{|S|}\}$ of $S$ is the \textit{reverse} of its perfect elimination ordering, i.e., such that $\{s_{|S|}, \ldots, s_{1}\}$ is the perfect elimination ordering.
\end{definition}
This way, the neighbors of $v$ that precede $v$ in the ordering form a clique.
Furthermore, when $S$ is a connected subgraph, any prefix $\{s_1, \ldots, s_{j\le |S|}\}$ of the canonical order induces a connected subgraph, because we can iteratively remove the last vertex, which is always simplicial. This means the canonical order satisfies condition~(\ref{item:canonical1}) of Definition~\ref{def:crecon}, in the case of both Maximal Induced Chordal Subgraphs and Maximal Connected Induced Chordal Subgraphs.
The neighboring function is defined as follows.
\begin{definition}[Neighboring function for Maximal (Connected) Induced\linebreak Chordal Subgraphs]\label{neighboring:chordal}~
We define $\textsc{neighbors}\xspace(S) = \bigcup\limits_{v\in V(G)\setminus S} \textsc{neighbors}\xspace(S,v)$.
\noindent For the non connected case we define
$\textsc{neighbors}\xspace(S,v)= \{\textsc{complete}\xspace(S \cup \{v\} \setminus ( N(v) \setminus Q )) :$
$Q \text{ is a maximal clique of } G[S \cup \{v\}] \text{ containing v} \}$
\noindent While for the connected case we define
$\textsc{neighbors}\xspace(S,v)=\{\textsc{complete}\xspace( \texttt{cc}\xspace_v(S \cup \{v\} \setminus ( N(v) \setminus Q )) ) :$
$Q \text{ is a maximal clique of } G[S \cup \{v\}] \text{ containing v} \}$
\end{definition}
Less formally, we add a vertex $v$ to $S$, then remove all its neighbors except one maximal clique $Q$ (meaning the \textit{removable} by definition of canonical reconstruction will be $N(v)\setminus Q$). In the connected case, we further remove vertices not in the connected component of $v$.
We can easily see that $S \cup \{v\} \setminus ( N(v) \setminus Q )$ is chordal, by showing a perfect elimination ordering: $v$ itself is simplicial as its neighbors form a clique, and can be removed; we can then complete the perfect elimination order as the remaining vertices form an induced subgraph of $S$, which is chordal as induced chordal subgraphs are hereditary.
We now need to prove the last condition; let $S$ and $S^*$ be two solutions, $S\ensuremath{\mathrel{\stackrel{\makebox[0pt]{\ensuremath{_\rightsquigarrow}}}{\cap}}}\xspace S^* = \{s^*_1, \ldots, s^*_h\}$ and $\ensuremath{\dot v}\xspace = s^*_{h+1}$ the earliest vertex in the canonical order of $S^*$ not in $S$.
By the canonical order, $N(\ensuremath{\dot v}\xspace) \cap (S\ensuremath{\mathrel{\stackrel{\makebox[0pt]{\ensuremath{_\rightsquigarrow}}}{\cap}}}\xspace S^*) = N(\ensuremath{\dot v}\xspace) \cap \{s^*_1, \ldots, s^*_h\}$ is a clique. When computing $\textsc{neighbors}\xspace(S,\ensuremath{\dot v}\xspace)$, as we try all maximal cliques, for some $Q$ we will have $N(\ensuremath{\dot v}\xspace) \cap (S\ensuremath{\mathrel{\stackrel{\makebox[0pt]{\ensuremath{_\rightsquigarrow}}}{\cap}}}\xspace S^*) \subseteq Q$. The resulting $S'$ will thus contain all neighbors of $\ensuremath{\dot v}\xspace$ in $S\ensuremath{\mathrel{\stackrel{\makebox[0pt]{\ensuremath{_\rightsquigarrow}}}{\cap}}}\xspace S^*$, and thus all of $S\ensuremath{\mathrel{\stackrel{\makebox[0pt]{\ensuremath{_\rightsquigarrow}}}{\cap}}}\xspace S^*$, plus \ensuremath{\dot v}\xspace, meaning that $|S'\ensuremath{\mathrel{\stackrel{\makebox[0pt]{\ensuremath{_\rightsquigarrow}}}{\cap}}}\xspace S^*|>|S\ensuremath{\mathrel{\stackrel{\makebox[0pt]{\ensuremath{_\rightsquigarrow}}}{\cap}}}\xspace S^*|$, which proves the correctness the $\textsc{neighbors}\xspace(\cdot)$ function.
Finally, $\textsc{neighbors}\xspace(\cdot)$ can indeed be computed in polynomial time.
We first need to list all maximal cliques containing $v$ in $G[S \cup \{v\}]$: these correspond exactly to the maximal cliques of $G[(S\cap N(v))\cup \{v\}]$; as $v$ is adjacent to \textit{all} vertices in $(S\cap N(v))$, we can further say that these correspond exactly to all maximal cliques of $G[(S\cap N(v))]$, to which we then add $v$ ($v$ can clearly be added to any clique of $G[(S\cap N(v))]$ since it is adjacent to all its vertices).
This correspondence is important, because $G[(S\cap N(v))]$ is an induced subgraph of $S$, and thus a chordal graph.
Recall now that a chordal graph has $O(n)$ cliques and they can be listed in $O(m)$ time (e.g., by computing a perfect elimination ordering~\cite{rose1976algorithmic}): $G[S\cap N(v)]$ has at most $|N(v)|$ vertices and $O(|N(v)|^2)$ edges, so we can list all the maximal cliques of $G[S\cap N(v)]$ --and thus all maximal cliques of $G[S \cup \{v\}]$ containing $v$-- in $O(|N(v)|^2)$ time, obtaining at most $|N(v)|$ maximal cliques.
The cumulative cost of listing all cliques for each $v\in V(G)\setminus S$ is thus bounded by $O(\sum\limits_{v\in V(G)\setminus S}|N(v)|^2) = O(mn)$ time, and the process yields $O(\sum\limits_{v\in V(G)\setminus S}|N(v)|) = O(m)$ maximal cliques. For each clique $Q$, we must further compute the corresponding $\textsc{complete}\xspace(\cdot)$ call: as the problem is hereditary, again we only need to test each vertex at most once for addition, and a chordality can be tested in $O(m)$ time, the cost $\mathcal{C}_t$ of a $\textsc{complete}\xspace(\cdot)$ is $O(mn)$ time (which dominates the time for checking membership in $\ensuremath{\mathcal{S}}\xspace$). Furthermore, the same bound applied to the connected case, as we simply need to consider vertices for addition only when they become adjacent to the current solution. Scanning the neighborhoods of the vertices that are added to the solution to find these candidates has an additional cost of $O(m)$ which does not affect the $O(mn)$ bound. The total cost will be $O(mn + m\cdot mn) = O(m^2n)$
We can thus state that:
\begin{theorem}
Maximal Induced Chordal Subgraphs and Maximal Connected Induced Chordal Subgraphs are proximity searchable, and can be listed with $O(m^2n)$ time delay.
\end{theorem}
\subsection{Maximal Edge-induced Chordal Subgraphs}
An algorithm for the edge version can be obtained by defining the canonical order for the edge-induced subgraph in the same way as for Bipartite Subgraphs, based on the canonical ordering of the vertices (see Definition~\ref{def:bip-non-canon}).
In this problem too, note how all Maximal Edge-induced Chordal Subgraphs of a connected graph are connected, as we can always add edges to a non-connected subgraph without creating cycles, so we do not need to separately consider the connected and non-connected case.
We can then devise a neighboring function $\textsc{neighbors}\xspace(S,(x,y))$ like the first one in Definition~\ref{neighboring:chordal}, where we use an edge $(x,y)$ as canonical extender.
When adding an edge $(x,y)$ to a maximal solution $S$, we try as $Q$ all maximal cliques containing either $x$ or $y$ in $G[S]$.
In any $S^*$ (for which $(x,y)$ is the canonical extender) one between $x$ and $y$ will occur later in the canonical ordering; wlog, let us say $y$.
As the canonical ordering is based on a reversed perfect elimination ordering, the neighbors of $y$ preceding $y$ in the canonical order of $S^*$ form a clique (including $x$ as well). Thus the neighboring funciton will eventually consider a clique $Q$ containing $y$ and all its preceding neighbors, and when this happens the proximity with $S^*$ is extended.
The number of neighboring solutions generated this way will be $O(\sum_{(x,y)\in E(G)}|N(x)|+N|(y)|) = O(mn)$.
The only further requirement is a polynomial time $\textsc{complete}\xspace(\cdot)$ function which needs to be applied to each neighboring solution: this follows from~\cite{HEGGERNES20091}, who prove that edge-induced chordal subgraphs are \textit{sandwich monotone}. In other words, if a edge-induced chordal subgraph $S\subseteq E(G)$ is \textit{not} maximal, then there is always a single edge $e\in E(G) \setminus S$ such that $S\cup \{e\}$ is a chordal subgraph.
This means $\textsc{complete}\xspace(\cdot)$ can be computed in a greedy way by testing, up to $m$ times, that any of the $O(m)$ remaining edges in the graph can be added, which takes $O(m)$ time, for a total cost $\mathcal{C}_t = O(m^3)$. The total cost follows.
\begin{theorem}
Maximal Edge-induced Chordal Subgraphs are proximity searchable, and can be listed with $O(m^4n)$ time delay.
\end{theorem}
\section{Maximal Induced Proper Interval Subgraphs}\label{sec:interval-exp}
Interval graphs are a well-known subclass of chordal graphs, whose vertices can be arranged as intervals on a line such that two vertices are adjacent if and only if their intervals intersect. In this section, we present a polynomial-delay enumeration algorithm for Maximal \textit{Proper} Interval Subgraphs, a subclass of interval subgraphs corresponding to interval graphs where no two intervals properly contain another.
Despite chordal graphs, interval graphs, and proper interval graphs being closely related to each other, it is interesting to observe how the three enumeration algorithms proposed here (chordal subgraphs, proper interval subgraphs) and in~\cite{Cao:arXiv:2020} (interval subgraphs) differ significantly.
Furthermore, an interesting open question would be to determine whether it is possible to enumerate Maximal Interval Subgraphs directly via proximity search, or whether there is an intrinsic difference in what can be achieved with retaliation-free paths.
\subsection{Maximal Connected Induced Proper Interval Subgraphs}
A \textit{proper} interval graph is an interval graph where, in the interval representation, no interval properly contains another. Equivalently, it can be defined as interval graphs that admit a \textit{unit-length} representation, i.e., where all intervals have length $1$~\cite{fulkerson1965incidence}. In this section we will adopt this latter definition, and all interval representations considered will be intended as unit-length.
In the following, we show how to enumerate Maximal (Connected) Proper Interval Subgraphs of a graph $G$. We show the connected version of the problem, and later remark how to adapt it to the non-connected case.
It is important to observe that every connected proper interval (sub)graph $S$ has two unique interval representation represented by a sequence $v_1, \ldots, v_{|S|}$, and its reverse.\footnote{Ambiguity may be caused by identical vertices, i.e., adjacent and with the same sets of neighbors, but it can be resolved by taking the smallest-id vertex first.} The \textit{canonical order} $v_1, \ldots, v_{|S|}$ of a Maximal Proper Interval Subgraph $S$ is defined as the sequence given by the interval representation of $S$ which has as $s_1$ the smallest among the two possible values.
A graphical example is given in Figure~\ref{fig:pinterval} (a),(b),(d),(e).
\begin{figure*}
\centering
\includegraphics[width=\textwidth]{img/pinterval.pdf}
\put (-505,145){\small(a)}
\put (-320,145){\small(b)}
\put (-140,145){\small(c)}
\put (-505,60){\small(d)}
\put (-320,60){\small(e)}
\put (-140,60){\small(f)}
%
%
\caption{\textbf{a}: a proper interval subgraph $S$ of the graph in Figure~\ref{fig:ex:bip}. \textbf{d}: another proper interval subgraph $T$. \textbf{b}: the unit interval representation of $S$, which induces the canonical order $1,2,4,3,8,7,9,12,11$. \textbf{e}: the representation of $T$, which induces the canonical order $1,2,4,5,11,12,9,10,8$.}
\label{fig:pinterval}
\end{figure*}
In order to generate neighboring solutions, we take each of the two representations, and for each we proceed as follows. Firstly, we try all vertices $v\in V(G)\setminus S$ as canonical extender.
Next, we want to identify the right way of inserting $v$ in the interval representation of $S$.
As the interval length is fixed to 1, there are $5|S|$ possibilities: for any other $w \in S$, we can place $v$ ending just before / just after the start of $w$, exactly overlapping $w$, or starting just before / just after the end of $w$. We can observe how these capture all possible ways to insert $v$, since given any other placement we can slide $v$ in any direction until it is about to gain or lose an overlap, i.e., one of the endpoints of $v$ will approach the endpoint of another interval, which puts us into one of the $5|S|$ cases above.
Finally, we have to make the representation consistent, by removing from $S\cup \{v\}$:
\begin{itemize}
\item All vertices (intervals) coming \emph{after} $v$ in this representation.
\item All neighbors of $v$ that do \emph{not} overlap its interval.
\item All non-neighbors of $v$ that do overlap its interval.
\item Finally, all vertices that have become disconnected from the connected component containing $v$ by performing the previous steps.
\end{itemize}
The resulting representation is consistent with a proper interval graph, so the resulting graph is clearly a proper interval graph. Let us call it $X$, and let us call $S' = \textsc{complete}\xspace(\texttt{cc}\xspace_{v}(X))$.
The following example illustrates these operations on the example graphs from Figure~\ref{fig:pinterval}.
\begin{example*}
Looking at the graphs in Figure~\ref{fig:pinterval} (\textbf{(a)}, \textbf{(d)}) and their canonical orderings (see \textbf{(b)},\textbf{(e)}) we can give an example of the operations performed by the neighboring function.
\textbf{(c)}: $5$ is added as canonical extender and placed ``just before'' $3$; we remove vertices the dashed vertices $3$ and $8$ (overlapping $5$ but not neighbors of $5$ in $G$), and $11$ and $12$ (not overlapping $5$ but neighbors of $5$ in $G$); $9$ and $7$ are also removed as not part of the same connected component as $5$. The resulting graph \textbf{(f)} is then is maximalized with $\textsc{complete}\xspace(\cdot)$, and has greater proximity with $b$ than $a$.
Indeed, $S\ensuremath{\mathrel{\stackrel{\makebox[0pt]{\ensuremath{_\rightsquigarrow}}}{\cap}}}\xspace T = \{1,2,3,4\}$, while the maximal solution obtained maximalizing \textbf{(f)} contains $1,2,3,4,5$.
\end{example*}
The set $\textsc{neighbors}\xspace(S)$ will be made of the $S'$ obtained by trying both representations of $S$, for each all $v\in V(G)\setminus S$, and for each all possible insertions, for a total of $O(n^2)$ neighboring solutions.
We now show that, given $S,T$, we always obtain some $S'\in \textsc{neighbors}\xspace(S)$ such that $|S'\ensuremath{\mathrel{\stackrel{\makebox[0pt]{\ensuremath{_\rightsquigarrow}}}{\cap}}}\xspace T| > |S\ensuremath{\mathrel{\stackrel{\makebox[0pt]{\ensuremath{_\rightsquigarrow}}}{\cap}}}\xspace T|$.
Let the proximity $S\ensuremath{\mathrel{\stackrel{\makebox[0pt]{\ensuremath{_\rightsquigarrow}}}{\cap}}}\xspace T$ be $t_1, \ldots, t_{i-1}$: this is a prefix of $T$, and since it is a connected subgraph, its vertices in the same order as in $T$ in one of the two representations of $S$. Furthermore, let $v$ be our canonical extender.
Looking at the canonical order of $T$, consider the placement of $t_i$ relative to the preceding intervals $t_1, \ldots, t_{i-1}$. Now, consider the case where $v = t_i$, the correct one between the two representations of $S$ is considered, and the placement of $v$ relative to $t_1, \ldots, t_{i-1}$ is the same as $t_i$ in $T$.
As we try all possible placements for $v = t_i$, and as $t_1, \ldots, t_{i-1}$ is a connected subgraph of $T$,
we will also try the placement of $t_i$ considered above.
The correct placement of $v$ tells us that, when we remove all intervals coming \emph{after} $v$ from $S\cup \{v\}$, we do not remove vertices from $t_1, \ldots, t_{i-1}$. The placement also tells us that all neighbors of $v$ in $t_1, \ldots, t_{i-1}$ overlap with $v$ in the interval representation, while all non-neighbors of $v$ in $t_1, \ldots, t_{i-1}$ do not, so vertices of $t_1, \ldots, t_{i-1}$ are not removed in the remaining steps.
It follows that the set $X$ obtained is a proper interval subgraph of $G$ containing $t_1, \ldots, t_{i}$, and that $S' = \textsc{complete}\xspace(X)$ is such that $|S'\ensuremath{\mathrel{\stackrel{\makebox[0pt]{\ensuremath{_\rightsquigarrow}}}{\cap}}}\xspace T| > |S\ensuremath{\mathrel{\stackrel{\makebox[0pt]{\ensuremath{_\rightsquigarrow}}}{\cap}}}\xspace T|$.
\subsection{Induced Proper Interval Subgraphs}
The non-connected version is similarly solved: as in previous sections, we define the canonical order by ordering each connected component as in the connected case, then ordering the components by smallest-id vertex.
The proximity is then defined by canonical reconstruction (Definition~\ref{def:proximity}).
Let $S_1,\ldots, S_j$ be the connected components of $S$, and $T_1, \ldots, T_k$ those of $T$, for some $T$.
We will have that $S\ensuremath{\mathrel{\stackrel{\makebox[0pt]{\ensuremath{_\rightsquigarrow}}}{\cap}}}\xspace T$ consists of some complete connected components of $T$, as well as a (possibly empty) subset of one component $T_i$. The canonical extender $t_i$ will be the earliest vertex of the canonical ordering of $T_i$ that is not in $S$.
We can also observe that each of these components is contained itself in some connected component of $S$.
When we select a canonical extender $v$, we also select a component $S_i$ (of course, trying all possibilities): we aim to find the component which contains the partially formed $T_i$.
As we add $v$ to $S$, we can immediately remove from $S$ all neighbors of $v$ in $S\setminus S_i$, as indeed as $t_i$ only has neighbors in $T_i$.
We then proceed as in the connected case, ordering $S_i$ in the two possible ways, and trying all $O(n)$ insert possibilities.
However, we must take care of the fact that $S_i$ may contain more than one connected component of $T$: components \textit{preceding} $v$ in the order are preserved (they cannot overlap $v$ as they precede the other elements of $T_i$, to which they are not adjacent), but there may be some \textit{following} $v$.
For this reason, we must introduce another ``guess'', that is we guess which of the intervals preceding $v$ is $t_{i-1}$ (as, to obtain the correct placement of $v$ in the interval representation, we may not have placed it just after $t_{i-1}$). Note how the number of possible $t_{i-1}$ is at most $|N(v)|$.
This step, which was not necessary in the connected version, allows us to effectively compute a safe way of detaching the intervals following $v$.
Specifically, we remove from $S_i \cup \{v\}$ the following vertices:
\begin{itemize}
\item All vertices (intervals) between $t_{i-1}$ and $v$.
\item All vertices (intervals) coming \emph{after} $v$ in this representation that are adjacent to $v$ or to $t_{i-1}$.
\item All neighbors of $v$ that do \emph{not} overlap its interval.
\item All non-neighbors of $v$ that do overlap its interval.
\end{itemize}
In this way, all the vertices removed could not be part of the proximity $S\ensuremath{\mathrel{\stackrel{\makebox[0pt]{\ensuremath{_\rightsquigarrow}}}{\cap}}}\xspace T$: the ones between $t_{i-1}$ and $t_i$ cannot be in the interval representation since $t_{i-1}$ and $t_i$ are consecutive in the interval representation of $T$; the others we remove were neighbors of $t_{i-1}$ or $t_i$, but did not precede them in the interval representation (i.e., in the canonical order), so they could not be part of $S\ensuremath{\mathrel{\stackrel{\makebox[0pt]{\ensuremath{_\rightsquigarrow}}}{\cap}}}\xspace T$. On the other hand, the remaining intervals following $v$ are now in a separate connected component, thus the interval representation of the resulting graph is consistent, contains $t_1, \ldots, t_i$ (for any $T$, when the correct choices are performed), and we can apply the $\textsc{complete}\xspace(\cdot)$ function to obtain a neighboring solution $S'$.
\subsection{Running time}
As for the complexity, the neighboring function tries $O(n)$ candidates for $v$, and for each, $5|S| = O(n)$ possible placements in $S$, in each of the $2$ representations.
For each, the cost of the procedure is dominated by the application of the $\textsc{complete}\xspace(X)$ call.
Since we can test whether a graph is a proper interval graph in $O(m)$ time~\cite{booth1975linear}, the $\textsc{complete}\xspace(X)$ function can be implemented in $O(mn)$ time as for chordal subgraphs, giving us a total cost per solution of $O(n^3m)$ time.
For the non-connected version, we must consider two additional phases: for each $v$, we firstly selected a connected component $S_i$ of $S$, and secondly we selected the possible $t_{i-1}$ among the neighbors of $v$. We can bound the number of connected components by $O(n)$, and, rather than adding another factor $O(n)$ for the choice of $t_{i-1}$, we can observe how each distinct pair $v,t_{i-1}$ corresponds to an edge, so the number of possible $v$ and $t_{i-1}$ pairs is $O(m)$, for a total complexity of $O(n^3m^2)$
\begin{theorem}
Maximal Induced Proper Interval Subgraphs and Maximal Connected Induced Proper Interval Subgraphs are proximity searchable, and can be listed with $O(n^3m^2)$ time delay and $O(n^3m)$ time delay, respectively.
\end{theorem}
\section{Maximal Obstacle-free Convex Hulls}
\newcommand{\phi\xspace}{\phi\xspace}
\newcommand{\textsc{moc}\xspace}{\textsc{moc}\xspace}
\newcommand{\textsc{moc}s\xspace}{\textsc{moc}s\xspace}
In application domains such as robotics planning and routing, a common problem is finding areas, typically convex, in a given environment which are free from obstacles (see, e.g.,~\cite{Deits2015,savin2017algorithm}).
In this section we solve the following formulation of the problem: let $V$ and $X$ be two sets of elements, which corresponds to points on a 2-dimensional plane. $V$ represents the point of interest for our application, and $X$ represents the obstacles. For short, let $|V|=j$ and $|X|=h$, and let $n = j+h$ be the total number of points. We are interested in listing all maximal obstacle-free convex hulls (\textsc{moc}s\xspace for short), where an obstacle-free convex hull is a set of elements $S\subseteq V$ such that the convex hull of $S$ does not contain any element of $X$.
This problem does not concern a graph, but its solutions are modeled as sets of elements, thus the technique may still be applied. Furthermore, we can naturally generalize the problem by adding a graph structure to $V$, i.e., adding edges between its points, and considering the problem of Maximal \textit{Connected} Obstacle-free Convex hulls.
\subsection{Maximal Obstacle-free Convex Hulls}
Again, note that the problem is hereditary, i.e., each subset $S'$ of a solution $S$ clearly also admits a convex hull which does not include elements of $X$ (since it will be contained in that of $S$).
It is worth observing that this is the only problem in this paper to which we do \textit{not} apply the canonical reconstruction strategy.
Consider a maximal solution $S$ and an element $v\in V\setminus S$. As $S$ is maximal, there is at least one element $x\in X$ included in the convex hull of $S\cup \{v\}$. This element $x$ casts two ``shadows'' $S_1$ and $S_2$ on $S$, seen by $v$: consider the straight line between $v$ and $x$, $S_1$ consists of all elements of $S$ above this line, and $S_2$ of all those below it. It is straightforward to see how both the convex hull of $S_1\cup \{v\}$ and that of $S_2\cup \{v\}$ do not contain $x$. Any element of $S$ that falls exactly on the line may not participate in any solution involving $v$.\footnote{Note that it may not fall between $v$ and $x$ otherwise the convex hull of $S$ would have included $x$.} Furthermore, any element $x'\in X$ above this line, and still in the convex hull of $S\cup \{v\}$, further casts two shadows on $S_1$, as any element below this line casts them on $S_2$. If we repeat this process for all elements of $X$ in the convex hull of $S\cup \{v\}$ we obtain a number of shadows of $S$ which is at most linear in the number of elements of $X$. Let $\phi\xspace(S,v)$ be the set of these shadows.
For each of these shadows $S_i\in \phi\xspace(S,v)$, we have that the convex hull of $S_i\cup \{v\}$ may not include elements of $X$, i.e., $S_i\cup \{v\}$ is a (possibly not maximal) solution.
The neighboring function is then obtained as follows.
\begin{definition}[Neighboring function for \textsc{moc}s\xspace]~
$$\textsc{neighbors}\xspace(S) = \bigcup\limits_{v\in V(G)\setminus S} \textsc{neighbors}\xspace(S,v)$$
Where
$$\textsc{neighbors}\xspace(S,v) = \{\textsc{complete}\xspace(S_i \cup \{v\}) : S_i \in \phi\xspace(S,v)\}$$
\end{definition}
Finally, for two solutions $S$ and $S^*$, we simply define $S \ensuremath{\mathrel{\stackrel{\makebox[0pt]{\ensuremath{_\rightsquigarrow}}}{\cap}}}\xspace S^*$ as the intersection $S\cap S^*$ between their elements.
Let $I = S\cap S^* = S\ensuremath{\mathrel{\stackrel{\makebox[0pt]{\ensuremath{_\rightsquigarrow}}}{\cap}}}\xspace S^*$, and $v$ any element in $S^*\setminus S$. Since $I\cup \{v\}$ is contained in a \textsc{moc}\xspace, $S^*$, its convex hull cannot contain any element of $X$. It follows that $I$ must be fully contained in a single $S_i \in \phi\xspace(S,v)$: indeed, if we take two points $u_i\in S_i$ and $u_j\in S_j$, it is evident by the definition of $\phi\xspace(S,v)$ that the convex hull of $\{v,u_i,u_j\}$ (or any superset of it) contains at least an element of $X$.
We have that the neighboring function will return $S'= \textsc{complete}\xspace(S_i\cup \{v\})$, with $I\cup\{v\} \subseteq S'$, which implies $|S'\ensuremath{\mathrel{\stackrel{\makebox[0pt]{\ensuremath{_\rightsquigarrow}}}{\cap}}}\xspace S^*| > |S\ensuremath{\mathrel{\stackrel{\makebox[0pt]{\ensuremath{_\rightsquigarrow}}}{\cap}}}\xspace S^*|$. The algorithm is thus correct.
As for the complexity, the problem is hereditary, so we may compute a $\textsc{complete}\xspace(S)$ call by testing each vertex in $V\setminus S$ once. The convex hull of $S$ can be computed in $O(|S|\log |S|)$ time~\cite{Chan1996}, and testing a solution consists in checking that each vertex of $X$ is not in this hull, which can trivially be done in $O(|S|\cdot h)$ time. The cost of $\textsc{complete}\xspace(\cdot)$ is thus $O(j(h+\log j))$ time.
For each candidate $v$, we have at most $h$ neighboring solutions, and since we need to consider at most $j$ candidates, the delay of the algorithm will be $j\cdot h$ times the cost of a $\textsc{complete}\xspace(\cdot)$ call.
We thus obtain an algorithm with the following complexity:
\begin{theorem}
Maximal Obstacle-free Convex Hulls are proximity searchable, and can be listed in $O(j^2h(h+\log j)) = O(n^4)$ time delay.
\end{theorem}
It could be argued that the neighboring function actually reports all solutions of the input-restricted problem in this instance, allowing us to induce a parent-child relationship with the structure of~\cite{lawler1980generating,Cohen20081147}, and reducing the space usage to $O(n)$ by using stateless iteration~\cite{DBLP:conf/icalp/ConteGMV16}. However, it is worth observing that proximity search required proving a weaker statement, and allows for an arguably simpler proof.
\subsection{Maximal Connected Obstacle-free Convex Hulls}
We now consider an extension of the problem where on top of $V$ and $X$ we have a graph structure $G = (V,E)$ on the points of $V$, and we are interested in listing all maximal set of points $S\subseteq V$ such that the convex hull of $S$ is obstacle-free, and $G[S]$ is connected.
We consider this a natural extension as, in the applications mentioned above, it could model requirements on the structure of the obstacle-free areas identified.
The algorithm is remarkably similar to the above version, as the neighboring function still considers $S_i \cup \{v\}$ for all $S_i\in \phi\xspace(S,v)$, but only keeps the connected component of $G[S_i \cup \{v\}]$ containing $v$.
\begin{definition}[Neighboring function for \textsc{moc}s\xspace]~
$$\textsc{neighbors}\xspace(S) = \bigcup\limits_{v\in V(G)\setminus S} \textsc{neighbors}\xspace(S,v)$$
Where
$$\textsc{neighbors}\xspace(S,v) = \{ \texttt{cc}\xspace_v(\textsc{complete}\xspace(S_i \cup \{v\}) ) : S_i \in \phi\xspace(S,v)\}$$
\end{definition}
For two solutions $S$ and $S^*$, we define $S \ensuremath{\mathrel{\stackrel{\makebox[0pt]{\ensuremath{_\rightsquigarrow}}}{\cap}}}\xspace S^*$ as the largest connected component of their intersection $S\cap S^*$. We now prove that there is $S'\in \textsc{neighbors}\xspace(S)$ such that $|S'\ensuremath{\mathrel{\stackrel{\makebox[0pt]{\ensuremath{_\rightsquigarrow}}}{\cap}}}\xspace S^*| > |S\ensuremath{\mathrel{\stackrel{\makebox[0pt]{\ensuremath{_\rightsquigarrow}}}{\cap}}}\xspace S^*|$.
Let $I = S\cap S^* = S\ensuremath{\mathrel{\stackrel{\makebox[0pt]{\ensuremath{_\rightsquigarrow}}}{\cap}}}\xspace S^*$, and $v$ any element in $S^*\setminus S$ such that $G[I\cup \{v\}]$ is connected.
Note that a suitable $v$ must exist, otherwise $I$ would not be connected to the elements of $S^*\setminus I$, contradicting the fact that $S^*$ is a connected solution.
Since $I\cup \{v\}$ is contained in $S^*$, its convex hull cannot contain any element of $X$. It follows that $I$ must be fully contained in a single $S_i \in \phi\xspace(S,v)$.
Furthermore, as $I\cup \{v\}$ is connected, it must be contained in $\texttt{cc}\xspace_v(S_i \cup\{v\})$, the connected component of $G[S_i \cup\{v\}]$ containing $v$.
Similarly to the above case, we have that the neighboring function will return $S'= \textsc{complete}\xspace(\texttt{cc}\xspace_v(S_i\cup \{v\}))$, with $I\cup\{v\} \subseteq S'$, which implies $|S'\ensuremath{\mathrel{\stackrel{\makebox[0pt]{\ensuremath{_\rightsquigarrow}}}{\cap}}}\xspace S^*| > |S\ensuremath{\mathrel{\stackrel{\makebox[0pt]{\ensuremath{_\rightsquigarrow}}}{\cap}}}\xspace S^*|$. The algorithm is thus correct.
The complexity can be also derived from the non-connected case: the only additional step is applying $\texttt{cc}\xspace_v(\cdot)$ before the $\textsc{complete}\xspace(\cdot)$ function. As $\texttt{cc}\xspace_v(\cdot)$ takes $O(m)$ time, where $m = |E(G)| = O(n^2)$, we can conclude the following:
\begin{theorem}
Maximal Connected Obstacle-free Convex Hulls are proximity\linebreak searchable, and can be listed in $O(jh (m + j(h+\log j))) = O(n^4)$ time delay.
\end{theorem}
\section{Maximal Connected Directed Acyclic Subgraphs}\label{sec:list:last}
\newcommand{\textsc{mcais}\xspace}{\textsc{mcais}\xspace}
In this section we consider a \textit{directed} graph, where each edge has a head and a tail, and its direction is from the tail to the head. We call $N^+(v)$ the \textit{out-neighbors} of the vertex $v$ and $N^-(v)$ its \textit{in-neighbors}.
The goal of this section is listing Maximal Induced Connected Acyclic Subgraphs (\textsc{mcais}\xspace hereafter) of a given directed graph $G$.
The problem is connected-hereditary,
and acyclicity can be tested in $O(m)$ time, thus $\textsc{complete}\xspace(\cdot)$ can be implemented in $O(mn)$ time.
For completeness, we remark that the non-connected version (Maximal Induced Directed Acyclic Subgraphs), corresponds to listing the complements of Minimal Feedback Vertex Sets in a directed graph, and is of no interest here as an output-polynomial algorithm is given in~\cite{schwikowski2002enumerating}. We thus address the connected version of the problem, which has no natural counterpart in terms of feedback vertex set.
Let us define the canonical order:
\begin{definition}[Canonical Order for Maximal Connected Induced Acyclic Subgraphs]\label{def:mcaisorder}
The canonical order of a \textsc{mcais}\xspace $S$ is the order $\{s_1, \ldots, s_{|S|}\}$ such that, for each $s_i$, $\{s_1, \ldots, s_{i}\}$ is connected, and either $\{s_1, \ldots, s_{i-1}\}\cap N^+(s_i) = \emptyset$ or $\{s_1, \ldots, s_{i-1}\}\cap N^-(s_i) = \emptyset$.
If multiple orders are possible let it be the lexicographically minimum.
\end{definition}
Our algorithm does not need to compute this order or $\ensuremath{\mathrel{\stackrel{\makebox[0pt]{\ensuremath{_\rightsquigarrow}}}{\cap}}}\xspace$, but we need to show that it always exists.
Firstly, recall that every acyclic graph has at least one source and one target, and let us observe an important property of acyclic graphs with a single source (whose proof trivially follows from the fact that any non-source vertex has a neighbor occurring before itself in the order):
\begin{lemma}\label{lem:topcon}
Let $G$ be a single-source acyclic connected graph, and $v_1, \ldots, v_n$ any topological order of $G$. Any prefix $v_1, \ldots, v_i$ of this order induces a connected subgraph.
\end{lemma}
Lemma~\ref{lem:topcon} also implies that the \textit{reversed} topological order (i.e., where vertices have no forward out-neighbors) of a single-target acyclic connected graph is such that every prefix induces a connected subgraph. We also remark that both these orders satisfy the intersection properties of Definition~\ref{def:mcaisorder}.
We now use this lemma to show that the defined canonical order exists for any \textsc{mcais}\xspace.
In the following, we define \textit{collapsing} a set of vertices $A \subseteq S$ into $x$ as replacing them with a single vertex $x$, whose in- and out-neighbors correspond to all vertices in $S\setminus A$ that were respectively in- and out-neighbors of some vertex in $A$.
\begin{lemma}
Every Directed Acyclic Graph allows a canonical order by Definition~\ref{def:mcaisorder}.
\end{lemma}
\begin{proof}
Let $S$ be a Directed Acyclic Graph. Let $v_1$ be a source of $S$, and $S_1$ be the set of vertices reachable by $v_1$, including $v_1$.
Let $s_{1,1}, \ldots, s_{1,|S_1|}$ a topological ordering of $S_1$.
No vertex in $S_1$ can have an out-neighbor outside of $S_1$ as otherwise said vertex would be in $S_1$ itself. Let instead $S_2$ be the set of all vertices in $S\setminus S_1$ that can reach some vertex of $S_1$.
If we collapse $S_1$ into a vertex $x$, we can observe that $S_2 \cup \{x\}$ is acyclic subgraph with $x$ being the only target. Let $x, s_{2,1}, \ldots, s_{2,|S_2|}$ be a reverse topological ordering of $S_2 \cup \{x\}$.
If we replace $x$ with the previously computed order of $S_1$, we obtain an order $s_{1,1}, \ldots, s_{1,|S_1|}, s_{2,1}, \ldots, s_{2,|S_2|}\}$ which respects Definition~\ref{def:mcaisorder}:
Each vertex in $s_{1,1}, \ldots, s_{1,|S_1|}$ has no backward out-neighbor by the topological ordering of $S_1$; each $s_{2,1}, \ldots, s_{2,|S_2|}$ has no backward in-neighbor by the reverse topological ordering of $S_2$, and because vertices of $S_1$ can not have out-neighbors outside $S_1$; finally, every prefix of $s_{1,1}, \ldots, s_{1,|S_1|}, s_{2,1}, \ldots, s_{2,j}$ is connected, as $x, s_{2,1}, \ldots, s_{2,j}$ is connected, meaning that all vertices in $s_{2,1}, \ldots, s_{2,j}$ are connected to some vertex in $S_1$, that is itself connected.
We may now repeat this step by collapsing $S_1 \cup S_2$ into a vertex $x'$, and since $x'$ will be a source, take $S_3$ as all vertices reached by $x'$ in $S\setminus (S_1 \cup S_2)$, and take a topological order of $S_3 \cup \{x'\}$, which we append to the order obtained so far (excluding $x'$).
By iterating steps, we obtain an ordering $s_{1,1}, \ldots, s_{1,|S_1|}, s_{2,1}, \ldots, s_{2,|S_2|},$\linebreak $s_{3,1}, \ldots, s_{3,|S_3|} \ldots, s_{k,1}, \ldots, s_{k,|S_k|}$, with $k\le |S|$, that contains all vertices of $S$, and such that any prefix will induce a connected subgraph, and any $s_{i,j}$ will have no backward out-neighbors if $i$ is odd, and no backward in-neighbors if $i$ is even, thus there exist an ordering satisfying Definition~\ref{def:mcaisorder} (if a feasible order exists, a lexicographically minimum one must exist too).
\end{proof}
Finally, the proximity $\ensuremath{\mathrel{\stackrel{\makebox[0pt]{\ensuremath{_\rightsquigarrow}}}{\cap}}}\xspace$ follows by Definition~\ref{def:proximity}.
We define the neighboring function as follows.
\begin{definition}[Neighboring Function for Maximal Connected Induced Acyclic Subgraphs]
For a solution $S$ and a vertex $v \in V(G)\setminus S$, we define
$$\textsc{neighbors}\xspace(S) = \bigcup\limits_{v\in V(G)\setminus S} \textsc{neighbors}\xspace(S,v)$$
Where
$\textsc{neighbors}\xspace(S,v) =\{ \textsc{complete}\xspace( \texttt{cc}\xspace_v(\{v\} \cup S\setminus N^+(v)) ), \textsc{complete}\xspace( \texttt{cc}\xspace_v(\{v\} \cup S\setminus N^-(v)) )\}$
\end{definition}
In other words, the function will add $v$ to $S$. $S\cup \{v\}$ is not acyclic, but all cycles must involve $v$, so we make it acyclic by removing either all the out-neighbors $N^+(v)$, which makes $v$ a target, or all its in-neighbors $N^-(v)$, which makes $v$ a source.
It then takes the connected component containing $v$ and feeds the result to $\textsc{complete}\xspace(\cdot)$, to surely obtain a \textsc{mcais}\xspace.
Consider now two solutions $S$ and $S^*$, and again let $\ensuremath{\dot v}\xspace$ be the first vertex in the canonical order of $S^*$ which is not in $S\ensuremath{\mathrel{\stackrel{\makebox[0pt]{\ensuremath{_\rightsquigarrow}}}{\cap}}}\xspace S^*$. More formally, let $S\ensuremath{\mathrel{\stackrel{\makebox[0pt]{\ensuremath{_\rightsquigarrow}}}{\cap}}}\xspace S^* = \{s^*_1, \ldots, s^*_h\}$ and $\ensuremath{\dot v}\xspace = s^*_{h+1}$.
Let $S' = \textsc{complete}\xspace( \texttt{cc}\xspace_{\ensuremath{\dot v}\xspace}(\{\ensuremath{\dot v}\xspace\} \cup S\setminus N^+(\ensuremath{\dot v}\xspace)) )$ and $S'' = \textsc{complete}\xspace( \texttt{cc}\xspace_{\ensuremath{\dot v}\xspace}(\{\ensuremath{\dot v}\xspace\} \cup S\setminus N^-(\ensuremath{\dot v}\xspace)))$ be the two solutions generated by $\textsc{neighbors}\xspace(S,\ensuremath{\dot v}\xspace)$.
By the canonical order of $S^*$, we have that $(S\ensuremath{\mathrel{\stackrel{\makebox[0pt]{\ensuremath{_\rightsquigarrow}}}{\cap}}}\xspace S^*)\cup \{\ensuremath{\dot v}\xspace\}$ is connected, and either $(S\ensuremath{\mathrel{\stackrel{\makebox[0pt]{\ensuremath{_\rightsquigarrow}}}{\cap}}}\xspace S^*) \cap N^+(\ensuremath{\dot v}\xspace) = \emptyset$ or $(S\ensuremath{\mathrel{\stackrel{\makebox[0pt]{\ensuremath{_\rightsquigarrow}}}{\cap}}}\xspace S^*) \cap N^-(\ensuremath{\dot v}\xspace) = \emptyset$.
It follows that if $(S\ensuremath{\mathrel{\stackrel{\makebox[0pt]{\ensuremath{_\rightsquigarrow}}}{\cap}}}\xspace S^*) \cap N^+(\ensuremath{\dot v}\xspace) = \emptyset$, then $(S\ensuremath{\mathrel{\stackrel{\makebox[0pt]{\ensuremath{_\rightsquigarrow}}}{\cap}}}\xspace S^*)\cup \{\ensuremath{\dot v}\xspace\} \subseteq \texttt{cc}\xspace_{\ensuremath{\dot v}\xspace}(\{\ensuremath{\dot v}\xspace\} \cup S\setminus N^+(\ensuremath{\dot v}\xspace)) \subseteq S'$, and otherwise we have $(S\ensuremath{\mathrel{\stackrel{\makebox[0pt]{\ensuremath{_\rightsquigarrow}}}{\cap}}}\xspace S^*) \cap N^-(\ensuremath{\dot v}\xspace) = \emptyset$, which means $(S\ensuremath{\mathrel{\stackrel{\makebox[0pt]{\ensuremath{_\rightsquigarrow}}}{\cap}}}\xspace S^*)\cup \{\ensuremath{\dot v}\xspace\} \subseteq \texttt{cc}\xspace_{\ensuremath{\dot v}\xspace}(\{\ensuremath{\dot v}\xspace\} \cup S\setminus N^-(\ensuremath{\dot v}\xspace))\subseteq S''$.
We thus have that either $|S'\ensuremath{\mathrel{\stackrel{\makebox[0pt]{\ensuremath{_\rightsquigarrow}}}{\cap}}}\xspace S^*| > |S\ensuremath{\mathrel{\stackrel{\makebox[0pt]{\ensuremath{_\rightsquigarrow}}}{\cap}}}\xspace S^*|$ or $|S''\ensuremath{\mathrel{\stackrel{\makebox[0pt]{\ensuremath{_\rightsquigarrow}}}{\cap}}}\xspace S^*| > |S\ensuremath{\mathrel{\stackrel{\makebox[0pt]{\ensuremath{_\rightsquigarrow}}}{\cap}}}\xspace S^*|$, which gives us the second necessary condition of proximity search.
Finally, it is straightforward to see that $\textsc{neighbors}\xspace(S)$ takes polynomial time, as its cost is bounded by $O(n)$ calls to $\textsc{complete}\xspace(\cdot)$, which can be implemented in $O(mn)$, meaning that all conditions of Definition~\ref{def:searchable} are satisfied. Theorem~\ref{thm:mcais} follows.
\begin{theorem}\label{thm:mcais}
Maximal Connected Induced Directed Acyclic Subgraphs are proximity searchable, and can be listed $O(mn^2)$ time delay.
\end{theorem}
\subsection{Maximal Connected Edge-induced Directed Acyclic Subgraphs}
\newcommand{\textsc{mcaes}\xspace}{\textsc{mcaes}\xspace}
We remark here that the structure can be adapted to the edge case, i.e., Maximal Connected Edge-induced Directed Acyclic Subgraphs (\textsc{mcaes}\xspace).
As the problem is still hereditary and acyclic subgraphs can be tested in linear time, we can implement the $\textsc{complete}\xspace(\cdot)$ function in $O(m^2)$ time. The canonical order is as follows.
\begin{definition}[Canonical order for \textsc{mcaes}\xspace]
Given a \textsc{mcaes}\xspace $S$, let the canonical ordering of the \textit{vertices} of $G[S]$ according to Definition~\ref{def:mcaisorder} be $v_1, \ldots, v_{|V[S]|}$.
The canonical ordering of $S$ is obtained by selecting the edges of $S$ by increasing order with respect to their \textit{later} endpoint in the vertex order, and breaking ties by increasing order of the other (earlier) endpoint.
\end{definition}
We obtain a canonical ordering $e_1, \ldots e_{|S|}$ of $S$ with the following properties: take an edge $e_i = \{v_j,v_k\}$, assuming wlog $j<k$. All edges whose latter endpoint comes earlier than $v_k$ in the vertex order are preceding $e_i$ in the order, thus all edges in the induced subgraph $G[\{v_1, \ldots, v_{k-1}\}]$ will be in the prefix $e_1, \ldots e_{i}$ of the canonical ordering of $S$. By Definition~\ref{def:mcaisorder} $G[\{v_1, \ldots, v_{k-1}\}]$ is connected. Finally, the only other edges in $e_1, \ldots, e_{i}$ are those whose latter endpoint is $v_k$, so their earlier endpoint is in $\{v_1, \ldots, v_{k-1}\}$. Thus each prefix $e_1, \ldots e_{i}$ forms a connected (edge) subgraph, which is also acyclic as it is a subgraph of the acyclic subgraph $S$.
Furthermore, it also holds that, for the latter endpoint $v_k$ of $e_i$, either\linebreak $\{v_1, \ldots, v_{k-1}\} \cap N^+(v_k) = \emptyset$ or $\{v_1, \ldots, v_{k-1}\} \cap N^-(v_k) = \emptyset$. This implies that either $\{e_1, \ldots, e_{i-1}\} \cap N_E^+(v_k) = \emptyset$, or $\{e_1, \ldots, e_{i-1}\} \cap N_E^-(v_k) = \emptyset$, which gives us our neighboring function:
\begin{definition}[Neighboring Function for \textsc{mcaes}\xspace]~
Let $S$ be a \textsc{mcaes}\xspace and $e = (v_t, v_h)$ a directed edge in $E(G)\setminus S$ directed \textit{from} its tail $v_t$ \textit{to} its head $v_h$. Furthermore, let $N_E^+(v_h)$ and $N_E^-(v_t)$ be the out-edges and in-edges of $v_h$ and $v_t$, respectively. We define $ \textsc{neighbors}\xspace(S, v_t, v_h) = \{\textsc{complete}\xspace(\texttt{cc}\xspace_{v_t}(\{e\} \cup (S\setminus N_E^-(v_t) )) , \textsc{complete}\xspace(\texttt{cc}\xspace_{v_h}(\{e\} \cup (S\setminus N_E^+(v_h) ))\}$
And thus
$$\textsc{neighbors}\xspace(S) = \bigcup\limits_{e= (v_t, v_h) \in E(G)\setminus S}\textsc{neighbors}\xspace(S,v_t,v_h) $$
\end{definition}
In other words, we add $e$ to $S$, and try each of the two possibilities to obtain the latter vertex in the canonical order of $S^*$: if it is the tail $v_t$ of the edge, surely its backward out-neighborhood in the canonical order of $S^*$ is not empty as it contains $v_h$, so it's in-neighborhood must be, thus we can safely remove $N_E^-(v_t)$ to make $S\cup \{e\}$ acyclic. Conversely, if it is the head $v_h$ we can safely remove $N_E^+(v_h)$. We thus obtain $|S'\ensuremath{\mathrel{\stackrel{\makebox[0pt]{\ensuremath{_\rightsquigarrow}}}{\cap}}}\xspace S^*| > |S\ensuremath{\mathrel{\stackrel{\makebox[0pt]{\ensuremath{_\rightsquigarrow}}}{\cap}}}\xspace S^*|$ for some $S'\in \textsc{neighbors}\xspace(S)$.
We can observe that the cost $\mathcal{C}_t$ of a $\textsc{complete}\xspace(X)$ call is $O(m^2)$ since we can test acyclicity in $O(m)$ time, which we do up to $m$ times, and finding and selecting the edges connected to $X$ take in total $O(m)$ time as well. As the neighboring function produces $O(m)$ solutions, we obtain:
\begin{theorem}\label{thm:mcaes}
Maximal Connected Edge-induced Directed Acyclic Subgraphs are proximity searchable, and can be listed $O(m^3)$ time delay.
\end{theorem}
\section{Proximity search in polynomial space}\label{sec:pspace-expl}\label{sec:pspace}
\newcommand{\ensuremath{\mathcal{F}}\xspace}{\ensuremath{\mathcal{F}}\xspace}
\newcommand{\kern0.75pt\mathcal{U}\xspace}{\kern0.75pt\mathcal{U}\xspace}
Proximity search consists in a graph traversal, where the number of nodes corresponds to that of solutions. If we store the set of visited nodes, as done in the algorithms presented until now, it follows that the space requirement of the algorithm becomes exponential in $n$.
Techniques such as reverse-search are able to turn this graph into a rooted tree, that can be traversed without keeping track of visited nodes, by means of a parent-child relationship among solutions, thus achieving polynomial space. However, known instances of reverse search have de facto relied on the problem at hand being hereditary, and the input-restricted problem being solvable in polynomial time (respectively, polynomial total time) to obtain polynomial delay (polynomial total time).
Recently, a generalization of reverse-search to non-hereditary properties has been proposed in~\cite{conte2019framework}: this allows us to induce a parent-child relationship for maximal solutions in any \textit{commutable} set system (a class of set systems which includes both hereditary and connected-hereditary properties), and obtain maximal listing algorithm with polynomial space, and whose delay is linked to the input-restricted problem.
In this section we show that, when suitable conditions are met, it is possible to get the best of both worlds: on one hand, using proximity search to overcome the burden of the input-restricted problem and achieve polynomial delay; on the other, using~\cite{conte2019framework} to induce a parent-child relationship among solutions and achieve polynomial space at the same time.
The final goal of the section is proving the following result.
\begin{theorem}
Let $(\kern0.75pt\mathcal{U}\xspace,\ensuremath{\mathcal{F}}\xspace)$ be a commutable set system, and $\textsc{neighbors}\xspace(S,s)$ a canonical reconstruction function for a proximity search algorithm (see Definition~\ref{def:crecon}).
If the canonical order relative to the function $\textsc{neighbors}\xspace(S,s)$ satisfies the properties of a prefix-closed order (Definition~\ref{def:prefix-closed-order}), the maximal solutions of $(\kern0.75pt\mathcal{U}\xspace,\ensuremath{\mathcal{F}}\xspace)$ can be enumerated without duplication in polynomial delay and polynomial space.
\end{theorem}
\subsection{Requirements and notation of~\cite{conte2019framework}}\label{sec:framework}
\newcommand{\textsc{seed}\xspace}{\textsc{seed}\xspace}
\newcommand{\textsc{start}\xspace}{\textsc{start}\xspace}
\newcommand{\textsc{parent}\xspace}{\textsc{parent}\xspace}
\newcommand{\textsc{pi}\xspace}{\textsc{pi}\xspace}
\newcommand{\textsc{core}\xspace}{\textsc{core}\xspace}
\newcommand{\textsc{restr}\xspace}{\textsc{restr}\xspace}
\newcommand{\textsc{gen}\xspace}{\textsc{gen}\xspace}
\newcommand{\textsc{children}\xspace}{\textsc{children}\xspace}
\newcommand{\textsc{cand}\xspace}{\textsc{cand}\xspace}
\newcommand{\textsc{p-check}\xspace}{\textsc{p-check}\xspace}
\newcommand{\textsc{choose}\xspace}{\textsc{choose}\xspace}
\newcommand{\textsc{r}\xspace}{\textsc{r}\xspace}
\let\oldnl\n
\newcommand{\nonl}{\renewcommand{\nl}{\let\nl\oldnl}
Let us briefly recall the requirements of~\cite{conte2019framework}.
In a set system $(\kern0.75pt\mathcal{U}\xspace,\ensuremath{\mathcal{F}}\xspace)$, $\kern0.75pt\mathcal{U}\xspace$ is the
\textit{ground set}, i.e., the elements constituting the solutions, and $\ensuremath{\mathcal{F}}\xspace$ defines the solutions, i.e., $S\in\ensuremath{\mathcal{F}}\xspace$ iff $S\subseteq \kern0.75pt\mathcal{U}\xspace$ satisfies the property at hand.
A set system is \textit{strongly accessible} if for any two distinct solutions $S,S'\in\ensuremath{\mathcal{F}}\xspace$ with $S \subset S'$, there exists an element $x\in S'\setminus S$ such that $S\cup \{x\}\in \ensuremath{\mathcal{F}}\xspace$. This is equivalent to saying that any non-maximal solution can be extended into a larger solution with a single element.
We say that a set system is \textit{commutable} if (i) it is strongly accessible, and (ii) it respects the \textit{commutable property}: for any $S,T\in \ensuremath{\mathcal{F}}\xspace$ with $S\subset T$, and any $a,b\in T\setminus S$, we have that $S\cup\{a\}\in\ensuremath{\mathcal{F}}\xspace \land S\cup\{b\}\in\ensuremath{\mathcal{F}}\xspace $ implies $S\cup\{a,b\}\in\ensuremath{\mathcal{F}}\xspace$.
As mentioned in~\cite{conte2019framework}, it is straightforward to see that both hereditary and connected-hereditary properties correspond to commutable set systems.
Furthermore, we call $Z$ the set of ``singleton solutions'', i.e., $Z = \{e\in \kern0.75pt\mathcal{U}\xspace : \{e\}\in\ensuremath{\mathcal{F}}\xspace\}$, and recall that in any strongly accessible set system $Z\cap S\ne \emptyset$ for any $S\in\ensuremath{\mathcal{F}}\xspace$. We also define, $S^+ = \{x : S\cup \{x\}\in\ensuremath{\mathcal{F}}\xspace \}$.
Given any commutable set system, we can obtain a maximal listing algorithm with two components. Firstly we need an efficient algorithm for solving the input-restricted problem. Secondly, to induce a parent-child structure we need what is called a \textit{family of prefix-closed orders} for the problem, satisfying the following properties:
\begin{definition}[Prefix-closed orders, from~\cite{conte2019framework}]
\label{def:prefix-closed-order}
Let $\Pi(X,v)$ be a family of orders parameterized by $X \in \ensuremath{\mathcal{F}}\xspace$ and $v \in X\cap Z$ such that $\Pi(X,v)$ yields a permutation of $X \cup X^+$.
For $X \in \ensuremath{\mathcal{F}}\xspace$ and $v \in X\cap Z$, let us denote by $x^v_1,\ldots,x^v_k$ the elements of $X$ ordered according to $\Pi(X,v)$.\footnote{Note that $x_1=v$ and that $x_i, x_{i+1} \in X$ are not necessarily consecutive in $\Pi(X,v)$ as some elements from $X^+$ can be interleaved with them.} We call the family $\Pi$ \emph{prefix-closed} if for all $X \in \ensuremath{\mathcal{F}}\xspace$ and $v \in X \cap Z$, and $i \in \{1,\ldots,k-1\}$, the following properties hold:
\begin{description}
\item[(first)] The minimal element is $v$, i.e., $x^v_1=v$.
\item[(prefix)] The $i$-th prefix $X_i=\{x^v_1,\ldots,x^v_i\}$ of $X$ is a solution, i.e., $X_i \in \ensuremath{\mathcal{F}}\xspace$.
\item[(greedy)] The element $x_{i+1}$ is the minimal element of $X_i^+\cap X$ with respect to the order $\Pi(X_i,v)$.
\end{description}
\end{definition}
As explained in~\cite{conte2019framework}, each subset $X$ of a maximal solution $S$ does not necessarily belong to $\ensuremath{\mathcal{F}}\xspace$ (as the set system is not necessarily hereditary). The \emph{first} property indicates that we can build $S$ starting from an element $v \in S \cap Z$, whereas the \emph{greedy} property indicates that we can iteratively expand $X =\{v\}$ by considering the elements of $X \cup X^+$ in a prefix-closed order, so that at any point, the prefix $\{x_1,\ldots,x_j\}$ found so far is a solution thank to the \emph{prefix} property.
We use the shorthand notation $\prec^{t}_{S}$ to represent $\Pi(S,t)$, where $a\prec^{t}_{S} b$ for any two elements $a,b\in\kern0.75pt\mathcal{U}\xspace$ means that $a$ occurs before $b$ in $\Pi(S,t)$.
Given a solution $S\in\ensuremath{\mathcal{F}}\xspace$ we define its \emph{seed}, $\textsc{seed}\xspace(S)$, as the element of smallest id in $S\cap Z$, i.e., the element $s$ of smallest id in $S$ such that $\{s\}\in\ensuremath{\mathcal{F}}\xspace$. Observe that every non-empty solution $S$ of a strongly accessible set system has a seed: since $\emptyset \subset S$, there is some $s\in S \setminus \emptyset$ such that $\emptyset\cup \{s\} = \{s\}\in \ensuremath{\mathcal{F}}\xspace$.
The simplified notations $\prec_{S}$ corresponds to $\prec^t_{S}$ with $t = \textsc{seed}\xspace(S)$. When $S$ is a maximal solution, $\prec_S$ defines an order $s_1,\ldots,s_{|S|}$ which is called the \emph{solution order} of $S$.
As in~\cite{conte2019framework}, we will also require a \textit{lexicographic} $\textsc{complete}\xspace(\cdot)$ function: for a solution $S$, $\textsc{complete}\xspace(S)$ must be obtained by iteratively adding to $S$ the smallest element in $S^+$ according to the order $\prec_S$ (i.e., the earliest in $\Pi(S,\textsc{seed}\xspace(S))$), until $S^+$ is empty. The resulting solution is maximal by definition of strongly accessible set systems. We remark that this alternative definition of $\textsc{complete}\xspace(S)$ still returns a maximal solution containing $S$, and is thus compatible with canonical reconstruction (Definition~\ref{def:crecon}).
Finally, given the canonical ordering $s_1, \ldots, s_{|S|}$ of $S$, the \textit{core} $\textsc{core}\xspace(S)$ of $S$ is the \textit{longest} prefix $s_1, \ldots, s_i$ of this order such that $\textsc{complete}\xspace(s_1, \ldots, s_i)\ne S$; its \textit{parent} is $\textsc{parent}\xspace(S) = \textsc{complete}\xspace(\textsc{core}\xspace(S)) = \textsc{complete}\xspace(s_1, \ldots, s_i)$; its parent index is $\textsc{pi}\xspace(S) = s_{i+1}$, i.e., the element following the last one of the core. It follows by definition of parent that $\textsc{complete}\xspace(\textsc{core}\xspace(S)\cup \{\textsc{pi}\xspace(S)\}) = \textsc{complete}\xspace(s_1, \ldots, s_{i+1}) = S$.
The function $\textsc{parent}\xspace(S)$ defines a forest among solutions, as every solution has a unique parent, except for the ones such that $\textsc{complete}\xspace(\textsc{seed}\xspace(S)) = S$ which are called \textit{roots}, and indeed correspond to the roots of the forest: these are linear in number (as each has a unique seed) and can be found by calling $\textsc{complete}\xspace(\{u\})$ for any $u\in \kern0.75pt\mathcal{U}\xspace$. The function $\textsc{children}\xspace(P,w)$ lets us perform a traversal of this structure, since it will find all $S$ such that $P=\textsc{parent}\xspace(S)$ and $w = \textsc{pi}\xspace(S)$.
\subsection{Combining proximity search with \cite{conte2019framework}}\label{sec:pcombination}
\newcommand{\mathit{prefix}\xspace}{\mathit{prefix}\xspace}
\begin{algorithm2e}[ht]
\caption{Polynomial-space proximity search}\label{alg:pspace}
\small%
\SetKwInOut{Input}{Input}
\SetKwInOut{Output}{Output}
\Input{Commutable set system $(\kern0.75pt\mathcal{U}\xspace,\ensuremath{\mathcal{F}}\xspace)$\\Prefix-closed order family $\preceq^s_S$\\$\textsc{neighbors}\xspace(S,s)$ for canonical reconstruction based on $\preceq^s_S$}
\Output{All maximal $X\in\ensuremath{\mathcal{F}}\xspace$}
\BlankLine
\ForEach{$S$ \textnormal{such that} $\textsc{complete}\xspace(\textsc{seed}\xspace(S)) = S$\label{ln:ps:roots}}{
$\textsc{spawn}\xspace( S )$
}
\SetKwProg{myproc}{Function}{}{}
\nonl \myproc{$\textsc{spawn}\xspace(X)$}{
\tcc{Output $X$ if depth is odd}
\ForEach{$w\in \kern0.75pt\mathcal{U}\xspace \setminus X$}{
\ForEach{$S\in \textsc{children}\xspace(X,w)$\label{ln:ps:rec}}{
$\textsc{spawn}\xspace(S)$
}
}
\tcc{Output $X$ if depth is even}
}
\SetKwProg{myproc}{Function}{}{}
\myproc{$\textsc{children}\xspace (P, w)$}{
\ForEach{$R\in \textsc{neighbors}\xspace(P,w)$\label{ln:ps:r}}{
\ForEach{$s \in (R\cap Z) \setminus \{w\}$\label{ln:ps:s}}{
$\mathit{prefix}\xspace \gets \{x\in R : x\preceq_{R}^s w\}$\label{ln:ps:pref}\;
$S \gets \textsc{complete}\xspace(\mathit{prefix}\xspace)$\;
\textbf{if} $\langle \textsc{parent}\xspace(S), \textsc{pi}\xspace(S), \textsc{seed}\xspace(S), \textsc{r}\xspace(S)\rangle = \langle P,w,s,R\rangle$
\textbf{then} \textbf{yield} $S$\label{ln:ps:pcheck}\;
}
}
}
\myproc(\tcc*[f]{finds the first $R$ that can generate $S$}){$\textsc{r}\xspace (S)$}{
$P\gets \textsc{parent}\xspace(S)$\;
$w\gets \textsc{pi}\xspace(S)$\;
$s\gets \textsc{seed}\xspace(S)$\;
\ForEach{$R\in \textsc{neighbors}\xspace(P,w)$}{
$\mathit{prefix}\xspace \gets \{x\in R : x\preceq_{R}^s w\}$\;
\lIf{$\textsc{complete}\xspace(\mathit{prefix}\xspace) = S$}{\textbf{return} $R$}
}
}
\end{algorithm2e}
When using proximity search in the \textit{canonical reconstruction} flavour, we use a canonical order to define the proximity by Definition~\ref{def:proximity}, and a suitable $\textsc{neighbors}\xspace(S,s)$ function such that together they satisfy Definition~\ref{def:searchable}.
In this section we show that we can combine proximity search and~\cite{conte2019framework} for commutable properties, if we can produce a canonical order for the canonical reconstruction that corresponds to the solution order induced by $\prec_S$.
We then show in Section~\ref{sec:pcord} that it is possible to meet these conditions for canonical orderings that are defined in a greedy way, e.g., by a BFS order like in bipartite subgraphs.
Assuming that we meet these conditions, i.e., we have a $\textsc{neighbors}\xspace(S,s)$ function that fits canonical reconstruction (Definition~\ref{def:crecon}), based on a canonical order defined by a prefix-closed order $\preceq_S$, we define a variant of~\cite{conte2019framework}, showed in Algorithm~\ref{alg:pspace}.
The main idea behind this combination comes from the following observation: the parent $P = \textsc{parent}\xspace(S) = \textsc{complete}\xspace(\textsc{core}\xspace(S))$ of $S$ is obtained from a prefix of $S$, and extending this prefix with $\textsc{pi}\xspace(S)$, then applying $\textsc{complete}\xspace(\cdot)$, gives us $\textsc{complete}\xspace(\textsc{core}\xspace(S)\cup\textsc{pi}\xspace(S)) = S$ (see definitions in Section~\ref{sec:framework}).
On the other hand, we will show that applying Definition~\ref{def:proximity}, $P \ensuremath{\mathrel{\stackrel{\makebox[0pt]{\ensuremath{_\rightsquigarrow}}}{\cap}}}\xspace S$ is exactly $\textsc{core}\xspace(S)$.
Relying on the neighboring function $\textsc{neighbors}\xspace(P, \textsc{pi}\xspace(S))$ of canonical reconstruction, and the \textit{core property} defined in~\cite{conte2019framework}, we are able to find the set $\textsc{core}\xspace(S)\cup\textsc{pi}\xspace(S)$, and finally obtain $S$.
We can now state:
\begin{theorem}
Given a commutable set system $(\kern0.75pt\mathcal{U}\xspace,\ensuremath{\mathcal{F}}\xspace)$, a prefix-closed order family $\preceq^s_S$ for $(\kern0.75pt\mathcal{U}\xspace,\ensuremath{\mathcal{F}}\xspace)$, and a function $\textsc{neighbors}\xspace(S,s)$ for canonical reconstruction (Definition~\ref{def:crecon}) based on $\preceq^s_S$, Algorithm~\ref{alg:pspace} enumerates all maximal solutions of $(\kern0.75pt\mathcal{U}\xspace,\ensuremath{\mathcal{F}}\xspace)$ without duplication in polynomial delay.
\end{theorem}
\begin{proof}
To prove the correctness, we show that any $S$ is found in $\textsc{children}\xspace(P,w)$ when $P=\textsc{parent}\xspace(S)$ and $w = \textsc{pi}\xspace(S)$.
We will first prove that there exists a solution $R \in \textsc{neighbors}\xspace(P, \textsc{pi}\xspace(S))$ (on Line~\ref{ln:ps:r}) such that $\textsc{core}\xspace(S)\cup\{\textsc{pi}\xspace(S)\}\subseteq R$.
Consider the proximity $P\ensuremath{\mathrel{\stackrel{\makebox[0pt]{\ensuremath{_\rightsquigarrow}}}{\cap}}}\xspace S$ by Definition~\ref{def:proximity}: the longer prefix of the solution order of $S$ that is completely in $P$ must include $\textsc{core}\xspace(S)$ since $P=\textsc{complete}\xspace(\textsc{core}\xspace(S))$. If $w\in P$ then $\textsc{neighbors}\xspace(S,w)$ returns $P$ by Definition~\ref{def:crecon}, and indeed $P\supseteq \textsc{core}\xspace(S)\cup\{w\}$.
Otherwise, $P$ does not include $\textsc{pi}\xspace(S)$, meaning that $P\ensuremath{\mathrel{\stackrel{\makebox[0pt]{\ensuremath{_\rightsquigarrow}}}{\cap}}}\xspace S = \textsc{core}\xspace(S)$ and that $w$ is the canonical extender for $P,S$.
Using the neighboring function $\textsc{neighbors}\xspace(P, \textsc{pi}\xspace(S))$ we obtain at least one solution $R \supseteq \textsc{core}\xspace(S)\cup\textsc{pi}\xspace(S)$.
Using the \textit{core property} defined in~\cite{conte2019framework}, we are able to use $R$ to retrieve $S$: It is proven that Lines~\ref{ln:ps:r}-\ref{ln:ps:pcheck} will find and output any solution $S$ such that $\textsc{core}\xspace(S)\cup\{\textsc{pi}\xspace(S)\}\subseteq R$, a condition which is guaranteed by what stated above.
The \textbf{if} on Line~\ref{ln:ps:pcheck} removes duplication: any $S$ is found only once out of all invocations of $\textsc{children}\xspace(P,w)$: when $P = \textsc{parent}\xspace(S)$, $w = \textsc{pi}\xspace(S)$, $s=\textsc{seed}\xspace(S)$, and $R = \textsc{r}\xspace(S)$. The function $\textsc{r}\xspace(S)$ simply aims at defining deterministically one single $R \supseteq \textsc{core}\xspace(S)\cup\{\textsc{pi}\xspace(S)\}$ once the other 3 variables have been fixed. It thus follows that this check is passed exactly once out of the whole execution of the algorithm for any solution (other than the roots, found on Line~\ref{ln:ps:roots}).
Line~\ref{ln:ps:roots} shows that, by definition, all the roots of the forest are explored by Algorithm~\ref{alg:pspace}. We just proved that Line~\ref{ln:ps:rec} discovers all the children of each visited node exactly once, which concludes the proof of the fact that Algorithm~\ref{alg:pspace} visits every maximal solution of $(\kern0.75pt\mathcal{U}\xspace,\ensuremath{\mathcal{F}}\xspace)$ without duplication.
\end{proof}
It is also straightforward to see that each recursive call uses polynomial space, and no solution dictionary $\ensuremath{\mathcal{S}}\xspace$ is maintained. However, the depth of the recursion tree is a factor in the space complexity too: to obtain a polynomial space guarantee, we further need to turn the recursive algorithm into a \textit{stateless} iterative one, as has been done in~\cite{conte2019framework}.
We can give a general bound with the following parameters: let $q$ be the maximum size of a solution; $\mathcal{R}_T$ be the time required to solve $\textsc{neighbors}\xspace(P,w)$; $\mathcal{R}_N$ a bound on the number of solutions returned by it; $\mathcal{C}_t$ be the time required to compute $\textsc{complete}\xspace(X)$ and $\mathcal{O}_T$ the time required to compute the canonical order of $X\cup X^+$. As these bounds are all assumed to be polynomial, we observe their space requirements will be polynomial as well.
Thanks to the alternative output technique, the delay will be bounded by the cost of one iteration of $\textsc{spawn}\xspace(X)$, that is, $O(|\kern0.75pt\mathcal{U}\xspace|)$ times the cost of $\textsc{children}\xspace(P,w)$. In turn, the cost of $\textsc{children}\xspace(P,w)$ is that of $\textsc{neighbors}\xspace(P,w)$, plus for each of the $O(\mathcal{R}_N)$ solutions $R$ returned, the cost of processing Lines~\ref{ln:ps:s}-\ref{ln:ps:pcheck}. \cite{conte2019framework} proved that this can be done in $O(q(\mathcal{O}_T+\mathcal{C}_t))$ time for the given definition of $\textsc{r}\xspace(S)$. However, our definition of $\textsc{r}\xspace(S)$ is different from the one in~\cite{conte2019framework}, and has a cost of $O(\mathcal{R}_T + \mathcal{R}_N\mathcal{O}_T)$ instead of $O(\mathcal{O}_T+\mathcal{C}_t)$. Thus, the total cost of processing Lines~\ref{ln:ps:s}-\ref{ln:ps:pcheck} is $O(q(\mathcal{R}_T + \mathcal{R}_N\mathcal{O}_T + \mathcal{C}_t))$.
We can thus claim the following:
\begin{theorem}\label{thm:prox-pspace-bound}
Given a commutable set system $(\kern0.75pt\mathcal{U}\xspace,\ensuremath{\mathcal{F}}\xspace)$, a prefix-closed order family $\preceq^s_S$ for $(\kern0.75pt\mathcal{U}\xspace,\ensuremath{\mathcal{F}}\xspace)$, and a function $\textsc{neighbors}\xspace(S,s)$ for canonical reconstruction (Definition~\ref{def:crecon}) based on $\preceq^s_S$,
the maximal solutions of $(\kern0.75pt\mathcal{U}\xspace,\ensuremath{\mathcal{F}}\xspace)$ can be enumerated in
$O(|\kern0.75pt\mathcal{U}\xspace|\mathcal{R}_T + |\kern0.75pt\mathcal{U}\xspace| \mathcal{R}_N q(\mathcal{R}_T + \mathcal{R}_N\mathcal{O}_T + \mathcal{C}_t))$ time delay and polynomial space.
\end{theorem}
\subsection{BFS-based canonical reconstruction}\label{sec:pcord}\label{sec:bfsproximity}
In this section, we provide a technique to implement the result of Section~\ref{sec:pcombination} (Theorem~\ref{thm:prox-pspace-bound}), i.e., a canonical reconstruction order that matches the prefix-closed order requirements, and can be applied to hereditary and connected-hereditary properties. We call this technique \textit{BFS-based canonical reconstruction}.
While it is possibly not the only way to obtain a suitable order, it is worth defining formally as we will apply it to several problems in the following sections.
We will first define the order for connected-hereditary property, then exploit it to cover the hereditary case.\footnote{Notably, this implies that a BFS-based canonical reconstruction algorithm for the non-connected case immediately follows from one for the connected case.}
\begin{definition}[canonical-BFS order for connected-hereditary properties]\label{def:canonBFSconn}
Let $S$ be a solution of a connected-hereditary set system, and $v$ any element in $S$.
The canonical order $\Pi(S,v) = s_1,\ldots ,s_{|S\cup S^+|}$ is the lexicographical order of the tuples $\langle d_{v}(s_i),s_i\rangle$, where $d_{v}(s_i)$ is the distance between $s_i$ and $v$ in $G[S\cup \{s_i\}]$.
\end{definition}
In other words, we order nodes first by $d_{v}(s_i)$, i.e., their distance from $v$ in $G[S]$, and break ties by vertex id.
The same logic applies to nodes $x$ of $S^+$, for which we use the distance from $v$ in $G[S\cup\{x\}]$.
This defines $\preceq^s_S$.
\begin{example*}
For the Maximal Connected Induced Bipartite Subgraph in Figure~\ref{fig:ex:bip} (b), the order $2,3,5,8,11,7,10$ (as defined in Section~\ref{sec:reconstruction}) is given by the tuples\linebreak $\langle0,2\rangle , \langle 1,3\rangle, \langle 1,5\rangle, \langle 2,8\rangle, \langle 2, 11\rangle, \langle 3,7\rangle, \langle 3,10\rangle$.
\end{example*}
We can observe how this canonical-BFS order $\Pi(S,v)$ satisfies the properties of Definition~\ref{def:prefix-closed-order}:\footnote{For completeness, we could equivalently observe that $d_{v}(s) = LAY^{v}_S(s)$ according to Definition~7 in~\cite{conte2019framework}.}
\begin{itemize}
\item[\textbf{(first)}] The first element $s^v_1$ of $\Pi(S,v)$ is $v$, as $d_{v}(v)=0$ and $d_{v}(\cdot)\ge1$ for any other vertex.
\item[\textbf{(prefix)}] Any prefix $S_i=\{s^v_1,\ldots,s^v_i\}$ of $\Pi(S,v)$ is connected (thus a solution), since for any $s_i$, the vertices on a shortest path in $G[S]$ to $s_1$ are at a smaller distance from $s_1$ and thus occur before $s_i$.
\item[\textbf{(greedy)}] For any $z \in S_i^+ \cap S$, let $k$ be the distance between $v$ and $z$ in $G[S_i \cup \{z\}]$. Since $z\in S_i^+$, there must be some $w \in S_i\cap N(z)$ at distance $k-1$ from $v$ (in $G[S_i \cup \{z\}]$). This means that the distance between $v$ and $z$ in $G[S]$ is still $k$: otherwise, there would be a vertex $y\in S\setminus S_i$, i.e., after $w$ in the canonical order, that is a neighbor of $z$ and has distance $\le k-2$ from $v$ in $G[S]$; this leads to contradiction since $y$ would then need to come \textit{before} $w$ in the BFS order.
\end{itemize}
It follows that the canonical-BFS order is a prefix-closed order.
now straightforward to see how this order satisfies the \textbf{(first)}, \textbf{(greedy)} and \textbf{(prefix)} properties of Definition~\ref{def:prefix-closed-order}, and essentially corresponds to the layer order defined in~\cite{conte2019framework}.
\begin{definition}[canonical-BFS order for hereditary properties]\label{def:canonBFSnon}
Let $S$ be a solution of a hereditary set system, and $v$ any element in $S$.
For each connected component $C_i$ of $G[S]$, we say the \textit{leader} of the component is $v$ if $C_i$ contains $v$, and otherwise the vertex of smallest id in $C_i$.
The canonical order $\Pi(S,v) = s_1,\ldots ,s_{|S\cup S^+|}$ (defined on $S\cup S^+$) is the lexicographical order of the tuples $\langle cid(S,s_i), d_l(S,s_i), s_i \rangle$, where for $s_i$ in the component $C_i$, $cid(S,s_i)$ is the id of the leader of $C_i$, or $0$ if this leader is $v$ (assuming wlog $0$ is smaller than any other id), and $d_l(S,s_i)$ is the distance from the leader of $C_i$ in $G[C_i]$. Observe how $s_1 = v$.
For a vertex $x$ in $S^+$, we use as $cid(S,x)$ and $d_l(S,x)$ the values obtained in $G[S\cup\{x\}]$.
\end{definition}
Less formally, we order each component by a BFS strategy as in the above case (since $G[C_i]$ is connected) using the leader as root (i.e., $s$ if the component contains $s$, or its smallest id vertex otherwise); then, we concatenate the sequences obtained by putting the one containing $s$ first, followed by the others ordered by id of their leader.
\begin{example*}
For the Maximal Induced Bipartite Subgraph in Figure~\ref{fig:ex:bip} (d), the order is $1,2,7,8,11,10$, given by the tuples \linebreak $\langle 1,0,1\rangle , \langle 1,1,2\rangle, \langle 7,0,7\rangle, \langle 7,1,8\rangle, \langle 7,1,11\rangle, \langle 7,2,10\rangle$.
\end{example*}
Before proving that this defines a prefix-closed order, let us prove this auxiliary lemma:
\begin{lemma}
\label{lem:samecid}
Let $X$ be a solution and $X_i$ any prefix of its canonical order. The following facts hold:
\begin{itemize}
\item $\forall z \in X_i^+\cap X, cid(X_i,z) \ge cid(X,z)$
\item $cid(X_i ,x_{i+1}) = cid(X,x_{i+1})$
\item $\forall z\in X_i^+\cap X, cid(X_i,z) = cid(X,z) \Rightarrow d_l(X_i,z) = d_l(X,z)$.
\end{itemize}
\end{lemma}
\begin{proof}
First, the leader of each connected component of $X_i$ is the same as the leader of the corresponding connected component of $X$ (since the leader is always the first element of the connected component in a solution order, and prefixes of components are connected as they are in a BFS order).
Moreover, an element $z$ in $X_i^+\cap X$ is either directly connected to a connected component of $X_i$, in which case it has the same leader in $X_i$ and in $X$ by what stated above, or it belongs to its own connected component in $G[X_i \cup \{z\}]$, in which case $z$ is its own leader in $G[X_i \cup \{z\}]$, meaning $cid(X_i, z) = z$. Since by definition $cid(X, z) \le z$, it follows that $cid(X, z) \le cid(X_i, z)$, proving the first statement.
We now prove that $cid(X_i, x_{i+1}) = cid(X, x_{i+1})$: Either $x_{i+1}$ is directly connected to the last connected component of $X_i$ (in which case we already proved the equality) or it isn't, in which case $cid(X_i, x_{i+1}) = x_{i+1}$. However, in this case $x_{i+1}$ must be its own leader by definition of the order, so it follows that $cid(X, x_{i+1}) = x_{i+1}$, proving the second statement.
Finally, consider $z\in X_i^+ \cap X$ such that $cid(X_i,z) = cid(X,z)$.
If $cid(X,z) = z$ then $d_l(X_i,z) = d_l(X,z) = 0$; otherwise, let $x_l$ be the leader of $z$ in $X_i\cup\{z\}$: $z$ is in the same connected component $C_z$ as $x_l$ in $X$, and $X_i$ contains a prefix of the canonical-BFS order of $C_z$; by the properties of the canonical-BFS order, the shortest path from $x_l$ to $z$ is in this prefix, implying the third statement.
\end{proof}
We can now observe how this order for hereditary properties also satisfies the properties of Definition~\ref{def:prefix-closed-order}
\begin{itemize}
\item[\textbf{(first)}] By definition $s_1$ is the first element.
\item[\textbf{(prefix)}] As this order is defined for hereditary properties, it follows that any subset (hence every prefix) is also a solution.
\item[\textbf{(greedy)}] We proved in Lemma~\ref{lem:samecid} that the tuple associated with each element of $X_i^+ \cap X$ with respect to $X_i$ is either the same or lexicographically greater than the tuple with respect to $X$. As the tuple for $x_{i+1}$ is the same, and since $x_{i+1}$ is the minimum of $X_i^+ \cap X$ with respect to the order in $X$, it follows that it's also the minimum of $X_i^+ \cap X$ with respect to the order in $X_i$.
\end{itemize}
We remark that it is possible to generalize this definition using different functions for $d(\cdot)$ and $d_l(\cdot)$, as long as monotone behaviour can be guaranteed, i.e., $d(X_i,x)$ (resp. $d_l(X_i,x)$) is less than or equal to $d(X,x)$ (resp. $d_l(X,x)$) when $X_i$ is a prefix of $X$.
\section{Polynomial space algorithms}\label{sec:pspace-algs}
In this section we apply the technique defined in Section~\ref{sec:pspace-expl}, and give polynomial-space-polynomial-delay proximity search algorithms, proving the bounds given in Theorem~\ref{thm:pspace}.
For the problems already solved in exponential space in the previous sections, we remark that it is simply necessary to define their canonical order as a canonical-BFS, then apply Theorem~\ref{thm:prox-pspace-bound}.
\subsection{Maximal Bipartite Subgraphs}
Looking at the canonical orders defined for Maximal Connected Induced Bipartite Subgraphs (Definition~\ref{def:bip-con-canon}) and Maximal Induced Bipartite Subgraphs (Definition~\ref{def:bip-non-canon}), we can see that their definitions match exactly those of canonical-BFS for connected-hereditary and hereditary properties (respectively, Definition~\ref{def:canonBFSconn} and Definition~\ref{def:canonBFSnon}).
We can thus immediately apply the polynomial space variant of the algorithm, and we proceed to compute its complexity.
The cost $\mathcal{O}_T$ for computing the canonical order will be $O(m)$ in all cases, as it corresponds to performing a BFS, while $\mathcal{C}_t$ corresponds to adding edges in a BFS order, which will take $O(m)$ on the connected version, but $O(m+n\iack(n))$ on the non-connected one due to the need to dynamically maintain the connected components.
The neighboring function for both cases produces a constant number of neighboring solutions, meaning $\mathcal{R}_N = O(1)$ and $\mathcal{R}_T = O(\mathcal{C}_t)$.
At the same time, all operations require no more than $O(m)$ space. Applying Theorem~\ref{thm:prox-pspace-bound}, we obtain:
\begin{theorem}
Maximal Connected Induced Bipartite Subgraphs and Maximal Induced Bipartite Subgraphs of a graph $G$ can be enumerated via BFS-based canonical reconstruction (Algorithm~\ref{alg:pspace}) in $O(m)$ space and, respectively, $O(qnm) = O(n^2m)$ and $O(qn(m+n\iack(n))) = O(n^2(m+n\iack(n))) $ time delay.
\end{theorem}
\subsection{Maximal Induced Trees and Forests}\label{sec:indtrees}
As defined above, a forest is an acyclic undirected graph, and a connected forest is called a tree.
These are a special cases of $k$-degenerate subgraphs: $1$-degenerate subgraphs are precisely forests, and connected $1$-degenerate subgraphs are trees. However, it is worth consider these problems separately, since we can obtain algorithms with lower delay and polynomial space.
It should be observed that listing Maximal Induced Forests corresponds to listing minimal feedback vertex sets in undirected graph: if $S\subset V$ is a Maximal Induced Forest, $V\setminus S$ is a minimal feedback vertex set. A polynomial-delay solution for the enumeration of feedback vertex sets (and thus Maximal Induced Forests) has been proposed in~\cite{schwikowski2002enumerating}. This result, however, requires exponential space, and does not extend to Maximal Induced Trees.
Furthermore, while the algorithms proposed could be extended to enumerate maximal edge-induced trees and forests, we do not consider it: these correspond to just the spanning trees of a graph, which are already known to be enumerable in polynomial delay and even constant amortized time~\cite{shioura1997optimal}.
\paragraph{Canonical order and neighboring function}
Let $S$ be a maximal induced tree.
We define its canonical as a canonical-BFS order (Definition~\ref{def:canonBFSconn}), i.e., the sequence $s_1, \ldots, s_{|S|}$ given by a BFS order of $G[S]$ rooted in the vertex $s_1$ of smallest id.
We then define the proximity by canonical reconstruction (Section~\ref{sec:reconstruction}), and we can immediately observe that this order meets the requirements of Section~\ref{sec:pspace}. Next, we focus on obtaining a suitable neighboring function.
\begin{definition}[Neighboring function for Maximal Induced Trees]~
We define $\textsc{neighbors}\xspace(S) = \bigcup\limits_{v\in V(G)\setminus S} \textsc{neighbors}\xspace(S,v)$.
Then, $\textsc{neighbors}\xspace(S,v)$ is defined as:
\noindent$\textsc{neighbors}\xspace(S,v) = \{\textsc{complete}\xspace(\texttt{cc}\xspace_v(S\setminus N(v)\cup \{w,v\}) ) : w\in N(v) \cap S\}$
\end{definition}
The key property here is that each vertex $s_i\in S$ has a single neighbor preceding it in the canonical order, corresponding to its parent in the BFS.
Given two solutions $S,T$, let $t_1,\ldots,t_{|T|}$ be the canonical order of $T$, and $t_i$ be the canonical extender for $S,T$, i.e., the vertex for which $S\ensuremath{\mathrel{\stackrel{\makebox[0pt]{\ensuremath{_\rightsquigarrow}}}{\cap}}}\xspace T = \{t_1,\ldots,t_{i-1}\}\subseteq S$ and $t_i\not\in S$. Furthermore, let $t_j$ be the parent of $t_i$ in the canonical BFS-order of $T$, observing that $t_j\in \{t_1,\ldots,t_{i-1}\}$.
To find a solution $S' \supseteq \{t_1,\ldots,t_{i}\}$, we can simply add $t_i$ to $S$, then remove all neighbors of $t_i$ \textit{except} $t_j$ so that we have again an acyclic subgraph, and finally discard every vertex not in the same connected component as $t_i$ (which will include $\{t_1,\ldots,t_{i}\}$). As we do not know which vertex is $t_j$, we of course try all $O(|N(t_i)|)$ possibilities, thus a suitable $S'$ is always found.
\paragraph{Complexity}
Firstly, we can use the $\textsc{neighbors}\xspace(S)$ function to build a proximity search algorithm whose delay is the cost of $\textsc{neighbors}\xspace(\cdot)$, and whose space is $O(\nsol \cdot n)$ (where $\nsol$ is the number of solutions).
We show the cost of $\textsc{neighbors}\xspace(\cdot)$ -and the delay of the algorithm- to be $O(m^2)$ time: Observe that the cost $\mathcal{C}_t$ a $\textsc{complete}\xspace(X)$ call is $O(m)$ time. We first compute the set of vertices adjacent to $X$, $P = \cup_{x\in X}N(x)$; for each vertex $v$, we simply need to check that it has exactly one neighbor in $X$, in $O(|N(v)|)$ time, and discard it otherwise. Whenever we add $v$ vertex to $X$, we add is neighbors to $P$ again in $O(|N(v)|)$ time. The total cost is $O(\sum_{v\in V(G)}|N(v)|) = O(m)$.
Now consider $\textsc{neighbors}\xspace(S,v)$: for each $w\in N(v)$, we must compute $\texttt{cc}\xspace_v(S\setminus N(v)\cup \{w,v\})$, which takes $O(m)$, then apply $\textsc{complete}\xspace(\cdot)$ which has the same complexity. The cost is thus $O(|N(v)|\cdot m)$. In turn, this means the cost of $\textsc{neighbors}\xspace(S)$ is $O(\sum_{v\in V(G)}|N(v)|\cdot m) = O(m^2)$.
Furthermore, as we are satisfying all conditions of Section~\ref{sec:pspace} (the order defined is a canonical BFS-order and the problem is connected-hereditary), we apply Theorem~\ref{thm:prox-pspace-bound} to obtain a BFS-based canonical reconstruction algorithm, with higher delay but polynomial space.
We observe that no component of the algorithm will require more than $O(m)$ space, and their time complexity is as follows: $\kern0.75pt\mathcal{U}\xspace = O(n)$, $\mathcal{R}_T = O(m\Delta)$, $\mathcal{R}_N = O(\Delta)$ (but as observed above, $|\kern0.75pt\mathcal{U}\xspace|\cdot \mathcal{R}_T$ can be better bounded by $O(m^2)$, and $|\kern0.75pt\mathcal{U}\xspace|\cdot \mathcal{R}_N$ can be bounded by $O(m)$), $q= O(n)$, $\mathcal{O}_T = O(m)$ and $\mathcal{C}_t = O(m)$.
The bound of Theorem~\ref{thm:prox-pspace-bound} thus resolves to
$O(m^2 + m q(m\Delta + \Delta m + m)) = O(m q(m\Delta)) = O(m^2n^2)$ time.
We can thus conclude the following:
\begin{theorem}
The Maximal Induced Trees of a graph $G$ can be enumerated in $O(m^2)$-time delay using $O(\nsol n)$ space, or alternatively in $O(m^2n^2)$-time delay and $O(m)$ space.
\end{theorem}
\subsection{Maximal Induced Forests}
As showed in Section~\ref{sec:bfsproximity}, a BFS-based canonical reconstruction algorithm for the non-connected case immediately follows from the connected one.
For completeness, we show how the algorithm for Maximal Induced Forests is obtained:
The canonical order is obtained by Definition~\ref{def:canonBFSnon}, i.e., a canonical-BFS order of each connected component, where different components are then sorted by their vertex of smallest id.
The neighboring function is essentially obtained from the connected case by removing the use of the $\texttt{cc}\xspace(\cdot)$ function (as we do not require solutions to be connected).
\begin{definition}[Neighboring function for Maximal Induced Forests]~
We define $\textsc{neighbors}\xspace(S) = \bigcup\limits_{v\in V(G)\setminus S} \textsc{neighbors}\xspace(S,v)$.
Then, $\textsc{neighbors}\xspace(S,v)$ is defined as:
\noindent$\textsc{neighbors}\xspace(S,v) = \{\textsc{complete}\xspace(S\setminus N(v)\cup w) : w\in N(v) \cap S\}$
\end{definition}
The complexity of the components of the algorithm is also inherently the same, with the only difference for the cost $\mathcal{C}_t$ of the $\textsc{complete}\xspace(\cdot)$ function: when we add a vertex, we need to make sure that it does not have two neighbors in the same connected component, and update the connected components as we add vertices. The cost of $\textsc{complete}\xspace(\cdot)$ will thus be $O(m + n\iack(n))$ time, obtained by the same logic as for Maximal Bipartite Subgraphs (see Section~\ref{sec:binon}), while the rest of the operations are exactly as in the connected case, thus bear the same cost.
We can conclude that $\textsc{neighbors}\xspace(S,v)$ takes $O(|N(v)|\cdot(m + n\iack(n))$ time, while $\textsc{neighbors}\xspace(S)$ takes $O(\sum_{v\in V(G)}|N(v)|\cdot (m + n\iack(n))) = O(m (m + n\iack(n)))$.
Again, we can obtain an exponential-space algorithm using canonical reconstruction proximity search whose delay is the cost of $\textsc{neighbors}\xspace(S)$, and a polynomial-space algorithm using BFS-based canonical reconstruction, whose delay is given by Theorem~\ref{thm:prox-pspace-bound}.
For the latter, the costs are obtained adapting the connected version with the new cost of $\textsc{complete}\xspace(\cdot)$: $\kern0.75pt\mathcal{U}\xspace = O(n)$, $\mathcal{R}_T = O((m + n\iack(n))\Delta)$, $\mathcal{R}_N = O(\Delta)$ (but $|\kern0.75pt\mathcal{U}\xspace|\cdot \mathcal{R}_T$ can be better bounded by $O(m(m + n\iack(n)))$, and $|\kern0.75pt\mathcal{U}\xspace|\cdot \mathcal{R}_N$ can be bounded by $O(m)$), $q= O(n)$, $\mathcal{O}_T = O(m)$ and $\mathcal{C}_t = O(m + n\iack(n))$. The bound of Theorem~\ref{thm:prox-pspace-bound} thus resolves to $O(m(m + n\iack(n)) + m q ( (m + n\iack(n)) + \Delta m + (m + n\iack(n)) ) ) = O( m q ( n\iack(n) + \Delta m ) )$ time, which we can again upper bound by $O(m^2n^2)$ time. We can thus conclude the following:
\begin{theorem}
The Maximal Induced Forests of a graph $G$ can be enumerated
in $O(m^2n^2)$-time delay and $O(m)$ space.
\end{theorem}
\section{Conclusions}\label{sec:concl}
We presented proximity search, a technique for the design of efficient enumeration algorithms, based on defining and traversing a solution graph with bounded out-degree.
We showed several application cases, considering problems that did not allow efficient algorithms by known methods, and showing that these allow polynomial delay algorithms by proximity search.
We have provided a guideline, called \textit{canonical reconstruction}, aimed at factorizing the most effective ways to apply our technique, and facilitating the design of efficient algorithms.
We have further shown a technique that, under suitable conditions, allows us to design proximity search algorithms that require only polynomial space. The results are polynomial-delay and polynomial-space algorithms for several problems whose input-restricted problem cannot be solved in polynomial time, including non-hereditary ones.
This paper ``breaks the barrier'' of the input-restricted problem, showing that its complexity does not imply lower bounds in terms of time or space, nor even a trade-off between the two. This closes questions left open since~\cite{Cohen20081147}, furthering our understanding on the complexity of enumeration in set systems.
At the same time, this reinvigorates the open question of which listing problems allow efficient algorithms and which do not, and to define a more complete theory of enumeration complexity.
On top of being a useful tool to design efficient algorithms for specific problems, we hope that this technique will be able to help us gain more insight into this general question.
\section*{acknowledgements}
We wish to thank the anonymous reviewers for their thorough analysis of the paper, which helped us improve both its content and presentation.
This work was partially supported by JST CREST, grant number JPMJCR1401, Japan and the Italian Ministry for Education and Research, under PRIN Project n. 20174LF3T8 AHeAD.
\bibliographystyle{plain}
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2,877,628,089,022 | arxiv | \section{Introduction} \label{sec:paper4_intro}
The study of the distribution of molecular gas in star-forming galaxies provides
us with an understanding of star formation processes and their relation to galactic
evolution.
In these studies carbon monoxide (CO) is used as a tracer of star-forming
regions and dust, since in these cold regions ($ T < 100$~K) H$_2$ is virtually
invisible.
The various rotational transitions of CO emit in the far-infrared (FIR)
spectrum, and are able to penetrate deep into clouds with high column densities,
which are otherwise opaque to visible light.
CO lines are usually optically thick and their emission emanates from the
C$^+$/C/CO transition zone \citep{Wolfire89-1}, with a small contribution to the
intensity from the deeper part of the cloud \citep{meijerink2007-1}. On the
other hand, other molecules, whose emission lines are optically thin beacuse of their
lower column densities, probe greater depths of the cloud compared to CO. These
less abundant molecules (e.g., $^{13}$CO~) have a weaker signal than CO, and a longer
integration time is required in-order to observe them. Since ALMA became
available, it has become possible to obtain well-resolved molecular emission
maps of star-forming galaxies in the local universe, due to its high
sensitivity, spatial and spectral resolution. In particular, many species have
been observed with ALMA, including the ones we consider in this paper, namely
CO, $^{13}$CO~, HCN, HNC, and ${\rm HCO}^+$~~ \citep[e.g.,][]{Imanishi13-1, Saito13-1,
Combes13-1, Scoville13-1, Combes14-1}.
Massive stars play an important role in the dynamics of the gas around the
region in which they form. Although the number of massive stars ($ M >
10$~${\rm M}_{\odot}$) is about 0.1\% of the total stellar population, they emit more than
99\% of the total ultraviolet (FUV) radiation.
This FUV radiation is one of the main heating mechanisms in the ISM of star-forming
regions. Such regions are referred to as photon-dominated regions (PDRs)
and they have been studied since the 1980s \citep{tielenshb1985,
Hollenbach1999}.
Since then, our knowledge of the chemical and thermal properties of these
regions has been improving.
Since molecular clouds are almost invariably accompanied by young luminous
stars, most of the molecular ISM forms in the FUV shielded region of a PDR, and
thus this is the environment where the formation of CO and other molecular
species can be studied.
In addition, the life span of massive stars is short, on the order of 10 Myrs,
thus they are the only ones that detonate as supernovae, liberating a
significant amount of energy into their surroundings and perturbing it. A small
fraction of the energy is re-absorbed into the ISM, which heats up the gas
\citep{Usero07-1, falgaron2007p, loenen2008}. In addition, starbursts (SB)
occur in centers of galaxies, where the molecular ISM can be affected by X-rays
of an accreting black hole (AGN) and enhanced cosmic ray rates or shocks
\citep[][among many others]{maloney96, komossa03, martin06-1, oka2005ApJ_1,
vandertak2006_1, pan2009-1, papadopoulos10, meijerink11, meijerink13-1,
Rosenberg14} that ionize and heat the gas.
By constructing numerical models of such regions, the various heating mechanisms
can be identified. However, there is no consensus about which combination of
lines define a strong diagnostic of the different heating mechanisms. This is
mainly due to the lack of extensive data which would probe the various
components of star-forming regions in extra-galactic sources. Direct and
self-consistent modeling of the hydrodynamics, radiative transfer and chemistry
at the galaxy scale is computationally challenging, thus some simplifying
approximations are usually employed. In the simplest case it is commonly assumed
that the gas has uniform properties, or is composed of a small number of uniform
components.
In reality, on the scale of a galaxy or on the kpc scale of starbursting
regions, the gas density follows a continuous distribution. Although the exact
functional form of this distribution is currently under debate
\citep[e.g.][]{Nordlund99-1}, it is believed that in SB regions, where the gas
is thought to be supersonically turbulent \citep{norman96-1, Goldman12-1} the
density distribution of the gas is a log-normal function \citep{Vazquez94-1,
Nordlund99-1, wada01-a, wada07-1, kritsuk11-1, ballesterosParedes11-1,
Burkhart13-1, hopkins13-1}.
This is a universal result, independent of scale and spatial location, although
the mean and the dispersion can vary spatially.
Self-gravitating clouds can add a power-law tail to the density PDF
\citep{kainulainen09-1,froebrich10-1,russeil13-1,AlvesdeOliveira14-1,Schneider14-1}.
However, \cite{kainulainen2013-1} claim that such gravitational effects are
negligible on the scale of giant molecular clouds, where the molecular emission
we are interested in emanates.
In \cite{mvk15-a}, we studied the effect of mechanical heating ($\Gamma_{\rm mech}$~~hereafter)
on molecular emission grids and identified some diagnostic line ratios to
constrain cloud parameters including mechanical heating. For example, we showed
that low-$J$ to high-$J$ intensity ratios of high density tracers will yield a
good estimate of the mechanical heating rate. In \cite{mvk15-b}, KP15b
hereafter, we applied the models by \cite{mvk15-b} to realistic models of the
ISM taken from simulations of quiescent dwarf and disk galaxies by
\cite{inti2009-1}. We showed that it is possible to constrain mechanical
heating just using $J < 4-3$ CO and $^{13}$CO~~ line intensity ratio from ground
based observations.
This is consistent with the suggestion by \cite{israel03} and
\cite{israel2009-1} that shock heating is necessary to interpret the high
excitation of CO and $^{13}$CO~~in star-forming galaxies. This was later verified by
\cite{loenen2008}, where it was shown that mechanically heated PDR models are
necessary to fit the line ratios of molecular emission of high density tracers
in such systems.
Following up on the work done by KP15b, we include high density gas ($n >
10^4$~cm$^{-3}$~) to produce more realistic synthetic emission maps of a simulated
disk-like galaxy, thus accounting for the contribution of this dense gas to the
molecular line emission. This is not trivial as global, galaxy wide models of
the star-forming ISM are constrained by the finite resolution of the simulations
in the density they can probe.
This paper is divided into two main parts. In the first part,
we present a new method to incorporate high density gas to account for its
contribution to the emission of the high density tracers, employing the
plausible assumption, on theoretical grounds \citep[e.g.][]{Nordlund99-1} that
the density field follows a log-normal distribution.
In the methods section, we describe the procedure with which the sampling of the
high density gas is accomplished. Once we have derived a re-sampled density
field we can employ the same procedure as in KP15b to model the line emission
of molecular species. In Section-\ref{subsec:paper4_emissionmaps}, we highlight
the main trends in the emission of the $J = 5-4$ to $J = 15-14$ transitions of
CO and $^{13}$CO~~ tracing the densest gas, along with the line emission of high
density tracers HCN, HNC and ${\rm HCO}^+$~ for transitions up to $J = 7-6$. In
Section-\ref{subsec:paper4_constrainig}, we fit emission line ratios using a
mechanically heated PDR (mPDR hereafter) and constrain the gas parameters of the
model disk galaxy. In the second part of the paper,
Section-\ref{section:constrainingpdf}, we will follow the reverse path and
examine what constraints can be placed on the PDF from molecular line emissions,
following the same modeling approach as in the first part. We discuss the
effect of the shape and width of the different density profiles on the emission
of high density tracers. In particular, we discuss the possibility of
constraining the dispersion and the mean of an assumed log-normal density
distribution using line ratios of high density tracers. We finalize with a
summary and general remarks.
\section{Methods \label{sec:paper4_methods}}
The numerical methods we implement in this paper are similar to those in KP15b.
We will focus exclusively on a single model disk galaxy, but the methods
developed here could be applied to other models. We implement a recipe for the
introduction of high density gas $n > 10^4$~cm$^{-3}$~, which is necessary to model
the emission of molecular lines with critical densities\footnote{We use the
following definition of the critical density, $n_{crit} \equiv k_{ij} / A_{ji}$,
where $k_{ij}$ is the collision coefficient from the level $i$ to the level $j$
and $A_{ji}$ is the Einstein coefficient (of spontaneous decay from level $j$ to
level $i$.} ($n_{\rm crit}$) in the range of $10^4 - 10^8$~cm$^{-3}$~ . In the
following, we summarize briefly the methods used in KP15b.
\subsection{Galaxy model, radiative transfer and subgrid modelling}
The model galaxy we use in this paper is the disk galaxy simulated by
\cite{inti2009-1}, with a total mass of $10^{10}$~${\rm M}_{\odot}$~and a gas content
representing 10\% of the total mass. This represents a typical example of a
quiescent star-forming galaxy. The code {\sc Fi} \citep{inti2009-1} was used to
evolve the galaxy to dynamical equilibrium (in total for $\sim 1$ Gyr).
The simulation code implements an Oct-tree algorithm which is used to compute
the self-gravity \citep{barnes86} and a TreeSPH method for evolving the
hydrodynamics \citep{monaghan92, springelHernquist02}, with a recipe for star
formation based on the Jeans mass criterion \citep{intiPhdT}. The adopted ISM
model is based on the simplified model by \cite{wolfire03}, where the local FUV
field in the neighborhood of the SPH particles is calculated using the
distribution of stellar sources and population synthesis models by
\cite{bruzualCharlot93, bruzualCharlot03} and \cite{parravano2003}. The local
average mechanical heating rate due to the self-interacting SPH particles is
derived from the prescription by \cite{springel05-1}: the local mechanical
heating rate is estimated using the local dissipation by the artificial
viscosity terms, which in this model ultimately derive mainly from the localized
supernova heating. For the work presented here the most important feature of
these simulations is that they provide us with the information necessary for
further subgrid modeling in post-processing mode. Details about this procedure
can be found in KP15b. In particular, the gas density $n$, the mean local
mechanical heating rate $\Gamma_{\rm mech}$~, the local FUV flux $G$ (measured in units of $G_0$~ $= 1.6\times 10^{-3}$erg cm$^{-2}$ s$^{-1}$~) and the mean $A_V$ of the
SPH particles are used. We refer to these as the main physical parameters of the
gas, which are essential to parametrize the state of PDRs that are used, to
obtain the emission maps. This method of computing the emission can be
generally applied to other simulations, as long as they provide these physical
parameters.
The PDR modeling \citep{meijerink2005-1} consists of a comprehensive set of
chemical reactions between the species of the chemical network by
\cite{umist1999}. The main assumption in the extended subgrid modeling, is
that these PDRs are in thermal and chemical equilibrium. We post-process an
equilibrium snapshot of the SPH simulation by applying these PDR models to the
local conditions sampled by each particle, and estimate the column densities of
the molecules, abundances of the colliding species, and the mean gas temperature
of the molecular clouds. These are the main ingredients necessary to compute
the emission emanating from an SPH particle. PDR models are non-homogeneous by
definition, as there are steep gradients in the kinetic temperature and the
abundances of chemical species. By assuming the large velocity gradient (LVG)
approximation \citep{sobolev1960} using {\sc Radex} \citep{radex2}, weighted
quantities from the PDR models were used as input to Radex to compute the
emission of the molecular species studied in this paper \citep{mvk15-a}. For
more details on computing the emission and constructing the emission maps we
refer to Section-2.3 of KP15b, where this method can be applied to any SPH
simulation of a star-forming galaxy that provides the ingredients mentioned
above. For the work in this paper, we do not include any AGN or enhanced cosmic
ray physics, since these are not relevant for the simulation we have used. XDR
and or enhanced cosmic ray models are necessary in modeling the ISM of ULIRGS,
as was discussed by, e.g., \cite{papadopoulos10} and \cite{meijerink11} and
shown in the application of the models to Arp 299 by \cite{2014A&A...568A..90R}.
\subsection{Sampling the high-density gas}
One of the main purposes of this paper is to constrain cloud parameters using
the molecular transitions mentioned earlier. The typical critical densities of
these transitions are between $\sim 10^5$ and $\sim 10^8$~cm$^{-3}$~. The highest
density reached in the SPH simulation is $~10^4$~cm$^{-3}$~, thus the rotational
levels we are interested in are subthermally populated in the non-LTE (local
thermal equilibrium) regime. Therefore the intensity of the emission associated
with these transitions is weak. In-order to make a more realistic
representation of the molecular line emission of the ISM of the simulated
galaxy, we resort to a recipe to sample particles up to $10^6$~cm$^{-3}$~. In what
follows, we describe the prescription we have used to sample particles to such
densities.
The sampling scheme we adopt is based on the assumption that the gas density PDF
of the cold neutral medium (CNM) and the molecular gas is a log-normal function,
given by Eq-\ref{eq:paper4_ln-func} \citep{nrcpp08}:
\begin{equation} \label{eq:paper4_ln-func}
\frac{dp}{d \ln{n}} = \frac{1}{\sqrt{2 \pi \sigma^2}} \exp{ \left( - \frac{1}{2} \left[ \frac{\ln{n} - \mu }{\sigma} \right]^2 \right) }
\end{equation}
\noindent where $\mu$ is the natural logarithm ($\ln$) of the median density
($n_{\rm med}$), and $\sigma$ is the width of the log-normal distribution. Such
PDFs are expected in the inner ($\sim$~kpc) ISM of galaxies. For example,
simulations by \cite{wada01-a} and \cite{wada01-b} reveal that the PDF of the
gas density is log-normally distributed over seven orders of magnitude for
densities ranging from $\sim 4$~cm$^{-3}$~~to $\sim 4 \times 10^7$~cm$^{-3}$~. Log-normal
density distributions have been discussed by other groups as well, and we refer
the reader to Section-\ref{sec:paper4_intro} for more references. The
simulations by Wada are evolved using a grid based code with a box size $\sim
1$~kpc and include AGN feedback, whereas our simulation was evolved with an SPH
code and without AGN feedback with a scale length of $\sim 10$~kpc. The major
assumption in our sampling is that the gas of our model disk galaxy follows a
log-normal distribution for $n > 10^{-2}$~cm$^{-3}$~. Using this assumption we can
sample and add high density gas beyond the maximum of $10^4$~cm$^{-3}$~~of the
simulated galaxy.
Of course such re-sampling does not provide a realistic spatial distribution of
high density gas, as the re-sampled particles are assumed to be placed randomly
within the smoothing length of SPH particles that are dense enough to support
star formation. We can use this simplification as long as we do not try to
construct maps resolving scales smaller than the original spatial resolution of
$\sim$ 0.05 kpc.
In Figure \ref{fig:paper4_T-n-hist}, we show the distributions of the gas
density of our simulation along with that of the temperature. The SPH
simulation consists of $N = 2 \times 10^6$ particles. The distribution of the
temperature has a peak at low temperature and a peak at high temperature,
similar to the PDF in Figure 4 in \cite{wada01-b}. The low temperature peak
around $T = 300$ K corresponds to gas that is thermally stable, whereas the
higher temperature peak around $T = 10000$ K corresponds to gas that is
thermally unstable \citep{Schwarz1972-1}.
On the other hand, the distribution of the density exhibits just one prominent
peak near $n \sim 1$~cm$^{-3}$~, with a small saddle-like feature at a lower density
of $\sim 10^{-2}$~cm$^{-3}$~~ (see the bottom panel of
Figure \ref{fig:paper4_T-n-hist}). The histogram of the density of the stable
gas in the bottom panel, shows that the density of this gas ranges between
$10^{-2} < n < 10^4$~cm$^{-3}$~. The lower bound of this interval is shifted towards
higher densities as the gas populations are limited to lower temperatures,
indicating efficient cooling of the gas for high densities. The temperature of
the SPH particles represents its peripheral temperature. Thus, although the
temperature might be too high for the formation of molecules at the surface,
molecules could form in the PDRs present in the subgrid modeling.
The 1000 K mark seems to be a natural boundary between the thermally stable and
the unstable population (see top panel of Figure \ref{fig:paper4_T-n-hist}).
Hence we use the minimum density range for this population, $n \sim
10^{-2}$~cm$^{-3}$~, as the lower limit for the fit of the PDF that we apply. This is
depicted in Figure \ref{fig:paper4_sampling-fit}, where we fit a log-normal
function to the gas density around the peak at $n = 1$~cm$^{-3}$~.
The density range of the fit is $n_{\rm low} = 10^{-2} < n < n_{high} =
10^2$~cm$^{-3}$~.
To find the best fit of the actual density PDF, we examined other values for
$n_{\rm low}$ and $n_{high}$. For example, choosing $n_{\rm low} =
10^{-1}$~cm$^{-3}$~~ causes the fit PDF to drop faster than the original distribution
for $n > 10^2$~cm$^{-3}$~, which reduces the probability of sampling particles with $n
> 10^5$~cm$^{-3}$~~ to less than one particle in a sample of one billion. On the
other hand, setting $n_{high} = 10^3$~cm$^{-3}$~~ does not affect the outcome of the
fit. Using the best fit density PDF, we notice that only the gas within the
$10^{-2} < n < 10^2$~cm$^{-3}$~~ range is log-normally distributed, where the original
distribution starts deviating from the log-normal fit for $n > 10^2$~cm$^{-3}$~~
(compare blue and red curves in Figure \ref{fig:paper4_sampling-fit}). The
deviation from a log-normal distribution is likely due to the resolution limit
of the SPH simulation.
For example, the PDF of a simulation with $4 \times 10^5$ SPH particles start to
deviate from log-normality for $n > 10^{1.5}$~cm$^{-3}$~.
A similar deviation in the density distribution fit by \cite{wada01-b} is
observed in the $n \gtrsim 10^6$~cm$^{-3}$~~ range, most likely due to similar
numerical
resolution constraints. Here we will assume that the gas density will be
distributed according to a log-normal function in the infinite $N$-limit.
For each SPH particle $i$, parent particle hereafter, we sample from the fit PDF
an ensemble $\{n_s\}$ of 100 sub-particles within the smoothing length of the
parent, where $n_s$ and $n_i$ are the gas number densities of the sampled and
parent particles respectively. The ensemble $\{n_s\}$ is chosen such that $n_s
> n_i$, which enforces having sampled particles that are denser than the parent.
The particles with $n > 10^2$~cm$^{-3}$~~ constitute 2\% of the gas mass, as their
total number is $\sim 40,000$. The PDF of the sampled particles is shown in
green in Figure \ref{fig:paper4_sampling-fit}, where the sampled PDF follows the
fit log-normal distribution very accurately for densities $n > 10^3$~cm$^{-3}$~. The
bias of sampling particles with $n_s > n_i$ affects the shape of the
distribution between $ n = 10^2$~cm$^{-3}$~~ to $10^3$~cm$^{-3}$~. This discrepancy is
resolved by adjusting the weights of the sampled and parent particles along bins
in density range such that the new weights match the fit PDF.
These weights are used to adjust the masses of the parent and sampled particles,
which ensures that the total mass of the system is conserved. The combined PDF
is overlaid as black crosses in Figure \ref{fig:paper4_sampling-fit}.
The re-sampling is restricted to particles with $n_i > 10^2$~cm$^{-3}$~~to ensure that
the sub-particles are sampled in regions that are likely to support
star formation. Thus, most of the far-IR molecular emission results from this
density range. In our sampling, we ignore any spatial dependence on the gas
density distribution, where the PDFs correspond to that of all the particles in
the simulation. Sampling new gas particles with $n > 10^2$~cm$^{-3}$~~ ensures that
these ensembles will lie in regions tracing CO and consequently H$_2$. At the
outskirts of the galaxy, the H$_2$ column density is at least 100 times lower
compared to the center of the galaxy. Moreover, since the density PDF is
log-normal, the number of SPH particles with $n > 10^4$~cm$^{-3}$~~in the outskirts
are so small that no emission from high density tracers is expected.
Thus, we would not have enhanced molecular emission due to the sampling, e.g.
for the high density tracers, at the edge of the galaxy, where it is not
expected to be observed. To address this point, we examine the shape of the gas
density PDF for increasing galactocentric distances. We find that the median of
the log-normal distribution is shifted to lower densities, while the dispersion
becomes narrower towards the edge of the galaxy. Consequently, the relative
probability of finding high density gas at the outer edge of the galaxy is
reduced compared to that of the central region. In fact, the PDF in the region
within $R < 2$~kpc is closely represented by the one we have shown in
Figure \ref{fig:paper4_sampling-fit}. The dispersion of the density PDF for
distances $R < 2$~kpc from the center is $\sigma = 2.05$ is very close to
$\sigma = 2.1$ of the fit PDF in Figure \ref{fig:paper4_sampling-fit}. On the
other hand, the dispersion is $\sigma = 1.2$ for $ 6 < R < 8$~kpc.
Thus, in this paper we focus on the central region of the galaxy, where the same
density PDF is used in sampling the gas particles within this region.
It is worth noting that after the sampling procedure, gas with $n >
10^4$~cm$^{-3}$~~constitutes only 0.5\% of the total gas mass. For example, this is
consistent with the estimate of the filling factor of the densest PDR component
derived from modeling the star-forming galaxy Arp 299 by
\cite{2014A&A...568A..90R}, even though our model galaxy is not necessarily
representative of Arp 299.
The log-normality feature of the density PDF reflects non-linearity in the
evolution of the gas, which we use as an argument for the cascade of turbulence
into small spatial scales where high density gas forms. Here we will take the
$\Gamma_{\rm mech}$~~of the sampled particles to be the same as their parent SPH particle,
although in \cite{mvk15-a}[Figure 2] we have shown that there is some
dependence of $\Gamma_{\rm mech}$~~on the gas density. However, this dependence is not trivial
\cite[see][]{Nordlund99-1}. We note that for our models $\Gamma_{\rm mech}$~~would never be the
dominant heating mechanism for high density gas ($>10^4$~cm$^{-3}$~~ ), while this
could change for modeling ULIRGs.
A similar assumption was made for the $A_V$ and $G$ of the sampled particles,
where the same values as that of the parent were used. As a check we picked a
typical region in the central kpc of the galaxy and quantified the mean and the
standard deviation of $\Gamma_{\rm mech}$~, $G$ and $A_V$ relative to the parent particle
compared to other gas particles within its smoothing length. We found that the
spread in $\Gamma_{\rm mech}$~~is within a factor of two, whereas this spread decreases to within
0.1 for $G$ and $A_V$. Therefore, the assumption of using the same parameters
as the parent for the sampled ones would have a small effect on the produced
emission maps.
\begin{figure}[!th]
\begin{minipage}[b]{1.0\linewidth}
\centering
\includegraphics[scale=1.0]{1.eps}
\end{minipage}
\caption{ (Top) Histogram of the kinetic temperature of the SPH particles.
(Bottom) Histogram of the gas density of the SPH particles. In this same panel,
the histograms for the gas densities of sub-populations of the gas are also
shown. The red, green and cyan histograms correspond to gas particles with
temperature below $T = 1000, 500, 100$~K respectively. These sub-populations of
the gas particles are thermally stable corresponding to the peak around $T =
300$~K in the top panel. The vertical axis refers to the number of SPH
particles within each bin.
\label{fig:paper4_T-n-hist}}
\end{figure}
\begin{figure}[!th]
\begin{minipage}[b]{1.0\linewidth}
\centering
\includegraphics[scale=1.0]{2.eps}
\end{minipage}
\caption{Probability density function of the SPH particles. The blue curve
corresponds to the PDF of gas density $n$ and it is proportional to $d N / d
\log(n)$; in other words this curve represents the probability of finding an SPH
particle within a certain interval of $\log(n)$. The dashed-red curve is the
log-normal fit of the PDF of the blue curve in the range $10^{-2}$ to $10^2$.
The green curve is the PDF of the sampled population from the original SPH
particles while keeping the samples with $n > 10^2$~cm$^{-3}$~.
The black crosses trace the combined PDF of the sampled and the original set of
the SPH particles. The median density and the dispersion are $n_{\rm med} =
1.3$~cm$^{-3}$~~ and $\sigma = 2.1$ respectively.
\label{fig:paper4_sampling-fit}}
\end{figure}
Various other authors have tried to overcome the resolution limit of their
simulations. For example, the sampling procedure by \cite{Narayanan2008-1} is
motivated by observations of GMCs. Their approach entails the sampling of GMCs
by assuming that half of the gas mass is represented by molecules. Moreover,
these GMCs are modeled as spherical and gravitationally bound with power-law
density profiles provided by \cite{rosolowsky05, blitz07} and
\cite{solomon1987-1}.
These sampling methods are similar to the work presented here, in the sense that
they make a (different) set of assumptions to probe high density gas not present
in the simulation. Our basic assumption is a physically motivated extension of
the log-normal distribution.
\section{Modelling dense gas in galaxy disks}
\subsection{Luminosity ladders and emission maps} \label{subsec:paper4_emissionmaps}
The contribution of the high density particles to the total luminosity of each
transitions is illustrated in Figure \ref{fig:paper4_total_luminosity}. The
luminosity of a SPH particle is approximated as the product of the line flux
($F$) with the projected area $(A)$ of the SPH particle. The total luminosity of
a certain emission line of the galaxy is computed as $L = \sum_{i=1}^{i=N} F_i
A_i$, where $i$ is the index of the SPH particles and $N$ is the number of SPH
particles. In the current implementation, the ``area'' of each sampled particle
is assumed to be $A_i / N_s$, where $N_s = 100$ is the number of the sampled
particles. This ensures that the total area of the sampled particles adds up
to the area of the parent particle. Such a normalization is consistent with the
SPH formalism, where all particles have the same mass. The luminosities of the
lines due to the sampling are marginally affected, with a weak dependence on the
transition and the species. The increase in the luminosity is due to the
dependence of the flux of the emission lines on the gas density, where in
general, the flux increases as a function of increasing density.
The dependence of the luminosity on density and its relationship to the density
PDF will be addressed in Section-\ref{section:constrainingpdf}.
The total luminosities as a function of $J$, luminosity ``ladders'', are
computed by considering all the SPH particles within a projected box of $16
\times 16$~kpc. In Figure \ref{fig:paper4_flux-maps} we show the flux maps of a
subset of the molecular lines. These maps give insight on the regions of the
galaxy where most of the emission emanates. In the next section we will use
line ratios in a similar manner to KP15b in order to constrain the gas
parameters within pixels in the central region of the galaxy. The rotational
emission lines we have used are listed in
Table \ref{tbl:paper4_line-ratio-combinations}.
\begin{figure}[!th]
\begin{minipage}[b]{1.0\linewidth}
\centering
\includegraphics[scale=1.0]{3.eps}
\end{minipage}
\caption{Total luminosity of all the molecular transitions for the model galaxy
over a region of $16 \times 16$~kpc. The dashed lines correspond to the original
set of particles, whereas the solid lines represent the combined luminosity of
the sampled and the original sets.
\label{fig:paper4_total_luminosity}}
\end{figure}
\begin{figure*}[!th]
\begin{minipage}[b]{1.0\linewidth}
\centering
\includegraphics[scale=0.68]{4.eps}
\end{minipage}
\caption{Flux maps of the model galaxy of a selection of the CO and $^{13}$CO~~ lines,
and all of the transitions of HCN, HNC, ${\rm HCO}^+$~.
\label{fig:paper4_flux-maps}}
\end{figure*}
\begin{table}[h]
\centering
\begin{tabular}{c c}
\hline
species & $J$-transitions\\
\hline
CO & 1-0 $\rightarrow$ 15-14\\
$^{13}$CO~ & 1-0 $\rightarrow$ 15-14\\
HCN & 1-0 $\rightarrow$ 7-6\\
HNC & 1-0 $\rightarrow$ 7-6\\
${\rm HCO}^+$~ & 1-0 $\rightarrow$ 7-6\\
\hline
\end{tabular}
\caption{Rotational lines used in constructing the flux maps
\label{tbl:paper4_line-ratio-combinations}}
\end{table}
\subsection{Constraining cloud parameters using line ratios} \label{subsec:paper4_constrainig}
With the synthetic luminosity maps at our disposal, we construct various line
ratios and use them to constrain the mechanical heating rate in addition to the
remaining gas parameters, namely $n$, $G$ and $A_V$, throughout the central $<
2$~kpc region of the galaxy. We follow the same approach by KP15b, where we
minimize the $\chi^2$ statistic of the line ratios of the synthetic maps against
these of a mechanically heated PDR model.
We fit the cloud parameters one pixel at a time and compare them to the mean
physical parameters of the gas in that pixel.
In Figure \ref{fig:paper4_fit1}, we show a sample fit for the central pixel of
the model galaxy that has a pixel size of $0.4 \times 0.4$~kpc, which is the
same pixel size that has been assumed by KP15b. For this fit we have considered
the luminosity ladders of CO, $^{13}$CO~, HCN, HNC and ${\rm HCO}^+$~ normalized to the
CO(1-0) transition. In addition to these, we have included the ladder of $^{13}$CO~~
normalized to $^{13}$CO~(1-0) transitions.
\begin{figure}[!th]
\begin{minipage}[b]{1.0\linewidth}
\centering
\includegraphics[scale=0.6]{5.eps}
\end{minipage}
\caption{Sample fit of the line ratios of the central pixel (0.4 x 0.4 kpc$^2$)
with a single PDR model. The points with error bars represent the line ratios
of the specified species from the synthetic maps. The dashed curves correspond
to the line ratios of the best fit PDR model. All the ratios are normalized to
CO(1-0) except for the red ratios which are normalized to $^{13}$CO~(1-0).
\label{fig:paper4_fit1}}
\end{figure}
The gas parameters derived from the fits for pixels of increasing distance from
the center are shown in Table \ref{tbl:paper4_fit1}. In addition to the fit
parameters, we show the mean values of these parameters in each pixel. This
allows us to compare the fit values to the average physical conditions, which is
not possible when such fits are applied to actual observations. We find that a
mPDR constrains the density and the mechanical heating rate to less than a
half-dex and the visual extinction to less than a factor of two. On the other
hand, the FUV flux ($G$) is largely unconstrained. We note that a pure PDR that
is not mechanically heated (not shown here), fails to fit most of the ratios
involving $J > 4-3$ transitions, and leads to incorrect estimates of the gas
parameters.
\begin{table*}[!tbh]
\centering
\begin{tabular}{c c | c c c c | c }
\hline
$R$ & & $\log_{10}[n]$ & $\log_{10}[G]$ & $\log_{10}[\Gamma_{\rm mech}]$ & $A_V$ &$\chi^2_{\rm red}$ \\
\hline
$ < 0.4$ & mPDR & $1.9 \pm 0.2$ & $-0.6 \pm 1.9$ & $-23.0 \pm 0.2$ & $12.2 \pm 0.4$ & $1.7 \pm 0.5$ \\
& Actual & $1.9 \pm 0.2$ & $1.8 \pm 0.1$ & $-23.4 \pm 0.2$ & $11.2 \pm 1.3$ & -- \\
\hline
$ \sim 0.8$ & mPDR & $1.9 \pm 0.3$ & $1.4 \pm 3.0$ & $-23.1 \pm 0.3$ & $12.8 \pm 2.3$ & $1.3 \pm 0.2$ \\
& Actual & $1.8 \pm 0.1$ & $1.7 \pm 0.1$ & $-23.3 \pm 0.1$ & $9.4 \pm 1.1$ & -- \\
\hline
$ \sim 1.0$ & mPDR & $1.9 \pm 0.2$ & $2.0 \pm 2.3$ & $-23.1 \pm 0.1$ & $12.0 \pm 4.4$ & $1.6 \pm 0.2$ \\
& Actual & $1.7 \pm 0.1$ & $1.7 \pm 0.1$ & $-23.3 \pm 0.1$ & $8.5 \pm 0.6$ & -- \\
\hline
$ \sim 1.5$ & mPDR & $1.7 \pm 0.4$ & $0.4 \pm 2.0$ & $-23.3 \pm 0.1$ & $8.2 \pm 1.3$ & $1.0 \pm 0.1$ \\
& Actual & $1.5 \pm 0.1$ & $1.3 \pm 0.1$ & $-23.7 \pm 0.1$ & $5.5 \pm 0.4$ & -- \\
\hline
\end{tabular}
\caption{Model cloud parameters fits for pixels of increasing distance $R$ (in
kpc) from the center of the galaxy. These parameters are $n$ (in cm$^{-3}$~), $G$ (in
units of $G_0$), $\Gamma_{\rm mech}$~~(in erg cm$^{-3}$ s$^{-1}$~) and $A_V$ (in mag). The values in the row
(mPDR) correspond to the average fit parameters and the dispersion for pixels at
a distance $R$ from the center. The actual mean values of the cloud parameters
are listed in the following row labeled ``Actual''.
We use 51 transitions in these fits and minimize the $\chi^2_{\rm red}$ by
varying four parameters, thus the DOF of the fit are 51-4 = 47. In the last
column we show the mean value of $\chi^2_{\rm red}$, which is the value of the
$\chi^2$ per degree of freedom minimizing the fits. Smaller $\chi^2_{\rm red}$
imply better fits but not necessarily good estimates of the actual values of the
cloud parameters.
\label{tbl:paper4_fit1}}
\end{table*}
The molecular gas is indirectly affected by the FUV flux via the dust. The dust
is heated by the FUV radiation, which in turn couples to the gas and heats it
up.
In addition to the FUV flux, this process depends also on the gas density,
where
it becomes very efficient for $n > 10^4$~cm$^{-3}$~~ and $G \gtrsim 10^3$. In our
case, the FUV flux is unconstrained mainly because the density of the gas is
significantly lower than the critical densities of the $J > 4-3$ transitions of
CO and $^{13}$CO~~ and the transitions of the high density tracers, where 99\% of the
gas has a density less than $10^3$~cm$^{-3}$~. These transitions are subthermally
excited and their fluxes depend strongly on density. On the other hand, the
emission grids of these transitions as a function of $n$ and $G$ show a weak
dependence on the FUV flux. For example in \cite{mvk15-a}[Figure 7] we
show that $\Gamma_{\rm mech}$~~plays a more important role in heating the gas in the molecular
zone compared to the heating due to the coupling of the dust to the gas.
Despite the fact that $G$ is not well constrained using the molecular lines we
have considered, which was also the case in KP15b, we did succeed to fit the
line ratios from the synthetic maps using a mechanically heated PDR.
We also learn from this exercise that it is possible to constrain $n$, $\Gamma_{\rm mech}$~~and
$A_V$ with high confidence within an order of magnitude using line ratios of
high density tracers as well as CO and $^{13}$CO~.
It is common to have large
degeneracies when using line ratios to constrain cloud parameters. Such
degeneracies arise mainly due to the small number of line ratios used in the
fits, while including additional line ratios reduces the degeneracy in the
parameter space. Another reason to having degeneracies in the interpretation
of line ratios is the assumption that the molecular lines used in these
ratios emanate on average from the same spatial region of the gas. This is
of-course not always true as it is evident in high resolution galactic
studies \citep[e.g.,][]{meier12-1, meier14-1}. The reduction in the
degeneracies in the parameter space is illustrated in
\cite{mvk15-a}[Figure 16], where using four independent line ratios of a sample of
LIRGS shrinks the degeneracy to less than half a dex in the $n - G$ parameter.
Possible ways of constraining $G$ will be discussed in
Section-\ref{sec:paper4_discussion}. The main reason why we manage to fit the
synthetic line ratios well is because the range in the densities where most of
the emission emanates is between 10 and 1000~cm$^{-3}$~~ (see
Figure \ref{fig:paper4_cumlum}). This narrow range in the density is due to
the narrow density PDF of our galaxy model. In the next section, we explore
the effect of increasing the median and the width of the density PDF on the
relative contribution of the whole span of densities to the flux of molecular
line emission of mechanically heated PDRs.
\begin{figure}[!th]
\begin{minipage}[b]{1.0\linewidth}
\centering
\includegraphics[scale=1.0]{6.eps}
\end{minipage}
\caption{Cumulative distribution of the densities and the luminosities of
CO(1-0), ${\rm HCO}^+$~(1-0) and HCN(1-0) for the whole galaxy. For example, $\sim$~90\% of
the gas particles have a density $n < 10^2$~cm$^{-3}$~, whereas 75\%, 40\%, 20\% of the
emission of CO(1-0), ${\rm HCO}^+$~(1-0) and HCN(1-0) emanate from gas whose density is
lower than $10^2$~cm$^{-3}$~.
\label{fig:paper4_cumlum}}
\end{figure}
\section{Constraining the gas density PDF} \label{section:constrainingpdf}
The gas density PDF is a physical property of the ISM that is of fundamental
interest because it gives information about the underlying physical processes,
such as the dynamics of the clouds and the cooling and heating mechanisms. A
log-normal distribution of the density is expected if the ISM is supersonically
turbulent \citep{Vazquez94-1, Nordlund99-1, wada01-a, wada07-1, Burkhart13-1,
hopkins13-1}, and this simple picture can explain, amongst others, the stellar
IMF, cloud mass functions, and correlation patterns in the star formation
\citep{hopkins12-1, hopkins12-2, hopkins13-1}. A log-normal distribution was
assumed in Section-\ref{sec:paper4_methods} which allows us to extrapolate the
density structure in the simulation beyond the resolution limit, in a
statistical sense.
The question remains whether it is possible to derive the properties of the
density PDF from observations. This is probably only doable for the molecular
ISM.
There is limited scope for deriving the density PDF of the warm and cold neutral
phase of the ISM, as the HI 21 cm line is not sensitive to variations in density.
For the ionized phases, we do not expect a simple log-normal density PDF, as the
turbulence may not be supersonic and or the length scales exceed the scale
lengths in the galaxy disk. For the colder gas, where we expect a simple
functional and relatively universal form, there is at least the prospect of
probing the density structure using the molecular line species, which come into
play at higher densities. Indeed, we may well reverse our approach taken in
Sections-\ref{subsec:paper4_constrainig}, and try to constrain the density PDF
(and the conditions in molecular clouds) using the emission of molecular
species. Such an approach is possible because we have assembled a large
database of PDR models and resulting line emission, covering a wide range of
parameter space.
In the remainder of this section, we will explore a limited part of the
parameter space to examine the contribution from different line species for
different density PDFs under some limiting assumptions. For completeness, we
recapitulate the assumptions (some of which are implicit): 1) simple functional
forms of the density PDF, where $G$, $A_V$ and~$\Gamma_{\rm mech}$~~are taken constant
\citep{Nordlund99-1}; 2) emission from each density bin in the PDF is assumed
to come from a PDR region at that density with a fixed $A_V$, with the density dependence
coming from emission line-width and cloud size relation \citep[e.g,
][]{larson1981}.
3) chemical and thermal equilibrium is assumed for the PDR models.
We have seen in Section-\ref{subsec:paper4_emissionmaps} that the high density
tracers, in the simulation of the quiescent disk galaxy, have a limited effect
on the emission ladders. That is because the density PDF drops off very fast in
the simulation.
This ultimately derives from the exponential decay in the assumed log-normal
distribution. For this reason, we will consider broader dispersions in the PDF.
In this paper, we adopt log-normal PDFs for the density, although a more relaxed
power law distribution could also be considered. Such a power law decay of the
density PDF is found in some models of supersonic turbulence when the effective
equation of state has a polytropic index, $\gamma$, smaller than one, and
temperatures strongly decrease with increasing density~\citep{Nordlund99-1}.
\subsection{Parameter study}
In this section, we compute the mean flux of molecular line emission emanating
from a volume of gas whose density is log-normally distributed. In the first
part, we explore the contribution of gas, of increasing density, to the mean
flux.
Particularly, we look for the necessary parameters of the PDF to obtain a
significant ($> 10$\%) contribution of the high density gas to the mean flux.
In the second part, we use line ratios of HCN(1-0), HNC(1-0) and ${\rm HCO}^+$~(1-0) for
star-forming galaxies to constrain the parameters of the density PDF of such
systems.
In computing the mean flux for the gas in the log-normal regime, the flux
emanating from gas within a certain density range, should be weighted by the
probability of finding it within that range. The mean flux is computed by
summing all these fluxes for all the density intervals in the log-normal regime.
In other words, the mean flux for a volume of gas is given by:
\begin{eqnarray} \label{eq:paper4_Ln}
\bar{F} & =& N \int_{n_1}^{n_2} {{\rm F}(n) \times \rm PDF}(n)~d \ln n \\
M & =&N \int_{n_1}^{n_2} {n \times \rm PDF}(n)~d \ln n,
\end{eqnarray}
\noindent where $N$ is a normalization factor which scales the flux ($\bar{F}$)
depending on the molecular gas mass ($M$) of the region, ${\rm PDF}(n)$ is the
gas density PDF we assume for the region considered, and ${\rm F}(n)$ is the
emission flux of a given line as a function of gas density (for a fixed $G$,
$\Gamma_{\rm mech}$~~and $A_V$).
Typically the bounds of the integrals are dictated by the density of the
molecular clouds, where the gas density is in the log-normal regime. A density
of 1~cm$^{-3}$~~ is a good estimate for the lower bound since no molecular emission is
expected for gas with $n$ less than that. The upper bound of the integral in
Eq-\ref{eq:paper4_Ln} can be as high as $10^6$-$10^8$~cm$^{-3}$~.
${\rm PDF}(n)$ is a Gaussian in log scale, which decays rapidly whenever the gas
density is 1-$\sigma$ larger than the mean. While ${\rm PDF}(n)$ is a
decreasing function of increasing density, ${\rm F}(n)$ is an increasing
function of increasing $n$.
Generally, the molecular gas mass in the central few kpc of star-forming
galaxies is on the order of $10^9-10^{10}$~${\rm M}_{\odot}$~\citep{Scoville91-1,
Bryant99-1}. This estimate, or a better one if available, can be used to
compute $N$ in Eq-\ref{eq:paper4_Ln}. We refer to the quantity ${\rm F}(n)
\times {\rm PDF}(n)~d \ln n$ as the weighted flux.
\subsection{Weighted fluxes}
The median density and the dispersion of the log-normal fit in
Section-\ref{sec:paper4_methods} are $n_{\rm med} = 1.3$~cm$^{-3}$~~and $\sigma =
2.1$ respectively, corresponding to a mean density of $\sim 10$~cm$^{-3}$~. The mean
density is much smaller than the critical densities of most of the transitions
of HCN, HNC and ${\rm HCO}^+$~. For this reason all of the emission of HCN(1-0) and
${\rm HCO}^+$~(1-0) originates from gas with $n < 10^4$~cm$^{-3}$~~ in
Figure \ref{fig:paper4_cumlum}. In Figure \ref{fig:paper4_syntetic_lum_PDF},
we show the weighted fluxes of gas of increasing density for HCN(1-0). The
fluxes are determined by computing $F$ in Eq-\ref{eq:paper4_Ln} for intervals in
$\log n$.
Similar distributions can be computed for other emission lines of high density
tracers, which are expected to be qualitatively similar. The rows in
Figure \ref{fig:paper4_syntetic_lum_PDF} correspond to the PDFs in
Figure \ref{fig:paper4_syntetic_pdfs}. Along the columns we vary the physical
cloud parameters over most of the expected physical conditions. $G$, $\Gamma_{\rm mech}$~~and
$A_V$ are varied in the first, second and last column, respectively.
In exploring the possible ranges in $G$ and $\Gamma_{\rm mech}$~~for the PDF of the SPH
simulation, we see that the peak of the emission is restricted to $n <
10^3$cm$^{-3}$~. The only situation where the peak is shifted towards densities
higher than $10^3$~cm$^{-3}$~~ occurs when the mean $A_V$ of the clouds in the galaxy
is $\sim 1$~mag. However, in this case, the flux would be too weak to be
observed. Hence with the combinations of these parameters, and with such a
log-normal density PDF, it is not possible to obtain a double peaked PDF,
or a gas density distribution with significant contribution from $n >
10^4$~cm$^{-3}$~~ gas. Thus, most of the emission of the high density tracers is from
gas with $n < 10^4$~cm$^{-3}$~. The analogous plots of
Figures-\ref{fig:paper4_syntetic_pdfs} and \ref{fig:paper4_syntetic_lum_PDF}
where the dispersion is varied for median densities of 12.5~cm$^{-3}$~~ and 100~cm$^{-3}$~~
are shown in the appendix.
\begin{figure}[!th]
\begin{minipage}[b]{1.0\linewidth}
\centering
\includegraphics[scale=1.0]{7.eps}
\end{minipage}
\caption{Effect of varying the median and the
dispersion of a density PDFs.
The blue curves represent PDFs that have the same median density as the one used
in SPH simulation, but with increasing dispersions. The green curve has the
same dispersion as that of the SPH simulation but with a higher median density.
Finally, the red curve corresponds to the density PDF by Wada (2001). The mean
densities corresponding to these PDFs are also listed in the legend.
\label{fig:paper4_syntetic_pdfs}}
\end{figure}
\begin{figure*}[!tbhp]
\begin{minipage}[b]{1.0\linewidth}
\centering
\includegraphics[scale=0.6]{8.eps}
\end{minipage}
\caption{Weighted fluxes for HCN(1-0) for different density PDFs. In the first
three rows the median density of the PDFs corresponds to that of the SPH
simulation, whereas the dispersion is increased from 2.1 to 2.7 to 3.5. In the fourth
row, the median density is increased by a factor of 100 compared to that of the
SPH simulation, but the dispersion is kept fixed at 2.1.
In the last row we show the weighted fluxes for the PDF obtained by the Wada
2001 simulation.
In the first column the FUV flux is varied, from 100 times the flux at the solar
neighborhood to $10^6$$G_0$~~corresponding to the FUV flux in extreme starbursts
($A_V = 10$~mag, $\log_{10}\Gamma_{\rm mech} = -22$). In the middle column we
vary $\Gamma_{\rm mech}$~~from the heating rate corresponding to quiescent disks to rates typical
to violent starbursts with a SFR of 1000 ${\rm M}_{\odot}$~per year ($A_V = 10$~mag,
$\log_{10} G = 2$). In the last column the $A_V$ is varied from 0.1 to 10 mag,
corresponding respectively to the typical values for a transition zone from
H$^+$ to H and for dark molecular clouds ($\log_{10} G = 2$,
$\log_{10}\Gamma_{\rm mech} = -22$). The onset of emission is determined mainly
by $\Gamma_{\rm mech}$~; for instance in looking at the middle column, we see that
this onset of emission corresponds to $n \sim 10^2$~cm$^{-3}$~, where for lower densities
H$_2$ does not form, which is essential for other molecules such as HCN to form.
For each curve in every panel we also plot (with filled circles)
the density, $n_{90}$, where 90\% of the emission emanates from $n < n_{90}$.
For example, in the top row, even for the most intense FUV flux $n_{90} \sim
10^{3.5}$~cm$^{-3}$~. On the other hand, in bottom row, $n_{90} > 10^5$~cm$^{-3}$~.
We note that the curves corresponding to $\Gamma_{\rm mech}$~~$10^{-50}$ (a pure PDR) and $10^{-30}$
erg cm$^{-3}$ s$^{-1}$~~in the second column overlap.
\label{fig:paper4_syntetic_lum_PDF}}
\end{figure*}
For density PDFs with broader dispersions, $\sigma =$~2.7 and 3.5, respectively,
it is possible to obtain luminosity distributions where at least 10\% of the
emission is from gas with densities $> 10^5$cm$^{-3}$~~ (see second, third and last
rows in Figure \ref{fig:paper4_syntetic_lum_PDF}). In multi-component PDF fits,
the filling factor of the ``denser'' component is on the order of a few percent
(by mass or by area). In the Figure \ref{fig:paper4_syntetic_lum_PDF}, we see
that it is necessary to have a broad dispersion in the PDF in order to have a
distribution of the luminosity where at least 10\% of it emanates from gas with
$n > 10^{5}$~cm$^{-3}$~.
When comparing the second and fourth\footnote{The PDF of the fourth row
corresponds to that of the simulation by Wada (2001)} rows of the weighted
fluxes in Figure \ref{fig:paper4_syntetic_lum_PDF}, we see that they are quite
similar.
Thus, it seems that a low median density and a broad dispersion (second row)
results in the same weighted fluxes as a PDF with a 100 times higher median
density and a narrower dispersion (fourth row). To check for such degeneracies
and constrain the PDF parameters using molecular emission of high density
tracers, we construct line ratio grids as a function of $n_{\rm med}$ and
$\sigma$.
It has been suggested that HCN(1-0) is a better tracer of star formation than
CO(1-0) because of its excitation properties
\citep{Gao04-1}.
As a proof of concept, in Figure \ref{fig:paper4_line_ratio_vs_density_PDF}, we
show a grid of line ratios of HCN(1-0)/HNC(1-0) and HCN(1-0)/${\rm HCO}^+$~(1-0) as a
function of the mean and the dispersion of the density PDF.
The line ratios of a sample of 117 LIRGS \cite{loenen2008}[Figures-1 and 2]
show that in most of these galaxies $1 \lesssim $~HCN(1-0)/HNC(1-0)$ \lesssim 4
$ and $0.5 \lesssim$~HCN(1-0)/${\rm HCO}^+$~(1-0)$ \lesssim 3$. The overlapping regions
in the line ratio grids in Figure \ref{fig:paper4_line_ratio_vs_density_PDF}
correspond to $2.54 < \sigma < 2.9$.
The width of a log-normal density PDF is related to the Mach number,
$\mathcal{M}$, in that medium via $\sigma^2 \approx \ln (1 + 4 \mathcal{M}^2 /
3)$ \citep{Krumholz07-1, hopkins12-2}. By applying this relationship to the
range in $\sigma$ constrained by the observations we find that $ 29 <
\mathcal{M} < 77$. This range in $\mathcal{M}$ is consistent with that of
violent starbursts that take place in extreme star-forming regions and galaxy
centers \citep{Downes98-1}.
\begin{figure}[!tbhp]
\begin{minipage}[b]{1.0\linewidth}
\centering
\includegraphics{9.eps}
\end{minipage}
\caption{{\bf Top} Line ratios of HNC(1-0)/HCN(1-0) as a function of the mean
and the dispersion of the PDF. {\bf Bottom} Line ratios of ${\rm HCO}^+$~(1-0)/HCN(1-0)
as a function of the mean and the dispersion of the PDF. The dark brown stripe represents
observational data. The overlapping
region of these two line ratios is colored in black in the bottom panel. For
all the grid points a typical $A_V = 10$~mag and $\Gamma_{\rm mech}$~~$=10^{-22}$~cm$^{-3}$~~ is used.
In both panels, Mach numbers are reported on the right axes and G is 100.
\label{fig:paper4_line_ratio_vs_density_PDF}}
\end{figure}
\section{Discussion} \label{sec:paper4_discussion}
The molecular emission of star-forming galaxies usually require more than one
PDR component to fit all the transitions. Typically, a low density PDR
components is needed to fit low-$J$ transitions, e.g., for CO and $^{13}$CO~, whereas
a high density $n > 10^4$~cm$^{-3}$~~ component is needed to fit the $J > 6-5$
transitions of these two species and the high density tracers. In the inner $<
0.1 - 1$~kpc a mechanically heated PDR and/or an XDR might be necessary in the
presence of an AGN or extreme starbursts.
In the first part of the paper, we sampled high density gas in our model
star-forming galaxy simulation by assuming the gas density is a log-normal
function, in order to account for the emission of the high density gas that was
missing in the model galaxy due to resolution constraints. Since the dispersion
of the density PDF of the model galaxy is narrow with $\sigma = 2.1$,
corresponding to a Mach number $\mathcal{M} \sim 10$, the gas with density $n >
10^4$~cm$^{-3}$~~ contributes $< 1\%$ of the total luminosity of each line.
Consequently we were able to recover $n$, $\Gamma_{\rm mech}$~~and $A_V$ of the gas parameters
within 2 kpc from line ratios of CO, $^{13}$CO~, HCN, HNC and ${\rm HCO}^+$~, reasonably well
using a one component mechanically heated PDR model.
The FUV flux was constrained less accurately, since $\Gamma_{\rm mech}$~~is a dominant heating
term at $A_V \gtrsim 1$~mag, where most of the molecular emission emanates. We
have seen in \cite{mvk15-a} that in the non-LTE regime, the molecular emission
is almost independent of the FUV flux. This is not the case for $n \gtrsim
10^4$~cm$^{-3}$~, however for such high densities the line ratio grids depend strongly
on $G$, thus whenever the mean gas density is $> 10^4$~cm$^{-3}$~~and $G > 10^4$, e.g.
in ULIRGS, we expect to constrain the FUV flux with high certainty. In this
case it might be necessary to model the emission with more than one PDR
component as was done by \cite{2014A&A...568A..90R}. In this paper, the CO
ladder of the ULIRG Arp 299 was fit using three PDR models and the FUV flux was
well constrained only for the densest component with $n = 10^{5.5}$cm$^{-3}$~~(See
\cite{2014A&A...568A..90R}[Figure 5]).
Another possible way of constraining $G$ is using diagnostic line ratios of
atomic fine-structure lines. Since atomic fine-structure emission originates
from $A_V< 1$~mag, it depends strongly on $G$. Consequently, these line ratios
show a strong dependence on $G$ as well \citep[see review by][and references
therein]{2013RvMP...85.1021T}.
This is valid even for the most extreme $\Gamma_{\rm mech}$~~rates (see Fig-A3 by
\cite{mvk15-a}).
Modeling the ISM as discrete components is not a realistic
representation, especially in ULIRGS, since the gas in such environments is
expected to be distributed log-normally and in some cases the distribution could
be a power-law (depending on the adiabatic index of the equation of state).
We have studied the contribution of the density function to the mean flux for
different parameters defining a log-normal probability functions and showed that
a broad dispersion is required to obtain significant emission for the $n >
10^4$~cm$^{-3}$~~ clouds. The main advantage of such modeling is in interpreting
un-resolved observations of star-forming galaxies where FUV heating and
mechanical heating may play an important role. This approach is more appropriate
than fitting the observations with one or more PDR components, since in such
modeling the gas is assumed to be uniformly distributed with discrete densities.
The number of free parameters arising from multi-component fits would be much
higher than fitting the parameters of the gas density PDF, which has just two.
This reduces the degeneracies and gives us information on the gas density PDF
and about the turbulent structure and the Mach number, which is directly related
to the density PDF and the mechanical heating rate.
The main caveat in our approach is the assumption that the density PDF for $n >
10^{-2}$~cm$^{-3}$~~is a log-normal function and that the mechanical heating rate used
in the PDR models is independent of the width of density PDF, A$_V$, $G$ and the
line-width. The reason for adopting this assumption is based on the fact that a
relationship between gas density, and consequently the PDF, and the mechanical
heating rate is lacking \citep{wheeler80-1,scilich96-1,freyer03-1,freyer06-1}.
Ultimately the mechanical heating rate derives from the cascade of turbulence to
the smallest scales due to a supernova event, where typically an energy transfer
efficiency of 10\% is assumed \citep{loenen2008}.
The outcome of recovering cloud parameters by independently varying them in the
fitting exercise is a good step towards probing possible relationships between
these parameters and mechanical heating.
We have demonstrated the possibility of constraining the density PDF using line
ratios of HCN(1-0), HNC(1-0) and ${\rm HCO}^+$~(1-0), where the derived Mach number is
consistent with previous predictions. Grids of diagnostics involving other
molecular species can also be computed \cite[see reviews by][and references
therein]{wolfire2011-1, bergin2011-1,aalto2014-1} in order to constrain the
properties of the PDF, but that is beyond the scope of this paper.
Moreover, we have used the flux ratios of molecular lines as diagnostics, but it
is also possible to use the ratio of the star formation rate (SFR) to the line
luminosity to constrain the PDF as is done by \cite{Krumholz07-1}.
This is in fact quite interesting, since the SFR can be related to the
mechanical heating rate as was done by \cite{loenen2008}. By doing so, a
tighter constraint on the mechanical heating rate can be imposed, instead of
considering it as a free parameter as we have done in our fitting procedure.
\section{Summary and Conclusion}
We have constructed luminosity maps of some molecular emission lines of a
disk-like galaxy model. These emission maps of CO, $^{13}$CO~, HCN, HNC, and
${\rm HCO}^+$~~have been computed using subgrid PDR modeling in post-processing mode.
Because of resolution limitations, the density of the simulation was restricted
to $n < 10^4$~cm$^{-3}$~. We demonstrated that the density PDF is log-normal for $n >
10^{-2}$~cm$^{-3}$~. Most of the emission of the high density tracers emanates from
the gas with densities $\sim 10^2$~cm$^{-3}$~~ for quiescent galaxies, which is at
least 1000 times lower than the critical density of a typical high
density tracer. We attribute this to the fact that the dispersion of the PDF is
narrow, and thus the probability of finding dense gas is low.
The main findings of this paper are:
\begin{itemize}
\item It is necessary to have a large dispersion in the density PDF ($\sigma >
2.7$) in order to have significant emission of high density tracers from $n >
10^4$~cm$^{-3}$~~ gas.
\item It is possible to constrain the shape of the PDF using line ratios of high
density tracers.
\item Line ratios of HCN(1-0), HNC(1-0), and ${\rm HCO}^+$~(1-0) for star-forming galaxies
and starbursts support the theory of supersonic turbulence.
\end{itemize}
A major caveat for this approach is the assumption concerning the thermal and
the chemical equilibrium. Care must be taken in interpreting and applying such
equilibrium models to violently turbulent environments such as starbursting
galaxies and galaxy nuclei.
Despite the appealing fact that the line ratios obtained from the example we
have shown in Figure \ref{fig:paper4_line_ratio_vs_density_PDF} favor high Mach
numbers ($ 29 < \mathcal{M} < 77$) consistent with previous prediction of
supersonic turbulence in starbursts, a time-dependent treatment might be
essential.
\begin{acknowledgements}
M.V.K is grateful to Alexander Tielens for useful comments and suggestions.
M.V.K also thanks the anonymous referee and Volker Ossenkopf for their
critical reviews on the manuscript that helped improve it significantly.
\end{acknowledgements}
\bibliographystyle{aa}
|
2,877,628,089,023 | arxiv | \section{Introduction}\label{sec:intro}
During the recent years, Generative Adversarial Networks (GANs)~\cite{goodfellow2014generative} have emerged as the dominating generative learning paradigm for the task of image synthesis, and they continue to dramatically change a wide and diverse range of disciplines, such as super-resolution~\cite{ledig2017photo}, face restoration~\cite{yang2021gan}, and editing~\cite{talktoedit2021iccv}, while they have also been incorporated in discriminative tasks~\cite{xu2021generative}. Despite their remarkable capability of capturing and modeling image data distributions through a semantically rich latent space, traversing a pre-trained GAN's latent space in an interpretable/controllable manner remains an open problem.
Exploring the latent space of a pre-trained GAN in an interpretable manner has drawn significant attention from the research community during the recent years~\cite{ganspace2020harkonen,voynov2020unsupervised,sefa2021cvpr,oldfield2021tensor,warpedganspace2021iccv,goetschalckx2019ganalyze,shen2020interpreting,plumerault20iclr,jahanian20iclr,abdal2021styleflow,wu2021stylespace,shen2020interfacegan}. Typically, these works first discover a set of linear or non-linear latent paths, in an unsupervised or (semi)supervised manner, and then try to label them based either on laborious manual annotation (e.g.,~\cite{voynov2020unsupervised,ganspace2020harkonen}) or by incorporating pre-trained detectors (e.g., smile detector)~\cite{warpedganspace2021iccv}. While using pre-trained detectors may label effectively the discovered latent paths, this process might ignore certain generative factors present in the GAN latent space since a) there is limited availability of pre-trained detectors for attributes of interest for the given pre-trained GAN, and b) pre-trained detectors are inherently limited to a closed set of visual concepts/attributes.
To address the above limitations, a few recent works have incorporated joint vision-language models due to their inherent capability of expressing a much wider set of visual concepts. TediGAN~\cite{xia2021tedigan} proposes a novel GAN inversion technique that can map multi-modal information (i.e., texts, sketches, labels) into a common latent space of a pre-trained StyleGAN~\cite{stylegan2_karras20cvpr}. Then, using visual-linguistic similarity learning, it provides an interactive system where the user provides input textual guidelines and the system generates diverse images given the same input text, allowing the user to edit the appearance of different attributes interactively. However, for doing so, it relies on both annotated training data and the hierarchical structure of the StyleGAN architecture. More specifically, the authors of~\cite{xia2021tedigan} introduce and use the Multi-Modal CelebA-HQ, a curated dataset of face images, that is augmented with a high-quality segmentation mask, sketch, and descriptive text. Then, they rely on the hierarchical characteristic of StyleGAN's $\mathcal{W}$-space (i.e., style-mixing) to learn the text-image matching by mapping the image and text into the same joint embedding space. By contrast, our method is not limited to the task of face editing, nor tied to a specific GAN architecture (i.e., StyleGAN) and, thus, can be applied to any latent space. Another recent work, closely related to ours, is StyleCLIP~\cite{patashnik2021styleclip}, which utilizes a CLIP~\cite{clip-radford2021icml}-based similarity loss in order to learn a StyleGAN-specific latent mapper network that infers a text-guided latent manipulation step for a given input image. To do so, StyleCLIP uses the CLIP text features of the given text prompt as the target vector and tries to align it with the CLIP image features of the manipulated image, adopting the standard text-image cosine similarity method (left part of Fig.~\ref{fig:standard_vs_proposed_sim}). However, the adopted loss criterion disregards a) the relative position of the manipulated and the original image in the image embedding, and b) the relative position of the image and the text embeddings since all are expressed in relation to the origin of the axes in the CLIP embedding space. This leads to abrupt image manipulations and quickly arrives at regions of low density and, thus, low image quality, as shown by the experimental results.
\begin{figure}[t]
\centering
\includegraphics[width=\textwidth]{figs/standard_vs_proposed_sim.pdf}
\caption{Methodological comparison between the proposed and the standard text-image cosine similarity loss calculation. Standard text-image similarity considers the features of a given image and the features of a given text prompt (encoded by CLIP) and calculates their cosine distance. We propose the use of local text directions along the non-linear paths that are induced by the warping of the text space due to a pair of contrasting sentences (semantic dipole).}
\label{fig:standard_vs_proposed_sim}
\end{figure}
In this work, we propose to learn non-linear interpretable paths in the latent space of a pre-trained GAN driven by \textit{semantic dipoles} given in natural language. More specifically, we define pairs of sentences, each pair corresponding to one trainable non-linear path in the GAN latent space and expressing contrasting meanings in natural language. When represented in the CLIP text space, the aforementioned pairs form ``dipoles'' that serve as the ``limits'' of the interpretation that we require by the optimised latent paths to encode. Then, we use the poles (i.e., the feature representation of each sentence in the pair) as the centres of an RBF-based warping function that endows the text space with a family of non-linear curves, providing end-to-end directional paths from the one pole/sentence to the other (Fig.~\ref{fig:summary}). By doing so, we decouple the supervisory text features from the origin of axes, establishing at the same time only local relations between the text and the image features. This leads to non-linear text paths that are smooth and allow for mild transition between the given point and the desired end. This is illustrated in the right part of Fig.~\ref{fig:standard_vs_proposed_sim}. As a result, transition to images described by the desired text prompts is smoother and longer than those of StyleCLIP~\cite{patashnik2021styleclip}, while traveling to regions of low density is significantly prevented, as will be shown in the experiments. The main contributions of this paper can be summarized as follows:
\begin{itemize}
\item We propose a method for traversing the latent space of a pre-trained GAN in an interpretable manner driven by semantic dipoles in natural language.
\item We model non-linear paths in the CLIP text space which allow for smooth transition from one pole to the other, preventing low-density regions/artifacts.
\item Our method is model-agnostic and not tied to any specific GAN architecture (e.g., StyleGAN), such as~\cite{xia2021tedigan} and~\cite{patashnik2021styleclip}.
\item We apply our method to two different GAN architectures (i.e., ProgGAN~\cite{proggan_karras18iclr}, and StyleGAN2~\cite{stylegan2_karras20cvpr}) and compare with GANSpace~\cite{ganspace2020harkonen}, WGS~\cite{warpedganspace2021iccv}, and StyleCLIP~\cite{patashnik2021styleclip}. We show that in comparison to state-of-the-art vision-language StyleCLIP~\cite{patashnik2021styleclip}, our method produces longer and more disentangled interpretable paths, generating images of higher quality with far better control of the generative factors.
\end{itemize}
\section{Related Work}\label{sec:related_work}
\subsection{Interpretable latent paths in pre-trained GAN generators}
Exploring the latent space of a pre-trained GAN in an interpretable manner has typically been approached by the research community by discovering a set of linear~\cite{voynov2020unsupervised,ganspace2020harkonen} or non-linear~\cite{warpedganspace2021iccv} latent paths, in an unsupervised or (semi)supervised manner, and then trying to label them based either on laborious manual annotation (e.g.,~\cite{voynov2020unsupervised,ganspace2020harkonen}) or by incorporating pre-trained detectors~\cite{warpedganspace2021iccv}. GANSpace~\cite{ganspace2020harkonen} performs PCA on deep features at the early layers of the generator and finds directions in the latent space that best map to those deep PCA vectors, arriving at a set of non-orthogonal directions in the latent space. Voynov and Babenko~\cite{voynov2020unsupervised} proposed an unsupervised method to discover linear interpretable latent space directions by requiring that the aforementioned directions lead to image transformations easily distinguishable from each other by a discriminator/reconstructor network. WarpedGANSpace (WGS)~\cite{warpedganspace2021iccv} extended this to the non-linear case by optimising a set of RBF-based functions, each giving rise to a family of interpretable paths in the latent space. However, whilst our method builds on them, the discovery of the paths in the latent space of GANs is instead driven by textual descriptions provided in natural language that define dipoles in the the CLIP text space.
\subsection{Vision-language models}
Cross-modal vision-language representations have recently drawn attention towards a plethora of related tasks, such as image captioning, visual question answering, and language-based image retrieval~\cite{li2020oscar,chen2020uniter,li2020unicoder,lu2019vilbert,sariyildiz2020learning,desai2021virtex}. This line of research has been revolutionized by the use of Transformers~\cite{vaswani2017attention,devlin2018bert}. DALL-E~\cite{ramesh2021zero}, a 12-billion parameter version of GPT-3~\cite{brown2020language}, has exhibited remarkable capabilities in generating and applying transformations to images guided by text, however, even in 16-bit precision, it requires over 24 GB of GPU VRAM. In~\cite{clip-radford2021icml}, Radford et al. introduced Contrastive Language-Image Pre-training (CLIP)~\cite{clip-radford2021icml}, a vision-language model pre-trained on 400 million image-text pairs collected from a variety of publicly available sources on the Internet. CLIP provides a vision-language embedding space that allows the estimation of the semantic similarity between given image-text pairs. Its powerful representation scheme (a Transformer~\cite{vaswani2017attention} as a Text Encoder and a Vision Transformer~\cite{dosovitskiy2020image} as an Image Encoder) exhibit state-of-the-art zero-shot image recognition performance. In this work, we use CLIP's joint text-image space, which we warp using RBF-based functions, in order to arrive at non-linear text paths that we use as a supervisory signal in order to optimize non-linear traversal paths in the GAN latent space.
\subsection{Text-guided image generation and manipulation}
Tracing back to~\cite{reed2016generative}, text-guided image generation has gained considerable attention. AttnGAN~\cite{xu2018attngan} incorporated an attention mechanism between text and image features in order to synthesize fine-grained details at different sub-regions of the image by paying attention to the relevant words in the natural language description. ManiGAN~\cite{li2020manigan} goes beyond AttnGAN by semantically editing parts of an image matching a given text that describes desired attributes (e.g., texture, colour, and background), while preserving other contents that are irrelevant to the text. In~\cite{talktoedit2021iccv} the authors propose Talk-to-Edit that performs interactive fine-grained attribute manipulation through dialog between the user and the system. To do so, they gather and curate a visual-language dataset of facial images, annotated with rich fine-grained labels, which classify one attribute into multiple degrees according to its semantic meaning, along with captions describing the attributes and a user editing request in natural language.
In a similar line of research, TediGAN~\cite{xia2021tedigan} proposes a novel GAN inversion technique that can map multi-modal information (i.e., texts, sketches, labels) into a common latent space of a pre-trained StyleGAN~\cite{stylegan2_karras20cvpr}. Then, using visual-linguistic similarity learning, it provides an interactive system where the user provides input textual guidelines and the system generates diverse images given the same input text, allowing the user to edit the appearance of different attributes interactively. However, for doing so, it relies on both annotated training data and the hierarchical structure of the StyleGAN architecture. More specifically, the authors of~\cite{xia2021tedigan} introduce and use the Multi-Modal CelebA-HQ, a curated dataset of face images, each having a high-quality segmentation mask, sketch, and descriptive text. Then, they use the hierarchical structure of StyleGAN's $\mathcal{W}$-space (i.e., style-mixing) to learn the text-image matching by mapping the image and text into the same joint embedding space. By contrast, our method is not limited to the task of face editing, is not tied to a specific GAN architecture (i.e., StyleGAN), and can be applied to any latent space.
Finally, a work closely related to ours, StyleCLIP~\cite{patashnik2021styleclip}, utilizes a CLIP-based similarity loss in order to learn a StyleGAN-specific latent mapper network that infers a text-guided latent manipulation step for a given input image. To do so, StyleCLIP uses the CLIP text features of the given prompt as the target vector and tries to align it with the CLIP image features of the manipulated image. This leads to abrupt image manipulations and quickly leads to regions of low density and, thus, low image quality, as shown by the experimental results. By contrast, our method is architecture-agnostic (i.e., not tied to a specific GAN architecture, such as StyleGAN) and models non-linear paths driven by semantic dipoles in the text feature space that are being used for aligning the desired non-linear latent paths (Fig.~\ref{fig:summary}), arriving at longer and more disentangled traversals and preventing traveling to regions of low density, leading to images of higher quality with far better control of the generative factors.
\begin{figure}[t]
\centering
\includegraphics[width=\textwidth]{figs/overview.pdf}
\caption{Overview of ContraCLIP -- The pre-trained CLIP~\cite{clip-radford2021icml} image-text space (given by the Image Encoder $\mathcal{C}_I$ and Text Encoder $\mathcal{C}_T$), warped due to \textit{semantic dipoles} of contrasting pairs of sentences given in natural language via the non-trainable warping network $\mathcal{W}^{\mathcal{C}}$, provides supervision to the optimisation of non-linear interpretable paths in the latent space of a pre-trained GAN $\mathcal{G}$ via the trainable warping network $\mathcal{W}^{\mathcal{G}}$.}
\label{fig:overview}
\end{figure}
\section{Proposed Method}\label{sec:proposed_method}
Fig.~\ref{fig:overview} gives an overview of the proposed method for learning non-linear paths in the latent space of a pre-trained GAN, driven by contrasting semantic dipoles given in natural language. To do so, we define pairs of sentences that convey contrasting meanings and express the limits of the interpretation that we require by the optimised latent paths to encode. Each such pair corresponds to one trainable path in the GAN latent space. Given $K$ semantic dipoles, we first represent them in the CLIP text space and then use them as the centres of the RBF-based warping functions $f_1^{\mathcal{C}},\ldots,f_{\kappa}^{\mathcal{C}},\ldots,f_K^{\mathcal{C}}$. This endows us with $K$ distinct vector fields (the functions' gradients $\nabla f_{\kappa}^{\mathcal{C}}$), which provide end-to-end directional paths from the one pole/sentence to the other and can be used as our supervisory signal that guides the trainable latent paths, for any given image and its transformed version along a certain latent path. In contrast to the standard text-image similarity (e.g.,~\cite{patashnik2021styleclip}), we do not try to blindly align the transformed image vectors with the text feature of the desired end. Instead, we decouple the supervisory text features from the origin of axes, by taking the gradient of the warping at any given point (i.e., image generated by the GAN and represented in the joint image-text CLIP space). By doing so, we establish only local relations between the text and the image features. This leads to smooth non-linear text paths that allow for mild transition between the given point and the desired end.
The rest of this section is organized as follows: a) in Sect.~\ref{sec:warpedganspace} we present our method for learning non-linear paths in the latent space of a pre-trained GAN using a modified version of the publicly available implementation of WGS~\cite{warpedganspace2021iccv} -- this is illustrated in the bottom-left part of Fig.~\ref{fig:overview}, b) in Sect.~\ref{sec:text_paths} we present the proposed method for modeling non-linear paths in the CLIP text space driven by semantic dipoles so as to derive a supervisory signal for learning the WGS latent paths -- this is illustrated in the right part of Fig.~\ref{fig:overview}, c) in Sect.~\ref{sec:loss} we discuss the proposed contrastive loss -- this is illustrated in the bottom-right part of Fig.~\ref{fig:overview} and in Fig.~\ref{fig:standard_vs_proposed_sim}.
\subsection{Non-linear latent paths on the GAN's latent space}\label{sec:warpedganspace}
In order to learn non-linear paths in the latent space of a pre-trained GAN, we build on
WGS~\cite{warpedganspace2021iccv}. WGS optimises a set of $K$ warpings, $f_1^{\mathcal{G}},\ldots,f_K^{\mathcal{G}}$, in the GAN's latent space, each parametrized by a set of RBF-based latent space warping functions, as follows
\begin{equation}\label{eq:wgs_f}
f_\kappa^{\mathcal{G}}(\mathbf{z})=
\sum_{i=1}^{N}
\left(
\exp\left(-\gamma_i^\kappa \lVert\mathbf{z}-\mathbf{q}_i^\kappa\rVert^2\right)
-
\exp\left(-\gamma_i^\kappa \lVert\mathbf{z}+\mathbf{q}_i^\kappa\rVert^2\right)
\right)
,\:\kappa\in\{1,\ldots,K\},
\end{equation}
where $\mathbf{q}_i^\kappa$ and $\gamma_i^\kappa$ denote the support vectors and the scaling parameters, respectively. Then, each $f_\kappa^{\mathcal{G}}$ gives rise to a family of non-linear paths via $\nabla f_\kappa^{\mathcal{G}}(\mathbf{z})$.
In~\cite{warpedganspace2021iccv}, the trainable parameters of the set of RBFs (i.e., $\mathbf{q}_i^\kappa$ and $\gamma_i^\kappa$) are optimised so that the images that are generated by codes along different paths, are easily distinguishable by a discriminator network. By contrast, in this work, the training objective is that the differences in the generated images are aligned with paths connecting the semantic dipoles expressed in natural language. This is schematically shown in Fig.~\ref{fig:overview} and will be detailed in the following section.
\subsection{Non-linear paths on the CLIP text space using semantic dipoles}\label{sec:text_paths}
\begin{figure}[t]
\centering
\includegraphics[width=\textwidth]{figs/dipole_betas.pdf}
\caption{Estimation of the $\gamma$ parameter for the dipole of sentence $(\mathbf{s}_\kappa^-,\mathbf{s}_\kappa^+)$. We require that the two RBFs overlap by a factor $\beta\in(0,1)$; i.e., $\exp\left(-\gamma_\kappa\lVert\mathbf{s}_\kappa^+-\mathbf{s}_\kappa^-\rVert^2\right)=\beta$, or $\gamma_\kappa=-\frac{\log(\beta)}{\lVert\mathbf{s}_\kappa^+-\mathbf{s}_\kappa^-\rVert^2}$. $\beta$ governs the non-linearity of the formed paths.}
\label{fig:dipole_betas}
\end{figure}
\begin{table}[t]
\centering
\caption{Example corpus of pairs of contrasting sentences in natural language. Each pair expresses the starting and the ending points of the latent path that needs to be learnt. Each such pair, when represented in the CLIP text space, forms a \textit{semantic dipole}, which via (\ref{eq:f_t}), is used to warp the text space and form non-linear directional paths from one pole to the other (see also Fig.~\ref{fig:overview}).}
\begin{tabular}{cc}
\hline
\multicolumn{2}{c}{$\mathbf{s}^-$ $\longrightarrow$ $\mathbf{s}^+$} \\ \hline\hline
\multicolumn{2}{c}{``a picture of a face in neutral expression.'' $\longrightarrow$ ``a picture of a smiling face.''} \\
\multicolumn{2}{c}{``a picture of a person with open eyes.'' $\longrightarrow$ ``a picture of a person with closed eyes.''} \\
\multicolumn{2}{c}{``a picture of a young person.'' $\longrightarrow$ ``a picture of an old person.''} \\
\multicolumn{2}{c}{``a picture of a male face.'' $\longrightarrow$ ``a picture of a female face.''} \\
\multicolumn{2}{c}{``a picture of a person with happy face.'' $\longrightarrow$ ``a picture of a person with fearful face.''} \\ \hline
\end{tabular}
\label{tab:corpus}
\end{table}
Given a set (corpus) of pairs of sentences, each pair conveying contrasting meanings, we first use the pre-trained CLIP Text Encoder ($\mathcal{C}_T$) in order to represent them in the text space $\mathcal{S}\subset\mathbb{R}^{512}$. That is, each pair of sentences in natural language is represented by a pair of vectors in the CLIP text space forming a dipole. An example of such corpus of sentences is given in Tab.~\ref{tab:corpus}.
Inspired by~\cite{warpedganspace2021iccv}, we propose the warping of the CLIP text space using RBF-based warping functions, each being the sum of two opposite RBFs centred at the text feature representations of the above dipoles. Each such warping of the text space endows it, via its gradient, with a family of non-linear curves as illustrated in Figs.~\ref{fig:summary},\ref{fig:overview}. That is, for any embedding $\mathbf{s}\in\mathcal{S}$, the gradient of the warping due to a certain semantic dipole gives a local direction vector, following which will eventually lead to one of the semantic poles. In other words, a warping function ensures that starting from any point in the text space, following the gradient of the function, one can --at will-- arrive at any of the two poles.
More formally, let $\{(\mathbf{s}_\kappa^-,\mathbf{s}_\kappa^+)\colon\mathbf{s}_\kappa^{\pm}\in\mathcal{S}, \kappa=1,\ldots,K\}$ be the corpus of $K$ pairs of sentences represented in the $512$-dimensional CLIP text space $\mathcal{S}$. For the $\kappa$-th pair, $(\mathbf{s}_\kappa^-,\mathbf{s}_\kappa^+)$, we define the following warping function
\begin{equation}\label{eq:f_t}
f_\kappa^{\mathcal{C}}(\mathbf{s}) =
\exp\left(-\gamma_\kappa \lVert\mathbf{s}-\mathbf{s}_\kappa^+\rVert^2\right)-
\exp\left(-\gamma_\kappa \lVert\mathbf{s}-\mathbf{s}_\kappa^-\rVert^2\right),
\end{equation}
where $\gamma_\kappa\in\mathbb{R}_+$ denotes the scale of each of the poles in the dipole and governs the non-linearity of the resulting paths. In Fig.~\ref{fig:dipole_betas} we illustrate the proposed approach for setting the $\gamma$ parameters of each pair of sentences of the given corpus. By requiring that the RBFs of each dipole will overlap by a given factor $\beta$, we arrive at warpings of the text feature space that have non-zero gradient and, thus, allow for traversing the space continuously from any given point to the desired end of the path. We note that $\gamma_\kappa$ depends on the relative positions of the sentences $\mathbf{s}_\kappa^-$ and $\mathbf{s}_\kappa^+$ and, thus, are in general different for each pair. The gradient of each warping function is given analytically as
\begin{equation}\label{eq:nabla_f_t}
\nabla f_\kappa^{\mathcal{C}}(\mathbf{s}) =
-2\gamma_\kappa
\left(
\exp\left(-\gamma_\kappa\lVert\mathbf{s}-\mathbf{s}_\kappa^+\rVert^2\right)\left(\mathbf{s}-\mathbf{s}_\kappa^+\right)
-
\exp\left(-\gamma_\kappa\lVert\mathbf{s}-\mathbf{s}_\kappa^-\rVert^2\right)\left(\mathbf{s}-\mathbf{s}_\kappa^-\right)
\right).
\end{equation}
\subsection{Proposed contrastive similarity loss}\label{sec:loss}
In this section we describe our objective for learning the trainable parameters of the set of RBFs, i.e., the support vectors $\mathbf{q}_i^\kappa$ and the scaling parameters $\gamma_i^\kappa$ in (\ref{eq:wgs_f}). We propose a contrastive loss term that imposes alignment between the vector that expresses the difference between the manipulated and the original image, and the local text vector (tangent to the non-linear curve) that is obtained in the CLIP text space by the non-trainable warping network $\mathcal{W}^{\mathcal{C}}$ (Fig.~\ref{fig:overview}).
Formally, given a mini-batch of $N$ images generated by the GAN generator and represented in the joint image-text CLIP space, the loss term introduced by the pair of images due to the $\kappa$-th latent path, i.e., by the original image $\mathbf{s}$ and the manipulated image $\mathbf{s}_{\kappa}$, is given by
\begin{equation}\label{eq:loss}
\ell_{\kappa}=
-\log\frac{\exp\left(\sfrac{p_{\kappa,\kappa}}{\tau}\right)}{\sum_{t=1}^{N}\exp\left(\sfrac{p_{\kappa,t}}{\tau}\right)},
\end{equation}
where $p_{\kappa,t}=\cos\left(\nabla f_{\kappa}^{\mathcal{C}}(\mathbf{s}),\mathbf{s}_t-\mathbf{s}\right)$ and $\tau$ denotes the temperature parameter. It is worth noting that the only trainable module of the proposed method is the set of the support vectors that warp the latent space of GAN (i.e., the trainable Warping Network $\mathcal{W}^{\mathcal{G}}$ shown in the bottom left part of Fig.~\ref{fig:overview}).
\section{Experiments}\label{sec:experiments}
In this section we will present the experimental evaluation of the proposed method and provide comparisons with state-of-the-art methods. We will first briefly present the types of GAN generators and the datasets that we use in Sect.~\ref{sec:gans_and_datasets}. We will then present ablation studies on various components of the proposed method in Sect.~\ref{sec:ablation_studies}, and we will show that using the semantic dipoles to warp the CLIP text space in order to find non-linear text paths is crucial with respect to the quality of the discovered latent paths (both in terms of image quality and disentanglement). More precisely, as shown in Figs.~\ref{fig:proggan_smiling},\ref{fig:proggan_makeup},\ref{fig:stylegan_expressions}, replacing the non-linear text paths with linear ones (e.g., the vector that connects the two poles) leads to significant deterioration of the discovered latent paths, both in terms of image quality and disentanglement, due to the abrupt image manipulations induced by the linear text directions. By contrast, non-linear text paths lead to smooth and more disentangled interpretable latent paths, generating images of higher quality and preserving attributes based on the given semantic dipoles. Moreover, due to the capability of traveling very long traversals in the latent space, using the proposed non-linear text paths leads to latent paths that exhibit far better control of the generative factors (Fig.~\ref{fig:ablation_traversal_lengths}). Finally, in Sect.~\ref{sec:soa_comparisons}, we will qualitatively and quantitatively show that the proposed method produces continuous and more disentangled latent path in comparison to state-of-the-art StyleCLIP~\cite{patashnik2021styleclip}, by preserving the identity (ID) significantly better, while exhibiting much lower correlation to facial attributes that are irrelevant to the given semantic dipoles (Tab.~\ref{tab:soa_dis}). We note that, in all experiments, in order to measure the ID preservation between the original image of each traversal sequence, i.e., the central image of the generated sequences across the various latent paths, and each of the rest on the path, we used the ID similarity score (i.e., a number in $[0,1]$) provided by ArcFace~\cite{deng2019arcface}. Finally, in order to measure the disentanglement of the discovered paths, we used the disentanglement metric proposed in~\cite{warpedganspace2021iccv}.
\subsection{Pre-trained GAN generators and datasets}\label{sec:gans_and_datasets}
In order to show how general our proposed approach is, we evaluate it using pre-trained GANs of different architectures, namely: a) ProgGAN~\cite{proggan_karras18iclr} trained on CelebA-HQ~\cite{celeba_liu15iccv}, and b) StyleGAN2~\cite{stylegan2_karras20cvpr} ($\mathcal{W}$-space) trained on FFHQ~\cite{stylegan2_karras20cvpr}. Additional experiments on StyleGAN2 trained on AFHQ Cats~\cite{choi2020stargan}, AFHQ Dogs~\cite{choi2020stargan}, and LSUN Cars~\cite{yu2015lsun} are given in the supplementary material.
\subsection{Ablation studies}\label{sec:ablation_studies}
In this section we will present our ablation studies on the length of the latent traversals and the type of text-image similarity measure used by our method. We also conducted ablation studies to show the robustness to the values of the $\beta$ parameter (Sect.~\ref{sec:text_paths}) and the temperature of the adopted contrastive loss (Sect.~\ref{sec:loss}), which we include in the supplementary material.
\begin{figure}[t!]
\centering
\includegraphics[width=\textwidth]{figs/ablation_traversal_lengths.pdf}
\caption{Ablation study on latent traversal length $L\in\{10.8,19.2,28.8\}$.}
\label{fig:ablation_traversal_lengths}
\end{figure}
\begin{figure}[t!]
\centering
\includegraphics[width=\textwidth]{figs/ablation_linearity_proggan_neutral2smiling.pdf}
\caption{Ablation study on the use of non-linear versus linear text paths for the semantic dipole ``\textit{a face in neutral expression.}'' $\rightarrow$ ``\textit{a smiling face.}'' (ProgGAN).}
\label{fig:proggan_smiling}
\end{figure}
\begin{figure}[t!]
\centering
\includegraphics[width=\textwidth]{figs/ablation_linearity_proggan_nomakeup2makeup.pdf}
\caption{Ablation study on the use of non-linear versus linear text paths for the semantic dipole ``\textit{a face without makeup.}'' $\rightarrow$ ``\textit{a face with makeup.}'' (ProgGAN).}
\label{fig:proggan_makeup}
\end{figure}
\begin{figure}[t!]
\centering
\includegraphics[width=\textwidth]{figs/ablation_linearity_stylegan_expressions.pdf}
\caption{Ablation study on the use of non-linear vs. linear text paths (StyleGAN2).}
\label{fig:stylegan_expressions}
\end{figure}
\subsubsection{Latent traversal length ablation}
In order to explore the limitations of the proposed method with respect to the length of the traversal along the discovered latent paths, we conducted the following ablation study. We calculated the latent traversals for a set of latent codes (in StyleGAN2's $\mathcal{W}$-space) for a various number of steps, leading to latent traversals of length $L\in\{10.8,19.2,28.8\}$. We show the results in Fig.~\ref{fig:ablation_traversal_lengths}. Surprisingly, the proposed non-linear text paths continue to semantically extend towards each end of the dipole without arriving at low-density areas on the latent space and, thus, low-quality generations. It is worth noting that by using linear text paths in the CLIP space (e.g., StyleCLIP~\cite{patashnik2021styleclip}), we arrive at heavy artifacts or extreme deformations of the images even after traveling for relatively few ($L\approx11$) units of length (as will be shown later in this section), while in our case we can travel for $L\approx29$ without introducing extreme artifacts, and at the same time we semantically approach even closer to the ends of the dipoles. This leads to better control over the discovered generative factors and prevents traveling to regions of low density.
\subsubsection{Non-linear vs linear text paths}
In order to assess the effectiveness of warping the CLIP text space and modeling non-linear path on it, we conducted an ablation study on the similarity criterion that the proposed method can employ. First, we considered linear text paths formed as the difference of dipoles ends -- this is denoted as ``ours (linear)''. Next, we considered the standard text-similarity calculation approach when a single text prompt is given (for fair comparisons, we learned latent paths for both ends of the semantic dipole separately). This approach is illustrated in the left part of Fig.~\ref{fig:standard_vs_proposed_sim} and since it is adopted by StyleCLIP~\cite{patashnik2021styleclip} we denote this as ``StyleCLIP\textsuperscript{\textasteriskcentered}''. We show the results for ProgGAN in Figs.~\ref{fig:proggan_smiling},~\ref{fig:proggan_makeup} and for StyleGAN2 in Figs.~\ref{fig:stylegan_expressions}. It is clear that adopting a similarity calculation approach that involves linear text directions leads to abrupt manipulations of the corresponding image, usually causing severe artifacts, or altering entirely the identity and/or facial attributes depicted on it. By contrast, the proposed non-linear text paths lead to latent traversals that are smooth, effective, and far more disentangled. Consequently, adopting a semantic dipole (instead of single text prompts) is not enough without warping the text space in order to obtain non-linear paths between the centres of the dipole.
\begin{figure}[t!]
\centering
\includegraphics[width=\textwidth]{figs/comp_soa_ganspace_wgs.pdf}
\caption{Comparison of the proposed method with GANSpace~\cite{ganspace2020harkonen} and WGS~\cite{warpedganspace2021iccv}.}
\label{fig:comp_soa_ganspace_warpedganspace}
\end{figure}
\begin{figure}[t!]
\centering
\includegraphics[width=\textwidth]{figs/comp_soa_styleclip.pdf}
\caption{Comparison of the proposed method with StyleCLIP~\cite{patashnik2021styleclip}.}
\label{fig:comp_soa_styleclip}
\end{figure}
\begin{figure}[t!]
\centering
\includegraphics[width=\textwidth]{figs/comp_soa_complex_dipoles.pdf}
\caption{Comparison of the proposed method with StyleCLIP~\cite{patashnik2021styleclip} for more complex and challenging semantic dipoles.}
\label{fig:comp_soa_complex}
\end{figure}
\begin{table}[t]
\centering
\caption{Comparison of the proposed ContraCLIP to state of the art methods in terms of the disentanglement metric proposed in~\cite{warpedganspace2021iccv}.}
\begin{tabular}{l||c|ccccc}
& ID ($\uparrow$) & Age ($\downarrow$) & Gender ($\downarrow$) & Skin ($\downarrow$) & Hair ($\downarrow$) & Beard ($\downarrow$) \\ \hline\hline
GANSpace~\cite{ganspace2020harkonen} & .23 & .95 & .63 & .57 & .89 & .33 \\
WarpedGANSpace~\cite{warpedganspace2021iccv} & .29 & .82 & .39 & .78 & .94 & .38 \\
StyleCLIP~\cite{patashnik2021styleclip} & .35 & .73 & .55 & .42 & .93 & .41 \\ \hline
\textbf{ContraCLIP (Ours)} & \textbf{.83} & \textbf{.12} & \textbf{.06} & \textbf{.30} & \textbf{.26} & \textbf{.14} \\ \hline
\end{tabular}
\label{tab:soa_dis}
\end{table}
\subsection{Comparison to state-of-the-art}\label{sec:soa_comparisons}
In this section, we first compare the proposed method, which discovers non-linear paths in the GAN's latent space and may adopt non-linear or linear paths in the text space, with GANSpace~\cite{ganspace2020harkonen} and WGS~\cite{warpedganspace2021iccv}, which discover linear or non-linear paths, respectively, in the GAN latent space in an unsupervised manner. In Fig.~\ref{fig:comp_soa_ganspace_warpedganspace} we show that using text guidelines leads to more disentangled interpretable paths (e.g., by preserving certain attributes such as hair style and facial expressions) and far less artifacts, without the need of labeling them using manual annotation~\cite{ganspace2020harkonen} or certain pre-trained detectors~\cite{warpedganspace2021iccv}.
In Figs.~\ref{fig:comp_soa_styleclip},\ref{fig:comp_soa_complex} we compare the proposed method with the state-of-the-art vision-language StyleCLIP~\cite{patashnik2021styleclip}. We observe that our method produces consistent and more disentangled latent paths (e.g., preserving certain attributes -- see also Tab.~\ref{tab:soa_dis}), which continuously traverses the latent space from a given semantic region to another, by preserving at the same time the identity significantly better than~\cite{patashnik2021styleclip} as is clear by the ID similarity values at the right hand side of each row. By contrast, StyleCLIP~\cite{patashnik2021styleclip} leads to inconsistent traversals that do not progress in a continuous manner and arrives at more unnatural generations (see also the more complex semantic dipoles shown in Fig.~\ref{fig:comp_soa_complex}).
Finally, in Tab.~\ref{tab:soa_dis} we show state-of-the-art results using the quantitative disentanglement metric proposed in~\cite{warpedganspace2021iccv}. The results are for the facial expressions transition experiment shown in Fig.~\ref{fig:comp_soa_styleclip}. In Tab.~\ref{tab:soa_dis} we show that in comparison to state-of-the-art methods (i.e., \cite{ganspace2020harkonen,warpedganspace2021iccv,patashnik2021styleclip}) we obtain much higher ID preservation and much lower correlation to five facial attributes that are irrelevant to facial expressions -- this indicates better disentanglement.
\section{Conclusions}\label{sec:conclusions}
In this paper, we presented our method for discovering non-linear paths in the latent space of pre-trained GANs driven by semantic dipoles. We do so by defining a set of contrasting pairs of sentences in natural language that represented in the CLIP text space give rise, via RBF-based warping functions, to non-linear text paths for traversing it from one semantic pole to the other. By defining an objective that discovers paths in the latent space of GANs that generate changes along the desired paths in the vision-language embedding space, we provide an intuitive and effective way of controlling the underlying generating factors.
\begin{appendices}
In this supplementary material we will present a) additional ablation studies with respect to hyperparameters of the proposed method in Sect.~\ref{supp:ablation_params}, b) quantitative experimental results on the preservation of the identity between the original sampled images and the generated image sequences across the various learnt latent paths in Sect.~\ref{supp:id_preservation}, and c) additional experimental results on non-facial datasets in Sect.~\ref{supp:other_datasets}.
\section{Ablation studies with respect to the hypeparameters $\beta$ and $\tau$}\label{supp:ablation_params}
In this section we will present additional ablation studies to show the robustness of the proposed method to the values of the $\beta$ parameter (see Sect.~\ref{sec:text_paths}) and the temperature of the adopted contrastive loss (see Sect.~\ref{sec:loss}).
\subsection{$\beta$ parameter}
As discussed in Sect.~\ref{sec:text_paths}, we introduce the parameter $\beta$ as a means of controlling the $\gamma$ parameter of each semantic dipole, or, in other words, a way of controlling the non-linearity of the curves that are induced by the warping of the CLIP text space due to function (\ref{eq:nabla_f_t}). In Fig.~\ref{fig:dipole_betas} we illustrate the proposed approach for setting the $\gamma$ parameters of each pair of sentences (semantic dipole) of a given corpus. More specifically, by requiring that the RBF centred on the one pole, evaluated on the other pole, has a value of $\beta\in(0,1)$, we arrive at warpings of the text feature space that have non-zero gradient and, thus, allow for traversing the space continuously from any given point to the desired end of the path. We note that $\gamma_t$ depends on the relative positions of the sentences $\mathbf{s}_t^-$ and $\mathbf{s}_t^+$ and, thus, are in general different for each pair.
A $\beta\to0$ leads to non-overlapping RBFs and thus greater flat regions around the poles and, thus, zero gradient (see the case of $\beta=0.01$ in Fig.~\ref{fig:dipole_betas}). In contrast, when $\beta\to1$ the RBFs largely overlap with each other by flattening the RBF ``bells''. We experimented with values of $\beta\in\{0.25, 0.75, 0.95\}$ and we show that the discovered latent paths are very similar to each other for the various choices of $\beta$ in this range, as shown in Fig.~\ref{fig:ablation_betas}. The visual differences one can observe are subtle in general, while the ID scores (i.e., an identity score for each image of the sequence that expresses the similarity between the original image -- central image of the sequence -- and each of the rest, using ArcFace~\cite{deng2019arcface}) are close to each other. Finally, it is worth noting that for smaller values of $\beta$, i.e., $\beta\in(0,0.2)$, the training process collapses since there are large flat regions around the poles which do not allow gradient to show to the desired direction.
\subsection{Contrastive temperature parameter $\tau$}
We also conducted a study on the temperature of the proposed contrastive loss (see Sect.~\ref{sec:loss}). In this section we will present our ablation study on the temperature $\tau$ and we will show that the proposed method is also robust with respect this parameter as shown in Fig.~\ref{fig:ablation_temperature}. We experimented with temperature values $\tau\in\{0.01,0.1,0.5,1.0,5.0\}$. Similarly to the previous section, we observe that the proposed method exhibits notable robustness with respect to the adopted temperature parameter.
\begin{figure}[t!]
\centering
\includegraphics[width=0.95\textwidth]{figs/ablation_betas.pdf}
\caption{Ablation study on the $\beta$ parameter.}
\label{fig:ablation_betas}
\end{figure}
\begin{figure}[t!]
\centering
\includegraphics[width=0.95\textwidth]{figs/ablation_temperature.pdf}
\caption{Ablation study on the contrastive loss temperature parameter $\tau$.}
\label{fig:ablation_temperature}
\end{figure}
\section{ID preservation}\label{supp:id_preservation}
\begin{figure}[t!]
\centering
\includegraphics[width=0.95\textwidth]{figs/stylegan_attributes_corpus.pdf}
\caption{Examples of the interpretable paths found with the proposed method for the dipoles: (a) ``a picture of a young person.'' $\longrightarrow$ ``a picture of an old person.'', (b) ``a picture of a shaved man.'' $\longrightarrow$ ``a picture of a man with beard.'', and (c) ``a picture of a face without makeup.'' $\longrightarrow$ ``a picture of a face with makeup.''.}
\label{fig:stylegan_various_corpus}
\end{figure}
As discussed in Sect.~\ref{sec:experiments}, for measuring the identity preservation between the original image of each traversal sequence, i.e., the central image of the generated sequences across the various latent paths, and each of the rest on the path, we used an ID similarity score (ID), i.e., a number in [0, 1], provided by ArcFace~\cite{deng2019arcface}.
In this section, we present quantitative results for a set of 100 randomly chosen latent codes (i.e., original images) and for a set of non-linear latent paths optimised for the semantic dipoles shown in Table~\ref{tab:corpus}. Examples of such latent paths for various latent codes and four semantic dipoles are shown in Fig.~\ref{fig:stylegan_various_corpus}. In this section we report results for the proposed non-linear method -- denoted as ``ContraCLIP'' and one that is adopted by the state-of-the-art StyleCLIP~\cite{patashnik2021styleclip} -- denoted as ``StyleCLIP\textsuperscript{\textasteriskcentered}'', as discussed in Sect.~\ref{sec:ablation_studies}. In Table~\ref{tab:id_results} we show the results of ID preservation (averaged over the 100 latent codes). Clearly, the proposed non-linear text paths lead to far better preservation of the identity of the original images (original latent codes) compared to linear ones -- see also Figs.~\ref{fig:proggan_smiling},\ref{fig:proggan_makeup},\ref{fig:stylegan_expressions}.
\begin{table}[t!]
\centering
\caption{Corpus of semantic dipoles (\textit{Facial Attributes}).}
\begin{tabular}{cc}
\hline
Semantic \\ Dipole & $\mathbf{s}^-$ $\longrightarrow$ $\mathbf{s}^+$ \\ \hline\hline
$D_1$ & ``a picture of a male face.'' $\longrightarrow$ ``a picture of a female face.'' \\
$D_2$ & ``a picture of a young person.'' $\longrightarrow$ ``a picture of an old person.'' \\
$D_3$ & ``a picture of a smiling face.'' $\longrightarrow$ ``a picture of a face in neutral expression.'' \\
$D_4$ & ``a picture of a person with black hair.'' $\longrightarrow$ ``a picture of a person with red hair.'' \\
$D_5$ & ``a picture of a man with hair.'' $\longrightarrow$ ``a picture of a bald man.'' \\
$D_6$ & ``a picture of a shaved man.'' $\longrightarrow$ ``a picture of a man with a beard.'' \\
$D_7$ & ``a picture of a face without makeup.'' $\longrightarrow$ ``a picture of a face with makeup.'' \\
$D_8$ & ``a picture of a person with open eyes.'' $\longrightarrow$ ``a picture of a person with closed eyes.'' \\
$D_9$ & ``a picture of a person with pale skin.'' $\longrightarrow$ ``a picture of a person with tanned skin.'' \\
$D_{10}$ & ``a picture of an angry face.'' $\longrightarrow$ ``a picture of a surprised face.'' \\ \hline
\end{tabular}
\label{tab:corpus}
\end{table}
\begin{table}[t!]
\centering
\caption{ID preservation results.}
\begin{tabular}{lcccccccccc}
\multirow{2}{*}{} & \multicolumn{10}{c}{Semantic Dipole} \\ \cline{2-11}
& $D_{1}$ & $D_{2}$ & $D_{3}$ & $D_{4}$ & $D_{5}$ & $D_{6}$ & $D_{7}$ & $D_{8}$ & $D_{9}$ & $D_{10}$ \\ \hline\hline
\multicolumn{1}{l|}{StyleCLIP\textsuperscript{\textasteriskcentered}} & .5624 & .5136 & .5790 & .6358 & .5304 & .5972 & .5705 & .5882 & .5642 & .5912 \\ \hline
\multicolumn{1}{l|}{ContraCLIP} & \textbf{.8662} & \textbf{.9245} & \textbf{.9099} & \textbf{.9559} & \textbf{.9372} & \textbf{.9407} & \textbf{.9402} & \textbf{.9426} & \textbf{.9344} & \textbf{.9567} \\ \hline
\end{tabular}
\label{tab:id_results}
\end{table}
\section{GAN pre-trained on non-facial datasets}\label{supp:other_datasets}
Besides the fact that the proposed method is model-agnostic and not tied to any specific GAN architecture (e.g., StyleGAN), such as~\cite{patashnik2021styleclip,xia2021tedigan}, it is also not limited to any specific type of imagery (i.e., facial images), such as~\cite{xia2021tedigan}. To support the former claim, we conducted and reported in Sect.~\ref{sec:experiments} experiments using different GAN architectures (i.e., ProgGAN~\cite{proggan_karras18iclr} and StyleGAN2~\cite{stylegan2_karras20cvpr}). To support the latter claim, we conducted and report in this section additional experiments using StyleGAN2 pre-trained on AFHQ Cats~\cite{choi2020stargan}, AFHQ Dogs~\cite{choi2020stargan}, and LSUN Cars~\cite{yu2015lsun}. We show the results in Fig.~\ref{fig:cats_dogs_cars}. We observe that the proposed method, i.e., the learnt non-linear latent paths driven by non-linear paths in the CLIP text space, arrive at high quality images and manipulations that are largely consistent with the semantics expressed in the natural language sentences that we defined.
\begin{figure}[t!]
\centering
\includegraphics[width=0.95\textwidth]{figs/cats_dogs_cars.pdf}
\caption{Results of the proposed method on non-facial GANs, i.e., StyleGAN2 pretrained on (a,b) AFHQ Cats~\cite{choi2020stargan}, (c,d) AFHQ Dogs~\cite{choi2020stargan}, and (e,f) LSUN Cars~\cite{yu2015lsun}.}
\label{fig:cats_dogs_cars}
\end{figure}
\end{appendices}
\bibliographystyle{splncs04}
|
2,877,628,089,024 | arxiv | \section{Introduction}
Over the past decades, the valuation of financial derivatives using stochastic processes has become an industry standard. The prevailing methods for valuing derivatives contracts are direct Monte Carlo methods for stochastic processes and finite difference methods for partial differential equations, which can be derived from stochastic processes via the Feynman-Kac theorem. While the parameters of any financial model can in principle be calibrated to reflect market conditions using these methods, for efficiency reasons almost all models that are practically used rely on (semi-)analytical formulas for the value of simple derivatives to facilitate the calibration. Naturally, the number of models that are (semi-)analytically solvable is very limited and (semi-)analytical calibration formulas that extend the scope of available models are highly desirable.
The most important model of quantitative finance is the Black-Scholes model, which assumes a log-normal behavior of the market. Over the past decades, several stochastic volatility models have been developed to overcome the shortcomings of the Black-Scholes model~\citep{BlackScholes1973}, such as a missing smile or skew of the volatility, i.e. its underestimation of probabilities for extreme events and the missing correlation between the direction of market moves and the volatility. Among these models, the Heston stochastic volatility model~\citep{Heston1993} plays an important role, since it reproduces market smiles and skews and can be calibrated rapidly using semi-analytical formulas.
However, to reproduce the term structure of volatility, the Heston model has to be extended to the case of time-dependent parameters. The most simple form of such time-dependence is given by piecewise constant sets of parameters assigned to subsequent time-intervals~\citep{Mikhailov2004, Elices2007, Putschoegl2010}. A piecewise constant time-dependence naturally reflects that the market quotes prices for instruments only at discrete maturities. The numerics of the semi-analytical calibration formulas for the Heston model in general has, however, been reported to be plagued by issues such as poor convergence of integrands~\citep{Mikhailov2004, Albrecher2007}. We intend to solve these issues and provide a simple and numerically stable scheme for the calibration of the Heston model and its extension to piecewise constant parameters, while preserving the semi-analytical nature of the calibration process. Of course, most of the components of our algorithm have been known previously. The aim of our paper, however, is to combine these ideas into a straightforward and easy-to-implement scheme.
Our manuscript is organized as follows: We start the method part by introducing the method of characteristic functions~\citep{CarrMadan1999, KahlJaeckel2005} and apply it to the example case of the Black-Scholes model. We then apply the same methodology to the case of the Heston model, which we extend to the case of piecewise constant parameters. Then we modify the relevant formulas with a Black-Scholes control variate~\citep{AndersenPiterbarg2010} that suppresses oscillations in the integrands of various integrals and, hence, leads to a computationally efficient implementation. Furthermore, we explain the general calibration strategy in the case of piecewise constant parameters.
The result part of the paper illustrates the benefits of the control variate method for numerical integration. We then show calibration results for the foreign exchange (FX) options market, in particular the calibrated volatility smile within the Heston model with piecewise constant parameters. The calibrated parameters are then used to price window-barrier options that are sensitive to the term-structure of the implied volatility surface. We conclude the paper with a summary of our results.
\section{Method}
\subsection{Characteristic function for the Black-Scholes model}
We start with the derivation of the characteristic function of the Black-Scholes (BS) model~\citep{BlackScholes1973}, which illustrates the general strategy for working with characteristic functions. Furthermore, we will later use the BS characteristic function as a control variate, when we solve the Heston model.
The stochastic process of the Black-Scholes model for the log-spot $x_t = \text{ln}[S(t)]$ is defined by the stochastic process
\begin{equation}
dx_t = \left(r_d - r_f - \frac{\sigma^2}{2}\right) dt + \sigma dW_t
,
\end{equation}
where $W_t$ is a Wiener process, $\sigma$ is the volatility and $r_d$ and $r_f$ are the interest rates for the domestic and foreign currency, respectively. Applying the Feynman-Kac theorem, see f.i. Ref.~\cite{Clark2010}, the partial differential equation (PDE) for options pricing within the Black-Scholes framework is given by
\begin{equation}
0 = \partial_t C + \frac{\sigma^2}{2} \partial_{xx} C + \left(r_d - r_f - \frac{\sigma^2}{2}\right) \partial_x C - r_d C ,
\label{eq:bspricing}
\end{equation}
where $C$ is the value of the product that we price and $t$ is the time from emission.
We define an ansatz to solve this equation for plain-vanilla call options, which is given by
\begin{equation}
C(x, \tau, K) = e^x P_1 (x, \tau, K) - e^{-r \tau} K P_2 (x, \tau, K) ,
\label{eq:bsansatz}
\end{equation}
where we have introduced the abbreviations $\tau = T-t$, which is the time to maturity, and $r = r_d - r_f$. The strike of the call option is denoted by $K$. The terms $P_1$ and $P_2$ can be interpreted as risk-neutral probabilities~\citep{CoxRoss1976, Heston1993}. Within the BS model these probabilities can be expressed in closed form as
\begin{subequations}
\begin{align}
P_j (x, \tau, K) &= \Phi (d_j), \quad (j=1,2) \\[2pt]
d_j &= \frac{1}{\sigma \sqrt{\tau}} \left[ x - \text{ln}(K) + \left( r + (-1)^{j-1} \frac{\sigma^2}{2} \right) \tau \right]
,
\end{align}
\label{eq:bsanalyticprob}
\end{subequations}
where $\Phi$ denotes the cumulative distribution function of the standard normal distribution. However, we will not immediately use these formulas, since we want to illustrate how to work with characteristic functions.
Therefore, we continue with calculating the derivatives of $C(K)$.
\begin{subequations}
\begin{align}
\partial_t C &= -\partial_\tau C = - e^{-x} \partial_\tau P_1 - r e^{-r \tau} K P_2 + e^{-r \tau} K \partial_\tau P_2\\[2pt]
\partial_x C &= e^x P_1 + e^x \partial_x P_1 - e^{-r \tau} K \partial_x P_2\\[2pt]
\partial_{xx} C &= e^x P_1 + 2 e^x \partial_x P_1 + e^x \partial_{xx} P_1 - e^{-r \tau} K \partial_{xx} P_2
.
\end{align}
\label{eq:ansatzderivpart1}
\end{subequations}
Substituting the ansatz (Eq.~\ref{eq:bsansatz}) into the BS pricing PDE (Eq.~\ref{eq:bspricing}) and collecting terms in $P_1$ and $P_2$, we find that these must satisfy the PDEs
\begin{equation}
\partial_\tau P_j = \frac{\sigma^2}{2} \partial_{xx} P_j + \left(r+ (-1)^{j-1}\frac{\sigma^2}{2}\right) \partial_x P_j - r_f P_j \quad (j = 1,2)
.
\label{eq:bsprobpde}
\end{equation}
At this point we introduce the characteristic function, which will later be used to solve the Heston model~\citep{Heston1993}. The characteristic function $f_j$ is related to the risk-neutral probabilities via
\begin{equation}
P_j (x, \tau, K) = \frac{1}{2} + \frac{1}{\pi} \int\limits_0^\infty d\phi \, \text{Re} \Bigg[ \frac{e^{-i\phi\text{ln}(K)}f_j(x, \tau, \phi)}{i\phi} \Bigg]
.
\label{eq:charfuncgeneral}
\end{equation}
The characteristic function $f_j$ satisfies the same PDE as the probability $P_j$~\citep{Heston1993}. Therefore, we make an ansatz for $f_j$, which is given by
\begin{equation}
f_j (x, \tau, \phi) = \text{exp} \left(D_j (\tau, \phi) + i \phi x\right)
,
\label{eq:bsansatzcf}
\end{equation}
where $D_j (\tau, \phi) $ is a function that has to be determined so that the ansatz for $f_j$ truly is a solution of the pricing PDE (Eq.~\ref{eq:bsprobpde}). We now obtain the derivatives of the characteristic function.
\begin{subequations}
\begin{align}
\partial_\tau f_j &= f_j \partial_\tau D_j\\[2pt]
\partial_x f_j &= i \phi f_j \\[2pt]
\partial_{xx} f_j &= - \phi^2 f_j
\end{align}
\end{subequations}
Substituting the ansatz for $f_j$ (Eq.~\ref{eq:bsansatzcf}) in place of the probabilities $P_j$ into Eq.~\ref{eq:bsprobpde}, we end up with an ordinary differential equation (ODE) for $D_j$.
\begin{equation}
\partial_\tau D_j = \left(r + (-1)^j \frac{\sigma^2}{2}\right) i \phi - \frac{\sigma^2}{2}\phi^2 - r_f
\end{equation}
Integrating this ODE in time with terminal condition $D(\tau = 0, \phi) = 0$~\citep{Heston1993} we obtain the solution
\begin{equation}
D_j (\tau, \phi) = \left[ \left(r + (-1)^j \frac{\sigma^2}{2}\right) i \phi - \frac{\sigma^2}{2}\phi^2 - r_f \right] \tau
.
\label{eq:bsDfunc}
\end{equation}
Using this expression together with Eqs.~\ref{eq:bsansatzcf}, \ref{eq:charfuncgeneral} and \ref{eq:bsansatz} we can in principle price plain-vanilla call options. Of course, this approach is much less efficient than directly evaluating the Black-Scholes formula, which expresses the probabilities $P_j$ in closed form. However, we will need the characteristic function of the Black-Scholes model later on to stabilize the numerics of the characteristic function in the Heston model, i.e. when using it as a control variate.
\subsection{Characteristic function for the Heston model with piecewise constant parameters}
A concise derivation of the characteristic function for the standard Heston model is given in Ref.~\cite{Rouah2013}. Here, we concentrate on a version of the model, in which the parameters of the process are time-dependent.
\begin{subequations}
\begin{align}
dx_t &= \left(r_d - r_f - \frac{v_t}{2} \right) dt + \sqrt{v_t} dW^x_t\\[2pt]
dv_t &= \kappa (t) \left[\theta (t) - v_t \right] dt + \xi(t) \sqrt{v_t} dW^v_t\\[2pt]
dW^x_t \cdot dW^v_t &= \rho(t) dt
\end{align}
\label{eq:hestonprocess}
\end{subequations}
Here, $x$ is again the log-spot $x_t = \text{ln}[S(t)]$, $v_t$ is the instantaneous variance, $W^x_t$ and $W^v_t$ are Wiener processes, $\rho (t)$ is the correlation between those processes, $\kappa (t)$ is the speed of mean-reversion, $\theta (t)$ is the long-term variance and $\xi (t)$ is the volatility of volatility.
For $\kappa (t)$, $\theta (t)$, $\rho(t)$ and $\xi (t)$ we assume that these are piecewise constant within the time-interval $[t_i, t_{i+1})$ and that parameters undergo a discrete jump at $t_{i+1}$, where the next time-interval with constant parameters begins. In addition to these piecewise constant parameters the model also needs the initial level of variance $v_0 = v(t=0)$ as an input, which we assume to be non-time-dependent, i.e. globally constant.
Again applying the theorem of Feynman and Kac, we obtain the pricing PDE for the Heston model with piecewise constant parameters
\begin{equation}
\begin{split}
0 =& \partial_t C + \frac{v}{2} \partial_{xx} C + \frac{\xi^2 (t)}{2} v \partial_{vv} C + \xi (t) \rho (t) v \partial_{xv} C \\
&+ \left( r_d - r_f - \frac{v}{2} \right) \partial_x C + \kappa (t) \left[\theta (t) - v\right] \partial_v C - r_d C
.
\end{split}
\label{eq:hestonpricing}
\end{equation}
In analogy to the previously presented BS case (Eq.~\ref{eq:bsansatz}) we use an ansatz to solve the pricing PDE.
\begin{equation}
C(x, \tau, K) = e^x P_1 (x, v, \tau, K) - e^{-r \tau} K P_2 (x, v, \tau, K)
\label{eq:hestonansatz}
\end{equation}
In addition to the previously calculated derivatives by $\tau$ and $x$ (see Eq.~\ref{eq:ansatzderivpart1}), we calculate the derivatives with respect to $v$.
\begin{subequations}
\begin{align}
\partial_v C &= e^x \partial_v P_1 - e^{-r\tau} K \partial_v P_2\\[2pt]
\partial_{vv} C &= e^x \partial_{vv} P_1 - e^{-r\tau} K \partial_{vv} P_2\\[2pt]
\partial_{xv} C &= e^x \partial_v P_1 + e^x \partial_{xv} P_1 - e^{-r\tau} K \partial_{xv} P_2
\end{align}
\end{subequations}
Substituting the ansatz from Eq.~\ref{eq:hestonansatz} into the pricing PDE (Eq.~\ref{eq:hestonpricing}) and collecting terms in $P_1$ and $P_2$ we find that in the Heston case these risk-neutral probabilities $P_j$, and hence the to be defined characteristic function $f_j$, must satisfy the PDEs
\begin{equation}
\begin{split}
\partial_\tau P_j =& \frac{v}{2} \partial_{xx} P_j + \frac{\xi^2 (t)}{2} v \partial_{vv} P_j + \xi (t) \rho (t) v \partial_{xv} P_j \\ &+ \left(r + (-1)^{j-1} \frac{v}{2}\right) \partial_x P_j + (a (t) - b_j (t) v) \partial_v P_j - r_f P_j
\quad (j = 1,2)
,
\end{split}
\label{eq:hestonprobpde}
\end{equation}
where the newly introduced coefficients $a$ and $b_j$ are given by $a(t) = \kappa (t) \theta (t) $, $b_1 (t) = \kappa (t) - \xi (t) \rho (t)$ and $b_2 (t) = \kappa (t)$.
Now we use the original ansatz for the characteristic function proposed in Ref.~\cite{Heston1993} to solve the PDEs for the risk-neutral probabilities. This ansatz is given by
\begin{equation}
f_j (x, v, \tau, \phi) = \text{exp}\left(C_j (\tau, \phi) + D_j (\tau, \phi) v + i \phi x \right)
.
\label{eq:hestonansatzcf}
\end{equation}
Functions $C_j (\tau, \phi)$ and $D_j (\tau, \phi)$ have to be determined so that the ansatz actually solves Eq.~\ref{eq:hestonprobpde}. To this end, we calculate the derivatives of the characteristic function.
\begin{subequations}
\begin{align}
\partial_\tau f_j &=f_j (\partial_\tau C_j + v \partial_\tau D_j)\\[2pt]
\partial_x f_j &= i \phi f_j\\[2pt]
\partial_{xx} f_j &= -\phi^2 f_j\\[2pt]
\partial_v f_j &= D_j f_j\\[2pt]
\partial_{vv} f_j &= D_j^2 f_j\\[2pt]
\partial_{xv} f_j &= i \phi D_j f_j
\end{align}
\end{subequations}
After inserting the ansatz for the characteristic function $f_j$ (Eq.~\ref{eq:hestonansatzcf}) in place of $P_j$ into Eq.~\ref{eq:hestonprobpde}, we end up with a number of terms that are linear in $v$ and others terms that do not depend on $v$. Therefore, following the original paper by Heston~\cite{Heston1993}, we write down separate ODEs for these groups of terms. Consequently, the linear factor $v$ drops out and the resulting systems of ODEs do not depend on the level of variance. Later on, when calculating the characteristic function numerically, we will set $v$ in Eq.~\ref{eq:hestonansatzcf} to $v_0 = v(t = 0)$. The set of ODEs we have to solve is now given by
\begin{subequations}
\begin{align}
\partial_\tau D_j &= \underbrace{\frac{\xi^2 (\tau)}{2}}_{= N(\tau)} D_j^2 + \underbrace{\left[\xi (\tau) \rho (\tau) i \phi - b_j (\tau)\right]}_{= - M_j (\tau, \phi)} D_j + \underbrace{\frac{(-1)^{j-1}}{2} i\phi - \frac{\phi^2}{2}}_{= L_j (\phi)} \\[2pt]
\partial_\tau C_j &= ri\phi - r_f + a (\tau) D_j
.
\end{align}
\end{subequations}
Now we consider a time interval $[\tau_0, \tau)$ in which the parameters $\kappa (t)$, $\theta (t)$, $\rho(t)$ and $\xi (t)$ are constant. We obtain the solution of the ODE for $D_j$ and subsequently insert the solution into the ODE for $C_j$. Defining the abbreviation $A_j (\tau, \phi) = \sqrt{4 L_j (\phi) N (\tau) - M_j^2 (\tau, \phi) }$ the solution of this system of ODEs with general initial condition $D_j (\tau = \tau_0, \phi) = D_{j0}$ and $C_j(\tau = \tau_0, \phi) = C_{j0}$ can be written as
\begin{subequations}
\begin{align}
\label{eq:hestonDfunc}
D_j (\tau, \phi) &= \frac{1}{2 N} \left[ M_j + A_j \text{tan} \left( \frac{1}{2} (\tau - \tau_0) A_j + \text{arctan} \left( \frac{2 N D_{j0} - M_j}{A_j} \right) \right) \right]\\[2pt]
\begin{split}
C_j (\tau, \phi) &= C_{j0} + \frac{1}{2N} \Bigg\{ (\tau - \tau_0) \big(a M_j + 2N(r i \phi - r_f)\big)\\[2pt]
& - a \Bigg[ \text{ln} \Bigg( 1 + \frac{(2N D_{j0} - M_j)^2}{A_j^2} \Bigg)\\[2pt]
& \quad + 2 \text{ln} \Big( \text{cos} \Big[ \frac{1}{2} A_j (\tau - \tau_0) + \text{arctan} \Big( \frac{2 N D_{j0} - M_j}{A_j} \Big) \Big] \Big) \Bigg] \Bigg\}
.
\label{eq:hestonCfunc}
\end{split}
\end{align}
\label{eq:hestonCDfunc}
\end{subequations}
Here, we have dropped the dependencies of $A_j$, $L_j$, $M_j$ and $N$ on $\tau$ and $\phi$ for notational brevity. While the dependency on $\phi$ still exists, the dependency on $\tau$ simply means that appropriate values, constant for the time interval under consideration, have to be inserted in place of the parameters $\kappa$, $\theta$, $\rho$ and $\xi$.
Let us now assume that we are given two intervals in time to maturity $\tau$, namely $[0, \tau_1)$ and $[\tau_1, T]$, and a related set of constant parameters for each time interval, $(\kappa_0, \theta_0, \rho_0, \xi_0)$ and $(\kappa_1, \theta_1, \rho_1, \xi_1)$. To calculate the characteristic function $f_j (\tau, \phi)$ at $\tau = T$, i.e. at the time of emission $t = 0$, we start by calculating $D_j(\tau_1, \phi)$ and $C_j(\tau_1, \phi)$ according to Eq.~\ref{eq:hestonCDfunc} with initial conditions $\tau_0 = 0$ and $D_{j0} = C_{j0} = 0$ and using parameters $(\kappa_0, \theta_0, \rho_0, \xi_0)$ in place of $\big(\kappa (t),\theta (t),\rho(t),\xi (t)\big)$. Subsequently, we obtain $D_j(T, \phi)$ and $C_j(T, \phi)$ by using Eq.~\ref{eq:hestonCDfunc} with initial conditions $\tau_0 = \tau_1$, $D_{j0} = D_j(\tau_1, \phi)$ and $C_{j0} = C_j(\tau_1, \phi)$ and using parameters $(\kappa_1, \theta_1, \rho_1, \xi_1)$ in place of $\big(\kappa (t),\theta (t),\rho(t),\xi (t)\big)$. After completing this iteration procedure, the characteristic function can be obtained by inserting the final $D_j(T, \phi)$ and $C_j(T, \phi)$ into Eq.~\ref{eq:hestonansatzcf}. Plain-vanilla call prices can then be calculated from the characteristic function using Eqs.~\ref{eq:charfuncgeneral} and \ref{eq:hestonansatz}.
\subsection{Black-Scholes control variate method for calculating plain-vanilla call prices within the Heston model with piecewise constant parameters}
An early treatment of the Heston model with piecewise constant parameters is presented in Ref.~\cite{Mikhailov2004}. The authors reached results similar to the ones presented in the previous section, but reported numerical problems with the necessary integrals and slow convergence behavior. Therefore, we extend the formalism once more, using the idea of a control variate~\citep{AndersenPiterbarg2010} to stabilize the numerics and, in particular, to make the integral that appears in the formula for the risk-neutral probabilities (Eq.~\ref{eq:charfuncgeneral}) converge faster.
We take the ansatz for the plain-vanilla call price within the Heston model with piecewise constant parameters (Eq.~\ref{eq:hestonansatz}) and add zero in the form of the Black-Scholes price of the same plain-vanilla call option calculated using the direct Black-Scholes formula for the risk-neutral probabilities $P_j$ subtracted by the Black-Scholes price expressed through the characteristic function (using Eqs.~\ref{eq:charfuncgeneral}, \ref{eq:bsansatzcf} and \ref{eq:bsDfunc}). For the Black-Scholes pricing we assume $\sigma = \sqrt{v_0}$. This results in a modified formula for the plain-vanilla call price within the Heston model with piecewise constant parameters.
\begin{equation}
\begin{split}
C(K) &= C^{BS} (K) + e^x \left[ P_1^H - P_1^{BS}\right] (x, v, \tau, K) - e^{-r \tau} K \left[ P_2^H - P_2^{BS} \right] (x, v, \tau, K) \\[2pt]
&= C^{BS} (K) + e^x \tilde P_1 (x, v, \tau, K) - e^{-r \tau} K \tilde P_2 (x, v, \tau, K)
\end{split}
\label{eq:hestonptildepricing}
\end{equation}
The quantity $\tilde P_j$ is the difference of the risk-neutral probabilities between the Heston model with piecewise constant parameters and the Black-Scholes model. Hence, $\tilde P_j$ itself is not a probability and can assume negative values. Using Eq.~\ref{eq:charfuncgeneral} for the risk-neutral probability expressed through the characteristic function $f_j$ we get an expression for $\tilde P_j$.
\begin{equation}
\tilde P_j (x, v, \tau, K) = \frac{1}{\pi} \int\limits_0^\infty d \phi \, \text{Re} \left[ \frac{e^{-i \phi \text{ln} (K)}}{i \phi} \left( f_j^H (x, v, \tau, K) - f_j^{BS} (x, \tau, K) \right) \right]
\label{eq:hestonptilde}
\end{equation}
Here, $f_j^{BS}$ is the characteristic function of the Black-Scholes model defined by Eq.~\ref{eq:bsansatzcf}, while $f_j^H$ is the characteristic function of the Heston model with piecewise constant parameters defined by Eq.~\ref{eq:hestonansatzcf}. In all BS terms we use $\sigma = \sqrt{v_0}$.
The crucial point about the control variate approach is that $C^{BS}(K)$ is calculated from the closed expression for the risk-neutral probabilities (Eqs.~\ref{eq:bsanalyticprob} and \ref{eq:bsansatz}). Therefore, $\tilde P_j$ just contains corrections of the piecewise constant Heston model with respect to the Black-Scholes model.
In practice, we calculate $\tilde P_j$ using the computationally efficient Gauss-Kronrod quadrature~\citep{Kahaner1989}. Note that this way we automatically avoid the evaluation of the integrand in Eq.~\ref{eq:hestonptilde} at the numerically problematic lower boundary ($\phi = 0$), since the quadrature nodes never coincide with the boundary points.
\subsection{Calibration of the Heston model with piecewise constant parameters to the foreign exchange options market}
For the calibration of the model defined by Eq.~\ref{eq:hestonprocess} we use a range of plain-vanilla call options with different strikes and maturities. Since FX options are conventionally quoted with respect to the option delta, we use five strikes equivalent to an option delta of $\Delta = (\pm 0.15, \pm 0.25, 0.5)$. The calibration of the Heston parameters is facilitated by numerically optimizing the option prices generated from Eq.~\ref{eq:hestonptildepricing} with respect to the market price using the Levenberg-Marquardt algorithm~\citep{Press2007}.
As a starting point for finding piecewise constant parameters, we first numerically optimize a global parameter set for the entire time interval $[0,T_N]$ up to the longest maturity of interest $T_N$. In this first optimization step the speed of mean-reversion is kept constant at $\kappa = 1.5$. After finding the globally optimal parameters, the initial variance $v_0$ is fixed to the value found in the global optimization step. The values for $\theta$, $\kappa$, $\rho$ and $\xi$ are used as initial values for the optimization when attempting to find time-dependent parameters.
The time-dependence of parameters $\theta$, $\kappa$, $\rho$ and $\xi$ is now introduced by bootstrapping parameter sets from the shortest to the longest maturity. We start with the shortest maturity $T_0$ and numerically optimize the parameters, which we then assign to the time interval $[0, T_0)$. Then we proceed to the next longer maturity $T_1$ and determine the parameters for the time interval $[T_0, T_1]$. We continue the iteration until parameters for all desired time intervals are determined.
Note that within the optimization process bounds must be imposed on the parameters. The correlation $\rho$ is limited to the interval $[-1:1]$, while the other parameters in general have to be positive. In practice, we impose bounds also on the other parameters to prevent the optimization algorithm from exploring regimes of excessively small or large parameters.
\section{Results}
\subsection{Effect of Black-Scholes control variate on integrands for risk-neutral probabilities}
Here, we show results for the integrand of the characteristic function of plain-vanilla call options in various settings to demonstrate the advantages of using a Black-Scholes control variate. The parameter sets we use are given in terms of a standard Heston model with constant parameters and were calibrated to the market on May 23rd, 2017. The calibration results are given in Table~\ref{tab:hestonparameterssimple}.
\begin{table}[bt]
\setlength\tabcolsep{10pt}
\caption{Plain Heston model parameters calibrated to market-quoted plain-vanilla call options on May 23rd, 2017.}
\begin{tabular}{r r r r r r r}
ccy pair & T & $v_0 \times100$ & $\theta \times10$ & $\kappa$ & $\rho$ & $\xi$ \\
\hline
EUR/USD & 2M & 0.675 & 0.199 & 1.04 & -0.276 & 0.313 \\
EUR/USD & 1Y & 0.674 & 0.166 & 0.92 & -0.239 & 0.190 \\
EUR/GBP & 2M & 0.697 & 0.177 & 1.09 & -0.032 & 0.290 \\
EUR/GBP & 1Y & 0.648 & 0.104 & 0.79 & 0.187 & 0.174 \\
EUR/JPY & 2M & 0.944 & 0.175 & 1.40 & -0.403 & 0.257 \\
EUR/JPY & 1Y & 0.893 & 0.148 & 0.89 & -0.265 & 0.240 \\
\end{tabular}
\label{tab:hestonparameterssimple}
\end{table}
In Fig.~\ref{fig:controlvar1y} we show the integrand of Eq.~\ref{eq:hestonptilde} and the logarithm of its absolute value with and without control variate plotted against the expansion parameter $\phi$ for an at-the-money call option with expiry after one year. The integrand is generally well-behaved and easy to integrate both with and without the control variate approach. Although the control variate approach reduces the magnitude of the integrand at the origin ($\phi = 0$) by several orders, it can lead to increases at larger $\phi$ as seen in the EUR/GBP case (see Fig.~\ref{fig:controlvar1y}(c,d)). When pricing at-the-money call options we did not find any numerical difficulties, even without the control variate approach. Compared to literature results~\citep{Mikhailov2004} this may be due to our slightly different formulas for the solution of the Heston Riccati ODEs (see Eq.~\ref{eq:hestonCDfunc}).
\begin{figure*}[t]
\includegraphics[width=\linewidth]{F01.pdf}
\caption{(Color online) Integrand (top row) and logarithm of the absolute value of the integrand (bottom row) of Eq.~\ref{eq:hestonptilde} without (w/o) and with (w) control variate (CV) as a function of expansion parameter $\phi$ for an at-the-money plain-vanilla call option in the Heston model. Parameters for maturity $T$=1Y are taken from Table~\ref{tab:hestonparameterssimple}.}
\label{fig:controlvar1y}
\end{figure*}
The power of the control variate approach is more evident when pricing options that are far out-of-the-money, where the integrand without control variate is much less well-behaved. In Fig.~\ref{fig:controlvar2m} we show results for the integrand of Eq.~\ref{eq:hestonptilde} for an out-of-the-money call option with strike $K = 1.3 S_0$. Without control variate, the integrand displays wild oscillations that are damped with increasing expansion parameter $\phi$. Using a control variate reduces the magnitude of the integrand at the origin by about eight orders and strongly suppresses the magnitude of oscillations. This results in a vast advantage when it comes to numerical integration, due to a significantly reduced number of function evaluations.
\begin{figure*}[t]
\includegraphics[width=\linewidth]{F02.pdf}
\caption{(Color online) Integrand (top row) and logarithm of the absolute value of the integrand (bottom row) of Eq.~\ref{eq:hestonptilde} without (w/o) and with (w) control variate (CV) as a function of expansion parameter $\phi$ for an out-of-the-money plain-vanilla call option with strike $K = 1.3 S_0$ in the Heston model. Parameters for maturity $T$=2M are taken from Table~\ref{tab:hestonparameterssimple}.}
\label{fig:controlvar2m}
\end{figure*}
Note, however, that our far out-of-the-money example represents a somewhat extreme case that is not likely to occur in real-world applications. Nevertheless, it demonstrates that the control variate approach is a simple way to stabilize the numerics of the Heston model when pricing options that involve less well-behaved integrands.
\subsection{Calibration of the Heston model with piecewise constant parameters to the term-structure of the foreign exchange options market}
In the following we want to price EUR/USD window barrier options with times to maturity 1M, 3M, 6M and 1Y. Therefore, we need to calibrate Heston models for those maturities. We subdivide each of the intervals into three subintervals (four in the case of 1Y to maturity) and calibrate piecewise parameters for each sub-interval using the methodology described above. The resulting parameter sets are shown in Table~\ref{tab:hestonparameterspiecewise}. The set indices (1,2,3,4) correspond to the times to maturity (1M, 3M, 6M, 1Y).
\begin{table}[bt]
\setlength\tabcolsep{10pt}
\caption{Piecewise Heston model parameters calibrated to market-quoted EUR/USD plain-vanilla call options on August 7th, 2013.}
\begin{tabular}{r r r r r r r r}
Set Index & From & To & $v_0 \times100$ & $\theta \times10$ & $\kappa$ & $\rho$ & $\xi$ \\
\hline
1 & 0 & 1W & 0.7 & 0.00 & 2.986 & -0.064 & 0.771 \\
& 1W & 3W & 0.7 & 0.29 & 6.000 & -1.000 & 0.484 \\
& 3W & 1M & 0.7 & 0.00 & 6.000 & -0.209 & 0.681 \\
\hline
2 & 0 & 1M & 0.6 & 0.13 & 1.539 & -0.322 & 0.281\\
& 1M & 2M & 0.6 & 0.12 & 4.948 & -0.764 & 0.173\\
& 2M & 3M & 0.6 & 0.02 & 1.735 & -0.288 & 0.367\\
\hline
3 & 0 & 2M & 0.6 & 0.09 & 1.859 & -0.374 & 0.232 \\
& 2M & 4M & 0.6 & 0.12 & 3.025 & -0.549 & 0.310 \\
& 4M & 6M & 0.6 & 0.05 & 2.855 & -0.477 & 0.203 \\
\hline
4 & 0 & 1M & 0.6 & 0.10 & 1.443 & -0.321 & 0.277\\
& 1M & 2M & 0.6 & 0.12 & 4.985 & -0.693 & 0.198\\
& 2M & 6M & 0.6 & 0.07 & 2.430 & -0.437 & 0.268\\
& 6M & 1Y & 0.6 & 0.16 & 1.613 & -0.503 & 0.328\\
\end{tabular}
\label{tab:hestonparameterspiecewise}
\end{table}
The stability of the calibration procedure is illustrated by the 0 to 1M sub-intervals in sets 2 and 4, which start the piecewise optimization from different globally optimal parameter sets, but should arrive at the same calibrated parameters if the algorithm is stable, since they calibrate to the same data. We observe that the deviations between different optimization runs are tiny and that the optimization is in general stable (see Table~\ref{tab:hestonparameterspiecewise}). For shorter maturities (sets 1 and 2) the quality of calibration is not very good and calibrated parameters are sometimes close to the imposed bounds ($\rho \in [-1:1]$, $\kappa \in (0, 6]$). The upper bound for $\kappa$ is somewhat artificial, but prevents the optimization algorithm from exploring excessively large speeds of mean-reversion, which can lead to slow calibration. This indicates that using very short time-intervals smaller than one month in the piecewise constant Heston model is not advisable.
For parameter set 4, which corresponds to an overall product time to maturity of 1Y, we show the calibrated and market-quoted implied volatilities for each time-slice in Fig.~\ref{fig:calibration1y}. Note the excellent quality of calibration despite the strong term structure of volatility contained in the market, which increases from short to longer maturities. In such cases, the Heston model with piecewise constant parameters is able to reproduce the market and improvements over a plain Heston model calibrated against maturity can be expected for products that are sensitive to the term-structure of volatility. Therefore, we continue with the pricing of window barrier options.
\begin{figure*}[t]
\includegraphics[width=\linewidth]{F03.pdf}
\caption{(Color online) Calibrated implied volatilities within the piecewise Heston model for EUR/USD plain-vanilla call options on August 7th, 2013. Note the strong term-structure of implied volatility, which increases from short maturities to longer ones. The values of volatility are given in percent and quoted versus the Delta of plain-vanilla put (10P, 25P) and plain-vanilla call options (25C, 10C). ATM stands for the at-the-money Delta. Lines serve as guides to the eye. The implied volatilities shown here correspond to parameter set No.~4 in Table~\ref{tab:hestonparameterspiecewise}.}
\label{fig:calibration1y}
\end{figure*}
\subsection{Pricing of window-barrier options within the Heston model with piecewise constant parameters}
Recently, Tian et al.~\cite{Tian2013} have presented a methodology to price window barrier options using a stochastic local volatility model, which shows good agreement with market prices. The reference prices contained in Ref.~\cite{Tian2013} correspond to actual traded options. Here, we cite the parameters of these options and the reference prices (Table~\ref{tab:windowbarrierspecs}).
\begin{table}[bt]
\caption{Window barrier options with reference prices taken from Ref.~\cite{Tian2013}. In the trigger column, L corresponds to a lower barrier, while U corresponds to an upper barrier.}
\setlength\tabcolsep{10pt}
\renewcommand{\arraystretch}{0.7}
\scalebox{0.7}{
\begin{tabular}{r r r r r r r r}
Maturity & Index & Trigger & Barrier start & Barrier end & Reference & H & HPW \\ \hline
1m, 2013/09/05 & 1 & L=1.26 & 2013/08/07 & 2013/08/21 & 0.0002 & 0.0005 & 0.0006 \\
& 2 & L=1.26 & 2013/08/21 & 2013/09/05 & 0.0017 & 0.0025 & 0.0024 \\
& 3 & L=1.26 & 2013/08/07 & 2013/09/05 & 0.0017 & 0.0025 & 0.0024 \\
& 4 & U=1.36 & 2013/08/07 & 2013/08/21 & 0.0036 & 0.0043 & 0.0043 \\
&5 & U=1.36 & 2013/08/21 & 2013/09/05 & 0.0079 & 0.0080 & 0.0081\\
&6 & U=1.36 & 2013/08/07 & 2013/09/05 & 0.0080 & 0.0081 & 0.0081 \\ \hline
3m, 2013/11/08 & 7 & L=1.23 & 2013/08/07 & 2013/08/21 & 0.0000 & 0.0003 & 0.0004 \\
&8 & L=1.23 & 2013/08/07 & 2013/11/01 & 0.0083 & 0.0093 & 0.0093 \\
&9 & L=1.23 & 2013/08/21 & 2013/11/08 & 0.0094 & 0.0100 & 0.0100 \\
&10 & L=1.23 & 2013/11/01 & 2013/11/08 & 0.0080 & 0.0085 & 0.0084 \\
&11 & L=1.23 & 2013/08/07 & 2013/11/08 & 0.0094 & 0.0100 & 0.0100 \\
&12 & U=1.39 & 2013/08/07 & 2013/08/21 & 0.0002 & 0.0012 & 0.0011 \\
&13 & U=1.39 & 2013/08/07 & 2013/11/01 & 0.0138 & 0.0134 & 0.0133 \\
&14 & U=1.39 & 2013/08/21 & 2013/11/08 & 0.0148 & 0.0140 & 0.0138 \\
&15 & U=1.39 & 2013/11/01 & 2013/11/08 & 0.0129 & 0.0130 & 0.0128 \\
&16 & U=1.39 & 2013/08/07 & 2013/11/08 & 0.0148 & 0.0140 & 0.0139 \\
\hline
6m, 2014/02/06 & 17 & L=1.20 & 2013/08/07 & 2013/08/21 &0.0000 & 0.0003 & 0.0002 \\
& 18 & L=1.20 & 2013/08/07 & 2013/11/01 & 0.0043 & 0.0056 & 0.0055 \\
& 19 & L=1.20 & 2013/08/07 & 2014/01/31 & 0.0159 & 0.0170 & 0.0171 \\
& 20 & L=1.20 & 2013/08/21 & 2014/02/06 & 0.0165 & 0.0170 & 0.0171 \\
& 21 & L=1.20 & 2013/11/01 & 2014/02/06 & 0.0164 & 0.0168 & 0.0168 \\
&22 & L=1.20 & 2014/01/31 & 2014/02/06 & 0.0135 & 0.0136 & 0.0136 \\
&23 & L=1.20 & 2013/08/07 & 2014/02/06 & 0.0165 & 0.0170 & 0.0171 \\
&24 & U=1.42 & 2013/08/07 & 2013/08/21 & 0.0000 & 0.0005 & 0.0004 \\
&25 & U=1.42 & 2013/08/07 & 2013/11/01 & 0.0065 & 0.0078 & 0.0076 \\
&26 & U=1.42 & 2013/08/07 & 2014/01/31 & 0.0189 & 0.0185 & 0.0183 \\
&27 & U=1.42 & 2013/08/21 & 2014/02/06 & 0.0195 & 0.0185 & 0.0183 \\
&28 & U=1.42 & 2013/11/01 & 2014/02/06 & 0.0193 & 0.0183 & 0.0181 \\
&29 & U=1.42 & 2014/01/31 & 2014/02/06 & 0.0160 & 0.0163 & 0.0161 \\
&30 & U=1.42 & 2013/08/07 & 2014/02/06 & 0.0195 & 0.0185 & 0.0183 \\ \hline
1y, 2014/08/07 & 31 & L=1.15 & 2013/08/07 & 2013/08/21 & 0.0000 & 0.0002 & 0.0002 \\
& 32 & L=1.15 & 2013/08/07 & 2013/11/01 & 0.0009 & 0.0023 & 0.0020 \\
&33 & L=1.15 & 2013/08/07 & 2014/01/31 & 0.0085 & 0.0111 & 0.0099 \\
& 34 & L=1.15 & 2013/08/07 & 2014/07/07 & 0.0253 & 0.0260 & 0.0258 \\
&35 & L=1.15 & 2013/08/21 & 2014/07/07 & 0.0253 & 0.0260 & 0.0258 \\
&36 & L=1.15 & 2013/11/01 & 2014/07/07 & 0.0253 & 0.0259 & 0.0257 \\
&37 & L=1.15 & 2014/01/31 & 2014/07/07 & 0.0250 & 0.0253 & 0.0251 \\
&38 & L=1.15 & 2013/08/07 & 2014/08/07 & 0.0282 & 0.0282 & 0.0285 \\
&39 & U=1.45 & 2013/08/07 & 2013/08/21 & 0.0000 & 0.0002 & 0.0002 \\
&40 & U=1.45 & 2013/08/07 & 2013/11/01 & 0.0024 & 0.0044 & 0.0038 \\
&41 & U=1.45 & 2013/08/07 & 2014/01/31 & 0.0117 & 0.0142 & 0.0123 \\
&42 & U=1.45 & 2013/08/07 & 2014/07/07 & 0.0314 & 0.0293 & 0.0285 \\
&43 & U=1.45 & 2013/08/21 & 2014/07/07 & 0.0314 & 0.0293 & 0.0285 \\
&44 & U=1.45 & 2013/11/01 & 2014/07/07 & 0.0313 & 0.0292 & 0.0285 \\
&45 & U=1.45 & 2014/01/31 & 2014/07/07 & 0.0310 & 0.0288 & 0.0281 \\
&46 & U=1.45 & 2013/08/07 & 2014/08/07 & 0.0345 & 0.0315 & 0.0312 \\
\end{tabular}
\label{tab:windowbarrierspecs}
\end{table}
We do not use a stochastic local volatility model, but price these options with a plain Heston model (abbreviated H) calibrated against maturity and using the Heston model with piecewise constant parameters (abbreviated HPW) as given in Table~\ref{tab:hestonparameterspiecewise}. The option pricing is done using the finite difference method, see f.i. Ref.~\cite{Clark2010}. The prices calculated using our implementations are given in Table~\ref{tab:windowbarrierspecs}. A comparison between the Heston model with and without time-dependence of parameters is given in Fig.~\ref{fig:barrierprices}.
The prices in the Heston model with piecewise constant parameters are generally consistent with those of a plain Heston model. Significant differences in price are only visible for options with a maturity of 6M or longer, since only then the term-structure of volatility actually takes effect. Strong improvements are visible for options with indices 33, 40 and 41 (see Table~\ref{tab:windowbarrierspecs}). All of those contain short barrier windows right at the beginning of the product term, which makes them especially sensitive to the much lower volatility in this time span compared to the rest of the product lifetime. Therefore, the plain Heston model calibrated against maturity uses a too large volatility at the beginning of the product term and overestimates the probability of the knock-in barrier being reached. Accordingly, it overestimates the value of the product.
In general, however, introducing piecewise constant parameters into the Heston model only leads to improvements, where products are very sensitive to the term-structure of volatility. Other problems of the Heston model, such as poor pricing for far out-of-the money barriers, like for option numbers 42 to 46, are not cured by the piecewise constant parameters. In such cases, a hybrid stochastic local volatility model has to be considered~\citep{Clark2010, Tian2013, vanDerStoep2014, Wyns2016}.
\begin{figure*}[t]
\includegraphics[width=\linewidth]{F04.pdf}
\caption{(Color online) Pricing error within the plain Heston model (H) and the Heston model with piecewise constant parameters (HPW) plotted against the index of the priced window barrier option. The option prices are calculated in EUR per USD.}
\label{fig:barrierprices}
\end{figure*}
\section{Conclusions}
We extended the calibration of the Heston stochastic volatility model via the method of characteristic functions to the case of piecewise constant parameters. In the numerical treatment of the resulting formulas we introduced the method of a control variate to suppress numerical instabilities. In combination with Gauss-Kronrod quadrature, this leads to a fast and reliable calibration strategy, even in the case of piecewise constant parameters.
We benchmarked the piecewise constant Heston model using window barrier options, which are sensitive to the term structure of volatility. For less sensitive options, the piecewise calibration does not improve upon a plain Heston model. However, we find that for maturities longer than 6M, the piecewise calibration leads to improved pricing of window barrier options with strong dependence on the term structure of volatility. In such cases, the Heston model with piecewise constant parameters in combination with our formalism for calibration offers rapid and reliable calibration, while the complexity of pricing is on the same level as for the standard Heston model.
|
2,877,628,089,025 | arxiv | \section{Introduction}
Actual causation has puzzled philosophers since at least the work by \citeN{lewis73}. One way of phrasing the problem is as follows: suppose we know the causal laws that govern some domain, and that we then observe a story that takes place in this domain; when should we now say that, in this particular story, one thing actually caused another? Recent work by \citeN{halpernpearl05a} has also garnered interest in this topic in the AI community. Their account (which I will refer to as {\em HP}) constructs a formal definition that tries to capture this intuition in the context of structural models \cite{pearl:book}. To be more concrete, it defines when some random variables $\vec{X}$ of the structural model having values $\vec{x}$ can be counted as an actual cause for $\vec{Y} = \vec{y}$.
In previous work, I have tried to show that the knowledge representation properties of Pearl's structural models can be improved by borrowing representations and techniques from logic programming. In particular, \citeN{vennekens04} introduced the probabilistic logic programming language of Logic Programming with Annotated Disjunctions, for which Riguzzi (2008; 2010) \nocite{riguzzi08,riguzzi10}
implemented SLD and SLG based resolution algorithms. Further analysis of this language has lead to a reformulation of its semantics, called {\em CP-logic}, which attempts to clarify its causal aspects and examine its relation to Pearl's work \cite{vennekens09}. A more recent paper \cite{vennekens:jelia} showed that Pearl's analysis of interventions and counterfactuals in the context of structural models can be elegantly redone in the context of CP-logic, yielding better results for a number of examples, most notably when cyclic causalities are involved.
The goal of this paper is to examine the notion of actual causation in the context of CP-logic. Section \ref{sec:mot} will start with some motivation, by explaining a few of the differences between structural models and CP-logic, and offering some hand-waving arguments for why CP-logic might offer a more appropriate setting for the study of actual causation. The semantics of CP-logic is briefly recalled in Section \ref{sec:prel}. Then, Section \ref{sec:hp} dives into the details of actual causation by discussing the HP definition, while Section \ref{sec:ac} gives my own account. The traditional way of testing such a definition is to run through a number of ``tricky'' examples and checking whether the obtained answers are intuitively plausible. In Section \ref{sec:ex}, I will follow suit. Finally, Section \ref{sec:imp} briefly comments on a naive implementation in Prolog that can be downloaded to play with my definitions.
\section{Motivation: structural models and CP-logic}\label{sec:mot}
A structural model is based on a set of random variables (RVs). Each RV has an associated domain of possible values. The simplest case are Boolean RVs, which have $\{{\bf t},{\bf f}\}$ as their domain, and can therefore be thought of as propositional symbols or ground atoms. Boolean RVs suffice for typical examples of actual causation, so I will from now on restrict attention to just these.
A structural model then consists of a set of equations $X := f(\vec{Y})$,
which define the value of the RV $X$ in terms of the RVs $\vec{Y}$ by a Boolean function $f$. The RVs that appear in the left-hand side of such an equation are called {\em endogenous}, and the other ones {\em exogenous}. Typically, these sets of equations are assumed to be acyclic. Their meaning is formalized by the obvious possible world semantics: each assignment of values to the RVs that satisfies all of the equations is a possible world. An acyclic set of equations has the useful property that an assignment of values to the exogenous RVs uniquely determines a single possible world.
Pearl uses structural models to represent causal relations: each equation $X := f(\vec{y})$ is taken to mean that the causes for $X$ taking value $x$ are all assignments $\vec{Y}=\vec{y}$ for which $f(\vec{y})=x$. This representation is used by Pearl to great effect, studying interventions, counterfactuals, and of course also actual causation.
Despite its successes, however, there is something peculiar about the structural model representation of causal relations: it does not take into account their dynamic nature. Suppose, for instance, that you make the causal claim that dropping a glass causes it to break. If I don't believe you, I might challenge you to prove your claim. How would you do this? Presumably, you would first hold out an unbroken glass. Then, you would drop it, so that I could watch it fall, hit the floor, and break. In other words, you would show me a transition from a state of the world in which the glass is whole to a one in which it is broken. If you can convince me that it was indeed your dropping the glass that initiated this transition, then you have proven your causal claim.
What this little thought experiment shows is that the idea of a transition from one state of the world to the next is inherently part of the way in which we interpret causal statements. However, structural models have nothing to do with such transitions. For instance, the causal claim about glasses breaking would just be represented by an equation $Break := Drop$, which determines two possible worlds: $\{ Break = {\bf t}, Drop ={\bf t}\}$ and $\{ Break = {\bf f}, Drop ={\bf f}\}$. In this sense, structural models make complete abstraction of the dynamic aspects of causality, until all that remains is a static picture of how the values of different random variables can be defined in terms of each other.
Other approaches to causality do not share this static worldview. For instance, \citeN{shafer:book} gives an explicitly dynamic account of causation. He represents causal systems by means of probability trees, in which edges represent transitions between states of the world. For example, in the following picture, the edge going from $N_1$ to $N_2$ represents a transition from a state in which Joe hasn't yet taken a swing at the ball to one in which he has and missed:
\begin{equation}
\begin{split}\label{transition}
\xymatrixrowsep{0.6cm}
\xymatrix{
&\ldots\ar[d]\\
& N_1 \ar[ld]_{\text{Joe misses ball}}^{0.75} \ar[rd]^{\text{Joe hits ball}}_{0.25}\\
N_2 \ar[d]& & N_3\ar[d]\\
\ldots && \ldots
}
\end{split}
\end{equation}
The edge $(N_1,N_3)$ represents a transition of the same state $N_1$ to a state where Joe has hit the ball. Together, these two edges represent a non-deterministic event\footnote{Note that I do not use the term ``event'' in its probability-theoretical meaning of ``a set of possible outcomes'', but rather in the common-sense meaning of ``a thing that happens''. }, namely that of Joe's taking a swing at the ball, which may result in one of these two outcomes.
The edges are labeled with the probabilities of the outcomes: the probability of Joe's swing missing is $0.75$ and that of it hitting is $0.25$.
This paper will use {\em CP-logic} as its formal language, which is essentially just a modular, syntactic representation for such Shaferian probability trees. A theory in CP-logic represents the causal structure of a domain by means of a set of {\em causal probabilistic laws} (CP-laws, for short). Each such CP-law is a blue-print for a class of non-deterministic events. For instance, the following CP-law:
\[ \forall p,b\ (Hit(p,b):0.25) \lor (Miss(p,b):0.75) \leftarrow Swing(p,b)\]
states that, for every player $p$ and ball $b$, player $p$'s taking at swing at ball $b$ causes a non-deterministic event, which has as one possible outcome that $p$ hits $b$ (and this happens with probability $0.25$), and as its other possible outcome that $p$ misses $b$ (which happens with probability $0.75$).
If $p$ and $b$ are instantiated to, respectively, a particular player and a particular ball, say Joe and the twelfth pitch, the we obtain a description of one particular event (that may or may not happen, depending on whether Joe decides to swing):
\begin{equation*} (Hit(Joe,12):0.25) \lor (Miss(Joe,12):0.75) \leftarrow Swing(Joe,12). \label{joeshit}
\end{equation*}
This instantiated CP-law can be seen as a textual representation of picture \eqref{transition}, provided that, of course, node $N_1$ represents a state of the domain where Joe has decided to take a swing at this particular pitch. In this way, each instantiation of a CP-law describes a piece of probability tree. As will be explained in more detail later, an entire CP-theory describes a class of probability trees, each of which can be constructed by putting these small pieces together.
Unlike structural models, the formal semantics of CP-logic therefore does provide mathematical objects that represent transitions between states of the world. As argued by \citeN{stonesoup}, these transitions are important for a study of actual causation. Indeed, when the goal is to figure out what caused what in a given {\em story}, it is obviously convenient to have a language whose formal semantics already offers objects that correspond in a natural way to stories. A branch of a Shaferian probability tree is precisely such an object, because, like a story, it is a description of a sequence of events that change the state of the world.
There is also a second argument in favour of CP-logic. The goal of actual causation is to explain why things happened. Typically, though, not everything is in need of explanation. A detective solving a murder case, for instance, will be interested in why the victim is dead, but he won't care about why he was ever alive in the first place. The detective's causal model will therefore list causes for dying (poison, gun shot, \ldots), but not for living (sexual intercourse, IVF, \ldots). In more technical terms, the detective considers living to be the {\em default} state of a person, and he is only interested in {\em deviations} from this default. Many authors, such as \citeN{hall04} or \citeN{hitchcock07}, have argued that actual causation should be studied under the assumption that each RV has such a default state.
Structural models make no distinction between the different values of a RV. Consequently, a RV $Alive$ with values $yes$ and $no$, and a RV $Dead$ with values $no$ and $yes$ are completely interchangeable. In CP-logic, this is not the case. Here, each RV (i.e., ground atom) has {{\bf f}} as its default value. This means that the mere existence of the atom $Alive(Adam)$ implies that the default condition is for $Adam$ to be dead, and that he can only come to life when there is a sufficient cause for this. By contrast, the detective's theories will contain atoms such as $Dead(Adam)$, indicating that living is the default and death is in need of causal explanation. In the probability trees generated by a CP-theory, an atom always starts out at its default value, and only deviates from this when it has sufficient cause to do so.
\section{Reminder: formal semantics of CP-logic}\label{sec:prel}
Lacking space for a full review of CP-logic, I will only summarize the main ideas and refer to \cite{vennekens09} for details. The general form of a CP-law is:
\begin{equation} \forall\vec{x}\ (A_1:\alpha_1)\lor\cdots\lor (A_n:\alpha_n)
\leftarrow \phi.\label{cplaw1}
\end{equation}
Here, $\phi$ is a first-order formula and the $A_i$ are atoms, such that the tuple of variables $\vec{x}$ contains all free variables in $\phi$ and the $A_i$. The $\alpha_i$ are non-zero probabilities with $\sum\alpha_i \leq 1$. Such a CP-law expresses that $\phi$ causes some (implicit) non-deterministic event, of which each $A_i$ is a possible outcome with probability $\alpha_i$. If $\sum_i \alpha_i = 1$, then at least one of the possible effects $A_i$ must result if the event caused by $\phi$ happens; otherwise, it is also possible that the event happens without any (visible) effect on the state of the world. For the purpose of this paper, the propositional fragment of CP-logic suffices, so I will from now on restrict attention to CP-laws in which the tuple of variables $\vec{x}$ is empty.
For a CP-law $r$, we refer to $\phi$ as the {\em body} of $r$, and to the sequence $(A_i,\alpha_i)_{i = 1}^{n}$ as the {\em head} of $r$. We denote these objects as $body(r)$ and $head(r)$, respectively, and also write $head_{At}(r)$ for the set of all $A_i$ for which there exists an $\alpha_i$ such that $(A_i,\alpha_i) \in head(r)$. For CP-laws that are vacuously caused, $body(r)$ may be omitted. If a CP-law has a deterministic effect, i.e., it is of the form $(A:1)\leftarrow\phi$, it is also written simply as $A\leftarrow\phi$.
A {\em CP-theory} is a finite set of CP-laws. Such a CP-theory describes the non-deterministic evolution of a domain, which is formally represented by a Shaferian probability tree. Initially, all RVs of this domain (i.e., all ground atoms) are in their default state. This means that we can describe the initial state of the domain, which corresponds to the root of the probability tree, by the interpretation that assigns {{\bf f}} to each of them. We then extend this root by picking a CP-law $r$ whose precondition $body(r)$ is satisfied according to this interpretation and creating a child node for each pair $(h_i: \alpha_i)$ in $head(r)$. The edge to child $i$ is labeled with the probability $\alpha_i$ and the corresponding new state of the domain is constructed from the previous state by switching $h_i$ to its deviant state {{\bf t}}. The CP-law $r$ has now happened, and will not happen again.
We repeat this process of adding children to one of the leaf nodes of the current tree, until this is no longer possible, i.e., until for all leaves $l$ of the current tree it is the case that all rules $r$ that have not yet happened in $l$ have a precondition $body(r)$ that is false in $l$. The resulting trees are called the {\em execution models} of the CP-theory. For a node $s$ of the tree, I denote by $\curly{I}(s)$ the interpretation that corresponds to the state of the world at that node, and, if $s$ is not a leaf, by $\curly{E}(s)$ the CP-law that was used to create the children of this node.
The construction of execution models is quite non-deterministic, in the sense that in any particular node of the tree, there can be many CP-laws that may be used to extend it. The question is now whether each of these trees actually reflects a sensible way in which a domain described by the CP-theory might evolve. The answer is a qualified ``yes'', and depends on precisely how we choose to interpret negation appearing in the body of a CP-law. Consider the following example:
\[
(Shatters: 0.9) \leftarrow \lnot DecidesNotToThrow(Suzy).
\]
We could take the body of this CP-law to mean that this transition may happen in any state where $DecidesNotToThrow$ is still at its default state {\bf f}, such as, by definition, the initial state in which Suzy has no yet made up her mind about throwing. Taking this view, every probability tree constructed according to the above principles can indeed be seen as a sensible description of how the domain might evolve. However, this is not very useful. As argued by \citeN{vennekens09}, it is more interesting to read negation in a slightly different way, namely, as not just saying that $DecidesNotToThrow$ is still at its default value in the current state, but that it can actually {\em never} deviate any more. In other words, according to this reading, the above CP-law will only be applicable {\em after} Suzy has decided that she will not refuse the throw. This idea is formalized in the semantics of CP-logic by means of a construction similar to the Gelfond-Lifschitz reduct. We use this to compute, for each state $s$, an overestimate $\curly{U}(s)\supseteq \curly{I}(s)$ of all atoms that can still be caused in this $s$. Only if an atom $a$ does not belong to $\curly{U}(s)$, do we then say that $\lnot a$ holds in $s$. If there are no loops containing double negation (i.e., some $\lnot P$ causing $Q$ and $\lnot Q$ causing $P$), then it is the case that, in any branch of a probability tree, each CP-law must either happen at some point, or else become impossible. \citeN{vennekens09} showed that there is a close connection between the resulting semantics and the well-founded model construction for a logic program.
Each probability tree $\curly{T}$ defines, in the obvious way, a probability distribution $\pi_\curly{T}$ over its leaves. For an execution model $\curly{T}$ of a CP-theory $C$, this distribution $\pi_\curly{T}$ induces a probabilistic possible world semantics: the probability $\pi_\curly{T}(S)$ of an interpretation $S$ is $\sum_{\curly{I}(l) = S} \pi_\curly{T}(l)$, where the sum is taken over leaves $l$ of $\curly{T}$. \citeN{vennekens09} showed that each execution model $\curly{T}$ of a CP-theory $C$ defines the same possible world semantics $\pi_\curly{T}$. For instance, the two trees shown on page \pageref{trees} are execution models of the same theory and, even though they are not isomorphic, they both define the same $\pi_\curly{T}$. In this way, each CP-theory $C$ defines a unique probability distribution, which is denoted as $\pi_C$. The probability of a formula $\phi$ can then be defined as $\pi_C(\phi) = \sum_{S \models \phi}\pi_C(S)$.
The fact that $\pi_C$ does not depend on the choice of any particular execution model $\curly{T}$ may help to explain why structural models choose to ignore the dynamic aspects of causality in the first place. Indeed, this result shows precisely that, for applications which only care about properties of the final state that the domain will eventually reach, the details of how this final state came about can be safely ignored. As I attempt to show in this paper, though, actual causation is {\em not} such an application.
Like structural models, CP-logic also makes a distinction between exogenous and endogenous random variables. With $X$ the set of all exogenous atoms, the semantics of a CP-theory now becomes relative to an interpretation $I$ for these atoms. In particular, an execution model for $C$ {\em given $I$} is defined as a execution model that starts not from a root in which {\em all} atoms are {{\bf f}}, but instead starts with only the endogenous atoms being {{\bf f}} and the exogenous atoms being interpreted by $I$. \citeN{vennekens09} have shown that for each interpretation $I$ for the exogenous predicates of a CP-theory $C$, all execution models $\curly{T}$ given $I$ define the same probability distribution $\pi_\curly{T}$, which is denoted as $\pi_C^I$.
\section{Actual causation in HP}\label{sec:hp}
This section briefly recalls the HP account. Their paper starts with this example:
\begin{quote}
Suppose that two arsonists drop lit matches in different parts of a dry forest, and that both cause trees to start burning, until the entire forest burns down. Both matches are necessary to burn down the forest; with only one match, the fire would die down.
\end{quote}
It is clear that both arsonists are an actual cause of the forest burning down. HP reach this conclusion as follows. To represent the causal structure of the example, they use a structural model consisting of a single equation:
\begin{equation}
Burn := Match_1 \land Match_2. \label{matches1}
\end{equation}
The particular story under consideration is then represented by the following assignment of values to the exogenous RVs: $\{ Match_1 = {\bf t}, Match_2 = {\bf t} \}$. This of course also uniquely determines the values of the endogenous RVs: $Burn ={\bf t}$.
The HP definition is now reproduced below. In it, $M$ is a structural model with endogenous RVs $\curly{V}$, $\vec{u}$ an assignment of values to the exogenous RVs, $\vec{X}$ a tuple of endogenous RVs, and $\phi$ a Boolean formula in the RVs. The notation $(M,\vec{u}) \models [\vec{X}\leftarrow\vec{x}] \phi$ means that $\phi$ holds in $(M,\vec{u})$ after the intervention of assigning $\vec{x}$ to $\vec{X}$ is performed, i.e., each $X_i \in \vec{X}$ has its defining equation removed from $M$ and replaced by $X_i := x_i$.
\begin{definition}[HP account of actual causation]\label{HP}
$\vec{X}=\vec{x}$ is an actual cause of $\phi$ in $(M, \vec{u})$ if the following three conditions hold.
\begin{itemize}
\item[AC1.] $(M, \vec{u}) \models (\vec{X}⃗ =\vec{x}) \land \phi.$ (That is, both $\vec{X} =\vec{x}$ and $\phi$ are true in the actual world.)
\item[AC2.] There exists a partition $(\vec{Z},\vec{W})$ of $\mathcal{V}$ with $\vec{X}\subseteq \vec{Z}$ and some setting $(⃗\vec{x}',\vec{w}')$ of the variables in $(\vec{X},\vec{W})$ such that if $(M, \vec{u}) \models Z = z^{*}$ for all $Z \in \vec{Z}$, then both of the following conditions hold:
\begin{itemize}
\item[(a)] $(M,\vec{u}) \models [\vec{X}\leftarrow \vec{x}', \vec{W}\leftarrow \vec{w}']\lnot\phi$. In words, changing $(\vec{X},\vec{W})$ from $(\vec{x},\vec{w})$ to $(\vec{x}',\vec{w}')$ changes $\phi$ from true to false.
\item [(b)] $(M,\vec{u}) \models [\vec{X}\leftarrow \vec{x}, \vec{W}' \leftarrow \vec{w}',\vec{Z}'\leftarrow\vec{z}^*]\phi$ for all subsets $\vec{W}'$ of $\vec{W}$ and all subsets $\vec{Z}'$ of $\vec{Z}$. In words, setting any subset of variables in $\vec{W}$ to their values in $\vec{w}'$ should have no effect on $\phi$, as long as $\vec{X}$ is kept at its current value $\vec{x}$, even if all the variables in an arbitrary subset of $\vec{Z}$ are set to their original values in the context $\vec{u}$.
\end{itemize}
\item[AC3.] $\vec{X}$ is minimal; no subset of $\vec{X}$ satisfies conditions AC1 and AC2. Minimality ensures that only those elements of the conjunction $\vec{X} = \vec{x}$ that are essential for changing $\phi$ in AC2(a) are considered part of a cause.
\end{itemize}
\end{definition}
With $\vec{X} = (Match_1)$, $\vec{Z} = (Match_1, Burn)$ and $\phi = Burn$, this definition provides the result that $Match_1$ actually caused $Fire$, since if we change $X$ to ${\bf f}$, while leaving $\vec{W} = (Match_2)$ at its original value (this trivially satisfies AC2(b)), we obtain $\lnot Burn$ as required by AC2(a). In other words, in this example, we get actual causation from a simple counterfactual dependency: if it hadn't been for $Match_1$, the forest wouldn't have burned down.
HP also consider a disjunctive variant of this example, where a single match already suffices to burn down the forest ($Burn := Match_1 \lor Match_2$). This causes the straightforward counterfactual criterion to fail, since stopping only one of the arsonists does not stop the forest burning down. This motivates the additional machinery of the above definition. By considering the context in which $Match_2 = {\bf f}$, that is $\vec{W} = (Match_2)$ and $w' = ({\bf f})$, we can re-establish the counterfactual dependency of $Burn$ on $\vec{X} = (Match_1)$.
\section{Actual causation in CP-logic}\label{sec:ac}
As we have seen, a question of actual causation can only be asked in the presence of two pieces of information: a causal model of a domain (the $M$ of Definition \ref{HP}) and a story that takes place in this domain (the $\vec{u}$). My definition will of course assume that the causal model is given in the form of a CP-theory $C$. In the context of CP-logic, the most obvious formal counterpart of a ``story'' is a branch of an execution model of $C$. Already, this allows us some more room for nuance than HP, as the following example from \citeN{hall04} shows.
\begin{quote}
Suzy and Billy might each decide to throw a rock at a bottle. If Suzy does so, her rock shatters the bottle with probability $0.9$. Billy's aim is slightly worse and he only hits with probability $0.8$.
\end{quote}
This domain corresponds to the following set of CP-laws, where $Throws(Suzy)$ and $Throws(Billy)$ are exogenous:
\begin{align} (Shatters: 0.9) &\leftarrow Throws(Suzy).\label{suzy}\\ (Shatters:0.8) &\leftarrow Throws(Billy).\label{billy}\end{align}
Assuming that Suzy and Billy both throw, there still exist two different execution models of the theory. Representing the states in which the bottle is broken by an empty circle, and those in which it is still whole by a full one, they look like this:\\ \label{trees}
\xymatrix@H=0.8cm{
&&\bullet \ar[ld]^{0.9}|(.3){\text{Suzy hits\hspace*{0.7cm}}} \ar[rd]_{0.1}|(.3){\text{\hspace*{0.6cm}misses}}\\
&\circ \ar[ld]_{0.8}|(.3){\text{Billy hits\hspace*{0.95cm}}} \ar[d]^{0.2}|(.3){\hspace*{0.95cm}\text{misses}} && \bullet \ar[d]_{0.8}|(.3){\text{Billy hits}\hspace*{0.95cm}} \ar[rd]^{0.2}|(.3){\hspace*{0.95cm}\text{misses}} \\
\circ&\circ&&\circ&\bullet
}
\hfill
\xymatrix@H=0.8cm{
&&\bullet \ar[ld]^{0.8}|(.3){\text{Billy hits\hspace*{0.7cm}}} \ar[rd]_{0.2}|(.3){\text{\hspace*{0.6cm}misses}}\\
&\circ \ar[ld]_{0.9}|(.3){\text{Suzy hits\hspace*{0.95cm}}} \ar[d]^{0.1}|(.3){\hspace*{0.95cm}\text{misses}} && \bullet \ar[d]_{0.9}|(.3){\text{Suzy hits}\hspace*{0.95cm}} \ar[rd]^{0.1}|(.3){\hspace*{0.95cm}\text{misses}} \\
\circ&\circ&&\circ&\bullet
}
\\
In the left execution model, Suzy's rock reaches the bottle before Billy's does, whereas in the right one, it is Billy's rock that gets there first. As discussed at the end of Section \ref{sec:prel}, this difference is irrelevant if we are only interested in the final outcomes that might be reached: the probability of the bottle shattering is $0.98$ in both models. However, the difference becomes relevant when we want to judge actual causation. Indeed, in the left execution model, it is possible for Suzy's rock to {\em actually} break the bottle even though Billy's also would have (in particular, this happens in the leftmost branch of the tree). According to the execution model on the right, however, this is impossible: here, Suzy's rock can only actually break the bottle if Billy's rock fails to do so.
\citeN{hall04} goes on to consider the following story:
\begin{quote} Suzy and Billy both pick up rocks and throw them at a bottle. Suzy's rock get there first, shattering the bottle. Since both throws are perfectly accurate, Billy's would have shattered the bottle had it not been preempted by Suzy's throw.
\end{quote}
This story tells us precisely that we are in the leftmost branch of the left execution model above. Hence, Suzy's rock should be the actual cause of the bottle breaking, and not Billy's. Before showing how I reach this conclusion in the context of CP-logic, let me first remark that things are more difficult for the HP account. Their paper first tries the following straightforward structural model:
\begin{align*} Shatters := (Throws(Suzy) \land Accurate(Suzy)) \lor (Throws(Billy) \land Accurate(Billy)).
\end{align*}
Here, there is no such thing as one execution in which Suzy's rock reaches the bottle first and one in which Billy's is first. Hall's story therefore seems to say nothing more than that all five RVs are ${\bf t}$, and the phrase ``Suzy's rock gets there first'' contributes nothing. Of course, because it is precisely this phrase that determines which rock actually broke the bottle, this causal model does not work.
HP fix the problem by introducing two new random variables: $Hits(Billy)$ (``Billy's rock hits the (unbroken) bottle'') and $Hits(Suzy)$. The order in which the two rocks actually reach the bottle can then be encoded {\em in the structure of the model}:
\[ Hits(Billy) := \lnot Hits(Suzy) \land Throws(Billy) \land Accurate(Billy)\]
To me, this does not seem the right way to go. The order in which the rocks arrive is a purely contingent matter, which belongs to the details of the particular story that is being told, and {\em not} to the general causal structure of the domain. Saying that Suzy's rock arrives before Billy's should not be placed on the same level of causal discourse as the statement that throwing rocks at bottles causes them to break. This is not just a matter of taste, but also has practical consequences. If we would want to know whether Suzy's rock would still have been the actual cause of the bottle breaking if Billy's rock had gotten there first, then---in the HP account---we would not just have to look at a different story in the same domain, but we would have to change the structure of our causal model. Such hand-tailoring of the causal model to the question under consideration is undesirable, and, as I will now show, it is not needed in CP-logic.
My definition too will be heavily based on the intuition of counterfactual dependency from a cause $C$ to an effect $E$. Therefore, I first formalize the following criterion:
\begin{center}
\hspace{\stretch{1}} \begin{minipage}{0.9\textwidth} If all events happen in the way they actually happened {\em with the exception that $C$ is somehow prevented from occurring}, then $E$ will no longer occur.
\end{minipage} \hspace{\stretch{1}} (*)
\end{center}
This requires some mathematical machinery. First, we need to be able to fix the outcome of certain events. For a CP-law $r$ of the form
$(A_1:\alpha_1) \lor \cdots \lor (A_n:\alpha_n)\leftarrow \phi$,
we write $r^{A_i}$ to denote the deterministic CP-law $A_i \leftarrow \phi$. If we now have a branch $b$ that tells us what actually happened, then we can define as follows a theory that fixes the outcome of all events that happened to their actual outcome.
\begin{definition}
Let $b = (s_0,\ldots,s_n)$ be a branch of an execution model of a CP-theory $T$. We define $T^b$ as the union of two disjoint sets $S_1$ and $S_2$, where $S_1$ contains all CP-laws from $T$ that did not happen in branch $b$, i.e., $ S_1 = T \setminus \{ \curly{E}(s_i) \mid 0 \leq i < n \}$,
and $S_2$ consists of all $r^A$ for which $r$ caused $A$ in $b$, i.e.,
\[ S_2 = \{ r^A \mid r \in T\text{ and for some } i: \curly{E}(s_i) = r\text{ and }\curly{I}(s_{i+1}) \setminus \curly{I}(s_i) = \{A\}\}.\]
\end{definition}
We also need an antonymical transformation, which prevents some $A_i$ from occurring. For an $r$ of the same form as above, we write $r^{\lnot{A_i}}$ for:
\[ (A_1:\alpha_1) \lor \cdots \lor (A_{i-1}:\alpha_{i-1}) \lor (A_{i+1}:\alpha_{i+1}) \lor \cdots \lor (A_n:\alpha_n)\leftarrow \phi.\]
To prevent an atom $A$ entirely, it now suffices to apply this transformation to all CP-laws that might cause it. Given a theory $T$, we therefore define $T^{\lnot A}$ as:
\[T^{\lnot A} = \{ r^{\lnot A} \mid r\in T\text{ and }A \in head_{At}(r) \} \cup \{ r \mid r\in T\text{ and } A \not \in head_{At}(r) \}.\]
By combining this transformation with the previous one, we can now construct a theory $(T^b)^{\lnot C}$ which corresponds precisely to the counterfactual eventuality that everything happens precisely as it did in branch $b$, with the exception that $C$ is somehow prevented from occurring. I thus formalize the counterfactual criterion (*), by expressing that, according to this new CP-theory $(T^b)^{\lnot C}$, $E$ will not occur.
\begin{definition} Let $b = (s_0,\ldots,s_n)$ be a branch of an execution model of a theory $T$.
For two atoms $C$ and $E$, such that both $C$ and $E$ hold in $\curly{I}(s_n)$, we say there is a {\em counterfactual dependency} from $C$ to $E$ if $\pi_{T'}^{I'}(E) = 0$ where $T' = (T^b)^{\lnot C}$ and, to cover the case where $C$ is exogenous, $I'$ is $\curly{I}(s_0) \setminus\{C\}$.
\end{definition}
Here, saying that $\pi_{T'}(E) = 0$ is of course equivalent to $E$ being false in each leaf $l$ of each execution model of $T'$.
This intuition of counterfactual dependency forms the core of the concept of actual causation, but as discussed above, it is in itself not enough. The additional aspect is the idea of {\em relevance}. A causal model might make provisions for a large number of eventualities, many of which may not have been relevant in the actual course of events. It is typical for judgments of actual causation that truly irrelevant causal mechanisms are ignored, even when they might appear to become relevant in a counterfactual context.
The typical case where this intuition manifests itself is when counterfactual dependencies are masked by {\em redundant causation}: there is some back-up mechanism waiting in the wings, which will ensures that the effect happens anyway, even if we preempt its actual cause. The example of Suzy and Billy is a good illustration of this. The reason why we nevertheless insist that Suzy is the actual cause of the bottle shattering is precisely a criterion of relevance: because Suzy's rock got to the bottle first, Billy's was irrelevant, so we ignore it.
\citeN{pearl:book} tried to formalize this same intuition by means of the concept of a {\em causal beam}, which is meant to encompass precisely the relevant parts of the causal model. However, the formal details proved hard to get right, and the refinement that eventually became part of the HP definition seems to be a significant source of complexity, which considerably clouds the otherwise rather simple idea of counterfactual dependency. In the explicitly dynamic context of CP-logic, something much more simple is possible.
Let us ask again why intuitions feels that Billy's rock is irrelevant if Suzy's rock gets to the bottle first and shatters it. I suggest the blindingly obvious answer: it just got there {\em too late}. By the time Billy's rock reached the bottle, the damage was already done, the bottle lay in pieces, and there was nothing left to shatter. In other words, one simply cannot cause what is already the case. My notion of relevance will comprise just this: whatever happened {\em after} the effect is irrelevant, and whatever happened {\em on the way to} the effect is counted as relevant. Of course, this is not yet a complete dichotomy, since it does not rule on the status of those events that did not happen at all. Recall that if some CP-law does not happen in a particular branch, this means that, somewhere along the way, its precondition must have become impossible. Whether an event that did not happen is considered relevant will depend on when its precondition became impossible: if this was {\em before} the effect arose, then it is relevant, otherwise not. This leads to the following definition.
\begin{definition}[Actual causation in a complete information setting] Let $b = (s_0,\ldots,s_n)$ be a branch of an execution model of a theory $T$. Let $C$ and $E$ be two atoms that both hold in the final state $s_n$ of $b$, i.e., $\{C,E\} \subseteq \curly{I}(s_n)$. $C$ is an {\em actual cause} of $E$ in branch $b$ if $\pi^{I'}_{T''}(E) = 0$ with $I' = \curly{I}(s_0) \setminus\{C\}$ and $T'' = ((T')^b)^{\lnot C}$, where $T'$ is constructed as follows. If $j$ is the smallest $k$ for which $E \in \curly{I}(s_k)$, then $T' = \{\curly{E}(s_i) \mid 0\leq i< j \} \cup \{ r \in T \mid \curly{U}(j-1) \models \lnot body(r)\}$. In words, $C$ is an actual cause of $E$ if there is a counterfactual dependency from $C$ to $E$, according to the theory $T '$ that consists of both those events that happened before $E$ was caused, and those events that had already become impossible by then. \label{def:actc}
\end{definition}
It is quite easy to check whether this definition is satisfied: you look at the given branch, find the place where $E$ first appeared, discard all events that had not yet happened then but still were possible, and check whether the remaining theory exhibits a counterfactual dependency between $C$ and $E$ or not. To illustrate, consider again the leftmost branch $(s_0,s_1,s_2)$ of the left execution model for the Billy and Suzy example. The bottle breaks in node $s_1$, i.e, $Shatters \in \curly{I}(s_1) \setminus \curly{I}(s_0)$. Therefore, $T' = \{\curly{E}(s_0)\} = \{
(Shatters : 0.9) \leftarrow Throws(Suzy)\}$ and $(T')^b = \{ Shatters \leftarrow Throws(Suzy)\}$. According to $(T')^b$, there now is indeed a counterfactual dependency from $Throws(Suzy)$ to $Shatters$, so the first is an actual cause of the second.
As this example illustrates, it is important that a branch $(s_0,\ldots,s_n)$ of an execution model not only records the successive states $\curly{I}(s_i)$ of the domain, but also the events $\curly{E}(s_i)$ that caused each of the state transitions.
Recall that the HP setting offers no mathematical objects that correspond to a complete story about what happened, so their definition is always just given the final outcome in the form of an assignment of values to the RVs. In this case, we cannot always say with certainty whether some potential cause actually caused an effect or not. Indeed, if we get only the final interpretation $\curly{I}(s_n)$ instead of the full branch $(s_o,\ldots,s_n)$, then the best we can do is this:
\begin{definition}[Actual causation in a partial information setting]
\label{partial} Let $T$ be a CP-theory and $I$ an interpretation for its vocabulary. Let $B(I)$ be the set of all branches of all execution models of $T$ that end in a state $s$ for which $\curly{I}(s) = I$. If $C$ actually causes $E$ in {\em at least one} branch $b \in B(I)$, we say that $C$ is a {\em possible} actual cause for $E$. If $C$ actually causes $E$ in {\em all} branches $b\in B(I)$, we say that $C$ is a {\em certain} actual cause for $E$.
\end{definition}
If, in the bottle breaking example, we are only told that eventually $Throws(Suzy)$, $Throws(Billy)$ and $Shatters$ all hold, we find ourselves faced with precisely the same problem as HP's first structural model: all that we can say is that both are possible actual causes, but neither is a certain actual cause. This is typical for redundant causation patterns, and fits well with intuition here: without knowledge about the order in which events happened, we cannot say which of the redundant causes actually ``got there first''.
So far, we have only considered actual causation as it applies to atoms causing atoms. Often, it is also interesting to wonder which omissions contributed to an effect (``did the doctor's failure to treat the patient cause his death?'') or why some effect was in fact not caused (``did the doctor's treatment prevent the patient's death?''). Extending the framework to also address such questions is easy enough:
\begin{itemize}
\item To extend our definition of actual causation to allow also literals $\lnot E$ to act as effects, we need to specify when such a $\lnot E$ ``happens'' for the first time, such that we may discard all later events when making counterfactual judgments to determine what caused $\lnot E$. The obvious cut-off point is when $E$ no longer belongs to the overestimate $\mathcal{U}(s)$.
\item To also allow literals $\lnot C$ to act as causes, we need to define precisely how we will check the counterfactual dependency in this case. To assume that $\lnot C$ was not the case, we need to assume that $C$ has somehow occurred, which we can do formally by just adding a new CP-law ``$C\leftarrow$'' that always causes $C$.
\end{itemize}
Due to space restrictions, formal details are left to the reader.
\section{Examples}\label{sec:ex}
There is a large literature about actual causation, with many examples, counterexamples, and counter-counterexamples. While e.g.~\citeN{stonesoup} have argued that the importance of such small examples should not be exaggerated, it nevertheless remains useful to check that my approach behaves sensibly for them. Due to space restrictions, I will limit myself to those examples that most clearly illustrate the difference between my approach and the HP account.
It is common practice in research on actual causation to formulate examples in terms of neuron diagrams. A neuron can be in one of two states, one is the default ``off'' state and the other is the deviant ``on'' state in which the neuron ``fires'' or ``is active''. Different kinds of links between two nodes define how the state of one affects the other. For instance, in the following figure, $E$ fires if and only if $B$ fires, and $B$ fires if at least one of $A$ or $C$ fires.
\begin{center}
\begin{pgfpicture}
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\pgfputat{\pgfnodecenter{e}}{\pgfbox[center,center]{E}}
\pgfputat{\pgfnodecenter{b}}{\pgfbox[center,center]{B}}
\color{black}
\pgfputat{\pgfnodecenter{c}}{\pgfbox[center,center]{C}}
\end{pgfpicture}
\end{center}
Neuron diagrams typically record not only this causal structure, but also the state of the neurons. In the figure above, nodes that are ``on'' are represented by full circles and nodes that are ``off'' are shown as empty circles. So, $A$, $B$ and $E$ all fire, whereas $C$ does not. In the language that we have developed so far, a neuron diagram therefore places us in the partial information setting of Definition \ref{partial}: we are given a causal model of a domain together with the final state that has been reached, but are not told precisely how this state has come about.
\citeN{hall07} shows a number of counterexamples to HP, and introduces an alternative account, which he formalizes for neuron diagrams only. One of his counterexamples concerns the following two diagrams:
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\pgfputat{\pgfnodecenter{d}}{\pgfbox[center,center]{D}}
\pgfputat{\pgfnodecenter{e}}{\pgfbox[center,center]{E}}
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\pgfputat{\pgfnodecenter{f}}{\pgfbox[center,center]{F}}
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In both diagrams, the edges from $B$ to $F$ and from $F$ to $E$ are {\em blocking} edges: if $B$ fires, then $F$ will never fire, regardless of its other incoming edges. In the left diagram, both $A$ and $C$ cause $E$: $A$ causes it directly and $C$ causes it by stopping $D$ from preventing $E$. In the right diagram, however, $C$ also causes the very ``threat'' to $E$ that it prevents. Therefore, Hall argues, in this diagram it should not be counted as a cause for $E$.
The HP account correctly handles the left diagram, but fails for the right one, since taking $\vec{X} = \{C\}$ and $\vec{W} = \{D\}$ allows us to create the context $D = {\bf t}$ in which there is a counterfactual dependency from $C$ to $E$.
To see how my definition fares, here are the obvious CP-logic versions. In the first, $A,C$ and $D$ are all exogenous, while in the second only $A$ and $C$ are. \\
\begin{minipage}{0.5\textwidth}
\begin{align}
\label{afe} E& \leftarrow A\land \lnot F.\\
\label{bdf} F& \leftarrow D\land \lnot B.\\
\label{cb} B& \leftarrow C.
\end{align}
\end{minipage}
\begin{minipage}{0.5\textwidth}
\begin{align}
\label{afe2}E& \leftarrow A\land \lnot F.\\
\label{bdf2} F& \leftarrow D\land \lnot B.\\
\label{cb2}B& \leftarrow C.\\
\label{cd2}D& \leftarrow C.
\end{align}
\end{minipage}
First, consider the left theory. Here, $E$ can only be caused after \eqref{afe} has already happened and both \eqref{bdf} and \eqref{cb} have become impossible. Therefore, all these CP-laws are relevant and we end up having to check whether there is a counterfactual dependency from $C$ to $E$ in the original theory. Clearly, this is the case, since no tree that starts from a root in which the exogenous predicates $D$ and $A$ are {{\bf t}} and $C$ is {{\bf f}} can produce $E$. In the second theory, the event \eqref{cd2} may either happen before $E$ is caused or after. This means we either have to check for a counterfactual dependency in the theory $\{\eqref{afe2},\eqref{bdf2},\eqref{cb2},\eqref{cd2}\}$ or in $\{\eqref{afe2},\eqref{bdf2},\eqref{cb2}\}$
. In neither theory we find a counterfactual dependency, so $C$ is correctly judged to certainly not be an actual cause of $E$.
HP's problems with this example are caused by the fact that, lacking an explicitly dynamic semantics, they have to resort to interventions to eliminate irrelevant events from consideration. As an undesired side-effect, they end up allowing the possibility that $D$ itself is relevant for judging the impact of $C$ on $E$, but the link between $C$ and $D$ is not.
The following is an example of {\em bogus prevention} \cite{hiddleston05}, taken from \citeN{hitchcock07}.
\begin{quote} \noindent Assassin is in possession of a lethal poison, but has a last minute change of heart and refrains from putting it in Victim's coffee. Bodyguard puts antidote in the coffee, which would have neutralized the poison had there been any. Victim drinks the coffee and survives.
\end{quote}
Here, HP, as well as others such as \citeN{hitchcock01}, erroneously designate the bodyguard's unnecessary antidote as an actual cause for Victim's survival. As I will now show, my account handles this correctly. Since the example states that Assassin has his $ChangeOfHeart$ before the $Antidote$ is administered, I will not make these exogenous atoms, but instead include them as endogenous atoms that are vacuously caused with some unknown (and irrelevant) probability.
\begin{align}
(Antidote:*) &\leftarrow. \label{anti}\\
(ChangeOfHeart:*)&\leftarrow.\label{coh} \\
Poison &\leftarrow \lnot ChangeOfHeart.\label{pois}\\
Death &\leftarrow Poison \land \lnot Antidote.\label{dth}
\end{align}
The example now tells the story that first event \eqref{coh} happens, which is then followed by \eqref{anti}. However, as soon as \eqref{coh} happens, both $Poison$ and $Death$ become impossible, so \eqref{anti} is considered irrelevant in the actual course of events and will not be part of the theory in which we check for a counterfactual dependency. Hence, preventing $Antidote$ in this theory has no effect whatsoever upon the Victim's survival, so it is not an actual cause of Victim's survival (but $ChangeOfHeart$ is). Note that if the antidote were administered before the assassin's change of heart, then it {\em would} be considered relevant, but still not an actual cause of Victim's survival because then \eqref{pois} would no longer be relevant. To make the antidote an actual cause of Victim's survival, it would have to be administered after the assassin has {\em failed} to have a change of heart.
\section{Implementation}\label{sec:imp}
A prototype implementation can be downloaded from the following URL:
\begin{center}\tt
\href{http://people.cs.kuleuven.be/~joost.vennekens/actcaus/act.pl}{http://people.cs.kuleuven.be/$\sim$joost.vennekens/actcaus/act.pl}
\end{center}
This small program computes whether an atom is a possible/certain actual cause for an effect in the partial information setting, or an actual cause in the complete information setting. It was written in SWI-prolog, but should also run in Sicstus or YAP. Currently, it only handles ground theories without disjunction in rule bodies.
In the partial information setting, this prototype performs a simple backtracking search over all branches that might generate the given observations. Obviously, this is not an approach that would scale well for larger examples. The goal of this prototype, however, is just to allow interested people to experiment with my definition, in order to see whether it corresponds to their intuition. As such, it is not meant to handle problems larger than the examples typically considered in the actual causation literature. Future work may investigate better algorithms, e.g., by means of an integration into Riguzzi's \citeyear{riguzzi10} query answering algorithm.
\section{Conclusion and related work}\label{sec:con}
This paper has tried to argue that the HP account of actual causation is flawed for two reasons, both of which stem from their choice of structural models as the formal language to express causal relations. First, structural models fail to make the distinction between default and deviant values, which has been argued by many authors to play a key role in a correct understanding of actual causation. Second, the static world-view of structural models is ill-suited to handling dynamic concepts, such as the stories that are part of the input to an actual causation problem.
Since the HP paper first appeared, it has received a great deal of attention among researchers interested in actual causation, and many counterexample and alternative approaches have been presented. Most of these, such as \citeN{hitchcock07} or \citeN{hall07}, recognize the importance of the deviant/default distinction. The problems caused by the mismatch between the static formalism of structural equations and the dynamic problem of actual causation have achieved less attention, even though they are recently also pointed out by \citeN{stonesoup}. Nevertheless, also these more recent approaches still use static formalisms such as neuron diagrams or variants of structural models. The main point I hope to make in this paper is that for problems that, like actual causation, require reasoning about the way in which a domain evolves, it pays to have a language with a formal semantics that contains mathematical objects that correspond to such evolutions.
I have tried to illustrate this by defining a notion of actual causation in the context of CP-logic, a probabilistic logic programming language which can be seen as a modular syntactic representation for Shaferian probability trees, which offer precisely the kind of dynamic representation that is perfectly suited for a study of actual causation. My definition is based on a counterfactual criterion similar to HP's, but is able to leverage the dynamic nature of CP-logic's semantical objects to come up with a very straightforward notion of relevance, namely, it only considers as relevant those events that happened (or became impossible) {\em before} the effect first arose. This is much simpler than the relevance criterion of HP, since I do not have to rely on complex manipulations by means of interventions.
While lacking space for an elaborate review of examples from the literature, I have shown that there are three examples where my definition beats HP: already for simple examples of redundant causation, it offers a more elegant account due to its ability to distinguish the complete and partial information settings; it is also able to detect fake causes that simply prevent themselves from preventing the effect; and it also handles bogus prevention. Of course, that is not to say my approach is perfect. For instance, the railroad switch example from the HP paper cannot be handled, because it contains a RV ($Destination$) whose default and deviant values switch in the middle of the story. I am also offering a prototype implementation of my definition, in the hope that it may help to find further examples where it does not correspond to intuition. Feedback will be appreciated.
\bibliographystyle{acmtrans}
|
2,877,628,089,026 | arxiv | \section{Introduction}
Conformal (or Weyl) transformations are widely used in
scalar-tensor theories of gravity \cite{ST}, the theory of a
scalar field
coupled non-minimally
to the Ricci curvature $R$, and in modified gravity theories in
which
terms non-linear in $R$ are added to the Einstein-Hilbert action
(due
perhaps to quantum corrections \cite{Stelle}). The present
acceleration of the universe discovered with the study of
supernovae of type Ia \cite{SN} calls either for an exotic form
of dark energy (in
Einstein
gravity or in scalar-tensor theories), or for modifications of gravity
described by terms non-linear in $R$ in the Lagrangian
\cite{modifiedgravity}, or the addition of terms containing the
invariants of the Riemann tensor $R_{ab}R^{ab} $ and
$R_{abcd}R^{abcd}$ \cite{PQ}. To
fix the ideas and the terminology, consider a scalar-tensor
theory of gravity,
described in the
Jordan frame by the action \cite{footnote1}
\begin{equation}
S=\int d^{4}x\; \sqrt{-g}\left[\frac{f(\phi)R}{2}
-\frac{\omega(\phi)}{2}g^{ab}\nabla_{a}\phi\nabla_{b}
\phi-V(\phi)
\right]+\alpha_{m}\mathcal{L}^{(m)}\left[g_{ab},\psi_{m}\right]\;,
\label{eq:1.1}
\end{equation}
where $f>0$, $S^{(m)}=\int d^{4}x\;\sqrt{-g}
\, \mathcal{L}^{(m)}$,
and $\mathcal{L}^{(m)}$ is the Lagrangian density describing
``ordinary''
matter (as opposed to the gravitational scalar field $\phi$, which
effectively plays the role of a form of non-conventional matter in
the field equations. Here $g_{ab}$ is the metric tensor with determinant
$g$, $f(\phi)$ and $\omega(\phi)$ are arbitrary coupling functions,
$\phi$ is the Brans-Dicke-like scalar field with potential
$V(\phi)$,
and $\psi_{m}$ collectively denotes the matter fields. The Jordan
frame in which the theory (\ref{eq:1.1}) is formulated is the set
of dynamical variables $(g_{ab},\phi)$ describing the gravitational
field. The effective gravitational coupling is
\begin{equation}
G_{eff}=\frac{1}{8\pi f(\phi)}\;,\label{eq:1.1bis}
\end{equation}
as can be immediately deduced from inspection of the action
(\ref{eq:1.1}).
However, in a Cavendish experiment the effective coupling is
instead \cite{Nordvedt,Willbook}
\begin{equation}
G_{eff}^{(*)}=\frac{2\omega f+2(\frac{df}{d\phi})^{2}}{8\pi
f\left[2\omega f+3(\frac{df}{d\phi})^{2}\right]}\;.
\end{equation}
This expression can also be derived from
cosmological perturbation theory \cite{Boisseauetal00}. Note that
in the Jordan frame description the Lagrangian density $\mathcal{L}^{(m)}\left[g_{ab},\psi_{m}\right]$
only depends on the metric $g_{ab}$ and the ``ordinary'' matter
fields $\psi_{m}$. As a consequence, this matter is described by
the stress-energy tensor
\begin{equation}
T_{ab}^{(m)}=\frac{-2}{\sqrt{-g}}\frac{\delta S^{(m)}}{\delta g^{ab}}\;,\label{eq:1.2}\end{equation}
and the invariance of $S^{(m)}$ under diffeomorphisms leads to the
covariant conservation of $T_{ab}^{(m)}$ \cite{Wald}
\begin{equation}
\nabla^{b}T_{ab}^{(m)}=0\;.
\end{equation}
As a consequence, test particles in the Jordan frame follow
geodesics, the weak equivalence principle \cite{Willbook} is
satisfied, and the theory (\ref{eq:1.1}) is metric. In this
frame the kinetic energy term of the scalar $\phi$, i.e.,
$- \omega(\phi)\nabla^{a}\phi\nabla_{a}\phi /2 $,
is non-canonical and has indefinite sign. The
experimental constraint $\left|\omega(\phi_{0})
\right| >40000 $, where $\phi_{0}$ is the present
value of the scalar field, applies \cite{BertottiIessTortora},
unless a potential $V(\phi)$ gives the field a very short
range.
Let us consider now the conformal transformation
\begin{equation}
g_{ab}
\longrightarrow \tilde{g}_{ab}=
\Omega^{2}g_{ab} \;,\;\;\;\;\;
\quad\Omega=\sqrt{f(\phi)}\;,\label{eq:1.4}
\end{equation}
and the scalar field redefinition
\begin{equation}
\phi \longrightarrow \tilde{\phi}=\int
\frac{d\phi}{f(\phi)}\sqrt{f(\phi)+\frac{3}{2}\left(\frac{df}{d\phi}
\right)^{2}}\;.\label{eq:1.5}
\end{equation}
This transformation brings the theory into the Einstein
conformal frame, i.e., to the set of variables
($\tilde{g}_{ab},\tilde{\phi}$)
in which the action (\ref{eq:1.1}) takes the form
\begin{equation}
S=\int d^{4}x\;\sqrt{-\tilde{g}}\left\{ \frac{\tilde{R}}{16\pi}
-\frac{1}{2}\,
\tilde{g}^{ab} \,
\tilde{\nabla}_{a}
\phi
\tilde{\nabla}_{b}\phi
-\tilde{U}(\tilde{\phi)}
+(G\phi)^{-2}\mathcal{L}^{(m)}\left[\tilde{g}_{ab},\psi_{m}\right]
\right\} \;,\label{eq:1.6}
\end{equation}
where
\begin{equation}
\tilde{U}(\tilde{\phi})=\frac{V\left[\phi(\tilde{\phi})\right]}{\Omega^{2}}\;,\label{eq:1.7}\end{equation}
and $\tilde{\nabla}_{a}$ is the covariant derivative operator of
the metric $\tilde{g}_{ab}$. Note that the ``new'' scalar
field $\tilde{\phi}$ exhibits a
canonical kinetic energy term and it couples minimally to the
Ricci
curvature $\tilde{R}$ of the ``new'' metric $\tilde{g}_{ab}$.
However, the action (\ref{eq:1.6}) does not describe simply general
relativity with an extra scalar field $\tilde{\phi}$, because $\tilde{\phi}$
couples explicitly to matter via the prefactor $\left[G\phi(\tilde{\phi})\right]^{-2}$
in front of the matter Lagrangian $\mathcal{L}^{(m)}$. The
exceptions are forms of conformal matter which obey equations
invariant under the conformal transformation
(\ref{eq:1.4}), such as the
Maxwell field, a radiation fluid, or a scalar field $\psi$
conformally coupled to $R$ and with zero
or quartic potential. The explicit coupling
of $\tilde{\phi}$ to all forms of non-conformal matter spoils
the equivalence principle in the Einstein frame. In this frame
all massive particles deviate from geodesics due to the
force, proportional to
$\tilde{\nabla}_{a}\tilde{\phi}$, exerted by $\tilde{\phi}$. By
contrast, zero mass particles still move along null geodesics. (This
can be realized by noting that the conformal transformation
(\ref{eq:1.4})
does not change the conformally invariant Maxwell equations in four
dimensions, which reduce to geometric optics in the high frequency
limit.)
The issue has been raised of ``which conformal frame is physical'',
i.e., should one regard the Jordan frame metric $g_{ab}$, or the
Einstein frame metric $\tilde{g}_{ab}$ as physical? This issue has
been the subject of much debate and is still contentious due to incorrect
formulations of this question. In fact, the question is answered,
to a large extent, by Dicke's paper \cite{Dicke} which
originally introduced the conformal transformation for
Brans-Dicke theory \cite{BransDicke},
the prototype of scalar-tensor gravity theories. The answer of
Ref.~\cite{Dicke} is that the two frames
are equivalent, provided that the units of mass, length, time, and
quantities derived therefrom scale with appropriate powers of the
conformal factor $\Omega$ in the Einstein frame. However, Dicke's
treatment is valid only at the classical level, while in modern cosmology
and in gravitational theories alternative to Einstein gravity, quantum
fields in curved space play a significant role and the equivalence
of the conformal frames is not clear at all - indeed there are
certain indications
that the equivalence breaks down at the quantum level. Of course,
nothing is known about this equivalence in quantum gravity due to
the lack of a definitive theory of quantum gravity.
In view of Dicke's paper, many authors consider the issue of which
conformal frame is physical a pseudo problem, and we agree with them
to a large extent, apart from the two problems mentioned above. However,
while the answer to the question of the physical equivalence of
conformal frames may be clear in principle, its
\emph{application} to practical situations
is a completely different matter. The scaling of units in the
Einstein frame is usually forgotten or not taken into account,
producing results that range from nonsensical to marginally
incorrect, to correct but it is not easy to understand if the
conformal transformation (\ref{eq:1.4})
and (\ref{eq:1.5}) is applied correctly. (It is worth noting that
Dicke himself applied the conformal transformation and the scaling
of units incorrectly in the simpler context of Einstein
gravity \cite{DickePeebles}). Misinterpretations of the
conformal
transformation
abound in the literature and fuel the debate on the issue (or pseudo-issue)
of the conformal frame, while other authors consider the problem a
closed one and sharply state that the Einstein frame is physical while
the Jordan frame should not be considered at all. The argument for
this choice is the positivity of the kinetic energy and the existence
of a ground state, but this argument is usually not explored in detail
for the specific theories considered. The existing review papers on
the subject \cite{MagnanoSokolowski,FGN} fail to
clarify this issue because they do not explicitly state the
assumptions made.
It appears that they refer to a version of scalar-tensor gravity
in the Einstein
frame in which the units of mass, length, time, etc. \emph{do not
scale} with powers of $\Omega$. This version of the theory has nothing
to do with the original Jordan frame and it is physically inequivalent
to it, but it has come to be implicitly accepted as a valid theory,
which adds to the confusion. It is our opinion that the issue deserves
some clarification and that the open problems (Cauchy problem, extension
to quantum matter) should be clearly formulated and addressed. In
this paper we state as clearly as possible what the problems are,
and we show how the divergence of opinions between different authors
is due to the fact that two physically different theories in the Einstein
frame (with or without scaling of units, respectively) are
considered by different
authors without realizing, or explicitly stating, which one is the
version under examination. As a consequence, much of the
existing debate becomes meaningless, the two opposite viewpoints
are both correct but they really refer to physically different
theories (one of which not as well motivated as the other),
while they are erroneously reported as pertaining to the same
physical theory. From a more conservative point of view,
instead, only the Einstein frame version of the theory
incorporating scaling units is physically motivated. Even
accepting this point of view, however, it is not always obvious
how to incorporate this scaling of units in a calculation,
for example computing the spectrum of inflationary perturbations
in scalar-tensor gravity, and this issue deserves some
attention.
Many (perhaps most) researchers in gravitation and cosmology are unaware
of the importance of scaling units in the Einstein frame, which is
neglected. This issue is discussed in Sec.~2, where it is shown
that the scaling of units is related to the ``anomalous'' coupling
of the scalar $\tilde{\phi}$ to matter in the Einstein frame, and
to the subsequent violation of the equivalence principle. In
Sec.~3 we examine some consequences of allowing or not the units
of fundamental quantities to scale in the Einstein frame, and we
resolve an apparent paradox in the literature regarding energy
conditions and singularity theorems in the two conformal frames
in Sec.~4. The cosmological constant problem and the
Cauchy problem are discussed in Sec.~5, while Sec.~6 contains
the conclusions.
\section{Conformal transformations, Jordan frame, and Einstein
frame}
\setcounter{equation}{0}
Here we recall the basic properties of the conformal
transformation to the Einstein frame, the transformation
properties of various geometrical
quantities, and the conservation equations for the matter stress-energy
tensor. The reader is referred to
Refs.~\cite{Synge,Wald,MagnanoSokolowski,FGN} for further
details.
Consider a spacetime ($M,g_{ab}$) where $M$ is a smooth
manifold with dimension $n>1$ and $g_{ab}$ is a Lorentzian
or Riemannian metric on $M$. The conformal
transformation
\begin{equation}
g_{ab}\longrightarrow
\tilde{g}_{ab} =\Omega^{2}g_{ab}\;,\label{eq:2.1}
\end{equation}
where $\Omega$ is a smooth, nowhere vanishing, function of the
spacetime point is a point-dependent rescaling of the metric. It
changes the length of timelike and spacelike intervals and
vectors, but it preserves their timelike or spacelike character.
Similarly, null intervals and null vectors according to the
``old'' metric $g_{ab}$ remain null
according to the ``new'' metric $\tilde{g}_{ab}$. The light
cones are not changed by the conformal transformation
(\ref{eq:2.1}) and the spacetimes ($M,g_{ab}$) and
($M,\tilde{g}_{ab}$) have the same causal structure; the
converse is also true \cite{Wald}. The inverse metric $g^{ab}$,
the metric determinant $g$, and the Christoffel
symbols transform according to \cite{Synge,Wald}
\begin{equation}
\tilde{g}^{ab}=\Omega^{-2}g^{ab}\;,\qquad\tilde{g}
=\Omega^{2n}g\;,\label{eq:2.2}
\end{equation}
\begin{equation}
\tilde{\Gamma}_{bc}^{a}=\Gamma_{bc}^{a}+\frac{1}{\Omega}
\left(\delta_{b}^{a}\nabla_{c}\Omega+\delta_{c}^{a}\nabla_{b}
\Omega-g_{bc}\nabla^{a}\Omega\right)\;,\label{eq:2.3}
\end{equation}
while the Riemann and Ricci tensor obey
\begin{eqnarray}
\tilde{ {R_{abc}}^d} &= & {R_{abc}}^d +2\delta_{\left[ a
\right.}^d
\nabla_{b \left. \right]} \nabla_c \left( \ln \Omega \right)
-2g^{de}\, g_{c\left[ \right.} \nabla_{b \left. \right]}
\nabla_e \left( \ln \Omega \right) \nonumber \\
&& \nonumber \\
& + & 2\nabla_{\left[ a \right.} \left( \ln \Omega \right)
\delta_{\left. b \right]}^d \nabla_c \left( \ln \Omega \right)
-2\nabla_{\left[ a \right.} \left( \ln \Omega \right) g_{b
\left. \right]c} \, g^{de}\nabla_e \left( \ln \Omega \right)
\nonumber \\
&& \nonumber \\
&-& 2 g_{c\left[ a \right.} \delta_{\left. b \right]}^d \,
g^{ef} \nabla_e \left( \ln \Omega \right) \nabla_f \left( \ln
\Omega \right) \;,
\label{eq:2.4}
\end{eqnarray}
\begin{eqnarray}
\tilde{R}_{ab} &=& R_{ab}-\left( n-2\right) \nabla_a \nabla_b
\left( \ln \Omega \right) -g_{ab}\, g^{ef} \nabla_f \nabla_e
\left( \ln \Omega \right) \nonumber \\
&& \nonumber \\
& + & \left( n-2 \right) \nabla_a \left( \ln \Omega \right)
\nabla_b \left( \ln \Omega \right) \nonumber \\
&& \nonumber \\
& - & \left( n-2 \right) g_{ab} \, g^{ef} \nabla_f \left( \ln
\Omega \right) \nabla_e \left( \ln \Omega\right) \;.
\label{eq:2.5}
\end{eqnarray}
For the Ricci curvature,
\begin{equation}
\tilde{R}=\tilde{g}^{ab}\tilde{R}_{ab} =
\frac{1}{\Omega^{2}}\left[R-2\left(n-1\right)
\square\left(\ln\Omega\right)-\left(n-1\right)
\left(n-2\right) \frac{g^{ab} \, \nabla_{a}\Omega
\nabla_{b}\Omega}{\Omega^{2}}\right]\;.\label{eq:2.6}
\end{equation}
In $n=4$ dimensions it is
\begin{equation}
\tilde{R}=\frac{1}{\Omega^{2}}\left(R-\frac{6\square\Omega}{\Omega}
\right)=\frac{1}{\Omega^{2}}\left[R-\frac{12\square\left(
\sqrt{\Omega}\right)}{\sqrt{\Omega}}+3 \,
\frac{g^{ab}
\, \nabla_{a}\Omega\nabla_{b}\Omega}{\Omega^{2}}\right]\;.
\label{eq:2.7}
\end{equation}
The Weyl tensor $ { C_{abc}}^d$ with the last
index raised is conformally invariant,
\begin{equation}
\tilde{ {C_{abc}}^d}= {C_{abc}}^d\;.\label{eq:2.8}
\end{equation}
However, the same tensor with the other indices raised or lowered
is not conformally invariant. Note that in the conformally rescaled
world the conformal factor $\Omega$ plays the role of a form of matter.
In fact, if the original metric is Ricci-flat ($R_{ab}=0$), the new
metric is not ($\tilde{R}_{ab}\neq0$).
If the Weyl tensor of $g_{ab}$ vanishes, also the Weyl tensor of
$\tilde{g}_{ab}$ in the conformally related frame vanishes (and vice-versa),
conformally flat metrics are mapped into conformally flat
metrics.
Let us consider covariant conservation for the matter
energy-momentum tensor $T_{ab}^{(m)}$. In the Jordan frame it is
\begin{equation}
\nabla^{b} T_{ab}^{(m)}=0\;;\label{eq:2.9}
\end{equation}
this equation is not conformally invariant and $T_{ab}$ scales
as \cite{Wald}
\begin{equation}
\tilde{T}_{(m)}^{ab}=\Omega^{s}
\, T_{(m)}^{ab}\;, \;\;\;\; \qquad\tilde{T}_{ab}^{(m)}
=\Omega^{s+4} \, T_{ab}^{(m)}\;,\label{eq:2.10}
\end{equation}
where $ s $ is a appropriate conformal weight. As a consequence,
the conservation equation in the conformally rescaled world is
\begin{equation}
\tilde{\nabla}_{a} \left(\Omega^{s} \, T_{(m)}^{ab}\right)=
\Omega^{s} \,
\nabla_{a}T_{(m)}^{ab}
+ \left(s+6\right)\Omega^{s-1}T_{(m)}^{ab}\nabla_{a}
\Omega-\Omega^{s-1}g^{ab }
\, T^{(m)}\nabla_{a}\Omega\;,\label{eq:2.11}
\end{equation}
in four spacetime dimensions \cite{Wald}. By conveniently
choosing $ s=-6 $ one obtains
\begin{equation}
\tilde{\nabla}_{a}\tilde{T}_{(m)}^{ab}=
-\tilde{T}^{(m)} \, \tilde{g}^{ab} \tilde{\nabla}_{a}
\left(\ln\Omega\right) \label{eq:2.12}
\end{equation}
and
\begin{equation}
\tilde{T}^{(m)}\equiv\tilde{g}^{ab}
\tilde{T}_{ab}^{(m)}=\Omega^{-4}
\, T^{(m)}\;.\label{eq:2.13}
\end{equation}
Hence, in the new conformal frame, the stress-energy tensor
$\tilde{T}_{ab}^{(m)}$ is not covariantly conserved unless it
describes conformally invariant matter with vanishing trace
$T^{(m)}=0$, in which case also $ \tilde{T}^{(m)}=0 $
and $\tilde{\nabla}^{b}\tilde{T}_{ab}^{(m)}=0 $.
It is well known that null geodesics of the Jordan metric
$g_{ab}$ are mapped into null geodesics of the Einstein frame
metric $\tilde{g}_{ab}$ \cite{Wald}. Timelike geodesics will be
considered in the next
section.
\section{Jordan frame, Einstein frame with running units, and
Einstein frame with fixed units}
\setcounter{equation}{0}
In this section only classical physics of spacetime
and matter is considered. Quantum matter will be discussed in
Sec.~6.
A viewpoint shared by many authors (see
\cite{MagnanoSokolowski,FGN} for references) states that the
Einstein and Jordan conformal frames
are physically equivalent. This viewpoint is generally correct
as shown below, and it is in open conflict with the viewpoint
that the Jordan frame should be abandoned in favour of the
Einstein frame because of the presence of negative energy.
\subsection{Einstein frame with running units}
The argument of the physical equivalence between the Jordan and Einstein
frames dates back to Dicke's 1962 paper introducing the conformal
transformation technique for Brans-Dicke theory \cite{Dicke}, a paper
often forgotten or misread. The basic idea is that the two
conformal frames are physically equivalent \emph{provided} that
in the Einstein
frame the units of time, length, mass, and derived quantities are
allowed to scale with appropriate powers of the conformal
factor $\Omega$. Physics must be invariant under a choice of
the units --- this includes not only transformations of units by
factors
which are the same everywhere in spacetime (``rigid'' changes
of units or ``dilatations''), but also changes of units that depend
on the spacetime point. A rescaling of the units of length and time
(and, on dimensional grounds, also of mass) is a conformal transformation.
Since physics is invariant under a change of units, it is invariant
under a conformal transformation provided that the units of length,
time, and mass $l_{u}$, $t_{u}$, and $m_{u}$ are scaled. The novelty
of Dicke's approach consists in allowing these units to be rescaled
by different factors at different spacetime points, with the change
in each unit being a smooth, nowhere vanishing, function of the spacetime
point. Instead of a system of units rigidly attached to the spacetime
manifold, the Einstein frame contains a system of units that
change
with the spacetime location. If one accepts this point of view, the
symmetry group of classical physics is enlarged to include conformal
transformations \emph{with the associated rescaling of units}.
It is shown in Ref.~\cite{Dicke} that $g_{ab}$ scales with the
dimensions of a time squared, and since
$ \tilde{g}_{ab}=\Omega^{2}g_{ab}$, it follows that times and
lengths scale with $\Omega$, so that
\begin{equation}
dt \rightarrow \tilde{dt}=\Omega \, dt\;, \;\;\;\;\; dx^{i}
\rightarrow \tilde{dx^{i}}=\Omega \,
dx^{i}\qquad(i=1,2,3)\;,\label{eq:3.4}
\end{equation}
while for masses
\begin{equation}
m\rightarrow\tilde{m}=\Omega^{-1}m\;,\label{eq:3.5}\end{equation}
on dimensional grounds.
Since the speed of light in vacuum $c$ is a ratio of space and time,
it is invariant and local Lorentz invariance is preserved. The Planck
constant, which has dimensions $\left[h\right]=\left[ML^{2}T^{-1}\right]$
is left unchanged, while energy with dimensions $\left[Mc^{2}\right]$
scales like a mass. In the Jordan frame of scalar-tensor gravity
the effective coupling (\ref{eq:1.1bis}) varies, while $h$, $c$,
the masses of elementary particles, and the coupling constants
of physics are true
constants, together with the units. The weak equivalence principle
holds and the theory is metric. On the contrary, in the Einstein frame
the gravitational coupling $G$ is constant and so are $h$ and $c$,
while the masses of elementary particles and the coupling ``constants''
of non-gravitational physics vary with time together with the units
of time, length, and mass $\tilde{t}_{u}$, $\tilde{l}_{u}$, and
$\tilde{m}_{u}$. In Dicke's viewpoint the Jordan and Einstein frames
are merely two equivalent representations of the same physics. One
can consider, for example, the proton mass, which has a constant value
$m_{p}$ in the Jordan frame. In the Einstein frame the proton mass
depends on $\Omega$ (or $\phi$) and is $\tilde{m}_{p}=\Omega^{-1}m_{p}$.
However, in an experiment one measures the \emph{ratio}
$ \tilde{m}_{p} / \tilde{m}_u $
between the proton mass and an arbitrarily chosen mass unit
$\tilde{m}_{u}$. Hence, in the Einstein frame it is not
$\tilde{m}_{p}$ that matters, but the ratio
\begin{equation}
\frac{\tilde{m}_{p}}{\tilde{m}_{u}}
=\frac{\Omega^{-1}m_{p}}{\Omega^{-1}m_{u}}=
\frac{m_{p}}{m_{u}} \label{eq:3.6}
\end{equation}
(in the Jordan frame as well, it is only $ m_{p} / m_u $
that is measured). A measurement of the proton mass with respect
to the chosen mass unit therefore yields the same value in the
Jordan and the Einstein frame. No preferred frame is selected by
such a measurement. The outcome of an experiment is the same
when analyzed in the Jordan or the Einstein frame. In this
context, the problem of which frame is physical is void of
content.
In the Einstein frame not only the masses of elementary
particles and the mass units, but also the coupling constants of
non-gravitational physics vary with $\phi$. To understand this
variation it is useful to consider the following example, to
which we will return later \cite{footnote2}.
\subsubsection{Examples}
Consider as an example Brans-Dicke theory \cite{BransDicke}
with a massive Klein-Gordon field $\psi$ as the only form of
matter, as described in the Jordan frame by the action
\begin{eqnarray}
S &=& S^{(BD)}+S^{(KG)}=\int d^4 x \sqrt{-g} \left( {\cal
L}^{(BD)}+\alpha_{KG}\, {\cal L}^{(KG)} \right) \nonumber \\
&& \nonumber \\
&=&
\int d^4 x \sqrt{-g} \left( \phi R -\frac{\omega}{\phi} \,
g^{ab} \nabla_a\phi\nabla_b\phi \right) -\frac{\alpha_{KG}}{2}
\int d^4 x \sqrt{-g} \left( g^{ab}\nabla_a\psi\nabla_b\psi
+m^2\psi^2 \right)\;, \nonumber \\
&& \label{eq:3.7}
\end{eqnarray}
where $\alpha_{KG}=16\pi G$ is the Klein-Gordon coupling
constant. The conformal transformation (\ref{eq:1.4}) and the
scalar field
redefinition (\ref{eq:1.5}) yield
\begin{equation}
\sqrt{-g}\left(\mathcal{L}^{(BD)} +\alpha_{KG}\mathcal{L}^{(KG)}
\right)=\sqrt{-\tilde{g}}\left\{ \tilde{\mathcal{L}}^{(GR)}
+\tilde{\alpha}_{KG}(\phi) \, \tilde{\mathcal{L}}^{(KG)}
\left[\tilde{g}_{ab},\psi\right]\right\} \;,\label{eq:3.8}
\end{equation}
where $ \tilde{\mathcal{L}}^{(GR)}
=
\tilde{R}-\frac{1}{2}\tilde{g}^{ab}
\tilde{\nabla}_{a}\tilde{\phi}\tilde{\nabla}_{b}\tilde{\phi} $
is the Einstein-Hilbert Lagrangian density with a canonical
scalar field $\tilde{\phi}$,
\begin{equation}
\tilde{\alpha}_{KG}(\tilde{\phi})=\Omega^{-2}\alpha_{KG}= 32\pi
G \exp\left( -8\sqrt{\frac{\pi
G}{2\omega+3}} \, \tilde{\phi} \right) \;,\label{eq:3.10}
\end{equation}
\begin{equation}
\tilde{\mathcal{L}}^{(KG)}\left[\tilde{g}_{ab} ,\phi\right]
= \frac{1}{2} \, \tilde{g}^{ab}
\,
\tilde{\nabla}_{a}\psi\tilde{\nabla}_{b}\psi+\frac{\tilde{m}^{2}}{2}\psi^{2}\;,\label{eq:3.10b}\end{equation}
and
\begin{equation}
\tilde{m}(\tilde{\phi})=\frac{m}{\Omega}= m
\exp\left( -4\sqrt{\frac{\pi
G}{2\omega+3}} \, \tilde{\phi} \right) \;,\label{eq:3.11}
\end{equation}
in accordance with eq.~(\ref{eq:3.5}). In the Einstein frame
the mass $\tilde{m}$ of the Klein-Gordon field $\psi$ and its
coupling constant $\tilde{\alpha}_{KG}$ acquire a dependence
from the Brans-Dicke scalar. This holds true for all forms of
matter except conformally invariant matter, which satisfies
conformally invariant equations. As an example of such matter,
consider the Maxwell field in four spacetime
dimensions, described by the matter action
\begin{eqnarray}
S^{(em)} & = & \int
d^{4}x\;\sqrt{-g}\, \alpha_{em} \, \mathcal{L}^{(em)}\nonumber
\\
& = & -\int d^{4}x\;\sqrt{-g} \, F_{ab}F^{ab}\;,\label{eq:3.12}
\end{eqnarray}
with $\alpha_{em}=4$ and $\mathcal{L}^{(em)}=-\frac{1}{4}F_{ab}F^{ab}$,
where $F_{ab}$ is the antisymmetric Maxwell tensor. The conformal
invariance can be directly verified by computing
\begin{eqnarray}
\sqrt{-g}\, \mathcal{L}^{(em)} & = & -\frac{1}{4}\sqrt{-g}\;
g^{ac}g^{bd}F_{ab}F_{cd}\nonumber \\
& = & -\frac{1}{4}\left(\Omega^{-4}\sqrt{-\tilde{g}}\right)\left(\Omega^{2}\tilde{g}^{ac}\right)\left(\Omega^{2}\tilde{g}^{bd}\right)F_{ab}F_{cd}\nonumber \\
& = & -\frac{1}{4}\sqrt{-\tilde{g}}\;\tilde{g}^{ac}\tilde{g}^{bd}\tilde{F}_{ab}\tilde{F}_{cd}\;,\label{eq:3.13}\end{eqnarray}
where $\tilde{F}_{ab}=F_{ab}$.
\subsubsection{Terminology}
Implementing the idea that physics should be conformally invariant
when units are rescaled leads to conflict with current terminology.
Consider, for example, a conformally coupled Klein-Gordon field $\psi$
with a non-zero mass, obeying the equation
\begin{equation}
\square\psi-\frac{R}{6}\psi-m^{2}\psi=0\;.\label{eq:3.14}
\end{equation}
According to standard terminology, the introduction of the mass $m$
breaks the conformal invariance that is present when $m=0$. One can,
however, generalize the notion of conformal invariance by allowing
the mass to vary with the scalar $\phi$. Upon the use of the relation
\begin{equation}
g^{ab} \nabla_{a}\nabla_{b}\psi-\frac{R}{6}\psi
=
\Omega^{3}\left[\tilde{g}^{ab}\tilde{\nabla}_{a}
\tilde{\nabla}_{b}\tilde{\psi}-\frac{\tilde{R}}{6}
\tilde{\psi}\right]\;,\label{eq:3.15}
\end{equation}
where $\tilde{\psi}\equiv\Omega^{-1}\psi$, one obtains from
eq.~(\ref{eq:3.14})
\begin{equation}
\tilde{\square}\tilde{\psi}-\frac{\tilde{R}}{6}\tilde{\psi}
-\tilde{m}^{2}\tilde{\psi}=0\;,\label{eq:3.16}
\end{equation}
where now $\tilde{m}\equiv\Omega^{-1}m$, in agreement with
eq.~(\ref{eq:3.5}).
Hence, the Klein-Gordon equation (\ref{eq:3.14}) is invariant in
form if the current definition of conformal transformation is enlarged
to include the notion that masses scale with eq.~(\ref{eq:3.5}).
However, eq.~(\ref{eq:3.14}) is not conformally invariant
according
to standard terminology.
\subsection{The equation of motion of massive particles in the
Einstein frame
with running units}
In the Einstein frame the equation of timelike geodesics receives
corrections and, as a result, massive particles do not follow
geodesics.
First, we want to find the transformation property of the four-velocity
$u^{a}= dx^a / d\lambda $ of a massive particle, where
$\lambda$
is a parameter along the geodesic, which is usually not discussed
in the literature. The Jordan frame normalization is $u^{a}u_{a}=-1$;
by assuming that $\tilde{u}_{a}=\Omega^{w}u_{a}$, where $w$ is an
appropriate conformal weight, and by imposing the Einstein frame
normalization $ \tilde{g}^{ab}\tilde{u}_{a} \tilde{u}_{b}=-1$,
one obtains $w=-1$,
or
\begin{equation}
\tilde{u}^{a}=\Omega^{-1} \,
u^{a}\;,\;\;\;\;\qquad\tilde{u}_{a}=\Omega \,
u_{a}\;.\label{eq:3.s1}\end{equation}
These relations can be used to find the relation between the parameters
$\lambda$ and $\tilde{\lambda}$ along the geodesic in the two conformal
frames. Since $u^{a}= dx^a / d\lambda $,
$ \tilde{u}^a= d\tilde{x}^a / d\tilde{\lambda} $,
and lengths scale as $d\tilde{x}^{a}=\Omega dx^{a}$, by setting $d\tilde{\lambda}=\Omega^{\alpha}d\lambda$
one obtains $\tilde{u}^{a}=\Omega^{1-\alpha}u^{a}$ which, compared
with eq.~(\ref{eq:3.s1}) yields $\alpha=2$, or
\begin{equation}
d\tilde{\lambda}=\Omega^{2}d\lambda\;,\label{eq:3.s.2}\end{equation}
which agrees with eq.~(D.6) of Ref.~\cite{Wald}. This relation
can
also be obtained from the fact that, in terms of proper times
$d\tau$
and $d\tilde{\tau}$, we have
\begin{equation}
d\tilde{s}^{2}=-d\tilde{\tau}^{2}=
\tilde{g}_{00}d\tilde{t}^{2}=
\left( \Omega^{2}g_{00}
\right) \left( \Omega^{2}dt^{2}\right)=-\Omega^{4}d\tau^{2}=
\Omega^{4}ds^{2}\;,
\end{equation}
which yields again $d\tilde{s}=\Omega^{2}ds$ for the parameter along
the timelike curve.
We are now ready to write the equation of motion of massive particles
in the Einstein frame. Under the conformal transformation (\ref{eq:2.1})
the Jordan frame geodesic equation $u^{a}\nabla_{a}u^{b}=0$ is
mapped to \cite{Wald}
\begin{equation}
u^{a}\tilde{\nabla}_{a}u^{b}=
2u^{b} \, \frac{u^{c} \nabla_{c}\Omega}{\Omega}
+\frac{g^{bd} \tilde{\nabla}_{d}\Omega}{\Omega}\;.
\label{eq:3.s.3}
\end{equation}
By rewriting this equation in terms of tilded quantities we
have
\begin{equation}
\tilde{u}^{a} \tilde{\nabla}_{a}\tilde{u}^{b}
=\left(\frac{\tilde{u}^{c}\tilde{\nabla}_{c}
\Omega}{\Omega}\right)\tilde{u}^{b}+
\frac{\tilde{g}^{bd} \tilde{\nabla}_{d}\Omega}{\Omega}\;.
\label{eq:3.s.4}
\end{equation}
The first term on the right hand side appears because the
equation is not expressed using an affine parameter, while the
second term proportional to the gradient
$\tilde{\nabla}_{a}\left(\ln\Omega\right)$
describes the direct coupling of the field $\phi$ to
non-conformal matter in the Einstein frame; it has been likened
to a fifth force violating the equivalence principle and making
scalar-tensor theory in the Einstein
frame non-metric. It is impossible to achieve an affine
parametrization
of this timelike curve and thus remove the first term on the right
hand side of eq.~(\ref{eq:3.s.4}). In fact, if this could be
achieved, the result would be incompatible with the
normalization $\tilde{u}^{a}\tilde{u}_{a}=-1$.
To prove this statement, note that the normalization implies that
the four-acceleration $\tilde{a}^{b}\equiv\tilde{u}^{a}\tilde{\nabla}_{a}\tilde{u}^{b}$
is orthogonal to the four-velocity $\left(\tilde{u}^{b}\tilde{a}_{b}=0\right)$,
a well known fact \cite{LandauLifschitz,Wald}. Then,
\begin{equation}
0=\tilde{u}^{b}\tilde{a}_{b}\equiv\tilde{u}^{b}\tilde{u}^{a}\tilde{\nabla}_{a}\tilde{u}_{b}=\tilde{u}^{b}\tilde{\nabla}_{b}\Omega\;,
\end{equation}
implying that the gradient of the conformal factor must be orthogonal
to $\tilde{u}^{a}$ for \emph{any} possible choice of
$\tilde{u}^{a}$:
this is clearly absurd. For example, in scalar-tensor cosmology where $\Omega=\Omega(t)$,
$t$ being the comoving time of a
Friedmann-Lemaitre-Robertson-Walker metric (FLRW),
and by choosing $\tilde{u}^{a}$ as the four-velocity of comoving
observers, it follows that $ \partial\Omega / \partial t=0 $,
which is impossible
\cite{footnote3}. Therefore, the term
$\left[ \tilde{u}^{c}\tilde{\nabla}_{c}
\left(\ln\Omega\right)\right]\tilde{u}^{b} $
in eq.~(\ref{eq:3.s.4}) can not be eliminated or, in other
words,
affine parametrization can not be achieved.
Equation~(\ref{eq:3.s.4})
can be rewritten using eq.~(\ref{eq:3.5}) as
\begin{equation}
\tilde{u}^{a}
\tilde{\nabla}_{a} \tilde{u}^{b}=
-\left(
\frac{ \tilde{u}^{c} \tilde{\nabla}_{c}\tilde{m}}{\tilde{m}}
\right)\tilde{u}^{b}-\frac{ \tilde{g}^{bd} \,\tilde{\nabla}_{d}
\tilde{m}}{\tilde{m}}\;.\label{eq:add1}
\end{equation}
Equation~(\ref{eq:add1}) suggests the interpretation that
massive particles
deviate from geodesics because their mass is a function of the spacetime
point, and this deviation is proportional to the mass gradient. The
impossibility of using an affine parametrization is then traced back
to the impossibility of eliminating the variation $\tilde{u}^{c}\tilde{\nabla}_{c}\tilde{m}$
of the mass $\tilde{m}$ along the direction of motion of the particle.
By introducing the three-dimensional metric on the 3-space
orthogonal to the four-velocity
$\tilde{u}^{a}$ of the particle,
\begin{equation}
\tilde{h}_{ab}\equiv\tilde{g}_{ab}+\tilde{u}_{a}\tilde{u}_{b}\;,
\end{equation}
where $h_{\: b}^{a}$ is the projection operator on the 3-space of
the observer $u^{a}$ (i.e., $ {h^a}_b \, u^{b}={h_a}^b
\, u^{a}=0$), eq.~(\ref{eq:add1}) is rewritten as
\begin{equation}
\tilde{u}^{a}\tilde{\nabla}_{a}\tilde{u}^{b}
=-\frac{\tilde{h}^{bd} \tilde{\nabla}_{d}\tilde{m}}{\tilde{m}}
\;,\label{eq:add2}
\end{equation}
which shows explicitly that the correction to the equation of
motion is given entirely by the variation of the particle mass
$\tilde{m}$ in the 3-space of an observer moving with the
particle. Removing the term $ -\left(
\tilde{u}^{c} \tilde{\nabla}_{c}\tilde{m} / \tilde{m} \right)
\tilde{u}^b $
from eq.~(\ref{eq:add1}) by means of affinely parametrizing the
curve would mean introducing corrections to the right hand side
of eq.~(\ref{eq:add2}) which are proportional to the derivative
of $\tilde{m}$ in the direction of motion $\tilde{u}^{a}$, and
this is impossible. It would mean that the right hand side of
eq.~(\ref{eq:add1}) could not be written explicitly as a
purely spatial vector, as is instead done in
eq.~(\ref{eq:add2}), and therefore it could not be the
four-acceleration $ a^{b}=u^{a}\nabla_{a}u^{b}$, which satisfies
$ u^{a}a_{a}=0 $.
Equation (\ref{eq:add2}) has consequences for cosmology. In the
FLRW metric
\begin{equation}
ds^{2}=-dt^{2}+a^{2}(t)\left[
\frac{dr^{2}}{1-Kr^{2}}+r^{2}\left(d\theta^{2} +\sin^{2}\theta
d\varphi^{2}\right)\right]\;,\label{eq:add3}
\end{equation}
let $u^{a}$ be the four-velocity of comoving observers. Since in
FLRW scalar-tensor cosmology the scalar
field depends only on the comoving
time in order to preserve spatial homogeneity, it is $\phi=\phi(t)$,
$\Omega=\Omega(t)$, and $\tilde{m}=\tilde{m}(t)$, which implies
that the spatial gradient $\tilde{h}^{bd}\tilde{\nabla}_{d}\tilde{m}$
vanishes identically, and the equation of motion of comoving observers,
which is the equation of timelike geodesics when a dust fluid with
pressure $P=0$ fills the universe, receives no correction in the
Einstein frame. Similarly, when there is pressure, the timelike
geodesic equation gets corrected by an extra term in $P$, but
no ``fifth force''
corrections $-\tilde{h}^{bd}
\tilde{\nabla}_{d}\left(\ln\tilde{m}\right)$ appear. The
equivalence
between Jordan and Einstein frames with respect to redshift,
Boltzmann equation, and particle physics reaction rates in the
early universe is discussed in Ref.~\cite{Catenaetal}.
The trajectories of particles with zero mass $\tilde{m}=m=0$ do
not receive corrections when going to the Einstein frame.
\subsection{Einstein frame with fixed units}
By now it is clear that if one performs the conformal
transformation (\ref{eq:1.4}) and (\ref{eq:1.5}) but does not
allow the units of length, time, and mass to scale with
$\Omega$ in the Einstein
frame (``fixed units''), one obtains a different physical
theory altogether. In this case, the conformal transformation is merely
a mathematical device relating the two conformal frames, and the Jordan
and Einstein frame are physically inequivalent. If the Jordan frame
and the Einstein frame \emph{with fixed units} were physically equivalent,
it would mean that the entire realm of (classical) physics is
conformally
invariant, according to current terminology. But, to quote an
example,
the Klein-Gordon field obeying eq.~(\ref{eq:3.14}) with $m\neq0$
is not conformally invariant in this sense. As another example, consider
conformally related metrics which are physically inequivalent such
as the Minkowski metric $\eta_{ab}$ and the
FLRW metric given by the line element
\begin{equation}
ds^{2}=g_{ab} \, dx^{a}dx^{b}=a^{2}(\eta)
\left(-d\eta^{2}+dx^{2}+dy^{2}+dz^{2}\right)\;,\label{eq:3.17}
\end{equation}
where $\eta$ is conformal time and $g_{ab}$ is manifestly
conformally flat: $g_{ab}=\Omega^{2}\eta_{ab}$ with
$\Omega(\eta)=a$. When $ da / d\eta >0$,
$g_{ab}$ describes an expanding universe with spacetime curvature,
cosmological redshift, possibly a Big Bang and/or other singularities,
and matter. By contrast, the conformally related metric $\eta_{ab}$
can not be associated to any of these spacetime features. The
two
metrics $g_{ab}$ and $\eta_{ab}$ are physically equivalent only
when the fundamental units are allowed to scale with $a(\eta)$ in
the spirit of Refs.~\cite{Dicke,DickePeebles}. Then, the
universe (\ref{eq:3.17}) appears flat when the units of time and
length scale as $ dt = a (\eta) d\eta$,
$d \tilde{x}^{i}=a(\eta)dx^{i}$, giving (see \cite{FGN} for
references)
\begin{equation}
ds^{2}=-d\tilde{t}^{2}+d\tilde{x}^{2}+d\tilde{y}^{2}+d\tilde{z}^{2}\;.\label{eq:3.18}\end{equation}
The recurring debate on the issue of which conformal frame is
physical arises from the fact that many authors refer to the
Einstein
frame
by keeping the fundamental units fixed in this frame. The result is
a theory, which we shall call ``Einstein frame with fixed units''
version, which is physically inequivalent to the Jordan frame version
of scalar-tensor gravity. These same authors often claim that
the Jordan frame
version and the ``Einstein frame with fixed units version'' are
equivalent, forgetting about the scaling of units and Dicke's paper.
The Einstein frame-with-fixed-units version does not share the physical
motivations that lead to its Jordan frame cousin. It can even be said
that the former arises from a mistake, but given the number of works
devoted to this ``wrong'' theory, we are perhaps facing an
(unintentional) new theory of gravity. We leave to the reader
the judgment
of whether there is enough physical motivation to pursue
``Einstein
frame with fixed units'' versions of gravitational theories, and
we content ourselves to
clarify the issue \cite{footnote4}.
Let us return for a moment to the Einstein frame with running units:
in this frame $\tilde{m}(\phi)=\Omega^{-1}m$ and the ratio of the
mass of a particle to the variable mass unit is constant,
\begin{equation}
\frac{\tilde{m}(\phi)}{\tilde{m}_{u}(\phi)}
=\mbox{constant}\;.
\end{equation}
This implies that
\begin{equation}
\frac{\tilde{\nabla}_{c}\tilde{m}}{\tilde{m}}=
\frac{\tilde{\nabla}_{c}\tilde{m}_{u}}{\tilde{m}_{u}}\;,
\end{equation}
and therefore the equation of motion of massive particles (\ref{eq:add2})
in the Einstein frame with running units can be written as
\begin{equation}
\tilde{u}^{a} \tilde{\nabla}_{a} \tilde{u}^{b}
= -\frac{ \tilde{h}^{bd} \tilde{\nabla}_d
\tilde{m}_{u}}{ \tilde{m}_{u}}\;;
\end{equation}
in other words, the correction to the equation of timelike geodesics
and the violation of the equivalence principle can be seen as arising
completely from the variation of the mass unit. Therefore, in the
Einstein frame with fixed units this correction vanishes and the equivalence
principle is satisfied unless, of course, one reintroduces these violations
by hand into the theory, but the latter now bears no relation to the
original Jordan frame one.
The running of fundamental units can also be seen as the fact
that
there is an anomalous coupling of $\tilde{\phi}$ to the matter sector
in the Einstein frame with running units. This will be clear at
the
end of this section. We now want to make contact with a different
notation that appeared recently in Ref.~\cite{Flanagan}. The
author E. Flanagan parametrizes different conformal frames of a
scalar-tensor
theory using three
different functions of the scalar field $A(\phi)$, $B(\phi)$, and
$\alpha(\phi)$. The action is written as
\begin{equation}
S=\int d^{4}x\;\sqrt{-g}\left[\frac{A(\phi)}{16\pi G}R-\frac{B(\phi)}{2}g^{ab}\nabla_{a}\phi\nabla_{b}\phi-V(\phi)\right]+S^{(m)}\left[e^{2\alpha(\phi)}g_{ab},\psi^{(m)}\right]\;.\label{eq:3.19}\end{equation}
A conformal transformation is described by
\begin{equation}
g_{ab} \longrightarrow\tilde{g}_{ab}=
e^{-2\gamma(\phi)}g_{ab}\;, \label{eq:3add1}
\end{equation}
\begin{equation}
\Phi\longrightarrow\tilde{\Phi}=
h^{-1}(\phi)\;,
\;\;\;\;\;
\mbox{or} \;\;\;\; \qquad\Phi=h(\tilde{\phi})\;,\label{eq:3add2}
\end{equation}
where $\gamma$ and $h$ are regular functions with $h'>0$. The
action can be rewritten as
\begin{equation}
S=\int d^{4}x\;\sqrt{-g}\left[\frac{\tilde{A}(\tilde{\phi})}{16\pi G}\tilde{R}-\frac{\tilde{B}(\tilde{\phi)}}{2}\tilde{g}^{ab}\tilde{\nabla}_{a}\tilde{\phi}\tilde{\nabla}_{b}\tilde{\phi}-\tilde{V}(\tilde{\phi})\right]+S^{(m)}\left[e^{2\tilde{\alpha}(\tilde{\phi})}\tilde{g}_{ab},\psi^{(m)}\right]\;,\label{eq:3add3}\end{equation}
where
\begin{eqnarray}
\tilde{\alpha}(\tilde{\phi})
&= &\alpha\left[h(\tilde{\phi})\right]
+\gamma(\tilde{\phi})\;,\label{eq:3add4} \\
&&\nonumber \\
\tilde{V}(\tilde{\phi})&=
&
e^{4\gamma(\tilde{\phi})}
V\left[h(\tilde{\phi})\right]\;,\label{eq:3add5} \\
&&\nonumber \\
\tilde{A}(\tilde{\phi})&=&e^{2\gamma(\tilde{\phi})}
A\left[h(\tilde{\phi})\right]\;,\label{eq:3add6} \\
&&\nonumber \\
\tilde{B}(\tilde{\phi})&=& e^{2\gamma(\tilde{\phi})}\left\{
h'(\tilde{\phi})B\left[h(\tilde{\phi})\right]
-\frac{3}{4\pi G}h'(\tilde{\phi})\gamma'(\tilde{\phi})
A'\left[h(\tilde{\phi})\right]-\frac{3}{4\pi G}
\left[\gamma'(\tilde{\phi})\right]^{2}A\left[h(\tilde{\phi})
\right]\right\} \;.\label{eq:3add7} \nonumber \\
&&\nonumber \\
&&
\end{eqnarray}
$\bullet$~~~In these notations the \emph{Jordan frame}
corresponds to the choice
\begin{equation}
\alpha=0 \;,\qquad B=1 \;,
\label{eq:3.20}
\end{equation}
and to the free functions $A(\phi)$ and $V(\phi)$. In our
notations this corresponds to identifying $A$ with $f$ and
(\ref{eq:3.19}) with the Jordan frame action
\begin{equation}
S=\int d^{4}x\;\sqrt{-g}\left[\frac{f(\phi)}{16\pi
G}R-\frac{1}{2}\nabla^{c}\phi\nabla_{c}\phi-V(\phi)\right]
+S^{(m)}\left[g_{ab},\psi^{(m)}\right]\;.\label{eq:3.21}
\end{equation}
In fact, Flanagan's Jordan frame action can be generalized to
arbitrary $B(\phi)$, which corresponds to our $\omega(\phi)$,
obtaining the
Jordan frame action
\begin{equation}
S=\int d^{4}x\;\sqrt{-g}\left[ \frac{f(\phi)R}{16\pi
G}-\frac{\omega(\phi)}{2}\nabla^{c}
\phi\nabla_{c}\phi-V(\phi)\right]
+S^{(m)}\left[g_{ab},\psi^{(m)}\right]\;.\label{eq:3.21b}
\end{equation}
$\bullet$~~~The \emph{Einstein frame with running units}
corresponds to the choice
\begin{equation}
A=1\;,\qquad B=1 \;,
\end{equation}
and to the free functions $\alpha(\phi)$ and $V(\phi)$. In our
notations with a tilde denoting Einstein frame quantities,
$\tilde{V}=\tilde{U}$
and $ e^{2\tilde{\alpha}}=\Omega^{-2}$ and the action
(\ref{eq:3.19})
corresponds to
\begin{equation}
S= \int d^{4}x\;\sqrt{-g}\left\{ \frac{\tilde{R}}{16\pi
G}-\frac{1}{2} \, \tilde{g}^{ab} \, \tilde{\nabla}_{a}
\tilde{\phi}\tilde{\nabla}_{b}\tilde{\phi}-\tilde{U}(
\tilde{\phi})+\Omega^{-2}\mathcal{L}^{(m)}
\left[\tilde{g}_{ab},\psi^{(m)}\right]\right\} \;,\label{eq:3.23}
\end{equation}
$\bullet$~~~The \emph{Einstein frame with fixed units}
corresponds to the
choice
\begin{equation}
A=1\;,\qquad B=1\;,\qquad\alpha=0 \;,
\end{equation}
and, in our notations, to the action
\begin{equation}
S=\int d^{4}x\;\sqrt{-g} \left\{ \frac{\tilde{R}}{16\pi
G}-\frac{1}{2}\tilde{g}^{cd}
\tilde{\nabla}_{c}
\tilde{\phi}\tilde{\nabla}_{d}\tilde{\phi}-\tilde{U}(
\tilde{\phi})+\mathcal{L}^{(m)}\left[\tilde{g}_{ab},
\psi^{(m)}\right]\right\} \;,\label{eq:3.25}
\end{equation}
in which there is no anomalous coupling of the scalar $\tilde{\phi}$
to matter $\left(\alpha=0\right)$. The difference between Einstein
frame with running units and with fixed units is in the choice of
the function $\alpha$. It could be said that in the Einstein frame
with fixed units the function $\alpha$ is not correctly transformed
according to eq.~(\ref{eq:3add4}), while the functions $A$ and
$B$
are transformed according to
eqs.~(\ref{eq:3add5})-(\ref{eq:3add7}).
Keeping the units fixed in the Einstein frame causes the masses to
remain constant. In our first example of Sec.~3.1.2, this would
correspond to replacing eq.~(\ref{eq:3.16}) with
\begin{equation}
\tilde{\square}
\tilde{\psi}-\frac{\tilde{R}}{6}\tilde{\psi}
-m^{2}\tilde{\psi}=0\;,\label{eq:3.26}
\end{equation}
with a constant mass $m$ introduced by hand. Of course, one can
\emph{postulate} this equation, which is debatable, but it
should at least be made clear that it can not be derived from
eq.~(\ref{eq:3.14}) by using Dicke's spacetime-dependent
rescaling of units. In other words, the first two terms in
eq.~(\ref{eq:3.26}) are obtained with a conformal
transformation while the third one is arbitrarily replaced by
$-m^{2}\tilde{\psi}$.
There are situations in cosmology in which the scalar field
$\phi$
is assumed to dominate the dynamics of the universe and ordinary
matter is ignored, setting $S^{(m)}=0$. In these situations
(corresponding to $ m=0$ in the example of eqs.~(\ref{eq:3.14})
and (\ref{eq:3.26})) the running of units does not matter as
long as only cosmological dynamics is studied. However, the
issue will resurface whenever massive test particles or test
fields, or non-conformal matter are introduced into this
picture, or when redshift or reaction rates are considered
\cite{Catenaetal}.
There is a way that is, in principle, consistent to obtain the
Einstein frame with fixed units: if one introduces in the Jordan
frame a factor that exactly compensates for the $\Omega^{-2}$
factor in front of the matter Lagrangian density when
conformally transforming to the Einstein frame, the scalar
$\tilde{\phi}$ will couple minimally to matter in this frame.
This is done, e.g., in Ref.~\cite{Ferraris}. However, the
price to pay is the non-minimal coupling of $\phi$ to
matter, and the violation of the equivalence principle, in the Jordan
frame, which is alien from the spirit of Brans-Dicke and other
scalar-tensor
theories.
\section{Energy conditions and singularity theorems}
\setcounter{equation}{0}
We now want to discuss the energy conditions in the Jordan and
Einstein frame. Let us consider, for the sake of illustration,
Brans-Dicke theory represented by the action
\begin{equation}
S=\int d^{4}x\; \sqrt{-g}\left[\phi
R-\frac{\omega}{\phi}\nabla^{c}\phi\nabla_{c}
\phi-V(\phi)\right]+S^{(m)} \label{eq:4.1}
\end{equation}
(the arguments proposed apply, however, to general scalar-tensor
theories).
The field equations can be written as
\begin{equation}
G_{ab}=\frac{8\pi}{\phi}T_{ab}^{(m)}+T_{ab}\left[\phi\right]\;,\label{eq:4.2}\end{equation}
\begin{equation}
\square\phi=\frac{1}{2\omega+3}\left[8\pi T^{(m)}+\phi\frac{dV }{d\phi}-2V\right]\;,\label{eq:4.3}\end{equation}
where \begin{equation}
T_{ab}\left[\phi\right]=\frac{\omega}{\phi^{2}}\left(\nabla_{a}\phi\nabla_{b}\phi-\frac{1}{2}g_{ab}\nabla^{c}\phi\nabla_{c}\phi\right)-\frac{V}{2\phi}g_{ab}+\frac{1}{\phi}\left(\nabla_{a}\nabla_{b}\phi-g_{ab}\square\phi\right)\;,\label{eq:4.4}\end{equation}
is often identified with an effective stress-energy tensor of the
scalar $\phi$. There are three possible ways of identifying an
effective stress-energy tensor for $\phi$
\cite{BellucciFaraoni,mybook} and the choice of
eq.~(\ref{eq:4.4})
has sometimes been criticized in
the literature \cite{SantiagoSilbergleit,mybook,Torres}. If
the choice (\ref{eq:4.4}) is accepted, as is common in the literature,
it is easy to see that the strong, weak, and dominant energy conditions
of general relativity \cite{Wald} can all be violated by the scalar
$\phi$ regarded as an effective form of matter. This is due to the
non-canonical form of $T_{ab}$; the last term in
eq.~(\ref{eq:4.4})
is linear in the second derivatives of $\phi$ instead of being quadratic
in the first derivatives, and it makes the sign of $T_{00}\left[\phi\right]$
indefinite, even causing negative energy densities. The possibility
of negative energy is regarded by certain authors as a criterion to
discard the Jordan frame \emph{a priori} as unphysical (see
\cite{MagnanoSokolowski,FGN} for references). Since we
know that Jordan frame and Einstein
frame with running units are physically equivalent, we argue that
these authors are left with the ``Einstein frame with fixed
units''
version of the theory, which is physically ill-motivated. Moreover,
it is not a negative kinetic energy that is worrisome, but rather
an energy that is unbounded from below, so that the system can decay
to lower and lower energy states {\em ad infinitum} (the
electron in the
hydrogen atom has negative total energy but there is a ground state
of the Hamiltonian which corresponds to a minimum for the
spectrum of energy eigenvalues). Hence, the mere possibility of
negative energies
is not, by itself, an argument to rule out the Jordan frame. As
pointed
out in Ref.~\cite{Flanagan}, the energy conditions differ in the
two frames but there is no physical observable corresponding to the
sign of $G_{ab}u^{a}u^{b}$ for all timelike vectors $u^{a}$, hence
there is no measurable inconsistency between the two frames.
Furthermore, a positive energy theorem has been shown to hold
for special scalar-tensor theories in the Jordan frame
\cite{Bertolamienergy}.
The validity of the energy conditions for the Einstein frame
scalar $\tilde{\phi}$ has been emphasized in relation with the Hawking-Penrose
singularity theorems \cite{HawkingEllis,Wald}. If the strong and
dominant energy conditions hold for $\tilde{\phi}$ and for ordinary
matter in the Einstein frame, the singularity theorems apply, even
though the same energy conditions are violated in the Jordan frame.
This situation is seen by some as the possibility to circumvent the
singularity theorems. In the cosmological context this would
imply
that it is possible to find solutions that are free of Big Bang
singularities just by going to the Jordan frame. This is clearly
impossible
if these two conformal frames are physically equivalent: the absence
of singularities in one frame and their occurrence in the conformally
rescaled theory has thus lead to an apparent paradox
\cite{529,725}. If, following Dicke \cite{Dicke}, the Jordan and
Einstein frame are equivalent, singularities occur in the
Einstein frame if and only if they occur
in the Jordan frame. The puzzle is quickly resolved as follows
(see also Ref.~\cite{mybook}): consider the
FLRW metric in the Jordan frame
\begin{equation}
ds^{2}=-dt^{2}+a^{2}(t)\left[\frac{dr^{2}}{1-Kr^{2}}
+r^{2}\left(d\theta^{2}+\sin^{2}\theta
\, d\varphi^{2}\right)\right]\;,
\end{equation}
and its Einstein frame cousin
\begin{equation}
d\tilde{s}^{2} = -d\tilde{t}^{2}+
\tilde{a}^{2}(\tilde{t})\left[\frac{dr^{2}}{1
-Kr^{2}}+r^{2}\left(d\theta^{2} +\sin^{2}\theta
d\varphi^{2}\right)\right]\;,
\end{equation}
with $ d\tilde{t}=\Omega \, dt$, $\tilde{a}=\Omega \, a$, and
proper length
$ d\tilde{l}=\tilde{a}
\, \left|d\underline{x}\right|=\Omega \,
a\left|d\underline{x}\right|=\Omega \, dl$. To ascertain
whether there is a Big Bang or other singularity in the
Einstein frame with running units it is not sufficient to
examine the behavior of the scale factor $\tilde{a}(\tilde{t})$
as $\tilde{t}\rightarrow0$. One must instead study the ratio of
a typical physical (proper) length
$\tilde{a}(\tilde{t})\left|d\underline{\tilde{x}}\right|$ to the
unit of length $\tilde{l}_{u}(\tilde{\phi})=\Omega \, l_{u}$,
where
$l_{u}$ is the fixed length unit in the Jordan frame and
$\left| d\underline{\tilde{x}}\right|$
is the (comoving) coordinate distance in the Einstein frame.
This ratio is
\begin{equation}
\frac{\tilde{a}(
\tilde{t}) \left|d\underline{\tilde{x}}\right|}{\tilde{l}_{u}(
\tilde{\phi})}=\frac{\Omega \, a(t)\left|d\underline{x}
\right|}{\Omega \,
l_{u}}=\frac{a(t)\left|d\underline{x}\right|}{l_{u}} \;.
\end{equation}
Therefore, $\frac{
\tilde{a}\left|d\underline{\tilde{x}}\right|}{\tilde{l}_{u}(
\tilde{\phi})}\rightarrow0$
if and only if $\frac{a\left|d\underline{x}\right|}{l_{u}}
\rightarrow0$, or
a singularity occurs in the Einstein frame if and only if it is
present in the Jordan frame. The argument is not yet complete,
because one has to make sure that the finite time at which the
singularity occurs (``initial time'' for a Big Bang singularity)
is not mapped into an infinite time in the other frame. This is
easily accomplished by examining the ratio of $\tilde{t}$ to
the varying unit of time $\tilde{t}_{u}(\tilde{\phi})=\Omega \,
t_{u}$ in the Einstein frame, where $t_{u}$ is the fixed unit
of time in the Jordan frame. This ratio is
\begin{equation}
\frac{\tilde{t}}{\tilde{t}_{u}}=\frac{\int_{0}^{t}
\Omega(\phi)dt'}{\Omega(\phi)t_{u}}\approx\frac{t}{t_{u}}\;,
\end{equation}
as $t\rightarrow0$ for an initial Big Bang singularity.
Therefore, $\tilde{t}\rightarrow0$ in the Einstein frame is
equivalent to $t\rightarrow0$ in the Jordan frame.
One can also check whether a singularity in the matter energy density
occurs in both frames. The energy density of the cosmic fluid transforms
as $\tilde{\rho}=\Omega^{-4}\rho$ on dimensional grounds (for a formal
derivation see, e.g., Ref.~\cite{mybook}). The Einstein frame
unit of
energy
is
\begin{equation}
\tilde{\rho}(\tilde{\phi})
\approx\frac{\tilde{m}}{\tilde{l}_{u}^{3}}
=\frac{\Omega^{-1}m_{u}}{\Omega^{3}l_{u}^{3}}
=\Omega^{-4}\rho_{u}\;,
\end{equation}
where $\rho_{u}\approx m / l_u^3 $ is the (constant) unit
of energy density in the Jordan frame. In a Big Bang singularity,
however, it is the ratio $ \tilde{\rho} / \tilde{\rho}_{u} $
that matters, not $\tilde{\rho}$. We have
\begin{equation}
\frac{\tilde{\rho}}{\tilde{\rho}_{u}}=\frac{\Omega^{-4}\rho}{\Omega^{-4}\rho_{u}}=\frac{\rho}{\rho_{u}}\;,
\end{equation}
hence $\frac{\tilde{\rho}}{\tilde{\rho}_{u}}\rightarrow\infty$ if
and only if $\frac{\rho}{\rho_{u}}\rightarrow\infty$, establishing
once again the equivalence of the two frames. If one were to consider
merely $\tilde{\rho}$ instead of $ \tilde{\rho} /
\tilde{\rho}_{u} $, one would erroneously conclude that
singularities occur in one conformal
frame but not in the other. This happens if the Einstein frame
with fixed units is considered, which is not physically
equivalent to the Jordan frame (if one wishes to regard it as a
physical theory).
\section{The $\Lambda$ problem and the Cauchy problem}
\setcounter{equation}{0}
In this section we briefly discuss other issues in the realm of
classical
physics in which the Jordan and the Einstein frame (with running units)
prove to be physically equivalent, in spite of claims to the
contrary.
These are the cosmological constant problem and the Cauchy problem.
It has been claimed that the issue of the conformal frame has
implications for the notorious cosmological constant problem
\cite{Weinberg} of why the cosmological constant energy
density is 120 orders
of magnitude smaller than what can be calculated with simple
quantum mechanics. The stress-energy tensor associated
with a cosmological constant $ T_{ab}^{(\Lambda)}=\Lambda
g_{ab}/\left( 8\pi G \right) $ provides a Jordan frame energy
density
$\rho_{\Lambda}= \Lambda / (8\pi G) $ which is constant, and a
conformal cousin $\tilde{\rho}_{\Lambda}=\Omega^{-4}\rho_{\Lambda}=e^{-\alpha\tilde{\phi}}\Lambda$
in the Einstein frame, where $\alpha>0$ is an appropriate constant.
Thus, $\tilde{\rho}_{\Lambda}$ represents a decaying cosmological
``constant''; the opinion is often voiced that the exponential
factor $e^{-\alpha\tilde{\phi}}$ multiplying $\Lambda$ in the Einstein
frame helps alleviating, if not outright solving, the
cosmological constant problem (see, e.g., Sec. 4.22 of
Ref.~\cite{Fujii}). Again, this would mean that the two
conformal frames are physically inequivalent
in contrast with the spirit of Dicke's paper
\cite{footnote5}.
It is easy to see that this argument fails to ease off the
cosmological constant problem. Again, what matters in the
Einstein frame is not
the form (or numerical value) of $\tilde{\rho}_{\Lambda}$, but the
ratio $ \tilde{\rho}_{\Lambda} / \tilde{\rho}_{u} $, where
$\tilde{\rho}_{u}=\Omega^{-4}\rho_{u}$
is the unit of energy density in the Einstein frame, and $\rho_{u}$
is the corresponding Jordan frame unit. The ratio
\begin{equation}
\frac{\tilde{\rho}_{\Lambda}}{\tilde{\rho}_{u}}=\frac{\Lambda e^{-\alpha\tilde{\phi}}}{8\pi G\rho_{u}e^{-\alpha\tilde{\phi}}}=\frac{\Lambda}{8\pi G\rho_{u}}\;,
\end{equation}
is the same in the Jordan and Einstein frame and, barring unforeseen
complications at the quantum level, the cosmological constant problem
is not alleviated a bit by choosing the Einstein frame with
running units.
Of course, one could then state that the Einstein frame with
fixed units solves the problem because then one would consider
$ \tilde{\rho}_{\Lambda}/ \rho_u \propto
e^{-\alpha\tilde{\phi}}$
instead of $
\tilde{\rho}_{\Lambda} / \tilde{\rho}_u=$constant;
this would be nonsense because the cosmological constant cannot
be calculated in one theory (where it is huge) and then mapped
into the Einstein frame with fixed units which bears no physical
relation with the original Jordan frame. $\Lambda$ should
be calculated directly in the Einstein frame with fixed units
and it is still huge. The argument presented here applies also
to situations in which the cosmological ``constant'' term
changes \cite{BertolamiNC}.
Finally, we want to comment on the Cauchy problem for
scalar-tensor gravity and its implications for the equivalence
of the two conformal frames. The folklore about the Cauchy
problem is that the mixing of the spin two and spin zero degrees
of freedom $g_{ab}$ and $\phi$ in the Jordan frame makes these
variables an inconvenient set for formulating the initial value
problem, which is not well posed in the Jordan frame: on the
other hand, the Einstein frame variables
$\left(\tilde{g}_{ab},\tilde{\phi}\right)$ admit a well-posed
Cauchy problem completely similar to that of general relativity
(see, e.g., the influential paper \cite{DamourFarese}). Were
this true, it
would appear that the Jordan and Einstein frame are physically
inequivalent in this respect. This position toward the Cauchy
problem, however, ignores two older references showing that the
Cauchy problem is well posed {\em in the Jordan frame} for
two specific scalar-tensor theories: Brans-Dicke
theory with a free scalar $\phi$ \cite{CockeCohen} and the
theory of a scalar field conformally coupled to the Ricci
curvature \cite{Noakes}. The task of studying the Jordan
frame Cauchy problem
for \emph{general} scalar-tensor theories has been taken on in
a recent paper \cite{Salgado} in which it is shown,
using generalized harmonic coordinates, that the Cauchy problem
is well posed, although further study is necessary for
implementing a full 3+1 formulation $\grave{a}$ la York
\cite{York} in practical (numerical)
applications \cite{Salgado}. This shows that, contrary to the
common lore, the Jordan and the Einstein frames are physically
equivalent
also with respect to the initial value problem. The issue of mapping
the details of the Jordan frame Cauchy problem into details of the
corresponding Einstein frame problem and, in particular, clarifying
the role played by running units, will be discussed elsewhere.
It is clear that the equivalence between the two conformal
frames breaks down when the conformal
transformation breaks down, i.e., when $f(\phi)=0$ or $f_1\equiv
2f+3\left( df/d\phi \right)^2=0$ (cf. eqs.~(\ref{eq:1.4}) and
(\ref{eq:1.5})). However, the Jordan frame initial value problem
may not be well posed as well when $f=0$, and requiring $f>0$
eliminates also the singularities $f_1=0$ (see
Refs.~\cite{Abramoetal,PRDsingularities} for a discussion of
these singularities and Ref.~\cite{Bronnikov} for conformal
continuation past these points).
\section{Conclusions}
It appears that, at the classical level, the Jordan and Einstein
frames are physically equivalent when the units of fundamental
and derived quantities are allowed to scale appropriately with
the conformal factor $\Omega$ in the Einstein frame. Previous
doubts on the physical equivalence with respect to the Cauchy
problem \cite{DamourFarese} seem to dissipate in the light of
recent work \cite{Salgado}, although a more comprehensive
picture is
desirable. The arguments against the equivalence of the two
frames raised in the past regard positivity of the energy in the
Einstein frame and the indefiniteness of its sign in the Jordan
frame (\cite{MagnanoSokolowski,FGN} and references therein).
This is particularly relevant at the quantum level: negative
energies do not allow a stable ground state and
the system would decay to a lower and lower energy states {\em
ad infinitum}. However, as was pointed out in \cite{Flanagan},
there is no physical observable corresponding to the sign of
$T_{ab}u^a u^b$ or $G_{ab}u^a u^b$, where $u^a$ is a timelike
four-vector, and specific examples of scalar-tensor theories
that are stable in the Jordan frame have been found
\cite{Bertolamienergy}. The relevant question to ask, at least
at the classical level, is not what the sign of the energy is,
but rather whether the
energy is bounded from below, which may well occur in
scalar-tensor gravity (see, e.g., \cite{PRDenergy}).
At the quantum level, the issue of the ground state becomes more
delicate, as there are more decay channels than at the
classical level. Although the conformal equivalence seems to hold
to some extent at the semiclassical level, in which the matter
fields are quantized while the variables $\left( g_{ab}, \phi
\right)$ are classical (see Ref.~\cite{Flanagan} for a brief
discussion and references), this equivalence definitely breaks
down when $\phi$ is quantized \cite{FGN}. When also $g_{ab} $ is
quantized
in full quantum gravity, inequivalent quantum theories have been
found
\cite{Flanagan,AshtekarCorichi,Elizaldeetal,Grumilleretal}.
This is not surprising because the conformal transformation can
be seen as a Legendre transformation \cite{MagnanoSokolowski}. A
similar Legendre transformation is used in the classical
mechanics of particles
to switch from the canonical coordinates $q$ of the Lagrangian
description to the variables $\left( q,p \right)$ of the
Hamiltonian formalism, and this Legendre map is an example of a
canonical transformation \cite{Goldsteinp385}. Now, it is well
known that Hamiltonians that are classically equivalent become
inequivalent when quantized: they exhibit different energy
spectra and scattering amplitudes \cite{CalogeroDegasperis}.
Therefore, we expect the analogous ``canonical
transformations'' between different conformal frames not to be
unitary and to yield physically inequivalent theories at the
quantum level \cite{footnote6}. A common objection to this
statement arising among particle physicists is based on the
equivalence theorem of Lagrangian field theory, which states
that the S-matrix is invariant under local (non-linear) field
redefinitions \cite{equivalence}. Since the conformal
transformation (\ref{eq:2.2}) and (\ref{eq:2.3}) is a local
non-linear redefinition of the fields $g_{ab}$ and $\phi$ , it
would seem that quantum physics is invariant under change of
the conformal frame. However, this is not true in general
because the field theory approach in which the equivalence
theorem is derived applies to gravity only in the perturbative
regime in which the fields describe small deviations from
Minkowski space. In this regime, tree level quantities can be
calculated in any conformal frame with the same results.
However, when the metric tensor is allowed full dynamical
freedom and is not restricted to be a small perturbation of
a fixed background, the field theory approach and the
equivalence theorem do not apply. It is plausible that the
equivalence theorem can be proved also for fixed backgrounds
that are curved and do not coincide with the Minkowski space of
effective field theory. However, we are not aware of such a
generalization in the literature on quantum field theory on
curved space (a proof of a generalized equivalence theorem will
be pursued elsewhere). Thus, it is clear that the equivalence
theorem fails in the non-perturbative regime; nevertheless, one
can consider semiclassical situations in which the metric is
classical and the full scalar $\phi$ is quantized, and it is
quite possible that the conformal transformation leaves the
quantum physics of $\phi$ unaffected --- after all, the physics
of the classical
metric is invariant under change of conformal frame and
quantization of a scalar field in a fixed background metric
poses no problems \cite{BirrellDavies}. Indeed, there are
examples in which such semiclassical theories related by a
conformal transformation seem to be equivalent
\cite{Bertolamisemiclassical}. A precise
and detailed understanding of the conformal (in)equivalence at
the quantum level, however, requires further work.
\section*{Acknowledgments}
We thank a referee for useful remarks and references. This work
was supported by the Natural Sciences and Engineering Research
Council of Canada ({\em NSERC}).
\clearpage
|
2,877,628,089,027 | arxiv | \section{Introduction}
Convolutional neural networks (CNNs) have been dominating the computer vision (CV) field since the renaissance of deep neural networks (DNNs). They have demonstrated effectiveness in numerous vision tasks from image classification \cite{DBLP:conf/cvpr/HeZRS16}, object detection\cite{DBLP:journals/pami/RenHG017}, to pixel-based segmentation \cite{DBLP:conf/iccv/HeGDG17}. Remarkably, despite the huge success of Transformer structure \cite{DBLP:conf/nips/VaswaniSPUJGKP17} in natural language processing (NLP) \cite{DBLP:conf/naacl/DevlinCLT19}, the CV society still focuses on the CNN structure for quite some time.
The transformer structure finally made its grand debut in CV last year.
Vision Transformer (ViT) \cite{DBLP:conf/iclr/DosovitskiyB0WZ21} showed that a pure Transformer applied directly to a sequence of image patches can perform very well on image classification tasks, if the training dataset is sufficiently large. DeiT \cite{DBLP:conf/icml/TouvronCDMSJ21} further demonstrated that Transformer can be successfully trained on typical-scale dataset, such as ImageNet-1K \cite{DBLP:conf/cvpr/DengDSLL009}, with appropriate data augmentation and model regularization.
Interestingly, before the heat of Transformer dissipated, the structure of multi-layer perceptrons (MLPs) was revived by Tolstikhin et al. in a work called MLP-Mixer \cite{DBLP:journals/corr/abs-2105-01601}. MLP-Mixer is based exclusively on MLPs applied across spatial locations and feature channels. When trained on large datasets, MLP-Mixer attains competitive scores on image classification benchmarks. The success of MLP-Mixer suggests that neither convolution nor attention are necessary for good performance. It sparked further research on MLP as the authors wished \cite{DBLP:journals/corr/abs-2105-08050,DBLP:journals/corr/abs-2107-00645}.
However, as the reported accuracy on image classification benchmarks continues to increase by new network designs from various camps, no conclusion can be made as which structure among CNN, Transformer, and MLP performs the best or is most suitable for vision tasks. This is partly due to the pursuit of high scores that leads to multifarious tricks and exhaustive parameter tuning. As a result, network structures cannot be fairly compared in a systematic way. The work presented in this paper fills this blank by conducting a series of controlled experiments over CNN, Transformer, and MLP in a unified framework.
We first develop a unified framework called SPACH as shown in Fig. \ref{fig:SPACH}. It is mostly adopted from current Transformer and MLP frameworks, since convolution can also fit into this framework and is in general robust to optimization. The SPACH framework contains a plug-and-play module called mixing block which could be implemented as convolution layers, Transformer layers, or MLP layers. Aside from the mixing block, other components in the framework are kept the same when we explore different structures. This is in stark contrast to previous work which compares different network structures in different frameworks that vary greatly in layer cascade, normalization, and other non-trivial implementation details. As a matter of fact, we found that these structure-free components play an important role in the final performance of the model, and this is commonly neglected in the literature.
\begin{figure*}[t]
\centering
\includegraphics[width=1\linewidth]{SPACH.pdf}
\caption{Illustration of the proposed experimental framework named SPACH.}
\label{fig:SPACH}
\end{figure*}
With this unified framework, we design a series of controlled experiments to compare the three network structures. The results show that all three network structures could perform well on the image classification task when pre-trained on ImageNet-1K. In addition, each individual structure has its distinctive properties leading different behaviors when the network size scales up. We also find several common design choices which contribute a lot to the performance of our SPACH framework. The detailed findings are listed in the following.
\begin{itemize}
\item Multi-stage design is standard in CNN models, but its effectiveness is largely overlooked in Transformer-based or MLP-based models. We find that the multi-stage framework consistently and notably outperforms the single-stage framework no matter which of the three network structures is chosen.
\item Local modeling is efficient and crucial. With only light-weight depth-wise convolutions, the convolution model can achieve similar performance as a Transformer model in our SPACH framework. By adding a local modeling bypass in both MLP and Transformer structures, a significant performance boost is obtained with negligible parameters and FLOPs increase.
\item MLP can achieve strong performance under small model sizes, but it suffers severely from over-fitting when the model size scales up. We believe that over-fitting is the main obstacle that prevents MLP from achieving SOTA performance.
\item Convolution and Transformer are complementary in the sense that convolution structure has the best generalization capability while Transformer structure has the largest model capacity among the three structures. This suggests that convolution is still the best choice in designing lightweight models but designing large models should take Transformer into account.
\end{itemize}
Based on these findings, we propose two hybrid models of different scales which are built upon convolution and Transformer layers. Experimental results show that, when a sweet point between generalization capability and model capacity is reached, the performance of these straightforward hybrid models is already on par with SOTA models with sophisticated architecture designs.
\section{Background}
CNN and its variants have dominated the vision domain. During the evolution of CNN models, useful experience about the architecture design has been accumulated. Recently, two types of architectures, namely Transformer \cite{DBLP:conf/iclr/DosovitskiyB0WZ21} and MLP \cite{DBLP:journals/corr/abs-2105-01601}, begin to emerge in the vision domain and have shown performance similar to the well-optimized CNNs. These results kindle a spark towards building better vision models beyond CNNs.
\textbf{Convolution-based vision models}
Since the entrance of deep learning era pioneered by AlexNet \cite{DBLP:conf/nips/KrizhevskySH12}, the computer vision community has devoted enormous efforts to designing better vision backbones. In the past decade, most work focused on improving the design of CNN, and a series of networks, including VGG \cite{DBLP:journals/corr/SimonyanZ14a}, ResNet \cite{DBLP:conf/cvpr/HeZRS16}, SENet \cite{DBLP:conf/cvpr/HuSS18}, Xception \cite{DBLP:conf/cvpr/Chollet17}, MoblieNet\cite{DBLP:journals/corr/HowardZCKWWAA17,DBLP:conf/cvpr/SandlerHZZC18}, and EfficientNet \cite{DBLP:conf/icml/TanL19,DBLP:conf/icml/TanL21}, are designed. They achieve significant accuracy improvements in various vision tasks.
A standard convolution layer learns filters in a 3D space, with two spatial dimensions and one channel dimension. Thus, the learning of spatial correlations and channel correlations are coupled inside a single convolution kernel. Differently, A depth-wise convolution layer only learns spatial correlations by moving the learning process of channel correlations to an additional 1x1 convolution. The fundamental hypothesis behind this design is that cross-channel correlations and spatial correlations are sufficiently decoupled that it is preferable not to map them jointly \cite{DBLP:conf/cvpr/Chollet17}. Recent work \cite{DBLP:conf/icml/TanL19,DBLP:conf/icml/TanL21} shows that depth-wise convolution can achieve both high accuracy and good efficiency, confirming this hypothesis to some extent. In addition, the idea of decoupling spatial and channel correlations is adopted in the vision Transformer. Therefore, this paper employs the spatial-channel decoupling idea in our framework design.
\textbf{Transformer-based vision models.}
With the success of Transformer in natural language processing (NLP) \cite{DBLP:conf/nips/VaswaniSPUJGKP17,DBLP:conf/naacl/DevlinCLT19}, many researchers start to explore the use of Transformer as a stand-alone architecture for vision tasks. They are facing two main challenges. First, Transformer operates over a group of tokens, but no natural tokens, similar to the words in natural language, exist in an image. Second, images have a strong local structure while the Transformer structure treats all tokens equally and ignores locality. The pioneering work ViT \cite{DBLP:conf/iclr/DosovitskiyB0WZ21} solved the first challenge by simply dividing an image into non-overlapping patches and treat each patch as a visual token. ViT also reveals that Transformer models trained on large-scale datasets could attain SOTA image recognition performance. However, when the training data is insufficient, ViT does not perform well due to the lack of inductive biases. DeiT \cite{DBLP:conf/icml/TouvronCDMSJ21} mitigates the problem by introducing a regularization and augmentation pipeline on ImageNet-1K.
Swin \cite{DBLP:journals/corr/abs-2103-14030} and Twins \cite{DBLP:journals/corr/abs-2104-13840} propose local ViT to address the second challenge. They adopt locally-grouped self-attention by computing the standard self-attention within non-overlapping windows. The local mechanism not only leads to performance improvement thanks to the reintroduction of locality, but also bring sufficient improvement on memory and computational efficiency. Thus, the pyramid structure becomes feasible again for vision Transformer.
There has been a blowout development in the design of Transformer-based vision models. Since this paper is not intended to review the progress of vision Transformer, we only briefly introduce some highly correlated Transformer models. CPVT \cite{chu2021conditional} and CvT \cite{DBLP:journals/corr/abs-2103-15808} introduce convolution into Transformer blocks, bringing the desired translation-invariance properties into ViT architecture. CaiT \cite{DBLP:journals/corr/abs-2103-17239} introduces a LayerScale approach to empower effective training of deeper ViT network. It is also discovered that some class-attention layers built on top of ViT network offer more effective processing than the class embedding. LV-ViT \cite{jiang2021all} proposes a bag of training techniques to build a strong baseline for vision Transformer. LeViT \cite{DBLP:journals/corr/abs-2104-01136} proposes a hybrid neural network for fast image classification inference.
\textbf{MLP-based vision models.}
Although MLP is not a new concept for the computer vision community, the recent progress on MLP-based visual models surprisingly demonstrates, both conceptually and technically, that simple architecture can achieve competitive performance with CNN or Transformer \cite{DBLP:journals/corr/abs-2105-01601}. The pioneering work MLP-Mixer proposed a Mixer architecture using channel-mixing MLPs and channel-mixing MLPs to communicate between different channels and spatial locations (tokens), respectively. It achieves promising results when trained on a large-scale dataset (i.e., JFT\cite{DBLP:conf/iccv/SunSSG17}). ResMLP \cite{DBLP:journals/corr/abs-2105-03404} built a similar MLP-based model with a deeper architecture. ResMLP does not need large-scale datasets and it achieves comparable accuracy/complexity trade-offs on ImageNet-1K with Transformer-based models. FF \cite{DBLP:journals/corr/abs-2105-02723} showed that simply replacing the attention layer in ViT with an MLP layer applied over the patch dimension could achieve moderate performance on ImageNet classification. gMLP \cite{DBLP:journals/corr/abs-2105-08050} proposed a gating mechanism on MLP and suggested that self-attention is not a necessary ingredient for scaling up machine learning models.
\section{A Unified Experimental Framework}
In order to fairly compare the three network structures, we are in need of a unified framework that excludes other performance-affecting factors. Since recent MLP-based networks have already shared a similar framework as Transformer-based networks, we build the unified experimental framework based on them and try to include CNN-based network in this framework as well.
\begin{table*}
\centering
\begin{tabular}{@{}ccccc@{}}
\toprule
Model & SPACH-XXS & SPACH-XS & SPACH-S \\
\midrule
Conv & $C=384,R=2.0,N=12$ & $C=384,R=2.0,N=24$ & $C=512,R=3.0,N=24$\\
\midrule
Transformer & $C=192,R=2.0,N=12$ & $C=384,R=2.0,N=12$ & $C=512,R=3.0,N=12$\\
\midrule
MLP & $C=384,R=2.0,N=12$ & $C=384,R=2.0,N=24$ & $C=512,R=3.0,N=24$\\
\midrule
Conv-MS & \thead{$C=64,R=2.0$\\$N_s=\{2,2,6,2\}$} & \thead{$C=96,R=2.0$\\$N_s=\{3,4,12,3\}$} & \thead{$C=128,R=3.0$\\$N_s=\{3,4,12,3\}$}\\
\midrule
Transformer-MS & \thead{$C=32,R=2.0$\\$N_s=\{2,2,6,2\}$} & \thead{$C=64,R=2.0$\\$N_s=\{3,4,12,3\}$} & \thead{$C=96,R=3.0$\\$N_s=\{3,4,12,3\}$}\\
\midrule
MLP-MS & \thead{$C=64,R=2.0$\\$N_s=\{2,2,6,2\}$} & \thead{$C=96,R=2.0$\\$N_s=\{3,4,12,3\}$} & \thead{$C=128,R=3.0$\\$N_s=\{3,4,12,3\}$}\\
\bottomrule
\end{tabular}
\caption{SPACH and SPACH-MS model variants. $C$: feature dimension, $R$: expansion ratio of MLP in $\mathcal{F}_c$, $N$: number of mixing blocks of SPACH, $N_s$: number of mixing blocks in the $i_{th}$ stage of SPACH-MS.}
\label{tab:configuration}
\end{table*}
\subsection{Overview of the SPACH Framework}
We build our experimental framework with reference to ViT \cite{DBLP:conf/iclr/DosovitskiyB0WZ21} and MLP-Mixer \cite{DBLP:journals/corr/abs-2105-01601}. Fig. \ref{fig:SPACH}(a) shows the single-stage version of the SPACH framework, which is used for our empirical study. The architecture is very simple and consists mainly of a cascade of mixing blocks, plus some necessary auxiliary modules, such as patch embedding, global average pooling, and a linear classifier. Fig. \ref{fig:SPACH}(b) shows the details of the mixing block. Note that the spatial mixing and channel mixing are performed in consecutive steps. The name SPACH for our framework is coined to emphasize the serial structure of SPAtial and CHannel processing.
We also enable a multi-stage variation, referred to as SPACH-MS, as shown in Fig. \ref{fig:SPACH}(c). Multi-stage is an important mechanism in CNN-based networks to improve the performance. Unlike the single-stage SPACH, which processes the image in a low resolution by down-sampling the image by a large factor at the input, SPACH-MS is designed to keep a high-resolution in the initial stage of the framework and progressively perform down-sampling. Specifically, our SPACH-MS contains four stages with down-sample ratios of 4, 8, 16, and 32, respectively. Each stage contains $N_s$ mixing blocks, where $s$ is the stage index. Due to the extremely high computational cost of Transformer and MLP on high-resolution feature maps, we implement the mixing blocks in the first stage with convolutions only. The feature dimension within a stage remains constant, and will be multiplied with a factor of 2 after down-sampling.
Let $I\in \mathbb{R}^{3\times h\times w}$ denotes an input image, where 3 is the RGB channels and $H\times W$ is the spatial dimensions. Our SPACH framework first passes the input image through a Patch Embedding layer, which is the same as the one in ViT, to convert $I$ into patch embeddings $X_p\in \mathbb{R}^{C\times \frac{h}{p} \times \frac{w}{p}}$. Here $p$ denotes patch size, which is 16 in the single-stage implementation and 4 in the multi-stage implementation.
After the cascades of mixing blocks, a classification head implemented by a linear layer is used for the supervised pre-training.
We list the hyper-parameters used in different model configurations in Table \ref{tab:configuration}. Three model size for each variations of SPACH are designed, namely SPACH-XXS, SPACH-XS and SPACH-S, by controlling the number of blocks, the number of channels, and the expansion ratio of channel mixing MLP $\mathcal{F}_c$. The model size, theoretical computational complexity (FLOPS), and empirical throughput are presented in Section \ref{sec:exp}. We measure the throughput using one P100 GPU.
\subsection{Mixing Block Design} \label{sec:3.1}
\begin{figure}[t]
\centering
\includegraphics[width=1.0\linewidth]{Mixing.pdf}
\caption{Three implementations of the spatial mixing module using convolution, Transformer, and MLP, respectively. P.E. denotes positional encoding, implemented by convolution in SPACH.}
\label{fig:mixing block}
\end{figure}
Mixing blocks are key components in the SPACH framework. As shown in Fig. \ref{fig:SPACH}(b), for an input feature $X\in\mathbb{R}^{C\times H\times W}$, where $C$ and $H\times W$ denote channel and spatial dimensions, it is first processed by a spatial mixing function $\mathcal{F}_s$ and then by a channel mixing function $\mathcal{F}_c$. $\mathcal{F}_s$ focuses on aggregating context information from different spatial locations while $\mathcal{F}_c$ focuses on channel information fusion. Denoting the output as $Y$, we can formulate a mixing block as:
\begin{equation}
Y = \mathcal{F}_s(\mathcal{F}_c(X)).
\end{equation}
Following ViT \cite{DBLP:conf/iclr/DosovitskiyB0WZ21}, we use an MLP with appropriate normalization and residual connection to implement $\mathcal{F}_c$. The MLP here can be also viewed as a 1x1 convolution (also known as point-wise convolution \cite{DBLP:conf/cvpr/Chollet17}) which is a special case of regular convolution. Note that $\mathcal{F}_c$ only performs channel fusion and does not explore any spatial context.
The spatial mixing function $\mathcal{F}_s$ is the key to implement different architectures. As shown in Fig. \ref{fig:mixing block}, we implement three structures using convolution, self-attention, and MLP. The common components include normalization and residual connection.
Specifically, the convolution structure is implemented by a 3x3 depth-wise convolution, as channel mixing will be handled separately in subsequent steps. For the Transformer structure, there is a positional embedding module in the original design. But recent research suggests that absolute positional embedding breaks translation variance, which is not suitable for images. In view of this and inspired by recent vision transformer design \cite{chu2021conditional,DBLP:journals/corr/abs-2103-15808}, we introduce a convolutional positional encoding (CPE) as a bypass in each spatial mixing module. The CPE module has negligible parameters and FLOPs.
For MLP-based network, the pioneering work MLP-Mixer does not use any positional embedding, but we empirically find that adding the very lightweight CPE significantly improves the model performance, so we use the same treatment for MLP as for Transformer.
The three implementations of $\mathcal{F}_s$ have distinctive properties as listed in Table \ref{tab:properites}. First, the convolution structure only involves local connections so that it is computational efficient. Second, the self-attention structure uses dynamic weight for each input instance so that model capacity is increased. Moreover, it has a global receptive field, which enables information to flow freely across different positions \cite{DBLP:conf/nips/VaswaniSPUJGKP17}. Third, MLP structure has a global receptive field just as the self-attention structure, but it does not use dynamic weight. In summary, these three properties seen in different architectures are all desirable and may have positive influence on the model performance or efficiency. We can find convolution and self-attention have complementary properties thus there is potential to build hybrid model to combine all desirable properties. Besides, MLP structure seems to be inferior to self-attention in this analysis.
\begin{table}
\centering
\begin{tabular}{@{}cccc@{}}
\toprule
Properties & Convolution & Self-Attention & MLP \\
\midrule
\thead{Sparse \\ Connectivity} & \checkmark & & \\
\midrule
\thead{Dynamic \\ Weight} & & \checkmark & \\
\midrule
\thead{Global \\ Receptive Field} & & \checkmark & \checkmark \\
\bottomrule
\end{tabular}
\caption{Three desired properties in network design are seen in different network structures.}
\label{tab:properites}
\end{table}
\section{Empirical Studies on Mixing Blocks} \label{sec:exp}
In this section, we design a series of controlled experiments to compare the three network structures. We first introduce the experimental settings in Section \ref{subsec:settings}, and then present our main findings in Section \ref{subsec:exp1}, \ref{subsec:exp2}, \ref{subsec:exp3}, and \ref{subsec:exp4}.
\subsection{Datasets and Training Pipelines} \label{subsec:settings}
We conduct experiments on ImageNet-1K (IN-1K) \cite{DBLP:conf/cvpr/DengDSLL009} image classification which has 1k classes. The training set has 1.28M images while the validation set has 50k images. The Top-1 accuracy on a single crop is reported. Unless otherwise indicated, we use the input resolution of 224x224. Most of our training settings are inherited from DeiT \cite{DBLP:conf/icml/TouvronCDMSJ21}. We employ an AdamW \cite{DBLP:journals/corr/abs-1711-05101} optimizer for 300 epochs with a cosine decay learning rate scheduler and 20 epochs of linear warm-up. The weight decay is 0.05, and the initial learning rate is $0.005\times \frac{\text{batchsize}}{512}$. 8 GPUs with mini-batch 128 per GPU are used in training, resulting a total batch-size of 1024. We use exactly the same data augmentation and regularization configurations as DeiT, including Rand-Augment \cite{DBLP:conf/nips/CubukZS020}, random erasing \cite{DBLP:conf/aaai/Zhong0KL020}, Mixup \cite{DBLP:conf/iclr/ZhangCDL18}, CutMix \cite{DBLP:conf/iccv/YunHCOYC19}, stochastic depth \cite{DBLP:conf/eccv/HuangSLSW16}, and repeated augmentation \cite{DBLP:journals/corr/abs-1902-05509,DBLP:conf/cvpr/HofferBHGHS20}. We use the same training pipeline for all comparing models. And the implementation is built upon PyTorch \cite{DBLP:conf/nips/PaszkeGMLBCKLGA19} and timm library \cite{rw2019timm}.
\begin{table*}
\centering
\setlength\tabcolsep{3pt}
\begin{tabular}{@{}c|c|ccc|c|ccc|c }
\toprule
\multirow{3}{*}{Network Scale} & \multirow{3}{*}{Model} & \multicolumn{4}{c}{Single-Stage} \vline & \multicolumn{4}{c}{Multi-Stage} \\ \cline{3-10}
& & \multirow{2}{*}{\#param.} & \multirow{2}{*}{FLOPs} & Throughput & IN-1K & \multirow{2}{*}{\#param.} & \multirow{2}{*}{FLOPs} & Throughput & IN-1K \\
& & & & (image/s) & Top-1 acc. & & & (image/s) & Top-1 acc. \\
\hline
\multirow{3}{*}{XXS}
& Conv & 8M & 1.4G & 1513 & 72.1 & 5M & 0.7G \textcolor{red}{(-0.7G)} & 1576 &73.3 \textcolor{red}{(+1.2)} \\
& Trans & 4M & 0.9G & 1202 & 68.0 & 2M & 0.5G \textcolor{red}{(-0.4G)} & 1558 & 65.4 \textcolor{green}{(-2.6)} \\
& MLP & 9M & 1.8G & 980 & 74.1 & 6M & 0.9G \textcolor{red}{(-0.9G)} & 1202 & 74.9 \textcolor{red}{(+0.8)} \\
\hline
\multirow{3}{*}{XS}
&Conv & 15M & 2.8G & 770 & 77.3 & 17M & 2.8G \textcolor{red}{(-0.0G)} & 602 & 80.1 \textcolor{red}{(+2.8)} \\
&Trans & 15M & 3.1G & 548 & 78.4 & 14M & 3.1G \textcolor{red}{(-0.0G)} & 441 & 80.1 \textcolor{red}{(+1.7)} \\
&MLP & 17M & 3.5G & 503 & 78.5 & 19M & 3.4G \textcolor{red}{(-0.1G)} & 438 & 80.7 \textcolor{red}{(+2.2)} \\
\hline
\multirow{3}{*}{S}
&Conv & 39M & 7.4G & 374 & 80.1 & 44M & 7.2G \textcolor{red}{(-0.2G)} & 328 & 81.6 \textcolor{red}{(+1.5)} \\
&Trans & 33M & 6.7G & 328 & 81.7 & 40M & 7.6G \textcolor{green}{(+0.9G)} & 246 & 82.9 \textcolor{red}{(+1.2)} \\
&MLP & 41M & 8.7G & 272 & 78.6 & 46M & 8.2G \textcolor{red}{(-0.5G)} & 254 & 82.1 \textcolor{red}{(+3.5)} \\
\bottomrule
\end{tabular}
\caption{Model performance of SPACH and SPACH-MS at three network scales.}
\label{tab:exp1.1}
\end{table*}
\begin{figure}[t]
\centering
\begin{subfigure}[b]{0.22\textwidth}
\begin{tikzpicture}[scale=0.48]
\begin{axis}[
xlabel={\#params (M)},
ylabel={ImageNet Top-1 Acc (\%)},
ymax=86,
height=3.5\linewidth,
width=2.2\textwidth,
legend pos=south east,
legend style={nodes={scale=1.0, transform shape}},
label style={font=\large}
]
\addplot[dash dot, smooth,color=red] plot coordinates {
(8, 72.1)
(15, 77.3)
(39, 80.1)
};
\addlegendentry{Conv }
\addplot[dash dot, smooth,color=black] plot coordinates {
(4, 68.0)
(15, 78.4)
(33, 81.7)
};
\addlegendentry{Trans }
\addplot[dash dot, smooth,color=blue] plot coordinates {
(9, 74.1)
(17, 78.5)
(41, 78.6)
};
\addlegendentry{MLP }
\addplot[solid, smooth,color=red] plot coordinates {
(5, 73.3)
(17, 80.1)
(44, 81.6)
};
\addlegendentry{Conv-MS }
\addplot[solid, smooth,color=black] plot coordinates {
(2, 65.4)
(14, 80.1)
(40, 82.9)
};
\addlegendentry{Trans-MS }
\addplot[solid, smooth,color=blue] plot coordinates {
(6, 74.9)
(19, 80.7)
(44, 82.1)
};
\addlegendentry{MLP-MS }
\end{axis}
\end{tikzpicture}
\end{subfigure}
~
\begin{subfigure}[b]{0.22\textwidth}
\begin{tikzpicture}[scale=0.48]
\begin{axis}[
xlabel={Throughput (image/s)},
ylabel={ImageNet Top-1 Acc (\%)},
ymax=86,
height=3.5\linewidth,
width=2.2\textwidth,
legend pos=north east,
legend style={nodes={scale=1.0, transform shape}},
label style={font=\large}
]
\addplot[dash dot, smooth,color=red] plot coordinates {
(1513, 72.1)
(770, 77.3)
(374, 80.1)
};
\addlegendentry{Conv }
\addplot[dash dot, smooth,color=black] plot coordinates {
(1202, 68.0)
(548, 78.4)
(328, 81.7)
};
\addlegendentry{Trans }
\addplot[dash dot, smooth,color=blue] plot coordinates {
(980, 74.1)
(503, 78.5)
(272, 78.6)
};
\addlegendentry{MLP }
\addplot[solid, smooth,color=red] plot coordinates {
(1576, 73.3)
(602, 80.1)
(328, 81.6)
};
\addlegendentry{Conv-MS }
\addplot[solid, smooth,color=black] plot coordinates {
(1558, 65.4)
(441, 80.1)
(246, 82.9)
};
\addlegendentry{Trans-MS }
\addplot[solid, smooth,color=blue] plot coordinates {
(1202, 74.9)
(438, 80.7)
(369, 82.1)
};
\addlegendentry{MLP-MS }
\end{axis}
\end{tikzpicture}
\end{subfigure}
\caption{The multi-stage models (named with -MS suffix) always achieve a better performance than their single-stage counterparts.}
\label{fig:exp1.1}
\end{figure}
\begin{table}
\centering
\setlength\tabcolsep{3pt}
\begin{tabular}{l|ccc|c}
\toprule
\multirow{2}{*}{Model} & \multirow{2}{*}{\#param.} & \multirow{2}{*}{FLOPs} & throughput & IN-1K \\
& & & (image/s) & Top-1 acc. \\
\hline
Trans-MS-S & 40M & 7.6G & 246 & 82.9 \\
$\text{Trans-MS-S}^{-}$ & 40M & 7.6G & 259 & 82.3 \\
\hline
MLP-MS-S & 46M & 8.2G & 254 & 82.1 \\
$\text{MLP-MS-S}^{-}$ & 46M & 8.2G & 274 & 80.1 \\
\bottomrule
\end{tabular}
\caption{Both Transformer structure and MLP structure benefit from local modeling at a very small computational cost. The superscription - indicates without local modeling.}
\label{tab:exp1.2}
\end{table}
\subsection{Multi-Stage is Superior to Single-Stage} \label{subsec:exp1}
Multi-stage design is standard in CNN models, but it is largely overlooked in Transformer-based or MLP-based models. Our first finding is that multi-stage design should always be adopted in vision models no matter which of the three network structures is chosen.
Table \ref{tab:exp1.1} compares the image classification performance between multi-stage framework and single-stage framework. For all three network scales and all three network structures, multi-stage framework consistently achieves better complexity-accuracy trade-off. For the sake of easy comparison, the changes of FLOPs and accuracy are highlighted in Table \ref{tab:exp1.1}. Most of the multi-stage models are designed to have slightly fewer computational costs, but they manage to achieve a higher accuracy than the corresponding single-stage models. An accuracy loss of 2.6 points is observed for the Transformer model at the XXS scale, but it is understandable as the multi-stage model happens to have only half of the parameters and FLOPs of the corresponding single-stage model.
In addition, Fig. \ref{fig:exp1.1} shows how the image classification accuracy changes with the size of model parameters and model throughput. Despite the different trends observed for different network structures, the multi-stage models always outperform their single-stage counterparts.
This finding is consistent with the results reported in recent work. Both Swin-Transformer \cite{DBLP:journals/corr/abs-2103-14030} and TWins \cite{DBLP:journals/corr/abs-2104-13840} adopt multi-stage framework and achieve a stronger performance than the single-stage framework DeiT \cite{DBLP:conf/icml/TouvronCDMSJ21}.
Our empirical study suggests that the use of multi-stage framework can be an important reason.
\subsection{Local Modeling is Crucial} \label{subsec:exp2}
Although it has been pointed out in many previous work \cite{DBLP:journals/corr/abs-2103-15808,chu2021conditional,DBLP:journals/corr/abs-2104-05707,DBLP:conf/icml/dAscoliTLMBS21,DBLP:journals/corr/abs-2103-14030} that local modeling is crucial for vision models, we will show in this subsection how amazingly efficient local modeling could be.
In our empirical study, the spatial mixing block of the convolution structure is implemented by a $3 \times 3$ depth-wise convolution, which is a typical local modeling operation. It is so light-weight that it only contributes to 0.3\% of the model parameter and 0.5\% of the FLOPs. However, as Table \ref{tab:exp1.1} and Fig. \ref{fig:exp1.1} show, this structure can achieve competitive performance when compared with the Transformer structure in the XXS and XS configurations.
It is due to the sheer efficiency of $3 \times 3$ depth-wise convolution that we propose to use it as a bypass in both MLP and Transformer structures. The increase of model parameters and inference FLOPs is almost negligible, but the locality of the models is greatly strengthened. In order to demonstrate how local modeling helps the performance of Transformer and MLP structures, we carry out an ablation study which removes this convolution bypass in the two structures.
Table \ref{tab:exp1.2} shows the performance comparison between models with or without local modeling. The two models we pick are the top performers in Table \ref{tab:exp1.1} when multi-stage framework is used and network scale is S. We can clearly find that the convolution bypass only slightly decreases the throughput, but brings a notable accuracy increase to both models. Note that the convolution bypass is treated as convolutional positional embedding in Trans-MS-S, so we bring back the standard patch embedding as in ViT \cite{DBLP:conf/iclr/DosovitskiyB0WZ21} in $\text{Trans-MS-S}^{-}$. For $\text{MLP-MS-S}^{-}$, we follow the practice in MLP-Mixer and do not use any positional embedding.
This experiment confirms the importance of local modeling and suggests the use of $3 \times 3$ depth-wise convolution as a bypass for any designed network structures.
\subsection{A Detailed Analysis of MLP} \label{subsec:exp3}
\begin{table}
\centering
\setlength\tabcolsep{3pt}
\begin{tabular}{c|ccc|c}
\toprule
\multirow{2}{*}{Model} & \multirow{2}{*}{\#param.} & \multirow{2}{*}{FLOPs} & throughput & IN-1K \\
& & & (image/s) & Top-1 acc. \\
\hline
MLP-S & 41M & 8.7G & 272 & 78.6 \\
+Shared & 39M & 8.7G & 274 & 80.2 \\
\hline
MLP-MS-S & 46M & 8.2G & 254 & 82.1 \\
+Shared & 45M & 8.2G & 244 & 82.5 \\
\bottomrule
\end{tabular}
\caption{The performance of MLP models are greatly boosted when weight sharing is adopted to alleviate over-fitting. }
\label{tab:exp3.2}
\end{table}
Due to the excessive number of parameters, MLP models suffer severely from over-fitting. We believe that over-fitting is the main obstacle for MLP to achieve SOTA performance. In this part, we discuss two mechanisms which can potentially alleviate this problem.
One is the use of multi-stage framework. We have already shown in Table \ref{tab:exp1.1} that multi-stage framework brings gain. Such gain is even more prominent for larger MLP models. In particular, the MLP-MS-S models achieves 2.6 accuracy gain over the single-stage model MLP-S. We believe this owes to the strong generalization capability of the multi-stage framework.
Fig. \ref{fig:exp3.1} shows how the test accuracy increases with the decrease of training loss. Over-fitting can be observed when the test accuracy starts to flatten.
These results also lead to a very promising baseline for MLP-based models. Without bells and whistles, MLP-MS-S model achieves 82.1\% ImageNet Top-1 accuracy, which is 5.7 points higher than the best results reported by MLP-Mixer \cite{DBLP:journals/corr/abs-2105-01601} when ImageNet-1K is used as training data.
The other mechanism is parameter reduction through weight sharing. We apply weight-sharing on the spatial mixing function $\mathcal{F}_s$. For the single-stage model, all $N$ mixing blocks use the same $\mathcal{F}_s$, while for the multi-stage model, each stage use the same same $\mathcal{F}_s$ for its $N_s$ mixing blocks. We present the results of S models in Table \ref{tab:exp3.2}. We can find that the shared-weight variants, denoted by "+Shared", achieve higher accuracy with almost the same model size and computation cost. Although they are still inferior to Transformer models, the performance is on par with or even better than convolution models.
Fig. \ref{fig:exp3.1} confirms that using shared weights in the MLP-MS model further delays the appearance of over-fitting signs. Therefore, we conclude that MLP-based models remain competitive if they could solve or alleviate the over-fitting problem.
\begin{figure}[t]
\centering
\begin{subfigure}[b]{0.45\textwidth}
\begin{tikzpicture}[scale=0.9]
\begin{axis}[
xlabel={Train loss},
ylabel={ImageNet Top-1 Acc (\%)},
height=0.55\linewidth,
width=1.1\columnwidth,
legend pos=north east,
legend style={nodes={scale=0.6, transform shape}},
label style={font=\footnotesize}
]
\addplot[solid, smooth,color=blue] plot coordinates {
(3.62, 74.1)
(3.16, 78.5)
(2.73, 78.6)
};
\addlegendentry{MLP }
\addplot[solid, smooth,color=black] plot coordinates {
(3.57, 74.9)
(2.84, 80.7)
(2.52, 82.2)
};
\addlegendentry{MLP-MS }
\addplot[only marks, mark=*, color=red] plot coordinates {
(2.58, 82.5)
};
\addlegendentry{MLP-MS-Shared }
\end{axis}
\end{tikzpicture}
\end{subfigure}
\caption{Illustration of the over-fitting problem in MLP-based models. Both multi-stage framework and weight sharing alleviate the problem.}
\label{fig:exp3.1}
\end{figure}
\subsection{Convolution and Transformer are Complementary} \label{subsec:exp4}
We find that convolution and Transformer are complementary in the sense that convolution structure has the best generalization capability while Transformer structure has the largest model capacity among the three structures we investigated.
Fig. \ref{fig:exp3.2} shows that, before the performance of Conv-MS saturates, it achieves a higher test accuracy than Trans-MS at the same training loss. This shows that convolution models generalize better than Transformer models. In particular, when the training loss is relatively large, the convolution models show great superiority against Transformer models. This suggests that convolution is still the best choice in designing lightweight vision models.
On the other hand, both Fig. \ref{fig:exp1.1} and Fig. \ref{fig:exp3.2} show that Transformer models achieve higher accuracy than the other two structures when we increase the model size and allow for higher computational cost.
Recall that we have discussed three properties of network architectures in Section \ref{sec:3.1}. It is now clear that the sparse connectivity helps to increase generalization capability, while dynamic weight and global receptive field help to increase model capacity.
\begin{figure}[t]
\centering
\begin{subfigure}[b]{0.45\textwidth}
\begin{tikzpicture}[scale=0.9]
\begin{axis}[
xlabel={Train loss},
ylabel={ImageNet Top-1 Acc (\%)},
height=0.55\linewidth,
width=1.1\columnwidth,
legend pos=north east,
legend style={nodes={scale=0.6, transform shape}},
label style={font=\footnotesize}
]
\addplot[solid, smooth,color=red] plot coordinates {
(3.78, 73.3)
(3.10, 80.1)
(2.73, 81.6)
};
\addlegendentry{Conv-MS }
\addplot[solid, smooth,color=black] plot coordinates {
(4.23, 65.4)
(3.04, 80.1)
(2.54, 82.9)
};
\addlegendentry{Trans-MS }
\end{axis}
\end{tikzpicture}
\end{subfigure}
\caption{Conv-MS has a better generalization capability than Trans-MS as it achieves a higher test accuracy at the same training loss before the model saturates.}
\label{fig:exp3.2}
\end{figure}
\section{Hybrid Models}
As discussed in Section \ref{sec:3.1} and \ref{subsec:exp3}, convolution and Transformer structures have complementary characteristics and have potential to be used in a single model. Based on this observation, we construct hybrid models at the XS and S scales based on these two structures. The procedure we used to construct hybrid models is rather simple. We take a multi-stage convolution-based model as the base model, and replace some selected layers with Transformer layers. Considering the local modeling capability of convolutions and global modeling capability of Transformers, we tend to do such replacement in later stages of the model. The details of layer selection in the two hybrid models are listed as follows.
\begin{itemize}
\item Hybrid-MS-XS: It is based on Conv-MS-XS. The last ten layers in Stage 3 and the last two layers in Stage 4 are replaced by Transformer layers. Stage 1 and 2 remain unchanged.
\item Hybrid-MS-S: It is based on Conv-MS-S. The last two layers in Stage 2, the last ten layers in Stage 3, and the last two layers in Stage 4 are replaced by Transformer layers. Stage 1 remains unchanged.
\end{itemize}
In order to unleash the full potential of hybrid models, we further adopt the \textit{deep} patch embedding layer (PEL) implementation as suggested in LV-ViT \cite{jiang2021all}. Different from \textit{default} PEL which uses one large (16x16) convolution kernel, the deep PEL uses four convolution kernels with kernel size $\{7,3,3,2\}$, stride $\{2,1,1,2\}$, and channel number $\{64,64,64,C\}$. By using small kernel sizes and more convolution kernels, deep PEL helps a vision model to explore the locality inside single patch embedding vector. We mark models with deep PEL as "Hybrid-MS-*+".
Table \ref{tab:exp4} shows comparison between our hybrid models and some of the state-of-the-art models based on CNN, Transformer, or MLP. All listed models are trained on ImageNet-1K. Within the section of our models, we can find that hybrid models achieve better model size-performance trade-off than pure convolution models or Transformer models. The Hybrid-MS-XS achieves 82.4\% top-1 accuracy with 28M parameters, which is higher than Conv-MS-S with 44M parameters and only a little lower than Trans-MS-S with 40M parameters. In addition, the Hybrid-MS-S achieve 83.7\% top-1 accuracy with 63M parameters, which has 0.8 point gain compared with Trans-MS-S.
The Hybrid-MS-S+ model we proposed achieves 83.9\% top-1 accuracy with 63M parameters. This number is higher than the accuracy achieved by SOTA models Swin-B and CaiT-S36, which have model size of 88M and 68.2M, respectively. The FLOPs of our model is also fewer than these two models. We believe Hybrid-MS-S can be serve as a strong yet simple baseline for future research on architecture design of vision models.
\begin{table}
\centering
\begin{tabular}{l|rr|c}
\toprule
\multirow{2}{*}{Model} & \multirow{2}{*}{\#param.} & \multirow{2}{*}{FLOPs} & IN-1K \\
& & & Top-1 acc. \\
\hline
\multicolumn{4}{c}{CNN} \\
\hline
RegNetY-4G \cite{DBLP:conf/cvpr/RadosavovicKGHD20} & 21M & 4.1G & 80.0 \\
RegNetY-8G \cite{DBLP:conf/cvpr/RadosavovicKGHD20} & 39M & 8.0G & 81.7 \\
RegNetY-16G \cite{DBLP:conf/cvpr/RadosavovicKGHD20} & 84M & 16.0G & 82.9 \\
\hline
\multicolumn{4}{c}{Transformer} \\
\hline
ViT-B/16* \cite{DBLP:conf/iclr/DosovitskiyB0WZ21} & 86M & - & 77.9 \\
DeiT-S \cite{DBLP:conf/icml/TouvronCDMSJ21} & 22M & 4.6G & 79.8 \\
DeiT-B \cite{DBLP:conf/icml/TouvronCDMSJ21} & 86M & 17.5G & 81.8 \\
Swin-T \cite{DBLP:journals/corr/abs-2103-14030} & 29M & 4.5G & 81.3 \\
Swin-S \cite{DBLP:journals/corr/abs-2103-14030} & 50M & 8.7G & 83.0 \\
Swin-B \cite{DBLP:journals/corr/abs-2103-14030} & 88M & 15.4G & 83.5 \\
CaiT-XS24 \cite{DBLP:journals/corr/abs-2103-17239} & 26.6M & 5.4G & 81.8 \\
CaiT-S36 \cite{DBLP:journals/corr/abs-2103-17239} & 68.2M & 13.9G & 83.3 \\
CvT-13 \cite{DBLP:journals/corr/abs-2103-15808} & 20M & 4.5G & 81.6 \\
CvT-21 \cite{DBLP:journals/corr/abs-2103-15808} & 32M & 7.1G & 82.5 \\
\hline
\multicolumn{4}{c}{MLP} \\
\hline
FF-Base \cite{DBLP:journals/corr/abs-2105-02723} & 62M & - & 74.9 \\
Mixer-B/16 \cite{DBLP:journals/corr/abs-2105-01601} & 79M & - & 76.4 \\
ResMLP-S24 \cite{DBLP:journals/corr/abs-2105-03404} & 30M & 6.0G & 79.4 \\
ResMLP-B24 \cite{DBLP:journals/corr/abs-2105-03404} & 45M & 23.0G & 81.0 \\
gMLP-S \cite{DBLP:journals/corr/abs-2105-08050} & 20M & 4.5G & 79.4 \\
gMLP-B \cite{DBLP:journals/corr/abs-2105-08050} & 73M & 15.8G & 81.6 \\
\hline
\multicolumn{4}{c}{Ours} \\
\hline
Conv-MS-XS & 17M & 2.8G & 80.1\\
Conv-MS-S & 44M & 7.2G & 81.6\\
Trans-MS-XS & 14M & 3.1G & 80.1\\
Trans-MS-S & 40M & 7.6G & 82.9\\
\hline
Hybrid-MS-XS & 28M & 4.5G & 82.4 \\
Hybrid-MS-XS+ & 28M & 5.6G & 82.8 \\
Hybrid-MS-S & 63M & 11.2G & 83.7 \\
Hybrid-MS-S+ & 63M & 12.3G & 83.9 \\
\bottomrule
\end{tabular}
\caption{Comparison of different models on ImageNet-1K classification. Compared models are grouped according to network structures, and our models are listed in the last, Most models are pre-trained with 224x224 images, except ViT-B/16*, which uses 384x384 images.}
\label{tab:exp4}
\end{table}
\section{Conclusion}
The objective of this work is to understand how the emerging Transformer and MLP structures compare with CNNs in the computer vision domain. We first built a simple and unified framework, called SPACH, that could use CNN, Transformer, or MLP as plug-and-play components. Under the SPACH framework, we discover with a little surprise that all three network structures are similarly competitive in terms of the accuracy-complexity trade-off, although they show distinctive properties when the network scales up. In addition to the analysis of specific network structures, we also investigate two important design choices, namely multi-stage framework and local modeling, which are largely overlooked in previous work. Finally, inspired by the analysis, we propose two hybrid models which achieve SOTA performance on ImageNet-1k classification without bells and whistles.
Our work also raises several questions worth exploring. First, realizing the fact that the performance of MLP-based models is largely affected by over-fitting, is it possible to design a high-performing MLP model that is not subject to over-fitting? Second, current analyses suggest that neither convolution nor Transformer is the optimal structure across all model sizes. What is the best way to fuse these two structures? Last but not least, do better visual models exist beyond the known structures including CNN, Transformer, and MLP?
{\small
\bibliographystyle{ieee_fullname}
|
2,877,628,089,028 | arxiv | \section{Motivation}
Experimental efforts to develop useful solid state quantum information processors have encountered a host of practical problems that have substantially limited progress. While the desire to reduce noise in solid state qubits appears to be the key factor that drives much of the recent work in this field, it must be acknowledged that there are formidable challenges related to architecture, circuit density, fabrication variation, calibration and control that also deserve attention. For example, a qubit that is inherently exponentially sensitive to fabrication variations with no recourse for in-situ correction holds little promise in any large scale architecture, even with the best of modern fabrication facilities. Thus, a qubit designed in the absence of information concerning its ultimate use in a larger scale system may prove to be of little utility in the future. In what follows, we present an experimental demonstration of a novel superconducting flux qubit \cite{fluxqubit} that has been specifically designed to address several issues that pertain to the implementation of a large scale quantum information processor. While noise is not the central focus of this article, we nonetheless present experimental evidence that, despite its physical size and relative complexity, the observed flux noise in this flux qubit is comparable to the quietest such devices reported upon in the literature to date.
It has been well established that rf-SQUIDs can be used as qubits given an appropriate choice of device parameters. Such devices can be operated as a flux biased phase qubit using two intrawell energy levels \cite{FluxBiasedPhaseQubit} or as a flux qubit using any pair of interwell levels \cite{fluxqubit}. This article will focus upon an experimental demonstration of a novel rf-SQUID flux qubit that can be tuned in-situ using solely {\it static} flux biases to compensate for fabrication variations in device parameters, both within single qubits and between multiple qubits. It is stressed that this latter issue is of critical importance in the development of useful large scale quantum information processors that could foreseeably involve thousands of qubits \cite{DiVincenzo}. Note that in this regard, the ion trap approach to building a quantum information processor has a considerable advantage in that the qubits are intrinsically identical, albeit the challenge is then to characterize and control the trapping potential with high fidelity \cite{Wineland}. While our research group's express interest is in the development of a large scale superconducting adiabatic quantum optimization [AQO] processor \cite{AQC,Santoro}, it should be noted that many of the practical problems confronted herein are also of concern to those interested in implementing gate model quantum computation [GMQC] processors \cite{GMQC} using superconducting technologies.
This article is organized as follows: In Section II, a theoretical argument is presented to justify the rf-SQUID design that has been implemented. It is shown that this design is robust against fabrication variations in Josephson junction critical current. Second, it is argued why it is necessary to include a tunable inductance in the flux qubit to account for differences in inductance between qubits in a multi-qubit architecture and to compensate for changes in qubit inductance during operation. Thereafter, the focus of the article shifts towards an experimental demonstration of the rf-SQUID flux qubit. The architecture of the experimental device and its operation are discussed in Section III and then a series of experiments to characterize the rf-SQUID and to highlight its control are presented in Section IV. Section V contains measurements of properties that indicate that this more complex rf-SQUID is indeed a flux qubit. Flux and critical current noise measurements and a formula for converting the measured flux noise spectral density into a free induction (Ramsey) decay time are presented in Section VI. A summary of key conclusions is provided in Section VII. Detailed calculations of rf-SQUID Hamiltonians have been placed in the appendices.
\section{rf-SQUID Flux Qubit Design}
The behavior of most superconducting devices is governed by three types of macroscopic parameters: the critical currents of any Josephson junctions, the net capacitance across the junctions and the inductance of the superconducting wiring. The Hamiltonian for many of these devices can generically be written as
\begin{equation}
\label{eqn:Hphase}
{\cal H}=\sum_i\left[\frac{Q_i^2}{2C_i}-E_{Ji}\cos(\varphi_i)\right]+\sum_{n}U_n\frac{\left(\varphi_n-\varphi_n^x\right)^2}{2} \; ,
\end{equation}
\noindent where $C_i$, $E_{Ji}=I_i\Phi_0/2\pi$ and $I_i$ denote the capacitance, Josephson energy and critical current of Josephson junction $i$, respectively. The terms in the first sum are readily recognized as being the Hamiltonians of the individual junctions for which the quantum mechanical phase across the junction $\varphi_i$ and the charge collected on the junction $Q_i$ obey the commutation relation $[\Phi_0\varphi_i/2\pi,Q_j]=i\hbar\delta_{ij}$. The index $n$ in the second summation is over closed inductive loops. External fluxes threading each closed loop, $\Phi_n^x$, have been represented as phases $\varphi_n^x\equiv 2\pi\Phi_n^x/\Phi_0$. The quantum mechanical phase drop experienced by the superconducting condensate circulating around any closed loop is denoted as $\varphi_n$. The overall potential energy scale factor for each closed loop is given by $U_n\equiv(\Phi_0/2\pi)^2/L_n$. Here, $L_n$ can be either a geometric inductance from wiring or Josephson inductance from large junctions \cite{vanDuzer}. Hamiltonian (\ref{eqn:Hphase}) will be used as the progenitor for all device Hamiltonians that follow.
\subsection{Compound-Compound Josephson Junction Structure}
\begin{figure}
\includegraphics[width=3.25in]{RFSSummary.pdf}
\caption{\label{fig:rfssummary} (color online) a) A single junction rf-SQUID qubit. b) Compound Josephson Junction (CJJ) rf-SQUID qubit. c) Compound-Compound Josephson Junction (CCJJ) rf-SQUID qubit. Junction critical currents $I_i$ and junction phases $\varphi_i$ ($1\leq i \leq 4$) as noted. Net device phases are denoted as $\varphi_{\alpha}$, where $\alpha\in\left(\ell ,r,q\right)$. External fluxes, $\Phi_n^x$, are represented as phases $\varphi_{n}^x\equiv2\pi\Phi_n^x/\Phi_0$, where $n\in\left(L,R,\text{cjj},\text{ccjj},q\right)$. Inductance of the rf-SQUID body, CJJ loop and CCJJ loop are denoted as $L_{\text{body}}$, $L_{\text{cjj}}$ and $L_{\text{ccjj}}$, respectively.}
\end{figure}
A sequence of rf-SQUID architectures are depicted in Fig.~\ref{fig:rfssummary}. The most primitive version of such a device is depicted in Fig.~\ref{fig:rfssummary}a, and more complex variants in Figs.\ref{fig:rfssummary} b and \ref{fig:rfssummary}c. For the single junction rf-SQUID (Fig.~\ref{fig:rfssummary}a), the phase across the junction can be equated to the phase drop across the body of the rf-SQUID: $\varphi_1=\varphi_q$. The Hamiltonian for this device can then be written as
\begin{subequations}
\begin{equation}
\label{eqn:1JHeff}
{\cal H}=\frac{Q_q^2}{2C_q}+V(\varphi_q) \; ;
\end{equation}
\vspace{-0.12in}
\begin{equation}
\label{eqn:1JV}
V(\varphi_q)=U_q\Big\{\frac{\left(\varphi_q-\varphi_q^x\right)^2}{2}-\beta\cos\left(\varphi_q\right)\Big\} \; ;
\end{equation}
\vspace{-0.12in}
\begin{equation}
\label{eqn:1Jbeta}
\beta=\frac{2\pi L_q I_q^c}{\Phi_0} \; ,
\end{equation}
\end{subequations}
\noindent with the qubit inductance $L_q\equiv L_{\text{body}}$, qubit capacitance $C_q\equiv C_1$ and qubit critical current $I_q^c\equiv I_1$ in this particular case. If this device has been designed such that $\beta>1$ and is flux biased such that $\varphi_q^x\approx\pi$, then the potential energy $V(\varphi_q)$ will be bistable. With increasing $\beta$ an appreciable potential energy barrier forms between the two local minima of $V(\varphi_q)$, through which the two lowest lying states of the rf-SQUID may couple via quantum tunneling. It is these two lowest lying states, which are separated from all other rf-SQUID states by an energy of order of the rf-SQUID plasma energy $\hbar\omega_p\equiv\hbar/\sqrt{L_qC_1}$, that form the basis of a qubit. One can write an effective low energy version of Hamiltonian (\ref{eqn:1JHeff}) as \cite{Leggett}
\begin{equation}
\label{eqn:Hqubit}
{\cal H}_{q}=-{\frac{1}{2}}\left[\epsilon\sigma_z+\Delta_q\sigma_x\right] \;\; ,
\end{equation}
\noindent where $\epsilon=2\left|I_q^p\right|\left(\Phi_q^x-\Phi_0/2\right)$, $\left|I_q^p\right|$ is the magnitude of the persistent current that flows about the inductive $q$ loop when the device is biased hard [$\epsilon\gg\Delta_q$] to one side and $\Delta_q$ represents the tunneling energy between the otherwise degenerate counter-circulating persistent current states at $\Phi^x_q=\Phi_0/2$.
\begin{figure}
\includegraphics[width=3.25in]{QubitComparison_ver2.pdf}
\caption{\label{fig:qubitcomparison} (color online) Depiction of the two lowest lying states of an rf-SQUID at degeneracy ($\epsilon=0$) with nomenclature for the energy basis ($\ket{g}$,$\ket{e}$) and flux basis ($\ket{\downarrow}$,$\ket{\uparrow}$) as indicated.}
\end{figure}
A depiction of the one-dimensional potential energy and the two lowest energy states of an rf-SQUID at degeneracy ($\Phi_q^x=\Phi_0/2$) for nominal device parameters is shown in Fig.~\ref{fig:qubitcomparison}. In this diagram, the ground and first excited state are denoted by $\ket{g}$ and $\ket{e}$, respectively. These two energy levels constitute the energy eigenbasis of a flux qubit. An alternate representation of these states, which is frequently referred to as either the flux or persistent current basis, can be formed by taking the symmetric and antisymmetric combinations of the energy eigenstates: $\ket{\downarrow}=\left(\ket{g}+\ket{e}\right)/\sqrt{2}$ and $\ket{\uparrow}=\left(\ket{g}-\ket{e}\right)/\sqrt{2}$, which yield two roughly gaussian shaped wavefunctions that are centered about each of the wells shown in Fig.~\ref{fig:qubitcomparison}. The magnitude of the persistent current used in Eq.~(\ref{eqn:Hqubit}) is then defined by $\left|I_q^p\right|\equiv\left|\bra{\uparrow}\left(\Phi_q-\Phi_0/2\right)/L_q\ket{\uparrow}\right|$. The tunneling energy is given by $\Delta_q=\bra{e}{\cal H}_q\ket{e}-\bra{g}{\cal H}_q\ket{g}$.
The aforementioned dual representation of the states of a flux qubit allows two distinct modes of operation of the flux qubit as a binary logical element with a logical basis defined by the states $\ket{0}$ and $\ket{1}$. In the first mode, the logical basis is mapped onto the energy eigenbasis: $\ket{0}\rightarrow\ket{g}$ and $\ket{1}\rightarrow\ket{e}$. This mode is useful for optimizing the coherence times of flux qubits as the dispersion of Hamiltonian (\ref{eqn:Hqubit}) is flat as a function of $\Phi_q^x$ to first order for $\epsilon\approx 0$, thus providing some protection from the effects of low frequency flux noise \cite{optimalpoint}. However, this is not a convenient mode of operation for implementing interactions between flux qubits \cite{parametriccoupling1,parametriccoupling2}. In the second mode, the logical basis is mapped onto the persistent current basis: $\ket{0}\rightarrow\ket{\downarrow}$ and $\ket{1}\rightarrow\ket{\uparrow}$. This mode of operation facilitates the implementation of inter-qubit interactions via inductive couplings, but does so at the expense of coherence times. GMQC schemes exist that attempt to leverage the benefits of both of the above modes of operation \cite{IBM,Oliver,NiftyItalianPaper}. On the other hand, those interested in implementing AQO strictly use the second mode of operation cited above. This, very naturally, leads to some interesting properties: First and foremost, in the coherent regime at $\epsilon=0$, the groundstate maps onto $\ket{g}=\left(\ket{0}+\ket{1}\right)/\sqrt{2}$, which implies that it is a superposition state with a fixed phase between components in the logical basis. Second, the logical basis is not coincident with the energy eigenbasis, except in the extreme limit $\epsilon/\Delta_q\gg 1$. As such, the qubit should not be viewed as an otherwise free spin-1/2 in a magnetic field, rather it maps onto an Ising spin subjected to a magnetic field with both a longitudinal ($B_z\rightarrow\epsilon$) and a transverse ($B_x\rightarrow\Delta_q$) component \cite{Ising}. In this case, it is the competition between $\epsilon$ and $\Delta_q$ which dictates the relative amplitudes of $\ket{\downarrow}$ and $\ket{\uparrow}$ in the groundstate wavefunction $\ket{g}$, thereby enabling logical operations that make {\it no} explicit use of the excited state $\ket{e}$. This latter mode of operation of the flux qubit has connections to the fields of quantum magnetism \cite{Anderson} and optimization theory \cite{Kirkpatrick}. Interestingly, systems of coupled flux qubits that are operated in this mode bear considerable resemblance to Feynman's original vision of how to build a quantum computer \cite{Feynman}.
While much seminal work has been done on single junction and the related 3-Josephson junction rf-SQUID flux qubit\cite{3JJfluxqubits,MooijMore3JJFluxQubits,MooijSuperposition,MooijCoherentDynamics,MooijCoupledSpectroscopy,OliverMachZehnder,OliverLandauZener,OliverAmplitudeSpectroscopy,ClarkeQubits,IPHT4Q,1OverFFluxQubit1,1OverFFluxQubit2}, it has been recognized that such devices would be impractical in a large scale quantum information processor as their properties are exceptionally sensitive to fabrication variations. In particular, in the regime $E_{J1}\gg\hbar\omega_p$, $\Delta_q\propto\exp(-\hbar\omega_p/E_{J1})$. Thus, it would be unrealistic to expect a large scale processor involving a multitude of such devices to yield from even the best superconducting fabrication facility. Moreover, implementation of AQO requires the ability to actively tune $\Delta_q$ from being the dominant energy scale in the qubit to being essentially negligible during the course of a computation. Thus the single junction rf-SQUID flux qubit is of limited practical utility and has passed out of favor as a prototype qubit.
The next step in the evolution of the single junction flux qubit and related variants was the compound Josephson junction (CJJ) rf-SQUID, as depicted in Fig.~\ref{fig:rfssummary}b. This device was first reported upon by Han, Lapointe and Lukens \cite{CJJ} and was the first type of flux qubit to display signatures of quantum superposition of macroscopic states \cite{LukensSuperposition}. The CJJ rf-SQUID has been used by other research groups\cite{ItaliansCJJ,NiftyItalianPaper,MoreNiftyItalianPaper} and a related 4-Josephson junction device has been proposed \cite{3JJfluxqubits,MooijMore3JJFluxQubits}. The CJJ rf-SQUID flux qubit and related variants have reappeared in a gradiometric configuration in more recent history \cite{HOMRT,IBM,Delft}. Here, the single junction of Fig.~\ref{fig:rfssummary}a has been replaced by a flux biased dc-SQUID of inductance $L_{\text{cjj}}$ that allows one to tune the critical current of the rf-SQUID in-situ. Let the applied flux threading this structure be denoted by $\Phi^x_{\text{cjj}}$. It is shown in Appendix A that the Hamiltonian for this system can be written as
\begin{subequations}
\begin{equation}
\label{eqn:2JHeff}
{\cal H}=\sum_n\left[\frac{Q_n^2}{2C_n}+U_n\frac{\left(\varphi_n-\varphi_n^x\right)^2}{2}\right]-U_q\beta_{\text{eff}}\cos\left(\varphi_q-\varphi_q^0\right) \; ,
\end{equation}
\noindent where the sum is over $n\in\left\{q,\text{cjj}\right\}$, $C_q\equiv C_1+C_2$, $1/C_{\text{cjj}}\equiv 1/C_1+1/C_2$ and $L_q\equiv L_{\text{body}}+L_{\text{cjj}}/4$. The 2-dimensional potential energy in Hamiltonian (\ref{eqn:2JHeff}) is characterized by
\begin{equation}
\label{eqn:2JBeff}
\beta_{\text{eff}}=\beta_+\cos\left(\frac{\varphi_{\text{cjj}}}{2}\right)\sqrt{1+\left[\frac{\beta_-}{\beta_+}\tan(\varphi_{\text{cjj}}/2)\right]^2} \; ;
\end{equation}
\vspace{-12pt}
\begin{equation}
\label{eqn:2JOffset}
\varphi_q^0\equiv 2\pi\frac{\Phi_q^0}{\Phi_0} =-\arctan\left(\frac{\beta_-}{\beta_+}\tan\left(\varphi_{\text{cjj}}/2\right)\right) \; ;
\end{equation}
\vspace{-12pt}
\begin{equation}
\label{eqn:2Jbetapm}
\beta_{\pm}\equiv 2\pi L_q\left(I_{1}\pm I_{2}\right)/\Phi_0 \; .
\end{equation}
\end{subequations}
\noindent Note that if $\cos\left(\varphi_{\text{cjj}}/2\right)<0$, then $\beta_{\text{eff}}<0$ in Hamiltonian (\ref{eqn:2JHeff}). This feature provides a natural means of shifting the qubit degeneracy point from $\varphi_q^x=\pi$, as in the single junction rf-SQUID case, to $\varphi_q^x\approx 0$. It has been assumed in all that follows that this $\pi$-shifted mode of operation of the CCJ rf-SQUID has been invoked.
Hamiltonian (\ref{eqn:2JHeff}) is similar to that of a single junction rf-SQUID modulo the presence of a $\varphi_{\text{cjj}}$-dependent tunnel barrier through $\beta_{\text{eff}}$ and an effective critical current $I_q^c\equiv I_1+I_2$.
For $L_{\text{cjj}}/L_q\ll 1$ it is reasonable to assume that $\varphi_{\text{cjj}}\approx 2\pi\Phi^x_{\text{cjj}}/\Phi_0$. Consequently, the CJJ rf-SQUID facilitates in-situ tuning of the tunneling energy through $\Phi^x_{\text{cjj}}$. While this is clearly desirable, one does pay for the additional flexibility by adding more complexity to the rf-SQUID design and thus more potential room for fabrication variations. The minimum achievable barrier height is ultimately limited by any so called {\it junction asymmetry} which leads to finite $\beta_{-}$. In practice, for $\beta_-/\beta_+=(I_{1}-I_{2})/(I_{1}+I_{2})\lesssim0.05$, this effect is of little concern. However, a more insidious effect of junction asymmetry can be seen via the change of variables $\varphi_q-\varphi_q^0\rightarrow\varphi_q$ in Eq.~(\ref{eqn:2JHeff}), namely an apparent $\Phi^x_{\text{cjj}}$-dependent flux offset: $\Phi^x_q\rightarrow\Phi^x_q-\Phi_q^0(\Phi^x_{\text{cjj}})$. If the purpose of the CJJ is to simply allow the experimentalist to target a particular $\Delta_q$, then the presence of $\Phi_q^0(\Phi^x_{\text{cjj}})$ can be readily compensated via the application of a static flux offset. On the other hand, any mode of operation that explicitly requires altering $\Delta_q$ during the course of a quantum computation \cite{IBM,Oliver,Kaminsky,Aharonov,NiftyItalianPaper,MoreNiftyItalianPaper} would also require active compensation for what amounts to a nonlinear crosstalk from $\Phi^x_{\text{cjj}}$ to $\Phi^x_q$. While it may be possible to approximate this effect as a linear crosstalk over a small range of $\Phi^x_{\text{cjj}}$ if the junction asymmetry is small, one would nonetheless need to implement precise {\it time-dependent} flux bias compensation to utilize the CJJ rf-SQUID as a flux qubit in any quantum computation scheme. While this may be feasible in laboratory scale systems, it is by no means desirable nor practical on a large scale quantum information processor.
A second problem with the CJJ rf-SQUID flux qubit is that one cannot homogenize the qubit parameters $\left|I_q^p\right|$ and $\Delta_q$ between a multitude of such devices that possess different $\beta_{\pm}$ over a broad range of $\Phi^x_{\text{cjj}}$. While one can accomplish this task to a limited degree in a perturbative manner about carefully chosen CJJ biases for each qubit \cite{synchronization}, the equivalence of $\left|I_q^p\right|$ and $\Delta_q$ between those qubits will be approximate at best. Therefore, the CJJ rf-SQUID does not provide a convenient means of accommodating fabrication variations between multiple flux qubits in a large scale processor.
Given that the CJJ rf-SQUID provides additional flexibility at a cost, it is by no means obvious that one can design a better rf-SQUID flux qubit by adding even more junctions. Specifically, it is desirable to have a device whose imperfections can be mitigated purely by the application of {\it time-independent} compensation signals. The novel rf-SQUID topology shown in Fig.~\ref{fig:rfssummary}c, hereafter referred to as the compound-compound Josephson junction (CCJJ) rf-SQUID, satisfies this latter constraint. Here, each junction of the CJJ in Fig.~\ref{fig:rfssummary}b has been replaced by a dc-SQUID, which will be referred to as left ($L$) and right ($R$) minor loops, and will be subjected to external flux biases $\Phi_L^x$ and $\Phi_R^x$, respectively. The role of the CJJ loop in Fig.~\ref{fig:rfssummary}b is now played by the CCJJ loop of inductance $L_{\text{ccjj}}$ which will be subjected to an external flux bias $\Phi^x_{\text{ccjj}}$. It is shown in Appendix B that if one chooses {\it static} values of $\Phi_L^x$ and $\Phi_R^x$ such that the net critical currents of the minor loops are equal, then it can be described by an effective two-dimensional Hamiltonian of the form
\begin{subequations}
\begin{equation}
\label{eqn:4JHeff}
{\cal H}=\sum_n\left[\frac{Q_n^2}{2C_n}+U_n\frac{\left(\varphi_n-\varphi_n^x\right)^2}{2}\right]-U_q\beta_{\text{eff}}\cos\left(\varphi_q-\varphi_q^0\right) \; ,
\end{equation}
\noindent where the sum is over $n\in\left\{q,\text{ccjj}\right\}$, $C_q\equiv C_1+C_2+C_3+C_4$, $1/C_{\text{ccjj}}\equiv 1/(C_1+C_2)+1/(C_3+C_4)$ and $L_q\equiv L_{\text{body}}+L_{\text{ccjj}}/4$. The effective 2-dimensional potential energy in Hamiltonian (\ref{eqn:4JHeff}) is characterized by
\begin{equation}
\label{eqn:4JBeffbalanced}
\beta_{\text{eff}}=\beta_+(\Phi^x_{L},\Phi^x_{R})\cos\left(\frac{\varphi_{\text{ccjj}}-\varphi^0_{\text{ccjj}}}{2}\right) \;\; ,
\end{equation}
\noindent where $\beta_+(\Phi^x_{L},\Phi^x_{R})=2\pi L_q I_q^c(\Phi^x_{L},\Phi^x_{R})/\Phi_0$ with
\begin{displaymath}
I_q^c(\Phi^x_{L},\Phi^x_{R})\equiv (I_1+I_2)\cos\left(\frac{\pi\Phi^x_{L}}{\Phi_0}\right)+(I_3+I_4)\cos\left(\frac{\pi\Phi^x_{R}}{\Phi_0}\right) \; .
\end{displaymath}
\noindent Given an appropriate choice of $\Phi^x_{L}$ and $\Phi^x_{R}$, the $q$ and ccjj loops will possess apparent flux offsets of the form
\begin{equation}
\label{eqn:qoffset}
\Phi_q^0=\frac{\Phi_0\varphi_q^0}{2\pi}=\frac{\Phi_{L}^0+\Phi_{R}^0}{2}\; ;
\end{equation}
\vspace{-12pt}
\begin{equation}
\label{eqn:ccjjoffset}
\Phi^0_{\text{ccjj}}=\frac{\Phi_0\varphi^0_{\text{ccjj}}}{2\pi}=\Phi_{L}^0-\Phi_{R}^0\; ,
\end{equation}
\end{subequations}
\noindent where $\Phi_{L(R)}^0$ is given by Eq.~(\ref{eqn:4JMinorOffset}), which is purely a function of $\Phi^x_{L(R)}$ and junction critical currents. As such, the apparent flux offsets are {\it independent} of $\Phi^x_{\text{ccjj}}$. Under such conditions, we deem the CCJJ to be {\it balanced}. Given that the intended mode of operation is to hold $\Phi_L^x$ and $\Phi_R^x$ constant, then the offset phases $\varphi_L^0$ and $\varphi_R^0$ will also be constant. The result is that Hamiltonian (\ref{eqn:4JHeff}) for the CCJJ rf-SQUID becomes homologous to that of an ideal CJJ rf-SQUID [$\beta_-=0$ in Eqs.~(\ref{eqn:2JBeff}) and (\ref{eqn:2JOffset})] with apparent {\it static} flux offsets. Such static offsets can readily be calibrated and compensated for in-situ using either analog control lines or on-chip programmable flux sources \cite{PCC}. For typical device parameters and junction variability on the order of a few percent, these offsets will be $\sim 1\rightarrow 10\,$m$\Phi_0$. Equations \ref{eqn:4JHeff}-\ref{eqn:ccjjoffset} with $\Phi_q^0=\Phi^0_{\text{ccjj}}=0$ will be referred to hereafter as the ideal CCJJ rf-SQUID model.
The second advantage of the CCJJ rf-SQUID is that one can readily accommodate for variations in critical current between multiple flux qubits. Note that in Eq.~(\ref{eqn:4JBeffbalanced}) that the maximum height of the tunnel barrier is governed by $\beta_+(\Phi^x_{L},\Phi^x_{R})\equiv\beta_L(\Phi^x_{L})+\beta_R(\Phi^x_{R})$, where $\beta_{L(R)}$ is given by Eq.~(\ref{eqn:4JMinorOffset}). One is free to choose any pair of $(\Phi^x_{L},\Phi^x_{R})$ such that $\beta_L(\Phi^x_{L})=\beta_R(\Phi^x_{R})$, as dictated by Eq.~(\ref{eqn:balancedapprox}). Consequently, $\beta_+=2\beta_R(\Phi^x_{R})$ in Eq.~(\ref{eqn:4JBeffbalanced}). One can then choose $\Phi^x_{R}$, which then dictates $\Phi^x_{L}$, so as to homogenize $\beta_+$ between multiple flux qubits. The results is a set of nominally uniform flux qubits where the particular choice of $(\Phi^x_{L},\Phi^x_{R})$ for each qubit merely results in unique static flux offsets $\Phi^0_q$ and $\Phi^0_{\text{ccjj}}$ for each device.
To summarize up to this point, the CCJJ rf-SQUID is robust against Josephson junction fabrication variations both within an individual rf-SQUID and between a plurality of such devices. The variations can be effectively tuned out purely through the application of {\it static} flux biases, which is of considerable advantage when envisioning the implementation of large scale quantum information processors that use flux qubits.
\subsection{$L$-tuner}
The purpose of the CCJJ structure was to provide a means of coming to terms with fabrication variations in Josephson junctions both within individual flux qubits and between sets of such devices. However, junctions are not the only key parameter that may vary between devices, nor are fabrication variations responsible for all of the potential variation. In particular, it has been experimentally demonstrated that the inductance of a qubit $L_q$ that is connected to other qubits via tunable mutual inductances is a function of the coupler configuration \cite{cjjcoupler}. Let the bare inductance of the qubit in the presence of no couplers be represented by $L_q^0$ and the mutual inductance between the qubit and coupler $i$ be represented by $M_{\text{co},i}$. If the coupler possesses a first order susceptibility $\chi_i$, as defined in Ref.~\onlinecite{cjjcoupler}, then the net inductance of the qubit can be expressed as
\begin{equation}
\label{eqn:LqNoTuner}
L_q=L_q^0-\sum_{i}M^2_{\text{co},i}\chi_i\; .
\end{equation}
\noindent Given that qubit properties such as $\Delta_q$ can be exponentially sensitive to variations in $L_q$, then it is undesirable to have variations in $L_q$ between multiple flux qubits or to have $L_q$ change during operation. This could have a deleterious impact upon AQO in which it is typically assumed that all qubits are identical and they are intended to be annealed in unison \cite{AQC}. From the perspective of GMQC, one could very well attempt to compensate for such effects in a CJJ or CCJJ rf-SQUID flux qubit by adjusting the tunnel barrier height to hold $\Delta_q$ constant, but doing so alters $\left|I_q^p\right|$, which then alters the coupling of the qubit to radiative sources, thus demanding further compensation. Consequently, it also makes sense from the perspective of GMQC that one find a means of rendering $L_q$ uniform between multiple qubits and insensitive to the settings of inductive coupling elements.
\begin{figure}
\includegraphics[width=2.5in]{LTuner.pdf}
\caption{\label{fig:LTuner} (color online) A CCJJ rf-SQUID with $L$-tuner connected to multiple tunable inductive couplers via transformers with mutual inductances $M_{\text{co},i}$ and possessing susceptibilities $\chi_i$. The $L$-tuner is controlled via the external flux bias $\Phi^x_{LT}$}
\end{figure}
In order to compensate for variations in $L_q$, we have inserted a tunable Josephson inductance \cite{vanDuzer} into the CCJJ rf-SQUID body, as depicted in Fig.~\ref{fig:LTuner}. We refer to this element as an inductance ($L$)-tuner. This relatively simple element comprises a dc-SQUID whose critical current vastly exceeds that of the CCJJ structure, thus ensuring negligible phase drop across the $L$-tuner. Assuming that the inductance of the $L$-tuner wiring is negligible, the $L$-tuner modifies Eq.~(\ref{eqn:LqNoTuner}) in the following manner:
\begin{equation}
\label{eqn:LqWithTuner}
L_q=L_q^0-\sum_{i}M^2_{\text{co},i}\chi_i + \frac{L_{J0}}{\cos(\pi\Phi^x_{LT}/\Phi_0)}\; ,
\end{equation}
\noindent where $L_{J0}\equiv\Phi_0/2\pi I^c_{LT}$, $I^c_{LT}$ is the net critical current of the two junctions in the $L$-tuner and $\Phi^x_{LT}$ is an externally applied flux bias threading the $L$-tuner loop. For modest flux biases such that $I^c_{LT}\cos(\pi\Phi^x_{LT}/\Phi_0)\gg I_q^c$, Eq.~(\ref{eqn:LqWithTuner}) is a reliable model of the physics of the $L$-tuner.
Given that the $L$-tuner is only capable of augmenting $L_q$, one can only choose to target $L_q>L_q^0-\sum_iM_{\text{co},i}^2\chi^{\text{AFM}}_i+L_{J0}$, where $\chi^{\text{AFM}}_i$ is the maximum antiferromagnetic (AFM) susceptibility of inter-qubit coupler $i$. In practice, we choose to restrict operation of the couplers to the range $-\chi_i^{\text{AFM}}<\chi_i<\chi_i^{\text{AFM}}$ such that the maximum qubit inductance that will be encountered is $L_q>L_q^0+\sum_iM_{\text{co},i}^2\chi^{\text{AFM}}_i+L_{J0}$. We then choose to prebias $\Phi^x_{\text{LT}}$ for each qubit to match the maximum realized $L_q\equiv L^{\text{max}}_q$ amongst a set of flux qubits. Thereafter, one can hold $L_q=L^{\text{max}}_q$ as couplers are adjusted by inverting Eq.~(\ref{eqn:LqWithTuner}) to solve for an appropriate value of $\Phi^x_{LT}$. Thus, the $L$-tuner provides a ready means of compensating for small variations in $L_q$ between flux qubits and to hold $L_q$ constant as inductive inter-qubit coupling elements are adjusted.
\section{Device Architecture, Fabrication and Readout Operation}
\begin{figure}
\includegraphics[width=3.25in]{Architecture_Schematic.pdf} \\
\includegraphics[width=3.25in]{Architecture_CrossSection.pdf} \\
\includegraphics[width=3.25in]{Architecture_OpticalQubit.pdf}
\caption{\label{fig:architecture} (color online) a) High level schematic of the analog devices on the device reported upon herein. Qubits are represented as light grey elongated objects and denoted as $q_0\ldots q_7$. One representative readout (RO), CCJJ and $L$-tuner ($LT$) each have been indicated in dashed boxes. Couplers (CO) are represented as dark objects located at the intersections of the qubit bodies. b) SEM of a cross-section of the fabrication profile. Metal layers denoted as BASE, WIRA, WIRB and WIRC. Insulating layers labeled as SiO$_2$. Topmost insulator has not been planarized in this test structure, but is planarized in the full circuit process. An example via (VIA), Josephson junction (JUNC, AlO$_x$/Al) and resistor (RESI) are also indicated. c) Optical image of a portion of a device completed up to WIRB. Portions of qubits $q_0\ldots q_3$ and the entirety of $q_4$ are visible.}
\end{figure}
To test the CCJJ rf-SQUID flux qubit, we fabricated a circuit containing 8 such devices with pairwise interactions mediated by a network of 16 in-situ tunable CJJ rf-SQUID inter-qubit couplers \cite{cjjcoupler}. Each qubit was also coupled to its own dedicated quantum flux parametron (QFP)-enabled readout \cite{QFP}. A high level schematic of the device architecture is shown in Fig.~\ref{fig:architecture}a. External flux biases were provided to target devices using a sparse combination of analog current bias lines to facilitate device calibration and an array of single flux quantum (SFQ) based on-chip programmable control circuitry (PCC) \cite{PCC}.
The device was fabricated from an oxidized Si wafer with Nb/Al/Al$_2$O$_3$/Nb trilayer junctions and four Nb wiring layers separated by planarized plasma enhanced chemical vapor deposited SiO$_{2}$. A scanning electron micrograph of the process cross-section is shown in Fig.~\ref{fig:architecture}b. The Nb metal layers have been labeled as BASE, WIRA, WIRB and WIRC. The flux qubit wiring was primarily located in WIRB and consisted of $2\,\mu$m wide leads arranged as an approximately $900\,\mu$m long differential microstrip located $200\,$nm above a groundplane in WIRA. CJJ rf-SQUID coupler wiring was primarily located in WIRC, stacked on top of the qubit wiring to provide inductive coupling. PCC flux storage loops were implemented as stacked spirals of 13-20 turns of $0.25\,\mu$m wide wiring with $0.25\,\mu$m separation in BASE and WIRA (WIRB). Stored flux was picked up by one-turn washers in WIRB (WIRA) and fed
into transformers for flux-biasing devices. External control lines were mostly located in
BASE and WIRA. All of these control elements resided below a groundplane in WIRC. The groundplane under the qubits and over the PCC/external control lines were electrically connected using extended vias in WIRB so as to form a nearly continuous superconducting shield between the analog devices on top and the bias circuitry below. To provide biases to target devices with minimal parasitic crosstalk, transformers for biasing qubits, couplers, QFPs and dc-SQUIDs using bias lines and/or PCC elements were enclosed in superconducting boxes with BASE and WIRC forming the top and bottom, respectively, and vertical walls formed by extended vias in WIRA and WIRB. Minimal sized openings were placed in the vertical walls through which the bias and target device wiring passed at opposing ends of each box.
An optical image of a portion of a device completed up to WIRB is shown in Fig.~\ref{fig:architecture}c. Qubits are visible as elongated objects, WIRB PCC spirals are visible as dark rectangles and WIRB washers are visible as light rectangles with slits. Note that the extrema of the CCJJ rf-SQUID qubits are terminated in unused transformers. These latter elements allow this 8-qubit unit cell to be tiled in a larger device with additional inter-qubit CJJ couplers providing the connections between unit cells.
\begin{figure}
\includegraphics[width=3.25in]{unipolarannealingwaveforms.pdf}
\caption{\label{fig:unipolarannealing} (color online) a) Schematic representation of a portion of the circuit reported upon herein. Canonical representations of all externally controlled flux biases $\Phi_{\alpha}^x$, readout current bias $i_{ro}$ and key mutual inductances $M_{\alpha}$ are indicated. b) Depiction of latching readout waveform sequence. c) Example QFP state population measurement as a function of the dc level $\Phi^x_{\text{qfp}}$ with no qubit signal. Data have been fit to Eq.~(\ref{eqn:transition}).}
\end{figure}
We have studied the properties of all 8 CCJJ rf-SQUID flux qubits on this chip in detail and report upon one such device herein. To clearly establish the lingua franca of our work, we have depicted a portion of the multi-qubit circuit in Fig.~\ref{fig:unipolarannealing}a. Canonical representations of the external flux biases needed to operate a qubit, a coupler and a QFP-enabled readout are labeled on the diagram. The fluxes $\Phi_L^x$, $\Phi_R^x$, $\Phi^x_{LT}$ and $\Phi_{\text{co}}^x$ were only ever subjected to dc levels in our experiments that were controlled by the PCC. The remaining fluxes and readout current biases were driven by a custom-built 128 channel room temperature current source. The mutual inductance between qubit and QFP ($M_{q-\text{qfp}}$), between QFP and dc-SQUID ($M_{\text{qfp-ro}}$), qubit and coupler ($M_{\text{co},i}$) and $\Phi^x_{\text{co}}$-dependent inter-qubit mutual inductance ($M_{\text{eff}}$) have also been indicated. Further details concerning cryogenics, magnetic shielding and signal filtering have been discussed in previous publications \cite{LOMRT,PCC,QFP,cjjcoupler}.
Since much of what follows depends upon a clear understanding of our novel QFP-enabled readout mechanism, we present a brief review of its operation herein. The flux and readout current waveform sequence involved in a single-shot readout is depicted in Fig.~\ref{fig:unipolarannealing}b. Much like the CJJ qubit \cite{LOMRT}, the QFP can be adiabatically {\it annealed} from a state with a monostable potential ($\Phi^x_{\text{latch}}=-\Phi_0/2$) to a state with a bistable potential ($\Phi^x_{\text{latch}}=-\Phi_0$) that supports two counter-circulating persistent current states. The matter of which persistent current state prevails at the end of an annealing cycle depends upon the sum of $\Phi^x_{\text{qfp}}$ and any signal from the qubit mediated via $M_{q-\text{qfp}}$. The state of the QFP is then determined with high fidelity using a synchronized flux pulse and current bias ramp applied to the dc-SQUID. The readout process was typically completed within a repetition time $t_{\text{rep}}<50\,\mu$s.
An example trace of the population of one of the QFP persistent current states $P_{\text{qfp}}$ versus $\Phi^x_{\text{qfp}}$, obtained using the latching sequence depicted in Fig.~\ref{fig:unipolarannealing}b, is shown in Fig.~\ref{fig:unipolarannealing}c. This trace was obtained with the qubit potential held monostable ($\Phi^x_{\text{ccjj}}=-\Phi_0/2$) such that it presented minimal flux to the QFP and would therefore not influence $P_{\text{qfp}}$. The data have been fit to the phenomenological form
\begin{equation}
\label{eqn:transition}
P_{\text{qfp}}=\frac{1}{2}\left[1-\tanh\left(\frac{\Phi^x_{\text{qfp}}-\Phi^0_{\text{qfp}}}{w}\right)\right]
\end{equation}
\noindent with width $w\sim 0.18\,$m$\Phi_0$ for the trace shown therein. When biased with constant $\Phi^x_{\text{qfp}}=\Phi^0_{\text{qfp}}$, which we refer to as the QFP degeneracy point, this transition in the population statistics can be used as a highly nonlinear flux amplifier for sensing the state of the qubit. Given that $M_{q-\text{qfp}}=6.28\pm0.01\,$pH for the devices reported upon herein and that typical qubit persistent currents in the presence of negligible tunneling $\left|I_q^p\right|\gtrsim 1\,\mu$A, then the net flux presented by a qubit was $2M_{q-\text{qfp}}\left|I_q^p\right|\gtrsim 6\,$m$\Phi_0$, which far exceeded $w$. By this means one can achieve the very high qubit state readout fidelity reported in Ref.~\onlinecite{QFP}. On the other hand, the QFP can be used as a linearized flux sensor by engaging $\Phi^x_{\text{qfp}}$ in a feedback loop and actively tracking $\Phi^0_{\text{qfp}}$. This latter mode of operation has been used extensively in obtaining many of the results presented herein.
\section{CCJJ rf-SQUID Characterization}
The purpose of this section is to present measurements that characterize the CCJJ, $L$-tuner and capacitance of a CCJJ rf-SQUID. All measurements shown herein have been made with a set of standard bias conditions given by $\Phi^x_{L}=98.4\,$m$\Phi_0$, $\Phi^x_{R}=-89.3\,$m$\Phi_0$, $\Phi^x_{\text{LT}}=0.344\,\Phi_0$ and all inter-qubit couplers tuned to provide $M_{\text{eff}}=0$, unless indicated otherwise. The logic behind this particular choice of bias conditions will be explained in what follows.
This section will begin with a description of the experimental methods for extracting $L_q$ and $I_q^c$ from persistent current measurements. Thereafter, data that demonstrate the performance of the CCJJ and $L$-tuner will be presented. Finally, this section will conclude with the determination of $C_q$ from macroscopic resonant tunneling data.
\subsection{High Precision Persistent Current Measurements}
The most direct means of obtaining information regarding a CCJJ rf-SQUID is to measure the persistent current $\left|I_q^p\right|$ as a function of $\Phi^x_{\text{ccjj}}$. A reasonable first approach to measuring this quantity would be to sequentially prepare the qubit in one of its persistent current states and then the other, and use the QFP in feedback mode to measure the difference in flux sensed by the QFP, which equals $2M_{q-\text{qfp}}\left|I_q^p\right|$. A fundamental problem with this approach is that it is sensitive to low frequency (LF) flux noise \cite{1OverF}, which can alter the background flux experienced by the QFP between the sequential measurements. For a typical measurement with our apparatus, the act of locating a single QFP degeneracy point to within $20\,\mu\Phi_0$ takes on the order of $1\,$s, which means that two sequential measurements would only be immune to flux noise below $0.5\,$Hz. We have devised a LF flux noise rejection scheme that takes advantage of the fact that such noise will generate a correlated shift in the apparent degeneracy points if the sequential preparations of the qubit can be interleaved with single-shot measurements that are performed in rapid succession. If these measurements are performed with repetition time $t_{\text{rep}}\sim 1\,$ms, then the measurements will be immune to flux noise below $\sim 1\,$kHz.
\begin{figure}
\includegraphics[width=3.25in]{iqplockinwaveforms.pdf}
\caption{\label{fig:iqplockin} (color online) a) Low frequency flux noise rejecting qubit persistent current measurement sequence. Waveforms shown are appropriate for measuring $\left|I_q^p\left(\Phi^x_{\text{ccjj}}\right)\right|$ for $-\Phi_0\leq\Phi^x_{\text{ccjj}}\leq 0$. The $\Phi^x_{\text{ccjj}}$ waveform can be offset by integer $\Phi_0$ to measure the periodic behavior of this quantity. Typical repetition time is $t_{\text{rep}}\sim 1\,$ms. b) Depiction of QFP transition and correlated changes in QFP population statistics for the two different qubit initializations.}
\end{figure}
A depiction of the LF flux noise rejecting persistent current measurement sequence is shown in Fig.~\ref{fig:iqplockin}a. The waveforms comprise two concatenated blocks of sequential annealing of the qubit to a target $\Phi^x_{\text{ccjj}}$ in the presence of an alternating polarizing flux bias $\pm\Phi_q^i$ followed by latching and single-shot readout of the QFP. The QFP flux bias is engaged in a differential feedback mode in which it is pulsed in alternating directions by an amount $\delta\Phi_m$ about a mean level $\Phi_m$. The two single-shot measurements yield binary results for the QFP state and the {\it difference} between the two binary results is recorded. Gathering a statistically large number of such differential measurements then yields a differential population measurement $\delta P_{\text{qfp}}$. Conceptually, the measurement works in the manner depicted in Fig.~\ref{fig:iqplockin}b: the two different initializations of the qubit move the QFP degeneracy point to some unknown levels $\Phi_m^0\pm\delta\Phi_m^0$, where $\Phi_m^0$ represents the true mean of the degeneracy points at any given instant in time and $2\delta\Phi_m^0$ is the true difference in degeneracy points that is independent of time. Focusing on flux biases that are close to the degeneracy point, one can linearize Eq.~(\ref{eqn:transition}):
\begin{equation}
\label{eqn:transitionlinear}
P_{\text{qfp},\pm}\approx\frac{1}{2}+\frac{1}{2w}\left[\Phi^x_{\text{qfp}}-\left(\Phi_m^0\pm\delta\Phi_m^0\right)\right] \; .
\end{equation}
\noindent Assuming that the rms LF flux noise $\Phi_n\ll w$ and that one has reasonable initial guesses for $\Phi_m^0\pm\delta\Phi_m^0$, then the use of the linear approximation should be justified. Applying $\Phi^x_{\text{qfp}}=\Phi_m\pm\delta\Phi_m$ and sufficient repetitions of the waveform pattern shown in Fig.~\ref{fig:iqplockin}a, the differential population will then be of the form
\begin{equation}
\label{eqn:diffpop}
\delta P_{\text{qfp}}=P_{\text{qfp},+}-P_{\text{qfp},-}=\frac{1}{w}\left[\delta\Phi_m+\delta\Phi_m^0\right]\; ,
\end{equation}
\noindent which is {\it independent} of $\Phi_m$ and $\Phi_m^0$. Note that the above expression contains only two independent variables, $w$ and $\delta\Phi_m^0$, and that $\delta P_{\text{qfp}}$ is purely a linear function of $\delta\Phi_m$. By sampling at three values of $\delta\Phi_m$, as depicted by the pairs of numbered points in Fig.~\ref{fig:iqplockin}b, the independent variables in Eq.~(\ref{eqn:diffpop}) will be overconstrained, thus readily yielding $\delta\Phi_m^0$. One can then infer the qubit persistent current as follows:
\begin{equation}
\label{eqn:iqplockin}
\left|I_q^p\right|=\frac{2\delta\Phi_m^0}{2M_{q-\text{qfp}}}=\frac{\delta\Phi_m^0}{M_{q-\text{qfp}}} \; .
\end{equation}
\begin{figure}[ht]
\includegraphics[width=3.25in]{QU1Lq0Extraction_ver2.pdf}
\caption{\label{fig:Lq0extraction} (color online) Example measurements of $\left|I_q^p\left(\Phi^x_{\text{ccjj}}\right)\right|$.}
\end{figure}
\noindent Example measurements of $\left|I_q^p\right|\left(\Phi^x_{\text{ccjj}}\right)$ are shown in Fig.~\ref{fig:Lq0extraction}. These data, for which $1.5\lesssim\left|\beta_{\text{eff}}\right|\lesssim 2.5$, have been fit to the ideal CCJJ rf-SQUID model by finding the value of $\varphi_q\equiv\varphi^{\text{min}}_q$ for which the potential in Eq.~(\ref{eqn:4JHeff}) is minimized:
\begin{equation}
\label{eqn:iqp2d}
\left|I_q^p\right|=\frac{\Phi_0}{2\pi}\frac{\left|\varphi^{\text{min}}_q-\varphi_q^x\right|}{L_q} \;\; .
\end{equation}
\noindent The best fit shown in Fig.~\ref{fig:Lq0extraction} was obtained with $L_q=265.4 \pm 1.0\,$pH, $L_{\text{ccjj}}=26\pm 1\,$pH and $I_q^c=3.103\pm 0.003\,\mu$A. For comparison, we had estimated $L_q=273\,$pH at the standard bias condition for $\Phi^x_{LT}$ and $L_{\text{ccjj}}=20\,$pH from design.
In practice, we have found that the LF flux noise rejecting method of measuring $\left|I_q^p\right|$ effectively eliminates any observable $1/f$ component in that measurement's noise power spectral density, to within statistical error. Finally, it should be noted that the LF flux noise rejecting method is applicable to any measurement of a difference in flux sensed by a linearized detector. In what follows herein, we have made liberal use of this technique to calibrate a variety of quantities in-situ using both QFPs and other qubits as flux detectors.
\subsection{CCJJ}
In this subsection, the CCJJ has been characterized as a function of $\Phi^x_{L}$ and $\Phi^x_{R}$ with all other static flux biases set to the standard bias condition cited above. Referring to Eq.~(\ref{eqn:4JQOffset}), it can be seen that the qubit degeneracy point $\Phi_q^0$ is a function of $\Phi^x_{\text{ccjj}}$ through $\gamma_0$ if the CCJJ has not been balanced. To accentuate this functional dependence, one can anneal the CCJJ rf-SQUID with $\Phi^x_{\text{ccjj}}$ waveforms of opposing polarity about a minimum in $\left|\beta_{\text{eff}}\right|$, as found at $\Phi^x_{\text{ccjj}}=-\Phi_0/2$. The expectation is that the {\it apparent} qubit degeneracy points will be antisymmetric about the mean given by setting $\gamma_0=0$ in Eq.~(\ref{eqn:4JQOffset}). The waveform sequence for performing a differential qubit degeneracy point measurement is depicted in Fig.~\ref{fig:bipolarlockin}. In this case, the QFP is used as a latching readout and the qubit acts as the linearized detector of its own apparent annealing polarization-dependent flux offset. As with the $\left|I_q^p\right|$ measurement described above, this LF flux noise rejecting procedure returns a {\it difference} in apparent flux sensed by the qubit and not the absolute flux offsets.
\begin{figure}[ht]
\includegraphics[width=3.25in]{bipolarannealingwaveforms.pdf}
\caption{\label{fig:bipolarlockin} (color online) Schematic of low frequency noise rejecting differential qubit degeneracy point measurement sequence. The qubit is annealed with a $\Phi^x_{\text{ccjj}}$ signal of opposing polarity in the two frames and the qubit flux bias is controlled via feedback.}
\end{figure}
To find balanced pairs of $\left(\Phi^x_{L},\Phi^x_{R}\right)$ in practice, we set $\Phi^x_{R}$ to a constant and used the LF flux noise rejecting procedure inside a software feedback loop that controlled $\Phi^x_{L}$ to null the difference in apparent degeneracy point to a precision of $20\,\mu\Phi_0$. Balanced pairs of $\left(\Phi^x_{L},\Phi^x_{R}\right)$ are plotted in Fig.~\ref{fig:balanced}a. These data have been fit to \ref{eqn:balancedapprox} using $\beta_-/\beta_+$ as a free parameter. The best fit shown in Fig.~\ref{fig:balanced}a was obtained with $1-\beta_{R,+}/\beta_{L,+}=(4.1\pm0.3)\times 10^{-3}$, which then indicates an approximately $0.4\%$ asymmetry between the pairs of junctions in the $L$ and $R$ loops.
\begin{figure}[ht]
\includegraphics[width=3.25in]{QU1MinorLobeBalancing.pdf} \\
\includegraphics[width=3.25in]{QU1BalancedIp_ver2.pdf} \\
\caption{\label{fig:balanced} (color online) a) Minor lobe balancing data and fit to Eq.~(\ref{eqn:balancedapprox}). The standard bias conditions for $\Phi^x_{L}$ and $\Phi^x_{R}$ are indicated by dashed lines. b) $\left|I_q^p(\Phi^x_{\text{ccjj}}=-\Phi_0)\right|$ versus $\Phi^x_{R}$, where $\Phi^x_{L}$ has been chosen using Eq.~(\ref{eqn:balancedapprox}). The data have been fit to the ideal CCJJ rf-SQUID model. The standard bias condition for $\Phi^x_{R}$ and the resultant $\left|I_q^p(\Phi^x_{\text{ccjj}}=-\Phi_0)\right|$ are indicated by dashed lines.}
\end{figure}
A demonstration of how the CCJJ facilitates tuning of $I_q^c$ is shown in Fig.~\ref{fig:balanced}b. Here, the measurable consequence of altering $I_q^c$ that was recorded was a change in $\left|I_q^p\right|$ at $\Phi^x_{\text{ccjj}}=-\Phi_0$. These data have been fit to the ideal CCJJ rf-SQUID model with the substitution
\begin{equation}
\label{eqn:Icbalanced}
I_q^c(\Phi^x_{R},\Phi^x_{L})=I_c^0\cos\left(\frac{\pi\Phi^x_{R}}{\Phi_0}\right)
\end{equation}
\noindent and using the values of $L_{\text{ccjj}}$ and $L_q$ obtained from fitting the data in Fig.~\ref{fig:Lq0extraction}, but treating $I_c^0$ as a free parameter. Here, $\Phi^x_{L}$ on the left side of Eq.~(\ref{eqn:Icbalanced}) is a function of $\Phi^x_{R}$ per the CCJJ balancing condition Eq.~(\ref{eqn:balancedapprox}). The best fit was obtained with $I_c^0=3.25\pm0.01\,\mu$A. This latter quantity agrees well with the designed critical current of four $0.6\,\mu$m diameter junctions in parallel of $3.56\;\mu$A. Thus, it is possible to target a desired $I_q^c$ by using Eq.~(\ref{eqn:Icbalanced}) to select $\Phi^x_{R}$ and then Eq.~(\ref{eqn:balancedapprox}) to select $\Phi^x_{L}$. The standard bias conditions for $\Phi^x_{L}$ and $\Phi^x_{R}$ quoted previously were chosen so as to homogenize $I_q^c$ amongst the 8 CCJJ rf-SQUIDs on this particular chip.
\subsection{$L$-Tuner}
To characterize the $L$-tuner, we once again turned to measurements of $\left|I_q^p(\Phi^x_{\text{ccjj}}=-\Phi_0)\right|$, but this time as a function of $\Phi^x_{LT}$. Persistent current results were then used to infer $\delta L_q=L_q(\Phi^x_{LT})-L_q(\Phi^x_{LT}=0)$ using the ideal CCJJ rf-SQUID model with $L_{\text{ccjj}}$ and $I_q^c$ held constant and treating $L_q$ as a free parameter. The experimental results are plotted in Fig.~\ref{fig:ltuner}a and have been fit to
\begin{equation}
\label{eqn:Ltunerfit}
\delta L_q=\frac{L_{J0}}{\cos\left(\pi\Phi^x_{LT}/\Phi_0\right)} \; ,
\end{equation}
\noindent and the best fit was obtained with $L_{J0}=19.60\pm0.04\,$pH. Modeling this latter parameter as $L_{q0}=\Phi_0/2\pi I^c_{LT}$, we estimate $I^c_{LT}=16.79\pm0.04\,\mu$A, which is close to the design value of $16.94\,\mu$A. The standard bias condition for $\Phi^x_{\text{LT}}$ was chosen so as to homogenize $L_q$ amongst the 8 CCJJ rf-SQUID flux qubits on this chip and to provide adequate bipolar range to accommodate inter-qubit coupler operation.
\begin{figure}[ht]
\includegraphics[width=3.25in]{QU1LTunerCalibration.pdf} \\
\includegraphics[width=3.25in]{QU1LTunerCompensationComparison_ver2.pdf} \\
\caption{\label{fig:ltuner} (color online) a) $L$-tuner calibration and fit to Eq.~(\ref{eqn:Ltunerfit}). The standard bias condition for $\Phi^x_{\text{LT}}$ and the resultant $\delta L_q$ are indicated by dashed lines. b) Observed change in maximum qubit persistent current with and without active $L$-tuner compensation and predictions for both cases.}
\end{figure}
To demonstrate the use of the $L$-tuner, we have probed a worst-case scenario in which four CJJ rf-SQUID couplers connected to the CCJJ rf-SQUID in question are tuned in unison. Each of the couplers had been independently calibrated per the procedures described in Ref.~\onlinecite{cjjcoupler}, from which we obtained $M_{\text{co},i}\approx 15.8\,$pH and $\chi_i\left(\Phi^x_{\text{co}}\right)$ ($i\in\left\{1,2,3,4\right\}$). Each of these devices provided a maximum AFM inter-qubit mutual inductance $M_{\text{AFM}}=M^2_{\text{co},i}\chi_{\text{AFM}}\approx 1.56\,$pH, from which one can estimate $\chi_{\text{AFM}}\approx 6.3\,$nH$^{-1}$. Measurements of $\left|I_q^p\right|$ with and without active $L$-tuner compensation as a function of coupler bias $\Phi^x_{\text{co}}$, as applied to all four couplers simultaneously, are presented in Fig.~\ref{fig:ltuner}b. The predictions from the ideal CCJJ rf-SQUID model, obtained by using $L_q=265.4\,\text{pH}$ (with compensation) and $L_q$ obtained from Eq.~(\ref{eqn:LqNoTuner}) (without compensation), are also shown. Note that the two data sets and predictions all agree to within experimental error at $\Phi^x_{\text{co}}=0.5\,\Phi_0$, which corresponds to the all zero coupling state ($M_{\text{eff}}=0$). The experimental results obtained without $L$-tuner compensation agree reasonably well with the predicted $\Phi^x_{\text{co}}$-dependence. As compared to the case without compensation, it can be seen that the measured $\left|I_q^p\right|$ show considerably less $\Phi^x_{\text{co}}$-dependence when $L$-tuner compensation is provided. However, the data suggest a small systematic deviation from the inductance models Eqs.~(\ref{eqn:LqNoTuner}) and (\ref{eqn:LqWithTuner}). At $\Phi^x_{\text{ccjj}}=-\Phi_0$, for which it is estimated that $\beta_{\text{eff}}\approx 2.43$, $\left|I_q^p\right|\propto 1/L_q$. Given that the data for the case without compensation are below the model, then it appears that we have slightly underestimated the change in $L_q$. Consequently, we have provided insufficient ballast inductance when the $L$-tuner compensation was activated.
\subsection{rf-SQUID Capacitance}
Since $I_q^c$ and $L_q$ directly impact the CCJJ rf-SQUID potential in Hamiltonian (\ref{eqn:4JHeff}), it was possible to infer CCJJ and $L$-tuner properties from measurements of the groundstate persistent current. In contrast, the rf-SQUID capacitance $C_q$ appears in the kinetic term in Hamiltonian (\ref{eqn:4JHeff}). Consequently, one must turn to alternate experimental methods that invoke excitations of the CCJJ rf-SQUID in order to characterize $C_q$. One such method is to probe macroscopic resonant tunneling (MRT) between the lowest lying state in one well into either the lowest order [LO, $n=0$] state or into a higher order [HO, $n>0$] state in the opposing well of the rf-SQUID double well potential \cite{HOMRT}. The spacing of successive HOMRT peaks as a function of rf-SQUID flux bias $\Phi^x_q$ will be particularly sensitive to $C_q$. HOMRT has been observed in many different rf-SQUIDs and is a well established quantum mechanical phenomenon \cite{HOMRT,Bennett,MRT3JJ}. LOMRT proved to be more difficult to observe in practice and was only reported upon relatively recently in the literature \cite{LOMRT}. We refer the reader to this latter reference for the experimental method for measuring MRT rates.
\begin{figure}
\includegraphics[width=3.25in]{QU1_HOMRT_Rate.pdf} \\
\includegraphics[width=3.25in]{QU1_HOMRT_GaussianWidth.pdf} \\
\includegraphics[width=3.25in]{QU1_HOMRT_PeakPosition.pdf}
\caption{\label{fig:HOMRT} (color online) a) HOMRT peaks fitted to Eq.~(\ref{eqn:HOMRTFit}). Data shown are for $\Phi^x_{\text{ccjj}}/\Phi_0=-0.6677$, $-0.6735$, $-0.6793$, $-0.6851$, $-0.6911$ and $-0.6970$, from left to right, respectively. Number of levels in target well $n$ as indicated. b) Best fit Gaussian width parameter $W_n$ as a function of $n$. c) Best fit peak position $\epsilon_p^n$ as a function of $n$.}
\end{figure}
Measurements of the initial decay rate $\Gamma\equiv dP_{\downarrow}/dt|_{t=0}$ versus $\Phi^x_q$ are shown in Fig.~\ref{fig:HOMRT}a with the order of the target level $n$ as indicated. The maximum observable $\Gamma$ was imposed by the bandwidth of the apparatus, which was $\sim 5\,$MHz. The minimum observable $\Gamma$ was dictated by experimental run time constraints. In order to observe many HO resonant peaks within our experimental bandwidth we have successively raised the tunnel barrier height in roughly equal intervals by tuning the target $\Phi^x_{\text{ccjj}}$. The result is a cascade of resonant peaks atop a monotonic background.
The authors of Ref.~\onlinecite{Bennett} attempted to fit their HOMRT data to a sum of gaussian broadened lorentzian peaks. It was found that they could obtain satisfactory fits within the vicinity of the tops of the resonant features but that the model was unable to correctly describe the valleys between peaks. We had reached the same conclusion with the very same model as applied to our data. However, it was empirically observed that we could obtain excellent fits to all of the data by using a model composed of a sum of purely gaussian peaks plus a background that varies exponentially with $\Phi^x_q$:
\begin{equation}
\label{eqn:HOMRTFit}
\frac{\Gamma(\Phi^x_q)}{\hbar}=\sum_{n}\sqrt{\frac{\pi}{8}}\frac{\Delta_n^2}{W_n}e^{-\frac{(\epsilon-\epsilon_p^n)^2}{2W_n^2}}+\Gamma_{\text{bkgd}}e^{\Phi^x_q/\delta\Phi_{\text{bkgd}}}\; ,
\end{equation}
\noindent where $\epsilon\equiv 2\left|I_q^p\right|\Phi^x_q$. These fits are shown in Fig.~\ref{fig:HOMRT}a. A summary of the gaussian width parameter $W_n$ in Fig.~\ref{fig:HOMRT}b is shown solely for informational purposes. We will refrain from speculating why there is no trace of lorentzian lineshapes or on the origins of the exponential background herein, but rather defer a detailed examination of HOMRT to a future publication.
For the purposes of this article, the key results to take from the fits shown in Fig.~\ref{fig:HOMRT}a are the positions of the resonant peaks, as plotted in Fig.~\ref{fig:HOMRT}c. These results indicate that the peak spacing is very uniform: $\delta\Phi_{\text{MRT}}=1.55\pm0.01\,$m$\Phi_0$. One can compare $\delta\Phi_{\text{MRT}}$ with the predictions of the ideal CCJJ rf-SQUID model using the previously calibrated $L_q=265.4\,$pH, $L_{\text{ccjj}}=26\,$pH and $I_q^c=3.103\,\mu$A with $C_q$ treated as a free parameter. From such a comparison, we estimate $C_q=190\pm 2\,$fF.
The relatively large value of $C_q$ quoted above can be reconciled with the CCJJ rf-SQUID design by noting that, unlike other rf-SQUID flux qubits reported upon in the literature, our qubit body resides proximal to a superconducting groundplane so as to minimize crosstalk. In this case, the qubit wiring can be viewed as a differential transmission line of length $\ell/2\sim 900\,\mu$m, where $\ell$ is the total length of qubit wiring, with the effective Josephson junction and a short on opposing ends. The transmission line will present an impedance of the form $Z(\omega)=-j Z_0\tanh(\omega\ell/2\nu)$ to the effective Josephson junction, with the phase velocity $\nu\equiv1/\sqrt{L_0C_0}$ defined by the differential inductance per unit length $L_0\sim 0.26\,$pH$/\mu$m and capacitance per unit length $C_0\sim 0.18\,$fF$/\mu$m, as estimated from design. If the separation between differential leads is greater than the distance to the groundplane, then $\ell/2\nu\approx\sqrt{L_{\text{body}}C_{\text{body}}/4}$, where $C_{\text{body}}\sim 640\,$fF is the total capacitance of the qubit wiring to ground. Thus, one can model the high frequency behavior of the shorted differential transmission line as an inductance $L_{\text{body}}$ and a capacitance $C_{\text{body}}/4$ connected in parallel with the CCJJ. Taking a reasonable estimated value of $40\,$fF/$\mu$m$^2$ for the capacitance per unit area of a Josephson junction, one can estimate the total capacitance of four $0.6\,\mu$m diameter junctions in parallel to be $C_J\sim 45\,$fF. Thus we estimate $C_q=C_J+C_{\text{body}}/4\sim 205\,$fF, which is in reasonable agreement with the best fit value of $C_q$ quoted above.
With all of the controls of the CCJJ rf-SQUID having been demonstrated, we reach the first key conclusion of this article: The CCJJ rf-SQUID is a robust device in that parametric variations, both within an individual device and between a multitude of such devices, can be accounted for using purely static flux biases. These biases have been applied to all 8 CCJJ rf-SQUIDs on this particular chip using a truly scalable architecture involving on-chip flux sources that are programmed by only a small number of address lines \cite{PCC}.
\section{Qubit Properties}
The purpose of the CCJJ rf-SQUID is to provide an as ideal as possible flux qubit \cite{fluxqubit}. By this statement, it is meant that the physics of the two lowest lying states of the device can be described by an effective Hamiltonian of the form Eq.~(\ref{eqn:Hqubit}) with $\epsilon=2\left|I_q^p\right|\left(\Phi_q^x-\Phi^0_q\right)$, $\left|I_q^p\right|$ being the magnitude of the persistent current that flows about the inductive loop when the device is biased hard to one side, $\Phi_q^0$ being a static flux offset and $\Delta_q$ representing the tunneling energy between the lowest lying states when biased at its degeneracy point $\Phi_q^x=\Phi^0_q$. Thus, $\left|I_q^p\right|$ and $\Delta_q$ are the defining properties of a flux qubit, regardless of its topology \cite{Leggett}. Given the complexity of a six junction device with five closed superconducting loops, it is quite justifiable to question whether the CCJJ rf-SQUID constitutes a qubit. These concerns will be directly addressed herein by demonstrating that measured $\left|I_q^p\right|$ and $\Delta_q$ agree with the predictions of the quantum mechanical Hamiltonian (\ref{eqn:4JHeff}) given independently calibrated values of $L_q$, $L_{\text{ccjj}}$, $I_q^c$ and $C_q$.
Before proceeding, it is worth providing some context in regards to the choice of experimental methods that have been described below. For those researchers attempting to implement GMQC using resonant electromagnetic fields to prepare states and mediate interactions between qubits, experiments that involve high frequency pulse sequences to drive excitations in the qubit (such as Rabi oscillations\cite{MooijSuperposition}, Ramsey fringes\cite{MooijSuperposition,1OverFFluxQubit1} and spin-echo\cite{MooijSuperposition,1OverFFluxQubit1,1OverFFluxQubit2}) are the natural modality for studying quantum effects. Such experiments are convenient in this case as the methods can be viewed as basic gate operations within this intended mode of operation. However, such methods are not the exclusive means of characterizing quantum resources. For those who wish to use precise dc pulses to implement GMQC or whose interests lie in developing hardware for AQO, it is far more convenient to have a set of tools for characterizing quantum mechanical properties that require only low bandwidth bias controls. Such methods, some appropriate in the coherent regime \cite{Greenberg,gsip} and others in the incoherent regime \cite{HOMRT,LOMRT,LZ}, have been reported in the literature. We have made use of such low frequency methods as our apparatuses typically possess 128 low bandwidth bias lines to facilitate the adiabatic manipulation of a large number of devices.
\begin{figure}
\includegraphics[width=3.25in]{QU1_LOMRT_ExampleTraces.pdf} \\
\includegraphics[width=3.25in]{QU1_LOMRT_FitParameters.pdf} \\
\caption{\label{fig:LOMRT} (color online) a) Example LOMRT peaks fitted to Eq.~(\ref{eqn:LOMRTFit}). Data shown are for $\Phi^x_{\text{ccjj}}/\Phi_0=-0.6621$, $-0.6642$ and $-0.6663$, from top to bottom, respectively. Data from the qubit initialized in $\ket{\downarrow}$ ($\ket{\uparrow}$) are indicated by solid (hollow) points. b) Energy scales obtained from fitting multiple LOMRT traces.}
\end{figure}
One possible means of probing quantum mechanical tunneling between the two lowest lying states of a CCJJ rf-SQUID is via MRT\cite{LOMRT}. Example LOMRT decay rate data are shown in Fig.~\ref{fig:LOMRT}a. We show results for both initializations, $\ket{\downarrow}$ and $\ket{\uparrow}$, and fits to gaussian peaks, as detailed in Ref.~\onlinecite{LOMRT}:
\begin{equation}
\label{eqn:LOMRTFit}
\frac{\Gamma(\Phi^x_q)}{\hbar}=\sqrt{\frac{\pi}{8}}\frac{\Delta_q^2}{W}e^{-\frac{(\epsilon-\epsilon_p)^2}{2W^2}} \;\; .
\end{equation}
\noindent A summary of the fit parameters $\epsilon_p$ and $W$ versus $\Phi^x_{\text{ccjj}}$ is shown in Fig.~\ref{fig:LOMRT}b. We also provide estimates of the device temperature using the formula
\begin{equation}
\label{eqn:TMRT}
k_BT_{\text{MRT}}=\frac{W^2}{2\epsilon_p} \; .
\end{equation}
\noindent As expected, $T_{\text{MRT}}$ shows no discernible $\Phi^x_{\text{ccjj}}$-dependence and is scattered about a mean value of $53\pm2\,$mK. A summary of $\Delta_q$ versus $\Phi^x_{\text{ccjj}}$ will be shown in conjunction with more experimental results at the end of this section. For further details concerning LOMRT, the reader is directed to Ref.~\onlinecite{LOMRT}.
A second possible means of probing $\Delta_q$ is via a Landau-Zener experiment \cite{LZ}. In principle, this method should be applicable in both the coherent and incoherent regime. In practice, we have found it only possible to probe the device to modestly larger $\Delta_q$ than we can reach via LOMRT purely due to the low bandwidth of our bias lines. Results from such experiments on the CCJJ rf-SQUID flux qubit will be summarized at the end of this section. We see no fundamental limitation that would prevent others with higher bandwidth apparatuses to explore the physics of the CJJ or CCJJ flux qubit at the crossover between the coherent and incoherent regimes using the Landau-Zener method.
In order to probe the qubit tunnel splitting in the coherent regime using low bandwidth bias lines, we have developed a new experimental procedure for sensing the expectation value of the qubit persistent current, similar in spirit to other techniques already reported in the literature \cite{gsip}. An unfortunate consequence of the choice of design parameters for our high fidelity QFP-enabled readout scheme is that the QFP is relatively strongly coupled to the qubit, thus limiting its utility as a detector when the qubit tunnel barrier is suppressed. One can circumvent this problem within our device architecture by tuning an inter-qubit coupler to a finite inductance and using a second qubit as a latching sensor, in much the same manner as a QFP. Consider two flux qubits coupled via a mutual inductance $M_{\text{eff}}$. The system Hamiltonian can then be modeled as
\begin{equation}
\label{eqn:H2Q}
{\cal H}=-\sum_{i\in\left\{q,d\right\}}\frac{1}{2}\left[\epsilon_i\sigma_z^{(i)}+\Delta_i\sigma_x^{(i)}\right]+J\sigma_z^{(q)}\sigma_z^{(d)} \; ,
\end{equation}
\noindent where $J\equiv M_{\text{eff}}|I_q^p||I_d^p|$. Let qubit $q$ be the flux source and qubit $d$ serve the role of the detector whose tunnel barrier is adiabatically raised during the course of a measurement, just as in a QFP single shot measurement depicted in Fig.~\ref{fig:unipolarannealing}. In the limit $\Delta_d\rightarrow 0$ one can write analytic expressions for the dispersion of the four lowest energies of Hamiltonian (\ref{eqn:H2Q}):
\begin{equation}
\label{eqn:E2Q}
\begin{array}{ccc}
E_{1\pm} & = & \pm\frac{1}{2}\sqrt{\left(\epsilon_q-2J\right)^2+\Delta_1^2}-\frac{1}{2}\epsilon_d \; ;\\
E_{2\pm} & = & \pm\frac{1}{2}\sqrt{\left(\epsilon_q+2J\right)^2+\Delta_1^2}+\frac{1}{2}\epsilon_d \; .
\end{array}
\end{equation}
\noindent As with the QFP, let the flux bias of the detector qubit be engaged in a feedback loop to track its degeneracy point where $P_{d,\downarrow}=1/2$. Assuming Boltzmann statistics for the thermal occupation of the four levels given by Eq.~(\ref{eqn:E2Q}), this condition is met when
\begin{equation}
\label{eqn:P2minus}
P_{d,\downarrow}=\frac{1}{2}=\frac{e^{-E_{2-}/k_BT}+e^{-E_{2+}/k_BT}}{\sum_{\alpha\in\left\{1\pm,2\pm\right\}}e^{-E_{\alpha}/k_BT}} \; .
\end{equation}
\noindent Setting $P_{d,\downarrow}=1/2$ in Eq.~(\ref{eqn:P2minus}) and solving for $\epsilon_2$ then yields an analytic formula for the balancing condition:
\begin{equation}
\label{eqn:HalfCondGeneral}
\epsilon_d= \frac{F(+)-F(-)}{2}+k_BT\ln\left(\frac{1+e^{-F(+)/k_BT}}{1+e^{-F(-)/k_BT}}\right) \; ;
\end{equation}
\vspace{-0.12in}
\begin{displaymath}
F(\pm)\equiv\sqrt{\left(\epsilon_q\pm 2J\right)^2+\Delta_1^2} \; .
\end{displaymath}
While Eq.~(\ref{eqn:HalfCondGeneral}) may look unfamiliar, it readily reduces to an intuitive result in the limit of small coupling $J\ll \Delta_1$ and $T\rightarrow 0$:
\begin{equation}
\label{eqn:HalfCondSmallJ}
\epsilon_d \approx M_{\text{eff}}|I_q^p|\frac{\epsilon_q}{\sqrt{\epsilon_q^2+\Delta_q^2}} = M_{\text{eff}}\bra{g}\hat{I}_q^p\ket{g} \; ,
\end{equation}
\noindent where $\ket{g}$ denotes the groundstate of the source qubit and $\hat{I}_q^p\equiv\left|I_q^p\right|\sigma_z^{(q)}$ is the source qubit persistent current operator. Thus Eq.~(\ref{eqn:HalfCondGeneral}) is an expression for the expectation value of the source qubit's groundstate persistent current in the presence of backaction from the detector and finite temperature. Setting $\epsilon_i=2|I_i^p|\Phi^x_{i}$ and rearranging then gives an expression for the flux bias of the detector qubit as a function of flux bias applied to the source qubit. Given independent calibrations of $M_{\text{eff}}=1.56\pm 0.01\,$pH for a particular coupler set to $\Phi^x_{\text{co}}=0$ on this chip, $T=54\pm 3\,$mK from LOMRT fits and $|I_d^p|=1.25\pm0.02\,\mu$A at the CCJJ bias where the LOMRT rate approaches the bandwidth of our bias lines, one can then envision tracing out $\Phi_d^x$ versus $\Phi_q^x$ and fitting to Eq.~(\ref{eqn:HalfCondGeneral}) to extract the source qubit parameters $|I_q^p|$ and $\Delta_q$ .
\begin{figure}
\includegraphics[width=3.25in]{QU1LargeDeltaTraceExample.pdf}
\caption{\label{fig:largedeltatrace} (color online) Example coupled flux trace taken at $\Phi^x_{\text{ccjj}}=-0.6513\,\Phi_0$ used to extract large $\Delta$ in the coherent regime. }
\end{figure}
An example $\Phi_d^x$ versus $\Phi_q^x$ data set for source CCJJ flux bias $\Phi^x_{\text{ccjj}}=-0.6513\,\Phi_0$ is shown in Fig.~\ref{fig:largedeltatrace}. The solid curve in this plot corresponds to a fit to Eq.~(\ref{eqn:HalfCondGeneral}) with a small background slope that we denote as $\chi$. We have confirmed from the ideal CCJJ rf-SQUID model that $\chi$ is due to the diamagnetic response of the source rf-SQUID to changing $\Phi_q^x$. This feature becomes more pronounced with increasing $C_q$ and is peaked at the value of $\Phi^x_{\text{ccjj}}$ for which the source qubit potential becomes monostable, $\beta_{\text{eff}}=1$. Nonetheless, the model also indicates that $\chi$ in no way modifies the dynamics of the rf-SQUID, thus the qubit model still applies. From fitting these particular data, we obtained $|I_q^p|=0.72\pm 0.04\,\mu$A and $\Delta_q/h=2.64\pm 0.24\,$GHz.
\begin{figure}
\includegraphics[width=3.25in]{deltawaveforms.pdf}
\caption{\label{fig:deltawaveforms} (color online) Depiction of large $\Delta_q$ measurement waveforms. The waveform sequence is similar to that of Fig.~\ref{fig:iqplockin}, albeit the source qubit's tunnel barrier is partially suppressed ($-\Phi_0/2<\Phi^x_{\text{ccjj}}<-\Phi_0$) and a second qubit (as opposed to a QFP) serves as the flux detector.}
\end{figure}
In practice we have found it inefficient to take detailed traces of $\Phi_d^x$ versus $\Phi_q^x$ as this procedure is susceptible to corruption by LF flux noise in the detector qubit. As an alternative approach, we have adapted the LF flux noise rejecting procedures introduced in the last section of this article to measure a series of three differential flux levels in the detector qubit. The waveforms needed to accomplish this task are depicted in Fig.~\ref{fig:deltawaveforms}. Here, the dc-SQUID and QFP connected to the detector qubit are used in latching readout mode while the detector qubit is annealed in the presence of a differential flux bias $\Phi_m\pm\delta\Phi_m$ which is controlled via feedback. Meanwhile, the source qubit's CCJJ bias is pulsed to an intermediate level $-\Phi_0<\Phi^x_{\text{ccjj}}<-\Phi_0/2$ in the presence of an initialization flux bias $\pm\Phi_q^i$. By choosing two appropriate pairs of levels $\pm\Phi_q^i$, as indicated by the solid points $1\pm$ and $2\pm$ in Fig.~\ref{fig:largedeltatrace}, one can extract $\left|I_q^p\right|$ and $\chi$ from the two differential flux measurements. In order to extract $\Delta_q$, we then choose a pair of $\pm\Phi_q^i$ in the centre of the trace, as indicated by the solid points $3\pm$, from which we obtain the central slope $d\Phi_d^x/d\Phi_q^x$. Taking the first derivative of Eq.~(\ref{eqn:HalfCondGeneral}) and evaluating at $\Phi_q^x=0$ yields
\begin{equation}
\label{eqn:centralslope}
\frac{d\Phi_d^x}{d\Phi_q^x}-\chi=\frac{2M_{\text{eff}}\left|I_q^p\right|^2}{\sqrt{\left(2J\right)^2+\Delta_q^2}}\tanh\left[\frac{\sqrt{\left(2J\right)^2+\Delta_q^2}}{2k_bT}\right] \; .
\end{equation}
\noindent Given independent estimates of all other parameters, one can then extract $\Delta_q$ from this final differential flux measurement.
\begin{figure}
\includegraphics[width=3.25in]{QU1IpSummary_ver6.pdf} \\
\includegraphics[width=3.25in]{QU1DeltaSummary_ver6.pdf}
\caption{\label{fig:DeltaAndIp} (color online) a) Magnitude of the persistent current $\left|I_q^p\right|$ as a function of $\Phi^x_{\text{ccjj}}$. b) Tunneling energy $\Delta_q$ between two lowest lying states of the CCJJ rf-SQUID as a function of $\Phi^x_{\text{ccjj}}$, as characterized by macroscopic resonant tunneling [MRT] and Landau-Zener [LZ] in the incoherent regime and coupled groundstate persistent current ($\bra{g}\hat{I}_q^p\ket{g}$) in the coherent regime. Solid curves are the predictions of the ideal CCJJ rf-SQUID model using independently calibrated $L_q$, $L_{\text{ccjj}}$, $I_q^c$ and $C_q$ with no free parameters.}
\end{figure}
A summary of experimental values of the qubit parameters $\left|I_q^p\right|$ and $\Delta_q$ versus $\Phi^x_{\text{ccjj}}$ is shown in Fig.~\ref{fig:DeltaAndIp}. Here, we have taken $\Delta_q$ from LOMRT and Landau-Zener experiments in the incoherent regime and from the LF flux noise rejecting persistent current procedure discussed above in the coherent regime. The large gap between the three sets of measurements is due to two reasons: First, the relatively low bandwidth of our bias lines does not allow us to perform MRT or Landau-Zener measurements at higher $\Delta_q$ where the dynamics are faster. Second, while the coherent regime method worked for $\Delta_q>k_BT$, it proved difficult to reliably extract $\Delta_q$ in the opposite limit. As such, we cannot make any precise statements regarding the value of $\Phi^x_{\text{ccjj}}$ which serves as the delineation between the coherent and incoherent regimes based upon the data shown in Fig.~\ref{fig:DeltaAndIp}b. Regulating the device at lower temperature would assist in extending the utility of the coherent regime method to lower $\Delta_q$. On the other hand, given that Eq.~(\ref{eqn:LOMRTFit}) predicts that $\Gamma\propto\Delta_q^2$, one would have to augment the experimental bandwidth by at least two orders of magnitude to gain one order of magnitude in $\Delta_q$ via either MRT or LZ experiments.
The solid curves in Fig.~\ref{fig:DeltaAndIp} were generated with the ideal CCJJ rf-SQUID model using the independently calibrated $L_q=265.4\,$pH, $L_{\text{ccjj}}=26\,$pH, $I_q^c=3.103\,\mu$A and $C_q=190\,$fF. Note that there are no free parameters. It can be seen that the agreement between theory and experiment is quite reasonable. Thus we reach the second key conclusion of this article: The CCJJ rf-SQUID can be identified as a flux qubit as the measured $\left|I_q^p\right|$ and $\Delta_q$ agree with the predictions of a quantum mechanical Hamiltonian whose parameters were independently calibrated.
\section{Noise}
With the identification of the CCJJ rf-SQUID as a flux qubit firmly established, we now turn to assessing the relative quality of this device in comparison to other flux qubits reported upon in the literature. In this section, we present measurements of the low frequency flux and critical current spectral noise densities, $S_{\Phi}(f)$ and $S_{I}(f)$, respectively. Finally, we provide explicit links between $S_{\Phi}(f)$ and the free induction (Ramsey) decay time $T^*_{2}$ that would be relevant were this flux qubit to be used as an element in a gate model quantum information processor.
\subsection{Flux Noise}
\begin{figure}
\includegraphics[width=3.25in]{QU1FluxNoise_ver1.pdf}
\caption{\label{fig:fluxnoise} (color online) Low frequency flux noise in the CCJJ rf-SQUID flux qubit. Data [points] have been fit to Eq.~(\ref{eqn:1OverF}) [solid curve].}
\end{figure}
Low frequency ($1/f$) flux noise is ubiquitous in superconducting devices and is considered a serious impediment to the development of large scale solid state quantum information processors \cite{1OverF}. We have performed systematic studies of this property using a large number of flux qubits of varying geometry \cite{1OverFGeometry} and, more recently, as a function of materials and fabrication parameters. These latter studies have aided in the reduction of the amplitude of $1/f$ flux noise in our devices and will be the subject of a forthcoming publication. Using the methods described in Ref.~\onlinecite{1OverFGeometry}, we have generated the one-sided flux noise power spectral density $S_{\Phi}(f)$ shown in Fig.~\ref{fig:fluxnoise}. These data have been fit to the generic form
\begin{equation}
\label{eqn:1OverF}
S(f)=\frac{A^2}{f^{\alpha}}+w_n\; ,
\end{equation}
\noindent with best fit parameters $\alpha=0.95\pm 0.05$, $\sqrt{w_n}=9.7\pm 0.5\,\mu\Phi_0/\sqrt{\text{Hz}}$ and amplitude $A$ such that $\sqrt{S_{\Phi}(1\,\text{Hz})}=1.3^{+0.7}_{-0.5}\,\mu\Phi_0/\sqrt{\text{Hz}}$. Thus we reach the third key conclusion of this article: We have demonstrated that it is possible to achieve $1/f$ flux noise levels with Nb wiring that are as low as the best Al wire qubits reported in the literature \cite{1OverF,1OverFFluxQubit1,1OverFFluxQubit2}. Moreover, we have measured similar spectra from a large number of identical flux qubits, both on the same and different chips, and can state with confidence that the $1/f$ amplitude reported herein is reproducible. Given the experimentally observed geometric scaling of $S_{\Phi}(1\,\text{Hz})$ in Ref.~\onlinecite{1OverFGeometry} and the relatively large size of our flux qubit bodies, we consider the prospects of observing even lower $1/f$ noise in smaller flux qubits from our fabrication facility to be very promising.
\subsection{Critical Current Noise}
A second noise metric of note is the critical current noise spectral density $S_I(f)$. This quantity has been studied extensively and a detailed comparison of experimental results is presented in Ref.~\onlinecite{vanHarlingen}. A recent study of the temperature and geometric dependence of critical current noise has been published in Ref.~\onlinecite{NewCriticalCurrentNoise}. Based upon Eq.~(18) of Ref.~\onlinecite{vanHarlingen}, we estimate that the $1/f$ critical current noise from a single $0.6\,\mu$m diameter junction, as found in the CCJJ rf-SQUID flux qubit, will have an amplitude such that $\sqrt{S_I(1\,\text{Hz})}\sim 0.2\,$pA$/\sqrt{\text{Hz}}$. Unfortunately, we were unable to directly measure critical current noise in the flux qubit. While the QFP-enable readout provided high fidelity qubit state discrimination when qubits are fully annealed to $\Phi^x_{\text{ccjj}}=-\Phi_0$, this readout mechanism simply lacked the sensitivity required for performing high resolution critical current noise measurements. In lieu of a measurement of $S_I(f)$ from a qubit, we have characterized this quantity for the dc-SQUID connected to the qubit in question. The dc-SQUID had two $0.6\,\mu$m junctions connected in parallel. A time trace of the calibrated switching current $I_{\text{sw}}\approx I_c$ was obtained by repeating the waveform sequence depicted in Fig.~\ref{fig:unipolarannealing}b except with $\Phi^x_{\text{latch}}=-\Phi_0/2$ at all time (QFP disabled, minimum persistent current) and $\Phi^x_{\text{ro}}=0$ to provide minimum sensitivity to flux noise. Assuming that the critical current noise from each junction is uncorrelated, the best that we could establish was an upper bound of $\sqrt{S_I(1\,\text{Hz})}\lesssim 7\,$pA$/\sqrt{\text{Hz}}$ for a single $0.6\,\mu$m diameter junction.
Given the upper bound cited above for critical current noise from a single junction, we now turn to assessing the relative impact of this quantity upon the CCJJ rf-SQUID flux qubit. It is shown in Appendix B that fluctuations in the critical currents of the individual junctions of a CCJJ generate apparent flux noise in the flux qubit by modulating $\Phi_q^0$. Inserting critical current fluctuations of magnitude $\delta I_c\lesssim 7\,$pA$/\sqrt{\text{Hz}}$ and a mean junction critical current $I_c=I_q^c/4\sim 0.8\,\mu$A into Eq.~(\ref{eqn:4JOffsetFluctuation}) yields qubit degeneracy point fluctuations $\left|\delta\Phi_q^0\right|\lesssim 0.1\,\mu\Phi_0/\sqrt{\text{Hz}}$. This final result is at least one order of magnitude smaller than the amplitude of $1/f$ flux noise inferred from the data in Fig.~\ref{fig:fluxnoise}. As such, we consider the effects of critical current noise in the CCJJ rf-SQUID to be tolerable.
\subsection{Estimation of $T^*_{2}$}
While measurements of noise power spectral densities are the most direct way of reporting upon and comparing between different qubits, our research group is frequently asked what is the dephasing time for our flux qubits. The answer presumably depends very strongly upon bias settings, for recall that we have measured properties of the CCJJ rf-SQUID flux qubit in both the coherent and incoherent regime. Given that our apparatuses contain only low bandwidth bias lines for enabling AQO, we are unable to measure dephasing within our own laboratory. Collaborative efforts to measure dephasing for our flux qubits are in progress. In the meantime, we provide a rough estimate below for our flux qubits if they were biased to the optimal point, $\Phi^x_q=\Phi_q^0$ based upon the measured $S_{\Phi}(f)$ and subjected to a free induction decay, or Ramsey fringe, experiment. Referring to Eq.~(33a) of Ref.~\onlinecite{Martinis} and key results from Ref.~\onlinecite{Schnirman}, the mean squared phase noise for a flux qubit at the optimal point will be given by
\begin{equation}
\label{eqn:dephasing1}
\left<\phi_n^2(t)\right>=\frac{1}{\hbar^2}\frac{\left(2\left|I_q^p\right|\right)^4}{2\Delta^2}\int^{\Delta/h}_{f_m}\! df S_{\Phi^2}(f)\frac{\sin^2(\pi f t)}{(\pi f)^2} \; ,
\end{equation}
\noindent where $S_{\Phi^2}(\omega)$ represents the quadratic flux noise spectral density and $f_m$ is the measurement cutoff frequency. Assuming that the first order spectral density $S_{\Phi}(\omega)=2\pi A^2/\omega$, then $S_{\Phi^2}(\omega)$ can be written as
\begin{eqnarray}
\label{eqn:sphisquared}
S_{\Phi^2}(\omega) & = & \frac{1}{2\pi}\int\! dt e^{-i\omega t}\left<\Phi_n^2(t) \Phi_n^2(0)\right> \nonumber\\
& = & \frac{1}{2\pi}\int\! dt e^{-i\omega t} \int d\omega^{\prime}\frac{2\pi A^2}{\omega^{\prime}}\int d\omega^{\prime\prime}\frac{2\pi A^2}{\omega^{\prime\prime}} \nonumber\\
& = & 8\pi^2 A^4\frac{\ln\left(\omega/\omega_{\text{ir}}\right)}{\omega} \; ,
\end{eqnarray}
\noindent where $\omega_{\text{ir}}\equiv 2\pi f_{\text{ir}}$ denotes an infrared cutoff of the $1/f$ noise spectral density. Inserting Eq.~(\ref{eqn:sphisquared}) into Eq.~(\ref{eqn:dephasing1}) and rendering the integral dimensionless then yields:
\begin{equation}
\label{eqn:dephasing2}
\left<\phi_n^2(t)\right>=\frac{t^2}{\hbar^2}\frac{\left(2\left|I_q^p\right| A\right)^4}{\pi\Delta^2}\int^{\Delta t/h}_{f_{\text{min}}t}\! dx \frac{\ln\left(x/f_{\text{ir}}t\right)\sin^2(\pi x)}{x^3} \; ,
\end{equation}
\noindent where $f_{\text{min}}=\max\left[\begin{array}{cc} f_m & f_{\text{ir}}\end{array}\right]$. We have numerically studied the behavior of the integral in Eq.~(\ref{eqn:dephasing2}). In the very long measurement time limit the integral is cut off by $f_{\text{ir}}$ and the integral varies as $1/t^2$, which then cancels the factor of $t^2$ in the numerator of Eq.~(\ref{eqn:dephasing2}). This means that the mean squared phase noise eventually reaches a finite limit. However, the more experimentally relevant limit is $f_m\gg f_{\text{ir}}$ , for which we found empirically that the integral varies roughly as $5\times\left[\ln\left(f_m/f_{\text{ir}}\right)\right]^2$ over many orders of magnitude in the argument of the logarithm. In this latter limit the result is independent of $t$, so Eq.~(\ref{eqn:dephasing2}) can be rewritten as $\left<\phi_n^2(t)\right>=t^2/(T^*_{2})^2$, which then yields the following formula for $T^*_{2}$:
\begin{equation}
\label{eqn:Tphi}
T^*_{2}\approx\left[\frac{1}{\hbar^2}\frac{\left(2\left|I_q^p\right| A\right)^4}{\pi\Delta^2}5\ln\left(f_m/f_{\text{ir}}\right)\right]^{-1/2} \; .
\end{equation}
Since flux noise spectra seem to obey the $1/f$ form down to at least $0.1\,$mHz and researchers are generally concerned with dephasing over times of order $1\,\mu$s, then it is fair to consider $f_m/f_{\text{ir}}\sim 10^{10}$. For a nominal value of $\Phi^x_{\text{ccjj}}$ such that the flux qubit is in the coherent regime, say $-0.652\,\Phi_0$, the qubit parameters are $\Delta_q/h\approx 2\,$GHz and $\left|I_q^p\right|\approx 0.7\,\mu$A. Substituting these quantities into Eq.~(\ref{eqn:dephasing2}) then yields $T^*_{2}\sim 150\,$ns. This estimate of the dephasing time is comparable to that observed in considerably smaller flux qubits with comparable $1/f$ flux noise levels \cite{1OverFFluxQubit1,1OverFFluxQubit2}.
\section{Conclusions}
One can draw three key conclusions from the work presented herein: First, the CCJJ rf-SQUID is a robust and scalable device in that it allows for in-situ correction for parametric variations in Josephson junction critical currents and device inductance, both within and between flux qubits using only static flux biases. Second, the measured flux qubit properties, namely the persistent current $\left|I_q^p\right|$ and tunneling energy $\Delta_q$, agree with the predictions of a quantum mechanical Hamiltonian whose parameters have been independently calibrated, thus justifying the identification of this device as a flux qubit. Third, it has been experimentally demonstrated that the low frequency flux noise in this all Nb wiring flux qubit is comparable to the best all Al wiring devices reported upon in the literature. Taken in summation, these three conclusions represent a significant step forward in the development of useful large scale superconducting quantum information processors.
We thank J.~Hilton, P.~Spear, A.~Tcaciuc, F.~Cioata, M.~Amin, F.~Brito, D.~Averin, A.~Kleinsasser and G.~Kerber for useful discussions. Siyuan Han was supported in part by NSF Grant No. DMR-0325551.
\begin{appendix}
\section{CJJ rf-SQUID}
Let the qubit and cjj loop phases be defined as
\begin{subequations}
\begin{equation}
\varphi_q\equiv\left(\varphi_1+\varphi_2\right)/2 \; ,
\end{equation}
\begin{equation}
\varphi_{\text{cjj}}\equiv\varphi_1-\varphi_2 \; ,
\end{equation}
\end{subequations}
respectively. Furthermore, assume that the CJJ loop has an inductance $L_{\text{cjj}}$ that is divided symmetrically between the two paths. Using trigonometric relations, one can write a Hamiltonian for this system in terms of modes in the $q$ and cjj loops that has the following form:
\begin{subequations}
\begin{eqnarray}
\label{eqn:2J2DH}
{\cal H} & = & \sum_n\left[\frac{Q_n^2}{2C_n}+U_n\frac{(\varphi_n-\varphi_n^x)^2}{2}\right] \nonumber\\
& & -U_q\beta_+\cos\left(\frac{\varphi_{\text{cjj}}}{2}\right)\cos\left(\varphi_q\right) \nonumber\\
& & +U_q\beta_-\sin\left(\frac{\varphi_{\text{cjj}}}{2}\right)\sin\left(\varphi_q\right) \; ;
\end{eqnarray}
\vspace{-12pt}
\begin{equation}
\beta_{\pm}=\frac{2\pi L_q\left(I_{1}\pm I_{2}\right)}{\Phi_0} \; ,
\end{equation}
\end{subequations}
\noindent where the sum is over $n\in\left\{q,\text{cjj}\right\}$, $C_q\equiv C_1+C_2$, $1/C_{\text{cjj}}\equiv 1/C_1+1/C_2$, $U_n\equiv (\Phi_0/2\pi)^2/L_n$, $L_q\equiv L_{\text{body}}+L_{\text{cjj}}/4$ and $[\Phi_0\varphi_n/2\pi,Q_n]=i\hbar$. The Josephson potential energy of Hamiltonian (\ref{eqn:2J2DH}) can be rearranged by defining an angle $\theta$ such that $\tan\theta=(\beta_-/\beta_+)\tan\left(\varphi_{\text{cjj}}/2\right)$. Further trigonometric manipulation then yields Eqs.~(\ref{eqn:2JHeff})-(\ref{eqn:2Jbetapm}).
\section{CCJJ rf-SQUID}
Following the same logic as for the CJJ rf-SQUID, one can define four orthogonal quantum mechanical degrees of freedom as follows:
\begin{subequations}
\begin{eqnarray}
\label{eqn:ccjjphases}
\varphi_L & \equiv & \varphi_1-\varphi_2 \; ;\\
\varphi_R & \equiv & \varphi_3-\varphi_4 \; ;\\
\varphi_{\text{ccjj}} & \equiv & \varphi_{\ell}-\varphi_r=\frac{\varphi_1+\varphi_2}{2}-\frac{\varphi_3+\varphi_4}{2} \; ;\\
\varphi_q & \equiv & \frac{\varphi_{\ell}+\varphi_r}{2}=\frac{\varphi_1+\varphi_2+\varphi_3+\varphi_4}{4}\; .
\end{eqnarray}
\end{subequations}
\noindent Using the same strategy as in Appendix A, one can use trigonometric identities to first express the Josephson potential in terms of the $L$ and $R$ loop modes:
\begin{subequations}
\begin{eqnarray}
\label{eqn:4J4DH}
{\cal H} & = & \sum_n\frac{Q_n^2}{2C_n}+\sum_mU_m\frac{(\varphi_m-\varphi_m^x)^2}{2} \nonumber\\
& & -U_q\beta_{L+}\cos\left(\frac{\varphi_L}{2}\right)\cos\left(\varphi_{\ell}\right) \nonumber\\
& & +U_q\beta_{L-}\sin\left(\frac{\varphi_L}{2}\right)\sin\left(\varphi_{\ell}\right) \nonumber \\
& & -U_q\beta_{R+}\cos\left(\frac{\varphi_R}{2}\right)\cos\left(\varphi_{r}\right) \nonumber\\
& & +U_q\beta_{R-}\sin\left(\frac{\varphi_R}{2}\right)\sin\left(\varphi_{r}\right) \; ;
\end{eqnarray}
\vspace{-12pt}
\begin{equation}
\label{eqn:betalrpm}
\beta_{L(R),\pm}\equiv\frac{2\pi L_q\left(I_{1(3)}\pm I_{2(4)}\right)}{\Phi_0} \; ,
\end{equation}
\end{subequations}
\noindent where the first sum is over $n\in\left\{L,R,\ell,r\right\}$ and the second sum is over closed inductive loops $m\in\left\{L,R,\text{ccjj},q\right\}$. As before, each of the modes obey the commutation relation $[\Phi_0\varphi_n/2\pi,Q_n]=i\hbar$. Here, $1/C_{L(R)}=1/C_{1(3)}+1/C_{2(4)}$, $C_{\ell(r)}=C_{1(3)}+C_{2(4)}$ and $U_m=(\Phi_0/2\pi)^2/L_m$.
We have found it adequate for our work to assume that $L_{L,R}/L_q\ll 1$, which then allows one to reduce the four dimensional system given in Hamiltonian (\ref{eqn:4J4DH}) to two dimensions. Consequently, we will substitute $\varphi_{L(R)}=\varphi^x_{L(R)}$ and ignore the $L$ and $R$ kinetic terms henceforth. Assuming that the inductance of the ccjj loop is divided equally between the two branches one can then write $L_q=L_{\text{body}}+L_{\text{ccjj}}/4$. With these approximations and the $\theta$ strategy presented in Appendix A, one can rearrange the Josephson potential terms to yield the following:
\begin{subequations}
\begin{eqnarray}
\label{eqn:4J2DHver1}
{\cal H} & = & \sum_n\frac{Q_n^2}{2C_n}+\sum_mU_m\frac{(\varphi_m-\varphi_m^x)^2}{2} \nonumber\\
& & -U_q\beta_{L}\cos\left(\varphi_{\ell}-\varphi_L^0\right) \nonumber\\
& & -U_q\beta_{R}\cos\left(\varphi_{r}-\varphi_R^0\right) \; ;
\end{eqnarray}
\vspace{-12pt}
\begin{eqnarray}
\label{eqn:betalr}
\beta_{L(R)} & = & \beta_{L(R),+}\cos\left(\frac{\varphi_{L(R)}^x}{2}\right) \\
& & \times\sqrt{1+\left[\frac{\beta_{L(R),-}}{\beta_{L(R),+}}\tan\left(\frac{\varphi_{L(R)}^x}{2}\right)\right]^2} \; ; \nonumber
\end{eqnarray}
\vspace{-12pt}
\begin{equation}
\label{eqn:4JMinorOffset}
\varphi_{L(R)}^0
=-\arctan\left(\frac{\beta_{L(R),-}}{\beta_{L(R),+}}\tan(\varphi_{L(R)}^x/2)\right)\; ,
\end{equation}
\end{subequations}
\noindent where the first sum is over $n\in\left\{\ell,r\right\}$ and the second sum is over $m\in\left\{\text{ccjj},q\right\}$. The Josephson potential is given by a sum of two cosines, as encountered in the CJJ rf-SQUID derivation of Hamiltonian (\ref{eqn:2J2DH}) from Hamiltonian (\ref{eqn:Hphase}). These two terms can be rewritten in the same manner by defining $\beta_{\pm}=\beta_L\pm\beta_R$. The result, similar to Hamiltonian (\ref{eqn:2J2DH}), can then be subjected to the $\theta$ strategy to yield
\begin{subequations}
\begin{eqnarray}
\label{eqn:4J2DHver2}
{\cal H} & = & \sum_n\left[\frac{Q_n^2}{2C_n}+U_n\frac{(\varphi_n-\varphi_n^x)^2}{2}\right] \nonumber\\
& & -U_q\beta_{\text{eff}}\cos\left(\varphi_q-\varphi_q^0\right) \; ,
\end{eqnarray}
\noindent where the sum is over $n\in\left\{q,\text{ccjj}\right\}$ and the capacitances are defined as $C_q=C_1+C_2+C_3+C_4$ and $1/C_{\text{ccjj}}=1/(C_1+C_2)+1/(C_3+C_4)$. The other parameters are defined as
\begin{equation}
\label{eqn:4JBeff}
\beta_{\text{eff}}=\beta_+\cos\left(\frac{\gamma}{2}\right)\sqrt{1+\left[\frac{\beta_-}{\beta_+}\tan\left(\frac{\gamma}{2}\right)\right]^2} \; ;
\end{equation}
\vspace{-12pt}
\begin{equation}
\label{eqn:4JQOffset}
\varphi_q^0=\frac{\varphi_L^0+\varphi_R^0}{2}+\gamma_0 \; ;
\end{equation}
\vspace{-12pt}
\begin{equation}
\label{eqn:4JCCJJPhase}
\gamma\equiv\varphi_{\text{ccjj}}-\left(\varphi_L^0-\varphi_R^0\right) \; ;
\end{equation}
\vspace{-12pt}
\begin{equation}
\label{eqn:4JCCJJMonkey}
\gamma_0\equiv -\arctan\left(\frac{\beta_-}{\beta_+}\tan(\gamma/2)\right) \; ;
\end{equation}
\begin{equation}
\label{eqn:betaccjjpm}
\beta_{\pm}\equiv \beta_L\pm\beta_R \; .
\end{equation}
\end{subequations}
Hamiltonian (\ref{eqn:4J2DHver2}) inherits much of its complexity from junction asymmetry both within the minor loops, which gives rise to $\varphi_{L(R)}^0$, and effective junction asymmetry between the minor loops, which gives rise to $\gamma_0$. For arbitrary external flux biases and nominal spread in junction critical current, the CCJJ rf-SQUID offers no obvious advantage over the CJJ rf-SQUID. However, upon choosing biases $\Phi_L^x$ and $\Phi_R^x$ such that
\begin{equation}
\label{eqn:balanced}
\beta_L=\beta_R \;\; ,
\end{equation}
\noindent then $\beta_-=0$, and consequently $\gamma_0=0$. With these substitutions, Hamiltonian (\ref{eqn:4J2DHver2}) yields Hamiltonian (\ref{eqn:4JHeff}). Note that for $\beta_{L(R),-}/\beta_{L(R),+}\ll 1$ and modest $\Phi^x_{L(R)}$ that the so-called CCJJ balancing condition given by Eqs.~(\ref{eqn:betalr}) and (\ref{eqn:balanced}) can be written approximately as
\begin{displaymath}
\beta_{L,+}\cos\left(\frac{\varphi_L^x}{2}\right) \approx \beta_{R,+}\cos\left(\frac{\varphi_R^x}{2}\right) \; ,
\end{displaymath}
\noindent which, upon solving for $\varphi_L^x$ yields
\begin{equation}
\label{eqn:balancedapprox}
\Phi^x_{L} = \frac{2\pi}{\Phi_0}\arccos\left[\frac{\beta_{R,+}}{\beta_{L,+}}\cos\left(\frac{\pi\Phi^x_{R}}{\Phi_0}\right)\right] \; .
\end{equation}
It is possible for critical current noise to couple into the $\varphi_q$ degree of freedom in any compound junction rf-SQUID qubit via modulation of the junction asymmetry-dependent apparent qubit flux offset $\Phi_q^0$. In the case of the CCJJ rf-SQUID, all three quantities on the right side of Eq.~(\ref{eqn:4JQOffset}) are ultimately related to the critical currents of the individual junctions. Given typical junction parameter spreads from our fabrication facility,
\begin{displaymath}
\left|\frac{\beta_{L(R),-}}{\beta_{L(R),+}}\right|=\left|\frac{I_{1(3)}-I_{2(4)}}{I_{1(3)}+I_{2(4)}}\right|\sim {\cal O}(0.01) \; ,
\end{displaymath}
\noindent so one can write an approximate expression for $\varphi^0_{L(R)}$ using Eq.~(\ref{eqn:4JMinorOffset}):
\begin{eqnarray}
\label{eqn:4JMinorOffsetApprox}
\varphi^0_{L(R)} & \approx & -\frac{I_{1(3)}-I_{2(4)}}{I_{1(3)}+I_{2(4)}}\tan\left(\frac{\varphi^x_{L(R)}}{2}\right) \nonumber\\
& \approx & -\frac{I_{1(3)}-I_{2(4)}}{2I_c}\tan\left(\frac{\varphi^x_{L(R)}}{2}\right) \; ,
\end{eqnarray}
\noindent and for $\gamma_0$ using Eqs.~(\ref{eqn:betalr}), (\ref{eqn:4JCCJJPhase}) and (\ref{eqn:4JCCJJMonkey}):
\begin{eqnarray}
\label{eqn:4JCCJJMonkeyApprox}
\gamma_0 & \approx & \frac{(I_3+I_4)\cos\left(\frac{\varphi_R^x}{2}\right)-(I_1+I_2)\cos\left(\frac{\varphi_L^x}{2}\right)}{(I_1+I_2)\cos\left(\frac{\varphi_L^x}{2}\right)+(I_3+I_4)\cos\left(\frac{\varphi_R^x}{2}\right)}\tan\left(\frac{\gamma}{2}\right) \nonumber\\
& \approx & \frac{(I_3+I_4)\cos\left(\frac{\varphi_R^x}{2}\right)-(I_1+I_2)\cos\left(\frac{\varphi_L^x}{2}\right)}{2I_c\left[\cos\left(\frac{\varphi_L^x}{2}\right)+\cos\left(\frac{\varphi_R^x}{2}\right)\right]}\tan\left(\frac{\gamma}{2}\right) , \nonumber\\
& &
\end{eqnarray}
\noindent where $I_c$ represents the mean critical current of a single junction. The CCJJ rf-SQUID is intended to be operated with only small flux biases in the minor loops, thus $\cos\left(\frac{\varphi_L^x}{2}\right)\approx\cos\left(\frac{\varphi_R^x}{2}\right)\approx 1$. It is also reasonable to assume that $\gamma\approx\varphi^x_{\text{ccjj}}$
as the corrections to $\tan(\gamma/2)$ from $\varphi^0_{L(R)}$ and from the effective two-dimensionality of the rf-SQUID potential will be very small. Inserting Eqs.~\ref{eqn:4JMinorOffsetApprox} and \ref{eqn:4JCCJJMonkeyApprox} into Eq.~(\ref{eqn:4JQOffset}) then yields
\begin{eqnarray}
\label{eqn:4JOffsetApprox}
\varphi^0_q & \approx & -\frac{I_1}{2I_c}\left[\tan\left(\frac{\varphi_L^x}{2}\right)+\frac{1}{2}\tan\left(\frac{\varphi^x_{\text{ccjj}}}{2}\right)\right] \nonumber\\
& & -\frac{I_2}{2I_c}\left[-\tan\left(\frac{\varphi_L^x}{2}\right)+\frac{1}{2}\tan\left(\frac{\varphi^x_{\text{ccjj}}}{2}\right)\right] \nonumber\\
& & -\frac{I_3}{2I_c}\left[\tan\left(\frac{\varphi_R^x}{2}\right)-\frac{1}{2}\tan\left(\frac{\varphi^x_{\text{ccjj}}}{2}\right)\right] \nonumber\\
& & -\frac{I_4}{2I_c}\left[-\tan\left(\frac{\varphi_R^x}{2}\right)-\frac{1}{2}\tan\left(\frac{\varphi^x_{\text{ccjj}}}{2}\right)\right] \; .
\end{eqnarray}
For the typical operating parameters described in this article, $\Phi^x_{L(R)}/\Phi_0\sim0.1$ and the device acts as a qubit for $\Phi^x_{\text{ccjj}}/\Phi_0\sim 0.65$. For these flux biases, the magnitude of the terms within the square braces in Eq.~(\ref{eqn:4JOffsetApprox}) are all of order 1. Therefore, for general flux bias conditions, the apparent qubit flux offset is roughly given by
\begin{displaymath}
\Phi_q^0 \approx -\frac{\Phi_0}{4\pi}\frac{(I_1+I_2)-(I_3+I_4)}{I_c} \; .
\end{displaymath}
\noindent Assume that each junction experiences critical current fluctuations of magnitude $\delta I_c$. If each junction's fluctuations are independent,
then the root mean square variation of the qubit degeneracy point $\left|\delta\Phi_q^0\right|$ will be
\begin{equation}
\label{eqn:4JOffsetFluctuation}
\left|\delta\Phi_q^0\right| \approx \frac{\Phi_0}{2\pi}\frac{\delta I_c}{I_c} \; .
\end{equation}
\noindent Thus, critical current fluctuations generate apparent flux noise in the CCJJ rf-SQUID flux qubit.
\end{appendix}
|
2,877,628,089,029 | arxiv | \section{Introduction}
In the study of random interface growth, universality is a ubiquitous topic. Informally, universality says that random growth models with similar physical properties will have identical behavior at long--time asymptotics. In particular, different models in the same universality class are expected to have the same growth exponents and limiting distributions. In this sense, the classical central limit theorem is a universality statement, where the growth exponent is $1/2$ and the limiting distribution is Gaussian, regardless of the distribution of each summand.
A different universality class, called the Kardar-Parisi-Zhang (KPZ) universality class (introduced in \cite{KPZ}), models a variety of real--world growth processes, such as turbulent liquid crystals \cite{TS} and bacteria colony growth \cite{WIMM}. If $h(\vec{x},t)$ is the height of the interface at location $\vec{x}$ and time $t$, then it satisfies the stochastic differential equation
$$
\frac{\partial h}{\partial t}=\nu \nabla^2 h + \frac{\lambda}{2} (\nabla h)^2 + \eta(\vec{x},t),
$$
where $\eta(\vec{x},t)$ is space--time white noise. Due to the non--linearity, however, this stochastic differential equation is not well--defined. A common mathematical approach has been to study exactly solvable models (i.e. where the finite--time probability distributions can be computed exactly) in the universality class, and then to take the long--time limits. Examples of such models include random matrix theory \cite{TW2}, the PNG droplet \cite{PS}, ASEP \cite{TW}, non--intersecting Brownian motions \cite{ABK}, and random partitions \cite{kn:BK0}. In all of these models, the growth exponent is $1/3$ and the limiting distribution is called the Airy process, demonstrating the universality of the KPZ equation. More recently, there have also been mathematically rigorous interpretations of a solution to the KPZ equation \cite{ACQ,H}.
The universality class considered in this paper is called anisotropic Kardar-Parisi-Zhang (AKPZ) with a wall. It is a variant of KPZ in two ways: there is anisotropy and the substrate acts as a reflecting barrier. As before, the stochastic differential equation is not well--defined, so we take the approach of analyzing exactly solvable models. So far, there have only been two models which have been proven to be in this universality class: a randomly growing stepped surface in $2+1$ dimensions \cite{kn:BK} and non--intersecting squared Bessel paths \cite{KMW}. In both cases, the limiting behavior near the critical point of the barrier has growth exponent $1/4$ and limiting process the Symmetric Pearcey process (defined in section \ref{SPK}).
The exactly solvable model considered here was introduced in \cite{kn:D}. It is a discrete-time interacting particle system with a wall which evolves according to geometric jumps with a parameter $q\in [0,1)$. In the $q\rightarrow 1$ limit, this model also has connections to a random matrix model. The main result of this paper is that in the long--time asymptotics near the wall, the symmetric Pearcey process appears after rescaling by $N^{1/4}$. This therefore helps to establish the universality of the growth exponent $1/4$ and the Symmetric Pearcey process in the AKPZ with a wall universality class. The approach is to show that when projected to a single level and to (finite) integer times, the particle system is identical to a previously studied family of determinantal point process. Taking asymptotics of the correlation kernel then yields the desired results.
We will also show that away from the critical point and at finite distances from the wall, the discrete Jacobi kernel appears in the long--time asymptotics. This kernel also appeared in the long--time limit in \cite{kn:BK}, but has not appeared anywhere else. In particular, it did not appear in non--intersecting squared Bessel paths \cite{KMW0}.
In section \ref{2}, we review the particle system from \cite{kn:D} and the determinantal point processes from \cite{kn:BK}. In section \ref{3}, we compute the correlation kernel for the particle system on one level by showing that the two processes are identical. In section \ref{4}, we take the large-time asymptotics.
The models in \cite{kn:D} and \cite{kn:BK} have connections to the representation theory of the orthogonal groups, but this paper is intended to be understandable without knowledge of representation theory.
It should also be true that given the initial conditions, the fixed-time distributions for the two models are identical without needing to restrict to a single level, but this is not pursued here.
\textbf{Acknowledgements}. The author would like to thank Alexei Borodin, Manon Defosseux, Ivan Corwin and the referees for helpful comments.
\section{Two Models}\label{2}
\subsection{Interacting Particle System}\label{IPS}
The interacting particle system in \cite{kn:D} arises from a Pieri-type formula for the (finite-dimensional) orthogonal groups. Here, we briefly describe the model.
The particles live on the lattice\footnote{$\mathbb{N}$ denotes the non--negative integers and $\mathbb{Z}_+$ denotes the positive integers.} $\mathbb{N}\times\mathbb{Z}_+$. The horizontal line $\mathbb{N}\times\{k\}$ is often called the $k$th level. There are always $\lfloor \tfrac{k+1}{2} \rfloor$ particles on the $k$th level, whose positions at time $n$ will be denoted $X^k_1(n)\geq X^k_2(n)\geq X^k_3(n)\geq \ldots\geq X^k_{\lfloor (k+1)/2 \rfloor}(n)\geq 0$. The time can take integer or half--integer values. For convenience of notation, $X^k(n)$ will denote $(X^k_1(n), X^k_2(n), X^k_3(n), \ldots, X^k_{\lfloor (k+1)/2 \rfloor}(n))$ $\in\mathbb{N}^{\lfloor (k+1)/2 \rfloor}$. More than one particle may occupy a lattice point. The particles must satisfy the \textit{interlacing property}
\begin{equation}\label{Interlacing}
X^{k+1}_{i+1}(n) \leq X^k_i(n) \leq X^{k+1}_i(n)
\end{equation}
for all meaningful values of $k$ and $i$. This will be denoted $X^k\prec X^{k+1}$. With this notation, the state space can be described as the set of all sequences $(X^1\prec X^2 \prec \ldots)$ where each $X^k\in \mathbb{N}^{\lfloor (k+1)/2 \rfloor}$. The initial condition is $X_i^k(0)=0$, called the \textit{densely packed} initial conditions. Now let us describe the dynamics.
For $n\geq 0,k\geq 1$ and $1\leq i \leq \lfloor \tfrac{k+1}{2} \rfloor$, define random variables
\[
\xi^k_i(n+1/2), \ \ \xi^k_i(n)
\]
which are independent identically distributed geometric random variables with parameter $q$. In other words, $\mathbb{P}(\xi^1_1(1/2)=x)=q^x(1-q)$ for $x\in\mathbb{N}$. Let $R(x,y)$ be a Markov kernel on $\mathbb{N}$ defined by
\[
R(x,y)= \frac{1-q}{1+q}\cdot\frac{q^{\vert x-y\vert}+q^{x+y}}{1+1_{y=0}},
\]
so that $R(x,\cdot)$ is the law of the random variable $\vert x+\xi_1^1(1)-\xi_1^1(\tfrac{1}{2})\vert$.
At time $n$, all the particles except $X^k_{({k+1})/{2}}(n)$ try to jump to the left one after another in such a way that the interlacing property is preserved. The particles $X^k_{({k+1})/{2}}(n)$ do not jump on their own. The precise definition is
\begin{eqnarray*}
X^k_{(k+1)/{2}}(n+\tfrac{1}{2})&=&\min(X^k_{(k+1)/{2}}(n), X_{(k-1)/2}^{k-1}(n+\tfrac{1}{2}))\ \ k \text{ odd}\\
X^k_i(n+\tfrac{1}{2})&=& \max(X^{k-1}_i(n), \min(X^k_i(n),X^{k-1}_{i-1}(n+\tfrac{1}{2})) - \xi^k_i(n+\tfrac{1}{2})),
\end{eqnarray*}
where $X^{k-1}_0(n+\tfrac{1}{2})$ is formally set to $+\infty$.
At time $n+\tfrac{1}{2}$, all the particles except $X^k_{({k+1})/{2}}(n+\tfrac{1}{2})$ try to jump to the right one after another in such a way that the interlacing property is preserved. The particles $X^k_{({k+1})/{2}}(n+\tfrac{1}{2})$ jump according to the law $R$. The precise definition is
\[
X^k_{(k+1)/{2}}(n+1)=\min(\vert X^k_{(k+1)/{2}}(n) + \xi^k_{(k+1)/2}(n+1) - \xi^k_{(k+1)/2}(n+\tfrac{1}{2}) \vert, X^{k-1}_{(k-1)/{2}}(n) )
\]
when $k$ is odd and
\[
X^k_i(n+1)= \min(X^{k-1}_{i-1}(n+\tfrac{1}{2}), \max(X^k_i(n+\tfrac{1}{2}),X^{k-1}_{i}(n+1)) + \xi^k_i(n+1)),
\]
where $X^{k-1}_0(n+1)$ is formally set to $+\infty$.
Let us explain the particle system. The particles preserve the interlacing property in two ways: by pushing particles above it, and being blocked by particles below it. So, for example, in the left jumps, the expression $\min(X^k_i(n),X^{k-1}_{i-1}(n+\tfrac{1}{2}))$ represents the location of the particle after it has been pushed by a particle below and to the right. Then the particle attempts to jump to the left, so the term $\xi^k_i(n+\tfrac{1}{2})$ is subtracted. However, the particle may be blocked a particle below and to the left, so we must take the maximum with $X^{k-1}_i(n)$.
While $X_i^k(n)$ is not simple, applying the shift $\tilde{X}_i^k(n)=X_i^k(n) + \lfloor\tfrac{k+1}{2}\rfloor -i$ yields a simple process. In other words, $\tilde{X}$ can only have one particle at each location.
Figure \ref{Jumping} shows an example of $\tilde{X}$. Additionally, an interactive animation can be found at \url{http://www.math.harvard.edu/~jkuan/DiscreteTimeWithAWall.html}
By drawing lozenges around the particles as in Figure \ref{3D}, one can see that the particle system can be interpreted as a two--dimensional stepped surface. This can be made rigorous by defining the height function at a point to be the number of particles to the right of that point. With this interpretation, the jumping of the particles corresponds to adding and removing sticks, and therefore the interacting particle system is equivalent to a randomly growing surface. The anisotropy is shown with the observation that only sticks of one type may be added or removed. The necessity of the interlacing condition is also visually apparent: it guarantees that the lozenges can be drawn in a way to make the figure three--dimensional.
\begin{center}
\begin{figure}
\caption{The figure on the left shows lozenges corresponding to the top figure in Figure \ref{Jumping}. The top right figure shows sticks that are never added or removed with each jump, while the bottom right figure shows sticks that are added or removed.}
\label{3D}
\includegraphics[height=2in]{ThreeDim.png}
\quad \quad \quad \quad
\includegraphics[height=2in]{Sticks2.png}
\end{figure}
\end{center}
\subsection{Determinantal Point Processes}\label{MC}
In \cite{kn:BK}, the authors introduce a family of determinantal point processes, indexed by a time parameter $n\in\mathbb{N}$, which arise from representations of the infinite-dimensional orthogonal group. (See \cite{kn:B} for background on determinantal point processes.) This family depends on a function $\phi\in C^1[-1,1]$. Each determinantal point process also lives on the lattice $\mathbb{N}\times\mathbb{Z}_+$, with exactly $\lfloor \tfrac{k+1}{2}\rfloor$ particles on the $k$th level, and the particles must also satisfy the interlacing property.
\textbf{Remark on notation.} It is convenient to re-label the levels. For $a=\pm 1/2$, one should think of $(r,a)$ as corresponding to the $2r+a+\tfrac{1}{2}$ level. Throughout this paper, the letter $k$ will denote the level and the letter $r$ will denote the number of particles. Set $\mathbb{J}_r$ to be the set of nonincreasing sequences of integers $(\lambda_1\geq\ldots\geq\lambda_r\geq 0)$. The superscript $\lambda^{(k)}$ will mean that $\lambda^{(k)}$ lives on the $k$th level. To save space, bold greek letters such as $\boldsymbol{\lambda}$ will denote
$\boldsymbol{\lambda} = (\lambda^{(1)}\prec\lambda^{(2)}\prec\ldots\prec\lambda^{(k)})$,
and similarly for $\boldsymbol{X}$, and $\mathbf{0}$ will denote the densely packed initial conditions.
Let $\boldsymbol{Y}(n)$ denote the positions of the particles in this determinantal point process at time $n$. Proposition 3.11 of \cite{kn:BK} establishes that there is a Markov chain\footnote{Strictly speaking, Proposition 3.11 proves \eqref{MP} without showing that $T^{\phi}$ has non--negative entries. A better term would be ``signed Markov chain,'' but this is not standard terminology. In any case, Proposition \ref{Prop} below will show that for the $\phi$ studied in this paper, $T^{\phi}$ is a bona--fide Markov chain.} $T^{\phi}$ connecting $\boldsymbol{Y}(n)$, in the sense that
\begin{equation}\label{MP}
\mathbb{P}(\boldsymbol{Y}(n+1)=\boldsymbol{\mu}) = \sum_{\boldsymbol{\lambda}} \mathbb{P}(\boldsymbol{Y}(n)=\boldsymbol{\lambda}) T^{\phi}(\boldsymbol{\lambda,\mu}).
\end{equation}
Now let us give the formula for $T^{\phi}$.
Let $\mathsf{J}_s^{(a,b)}(x)$ denote the (normalized) $s$-th Jacobi polynomial with parameters $a,b$. These are polynomials of degree $s$ which are orthogonal with respect to the measure $(1-x)^a(1+x)^bdx$ on $[-1,1]$. In this paper, we just need the equations
\begin{align*}
\mathsf{J}_s^{(1/2,-1/2)}\left(\frac{z+z^{-1}}{2}\right) =& \frac{z^{s+1/2}-z^{-s-1/2}}{z^{1/2}-z^{-1/2}},\\
\mathsf{J}_s^{(-1/2,-1/2)}\left(\frac{z+z^{-1}}{2}\right) =& \frac{z^s+z^{-s}}{2}.
\end{align*}
Also define
\[
W^{(a,b)}(s)=
\begin{cases}
2,\ \ \text{if}\ \ s>0,a=b=-\frac{1}{2}\\
1,\ \ \text{if}\ \ s=0,a=b=-\frac{1}{2}\\
1,\ \ \text{if}\ \ s\geq 0,a=\frac{1}{2},b=-\frac{1}{2}
\end{cases}
\]
For a function $\phi\in C^1[-1,1]$, define
\[
I_a^{\phi}(l,s)=\frac{W^{(a,-1/2)}(s)}{\pi}\int_{-1}^1 \mathsf{J}_s^{(a,-1/2)}(x) \mathsf{J}_l^{(a,-1/2)}(x) \phi(x) (1-x)^a(1+x)^{-1/2}dx.
\]
For $a=\pm\tfrac{1}{2}$, define the matrix $T_{r,a}^{\phi}$ with nonnegative entries, and rows and columns paramterized by $\mathbb{J}$:
\[
T_{r,a}^{\phi}(\mu,\lambda)=\det[I_a^{\phi}(\mu_i-i+r,\lambda_j-j+r)]_{1\leq i,j\leq r}\frac{\dim_{2r+1/2+a}\lambda}{\dim_{2r+1/2+a}\mu}.
\]
Here $\dim$ is the dimension of the corresponding representation of $SO(2r+1/2+a)$ -- but for the purposes of this paper, it suffices just to know that $\dim$ is a positive integer. In the proof of Proposition \ref{Prop}, the $\dim$ terms will cancel immediately anyway. Set
\[
T_k^{\phi}=\begin{cases} T^{\phi}_{\lfloor(k+1)/2\rfloor,1/2}, \ \ k\ \ \textit{even} \\ T^{\phi}_{\lfloor (k+1)/2\rfloor,-1/2}, \ \ k\ \ \textit{odd} \end{cases}
\]
For $\lambda$ on the $k$th level and $\mu$ on the $k-1$ level, let $\varkappa_{k-1}^k$ be
\[
\varkappa^k_{k-1}(\lambda,\mu)=
\begin{cases}
0,\ \ \mu\not\prec\lambda \\
1, \ \ \mu\prec\lambda\ \ \textit{and k odd} \\
1,\ \ \mu\prec\lambda,\ \mu_{r/2}=0, \ \ \textit{and k even}\\
2, \ \ \mu\prec\lambda,\ \mu_{r/2}>0,\ \ \textit{and k even}
\end{cases}
\]
and $T^k_{k-1}$ be
\[
T^k_{k-1}(\lambda,\mu)=\frac{\dim_{k}\mu}{\dim_{k+1}\lambda}\varkappa^k_{k-1}(\lambda,\mu)
\]
and
\[
\Delta^k_{k-1}(\lambda,\mu)=\sum_{\nu} T_k(\lambda,\nu)T^k_{k-1}(\nu,\mu)
\]
The matrix of transition probabilities is
\[
T^{\phi}(\boldsymbol{\mu},\boldsymbol{\lambda})
=T_1^{\phi}(\mu^{(1)},\lambda^{(1)})\prod_{j=2}^k \frac{T^{\phi}_j(\mu^{(j)},\lambda^{(j)})T^j_{j-1}(\lambda^{(j)},\lambda^{(j-1)})}{\Delta_{j-1}^j(\lambda^{(j)},\lambda^{(j-1)})}.
\]
For the densely packed initial conditions, $T^{\phi}$ satisfies a semingroup property. More specifically, if $T^{\phi_1}T^{\phi_2}$ denotes matrix multiplication, then (\cite{kn:BK})
\[
T^{\phi_1\phi_2}(\mathbf{0},\boldsymbol{\mu})=[T^{\phi_1}T^{\phi_2}](\mathbf{0},\boldsymbol{\mu}).
\]
When projected to the $k$th level, the matrix of transition probabilities is just $T_k^{\phi}$.
Certain functions $\phi$ arise naturally from the representations of $O(\infty)$ -- see section 2.1 of \cite{kn:BK}. For our purposes, it suffices to consider the function:
$$
\phi_{\alpha}(x)=(1+\alpha(1-x)+\alpha^2(1-x)/2)^{-1}, \ \ \alpha\geq 0.
$$
By Proposition 4.1 from \cite{kn:BK}, $\boldsymbol{Y}(n)$ is determinantal with correlation kernel $K(r_1,a_1,s_1;r_2,a_2,s_2)$ equal to
\begin{multline}\label{Kernel}
1_{2r_1+a_1\geq 2r_2+a_2}\frac{W^{(a_1,-1/2)}(s_1)}{\pi}\int_{-1}^1\mathsf{J}_{s_1}^{(a_1,-1/2)}(x)\mathsf{J}_{s_2}^{(a_2,-1/2)}(x)(x-1)^{r_1-r_2}(1-x)^{a_1}(1+x)^{-1/2}dx\\
+\frac{W^{(a,-1/2)}(s_1)}{\pi}\frac{1}{2\pi i}\int_{-1}^1\oint_C \frac{\phi_{\alpha}(x)^n}{\phi_{\alpha}(u)^n} \mathsf{J}_{s_1}^{(a_1,-1/2)}(x) \mathsf{J}_{s_2}^{(a_2,-1/2)}(u)\\
\times \frac{(x-1)^{r_1}}{(u-1)^{r_2}}\frac{(1-x)^{a_1}(1+x)^{-1/2}dudx}{x-u}.
\end{multline}
\subsection{Finite--time distributions}
The next theorem, which will be proved in the next section, establishes that $\tilde{X}^k$ is a determinantal point process.
\begin{theorem}\label{MainTheorem} Let $\alpha=2q/(1-q)$. Then $\tilde{X}^k(n)=Y^k(n).$ In particular, $\tilde{X}^k(n)$ is a determinantal point process on $\mathbb{N}$ with kernel $K(r,a,s_1;r,a,s_2)$, where $2r+1/2+a=k$.
\end{theorem}
Numerical calculations made by the author indicate that the fixed time marginals on multiple levels should also be identical, assuming the densely packed initial conditions. The exact statement is below:
\begin{conjecture}\label{Conjecture}
For any time $n\geq 0$,
\[
\mathbb{P}(\boldsymbol{X}(n)=\boldsymbol{\lambda})=T^{\phi_{\alpha}^n}(\boldsymbol{0},\boldsymbol{\lambda}).
\]
\end{conjecture}
Note that without the fixed-time assumption, the conjecture is false. For example,
\[
\mathbb{P}(\boldsymbol{X}(n+1)=(0,0,(0,0)) \vert \boldsymbol{X}(n) = (0,1,(1,0)))=0,
\]
by the fact that $X_1^2$ prevents $X_1^3$ from jumping to $0$. However,
\[
T^{\phi_{\alpha}}((0,1,(1,0)),(0,0,(0,0)))\neq 0,
\]
since none of the terms in the definition of $T^{\phi_{\alpha}}$ is zero.
\section{Proof of theorem \ref{MainTheorem}}\label{3}
Let $P_k(\lambda,\beta)$ denote the transition kernel of $X$ on the $k$th level. In other words
\[
P_k(\lambda,\beta)=\mathbb{P}(X^k(n+1)=(\beta_1,\beta_2,\ldots,\beta_{\lfloor (k+1)/2\rfloor}) \vert X^k(n)= (\lambda_1,\lambda_2,\ldots,\lambda_{\lfloor (k+1)/2\rfloor}))
\]
By Theorem 7.1 of \cite{kn:D},
\[
P_{2r}(\lambda,\beta)=\sum_{c\in\mathbb{N}^r, c\prec\lambda,\beta} (1-q)^{2r}\frac{\dim_{2r+1}\beta}{\dim_{2r+1}\lambda} q^{\sum_{i=1}^r \lambda_i+\beta_i-2c_i}\left(1_{c_r>0}+\frac{1_{c_r=0}}{1+q}\right)
\]
\[
P_{2r+1}(\lambda,\beta)=\sum_{c\in \mathbb{N}^{r}, c\prec\lambda,\beta}(1-q)^{2r+1}\frac{\dim_{2r+2}\beta}{\dim_{2r+2}\lambda} q^{\sum_{i=1}^{r} \lambda_i+\beta_i-2c_i} R(\lambda_{r+1},\beta_{r+1}).
\]
In this section, we prove Theorem \ref{MainTheorem}. Set $\phi=\phi_{\alpha}$, where $\alpha=\tfrac{2q}{1-q}$. Since $\boldsymbol{X}(0)=\boldsymbol{Y}(0)=0$, the following proposition suffices.
\begin{proposition}\label{Prop}
For $a=\pm\tfrac{1}{2}$, $T_{k}^{\phi_{\alpha}}=P_{k}$.
\end{proposition}
We start with a few lemmas.
\begin{lemma}\label{FirstLemma} Let
\[
\phi(x)=\frac{1}{1+\alpha(1-x)+\frac{\alpha^2}{2}(1-x)},\ \ \alpha=\frac{2q}{1-q}.
\]
Then $I_{-1/2}^{\phi}(l,k)=R(l,k)$ and
\[
I_{1/2}^{\phi}(l,k)=\frac{q-1}{q+1}\left(q^{k+l+1}-q^{\vert k-l \vert}\right)
\]
\end{lemma}
\begin{proof}
Substitute $x=(z+z^{-1})/2$. Then
\[
I_{-1/2}^{\phi}(l,k)=\frac{W^{(-1/2,-1/2)}(k)}{2\pi i}\oint_{\vert z\vert=1}\frac{z^k+z^{-k}}{2} \frac{z^l+z^{-l}}{2}\frac{(1-q)^2}{(1-qz)(z-q)}dz,
\]
which has residues at $q$ and $0$. The residue at $z=q$ is
\[
W^{(-1/2,-1/2)}(k)\frac{q^k+q^{-k}}{2} \frac{q^l+q^{-l}}{2}\frac{1-q}{1+q}.
\]
Using the expansion
\[
\frac{1}{(1-qz)(z-q)}=\sum_{m=0}^{\infty}\frac{q^{m+1}-q^{-m-1}}{1-q^2}z^m,
\]
the residue at $z=0$ is
\[
W^{(-1/2,-1/2)}(k)\frac{1-q}{1+q}\left(\frac{q^{k+l}-q^{-k-l}}{4} + \frac{q^{\vert k-l\vert}-q^{-\vert k-l\vert}}{4} \right),
\]
so the total contribution is
\[
I_{-1/2}^{\phi}(l,k)=\frac{W^{(-1/2,-1/2)}(k)}{2}\frac{1-q}{1+q}(q^{k+l}+q^{\vert k-l\vert})=R(l,k).
\]
For $a=1/2$,
\[
-\frac{1}{4\pi i}\oint_{\vert z\vert=1}(z^{k+1/2}-z^{-k-1/2}) (z^{l+1/2}-z^{-l-1/2})\frac{(1-q)^2}{(1-qz)(z-q)}dz.
\]
The residue at $z=q$ is
\[
-\frac{1}{2}(q^{k+1/2}-q^{-k-1/2}) (q^{l+1/2}-q^{-l-1/2})\frac{1-q}{1+q}
\]
and the residue at $z=0$ is
\[
-\frac{1}{2}\frac{1-q}{1+q}\left(q^{k+l+1}-q^{-k-l-1} - q^{\vert k-l\vert} + q^{-\vert k-l\vert} \right).
\]
\end{proof}
\begin{lemma}\label{Interlace}
Let $c=(c_1\geq c_2\geq \ldots \geq c_r)$ and $\lambda=(\lambda_1\geq \lambda_2 \geq \ldots \geq \lambda_r)$. Set
\[
\psi_m(s,l)=
\begin{cases}
m, \ \ \text{if}\ l\geq s=0\\
1, \ \ \text{if}\ l\geq s>0\\
0, \ \ \text{if}\ l<s.
\end{cases}
\]
Then
\[
\det[\psi_m(c_i-i+r,\lambda_j-j+r)]=
\begin{cases}
m, \ \ \text{if}\ c\prec\lambda,\ c_r=0\\
1, \ \ \text{if}\ c\prec\lambda,\ c_r>0\\
0, \ \ \text{if}\ c\not\prec\lambda.
\end{cases}
\]
\end{lemma}
\begin{proof}
The proof is standard, see e.g. Lemma 3.8 of \cite{kn:BK}.
\end{proof}
Now return to the proof of Theorem \ref{MainTheorem}. Start with the odd case. By the lemma, we can write
\[
P_{2r+1}(\lambda,\beta)=(1-q)^{2r}\frac{\dim_{2r+1}\beta}{\dim_{2r+1}\lambda}\sum_{s_1>\ldots>s_r\geq 0} \det[f_{s_i,1}(\lambda_j-j+r)] \det[f_{s_i,\frac{1}{1+q}}(\beta_j-j+r)],
\]
where
\[
f_{s,m}(l)=q^{l-s}\psi_m(s,l).
\]
Thus, by Lemma 2.1 of \cite{kn:BK},
\[
P_{2r+1}(\lambda,\beta)=(1-q)^{2r}\frac{\dim_{2r+1}\beta}{\dim_{2r+1}\lambda}\det\left[\sum_{s=0}^{\infty}f_{s,1}(\lambda_i-i+r)f_{s,\frac{1}{1+q}}(\beta_j-j+r)\right].
\]
A simple calculation shows that
\[
\sum_{s=0}^{\infty}f_{s,1}(x)f_{s,\frac{1}{1+q}}(y)=
\begin{cases}
\dfrac{1}{1+q}q^{x+y}, \ \text{if}\ \ \min(x,y)=0 , \\
\dfrac{q^{x+y+1}-q^{\vert x-y\vert}}{q^2-1}, \ \text{otherwise},
\end{cases}
\]
which, by Lemma \ref{FirstLemma}, equals $(1-q)^{-2}I_{1/2}^{\phi}(x,y)$.
Now proceed to the even case. Lemma 2.1 from \cite{kn:BK} is not immediately applicable, because we are summing over elements of $\mathbb{N}^{r-1}$ while the determinants are of size $r$. Notice, however, that $c\prec\lambda,\beta$ if and only if $c\prec\lambda_{\text{red}},\beta_{\text{red}}$ (where $\lambda_{\text{red}},\beta_{\text{red}}$ denote $(\lambda_1,\ldots,\lambda_{r-1}),(\beta_1,\ldots,\beta_{r-1})$) and $c_{r-1}\geq\max(\lambda_r,\beta_r)$. Thus
\begin{multline*}
(1-q)^{2r-1}\frac{\dim_{2r}\beta}{\dim_{2r}\lambda}\frac{q^{\vert\lambda_r-\beta_r\vert}+q^{\lambda_r+\beta_r}}{1+1_{\beta_r=0}}\\
\times\sum_{s_1>s_2>\ldots> s_{r-1}\geq\max(\lambda_r,\beta_r)} \det[f_{s_i,1}(\lambda_j-j+r-1)]_1^{r-1}\det[f_{s_i,1}(\beta_j-j+r-1)]_1^{r-1}\\
=(1-q)^{2r-2}R(\lambda_r,\beta_r)\frac{\dim_{2r}\beta}{\dim_{2r}\lambda}\det\left[\sum_{s=\max(\lambda_r,\beta_r)}^{\infty}f_{s,1}(\lambda_i-i+r-1)f_{s,1}(\beta_j-j+r-1)\right]_1^{r-1}.
\end{multline*}
A straightforward calculation shows that if $\max(\lambda_r,\beta_r)\leq\min(x,y)$, then
\[
\sum_{s=\max(\lambda_r,\beta_r)}^{\infty}f_{k,1}(x)f_{k,1}(y)=\frac{q^{x+y-2\max(\lambda_r,\beta_r)+2}-q^{\vert x-y\vert}}{q^2-1}.
\]
To deal with the case $\max(\lambda_r,\beta_r)>\min(x,y)$, we use the following lemma.
\begin{lemma}
If $\max(\lambda_r,\beta_r)>\min(\lambda_{r-1},\beta_{r-1})$, then $P_{2r-1}(\lambda,\beta)=T^{\phi}_{2r-1}(\lambda,\beta)=0$.
\end{lemma}
\begin{proof}
The fact that $P_{2r-1}(\lambda,\beta)=0$ follows immediately from the description of the interacting particle system, or from the fact that $\{c\in\mathbb{N}^{r-1}:c\prec\lambda,\beta\}$ is empty.
Now it remains to show that $T^{\phi}_{r,-1/2}=0$. If $\lambda_r>\beta_{r-1}$, then $\lambda_1\geq\lambda_2\geq\ldots\geq\lambda_r>\beta_{r-1}\geq\beta_r$, so
\[
(r-1)\text{th column} = \frac{R(\lambda_{r-1}+1,\beta_{r-1}+1)}{R(\lambda_r,\beta_r)}(r\text{th column}),
\]
implying the determinant is zero. An identical argument holds if $\beta_r>\lambda_{r-1}$.
\end{proof}
For the rest of the proof, assume that $\max(\lambda_r,\beta_r)\leq\min(\lambda_{r-1},\beta_{r-1})$.
Notice now that the determinant in $T_{2r-1}^{\phi}$ is of size $r$, which needs to be compared to a determinant of size $r-1$. To show that the larger determinant is $(1-q)^{2r-2}R(\lambda_r,\beta_r)$ times the smaller determinant, we perform a sequence of operations to the smaller matrix. These operations are slightly different for $\lambda_r>\beta_r$ and $\lambda_r\leq\beta_r$. Consider $\lambda_r>\beta_r$ for now.
First, add a row and a column to the matrix of size $r-1$. The $r$th column is $[0,0,0,\ldots,0,R(\lambda_r,\beta_r)]$ and the $r$th row is $[R(\lambda_r,\beta_1-1+r),R(\lambda_r,\beta_2-2+r),\ldots,R(\lambda_r,\beta_{r-1}+1),R(\lambda_r,\beta_r)]$.
This multiplies the determinant by $R(\lambda_r,\beta_r)$.
Second, for $1\leq i\leq r-1$, perform row operations by replacing the $i$th row with
\[
i\text{th row} + \frac{1}{(q-1)^2}\frac{R(\lambda_i-i+r,\beta_r)}{R(\lambda_r,\beta_r)}(r\text{th row}).
\]
For $1\leq i,j\leq r-1$ and letting $(x,y)=(\lambda_i-i+r,\beta_j-j+r)$, the $(i,j)$ entry is
\begin{eqnarray*}
&&\frac{q^{x+y-2\max(\lambda_r,\beta_r)}-q^{\vert x-y\vert}}{q^2-1}+ \frac{1}{(q-1)^2}\frac{R(x,\beta_r)}{R(\lambda_r,\beta_r)}R(\lambda_r,y)\\
&=&\frac{q^{x+y-2\max(\lambda_r,\beta_r)}-q^{\vert x-y\vert}}{q^2-1}-\dfrac{q^{x-\lambda_r}}{(1+1_{y=0})(q^2-1)}( q^{\lambda_r+y} + q^{\vert \lambda_r-y\vert})\\
&=&\frac{-q^{x+y}-q^{\vert x-y\vert}}{q^2-1}=(1-q)^{-2}R(x,y).
\end{eqnarray*}
Here, we used the fact that $y\geq\beta_{r-1}\geq\min(\lambda_{r-1},\beta_{r-1})\geq\lambda_r$ and $y\geq\lambda_r>\beta_r\geq 0$. For $j=r$, $\lambda_r>y=\beta_r$, so the $(i,j)$ entry is
\[
\frac{-q^{x+y}-q^{\vert x-y\vert}}{(1+1_{y=0})(q^2-1)}=(1-q)^{-2}R(x,y).
\]
Thus, the larger determinant is $(1-q)^{2(r-1)}R(\lambda_r,\beta_r)$ times the larger determinant.
Now consider $\lambda_r\leq\beta_r$. First, add the $r$th row, which is $[0,0,\ldots,0,R(\lambda_r,\beta_r)]$, and add the $r$th column which is $[R(\lambda_1-1+r,\beta_r),R(\lambda_2-2+r,\beta_r),\ldots,R(\lambda_r,\beta_r)]$. This multiplies the determinant by $R(\lambda_r,\beta_r)$.
Second, for $1\leq j\leq r-1$, perform column operations by replacing tSecond, for $1\leq j\leq r-1$, perform column operations by replacing the $j$th column with
\[
j\text{th column} + \frac{1}{(q-1)^2}\frac{R(\lambda_r,\beta_j-j+r)}{R(\lambda_r,\beta_r)}(r\text{th column}).
\]
Once again, this yields a matrix whose entries are $(1-q)^{-2}R(\lambda_i-i+r,\beta_j-j+r)$, except for the last column, which is $R(\lambda_i-i+r,\beta_r)$.
\section{Asymptotics}\label{4}
Thus far, we have shown that $\tilde{X}^k(n)$ is determinantal with correlation kernel $K(r,a,s_1;r,a,s_2)$. In this section, we will take asymptotics of $K(r_1,a_1,s_1;r_2,a_2,s_2)$, with $(r_1,a_1)$ not necessarily equal to $(r_2,a_2)$. This is because the asymptotic analysis is not much more difficult, and this would be the appropriate limit if Conjecture \ref{Conjecture} were true. Recall that $(r,a)$ corresponds to the $2r+1/2+a$ level.
\subsection{Discrete Jacobi Kernel}
For $-1< u < 1$ and $a_1,a_2=\pm\tfrac{1}{2}$, define the \textit{discrete Jacobi kernel}
$L(r_1,a_1,s_1,r_2,a_2,s_2;u)$ as follows. If $2r_1+a_1\geq 2r_2+a_2$, then
\begin{multline*}
L(r_1,a_1,s_1,r_2,a_2,s_2;u)\\
=\frac{W^{(a_1,-1/2)}(s_1)}{\pi}\int_u^1 \mathsf{J}_{s_1}^{(a_1,-1/2)}(x)\mathsf{J}_{s_2}^{(a_2,-1/2)}(x)(x-1)^{r_1-r_2}(1-x)^{a_1}(1+x)^{-1/2}dx.
\end{multline*}
If $2r_1+a_1< 2r_2+a_2$, then
\begin{multline*}
L(r_1,a_1,s_1,r_2,a_2,s_2;u)\\
=-\frac{W^{(a_1,-1/2)}(s_1)}{\pi}\int_{-1}^u
\mathsf{J}_{s_1}^{(a_1,-1/2)}(x)\mathsf{J}_{s_2}^{(a_2,-1/2)}(x)(x-1)^{r_1-r_2}(1-x)^{a_1}(1+x)^{-1/2}dx.
\end{multline*}
\begin{theorem}
Let $n$ depend on $N$ in such a way that $n/N\rightarrow t$. Let $r_1,\ldots,r_l$ depend on $N$ in such a way that $r_i/N\rightarrow l$ and their differences $r_i-r_j$ are fixed finite constants. Here, $t,l>0$. Fix $s_1,s_2,\ldots,s_l$ to be finite constants. Let
\[
\theta=1+\frac{2l}{(l-t)(2\alpha+\alpha^2)}
\]
Then
\begin{multline*}
\lim_{N\rightarrow\infty}\det[K(r_i,a_i,s_i,r_j,a_j,s_j)]_{i,j=1}^l \\
=\begin{cases}
1,\ \ &l\geq (1-(1+\alpha)^{-2})t\\
\det[L(r_i,a_i,s_i,r_j,a_j,s_j;\theta)]_{i,j=1}^l,\ \ &l<(1-(1+\alpha)^{-2})t
\end{cases}
\end{multline*}
\end{theorem}
\begin{proof}
First consider the case when $l<t$. Let $A(z)=-t\log(1+\alpha(1-z)+\alpha^2/2\cdot(1-z))+l*\log(z-1)$. Then the kernel asymptotically equals
\begin{multline*}
1_{2r_1+a_1\geq 2r_2+a_2}\frac{W^{(a_1,-1/2)}(s_1)}{\pi}\int_{-1}^1\mathsf{J}_{s_1}^{(a_1,-1/2)}(x)\mathsf{J}_{s_2}^{(a_2,-1/2)}(x)(x-1)^{r_1-r_2}(1-x)^{a_1}(1+x)^{-1/2}dx\\
+\frac{W^{(a_1,-1/2)}(s_1)}{\pi}\frac{1}{2\pi i}\int_{-1}^1\oint_C \frac{e^{N(A(x)-A(\theta))}}{e^{N(A(u)-A(\theta))}}\mathsf{J}_{s_1}^{(a_1,-1/2)}(x)\mathsf{J}_{s_2}^{(a_2,-1/2)}(u)\\
\times (x-1)^{r_1-r_2}\frac{(1-x)^{a_1}(1+x)^{-1/2}dudx}{x-u}.
\end{multline*}
Deform the contours as in Figure \ref{Contours}. With these deformations, the double integral converges to zero, but residues are picked up at $u=x$. For $l>(1-(1+\alpha)^{-2})t$, the parameter $\theta$ is less than $-1$, so no residues are picked up. We arrive at a triangular matrix with diagonal entries equal to $1$, so the determinant is $1$. For $l<(1-(1+\alpha)^{-2})t$, the parameter $\theta$ is in $(-1,1)$, and the residues give the discrete Jacobi kernel.
\begin{center}
\begin{figure}[htp]
\caption{Shaded regions indicate $\Re(A(z)-A(\theta))>0$ and white regions indicate $\Re(A(z)-A(\theta))<0$. The double zero occurs at $\theta$.}
\begin{center}
\includegraphics[height=2in]{Contour1.png}\ \ \ \ \ \ \includegraphics[height=2in]{NewContour2.png}
\end{center}
\label{Contours}
\end{figure}
\end{center}
For $l>t$, the situation is quite different, due to the discontinuity in $\theta$ at $l=t$. Make the substitutions $x=(z+z^{-1})/2$ and $u=(v+v^{-1})/2$. Now the $x$-contour is the unit circle and the $v$ contour is a simple loop that goes outside the unit circle. After deforming as shown in Figure \ref{Contours2}, the double integral converges to $0$, with no residues picked up. Again, we obtain a triangular matrix with diagonal entries equal to $1$.
\begin{center}
\begin{figure}[htp]
\caption{Shaded regions indicate $\Re(A\left(\frac{z+z^{-1}}{2}\right)-A(-1))>0$ and white regions indicate $\Re(A(z)-A(-1))<0$. The double zero occurs at $-1$.}
\begin{center}
\includegraphics[height=2in]{Contours3.png}\ \ \ \ \ \ \includegraphics[height=2in]{NewContours4.png}
\end{center}
\label{Contours2}
\end{figure}
\end{center}
\end{proof}
\subsection{Symmetric Pearcey Kernel}\label{SPK}
Define the \textit{symmetric Pearcey kernel} $\mathcal{K}$ on $\mathbb{R}_+\times\mathbb{R}$ as follows. In the expressions below, the $u$-contour is integrated on rays from $\infty e^{i\pi/4}$ to $0$ to $\infty e^{-i\pi/4}$. Let
\begin{multline*}\label{GaussianLikeKernel}
\mathcal{K}(\sigma_1,\eta_1,\sigma_2,\eta_2)= \\
\frac{2}{\pi^2 i} \int\int_0^{\infty}\exp(-\eta_1 x^2 + \eta_2 u^2 + u^4 - x^4)\cos(\sigma_1 x) \cos(\sigma_2 u) \frac{u}{u^2-x^2}dxdu\\
-\frac{1_{\eta_2<\eta_1}}{2\sqrt{\pi(\eta_1-\eta_2)}}\left(\exp\frac{(\sigma_1+\sigma_2)^2}{4(\eta_2-\eta_1)}+\exp\frac{(\sigma_1-\sigma_2)^2}{4(\eta_2-\eta_1)}\right).
\end{multline*}
\begin{theorem}
Let $c_{\alpha}$ be the constant $(1+\alpha)(\alpha(2+\alpha))^{-1/4}$. Let $s_1$ and $s_2$ depend on $N$ in such a way that $s_i/N^{1/4}\rightarrow 2^{-5/4}\sigma_ic_{\alpha}^{-1}>0$ as $N\rightarrow\infty$. Let $n$ and $r_1,r_2$ also depend on $N$ in such a way that $n/N\rightarrow 1$ and $(r_j- (1-(1+\alpha)^{-2})N)/\sqrt{N}\rightarrow 2^{-1/2}\eta_j$. Then
\[
(-2)^{r_2-r_1}(-1)^{s_1-s_2}\frac{N^{1/4}}{c_{\alpha}2^{5/4}}K(r_1,a_1,s_1,r_2,a_2,s_2)\rightarrow \mathcal{K}(\sigma_1,\eta_1,\sigma_2,\eta_2).
\]
\end{theorem}
\begin{proof}
Since the proof is almost identical to the proof of Theorem 5.8 from \cite{kn:BK}, the details will be omitted. The only difference is that now
\[
A(z)=\log\phi_{\alpha}(z)+(1-(1+\alpha)^{-2})\log(z-1),
\]
with asymptotic expansion
\[
A(z)-A(-1)=-\frac{\alpha(2+\alpha)}{8(1+\alpha)^4}(z+1)^2+O((z+1)^3).
\]
\end{proof}
\bibliographystyle{plain}
|
2,877,628,089,030 | arxiv | \section{Introduction}
Many scenarios involve classification systems constrained by measurement acquisition budget. In this setting, a collection of sensor modalities with varying costs are available to the decision system. Our goal is to learn adaptive decision rules from labeled training data that when presented with a new unseen example would select the most informative and cost-effective strategy for the example. In contrast to non-adaptive methods \cite{efron2004least,xu2012greedy}, which attempt to identify a common sparse subset of sensors that can work well for all examples, our goal is an adaptive method that can classify typical cases using inexpensive sensors and using expensive sensors only for atypical cases.
We propose to learn a sensor tree using labeled training examples for making decisions on unseen test examples. The learned sensor tree is composed of internal node decision rules. Given an example these decision rules select sensors and guide an example along a particular path terminating at a leaf where it is classified with a classifier. We pose the problem as a global empirical risk minimization (ERM) over the choice of tree structures, node decision rules and leaf classifiers.
The general problem is a highly coupled problem consisting of combinatorial (sensor tree structure) and continuous components (decision rules to generalize to unseen examples) and difficult to optimize. To gain further insight we abstract away the generalization aspect and observe that the resulting combinatorial problem, a special case of ours, is known to be NP hard and requires greedy approximations~\cite{chakr07,cicalese}.
The combinatorial issue can be circumvented in cases where expert knowledge exists, only a few sensors (as in \cite{trapeznikov:2013,trapeznikov:2013b,wang2014lp}) exist or for small depth trees. For these latter two cases we contruct an exhaustive tree and globally learn decision functions using a linear program by generalizing the cascade structures presented in \cite{trapeznikov:2013b,wang2014lp} to binary trees, resulting in more flexible decision systems. Convex surrogates of products of indicators have previously been studied for supervised learning \cite{wang2013locally}.
For more general cases we propose a two-step approach to decouple the issue of sensor structure from the decision rule design.
\begin{itemize}
\item We greedily solve for the combinatorial tree structure and obtain feature/sensor sub-collections efficiently by a greedy approximation to the NP-hard problem. From these subsets, we construct a binary tree using hierarchical clustering of feature subsets.
\item On the learned tree structure, our problem now reduces to the ERM problem discussed above for a fixed tree where we apply a novel surrogate, allowing us to jointly learn the decision functions in the tree by solving a linear program.
\end{itemize}
In the experiments, we demonstrate performance of our approach both for when feature subsets and tree structure is given and when feature subsets and tree structure must be learned. We show on real world data that our approach outperforms previously proposed approaches to budgeted learning.
\subsection{Related Work}
There is an extensive literature on adaptive methods for sensor selection for reducing test-time costs. It arguably originated with detection cascade (see \cite{zhang:2010,chen:2012} and references therein), which is a popular method in reducing computation cost in object detection for cases with highly skewed class imbalance and generic features. Computationally cheap features are used at first to filter out negative examples and the more expensive features are used in the later stages
Our technical approach is closely related to \cite{trapeznikov:2013b} and \cite{wang2014lp}. Like us they formulate an ERM problem and generalize detection cascades to classifier cascades and handle balanced and/or multi-class scenarios. Like us, \cite{wang2014lp} construct convex surrogates for their empirical risk functions and propose efficient LP solutions. Unlike us their approach is limited to cascades of known structure and cannot handle trees and unknown sensor structures.
Conceptually, our work is closely related to ~\cite{xu2013cost} and \cite{kusner2014feature}, who introduced cost-sensitive tree of classifiers (CSTC) for reducing test time costs. Like our paper they proposed a global ERM problem. They solve for the tree structure, internal decision rules and leaf classifiers jointly using alternative minimization techniques. Recently, \cite{kusner2014feature} propose a more efficient version of CSTC. In contrast we decompose our global objective and separately solve the individual parts. The disadvantage of our decoupled approach is somewhat offset by globally convergent solution for the decision rules once a structure is determined. Nevertheless, which approach is better is an important question that must be addressed but outside the scope of this work.
\nocite{wang2014model}
The subject of this paper is broadly related to other adaptive methods in the literature but unlike us these methods do not learn sensor trees but learn policies from training data. Generative methods pose the problem as a POMDP, learn conditional probability models \cite{zubek:2002,Sheng06featurevalue,bilgic:2007,ji:2007,kanani:2008,kapoor:2009,gao2011active}
and myopically select features based information gain of unknown features.
MDP-based methods ~\cite{karayev2013dynamic}, ~\cite{dulac2011datum}, \cite{he2012imitation}, \cite{busa2012fast}
encode current observations as state, unused features as action space, and formulate various reward functions to account for classification error and costs.
He et. al. \cite{he2012imitation} apply imitation learning of a greedy policy with a single classification step as actions. ~\cite{dulac2011datum} and \cite{karayev2013dynamic} apply reinforcement learning to solve this MDP. \cite{busa2012fast} propose classifier cascades within an MDP framework. They consider a fixed-ordering of features and extend sequential boosted classifier with an additional skip action.
\begin{comment}
{\bf Related Work:} Learning adaptive sensor selection systems has been explored by several researchers in the past.
In a Bayesian setting, \cite{ji:2007,kapoor:2009} model the problem of learning with test time budgets as a POMDP, \cite{Sheng06featurevalue,bilgic:2007,zubek:2002} study cost sensitive decision trees, and \cite{kanani:2008} use an expected utility criteria. However, all these methods require estimating a probability likelihood that a certain feature value occurs given the features collected so far. In contrast, our problem domain deals with high dimensional measurements (such as images consisting of thousands of pixels), so estimating probability densities reliably is not possible.
To avoid estimating probability models within an MDP framework, \cite{he2012imitation} applies an imitation learning (IL) algorithm introduced by \cite{ross2010efficient} to the problem of feature selection. IL learns decision functions that mimics an \emph{oracle} on training data. Many issues arise in this context. We generally do not have access to an oracle for a feature selection problem and so may have to learn it from training data. This can be cumbersome. He et. al. \cite{he2012imitation} generate arbitrary collection of states (candidate feature subsets) from training data to ensure a sufficiently rich collection of state-actions to mimic. This means that an oracle must be capable of providing decisions on any arbitrary subset of features with full foresight of future costs. For this reason a myopic oracle is often used. Allowing arbitrary subset of features introduces new issues: the need for training combinatorially many classifiers customized to different feature subsets. Since this is impossible a low-quality missing data classifier must be used. In contrast the key difference of our framework is that we only require a sensing architecture rather than an oracle capable of making decisions. Also we can utilize/train high-performance classifiers due to the relatively small number of leaves on our sensing tree.
Related to the IL approach is another direction based on reinforcement learning \cite{karayev13,busa2012fast,dulac2011datum}. Here, the state encodes current observations, and the action space consists of all unutilized features. In lieu of an oracle the authors linearly parametrize a value function and estimate it with standard RL techniques. Again several issues arise in our context. First, the missing data classifier issue remains as in IL. Second, this framework imposes a notion of stationarity; same policy function is applied regardless of the observation state. RL is difficult to justify for feature selection because of the deterministic nature of state transitions when conditioned on current state and action.
A special case of sensor trees, namely, cascade architectures with sensor acquired in a fixed order has also been studied. The detection cascade (see \cite{viola01,zhang:2010,chen:2012} and references therein), a popular method in reducing computation cost in object detection, where computationally cheap features are used at first to filter out negative examples and the more expensive features are used in the later stages of the cascade. \cite{trapeznikov:2013b}, generalize the cascade to a multi-class scenario. Other related work includes web page ranking problems, \cite{xu2013cost} introduced a general tree of cost sensitive classifiers, where each node is parametrized with boosted weak learners. However, the optimization problems described in these papers are inherently non-convex and most solutions resort to alternative minimization schemes \cite{chen:2012,trapeznikov:2013,trapeznikov:2013b,xu2013cost}. While experimental results demonstrate good performance, they are computationally demanding and the lack of global optimality prevents theoretical guarantees for the algorithms and the solutions.
\end{comment}
\section{Problem Formulation: Global Empirical Risk Minimization Objective}\label{sec.struct}
We consider learning an adaptive decision system with training examples $(x_1,y_1),\allowbreak \ldots,\allowbreak(x_N,y_N)$ with $L$ sensors and acquisition cost $c_m,\,m=1,\,2,\ldots,L$. We can pose the problem of learning a rooted sensor tree as an ERM problem.
Our system is composed of three components, a tree, $T$, decision rules $\mathbf{g}_J=\{g_j\}_{j=1}^J,\, \mathbf g_j \in {\cal G}_j$ associated with $J$ internal nodes, and classifier $\mathbf{f}_K=\{f_k\}_{k=1}^K,\,f_k \in {\cal F}_k$ associated with leaves. Each internal node of the tree is associated with a sensor and its children denote available sensor choices. Each leaf, $k$, is associated with a sensor subset, $S_k$, and corresponds to the unique path from the root to the leaf. The decision rule, $g_j$, associated with node $j$ acts upon acquired measurements for an example and routes it to one of its children. By uniquely associating each leaf, $k$, with the sensor outputs $b_k$ we can write ERM as
\begin{align}\label{eq.globalerm}
L({\cal S},\mathbf{f},\mathbf{g})=\sum_{i=1}^{N}\sum_{k=1}^{K}\left(\mathbbm{1}_{f_k(x_i)\neq y_i}+\alpha\sum_{m \in S_k}c_m\right)\mathbbm{1}_{\mathbf{g}_J(x_i)=b_k}\notag\\ \longrightarrow \min_{{\cal S}} \min_{\mathbf{f}_K} \min_{\mathbf g_J} L({\cal S},\mathbf{f},\mathbf{g})
\end{align}
where $\alpha$ is a trade-off parameter balancing classification performance with sensor acquisition cost and ${\cal S}$ is the set of paths. An instance of this general problem has been considered in the literature and shown to be NP hard~\cite{chakr07} for the special case of discrete valued sensor measurements, arbitrarily powerful decision rules, $\mathbf g_J$ and with separable leaves, i.e, features acquired corresponding to the leaf path uniquely \& correctly identifies the class. The authors develop greedy algorithms for approximating their solution.
In our setting we allow for continous valued high-dimensional measurements and so we cannot discretize our space. Furthermore we are not in a separable situation and the Bayes error is not zero. However, our general ERM problem is highly coupled, difficult to optimize. This motivates our proposed decomposition approach described below.
Indeed, assuming arbitrary families ${\cal G}_j$ in Eq.~\ref{eq.globalerm} leads to new insights that motivates our decomposition approach. The general problem reduced to a purely combinatorial structure learning with arbitrary ${\cal G}_j$ when we also have access to an oracle classifier $\mathbf{f}_K$ capable of classifying with any subset of acquired features.
The resulting problem while NP hard as before is amenable to greedy strategies. Alternatively, given a tree structure and the oracle classifier the optimization objective takes a multilinear form as evidenced by Eq.~\ref{eq.globalerm} which also lends itself to optimization strategies. The only issue remaining is that of an oracle classifier, which can be circumvented for our approach.
This overall perspective justifies our approach:
\begin{enumerate}
\item Learn the tree structure assuming powerful decision rules.
\item Learn decision functions $g_1,\ldots,g_{K-1}$ for the tree structure learned in Step 1.
\end{enumerate}
\subsection{Learning Tree Structures Greedily}
For simplicity we assume a binary sensor tree with $K$ leaves and $K-1$ internal nodes. Motivated by previous methods~\cite{gao2011active,xu2013cost} we assume that the number of leaves $K$ is small relative to the feature dimension. Our approach identifies a sub-collection of subsets of features ${\cal S}=\{S_1,\ldots,S_K\}$ from training data, and as such, the tree structure, $T$. Assuming arbitrarily powerful decision functions, $g_1,\ldots,g_{K-1}$, effectively implies we can route each example to its optimal subset. Furthermore, assuming that we have access to an oracle classifer, $f_j$'s we can predicted the class on any subset of features. Then the optimization loss of Eq.~\ref{eq.globalerm} associated with the subcollection ${\cal S}$ reduces to:
\begin{align}\label{eq.subsets_min}
L(S_1,\ldots,S_K)= \frac{1}{N}\sum_{i=1}^{N}\min_{j \in \{1,\ldots,K\}}\left[\mathbbm{1}_{f_{s_j}(x_i)\neq y_i} + \alpha \sum_{k \in S_j}c_k\right]
\end{align}
Even in the absence of noise, as noted before the problem of minimizing this loss is NP-hard and motivates greedy strategies. Additionally, we overcome the issue of an oracle classifier by learning it as we grow the tree greedily. While many greedy strategies exist in the literature for related objectives, they are not directly applicable to our setting\footnote{For instance~\cite{cicalese} proposes a submodular surrogate to leverage properties of Wolsey greedy algorithm but their surrogates require discrete sensor measurements.}. Our greedy algorithm is related to \cite{awasthi13}, who learn sparse trees/polynomials in the separable PAC setting and provide statistical and computational guarantees. We adapt their approach to our setting as follows: given a sub-collection of features we expand this subcollection if there is a sensor that can further reduce our loss. If no sensor is found we restart the process and look for a new subset of sensors.
\begin{algorithm}[H]
\begin{algorithmic}
\STATE {\bfseries Input:} Number of Subsets $K$
\STATE {\bfseries Output:} Feature subsets, $s_1,\ldots,s_K$
\STATE {\bfseries Initialize:} $s_1,\ldots,s_K=\emptyset$
\FOR{k=1,\ldots,K}
\STATE $j=\operatornamewithlimits{argmin}_{j \in s_k^C}\mathcal{L}(s_1,...,s_k\cup j,...,s_K)$
\WHILE{$L(s_1,...,s_k\cup j,...,s_K)<\mathcal{L}(s_1,...,s_k,...,s_K)$}
\STATE $s_k=s_k\cup j$
\STATE $j=\operatornamewithlimits{argmin}_{j \in s_k^C}\mathcal{L}(s_1,...,s_k\cup j,...,s_K)$
\ENDWHILE
\ENDFOR
\end{algorithmic}
\caption{Sensor Subset Selection}
\label{greedy_subset_algorithm}
\end{algorithm}
\textbf{Tree Structure:} Given the set of sensor subsets $s_1,\ldots,s_K$, the problem of choosing a tree structure partitioning between these sensor subsets arises. We propose a hierarchical clustering approach, where subsets are grouped based on the number of common elements. Given a set of feature subsets, the two subsets with the highest number of common elements are grouped together and replaced in the set with the intersection of their elements. This is recursively repeated until only a single subset exists, resulting in a binary tree structure. This can be viewed as a generalization of the cascade approach, as given the set of feature subsets where an additional sensor is added to each previous subset, a cascade structure is always recovered.
Once a tree structure, $T$ is learned we need to populate it with decision functions so that we can generalize to unseen examples. Note that the learned sensor structure provides us with possible choices but does not tell us what choice to make on an unseen example. This motivates the following section.
\subsection{Empirical Risk Problem for a Fixed Tree}
\begin{figure}[ht]
\begin{minipage}{.5\textwidth}
\centering
\includegraphics[height = .5 \linewidth]{treeNode.eps}
\end{minipage}%
\begin{minipage}{.5\textwidth}
$$
\begin{array}{c}
\mathbf P \\
\begin{bmatrix}
0 & 0 & 0 \\
0 & 1 & 0 \\
1 & 0 & 0 \\
1 & 0 & 1
\end{bmatrix}
\end{array}
- \begin{array}{c}
\mathbf N\\
\begin{bmatrix}
1 & 1 & 0 \\
1 & 0 & 0 \\
0 & 0 & 1 \\
0 & 0 & 0
\end{bmatrix}
\end{array} =
\begin{array}{c}
\\
\begin{bmatrix}
-1 & -1 & ~0 \\
-1 & +1 & ~0 \\
+1 & ~0 & -1 \\
+1 & ~0 & +1
\end{bmatrix}
\end{array}
$$
\end{minipage}
\caption{An example decision system of depth two: node $g_1(x_1)$ selects either to acquire sensor $2$ for a cost $c_2$ or $3$ for a cost $c_3$. Node $g_2(x_1,x_2)$ selects either to stop and classify with sensors $\{1,2\}$ or to acquire 3 for $c_3$ and then stop. Node $g_3(x_1,x_3)$ selects to classify with $\{1,3\}$ or with $\{1,2,3\}$.}
\label{fig:tree}
\end{figure}
We represent our decision system as a binary tree. The binary tree is composed of $K$ leafs and $K-1$ internal nodes. At each internal node, $j=1, \ldots, K-1$, is a binary decision function, $\mbox{sign} [g_j(\mathbf{x})] \in \{+1,-1\}$. This function determines which action should be taken for a given example. The binary decisions, $g_j(\mathbf{x})$'s, represent actions from the following set: stop and classify with the current set of measurements, or choose which sensor to acquire next. Each leaf node, $k=1,\ldots K$, represents a terminal decision to stop and classify based on the available information\footnote{For notational simplicity, we denote applying a decision node and a leaf classifier as $g_j(\mathbf{x})$ and $f_k(\mathbf{x})$ respectively.} We assume that the leaf classifiers, $f_j$ are known and fixed\footnote{Note that the classifiers at each leaf, $f_k(\mathbf{x}) \in \mathcal Y$, can be learned for the $K$ leaves once the tree structure, $T$, has been determined from the previous section.}. However note, the functions implicitly operate only on the sensors that have been acquired along the path to the corresponding node. The objective is to learn the internal decision functions: $g_j(\mathbf{x})$'s. We define the system risk:
\begin{equation}\label{eq.point_risk}
R(\mathbf g,\mathbf{x},y)=\sum_{k=1}^{K}R_k(f_k,\mathbf{x},y)G_k(\mathbf g,\mathbf{x})
\end{equation}
Here, $\mathbf g= \{g_1, \ldots g_{K-1}\}$ is the set of decision functions. $R_k(f_k,\mathbf{x},y)=\mathds{1}_{f_k(\mathbf{x})\neq y}+\alpha \sum_{m \in S_k}c_m$ is the risk of making a decision at a leaf $k$. It consists of two terms: error of the classifier at the leaf and the cost of sensors acquired along the path from the root node to the leaf. $S_k$ is this set of sensors, and $\alpha$ is a parameter that controls trade-off between acquisition cost and classification error.
$G_k(\mathbf g,\mathbf{x})\in\{0,1\}$ is a binary state variable indicating if $\mathbf{x}$ is classified at the $k$th leaf. We compactly encode the path from the root to every leaf in terms of internal decisions, $g_j(\mathbf{x})$'s, by two auxiliary binary matrices: $\mathbf P$, $\mathbf N$ $\in \{0,1\}^{K \times K-1}$. If $\mathbf P_{k,j}=1$ then, on the path to leaf $k$, a decision node $j$ must be positive: $g_j > 0$. If $\mathbf N_{k,j}=1$ then on the path to leaf $k$, a decision at node $j$ must be negative: $g_j \leq 0$. A $k$th row in $\mathbf P$ and $\mathbf N$ jointly encode a path from the root node to a leaf $k$. The sign pattern for each path is obtained by $\mathbf P-\mathbf N$. For an example refer to Fig. \ref{fig:tree}. Using this path matrix, the state variable can be defined:
$G_k(\mathbf g,\mathbf{x})= \prod_{j=1}^{K-1}[\mathds{1}_{g_j(\mathbf{x})>0}]^{\mathbf P_{k,j}}[\mathds{1}_{g_j(\mathbf{x})\leq 0}]^{\mathbf N_{k,j}}$.
Our goal is to learn decision functions $g_1,\ldots,g_{K-1}$ that minimize the expected system risk:
$\min_{\mathbf g}\mathds{E}_{\mathcal{D}}\left[R(\mathbf g,\mathbf{x},y)\right]$
However, the model $\mathcal{D}$ is assumed to be unknown and cannot be estimated reliably due to potentially high-dimensional nature of sensor outputs. Instead, we are given a set of $N$ training examples with full sensor measurements, $(\mathbf{x}_1,y_1),\ldots,(\mathbf{x}_N,y_N)$. We approximate the expected risk by a sample average over the data and construct the following ERM problem:
\begin{align}
\min_{\mathbf g}\sum_{i=1}^{N}R(\mathbf g,\mathbf{x}_i,y_i)= \label{eq.erm_prod_ind} \sum_{i=1}^{N}\sum_{k=1}^{K} \overbrace{R_k(f_k,\mathbf{x}_i,y_i)}^{\text{risk of leaf $k$}}\underbrace{\prod_{j=1}^{K-1}[\mathds{1}_{g_j(\mathbf{x}_i)>0}]^{\mathbf P_{k,j}}[\mathds{1}_{g_j(\mathbf{x}_i)\leq 0}]^{\mathbf N_{k,j}}}_{\text{$G_k(\cdot)=$ state of $\mathbf{x}_i$ in a tree}}
\end{align}
Note that by the definition of risk in \eqref{eq.point_risk}, the ERM problem can be viewed as a minimization over a function of indicators with respect to decisions: $g_1(\mathbf{x}),\allowbreak \ldots, \allowbreak g_{K-1}(\mathbf{x})$.
\section{Convex Re-formulation and Solution by Linear Programming}
A popular approach to solving ERM problems is to substitute indicators with convex upper-bounding surrogates, $\phi(z) \geq \indicator{z}$ and then to minimize the resulting surrogate risk. However, such strategy generally leads to a non-convex, multi-linear optimization problem. Previous attempts to solve problems of this form have focused on computationally costly alternating optimization approaches with guarantees of convergence only to a local minimum \cite{trapeznikov:2013b,nips_paper}. Rather than attempting to solve this non-convex surrogate problem, we instead reformulate the indicator empirical risk in \eqref{eq.erm_prod_ind} as a maximization over sums of indicators before introducing convex surrogate. Our approach yields a globally convex upper-bounding surrogate of the empirical loss function. In the next section, we derive this reformulation for a binary tree of depth 2 in Fig. \ref{fig:tree} before generalizing to arbitrary trees.
\subsection{Simple Tree Example}
Consider the decision system shown in Fig. \ref{fig:tree}. The goal is to learn the decision functions $g_1$, $g_{2}$, and $g_{3}$ that minimize the empirical risk \eqref{eq.erm_prod_ind}.
In reformulating the risk, it is useful to define the ''savings" for an example. The \emph{savings}, $\pi_k^i$, for an example $i$, represents the difference between the worst case outcome, $R_{max}$ and the risk $R_k(f_k,\mathbf{x}_i,y_i)$ for terminating and classifying at the $k$th leaf. The worst case risk is acquiring all sensors and incorrectly classifying: $R_{max}=1+\alpha \sum_{m} c_m$.
\begin{align}
\pi_k^i= R_{max}- R_k(f_k,\mathbf{x}_i,y_i)
= \mathds{1}_{f_k(\mathbf{x}_i)= y_i}+ \alpha \displaystyle \sum_{m \in S_k^C}c_m \label{eq.savings}
\end{align}
Here, $S_k^C$ is the complement set of sensors acquired along the path to leaf $k$ (the sensors not acquired on the path to leaf $k$). Note that the savings do not depend on the decisions, $g_j's$, that we are interested in learning.
For our example, there are only 4 leaf nodes and the state of terminating in a leaf is a encoded by a product of two indicators. For instance, to terminate in Leaf 1, $g_1(\mathbf{x}_i) \leq 0$ and $g_2(\mathbf{x}_i) \leq 0$. This empirical risk can be formulated by enumerating over the leaves and their associated risks:
\begin{align} \nonumber
R&(\mathbf g,\mathbf{x}_i,y_i)= \overbrace{\Big(R_{max}-\pi_1^i\Big)\mathds{1}_{g_1(\mathbf{x}_i)\leq 0}\mathds{1}_{g_{2}(\mathbf{x}_i)\leq 0}}^\text{Leaf 1} + \overbrace{\Big(R_{max}-\pi_2^i\Big)\mathds{1}_{g_1(\mathbf{x}_i)\leq 0}\mathds{1}_{g_{2}(\mathbf{x}_i)> 0}}^\text{Leaf 2}\\ &+ \underbrace{\Big(R_{max}-\pi_3^i\Big)\mathds{1}_{g_1(\mathbf{x}_i)> 0}\mathds{1}_{g_{3}(\mathbf{x}_i)\leq 0}}_\text{Leaf 3} + \underbrace{\Big(R_{max}-\pi_4^i\Big)\mathds{1}_{g_1(\mathbf{x}_i)> 0}\mathds{1}_{g_{2}(\mathbf{x}_i)> 0}}_\text{Leaf 4} \label{eq.example_prod}
\end{align}
Directly replacing every $\indicator{z}$ with an upper bounding surrogate such as a hinge loss, $\max[0,1+z] \geq \indicator{z}$, produces a non-convex bilinear objective due the indicator product terms. Bilinear optimization is computationally intractable to solve globally.
Rather than directly substituting surrogates and solving the non-convex minimization problem, we reformulate the empirical risk with respect to the indicators in the following theorem:
\begin{thm}\label{thm.erm_reform}
The empirical risk in \eqref{eq.example_prod} is equal to \eqref{eq.example_max}.
\begin{align}
R(g_1,g_{2},g_{3},\mathbf{x}_i,y_i)= R_{max}-\sum_{j=1}^4\pi_j^i&+
\max \Big [(\pi_3^i+\pi_4^i) \mathds{1}_{g_1(\mathbf{x}_i)\leq 0}+\pi_2^i\mathds{1}_{g_{2}(\mathbf{x}_i)\leq 0},\notag\\
(\pi_3^i+\pi_4^i)\mathds{1}_{g_1(\mathbf{x}_i)\leq 0}+\pi_1^i\mathds{1}_{g_{2}(\mathbf{x}_i)> 0},&
(\pi_1^i+\pi_2^i)\mathds{1}_{g_1(\mathbf{x}_i)> 0}+\pi_4^i\mathds{1}_{g_{2}(\mathbf{x}_i)\leq 0},\notag\\
&(\pi_1^i+\pi_2^i)\mathds{1}_{g_1(\mathbf{x}_i)> 0}+\pi_3^i\mathds{1}_{g_{3}(\mathbf{x}_i)> 0}\Big]\label{eq.example_max}
\end{align}
\end{thm}
\begin{proof}
Here, we provide a brief sketch of the proof. For full details please refer to the Appendix. We utilize the following two identities: $\indicator{A}\indicator{B} = \min [ \indicator{A}, \indicator{B}]$ and $\indicator{A}=1-\indicator{\bar A}$ and express the risk in \eqref{eq.example_prod} in terms of maximizations:
\begin{align}\label{eq.proofstep}
R \left(g_1,g_{2},g_{3},\mathbf{x}_i,y_i\right)=R_{max} -\sum_{j=1}^4\pi_j^i
+\pi_1^i \max\left(\mathds{1}_{g_1(\mathbf{x}_i)> 0},\mathds{1}_{g_{2}(\mathbf{x}_i)> 0}\right)\notag\\
+\pi_2^i \max\left(\mathds{1}_{g_1(\mathbf{x}_i)> 0},\mathds{1}_{g_{2}(\mathbf{x}_i)\leq 0}\right)
+\pi_3^i \max\left(\mathds{1}_{g_1(\mathbf{x}_i)\leq 0},\mathds{1}_{g_{3}(\mathbf{x}_i)> 0}\right)\notag\\
+\pi_4^i \max\left(\mathds{1}_{g_1(\mathbf{x}_i)\leq 0},\mathds{1}_{g_{3}(\mathbf{x}_i)\leq 0}\right)
\end{align}
Recall that the signs of $g_1,g_2,g_3$ encode a unique path for $\mathbf{x}_i$. So let us consider sign patterns for each path. For instance, to reach leaf 1, $g_1 \leq 0$ and $g_2 \leq 0$. In this case, by inspection of \eqref{eq.proofstep}, the risk is $(\pi_3^i+\pi_4^i) \indicator{ g_1 (\mathbf{x}_i) \leq 0} + \pi_2^i \indicator{g_2(\mathbf{x}_i) \leq 0}+$ constants. This is exactly the first term in the maximization in \eqref{eq.example_max}. We can perform such computation for each leaf (term in the max) in a similar fashion. And due to the interdependencies in \eqref{eq.proofstep}, the term corresponding to a valid path encoding will be the maximizer in \eqref{eq.example_max}.
\end{proof}
{\bf Risk Interpretability:} Intuitively, in the reformulated empirical risk in \eqref{eq.example_max}, each term in the maximization encodes a path to one of the $K$ leaves. The largest (active) term correspond to the path induced by the $g_j$'s for an example $\mathbf{x}_i$. Additionally, the weights on the indicators in \eqref{eq.example_max} represent the \emph{savings lost} if the argument of the indicator is active. For example, if the decision function $g_1(\mathbf{x}_i)$ is negative, leaves $3$ and $4$ cannot be reached by $\mathbf{x}_i$, and therefore $\pi_3^i$ and $\pi_4^i$, the savings associated with leaves $3$ and $4$, cannot be realized and are lost.
A distinct advantage of the reformulated risk in \eqref{eq.example_max} arises when replacing indicators with convex upper-bounding surrogates of the form $\phi(z)\geq \mathds{1}_{z\leq0}$. Introducing such surrogates in the original risk in \eqref{eq.example_prod} produces a bilinear function for which a global optimum cannot be efficiently found. In contrast, introducing convex surrogate functions in \eqref{eq.example_max} produces a convex upper-bound for the empirical risk.
\subsection{Extension to Arbitrary Binary Trees}
In this section, we generalize the empirical risk reformulation for any binary tree and present a convex surrogate. Consider a binary tree, $\mathcal T$, composed of $K-1$ internal nodes and $K$ leaves. As defined in \eqref{eq.savings}, each leaf has a corresponding savings $\pi_k^i$ that captures the difference between the worst case risk and the risk of classifying at leaf $k$.
Note that in the previous example, the risk in \eqref{eq.erm_prod_ind} consists of a max of $K$ terms. Each term is a weighted linear combinations of indicators, and each weight corresponds to the \emph{savings lost} if the decision inside the indicator argument is true. For an arbitrary binary tree of $K$ leaves, the risk has an analogous form.
Before stating the result, we define the weights for the linear combination in each term of the max. For an internal node $j$, we denote $C_j^n$ as the set of leaf nodes in a subtree corresponding to a negative decision $g_j(\mathbf{x})\leq 0$. And $C_j^p$ is the set of leaf nodes in a subtree corresponding to a positive decision. For instance in Fig. \ref{fig:tree}, $C_1^p=\{Leaf~3,Leaf~4\}$, and in our example \eqref{eq.example_max}, the weight multiplying $\indicator{g_1(\mathbf{x}_i) \leq 0}$ is the sum of these savings for leaves 3 and 4 (i.e. savings lost if $g_1 \leq 0$). Therefore, sets $C^{p/n}_j$ define which $\pi^i_k$'s contribute to a weight for a decision term.
For a compact representation, recall that the $k$th rows in matrices $\mathbf P$ and $\mathbf N$ define a path to leaf $k$ in terms of $g_1, \ldots, g_{K-1}$, and a non-zero $\mathbf P/\mathbf N_{k,j}$ indicates if $g_j \lessgtr 0$ is on the path to leaf $k$. So for each $\mathbf{x}_i$ and each leaf $k$, we introduce two positive weight row vectors of length $K-1$:
\begin{align*}
\mathbf w^i_{n,k} =
\left [\mathbf N_{k,1} \sum_{l \in C_1^p}\pi_l^i,\ldots, \mathbf N_{k,K-1} \sum_{l \in C_{K-1}^p}\pi_l^i \right ]\\
\mathbf w^i_{p,k} = \left [ \mathbf P_{k,1} \sum_{l \in C_1^n} \pi_l^i, \ldots, \mathbf P_{k,K-1} \sum_{l \in C_{K-1}^n} \pi_l^i \right ]
\end{align*}
The $j$th component of $\mathbf w^i_{n,k}$ multiplies $\indicator{g_j(\mathbf{x}_i) \leq 0}$ in the term corresponding to the $k$th leaf. For instance in our $4$ leaf example in \eqref{eq.example_max}, $\left ( \mathbf w^i_{n,1} \right)_1=\pi^i_3+\pi^i_4$. If $\mathbf P/\mathbf N_{k,j}$ is zero then decision $g_j \gtrless 0$ is not on the path to leaf $k$ and the weight is zero. Using these weight definitions, the empirical risk in \eqref{eq.example_max} can be extended to arbitrary binary trees:
\begin{cor}\label{cor:tree_emp_risk}
The empirical risk of tree $\mathcal T$ is:
\begin{align}\label{eq.erm_tree}
R(\mathbf g,\mathbf{x}_i,y_i)=R_{max}-\sum_{k=1}^K \pi^i_k \notag\\
+\max_{k \in \{1, \ldots, K\}} \mathbf w_{p,k}^i \begin{bmatrix} \mathds{1}_{g_1(\mathbf{x}_i)>0}\\ \vdots \\ \mathds{1}_{g_{K-1}(\mathbf{x}_i)>0} \end{bmatrix}
+\mathbf w_{n,k}^i \begin{bmatrix}
\mathds{1}_{g_1(\mathbf{x}_i)\leq 0}\\ \vdots \\ \mathds{1}_{g_{K-1}(\mathbf{x}_i)\leq 0}
\end{bmatrix}
\end{align}
\end{cor}
The proof of this corollary is included in the Appendix and follows the same steps as Thm. \ref{thm.erm_reform}.
The empirical risk in \eqref{eq.erm_tree} represents a scan over the paths to each leaf ($k=1,\ldots,K$), and the active term in the maximization corresponds to the leaf to which an observation is assigned by the decision functions $g_1,\ldots,g_{K-1}$. An important observation is that each term in the max in \eqref{eq.erm_tree} is a linear combination of indicators instead of a product as in \eqref{eq.erm_prod_ind}. This transformation enables us to upper-bound each indicator function with a convex surrogate, $\phi(z)$:
$\phi[g_j(\mathbf{x})] \geq \indicator{g_j(\mathbf{x}) >0}~,~\phi[-g_j(\mathbf{x})] \geq \indicator{g_j(\mathbf{x}) \leq 0}$
. And the result is a novel convex upper-bound on the empirical risk in \eqref{eq.erm_tree}. We denote this risk as ${R}_\phi(\mathbf g)$. And the optimization problem over a set of training examples, $\{\mathbf{x}_i,y_i\}_{i=1}^N$ and a family of decision functions $\mathcal{G}$:
$
\min_{\mathbf g \in \mathcal{G}} \sum_{i=1}^{N}{R}_\phi\left(\mathbf g,\mathbf{x}_i,y_i\right) \label{eq.erm_min}
$.
\subsection{Linear Programming}
There are many valid choices for the surrogate $\phi(z)$. However, if a hinge loss is used as an upper bound and $\mathcal G$ is a family of linear functions of the data then the optimization problem in \eqref{eq.erm_min} becomes a linear program (LP).
\begin{prop}\label{prop.lp_tree}
For $\phi(z)=\max(1-z,0)$ and linear decision functions $g_1,\allowbreak \ldots, \allowbreak g_{K-1}$, the minimization in \eqref{eq.erm_min} is equivalent to the following linear program:
\begin{align}
&\min_{\substack{g_1,\ldots,g_{K-1},\gamma^1,\ldots,\gamma^N \\ \alpha_1^1,\ldots,\alpha_{K-1}^N,\beta_1^1, \ldots, \beta_{K-1}^N}}\sum_{i=1}^{N} \gamma^i \label{eq.tree_lp}\\
\text{subject to:}\notag\\
&\gamma^i\geq \mathbf w^i_{p,k} \begin{bmatrix}\alpha_1^i\\ \vdots \\ \alpha_{K-1}^i \end{bmatrix} + \mathbf w^i_{n,k} \begin{bmatrix} \beta_1^i \\ \vdots \\ \beta_{K-1}^i \end{bmatrix}, \,\,\,\, \begin{matrix}i \in [N]\\k \in [K]\end{matrix},\notag\\
&\begin{matrix}
1+g_j(\mathbf{x}_i)\leq \alpha_j^i,\\
1-g_j(\mathbf{x}_i)\leq \beta_j^i,\\
\alpha_j^i\geq 0,\beta_j^i\geq 0,\\
\end{matrix} \,\,\,\,\,\, \begin{array}{l} i \in [N]\\k \in [K-1]\end{array}.\notag
\end{align}
\end{prop}
We introduce the variable $\gamma^i$ for each example $\mathbf{x}_i$ to convert from a maximization over leaves to a set of linear constraints. Similarly, the maximization within each hinge loss is converted to a set of linear constraints. The variables $\alpha_j^i$ upper-bound the indicator $\mathds{1}_{g_j(x_i)> 0}$ and the variables $\beta_j^i$ upper-bound the indicator $\mathds{1}_{g_j(x_i)\leq 0}$. Additionally, the constant terms in the risk are removed for notational simplicity, as these do not effect the solution to the linear program. For details please refer to Appendix.
{\bf Complexity:} Linear programming is a relatively well-studied problem, with efficient algorithms previously developed. Specifically, for $K$ leaves, $N$ training points, and a maximum feature dimension of $D$, we have $O(KD+KN)$ variables and $O(KN)$ constraints. The state of the art primal-dual methods for LP are fast in practice, with an expected number of iterations $O(\sqrt{n} \log n)$, where $n$ is the number of variables \cite{LP_probabilisty_complexity}.
\begin{comment}
The proposed linear programming approach in Prop. \ref{prop.lp_tree} allows for efficient training of sequential decision functions in trees, however assumes a fixed tree and sensor structure. One natural approach to learning this problem is to build the tree in a greedy fashion, starting with a single root node and iteratively expanding the tree. We propose expansion of the tree at each iteration by appending a wide variety of new leaves to each previous iteration and enforcing a sparsity constraint over the newly added leaves. Consider the tree $\mathcal{T}$ to which we add a new set of leaves, $S$, introducing additional decision functions $g_K,\ldots,g_{K+|S|}$. Introducing a group sparsity term over the leaves, the empirical risk minimization problem on the functions $\mathbf g = \{g_1,\ldots,g_{K+|S|}\}$ can be expressed:
\begin{align}\label{eq.erm_newleaves}
\min_{\mathbf g \in \mathcal{G}} \underbrace{\left(\sum_{i=1}^{N}R(\mathbf g,\mathbf{x}_i,y_i)\right)}_{\text{Empirical risk of the expanded tree $\mathcal{T}\cup S$}}+\lambda \underbrace{ \sum_{p \in S}\left(\max_{i \in \{1,\ldots,N\}}\sum_{k=1}^{K+|S|}\left(\mathbf{N}_{p,k}\mathbbm{1}_{g_k(x_i)\leq0}+\mathbf{P}_{p,k}\mathbbm{1}_{g_k(x_i)>0}\right)\right)}_{\text{Regularization over leaves in set $S$.}}
\end{align}
where $R(\mathbf g,\mathbf{x}_i,y_i)$ is the loss as defined in \ref{eq.erm_tree}. The $\ell_{\infty,1}$ regularization term in \eqref{eq.erm_newleaves} over the appended leaves in set $S$ enforces group sparsity over the leaves, with the regularization term $\lambda$ controlling the number of active new leaves from the set $S$. As in the case of a fixed tree structure, hinge-losses can be introduced to upper-bound the indicator functions, yielding a linear program:
\begin{align}
&\min_{\substack{g_1,\ldots,g_{K+|S|},\gamma^1,\ldots,\gamma^N \\ \alpha_1^1,\ldots,\alpha_{K+|S|}^N,\beta_1^1, \ldots, \beta_{K+|S|}^N\\ \kappa_{K},\ldots,\kappa_{K+|S|}}}\sum_{i=1}^{N} \gamma^i +\lambda\sum_{k=K}^{K+|S|}\kappa_k \label{eq.lp_tree_expand} \\
&\text{subject to:} ~~ \gamma^i\geq \mathbf w^i_{p,k} \begin{bmatrix}\alpha_1^i\\ \vdots \\ \alpha_{K-1}^i \end{bmatrix} + \mathbf w^i_{n,k} \begin{bmatrix} \beta_1^i \\ \vdots \\ \beta_{K-1}^i \end{bmatrix}, i \in [N], ~k \in [K] \notag\\
&1+g_j(\mathbf{x}_i)\leq \alpha_j^i,~~1-g_j(\mathbf{x}_i)\leq \beta_j^i,~~~~\alpha_j^i\geq 0,~~\beta_j^i\geq 0, \notag\\ & j \in [K+|S|], i \in [N] \notag\\
& \sum_{k=1}^{K+|S|}\mathbf{N}_{p,k}\beta_k^i+\mathbf{P}_{p,k}\alpha_k^i\leq \kappa_p, p \in S, j \in [N] \notag
\end{align}.
Starting with a single node, an iterative greedy algorithm follows. At each iteration, the current tree is expanded along each leaf to allow for any sensor to be subsequently acquired. The linear program in \eqref{eq.lp_tree_expand} is then solved for a predetermined number of active leaves in $S$ and inactive leaves are pruned, with the process repeated until a stopping criteria (maximum tree size or minimum performance improvement) is reached.
\end{comment}
\section{Experiments}
\begin{table}
\caption{Small dataset descriptions}
\label{table.dataset_info}
\begin{tabular}{llllll}
\hline\noalign{\smallskip}
Name & Classes & Stage 1 & Stage 2 & Stage 3 & Stage 4\\
\noalign{\smallskip}\hline\noalign{\smallskip}
letter & 26 & Pixel Count & Moments & Edge Features & -\\
landsat & 6 & Band 1 & Band 2 & Band 3 & Band 4\\
SUN Mech. Turk & 16 & Function & Materials & Surf./Spat. Prop.& -\\
Image Seg. & 7 & Location & Pixel Int. & Color & $\cdots$ \\
\noalign{\smallskip}\hline
\end{tabular}
\end{table}
We demonstrate performance of our proposed approach on two types of real world data sets. First, we demonstrate performance of the LP formulation on examples with fixed structures. In these cases, sensor subsets do not need to be learned and an exhaustive tree can be constructed over all subsets of sensors. We compare the performance of our approach to the alternating optimization scheme presented in \cite{trapeznikov:2013b} applied to the same tree, demonstrating efficiency and performance of our LP formulation. Next, we apply our proposed approach to data sets where dimensionality prevents exhaustive search through feature subsets and both feature subsets and tree structure must be learned along with decision functions in the tree. We show performance on classification data sets presented in \cite{kusner2014feature} and compare performance with Cost Sensitive Trees of Classifiers (CSTC) \cite{xu2013cost} and Approximately Submodular Trees of Classifiers (ASTC) \cite{kusner2014feature}.
For all feature subset classifiers used in our proposed approach and the alternating optimization approach, classifiers are trained using logistic loss on $2^{nd}$-order homogeneous polynomial expanded basis on the entire training set.
\textbf{Learning Decision Functions in Fixed Structures:} Fig. \ref{fig:exp_perf} shows performance of our proposed approach on 5 data sets where few sensors are used and fixed structure can be easily found. For the letter, landsat, and SUN Mechanical Turk data sets \cite{uci_repository,SUN_mechanical_turk}, a tree can be easily constructed using all possible feature subsets. For the image segmentation data set, \cite{uci_repository}, we fix a greedily constructed tree with 8 leaves.
For comparison, we use the alternating minimization approach proposed in \cite{trapeznikov:2013b}. Additionally, we also show performance of a simple myopic strategy for a baseline comparison on these example. The LP approach generally performs comparably to the non-convex alternating optimization approach. Additionally, as shown in Table \ref{table.performance}, the LP is dramatically faster during training time with comparable performance to the alternating training approach.\footnote{All computations were performed on an Intel I5 M430 CPU @ 2.27 GHz with 4 cores.} Note that we do not compare performance to ASTC or CSTC for these experiments. The purpose of these examples is to show the efficacy of the LP for training sequential decision functions independent of feature subsets, classifier design, or tree structure. Both CSTC and ASTC simultaneously learn the feature subsets, classifiers, and decision functions, limiting comparison.
\begin{figure*}[htb!]
\centering
\subfigure[letter]{\includegraphics[width=.49\linewidth]{letter_perf.eps}}
\subfigure[landsat]{\includegraphics[width=.49\linewidth]{landsat_perf.eps}}\\
\subfigure[Image Seg.]{\includegraphics[width=.49\linewidth]{image_seg_perf.eps}}
\caption{Comparison of error vs. average budget trade-off between a myopic, AM~\cite{trapeznikov:2013b}, our LP method. LP clearly out performs the myopic approach, and generally matches or exceeds the non-convex approach with the added benefit of reduced computational cost, repeatability, and guaranteed convergence.}
\label{fig:exp_perf}
\end{figure*}
\begin{table}
\caption{Average percentage of the budget required to achieve a desired error rate chosen to be close to the error achieved using the entire set of features (approximately $95\%$ of the improvement gained using all features compared the initial features). The percentage of the budget required is with respect to the maximum budget. The training time is the amount of time (in seconds) required to learn a policy for a fixed budget trade-off parameter $\alpha$.}
\label{table.performance}
\begin{tabular}{lllllll}
\hline\noalign{\smallskip}
Dataset &\begin{tabular}{c}Target\\ Errors\end{tabular}& Myopic & AM & LP & \begin{tabular}{c}AM Train\\ Time(sec)\end{tabular}& \begin{tabular}{c}LP Train\\ Time(sec)\end{tabular}\\
\noalign{\smallskip}\hline\noalign{\smallskip}
letter &$40\%$& $73\%$ & $48\%$ & $49\%$ &$93.56$ & $57.03$ \\
landsat &$15\%$& $100\%$ & $75\%$ & $75\%$ & $186.0$ &$108.7$ \\
SUN Mech. Turk &$40.4\%$& $99\%$ & $90\%$ & $90\%$ & $2818.9$ & $71.08$\\
Image Seg. &$9\%$& $56\%$ & $21\%$ & $26\%$ & $46.26$ & $16.46 $\\
\noalign{\smallskip}\hline
\end{tabular}
\end{table}
\textbf{Learning Tree Structure:} We next applied our proposed approach to three data sets where learning a tree over all feature subsets is infeasible. For all three datasets, we learn the set of subsets as described in Section \ref{sec.struct}, with a total of 16 subsets of features (leaves of the tree) used. For the MiniBooNE and forest data sets, the proposed approach outperforms CSTC, with performance exceeding ASTC for the forest data set and matching ASTC for MiniBooNE. On the CIFAR dataset, the proposed LP approach matches CSTC when using 50 features, but otherwise is generally outperformed by both CSTC and ASTC. We attribute this to the limited complexity ($2^{nd}$ order homogeneous polynomial) of the classification functions, which requires more features to gain flexibility to accurately partition the data.
\begin{figure*}[htb!]
\centering
\subfigure[Forest]{\includegraphics[width=.49\linewidth]{forest_tree_perf.eps}}
\subfigure[MiniBooNE]{\includegraphics[width=.49\linewidth]{miniboone_tree_perf.eps}}\\
\subfigure[CIFAR]{\includegraphics[width=.49\linewidth]{cifar_tree_perf.eps}}
\caption{Plot of our proposed approach (LP tree) and CSTC on three real world data sets. On all three data sets, LP tree generally outperforms CSTC, producing high levels of classification accuracy with very low budgets.}
\label{fig:exp_perf2}
\end{figure*}
\bibliographystyle{spbasic}
\section{Conclusion}
|
2,877,628,089,031 | arxiv |
\section{Introduction}
\emph{Cyclic} and \emph{non-wellfounded} proofs have been studied by a number of authors as an alternative to proofs by induction.
This includes cyclic systems for fragments of the modal $\mu$-calculus, e.g.\ \cite{NiwWal96:games-mucalc,SprDam03:mucalc,DaxHofLan06:multl,BaeDouHirSau16:multl,Dou17:multl,AfsLei17:cut-free-mucalc}, structural proof theory for logics with fixed-points, e.g.\ \cite{San02:circ-proofs-categorical-semantics,ForSan13:cuts-circ-proofs-sem-cut-elim,For14:phdthesis,BaeDouSau16:cut-elim}, (automated) proofs of program termination in separation logic, e.g.\ \cite{BroBorCal08:cyc-proofs-term,BroDisPet11:aut-cyc-ent,BroRow17:aut-cyc-term} and, in particular, cyclic systems for first-order logic with inductive definitions, e.g.\ \cite{Bro05:cyc-proofs-folid,Bro06:phdthesis,BroSim07:comp-seq-calc-ind-inf-desc,BroSim11:seq-calc-ind-inf-desc}.
Due to the somewhat implicit nature of invariants they define, cyclic systems can be advantageous for metalogical analysis, for instance offering better algorithms for proof search,~e.g.~\cite{BroGorPet12:gen-cyc-prover,DasPou17:cut-free-cyc-prf-sys-kl-alg}.
Cyclic proofs may be seen as more intuitively analogous to proofs by `infinite descent' than proofs by induction (see, e.g., \cite{Sim17:cyclic-arith}); this subtle difference is enough to make inductive invariants rather hard to generate from cyclic proofs.
Indeed it was recently shown that simulating cyclic proofs using induction is not possible for some sub-arithmetic languages \cite{BerTat17:cfolid-neq-folid}, but becomes possible once arithmetic reasoning is available \cite{Sim17:cyclic-arith,BerTat17:lics}.
\emph{Cyclic arithmetic} was proposed as a general subject of study by Simpson in \cite{Sim17:cyclic-arith}.
Working in the language of arithmetic, it replaces induction by non-wellfounded proofs with a certain `fairness' condition on the infinite branches.
The advantage of this approach to infinite proof theory as opposed to, say, infinite well-founded proofs via an $\omega$-rule (see, e.g., \cite{Sch77:prf-th}), is that it admits a notion of \emph{finite proofs}: those that have only finitely many distinct subproofs, and so may be represented by a finite (possibly cyclic) graph.
Cyclic arithmetic itself is to cyclic proofs what Peano arithmetic is to traditional proofs: it provides a general framework in which many arguments can be interpreted and/or proved in a uniform manner, and this is one reason why it is an interesting subject of study.
This is already clear from, say, the results of \cite{BerTat17:lics}, where the study of cyclic proofs for first-order logic with inductive definitions relied on an underlying arithmetic framework.
We elaborate further on this in Sect.~\ref{sect:conc}.
\subsection*{Contribution}
In \cite{Sim17:cyclic-arith}, Simpson showed that Peano Arithmetic ($\mathsf{PA}$) is able to simulate cyclic reasoning by proving the {soundness} of the latter in the former. (The converse result is obtained much more easily.)
Nonetheless, several open questions remain from \cite{Sim17:cyclic-arith}, concerning constructivity, normalisation, logical complexity and proof complexity for cyclic and non-wellfounded proofs.
In this work we address the \emph{logical complexity} and \emph{proof complexity} of proofs in Cyclic Arithmetic ($\mathsf{CA}$), as compared to $\mathsf{PA}$.
Namely, we study how quantifier alternation of proofs in one system compares to that in the other, and furthermore how the size of proofs compare.
Writing $\CSn{n}$ for (the logical consequences of) cyclic proofs containing only $\Sin{}{n}$ formulae, we show, for $n \geq 0$:
\begin{enumerate}[start=1,label={(\arabic*)}]
\item\label{item:cyc-sim-ind} $\ISn{n+1} \subseteq \CSn n$ over $\Pin{}{n+1}$ theorems (Sect.~\ref{sect:cyc-sim-ind}, Thm.~\ref{thm:cyclic-sim-ind}).
\item\label{item:proof-complexity} $\mathsf{CA}$ and $\mathsf{PA}$ proofs of the same theorem differ only exponentially in size (Sect.~\ref{sect:ind-sim-cyc}, Thm.~\ref{thm:elementary-simulation}).
\item\label{item:ind-sim-cyc} $\CSn n \subseteq \ISn{n+1}$ over all theorems (Sect.~\ref{sect:nonuniform-ind-sim-cyc}, Thm.~\ref{thm:nonuniform-ind-sim-cyc}).
\end{enumerate}
\ref{item:cyc-sim-ind} is obtained by proof theoretic techniques, relying on normal forms and structural manipulations of Peano Arithmetic proofs. It improves upon the natural result that $\ISn n \subseteq \CSn n$, although induces a non-elementary blowup in the size of proofs.
\ref{item:proof-complexity} is obtained via a certain `uniformisation' of the approach of \cite{Sim17:cyclic-arith}. In particular, by specialising the key intermediate results to the case of cyclic proofs, we are able to extract small $\mathsf{PA}$ proofs of some required properties of infinite word automata from analogous ones in `second-order' (SO) arithmetic.
Finally, \ref{item:ind-sim-cyc} is obtained by calibrating the argument of \ref{item:proof-complexity}
with recent results on the reverse mathematics of B\"uchi's theorem \cite{KMPS16:buchi-reverse},
allowing us to bound the logical complexity of proofs in the simulation.
Together, these results almost completely characterise the logical and proof complexity theoretic strength of cyclic proofs in arithmetic, answering the questions (ii) and (iii), Sect.~7 of \cite{Sim17:cyclic-arith}.
After demonstrating these results, we give a metamathematical analysis of provability in cyclic theories, in particular showing that the consistency of $\CSn n $ is provable in $\ISn{n+2}$ but not $\ISn{n+1}$, by appealing to a form of \emph{G\"odel incompleteness} for cyclic theories.
We also use these observations to show that certain formulations of McNaughton's theorem, that every nondeterministic B\"uchi automaton has an equivalent deterministic parity (or Rabin, Muller, etc.) automaton, are not provable in the SO theory $\mathsf{RCA}_0$.
This partially resolves the question of the logical strength of McNaughton's theorem left open in \cite{KMPS16:buchi-reverse}.
\subsection*{Structure of the paper}
In Sects.~\ref{sect:prelims-pa-pt} and \ref{sect:prelims-ca-aut} we introduce some preliminaries on Peano Arithmetic, proof theory, cyclic proofs and automaton theory.
In Sect.~\ref{sect:cyc-sim-ind} we present \ref{item:cyc-sim-ind} and give an example of the translation in App.~\ref{sect:php-case-study}. The contents of Sects.~\ref{sect:prelims-pa-pt}, \ref{sect:prelims-ca-aut} and \ref{sect:cyc-sim-ind} and App.~\ref{sect:php-case-study} are more-or-less self contained and should be accessible to the general proof theorist.
We briefly introduce some SO theories of arithmetic in Sect.~\ref{sect:so-theories} that are conservative over the fragments of $\mathsf{PA}$ we need in order to conduct some of the intermediate arguments on infinite word automata.
In Sect.~\ref{sect:ind-sim-cyc} we present \ref{item:proof-complexity}, and in Sect.~\ref{sect:nonuniform-ind-sim-cyc} we adapt the argument to obtain \ref{item:ind-sim-cyc}.
In Sect.~\ref{sect:further-metalogical} we give our metamathematical analysis of cyclic theories, and in Sect.~\ref{sect:red-to-det} we explain their consequences for the logical strength of certain forms of McNaughton's theorem.
We conclude with some further remarks and perspectives in Sect.~\ref{sect:conc}, including a comparison with the results of \cite{Sim17:cyclic-arith} and \cite{BerTat17:lics}.
For Sects.~\ref{sect:so-theories}, \ref{sect:ind-sim-cyc}, \ref{sect:nonuniform-ind-sim-cyc} and \ref{sect:red-to-det} it would be helpful for the reader to have some background in subsystems of second-order arithmetic (see e.g.~\cite{Sim09:reverse-math,Hir14:reverse-math}) and $\omega$-automaton theory (see e.g.~\cite{Tho97:aut-chapt}).
For Sect.~\ref{sect:further-metalogical} it would be helpful for the reader to have some background in the metamathematics of first-order arithmetic (see e.g.~\cite{HajPud:93}).
Nonetheless we aim to give sufficient details for the general proof theorist to appreciate all the content herein.
\section{Preliminaries on first-order arithmetic proof theory}
\label{sect:prelims-pa-pt}
We present only brief preliminaries, but the reader is encouraged to consult, e.g., \cite{Bus98:handbook-of-pt} for a more thorough introduction to first-order arithmetic.
We work in first-order (FO) logic with equality, $=$, with variables written $x,y,z $ etc., terms written $s,t,u$ etc., and formulae written $\phi, \psi$ etc., construed over the logical basis $\{ \neg, \vee ,\wedge , \exists , \forall \}$.
We will usually assume formulae are in \textbf{De Morgan normal form}, with negation restricted to atomic formulae.
Nonetheless, we may write $\neg \phi$ for the De Morgan `dual' of $\phi$, defined as follows:
\[
\neg \neg \phi := \phi
\quad
\begin{array}{rcl}
\neg (\phi \wedge \psi) &:= &\neg \phi \vee \neg \psi
\\
\neg (\phi \vee \psi) &:= &\neg \phi \wedge \neg \psi
\end{array}
\quad
\begin{array}{rcl}
\neg \forall x . \phi &:= &\exists x . \neg \phi
\\
\neg \exists x . \phi &:= &\forall x . \neg \phi
\end{array}
\]
We also write $\phi \supset \psi $ for $\neg \phi \vee \psi$ and $\phi \equiv \psi $ for $(\phi \supset \psi ) \wedge (\psi \supset \phi)$.
FO logic has equality `built-in', i.e.\ we always assume the following axioms are present:
\begin{enumerate}[start=1,label={(eq\arabic*)}]
\item\label{item:eq-ax-refl} $\forall x . x = x $.
\item\label{item:eq-ax-fn} $\forall \vec x , \vec y . ( (x_1 = y_1 \wedge \cdots \wedge x_k = y_k) \supset f(\vec x ) = f(\vec y)$, for each $k\in \mathbb{N}$ and each function symbol $f$ of arity $k$.
\item\label{item:eq-ax-pred} $\forall x,y . (((x_1 = y_1 \wedge \cdots \wedge x_k = y_k)\wedge P(\vec x)) \supset P(\vec y))$, for each $k \in \mathbb{N}$ and each predicate symbol $P$ of arity $k$.
\end{enumerate}
\noindent
Following \cite{Sim17:cyclic-arith}, the \textbf{language of arithmetic} (with inequality) is formulated as $\{ 0, \succ, +, \times , < \}$, with their usual interpretations over $\mathbb{N}$.
A \textbf{theory} is a set $T$ of closed formulae over this language.
We write $T\vdash \phi$ if $\phi$ is a logical consequence of $T$.
We write $T_1 \subseteq T_2 $ if $T_1 \vdash \phi$ implies $T_2 \vdash \phi$, and $T_1 = T_2$ if $T_1 \subseteq T_2$ and $T_2 \subseteq T_1$.
The theory of \textbf{Robinson arithmetic} (with inequality), written $\mathsf{Q}$, is axiomatised by:
\begin{enumerate}[start=1,label={(Q\arabic*)}]
\item $\forall x . \succ x \neq 0 $.
\item $\forall x, y. (\succ x = \succ y \supset x = y)$.
\item\label{item:rob-succ-non-zer} $\forall x . (x \neq 0 \supset \exists y . x = \succ y)$.
\item $\forall x .\ x + 0 = x$.
\item $\forall x , y .\ x + \succ y = \succ (x + y)$.
\item $\forall x .\ x \cdot 0 = 0 $.
\item $\forall x , y .\ x \cdot \succ y = x\cdot y + x$.
\medskip
\item\label{item:rob-ineq-dfn} $\forall x , y . (x<y \equiv \exists z . (x + \succ z = y) )$
\end{enumerate}
\noindent
Notice that, above and elsewhere, we may write $\cdot$ instead of $\times $ in terms, or even omit the symbol altogether, and we assume it binds more strongly than $+$.
We also write $\forall x< t. \phi$ and $\exists x < t. \phi$ as abbreviations for $\forall x . (x< t \supset \phi)$ and $\exists x. ( x < t \wedge \phi )$ resp.
Formulae with only such quantifiers are called \textbf{bounded}.
As usual, we may assume that $\mathsf{Q}$ is axiomatised by the universal closures of bounded formulae. In particular the existential quantifiers in axioms \ref{item:rob-succ-non-zer} and \ref{item:rob-ineq-dfn} above may be bounded by $x$ and $y$ resp., provably under quantifier-free induction.
We will implicitly assume this bounded axiomatisation for the sequent calculus formulation of arithmetic later.
\begin{rem}
\label{rmk:diff-ax-simpson}
Our basic axioms and, later, our inference rules differ slightly from those in \cite{Sim17:cyclic-arith}, however it is routine to see that the theories $\mathsf{PA}$ and $\mathsf{CA}$ defined in this work coincide with those of \cite{Sim17:cyclic-arith}.
In particular the axiomatisations are equivalent once even open induction is present (this is weaker than any theory we will consider).
We chose a slightly different presentation so that we could readily apply certain metalogical results, such as Thm.~\ref{thm:free-cut-elim}, with no intermediate proof manipulation.
\end{rem}
\begin{defi}
[Arithmetical hierarchy]
For $n\geq 0$, we define:
\begin{itemize}
\item $\Din{}{0} = \Pin{}{0} = \Sin{}{0}$ is the class of bounded formulae.
\item $\Sin{}{n+1}$ is the class of formulae of the form $\exists \vec x . \phi$, where $\phi \in \Pin{}{n}$.
\item $\Pin{}{n+1}$ is the class of formulae of the form $\forall \vec x . \phi$, where $\phi \in \Sin{}{n}$.
\end{itemize}
\end{defi}
\noindent
Notice in particular that, by definition of De Morgan normal form, if $\phi \in \Sin{}{n}$ then $\neg \phi \in \Pin{}{n}$ and vice-versa.
{In practice we often consider these classes of formulae up to logical equivalence.}
We say that a formula is in $\Din{}{n}$ (in a theory $T$) if it is equivalent (resp.\ provably equivalent in $T$) to both a $\Sin{}{n}$ and $\Pin{}{n}$ formula.
\begin{defi}
[Arithmetic]
\textbf{Peano Arithmetic} ($\mathsf{PA}$) is axiomatised by $\mathsf{Q}$ and the axiom schema of \textbf{induction}:
\begin{equation}
\label{eqn:ind-ax}
( \phi(0) \wedge \forall x . (\phi(x) \supset \phi(\succ x)) ) \supset \forall x . \phi(x)
\end{equation}
For a class of formulae $\Phi$, we write $\CIND{\Phi}$ for the set of induction axiom instances when $\phi \in \Phi$ in \eqref{eqn:ind-ax}.
We write $\IC{\Phi}$ for the theory $\mathsf{Q} + \CIND{\Phi}$.
%
\end{defi}
The following is a classical result:
\begin{prop}
[See e.g.~\cite{Bus98:handbook-of-pt,Kay91:models-of-pa}]
\label{prop:ipn-equals-isn}
For $n\geq 0$, we have $\ISn{n} = \IPn{n}$.
\end{prop}
\subsection{A sequent calculus presentation of $\mathsf{PA}$}
\begin{figure}
\[
\begin{array}{ccc}
\vlinf{
\mathit{id}
}{}{\Gamma, \phi \Rightarrow \Delta, \phi}{}
\ & \
\vlinf{=_1}{}{\Gamma \Rightarrow \Delta , t=t}{}
\ & \
\vlinf{=_2}{}{\Gamma, s_1 = t_1 , \dots , s_k = t_k \Rightarrow \Delta, f(\vec s) = f(\vec t)}{}
\\
\vlinf{\lefrul\neg}{}{\Gamma , \phi, \neg \phi \Rightarrow \Delta}{}
\ & \
\vlinf{\rigrul \neg}{}{\Gamma \Rightarrow \Delta, \phi, \neg \phi}{}
\ & \
\vlinf{=_3}{}{\Gamma, s_1 = t_1 , \dots , s_k = t_k , P(\vec s)\Rightarrow \Delta, P(\vec t)}{}
\end{array}
\]
\medskip
\[
\begin{array}{ccc}
\vlinf{
\theta\text{-}
\mathit{sub}
}{}{\theta(\Gamma) \Rightarrow \theta(\Delta)}{\Gamma \Rightarrow \Delta}
\ & \
\vliinf{
\mathit{cut}
}{}{\Gamma \Rightarrow \Delta}{\Gamma \Rightarrow \Delta , \phi}{\Gamma, \phi \Rightarrow \Delta}
\ & \
\vlinf{
\mathit{wk}
}{}{\Gamma, \Gamma'\Rightarrow \Delta , \Delta'}{\Gamma \Rightarrow \Delta}
\end{array}
\]
\medskip
\[
\hspace{-1em}
\begin{array}{cccc}
\vliinf{\lefrul \vee}{}{\Gamma, \phi \vee \psi \Rightarrow \Delta }{ \Gamma , \phi \Rightarrow \Delta }{\Gamma, \psi \Rightarrow \Delta}
&
\vlinf{\lefrul \wedge}{
}{\Gamma , \phi_0 \wedge \phi_1 \Rightarrow \Delta}{\Gamma, \phi_i \Rightarrow \Delta }
&
\vlinf{\lefrul \exists}{
}{\Gamma , \exists x . \phi \Rightarrow \Delta}{\Gamma , \phi [a/x] \Rightarrow \Delta}
& \vlinf{\lefrul \forall}{}{\Gamma , \forall x . \phi \Rightarrow \Delta}{\Gamma, \phi [t/x] \Rightarrow \Delta}
\\
\noalign{\medskip}
\vliinf{\rigrul \wedge}{}{\Gamma \Rightarrow \Delta, \phi \wedge \psi}{\Gamma \Rightarrow \Delta, \phi }{\Gamma \Rightarrow \Delta, \psi }
&
\vlinf{\rigrul \vee}{
}{\Gamma\Rightarrow \Delta , \phi_0 \vee \phi_1}{ \Gamma \Rightarrow \Delta , \phi_i}
&
\vlinf{\rigrul \forall}{
}{\Gamma \Rightarrow \Delta, \forall x . \phi}{\Gamma \Rightarrow \Delta, \phi [a/x] }
& \vlinf{\rigrul \exists}{}{\Gamma \Rightarrow \Delta, \exists x. \phi}{\Gamma\Rightarrow \Delta, \phi [t/x] }
\end{array}
\]
\caption{The sequent calculus for FO logic with equality, where $a$ occurs only as indicated and $i \in \{0,1\}$.}
\label{fig:seq-calc}
\end{figure}
We will work with a standard sequent calculus presentation of FO logic, given in Fig.~\ref{fig:seq-calc}, where $i \in \{0,1\}$ and $a$, known as the `eigenvariable', is fresh, i.e.\ does not occur free in the lower sequent.
Two important considerations are that we work with cedents as \emph{sets}, i.e.\ there is no explicit need for contraction rules, and that we have an explicit \textbf{substitution} rule.
{In the $\theta$-$\mathit{sub}$ rule the `substitution' $\theta$ is a mapping from variables to terms, which is extended in the natural way to cedents.}
Substitution is important for the definition of a cyclic arithmetic proof in the next section, but does not change provability in usual proofs.
The sequent calculus for $\mathsf{Q}$ is obtained from the FO calculus in the language of arithmetic by adding appropriate initial sequents for each instantiation of an axiom of $\mathsf{Q}$ by terms.
For theories extending $\mathsf{Q}$ by (at least quantifier-free) induction, we assume that these intial sequents contain only $\Din{}{0}$ formulae by appropriately bounding the existential quantifiers. The schema $\CIND{\Phi}$, for $\Phi$ closed under subformulas and substitution, is implemented in the calculus by adding the induction rule,
\[
\vliinf{\mathit{ind}}{}{\Gamma \Rightarrow \phi(t), \Delta}{\Gamma \Rightarrow \phi (0), \Delta}{\Gamma , \phi(a) \Rightarrow \phi(\succ a), \Delta }
\]
for formulae $\phi \in \Phi$.
{Here we require $a$ to not occur free in the lower sequent.} {Notice that this satisfies the subformula property, in the `wide' sense of FO logic, i.e.\ up to substitution.}
For fragments of $\mathsf{PA}$ with induction axioms of bounded logical complexity, we also have the \emph{bounded quantifier rules}:
\[
\begin{array}{cc}
\vlinf{}{}{\Gamma,\exists x < s . \phi(x) \Rightarrow \Delta}{\Gamma, a < s , \phi(a) \Rightarrow \Delta}
\quad &\quad
\vlinf{}{}{\Gamma, t< s \Rightarrow \Delta, \exists x < s . \phi(x)}{\Gamma \Rightarrow \Delta , \phi(t)}
\\
\noalign{\bigskip}
\vlinf{}{}{ \Gamma \Rightarrow \Delta , \forall x < s . \phi(x) }{ \Gamma,a< s \Rightarrow \Delta , \phi(a) }
\quad &
\quad
\vlinf{}{}{\Gamma , t< s , \forall x < s . \phi(x) \Rightarrow \Delta}{\Gamma , \phi(t) \Rightarrow \Delta}
\end{array}
\]
In all cases the eigenvariable $a$ occurs only as indicated,
The following normalisation result is well-known in the proof theory of arithmetic, and will be one of the main structural proof theoretic tools in this work:
\begin{thm}
[Free-cut elimination, e.g.\ \cite{Bus98:handbook-of-pt}]
\label{thm:free-cut-elim}
Let $\mathcal S$ be a sequent system extending FO by the induction rule and some other nonlogical rules/axioms closed under substitution.
Then any $\mathcal S$-proof can be effectively transformed into one of the same conclusion containing only (substitution instances of) subformulae of the conclusion, an induction formula or a formula occurring in another nonlogical step.
\end{thm}
Naturally, this applies to the various fragments of $\mathsf{PA}$ that we consider. In particular, notice that a free-cut free proof in $\ISn{n}$ or $\IPn{n}$ of sequents containing only $\Sin{}{n}$ or $\Pin{}n$ formulae, resp., contains just $\Sin{}{n}$ or $\Pin{}{n}$ formulae, resp.
It is well known that Thm.~\ref{thm:free-cut-elim} can itself be proved within $\ISn 1$ and even weaker theories (see, e.g., \cite{HajPud:93}), under an appropriate coding of mathematical objects. We use this observation later in Sect.~\ref{sect:further-metalogical}.
\medskip
We say that a sequent is $\Sin {} n$ (or $\Pin{}{n}$) if it contains only $\Sin{}{n}$ (resp.\ $\Pin {} n$) formulae.
A slight issue that will be relevant later in Sect.~\ref{sect:cyc-sim-ind} is that we have not defined $\Sin{}{n}$ and $\Pin{}{n}$ as being syntactically closed under positive Boolean combinations, even if semantically we know that they are.
In fact, this does not cause a problem for the result above, since we can always prenex `on the fly' in a proof by cutting against appropriate derivations.
For instance, in a proof, a step of the form,
\[
\vliinf{\wedge}{}{ \Gamma \Rightarrow \Delta , \forall x. \phi \wedge \forall y . \psi }{\Gamma \Rightarrow \Delta, \forall x . \phi}{\Gamma \Rightarrow \Delta, \forall y . \psi}
\]
may be locally replaced by a derivation of the form:
\[\vlderivation{
\vliin{\mathit{cut}}{}{ \Gamma \Rightarrow \Delta, \forall x , y. (\phi \wedge \psi) }{
\vlhy{\Gamma \Rightarrow \Delta, \forall x . \phi}
}{
\vliin{\mathit{cut}}{}{ \Gamma , \forall y . \psi \Rightarrow \Delta, \forall x,y . (\phi \wedge \psi) }{
\vlhy{\Gamma \Rightarrow \Delta , \forall y . \psi}
}{
\vliq{}{}{\forall x . \phi , \forall y . \psi \Rightarrow \forall x , y . (\phi \wedge \psi) }{\vlhy{}}
}
}
}
\]
In a similar way we will often assume that a `block' of existential or universal quantifiers is coded by a single quantifier, using pairings and G\"odel $\beta$ functions, whose basic properties are all formalisable already in $\IDn 0$ (see, e.g., \cite{Bus98:handbook-of-pt}).
\section{Preliminaries on cyclic arithmetic and automata}
\label{sect:prelims-ca-aut}
Before presenting `cyclic arithmetic', we will present the general notion of non-wellfounded proofs in arithmetic, from \cite{Sim17:cyclic-arith}.
By convention, we say \emph{binary tree} to mean a nonempty set $T\subseteq \{0,1\}^*$ that is prefix-closed, i.e.\ if $\sigma i \in T$ then $\sigma \in T$.
We construe such $T$ as a bona fide tree with nodes $T$ and directed edges from $\sigma$ to $\sigma i $, if $\sigma i \in T$, for $i \in \{0,1\}$.
The empty word, $\varepsilon$, is the \emph{root} of $T$.
\begin{defi}
A \textbf{preproof} is a possibly infinite binary tree labelled by sequents and rules in a locally correct manner in the calculus for $\mathsf{Q}$. {Following \cite{Sim17:cyclic-arith}, we treat inference steps as nodes of the tree and sequents as edges.}
A preproof is \textbf{regular} if it has only finitely many distinct (labelled) subtrees or, equivalently, if it is the unfolding of a finite labelled directed graph, possibly with cycles.
\end{defi}
The following notions are variants of those from Dfns.~1 and 2 in \cite{Sim17:cyclic-arith}:
\begin{defi}
[Precursors, traces, $\infty$-proofs]
Let $(\Gamma_i \Rightarrow \Delta_i )_{i \geq 0} $ be an infinite branch through a preproof.
For terms $t, t'$ we say that $t'$ is a \dfntrm{precursor} of $t$ at $i$ if one of the following holds:
\begin{enumerate}[label={\roman*)}]
\item\label{item:prec-sub} $\Gamma_i \Rightarrow \Delta_i$ concludes a $\theta$-$\mathit{sub}$-step and $t $ is $ \theta (t')$.
\item\label{item:prec-eq} $\Gamma_i \Rightarrow \Delta_i$ concludes any other step and $t' = t$ occurs in $ \Gamma_i$.
\item\label{item:prec-id} $\Gamma_i \Rightarrow \Delta_i$ concludes any other step and $t'$ is $t$.
\end{enumerate}
\noindent
A \dfntrm{trace} along $(\Gamma_i \Rightarrow \Delta_i)_{i \geq 0}$ is a sequence $(t_i)_{i \geq n}$, for some $n\geq 0$, such that whenever $i\geq n$ the term $t_i$ occurs in $\Gamma_i \Rightarrow \Delta_i $ and,
\begin{enumerate}[label={(\alph*)}]
\item
$t_{i+1}$ is a precursor of $t_i$ at $i$; or
\item\label{item:progress-point}
the atomic formula $t_{i+1} < t $ occurs in $\Gamma_{i+1}$, where $t$ is a precursor of $t_i$ at $i$.
\end{enumerate}
When
\ref{item:progress-point}
holds, we say that the trace \dfntrm{progresses} at $i+1$.
An \dfntrm{$\infty$-proof} is a preproof for which any infinite branch has a trace that progresses infinitely often.
If it is regular then we simply call it a \dfntrm{cyclic proof}.
$\mathsf{CA}$ is the theory induced by cyclic proofs in the calculus for $\mathsf{Q}$.
\end{defi}
\begin{rem}
\label{rmk:traces-occurrences}
When defining explicit traces, for the precursor case \ref{item:prec-id}, we will typically not worry about whether the term $t_i $ in a trace occurs in the sequent or not. All that matters is that, if the current step is $\lefrul \exists$, $\rigrul \forall$ or $\mathit{sub}$, $t_{i}$ does not contain the associated eigenvariables.\footnote{We say that $a$ is an eigenvariable of a $\theta$-$\mathit{sub}$ step if it is in the support of the substitution $\theta$.} As long as we satisfy this constraint we may simply consider an equivalent proof that prepends $t_i=t_i$ to the antecedent to make sure that $t_i$ `occurs'. We use this assumption implicitly in the remainder of this work.
\end{rem}
The reader may consult \cite{Sim17:cyclic-arith} for several examples of $\infty$-proofs.
Notably, $\infty$-proofs are sound and {complete} for the standard model $\mathbb{N}$ (Thm.~4, \cite{Sim17:cyclic-arith}).
(Similar results for other logics, with respect to standard models, were known before \cite{Bro06:phdthesis,BroSim11:seq-calc-ind-inf-desc}.)
We recall the proof of soundness since we will have to formalise a variant of it in Sect.~\ref{sect:ind-sim-cyc}, and also since the quantifier case in the argument of \cite{Sim17:cyclic-arith} is omitted, whereas this subtlety will need some consideration when it is formalised.
\begin{prop}
[Soundness of $\infty$-proofs]
\label{prop:sound-cyclic}
If $\pi$ is an $\infty$-proof of $\phi$, then $\mathbb{N} \models \phi$.
\end{prop}
\begin{proof}
Suppose otherwise, i.e.\ $\mathbb{N} \models \neg \phi$.
We will inductively construct an infinite branch $(\Gamma_i \Rightarrow \Delta_i)_{i \geq 0}$ of $\pi$ and associated assignments $\rho_i$ of natural numbers to each sequent's free variables, such that $\mathbb{N} , \rho_i \nvDash \Gamma_i \Rightarrow \Delta_i$.
Assuming $\phi$ is closed (by taking its universal closure), we set $\Gamma_0 \Rightarrow \Delta_0 $ to be $\Rightarrow \phi$ and $\rho_0 = \emptyset$.
Each step except for substitution, $\rigrul{\forall}$ and $\lefrul{\exists}$ constitutes a true implication, so if $\mathbb{N}, \rho_i \nvDash \Gamma_i \Rightarrow \Delta_i$ then $\rho_i$ also must not satisfy one of its premisses.
We may thus choose one such premiss as $\Gamma_{i+1} \Rightarrow \Delta_{i+1}$ and set $\rho_{i+1} = \rho_i$.
If $\Gamma_i \Rightarrow \Delta_i$ concludes a $\theta$-$\mathit{sub}$ step, we may set $\rho_{i+1} = \rho_i \circ \theta$.
If $\Gamma_i \Rightarrow \Delta_i$ concludes a $\rigrul{\forall}$ step, let $\forall x . \phi $ be the principal formula and assume $x $ does not occur free in the conclusion.
Since $\mathbb{N}, \rho_i \nvDash \Gamma_i \Rightarrow \Delta_i$, we must have that $\mathbb{N} , \rho_i \models \exists x . \neg \phi$. We choose a value $k \in \mathbb{N}$ witnessing this existential and set $\rho_{i+1} = \rho_i \cup \{ x \mapsto k \}$.
The $\lefrul{\exists}$ case is dealt with similarly.
This infinite branch must have an infinitely progressing trace, say $(t_i)_{i \geq n}$, by the definition of $\infty$-proof.
However notice that, for $i \geq n$, $\rho_i (t_i) \geq \rho_{i+1} (t_{i+1})$ and, furthermore, at a progress point along the trace, $\rho_i (t_i) > \rho_{i+1} ( t_{i+1})$.
Thus, $(\rho_i (t_i))_{i \geq n}$ is a monotone decreasing sequence of natural numbers that does not converge, contradicting the fact that $\mathbb{N}$ is well-ordered.
\end{proof}
\noindent
Later, in Sect.~\ref{sect:ind-sim-cyc}, we will use the fact that the choices for generating an invalid branch in the proof above can be made \emph{uniformly} in an arithmetic setting.
\subsection{Defining $\CSn{n}$}
Simpson proposes in \cite{Sim17:cyclic-arith} to study systems of cyclic proofs containing only $\Sin{}{n}$ formulae, and to compare such systems to $\ISn{n}$.
This is rather pertinent in light of the free-cut elimination result we stated, Thm.~\ref{thm:free-cut-elim}: any $\ISn{n}$-proof of a $\Sin{}{n}$-sequent can be assumed to contain just $\Sin{}{n}$ formulae (possibly at a non-elementary cost in proof size), whence the comparison.
However, in order to be able to admit routine derivations of more complex formulae, {e.g.\ the $\Sin{}{n+1}$ law of excluded middle or the universal closure of a $\Sin{}{n}$ sequent}, we will close this notion under \emph{logical consequence}.
\begin{defi}
\label{dfn:cphi}
Let $\Phi$ be a set of formulae closed under subformulae and substitution.
$\CC{\Phi}$ is the first-order theory axiomatised by the universal closures of conclusions of cyclic proofs containing only $\Phi$-formulae.
\end{defi}
Notice that, by the free-cut elimination result, Thm.~\ref{thm:free-cut-elim}, and the subformula property, any $\CSn n$ proofs of $\Sin{}{n}$-sequents contain only $\Sin{}{n}$-sequents anyway, without loss of generality.
This more `robust' definition allows us to easily compare fragments of cyclic arithmetic.
For instance, we have the following:
\begin{prop}
\label{prop:csn-eq-cpn}
$\CSn n = \CPn n$, for $n \geq 0$.
\end{prop}
\begin{proof}
For the left-right inclusion, replace each $\Sin{}{n}$ sequent $\vec p ,\Gamma \Rightarrow \Delta$ with the sequent $\vec p, \compl \Delta \Rightarrow \compl \Gamma$, where $\compl \Gamma$ and $ \compl \Delta $ contain the De Morgan dual formulae of $\Gamma $ and $ \Delta$ resp.\ and $\vec p$ exhausts the atomic formulae of the antecedent.
Any traces will be preserved and the proof can be made correct by locally adding some logical steps.
The converse implication is proved in the same way.
\end{proof}
Using a standard technique, e.g.\ from \cite{Bro06:phdthesis}, we also can rather simply show the following result, which we will later strengthen in Sect.~\ref{sect:cyc-sim-ind}:\footnote{A similar result was given in \cite{Sim17:cyclic-arith}, but that argument rather shows that $\ISn n \subseteq \CSn{n+1}$.}
\begin{prop}
\label{prop:isn-in-csn}
$\CSn n $ proves any $\Pin{}{n+1}$ theorem of $\ISn n$, for $n\geq 0$.
\end{prop}
\begin{proof}
[Proof sketch]
Suppose $\ISn n $ proves $\forall \vec x . \psi(\vec x)$ where $\phi$ is $\Sin{}{n}$.
Let $\pi$ be a free-cut free $\ISn n$ proof of $\psi(\vec a)$, so in particular contains only $\Sin{}n$ formulas, by the subformula property.
We may now construct a $\CSn n $ proof of $\phi(\vec a)$ by simply simulating every local inference step of $\pi$; the only nontrivial case is the induction rule:
\[
\vliinf{\mathit{ind}}{}{\Gamma \Rightarrow \phi(t), \Delta}{\Gamma \Rightarrow \phi(0), \Delta}{ \Gamma, \phi(a) \Rightarrow \phi(\succ a), \Delta}
\]
This is simulated by the following cyclic derivation (omitting some routine proof steps),
\[
\vlderivation{
\vlin{\mathit{sub}}{}{\Gamma \Rightarrow \phi(t), \Delta}{
\vliin{\mathit{cut}}{\bullet}{\Gamma \Rightarrow \phi(b), \Delta}{
\vlin{=}{}{0=b, \Gamma \Rightarrow \phi(b), \Delta}{
\vlhy{\Gamma \Rightarrow \phi(0), \Delta}
}
}{
\vlin{}{}{0< b , \Gamma \Rightarrow \phi(b), \Delta}{
\vlin{}{}{b= \succ a, \Gamma \Rightarrow \phi(b), \Delta}{
\vliin{\mathit{cut}}{}{\underline{a< b}, \Gamma \Rightarrow \phi(\succ a), \Delta}{
\vlin{\mathit{sub}}{}{\Gamma \Rightarrow \phi(a) , \Delta}{
\vlin{\mathit{cut}}{\bullet}{\Gamma \Rightarrow \phi(b), \Delta}{\vlhy{\vdots}}
}
}{
\vlhy{\Gamma, \phi(a) \Rightarrow \phi(\succ a), \Delta}
}
}
}
}
}
}
\]
where we have written $\bullet$ to mark roots of identical subtrees.
An infinite branch that does not have a tail in the proofs of the two premisses of $\mathit{ind}$ must eventually loop on $\bullet$. Therefore it admits an infinitely progressing trace alternating between $a$ and $b$, with the progress point underlined above.
Now the proposition follows by simple application of $\rigrul \forall$.
\end{proof}
Following Rmk.~\ref{rmk:traces-occurrences}, notice that, e.g.\ in the simulation of induction above, traces need not be connected in the graph of ancestry of a proof. This deviates from other settings where it is \emph{occurrences} that are tracked, rather than terms, e.g.\ in \cite{BaeDouHirSau16:multl,BaeDouSau16:cut-elim,Dou17:multl}.
\subsection{B\"uchi automata: checking correctness of cyclic proofs}
\label{sect:sect:automata-prelims}
A cyclic preproof can be effectively checked for correctness by reduction to the inclusion of `B\"uchi automata', yielding a $\mathbf{PSPACE}$ bound. {As far as the author is aware, this is the best known upper bound, although no corresponding lower bound is known.
As we will see later in Sect.~\ref{sect:ind-sim-cyc}, this is one of the reasons why we cannot hope for a `polynomial simulation' of cyclic proofs in a usual proof system, and so why elementary simulations are more pertinent.}
\begin{defi}
A \dfntrm{nondeterministic B\"uchi automaton} (NBA) $\mathcal A$ is a tuple $(A, Q, \delta , q_0 , F)$ where:
$A$ is a finite set, called the \dfntrm{alphabet},
$Q$ is a finite set of \dfntrm{states},
$\delta \subseteq (Q \times A ) \times Q$ is the \dfntrm{transition relation},
$q_0 \in Q$ is the \dfntrm{initial} state,
and
$F\subseteq Q$ is the set of \dfntrm{final} or \dfntrm{accepting} states.
We say that $\mathcal A$ is \dfntrm{deterministic} (a DBA) if $\delta $ is (the graph of) a function $Q \times A \to Q$.
A `word' $(a_i)_{i \geq 0} \in A^\omega $ is \dfntrm{accepted} or \dfntrm{recognised} by $\mathcal A$ if there is a sequence $(q_i)_{i \geq 0} \in Q^\omega$ such that:
for each $i\geq 0$, $(q_i, a_i , q_{i+1}) \in \delta$,
and
for infinitely many $i\geq 0$ we have $q_i \in F$.
We write $\mathcal{L} (\mathcal A)$ for the set of words in $A^\omega$ accepted by $\mathcal A$.
\end{defi}
From a cyclic preproof $\pi$ we can easily define two automata, say $\mathcal A^\pi_b$ and $\mathcal A^\pi_t$,\footnote{These are rather called $B_p$ and $B_t$ respectively in \cite{Sim17:cyclic-arith}.}
respectively accepting just the infinite branches and just the infinite branches with infinitely progressing traces. {See \cite{Sim17:cyclic-arith} for a construction of $\mathcal A^\pi_t$.}
We point out that $\mathcal A^\pi_b$ is essentially just the dependency graph of $\pi$ with all states final, and so is in fact deterministic;\footnote{Technically the transition relation here is not total, but this can be `completed' in the usual way by adding a non-final `sink' state for any outstanding transitions.}
we will rely on this observation later in Sects.~\ref{sect:ind-sim-cyc}, \ref{sect:nonuniform-ind-sim-cyc} and \ref{sect:red-to-det}.
We now state the well-known `correctness criterion' for cyclic proofs:
\begin{propC}
[\cite{Sim17:cyclic-arith}]
\label{prop:reg-proog-correctness-condition}
A cyclic preproof $\pi$ is a $\infty$-proof iff $\mathcal{L}(\mathcal A^\pi_b) \subseteq \mathcal{L} (\mathcal A^\pi_t) $.
\end{propC}
\section{A translation from $\ISn{n+1}$ to $\CSn n$, over $\Pin{}{n+1}$-theorems}
\label{sect:cyc-sim-ind}
We show in this section our first result, that cyclic proofs containing only $\Sin{}{n}$-formulae are enough to simulate $\ISn{n+1}$ over not-too-complex formulae:
\begin{thm}
\label{thm:cyclic-sim-ind}
$\ISn{n+1} \subseteq \CSn{n}$, over $\Pin{}{n+1}$ theorems, for $n\geq 0$.
\end{thm}
{One example of such logical power in cyclic proofs was given in \cite{Sim17:cyclic-arith}, in the form of $\CSn 1$ proofs of the totality of the Ackermann-P\'eter function. This already separates it from $\ISn 1$, which only proves the totality of the primitive recursive functions \cite{Par72:n-quant-ind}.}
To prove the theorem above, we will rather work in $\IPn{n+1}$, cf.~Prop.~\ref{prop:ipn-equals-isn}, since the exposition is more intuitive.
We first prove the following intermediate lemma.
\begin{lem}
\label{lem:ind-to-cyc-trans}
Let $\pi $ be a $\IPn{n+1}$ proof, containing only $\Pin{}{n+1}$ formulae, of a sequent,
\begin{equation}
\label{eqn:pitwo-conc}
\Gamma, \forall x_1 . \phi_1 , \dots , \forall x_l . \phi_l \Rightarrow \Delta, \forall y_1 . \psi_1 , \dots , \forall y_m . \psi_m
\end{equation}
where $\Gamma, \Delta, \phi_i, \psi_j$ are $\Sin{}{n}$ and $x_i , y_j$ occur only in $ \phi_i, \psi_j$ respectively.
Then there is a $\CSn n$ derivation $\lift \pi$ of the form:
\[
\toks0={.5}
\vlderivation{
\vltrf{\lift \pi}{\Gamma \Rightarrow \Delta, \psi_1 , \dots , \psi_m }{\vlhy{}}{\vlhy{\left\{ \Gamma \Rightarrow \Delta, \phi_i \right\}_{i\leq l}}}{\vlhy{}}{\the\toks0}
}
\]
Moreover, no free variables of \eqref{eqn:pitwo-conc} occur as eigenvariables for $\lefrul \exists$, $\rigrul \forall$ or $\mathit{sub}$ steps in $\lift \pi$.
\end{lem}
\begin{proof}
We proceed by induction on the structure of $\pi$.
Notice that we may assume that any $\Pin{}{n+1}$ formulae occurring have just a single outermost $\forall$ quantifier, by interpreting arguments as pairs and using G\"odel's $\beta$ functions. {(This introduces only cuts on formulae of the same form.)}
We henceforth write $\vec \phi$ for $\phi_1 , \dots , \phi_l$ and $\vec \psi$ for $\psi_1 , \dots , \psi_m$ and, as an abuse of notation, $\forall \vec x . \vec \phi $ and $\forall \vec y . \vec \psi$ for $\forall x_1 . \phi_1 , \dots , \forall x_l . \phi_l $ and $\forall y_1 . \psi_1 , \dots , \forall y_m . \psi_m $ respectively. {(Notice that this is a reasonable abuse of notation, since $\forall$s can be prenexed outside conjunctions and disjunctions already in pure FO logic.)}
Propositional logical steps are easily dealt with, relying on invertibility and cuts, with possible structural steps.
Importantly, due to the statement of the lemma, such steps apply to only $\Sin{}{n}$ formulae (recall the discussion at the end of Sect.~\ref{sect:prelims-pa-pt}).
For instance, if $\pi$ extends a proof $\pi'$ by a $\wedge$-left step,
\[
\vlinf{\lefrul \wedge}{}{\Gamma , \chi_0 \wedge \chi_1, \forall \vec x . \vec \phi \Rightarrow \Delta , \forall \vec y . \vec \psi}{\Gamma , \chi_0, \chi_1, \forall \vec x . \vec \phi \Rightarrow \Delta , \forall \vec y . \vec \psi}
\]
then we define $\lift \pi$ as,
\[
\toks0={.25}
\vlderivation{
\vlin{\lefrul\wedge}{}{\Gamma, \chi_0\wedge \chi_1 \Rightarrow \Delta , \vec \psi}{
\vltrf{ \lift{\pi'} }{ \Gamma, \chi_0 , \chi_1 \Rightarrow \Delta , \vec \psi }{\vlhy{}}{
\vlhy{
\left\{
\vlderivation{
\vliin{\mathit{cut}}{}{\Gamma, \chi_0 , \chi_1 \Rightarrow \Delta, \phi_i }{
\vliq{}{}{\chi_0 , \chi_1 \Rightarrow \chi_0 \wedge \chi_1}{\vlhy{}}
}{ \vlhy{\Gamma, \chi_0 \wedge \chi_1 \Rightarrow \Delta, \phi_i} }
}
\right\}_{i\leq l}
}
}{\vlhy{}}{\the\toks0}
}
}
\]
and if $\pi$ extends proofs $\pi_0$ and $ \pi_1$ by a $\wedge$-right step,
\[
\vliinf{\rigrul \wedge}{}{ \Gamma , \forall \vec x . \vec \phi \Rightarrow \Delta , \chi_0 \wedge \chi_1 , \forall \vec y . \vec \psi }{\Gamma , \forall \vec x . \vec \phi \Rightarrow \Delta , \chi_0 , \forall \vec y . \vec \psi}{\Gamma , \forall \vec x . \vec \phi \Rightarrow \Delta , \chi_1 , \forall \vec y . \vec \psi}
\]
then we define $\lift \pi $ as:
\[
\toks0={.3}
\vlderivation{
\vlin{\rigrul \wedge}{\forall j <2}{\Gamma \Rightarrow \Delta, \chi_0 \wedge \chi_1, \vec \psi}{
\vltrf{\lift{\pi_j}}{ \Gamma \Rightarrow \Delta, \chi_j, \vec \psi }{\vlhy{}}{\vlhy{
\left\{
\vlderivation{
\vliin{\mathit{cut}}{}{\Gamma \Rightarrow \Delta, \chi_j, \phi_i}{\vlhy{\Gamma \Rightarrow \Delta, \chi_0 \wedge \chi_1, \phi_i}}{
\vliq{}{}{ \chi_0 \wedge \chi_1 \Rightarrow \chi_j }{\vlhy{}}
}
}
\right\}_{i \leq l}
}}{\vlhy{}}{\the\toks0}
}
}
\]
If $\pi$ extends a proof $\pi'$ by a thinning step,
\[
\vlinf{\mathit{wk}}{}{ \Gamma', \Pi, \Gamma, \forall \vec x . \vec \phi \Rightarrow \Delta , \forall \vec y . \vec \psi , \Delta', \forall \vec z . \vec \chi}{\Gamma, \forall \vec x . \vec \phi \Rightarrow \Delta , \forall \vec y . \vec \psi}
\]
where $\Gamma', \Delta', \vec \chi$ are $\Sin{}{n}$ and $\Pi$ is $\Pin{}{n+1}$, then we define $\lift \pi$ as:
\[
\toks0={0.5}
\vlderivation{
\vlin{\mathit{wk}}{}{\Gamma', \Gamma \Rightarrow \Delta, \vec \psi , \Delta' , \vec \chi}{
\vltrf{\Gamma',\lift{\pi'}, \Delta'}{ \Gamma', \Gamma \Rightarrow \Delta, \vec \psi , \Delta'}{\vlhy{}}{\vlhy{\left\{ \Gamma',\Gamma \Rightarrow \Delta, \phi_i , \Delta' \right\}_{i \leq l} } }{\vlhy{}}{\the\toks0}
}
}
\]
where $\Gamma', \lift{\pi'}, \Delta'$ is obtained from $\lift{\pi'}$ by prepending $\Gamma'$ and appending $\Delta'$ to each sequent.
For this we might need to rename some free variables in $\pi'$ so that eigenvariable conditions are preserved after the transformation; this does not affect the cedents $\Gamma, \Delta$ by the assumption from the inductive hypothesis.
{Notice that we are simply ignoring the extra premisses due to $\Pi$.}
%
If $\pi$ extends proofs $\pi_0 $ and $\pi_1$ by a $\mathit{cut}$ step on a $\Pin{}{n+1}$ formula,
\[
\vliinf{\mathit{cut}}{}{\Gamma , \forall \vec x . \vec \phi \Rightarrow \Delta , \forall \vec y . \vec \psi }{\Gamma , \forall \vec x . \vec \phi \Rightarrow \Delta , \forall \vec y . \vec \psi , \forall z . \chi}{\Gamma , \forall \vec x . \vec \phi , \forall z . \chi \Rightarrow \Delta , \forall \vec y . \vec \psi}
\]
then we define $\lift \pi$ as:
\[
\toks0={0.4}
\toks1={0.3}
\vlderivation{
\vlid{}{}{\Gamma \Rightarrow \Delta , \vec \psi}{
\vltrf{\lift{\pi_1}, \vec \psi}{ \Gamma \Rightarrow \Delta , \vec \psi , \vec \psi }{
\vlhy{\vlderivation{\vltrf{\lift{\pi_0}}{\Gamma \Rightarrow \Delta, \vec \psi , \chi }{\vlhy{}}{\vlhy{ \left\{ \Gamma \Rightarrow \Delta , \phi_i \right\}_{i \leq l} }}{\vlhy{}}{\the\toks0}}}
}{
\vlhy{}
}{
\vlhy{
\left\{
\vlderivation{
\vlin{\mathit{wk}}{}{\Gamma \Rightarrow \Delta , \vec \psi , \phi_i }{\vlhy{ \Gamma \Rightarrow \Delta, \phi_i } }
}
\right\}_{i \leq l}
}
}{
\the\toks1
}
}
}
\]
The final dotted `contraction' step is implicit, since we treat cedents as sets.
Again,
we might need to rename some variables in $\pi_1$.
If instead the cut formula were $\Sin{}n$, say $\chi$, we would define $\lift \pi $ as:
\[
\toks0={0.55}
\toks1={0.4}
\vlderivation{
\vliin{\mathit{cut}}{}{ \Gamma \Rightarrow \Delta , \vec \psi }{
\vltrf{\lift{\pi_0}}{ \Gamma \Rightarrow \Delta , \chi , \vec \psi }{\vlhy{}}{
\vlhy{\{ \Gamma \Rightarrow \Delta , \phi_i \}_{i\leq l}}
}{\vlhy{}}{\the\toks0}
}{
\vltrf{\lift{\pi_1}}{ \Gamma, \chi \Rightarrow \Delta , \vec \psi}{\vlhy{}}{
\vlhy{\left\{
\vlderivation{
\vlin{\mathit{wk}}{}{\Gamma, \chi \Rightarrow \Delta, \phi_i}{\vlhy{\Gamma \Rightarrow \Delta , \phi_i}}
}
\right\}_{i \leq l}}
}{\vlhy{}}{\the\toks1}
}
}
\]
If $\pi$ extends a proof $\pi'$ by a $\forall$-left step,
\[
\vlinf{\lefrul \forall}{}{\Gamma , \forall z. \chi(z), \forall \vec x . \vec \phi \Rightarrow \Delta, \forall \vec y. \vec \psi}{\Gamma , \chi(t), \forall \vec x . \vec \phi \Rightarrow \Delta, \forall \vec y. \vec \psi}
\]
where $\forall z . \chi(z)$ is $\Pin{}{n+1}$, we define $\lift \pi$ as follows:
\[
\toks0={0.3}
\vlderivation{
\vliin{\mathit{cut}}{}{\Gamma \Rightarrow \Delta, \vec \psi}{
\vlin{\mathit{sub}}{}{\Gamma \Rightarrow \Delta , \chi(t)}{\vlhy{\Gamma \Rightarrow \Delta, \chi(z)}}
}{
\vltrf{\lift{\pi'}}{\Gamma , \chi(t) \Rightarrow \Delta, \vec \psi}{\vlhy{}}{
\vlhy{
\left\{ \vlinf{\mathit{wk}}{}{\Gamma, \chi(t) \Rightarrow \Delta , \phi_i}{\Gamma \Rightarrow \Delta, \phi_i} \right\}_{i\leq l}
}
}{\vlhy{}}{\the\toks0}
}
}
\]
(Notice that, although $z$ occurs as an eigenvariable for a $\mathit{sub}$ step here, it is already bound in the conclusion of $\pi$, so we preserve the inductive hypothesis.)
If $\pi$ extends a proof $\pi'$ by a $\forall$-right step,
\[
\vlinf{\rigrul \forall}{}{\Gamma , \forall \vec x . \vec \phi \Rightarrow \Delta, \forall \vec y . \vec \psi , \forall z . \chi}{\Gamma , \forall \vec x . \vec \phi \Rightarrow \Delta, \forall \vec y . \vec \psi , \chi}
\]
where $\forall z .\chi$ is $\Pin{}{n+1}$, then we define $\lift \pi$ as:
\[
\toks0={0.5}
\vlderivation{
\vltrf{\lift{\pi'}, \chi}{ \Gamma \Rightarrow \Delta , \vec \psi , \chi }{\vlhy{}}{
\vlhy{\left\{\vlderivation{\vlin{\mathit{wk}}{}{\Gamma \Rightarrow \Delta , \phi_i , \chi}{\vlhy{\Gamma \Rightarrow \Delta, \phi_i}} } \right\}_{ i \leq l} }
}{\vlhy{}}{\the\toks0}
}
\]
%
%
If $\pi$ extends a proof $\pi'$ by a $\exists$-right step,
\[
\vlinf{\rigrul \exists}{}{ \Gamma , \forall \vec x . \vec \phi \Rightarrow \Delta, \forall \vec y . \vec \psi , \exists z . \chi(z) }{ \Gamma , \forall \vec x . \vec \phi \Rightarrow \Delta, \forall \vec y . \vec \psi , \chi(t) }
\]
where $\exists z . \chi(z)$ is $\Sin{}{n}$, then we define $\lift \pi$ as:
\[
\toks0={0.3}
\vlderivation{
\vlin{\rigrul \exists}{}{ \Gamma \Rightarrow \Delta, \vec \psi , \exists z . \chi(z) }{
\vltrf{\lift{\pi'}, \exists z . \chi(z)}{ \Gamma \Rightarrow \Delta , \vec \psi , \chi(t) , \exists z . \chi(z) }{\vlhy{}}{
\vlhy{
\left\{
\vlinf{\mathit{wk}}{}{\Gamma \Rightarrow \Delta , \phi_i , \chi(t) , \exists z . \chi(z)}{\Gamma \Rightarrow \Delta , \phi_i , \exists z . \chi(z)}
\right\}_{i\leq l}
}
}{\vlhy{}}{\the\toks0}
}
}
\]
Again, some eigenvariables of $\pi'$ might have to be renamed.
Any other quantifier steps are dealt with routinely.
Finally, if $\pi$ extends proofs $\pi_0$ and $\pi'$ by an induction step,
\[
\vliinf{\mathit{ind}}{}{\Gamma , \forall \vec x . \vec \phi \Rightarrow \Delta, \forall \vec y . \vec \psi , \forall z . \chi(t)}{\Gamma , \forall \vec x . \vec \phi \Rightarrow \Delta, \forall \vec y . \vec \psi , \forall z . \chi(0)}{ \Gamma , \forall \vec x . \vec \phi , \forall z . \chi(c) \Rightarrow \Delta, \forall \vec y . \vec \psi , \forall z . \chi(\succ c) }
\]
we define $\lift \pi$ to be the following cyclic proof,
\[
\toks0={0.3}
\toks1={0.5}
\vlderivation{
\vlin{\mathit{sub}}{}{\Gamma \Rightarrow \Delta, \vec \psi , \chi(t) }{
\vliin{}{\bullet}{\Gamma \Rightarrow \Delta , \vec \psi , \chi(d)}{
\vlin{}{}{d=0, \Gamma \Rightarrow \Delta , \vec \psi , \chi(d)}{
\vltrf{\lift{\pi_0}}{\Gamma \Rightarrow \Delta, \vec \psi , \chi(0)}{\vlhy{}}{\vlhy{ \left\{ \Gamma \Rightarrow \Delta, \phi_i \right\}_{i\leq l} }}{\vlhy{}}{\the\toks1}
}
}{
\vlin{}{}{\underline{c<d}, d = \succ c , \Gamma \Rightarrow \Delta, \vec \psi , \chi(d)}{
\vltrf{\lift{\pi'},\vec \psi }{ \Gamma \Rightarrow \Delta , \vec \psi , \chi(\succ c)}{
\vlin{\mathit{sub}}{}{\Gamma \Rightarrow \Delta, \vec \psi, \chi(c)}{
\vlin{}{\bullet}{\Gamma \Rightarrow \Delta ,\vec \psi , \chi(d) }{\vlhy{\vdots}}
}
}{
\vlhy{}
}{
\vlhy{\left\{ \vlinf{\mathit{wk}}{}{\Gamma \Rightarrow \Delta, \vec \psi , \phi_i}{\Gamma \Rightarrow \Delta , \phi_i} \right\}_{i\leq l} }
}{
\the\toks0
}
}
}
}
}
\]
where we have written $\bullet$ to mark roots of identical subtrees.
Notice that any branch hitting $\bullet$ infinitely often will have an infinitely progressing trace alternating between $c$ and $d$, by the underlined progress point $c< d$: thanks to the assumption from the inductive hypothesis, $c$ will not occur in $\lift{\pi'}$ as an eigenvariable for $\lefrul \exists$, $\rigrul \forall$ or $\mathit{sub}$ steps so the trace along $c$ in $\lift{\pi ' }$ remains intact, cf.~Rmk.~\ref{rmk:traces-occurrences}.
Any other infinite branch has a tail that is already in $\lift{\pi'}$ or $\lift{\pi_0}$ and so has an infinitely progressing trace by the inductive hypothesis.
\end{proof}
The lemma above gives us a simple proof of the main result of this section:
\begin{proof}
[Proof of Thm.~\ref{thm:cyclic-sim-ind}]
Let $\pi$ be a $\IPn{n+1} $ proof of a sequent $\Rightarrow \forall x . \phi$, where $\phi \in \Sin{}n$, under Prop.~\ref{prop:ipn-equals-isn}.
By Thm.~\ref{thm:free-cut-elim} we may assume that $\pi$ contains only $\Pin{}{n+1}$ cuts, whence we may simply apply Lemma~\ref{lem:ind-to-cyc-trans} to obtain
a $\CSn n $ proof of $\Rightarrow \phi$.
(Notice that there are no assumption sequents after applying the lemma since the antecedent is empty.)
Now the result follows simply by an application of $\rigrul \forall$.
%
%
\end{proof}
For the interested reader, we have given an example of this translation in action in App.~\ref{sect:php-case-study}, for a `relativised' version of arithmetic with an uninterpreted function symbol.
\section{Second-order theories for reasoning about automata}
\label{sect:so-theories}
We now consider a two-sorted, or `second-order' (SO), version of FO logic, with variables $X,Y,Z, $ etc.\ ranging over sets of individuals, and new atomic formulae $t \in X$, sometimes written $X(t)$.
We also have SO quantifiers binding the SO variables with the natural interpretation.
Again, we give only brief preliminaries, but the reader is encouraged to consult the standard texts \cite{Sim09:reverse-math} and \cite{Hir14:reverse-math}.
We write $\mathsf{Q}_2$ for an appropriate extension of $\mathsf{Q}$ by basic axioms governing sets (see, e.g., \cite{Sim09:reverse-math} or \cite{Hir14:reverse-math}), and write $\Sin{0}{n}$ and $\Pin{0}{n}$ for the classes $\Sin{}{n}$ and $\Pin{}{n}$ respectively, but now allowing free set variables to occur.
\begin{defi}
The \textbf{recursive comprehension} axiom schema is the following:\footnote{Notice that there is an unfortunate coincidence of the notation $\mathsf{CA}$ for `comprehension axiom' and `cyclic arithmetic', but the context of use should always avoid any ambiguity.}
\[
\CCA{\Din 0 1}
\ :\
\forall \vec y , \vec Y . (\forall x . ( \phi(x,\vec y , \vec Y) \equiv \neg \psi(x, \vec y , \vec Y ) ) \supset \exists X . \forall x. (X(x) \equiv \phi (x)))
\]
where $\phi,\psi$ are in $\Sin 0 1 $ and $X$ does not occur free in $\phi$ or $\psi$.
From here, the theory $\mathsf{RCA}_0$ is defined as $\mathsf{Q}_2 + \CCA{\Din 0 1} + \CIND{\Sin 0 1 }$.
%
%
%
\end{defi}
Since we will always work in extensions of $\mathsf{RCA}_0$, which proves the totality of primitive recursive functions, we will conservatively add function symbols for primitive recursive functions on individuals whenever we need them.
We will also henceforth consider FO theories extended by `oracles', i.e.\ uninterpreted set/predicate variables, in order to access `uniform' classes of FO proofs. We write $\ISn n(X)$ for the same class of proofs as $\ISn n$ but where $X$ is allowed to occur as a predicate symbol. The usefulness of a $\ISn{n}(X)$ proof is that we may later substitute $X$ for a FO formula, say $\phi(-) \in \Din{}{m+1}$, to arrive at a $\ISn{m+n}$ proof of size $O(|\phi|)$. This `parametrisation' of a FO proof allows us to avoid unnecessary blowups in proof size induced by `non-uniform' translations from second-order theories; we implicitly use this observation for proof complexity bounds later, particularly in Sect.~\ref{sect:ind-sim-cyc}.
\smallskip
The following result is an adaptation of well known conservativity results, e.g.\ as found in \cite{Sim09:reverse-math,Hir14:reverse-math}, but we include a proof anyway for completeness.
\begin{prop}
\label{prop:conservativity}
$\mathsf{RCA}_0 + \CIND{\Sin 0 n}$ is conservative over $\ISn n (X)$.
\end{prop}
\begin{proof}
[Proof sketch]
First we introduce countably many fresh set symbols $X^{\vec t, \vec Y}_{\phi, \psi}$, indexed by $\Sin 0 1$ formulae $\phi(x, \vec x, \vec X), \psi(x, \vec x , \vec X)$ with all free variables indicated, FO terms $\vec t$ with $|\vec t | = | \vec x|$ and SO variables $\vec Y$ with $|\vec Y| = | \vec X|$.
These will serve as witnesses to the sets defined by comprehension.
We replace the comprehension axioms by initial sequents of the form:
\begin{equation}
\label{eqn:ca-init-pos}\vlinf{}{}{\Gamma, \forall x . ( \phi(x,\vec t , \vec Y) \equiv \neg\psi(x, \vec t , \vec Y ) ),\phi(t,\vec t , \vec Y) \Rightarrow t \in X^{\vec t, \vec Y}_{\phi, \psi} , \Delta }{}
\end{equation}
\begin{equation}
\label{eqn:ca-init-neg}
\vlinf{}{}{\Gamma, \forall x . ( \phi(x,\vec t , \vec Y) \equiv \neg\psi(x, \vec t , \vec Y ) ), t \in X^{\vec t, \vec Y}_{\phi, \psi} \Rightarrow \phi(t,\vec t , \vec Y) ,\Delta }{}
\end{equation}
\medskip
\noindent
It is routine to show that these new initial sequents are equivalent to the comprehension axioms for $\phi,\psi$.
Now we apply free-cut elimination, Thm.~\ref{thm:free-cut-elim}, to a proof in such a system and replace every occurrence of $t \in X^{\vec t , \vec Y}_{\phi, \psi}$ with $\phi(t, \vec t , \vec Y)$,
and every occurrence of $t \notin X^{\vec t , \vec Y}_{\phi,\psi}$ with $\psi(t, \vec t , \vec Y)$. {(Recall here that we assume formulae are in De Morgan normal form.)}
Any comprehension initial sequents affected by this replacement become
purely logical theorems.
%
Furthermore, any induction formulae remain $\Sin{0}{n}$, provably in pure logic, thanks to our consideration of whether $ X^{\vec t , \vec Y}_{\phi, \psi}$ occurs positively or negatively.
Any extraneous free set variables in induction steps (except $X$), e.g.\ $Y$, may be safely dealt with by replacing any atomic formula $Y(s)$ with $\top$.
The resulting proof is in $\ISn n (X)$.
\end{proof}
It is worth pointing out that, in general, the transformation from a SO proof to a FO proof can yield a possibly non-elementary blowup in the size of proofs, due to, e.g., the application of (free-)cut elimination.
\subsection{Formalisation of B\"uchi acceptance}
From now on we will be rather informal when talking about finite objects, e.g.\ automata, finite sequences, or even formulae.
In particular we may freely use such meta-level objects within object-level formulae when, in fact, we are formally referring to their `G\"odel numbers'.
Also, statements inside quotations, ``-'', will usually be (provably) recursive in any free variables occurring, i.e.\ $\Din 0 1 $.
This way quantifier complexity is (usually) safely measured by just the quantifiers outside quotations.
We often treat a set symbol $X$ as a binary predicate by interpreting its argument as a pair and using G\"odel's `$\beta$ functions' to primitive-recursively extract its components.
We use such predicates to encode sequences by interpreting $X(x,y)$ as ``the $x$\textsuperscript{th} symbol of $X$ is $y$''; this interpretation presumes we already have the totality and determinism of $X$ as a binary relation.
Formally, for a set $S$ and a set symbol $X$ treated as a binary predicate,
we will write $X \in S^\omega$ for the conjunction of the following two formulae,
\begin{equation}
\label{eqn:tot-oracle}
\forall x . \exists y \in S . X(x,y)
\end{equation}
\begin{equation}
\label{eqn:det-oracle}
\forall x , y , z . ((X(x,y) \wedge X(x,z)) \supset y=z)
\end{equation}
i.e.\ $X$ is, in fact, the graph of a function $\mathbb{N} \to S$.
When we know that these formulae hold true for $X$, we may construe the expression $X(x)$ as a term in formulae, for instance writing $\phi(X(x))$ as shorthand for $\exists y . (X(x,y) \wedge \phi(y))$ or, equivalently, $\forall y . (X(x,y) \supset \phi(y))$.
\begin{defi}
[Language membership]
\label{dfn:lang-memb-so}
Let $\mathcal A = ( A, Q, \delta, q_0 , F )$ be a NBA and treat $X$ as a binary predicate symbol.
We define the formula
$X \in \mathcal{L} (\mathcal A)$ as:
\begin{equation}
\label{eqn:nd-memb}
X \in A^\omega \ \wedge \ \exists Y \in Q^\omega .
\left(
\begin{array}{rl}
& Y(0, q_0) \\
\wedge & \forall x . \ (Y(x), X(x) , Y(\succ x) ) \in \delta \\
\wedge & \forall x . \exists x' > x . \ Y(x') \in F
\end{array}
\right)
\end{equation}
%
If $\mathcal A$ is deterministic and $X \in A^\omega$, we write $q_X (x,y)$ for ``$y$ is the $x$\textsuperscript{th} state of the run of $X$ on $\mathcal A$'', which is provably recursive in $\mathsf{RCA}_0$.
Similarly to
before,
we may write $\phi (q_X(x))$ as shorthand for $\exists y . (q_X(x,y) \wedge \phi (y))$ or, equivalently in $\mathsf{RCA}_0$, for $\forall y . (q_X(x,y) \supset \phi(y) )$.
For DBA, we alternatively define $X \in \mathcal{L} (\mathcal A)$ as:
\begin{equation}
\label{eqn:det-acc}
X \in A^\omega
\ \wedge \
\forall x . \exists x'> x .\ q_X(x') \in F
\end{equation}
\end{defi}
\noindent
This `double definition' will not be problematic for us, since $\mathsf{RCA}_0$ can check if an automaton is deterministic or not and, if so, even prove the equivalence between the two definitions:
\begin{prop}
$\mathsf{RCA}_0 \vdash \forall \text{ DBA } \mathcal A . (\eqref{eqn:nd-memb} \equiv \eqref{eqn:det-acc})$.
\end{prop}
\begin{proof}
[Proof sketch]
Let $\mathcal A = (A, Q , \delta, q_0, F )$ be a deterministic automaton.
For the left-right implication let $Y\in Q^\omega$ be an `accepting run' of $X$ on $\mathcal A$ and use induction to show that $Y(x, q_X(x))$.
For the right-left implication, we use comprehension to define an `accepting run' $Y\in Q^\omega$ by:
\(
Y(x,q)
\ \equiv \
q_X (x,q)
\).
Clearly the definition of $Y$ is $\Din 0 1$, and we can show that such $Y$ is a `correct run' by induction on $x$.
\end{proof}
Notice that, for a deterministic automaton, the formula for acceptance is \emph{arithmetical} in $X$, i.e.\ there are no SO quantifiers.
This will be rather important for uniformity in the simulation of cyclic proofs in the next section.
\subsection{Formalisations of some automaton constructions}
Recall that we may freely add symbols for primitive recursive functions to our language.
Since we rely on various results from \cite{KMPS16:buchi-reverse} as the `engine' behind some of our proofs, we will use their notions for manipulating automata.
For NBA $\mathcal A , \mathcal A'$, we write $\mathcal A^c$ and $\mathcal A \sqcup \mathcal A'$ to denote the complement and union constructions of automata from \cite{KMPS16:buchi-reverse} (Sects.~5 and 6 resp.).
We also write $\mathrm{Empty}(\mathcal A)$ for the recursive algorithm from \cite{KMPS16:buchi-reverse} (Sect.~6), expressed as a $\Sin{}1$ formula in $\mathcal A$, determining whether $\mathcal A$ computes the empty language.
It will also be useful for us later, in order to bound logical and proof complexity, to notice that DBA can already be complemented in $\mathsf{RCA}_0$. This is a rather unsurprising result but does not appear in \cite{KMPS16:buchi-reverse}, so we give it here.
For a DBA $\mathcal A = (A, Q , \delta , q_0 , F)$, we define a complementary NBA $ \mathcal A^c $ as follows,
$$\mathcal A^c \ := \ (A, (Q \times \{0\}) \cup ((Q\setminus F) \times \{1\}) , \delta^c ,(q_0 , 0), (Q \setminus F) \times \{1\} )$$
where $\delta^c \subseteq (Q^c \times A) \times Q^c $ (writing $Q^c$ for $Q \times \{0\} \cup (Q\setminus F) \times \{1\}$) is defined as:
\[
\begin{array}{rl}
& \{ ((q,0), a , (q',0)) \ : \ (q,a,q') \in \delta \} \\
\cup & \{ ( (q,i), a , (q',1) ) \ : \ (q,a,q') \in \delta, i = 0,1 , q' \in Q\setminus F \}
\end{array}
\]
The idea behind this construction is that a run of $\mathcal A^c$ follows $\mathcal A$ freely for some finite time (in the `$0$' component), after which it may no longer visit final states of $\mathcal A$ (once in the `$1$' component). The determinism of $\mathcal A$ guarantees that such a word is not accepted by it.
By directly inspecting the definitions from \cite{KMPS16:buchi-reverse}, and DBA complementation above, we have the following properties:
\begin{obs}
\label{obs:complexity-aut-constructions}
For NBA $\mathcal A, \mathcal A'$ we have that:
\begin{enumerate}
\item $\mathrm{Empty}(\mathcal A)$ is a polynomial-time predicate in $\mathcal A$.
\item $\mathcal A \sqcup \mathcal A' $ is constructible in polynomial-time from $\mathcal A$ and $\mathcal A'$.
\item $\mathcal A^c$ is constructible in exponential-time from $\mathcal A$.
\end{enumerate}
For a DBA $\mathcal A$, we have that:
\begin{enumerate}
\setcounter{enumi}{3}
\item $\mathcal A^c$ is constructible in polynomial-time from $\mathcal A$.
\end{enumerate}
\end{obs}
\noindent
None of these bounds are surprising, due to known bounds on the complexity of union, complementation and emptiness checking for (non)deterministic B\"uchi automata.
Nonetheless it is important to state them for the particular constructions used in this work for bounds on proof complexity later.
\begin{lem}
\label{lem:aut-clos-props-in-so-arith}
From \cite{KMPS16:buchi-reverse} we have the following:
\begin{enumerate}
\item\label{item:emptiness-rca} $ \mathsf{RCA}_0\vdash \forall \text{ NBA }\mathcal A . (\mathrm{Empty} (\mathcal A) \equiv \forall X \in A^\omega. X \notin \mathcal{L} (\mathcal A))$.
\item\label{item:union-rca} $ \mathsf{RCA}_0\vdash \forall \text{ NBA } \mathcal A_1 , \mathcal A_2 . (X \in \mathcal{L} (\mathcal A_1 \sqcup \mathcal A_2) \equiv (X \in \mathcal{L} (\mathcal A_1) \vee X \in \mathcal{L} (\mathcal A_2) ))$.
\item\label{item:compl-rca-s2ind} $\mathsf{RCA}_0 + \CIND{\Sin 0 2} \vdash \forall \text{ NBA } \mathcal A . ( X \in A^\omega \supset (X \in \mathcal{L} (\mathcal A^c) \equiv X \notin \mathcal{L} (\mathcal A)))$.
\end{enumerate}
We also have that:
\begin{enumerate}
\setcounter{enumi}{3}
\item\label{item:compl-dba-rca} $\mathsf{RCA}_0 \vdash \forall \text{ DBA } \mathcal A .\ ( X\in A^\omega \supset (X \in \mathcal{L} (\mathcal A^c) \equiv X \notin \mathcal{L} (\mathcal A ) )) $.
\end{enumerate}
\end{lem}
\begin{proof}
\ref{item:emptiness-rca}, \ref{item:union-rca} and \ref{item:compl-rca-s2ind} follow from \cite{KMPS16:buchi-reverse}, namely from Prop.~6.1 and Lemma~5.2,
so we give a proof of \ref{item:compl-dba-rca}.
Working in $\mathsf{RCA}_0$, let $\mathcal A = (A, Q , \delta , q_0 , F)$ be a DBA.
For the right-left implication,
if $X \notin \mathcal{L} (\mathcal A)$ then $\exists x . \forall x' > x .\ q_X(x) \notin F$, so let $x_0$ witness this existential.
Now, define by comprehension the run $Y \in (Q^c)^\omega$ as follows:
\[
Y(x,y)
\ \equiv \
((x \leq x_0 \wedge y = (q_X(x), 0 )) \vee ( x > x_0 \wedge y = (q_X (x), 1) ) )
\]
Now, indeed $Y(0, (q_0, 0))$, since $q_X (0) = q_0$, and $Y$ is a correct run of $X$ on $\mathcal A^c$ by considering separately the cases $x< x_0$, $x=x_0$ and $x> x_0$. Finally, for any $x$, $Y$ hits a final state at $\max (x, x_0) + 1 > x$.
For the left-right implication, suppose $X \in \mathcal{L} (\mathcal A^c)$ and let $Y\in (Q^c)^\omega$ be an accepting run.
By induction we have $\forall x . (Y(x) = (q_X (x), 0) \vee Y(x) = (q_X (x), 1) )$.
Now, $Y$ must eventually hit an accepting state of $\mathcal A^c$, i.e.\ in the $1$-component, say at position $x_0$.
Again by induction, we may show that $Y$ remains in the $1$-component of $\mathcal A^c$ after $x_0$, and hence $q_X(x) \notin F$ for $x\geq x_0$, as required.
\end{proof}
\section{An exponential simulation of $\mathsf{CA}$ in $\mathsf{PA}$}
\label{sect:ind-sim-cyc}
In this section we will adapt Simpson's approach in \cite{Sim17:cyclic-arith} for showing that $\mathsf{CA} \subseteq \mathsf{PA}$ into a \emph{uniform} result in $\mathsf{PA}$.
This essentially constitutes a formalisation of the soundness argument, Prop.~\ref{prop:sound-cyclic}, in a SO theory conservative over the target fragment of $\mathsf{PA}$.
The `uniformity' we aim for ensures that the possibly non-elementary blowup translating from SO proofs to FO proofs occurs once and for all for a single arithmetical theorem.
Only then do we instantiate the theorem (inside $\mathsf{PA}$) by the cyclic proof in question, leading to an only elementary blowup.
To give an idea of how the result is obtained, and how our exposition refines that of \cite{Sim17:cyclic-arith}, we take advantage of the following aspects of the soundness argument for cyclic proofs:
\begin{enumerate}[label={(\alph*)}]
\item\label{item:branch-aut-det} The B\"uchi automaton accepting all infinite branches of a cyclic proof is, in fact, deterministic, and so we can express acceptance of an $\omega$-word in this automaton arithmetically.
\item\label{item:invalid-branch-uniform} A branch of invalid sequents and corresponding assignments, as in the proof of Prop.~\ref{prop:sound-cyclic}, can be uniformly generated from an initial unsatisfying assignment by an arithmetical formula.
\item\label{item:finite-closure-ordinals} Since all inductions are only up to $\omega$, we need only \emph{arbitrarily often} progressing traces, rather than explicit infinitely progressing traces.
\end{enumerate}
\noindent
Together, these properties give us just enough `wiggle room' to carry out the soundness argument in a sufficiently uniform way.
Throughout this section we will also carefully track how much quantifier complexity is used in theorem statements, since we will later modify this argument to obtain a converse result to Thm.~\ref{thm:cyclic-sim-ind}.
\subsection{An arithmetically uniform treatment of automata}
Referring to \ref{item:finite-closure-ordinals} above, we define an arithmetical corollary of NBA acceptance that is nonetheless sufficiently strong to formalise the soundness argument for cyclic proofs:
\begin{defi}
[Arithmetic acceptance]
\label{dfn:aracc}
Let $\mathcal A = (A,Q,\delta, q_0, F)$ be a NBA and $X\in A^\omega$, and temporarily write:
\begin{itemize}
\item $F(x) := $ ``$x$ is a finite run of $X$ on $\mathcal A$ ending at a final state''.
\item $E(z,x,y ) := $ ``$z$ extends $x$ to a finite run of $X$ on $\mathcal A$ hitting $\geq y$ final states''
\end{itemize}
We define:
\begin{equation}
\label{eqn:aracc-dfn}
\mathrm{ArAcc} (X, \mathcal{A})
\ := \
X\in A^\omega \wedge
\exists x .
\left(
F(x)
\wedge
\forall y . \exists z . E(z,x,y)
\right)
\end{equation}
\end{defi}
\noindent
{For intuition, we may consider $\omega$-regular expressions rather than automata, which are of the form $\sum\limits_{i< n} e_i \cdot f_i^\omega$, for some $n\in \mathbb{N}$, without loss of generality.
The formula $\mathrm{ArAcc}$ for this expression essentially recognises infinite words that have prefixes of the form $\sigma\tau_k$ for some $\sigma \in \mathcal{L}(e_i)$, for some $i<n$, and $\tau_k \in \mathcal{L} (f_i^k)$ for each $k \in \mathbb{N}$.
Clearly the condition $\mathrm{ArAcc}$ is a (provable) consequence of acceptance itself:
\begin{prop}
\label{prop:arith-acc}
$\mathsf{RCA}_0 \vdash \forall \mathcal A . (X \in \mathcal{L} (\mathcal A) \supset \mathrm{ArAcc} (X, \mathcal A))$.
\end{prop}
\begin{proof}
Working in $\mathsf{RCA}_0$, fix $\mathcal A = (A, Q , \delta, q_0, F)$ and suppose $X \in \mathcal{L}(\mathcal A)$.
Let $Y\in Q^\omega$ be an `accepting run' of $X$ on $\mathcal A$, cf.~\eqref{eqn:nd-memb}.
We may show that,
\begin{equation}
\label{eqn:fin-pref-run}
\exists z \in Q^* . \text{``$z$ is a finite prefix of $Y$ hitting $\geq y$ final states in $\mathcal A$''}
\end{equation}
by $ {\Sin{0}{1}} $-induction on $y$, appealing to the unboundedness of final states in $Y$
for both the base case and the inductive steps.
Now, in the definition of $\mathrm{ArAcc}$ in \eqref{eqn:aracc-dfn}, we set $x$ to be the least such $z$ for which \eqref{eqn:fin-pref-run}$[1/y]$ holds (again by induction), so that $F(x)$ from \eqref{eqn:aracc-dfn} holds.
Thus, for any $y \in \mathbb{N}$, we may find an appropriate $z$ making $E(z,x,y)$ in \eqref{eqn:aracc-dfn} true by appealing to \eqref{eqn:fin-pref-run}.
The fact that $z$ extends $x$ follows from leastness of $x$ and that $Y$ is a sequence, cf.~\eqref{eqn:tot-oracle} and \eqref{eqn:det-oracle}.
\end{proof}
%
Let us write $\mathcal A_1 \sqsubseteq \mathcal A_2$ for $\mathrm{Empty} ( ( \mathcal A_1^c \sqcup \mathcal A_2 )^c )$.
%
%
We may now present our main `uniform' result needed to carry out our soundness proof in FO theories.
\begin{thm}
\label{thm:arithmetisation-of-correctness}
$\mathsf{RCA}_0 + \CIND{\Sin 0 2} $ proves:
\begin{equation}
\label{eqn:arith-form-prog-traces}
\forall\ \text{DBA}\ \mathcal A_1 , \forall \ \text{NBA}\ \mathcal A_2 .\
\left(
(\mathcal A_1 \sqsubseteq \mathcal A_2 \wedge X \in \mathcal{L}(\mathcal A_1) )
\supset
\mathrm{ArAcc} (X, \mathcal A_2 )
\right)
\end{equation}
\end{thm}
\begin{proof}
Working in $\mathsf{RCA}_0 + \CIND{\Sin 0 2}$, let $\mathcal A_1$ be a DBA and $\mathcal A_2$ be a NBA such that $X \in \mathcal{L} (\mathcal A_1)$ and $\mathcal A_1 \sqsubseteq \mathcal A_2$.
We have:
\[
\begin{array}{rll}
& \mathrm{Empty}( (\mathcal A_1^c \sqcup \mathcal A_2)^c ) & \text{since $\mathcal A_1 \sqsubseteq \mathcal A_2$} \\
\implies & \forall Y \in A^\omega.\ Y\notin \mathcal{L} ( (\mathcal A_1^c \sqcup \mathcal A_2)^c ) & \text{by Lemma~\ref{lem:aut-clos-props-in-so-arith}.\ref{item:emptiness-rca}} \\
\implies & \forall Y \in A^\omega.\ Y \in \mathcal{L}(\mathcal A_1^c \sqcup \mathcal A_2) & \text{by Lemma~\ref{lem:aut-clos-props-in-so-arith}.\ref{item:compl-rca-s2ind}} \\
\implies & \forall Y \in A^\omega. (Y \in \mathcal \mathcal{L}(A_1^c) \vee Y \in \mathcal{L}(\mathcal A_2)) & \text{ by Lemma \ref{lem:aut-clos-props-in-so-arith}.\ref{item:union-rca}}\\
\implies & \forall Y \in A^\omega . ( Y \in \mathcal{L} (\mathcal A_1) \supset Y \in \mathcal{L} (\mathcal A_2)) & \text{by Lemma \ref{lem:aut-clos-props-in-so-arith}.\ref{item:compl-dba-rca}}\\
\implies & X \in \mathcal{L} (\mathcal A_2) & \text{since $X\in \mathcal{L}(\mathcal A_1)$}\\
\implies & \mathrm{ArAcc}(X, \mathcal A_2) & \text{by Prop.~\ref{prop:arith-acc}.} \qedhere
\end{array}
\]
\end{proof}
\noindent
Noticing that DBA acceptance is also purely arithmetical in $X$ (cf.~\ref{item:branch-aut-det}), by the conservativity result Prop.~\ref{prop:conservativity}, we have:
\begin{cor}\label{cor:is2-prov-aracc}
$\ISn 2 (X)$ proves \eqref{eqn:arith-form-prog-traces}.
\end{cor}
\subsection{Formalising the soundness argument for cyclic proofs}
At this point we are able to mostly mimic the formalisation of the soundness argument from \cite{Sim17:cyclic-arith}, although we must further show that a branch of invalid sequents, cf.~the proof of Prop.~\ref{prop:sound-cyclic}, is uniformly describable (cf.~\ref{item:invalid-branch-uniform}).
%
\smallskip
For $n\geq 0$, let $\mathbb{N}, \rho \models_n \phi$ be an appropriate $\Din{}{n+1}$ formula (provably in $\ISn{n+1}$)
asserting that a formula $\phi$ is true in $\mathbb{N} $ under the assignment $\rho$ of its free variables to natural numbers, as long as $\phi$ is a Boolean combination of $\Sin{}{n}$ (or $\Pin{}{n}$) formulae.\footnote{If $\phi$ is not a Boolean combination of $\Sin{}n$ formulae then $\mathbb{N}, \rho \models_n \phi$ crashes and returns $\bot$.}
%
Formally, the formula $\mathbb{N}, \rho \models_n \phi$ takes as arguments the \emph{codes} of $\rho$ and $\phi$, i.e.\ their G\"odel numbers; the construction of such a formula for $\models_n$ is standard (see, e.g.,~\cite{Bus98:handbook-of-pt,Kay91:models-of-pa,HajPud:93}) and it has size polynomial in $n$.
%
%
Importantly, there are $\ISn{n+1}$ proofs that $\models_n$ satisfies `Tarski's truth conditions'.
Writing $\mathrm{Bool}(\Phi)$ for the class of Boolean combinations of $\Phi$-formulae, we have:
\begin{prop}
[Properties of $\models_n$, see e.g.\ \cite{HajPud:93}]
\label{prop:tarski}
For $n\geq 0$, the following $\Pin{}{n+1}$ formulae have $\ISn{n+1}$ proofs of size polynomial in $n$:
\begin{enumerate}
\item\label{item:tarski-not} $\forall \phi \in \mathrm{Bool} (\Sin{}n). \forall \rho. \ (\mathbb{N}, \rho \models_n \neg \phi \ \equiv \ \mathbb{N}, \rho \nvDash_n \phi)$.
\item\label{item:tarski-or} $\forall \phi,\psi \in \mathrm{Bool} (\Sin{}n). \forall \rho. \ (\mathbb{N}, \rho \models_n (\phi \vee \psi) \ \equiv \ (\mathbb{N}, \rho \models_n \phi \ \vee \ \mathbb{N}, \rho \models \psi))$.
\item\label{item:tarski-and} $\forall \phi,\psi \in \mathrm{Bool} (\Sin{}n). \forall \rho. \ (\mathbb{N}, \rho \models_n (\phi \wedge \psi) \ \equiv \ (\mathbb{N}, \rho \models_n \phi \ \wedge \ \mathbb{N}, \rho \models \psi))$.
\item\label{item:tarski-exists} $\forall \phi \in \Sin{}{n}. \forall \rho. \ ( \mathbb{N}, \rho \models_n \exists x . \phi \ \equiv \ \exists y . (\mathbb{N}, \rho \cup \{ x \mapsto y \} \models_n \phi ) )$.
\item\label{item:tarski-forall} $\forall \phi \in \Pin{}{n}. \forall \rho. \ ( \mathbb{N}, \rho \models_n \forall x . \phi \ \equiv \ \forall y . (\mathbb{N}, \rho \cup \{ x \mapsto y \} \models_n \phi ) )$.
\end{enumerate}
\smallskip
\noindent
We also have $\ISn{n+1}$ proofs of size polynomial in $n$ of the \emph{substitution property}:
\begin{enumerate}
\setcounter{enumi}{5}
\item\label{item:tarski-subst} $\forall \phi \in \mathrm{Bool} (\Sin{}n) . \forall \rho. \forall \text{ terms } t. \ ( \mathbb{N}, \rho \cup \{ a \mapsto \rho(t) \} \models_n \phi \ \equiv \ \mathbb{N}, \rho \models_n \phi[t/a] )$.
\end{enumerate}
\end{prop}
\noindent
In particular we have the \emph{reflection property}:
\begin{prop}
[Reflection]
\label{prop:reflection}
For $n\geq 0$ we have $\ISn{1} \vdash \phi \equiv (\mathbb{N}, \emptyset \models_n \phi)$ with proofs of size polynomial in $n$ and $|\phi|$, for any closed formula $\phi \in \Sin{}n \cup \Pin{}n$.
\end{prop}
\noindent
Henceforth, all our proof complexity bounds in $n$ follow from the fact that proofs are parametrised by $\models_n$ and its basic properties from Prop.~\ref{prop:tarski} above.
\begin{defi}
[Uniform description of an invalid branch]
Let $\pi$ be a $\mathsf{CA}$ preproof of a sequent $\Gamma \Rightarrow \Delta$, and let $n\in \mathbb{N}$ be such that all formulae occurring in $\pi$ are $\Sin{}{n}$.
Let $\rho_0$ be an assignment such that $\mathbb{N}, \rho_0 \models_n \bigwedge \Gamma$ but $\mathbb{N} , \rho_0 \nvDash_n \bigvee \Delta$.
%
The branch of $\pi$ \dfntrm{generated} by $\rho_0$ is the invalid branch as constructed in the proof of Prop.~\ref{prop:sound-cyclic}, where at each step that there is a choice of premiss the leftmost one is chosen, and at each step when there is a choice of assignment of a natural number to a free variable the least one is chosen.
We write $\mathrm{Branch}_n(\pi, \rho_0, x, y)$ for the following predicate:
\[
\text{``the $x$\textsuperscript{th} element of the branch generated by $\rho_0$ in $\pi$ is $y$''}
\]
To be precise, the `element' $y$
is given as a pair $\pair{\rho_x}{\Gamma_x\Rightarrow \Delta_x}$ consisting of
a sequent $\Gamma_x \Rightarrow \Delta_x$ and an assignment $\rho_x$
that invalidates it.
\end{defi}
Notice that $\mathrm{Branch}_n(\pi, \rho_0 , x , y)$ is recursive w.r.t.\ the oracle $\models_n$, and so is expressible by a $\Din{}{1} ( \models_n )$ formula, making it altogether $\Din{}{n+1} $ in its arguments.
In fact, this is demonstrably the case in $\ISn{n+1}$, which can prove that $\mathrm{Branch}_n (\pi, \rho_0 , - , - )$ is the graph of a \emph{function}, as shown in Prop.~\ref{prop:inv-branch-isn} below.
%
Let us
write $\mathsf{conc} (\pi)$ for the conclusion of a $\mathsf{CA}$ proof $\pi$ and, as in Sect.~\ref{sect:sect:automata-prelims}, $\mathcal A^\pi_b$ and $\mathcal{A}^\pi_t$ for its branch and trace automata, resp.
When we write $\mathbb{N}, \rho \models_n( \Gamma \Rightarrow \Delta)$ we mean the $\Din{}{n+1}$ formula $(\mathbb{N}, \rho \nvDash_n \bigwedge\Gamma) \vee (\mathbb{N}, \rho \models_n \bigvee \Delta)$.
\begin{prop}
\label{prop:inv-branch-isn}
For $n\geq 0$, there are $\ISn{n+1}$ proofs of size polynomial in $n$ of:
\begin{equation}
\label{eqn:branch-is-total}
\begin{array}{l}
\forall \pi \text{ a $\mathsf{CA}$ preproof containing only $\Sin{} n$ formulae}. \\
\forall \rho_0 . \ ((\mathbb{N}, \rho_0 \nvDash_n \mathsf{conc} (\pi) )\supset \mathrm{Branch}_n (\pi, \rho_0 , - , -) \in \mathcal{L} ( \mathcal A^\pi_b ))
\end{array}
\end{equation}
\end{prop}
\begin{proof}
Working in $\ISn{n+1}$, let $\pi$ and $\rho_0$ satisfy the hypotheses of \eqref{eqn:branch-is-total} above.
The fact that $\mathrm{Branch}_n(\pi, \rho_0 , - , - )$ is deterministic, cf.~\eqref{eqn:det-oracle}, follows directly by induction on the position of the branch.
The difficult part is to show that $\mathrm{Branch}_n (\pi, \rho_0 , - , - )$ is total, cf.~\eqref{eqn:tot-oracle}, i.e.~that it never reaches a deadlock. For this we show,
\begin{equation}
\label{eqn:branch-total-invalid}
\forall x . \exists \pair {\rho_x} {\Gamma_x\Rightarrow \Delta_x} . \ (\mathrm{Branch}_n(\pi, \rho_0 , x , \pair {\rho_x} {\Gamma_x\Rightarrow \Delta_x} ) \wedge \ \mathbb{N}, \rho_x \nvDash_n (\Gamma_x\Rightarrow \Delta_x) )
\end{equation}
by ${\Sin{}{n+1}}$-induction on $x$.
The base case, when $x=0$, follows by assumption, so we proceed with the inductive case.
For a given $x$ let $\pair{\rho_x}{\Gamma_x \Rightarrow \Delta_x}$ witness \eqref{eqn:branch-total-invalid} above and let $\mathsf{r}$ be the rule instance in $\pi$ that $\Gamma_x \Rightarrow \Delta_x$ concludes.
If $\mathsf{r}$ is a $ \rigrul \exists$ step with associated term $t$, then there is only one premiss which we show remains false in the current assignment. This follows from:
\[
\begin{array}{rcll}
\mathbb{N}, \rho \nvDash_n \exists x . \phi & \implies & \mathbb{N}, \rho \models_n \forall x . \neg \phi & \text{by Prop.~\ref{prop:tarski}.\ref{item:tarski-not}} \\
& \implies & \forall y . (\mathbb{N}, \rho \cup \{ x \mapsto y \} \models_n \neg \phi ) & \text{by Prop.~\ref{prop:tarski}.\ref{item:tarski-forall}} \\
& \implies & \mathbb{N}, \rho \cup \{ x \mapsto \rho(t) \} \models_n \neg \phi & \text{by pure logic} \\
& \implies & \mathbb{N}, \rho \models_n \neg \phi [t/x] & \text{by Prop.~\ref{prop:tarski}.\ref{item:tarski-subst}} \\
& \implies & \mathbb{N}, \rho \nvDash_n \phi[t/x] & \text{by Prop.~\ref{prop:tarski}.\ref{item:tarski-not}.}
\end{array}
\]
If $\mathsf{r}$ is a $\rigrul{\forall}$ step
then there is only one premiss, for which we show that the appropriate invalidating assignment exists. This follows from,
\[
\begin{array}{rcll}
\mathbb{N}, \rho \nvDash_n \forall x . \phi & \implies & \mathbb{N}, \rho \models_n \exists x . \neg \phi & \text{by Prop.~\ref{prop:tarski}.\ref{item:tarski-not}} \\
& \implies & \exists y . ( \mathbb{N}, \rho \cup\{ x \mapsto y \} \models_n \neg \phi ) & \text{by Prop.~\ref{prop:tarski}.\ref{item:tarski-exists}} \\
& \implies & \exists \text{ least } y . ( \mathbb{N}, \rho \cup\{ x \mapsto y \} \models_n \neg \phi ) & \text{by $\CIND{\Sin{}{n+1}}$
}\\
& \implies & \exists \text{ least } y . ( \mathbb{N}, \rho\cup \{ x \mapsto y \} \nvDash_n \phi ) & \text{by Prop.~\ref{prop:tarski}.\ref{item:tarski-not}.}
\end{array}
\]
where, in the penultimate implication, we rely on the fact that the appropriate `minimisation' property is provable in $\CIND{\Sin{}{n+1}}$ (see, e.g., \cite{Bus98:handbook-of-pt}).
If $\mathsf{r}$ is a left quantifier step then it is treated similarly to the two right quantifier cases above by De Morgan duality. If $\mathsf{r} $ is a propositional step then the treatment is simple, following directly from Prop.~\ref{prop:tarski}. If $\mathsf{r}$ is an initial sequent we immediately hit a contradiction, since all of the axioms are provably true in all assignments.
If $\mathsf{r} $ is a substitution step, then the existence of the appropriate assignment follows directly from the substitution property, Prop.~\ref{prop:tarski}.\ref{item:tarski-subst}.
Finally, we may show that every state of the run of $\mathrm{Branch}_n (\pi, \rho_0 ,- , - )$ on $\mathcal A^\pi_b$ is final, by $\Sin{}{n+1}$-induction, since it always correctly follows a branch of $\pi$. Thus we have that $\mathrm{Branch}_n (\pi, \rho_0, -, - ) \in \mathcal{L} (\mathcal A^\pi_b)$.
\end{proof}
Now we can give a formalised proof of the soundness of cyclic proofs:
\begin{thm}
[Soundness of cyclic proofs, formalised]
\label{thm:soundness-formalised}
For $n \geq 0$, there are $\ISn{n+2}$ proofs of size polynomial in $n$ of:
\begin{equation}
\label{eqn:soundness-polysize}
\begin{array}{l}
\forall \pi \text{ a $\mathsf{CA}$ preproof containing only $\Sin{} n$ formulae}. \\
(\mathcal A^\pi_b \sqsubseteq \mathcal A^\pi_t
\ \supset\
\forall \rho_0 . \ \mathbb{N}, \rho_0 \models_n \mathsf{conc} (\pi))
\end{array}
\end{equation}
\end{thm}
\begin{proof}
First, instantiating $X$ in Cor.~\ref{cor:is2-prov-aracc} with a $\Din{}{n+1}$ formula $\phi_n$ yields $O(|\phi_n|)$-size $\ISn{n+2}$ proofs of \eqref{eqn:arith-form-prog-traces}$[\phi_n/X]$.
Hence, setting $\phi_n$ to be $\mathrm{Branch}_n (\pi, \rho_0 , - , - )$ and appealing to Prop.~\ref{prop:inv-branch-isn} above, we arrive at $\ISn{n+2}$ proofs of size polynomial in $n$ of:
\begin{equation}
\label{eqn:rfn-princ}
\begin{array}{l}
\forall \pi \text{ a $\mathsf{CA}$ preproof containing only $\Sin{} n$ formulae}. \\
\mathcal A^\pi_b \sqsubseteq \mathcal A^\pi_t
\supset
\forall \rho_0 .
(\mathbb{N}, \rho_0 \nvDash_n \mathsf{conc} (\pi)
\supset
\mathrm{ArAcc}( \mathrm{Branch}_n (\pi, \rho_0 , - , - ) , \mathcal A^\pi_t )
)
\end{array}
\end{equation}
Now, working in $\ISn{n+2}$, to prove \eqref{eqn:soundness-polysize} let $\pi $ satisfy $\mathcal A^\pi_b \sqsubseteq \mathcal A^\pi_t$.
For contradiction
assume, for some $\rho_0$, that $\mathbb{N}, \rho_0 \nvDash_n \mathsf{conc} (\pi)$.
By Prop.~\ref{prop:inv-branch-isn} we have
$\mathrm{Branch}_n (\pi, \rho_0 , - , - ) \in \mathcal{L} (\mathcal A^\pi_b)$,
so we henceforth write $\Gamma_x \Rightarrow \Delta_x$ and $\rho_x$ for the sequent and assignment at the $x$\textsuperscript{th} position of $\mathrm{Branch}_n (\pi, \rho_0 , -, - )$.
By \eqref{eqn:rfn-princ} above we have $\mathrm{ArAcc}( \mathrm{Branch}_n (\pi, \rho_0 , - , - ) , \mathcal A^\pi_t )$, so let $x$ witness its outer existential, cf.~\eqref{eqn:aracc-dfn}.
Now, let $y$ be the maximum value of $\rho_x (t)$ for all terms $t$ occurring in $\Gamma_x \Rightarrow \Delta_x$.
Again by $\mathrm{ArAcc}( \mathrm{Branch}_n (\pi, \rho_0 , - , - ) , \mathcal A^\pi_t )$, we have that there is some (finite) trace $z$ beginning from $\Gamma_x \Rightarrow \Delta_x $ that progresses $y+1$ times.
Writing $z(i)$ to denote the $i$\textsuperscript{th} term in the trace $z$, we may show by induction on $i\leq |z|$
that, if there are $j$ progress points between $z(0)$ and $z(i)$, then we have that $\rho_x (z(0)) \geq \rho_{x+i} (z(i)) +j$.
In particular, $y \geq \rho_x (z(0)) \geq \rho_{x + |z|} (z(|z|)) + (y +1) \geq y+1$, yielding a contradiction.
\end{proof}
\subsection{$\mathsf{PA}$ exponentially simulates $\mathsf{CA}$}
We can now give our main proof complexity result:
\begin{thm}
\label{thm:elementary-simulation}
If $\pi $ is a $\mathsf{CA}$ proof of $\phi$, then we can construct a $\mathsf{PA}$ proof of $\phi$ of size exponential in $|\pi|$.
\end{thm}
\begin{proof}
Take the least $n\in\mathbb{N}$ such that $\pi$ contains only $\Sin{}{n}$ formulae; in particular $n\leq |\pi|$.
Since $\pi$ is a correct cyclic proof, there is a $\mathsf{PA}$ proof of $\mathcal A^\pi_b \sqsubseteq \mathcal A^\pi_t$, by exhaustive search.
In fact, such a proof in $\mathsf{Q}$ may be constructed in exponential time in $|\pi|$, thanks to Obs.~\ref{obs:complexity-aut-constructions} and $\Sigma_1$-completeness of $\mathsf{Q}$ (see, e.g., \cite{HajPud:93}).
Hence, by instantiating $\pi$ in Thm.~\ref{thm:soundness-formalised}, we have $\ISn{n+2}$ proofs of $\mathbb{N}, \emptyset \models_n \phi$ of size exponential in $|\pi|$.
Finally by the reflection property, Prop.~\ref{prop:reflection}, we have that $\ISn{n+2} \vdash \phi$ with proofs of size exponential in $|\pi|$.
\end{proof}
%
Notice that we already have a converse polynomial simulation of $\mathsf{PA}$ in $\mathsf{CA}$ by the results of \cite{Sim17:cyclic-arith} or, alternatively, by Prop.~\ref{prop:isn-in-csn}.
\section{$\ISn{n+1}$ contains $\CSn n$}
\label{sect:nonuniform-ind-sim-cyc}
In fact the proof method we developed in the last section allows us to recover a result on logical complexity too.
By tracking precisely all the bounds therein, we obtain that $\CSn n$ is contained in $\ISn{n+2}$, which is already an improvement to Simpson's result (see Sect.~\ref{sect:conc} for a comparison).
To derive such bounds, in this section we concern ourselves only with cyclic proofs containing $\Sin{}{n}$ formulae.
The universal closures of the conclusions of such proofs axiomatise $\CSn{n}$, cf.~Dfn.~\ref{dfn:cphi}, so more complex theorems of $\CSn{n}$ are thence derivable by pure logic.
In fact, we may actually improve this logical bound and arrive at an optimal result (given Thm.~\ref{thm:cyclic-sim-ind}).
By more carefully analysing the proof methods of \cite{KMPS16:buchi-reverse}, namely an inspection of the proofs of Thms.~5 and 12 in that work, we have that:
\begin{prop}
[Implicit in \cite{KMPS16:buchi-reverse}]
\label{prop:nonuniform-compl}
$\mathsf{RCA}_0 \vdash \forall X \in A^\omega . (X\in \mathcal{L}(\mathcal A^c) \equiv X \notin \mathcal{L} (\mathcal A) )$, for any NBA $\mathcal A$.
\end{prop}
\noindent
Notice here that the universal quantification over NBA is \emph{external}, so that the complementation proofs are not necessarily uniform.
This is not a trivial result, since it relies on a version of Ramsey's theorem, the \emph{additive Ramsey theorem}, which can be proved by induction on the number of `colours'.
Usual forms of Ramsey's theorem are not proved by such an argument, and in fact it is well known that $\mathsf{RCA}_0$ cannot even prove Ramsey's theorem for pairs with only two colours (see, e.g., \cite{Hir14:reverse-math}).
We include in App.~\ref{sect:complementation-nonuniform} a self-contained (and somewhat simpler) proof of Prop.~\ref{prop:nonuniform-compl} above, for completeness.
%
This allows us to `un-uniformise' the results of the previous section, using Prop.~\ref{prop:nonuniform-compl} above instead of Lemma~\ref{lem:aut-clos-props-in-so-arith}.\ref{item:compl-rca-s2ind}, in order to `trade off' proof complexity for logical complexity:
\begin{prop}
[Soundness of cyclic proofs, non-uniformly formalised]
\label{prop:soundness-nonunif}
Let $n\geq 0 $ and $\pi$ be a $\mathsf{CA} $ proof containing only $\Sin{}{n}$ formulae.
$\ISn{n+1} \vdash \forall \rho_0. (\mathbb{N}, \rho_0 \models_n \mathsf{conc} (\pi))$.
\end{prop}
\begin{proof}
[Proof sketch]
We mimic the entire argument of Thm.~\ref{thm:soundness-formalised} by instantiating the fixed proof $\pi$ and using Prop.~\ref{prop:nonuniform-compl} above instead of Lemma~\ref{lem:aut-clos-props-in-so-arith}.\ref{item:compl-rca-s2ind}.
In particular, the required `non-uniform' versions of Thm.~\ref{thm:arithmetisation-of-correctness} and Cor.~\ref{cor:is2-prov-aracc} become derivable in $\mathsf{RCA}_0$ and $\ISn 1 (X)$ resp., thus reducing the global induction complexity by one level.
Hence we arrive at a `non-uniform' version of Thm.~\ref{thm:soundness-formalised}, peculiar to the fixed proof $\pi$ we began with, proved entirely within $\ISn {n+1}$, as required.
\end{proof}
\begin{thm}
\label{thm:nonuniform-ind-sim-cyc}
For $n\geq 0$, we have that $\CSn n \subseteq \ISn{n+1}$.
\end{thm}
\begin{proof}
By the definition of $\CSn n$, cf.~Dfn.~\ref{dfn:cphi}, it suffices to derive in $\ISn{n+1}$ just the (possibly open) conclusions of $\CSn{n}$ proofs containing only $\Sin {} n$ formulae,
and so the result follows directly from Props.~ \ref{prop:soundness-nonunif} and \ref{prop:reflection}.
\end{proof}
\subsection{On the proof complexity of $\CSn n$}
\label{sect:sect:prf-comp-csn}
One might be tempted to conclude that the elementary simulation of $\mathsf{CA}$ by $\mathsf{PA}$ should go through already for $\CSn n$ by $\ISn{n+2}$ (independently of $n$), due to the bounds implicit in the proof of Thm.~\ref{thm:elementary-simulation}.
Furthermore, if we are willing to give up a few more exponentials in complexity, one may even bound the size of $\ISn 1$ proofs arising from Prop.~\ref{prop:nonuniform-compl} by an appropriate elementary function (though this analysis is beyond the scope of this paper).
%
However, we must be conscious of the `robustness' of the definition of $\CSn n$ proofs in terms of complexity.
The one we gave, which essentially requires cyclic proofs to contain only $\Sin{}{n}$-formulae, is more similar to `free-cut free' $\ISn n $ proofs than general ones, cf.~Thm.~\ref{thm:free-cut-elim}, so it seems unfair to compare these notions of proof in terms of proof complexity.
In fact we may define a more natural notion of a $\CSn n$ proof from the point of view of complexity, while inducing the same theory.
\smallskip
First, let us recall some notions from, e.g., \cite{Bro06:phdthesis,BroSim11:seq-calc-ind-inf-desc}.
Any cyclic proof can be written in `cycle normal form', where any `backpointer' (e.g., the conclusions of upper sequents marked $\bullet$ until now) points only to a sequent that has occurred below it in the proof.
Referring to the terminology of \cite{Bro06:phdthesis,BroSim11:seq-calc-ind-inf-desc} etc., we say that a sequent is a \emph{bud} in a cyclic proof if it has a backpointer pointing to an identical sequent (called the \emph{companion}).
\begin{prop}
\label{prop:cyclic-sn-general-dfn}
If $\phi $ has a $\mathsf{CA}$ proof whose buds and companions contain only $\Sin {} n$ formulae then $\CSn n \vdash \phi$.
\end{prop}
\begin{proof}
[Proof idea]
Once again, we simply apply free-cut elimination, Thm.~\ref{thm:free-cut-elim}, treating any backpointers as `initial sequents' for a proof in cycle normal form.
\end{proof}
%
This result again shows the robustness of the definition of $\CSn n$ as a theory, and we would further argue that, from the point of view of proof complexity, the backpointer-condition induced by Prop.~\ref{prop:cyclic-sn-general-dfn} above constitutes a better notion of `proof' for $\CSn n$.
One could go yet further and argue that an even better notion of `proof' would allow pointers to any other identical sequent in the proof (not just those below it), which could potentially make for an exponential improvement in proof complexity.
%
At the same time we see that it is not easy to compare the proof complexity of such systems for $\CSn n $ with those for $\ISn{n+1}$, due to the fact that we have used a free-cut elimination result for the simulations in both directions, inducing a possibly non-elementary blowup in proof size.
It would be interesting if a more fine-grained result regarding the relative proof complexity of $\ISn{n+1}$ and $\CSn{n}$ could be established, but this is beyond the scope of the current work.
\section{Some metamathematical results}
\label{sect:further-metalogical}
In this section we make some further observations on various properties of the cyclic theories in this work, which are later applied in Sect.~\ref{sect:red-to-det}. The exposition we give is brief, since we follow standard methods, but we provide appropriate references for the reader.
\subsection{Provably recursive functions of $\CDn 0$}
\label{sect:sect:prov-rec-fns-cd0}
As a corollary of the Thms.~\ref{thm:cyclic-sim-ind} and \ref{thm:nonuniform-ind-sim-cyc} we have that, for $n\geq 1$, the provably recursive functions of $\CSn n $ coincide with those of $\ISn{n+1}$. Such functions were characterised by Parsons in \cite{Par72:n-quant-ind} as certain fragments of G\"odel's system $\mathsf T$ or, equivalently, by recursion up to suitably bounded towers of $\omega$, e.g.\ in \cite{Buss1995witness}.
This apparently leaves open a gap for the case of $\CDn 0 $ (a.k.a.\ $ \CSn 0$).
However notice that from the definition of $\CSn n $ and Thm.~\ref{thm:cyclic-sim-ind} we have:
\begin{cor}
[of Thm.~\ref{thm:cyclic-sim-ind}]
\label{cor:pi-axiomatised}
$\CSn n$ is axiomatised by the $\Pin{}{n+ 1}$-consequences of $\ISn {n+1}$, for $n\geq 0$.
\end{cor}
\noindent
In particular, since $\IDn 0 $ is known to be $\Pin{}1$-axiomatised (see, e.g., Thm.~1.27 in \cite{HajPud:93}), we have that in fact $\CDn 0 \supseteq \IDn 0 $ (i.e.\ over \emph{all} theorems).
Thus $\CDn 0 $ can prove recursive \emph{at least} the functions (bitwise computable) in the linear-time hierarchy, from the analogous result for $\IDn 0$ (see, e.g., Theorem~III.4.8 in \cite{Cook:2010:LFP:1734064}).
Conversely, since $\CDn 0 $ is also $\Pin{}1$-axiomatised by the above observation, it has no more `computational content' than $\IDn 0 $, as we will now see.
Recall that the {linear-time hierarchy} ($\mathbf{LTH}$) is the class of predicates expressible by a $\Din{}0$ formula in the language of arithmetic.\footnote{Equivalently it is the class of predicates recognised by an alternating Turing machine with random access in a linear number of steps with only boundedly many alternations. See, e.g., \cite{CloKra02} for more details.}
A function is said to be in $\mathbf{FLTH}$ just if it has linear growth rate (in terms of the bit string representation) and is bitwise computable in $\mathbf{LTH}$.
\begin{prop}
\label{prop:cd0-lth}
If $\CDn 0 \vdash \forall \vec x . \exists ! y . \phi(\vec x , y)$ for a $\Sin{}1$-formula $\phi$, then $\phi$ computes the graph of a function in $\mathbf{FLTH}$.
\end{prop}
\begin{proof}
[Proof sketch]
Suppose $\phi $ is $\exists \vec y . \phi_0$ with $\phi_0 \in \Din{} 0 $.
By `Parikh's theorem' for $\Pin{}1$-axiomatised theories (see, e.g., Thm.~III.2.3 in \cite{Cook:2010:LFP:1734064}) we moreover have $\CDn 0 \vdash \forall \vec x . \exists! y < t . \exists \vec y < \vec t . \phi_0$ for some terms $t, \vec t$.
This means we may simply search for a witness $y<t$ of linear size verifying $\exists \vec y < \vec t . \phi_0 $, an $\mathbf{LTH}$ property, and thus $\exists \vec y < \vec t . \phi_0 $ computes the graph of a function in $\mathbf{FLTH}$.
\end{proof}
Interestingly, we cannot strengthen the above proposition to ``$\CDn 0 $ and $\IDn 0 $ prove the same $\Pin{}2$ theorems''. This is because $\ISn 1$ proves the \emph{consistency} of $\IDn 0 $, a $\Pin{} 1$ sentence, so $\CDn 0 $ does too by Thm.~\ref{thm:cyclic-sim-ind}.
Thus the aforementioned stronger statement would contradict G\"odel's second incompleteness theorem for $\IDn 0 $.
\smallskip
See, e.g., \cite{Bus98:handbook-of-pt,Cook:2010:LFP:1734064} for further discussions on the provably recursive functions of fragments of (bounded) arithmetic, and see, e.g., \cite{CloKra02} for more details on relationships between the language of arithmetic and recursive function classes.
\subsection{Failure of cut-admissibility}
\label{sect:sect:cut-ad-fails}
As a corollary of our results, we may formally conclude that the cut rule is not admissible in $\mathsf{CA}$, or indeed any of its fragments $\CSn n$.\footnote{This observation was pointed out to me by Stefano Berardi.}
In fact the situation is rather worse than that:
\begin{cor}
[of Thms.~\ref{thm:cyclic-sim-ind} and \ref{thm:nonuniform-ind-sim-cyc}]
\label{prop:cut-non-admiss}
Let $n\geq 1$. The class of $\mathsf{CA}$ proofs with only $\Sin {} {n-1} $ cuts is not complete for even the $\Pin{}1$ theorems of $\CSn {n}$.
\end{cor}
\begin{proof}
For a recursively axiomatised theory $T$, let $\cons T$ be an appropriate $\Pin{}{1}$ sentence expressing that ``$T$ does not prove $0=1$''.
It is well-known that $\ISn{n+1} \vdash \cons {\ISn n}$ (see, e.g., \cite{Kay91:models-of-pa,Bus98:handbook-of-pt,HajPud:93}), so also $\CSn n \vdash \cons {\ISn n}$ by Thm.~\ref{thm:cyclic-sim-ind}.
For contradiction, suppose $\cons{\ISn n }$ concludes some $\mathsf{CA}$ proof with only $\Sin{}{n-1}$ cuts; then in fact $\CSn{n-1} \vdash \cons {\ISn n}$ by degeneralising (for the case $n=1$) and the subformula property. However this implies $\ISn n \vdash \cons{\ISn n }$ by Thm.~\ref{thm:nonuniform-ind-sim-cyc}, which is impossible by G\"odel's second incompleteness theorem for $\ISn n $.
\end{proof}
See, e.g., \cite{Bus98:handbook-of-pt,Kay91:models-of-pa,HajPud:93} for further discussions on the provability of consistency principles for fragments of arithmetic.
\subsection{Reflection and consistency}
\label{sect:sect:rfn-cons}
Thanks to the uniformity of the results from Sect.~\ref{sect:ind-sim-cyc}, we can give some fundamental metalogical properties regarding provable soundness and consistency of cyclic proofs.
First we will fix our formalisation of $\CSn n $-provability (of arbitrary sequents, not just $\Sin{}{n}$) in the language of arithmetic.
\smallskip
Let $n\geq 0$.
We will fix some appropriate formula $\mathsf{Prf}_n (\pi,\phi)$ expressing that $\pi$ is a $\CSn n $ proof of $\phi$.
We suppose that proofs are written as usual derivations (finite trees or dags)
whose leaves are either axiom instances from $\mathsf{Q}$, or otherwise some (possibly open) $\Sin{}n$ sequent labelled by an associated cyclic proof that derives it (containing only $\Sin{}{n}$ formulae).
Descriptively, $\mathsf{Prf}_n (x,y)$ checks that $\pi$ is a proof of $\phi$ by first checking that it is a well-formed derivation, then checking that each premiss is either an axiom instance from $\mathsf{Q}$ or is labelled by a correct cyclic proof deriving it.
In the latter case it must search for a certificate verifying that the cyclic proof satisfies the automaton-inclusion condition, i.e.\ that $\mathcal A_b \sqsubseteq \mathcal A_t$.
%
While $\mathsf{Prf}_n (\pi,\phi)$ is recursive in $\pi$ and $\phi$, this may not be provably the case in weak theories such as $\IDn 0 $.
%
Thus we fix $\mathsf{Prf}_n $ to be an appropriate $\Sin{}1$ formula, as described above,
%
and we write $\square_n \phi$ for $\exists \pi . \mathsf{Prf}_n (\pi, {{\phi}})$.
%
We write $\Rfn{\Pin{}{k}}{\CSn n }$ for the (local) \emph{$\Pin{}{k}$-reflection principle} of $\CSn n $. I.e.\
\[
\Rfn{\Pin{}{k}}{\CSn n }
\ := \
\{\square_n \phi \supset \phi : \phi \in \Pin{}{k} \}
\]
\begin{cor}
[of Thm.~\ref{thm:soundness-formalised}]
\label{cor:reflection}
For $n\geq 0$, we have $\ISn{n+2}\vdash \Rfn{\Pin{}{n+1}}{\CSn n}$.
\end{cor}
\begin{proof}
Let $\phi(\vec x )$ be a $\Sin{}{n}$ formula. Working in $\ISn{n+2} $, suppose that $\square_n \forall \vec x . \phi$ (so that $\CSn n \vdash \forall \vec x . \phi (\vec x)$).
We may assume that every formula occurring in a $\CSn{n}$ proof of the sequent $\Rightarrow \phi(\vec x)$ is $\Sin{}{n}$, thanks to free-cut elimination, Thm.~\ref{thm:free-cut-elim} (recall that this result is provable already in $\ISn 1$).
Thus we have $\mathbb{N}, \emptyset \models_{n+1} \forall \vec x . \phi(\vec x)$ by Thm.~\ref{thm:soundness-formalised} and Prop.~\ref{prop:tarski}.\ref{item:tarski-forall}, whence the result follows by the reflection property, Prop.~\ref{prop:reflection}.
\end{proof}
\noindent
Notice that, while the statement of Cor.~\ref{cor:reflection} above is peculiar to the current formulation of a $\CSn n$ proof, $\mathsf{Prf}_n$, it holds also under any other notion of proof that is provably equivalent in $\ISn{n+2}$.
In particular, in Sect.~\ref{sect:sect:prf-comp-csn} we discussed another notion of proof for $\CSn{n}$ which, morally, allowed ``free cuts'' to occur inside cycles.
Since the equivalence of the two formulations, Prop.~\ref{prop:cyclic-sn-general-dfn}, is proved using only free-cut elimination and basic reasoning, all formalisable in $\ISn 1$,
the version of Cor.~\ref{cor:reflection} for that more liberal notion of a $\CSn n $ proof holds too.
\smallskip
As usual, we may see $\Pin{}{1}$-reflection as another formulation of `consistency'.
Let us write $\cons{\CSn n }$ for the sentence $\neg \square_n 0=1$ (notice that this is a $\Pin{}{1}$ sentence).
\begin{cor}
[of Thm.~\ref{thm:soundness-formalised}]
\label{cor:isn+2-prv-cons-csn}
For $n\geq 0$, we have $\ISn{n+2} \vdash \cons{\CSn n}$
\end{cor}
\begin{proof}
Follows immediately from Cor.~\ref{cor:reflection} above by substituting $0=1$ for $\phi$.
\end{proof}
\noindent
We will see in the next subsection that this result is, in fact, optimal with respect to logical complexity.
\subsection{Incompleteness}
\label{sect:sect:incompleteness-csn}
Unsurprisingly, all the theories $\CSn n$ suffer from G\"odel's incompleteness theorems.
Even though $\CSn n$ is not explicitly defined axiomatically, it does have a recursively enumerable notion of provability, namely $\square_n $, and so must be incomplete with respect to this notion (see, e.g., Thm.~2.21 in \cite{HajPud:93}):
%
\begin{thm}
[G\"odel's second incompleteness theorem, for cyclic theories]
\label{thm:goedel-second-incompleteness}
For $n\geq 0$, as long as $\CSn n $ is consistent (i.e.\ $\CSn n \nvdash 0=1$), we have $\CSn{n} \nvdash \cons{\CSn n }$.
\end{thm}
%
\noindent
Consequently we have that Cor.~\ref{cor:isn+2-prv-cons-csn} is, in fact, optimal in terms of logical complexity:
\begin{cor}
\label{cor:isn+1-nprv-cons-csn}
For $n\geq 0 $, we have $\ISn{n+1} \nvdash \cons{\CSn n}$.
\end{cor}
\begin{proof}
Suppose otherwise. Then also $\CSn n \vdash \cons{\CSn n }$ by $\Pin{}{1}$-conservativity, cf.\ Thm.~\ref{thm:cyclic-sim-ind}, which contradicts G\"odel's second incompleteness above, Thm.~\ref{thm:goedel-second-incompleteness}.
\end{proof}
\noindent
We will see in the next section that this has a curious consequence for the reverse mathematics of results in $\omega$-automaton theory.
\section{On the logical strength of McNaughton's theorem}
\label{sect:red-to-det}
In this section we show how the results of this work yield an unexpected corollary: certain formulations of \emph{McNaugton's theorem}, that every NBA has an equivalent deterministic `parity' or `Muller' automaton, are not provable in $\mathsf{RCA}_0$. The general question of the logical strength of McNaughton's theorem was notably left open in the recent work \cite{KMPS16:buchi-reverse}.
Our result is non-uniform in the sense that unprovability holds for any \emph{explicit} primitive recursive determinisation construction. As far as the author is aware this accounts for all known proofs of McNaughton's theorem, suggesting that it is unlikely to be provable at all, in its usual uniform version, in $\mathsf{RCA}_0$.
That said, we point out that the statement of McNaughton's theorem itself is arguably not so well-defined in the context of reverse mathematics: it is not clear in $\mathsf{RCA}_0$ that different versions of the theorem coincide, namely with respect to the choice of (a) acceptance conditions (parity, Muller, etc.) and (b) formulation of the set of states infinitely often hit during a run (negative, $\forall$, vs.\ positive, $\exists$).
Our argument is based on an alternative route to proving the soundness of $\CSn n$. Assuming that an appropriate version of McNaughton's theorem is indeed provable in $\mathsf{RCA}_0$, we are in fact able to formalise the soundness argument for $\CSn n $ already in $\ISn{ n+1}$. However, consequently we have that $\ISn {n+1}$ proves the consistency of $\CSn n $, and so $\CSn n $ proves its own consistency by $\Pin{}1$-conservativity, cf.~Thm.~\ref{thm:cyclic-sim-ind}, which is absurd by G\"odel's second incompleteness theorem for $\CSn n$, Thm.~\ref{thm:goedel-second-incompleteness}.
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%
\subsection{Deterministic parity automata and universality}
Due to space considerations, we only briefly present the details of parity automata. The reader is encouraged to consult, e.g., \cite{Tho97:aut-chapt}, for further details on automaton theory for $\omega$-languages.
A (non-deterministic) \textbf{Rabin} or \textbf{parity} automaton (NRA) is a just a NBA where, instead of a set of final states $F$, we have a function $c: Q \to \mathbb{N}$, called a \textbf{colouring}.
A word is accepted by a NRA if it has a run in which the least colour of a state occurring infinitely often is even.
The notion of deterministic parity automaton (DRA) is analogous to that of a DBA, i.e.\ requiring the transition relation to be deterministic and total.
\begin{thmC}
[McNaughton, \cite{McN66:determinisation}]
For every NBA $\mathcal A$, we can effectively construct a DRA accepting the same language.
\end{thmC}
\noindent
Actually, McNaughton gave this result for deterministic \emph{Muller} automata rather than parity automata. The equivalence of these two models is well-known though, as we previously mentioned, it is not clear whether $\mathsf{RCA}_0$ can prove their equivalence.
The fact that we use parity automata here is arbitrary; we believe a similar exposition could be carried out for Muller automata.
\smallskip
%
As for DBA, we may naturally express language acceptance for a DRA $\mathcal A = (A, Q, \delta, q_0, c)$ by an {arithmetical} formula, i.e.\ without SO quantifiers.
For our purposes, it will be useful to take a `negative' formulation of acceptance:
\[
X \in \mathcal{L} (\mathcal A)
\ := \
\forall q \in Q .
\left(
\left(
\begin{array}{rl}
& \forall x . \exists x' > x .\ q_X (x') = q \\
\wedge & \exists x . \forall x'> x .\ c(q_X (x')) \geq c(q)
\end{array}
\right)
\supset \text{``$c(q)$ is even''}
\right)
\]
%
%
%
%
%
We write $\sigma : q_1 \trarr * \delta q_2$ if a word $\sigma \in A^*$ determines a path along $\delta $ starting at $q_1$ and ending at $q_2$. We write $\trarr + \delta$ when the path is nonempty.
A \emph{simple loop} about a state $q\in Q$ is a nonempty path along $\delta$ beginning and ending at $q$ that visits no intermediate state more than once.
Recall that we call an $\omega$-automaton \emph{universal} if it accepts all $\omega$-words over its alphabet.
We write $\mathrm{Univ} (\mathcal A)$ for a standard recursive procedure testing {universality} of a DRA $\mathcal A$: ``for every odd-coloured state $q$ reachable from $q_0$, any simple loop about $q$ contains a state coloured by an even number $<c(q)$''. More formally, writing $\sigma' \leq \sigma$ if $\sigma'$ is a prefix of $\sigma$:
$$
\mathrm{Univ} (\mathcal A)
\ := \
\begin{array}{l}
\ \ \forall q \trrra * \delta q_0 . \ \forall \sigma : q \trarr + \delta q. \\
\left(
\begin{array}{rl}
&
\text{``$\sigma$ is a simple loop''} \wedge \text{``$c(q) $ is odd''} \\
\supset &
\exists \sigma' \leq \sigma . \ \exists q'\in Q . \ (\sigma' : q \trarr + \delta q' \ \wedge \text{``$c(q') $ even''} \wedge c(q')<c(q))
\end{array}
\right)
\end{array}
$$
Clearly this formula is provably $\Din 0 1 $ in $\mathsf{RCA}_0$.
%
%
%
Furthermore:
\begin{prop}
\label{prop:rca-prov-corr-univ-dra}
$\mathsf{RCA}_0 \vdash \forall \text{ DRA } \mathcal A . \ (\mathrm{Univ}(\mathcal A) \equiv \forall X \in A^\omega . X \in \mathcal{L} (\mathcal A))$.
\end{prop}
\begin{proof}
Working in $\mathsf{RCA}_0$, let $\mathcal A = (A, Q, \delta, q_0, c)$ be a DRA.
For the left-right implication, suppose there is some $X \in A^\omega$ such that $X \notin \mathcal{L} (\mathcal A)$. Thus we have some $q \in Q$ such that $c(q)$ is odd and the following hold:
\begin{equation}
\label{eqn:inf-occ}
\forall x . \exists x' > x .\ q_X (x') = q
\end{equation}
\begin{equation}
\label{eqn:event-lb}
\exists x . \forall x'> x.\ c(q_X (x') ) \geq c(q)
\end{equation}
Let $x_0$ be a witness to \eqref{eqn:event-lb}, and let $x_0 < x_1 < x_2$ such that $q_X (x_1) = q_X (x_2) = q$, by two applications of \eqref{eqn:inf-occ}.
We will need the following intermediate
(arithmetical)
result,
\[
\text{If $\sigma: q \trarr + \delta q$, there is a subsequence $\sigma'$ of $\sigma$ that is a simple loop on $q$.}
\]
\noindent
which follows directly by induction on $|\sigma|$, eliminating intermediate loops at each inductive step in the case of non-simplicity.
Now we apply this result to the sequence $(X(x))^{x_2}_{x = x_1}$ to obtain a simple loop about $q$; moreover since this will be a subsequence of $(X(x))^{x_2}_{x = x_1}$, we have that any even-coloured state occurring in it is coloured $> c(q)$, since $x_0$ witnesses \eqref{eqn:event-lb} and $x_0 < x_1 < x_2 $, so $\neg \mathrm{Univ} (\mathcal A)$.
For the right-left implication, we proceed again by contraposition. Suppose $\neg \mathrm{Univ} (\mathcal A)$, and let $\sigma: q_0 \trarr * \delta q$ and $\tau : q \trarr + \delta q$ such that $c(q)$ is odd, and $\tau$ is a simple loop containing no states coloured $< c(q)$.
We may now set $X = \sigma \tau^\omega$ (which is easily defined by comprehension) and show that $X\notin \mathcal{L} (\mathcal A)$.
For this it suffices to show \eqref{eqn:inf-occ} and \eqref{eqn:event-lb} above. For the former, given $x$ we set $x ' = |\sigma | + m|\tau| > x$, for some sufficiently large $m$.
For the latter, we set $x = |\sigma|$ as the witness to the outer existential, whence \eqref{eqn:event-lb} follows by construction of $\tau$.
%
%
%
%
\end{proof}
\subsection{Reducing soundness of $\CDn 0 $ to a version of McNaughton's theorem}
Henceforth we may write $\mathcal{L} (\mathcal A) = \mathcal{L} (\mathcal B)$ as shorthand for $\forall X . (X \in \mathcal{L} (\mathcal A) \equiv X\in \mathcal{L}(\mathcal B))$, where $\mathcal A$ and $\mathcal B$ may be any type of automaton thus far encountered with respect to their associated notions of membership.
Based on our `negative' formulation of DRA acceptance, we define for a (definable) function $d$: $$\mathsf{McNaughton}^-_d \ := \
\forall \text{ NBA }\mathcal A .
(
\text{``$d(\mathcal A)$ is a DRA''}
\wedge
\mathcal{L} (\mathcal A) = \mathcal{L} (d(\mathcal A))
$$
Assuming this is provable in $\mathsf{RCA}_0$ for some primitive recursive function $d$, we will reproduce a version of Thm.~\ref{thm:arithmetisation-of-correctness} in $\ISn{n+1}$.
%
The idea is that, rather than expressing the fact that $\mathcal{L} (\mathcal A_1 )\subseteq \mathcal{L}(\mathcal A_2)$ by saying ``$(\mathcal A_1^c \sqcup \mathcal A_2 )^c $ is empty'', as we did in Sects.~\ref{sect:ind-sim-cyc} and \ref{sect:nonuniform-ind-sim-cyc}, we may rather express it as ``$\mathcal A_1^c \sqcup \mathcal A_2$ is universal'', relying on
$\mathsf{McNaughton}^-_d$ and Prop.~\ref{prop:rca-prov-corr-univ-dra} above.
\medskip
For a DBA $\mathcal A_1 $ and a NBA $\mathcal A_2$ we define $\mathcal A_1 \sqsubseteq_d \mathcal A_2$ as
\(
\mathrm{Univ} (d(\mathcal A_1^c \sqcup \mathcal A_2 )))
\).
Let us write $(\ref*{eqn:arith-form-prog-traces}_d)$ for the equation \eqref{eqn:arith-form-prog-traces} with $\sqsubseteq$ replaced by $\sqsubseteq_d$, i.e.:
\begin{equation}
\tag{$\ref*{eqn:arith-form-prog-traces}_d$}
\label{eqn:arith-form-prog-traces'}
\forall\ \text{DBA}\ \mathcal A_1 , \forall \ \text{NBA}\ \mathcal A_2 .\
\left(
(\mathcal A_1 \sqsubseteq_d \mathcal A_2 \wedge X \in \mathcal{L}(\mathcal A_1) )
\supset
\mathrm{ArAcc} (X, \mathcal A_2 )
\right)
\end{equation}
We have the following analogue to Thm.~\ref{thm:arithmetisation-of-correctness}:
\begin{prop}
\label{prop:rca+mcn-prov-aracc'}
$\mathsf{RCA}_0 + \mathsf{McNaughton}^-_d\vdash \eqref{eqn:arith-form-prog-traces'}$.
\end{prop}
\begin{proof}
Mimicking the proof of Thm.~\ref{thm:arithmetisation-of-correctness}, we work in $\mathsf{RCA}_0 $ and suppose $X\in \mathcal{L} (\mathcal A_1)$ and $\mathcal A_1 \sqsubseteq_d \mathcal A_2$.
We have:
\[
\begin{array}{rll}
& \mathrm{Univ} (d(\mathcal A_1^c \sqcup \mathcal A_2)) & \text{since $\mathcal A_1 \sqsubseteq_d \mathcal A_2$} \\
\implies & \forall Y \in A^\omega . Y \in \mathcal{L} (\mathcal A_1^c \sqcup \mathcal A_2) & \text{by
Prop.~\ref{prop:rca-prov-corr-univ-dra} and
$\mathsf{McNaughton}^-_d$} \\
\implies & \forall Y \in A^\omega. ( Y \in \mathcal{L} (\mathcal A^c_1) \vee Y \in \mathcal{L} (\mathcal A_2) ) & \text{by Lemma~\ref{lem:aut-clos-props-in-so-arith}.\ref{item:union-rca} } \\
\implies &\forall Y \in A^\omega. ( Y \in \mathcal{L} (\mathcal A_1) \supset Y \in \mathcal{L} (\mathcal A_2) ) & \text{by Lemma~\ref{lem:aut-clos-props-in-so-arith}.\ref{item:compl-dba-rca}} \\
\implies & X \in \mathcal{L} (\mathcal A_2) & \text{since $X \in \mathcal{L}(\mathcal A_1)$} \\
\implies & \mathrm{ArAcc} (X, \mathcal A_2) & \text{by Prop.~\ref{prop:arith-acc}.} \qedhere
\end{array}
\]
\end{proof}
We may use this result to reconstruct the entire formalised soundness argument for $\CSn n$ of Sect.~\ref{sect:ind-sim-cyc} in $\ISn{n+1}$ instead of $\ISn{n+2}$, assuming $\mathsf{McNaughton}^-_d$. In particular, using Prop.~\ref{prop:rca+mcn-prov-aracc'} above instead of Thm.~\ref{thm:arithmetisation-of-correctness}, we may recover versions of Cor.~\ref{cor:is2-prov-aracc}, Thm.~\ref{thm:soundness-formalised} and Cor.~\ref{cor:reflection} for $\ISn{n+1}$ instead of $\ISn{n+2}$, with respect to \eqref{eqn:arith-form-prog-traces'} instead of \eqref{eqn:arith-form-prog-traces}.
%
Formally, let $\Rfnd{\Pin{}{k}}{\CSn n }$ denote the formulation of the $\Pin{}{k}$-reflection principle for $\CSn n $ induced by using $\sqsubseteq_d$ instead of $\sqsubseteq$ throughout Sect.~\ref{sect:sect:rfn-cons}, with respect to the definitions of $\mathsf{Prf}_n$ and $\square_n $.
Similarly, let $\consd{\CSn n }$ be the induced consistency principle.
\begin{prop}
\label{prop:if-mcn-then-rfn}
For $n\geq 0$, if $\mathsf{RCA}_0 \vdash \mathsf{McNaughton}^-_d$ for some primitive recursive function $d$, then $\ISn{n+1} \vdash \Rfnd{\Pin{}{n+1}}{\CSn n} $, so
in particular $\ISn{n+1} \vdash \consd{\CSn n }$.
\end{prop}
\begin{proof}
[Proof sketch]
The argument goes through just like that of Cors.~\ref{cor:reflection} and \ref{cor:isn+2-prv-cons-csn} of Thm.~\ref{thm:soundness-formalised}, except that we use Prop.~\ref{prop:rca+mcn-prov-aracc'} instead of Thm.~\ref{thm:arithmetisation-of-correctness}. Appealing to the assumption that $\mathsf{RCA}_0 \vdash \mathsf{McNaughton}^-_d$, this version of the argument requires only $\CIND{\Sin{0}{1}}$ instead of $\CIND{\Sin{0}{2}}$, thus yielding $\ISn{n+1}$ proofs overall once we substitute the appropriate formulae for $X$.
\end{proof}
\begin{thm}
$\mathsf{RCA}_0 \nvdash \mathsf{McNaughton}^-_d$, for any primitive recursive function $d$.
\end{thm}
\begin{proof}
[Proof sketch]
The same argument as Cor.~\ref{cor:isn+1-nprv-cons-csn} holds for our revised notion of consistency; in particular we have that $\ISn 1 \nvdash \consd{\CDn 0 }$.
The result now follows immediately by the contraposition of Prop.~\ref{prop:if-mcn-then-rfn} above, for $n=0$.
\end{proof}
\section{Conclusions and further remarks}
\label{sect:conc}
In this work we developed the theory of cyclic arithmetic by studying the logical complexity of its proofs.
We showed that inductive and cyclic proofs of the same theorems require similar logical complexity, and obtained tight quantifier complexity bounds in both directions.
We further showed that the proof complexity of the two frameworks differs only elementarily, although it remains unclear how to properly measure proof complexity for the fragments $\CSn n$, even if the theory itself seems well-defined and robust.
Many of these issues constitute avenues for further work.
\subsection{Comparison to the proofs of \cite{BerTat17:lics} and \cite{Sim17:cyclic-arith}}
One reason for our improved quantifier complexity compared to \cite{Sim17:cyclic-arith}, is that Simpson rather relies on \emph{Weak K\"onig's Lemma} ($\mathsf{WKL}$) to obtain an infinite branch when formalising the soundness argument for cyclic proofs.
This, \emph{a priori}, increases quantifier complexity of the argument, since $\mathsf{WKL}$ is known to be unprovable in $\mathsf{RCA}_0$ even in the presence of $\CIND{\Sin{0}{2}}$;
in fact, it is incomparable to $\CIND{\Sin{0}{2}}$ (see, e.g., \cite{KMPS16:buchi-reverse}).
That said, we believe that the `bounded-width $\mathsf{WKL}$' ($\mathsf{bwWKL}$) of \cite{KMPS16:buchi-reverse} should suffice to carry out Simpson's proof, and this principle is provable already in $\mathsf{RCA}_0 + \CIND{\Sin 0 2}$.
Applying this strategy to his proof yields only that $\CSn n \subseteq \ISn{n+3}$, since $\mathsf{bwWKL}$ is applied to a $\Pin{0}{n+1}$ set, though this should improve to a $\ISn{n+2}$ bound by using the non-uniform version of NBA complementation implicit in \cite{KMPS16:buchi-reverse}, cf.~Prop.~\ref{prop:nonuniform-compl}.
We reiterate that the main improvement here is in giving a uniform formulation of those results; not only does this lead to a better proof complexity result, cf.~Thm.~\ref{thm:elementary-simulation}, but we also recover a metamathematical account of the theories $\CSn n$, cf.~\ref{sect:further-metalogical}.
%
%
%
%
Berardi and Tatsuta's approach, \cite{BerTat17:lics}, is rather interesting since it is arguably more `structural' in nature, relying on proof-level manipulations rather than reflection principles.
That said there are still crucial sources of logical complexity blowup, namely in an `arithmetical' version of \emph{Ramsey's theorem} (Thm.~5.2) and the consequent \emph{Podelski-Rybalchenko termination theorem} (Thm.~6.1).
Both of these apparently increase quantifier complexity by several levels, and so their approach does not seem to yield comparable logical bounds to this work.
%
Since proof complexity is not a primary consideration of their work, it is not simple to track the precise bounds in \cite{BerTat17:lics}. There are some apparent sources of exponential blowups,\footnote{For instance, Lemma 8.4 in that work yields a set of apparently exponential size in the worst case, and this bounds from below the size of the overall translation, e.g.\ as in Lemma 8.7.}
%
though it seems that the global simulation is elementary.
As before, we reiterate that the major improvement in the present work is in the uniformity of our exposition: the approach of \cite{BerTat17:lics} is fundamentally non-uniform so does not yield any metamathematical account of cyclic arithmetic.
\subsection{On the correctness criteria for cyclic proofs}
Since the algorithms used to check correctness of a cyclic preproof reduce to the inclusion of B\"uchi automata, the exponential simulation of $\mathsf{CA}$ by $\mathsf{PA}$ is optimal, unless there is a nondeterministic subexponential-time algorithm for $\mathbf{PSPACE}$ or, more interestingly, there is an easier way to check cyclic proof correctness. (In fact, technically, it would suffice to have an easier criterion for a \emph{larger} class of preproofs that were, nonetheless, sound.)
%
As far as we know, $\mathbf{PSPACE}$ remains the best known upper bound for checking the correctness of general cyclic preproofs, although efficient algorithms have recently been proposed for less general correctness criteria, cf.~\cite{Stratulat17,Nol-Sau-Tas:18:local-validity}. Thus, it would be interesting to prove a corresponding lower bound or otherwise improve the upper bound.
Conditional such results could be obtained via, say, certain polynomial upper bounds on proof complexity in $\mathsf{CA}$: for instance, if $\mathsf{CA}$ were to have polynomial-size proofs of each correct B\"uchi inclusion then cyclic proof correctness would not be polynomial-time checkable, unless $\mathbf{NP} =\mathbf{PSPACE}$.
Unfortunately na\"ive attempts at this approach fail, but the general question of whether $\mathsf{PA}$ and $\mathsf{CA}$ are exponentially separated seems pertinent.
On the other hand, the translation of Lemma~\ref{lem:ind-to-cyc-trans} from inductive proofs to cyclic proofs is rather structured. In light of the converse result in Sect.~\ref{sect:ind-sim-cyc} it might make sense in further work, from the point of view of logical complexity, to consider only cyclic proofs accepted by some weaker more efficiently verified criterion, such as \cite{Stratulat17,Nol-Sau-Tas:18:local-validity}.
\subsection{Interpreting ordinary inductive definitions in arithmetic}
\label{sect:sect:interp-folid-arith}
%
%
%
%
%
%
In earlier work by Brotherston and Simpson, cyclic proofs were rather considered over a system of FO logic extended by `ordinary' \emph{Martin-L\"of} inductive definitions \cite{Pml71:haupt-int-iid}, known as $\mathrm{FOL}_{\mathit{ID}}$ \cite{Bro06:phdthesis,BroSim07:comp-seq-calc-ind-inf-desc,BroSim11:seq-calc-ind-inf-desc}.
Berardi and Tatsuta showed in \cite{BerTat17:lics} that the cyclic system $\mathsf{CLKID}^\omega$ for $\mathrm{FOL}_{\mathit{ID}}$ is equivalent to the inductive system $\mathsf{LKID}$, when at least arithmetic is present, somewhat generalising Simpson's result \cite{Sim17:cyclic-arith}.
%
We point out that
ordinary Martin-L\"of inductive definitions can be \emph{interpreted} in arithmetic
in the usual way by a $\Sin{}{1}$ inductive construction of `approximants', and a proof of $\mathsf{CLKID}^\omega$ may be similarly interpreted line-by-line in $\mathsf{CA}$.
{(This is similar to the role of the `stage number predicates' in \cite{BerTat17:lics}.)}
In particular, this means that $\mathsf{CLKID}^\omega (+ \mathsf{PA})$ is \emph{conservative} over $\mathsf{CA}$.
We reiterate that the interest behind the results of \cite{BerTat17:lics} is rather the structural nature of the transformations, but this observation also exemplifies why $\mathsf{CA}$ is a natural and canonical object of study, as argued in \cite{Sim17:cyclic-arith}.
%
\subsection{Cyclic propositional proof complexity}
One perspective gained from this work comes in the setting of \emph{propositional proof complexity} (see, e.g., \cite{Cook:2010:LFP:1734064,Krajicek:1996:BAP:225488}).
Thm.~\ref{thm:cyclic-sim-ind} of Sect.~\ref{sect:cyc-sim-ind} should relativise to theories with oracles too. For instance, we may formalise in $\CDn 0 (f)$, where $f$ is a fresh (uninterpreted) function symbol, a proof of the relativised version of the (finitary) pigeonhole principle (see App.~\ref{sect:php-case-study}).
This formula is known to be unprovable in $\IDn 0 (f)$ due to lower bounds on propositional proofs of bounded depth \cite{KraPudWood:95:ExpPHPbdF,PitBeaImp:93:ExpLBPHP}.
At the same time the `Paris-Wilkie' translation \cite{paris19810}, which fundamentally links $\IDn{0}(f)$ to bounded-depth propositional proofs, works locally on an arithmetic proof, at the level of formulae.
Consequently one may still apply the translation to the lines of a $\CDn{ 0}(f)$ proof to obtain small `proof-like' objects containing only formulae of bounded depth, and a cyclic proof structure.
One would expect that this corresponds to some strong form of `extension', since it is known that adding usual extension to bounded systems already yields full `extended Frege' proofs.
However at the same time, some of this power has been devolved to the proof structure rather than simply at the level of the formula, and so could yield insights into how to prove simulations between fragments of Hilbert-Frege systems with extension.
We point out that recent work, \cite{AtsLaur:18}, relating cyclic proof structures to proof complexity has already appeared, albeit with a different correctness criterion.
\section*{Acknowledgments}
\noindent
I am indebted to Alex Simpson for encouraging me to pursue this work and for his valuable feedback. Similarly, I would like to thank Stefano Berardi for several illuminating discussions on metalogical matters regarding cyclic proofs.
Finally, I would like to thank James Brotherston, Guilhem Jaber, Alexis Saurin and the anonymous reviewers for this and previous versions of this work for all their helpful comments and insights.
\bibliographystyle{alpha}
|
2,877,628,089,032 | arxiv | \section{Introduction}\label{intro}
Let consider the following problem governed by the parabolic variational inequality
\begin{equation}\label{eq1}
\langle \dot{u}(t) \,, \, v-u(t)\rangle + a(u(t) \, ,\, v-u(t)) + \Phi(v) - \Phi(u(t)) \geq <g(t) \,,\, v- u(t) > \quad
\forall v\in K,
\end{equation}
a.e. $t\in ]0 , T[$, with the initial condition
\begin{equation}\label{ic1}
u(0)= u_{b},
\end{equation}
where, $a$ is a symmetric continuous and coercive bilinear form on the Hilbert space $V\times V$, $\Phi$ is a proper and convex function from $V$ into $\mathbb{R}$ and is lower semi-continuous for the weak topology on $V$,
$< \cdot , \cdot>$ denotes the duality brackets between $V'$ and $V$,
$K$ is a closed convex non-empty subset of $V$, $u_{b}$ is an initial value in another Hilbert space $H$ with $V$ being densely and continuously imbedded in $H$, and $g$ is a given function in the space $L^{2}(0 , T , V')$.
It is well known \cite{Brezis68, Brezis72, Chipot2000, DL1972} that, there exists a unique solution
$$u\in {\cal C}(0 , T , H)\cap L^{2}(0 , T , V) \quad {\rm with }
\quad \dot{u}= \dfrac{\partial u}{\partial t}\in L^{2}(0 , T , H)
$$
to (\ref{eq1})-(\ref{ic1}). So we can consider $g \mapsto u_{g}$ as a function from
$L^{2}(0 , T , H)$ to ${\cal C}(0 , T , H)\cap L^{2}(0 , T , V)$. Then we can consider \cite{KeMu2008, JLL, NST2006}
the cost functional $J$ defined by
\begin{eqnarray}\label{e4.1}
J(g)= {1\over 2}\|u_{g}\|^{2}_{L^{2}(0 , T , H)} + {M\over 2}\|g\|^{2}_{L^{2}(0 , T , H)},
\end{eqnarray}
where $M$ is a positive constant, and $u_{g}$ is the unique solution to (\ref{eq1})-(\ref{ic1}), corresponding to the control $g$.
One of our main purposes is to prove the existence and uniqueness of the optimal control problem
\begin{equation}\label{P}
\mbox{Find } g_{op}\in L^{2}(0 , T, H) \quad\mbox{such that}
\quad J(g_{op})= \min_{g\in L^{2}(0 , T, H)} J(g).
\end{equation}
This can be reached if we prove the strictly convexity of the cost functional $J$, which follows
(see Theorem \ref{th4.2}) from
the following monotony property : {\it for any two control $g_{1}$ and $g_{2}$ in $L^{2}(0 , T , H)$, }
\begin{eqnarray}\label{mopr}
u_{4}(\mu)\leq u_{3}(\mu) \qquad \forall \mu\in [0 , 1],
\qquad\label{u3}
\end{eqnarray}
where
\begin{eqnarray}\label{u34}
u_{3}(\mu)= \mu u_{1}+ (1-\mu)u_{2}, \qquad u_{4}(\mu)= u_{g_{3}(\mu)}, \quad {\rm with} \quad g_{3}(\mu)= \mu g_{1} + (1-\mu)g_{2}.
\end{eqnarray}
In Section {\rm\ref{sec-3}}, we establish first in Theorem \ref{th1}, the error
estimate between $u_{3}(\mu)$ and $u_{4}(\mu)$.
This result generalizes our previous result obtained in \cite{MB-DT1} for the elliptic variational inequalities.
We deduce in Corollary \ref{cor22} a condition on the data to get
$u_{3}(\mu)= u_{4}(\mu)$ for all $\mu\in [0 , 1]$.
Then we assume, that the convex $K$ is a subset of $V=H^{1}(\Omega)$ and consider the parabolic variational problems ($P$) and ($P_{h}$).
So, using a regularization method, we prove in Theorem \ref{la} this monotony property {\rm(\ref{mopr})}, for the solutions of the two problems
($P$) and ($P_{h}$).
This result with a new proof and simplified, generalizes that obtained by \cite{Mingot1} for elliptic variational inequalities.
In Subsection \ref{sub-3.2} we also obtain some properties of dependency solutions based on the data
$g$ and on a positive parameter $h$ for the parabolic variational inequalities (\ref{eq1}) and (\ref{iv2}),
see Propositions \ref{l2.3}, \ref{l2.3h} and \ref{l2.3h2}.
In Section \ref{secOPC}, we consider the family of distributed optimal control problems $(P_{h})_{h>0}$,
\begin{equation}\label{Ph}
\mbox{Find } g_{op_{h}}\in L^{2}(0 , T , H) \quad \mbox{such that }
\quad J(g_{op_{h}})= \min_{g\in L^{2}(0 , T , H)} J_{h}(g),
\end{equation}
with the cost functional
\begin{equation}\label{Jh}
J_{h}(g)= {1\over 2}\|u_{g_{h}} \|_{L^{2}(0 , T , H)}^{2}+ {M\over 2}\|g\|_{L^{2}(0 , T , H)}^{2},
\end{equation}
where $u_{g_{h}}$ is the unique solution of (\ref{iv2})-(\ref{ic1}), corresponding to the control $g$ for each $h>0$,
and the distributed optimal control problems
\begin{equation}\label{P}
\mbox{Find } g_{op}\in L^{2}(0 , T , H) \quad \mbox{such that }
\quad J(g_{op})= \min_{g\in L^{2}(0 , T , H)} J(g),
\end{equation}
with the cost functional (\ref{e4.1}) where
$u_{g}$ is the unique solution to (\ref{eq1})-(\ref{ic1}), corresponding to the control $g$.
Using Theorem \ref{la} with its crucial property of monotony (\ref{u3}),
we prove the strict convexity of the cost functional (\ref{e4.1}) and also of the cost functional (\ref{Jh}),
associated to the problems (\ref{P}) and (\ref{Ph}) respectively.
Then, the existence and uniqueness of solutions to the optimal controls problems (\ref{P}) and (\ref{Ph}) follows from \cite{JLL}.
In general see for example \cite{CaJa1959}
the relevant physical condition, to impose on the boundary, is Newton's law, or Robin's law, and not Dirichlet's.
Therefore, the objective of this work is to approximate the optimal control problem (\ref{P}), where the state is the solution to parabolic
variational problem (\ref{eq1})-(\ref{ic1}) associated with the Dirichlet condition (\ref{pbc1}),
by a family indexed by a factor $h$ of optimal control problems (\ref{iv2})-(\ref{ic1}),
where states are the solutions to parabolic variational problems,
associated with the boundary condition of Newton (\ref{pbc3}).
Moreover, from a numerical analysis point of view
it maybe preferable to consider approximating Neumann problems in
all space $V$ (see (\ref{iv2})-(\ref{ic1})), with
parameter $h$, rather than the Dirichlet problem in a subset of
the space $V$ (see (\ref{eq1})-(\ref{ic1})).
So the asymptotic behavior can be considered very important in the optimal control.
In the last subsection \ref{secLim}, which is also the goal of our paper, we prove that the optimal control $g_{op_{h}}$
(unique solution of the optimization problem (\ref{Ph}))
and its corresponding state $u_{g_{op_{h}h}}$ (the unique solution of the parabolic variational problem
(\ref{iv2})-(\ref{ic1})) for each $h>1$, are strongly convergent to $g_{op}$ (the unique solution of the optimization
problem (\ref{P})), and $u_{g_{op}}$ (the unique solution of the parabolic variational problem (\ref{eq1})-(\ref{ic1}))
in $L^{2}([0 , T]\times\Omega)$ and $L^{2}(0 , T , H^{1}(\Omega))$ respectively when $h\to +\infty$.
This paper generalizes the results obtained in \cite{GT}, for elliptic variational equalities, and
in \cite{MT} for parabolic variational equalities, to the case of parabolic variational inequalities of second kind.
Various problems with distributed optimal control, associated with elliptic variational inequalities are given see for example
\cite{A2006, Barbu84a}, \cite{B1997a}-\cite{BM2000}, \cite{Ca2000, IK2000},
\cite{Mingot1}-\cite{NPS2006}, \cite{YC2004} and for the parabolic case see for example
\cite{Amassad2008, Barbu84a, Barbu84c}, \cite{BerMer2000}-\cite{Bonas1986}, \cite{NT88, NT},
\cite{pironneau1984}.
\section{On the property of monotony}\label{sec-3}
As we can not prove the property of monotony {\rm(\ref{mopr})} for any convex set $K$.
Let $\Omega$ a bounded open set in $\br^{N}$ with smooth boundary $\partial\Omega=\Gamma_{1}\cup\Gamma_{2}$.
We assume that $\Gamma_{1}\cap\Gamma_{2}=\O$, and $meas(\Gamma_{1})>0$. Let $H= L^{2}(\Omega)$, $V= H^{1}(\Omega)$.
We can prove the property of monotony {\rm(\ref{mopr})} for any convex subset of $V$.
Let
$$K=\{v\in V : \quad v_{|\Gamma_{1}}=0\}, \quad and \quad K_{b}=\{v\in V : \quad v_{|\Gamma_{1}}= b\}. $$
So we consider the following variational problems with such convex subset.
\noindent{\bf Problem ($P$)}
Let given $b\in L^{2}(]0 , T[\times\Gamma_{1})$, $g\in L^{2}(0 , T , H)$ and $q\in L^{2}(]0 , T[\times \Gamma_{2})$, $q>0$.
Find $u$ in ${\cal C}([0 , T] , H)\cap L^{2}(0 , T, K_{b})$ solution of
the parabolic problem (\ref{eq1}), where $< \cdot , \cdot >$
is only the scalar product $( \cdot , \cdot )$ in $H$, with the initial condition (\ref{ic1}),
and $\Phi(v)=\int_{\Gamma_{2}} q|v| ds.$
\smallskip
\noindent{\bf Problem ($P_{h}$)}
Let given $b\in L^{2}(]0 , T[\times\Gamma_{1})$, $g\in L^{2}(0 , T , H)$ and $q\in L^{2}(]0 , T[\times \Gamma_{2})$, $q>0$.
For all coefficient $h>0$, find $u\in {\cal C}(0 , T, H)\cap L^{2}(0 , T , V)$ solution of
the parabolic variational inequality
\begin{eqnarray}\label{iv2}
\langle \dot{u}(t) \, , \, v-u(t)\rangle + a_{h}(u(t) \, ,\, v-u(t))
+ \Phi(v)- \Phi(u(t))\geq
( g(t) , v-u(t)) \nonumber\\
+ h\int_{\Gamma_{1}}b(t)(v-u(t))ds\quad \forall v\in V,
\end{eqnarray}
and the initial condition {\rm(\ref{ic1})},
where
$
a_{h}(u , v)= a(u , v)+ h\int_{\Gamma_{1}}u v ds.
$
It is easy to see that the problem ($P$) is with the Dirichlet condition
\begin{eqnarray}
u= b \quad on \quad\Gamma_{1}\times]0 , T[, \label{pbc1}
\end{eqnarray}
and the problem ($P_{h}$) is with the following Newton-Robin's type condition
\begin{eqnarray}
-\dfrac{\partial u}{ \partial n}= h(u-b)\quad on \quad
\Gamma_{1}\times]0 , T[. \label{pbc3}
\end{eqnarray}
where $n$ is the exterior unit vector normal to the boundary. The integal on $\Gamma_{2}$ in the expression
of $\Phi$ comes from the Tresca boundary condition (see
\cite{mb-Ls1}-\cite{sf1},\cite{DL1972}) with $q$ is the Tresca friction coefficient on $\Gamma_{2}$.
Note that only for the proof of Theorem \ref{la} we have need to specify an expression of the functional $\Phi$.
By assumption there exists $\lambda >0$ such that
$
\lambda \|v\|_{V}^{2}\leq a( v \, , \, v)\quad \forall v\in V$.
Moreover, it follows from \cite{DT, Ta1979} that there exists $\lambda_{1}>0$ such that
$$ a_{h}(v , v)\geq \lambda_{h}\|v\|_{V}^{2} \quad \forall v\in V,
\quad \mbox{ with }\lambda_{h}= \lambda_{1}\min\{1 \,, \, h\}
$$
so $a_{h}$ is a bilinear, continuous, symmetric and coercive form on $V$. So there exists an unique solution to each of the two problems
($P$) and ($P_{h}$).
We recall that $u_{g}$ is the unique solution of the parabolic variational problem ($P$),
corresponding to the control $g\in L^{2}(0 , T , H)$, and also that $u_{g_{h}}$ is
the unique solution of the parabolic variational problem ($P_{h}$),
corresponding to the control $g\in L^{2}(0 , T , H)$.
\begin{proposition}\label{r2.3.2}
Assume that $g\geq 0$ in $\Omega\times ]0 , T[$, $b\geq 0$ on $\Gamma_{1}\times ]0 , T[$,
$u_{b}\geq 0$ in $\Omega$.
Then as $q>0$, we have $u_{g}\geq 0$. Assuming again that $h>0$, then $u_{g_{h}}\geq 0$ in $\Omega\times ]0 , T[$.
\end{proposition}
\begin{proof} For $u=u_{g_{h}}$, it is enough to take $v=u^{+}$ in {\rm(\ref{iv2})}, to get
\begin{eqnarray}\label{pmx}
\|u^{-}(T)\|^{2}_{L^{2}(\Omega)} + \lambda \int_{0}^{T} \|u^{-}(t)\|_{V}^{2} dt
+ h\int_{0}^{T}\int_{\Gamma_{1}}(u^{-}(t))^{2}ds dt
+ \leq - \int_{0}^{T}( g(t) , u^{-}(t)) dt \nonumber\\
-\int_{0}^{T}\int_{\Gamma_{2}}q (|u(t)|-|u^{+}(t)|)dsdt
-h\int_{0}^{T}\int_{\Gamma_{1}}b(t) u^{-}(t) ds dt + \|u^{-}(0)\|^{2}_{L^{2}(\Omega)} \quad
\end{eqnarray}
so the result follows.
\end{proof}
\begin{theorem}\label{th1}
Let $u_{1}$ and $u_{2}$ be two solutions of the parabolic variational inequality {\rm(\ref{eq1})}
with the same initial condition, and corresponding to the two control $g_{1}$ and $g_{2}$ respectively. We have the following estimate
\begin{eqnarray*}\label{eq3.1}
{1\over 2}\|u_{4}(\mu) -u_{3}(\mu)\|^{2}_{L^{\infty}(0 , T, H)}
+ \lambda \|u_{4}(\mu) -u_{3}(\mu)\|^{2}_{L^{2}(0 , T, V)}
+ \mu {\cal I}_{14}(\mu)(T) + (1-\mu) {\cal I}_{24}(\mu)(T)
\nonumber\\
+\mu\Phi(u_{1}) + (1-\mu)\Phi(u_{2})- \Phi(u_{3}(\mu))
\leq \mu(1-\mu)({\cal A}(T , g_{1}) + {\cal B}(T, g_{2})) \quad \forall \mu\in [0 , 1],
\end{eqnarray*}
where
$${\cal I}_{j4}(\mu)(T) = \int_{0}^{T} I_{j4}(\mu)(t) dt \quad{\rm for \,} j=1 , 2,
\quad
{\cal A}(T, g_{1}) = \int_{0}^{T}\alpha(t) dt, \quad {\cal B}(T, g_{2}) = \int_{0}^{T}\beta (t) dt,$$
\begin{eqnarray*}
I_{j4}(\mu)= \langle \dot{u}_{j} \,, \, u_{4}(\mu) -u_{j} \rangle + a(u_{j}\, , \, u_{4}(\mu) - u_{j}) + \Phi(u_{4}(\mu)) -\Phi(u_{j})
-\langle g_{j} , u_{4}(\mu) - u_{j}\rangle \geq 0,
\end{eqnarray*}
\begin{gather}\label{alp}
\alpha = \langle \dot{u}_{1} \,, \, u_{2} -u_{1} \rangle+ a(u_{1}\, , \, u_{2} - u_{1} )+ \Phi(u_{2}) -\Phi(u_{1})
-\langle g_{1} , u_{2} - u_{1}\rangle \geq 0, \\
\label{bet}
\beta = \langle \dot{u}_{2} \,, \, u_{1} -u_{2}\rangle + a(u_{2}\, , \, u_{1} - u_{2} )+ \Phi(u_{1}) -\Phi(u_{2})
-\langle g_{2} , u_{1} - u_{2}\rangle \geq 0.
\end{gather}
\end{theorem}
\begin{proof}
As $u_{3}(\mu)(t) \in K$ so with $v= u_{3}(\mu)(t)$, in the variational inequality (\ref{eq1}) where $u=u_{4}(\mu)$ and $g=g_{3}(\mu)$, we obtain
\begin{eqnarray*}
\langle \dot{u}_{4}(\mu) \,, \, u_{3}(\mu) -u_{4}(\mu) \rangle + a(u_{4}(\mu) \, , \, u_{3}(\mu)-u_{4}(\mu))
+ \Phi(u_{3}(\mu)) -\Phi(u_{4}(\mu))\nonumber\\
\geq\langle g_{3}(\mu) , u_{3}(\mu)- u_{4}(\mu)\rangle \quad a.e. \, t\in ]0 , T[,
\end{eqnarray*}
then
\begin{eqnarray*}
\langle \dot{u}_{4}(\mu) - \dot{u}_{3}(\mu) \,, \, u_{4}(\mu) -u_{3}(\mu) \rangle + a(u_{4}(\mu)- u_{3}(\mu) \, , \, u_{4}(\mu)-u_{3}(\mu)) \qquad
\nonumber\\
\leq \langle \dot{u}_{3}(\mu) \,, \, u_{3}(\mu) -u_{4}(\mu) \rangle +
a(u_{3}(\mu) \, , \, u_{3}(\mu)-u_{4}(\mu))\nonumber\\
+ \Phi(u_{3}(\mu)) -\Phi(u_{4}(\mu)(t))
-\langle g_{3}(\mu) , u_{3}(\mu)- u_{4}(\mu)\rangle \quad a.e. \, t\in ]0 , T[,
\end{eqnarray*}
thus
\begin{eqnarray*}
{1\over 2}{\partial \over\partial t}\left(\|u_{4}(\mu) -u_{3}(\mu)\|^{2}_{H}\right) + \lambda\|u_{4}(\mu)- u_{3}(\mu)\|_{V}^{2}
\leq
\langle \dot{u}_{3}(\mu) \,, \, u_{3}(\mu) -u_{4}(\mu) \rangle \nonumber\\
+ a(u_{3}(\mu) \, , \, u_{3}(\mu)-u_{4}(\mu))+ \Phi(u_{3}(\mu)) -\Phi(u_{4}(\mu))
\nonumber\\-
\langle g_{3}(\mu) \, ,\, u_{3}(\mu)- u_{4}(\mu)\rangle, \quad a.e. \, t\in ]0 , T[,
\end{eqnarray*}
using that $u_{3}(\mu)=\mu(u_{1} -u_{2})+ u_{2}$, $g_{3}(\mu)=\mu(g_{1} -g_{2})+ g_{2}$
we get
\begin{eqnarray*}
{1\over 2}{\partial \over\partial t}\left(\|u_{4}(\mu) -u_{3}(\mu)\|^{2}_{H}\right) + \lambda\|u_{4}(\mu)- u_{3}(\mu)\|_{V}^{2} +\mu\Phi(u_{1}) + (1-\mu)\Phi(u_{2})- \Phi(u_{3}(\mu)
\nonumber\\ \leq
\mu(1-\mu)(\alpha + \beta)- \mu I_{1 4}(\mu) - (1-\mu) I_{2 4}(\mu) \quad a.e. \, t\in ]0 , T[,
\end{eqnarray*}
so by integration between $t=0$ and $t=T$, we deduce the required result.
\end{proof}
\begin{corollary}\label{cor22} From Theorem \rm{\ref{th1}} we get $a.e. \, t\in [0 , T]$
\begin{eqnarray*}
{\cal A}(T,g_{1})={\cal B}(T , g_{2})= 0 \Rightarrow
\left\{
\begin{array}{ll}
&u_{3}(\mu) =u_{4}(\mu) \qquad \forall \mu\in [0 , 1],
\\
&I_{14}(\mu)= I_{24}(\mu)=0 \qquad \forall \mu\in [0 , 1],
\\
& \Phi(u_{3}(\mu))=\mu\Phi(u_{1}) + (1-\mu)\Phi(u_{2})\qquad \forall \mu\in [0 , 1].
\end{array}
\right.
\end{eqnarray*}
\end{corollary}
\begin{lemma}\label{l1}
Let $u_{1}$ and $u_{2}$ be two solutions of the parabolic variational inequality of second kind {\rm(\ref{eq1})}
with respectively as second member $g_{1}$ and $g_{2}$, then we get
\begin{equation}\label{3.4}
\|u_{1} -u_{2}\|^{2}_{L^{\infty}(0 , T, H)} +
\lambda\|u_{1} -u_{2}\|^{2}_{L^{2}(0 , T, V)}
\leq {1\over \lambda }\|g_{1}-g_{2}\|^{2}_{L^{2}(0 , T, V')},
\end{equation}
Where $\lambda$ is the coerciveness constant of the biliear form $a$.
\end{lemma}
\begin{proof}
Taking $v=u_{2}$ in (\ref{eq1}) where $u=u_{1}$ and
$g= g_{1}$; then $v=u_{1}$ in (\ref{eq1}) where $u=u_{2}$
and $g= g_{2}$, so by addition
(\ref{3.4}) holds.
\end{proof}
We generalize now in our case the result on a monotony property, obtained by \cite{Mingot1}
for the elliptic variational inequality. This theorem is the cornestone to prove the strict convexity
of the cost functional $J$ defined in Problem (\ref{P}) and the cost functional $J_{h}$ defined in Problem (\ref{Ph}).
Remark first that with the duality bracks $< \cdot , \cdot >$ defined by
$$ < g(t) , \varphi >= (g(t) , \varphi) + h\int_{\Gamma_{1}} b(t) \varphi ds$$
(\ref{iv2}) leads to ({\ref{eq1}).
We prove the following theorem for $\Phi$ such that $\Phi(v)=\int_{\Gamma_{2}} q|v| ds$.
\begin{theorem}\label{la}
For any two control $g_{1}$ and $g_{2}$ in $L^{2}(0, T, H)$, it holds that
\begin{equation}\label{eq3.4}
u_{4}(\mu) \leq u_{3}(\mu)\quad in \quad \Omega\times[0 , T],
\quad \forall \mu\in [0 , 1].
\end{equation}
Here $u_{4}(\mu)= u_{\mu g_{1}+(1-\mu) g_{2}}$, $u_{3}(\mu)= \mu u_{g_{1}}+(1-\mu) u_{g_{2}}$,
$u_{1}=u_{g_{1}}$ and $u_{2}=u_{g_{2}}$ are the unique solutions of the variational problem $P$,
with $g=g_{1}$ and $g=g_{2}$ respectively, and
for the same $q$, and the same initial condition {\rm(\ref{ic1})}.
Moreover, it holds also that
\begin{equation}\label{eq3.4h}
u_{h4}(\mu) \leq u_{h3}(\mu)\quad in \quad \Omega\times[0 , T],
\quad \forall \mu\in [0 , 1].
\end{equation}
Here $u_{4h}(\mu)= u_{\mu g_{1h}+(1-\mu) g_{2h}}$, $u_{3h}(\mu)= \mu u_{g_{1h}}+(1-\mu) u_{g_{2h}}$,
$u_{1h}=u_{g_{1h}}$ and $u_{h2}=u_{g_{h2}}$ are the unique solutions of the variational problem {\rm$P_{h}$},
with $g=g_{1}$ and $g=g_{2}$ respectively, and
for the same $q$, $h$, $b$ and the same initial condition \rm{(\ref{ic1})}.
\end{theorem}
\begin{proof}
The main difficulty, to prove this result comes from the fact that the functional $\Phi$ is not differentiable.
To overcome this difficulty, we use the regularization method and consider for $\varepsilon >0$ the following approach of $\Phi$
$$\Phi_{\varepsilon}(v)= \int_{\Gamma_{2}}q\sqrt{\varepsilon^{2} + |v|^{2}} ds, \qquad \forall v\in V,$$
which is Gateaux differentiable, with
$$\langle \Phi'_{\varepsilon}(w) \,, \, v\rangle = \int_{\Gamma_{2}}
{qw v \over \sqrt{\varepsilon^{2} +|w|^{2}}} ds \qquad \forall (w , v)\in V^{2}.
$$
Let $u^{\varepsilon}$ be the unique solution of the variational inequality
\begin{eqnarray}\label{ive}
\langle \dot{u}^{\varepsilon}\, , \, v- u^{\varepsilon} \rangle + a(u^{\varepsilon}\, , \, v- u^{\varepsilon})
+ \langle \Phi'_{\varepsilon}(u^{\varepsilon}) \,, \, v-u^{\varepsilon} \rangle
\geq \langle g \,, \, v-u^{\varepsilon} \rangle \quad a.e. \, t\in [0 , T]
\nonumber\\
\forall v\in K, \mbox{ and } u^{\varepsilon}(0)= u_{b}.
\end{eqnarray}
Let us show first that for all $\mu\in [0 , 1]$ $u^{\varepsilon}_{4}(\mu)\leq u^{\varepsilon}_{3}(\mu)$,
then that $u^{\varepsilon}_{3}(\mu) \to u_{3}(\mu)$ and $u^{\varepsilon}_{4}(\mu) \to u_{4}(\mu)$ strongly in $L^{2}(0 , T; H)$ when $\varepsilon\to 0$.
Indeed for all $\mu\in [0 , 1]$, let consider $U_{\varepsilon}(\mu)=u^{\varepsilon}_{4}(\mu) - u^{\varepsilon}_{3}(\mu)$ thus $u^{\varepsilon}_{4}(\mu)(t) -U_{\varepsilon}^{+}(\mu)(t)$ is in $K$. So we can take $v= u^{\varepsilon}_{4}(\mu)(t) -U_{\varepsilon}^{+}(\mu)(t)$ in (\ref{ive}) where $u^{\varepsilon}=u^{\varepsilon}_{4}(\mu)$ and $g=g_{3}(\mu)=\mu (g_{1}-g_{2})+g_{2}$.
We also can take $v=u^{\varepsilon}_{1}(t) +U_{\varepsilon}^{+}(\mu)(t)$ in (\ref{ive}) where $u^{\varepsilon}=u^{\varepsilon}_{1}$ and $g=g_{1}$, and we multiply the two sides of the obtained inequality by $\mu$
then we take $v=u^{\varepsilon}_{2} +U_{\varepsilon}^{+}(\mu)$ in (\ref{ive}) where $u^{\varepsilon}=u^{\varepsilon}_{2}$ and $g=g_{2}$ and we multiply the two sides of the obtained inequality by $(1-\mu)$.
By adding the three obtained inequalities we get $a.e. \, t\in ]0 , T[$,
\begin{eqnarray*}
{1\over 2}{\partial \over\partial t}(\|U_{\varepsilon}^{+}(\mu)\|_{H}^{2})
+ \lambda \|U_{\varepsilon}^{+}(\mu)\|_{V}^{2}\leq
\langle \mu \Phi'_{\varepsilon}(u^{\varepsilon}_{1})
+ (1-\mu) \Phi'_{\varepsilon}(u^{\varepsilon}_{2}) -\Phi'_{\varepsilon}(u^{\varepsilon}_{4}(\mu)) \,, \, U^{+}_{\varepsilon} (\mu)\rangle,
\end{eqnarray*}
hence as $U_{\varepsilon}^{+}(\mu)(0)=0$, by integration from $t=0$ to $t=T$ we obtain a.e. $t\in ]0 , T[$
\begin{eqnarray*}
{1\over 2}\|U_{\varepsilon}^{+}(\mu)(T)\|^{2}_{H}+ \lambda\int_{0}^{T}
\| U_{\varepsilon}^{+}(\mu)(t)\|^{2}_{V} dt \leq\qquad \qquad
\nonumber\\
\leq
\int_{0}^{T}\langle \mu \Phi'_{\varepsilon}(u^{\varepsilon}_{1}(t))
+ (1-\mu) \Phi'_{\varepsilon}(u^{\varepsilon}_{2}(t)) -\Phi'_{\varepsilon}(u^{\varepsilon}_{4}(\mu)(t)) \,, \, U^{+}_{\varepsilon}(\mu)(t) \rangle dt.
\end{eqnarray*}
As
\begin{eqnarray*}
&&< \mu \Phi'_{\varepsilon}(u^{\varepsilon}_{1})
+ (1-\mu) \Phi'_{\varepsilon}(u^{\varepsilon}_{2}) -\Phi'_{\varepsilon}(u^{\varepsilon}_{4}(\mu)) \,, \, U^{+}_{\varepsilon}(\mu) > =
\nonumber\\\nonumber\\ &=&
\int_{\Gamma_{2}'}{q\mu u^{\varepsilon}_{1} U^{+}_{\varepsilon}(\mu) \over \sqrt{\varepsilon^{2} + |u^{\varepsilon}_{1}|^{2}}} ds
+ \int_{\Gamma_{2}'}{q(1-\mu) u^{\varepsilon}_{2} U^{+}_{\varepsilon}(\mu) \over \sqrt{\varepsilon^{2} + |u^{\varepsilon}_{2}|^{2}}} ds
-\int_{\Gamma_{2}'}{qu^{\varepsilon}_{4}(\mu) U^{+}_{\varepsilon}(\mu) \over \sqrt{\varepsilon^{2} + |u^{\varepsilon}_{4}|^{2}}} ds
\end{eqnarray*}
where $\Gamma_{2}'= \Gamma_{2}\cap\{u^{\varepsilon}_{4}(\mu)> u^{\varepsilon}_{3}(\mu)\}$. The function $x\mapsto \psi(x)= \dfrac{x}{\sqrt{\varepsilon^{2} + x^{2}}}$ for $x\in \br$ is increasing ($\psi'(x) = \varepsilon^{2} (\varepsilon^{2} + x^{2})^{-3\over 2} >0$) so
\begin{eqnarray*}
&& \int_{\Gamma_{2}'}{q\mu u^{\varepsilon}_{1} U^{+}_{\varepsilon}(\mu) \over \sqrt{\varepsilon^{2} +\|u^{\varepsilon}_{1}\|_{\br^{N}}^{2}}} ds
+ \int_{\Gamma_{2}'}{q(1-\mu) u^{\varepsilon}_{2} U^{+}_{\varepsilon}(\mu) \over \sqrt{\varepsilon^{2} + |u^{\varepsilon}_{2}|^{2}}} ds
-\int_{\Gamma_{2}'}{qu^{\varepsilon}_{4}(\mu) U^{+}_{\varepsilon}(\mu) \over \sqrt{\varepsilon^{2} +
|u^{\varepsilon}_{4}|^{2}}}ds
\nonumber\\\nonumber\\
&\leq&
\int_{\Gamma_{2}'}{q\mu u^{\varepsilon}_{1} U^{+}_{\varepsilon}(\mu) \over \sqrt{\varepsilon^{2} +
|u^{\varepsilon}_{1}|^{2}}} ds
+ \int_{\Gamma_{2}'}{q(1-\mu) u^{\varepsilon}_{2} U^{+}_{\varepsilon}(\mu) \over \sqrt{\varepsilon^{2} +
|u^{\varepsilon}_{2}|^{2}}}ds
-\int_{\Gamma_{2}'}{qu^{\varepsilon}_{3}(\mu) U^{+}_{\varepsilon}(\mu) \over \sqrt{\varepsilon^{2} +
|u^{\varepsilon}_{3}|^{2}}} ds.
\end{eqnarray*}
Moreover the function $\psi$ is concave on $\br^{+}\setminus\{0\}$
($\psi''(x)= -3\varepsilon^{2} x (\varepsilon^{2} + x^{2})^{-5\over 2} <0$) thus
\begin{eqnarray}\label{eq3.12}
{1\over 2}\|U^{+}(\mu)(T)\|^{2}_{H}+ \lambda\int_{0}^{T}
\| U^{+}(\mu)(t)\|^{2}_{V} dt \leq 0.
\end{eqnarray}
As $U^{+}_{\varepsilon}(\mu)=0$ on $\{\Gamma_{2}\times[0 , T]\}\cap\{u^{\varepsilon}_{4}(\mu)\leq u^{\varepsilon}_{3}(\mu)\}$ so
\begin{eqnarray}\label{eq2.10}
u^{\varepsilon}_{4}(\mu)\leq u^{\varepsilon}_{3}(\mu) \quad \forall \mu\in [0 , 1].
\end{eqnarray}
Now we must prove that $u^{\varepsilon}_{3}(\mu) \to u_{3}(\mu)$ and $u^{\varepsilon}_{4}(\mu) \to u_{4}(\mu)$ strongly in $L^{2}(0 , T ; H)$ when $\varepsilon \to 0$.
Taking in (\ref{ive}) $v= u_{b}\in K$ with $u^{\varepsilon}= u^{\varepsilon}_{i}$ ($i=1 , 2$), we deduce that
\begin{eqnarray*}
\langle \dot{u}^{\varepsilon}_{i} \, , \, u^{\varepsilon}_{i}-u_{b}\rangle+ a( u^{\varepsilon}_{i}-u_{b} \, , \, u^{\varepsilon}_{i}-u_{b}) +
\langle \Phi'_{\varepsilon}(u^{\varepsilon}_{i}) \,, \, u^{\varepsilon}_{i}
\rangle
\leq
a( u_{b} \, , \, u_{b}-u^{\varepsilon}_{i})
\nonumber\\ +
\langle \Phi'_{\varepsilon}(u^{\varepsilon}_{i}) \,, \, u_{b}\rangle - \langle g_{i} \,, \, u_{b}-u^{\varepsilon}_{i} \rangle.
\end{eqnarray*}
As
$$
\langle \Phi'_{\varepsilon}(u^{\varepsilon}_{i}) \,, \, u^{\varepsilon}_{i} \rangle
\geq 0 \quad \mbox{ and }\quad
|\langle \Phi'_{\varepsilon}(u^{\varepsilon}_{i}) \,, \, u_{b}\rangle|
\leq \int_{\Gamma_{2}}q|u_{b}|ds $$
we deduce, using the Cauchy-Schwartz inequality, that
$\|u^{\varepsilon}_{i}\|_{L^{2}(0 , T; V)}$ so also $\|u^{\varepsilon}_{3}(\mu)\|_{L^{2}(0 , T; V)}$
are bounded independently from $\varepsilon$. By Theorem \ref{th1} we get
\begin{eqnarray*}
{1\over 2}\|u^{\varepsilon}_{3}(\mu)-u^{\varepsilon}_{4}(\mu)\|_{L^{\infty}(0 , T; H)} + \lambda\|u^{\varepsilon}_{3}(\mu)-u^{\varepsilon}_{4}(\mu)\|_{L^{2}(0 , T; V)}\leq \mu(1-\mu)({\cal A}^{\varepsilon} (T,g_{1}) + {\cal B}^{\varepsilon}(T,g_{2}))
\nonumber\\
\leq \mu(1-\mu){1\over 2}\left(\|g_{1}-g_{2}\|^{2}_{L^{2}(0 , T; H)} + \|u^{\varepsilon}_{1}-u^{\varepsilon}_{2}\|^{2}_{L^{2}(0 , T; H)}\right)
\quad \forall \mu\in [0 , 1],
\end{eqnarray*}
thus $\|u^{\varepsilon}_{4}(\mu)\|_{L^{2}(0 , T; V)}$ is also bounded independently from $\varepsilon$.
So there exists $l_{i}\in V$, for $i=1, \cdots, 4$, such that
\begin{eqnarray}\label{uib}
u^{\varepsilon}_{i}\tow l_{i} \mbox{ in } L^{2}(0 , T; V)\mbox{ weak},
\mbox{ and in } L^{\infty}(0 , T; H) \mbox{ weak star}.
\end{eqnarray}
We check now that $l_{i}= u_{i}$. Indeed for $i=1, 2$ or $4$ and as $\Phi$ is convex functional we have,
\begin{eqnarray*}\label{ive1}
\langle \dot{u}^{\varepsilon}_{i}\, , \, v -u^{\varepsilon}_{i} \rangle + a(u^{\varepsilon}_{i}\, , \, v -u^{\varepsilon}_{i} ) + \Phi_{\varepsilon}(v) - \Phi_{\varepsilon}(u^{\varepsilon}_{i}) \geq \qquad \qquad \qquad
\nonumber\\
\langle \dot{u}^{\varepsilon}_{i}\, , \, v -u^{\varepsilon}_{i} \rangle + a(u^{\varepsilon}_{i}\, , \, v -u^{\varepsilon}_{i} ) +
\langle \Phi'_{\varepsilon}(u^{\varepsilon}_{i}) \,, \, v -u^{\varepsilon}_{i} \rangle
\geq \langle g_{i} \,, \, v -u^{\varepsilon}_{i} \rangle, \quad a.e.\, t\in ]0 , T[
\end{eqnarray*}
thus
\begin{eqnarray}\label{i1}
\langle \dot{u}^{\varepsilon}_{i}\, , \, v -u^{\varepsilon}_{i} \rangle + a(u^{\varepsilon}_{i}\, , \, v -u^{\varepsilon}_{i} ) + \Phi_{\varepsilon}(v) - \Phi_{\varepsilon}(u^{\varepsilon}_{i})
\geq \langle g_{i} \,, \, v -u^{\varepsilon}_{i} \rangle, \quad a.e.\, t\in ]0 , T[.
\end{eqnarray}
Taking $v=u^{\varepsilon}_{i}\pm \varphi$, in (\ref{i1}) we have
\begin{eqnarray}\label{dib}
\langle \dot{u}^{\varepsilon}_{i}\, , \, \varphi \rangle= -
a(u^{\varepsilon}_{i} \, , \, \varphi) + \langle g_{i} \, , \, \varphi\rangle
, \quad \forall \varphi\in L^{2}(0 , T, H^{1}_{0}(\Omega)).
\end{eqnarray}
As $H^{1}_{0}(\Omega)\subset V$ with continuous inclusion but not dense, so $V'$ (the topological dual of the space $V$) is not identifiable with a subset of $H^{-1}(\Omega)$.
However, following \cite{MT} we can use the Hahn-Banach Theorem in order to extend any element in
$H^{-1}(\Omega)$ to an element of $V'$ preserving its norm.
So from (\ref{uib}) and (\ref{dib}) we conclude that
\begin{eqnarray}\label{convui}
\left.
\begin{array}{ll}
u^{\varepsilon}_{i}\tow l_{i} \mbox{ in } L^{2}(0 , T, V) \mbox{ weak},
\mbox{ in } L^{\infty}(0 , T, H)
\mbox{ weak star}, \\
\mbox{ and }
\dot{u}^{\varepsilon}_{i}\tow \dot{l}_{i} \mbox{ in } L^{2}(0 , T, V')
\mbox{ weak}.
\end{array}
\right\}
\end{eqnarray}
Then from (\ref{i1}), and following (\cite{DL1972, Ta1982}) we can write
\begin{eqnarray*}\label{}
\int_{0}^{T} \left\{
\langle \dot{u}^{\varepsilon}_{i}\, , \, v \rangle + a(u^{\varepsilon}_{i}\, , \, v )
+ \Phi_{\varepsilon}(v) -\langle g_{i} \,, \, v -u^{\varepsilon}_{i} \rangle\right\} dt
\geq
\int_{0}^{T} \left\{
\langle \dot{u}^{\varepsilon}_{i}\, , \, u^{\varepsilon}_{i} \rangle + a(u^{\varepsilon}_{i}\, , \, u^{\varepsilon}_{i} )
+ \Phi_{\varepsilon}(u^{\varepsilon}_{i}) \right\} dt
\nonumber\\
={1\over 2}\|u^{\varepsilon}_{i}(T)\|^{2}_{H}- {1\over 2}\|u_{b}(T)\|^{2}_{H} +
\int_{0}^{T} \left\{a(u^{\varepsilon}_{i}\, , \, u^{\varepsilon}_{i} ) + \Phi_{\varepsilon}(u^{\varepsilon}_{i}) \right\} dt.
\qquad
\end{eqnarray*}
Using the property of $\Phi_{\varepsilon}$ we have $\liminf_{\varepsilon\to 0} \Phi_{\varepsilon}(u^{\varepsilon}_{i})\geq \Phi(l_{i})$, and (\ref{convui}) we obtain
\begin{eqnarray}\label{int2}
\int_{0}^{T} \left\{
\langle \dot{l}_{i}\, , \, v \rangle + a(l_{i}\, , \, v ) + \Phi(v) -\langle g_{i} \,, \, v -l_{i} \rangle\right\} dt
\geq
\int_{0}^{T} \left\{
\langle \dot{l}_{i}\, , \, l_{i} \rangle + a(l_{i}\, , \, l_{i} ) + \Phi(l_{i}) \right\} dt. \quad
\end{eqnarray}
Let $w\in K$ and any $t_{0}\in ]0 , T[$ then we consider the open interval
${\cal O}_{j}= ]t_{0}-{1\over j} , t_{0}+{1\over j}[\subset ]0 , T[$
for $j\in \bn^{\star}$ sufficiently large we take in (\ref{int2})
$ v=
\left\{
\begin{array}{ll}
w \mbox{ if } t\in {\cal O}_{j},\\
l_{i}(t) \mbox{ if } t\in ]0 , T[\setminus{\cal O}_{j}
\end{array}
\right.
$
to get
\begin{eqnarray}\label{int1}
\int_{{\cal O}_{j}} \left\{
\langle \dot{l}_{i}\, , \, w- l_{i}\rangle + a(l_{i}\, , \, w- l_{i}) + \Phi(w) - \Phi(l_{i})\right\} dt \geq \int_{{\cal O}_{j}} \langle g_{i} \,, \, w- l_{i} \rangle dt.
\end{eqnarray}
We use now the Lebesgues Theorem to obtain, when $j\to +\infty$
\begin{eqnarray}\label{e1}
\langle \dot{l}_{i}\, , \, w- l_{i}\rangle + a(l_{i}\, , w- l_{i} )+ \Phi(w) - \Phi(l_{i})
\geq \langle g_{i} \,, \, w -l_{i} \rangle, \quad a.e. \, t\in ]0 , T[.
\end{eqnarray}
So by the uniqueness of the solution of the parabolic variational inequality of second kind (\ref{eq1}), we deduce that $l_{i}= u_{i}$.
To finish the proof we check the strong convergence of $u^{\varepsilon}_{i}$ to $u_{i}$.
Indeed for $i=1,2$ or $4$ taking $v=u_{i}(t)$ in (\ref{eq1}) where $u= u^{\varepsilon}_{i}$ then
$v= u^{\varepsilon}_{i}(t)$ in (\ref{eq1}) where $u=u_{i}$,
then by addition, and integration over the time interval $[0 , T]$ we obtain
\begin{eqnarray}\label{2.10}
{1\over 2} \|u_{i}(T) -u^{\varepsilon}_{i}(T)\|_{H}^{2}+ \int_{0}^{T} a(u_{i}(t)-u^{\varepsilon}_{i}(t)\, , u_{i}(t)- u^{\varepsilon}_{i}(t)) dt
\nonumber\\
\leq \int_{0}^{T}\Phi_{\varepsilon}(u_{i}(t))- \Phi(u_{i}(t))+
\Phi(u^{\varepsilon}_{i}(t)) -\Phi_{\varepsilon}(u^{\varepsilon}_{i}(t)) dt
\end{eqnarray}
as
$$\Phi_{\varepsilon}(v) - \Phi(v) = \int_{\Gamma_{2}} q(\sqrt{\varepsilon^{2} + |v|^{2}} - |v|) ds\leq \varepsilon \sqrt{|\Gamma_{2}|}\|q\|_{L^{2}(\Gamma_{2})},$$
so from (\ref{2.10})
\begin{eqnarray*}
{1\over 2} \|u_{i} -u^{\varepsilon}_{i}\|_{L^{\infty}(0 , T , H)}^{2}+ \int_{0}^{T} a(u_{i}(t)-u^{\varepsilon}_{i}(t)\, , u_{i}(t)- u^{\varepsilon}_{i}(t)) dt
\leq 2T\varepsilon \sqrt{|\Gamma_{2}|}\|q\|_{L^{2}(\Gamma_{2})}
\end{eqnarray*}
thus
\begin{eqnarray}\label{2.11}
u^{\varepsilon}_{i}\to u_{i} \mbox{ strongly in } L^{2}(0 , T; V)\cap L^{\infty}(0 , T; H) \mbox{ for } i= 1, 2, 4
\end{eqnarray}
then also
\begin{eqnarray}\label{2.12}
u^{\varepsilon}_{3}(\mu)= \mu u^{\varepsilon}_{1} + (1-\mu)u^{\varepsilon}_{2} \to u_{3} \mbox{ strongly in } L^{2}(0 , T; V)
\cap L^{\infty}(0 , T; H).
\end{eqnarray}
from (\ref{eq2.10}), (\ref{2.11}) and (\ref{2.12}) we get (\ref{eq3.4}).
As the proof is given for any two control $g=g_{1}$ and $g=g_{2}$ in $L^{2}(0 , T , H)$, but for
the same $q$, $h$, $b$ and the same initial condition \rm{(\ref{ic1})}, so we get also (\ref{eq3.4h}).
\end{proof}
\subsection{Dependency of the solutions on the data}\label{sub-3.2}
Note that this Subsection is not needed in the last Section.
We just would like to establish three propositions which allow us to deduce some additional and interesting properties on the solutions of
the variational problems $P$ and $P_{h}$.
\begin{proposition}\label{l2.3} Let $u_{g_{n}}$, $u_{g}$ be two solutions of Problem {\rm $P$}, with
$g=g_{n}$ and $g=g$ respectively.
Assume that $g_{n}\tow g\quad in\quad L^{2}(0 , T , H)$ (weak), we get
\begin{equation}\label{eq3.5}
u_{g_{n}}\to u_{g}\quad in \quad L^{2}(0 , T , V)\cap L^{\infty}(0 , T , H)\quad (strong)
\end{equation}
\begin{equation}\label{eq3.5x}
\dot{u}_{g_{n}}\to \dot{u}_{g}\quad in \quad L^{2}(0 , T , V')\quad (strong).
\end{equation}
Moreover
\begin{equation}\label{eq3.6}
g_{1}\geq g_{2} \quad in \quad \Omega\times[0 , T] \quad then \quad u_{g_{1}}\geq u_{g_{2}}\quad in \quad \Omega\times[0 , T].
\end{equation}
\begin{equation}\label{eq3.7}
u_{min(g_{1} , g_{2})} \leq u_{4}(\mu) \leq u_{max(g_{1} , g_{2})}, \qquad \forall \mu\in [0 , 1].
\end{equation}
Let $u_{g_{1}h}$, $u_{g_{2}h}$ be two solutions of Problem {\rm$P_{h}$}, with
$g=g_{1}$ and $g=g_{2}$ respectively for all $h>0$, we get
\begin{equation}\label{eq3.6new}
g_{1}\geq g_{2} \quad in \quad \Omega\times[0 , T] \quad then \quad u_{g_{1}h}\geq u_{g_{2}h}\quad in \quad \Omega\times[0 , T].
\end{equation}
\begin{equation}\label{eq3.7h}
u_{min(g_{1} , g_{2})h} \leq u_{h4}(\mu)
\leq
u_{max(g_{1} , g_{2})h} \qquad \forall \mu\in [0 , 1].
\end{equation}
\end{proposition}
\begin{proof}
Let $g_{n}\tow g$ in $L^{2}(0 , T , H)$, $u_{g_{n}}$ and $u_{g}$ be in $L^{2}(0 , T , K)$ such that
\begin{eqnarray}\label{gh}
\langle \dot{u}_{g_{n}} , v-u_{g_{n}}\rangle + a(u_{g_{n}} , v-u_{g_{n}})
+\Phi(v)- \Phi(u_{g_{n}})
\geq ( g_{n} , v-u_{g_{n}}) \nonumber\\ \qquad \forall v\in K , \quad a.e. \, t\in ]0 , T[.
\end{eqnarray}
Remark also that $V_{2}= \{ v \in V : \quad v_{|_{\Gamma_{2}}}= 0\}\subset V$ with continuous inclusion
but not dense, so $V'$ is not identifiable
with a subset of $V'_{2}$. However, following again \cite{MT} we can use the Hahn-Banach Theorem in
order to extend any element in $V'_{2}$ to an element of $V'$ preserving its norm.
So with the same arguments as in (\ref{i1})- (\ref{e1}), we conclude that there exists $\eta$
such that (eventually for a subsequence)
\begin{eqnarray}\label{eqW}
\left.\begin{array}{ll}
u_{g_{n}} \tow \eta \mbox{ in } L^{2}(0 , T, V) \mbox{ weak},
\mbox{ in } L^{\infty}(0 , T, H)
\mbox{ weak star, } \\
\mbox{ and }
\dot{u}_{g_{n}}\tow \dot{\eta} \mbox{ in } L^{2}(0 , T, V')\mbox{ weak}
\end{array}\right\}
\end{eqnarray}
Using (\ref{eqW}) and taking $n\to +\infty$ in (\ref{gh}), we get
\begin{equation}\label{qh}
\langle \dot{\eta} , v-\eta\rangle + a(\eta , v-\eta ) +\Phi(v)- \Phi(u_{\eta})\geq ( g , v-\eta), \qquad \forall v\in K, \quad a.e. \, t\in ]0 , T[,
\end{equation}
by the uniqueness of the solution of (\ref{eq1}) we obtain that $\eta= u_{g}$.
Taking now $v= u_{g}(t)$ in (\ref{gh}) and $v= u_{g_{n}}(t)$ in (\ref{qh}), we get by addition
and integration over $[0 , T]$ we obtain
\begin{equation*}
{1\over 2}\|u_{g_{n}}(T) - u_{g}(T)\|^{2}_{H} +\lambda\|u_{g_{n}} - u_{g}\|_{L^{2}(0, T , V)}^{2}\leq \int_{0}^{T}(g_{n}(t) -g(t)\, , \, u_{g_{n}}(t)- u_{g}(t))dt,
\end{equation*}
so from the above inequality and (\ref{eqW}) we deduce (\ref{eq3.5}).
To prove (\ref{eq3.6}) we take first $v=u_{1}(t)+(u_{1}(t)-u_{2}(t))^{-}$ (which is in $K$) in (\ref{eq1}) where $u=u_{1}$ and
$g= g_{1}$, then taking $v=u_{2}(t) - (u_{1}(t)-u_{2}(t))^{-}$ (which also is in $K$) in (\ref{eq1}) where $u=u_{2}$
and $g= g_{2}$, we get
$$
{1\over 2}\|(u_{1}(T) - u_{2}(T))^{-}\|^{2}_{H} + \lambda\|(u_{1} - u_{2})^{-}\|_{L^{2}(0, T , V)}^{2}
\leq \int_{0}^{T}(g_{2}(t)-g_{1}(t)\, ,\, (u_{1}(t)-u_{2}(t))^{-} )dt
$$
as
$$\Phi(u_{1}) - \Phi(u_{1} +(u_{1}-u_{2})^{-})+ \Phi(u_{2})- \Phi(u_{2} -(u_{1}-u_{2})^{-})=0.$$
So if $g_{2}-g_{1}\leq 0$ in $\Omega\times[0 , T]$ then
$\|(u_{1}-u_{2})^{-}\|_{L^{2}(0 , T, V)}=0$, and as
$(u_{1}-u_{2})^{-}=0$ on $\Gamma_{1}\times]0 , T[$ we have by the Poincar\'e inequality that $u_{1}-u_{2}\geq 0$ in $\Omega\times[0 , T]$. Then (\ref{eq3.7}) follows from (\ref{eq3.6}) because
$$ min\{g_{1} , g_{2}\} \leq \mu g_{1} +(1-\mu) g_{2} \leq max\{g_{1} , g_{2}\} \quad \forall \mu \in [0 , T].
$$
Similarly taking $v=u_{g_{1}h}(t)+(u_{g_{1}h}(t)-u_{g_{2}h}(t))^{-}$ (which is in $V$) in (\ref{iv2}) where $u=u_{g_{1}h}$ and
$g= g_{1}h$, then taking $v=u_{g_{2}h}(t) - (u_{g_{1}h}(t)-u_{g_{2}h}(t))^{-}$ (which also is in $V$) in (\ref{iv2}) where $u=u_{g_{2}h}$
and $g= g_{2}h$, we get
\begin{eqnarray*}
{1\over 2}\|(u_{g_{1}h}(T) - u_{g_{2}h}(T))^{-}\|^{2}_{H} + \lambda\|(u_{g_{1}h} - u_{g_{2}h})^{-}\|_{L^{2}(0, T , V)}^{2}
+ h\|(u_{g_{1}h} - u_{g_{2}h})^{-}\|_{L^{2}(0, T , L^{2}(\Gamma_{1}))}^{2}\nonumber\\
\leq \int_{0}^{T}(g_{2}(t)-g_{1}(t)\, ,\, (u_{1}(t)-u_{2}(t))^{-} )dt
\end{eqnarray*}
so we get also (\ref{eq3.6new}), then (\ref{eq3.7h}) follows.
\end{proof}
The following propositions \rm{\ref{l2.3h}} and \rm{\ref{l2.3h2}} are to give, with some assumptions, a first information that
the sequence $(u_{g_{h}})_{ h> 0}$ is increasing and
bounded, therefore it is convergent in some sense. Remark from (\ref{pmx}) that $u_{g_{h}} \geq 0$ although $g <0$,
provided to take the parameter $h$ sufficiently large.
\begin{proposition}\label{l2.3h} Assume that $h>0$ and is sufficiently large,
$b$ is a positive constant, $q\geq 0$ on $\Gamma_{2}\times[0 , T]$, then we have
\begin{equation}\label{eq3.71}
g\leq 0 \mbox{ in } \Omega\times[0 , T] \Longrightarrow 0\leq u_{g_{h}}\leq b
\mbox{ in } \Omega\cup\Gamma_{1}\times[0 , T],
\end{equation}
\end{proposition}
\begin{proof}
Taking in (\ref{iv2}) $u= u_{g_{h}}(t)$ and $v= u_{g_{h}}(t)- (u_{g_{h}}(t)-b)^{+}$, we get
\begin{eqnarray*}\label{}
\langle \dot{u}_{g_{h}}\, ,\, (u_{g_{h}}-b)^{+}\rangle
+ a_{h}(u_{g_{h}} \, ,\, (u_{g_{h}}-b)^{+})
- \Phi(u_{g_{h}}- (u_{g_{h}}-b)^{+})+ \Phi(u_{g_{h}})
\nonumber\\ \leq
(g \, ,\, (u_{g_{h}}-b)^{+}) + h\int_{\Gamma_{1}}b(u_{g_{h}}-b)^{+} ds,\quad a.e.\, t\in
]0 , T[
\end{eqnarray*}
as $b$ is constant we have
$a(b \, ,\, (u_{g_{h}}(t)-b)^{+}) =0$ so a.e. $t\in ]0 , T[$
\begin{eqnarray*}
{1\over 2}{\partial \over\partial t}\left(\|(u_{g_{h}}(t)-b)^{+}\|^{2}_{H}\right)
+ a((u_{g_{h}}-b)^{+} \, ,\, (u_{g_{h}}-b)^{+})
+ h\int_{\Gamma_{1}}u_{g_{h}}(u_{g_{h}}-b)^{+} ds\nonumber\\
\leq (g \, ,\, (u_{g_{h}}-b)^{+})
+h\int_{\Gamma_{1}}b(u_{g_{h}}-b)^{+} ds
+\Phi(u_{g_{h}}- (u_{g_{h}}-b)^{+})- \Phi(u_{g_{h}}),
\end{eqnarray*}
as $u_{g_{h}}(0)=b$ and
\begin{eqnarray*}
\Phi(u_{g_{h}}- (u_{g_{h}}-b)^{+})- \Phi(u_{g_{h}})
&=& \int_{\Gamma_{2}}q ( |u_{g_{h}}- (u_{g_{h}}-b)^{+}|- |u_{g_{h}}|) ds\leq 0,
\end{eqnarray*}
so
\begin{eqnarray*}
{1\over 2}\|(u_{g_{h}}(T)-b)^{+}\|^{2}_{H}
+ \int_{0}^{T}a_{h}((u_{g_{h}}(t)-b)^{+} \, ,\, (u_{g_{h}}(t)-b)^{+}) dt
\leq \nonumber\\
\leq\int_{0}^{T}(g(t) \, ,\, (u_{g_{h}}(t)-b)^{+})dt \leq 0,
\end{eqnarray*}
thus (\ref{eq3.71}) holds.
\end{proof}
\begin{proposition}\label{l2.3h2}
Assume that $h>0$ and is sufficiently large.
Let $g$, $g_{1}$, $g_{2}$ in $L^{2}(0 , T , H)$, $q\in L^{2}(0 , T , L^{2}(\Gamma_{2}))$ and $b$ is a positive constant,
we have
\begin{equation}\label{eq3.72}
g_{2}\leq g_{1}\leq 0 \mbox{ in } \Omega\times[0 , T] \quad and
\quad h_{2}\leq h_{1}
\Longrightarrow 0\leq u_{g_{2}h_{2}}\leq u_{g_{1} h_{1}} \mbox{ in } \Omega\times[0 , T],
\end{equation}
\begin{equation}\label{eq3.73}
g \leq 0 \mbox{ in } \Omega\times[0 , T]
\Longrightarrow 0\leq u_{g_{h}}\leq u_{g} \mbox{ in } \Omega\times[0 , T], \quad \forall h>0.
\end{equation}
\begin{equation}\label{eq3.74}
h_{2}\leq h_{1}
\Longrightarrow \|u_{g_{h_{2}}}- u_{g_{h_{1}}}\|_{L^{2}(0 , T , V)}\leq {\|\gamma_{0}||\over \lambda_{1}\min(1 , h_{2})}\|b-u_{g_{h_{1}}}\|_{L^{2}(0 , T , {\bf L}^{2}(\Gamma_{1}))} (h_{1}-h_{2})
\end{equation}
\end{proposition}
\begin{proof}
To check (\ref{eq3.72}) we take first $v=u_{g_{1}h_{1}}(t)+(u_{g_{2}h_{2}}(t)-u_{g_{1}h_{1}}(t))^{+}$,
for $t\in [0 , T]$, in (\ref{iv2}) where $u= u_{g_{1}h_{1}}$, $g=g_{1}h_{1}$ and $h=h_{1}$,
then taking
$v= u_{g_{2}h_{2}}(t)-(u_{g_{2}h_{2}}(t)-u_{g_{1}h_{1}}(t))^{+}$ in (\ref{iv2}) where $u= u_{g_{2}h_{2}}$, $g=g_{2}h_{2}$ and $h=h_{2}$,
adding the two obtained inequalities, as
\begin{eqnarray*}
\Phi(u_{g_{1}h_{1}}+(u_{g_{2}h_{2}}-u_{g_{1}h_{1}})^{+})
-\Phi(u_{g_{1}h_{1}})
+\Phi(u_{g_{2}h_{2}}-(u_{g_{2}h_{2}}-u_{g_{1}h_{1}})^{+}))-\Phi(u_{g_{2}h_{2}})= 0
\end{eqnarray*}
we get
\begin{eqnarray*}
-{1\over 2}{\partial \over\partial t} \left(\|(u_{g_{2}h_{2}}-u_{g_{1}h_{1}})^{+}\|^{2}_{H}\right)
-a(u_{g_{2}h_{2}} -u_{g_{1}h_{1}} \, ,\, (u_{g_{2}h_{2}}-u_{g_{1}h_{1}})^{+} ) \nonumber\\
+ \int_{\Gamma_{1}} (h_{1} u_{g_{1}h_{1}} - h_{2}u_{g_{2}h_{2}}) (u_{g_{2}h_{2}}-u_{g_{1}h_{1}})^{+} ds \geq
(g_{1} - g_{2} \, ,\, (u_{g_{2}h_{2}}-u_{g_{1}h_{1}})^{+} )
\nonumber\\
+
(h_{1}-h_{2}) \int_{\Gamma_{1}}b (u_{g_{2}h_{2}}-u_{g_{1}h_{1}})^{+} ds,
\quad a.e.\, t\in ]0 , T[,
\end{eqnarray*}
so by integration on $]0 , T[$, we deduce
\begin{eqnarray*}
{1\over 2}\|(u_{g_{2}h_{2}}(T)-u_{g_{1}h_{1}}(T))^{+}\|^{2}_{H}+
\int_{0}^{T} a_{h_{2}} ((u_{g_{2}h_{2}}-u_{g_{1}h_{1}})^{+} \, ,\,
(u_{g_{2}h_{2}}-u_{g_{1}h_{1}}(t))^{+} ) dt \leq
\nonumber\\
\int_{0}^{T}(g_{2} -g_{1}\, ,\, (u_{g_{2}h_{2}}(t)-u_{g_{1}h_{1}})^{+} )dt
+(h_{1}- h_{2}) \int_{0}^{T}\int_{\Gamma_{1}}(u_{g_{1}h_{1}}-b) (u_{g_{2}h_{2}}-u_{g_{1}h_{1}})^{+} dsdt,
\end{eqnarray*}
and from (\ref{eq3.71}) we get (\ref{eq3.72}).
To check (\ref{eq3.73}), let $W= u_{g_{h}}(t)- u_{g}(t)$, and choose, in (\ref{iv2}), $v= u_{g_{h}}(t) -W^{+}(t)$, so a.e. $t\in ]0 , T[$
\begin{eqnarray*}
\langle \dot{u}_{g_{h}} \, ,\, W^{+}\rangle + a_{h}(u_{g_{h}} \, ,\, W^{+}) \leq
+\Phi(u_{g_{h}} -W^{+})-\Phi(u_{g_{h}})+
(g \, ,\, W^{+}) + h\int_{\Gamma_{1}}bW^{+} ds,
\end{eqnarray*}
as $u_{g}= b$ on $\Gamma_{1}\times[0 , T]$ we obtain a.e. $t\in ]0 , T[$
\begin{eqnarray}\label{e5.11}
\langle \dot{u}_{g_{h}} \, ,\, W^{+}\rangle + a(u_{g_{h}} \, ,\, W^{+}) + h\int_{\Gamma_{1}} |W^{+}|^{2}ds
\leq ( g \, ,\, W^{+} ) +\Phi(u_{g_{h}} -W^{+})-\Phi(u_{g_{h}}).
\end{eqnarray}
Then we choose, in (\ref{eq1}), $v= u_{g}(t) +W^{+}(t)$, which is in $K$ because from (\ref{eq3.71}) we have $W^{+}= 0$ on $\Gamma_{1}\times[0 , T]$, so
\begin{eqnarray}\label{e5.12}
\langle \dot{u}_{g} , W^{+}(t) \rangle + a( u_{g} , W^{+}) \geq ( g \, ,\, W^{+} ) -\Phi(u_{g} +W^{+})+\Phi(u_{g}),
\quad a.e.\, t\in ]0 , T[.
\end{eqnarray}
So from (\ref{e5.11}) and (\ref{e5.12}) we deduce that
\begin{eqnarray*}
{1\over 2}\|W^{+}(T)\|^{2}_{H}+ \int_{0}^{T}a(W^{+} , W^{+})dt + h\int_{\Gamma_{1}} |W^{+}|^{2}ds\nonumber\\
\leq \Phi(u_{g_{h}} -W^{+})-\Phi(u_{g_{h}})+\Phi(u_{g} +W^{+})-\Phi(u_{g})= 0.
\end{eqnarray*}
Then (\ref{eq3.73}) holds.
To finish the proof we must check (\ref{eq3.74}).
We choose $v= u_{g_{h_{1}}}(t)$ in (\ref{iv2}) where $u=u_{g_{h_{2}}}(t)$, then
choosing $v= u_{g_{h_{2}}}(t)$ in (\ref{iv2}) where $u=u_{g_{h_{1}}}(t)$,
we get
\begin{eqnarray*}\label{e5.15}
-\langle \dot{u}_{g_{h_{2}}} - \dot{u}_{g_{h_{1}}}\, ,\, u_{g_{h_{2}}}-u_{g_{h_{1}}}\rangle
- a(u_{g_{h_{2}}} - u_{g_{h_{1}}}\, ,\, u_{g_{h_{2}}}-u_{g_{h_{1}}})
\nonumber\\
-h_{2} \int_{\Gamma_{1}} u_{g_{h_{2}}}(u_{g_{h_{2}}}-u_{g_{h_{1}}}) ds
+ h_{1}\int_{\Gamma_{1}}u_{g_{h_{1}}}(u_{g_{h_{2}}}-u_{g_{h_{1}}})ds
\geq \nonumber\\
-(h_{2} -h_{1})\int_{\Gamma_{1}}b(u_{g_{h_{2}}}-u_{g_{h_{1}}}) ds,
\quad a.e.\, t\in ]0 , T[,
\end{eqnarray*}
then
\begin{eqnarray*}\label{e5.15}
{1\over 2}\|u_{g_{h_{2}}}(T)-u_{g_{h_{1}}}(T)\|_{H}^{2} +
\int_{0}^{T}a_{h_{2}}(u_{g_{h_{2}}} - u_{g_{h_{1}}}\, ,\, u_{g_{h_{2}}}-u_{g_{h_{1}}}) dt
\nonumber\\ \leq
(h_{1} -h_{2})\int_{0}^{T}\int_{\Gamma_{1}}(u_{g_{h_{1}}}-b) (u_{g_{h_{2}}}-u_{g_{h_{1}}}) dsdt.
\end{eqnarray*}
So
\begin{eqnarray*}\label{e5.15}
{1\over 2}\|u_{g_{h_{2}}}-u_{g_{h_{1}}}\|_{L^{\infty}(0 , T , H)}^{2}
+\lambda_{1}\min\{1 , h_{2}\} \|u_{g_{h_{2}}} - u_{g_{h_{1}}}\|_{L^{2}(0 , T , V)}^{2}
\nonumber\\
\leq\|\gamma_{0}\| (h_{1} -h_{2})\|b- u_{g_{h_{1}}}\|_{L^{2}(0 , T , {{\bf L}^{2}(\Gamma_{1}))}}
\|u_{g_{h_{2}}}-u_{g_{h_{1}}}\|_{L^{2}(0 , T , V)}
\end{eqnarray*}
where $\gamma_{0}$ is the trace embedding from $V$ to $L^{2}(\Gamma_{1})$.
Thus (\ref{eq3.74}) holds.
\end{proof}
\section{Optimal Control problems and convergence for $h\to +\infty$}\label{secOPC}
In this section, $b$ is not constant but a given function in $L^{2}(]0 , T[\times\Gamma_{1})$. We prove first the existence and uniqueness of the solution for the optimal control problem associated to the parabolic variational inequalities of second kind (\rm{\ref{eq1}}), and for the optimal control problem associated also to
(\rm{\ref{iv2}}), then in Subsection \ref{secLim} we prove (see Lemma
\rm{\ref{l6.1}} and Theorem \rm{\ref{th6.1}}) the convergence of the state $u_{{g_{op}}_{h}h}$ and
the optimal control ${g_{op}}_{h}$,
when the coefficient $h$ on $\Gamma_{1}$, goes to infinity.
The existence and uniqueness of the solution to the parabolic variational inequalities of second kind
(\ref{eq1}) and (\ref{iv2}),
with the initial condition (\ref{ic1}), allow us to consider $g\mapsto u_{g}$ and
$g\mapsto u_{g_{h}}$ as functions from $L^{2}(0 , T, H)$ to $L^{2}(0 , T, V)$, for all $h > 0$.
Using the monotony property (\ref{eq3.4}) and (\ref{eq3.4h}), established in Theorem \ref{la}, we prove in the following that $J$ and $J_{h}$, defined by (\ref{e4.1}) and (\ref{Jh}), are strictly convex applications on $L^{2}(0 , T , H)$, so \cite{JLL} there exists a unique solution $g_{op}$ in $L^{2}(0 , T , H)$ of the Problem \rm{(\ref{P}), and there exists also a unique solution $g_{op_{h}}$ in $L^{2}(0 , T , H)$ of Problem \rm{(\ref{Ph})} for all $h>0$.
\begin{theorem}\label{th4.2}
Assume the same hypotheses of Proposition \ref{r2.3.2}. Then $J$ and $J_{h}$, defined by (\ref{e4.1}) and (\ref{Jh}) respectively, are strictly convex applications on $L^{2}(0 , T , H)$, so there exist unique solutions $g_{op}$ and
$g_{op_{h}}$ in $L^{2}(0 , T, H)$ respectively of the Problems {\rm(\ref{P})} and
{\rm(\ref{Ph})}.
\end{theorem}
\begin{proof}
Let $u=u_{g_{i}}$ and $u_{g_{i}h}$ be respectively the solution of
the variational inequalities {\rm(\ref{eq1})} and {\rm(\ref{iv2})}
with $g=g_{i}$ for $i=1 , 2$.
We have
\begin{eqnarray*}
\|u_{3}(\mu)\|_{L^{2}(0 , T, H)}^{2} = \mu^{2} \|u_{g_{1}}\|_{L^{2}(0 , T, H)}^{2} + (1-\mu)^{2}\|u_{g_{2}}\|_{L^{2}(0 , T, H)}^{2}
+ 2 \mu(1-\mu)(u_{g_{1}} , u_{g_{2}})
\end{eqnarray*}
then the following equalities hold
\begin{eqnarray}\label{4.9}
\|u_{3}(\mu)\|_{L^{2}(0 , T, H)}^{2} = \mu \|u_{g_{1}}\|_{L^{2}(0 , T, H)}^{2} +
(1-\mu)\|u_{g_{2}}\|_{L^{2}(0 , T, H)}^{2} \nonumber\\
-\mu(1-\mu)\|u_{g_{2}}-
u_{g_{1}}\|_{L^{2}(0 , T, H)}^{2},
\end{eqnarray}
\begin{eqnarray}\label{4.9h}
\|u_{3h}(\mu)\|_{L^{2}(0 , T, H)}^{2} = \mu \|u_{g_{1}h}\|_{L^{2}(0 , T, H)}^{2} + (1-\mu)\|u_{g_{2}h}\|_{L^{2}(0 , T, H)}^{2}
\nonumber\\
-\mu(1-\mu)\|u_{g_{2}h}- u_{g_{1}h}\|_{L^{2}(0 , T, H)}^{2}.
\end{eqnarray}
Let now $\mu\in [0 , 1]$ and $g_{1}, g_{2} \in L^{2}(0 , T, H)$ so
\begin{eqnarray*}
\mu J(g_{1})+ (1-\mu)J(g_{2})- J(g_{3}(\mu))= {\mu \over
2}\|u_{g_{1}}\|_{L^{2}(0 , T, H)}^{2} + {(1-\mu) \over 2}\|u_{g_{2}}\|_{L^{2}(0 , T, H)}^{2}
\nonumber\\
-{1 \over 2}\|u_{4}(\mu)\|_{L^{2}(0 , T, H)}^{2} +{M \over 2}\left\{\mu
\|g_{1}\|_{L^{2}(0 , T, H)}^{2} +(1-\mu)\|g_{2}\|_{L^{2}(0 , T, H)}^{2}
-\|g_{3}(\mu)\|_{L^{2}(0 , T, H)}^{2}\right\}
\end{eqnarray*}
using (\ref{4.9}) and $g_{3}(\mu)= \mu g_{1} + (1-\mu)g_{2}$ we obtain
\begin{eqnarray}\label{str}
\mu J(g_{1})+ (1-\mu)J(g_{2})- J(g_{3}(\mu))=
{1 \over 2}\left(\|u_{3}(\mu)\|_{L^{2}(0 , T, H)}^{2} -\|u_{4}(\mu)\|_{L^{2}(0 , T, H)}^{2}\right)
\nonumber\\
+{1 \over 2}\mu(1-\mu) \|u_{1}-u_{2}\|_{L^{2}(0 , T, H)}^{2}
+{M \over 2}\mu(1-\mu) \|g_{1}-g_{2}\|_{L^{2}(0 , T, H)}^{2},
\end{eqnarray}
for all $\mu\in ]0 , 1[$ and for all $g_{1}, g_{2}$ in $L^{2}(0 , T, H)$.
From Proposition \ref{r2.3.2} we have $u_{4}(\mu) \geq 0$ in $\Omega\times [0 , T]$ for all $\mu\in [0 , 1]$,
so using the monotony property (\ref{eq3.4}) (Theorem \ref{la})
and we deduce
\begin{eqnarray}\label{n1}
\|u_{4}(\mu)\|_{L^{2}(0 , T, H)}^{2}\leq \|u_{3}(\mu)\|_{L^{2}(0 , T, H)}^{2}.
\end{eqnarray}
Finally from {\rm(\ref{str})} the cost functional $J$ is strictly convex, thus \cite{JLL} the
uniqueness of the optimal control of the problem (\ref{P}) holds.
The uniqueness of the optimal control of the problem (\ref{Ph}) follows using the analogous
inequalities (\ref{str})-(\ref{n1}) for any $h>0$.
\end{proof}
\subsection{Convergence when $h\to +\infty$}\label{secLim}
In this last subsection we study the convergence of the state $u_{{g_{op}}_{h}h}$ and the optimal control ${g_{op}}_{h}$,
when the coefficient $h$ on $\Gamma_{1}$, goes to infinity. For a given $g$ in $L^{2}(0 , T , H)$ we have first the
following estimate which generalizes \cite{DT, Ta1979}.
\begin{lemma}\label{l6.1} Let $u_{g_{h}}$ be the unique solution of the parabolic variational inequality
{\rm(\ref{iv2})}
and $u_{g}$ the unique solution of the parabolic variational inequality {\rm(\ref{eq1})}, then
$$u_{g_{h}}\to u_{g}\in L^{2}(0 , T , V) \mbox{ strongly as } h\to +\infty, \qquad \forall g\in L^{2}(0 , T , H).$$
\end{lemma}
\begin{proof}
We take $v=u_{g}(t)$ in (\ref{iv2}) where $u= u_{g_{h}}$, and recalling that $u_{g}(t)= b$ on $\Gamma_{1}\times]0 , T[$,
taking $ u_{g_{h}}(t)- u_{g}(t)= \phi_{h}(t)$ we obtain for $h>1$,
a.e. $t\in ]0 , T[$
\begin{eqnarray*}
\langle \dot{\phi_{h}}, \phi_{h}\rangle +
a_{1}(\phi_{h} \, , \, \phi_{h}) + (h-1)\int_{\Gamma_{1}} |\phi_{h}|^{2}ds
\leq
-\langle \dot{u}_{g} , \phi_{h}\rangle
-a(u_{g}, \phi_{h})
+ ( g , \phi_{h})
+\Phi(\phi_{h}),
\end{eqnarray*}
so we deduce that
$${1\over 2}\| \phi_{h}\|^{2}_{L^{\infty}(0 , T, H)}
+ \|\phi_{h}\|^{2}_{L^{2}(0 , T , V)}+ (h-1)\|\phi_{h}\|^{2}_{L^{2}(0 , T, L^{2}(\Gamma_{1}))}$$
is bounded for all $h>1$,
then $\|u_{g_{h}}\|_{L^{2}(0 , T , V)}\leq\|\phi_{h}\|_{L^{2}(0 , T , V)}+ \|u_{g}\|_{L^{2}(0 , T , V)}$
is also bounded for all $h>1$. So there exists $\eta\in L^{2}(0 , T , V)$
such that
$u_{g_{h}}\tow \eta \mbox{ weakly in } L^{2}(0 , T , V)$ and
$u_{g_{h}}\to b$ strongly on $\Gamma_{1}$ when $h\to +\infty$ so $\eta(0)= b$.
Let $\varphi\in L^{2}(0 , T , V_{2})$ and taking in (\ref{iv2}) where
$u= u_{g_{h}}$,
$v= u_{g_{h}}(t) \pm \varphi(t)$, we obtain
\begin{eqnarray*}
\langle \dot{u}_{g_{h}} , \varphi\rangle = -a(u_{g_{h}} , \varphi)
+(g , \varphi) \quad a.e. \, t\in ]0 , T[.
\end{eqnarray*}
As $\|u_{g_{h}}\|_{L^{2}(0 , T , V)}$
is bounded for all $h>1$, we deduce that
$\|\dot{u}_{g_{h}}\|_{L^{2}(0 , T , V'_{2})}$ is also bounded for all $h>1$.
Following the proof of Lemma 2.3, we conclude that
\begin{eqnarray}\label{eqW2}
\left.
\begin{array}{ll}
u_{g_{h}} \tow \eta \mbox{ in } L^{2}(0 , T, V)\mbox{ weak, }
\mbox{ and in } L^{\infty}(0 , T, H)
\mbox{ weak star,} \\
\mbox{ and }
\dot{u}_{g_{n}}\tow \dot{\eta} \mbox{ in } L^{2}(0 , T, V') \mbox{ weak}.
\end{array}
\right\}
\end{eqnarray}
From (\ref{iv2}) and taking $v\in K$ so $v= b$ on $\Gamma_{1}$, we obtain
\begin{eqnarray*}
\langle \dot{u}_{g_{h}}, v- u_{g_{h}}\rangle +
a(u_{g_{h}} , v- u_{g_{h}}) - h\int_{\Gamma_{1}} |u_{g_{h}}- b|^{2}ds \geq
\nonumber\\ \Phi(u_{g_{h}}) - \Phi(v) + ( g , v - u_{g_{h}})
\qquad \forall v\in K, \quad a.e. \, t\in ]0 , T[,
\end{eqnarray*}
then
\begin{eqnarray}\label{eq6.1}
\langle \dot{u}_{g_{h}}, v- u_{g_{h}}\rangle +
a(u_{g_{h}} , v- u_{g_{h}}) \geq
\Phi(u_{g_{h}}) - \Phi(v) + ( g , v - u_{g_{h}})
\quad \forall v\in K, \, a.e. \, t\in ]0 , T[. \quad
\end{eqnarray}
So with (\ref{eqW2}) and the same arguments as in (\ref{i1})- (\ref{e1}), we obtain
\begin{eqnarray*}
\langle \dot{\eta}, v- \eta\rangle +
a(\eta , v- \eta) +\Phi(v)- \Phi(\eta) \geq ( g , v - \eta)
\quad \forall v\in K , \quad a.e. \, t\in ]0 , T[.
\end{eqnarray*}
and $\eta(0)=b$.
Using the uniqueness of the solution of (\ref{eq1})-(\ref{ic1}) we get that $\eta= u_{g}$.
To prove the strong convergence, we take $v= u_{g}(t)$ in (\ref{iv2})
\begin{eqnarray*}
\langle \dot{u}_{g_{h}} , u_{g} - u_{g_{h}} \rangle +
a_{h}(u_{g_{h}} , u_{g} - u_{g_{h}} ) +\Phi(u_{g})- \Phi(u_{g_{h}}) \geq
( g , u_{g} - u_{g_{h}} )
\nonumber\\
+ h\int_{\Gamma_{1}} b (u_{g} - u_{g_{h}} )ds, \quad a.e. \, t\in ]0 , T[
\end{eqnarray*}
thus as $u_{g}= b$ on $\Gamma_{1}\times]0 , T[$,
we put $u_{g_{h}} - u_{g}= \phi_{h}$, so a.e. $t\in ]0 , T[$
\begin{eqnarray*}
\langle \dot{\phi_{h}} \, ,\, \phi_{h}\rangle +
a(\phi_{h} , \phi_{h})
+ h \int_{\Gamma_{1}} |\phi_{h}|^{2}ds + \Phi(u_{g_{h}})-\Phi(u_{g})\leq
\langle \dot{u}_{g} , \phi_{h}\rangle
+
a( u_{g} \, ,\, \phi_{h}) + ( g , \phi_{h}),
\end{eqnarray*}
so
\begin{eqnarray*}
&&{1\over 2}\|\phi_{h}\|_{L^{\infty}(0 , T, H)}^{2} +
\lambda_{h}\|\phi_{h}\|_{L^{2}(0 , T , V)}^{2}
+ \Phi(u_{g_{h}})-\Phi(u_{g})\leq
-\int_{0}^{T}\langle \dot{u}_{g}(t), \phi_{h}(t)\rangle dt
\nonumber\\
&&-\int_{0}^{T} a(u_{g}(t), \phi_{h}(t)dt + \int_{0}^{T}( g(t) , \phi_{h}(t)dt,
\end{eqnarray*}
using the weak semi-continuity of $\Phi$ and the weak convergence (\ref{eqW}) the right side
of the just above inequality tends to zero when $h\to +\infty$, then we deduce the strong convergence
of $\phi_{h}=u_{g_{h}}-u_{g}$ to $0$ in $L^{2}(0 , T , V)\cap L^{\infty}(0 , T, H) $, for all
$g\in L^{2}(0 , T , H)$.
This ends the proof.
\end{proof}
We give now, without need to use the notion of adjoint states \cite{JLL}, the convergence
result which generalizes the result obtained
in \cite{MT} for a parabolic variational equations (see also
\cite{arada2000, belgacem2003, GT, GT2008}).
\begin{theorem}\label{th6.1}
Let $u_{{g_{op}}_{h}h}$, ${g_{op}}_{h}$ and $u_{g_{op}}$, $g_{op}$ be respectively
the states and the optimal control defined in the problems {\rm(\ref{P})}
and {\rm(\ref{Ph})}.
Then
\begin{eqnarray}\label{6.1}
\lim_{h\to +\infty}\|u_{g_{op_{h}h}}-u_{g_{op}}\|_{L^{2}(0 , T , V)}&=&
\lim_{h\to +\infty}\|u_{g_{op_{h}h}}-u_{g_{op}}\|_{L^{\infty}(0 , T , H)} ,
\nonumber\\
&=& \lim_{h\to +\infty}\|u_{g_{op_{h}h}}-u_{g_{op}}\|_{L^{2}(0 , T , L^{2}(\Gamma_{1}))}= 0,
\end{eqnarray}
\begin{equation}\label{6.2}
\lim_{h\to +\infty}\|g_{op_{h}}-g_{op}\|_{L^{2}(0 , T , H)}= 0.
\end{equation}
\end{theorem}
\begin{proof}
We have first
\begin{eqnarray*}
J_{h}(g_{op_{h}})= {1\over 2}\|u_{g_{op_{h}h}}\|_{L^{2}(0 , T , H)}^{2} + {M\over 2}\|g_{op_{h}}\|_{L^{2}(0 , T , H)}^{2}
\leq {1\over 2}\|u_{g_{h}}\|_{L^{2}(0 , T , H)}^{2} + {M\over 2}\|g\|_{L^{2}(0 , T , H)}^{2},
\end{eqnarray*}
for all $g\in L^{2}(0 , T , H)$,
then for $g=0\in L^{2}(0 , T , H)$ we obtain that
\begin{eqnarray}\label{e6.3}
J_{h}(g_{op_{h}})= {1\over 2}\|u_{g_{op_{h}h}}\|_{L^{2}(0 , T , H)}^{2} + {M\over 2}\|g_{op_{h}}\|_{L^{2}(0 , T , H)}^{2}\leq {1\over 2}\|u_{0_{h}}\|_{L^{2}(0 , T , H)}^{2}
\end{eqnarray}
where $u_{0_{h}}\in L^{2}(0 , T , V)$ is the solution of the following parabolic variational inequality
$$
\langle \dot{u}_{0_{h}} , v- u_{0_{h}}\rangle +
a_{h}( u_{0_{h}} , v- u_{0_{h}}) +\Phi(v)- \Phi(u_{0_{h}})
\geq h\int_{\Gamma_{1}} b(v- u_{0_{h}})ds, \quad a.e. \, t\in ]0 , T[
$$
for all $v\in V$ and $u_{0_{h}}(0)= u_{b}$. Taking $v= u_{b}\in K$ we get that
$\|u_{0_{h}}- u_{b}\|_{L^{2}(0 , T , V)}$ is bounded independently of $h$, then
$\|u_{0_{h}}\|_{L^{2}(0 , T , H)}$ is bounded independently of $h$. So we deduce with (\ref{e6.3}) that
$\|u_{g_{op_{h}h}}\|_{L^{2}(0 , T , H)}$ and $\|g_{op_{h}}\|_{L^{2}(0 , T , H)}$ are also bounded independently
of $h$. So there exists $f$ and $\eta$ in $L^{2}(0 , T , H)$ such that
\begin{eqnarray}\label{6.5}
g_{op_{h}} \tow f \quad in \quad L^{2}(0 , T , H) \quad (weak) \quad {\rm and }\quad u_{g_{op_{h}h}}\tow\eta
\quad in \quad L^{2}(0 , T , H) \quad (weak).
\end{eqnarray}
Taking now $v=u_{g_{op}}(t)\in K$ in (\ref{iv2}), for $t\in ]0 , T[$, with $u= u_{g_{op_{h}}h}$
and $g=g_{op_{h}}$, we obtain
\begin{eqnarray*}
\langle \dot{u}_{g_{op_{h}}h} , u_{g_{op}} - u_{g_{op_{h}h}}\rangle +
a_{1}( u_{g_{op_{h}}h} , u_{g_{op}} - u_{g_{op_{h}h}})
\nonumber\\
+(h-1) \int_{\Gamma_{1}}u_{g_{op_{h}h}} (u_{g_{op}} - u_{g_{op_{h}h}} )ds
+ \Phi(u_{g_{op}})- \Phi(u{g_{op_{h}h}})
\geq
\nonumber\\
( g_{op_{h}} , u_{g_{op}} - u_{g_{op_{h}}h} )
+
h\int_{\Gamma_{1}} b (u_{g_{op}} - u_{g_{op_{h}}h} )ds, \quad a.e. \, t\in ]0 , T[
\end{eqnarray*}
as $ u_{g_{op}}= b$ on $\Gamma_{1}\times[0 , T]$, taking
$u_{g_{op}} - u_{g_{op_{h}}h}=\phi_{h}$
we obtain
\begin{eqnarray*}
\langle \dot{\phi_{h}} , \phi_{h}\rangle
+a_{1}(\phi_{h} , \phi_{h})
+(h-1) \int_{\Gamma_{1}} |\phi_{h}|^{2}ds
\leq -(g_{op_{h}} , \phi_{h} ) \nonumber\\
+ \int_{\Gamma_{2}} q |\phi_{h}| ds
+\langle \dot{u}_{g_{op}} , \phi_{h}\rangle +
a( u_{g_{op}} ,\phi_{h}), \quad a.e. \, t\in ]0 , T[
\end{eqnarray*}
then
\begin{eqnarray*}
{1\over 2}\|\phi_{h}\|_{L^{\infty}(0 , T, H)}^{2}
+ \lambda_{1}\|\phi_{h}\|_{L^{2}(0 , T , V)}^{2}
+(h-1) \int_{0}^{T}\int_{\Gamma_{1}}|\phi_{h}(t)|^{2}ds dt
\nonumber\\ \leq
-\int_{0}^{T}(g_{op_{h}}(t) , \phi_{h}(t)) dt
+ \int_{0}^{T}\int_{\Gamma_{2}} q |\phi_{h}(t)|dsdt
+\int_{0}^{T}\langle \dot{u}_{g_{op}}(t) , \phi_{h}(t)\rangle dt
\nonumber\\+
\int_{0}^{T}a( u_{g_{op_{h}}h}(t) , \phi_{h}(t))dt.
\end{eqnarray*}
There exists a constant $C>$ which does not depend on $h$ such that
\begin{eqnarray*}\label{6.q}
\|\phi_{h}\|_{L^{2}(0 , T , V)}= \|u_{g_{op_{h}}h} -u_{g_{op}}\|_{L^{2}(0 , T , V)}\leq C,
\quad \|\phi_{h}\|_{L^{\infty}(0 , T, H)}\leq C
\nonumber\\
\mbox{ and }
(h-1)\int_{0}^{T}\int_{\Gamma_{1}} |u_{g_{op_{h}}h}- b|^{2}ds dt \leq C,
\end{eqnarray*}
then $\eta \in L^{2}(0 , T , V)$ and
\begin{eqnarray}\label{6.6}
u_{g_{op_{h}h}} \tow \eta \quad in \quad L^{2}(0 , T , V) \quad weak \mbox{ and in } L^{\infty}(0 , T , H)
\mbox{ weak star }
\end{eqnarray}
\begin{eqnarray}\label{6.7}
u_{g_{op_{h}h}} \to b\quad in \quad L^{2}(0 , T , L^{2}(\Gamma_{1})) \quad strong,
\end{eqnarray}
so $\eta(t)\in K$ for all $t\in [0 , T]$.
Now taking $v\in K$ in (\ref{iv2}) where $u= u_{g_{op_{h}}h}$ and $g=g_{op_{h}}$ so
\begin{eqnarray*}\label{}
\langle \dot{u}_{g_{op_{h}h}} , v- u_{g_{op_{h}h}} \rangle +
a_{h}(u_{g_{op_{h}h}} , v- u_{g_{op_{h}h}} )
+\Phi(v)- \Phi(u_{g_{op_{h}h}})
\geq ( g_{op_{h}} , v- u_{g_{op_{h}}h} )
\nonumber\\
+ h\int_{\Gamma_{1}} b(v- u_{g_{op_{h}}h} )ds, \quad a.e. \, t\in ]0 , T[
\end{eqnarray*}
as $v\in K$ so $v=b$ on $\Gamma_{1}$, thus we have
\begin{eqnarray*}\label{}
\langle \dot{u}_{g_{op_{h}h}} , u_{g_{op_{h}h}} -v \rangle + a(u_{g_{op_{h}h}} , u_{g_{op_{h}h}} -v) +
h \int_{\Gamma_{1}}|u_{g_{op_{h}h}}-b|^{2}ds + \Phi(u_{g_{op_{h}h}})
-\Phi(v) \nonumber\\ \leq
\langle
-( g_{op_{h}} , v- u_{g_{op_{h}h}})
\quad a.e. \, t\in ]0 , T[.
\end{eqnarray*}
Thus
\begin{eqnarray*}\label{}
\langle \dot{u}_{g_{op_{h}h}} , u_{g_{op_{h}h}} -v \rangle + a(u_{g_{op_{h}h} }, u_{g_{op_{h}h}} -v)
+ \Phi(u_{g_{op_{h}h}}) -\Phi(v)
\leq
-( g_{op_{h}} , v- u_{g_{op_{h}h}}) \nonumber\\
\quad a.e. \, t\in ]0 , T[.
\end{eqnarray*}
Using (\ref{6.5}) and (\ref{6.6}) and the same arguments as in (\ref{i1})- (\ref{e1}), we deduce that
\begin{eqnarray*}\label{}
\langle \dot{\eta} , v - \eta \rangle + a(\eta , v- \eta)
+\Phi(v)- \Phi(\eta)
\geq (f , v -\eta), \quad \forall v\in K, \quad a.e. \, t\in]0 , T[,
\end{eqnarray*}
so also by the uniqueness of the solution of (\ref{eq1}) we obtain that
\begin{eqnarray}\label{xi}
u_{f}= \eta.
\end{eqnarray}
We prove that $f= g_{op}$. Indeed we have
\begin{eqnarray*}\label{}
J(f)&=&{1\over 2} \|\eta \|_{L^{2}(0 , T; H)}^{2} + {M\over 2} \|f\|_{L^{2}(0 , T; H)}^{2}
\nonumber\\
&\leq& \liminf_{h\to +\infty} \left\{{1\over 2} \|u_{g_{op_{h}}h}\|_{L^{2}(0 , T; H)}^{2} + {M\over 2} \|g_{op_{h}}\|_{L^{2}(0 , T; H)}^{2} \right\}
=\liminf_{h\to +\infty} J_{h}(g_{op_{h}})
\nonumber\\
&\leq& \liminf_{h\to +\infty} J_{h}(g)
=\liminf_{h\to +\infty}
\left\{{1\over 2} \|u_{g_{h}}\|_{L^{2}(0 , T; H)}^{2} + {M\over 2} \|g\|_{L^{2}(0 , T; H)}^{2} \right\}
\end{eqnarray*}
using now the strong convergence $u_{g_{h}}\to u_{g}$ as $h\to
+\infty,\; \forall \; g\in H$ (see Lemma \ref{l6.1}), we obtain that
\begin{eqnarray}\label{6.9}
J(f)\leq \liminf_{h\to +\infty} J_{h}(g_{op_{h}}) \leq
{1\over 2} \|u_{g}\|_{L^{2}(0 , T; H)}^{2} + {M\over 2} \|g\|_{L^{2}(0 , T; H)}^{2}= J(g),
\quad \forall g\in L^{2}(0 , T; H)
\end{eqnarray}
then by the uniqueness of the optimal control problem (\ref{P}) we get
\begin{eqnarray}\label{f}
f= g_{op}.
\end{eqnarray}
Now we prove the strong convergence of $u_{g_{op_{h}}h}$ to $\eta=u_{f}$ in
$L^{2}(0 , T , V)\cap L^{\infty}(0 , T , H)\cap L^{2}(0 , T , L^{2}(\Gamma_{1}))$,
indeed taking $v=\eta$ in (\ref{iv2}) where $u=u_{g_{op_{h}}h}$ and $g= g_{op_{h}}$,
as $\eta(t)\in K$ for $t\in [0 , T]$, so $\eta=b$ on $\Gamma_{1}$, we obtain
we get
\begin{eqnarray*}\label{}
\langle \dot{u}_{g_{op_{h}h}}- \dot{\eta} , u_{g_{op_{h}h}}-\eta \rangle +
a_{1}( u_{g_{op_{h}h}} -\eta , u_{g_{op_{h}h}}- \eta )
+(h-1) \int_{\Gamma_{1}}|u_{g_{op_{h}h}} - \eta|^{2}ds
\nonumber\\+ \Phi(u_{g_{op_{h}h}})- \Phi(\eta)
\leq (g_{op_{h}} , u_{g_{op_{h}h}}-\eta )
+ \langle \dot{\eta} , u_{g_{op_{h}h}}-\eta \rangle
+ a(\eta , u_{g_{op_{h}h}}- \eta )
\end{eqnarray*}
thus
\begin{eqnarray*}
{1\over 2} \|u_{g_{op_{h}h}} -\eta\|_{L^{\infty}(0 , T; H)}^{2} + \lambda_{1}\|u_{g_{op_{h}h}}
-\eta\|_{L^{2}(0 , T , V)}^{2}
\nonumber\\
+\int_{0}^{T} \{\Phi(u_{g_{op_{h}h}})- \Phi(\eta)\} dt
+(h-1) \|u_{g_{op_{h}h}} - \eta\|_{L^{2}(0 , T , L^{2}(\Gamma_{1}))} ^{2}
\nonumber\\
\leq \int_{0}^{T}(g_{op_{h}}(t) , u_{g_{op_{h}h}}(t)-\eta(t))dt
+ \int_{0}^{T}\langle \dot{\eta} , u_{g_{op_{h}h}}-\eta \rangle dt
\nonumber\\
+ \int_{0}^{T}a(\eta(t) , \eta(t) - u_{g_{op_{h}h}}(t))dt.
\end{eqnarray*}
Using (\ref{6.6}) and the weak semi-continuity of $\Phi$ we deduce that
\begin{eqnarray*}\label{}
\lim_{h\to +\infty}\|u_{g_{op_{h}h}} -\eta\|_{L^{\infty}(0 , T; H)}
&=&\lim_{h\to +\infty}\|u_{g_{op_{h}}h} -\eta\|_{L^{2}(0 , T , V)} \nonumber\\
&=& \|u_{g_{op_{h}h}} - \eta\|_{L^{2}(0 , T , L^{2}(\Gamma_{1}))} = 0,
\end{eqnarray*}
and with (\ref{xi}) and (\ref{f}) we deduce (\ref{6.1}).
As $f\in L^{2}(0 , T , H)$, then from (\ref{6.9}) with $g=f$ and (\ref{f}) we can write
\begin{eqnarray}\label{eq6.11}
J(f)&=& J(g_{op}) ={1\over 2}\|u_{g_{op}}\|_{L^{2}(0 , T , H)}^{2} + {M\over 2}\|g_{op}\|_{L^{2}(0 , T , H)}^{2}\nonumber\\
&\leq& \liminf_{h\to+\infty} J_{h}(g_{op_{h}})
= \liminf_{h\to+\infty} \left\{{1\over 2}\|u_{g_{op_{h}h}}\|_{L^{2}(0 , T , H)}^{2} + {M\over 2}\|g_{op_{h}}\|_{L^{2}(0 , T , H)}^{2}\right\}\nonumber\\
&\leq& \lim_{h\to+\infty} J_{h}(g_{op}) = J((g_{op})
\end{eqnarray}
and using the strong convergence (\ref{6.1}), we get
\begin{eqnarray}\label{eq6.13}
\lim_{h\to+\infty}\|g_{op_{h}}\|_{L^{2}(0 , T , H)}= \|g_{op}\|_{L^{2}(0 , T , H)}.
\end{eqnarray}
Finally as
\begin{eqnarray}\label{eq6.14}
\|g_{op_{h}}- g_{op}\|_{L^{2}(0 , T; H)}^{2} =
\|g_{op_{h}}\|_{L^{2}(0 , T; H)}^{2}+ \|g_{op}\|_{L^{2}(0 , T; H)}^{2}
-2(g_{op_{h}} , g_{op})
\end{eqnarray}
and by the first part of (\ref{6.5}) we have
$$\lim_{h\to +\infty}\left(g_{op_{h}} , g_{op}\right) = \|g_{op}\|_{L^{2}(0 , T , H)}^{2},$$
so from (\ref{eq6.13}) and (\ref{eq6.14}) we get \mbox{(\ref{6.2})}. This ends the proof.
\end{proof}
\noindent{\bf Acknowledgements:}
This work was realized while the
second author was a visitor at Saint Etienne University
(France) and he is grateful to this institution for its hospitality.
|
2,877,628,089,033 | arxiv | \subsection*{Acknowledgements}
We are thankful to Abhishek Dutta and Ashish Thandavan for their great support.
We thank Rajan and his team of annotators at Elancer for their precise work.
We are grateful for support from the AWS Machine Learning Research Awards (MLRA), EPSRC Centre for Doctoral Training in Autonomous Intelligent Machines \& Systems [EP/L015897/1], the Qualcomm Innovation Fellowship, a Royal Society Research
Professorship, and the EPSRC Programme Grant VisualAI EP/T028572/1.
C.~R.~is supported by Innovate UK (project 71653) on behalf of UK Research and Innovation (UKRI) and by the European Research Council (ERC) IDIU-638009.
{\small\bibliographystyle{plainnat}
\section{Introduction \label{sec:intro}}
The development of modern machine learning could not have happened without the availability of increasingly large and diverse research datasets.
Consider computer vision for example: algorithms were initially developed using small datasets collected in laboratory conditions and, as a consequence, almost no method worked in the real world.
This cycle was broken only once researchers adopted datasets such as PASCAL VOC, MS COCO and ImageNet, which are collections of images sampled from the Internet.
These collections are not only much larger than prior datasets, but they also provide a much better representation of the statistics of natural images because they arise from the real world.
Because of this, algorithms became vastly more robust and were able to better generalize to the real world.
Furthermore, these datasets have acquired fundamental scientific functions as well: they allow reproducible and quantitative comparison of algorithms and they enable researchers to efficiently build on each other's work.
\begin{figure}[t]
\centering
\includegraphics[width=0.99\textwidth]{figs/pass_0.png}
\caption{\textbf{{\color{ForestGreen}\textbf{PASS}}\xspace\!: Pictures without humAns for Self-Supervision.}
We propose a new dataset of $1.28$M Internet images with CC-BY license that do not contain humans or body parts at all.
The dataset is collected randomly without using search engines and has no labels, which makes it particularly suitable for self-supervised learning.
We show that this dataset can largely replace ImageNet for the purpose of model pretraining.
While this does \emph{not} make ImageNet obsolete, it does remove the need for using ImageNet for one of its most common applications.
Individual image attributions for this figure are given in the Appendix.}\label{fig:splash}
\end{figure}
However, for all their benefits, these datasets have technical, ethical and legal shortcomings.
One issue is \emph{copyright}.
While some datasets contain images explicitly licensed for reuse (e.g. Creative Commons), some do not.
Some national laws contain copyright exceptions or otherwise allow the usage of copyrighted works for research and/or for training machine learning models, but this is not a universal, nor always firmly established, fact.
Another issue is \emph{data protection}.
The vast majority of images are collected by humans for human consumption in human-populated areas.
As a natural consequence, a large fraction of Internet images contain people.
Because it is nearly impossible to obtain consent for all the people in these images, this data is collected without consent~\citep{birhane2021large,yang20towards}.
This is an ethical issue as well as a legal one, as personal data is protected by legislation such as the EU and UK General Data Protection Regulation (GDPR).
Data protection legislation may still allow the usage of such data for research purposes, but this generally requires the data to be minimal, i.e.~required for the specific research one is conducting.
While the copyright issue can sometimes be addressed by choosing liberally-licensed images, the data protection issue is much more challenging.
The key question here is whether computer vision can simply avoid having any images containing people.
When the goal is to extract information about people (e.g., pose recognition), the answer is obviously no.
However, datasets such as ImageNet are often used for model pretraining even if the final application is not about people at all.
In this case, it is legitimate to ask if pretraining could be done on data that does not contain people.
Note that blurring people as recently done in~\cite{yang2021study}, while helpful, is not enough to remove privacy concerns; for instance, based on GDPR such images are still personal data insofar as the blurred individual can be recognized due to e.g.~contextual cues, which can also be picked up by reverse image search engines~\citep{pyrrhic-blog}.
We look instead at whether people can be excluded \emph{completely}.
Rather than just filtering ImageNet, however, we take a different route.
Many datasets such as ImageNet, which were designed for supervised object classification, contain undesirable \emph{biases}.
For example, \citep{birhane2021large} highlights harmful depictions in the popular ImageNet 2012 split, and its collection and annotation processes have been criticized for leading to stereotyped and problematic depictions of categories~\citep{shankar2017classification,birhane2021large,steed2021biases,stock2018convnets}.
For ImageNet, we therefore identify as main sources of bias its collection by scraping with search engines, and its selection of 1000 labels.
On the other hand, we note that the current state-of-the-art model pretraining uses self-supervised learning (SSL) and thus does \emph{not} require labels at all.
Motivated by this, we thus consider forming a dataset \emph{without using labels}, significantly increasing diversity and removing the search engine selection bias.
Because we \emph{remove images with humans}, we further significantly reduce the risk of including contextual biases linked to the appearance of people.
Furthermore, due to its more random and unsupervised nature, this dataset also serves as a better benchmark for SSL to study scaling to natural images that are not curated to a pre-defined set of class labels, addressing a technical shortcoming of current evaluations.
Concretely, our first contribution is {\color{ForestGreen}\textbf{PASS}}\xspace\!, a large collection of images (1.28M) excluding humans (along with other identifiable information such as license plates, signatures, or handwriting and NSFW images).
We do so by starting from a large-scale (100 million random flickr images) dataset---YFCC100M~\citep{thomee2016yfcc100m}---meaning that the data is better randomized\footnote{Of course, this does not remove all biases, as we discuss in the limitations section.} and identify a `safer' subset within it.
We also focus on data made available under the most permissive Creative Common license (CC-BY) to address copyright concerns.
Given this data, we then conduct an extensive evaluation of SSL methods, discussing performance differences when these are trained using ImageNet and {\color{ForestGreen}\textbf{PASS}}\xspace\!\@. Compared to ImageNet, there are three essential differences with the {\color{ForestGreen}\textbf{PASS}}\xspace dataset:
the lack of class-level curation and search; the lack of `community optimization' on this dataset; and, of course, the lack of humans.
We study via further ablation the contribution of these effects.
We find that:
(i) self-supervised approaches such as MoCo, SwAV and DINO train well on our dataset, yielding strong image representations;
(ii) excluding images with humans during pretraining has almost no effect on downstream task performances, even if this is done in ImageNet;
(iii) in 8/13 frozen encoder evaluation benchmarks, performance of models trained on {\color{ForestGreen}\textbf{PASS}}\xspace yield better results than pretraining on ImageNet, ImageNet without humans, or Places205, when transferred to other datasets; and for finetuning evaluations, such as detection and segmentation, {\color{ForestGreen}\textbf{PASS}}\xspace pretraining yields results within $\pm1\%$ mAP and AP50 on COCO\@.
(iv) Even on tasks involving humans, such as dense pose prediction, pretraining on our dataset yields performance on par with ImageNet pretraining.
\section{Related Work}
\paragraph{Image datasets for model pretraining.}
By far the most widely used pretraining dataset is ImageNet ILSVRC12 (IN-1k)~\citep{deng09imagenet:}, containing 1.28M images covering 1000 object categories.
This is followed by MIT-Places 205 and 365 datasets~\citep{zhou14learning,zhou2017places}, containing 2M and upto 10M images with labels, respectively, Webvision~\citep{li2017webvision} containing the same labels as IN-1k, as well as Taskonomy~\citep{zamir18taskonomy:}, containing 4M images with scene attributes.
YFCC-100M~\citep{thomee2016yfcc100m} is a much larger dataset with 99M images with licence information and other metadata. OpenImages~\citep{kuznetsova2020open,openimages,benenson2019large} contains 6M images and 20K classes.
These datasets have been further combined and relabeled to yield further datasets, including Tencent ML-images~\citep{tencentmlimages} and Conceptual Captions~\citep{sharma2018conceptual}.
When measured on downstream task performance, supervised pretraining on ImageNet (IN-1k) has nowadays been surpassed by unsupervised pretraining, but often these methods use the same images for both pretraining and evaluation (with labels).
We argue that SSL methods should instead be trained on more diverse and less-curated datasets, so as to disentangle the performance gains from the potential requirements on the underlying data.
This is particularly true if ImageNet is \emph{also} used for evaluation.
\paragraph{Bias and privacy.}
Biases contained in datasets have been investigated in various works~\citep{hendricks2018women,buolamwini2018gender,zhao2017men,excavating,dulhanty2019auditing,yang20towards,steed2021biases,linesofsight,birhane2021large,paullada2020data} and in particular its annotation practices~\citep{ghai2020measuring,hube2019understanding,miltenburg2016stereotyping,misra2016seeing,miceli2020between,aroyo2015truth,tsipras2020from,recht2019imagenet,beyer2020we}.
We focus specifically on ImageNet as this is the defacto standard dataset used for pretraining representations\footnote{Although these issues are likely to be present in most other web images datasets too: following the methodology of \citep{birhane2021large}, we easily find several pornographic and troubling images even in Places205~\citep{zhou16places:}. Better filtering for such content is necessary.}.
Notably, \citep{excavating} shows that the larger version of ImageNet had stereotypes/slurs as class labels, and further was biased with regards to gender-biased depictions~\citep{dulhanty2019auditing}, which led to the removal of the person categories from the dataset~\citep{yang20towards}.
More recently, \citet{birhane2021large} finds biased and NSFW images in ImageNet LSVRC-12 and criticises its labelling process.
This paper and its subsequent blog-post~\citep{pyrrhic-blog} also outline problematic categories, such as the `bikini' class.
Another issue is that ImageNet, as well as other datasets, contain biases inherited from the search engines used to collect them, which sometimes results in racist, sexualized, skewed or stereotyped representations~\citep{linesofsight}.
These biases are passed down to models trained on this data, e.g.~as shown in the image completion work~\citep{steed2021biases}, or face upsampling~\citep{menon2020pulse}.
Furthermore, the labels included in ImageNet are also imperfect: when~\citep{recht2019imagenet} reproduced ImageNet's annotation process, they noted highly-variable results yielding up to 11\%--14\% drops in accuracy.
For a more in-depth review of datasets, their collection practises and current issues we refer to \citet{paullada2020data}.
In response to some of these issues,
a version of ImageNet with blurred faces has now been published~\citep{yang2021study}, in which the authors show that this only marginally affects object classification performance.
However, as~\citep{pyrrhic-blog} points out, this does not eliminate the problem, as even after blurring faces, reverse image search can be used to retrieve the original picture, thus leaving the privacy concerns unsolved.
In this work, we focus on alleviating some of these privacy and bias issues by constructing a dataset that is completely free of humans, does not rely on search engines or labels, and only contains images with a permissive license (CC-BY) and attribution information.
\paragraph{Self-supervised learning.}
Starting from early works such as~\citep{hinton93autoencoders,pathak16context}, self-supervised representation learning methods methods are nowadays mainly based on clustering~\citep{caron18deep,asano20self-labelling,gidaris2020learning,PCL,gidaris2020online} and/or noise-contrastive instance discrimination~\citep{hadsell06dimensionality,misra2020pirl,wu2018unsupervised,he19momentum,chen20a-simple,grill20bootstrap,dwibedi2021little}.
In particular, MoCo~\citep{he19momentum} has sparked several further implementations~\citep{kalantidis2020hard,xie2020propagate, chen2020exploring,PCL,ayush2020geography,shen2020mix,zhu2020eqco} and therefore we use this model for most of the experiments in this paper.
Several papers~\citep{Ericsson2021HowTransfer, zhao2021makes,kotar2021contrasting} also compare SSL methods in detail and find among other results that, downstream task performance is dependent on the pretraining dataset, and that self-supervised methods struggle on fine-grained classification tasks compared to their supervised counterparts.
We therefore evaluate various distributional splits of our dataset and evaluate also on finegrained datasets.
\section{PASS: images minus people \label{sec:pass}}
\input{figs/schematic}
\input{figs/loc_date}
The PASS dataset is obtained as a subset of the YFCC-100M dataset~\citep{thomee2016yfcc100m}.
The latter is freely available on the AWS public datasets page~\citep{awspublicdatasets}, and has complete metadata that includes the individual licenses and creators, thus satisfying our requirement of CC-BY\@.
In case this dataset ceases to exist on the public datasets of AWS, we have made necessary backups, for details see the `Datasheets of Datasets' section in the Appendix.
\paragraph{Collection modality.}
PASS was obtained as follows (see also~\cref{fig:schematic}).
First, the YFCC-100M metadata was used to select images with CC-BY licence, which left us with 17M images.
Once, these images were downloaded, corrupted or single-color images\footnote{For a surprisingly large fraction of the images all pixels have the same color.} were removed (leaving 10M).
Pretrained RetinaFace~\citep{deng2019retinaface} and Cascade-RCNN (3x)~\citep{cai2018cascade} models were used to filter out images containing human faces and humans (6M).
All details are provided in the Appendix.
In YFCC-100M, the distribution of images per photographer is highly skewed.
Thus, to increase dataset diversity, we set a maximum threshold of 80 images per photographer and uniformly sampled 1.4M images.
Finally, these images were submitted for human labelling\footnote{Prior to human verification we further checked the most highly ranked 1K images according to a NSFW classifier~\citep{nsfwmodel} for possible harmful content and found none.}.
\paragraph{Manual Filtering.}
The annotators were asked to identify images that contain people or body parts, as well as personal information such as IDs, names, license plates, signatures etc.
Additionally, the annotators were asked to flag images with problematic content such as drugs, nudity (mostly included in the ``no people'' rule), blood and other offensive content.
The annotations were performed over the course of three weeks by an annotation company whose annotators were paid 150\% of the minimum wage.
During the manual annotation process, around 2\% of the images were flagged and subsequently removed.
From the remaining images (1.46M) we further removed duplicates (see Appendix~\ref{appx:duplicates}) and randomly selected a subset with approximately the same size as IN-1k (1{.}440{.}191 images).
\paragraph{Statistics.}
We first provide some descriptive results of our dataset.
Using the meta-data provided, we plot the GPS location of the images in \cref{fig:geo}.
We find that the subset of the dataset that contains GPS coordinates covers a wide area of the world, but is focused on western countries such as the US, Europe and Japan.
This likely reflects the skew of the Flickr user database and is a form of bias that can limit generalisation capabilities~\citep{de2019does}, as for example images taken in less developed countries might be uploaded by tourists that take images of stereotypes~\citep{shankar2017classification,revisetool}.
With this we wish to stress that the {\color{ForestGreen}\textbf{PASS}}\xspace dataset is by no means bias-free, and its misunderstanding as such might lead to spurious claims of fairness.
While the dataset comes with some textual annotations such as tags and descriptions, these are discarded to avoid further biases~\citep{miltenburg2016stereotyping} since they are not necessary for self-supervised learning.
\input{figs/in21_places}
In \cref{fig:in21k-places} we show the results of running pretrained classification models on our dataset and compare the output diversity against IN-1k.
Note that in \cref{fig:in21k-dist}, the distribution for IN-1k objects should technically be zero for classes after 1000, as IN-1k is a subset of IN-21k.
We attribute this deviation to the quality of the model---classifiers with a very high number of classes (here: 21,000) are still very noisy.
Nonetheless, for a relative comparison of two datasets, even a noisy classifier is sufficient.
We find that, compared to IN-1k, our dataset contains more variety in terms of places and objects, as shown by the higher tail of the distributions.
\section{Experimental setup}
\input{tabs/benchmarks}
We evaluate {\color{ForestGreen}\textbf{PASS}}\xspace by pretraining models on the {\color{ForestGreen}\textbf{PASS}}\xspace data using self-supervision and then testing the quality of the resulting models on a range of downstream tasks outlined in \cref{tab:evaluations}.
\paragraph{SSL pretraining methods.}
We pretrain models using MoCo-v2~\citep{chen2020mocov2} as it has become a staple SSL method, can be extended in various ways~\citep{kalantidis2020hard, xie2020propagate, chen2020exploring,PCL,ayush2020geography,shen2020mix,zhu2020eqco} and, with updates to the learning schedule and most recently its implementation of the loss~\citep{chen2021empirical} remains state-of-the-art in feature learning.
This makes it an ideal candidate to probe the pretraining performance of our dataset.
We therefore conduct most of our experiments using the MoCo-v2 public implementation with a 200 epoch pretraining schedule.
We further train and evaluate models trained with SwAV~\citep{caron20unsupervised}, another state-of-the-art representation learning method, but one which is based on unsupervised clustering.
Finally, we also pretrain using DINO~\citep{caron2021emerging}, a recent method that is tailored to vision transformer~\citep{dosovitskiy2020image} SSL\@.
For downstream evaluation, we set all training parameters as proposed in existing literature (which, in practice, means they are tuned for IN-1k).
Specific tuning for each pretraining dataset would probably improve performance.
For pretraining MoCo-v2 and MoCo-v2 CLD, we follow the official implementation and adhere to the 200 epoch schedule.
We use the 200 epoch configuration with multi-crop schedule for SwAV which takes around double the GPU hours as MoCo's 200 epochs.
For DINO, we follow their 100 epoch training code, which also includes multi-crop.
Further experimental details are given in the Appendix.
\paragraph{Pretraining Data.}
Using MoCo-v2, we compare {\color{ForestGreen}\textbf{PASS}}\xspace against other datasets for pretraining, such as:
\textbf{(a)} IN-1k: ImageNet-2012 split~\citep{deng09imagenet:}, the current standard for SSL pretraining;
\textbf{(b)} IN-1k$^*$: ImageNet-2012 with humans and human parts removed, as an ablation of IN-1k;
\textbf{(c)} Places-205~\citep{zhou16places:}: to compare the effect of the data distribution on pretraining, as in~\citep{zhao2021makes, kotar2021contrasting}.
We construct (b) by first removing all images with faces~\citep{yang2021study} and subsequently running our automated human and human part detection pipeline.
The selected image IDs are provided in the supplementary material.
We do not include manual verification because this dataset is only meant to measure the effect of removing humans from ImageNet.
We also compare against supervised pretraining using datasets (a) and (b) for which labels are available.
\paragraph{Frozen encoder evaluation.}
Frozen encoder evaluations are seen as a gold-standard for evaluating representation learning performance due to their simplicity and ease of use.
We include: the two standard evaluation datasets ImageNet~\citep{deng09imagenet:} and Places~\citep{zhou14learning}, as well as ObjectNet~\citep{barbu2019objectnet}, CIFAR-10~\citep{krizhevsky09learning} as an example of a low-resolution dataset (though we note that its classes are contained within IN-1k), and the fine-grained classification datasets Oxford Flowers~\citep{nilsback08automated} and Herbarium 2019~\citep{tan2019herbarium}.
We use linear probing, low-shot classification and clustering as tasks to evaluate the frozen encoders.
For linear probing, we use MoCo-v2's default linear evaluation code, without any adjustments\footnote{For ObjectNet, which is a test set only, we transfer the linear classifier trained on IN-1k and report performance on the overlapping classes.}.
For low-shot classification we follow the public implementation of~\citep{PCL}, again without tuning hyperparameters.
For clustering, $K$-means is run on the embedded features of the validation set and their clustering is compared against the ground-truth labels, as in~\citep{sariyildiz2020concept,zheltonozhskii2020selfsupervised}.
Here $K$ is simply set to be equal to the number of ground-truth classes.
This evaluation does not contain any hyperparameters.
\paragraph{Finetuning evaluation.}
For finetuning, we follow standard evaluation practices of the self-supervised literature: we evaluate object segmentation and detection.
Beyond this, we also include human-focused efficient keypoint detection and pose estimation on COCO~\citep{coco} to measure the effect of removing images with humans on person-centric tasks.
Finally, we also finetune for object detection on PASCAL VOC~\citep{everingham12the-pascal} and long-tailed object segmentation on LVIS-v1~\citep{gupta19lvis:}.
Specific details such as number of epoch, learning rates and schedules are provided in the Appendix. Additionally we include all code to reproduce the evaluations.
\section{Results \label{sec:results}}
\input{tabs/frozen}
\input{tabs/finetune}
When comparing pretraining datasets, a key factor is whether the pretraining and target datasets used for evaluation coincide or not.
The realistic usage scenario is that they differ, which amounts to transfer learning. Yet, for benchmarking SSL methods, the non-transfer setting is often used instead. For example, both pretraining \textit{and} evaluating on IN-1k does not test whether the features generalize beyond ImageNet.
In the tables, we clearly distinguish these two cases by \emph{graying the non-transfer learning results}.
Naturally, there is a performance gap when training on {\color{ForestGreen}\textbf{PASS}}\xspace compared to matching pretraining and target datasets.
However, for transfer learning we show that {\color{ForestGreen}\textbf{PASS}}\xspace performs on par to pretraining on ImageNet or similar datasets, even on downstream tasks involving humans, and for different SSL methods.
Furthermore, we ablate the effect of using different pretraining splits and pretraining augmentations on downstream performances.
\paragraph{Frozen encoder evaluation.}
For the clustering evaluation in \cref{tab:cluster}, pretraining on {\color{ForestGreen}\textbf{PASS}}\xspace yields generalisable features that can outperform IN-1k and IN-1k$^*$ pretraining in the transfer learning setting.
The linear probing benchmark in \cref{tab:linear} shows a similar pattern: {\color{ForestGreen}\textbf{PASS}}\xspace yields the best transfer performance on IN-1k, ObjectNet, Places205 and Herbarium19 (HT19). However, for CIFAR-10 transfer from ImageNet works better than Places and {\color{ForestGreen}\textbf{PASS}}\xspace\!.
This is most likely due to the semantic overlap of these two datasets: all CIFAR-10 classes are contained in ImageNet, reducing the transfer gap.
Finally, we report the low-shot classification results in \cref{tab:lowshot}.
Here, we find that {\color{ForestGreen}\textbf{PASS}}\xspace pretraining performs close to IN-1k and surpasses IN-1k$^*$ when transferring to Places205.
For PASCAL VOC, {\color{ForestGreen}\textbf{PASS}}\xspace is worse than both ImageNet variants, but outperforms Places205 pretraining.
The gap to IN-1k pretraining might be due to the insufficient tuning of the SVM cost hyperparameter, as well as the semantic similarity of PascalVOC with ImageNet (common objects, object-centric images and indeed overlapping object categories). However, as shown below, different pretraining methods such as DINO can yield significant improvements.
For Herbarium19, {\color{ForestGreen}\textbf{PASS}}\xspace yields the strongest pretraining performance, and is the only example where the self-supervised representation surpasses the supervised baselines for this benchmark.
\paragraph{Finetuning evaluation.}
Next we analyse the performance of the pretrained representations when they are used as initialisation for downstream tasks (c.f.~\cref{tab:finetune}).
For Mask-RCNN based detection and segmentation on COCO (\cref{tab:coco}, we find that the gaps between the different pretraining datasets are overall small and within $\pm1\%$ for the AP, AP50 and AP75 measures for both bounding-boxes ($\text{bb}$) and mask ($\text{mk}$) evaluations.
We also report the performance on the `person' object category (\Strichmaxerl[1.1]) and find that the encoder pretrained on {\color{ForestGreen}\textbf{PASS}}\xspace still matches and surpasses the ones obtained with supervision on IN-1k and IN-1k$^*$. In \cref{tab:densepose} we find similar trend for dense pose estimation, an inherently human-focused task, where the differences between pretraining datasets are even smaller.
We provide further experimental results in the Appendix.
\input{tabs/compare_ssl}
\input{tabs/compare_augmentation}
\input{tabs/compare_splits}
\paragraph{Effect of data content.}
In this section we study the effect of selecting different types of visual content, using the splits specified in \cref{tab:splits}.
The first question is \emph{whether humans matter or not} for pretraining.
In the Humans split, we proceed as in {\color{ForestGreen}\textbf{PASS}}\xspace but skip the filtering step (in this case, 57\% of the images contain humans and there is a 27\% overlap with the filtered version).
This leads to a small improvement in object detection 4-shot classification and a small decrease in performance for HT19, Flowers, and ObjectNet, but mostly staying within $\pm 2\%$.
The second question is whether the distribution of \emph{other} classes matters or not.
We test biasing the selection of images by using an IN-1k pretrained classifier to match the class statistics of IN-1k as well as possible.
The $\approx$IN800 split is obtained from the 6M images by retaining the most frequent 800 classes and retaining at most 2K samples per class, followed by subsequent sampling to a size of 1.28M.
The $\approx$IN800 split leads to a large increase in performance for IN-1k and ObjectNet linear probing, echoing previous findings that the closeness of the pretraining distribution to the downstream task matters~\citep{kotar2021contrasting,reed2021selfsupervised}.
However, it also leads to small decreases on datasets such as Places, H19 and Flowers.
The third question is whether capping the number of pictures per photographer matters.
The Random6M split contains all 6M images left after cleaning and removing humans, as determined by pretrained models, but skipping the photographer balancing and manual cleanup step (the latter is for safety and only changes the dataset composition marginally).
The Random split further restricts that to 1.28M.
While the Random split generally performs less well than the other splits, Random6M achieves strong performances on many tasks.
While this result is also partially due to the larger number of optimization steps (all methods were trained with 200 epochs), it might show how SSL methods can scale with more data.
Overall, {\color{ForestGreen}\textbf{PASS}}\xspace yields overall good performance without either containing humans, or the need for a pretrained classifier and is of size comparable to IN-1k.
\paragraph{Effect of pretraining method.}
Finally, we pretrain other self-supervised learning methods on our dataset and show results in \cref{tab:ssl}.
First, we compare MoCo-v2 against MoCo-v2-CLD~\citep{wang2020unsupervised}, and find improvements similar in scale to the ones reported in~\citep{wang2020unsupervised}.
Next, from the results of SwAV~\citep{caron20unsupervised} and BYOL~\citep{grill20bootstrap}, we find that methods that performed better on IN-1k also tend to perform better on our dataset, show that {\color{ForestGreen}\textbf{PASS}}\xspace can be used instead of IN-1k to compare models.
\paragraph{A note on augmentations.}
In this paper we have solely used the hyperparameters that come with the methods, and as such, have been optimised by the computer vision community for the last few years on IN-1k.
Augmentations play an important role for self-supervised learning~\citep{chen20a-simple,asano20a-critical,chen2020mocov2}, and strongly influence the final performance.
We cannot replicate several years of augmentation tuning for our {\color{ForestGreen}\textbf{PASS}}\xspace dataset but can expect that with time, better settings can be found.
To support this claim, we show one example of such adaption in \cref{tab:aug} that we were able to find.
By changing the minimum-size parameter used in random-resized crops from $0.2$ to $0.1$, we observe an improvement in performance in almost every benchmark.
While small, the improvement might be explained by the fact that our dataset is less object-centric than ImageNet, and so stronger crops can be used, as there is no need for the crop to cover an object.
\section{Discussion}\label{s:discussion}
In the introduction, we have motivated our new dataset {\color{ForestGreen}\textbf{PASS}}\xspace from a technical, ethical and legal perspective.
By using CC-BY images, we greatly reduce the risk of using images in a manner incompatible with copyright.
By avoiding the usage of search engines and labels to form a dataset, we avoid introducing corresponding biases.
By excluding all images that contain humans, as well as other identifiable information and NSFW images, we significantly reduce data protection and other ethics risks for the data subjects.
We have shown that, despite these changes, we can effectively pretrain neural networks using this data.
By conducting extensive downstream task evaluations, we have shown that pretrained networks obtained using self-supervised training on this dataset are competitive to ImageNet on transfer settings and even on downstream tasks that involve humans.
However, several limitations remain.
First, while we put care in filtering the images, automatically and manually, some harmful content might have slipped through.
Second, sampling images randomly from an uncurated large collection removes specific biases such as search engine selection but not others, for example the geographic bias.
Furthermore, we added one significant bias: there are no people in these pictures, despite the fact that a large fraction of all images in existence contain people.
While this appears to be acceptable for model \textit{pretraining}, {\color{ForestGreen}\textbf{PASS}}\xspace cannot be used to learn models of people, such as for pose recognition.
Thirdly, since {\color{ForestGreen}\textbf{PASS}}\xspace contains no labels, {\color{ForestGreen}\textbf{PASS}}\xspace (in contrast to ImageNet) cannot be used alone for training and benchmarking.
For this, curated datasets remain necessary, which continue to carry many of the issues of privacy and copyright described in the paper.
Despite these limitations, we believe that {\color{ForestGreen}\textbf{PASS}}\xspace is an important step towards curating and improving our datasets to reduce ethical and legal risks for many tasks and applications and at the same time to challenge the SSL community with a new, more realistic training scenario of utilizing images not obtained from a labeled dataset.
\section{Conclusion}
In the introduction, we have motivated our new dataset PASS from a scientific, ethical and legal perspective.
By using CC-BY images, we greatly reduce the risk of using images in a manner incompatible with copyright.
By avoiding the usage of search engines and labels to form a dataset, we avoid introducing corresponding biases.
By excluding all images that contain humans, as well as other identifiable information and NSFW images, we significantly reduce data protection and other ethics risks for the data subjects~\cite{prabhu20large}.
We have shown that, despite or because of these changes, we can effectively pre-train neural networks using this data.
However, several limitations remain.
First, while we put care in filtering the images, automatically and manually, some harmful content might have slipped through.
Second, sampled images randomly from an uncurated large collection removes specific biases such as search engine selection but not others.
Furthermore, we added one significant bias: there are no people in these pictures, despite the fact that the vast majority of images in existence contain people.
While this appears to be acceptable for model pretraining, {\color{ForestGreen}\textbf{PASS}}\xspace cannot be used to learn models of people, such as for pose recognition.
Thirdly, pretraining on {\color{ForestGreen}\textbf{PASS}}\xspace is equivalent to ImageNet or other such datasets only in the transfer learning setting.
Pretraining in a non-transfer learning setting is not very useful in applications, but is used for benchmarking.
Fourthly, since {\color{ForestGreen}\textbf{PASS}}\xspace contains no labels, {\color{ForestGreen}\textbf{PASS}}\xspace cannot be used alone for benchmarking.
For this, curated datasets remain necessary.
There is also the issue of reproducibility and model comparison: traditional benchmarks such as ImageNet are likely to remain essential to allow researchers to compare models.
Nonetheless, we believe that {\color{ForestGreen}\textbf{PASS}}\xspace is an step towards curating and improving our dastasets to reduce scientific, ethical and legal risks for many tasks and applications.
\part{Appendix}
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\parttoc
}
\bigskip
\section{Dataset Access}
During review, we had provided instructions for the reviewers to view the dataset in the OpenReview submission.
The dataset is hosted by Zenodo~\cite{zenodo} and can be found under this URL:
\url{https://zenodo.org/record/5528345/}
\section{Image attributions}
The creators for the images in \cref{fig:splash}, are (top-left to bottom-right):
\textit{RichTatum, chongeileen, Vlad Iorsh, millstastic, Ross\_Angus, nfeli777, semper\_fi\_brother, jj-photo, BluEyedA73, Pazit Polak, florianplag, Johan Lange, kimadababe, Alzheimer's Association - Greater Illinois, Esme\_Vos, babeltravel, Henrique B Costa, Dude of Lego, matthewreid, Russ Dill, theunwiseman, Wenkan, at8eqeq3, Larry1732, Haroldo Kennedy, DGriebeling, Tez\_kuma, Overman Alawami, FreeCat, fczuardi, Norisa1, pescatello, jasonlsraia, semanticwebcompany, jumblejet, Rain San Martin, crackerbelly, Problemkind, M\_Hartman Photography, antaean, andyket, mirven, Quinn Rossi, MARCO\_QUARANTOTTI, -Tripp-, JulieHagenbuch, frank drewett, MeRyan, Sharib4rd, Schlusselbein2007, BBR1245, Officer Phil, cadyellow, Andreas Joensen, FuFuWolf, zerojay, Spoon Monkey, Tatters80, gypsygirl.photography, ChristinaT, Irene Vlachou, allenreichert, rs-foto, Sara Cimino, JonoMueller, Lock The Gate, m01229, ebis50, Speculum Mundi, fontosiskola, rakh1, dreamcicle19772006 ON OFF
}.
All images in this paper and the dataset are licensed by the \textbf{CC-BY }(\url{https://creativecommons.org/licenses/by/2.0/}) license and contain information on their creator.
On the above url, this license is described as follows:
\begin{enumerate}
\item Copy and redistribute the material in any medium or format
\item Remix, transform, and build upon the material
for any purpose, even commercially.
\item This license is acceptable for Free Cultural Works.
\item The licensor cannot revoke these freedoms as long as you follow the license terms.
\end{enumerate}
\section{Dataset Generation Details \label{sec:app_dataset}}
\subsection{Automated pipeline}
For the detection of faces we use RetinaFace~\citep{deng2019retinaface} and the MobileNet0.25 model from the PyTorch RetinaFace repository (\url{https://github.com/biubug6/Pytorch_Retinaface}).
We keep all parameters, including the detection threshold value of $0.02$.
For the automated detection of humans and human parts we use the highly performant Cascade-RCNN trained using the 3x schedule on COCO, available from the detectron2 library~\citep{wu2019detectron2}.
The threshold for detections is set to its default value of $0.5$.
Finally, before passing the images to the professional human annotators, we run the a model pretrained to detect ``not safe for work'' (NSFW) content on the set of images from (\citep{nsfwmodel}).
After visually inspecting the top 500 entries for both the `hentai' and the `porn' categories, and not finding any positive results, we pass the data to the human annotators.
\subsection{Human verification}
In \cref{fig:verification-instruction}, we provide two screenshots of the verification instructions given to the annotators.
We have manually assessed the quality of these annotations using two independent samples of size 1000 each and only found 15 false positives and 2 minor false negatives.
We informed the annotation team of our findings to further improve the quality.
The important error type here are false negatives, as this would mean humans or personal information remains undetected.
After human verification 2\% of images are flagged and subsequently removed.
The vast majority of flagged images stem from the presence of car licence plates.
\begin{figure}
\centering
\includegraphics[width=0.45\textwidth]{figs/safe_instructions1.jpeg}
\includegraphics[width=0.45\textwidth]{figs/safe_instructions2.jpeg}
\caption{\textbf{Human verification instructions.} We have blurred faces and other personally identifiable information solely for in these screenshots. The professional annotators saw unblurred versions to provide a clearer understanding of what images require flagging.}
\label{fig:verification-instruction}
\end{figure}
\subsection{Removal of duplicates \label{appx:duplicates}}
We also conduct a basic removal of duplicates process using the Find Identical Images tool~\citep{dutta2021fii}.
Using this we find and remove $2529$ self-duplicates, $114$ duplicates with MS COCO, and $9$ duplicates with ImageNet LSVRC-12. We further checked against duplicates with Places-205, but found none.
\section{Implementation Details}
\subsection{Representation learning experiments}
\paragraph{MoCo-v2.} We pretrain the MoCo-v2 models using its default settings for 200 epoch using the publicly available repository (\url{https://github.com/facebookresearch/moco}).
This includes training using the cosine learning rate schedule, augmentations consisting of random resized crops, colorjitter, random horizontal flipping and blurring with a batch-size of 256 and a learning rate of $0.03$.
On two RTX6000 GPUs this training requires around one week to complete.
\paragraph{SwAV.} We pretrain SwAV~\citep{caron20unsupervised} using the fixed hyperparameters for 200 epoch pretraining with a batch-size of 256 found at the publicly available repository (\url{https://github.com/facebookresearch/swav}).
For this we use 4 RTX6000 GPUs and this requires training for a bit more than a week.
SwAV's augmentations include those of MoCo-v2 and in addition Multi-Crop, in which in addition to the two 224-sized crops, six 96x96-sized crops are extracted and used for computing the loss.
\paragraph{DINO.} We pretrain DINO~\citep{caron2021emerging} using for 100 epochs with a batch-size of 256 using the publicly available repository (\url{https://github.com/facebookresearch/dino}).
Training augmentations include those of SwAV, including Multi-Crop, as well as solorization, introduced in \citep{grill20bootstrap}.
Here, the architecture trained is ViT-B~\citep{dosovitskiy2020image} and training takes around 2 days on 8 GPUs.
\paragraph{Pretrained models.}
We utilise pretrained models for the MoCo-v2 models trained on Places and ImageNet from the MoCo-v2 repository and from the repository of \citep{zhao2021makes}.
In addition we use PyTorch's publicly available models: a supervisedly pretrained ResNet50 model for benchmarking and a pretrained MobileNet-v3 for generating the $\approx$IN800 split.
\subsection{Downstream tasks}
\paragraph{Cluster evaluation.} We conduct these experiments on the test sets of the datasets by resizing the images to smaller-side 256pixels and taking a centered 224x224 crop.
These images' embeddings after global average pooling are computed and input in k-means, implemented by the FAISS~\citep{JDH17} library. For k-means, we use 50 iterations and 5 restarts and set K to be equal to the number of ground-truth classes. NMI, aRI calculations are computed by scikit-learn~\citep{scikit-learn}.
\paragraph{Linear probing.} We exactly follow MoCo-v2's implementation. For ease, we describe it here:
Training is done for 100 epochs on the global average pooled features of the encoders without the use of an additional BatchNorm layer.
The learning rate is dropped by a factor of 0.1 at epochs 60, 80 starting at a value of 30.0. The batchsize is 256, and we use a single GPU.
The augmentations during training consist of random horizontal flipping and RandomResizedCrops to size 224x224 with the default parameters (crops of sizes (0.08, 1.0) of the original size and variations in aspect ratio of (3/4, 4/3)).
For linear probing on the Vision Transformer~\citep{dosovitskiy2020image}, we use the 100 epoch linear evaluation code of \citep{caron2021emerging}.
\paragraph{Data-efficient SVM classification.} We exactly follow the publicly available implementation of PCL~\citep{PCL} (\url{https://github.com/salesforce/PCL}).
Here, an one-vs-all SVM is fit on the data using a cost parameter $c=0.5$, which we keep at its default value.
The features for the SVM are obtained by smaller-side-resizing to 256 and taking a center crop of 224x224.
\paragraph{Finetuning.} We follow the publicly available implementation of MoCo~\citep{he19momentum}.
For the R50-C4 architecture used in the Pascal VOC experiments, this includes the use of an additional normalisation layer and training for 24k steps.
For FPN architectures (used for COCO detection, segmentation, keypoint, densepose and LVIS segmentation) this includes the use of SyncBatchNorm and training using the `1x' schedule for which trains for 90K steps and has a learning rate warmup for the first 1000 steps from 0.02/100 and subsequent training with an initial learning rate of 0.02 being divided by 10 at steps 60K and 80K.
For the data-efficient keypoint detection we use the R50-FPN with a fixed split containing 5\% of the annotations of COCO2017's train split, train for 9K steps and evaluate on the full validation split.
For the dense pose estimation on COCO~\citep{coco} we use an R50-FPN with chart-based embeddings~\citep{Guler2018DensePose} and the `s1x' schedule, which decays the learning rate at steps 100K, 120K and trains for 130K steps in total.
Training is done on the 2014 densepose training split and the 2014 `valminusminival' split, and performance tested on the 2014 `minival' split.
Note that the default `1x' schedule of LVIS includes 180K iterations, so in these terms our LVIS training can be referred to as `0.5x'.
Instead of a batch-size of 16 we use a batch-size of 8 on two GPUs and divide the learning rates by half and the multiply the learning-rate schedules by two. We use the detectron2 library~\citep{wu2019detectron2}.
Following the MoCo-v2 implementation and experiments, we evaluate object detection and segmentation on COCO~\citep{coco} using a Mask-RCNN~\citep{he17mask} with the R50-FPN backbone~\citep{liun17feature} with the `1x' schedule and object detection on Pascal VOC07+12~\citep{everingham12the-pascal} using a R50-C4 using 24k steps.
\subsubsection{A note on evaluation hyperparameters}
\input{tabs/lineardelta}
Finally, we note that comparing different self-supervised methods using linear evaluation is non-trivial, as the optimal settings (learning rate and schedule) can depend on the encoder's pretraining method.
As we show in \cref{tab:linear-delta}, both MoCo-v2 and SwAV suffer strongly when instead of their linear evaluation code, the other's is used.
For completeness, we report the linear probing numbers of SwAV when their linear evaluation code is used in \cref{tab:swav}.
\begin{table}[tb]
\footnotesize
\caption{SwAV linear probing performance.
We report the linear evaluation performance of the SwAV pretrained model when using SwAV's linear evaluation code. \label{tab:swav}}
\vspace{1em}
\centering
\footnotesize
\setlength{\tabcolsep}{2pt}
\begin{tabular}{l ccccc}
\toprule
& \multicolumn{5}{c}{\textbf{Linear probing}} \\
\cmidrule{2-6}
\textbf{SSL method} & {IN-1k} & ObjN & {Plcs} & Flwrs & HT19 \\
\midrule
SwAV & 60.4 & 10.4 & 55.0 & 88.3 & 38.7 \\
\bottomrule
\end{tabular}
\end{table}
While outside of this work, frozen evaluation of methods which do not require further finetuning and are close to deployment also include the clustering evaluation in the paper (which can be combined with a Hungarian algorithm to yield label predictions), kNN evaluations (as done in e.g. \citep{caron2021emerging}) and the normalisation of feature statistics before training the linear layer, as in~\citep{koohpayegani2021mean}.
\subsubsection{Dataset details}
For detection and segmentation in \cref{tab:coco} we use COCO2017, which contains 80 labeled objects with segmentation masks and boxes. The training set contains 118K images with 850K annotations, and the validation set 5K images with 36K annotations.
For dense human pose estimation in \cref{tab:densepose}, COCO2014 (a subset of COCO2017) is used, which contains masks of people and their bodies’ 3D surface on a mesh. The training and validation sets contain 32K and 1.5K images with 99K and 25K persons instances, respectively.
For object detection in \cref{tab:pvoc} we use the common Pascal VOC “07+12” split containing 20 object categories. This split uses the VOC7’s test set (2.5K images and 15K annotations) for evaluation and merges the two trainval sets for finetuning (16.5K images and 47K annotations).
For data-efficient keypoint detection in \cref{tab:kp}, we generate a random, fixed split of COCO2017, resulting in 1K images with 2K annotations for training and the full test set of 5K images with 10K annotations.
For detection and segmentation in \cref{tab:lvis}, we use LVIS-v1~\citep{gupta19lvis:}, which contains 1203 labeled objects with segmentation masks and boxes. The training set contains 100K images with 1.2M annotations, and the validation set 5K images with 244K annotations.
For cross-domain transfer in \cref{tab:cross}, we use multiple smaller-scale datasets taken from the Visual Task Adaptation Benchmark (VTAB)~\citep{zhai2019large}:
CLEVR-count~\citep{clevr} contains synthetic rendered images of objects and the task is to count the number of objects via 8-way classification and contains 70K, 15K images for training and testing.
Our construction of the dataset follows the code from the VISSL repository~\citep{vissl}.
DTD~\citep{cimpoi14describing} contains 47 classes of different describable textures (\textit{e.g.} `honeycombed' or `sprinkled'), and we use the first train and test-split supplied in the paper, containing around 2K images each.
Both Eurosat~\citep{helber2019eurosat} and Resisc45~\citep{Cheng_2017} are remote sensing image classification datasets with 10 and 45 classes respectively (\textit{e.g.} `baseball diamond', or `river').
They do not come with a natural train/test split, so we create a fixed training split using 60\% and 20\% of the data, following VTAB~\citep{zhai2019large}.
This yields around 21K and 25K images for training and 5K and 6K for testing, for Eurosat and Resisc45, respectively.
\section{Additional Experimental Results \label{sec:app_results} }
\begin{table}[htb]
\caption{\textbf{Finetuning representation evaluations.} a) PascalVOC07+12 detection and (b) human keypoint estimation and c) Object segmentation on LVIS. AP denotes COCO's AP on IoUs [0.5:0.95:0.05]. \label{tab:finetune-appendix}}
\footnotesize
\setlength{\tabcolsep}{0.8em}
\begin{subtable}[b]{0.4\textwidth}
\setlength{\tabcolsep}{2.5pt}
\centering
\input{tabs/_detection_pvoc}
\end{subtable}
\hfill
\begin{minipage}[b]{0.5\textwidth}
\begin{subtable}[b]{\textwidth}
\setlength{\tabcolsep}{5pt}
\input{tabs/_efficient_keypoint}
\end{subtable}
\\
\begin{subtable}[b]{\textwidth}
\setlength{\tabcolsep}{5pt}
\input{tabs/_lvis}
\end{subtable}
\end{minipage}
\end{table}
\subsection{Object detection on PascalVOC}
In \cref{tab:pvoc} we report the performances of object detection using R50-C4 on Pascal VOC with 24k steps (as in MoCo).
As reported in previous works~\citep{zhao2021makes,he19momentum}, the self-supervised baselines outperform supervised pretraining especially the more strict AP and AP75 measures.
We find that pretraining on {\color{ForestGreen}\textbf{PASS}}\xspace falls within 1\% of the performance of pretraining on IN-1k without humans.
\subsection{Data-efficient keypoint detection on COCO}
In \cref{tab:kp}, we show the results of finetuning the encoders using a small sample of ground-truth annotations from the COCO 2014 keypoint annoations.
While using 100\% of the data results in almost no gains between pretrained and random initialisations (as shown in \citep{he19momentum}), we see a large difference when only 5\% of the annotations are considered.
We find that pretraining on {\color{ForestGreen}\textbf{PASS}}\xspace yields the strongest keypoint detection results in terms of AP and AP75, surpassing even supervised IN-1k pretraining.
\subsection{Long-tailed instance segmentation on LVIS-v1}
In \cref{tab:lvis}, we show the results of finetuning for object segmentation using Mask-RCNN with R50-FPN on LVIS-v1~\citep{gupta19lvis:}, which contains 1203 classes.
We find that different to object detection on COCO in \cref{tab:coco}, the supervisedly trained baseline performs best with an overall AP of $18.9$, and the MoCo-v2 pretrained on {\color{ForestGreen}\textbf{PASS}}\xspace yielding an AP of $16.6$, and large gap in the `rare' categories, as measured by AP$_\text{r}$.
\subsection{Cross-domain transfer}
\begin{table}[htb]
\centering
\caption{\textbf{Cross-domain transfer}. Linear-probing Top-1 accuracies are reported for various datasets whose domain is further from typical web-images. \label{tab:cross}}
\begin{tabular}{l cccc}
\toprule
\textbf{Model} & Resisc45 & Clevr-cnt & Eurosat & DTD \\
\midrule
sup. IN-1k & 90.2 & 61.8 & \textbf{96.6} & 64.5 \\
MoCo-v2 Places & 91.0 & 70.4 & 93.5 & 62.9 \\
MoCo-v2 IN-1k & 90.7 & 72.6 & 96.5 & 66.7 \\
\midrule
MoCo-v2 {\color{ForestGreen}\textbf{PASS}}\xspace & \textbf{91.4} & \textbf{73.1} & 95.4 & \textbf{66.8} \\
\bottomrule
\end{tabular}
\end{table}
In~\cref{tab:cross}, we report cross-domain transfer results on the following datasets and tasks taken from VTAB~\citep{zhai2019large}: CLEVR-count~\citep{clevr} (object counting), DTD~\citep{cimpoi14describing} (texture classification) and Eurosat~\citep{helber2019eurosat} (satellite image land-cover classification) Resisc45~\citep{Cheng_2017} (remote sensing image classification).
We use the same linear-evaluation protocol as in the remainder of the paper, i.e. the settings from MoCo-v2, without any hyperparameter tuning.
From~\cref{tab:cross} we find that MoCo-v2 trained on our {\color{ForestGreen}\textbf{PASS}}\xspace dataset does very well across those datasets, surpassing both IN-1k supervised and MoCo-v2 on IN-1k for three out of four cases.
This shows that pretraining on {\color{ForestGreen}\textbf{PASS}}\xspace is a viable strategy even if the downstream task domain is very different.
\clearpage
\section{Dataset Documentation: Datasheets for Datasets}
Here we answer the questions outlined in the datasheets for datasets paper by \citet{gebru2020datasheets}.
\subsection{Motivation}
\paragraph{For what purpose was the dataset created?}
Neural networks pretrained on large image collections have been shown to transfer well to other visual tasks where there is little labelled data,
i.e. transferring a model works better than starting with a randomly initialized network every time for a new task, as many visual features can be repurposed.
This dataset has as its goal to provide a safer large-scale dataset for such pretraining of visual features.
In particular, this dataset does not contain any humans or human parts and does not contain any labels.
The first point is important, as the current standard for pretraining, ImageNet and its face-blurred version only provide pseudo-anonymity and furthermore do not provide correct licences to the creators.
The second point is relevant as pretraining is moving towards the self-supervised paradigm, where labels are not required.
Yet most methods are developed on the highly curated ImageNet dataset, yielding potentially non-generalizeable research.
\paragraph{Who created the dataset (e.g., which team, research group) and on behalf of which entity (e.g., company, institution, organization)?}
The dataset has been constructued by the research group ``Visual Geometry Group'' at the University of Oxford at the Engineering Science Department.
\paragraph{Who funded the creation of the dataset?}
The dataset is created for research purposes at the VGG research group.
Individual researchers have been funded by AWS Machine Learning Research Awards (MLRA), EPSRC Centre for Doctoral Training in Autonomous Intelligent Machines \& Systems [EP/L015897/1], the Qualcomm Innovation Fellowship, Innovate UK (project 71653) on behalf of UK Research and Innovation (UKRI) and by the European Research Council (ERC) IDIU-638009.
\subsection{Composition}
\paragraph{What do the instances that comprise the dataset represent (e.g., documents, photos, people, countries)?}
This dataset only contains photos.
In addition we provide tabular meta-data for these images, which contain information such as the creator's username and image capture date.
\paragraph{How many instances are there in total (of each type, if appropriate)?}
The dataset contains 1.4M images, resulting in 181GB as a tar file.
\paragraph{Does the dataset contain all possible instances or is it a sample (not necessarily random) of instances from a larger set?}
The dataset is a sample of a larger set---all possible digital photographs.
As outlined in \cref{sec:pass} we start from an existing dataset, YFCC-100M, and stratify the images (removing images with people and personal information, removing images with harmful content, removing images with unsuitable licenses, each user contributes at most 80 images to the dataset).
This leaves 1.6M images, out of which we take a random sample of 1.28M images to replicate the size of the ImageNet dataset.
While this dataset can thus be extended, this is the set that we have verified to not contain humans, human parts and disturbing content.
\paragraph{What data does each instance consist of?}
Digital photographs uploaded by users of the flickr platform.
\paragraph{Is there a label or target associated with each instance?
No. Our dataset deliberately does not contain labels.
\paragraph{Is any information missing from individual instances?}
Not from the dataset. Note however that the meta-data that we additionally provide is not complete and might have non-uniform missing values.
\paragraph{Are relationships between individual instances made explicit (e.g., users’ movie ratings, social network links)?}
Not applicable: each image stands on its own and we do not provide relationships between these.
\paragraph{Are there recommended data splits (e.g., training, development/validation, testing)?}
As outlined in the intended usecases, this dataset is meant for pretraining representations.
As such, the models derived from training on this dataset need to be evaluated on \emph{different datasets}, so called down-stream tasks. Thus the recommended split is to use all samples for training.
\paragraph{Are there any errors, sources of noise, or redundancies in the dataset?}
No.
\paragraph{Is the dataset self-contained, or does it link to or otherwise rely on external resources (e.g., websites, tweets, other datasets)?}
No.
The dataset contains links to the publicly hosted mirror of the YFCC dataset on Amazon Web Services.
\paragraph{Does the dataset contain data that might be considered confidential (e.g., data that is protected by legal privilege or by doctor-patient confidentiality, data that includes the content of individuals’ non-public communications)?}
No.
\paragraph{Does the dataset contain data that, if viewed directly, might be offensive, insulting, threatening, or might otherwise cause anxiety?}
No. Besides checking for human presence in the images, the annotators were also given the choice of flagging images for disturbing content, which once flagged was removed.
\paragraph{Does the dataset relate to people? If not, you may skip the remaining questions in this section.}
No.
\paragraph{Does the dataset identify any subpopulations (e.g., by age, gender)?} NA
\paragraph{Is it possible to identify individuals (i.e., one or more natural persons), either directly or indirectly (i.e., in combination with other data) from the dataset?} NA
\paragraph{Does the dataset contain data that might be considered sensitive in any way (e.g., data that reveals racial or ethnic origins, sexual orientations, religious beliefs, political opinions or union memberships, or locations; financial or health data; biometric or genetic data; forms of government identification, such as social security numbers; criminal history)?} NA
\subsection{Collection process}
\paragraph{How was the data associated with each instance acquired?}
The data was collected from the publicly available dataset YFCC-100M which is hosted on the AWS public datasets platform.
We have used the meta-data, namely the copyright information to filter only images with the CC-BY licence and have downloaded these using the aws command line interface, allowing for quick and stable downloading.
In addition, all files were subsequently scanned for viruses using Sophos SAVScan virus detection utility, v.5.74.0.
\paragraph{What mechanisms or procedures were used to collect the data (e.g., hardware apparatus or sensor, manual human curation, software program, software API)?}
Our dataset is a subset of the YFCC-100M dataset.
The YFCC-100M dataset itself was created by effectively randomly selecting publicly available images from flickr, resulting in approximately 98M images.
\paragraph{If the dataset is a sample from a larger set, what was the sampling strategy (e.g., deterministic, probabilistic with specific sampling probabilities)?}
See the similar question in the Composition section.
\paragraph{Who was involved in the data collection process (e.g., students, crowdworkers, contractors) and how were they compensated (e.g., how much were crowdworkers paid)?}
As described, the data was collected automatically by simply downloading images from a publicly hosted S3 bucket.
The human verification was done using a professional data annotation company that pays 150\% of the local minimum wage.
\paragraph{Over what timeframe was the data collected?}
The images underlying the dataset were downloaded between March and June 2021 from the AWS public datasets' S3 bucket, following the download code provided in the repo.
However the images contained were originally and taken anywhere from 2000 to 2015, with the majority being shot between 2010-2014.
\paragraph{Were any ethical review processes conducted (e.g., by an institutional review board)?}
No.
\paragraph{Does the dataset relate to people? If not, you may skip the remainder of the questions in this section.}
No, this is one of the specific purposes of this dataset.
\subsection{Preprocessing/cleaning/labeling}
\paragraph{Was any preprocessing/cleaning/labeling of the data done (e.g., discretization or bucketing, tokenization, part-of-speech tagging, SIFT feature extraction, removal of instances, processing of missing values)?}
After the download of approx. 17M images, the corrupted, or single-color images were removed from the dataset prior to the generation of the dataset(s) used in the paper.
The images were not further preprocessed or edited.
\paragraph{Was the “raw” data saved in addition to the preprocessed/cleaned/labeled data (e.g., to support unanticipated future uses)?}
Yes. The creators of the dataset maintain a copy of the 17M original images with the CC-BY licence of YFCC100M that sits at the start of our dataset creation pipeline.
\paragraph{Is the software used to preprocess/clean/label the instances available?}
We have only used basic Python primitives for this. For the annotations we have used VIA~\citep{dutta2019vgg,dutta16vgg-image}
\subsection{Uses}
\paragraph{Has the dataset been used for any tasks already?}
In the paper we show and benchmark the intended use of this dataset as a pretraining dataset.
For this the dataset is used an unlabelled image collection on which visual features are learned and then transferred to downstream tasks.
We show that with this dataset it is possible to learn competitive visual features, without any humans in the pretraining dataset and with complete license information.
\paragraph{Is there a repository that links to any or all papers or systems that use the dataset?}
We will be listing these at the repository.
\paragraph{What (other) tasks could the dataset be used for?}
We believe this dataset might allow researchers and practitioners to further evaluate the differences that pretraining datasets can have on the learned features.
Furthermore, since the meta-data is available for the images, it is possible to investigate the effect of image resolution on self-supervised learning methods, a domain largely underresearched thus far, as the current de-facto standard, ImageNet, only comes in one size.
\paragraph{Is there anything about the composition of the dataset or the way it was collected and preprocessed/cleaned/labeled that might impact future uses?}
Given that this dataset is a subset of a dataset that randomly samples images from flickr, the image distribution is biased towards European and American creators.
As in the main papers discussion, this can lead to non-generalizeable features, or even biased features as the images taken in other countries might be more likely to further reflect and propagate stereotypes~\citep{revisetool}, though in our case these do not refer to sterotypes about humans.
\paragraph{Are there tasks for which the dataset should not be used?}
This dataset is meant for research purposes only.
The dataset should also not be used for, e.g. connecting images and usernames, as this might risk de-anonymising the dataset in the long term.
The usernames are solely provided for attribution.
\subsection{Distribution}
\paragraph{Will the dataset be distributed to third parties outside of the entity (e.g., company, institution, organization) on behalf of which the dataset was created?}
No.
\paragraph{How will the dataset will be distributed (e.g., tarball on website, API, GitHub)?}
The dataset will be provided as a csv along with code hosted on GitHub that allows the user to download the images in our dataset.
In addition, we hope to also host it as a single tarball on our servers.
\paragraph{When will the dataset be distributed?}
Starting from July 2021.
\paragraph{Will the dataset be distributed under a copyright or other intellectual property (IP) license, and/or under applicable terms of use (ToU)?}
CC-BY.
\paragraph{Have any third parties imposed IP-based or other restrictions on the data associated with the instances?}
No.
\paragraph{Do any export controls or other regulatory restrictions apply to the dataset or to individual instances?}
Not that we are are of. Regular UK laws apply.
\subsection{Maintenance}
\paragraph{Who is supporting/hosting/maintaining the dataset?}
The dataset is supported by the authors and by the VGG research group.
The main contact person is Yuki M. Asano.
We host the dataset on zenodo: \url{https://zenodo.org/record/5528345}.
\paragraph{How can the owner/curator/manager of the dataset be contacted (e.g., email address)?}
The authors of this dataset can be reached at their e-mail addresses: \textit{\{yuki,chrisr,vedaldi,az\}@robots.ox.ac.uk}.
In addition, we have added a contact form in which we can be contacted anonymously at \url{https://forms.gle/tkZugt2DJnFdCE1i6}.
\paragraph{Is there an erratum?}
If errors are found and erratum will be added to the website.
\paragraph{Will the dataset be updated (e.g., to correct labeling errors, add new instances, delete instances)?}
Yes, updates will be communicated via the website. The dataset will be versioned.
\paragraph{If the dataset relates to people, are there applicable limits on the retention of the data associated with the instances (e.g., were individuals in question told that their data would be retained for a fixed period of time and then deleted)?}
Not applicable.
\paragraph{Will older versions of the dataset continue to be supported/hosted/maintained?}
If even after our verification we find further images that contain humans or problematic content, we will remove those further images from existing splits to preserve the goal of this dataset.
\paragraph{If others want to extend/augment/build on/contribute to the dataset, is there a mechanism for them to do so?}
Others are free to reach out to us if their ideas can build on this dataset. All code will be made available.
\subsection{Other questions}
\paragraph{Is your dataset free of biases?}
No. There are many kinds of biases that can either be quantified, e.g. geo-location (most images originate from the US and Europe) or camera-model (most images are taken with professional DSLR cameras not easily affordable), there are likely many more biases that this dataset does contain.
The only thing that this dataset does not contain are humans and parts of humans, as far as our validation procedure is accurate.
\paragraph{Can you guarantee compliance to GDPR?}
No, we cannot comment on legal issues.
\subsection{Author statement of responsibility}
The authors confirm all responsibility in case of violation of rights and confirm the licence associated with the dataset and its images.
\section*{Checklist}
\begin{enumerate}
\item For all authors...
\begin{enumerate}
\item Do the main claims made in the abstract and introduction accurately reflect the paper's contributions and scope?
\answerYes{We propose a dataset for pretraining and evaluate it extensively in \cref{tab:linear,tab:finetune,tab:ssl,tab:aug,tab:splits}. }
\item Did you describe the limitations of your work?
\answerYes{See in particular \cref{s:discussion}}
\item Did you discuss any potential negative societal impacts of your work?
\answerYes{We discuss the issue of our dataset still being US and Euro-centric in \cref{sec:pass}.}
\item Have you read the ethics review guidelines and ensured that your paper conforms to them?
\answerYes{}
\end{enumerate}
\item If you are including theoretical results...
\begin{enumerate}
\item Did you state the full set of assumptions of all theoretical results?
\answerNA{}
\item Did you include complete proofs of all theoretical results?
\answerNA{}
\end{enumerate}
\item If you ran experiments (e.g. for benchmarks)...
\begin{enumerate}
\item Did you include the code, data, and instructions needed to reproduce the main experimental results (either in the supplemental material or as a URL)?
\answerYes{Code is attached as supplementary material and will be released together with the dataset.}
\item Did you specify all the training details (e.g., data splits, hyperparameters, how they were chosen)?
\answerYes{See beginning of \cref{sec:results} and further details in Appendix.}
\item Did you report error bars (e.g., with respect to the random seed after running experiments multiple times)?
\answerNo{Pretraining is very compute intensive. We could only run pretraining once per experiment.}
\item Did you include the total amount of compute and the type of resources used (e.g., type of GPUs, internal cluster, or cloud provider)?
\answerYes{These details are provided in the implementation details in the Appendix.}
\end{enumerate}
\item If you are using existing assets (e.g., code, data, models) or curating/releasing new assets...
\begin{enumerate}
\item If your work uses existing assets, did you cite the creators?
\answerYes{We mainly use YFCC-100M~\citep{thomee2016yfcc100m} and mention them in \cref{sec:intro}, \cref{sec:pass}. All datasets used for evaluation are cited in \cref{sec:results}.}
\item Did you mention the license of the assets?
\answerYes{The license is CC-BY, which is mentioned in \cref{sec:intro} and \cref{sec:pass}.}
\item Did you include any new assets either in the supplemental material or as a URL?
\answerYes{We propose a new dataset, which is a subset of YFCC-100M. Download is made possible via code in the Supplementary Material and a link for reviewers to inspect it is provided in \cref{sec:app_dataset}}
\item Did you discuss whether and how consent was obtained from people whose data you're using/curating?
\answerYes{We discuss this in \cref{sec:pass}.}
\item Did you discuss whether the data you are using/curating contains personally identifiable information or offensive content?
\answerYes{The data is created to explicitly not contain this information, as stated in \cref{sec:intro,sec:pass}.}
\end{enumerate}
\item If you used crowdsourcing or conducted research with human subjects...
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\item Did you include the full text of instructions given to participants and screenshots, if applicable?
\answerYes{See \cref{sec:app_dataset}.}
\item Did you describe any potential participant risks, with links to Institutional Review Board (IRB) approvals, if applicable?
\answerNA{We filtered images for pornographic and disturbing content even before sending the images to the annotators.}
\item Did you include the estimated hourly wage paid to participants and the total amount spent on participant compensation?
\answerYes{See \cref{sec:pass}.}
\end{enumerate}
\end{enumerate}
|
2,877,628,089,034 | arxiv | \section{Introduction}\label{sec:introduction}}
\IEEEPARstart {W}{ith} the rapid growth of location-based services and the wide spreading of GPS-equipped devices, recent years witness increasing availability of trajectory data, including taxi trajectories, check-ins to point-of-interests, cellular signaling sequences, etc. As a result, mining spatial-temporal trajectory data has been extensively studied, such as predicting future trajectories \cite{kong2018hst}, estimating the time of arrival \cite{li2019learning,hong2020heteta} and trajectory clustering \cite{yao2017trajectory,liu2020online}.
Among these researches, learning embeddings of trajectories is a fundamental and critical issue, since the accuracy and comprehensiveness of embeddings are of crucial importance for achieving good performances in downstream tasks. Existing trajectory mining methods mostly adopt the end-to-end embedding strategy, which usually trains a trajectory encoder with task-specific learning objectives \cite{zhao2019go}.
Yet, we argue that the pre-training trajectory embedding methods deserve more attention. Firstly, they can make use of widely existing unlabeled trajectory datasets, and explore the universal spatial-temporal patterns underling trajectories. These patterns can promote the prediction accuracy and generalization performance of downstream tasks.
Secondly, they can be fine-tuned to suit a wide variety of downstream tasks with just a small-scale labeled dataset, or directly applied to unsupervised tasks. This can make the downstream prediction models train faster, and improve the overall computational efficiency.
The core idea of pre-training spatial-temporal trajectory embeddings is to design a self-supervised pretext task to pre-train a trajectory encoder, so that the embedding method can extract high-level travel semantics from unlabeled trajectory datasets, including users or vehicles' spatial-temporal moving patterns and travel purposes. This information can then be utilized by downstream models to perform prediction or classification tasks.
In recent years, some methods have been proposed to pre-train general trajectory embeddings for various downstream tasks. t2vec \cite{li2018deep} adopts a generative auto-encoding-based objective to pre-train a recurrent neural network (RNN) \cite{hochreiter1997long} trajectory encoder. The encoder can then be used for generating latent embedding vectors for trajectories. GM-VSAE \cite{liu2020online} extends the above idea by designing a variational encoder to cast trajectory embeddings into a multidimensional Gaussian space, so that a high-performance online anomalous trajectory detection task can be performed. traj2vec \cite{yao2017trajectory} and TremBR \cite{fu2020trembr} further incorporate some aspects of spatial-temporal information in trajectories into the embeddings by special trajectory encoder designs. These researches demonstrate that the pre-trained trajectory embeddings are indeed beneficial for downstream tasks.
Despite the existing efforts, there are still some challenges lie in pre-training spatial-temporal trajectory embeddings.
\textbf{First, generative self-supervised pretext tasks aim to recover all the details of a trajectory, so that they fail to extract the high-level semantics of trajectories. }
Since trajectories are sequences, existing pre-training trajectory embedding methods mostly design their pretext tasks following the language models in natural language processing (NLP) \cite{dai2015semi}, which uses generative auto-regressive or auto-encoding self-supervised objectives. Specifically, the input trajectory sequence is encoded into a latent representation through a RNN-based encoder, then is recovered to its original form relying on the latent representation. We argue that this type of pretext tasks is not suitable for pre-training trajectory embedding vectors. As shown in Figure \ref{fig:pretext-task}(a), the process of recovering the original input sequence will force the result embedding focus on reflecting all the details contained in a trajectory \cite{oord2018representation}, which violates trajectory embedding learning's goal, i.e., modeling high-level semantics. Generative pretext tasks can also lead the embeddings be sensitive to noise, which is common in real-world trajectory datasets.
By comparison, the contrastive pretext task aims to model the relationship between trajectory samples by contrasting them in the high-level embedding space, as demonstrated in Figure \ref{fig:pretext-task}(b). Theoretically, it can effectively extract trajectories' high-level travel semantics and is robust to the widely existing noise. Yet, the contrastive learning framework is not implemented in existing pre-training trajectory embedding methods.
\begin{figure}[h]
\centering
\includegraphics[width=0.9\linewidth]{figure/pretext-task.pdf}
\caption{Robustness of generative and contrastive pretext tasks.}
\label{fig:pretext-task}
\end{figure}
\textbf{Second, data augmentation methods that can preserve high-level semantics is crucial for constructing contrastive pretext task. Yet, existing data augmentation methods do not apply well to trajectories.} As shown in Figure \ref{fig:pretext-task}(b), data augmentation can decide the quality of feedback information in contrastive learning. Contrastively pre-training embedding methods in computer vision (CV) \cite{bachman2019learning} and Natural Language Processing (NLP) \cite{giorgi2021declutr} have developed several data augmentation methods to extract high-level information from unlabeled datasets, including adding random noise, cropping, reversing and so on \cite{tian2020contrastive}. Yet, as demonstrated in Figure \ref{fig:existing-augmentation}, when applied to trajectories, these augmentation methods will fundamentally change the high-level information of trajectories, which indicates that they are not suitable for trajectory data. How to design a data augmentation method that can preserve and extract high-level semantics and the spatial-temporal patterns embedded in trajectories is challenging.
\begin{figure}[h]
\centering
\includegraphics[width=0.9\linewidth]{figure/cv-augmentation.pdf}
\caption{Existing data augmentation methods applied to trajectories.}
\label{fig:existing-augmentation}
\end{figure}
\textbf{Third, trajectories contain long-term spatial-temporal correlations, which are not sufficiently modeled.} Trajectories are often long sequences, with complex spatial-temporal correlations. Comprehensively modeling these correlations through the specific design of the trajectory encoder is essential for trajectory embedding methods to gain satisfactory results. As shown in Figure \ref{fig:spatial-temporal-information}, we can consider the spatial-temporal correlations in trajectories from two aspects, i.e., absolute and relative, and both aspects can infer the trajectories' moving patterns and travel purposes. Yet, existing methods fail to explicitly incorporate both the two aspects of spatial-temporal correlations into consideration. traj2vec \cite{yao2017trajectory} and TremBR \cite{fu2020trembr} only incorporate relative spatial-temporal properties by calculating the differences in a set of attributes between consecutive trajectory points. This limits the comprehensiveness of the learned trajectory embeddings.
On the other hand, the most widely adopted RNN-based trajectory encoder often suffers from the vanishing gradient issue \cite{linzen2016assessing}, while the vanilla Transformer encoder \cite{vaswani2017attention} has squared time and memory complexity. Thus, they are not suitable for modeling long-term correlations in trajectories. How to build a comprehensive trajectory encoder with the characteristics of spatial-temporal trajectories in mind is another challenge.
\begin{figure}[h]
\centering
\includegraphics[width=1\linewidth]{figure/absolute-relative-st.pdf}
\caption{Two aspects of spatial-temporal information embedded in trajectories. }
\label{fig:spatial-temporal-information}
\end{figure}
In order to tackle the above-discussed challenges in pre-training trajectory embeddings, we aim to utilize the contrastive learning framework \cite{tian2020contrastive,chen2020simple} to construct a self-supervised model for efficiently pre-training comprehensive trajectory embeddings. In this paper, we propose a novel \textit{Contrastive Spatial-Temporal Trajectory Embedding} (\textbf{CSTTE}) model. To make the embedding model better at extracting high-level travel semantics of trajectories and more robust to noise, we construct our self-supervised pretext task under the contrastive learning framework, coupled with a specially designed data augmentation method for trajectories. We then propose an attention-based trajectory encoder with a spatial-temporal encoding layer and globally-shared attention anchors as the pre-training target. This makes our embedding method capable of modeling long-term spatial-temporal correlations in trajectories, while still being computationally efficient. The main contributions of this paper are summarized as follows:
\begin{itemize}
\item We propose a pre-training spatial-temporal trajectory embedding model CSTTE based on the contrastive learning framework. Its self-supervised pretext task is robust to noise and can extract high-level travel semantics from trajectories.
\item A data augmentation method is designed to preserve the high-level travel semantics embedded in trajectories. It samples sub-trajectories from the trajectory dataset for the embedding method to be trained contrastively.
\item An efficient spatial-temporal trajectory encoder is built to comprehensively and efficiently model the long-term spatial-temporal correlations in trajectories. It can generate latent embeddings for trajectories by incorporating comprehensive information to serve downstream tasks.
\item We apply our pre-training embedding method to two downstream tasks, and conduct experiments on three real-world trajectory datasets. The experimental results prove that the embeddings generated from CSTTE can indeed help downstream models to improve the prediction performance.
\end{itemize}
\section{Related Work}
\subsection{Learning Embeddings for Spatial-Temporal Data Mining}
Embedding learning is a fundamental research topic in neural network data mining, for most neural network models are feature-based, and require the input targets to be represented by latent embedding tensors.
In spatial-temporal data mining, embedding learning is also a crucial question. Spatial-temporal objects, like locations, road networks and trajectories, often contain complex spatial-temporal information, while geographically and dynamically correlated with other objects. How to capture and fuse the above-mentioned spatial-temporal information into embedding learning methods is challenging.
Locations are fundamental units in spatial-temporal datasets, and location embedding techniques are widely used in spatial-temporal data mining models. A straightforward idea is to use an index-fetching embedding layer \cite{kong2018hst,zhao2019go} since locations are often presented by discrete indices such as POIs and road segment IDs. Yet, this kind of embedding is trained through task-specific supervision, cannot properly capture the comprehensive spatial-temporal correlations between locations, and often require a large-scale labeled dataset for optimum performance.
To tackle the above problems, DeepMove \cite{zhou2018deepmove} implements word2vec \cite{Mikolov2013Efficient} on unlabeled movement datasets to pre-train a set of embedding vectors for POIs. POI2Vec \cite{feng2017poi2vec} and TALE \cite{wan2021pre} further incorporates spatial and temporal correlations between locations into the embedding vectors and gain better results. CTLE \cite{lin2021pre} migrates the idea of contextual embedding in language models into location embedding. The above pre-training methods can utilize large-scale unlabeled spatial-temporal datasets to obtain general location embedding vectors, and apply them to a wide variety of tasks, including location recommendation \cite{feng2015personalized,zhou2018atrank} and location classification \cite{yao2018representing,shimizu2020learning}.
Road segments or intersections in road networks can also be seen as locations, yet they contain more complex information due to the topology structure of road networks. Only using index-fetching embedding layers for road network representation will discard its intricate yet important topology information. IRN2Vec \cite{wang2019learning} proceeds random walking on road networks coupled with word2vec to incorporate topology correlations into road segment embedding. Toast \cite{chen2021robust} further utilizes real-world trajectories to fuse network topology with traffic patterns. HRNR \cite{wu2020learning} proposes a hierarchical graph neural network and an auto-encoding pre-training objective to learn comprehensive embedding vectors for road segments. Akin to pre-trained location embeddings, these pre-trained road network embeddings can be applied to various tasks, such as route planning \cite{dai2015personalized,wang2019empowering} and estimated time of arrival \cite{li2018multi,li2019learning}.
In this paper, we focus on pre-training spatial-temporal trajectory embeddings. Compared to location embeddings and road network embeddings which aim to learn information for discrete units in spatial-temporal datasets, trajectory embeddings pay more attention to the high-level semantic information reflected in a series of movements as a whole, including moving patterns and travel purposes. The next section gives a summary of existing works in pre-training trajectory embeddings.
\subsection{Pre-training Trajectory Embedding}
The quality of trajectory embeddings is essential for achieving good performance in trajectory mining tasks. The most widely applied strategy for trajectory embedding is to construct an end-to-end sequence encoder for generating latent representations \cite{kong2018hst,zhao2019go}. Yet, trajectory embedding vectors generated this way often lack generalization, thus hard to migrate to other models or tasks. This strategy can also be unworkable when lacks labeled data, such as in trajectory similarity measurement.
To tackle the above problems, there is a rising interest in pre-training trajectory embeddings with self-supervised learning objectives. t2vec \cite{li2018deep} and GM-VSAE \cite{liu2020online} build an RNN-based trajectory encoder and a generative pre-training objective to infer the probability distribution of trajectories based on sequential correlations of visited locations. Trajectories also contain complex spatial-temporal properties which shouldn't be ignored. TremBR \cite{fu2020trembr} incorporates temporal information by concatenating the travel time of each trajectory point with its latent embedding vector. traj2vec \cite{yao2017trajectory} considers relative spatial-temporal properties by taking the differences in a set of attributes between consecutive trajectory points as the input features. Experimental results show the spatial-temporal information is indeed helpful for generating higher quality trajectory embedding vectors. Yet, considering that there are complex long-term spatial-temporal correlations embedded in trajectories, existing methods are not comprehensive enough.
One of the core components of trajectory embedding methods is the pretext task. Most existing methods design their pretext tasks using generative objectives, including auto-encoding \cite{hinton2006reducing} and auto-regressive \cite{pauls2011faster}. Yet, this type of objective involves reconstructing input features, which will lead the embeddings to reflect all the details contained in trajectories \cite{oord2018representation}. Contrastive learning \cite{tian2020contrastive} in computer vision inspired us to propose a self-supervised learning objective that can better extract the high-level sequential information from trajectories.
\section{Preliminaries}
\begin{definition}
\textbf{Visiting Record.}
In location-based services and GPS record datasets, a person or a vehicle's visit to a certain location is represented by a visiting record $r=(l,t,c_x,c_y)$, which indicates that location $l$ is visited at time $t$. $l$ is a discrete location index indicator, such as a POI index or a grid index. $c_x$ and $c_y$ are the longitude and latitude of the visited location, and $t$ is the visited timestamp.
\end{definition}
\begin{definition}
\textbf{Spatial-Temporal Trajectory.}
The movements of an object during a certain period can be represented by a list of sequential visiting records, which we called a spatial-temporal trajectory. We denote a trajectory as $s=\{r_1, r_2, \dots, r_N\}$, where the visiting records are ordered by their visited time, and $N$ is the length of the trajectory. We denote the set of all trajectories in a dataset as $S$.
\end{definition}
\noindent \textbf{Problem Statement.}
\textit{Pre-training Trajectory Embeddings.} Given a set of spatial-temporal trajectories $S$, we aim to pre-train a trajectory encoder $f$ to generate a embedding vector $\emph{\textbf{e}}_s$ given the trajectory $s$, i.e., $\emph{\textbf{e}}_s=f(s)$. This encoder should be pre-trained with a pretext task and a data augmentation method with no requirement for task-specific labels.
\begin{figure*}[ht]
\centering
\includegraphics[width=.85\linewidth]{figure/framework.pdf}
\caption{The overall framework of our proposed model.}
\label{fig:framework}
\end{figure*}
\section{The CSTTE Model}
\subsection{Overall Contrastive Framework}
The overall framework of our proposed model is demonstrated in Figure \ref{fig:framework}. Following the contrastive learning framework \cite{chen2020simple}, the overall model aims to achieve a self-supervised pretext task. That is, to maximize the similarity of embeddings between trajectory samples that represent similar high-level travel semantics, and to maximize the difference of embeddings between those that represent dissimilar semantics.
To generate trajectory samples that represent similar and dissimilar high-level travel semantics, we design a trajectory data augmentation method to sample sub-trajectories from the original trajectory dataset. Two sub-trajectories sampled from the same trajectory are regarded as the query and the positive sample, respectively; sub-trajectories sampled from other trajectories are defined as the negative samples with regard to the query.
Finally, to effectively and efficiently model the long-term spatial-temporal correlations embedded in trajectories, we build a trajectory encoder to cast the input trajectory record sequences into fixed-length embedding vectors. During self-supervised pre-training, the encoder is fed with the above sub-trajectory samples to calculate embeddings that correspond to these samples. The contrastive learning-based pretext task is then applied to guide the training of the trajectory encoder.
After the pre-training, the result trajectory encoder can be used to generate spatial-temporal trajectories' embeddings for downstream tasks. The following content explains our proposed model in detail.
\subsection{Contrastive Pretext Task} \label{sec:contrastive-pretext-task}
Self-supervised pretext tasks are the core of pre-training embedding methods. To learn high-quality trajectory embeddings, it is essential for the pretext task to extract high-level travel semantics from trajectories. Most existing pre-training trajectory embedding methods follow the language models in NLP, and design their pretext tasks based on generative objectives like auto-encoding and auto-regressive. Yet, generative pretext tasks are not suitable for extracting high-level semantics from trajectories, rather they will lead the learned embeddings to reflect all the details contained in the trajectories, as shown in Figure \ref{fig:pretext-task}(a). To this end, we demonstrate our pretext task, which is based on the contrastive learning framework \cite{chen2020simple}.
To extract high-level travel semantics from trajectories and embed them into the learned embeddings, we aim to design a pretext task that minimizes the distance between trajectory samples that represent similar high-level semantics in the embedding space, and maximizes the distance between those that represent dissimilar high-level semantics, as shown in Figure \ref{fig:framework}.
Formally, we define three types of trajectory samples: the query $s^q$, the positive sample $s^{k_+}$, and the negative samples $s^{k_i}, i\in \{1, 2, \dots, n_{\mathrm{neg}}\}$, where $n_{\mathrm{neg}}$ is the number of negative samples.
Ideally, the positive sample shares similar high-level travel semantics with the query, while the negative samples represent different semantics from the query. Since we aim to model their relationships in the embedding space, a trajectory encoder $f$ is utilized to encode these samples into their corresponding embeddings:
\begin{equation}
\emph{\textbf{q}}=f(s^q), \emph{\textbf{k}}_+=f(s^{k_+}), \emph{\textbf{k}}_i=f(s^{k_i}),
\label{eq:encode-samples}
\end{equation}
where $\emph{\textbf{q}},\emph{\textbf{k}}_+,\emph{\textbf{k}}_i\in \mathbb R^{d}$, $d$ is the dimension of embedding vectors. Next, to capture the relationships between these embeddings and to implement our contrastive pretext task, we adopt the InfoNCE loss \cite{oord2018representation}:
\begin{equation}
\mathcal L=-\log \frac{\exp(\emph{\textbf{q}}^{\top} \emph{\textbf{k}}_+/\tau)}{\sum_\emph{\textbf{k}}\exp(\emph{\textbf{q}}^{\top}\emph{\textbf{k}}/\tau)}, \emph{\textbf{k}}\in \{\emph{\textbf{k}}_+, \emph{\textbf{k}}_1, \emph{\textbf{k}}_2, \dots, \emph{\textbf{k}}_{n_\mathrm{neg}}\},
\end{equation}
where $\tau$ is the temperature hyper-parameter. The InfoNCE loss can be viewed as discriminating positive samples from the set of all samples with regard to the query. In this way, the learned embeddings can reflect the similarity or dissimilarity of trajectory samples' high-level semantics, thus leading the embedding method to extract the semantic information from trajectories.
After the contrastive pre-training, we can utilize the learned trajectory encoder for generating trajectory embeddings as $\emph{\textbf{e}}_s=f(s)$.
To complete the contrastive pretext task, two questions remain:
1. How to create trajectory samples $s^q$, $s^{k_+}$ and $s^{k_i}$, i.e., what data augmentation method to use on the trajectory dataset;
2. How to implement the trajectory encoder $f$.
In the following sections, we will answer these questions by introducing the data augmentation method we used for creating high-level semantic-preserved trajectory samples, and designing our trajectory encoder for modeling long-term spatial-temporal correlations.
\subsection{Trajectory Data Augmentation}
The data augmentation method is one of the basic components of the contrastive learning framework. Trajectory augmentation methods need to preserve the high-level travel semantics of trajectories so that they can be incorporated during pre-training. Yet, as we can see in Figure \ref{fig:existing-augmentation}, popular data augmentation methods will fundamentally change the high-level semantics of trajectories. In this section, we introduce our proposed trajectory data augmentation method that can better suit the scenario of contrastive trajectory embedding learning.
In the real-world, the movement of an object always follows a continuous path, while visiting records in trajectory datasets can be seen as discrete samples of the path \cite{li2018deep}.
The underlying path of a trajectory reflects its high-level travel semantics, regardless of the sample rates.
Thus, multiple sub-trajectories re-sampled from one trajectory correspond to the same underlying path and point to similar high-level semantics, as demonstrated in Figure \ref{fig:underlining-path}.
Based on the above analysis, we design our trajectory data augmentation method based on the 2-hop sampling. Formally, given a trajectory $s=\{r_1, r_2, \dots, r_N\}$, we generate the query $s^q$ and the corresponding positive sample $s^{k_+}$ by:
\begin{equation}
\begin{aligned}
s^q=&\{r_1, r_3, r_5, \dots, r_{2\lfloor N/2 \rfloor +1}\}, \\
s^{k_+}=&\{r_2, r_4, r_6, \dots, r_{2\lfloor N/2 \rfloor}\}.
\end{aligned}
\end{equation}
\begin{figure}[h]
\centering
\includegraphics[width=1.0\linewidth]{figure/sampled-trajectory.pdf}
\caption{Two re-sampled sub-trajectories correspond to the sample underlying path.}
\label{fig:underlining-path}
\end{figure}
While the negative samples $s^{k_i}$ corresponds to the query $s^q$ is randomly selected from the queries and the positive samples corresponds to the other trajectories $s'$ in the batch, denoted as:
\begin{equation}
s^{k_i}\in \{s'^{q}, s'^{k_+}~\mathrm{where}~s'\in S, s'\neq s\}.
\end{equation}
The idea behind the above augmentation method is that the sub-trajectories sampled from the same trajectory represent similar high-level travel semantics, while the ones sampled from different trajectories represent dissimilar semantics. Coupling this high-level semantics-preserved augmentation method with the contrastive pretext task introduced in Section \ref{sec:contrastive-pretext-task}, the learned embeddings are able to extract and incorporate the semantic information embedded in the trajectories.
\begin{figure*}[t]
\centering
\includegraphics[width=1\linewidth]{figure/trajectory-encoder.pdf}
\caption{The architecture of our proposed efficient spatial-temporal trajectory encoder and its components.}
\label{fig:trajectory-encoder}
\end{figure*}
\subsection{Efficient Spatial-Temporal Trajectory Encoder}
In pre-training trajectory embedding methods, trajectory encoders are built to model the embedded spatial-temporal correlations, and to cast the input trajectory sequences into their corresponding embeddings. Trajectories are often long sequences, and possess complex spatial-temporal correlations that can be viewed from absolute and relative aspects. These characteristics impose challenges to trajectory encoder design. To tackle these challenges, we build a trajectory encoder $f$ that can efficiently and comprehensively model long-term spatial-temporal correlations in trajectories. As shown in Figure \ref{fig:trajectory-encoder}(b), our proposed trajectory encoder mainly consists of two types of layers: the spatial-temporal encoding layer, and the induced attentive layer. The following sections present its construction in detail.
\subsubsection{Spatial-Temporal Encoding Layer}
As demonstrated in Figure \ref{fig:spatial-temporal-information}, spatial-temporal information embedded in trajectories can be viewed from two aspects: absolute and relative. Absolute spatial-temporal information, including the discrete indices and geographical positions of visited locations, plus the time periods of trajectory records, can reveal trajectories' corresponding travel purposes. Relative spatial-temporal information, including the time difference and relative position between trajectory records, indicates the movement direction and speed of a user or a vehicle, thus representing moving patterns. These two aspects of spatial-temporal information are both complicated and essential for trajectory embedding. Directly feeding raw features into the trajectory encoder is not the optimal choice, since spatial-temporal features don't scale linearly like some physical variables. We propose to more properly incorporate these raw features through a specially designed spatial-temporal encoding layer with learnable parameters, and encode trajectories into their corresponding input latent sequences. In this way, both absolute and relative aspects of spatial-temporal information in trajectories can be properly incorporated.
\paratitle{Encoding Absolute Spatial-temporal Information.}
The absolute spatial-temporal information of trajectories includes the timestamp of records, the indices and the geographical coordinates of visited locations. In other words, we aim to encode each record $r=(l,t,c_x,c_y)$ into a latent vector $\emph{\textbf{z}}_r\in \mathbb R^{d_L}$ to extract the absolute spatial-temporal information embedded in the record. $d_L$ denotes the size of the input latent vector.
For the discrete location index $l$, we implement an index fetching embedding module. Suppose there is a total of $N_l$ locations in the dataset. We initialize an embedding matrix $\emph{\textbf{E}}\in \mathbb R^{N_l\times d_L}$, and fetch its $l$-th row vector $\emph{\textbf{E}}_l$ as the encoding vector of location index $l$:
\begin{equation}
\emph{\textbf{z}}_l=\emph{\textbf{E}}_l.
\end{equation}
In this way, every location is represented by a learnable latent vector.
For the three continuous features $t$, $c_x$ and $c_y$, directly regarding them as input features will lose information, since they don't scale linearly in the feature space. Akin to the location index, we aim to cast these features into learnable latent vectors, while also emphasizing their characteristics, such as periodicity. Inspired by the positional encoding introduced in the Transformer \cite{vaswani2017attention} and the temporal encoding proposed in some recent works \cite{DBLP:conf/iclr/XuRKKA20,lin2021pre}, we construct a continuous encoding module as:
\begin{equation}
\varPsi(v) = [\cos(\omega_1 v),\sin(\omega_1 v),\dots,\cos(\omega_{d_v/2} v),\sin(\omega_{d_v/2} v)],
\label{eq:numerical-encoding-module}
\end{equation}
where $\varPsi$ denotes the continuous encoding module, $v$ denotes the input numerical feature, ${\{\omega_1,\omega_2,\dots,\omega_{d_v/2}\}}$ is a set of learnable parameters, $d_v$ is the dimension of the encoding vector. By using trigonometric functions, we can preserve the periodicity of space and time. By using learnable parameters, the information embedded in the continuous features can be learned by the model during training. The three continuous features' corresponding encoding vectors are calculated as:
\begin{equation}
\begin{aligned}
\emph{\textbf{z}}_t=&\varPsi(t)\in \mathbb R^{d_L},\\
\emph{\textbf{z}}_{c_x}=&\varPsi(c_x)\in \mathbb R^{d_L/2},\\
\emph{\textbf{z}}_{c_x}=&\varPsi(c_x)\in \mathbb R^{d_L/2}.
\end{aligned}
\end{equation}
Finally, we construct the spatial-temporal encoding layer by fusing all encoding vectors to gain the input latent vector of record $r$:
\begin{equation}
\emph{\textbf{z}}_r = \mathrm{ST}(r)
= \emph{\textbf{z}}_l + \emph{\textbf{z}}_t + [\emph{\textbf{z}}_{c_x} || \emph{\textbf{z}}_{c_y}],
\label{eq:fusing-encoding-vectors}
\end{equation}
where $\mathrm{ST}$ denotes the spatial-temporal encoding layer, $||$ denotes concatenation along feature dimension. We illustrate the structure of $\mathrm{ST}$ in Figure \ref{fig:trajectory-encoder}(a).
\paratitle{Encoding Relative Spatial-temporal Information.}
To extract the relative spatial-temporal information, we want the model to formulate and parameterize the difference or the distance between the continuous features of two records. Due to one characteristic of the trigonometric functions in $\varPsi$, there is no need to explicitly encode the difference value of two features into a latent vector. To explain, we give the dot-product of two encoding vectors $\varPsi(v)$ and $\varPsi(v+\delta)$ as:
\begin{equation}
\varPsi(v)\cdot\varPsi(v+\delta)
= \cos(\omega_1\delta) + \cos(\omega_2\delta) + \dots + \cos(\omega_{d_v/2}\delta),
\label{eq:dot-product-of-numerical-encoding}
\end{equation}
which means the distance (measured by dot-product) between $\varPsi(v)$ and $\varPsi(v+\delta)$ is only dependent on the set of learnable parameters and the difference $\delta$, not on the offset $v$. In other words, the relative information is only related to the difference between two features, and can be learned by the model during training. Since we use dot-product attention in our trajectory encoder, as denoted later in Equation \ref{eq:summarizer-layer}, the dot-product between the encoding vectors of two records' timestamps or coordinates can be easily incorporated by the encoder.
Thus, information embedded in the relative differences between spatial-temporal features can be captured and learned.
\paratitle{Forming Input Latent Sequence.}
Finally, the input latent sequence $\emph{\textbf{Z}}_s$ of a trajectory $s$ is formed by the input latent vectors of every record in the trajectory. Formally,
\begin{equation}
\begin{split}
\mathrm{Given}\ s &= \{r_1,r_2,\dots,r_N\}, \\
\emph{\textbf{Z}}_s &= \mathrm{ST}(s)\\
&=\mathrm{ST}(\{r_1, r_2, \dots, r_N\}) \\
&= \{\emph{\textbf{z}}_{r_1}, \emph{\textbf{z}}_{r_2}, \dots, \emph{\textbf{z}}_{r_N}\}.
\end{split}
\end{equation}
\subsubsection{Induced Attentive Layer}
Trajectories contain long-term correlations, which means that all visiting records in a trajectory can be useful for revealing the trajectory's high-level semantic information, regardless of their position. For example, starting a trajectory from hotels or ending one at scenic spots can both indicate that the corresponding mobile user is traveling.
The Recurrent Neural Network (RNN) \cite{hochreiter1997long} used in existing trajectory embedding methods suffers from the vanishing gradient issue \cite{linzen2016assessing}, thus not powerful enough for modeling long-term correlations.
On the other hand, different visiting records have different importance with regard to the trajectory. For instance, in a taxi trajectory, the staying points contain more semantic information than the passing-by points.
To solve the vanishing gradient issue of RNNs and to dynamically model the importance of different visiting records, a straightforward idea is to construct the trajectory encoder based on self-attention \cite{vaswani2017attention} and pooling layers. Yet, self-attention will lead to square time and memory complexity, which hinders the model's scalability on long trajectories. Plus, max and mean pooling layers are unable to dynamically calculate an importance score for each item in a sequence.
Inspired by recent advances in sequential models in neural language processing \cite{lee2019set}, we utilize the attention mechanism for better incorporation of long-term correlations, and use globally-shared anchor sequences for attention weight calculation to reduce computational complexity. We call this layer the induced attentive layer for its calculation process can be viewed as inducing the input sequence into the anchor sequence through the attention mechanism. Formally, we implement each induced attentive layer based on attention and feed-forward networks:
\begin{equation}
\begin{aligned}
\mathrm{IA}(\emph{\textbf{X}},\emph{\textbf{A}}) &= \mathrm{Norm}(\emph{\textbf{H}} + \mathrm{FFN}(\emph{\textbf{H}})), \\
\mathrm{where}\ \emph{\textbf{H}} &= \mathrm{Norm}(\emph{\textbf{A}} + \mathrm{Att}(\emph{\textbf{A}},\emph{\textbf{X}},\emph{\textbf{X}})),
\end{aligned}
\label{eq:summarizer-layer}
\end{equation}
where $\mathrm{IA}$ denotes one induced attentive layer. $\mathrm{Norm}$, $\mathrm{FFN}$, $\mathrm{Att}$ represents the layer normalization \cite{ba2016layer}, the feed-forward network and the multi-head dot-product attention, respectively. $\emph{\textbf{X}}\in \mathbb R^{N_X\times d_L}$ is the input sequence for this layer, $\emph{\textbf{A}} \in \mathbb R^{N_A\times d_L}$ represents the anchor sequence with length $N_A$, and can be viewed as a set of learnable parameters of this layer. During attention calculation, $\emph{\textbf{A}}$ is regarded as the query, $\emph{\textbf{X}}$ is regarded as the key and value.
In conclusion, Equation \ref{eq:summarizer-layer} can be viewed as inducing the input sequence $\emph{\textbf{X}}$ into the anchor sequence $\emph{\textbf{A}}$. The structure of $\mathrm{IA}$ can be seen in Figure \ref{fig:trajectory-encoder}(c).
Because the attention scores of all records in one input sequence are synchronously calculated, the induced attentive layer can avoid the vanishing gradient issue \cite{linzen2016assessing} found in RNNs, thus better at modeling the long-term correlations of trajectories.
On the other hand, sharing the anchor sequences across the whole dataset not only can significantly reduce the parameter numbers, but also is useful to learn the most prominent patterns among all the trajectories.
\subsubsection{Stacking Multiple Layers}
Finally, our trajectory encoder is constructed by stacking multiple induced attentive layers on top of one spatial-temporal encoding layer, as shown in Figure \ref{fig:trajectory-encoder}(b). We give an example of a trajectory encoder with two induced attentive layers as:
\begin{equation}
\begin{aligned}
f(s) &= \mathrm{cat}(\mathrm{IA}_2(
\mathrm{IA}_1(\mathrm{ST}(s),\emph{\textbf{A}}_1),
\emph{\textbf{A}}_2)), \\
&=\mathrm{cat}(\mathrm{IA}_2(
\mathrm{IA}_1(\emph{\textbf{Z}}_s,\emph{\textbf{A}}_1),
\emph{\textbf{A}}_2)),
\end{aligned}
\label{eq:summarizer}
\end{equation}
where $\mathrm{IA}_1$ and $\mathrm{IA}_2$ denotes the first and the second induced attentive layers, $\emph{\textbf{A}}_1 \in \mathbb R^{N_{A_1}\times d_L}$, $\emph{\textbf{A}}_2 \in \mathbb R^{N_{A_2}\times d_L}$ are the anchor sequences of two induced attentive layers, $f$ denotes our proposed efficient spatial-temporal trajectory encoder introduced in Equation \ref{eq:encode-samples}, $\emph{\textbf{Z}}_s$ is the input latent sequence of trajectory $s$. $\mathrm{cat}$ denotes concatenation along the feature dimension. In other words, we concatenate the output from $\mathrm{IA}_2$ to get the result trajectory embedding vector. It is clear that the size of the result embedding vector is related to the length of the anchor sequence in $\mathrm{IA}_2$. We denote the size as $d_O$, and $d_O=N_{A_2} * d_L$.
By compressing the input trajectories into corresponding embeddings utilizing the attention mechanism, the importance of each visiting record in a trajectory is efficiently and adaptively estimated. Meanwhile, the spatial-temporal encoding layer with learnable parameters enables our encoder to comprehensively model the spatial-temporal information embedded in trajectories.
\section{Experiments}
In order to evaluate the quality of trajectory embedding vectors generated by our model, we incorporate these vectors into two downstream trajectory mining tasks, and compare the results with other methods on three real-world datasets.
\subsection{Datasets}
We conduct our experiments on three real-world datasets. Two of which contain taxi trajectories collected from the center areas of Chengdu and Xian by Didi, denoted as Taxi-CD and Taxi-XA. These trajectories are reported by GPS equipments on taxes during journeys. The other one contains mobile signaling data collected from Shenyang, denoted as Mobile-SY. It records mobile phone users' switching events between telecommunication towers.
For taxi trajectories, we re-sample the original data to modify the sample rate to roughly 1 minute, and filter out trajectories that contain less than 20 records. We then choose the trajectories that started from November 1st to 7th, 2018. Finally, we use a squared grid with 250 meter-long edges to assign each coordinate point in the trajectories with a discrete location index. For mobile signaling data, we regard telecommunication towers as locations, and a mobile user's switch records within one day as one trajectory. We then remove records with staying time below 1 minute, and trajectories with less than 20 records. The statistics of datasets after preprocessing are shown in Table \ref{table:dataset-statistics}.
\begin{table}[h]
\centering
\caption{Statistics of datasets.}
\scalebox{1}{
\begin{tabular}{ccccc}
\toprule
Dataset & \#Trajectories & \#Locations & \#Records & Time span \\
\midrule
Taxi-CD & 44,551 & 1,443 & 1,555,042 & 7 days \\
Taxi-XA & 70,222 & 1,469 & 2,597,066 & 7 days \\
Mobile-SY & 64,328 & 7,456 & 1,338,838 & 11 days \\
\bottomrule
\end{tabular}
}
\label{table:dataset-statistics}
\end{table}
\subsection{Comparison Methods}
To prove the superiority of our proposed model, we include two end-to-end embedding methods that are widely adopted by trajectory mining models, and some state-of-the-art pre-training trajectory embedding methods as baselines.
\begin{itemize}
\item \textbf{Mean}: a permutation equivariant embedding method, which applies a mean pooling operation on a trajectory's sequence of input features, followed by a linear transform layer to generate its latent embedding vector.
\item \textbf{RNN} \cite{hochreiter1997long}: aggregates a trajectory's sequence of input features using a recurrent neural network, and regards the output hidden state as the latent embedding vector of the trajectory.
\item \textbf{traj2vec} \cite{yao2017trajectory}: constructs feature sequences by calculating the difference in spatial-temporal attributes between consecutive points, and applies an RNN encoder to aggregate them.
\item \textbf{t2vec} \cite{li2018deep}: pre-trains an RNN trajectory encoder based on de-noise auto-encoding, aims to recover the underlying route of trajectories.
\item \textbf{GM-VSAE} \cite{liu2020online}: casts trajectories into a multi-dimension Gaussian space by coupling an RNN trajectory encoder with variational fully-connections.
\item \textbf{TremBR} \cite{fu2020trembr}: incorporates temporal information by concatenating the travel time with location embedding vectors to form the input features.
\end{itemize}
\begin{table*}[!t]
\centering
\caption{Downstream prediction performance comparison of different approaches on similar trajectory search.}
\scalebox{1}{
\begin{tabular}{c|c|ccccc}
\toprule
\multicolumn{2}{c|}{Metric} &
\multirow{2}{*}{Acc@1 (\%)} &
\multirow{2}{*}{Acc@5 (\%)} &
\multirow{2}{*}{Acc@10 (\%)} &
\multirow{2}{*}{Acc@20 (\%)} &
\multirow{2}{*}{macro-F1 (\%)}
\\ \cline{1-2}
Dataset &
Embedding & & & & &
\\
\midrule
& DTW & 11.947 & 30.089 & 43.805 & 56.637 & 8.563 \\
&
Mean & 10.009$\pm$0.19 & 33.281$\pm$1.27 & 43.559$\pm$1.73 & 57.361$\pm$1.76 & 8.700$\pm$0.37 \\
&
RNN & 19.962$\pm$0.24 & 46.589$\pm$0.13 & 58.977$\pm$0.51 & 70.994$\pm$1.06 & 15.416$\pm$0.23 \\
Taxi-CD &
traj2vec & 26.638$\pm$1.52 & 56.003$\pm$0.59 & 69.199$\pm$0.02 & 82.866$\pm$0.43 & 17.855$\pm$1.08 \\
&
t2vec & 30.969$\pm$0.63 & 60.806$\pm$0.46 & 74.158$\pm$1.67 & 85.435$\pm$2.03 & 24.000$\pm$0.68 \\
&
GM-VSAE & 28.097$\pm$1.98 & 60.368$\pm$3.41 & 74.237$\pm$2.60 & 86.378$\pm$1.16 & 21.494$\pm$2.13 \\
&
TremBR & \underline{34.706$\pm$0.43} & \underline{65.934$\pm$1.49} & \underline{78.714$\pm$1.16} & \underline{88.936$\pm$0.92} & \underline{25.818$\pm$0.32} \\
&
\textbf{CSTTE} & \textbf{40.911}$\pm$\textbf{1.87} & \textbf{72.969}$\pm$\textbf{0.43} & \textbf{83.802}$\pm$\textbf{0.66} & \textbf{92.157}$\pm$\textbf{0.36} & \textbf{33.723}$\pm$\textbf{1.93} \\
\midrule
& DTW & 24.684 & 50.633 & 63.291 & 77.215 & 19.421 \\
&
Mean & 13.570$\pm$0.66 & 33.106$\pm$1.61 & 41.250$\pm$2.84 & 52.442$\pm$3.16 & 10.451$\pm$0.44 \\
&
RNN & 35.256$\pm$0.52 & 64.161$\pm$2.64 & 76.164$\pm$2.38 & 86.644$\pm$2.42 & 26.595$\pm$0.76 \\
Taxi-XA &
traj2vec & 42.589$\pm$1.91 & 73.323$\pm$2.74 & 82.629$\pm$2.36 & 89.278$\pm$1.96 & 35.893$\pm$1.11 \\
&
t2vec & 51.004$\pm$1.73 & 81.966$\pm$0.69 & 90.196$\pm$0.63 & 94.511$\pm$1.20 & 43.699$\pm$2.05 \\
&
GM-VSAE & 48.384$\pm$1.95 & 76.022$\pm$2.47 & 84.451$\pm$3.78 & 90.545$\pm$2.37 & 41.665$\pm$1.40 \\
&
TremBR & \underline{54.948$\pm$0.52} & \underline{85.348$\pm$2.64} & \underline{92.852$\pm$2.38} & \underline{96.839$\pm$2.42} & \underline{47.660$\pm$0.26} \\
&
\textbf{CSTTE} & \textbf{63.391}$\pm$\textbf{2.17} & \textbf{90.645}$\pm$\textbf{0.90} & \textbf{95.093}$\pm$\textbf{0.56} & \textbf{97.314}$\pm$\textbf{0.86} & \textbf{56.195}$\pm$\textbf{2.66} \\
%
\midrule
& DTW & 5.185 & 18.221 & 23.644 & 31.887 & 3.799 \\
&
Mean & 4.055$\pm$0.17 & 14.625$\pm$2.39 & 23.755$\pm$1.84 & 33.102$\pm$2.36 & 2.291$\pm$0.15 \\
&
RNN & 5.106$\pm$0.33 & 15.068$\pm$1.32 & 23.988$\pm$1.77 & 33.546$\pm$2.29 & 2.848$\pm$0.66 \\
Mobile-SY &
traj2vec & 7.145$\pm$0.40 & 20.416$\pm$1.60 & 29.631$\pm$2.94 & 40.068$\pm$4.47 & 4.437$\pm$0.32 \\
&
t2vec & 9.916$\pm$1.45 & 24.175$\pm$2.67 & 32.628$\pm$2.45 & 42.762$\pm$1.46 & 6.497$\pm$1.22\\
&
GM-VSAE & 11.800$\pm$0.36 & 27.335$\pm$0.31 & 36.379$\pm$0.77 & 46.404$\pm$0.43 & 8.258$\pm$0.95 \\
&
TremBR & \underline{12.842$\pm$0.48} & \underline{30.542$\pm$2.22} & \textbf{41.477}$\pm$\textbf{1.05} & \textbf{52.249}$\pm$\textbf{1.82} & \underline{9.027$\pm$0.18} \\
&
\textbf{CSTTE} & \textbf{13.434}$\pm$\textbf{1.32} & \textbf{30.612}$\pm$\textbf{1.75} & \underline{40.528$\pm$2.50} & \underline{50.615$\pm$2.63} & \textbf{9.396}$\pm$\textbf{1.23} \\
%
\bottomrule
\end{tabular}
}
\label{tab:similar-trajectory-search-result}
\end{table*}
\begin{table*}[!t]
\centering
\caption{Downstream prediction performance comparison of different approaches on destination prediction.}
\scalebox{1}{
\begin{tabular}{c|c|ccccc}
\toprule
\multicolumn{2}{c|}{Metric} &
\multirow{2}{*}{Acc@1 (\%)} &
\multirow{2}{*}{Acc@5 (\%)} &
\multirow{2}{*}{Acc@10 (\%)} &
\multirow{2}{*}{Acc@20 (\%)} &
\multirow{2}{*}{macro-F1 (\%)}
\\ \cline{1-2}
Dataset &
Embedding & & & & &
\\
\midrule
& MC & 11.491 & 26.509 & 37.815 & 43.663 & 0.876 \\
&
Mean &
5.364$\pm$0.25 & 17.527$\pm$0.96 & 26.728$\pm$0.83 & 37.377$\pm$0.81 & 0.581$\pm$0.08 \\
&
RNN &
15.215$\pm$0.57 & 32.148$\pm$0.86 & 41.427$\pm$0.59 & 51.346$\pm$0.78 & 0.888$\pm$0.06 \\
Taxi-CD &
traj2vec &
10.660$\pm$0.40 & 23.676$\pm$0.75 & 31.486$\pm$0.56 & 40.126$\pm$0.54 & 0.649$\pm$0.04 \\
&
t2vec &
17.549$\pm$0.92 & 34.089$\pm$1.30 & 42.908$\pm$1.41 & 52.962$\pm$2.17 & 1.273$\pm$0.23 \\
&
GM-VSAE &
16.786$\pm$0.71 & 33.707$\pm$1.03 & 41.809$\pm$1.05 & 51.167$\pm$1.22 & 0.973$\pm$0.06 \\
&
TremBR &
\underline{20.007$\pm$0.40} & \underline{37.264$\pm$1.13} & \underline{45.433$\pm$1.16} & \underline{54.275$\pm$1.51} & \underline{1.407$\pm$0.07} \\
&
\textbf{CSTTE} &
\textbf{22.913}$\pm$\textbf{1.24} & \textbf{41.394}$\pm$\textbf{0.90} & \textbf{50.707}$\pm$\textbf{0.36} & \textbf{60.536}$\pm$\textbf{0.75} & \textbf{1.714}$\pm$\textbf{0.15} \\
\midrule
& MC & 13.031 & 31.287 & 36.178 & 49.715 & 0.879 \\
&
Mean &
6.386$\pm$0.39 & 19.806$\pm$0.60 & 29.075$\pm$0.57 & 38.157$\pm$1.25 & 0.651$\pm$0.07 \\
&
RNN &
16.203$\pm$0.50 & 33.348$\pm$0.94 & 40.977$\pm$0.88 & 49.964$\pm$0.80 & 0.889$\pm$0.12 \\
Taxi-XA &
traj2vec &
11.754$\pm$0.78 & 26.242$\pm$1.22 & 34.700$\pm$1.32 & 43.678$\pm$1.60 & 0.820$\pm$0.08 \\
&
t2vec &
18.467$\pm$0.50 & 38.744$\pm$1.34 & 43.728$\pm$1.94 & 53.040$\pm$2.17 & 1.117$\pm$0.13 \\
&
GM-VSAE &
18.425$\pm$0.92 & 35.241$\pm$1.27 & 41.172$\pm$1.83 & 49.722$\pm$1.94 & 0.957$\pm$0.06 \\
&
TremBR &
\underline{20.953$\pm$0.21} & \underline{39.307$\pm$0.53} & \underline{46.853$\pm$1.62} & \underline{55.204$\pm$1.87} & \underline{1.326$\pm$0.20} \\
&
\textbf{CSTTE} &
\textbf{24.050}$\pm$\textbf{0.38} & \textbf{43.037}$\pm$\textbf{0.31} & \textbf{51.260}$\pm$\textbf{0.16} & \textbf{59.853}$\pm$\textbf{0.11} & \textbf{1.430}$\pm$\textbf{0.10} \\
%
\midrule
& MC & 3.0044 & 14.489 & 22.351 & 35.181 & 1.246 \\
&
Mean &
1.276$\pm$0.09 & 5.666$\pm$0.15 & 10.048$\pm$0.17 & 16.462$\pm$0.36 & 1.127$\pm$0.06 \\
&
RNN &
4.125$\pm$0.22 & 15.894$\pm$0.08 & 24.782$\pm$0.94 & 35.788$\pm$0.32 & 1.498$\pm$0.10 \\
Mobile-SY &
traj2vec &
3.923$\pm$0.13 & 14.306$\pm$0.15 & 22.408$\pm$0.65 & 33.562$\pm$0.57 & 1.571$\pm$0.08 \\
&
t2vec &
4.732$\pm$0.28 & 15.707$\pm$0.96 & 24.408$\pm$1.10 & 34.760$\pm$1.49 & 1.792$\pm$0.29 \\
&
GM-VSAE &
\underline{5.168$\pm$0.20} & 17.341$\pm$1.02 & 26.728$\pm$1.54 & 39.072$\pm$1.86 & \underline{2.103$\pm$0.24} \\
&
TremBR &
5.164$\pm$0.41 & \underline{19.100$\pm$1.29} & \underline{28.502$\pm$1.02} & \underline{41.687$\pm$0.75} & 1.972$\pm$0.36 \\
&
\textbf{CSTTE} &
\textbf{6.157}$\pm$\textbf{0.21} & \textbf{21.069}$\pm$\textbf{0.19} & \textbf{31.522}$\pm$\textbf{0.31} & \textbf{43.501}$\pm$\textbf{0.39} & \textbf{2.344}$\pm$\textbf{0.14} \\
%
\bottomrule
\end{tabular}
}
\label{tab:destination-prediction-result}
\end{table*}
\subsection{Downstream Trajectory Mining Tasks}
We evaluate the performance of trajectory embedding methods through two downstream mining tasks: similar trajectory search and destination prediction.
\subsubsection{Similar Trajectory Search} We aim to rank similar trajectories by utilizing their distances in the latent embedding space. Due to the lack of ground truth, we implement the strategy used in TremBR \cite{fu2020trembr}. Specifically, for each trajectory $s$ in the dataset $S$, we take odd-numbered and even-numbered points to create two sub-trajectories $s^{(a)}$ and $s^{(b)}$, thus forming two trajectory sets $S^{(a)}$ and $S^{(b)}$. While performing the task, given one odd-sampled trajectory $s_i^{(a)} \in S^{(a)}$, we rank even-sampled trajectories $s_j^{(b)} \in S^{(b)}$ in terms of their embedding vectors' dot-product value with $s_i^{(a)}$. Technically, the corresponding trajectory $s_i^{(b)}$ should have the highest dot-product value, and be ranked at the top. For this task, we include a classic sequence similarity measurement method \textbf{DTW} (Dynamic Time Warping) \cite{muller2007dynamic} as an additional baseline.
\subsubsection{Destination Prediction} We intend to predict the destination for each trajectory given its embedding vector. The embedding methods are fed with the target trajectory excluding its last record to generate an embedding vector, which is then brought into a fully connected network to predict the location index of the last record. For this task, we include a classic sequential correlation modeling method \textbf{MC} (Markov Chain) \cite{brooks1998markov} as an additional baseline.
\subsection{Settings}
For all datasets, we sort trajectories by their starting time, and split the whole trajectory set into the training, evaluation and testing sets by 8:1:1. Both embedding models and downstream models are trained with the training set, early-stopped on the evaluation set, and calculated final metrics on the testing set. We choose Top-N accuracy (i.e., Acc@$N$, $N\in [1, 5, 10, 20]$) and macro-F1 as the evaluation metrics for both tasks.
We implement all models using PyTorch \cite{paszke2019pytorch}.
\footnote{The code and datasets are attached in the supplementary material, and will be published on GitHub after the review.}
The size of input latent vector $d_L$ is set to 64, the target dimension of the result trajectory embedding vectors $d_O$ and the hidden size of feed-forward networks are set to 128. For our proposed model, we construct the encoder using 1 encoder layer, and set the length of anchor sequence $N_{A_1}$ to 2, the number of attention heads $N_H$ to 8. The number of negative samples $n_\mathrm{neg}$ is set to 2, and the temperature $\tau$ is set to 0.07. During optimization, we choose the Adam optimizer and an initial learning rate of 0.001 across the board.
All experiments were run on a GPU cluster with 4 workers, where each worker equipped with one Intel Xeon Silver 4210 CPU, and 4 NVIDIA RTX2080Ti GPUs.
\subsection{Experimental Results}
\subsubsection{Overall Performance}
Table \ref{tab:similar-trajectory-search-result} and Table \ref{tab:destination-prediction-result} demonstrates the performance comparison of different approaches for similar trajectory search and destination prediction, respectively. Our proposed method consistently shows its performance superiority over other trajectory embedding methods except on the similar trajectory search task, Mobile-SY dataset.
The two classic machine learning methods, DTW and MC specifically solve problems through feature engineering and kernel designs. Compared to embedding methods that are widely used in deep learning, they are struggling to model complex spatial-temporal features in trajectories. Yet, we can see that they can still beat the permutation equivalent embedding method Mean, for they take sequential correlation into consideration.
The two end-to-end embedding methods, Mean and RNN only trained on specific prediction objectives, which makes it hard to share embedding vectors between different tasks, and can lead to over-fitting on small datasets.
By comparison, the two RNN-based pre-training methods t2vec and GM-VSAE show performance superiority, which proves the effectiveness of the self-supervised pre-training strategy.
Incorporating spatial-temporal information is essential for generating high-quality trajectory embeddings. t2vec and GM-VSAE only implement index-fetching embedding layers to represent location indices, thus failing to incorporate temporal information. TremBR takes relative temporal information into consideration on top of location indices. traj2vec constructs relative features by calculating the difference in spatial-temporal attributes between consecutive visiting records in trajectories. Yet, none of the aforementioned trajectory embedding methods explicitly incorporate both relative and absolute aspects of spatial-temporal information, which will limit their comprehensiveness. By contrast, the spatial-temporal encoding layer in our proposed CSTTE method can extract relative and absolute aspects of spatial-temporal information from trajectories, and fuses them into the trajectory encoder.
In addition, we propose an efficient spatial-temporal trajectory encoder to dynamically summarize trajectories into their embedding vectors, and train the encoder under the contrastive learning framework. Compared to the RNN encoder and generative pretext task used in the baseline trajectory embedding methods, our method is more capable of digging high-level travel semantics and long-term correlations from trajectories. As we can see from Table \ref{tab:similar-trajectory-search-result} and Table \ref{tab:destination-prediction-result}, these designs result in higher quality trajectory embedding vectors, thus helping downstream prediction models to achieve better performance.
\subsubsection{Influence of Contrastive Pretext Task and Spatial-Temporal Encoding}
To investigate the effectiveness of the contrastive pretext task, and the modules of our proposed spatial-temporal encoding layer in the trajectory encoder, we design five variants of the model:
\begin{enumerate}
\item Generative: uses a mirrored trajectory decoder to train the encoder in a generative auto-encoding manner.
\item -Discrete: removes the index fetching embedding module $\emph{\textbf{E}}$. In other words, removes the encoding of discrete location indices, only uses the encoding of continuous features $\emph{\textbf{z}}_t$, $\emph{\textbf{z}}_{c_x}$ and $\emph{\textbf{z}}_{c_y}$.
\item -Continuous: removes the continuous encoding module $\varPsi$, only uses the encoding of discrete location index $\emph{\textbf{z}}_l$, and the original positional encoding from the vanilla Transformer \cite{vaswani2017attention}.
\item -Time: removes the encoding vectors for the visited time $\varPsi(t)$ in the continuous encoding module.
\item -Coordinate: removes the encoding vectors for the coordinate $\varPsi(c_x), \varPsi(c_y)$ in the continuous encoding module.
\end{enumerate}
We compare these variants with the full model on all three datasets, destination prediction task. As Figure \ref{fig:component-analysis} illustrates, the CSTTE model will receive a performance penalty if the encoder is trained with the generative pretext task rather than the contrastive one. This proves the effectiveness of the contrastive learning framework. As demonstrated in Figure \ref{fig:pretext-task}, generative pretext task can lead the learned embeddings to be sensitive to noise, while contrastive one focuses on extracting high-level semantics and is more robust.
The spatial-temporal encoding layer in the trajectory encoder incorporates both absolute and relative aspects of spatial and temporal information into the model, by encoding the discrete location indices and three continuous features of trajectory records into their latent vectors. We can see from Figure \ref{fig:component-analysis} that both discrete location indices and continuous features are essential for comprehensively describing the spatial-temporal properties of trajectories. The spatial-temporal encoding layer in our final model combines these features to gain the optimum results.
\begin{figure}[!h]
\centering
\includegraphics[width=1\linewidth]{figure/component-analysis.pdf}
\caption{Comparison of the generative pretext task variant and four spatial-temporal encoding variants with the full model on three datasets, destination prediction task.}
\label{fig:component-analysis}
\end{figure}
\subsubsection{Influence of Trajectory Data Augmentation Methods}
We further investigate the effectiveness of the trajectory data augmentation methods used in generating samples for the contrastive pretext task. We implement four sequence sampling methods that are frequently used \cite{giorgi2021declutr}:
\begin{enumerate}
\item Random: for each sample, select a portion of the trajectory by randomly deciding whether to choose each trajectory record.
\item Adjacent: select two sub-trajectories that are adjacent, yet not overlapping with each other.
\item Overlap: select two sub-trajectories that are overlapping with each other, yet neither is totally subsumed by the other.
\item Subsume: select two sub-trajectories where the long one subsumes the short one.
\end{enumerate}
We compare these methods with the augmentation method we used in CSTTE (denoted as 2-Hop) on all three datasets, destination prediction task. As we can see in Figure \ref{fig:sampler-analysis}, the 2-Hop method constantly gains the best result compared with other augmentation methods. This shows that choosing an optimum data augmentation method is important under the contrastive learning framework, and our proposed augmentation method is effective on trajectories.
\begin{figure}[!h]
\centering
\includegraphics[width=1\linewidth]{figure/sampler-analysis.pdf}
\caption{Comparison of the trajectory data augmentation methods on three datasets, destination prediction task.}
\label{fig:sampler-analysis}
\end{figure}
\subsubsection{Choosing the Optimum Input and Output Embedding Size} \label{sec:optimum-embedding-size}
The input embedding size or the size of input latent vector $d_L$, and the output embedding size or the size of the result trajectory embedding vector $d_O$, will influence the model capacity of the spatial-temporal encoding layers and the downstream prediction models. In our CSTTE model, they are also related to the length of the anchor sequence in the last induced attentive layer $N_A$, since $N_A = d_O / d_L$.
We proceed a set of hyper-parameter experiments on all three datasets, destination prediction task to find the optimum choice for $d_L$ and $d_O$. While testing on $d_L$, we fix $d_O$ to 128; while testing on $d_O$, we fix $d_L$ to 64. Figure \ref{fig:input-size} and Figure \ref{fig:output-size} illustrate the results.
As we can see in the results, in most scenarios, the optimum input embedding size $d_L$ will be 64. As for the output size $d_O$, setting it to larger than 128 can gain some improvements, yet the improvements on the prediction metrics are limited, and it comes with higher computational complexity. Thus, we set $d_O$ to 128 for our final model.
\begin{figure}[!h]
\centering
\includegraphics[width=1\linewidth]{figure/input-size.pdf}
\caption{Prediction performance w.r.t. the input embedding size on three datasets, destination prediction task.}
\label{fig:input-size}
\end{figure}
\begin{figure}[!h]
\centering
\includegraphics[width=1\linewidth]{figure/output-size.pdf}
\caption{Prediction performance w.r.t. the output embedding size on three datasets, destination prediction task.}
\label{fig:output-size}
\end{figure}
\subsubsection{Choosing the Optimum Number of Encoder Layers}\label{sec:optimum-num-encoder-layers}
We also test the optimum number of encoder layers for the trajectory encoder on all three datasets, destination prediction task. For the encoder with one layer, the length of anchor sequence $N_{A_1}$ is set to 2; for the encoder with two layers, $N_{A_1}, N_{A_2} = 8, 2$; for the encoder with three layers, $N_{A_1}, N_{A_2}, N_{A_3} = 16, 8, 2$. Figure \ref{fig:num-layers} demonstrates the results, and it is clear that on these datasets, using only one encoder layer is the best.
\begin{figure}[!h]
\centering
\includegraphics[width=1\linewidth]{figure/num-layers.pdf}
\caption{Prediction performance w.r.t. the number of trajectory encoder layers on three datasets, destination prediction task.}
\label{fig:num-layers}
\end{figure}
\section{Conclusion}
In this paper, we propose a novel spatial-temporal aware trajectory embedding method CSTTE.
It can effectively extract high-level travel semantics from trajectories, while also incorporating long-term spatial-temporal information into the embeddings both comprehensively and efficiently.
The experimental results prove the effectiveness and superiority of our method for pre-training high-quality trajectory embeddings.
\ifCLASSOPTIONcompsoc
\section*{Acknowledgments}
\else
\section*{Acknowledgment}
\fi
This work was supported by the Fundamental Research Funds for the Central Universities (Grant No. 2021YJS030).
\ifCLASSOPTIONcaptionsoff
\newpage
\fi
\bibliographystyle{IEEEtran}
|
2,877,628,089,035 | arxiv | \section{Introduction} \label{Introduction}
A fundamentally important result in heavy-traffic queueing theory is the validity of diffusion approximations, that is, the interchange of limits property.
It provides an approximation of the steady state distribution of the queueing processes in a heavy-traffic regime by using the invariant measure of the limiting diffusion.
For instance, the interchange of limits results are proved for stochastic networks in \cite{BL,GZ,G14,YY16,YY18} and for many server queues \cite{AHP1,DDG1,GS,HAP22-OR,HAP22,S15}.
For some queueing models, the invariant measures of the limiting diffusions can be explicitly characterized \cite{DDG1,HW87}.
When this is impossible, numerical schemes are often drawn upon to compute the invariant measures, for example, computation of invariant measures of reflected Brownian motions in \cite{BCR1,DH92,DH1}. Our paper is of similar flavor as Budhiraja et al. \cite{BCR1} where a Euler scheme approximation is developed for the constrained diffusions arising as scaling limits of stochastic networks.
In this paper, we focus on $G/Ph/n+GI$ queues in the Halfin--Whitt regime.
For many-server queues with exponential services, the limiting diffusions of the scaled queueing processes are one-dimensional with a piecewise-linear drift, whose steady state distributions have explicit expression as shown in \cite{BW,DDG1}. However, for many-server queues with phase-type service time distributions, the limiting diffusions are multidimensional with a piecewise-linear drift, as shown in \cite{DHT1,PR}. Although the validity of diffusion approximations is proved for the $GI/Ph/n+M$ queues with renewal arrivals and exponential patience times in the Halfin--Whitt regime in Dai et al. \cite{DDG1}, the multi-dimensional limiting diffusion does not have an explicit invariant measure as shown in Dieker and Gao \cite{DG1}.
In fact, characterization of multi-dimensional piecewise diffusions has been left as an open problem thus far in Browne et al. \cite{BW}. The objective in this paper is to provide an approximation for the invariant measure of the limiting diffusion of $G/Ph/n+GI$ queues in the Halfin-Whitt regime. As a consequence, our result also provides an approximation for the steady-state of the diffusion-scaled queueing processes for $GI/Ph/n+M$ queues in the Halfin--Whitt regime, given the justification of interchange of limits in Dai et al. \cite{DDG1}. Moreover,
an approximation error bound is obtained for the steady-state of the $M/Ph/n+M$ queues by applying the results in Gurvich \cite{Gur1} and Braverman and Dai \cite{BD1} (see Remark \ref{rem-prelimit-approx}).
\subsection{Summary of results and contributions}
The limiting diffusion $(X_t)_{t\geq 0}$ satisfies the following stochastic differential equation (for short, SDE):
\begin{eqnarray}\label{hSDEg}
\mathrm{d} X_{t} &=& g (X_{t} ) \mathrm{d} t + \sigma \mathrm{d} B_t
\end{eqnarray}
with $(B_t)_{t\geq 0}$ being a $d$-dimensional standard Brownian motion and
\begin{eqnarray*}
g(x) &=& -\beta p-Rx+(R-\alpha I)p({\rm e}' x)^+, \ \ \forall x\in \mathbb{R}^d\,.
\end{eqnarray*}
Here $y^+=\max\{0,y\}$ for all $y\in \mathbb{R}$, $\alpha> 0$ is the patience rate, $\beta$ is the slack in the arrival rate relative to a critically loaded system, $p \in \mathbb{R}^d$ is a vector of non-negative entries whose sum is equal to one, ${\rm e}=(1,1,\cdots,1)'$ with $'$ denoting the transpose, $I$ is the identity matrix,
\begin{eqnarray*}
R \ = \ (I-P') \textrm{diag}(v), \quad \frac{1}{\zeta} \ = \ {\rm e}' R^{-1}p, \quad
\gamma \ = \ \zeta R^{-1}p,
\end{eqnarray*}
where $v=(v_1,\cdots,v_d)$ with $v_k$ being the service rate in phase $k$,
and $P$ be a sub-stochastic matrix describing the transitions between service phases such that $P_{ii}=0$ for $i=1,\cdots,d$, and $I-P$ being invertible (see Dai et al. \cite[Section 2.2]{DHT1}).
Assume that the Ph phase distribution has mean $1$, that is, $\zeta=1$. It is easy to check ${\rm e}' \gamma=1$.
$\sigma \sigma'$ has the following form:
\begin{eqnarray*}
\sigma \sigma' &=& \textrm{diag} (p)c_a^2 + H^{(0)} + \sum_{k=1}^d \gamma_k v_k H^{(k)} + (I- P')\textrm{diag}(v) \textrm{diag}(\gamma)(I-P),
\end{eqnarray*}
where $c_a^2>0$ is a constant, $\gamma=(\gamma_1,\cdots,\gamma_d)$, $H^{(k)}=(H^{(k)}_{ij})_{1\leq i, j \leq d}\in \mathbb{R}^{d\times d}$ with $H_{ii}^{(k)} = P_{ki}(1-P_{ki})$ and $H_{ij}^{(k)}=-P_{ki}P_{kj}$ for $j\neq i$ and $k =1, \dots, d$, and $H^{(0)}_{ii} = p_i(1-p_i)$ for $i=1,\dots,d$ and $H^{(0)}_{ij} = -p_ip_j$ for $i\neq j$. Throughout this paper, we assume that there exists some constant $c>0$ such that $\xi^{'}\sigma\sigma^{'}\xi \geq c\xi^{'}\xi$ for all $\xi \in \mathbb{R}^d$. It is shown in Dieker and Gao \cite[Theorem 3]{DG1} that the diffusion $(X_t)_{t\geq 0}$ is exponentially ergodic and admits a unique invariant measure $\mu$. It is well known that for every $x\in \mathbb{R}^d$, SDE (\ref{hSDEg}) has a unique solution $(X^{x}_t)_{t\ge 0}$ starting from $x$ and the solution is nonexplosive from \eqref{e:AV} below and Meyn and Tweedie \cite[Theorem 2.1]{MT2}.
\begin{remark}
This diffusion $X_t$ is shown to be the limit of the diffusion-scaled queueing processes at all phases of the $G/Ph/n+GI$ queues in the Halfin--Whitt regime \cite{DHT1} assuming that the patience time distribution $F$ satisfies $F(0)=0$ and $\alpha :=\lim_{x\downarrow 0} F(x)/x<\infty$ and the service rate equals to 1.
In the covariance $\sigma \sigma'$, $c_a^2$ captures the variability in the arrival process, specifically, when the arrival process is renewal, then $c_a^2$ is equal to the squared coefficient of variation of the interarrival times. The explicit form of $\sigma$ is not important for the result in Dieker and Gao \cite{DG1} and in our work.
\end{remark}
\smallskip
The {\bf Euler--Maruyama scheme} that we design to approximate the invariant measure $\mu$ of $(X_t)_{t\geq 0}$ reads as the following:
\begin{eqnarray}\label{e:XD}
\tilde{X}_{k+1}^{\eta}
&=& \tilde{X}_{k}^{\eta}+g(\tilde{X}_{k}^{\eta}) \eta + \sqrt{\eta} \sigma \xi_{k+1},
\end{eqnarray}
where $k\in \mathbb{N}_0\triangleq \mathbb{N}\cup\{0\}$, $\tilde{X}^{\eta}_{0}$ is the initial value and $\{\xi_k\}_{k\in \mathbb{N}}$ are the independent standard $d$-dimensional Gaussian random variables. $\tilde{X}^{\eta,x}_k$ is the value of $k$-step with initial value $\tilde{X}^{\eta}_0=x$.
Our first main result is the following theorem about this Euler--Maruyama (EM) scheme, which provides a non-asymptotic estimate for the error between the ergodic measures of the SDE and its EM scheme.
\begin{theorem}\label{thm:DDE}
$(\tilde{X}_{k}^{\eta})_{k\in \mathbb{N}_0}$ defined by \eqref{e:XD} admits a unique invariant measure $\tilde \mu_{\eta}$ and is exponentially ergodic. Moreover, the following two statements hold.
(i) For any small enough $\varsigma \in (0,1)$, there exists some positive constant $C_\varsigma$, depending on $\varsigma$ but not on $\eta$, such that
\begin{eqnarray*}
d_{W}(\mu,\tilde \mu_{\eta}) \ \leq \ C_\varsigma \eta^{\frac{1-\varsigma}{2}},
\end{eqnarray*}
where $d_W$ is the Wasserstein-1 distance, see \eqref{e:dW} below for the definition.
(ii) For any (small) error $\delta>0$ and any small enough $\varsigma\in (0,1)$ in (i), taking $\eta=\delta^\frac{2}{1-\varsigma}$, we can run the EM algorithm $N:=O( \delta^{\frac{2}{\varsigma-1}} |\log \delta|)$ steps so that the law of $\tilde{X}^\eta_N$, denoted as $\mathcal L(\tilde{X}^\eta_N)$, satisfies
$$d_W(\mathcal L(\tilde{X}^\eta_N),\mu) \ \leq \ \delta.$$
\end{theorem}
Our second set of main results are the central limit theorem (CLT), the moderate deviation principle (MDP) for the long term behavior of $(X_t)_{t\geq 0}$ and $(\tilde{X}^{\eta}_k)_{k\in \mathbb{N}_0}$. For any $x \in \mathbb{R}^d$ and $T>0$, the empirical measure $\mathcal E_T^x$ of $(X^x_t)_{t \ge 0}$ is defined by
\begin{eqnarray*}
\mathcal E_T^x(A) \ = \ \frac 1T \int_0^T \delta_{X^x_s}(A) \mathrm{d} s, \ \ \ \ \ A \in \mathcal{B}(\mathbb{R}^d),
\end{eqnarray*}
where $\mathcal{B}(\mathbb{R}^d)$ is the collection of Borel sets on $\mathbb{R}^{d}$, $\delta_y(\cdot)$ is a delta measure, that is, $\delta_y(A)=1$ if $y \in A$ and $\delta_y(A)=0$ if $y \notin A$. It is easy to check that for any measurable function $h: \mathbb{R}^d \rightarrow \mathbb{R}$,
$$\mathcal E_T^x(h) \ = \ \frac 1T \int_0^T h(X^x_s) \mathrm{d} s.$$
For any $x \in \mathbb{R}^d$ and $n \in \mathbb{N}$, the empirical measure $ \mathcal E_n^{\eta,x} $ of $(\tilde{X}^{\eta,x}_k)_{k\in \mathbb{N}_0}$ is defined by
\begin{eqnarray*}
\mathcal E_n^{\eta,x}(A) \ = \ \frac{1}{n} \sum_{k=1}^n \delta_{\tilde{X}^{\eta,x}_k}(A), \ \ \ \ \ A \in \mathcal{B}(\mathbb{R}^d).
\end{eqnarray*}
It is easy to check that for any measurable function $h: \mathbb{R}^d \rightarrow \mathbb{R}$
$$\mathcal E_n^{\eta,x}(h) \ = \ \frac{1}{n} \sum_{k=1}^n h(\tilde{X}^{\eta,x}_k).$$
In order to state our theorems about CLT and MDP, for a given probability measure $\nu$, we define $L^2(\nu)$ as
the Hilbert space induced by $\nu$ with inner product
$$\langle f_1, f_2\rangle_{\nu}=\int f_1 f_2 d \nu \ \ \ \ \ {\rm for} \ \ \ f_1, f_2 \in L^2(\nu).$$
For further use, we denote by $\mathcal{B}_b(\mathbb{R}^d,\mathbb{R})$ the set of bounded measurable functions from $\mathbb{R}^d$ to $\mathbb{R}$, and denote by ${\rm Lip}(\mathbb{R}^d,\mathbb{R})$ the set of globally Lipschitz functions from $\mathbb{R}^d$ to $\mathbb{R}$.
\begin{theorem} [CLT] \label{thm:CLT}
For any $h\in \mathcal{B}_b(\mathbb{R}^d,\mathbb{R})$ and $x \in \mathbb{R}^d$, $\sqrt{t} \left[\mathcal E^x_t(h)-\mu(h)\right]$ converges weakly to Gaussian distribution $\mathcal{N}(0,\mu(|\sigma^{\prime} \nabla f|^2))$ as $t \rightarrow \infty$, where $f$ is the solution to the Stein's equation \eqref{e:SE} below. Furthermore, $\mu(|\sigma' \nabla f|^2)\leq C\|h\|^2_{\infty}<\infty$ for some constant $C>0$.
\end{theorem}
\begin{theorem}[MDP]\label{thm:MDP}
For any $h\in \mathcal{B}_b(\mathbb{R}^d,\mathbb{R})$, $x \in \mathbb{R}^d$ and measurable set $A\subset \mathbb{R}$, one has
\begin{eqnarray*}
-\inf_{z \in A^{ {\rm o} } } \frac{ z^2 }{ 2 \mu(| \sigma^{\prime} \nabla f |^2) }
\ &\leq& \
\liminf_{t\to\infty}\frac{1}{a_t^2}\log \mathbb{P} \left( \frac{\sqrt{t}}{a_t} \left[\mathcal E^x_t(h)-\mu(h)\right] \in A \right) \\
\ &\leq& \
\limsup_{t\to\infty}\frac{1}{a_t^2}\log \mathbb{P} \left( \frac{\sqrt{t}}{a_t} \left[\mathcal E^x_t(h)-\mu(h)\right] \in A \right)
\ \leq \ -\inf_{z \in \bar{A}} \frac{ z^2 }{ 2 \mu(| \sigma^{\prime} \nabla f |^2) },
\end{eqnarray*}
where $\bar{A}$ and $A^{ {\rm o} }$ are the closure and interior of set $A$, respectively, and $a_t$ satisfies $a_t \to \infty$ and $\frac{a_t}{\sqrt{t}} \to 0$ as $t\to \infty$ and $f$ is the solution to the Stein's equation \eqref{e:SE} below.
\end{theorem}
\begin{theorem}[CLT and MDP]\label{thm:EMMDP}
(i). For any $h\in \mathcal{B}_b(\mathbb{R}^d,\mathbb{R})$, $\sqrt{n} \big[ \mathcal E_n^{\eta, x}(h) - \tilde{\mu}_{\eta}(h) \big]$ converges weakly to Gaussian distribution $\mathcal{N}(0,\tilde{ \mathcal V}(h))$ as $n\to \infty$ with
\begin{equation*}
\tilde{ \mathcal V} (h) \ \ = \ \ \langle f, f \rangle_{\tilde{\mu}_{\eta}}-\langle \tilde{\mathcal{P}}_{\eta} f, \tilde{\mathcal{P}}_{\eta} f \rangle_{\tilde{\mu}_{\eta}},
\end{equation*}
where $f$ is the solution to the third Stein's equation \eqref{e:Stein-3} below and $\tilde{\mathcal P}_\eta f(x)=\mathbb{E} f(\tilde X_1^{\eta,x})$. Moreover,
\begin{eqnarray}\label{e:EMVh}
\tilde{ \mathcal V} (h) \ &=& \ \langle h-\tilde{\mu}_{\eta}(h) , h-\tilde{\mu}_{\eta}(h) \rangle_{\tilde{\mu}_{\eta}} + \sum_{k=1}^{\infty} \langle \tilde{\mathcal{P}}^k_{\eta} h, h-\tilde{\mu}_{\eta}(h) \rangle_{\tilde{\mu}_{\eta}}.
\end{eqnarray}
(ii). For any $h\in \mathcal{B}_b(\mathbb{R}^d,\mathbb{R})$, $x \in \mathbb{R}^d$ and measurable set $A\subset \mathbb{R}$, one has
\begin{eqnarray*}
-\inf_{z \in A^{ {\rm o} } } \frac{ z^2 }{2 \tilde{ \mathcal V} (h)}
\ &\leq& \
\liminf_{n \to\infty}\frac{1}{a_n^2}\log \mathbb{P} \left( \frac{\sqrt{n}}{a_n} \left[ \mathcal E_n^{\eta,x}(h) - \tilde{\mu}_{\eta}(h)\right] \in A \right) \\
\ &\leq& \
\limsup_{n \to\infty}\frac{1}{a_n^2}\log \mathbb{P} \left( \frac{\sqrt{n}}{a_n} \left[ \mathcal E_n^{\eta,x}(h) - \tilde{\mu}_{\eta}(h)\right] \in A \right)
\ \leq \ -\inf_{z \in \bar{A}} \frac{ z^2 }{ 2 \tilde{ \mathcal V} (h) },
\end{eqnarray*}
where $\bar{A}$ and $A^{ {\rm o} }$ are the closure and interior of set $A$, respectively, and $a_n$ satisfies $a_n \to \infty$ and $\frac{a_n}{\sqrt{n}} \to 0$ as $n\to \infty$.
\end{theorem}
\begin{remark}
We shall see below that Stein's equation will play an important role in proving Theorems \ref{thm:DDE}, \ref{thm:CLT} and \ref{thm:MDP}. The proof of Theorem \ref{thm:EMMDP} is a direct application of Jones \cite[Theorem 9]{jones2004markov} and Wu \cite[Theorem 2.1]{WLM1}, but we can see that a discrete Stein's equation \eqref{e:Stein-3} below plays an important role as well from the statement of the theorem.
\end{remark}
\subsection{Related works}
Our paper is relevant to the following streams of works in the literature.
{\it (a) Steady state analysis of many-server queues.}
A few significant results have been obtained for understanding the steady-state of many-server queues, see, e.g., \cite{AR20,AHP1,AHPS1,APS1,BD1,BDF,GG13b,GG13a,Gur1,HAP22-OR,HAP22} and references therein.
The most relevant to us is the recent development using Stein's method to analyze the steady state of queueing processes via diffusion approximations.
Gurvich \cite{Gur1} provides a framework of analyzing the steady states via direct diffusion approximations (rather than diffusion limits) for a family of continuous-time exponentially ergodic Markov processes,
in particular, the gap between the steady-state moments of the diffusion models and those of the Markov processes is characterized. This result can be applied to Markovian many-server queueing systems.
Braverman et al. \cite{BDF} introduced the Stein's method framework formally, proving Wasserstein and Kolmogorov distances between the steady-state distributions of the queueing processes and approximate diffusion models, and applied to the classical Erlang A and C models, where the bound is characterized by the system size.
Braverman and Dai \cite{BD1} then extended this approach for the $M/Ph/n+M$ queues.
Braverman et al. \cite{BDF20} recently studied high order steady-state approximations of 1-dimensional Markov chains and applied to Erlang C models.
As discussed before, the invariant measure of the limiting diffusion of $G/Ph/n+GI$ queues in the Halfin--Whitt regime lacks an explicit expression and is difficult to compute directly. Dai and He \cite{DH1} proposed a numerical method to compute the invariant measure of the limiting diffusion by solving the basic adjoint relationship, however, their work does not give theoretical error bounds.
In this work, we provide a stochastic algorithm based on the EM scheme to compute the invariant measure for the limiting diffusion. More importantly, our work characterizes the non-asymptotic error bound in terms of the step size in the algorithm.
Although this approach uses discretization, in comparison with the discrete event simulation approach, the EM scheme is much more efficient computationally, especially when the number of servers $n$ is large in the queueing model.
It is worth noting that for the models with the one-dimensional limiting diffusions of linear or piecewise-linear drifts, since the invariant measure has an explicit density, when studying the steady-state approximation problem by Stein's method, one can explicitly solve Stein's equation and obtain the desired regularity properties easily. That is very similar to the one dimensional normal approximation case. It is well known that the generalization of Stein's method from one to multi-dimensional approximations is highly nontrivial \cite{CM08,RR-09}.
Our problem is a multidimensional diffusion approximation.
\smallskip
{\it (b) Error estimates of EM schemes for diffusions.}
Let us recall the results concerning the error estimates between the ergodic measures of SDEs and their EM scheme.
For the ease of stating and comparing the results in the literatures below, we denote in this subsection by $(Y_{t})_{t \ge 0}$ and $(\tilde{Y}_{n})_{n \in \mathbb{N}_{0}}$ the stochastic processes associated to SDEs and their EM scheme respectively, and by $\pi_{sde}$ and $\pi_{em}$ their ergodic measures respectively.
There have been many results concerning the error estimates between the ergodic measures of $(Y_{t})_{t \ge 0}$ and $(\tilde{Y}_{n})_{n \in \mathbb{N}_{0}}$, see for instance \cite{BBC1,BDMS1, BCR1, DM2, DM1, KT1, LP1, P1, P2, T3}, but most of them are
asymptotic type. For asymptotic results, we recall those in the literatures \cite{BCR1,GHL1, P1, P2} whose settings are close to ours. An empirical measure $\Pi^{n}_{em}$ of $\pi_{em}$ for a class of SDEs driven by multiplicative L\'evy noises is considered in \cite{GHL1, P1, P2}, and it is shown that $\frac1{\sqrt{\Gamma_n}}(\Pi^{n}_{em}(f)-\pi_{sde}(f))$ converges to a normal distribution as $n \rightarrow \infty$ for $f$ in a certain high order differentiable function family ($\Gamma_n$ has the same order as $n$).
A similar type CLT is obtained in Budhiraja et al. \cite{BCR1} for a reflected SDE driven arising in queueing systems. All these works need strong dissipation and high order differentiability conditions on the drift of SDEs, which do not hold for the limiting diffusions of our queueing systems.
Among the few non-asymptotic results, the works in \cite{BDMS1,DM2,DM1}, arising from the sampling of Langevin dynamics, are probably most close to ours. Their SDEs are gradient systems, i.e., the drift is the gradient of a potential $U$, thus analytical tools such as concentration inequalities are available.
Under certain conditions on the drift, they prove non-asymptotic bounds for the total variation or Wasserstein-2 distance between $\pi_{sde}$ and $\pi_{em}$. Their analysis heavily depends on the gradient form of the drift, and is not easily seen to be extended to a non-gradient system. Our SDE is not a gradient system in that its drift $g(x)$ can not be represented as a gradient of a potential, and what is worse is that $g(x)$ is even not differentiable.
There are some works (see \cite{BBC1,T3} and references therein) giving non-asymptotic results for the difference between the law of $\tilde{Y}_n$ and $\pi_{sde}$ for large $n$. Most of these works need strong dissipation and high order differentiability assumptions on the drift of the SDE, and their estimates are in the form
$|\mathbb{E} h(\tilde{Y}_n)-\pi_{sde}(h)|$ or $|\frac{1}n \sum_{i=1}^n h(\tilde{Y}_i)-\pi_{sde}(h)|$ for $h$ in a certain high order differentiable function family, from which one usually cannot derive a bound between the law of $\tilde{Y}_n$ and $\pi_{sde}$ in a Wasserstein distance.
\smallskip
{\it (c) CLT and MDP with respect to ergodic measures.} Dieker and Gao \cite{DG1} proved that SDE (\ref{hSDEg}) is exponentially ergodic with ergodic measure $\mu$, it implies that Birkhoff ergodic theorem holds true for the empirical measure $\mathcal{E}_T$ of the process $X_t$, i.e. $\lim_{T \rightarrow \infty} \mathcal{E}_T=\mu$ a.s., see for instance Prato and Zabczyk \cite{da1996ergodicity}.
It is natural to consider the CLT and MDP with respect to $\mathcal{E}_T$.
Using the method in \cite{WXX1,WLM1}, we will establish the CLT and MDP for $\mathcal{E}_T$, in which the related variances can be determined by a Stein's equation. Because $\mathcal{E}_T$ is a random measure, it is natural to choose $\mathcal B_b(\mathbb{R}^d,\mathbb{R})$ rather than ${\rm Lip}(\mathbb{R}^d,\mathbb{R})$ as the test functions family, which makes the regularity results in Gurvich \cite[Theorem 4.1]{Gur1,Gur11C} not applicable. Alternatively, we apply Malliavin calculus to study the regularity of the Stein's equation.
We also prove the CLT and MDP for the EM scheme in which the variance is determined by another Stein's equation. There exist very few results for studying CLT and MDP of the EM scheme, see \cite{fukasawa2020efficient,lu2020central}.
\subsection{Organization of the paper}
In the remainder of this section, we introduce notations which will be frequently used. Section \ref{sec-proof strategy} gives the proof of Theorem \ref{thm:DDE}, while Sections \ref{s:CLTMDP} and \ref{sec:EMCLTMDP} provide the proofs for the CLTs and MDPs with respect to the process $(X_t)_{t \ge 0}$ and the EM scheme $(\tilde X^{\eta}_k)_{k \in \mathbb{N}_0}$ respectively. We prove in Appendix \ref{App:GeneralErgodicEM} the ergodicity of the EM scheme, and prove in Appendix \ref{sec:AAS} the propositions and Lemmas in Section \ref{sec:EMCLTMDP}.
\subsection{Notations}
Let $\mathbb{R}$ and $\mathbb{C}$ be real numbers and complex numbers respectively. The Euclidean metric is denoted by $| \cdot |$. For matrixes $A=(A_{ij})_{d\times d}$ and $B=(B_{ij})_{d\times d}$, denote $\langle A, B \rangle_{ {\rm HS} }=\sum_{i,j=1}^d A_{ij} B_{ij}$ and Hilbert Schmidt norm is $ \| A \|_{ {\rm HS} } = \sqrt{ \sum_{i,j=1}^d A_{ij}^2 } $ and operator norm is $\| A \|_{ {\rm op} } = \sup_{ |u|=1 } |Au|$.
We write a symmetric matrix $A>0\, (A<0)$ if $A$ is a positive (negative) definite matrix, and write $A\ge 0 \, (A \leq0)$ if $A$ is a positive (negative) semi-definite matrix. $\langle x, y \rangle$ means the inner product, that is, $\langle x, y \rangle = x^{\prime} y$ for $x,y\in \mathbb{R}^d$. $\otimes$ is the outer product, that is, for vector $u=(u_1,\cdots,u_d)$ and matrix $A=(A_{ij})_{d\times d}$, then $(u \otimes A)_{ijk}=u_{i}A_{jk}$ for $1 \le i, j, k \le d$.
$\mathcal{C}^k(\mathbb{R}^d, \mathbb{R}_+)$ means $\mathbb{R}_+$-valued $k$-times continuous derivatives functions defined on $\mathbb{R}^d$ with $k\in \mathbb{N}$ and $\mathbb{R}_+=[0,\infty)$. $ \mathcal{C}_b(\mathbb{R}^d, \mathbb{R})$ is $\mathbb{R}$-valued continuous bounded functions defined on $\mathbb{R}^d$.
Denote $\| f \|_{\infty}= {\rm ess}\sup_{x\in \mathbb{R}^d} |f(x)|$ for $f\in\mathcal{B}_b(\mathbb{R}^d, \mathbb{R} )$. For $f\in \mathcal{C}^2(\mathbb{R}^d, \mathbb{R})$, denote $\nabla f=(\partial_1 f, \partial_2 f, \cdots, \partial_d f) \in \mathbb{R}^d$ and $\nabla^2 f= ( \partial_{ij} f)_{1\leq i,j \leq d} \in \mathbb{R}^{d\times d}$ the gradient and Hessian matrix for function $f$. For $f\in \mathcal{C}^1(\mathbb{R}^d, \mathbb{R})$ and $u, x\in \mathbb{R}^d$, the directional derivative $\nabla_{u} f (x)$ is defined by
\begin{eqnarray*}
\nabla_{u} f(x) &=& \lim_{\varepsilon_1 \to 0} \frac{f(x+\varepsilon_1 u) - f(x)}{\varepsilon_1}.
\end{eqnarray*}
We know $\nabla f (x) \in \mathbb{R}^d$ for each $x \in \mathbb{R}^{d}$ and $\nabla_{u} f (x) =\langle \nabla f(x), u \rangle$. For $f \in \mathcal{C}^2(\mathbb{R}, \mathbb{R})$, $\dot{f}$ and $\ddot{f}$ are the first and second derivatives of function $f$, respectively. For any probability measure $\nu$, denote $\nu(f) = \int f(x) \nu(\mathrm{d} x)$.
$B(y,r)$ means the ball in $\mathbb{R}^d$ with centre $y \in \mathbb{R}^d$ and radius $r>0$, that is, $B(y,r)=\{z\in \mathbb{R}^d:|z-y|\leq r\}$.
$\mathcal{N}(a,A)$ with $a\in \mathbb{R}^d$ and $A\in \mathbb{R}^{d\times d}$ denotes Gaussian distribution with mean $a$ and covariance matrix $A$.
A sequence of random variables $\{Y_n, n\geq 1 \}$ is said to converge weakly or converge in distribution to a limit $Y_{\infty}$, that is, $Y_n \Rightarrow Y_{\infty}$ if $\lim_{n \rightarrow \infty} \mathbb{E} f(Y_{n})=\mathbb{E} f(Y_{\infty})$ for all bounded continuous function $f$. In addition, $Y_n \stackrel{p}{\longrightarrow} Y_{\infty}$ means convergence in probability, namely, $\lim_{n\to \infty} \mathbb{P}(|Y_n - Y_{\infty}|>\delta)=0$ for all $\delta \geq 0$. $Y_n \stackrel{L^p}{\longrightarrow} Y_{\infty}$ means the $L^p$ convergence, that is, $\lim_{n\to \infty}\mathbb{E} |Y_n-Y_{\infty}|^p=0$.
Denote $X^x_t$ the process $X_t$ given $X_0=x$. Denote by $P_{t}(x,\cdot)$ the transition probability of $X_{t}$ given $X_{0}=x$. Then the associated Markov semigroup $(P_t)_{t\ge 0}$ is given by, for all $x\in\mathbb{R}^d$ and $f\in\mathcal{B}_b(\mathbb{R}^d,\mathbb{R} )$
\begin{eqnarray*}
P_tf(x)
\ = \ \mathbb{E} f(X^x_t)
\ = \ \int_{\mathbb{R}^{d}} f(y) P_{t}(x, \mathrm{d} y), \quad \forall t\ge0.
\end{eqnarray*}
The generator $\mathcal{A}$ of $(X_t)_{t\geq 0}$ is given by, for $y\in\mathbb{R}^d$,
\begin{eqnarray}\label{e:A}
\mathcal{A} f(y) \ = \ \langle \nabla f(y), g(y)\rangle+\frac{1}{2} \langle \sigma\sigma^{\prime}, \nabla^2 f(y) \rangle_{{\rm HS}}, \quad f \in \mathcal{D}(\mathcal{A}),
\end{eqnarray}
where $\mathcal{D}(\mathcal{A})$ is the domain of $\mathcal{A}$, whose exact form is determined by the function space where the semigroup $(P_{t})_{t \ge 0}$ is located.
Denote by $\tilde{\mathcal{P}}_{\eta}(x,\cdot)$ the one step transition probability for the Markov chain $(\tilde{X}_k^{\eta})_{k\in \mathbb{N}_0}$ with $\tilde{X}_0^{\eta}=x$, that is, for $f\in \mathcal{B}_b(\mathbb{R}^d,\mathbb{R} )$ and $x\in \mathbb{R}^d$, one has
\begin{eqnarray*}
\tilde{\mathcal{P}}_{\eta}f(x) \ = \ \int_{\mathbb{R}^d} f(y)\tilde{\mathcal{P}}_{\eta}(x,\mathrm{d} y),
\end{eqnarray*}
and denote $\tilde{\mathcal{P}}_{\eta}^k=\tilde{\mathcal{P}}_{\eta} \circ \tilde{\mathcal{P}}_{\eta}^{k-1}$ for integers $k\geq 2$.
We denote by $\mathbb{E}^{\mu}$ the conditional expectation given that $X_{0}$ has a distribution $\mu$. If $\mu=\delta_{x}$, we write $\mathbb{E}^{x}=\mathbb{E}^{\delta_{x}}$. $\mathbb{E}_{\mathbb{P}}$ and $\mathbb{E}_{\mathbb{Q}}$ mean expectations under probability spaces $\mathbb{P}$ and $\mathbb{Q}$, respectively.
Recall that the following measure distances. The Wasserstein-1 distance between two probability measures $\mu_1$ and $\mu_2$ is defined as (see Hairer and Mattingly \cite[p. 2056]{HM1})
\begin{eqnarray}\label{e:dW}
d_{W}(\mu_1,\mu_2)
&=& \sup_{h \in {\rm Lip(1)}}\left\{\int h(x) \mu_1 (\mathrm{d} x) - \int h(x) \mu_2 (\mathrm{d} x) \right \} \nonumber \\
&=& \sup_{h \in {\rm Lip_0(1)}} \left \{\int h(x) \mu_1 (\mathrm{d} x) - \int h(x) \mu_2 (\mathrm{d} x) \right \} \nonumber \\
&=& \sup_{h \in {\rm Lip(1)}} \left \{\int h(x) \mu_1 (\mathrm{d} x) - \int h(x) \mu_2 (\mathrm{d} x), \ \ |h(x)| \le |x| \right \},
\end{eqnarray}
where ${\rm Lip(1)}$ is the set of Lipschitz function with Lipschitz constant $1$, that is, ${\rm Lip(1)}=\{ h: |h(x)-h(y)|\leq |x-y|$ for all $x,y \in \mathbb{R}^d \}$, and ${\rm Lip_0(1)}:=\{h \in {\rm Lip(1)}: h(0)=0\}$.
The total variation distance (see Hairer \cite[p. 57]{MH2}) between two measures $\mu_1$, $\mu_2$ is defined by
\begin{eqnarray*}
\|\mu_1-\mu_2\|_{\rm{TV}}
&=& \sup_{ \substack{h \in \mathcal{B}_b(\mathbb{R}^d, \mathbb{R}), \, \| h \|_{\infty}\leq1} } \left \{ \int_{\mathbb{R}^d} h(x) \mu_1(\mathrm{d} x) - \int_{\mathbb{R}^d} h(x) \mu_2(\mathrm{d} x) \right \} .
\end{eqnarray*}
Let $V:\mathbb{R}^d \rightarrow \mathbb{R}_+$ be a measurable function, define a weighted supremum norm on measurable functions (see Hairer \cite[p. 57]{MH2}) by
\begin{eqnarray*}
\| \varphi \|_{V} &=& \sup_{x\in \mathbb{R}^d } \frac{ | \varphi(x) | }{ 1+ V(x) },
\end{eqnarray*}
as well as the dual norm of measures by
\begin{eqnarray*}
\|\mu_1-\mu_2 \|_{\rm{TV}, \rm{V}}
&=& \sup \left\{ \int \varphi(x) \mu_1(\mathrm{d} x) -\int \varphi(x) \mu_2(\mathrm{d} x) : \| \varphi \|_V \leq 1\right\}.
\end{eqnarray*}
An alternative expression for the weighted total variation norm is given by
\begin{eqnarray}\label{normTVV}
\|\mu_1-\mu_2\|_{\rm{TV}, \rm{V}}
&=& \int_{\mathbb{R}^d} (1+V(x)) |\mu_1-\mu_2 | (\mathrm{d} x),
\end{eqnarray}
where $\mu_1-\mu_2$ is a signed measure and $|\mu_1-\mu_2|$ is the absolute value of $\mu_1-\mu_2$. Under $V\ge 0$, one has the relation $\|\mu_1-\mu_2\|_{\rm{TV}} \leq \|\mu_1-\mu_2\|_{\rm{TV}, \rm{V}} $. If $1+V(x) \ge 1+c |x|^2 \ge c'|x|$ for some constants $c, c'>0$, it follows from \eqref{e:dW} and \eqref{normTVV} that there exists some constant $C>0$ such that
\begin{equation} \label{e:dWandTV}
d_W(\mu_1,\mu_2) \ \leq \ C \|\mu_1-\mu_2\|_{\rm{TV}, \rm{V}}.
\end{equation}
Let $P_t^*$ be the dual operator of $P_t$ for all $t\geq 0$, that is, for some measurable set $A$ and measure $\mu_1$, one has
\begin{eqnarray*}
(P_t^* \mu_1 )(A) \ = \ \int_{\mathbb{R}^d} P_t(x,A) \mu_1(\mathrm{d} x).
\end{eqnarray*}
We use the letter $C$ to represent a positive constant, which may be different from line to line. Denote
\begin{eqnarray}
C_{\rm op} \ & = & \ \|R\|_{\rm op}+\|(R-\alpha I)p {\rm e}'\|_{\rm op}, \label{Cop} \\
\tilde{C}_{\rm op} \ & = & \ C_{\rm op}+\| \sigma\sigma^{\prime} \|_{\rm HS}+1+\|R-\alpha I\|_{\rm op}+|\beta|, \label{tlCop} \\
C_m \ & = & \ 2m^2\tilde{C}_{{\rm op}} {\rm \ for \ integers \ } m\geq 2. \label{Cm}
\end{eqnarray}
\section{Proof of Theorem \ref{thm:DDE} } \label{sec-proof strategy}
We give the proof of Theorem \ref{thm:DDE} by the help of Proposition \ref{p:GeneralErgodicEM} below and the Stein's method developed in Fang et al. \cite{FSX1}. Proposition \ref{p:GeneralErgodicEM} is on the existence of a unique invariant measure and the exponential ergodicity of the Markov chain $(\tilde{X}_{k}^{\eta})_{k\in \mathbb{N}_0}$ in \eqref{e:XD} in the EM scheme, while we use the Stein's method via solving a Stein's equation. Note that even if an SDE is ergodic, its EM scheme may blow up (see \cite{FG1,MSH1}), so a careful study of these ergodicity properties is necessary. Its proof will be given in Appendix \ref{App:GeneralErgodicEM}.
\begin{proposition} \label{p:GeneralErgodicEM}
$(\tilde{X}_{k}^{\eta})_{k\in \mathbb{N}_0}$ in \eqref{e:XD} admits a unique invariant measure $\tilde \mu_{\eta}$ and is exponentially ergodic. More precisely, for any $k\in \mathbb{N}_0$ and measure $\nu$ satisfying $\nu(|\cdot|^2)<\infty$, one has
\begin{eqnarray*}
d_W( (\tilde{\mathcal{P}}_{\eta}^k)^* \nu, \tilde{\mu}_{\eta})
\ &\leq& \ C\eta^{-1} e^{-c k\eta} , \\
\| (\tilde{\mathcal{P}}_{\eta}^k)^* \nu- \tilde{\mu}_{\eta} \|_{\rm TV}
\ &\leq& \ C\eta^{-1} e^{-c k\eta},
\end{eqnarray*}
where $C, c$ are positive constants independent of $k$ and $\eta$. Moreover, for integers $\ell \geq 2$, there exists some positive constant $C$ depending on $\ell$ but not on $\eta$ such that
\begin{eqnarray*}
\tilde \mu_{\eta}(|\cdot|^{\ell}) \ \ \leq \ \ C, \ \ \ \ \ell \ge 2.
\end{eqnarray*}
\end{proposition}
\subsection{The first Stein's equation} In order to apply the Stein's method developed in Fang et al. \cite{FSX1}, we consider our first Stein's equation as follows: for a Lipschitz function $h: \mathbb{R}^d \to \mathbb{R} $ with $\| \nabla h \|_{\infty}<\infty$,
\begin{eqnarray}\label{e:PoiLip}
\mathcal{A} f(x) \ = \ h(x) - \mu(h),
\end{eqnarray}
where $\mathcal{A}$ is defined as in \eqref{e:A}, and $\mu$ is the invariant measure for the process $(X_t)_{t\geq 0}$ in \eqref{hSDEg} with semigroup $(P_{t})_{t\geq 0}$. Without loss of generality, we assume that $h\in {\rm Lip}_0(1)$. Then we can get the regularity for the solution to Stein's equation \eqref{e:PoiLip} from {Gurvich \cite[Theorem 4.1]{Gur1,Gur11C} (see also Braverman and Dai \cite[Lemma 1]{BD1}).}
The regularity results of the equation \eqref{e:PoiLip} in the following lemma will be used in the proof of Theorem \ref{thm:DDE}.
\begin{lemma}\label{lem:Lipregf}
Let $h\in {\rm Lip}_{0}(1)$ and $f$ be the solution to \eqref{e:PoiLip}.
There exists some positive constant $C$ such that for $1\leq i,j\leq d$,
\begin{eqnarray}
|f(x)| \ &\leq& \ C(1+|x|), \nonumber \\
|\partial_i f(x)| \ &\leq& \ C(1+|x|^2), \nonumber \\
|\partial_{ij} f(x)| \ &\leq& \ C(1+|x|^3), \label{e:2f}
\end{eqnarray}
where $\nabla f=(\partial_1 f, \partial_2 f, \cdots, \partial_d f) \in \mathbb{R}^d$ and $\nabla^2 f= ( \partial_{ij} f )_{1\leq i,j \leq d} \in \mathbb{R}^{d\times d}$ are the gradient and Hessian matrix for $f$, respectively.
For any small enough $\varsigma\in (0,1)$, there exists some positive constant $C_\varsigma$ depending on $\varsigma$ such that for $1\leq i,j\leq d$,
\begin{equation}
\sup_{y\in \mathbb{R}^d: |y-x|<1} \frac{ | \partial_{ij} f(y)-\partial_{ij} f(x)| }{|y-x|^{1-\varsigma}} \ \leq \ C_\varsigma (1+|x|^4). \label{e:3f}
\end{equation}
\end{lemma}
\subsection{{Proof of Theorem \ref{thm:DDE} }}
\begin{proof}[Proof of Theorem \ref{thm:DDE}] (i) By Proposition \ref{p:GeneralErgodicEM}, we know $(\tilde{X}_{k}^{\eta})_{k\in \mathbb{N}_0}$ in \eqref{e:XD} admits a unique invariant measure $\tilde{\mu}_{\eta}$. Let the initial value $\tilde{X}^{\eta}_{0}$ be the invariant measure $\tilde \mu_{\eta}$.
We know $(\tilde{X}^{\eta}_{k})_{k \in \mathbb{N}_0}$ is a stationary Markov chain. Denote $W=\tilde{X}_{0}^{\eta}$, $W'=\tilde{X}_{1}^{\eta}$ and $\delta = W'-W$. It is easy to see that
\begin{eqnarray} \label{e:EDelW}
\mathbb{E}[\delta|W] \ = \ g(W) \eta \text{ \ \ and \ \ }
\mathbb{E}[\delta \delta^{\prime}|W] \ =\ g(W) g^{\prime}(W) \eta^2 +\sigma \sigma^{\prime}\eta.
\end{eqnarray}
Let $f$ be the solution to Eq. \eqref{e:PoiLip} with $h\in {\rm Lip}_0(1)$. Since $W$ and $W'$ have the same distribution, we have
\begin{eqnarray}\label{hMC51e}
0&=& \mathbb{E}f(W')-\mathbb{E}f(W) \nonumber \\
&=& \mathbb{E}[ \langle \delta, \nabla f(W) \rangle ]
+\mathbb{E} \int_0^1 \int_0^1 r \langle \delta \delta^{\prime}, \nabla^2 f (W+\tilde{r} r \delta ) \rangle_{\textrm{HS}} \mathrm{d} \tilde{r} \mathrm{d} r \nonumber \\
&=&\mathbb{E}[ \langle g(W),\nabla f(W) \rangle ]\eta+\mathbb{E} \int_0^1 \int_0^1 r \langle \delta \delta^{\prime}, \nabla^2 f(W+\tilde{r} r \delta ) \rangle_{\textrm{HS}} \mathrm{d} \tilde{r} \mathrm{d} r, \nonumber
\end{eqnarray}
where the last inequality holds from below
\begin{eqnarray}\label{hMC52e}
\mathbb{E}[ \langle \delta, \nabla f(W) \rangle ]
\ = \ \mathbb{E}[ \langle \mathbb{E}(\delta|W),\nabla f(W)\rangle ]
\ = \ \mathbb{E}[ \langle g(W),\nabla f(W) \rangle ]\eta. \nonumber
\end{eqnarray}
In addition, one has
\begin{eqnarray*}
&& \mathbb{E} \int_0^1 \int_0^1 r \langle \delta \delta^{\prime}, \nabla^2 f(W+\tilde{r} r \delta ) \rangle_{\textrm{HS}} \mathrm{d} \tilde{r} \mathrm{d} r \nonumber \\
&=& \frac{1}{2}\mathbb{E} \langle \delta \delta^{\prime}, \nabla^2 f(W) \rangle_{\textrm{HS}}+
\mathbb{E} \int_0^1 \int_0^1 r \langle \delta \delta^{\prime}, \nabla^2 f(W+\tilde{r} r \delta )-\nabla^2 f(W) \rangle_{\textrm{HS}} \mathrm{d} \tilde{r} \mathrm{d} r \nonumber \\
&=& \frac{\eta}{2} \mathbb{E} [ \langle \sigma \sigma^{\prime}, \nabla^2 f(W) \rangle_{\textrm{HS}} ] +\frac{\eta^2}{2} \mathbb{E} [ \langle g(W) g^{\prime}(W), \nabla^2 f(W) \rangle_{\textrm{HS}} ] \nonumber \\
&\ & \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ +\mathbb{E} \int_0^1 \int_0^1 r \langle \delta \delta^{\prime}, \nabla^2 f (W+\tilde{r} r \delta )-\nabla^2 f(W) \rangle_{\textrm{HS}} \mathrm{d} \tilde{r} \mathrm{d} r,
\end{eqnarray*}
where the second equality is by the relation $\mathbb{E}[\langle \delta \delta^{\prime}, \nabla^2 f(W) \rangle_{\textrm{HS}}]=\mathbb{E}[\langle \mathbb{E}[\delta \delta^{\prime}|W], \nabla^2 f(W) \rangle_{\textrm{HS}}]$ and \eqref{e:EDelW}.
Collecting the previous relations, we obtain
\begin{eqnarray}\label{e:Afh}
\mathbb{E}[\mathcal{A}f(W) ] \ = \ \frac \eta 2 {\rm I}+\frac{1}{\eta} {\rm II},
\end{eqnarray}
where
\begin{align*}
{\rm I} &\ = \ -\mathbb{E} [ \langle g(W) g^{\prime}(W), \nabla^2 f(W) \rangle_{\textrm{HS}} ], \\
{\rm II} &\ = \ -\mathbb{E} \int_0^1 \int_0^1 r \langle \delta \delta^{\prime}, \nabla^2 f(W+\tilde{r} r \delta )-\nabla^2 f(W) \rangle_{\textrm{HS}} \mathrm{d}\tilde{r} \mathrm{d} r.
\end{align*}
From the estimate $|g(x)| \leq \tilde{C}_{\rm op}(1+|x|)$ for all $x\in \mathbb{R}^d$ with $\tilde{C}_{{\rm op}}$ in \eqref{tlCop} and using inequality \eqref{e:2f} in Lemma \ref{lem:Lipregf}, one has
\begin{eqnarray}\label{e:Afh-1}
|{\rm I}|
&=&\mathbb{E}|\langle g(W) g^{\prime}(W), \nabla^2 f(W) \rangle_{\textrm{HS}} |
\ \leq \ C(1+\mathbb{E}|W|^5)
\ \leq \ C,
\end{eqnarray}
where the last inequality is by Proposition \ref{p:GeneralErgodicEM}.
We claim that for any small enough $\varsigma \in (0,1)$, there exists some positive constant $C_\varsigma$, depending on $\varsigma$ but not on $\eta$ such that
\begin{eqnarray}\label{e:claim-1}
\mathbb{E} \left| \int_0^1 \int_0^1 r \langle \delta \delta^{\prime}, \nabla^2 f(W+\tilde{r} r \delta )-\nabla^2 f(W) \rangle_{\textrm{HS}} 1_{\{|\delta|< 1\} } \mathrm{d}\tilde{r} \mathrm{d} r \right|
\ &\leq& \ C_{\varsigma} \eta^{\frac{3-\varsigma}{2}},
\end{eqnarray}
and there exists some positive constant $C$, independent of $\eta$ such that
\begin{eqnarray}\label{e:claim-2}
\mathbb{E} \left| \int_0^1 \int_0^1 r \langle \delta \delta^{\prime}, \nabla^2 f(W+\tilde{r} r \delta )-\nabla^2 f(W) \rangle_{\textrm{HS}} 1_{ \{ |\delta|\geq 1 \} } \mathrm{d}\tilde{r} \mathrm{d} r \right|
\ &\leq&\ C\eta^{\frac{3}{2}}\,.
\end{eqnarray}
Combining \eqref{e:claim-1} and \eqref{e:claim-2}, we know for any small enough $\varsigma \in (0,1)$, there exists some positive constant $C_\varsigma$, depending on $\varsigma$ but not on $\eta$ such that
\begin{eqnarray}\label{e:Afh-2}
|{\rm II}| \ \leq \ C_{\varsigma} \eta^{\frac{3-\varsigma}{2}}.
\end{eqnarray}
Combining \eqref{e:Afh}, \eqref{e:Afh-1}, \eqref{e:Afh-2} and Eq. \eqref{e:PoiLip}, we know for any small enough $\varsigma \in (0,1)$, there exists some positive constant $C_\varsigma$, depending on $\varsigma$ but not on $\eta$ such that
\begin{eqnarray*}
d_{W}(\tilde \mu_{\eta},\mu)
&=& \sup_{h\in {\rm Lip_0(1)}} |\mathbb{E} h(W)-\mu(h)|
\ = \ \sup_{h\in {\rm Lip_0(1)} } | \mathbb{E}\mathcal{A}f(W)|
\ \leq \ C_{\varsigma}\eta^{\frac{1-\varsigma}{2}}.
\end{eqnarray*}
It remains to show that inequalities \eqref{e:claim-1} and \eqref{e:claim-2} hold. It follows from \eqref{e:3f} in Lemma \ref{lem:Lipregf} and Proposition \ref{p:GeneralErgodicEM} that for any small enough $\varsigma \in (0,1)$, there exists some positive constant $C_\varsigma$, depending on $\varsigma$ but not on $\eta$ such that
\begin{eqnarray*}
&& \mathbb{E} \left| \int_0^1\int_0^1 r \langle \delta \delta^{\prime}, \nabla^2 f(W+\tilde{r} r \delta )-\nabla^2 f(W) \rangle_{\textrm{HS}} 1_{\{|\delta|< 1\} } \mathrm{d}\tilde{r} \mathrm{d} r \right| \\
&\leq& C_\varsigma \mathbb{E}[ |\delta|^{3-\varsigma} (1+|W|^4) 1_{\{|\delta|< 1\} } ] \\
&\leq& C_\varsigma \mathbb{E}[ |\eta g(W) + \sigma \eta^{\frac{1}{2}} \xi_1|^{3-\varsigma} (1+|W|^4) ] \\
&\leq& C_\varsigma \eta^{\frac{3-\varsigma}{2}} \mathbb{E}[1+|W|^7] \\
&\leq& C_\varsigma \eta^{\frac{3-\varsigma}{2}},
\end{eqnarray*}
where the third inequality holds from small $\eta<1$. Thus, inequality \eqref{e:claim-1} holds.
It follows from \eqref{e:2f} in Lemma \ref{lem:Lipregf} and Proposition \ref{p:GeneralErgodicEM} that for small $\eta<1$, there exists some positive constant $C$, independent of $\eta$ such that
\begin{eqnarray*}
&& \mathbb{E} \left| \int_0^1 \int_0^1 r \langle \delta \delta^{\prime}, \nabla^2 f(W+\tilde{r} r \delta )-\nabla^2 f(W) \rangle_{\textrm{HS}} 1_{ \{ |\delta|\geq 1 \} } \mathrm{d}\tilde{r} \mathrm{d} r \right| \\
&\leq& C \mathbb{E} \int_0^1 \int_0^1 r |\delta|^2 |\nabla^2 f(W+\tilde{r} r \delta )| 1_{ \{ |\delta|\geq 1 \} } \mathrm{d}\tilde{r} \mathrm{d} r \\
& & \quad + C \mathbb{E} \int_0^1 \int_0^1 r |\delta|^2 |\nabla^2 f(W)| 1_{ \{ |\delta|\geq 1 \} } \mathrm{d}\tilde{r} \mathrm{d} r \\
&\leq& C\mathbb{E}[ |\delta|^2 (1+|W|^3+|\delta|^3) 1_{ \{ |\delta|\geq 1 \} } ] \\
&\leq& C(\mathbb{E}[ |\delta|^4])^{\frac{1}{2}} \mathbb{P}^{\frac{1}{2}} ( |\delta|\geq 1 )
+C (\mathbb{E}[|W|^6])^{\frac{1}{2}} ( \mathbb{E}[ |\delta|^8] )^{\frac{1}{4}} \mathbb{P}^{\frac{1}{4}}( |\delta|\geq 1 )
+C\mathbb{E}[ |\delta|^5 ] \\
&\leq& C\eta^{\frac{3}{2}},
\end{eqnarray*}
where the last inequality holds from Chebyshev's inequality and Proposition \ref{p:GeneralErgodicEM}, that is,
\begin{eqnarray*}
\mathbb{P}( |\delta|\geq 1)
\ \leq \ \mathbb{E}|\delta|^k
\ \leq \ C\eta^{\frac{k}{2}} \mathbb{E}[1+|W|^k]
\ \leq \ C \eta^{\frac{k}{2}}
\end{eqnarray*}
for any integers $k\geq 1$. Thus, inequality \eqref{e:claim-2} holds.
(ii) By triangle inequality and using Proposition \ref{p:GeneralErgodicEM}, for any small enough $\varsigma \in (0,1)$, there exists some positive constant $C_\varsigma$, depending on $\varsigma$ but not on $\eta$ such that
\begin{eqnarray*}
d_W( \mathcal{L}(\tilde{X}_N^{\eta}),\mu)
&\leq& d_W( \mathcal{L}(\tilde{X}_N^{\eta}),\tilde{\mu}_{\eta})
+d_W(\tilde{\mu}_{\eta},\mu)
\ \leq \ C\eta^{-1} e^{-c N\eta} + C_\varsigma \eta^{\frac{1-\varsigma}{2}}.
\end{eqnarray*}
Taking $\eta=\delta^{\frac{2}{1-\varsigma}}$ and $N:=O(\delta^{ \frac{2}{\varsigma-1} } |\log \delta|)$, one has $\eta^{-1} e^{-cN\eta} \leq \delta$, it implies that
\begin{eqnarray*}
d_W( \mathcal{L}(\tilde{X}_N^{\eta}),\mu)
\ \leq \ \delta.
\end{eqnarray*}
The proof is complete.
\end{proof}
\begin{remark} \label{rem-prelimit-approx}
Recall the steady-state approximation of the $M/Ph/n+M$ model was studied in Braverman and Dai \cite{BD1}. The process $X_t$ is the limiting diffusion of the diffusion-scaled process $\hat{X}^n(t) = \frac{X^n(t) - n\gamma}{\sqrt{n}}$. The process $\hat{X}^n(t)$ admits a unique ergodic measure $\hat \mu^n$ by Gurvich \cite{Gur1}. The result in Braverman and Dai \cite{BD1}, implies that for any small enough $\varsigma \in (0,1)$, there exists some positive constant $C_\varsigma$, depending on $\varsigma$ but not on $n$ such that
\begin{equation} \label{e:BrDa-0}
d_W(\hat \mu^n, \mu) \ \leq \ C_\varsigma n^{\frac{\varsigma-1}{2}}.
\end{equation}
We apply the EM scheme developed in this paper to provide an approximation for the steady-state of $\hat{X}^n(t)$.
By Theorem \ref{thm:DDE} (i) and \eqref{e:BrDa-0}, there exists some positive constant $C_\varsigma$, depending on $\varsigma$ but not on $n$ such that
\begin{eqnarray*}
d_W(\hat \mu^n, \tilde{\mu}_\eta) \ \leq \ C_\varsigma (n^{\frac{\varsigma-1}{2}} + \eta^{\frac{1-\varsigma}{2}}) \ \leq \ C_\varsigma n^{\frac{\varsigma-1}{2}},
\end{eqnarray*}
as $\eta$ is sufficiently small (say $\eta=n^{-1}$). Moreover, by Theorem \ref{thm:DDE} (ii), we can only run the EM scheme $N=O(n \log n)$ steps and obtain that there exists some positive constant $C_\varsigma$, depending on $\varsigma$ but not on $n$ such that
\begin{eqnarray*}
d_W( \mathcal{L}(\tilde{X}_N^{\eta}),\hat \mu^n) \ \leq \ C_\varsigma n^{\frac{\varsigma-1}{2}}.
\end{eqnarray*}
We expect a similar result holds for the general $GI/Ph/n+GI$ queues, if the result in \eqref{e:BrDa-0} can be established for non-Markovian queues.
\end{remark}
\section{Proofs of Theorems \ref{thm:CLT} and \ref{thm:MDP}} \label{s:CLTMDP}
In order to determine the variance of CLT and MDP in Theorems \ref{thm:CLT} and \ref{thm:MDP}, we need to consider a new Stein's equation, whose regularity need to be studied by Malliavin calculus. All the proofs for the lemmas and propositions in this section are postponed to Appendix \ref{sec:AAS}.
\subsection{The second Stein's equation}
We consider the following Stein's equation: for $h\in \mathcal{B}_b(\mathbb{R}^d,\mathbb{R})$,
\begin{eqnarray}\label{e:SE}
\mathcal{A} f(x) \ = \ h(x) - \mu(h),
\end{eqnarray}
where $\mathcal{A}$ is defined in \eqref{e:A}, and $\mu$ is the invariant measure for the process $(X_t)_{t\geq 0}$ in \eqref{hSDEg} with semigroup $(P_{t})_{t\geq 0}$.
The following proposition plays a crucial role in the proof Theorems \ref{thm:CLT} and \ref{thm:MDP}.
\begin{proposition}\label{lem:regf}
Let $h\in \mathcal{B}_b(\mathbb{R}^d, \mathbb{R})$ and $f$ be the solution to the Stein's equation \eqref{e:SE}.
There exists some positive constant $C$ such that
\begin{eqnarray*}
|f(x)| \ &\leq& \ C\|h\|_\infty(1+|x|^2), \\
|\nabla f(x)| \ &\leq& \ C \|h\|_{\infty}(1+ |x|^2).
\end{eqnarray*}
\end{proposition}
Let us sketch the strategy for the proof of Proposition \ref{lem:regf} as the following. Unlike Stein's equation \eqref{e:PoiLip} above, the function $h$ on the right hand side of the equation \eqref{e:SE} belongs to $\mathcal B_b(\mathbb{R}^d,\mathbb{R})$, whereby
the regularity result in \cite[Theorem 4.1]{Gur1,Gur11C} cannot be applied to prove Proposition \ref{lem:regf}. Alternatively, we first prove the following two representations for
the solution $f$:
\begin{lemma}\label{prop:ST}
For any function $h\in \mathcal{B}_b(\mathbb{R}^d,\mathbb{R})$, the following statements hold.
(i) A solution to \eqref{e:SE} is given by
\begin{eqnarray}\label{e:SE1}
f(x) \ = \ - \int_0^{\infty} P_t[ h(x) - \mu(h) ] \mathrm{d} t.
\end{eqnarray}
(ii) The solution to \eqref{e:SE} is also given by
\begin{eqnarray*}
f(x) \ = \ \int_0^{\infty} e^{-\lambda t} P_t[ \lambda f(x) - h(x) + \mu(h) ] \mathrm{d} t, \quad \forall \lambda>0.
\end{eqnarray*}
\end{lemma}
Thanks to Lemma \ref{prop:ST}, to bound $\nabla f$ we only need to bound $\nabla P_t[ \lambda f(x) - h(x) + \mu(h)]$, whereby it is natural to see that the powerful Bismut-Elworthy-Li formula can be applied. Due to the non-differentiability of $g$, we first mollify the drift $g(x)$ by $\varepsilon$, then apply Malliavin calculus to the consequent mollified SDE \eqref{hSDEgApp} below, and finally let $\varepsilon \rightarrow 0$ to obtain the following Bismut-Elworthy-Li type formula:
\begin{lemma}\label{hLef2}
Let $\psi \in \mathcal{C}^1(\mathbb{R}^d,\mathbb{R})$ be such that $\| \nabla \psi \|_{\infty} < \infty$. For every $t>0$, $x, u \in \mathbb{R}^d$, we have
\begin{eqnarray*}
\nabla_{u} \mathbb{E} [\psi (X^{x}_t)] \ = \ \mathbb{E}[\psi(X^{x}_t) \mathcal{I}_{u}^x(t)],
\end{eqnarray*}
where $\mathcal{I}_{u}^x(t)$ is defined as the second relation in \eqref{e:Iuxt} below.
\end{lemma}
The aforementioned strategy for the proof of Lemma \ref{hLef2} is detailed in Section \ref{sub:MolDif} below. We will see that the weighted occupation time plays a crucial role.
Combining Lemmas \ref{prop:ST} and \ref{hLef2}, we prove Proposition \ref{lem:regf} with the details addressed in Appendix \ref{sec:AAS}. All the proofs for the lemmas and propositions in this section will be given in Appendix \ref{sec:AAS}.
\subsection{A mollified diffusion} \label{sub:MolDif}
We shall use the Malliavin calculus to get the regularity in Lemma \ref{lem:regf} for $h\in \mathcal{B}_b(\mathbb{R}^d,\mathbb{R})$. Since $g(x)$ is not differentiable, we consider an approximation of SDE \eqref{hSDEg} by mollifying the drift:
\begin{eqnarray}\label{hSDEgApp}
\mathrm{d} X^{\varepsilon}_t \ = \ g_\varepsilon (X^{\varepsilon}_t) \mathrm{d} t +\sigma \mathrm{d} B_t,
\end{eqnarray}
where
\begin{eqnarray*}
g_{\varepsilon} (x) \ = \ -\beta p -Rx+\rho_{\varepsilon}({\rm e}^{\prime} x) (R-\alpha I)p,
\end{eqnarray*}
and $\rho_{\varepsilon}$ is defined as below: for $0<\varepsilon<1$,
$$\rho_{\varepsilon}(y) \ = \ \begin{cases}
0, & \quad y<-\varepsilon, \\
y, & \quad y>\varepsilon, \\
\frac{3\varepsilon}{16} - \frac{ 1 }{ 16 \varepsilon^3} y^4 + \frac{ 3 }{8 \varepsilon} y^2 + \frac{1}{2} y, & \quad |y| \le \varepsilon. \\
\end{cases}$$
It is easy to check that $\|\dot{\rho}_{\varepsilon}\|_{\infty} \leq 1$ and that
\begin{eqnarray}
g_{\varepsilon} \ & \in & \ \mathcal{C}^2(\mathbb{R}^d,\mathbb{R}^d), \nonumber \\
\nabla g_{\varepsilon}(x) \ & = & \ -R +\dot{\rho}_{\varepsilon} ({\rm e}^{\prime} x) (R-\alpha I)p {\rm e}^{\prime} , \label{e:Nge} \\
\nabla^2 g_{\varepsilon}(x) \ & =& \ \ddot{\rho}_{\varepsilon}({\rm e}^{\prime} x){\rm e}\otimes(R-\alpha I) p{\rm e}^{\prime} . \nonumber
\end{eqnarray}
Moreover, we can see that $ \lim\limits_{\varepsilon \rightarrow 0}g_{\varepsilon}(x)=g(x)$ for all $x\in \mathbb{R}^d$ and
$$\lim_{\varepsilon \rightarrow 0}\nabla g_{\varepsilon}(x) \ = \ \begin{cases}-R +(R-\alpha I)p {\rm e}^{\prime} , & \quad {\rm e}^{\prime} x>0,
\\
-R, & \quad {\rm e}^{\prime} x<0, \\
-R + \frac{1}{2}(R-\alpha I)p{\rm e}^{\prime} , & \quad {\rm e}^{\prime} x=0.
\end{cases}$$
\subsubsection{Moment estimates and Jacobi flows}
\begin{lemma}\label{lem:XXem}
For all $x \in \mathbb{R}^{d}$, $t\geq 0$ and intergers $m \geq 2$, we have
\begin{eqnarray} \label{e:XXem}
\mathbb{E} |X^{\varepsilon,x}_t|^{m}, \mathbb{E}|X^{x}_{t}|^{m} \ &\leq& \ e^{C_m t}(|x|^{m}+1),
\end{eqnarray}
where $C_m$ is in \eqref{Cm}.
Moreover, we have as $\varepsilon \to 0$,
\begin{eqnarray}
\mathbb{E}|X^{\varepsilon,x}_{t}-X^{x}_{t}|^m \rightarrow 0, \qquad t \geq 0. \label{e:XeCon-1}
\end{eqnarray}
\end{lemma}
We consider the derivative of $X^{\varepsilon,x}_t$ with respect to initial value $x$, which is called the Jacobi flow. Let $u \in \mathbb{R}^d$ and the Jacobi flow $\nabla_{u} X^{\varepsilon,x}_{t}$ along the direction $u$ be defined as
\begin{eqnarray*}
\nabla_{u} X^{\varepsilon,x}_{t} &=& \lim_{\varepsilon_1 \rightarrow 0} \frac{ X^{\varepsilon,x+\varepsilon_1 u}_t - X^{\varepsilon,x}_t }{\varepsilon_1}, \quad t\geq 0.
\end{eqnarray*}
The above limit exists and satisfies
\begin{eqnarray*}
\frac{\mathrm{d}}{\mathrm{d} t}\nabla_{u} X^{\varepsilon,x}_{t} &=&\nabla g_{\varepsilon} (X^{\varepsilon,x}_t) \nabla_{u} X^{\varepsilon,x}_{t}, \quad \nabla_u X_{0}^{\varepsilon,x}=u.
\end{eqnarray*}
Define
\begin{eqnarray*}
J_{s,t}^{\varepsilon,x}:&=&\exp \left( \int_s^t \nabla g_{\varepsilon} (X_{r}^{\varepsilon,x}) \mathrm{d} r \right), \ \ \ \ \ \ 0 \le s \le t<\infty.
\end{eqnarray*}
It is called the Jacobian between $s$ and $t$. For notational simplicity, denote $J^{\varepsilon,x}_{t}=J^{\varepsilon,x}_{0,t}$. Then we have
\begin{eqnarray}\label{hJaF1}
\nabla_{u} X^{\varepsilon,x}_{t} &=& J_{t}^{\varepsilon,x} u .
\end{eqnarray}
Define
$$\nabla \widetilde{g}(x):\ = \ -R +1_{\{{\rm e}^{\prime} x>0\}} (R-\alpha I)p {\rm e}^{\prime}.$$
It is easy to see that $\lim_{\varepsilon \rightarrow 0} \nabla g_{\varepsilon}(x)=\nabla \widetilde{g}(x)$ for all ${\rm e}^{\prime} x \ne 0$.
Because $g(x)$ is not differentiable for ${\rm e}^{\prime} x = 0$, it is necessary for us to define the above $\widetilde{g}(x)$ which takes the same value as $\nabla g(x)$ for ${\rm e}^{\prime} x \ne 0$ and has a definition on ${\rm e}^{\prime} x=0$. Define
$$J^{x}_{s,t}:\ = \ \exp\left(\int_{s}^{t} \nabla \widetilde{g} (X^{x}_{r}) \mathrm{d} r\right), \ \ \ \ \ \ x \in \mathbb{R}^{d}, \ \ 0 \le s \le t<\infty.$$
We also denote $J^{x}_{t}=J^{x}_{0,t}$.
Then we have the following lemma.
\begin{lemma} \label{l:XeCon}
For any $x \in \mathbb{R}^{d}$, as $\varepsilon \rightarrow 0$, the following holds:
\begin{eqnarray*}
\|J_{s,t}^{\varepsilon,x}-J^{x}_{s,t}\|_{{\rm op}}\ {\longrightarrow} \ 0, \ \ \ \ \ \ \ & & 0 \le s \le t<\infty, \ \ \ \ \ { \rm a.s. }
\end{eqnarray*}
\end{lemma}
Now we give estimates for $\|J^{\varepsilon,x}_{s,t}\|_{ {\rm op} }$ and $\|J^{x}_{s,t}\|_{ {\rm op} }$. By \eqref{e:Nge}, we can easily see that
\begin{eqnarray} \label{e:Ngeop}
\|\nabla g_{\varepsilon}(x)\|_{ {\rm op} } \ \le \ \|R\|_{ {\rm op} }+\|(R-\alpha I) p {\rm e}^{\prime} \|_{ {\rm op} } \ = \ C_{{\rm op}},
\end{eqnarray}
from which we obtain
\begin{eqnarray*}
\|J^{\varepsilon,x}_{s,t}\|_{ {\rm op} }
& \le & \exp\left(\int_s^{t} \|\nabla g_{\varepsilon}(X^{\varepsilon,x}_{r})\|_{ {\rm op} } \mathrm{d} r\right) \ \le \ e^{C_{{\rm op}}(t-s)}.
\end{eqnarray*}
So for all $0 \le s \le t<\infty$, we have
\begin{eqnarray}\label{e:JJe}
\|J^{x}_{s,t}\|_{ {\rm op} },\|J^{\varepsilon,x}_{s,t}\|_{ {\rm op} } \ \le \ e^{C_{{\rm op}}(t-s)},
\end{eqnarray}
where the bound of $\|J^{x}_{s,t}\|_{ {\rm op} }$ comes from the same argument since the bound in \eqref{e:Ngeop} also holds for $\nabla \widetilde{g}(x)$. Observe that the above estimates immediately implies that for $u\in\mathbb{R}^d$,
\begin{eqnarray*}
|\nabla_u X_t^x|,\,|\nabla_u X_t^{\varepsilon,x}| \ \leq \ e^{C_{{\rm op}}t}|u|.
\end{eqnarray*}
\subsubsection{Bismut's formula of Malliavin calculus for the mollified diffusion}
Let $v\in L^2_{\rm{loc}} ([0,\infty) \times (\Omega, \mathcal{F}, \mathbb{P}), \mathbb{R}^d)$, that is, $\mathbb{E} \int_0^t |v(s)|^2 \mathrm{d} s < \infty$ for all $t >0$. Assume that $v$ is adapted to the filtration $ (\mathcal{F}_t)_{t\geq 0}$ with $\mathcal{F}_t= \sigma(B_s: 0\leq s \leq t)$, that is, $v(t)$ is $\mathcal{F}_t$ measurable for $t\geq 0$. Define
\begin{eqnarray}\label{hMCV}
\mathbb{V}(t) \ = \ \int_0^t v(s) \mathrm{d} s, \quad t\geq 0.
\end{eqnarray}
For $t>0$, let $F_t: \mathcal{C}([0,t],\mathbb{R}^d) \rightarrow \mathbb{R}^d$ be a $\mathcal{F}_t$ measurable map. If the following limit exists
\begin{eqnarray*}
D_{\mathbb{V}} F_t(B) \ = \ \lim_{\varepsilon_1 \rightarrow 0} \frac{ F_t(B+\varepsilon_1 \mathbb{V}) - F_t(B) }{\varepsilon_1 }
\end{eqnarray*}
in $L^2 ( (\Omega, \mathcal{F}, \mathbb{P}), \mathbb{R}^d ) $, then $F_t(B)$ is said to be Malliavin differentiable and $D_{\mathbb{V}} F_t (B)$ is called the Malliavin derivative of $F_t(B)$ in the direction $\mathbb{V}$.
\textbf{Bismut's formula.} For Malliavin differentiable $F_t(B)$ such that $F_t(B), D_{\mathbb{V}} F_t(B) \in L^2((\Omega,\mathcal{F},\mathbb{P}), \mathbb{R}^d)$, we have
\begin{eqnarray}\label{e:BisFor}
\mathbb{E} [D_{\mathbb{V}} F_t(B)] \ = \ \mathbb{E} \left[ F_t(B) \int_0^t \langle v(s), \mathrm{d} B_s \rangle \right].
\end{eqnarray}
The following Malliavin derivative of $X^{\varepsilon,x}_t$ along the direction $\mathbb{V}$ exists in $L^2((\Omega,\mathcal{F},\mathbb{P}), \mathbb{R}^d)$ and is defined by
\begin{eqnarray}\label{MaC1}
D_{\mathbb{V}} X^{\varepsilon,x}_t &=& \lim_{\varepsilon_1 \rightarrow 0 }\frac{ X^{\varepsilon,x}_t(B+\varepsilon_1 \mathbb{V}) - X^{\varepsilon,x}_t(B) }{\varepsilon_1}.
\end{eqnarray}
It satisfies the following equation
\begin{eqnarray*}
D_{\mathbb{V}} X^{\varepsilon,x}_t &=& \sigma \mathbb{V}(t) + \int_0^t \nabla g_{\varepsilon} (X_{s}^{\varepsilon,x}) D_{\mathbb{V}} X_{s}^{\varepsilon,x} \mathrm{d} s, \quad D_{\mathbb{V}} X_{0}^{\varepsilon,x}=0,
\end{eqnarray*}
which is solved by
\begin{eqnarray*}
D_{\mathbb{V}} X^{\varepsilon,x}_t &=& \int_0^t J_{r,t}^{\varepsilon,x} \sigma v(r) \mathrm{d} r.
\end{eqnarray*}
Taking $v(r)=\frac{\sigma^{-1}}{ t} J^{\varepsilon,x}_{r}u$ for $0\leq r \leq t$, by \eqref{hJaF1}, we get
\begin{eqnarray} \label{e:DVNu}
D_{\mathbb{V}} X^{\varepsilon,x}_t&=& \nabla_{u} X^{\varepsilon,x}_{t} .
\end{eqnarray}
With the same $v$, a similar straightforward calculation gives that
\begin{eqnarray*}
D_{\mathbb{V}} X_{s}^{\varepsilon,x} & = &\frac{s}{t} \nabla_u X_{s}^{\varepsilon,x}, \quad 0 \le s \le t.
\end{eqnarray*}
For further use, for $x, u\in \mathbb{R}^d$, we define
\begin{eqnarray} \label{e:Iuxt}
\mathcal{I}_{u}^{\varepsilon,x}(t) \ : = \ \frac{1}{ t } \int_0^t \langle \sigma^{-1} J^{\varepsilon,x}_{r} u, \mathrm{d} B_r \rangle, \ \ \ \ \
\mathcal{I}_{u}^{x}(t) \ : = \ \frac{1}{ t } \int_0^t \langle \sigma^{-1} J^{x}_{r} u, \mathrm{d} B_r \rangle.
\end{eqnarray}
Now we are at the position to state the following lemmas.
\begin{lemma}\label{lem:EIm}
For all $x, u \in \mathbb{R}^d$, $m \ge 2$ and $t>0$, we have
\begin{eqnarray}
\mathbb{E} \left|\mathcal{I}_{u}^{\varepsilon,x}(t)\right|^m, \mathbb{E} \left|\mathcal{I}_{u}^{x}(t)\right|^m \ &\le& \ \frac{C |u|^{m}}{t^{m/2}} e^{mC_{ {\rm op} }t}, \label{e:IuexEst} \\
\lim_{\varepsilon \rightarrow 0} \mathbb{E} \left|\mathcal{I}_{u}^{\varepsilon,x}(t)- \mathcal{I}_{u}^{x}(t)\right|^m&=&0. \label{e:IueCon}
\end{eqnarray}
\end{lemma}
\subsubsection{Moment estimate for weighted occupation time} \label{app:occupation}
We introduce in this section a weighted occupation time for the limiting diffusion $X_t^x$, and study its moment estimate.
This result is used in the proof of Lemma \ref{l:XeCon}, and we expect that the method in this section may be used in future research.
For any $\epsilon>0$, the weighted occupation time $L_t^{\epsilon,x}$ is defined as
\begin{eqnarray*}
L_t^{\epsilon,x} &=& \int_0^t \left[-\frac{1}{\epsilon^2} ({\rm e}^{\prime} X_s^x)^2 +1 \right] 1_{ \{ |{\rm e}^{\prime} X_s^x| \leq \epsilon \} } \mathrm{d} s.
\end{eqnarray*}
We know $L_t^{\epsilon,x} \geq 0$ for all $t\geq 0$. We call $L_t^{\epsilon,x}$ weighted occupation time because it can be represented as an integral over an occupation measure associated with the limiting diffusion $X_t^x$.
Namely, we have
\begin{eqnarray*}
L^{\epsilon,x}_t & = & \int_0^t \psi({\rm e}' X^x_s) 1_{\{|{\rm e}' X^x_s| \le \epsilon \}} \mathrm{d} s \ = \ \int_{|y| \le \epsilon} \psi(y) A^{x}_t(\mathrm{d} y),
\end{eqnarray*}
where $\psi(y)=1-\frac{y^2}{\epsilon^2} $ and $A^{x}_t(\cdot):=\int_0^t \delta_{ {\rm e}' X^{x}_s} (\cdot) \mathrm{d} s$ with $\delta_z(\cdot)$ being the delta function of a given $z$. $A^x_t(\cdot)$ is called the occupation measure of ${\rm e}' X^x_s$ over time $[0,t]$.
If $\psi(x)=1$, then $L^{\epsilon,x}_t$ will be the occupation time of $({\rm e}' X^x_s)_{0 \le s \le t}$ on the set $\{|y| \le \epsilon \}$.
\begin{proposition}\label{lem:occupation}
For $L_t^{\epsilon,x}$ defined above, there exist some positive constants $C$ and $c$, independent of $\epsilon$ and $t$, such that
\begin{eqnarray*}
\mathbb{E} L_t^{\epsilon,x} &\leq& C\epsilon e^{c t} (1+|x|)(1+t).
\end{eqnarray*}
\end{proposition}
\subsection{Proofs of Theorems \ref{thm:CLT} and \ref{thm:MDP}}
We first prove Theorem \ref{thm:CLT} by It\^{o}'s formula and martingale CLT, and then Theorem \ref{thm:MDP} by a criterion from Wu \cite{WLM1}.
\begin{proof}[Proof of Theorem \ref{thm:CLT}]
For $(X_t)_{t\geq 0} $ in SDE \eqref{hSDEg} with $X_0=x$, using It\^{o}'s formula to $f$ which is the solution to Stein's equation \eqref{e:SE}, we have
\begin{eqnarray*}
f(X_t^x)-f(x)
&=&\int_0^t \mathcal{A}f(X_s^x)\mathrm{d} s+\int_0^t (\nabla f(X_s^x))^{\prime} \sigma \mathrm{d} B_s \nonumber \\
&=&\int_0^t [ h(X_s^x)-\mu(h)] \mathrm{d} s+\int_0^t (\nabla f(X_s^x))^{\prime} \sigma \mathrm{d} B_s,
\end{eqnarray*}
which implies that
\begin{eqnarray*}
\sqrt{t} \left[\frac{1}{t}\int_0^t \delta_{X_s^{x}}(h)\mathrm{d} s-\mu(h) \right]
&=&\frac{1}{\sqrt{t}}\int_0^t [ h(X_s^{x}) -\mu(h) ] \mathrm{d} s \\
&=&\frac{1}{\sqrt{t}} [ f(X_t^x)-f(x)]-\frac{1}{\sqrt{t}} \int_0^t (\nabla f(X_s^x))^{\prime} \sigma \mathrm{d} B_s.
\end{eqnarray*}
From Lemma \ref{lem:regf} and the estimate for $\mathbb{E} V(X_t^x)$ in \eqref{e:Vm}, we obtain
\begin{eqnarray*}
\mathbb{E} \left| \frac{1}{\sqrt{t}} [ f(X_t^{x})-f(x)] \right|
&\to& 0 { \ \ \rm as \ \ } t \to \infty.
\end{eqnarray*}
It follows from Lemmas \ref{lem:regf} and \ref{lem:AV2} and inequality \eqref{e:BV} that for some constant $C>0$ such that
\begin{eqnarray*}
\mu( |\sigma^{\prime} \nabla f|^2 ) \ \leq \ C\mu(V^4) \ < \ \infty .
\end{eqnarray*}
We write
\begin{eqnarray*}
\frac{1}{\sqrt{t}} \int_0^t (\nabla f(X_s^x))^{\prime} \sigma \mathrm{d} B_s
\ = \
\frac{1}{\sqrt{t}} \left(\int_0^1 + \int_1^2 +\cdots + \int_{\lfloor t \rfloor-1}^{\lfloor t \rfloor} + \int_{\lfloor t \rfloor}^t \right) (\nabla f(X_s^x))^{\prime} \sigma \mathrm{d} B_s,
\end{eqnarray*}
and denote
\begin{eqnarray*}
U_i \ = \ \int_{i-1}^i (\nabla f(X_s^x))^{\prime} \sigma \mathrm{d} B_s \text{ \ \ for \ \ } i=1,2,\cdots,\lfloor t \rfloor
\end{eqnarray*}
and
\begin{eqnarray*}
U_{\lfloor t \rfloor+1} \ = \ \int_{\lfloor t \rfloor}^t (\nabla f(X_s^x))^{\prime} \sigma \mathrm{d} B_s.
\end{eqnarray*}
Then one obtains that $U_i$'s are martingale differences and $\mathbb{E} U_i^2 <\infty$ for all $i=1,2,\cdots,\lfloor t \rfloor+1$. We claim that
\begin{eqnarray}
\lim_{t \to \infty} \mathbb{E}\left( \max_{1\leq i \leq \lfloor t \rfloor +1 } \frac{1}{\mu( |\sigma^{\prime} \nabla f|^2 ) t } |U_i|^2 \right) \ = \ 0, \label{e:CLT1} \\
\lim_{t \to \infty} \mathbb{E} \left| \sum_{i=1}^{\lfloor t \rfloor +1 } \frac{1}{\mu( |\sigma^{\prime} \nabla f|^2 ) t } |U_i|^2 -1 \right|^2 \ = \ 0. \label{e:CLT2}
\end{eqnarray}
We observe that equalities \eqref{e:CLT1} and \eqref{e:CLT2} imply
\begin{eqnarray*}
\mathbb{E}\left( \max_{1\leq i \leq \lfloor t \rfloor +1 } \frac{1}{ \sqrt{\mu( |\sigma^{\prime} \nabla f|^2 ) t} } |U_i| \right) \to 0
\text{ \ \ and \ \ }
\sum_{i=1}^{\lfloor t \rfloor +1 } \frac{1}{\mu( |\sigma^{\prime} \nabla f|^2 ) t } |U_i|^2 \stackrel{p}{\longrightarrow} 1
\end{eqnarray*}
as $t$ goes to infinity. Using the martingale CLT in Sethuraman \cite[Theorem 2]{SS1} due to McLeish \cite{MDL1}, one has
\begin{eqnarray*}
\frac{1}{\sqrt{\mu(|\sigma^{\prime} \nabla f|^2) t}}\sum_{i=1}^{\lfloor t \rfloor+1} U_i
\ = \ \frac{1}{\sqrt{\mu(|\sigma^{\prime} \nabla f|^2) t}} \int_0^t (\nabla f(X_s^x))^{\prime} \sigma \mathrm{d} B_s \ \Rightarrow \ \mathcal{N} (0,1) \text{ \ \ as \ \ } t \to \infty.
\end{eqnarray*}
Then, we obtain
\begin{eqnarray*}
\sqrt{t} \left[ \frac{1}{t}\int_0^t \delta_{X_s^{x}}(h)\mathrm{d} s-\mu(h) \right]
\ \Rightarrow \ \mathcal{N} (0,\mu(|\sigma^{\prime} \nabla f|^2))
\text{ \ \ as \ \ } t \to \infty.
\end{eqnarray*}
It remains to show that equalities \eqref{e:CLT1} and \eqref{e:CLT2} hold. For \eqref{e:CLT1}, one has
\begin{eqnarray*}
\mathbb{E}\left( \max_{1\leq i \leq \lfloor t \rfloor +1} |U_i|^2 \right)
&=& \mathbb{E}\left( \max_{1\leq i \leq \lfloor t \rfloor +1} (|U_i|^2 1_{ \{ |U_i|^2\leq \sqrt{t} \} } + |U_i|^2 1_{ \{|U_i|^2>\sqrt{t} \}}) \right) \nonumber \\
&\leq& \mathbb{E}\left( \max_{1\leq i \leq \lfloor t \rfloor+1} |U_i|^2 1_{ \{ |U_i|^2\leq \sqrt{t} \} } \right)
+ \mathbb{E}\left( \max_{1\leq i \leq \lfloor t \rfloor+1} |U_i|^2 1_{ \{|U_i|^2> \sqrt{t} \}} \right) \nonumber \\
&\leq& \sqrt{t} + \mathbb{E} \left( \max_{1\leq i \leq \lfloor t \rfloor +1} |U_i|^2 1_{ \{|U_i|^2> \sqrt{t} \}} \right) \nonumber \\
&\leq& \sqrt{t} +(\lfloor t \rfloor+1) \max_{1\leq i \leq \lfloor t \rfloor+1} \mathbb{E} (|U_i|^2 1_{ \{|U_i|^2> \sqrt{t} \}} ),
\end{eqnarray*}
where the last inequality holds from that
\begin{eqnarray*}
\mathbb{E} \left( \max_{1\leq i \leq \lfloor t \rfloor +1} |U_i|^2 1_{ \{|U_i|^2> \sqrt{t} \}} \right) \ &\leq& \ \sum_{i=1}^{ \lfloor t \rfloor +1} \mathbb{E} (|U_i|^2 1_{ \{|U_i|^2> \sqrt{t} \}} ).
\end{eqnarray*}
Thus, we obtain
\begin{eqnarray*}
&& \mathbb{E} \left( \max_{1\leq i \leq \lfloor t \rfloor+1 } \frac{1}{ \mu( |\sigma^{\prime} \nabla f|^2 ) t} | U_i |^2 \right) \\
&\leq& \frac{1}{ \mu( |\sigma^{\prime} \nabla f|^2 ) t} \left[ \sqrt{t} +(\lfloor t \rfloor+1) \max_{1\leq i \leq \lfloor t \rfloor +1} \mathbb{E} (|U_i|^2 1_{ \{|U_i|^2> \sqrt{t} \}} ) \right] \\
&\to & 0 \text{ \ \ as \ \ } t\to \infty.
\end{eqnarray*}
For \eqref{e:CLT2}, we have
\begin{eqnarray*}
&& \mathbb{E} \left| \sum_{i=1}^{ \lfloor t \rfloor +1 } \frac{1}{\mu( |\sigma^{\prime} \nabla f|^2 ) t } |U_i|^2 -\frac{ \lfloor t \rfloor +1 }{t} \right|^2
\ = \ \mathbb{E} \left| \frac{1}{t} \sum_{i=1}^{ \lfloor t \rfloor +1 } \left( \frac{1}{\mu( |\sigma^{\prime} \nabla f|^2 ) } |U_i|^2 - 1 \right) \right|^2 \\
&=& \frac{1}{t^2} \sum_{i=1}^{ \lfloor t \rfloor +1 }\mathbb{E}\left( \frac{1}{\mu( |\sigma^{\prime} \nabla f|^2 ) } |U_i|^2 - 1 \right)^2 \\
&&+\frac{2}{t^2}\sum_{i<j}\mathbb{E}\left[ ( \frac{1}{\mu( |\sigma^{\prime} \nabla f|^2 ) } |U_i|^2 - 1 ) ( \frac{1}{\mu( |\sigma^{\prime} \nabla f|^2 ) } |U_j|^2 - 1 ) \right] \\
&=:& {\rm I} + {\rm II}.
\end{eqnarray*}
Using the Burkholder-Davis-Gundy inequality, there exists some positive constant $C$ such that
\begin{eqnarray*}
\mathbb{E}|U_i|^4
&=& \mathbb{E} \left| \int_{i-1}^i (\nabla f(X_s^x))^{\prime} \sigma \mathrm{d} B_s \right|^4
\ \leq \ C \mathbb{E} \left| \int_{i-1}^i |(\nabla f(X_s^x))^{\prime} \sigma|^2 \mathrm{d} s \right|^2.
\end{eqnarray*}
Combining with Lemma \ref{lem:regf}, we know
\begin{eqnarray*}
\mathbb{E}\left( \frac{1}{\mu( |\sigma^{\prime} \nabla f|^2 ) } |U_i|^2 - 1 \right)^2
&=& \mathbb{E} \left[ \frac{1}{\mu^2(|\sigma^{\prime} \nabla f|^2)} |U_i|^4 - \frac{2}{\mu( |\sigma^{\prime}\nabla f|^2)}|U_i|^2+1 \right] \\
&\leq & C(1+\mathbb{E} |X_i^x|^8).
\end{eqnarray*}
Combining this with \eqref{e:Vm} in Lemma \ref{lem:AV2} and inequality \eqref{e:BV}, we know
\begin{eqnarray*}
{\rm I}
\ &=& \ \frac{1}{t^2} \sum_{i=1}^{ \lfloor t \rfloor +1 } \mathbb{E}\left( \frac{1}{\mu( |\sigma^{\prime} \nabla f|^2 ) } |U_i|^2 - 1 \right)^2
\ \leq \ \frac{C}{t^2} \sum_{i=1}^{ \lfloor t \rfloor +1 } (1+\mathbb{E} |X_i^x|^8)
\to 0, {\ \ \rm as \ \ } t \to \infty.
\end{eqnarray*}
We also have
\begin{eqnarray*}
&& {\rm II}
\ = \ \frac{2}{t^2}\sum_{i<j}\mathbb{E} \left[ \left( \frac{1}{\mu( |\sigma^{\prime} \nabla f|^2 ) } |U_i|^2 - 1 \right) \left( \frac{1}{\mu( |\sigma^{\prime} \nabla f|^2 ) } |U_j|^2 - 1 \right) \right] \\
&=& \frac{2}{t^2}\sum_{i=1}^{\lfloor t \rfloor } \sum_{j=i+1}^{\lfloor t \rfloor +1} \mathbb{E} \left\{ \left(\frac{1}{\mu(|\sigma^{\prime} \nabla f|^2)}|U_i|^2-1\right) \mathbb{E} \left[ \left ( \frac{1}{\mu( |\sigma^{\prime} \nabla f|^2 ) } |U_j|^2 - 1 \right) |\mathcal{F}_i \right] \right\} \\
&=& \frac{2}{\mu( |\sigma^{\prime} \nabla f|^2 ) t^2}\sum_{i=1}^{\lfloor t \rfloor } \sum_{j=i+1}^{\lfloor t \rfloor +1} \mathbb{E} \left\{ \left(\frac{1}{\mu(|\sigma^{\prime} \nabla f|^2)}|U_i|^2-1\right) \mathbb{E}[(|U_j|^2 - \mu( |\sigma^{\prime} \nabla f|^2 ) ) |\mathcal{F}_i ] \right\} \\
&=& \frac{2}{\mu( |\sigma^{\prime} \nabla f|^2 ) t^2}\sum_{i=1}^{\lfloor t \rfloor } \sum_{j=i+1}^{\lfloor t \rfloor +1} \mathbb{E}\left\{ \left(\frac{1}{\mu(|\sigma^{\prime} \nabla f|^2)}|U_i|^2-1\right) \int_{j-1}^j [ \mathbb{E}|\sigma^{\prime} \nabla f(X_s^{X_i^x})|^2 - \mu( |\sigma^{\prime} \nabla f|^2 ) ] \mathrm{d} s \right\} \\
&=& \frac{2}{\mu( |\sigma^{\prime} \nabla f|^2 ) t^2}\sum_{i=1}^{\lfloor t \rfloor } \mathbb{E} \left\{ \left(\frac{1}{\mu(|\sigma^{\prime} \nabla f|^2)}|U_i|^2-1\right) \int_{i}^{\lfloor t \rfloor+1} [ \mathbb{E}|\sigma^{\prime} \nabla f(X_s^{X_i^x})|^2 - \mu( |\sigma^{\prime} \nabla f|^2 ) ] \mathrm{d} s \right\}.
\end{eqnarray*}
It follows from Lemma \ref{lem:AV2} that
\begin{eqnarray*}
{\rm II} &\leq& \frac{C}{t^2}\sum_{i=1}^{\lfloor t \rfloor } \mathbb{E} \left\{ \left|\frac{1}{\mu(|\sigma^{\prime} \nabla f|^2)}|U_i|^2-1 \right| \int_{0}^{\lfloor t \rfloor+1} (1+V^2(X_i^x))e^{-cs} \mathrm{d} s \right\} \\
&\leq& \frac{C}{t^2}\sum_{i=1}^{\lfloor t \rfloor } \mathbb{E} \left\{ \left|\frac{1}{\mu(|\sigma^{\prime} \nabla f|^2)}|U_i|^2-1 \right| (1+V^2(X_i^x)) \right\} \\
&\leq& \frac{C}{t^2}\sum_{i=1}^{\lfloor t \rfloor } \left[\mathbb{E} \left|\frac{1}{\mu(|\sigma^{\prime} \nabla f|^2)}|U_i|^2-1\right|^2\right]^{\frac{1}{2}} [1+\mathbb{E} V^4(X_i^x)]^{\frac{1}{2}} \\
&\to& 0, {\ \ \rm as \ \ } t \to \infty.
\end{eqnarray*}
Combining estimates for ${\rm I}$ and ${\rm II}$, we obtain
\begin{eqnarray*}
\lim_{t \to \infty} \mathbb{E} \left| \sum_{i=1}^{ \lfloor t \rfloor +1 } \frac{1}{\mu( |\sigma^{\prime} \nabla f|^2 ) t } |U_i|^2 -\frac{ \lfloor t \rfloor +1 }{t} \right|^2 \ = \ 0.
\end{eqnarray*}
Thus, \eqref{e:CLT2} holds. The proof is complete.
\end{proof}
\medskip
\begin{proof}[Proof of Theorem \ref{thm:MDP}]
It follows from Lemma \ref{lem:AV2} that for any function $\tilde{f}$ satisfying $\tilde{f}(x)\leq 1+V(x)$ for all $x\in \mathbb{R}^d$ with $V$ in \eqref{e:Lypfun}, there exist some positive constants $C$ and $c$ such that
\begin{eqnarray*}
| P_t \tilde{f}(x) - \mu(\tilde{f}) | \ \leq \ C(1+V(x)) e^{-c t}.
\end{eqnarray*}
It follows from Wu \cite[Remark (2.17)]{WLM2} that $P_t$ has a spectral gap near its largest eigenvalue $1$ which implies that $1$ is an isolate eigenvalue. Since $P_t c = c$ for all $t>0$, one has the property that $1$ is an eigenvalue of $P_t$ for all $t>0$. If there exists some function $\hat{f}$ satisfying $0\neq \hat{f}(x) \leq 1+V(x)$ for all $x\in \mathbb{R}^d$ such that $P_{t_1} \hat{f} = \lambda \hat{f}$ for some $t_1 > 0$ and $\lambda \in \mathbb{C}$ with $|\lambda|=1$, then $\lambda^{\tilde{n}} \hat{f} = P_{\tilde{n} t_1} \hat{f} \to \mu(\hat{f})$ as $\tilde{n} \to \infty$, so that $\hat{f}$ has to be constant and $\lambda=1$. Thus, $1$ is a simple and the only eigenvalue with modulus $1$ for $P_t$.
For $h\in \mathcal{B}_b(\mathbb{R}^d, \mathbb{R})$, it follows from Wu \cite[Theorem 2.1]{WLM1} that
$$\mathbb{P}\left( \frac{1}{a_t\sqrt{t}}\int_0^t [ h(X_s^x) - \mu(h) ] \mathrm{d} s \in \,\, \cdot \,\, \right)$$
satisfies the large deviation principle with speed $a_t^{-2}$ and rate function $I_h(z)=\frac{z^2}{2\mathcal{V}(h)}$ with
$$\mathcal{V}(h) \ = \ 2\int_0^{\infty} \langle P_t h, h-\mu(h) \rangle_{\mu} \mathrm{d} t,$$
that is,
\begin{eqnarray*}
-\inf_{z \in A^{ {\rm o} } } I_h (z)
&\leq& \liminf_{t\to\infty}\frac{1}{a_t^2}\log \mathbb{P} \left( \frac{\sqrt{t}}{a_t} \left[\mathcal E^x_t(h)-\mu(h)\right] \in A \right) \\
&\leq& \limsup_{t\to\infty}\frac{1}{a_t^2}\log \mathbb{P} \left( \frac{\sqrt{t}}{a_t} \left[\mathcal E^x_t(h)-\mu(h)\right] \in A \right)
\ \leq \ -\inf_{z \in \bar{A}} I_h (z),
\end{eqnarray*}
where $\bar{A}$ and $A^{ {\rm o} }$ are the closure and interior of set $A$ respectively.
We claim that
\begin{eqnarray}\label{e:Vh}
\mathcal{V}(h) \ = \ \mu(|\sigma^{\prime}\nabla f|^2 ),
\end{eqnarray}
where $f$ is the solution to Stein's equation \eqref{e:SE}. Then the desired result holds.
Now, we show that the claim \eqref{e:Vh} holds. With some calculations, one has
\begin{eqnarray*}
&& \frac{1}{t}\mathbb{E}^{\mu} \left( \int_0^t [h(X_s^x) - \mu(h)] \mathrm{d} s \right)^2 \\
&=& \frac{1}{t}\mathbb{E}^{\mu} \left( \int_0^t \int_0^t [h(X_s^x)-\mu(h)][h(X_u^x)-\mu(h)] \mathrm{d} u \mathrm{d} s \right) \\
&=& \frac{2}{t}\mathbb{E}^{\mu} \left( \int_0^t \int_0^u [h(X_s^x)-\mu(h)][h(X_u^x)-\mu(h)] \mathrm{d} s \mathrm{d} u \right) \\
&=& \frac{2}{t} \int_0^t \int_0^u \mathbb{E}^{\mu} \{ [h(X_s^x)-\mu(h)] \mathbb{E} \{ [h(X_u^x)-\mu(h)]| X_s^x \} \} \mathrm{d} s \mathrm{d} u \\
&=& \frac{2}{t} \int_0^t \int_0^u \mathbb{E}^{\mu} \{ h(X_s^x) \mathbb{E} \{ [h(X_u^x)-\mu(h)]| X_s^x \} \} \mathrm{d} s \mathrm{d} u,
\end{eqnarray*}
where the third equality holds from conditional probability and the last equality holds from
\begin{eqnarray*}
\int_0^t \int_0^u \mathbb{E}^{\mu} \{ \mu(h) \mathbb{E} \{ [h(X_u^x)-\mu(h)]| X_s^x \} \} \mathrm{d} s \mathrm{d} u
&=& \mu(h) \int_0^t \int_0^u \mathbb{E}^{\mu} \{ h(X_u^x)-\mu(h) \} \mathrm{d} s \mathrm{d} u \\
&=&0.
\end{eqnarray*}
Furthermore, for all $0\leq s \leq u<\infty$, one has
\begin{eqnarray*}
\mathbb{E}^{\mu} \{ h(X_s^x) \mathbb{E} \{ [h(X_u^x)-\mu(h)]| X_s^x \} \}
&=& \mathbb{E}^{\mu} \{ h(X_s^x) [ P_{u-s}h(X_s^x) - \mu(h) ] \} \\
& = & \int_{\mathbb{R}^d} h(y)[ P_{u-s}h(y)-\mu(h) ] \mu(\mathrm{d} y),
\end{eqnarray*}
which implies that
\begin{eqnarray*}
&& \frac{2}{t} \int_0^t \int_0^u \mathbb{E}^{\mu} \{ h(X_s^x) \mathbb{E} \{ [h(X_u^x)-\mu(h)]| X_s^x \} \} \mathrm{d} s \mathrm{d} u \\
&=& \frac{2}{t} \int_0^t\int_0^u \int_{\mathbb{R}^d} h(y)[ P_{u-s}h(y)-\mu(h) ] \mu(\mathrm{d} y) \mathrm{d} s \mathrm{d} u \\
&=& \int_{\mathbb{R}^d} h(y) \frac{2}{t} \int_0^t\int_0^u [ P_{u-s}h(y)-\mu(h) ] \mathrm{d} s \mathrm{d} u \mu(\mathrm{d} y) \\
&=& \int_{\mathbb{R}^d} h(y) \frac{2}{t} \int_0^t\int_0^u [ P_{\tilde{s}}h(y)-\mu(h) ] \mathrm{d} \tilde{s} \mathrm{d} u \mu(\mathrm{d} y).
\end{eqnarray*}
Using the Hospital's rule, one has
\begin{eqnarray*}
&& \lim_{t\to\infty}\int_{\mathbb{R}^d} h(y) \frac{2}{t} \int_0^t\int_0^u [ P_{\tilde{s}}h(y)-\mu(h) ] \mathrm{d} \tilde{s} \mathrm{d} u \mu(\mathrm{d} y) \\
&=&\lim_{t\to\infty}2 \int_{\mathbb{R}^d} h(y) \int_0^{t} [P_{\tilde{s}} h(y) - \mu(h)] \mathrm{d} \tilde{s} \mu(\mathrm{d} y) \\
&=& 2 \int_{\mathbb{R}^d} h(y) \int_0^{\infty} [P_{\tilde{s}} h(y) - \mu(h)] \mathrm{d} \tilde{s} \mu(\mathrm{d} y).
\end{eqnarray*}
From the expression of $\mathcal{V}(h)$, we know
\begin{eqnarray*}
\mathcal{V}(h)
&=& 2\int_0^{\infty} \langle P_t h, h-\mu(h) \rangle_{\mu} \mathrm{d} t
\ = \ 2\int_0^{\infty} \langle P_t h - \mu(h), h \rangle_{\mu} \mathrm{d} t,
\end{eqnarray*}
so that,
\begin{eqnarray*}
\frac{1}{t}\mathbb{E}^{\mu} \left( \int_0^t [h(X_s^x) - \mu(h)] \mathrm{d} s \right)^2
&\to& \mathcal{V}(h)
\text{ \ \ as \ \ } t \to \infty.
\end{eqnarray*}
Using It\^{o}'s formula and Stein's equation \eqref{e:SE}, we have
\begin{eqnarray*}
f(X_t^x)-f(x)
&=&\int_0^t \mathcal{A}f(X_s^x)\mathrm{d} s+\int_0^t (\nabla f(X_s^x))^{\prime} \sigma \mathrm{d} B_s \nonumber \\
&=&\int_0^t [ h(X_s^x)-\mu(h)] \mathrm{d} s+\int_0^t (\nabla f(X_s^x))^{\prime} \sigma \mathrm{d} B_s,
\end{eqnarray*}
which implies that
\begin{eqnarray*}
\frac{1}{t}\mathbb{E}^{\mu} \left( \int_0^t [h(X_s^x) - \mu(h)] \mathrm{d} s \right)^2
&=& \mathbb{E}^{\mu} \left( \frac{1}{\sqrt{t}} [ f(X_t^x)-f(x)] - \frac{1}{\sqrt{t}} \int_0^t (\nabla f(X_s^x))^{\prime} \sigma \mathrm{d} B_s \right)^2 \\
&\to& \mu(|\sigma^{\prime} \nabla f |^2) \text{ \ \ as \ \ } t \to \infty.
\end{eqnarray*}
Thus, we know $\mathcal{V}(h) = \mu( |\sigma^{\prime} \nabla f|^2 )$. The claim \eqref{e:Vh} holds. The proof is complete.
\end{proof}
\section{ Proof of Theorem \ref{thm:EMMDP} } \label{sec:EMCLTMDP}
In order to determine the variance in Theorem \ref{thm:EMMDP}, we need to study the third Stein's equation as the following
\begin{equation} \label{e:Stein-3}
\mathcal A_\eta f(x)=h(x)-\tilde{\mu}_\eta(h),
\end{equation}
where $\mathcal A_\eta f(x)=\tilde{\mathcal{P}}_{\eta} f(x)-f(x)$ with $\tilde{\mathcal{P}}_{\eta} f(x)=\mathbb{E} f(\tilde X^{\eta,x}_1)$ for all $x \in \mathbb{R}^d$. It is easy to verify that the solution of the equation \eqref{e:Stein-3} is
\begin{eqnarray} \label{e:fRep}
f(x) \ &=&-\sum_{k=0}^{\infty} \tilde{\mathcal{P}}^k_{\eta} \left( h(x)-\tilde{\mu}_{\eta}(h) \right),
\end{eqnarray}
because of $ \tilde{\mathcal{P}}_{\eta} f(x) = -\sum_{k=1}^{\infty} \tilde{\mathcal{P}}^k_{\eta} \left( h(x)-\tilde{\mu}_{\eta}(h) \right)$.
\begin{proof}[Proof of Theorem \ref{thm:EMMDP}]
(i) From Proposition \ref{p:GeneralErgodicEM}, we know the Markov chain $(\tilde{X}_{k}^{\eta})_{k\in \mathbb{N}_0}$ is exponentially ergodic under total variation distance and $\tilde{\mu}_{\eta}(V^{\ell})\leq C$ for any integers $\ell$ and $V$ in \eqref{e:Lypfun}. It follows from Jones \cite[Theorem 9]{jones2004markov} that for any function $h$ satisfying $|h|\leq V^{\ell}$ with some integer $\ell$ and any initial distribution $\tilde{X}_0^{\eta}$, one has
\begin{eqnarray*}
\sqrt{n}\left[ \mathcal E_n^{\eta,x}(h) - \tilde{\mu}_{\eta}(h) \right] \ \Rightarrow \ \mathcal{N}(0, \sigma_h^2)
\end{eqnarray*}
with
\begin{eqnarray}\label{e:sigmaf2}
\sigma_h^2
&=& \langle h-\tilde{\mu}_{\eta}(h), h-\tilde{\mu}_{\eta}(h) \rangle_{\tilde{\mu}_{\eta}} + 2\sum_{k=1}^{\infty} \langle \tilde{\mathcal{P}}^k_{\eta} h, h-\tilde{\mu}_{\eta}(h) \rangle_{\tilde{\mu}_{\eta}}.
\end{eqnarray}
We give some calculations for the variance of $\sqrt{n}\left[ \mathcal E_n^{\eta,x}(h) - \tilde{\mu}_{\eta}(h) \right]$. Since $\mathbb{E} [\sqrt{n} ( \mathcal E_n^{\eta,x}(h) - \tilde{\mu}_{\eta}(h) ) ]=0$, one has
\begin{eqnarray}\label{e:var-d1}
\mathbb{E} |\sqrt{n}\left[ \mathcal E_n^{\eta,x}(h) - \tilde{\mu}_{\eta}(h) \right]|^2
&=& \mathbb{E}| \frac{1}{\sqrt{n}} \sum_{k=0}^{n-1} [h(\tilde{X}^{\eta,x}_k) - \tilde{\mu}_{\eta}(h) ] |^2 \nonumber \\
&=& \frac{1}{n} \sum_{k=0}^{n-1}\mathbb{E} | h(\tilde{X}^{\eta,x}_k) - \tilde{\mu}_{\eta}(h) |^2 \nonumber \\
&& + \frac{2}{n} \sum_{i=0}^{n-1} \sum_{j=0}^{i-1}
\mathbb{E} \{ [h(\tilde{X}^{\eta,x}_i) - \tilde{\mu}_{\eta}(h) ] [h(\tilde{X}^{\eta,x}_j) - \tilde{\mu}_{\eta}(h) ] \}.
\end{eqnarray}
We aim to the second term on the right hand of above equality \eqref{e:var-d1}. By the conditional expectation, one has
\begin{eqnarray}\label{e:var-d2}
&& \frac{2}{n} \sum_{i=0}^{n-1} \sum_{j=0}^{i-1} \mathbb{E} \{ [h(\tilde{X}^{\eta,x}_i) - \tilde{\mu}_{\eta}(h) ] [h(\tilde{X}^{\eta,x}_j) - \tilde{\mu}_{\eta}(h) ] \} \nonumber \\
&=& \frac{2}{n} \sum_{i=0}^{n-1} \sum_{j=0}^{i-1} \mathbb{E} \{ [h(\tilde{X}^{\eta,x}_j) - \tilde{\mu}_{\eta}(h) ] \mathbb{E}[ (h(\tilde{X}^{\eta,x}_i) - \tilde{\mu}_{\eta}(h) ) | \tilde{X}^{\eta,x}_j ] \} \nonumber \\
&=& \frac{2}{n} \sum_{j=0}^{n-2} \mathbb{E} \{ [h(\tilde{X}^{\eta,x}_j) - \tilde{\mu}_{\eta}(h) ] \mathbb{E}[ \sum_{i=j+1}^{n-1} (h(\tilde{X}^{\eta,x}_i) - \tilde{\mu}_{\eta}(h) ) | \tilde{X}^{\eta,x}_j ] \} \nonumber \\
&=& \frac{2}{n} \sum_{j=0}^{n-2} \mathbb{E} \{ [h(\tilde{X}^{\eta,x}_j) - \tilde{\mu}_{\eta}(h) ] \mathbb{E}[ \sum_{k=1}^{n-1-j} \tilde{P}_{\eta}^k (h(\tilde{X}^{\eta,x}_j) - \tilde{\mu}_{\eta}(h) ) ] \} \nonumber \\
&=& \frac{2}{n} \sum_{j=0}^{n-2} \mathbb{E} \{ [h(\tilde{X}^{\eta,x}_j) - \tilde{\mu}_{\eta}(h) ] [ \sum_{k=1}^{n-1-j} \tilde{P}_{\eta}^k (h(\tilde{X}^{\eta,x}_j) - \tilde{\mu}_{\eta}(h) ) ] \} \nonumber \\
&\to& 2\sum_{k=1}^{\infty} \langle \tilde{\mathcal{P}}^k_{\eta} [h(x)-\tilde{\mu}_{\eta}(h)], h(x)-\tilde{\mu}_{\eta}(h) \rangle_{\tilde{\mu}_{\eta}}, \quad {\rm as \ \ } n\to \infty.
\end{eqnarray}
Combining \eqref{e:var-d1} and \eqref{e:var-d2}, we know
\begin{eqnarray*}
\mathbb{E} |\sqrt{n}\left[ \mathcal E_n^{\eta,x}(h) - \tilde{\mu}_{\eta}(h) \right]|^2
&\to& \langle h-\tilde{\mu}_{\eta}(h), h-\tilde{\mu}_{\eta}(h) \rangle_{\tilde{\mu}_{\eta}} + 2\sum_{k=1}^{\infty} \langle \tilde{\mathcal{P}}^k_{\eta} h, h-\tilde{\mu}_{\eta}(h) \rangle_{\tilde{\mu}_{\eta}},
\end{eqnarray*}
as $n\to \infty$, which implies the equality \eqref{e:sigmaf2} holds.
Since $\langle \tilde{\mu}_{\eta}(h), h-\tilde{\mu}_{\eta}(h) \rangle_{\tilde{\mu}_{\eta}}=0$ and \eqref{e:fRep}, it is easy to verify that
\begin{eqnarray*}
\tilde{ \mathcal V} (h) &=& \langle f, f \rangle_{\tilde{\mu}_{\eta}}-\langle \tilde{\mathcal{P}}_{\eta} f, \tilde{\mathcal{P}}_{\eta} f \rangle_{\tilde{\mu}_{\eta}} \\
&=& \langle h-\tilde{\mu}_{\eta}(h), h-\tilde{\mu}_{\eta}(h) \rangle_{\tilde{\mu}_{\eta}} + 2\sum_{k=1}^{\infty} \langle \tilde{\mathcal{P}}^k_{\eta} h, h-\tilde{\mu}_{\eta}(h) \rangle_{\tilde{\mu}_{\eta}}
\ \ = \ \ \sigma_h^2.
\end{eqnarray*}
(ii) It follows from the proof of Proposition \ref{p:GeneralErgodicEM} that there exist some positive constants $C$ and $c$ such that
\begin{eqnarray*}
\|\tilde{\mathcal{P}}_{\eta}^n(x,\cdot) -\tilde{\mu}_{\eta} \|_{1+V} \ := \ \sup_{|f|\leq 1+V} | \tilde{\mathcal{P}}_{\eta}^n f(x)-\tilde{\mu}_{\eta}(f)|
\ \leq \ C\eta^{-1} e^{-c n\eta}.
\end{eqnarray*}
It follows from Wu \cite[Remark (2.17)]{WLM2} that $\tilde{\mathcal{P}}^n_{\eta}$ has a spectral gap near its largest eigenvalue $1$ which implies that $1$ is an isolate eigenvalue. Since $\tilde{\mathcal{P}}^k_{\eta} c = c$ for all $k \in \mathbb{N}_0$, one has that $1$ is an eigenvalue of $\tilde{\mathcal{P}}^k_{\eta}$ for all $k \in \mathbb{N}_0$. If there exists some function $\hat{f}$ satisfying $0\neq \hat{f}(x) \leq 1+V(x)$ for all $x\in \mathbb{R}^d$ such that $\tilde{\mathcal{P}}^{k_1}_{\eta} \hat{f} = \lambda \hat{f}$ for some $k_1 \in \mathbb{N}_0$ and $\lambda \in \mathbb{C}$ with $|\lambda|=1$, then $\lambda^{\tilde{n}} \hat{f} = \tilde{\mathcal{P}}^{\tilde{n} k_1}_{\eta} \hat{f} \to \mu(\hat{f})$ as $\tilde{n} \to \infty$, so that $\hat{f}$ has to be constant and $\lambda=1$. Thus, $1$ is a simple and the only eigenvalue with modulus $1$ for $\tilde{\mathcal{P}}_{\eta}^n$.
For $h\in \mathcal{B}_b(\mathbb{R}^d, \mathbb{R})$, it follows from Wu \cite[Theorem 2.1]{WLM1} that
$$\mathbb{P}\left( \frac{\sqrt{n}}{a_n} \left[ \mathcal E_n^{\eta,x}(h) - \tilde{\mu}_{\eta}(h)\right] \in \,\, \cdot \,\, \right)$$
satisfies the large deviation principle with speed $a_n^{-2}$ and rate function $I_h(z)=\frac{z^2}{2\tilde{ \mathcal V} (h)}$ with $\tilde{ \mathcal V}(h)$ in \eqref{e:EMVh}, that is,
\begin{eqnarray*}
-\inf_{z \in A^{ {\rm o} } } \frac{ z^2 }{2 \tilde{ \mathcal V}(h)}
\ &\leq& \
\liminf_{n \to\infty}\frac{1}{a_n^2}\log \mathbb{P} \left( \frac{\sqrt{n}}{a_n} \left[ \mathcal E_n^{\eta,x}(h) - \tilde{\mu}_{\eta}(h)\right] \in A\right) \\
\ &\leq& \
\limsup_{ n \to\infty}\frac{1}{a_n^2}\log \mathbb{P} \left( \frac{\sqrt{n}}{a_n} \left[ \mathcal E_n^{\eta,x}(h) - \tilde{\mu}_{\eta}(h)\right] \in A\right)
\ \leq \ -\inf_{z \in \bar{A}} \frac{ z^2 }{ 2 \tilde{ \mathcal V}(h) },
\end{eqnarray*}
where $\bar{A}$ and $A^{ {\rm o} }$ are the closure and interior of set $A$ respectively. The proof is complete.
\end{proof}
\begin{appendix}
\section{ Exponential ergodicity for Markov chain $(\tilde{X}_k^{\eta})_{k\in \mathbb{N}_0}$ } \label{App:GeneralErgodicEM}
It follows from Dieker and Gao \cite[Theorem 1]{DG1} that there exists a positive definite matrix $\tilde{Q}=( \tilde{Q}_{ij} )_{d\times d}$ with $\sum_{i,j=1}^d |\tilde{Q}_{ij}|=1$ such that
\begin{eqnarray*}
\tilde{Q}(-R) + (-R)^{\prime} \tilde{Q} \ &<& \ 0, \\
\tilde{Q}(-(I-p{\rm e}^{\prime})R) +(-R^{\prime}(I-{\rm e}p^{\prime}))\tilde{Q}
\ &\leq& \ 0.
\end{eqnarray*}
Let the function $\tilde{V}\in \mathcal{C}^2(\mathbb{R}^d,\mathbb{R}_+)$ be constructed in Dieker and Gao \cite[Eq. (5.24)]{DG1}, that is,
\begin{eqnarray*}
\tilde{V}(y)&=&({\rm e}^\prime y)^2+\kappa[y-p\phi({\rm e}^\prime y)]^\prime \tilde{Q}[y-p\phi({\rm e}^\prime y)], \qquad \forall y\in \mathbb{R}^d,
\end{eqnarray*}
where $\kappa$ is a positive constant and $\phi\in \mathcal{C}^2(\mathbb{R}, \mathbb{R})$ is a real-valued function which is defined as below:
\begin{eqnarray*}
\phi(z) &=&
\left\{
\begin{array}{lll}
z, & \text{if } z \ge 0, \\
-\frac{1}{2}, & \text{if } z \leq -1, \\
-\frac{1}{2}z^4 - z^3 + z, &\text{if } -1<z<0,
\end{array}
\right.
\end{eqnarray*}
it is easy to check that
\begin{eqnarray*}
(\nabla \tilde{V}(y) )^{\prime} &=& 2({\rm e}^{\prime} y) {\rm e}^{\prime} + 2\kappa [ y^{\prime} - p^{\prime} \phi ({\rm e}^{\prime} y) ] \tilde{Q} [ I - p {\rm e}^{\prime} \dot{\phi}({\rm e}^{\prime} y)] \text{ \ \ for all \ \ } y \in \mathbb{R}^d.
\end{eqnarray*}
Then there exists some positive constant $C$ such that
\begin{eqnarray}\label{e:NtlV}
|\nabla \tilde{V}(y)|
\ &\leq& \ C(1+|y|) \text{ \ \ for all \ \ } y \in \mathbb{R}^d.
\end{eqnarray}
The Lyapunov condition holds for function $\tilde{V}$ from Dieker and Gao \cite[proof of Theorem 3]{DG1}, that is, there exist some positive constants $c_1$ and $\check{c}_1$ satisfying
\begin{eqnarray}\label{e:AtlV}
\mathcal{A}\tilde{V}(y)
\ &\leq& \ -c_1 \tilde{V}(y)+\check{c}_1
\text{ \ \ for any \ \ }y \in \mathbb{R}^d,
\end{eqnarray}
where $\mathcal{A}$ is in \eqref{e:A}. Also, it follows from Dieker and Gao \cite[proof of Theorem 3]{DG1} that there exist some positive constants $\hat{C}_1, \hat{C}_2$, $\hat{c}_1$ and $\hat{c}_2$ such that
\begin{eqnarray}\label{e:BtlV}
\hat{c}_1|y|^2 - \hat{c}_2
\ \leq \
\tilde{V}(y)
\ \leq \ \hat{C}_1|y|^2+\hat{C}_2
\text{ \ \ for all \ \ } y \in \mathbb{R}^d.
\end{eqnarray}
Let the Lyapunov function $V$ be defined as
\begin{eqnarray}\label{e:Lypfun}
V(y) &=& \tilde{V}(y)+\hat{c}_2 \nonumber \\
&=& ({\rm e}^\prime y)^2+\kappa[y-p\phi({\rm e}^\prime y)]^\prime \tilde{Q}[y-p\phi({\rm e}^\prime y)] + \hat{c}_2, \qquad \forall y\in \mathbb{R}^d,
\end{eqnarray}
where $\hat{c}_2$ is in \eqref{e:BtlV}. Then one has
\begin{eqnarray}\label{e:BV}
\hat{c}_1|y|^2
\ \leq \
V(y)
\ \leq \ \hat{C}_1|y|^2+\hat{C}_2+\hat{c}_2
\text{ \ \ for all \ \ } y \in \mathbb{R}^d.
\end{eqnarray}
Furthermore, we know
\begin{eqnarray*}
\nabla V(y) \ = \ \nabla \tilde{V}(y),
\qquad
\nabla^2 V(y) \ = \ \nabla^2 \tilde{V}(y)
\text{ \ \ for all \ \ } y \in \mathbb{R}^d .
\end{eqnarray*}
Thus, \eqref{e:NtlV} also holds for function $V$, that is,
\begin{eqnarray}\label{e:NV}
|\nabla V(y)|
&\leq& C(1+|y|) \text{ \ \ for all \ \ } y \in \mathbb{R}^d,
\end{eqnarray}
and $\mathcal{A} V(y)=\mathcal{A}\tilde{V}(y)$ for all $y \in \mathbb{R}^d$.
Combining with \eqref{e:AtlV}, the Lyapunov condition also holds for the function $V$, that is,
\begin{eqnarray}\label{e:AV}
\mathcal{A} V(y)
&\leq& -c_1 (\tilde{V}(y)+\hat{c}_2)+\check{c}_1 +c_1\hat{c}_2
\ \leq \ -c_1 V(y)+\breve{c}_1
\text{ \ \ for any \ \ }y \in \mathbb{R}^d
\end{eqnarray}
with $\breve{c}_1=\check{c}_1 +c_1\hat{c}_2$.
\begin{lemma}\label{lem:AV2}
For $\mathcal{A}$ in \eqref{e:A}, $V$ in \eqref{e:Lypfun} and integers $\ell\geq 1$, there exists some positive constant $\breve{c}_{\ell}$ depending on $\ell$ such that
\begin{eqnarray*}
\mathcal{A} V^{\ell}(x)
&\leq& -c_1 V^{\ell}(x)+\breve{c}_{\ell},
\end{eqnarray*}
and
\begin{eqnarray}\label{e:Vm}
\mathbb{E} V^{\ell}(X_t^{x})
\ &\leq& \ e^{-c_1 t }V^{\ell}(x)+\frac{\breve{c}_{\ell}(1-e^{-c_1t})}{c_1},
\quad \forall t\geq 0.
\end{eqnarray}
In addition, $\mu(V^{\ell})\leq \frac{\breve{c}_{\ell}}{c_1}$ and $\mu(|\cdot|^{2\ell})\leq C$ where the constant $C$ depends on $\ell$. Furthermore, for any positive integers $\ell$ and probability measure $\nu$ satisfying $\nu(V^{\ell})<\infty$, there exist some positive constants $C$ and $c$, independent of $t$ such that
\begin{eqnarray*}
d_W(P_{t}^* \nu, \mu) \ &\leq& \ C(1+\nu(V)) e^{-c t}, \\
\| P_{t}^* \nu - \mu\|_{{\rm TV}} \ &\leq& \ \| P_{t}^* \nu - \mu\|_{{\rm TV, V^{\ell}}} \ \leq \ C(1+\nu(V^{\ell})) e^{-c t}.
\end{eqnarray*}
\end{lemma}
In order to prove Proposition \ref{p:GeneralErgodicEM}, we first show that $(\tilde{X}_k^{\eta})_{k\in \mathbb{N}_0}$ in \eqref{e:XD} is strong Feller and irreducible and the Lyapunov condition holds, and then we can get the exponential ergodicity from \cite{DFMS1, TT1}. We note that rather than using the classical Lyapunov function criterion, it is more convenient to use a modified version of the criterion in Douc et al. \cite{DFMS1} to establish the exponential ergodicity.
Rewrite \eqref{e:XD} as
\begin{eqnarray}\label{e:reXD}
\tilde{X}_{k+1}^{\eta}
&=& \tilde{X}_{k}^{\eta}+g(\tilde{X}_{k}^{\eta}) \eta + \sigma (B_{(k+1){\eta}} -B_{k\eta}),
\end{eqnarray}
where $k\in \mathbb{N}_0$ and $\tilde{X}^{\eta}_{0}$ is the initial value.
\begin{lemma}\label{lem:GePe}
For $\tilde{\mathcal{P}}_{\eta}$ defined before and $f\in \mathcal{B}_b(\mathbb{R}^d,\mathbb{R})$, one has
\begin{eqnarray*}
\| \nabla \tilde{\mathcal{P}}_{\eta} f \|_{\infty} &\leq& \| f \|_{\infty} (1+C_{\rm op}\eta) \| \sigma^{-1} \|^2_{\rm op} \| \sigma \|_{\rm op}\eta^{-\frac{1}{2}}d^{\frac{1}{2}},
\end{eqnarray*}
which implies $(\tilde{X}_k^{\eta})_{k\in \mathbb{N}_0}$ is strong Feller. Furthermore, $(\tilde{X}_{k}^{\eta})_{k\in \mathbb{N}_0}$ is irreducible.
\end{lemma}
\begin{lemma}\label{lem:Xgesm}
For $(X^{x}_t)_{t\geq 0}$ and $(\tilde{X}_{k}^{\eta,x})_{k\in \mathbb{N}_0}$ in \eqref{hSDEg} and \eqref{e:reXD}, respectively, and integers $\ell\geq 1$, there exists some positive constant $\tilde{C}_{\ell}$ depending on $\ell$ not on $\eta$ such that for $0\leq s <1$,
\begin{eqnarray*}
\mathbb{E} |X^{x}_s-x|^{2\ell} \ \leq \ \tilde{C}_{\ell}(1+V^{\ell}(x))s^{\ell}
{\rm \ \ \ and \ \ \ }
\mathbb{E} |X^{x}_{\eta} - \tilde{X}^{\eta,x}_{1}|^{2\ell} \ \leq \ \tilde{C}_{\ell} (1+V^{\ell}(x))\eta^{3\ell},
\end{eqnarray*}
where $V$ is in \eqref{e:Lypfun}.
\end{lemma}
\begin{lemma}\label{lem:PXeD}
For $(\tilde{X}^{\eta,x}_{k})_{k\in \mathbb{N}_0}$ in \eqref{e:reXD}, $V$ in \eqref{e:Lypfun} and integers $\ell\geq 1$, there exist some constants $\check{\gamma}_{\ell} \in(0,1)$ and $\check{K}_{\ell} \in[0,\infty)$ such that
\begin{eqnarray*}
\tilde{\mathcal{P}}_{\eta} V^{\ell}(x) &\leq& \check{\gamma}_{\ell} V^{\ell}(x)+\check{K}_{\ell},
\end{eqnarray*}
where $\check{\gamma}_{\ell}=e^{-c_1 \eta} + \tilde{C}_{\ell} \eta^{\frac{3}{2}}$ and $\check{K}_{\ell}= \frac{\breve{c}_{\ell}}{c_1}(1-e^{-c_1\eta}) + \tilde{C}_{\ell} \eta^{\frac{3}{2}}$ with $c_1$ and $\breve{c}_{\ell}$ in Lemma \ref{lem:AV2} and the constant $ \tilde{C}_{\ell} $ depends on $\ell$ but not on $\eta$.
\end{lemma}
We give proofs for Lemmas \ref{lem:AV2}, \ref{lem:GePe} \ref{lem:Xgesm} and \ref{lem:PXeD} at the end of Appendix \ref{App:GeneralErgodicEM}.
\begin{proof}[Proof of Proposition \ref{p:GeneralErgodicEM}]
The process $(\tilde{X}_k^{\eta})_{k\in \mathbb{N}_0}$ is strong Feller and irreducible from Lemma \ref{lem:GePe}. Then $(\tilde{X}_k^{\eta})_{k\in \mathbb{N}_0}$ has at most one invariant measure from Peszat and Zabczyk \cite[Theorem 1.4]{PZ1}. Combining Feller property in Lemma \ref{lem:GePe} and Lyapunov condition in Lemma \ref{lem:PXeD}, we know $(\tilde{X}_k^{\eta})_{k\in \mathbb{N}_0}$ is ergodic with unique invariant measure $\tilde \mu_{\eta}$ from Meyn and Tweedie \cite[Theorem 4.5]{MT2}. Thus, we obtain the exponential ergodicity from \cite{DFMS1, TT1}.
For any $n\in \mathbb{N}_0$ and $x\in \mathbb{R}^d$, let $V$ be in \eqref{e:Lypfun},
\begin{eqnarray*}
V_{n}(x) \ = \ e^{\frac{c_1}{8} n\eta}(1+V(x)), \quad
r(n) \ = \ \frac{c_1}{8}\eta e^{\frac{c_1}{8}n\eta}, \quad
\Psi(x) \ = \ 1+V(x),
\end{eqnarray*}
and
\begin{eqnarray*}
\mathscr{C}\ = \ \{x: V(x) \ \leq \ \frac{8 e^{\frac{c_1}{8}\eta}}{c_1 \eta}(1+\check{K}_{1}-\check{\gamma}_{1})-1 \},
\quad
b\ = \ \frac{8e^{\frac{c_1}{8}\eta}} {c_1\eta}(1+\check{K}_{1}-\check{\gamma}_{1}),
\end{eqnarray*}
where $\check{\gamma}_{1}=e^{-c_1 \eta} + \tilde{C}_{1} \eta^{\frac{3}{2}}$, $\check{K}_{1}= \frac{\breve{c}_{1}}{c_1}(1-e^{-c_1\eta}) + \tilde{C}_{1} \eta^{\frac{3}{2}}$ with $c_1$ and $\breve{c}_{1}$ in \eqref{e:AV}.
The set $\mathscr{C}$ is compact from \eqref{e:BV}.
It follows from Lemma \ref{lem:PXeD} that
\begin{eqnarray*}
&& \tilde{\mathcal{P}}_{\eta} V_{n+1}(x) + r(n)\Psi(x) \\
&\leq& \check{\gamma}_{1}e^{\frac{c_1}{8} \eta} V_n(x) +e^{\frac{c_1}{8}(n+1)\eta} (1+\check{K}_{1}-\check{\gamma}_{1})
+\frac{c_1}{8}\eta V_n(x) \\
&=& V_n(x)+\left( \check{\gamma}_{1}e^{\frac{c_1}{8}\eta} -1+\frac{c_1}{8}\eta \right) e^{\frac{c_1}{8}n\eta}(V(x)+1) +e^{\frac{c_1}{8}(n+1)\eta} (1+\check{K}_{1}-\check{\gamma}_{1}) \\
&=& V_n(x)+ \frac{c_1}{8} \eta e^{\frac{c_1}{8}n\eta} \left( \frac{ \check{\gamma}_{1}e^{\frac{c_1}{8}\eta}-1 +\frac{c_1}{8}\eta}{ \frac{c_1}{8} \eta} (V(x)+1) + \frac{e^{\frac{c_1}{8}\eta}}{\frac{c_1}{8}\eta} (1+\check{K}_{1}-\check{\gamma}_{1}) \right) \\
&\leq& V_n(x) + b r(n)1_{\mathscr{C}}(x),
\end{eqnarray*}
where the last inequality holds from that
$\check{\gamma}_{1}e^{\frac{c_1}{8}\eta} -1+\frac{c_1}{8}\eta\leq -\frac{1}{8} c_1 \eta$ for small enough $\eta>0$.
We claim that the compact set $\mathscr{C}$ is petite. It follows from Tuominen and Tweedie \cite[Theorem 2.1]{TT1} or Douc et al. \cite[Theorem 1.1]{DFMS1} that
\begin{eqnarray*}
\lim_{n\to \infty} r(n) \|\tilde{\mathcal{P}}_{\eta}^n(x,\cdot) -\tilde{\mu}_{\eta} \|_{\Psi} \ = \ 0,
\end{eqnarray*}
where $\|\tilde{\mathcal{P}}_{\eta}^n(x,\cdot) -\tilde{\mu}_{\eta} \|_{\Psi}= \sup_{|h|\leq \Psi} | \tilde{\mathcal{P}}_{\eta}^n h(x)-\tilde{\mu}_{\eta}(h)|$.
Since $h\in {\rm Lip}_0(1)$, it follows from the inequality \eqref{e:BV} that $h(x)\leq C(1+V(x))= C \Psi(x)$ for all $x\in \mathbb{R}^d$ with some constant $C\geq 1$. Then for any integers $k\geq 1$ and measure $\nu$ with $\nu(V)<\infty$, one has
\begin{eqnarray*}
d_W( (\tilde{\mathcal{P}}_{\eta}^k)^* \nu, \tilde{\mu}_{\eta})
\ &\leq& \ C\eta^{-1} e^{-c k\eta}, \\
\| (\tilde{\mathcal{P}}_{\eta}^k)^* \nu- \tilde{\mu}_{\eta} \|_{\rm TV}
\ &\leq& \ C\eta^{-1} e^{-c k\eta}.
\end{eqnarray*}
It follows from Lemma \ref{lem:PXeD} that
\begin{eqnarray*}
\int_{\mathbb{R}^d} \tilde{\mathcal{P}}_{\eta} V^{\ell}(x) \tilde{\mu}_{\eta}(\mathrm{d} x)
\ &\leq& \ \int_{\mathbb{R}^d} \check{\gamma}_{\ell} V^{\ell}(x) \tilde{\mu}_{\eta}(\mathrm{d} x) +\check{K}_{\ell},
\end{eqnarray*}
such that
\begin{eqnarray*}
\tilde{\mu}_{\eta}(V^{\ell})
\ &\leq& \ \check{\gamma}_{\ell} \tilde{\mu}_{\eta}(V^{\ell}) +\check{K}_{\ell},
\end{eqnarray*}
that is,
\begin{eqnarray*}
\tilde{\mu}_{\eta}(V^{\ell})
\ &\leq& \ \frac{\check{K}_{\ell}}{1-\check{\gamma}_{\ell}}
\ = \ \frac{\frac{\breve{c}_{\ell}}{c_1}(1-e^{-c_1\eta}) + \tilde{C}_{\ell} \eta^{\frac{3}{2}}}{1-e^{-c_1 \eta}- \tilde{C}_{\ell}\eta^{\frac{3}{2}}}
\ \leq \ \frac{2\frac{\breve{c}_{\ell}}{c_1} c_1 \eta }{\frac{1}{2}c_1 \eta}
\ = \ \frac{4\breve{c}_{\ell}}{c_1},
\end{eqnarray*}
where the last inequality holds from Taylor expansion for $e^{-c_1 \eta}$ with small $\eta>0$. This implies the desired inequality from the relationship between $V$ and $|\cdot|^2$ in \eqref{e:BV}.
To show the compact set $\mathscr{C}$ is petite. It suffices to show that
\begin{eqnarray}\label{e:petite}
p(\eta,x,z) \ \geq \ c \nu(z), \quad \forall x\in \mathscr{C},
\end{eqnarray}
where $p(\eta,x,z)$ is the density of $\tilde{X}_1^{\eta,x}$, $c$ is some positive constant and $\nu$ is a probability measure from Tuominen and Tweedie \cite[p. 778]{TT1}. Since
\begin{eqnarray}\label{e:pe}
p(\eta,x,z)
&=& ((2\pi)^d \eta^d {\rm det}(\sigma\sigma^{\prime}) )^{-\frac{1}{2}} \exp\left(-(z-x-\eta g(x))^{\prime} \frac{(\sigma\sigma^{\prime})^{-1}} {2\eta}(z-x-\eta g(x)) \right), \nonumber \\
\end{eqnarray}
denoting $\lambda_M$ and $\lambda_m$ as the maximum and minimum eigenvalues of matrix $\sigma \sigma^{\prime}$, respectively, and from the fact
$$
|z-x-\eta g(x)|^2 \ \leq \ 2|z|^2 + 4|x|^2 + 8\tilde{C}_{\rm op}^2 \eta^2(1+|x|^2), \quad \forall x, z \in \mathbb{R}^d,
$$
we obtain that $p(\eta,x,z)$ is bigger than
\begin{eqnarray*}
\left( (2\pi)^d \eta^d (\frac{1}{2}\lambda_m)^d \right)^{-\frac{1}{2}} \exp\left(-\frac{|z|^2} {\lambda_m\eta}\right)
\left(\frac{2\lambda_M}{\lambda_m} \right)^{-\frac{d}{2}} \exp\left(-\frac{\lambda_m^{-1}} {2\eta}(4|x|^2 + 8\tilde{C}_{\rm op}^2 \eta^2(1+|x|^2)) \right).
\end{eqnarray*}
Thus, inequality \eqref{e:petite} holds by taking
\begin{eqnarray*}
\nu(z) &=& \left((2\pi)^d \eta^d (\frac{1}{2}\lambda_m)^d \right)^{-\frac{1}{2}} \exp\left(-\frac{|z|^2} {\lambda_m\eta}\right),
\end{eqnarray*}
and
\begin{eqnarray*}
c&=& \inf_{ x \in \mathscr{C}} \left\{ \left( \frac{2\lambda_M}{\lambda_m} \right)^{-\frac{d}{2}} \exp\left(-\frac{\lambda_m^{-1}} {2\eta}(4|x|^2 + 8\tilde{C}_{\rm op}^2 \eta^2(1+|x|^2))\right) \right \} >0
\end{eqnarray*}
for compact set $\mathscr{C}$. The proof is complete.
\end{proof}
\begin{proof}[Proof of Lemma \ref{lem:AV2}]
(i) Recall that $\mathcal{A} V(x)\leq -c_1V(x)+\breve{c}_1$ for all $x\in \mathbb{R}^d$ in \eqref{e:AV} and the function $V$ in \eqref{e:Lypfun}.
Combining \eqref{e:BV}, \eqref{e:NV} and using the Young's inequality, we obtain that there exist some positive constants $\breve{c}_{\ell}$ such that for all $x\in \mathbb{R}^d$ and integers $\ell \geq 1$,
\begin{eqnarray*}
\mathcal{A}V^{\ell}(x)
&=& \ell V^{\ell-1}(x) \mathcal{A}V(x) + \frac{\ell(\ell-1)}{2}V^{\ell-2}(x) \langle \nabla V(x)(\nabla V(x))^{\prime}, \sigma \sigma^{\prime} \rangle_{\rm HS} \\
&\leq & -c_1 \ell V^{\ell}(x)+\breve{c}_1 \ell V^{\ell-1}(x) + \frac{\ell(\ell-1)}{2}V^{\ell-2}(x) \langle \nabla V(x)(\nabla V(x))^{\prime}, \sigma \sigma^{\prime} \rangle_{\rm HS} \\
&\leq& -c_1 V^{\ell}(x) + \breve{c}_{\ell}.
\end{eqnarray*}
Using It\^{o}'s formula, we know for all $t\geq 0$,
\begin{eqnarray*}
\mathbb{E} V^{\ell}(X_t^{x})
&=& V^{\ell}(x)+\int_0^t\mathbb{E} \mathcal{A}V^{\ell}(X_s^{x}) \mathrm{d} s
\ \leq \ V^{\ell}(x)+\int_0^t ( -c_1\mathbb{E} V^{\ell}(X_s^{x})+\breve{c}_{\ell}) \mathrm{d} s.
\end{eqnarray*}
This implies that (see Gurvich \cite[proof of Lemma 7.2]{Gur1})
\begin{eqnarray*}
\mathbb{E} V^{\ell}(X_t^{x})
&\leq& e^{-c_1 t }V^{\ell}(x)+\frac{\breve{c}_{\ell}(1-e^{-c_1t})}{c_1},
\quad \forall t\geq 0.
\end{eqnarray*}
Let $\chi: [0,\infty) \to [0,1]$ be a continuous function such that $\chi(r)=1$ for $0\leq r \leq 1$ and $\chi(r)=0$ for $r\geq 2$ and $L>0$ be a large number. It follows from \eqref{e:Vm} that
\begin{eqnarray*}
\mathbb{E} \left[ V^{\ell}(X_t^{x}) \chi\left(\frac{|X_t^{x}|}{L}\right) \right]
&\leq& e^{-c_1 t }V^{\ell}(x)+ \frac{\breve{c}_{\ell} (1-e^{-c_1 t})}{c_1}.
\end{eqnarray*}
First, taking $t\to \infty$, we have
\begin{eqnarray*}
\int_{\mathbb{R}^d} V^{\ell}(x) \chi\left(\frac{|x|}{L}\right) \mu(\mathrm{d} x)
&\leq& \frac{\breve{c}_{\ell}}{c_1},
\end{eqnarray*}
then as $L \to \infty$, one knows
\begin{eqnarray*}
\int_{\mathbb{R}^d} V^{\ell}(x) \mu(\mathrm{d} x) &\leq& \frac{\breve{c}_{\ell}}{c_1}.
\end{eqnarray*}
Combining with \eqref{e:BV}, we can get the inequality $\mu(|\cdot|^{2\ell})\leq C$ where $C$ depends on $\ell$.
(ii) With similar calculations from Dieker and Gao \cite[proof of Theorem 3]{DG1} and combining \eqref{e:Vm}, we obtain that there exist some positive constants $c$ and $C$, independent of $t$, such that for any positive integers $\ell$,
\begin{eqnarray*}
\| P^*_{t}\nu-\mu \|_{\rm{TV}, \rm{V}^{\ell} }
&\leq& C (1+\nu(V^{\ell}))e^{-c t}.
\end{eqnarray*}
Thus, the exponential ergodicity for $(X_t)_{t\geq 0}$ in Wasserstein-1 distance holds from \eqref{e:dWandTV} and \eqref{e:BV} by taking $\ell=1$. Furthermore, one has the following inequality $\| P_{t}^* \nu-\mu \|_{ \rm{TV}} \leq \| P_{t}^* \nu-\mu \|_{ \rm{TV}, \rm{V} }$ under $V(x)\geq 0$ for all $x\in \mathbb{R}^d$. The proof is complete.
\end{proof}
\begin{proof}[Proof of Lemma \ref{lem:GePe}]
(i) Since $\tilde{X}_{1}^{\eta,x}$ has the same law as Gaussian distribution $\mathcal{N}(x+g(x)\eta, \eta\sigma\sigma^{\prime})$ with the density function $p(\eta,x,z)$ in \eqref{e:pe}, one has
\begin{eqnarray*}
\tilde{\mathcal{P}}_{\eta}f(x)
&=& \mathbb{E} f(\tilde{X}_{1}^{\eta,x})
= \int_{\mathbb{R}^d} f(z)p(\eta, x, z) \mathrm{d} z.
\end{eqnarray*}
Thus,
\begin{eqnarray*}
\nabla \tilde{\mathcal{P}}_{\eta}f(x)
&=& \int_{\mathbb{R}^d} \eta^{-1} f(z)(I+\nabla g(x)\eta)(\sigma\sigma^{\prime})^{-1} (z-x-g(x)\eta) p(\eta,x,z) \mathrm{d} z.
\end{eqnarray*}
It implies that
\begin{eqnarray*}
|\nabla \tilde{\mathcal{P}}_{\eta}f(x)|
&\leq& \int_{\mathbb{R}^d} \eta^{-1} |f(z)| \|I+\nabla g(x)\eta\|_{\rm op} \| \sigma^{-1} \|^2_{\rm op}|z-x-g(x)\eta| p(\eta,x,z) \mathrm{d} z \\
&\leq& \| f \|_{\infty} (1+C_{\rm op}\eta) \| \sigma^{-1} \|^2_{\rm op} \| \sigma \|_{\rm op}\eta^{-\frac{1}{2}}d^{\frac{1}{2}},
\end{eqnarray*}
where the second inequality holds from that $\tilde{X}_{1}^{\eta,x}-x-g(x)\eta$ has the same law as Gaussian distribution $\mathcal{N}(0,\eta\sigma\sigma^{\prime})$.
(ii) For any $x,y\in \mathbb{R}^d$ and $r>0$, it follows from \eqref{e:reXD} that
\begin{eqnarray*}
\tilde{\mathcal{P}}_{\eta}(x,B(y,r))
&=& \mathbb{P}(\tilde{X}_{1}^{\eta,x} \in B(y,r) )
\ = \ \mathbb{P}(x+g(x)\eta+\sigma B_{\eta} \in B(y,r)) \\
&=&\mathbb{P}(\sigma B_{\eta} \in B(y-x- g(x)\eta,r))>0,
\end{eqnarray*}
where $B_{\eta}$ has the same law as Gaussian distribution $\mathcal{N}(0,\eta I)$.
Assuming that $\tilde{\mathcal{P}}_{\eta}^k(x,B(y,r))>0$ for any $x,y\in\mathbb{R}^d$, $r>0$ and some integer $k$, one has
\begin{eqnarray*}
\tilde{\mathcal{P}}_{\eta}^{k+1}(x,B(y,r))
=\int_{\mathbb{R}^d} \tilde{\mathcal{P}}_{\eta}^{k}(x,z) \tilde{\mathcal{P}}_{\eta}(z,B(y,r)) \mathrm{d} z
>0.
\end{eqnarray*}
Thus, the irreducibility holds by induction.
\end{proof}
\begin{proof}[Proof of Lemma \ref{lem:Xgesm}]
(i) Combining with Lemma \ref{lem:AV2} and with similar calculations for \eqref{e:Vm}, one has
\begin{eqnarray}\label{e:V4}
\mathbb{E} V^{\ell}(X^{x}_{\eta})
&\leq& e^{-c_1 \eta} V^{\ell}(x) + \frac{\breve{c}_{\ell}}{c_1}(1-e^{-c_1 \eta}).
\end{eqnarray}
From $(X_t)_{t \geq 0}$ in SDE \eqref{hSDEg} and $0<s<1$, one has
\begin{eqnarray*}
X^{x}_s &=& x + \int_0^s g(X^{x}_u) \mathrm{d} u +\sigma B_s.
\end{eqnarray*}
Using the H\"{o}lder inequality and the estimate $|g(x)| \leq \tilde{C}_{{\rm op}}(1+|x|)$ for all $x\in \mathbb{R}^d$ with $\tilde{C}_{{\rm op}}$ in \eqref{tlCop}, one obtains that there exists some positive constant $\tilde{C}_{\ell}$ depending on $\ell$ such that
\begin{eqnarray*}
|X^{x}_s-x|^{2\ell}
&=& \left| \int_0^s g(X^{x}_u)\mathrm{d} u+\sigma B_s \right|^{2\ell}
\ \leq \ \tilde{C}_{\ell} s^{2\ell-1} \int_0^s (1+V^{\ell}(X^{x}_u) ) \mathrm{d} u + \tilde{C}_{\ell}| B_s |^{2\ell},
\end{eqnarray*}
where the last inequality holds from \eqref{e:BV}. Combining with \eqref{e:V4}, there exists some positive constant $\tilde{C}_{\ell}$ depending on $\ell$ such that
\begin{eqnarray}\label{e:Exe}
\mathbb{E} |X^{x}_s -x |^{2\ell}
&\leq& \tilde{C}_{\ell} s^{2\ell-1} \int_0^s (1+\mathbb{E} V^{\ell}(X^{x}_u) ) \mathrm{d} u + \tilde{C}_{\ell} \mathbb{E} | B_s |^{2\ell} \nonumber \\
& \leq & \tilde{C}_{\ell} s^{\ell}(1+V^{\ell}(x)),
\end{eqnarray}
where the second inequality holds from \eqref{e:V4} and the last inequality holds for $0<s<1$.
(ii) From $(X_t)_{t\geq 0}$ and $(\tilde{X}^{\eta}_{k})_{k\in \mathbb{N}_0}$ in \eqref{hSDEg} and \eqref{e:reXD}, respectively, one has
\begin{eqnarray*}
X^{x}_{\eta} - \tilde{X}^{\eta,x}_{1}
&=& \int_0^{\eta} ( g(X^{x}_s) - g(x) ) \mathrm{d} s.
\end{eqnarray*}
Using the H\"{o}lder inequality and the estimate $\| \nabla g(x) \|_{{\rm op}}\leq C_{{\rm op}}$ for all $x\in \mathbb{R}^d$ with $C_{{\rm op}}$ in \eqref{Cop}, one obtains that there exists some positive constant $\tilde{C}_{\ell}$ depending on $\ell$ not on $\eta$ such that
\begin{eqnarray*}
|X^{x}_{\eta} - \tilde{X}^{\eta,x}_{1} |^{2\ell}
&\leq& \left| \int_0^{\eta} ( g(X^{x}_s) - g(x) ) \mathrm{d} s \right|^{2\ell}
\ \leq \ \tilde{C}_{\ell} \eta^{2\ell-1} \int_0^{\eta} |X^{x}_s - x|^{2\ell} \mathrm{d} s .
\end{eqnarray*}
Combining this with \eqref{e:Exe}, one obtains that there exists some positive constant $\tilde{C}_{\ell}$ depending on $\ell$ not on $\eta$ such that
\begin{eqnarray*}
\mathbb{E} |X^{x}_{\eta} - \tilde{X}^{\eta,x}_{1} |^{2\ell}
&\leq& \tilde{C}_{\ell} \eta^{2\ell-1} \int_0^{\eta} s^{\ell}(1+V^{\ell}(x)) \mathrm{d} s
\ \leq \ \tilde{C}_{\ell} (1+V^{\ell}(x))\eta^{3\ell}.
\end{eqnarray*}
The proof is complete.
\end{proof}
\begin{proof}[Proof of Lemma \ref{lem:PXeD}]
We use the method from Mattingly et al. \cite[Theorem 7.2]{MSH1} to get the desired inequality. For $V$ in \eqref{e:Lypfun}, one has
\begin{eqnarray*}
\mathbb{E} V^{\ell}(\tilde{X}^{\eta,x}_{1})
&\leq& \mathbb{E} V^{\ell}(X^{x}_{\eta})+\mathbb{E}| V^{\ell}(\tilde{X}^{\eta,x}_{1})-V^{\ell}(X^{x}_{\eta})|.
\end{eqnarray*}
Let $\tilde{C}_{\ell}$ be some constants depending on $\ell$ but not on $\eta$, whose values may vary from line to line.
We claim that for small $\eta \in (0,e^{-1})$,
\begin{eqnarray}\label{e:V-V4}
\mathbb{E} |V^{\ell}(\tilde{X}^{\eta,x}_{1}) -V^{\ell}(X^{x}_{\eta})|
&\leq& \tilde{C}_{\ell} (1+V^{\ell}(x))\eta^{\frac{3}{2}}.
\end{eqnarray}
Combining this with \eqref{e:V4}, one has
\begin{eqnarray*}
\mathbb{E} V^{\ell}(\tilde{X}^{\eta,x}_{1})
&\leq& \mathbb{E} V^{\ell}(X^{x}_{\eta}) + \mathbb{E} | V^{\ell}(\tilde{X}^{\eta,x}_{1}) -V^{\ell}(X^{x}_{\eta})| \\
&\leq& e^{-c_1 \eta} V^{\ell}(x) + \frac{\breve{c}_{\ell}}{c_1}(1-e^{-c_1\eta}) + \tilde{C}_{\ell} \eta^{\frac{3}{2}} (1+V^{\ell}(x)) \\
&\leq& (e^{-c_1 \eta} + \tilde{C}_{\ell} \eta^{\frac{3}{2}} ) V^{\ell}(x) + \frac{\breve{c}_{\ell}}{c_1}(1-e^{-c_1\eta}) + \tilde{C}_{\ell} \eta^{\frac{3}{2}},
\end{eqnarray*}
which implies
\begin{eqnarray*}
\tilde{\mathcal{P}}_{\eta} V^{\ell}(x)
&\leq& \check{\gamma}_{\ell} V^{\ell}(x)+\check{K}_{\ell},
\end{eqnarray*}
with $\check{\gamma}_{\ell}=e^{-c_1 \eta} + \tilde{C}_{\ell} \eta^{\frac{3}{2}}$ and $\check{K}_{\ell}= \frac{\breve{c}_{\ell}}{c_1}(1-e^{-c_1\eta}) + \tilde{C}_{\ell} \eta^{\frac{3}{2}}$ for small enough $\eta\in (0,e^{-1})$.
It remains to show the claim \eqref{e:V-V4} holds.
Since
\begin{eqnarray*}
&& V^{\ell}(\tilde{X}^{\eta,x}_{1})-V^{\ell}(X^{x}_{\eta}) \\
&=& \ell \int_0^1 (\tilde{X}^{\eta,x}_{1}-X^{x}_{\eta})^{\prime} \nabla V( X^{x}_{\eta} + r(\tilde{X}^{\eta,x}_{1}-X^{x}_{\eta}) ) V^{\ell-1}( X^{x}_{\eta} + r(\tilde{X}^{\eta,x}_{1}-X^{x}_{\eta}) ) \mathrm{d} r,
\end{eqnarray*}
it follows from \eqref{e:BV} that
\begin{eqnarray*}
|V^{\ell}(\tilde{X}^{\eta,x}_{1})-V^{\ell}(X^{x}_{\eta})|
&\leq& \tilde{C}_{\ell} \int_0^1 [1+|X^{x}_{\eta} + r(\tilde{X}^{\eta,x}_{1}-X^{x}_{\eta})|]^{2\ell-1} |\tilde{X}^{\eta,x}_{1}-X^{x}_{\eta}| \mathrm{d} r \\
& \leq & \tilde{C}_{\ell} (1+|X^{x}_{\eta}|^{2\ell-1})| \tilde{X}^{\eta,x}_{1}-X^{x}_{\eta}| +\tilde{C}_{\ell} |\tilde{X}^{\eta,x}_{1}-X^{x}_{\eta}|^{2\ell}.
\end{eqnarray*}
By the H\"{o}lder inequality and \eqref{e:BV}, one has
\begin{eqnarray*}
&& \mathbb{E} |V^{\ell}(\tilde{X}^{\eta,x}_{1}) -V^{\ell}(X^{x}_{\eta})| \\
&\leq& \tilde{C}_{\ell}(\mathbb{E} [1+|X^{x}_{\eta}|^{2\ell-1}] ^{\frac{2\ell}{2\ell-1}})^{\frac{2\ell-1}{2\ell}} [\mathbb{E} |\tilde{X}^{\eta,x}_{1}-X^{x}_{\eta}|^{2\ell}] ^{\frac{1}{2\ell}}+ \tilde{C}_{\ell} \mathbb{E} |\tilde{X}^{\eta,x}_{1}-X^{x}_{\eta}| ^{2\ell} \\
& \leq& \tilde{C}_{\ell}(\mathbb{E} [1+V^{\ell}(X^{x}_{\eta})]) ^{\frac{2\ell-1}{2\ell}} [\mathbb{E} |\tilde{X}^{\eta,x}_{1}-X^{x}_{\eta}|^{2\ell}] ^{\frac{1}{2\ell}}
+ \tilde{C}_{\ell} \mathbb{E} |\tilde{X}^{\eta,x}_{1}-X^{x}_{\eta}|^{2\ell}.
\end{eqnarray*}
Combining this with \eqref{e:V4} and Lemma \ref{lem:Xgesm}, for small $\eta \in (0,e^{-1})$, we have
\begin{eqnarray*}
&& \mathbb{E} |V^{\ell}(\tilde{X}^{\eta,x}_{1})-V^{\ell}(X^{x}_{\eta})| \\
&\leq& \tilde{C}_{\ell} \left(1+ e^{-c_1 \eta}V^{\ell}(x) + \frac{\breve{c}_{\ell}}{c_1} \right)^{\frac{2\ell-1}{2\ell}} \eta^{\frac{3}{2}}(1+V^{\ell}(x))^{\frac{1}{2\ell}} + \tilde{C}_{\ell} (1+V^{\ell}(x))\eta^{3\ell} \\
&\leq&\tilde{C}_{\ell} (1+V^{\ell}(x)) \eta^{\frac{3}{2}}.
\end{eqnarray*}
The proof is complete.
\end{proof}
\section{The proofs for the lemmas and propositions in Section \ref{s:CLTMDP}} \label{sec:AAS}
\subsection{The proof of Proposition \ref{lem:regf}}
\begin{proof}[Proof of Proposition \ref{lem:regf}]
(i) It follows from Eq. \eqref{e:SE1} and Lemma \ref{lem:AV2} that
\begin{eqnarray}\label{e:SE2}
|f(x)| & \leq & \int_0^{\infty} |P_t h(x) - \mu(h) | \mathrm{d} t
\ \leq \ \|h\|_{\infty} \int_0^{\infty} \| P^*_{t}\delta_x - \mu \|_{ \rm{TV}, \rm{V} } \mathrm{d} t \nonumber \\
&\leq& C \|h\|_{\infty}(1+V(x) )
\ \leq \ C \|h\|_{\infty}(1+|x|^2),
\end{eqnarray}
where the last inequality holds from the relationship between $V$ and $|\cdot|^2$ in \eqref{e:BV}.
(ii) First, let $h \in \mathcal{C}_b^1(\mathbb{R}^d,\mathbb{R})$, we have $\nabla_u\mathbb{E}[h(X_t^x)]=\mathbb{E}[\nabla_u h(X_t^x)]$ by Lebesgue's dominated convergence theorem. Then we consider the term $ \nabla_u [e^{-\lambda t} \mathbb{E} f(X_t^x)]$.
It follows from Lemma \ref{hLef2} that
\begin{eqnarray*}
\mathbb{E} [\nabla_u f(X_t^x)]
&=& \mathbb{E}[\nabla f(X_t^x) \nabla_u X_t^x]
\ = \ \mathbb{E}[\nabla f(X_t^x) D_{\mathbb{V}} X_t^x] \\
&= & \mathbb{E}[D_{\mathbb{V}} f(X_t^x)]
\ = \ \mathbb{E}[f(X_t^x)\mathcal{I}_{u}^x(t)].
\end{eqnarray*}
From \eqref{e:SE2} and the estimate for $\mathbb{E} V^2(X_t^x)$ in \eqref{e:Vm} of Lemma \ref{lem:AV2}, one has
\begin{eqnarray}\label{e:f2}
[ \mathbb{E} |f(X_t^x)|^2 ]^{\frac{1}{2}}
\ \leq \ C [ 1+ \mathbb{E} V^2(X_t^x)] ^{\frac{1}{2}}
\ \leq \ C e^{\frac{C_4}{2} t}(1+|x|^2).
\end{eqnarray}
Combining this with the estimate for $\mathbb{E} | \mathcal{I}_{u}^x(t)|^2$ in \eqref{e:IuexEst} and using the H\"{o}lder inequality, one has
\begin{eqnarray*}
\mathbb{E}|f(X_t^x)\mathcal{I}_{u}^x(t)|
&\leq& [\mathbb{E} f^2 (X_t^x) ]^{\frac{1}{2}} [\mathbb{E} |\mathcal{I}_{u}^x(t) |^2 ]^{\frac{1}{2}}
\ \leq \ C e^{\frac{C_4}{2} t }(1+ |x|^2) |u| t^{-\frac{1}{2}} e^{C_{{\rm op}}t}
\ < \ \infty.
\end{eqnarray*}
With similar calculations, one has
\begin{eqnarray*}
\mathbb{E}|h(X_t^x)\mathcal{I}_{u}^x(t)|
&\leq& \| h \|_{\infty} \mathbb{E} | \mathcal{I}_{u}^x(t) |
\ < \ \infty.
\end{eqnarray*}
Here we fix $x$ and $t>0$. Then by Lebesgue's dominated convergence theorem, we have
\begin{eqnarray*}
\nabla_u\mathbb{E}[e^{-\lambda t}f(X_t^x)] \ =\ \mathbb{E}[e^{-\lambda t}\nabla_u f(X_t^x)].
\end{eqnarray*}
It follows from the H\"{o}lder inequality that
\begin{eqnarray}\label{e:Nfx0}
&& \int_0^{\infty} \left|e^{-\lambda t} ( \lambda \nabla_{u}\mathbb{E} [ f (X_t^x) ] - \nabla_u\mathbb{E} [ h(X_t^x)] )\right| \mathrm{d} t \nonumber \\
&=& \int_0^{\infty}\left| e^{-\lambda t} ( \lambda \mathbb{E} [ f (X_t^x) \mathcal{I}_{u}^x(t) ] - \mathbb{E} [ h(X_t^x) \mathcal{I}_u^x(t) ] ) \right|\mathrm{d} t \nonumber \\
&\leq& \int_0^{\infty} e^{-\lambda t} \lambda [ \mathbb{E} f^2 (X_t^x) ]^{\frac{1}{2}} [ \mathbb{E} | \mathcal{I}_{u}^x(t) |^2 ]^{\frac{1}{2}} \mathrm{d} t
+ \int_0^{\infty} e^{-\lambda t} \| h \|_{\infty} \mathbb{E} [ |\mathcal{I}_u^x(t)| ] \mathrm{d} t.
\end{eqnarray}
From estimates for $\mathbb{E} | \mathcal{I}_{u}^x(t) |^2$ in \eqref{e:IuexEst} and $[ \mathbb{E} |f(X_t^x)|^2 ]^{\frac{1}{2}} $ in \eqref{e:f2}, one has
\begin{eqnarray}\label{e:Nfx1}
\int_0^{\infty} e^{-\lambda t} \lambda [ \mathbb{E} f^2 (X_t^x) ]^{\frac{1}{2}} [ \mathbb{E} | \mathcal{I}_{u}^x(t) |^2 ]^{\frac{1}{2}} \mathrm{d} t
&\leq& \int_0^{\infty} \lambda (1+ |x|^2) e^{\frac{C_4}{2} t } |u| t^{-\frac{1}{2}} e^{ (-\lambda + C_{ {\rm op} })t } \mathrm{d} t \nonumber \\
&\leq& C (1+ |x|^2) |u|,
\end{eqnarray}
where the last inequality holds by taking $\lambda\geq \frac{C_4}{2} + C_{ { \rm op} }+1$.
From estimate for $\mathbb{E} | \mathcal{I}_{u}^x(t) |$ in \eqref{e:IuexEst}, one has
\begin{eqnarray}\label{e:Nfx2}
\int_0^{\infty} e^{-\lambda t} \| h \|_{\infty} \mathbb{E} [ | \mathcal{I}_u^x(t) | ] \mathrm{d} t
&\leq& \int_0^{\infty} e^{-\lambda t} \| h \|_{\infty} \frac{C|u|}{ t^{1/2} } e^{C_{ \textrm{op} } t} \mathrm{d} t
\ \leq \ C\|h\|_{\infty} |u|,
\end{eqnarray}
where the last inequality holds by taking $\lambda \geq C_{ \textrm{op} }+1$.
Since
\begin{eqnarray*}
f(x)= \int_0^{\infty} e^{-\lambda t} P_t[ \lambda f(x) - h(x) + \mu(h) ] \mathrm{d} t, \quad \forall \lambda>0,
\end{eqnarray*}
by Lebesgue's dominated convergence theorem, we have
\begin{eqnarray}\label{e:nuf}
\nabla_u f(x)&=& \int_0^{\infty} \nabla_u [\lambda e^{-\lambda t} \mathbb{E} f(X_t^x) - e^{-\lambda t} \mathbb{E} h(X_t^x)] \mathrm{d} t \nonumber \\
&=& \int_0^{\infty} e^{-\lambda t} ( \lambda \nabla_{u}\mathbb{E} [ f (X_t^x) ] - \nabla_u\mathbb{E} [ h(X_t^x)] ) \mathrm{d} t \nonumber \\
&=&\int_0^{\infty} e^{-\lambda t} ( \lambda \mathbb{E} [ f (X_t^x) \mathcal{I}_{u}^x(t) ] - \mathbb{E} [ h(X_t^x) \mathcal{I}_{u}^x(t) ] ) \mathrm{d} t.
\end{eqnarray}
Taking $\lambda= \frac{C_4}{2} + C_{ \textrm{op} } +1$, combining \eqref{e:Nfx0}, \eqref{e:Nfx1} and \eqref{e:Nfx2}, one has
\begin{eqnarray*}
|\nabla_{u} f(x)|
&\leq& C \| h \|_{\infty} (1+ |x|^2) |u|.
\end{eqnarray*}
Second, we extend $h\in \mathcal{C}_b^1(\mathbb{R}^d,\mathbb{R})$ to $h\in \mathcal{B}_b(\mathbb{R}^d,\mathbb{R})$ by the standard approximation (see Fang et al. \cite[pp. 968-969]{FSX1}). Let $h\in \mathcal{B}_b(\mathbb{R}^d,\mathbb{R})$, and define
\begin{eqnarray*}
h_{\delta}(x) = \int_{\mathbb{R}^d} \varphi_{\delta}(y) h(x-y) \mathrm{d} y, \quad \delta>0,
\end{eqnarray*}
where $\varphi_{\delta}$ is the density function of the normal distribution $\mathcal{N}(0,\delta^2 I)$. Thus $h_{\delta}$ is smooth, $\| h_{\delta} \|_{\infty} \leq \| h\|_{\infty}$ and the solution to the Poisson equation \eqref{e:SE} with $h$ replaced by $h_{\delta}$, is
\begin{eqnarray*}
f_{\delta}(x)= - \int_0^{\infty} P_t[ h_{\delta}(x) - \mu(h_{\delta}) ] \mathrm{d} t.
\end{eqnarray*}
Denote $\hat{h}_{\delta} = -h_{\delta} + \mu(h_{\delta})$. Since $h_{\delta} \in \mathcal{C}_b^1(\mathbb{R}^d,\mathbb{R})$, one has $h_{\delta}(x) \leq \|h_{\delta}\|_{\infty}( 1+ V(x))$ for all $x\in \mathbb{R}^d$. From Lemma \ref{lem:AV2}, one has
\begin{eqnarray*}
|P_t h_{\delta}(x) - \mu(h_{\delta}) |
&\leq& \|h_{\delta}\|_{\infty} \| P^*_{t}\delta_x - \mu \|_{\rm{TV},\rm{V}}
\ \leq \ C \|h_{\delta}\|_{\infty}(1+V(x)) e^{-ct}.
\end{eqnarray*}
With similar calculations in the proof for Lemma \ref{prop:ST}, one has
\begin{eqnarray*}
|f_{\delta}(x)|
&\leq& C \|h_{\delta}\|_{\infty}(1+ |x|^2 ).
\end{eqnarray*}
By the dominated convergence theorem, one has
\begin{eqnarray*}
\lim_{\delta \to 0} f_{\delta}(x) \ = \ - \int_0^{\infty} P_t[ h(x) - \mu(h) ] \mathrm{d} t
\ = \ f(x).
\end{eqnarray*}
From \eqref{e:SE2} and $\| h_{\delta} \|_{\infty} \leq \| h \|_{\infty}$, we know
\begin{eqnarray*}
| f_{\delta}(x)|
&\leq& C \|h_{\delta}\|_{\infty}( 1 + |x|^2 )
\ \leq \ C \|h\|_{\infty}(1+ |x|^2).
\end{eqnarray*}
Let $\delta \to 0$, we know
\begin{eqnarray*}
| f(x)|
\ \leq \ C \|h\|_{\infty}( 1 + |x|^2 ).
\end{eqnarray*}
From calculations for \eqref{e:nuf}, one has
\begin{eqnarray*}
\nabla_u f_{\delta}(x)
&=&\int_0^{\infty} e^{-\lambda t} ( \lambda \mathbb{E} [ f_{\delta} (X_t^x) \mathcal{I}_{u}^x(t) ] - \mathbb{E} [ h_{\delta}(X_t^x) \mathcal{I}_{u}^x(t) ] ) \mathrm{d} t.
\end{eqnarray*}
With similar calculations for the proof of \eqref{e:SE2}, one has
\begin{eqnarray*}
|\nabla_{u} f_{\delta}(x)|
&\leq& C \| h_{\delta} \|_{\infty} (1+ |x|^2) |u|
\ \leq \ C \| h \|_{\infty}(1+|x|^2) |u|.
\end{eqnarray*}
Since the operator $\nabla$ is closed (see Partington \cite[Theorem 2.2.6]{PJR1}), it follows from the dominated convergence theorem that
\begin{eqnarray*}
\lim_{\delta \to 0} \nabla_u f_{\delta}(x)
&=& \lim_{\delta \to 0} \int_0^{\infty} e^{-\lambda t} ( \lambda \mathbb{E} [ f_{\delta} (X_t^x) \mathcal{I}_{u}^x(t) ] - \mathbb{E} [ h_{\delta}(X_t^x) \mathcal{I}_{u}^x(t) ] ) \mathrm{d} t \\
&=& \int_0^{\infty} e^{-\lambda t} ( \lambda \mathbb{E} [ f (X_t^x) \mathcal{I}_{u}^x(t) ] - \mathbb{E} [ h(X_t^x) \mathcal{I}_{u}^x(t) ] ) \mathrm{d} t \\
&=& \nabla_u f(x).
\end{eqnarray*}
Letting $\delta \to 0$, we obtain
\begin{eqnarray*}
|\nabla_{u} f(x)|
&\leq& C \| h \|_{\infty} (1+ |x|^2) |u|.
\end{eqnarray*}
The proof is complete.
\end{proof}
\begin{proof}[Proof of Lemma \ref{prop:ST}]
We first show that $\int_0^{\infty}[P_t h(x) - \mu(h)] \mathrm{d} t$ is well defined. Denote $\hat{h} = -h + \mu(h)$. For any $h \in \mathcal{B}_b(\mathbb{R}^d,\mathbb{R})$, we have that $h(x) \leq \|h\|_{\infty}( 1+ V(x))$ for all $x\in \mathbb{R}^d$, and from Lemma \ref{lem:AV2} that
\begin{eqnarray*}
|P_t h(x) - \mu(h) | \ \leq \ \|h\|_{\infty} \| P^*_{t}\delta_x - \mu \|_{ \rm{TV}, \rm{V} }
\ \leq \ C \|h\|_{\infty}(1+ V(x)) e^{-c t}.
\end{eqnarray*}
This implies that
\begin{eqnarray*}
\left| \int_0^{\infty}[P_t h(x)-\mu(h)] \mathrm{d} t \right|
\ \leq \
C \|h\|_{\infty}(1+V(x))
\ < \ \infty.
\end{eqnarray*}
The reminder is similar to that of {Fang et al. \cite[Proposition 6.1]{FSX1}.} The proof is complete.
\end{proof}
\medskip
\subsection{Properties of the mollified diffusion}
\begin{proof}[Proof of Lemma \ref{lem:XXem}]
(i) Using It\^o's formula, for any integers $m \geq 2$, we have
\begin{eqnarray*}
\mathbb{E} |X^{x}_{t}|^{m}&=&|x|^{m}+m \mathbb{E} \int_{0}^{t} |X^{x}_{s}|^{m-2} (X^{x}_{s})' g(X^{x}_{s}) \mathrm{d} s \\
&&+\frac m2 \mathbb{E} \int_{0}^{t} |X^{x}_{s}|^{m-4}\left[(m-2)|\sigma' X^{x}_{s}|^{2}+ { \rm tr} (\sigma \sigma') |X^{x}_{s}|^{2} \right] \mathrm{d} s.
\end{eqnarray*}
By the bound $|g(x)| \leq \tilde{C}_{\rm op}(1+|x|)$ for all $x\in \mathbb{R}^d$ with $\tilde{C}_{{\rm op}}$ in \eqref{tlCop}, we further get
\begin{eqnarray*}
\mathbb{E} |X^{x}_{t}|^{m}& \leq &|x|^{m}+\tilde{C}_{\rm op} \left( m\int_{0}^{t} \mathbb{E} |X^{x}_{s}|^{m} \mathrm{d} s+ m\int_{0}^{t} \mathbb{E} |X^{x}_{s}|^{m-1} \mathrm{d} s+ m^2\int_{0}^{t} \mathbb{E} |X^{x}_{s}|^{m-2} \mathrm{d} s\right) \nonumber \\
&\le & |x|^{m}+2m^2 \tilde{C}_{\rm op} \left( \int_{0}^{t} \mathbb{E} |X^{x}_{s}|^{m} \mathrm{d} s+t \right),
\end{eqnarray*}
where the second inequality is by the Young's inequality. Thus, we have
\begin{eqnarray*}
\mathbb{E} |X^{x}_t|^{m} \ \le \ e^{C_m t}(|x|^{m}+1),
\end{eqnarray*}
where $C_m=2m^2 \tilde{C}_{\rm op}$. Since $|g_\varepsilon(x)| \leq \tilde{C}_{\rm op}(1+|x|)$ uniformly for $\varepsilon \in (0,1)$, the moment estimates $\mathbb{E} |X^{\varepsilon,x}_t|^{m}$ can be obtained similarly.
Thus, inequality \eqref{e:XXem} holds.
(ii) Consider $X^{\varepsilon,x}_{t}-X_{t}^{x}$, which satisfies the following equation
\begin{eqnarray*}
\frac{\mathrm{d}}{\mathrm{d} t} \left(X^{\varepsilon,x}_{t}-X_{t}^{x}\right)&=&g_\varepsilon (X^{\varepsilon,x}_t)-g(X^{x}_t) \\
&=&g_\varepsilon (X^{\varepsilon,x}_t)-g_{\varepsilon}(X^{x}_t)+g_{\varepsilon}(X^{x}_t)-g(X^{x}_{t}), \\
&=&\nabla g_{\varepsilon}(\theta_{t}) \left(X^{\varepsilon,x}_{t}-X_{t}^{x}\right)+g_{\varepsilon}(X^{x}_t)-g(X^{x}_{t}),
\end{eqnarray*}
where $\theta_{t}$ is between $X^{x}_t$ and $X^{\varepsilon,x}_t$. The above equation can be solved by
\begin{eqnarray*}
X^{\varepsilon,x}_{t}-X_{t}^{x}&=&\int_{0}^{t} \exp\left(\int_{s}^{t} \nabla g_{\varepsilon}(\theta_{r}) \mathrm{d} r \right) (g_{\varepsilon}(X^{x}_s)-g(X^{x}_{s})) \mathrm{d} s.
\end{eqnarray*}
Since $\|\nabla g_{\varepsilon}(x)\|_{ {\rm op} } \le C_{{\rm op}}$ for all $x \in \mathbb{R}^d$, the relation $|g_{\varepsilon}(x)-g(x)| \le C_{{\rm op}}\varepsilon$ for all $x \in \mathbb{R}^d$ immediately gives us \eqref{e:XeCon-1}.
\end{proof}
\begin{proof}[Proof of Lemma \ref{l:XeCon}]
Denote the event
$$N=\left\{\int_{0}^{\infty} \|\nabla g_{\varepsilon}(X^{\varepsilon,x}_{s})\|_{ {\rm op} }1_{\{{\rm e}^{\prime} X^{x}_{s}=0\}} \mathrm{d} s\ne 0\right\}.
$$
We claim that
\begin{equation} \label{e:PN=0}
\mathbb{P}(N) \ = \ 0.
\end{equation}
Indeed, for any $T>0$, by $\|\nabla g_{\varepsilon}(x)\|_{{\rm op}} \le C_{{\rm op}}$ for all $x \in \mathbb{R}^d$ and $\varepsilon$, we have
\begin{eqnarray}\label{e:ET}
\mathbb{E} \int_{0}^{T} \|\nabla g_{\varepsilon}(X^{\varepsilon,x}_{s})\|_{ {\rm op} }1_{\{{\rm e}^{\prime} X_{s}^x=0\}} \mathrm{d} s
&=& \int_{0}^{T} \mathbb{E}[\|\nabla g_{\varepsilon}(X^{\varepsilon,x}_{s})\|_{{\rm op}}1_{\{{\rm e}^{\prime} X_{s}^x=0\}}] \mathrm{d} s \nonumber \\
& \leq & \int_{0}^{T} C_{{\rm op}} \mathbb{E} 1_{\{{\rm e}^{\prime} X_{s}^x=0\}} \mathrm{d} s \nonumber \\
&=& 0,
\end{eqnarray}
where the last equality is by Proposition \ref{lem:occupation} and the fact that for any small $\epsilon>0$,
\begin{eqnarray*}
\mathbb{E}\int_0^T 1_{\{{\rm e}^{\prime} X_{s}^x=0\}} \mathrm{d} s
& \leq & \mathbb{E} L_T^{\epsilon,x}
\ \leq \ C\epsilon e^{\frac{C_2}{2} T} (1+|x|)(1+T),
\end{eqnarray*}
while $L_t^{\epsilon,x} = \int_0^t [-\frac{1}{\epsilon^2} ({\rm e}^{\prime} X_s^x)^2 +1 ] 1_{ \{ |{\rm e}^{\prime} X_s^x| \leq \epsilon \} } \mathrm{d} s$. The above inequality holds for any $\epsilon>0$, we know $ \mathbb{E}\int_0^T 1_{\{{\rm e}^{\prime} X_{s}^x=0\}} \mathrm{d} s = 0$.
Since \eqref{e:ET} holds for all $T>0$, we see that
\begin{eqnarray*}
\mathbb{E} \int_{0}^{\infty} \|\nabla g_{\varepsilon}(X^{\varepsilon,x}_{s})\|_{ {\rm op} }1_{\{{\rm e}^{\prime} X^{x}_{s}=0\}} \mathrm{d} s
& = & 0,
\end{eqnarray*}
hence \eqref{e:PN=0} holds.
Recall the definition of $J^{\varepsilon,x}_{s,t}$ and define
$$\hat J^{\varepsilon,x}_{s,t}:=\exp \left(\int_s^t \nabla g_{\varepsilon} (X_{r}^{\varepsilon,x}) 1_{\{{\rm e}^{\prime} X^{x}_{r} \ne 0\}}\mathrm{d} r \right).
$$
It is easy to verify that
\begin{equation} \label{e:JstConN}
\lim_{\varepsilon \rightarrow 0} \hat J^{\varepsilon,x}_{s,t}=J^{x}_{s,t}, \ \ \ \ \ \ 0 \le s \leq t<\infty.
\end{equation}
For any $\omega \notin N$, we know $\int_{0}^{\infty} \nabla g_{\varepsilon}(X^{\varepsilon,x}_{s})1_{\{{\rm e}^{\prime} X^{x}_{s}=0\}} \mathrm{d} s=0$ and thus
$$\exp\left(\int_{s}^{t} \nabla g_{\varepsilon}(X^{\varepsilon,x}_{r})1_{\{{\rm e}^{\prime} X^{x}_{r}=0\}} \mathrm{d} r \right)=I, \ \ \ \ 0 \le s \leq t<\infty.$$
Since $I$ commutes with any matrix, for all $\omega \notin N$, we get
$$\hat J^{\varepsilon,x}_{s,t}=\hat J^{\varepsilon,x}_{s,t} \exp \left(\int_{s}^{t} \nabla g_{\varepsilon}(X^{\varepsilon,x}_{r})1_{\{{\rm e}^{\prime} X^{x}_{r}=0\}} \mathrm{d} r \right)=J^{\varepsilon,x}_{s,t}, \ \ \ \ \ 0 \le s \leq t<\infty.$$
This, combining with \eqref{e:JstConN}, implies that for all $\omega \notin N$,
$$\lim_{\varepsilon \rightarrow 0} J^{\varepsilon,x}_{s,t}=J^{x}_{s,t}, \ \ \ \ \ \ 0 \le s \leq t<\infty.$$
Note that $ J^{\varepsilon,x}_{s,t}$ and $J^{x}_{s,t}$ are matrices, hence the above pointwise convergence implies the convergence in operator.
\end{proof}
\subsection{Bismut's formula}
\begin{proof}[Proof of Lemma \ref{lem:EIm}]
Using the Burkholder-Davis-Gundy inequality, we have
\begin{eqnarray*}
\mathbb{E} \left|\mathcal{I}_{u}^{\varepsilon,x}(t)\right|^m
&\leq &\frac{C}{t^m}\mathbb{E} \left(\int_0^t |\sigma^{-1}J^{\varepsilon,x}_{r} u|^2 \mathrm{d} r \right)^{m/2}
\le \frac{ C }{t^m} \mathbb{E} \left(\int_0^t \|\sigma^{-1}\|^{2}_{ {\rm op} } \|J^{\varepsilon,x}_{r}\|^{2}_{ {\rm op} } |u|^{2}\mathrm{d} r \right)^{m/2},
\end{eqnarray*}
which, together with \eqref{e:JJe}, immediately gives \eqref{e:IuexEst}.
For the second relation, by Burkholder-Davis-Gundy inequality, we have
\begin{eqnarray*}
\mathbb{E} |\mathcal{I}_{u}^{\varepsilon,x}(t)- \mathcal{I}_{u}^{x}(t)|^m
&=& \mathbb{E} \left|\frac1t\int_0^t \langle \sigma^{-1}(J^{\varepsilon,x}_{r}-J^{x}_{r}) u, \mathrm{d} B_r\rangle \right|^m \\
&\leq &\frac{ C }{t^m} \mathbb{E} \left(\int_0^t |\sigma^{-1}(J^{\varepsilon,x}_{r}-J^{x}_{r}) u|^2 \mathrm{d} r \right)^{m/2} \ \rightarrow \ 0 { \ \ \rm as \ \ } \varepsilon \rightarrow 0,
\end{eqnarray*}
where the limit is by dominated convergence theorem (with a notice of Lemma \ref{l:XeCon}).
\end{proof}
\medskip
\begin{proof}[Proof of Lemma \ref{hLef2}]
If $\psi\in \mathcal{C}^1(\mathbb{R}^d,\mathbb{R})$, then by \eqref{e:DVNu} and the Bismut's formula \eqref{e:BisFor}, we have
\begin{eqnarray*}
\nabla_{u} \mathbb{E}[\psi(X^{\varepsilon,x}_t)] &=& \mathbb{E}[\nabla \psi(X^{\varepsilon,x}_t) \nabla_{u} X^{\varepsilon,x}_t ]
\ \ = \ \ \mathbb{E}[\nabla \psi(X^{\varepsilon,x}_t) D_{\mathbb{V}}X^{\varepsilon,x}_t] \\
&=& \mathbb{E}[D_{\mathbb{V}} \psi(X^{\varepsilon,x}_t)]
\ \ = \ \ \mathbb{E} [\psi(X^{\varepsilon,x}_t) \mathcal{I}_{u}^{\varepsilon,x}(t)],
\end{eqnarray*}
where $\mathbb{V}$ is the direction of Malliavin derivative.
Since the operator $\nabla$ is closed and by the well known property of closed operators (see Partington \cite[Proposition 2.1.4]{PJR1}), as long as it is shown that
\begin{eqnarray} \label{e:EPhi}
&& \lim_{\varepsilon \rightarrow 0} \mathbb{E}[\psi(X^{\varepsilon,x}_t)] \ = \ \mathbb{E}[\psi(X^{x}_t)],
\ \ \lim_{\varepsilon \rightarrow 0} \mathbb{E} [\psi(X^{\varepsilon,x}_t) \mathcal{I}_{u}^{\varepsilon,x}(t)]\ =\ \mathbb{E} [\psi(X^{x}_t) \mathcal{I}_{u}^{x}(t)],
\end{eqnarray}
then we know that $\nabla_u\mathbb{E}[\psi(X^x_t)]$ exists and has its value as $\mathbb{E} [\psi(X^{x}_t) \mathcal{I}_{u}^{x}(t)]$. Hence, the first relation is proved.
Before proving \eqref{e:EPhi}, let us show that for all $m \ge 1$,
\begin{eqnarray} \label{e:PhiPowM}
\mathbb{E} |\psi(X^{\varepsilon,x}_t)|^{m}, \mathbb{E} |\psi(X^{x}_t)|^{m} & \le & \hat{C}_1 e^{C_m t} (|x|^{m}+1),
\end{eqnarray}
where $\hat{C}_1$ does not depend on $\varepsilon$, $t$ and $x$. (Without loss of generality, we can take $\hat{C}_1$ depending on $m$, $\| \nabla \psi \|_{\infty}$ and $\psi(0)$.) Indeed, it is easily seen that
\begin{eqnarray*}
\mathbb{E} |\psi(X^{\varepsilon,x}_t)|^{m}
& \le & \hat{C}_2 \left(|\psi(0)|^{m}+ \| \nabla \psi \|^{m}_{\infty} \mathbb{E} |X^{\varepsilon,x}_t|^{m} \right),
\end{eqnarray*}
where $\hat{C}_2$ only depends on $m$ and this, together with \eqref{e:XXem}, immediately yields the aimed inequality.
For the first limit in \eqref{e:EPhi}, for all $m \ge 1$, by \eqref{e:XeCon-1} we have
\begin{eqnarray}
\mathbb{E} |\psi(X^{\varepsilon,x}_t)-\psi(X^{x}_{t})|^{m} & \le & \| \nabla \psi \|^{m}_{\infty} \mathbb{E} |X^{\varepsilon,x}_t-X^{x}_t |^{m} \ \ \rightarrow \ 0 \textrm{ \ \ \rm as \ \ } \varepsilon \to 0. \label{e:PhiXXe}
\end{eqnarray}
For the second one, we have
\begin{eqnarray*}
&& |\mathbb{E} [\psi(X^{\varepsilon,x}_t) \mathcal{I}_{u}^{\varepsilon,x}(t)]-\mathbb{E} [\psi(X^{x}_t) \mathcal{I}_{u}^{x}(t)]| \\
&\leq & \mathbb{E} [|\psi(X^{\varepsilon,x}_t)| |\mathcal{I}_{u}^{\varepsilon,x}(t)-\mathcal{I}_{u}^{x}(t)|]+\mathbb{E} [|\psi(X^{\varepsilon,x}_t)-\psi(X^{x}_t)||\mathcal{I}_{u}^{x}(t)|] \\
& \rightarrow & 0 { \ \ \rm as \ \ } \varepsilon \rightarrow 0,
\end{eqnarray*}
where the convergence is by the following argument: applying the Cauchy-Schwarz inequality on the two expectations, and using \eqref{e:IuexEst}, \eqref{e:IueCon}, \eqref{e:PhiPowM} and \eqref{e:PhiXXe}.
\end{proof}
\subsection{Weighted occupation time}
\begin{proof}[Proof of Proposition \ref{lem:occupation}]
Denote
\begin{eqnarray}\label{e:rhobar0}
\phi_{\epsilon}(y)=
\left\{
\begin{array}{lllll}
\frac{2}{3}\epsilon y -\frac{1}{4}\epsilon^2,& \text{if } y>\epsilon,\\
-\frac{1}{12\epsilon^2}y^4 + \frac{1}{2}y^2, & \text{if } -\epsilon \leq y\leq \epsilon, \\
-\frac{2}{3}\epsilon y-\frac{1}{4}\epsilon^2, & \text{if } y <-\epsilon.
\end{array}
\right.
\end{eqnarray}
It is easy to check that
$$\ddot{\phi}_{\epsilon}(y) \ = \ \left[ \frac{-1}{\epsilon^2}y^2+1 \right]1_{ \{ |y|\leq \epsilon \} },$$
$|\phi_{\epsilon}(y)|\leq C\epsilon |y|+C\epsilon^2$, $|\dot{\phi}_{\epsilon}(y)|\leq C\epsilon$ and $|\ddot{\phi}_{\epsilon}(y)|\leq 1$ for all $y\in \mathbb{R}$ and the positive constant $C$ is independent of $\epsilon$.
Applying It\^{o}'s formula to the function $\phi_{\epsilon}$, one has
\begin{eqnarray*}
\phi_{\epsilon}( {\rm e^{\prime}} X_t^x)
&=& \phi_{\epsilon}( {\rm e^{\prime} } x)
+\int_0^t \dot{\phi}_{\epsilon}({\rm e^{\prime}} X_s^x) {\rm e^{\prime}}g(X_s^x) \mathrm{d} s + \int_0^t \dot{\phi}_{\epsilon}( {\rm e^{\prime}}X_s^x ) {\rm e^{\prime}}\sigma \mathrm{d} B_s + \frac{|\sigma^{\prime} {\rm e}|^2}{2} L_t^{\epsilon,x},
\end{eqnarray*}
which implies that
\begin{eqnarray*}
\mathbb{E} L_t^{\epsilon,x} &\leq& C \left[ \mathbb{E} |\phi_{\epsilon}( {\rm e^{\prime}} X_t^x)|
+|\phi_{\epsilon}( {\rm e^{\prime} } x)|
+\mathbb{E} \left|\int_0^t \dot{\phi}_{\epsilon}({\rm e^{\prime}} X_s^x) {\rm e^{\prime}}g(X_s^x) \mathrm{d} s \right| \right] \\
&\leq& C\epsilon e^{\frac{C_2}{2}t}(1+|x|)(1+t),
\end{eqnarray*}
where the last inequality holds from Lemma \ref{lem:XXem}. The proof is complete.
\end{proof}
\end{appendix}
\section*{Acknowledgments.}
L. Xu is supported in part by NSFC grant (No. 12071499), Macao S.A.R grant FDCT 0090/2019/A2 and University of Macau grant MYRG2018-00133-FST.
G. Pang is supported in part by the US National Science Foundation grants DMS-1715875 and DMS-2216765.
X. Jin was supported in part by the Fundamental Research Funds for the Central Universities grant (JZ2022HGQA0148).
|
2,877,628,089,036 | arxiv | \section{Introduction}
Recently, a notorious open problem in quantum information theory known as the additivity of the Holevo capacity was finally resolved~\cite{counterexSmin1} with the negative answer. The article culminated a long period of waiting for the answer to the question (later a conjecture) which appeared shortly after people started to ask about the role of quantum correlations for information theory~\cite{HolevoCapAdd}. The former conjecture states that entangled states do not improve the classical capacity of quantum channels. Quantum channel $\mathcal{N}$ is a completely positive (CP) map $\mathcal{N}:\mathscr{F}\big(\HH^{(I)}_{\rm in}\big)\to\mathscr{F}\big(\HH^{(O)}_{\rm out}\big)$. $\mathscr{F}\big(\HH^{(K)}\big)$ is the state space for a $K-$dimensional Hilbert space $\HH^{(K)}$ occupied by Hermitean operators of trace one. The ultimate formula for the classical capacity is $C=\lim_{n\to\infty}{{1\over n}C_{\rm Hol}\left(\mathcal{N}^{\otimes n}\right)}$. $C_{\rm Hol}(\mathcal{N})$ is the Holevo capacity of a channel~$\mathcal{N}$~\cite{HSW} defined as
\begin{equation}\label{HolevoCap}
C_{\rm Hol}(\mathcal{N})\stackrel{\rm df}{=}\sup_{\{p_i\varrho_i\}}{\bigg\{S\bigg(\sum_ip_i\mathcal{N}(\varrho_i)\bigg)-\sum_i p_iS\bigg(\mathcal{N}(\varrho_i)\bigg)\bigg\}},
\end{equation}
where $\{p_i\varrho_i\}$ is the input ensemble $\varrho=\sum_ip_i\varrho_i$ and $S(\sigma)=-\Tr{}\sigma\log\sigma$ is the von Neumann entropy~\footnote{$\log$ gives the logarithm to base two.}.
The calculation of $C$ appears to be an intractable problem. The conjecture claimed that $C_{\rm Hol}(\mathcal{N}_1\otimes\mathcal{N}_2)=C_{\rm Hol}(\mathcal{N}_1)+C_{\rm Hol}(\mathcal{N}_2)$ for arbitrary channels $\mathcal{N}_1,\mathcal{N}_2$. This condition is slightly stronger (strong additivity) than if $\mathcal{N}_1=\mathcal{N}_2$ (weak additivity). One can immediately see how the calculation of $C$ might have been much simpler if the conjecture had been correct. Let us stress, however, that even if the conjecture does not hold in general there are important classes of channels for which it holds~\cite{DepolCh,TransDepol}.
The final disproof of the conjecture would not be possible without many important intermediate results. First, it was shown that the additivity of the Holevo capacity is globally (that is, not for a particular channel) equivalent to other additivity questions~\cite{ConjectsEquiv}, particularly to the additivity of the minimum output entropy (MOE)~\cite{Smin}. The MOE belongs to the more general class of entropies known as the minimum output R\'enyi entropy (MORE). The MORE of a channel $\mathcal{N}$ is defined
\begin{equation}\label{Smin}
\Sminp{\mathcal{N}}\stackrel{\rm df}{=}\min_{\varrho}{\{S_p(\mathcal{N}(\varrho))\}},\hspace{.567cm}\varrho\in\mathscr{F}(\HH_{\rm in}),
\end{equation}
where $S_p(\varrho)=(1-p)^{-1}\log{\Tr{}\varrho^p}$ is the R\'enyi entropy (for $p\to1^+$ we get the von Neumann entropy). The MORE conjecture was disproved for various intervals of $p$ (for $p>1$ in~\cite{counterexsSminp_g1} and for $p\to0$ in~\cite{counterexsSminp_0}) and, as indicated, at last also for $p=1$~\cite{counterexSmin1}. Note that by the concavity of entropy we may restrict ourselves to the minimization over input pure states.
The question of additivity of the classical and quantum capacity comes from the analysis of an infinite-dimensional channel which appears in the context of quantum field theory in curved spacetime -- the Unruh channel~\cite{UnruhCh}. The Unruh channel occupies an important place in the field of relativistic quantum information and quantum field theory due to its close relationship to the process known as black hole evaporation or more generally the process of black hole stimulated emission. Interestingly, the Unruh channel decomposes into a sequence of finite-dimensional channels which is closely related to the channels arising from universal quantum cloning machines (UQCM) for qubits~\cite{cloners}. We will call them cloning channels and we will prove that there exists a single-letter formula for the classical capacity for all of them. It is known that cloning channels are conjugate degradable. Channels are called conjugate degradable by virtue of existence of a conjugate degrading map transforming the output of the channel to its complementary output up to complex conjugation. It has been recently shown that the optimized coherent information of conjugate degradable channels is additive~\cite{ConjDeg}. Therefore, as a result of this paper we obtain an infinite sequence of channels for which both the classical and quantum capacity can be calculated in an easy way. As an aside we will show that all studied cloning channels are not only conjugate-degradable but also degradable. Originally, this seemed to be a difficult task to directly~\cite{cloners} prove it. Here we found a relatively straightforward way to show this fact and conjugate degradability of cloning channels again played an important role in the proof. The second main result of this paper is the proof of additivity of the Holevo capacity for the Unruh channel itself. It provides us with a non-trivial example of an infinite-dimensional channel for which both the classical and quantum capacity are known and easily calculable.
In section~\ref{sec:Unruhchannel} we briefly recall the properties of the Unruh channel~\cite{UnruhCh} and present its decomposition into a sequence of finite-dimensional channels. Section~\ref{sec:infseq} contains the main result of the paper. Using the structure of cloning channels we prove in the first part that (i) the Holevo capacity of cloning channels is additive and (ii) all cloning channels are degradable by showing that their complementary channels are entanglement-breaking. In the second part of section~\ref{sec:infseq} we prove the additivity of the Holevo capacity for the infinite-dimensional Unruh channel. We conclude the paper with a technical tool to actually determine the form of degrading channels for cloning channels and illustrate it on a few examples.
If not stated otherwise, note that in this paper by additivity of a channel we mean strong additivity of the Holevo capacity.
\section{Unruh channel}\label{sec:Unruhchannel}
In this section we briefly review the definition and properties of the Unruh channel~\cite{UnruhCh}. The channel naturally appears as the transformation of a photonic qubit prepared by a stationary Minkowski observer if it is detected by a uniformly accelerated observer. It is well known that inertial and non-inertial observers cannot agree on the notion of a particle. The most dramatic example is the Minkowski vacuum seen by an non-inertial observer as a thermally populated state~\cite{Unruh76}.
In the same spirit, a pure qubit prepared in the Hilbert space of a Minkowski observer is seen as an infinite-dimensional mixed state in the Hilbert space of the accelerated observer. The responsible transformation reads
\begin{multline}\label{squeeze}
U_{abcd}(r)=\smfrac{1}{\cosh^2{r}}e^{\tanh{r}(a^\dagger c^\dagger+b^\dagger d^\dagger)}\\
\times e^{-\ln{\cosh{r}}(a^\dagger a+b^\dagger b+c^\dagger c+d^\dagger d)}e^{-\tanh{r}(ac+bd)},
\end{multline}
where $r$ is the proper acceleration of the inertial observer. For an input state $\ket{\psi} = (\beta b^\dagger + \alpha a^\dagger)\ket{\mathsf{vac}}$, we can further simplify $\ket{\phi}=U_{abcd}(r)\ket{\psi}$ as
\begin{equation}\label{simp_squeeze}
\ket{\phi}=1/\cosh^3{r}(\beta b^\dagger+\alpha a^\dagger)\exp{[\tanh{r}(a^\dagger c^\dagger+b^\dagger d^\dagger)]}\ket{\mathsf{vac}}.
\end{equation}
From a physical point of view, the modes $c,d$ appear beyond the event horizon of the accelerated observer and are therefore unobservable. Tracing over them, we get a state with an interesting structure, further investigated in~\cite{UnruhCh}. If we reorder the basis according to the total number of incoming photons in modes $a$ and $b$, we obtain an infinite-dimensional block-diagonal density matrix
\begin{equation}\label{blockdiag}
\sigma=1/2(1-z)^3\bigoplus_{\ell=2}^\infty \ell(\ell-1)z^{\ell-2}\varepsilon_\ell,
\end{equation}
where $0\leq z<1,z=\tanh^2{r}$. The states $\varepsilon_\ell$ and the corresponding quantum channel will be studied in the next section.
The transformation leading to Eq.~(\ref{blockdiag}) has already been studied before in a slightly different context. The authors of Ref.~\cite{blackhole_clone} analyzed the process of black hole stimulated emission induced by impinging photonic qubits. The stimulated emission dynamics is governed by exactly the same Hamiltonian as the one leading to the unitary operator $U_{abcd}$. The reason for this formal similarity lies in the linear relations known as Bogoliubov transformation~\cite{bogo}. Bogoliubov transformation connects the creation and annihilation operators of the Hilbert space of a Minkowski observer and a uniformly accelerating observer in our case and similarly the Hilbert space of a freely falling observer and an observer in a distant future in case of Ref.~\cite{blackhole_clone}. In the former case the physical parameter of the evolution operator is the proper acceleration $r$ and in the latter case it is the black hole surface gravity~\footnote{Note that the particle-antiparticle basis used in~\cite{blackhole_clone} exactly corresponds to the dual-rail encoding in which an input state $\ket{\psi}$ is written.}. Even more interestingly, as observed in~\cite{blackhole_clone}, the same Hamiltonian is closely related to the $N\to M$ universal cloning machine for qubits~\cite{cloners} ($\ell=M+1$). In other words, if an observer throws an $N-$qubit photonic state into a black hole (the state is already symmetrized due to the bosonic nature of the photons) another observer in a distant future gets $M$ approximate copies depending on the total number $M$ of photons he measures. We will study the explicit output of Eq.~(\ref{blockdiag}) which corresponds to the case of $1\to(\ell-1)$ cloning machines.
\section{Additivity of the classical capacity}\label{sec:infseq}
\subsection{The classical capacity of cloning channels}\label{subsec:infseq}
The previous section served as a physical motivation for the appearance of UQCMs for qubits. In this section we observe that the cloning channels `constitute' the corresponding Unruh channel in a very specific way. Namely, we will show that the additivity of the Holevo capacity for the $1\to2$ cloning channel implies the additivity of the Holevo capacity for all $1\to(\ell-1)$ cloning channels (that is for all $\ell>3$). Another consequence will be the proof of additivity of the Holevo capacity for the Unruh channel itself.
We first recall the definition of {\it unitarily covariant channels} introduced in~\cite{CovCh}.
\begin{defi}\label{def:covariance}
Let $G$ be a unitary compact group of Lie type and let $r_1(g)\in\HH_{\rm in},r_2(g)\in\HH_{\rm out}$ be irreps of $g\in G$. A channel $\mathcal{N}:\mathscr{F}\big(\HH_{\rm in}\big)\to\mathscr{F}\big(\HH_{\rm out}\big)$ is unitarily covariant if
\begin{equation}\label{covcond}
\mathcal{N}\left(r_1(g)\varrho r_1(g)^\dagger\right)=r_2(g)\mathcal{N}(\varrho)r_2(g)^\dagger
\end{equation}
holds for all $\varrho$.
\end{defi}
In the following text, by covariant we mean unitarily covariant. It has been shown that for any covariant channel the following equivalence condition holds
\begin{equation}\label{equivlocal}
C_{\rm Hol}(\mathcal{N})=\log{f}-\Smin{\mathcal{N}},
\end{equation}
where $f=\dim{\HH_{\rm out}}$. Nevertheless, for Eq.~(\ref{equivlocal}) to hold the conditions in Definition~\ref{def:covariance} are not necessary and can be relaxed~\cite{WolfEisert}.
Let us stress that we will leave the domain of the Fock space and adopt new notation. From now on, $\ket{n}$ represents a qudit living in an abstract Hilbert space~$\HH$ and not a Fock state of $n$ photons like in Section~\ref{sec:Unruhchannel}. The reason is that the Fock space formalism is a bit clumsy for the quantum information considerations which will follow. We will make occasional connections from one formalism to another to avoid possible confusion.
Let $W$ be the Hilbert space isometry $W:A\hookrightarrow EH$ such that $W(\varphi)=U_{EH}^{(K)}(\ket{\varphi}_A,\ket{0})$ where $U_{EH}^{(K)}$ is a $K-$dimensional unitary transformation defined by its action on an input pure state $\ket{\varphi}=\alpha\ket{0}+\beta\ket{1}$ and an ancilla $\ket{0}$
\begin{multline}\label{unitary}
\ket{\varphi}_A\ket{0}_{anc}\xrightarrow{U_{EH}^{(K)}} {\sqrt{2\over(k+1)(k+2)}}\\
\times\Biggl(\sum_{n=0}^k\alpha\sqrt{k-n+1}\ket{k-n}_E\ket{n}_H\\
+\beta\sqrt{n+1}\ket{k-n+1}_E\ket{n}_H\Biggr),
\end{multline}
(thus $K=2(k+1)$). This unitary operation induces a class of CP maps $\varepsilon_\ell\stackrel{\rm df}{=}\Tr{H}\left[W(\varphi)\right]=\Cl{\ell-1}(\varphi)$ which we will call $1\to(\ell-1)$ cloning channels ($\ell=k+2$). The explicit output of $\Cl{\ell-1}(\varphi)$ and the corresponding complementary channel $\bar\kappa_{\ell-1}\stackrel{\rm df}{=}\mathcal{S}^c_{\ell-1}(\varphi)=\Tr{E}\left[W(\varphi)\right]$ for an input qubit $\varphi=\mathbb{1}/2+\vec{n}\cdot\vec{J}^{(2)}$ read
\begin{eqnarray}\label{StokesplusConjugateStokes1}
\varepsilon_\ell&=&{2\over\ell(\ell-1)}\Big(\mathbb{1}^{(\ell)}(\ell-1)/2+\sum_{i=x,y,z}n_iJ_i^{(\ell)}\Big)\\
\label{StokesplusConjugateStokes2}
\bar\kappa_{\ell-1}&=&{2\over\ell(\ell-1)}\Big(\mathbb{1}^{(\ell-1)}\ell/2+\sum_{i=x,y,z}\tilde n_iJ_i^{(\ell-1)}\Big),
\end{eqnarray}
where $J_i^{(\ell)}$ are related to the $\ell-$dimensional generators of the $su(2)$ algebra. The $su(2)$ algebra generators are defined~\footnote{More precisely, the $su(2)$ algebra is a compact real form of the special linear algebra $sl(2,{\mathbb C})$.} by
$\big[J^{(\ell)}_+,J^{(\ell)}_-\big]=2J^{(\ell)}_z,\big[J^{(\ell)}_z,J^{(\ell)}_{\pm}\big]=\pm J^{(\ell)}_\pm$ and in the above equations we use $J_x^{(\ell)}=1/2\big(J^{(\ell)}_++J^{(\ell)}_-\big),J^{(\ell)}_y=-i/2\big(J^{(\ell)}_+-J^{(\ell)}_-\big)$. We also defined $n_x=\alpha\bar \beta+\bar \alpha\beta,n_y=i(\alpha\bar \beta-\bar \alpha\beta),n_z=|\alpha|^2-|\beta|^2$ and $\tilde n_x=n_x, \tilde n_y=-n_y,\tilde n_z=n_z$. For the purposes of this paper we consider only input pure states $\|\vec{n}\|_2=1$. Note that barred operator $\bar\kappa_{\ell-1}$ indicates its entry-wise complex conjugation which results in transposition for density matrices.
States in Eq.~(\ref{StokesplusConjugateStokes1}) are exactly those from Eq.~(\ref{blockdiag}) but stripped of all quantum-optical interpretations. However, one could get the same matrix form from the $1\to (\ell-1)$ UQCM if the channel output was rewritten in the completely symmetric (fixed) basis of $\ell-1$ qubits. Henceforth, contrary to the definition of UQCMs we consider cloning channels $\Cl{\ell-1}$ to be CP maps whose output is composed of all $\ell-1$ clones.
Comparing an input state $\varphi$ with an output $\varepsilon_\ell$ of $\Cl{\ell-1}$ we see that the transformation preserves the Stokes parameters $n_i$, even as the dimension of the algebra representation changes. We may interpret $\Cl{\ell-1}$ as an input state representation-changing channel. Similarly, the complementary channels $\mathcal{S}^c_{\ell-1}$ also change the representation of the input state accompanied by transposition (complex conjugation). We will make use of this intriguing interpretation of these channels in the proof of Lemma~\ref{lem:S_ell-entbreaking}. Finally, we observe that $\Cl{\ell-1}$ is a covariant channel (all UQCMs are by definition covariant) and so is $\mathcal{S}^c_{\ell-1}$.
Recall that for $\ell=2$, $\Cl{1}$ is an identity map and the complementary map $\mathcal{S}_1^c$ is just an ordinary trace map. Some interesting things start to happen for $\ell=3$ where $\mathcal{S}^c_{2}(\varphi)=1/3(\bar\varphi+\mathbb{1})$. This is an instance of the transpose depolarizing channel (alias the optimal transposition map for qubits) whose Holevo capacity is known to be strongly additive~\cite{TransDepol}. It follows that its complement $\Cl{2}$ is strongly additive too~\cite{ComplCh}.
\begin{thm}\label{thm1}
Cloning channels $\Cl{\ell-1}$ are additive for all~$\ell\geq2$.
\end{thm}
Before proving the theorem we first introduce the concept of conjugate degradability followed by a useful lemma. The definition of conjugate degradability~\cite{ConjDeg} resembles the one of degradability~\cite{DegCh} which we present for the sake of completeness.
\begin{defi}
(i) A channel $\mathcal{N}$ is degradable if there exists a map $\mathcal{D}$ called a degrading map which degrades the channel to its complementary channel $\mathcal{N}^c$
\begin{equation}\label{defCD}
\mathcal{D}\circ\mathcal{N}=\mathcal{N}^c.
\end{equation}
We say that a channel is anti-degradable if its complementary channel is degradable.\\
(ii) A channel $\mathcal{N}$ is conjugate degradable if there exists a map ${\check{\D}}$ called a conjugate degrading map which degrades the channel to its complementary channel $\mathcal{N}^c$ up to complex conjugation $\mathcal{C}$
\begin{equation}\label{defCD}
{\check{\D}}\circ\mathcal{N}=\mathcal{C}\circ\mathcal{N}^c.
\end{equation}
\end{defi}
A single-letter quantum capacity formula exists for all degradable and conjugate degradable channels~\cite{DegCh,ConjDeg}.
\begin{lem}\label{lem:S_ell-entbreaking}
The complementary channels of all cloning channels $\Cl{\ell-1}$ are entanglement-breaking.
\end{lem}
\begin{figure}[t]
\begin{diagram}[labelstyle=\textstyle,size=1.9em,objectstyle=\textstyle]
& A & \rTo^{\Cl{\ell-1}} & E & & \\
& & \rdTo_{\mathcal{S}^c_{\ell-1}}(2,4)\; & & \rdTo^{{\!\!\check{\,\,\D_\ell}}} & \\
& & & & & E^\prime \\
& & & & \ldDotsto\ruDotsto_{\mathcal{C}} & \\
& & & H & &
\end{diagram}
\caption{\label{diag:chnl_struct}In the diagram, $\Cl{\ell-1}$ is a $1\to (\ell-1)$ cloning channel with a conjugate degrading map ${\!\!\check{\,\,\D_\ell}}$. The dotted line signalizing a non-CP map is complex conjugation $\mathcal{C}:\bar\kappa_{\ell-1}\leftrightarrow\kappa_{\ell-1}$.}
\end{figure}
Looking at the diagram in Fig.~\ref{diag:chnl_struct} we recall an observation made in~\cite{ConjDeg}. The complementary channel of a conjugate degradable channel is either entanglement-breaking or entanglement-binding~\cite{entbind}. The reason is that the transposed output of the complementary channel is by definition a positive operator and this condition is satisfied only by the two classes of channels. If we show that the complementary channels of cloning channels $\Cl{\ell-1}$ are entanglement-breaking then the Holevo capacity of cloning channels is additive too. It follows from the fact that entanglement-breaking channels single-letterize the classical capacity~\cite{entbreak} together with the result of Ref.~\cite{ComplCh} showing that a channel is additive if and only if its complementary channel is additive.
\begin{proof}[Proof of Lemma~\ref{lem:S_ell-entbreaking}]
Let us first take a look at Fig.~\ref{fig:chnl_struct} representing the intricate mutual dependence of $\Cl{\ell-1}$ for all $\ell\geq3$. By a direct calculation we verify that $\mathcal{S}_2^c$ is entanglement-breaking since $R_2=(\mathbb{1}\otimes\mathcal{S}_2^c)(\Phi^+)$ is a PPT state which stands for
positive partial transpose. Therefore, it is a separable state (the input and output Hilbert space is two-dimensional). As a consequence we may write
\begin{equation}\label{eq:entbreakoutput2}
R_2=\sum_iq_i\chi_i\otimes\upsilon_i=\sum_i\sum_{k,l}\mu_{ikl}J_k^{(2)}\otimes J_l^{(2)},
\end{equation}
where $\chi_i,\upsilon_i$ are positive operators and $0\leq q_i\leq1,\sum_iq_i=1$. The second equation is valid in general since the $su(n)$ algebra generators form an orthogonal basis. To continue let us recall how we determine the output of the rest of complementary channels $\mathcal{S}_{\ell-1}^c$. We found the answer in Eq.~(\ref{StokesplusConjugateStokes2}). We get the output state by a mere exchange of the $J^{(2)}$ generators of the $su(2)$ algebra for higher-dimensional generators $J^{({\ell-1})}$. The Stokes coefficients $\tilde n_i$ stay preserved and $\bar\kappa_{\ell-1}$ is a density operator for all $\ell$. Hence if we write $R_{\ell-1}=(\mathbb{1}\otimes\mathcal{S}_{\ell-1}^c)(\Phi^+)$ for the corresponding (higher-dimensional) maximally entangled state $\Phi^+$ then
\begin{equation}\label{eq:entbreakoutput_ell}
R_{\ell-1}={1\over c}\sum_i\sum_{k,l}\mu_{ikl}J_k^{(2)}\otimes J_l^{({\ell-1})},
\end{equation}
is again a valid quantum state (up to the normalization constant $c$). Moreover, the separable form from the middle of Eq.~(\ref{eq:entbreakoutput2}) is preserved since the Stokes coefficients hidden in $\mu_{ikl}$ stay preserved too. It follows that all $\mathcal{S}_{\ell-1}^c$ are entanglement-breaking.
\end{proof}
\begin{figure}[h]
\begin{center}
\resizebox{8cm}{4.4cm}{\includegraphics{stclonechnl_struct.eps}}
\caption{The relation among various cloning channels $\Cl{\ell-1}$ is sketched here. Numbers indicate the dimension of the particular Hilbert space $\HH$ for the output of $\Cl{\ell-1}$ and the output of their complementary channels $\mathcal{S}_{\ell-1}^c$. In the lower row we further distinguish between the space occupied by the actual output $\bar\kappa_{\ell-1}$ of the channel $\mathcal{S}_{\ell-1}^c$ (barred numbers) and its complex conjugated version (unbarred numbers). $\mathcal{P}_\ell^\lambda$ is a depolarizing channel where $\lambda=(\ell-1)/(\ell+1)$ and ${\!\!\check{\,\,\D_\ell}}$ is a conjugate degrading map.}
\label{fig:chnl_struct}
\end{center}
\end{figure}
\begin{proof}[Proof of Theorem~\ref{thm1}]
All entanglement-breaking channels single-letterize the classical capacity quantity~\cite{entbreak}. Since the complementary channels of cloning channels $\Cl{\ell-1}$ are entanglement-breaking $\Cl{\ell-1}$ are therefore also additive.
\end{proof}
\begin{cor}
The previous theorem further shed some light on the properties of $1\to(\ell-1)$ cloning channels. Invoking the result of Cubitt et al.~\cite{degch_study} stating that all entanglement-breaking channels are anti-degradable it follows that all cloning channels $\Cl{\ell-1}$ are also degradable. Degradable channels are known to possess a single-letter formula for the quantum capacity so this result confirms the same findings from Ref.~\cite{ConjDeg} based on the property of conjugate degradability.
\end{cor}
\begin{cor}
We are now able to explicitly write down the formula for the classical capacity. Since all $\Cl{\ell-1}$ are covariant we suitably choose the coefficients $\alpha,\beta$ such that states $\varepsilon_\ell$ from Eq.~(\ref{StokesplusConjugateStokes1}) are diagonal ($\alpha=1,\beta=0$). Then
$$
\varepsilon_\ell={1\over\Delta_\ell}\sum^{\ell-1}_{k=0}k\kb{k}{k},
$$
where $\Delta_\ell=\ell(\ell-1)/2$. Hence, considering $\log{f}=\log{\ell}$ in Eq.~(\ref{equivlocal}), we get
\begin{equation}\label{eq:cloningchannelclasscap}
C(\Cl{\ell-1})=1-\log{(\ell-1)}+{1\over\Delta_\ell}\sum_{k=0}^{\ell-1}k\log{k}.
\end{equation}
\end{cor}
A potentially useful consequence of Lemma~\ref{lem:S_ell-entbreaking} is the fact that a composition of a cloning channel and a depolarizing channel $\mathcal{P}^\lambda_{\ell}\circ\Cl{\ell-1}=\Cl{\ell-1}\circ\mathcal{P}^\lambda_2$ (holds for $\lambda=(\ell-1)/(\ell+1)$) is weakly additive (this is not needed for the purpose of this paper). Recall the definition of the depolarizing channel $\mathcal{P}^\lambda_\ell(\varrho)=\lambda\varrho+(1-\lambda)\mathbb{1}^{(\ell)}/\ell$ for ${-1/(\ell^2-1)}\leq\lambda\leq1$ To prove the claim we make use of the following lemma.
\begin{lem}
Let $\mathcal{N}$ be an additive channel (weakly or not). Then if $\mathcal{M}=\mathcal{C}\circ\mathcal{N}$ is another channel, where $\mathcal{C}$ denotes complex conjugation, $\mathcal{M}$ is weakly additive as well.
\end{lem}
\begin{rem}
The map $\mathcal{M}$ might not be always a CP map (for instance, if $\mathcal{N}$ is an identity channel).
\end{rem}
\begin{proof}
Using the fact that $S(\varrho)=S(\bar\varrho)$ we see that $C_{\rm Hol}(\mathcal{N})=C_{\rm Hol}(\mathcal{C}\circ\mathcal{N})$. Hence if $C_{\rm Hol}(\mathcal{N}^{\otimes n})=nC_{\rm Hol}(\mathcal{N})$ then $C_{\rm Hol}(\mathcal{M}^{\otimes n})=C_{\rm Hol}((\mathcal{C}\circ\mathcal{N})^{\otimes n})=nC_{\rm Hol}(\mathcal{C}\circ\mathcal{N})$ because we are trying to maximize eigenvalues given by the same characteristic equation. But by definition the maximum for $C_{\rm Hol}(\mathcal{N}^{\otimes n})$ corresponds to a factorized input state and so such a state also maximizes $C_{\rm Hol}(\mathcal{M}^{\otimes n})$.
\end{proof}
Looking at Fig.~\ref{fig:chnl_struct} we can see why additivity of $\mathcal{S}_{\ell}^c$ for $\ell\geq3$ implies additivity of $\mathcal{P}^\lambda_{\ell}\circ\Cl{\ell-1}$. The reason lies in the fact that $\mathcal{P}^\lambda_{\ell}\circ\Cl{\ell-1}$ composed with complex conjugation is equal to $\mathcal{S}_{\ell}^c$ for all $\ell$.
\subsection{The classical capacity of the Unruh channel}
The output of the Unruh channels is a weighted direct sum of outputs of cloning channels $\Cl{\ell-1}$ for all $\ell$. From Theorem~\ref{thm1} we know that the Holevo capacity of all of them is additive. This directly leads to the proof of additivity of the Holevo capacity for the Unruh channels itself.
\begin{thm}\label{thm2}
The infinite-dimensional Unruh channel studied in~\cite{UnruhCh} is additive
\end{thm}
First, let us present a lemma.
\begin{lem}\label{lem:Smin_directsum}
Let $\mathcal{A},\mathcal{B}$ be additive and covariant but otherwise arbitrary finite-dimensional channels whose input Hilbert spaces are of the same dimension. Then a channel $\mathcal{G}:\mathscr{F}\big(\HH\big)\to\mathscr{F}\big(\HH_\mathcal{A}\oplus\HH_\mathcal{B}\big)$ is additive for any ensemble $\{q_\mathcal{A},q_\mathcal{B}\}$.
\end{lem}
\begin{proof}
The channel output is unitarily equivalent to
\begin{equation}\label{channleaction}
\varrho\xrightarrow{\mathcal{G}}q_\mathcal{A}\varrho_\mathcal{A}\oplus q_B\varrho_B\equiv\kb{0}{0}\otimes q_\mathcal{A}\varrho_\mathcal{A}+\kb{1}{1}\otimes q_\mathcal{B}\varrho_\mathcal{B}.
\end{equation}
Defining $\mathcal{T}$ to be an arbitrary channel we see that for any input pure state $\omega$ of the channel $\mathcal{G}\otimes\mathcal{T}$ the output state is a block-diagonal matrix $\sigma=q_\mathcal{A}(\mathcal{A}\otimes\mathcal{T})(\omega)\oplus q_\mathcal{B}(\mathcal{B}\otimes\mathcal{T})(\omega)$. Thus, $S(\sigma)=S(\{q_\mathcal{A},q_\mathcal{B}\})+q_\mathcal{A} S((\mathcal{A}\otimes\mathcal{T})(\omega))+q_\mathcal{B} S((\mathcal{B}\otimes\mathcal{T})(\omega))$.
Hence
\begin{eqnarray}
S^{\rm min}(\mathcal{G}\otimes\mathcal{T})&=&S(\{q_\mathcal{A},q_\mathcal{B}\})+q_\mathcal{A}\min_\omega\{S((\mathcal{A}\otimes\mathcal{T})(\omega))\}\nonumber\\
&+&q_\mathcal{B}\min_{\omega^\prime}\{S((\mathcal{B}\otimes\mathcal{T})(\omega^\prime))\}\nonumber\\
&=&S(\{q_\mathcal{A},q_\mathcal{B}\})\nonumber\\
&+&q_\mathcal{A} S(\mathcal{A}(\varphi))+q_\mathcal{B} S(\mathcal{B}(\varphi))+\Smin{\mathcal{T}}\nonumber\\
&\equiv& S(\mathcal{G}(\varphi))+\Smin{\mathcal{T}}
\end{eqnarray}
using the properties of $\mathcal{A}$ and $\mathcal{B}$.
\end{proof}
\begin{proof}[Proof of Theorem~\ref{thm2}]
The proof is a direct application of the previous lemma since the Unruh channel happens to be $\mathcal{U}(\varphi)=\bigoplus_{\ell=2}^\infty p_\ell\Cl{\ell-1}(\varphi)$ where $p_\ell=(1-z)^3z^{\ell-2}(\ell-1)\ell/2,0\leq z<1$. The channel $\mathcal{U}(\varphi)$ is the same channel as in Eq.~(\ref{squeeze}) where it is written in the Fock space representation.
We show using the block-diagonal structure of the output state and the unitary covariance of the Unruh channel that the inductive process described above approximates the channel output with an arbitrary precision for any input qubit.
Namely, let us denote a partial sum $c_K=\sum_{\ell=2}^Kp_\ell$. We get
\begin{equation}
c_K={1\over2}\big(2+2(K^2-1)z^K-K(K+1)z^{K-1}-K(K-1)z^{K+1}\big)
\end{equation}
and so $\lim_{K\to\infty}{c_K}=1$ for all $0\leq z<1$.
\end{proof}
\begin{rem}
Note that the channel input for otherwise infinite-dimensional Unruh channel is naturally energy constrained since the set of input states is limited to qubits.
\end{rem}
\begin{cor}
Theorem~\ref{thm2} enables us to bring up the formula for the classical capacity of the Unruh channel. Using Eq.~(\ref{HolevoCap}) and the covariance of the Unruh channel we get
\begin{equation}\label{capUnruh}
C(\mathcal{U})=1-\sum_{\ell=2}^\infty p_\ell\log{(\ell-1)} + \sum_{\ell=2}^\infty {p_\ell\over\Delta_\ell}\sum_{k=0}^{\ell-1} k\log{k}.
\end{equation}
The plot in Fig.~\ref{fig:chnl_UnClCap} depicts the Holevo capacity as a function of the parameter $z$.
\end{cor}
\begin{figure}[t]
\begin{center}
\resizebox{9.4cm}{6cm}{\includegraphics{stclonechnl_UnruhClCap.eps}}
\caption{The Holevo capacity for the Unruh channel as a function of $z$. Recall that $z$ itself is a function of the proper acceleration of a non-inertial observer. Note that in the limit of infinite acceleration ($z\to1$) the capacity converges to a non-zero value. This value is the same as the capacity for $\Cl{\ell-1}$ for $\ell\to\infty$ from Eq.~(\ref{eq:cloningchannelclasscap}) since the peak of probability distribution is `moving' towards infinity with growing acceleration.}
\label{fig:chnl_UnClCap}
\end{center}
\end{figure}
\section{Degrading map construction}\label{sec:degrad}
Let us attempt to construct degrading maps for several low-dimensional cloning channels $\Cl{\ell-1}$ we studied in the previous section.
We first analyze the case $\ell=3$. Looking at Eqs.~(\ref{StokesplusConjugateStokes1}) and (\ref{StokesplusConjugateStokes2}) we see that the complementary output of every $\Cl{\ell-1}$ is effectively conjugated with respect to the channel output. This fact together with the unitary covariance of cloning channels leads to the condition similar to Eq.~(\ref{covcond})
\begin{equation}\label{channelcovariance}
\overline{{\D_3}(r_1(g)\varrho\,r_1(g)^\dagger)}=r_2\,\overline{{\D_3}(\varrho)}\,r_2^\dagger,
\end{equation}
where the presence of bars is the result of complex conjugation. In this case, $r_2$ and $r_1$ is the two- and three-dimensional irrep of $g\in G=SU(2)$, respectively. This is, however, the same as the contravariance condition
\begin{equation}\label{channelcontracovariance}
{\D_3}(r_1\varrho\,r_1^\dagger)=\overline{r_2(g)}\,{\D_3}(\varrho)\,r_2(g)^T.
\end{equation}
By rephrasing this condition within the Choi-Jamio\l kowski isomorphism~\cite{jamiolk} we get
\begin{equation}\label{jamicovariance}
\big[\,\overline{R_{\D_3}},r_2\otimes r_1\big]=0
\end{equation}
when $\overline{R_{\D_3}}$ is a positive semidefinite matrix corresponding to the CP map ${\D_3}$. One of Schur's lemmas dictates $\overline{R_{\D_3}}=\bigoplus_ic_i\Pi_i$ where $c_i\geq0$ and $\Pi_i$ are projectors into the subspaces of the split product $[2]\otimes[3]=[2]\oplus[4]$ and thus $\overline{R_{\D_3}}\equiv R_{\D_3}$. We insert $R_{\D_3}$ into ${\D_3}(\varepsilon_{3_{in}})=\Tr{in}\big[\left(\mathbb{1}_{out}\otimes\bar\varepsilon_{3_{in}}\right)R_{\D_3}\big]$ since we are looking for such $R_{\D_3}$ that $\bar\kappa_2={\D_3}(\varepsilon_{3})$ where (index $in$ omitted)
\begin{gather}\label{inputoutput_ell3}
\varepsilon_3={1\over3}
\begin{pmatrix}
2|\alpha|^2 & \sqrt{2}\alpha\bar \beta & 0 \\
\sqrt{2}\bar \alpha\beta & 1 & \sqrt{2}\alpha\bar \beta \\
0 & \sqrt{2}\bar \alpha\beta & 2|\beta|^2 \\
\end{pmatrix},\\
\bar\kappa_2={1\over3}
\begin{pmatrix}
|\alpha|^2+1 & \bar \alpha\beta \\
\alpha\bar \beta & |\beta|^2+1\\
\end{pmatrix}.
\end{gather}
In other words, we maximize the fidelity between these two states checking whether it reaches one for some $c_1,c_2$ considering the constraints $c_{1,2}\geq0$ and $\Tr{out}\big[R_{\D_3}\big]=\mathbb{1}^{(3)}$. Because we are dealing with mixed states, we use the fidelity expression due to Bures which simplifies for two-dimensional matrices~\cite{FidelityDim2} as
\begin{equation}\label{Buresfid}
F({\D_3}(\varepsilon_{3}),\bar\kappa_2)=\Tr{}[{\D_3}(\varepsilon_{3})\bar\kappa_2]+2\sqrt{\mathop{{\mathrm{Det}}}[{\D_3}(\varepsilon_{3})]\mathop{{\mathrm{Det}}}[\bar\kappa_2]}.
\end{equation}
As expected from the results in section~\ref{subsec:infseq}, the fidelity reaches one. In general, the decomposition $\overline{R_{\D_\ell}}=\bigoplus_ic_i\Pi_i$ might be difficult to determine. Nevertheless, the good news is that an ansatz can be made. Following the lowest-dimensional exact solutions for the form of the degrading maps of $\Cl{2}$ and $\Cl{3}$ we observe that the only surviving coefficient $c_i$ from the expression for the Jamio\l kowski matrices is the one accompanying the highest irrep of the $SU(2)$ tensor product. Indeed, applying this guess on a few more $1\to (\ell-1)$ cloning channels ($\ell=5,6,7$) it always yields the sought degrading map. So we know that $\Cl{\ell-1}$ are degradable and the construction of the degrading maps might be hard for large $\ell$ but verification of the ansatz is very fast even for large $\ell$.
\section{Conclusions}
The general non-additivity result for the classical capacity of quantum channels is in some sense very satisfactory. Not only did entanglement prove to be useful for the transmission of classical information but it will spark even more effort to find out what makes a channel (non-)additive. Also, some novel strategies may be found to prove (non-)additivity for particular channels as it is now known that there is no general proof. In this paper we investigated an infinite family of channels we call $1\to (\ell-1)$ cloning channels ($\ell=2\dots\infty$) which are the incarnations of universal quantum cloning machines for qubits.
To prove additivity of the Holevo capacity for cloning channels we used the fact that cloning channels enjoy the property of being conjugate degradable channels. Conjugate degradable channels already prove to be a useful concept since it is known that their quantum capacity has a single-letter formula. We have therefore found an infinite family of channels for which both the classical and quantum capacity is easily calculable. Also, we were able to prove that $1\to (\ell-1)$ cloning channels are degradable. Furthermore, using the fact that cloning channels are intimately related to an infinite-dimensional channel called the Unruh channel we were also able to present the additivity proof of the Unruh channel which otherwise seems intractable. The infinite-dimensional Unruh channel is now a member of a rare family of channels with both capacities easily calculable (together with a dephasing channel) since the existence of a single-letter quantum capacity formula has been proved elsewhere. This result might find an important future application in quantum field theory in curved spacetime considering the prominent role the Unruh channel plays in this branch of modern physics.
\section*{acknowledgments}
The author is grateful to Patrick Hayden for comments and discussions. The comments on early versions of the manuscript made by Dave Touchette, Min-Hsiu Hsieh and in particular Mark Wilde are greatly appreciated. The work was supported by QuantumWorks and by a grant from the Office of Naval Research (N000140811249).
\bibliographystyle{IEEEtran}
|
2,877,628,089,037 | arxiv | \section{Introduction}
\label{secintro}
Ultra-High Energy Cosmic Rays (UHECRs) are protons or light
nuclei with energies greater than about 10$^{19}$ eV (10 EeV).
When these highly energetic particles enter the earth's atmosphere
they produce a shower of secondary particles and excite
the Nitrogen molecules in the atmosphere; both effects can be
detected from the ground. Recent advances in ground shower
detectors allow a precise measurement of both the initial energy and the
arrival direction of the UHECR above the earths atmosphere,
e.g. AGASA \citep{taket99} and
the Pierre Auger Observatory \citep{abret04,demet07}.
While UHECRs are rare they have a unique astronomical potential:
UHECRs with energies $\gtrsim$50~EeV are not expected to be
deflected significantly by Galactic or intergalactic magnetic
fields. Their measured arrival directions can therefore be traced
directly back to the originating source within the observatory
measurement uncertainties.
Suspected sources of high energy cosmic rays include
core collapse Supernovae (SN II),
Supernova Remnants,
Pulsars,
Active Galaxies - especially radio jets and radio lobes, and
gamma rays bursts (for reviews and lectures see
\citet{kat08,hil99} and references therein).
In these sources, the cosmic rays are posited to undergo
successive accelerations via scattering off energetic
charged particles and/or in shocks.
More exotic explanations include an origin in dark matter anhilation.
The main energy loss for UHECR propagating over cosmological
distances is expected to be pion-production, the so called
Greisen Zatsepin Kuzmin (GZK) effect \citep{gre66,zatkuz66}.
In this process, a UHECR interacts with a CMB photon and
loses an estimated $\sim$30\% of its energy.
Both the energy loss and the mean free path between interactions
depend on several details; energy loss length predictions vary
between 20 and 100~Mpc (see e.g. \citet{sta04},\citet{kat08},PA08). If the
GZK effect is at work then UHECRs with energy above
$\sim$50~EeV are expected to originate primarily in sources closer
than this predicted energy loss length.
The Pierre Auger Observatory (hereafter PAO) in Argentina \citep{abret04,demet07},
an array of 1600 Cerenkov detectors spread over 3000 km$^2$ plus
six optical telescopes, is designed
to measure arrival directions and energies of cosmic rays via
their secondary particle showers and their atmospheric flouresence.
Latest details on the instrumentation and exciting science results
being produced by the Auger Observatory and Collaboration can be
found at the Auger web-site http://www.auger.org.
The array has been in partial operation since January 2004.
The Auger collaboration has
reported 81 events between 2004 January 1 and 2007 August 31
with reconstructed energies above 40~EeV and
zenith angles smaller than 60$^{\circ}$; of these, 27 have energies
above 56~EeV \citep[][hereafter PA07,PA08]{abret07,abret08}.
The origins of the latter 27 UHECRs are the focus of this paper.
The Auger Collaboration found that the arrival directions of
the 27 UHECRs with energies above 56~EeV are
not isotropic at a 99\% significance level (PA07,PA08).
Instead, they found that the arrival directions are
correlated with the positions of AGNs within
$\sim$71~Mpc (PA07,PA08,\citet{molet07}).
For this analysis they used
all AGNs (and galaxies with \ion{H}{II}\ nuclei)
in the catalog of V{\'e}ron-Cetty \& V{\'e}ron, 12th
Edition \citep{verver06}.
In their statistical correlation of all 81 UHECRs at
energy above 40~EeV with the V{\'e}ron-Cetty \& V{\'e}ron catalog of AGNs,
PA07,PA08 found that the highest correlation between UHECRs and
AGNs was obtained for a maximum AGN distance of 71~Mpc,
an angular separation of 3.2$^{\circ}$\ between ``matched'' AGNs and UHECRs,
and an energy threshold for UHECRs of 56~EeV.
This maximum distance, 71~Mpc, is in line with the expectations of
the GZK effect.
\citet{goret07}, in an astro-ph comment, disagreed with this
finding, pointing out that weighting each AGN with its distance
would predict a very different distribution of UHECRs than that found
by the PAO.
Our interest in this topic was
provoked when a quick examination of the 27 UHECR events
showed that the direction of at least five of the 27
UHECR events were directly in the line of sight to
nearby extended radiogalaxies.
In this article, we attempt a more comprehensive correlation
between the arrival directions of the 27 UHECRs with energies
above 56~EeV and catalogs of potential sources of UHECRs,
both Galactic and Extragalactic.
Sec. \ref{secdata} describes the data used in our study,
Sec. \ref{secres} describes the principle results obtained,
and Sec. \ref{secdis} contains a
brief discussion and conclusions of our study.
Distances to galaxies are calculated using 72 ${\rm ~km~s^{-1}~Mpc^{-1}}$, except for
relatively nearby galaxies for which we use
distances as referenced.
\section{Data}
\label{secdata}
Positions and energies of the 27 UHECRs detected by the PAO
with energies $\geq$~56~EeV
were taken from PA08. The figures in this publication are slightly
different from those of the Auger collaboration: PA07 and PA08
used the Aitoff projection (an equidistant projection with 2:1 aspect ratio),
while we use the Aitoff-Hammer projection (an equal area projection). The
latter is more commonly used in astronomy and is typically referred to as
simply the `Aitoff' projection, e.g. the IDL procedure aitoff.pro.
Positions and energies of the UHECRs detected by AGASA
are taken from \citet{hayet00} which is an updation of the data listed
in \citet{taket99}.
Our catalogs of astronomical sources are drawn from various
publications and web-based source lists.
In all cases we attempted to assemble as complete a
sample as possible. Further, following the results of PA07, PA08,
and the expectations from the GZK effect, we considered
only objects within a distance of $\sim$200~Mpc. For matches
to the PAO UHECRs we consider only those
sources which rise above elevation 30$^{\circ}$\ at PAO
since the PAO UHECR list includes only UHECRs detected above this
elevation.
Our list of extragalactic
supernova remnants, galaxies, and galaxy clusters
were drawn from NED \citep{madet92} using the `All Sky Search' option.
The list of Gamma Ray Bursts (GRBs) was taken from the Gamma-ray Burst Real-Time
Sky Map{\footnote{http://grb.sonoma.edu}}.
For Galactic Supernova Remnants we used the list maintained by D.
Green{\footnote{http://www.mrao.cam.ac.uk/surveys/snrs/snrs.data.html}}.
Extragalactic Radio Supernova were taken from Kurt
Weiler's list at the web-site of the Naval Research
Laboratory{\footnote{http://rsd-www.nrl.navy.mil/7213/weiler/kwdata/RSNtable.txt}}.
For extragalactic supernova, we used the Sternberg Astronomical Institute
(SAI) Supernova
Catalog{\footnote{http://www.sai.msu.su/sn/sncat/}} maintained by
Tsvetkov et al.
In the case of radiogalaxies and radio sources we used
several surveys and catalogs, including NVSS \citep{conet98},
SUMMS \citep{bocet99}, and NED. The list of extragalactic jets is based
on that compiled by \citet{liuzha02}; to this list we added data on the galaxies
as given in NED, and re-measured the total flux and total extent
of the radio emission from NVSS and SUMMS maps.
The total extent of the radio emission therefore includes both
radio jets and any radio lobes. This extent is typically referred to as
the Largest Angular Size (LAS) or Largest Linear Size (LLS). In this work
we added the LLS of the individual jets and lobes as scalars instead of vectors
in order to discount the effects of jet and lobe bending.
To recreate the results of the Auger Collaboration (PA07,PA08), we
used the 12th edition of the \citet{verver06} catalog of AGNs.
The list of AGNs (agn.dat) in this catalog includes nuclei with
\ion{H}{II}\ spectra which were previously misclassified as AGNs. These
should ideally be left out of the correlation analysis. From the
AGN statistics quoted in PA07,PA08, it appears
that the Auger Collaboration included these \ion{H}{II}\ nuclei in their
analysis, so for consistency we do the same.
Additionally, the list of QSOs (qso.dat)
lists seven galaxies with redshift=0. Four of these are
truly nearby galaxies, and the other three are probably high redshift
quasars. Again, for consistency with PA07 and PA08 we keep the
latter three in our analysis.
We have also used the Fanaroff-Riley (FR) classification for radiogalaxies
\citep{fanril74}. This division is based on whether the jets increase
(FR~II) or decrease (FR~I) in brightness with increasing distance
from the nucleus. A clear division in radio luminosity is found between the
two classes, with FR~IIs being more luminous in the radio
than FR~Is \citep{oweled94}.
\section{Results}
\label{secres}
A comparison of the arrival directions of the 27 UHECRs
and the positions of nearby galaxies with radio jets
\citep[all galaxies listed in][]{liuzha02}
is shown in Fig.~\ref{figjetned}.
We have divided the galaxies with radio jets into three redshift
bins: ``nearby'' (D $\leq$ 75~Mpc), ``intermediate'' (75~Mpc $<$ D $\leq$ 150~Mpc),
and ``distant'' (150~Mpc $<$ D $\leq$ 210~Mpc). In the closest
redshift bin, D $\leq$ 75~Mpc,
we distinguish between galaxies with radio structures more extended than 180~kpc
(``extended'') and those with radio structures less extended than 180~kpc
(``compact'').
We calculated the total extent of the radio structure of any given galaxy by
summing the lengths of the two radio jets and/or lobes as scalars instead of vectors.
The total radio extent includes radio jets and any radio lobes.
There is no intrinsic reason to use a radius $\psi=$~3.5$^{\circ}$\ for
a ``match'' between UHECR arrival direction and the position of an
astronomical source. The instrumental resolution of the PAO
is about 1$^{\circ}$\ at these UHECR energies, to which one
must add a small angle -- perhaps up to a few degrees \citep{sta04} -- to account
for the deflection of the UHECR by magnetic fields.
The best correlation between UHECRs and nearby AGNs found by PA07, PA08
is based on considering $\psi \sim$3.2$^{\circ}$ as a match. However, as we
show later (e.g. Fig.~\ref{figscanb}), the value of $\psi$ in the
analysis of PA07,PA08 should really be $\psi \sim$ 3.1$^{\circ}$--4.3$^{\circ}$.
For the above reasons, we consider two values of $\psi$ for a match:
$\psi=$~1.5$^{\circ}$\ and $\psi=$~3.5$^{\circ}$.
The distance limit for `nearby' galaxies - 75~Mpc - is similar to
the value d$_{\rm max} \sim$ 71~Mpc found by PA07 and PA08.
The division between `compact' and `extended' radio structures
- 180~kpc - is the one `new' free parameter in our analysis. We note that
this value is close to the minimum linear size of the radio structures of a
typical radio galaxy \citep[e.j.][]{haret98}.
Of all 27 UHECRs with energy above 56~EeV detected by the PAO
four (eight) are within $\psi=$~1.5$^{\circ}$\ ($\psi=$~3.5$^{\circ}$)
of the radio structures of nearby extended radiogalaxies in the
field of view of the PAO.
Conversely, of all ten nearby extended radiogalaxies in the
field ov view of the PAO three (six)
have radio structures which
can be matched to within 1.5$^{\circ}$\ (3.5$^{\circ}$) of a UHECR event. Of the remaining
four `unmatched' radiogalaxies, three - NGC~1316, NGC~4261 and NGC~4760 - fall outside
the area of maximum exposure at the PAO.
For the intermediate and distant redshift bins
there is little match between extragalactic radio jets and
UHECRs; the only two matches within 3.5$^{\circ}$\ are CGCG~403-019, a
Seyfert/BL~Lac
at 112~Mpc, and Mrk~612, a Seyfert galaxy at 85~Mpc.
The match between a UHECR and CGCG~403-019 deserveres
furter comment.
CGCG~403-019 is a BL~Lac with a Seyfert~1 spectrum
\citep{verver93}. In the 12th AGN Catalog of
\citet{verver06} it is listed by its alias
PKS~2201+04 and is classified as a Seyfert 1.
It is a poweful radio source with a core-jet
structure with total extent $\sim$6{\arcmin} or
$\sim$190~kpc \citep{ulvjoh84,lauet93}. Within
the unified scheme of radiogalaxies, BL~Lacs are
posited to be pole-on FR~Is, implying a much
larger deprojected radio extent.
We therefore consider CGCG~403-019 as an `intermediate'
distance `extended' radiogalaxy.
Further, CGCG~403-019 is unique in being the
nearest BL~Lac with extended radio structure.
The 12th AGN Catalog of \citet{verver06} includes only
three confirmed BL~Lacs - V~Zw~331, RXS J05055+0416, and
TEX 0554+534 - and only one probable BL~Lac -
RXS J21231-1036 - at D $\leq$ 150~Mpc. All four show
compact radio emission in NVSS radio maps.
The Auger Collaboration noted that the region around Cen~A has
an unusually large number of UHECRs; this is also the area
with the maximum number of nearby extended radiogalaxies.
As seen in Fig.~\ref{figsjetned}, the two UHECRs centered on
Cen~A can be explained as originating in either of Cen~A,
NGC~5090 (D=48~Mpc; total radio extent $\sim$187~kpc), or even
PKS~1308-441 (D=211~Mpc; total radio extent $\sim$1140~kpc).
The event just further south can be matched to the
southern jet of Cen~A, or to the relatively distant PKS~1308-441.
Cen~B (D=54~Mpc; total radio extent $\sim$250~kpc) is almost
perfectly centered on a UHECR, and
WKK~4552 (D=67~Mpc; total radio extent $\sim$360~kpc)
is within $\sim$3$^{\circ}$\ of another UHECR. The
UHECR just to the north of Cen~A is bracketed by
IC~4296 (D=52~Mpc; total radio extent $\sim$1560~kpc)
and
ESO~443-G024 (D=71~Mpc; total radio extent $\sim$60~kpc). The
very extended radio structure of the former actually intersects with
the 3.5$^{\circ}$\ circle around the UHECR event.
Are the nearby extended radiogalaxies matched to PAO UHECRs different
from those not matched to PAO UHECRs?
A comparison of the relevant properties of the ten nearest
extended radiogalaxies is given in Table~1, and their
radio contour maps in Fig.~\ref{figcontour}. Table~1 also lists
the relevant radio properties of CGCG~403-019 and Mrk~0612, the
two intermediate distance AGNs which match to a UHECR.
There is some difference in the combination of two factors:
the radiogalaxies in the direction to UHECRs include all of
the subset which have radio jet morphologies closer to FR~II,
and are all in the area where
the PAO has obtained a higher exposure
time. In any case, given the small number statistics, i.e. typically
1$\pm$1 UHECR events per matched radiogalaxy, it is not
surprising that UHECRs are not yet detected towards the
remaining four radiogalaxies. Clearly more UHECRs are needed
to form any firm conclusion.
Fig.~\ref{figcluster} compares the arrival directions of
UHECRs with the distribution of nearby groups and clusters of galaxies.
While there is an over-density of galaxies, clusters of
galaxies and UHECRs around the region of Cen~A, the overall
distributions of nearby galaxies and nearby galaxy clusters
do not correlate with the arrival directions of UHECRs.
This argues against an origin of UHECRs in material associated
with large mass haloes, e.g. Dark Matter.
The more detailed analysis of \citet{goret07} also predicts
a different distribution of UHECRs when the distances to
galaxies and clusters are taken into account.
We have also compared the arrival directions of the
27 UHECRs with the positions of Galactic and
nearby extragalactic supernovae and supernova
remnants,
of nearby extragalactic radio supernovae, of
nearby BL~Lacs, and of gamma-ray bursts.
In all cases there appears to be no
more than a random overlap.
The analysis of PA07 and PA08 considered a single source
population for the UHECRs. How would their results
change if one were to consider multiple source populations,
at least one of which is nearby extended radiogalaxies?
Fig.~\ref{fighist} compares the histograms of the
angular distribution of the 27 UHECRs with energies above
56~EeV, the subset of 19 UHECRs not towards nearby
extended radiogalaxies,
and the expectations from a random distribution of 27 UHECRs.
The deviation from isotropy discussed by
PA07 and PA08 (dashed line in the figure)
comes primarily from the clustering of events
around Cen~A. Deleting the UHECRs which lie within 3.5$^{\circ}$\ of
nearby radiogalaxies (solid line in the figure) gives a distribution
in which isotropy cannot be ruled out at high significance.
The Auger Collaboration cross correlated the positions of the 27 UHECRs
with the AGNs in the 12th catalog of \citep{verver06} by
varying the maximum AGN distance (z$_{\rm max}$),
the angular separation between AGN and UHECR to be considered a match
($\psi$), and the threshold energy of UHECRs.
They then looked for the minimum cumulative probability of chance coincidences.
Note that they
apparently included \ion{H}{II}\ nuclei from the catalog in their analysis.
In this process (see PA08 for details) they found that
the best match was found for z$_{\rm max}$ = 0.017 (D $\sim$ 71~Mpc)
and $\psi$ = 3.2$^{\circ}$.
We repeated their cumulative probability scans in
the parameters z$_{\rm max}$ and $\psi$;
we were unable to scan in energy as we do not have
access to the position and energy data on the UHECRs with energies less than
56~EeV. The resulting scans are shown in Fig.~\ref{figscana}. The dashed line
shows the result if all 27 UHECRs are used, i.e. the result
obtained by PA08. In this case 19 of the 27 UHECRs match to some
AGN or \ion{H}{II}\ nuclei in the catalog.
The middle solid line of each panel shows the result of considering only the 23 UHECRs
remaining after deleting the four UHECRs which best match (within 1.5$^{\circ}$)
the radio structures of nearby extended radiogalaxies
(2 UHECRs for Cen~A, and one each for Cen~B and NGC~7626). In this case
13 of the 23 UHECRs match to some AGN or \ion{H}{II}\ nuclei in the catalog.
The upper solid lines show the results of considering only the 19 UHECRs remaining
after deleting the four UHECRs mentioned above, plus the UHECRs
matched within 3.5$^{\circ}$\ to the South jet of Cen~A, to CGCG~403-019, to
WKK~4452, and to IC~4296.
In this case 11 of the 19 UHECRs match to some
AGN or \ion{H}{II}\ nuclei in the catalog.
While the overall shape of the scans in $\psi$ and z$_{\rm max}$
remain roughly the same, the significance of the results decrease
significantly. In the case of the scan in $\psi$, the minimum at
around 3.2$^{\circ}$\ is no longer a clear and unique minimum.
We repeated the above exercise after making two modifications
to the AGN list: we deleted all galaxies with H~II type nuclei
(50 galaxies at z $\le$ 0.024)
and the three probable high redshift QSOs which are listed as having
z=0 in the AGN Catalog 12th. ed.. The results are shown in Fig.~\ref{figscanb}.
The main difference with the previous figure is the broader valley
in the $\psi$ scan. Instead of a clearly defined minimum at $\psi$ = 3.2$^{\circ}$
one now has a greater range $\psi \sim$ 3.2$^{\circ}$--~4.3$^{\circ}$.
\section{Discussion \& Concluding Remarks}
\label{secdis}
The arrival directions a subset of
the 27 UHECRs with energies above 56~EeV detected by the
PAO are statistically most closely related
to nearby radiogalaxies with extended radio jets and/or lobes.
It is thus likely that nearby (due to the GZK effect)
radiogalaxies with extended (radio extent $\ge$ 180~kpc), two sided
jets and lobes
are the source population of a subset of the UHECRs detected by the
PAO. Additionally, there is weak evidence
that hosting a jet with morphology closer to FR~II makes a
radiogalaxy more likely
to be a UHECR source. Interestingly, all these factors are
consistent with the lack of UHECR detections towards M~87.
The results of the previous section were based only on
UHECRs detected by the PAO. We have also tested for correlations
against the UHECRs with energy above 56~EeV detected by AGASA
(red circles with radius 3.5$^{\circ}$\ in Fig.\ref{figjetned}. Among the
nearby (D $\leq$ 75~Mpc) galaxies with extended radio jets, the only clear match
is to CGCG~514-050, a radiogalaxy at D=72.2~Mpc with radio extent
$\sim$250~kpc. The remaining three nearby galaxies with extended radio galaxies,
NGC~315, NGC~383, and NGC~5127, are not matched to within 3.5$^{\circ}$\
of any AGASA UHECR event, though there are several UHECRs at slightly
larger distances. Additionally, One AGASA UHECR is close to the location of
NGC~7626 and another is centered on the radiogalaxy 3C~120, at D=135~Mpc
and total radio extent $\sim$315~kpc.
Among the UHECRs detected by PAO,
an isotropic distribution of the $\sim$20 UHECRs which are not in the
line of sight to nearby extended radiogalaxies cannot be ruled out at
high significance.
The significance of the correlation found by PA07 and PA08 between
UHECR arrival directions and nearby AGNs from the catalog
of V{\'e}ron-Cetty \& V{\'e}ron is much lower when one only
considers the $\sim$ 20 UHECRs not matched to nearby extended
radiogalaxies and even lower when H~II nuclei in the
the AGN Catalog are deleted. The findings of PA07,PA08
were probably largely influenced by
the correlation between UHECRs and nearby radiogalaxies
which we have shown above.
The UHECR arrival directions are not strongly correlated
with either supernovae, extragalactic radio-supernovae,
or nearby groups and clusters of galaxies.
The main difference between the results of the Pierre Auger
Collaboration and those from previous studies of UHECRs, e.g.
AGASA, could primarily be that the southern location of the
PAO is more privileged with respect
to nearby extended radiogalaxies.
\begin{acknowledgements}
NN dedicates this work to his thesis advisor and collaborator
Andrew S. Wilson.
We acknowledge funding from ALMA 3105000 and 3016013, and
the FONDAP Center for Astrophysics.
This research has made use of the NASA/IPAC Extragalactic Database
(NED) which is operated by the Jet Propulsion Laboratory, California
Institute of Technology, under contract with the National Aeronautics
and Space Administration.
\end{acknowledgements}
|
2,877,628,089,038 | arxiv | \section{Constraints on ALPs with hypercharge coupling}\label{app:hypercharge}
\begin{figure*}[t]
\centering
\hspace{-0.5cm}
\includegraphics[width=0.5 \textwidth ]{fig4a.pdf}
\includegraphics[width=0.5 \textwidth ]{fig4b.pdf}
\caption{Similar to Figs.~\ref{fig:BtoKa_reach}-\ref{fig:invisibleALP}, except we include a coupling of the ALP to hypercharge gauge bosons and set $g_{aY} = g_{aW}$.
}
\label{fig:ALPgY}
\end{figure*}
In Fig.~\ref{fig:ALPgY}, we show the analogous results for ALPs that additionally couple to hypercharge gauge bosons. We take $g_{aY} = g_{aW}$ as a benchmark, such that the coupling of the ALP to photons is $g_{aW}$. For $a \to \gamma \gamma$, the reach for rare meson decays is strengthened at low $M_a$, where the larger ALP decay rate leads to a larger fraction of decays within the detector volume. The LEP constraint on $Z \to a \gamma$, shown in Fig.~2, is significantly weaker for $g_{aY} = g_{aW}$. Instead, we show the stronger LEP-II constraint on $e^+ e^- \to a \gamma$ taken from Ref.~\cite{Knapen:2016moh}. Even with the additional ALP coupling to hypercharge, however, we see that the projections and limits from rare meson decays can exceed the sensitivity of projected searches based on photon couplings alone.
|
2,877,628,089,039 | arxiv | \section{Introduction}
Currently, the general understanding of galaxy evolution suggests that
interactions play a significant role. However, the dominant processes
are still not clear. Galaxy clusters provide a good laboratory in which
to examine the effects of interactions on the evolution of individual
galaxies within the cluster. The systematic differences in optical
morphology between galaxies observed in the dense cores of clusters and
those from the field regions \citep[i.e.\ the morphology--density
relation ][]{dressler80} suggest that galaxy evolution is strongly driven
by the environment. Effects of the various forms of interactions
have been observed in detail in the Virgo Cluster where it is possible
to spatially resolve regions of dust, gas and stars in individual galaxies.
\citet{koopmann04} used the spatial distribution of H$\alpha$ to study
the environmental effects on star formation in the Virgo Cluster galaxies.
However, H$\alpha$ observations are compromised by dust extinction
effects. Without accounting for dust extinction effects, it is difficult to
answer important questions about star formation such as: i) the existence of
enhanced star-forming regions due to ram pressure or gravitational effects,
and ii) the comparison between extraplanar and disk star formation rates.
Recently, it has been shown that a combination of H$\alpha$ and 24 $\mu$m
emission has been found to be a reliable indicator of total star formation
\citep{calzetti07}, since the 24 $\mu$m emission is presumed to consist
mostly of re-radiated dust emission from the extincted star-forming regions.
Hence, we intend to use a combination of H$\alpha$ and 24 $\mu$m observations
to explore how the type and phase of interaction experienced relates to the
observed spatial variations of star formation within galaxies and the global
variations between galaxies.
We present the preliminary study of three Virgo
galaxies known to be three of the best examples of cluster galaxies
experiencing ram pressure; NGC 4522, NGC 4402 and NGC 4501.
Our preliminary study of these three galaxies
is discussed in Section 2. Section 3 provides a summary.
\section{Preliminary study on NGC 4522, NGC 4402 and NGC 4501}
NGC 4522, NGC 4402 and NGC 4501 are three of the best examples of ram
pressure stripping in action \citep{kenney04,crowl05,vollmer08,vollmer07}.
The \mbox{\sc H{i}}\ deficiencies of these
three galaxies indicate that these galaxies have approximately three to four times
less \mbox{\sc H{i}}\ as one would expect from a `normal' galaxy. On the other hand, all three
galaxies have very different gas truncation radii. NGC 4522 has a smaller gas truncation
radius \citep[0.35 $R_{25}$; ][]{kenney04} compared to that of NGC 4402 \citep[0.6
$R_{25}$; ][]{crowl05} because a significant amount of \mbox{\sc H{i}}\ in NGC 4522 is
extraplanar. And since NGC 4501 is still in a pre-peak phase of ram pressure
stripping \citep{vollmer08}, its \mbox{\sc H{i}}\ disk is truncated within the stellar disk
($\sim R_{25}$).
The effect of the ICM--galaxy interaction is best illustrated by comparing
the optical stellar morphology to the \mbox{\sc H{i}}\ or radio continuum morphology
of each galaxy. As shown in Figure~\ref{oldimages}, the \mbox{\sc H{i}}\ morphology of
NGC 4522 is distorted and is shaped roughly like a bow-shock compared to the
undisturbed stellar disk observed in the optical $R$-band \citep{kenney04}.
The \mbox{\sc H{i}}\ morphology clearly shows the neutral gas being stripped out of the disk as the galaxy
travels through the ICM. Similarly, the radio continuum morphologies of
NGC 4402 and NGC 4501 show compression on the sides of the leading edges
as well as extended tails on the trailing sides \citep{crowl05,vollmer07,vollmer08}.
Evidence of compression and shear are also shown by the total polarization
vectors (marked by the white vectors in Figure~\ref{oldimages}).
\begin{figure*}
\begin{center}
\includegraphics[scale=0.21]{wongo_fig1_v1.ps}
\end{center}
\caption{\footnotesize{Columns (a), (b), (c) correspond to NGC 4522,
NGC 4402 and NGC 4501, respectively. Each columns consists of 5 panels;
which show (from top to bottom) the grayscale $R$-band image overlaid
with \mbox{\sc H{i}}\ or radio continuum contours as well as polarization vectors,
the H$\alpha$ image, the 24 $\mu$m image, the H$\alpha$$+24$ $\mu$m
star formation map and the H$\alpha$$/24$ $\mu$m ratio map.}}
\label{oldimages}
\end{figure*}
Figure~\ref{oldimages} presents the H$\alpha$ and 24 $\mu$m observations
for NGC 4522 (column a), NGC 4402 (column b) and NGC 4501 (column c). Each
column consists of five panels which (from top to bottom) show (i) the grayscale
$R$-band image overlaid with the \mbox{\sc H{i}}\ or radio continuum contours as well as
polarization vectors, (ii) the H$\alpha$ map, (iii) the $24$ $\mu$m map,
(iv) the H$\alpha$$/24$ $\mu$m ratio map, and (v) the total star formation
rate map determined from the observed H$\alpha$$+24$ $\mu$m emission.
Section 2.1 describes the star-forming regions and the total star formation rate
observed in the three galaxies and the ratios of the H$\alpha$ to 24 $\mu$m
emission are discussed in Section 2.2.
\subsection{Star formation in NGC 4522, NGC 4402 and NGC 4501}
We aim to identify regions of ram pressure-induced star formation which
might plausibly be occuring since all three galaxies show evidence of active pressure.
Using the new 24 $\mu$m data from the Spitzer Survey of Virgo (SPITSOV; see
Kenney et al.\ in these proceedings) in combination with previous H$\alpha$
observations, we examine the total star formation rate within these galaxies.
The total star formation rate can be derived using a linear combination of the
observed H$\alpha$ and 24 $\mu$m emission \citep{calzetti07}:
\begin{equation}
SFR (\mathrm{M}_{\odot} \; yr^{-1}) = 5.3 \times 10^{-42}\, [L(\mathrm{H}\alpha)+ 0.031\,L(24 \mu\mathrm{m})]
\end{equation}
\begin{table}
\footnotesize{
\caption{\footnotesize{Total star formation rates of
NGC 4402, NGC 4501 and NGC 4522.}}
\label{sfr}
\begin{center}
\begin{tabular}{lccc}
\tableline \tableline
Galaxy & Inclination &\multicolumn{2}{c}{SFR} \\
& &H$\alpha$ & H$\alpha$+24 $\mu$m\\
\tableline
NGC 4402 & 80$^{\circ}$ & 0.28 M$_{\odot}$ & 0.60 M$_{\odot}$ \\
NGC 4501 & 62$^{\circ}$ & 1.67 M$_{\odot}$ & 2.54 M$_{\odot}$ \\
NGC 4522 & 79$^{\circ}$ & 0.10 M$_{\odot}$ & 0.16 M$_{\odot}$ \\
\tableline \tableline
\end{tabular}
\end{center}}
\end{table}
Estimates of the global star formation rates from the combination of
H$\alpha$ and 24 $\mu$m emission were found to be 1.5, 1.6 and 2.1 times that of the
star formation rates derived from the H$\alpha$ emission alone \citep[using the
calibration from ][]{kennicutt98} for NGC 4501, NGC 4522 and NGC 4402,
respectively (see Table~\ref{sfr}). The greatest difference is observed in
the most inclined galaxy ($i=80^{\circ}$), NGC 4402. Therefore, the difference
between the total star formation rates derived using the two methods
appears to be correlated with the observed inclination of the galaxy. This suggests
that more inclined galaxies with more dust will have a greater 24 $\mu$m
contribution to the total star formation rate.
From the total star formation rate maps of all three galaxies (in Figure~\ref{oldimages}),
the regions of most intense star formation are found in the galaxy centers and inner
disk regions. We do not find any strongly-enhanced star-forming regions along the
leading edges of interaction. However, extraplanar star formation is clearly observed
in NGC 4522 where stars have been formed from the stripped gas and are entering the
galaxy halo or intracluster space. By comparing the H$\alpha$ and 24 $\mu$m fluxes
in the disk and extraplanar regions of NGC 4522, we find that $\sim$18\% of the total
24 $\mu$m flux and $\sim$11\% of the total H$\alpha$ emission originated from the
extraplanar regions of NGC 4522. This implies that the extraplanar star-forming regions
are more obscured, on average, than the disk regions. It is possible that the dust is
being pushed between the observer and the star-forming regions as NGC 4522 travels away
from the observer through the ICM.
\subsection{Spatial distribution of H$\alpha$ to 24 $\mu$m emission ratios}
For a galaxy moving towards the observer, one might expect ram pressure
to sweep the dust behind the observed star-forming regions causing a lower
H$\alpha$ attenuation. Conversely, a galaxy moving
away from the observer may display a higher H$\alpha$ attenuation as more
stripped dust is pushed between the observer and the star-forming regions.
Using the H$\alpha$/24 $\mu$m ratios, we investigate the spatial distribution
of H$\alpha$ attenuation in our sample of cluster galaxies where ram pressure
is active. It should be noted that both NGC 4501 and NGC 4522 are moving
away from the observer, while NGC 4402 is moving towards the observer.
As expected, the most obscured regions of star formation are found in the
galaxy centers (see the H$\alpha$/24 $\mu$m ratio maps in Figure~\ref{oldimages}),
whereas the least obscured star-forming regions are found in the outer edges of all
three galaxies. In particular, regions of high H$\alpha$/24 $\mu$m ratios are observed
(from Figure~\ref{oldimages}) near the leading edge of interaction of NGC 4522.
The global H$\alpha$$/24$ $\mu$m ratios of our three galaxies are smaller
than the median H$\alpha$$/24$ $\mu$m ratio (0.034) of the SINGS sample used
by \citet{calzetti07}. We find H$\alpha$$/24$ $\mu$m ratios of 0.029, 0.023 and
0.008 for NGC 4501, NGC 4522 and NGC 4402, respectively. It should be noted that
the SINGS sample used by \citet{calzetti07} consists of galaxies spanning the full
range of inclinations, while our galaxies have inclinations ranging from $62^{\circ}$
to $80^{\circ}$. Our current results suggest that a greater inclination may
be correlated with a smaller global H$\alpha$$/24$ $\mu$m ratio.
\begin{figure*}
\begin{center}
\includegraphics[scale=0.175]{wongo_fig2_v1.eps}
\end{center}
\caption{\footnotesize{ The 24 $\mu$m/H$\alpha$ ratio of 500 pc regions
in NGC 4522. The H$\alpha$ and 24 $\mu$m observations overlaid with
regions of 500 pc diameter apertures are shown in panels (a) and (b),
respectively. The regions are divided into three categories; regions within
the inner disk (green), regions on the leading edge of the ISM--ICM interaction
(red) and outer disk or extraplanar regions (blue). Panel (c) shows the
24 $\mu$m/H$\alpha$ ratio of each 500 pc region as a function of the
projected radius from the center of the galaxy. The regions circled
in blue, red and green correspond to the blue, red and green dots in the
right panel respectively.}}
\label{n4522}
\end{figure*}
\begin{figure*}
\begin{center}
\includegraphics[scale=0.575]{wongo_fig3_v1.ps}
\end{center}
\caption{\footnotesize{ The 24 $\mu$m/H$\alpha$ ratio of 500 pc regions
in NGC 4402. See figure~\ref{n4522} for more details.}}
\label{n4402}
\end{figure*}
\begin{figure*}
\begin{center}
\includegraphics[scale=0.208]{wongo_fig4_v1.eps}
\end{center}
\caption{\footnotesize{The 24 $\mu$m/H$\alpha$ ratio of 500 pc regions
in NGC 4501. See figure~\ref{n4522} for more details.}}
\label{n4501}
\end{figure*}
Recently, \citet{prescott07} found decreasing radial trends in H$\alpha$
attenuation by dust for a majority of the SINGS galaxies. Similar to \citet{prescott07},
we selected regions with circular apertures of 500 pc and measured the ratios of
$\nu L_{\nu}(24)/L(H\alpha)$ as a function of the projected radius from the centers
of our galaxies. However unlike \citet{prescott07}, our regions were selected
from the H$\alpha$ observations as well as the 24 $\mu$m observations in order
to probe the outer star-forming regions more effectively and avoid biasing against
very unobscured regions. Although an aperture size of 500 pc is not optimal for
separating individual star-forming regions, it is comparable to the spatial
resolution of the 24 $\mu$m observations.
Figures~\ref{n4522},~\ref{n4402} and~\ref{n4501} show the
regions selected in NGC 4522, NGC 4402 and NGC 4501,
respectively. The first two panels of each figure (from the left) show
the H$\alpha$ and the 24 $\mu$m map of the galaxy with the selected
regions overlaid. The final panel of each figure (c) plots
$\nu L_{\nu}(24)/L(H\alpha)$ as a function of projected radius from the
galaxy center. The selected regions are loosely divided
into three categories; regions on the side of the leading edge (red),
extraplanar or outer disk regions (blue) and regions in the inner
disk (green). These categories are color-coded for easier
identification and the color of the apertures shown on the maps correspond
directly to the color of the dots on the plot in panel (c).
The average 24 $\mu$m/H$\alpha$ ratios measured from the inner disk regions (green
regions) of the three galaxies are greater than those measured from the
outer disk or extraplanar regions (blue regions). The average ratios for the
inner disk regions of all three galaxies are very similar (with values between 2.52
and 2.64) but the difference in the 24 $\mu$m/H$\alpha$ ratio between the inner
and outer regions varies significantly for each galaxy. The minimum difference
in average ratios between the inner and outer disk is $\sim1.6\%$ for NGC 4501 and
the maximum difference between the outer and inner disk is $\sim22.9\%$ for NGC 4402.
Since NGC 4402 is highly inclined, the
24 $\mu$m observations will be biased by the dusty disk of the galaxy. However,
the motion of NGC 4402 towards the observer is able to push the dust from
the outer disk (or extraplanar) regions behind star-forming regions as these
regions are not as affected by the dusty center. This may explain the fact that
a decrease in the 24 $\mu$m/H$\alpha$ ratio is observed only at the two extreme ends
and the extraplanar \mbox{\sc H{ii}}\ region of NGC 4402.
Of the three galaxies analysed, NGC 4402 exhibits the most convincing radial
decrease in the 24 $\mu$m/H$\alpha$ ratio (see Figure~\ref{n4402}). Its
24 $\mu$m/H$\alpha$ ratio is highest in the central 0.5 kpc and then decreases
very gradually to 4 kpc. The two outer disk regions on both sides of the disk
(as represented by the two blue points at radius $\sim$4--5 kpc in panel (c) of
Figure~\ref{n4402}) exhibit a clear decrease in the 24 $\mu$m/H$\alpha$ ratios.
In addition, the region at the largest projected radius is an \mbox{\sc H{ii}}\ region,
{\em{122603+130724}}, which appears to be the {\em{most unobscured}} star-forming
region found in the current analysis. This extraplanar \mbox{\sc H{ii}}\ region is thought to
have formed from enriched material stripped from NGC 4402 \citep{cortese04}.
It should be noted that these three outermost regions in NGC 4402 have fairly low
H$\alpha$ luminosities (i.e.\ they are not high luminosity \mbox{\sc H{ii}}\ regions with low
obscurity).
Even though bright star-forming regions have been observed near the leading
edge in the optical images of NGC 4402 \citep{crowl05}, the 24 $\mu$m/H$\alpha$
ratios in this part of the galaxy do not appear unusual. The
24 $\mu$m/H$\alpha$ ratios of NGC 4402 are comparable to the the ratios
obtained by \citet{prescott07} for the highly-inclined SINGS sample. The lowest
log $[\nu L_{\nu}(24)/L(H\alpha)]$ ratio found by \citet{prescott07} for the
highly-inclined SINGS sample is $\sim$0.6, while the most unobscured
extraplanar \mbox{\sc H{ii}}\ region in NGC 4402 is $\sim$0.8. On scales of 500 pc, it is
not possible to identify leading edge regions in this galaxy due to resolution constraints.
Unlike NGC 4402, we do not find significant radial decreases in the H$\alpha$
attenuation for NGC 4501 and NGC 4522. A very slight radial decrease in the
H$\alpha$ attenuation is observed in NGC 4522 (Figure~\ref{n4522}) but there
remains a large scatter in the ratios measured from the outer disk (or extraplanar)
regions. Unlike Figure~\ref{oldimages}, we do not find any evidence for
highly-unobscured star formation along the leading edge of NGC 4522 in
Figure~\ref{n4522}. This may be due to the fact that the regions of low H$\alpha$
attenuation in NGC 4522 are located in small regions with low luminosities.
Consequently, the effect of decreased H$\alpha$ attenuation from these regions
is diminished when the ratios are averaged across 500 pc apertures. In addition,
the obscurity of the extraplanar regions of this galaxy is enhanced due to its
direction of motion away from the observer through the ICM.
Similar to NGC 4522, only a slight decrease is found in NGC 4501 (Figure~\ref{n4501}).
There are two possible explanations which may account for the lack of an observed
decrease in the H$\alpha$ attenuation at the leading edge. Firstly, NGC 4501 is
travelling through the ICM away from the observer which will result in a displacement
of the gas and dust from the regions close to the leading edge towards the observer.
Secondly, \citet{vollmer08} concluded that NGC 4501 is in a pre-peak
stripping stage and that ram pressure will only reach its maximum in $\sim100$ Myr.
Hence, it is possible that the current ram pressure is not strong enough
to remove enough dust at the leading edge even though there is enough
pressure to strip off the diffuse gas.
\section{Summary}
Using the 24 $\mu$m observations obtained from the Spitzer Survey of Virgo (SPITSOV)
in combination with H$\alpha$ observations, we explored the
distribution of star-forming regions within three Virgo Cluster galaxies (NGC
4402, NGC 4501 and NGC 4522) which are experiencing active ram pressure.
The combination of the 24 $\mu$m and H$\alpha$ observations allowed us
to uncover regions of significant dust obscuration. From this we are able to
produce total star formation maps without the usual bias caused by dust extinction.
From our preliminary analysis of NGC 4522, NGC 4402 and NGC 4501, we
find evidence neither for highly obscured star forming regions nor for
strongly enhanced star formation along the leading edges of interaction. The
H$\alpha$ attenuation (ratio between H$\alpha$ and 24 $\mu$m emission) of these
three galaxies are comparable to the non-cluster SINGS galaxies studied by
\citet{calzetti07} and \citet{prescott07}. However, NGC 4402 exhibits
unobscured outer disk and extraplanar star-forming regions, while, the outer
disk and extraplanar star-forming regions of NGC 4522 appears moderately
obscured. We attribute the differences between these two galaxies to the direction
of motion of each galaxy through the ICM.
\acknowledgments
This work is based on observations made with the Spitzer Space Telescope, which is
operated by the Jet Propulsion Laboratory, California Institute of Technology
under a contract with NASA. Support for this work was provided by NASA through
an award issued by JPL/Caltech.
|
2,877,628,089,040 | arxiv | \section{Introduction}
The importance of the centre-of-mass and linear-momentum integrals in the classical
$N$-body problem and its generalization to quasihomogeneous potentials, \cite{Diacu0}, is never enough emphasized. From integrable systems, to the Smale and Saari conjectures, \cite{Saari}, \cite{Smale}, to the study of singularities, \cite{Diacu1}, \cite{Diacu5}, these integrals have played an essential role towards answering the most basic questions of particle dynamics, in general, and celestial mechanics, in particular.
But they seem to characterize only the Euclidean space. Away from zero curvature, they disappear, probably because of the symmetry loss that occurs in the ambient space. We will show here that they don't exist in the curved $N$-body problem (the natural extension of the Newtonian $N$-body problem to spaces of constant curvature).
This note follows from a discussion we had with Ernesto P\'erez-Chavela and Guadalupe Reyes Victoria of Mexico City. Given the absence of the above mentioned integrals in discretizations of Einstein's field equations, such as those of Levi-Civita, \cite{Civ}, \cite{Civita}, Einstein, Infeld, Hoffmann, \cite{Ein}, and Fock, \cite{Fock}, we had taken this property for granted in the curved $N$-body problem. But as we could not immediately support this claim with a counterexample, our colleagues asked if there might always exist centre-of-mass-like points in the 3-sphere, $\mathbb S^3$, and the hyperbolic 3-sphere, $\mathbb H^3$, i.e.\ space points that are either at rest or move uniformly along geodesics during the motion of the particles, as in the classical case.
All the examples that came to mind had centre-of-mass-like points (see Section 3), so the property we had taken for granted wasn't obvious. Therefore we began to seek orbits that failed to exhibit this feature. But before presenting the outcome of our attempts to answer this question, we need to lay some background.
The curved $N$-body problem has its roots in the work of J\'anos Bolyai, \cite{Bol},
and Nikolai Lobachevsky, \cite{Lob}. These co-founders of non-Euclidean geometry proposed, independently of each other, a 2-body problem in hyperbolic space for which the attraction is proportional to the area of the sphere of radius equal to the hyperbolic distance between the bodies. In 1870, Ernst Schering showed that a potential involving the hyperbolic cotangent of the distance agrees with this law, \cite{Sche}. Wilhelm Killing extended this idea to the sphere $\mathbb S^3$ by taking the circular tangent of the spherical distance, \cite{Kil1}, \cite{Kil10}. The topic began to attract more attention after Heinrich Liebmann proved two properties that agreed with the classical Kepler problem: the potential is a harmonic function in 3-space and every bounded orbit is closed, \cite{Ber}, \cite{Lie1}, \cite{Lie2}, \cite{Lie3}. Then Ernest Schr\"odinger and his followers exploited these ideas at the quantum level, \cite{Schr}, \cite{Inf}, \cite{InfS}. Starting with the 1990s, the Russian school of celestial mechanics further developed the 2-body case, \cite{Koz}, \cite{Koz2}, \cite{Shc}, \cite{Shc1}, \cite{Shc2}.
In some recent work, we generalized the Schering-Killing potential to any number $N\ge 2$ of bodies, \cite{Diacu1}, \cite{Diacu3}, \cite{Diacu-mono}, \cite{Diacu-memoirs}, \cite{Diacu2}, \cite{Diacu4}, \cite{Diacu5}, \cite{Diacu6}, and used the results obtained there to argue that, for distances comparable to those of our solar system, the geometry of the physical space is Euclidean. Our method exploited the connection between geometry and dynamics, thus going beyond Gauss's attempt to determine the nature of the physical space through topographic measurements. For more details on this subject and an extensive bibliography, see \cite{Diacu-mono}, \cite{Diacu-memoirs}.
In this note we prove that, for a certain class of solutions of the curved $N$-body
problem in $\mathbb S^3$, there are no centre-of-mass-like points. In other words,
we show that forces cannot keep any such point at rest or make it move uniformly along a great circle of $\mathbb S^3$, results that occur in Sections \ref{counterexamples} and \ref{class}. For positive curvature, our choice of $\mathbb S^3$ is not restrictive since a suitable coordinate and time-rescaling transformation makes the curvature parameter vanish from the equations of motion, \cite{Diacu-mono}, \cite{Diacu7}. In Section \ref{motion} we introduce these differential equations and the known integrals of motion, in Section \ref{examples} we give examples of orbits with centre-of-mass-like points, and in Section \ref{compl} we set the geometric background, necessary for understanding the orbits we describe in Sections \ref{counterexamples} and \ref{class}.
Unfortunately, a similar example in $\mathbb H^3$ still eludes us, and we don't know the size of the set of orbits that have centre-of-mass-like points in $\mathbb S^3$ or $\mathbb H^3$, but expect it to be negligible from both the measure theoretical and the topological point of view. So apart from the result presented in this note, which proves the non-existence of these integrals for positive curvature, the issue my colleagues raised leads to several interesting questions, all of which still await answers.
\section{Equations of motion}\label{motion}
Consider $N$ point particles (bodies) of masses $m_1,\dots, m_N>0$ moving in
$\mathbb{S}^3$ (thought as embedded in the Euclidean space $\mathbb R^4$) or $\mathbb H^3$ (thought as embedded in the Minkowski space $\mathbb R^{3,1}$),
where
$$
{\mathbb S}^3=\{(w,x,y,z)\ | \ w^2+x^2+y^2+z^2=1\},
$$
$$
{\mathbb H}^3=\{(w,x,y,z)\ | \ w^2+x^2+y^2-z^2=-1, \ z>0\}.
$$
Then the configuration of the system is described by the $4N$-dimensional vector
$$
{\bf q}=({\bf q}_1,\dots,{\bf q}_N),
$$
where ${\bf q}_i=(w_i,x_i,y_i,z_i), i=\overline{1,N}$, denote the position vectors of the bodies. The equations of motion are given by the second-order system
\begin{equation}
\label{both}
\ddot{\bf q}_i=\sum_{j=1,j\ne i}^N\frac{m_j[{\bf q}_j-\sigma({\bf q}_i\odot {\bf q}_j){\bf q}_i]}{[\sigma-\sigma({\bf q}_i\odot {\bf q}_j)^2]^{3/2}}-\sigma(\dot{\bf q}_i\odot \dot{\bf q}_i){\bf q}_i, \ \ i=\overline{1,N},
\end{equation}
with constraints
\begin{equation}
{\bf q}_i\odot{\bf q_i}=\sigma, \ \ {\bf q}_i\odot\dot{\bf q}_i=0, \ \ i=\overline{1,N},
\end{equation}
where $\odot$ is the standard inner product $\cdot$ of signature $(+,+,+,+)$ in $\mathbb S^3$ or the Lorentz inner product $\boxdot$ of signature $(+,+,+,-)$ in $\mathbb H^3$, and
$$
\sigma=
\begin{cases}
+1,\ \ {\rm in}\ \ {\mathbb S^3},\cr
-1,\ \ {\rm in}\ \ {\mathbb H^3},
\end{cases}
$$
denotes the signum function, \cite{Diacu-mono}, \cite{Diacu-memoirs}. System \eqref{both} has dimension $6N$. The force acting on each body has a gravitational component (the above sum) and a term (involving the velocities) that corresponds to the constraints.
As a consequence of Noether's theorem, system \eqref{both} has the scalar integral of energy,
$$
T({\bf q},\dot{\bf q})-U({\bf q})=h,
$$
where
$$U({\bf q})=\sum_{1\le i<j\le N}\frac{\sigma m_im_j{\bf q}_i\odot{\bf q}_j}{[\sigma-\sigma({\bf q}_i\odot{q}_j)^2]^{3/2}}$$
is the force function ($-U$ representing the potential),
$$
T({\bf q},\dot{\bf q})=\frac{1}{2}\sum_{i=1}^Nm_i(\dot{\bf q}_i\odot\dot{\bf q}_i)
(\sigma{\bf q}_i\odot{\bf q}_i)
$$
is the kinetic energy, with $h$ representing an integration constant; and the 6-dimensional integral of the total angular momentum,
$$
\sum_{i=1}^Nm_i{\bf q}_i\wedge\dot{\bf q}_i={\bf c},
$$
where $\wedge$ is the wedge product and ${\bf c}=(c_{wx},c_{wy},c_{wz},c_{xy},c_{xz},c_{yz})$ denotes an integration vector, each component measuring the rotation of the system about the origin of the frame relative to the plane corresponding to the bottom indices.
But it is far from clear whether the centre-of-mass and linear-momentum integrals exist for system \eqref{both}. In Euclidean space, these
integrals are obtained by showing that $\sum_{i=1}^Nm_i\ddot{\bf q}_i={\bf 0}$. This identity is not satisfied in the curved $N$-body problem because $\frac{1}{M}\sum_{i=1}^Nm_i{\bf q}_i$,
where $M=m_1+\dots+m_N$, does not even belong to $\mathbb S^3$ or $\mathbb H^3$. However, it may be possible that there are points in $\mathbb S^3$ or $\mathbb H^3$ that behave like a centre of mass, i.e.\ the forces acting on them cancel each other within the manifold or make those points move uniformly along a geodesic. The next section will provide a couple of examples in which such centre-of-mass-like points exist.
\section{Orbits with centre-of-mass-like points} \label{examples}
The orbits with centre-of-mass-like points we will further present are the Lagrangian
elliptic relative equilibria of equal masses in $\mathbb S^3$ and the Eulerian
hyperbolic relative equilibria of equal masses in $\mathbb H^3$. More examples
having this property can be found in \cite{Diacu-mono}, \cite{Diacu-memoirs}.
\subsection{Lagrangian elliptic relative equilibria in $\mathbb S^3$}
A straightforward computation shows that for $N=3$, $\sigma=1$, and $m_1=m_2=m_3=:m$, system \eqref{both} has orbits of the form
\begin{equation}
{\bf q}=({\bf q}_1,{\bf q}_2,{\bf q}_3), \ {\bf q}_i=(w_i,x_i,y_i,z_i),\ i=1,2,3,
\label{0elliptic1}
\end{equation}
\begin{align*}
w_1(t)&=r\cos\omega t,& x_1(t)&=r\sin\omega t,\\
y_1(t)&=y\ ({\rm constant}),& z_1(t)&=z\ ({\rm constant}),\\
w_2(t)&=r\cos(\omega t+2\pi/3),& x_2(t)&=r\sin(\omega t+2\pi/3),\\
y_2(t)&=y\ ({\rm constant}),& z_2(t)&=z\ ({\rm constant}),\\
w_3(t)&=r\cos(\omega t+4\pi/3),& x_3(t)&=r\sin(\omega t+4\pi/3),\\
y_3(t)&=y\ ({\rm constant}),& z_3(t)&=z\ ({\rm constant}),
\end{align*}
with $r^2+y^2+z^2=1, r\in(0,1),$ and
$$
\omega^2=\frac{8m}{\sqrt{3}r^3(4-3 r^2)^{3/2}}.
$$
These are Lagrangian elliptic relative equilibria, i.e.\ orbits for which the bodies lie at the vertices of a rotating equilateral triangle, such that the mutual distances between bodies remain constant during the motion. The forces acting on points of the form ${\bf q}_0=(0,0,y_0,z_0)\in\mathbb S^3$, i.e.\ with $y_0^2+z_0^2=1$, cancel each other within $\mathbb S^3$. Moreover, the spherical distance, $d_{\mathbb S^3}$, from ${\bf q}_0$ to $m_1, m_2$, and $m_3$ is the same:
$$
d_{\mathbb S^3}({\bf q}_0,{\bf q}_i)=\cos^{-1}({\bf q}_0\cdot{\bf q}_i)=\cos^{-1}(y_0y+z_0z), \ i=1,2,3.
$$
Therefore all these points (infinitely many) are centre-of-mass-like. The condition that the masses are equal is sharp: such equilateral orbits don't exist if the masses are distinct, but it can be shown that orbits of nonequal masses exist for scalene triangles, \cite{Diacu3}.
If we restrict the motion to the great 2-sphere
$$
{\bf S}_w^2=\{(0,x,y,z)\ | \ x^2+y^2+z^2=1\},
$$
then there are only two centre-of-mass-like points, the north pole $(0,1,0,0)$
and the south pole $(0,-1,0,0)$ of ${\bf S}_w^2$. Nevertheless, the situation is still different from the Euclidean case, where the centre of mass is unique.
\subsection{Eulerian hyperbolic relative equilibria in $\mathbb H^3$}
A straightforward computation shows that for $N=3$, $\sigma=-1$, and $m_1=m_2=m_3=:m$, system \eqref{both} has orbits of the form
\begin{equation}
{\bf q}=({\bf q}_1, {\bf q}_2, {\bf q}_3), \ {\bf q}_i=(w_i,x_i,y_i,z_i), \ i=1,2,3,
\label{hyp-4}
\end{equation}
\begin{align*}
w_1&=0,& x_1&=0,& y_1&=\sinh\beta t,& z_1&=\cosh\beta t,\\
w_2&=0,& x_2&=x\ {\rm (constant)},& y_2&=\eta\sinh\beta t,& z_2&=\eta\cosh\beta t,\\
w_3&=0,& x_3&=-x\ {\rm (constant)},& y_3&=\eta\sinh\beta t,& z_3&=\eta\cosh\beta t,
\end{align*}
with $x^2-\eta^2=-1,\ \eta>1$, and
$$
\beta^2=\frac{1+4\eta^2}{4\eta^3(\eta^2-1)^{3/2}}.
$$
These are Eulerian hyperbolic relative equilibria, orbits lying on a geodesic of $\mathbb H^3$ that rotates hyperbolically, which means that the mutual distances between bodies remain constant during the motion.
Points of the form ${\bf q}_*=(w_*,0,\rho_*\sinh\beta t,\rho_*\cosh\beta t)$, with $w_*^2-\rho_*^2=-1$, move uniformly along the geodesic $w=w_*, x=0$ in $\mathbb H^3$. The hyperbolic distance, $d_{\mathbb H^3}$, between ${\bf q}_*$ and $m_1$ is constant along the motion because
$$
d_{\mathbb H^3}({\bf q}_*,{\bf q}_1)=\cosh^{-1}(-{\bf q}_*\boxdot{\bf q}_1)=\cosh^{-1}\rho_*.
$$
The same holds for the distance between ${\bf q}_*$ and $m_2$ or $m_3$,
$$
d_{\mathbb H^3}({\bf q}_*,{\bf q}_i)=\cosh^{-1}(-{\bf q}_*\boxdot{\bf q}_i)=\cosh^{-1}\eta\rho_*, \ i=2,3.
$$
Consequently ${\bf q}_*$ is centre-of-mass-like. For $w_*=0$, which yields $\rho_*=1$, the point ${\bf q}_*$ overlaps with $m_1$. This fact shows that when we restrict the motion to the great hyperbolic 2-sphere,
$$
{\bf H}_w^2=\{(0,x,y,z)\ | \ x^2+y^2-z^2=-1\},
$$
there is only one centre-of-mass-like point, which is identical to $m_1$.
\section{Complementary circles in $\mathbb S^3$}\label{compl}
In order to produce a class of orbits with no centre-of-mass-like points, we will introduce the following geometric concept. Two great circles $C_1$ and $C_2$ of two distinct great 2-spheres of $\mathbb S^3$ are called complementary if there is a coordinate system in which they can be represented as
$$
C_1=\{(w,x,y,z)\ | \ w=x=0\ {\rm and} \ y^2+z^2=1\},
$$
$$
C_2=\{(w,x,y,z)\ | \ y=z=0\ {\rm and} \ w^2+x^2=1\}.
$$
In topological terms, $C_1$ and $C_2$ form a Hopf link in a Hopf fibration, which is a map
$$
{\mathcal H}\colon{\mathbb S}^3\to{\mathbb S}^2, \ h(w,x,y,z)=(w^2+x^2-y^2-z^2,2(wz+xy),2(xz-wy))
$$
that takes circles of $\mathbb S^3$ to points of $\mathbb S^2$,
\cite{Hopf}, \cite{Lyons}. In particular, ${\mathcal H}(C_1)=(1,0,0)$ and ${\mathcal H}(C_2)=(-1,0,0)$. Using the stereographic projection, it can be shown that the circles $C_1$ and $C_2$ are linked (like the adjacent rings of a chain), hence the name of the pair, \cite{Lyons}.
The spherical distance between two complementary circles is constant. Indeed, if ${\bf a}=(0,0,y,z)\in C_1$ and ${\bf b}=(w,x,0,0)\in C_2$, then ${\bf a}\cdot{\bf b}=0$, so
$d_{\mathbb S^3}({\bf a},{\bf b})=\pi/2.$
This unexpected fact turns out to be even more surprising from the dynamical point of view: the magnitude of the gravitational force (but not its direction) between a body lying on a great circle and a body lying on the complementary great circle is the same, no matter where the bodies are on their respective circles.
\section{Non-existence of linear-momentum and centre-of-mass integrals}\label{counterexamples}
We can now construct an example in the 6-body problem in $\mathbb S^3$ in which three bodies of equal masses move along a great circle at the vertices of an equilateral triangle, while the other 3 bodies of masses equal to those of the previous bodies move along a complementary circle of another great sphere, also at the vertices of an equilateral triangle. So take the masses
$m_1=m_2=m_3=m_4=m_5=m_6=:m>0$ and the frequencies $\alpha, \beta\ne 0$, which, in general, are distinct, $\alpha\ne\beta$.
Then a candidate for a solution as described above has the form
\begin{equation}
{\bf q}=({\bf q}_1,{\bf q}_2, {\bf q}_3, {\bf q}_4, {\bf q}_5, {\bf q}_6),\ {\bf q}_i=(w_i,x_i,y_i,z_i),\ i=\overline{1,6},
\label{sol}
\end{equation}
\begin{align*}
w_1&=\cos\alpha t, & x_1&=\sin\alpha t,\\
y_1&=0, & z_1&=0,\displaybreak[0]\\\
w_2&=\cos(\alpha t+2\pi/3), & x_2&=\sin(\alpha t +2\pi/3),\\ y_2&=0, & z_2&=0,\displaybreak[0]\\
w_3&=\cos(\alpha t+4\pi/3), & x_3&=\sin(\alpha t +4\pi/3),\\ y_3&=0, & z_3&=0,\displaybreak[0]\\
w_4&=0, & x_4&=0, \\ y_4&=\cos\beta t, & z_4&=\sin\beta t,\displaybreak[0]\\
w_5&=0, & x_5&=0,\\ y_5&=\cos(\beta t+2\pi/3), & z_5&=\sin(\beta t+2\pi/3),\displaybreak[0]\\
w_6&=0, & x_6&=0,\\ y_6&=\cos(\beta t+4\pi/3), & z_6&=\sin(\beta t+4\pi/3).
\end{align*}
For $t=0$, we obtain a fixed-point configuration, ${\bf q}(0)$, specific to ${\mathbb S}^3$, in the sense that there is no 2-sphere that contains it:
\begin{align*}
w_1&=0,& x_1&=1,& y_1&=0,& z_1&=0,\displaybreak[0]\\\
w_2&=-\frac{1}{2},& x_2&=\frac{\sqrt{3}}{2},& y_2&=0,& z_2&=0,\displaybreak[0]\\
w_3&=-\frac{1}{2},& x_3&=-\frac{\sqrt{3}}{2},& y_3&=0,& z_3&=0,\displaybreak[0]\\
w_4&=0,& x_4&=0,& y_4&=1,& z_4&=0,\displaybreak[0]\\
w_5&=0,& x_5&=0,& y_5&=-\frac{1}{2},& z_5&=\frac{\sqrt{3}}{2},\displaybreak[0]\\
w_6&=0,& x_6&=0,& y_6&=-\frac{1}{2},& z_6&=-\frac{\sqrt{3}}{2}.
\end{align*}
A long but straightforward computation shows that \eqref{sol} is solution
of system \eqref{both} for $N=6$, $\sigma=1$, and any $\alpha,\beta\ne 0$. If $\alpha$ and $\beta$ are rational multiples of $2\pi$, a set of frequency pairs that has measure zero in $\mathbb R^2$, the corresponding orbits are periodic. But in general $\alpha/\beta$ is a irrational, so the orbits are quasiperiodic.
The angular momentum constants are
$$
c_{wx}=3m\alpha\ne 0,\ \ c_{yz}=3m\beta\ne 0, \ \
c_{wy}=c_{wz}=c_{xy}=c_{xz}=0,
$$
which means that the rotation of the system takes place around the origin of the coordinate system only relative to the planes $wx$ and $yz$.
We can now state and prove the main result of this note.
\begin{proposition}
For a solution \eqref{sol} of system \eqref{both}, with $N=6,\ \sigma=1$, and $\alpha/\beta$ irrational, there is no point in $\mathbb S^3$ such that the forces acting on it vanish in $\mathbb S^3$\! or make it move uniformly along a geodesic.
\end{proposition}
\begin{proof}
A solution \eqref{sol} of system \eqref{both} can be viewed as obtained from the action of an element of the Lie group $SO(4)$ on the fixed-point configuration ${\bf q}(0)$. Without loss of generality, we can choose a suitable basis to give this element of the group the matrix representation
\begin{equation}\label{matrix}
A=\begin{pmatrix}
\cos\gamma t & -\sin\gamma t & 0 & 0\\
\sin\gamma t & \cos\gamma t & 0 & 0\\
0 & 0 & \cos\delta t & -\sin\delta t\\
0 & 0 & \sin\delta t & \cos\delta t
\end{pmatrix},
\end{equation}
with $\gamma,\delta\ne 0$ compatible with $\alpha,\beta\ne 0$. Then the matrix $A$ rotates all the points of $\mathbb S^3$, including the bodies $m_1,\dots, m_6$, such that the latter form a solution of the equations of motion.
The only point of $\mathbb R^4$ that stays put under the action of $A$, when $\gamma,\delta\ne 0$, is the origin, $\bf 0$, of the coordinate system. But ${\bf 0}\notin\mathbb S^3$, so no point in $\mathbb S^3$ stays fixed.
We still need to prove that no candidate for a centre-of-mass-like point of this 6-body problem can move uniformly along a geodesic of $\mathbb S^3$ under the action of $A$. For this purpose, consider a point ${\mathfrak q}_0=(\mathfrak w_0, \mathfrak x_0, \mathfrak y_0, \mathfrak z_0)$
in $\mathbb S^3$.
Then the components of the vector $A{\mathfrak q}_0^T$, where the upper $^T$\! represents the transposed, are:
\begin{align*}
\mathfrak w(t)&=\mathfrak w_0\cos\gamma t -\mathfrak x_0\sin\gamma t, & \mathfrak x(t)&=\mathfrak w_0\sin\gamma t + \mathfrak x_0\cos\gamma t,\\
\mathfrak y(t)&=\mathfrak y_0\cos\delta t -\mathfrak z_0\sin\delta t, & \mathfrak z(t)&=\mathfrak y_0\sin\delta t + \mathfrak z_0\cos\delta t.
\end{align*}
But since $\gamma$ and $\delta$ are compatible with $\alpha$ and $\beta$,
we have
$$
\mathfrak w(t)=r_0\cos\alpha t,\ \mathfrak x(t)=r_0\sin\alpha t,\ \mathfrak y(t)=\rho_0\cos\beta t, \ \mathfrak z(t)=\rho_0\sin\beta t,
$$
where $r_0=(\mathfrak w_0^2+ \mathfrak x_0^2)^{1/2}$ and $\rho_0=(\mathfrak y_0^2+ \mathfrak z_0^2)^{1/2}$, with
$r_0^2+\rho_0^2=1$.
To determine when the curve ${\mathfrak q}=(\mathfrak{w,x,y,z})$, described by the point $
(\mathfrak w_0, \mathfrak x_0, \mathfrak y_0, \mathfrak z_0)$ under the action of $A$, describes a geodesic, it is necessary that the point moves on a great sphere of $\mathbb S^3$, i.e.\ the function $\mathfrak q$ that draws the trajectory satisfies the conditions
$$
\mathfrak w^2+ \mathfrak x^2+ \mathfrak y^2+ \mathfrak z^2=1\ \ {\rm and}\ \ A\mathfrak w+B \mathfrak x+C \mathfrak y+D\mathfrak z=0,
$$
where not all the real coefficients $A,B,C,D$ are zero. The first condition, which asks that $\mathfrak q$ lies
in $\mathbb S^3$, is obviously satisfied. The second condition, which requires that $\mathfrak q$ lies in a hyperplane passing through the origin of the coordinate system, translates into
$$
Ar_0\cos\alpha t + Br_0\sin\alpha t + C\rho_0\cos\beta t + D\rho_0\sin\beta t =0.
$$
Since $\alpha/\beta$ is not rational, the above equality holds for all $t$ only if
$$
r_0(A\cos\alpha t + B\sin\alpha t)=\rho_0(C\cos\beta t + D\sin\beta t)=0,
$$
which is equivalent to either
(i) $r_0=0$ and $C=D=0$ or
(ii) $\rho_0=0$ and $A=B=0$.
\noindent As $r_0=0$ implies $w_0=x_0=0$ and $\rho_0=1$, and $\rho_0=0$ implies $y_0=z_0=0$ and $r_0=1$, the trajectory $\mathfrak q$ of ${\mathfrak q}_0$ under the action of $A$ can be either in agreement with (i) or with (ii), but not with both, i.e.\ ${\mathfrak q}(t)$ has the form
$$
\begin{pmatrix}
0\\
0\\
\cos\beta t\\
\sin\beta t
\end{pmatrix}
\ \ {\rm or}\ \
\begin{pmatrix}
\cos\alpha t\\
\sin\alpha t\\
0\\
0
\end{pmatrix}.
$$
Therefore the only geodesics lying on great spheres that remain invariant (globally, not point-wise) under the action of $A$, with $\gamma,\delta\ne 0$, are the complementary great circles
$$
C_1=\{(0,0,y,z)\ | \ y^2+z^2=1\}\ \ {\rm and}\ \ C_2=\{(w,x,0,0)\ | \ w^2+x^2=1\}.
$$
Any point moving uniformly on $C_1$ is equidistant from any point of $C_2$, so it could act like a centre of mass for the bodies rotating on $C_2$, but it does not qualify as
a centre of mass for the bodies rotating on $C_1$, so it cannot be a centre of mass for the entire system. The same is true if the roles of $C_1$ and $C_2$ are interchanged. Since no other points rotate on geodesics, these remarks complete the proof.
\end{proof}
\section{A class of counterexamples}\label{class}
The example presented in Section 5 can be extended to a larger class of orbits with no centre-of-mass-like points. Consider first the $(N+M)$-body problem (with $N,M\ge 3$ and odd) of equal masses, in which $N$ bodies rotate along a great circle of a great sphere at the vertices of a regular $N$-gon, while the other $M$ bodies rotate along a complementary great circle of another great sphere at the vertices of a regular $M$-gon. The same as in the 6-body problem discussed above, the rotation takes place around the origin of the coordinate system only relative to two out of six reference planes. The condition that $N$ and $M$ are odd is imposed to avoid antipodal configurations, which introduce singularities, \cite{Diacu1}, \cite{Diacu5}. The existence of the orbits we just described follows from \cite{Diacu3}, \cite{Diacu-mono}, \cite{Diacu-memoirs}. The mass-equality condition can be relaxed, a case in which the polygons are not regular anymore. For suitable polygonal shapes and well chosen masses, we can also eliminate the condition that $M$ and $N$ are odd. The non-existence proof given in Proposition 1 works similarly in the general case. Consequently the above class of orbits covers any finite number of bodies, and shows that, in the curved $N$-body problem in $\mathbb S^3$ there are no centre-of-mass and linear-momentum integrals.
\medskip
\noindent{\bf Acknowledgment.} The research presented in this note was supported in part by a Discovery Grant the author received from NSERC of Canada.
|
2,877,628,089,041 | arxiv | \section{Introduction}
Stars are born out of collapsing molecular clouds, which determine their masses, initial chemical composition, and initial angular momentum (hereafter AM) content. It has become clear in the past few decades that rotation can strongly influence the stellar evolution \citep{ES76,MaederMeynetARAA2000,Maederbook}, which
makes understanding the rotational evolution of stars one of the greatest challenges in stellar physics.\\
Particular interest has been given to the rotational evolution of solar-mass stars. Photometric surveys of open clusters and stellar associations \citep[see e.g.][for a recent review]{BouvierPPVI} provide a wealth of rotation periods from ages of 1 Myr to the age of the Sun. The patterns revealed by the rotational periods indicate that a large dispersion exists among stars in the mass range 0.9-1.1 M$_\odot$ during the pre-main sequence (PMS) that narrows as they evolve, and converges towards a single peaked distribution by the time they reach an age of about 1 Gyr. The main trends of the rotational evolution of solar-type stars through the PMS, zero-age main sequence (ZAMS), and main sequence (MS) can be described by considering three types of initial rotation, as suggested by statistical analyses of the observed samples \citep{IrwinBouvier2009,GB13}: fast, medium, and slow rotation. \\ Several models have been built to account for the observed evolution of surface velocity \citep[e.g.][]{ES81,MB1991,KPBS97,MacGregor2000,Denissenkov2010,Spadaetal2011,Charbonnel2013,GB15,LS15}, which all include three fundamental processes: (1) a star-disc interaction phase, often modelled as a constant surface angular velocity phase,
(2) a coupling mechanism between the radiative interior and the convective envelope that regulates the angular momentum exchange between these two reservoirs during the evolution, and (3) a braking torque expression that is due to the loss of angular momentum extracted by the stellar winds and leads to a decrease of the surface angular velocity mostly during the MS evolution. All these studies have highlighted that the torque and the internal coupling processes are important for reproducing the observations.
In the present paper, we extensively test the different prescriptions available to date for both stellar wind torque and the turbulence modelling within the self-consistent theoretical framework provided by \citet{Zahn1992} and \cite{MaederZahn1998} to treat the internal transport of angular momentum by meridional circulation and shear-induced turbulence in 1.5D \citep[and reference therein]{DecressinMathis2009}. These later internal processes successfully account in a consistent way for the rotational behaviour and the evolution of the surface abundances of light elements for dwarf and subgiant stars with masses above $\sim$ 1.4 M$_\odot$ \citep[the blue side of the so-called lithium dip;][]{CT99,PTCF2003}. However, their efficiency in transporting AM has been shown to be too low in solar-type stars to reproduce the present-day solar internal rotation as inferred from helioseismic data and the Li abundance in solar type stars \citep{TalonCharbonnel1998}; for these objects, additional AM transport processes may be required to better fit the data, such as internal gravity waves or magnetic fields \citep{CharbonnelTalon2005Science,Eggenberger2005}. Despite this assessment concerning lithium and helioseismology, AM extraction and rotation-induced mechanisms have become the basic processes upon which more elaborate models can be built \citep{TalonCharbonnel2003,Lagarde2012,Charbonnel2013}. It is thus important to clarify their exact efficiency and also, if possible, the more appropriate combination of prescriptions that should be used to model solar-mass stars, in a similar way as \citet{Meynet2013} did for the case of massive stars.
Unlike massive stars, AM transport mechanism in solar-like stars are driven by the surface extraction of AM. New magnetic torque prescriptions have recently been developed and have to be accounted for to find a more appropriate combination of prescriptions.
We present the different formalisms that we test in Sect.~\ref{sec:formalism} and recall the overall physics of the models in Sect.~\ref{sec:models}. Next, we describe our reference model in Sect.~\ref{sec:ref} and then explore the parameter space in Sect.~\ref{sec:explore}. In Sect.~\ref{sec:bizone} we compare our results to the recent bi-zone models by \citet{GB13,GB15}. Finally, we briefly compare our models to the additional constraints given by helioseismology and the surface abundances of lithium in Sect.~\ref{sec:other}. A summary of our results and some perspectives are given in Sect.~\ref{sec:end}.
\section{Formalism}\label{sec:formalism}
\subsection{Disc coupling}
A solar-mass star undergoes a global contraction during the first few million years of its evolution. During this period the star magnetically interacts with its circumstellar accretion disc, and this interaction modifies the stellar angular momentum.
The final stellar velocity results from the balance between the increase of AM from the accreted matter and the strong loss that can be due either to accretion-powered stellar winds \citep{MattPudritz2005} or to the so-called disc-locking process\citep{Koenigl1991,GL79}.
While these mechanisms still lack an accepted physical description, it is observationally evident \citep[e.g.][and references therein]{Rebull2004} that this interaction is very effective during the whole disc lifetime and compensates for the increase of angular velocity that is due to the stellar contraction.
In this configuration, the star will, on average, maintain a constant angular velocity as long as it is coupled with the disc. This is confirmed by the analysis of rotation periods in young open clusters and associations \citep{GB13}.
The duration of this period of coupling varies with both the mass \citep{KennedyKenyon2009} and the initial angular momentum of the star.
To evaluate the duration of this strong star-disc interaction phase, it is necessary to rely on observations since there is a degeneracy between the initial angular momentum content of the star and the disc-coupling time. A star with a large initial angular momentum content that will maintain a coupling with its circumstellar disc for a relatively long time is expected to undergo the same rotational evolution for the remaining evolution as a star with a smaller initial angular momentum content coupled to a shorter lived disc. \citet{Edwardsetal1993} showed that the observed rotation periods of stars in very young clusters demand shorter disc lifetimes for fast rotators than for slow rotators. This has been confirmed with more observational evidence by \citet{Rebull2004} and \citet{Bouvier2008}.
In the present study, we assume a 2.5 Myr and 5 Myr star-disc interaction for fast and medium or slow rotators, respectively, based on observations \citep[e.g.][]{Belletal2013}. Throughout this coupling, the surface rotation period is assumed to be constant.
\subsection{Stellar wind torque}\label{sec:torques}
\begin{figure}
\includegraphics[width=0.45\textwidth]{FIGURES/Figure1.eps}
\caption{Stellar wind torque as a function of the surface angular velocity for the different prescriptions as labelled in the plot. The parameter values used in each case are given in Table.~\ref{tab:brake}.}
\label{Fig:torqueomega}
\end{figure}
For the past two decades, many different more or less complex
prescriptions of stellar wind braking have been developed.
Since \cite{Schatzman1962} and \citet{WD67}, all the processes playing a role in the coupling between wind-driven mass-loss and the magnetic field have been extensively studied and improved.
In a general manner, the net torque exerted on the star can be written as
\begin{equation}
\frac{{\rm d}J}{{\rm d}t} = \ensuremath{\dot{M}} \Omega_\star r_A^2
,\end{equation}
where $J$ is the total angular momentum of the star, $\ensuremath{\dot{M}}$ is the mass-loss rate, $\Omega_\star$ the average angular velocity, and $r_A$ the average Alfven radius. The most difficult term to constrain in this equation, which is also the most important one, is the Alfven radius, which depends on several parameters such as the stellar radius, effective temperature, and magnetic field.\\
In the following, we present the different prescriptions we used to evaluate their impact on the evolution of the surface velocity and internal rotation profile of a single 1\ensuremath{M_\odot}~star. All the prescriptions detailed below are semi-empirical, based on theoretical and observational properties. Except for the expression from \cite{RM12} (detailed in part \ref{RM12}), they all lead to a rotational evolution that follows the empirical relationship
of \cite{Skumanich72}, $\Omega_S \propto t^{-1/2}$ , beyond the spin-down phase of the model stars (after at least 1 Gyr of evolution). We calibrated the constants of each braking law so that it leads to the solar rotation rate at the age of the Sun for a 1~\ensuremath{M_\odot}~model with our reference prescription. The constants are summarised in Table~\ref{tab:brake}. Figure~\ref{Fig:torqueomega} shows the torque evolution as a function of the surface angular velocity on the main sequence for each of the prescriptions.
\subsubsection{Kawaler (1988) - Chaboyer et al. (1995a)}\label{K88}
\cite{Kawaler88} derived a braking law that takes into account the magnetic field strength and topology and the mass-loss rate. The expression can easily be incorporated in stellar evolutionary codes accounting for rotation \citep[e.g.][]{Pinsonneaultetal1989}
\begin{equation}
\frac{dJ}{dt} = -K_W\Omega^{1+(4an/3)}\left(\frac{R}{\ensuremath{R_\odot}}\right)^{2-n}\left(\frac{\ensuremath{\dot{M}}}{-10^{-14}\ensuremath{M_\odot}/yr^{-1}}\right)^{1-(2n/3)}\left(\frac{M}{\ensuremath{M_\odot}}\right)^{-n/3}.
\end{equation}
Here, the angular momentum loss is directly related to the rotation rate $\Omega$. It only depends on three parameters: the exponent $a$ that depicts the total magnetic field strength dependence on the rotation rate ($B_0 = K_B\left(R/\ensuremath{R_\odot}\right)^{-2}\Omega^a$ , with $B_0$ the surface magnetic field strength and $K_B$ constant) fixed equal to $1$; the wind-index factor $n$ that varies with the magnetic field topology and is typically assumed to be equal to $3/2$ to reproduce the Skumanich relationship; and the parameter $K_W$. The latter consists, in the initial paper, of two components linked on one hand to the dependence of the magnetic field generation to the convection zone depth and on the other hand to the structure of the stellar wind. It is here assumed to be a single constant that is calibrated to reproduce the solar surface velocity at the age of the Sun.
Taking $n = 3/2$ simplifies the problem by suppressing the dependence
on mass loss rate, leading to
\begin{equation}
\frac{dJ}{dt} = -K_W\Omega^{(1+2a)}\left(\frac{R}{\ensuremath{R_\odot}}\right)^{1/2}\left(\frac{M}{\ensuremath{M_\odot}}\right)^{-1/2},
\label{Eq:K88a}
\end{equation}
where $K_W = 2.10^{48}$ is calibrated in cgs units to reproduce the solar case, and $a = 0$ or $1$ for a saturated or unsaturated magnetic field, respectively.
Following \cite{StaufferHartmann87}, \cite{Chaboyeretal1995a} assumed that the magnetic field reaches saturation above a certain value of the surface angular velocity. As a result, this saturation modifies the braking law and the dependence of the angular momentum loss on the rotation rate.
It is not yet clear whether this saturation is due to the saturation of the dynamo itself or to coronal processes \citep{Wright2011}. However, the magnetic field strength seems to stop increasing even if the star still spins up \citep{Vilhu84,Odelletal95} and $a = 0$ in the expression of $B_0$.
Equation \ref{Eq:K88a} then becomes
\begin{equation}
\frac{dJ}{dt} = -K_W\Omega\Omega_{sat}^2\left(\frac{R}{\ensuremath{R_\odot}}\right)^{1/2}\left(\frac{M}{\ensuremath{M_\odot}}\right)^{-1/2} \qquad {\rm for} \; \Omega \geq \Omega_{sat}
\end{equation}
\begin{equation}
\frac{dJ}{dt} = -K_W\Omega^{3}\left(\frac{R}{\ensuremath{R_\odot}}\right)^{1/2}\left(\frac{M}{\ensuremath{M_\odot}}\right)^{-1/2} \qquad {\rm for} \; \Omega < \Omega_{sat}
,\end{equation}
which is the formulation that we use in our stellar evolution code.
\subsubsection{\cite{Matt2012}}\label{Matt12}
\cite{Matt2012} proposed a braking law based on current 2D magnetohydrodynamical (MHD) simulations for stellar winds.
They varied the magnetic field strength relative to the mass-loss rate and surface gravity to derive the most complete stellar wind torque formula available at the time for solar-type stars,
\begin{equation}
\frac{dJ}{dt} = - \frac{K_1^2}{\left(2G\right)^m}\bar{B}^{4m}\ensuremath{\dot{M}}^{1-2m}_W\frac{R^{5m+2}}{M^m}\frac{\Omega}{\left(K^2_2+0.5u^2\right)^m},
\label{eq:Matt12}
\end{equation}
with $u$ being the ratio of the surface velocity to the brake-up velocity, and $K_1$, $K_2$ and $m$ tunable (to a certain point) parameters to fit the observations. The adopted values were taken from \citet{GB13} and are given in Table~\ref{tab:brake}.
\begin{table}
\caption{Parameters used for the prescriptions from \cite{Kawaler88} and \cite{Chaboyeretal1995a} ({\sf K88}), \cite{Matt2012} ({\sf Matt12}), \cite{Mattetal2015} ({\sf Matt15}), and \cite{RM12} ({\sf RM12}).}
\begin{tabular}{ c | c | c | c }
\hline
\hline
{\sf K88}& {\sf Matt12} & {\sf Matt15} & {\sf RM12}\\
\hline
$K_W = 2.10^{48} $ & $K_1 =6.7$ & $K = 5.10^{31}$ & $\mathcal{C} = 10^{39}$ \\
$n = 3/2$ & $K_2 = 0.506$ & $m = 0.22$ & $\Omega_{sat} = 3\Omega_\odot$ \\
$\Omega_{sat} = 10 \Omega_\odot$ & $m = 0.17$ & $p = 1.7$ \\
& & $\chi = 10$\\
\hline
\end{tabular}
\label{tab:brake}
\end{table}
We directly included a modified version of the BOREAS subroutine \citep{CS2011,GB13} in our evolution code STAREVOL (see the Appendix for details concerning the applied modifications to the original distributed version of BOREAS) to obtain the mean magnetic field $\bar{B} = f_* B_*$ at each time step, $f_*$ being the filling factor expressing the magnetized fraction of the stellar surface \citep{Saar1996}.
We had to recalibrate the expression of $f_*$ to reach the solar mass-loss value at the age of the Sun for solid-body rotating models,
\begin{equation}
f_\star = \frac{0.4}{[1+\left(x/0.16\right)^{2.3}]^{1.22}},\end{equation}
with $x$ being the normalised Rossby number\footnote{$\tau_{conv}$ and $\tau_{conv_\odot}$ are determined as a function of the effective temperature as in \citet{CS2011}.} $x = \frac{P_{rot}}{P_{rot_\odot}}\frac{\tau_{conv_\odot}}{\tau_{conv}}$.
With this expression, the magnetic field reaches a saturation threshold for $\Omega_{sat} \approx 10 - 15 \Omega_\odot$.\\
We also used the mass loss accounting for magnetohydrodynamic turbulence and Alfv\'en waves that is an output of the BOREAS routine. We accounted for mass loss from the end of the disc-coupling phase during the pre-main sequence to the end of the main sequence considering that the mechanisms responsible for mass loss remain the same during these phases.
As shown by \cite{GB13}, Eq.~\ref{eq:Matt12} combined with the prescriptions of \cite{CS2011} as computed with the modified BOREAS routine allows reproducing the observed rotation periods well in the validity domain of \citet{CS2011}'s work, which is in particular the case of stars of about 1~\ensuremath{M_\odot}. \cite{VSP2013} extended the application to more massive stars (1.1-1.5\ensuremath{M_\odot}) by combining this prescription with the mass-loss rate of \citet{Woodetal2005}.
\subsubsection{\cite{RM12}}\label{RM12}
\citet{RM12} introduced a formalism in which the magnetic field \textit{strength} is accounted for as directly depending on the Rossby number in the unsaturated regime (while it is the magnetic \textit{flux} that depends on the Rossby number in the approach of \citet{Matt2012}),
\begin{equation}
\frac{dJ}{dt} = -\mathcal{C} \left(\frac{R^{16}}{M^2}\right)^{1/3} \Omega \qquad {\rm for }\; \Omega \geq \Omega_{sat},
\end{equation}
\begin{equation}
\frac{dJ}{dt} = -\mathcal{C} \left(\frac{R^{16}}{M^2}\right)^{1/3} \left(\frac{\Omega}{\Omega_{sat}}\right)^4 \Omega \qquad {\rm for }\; \Omega < \Omega_{sat},
\label{eq:RM12}
\end{equation}
with
\begin{equation}
\mathcal{C}=\frac{2}{3}\left(\frac{B^8_{crit}}{G^2K_V^4\ensuremath{\dot{M}}}\right)^{1/3} \nonumber
\end{equation}
being a constant because each parameter on the right-hand side is assumed to adjust to keep the overall product constant throughout the evolution.
The main consequence of this change of paradigm is a torque that strongly depends on the stellar radius (${\rm d}J/{\rm d}t\propto R^{16/3}$!) and hence on the stellar mass and evolutionary stage. As shown in Fig. \ref{Fig:torqueomega}, the saturation value is reached at a very low rotation rate, which drastically changes the behaviour of surface rotation rate. The extraction of AM can be one order of magnitude higher for rotation rates between 1 and 10 times the solar value with this braking law.
This prescription shows some important weaknesses in reproducing the rotation rates of stars in open clusters as a function of mass, as shown by \citet[][their Fig.4]{RM12}. At the age of the Hyades, they obtain in particular a decrease of $P_{rot}$ with decreasing mass for M $\ge 0.3$ M$_\odot$, which contradicts observations. Reiners and Mohanty\textcolor[rgb]{1,0.501961,0}{} interpreted this discrepancy as the result of a possible core-envelope decoupling that would be proportional to the size of the radiative core (which increases with increasing mass at a given age). To test this hypothesis, we used this prescription for the stellar wind torque in our self-consistent differentially rotating 1~\ensuremath{M_\odot}~models.
\subsubsection{\cite{Mattetal2015}}
The main improvement reached by \citet{Mattetal2015} over \cite{Matt2012} is to highlight the Rossby number as a critical parameter, which was first reported by \citet{Soderblom85} and \citet{Vilhu86}. Compared to \citet{Matt2012}, \cite{Mattetal2015} expressed the torque as a simple function of mass, radius, angular velocity, and turnover timescale of the convective envelope\footnote{Determined as a function of T$_{eff}$ as in \citet{CS2011}.}.
Starting from a general expression of the torque from \cite{Matt2012},
\begin{equation}
\frac{dJ}{dt} = \left(\frac{dJ}{dt}\right)_\odot \left(\frac{M_\star}{M_\odot}\right)^{-m} \left(\frac{R_\star}{R_\odot}\right)^{5m+2} \left(\frac{B_\star}{B_\odot}\right)^{4m} \left(\frac{\ensuremath{\dot{M}}_\star}{\ensuremath{\dot{M}}_\odot}\right)^{1-2m} \left(\frac{\Omega_\star}{\Omega_\odot}\right),
\end{equation}
they expressed the more uncertain terms, namely $\ensuremath{\dot{M}}_\star$ and $B_\star$, as a function of the Rossby number. This results in some equations according to the regime of the magnetic field (saturated or not),
\begin{equation}
\frac{dJ}{dt} = -\mathcal{T}_0 \left(\frac{\tau_{CZ}}{\tau_{CZ\odot}}\right)^{p} \left(\frac{\Omega_\star}{\Omega_\odot}\right)^{p+1} \rightarrow {\rm unsaturated},
\end{equation}
\begin{equation}
\frac{dJ}{dt} = -\mathcal{T}_0 \chi^p \left(\frac{\Omega_\star}{\Omega_\odot}\right) \rightarrow {\rm saturated},
\end{equation}
with
\begin{equation}
\mathcal{T}_0 = K \left(\frac{R_\star}{R_\odot}\right)^{3.1} \left(\frac{M_\star}{M_\odot}\right)^{0.5}\gamma^{2m} ,
\end{equation}
where $\gamma = \sqrt{1+(u/0.072)^2}$ comes from Eq. $(8)$ of \cite{Matt2012}, and $u$ has the same meaning as above. $\chi = \frac{Ro_\odot}{Ro_{\rm sat}}$ is a constant. Considering that for $Ro \leq Ro_{\rm sat}$ the magnetic activity stops increasing and saturates at an approximately constant value.
The calibrated values of the parameters $K,p,m$ and $\chi$ are given in Table~\ref{tab:brake}.
\subsection{Transport of angular momentum in the stellar interior}
\subsubsection{Global equations}
We treated the angular momentum evolution from the first model on the PMS following the formalism developed by \citet{Zahn1992} and \citet{MaederZahn1998}. This formalism assumes a strong anisotropy in turbulence, the horizontal turbulence (i.e. along isobars) being much stronger than the vertical one (i.e. perpendicular to the isobars), thus enforcing a shellular rotation. The transport of AM in the stellar interior follows the advection/diffusion equation,
\begin{equation}
\rho \frac{{\rm d}}{{\rm d}t}\left(r^2\Omega\right)= \frac{1}{5r^2}\ddr{} \left(\rho r^4 \Omega U_r\right) + \frac{1}{r} \ddr{} \left(r^4\rho \nu_v \ddr{\Omega}\right),
\label{eq:general}
\end{equation}
where $\rho$, $ r$, $\nu_v$ and $U_r$ are the density, radius, vertical component of the turbulent viscosity, and the meridional circulation velocity on a given isobar, respectively.
This equation applies to radiative regions, while convective zones are assumed to rotate as solid bodies. The torque discussed in the previous section is applied at the upper convective boundary \citep[see][]{PTCF2003},
\begin{displaymath}
\begin{array}{ll}
{\displaystyle \frac{\partial}{\partial t}
\left[ \Omega \int _{r_t}^R r^4 \rho \, {\rm d} r \right] =
-\frac{1}{5} r^4 \rho \Omega U + {\cal F}_\Omega } & ~~~{\rm for} ~ r=r_t \vspace{0.2cm}\\
\end{array}
\end{displaymath}
where ${\cal F}_\Omega$ is the torque, and $r_t$ is the radius at the lower edge of the convective envelope.\\
By integrating the angular momentum transport Eq. (\ref{eq:general}) over the surface of radius $r_{cz}$, $r_{cz}$ being the radius of the inner convective zone boundary, we obtain a flux equation,
\begin{equation}
F_{\rm tot} = F_{S}(r_{\rm cz}) + F_{MC}(r_{\rm cz})
\label{eq:fluxeq}
,\end{equation}
with
\begin{equation}
F_{S}(r_{\rm cz}) = \frac{\rm{d} J_{S}}{\rm{d} t} \bigg|_{r=r_{cz}} = -\rho r^4 \nu_v \ddr{\Omega}\bigg|_{r=r_{cz}}
\end{equation}
the flux carried by shear-induced turbulence from the radiative zone to the convective envelope, and
\begin{equation}
F_{MC}(r_{\rm cz}) = \frac{\rm{d} J_{MC}}{\rm{d} t} \bigg|_{r=r_{cz}}= -\frac{1}{5} \rho r_{cz}^4 \Omega U_{r_{cz}}
\end{equation}
the flux carried by meridional circulation. A detailed derivation of the AM fluxes is given in
\cite{DecressinMathis2009} as part of a set of tools for assessing the relative importance of the processes involved in AM transport in stellar radiative interiors.
\begin{table}[h]
\caption{Different prescriptions used for the turbulent diffusion coefficients in our models.}
\begin{center}
\begin{tabular}{| c | c | c }
\hline\hline
\rowcolor{Gray}Prescription & $D_h \equiv \nu_h$\\ \hline\hline
& \\
\citet{MPZ2004} & $r\sqrt{\left[Cr\overline\Omega \vert 2 V_r-\alpha U_r\vert\right]}$\\
({\sf MPZ04})& with $ \alpha={\frac{1}{2}}\,\frac{{\partial}\ln(r^2\overline\Omega)}{{\partial}\ln r}$ \\
& and $C = 1.6\times10^{-6}$ \\
& \\\hline
& \\
\citet{Zahn1992} & $\frac{r}{C_{h}}\vert2 V_r-\alpha U_r\vert$\\
({\sf Zahn92})& with $C_h$ = 1\\
& \\\hline \hline
\rowcolor{Gray}Prescription & $D_v \equiv \nu_v$\\ \hline\hline
& \\
\citet{TZ97} & $\frac{Ri_{\rm c}}{N^2_{T} /(K_{T} + D_{h}) + N^{2}_{\mu}/ D_{h}} \left(r\frac{\partial \overline\Omega}{\partial r}\right)^2\,$\\
({\sf TZ97})& with $Ri_c = 0.25$\\
& \\ \hline
& \\
\citet{Maeder1997} & $f_\text{energ} \frac{H_P}{g\delta}\frac{K}{\left[\frac{\varphi}{\delta}\nabla_\mu
+ \left( \nabla_\text{ad} - \nabla_\text{rad} \right)\right]}
\left( \frac{9\pi}{32}\ \Omega\ \frac{\text{d} \ln \Omega}{\text{d} \ln r} \right)^2$\\
({\sf Maeder97}) & with $K = \frac{4ac}{3\kappa}\frac{T^4\nabla_\text{ad}}{ \rho P \delta}$\\
& $f_\text{energ} = 1$ and $\varphi = \left( \frac{\text{d}\ln\rho}{\text{d}\ln\mu} \right)_{P,T} = 1$\\
\hline
\end{tabular}
\end{center}
\label{tab:coeffdiff}
\end{table}
Equation~\ref{eq:general} is complemented by the evolution equation for the relative mean molecular weight variations over an isobar $\Lambda = \tilde{\mu}/\bar{\mu}$ ,
\begin{equation}
\frac{{\rm d}\Lambda}{{\rm d}t} - \frac{{\rm d} \ln {\overline \mu}}{{\rm d}t} \Lambda = \frac{N_{\mu}^{2}}{{\overline g}\varphi}U_r-\frac{6}{r^2}\nu_h\Lambda\, ,
\label{Lambda}
\end{equation}
where $\overline \mu$ is the mean molecular weight over an isobar, $\nu_h$ is the diffusion coefficient associated with the horizontal shear instability, and $N_{\mu}$, the chemical part of the Brunt-V\"ais\"al\"a
frequency is given by $N_{\mu}^2=\left({\overline
g}\varphi/H_{P}\right)\nabla_{\mu}$ with $\nabla_{\mu}={\partial \ln
\overline{\mu}}/{\partial \ln P}$. \\
\subsubsection{Turbulence modelling}
As in \citet[e.g.][]{Zahn1992}, the shear turbulence in the horizontal and vertical directions is represented as a diffusive process. We assumed that the diffusion coefficients are well represented by the viscosities, that is, $D_v \approx \nu_v$ and $D_h \approx \nu_h$.\\ Several prescriptions exist for these viscosities, and we explored the effect of four different prescriptions on the angular momentum evolution of a 1\ensuremath{M_\odot}~model. These prescriptions are summarised in Table~\ref{tab:coeffdiff}, and we refer to the associated papers for more details on how they were derived.\\
For the vertical diffusion coefficient, the prescription of \citet{TZ97} has been used for all the rotating models computed with the STAREVOL code so far \citep{PTCF2003,PalaciosCharbonnel2006,DecressinMathis2009,Lagarde2012}, while the prescription of \citet{Maeder1997} has been systematically used for the rotating models computed with the Geneva code \citep{Eggenbergeretal12,Georgy2013}. \citet{Meynet2013} have shown that the choice of the turbulent transport prescriptions may dramatically affect predictions for the structural, chemical, and rotational evolution of massive stars, but they were unable to clearly determine a best combination
for fitting the observational chemical constraints. In Sect.~\ref{sec:explore} we discuss this in a similar way within the framework of the rotational evolution of solar-type stars.
\subsection{Transport of chemicals}
The transport of chemical species in radiative region is computed as a purely diffusive process \citep{ChaboyerZahn92} but accounts for vertical advection and a strong horizontal diffusion. For a chemical species $i$, the concentration $c_i$ follows the equation
\begin{equation}
\frac{{\rm d} c_i}{{\rm d}t} = \dot{c}_i + \frac{1}{\rho r^2}\ddr{}\left(\rho r^2 D_{tot} \ddr{c_i}\right),
\end{equation}
with $D_{tot} = D_{eff}+D_v$ the total diffusion coefficient, $D_{eff}$ given by
\begin{equation}
D_{eff} = \frac{\vert rU(r)\vert^2}{30D_h},
\end{equation}
where $D_v$ and $D_h$ were defined previously (see Table \ref{tab:coeffdiff}). Finally, the term $\dot{c}_i$ refers to the temporal evolution of the concentration of chemical species $i$ due to nuclear burning.
\begin{figure}[t]
\includegraphics[width=0.45\textwidth]{FIGURES/Figure2.eps}
\caption{Evolution of the angular velocity as a function of time for the reference models. Plotted on the left as the rotational periods in days and during the right as the angular velocity in solar units with $\Omega_\odot = 2.86.10^{-6}$s$^{-1}$. Each cross represents a measurement for a star belonging to an open cluster whose age has been taken from the literature. All observational data are taken from \citet[][and references therein]{GB15} except the "OD" black frame between 8 and 10 Gyr, which corresponds to photometric data for stars observed with the {\sf KEPLER} satellite that belong to the disc of the Milky Way \citep{McQuillan2013}.
Solid lines show the evolution of surface rotation for the slow (magenta), medium (green), and fast (blue) rotating models, while short dashed lines represent the associated averaged angular velocity of the radiative interior.}
\label{Fig:refprot}
\end{figure}
\section{Physics of the models}\label{sec:models}
For all the models reported in Table~\ref{tab:mods}, the basic input physics (equation of state, nuclear reaction, opacities) can be found in \cite{Lagarde2012}. The initial abundances and mixing length parameter are calibrated without microscopic diffusion to reproduce a non-rotating Sun with respect to the \cite{AsplundGrevesse2009} solar mixture with a $10^{-5}$ precision for luminosity and radius at the age of the Sun. We used a mixing length parameter $\alpha_c = 1.7020$, an initial helium abundance $ Y = 0.2689 $, and an initial metal abundance $ Z = 0.0134 $. These values differ slightly from \citet{Lagarde2012} because we included a non-grey atmosphere treatment based on \cite{KrishnaSwamy}, which better agrees with observations for Sun-like stars \citep[e.g.][]{Vandenberg2007}.
In addition to the treatment of transport and loss of angular momentum and mass as described in Sect.~2, we included the modification of the effective gravity by the centrifugal forces and its effect on the stellar structure equations following \citet{ES76}. This effect is slightly visible on the evolutionary track in the Hertzsprung-Russell diagram when the stars rotate fast.
Rotation can have a non-negligible impact on the effective temperature for the fastest rotators around the ZAMS.
\begin{figure*}\includegraphics[width=0.95\textwidth]{FIGURES/Figure3.eps}
\caption{Angular velocity profile of the reference models at different epochs (solid: 10 Myr; dotted: 45 Myr; dashed: 100 Myr; dot-dashed: 250 Myr; long-dashed: 1 Gyr; long-dash-dotted: 4.57 Gyr) as a function of the relative mass fraction. The left, central, and right plots correspond to cases with slow, medium, and fast rotation (models $1s$, $1m$, and $1f$).}
\label{Fig:refprofomega}
\end{figure*}
\section{Reference model}\label{sec:ref}
The statistical analysis of the distributions of rotation periods in open clusters and associations from 1 Myr to 2.5 Gyr performed by \citet{GB13,GB15} focuses on three types of rotators: the slow, medium, and fast rotators, associated to the 25th, 50th,
and 90th percentiles of the statistical sample in each of the clusters. We focused on the same types of rotators and defined our reference models as those that reproduce their evolutionary paths, as shown in Fig.~\ref{Fig:refprot}.
They are labelled $1s$, $1m,$ and $1f$ in Table~\ref{tab:mods}. They are characterised by the following set of prescriptions: {\sf Matt+15} for the stellar wind torque, {\sf TZ97} for $\nu_v$ , and {\sf MPZ04} for $\nu_h$. We preferred the {\sf Matt+15} prescription for the torque over the formulation of {\sf Matt+12} , even though the latter directly takes into account magnetic field and mass-loss rate, precisely for sake of the consistency. Those are multi-dimensional processes that are still implemented in 1D models with many uncertainties. We show in Sect. \ref{sec:models} that both fit the observed rotation periods, but we prefer a simpler approach that includes fewer uncertain processes to ensure that we will not miss any effects.
\begin{table}[h]
\caption{Parameters and assumptions of the different models computed in this study: MPZ04 refers to \cite{MPZ2004}, TZ97 to \cite{TZ97}, Matt+15 to \cite{Mattetal2015}, Matt+12 to \cite{Matt2012}, R\&M 12 to \cite{RM12}, and GB13 to \cite{GB13}. SB is for solid-body rotation.}
\begin{center}
\begin{tabular}{ c | c | c | c c }
\hline
\hline
Initial P$_{rot}$, & $\nu_h$ & $\nu_v$& Braking law & Ref. \\
DL time & prescrip. & prescrip. & & \\
\hline
\hline
7 days & MPZ04 & TZ97 & Matt+15 & 1s \\
5 Myr & MPZ04 & Maeder97 & Matt+15 & 2s \\
& Zahn92 & TZ97 & Matt+15 & 3s \\
& Zahn92 & Maeder97 & Matt+15 & 4s \\
& MPZ04 & TZ97 & R\&M 12 & 5s \\
& MPZ04 & TZ97 & Matt+12 & 6s \\
& MPZ04 & TZ97 & K88 & 7s \\
& SB & SB & Matt+15 & 8s \\
& MPZ04 & TZ97 & GB13 & 9s \\
\hline
3.7 days & MPZ04 & TZ97 & Matt+15 & 1m \\
5 Myr & MPZ04 & TZ97 & R\&M 12 & 5m \\
& MPZ04 & TZ97 & Matt+12 & 6m \\
& MPZ04 & TZ97 & K88 & 7m \\
& SB & SB & Matt+15 & 8m \\
& MPZ04 & TZ97 & GB13 & 9m \\
\hline
1.4 days & MPZ04 & TZ97 & Matt+15 & 1f \\
3 Myr & MPZ04 & Maeder97 & Matt+15 & 2f \\
& Zahn92 & TZ97 & Matt+15 & 3f \\
& Zahn92 & Maeder97 & Matt+15 & 4f \\
& MPZ04 & TZ97 & R\&M 12 & 5f \\
& MPZ04 & TZ97 & Matt+12 & 6f \\
& MPZ04 & TZ97 & K88 & 7f \\
& SB & SB & Matt+15 & 8f \\
& MPZ04 & TZ97 & GB13 & 9f \\
\hline
\end{tabular}
\label{tab:mods}
\end{center}
\end{table}
\subsection{Evolution of surface angular velocities}
Figure~\ref{Fig:refprot} shows the surface angular velocity evolution (in terms of rotation period) as a function of time for the three reference models ($1s$, $1f,$ and $1m$) as solid lines.
Depending on their initial velocity, these models present very different behaviours, but they are qualitatively similar to those obtained by \citet{KPBS97}. The slow and medium rotators (models $1s$ and $1m$) experience a stronger stellar wind braking related to their initial angular momentum during the early evolution than do the fast rotator (model $1f$) towards the age of the present Sun. We also observe a convergence of the averaged rotation rate of the radiative interiors (dashed lines); this result is similar to previous work done with a slightly different treatment of AM transport \citep[e.g.][]{P90}. Consequently, all the models reach the age of the Sun with the same total amount of angular momentum independently of their initial content. We detail the evolution of the radiative region angular velocity profile to understand which processes are involved.
\begin{figure}[t]
\includegraphics[angle=270,width=0.47\textwidth]{FIGURES/Figure4.eps}
\includegraphics[angle=270,width=0.47\textwidth]{FIGURES/Figure4b.eps}%
\caption{Angular momentum fluxes carried by meridional circulation (dashed red), shear (dotted blue), and the total flux (solid black) in the fast and slowly rotating models (top and bottom, respectively). Hatched areas in all plots indicate the convective regions.}
\label{Fig:reffluxAM}
\end{figure}
\subsection{Rotation profile and differential rotation}
Figure \ref{Fig:refprofomega} shows the evolution of the internal angular velocity profiles as a function of the mass coordinate for our reference models $1s$ ({\em left}), $1m$ ({\em central}), and $1f$ ({\em right}). We observe that for the special combination of braking and transport prescriptions adopted, all the models end with the same rotation profile (long-dash-dotted cyan profile), regardless of their initial angular momentum content. In all three cases, the differential rotation first increases (the core spins up) during the PMS evolution up to 45 Myr due to the contraction of the stars as they evolve along the Henyey track. Beyond 45 Myr, the core spins down in all models as a result of the meridional circulation that extracts AM from the core to compensate for the loss generated at the surface by the torque (see right panels of Fig.~\ref{Fig:reffluxAM} and below). The main difference between the three cases is the efficiency of the meridional circulation, which adjusts to the intensity of the torque: the larger the torque (for the fast rotators), the larger the angular velocity gradient and the more efficient the meridional circulation.
Moreover, since the efficiency of these angular momentum transport processes is proportional to the surface angular velocity, we observe a weak coupling between the radiative core and the envelope in models $1s$ and $1m$ at the same epochs (radiative interior rotating on average much faster than the convective envelope; see solid and dashed lines). Thus, the surface velocity evolution is already shaped by the torque before the stars reach the ZAMS (at $\approx$ 60 Myr), leading to their early spin down.\\
In contrast, the fast rotating model strongly couples the radiative and convective zone as long as the star contracts (spins up), and the internal angular momentum is carried outward. During these early phases, model $1f$ is in the saturated dynamo regime. At its arrival on the ZAMS, the star stops spinning up and the surface is strongly braked on a relatively short timescale. The surface velocity is divided by a factor of 10 in 200 Myr, until the star spins down enough to enter the unsaturated-dynamo regime and further decreases by another factor of 10 for the following 4.3 Gyr.
For clusters older than the Hyades (beyond $\approx$ 700 Myr), the observed distributions of rotation periods strongly narrow.
By the time the models reach 1 Gyr, the torque has become much less efficient and the angular velocity profile of the fast rotator is similar to that of the median and slow rotators. The three models later follow the same evolution, which is dictated by the meridional circulation and shear turbulence until the age of the Sun. The surface angular velocity and the total angular momentum content of the slow, medium, and fast rotators are similar beyond 1 Gyr. The angular velocity profiles also share a similar shape and evolution in all three cases.
\begin{figure}
\includegraphics[width=0.45\textwidth]{FIGURES/Figure5.eps}
\caption{Same as Fig.~\ref{Fig:refprot} for fast and slow rotator models computed with four different transport prescriptions. Among the slow rotators models $1s$, $2s$, $3s$, and $4s$ correspond to the tracks from top to bottom for an age of 500 Myr. The fast rotating models $1f$, $2f$, $3f$, and $4f$ follow he same order for an age of 60 Myr. The description of the physics used in each of these models is given in Table~\ref{tab:mods}.}
\label{Fig:protmixing}
\end{figure}
\subsection{Transport of AM}
Here we examine the processes that drive the internal transport of AM in our reference models. To do so, we followed the diagnostic procedure proposed by \citet{DecressinMathis2009}, which relies on the analysis of the AM fluxes associated with each transport process that is accounted for. The transport of angular momentum is mainly driven by the meridional circulation as long as the star is rotating relatively fast (above the saturation value), as shown in the left panels of Fig. \ref{Fig:reffluxAM}, and this independently of the initial velocity. After the ZAMS ($\approx$ 60 Myr), in all cases meridional circulation weakens because the star does not contract any longer and the surface velocity has already decreased. Angular velocity gradient increases at the base of the convective envelope, leading to a predominance of the shear turbulence over the meridional circulation in this region. For the slower rotators, the shear turbulence takes over the transport of AM in the outer radiative zone below the envelope and efficiently extracts (acting along the angular velocity gradient) the angular momentum,
ensuring a continuous decrease of the surface angular velocity from the ZAMS until the age of the Hyades ($\approx$ 700 Myr) (see lower panel in Fig.~\ref{Fig:reffluxAM}). On the other hand, in the fast rotator, the meridional circulation is very efficient as long as the star contracts (2 orders of magnitude larger than in the slow rotator) and efficiently couples the interior to the convective envelope up to the ZAMS. Beyond this age, its efficiency decreases rapidly, and the shear generated by the torque dominates below the convective envelope, although it is not sufficient to maintain an efficient transfer of AM from the core to the envelope.\\
The overall transport of AM by these processes tends to become less efficient as the star evolves on the main sequence as a
result of the decrease of the angular velocity gradient and of the torque.
Around the age of the Sun, some very tight reversals of the circulation that are due to the turbulence at very slow rotation rate appear close to the convective envelope bottom. The earlier evolution is dominated by an outward loop carrying angular momentum from the core to the surface to compensate for the loss that
is due to stellar wind.
\begin{figure}[t]
\includegraphics[width=0.45\textwidth]{FIGURES/Figure6.eps}
\includegraphics[width=0.45\textwidth]{FIGURES/Figure6b.eps}
\caption{Effect of modelling the vertical turbulent shear diffusion coefficient $D_v$ on the angular velocity profiles. Same as Fig.~\ref{Fig:refprofomega} for models $2s$ (dashed lines) and $1s$ (solid lines) in the top, and for models $2f$ (dashed lines) and $1f$ (solid lines) in the bottom panels. The description of the physics used in each of these models is given in Table~\ref{tab:mods}.}
\label{Fig:profomegaMaeder}
\end{figure}
\section{Exploring the physical parameter space}\label{sec:explore}
The description of our reference models points out the different roles played by the meridional circulation, the shear turbulence, and the torque in shaping the internal angular velocity profiles throughout the evolution. As indicated in Table~\ref{tab:mods} and Sect.~\ref{sec:torques}, several prescriptions exist to account for the transport of AM in the radiative interiors and to describe the stellar wind torque exerted at the surface. In this section, we analyse the effect of the prescription choice on the angular velocity and momentum evolution in solar-mass models. To do so, we use the set of diagnostic tools developed by \cite{DecressinMathis2009}.\\
\begin{figure}[t]
\includegraphics[width=0.45\textwidth]{FIGURES/Figure7.eps}
\includegraphics[width=0.45\textwidth]{FIGURES/Figure7b.eps}
\caption{Effect of the modelling of the horizontal turbulent shear diffusion coefficient $D_h$ on the angular velocity profiles. Same as Fig.~\ref{Fig:refprofomega} for models $3s$ (dashed lines) and $1s$ (solid lines) in the top, and for models $3f$ (dashed lines) and $1f$ (solid lines) in the bottom panels. The description of the physics used in each of these models is given in Table~\ref{tab:mods}.}
\label{Fig:profomegaZahn}
\end{figure}
\subsection{Transport of angular momentum in the stellar interior}
Within the paradigm described by \cite{Zahn1992} for the transport of angular momentum in radiative regions, several prescriptions have been developed to describe the shear turbulent viscosity in both the horizontal and vertical directions. While different groups prefer different combinations according to the type of stars studied, it was not until the study by \citet{Meynet2013} that the actual impact of choosing a specific combination of prescriptions on the evolution (structural, chemical, and rotational) was explored. These authors modelled massive stars ($9-40$M$_\odot$) and showed large differences in the HR diagram, the evolution
of surface chemical abundances, and the rotational evolution. However, the available observational data did not allow them to favour one of the combinations.
Here we analyse the effect of two different prescriptions for both $\nu_h$ and for $\nu_v$ on the rotational evolution of a 1 ~\ensuremath{M_\odot}~ model. The overall effect on the surface and internal averaged angular velocity is shown in Fig.~\ref{Fig:protmixing}.\\
\subsection{Effect of the prescription for vertical shear}
We first compare models with the {\sf MPZ04} prescription for $D_h$ but with different prescriptions for the vertical shear ({\sf TZ97} or {\sf Maeder97}). The use of the {\sf Maeder97} prescription (given in Table~\ref{tab:coeffdiff}) for $\nu_v$ hinders the development of the vertical turbulent shear instability in a larger portion of the radiative interior, and the shear only contributes very little to the transport of AM throughout the evolution. For the slow rotator (model $2s$ in Fig.~\ref{Fig:protmixing} and top panel in Fig.~\ref{Fig:profomegaMaeder}), this leads to a sharp decrease of the surface angular velocity from the ZAMS after the contraction of the star has stopped and the meridional circulation is weaker. Between 100 Myr and 1 Gyr, the efficiency of the meridional circulation to transport AM remains almost constant in the inner radiative zone and ensures the transport of AM from the core to the envelope to compensate for the extraction of AM at the surface by the stellar wind torque. Therefore this leads to an almost constant surface angular velocity over this period of time.
For the fast rotators (model $2f$ in Fig.~\ref{Fig:protmixing} and lower panel in Fig.~\ref{Fig:profomegaMaeder}), on the other hand, the prescription of $\nu_v$ has a very weak influence on the surface velocity because the transport is always dominated by the meridional circulation, which is of similar amplitude as in the reference model. The weak effect of the vertical shear diffusion prescription on the AM evolution also appears when looking at the evolution of the total specific angular momentum as a function of time shown in Fig.~\ref{Fig:momspec} (dotted line, left panel).\\
\subsection{Effect of the prescription for horizontal turbulence}
The effect of the choice of the prescription for the horizontal shear turbulence is much more obvious: in Fig.~\ref{Fig:protmixing} the two models with $\nu_h$ from {\sf Zahn92} (models $3s, 3f, 4s,$ and $4f$
) present a very high ratio $\Omega_{interior}/\Omega_{surface}$ compared to the models $1s$ (resp. $1f$) and $2s$ (resp. $2f$) computed with $\nu_h$ from {\sf MPZ04} (see Table~\ref{tab:mods}).
Using the expression of {\sf Zahn92} for $\nu_h$ leads to a meridional circulation that is overall slower than for $\nu_h$ of {\sf MPZ04}.
The turbulence can develop and disturb the meridional circulation in the radiative interior if the Richardson criterion is fulfilled. In our reference model, the circulation is strong enough to keep its original shape even in presence of shear turbulence, but this is not always the case. Thus, in models $3s$($f$) and $4s$($f$), using $\nu_{h,Zahn92}$, the angular velocity profiles are much steeper, as shown in Fig.~\ref{Fig:profomegaZahn}, and trigger a much stronger mean shear-induced turbulence. The loop that was carrying AM outward then breaks into several reverse cells as soon as the Richardson criterion is fulfilled (shown in Fig.~\ref{Fig:2dcirc_Dh}) and annihilates AM transport locally. Consequently, a break appears in the angular velocity profile that can be seen in Fig.~\ref{Fig:profomegaZahn} around 0.3 M$_r$ (resp. 0.2 M$_r$) at 1 Gyr and around 0.5 M$_r$ (resp. 0.4 M$_r$) at the solar age for the slow (resp. fast) rotator.
In addition, the surface velocity at the ZAMS and during the early evolution on the main sequence is lower than in the reference models, leading to a less efficient torque (which is directly proportional to the surface angular velocity) and to a global larger amount of specific angular momentum when the star reaches the age of the Sun (see Fig.~\ref{Fig:momspec}).
\begin{figure}
\includegraphics[width=0.48\textwidth]{FIGURES/Figure8.eps}
\caption{ Same legend as Fig.~\ref{Fig:reffluxAM} for two slow rotators at 1 Gyr computed with $\nu_h$ from {\sf MPZ04} (left) and {\sf Zahn92} (right).}
\label{Fig:2dcirc_Dh}
\end{figure}
\subsection{Stellar wind torque}
We now explore the effect of the stellar wind torque prescriptions on the overall angular momentum and angular velocity evolution of solar-mass stars.\\ First, we calibrated each of the prescriptions detailed in Sect.~\ref{sec:torques} to reproduce the surface angular velocity of the present Sun at 4.57 Gyr. To do so, we calibrated one parameter, $\mathcal{C}, K_1$ and $K_W$ for breaking laws of {\sf R\&M12, Matt+12,} and {\sf K88}, respectively. Second, we adjusted a second parameter\footnote{$\Omega_{\rm sat}$, $m$ and $\Omega_{\rm sat}$ for ${\rm R\&M12}$, ${\rm Matt+12,}$ and ${\rm K88, respectively.}$} to adjust the data in younger clusters, if possible. The obtained values are given in Table~\ref{tab:brake}, and for every model with a given braking law, the same set of parameters was used, regardless of the initial angular momentum content. \\
\begin{figure*}[t]
\includegraphics[width=0.95\textwidth]{FIGURES/Figure9.eps}
\caption{Same legend as Fig. \ref{Fig:refprot} with different braking laws than in our reference model. The references of the models are given in Table~\ref{tab:mods}. The reference model corresponds to the black tracks.}
\label{Fig:prot}
\end{figure*}
Figure \ref{Fig:prot} presents the evolution of the surface rotation for slow, medium, and fast rotators with different torque prescriptions, but with the same internal AM transport description ({\sf MPZ04, TZ97}).
We note that in all cases, the mean angular velocity of the radiative core reaches the same rotation period, regardless of the stellar wind torque prescription used, as shown in Fig. \ref{Fig:momspec}.
\subsubsection*{\sf RM12}
The surface rotation evolution using this prescription is compared to the reference models in the upper row of Fig. \ref{Fig:prot}. The strong radius dependency of the breaking law reported in
\citet{RM12} explains the very high loss of angular momentum observed in Fig. \ref{Fig:couple} ($5f$ track in the upper right panel) during the PMS, when the star is still contracting. The angular velocity increases very slowly during the PMS compared to the reference model, and sharply drops at the ZAMS. As soon as the velocity drops below the saturation value ($\Omega_{sat,RM12} = 3\Omega_\odot$), the extraction of angular momentum almost stops and the surface angular velocity evolution stabilises to converge to $\Omega(t) \propto t^{-0.25}$ (see Eq. \ref{eq:RM12}).
\subsubsection*{\sf Matt+12}
The surface rotation evolution using this prescription is compared to the reference models in the middle row of Fig. \ref{Fig:prot}. The angular velocity evolution of fast rotators is the same as in the reference model, despite the absence of explicit saturation value in {\sf Matt+12} prescription. It appears in the BOREAS routine as an asymptotic regime of the magnetic field filling factor $f_\star$ that is perfectly mimicked by the saturation coefficient $\chi$ we took for the reference model using {\sf Matt+15} prescription.
A difference appears around the ZAMS for slow rotators, and because of this difference of saturation, models with the torque according
to {\sf Matt+12} reach higher rotation periods. Nevertheless, the discrepancy remains very small and as we lack data for these ages, there is no real constraint to determine which one is better.
\subsubsection*{\sf K88}
The surface rotation evolution using this prescription is compared to the reference models in the lower row of Fig. \ref{Fig:prot}. Both slow and medium rotators braked with {\sf K88} behave as the reference model in the unsaturated regime. Still, to obtain a solar surface angular velocity at the age of the Sun, we need to take a high $K_W$ value. The extraction of angular momentum is therefore three times higher at the ZAMS than for the reference model (see right panel of Fig. \ref{Fig:couple}).
This strong braking prevents the models from achieving rapid rotation at the age of the Pleiades with the {\sf K88} torque. This result is similar to what was obtained in previous work \citep[e.g.][]{Pinsonneaultetal1989}.
\section{Comparison to bi-zone models}\label{sec:bizone}
\begin{figure}
\includegraphics[width=0.45\textwidth]{FIGURES/Figure10.eps}
\caption{Evolution of the total specific angular momentum as a function of time. ({\em Left}) Effect of the AM transport prescriptions. ({\em Right}) Effect of the torque prescriptions.}
\label{Fig:momspec}
\end{figure}
\begin{figure}[t]
\includegraphics[width=0.45\textwidth]{FIGURES/Figure11.eps}
\caption{Same as Fig.~\ref{Fig:momspec} for the evolution of the ratio between the total angular momentum and the angular momentum loss due to wind alone (disc-coupling loss not accounted for).}
\label{Fig:couple}
\end{figure}
In this section we compare our predictions with those of the bi-zone models of \citet{GB13}.
Their approach splits the star into two regions, each of them rotating as a solid body. The external region corresponds to the convective envelope, the inner region to the inside layers of the star (including the radiative region and, if exists, the convective core).
Exchange of AM between these two zones is assumed to occur at a given rate along the whole evolution.
The empirical coupling timescale is constrained to reproduce the observed surface rotation periods.
For a comparison between the two approaches, we deduced a coupling timescale between the convective envelope and the rest of the star for our full evolutionary models. This is possible thanks to the determination of the angular momentum flux transferred by meridional circulation and shear turbulence between the external convective region and the radiative part of the star (see Eq. \ref{eq:fluxeq} in Sect. \ref{sec:formalism}).
This flux was then introduced into the formulation of \cite{MB1991} to determine a timescale,
\begin{equation}
\tau_c = \Delta J \times \frac{1}{F_{\rm tot}}
,\end{equation}
where
\begin{equation}
\Delta J = \frac{I_{\rm conv}J_{\rm rad} -I_{\rm rad}J_{\rm conv}}{I_{\rm rad} + I_{\rm conv}}
,\end{equation}
with $I_{\rm conv}$, $J_{\rm conv}$ and $I_{\rm rad}$, $J_{\rm rad}$ being the inertial and angular momentum of the convective envelope and radiative zone (and if exists, the convective core).
Because the amount of AM that is transported is mainly dominated by the meridional circulation, its corresponding timescale is two orders of magnitude shorter than the one associated with diffusive turbulence. The timescales associated with both meridional circulation and shear turbulence are much longer than the constant coupling time-scale used in bi-zone models \citep[e.g.][]{GB13,GB15}.
These models are calibrated to reproduce the open cluster observations and the surface velocity, the total angular momentum, and the mass-loss rate of the present Sun. The assumed coupling does not necessarily depend on time, rotation rate, or differential rotation rate.
To properly compare our models to the bi-zone models of \citet{GB13,GB15}, we computed a slowly rotating model with exactly the same braking law as described in Table 2 of \citet{GB13} (model {\em 9s}), and a bi-zone model with the same parameters for the torque and for the coupling timescale as in \citet{GB13}, but based on the structural evolution of our STAREVOL models (model {\em GB13})\footnote{In \citet{GB13}, the evolution of the structural quantities, e.g. M, R, inertia momenta, are taken from \citet{Baraffe98} grid. }.
Thus, the only differences between the two models are the transport of angular momentum in the radiative core and the exchange with the convective region. The evolution of the surface velocity of these models as a function of time is shown in Fig.~\ref{Fig:prot2}.
Model $9s$ solves a self-consistent time-dependent AM exchange between the radiative interior and the convective envelope, with an evolving associated timescale, while the $GB13$ model assumes a constant transfer of AM between the interior and the convective envelope, leading to a constant timescale throughout the evolution, as shown in Fig. \ref{Fig:Timescales}.
\begin{figure*}[t]
\includegraphics[width=0.95\textwidth]{FIGURES/Figure12.eps}
\caption{Surface rotation evolution as a function of time in the reference slow ({\em left}) and fast ({\em right}) models compared to bi-zone models ({\sf GB13}). Data points are as described in Fig. \ref{Fig:refprot}. The references of the models are given in Table~\ref{tab:mods}.}
\label{Fig:prot2}
\end{figure*}
This can qualitatively be transcribed as a forced transfer of AM from the radiative to the convective region that changes the surface velocity and therefore the angular velocity-dependent AM extraction at the surface, while in our self-consistent model the AM extraction triggers the AM transport below the convective region and thus changes the core-envelope coupling.
At the very beginning of the evolution, when the star is still coupled to its disc but is already partially radiative, the forcing of the disc on the stellar surface leads to a very strong meridional circulation, and the corresponding AM transport timescale is very short compared to the rest of the evolution.
After the star decouples from the disc, the forcing stops and the AM transfer timescale increases sharply. Beyond this point, the coupling timescale in model $9s$(f) is always much longer than the timescale assumed in the bi-zone model $GB13$ (fast or slow). On the MS the structure of the star stabilizes and the core-envelope coupling follows both the evolution of the surface angular velocity and the differential rotation.
We can note that the fast rotators experience a much stronger coupling during the whole evolution. This is especially evident around 200 Myr when the star is still a fast rotator and the differential rotation quickly increases because of the strong braking that slows down the surface. Since both the absolute and the differential rotation rate play a role in the efficiency of AM transport, the coupling timescale reaches a minimum. Even though the convective and radiative regions of the fast rotator are more strongly coupled than for the slow rotator, the coupling timescale remains more than one order of magnitude higher than the constant coupling timescale set for the two-zones model.
Figure \ref{Fig:TorqueGB13} shows the extraction of angular momentum by magnetized winds in both cases. As expected, the evolution is similar during the PMS because the evolution of angular velocity is driven by structural effects that are identical in both models. After the star stops contracting, the internal angular momentum transport becomes relevant and a difference appears between the two models. As more AM is brought to the surface, the angular velocity is higher with a stronger coupling in model $GB13$. When the angular velocity is higher for the same structure, the extraction by the winds is stronger during this phase and pumps out most of the radiative core AM reservoir. All the AM brought to the convective region is immediately removed by stellar winds
because a certain equilibrium is reached between the core-envelope coupling and the surface extraction.
Around 1 Gyr, the model with $GB13$ parameters rotates almost as a solid body, the transfer of AM from the core to the external envelope becomes very weak and causes the radiative core to rotate at the same angular velocity as the convective region. At the same time, model $9s$ loses AM faster, but the rate is still two orders of magnitude lower than at the ZAMS. All the AM loss that matters for the AM content at the age of the Sun occurs before 1 Gyr.
\begin{figure}
\includegraphics[width=0.42\textwidth]{FIGURES/Figure13.eps}
\caption{Coupling timescale from the AM exchange between the radiative core and the convective envelope.}
\label{Fig:Timescales}
\end{figure}
\begin{figure}
\includegraphics[width=0.42\textwidth]{FIGURES/Figure14.eps}
\caption{Specific angular momentum loss per year as a function of time.}
\label{Fig:TorqueGB13}
\end{figure}
\section{Discussion with respect to additional constraints}\label{sec:other}
\subsection{Constraint on internal rotation from helioseismology}
Asteroseismology, and more specifically helioseismology, has
delivered very accurate data that allow probing the internal structure of stars. \citet{ThompsonScience1996} showed that the solar rotation profile can be inverted deep in the radiative zone, down to about 0.2 $R_\odot$, using the information carried by the rotational splitting of high-order low-degree $p$-modes.
More recently, and as shown in Fig. \ref{Fig:Sunprof}, \citet{Garcia2011} used rotational splitting of candidate $g$-modes to deduce the rotation profile of the Sun in its very core, below $0.2 R_\odot$.
The $p$-mode analysis shows that the Sun is rotating almost as a solid body in the radiative region down to about $0.2 R_\odot$, while candidate g-modes seem to indicate an increase of the angular velocity from $\simeq 430$ nHz to 2 $\mu$Hz in the innermost regions. This latest result is still debated, however. We here neglect a number of AM transport processes that are expected to modify the angular velocity profile in the radiative interior \citep{CharbonnelTalon2005Science,Eggenberger2005,Charbonnel2013}, so that we do not expect our models to fit the helioseismic profile.
When plotted against helioseismic data, all the models discussed previously are, as expected, far from observations in the radiative region. Indeed, even the model with the flatter rotation profile that we obtain at the age of the Sun still spins four times faster in the central region and does not reproduce the solid-body region between 0.2 R$_\odot$ and the bottom of the convective envelope
at all.
Internal gravity waves \citep{CharbonnelTalon2005Science} and magnetic fields \citep{Eggenberger2005} have shown very promising results in reproducing the solar rotation profile. However, these processes appear to act on very different timescales, and incorporating
them is beyond the scope of the present study, where we focus on hydrodynamical processes.
\begin{figure}
\includegraphics[width=0.45\textwidth]{FIGURES/Figure15.eps}
\caption{Comparison of our models with the rotation profile of the Sun obtained from helioseismology by \citet{Garcia2011}}
\label{Fig:Sunprof}
\end{figure}
\subsection{Constraint from lithium abundances}
Because lithium is prone to destruction by proton captures at temperatures $T \geq 2.5 \times 10^6$ K, its abundance at the stellar surface is a good proxy to evaluate the depth and efficiency of mixing processes that may connect the low-temperature convective envelopes of stars to deeper radiative regions where lithium is destroyed \citep[e.g.][]{Deliyannisetal00}.
Considering that meridional circulation and turbulent shear instability not only transport AM but also matter, it is therefore interesting to look at the predictions of our models concerning the evolution of the lithium surface abundance. It worth noting that in our self-consistent approach, the lithium depletion predicted by the models is a direct consequence of the evolution angular rotation profile driven by the extraction and internal transport of AM (i.e. we do not have any adjustable parameter to reduce the efficiency of the mixing to fit the Li data). \\ The results are shown in Fig.~\ref{Fig:Li}, where we compare the predictions of the models to the observed range of Li abundances determined for solar-type stars in different open clusters from 5 Myr to the age of the Sun as compiled by \citet{SestitoRandich2005}\footnote{To establish these ranges, we selected the stars in each cluster that have the temperature of a solar-type star of the same age according to stellar evolution models.}. As expected from previous works, our reference models (both the slow and the fast rotators) selected to best reproduce the angular velocity evolution of solar-type stars fail to simultaneously reproduce the observed lithium abundance evolution of solar-type stars on the main sequence (beyond 250 Myr (resp. 100 Myr) for the slow (resp. fast) rotators). This is due to the efficient transport of lithium by the shear turbulent instability in the radiative interior of our models. The lithium destruction occurs earlier in the fast rotators because of the correlation between the shear mixing efficiency and the angular velocity gradient, which is directly related to the torque. \citet{CharbonnelTalon2005Science} obtained similar results and showed that the introduction of internal gravity waves could reconcile the models predictions with observations by flattening out the angular velocity profile and thus reducing the efficiency of the turbulent shear mixing.\\
We point out the unexpected behaviour of model {\em 2s} that is computed using \citet{Maeder1997} prescription for the vertical turbulent shear diffusivity. In this case, the effective diffusion coefficient representing the transport of nuclides by the meridional circulation is the same as in our reference case {\em 1s} because we used the same prescription for the $D_h$ expression. However, $D_v$ from {\sf Maeder97} is significantly lower than that predicted by {\sf TZ97}, in particular in regions where the mean molecular weight gradients are large. As a result, the transport of nuclides is less efficient in model {\em 2s} and the surface abundance of lithium decreases much more slowly than in our reference model {\em 1s}.
In summary, none of the present models that account for the rotational behaviour of solar-type stars is able to simultaneously account for the lithium evolution and the helioseismic constraints, as already anticipated in previous works of our group.
\begin{figure*}
\includegraphics[width = 0.45\textwidth]{FIGURES/Figure16.eps}%
\includegraphics[width = 0.45\textwidth]{FIGURES/Figure16b.eps}
\caption{Evolution of the surface lithium abundances in the slow ({\em left}) and fast ({\em right}) rotating models as a function of time. The open boxes represent the ranges of lithium abundances determined by \citet{SestitoRandich2005} for the following clusters and associations: NGC2264 at 5 Myr, IC2391/IC2602/IC4665 at 30 Myr, Pleiades/Blanco1 at 80 Myr, NGC2516 at 150 Myr, M34/M35/NGC6475 at 250 Myr, Hyades/Praesepe/Coma Ber/NGC6633 at 600 Myr, NGC752/NGC3680/IC4651 at 2 Gyr and M67 at 4.5 Gyr. The Sun is also indicated with the solar symbol $\odot$.}
\label{Fig:Li}
\end{figure*}
\section{Summary and global picture}\label{sec:end}
Observations have shown that the AM of solar-mass stars is reduced by two orders of magnitude during the first few Myr of their interaction with their disc. After this phase, they need to decrease it by two more orders of magnitude to reach the solar angular momentum content. This loss of AM is thought to occur through stellar wind coupled with the large-scale stellar magnetic field. However, since young stars are not expected to rotate as solid bodies, the transport of angular momentum in the stellar radiative interiors should modify the evolution of their surface velocity at every age. Hydrodynamical processes transporting AM are driven by the forcing induced by the extraction of AM at the surface by stellar winds. Thus, depending on the wind-braking prescription used in the models, the efficiency of the transport will be different.
Different prescriptions for the AM extraction and transport processes exist in the literature, with different degrees of approximation.
We here showed that self-consistent 1 M$_\odot$ rotating evolutionary models that account for the transport of AM by meridional circulation and shear turbulence alone and that use a realistic AM wind-driven extraction can reproduce the evolution of the surface rotation of open clusters from the disc-coupling phase to the age of the Sun. Using different prescription sets for the horizontal and vertical diffusion coefficients essentially impacts the AM distribution inside the star.
The surface extraction of AM and the AM transport in the radiative region interact in a complex way, the first triggering the later and the later enforcing the first.\\
We find that the stellar wind prescriptions from \citet{Matt2012,Mattetal2015} provide a really good match with observations of rotation periods in open clusters when they are combined with the adapted AM transport, meaning here {\sf MPZ04} and {\sf TZ97} for the prescriptions of the horizontal diffusion and vertical diffusion coefficients, respectively. None of the other braking laws lead to as good results, independently of the chosen prescriptions for internal AM transport.
This led us to select the following combination for our reference model: ($D_v$; $D_h$; $dJ/dt$) $\equiv$ ({\sf TZ97, MPZ04, Matt15}). With this set of prescriptions, our models all reached the same internal rotation profile at the age of the Sun, regardless of the assumed initial angular momentum content.
Unlike in bi-zone models, the angular momentum content of the present Sun cannot be retrieved with our models. Even using the most favourable combination of the available prescriptions, meridional circulation and shear-induced turbulence are therefore
just efficient enough to reproduce the observed $P_{rot}$ data, but additional processes are still needed to reproduce asteroseismic and surface chemical abundances observations. Regardless of the set of AM transport prescriptions we used, we were unable to reproduce these observed data.
In the recent years, the asteroseismic data mostly gathered by the \textit{Kepler} mission for subgiant and red giant stars also revealed the need to go beyond the current description of rotational mixing to be able to account for the rotational properties (surface and core rotation) of evolved low-mass stars \citep{Cellieretal2012,VSP2013,Garciaetal2014,Deheuvelsetal2014,Deheuvelsetal2015}.
The dominant processes that shape the AM evolution may differ according to the evolutionary phase. However, observational evidence now exists, for stars ranging from the PMS to the red clump, pointing towards the need of additional AM transport mechanisms that would efficiently couple the core and the envelope of solar-type stars.
As predicted by \citet{MacGregor2000} and tested in main-sequence evolutionary models by \citet{CharbonnelTalon2005Science} and \citet{Eggenberger2005}, magnetic field and internal gravity waves are very good candidates to provide additional angular momentum transport and should be investigated in a further study.\\ It is also important to investigate whether our conclusions also apply to other low-mass stars in the mass range 0.5~\ensuremath{M_\odot}~ to 2~\ensuremath{M_\odot}~. The structural and rotational evolution is expected to differ depending on the initial mass in this mass range \citep[see e.g.][]{GB15}, which may have a significant effect on the efficiency of possible transport processes, as was discussed for instance by \citet{TalonCharbonnel2003}.
\begin{acknowledgements}
This study was supported by the grant ANR
2011 Blanc SIMI5-6 020 01 “Toupies: Towards understanding the spin
evolution of stars” (http:\/\/ipag.osug.fr\/Anr\_Toupies\/ ).
C. Charbonnel and F. Gallet acknowledge support from the European COST Action TD 1308 Origins and the SEFRI project C14.0049 of the Conf\'ed\'eration Suisse.
\end{acknowledgements}
|
2,877,628,089,042 | arxiv | \section{INTRODUCTION}
Robots need artificial perception to be able to plan for the future, learn from the past and make intelligent judgements in the present~\cite{Tani}, furthermore temporal concepts are needed for an agent to comprehend its environment and to successfully communicate with humans in a meaningful manner~\cite{Lingodroid}. Time perception covers several essential concepts which are needed for many tasks: duration of a task, perceived simultaneity of events with a small delta of time between them and ordering of events~\cite{Tani}. For example, to enable robots to switch tasks quickly between two (or more) different behaviours~\cite{Tani}, the robot needs some concept of different rules at different time states.
There has not been a great deal of work on artificial time perception and in studying human time perception we must turn to both neuroscience and philosophy. Robotics has drawn from these two areas, with a recent paper looking at the underlying structure of neural nets with regards to time-perception and rule switching plasticity~\cite{Tani}. Wittgenstein famously argued that a thought was impossible without a language containing that thought's concepts~\cite{Wittgenstein} and another recent paper looked at getting robots to develop their own language for time concepts~\cite{Lingodroid}, which interestingly involved errors due to individual robot's map not being entirely congruent with each other (a concept behind many human misunderstandings and conflict). In this paper, we will take the neuroscience view and consider the structure of an artificial `brain' that could understand temporal concepts.
In the brain, the time-perception tends to `ride along' with other mental processes: there is no part of the brain that is specifically evolved to deal with time perception~\cite{Tani} and different parts are associated with temporal aspects of behaviour on different timescales~\cite{TimePerception}. Instead, time seems to be perceived relative to internal neurological changes~\cite{Sumbre} as is evidenced by how it can be disrupted by disease~\cite{Alz}. The operation of time perception thus seems related to the network dynamics of the brain.
The memristor is the 4$^{\mathrm{th}}$ fundamental circuit element~\cite{Chua1971} which is essentially a resistor with memory, and which, in 1971, was predicted (based on both electromagnetic and circuit theory) to be a two-terminal circuit element with a constitutive relation that would relate magnetic flux to charge. Although the constitutive relation technically covers everything about a circuit element's operation, this theory offered little clue of how to build such a device. Thus, the memristor was only related to an actual device in 2008~\cite{Strukov}, even though memristors had been previously experimentally studied and commercially investigated under the moniker of ReRAM. There are two different theories that model experimental memristor's operation: the phenomenological model~\cite{Strukov}, which is based on a 1-D model of variable resistors and which has been the basis of more complex models (such as those which include non-linear drift~\cite{94} or window functions) and many simulations (such as~\cite{84}); and the memory-conservation model (see ~\cite{F0} for a full description or~\cite{NM} for a summary) which is based on the electrodynamics of a 3-D model of variable resistors and fits with the constitutive relation. Recent simulations have shown that memristors can be used as synapses with artificial spiking neurons~\cite{David,STDP1}, theoretical results have demonstrated that action potential transport in real neurons can be modelled using memristors~\cite{Chua2013} and finally recent work~\cite{ICNAAM} has highlighted the memristors native spiking ability, all of which suggest memristors could be the basis of synthetic neuron analogues for use in an artificial brain.
In this short paper we will summarise some recent relevant memristor results and discuss how memristor networks might provide a route to incorporating time perception into an artificial brain/intelligence in a bio-mimetic or even human-like manner.
\section{MEMRISTOR'S SHORT-TERM MEMORY}
Memristors are commonly thought to be a.c. components. The pinched hysteresis loop used to identify a memristor is usually plotted in $V-I$ space, however this description is not complete without inclusion of the aspect of time-dependence. In a.c. systems this is evidenced as the dependence of the Lissajous curve lobe size on the voltage waveform frequency (and is why memristor papers now tend to include a graph showing this). This time dependence is due to the memory property of the device responding slower than the frequency of the voltage change (see~\cite{F0} for a discussion of what this memory property might be) and this memory property must have some characteristic time scale, $\tau$, associated with it that relates to the fundamental frequency, $\omega_0$ at which the a.c. voltage input produces the maximum hysteresis in the memristor $I-V$ curve.
In steady voltage circuits (i.e. d.c. voltage input), we suggest that the time dependence is the commonly-observed current transients as seen when the voltage changes or is switched on or off, $\Delta V$. Figure~\ref{fig:ExampleSpike} taken from~\cite{ICNAAMJournal} shows an example current spike response to a step voltage. The characteristic timescale is related to the time taken for the spike to decay. As $\Delta V \rightarrow \delta V$ and we go from d.c. steps to an a.c. smooth curve with a set frequency, we can see that this characteristic timescale response is related to the hysteretic lag.
\begin{figure}[thpb]
\centering
\includegraphics[scale=0.4]{ExampleSpike.png}
\caption{An example of a typical $I-t$ spike profile as taken from~\cite{ICNAAMJournal}. The characteristic timescale, $\tau$, is around 3.5-4s, marked on are the timescales for decay to a percentage of peak height: red line $\tau_{50}$ (decay to 50\%), orange line $\tau_{90}$, green $\tau_{95}$ and grey $\tau_{99}$.}
\label{fig:ExampleSpike}
\end{figure}
Whilst the memristor is responding to a voltage change, it is in a different state to a memristor that has responded: this is a form of memory. Specifically, it is a short term memory rather than a long-term memory or stateful response, this short-term memory could be used as a form of working memory. If a second spike is input into the system, it reacts differently as a result of the previous spike if and only if the second spike is happens within time $\tau$.
The memory-conservation model of memristance~\cite{F0} introduces the concept of a second charge carrier, the ionic charge carrier, in addition to the electrons. This ionic charge carrier has a different mobility, speed and inertia to electrons and therefore takes longer to respond to voltage changes. Thus, in~\cite{F0} it is claimed that the lag which causes the memristor hysteresis under a.c. is due to the slower response time of the ionic charge carriers, and this is investigated in a forthcoming paper. The slow ionic charge response is apparent in the d.c. response as the decay of the $I-t$ curve. The characteristic timescale is a measure of the ionic charge carrier's slower response to a voltage change (which we expect will be related to it's ionic mobility) and it measures when the ionic effect is negligible. Therefore, if a second spike is received within time $\tau$, the ionic charge carrier hasn't recovered and responds at a different level than would be otherwise expected.
\section{NETWORKS OF MEMRISTORS}
\subsection{SIMULATION}
What could such a short term memory be used for? A recent simulation based on the experimental observations outlined earlier showed that networks of memristors can learn, change and adapt~\cite{Mu0}. Memristor networks were designed and simulated with the purpose of composing and performing music. Each network node was a note (in terms of function) and a source drain or sink (in terms of modelled component), each connection a transition from one note to the other (function) and a pair of antiparallel memristors (component). The normalised conduction profile of a memristor under d.c. voltage as modelled using the memory-conservation model~\cite{F0} was descretized and used as a look-up table of how connection weight changed each time a connection was used.
The network was capable of being seeded to produce music similar to the seed genre. Crucially, the memristor network continued to change and adapt as it was used. This shows a similar plasticity to the brain and a very simple type of Hebbian learning. This is different to evolutionary techniques because the learning is a result of using the networks rather than being directed by similarity to a fitness function or desired output.
Although this experiment serves to demonstrate how a composing memristor machine could be built, there are similarities in the structure to the brain, which is also a learning network. It has been suggested that building a creative computer could require the almost accidental building of a brain-like computer.
\subsection{EXPERIMENT}
\begin{figure}[thpb]
\centering
\includegraphics[scale=0.4]{ThreeMemsCircuit8.png}
\caption{The three memristor circuit used to generate the output in figure~\ref{fig:Brainwave}}
\label{fig:ThreeMems}
\end{figure}
\begin{figure}[thpb]
\centering
\includegraphics[scale=0.4]{101012MT1p1.png}
\caption{Brainwave like oscillations and spike trains that emerged from the circuit in figure~\ref{fig:ThreeMems}, as taken from~\cite{Mu0}. Utilising these dynamics could provide a route build a neuromorphic control computer for a robot.} \label{fig:Brainwave}
\end{figure}
Experimental data of highly simplified TiO$_2$ sol-gel memristor networks show some intriguing similarities to the dynamics of the brain. A simple `network' of just three memristors (arranged as in figure~\ref{fig:ThreeMems}) put under a constant positive d.c. voltage shows a current response similar to that shown in figure~\ref{fig:Brainwave}. Here we see sudden and large spiking responses against a background of an emergent oscillation, this is interesting as it even includes the measurement of negative currents under a positive driving voltage. These oscillations may arise due to the interaction within the network of the spiking components. As the brain also consists of spiking components and shows emergent mass synchronisation across certain frequencies (i.e. brainwaves) this result could show that we are on the right track in attempting to make neuromorphic (brain-like) computers with memristor networks. As complex behaviour (and learning) is seen in other networks of individually `simple' components, such as \textit{Physarum polycephalum}, a eukaryotic mould that can perform simple learning via interaction of its many nuclei, it suggests that the network structure may be of more importance than the precise components or measurables.
Regardless of where these brain-wave-like dynamics emerge from, they have a use regarding time-perception. These oscillations can be used as an internal clocking signal, in fact, there is some evidence that this may be part of what synchronised spiking responses might be used for within the mammalian brain~\cite{Sumbre}.
\section{DESIGN IMPLICATIONS FOR ROBOTIC CONTROL SYSTEMS}
Let us imagine a robot `brain' built from a complex memristor network in order to discuss how order and task switching might be encoded. Consider a standard test of asking a robot to navigate a T-maze under two different reward states, namely: I. reaching the left hand top of the `T' cross-bar; II. reaching the right-hand top of the `T' cross-bar (see~\cite{David,Tani} as relevant examples). We shall assume that after training different groups of memristors spike in different ways for the two solutions (as was seen with artificial neural networks in~\cite{Tani}). If the robot was operating under the rule turn left, we could see that continual brain-wave activity across the network could be used to keep the memristors in the correct short-term memory state for the robot to respond to external stimuli by turning left. Should we switch the reward state, we would expect the spike patterns to change (as seen in~\cite{Tani}), causing the memristor network to switch to the other behaviour. If pre-trained and plastic with distributed activity (i.e. the cause of the brainwaves) we can see that this function would allow the whole network to be switched due to the oscillations across it rather than waiting for each memristor to switch in turn as the robot processes the rule change and this may cause the robot to switch rules faster. This mechanism also allows the robot brain to have greater plasticity, as previous mechanisms can be co-opted to encode different responses applicable to situations outside those the robot has been trained for. Thus, an artificial brain based on memristor networks may offer a time-perception functionality due to the plasticity, learning and synchronisation properties of the network.
\section*{ACKNOWLEDGMENT}
E.G. thanks David Howard, Ioannis Georgialas and Oliver Matthews for helpful discussions.
|
2,877,628,089,043 | arxiv | \section{\label{sec:level1}First-level heading}
\paragraph{Introduction.}
Finding the ground states of a physical system is hard. This is true not only in a practical sense but also in a formal, computational, sense. A prototypical example is that of a classical spin-glass system for which the problem of finding the ground states is $\NP$-hard \cite{Barahona_1982}. This implies there is no efficient algorithm for finding ground states, assuming standard conjectures in computational complexity theory. A number of deep connections between the theory of computational complexity and statistical mechanics systems have been pointed out in \cite{Kirkpatrick671,Fu_1986,NP_PhaseTransitions,aaronson2005npcomplete,CubittUndec}.
Supersymmetry is a symmetry relating bosonic and fermionic states of a system. Supersymmetric systems are often more amenable to analysis and various analytic and exact results are possible. Indeed, the ground states--or some of their properties--can be found analytically in various nontrivial supersymmetric systems. This raises the question of the computational complexity associated to the ground states of supersymmetric systems. Although many exact results are known in {\it specific} supersymmetric systems, we show that the ground state problem for supersymmetric theories remains computationally hard.
Supersymmetry was first proposed as a possible symmetry of relativistic quantum field theory. However, its applications extend to a number of areas in mathematics including, most famously, Morse theory, mirror symmetry, and generalized complex geometry.
In this letter, we focus on {\it supersymmetric quantum mechanics} \cite{Nicolai:1976xp,Witten:1981nf,Witten:1982im,Witten:1982df} and bring the attention to the interplay between supersymmetry and the theory of quantum computation. As we discuss, a natural setting for incorporating supersymmetry in quantum computation is the fermionic model of quantum computation \cite{2002AnPhy.298..210B,Ortiz:2000gc}. Defining supersymmetry operators in the fermionic model, one can then map these to qubit space via a standard spin-$\tfrac12$ Jordan-Wigner transformation or its generalizations. Having defined the action of supersymmetry in the space of qubits, we then define the notion of \emph{supersymmetric quantum circuits}, and use these to design quantum algorithms associated to certain supersymmetric physical observables.
The intuition from the physics of supersymmetric systems is that this should define a nontrivial subclass of quantum algorithms, with advantageous properties (e.g., invariance under certain deformations) over non-supersymmetric ones, and which capture highly non-trivial problems of both physical as well as mathematical interest. As we discuss, this intuition bears out.
We emphasize that this definition of supersymmetry holds for any system of qubits and does not require supersymmetry to be realized at a fundamental level in nature. In particular, any implementation of a quantum computer can be made supersymmetric in this sense \footnote{It is tempting to call such a device a ``supercomputer'' but this would admittedly lead to many confusions.}. \\
\paragraph{${\cal N}=2$ quantum mechanics. }
The Hilbert space of any quantum mechanical theory can be decomposed as ${\cal H}={\cal H}^{B}\oplus {\cal H}^{F}$ where each factor refers to the subspace of bosonic and fermionic states. These are distinguished by the operator $(-1)^{F}$, acting as $+1$ on bosonic states and as $-1$ on fermionic states. By definition, in a theory with ${\cal N}=2$ supersymmetry there exists a complex Grassmann operator ${\cal Q}$, sending states in ${\cal H}^{B}$ into states in ${\cal H}^{F}$ and vice versa, and satisfying the algebra \cite{Nicolai:1976xp,Witten:1981nf,Witten:1982im,Witten:1982df},
\equ{\label{algebra}
H=\{{\cal Q},{\cal Q}^{\dagger}\}\,,\quad {\cal Q}^{2}=({\cal Q}^{\dagger})^{2}=0\,,
}
where $H$ is the Hamiltonian of the system and $\{(-1)^{F},{\cal Q}\}=0$. One says an operator is bosonic or fermionic if it commutes or anticommutes with $(-1)^{F}$, respectively. The supercharge is thus a fermionic operator and the Hamiltonian bosonic. It follows directly from \eqref{algebra} that the spectrum of supersymmetric systems is positive semidefinite, $E\geq 0$, and that a state $\ket{\Omega}$ has $E=0$ iff ${\cal Q}\ket{\Omega}={\cal Q}^{\dagger}\ket{\Omega}=0$. All such states, which may be bosonic or fermionic, are called supersymmetric ground states. A crucial property of excited states is that they are paired: for every bosonic state with energy $E>0$ there is a corresponding fermionic state with the same energy. This is not necessarily the case for supersymmetric ground states. In fact, a quantity of particular interest is the Witten index, defined as the difference in the number of bosonic and fermionic supersymmetric ground states \cite{Witten:1982df}:
\equ{\label{WI}
{\cal I}\equiv n_{E=0}^{B}-n_{E=0}^{F}=\text{Tr}_{{\cal H}}\, (-1)^{F}\,,
}
where in the last equality one uses the fact that excited states are paired and thus do not contribute to the trace. The Witten index gives a lower bound on the total number of supersymmetric ground states via the inequality $ n_{E=0}^{B}+n_{E=0}^{F}\geq \abs{{\cal I}}$. In particular, if ${\cal I}\neq 0$ the system must have supersymmetric ground states.
Since ${\cal Q}^{2}=0$, supersymmetry defines the $\Bbb Z_{2}$-graded complex of vector spaces,
\equ{
C:\quad {\cal H}^{F} \xrightarrow{{\cal Q}}{\cal H}^{B}\xrightarrow{{\cal Q}}{\cal H}^{F} \xrightarrow{{\cal Q}}{\cal H}^{B} \,,
}
and the Euler characteristic of $C$ coincides with the Witten index. It is this topological nature of the Witten index that makes it a robust quantity and, in some situations, easily calculable.
In general, the difficulties in computing general physical observables in supersymmetric theories can be as formidable as in non-supersymmetric systems. However, there are a subset of physical observables, ``supersymmetric observables,'' which have special properties and can often be computed exactly, the Witten index being an example. These relate to an important set of operators called supersymmetric, or ${\cal Q}$-closed, operators. A bosonic operator ${\cal O}$ is ${\cal Q}$-closed if
\equ{
[{\cal Q},{\cal O}]=0\,.
}
Among these, ${\cal Q}$-exact operators are defined as those which can be written as ${\cal E}=\{{\cal Q},\Psi'\}$, for some fermionic $\Psi'$. By nilpotency, all ${\cal Q}$-exact operators are ${\cal Q}$-closed. Not all ${\cal Q}$-closed operators, however, are necessarily ${\cal Q}$-exact; whether or not this is the case is determined by the ${\cal Q}$-cohomology of operators. Two supersymmetric operators ${\cal O}$ and ${\cal O}'$ are said to be in the same cohomology class if ${\cal O}'={\cal O}+{\cal E}$. The analogous definitions hold for fermionic ${\cal Q}$-closed and exact operators, exchanging commutators and anticommutators.
An important set of physical observables is given by the correlation function of supersymmetric operators in a supersymmetric ground state:
\equ{\label{corr}
\langle {\cal O}_{1}\cdots{\cal O}_{n} \rangle_{\Omega}\equiv \bra{\Omega} {\cal O}_{1}(t_{1})\cdots{\cal O}_{n}(t_{n})\ket{\Omega}\,,
}
where the $t_{i}$ are insertion points in Lorentzian time $t\in \Bbb R$, and can be expressed by a standard path integral.
Another set of observables is given by a refined or generalized Witten index \eqref{WI}, obtained by the insertion of supersymmetric operators into the trace:
\equ{\label{indexdef}
Z_{\text{P}}[{\cal O}_{1}\,,\cdots \,,{\cal O}_{n}]\equiv \text{Tr}_{{\cal H}}\, \left[(-1)^{F}{\cal O}_{1}(\tau_{1})\cdots {\cal O}_{n}(\tau_{n})\right]\,.
}
This can be thought of as the insertion of operators in the Euclidean path integral of the theory, with periodic boundary conditions for fermions along a compactified Euclidean time direction $\tau=i t$. An important property of the observables \eqref{corr} and \eqref{indexdef} is that they are invariant under exact deformations,
\equ{\label{exactdef}
{\cal O}_{k}\to {\cal O}_{k}+{\cal E}_{k}\,,
}
as can be easily checked. For the former, this follows from properties of the supersymmetric ground state and, for the latter, from cyclicity of the trace. Thus, these observables are sensitive only to the cohomology class of supersymmetric operators. These robustness properties will be relevant to our discussion of quantum computation below.
We will focus on systems with a finite-dimensional Hilbert space ${\cal H}$, given by a certain subspace of $(\Bbb C^{2})^{\otimes N}$, as ${\cal H}=\{\ket{s}\in \{0,1\}^{N}\, | \, {\cal P} \ket{s} = \ket{s}\}$, where ${\cal P}$ is a projector which needs to be specified. For compatibility with supersymmetry, we require $[{\cal Q},{\cal P}]=0$. Note that for systems with ${\cal P}=\Bbb I$ the Witten index vanishes, but the number of ground states or the generalized Witten may not. \\
\paragraph{Computational complexity of supersymmetric systems.}
Let us briefly review relevant concepts of complexity theory (see, e.g., \cite{sipser13}). The complexity class $\P$ is the class of decision problems (with a ``yes/no'' answer) which can be solved by a deterministic Turing machine in polynomial time. The class $\NP$ is the class of decision problems for which the problem instances which give ``yes'' can be \emph{checked} in polynomial time. The complexity class $\#\P$ is the set of \emph{counting} problems associated to decision problems in $\NP$. For example, whereas the problem of deciding if a boolean formula has a satisfying instance is a problem in $\NP$, the problem of counting how many satisfying instances it has is a problem in $\#\P$. A problem {\sc H} is said to be $\#\P$-hard if it is at least as hard as any problem in $\#\P$ or, more precisely, if any problem in $\#\P$ can be reduced to {\sc H} in polynomial time. A problem is said to be $\#\P$-complete if it is $\#\P$-hard and belongs to the class $\#\P$.
The Witten index \eqref{WI} can sometimes be computed exactly and with little dynamical information. In particular, supersymmetry ensures that the contribution of all excited states in the trace in \eqref{WI} cancel out and thus the index can be computed with no knowledge of the supersymmetric ground states themselves. Furthermore, under certain conditions the index is invariant under small, supersymmetric, deformations of the system \cite{Witten:1982df} which can sometimes be exploited to bring the system to a weakly coupled point, where the Witten index can be efficiently computed in perturbation theory. Although this is often the case in \emph{specific} supersymmetric systems, we show next that there can be no efficient algorithm for computing the Witten index for generic supersymmetric systems. This follows from the following result:
\begin{thm}\label{thmWI}
Given a quantum mechanical system with a finite-dimensional Hilbert space and ${\cal N}\leq 2$ supersymmetry, specified by a projector ${\cal P}$ and a representation of the supercharge ${\cal Q}$, the problem of computing the Witten index is $\#\P$-complete.
\end{thm}
\begin{proof}
To show that the problem is $\#\P$-hard, it is enough to show that it is hard in a specific instance. We then consider the statistical models of hard-core fermions on a graph $G$ with ${\cal N}=2$ supersymmetry of \cite{Fendley:2002sg}. In these models the projector ${\cal P}$ and supercharge ${\cal Q}$ are uniquely specified by, and computed efficiently from, $G$. The Hilbert space is in one-to-one correspondence with the collection of independent sets of $G$ and the Witten index equals the Euler characteristic of the independence complex of $G$ (see \cite{huijse2009supersymmetry,Huijse:2011aa} and references therein). On the other hand, it was shown in \cite{ROUNE2013170}
that, given a graph $G$ as input, computing the Euler characteristic of the independence complex is $\#\P$-hard. To show the problem is in $\#\P$ one can adapt the arguments in \cite{ROUNE2013170} to the current setting.
\end{proof}
Thus, assuming standard computational complexity conjectures, no efficient algorithm for the Witten index is expected to exist. As a corollary, no efficient algorithm for finding or counting all bosonic and fermionic ground states of ${\cal N}\leq 2$ systems is expected to exist either; if it did exist the Witten index could then be computed with this information in polynomial time. It also follows that the problem of computing the more general \eqref{indexdef} is $\#\P$-hard.
Having discussed a consequence of the theory of computation in supersymmetric systems we now discuss some consequences of supersymmetric systems in the theory of quantum computation. \\
\paragraph{Supersymmetry in qubit space.} The Hilbert space of $N$ qubits is spanned by the set of $N$-bit strings. It comes with a natural $\Bbb Z_{2}$-grading, ${\cal H}={\cal H}^{+}\oplus{\cal H}^{-}$, where ${\cal H}^{+}$ and ${\cal H}^{-}$ are spanned by the set of $N$-bit strings with an even and odd number of 1's, respectively. The basic observation we make here is that these can be consistently identified with ``bosonic'' and ``fermionic'' subspaces, with $+1$ and $-1$ parity under $(-1)^{F}=\otimes_{i=1}^{N} \sigma_{z}^{i}$, respectively. Then, we define ${\cal N}=2$ supersymmetry in the space of qubits as a nilpotent map ${\cal Q}$ sending ${\cal H}^{+}$ into ${\cal H}^{-}$ and vice versa.
One way to construct such a map is to recall that the Hilbert space of $N$ qubits is isomorphic to the Hilbert space of $N$ (spinless) fermions, which is exploited in the fermionic model of quantum computation \cite{2002AnPhy.298..210B,Ortiz:2000gc}. In this model one considers $N$ vertices of a graph $G$, each of which can be occupied by 0 or 1 spinless fermions. A fermion at vertex $i$ is created by an operator $a_{i}^{\dagger}$ and annihilated by $a_{i}$, which satisfy the standard anticommutation relations
\equ{
\{a_i,a_j^\dagger\}= \delta_{ij}\,,\qquad i,j\in\{1,\ldots,N\}\,,
}
with all other anticommutators vanishing. The $2^{N}$-dimensional Fock space is constructed by acting with creation operators on the vacuum state with no fermions, $\ket{0\cdots0}_{f}$, and is in one-to-one correspondence with the space of $N$ qubits:
\equ{
\ket{n_{1}\cdots n_{N}}_{q}\leftrightarrow (a_{1}^{\dagger})^{n_{1}}\cdots (a_{N}^{\dagger})^{n_{N}}\ket{0\cdots0}_{f}\,.
}
Operators in qubit space are obtained from operators in Fock space via a spin-$\tfrac12$ Jordan-Wigner transformation,
\equ{\label{asigma}
a_i \rightarrow K_{i}\, \sigma_+^i \,, \quad a_i^\dagger \rightarrow \sigma_-^i K_{i}^{\dagger} \,,
}
where $\sigma_{\pm}=\tfrac12(\sigma_{x}\pm i \sigma_{y})$ and the $K_{i}$ are non-local operators, which depend on the graph. In the case of a 1d graph $K_{i}=\prod_{j=1}^{i-1}(-1)^{n_{i}}$ (see, e.g., \cite{Ortiz:2000gc} and references therein for generalizations). The first example of a system of spinless fermions on a graph with ${\cal N}=2$ supersymmetry was constructed by Nicolai \cite{Nicolai:1976xp}, in which the supercharges are cubic functions of the creation and anihilation operators. As a generalization, we consider the ansatz,
\equ{\label{ansatzQ}
{\cal Q} = \sum_{i} a_{i}^{\dagger} \, B_{i}(a,a^{\dagger})\,, \quad {\cal Q}^{\dagger} = \sum_{i} B^{\dagger }_{i}(a,a^{\dagger})\, a_{i}\,,
}
where the $B_{i}$ are a set of bosonic operators built out of the creation/annihilation operators. Nilpotency constrains the choice of $B_{i}$ by requiring
\equ{\label{nilpQ}
{\cal Q}^{2}= \sum_{i,j} a_{i}^{\dagger} B_{i}(a,a^{\dagger})\, a_{j}^{\dagger} B_{j}(a,a^{\dagger})=0\,.
}
Different solutions to this constraint amount to different realizations of supersymmetry on a statistical mechanical system of $N$ fermions. We discuss some examples below. For now, we keep the discussion general and do not specify the choice of supercharges. Now, applying \eqref{asigma} we have
\equ{\label{QQbits}
{\cal Q} =\sum_i \sigma_-^i K_{i}^{\dagger}\,B_{i}\,, \quad {\cal Q}^\dagger =\sum_i B_i^\dagger\, K_{i}\sigma_+^i \,,
}
where $B_{i}=B_{i}(K\sigma_{+},\sigma_{-}K)$. This defines the action of ${\cal N}=2$ supersymmetry on the space of qubits.
Note that the supersymmetry operators \eqref{QQbits} could have been defined directly in the space of qubits, with no reference to the fermionic model, though this would be rather unnatural. Thus, although the fermionic system does not play a fundamental role, it provides a setting where supersymmetry--and the associated computational problems--are naturally defined. \\
\paragraph{Supersymmetric circuits.}
Let us call a circuit $U$ bosonic if it preserves the parity of the state it acts on, $[(-1)^{F},U]=0$, and fermionic if it flips it, $\{(-1)^{F},U\}=0$. The notions of closedness and exactness in Fock space translate directly into the corresponding notions in qubit space. We define a bosonic \emph{supersymmetric quantum circuit} as a bosonic quantum circuit $U_{S}$, which is closed with respect to the supercharge ${\cal Q}$, i.e.,
\equ{\label{QU}
[{\cal Q},U_{S}]=0\,.
}
An obvious example of a supersymmetric circuit is time evolution by the supersymmetric Hamiltonian, as $U_{S}=e^{- i t \{{\cal Q},{\cal Q}^{\dagger}\}}$ commutes with ${\cal Q}$ (and also with ${\cal Q}^{\dagger}$ in this special case). A fermionic supersymmetric circuit is similarly a fermionic circuit satisfying $\{{\cal Q},U_{S}\}=0$. Note the composition of supersymmetric circuits by matrix multiplication is supersymmetric. In fact, it is straightforward to see that, for a given ${\cal Q}$, the collection of supersymmetric circuits form a group. Similarly, we call a qubit state ${\cal Q}$-closed if ${\cal Q}\ket{s}_{q}=0$ and a supersymmetric ground state if it is closed with respect to both supercharges, ${\cal Q}\ket{\Omega}_{q}={\cal Q}^{\dagger}\ket{\Omega}_{q}=0$.
From now on we drop the subscript $q$, with the understanding that all states refer to qubit states.
Thus, for a given supercharge ${\cal Q}$, Eq.~\eqref{QU} imposes a constraint on the class of quantum circuits we consider. The next question is which states we allow as inputs into supersymmetric circuits. Here we let the supersymmetric observables reviewed above guide us, which suggest two natural ``modes'' of computation. \\
\paragraph{The supersymmetric Hadamard test.}
\begin{figure}
\centering
\begin{quantikz}
\lstick{$\ket{0}$} & \gate{H} & \ctrl{1} & \gate{H}& \meter{}\\
\lstick{$\ket{\Omega}$}& \qw \qwbundle{}& \gate{U_{S}+{\color{blue!60}{\cal E}}}& \qw \qwbundle{} &
\end{quantikz}
\caption{The Hadamard test for a supersymmetric circuit. If the input states are supersymmetric ground states $\ket{\Omega}$, the deformation ${\cal E}$ does not affect the outcomes of the ancilla qubit provided ${\cal E}= \{{\cal Q},\Psi\}$. }
\label{fig:Had_Circuit}
\end{figure}
The first mode of computation, suggested by the correlation function \eqref{corr}, is to take as input supersymmetric ground states and calculate matrix elements of $U_{S}$. The standard technique for estimating matrix elements is the Hadamard test \cite{aharonov2005polynomial} which, given a state $\ket{s}$ and a unitary matrix $U$, estimates $\bra{s}U\ket{s}$. The procedure consists of adding an ancilla qubit which is initialized to $\ket{0}$, then Hadamarded to $\ket{+}$, and used as a control qubit for unitary evolution of the state $\ket{s}$ by the circuit $U$. Finally, the ancilla qubit is Hadamarded one more time and measured in the computational basis. Applying this to a supersymmetric circuit in a supersymmetric ground state, the probability of measuring $\ket{0}$ in the ancilla qubit is
\equ{
\label{p0Had}
p(0) = \frac{1}{2}\(1+\Re \bra{\Omega}U_{S}\ket{\Omega}\)\,.
}
Repeated measurements of the ancilla qubit then leads to an estimate of $\Re \bra{\Omega}U_{S}\ket{\Omega}$. (The imaginary part is obtained by changing the initial state of the ancilla.) What is particular about the supersymmetric setting is that the robustness property of supersymmetric systems translates directly into a corresponding property of the quantum circuit. Namely, the outcome probability \eqref{p0Had} of the ancilla qubit is unaffected by a deformation of the circuit of the form
\equ{\label{exactdefqubit}
U_{S}\to U_{S}+{\cal E}\,,
}
with ${\cal E}=\{{\cal Q},\Psi\}$ for some $\Psi$. This is shown schematically in Fig.~\ref{fig:Had_Circuit}. More generally, for $U_{S}=U_{S}^{(1)}\cdots U_{S}^{(n)}$ a product of supersymmetric circuits the outcome probability is unaffected by an exact deformation of any of the circuits, $U_{S}^{(k)}\to U_{S}^{(k)}+{\cal E}^{(k)}$. That is, such deformations do not propagate, as long as each circuit is supersymmetric. This is the first example of ``robustness'' of supersymmetric quantum circuits. If one is interested in computing correlation functions of operators as in \eqref{corr}, not necessarily unitary, one can apply the approach of \cite{Ortiz:2000gc} to expand the operators as a linear combination of unitaries and apply the algorithm above to each term.
To run this algorithm the supersymmetric ground states $\ket{\Omega}$ must be either known in advance or be prepared. In principle, supersymmetric ground states can be prepared by existing methods, applied to the supersymmetric Hamiltonian $H=\{{\cal Q},{\cal Q}^{\dagger}\}$. However, as discussed above the problem of finding supersymmetric ground states is computationally hard and thus we do not expect that this can always be done efficiently, even with a quantum computer. In fact, one of the physically important questions is to determine whether a given system has supersymmetric ground states to begin with. Thus, rather than limiting ourselves to inputing supersymmetric ground states, we would like to design a supersymmetric circuit that gives us information on possible supersymmetric ground states. \\
\paragraph{The generalized Witten index algorithm.} The second mode of computation, suggested by the refined Witten index \eqref{indexdef}, is to allow for mixed states and take this to be the maximally mixed state, $\rho_{{\cal H}}=\mathbb{I}/\dim {\cal H}$. This mode of computation has the advantage that there is no need to prepare supersymmetric ground states.
\begin{figure}
\centering
\begin{quantikz}
\lstick{$\ket{0}$} & \gate{H} &\ctrl{1} & \gate{H}& \meter{}\\
\lstick{$\rho_{{\cal H}}$}& \qw \qwbundle{} & \gate{(-1)^{F}(U_{S}+{\color{blue!60}{\cal E}})} & \qw \qwbundle{} &
\end{quantikz}
\caption{The trace estimation algorithm for the generalized Witten index $\text{Tr}\left[(-1)^{F}U_{S}\right]$. The average outcome of the ancilla qubit is insensitive to exact deformations ${\cal E}$. }
\label{fig:WittenAlgoDef}
\end{figure}
The standard method for calculating the trace of a unitary matrix acting on $M$ qubits is trace estimation \cite{PhysRevLett.81.5672}. This is basically the same as the Hadamard test above except that, rather than inputing a pure state $\ket{s}$, the input is the maximally mixed state, $\rho_{{\cal H}}=\frac{\mathbb{I}}{2^{M}}$. This is known as the one-clean-qubit model of quantum computation \cite{PhysRevLett.81.5672}. One then applies a controlled unitary evolution of the density matrix, $\rho_{{\cal H}}\to U \rho_{{\cal H}}U^{\dagger}$, and measures the ``clean'' ancilla qubit. Applying this to a circuit of the form $U=(-1)^{F}U_{S}$, the probability of measuring $\ket{0}$ in the ancilla qubit is given by
\equ{\label{p0}
p(0)= \frac12\(1+\frac{1}{2^{M}} \Re \text{Tr}_{{\cal H}} \left[ (-1)^{F} U_{S}\right]\)\,,
}
and repeated measurement of the ancilla qubit approximates the generalized Witten index. Once again, we note that the outcome probability \eqref{p0} is unaffected by the exact deformation \eqref{exactdefqubit} (see Fig.~\ref{fig:WittenAlgoDef}). More generally, for $U_{S}$ a product of supersymmetric circuits the outcomes are unaffected by an exact deformation of each circuit. Note that since this robustness property relies crucially on the cyclicity of the trace, there is an important distinction relative to the robustness property of \eqref{p0Had}. Inputing the maximally mixed state $\rho_{{\cal H}}$ amounts to taking an average over measurements on pure states, which are drawn uniformly from the full Hilbert space. For each such measurement, the outcome probability of the ancilla qubit {\it is} affected by the deformation; it is only when one takes the average that the deformation averages to zero (assuming ${\cal E}$ does not significantly change in the process). If one is interested in $\text{Tr}_{{\cal H}}\,[(-1)^{F}{\cal O}]$ with ${\cal O}$ not necessarily unitary one may expand ${\cal O}$ as sum of unitary supersymmetric matrices. \\
Another interesting property arises when the circuit is closed with respect to both supercharges. Consider the supersymmetric circuit $U_{S}=\hat U_{S} U_{{\cal E}}$, with $[{\cal Q},\hat U_{S}]=[{\cal Q}^{\dagger},\hat U_{S}]=0$ and $U_{{\cal E}}\equiv e^{i \text{Re}\, {\cal E}}$. Then, one can check that
\equ{
\text{Tr}_{{\cal H}} \left[(-1)^{F} \hat U_{S}\, U_{{\cal E}}\right] = \text{Tr}_{{\cal H}} \left[ (-1)^{F} \hat U_{S}\right]\,.
}
Thus, circuits of the form of $U_{{\cal E}}$ can be completely removed from such quantum algorithms (see Fig.~\ref{fig:WittenAlgoExactCirc}). We emphasize that this is the case only in this mode of computation, with the maximally mixed state as an input, as the property above relies crucially on cyclicity of the trace.
\begin{figure}
\centering
\begin{quantikz}
\lstick{$\ket{0}$} & \gate{H} & \ctrl{1} & \ctrl{1} & \gate{H} & \meter{}\\
\lstick{$\rho_{{\cal H}}$} & \qw \qwbundle{} & \gate{(-1)^{F}\hat U_{S}} &\gate[style={fill=blue!30}]{\parbox[c][0.52cm]{1cm}{\centering $e^{i \Re {\cal E}}$}}& \qw \qwbundle{} &
\end{quantikz}
\caption{ The unitary circuit $e^{i\Re {\cal E}}$ can be dropped from the algorithm, without affecting the measured outcomes, provided the input state $\rho_{{\cal H}}$ is the maximally mixed state and $[{\cal Q},\hat U_{S}]=[{\cal Q}^{\dagger},\hat U_{S}]=0$. }
\label{fig:WittenAlgoExactCirc}
\end{figure}
It is important to note that the algorithm just described provides only an approximation to the (generalized) Witten index and \emph{not} an exact result which, as implied by Theorem~\ref{thmWI}, is intractable even for a quantum computer. Indeed, for the trace estimation algorithm to provide an ${\cal O}(1)$ approximation to the trace, an exponential number of measurements is required, which is still the case for the Witten index. In addition, the preparation of $\rho_{{\cal H}}$ can be a challenging computational problem. We discuss this issue in the hard-core model and provide a resolution in the Appendix. \\
\paragraph{Examples.} A number of supersymmetric models of fermions on a graph have been constructed, starting with \cite{Nicolai:1976xp}. A generalization, which is relevant to the study of holography and black holes, is the supersymmetric version of the Sachdev-Ye-Kitaev (SYK) model \cite{Fu:2016vas}, corresponding to ${\cal P}=\Bbb I$ and
\equ{
{\cal Q}=i\sum_{i_{1}<i_{2}<\ldots<i_{q}} C_{i_{1}i_{2}\ldots i_{q}} \, a^{\dagger}_{i_{1}}a^{\dagger}_{i_{2}}\ldots a^{\dagger}_{i_{q}}\,,
}
where each index takes values $i_{n}\in\{1,\ldots,N\}$ and $C_{i_{1}i_{2}\ldots i_{q}}$ is a totally antisymmetric tensor. A refined Witten index can be computed exactly in this model, giving
\equ{
\text{Tr}\, \left[ (-1)^{F} g^{r}\right] = e^{i N\pi (\frac{r}{q}-\frac12)}\left[2 \sin \frac{\pi r}{q}\right]^{N}\,,
}
where $g=e^{i\pi n/q}$, $n\in \Bbb N$, is a generator of a discrete $Z_{q}$ symmetry of the system which commutes with ${\cal Q}$ and ${\cal Q}^{\dagger}$, and $r\in \Bbb R$ is a chemical potential for this symmetry, and the trace is over all $N$-bit strings. This quantity is computed by setting $\hat U_{S}=g^{r}$ in the algorithm above. An interesting question is whether this quantum computational perspective on SYK and its refined Witten index could lead to further insights into the physics of supersymmetric black holes or AdS$_{2}$ holography (see \cite{Sarosi_2018,Rosenhaus:2018dtp} for reviews). Although the exact answer is known, and thus there is no need for an algorithm, this result may serve to benchmark the performance of quantum computers, perhaps along the lines of similar tests based on the Jones polynomial on IBM Q devices \cite{gkta2019benchmarking}.
A second rich class of examples is provided by the fermionic hard-core models of \cite{Fendley:2002sg}. As already mentioned, in this model the Hilbert space is in one-to-one correspondence with independent sets of $G$ and the Witten index equals the Euler characteristic of the independence complex of $G$. The Witten index is known in the case of a 1d graph \cite{Fendley:2002sg} and for some simple 2d graphs, e.g., \cite{huijse2009supersymmetry,Huijse:2011aa}, which may also be used as benchmarks. However, there is no known result for a generic graph. In fact, this is expected as the problem of computing the Euler characteristic of the independence complex is hard \cite{ROUNE2013170}, a fact already used in the proof of Theorem~\ref{thmWI}. We show in the Appendix that a simple extension of the algorithms in \cite{chowdhury2019computing} can be used to approximate the generalized Witten index in these models. This is a good example of a discrete supersymmetric system capturing quantities of mathematical and computational interest. Generalizations of this model, in which up to $k$ consecutive vertices may be occupied, denoted $M_{k}$, were introduced in \cite{Fendley_2003}. Interestingly, these describe a discretization of ${\cal N}=2$ superconformal minimal models at level $k$ \cite{SciPostPhys.3.1.004}. See \cite{huijse:10} for a review of these models. The observations here may thus provide new tools for exploring aspects of superconformal field theories. \\
\paragraph{Outlook.} A number of directions for future research are suggested. At the level of classical complexity theory, a question is whether the problem of computing the exact Witten index remains hard for systems with ${\cal N}>2$ supersymmetry.
At the level of quantum complexity theory, it is well known that the problem of finding the ground states of a quantum Hamiltonian is $\mathsf{QMA}$-complete \cite{2004quant.ph..6180K}. A natural question is whether this is still the case for supersymmetric Hamiltonians. A physical intuition on why this problem may remain hard is that the supersymmetric hard-core models typically exhibit frustration \cite{Huijse_2008}. On the other hand, supersymmetry is a rich and often very constraining structure. Similarly, a natural question is whether trace estimation, known to be $\mathsf{DQC1}$-hard \cite{shor2007estimating}, remains hard for matrices of the form $U=(-1)^{F}U_{S}$, with $U_{S}$ supersymmetric. We hope to return to these questions in the future.
We note that the robustness properties discussed above suggest the possibility that these may be exploited to design algorithms that are intrinsically robust under certain ``errors.'' The crucial point here is that the supercharge ${\cal Q}$ is not dictated by the environment but rather one is free to choose it. Then, given an error ${\cal E}$ one wishes to protect against, the question is whether there is a choice of supercharge ${\cal Q}$ and an operator $\Psi$ such that ${\cal E}=\{{\cal Q},\Psi\}$. If so, and the ${\cal Q}$-cohomology of unitary operators is nontrivial, this will lead to a family of quantum algorithms which are robust under these errors.
Interesting computational problems also arise in continuum supersymmetric quantum mechanical systems, including the calculation of the Morse index of functions and the Euler characteristic of manifolds \cite{Witten:1982im,Witten:1982df} (see also \cite{Hori:2003ic} for various applications of supersymmetric quantum mechanics). It would be interesting if the some of the ideas presented here could be applied to a discretized version of these systems. \\
Finally, we hope that our discussion brings the attention to the role of supersymmetry in quantum computation and quantum information more broadly, a subject much underexplored.
\
\begin{acknowledgments} The author thanks Jan de Boer, Chris Cade, Irina Kostitsyna, and Kareljan Schoutens for discussions. This work was supported by Nederlandse Organisatie voor Wetenschappelijk Onderzoek (NWO) via a Vidi grant and is also part of the Delta ITP consortium, a program of the NWO that is funded by the Dutch Ministry of Education, Culture and Science (OCW), and by the EU's Horizon 2020 Research Council grant 724659 Massive-Cosmo ERC-2016-COG and the STFC grant ST/T000791/1.
\end{acknowledgments}
\bibliographystyle{apsrev4-1}
|
2,877,628,089,044 | arxiv | \section{Introduction}
\setcounter{equation}{0}
Recent exciting work by Cachazo, He, and
Yuan \cite{Cachazo:2013iaa,Cachazo:2013gna,Cachazo:2013hca,Cachazo:2013iea}
has shed new light on the color-kinematic duality of gauge-theory amplitudes
and the double-copy construction that relates gauge-theory and gravity amplitudes \cite{Bern:2008qj,Bern:2010ue}.
Bern, Carrasco, and Johansson showed that tree-level gauge-theory
and gravity amplitudes may be expressed as sums over cubic diagrams
\begin{equation}
{\cal A} ~=~ \sum_i {c_i ~ n_i \over d_i }, \qquad\qquad
{\cal M} ~=~ \sum_i { \tilde{n} _i ~ n_i \over d_i }
\label{amps}
\end{equation}
where the kinematic numerators $n_i$ are chosen to possess the same symmetries as
the color factors $c_i$.
Cachazo et al. \cite{Cachazo:2013iea}
extended the gauge-gravity nexus to include
amplitudes of the theory of massless scalars in the adjoint representation
of $U(N) \times U(\tilde{N})$ with cubic interactions
\begin{equation}
{\cal A}^{{\rm scalar}} ~=~ \sum_i { \tilde{c} _i ~ c_i \over d_i }
\end{equation}
and they showed that tree-level $n$-point scattering amplitudes for all three theories may be
expressed in terms of an integral over the space of $n$ marked points on a sphere, $\{ \sigma_j ~|~ j=1, \cdots n\}$.
For example, the double-color scalar theory amplitudes may be expressed as\footnote{The definition of $m(\alpha|\beta)$
here differs from that of ref.~\cite{Cachazo:2013iea} by an overall sign when $n$ is even.}
\begin{eqnarray}
{\cal A}^{{\rm scalar}}
&=&
\sum_{\alpha, \beta}
\mathop{\rm Tr}\nolimits[\alpha]
~m(\alpha|\beta) ~
\widetilde{\mathop{\rm Tr}\nolimits}[\beta]
\\
\mathop{\rm Tr}\nolimits [\alpha]
&=&
{\rm Tr}({T}^{\textsf{a}_{\alpha(1)}}{T}^{\textsf{a}_{\alpha(2)}}\cdots {T}^{\textsf{a}_{\alpha(n)}})
\\
m(\alpha|\beta)\ &=& (-1)^{n-1} ~\int \frac{d\,^n\sigma}{\textrm{vol}\,\mathrm{SL}(2,\mathbb{C})}
\frac{\prod'_j
~ \delta \left(\sum_{k\neq j} \frac{s_{jk}}{\sigma_{j,k}} \right)
}{(\sigma_{\alpha(1),\alpha(2)}\cdots\sigma_{\alpha(n),\alpha(1)})
(\sigma_{\beta(1),\beta(2)}\cdots\sigma_{\beta(n),\beta(1)})}
\label{local}
\end{eqnarray}
where $\sigma_{j,k}= \sigma_j - \sigma_k$
and $s_{jk}=( p_j + p_k)^2$ with $p_j$ the momenta of the external particles.
The delta function localizes the integral (\ref{local}) on the solutions of the scattering
equations \cite{Cachazo:2013iaa,Cachazo:2013gna}
\begin{equation}
\sum_{k\neq j} \frac{s_{jk}}{\sigma_{j,k}} ~=~ 0 \quad {\rm for} \quad j=1, \cdots n \,.
\label{scatt}
\end{equation}
As these equations have, up to $\mathrm{SL}(2,\mathbb{C})$ transformations,
$(n-3)!$ solutions $\{\sigma_j^{(I)}\}$,
the double-partial amplitude $m(\alpha|\beta)$ may
expressed as a sum over solutions
\begin{equation}
m(\alpha|\beta)=
~(-1)^{n-1}~\sum_{I=1}^{(n-3)!} \frac{1}{(\sigma^{(I)}_{\alpha(1),\alpha(2)}\cdots\sigma^{(I)}_{\alpha(n),\alpha(1)})(\sigma^{(I)}_{\beta(1),\beta(2)}\cdots\sigma^{(I)}_{\beta(n),\beta(1)}){\det}'\Phi(\sigma^{(I)})}
\label{sumoversolutions}
\end{equation}
demonstrating that $m(\alpha|\beta)$ has rank $(n-3)!$.
The amplitudes (\ref{amps}) for gauge theory and gravity
may also be succinctly expressed in terms of sums over
solutions\cite{Cachazo:2013hca,Cachazo:2013iea}.
Related work on the scattering equations includes
refs.~\cite{Litsey:2013jfa,Adamo:2013tca,Monteiro:2013rya,Mason:2013sva,Chiodaroli:2013upa,Dolan:2013isa,Adamo:2013tsa,Gomez:2013wza,Kalousios:2013eca,Stieberger:2014hba,Yuan:2014gva,Weinzierl:2014vwa,Dolan:2014ega,Bjerrum-Bohr:2014qwa,He:2014wua,Kol:2014yua,Kol:2014zca,Geyer:2014fka,Schwab:2014xua}.
\\ [-2mm]
Because the color factors $c_i$ in \eqn{amps} satisfy Jacobi identities
and hence are not linearly independent, the kinematic numerators $n_i$
for a given $n$-point gauge-theory amplitude are not uniquely determined.
This generalized gauge freedom can be used to require that $n_i$ satisfy
the same Jacobi identities as the color factors \cite{Bern:2008qj}.
As a consequence all kinematic numerators $n_i$ may be expressed in terms of
an independent set ${\bf n}_{1\gamma n}$ associated with half-ladder diagrams.
Defining color-ordered amplitudes $ A_\alpha$ as
the coefficients of the
gauge-theory amplitude expressed in the trace basis
\begin{equation}
{\cal A}
= \sum_{\alpha} \mathop{\rm Tr}\nolimits[ \alpha] ~ A_{\alpha}
\label{trace}
\end{equation}
it can then be shown that the half-ladder numerators must satisfy
\begin{equation}
A_\alpha ~=~ \sum_{\gamma} m( \alpha| 1 \gamma n) ~{\bf n}_{1 \gamma n } \,.
\end{equation}
Because there are $(n-2)!$ independent half-ladder numerators
${\bf n}_{1 \gamma n}$ and because rank $m = (n-3)!$,
this equation cannot be uniquely inverted, meaning that
${\bf n}_{1 \gamma n }$ still possesses some residual generalized gauge freedom
even after color-kinematic duality is imposed.
Many different representations of the kinematic numerators are possible,
{e.g.}, see refs.~\cite{Kiermaier,BjerrumBohr:2010hn,Mafra:2011kj}.
\\ [-2mm]
In an effort to define an economical and natural representation for the numerators,
Broedel and Carrasco \cite{Broedel:2011pd} enumerated
three virtues that kinematic numerators would ideally possess:
(1) color-kinematic duality (numerators obey the same symmetries
as the associated color factors),
(2) amplitude-encoding (external-state dependence is expressed in terms of
color-ordered amplitudes $A_\alpha$), and
(3) symmetry (numerator functions corresponding to diagrams with the same topology
but different labelings of the external legs
are all related by permutations of their arguments).
They then proceeded to construct virtuous numerators for four- and five-point
tree-level amplitudes, and for six-point MHV amplitudes in four dimensions,
by assuming a general ansatz for the numerators and imposing functional constraints.
Their approach becomes impracticable, however, for larger values of $n$,
and they voiced the hope that a constructive procedure for virtuous numerators
for arbitrary $n$-point amplitudes could be found.
\\ [-2mm]
In sec.~3 of this paper, we present a procedure to produce $n$-point kinematic numerators
satisfying all three virtues of Broedel and Carrasco.
We begin with a specific set of nonsymmetric numerators from
ref.~\cite{Cachazo:2013iea} obtained using the properties of $m(\alpha|\beta)$.
By applying arbitrary permutations to the external legs,
we derive other nonsymmetric sets of numerators.
Finally, we generate a symmetric numerator by summing over all such representations.\footnote{Recently,
Fu, Du, and Feng \cite{Fu:2014pya}
presented a different but apparently equivalent algorithm by obtaining symmetric numerators
from a KLT expression for the gauge-theory amplitude.}
Our expression manifestly satisfies all Jacobi identities and diagram symmetries
for the numerators without having to invoke the BCJ relations for the color-ordered amplitudes.
We give explicit results for four- and five-point amplitudes, but the procedure is
completely general.
\\ [-2mm]
The double-copy construction relates gauge-theory to gravity amplitudes by replacing
the color factor $c_i$ in \eqn{amps} with a function $ \tilde{n} _i$ of kinematic variables \cite{Bern:2008qj}.
Bern and Dennen \cite{Bern:2011ia}
suggested that, in a similar way, the gravity amplitude could be
obtained from the gauge-theory amplitude (\ref{trace})
by replacing the color trace factor $\mathop{\rm Tr}\nolimits[\alpha]$
with a dual-trace function $\tau_\alpha$ of the kinematic variables
\begin{equation}
{\cal M}
= \sum_{\alpha} \tau_\alpha ~ A_{\alpha} \,.
\end{equation}
It can then be shown that
\begin{equation}
A_\alpha ~=~ \sum_{\beta} m( \alpha| \beta) \tau_\beta \,.
\label{AYMmtau}
\end{equation}
Once again, because there are $(n-1)!/2$ independent $\tau_\alpha$ and rank $m = (n-3)!$, this
equation cannot be uniquely inverted; many choices of dual-trace functions are possible.
\\ [-2mm]
In ref.~\cite{Bern:2011ia}, Bern and Dennen
presented explicit expressions for $\tau_\alpha$
in terms of ${\bf n}_{1 \gamma n }$
for four-, five-, and six-point amplitudes,
and verified the existence of such expressions through nine points,
but did not present a general procedure for arbitrary $n$.
Further progress was achieved by Du et al. \cite{Du:2013sha,Fu:2013qna}.
\\ [-2mm]
In sec.~4 of this paper, we present a procedure to generate
$n$-point dual-trace functions $\tau_\alpha$ that possess
the three virtues identified by Broedel and Carrasco:
(1) they automatically lead to kinematic numerators that obey color-kinematic duality,
(2) they are expressed in terms of $A_\alpha$, {i.e.}, they are amplitude-encoded, and
(3) they are symmetric: a single function $\tau$ suffices to determine the
full set via permutations of its arguments.
We first identify a specific set of nonsymmetric dual-trace functions
that satisfy \eqn{AYMmtau},
apply arbitrary permutations to the external legs to obtain other representations,
and then generate a symmetric dual-trace function by summing over all such representations.
We present explicit expressions for four- and five-point functions,
but again the procedure is completely general.
\\ [-2mm]
The expressions for $\tau_\alpha$ obtained in ref.~\cite{Bern:2011ia}
were observed to obey Kleiss-Kuijf relations,
and the authors argued that this was a necessary condition
for symmetric dual-trace functions expressed in terms of kinematic numerators.
The results for $\tau_\alpha$ obtained in this paper do not satisfy Kleiss-Kuijf relations;
they are expressed directly in terms of color-ordered amplitudes,
and so are able to avoid this requirement.
Thus, even after the virtues of Broedel and Carrasco are imposed,
there remains some freedom in the dual-trace function.
\\ [-2mm]
This paper is structured as follows.
Section 2 reviews the tree-level amplitudes of
gauge, gravity, and double-color scalar theories,
and the color-kinematic dualities that relate them.
Section 3 describes the criteria of Broedel and Carrasco
for virtuous numerators, outlines our procedure for constructing them,
and presents explicit results for four- and five-point amplitudes.
Section 4 reviews the properties of the dual-trace functions of
Bern and Dennen, and then outlines our procedure for constructing them,
again presenting explicit results for four- and five-point amplitudes.
Section 5 discusses open questions,
and an appendix contains the proof of the procedure used
to produce symmetric kinematic numerators and dual-trace functions.
\section{Review of color-kinematic duality}
\setcounter{equation}{0}
In this section, we review tree-level
amplitudes of gauge, gravity, and double-color scalar theories,
and the color-kinematic dualities that relate these various
amplitudes. \\ [-2mm]
We begin with the tree-level $n$-gluon amplitude
$ {\cal A} ( 1,2, \cdots, n )$, where the arguments denote
the momentum $p_j$, polarization $\varepsilon_j$, and color $\textsf{a}_j$
of the external particles $j=1, \cdots, n$.
The color dependence can be specified in terms of a set of
color factors
$c_i$ obtained by sewing together cubic
vertices $f_{\textsf{abc}} \equiv \mathop{\rm Tr}\nolimits(T^\textsf{a} [ T^\textsf{b}, T^\textsf{c}] )$,
where $T^\textsf{a}$ denote generators in the fundamental representation of the
color group.
The $n$-point amplitude is then expressed as a sum over all cubic diagrams
\cite{Bern:2008qj}
\begin{equation}
{\cal A} ( 1,2, \cdots, n )
~=~ \sum_i
{c_i ~ n_i ( 1,2, \cdots, n )
\over d_i ( 1,2, \cdots, n )} \,.
\label{gluonamp}
\end{equation}
The denominator $d_i ( 1,2, \cdots, n )$
associated with the color diagram $c_i$
is a product of the diagram propagators,
and is a function of the external momenta $p_j$.
The kinematic numerator $n_i ( 1,2, \cdots, n )$ associated with $c_i$
is a function of both $p_j$ and $\varepsilon_j$.
(As usual, terms in the Feynman diagram expansion arising
from quartic vertices can be parceled into terms involving only cubic vertices.) \\ [-2mm]
The color factors $c_i$ can be expanded in a trace basis
\begin{equation}
c_i ~=~
\sum_{\alpha \in {S_n /\mathbb{Z} } } M_{i \alpha} \mathop{\rm Tr}\nolimits[\alpha] , \qquad \qquad
\mathop{\rm Tr}\nolimits [\alpha] ~\equiv~
{\rm Tr}({T}^{\textsf{a}_{\alpha(1)}}{T}^{\textsf{a}_{\alpha(2)}}\cdots {T}^{\textsf{a}_{\alpha(n)}}) \,.
\label{colortrace}
\end{equation}
The trace basis is independent provided we restrict the sum to permutations
of $n$ indices modulo cyclic permutations.
The gauge-theory amplitude (\ref{gluonamp}) can be decomposed in this basis
\begin{eqnarray}
{\cal A} ( 1,2, \cdots, n )
&=& \sum_{\alpha \in {S_n /\mathbb{Z} } } \mathop{\rm Tr}\nolimits[ \alpha] ~ A_{\alpha} ( 1,2, \cdots, n ) ,
\label{tracedecomp}
\\
A_{\alpha} ( 1,2, \cdots, n )
&=& \sum_i
{M_{i \alpha} ~ n_i ( 1,2, \cdots, n )
\over d_i ( 1,2, \cdots, n )}
\label{colorordered}
\end{eqnarray}
where the arguments of the color-ordered amplitudes
$ A_{\alpha} ( 1,2, \cdots, n )$ denote $p_j$ and $\varepsilon_j$.
Since the trace basis is independent, the color-ordered amplitudes
are well-defined and gauge invariant.
Because of the invariance of the full amplitude $ {\cal A} ( 1,2, \cdots, n )$ under
permutations of the arguments and the independence of the
trace basis,\footnote{See discussion in the appendix.}
the color-ordered amplitudes are all related to one another
via
\begin{equation}
A_\alpha ( 1,2, \cdots, n ) ~=~ A ( \alpha(1), \cdots, \alpha(n))
\end{equation}
where $ A (i,j,\cdots) \equiv A_{12\cdots n}(i, j, \cdots) $,
the coefficient of
${\rm Tr}(T^{\textsf{a}_{1}}T^{\textsf{a}_{2}}\cdots T^{\textsf{a}_{n}})$.
The color-ordered amplitudes are thus ``symmetric'' functions,
in the sense of Broedel and Carrasco\cite{Broedel:2011pd}. \\ [-2mm]
The color factors $c_i$ in \eqn{gluonamp}
are not independent but satisfy various Jacobi identities
\begin{equation}
c_i + c_j + c_k ~=~ 0
\label{cjacobi}
\end{equation}
which can be expressed
\cite{Naculich:2011ep,Edison:2011ta,Edison:2012fn}
as $\sum_i \ell_i c_i = 0$,
where $\ell_i$ are left null vectors of the matrix $M_{i\alpha}$:
$\sum_i \ell_i M_{i\alpha}=0$.
The matrix $M_{i\alpha}$ has rank $(n-2)!$ so there are $(n-2)!$ independent
color factors.
The matrix $M_{i\alpha}$ also possesses a set of right null vectors
$\sum_\alpha M_{i\alpha} r_\alpha=0$.
By \eqn{colorordered}, these give
rise\cite{Naculich:2011ep,Edison:2011ta,Edison:2012fn}
to a set of constraints
$\sum_\alpha A_{\alpha} r_\alpha=0$
on the color-ordered amplitudes, {viz.},
the Kleiss-Kuijf relations \cite{Kleiss:1988ne}. \\ [-2mm]
Because of the linear dependence of the color factors $c_i$,
the kinematic numerators $n_i ( 1,2, \cdots, n )$ in \eqn{gluonamp}
are not uniquely determined but can undergo
what are termed generalized gauge transformations \cite{Bern:2010yg}
without altering the amplitudes.
The insight of Bern, Carrasco, and Johannson \cite{Bern:2008qj}
is that there exists a generalized gauge choice
for which the numerators satisfy the same Jacobi identities as the color factors
\begin{equation}
n_i ( 1,2, \cdots, n ) ~+~ n_j ( 1,2, \cdots, n ) ~+~ n_k ( 1,2, \cdots, n ) ~=~ 0 \,.
\label{njacobi}
\end{equation}
We will refer to a set of kinematic numerators satisfying \eqn{njacobi} as BCJ numerators.
Such a choice is not unique: there remain residual generalized
gauge transformations that preserve the Jacobi identities (\ref{njacobi}).
\\ [-2mm]
A subset of the color factors $c_i$ are the half-ladder diagrams,
labeled by $ \alpha \in S_n$:
\begin{equation}
{\bf c}_{\alpha} ~=~ \sum_{\textsf{b}_1,\ldots,\textsf{b}_{n{-}3}}
f_{\textsf{a}_{\alpha(1)} \textsf{a}_{\alpha(2)} \textsf{b}_1}\cdots f_{\textsf{b}_{n{-}3} \textsf{a}_{\alpha(n{-}1)} \textsf{a}_{\alpha(n)}}
~=~ \mathop{\rm Tr}\nolimits(
T^{\textsf{a}_{\alpha(1)}}
[T^{\textsf{a}_{\alpha(2)}} ,
[ \cdots [
T^{\textsf{a}_{\alpha(n-1)}} ,
T^{\textsf{a}_{\alpha(n)}} ] \cdots ]] ) \,.
\label{c-basis}
\end{equation}
An independent set of color factors \cite{DelDuca:1999ha,DelDuca:1999rs}
consists of those half ladders with $\alpha(1)=1$ and $\alpha(n)=n$:
\begin{equation}
{\bf c}_{1 \gamma n } ~\equiv~
{\bf c}_{1 \gamma(2) \cdots \gamma(n-1) n }
, \qquad\qquad \gamma \in S_{n-2}
\end{equation}
whose expansion in the trace basis begins
\begin{equation}
{\bf c}_{1 \gamma n }
~=~ \mathop{\rm Tr}\nolimits[ 1 \gamma(2) \cdots \gamma(n-1) n]
~+~ (-1)^n \mathop{\rm Tr}\nolimits[ n \gamma(n-1) \cdots \gamma(1) 1] ~+~ \cdots
\label{halfladderexpand}
\end{equation}
where $+ \cdots$ denotes traces without $1$ and $n$ adjacent.
That the set ${\bf c}_{1 \gamma n }$ is independent
follows from \eqn{halfladderexpand} together with the independence of the trace basis
(modulo cyclic permutations).
That the set is complete was shown in ref.~\cite{DelDuca:1999rs}
where an arbitrary color diagram was reduced to a linear combination
$c_i = \sum_i \lambda_{i,\gamma} {\bf c}_{1 \gamma n }$
using the Jacobi identities.
By matching the coefficients of $\mathop{\rm Tr}\nolimits[ 1 \gamma(2) \cdots \gamma(n-1) n] $
on both sides of this equation,
one then establishes that
\begin{equation}
c_i ~=~ \sum_{\gamma \in S_{n-2}} M_{i, 1\gamma n} {\bf c}_{1 \gamma n } \,.
\label{cMc}
\end{equation}
Then \eqns{gluonamp}{colorordered} imply \cite{DelDuca:1999ha,DelDuca:1999rs}
\begin{equation}
{\cal A} ( 1,2, \cdots, n ) ~=~ \sum_{\gamma \in S_{n-2}}
{\bf c}_{1 \gamma n } A(1, \gamma(2), \cdots, \gamma(n-1), n) \,.
\label{cgammaA}
\end{equation}
\subsection{Double-color scalar amplitudes}
In their seminal paper \cite{Bern:2008qj},
Bern, Carrasco, and Johansson showed that,
given a set of BCJ numerators $n_i( 1,2, \cdots, n )$,
one can obtain tree-level gravity amplitudes\footnote{
Up to an overall factor depending on coupling strengths
that we suppress throughout this paper}
by replacing $c_i$ in \eqn{gluonamp} with the kinematic
numerators $ \tilde{n} _i( 1,2, \cdots, n )$ of a second gauge theory (the double-copy procedure)
\begin{equation}
{\cal M} ( 1,2, \cdots, n )
~=~ \sum_i
{ \tilde{n} _i ( 1,2, \cdots, n ) ~ n_i ( 1,2, \cdots, n )
\over d_i ( 1,2, \cdots, n )}
\label{gravamp}
\end{equation}
where the $ \tilde{n} _i( 1,2, \cdots, n )$ need not themselves satisfy the Jacobi
identities \cite{Bern:2010yg}.
\\ [-2mm]
An alternative double-copy procedure replaces the $n_i( 1,2, \cdots, n )$ in
\eqn{gluonamp} with the color factors $ \tilde{c} _i$ of a second color group
\begin{equation}
{\cal A}^{{\rm scalar}} ( 1,2, \cdots, n )
~=~ \sum_i
{ \tilde{c} _i ~ c_i \over d_i ( 1,2, \cdots, n )} \,.
\label{scalaramp}
\end{equation}
Such an expression corresponds \cite{Cachazo:2013iea}
to the $n$-point amplitude of a theory of massless scalar
particles $ \phi^{\textsf{a}\textsf{a'}}$
in the adjoint of the color group $U(N) \times U(\tilde{N})$
with cubic interactions of the form
\begin{equation}
f_{\textsf{abc}}\tilde f_{\textsf{a'b'c'}}\phi^{\textsf{a}\textsf{a'}}\phi^{\textsf{b}\textsf{b'}}\phi^{\textsf{c}\textsf{c'}}
\end{equation}
where
$f_{\textsf{abc}}$ and $\tilde f_{\textsf{a'b'c'}}$
are the structure constants of $U(N)$ and $U(\tilde{N})$.
Using
\begin{equation}
\tilde{c} _i ~=~
\sum_{\alpha \in {S_n /\mathbb{Z} } } M_{i \alpha} \widetilde{\mathop{\rm Tr}\nolimits}[\alpha] , \qquad \qquad
\widetilde{\mathop{\rm Tr}\nolimits} [\alpha] ~\equiv~
{\rm Tr}({\tilde T}^{\textsf{a}_{\alpha(1)}}{\tilde T}^{\textsf{a}_{\alpha(2)}}\cdots {\tilde T}^{\textsf{a}_{\alpha(n)}})
\end{equation}
together with \eqn{colortrace}, the double-color amplitude (\ref{scalaramp}) can be written as
\begin{equation}
{\cal A}^{{\rm scalar}} ( 1,2, \cdots, n )
~=~
\sum_{\alpha \in {S_n /\mathbb{Z} } } \sum_{\beta \in {S_n /\mathbb{Z} } }
\mathop{\rm Tr}\nolimits[\alpha]
~m(\alpha|\beta) ~
\widetilde{\mathop{\rm Tr}\nolimits}[\beta]
\label{TmT}
\end{equation}
where\footnote{This definition of $m(\alpha|\beta)$ differs by a
sign from that of ref.~\cite{Cachazo:2013iea} when $n$ is even.}
\begin{equation}
m(\alpha| \beta)
~=~ \sum_i
{M_{i\alpha} M_{i\beta} \over d_i( 1,2, \cdots, n ) },
\qquad \alpha, \beta \in S_n \,.
\label{doublepartial}
\end{equation}
The coefficients $A_\alpha$ of $\mathop{\rm Tr}\nolimits[\alpha]$ of the gauge-theory amplitude (\ref{tracedecomp})
are sometimes termed ``partial amplitudes.''
In ref.~\cite{Cachazo:2013iea},
Cachazo, He, and Yuan dubbed $m(\alpha|\beta)$,
the coefficients of
$\mathop{\rm Tr}\nolimits[\alpha] \widetilde{\mathop{\rm Tr}\nolimits}[\beta]$ in \eqn{TmT},
``double-partial amplitudes.''
They showed that $m(\alpha|\beta)$ computes the sum of all trivalent scalar diagrams
that can be regarded both as $\alpha$-color-ordered and $\beta$-color-ordered,
where each diagram's contribution is given by the product of its propagators.
The double-partial amplitudes satisfy (on both sides)
Kleiss-Kuijf relations
$ \sum_\alpha r_\alpha m(\alpha|\beta) = \sum_\alpha m(\alpha|\beta) r_\beta = 0$
where $r_\alpha$ are right null vectors of the rank $(n-2)!$ matrix $M_{i\alpha}$.
\\ [-2mm]
Using \eqn{cMc}, the double-color amplitude (\ref{scalaramp}) can also be expressed as
\begin{equation}
{\cal A}^{{\rm scalar}} ( 1,2, \cdots, n )
~=~ \sum_{\gamma \in S_{n-2}} \sum_{\delta \in S_{n-2}}
{\bf \tilde{c} }_{1 \gamma n } ~m( 1 \gamma n| 1 \delta n) ~{\bf c}_{1 \delta n } \,.
\label{cmc}
\end{equation}
The double-partial amplitudes in this equation,
$m( 1 \gamma n| 1 \delta n)$,
are essentially the entries in the $(n-2)! \times (n-2)!$ matrix
considered in ref.~\cite{Vaman:2010ez}.
There it was shown by consideration of low values of $n$ that,
as a result of momentum conservation,
the rank of this matrix is $(n-3)!$.
Cachazo et al. showed that the double-partial amplitudes could alternatively be expressed
as \eqn{sumoversolutions}, which shows explicitly that
$m(\alpha|\beta)$ has rank $(n-3)!$.
It consequently possesses $(n-3)!-(n-2)!$ additional null vectors,
dependent on kinematic invariants,
which implies that the double-partial amplitudes satisfy
BCJ relations \cite{Bern:2008qj} in addition to Kleiss-Kuijf relations.
\subsection{Gauge-theory amplitudes}
Just as the Jacobi identities for the color factors (\ref{cjacobi})
imply that they can be expressed as a linear combination of
independent half-ladder color factors (\ref{cMc}),
so the Jacobi identities (\ref{njacobi})
imply that the kinematic numerators can be written
in terms of an independent basis of numerators
${\bf n}_{1 \gamma n } ( 1,2, \cdots, n ) $ associated with ${\bf c}_{1 \gamma n } $
\begin{equation}
n_i ( 1,2, \cdots, n ) ~=~ \sum_{\gamma \in S_{n-2}} M_{i, 1\gamma n}
~{\bf n}_{1 \gamma n } ( 1,2, \cdots, n ) \,.
\label{nMn}
\end{equation}
Then, analogous to \eqn{cmc} for double-color scalar amplitude,
we can write the gauge-theory amplitude (\ref{gluonamp})
as
\begin{equation}
{\cal A} ( 1,2, \cdots, n )
~=~ \sum_{\gamma \in S_{n-2}} \sum_{\delta \in S_{n-2}}
{\bf c}_{1 \gamma n } ~m( 1 \gamma n| 1 \delta n) ~{\bf n}_{1 \delta n } \,.
\label{cmn}
\end{equation}
We can also substitute \eqn{nMn} into \eqn{colorordered}
to obtain the color-ordered amplitudes
\begin{equation}
A_\alpha ( 1,2, \cdots, n )
~=~ \sum_{\gamma \in S_{n-2}}
m( \alpha| 1 \gamma n)
~{\bf n}_{1 \gamma n } ( 1,2, \cdots, n ) \,.
\label{Amn}
\end{equation}
As noted above,
the matrix $m( \alpha| 1 \gamma n) $ has rank $(n-3)!$
and has $(n-2)! - (n-3)!$ additional\footnote{In addition,
that is, to the null vectors of $M_{i\alpha}$
which give rise to the Kleiss-Kuijf relations.}
null vectors (dependent on kinematic invariants)
which give rise \cite{Vaman:2010ez,Boels:2012sy,Litsey:2013jfa}
to the BCJ relations \cite{Bern:2008qj} among the color-ordered amplitudes.
\\ [-2mm]
The kernel of the matrix corresponds to
$(n-2)! - (n-3)!$ degrees of freedom of
residual generalized gauge transformations of the BCJ numerators
(i.e. generalized gauge transformations that preserve the Jacobi constraints (\ref{njacobi})).
Because of this residual gauge freedom,
\eqn{Amn} cannot be inverted
to obtain unique expressions for BCJ numerators
in terms of color-ordered amplitudes.
To invert \eqn{Amn}, we must first make a choice of gauge.
One possible gauge choice is to set
\begin{equation}
{\bf n}_{1\gamma(2)\cdots\gamma(n-1)n} = 0, \qquad \qquad \gamma(n-1)\neq n-1
\label{zeronumerators}
\end{equation}
which allows us to restrict the sum in \eqn{Amn} to $S_{n-3}$.
\\ [-2mm]
Now consider the subset of color-ordered amplitudes
\begin{equation}
A (1, \beta(2), \cdots, \beta(n-2), n, n-1)
~=~ \sum_{\gamma \in S_{n-3}}
m( 1 \beta n, n-1 | 1 \gamma n-1,n)
~ {\bf n}_{1\gamma(2)\cdots\gamma(n-2)n-1,n} ( 1,2, \cdots, n ) \,.
\label{mbetagamma}
\end{equation}
It was shown in ref.~\cite{Cachazo:2013iea}
by using KLT orthogonality \cite{Cachazo:2013gna} that
the $(n-3)! \times (n-3)!$ submatrix
appearing in \eqn{mbetagamma} is invertible,
with the inverse given by
the (negative of the) momentum kernel\footnote{Here $\gamma,\beta\in S_{n-3}$ are permutations acting on
labels $2,3,\ldots,n{-}2$; $\theta(r,s)_\beta=1$ if the ordering of $r,s$
is the same in both sequences of labels, $\gamma(2),\ldots,\gamma(n{-}2)$
and $\beta(2),\ldots,\beta(n{-}2)$, and zero otherwise.} \cite{BjerrumBohr:2010ta,BjerrumBohr:2010zb,BjerrumBohr:2010yc,BjerrumBohr:2010hn}
\begin{equation}
S[\gamma|\beta]~=~
\prod^{n{-}2}_{i=2}\left[s_{1, \gamma(i)}+\sum^{i{-}1}_{j=2} \theta(\gamma(j), \gamma(i))_{\beta} s_{\gamma(j),\gamma(i)}\right].
\end{equation}
Thus the nonzero members of the independent basis of numerators
can be expressed as \cite{Cachazo:2013iea}
\begin{equation}
{\bf n}_{1\gamma(2)\cdots\gamma(n-2)n-1,n} ( 1,2, \cdots, n )
~=~ -~\sum_{\beta\in S_{n-3}}S[\gamma|\beta]
A (1,\beta(2), \cdots, \beta(n-2) ,n,n-1) \,.
\label{BCJnumerator}
\end{equation}
The full set of (BCJ) numerators,
including the half-ladders not included in the independent set,
can then be obtained from \eqns{zeronumerators}{BCJnumerator} via \eqn{nMn}.
\\ [-2mm]
\subsection{Gravity amplitudes}
By the double-copy construction, we can replace
${\bf c}_{1 \gamma n }$ in \eqns{cgammaA}{cmn} with
${\bf n}_{1 \gamma n } ( 1,2, \cdots, n )$
to obtain the gravity amplitude
\begin{eqnarray}
{\cal M} ( 1,2, \cdots, n )
&=& \sum_{\gamma \in S_{n-2}}
{\bf n}_{1 \gamma n } ( 1,2, \cdots, n )
A (1, \gamma(2), \cdots, \gamma(n-1),n)
\nonumber\\
&=& \sum_{\gamma \in S_{n-2}} \sum_{\delta \in S_{n-2}}
{\bf n}_{1 \gamma n } ( 1,2, \cdots, n ) ~m( 1 \gamma n| 1 \delta n) ~{\bf n}_{1 \delta n } ( 1,2, \cdots, n ) \,.
\end{eqnarray}
The specific choice of gauge in \eqns{zeronumerators}{BCJnumerator} can then
be used to write
\begin{eqnarray}
&& {\cal M} ( 1,2, \cdots, n )
\nonumber\\
&& ~~~
=
\sum_{\gamma \in S_{n-3}}
\sum_{\delta \in S_{n-3}}
A (1, \gamma(2), \cdots, \gamma(n-2),n-1,n)
S[\gamma|\delta]
A (1,\delta(2), \cdots, \delta(n-2) ,n,n-1)
\nonumber\\
\end{eqnarray}
which is one possible form of the field-theory limit of the KLT relation \cite{Kawai:1985xq,Bern:1998sv,BjerrumBohr:2010hn}.
\\ [-2mm]
\section{Virtuous kinematic numerators}
\setcounter{equation}{0}
As described in the previous section,
even after requiring the kinematic numerators to satisfy color-kinematic duality,
a certain amount of generalized gauge freedom remains.
To fix this residual gauge freedom in a natural, economical,
and possibly unique way,
Broedel and Carrasco \cite{Broedel:2011pd} identified three
desirable features that a set of numerators should possess:
\begin{enumerate}
\item
color-kinematic duality:
BCJ numerators obey the same symmetries as their associated color factors.
Such numerators can be used to construct gravity amplitudes using the double-copy procedure.
\item
amplitude-encoding: the external state dependence ({e.g.}, helicities) of
the numerators is expressed in terms of color-ordered amplitudes.
Such a representation would also be independent of the number of space-time dimensions.
\item
symmetry:
all of the numerators $n_i$ for a given diagram topology can
be expressed via permutations of the arguments of a single function.
This virtue is thus one of economy, and also makes the gravity amplitudes
constructed from these numerators manifestly invariant under permutations
of the external legs.
\end{enumerate}
Kinematic numerators that satisfy all three features are dubbed ``virtuous.''
\\ [-2mm]
As we saw in the previous section, numerators that satisfy the
first virtue are expressed in terms
of an independent basis of half-ladder numerators
${\bf n}_{1 \gamma n } ( 1,2, \cdots, n )$ via \eqn{nMn}.
Color-ordered amplitudes are expressed as in terms of these as
\begin{equation}
A_\alpha ( 1,2, \cdots, n )
~=~ \sum_{\gamma \in S_{n-2}}
m( \alpha| 1 \gamma n)
~{\bf n}_{1 \gamma n } ( 1,2, \cdots, n ) \,.
\label{Amnrepeat}
\end{equation}
Were we able to invert this equation, we would possess numerators
that also satisfy the second virtue of amplitude-encoding.
Because $\det m= 0$, \eqn{Amnrepeat} has no unique inverse;
there exists rather a family of generalized inverses \cite{BenIsrael,Boels:2012sy},
each corresponding to a particular gauge choice for ${\bf n}_{1 \gamma n }$.
One such choice is given by \cite{Cachazo:2013iea}
\begin{eqnarray}
{\bf n}_{1\gamma(2)\cdots\gamma(n-1)n}( 1,2, \cdots, n ) &=& 0, \qquad \qquad \gamma(n-1)\neq n-1
\nonumber\\
{\bf n}_{1\gamma(2)\cdots\gamma(n-2)n-1,n} ( 1,2, \cdots, n ) &=&
-~\sum_{\beta\in S_{n-3}}S[\gamma|\beta]
A (1,\beta(2), \cdots, \beta(n-2) ,n,n-1)\,.
\label{gaugechoice}
\end{eqnarray}
This choice, however, does not satisfy the third virtue of symmetry.
\\ [-2mm]
Next, we describe how one can generate a symmetric representation of
numerators starting from a nonsymmetric representation,
such as that given in \eqn{gaugechoice}.\footnote{We thank Freddy Cachazo for
suggesting this approach.}
We show in the appendix that,
given one set of numerators
${\bf n}_{\alpha} ( 1,2, \cdots, n )$,
one can use an arbitrary permutation $\beta$ acting on the
external legs to generate another set of valid numerators
\begin{equation}
{\bf n}'_{\alpha} ( 1,2, \cdots, n )
~=~ {\bf n}_{\beta^{-1} \alpha} (\beta(1), \cdots, \beta(n) )
\end{equation}
{i.e.}, these also satisfy \eqn{Amnrepeat}.
If we average over all permutations $\beta \in S_n$,
the resulting function
\begin{equation}
{\bf n} ( 1,2, \cdots, n )
~\equiv~
{\bf n}^{\rm sym} ( 1,2, \cdots, n )
~=~ {1 \over n!} \sum_{\beta \in S_n} {\bf n}_{\beta^{-1} } (\beta(1), \cdots, \beta(n) )
\label{symmetrized}
\end{equation}
will be symmetric; {i.e.}, all half-ladder numerators are given by permutations
of the arguments of the single function ${\bf n} ( 1,2, \cdots, n ) $.
Symmetric numerator functions for topologies other than the half-ladder are then
obtained from ${\bf n} ( 1,2, \cdots, n ) $ via \eqn{nMn}.
Since the individual sets of numerators
${\bf n}'_{\alpha} ( 1,2, \cdots, n )$ satisfy \eqn{Amnrepeat},
so does the average (\ref{symmetrized}):
\begin{equation}
A (\alpha(1), \cdots, \alpha(n))
~=~
\sum_{\gamma \in S_{n-2}}
~m( \alpha| 1 \gamma n)
~{\bf n} (1, \gamma(2), \cdots, \gamma(n-1), n )\,.
\label{validvirtue}
\end{equation}
Hence we have constructed a representation of kinematic numerators satisfying all three virtues of Broedel and Carrasco.
\\ [-2mm]
We apply this procedure in the following subsections to obtain
explicit expressions for virtuous numerators for four- and five-point amplitudes.
The expressions become increasingly lengthy for higher-point amplitudes, but
the important point is that there exists a constructive proof of the existence of
virtuous numerators for all tree-level $n$-gluon amplitudes.
Whether such numerators are unique remains an open question.
\\ [-2mm]
In a recent paper \cite{Fu:2014pya}, Fu, Du, and Feng
also gave a prescription for obtaining virtuous numerators
based on a similar symmetrization strategy applied to
a KLT expression for the gauge-theory amplitude.
It seems likely that this gives the same results as \eqn{symmetrized}.
We will compare specific results below.
\subsection{Four-point symmetric kinematic numerators}
For the four-point amplitude, there is only one topology
for color factors, the half ladder.
A symmetric half-ladder numerator ${\bf n}(1,2,3,4)$ must satisfy \eqn{validvirtue}
which in this case becomes
\begin{equation}
A(1,2,3,4) ~=~ \left( {1 \over s_{12}} + {1 \over s_{14}} \right) {\bf n}(1,2,3,4)
~+~ {1 \over s_{14}} {\bf n}(1,3,4,2) \,.
\label{fourpointcolorordered}
\end{equation}
\Eqn{gaugechoice} yields
a nonsymmetric amplitude-encoded BCJ representation
for independent half-ladder numerators
\begin{eqnarray}
{\bf n}_{1234}(i,j,k,l) &=& - s_{ij} A (i,j,l,k)\,, \nonumber\\
{\bf n}_{1324}(i,j,k,l) &=& 0
\label{fourpointnonsym}
\end{eqnarray}
with the remaining numerators given by \eqn{nMn}, {i.e.}, by Jacobi identities and numerator symmetries.
We obtain a virtuous representation by summing over all permutations (\ref{symmetrized})
\begin{equation}
{\bf n}(1,2,3,4) ~=~
{1 \over 12} {(s_{12} + s_{34})} \left[
A(1,2,3,4) -A(1,3,4,2)
\right]
+{1 \over 12} {(s_{13} +s_{24} -s_{14} - s_{23})}
A(1,4,2,3) \,.
\label{fourpointvirtue}
\end{equation}
We have reduced the number of terms by using
the cyclic invariance of $A(1,2,3,4)$
as well as the reversal symmetry $A(1,2,3,4)= A(4,3,2,1)$.
This expression manifestly obeys the Jacobi identities
\begin{equation}
{\bf n}(1,2,3,4) + {\bf n}(1,3,4,2) + {\bf n}(1,4,2,3) ~=~ 0
\end{equation}
as well as the dihedral symmetries of the half-ladder diagram
\begin{equation}
{\bf n}(1,2,3,4) ~=~ - {\bf n}(2,1,3,4) ~=~ - {\bf n}(1,2,4,3) ~=~ {\bf n}(4,3,2,1) \,.
\end{equation}
\Eqn{fourpointvirtue} also satisfies \eqn{fourpointcolorordered}
provided that the color-ordered amplitudes satisfy the four-point BCJ relations
\begin{equation}
s_{ij} A(i,j,k,l) ~=~ s_{ik} A(i,l,j,k) \,.
\label{fourpointBCJ}
\end{equation}
Our expression (\ref{fourpointvirtue}) agrees with
that recently obtained in ref.~\cite{Fu:2014pya}.
It is also in agreement with the shorter expression in ref.~\cite{Broedel:2011pd}
\begin{equation}
{\bf n}(1,2,3,4) ~=~ {1 \over 3} \left[ s_{12} A (1, 2, 3, 4) - s_{14} A(1, 4, 2, 3) \right]
\label{fourpointBC}
\end{equation}
once momentum conservation and the Kleiss-Kuijf (subcyclic) identity
$ A(1,2,3,4) + A(1,3,4,2) + A(1, 4, 2, 3) = 0 $
are imposed.
In fact, the three virtues listed above are sufficient to uniquely determine
the four-point numerator.
\\ [-2mm]
The BCJ relations (\ref{fourpointBCJ}) can be used to rewrite \eqn{fourpointBC} as
\begin{equation}
{\bf n}(1,2,3,4) ~=~ {1 \over 3} s_{12} \left[ A(1,2,3,4) - A(1, 3, 4, 2) \right] \,.
\end{equation}
We could further use the BCJ relations to write
${\bf n}(1,2,3,4)$
in terms of a single color-ordered amplitude,
but only at the price of having kinematic invariants in the denominator.
\subsection{Five-point symmetric numerators}
For the five-point amplitude, there is again only one topology
for the color factors, the half ladder.
A symmetric half-ladder numerator ${\bf n}(1,2,3,4,5)$ satisfies \eqn{validvirtue},
which after using Jacobi identities is equivalent to
\begin{equation}
A(1,2,3,4,5) ~=~
{{\bf n}(1,2,3,4,5) \over s_{12}s_{45}}
~+~{{\bf n}(2,3,4,5,1) \over s_{23}s_{51}}
~+~{{\bf n}(3,4,5,1,2) \over s_{34}s_{12}}
~+~{{\bf n}(4,5,1,2,3) \over s_{45}s_{23}}
~+~{{\bf n}(5,1,2,3,4) \over s_{51}s_{34}} \,.
\label{fivepointcolorordered}
\end{equation}
\Eqn{gaugechoice} yields
a nonsymmetric amplitude-encoded BCJ representation
for independent half-ladder numerators
\begin{eqnarray}
{\bf n}_{12345}(i,j,k,l,m) &=& - s_{ij} (s_{ik}+s_{jk} ) A(i,j,k,m,l) - s_{ij} s_{ik} A (i,k,j,m,l)\,, \nonumber\\
{\bf n}_{13245}(i,j,k,l,m) &=& - s_{ij} s_{ik} A(i,j,k,m,l) - s_{ik} (s_{ij}+s_{jk} ) A(i,k,j,m,l)\,, \\
{\bf n}_{14235}(i,j,k,l,m) &=& 0\,, \nonumber\\
{\bf n}_{12435}(i,j,k,l,m) &=& 0\,, \nonumber\\
{\bf n}_{14235}(i,j,k,l,m) &=& 0\,, \nonumber\\
{\bf n}_{14325}(i,j,k,l,m) &=&0
\nonumber
\end{eqnarray}
where the other half-ladder numerators are obtained using \eqn{nMn}.
Again, we obtain a virtuous representation by summing over all permutations (\ref{symmetrized})
\begin{eqnarray}
{\bf n}(1,2,3,4,5) &=&
{1 \over 60} \Big[
\left(s_{1 2} s_{1 3} + 2 s_{1 2} s_{2 3} + 2 s_{3 4} s_{4 5} + s_{3 5} s_{4 5}
\right) A(1, 2, 3, 4, 5) \nonumber\\
&+&\left(s_{1 3} s_{1 4} + s_{1 2} s_{1 5} + s_{2 3} s_{2 4} + 2 s_{2 3} s_{3 4} + s_{2 4} s_{3 4} + s_{2 5} s_{3 5} + s_{1 5} s_{4 5}
\right) A(1, 4, 3, 2, 5) \nonumber\\
&+&\left(s_{1 3} s_{1 4} + s_{2 3} s_{2 4} + s_{1 2} s_{2 5} + 2 s_{1 3} s_{3 4} + s_{1 4} s_{3 4} + s_{1 5} s_{3 5} + s_{2 5} s_{4 5}
\right) A(1, 3, 4, 2, 5) \nonumber\\
&+&\left(s_{1 2} s_{1 4} - 2 s_{1 2} s_{1 5} + 2 s_{1 2} s_{2 4} - s_{1 2} s_{2 5} - s_{3 4} s_{4 5} + s_{3 5} s_{4 5}
\right) A(1, 2, 4, 3, 5) \nonumber\\
&+&\left(s_{1 2} s_{1 4} + s_{1 3} s_{1 5} + s_{2 3} s_{2 5} + s_{2 4} s_{3 4} + 2 s_{2 3} s_{3 5} + s_{2 5} s_{3 5} + s_{1 4} s_{4 5}
\right) A(1, 4, 2, 3, 5) \nonumber\\
&+&\left(s_{1 2} s_{1 3} - s_{1 2} s_{2 3} - s_{1 4} s_{4 5} - 2 s_{1 5} s_{4 5} + 2 s_{2 4} s_{4 5} + s_{2 5} s_{4 5}
\right) A(1, 3, 2, 4, 5) \nonumber\\
&+&\left(2 s_{1 2} s_{1 3} + s_{1 2} s_{2 3} + 2 s_{3 4} s_{4 5} + s_{3 5} s_{4 5}
\right) A(1, 2, 5, 4, 3) \nonumber\\
&+&\left(s_{1 2} s_{1 3} - s_{1 2} s_{2 3} - 2 s_{1 4} s_{4 5} - s_{1 5} s_{4 5} + s_{2 4} s_{4 5} + 2 s_{2 5} s_{4 5}
\right) A(1, 4, 5, 2, 3) \nonumber\\
&+&\left(-s_{1 3} s_{1 5} - s_{1 2} s_{2 4} - s_{2 3} s_{2 5} - s_{1 4} s_{3 4} - 2 s_{1 3} s_{3 5} - s_{1 5} s_{3 5} - s_{2 4} s_{4 5}
\right) A(1, 3, 5, 2, 4) \nonumber\\
&+&\left(2 s_{1 2} s_{1 4} - s_{1 2} s_{1 5} + s_{1 2} s_{2 4} - 2 s_{1 2} s_{2 5} - s_{3 4} s_{4 5} + s_{3 5} s_{4 5}
\right) A(1, 2, 5, 3, 4) \nonumber\\
&+&\left(s_{1 2} s_{1 3} + 2 s_{1 2} s_{2 3} + s_{3 4} s_{4 5} + 2 s_{3 5} s_{4 5}
\right) A(1, 4, 5, 3, 2) \nonumber\\
&+&\left(2 s_{1 2} s_{1 3} + s_{1 2} s_{2 3} + s_{3 4} s_{4 5} + 2 s_{3 5} s_{4 5}
\right) A(1, 3, 5, 4, 2)
\Big]
\label{fivepointvirtue}
\end{eqnarray}
where we have used the reversal property
$ A(1,2,3,4,5)= -A(5,4,3,2,1) $
together with cyclic invariance of $A(1,2,3,4,5)$.
This expression automatically obeys the dihedral symmetry of the half-ladder diagram
\begin{equation}
{\bf n}(1,2,3,4,5) ~=~ - {\bf n}(2,1,3,4,5) ~=~ - {\bf n}(1,2,3,5,4) ~=~ - {\bf n}(5,4,3,2,1)
\label{fivepointdihedral}
\end{equation}
as well as all the Jacobi identities.
\\ [-2mm]
\Eqn{fivepointvirtue} agrees with the shorter expression
given in Broedel and Carrasco \cite{Broedel:2011pd}
after imposing momentum conservation and BCJ relations
on the color-ordered amplitudes.
The Broedel-Carrasco expression is virtuous,
but to show that it obeys \eqn{fivepointdihedral}
requires the imposition of momentum conservation and BCJ relations,
whereas for \eqn{fivepointvirtue} the dihedral symmetry is manifest.
Both the Broedel-Carrasco expression and \eqn{fivepointvirtue}
satisfy \eqn{fivepointcolorordered} only after momentum conservation and the BCJ relations
are imposed.
\\ [-2mm]
We can re-express \eqn{fivepointvirtue} in the Kleiss-Kuijf basis
to find
\begin{eqnarray}
{\bf n}(1,2,3,4,5) &=&
{1 \over 10} \Big[
s_{12} \left(s_{13}+s_{23}\right)+s_{45}\left(s_{34}+s_{35}\right)
\Big] A(1, 2, 3, 4, 5)
\nonumber\\ &+&
{1 \over 60} \Big[
s_{12} \left(s_{13}+2 s_{14}-s_{23}-2 s_{25}\right)
+s_{13} \left(s_{14}-s_{15}-2 s_{35}\right)
+s_{23} \left(s_{24}-s_{25}+2 s_{34}\right)
\nonumber\\ &&
+s_{34} \left( s_{24}-s_{14} \right)
+s_{35} \left(s_{25}-s_{15}\right)
+s_{45} \left( -2 s_{14}+2 s_{25}-s_{34}+s_{35} \right)
\Big]
A(1, 4, 3, 2, 5)
\nonumber\\ &+&
{1 \over 60} \Big[
s_{12} \left(3 s_{13}-s_{24}+s_{25}\right)+s_{13} \left(s_{14}-s_{15}+2 s_{34}-2 s_{35}\right)
+ s_{23} \left( s_{24}-s_{25} \right)
\nonumber\\ &&
+ s_{45} \left(-2 s_{14}-s_{15}+3 s_{25}+2 s_{34}+s_{35}\right)
\Big]
A(1, 3, 4, 2, 5)
\nonumber\\ &+&
{1 \over 20} \Big[
s_{12} \left(s_{13}+s_{14}-s_{15}+s_{23}+s_{24}-s_{25}\right)+2 s_{35} s_{45}
\Big]
A(1, 2, 4, 3, 5)
\nonumber\\ &+&
{1 \over 60} \Big[
s_{12} \left(s_{13}+3 s_{14}-s_{15}+2 s_{23}-2 s_{25}\right)
+s_{14} \left(s_{45}-s_{34}\right)
\nonumber\\ &&
+ s_{24} \left(s_{34}-s_{45}\right)+s_{35} \left(-2 s_{13}-s_{15}+2 s_{23}+s_{25}+3 s_{45}\right)
\Big]
A(1, 4, 2, 3, 5)
\nonumber\\ &+&
{1 \over 20} \Big[
2 s_{12} s_{13}+
s_{45}\left(-s_{14}-s_{15}+s_{24}+s_{25}+s_{34}+s_{35}\right)
\Big]
A(1, 3, 2, 4, 5) \,.
\label{kkbasis}
\end{eqnarray}
This expression differs from the result given in ref.~\cite{Fu:2014pya},
but that result apparently contains some typographical errors which will be corrected
in a revised version\footnote{Private communication}.
We emphasize that \eqn{kkbasis} manifestly obeys Jacobi identities and dihedral symmetries
using only the Kleiss-Kuijf relations and the cyclic and reversal properties of the color-ordered
amplitudes (but not momentum conservation or BCJ relations).
\\ [-2mm]
Fu et al. \cite{Fu:2014pya} also give explicit, rather lengthy, expressions
for virtuous numerators for six-point amplitudes.
\section{Virtuous dual-trace functions}
\setcounter{equation}{0}
The color factors $c_i$ and the kinematic numerators $n_i( 1,2, \cdots, n )$
play dual roles in the gauge-theory amplitude (\ref{gluonamp}).
Bern and Dennen \cite{Bern:2011ia} proposed that a role dual
to the traces of generators $\mathop{\rm Tr}\nolimits[\alpha]$ in \eqn{tracedecomp} could be played
by a function $\tau_\alpha( 1,2, \cdots, n )$ of the variables $p_j$ and $\varepsilon_j$
that is related to $n_i( 1,2, \cdots, n )$
in the same way (\ref{colortrace}) that $\mathop{\rm Tr}\nolimits[\alpha]$ is related to $c_i$:
\begin{equation}
n_i ( 1,2, \cdots, n ) ~=~
\sum_{\alpha \in {S_n /\mathbb{Z} } }
M_{i \alpha} \, \tau_\alpha ( 1,2, \cdots, n ) \,.
\label{nMtau}
\end{equation}
Just as the Jacobi identities for the color factors (\ref{cjacobi})
can be expressed as $\sum_i \ell_i c_i = 0$,
where $\ell_i$ are left null vectors of the rank $(n-2)!$ matrix $M_{i\alpha}$,
so the Jacobi identities for the kinematic numerators (\ref{njacobi})
are expressed as $\sum_i \ell_i n_i = 0$,
and will therefore be automatically satisfied by \eqn{nMtau}.
Using \eqn{nMtau}, together with \eqns{colorordered}{doublepartial}, the color-ordered amplitudes can be written
in terms of the dual-trace functions as
\begin{equation}
A_\alpha ( 1,2, \cdots, n ) ~=~
\sum_{\beta \in {S_n /\mathbb{Z} } }
m(\alpha|\beta)
~ \tau_{\beta} ( 1,2, \cdots, n ) \,.
\label{Amtau}
\end{equation}
If we could invert this equation, we would have
a prescription for an amplitude-encoded dual-trace function,
i.e. in which the dependence on the external states is expressed
through the color-ordered amplitudes.
Since $m(\alpha|\beta)$ has rank $(n-3)!$,
there is no unique inverse, but rather a family of generalized inverses,
each corresponding to a particular
gauge choice imposed on the dual-trace functions.
\\ [-2mm]
First we consider the symmetries of the dual-trace functions $\tau_\alpha$.
Because $M_{i\alpha} = M_{i\alpha'}$
when $\alpha$ and $\alpha'$ are related by cyclic permutations,
and $ M_{i\alpha} = (-1)^n M_{i\alpha'} $
when $\alpha'$ is the reverse of $\alpha'$,
we can impose these properties on $\tau_\alpha$ without loss of generality
\begin{eqnarray}
\tau_{\alpha(1)\alpha(2)\cdots \alpha(n)} ( 1,2, \cdots, n )
&= &\tau_{\alpha(2)\alpha(3)\cdots \alpha(1)} ( 1,2, \cdots, n ) \,,
\label{taucyclic}
\\[2 mm]
\tau_{\alpha(1) \alpha(2)\cdots \alpha(n)} ( 1,2, \cdots, n )
&=& (-1)^n \tau_{\alpha(n) \cdots \alpha(2) \alpha(1)} ( 1,2, \cdots, n )
\label{taureverse}
\end{eqnarray}
leaving $(n-1)!/2$ independent dual-trace functions to be determined.
These must satisfy \eqn{nMtau}, which for the independent half-ladder numerators
takes the form
\begin{eqnarray}
{\bf n}_{1 \gamma(2) \cdots \gamma(n-1) n} ( 1,2, \cdots, n ) &=& \tau_{ 1[ \gamma(2), [ \cdots, [\gamma(n-1), n] \cdots ]] } ( 1,2, \cdots, n )
\label{ntau}
\\ [2mm]
&=& \tau_{1 \gamma(2) \cdots \gamma(n-1) n} ( 1,2, \cdots, n ) + (-1)^n
\tau_{1 n \gamma(n-1) \cdots \gamma(2)} ( 1,2, \cdots, n ) + \cdots
\nonumber
\end{eqnarray}
where $+\cdots$ consists of terms
$\tau_{1\gamma(2)\cdots\gamma(n)}$
for which $\gamma(2)\neq n$ and $\gamma(n)\neq n$.
Since there are only $(n-2)!$ kinematic numerators
${\bf n}_{1 \gamma n} $
and $(n-1)!/2$ dual-trace functions $\tau_\alpha$,
there remains a great deal of (gauge) freedom in choosing $\tau_\alpha$.
\\ [-2mm]
In refs.~\cite{Bern:2011ia,Du:2013sha}, Kleiss-Kuijf relations $\sum_\alpha r_\alpha \tau_\alpha = 0$
were imposed on $\tau_\alpha$, reducing the number of independent dual-trace functions to $(n-2)!$.
This is an optional\footnote{
Recall that Kleiss-Kuijf relations follow from the existence of right null
vectors
$\sum_\alpha M_{i\alpha}r_\alpha = 0$
of the matrix $M_{i\alpha}$.
By virtue of \eqn{colorordered},
the color-ordered amplitudes $A_\alpha$ satisfy the
relations $\sum_\alpha r_\alpha A_\alpha =0$.
Kleiss-Kuijf relations do not apply to the trace basis $\mathop{\rm Tr}\nolimits[\alpha]$
and therefore one is not required to impose them on $\tau_\alpha.$
For example, see ref.~\cite{Fu:2013qna}.}
gauge choice, however,
and we choose instead to set to zero all terms of the form
$\tau_{1\gamma(2)\cdots\gamma(n)} ( 1,2, \cdots, n )$
except
\begin{eqnarray}
\tau_{1\gamma(2)\cdots\gamma(n-2)n-1,n} ( 1,2, \cdots, n )
&=& -{1\over 2}\sum_{\beta\in S_{n-3}}S[\gamma|\beta]A(1,\beta(2), \cdots, \beta(n-2) ,n,n-1),
\nonumber\\
\tau_{1,n,n-1\gamma(n-2)\cdots\gamma(2)} ( 1,2, \cdots, n )
&=& ( -1)^n \tau_{1\gamma(2)\cdots\gamma(n-2)n-1,n} ( 1,2, \cdots, n ) \,.
\label{tauchoice}
\end{eqnarray}
All remaining dual-trace functions follow from cyclic invariance (\ref{taucyclic}).
Note that our gauge choice for $\tau_\alpha$ also implies a particular gauge choice for
${\bf n}_{1 \gamma n} $, namely, \eqn{gaugechoice}.
According to ref.~\cite{Bern:2011ia}, the ability to express the dual-trace functions in
terms of kinematic numerators requires us to impose Kleiss-Kuijf relations on $\tau_\alpha$.
If, however, our goal is to write amplitude-encoded dual-trace functions, then this restriction
is not necessary, as we will see explicitly below.
\\ [-2mm]
The dual-trace functions defined in \eqn{tauchoice}
are amplitude-encoded and satisfy \eqn{Amtau},
but they are not symmetric functions in the sense of Broedel and Carrasco.\footnote{In the language of
ref.~\cite{Du:2013sha}, they do not have a natural relabeling property.}
In the appendix, it is shown that we can
follow the same procedure as in the previous section
to generate from \eqn{tauchoice} a symmetric dual-trace function
\begin{equation}
\tau ( 1,2, \cdots, n ) ~\equiv~
{1 \over n!} \sum_{\beta \in S_n} \tau_{\beta^{-1} } (\beta(1), \cdots, \beta(n)) \,.
\label{symmetrictau}
\end{equation}
This then provides a constructive definition
for a virtuous dual-trace function for tree-level $n$-point amplitudes,
proving that such a representation exists for all $n$.
The symmetric dual-trace function can be used to express the $n$-gluon amplitude as
\begin{equation}
{\cal A} ( 1,2, \cdots, n )
~=~
\sum_{\alpha \in {S_n /\mathbb{Z} } } \sum_{\beta \in {S_n /\mathbb{Z} } }
\mathop{\rm Tr}\nolimits[\alpha]
~m(\alpha|\beta)
~\tau (\beta)
\label{Tmtau}
\end{equation}
where $\tau(\beta) \equiv \tau(\beta(1), \beta(2), \cdots, \beta(n))$.
\\ [-2mm]
By substituting \eqn{nMtau} into (\ref{gravamp}),
the scattering amplitude for gravitons may be written
\begin{equation}
{\cal M} ( 1,2, \cdots, n )
~=~
\sum_{\alpha \in {S_n /\mathbb{Z} } } \sum_{\beta \in {S_n /\mathbb{Z} } }
\tau (\alpha) ~ m(\alpha|\beta) ~ \tau (\beta) \,.
\end{equation}
This may equivalently be obtained by replacing $\mathop{\rm Tr}\nolimits[\alpha]$ with $\tau_\alpha$ in \eqn{Tmtau}.
The gravity amplitude may also be written \cite{Bern:2011ia}
\begin{equation}
{\cal M} ( 1,2, \cdots, n )
~=~
\sum_{\alpha \in {S_n /\mathbb{Z} } }
\tau (\alpha) ~ A (\alpha)
\end{equation}
which has the nice feature of demonstrating that
the gravity amplitude is {\it manifestly} invariant under an arbitrary
permutation $\beta \in S_n$:
\begin{equation}
{\cal M} (\beta )
~=~
\sum_{\alpha \in {S_n /\mathbb{Z} } }
\tau (\beta \alpha)
~ A (\beta \alpha)
~=~
\sum_{\alpha' \in {S_n /\mathbb{Z} } }
\tau (\alpha')
~ A (\alpha')
~=~
{\cal M} ( {\mathsf{1}\kern -3pt \mathsf{l} } )\,.
\end{equation}
\\ [-2mm]
In the following subsections, we compute the symmetric dual-trace function (\ref{symmetrictau})
explicitly for four- and five-point amplitudes.
Despite being symmetric, our expressions do not satisfy the Kleiss-Kuijf relations,
illustrating that the virtues of Broedel and Carrasco
are not sufficient to single out a unique dual-trace function.
\\ [-2mm]
\subsection{Four-point symmetric dual-trace functions}
For the four-gluon amplitude, \eqn{tauchoice} yields a nonsymmetric representation
for the dual-trace function
\begin{eqnarray}
\tau_{1234}(i,j,k,l) &=& - {1 \over 2} s_{ij} A (i,j,l,k) \,,
\nonumber\\
\tau_{1342}(i,j,k,l) &=& 0 \,,
\nonumber\\
\tau_{1423}(i,j,k,l) &=& 0
\label{fournonsymtau}
\end{eqnarray}
with the remaining $\tau_\alpha$ given by \eqn{taucyclic}.
We now use \eqn{symmetrictau} to generate a symmetric dual-trace function, obtaining
\begin{equation}
\tau(1,2,3,4) ~=~
- \frac{1}{24}
\Big[
\left(s_{12}+s_{34}\right) A(1,3,4,2) + \left(s_{14}+s_{23}\right)
A(1,4,2,3)
\Big]
\label{fourpointtau}
\end{equation}
where we have used
$A(1,2,3,4)= A(4,3,2,1)$
and the cyclic invariance of $A(1,2,3,4)$.
\Eqn{fourpointtau} manifestly obeys
$\tau(1,2,3,4)= \tau(2,3,4,1)$ and $\tau(1,2,3,4)= \tau(4,3,2,1)$;
however,
$\tau(1,2,3,4) + \tau(1,3,4,2) + \tau(1,4,2,3)$ does not vanish;
{i.e.},
\eqn{fourpointtau} does not satisfy the Kleiss-Kuijf relations.
When substituted into \eqn{ntau}, which takes the form
\begin{eqnarray}
{\bf n}(1,2,3,4) &=& 2 \left[ \tau(1,2,3,4) - \tau(1,2,4,3) \right] \,,
\label{fourntau}
\end{eqnarray}
\eqn{fourpointtau} yields precisely the symmetric kinematic numerator found in \eqn{fourpointvirtue}.
\Eqn {fourpointtau} can be written more briefly as
\begin{equation}
\tau (1,2,3,4)
~=~ - {1 \over 6} s_{12} A(1, 3, 4,2)
\end{equation}
by using momentum conservation and the BCJ relations (\ref{fourpointBCJ}).
\\ [-2mm]
Bern and Dennen proposed that the four-point dual-trace function takes the form \cite{Bern:2011ia}
\begin{equation}
\tau_{BD} (1,2,3,4) ~=~
{1 \over 6} \left[ {\bf n}(1,2,3,4) + {\bf n}(2,3,4,1) \right] \,.
\label{BernDennentau}
\end{equation}
The expression for $\tau$ given in ref.~\cite{Du:2013sha} is equivalent to \eqn{BernDennentau}.
\Eqn{BernDennentau}
manifestly obeys
$\tau(1,2,3,4)= \tau(2,3,4,1)$ and $\tau(1,2,3,4)= \tau(4,3,2,1)$,
and in addition satisfies the Kleiss-Kuijf relation.
To compare our expression with that of Bern and Dennen,
we substitute the symmetric numerator (\ref{fourpointvirtue}) into \eqn{BernDennentau} to find
\begin{eqnarray}
&&\tau_{BD} (1,2,3,4) ~-~ \tau(1,2,3,4)
\\
&&~~~~ = ~\frac{1}{36}
\Big[
\left(s_{12}+s_{14}\right) A(1,2,3,4) +
\left(s_{12}+s_{13}\right) A(1,3,4,2) +
\left(s_{13}+s_{14}\right) A(1,4,2,3)
\Big] \,.
\nonumber
\end{eqnarray}
The difference is ``pure gauge'': it vanishes when substituted into \eqn{fourntau},
and therefore does not contribute to ${\bf n}(1,2,3,4)$.
Hence we see that imposing the three virtues of Broedel and Carrasco
on the dual-trace function is not sufficient to
determine it uniquely.
\\ [-2mm]
To further elucidate the difference between \eqns{fourpointtau}{BernDennentau},
we rewrite \eqn{Amtau}
as a matrix equation
\begin{equation}
\begin{pmatrix}
A_{1234} \\
A_{1342} \\
A_{1423}
\end{pmatrix}
~=~ m
\begin{pmatrix}
\tau_{1234} \\
\tau_{1342} \\
\tau_{1423}
\end{pmatrix},
\qquad\qquad
m ~=~-~{2\over s t u} \begin{pmatrix} u \\ t \\ s \end{pmatrix} \begin{pmatrix} u &t & s\end{pmatrix}
\label{matrix}
\end{equation}
where $s=s_{12}$, $t=s_{14}$, and $u=s_{13}$.
As expected, $m$ has rank $(4-3)! = 1$.
It therefore does not possess a unique inverse, but rather a
family of generalized inverses $m^+$
\begin{equation}
\begin{pmatrix}
\tau_{1234} \\
\tau_{1342} \\
\tau_{1423}
\end{pmatrix}
~=~ m ^+
\begin{pmatrix}
A_{1234} \\
A_{1342} \\
A_{1423}
\end{pmatrix},
\qquad\qquad m m^+ m= m \,.
\end{equation}
A generalized inverse must satisfy $m m^+ m= m$ \cite{BenIsrael,Boels:2012sy},
which guarantees that the resulting $\tau_\alpha$ will satisfy \eqn{matrix}.
\\ [-2mm]
Different gauge choices for $\tau$ correspond to different generalized inverses.
The nonsymmetric representation (\ref{fournonsymtau})
corresponds to
\begin{equation}
m_{\rm nonsym}^+ ~=~-~{1\over 2} \begin{pmatrix} 0 & s & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{pmatrix} \,.
\end{equation}
The symmetric dual-trace function (\ref{fourpointtau}) corresponds to
\begin{equation}
m_{\rm sym}^+ ~=~-~{1\over 12} \begin{pmatrix} 0 & s & t \\ s & 0 & u \\ t & u & 0 \end{pmatrix}
\end{equation}
whereas the Bern-Dennen dual-trace function corresponds to
\begin{equation}
m_{BD}^+ ~=~{1\over 36}
\begin{pmatrix}
-u & -3 s-t & -3t-s \\
-3 s-u & -t & -3u-s \\
-3 t-u & -3u-t u & -s \\
\end{pmatrix} \,.
\end{equation}
All three generalized inverses satisfy $m m^+ m= m$.
\subsection{Five-point symmetric dual-trace functions}
For the five-gluon amplitude, \eqn{tauchoice} yields a nonsymmetric representation
\begin{eqnarray}
\tau_{12345}(i,j,k,l,m) &=&
- {1\over 2} \left[ s_{ij} (s_{ik}+s_{jk} ) A(i,j,k,m,l) + s_{ij} s_{ik} A (i,k,j,m,l)\right]\,, \nonumber\\
\tau_{13245}(i,j,k,l,m) &=&
- {1\over 2} \left[ s_{ij} s_{ik} A(i,j,k,m,l) + s_{ik} (s_{ij}+s_{jk} ) A(i,k,j,m,l) \right]
\label{fivepointnonsymtau}
\end{eqnarray}
with all remaining dual-trace functions set to zero, except for those related to \eqn{fivepointnonsymtau} by \eqn{taucyclic}.
We now use \eqn{symmetrictau} to generate a symmetric dual-trace function, obtaining
\begin{eqnarray}
\tau(1,2,3,4,5) &=&
\frac{1}{120}
\Big[
(s_{23} s_{24}+s_{34} s_{24}+2 s_{23} s_{34}) A(1, 4, 3, 2, 5)
+(s_{23} s_{24}+s_{12} s_{25}) A(1, 3, 4, 2, 5)
\nonumber\\ &-&
(2 s_{12} s_{15}+s_{25} s_{15}+s_{12} s_{25}) A(1, 2, 4, 3, 5)
+(s_{24} s_{34}+s_{14} s_{45}) A(1, 4, 2, 3, 5)
\nonumber\\ &-&
(s_{14} s_{15}+2 s_{45} s_{15}+s_{14} s_{45}) A(1, 3, 2, 4, 5)
+(s_{34} s_{35}+s_{45} s_{35}+2 s_{34} s_{45}) A(1, 2, 5, 4, 3)
\nonumber\\ &+&
(s_{12} s_{13}+s_{14} s_{15}) A(1, 4, 5, 2, 3)
+(s_{15} s_{25}+s_{35} s_{45}) A(1, 2, 5, 3, 4)
\nonumber\\ &+&
(s_{12} s_{13}+s_{23} s_{13}+2 s_{12} s_{23}) A(1, 4, 5, 3, 2)
-(s_{13} s_{23}+s_{34} s_{35}) A(1, 3, 5, 4, 2)
\Big]
\label{fivepointtau}
\end{eqnarray}
where we have used
$ A(1,2,3,4,5)= -A(5,4,3,2,1) $
together with cyclic invariance of $A(1,2,3,4,5)$.
This expression manifestly satisfies $\tau(1,2,3,4,5)=\tau(2,3,4,5,1)$
and $\tau(1,2,3,4,5) = - \tau(5,4,3,2,1)$, but not the Kleiss-Kuijf relations.
It therefore differs from the expressions given
in refs. \cite{Bern:2011ia,Du:2013sha},
which do satisfy the Kleiss-Kuijf relations.
\\ [-2mm]
\Eqn{fivepointtau} yields precisely the symmetric kinematic numerator \eqn{fivepointvirtue}
when substituted into \eqn{fiventau}, which takes the form
\begin{eqnarray}
{\bf n}(1,2,3,4,5) &=&
2 \left[\tau(1,2,3,4,5) + \tau(1,2,5,4,3) + \tau(1,4,5,3,2) + \tau(1,3,5,4,2) \right] \,.
\label{fiventau}
\end{eqnarray}
Finally, the Kleiss-Kuijf relations for the color-ordered amplitudes can be used to rewrite \eqn{fivepointtau}
as
\begin{eqnarray}
{\bf \tau}(1,2,3,4,5) &=&
{1 \over 120} \Big[
(
s_{12} s_{13}+2 s_{12} s_{23}+2 s_{34} s_{45}+s_{35} s_{45}
) A(1, 2, 3, 4, 5)
\nonumber\\ &+&
(
s_{12} s_{13}+s_{14} s_{15}+s_{23} s_{24}+s_{15} s_{25}+2 s_{23} s_{34}+s_{24} s_{34}+s_{35} s_{45}
)A(1, 4, 3, 2, 5)
\nonumber\\ &+&
(
s_{12} s_{13}+s_{14} s_{15}+s_{23} s_{24}+s_{12} s_{25}+s_{34} s_{35}+2 s_{34} s_{45}+s_{35} s_{45}
)A(1, 3, 4, 2, 5)
\nonumber\\ &+&
(
s_{12} s_{13}-2 s_{12} s_{15}+2 s_{12} s_{23}-s_{12} s_{25}-s_{34} s_{35}+s_{35} s_{45}
) A(1, 2, 4, 3, 5)
\nonumber\\ &+&
(
s_{12} s_{13}+s_{23} s_{13}+2 s_{12} s_{23}+s_{15} s_{25}+s_{24} s_{34}+s_{14} s_{45}+s_{35} s_{45}
)A(1, 4, 2, 3, 5)
\nonumber\\ &+&
(
s_{12} s_{13}-s_{23} s_{13}-s_{14} s_{45}-2 s_{15} s_{45}+2 s_{34} s_{45}+s_{35} s_{45}
)A(1, 3, 2, 4, 5)
\Big]\,.
\nonumber\\
\end{eqnarray}
\section{Discussion}
\setcounter{equation}{0}
In this paper, we have offered a constructive procedure for computing
virtuous kinematic numerators for $n$-point gauge-theory scattering amplitudes;
that is, numerators that simultaneously satisfy color-kinematic duality,
are expressed in terms of color-ordered amplitudes,
and are symmetric in the sense of Broedel and Carrasco.
We have presented explicit expressions
for four- and five-point amplitudes,
which although somewhat lengthy,
have the advantage of manifestly obeying the Jacobi identities and
diagram symmetries without having to invoke the BCJ relations
for the color-ordered amplitudes.
Our results are equivalent (upon using the BCJ relations)
to other virtuous expressions in the literature,
suggesting the possibility that the three virtues of Broedel and Carrasco
are sufficient to uniquely determine the kinematic numerators,
although we do not have a proof of this.
\\ [-2mm]
We have also applied this procedure to
compute symmetric, amplitude-encoded dual-trace functions $\tau$ for $n$-point amplitudes,
presenting explicit expressions for four- and five-point amplitudes.
In this case, the results are not uniquely determined by these criteria alone.
In particular, symmetric expressions for $\tau$ obtained by other authors
additionally satisfy (optional) Kleiss-Kuijf relations, whereas our results do not.
While it is possible that a constructive procedure could be found to
generate virtuous dual-trace functions
for $n$-point amplitudes that also satisfy Kleiss-Kuijf relations,
it is also possible that an alternative criterion
to single out a unique dual-trace function
may be more useful or natural.
\section*{Acknowledgments}
I am grateful to Johannes Broedel for useful correspondence.
I especially wish to thank Freddy Cachazo
and Ellis Yuan for crucial conversations
at the outset of this project.
This research was supported in part by the NSF under grant no. PHY10-67961.
|
2,877,628,089,045 | arxiv | \chapter*{#1}
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\begin{document}
\rhead{\bfseries Solar Neutrinos}
\mchapter{Solar Neutrinos}
{Authors:\ V. Antonelli$^a$, L. Miramonti$^a$, C. Pe\~na-Garay$^b$ and A. Serenelli$^c$}
\label{ch-14:mycontribution}
\vspace{0.5cm}
\begin{center}
$^a$ {\it Dipartimento di Fisica, Universit\'a degli Studi di Milano and INFN Milano, Via Celoria 16,\\
I-20133 Milano, Italy} \\ [6pt]
$^b$ {\it Instituto de Fisica Corpuscular, CSIC-UVEG, Valencia E-46071, Spain}\\ [6pt]
$^c$ {\it Instituto de Ciencias del Espacio (CSIC-IEEC), Facultad de Ciencias, Campus UAB, Bellaterra, 08193, Spain}
\end{center}
\vspace{3cm}
\begin{center}
{\bf Abstract}
\end{center}
The study of solar neutrinos has given since ever a fundamental contribution both to
astroparticle and to elementary particle physics, offering an ideal test of solar models
and offering at the same time relevant indications on the fundamental interactions among
particles. After reviewing the striking results of the last two decades, which were determinant
to solve the long standing solar neutrino puzzle and refine the Standard Solar Model,
we focus our attention on the more recent results in this field and on the experiments
presently running or planned for the near future. The main focus at the moment is
to improve the knowledge of the mass and mixing pattern and especially to study in detail
the lowest energy part of the spectrum, which represents most of solar neutrino spectrum
but is still a partially unexplored realm. We discuss this research project and the
way in which present and future experiments could contribute to make the
theoretical frawemork more complete and stable, understanding the origin of some
``anomalies'' that seem to emerge from the data and contributing to answer some present
questions, like the exact mechanism of the vacuum to matter transition and the
solution of the so called solar metallicity problem.
\section{Motivations for the solar neutrino study}
The analysis of neutrinos emitted in the fusion processes inside the Sun is
one of most significant examples of the relevant role played by the study of
neutrino properties in elementary particle physics and astrophysics and in
creating a link between these two sectors. The pioneering work in the
sixties \cite{14-Davis1968cp} had the main goal of
understanding better the way in which our star shines and to test solar
models. But, the surprising result of an apparent deficit in the electron
neutrino flux reaching the detector marked the raise of the so called solar
neutrino puzzle, and opened a whole new field of research that has been
central in elementary particle physics for many decades.
The experimental results obtained using different techniques in more than
thirty years and the parallel theoretical advancements confirmed at the end
the validity of Pontecorvo's revolutionary idea of neutrino oscillation
\cite{14-Pontecorvo}, proving in a crystal clear way that neutrinos are
massive and oscillating particles. This is one of the first pieces of clear
evidence of the need to go beyond the Standard Model of electroweak
interactions and the attempt to accommodate the experimental results about
neutrino masses and mixing is a test every theory ``beyond the Standard
Model'' has to pass. Therefore, it is clear why these results have had a great
impact on elementary particle physics and also on cosmological models. At the
same time, the possibility of measuring directly at least some components of
the solar neutrino spectrum and of recovering in an indirect way the value of
total solar neutrino flux have been fundamental for the progressive refinement
of the Standard Solar Model (SSM), which has evolved during these years and
is now in a general good agreement with the solar neutrino experiments.
Despite the fundamental steps forward made in the last decades, many questions
are still open about the real nature and the main properties of neutrinos and
the exact mixing mechanism, e.g. are neutrinos Majorana or Dirac fermions,
determination of mass hierarchy and exact mass values, accurate determination
of the mixing angles, presence of CP violation. The solar neutrino experiments
presently running or planned for the future can contribute to solve at least
some of these puzzles. The new frontier in this field is the study of the low
energy part of the solar neutrino spectrum which represents the great majority
of the spectrum, and is still an almost unexplored realm. Some of the
challenges ahead are: reducing significantly the indetermination on {\it pep}
and CNO neutrinos and attaching the {\it pp} solar neutrino measurement.
This would be essential to test the stability and consistency of the standard
explanation of the oscillation mechanism, confirming or definitely disproving
the presence of discrepancies between theory and experiments, which has lately
stimulated a flourishing of models introducing the so called ``Non Standard
Interactions'' (Section~\ref{14-sec:status-mixing}).
Once more, these results would be of great interest to
improve the knowledge both of elementary particle properties and interactions
and of the astrophysical models of the Sun. They could help also to
discriminate between different versions of the solar models, for instance of
the so-called ``solar abundance problem'', and to deepen the comparison with
the results coming from other studies of solar properties, e.g. from
helioseismology. This research project would of course imply a further
improvement of the already known detection techniques and the introduction of
new ones (see, for instance, the section \ref{14-sec:future}). Also from this
point of view, solar neutrino physics will continue to give a stimulating
contribution both to elementary particle physics and to astrophysics.
\section{Brief history and solution of the solar neutrino problem}
\subsection{From Homestake to Super-Kamiokande}\
The first experiment built to detect solar neutrinos took place in the
Homestake gold mine in South Dakota \cite{14-Davis1968cp}. The detector
consisted of a large tank containing 615 tons of liquid perchloroethylene,
chosen because it is rich in chlorine and the experiment operated continuously
from 1970 until 1994. Neutrinos were detected via the reaction:
\begin{equation}
\nu_{e} + {\rm ^{37}Cl} \rightarrow {\rm ^{37}Ar} + e^{-} \, .
\label{eq:homestake}
\end{equation}
The energy threshold of this reaction, $E_{th} = 814 \: \rm{keV}$, allowed the
detection of $^{7}$Be and $^{8}$B (and a small signal from the CNO and
{\it pep}) but not that of {\it $pp$} neutrinos, because of their low maximal
energy of $0.42 \: \rm{MeV}$. The radioactive $^{37}Ar$ isotopes decay by
electron capture with a $\tau_{1/2}$ of about 35 days into ${\rm ^{37}Cl^{*}}$:
\begin{equation}
{\rm ^{37}Ar} + e^{-} \rightarrow {\rm ^{37}Cl^{*}} + \nu_{e} \, . \label{eq:argon37}
\end{equation}
Once a month, after bubbling helium through the tank, the $^{37}$Ar atoms were
extracted and counted. The number of atoms created was only about $5$ atoms of
$^{37}$Ar per month in 615 tons ${\rm C_{2}Cl_{4}}$. The number of detected
neutrinos was about 1/3 lower than expected by the Solar Standard Model.
This discrepancy is the essence of the Solar Neutrino Problem, which
has been for many years an important puzzle among physicists.
There were three possible explanations to the Solar Neutrino Problem. The
first one was to consider that Homestake could be wrong, i.e. the Homestake
detector could be inefficient and, in this case, its reactions would not have
been cpredicted correctly. After all, to detect a handful of atoms per week in
more than 600 tons of material is not an easy task. The second one was to
consider that the SSM was not correct, but as helioseismology\footnote{The science
that studies the interior of the Sun by looking at its vibration modes.} started
to provide independent tests of solar models the SSM
passed all tests. Indeed, non-standard solar models constructed
ad-hoc to resolve the Solar Neutrino Problem seemed very unlikely when
scrutinized under the light of helioseismology. The third one, and the
strangest hypothesis, was to consider that something happens to the neutrinos
while traveling from the core of the Sun to the Earth.
The first real time solar neutrino detector, Kamiokande, was built in Japan in
1982-83 \cite{14-Kamiokande89}. It consisted of a large water \v{C}erenkov
detector with a total mass of 3000 tons of pure water. In real time neutrino
experiments scientists study the bluish light produced by the electrons
scattered by an impinging neutrino according to the following equation:
\begin{equation}
\nu_{x} + e^{-} \rightarrow \nu_{x} + e^{-} \, .
\label{eq:ES}
\end{equation}
In the Kamiokande detector light is recorded by 1000 photomultiplier tubes
(PMT) and the energy threshold of the reaction is $E_{th} = 7.5\; {\rm MeV}$;
therefore only $^{8}$B and $hep$ neutrinos are detected. At the beginning of
the '90s a much larger version of the detector was built, Super-Kamiokande,
where the active mass was 50000 tons of pure water viewed by 11200 PMTs. In
Super-Kamiokande the energy threshold was lowered to $E_{th} = 5.5\; {\rm
MeV}$ \cite{14-SK99}.
Radiochemical experiments integrate in time and in energy because they are
slow and need time to produce measurable results. This causes the loss of
information about single individual energy values. In real time experiments,
instead, it is possible to obtain single values and therefore a spectrum
energy to distinguish the different neutrino contributions. Furthermore,
given that the scattered electron maintains the same direction of the
impinging neutrino, it is possible to infer the direction of the incoming
neutrino and therefore to point at its source. This proved that the detected
neutrinos actually came from the Sun. The number of detected neutrinos was
about 1/2 lower than the number of expected ones, aggravating the Solar
Neutrino Problem.
Until 1990 there were no observations of the initial reaction in the nuclear
fusion chain, i.e. the detection of ${\it pp}$ neutrinos, which are less
model-dependent and hence more significant to test the hypothesis that fusion
of hydrogen powers the Sun. Two radiochemical experiments were built in order
to detect solar ${\it pp}$ neutrinos, both employing the reaction:
\begin{equation}
\nu_{e} + {\rm ^{71}Ga} \rightarrow {\rm ^{71}Ge} + e^{-} \, .
\label{eq:gallex-sage}
\end{equation}
which has a threshold of $E_{th} = 233\; {\rm keV}$.
In the Gallex experiment, located at the Gran Sasso underground laboratory in
Italy, 30 tons of natural gallium were employed
\cite{14-Hamp99,14-Alt05},\ while in the soviet-american experiment (SAGE),
located in the Baksan underground laboratory, there were 50 tons of metallic
gallium \cite{14-Abd99}. Calibration tests with an artificial neutrino
source, $^{51}$Cr, confirmed the efficiency of both detectors. Once again the
measured neutrino signal was smaller than predicted by the SSM ($\approx\;
60\%$).
All experiments detected fewer neutrinos than expected from the SSM. Table
\ref{tab:observed vs expected ratio in the four experiments (before SNO)}
summarizes the observed vs expected ratio for all experiments.
\begin{table}[h]
\centering
\vspace{0.5truecm}
\begin{tabular}{c c}
\hline \hline
Homestake & $0.34 \pm 0.03$ \\
Super-K & $0.46 \pm 0.02$ \\
SAGE & $0.59 \pm 0.06$ \\
Gallex and GNO & $0.58 \pm 0.05$ \\
\hline \hline
\end{tabular}
\caption{Observed vs expected ratio in the four experiments (before SNO, see
later).}
\label{tab:observed vs expected ratio in the four experiments (before SNO)}
\end{table}
\vspace{1cm}
\subsection{The advent of SNO and Kamland: the solution of the Solar Neutrino
Problem}\
The real breakthrough in solar neutrino physics was due to the advent of the
SNO (Sudbury Neutrino Observatory) experiment. It had the peculiarity to
measure simultaneously, by means of a deuterium \v{C}erenkov detector, three
different interaction channels for neutrinos: the neutral current (NC: $\nu_X
+ d \to \nu_X + p^+ + n$), receiving contributions from all active flavors,
the elastic scattering (ES: $\nu_X + e^- \to \, \nu_X + e^- $) and the charged
current (CC: $\nu_e + d \to e^- + p^+ + p^+$), that is sensitive only to
electronic neutrinos. In this way it has been possible to prove in a clear
and direct way that the measured total neutrino flux was in very good
agreement with the SSM predictions, but only a fraction of these neutrinos had
conserved its flavor during their way from the production point in the Sun to
the detector.
The first SNO data \cite{14-SNOES}, including elastic scattering and charged
current analysis, published in 2001, confirmed the results obtained by
previous solar neutrino experiments, mainly by Super-Kamiokande
\cite{14-SK2001}, providing a significant evidence (at the $3.3 \, \sigma$
level) of the presence of a non-electronic active neutrino component in the
solar flux. For the first time it was possible to indicate the Large Mixing
Angle (LMA) as the preferred solution of the solar neutrino puzzle, even if
different alternative possibilities (and in particular the low probability,
low mass -LOW- solution) were still surviving \cite {14-Global2001}. In the
following years, the SNO experiment measured also the neutral current channel,
using different techniques. The data of these different ``phases'' of the
experiment are usually reported as SNO I \cite{14-SNOI}, SNO II
\cite{14-SNOII} (characterized by the addition of salt to improve the
efficiency of neutral current detection) and SNO III \cite{14-SNOIII} (with
the use of helium chamber proportional counters).
The year 2002 is very often denoted as the ``annus mirabilis'' of solar
neutrino physics: on April the first SNO results including neutral current
detection \cite{14-SNOI,14-Ahmad2002ka} marked a turning point in the history
of the solar neutrino problem, in October the Nobel prize for physics was
awarded to R. Davis Jr. \cite{14-Davis2002fb} and M.Koshiba (for their
pioneering work on the detection of cosmic neutrinos) and on December of the
same year the first results of the Kamiokande Liquid scintillator AntiNeutrino
Detector (KamLAND) \cite{14-firstKL} offered the first clear terrestrial
confirmation of the validity of the oscillation solution to the solar neutrino
problem.
The total $^8$B neutrino flux,
$\phi_{NC}= 5.09^{+0.44}_{-0.43}(\rm stat)^{+0.46}_{-0.43}(syst) \times 10^6
\, cm^{-2}\ s^{-1}$,
measured by SNO with neutral currents was in very good agreement
with the SSM
\cite{14-bahcall:2001}. Assuming the standard shape for the component of the
solar neutrino flux (undistorted spectrum hypothesis), the SNO collaboration
recovered also a value of the non-electronic component of the flux
which was $5.3 \, \sigma$ different from zero, providing a direct proof of the
validity of the oscillation hypothesis. These data were also decisive to
indicate the LMA region as the solution to the solar neutrino puzzle.
Looking at the oscillation probability\footnote{For instance, in a simple 2
flavor analysis, the flavor transition probability is given by the
expression \hbox{$P_{12} = \sin{^2 (2 \theta_{12})} \sin{^2
\left(\frac{\Delta m_{12}^2 ({\rm eV^2}) \, L({\rm km})}{4 E ({\rm
GeV})}\right)}$}, where $\theta_{12} $ is the mixing angle between
the two flavors, $\Delta m^2_{\rm ij}\equiv m^2_{\rm 1}-m^2_{2}$ the
difference of the masses squared, $L$ the distance traveled, and $E$ the
neutrino energy.}, it is apparent that the reactor experiments that run
before KamLAND, and used neutrino energy beams of the order of the MeV with a
baseline of the order of 1 km, could test only values of $\Delta m^2$ above
$10^{-3}$ $\rm{eV}^2$. The KamLAND experiment, instead, with an average
baseline of about 180 km, was ideal to probe the LMA region, which corresponds
to values of $\Delta \rm{m}^2$ of the order $10^{-5}-10^{-4} \, \rm{eV}^2$
\cite{14-SK_KLpotentialities}. The KamLAND experiment studied the ratio of
the number of inverse $\beta$ decay events (due to reactor $\bar{\nu}_e$ with
an energy threshold of 3.4 MeV) to the expected number of events without
disappearance and also the spectrum shape \cite{14-firstKL}. The observed
deficit of events was inconsistent with the expected rate in absence of
oscillation at the $99.95 \%$ confidence level.
Since one would expect a negligible reduction of the $\bar{\nu}_e$ flux from
the SMA, LOW and vacuum solar neutrino solutions, the LMA was the only
oscillation solution compatible with KamLAND results and CPT invariance. This
evidence were further reinforced by the data published by the collaboration in
the following years (with greater statistical precisions and reduced
systematic errors), which showed also a spectral distortion in very good
agreement with the oscillation solution \cite{14-KL2004,14-KLfollowing2,
14-KLfollowing3}. KamLAND data also restricted the allowed LMA region in a
significant way. The preferred values for $\Delta m^2_{12}$ and $\theta_{12}$
are slightly higher than the ones corresponding to the best fit solution of
the solar neutrino experiments, but this small tension can be explained by
taking into account the experimental uncertainties. Moreover, the difference
on the $\Delta m^2_{12}$ parameter has been reduced by the more recent solar
neutrino data.
\section{Standard Solar Model}
\label{14-section:SSM}
SSMs have to be understood, primarily, as a framework within which solar models can
be constructed and clear predictions can be made with respect to the properties of
the solar interior, including the production of solar neutrinos. The defining
characteristics are simple: the SSM is the result of the evolution of a 1~${\rm M_{\odot}}$
star since its formation and, the evolutionary models have to include the
physical ingredients considered {\it standard} in stellar structure and evolution
models (here, {\it standard} also implies trying to keep to a minimum the number of
free tunable parameters -knobs- in the model). SSMs are therefore progressively
refined as our understanding of stellar physics progresses.
In practice, a SSM is constructed as follows. An initial chemically homogeneous
model of a 1~${\rm M_{\odot}}$ stellar model on the \hbox{pre-main} sequence is constructed
with a composition determined by a guess (educated one) for the initial mass
fractions of hydrogen $X_{\rm ini}$,
helium $Y_{\rm ini}$, and metals $Z_{\rm ini}$ ($X_{\rm ini}+Y_{\rm ini}+Z_{\rm ini}=1$); additionally, a third
free parameter has to be specified, the mixing length parameter $\alpha_{\rm MLT}$
of convection. This model is then evolved up to the
solar system age \hbox{$\tau_\odot= 4.57$~Gyr} \cite{14-bahcall:1982,
14-bahcall:1989}. At this age the model is required to match the present-day
solar
luminosity ${\rm L_{\odot}}$ and radius ${\rm R_{\odot}}$, as well as the surface
metal-to-hydrogen abundance ratio $\left(Z/X\right)_\odot$. The initial and final surface
\hbox{metal-to-hydrogen} ratios differ by about 10 to 15\% due to the effects
of gravitational settling. In general, the SSM constructed with the first set of
guesses for
$\alpha_{\rm MLT}$, $Y_{\rm ini}$, and $Z_{\rm ini}$ will not lead to a satisfactory
agreement with the surface constraints, and an iterative procedure is used to refine
the free parameters until the right surface conditions are achieved at $\tau_\odot$.
In general, surface conditions are matched to one part in $10^5$ or $10^6$ within two
or three iterations. It is important to keep in mind
that the SSM is not just a snapshot aimed at representing the present-day
structure of the Sun, but actually the result of taking into account all its previous
history. There are alternative ways to construct a model of the
present-day solar structure using, for example, helioseismic constraints.
These kind of models are constructed `ad-hoc' to match helioseismic data and
have, therefore, limited predictive power.
The internal structure of a SSM depends on the values adopted for the three
constraints mentioned above and, of course, on the physical inputs of the
models such as the radiative opacities, cross sections of nuclear reactions
and others. Next we describe the changes/updates that have occurred during
the last decade that impact predictions of solar models.
\subsection{Input physics and parameters}
\subsubsection{Solar Surface Composition}\label{14-sec:solarcompo}
The constraint imposed by the surface metallicity of the Sun or, more precisely, the
surface metal-to-hydrogen ratio $\left(Z/X\right)_\odot$, is critical in the construction of
solar models. The reason is that, aside from the 10 to 15\% change in this value due
to the action of gravitational settling, $\left(Z/X\right)_\odot$ determines almost directly the
metallicity of solar models. As for any other star, the metal content in the Sun
has a fundamental role in its structure through its contribution to the radiative
opacity $\kappa$, which determines, in turn, the temperature gradient in the
radiative solar interior. It is important, in fact, that the abundance of
individual metals are accurately determined, because different
elements contribute to the radiative opacities in different regions of the Sun.
The abundance of metals in the solar surface has to be determined or inferred
from a variety of sources: photospheric abundances from solar
spectra, chemical analysis of primitive meteorites, emission lines from the
solar corona, composition of the solar wind \cite{14-lodders:2009}. While
meteoritic abundances are the most precisely determined, at least 2/3 of the
solar metallicity is composed by the volatile elements C, N, and O and can
only be determined from analysis of the solar spectrum.
Over the last decade, the development of three-dimensional radiation
hydrodynamic (3D RHD) models of the solar atmosphere has prompted a thorough
revision of the solar composition determined from the solar spectrum. These 3D
RHD models of the solar atmosphere capture the dynamics of convection and its
interaction with the radiation field, and are able to reproduce features such
as the solar granulation pattern, observed limb-darkening, asymmetries in the
shapes of spectral lines \cite{14-beeck:2012}. The structure of the solar
model atmospheres derived by different groups are nicely consistent with each
other, adding to the credibility of the models. Newly derived spectroscopic
abundances rely on the 3D atmosphere model, or more appropriately on a one-
dimensional model obtained from a suitably averaged 3D model,
as the background on top of which detailed radiative
transfer and line formation calculations are performed {\it a posteriori}.
It is this second step that leads, finally, to
the determination of the abundances of the different elements. The most
thorough and consistent determination of the solar photospheric abundances
based on 3D model atmospheres has been presented by Asplund and collaborators
\cite{14-asplund:2009, 14-asplund:2005}, although revision on key elements
like oxygen were initially published already in 2001 \cite{14-allendeprieto:2001}.
In addition to using 3D RHD atmosphere models, non-local thermodynamic
equilibrium has been taken into account when computing line formation for some key
elements such as C, N, and O. Also, and this is of particular importance for
oxygen, blends in the solar spectrum that had been previously unnoticed were
identified and taken into account in the determination of abundances. The
most relevant result in the context of solar models and neutrinos is that
abundances of CNO elements (also Ne, but this is mostly because its abundance ratio
to oxygen is assumed fixed) have been revised down by 30 to 40~\%. Combining the
abundance of all
metals, the present-day metal-to-hydrogen ratio that has been obtained is
$\left(Z/X\right)_\odot=0.0178$ \cite{14-asplund:2009}. This represents a large decrease in
comparison with
previously accepted values, 0.0245 \cite{14-grevesse:1993} and 0.0229
\cite{14-grevesse:1998}, that have been widely used in solar modeling. We
note, however, that results by Asplund have not been unchallenged. In fact,
also based on 3D RHD model solar atmospheres, larger CNO abundances have been
derived \cite{14-caffau:2011} to yield $\left(Z/X\right)_\odot= 0.0209$, much closer to older
determinations. Discrepancies between authors seem to have their origin at the
preferred set of spectral lines each group uses and on using either a spectral
synthesis or equivalent width techniques to determine the final abundances.
In the last decade there have been two flavors in SSM calculations. In one
case a {\em high} solar metallicity from older determinations
\cite{14-grevesse:1993,14-grevesse:1998} is adopted; we will generically refer
to these models
as {\em high-Z} solar models. In the other case a {\em low} $\left(Z/X\right)_\odot$
\cite{14-asplund:2009,14-asplund:2005} is taken from and we refer to these,
not surprisingly, as the {\em low-Z} solar models. Differences in the
structure of {\em high-Z} and {\em low-Z} models are readily noticeable in
quantities such as the internal sound speed and density profiles, the depth of
the solar convective envelope, and the surface helium abundance among
others. The deficit that {\em low-Z} models have in matching helioseismic
constraints has been named the solar abundance problem in the literature, in clear
analogy to the solar neutrino problem. We discuss it in some detail in Section~\ref
{14-sec:helios}.
\subsubsection{Radiative Opacities}
The most widely used calculations of atomic radiative opacities, appropriate
for solar interiors, are those from OPAL \cite{14-iglesias:1996}. However, the
Opacity Project (OP) released in 2005 a completely independent set of
atomic radiative opacities for stellar interiors \cite{14-badnell:2005}. In
the case of the solar radiative interior, differences between OPAL and OP
Rosseland mean opacities are of the order of a few percent, with OP being
larger by about 3\% at the base of the convective zone and 1 to 2\% smaller in
the central regions (see Fig. 7 in \cite{14-badnell:2005}).
At low temperatures, at which molecules can form, neither OP or OPAL atomic
opacities are adequate and have to be complemented by low-temperature
opacities \cite{14-ferguson:2005}. Due to the relatively high solar
temperature, their influence in the properties of solar models is rather
limited.
\subsubsection{Nuclear reactions cross sections}
Experimental and theoretical work on the determination of nuclear cross
sections have been very active fields with a strong impact on solar
model predictions of solar neutrino fluxes. Recently, a set of recommended rates
and uncertainties, expressed through the $S$-factor\footnote{A
non-resonant charged-particle induced reaction cross section can be written as
$\sigma (E)=\frac{S(E)}{E}
{\rm exp \left[ -2 \pi \eta(E)\right]}$ where $\eta(E)= Z_1 Z_2 \alpha/v$ is
the Sommerfeld parameters, $v=\sqrt(2E/\mu)$, $\alpha$ the fine structure
constant in natural units, and $\mu$ the reduced mass of the interacting
nuclei. The nuclear physics is isolated in $S(E)$, the
astrophysical or S-factor, a slowly
varying function of energy that can be more accurately extrapolated from
experimental data down to the energy of the Gamow peak.}, for
all the reactions both in the pp-chains and CNO-bicycle that are relevant to
solar modeling and neutrino production, has been
published (Solar Fusion II, \cite{14-adelberger:2011}, hereafter SFII). The
results presented in SFII reflect the progress made in laboratory and
theoretical nuclear astrophysics over the last decade, since the publication
of the seminal Solar Fusion I (SFI) article \cite{14-adelberger:1998} .
Unfortunately, for reasons of space, here we cannot review in detail every
reaction. Instead, we provide in Table~\ref{14-tab:nucrates} the standard
$S$-factors at zero energy, $S(0)$, and the uncertainties recommended in SFII
for the most relevant reactions. For comparison, with results from SFI are also
shown. The impact of changes in key reactions on the production of neutrino
fluxes is discussed in Section~\ref{14-sec:fluxes}. The reader is referred to the
SFII paper and references therein for details on the experimental and theoretical
developments in nuclear astrophysics related to the Sun during the last
decade.
\begin{table}[h]
\begin{center}
\begin{tabular}{llcc}
\hline \hline
\multicolumn{2}{c}{Reaction} & SFII & SFI \\
& & $S(0) \ \left[\hbox{keV b}\right]$ & $S(0) \ \left[\hbox{keV b}\right]$ \\
\hline
${\rm S_{11}}$ & $p(p,{\rm e}^+\nu_e)d$ & $4.01\times 10^{-22} \left(1 \pm
0.010\right)$ & $4.00\times 10^{-22} \left(1 \pm 0.005\right)$ \\
${\rm S_{33}}$ & ${\rm ^3He(^3He},2p){\rm ^4He}$ & $5.21\times 10^{3} \left(1
\pm 0.052\right)$ & $5.4\times 10^{3} \left(1 \pm 0.074\right)$ \\
${\rm S_{34}}$ & ${\rm ^3He(^4He},\gamma){\rm ^7Be}$ & $5.6\times 10^{-1}
\left(1 \pm 0.054\right)$ & $5.3\times 10^{-1} \left(1 \pm 0.094\right)$ \\
${\rm S_{hep}}$ & ${\rm ^3He}(p,{\rm e}^+\nu_e){\rm ^4He}$ & $8.6\times 10^{-20}
\left(1 \pm 0.30\right)$ & $2.3\times 10^{-20}$ \\
${\rm S_{17}}$ & ${\rm ^7Be}(p,\gamma){\rm ^8B}$ & $2.08\times 10^{-2} \left(1
\pm 0.077\right)$ & $1.9\times 10^{-2} \left(1 ^{+ 0.20}_{-0.10}\right)$ \\
${\rm S_{1,14}}$ & ${\rm ^{14}N}(p,\gamma){\rm ^{15}O}$ & $1.66 \left(1 \pm
0.072\right)$ & $3.5 \left(1 ^{+0.11}_{-0.46} \right)$ \\
\hline
\end{tabular}
\caption{Standard astrophysical factors and uncertainties for key nuclear
reactions in the pp-chains and CNO-bicycle. SFII represents the
state-of-the-art \cite{14-adelberger:2011}; SFI \cite{14-adelberger:1998} shows,
for comparison, the situation
around 1998. \label{14-tab:nucrates}}
\end{center}
\end{table}
\subsection{Solar Models: Helioseismology}\label{14-sec:helios}
Helioseismology, the study of the natural oscillations of the Sun, provides a
unique tool to determine the structure of the solar interior. The '90s
witnessed a rapid development of helioseismic observations and analysis
techniques, which led, in very few years, to an accurate characterization of the solar
interior \cite{14-jcd:2002}. The agreement between SSMs and helioseismic
inferences of the solar structure \cite{14-jcd:1996,14-bahcall:2001} provided
a strong support to the accuracy with which SSMs could predict the $^8$B
neutrino flux and, therefore, a strong indication, before Kamland and SNO
results found evidence of neutrino flavor oscillations, that the solution to
the solar neutrino problem had to be found in the realm of particle physics.
In the context of the present article, the most relevant results from
helioseismology are the following. The depth of the convective
envelope\footnote{The Sun is characterized by an outer region where energy is
transported by convection. The boundary between this region, located at
$R_{\rm CZ}$, and the radiative interior can be accurately located by
helioseismology because the discontinuity in the slope of the temperature
gradient accross this boundary leaves its imprint in the solar sound speed
profile. The depth of the envelope can be located by helisoeismology
because properties of solar oscillations are sensitive to the derivative of
the sound speed as a function of depth.} is $R_{\rm CZ}=0.713 \pm 0.001\, {\rm R_{\odot}}$
\cite{14-basu:1997a} and the surface helium abundance $Y_{\rm S}=0.2485 \pm
0.0034$ \cite{14-basu:2004}. The sound speed differences between the Sun
and a reference solar model can be obtained by inversion from the oscillation
frequencies with a formal error of a few parts per $10^{-4}$ for most of the
solar interior $0.07 \lesssim R/{\rm R_{\odot}} \lesssim 0.95$
\cite{14-basu:1997b,14-kosovichev:1997}. Most recently, using a time series
4752 days-long from the Birmingham Solar Oscillation Network, improved
results on the sound speed in the solar core have been obtained
\cite{14-basu:2009}. The density profile can also be determined from
inversion of frequencies, but with worse precision than for the sound speed,
and we therefore assign to it a secondary role in constraining the solar
structure.
As mentioned previously, metals determine to a large extent the radiative
opacity in the solar interior and, in this way, define the temperature
stratification from below of convective envelope inwards, to the solar center.
At the base of the convective zone, for example, metals are responsible of
about 70\% of the total radiative opacity with O, Fe and Ne being the main
contributors. In the solar core, where light metals are completely ionized,
the contribution from Fe and, to a lesser extent Ni, Si and S, is still above
30\%. In view of this, it is not surprising that the low CNO and Ne
abundances determined from 3D model atmospheres have a strong impact in the
structure of the solar interior.
It has been clear since initial works where {\em low-Z} SSMs were presented
that low $\left(Z/X\right)_\odot$ values posed a problem, later named the solar abundance problem,
for solar modeling \cite{14-montalban:2004, 14-turckchieze:2004, 14-basu:2004,
14-bahcall:2005a}. In short, all helioseismic predictions of these models
are in disagreement with observations. On the other hand, {\em high-Z} SSMs
have consistently reproduced earlier success \cite{14-jcd:1996}. The solar
abundance problem represents the incompatibility between the best solar
atmosphere and interior models available \cite{14-delahaye:2006}. In this
review, we will base the presentation and discussion of results on the
most up-to-date
standard solar models that we identify as SFII-GS98 and SFII-AGSS09
\cite{14-serenelli:2011}, representative of {\em high-Z} and {\em low-Z} SSM
families defined in Section~\ref{14-sec:solarcompo} respectively. With the
exception made on small quantitative variations, results based on these models
are extensible to results for all SSMs available in the literature
corresponding to each of the two families.
The most important characteristics of the SFII-GS98 and SFII-AGSS09 models are
summarized in Table~\ref{14-tab:heliossm}. Helioseismic constraints are also
included for comparison when appropriate. The disagreement between SFII-AGSS09
and helioseismic data is evident in the surface metallicity and helium
abundances, $Z_{\rm S}$ and $Y_{\rm S}$, and in the depth of the convective envelope
$R_{\rm CZ}$. When model uncertainties are included, the discrepancy between
SFII-AGSS09 and seismic results are, for each of the quantities mentioned
above, of the order 3 to $4-\sigma$ \cite{14-serenelli:2009}. On the
contrary, the SFII-GS98 model performs very well, within $1-\sigma$ when model
uncertainties are accounted for.
\begin{table}[h]
\begin{center}
\begin{tabular}{lccc}
\hline \hline
& SFII-GS98 & SFII-AGSS09 & Helioseismology\\
\hline
$\left(Z/X\right)_\odot$ & 0.0229 & 0.0178 & --- \\
$Z_{\rm S}$ & 0.0170 & 0.0134 & $0.0172 \pm 0.002$ \cite{14-antia:2006}\\
$Y_{\rm S}$ & 0.2429 & 0.2319 & $0.2485 \pm 0.0034$ \cite{14-basu:2004} \\
$R_{\rm CZ}/{\rm R_{\odot}}$ & 0.7124 & 0.7231 & $0.713 \pm 0.001$ \cite{14-basu:1997a} \\
$\left< \delta c / c \right> $ & 0.0009 & 0.0037 & --- \\
$\left< \delta \rho / \rho \right> $ & 0.011 & 0.040 & --- \\
$Z_{\rm C}$ & 0.0200 & 0.0159 & --- \\
$Y_{\rm C}$ & 0.6333 & 0.6222 & --- \\
$\left< \mu_C \right> $ & 0.7200 & 0.7136 & $0.7225 \pm 0.0014$
\cite{14-chaplin:2007} \\
$Z_{\rm ini}$ & 0.0187 & 0.0149 & --- \\
$Y_{\rm ini}$ & 0.2724 & 0.2620 & --- \\
\hline
\end{tabular}
\caption{Main characteristics of SSMs representative of {\em high-Z} (GS98)
and {\em low-Z} (AGSS09) solar compositions. Models have been computed
including the most up-to-date input physics
\cite{14-serenelli:2011}. Helioseismic constraints are given when
available. See text for details.
\label{14-tab:heliossm}}
\end{center}
\end{table}
Very explicit manifestations of the solar abundance problem are shown in the
plots in Figure~\ref{14-fig:inversions}, where degradation in the sound speed
and density profiles found in {\em low-Z} SSMs are clearly
evident. Particularly the peak in the sound speed profile differences found
right below the convective zone is 4 times larger in the {\em low-Z}
SFII-AGSS09 than in the {\em high-Z} SFII-GS98 model. The reason is the wrong
location of $R_{\rm CZ}$ in the model, caused by the lower opacity which, in turn,
is due to the low abundance of metals. The density profile also shows very
large discrepancies, but they are less telling. Density inversions include as
a constraint the known value of the solar mass and for this reason
small differences in
the core, where density is large, translate into the large difference seen in
the outer envelope. The average rms in the sound speed and density
differences, $\left< \delta c / c\right>$ and $\left< \delta \rho / \rho
\right>$, also show that {\em low-Z} models are about 4 times worse than {\em
high-Z} models.
\begin{figure}[h!]
\begin{center}
\hspace{-2truecm}
\includegraphics[width=16.5cm]{14-inversions.pdf}
\caption{Sound speed and density relative differences between solar models
and the Sun as determined from helioseismic inversions
\cite{14-serenelli:2009}. The convective envelope is depicted by the grey
area. \label{14-fig:inversions}}
\end{center}
\end{figure}
Low-degree helioseismology provides useful information about the solar
innermost regions. Specific combinations of mode frequencies enhance the
signal that the structure of the solar core imprints on the oscillation
pattern \cite{14-roxburgh:2003}. This has been used to determine the mean
molecular weight averaged over the innermost 20\% solar core
\cite{14-chaplin:2007}, $\left< \mu_C \right>$ in Table~\ref{14-tab:heliossm}.
Comparison with SSMs results shows that $\left< \mu_C \right>$ is too low in
{\em low-Z} models as a result of the lower helium abundance ($Y_{\rm C}$).
This is due to the lower temperature in the solar core and the
constraint imposed by the solar luminosity. The decreased nuclear energy
production originated by a smaller core temperature has to be compensated by
an increased hydrogen mass fraction, therefore leading to a lower molecular
weight. It is interesting to note this puts a stringent constraint in the
amount of rotational mixing that can take place in the solar core if the {\em
low-Z} abundances are correct, since any mixing would lower the molecular
weight even more, by bringing fresh hydrogen from outer regions, and make the
agreement with helioseismic data worse.
The current situation regarding SSMs and their performance against
helioseismic inferences on the solar structure can be summarized as follows.
SSMs that use solar abundances derived from 1D model atmospheres
\cite{14-grevesse:1993, 14-grevesse:1998}, i.e. {\em high-Z} models, reproduce
overall the most important seismic constraints. Improvements in the input
physics, e.g. radiative opacities and nuclear reaction rates, that have
occurred over the last 10 years introduce only small changes to the solar
structure as seen by helioseismology. On the other hand, the solar abundance
problem arises if the solar surface composition used to construct SSMs are
derived from the most sophisticated 3D RHD solar model atmospheres. The
family of {\em low-Z} SSMs does not match any helioseismic constraint.
Have we reached the limit where the paradigm of the SSM is not good enough as
a model of the solar interior? Are the 3D-based determinations of solar
abundances systematically underestimating the metallicity of the solar surface?
Does the microscopic input physics in solar models, e.g. radiative opacities,
need to be thoroughly revised? It is not possible to advance answers to these
questions, but solar neutrino experiments can play an important role in
guiding research towards the solution of the solar abundance problem. In the
next section we discuss the current status on the theoretical predictions of
solar neutrino fluxes and the prospects of using solar neutrinos to constraint
the properties of the solar core.
\subsection{Solar Models: Neutrino Fluxes}\label{14-sec:fluxes}
\subsubsection{Production}
Based on theoretical arguments and indirect evidence, it has long been
believed that the source of energy of the Sun is the conversion of protons
into helium, $4{\rm p \longrightarrow ^4He + 2e^+ + 2\nu_e +\gamma}$. The
original quest for solar neutrinos was indeed the search for the experimental
confirmation of this hypothesis. In more detail, hydrogen burning in the Sun
(and in all other hydrogen-burning stars) takes place either through the
pp-chains or the CNO-bicycle\footnote{Under peculiar conditions reached in
advanced phases of stellar evolution, hydrogen can be converted into helium
by other cycles like the NaMg-cycle. While important for nucleosynthesis or
intermediate mass elements, these processes are not energetically relevant}
\cite{14-clayton:1984, 14-bahcall:1989}. Proton fusion through the pp-chains
is a primary process because only protons need be present in the star. On the
contrary, the CNO-bicyle is secondary because proton fusion relies on, and is
regulated by, the abundance of C, N, and O which act as catalyzers. This
qualitative difference is very important, since it renders neutrino fluxes
from the CNO-bicyle a very good diagnostic tool to study properties of the
solar core, particularly its composition, as it will be discussed below. A
general discussion on the production of solar neutrinos is out of the scope of
the present review, but can be found elsewhere \cite{14-bahcall:1989}.
SSM calculations of neutrino fluxes have been affected by developments in the
input physics discussed in previous sections. The two areas that have the
strongest impact on the neutrino fluxes predicted by models are: changes in
nuclear cross sections, and the new solar composition. In
Table~\ref{14-tab:nufluxes} we list the results for neutrino fluxes for the
up-to-date SSMs SFII-GS98 and SFII-AGSS09. For comparison we include, in the
last column, results from the BP04 SSM \cite{14-bahcall:2004}.
\begin{table}[ht]
\begin{center}
\begin{tabular}{lcccc}
\hline \hline
Flux & SFII-GS98 & SFII-AGSS09 & Solar & BP04 \\
\hline
pp & $5.98(1 \pm 0.006)$ & $6.03(1 \pm 0.006)$ & $6.05(1^{+0.003}_{-0.011})$ &
$5.94(1\pm 0.01 )$\\
pep & $1.44(1 \pm 0.012)$ & $1.47(1 \pm 0.012)$ & $1.46(1^{+0.010}_{-0.014})$
& $1.40 (1 \pm 0.02)$\\
hep & $8.04(1 \pm 0.30)$ & $8.31(1 \pm 0.30)$ & $18(1^{+0.4}_{-0.5})$ &
$7.8 (1 \pm 0.16) $ \\
$^7$Be & $5.00(1 \pm 0.07)$ & $4.56(1 \pm 0.07)$ & $4.82(1^{+0.05}_{-0.04})$ &
$4.86 (1 \pm 0.12) $ \\
$^8$B & $5.58(1 \pm 0.13)$ & $4.59(1 \pm 0.13)$ & $5.00(1\pm 0.03)$ &
$5.79 (1 \pm 0.23 )$\\
$^{13}$N & $2.96(1 \pm 0.15)$ & $2.17(1 \pm 0.13)$ &$\leq 6.7$ &
$5.71 (1 \pm 0.36)$ \\
$^{15}$O & $2.23(1 \pm 0.16)$ & $1.56(1 \pm 0.15) $ &$\leq 3.2$ &
$5.03 (1 \pm 0.41)$ \\
$^{17}$F & $5.52(1 \pm 0.18)$ & $3.40(1 \pm 0.16)$ & $\leq 59.$ &
$5.91 (1 \pm 0.44) $\\
\hline
$\chi^2/P^{\rm agr}$& 3.5/90\% & 3.4/90\%& --- & --- \\
\hline
\end{tabular}
\caption{SSM predictions for solar neutrino fluxes (second and third columns)
and solar neutrino fluxes (fourth column) inferred from all available
neutrino data. Units are, in ${\rm cm^{-2} s^{-1}}$, as usual: $10^{10}$
(pp), $10^9$ ($^7$Be), $10^8$ (pep, $^{13}$N, $^{15}$O) $10^6$ ($^8$B,
$^{17}$F), and $10^3$ (hep).
\label{14-tab:nufluxes}}
\end{center}
\end{table}
The most striking difference is the large reduction in the $^{13}$N and
$^{15}$O fluxes between the SFII-GS98 and BP04 models, which use the same
solar composition. This reduction comes as a result of the new determination of
${\rm S_{1,14}}$, mostly by the LUNA experiment \cite{14-formicola:2004,
14-marta:2008}, that has halved its value with respect to previous results
(Table~\ref{14-tab:nucrates}). If correct, the new expectation value of the
combined $^{13}$N+$^{15}$O fluxes poses an even more challenging task for
neutrino experiments to detect CNO fluxes.
By comparing fluxes in Table~\ref{14-tab:nufluxes} for models computed with
the same solar composition (SFII-GS98 and BP04), it can be seen that in terms
of flux values, those associated with the pp-chains have not changed much
since 2004, despite improvements in the input physics entering solar model
calculations. Few percent changes are present and are the result of changes in
the nuclear cross sections discussed before and also of the new OP radiative
opacities. This is an encouraging situation; it implies that neutrino fluxes
are robust predictions of solar models and, as experimental data on solar
neutrinos accumulate, it will be possible to start fulfilling the initial goal
posed by Davis and Bahcall: to use solar neutrinos to learn about the solar
interior.
\begin{figure}
\includegraphics[width=15cm]{14-nudistri.pdf}
\caption{Normalized production profiles of solar neutrinos as a function of
solar radius. \label{14-fig:nuprofiles}}
\end{figure}
In Figure~\ref{14-fig:nuprofiles} we show the
distribution of the solar neutrino fluxes as a function of solar radius. Together
with the electron density profile, provided also by solar models (and neutron density
profiles for sterile neutrino studies), these quantities are of fundamental
importance for neutrino oscillation studies. It is
worth noting that the $^{13}$N flux has two components. The larger one is
associated with the operation in quasi steady-state of the CN-cycle in the
innermost solar core ($R < 0.1\, {\rm R_{\odot}}$), and for this reason coincides with
the production region of the $^{15}$O flux (right panel in
Figure~\ref{14-fig:nuprofiles}, blue and black curves respectively). This
component of the $^{13}$N flux, as well as the total $^{15}$O flux, is
linearly dependent on ${\rm S_{1,14}}$. The additional component of the
$^{13}$N flux comes from the residual burning of $^{12}$C by
the reactions ${\rm ^{12}C(p,\gamma)^{13}N(\beta^+)^{13}C}$ at temperatures
not high enough to close the CN-cycle with a proton capture on $^{14}$N.
This component is
completely independent of ${\rm S_{1,14}}$. The careful
reader will notice that the ratio of $^{13}$N and $^{15}$O fluxes is different
in the SFII-GS98 and BP04 models, despite having the same solar composition.
Whereas the added $^{13}$N+$^{15}$O neutrino flux is linearly proportional to the C+N
abundance in the solar core and also linearly proportional to ${\rm S_{1,14}}$, this
degeneracy can be broken, at least theoretically, if the two fluxes can be
experimentally isolated from one another.
The impact of the {\em low-Z} solar composition on the production of solar
neutrinos can be grasped by comparing results of models SFII-GS98 and
SFII-AGSS09 shown in Table~\ref{14-tab:nufluxes}. As stated before,
metals shape the solar structure through the radiative opacity. The lower
abundance of metals in the AGSS09 composition is responsible for a reduction
of the temperature in the solar core of about 1\%. Because of the
extreme temperature sensitivity of some of the neutrino fluxes this is enough
to produce large changes in the total fluxes. The most extreme case is, of
course, $^8$B, with the SFII-AGSS09 value being $\sim 20\%$ smaller. For
$^7$Be the reduction is of $\sim 9\%$.
Given the small uncertainties in the
experimental determination of these fluxes, it would be tempting to think these
neutrino fluxes have
the potential to discriminate between the two flavors of solar composition and
contribute, in this way, to the solution of the solar abundance problem. As
can be seen in Table~\ref{14-tab:nufluxes}, unfortunately, the $^7$Be and
$^8$B fluxes determined from experiments lie almost right in between the
{\em high-Z} and {\em low-Z} models.
In any case, since it is known that {\em low-Z} solar models do not reproduce
well the solar structure as discussed in the previous section, it is dangerous
to extract conclusions from comparing neutrino fluxes of this model to
experimental results. Regardless
of what the solution to the solar abundance problem is, since it will modify
the solar interior structure, it will also change the expected values for the
neutrino fluxes. In this regard, CNO fluxes are particularly interesting.
Although
they are of course affected by temperature variations to a comparable degree
as the $^8$B flux is, they carry an extra linear dependence on the solar
composition that is not related to temperature variations. Of particular
interest is the linear dependence of the $^{13}$N and $^{15}$O fluxes on the
combined C+N abundance\footnote{The $^{17}$F flux is linearly dependent on O,
but unfortunately the flux is too low to be detectable with current
experimental capabilities.}. It is this dependence that enhances their
capability as a diagnostic tool. In fact, differences between SFII-GS98 and
SFII-AGSS09 models for these two fluxes are of the order of 30\% (taking
SFII-GS98 as reference) and, what is more important, a large contribution to
these differences does not have an origin on temperature differences between
the models.
The last row in Table~\ref{14-tab:nufluxes} shows the results of a $\chi^2$
test for the two models against the solar fluxes also shown in the table. It
is clear that both SSMs give very good agreement with current data. We
emphasize again, however, that the four fluxes that are currently well
determined from data and the luminosity constraint, depend on the solar
composition only in an indirect manner. Experimental determination of the
combined $^{13}$N+$^{15}$O flux will therefore provide qualitatively new
information on the solar structure and composition. In fact, one can
take advantage of the similar response to temperature variations that CNO
fluxes and the $^8$B flux has. This has been exploited
\cite{14-haxton:2008} to develop a very simple method to determine the solar
core C+N abundance that minimizes environmental uncertainties in solar models
(that is, sources of uncertainty that affect the solar core temperature). The
idea is simple: the temperature dependences are cancelled out by using an
appropriate ratio between the $^8$B and the combined $^{13}$N+$^{15}$O fluxes
where SSM fluxes only act as normalization values and the overall scale is
determined by an actual $^8$B flux measurement. The only additional
requirement is that a measurement of the combined $^{13}$N+$^{15}$O flux
becomes available. The current upper limit on this combined flux from
Borexino \cite{14-pepBX} places an upper limit on the C+N central mass fraction of
$X_{\rm C+N} < 0.072$. Results for the SFII-GS98 and SFII-AGSS09 models are
$X_{\rm C+N}=0.048$ and 0.039 respectively.
\subsubsection{Uncertainties}
Uncertainties in the model predictions of solar neutrino fluxes are given in
Table~\ref{14-tab:nufluxes}. For deriving the total uncertainty basically two
approaches can be used. On one hand, all contributions of uncertainty can be
treated simultaneously by doing a Monte Carlo simulation
\cite{14-bahcall:2006}. The advantage is that intrinsic non-linearities are
captured in the total error. The disadvantage is that individual
contributions to the total uncertainty are hardwired in the final result and
can not be disentangled. Fortunately, for the current level of uncertainties
entering SSM calculations, non-linearities seem to be negligible and the total
uncertainty in neutrino fluxes can be obtained (adding quadratically) from
individual contributions. To compute the latter, the expansion
of fluxes as a product of power-laws in the input parameters
\cite{14-bahcall:1989} around central values is a widely used, practical,
insightful and accurate approach. Uncertainties in the model fluxes listed in
Table~\ref{14-tab:nufluxes} have been obtained in this way.
The most important change introduced in the estimation of uncertainties is
related to the treatment of the solar composition. Up until the BP04 model
\cite{14-bahcall:2004}, the uncertainty in the solar composition was taken
into account by considering variations of the total solar metallicity (to be
more precise, changes in the $\left(Z/X\right)_\odot$ value used to construct SSMs). This
leads to an overestimation of the neutrino uncertainties. The reason is that
metals dominating the error budget in $\left(Z/X\right)_\odot$ (C, N, O, and Ne) have, at
most, a moderate impact on the neutrino fluxes because of their small
contribution to the radiative opacity, and therefore a rather small impact on
temperature, in the region where most neutrinos are produced. On the other
hand, elements such as Fe, S and Si are second order in determining $\left(Z/X\right)_\odot$
but play a fundamental role as sources of opacity in the solar core. It is
important, therefore, to treat metal uncertainties individually
\cite{14-bahcall:2005c}. Of course, in the case of the CNO fluxes the
situation is different because CNO elements catalyze the CNO-bicycle and this
overimposes an almost linear dependence of the $^{13}$N and $^{15}$O on the
C+N content of the solar and a similar dependence of $^{17}$F on the O
abundance. The uncertainties in the neutrino fluxes given for the SFII-GS98
and SFII-AGSS09 SSMs have been computed using the uncertainties for each
relevant element given in the original publications \cite{14-grevesse:1998,
14-asplund:2009}. As a result, for either family of solar models, i.e. {\em
high-Z} or {\em low-Z} models, the solar composition is not the dominant
source of uncertainty for any of the fluxes of the pp-chains. In the case of
the CNO fluxes, the linear dependence mentioned above is the dominant source
of uncertainty: the combined C+N abundance contributes to a 12\% uncertainty
for both the $^{13}$N and the $^{15}$O fluxes, and the O abundance to 15\% in
the $^{17}$F flux.
In the case of the non-composition uncertainties, the situation has improved
in some cases thanks to more precise measurements of nuclear reaction
rates. This is the case, in particular, for the ${\rm ^3He(^4He},\gamma){\rm
^7Be}$ reaction, which now contributes only 4.7\% and 4.5\% of the total
uncertainty in the $^7$Be and $^8$B fluxes respectively. For comparison, the
analogous contributions in the BP04 model were 8.0\% and 7.5\%
\cite{14-bahcall:2004}. Significant progress has also been achieved regarding
${\rm ^{14}N}(p,\gamma){\rm ^{15}O}$, which now introduces uncertainties of
only 5\% and 7\% in the $^{13}$N and $^{15}$O fluxes, half the amount it did
in 2004. An important contribution to the uncertainty in the $^8$B flux now
comes from ${\rm ^7Be}(p,\gamma){\rm ^8B}$ because the uncertainty of this
reaction has been revised upwards \cite{14-adelberger:2011}. Even if the
uncertainty in this rate is now smaller than in SFI (see
Table~\ref{14-tab:nucrates}), it is larger than that used for the BP04 model,
which was taken considering only one experimental result for this reaction.
While progress has been done in some cases, others have not seen much
development, particularly diffusion and the delicate issue of radiative
opacities. In Table~\ref{14-tab:fluxuncert} we give the individual
contributions to flux uncertainties for the most relevant sources. The reader
can compare directly to the situation in 2004 \cite{14-bahcall:2004}.
\begin{table}
\begin{center}
\begin{tabular}{lccccccc}
\hline \hline
& ${\rm S_{11}}$& ${\rm S_{33}}$& ${\rm S_{34}}$& ${\rm S_{17}}$
& ${\rm S_{1,14}}$ & Opac & Diff \\
\hline
pp & 0.1 & 0.1 & 0.3 & 0.0 & 0.0 & 0.2 & 0.2 \\
pep & 0.2 & 0.2 & 0.5 & 0.0 & 0.0 & 0.7 & 0.2 \\
hep & 0.1 & 2.3 & 0.4 & 0.0 & 0.0 & 1.0 & 0.5 \\
$^7$Be & 1.1 & 2.2 & 4.7 & 0.0 & 0.0 & 3.2 & 1.9 \\
$^8$B & 2.7 & 2.1 & 4.5 & 7.7 & 0.0 & 6.9 & 4.0 \\
$^{13}$N & 2.1 & 0.1 & 0.3 & 0.0 & 5.1 & 3.6 & 4.9 \\
$^{15}$O & 2.9 & 0.1 & 0.2 & 0.0 & 7.2 & 5.2 & 5.7 \\
$^{17}$F & 3.1 & 0.1 & 0.2 & 0.0 & 0.0 & 5.8 & 6.0 \\
\hline
\end{tabular}
\caption{Percentage contribution of selected individual sources of uncertainty
to the neutrino fluxes. \label{14-tab:fluxuncert}}
\end{center}
\end{table}
\section{Neutrino flavor conversion in vacuum and matter}
Neutrino flavor conversion has been reviewed by A. Yu. Smirnov in this volume
and we refer the reader for a detailed physics discussion and references to
his article. Here we just summarize the basic features and formulae of flavor
conversion relevant to solar neutrinos.
We consider mixing of the three flavor
neutrinos. The description of flavor conversion of solar neutrinos traveling
through a medium is simplified because a) the hierarchy in mass splittings
determined by solar and atmospheric data leads to a reduction of the three
neutrino flavor conversion to an effective two flavor problem, b) the neutrino
parameters, the mixings and solar mass splitting, lead to adiabatic flavor
conversion in solar matter and to cancel the interference term by averaging
out. Therefore, the physics of the flavor conversion of solar neutrinos is
described by simple expressions with very good accuracy. In practice, the
survival probability is computed numerically to correctly include the number
density of scatterers along the trajectory of neutrinos from production to
detection and to average over the neutrino production region. In solar
neutrino flavor conversion, $\nu_\mu$ and $\nu_\tau$ are indistinguishable and
therefore the survival probability of electron neutrinos is the only function
needed to describe the flavor composition of the solar neutrino flux.
Solar neutrino survival or appearance probabilities depend on three
oscillation parameters: the solar oscillation parameters ($\theta_{12}$,
$\Delta m^2_{21}$), and $\theta_{13}$. The survival probability in the absence
of Earth--matter effects, i.e., during the day, is well described by
\begin{eqnarray}
P_{ee}^{D} = \cos^4\theta_{13} \left( \frac{1}{2} + \frac{1}{2} \cdot
\cos2\theta_S \cdot \cos2\theta_{12} \right) + \sin^4\theta_{13}.
\label{Pday}
\end{eqnarray}
Here $\theta_S$ is the mixing angle at the production point inside the Sun:
\begin{eqnarray}
\cos 2\theta_S \equiv \cos2\theta_m(\rho_S)
\label{eq}
\end{eqnarray}
where $\theta_m(\rho)$ is the mixing angle in matter of density $\rho_{S}$,
\begin{eqnarray}
\cos2\theta_S = { \cos2\theta_{12} - \xi_{S} \over( 1 -2\xi_{S} \cos 2\theta_{12} +
\xi_{S}^2 )^{1/2}}.
\label{cos2t12}
\end{eqnarray}
In (\ref{cos2t12}), $\xi_{S}$ is defined as the ratio of the neutrino
oscillation length in vacuum, $l_\nu$, to the refraction length in matter,
$l_0$:
\begin{eqnarray}
\xi_{S} \equiv \frac{l_\nu}{l_0} &=&
\frac{2 \sqrt{2} G_F \rho_{S} Y_e \cos^2\theta_{13}}{m_N}
\frac{E}{\Delta m^2}
\nonumber \\
&=&
0.203 \times
\cos^{2} \theta_{13}
\left( \frac{E}{ \rm{ 1 MeV} } \right)
\left( \frac{ \rho_{S} Y_e }{ \mbox{\rm 100 g\ cm}^{-3} } \right),
\label{xi-def}
\end{eqnarray}
where
\begin{eqnarray}
l_\nu\equiv\frac{4\pi E}{\Delta m^2},
\hspace{10mm}
l_0\equiv \frac{2\pi m_N}{\sqrt2 G_F \rho_{S} Y_e \cos^2\theta_{13}}.
\label{length}
\end{eqnarray}
In (\ref{xi-def}) and (\ref{length}), $\rho_{S}$ is the solar matter density,
$Y_{e S}$ is the number of electrons per nucleon, and $m_N$ is the nucleon
mass. The electron solar density and neutrino production distribution of the
neutrino fluxes are derived from solar models as discussed in previous
section. In the last line in (\ref{xi-def}) we have used the best fit values
of the global analysis $\Delta m^2= 7.5 \times 10^{-5}$ eV$^2$. The ratio of
the parameter $\rho_{S}$ to $\cos 2\theta_{12}$, separates the region where
the flavor conversion corresponds to vacuum averaged oscillations from the one
of matter dominated conversion.
The $\nu_{e}$ survival probability at night during which solar
neutrinos pass through the Earth can be written as
\begin{eqnarray}
P_{ee}^{N} =
P_{ee}^{D} - \cos 2\theta_{S} \cos^2 \theta_{13} \langle f_{reg}
\rangle_{\rm{zenith}}
\label{Pnight}
\end{eqnarray}
where $P_{ee}^{D} $ is the one given in (\ref{Pday}). $f_{reg}$ denotes the
regeneration effect in the Earth, and is given as $f_{reg} = P_{2e} -
\sin^2\theta_{12} \cos^2 \theta_{13} $, where $P_{2e}$ is the transition
probability of second mass eigenstate to $\nu_{e}$.
Under the constant density approximation in the Earth, $f_{reg}$ is given by
\begin{eqnarray}
f_{reg} = \xi_{E} \cos^2 \theta_{13} \sin^2 2\theta_{E}
\sin^2 \left[ a_{E} \cos^2 \theta_{13} (1-2 \xi_{E}^{-1}\cos^2
\theta_{12}+\xi_{E}^{-2})^{\frac{1} {2} }
\left( \frac{L}{2} \right) \right]
\label{freg}
\end{eqnarray}
for passage of distance $L$, where we have introduced
$a_{E} \equiv \sqrt{2} G_{F} n_{e}^{Earth} = \frac{ \sqrt{2} G_{F} \rho_{E} Y_{e
E} } {m_N} $.
In ($\ref{freg}$), $\theta_{E}$ and $\xi_{E}$ stand for the mixing angle and
the $\xi$ parameter [see (\ref{xi-def})] with matter density $\rho_{E}$ in the
Earth. Within the range of neutrino parameters allowed by the solar neutrino
data, the oscillatory term averages to $\frac{1}{2}$ in a good approximation
when integrated over zenith angle. Then, the equation simplifies to
\begin{eqnarray}
\langle f_{reg} \rangle_{\rm{zenith}} =
\frac{1} {2} \cos^2 \theta_{13} \xi_{E} \sin^2 2\theta_{E}.
\label{freg-ave}
\end{eqnarray}
At $E=7$ MeV, which is a typical energy for $^8$B neutrinos, $\xi_{E}=3.98
\times 10^{-2}$ and $\sin 2\theta_{E} = 0.940$ for the average density
$\bar{\rho}_{E} = 5.6 \rm{g/cm}^3$ and the electron fraction $Y_{e E} = 0.5$
in the Earth. Then, $\langle f_{reg} \rangle_{\rm{zenith}}$ is given as $
\langle f_{reg} \rangle_{\rm{zenith}} = 1.76 \times 10^{-2}$ for the best fit
neutrino parameters. This result is in reasonable agreement with the computed
Earth-matter factor using the best estimates on the Earth-matter density.
\section{Recent solar neutrino measurements}
\subsection{The SNO and SK legacy}\
After the results and analyses from 2002, it was clear that the LMA
oscillation was the right solution of the long standing solar neutrino puzzle
\cite{14-afterSNO2002}, but the activity of the SNO and SK experiments
continued in the following years. The data obtained from these experiments
were very important in making the LMA solution more robust and in improving
the accuracy and precision of the mixing parameters determination.
The so-called SNO II experiment began in June of 2001 with the addition of
2000~kg of NaCl to the 1000 tons of $D_2 O$ and ended in October 2003 when
the NaCl was removed. The addition of salt significantly increased SNO's
efficiency (by a factor
$\sim$\,3 with respect to the pure $D_2 O$ phase) in the detection of neutrons
produced in the neutral current (NC)
disintegration of deuterons by solar neutrinos and, by enhancing the energy of
the $\gamma$-ray coming from neutron capture, allowed a more precise
measurement of this interaction channel, well above the low-energy radioactive
background. Moreover, the isotropy of the multiple $\gamma$-ray emission by
neutron capture on $^{35}{\rm Cl}$ is different from the one of the \v{C}erenkov
light emitted by the single electron of the charged current interaction;
therefore, by studying the event isotropy, it has been possible to separate the
neutral from the charged current events without any additional assumption on
the neutrino energy spectrum. The salt phase results have been reported in
two main publications. In \cite{14-SNOII}, referring to the first 254 live
days, a global analysis including all the solar and reactor neutrino results
rejected the maximal mixing hypothesis at a $5.4 \, \sigma$ level and gave a
value of the $^8$B neutrino flux in agreement with previous measurements and
with SSMs. These results were essentially confirmed (even if
with a small shift towards larger values of the mixing angle) by the second
publication \cite{14-SNOII_2005}, which included the full data of the salt
phase (391 live days), analyzed in terms of the CC spectra (starting from
$5.5 \, \rm{MeV}$
kinetic energy) and NC and ES integrated fluxes separately for day and
night. The day-night asymmetry in the neutral current rate, which would be an
indication of oscillation to sterile neutrinos or non standard interaction
with matter in the earth, came out to be consistent with zero.
This result confirmed also the outcome of the study performed for elastic
scattering (ES) interaction above $5 \, \rm{MeV}$ by the Super-Kamiokande
collaboration \cite{14-SK2003SMY}. The full SK-I low energy data,
corresponding to 1496 live days until July 2001, were investigated analyzing
the time variations of the ES rates and fitting them to the variations
expected from active two neutrino oscillations. The day-night asymmetry turned
out to be $A_{DN} = \frac{2 (D-N)}{D + N} = -0.021 \pm 0.020 \, {\rm (stat.)}
\, ^{+0.013}_{-0.012} \, {\rm (syst.)}$, which is consistent with zero within
$0.9 \, \sigma$. This value was in good agreement also with the LMA
oscillation solution, which (for the best fit parameter) predicted
\cite{14-SK2003SMY} $A_{D N}= -0.018 \pm 0.016 \, {\rm (stat.)} \,
^{+0.013}_{-0.012} \, {\rm (syst.)}$. The SK analysis
\cite{14-SK2003SMY,14-SKI-2005} also showed that the energy spectrum of the
recoiling electron was consistent with an undistorted solar $^8$B neutrino
spectrum and did not find any anomalous periodic time variation of the rates,
apart from the expected seasonal variation due to the Earth's orbit
eccentricity. The SK best fit point was in quite a good agreement with the
SNO results, even if SK would favor slightly larger values of ${\rm tan}^2\theta$.
A SNO-only analysis gave the following best fit
parameters\cite{14-SNOII_2005}: $\Delta m_{12}^2 = 5.0 \times 10^{-5} \,
\rm{eV}^2$, $\rm{tan}^2 \theta_{12} =0.45$. Including all the other solar
neutrino and the KamLAND results the best fit was obtained for $\Delta
m_{12}^2 = 8.0^{+ 0.6}_{-0.4} \times 10^{-5} \, \rm{eV}^2$, $\rm{tan}^2
\theta_{12} = 0.452^{+0.088}_{-0.070}$. The effect of KamLAND data was mainly
to increase the value of $\Delta m^2$ and to restrict the allowed region in the
mixing parameter plane. The main difference of the global analysis done with
the SNO salt phase data with respect to previous studies was the possibility
to exclude at $95 \% \rm{C.L.}$ the secondary region at even larger values of
the mass differences (the so called LMA II solution, with $\Delta m_{12}^2 >
10^{-4} \, \rm{eV}^2$).
In the third SNO phase (November 2004-November 2006) the neutral current
signal neutrons were mainly detected by means of an array of $^3 He$
proportional counters deployed in the $D_2 O$ and looking at the gas
ionization induced by neutron capture on $^3$He. In this way the fluxes
correlation was reduced and the accuracy in the mixing angle determination was
improved. The total active $^8$B neutrino flux was found \cite{14-SNOIII} to
be $5.54^{+0.33}_{-0.31} (\rm{stat}) ^{+0.36}_{-0.34} (\rm{syst}) \times 10^6
\, {\rm cm^{-2}} {\rm s}^{-1}$, in agreement with previous measurements and
SSMs. The ratio of the $^8$B neutrino flux measured with
CC and NC reaction was ${\rm \Phi^{SNO}_{CC}/\Phi_{NC}^{SNO}} = 0.301 \pm 0.033$.
The global solar neutrino experiment analysis included, in this case, also the
first results coming from the Borexino experiment \cite{14-Arpesella2008}, that we
discuss in subsection $\ref{Borexino}$. The best fit point moved
to $\Delta m_{12}^2 = 4.90 \times 10^{-5} \rm{eV}^2$, $\rm{tan}^2 \theta_{12}
= 0.437$ and the uncertainty in the mixing parameter plane was still quite
large. Adding the KamLAND data, the allowed region was significantly
restricted (mainly for $\Delta m^2$) and the marginalized $1 \, \sigma$
regions were $\Delta m_{12}^2 = 7.59^{+0.19}_{-0.21} \times 10^{-5}
\rm{eV}^2$, $\rm{tan}^2 \theta_{12} = 0.469_{-0.041}^{+0.047}$.
A subsequent joint reanalysis of SNO I and SNO II data, known as LETA (Low
Energy Threshold Analysis) \cite{14-LETA}, succeeded, with improved
calibration and analysis techniques, in lowering the energy threshold, with
respect to previous analyses (\cite{14-SNOphaseI_analisi2007,14-SNOII_2005}),
down to an effective electron kinetic energy of $T_{\rm eff} = 3.5 \, {\rm
MeV}$. The main effect was to increase the statistics of CC and ES and, above
all, of NC events, and to increase significantly the precision on both the total
$^8$B neutrino flux and the neutrino mixing parameters. The value for the
total $^8$B neutrino flux extracted from neutral current was $\Phi_{NC} =
5.14^{+0.21}_{-0.20} \times 10^6 \, {\rm cm^{-2}} {\rm s}^{-1}$, where the
error, obtained by summing in quadrature the statistic and systematic
contributions, was reduced by more than a factor of two with respect to
previous publications. For SNO data alone (LETA plus SNO III) the best fit
point moved to the LOW region of parameter space, but the significance level
was very similar to the one of the usual LMA solution. A global fit, including
all the solar and the KamLAND data, essentially confirmed, instead, the
previous results \cite{14-SNOIII} for $\Delta m_{12}^2$ and it made possible a
further improvement in the angle determination, giving, in a 2 flavor
analysis, $\rm{tan}^2 \theta_{12} = 0.457_{-0.028}^{+0.041}$.
In the last five years also the Super-Kamiokande collaboration presented new
analyses, including the data of the different working phases of this
experiment: Super-Kamiokande II \cite{14-SKII} (from December 2002 to October
2005) and Super-Kamiokande III (from July 2006 to August 2008)
\cite{14-SKIII}. Due to the 2001 accident, which damaged some of the
photomultiplier tubes, the detector sensitivity was reduced with respect to
SK-I and therefore it was important to improve the methods adopted for data
collection (particularly for vertex event reconstruction, angular resolution
and background reduction) and analysis. In this way, during the 548 days of
SK-III a $2.1 \%$ systematic uncertainty on the total flux (corresponding
roughly to two thirds of the SK-I value) was reached. The second and third
Super-Kamiokande phases essentially confirmed the SK-I results, for what
concerns the absence of significant spectral distortion, the total $^8$B
measured flux and the day-night asymmetry.
Since September 2008, Super-Kamiokande is running with modernized data
acquisition system (DAQ) and electronics, which allow a wider dynamic range in
the measured charge and is read out via Ethernet. This phase of the experiment
is denoted as Super-Kamiokande-IV \cite{14-SuperKamiokandeIV}. Thanks to the
fast DAQ every hit can be recorded and the resulting data stream analyzed by
an online computer system that finds timing coincidences which are saved as
triggers. As a consequence, Super-Kamiokande's energy threshold is now only
limited by computing speed and the event reconstruction. The present event
reconstruction is able to reconstruct electrons with a total energy of 3 MeV
or more. The computing speed limits the energy threshold to ~4.2 MeV which is
just below the threshold of Super-Kamiokande-I and III (4.5 MeV). The
same water flow techniques developed during Super-Kamiokande-III result in an
observed solar neutrino elastic scattering peak between 4 and 4.5 MeV total
recoil electron energy. Special techniques are developed to discriminate the
signal from the background, taking advantage from the fact that the background is
mainly due to $\beta$ emission from $^{214}$Bi and it is characterized by a
larger Coulomb multiple scattering. This makes possible a reduction of about
$10-15 \%$ of the statistical uncertainty and this method can also be applied
to previous phases of the experiment. The additional systematic uncertainty of
this method is under investigation.
\subsection{The impact of KamLAND results on solar neutrino physics}\
Even if it is based on the analysis of a reactor antineutrino beam, the KamLAND
experiment played a fundamental role in the solution of the long standing
solar neutrino puzzle. In fact, the first KamLAND data \cite{14-firstKL} were
determinant, in conjunction with the previous solar neutrino experiments (and
mainly with SNO) and assuming CPT invariance, to prove the validity of the
oscillation hypothesis and to select the LMA solution as the correct one.
Between March 2002 and January 2004 a new set of data were collected and
the KamLAND collaboration performed a study including also a re-analysis of the
previous data. During the 2002-2004 campaign important upgrades were done
both on the central detector (increasing the photocatode coverage and
improving the energy resolution) and in the analysis techniques (reduction of
the background with better techniques in the event selection cuts based on the
time, position and geometry of the events). The number of
antineutrino events above $2.6 \, {\rm MeV}$ expected in absence of
antineutrino disappearance was $365.2 \pm 23.7 {\rm(syst)}$ and the 258
observed events corresponded to a $\bar{\nu_e}$ survival probability equal
to $0.658 \pm 0.044 ({\rm stat}) \pm 0.047 ({\rm syst})$. The energy spectrum
analysis was in disagreement with the no oscilation hypothesis at 99.6$\%$
statistical significance. In \cite{14-KL2004} the KamLAND collaboration,
looking at the $L_0/E$ spectrum dependence (where $L_0$ is the source-detector
distance and $E$ the $\bar{\nu}_e$ energy), performed also an interesting study of
other alternative hypotheses (like decoherence and decay) for neutrino
disappearance. The oscillation hypothesis offered by far the best explanation
of the spectrum shape, as one can see from Fig.(\ref{fig-14:1}).
\begin{figure}[h!!]
\begin{center}
\includegraphics[width=8cm,height=5.7cm,angle=0]{LE-Plot.pdf}
\caption{Ratio of the observed $\bar{\nu}_e$ spectrum to the expectation for
no-oscillation versus L$_{0}$/E. The curves show the expectation for the
best-fit oscillation, best-fit decay and best-fit decoherence models, taking
into account the individual time-dependent flux variations of all reactors
and detector effects.
Taken from \cite{14-KL2004}.
\label{fig-14:1}}
\end{center}
\end{figure}
As shown in Fig.(\ref{fig-14:2}A), the best fit obtained from the data
analysis was in the so-called LMAI region (with values of $\Delta m_{12}^2$
around $8 \cdot 10^{-5} \, {\rm eV}^2$) and the alternative solution at higher
$\Delta m_{12}^2$ (around $2 \cdot 10^{-4} \, \rm{eV}^2$) was strongly
disfavoured, at $98 \%$ C.L., mainly due to the spectrum distorsions. The
KamLAND data alone were not sufficient to solve completely the ambiguity on
the mixing angle values and to exclude maximal mixing. However, including in
the analysis also the results coming from solar neutrino experiments, the
allowed values of the angle were significantly restricted (see
Fig.($\ref{fig-14:2}$B)) and the two flavor combined analysis gave
$\Delta m_{12}^2 = 7.9^{+0.6}_{-0.5}
\cdot 10^{-5} \, \rm{eV}^2$ , \, $\rm{tan}^2 \theta_{12} =
0.40^{+0.10}_{-0.07}$ at a $1 \, \sigma$ level.
\begin{figure}[h!t]
\begin{center}
\vspace{-10.5truecm}
\includegraphics[width=14cm,height=18cm,angle=0]{contourplot04.pdf}
\caption{(A) Allowed region of the neutrino oscillation parameter from KamLAND
anti-neutrino data (colored regions) and solar neutrino experiments (lines)
\cite{14-SNOII}. (B) Result of a combined two-neutrino oscillation analysis
of KamLAND and the observed solar neutrino fluxes under the assumption of
CPT invariance. Taken from \cite{14-KL2004}.
\label{fig-14:2}
}
\end{center}
\end{figure}
The next KamLAND analysis \cite{14-KLfollowing2} included also, in addition to the
one of \cite{14-firstKL,14-KL2004}, the new data collected up to May
2007. The increase in data collection was significant (also thanks to the
enlarging the radius of the fiducial volume from 5.5 to 6 m) and there was a reduction
of systematic uncertainties, in the number of target protons and the
background. The total uncertainty on $\Delta m_{21}^2$ was around $2 \%$,
mainly due to the distortion of the energy scale in the detector. The total
uncertainty, $4.1 \%$, on the expected event rate was due to different sources
(above all the definition of the detector fiducial volume and energy
threshold, the $\bar{\nu}_e$ spectra and the reactor power) and it affected
primarily the mixing angle determination.
The different background sources were studied and reduced further. The most
important one was the ${\rm ^{13} C (\alpha,n) ^{16}O}$ reaction, made possible
by the $\alpha$ decay of $^{210}$Po (a daughter of $^{222}$Rn)
introduced in the liquid scintillator during the construction, which produces
neutrons with energies up to $7.3 \, {\rm MeV}$.
The results of the statistical analysis are reported in Fig.(\ref{fig-14:3}),
taken from \cite{14-KLfollowing2}. The allowed oscillation parameter values
were $\Delta m_{21}^2 = 7.58^{+0.14}_{-0.13} (\rm{stat}) ^{+0.15}_{-0.15}
(\rm{syst}) \cdot 10^{-5} \, \rm{eV}^2$ for the mass eigenvalues and
$ {\rm tan}^2 \theta_{12} =0.56^{+0.10}_{-0.07} ({\rm stat}) ^{+0.10}_{-0.06} ({\rm syst})$,
for ${\rm tan}^2 \theta_{12} \, < \, 1$ and the no oscillation hypothesis was excluded
at $5 \, \sigma$. The extension to the three neutrino oscillation analysis had
the main effect to enlarge the uncertainty on $\theta_{12}$, leaving $\Delta
m_{12}^2$ substantially unchanged. Figure (\ref{fig-14:3}), taken from
\cite{14-KLfollowing2}, shows that the effect of the inclusion in the analysis
of the data from SNO \cite{14-SNOII_2005} and previous solar neutrino
experiments was essentially to reduce the interval of allowed $\theta_{12}$
values and also to move the best fit point towards slightly lower values of the
mixing angle.
\begin{figure}[h!!]
\begin{center}
\vspace{-2.5cm}
\includegraphics[width=11cm,height=12cm,angle=0]{contourplot08.pdf}
\caption{ Allowed region for neutrino oscillation parameters from KamLAND and
solar neutrino experiments. The side-panels show the $\Delta
\chi^{2}$-profiles for KamLAND (dashed) and solar experiments (dotted)
individually, as well as the combination of the two (solid). Taken from
\cite{14-KLfollowing2}.
\label{fig-14:3}}
\end{center}
\end{figure}
Figure (\ref{fig-14:4}) (taken from \cite{14-KLfollowing2}) illustrates,
instead, the $\bar{\nu}_e$ survival probability, as a function of the ratio
$L_0/E$ between the average baseline and the antineutrino energies. One can
notice that the observed spectrum (after subtraction of background and
geo-neutrino signals), reproduces correctly the general shape of the expected
oscillation cycle, with a slight excess of low energy antineutrinos, that
could be interpeted as geo-neutrinos.
\begin{figure}[h!!]
\begin{center}
\vspace{-1cm}
\includegraphics[width=9.5cm,height=8.0cm,angle=0]{LLE.pdf}
\vspace{-0.5truecm}
\caption{ Ratio of the background and geo-neutrino-subtracted $\bar\nu_e$
spectrum to the expectation for no-oscillation as a function of
$L_{0}/E$. $L_{0}$ is the effective baseline taken as a flux-weighted
average ($L_{0}$\,=\,180\,km). The energy bins are equal probability bins of
the best-fit including all backgrounds. The histogram and curve show the
expectation accounting for the distances to the individual reactors,
time-dependent flux variations and efficiencies. The error bars are
statistical only and do not include, for example, correlated systematic
uncertainties in the energy scale. Taken from \cite{14-KLfollowing2}.
\label{fig-14:4}}
\end{center}
\end{figure}
\subsection{Toward the sub-MeV analysis: the Borexino detector and its measurements}\label{Borexino}\
In the last decade significant steps forward have been done in the knowledge
of solar neutrino properties, thanks mainly to the results obtained by the
kiloton scale \v{C}erenkov detectors (SK and SNO) and by advent of the reactor
neutrino experiment KamLAND. However, these experiments investigated only the
energy part of solar neutrino spectrum above 5~MeV, which represents
a small fraction of the full spectrum. The single components of the neutrino
spectrum cannot be determined by such techniques at low energies and,
therefore, up to the last four years, low energy neutrinos had been observed
only via radiochemical methods. A significant change took place with the
advent of Borexino, a real time experiment which opened the way to the
investigation of the sub-MeV region and isolated for the first time the
neutrinos corresponding to the monochromatic berillium line.
\subsubsection{ The Borexino detector}
\label{Borexinodetector}
Borexino is an ultra-high radiopure large volume liquid scintillator detector
(using pseudocumene -PC-\footnote{1,2,4-trimethylbenzene} as aromatic
scintillation solvent, and PPO\footnote{2,5-diphenyloxazole} as solute at a
concentration of 1.5~g/l) located underground at the italian Gran Sasso
National Laboratories (LNGS), under about 1400 m of rock (3800 mwe)
\cite{14-Bor09}. The employment of a liquid scintillator as target mass
assures a light production sufficient to observe low energy neutrino events
via elastic scattering by electrons. This reaction is sensitive to all
neutrino flavors, through the neutral current interaction, but the cross
section for $\nu_e$ is larger than $\nu_\mu$ and $\nu_\tau$ by a factor of
5-6, due the combination of charged and neutral currents. The main goal of
Borexino is the measurement of the mono-energetic ($0.862 \, {\rm MeV}$) $^7$Be neutrinos, which have the basic signature of the Compton-like edge of the
recoil electrons at 665~keV (see Fig.~\ref{fig-14:5}).
\begin{figure}[h!!]
\begin{center}
\vspace{-4truecm}
\includegraphics[width=10.0cm,height=10.5cm,angle=0]{teoflux7BeBX.pdf}
\caption{Neutrino spectra expected in Borexino (accounting for the detector's
energy resolution). The upper line represents the neutrino signal rate in
Borexino according to the most recent predictions of the Standard Solar
Model~\cite{14-carlos}
including neutrino oscillations with the LMA-MSW parameters. The lower line
illustrates the contribution due to ${^7}$Be~neutrinos. The {\it pp} neutrinos
contribute to the spectrum below 0.3~MeV and the edge at 1.2~MeV is due to
{\it pep} neutrinos (from \cite{14-bxfirstresults}).
\label{fig-14:5}}
\end{center}
\end{figure}
The high light yield typical of a liquid scintillator makes it possible to reach
a low energy threshold, a good energy resolution of about $5\%$ at 1 MeV and a
pulse shape discrimination between $\alpha$ and $\beta$ decays. On the other hand,
no directionality is possible and it is also not possible to distinguish
neutrino
scattered electrons from electrons due to natural radioactivity. For this
reason, an extremely low level of radioactive contamination is compulsory and
this has been one of the main tasks and technological achievements of the
experiment. The background due to the presence of $\beta$~decay of $^{14}$C
($\beta_{end-point}$ 156~keV), intrinsic to the scintillator, limits neutrino
observation to energies above 200~keV. Techniques for the scintillator
purification are based mainly on methods developed and tested in earlier
studies with the Counting Test Facility (CTF), a 4-ton prototype of Borexino
which demonstrated for the first time the feasibility of achieving the low
backgrounds needed to detect solar neutrinos in a large scale
scintillator~\cite{14-ctf1,14-ctf2,14-ctf3}. For Borexino, a larger
purification plant was developed similar to the CTF system, but with several
improved features including the use of high vacuum and precision cleaning
techniques.
The design of Borexino is based on the principle of graded shielding
(onion-like structure - see Fig.~\ref{fig-14:Borexino_detector}).
\begin{figure}[h]
\begin{center}
\begin{minipage}{0.8\textwidth}
\centering{\includegraphics[width=0.6\textwidth]{borexino_shematic.pdf}}
\caption{Schematic view of the Borexino detector.}\label{fig-14:Borexino_detector}
\end{minipage}
\end{center}
\end{figure}
The scintillator ($\approx$~300~tons) is contained in a thin nylon Inner
Vessel (IV), of radius 4.25~m, at the center of a set of concentric shells of
increasing radiopurity and it is surrounded by an outer vessel (OV),
filled with PC and 5.0~g/l DMP \footnote{dimethylphthalate}, a material
which is able to quench the residual scintillation of PC and acts as a passive
shield against radon and other background contaminations originating from the
external parts. A third more external vessel is composed of a stainless steel
sphere (SSS),
enclosing the passive shield (PC-DMP), and the entire detector is contained in
a dome-shape structure 16.9~m high with a radius of 9~m, filled with
ultra-pure water, denominated Water Tank (WT).
The scintillation light is recorded by 2212 8-inches photomultipliers
distributed on the inner part of the SSS~\cite{14-pmts1,14-pmts2}; 1828 of
them are equipped with aluminum light concentrators designed to increase the
light collection efficiency~\cite{14-cones}. \v{C}erenkov light and residual
background scintillation in the buffer are thus reduced. The others 384
photomultipliers without concentrators are used to study this background and
to identify muons that cross the buffer and not the Inner Vessel. The Water
Tank is equipped with~208 8-inches photomultipliers and acts as a \v{C}erenkov
muon detector.
Although the muon flux is reduced by six order of magnitude by
the~3800~m.w.e. depth of the Gran Sasso Laboratory, is still significant
(1.1~$\mu$~m$^{-2}$~h$^{-1}$). An additional reduction, of the order of
about~$10^{4}$, has been necessary; for more details see
Ref.~\cite{14-bxdetector}.
In order to remove contaminants from dust (U, Th, K), air ($^{39}$Ar,
$^{85}$Kr) and cosmogenically produced isotopes (${^7}$Be), different
purification techniques were applied, such as distillation, water extraction,
nitrogen stripping and ultra-fine filtration. The pseudocumene was distilled
in-line during the detector filling at 80~mbar and at a temperature of about
90--95\,$^{\circ}$C. Distilled pseudocumene was stripped in a 8~m-high (15~cm
in diameter) packed column with specially prepared ultra-low Ar/Kr nitrogen
(0.005~ppm Ar and 0.06~ppt Kr, see Ref.~\cite{14-lakn}). Position
reconstruction of the events, as obtained from the photomultipliers timing
data via a time-of-flight algorithm, allowed to define a fiducial spherical
volume, corresponding approximately to 1/3 (i.e. about 100 tons) of the
scintillator volume in order to reject external~$\gamma$~background. The
others 2/3 of the scintillator act as an active shield.
\subsubsection{ The measurement of the ${^7}$Be line}\
The Borexino collaboration started taking data in May~2007 and after only 3
months (47.4~live days) it was able to extract the ${^7}$Be signal from the
background. The best value estimate for the rate was $47 \pm 7 \, ({\rm
stat}) \pm 12 \, ({\rm syst})$~counts/(day~$\cdot$~100~ton), where the
systematic error is mainly due to the fiducial mass determination
\cite{14-bxfirstresults}. An update of the ${^7}$Be signal was reported after
9 months from an analysis of 192 live days (from May $16^{th}$ 2007 to April
$12^{th}$ 2008), corresponding to 41.3~ton$\cdot$yr fiducial exposure to solar
neutrinos.
The severe cuts that had to be passed by the events in order to be selected
and enter the analysis were mainly designed to avoid pile up of multiple
events, reject the events originated by muons and their daughters and the ones
due to radon daughters preceding the $\alpha-\beta$ Bi-Po delayed
coincidences. Moreover, severe cuts (radial and based on the z-coordinates)
were finalized to reduce the external $\gamma$ background. The remaining
fiducial mass was of 78.5 tons. Important background sources were the fast
coincidence decays from the $^{238}$U chain (contamination level of $(1.6 \pm
0.1) \, 10^{-17}$ g/g) and the $^{232}$Th chain (contamination level of $(6.8
\pm 1.5) \, 10^{-18}$ g/g) and the $^{85}$Kr contained in the scintillator that produces the rare decay sequence ${\rm ^{85}Kr} \to \, {\rm ^{85 m}Rb} \, + \,
e^+ \, + \nu_e \, , \, {\rm ^{85 m}Rb} \to \, {\rm ^{85}Rb} + \gamma$.
The total estimated systematic error was 8.5\% \cite{14-Arpesella2008},
mainly determined by two sources, introducing an uncertainty of 6\% each:
the total uncertainty on the fiducial mass and the one on the response
function. The best value for the interaction rate of the 0.862~MeV $^7$Be
solar neutrinos was $49 \, \pm 3 ({\rm stat}) \pm 4 ({\rm
syst})$~counts/(day$\cdot$100~ton). This result excludes at the $4 \,
\sigma$ C.L. the no oscillation hypothesis for $^7$Be solar neutrinos, which
in the high metallicity SSM \cite{14-SSM06,14-carlos} would
imply $74 \, \pm 4$ counts/(day$\cdot$100~ton). The Borexino result is,
instead, in very good agreement with the predictions of the LMA oscillation
solution: $48 \, \pm 4$~counts/(day$\cdot$100~ton).\\
In order to reduce the systematic uncertainties and to tune the reconstruction
algorithm and Monte Carlo simulations, a calibration campaign was performed in
2009 introducing inside the Borexino detector several internal radiosources
$\alpha$'s, $\beta$'s, $\gamma$'s, and neutrons, at different energies and in
hundreds of different positions, which were determined with a precision better
than 2~cm. The previous systematic error on $^7$Be solar neutrino flux was
estimated to be \cite{14-Arpesella2008} at the level of $6 \%$ for both the
fiducial volume and the energy scale. In the calibration campaign, the
detector energy response was studied with eight $\gamma$ sources and Am-Be
neutron source\footnote{When thermal neutrons are captured by protons a 2.2~MeV
$\gamma$-ray is generated.} and comparing the calibration data and Monte
Carlo simulations at different energies within the solar neutrinos energy
region.
The energy scale uncertainty, obtained with these studies, was determined to
be less than $1.5\%$.
The inaccuracy of the position (reduced by means of studies with $\alpha$ and
$\beta$ events) was less than 3 cm, equivalent to a systematic error of $1.3
\%$ for the overall fiducial volume in the $^7$Be solar neutrino energy region.
The
analyzed data set run from May 2007 to May 2010, with a fiducial exposure
equivalent to 153.6 ton$\cdot$year. In order to extract the $^7$Be solar
neutrino signal, the spectral fit was applied assuming all the intrinsic
background components such as $^{85}$Kr, $^{210}$Bi, $^{14}$C, $^{11}$C. The
$^{7}$Be solar neutrino rate was evaluated to be ${\rm 46.0 \pm 1.5 (stat) \pm
1.3(syst)}$~counts/day$\cdot$100~ton \cite{14-Bellini2011rx}. Thanks to the
calibration campaign, the systematic error was reduced to $2.7\%$ and the
total uncertainty to $4.3\%$.
\begin{figure}[h!]
\vspace{-0.5truecm}
\hspace{-3.0cm}
\includegraphics[width=11.0cm,height=11.5cm,angle=0]{MC_final.pdf}
\hspace{-2.0cm}
\includegraphics[width=11.0cm,height=11.5cm,angle=0]{M4-LNGS_final.pdf}
\vspace{-3.5truecm}
\caption{ Examples of fitted spectra; the fit results in the legends have
units [counts/(day$\cdot$100\,ton)]. Left panel: A Monte Carlo based fit
over the energy region 270--1600~keV to a spectrum from which some, but not
all, of the $\alpha$ events have been removed using a PSA cut, and in which
the event energies were estimated using the number of photons detected by
the PMT array. Right panel: An analytic fit over the 290--1270~keV energy
region to a spectrum obtained with statistical $\alpha$ subtraction and in
which the event energies were estimated using the total charge collected by
the PMT array. In all cases the fitted event rates refer to the total rate
of each species, independently from the fit energy window (from
\cite{14-Bellini2011rx}).
\label{fig-14:6}}
\end{figure}
\subsection{The {\mbox{\it pep}} and {\mbox{CNO}} neutrinos measurement in
Borexino}\
In the SSM, due to the solar luminosity constraint and their intimate link to
the {\it pp} neutrinos \cite{14-bahcall:1989,14-adelberger:1998}, the
mono-energetic 1.44\,MeV \mbox{\it pep} neutrinos have one of the smallest
uncertainties (1.2\%) \cite{14-serenelli:2011}. For this reason, after
the \mbox{\it pp} neutrinos, they constitute the ideal probe to test SSM
hypotheses. On
the other hand, the detection of neutrinos within the CNO-bicycle is
central to probe the solar core metallicity and contribute in this way to the
solution of the solar metallicity problem~\cite{14-metallicity, 14-serenelli:2011}. Also, they are believed to fuel massive stars with mass
greater than $\sim 1.2\, {\rm M_{\odot}}$ during main sequence evolution and also
stars with lower masses in more advanced stages of evolution. The energy spectrum
of neutrinos from the CNO-bicycle is the result of three continuous spectra with
end point
energies of 1.19\,MeV ($^{13}$N), 1.73\,MeV ($^{15}$O) and 1.74\,MeV
($^{17}$F). Despite their relevance, until 2011, no \mbox{\it pep} and
\mbox{CNO} neutrinos had been detected directly.
The electron recoil energy spectrum from \mbox{\it pep} neutrino interactions
in Borexino is a Compton-like shoulder with end point of 1.22\,MeV, as one can
see from Fig.(\ref{fig-14:7}), showing the \mbox{\it pep} and \mbox{CNO}
contribution in Borexino .
\begin{figure}[h!!]
\begin{center}
\hspace{-1truecm}
\includegraphics[width=12.0cm,height=10.5cm,angle=0]{pepCNO.pdf}
\vspace{-2.5truecm}
\caption{ The neutrino-induced electron recoil spectra expected in
Borexino. The total rates are those predicted by the latest {\em high-Z}
solar model \cite{14-serenelli:2011}. The {\it pep} and CNO neutrinos recoil
spectra with end points in the region 1.2-1.5~MeV are shown. Also the $^7$Be
neutrinos (measured in \cite{14-Bellini2011rx}), with a count rate about
10 times larger, are shown for comparison. Note that the variable on the x
axis is not directly the energy value. Taken from \cite{14-Davini}.
\label{fig-14:7}}
\end{center}
\end{figure}
As already mentioned, very low background
levels~\cite{14-bxfirstresults,14-Arpesella2008} are required to detect
$^{7}$Be neutrinos; the detection of \mbox{\it pep} and \mbox{CNO} neutrinos
is even more challenging, as their expected interaction rates are
$\sim$10~times lower. The expected rate is on the order of a few counts per day
in a 100\,ton target. To detect \mbox{\it pep} and \mbox{CNO} neutrinos the
Borexino Collaboration adopted a novel analysis procedure to suppress the
dominant background in the 1--2 MeV energy range, due to the
cosmogenic $\beta^+$-emitter \mbox{$^{11}$C}
produced within the scintillator
by muon interactions with {\mbox{$^{12}$C}} nuclei. The muon flux crossing
the Borexino detector, $\sim$4300\,$\mu$/day, yields a \mbox{$^{11}$C} production
rate of $\sim$27 \mbox{counts/(day$\cdot$100\,ton)}. This background can be
reduced by performing a space and time veto following coincidences between signals
from the muons and the cosmogenic neutrons \cite{14-deutsch, 14-pep-ctf},
discarding exposure that is more likely to contain \mbox{$^{11}$C} due to the
correlation between the parent muon, the neutron\footnote{In 95\%~of the
cases at least one free neutron is spalled in the \mbox{$^{11}$C} production
process \cite{14-c11cris}, and then captured in the scintillator with a mean
time of 255\,$\mu$s \cite{14-bxmuon}.} and the subsequent \mbox{$^{11}$C}
decay (the Three-Fold Coincidence, TFC). The TFC technique is based on the
reconstructed track of the muon and the reconstructed position of the
neutron-capture $\gamma$-ray \cite{14-bxmuon}. The criteria of rejection were
applied to obtain the best compromise between \mbox{$^{11}$C} rejection and
preservation of fiducial exposure, resulting in a \mbox{$^{11}$C} rate of
(2.5$\pm$0.3) count per day, (9$\pm$1)$\%$ of the original rate, while
preserving 48.5\% of the initial exposure.
Figure~(\ref{fig-14:8}) shows the resulting spectrum obtained with data
collected between January 2008 and May 2010, corresponding to a fiducial
exposure of 20409 ton$\cdot$day \cite{14-pepBX}. Despite the TFC veto, the
number of \mbox{$^{11}$C} surviving events still constituted a significant
background.
\begin{figure}[h!!]
\begin{center}
\includegraphics[width=9.0cm,height=8.0cm,angle=0]{tfc.pdf}
\caption{Energy spectra of the events in the FV before and after the TFC veto
is applied. The solid and dashed blue lines show the data and estimated
\mbox{$^{11}$C} rate before any veto is applied. The solid black line shows
the data after the procedure, in which the \mbox{$^{11}$C} contribution
(dashed) has been greatly suppressed. The next largest background,
\mbox{$^{210}$Bi}, and the electron recoil spectra of the best estimate of
the \mbox{\it pep} neutrino rate and of the upper limit of \mbox{CNO}
neutrino rate are shown for reference. Rate values in the legend are quoted
in counts/(day $\cdot$ 100 ton) from \cite{14-pepBX}.
\label{fig-14:8}}
\end{center}
\end{figure}
To discriminate \mbox{$^{11}$C} $\beta^+$ decays from neutrino-induced $e^-$
recoils and $\beta^-$ decays the pulse shape differences between $e^-$ and
$e^+$ interactions in organic liquid scintillators \cite{14-annihilation,
14-positronium} were exploited. In fact a small difference in the time
distribution of the scintillation signal arises from the finite lifetime of
ortho-positronium as well as from the presence of annihilation $\gamma$-rays,
which present a distributed, multi-site event topology and a larger average
ionization density than electron interactions. The Borexino Collaboration
employed an optimized pulse shape parameter using a boosted-decision-tree
algorithm \cite{14-tmva}, trained with a TFC-selected set of \mbox{$^{11}$C}
events ($e^+$) and \mbox{$^{214}$Bi} events ($e^-$) selected by the fast
\mbox{$^{214}$Bi-$^{214}$Po} $\alpha$-$\beta$ decay sequence. In a work
published in 2012 \cite{14-pepBX} the Borexino Collaboration presented the
results of an analysis based on a binned likelihood multivariate fit performed
on the energy, pulse shape, and spatial distributions of selected
scintillation events whose reconstructed position is within the fiducial
volume\footnote{less than 2.8\,m from the detector center and with a
vertical position relative to the detector center between -1.8\,m and
2.2\,m.}.
The energy spectra and spatial distribution of the external $\gamma$-ray
backgrounds have been obtained from a full, Geant4-based Monte Carlo
simulation, and validated with calibration data from a high-activity
$^{228}$Th source~\cite{14-maneschg} deployed in the outermost buffer region,
outside the active volume. $\alpha$ events were removed from the energy
spectrum by the statistical subtraction method~\cite{14-bxfirstresults}. In
the energy region of interest of the fit procedure all background species
whose rates were estimated to be less than 5\% of the predicted rate from
\mbox{\it pep} neutrinos have been excluded. All rates were constrained to
positive values and thirteen species were left free in the
fit\footnote{electron recoils from $^{7}$Be, \mbox{\it pep}, and \mbox{CNO}
solar neutrinos, internal radioactive backgrounds $^{210}$Bi, $^{11}$C,
$^{10}$C, $^{6}$He, $^{40}$K, $^{85}$Kr, and $^{234m}$Pa, and external
$\gamma$-rays from $^{208}$Tl, $^{214}$Bi, and $^{40}$K.}. The rate of the
radon daughter $^{214}$Pb was fixed using the measured rate of
\mbox{$^{214}$Bi-$^{214}$Po} delayed coincidence events. The contribution from
\mbox{\it pp} solar neutrinos was fixed to the SSM assuming MSW-LMA with
$\tan^2\theta_{12}$=0.47$^{+0.05}_{-0.04}$, $\Delta
m^2_{12}$={(7.6$\pm$0.2)}$\cdot 10^{-5}$\,eV$^2$~\cite{14-pdg2010}, and the
contribution from $^{8}$B neutrinos to the rate from the measured
flux~\cite{14-LETA,14-SNOI+II+III}.
In Table~\ref{tab:results-summary} the results for the \mbox{\it pep} and
\mbox{CNO} neutrino interaction rates are shown. The absence of a \mbox{\it
pep} neutrino signal was rejected at 98\%~C.L. Concerning the \mbox{CNO}
neutrinos flux, its electron-recoil spectrum is similar to the spectral shape
of $^{210}$Bi, but the last one is about 10 times greater; therefore it has
only been possible to provide an upper limit on the \mbox{CNO} neutrino
interaction rate. The 95\%~C.L. limit reported in
Table~\ref{tab:results-summary} has been obtained from a likelihood ratio
test with the \mbox{\it pp} neutrino rate fixed to the SSM
prediction~\cite{14-serenelli:2011} under the assumption of MSW-LMA,
(2.80$\pm$0.04)\,\mbox{counts/(day$\cdot$100\,ton)}.
\begin{table}[!ht]
\begin{center}
\begin{tabular}{lccc}
\hline \hline $\nu$ &Interaction rate &Solar-$\nu$ flux
&Data/SSM\\ &[\mbox{counts/(day$\cdot$100\,ton)}] &[$10^{8} cm^{-2} s^{-1}$]
&ratio\\ \hline \mbox{\it pep} &$3.1 \pm 0.6_{\rm stat} \pm$ 0.3$_{\rm syst}$
&$1.6\pm0.3$ &$1.1\pm0.2$\\ \mbox{CNO} &$<7.9$ ($<7.1_{\rm stat\,only}$)
&$<7.7$ &$<1.5$\\ \hline \hline
\end{tabular}
\end{center}
\caption{ Best estimates for the \mbox{\it pep} and \mbox{CNO} solar
neutrino interaction rates. For the results in the last two columns both
statistical and systematic uncertainties are considered. Total fluxes have
been obtained assuming MSW-LMA and using the scattering cross-sections from
\cite{14-BahcallRadiativeCorrection, 14-pdg2010, 14-erlerRadCorr} and a
scintillator $e^-$ density of (3.307$\pm$0.003)$\cdot 10^{29}$ \,ton$^{-1}$.
The last column gives the ratio between our measurement and the
{\em high-Z} (GS98) SSM~\cite{14-serenelli:2011}. Table taken from \cite{14-pepBX}.}
\label{tab:results-summary}
\end{table}
\section{Phenomenological analysis}\
\label{14-sec:pheno-analysis}
\subsection{Status of the determination of the mixing parameters in a 3 flavor
analysis}
\label{14-sec:status-mixing}
Recently, the SNO collaboration performed a combined analysis of all the three
working phases of the experiment \cite{14-SNOI+II+III} based on
a fit to Monte Carlo derived probability density functions (PDFs) for each of
the possible signals and backgrounds, and also introduced a new way to
parametrize the $^8$B neutrino signal. Figure~(\ref{fig-14:9}),
reporting the results of the two flavour (with the assumption $\theta_{13} =
0$) SNO only analysis, shows the further improvement in the mixing parameters
accuracy, but, at the same time, it confirms that the SNO results alone would
not be sufficient to completely exclude the LOW solution.
\begin{figure}[h!!]
\begin{center}
\includegraphics[width=7.5cm,height=7.0cm,angle=0]{SNO2011alone.pdf}
\caption{Two-flavor neutrino oscillation analysis contour using only SNO data \,
(taken from \cite{14-SNOI+II+III}).
\label{fig-14:9}}
\end{center}
\end{figure}
This ambiguity was definitely removed, as shown in Figure~(\ref{fig-14:10}),
by including in the analysis the results of all previous solar neutrino
experiments \cite{14-Gallium09,14-Cleveland1998,14-SKI-2005,14-SKII,14-SKIII},
the $^7$Be solar neutrino rate measured by Borexino
\cite{14-Bellini2011rx}, the $^8$B neutrino spectra
\cite{14-Bellini2008mr} and the KamLAND data\footnote{The KamLAND data were
obtained in a completely independent
experiment and, therefore, the corresponding $\chi^2$ values, as functions
of the mixing parameters, were directly summed to the $\chi^2$ values
computed by direct solar neutrino analysis.} \cite{14-KLfollowing3}.
\begin{figure}[h!!]
\begin{center}
\includegraphics[width=8.5cm,height=8.0cm,angle=0]{SNO2011_global_2nu.pdf}
\caption{Two-flavor neutrino oscillation analysis contour using both solar neutrino
and KamLAND results \,(taken from \cite{14-SNOI+II+III}).
\label{fig-14:10}}
\end{center}
\end{figure}
The higher values of $\Delta m_{12}^2$ in the LMA region were excluded,
together with the full LOW solution, thanks mainly to the large discrimination
power of KamLAND. This experiment, however, did not contribute significantly to
improve the mixing angle determination and the accuracy on this parameter
remained quite high. The results of the two flavor analysis are reported in
Table~(\ref{tab:global2nu}) (taken from \cite{14-SNOI+II+III}).
\begingroup
\begin{table}
\centering
\begin{tabular}{lccc}
\hline\hline
Analysis & $\tan^{2} \theta_{12}$ & $\Delta m_{21}^2 [{\rm eV^{2}}]$ & $\chi^{2}/{\rm NDF}$\\
\hline
SNO only (LMA)&$0.427^{+0.033}_{-0.029}$& $5.62^{+1.92}_{-1.36}\times 10^{-5}$& $1.39/3$\\
SNO only (LOW) &$0.427^{+0.043}_{-0.035}$& $1.35^{+0.35}_{-0.14}\times 10^{-7}$& $1.41/3$ \\
Solar &$0.427^{+0.028}_{-0.028}$& $5.13^{+1.29}_{-0.96}\times 10^{-5}$& $108.07/129$\\
Solar+KamLAND&$0.427^{+0.027}_{-0.024}$& $7.46^{+0.20}_{-0.19}\times 10^{-5}$& \\
\hline\hline
\end{tabular}
\caption{Best-fit neutrino oscillation parameters from a two-flavor neutrino
oscillation analysis. Uncertainties listed are $1 \, \sigma$ after the
$\chi^2$ was minimized with respect to all other parameters (taken from
\cite{14-SNOI+II+III}).}
\label{tab:global2nu}
\end{table}
\endgroup
The slight tension between the solar neutrino experiments and KamLAND was
significantly reduced by extending the analysis to the 3 flavor oscillation case
as shown in Figure~(\ref{fig-14:11}), from which it is clear that the best global
fit is obtained for values of $\theta_{13}$ different from zero.
\begin{figure}[h!!]
\begin{center}
\hspace{-0.50cm}
\includegraphics[width=7.6cm,height=7.6cm,angle=0]{contour_global_3nu_12.pdf}
\hspace{0.1cm}
\includegraphics[width=7.6cm,height=7.6cm,angle=0]{contour_global_3nu_13.pdf}
\caption{Three-flavor neutrino oscillation analysis contour using both solar
neutrino and KamLAND results. Taken from \cite{14-SNOI+II+III}.
\label{fig-14:11}}
\end{center}
\end{figure}
A detailed analysis of the $\chi^2$ behavior proved also that the combination
of solar experiments and KamLAND enables to improve significantly the
discriminating power on the $\theta_{13}$ mixing parameter (see Figure~
\ref{fig-14:12} and Table~\ref{tab:chi3nu}).
\begin{figure}[h!!]
\begin{center}
\includegraphics[width=7.5cm,height=7.5cm,angle=0]{proj_global_3nu_th13_1s.pdf}
\vspace{-0.5cm}
\caption{Projections of the three-flavor neutrino oscillation parameters.
The horizontal lines represent the $\Delta\chi^2$ for a particular confidence
level. Taken from \cite{14-SNOI+II+III}.
\label{fig-14:12}}
\end{center}
\end{figure}
\begingroup
\begin{table}
\begin{tabular}{lccc}
\hline\hline
Analysis & $\tan^2 \theta_{12}^2$ & $\Delta m_{12}^2 [{\rm eV^{2}}]$ & $\sin^2
\theta_{13} \times 10^{-2}$\\
\hline
Solar & $0.436^{+0.048}_{-0.036}$ & $5.13^{+1.49}_{-0.98}\times 10^{-5}$ & $ <5.8$ (95\% C.L.)\\
Solar+KL& $0.446^{+0.030}_{-0.029}$ &$7.41^{+0.21}_{-0.19}\times 10^{-5}$& $2.5^{+1.8}_{-1.5}$\\
& & & $<5.3$ (95\% C.L.)\\
Global & & & $2.02^{+0.88}_{-0.55}$\\
\hline\hline
\end{tabular}
\caption{Best-fit neutrino oscillation parameters from a three-flavor neutrino
oscillation analysis. Uncertainties listed are $\pm 1\, \sigma$ after the
$\chi^2$ was minimized with respect to all other parameters. The global
analysis includes Solar+KL+ATM+LBL+CHOOZ.}
\label{tab:chi3nu}
\end{table}
\endgroup
The indication in favor of $\theta_{13}$ being different from zero was in
agreement with the recent results from the long-baseline experiments T2K
\cite{14-T2K2011} and MINOS \cite{14-MINOS2011}, and with the combined analysis
performed in \cite{14-FogliLisi2011}, including also the atmospheric neutrino
and the CHOOZ \cite{14-CHOOZ03} data. Moreover the validity of this hint has
been corroborated by the data obtained this year by the short baseline neutrino
reactor experiments \cite{14-DoubleChooz, 14-DayaBay, 14-RENO}, which
established that $\theta_{13} > 0$ at about $5 \sigma$ (and even more in the
Daya Bay case \cite{14-Daya_Bay_neu2012}). These experiments
found values of $\sin^2
\theta_{13}$ centered between $0.020$ and $0.030$; very promising results for
future experiments looking for leptonic CP violation \cite{14-Fogli2012ua}.
The impact and the possible consequences of these recent results have been
discussed, among the others, in the following papers \cite{14-Fogli2012ua,
14-Tortola2012te,14-Schwetz2012}. The different accuracy that can be
reached in the determination of the mixing angle between the first and third
generation, according to the different kind of neutrino experiments included
in the analysis, is represented in Figure~(\ref{fig-14:13}).
\begin{figure}[h!!]
\begin{center}
\includegraphics[width=7.5cm,height=6.8cm,angle=0]{proj_th13_global.pdf}
\caption{Projection over $\sin^{2}\theta_{13}$ combining the projections
obtained by analyzing data from all neutrino sources. The data from
atmospheric, short-baseline experiments and long-baseline experiments
(ATM+LBL+CHOOZ) was determined from Figure~2 (left panel) in
\cite{14-FogliLisi2011} which already includes the latest
T2K~\cite{14-T2K2011} and MINOS~\cite{14-MINOS2011} results.
\label{fig-14:13}}
\end{center}
\end{figure}
The combined analysis of the different SNO phases was also very useful to
obtain a precise determination of the $^8$B solar neutrino flux, $\Phi_{^8
B} = 5.25 \pm 0.16 (\rm stat)^{+0.11}_{-0.13} (\rm syst) \times 10^6
\,{\rm cm^{-2} \, s^{-1}}$, with an important reduction of the systematic
uncertainty.
This result was consistent with, but more precise than, both the {\em high-Z}
BPS09(GS), $\Phi = (5.88 \pm 0.65) \times 10^6 {\rm \, cm^{-2} \, s^{-1}}$, and
{\em low-Z} BPS09(AGSS09),
$\Phi = (4.85 \pm 0.58) \times 10^6 {\rm \, cm^{-2} \, s^{-1}}$, solar
model predictions \cite{14-Serenelli2009yc}.
The combination of the LETA analysis by the SNO collaboration \cite{14-LETA} and
of the Borexino measurements \cite{14-Bellini2008mr} made possible a detailed
study of the low energy part of the $^8$B solar neutrino spectrum. Even if
characterized by a larger uncertainty (mainly due to a more limited
statistics), Borexino data confirm the LETA indication of low energy data
points lower than the theoretical expections based on matter enhanced
oscillation and solar models as shown in Figure~\ref{fig-14:14} (taken from
\cite{14-Bellini2008mr}). These results agreed also with the
Super-Kamiokande observation \cite{14-SKI-2005} of flat spectrum, consistent
with the undistorted spectrum hypothesis. The emergence of this slight tension
between theory and experiments seems to indicate the presence of new
subdominant effects and also suggests the possibility of non-standard
neutrino interactions (like those studied in \cite{14-Friedland:2004pp}) or
the mixing with a very light sterile neutrino \cite{14-deHolanda2010am}.
Future solar neutrino experiments, like SNO+, could shed more light on this
subject, by performing precision measurements of lower energies solar neutrinos
(like the \mbox{\it pep} neutrinos).
\begin{figure}[h!!]
\begin{center}
\includegraphics[width=8.5cm,height=7.0cm,angle=0]{Borex8B.pdf}
\caption{
Taken from (\cite{14-Bellini2008mr}).
\label{fig-14:14}}
\end{center}
\end{figure}
\subsection{Free flux analyses}
\label{14-subsection:freeflux}
The increasing data of solar neutrinos allow to independently test the astrophysics of the solar interior
and the physics of neutrino propagation. The analysis discussed in previous sections can be modified by also varying the solar neutrino fluxes
in order to accommodate all neutrino data, while all the functional dependences are maintained as predicted
by the standard model dependences. A key step in this kind of analysis is the
imposition of the luminosity constraint~\cite{14-spirovignaud,14-luminosity}, which implements in a global way for the Sun the constraint of conservation of energy for nuclear fusion among light elements. Each neutrino flux is associated with a
specific amount of energy released to the
star and therefore a particular linear combination of the solar neutrino fluxes is equal to the solar
luminosity (in appropriate units). One can write the luminosity constraint as
\begin{equation}
{L_\odot\over 4\pi (A.U.)^2} = \sum\limits_i \alpha_i \Phi_i~, \label{eq:genconstraint}
\end{equation}
where $\L_\odot$ is the solar luminosity measured at the earth's surface, 1 $A.U.$ is the average
earth-sun distance, and the coefficient $\alpha_i$ is the amount of energy provided to the star by
nuclear fusion reactions associated with each of the important solar neutrino fluxes, $\Phi_i$. The
coefficients $\alpha_i$ are calculated accurately in ref.~\cite{14-luminosity}.
The model independent determination of the solar neutrino fluxes \cite{14-concha:2010, 14-roadmap} shows that
present solar neutrino data leads to accurate results for four fluxes and also the correlations between them.
This information allows for a consistent global comparison of SSM fluxes with the inferred fluxes by neutrino
data. Present data leads to the values for the inferred solar neutrino fluxes
reported in the fourth column (labelled as ``Solar'') of
Table~\ref{14-tab:nufluxes} in Section~\ref{14-section:SSM}.
The precision of the $^7$Be and $^8$B neutrino fluxes is driven by the Borexino and SNO (SK) neutrino experiments,
while the precision of the pp and pep neutrino fluxes mainly comes by the imposition of the luminosity constraint.
The neutrino data directly demonstrates that the Sun shines by the pp chain. The CNO cycle only contributes to the
total luminosity at the percent level.
The reader may wonder how much these inferences are affected by the luminosity constraint. The idea that the Sun
shines because of nuclear fusion reactions can be tested accurately by comparing the observed photon luminosity of
the Sun with the luminosity inferred from measurements of solar neutrino fluxes. Moreover, this same comparison will
test a basic result of the standard solar model, namely, that the Sun is in a quasi-steady state in which the current
energy generation in the interior equals the current luminosity at the solar surface. The free flux analysis, without
imposing luminosity constraint, permits an estimation of the solar luminosity inferred by neutrino data,
which agrees with the directly measured one within 15 \% (1 $\sigma$).
\section{Future solar neutrino experiments}\label{14-sec:future}
\subsection{The near future: improvement of {\it pep} measurements and CNO
detection}
In the last decades the intensive study of $^8$B and, more
recently, $^7$Be solar neutrinos made possible fundamental steps forward in
the solution of the solar neutrino puzzle and the determination of the neutrino
mixing parameters. Nevertheless, many key features of the oscillation models (like
the transition between the vacuum dominated sub-MeV region and the spectral region
between 1 and 3 MeV, where matter effects become
relevant) still have to be tested or verified with better accuracy and precision
(see Figure~\ref{fig-14:15}, taken from \cite{14-Chavarria2012sd}).
\begin{figure}[h!!]
\begin{center}
\hspace{-4truecm}
\includegraphics[width=14.0cm,height=12.0cm,angle=0]{chavarria.pdf}
\vspace{-3.2truecm}
\caption{The $\nu_e$ survival probability is represented as a function of
neutrino energy. The gray band represent the MSW-LMA prediction. The higher
survival probability region at low energies is where vacuum-dominated
oscillations occur. As the neutrino energy increases, matter effects become
important and the lower survival probability at high energies is due to
matter-enhanced oscillations. The reported data correspond to solar nuetrino
flux measurements performed by different experiments. Taken from
(\cite{14-Chavarria2012sd}).}
\label{fig-14:15}
\end{center}
\end{figure}
The apparent partial deficit of events in the low energy part of the $^8$B
spectrum suggested the introduction of new theoretical models (as discussed in
section \ref{14-sec:pheno-analysis}). Also for these reasons, the experimental
efforts in the last years focused on the detection of neutrinos of ever
decreasing energies, to fully confirm the validity of the MSW-LMA solution and
verify the fluxes predicted by SSMs, discriminating between
different version of these models. The fluxes of the medium and high energy
neutrinos of the {\it pp} chains ($^7$Be, $^8$B and hep) are predicted with
quite large uncertainties, mainly due to the uncertainties in nuclear cross
sections and solar opacity (Table~\ref{14-tab:fluxuncert}). The {\it pp} and {\it
pep} fluxes,
instead, are strongly correlated between themselves and their values are
predicted with the highest precision because SSMs predict that {\it pp} chain
reactions are responsible for more than 99\% of the energy powering the Sun
\cite{14-bahcall:2006}.
Therefore, the measurements of these components
would be the most stringent test of the SSM.
The tight correlation between {\it pep} and {\it pp} neutrinos is theoretically
well established and, therefore, even in the pessimistic hypothesis that
{\it pp} neutrinos could not be measured with the desired accuracy, a
significant improvement in the {\it pep} neutrinos measurement with respect to
data presently available would make possible to reduce significantly the
$ 15 \%$ indetermination on the solar luminosity (see subsection \ref{14-subsection:freeflux}) and to test
indirectly the SSM's predictions that almost $100 \%$ of solar energy is
produced by nuclear burning.
As already mentioned, water \v{C}erenkov detectors, which played a fundamental
role in the solution
of the solar neutrino problem, are characterized by a low photon yield
\cite{14-Boger1999bb, 14-Fukuda2003s} and therefore can detect only the
higher part of the spectrum (hep and $^8$B neutrinos with a threshold around
$3.5 \, \rm{MeV}$). The radiochemical experiments \cite{14-Cleveland1998,
14-Abdurashitov1999zd} are limited, instead, by their ability to measure
only the
integrated neutrino rate above the charge-current interaction threshold (down
to $0.23 \, \rm{MeV}$ for the Gallium experiments), without the possibility to
discriminate between the different spectrum components. Therefore, an
important contribution should come from the present and future organic liquid
scintillator detectors, planned to perform low energy solar neutrino
spectroscopy. To reach this goal, they will take advantage from the high
values of light yield (about $10^4$ photons per MeV of deposited energy) and
from the possibility to assemble very large masses of high purity material.
The excellent levels of radiopurity, reached for instance at Borexino, and the
typical geometry of these detectors (which are unsegmented and can be easily
adapted to the definition of a fiducial volume) are fundamental to reduce the
impact of the background, that is so critical due to the feebleness of the
low energy signal.
In the near future significant contributions are expected from Borexino and
SNO$+$ \cite{14-Kraus2010zzb} experiments. Borexino has already proved its
importance in this kind of analysis performing the first measurements of {\it
pep} and CNO neutrinos (even if the level accuracy is not yet the
desired one) and further reducing, with the purification campaign started
since July 2010, the level of contamination from almost all of the main
radioactive background sources\footnote{The main problem still surviving seems
to be the reduction of $^{210}$Pb.}. The purification efforts are still
ongoing and should make possible a further improvement on the accuracy
of the signal extraction. The SNO+ experiment, that should start taking data
soon in the SNOLAB, should take advantage from the location (about two times deeper
underground than the Gran Sasso laboratory), with the consequent lower muon flux
and a strongly reduced $^{11}$C rate. Moreover, thanks to the detector mass
(about three times larger than in Borexino), it should be able to reach
a higher counting rate. This could determine a fundamental improvement at
least in the case of the {\it pep} neutrino measurement, where a $5 \%$
uncertainty is expected, to a level that should make possible a significant test
of the MSW transition region.
In the more optimistic scenarios it may be also possible to attach the main
problem of measuring lowest energy parts of the solar neutrino spectrum, that
is the {\it pp} neutrinos and the $0.38 \, {\rm MeV}$ Berillium line. In any
case the presence in organic scintillators of an intrinsic $^{14}$C
background will make this very low energy measurements an extremely hard task
and they may require the introduction of new techniques, like the ones we are
going to describe in the next subsections.
\subsection{The far future: experimental challenges}
The challenge for all future experiments aimed at measuring the low energy
part of solar neutrino spectrum is that of assembling experimental devices
with low energy thresholds suitable to detect a low rate signal in a region
characterized by different potential sources of radioactive background. This
difficult experimental task is common also to the experiments looking for
neutrinoless double $\beta$ decay or for dark matter signals (search for
signatures of WIMPs, a stable or long-lived weakly interacting
elementary particle, produced in the early Universe, whose existence is
predicted in extensions of the Standard Model). In fact, some of the solar
neutrino
experiments planned for the future are multipurpose experiments designed also
for the other above-quoted topics.
They are all characterized by a very large detector target mass and by the
need to reach very high levels of radiopurity. The common feature is that of
using scintillator detectors, but they differ for the chosen active
scintillator material, which can vary from traditional organic scintillators
(developed with the use of innovative technological devices) to new materials,
like the noble gases.
\subsubsection{Noble liquid detectors: CLEAN and XMASS}\
One of the possible future frontiers is the idea to use scintillation
detectors with liquid noble gases, like xenon, argon and neon. These
materials have the advantage of being relatively inexpensive, easy to obtain
and dense and it is not too difficult to build large homogeneous detectors of
this kind; moreover, they can be quite easily purified, offer very high
scintillation yields (about $30-40$ photons/keV) and do not absorb their own
scintillation light.
\begin{figure}[h!]
\begin{center}
\includegraphics[width=8.5cm,height=7.3cm,angle=0]{CLEAN-figure.pdf}
\caption{Scheme of the CLEAN detector. Taken from \cite{14-CLEAN2004}.
\label{fig-14:CLEAN}}
\end{center}
\end{figure}
A first example is offered by the CLEAN/DEAP family, a series of
detectors based entirely on scintillation in liquid neon (LNe) and liquid
argon (LAr). They have been realized using a scaleable technology in order to
reach increasing sensitivities in the different prototypes realized and
installed in the SNOLAB (Pico-CLEAN, Micro-CLEAN, DEAP-I, Mini-CLEAN and
CLEAN/DEAP) with the aim to search for dark matter and to perform (through the
analysis of elastic neutrino-electron and neutrino-nucleus scattering) a real
time measurement of the ${\it pp}$ solar neutrino flux. The final detector
CLEAN (Cryogenic Low Energy Astrophysics with Noble gases) \cite{14-CLEAN2004}
(see figure \ref{fig-14:CLEAN}) will be made by a stainless steel tank, of
about 6 meters of diameter, filled with 100~tons of cryogenic liquid neon;
only the central part of it, surrounded isotropically by a series of
photomultipliers, will constitute the detector fiducial volume. An external
tank of water, 10~metres wide and 12~metres high, will act as $\gamma$-ray
shielding, neutron shielding and muon veto. According to Monte Carlo
simulations, there should be a production of 15000 photons/MeV and it should be
possible to reach a 100\% photon wavelength shifter efficiency and a
statistical uncertainty on the ${\it pp}$ measurements of the order of $1 \,
\%$.
A precise measurement of the ${\it pp}$ component and of the ratio between
${\it pp}$ and $^7$Be fluxes would be essential to test the predictions of
SSMs. A high accuracy on the ${\it pp}$ neutrino flux would
also make possible a better determination of the $\theta_{12}$ mixing angle,
which, complemented with the results from previous solar neutrino experiments
and from KamLAND (essential for the $\Delta m_{12}^2$ measurement), would
be fundamental to test the consistency of the LMA solution also in the region of
transition between vacuum dominated and matter enhanced oscillations. Finally,
CLEAN could in principle try to measure also the CNO neutrino flux, through
the analysis of neutrino spectrum from 0.7 to 1.0 MeV, with an estimated
accuracy between 10 and 15\%.
An interesting alternative to the use of neon is offered by liquid xenon
scintillator detectors \cite{14-Aprile2009dv}, which take advantage of the
fact that among liquid rare gases xenon has the highest stopping power for
penetrating radiation (thanks to its high atomic number, $A \simeq 131$ and
density, $\rho = 3 {\rm g/cm^3}$) and also the highest ionization and
scintillation yield. The technological improvements of the last twenty to
thirty years made
possible significant improvements in the cooling and purification techniques
of this kind of detectors and in the possibility of assembleing large mass
detectors, of the order of some tons (like in the case of MEG \cite{14-MEG}
experiment, studying the $\mu \rightarrow e \gamma$ decay).\\
The XMASS experiment (see Figure \ref{fig-14:XMASS}) is a multipurpose low
background and low energy threshold experiment that will use a large massive
liquid xenon detector and has been designed to look for WIMPs (dark matter
candidates), search for neutrinoless double
$\beta$ decay and study the ${\it pp}$ and the $^7$Be solar neutrinos. After
two preliminary phases, during which smaller prototypes have been realized and
installed in the Kamioka mine \cite{14-Moriyama2011zz}, and the first data on
double beta decay and dark matter have been taken, the full XMASS detector
(that will measure also solar neutrinos) will have a total mass of 20 tons,
with a fiducial volume of 10 tons. Special efforts are required mainly to
lower the background, by reducing the radioactive contamination in the parts
used for detector construction (with special attention to the photomultipliers
and the copper material used for PMT holder), constructing a larger pure water
active shield (for muons and mainly neutrons and $\gamma$ rays) and, above
all, developing a distillation system for xenon in order to reduce the
contamination by $^{85}$Kr, the major source of radioactive background inside
the detector.
\begin{figure}
\begin{center}
\includegraphics[width=14.5cm,height=12.5cm,angle=0]{XMASS-figure.pdf}
\vspace{-4.2truecm}
\caption{Schematic view of the full XMASS facilty (left) and a detail of the inner
detector (right panel), from which one can see the particular configuration
of the hexagonal photomultiplier tubes. Taken from \cite{14-Moriyama2011zz}
and \cite{14-XMASStalk}.
\label{fig-14:XMASS}}
\end{center}
\end{figure}
Another interesting experimental project based on the noble gases liquid
scintillator technique is that of DARWIN (DARk matter WImp search with Noble
liquids) \cite{14-DARWIN}, which brings together differen European and US
research groups working on existing experiments and on the study for a future
multi-ton scale LAr and LXe dark matter search facility in Europe. The main
goal of the experiment is to look for a WIMP signal and to demonstrate its dark
matter nature, taking advantage from the fact of performing the measurement
with multiple different targets operating under similar conditions. In this
way, it should be possible to estimate the dependence of the rate with the
target material and, therefore, to better determine the WIMP candidate mass
and to distinguish between spin independent and spin dependent couplings. The
energy region of the nuclear recoil spectrum, below 200~keV, that should be
investigated by this future experiment is of particular interest also for the
study of the ${\it pp}$ solar neutrinos and, in fact, the elastic scattering on
electrons by the low energy component of the neutrino spectrum would be one
of the main background sources for WIMP searches in liquid xenon detectors, as
shown in Figure~\ref{fig-14:DARWIN}.
\begin{figure}[h!]
\begin{center}
\vspace{1cm}
\includegraphics[width=7.5cm,height=6.5cm,angle=0]{DARWIN-spectrum.pdf}
\caption{Expected nuclear recoil spectrum from WIMP scatters in LXe for a
spin-independent WIMP-nucleon cross section of 10$^{-47}$\,cm$^2$ (red
solid) and 10$^{-48}$\,cm$^2$ (red dashed) and a WIMP mass of 100\,GeV/c$^2$,
along with the differential energy spectrum for pp (blue) and $^{7}$Be
(cyan) neutrinos, and the electron recoil spectrum from the double beta
decay of $^{136}$Xe (green).
Assumptions are: 99.5\% discrimination of electronic recoils, 50\% acceptance
of nuclear recoils, 80\% flat analysis cuts acceptance. Taken from the second
paper of \cite{14-DARWIN}.
\label{fig-14:DARWIN}}
\end{center}
\end{figure}
DARWIN officially started in 2010; a technical design study should be ready in
Spring 2013 and the start of the first physics run is expected by mid 2017.
\subsubsection{Multi kiloton scale liquid scintillators: example LENA}\
The Borexino experiment demonstrated the great potential of the
liquid-scintillator technique for the detection of low energy solar neutrinos.
Thanks to this experience, a next-generation neutrino detector has been
proposed: LENA (Low Energy Neutrino Astronomy) \cite{14-LENA}. LENA is a
multipurpose detector aiming to study supernova neutrinos, diffuse supernova
neutrino background, proton decay, atmospheric neutrinos, long-baseline
neutrino beams, geoneutrinos and, last but not least, solar neutrinos. The LENA
project foresees a cylindrical detector with a diameter of 30~m and a
length of about 100~m. Inside the detector is foreseen an internal part (with
a diameter of about 26 m) containing about 50 kilotons of liquid scintillator,
separated from a non-scintillating buffer region by a nylon barrier. Outside,
a tank (made in steel or concrete) separates the inner detector from an outer
water tank; it is used both for shielding and as an active muon veto. To
collect the scintillation light, about 45,000 photomultipliers (with a
diameter of 20 cm) are mounted to the internal walls of the detector. To
increase the optically active area, the photomultipliers tubes are equipped
with conic mirrors, the corresponding surface coverage is about 30\%. Figure
(\ref{fig-14:16}) shows a schematic overview of the current LENA design.
\begin{figure}[h!!]
\begin{center}
\includegraphics[width=8.8cm,height=8.0cm,angle=0]{LENA.pdf}
\caption{Schematical view of the LENA detector. From \cite{14-LENA}
\label{fig-14:16}}
\end{center}
\end{figure}
Among the favored solvent for the liquid scintillator in LENA, the LAB (linear
alkylbenzene) is currently the preferred one. It has a high light yield and
large attenuation length and it has also the advantage of being a non-hazardous
liquid. The attenuation lengths is on the order of 10 to 20~m (at a wavelength of
430~nm) and the photoelectron yield could be greater than 200 photoelectrons per
MeV (with a scintillator mixture containing 2g/l PPO and 20 mg/l bisMSB as
wavelength shifters). Studies have been carried out to test the large-scale
light transport and the differences in scintillator response for $\alpha$,
$\beta$ and $\gamma$ particles. An alternative solvent option is the well studied
PXE \cite{14-PXE} or a mixture of PXE and dodecane.
As already pointed out, Borexino has splendidly demonstrated the potential of
the detection technique with liquid scintillator based detectors for solar
neutrino detection. This technique offers the opportunity for a spectrally
resolved measurement of the solar neutrino spectrum in the all energy range.
Because the smaller ratio of surface to volume compared to the Borexino
detector\footnote{A smaller ratio of surface to volume decreases the chance
that the scintillator is contaminated with radioimpurities}, in LENA it is
very likely to reach the excellent background conditions of Borexino. Monte
Carlo simulations of the gamma background due to the uranium, thorium and
potassium from the photomultipliers glass shows that a fiducial volume of the
order of 30~ktons is achievable for solar neutrino studies; LENA will be able
to address topics both in neutrino oscillations and in solar physics thanks to
its unprecedented statistics. A high statistics can be obtained in short times
and in both Pyhsalmi and Frejus underground laboratories, where the detector
could be hosted, where the cosmogenic background of $^{11}$C will be significantly
lower than in Borexino.
Monte Carlo simulations show that for {\it pep}, CNO and low-energy
$^{8}$B-$\nu$s detection a fiducial mass of $\sim$30\,kton is
necessary, while the fiducial mass for $^{7}$Be-$\nu$s and high-energy
($E>5$\,MeV) $^{8}$B-$\nu$s could be enlarged to 35\,kton or more.
In Table~\ref{tab:ratesinLENA} are reported the expected rates in 30\,kton for
the neutrinos emitted in the {\it pp} chain and the CNO-bicycle, using the
most recent solar model predictions. This evaluation refers to a detection
threshold set at about 250\,keV.
\begin{table}
\begin{tabular}{lcccr}
\hline Source & EW [MeV] & $m_\textrm{fid}$ [kt] & Rate [cpd] \\ \hline pp &
$>$0.25 & 30 & 40 \\ pep & 0.8$-$1.4 & 30 & 2.8$\times$10$^2$ \\ $^{7}$Be &
$>$0.25 & 35 & 1.0$\times$10$^4$ \\ $^{8}$B & $>$2.8 & 35 & 79 \\ CNO &
0.8$-$1.4 & 30 & 1.9$\times$10$^2$ \\ \hline
\end{tabular}
\caption{Expected solar neutrino rates in LENA (channel $\nu e\to e\nu$). The
estimates are derived from the existing Borexino analyses
\cite{14-Arpesella2008mt,14-Bellini2008mr} as well as expectation values for
the respective energy windows (EW) of observation
\cite{14-wur09phd,14-Dangelo2006,14-Ianni2005ki}. The quoted fiducial masses,
$m_\textrm{fid}$, in LAB are based on a Monte Carlo simulation of the
external $\gamma$-ray background in LENA. Table taken and adapted from
\cite{14-LENA}. \label{tab:ratesinLENA}}
\end{table}
\subsubsection{New techniques with organic scintillators: LENS}\
The main goal of the Low Energy Neutrino Spectroscopy (LENS) detector is the
real time measurement of solar neutrinos as a function of their energy,
focusing, in particular, in the analysis of the lowest energy neutrinos
coming from proton-proton fusion (i.e. the {$\it pp$} neutrinos), which
represent the main contribution and the less known component of the pp-chain
of fusion reactions inside the Sun.
In order to make an energy spectrum measurement on low energy neutrinos, it is
necessary to reach a low threshold for the charged current (CC) process and to
be able to discriminate the background from radioactive decays. The CC
process employed in LENS is the neutrino induced transition of $^{115}$In to
an excited state of $^{115}$Sn:
\begin{equation}
\nu_{e} + ^{115}{\rm In} \rightarrow ^{115}{\rm Sn}^* + e^{-} \ ({\rm E=E_\nu -114 keV})\, .
\label{eq:lens1}
\end{equation}
\begin{equation}
^{115}{\rm Sn}^* (\tau = 4.76 \mu s) \rightarrow ^{115}{\rm Sn + \gamma(498 keV) + \gamma(116 keV)}\, .
\label{eq:lens2}
\end{equation}
Thanks to that it is possible to detect low energy neutrinos with a threshold
of 114 keV and measure their energy, following an idea that has been
investigated since the 1970's \cite{14-Raghavan1976yc}.
The primary interaction and the secondary cascade enable a triple coincidence,
correlated in space and time. LENS employs as detection medium a liquid
scintillator chemically doped with natural indium ($^{115}$In = 95.7\%). In
order to exploit the spatial correlation, the volume of the detector is
segmented into cubic cells (7.5 cm) by clear foils (Teflon FEP) that have a
lower index of refraction than the liquid scintillator. By internal
reflection, the scintillation light produced in a cell is channeled in the
directions of the 6 cell faces. The collected channeled light is read-out at
the edge of the detector by photomultiplier tubes.
LENS should be able to determine the low energy solar neutrino
fluxes with an accuracy $\leq 4 \%$, testing neutrino and solar physics with a
global precision better than the present one and also looking for any
inconsistency in the LMA conversion mechanism \cite{14-Raghavan2004}.
\section{Open questions in solar neutrino physics}\
\subsection{The metallicity problem}\label{14-sec:metalproblem}
The solar abundance or solar metallicity problem has been around for some time now.
In analogy with the solar neutrino problem, there have been attempts
(although in most cases, it is fair to say, of somewhat less radical nature)
to solve it by introducing modifications to the input physics of
SSMs. To mention a few of them, we could remember:\\
- large enhancement of Ne abundance
\cite{14-antia:2005, 14-bahcall:2005}, important because of its contribution
uncertainty and a weak bond in solar abundances because its abundance is
determined rather indirectly \cite{14-lodders:2009};\\
- increased element
diffusion rates \cite{14-montalban:2004, 14-guzik:2005};\\
- accretion of
metal-poor material leading to a `two-zone' solar model in terms of
composition \cite{14-guzik:2005, 14-castro:2007, 14-serenelli:2011}.\\
Also solar models including some sort of prescriptions to account for rotation and
other dynamical effects have been put forward, however their performance is
quite poor.
So far, all attempts of finding a solution to all the manifestations of
the solar abundance problem have failed. In some cases $Y_{\rm S}$ can be brougth into
agreement with helioseismology, in other cases $R_{\rm CZ}$ and the sound speed
profile, but a simultaneous solution to all the
problems has not yet being found.
The exceptions are the two obvious one: a) the {\em low-Z} solar abundances
actually underestimate the true metal content of the Sun; b) an increase of
radiative opacities by the right amount (15\% to 20\% at the base of the
convective zone down to about 3\% in the solar core) to compensate for the
decrease induced by the {\em low-Z} abundances. The drawback to this idea is
that current state-of-the-art radiative opacity calculations differ by only 2
to 3\% at the base of the convective envelope, much lower from what would be
required by {\em low-Z} models.
It has indeed been shown that by increasing the radiative opacity in
{\rm low-Z} SSMs the
agreement with helioseismology can be restored to match results from {\em
high-Z} SSMs \cite{14-jcd:2009}. Addtionally, the $^7$Be and $^8$B fluxes of
a {\em low-Z} SSMs with increased opacitiy coincide with those from a {\em
high-Z} model \cite{14-serenelli:2010}. As good as this may seem, it shows
the intrinsic degeneracy between composition and opacities.
Recently, a novel approach, the Linear Solar Models (LSM), that
relate changes in solar observables to modifications in the input physics by
the calculation of kernels based on SSMs has been developed by
\cite{14-villante:2010a}. LSMs\footnote{We remark that LSMs offer an efficient
way of studying the response of the solar
structure to changes in any of the physical ingredients entering solar model
calculations that does not require the construction of solar models with the varied
physics.} have been applied in particular to the solar
abundance problem and the changes required in the radiative opacity to restore
the agreement between {\it low-Z} models and helioseismology
\cite{14-villante:2010b}. Quantitively, results similar to those quoted above.
By using $^8$B and now $^7$Be as thermometers of the solar core
\cite{14-haxton:2008}, CNO neutrinos represent a unique way to break this
degeneracy and provide an independent determination of the CNO abundances,
particularly the C+N abundance in the solar core. Keeping in mind the
antagonism
between solar interior and solar atmosphere models that the solar abundance
problem has established, results from CNO fluxes will be of the outmost
relevance for solar, and by extension stellar, physics.
\subsection{The vacuum to matter transition?}
Solar neutrino experiments already measured the two extreme flavor conversion regimes,
the vacuum term domination and matter term domination. There is no direct experimental evidence
of the transition from one to the other. In fact, the lower energetic $^8$B neutrinos are sensitive
to the rise of the spectrum from matter combination towards vacuum, but the data
(still very uncertain) seems not to show it. More data coming from Super-Kamiokande, Borexino
and SNO+ experiments will further explore the conversion in this regime.
The precise measurement of low energy neutrinos like pep, exploiting the fact that are more energetic than
$^7$Be neutrinos, will also help to see small solar matter effects in the flavor conversion. This matter effects
will be more precisely determined by the comparison of pep and pp neutrino measurements. In fact, the
low energy neutrinos that are better suited to test matter effects are the CNO neutrinos. While the CNO neutrinos
energy is around the pep neutrinos energy, the former are produced at higher temperatures and therefore
at higher densities. The larger matter density where neutrinos are produced
leads to larger matter effects
for CNO neutrinos than for pep neutrinos. In fact, matter effects produce a significant spectral tilt of CNO
neutrinos ($ \sim 10 \%$), which might be a good handle to separate the signal for background.
The determination of the vacuum to matter transition has a significant impact on the determination of the
solar mass splitting derived by solar data, which adds to the implications of earth matter effects measured by
comparing the neutrino fluxes during the day and night. The good match of the independently determined solar
mass splitting by solar neutrino experiments and by reactor experiments leads to the best test on
non-standard neutrino physics to solar neutrinos. There are many possibilities but the two scenarios more
studied are the addition of new neutral current interactions \cite{14-Friedland:2004pp,14-Barranco:2007ej} which
modify the amplitude of matter effects and therefore
shift the effective mass splitting and the existence of a sterile neutrino which adds a new state with
the appropriate mass splitting \cite{14-deHolanda2010am} to produce deviations of the flavor conversion
in the 1-3 MeV range.
\subsection{What else can we learn from CNO fluxes?}
The most fundamental information the CNO fluxes carry is the most obvious one:
that the CNO-bicycle operates in stars and it is a viable process for hydrogen
fusion. It must not be forgotten that neutrinos are the only direct evidence
of nuclear reactions being the source of energy in solar (stellar) interiors.
For the Sun, models predict a marginal contribution to the total energy budget
from CNO reactions, 0.8\% and 0.4\% for {\em high-Z} and {\em low-Z} solar
models. However, CNO becomes the dominant mode for hydrogen burning in stars
with masses right above the solar value. Detection of CNO neutrinos will provide
direct evidence that CNO reactions actually take place in nature, as originally
envisioned by Hans Bethe \cite{14-bethe:1939}; it has been a long wait.
The second important aspect of CNO neutrinos is the information they provide
about the abundance of metals in the solar core. Knowing the abundance of CNO
elements in the solar core is important by itself. In particular, a `perfect'
measurement of the combined $^{13}$N+$^{15}$O flux translates into a
determination of the solar C+N abundance with $\sim 10\%$ uncertainty
\cite{14-haxton:2008}, and the dominant sources of uncertainties are
experimental and can be potentially reduced. Assuming we know the solar
surface abundance of the same elements, i.e. let us forget for the time being
about the solar abundance problem, we can then put constraints on mixing
mechanisms that may have created composition gradients during the evolution of
the Sun. SSMs predict that the number density of C+N is enhanced in the solar
core, at present-day, by $\sim 16\%$ with respect to the surface due to the
effects of microscopic diffusion. And, although helioseismology shows that models
with diffusion work much better than models without, there is no direct
evidence of how efficient diffusion is. In fact, there have been suggestions
that the standard prescription \cite{14-thoul:1994} may be too efficient in
the Sun \cite{14-pinsonneault:2010} and that diffusion rates should be lowered
by $\sim 15 \%$. Solar CNO neutrinos could provide a test for the efficiency
of diffusion.
There are other possibilities that might create a contrast between the solar
core and surface composition. Recently, it has been shown that the Sun has a
peculiar composition when compared to `solar twins', that is stars almost
identical to the Sun in their surface properties \cite{14-melendez:2009,
14-ramirez:2010}. The authors found that the Sun is enhanced in volatile
elements with respect to the solar twins that show no sign of harbouring
planets by about 20\%. In fact, they have associated this fact to the
presence of rocky planets in the Solar System, where refractory elements are
locked, and the occurrence of an accretion episode of volatile-enriched
material
\cite{14-ramirez:2010} after rocky cores are formed in the protoplanetary disk. If
this were true, then the Sun would have an
envelope that is richer in CNO than its interior. If a measurement of the
$^{13}$N+$^{15}$O flux would yield as a result a core composition where the
abudance of C+N would be comparable or less than the surface value, then we
would have an extremely exciting piece of evidence about the earlier phases of
planet formation in the solar system \cite{14-haxton:2008}.
|
2,877,628,089,046 | arxiv | \section{Systems Without Feedback}
Let $S$ be an arbitrary system that maps a real-valued time-varying input $x(t)$ to a real-valued time-varying output $y(t)$. In the simplest case, the output is a deterministic function of the input so that $y(t) = \Sof{x(t)}$.
\begin{center}
\begin{tikzpicture}[node distance=2.5cm,auto,>=latex']
\node [minimum size=3em, draw] (a) {$S$};
\node (b) [left of=a,node distance=6em, coordinate] {$x(t)$};
\node (c) [right of=a,node distance=6em, coordinate] {$y(t)$};
\path[->] (b) edge node {$x(t)$} (a);
\path[->] (a) edge node {$y(t)$} (c);
\end{tikzpicture}
\end{center}
The input $x(t)$ is assumed to be known and controlled (perhaps by an experimenter or control system).
When convenient, we may omit the time parameter without changing the meaning, as in $ \Sof{x} = y $.
\begin{definition}
The system without feedback $\Sof{x(t)}$ is linear iff
\begin{gather}
\Sof{x_1(t)} = y_1(t) \qquad\text{and}\qquad \Sof{x_2(t)} = y_2(t) \nonumber \\[0.5em]
\text{implies that} \nonumber \\[0.5em]
\Sof{\alpha x_1(t) + \beta x_2(t)} = \alpha y_1(t) + \beta y_2(t) \label{linear}
\end{gather}
The system without feedback $\Sof{x(t)}$ is time-invariant iff
\begin{gather}
\Sof{x(t)} = y(t) \nonumber \\[0.5em]
\text{implies that} \nonumber \\[0.5em]
\Sof{ x(t + \delta) } = y(t + \delta) \label{time-invariant}
\end{gather}
A system is linear time-invariant (LTI) iff the system is both linear and time-invariant \cite{ogata_modern_1970, lathi_signal_1998, sauro_control_2019}. A simple but important corollary of (\ref{linear}) is that $\Sof{0} = 0$ for any LTI system $S$.
\end{definition}
Determining linearity is straightforward when there is no feedback. For example, consider the system
$$y = \Sof{x} = ax $$\\[-0.5em]
We can confirm mechanically that $S$ is LTI by checking that (\ref{linear}) and (\ref{time-invariant}) are satisfied. Let $\alpha$ and $\beta$ be arbitrary constants, and let $x_1(t)$ and $x_2(t)$ be arbitrary inputs.
$$\Sof{\alpha x_1 + \beta x_2} = a(\alpha x_1 + \beta x_2) = a \alpha x_1 + a \beta x_2 = \alpha y_1 + \beta y_2$$
$$ \Sof{x(t + \delta)} = ax(t + \delta) = y(t + \delta) $$
Furthermore, a system can be shown to be non-LTI by finding a single counterexample to (\ref{linear}) or (\ref{time-invariant}). For example, consider the system
\begin{equation} y = \Sof{x} = ax + b \label{notlti} \end{equation}
This system is not LTI because for $x_1(t) = 3$ and $x_2(t) = 4$, we find that
$$ \Sof{x_1 + x_2} = a(x_1 + x_2) + b = a(3 + 4) + b = 7a + b $$
which, in violation of (\ref{linear}) does not equal
$$ \Sof{x_1} + \Sof{x_2} = a(x_1) + b + a(x_2) + b = 3a + b + 4a + b = 7a + 2b $$
Note that although $y = ax + b$ is frequently referred to as ``linear'' in other contexts because it describes a line with intercept $b$, strictly speaking this is an affine transformation; it is \emph{not} a linear system by our definition.
\section{Systems With Feedback}
It is also possible for the system to exhibit some feedback, so that the output depends on both the input and the output itself $y(t) = \Sof{x(t), y(t)}$.
\begin{center}
\begin{tikzpicture}[node distance=2.5cm, auto, >=latex']
\node [minimum size=3em, draw] (a) {$S$};
\node (b) [left of=a,node distance=6em, coordinate] {$x(t)$};
\node (c) [right of=a,node distance=6em, coordinate] {$y(t)$};
\path[->] (b) edge node {$x(t)$} (a);
\path[->] (a) edge node {$y(t)$} (c);
\node [right of=a, node distance=3.5em, coordinate] (ac) {yo};
\node (fb) [below of=ac, node distance=3em, coordinate] {};
\path[-] (ac) edge node {} (fb);
\node (fbb) [left of=fb, node distance=3.5em, coordinate] {};
\path[-] (fb) edge node {} (fbb);
\path[->] (fbb) edge node {} (a);
\end{tikzpicture}
\end{center}
It is no longer immediately clear how to apply \eqref{linear} and \eqref{time-invariant} to test if $S$ is LTI. One solution is to ``unroll'' the feedback to obtain a closed-form expression for $y$ in terms of only x. For example, consider the following system with feedback:
$$y(t) = \Sof{x(t), y(t))} = a y(t) + b x(t)$$
We can test if the system is LTI by computing an equivalent system $T$ so that $y(t) = \Tof{x(t)}$. Solving the equation above for $y(t)$ we now obtain:
$$y(t) = \Tof{x(t)} = \frac{b}{1-a} x(t)$$
Clearly $S$ and $T$ have the same behavior since the relationship between $y$ and $x$ is identical for all times. Since we can easily confirm that $T$ is LTI by applying \eqref{linear} and \eqref{time-invariant}, we can also say that $S$ is LTI.
However, this strategy of ``unrolling'' a system with feedback is not always a simple algebraic manipulation. Consider, for example, the following system with feedback:
\begin{equation} y(t) = \Sof{x(t), y(t))} = a \dv{y}{t} (t) + b x(t) \label{simple-ode} \end{equation}
Applying the ``unrolling'' trick now requires solving this ordinary differential equation (ODE). The general solution is
\begin{equation} y(t) = \Tof{x(t)} = e^{\frac{t}{a}} \left( y_0 - \tfrac{b}{a} \int_0^t e^{-\frac{\tau}{a}} x(\tau) \dd\tau \right)
\label{simple-ode-sol} \end{equation}
where $y_0 = y(0)$ is the initial condition of the system. However, the explicit reliance on the initial condition becomes problematic when we try to apply Definition 1. The parameter $y_0$ produces an entire family of different ``unrolled'' systems that satisfy the original equation \eqref{simple-ode}. We now wish to determine if any members of this family are LTI.
To satisfy linearity, recall that every LTI system must have $S[0] = 0$. Applying this constraint to $T$, we find that
$$\Tof{0} = y_0 e^{at} = 0$$
which is only true when $y_0 = 0$. This leaves us with the following specific solution to \eqref{simple-ode} which now has no parameters.
$$\Tof{x(t)} = - \tfrac{b}{a} e^{at} \int_0^t e^{-a\tau} x(\tau) \dd\tau$$
It is easy to show that this solution is linear; however, we find that it fails the time-invariance test, since
$$ \Tof{x(t+\delta)} = - \tfrac{b}{a} e^{at} \int_0^t e^{-a\tau} x(\tau+\delta) \dd\tau \neq -\tfrac{b}{a} e^{a(t+\delta)} \int_0^{t+\delta} e^{-a\tau} x(\tau) \dd\tau =\Tof{x(t)}(t+\delta)$$
Therefore, there does not exist a particular closed-form solution to \eqref{simple-ode} that satisfies both \eqref{linear} and \eqref{time-invariant}. In particular, the linearity constraint requires us to set the initial conditions to zero; however, fixing a particular initial condition violates the requirement of time-invariance. To preserve time-invariance, we would need to both shift $x(t)$ and $y(t)$ in time, and also appropriately update our choice of $y_0$; however, doing so results in a different formula for $T$. Does this mean that system \eqref{simple-ode} is not an LTI system? To the contrary, we will find that system \eqref{simple-ode} in fact \emph{is} an LTI system; we have simply failed to find an LTI transformation $T$ \emph{without feedback} that is equivalent to the system $S$ \emph{with feedback}. Although the ``unrolling'' strategy has not succeeded, we would still like to determine whether $S$ is LTI. Thus, we can extend the definitions of linearity and time-invariance to better accommodate systems with feedback.
\begin{definition}
The system with feedback $\Sof{x, y}$ is linear iff
\begin{gather}
\Sof{x_1, y_1} = y_1 \qquad\text{and}\qquad \Sof{x_2, y_2} = y_2 \nonumber \\[0.5em]
\text{implies that} \nonumber \\[0.5em]
\Sof{\alpha x_1 + \beta x_2,\ \alpha y_1 + \beta y_2} = \alpha y_1 + \beta y_2 \label{linear-fb}
\end{gather}
The system with feedback $\Sof{x, y}$ is time-invariant iff
\begin{gather}
\Sof{x(t), y(t)} = y(t) \nonumber \\[0.5em]
\text{implies that} \nonumber \\[0.5em]
\Sof{ x(t + \delta), y(t + \delta) } = y(t + \delta) \label{time-invariant-fb}
\end{gather}
\end{definition}
Using this new definition, it is straightforward to show that system \eqref{simple-ode} is LTI. Suppose that there exist two arbitrary solutions to the ODE:
$$ y_1 = a \dv{y_1}{t} + b x_1 \qquad\text{and}\qquad y_2 = a \dv{y_2}{t} + b x_2$$
Applying \eqref{linear-fb} directly, we obtain
\begin{align*}
\Sof{\alpha x_1 + \beta x_2,\ \alpha y_1 + \beta y_2}
&= a \dv{t} \left(\alpha y_1 + \beta y_2\right) + b \left(\alpha x_1 + \beta x_2\right) \\
&= \alpha \left(a\dv{y_1}{t} + bx_1\right) + \beta \left(a\dv{y_1}{t} + bx_2\right) \\
&= \alpha y_1 + \beta y_2 \label{linear-fb}
\end{align*}
which confirms that \eqref{simple-ode} is linear. Now applying \eqref{time-invariant-fb}, we find
$$ \Sof{x(t+\delta), y(t+\delta)} = a \dv{y}{t} (t+\delta) + b x(t + \delta) = y(t+\delta) $$
which confirms that \eqref{simple-ode} is time-invariant. In this way, we have shown that the ODE is LTI without needing to solve it.
We can likewise demonstrate that a system with feedback is not LTI by finding a counterexample to either \eqref{linear-fb} or \eqref{time-invariant-fb}. Consider the following system with feedback:
$$ y = \Sof{x, y} = \dv{y}{t} + x + a $$
where $a$ is an arbitrary constant. This system is not linear, since
$$ \Sof{\alpha x, \alpha y} = \alpha \dv{y}{t} + \alpha x + a \neq \alpha \left( \dv{y}{t} + x + a \right) = \alpha y $$
\section{Linearity and Time-Invariance of ODE-Based Systems}
By the new definition, we can now state more generally that any system written as a linear combination of derivatives of the input and output is LTI. Consider an arbitrary system with the following form.
$$ \Sof{x, y} = a_1 \dv{y}{t} + a_2 \dv[2]{y}{t} + \ldots + a_n \dv[n]{y}{t} + b_0 x + b_1 \dv{x}{t} + \ldots b_m \dv[m]{x}{t} = \sum_{i=1}^n a_i \dv[i]{y}{t} + \sum_{j=0}^m b_j \dv[j]{x}{t} $$
In an analysis similar to that above, it follows that \eqref{linear-fb} and \eqref{time-invariant-fb} both hold for this system; thus, it is LTI.
There may be multiple equivalent representations for a single system with feedback. For example, the following two systems are equivalent:
\begin{align*}
y &= \Sof{x, y} = \dv{y}{t} + \int y \dd{t} + x\\
y &= \Sof{x, y} = -\dv[2]{y}{t} + \dv{y}{t} - \dv{x}{t}
\end{align*}
The equivalence follows by differentiating the top system and solving for $y(t)$. In this way, any system described by a linear combination of derivatives and integrals of $y$ can be re-written to only contain derivatives terms. However, this does not work in general for systems with integrals of $x$. For example, consider the following.
\begin{equation} y = \Sof{x, y} = \dv{y}{t} + \int x \dd{t} \label{untangleable} \end{equation}
Eliminating the integral from this equation eliminates the zeroth order term in $y$, preventing us from being able to solve the equation for $y$. Instead, the best we can do is this:
\begin{equation} \dv{y}{t} = \dv[2]{y}{t} + x \label{untangled} \end{equation}
By the linearity of differentiation and integration, we can show that \eqref{untangleable} is LTI. However, when we re-write it in the form of an ODE, we cannot solve for $y$ to write out $\Sof{x, y}$ as we have done before. Nonetheless, we can still understand \eqref{untangled} to describe an LTI system even though it is not written with $y$ separated on its own. By allowing ourselves this flexibility in notation, we can now claim generally that {\bf every linear ODE describes an LTI system}. Indeed, a wide array of LTI systems with scientific and engineering applications are described in the form of ODEs.
\section{Conclusion}
We have presented an extension to the definition of linear time-invariance that accommodates systems with feedback. We presented methods for proving and disproving the linear time-invariance of systems with and without feedback. Finally, we showed that every linear ODE describes an LTI system.
\bibliographystyle{unsrt}
|
2,877,628,089,047 | arxiv | \section{Introduction}
\label{sect_intro}
\begin{figure*}[ht]
\centering
\includegraphics[trim = 45mm 5mm 35mm 10mm, clip, width=16.5cm]{Spectrum-03b.eps}
\caption{Two isomers of vinyl isocyanate and their characteristic features in the room-temperature millimeter wave spectrum. The spectrum reveals compact groups of $a$-type $R$-branch transitions of the \textit{trans} species in the ground state and $v_{18}=1$, $v_{13}=1$, and $v_{18}=2$ excited vibrational states. Weaker $a$-type $R$-branch transitions of the less stable \textit{cis} isomer are at the present scale observed in the form of significantly more diffuse groups.}
\label{spectrum}
\end{figure*}
A few isocyanate bearing molecules have been detected in the interstellar medium (ISM).
The simplest one, isocyanate radical (NCO), has been observed relatively recently in the line survey of L483 \citep{Marcelino2018},
while isocyanic acid (HNCO) belongs among the early molecules detected in the universe dating back to 1972 \citep{Snyder1972,Buhl1972}. HNCO is a well established interstellar molecule observed in the direction of a variety of sources such as TMC-1 \citep{Brown1981}, IRAS~16293-2422 \citep{Bisschop2008,Coutens2016}, L483 \citep{Marcelino2018}, NGC~6334I \citep{Ligterink2020}, G31.41+0.31 \citep{Colzi2021}, Serpens~SMM1 \citep{Ligterink2021}, and G331.512-0.103 \citep{Canelo2021}.
Its protonated form, H$_2$NCO$^+$, had been tentatively observed in Sgr~B2 by \cite{Gupta2013}, but later on its presence in space has been confirmed by \cite{Marcelino2018}.
H$_2$NCO$^+$ has been recently detected toward the molecular cloud G+0.693-0.027 \citep{Rodriguez-Almeida21}
where NCO, cyanogen isocyanate (NCNCO), and ethynyl isocyanate (HCCNCO) were also searched for.
Methyl derivative of HNCO, methyl isocyanate (CH$_3$NCO), has been first detected in the Sgr~B2(N) and Orion~KL star-forming regions \citep{Halfen15,Cernicharo16,Belloche17}. Further observations have shown that it is also present in other sources such as IRAS~16293-2422 \citep{Ligterink17,Martin-Domenech2017}, G31.41+0.31 \citep{Gorai2021,Colzi2021}, Serpens~SMM1 \citep{Ligterink2021}, G10.47+0.03 \citep{Gorai2020}, G+0.693-0.027 \citep{Zeng2018}, and G328.2551-0.5321 \citep{Csengeri2019}. Very recently, the even more complex ethyl isocyanate (C$_2$H$_5$NCO) has been discovered toward the G+0.693-0.027 molecular cloud by \cite{Rodriguez-Almeida21}. Isocyanates are therefore quite widespread across the Galaxy and more such species might be detected in future. In addition, new molecular discoveries are reinforced by increasing the sensitivity and detector bandwidths of astronomical observing capabilities \citep{Jorgensen2020,Tercero2021} as demonstrated by some of the latest astronomical detections \citep{Cernicharo2021,Rivilla2021,McCarthy2021} which contributed to the recent huge jump in the total number of detected compounds \citep{McGuire2022}.
The aforementioned discoveries opened unique possibilities to uncover the fundamentals of the chemistry of isocyanates in space and their possible role as precursors in the formation of other compounds \citep{Majumdar2017,Zeng2018}. For example, H$_2$NCO$^+$ has been considered as a candidate reactant partner of HC$_3$N in the synthesis of nucleobases \citep{Choe2021} and CH$_3$NCO as a precursor for the synthesis of N-methylformamide (CH$_3$NHCHO, \citealt{Belloche17}). The latter has been observed toward Sgr~B2 and NGC~6334I star-forming regions \citep{Belloche17,Belloche19,Ligterink2020} and very recently also toward the hot core G31.41+0.31 \citep{Colzi2021}. This possible link with prebiotic chemistry further increases an interest in isocyanates as targets for laboratory spectroscopic studies in view of future observations.
In the present work, the molecule in question is vinyl isocyanate (C$_2$H$_3$NCO),
the most thermodynamically stable isomer with a C$_3$H$_3$NO formula \citep{Fourre2020}.
It is more complex than CH$_3$NCO and at the same time it contains two hydrogen atoms fewer than recently detected C$_2$H$_5$NCO. In addition, it bears in its backbone the vinyl functional group which is a common structural motif in several other interstellar compounds such as vinyl cyanide \citep[C$_2$H$_3$CN;][]{Gardner1975}, vinylacetylene \citep[C$_2$H$_3$CCH;][]{Cernicharo2021c}, vinylcyanoacetylene \citep[C$_2$H$_3$C$_{3}$N;][]{Kelvin_Lee_2021}, and vinylamine \citep[C$_2$H$_3$NH$_{2}$;][]{Zeng2021}. A search for vinyl isocyanate in the line survey of the G+0.693-0.027 molecular cloud has been reported very recently \citep{Rodriguez-Almeida21} on the basis of the only rotational spectrum reported so far below 40~GHz \citep{Bouchy1977,Kirby1978,Bouchy1979}. On the other hand, no such trials have been performed in the millimeter wave surveys of other interstellar sources very likely due to the lack of laboratory information in the millimeter wave spectral region.
The aim of the present work is therefore two-fold. First, we want to extend the laboratory rotational spectrum of vinyl isocyanate up to 330 GHz and analyze the ground state rotational transitions of its two stable planar forms: \textit{trans} and \textit{cis} (see Fig. \ref{spectrum}). The latter has been found by \cite{Kirby1978} to be less stable by 4.99(24)~kJ~mol$^{-1}$ or 417(20)~cm$^{-1}$ or 600(29)~K, where the numbers in parentheses represent uncertainties in units of the last decimal digits.
These new measurements and analyses then lay a foundation to accomplish a search for interstellar signatures of vinyl isocyanate by millimeter wave astronomy.
We target the high-mass star-forming region Sgr B2(N) which reveals an astonishingly rich collection of molecular species, including isocyanates.
\begin{table*}
\caption{Spectroscopic constants for the \textit{trans} and \textit{cis} isomers of vinyl isocyanate in their ground vibrational states ($A$-reduction, I$^{\text{r}}$-representation) in comparison with previously published results and quantum-chemical calculations.}
\label{constants}
\begin{center}
\begin{footnotesize}
\setlength{\tabcolsep}{6.0pt}
\begin{tabular}{ l r r r r r r r}
\hline\hline
\vspace{-0.3cm}\\
& \multicolumn{3}{c}{\textit{Trans}} & & \multicolumn{3}{c}{\textit{Cis}} \\
\cline{2-4}
\cline{6-8}
\vspace{-0.2cm}\\
& This work & \cite{Kirby1978} & Calculated\tablefootmark{a} & & This work & \cite{Kirby1978} & Calculated\tablefootmark{a} \\
\hline
\vspace{-0.2cm}\\
$A $ /MHz & 62586.3098 (25)\tablefootmark{b} & 62584.051 (35) & 62314.45 & & 20144.090 (41) & 20146.8 (10) & 20193.42 \\
$B $ /MHz & 2437.747011 (86) & 2437.730 (3) & 2438.70 & & 3107.45256 (44) & 3107.267 (20) & 3096.35 \\
$C $ /MHz & 2346.477545 (88) & 2346.507 (1) & 2346.96 & & 2689.42677 (34) & 2689.513 (25) & 2682.28 \\
$\mathit{\Delta_{J}} $ /kHz & 0.266716 (35) & 0.321 (9) & 0.2695 & & 3.03750 (13) & 2.23 (26) & 3.078 \\
$\mathit{\Delta_{JK}} $ /kHz & --14.3999 (20) & --14.30 (7) & --18.48 & & --80.8627 (28) & --80.26 (71) & --80.49 \\
$\mathit{\Delta_{K}} $ /kHz & 2270.62 (24) & ... & 2343 & & 841.7 (15) & ... & 780.0 \\
$\delta_{J} $ /kHz & 0.0182857 (54) & ... & 0.02243 & & 0.830560 (99) & ... & 0.8380 \\
$\delta_{K} $ /kHz & 10.3755 (90) & ... & 8.657 & & 13.792 (22) & ... & 11.62 \\
$\mathit{\Phi_{J}} $ /Hz & 0.0002564 (41) & ... & 0.0002777 & & 0.008427 (26) & ... & 0.009854 \\
$\mathit{\Phi_{JK}} $ /Hz & --0.04791 (43) & ... & --0.07105 & & 0.2450 (49) & ... & 0.2170 \\
$\mathit{\Phi_{KJ}} $ /Hz & --2.129 (69) & ... & --2.487 & & --14.050 (32) & ... & --12.44 \\
$\mathit{\Phi_{K}} $ /Hz & --471.8\tablefootmark{c} & ... & --471.8 & & 289 (24) & ... & 130.8 \\
$\phi_{J} $ /mHz & 0.03926 (87) & ... & 0.06060 & & 3.583 (16) & ... & 4.221 \\
$\phi_{JK} $ /Hz & 0.0311 (20) & ... & 0.01395 & & --0.0271 (43) & ... & 0.01708 \\
$\phi_{K} $ /Hz & 21.58\tablefootmark{c} & ... & 21.58 & & 12.40 (36) & ... & 10.60 \\
$ L_{JK} $ /mHz & ... & ... & ... & & --0.16472 (85) & ... & ... \\
$ L_{KKJ} $ /mHz & ... & ... & ... & & 5.267 (96) & ... & ... \\
$ P_{KKKJ} $ /mHz & ... & ... & ... & & --0.00169 (11) & ... & ... \\
$\Delta E_{\text{ZPE}} $\tablefootmark{d} /cm$^{-1}$ & ... & ... & 0 & & ... & 417 (20) & 301 \\
$J_{\text{min}}/J_{\text{max}}$ & 4 / 80 & 4 / 19 & ... & & 1 / 61 & 4 / 6 & ... \\
$K_{a}^{\text{min}}/K_{a}^{\text{max}}$ & 0 / 7 & 0 / 7 & ... & & 0 / 21 & 0 / 5 & ... \\
$N$\tablefootmark{e} & 464 & 30 & ... & & 608 & 18 & ... \\
$\sigma_{\text{fit}}$\tablefootmark{f}/MHz & 0.024 & ... & ... & & 0.030 & ... & ... \\
$\sigma_{\text{w}}$\tablefootmark{g} & 0.90 & ... & ... & & 0.88 & ... & ... \\
\hline
\end{tabular}
\end{footnotesize}
\end{center}
\tablefoot{
\tablefoottext{a}{Calculated at CCSD/cc-pVTZ level of theory.}
\tablefoottext{b}{The numbers in parentheses are the parameter uncertainties in units of the last decimal digits. Their values are close to 1$\sigma$ standard uncertainties (67\% confidence level) because the unitless (weighted) deviation of the fit is close to 1.0. SPFIT/SPCAT program package \citep{Pickett1991} was used for the analysis.}
\tablefoottext{c}{Fixed to the calculated value, which is usually a preferred constraint over the zero or poorly determined value \citep{Urban1990,Koucky2013}.}
\tablefoottext{d}{Relative energy with respect to the global minimum, taking into account the zero-point energy (ZPE).}
\tablefoottext{e}{Number of distinct frequency lines in the fit.}
\tablefoottext{f}{Root mean square deviation of the fit.}
\tablefoottext{g}{Unitless (weighted) deviation of the fit.}}
\end{table*}
\section{Experimental details}
{\label{s:experiments}}
\subsection{Synthesis}
The sample of vinyl isocyanate was prepared by Curtius rearrangement of acryloyl azide under vacuum (0.1~mbar) using a modified synthesis of \cite{Kirby1978}.
Briefly, sodium azide NaN$_{3}$ (3.25~g, 50~mmol) was mixed with 30~mL of diethylene glycol dibutyl ether in a three-necked flask equipped
with a magnetic stirring bar and a stopcock. The flask was evacuated to a pressure of about 0.1~mbar. Keeping the stopcock closed, the flask was immersed in a bath at --20~$^{\circ}$C. In the next step, acryloyl chloride (2.85~g, 31~mmol) was added in five portions (5 x 500~$\mu$L) through a septum. The mixture was stirred for two hours at 0~$^{\circ}$C allowing
the formation of acryloyl azide. The stopcock was then opened and all volatiles passed into the vacuum line containing a quartz tube in an oven heated approximately to 500~$^{\circ}$C and two successive cold U-tubes. The first U-tube was immersed in a bath at --70~$^{\circ}$C to remove impurities and the second one in liquid nitrogen bath to collect
vinyl isocyanate. The final product was used without any further purification. The main advantage of this experimental procedure was to avoid isolation of the potentially explosive acryloyl azide.
\subsection{Spectroscopic measurements}
The rotational spectrum of vinyl isocyanate was recorded in the frequency regions 127.5--218 and 282--330~GHz using the upgraded Prague semiconductor millimeter wave spectrometer. The spectrometer is based on a sequential multiplication of the fundamental synthesizer frequency (lower than 50 GHz) by a set of active and passive multipliers and a phase-sensitive detection as described in \cite{Kania2006}. The 2.8 and 2.3 meters long Pyrex glass free-space cells were used for the measurements. The optical path lengths were doubled to 5.6 and 4.6~m by roof-top mirrors. The millimeter-wave radiation was modulated at the modulation frequency of 28 kHz
and the detected signal was demodulated by means of a lock-in amplifier working at twice the modulation frequency.
All spectra were registered by upward and downward frequency scanning and averaged. The sample was kept at room temperature and a pressure of around 20~$\mu$bar during the experiments.
\section{Quantum-chemical calculations}
{\label{s:theory}}
Although vinyl isocyanate has been the subject of some computational studies \citep{Badawi2001,Olsen1979}, we have undertaken our own calculations
in order to obtain a reasonable estimation for the spectroscopic parameters relevant to this work.
We used coupled-cluster approximation on the level of coupled-cluster-single-double (CCSD) model \citep{ccsd}
as it is implemented in CFOUR program package \citep{cfour}, in conjunction with Dunning's correlation consistent triple-$\zeta$ (cc-pVTZ) basis set \citep{Dunning1989}.
The convergence criteria for the HF-SCF equations, the CC amplitude equations and
the linear equations were set to 10$^{-8}$ atomic units.
This means the convergence structure calculations and the analytical second derivatives were followed by finite difference techniques to obtain the full
cubic force field. These harmonic and anharmonic force field calculations yielded the rotational and centrifugal distortion constants listed in Table \ref{constants} together
with the energy difference between the \textit{cis} and \textit{trans} isomer of 3.6~kJ~mol$^{-1}$ (301~cm$^{-1}$ or 433~K).
The optimized geometries of both species are shown in Fig. \ref{spectrum} and their harmonic and anharmonic vibrational frequencies are provided in Table \ref{vib-modes}.
\begin{figure}[ht]
\centering
\includegraphics[trim = 0mm 20mm 0mm 25mm, clip, width=8.5cm]{K-structure.eps}
\caption{Perturbations in the ground state rotational spectrum of \textit{trans} vinyl isocyanate.
The experimental spectrum is compared with predictions based on the analysis up to $K_a=6$.
All the assigned lines correspond to $a$-type $R$-branch transitions except that marked with a star which is the $b$-type $40_{0,40}\leftarrow 39_{1,39}$ transition. The blue diamond symbols highlight the transitions included in the fit while the red question marks indicate perturbed transitions that could not be confidently assigned.}
\label{perturbations}
\end{figure}
\section{Rotational spectra and analyses}
{\label{s:analysis}}
\subsection{Trans isomer}
Since the \textit{trans} isomer represents the most stable species on the potential energy landscape of vinyl isocyanate, it is the most relevant
target for astronomical observations as it presents the strongest lines in the millimeter wave spectrum in Fig. \ref{spectrum}.
The most visible features in the spectrum are compact groups of $a$-type $R$-branch transitions arising from the near-prolate character of the molecule ($\kappa$ = --0.997) and its large dipole moment component along the $a$ principal inertial axis ($|\mu_{a}|=$ 2.047(6)~D and $|\mu_{b}| =$ 0.824(9)~D from \citealt{Kirby1978}).
The analysis was commenced by refitting the data set from \cite{Kirby1978} and generating spectral line predictions.
The predicted transitions were searched for with the help of the Loomis-Wood-type plot technique \citep{LW} implemented in our own program \citep{Vavra2020}.
The program is written in the Matlab software \citep{MATLAB:2020} and allows for the line assignments to quantum numbers, precise determination of the line frequencies employing the Voight profile function, and saving the data in a line list among other features. It generates input files for the SPFIT/SPCAT program package \citep{Pickett1991} which is executed directly from the user interface, allowing a straightforward analysis of the assigned lines. On this basis, we easily expanded the assignments for $K_{a}=$ 0--6 transitions up to $J = 69$ and identified weak $b$-type $R$-branch and $Q$-branch transitions.
Difficulties were encountered in assignments and fitting $K_{a}\geq 7$ transitions. In the absence of perturbations, these transitions are expected to progressively run to higher frequencies as shown in the lower panel of Fig. \ref{perturbations}. Instead, we observed irregular shifts with respect to their predicted positions that could not be treated within the scope of classical semi-rigid rotor Hamiltonian.
Some of these transitions even could not be confidently assigned (see Fig. \ref{perturbations}).
Consequently, we limited the analysis of $a$-type $R$-branch transitions to $K_{a}=6$.
These transitions were merged with $b$-type $R$-branch and $Q$-branch transitions which, due to low intensities and the large value of the $A$ rotational constant, were limited to $K_{a}=3$ and $J=80$.
Our data set was finally combined with microwave transitions from \cite{Kirby1978} and analyzed using Watson's $A$-reduced Hamiltonian in I$^{\text{r}}$-representation \citep{Watson1977} with terms up to the sixth power in the angular momentum. The only exceptions were $\Phi_K$ and $\phi_K$ which were fixed at the values estimated from quantum-chemical calculations. We note that all transitions from \cite{Kirby1978} were found to be fully consistent with our measurements and could be perfectly fitted. Results from this joint analysis are provided in Table \ref{constants} and the list of measured transitions in Table \ref{transitions-trans}. Watson’s $S$-reduced Hamiltonian led to results of similar quality.
The origin of perturbations in the rotational spectrum of \textit{trans} vinyl isocyanate is further discussed in Sect. \ref{ss:lab}.
\begin{table*}
\caption{Partition functions and abundances for the two isomers of vinyl isocyanate.}
\label{part-fce}
\begin{center}
\begin{footnotesize}
\setlength{\tabcolsep}{7.0pt}
\begin{tabular}{ r r r r r r r r r r r}
\hline\hline
\vspace{-0.3cm}\\
& \multicolumn{3}{c}{\textit{Trans} isomer} & & \multicolumn{3}{c}{\textit{Cis} isomer} & & & \\
\cline{2-4}
\cline{6-8}
\vspace{-0.3cm}\\
$T$ (K) & $Q_{\text{rot}}$ & $Q_{\text{vib}}$ & (\%)\tablefootmark{a} & & $Q_{\text{rot}}$\tablefootmark{b} & $Q_{\text{vib}}$ & (\%) & & $Q_{\text{rot}}^{trans+cis}$ & $Q_{\text{vib}}^{trans+cis}$ \\
\vspace{-0.3cm}\\
\hline
\vspace{-0.3cm}\\
300.000 & 46450.52 & 9.03 & 74 & & 16049.82 & 9.22 & 26 & & 62500.34 & 9.08 \\
225.000 & 30146.59 & 4.89 & 82 & & 6435.09 & 5.00 & 18 & & 36581.68 & 4.91 \\
150.000 & 16397.84 & 2.65 & 92 & & 1336.05 & 2.69 & 8 & & 17733.89 & 2.65 \\
75.000 & 5794.43 & 1.39 & 100 & & 26.29 & 1.38 & 0 & & 5820.71 & 1.39 \\
37.500 & 2048.88 & 1.06 & 100 & & 0.03 & 1.05 & 0 & & 2048.91 & 1.06 \\
18.750 & 724.98 & 1.00 & 100 & & 0.00 & 1.00 & 0 & & 724.98 & 1.00 \\
9.375 & 256.81 & 1.00 & 100 & & 0.00 & 1.00 & 0 & & 256.81 & 1.00 \\
5.000 & 100.37 & 1.00 & 100 & & 0.00 & 1.00 & 0& & 100.37 & 1.00 \\
2.725 & 40.65 & 1.00 & 100 & & 0.00 & 1.00 & 0& & 40.65 & 1.00 \\
\hline
\end{tabular}
\end{footnotesize}
\end{center}
\tablefoot{
\tablefoottext{a}{Isomer abundance calculated as $Q_{\text{tot}}^{\mathit{trans}}/Q_{\text{tot}}^{\mathit{trans+cis}}$ where $Q_{\text{tot}}=Q_{\text{rot}}\times Q_{\text{vib}}$.}
\tablefoottext{b}{Corrected for the energy difference between the \textit{trans} and \textit{cis} isomer, i.e. the $0_{0,0}$ level is set to 301~cm$^{-1}$.}
}
\end{table*}
\begin{table*}[!ht]
\begin{center}
\caption{
Parameters of our best-fit LTE model of methyl isocyanate toward Sgr~B2(N1S) and upper limits for ethyl isocyanate and vinyl isocyanate.
}
\label{t:coldens}
\vspace*{-1.2ex}
\begin{tabular}{lcrccccccr}
\hline\hline
\multicolumn{1}{c}{Molecule} & \multicolumn{1}{c}{Status\tablefootmark{a}} & \multicolumn{1}{c}{$N_{\rm det}$\tablefootmark{b}} & \multicolumn{1}{c}{Size\tablefootmark{c}} & \multicolumn{1}{c}{$T_{\mathrm{rot}}$\tablefootmark{d}} & \multicolumn{1}{c}{$N$\tablefootmark{e}} & \multicolumn{1}{c}{$F_{\rm vib}$\tablefootmark{f}} & \multicolumn{1}{c}{$\Delta V$\tablefootmark{g}} & \multicolumn{1}{c}{$V_{\mathrm{off}}$\tablefootmark{h}} & \multicolumn{1}{c}{$\frac{N_{\rm ref}}{N}$\tablefootmark{i}} \\
& & & \multicolumn{1}{c}{\small ($''$)} & \multicolumn{1}{c}{\small (K)} & \multicolumn{1}{c}{\small (cm$^{-2}$)} & & \multicolumn{1}{c}{\small (km~s$^{-1}$)} & \multicolumn{1}{c}{\small (km~s$^{-1}$)} & \\
\hline
CH$_3$NCO\tablefootmark{(j)}$^\star$ & d & 51 & 2.0 & 200 & 2.5 (17) & 1.00 & 5.0 & 0.0 & 1 \\
\hline
C$_2$H$_5$NCO, $\varv=0$ & n & 0 & 2.0 & 200 & $<$ 8.1 (16) & 10.1 & 5.0 & 0.0 & $>$ 3.1 \\
\hspace*{10ex} $\varv_{\rm t}=1$ & n & 0 & 2.0 & 200 & $<$ 8.1 (16) & 10.1 & 5.0 & 0.0 & $>$ 3.1 \\
\hline
\textit{trans}-C$_2$H$_3$NCO, $\varv=0$ & n & 0 & 2.0 & 200 & $<$ 2.4 (16) & 4.00 & 5.0 & 0.0 & $>$ 11 \\
\textit{cis}-C$_2$H$_3$NCO, $\varv=0$ & n & 0 & 2.0 & 200 & $<$ 5.0 (17) & 4.00 & 5.0 & 0.0 & $>$ 0.50 \\
\hline
\end{tabular}
\end{center}
\vspace*{-2.5ex}
\tablefoot{
\tablefoottext{a}{d: detection, n: nondetection.}
\tablefoottext{b}{Number of detected lines \citep[conservative estimate, see Sect.~3 of][]{Belloche16}. One line of a given species may mean a group of transitions of that species that are blended together.}
\tablefoottext{c}{Source diameter (\textit{FWHM}).}
\tablefoottext{d}{Rotational temperature.}
\tablefoottext{e}{Total column density of the molecule. $x$ ($y$) means $x \times 10^y$.}
\tablefoottext{f}{Correction factor that was applied to the column density to account for the contribution of vibrationally excited states, in the cases where this contribution was not included in the partition function of the spectroscopic predictions.}
\tablefoottext{g}{Linewidth (\textit{FWHM}).}
\tablefoottext{h}{Velocity offset with respect to the assumed systemic velocity of Sgr~B2(N1S), $V_{\mathrm{sys}} = 62$ km~s$^{-1}$.}
\tablefoottext{i}{Column density ratio, with $N_{\rm ref}$ the column density of the previous reference species marked with a $\star$.}
\tablefoottext{j}{The parameters were derived from the ReMoCA survey by \citet{Belloche19}.}
}
\end{table*}
\subsection{Cis isomer}
The spectroscopic constants from \cite{Kirby1978} were used for the first prediction of the rotational spectrum of \textit{cis} vinyl isocyanate in the millimeter wave region.
Only $a$-type $R$-branch transitions were observed for this isomer in agreement with the dipole moment components of $|\mu_{a}|=$ 2.14~(2)~D and $|\mu_{b}| =$~0.09~(2)~D as determined by Stark spectroscopy \citep{Bouchy1979}.
It was possible to identify and iteratively fit the rotational transitions up to $J=61$ and $K_a=21$.
We were not able to assign the transitions with higher values of $K_a$ because of their low intensities, which caused them to disappear in the spectral confusion,"weeds", of lines from the most stable \textit{trans} isomer.
Our data set was combined with low-frequency transitions from \cite{Kirby1978} and \cite{Bouchy1977} and was
fit to matrix elements of Watson's $A$-reduced effective rotational Hamiltonian \citep{Watson1977}. Some transitions from previous microwave works revealed larger residuals.
Larger uncertainties were thus assigned to these transitions and four of them were excluded from the fit.
The complete list of treated transitions is provided in Table \ref{transitions-cis}.
and the molecular constants determined from the analysis are reported in Table~\ref{constants}.
\begin{figure*}
\centerline{\resizebox{0.75\hsize}{!}{\includegraphics[angle=0]{c2h3nco-t_ve0_n1s_1.eps}}}
\caption{Transitions of vinyl isocyanate \textit{trans}-C$_2$H$_3$NCO covered
by the ReMoCA survey. The LTE synthetic spectrum used to derive the upper limit
on the column density of \textit{trans}-C$_2$H$_3$NCO, $\varv = 0$ is
displayed in red and overlaid on the observed spectrum of Sgr~B2(N1S) shown in
black. The blue synthetic spectrum contains the contributions of all molecules
identified in our survey so far, but does not include the contribution of the
species shown in red. The central frequency is indicated in MHz below each
panel as well as the half-power beam width on the left, the width of each
panel in MHz in parentheses, and the continuum level of the
baseline-subtracted spectra in K in brackets. The y-axis is labeled in
brightness temperature units (K). The dotted line indicates the $3\sigma$
noise level. The figure only shows the transitions of \textit{trans}-C$_2$H$_3$NCO for which the red
synthetic spectrum has a significant peak temperature (compared to the noise
level) and which are not too heavily blended with much stronger emission of
other molecules.}
\label{f:spec_c2h3nco-t_ve0}
\end{figure*}
\subsection{Partition functions}
The spectroscopic constants from Table \ref{constants} were used to evaluate the rotational partition function ($Q_{\text{rot}}$) for both isomers.
We used the SPCAT program \citep{Pickett1991} to undertake the numerical summation over the ground state energy levels up to $J=240$ and $K_{a}=46$ for the \textit{trans} isomer and $J=210$ and $K_{a}=72$ for the \textit{cis} isomer. In addition, for the \textit{cis} form, this summation was corrected for the energy difference between the \textit{cis} and \textit{trans} form.
We used $\Delta E_{\text{ZPE}}=301$~cm$^{-1}$ from our quantum-chemical calculations. This value reproduces quite well the experimental intensities of the \textit{cis} isomer lines with respect to neighboring lines of the \textit{trans} isomer in our spectrum. On the other hand, the experimental value 417(20)~cm$^{-1}$ from \cite{Kirby1978} underestimates these \textit{cis} isomer lines relative intensities.
For this reason we consider our calculated value for the energy difference as more reliable.
Obtained rotational partition functions at nine different temperatures are provided in Table \ref{part-fce}. Their values represent the individual contributions from the \textit{trans} and \textit{cis} isomer to the rotational partition function of the molecule as a whole which is also given in Table \ref{part-fce}.
For completeness, we provide in Table \ref{part-fce-cisE0} the rotational partition functions of the \textit{cis} isomer also as a separate species, i.e. with the $0_{0,0}$ level set to 0~cm$^{-1}$.
The vibrational partition functions ($Q_{\text{vib}}$) were estimated using Eq. 3.60 of \cite{Gordy1970} by taking into account the anharmonic frequencies of the eighteen normal vibrational modes from Table~\ref{vib-modes} and are listed in Table \ref{part-fce}.
The same table also shows that the \textit{cis} isomer represents an important fraction of the room-temperature population of
vinyl isocyanate while its abundance is estimated to only 8~\% at 150~K.
\begin{figure*}
\centerline{\resizebox{0.75\hsize}{!}{\includegraphics[angle=0]{c2h3nco-c_ve0_n1s_1.eps}}}
\caption{Same as Fig.~\ref{f:spec_c2h3nco-t_ve0} but for
\textit{cis}-C$_2$H$_3$NCO, $\varv$~=~0.}
\label{f:spec_c2h3nco-c_ve0}
\end{figure*}
\begin{figure*}
\centerline{\resizebox{0.75\hsize}{!}{\includegraphics[angle=0]{c2h5nco_ve0_n1s_1.eps}}}
\caption{Same as Fig.~\ref{f:spec_c2h3nco-t_ve0} but for C$_2$H$_5$NCO,
$\varv$~=~0.}
\label{f:spec_c2h5nco_ve0}
\end{figure*}
\section{Search for vinyl isocyanate toward Sgr~B2(N1)}
\label{s:astro}
\subsection{Observations}
\label{ss:obs_remoca}
The imaging spectral line survey Reexploring Molecular Complexity with ALMA
(ReMoCA) was performed toward the high-mass star forming protocluster
Sgr~B2(N) with the Atacama Large Millimeter/submillimeter Array (ALMA). A
detailed description of the observations and data reduction can be found in
\citet{Belloche19}. We summarize the main features of the survey here. The
phase center is located at the equatorial position
($\alpha, \delta$)$_{\rm J2000}$=
($17^{\rm h}47^{\rm m}19{\fs}87, -28^\circ22'16{\farcs}0$) which is halfway
between the two hot molecular cores Sgr~B2(N1) and Sgr~B2(N2). We defined five
frequency tunings to cover the frequency range from 84.1~GHz to
114.4~GHz with a spectral resolution of 488~kHz (1.7 to 1.3~km~s$^{-1}$).
The survey has a sensitivity per spectral channel that varies
between 0.35~mJy~beam$^{-1}$ and 1.1~mJy~beam$^{-1}$ (rms) depending on the
setup, with a median value of 0.8~mJy~beam$^{-1}$. The angular resolution
(HPBW) ranges from $\sim$0.3$\arcsec$ to $\sim$0.8$\arcsec$ with a median
value of 0.6$\arcsec$ that corresponds to $\sim$4900~au at the distance
of Sgr~B2 \citep[8.2~kpc,][]{Reid19}. An improved version of the
data reduction as described in \citet{Melosso20} was used for this work.
A detailed description of the procedure that was followed to subtract the
continuum emission can be found in Sect. 2.2 of \citet{Belloche19},
complemented by Sect. 4.1 of \citet{Melosso20}. It is difficult to estimate
the uncertainty on the subtracted continuum level in a robust way because of
spectral confusion. From our experience in modeling the line spectra of the
ReMoCA survey, we believe that this uncertainty may be in some cases on the
same order as the noise level itself. We emphasize however that our continuum
subtraction procedure is applied to each spectral window of 1.8~GHz width as a
whole, therefore it is not affected by spectral confusion that arises at
scales of tens or hundreds of MHz.
\begin{figure*}
\centerline{\resizebox{0.75\hsize}{!}{\includegraphics[angle=0]{c2h5nco_ve1_n1s_1.eps}}}
\caption{Same as Fig.~\ref{f:spec_c2h3nco-t_ve0} but for C$_2$H$_5$NCO,
$\varv_{\rm t}$~=~1.}
\label{f:spec_c2h5nco_ve1}
\end{figure*}
We analyzed the spectrum obtained toward the position Sgr~B2(N1S) at
($\alpha, \delta$)$_{\rm J2000}$=
($17^{\rm h}47^{\rm m}19{\fs}870$, $-28^\circ22\arcmin19{\farcs}48$) following the
strategy employed by \citet{Belloche19}. This position is offset by about
1$\arcsec$ to the south of the main hot core Sgr~B2(N1) and has a lower
continuum opacity compared to the peak of the hot core. We assumed local
thermodynamic equilibrium (LTE) and produced synthetic spectra with the
astronomical software Weeds \citep[][]{Maret11} in order to analyze the
observed spectrum. The LTE assumption is justified by the high
densities of the regions where hot-core emission is detected in Sgr~B2(N)
\citep[$>1 \times 10^{7}$~cm$^{-3}$, see][]{Bonfand19}.
We derived a best-fit synthetic spectrum for each molecule separately, and
then added the contributions of all identified molecules together. We used a
set of five parameters to model the contribution of each species: size of the
emitting region
($\theta_{\rm s}$), column density ($N$), temperature ($T_{\rm rot}$), linewidth
($\Delta V$), and velocity offset ($V_{\rm off}$) with respect to the assumed
systemic velocity of the source, $V_{\rm sys}=62$~km~s$^{-1}$.
For nondetected species, the
synthetic spectra that were used to derive their column density upper limits
(red spectra in the figures) are conservative in the sense that they have
synthetic peak temperatures that are sometimes somewhat higher than the
3$\sigma$ noise level (dotted lines in the figures) or slightly above the
signals detected by ALMA (black spectra in the figures), implicitly accounting
for the additional (uncertain) uncertainty that affects the baseline level. In
this way, we are confident that the upper limits do not underestimate the
actual column densities of the nondetected species. For molecules that are detected (see, e.g., \citealt{Belloche19}), the
emission size is measured with Gaussian fits to the integrated intensity maps
of their uncontaminated transitions. The velocity offset and velocity width
are determined directly from the individual line profiles. The rotation
temperature is estimated from a population diagram. Finally, the only
remaining free parameter, the column density, is adjusted manually until a
good visual match between the synthetic and observed spectra is obtained. For
an undetected species, the first four parameters are fixed to values obtained
for a related species and the column density is varied until discrepancies at
the $\sim$3$\sigma$ level appear between the synthetic and observed spectra.
This yields the column density upper limit of the undetected species.
\subsection{Nondetection of vinyl isocyanate}
\label{ss:nondetection_remoca}
In order to search for vinyl isocyanate, C$_2$H$_3$NCO, toward Sgr~B2(N1S),
we relied on the LTE parameters derived for methyl isocyanate, CH$_3$NCO,
toward the same source by \citet{Belloche19} with the ReMoCA survey. These
parameters are listed in Table~\ref{t:coldens}. Assuming that the more complex
molecule vinyl isocyanate traces the same region as methyl isocyanate, we
produced LTE synthetic spectra for the former species adopting the same
parameters as for the latter with only the column density left as a free
parameter. We employed the spectroscopic predictions derived for the
\textit{trans} and \textit{cis} isomers of vinyl isocyanate in
Sect.~\ref{s:analysis} to compute their LTE synthetic spectra. None of these
conformers is detected toward Sgr~B2(N1S), as illustrated in
Figs.~\ref{f:spec_c2h3nco-t_ve0} and \ref{f:spec_c2h3nco-c_ve0}. The upper
limit on the total column density of vinyl isocyanate derived from each
conformer is reported in Table~\ref{t:coldens},
after accounting for the vibrational partition function $Q_{\rm
vib}^{trans+cis}$ provided in Sect.~\ref{s:analysis}.
We also report in Table~\ref{t:coldens} the column density upper limit that we
obtained with the ReMoCA survey for ethyl isocyanate, C$_2$H$_5$NCO. The
nondetection of this molecule, both in its torsional ground state and its
first torsionally excited state, is illustrated in
Figs.~\ref{f:spec_c2h5nco_ve0} and \ref{f:spec_c2h5nco_ve1}, respectively.
We employed the spectroscopic entry 71508 (version 1) of the Cologne Database
for Molecular Spectroscopy\footnote{https://cdms.astro.uni-koeln.de/}
\citep[CDMS,][]{Mueller05} to compute the LTE synthetic spectra of the
torsional ground state used to derive the upper limit to the column density of
ethyl isocyanate. This CDMS entry is mainly based on the measurements reported
in \citet{Kolesnikova18}. For the first torsionally excited state, we used
spectroscopic predictions from \citet{Kolesnikova18} prepared in electronic
format by one of us. The upper limit given in Table~\ref{t:coldens} accounts
for the (substantial) vibrational correction that was estimated using the
energies of the vibrational modes derived by \citet{Durig10} for ethyl
isocyanate. This upper limit holds under the assumption that a single conformation
exists for this species. A single form with quite a large number of molecules
in excited torsional states is suggested from the room-temperature microwave
and millimeter wave spectroscopy of gas phase samples \citep{Sakaizumi1976,Kolesnikova18}. Infrared spectroscopic data of
ethyl isocyanate dissolved in liquid noble gases were, on the other hand, interpreted
as a mixture of the \textit{cis} and \textit{trans} forms on the basis of quantum-chemical calculations \citep{Durig10}.
These calculations are, however, strongly dependent on the basis set used and
predict very low barrier for the conformational interchange which could even
fall into a calculation error.
The most stringent constraint on the column density of vinyl isocyanate is
obtained from its \textit{trans} conformer. We find that vinyl isocyanate is
at least 11 times less abundant than methyl isocyanate toward Sgr B2(N1S).
This is a factor $\sim$2 less stringent than the limit found on the basis of
the \textit{trans} conformer of vinyl isocyanate by \cite{Rodriguez-Almeida21}
toward G+0.693-0.027, a shocked region located close to Sgr B2(N)
(CH$_3$NCO/\textit{trans}-C$_2$H$_3$NCO > 26). For the \textit{cis} conformer, considered as an independent
species, they obtained a lower limit of 7 for CH$_3$NCO/\textit{cis}-C$_2$H$_3$NCO but this cannot
be compared directly to the ratio we report in Table \ref{t:coldens} because, for the high
densities of Sgr B2(N1S), we assume an LTE distribution of both conformers,
and not independent species.
Because of its large vibrational partition function, the
upper limit we obtained for ethyl isocyanate toward Sgr B2(N1S) is less stringent than for vinyl isocyanate: we find that ethyl
isocyanate is at least 3 times less abundant than methyl isocyanate. For
comparison, propanal, C$_2$H$_5$CHO, and ethylamine, C$_2$H$_5$NH$_2$, were
both found to be at least 5 times less abundant than acetaldehyde, CH$_3$CHO,
and methylamine, CH$_3$NH$_2$, respectively, toward Sgr~B2(N1S) with the ReMoCA
survey \citep[][]{SanzNovo22,Margules22}. Therefore, it is likely that ethyl
isocyanate is at least twice less abundant than the upper limit derived above, which would be in line with the ratio
CH$_3$NCO/C$_2$H$_5$NCO of 8 found by \cite{Rodriguez-Almeida21} toward
G+0.693-0.027.
In contrast, the upper limit obtained for vinyl isocyanate toward Sgr~B2(N1S) is a bit more
stringent than the one derived toward the same source for vinylamine,
C$_2$H$_3$NH$_2$, that was found to be at least 8 times less abundant than
methylamine \citep[][]{Margules22}.
\section{Discussion}
\label{s:discussion}
\subsection{Laboratory spectroscopy of vinyl isocyanate}
\label{ss:lab}
Isocyanates are rather flexible molecules with plenty of anomalies in their rotational spectra \citep{Yamada1980,Koput1984,Cernicharo16,Pienkina2017,Kolesnikova18,Kolesnikova2019} and for vinyl isocyanate this has also proven to be the case. The ground state rotational spectrum of the \textit{trans} isomer suffered from strong perturbations which allowed only transitions involving levels up to $K_{a}=6$ to be analyzed without the loss of physical meaning of the fitted constants.
The origin of these perturbations can be understood once excited vibrational states are taken into consideration and the reduced energy level diagram (Fig. \ref{Ered}) is plotted.
Three excited vibrational states, namely $v_{18}=1$, $v_{13}=1$, and $v_{18}=2$, fall into the energy window of 200~cm$^{-1}$. Figure~\ref{spectrum} shows that our spectrum exhibits noticeable patterns assignable to these states on the basis of the spectroscopic constants from \cite{Kirby1978}, nevertheless, their low $K_{a}$ transitions were already heavily perturbed.
\cite{Kirby1978} estimated the frequencies of the associated vibrational modes $\nu_{18}$ and $\nu_{13}$ to 77(10)~cm$^{-1}$ and 200(20)~cm$^{-1}$, respectively, from microwave relative intensity measurements. We find a remarkable agreement between the literature value and the anharmonic frequency of 78~cm$^{-1}$ for the $\nu_{18}$ vibrational mode from our quantum-chemical calculations.
On the other hand, our computations indicated a significantly lower frequency for the $\nu_{13}$ mode (136~cm$^{-1}$) than previously reported.
This lower value leads to better agreement with our experimental spectrum; the rotational transitions in $v_{13}=1$ have on average slightly higher relative intensities than the same transitions in $v_{18}=2$ which is estimated to lie at 155~cm$^{-1}$.
The excited vibrational states $v_{18}=1$ and $v_{13}=1$ are in strong non-resonant Coriolis interaction, repelling each other, as evidenced by their $A$ rotational constants (54~601 and 70~532~MHz for $v_{18}=1$ and $v_{13}=1$, respectively, \citealt{Kirby1978}). The rotational energy levels in $v_{18}=1$ are thus pushed down. Since the ground state rotational constant $A$ is considerable ($\sim2$~cm$^{-1}$ or 62~586~MHz) in comparison with the energy of $v_{18}=1$, the rotational levels in the ground state quickly reach in energy the levels in $v_{18}=1$ and rovibrational interactions might appear already for $K_{a}=7$ of the ground state (see Fig.~\ref{Ered}). Resonant interactions between $v_{18}=1$ and $v_{13}=1$ (e.g., between $K_{a}=8$ and 5, $K_{a}=9$ and 6, etc.) will further complicate the situation.
Figure~\ref{Ered} also shows that the $v_{13}=1$ state lies close in energy to $v_{18}=2$ and their rotational energy levels cross at $K_{a}=5$ and 6. Effects of this Fermi interaction
might be non-negligible as shown, for example, in \textit{n}-propyl cyanide \citep{Liu2019}.
All in all, the ground vibrational state cannot be completely analyzed without $v_{18}=1$ which cannot be treated without $v_{13}=1$ which in turn cannot be analyzed without $v_{18}=2$. Similar interactions among multiple vibrational states were observed in the rotational spectra of quasi-linear molecules such as HNCO \citep{Yamada1980} and hydrazoic acid (HN$_3$, \citealt{Hegelund1987,Vavra2017}) and other near-prolate species such as vinyl cyanide (C$_2$H$_3$CN, \citealt{Kisiel2009,Kisiel2012}) and \textit{n}-propyl cyanide (\textit{n}-C$_3$H$_7$CN, \citealt{Liu2019}).
The above network of interacting states is already very complicated and is expected to be even more complex due to possible coupling with other excited vibrational states which are not included in Fig.~\ref{Ered} for simplicity. In particular, $v_{18}=3$, ($v_{18}=1$, $v_{13}=1$), and $v_{13}=2$ might be at play and would make the analysis extremely challenging. A logical step toward the understanding of these interactions would be the measurement and analysis
of a high-resolution vibrational spectrum of vinyl isocyanate. For the time being, we prefer to make our laboratory data available for astrophysical applications even though our data set is limited in terms of $K_{a}$ in comparison with "well-behaved" molecular systems.
We emphasize that given the large value of the $A$ rotational constant, $K_{a}=6$ is more than enough to identify the molecule in fairly warm environments such as hot cores.
Of course, our results are also perfectly suitable for a search of the molecule in colder interstellar sources.
For CH$_3$NCO, only $K_{a}\leq 3$ transitions were experimentally accessible at the time of its search in space and led to its detection \citep{Halfen15,Cernicharo16}.
\begin{figure}[ht]
\centering
\includegraphics[trim = 0mm 5mm 0mm 0mm, clip, width=8.9cm]{Graph1.eps}
\caption{Diagram of reduced energies for the ground state (black), $v_{18}=1$ (red), $v_{13}=1$ (blue), and $v_{18}=2$ (green) excited vibrational states in \textit{trans} vinyl isocyanate. The reduced energies $E_{\text{red}}$ are calculated as $E_{\text{red}}=E - (B+C)J(J+1)/2$ where $E$ corresponds to the energy of rotational levels. Their values are obtained from the experimental spectroscopic constants and calculated vibrational energies.
}
\label{Ered}
\end{figure}
The set of the spectroscopic constants for \textit{trans} vinyl isocyanate obtained in this work is definitely more complete and accurate in comparison with the original work of \cite{Kirby1978}.
Table~\ref{constants} illustrates that the previous values for the rotational and quartic centrifugal distortion constants are improved by up to two orders of magnitude. Furthermore, the physical meaning of those newly determined quantities is corroborated by the agreement between the fitted values and their quantum-chemical counterparts in Table~\ref{constants}.
Spectral predictions for $K_{a} < 7$ generated using these constants thus serve as
an accurate observational reference, at least in the case when these are interpolations within the measured data sets. Although some transitions may reveal satisfactory predictive power outside the present data region, we do not recommend such extrapolations due to the amount and complexity of perturbations expected to appear at higher frequencies.
Final remarks concern the spectroscopic constants for \textit{cis} vinyl isocyanate.
We can infer from Table~\ref{constants} that their values are significantly better determined than previously; some of them by two orders of magnitude.
In addition, many centrifugal distortion constants are determined for the first time.
Table~\ref{constants} further illustrates that the \textit{cis} isomer analysis called for inclusion of centrifugal distortion terms up to 10$^{\text{th}}$ power in the angular momentum. The requirement for such a high expansion of the rotational Hamiltonian might be an indicative of rovibrational interactions with low-lying excited vibrational states. Here, the low-energy states $v_{18}=1$ and $v_{13}=1$ are predicted by our quantum-chemical calculations at 90 and 110 ~cm$^{-1}$, respectively. Since the rotational constant $A$ is three times smaller than that of the \textit{trans} isomer, such interaction would affect the rotational energy levels with higher $K_{a}$. To assess whether the interaction results in contributions to the centrifugal distortion constants, one can compare their experimental and quantum-chemical values.
Table \ref{constants} shows good correspondence between our quartic and sextic constants and those calculated at CCSD/cc-pVTZ level of theory except for $\phi_{JK}$ which is opposite in sign.
Thus, if there existed an interaction with excited vibrational states it would probably become noticeable at the sextic and higher-order level of the Hamiltonian. Unfortunately, we did not find the rotational transitions in excited vibrational states in our spectra that could address this issue.
\subsection{The chemistry of methyl, vinyl, and ethyl isocyanate}
\label{s:chemistry}
\subsubsection{Methyl isocyanate}
While methyl isocyanate is the only one of the three molecules listed in
Table~\ref{t:coldens} to have been definitively detected in our survey,
the chemistry by which it forms is still uncertain.
\citet{Halfen15} proposed gas-phase reactions between the methyl radical,
CH$_3$, and either HNCO (isocyanic acid) or HOCN (cyanic acid) as plausible
mechanisms:
\begin{equation}
{\rm CH}_3 + {\rm HNCO} \rightarrow {\rm CH}_3{\rm NCO} + {\rm H} \label{reac1}
\end{equation}
\begin{equation}
{\rm CH}_3 + {\rm HOCN} \rightarrow {\rm CH}_3{\rm NCO} + {\rm H} \label{reac2}
\end{equation}
While HNCO is substantially more abundant than HOCN in the ISM
\citep[][]{Bruenken10}, it is also the more stable structure; as noted by
\citet{Halfen15}, reaction \ref{reac1} should be endothermic. However, a
precise determination of this is challenging, as the enthalpy of formation of
methyl isocyanate is poorly defined in the literature. The CRC Handbook of
Chemistry and Physics \citep[][]{Haynes17} provides only a value for the
liquid phase ($\Delta_{f}H^{0}(l)=-92.0$~kJ~mol$^{-1}$) rather than the gas
phase. But even crudely adjusting this value to take account of the enthalpy
of vaporization, it appears almost certain that reaction \ref{reac1} would be
highly endothermic (by more than 100~kJ~mol$^{-1}$).
Calculations by \citet{Majumdar2017} indeed indicate the reaction to be endothermic by
77~kJ~mol$^{-1}$, with a transition state that lies 83~kJ~mol$^{-1}$ above the entrance level.
Furthermore, it is likely
that even reaction \ref{reac2} is somewhat endothermic. Substantial activation
energy barriers might also be expected for both reactions, suggesting that
either route would be quite inefficient in the ISM.
\citet{Halfen15} also proposed ion-molecule reactions between HNCO/HOCN and
CH$_5^+$ to produce protonated methyl isocyanate; however, such processes
involving CH$_5^+$ typically result in proton transfer in laboratory
experiments, indicating that protonated HNCO/HOCN would be the preferred
products. It therefore remains unclear whether there is a plausible gas-phase
mechanism that could produce CH$_3$NCO in sufficient quantities to explain
observations. As noted by \citet{Cernicharo16}, the detection of CH$_3$NCO
only in high-temperature regions may indicate that methyl isocyanate
originates on dust-grain surfaces.
Chemical models have been used to study the formation of CH$_3$NCO and related
species on interstellar dust grains. \citet{Belloche17} considered a direct
radical-radical association reaction:
\begin{equation}
{\rm CH}_3 + {\rm OCN} \rightarrow {\rm CH}_3{\rm NCO} \label{reac3}
\end{equation}
The radicals would be produced by the photodissociation of, or
chemical H-abstraction from, CH$_4$ and HNCO in the dust-grain ice mantles,
while CH$_3$ could also be produced by repetitive H addition to atomic carbon
on the grain/ice surfaces. Experiments presented by \citet{Ligterink17} indeed
show that CH$_3$NCO may be formed by UV irradiation of mixed CH$_4$:HNCO ices.
The precise mechanism of production was not determined, but was assumed to be
either reaction \ref{reac3} or reaction \ref{reac1}, which would be
endothermic also in the solid phase. As with reaction \ref{reac1}, the
alternative branch CH$_3$ + HNCO $\rightarrow$ CH$_4$ + OCN is endothermic.
However, the production of excited CH$_3$ as a photoproduct of CH$_4$ might be
sufficient to remove these difficulties in both cases.
\citeauthor{Ligterink17} also proposed that the OCN$^-$ ion, which was
produced abundantly in the experiments, could be active in forming CH$_3$NCO.
In their observational study, \citet{Ligterink2021} concluded that reaction \ref{reac3}, occurring during the dark cloud stage of evolution, would be a plausible explanation for the uniform CH$_3$NCO/HNCO ratio of around 10\% observed toward various sources.
The astrochemical model of \citet{Belloche17} indicated that, given an
appropriate degree of grain-surface formation of HNCO, methyl isocyanate could
also be formed on the grains during the warm-up phase of a hot core, via
reaction \ref{reac3}. The radicals would become mobile, and thus reactive, at
elevated temperatures; the model results provided an acceptable match to the
observed abundances.
However, the more recent models of \citet{Garrod22} have challenged the purely
diffusive picture of hot-core grain-surface chemistry, by including so-called
non-diffusive reactions on the grain surfaces and within the bulk ices
\citep[][]{Jin20}. Furthermore, these models restrict the mobility of most
chemical species in the bulk, allowing only H and H$_2$ to diffuse internally.
Larger species may diffuse only on the grain/ice surfaces themselves. The
non-diffusive reaction mechanisms in the model nevertheless allow
radical-radical reactions in the bulk ice to occur, but with rates driven by
the processes that initiate the radical production that precedes them. In this
scenario, the efficiency of bulk-ice chemistry is only weakly determined by
the ice temperature, while radical reactions in the bulk can occur in
principle at any temperature if, for example, photodissociation of a stable
molecule in the bulk ice occurs in the presence of a radical with which one of
the photoproducts may rapidly react.
Unfortunately, in these new models, CH$_3$NCO is severely underproduced on the
grains; OCN radicals tend to recombine quickly with mobile H atoms, both on
the surface and in the bulk ice, providing little opportunity for reaction
with CH$_3$. Furthermore, the endothermicity of the H-abstraction reaction
H + HNCO $\rightarrow$ OCN + H$_2$ removes an alternative means of reforming
OCN and thus raising its abundance on the grains. Meanwhile, a competing
process, H + HNCO $\rightarrow$ NH$_2$CO, is exothermic with only a modest
barrier \citep[1390 K;][]{Nguyen96}. The removal of the bulk diffusion
mechanisms for OCN and CH$_3$ also inhibits the diffusive reactions that
produced CH$_3$NCO effectively at higher temperatures in the older models.
HNCO, however, is itself formed effectively on the grain surfaces at very low
temperatures in these models \citep[see][]{Garrod22}, through the
barrier-mediated reaction NH + CO $\rightarrow$ HNCO.
Thus, while photodissociation of CH$_4$ and HNCO, followed by radical addition,
still appears to be the main production mechanism for methyl isocyanate in the
newest astrochemical models, the resulting abundances are several orders of
magnitude lower than the observed values.
Several other gas-grain modeling studies have been conducted to explain observed gas-phase CH$_3$NCO abundances toward cold sources and warm star-forming regions \citep{Martin-Domenech2017,Quenard2018}. Those models adopted the gas-phase and grain-surface reaction mechanisms noted above; they concluded that reactions \ref{reac1} and \ref{reac2} are dominant contributors to CH$_3$NCO production following ice mantle desorption, based on the reactions occurring at the collisional rate, i.e. somewhere on the order of 10$^{-10}$ cm$^{3}$~s$^{-1}$. However, as noted above, the substantial endothermicities of those reactions would likely render them highly inefficient; some alternative formation mechanism would therefore still be required, whether in the gas phase or on the grains.
Noting the problems with the suggested gas-phase reactions, \citet{Majumdar2017} proposed two new mechanisms for grain-surface CH$_3$NCO production. The first involves the reaction of an H atom with an HCN molecule while the latter is in a van der Waals complex with a CO molecule; i.e.
\begin{equation}
{\rm H} + {\rm HCN}...{\rm CO} \rightarrow {\rm H}_2{\rm CN}...{\rm CO} \rightarrow {\rm CH}_2{\rm NCO} \label{reac_maj1}
\end{equation}
The energy produced in the initial reaction would allow the activation energy barrier to be overcome for an immediate follow-on reaction, producing CH$_2$NCO; the latter radical could then be rapidly hydrogenated by H to form methyl isocyanate. These authors' other mechanism involves a barrieless reaction between atomic N and the radical CH$_3$CO, for which they calculate that the production of CH$_3$NCO is the most energetically favorable outcome:
\begin{equation}
{\rm N} + {\rm CH}_3{\rm CO} \rightarrow {\rm CH}_3{\rm NCO} \label{reac_maj2}
\end{equation}
\citet{Majumdar2017} find that reactions \ref{reac_maj1} and \ref{reac_maj2} contribute to the production CH$_3$NCO sufficiently to explain observational values. However, the efficiency of reaction \ref{reac_maj1} is not well defined and is dependent on the precise reaction dynamics involved, as well as on the choice of barrier to the initial reaction. Reaction \ref{reac_maj2} relies on the hydrogenation of ketene (CH$_2$CO) to form the necessary radical, which is also a barrier-dependent process. The results are encouraging, although further testing of these mechanisms is likely required to determine their true effectiveness.
One further mechanism might be plausible for grain-surface production of
CH$_3$NCO. \citet{Garrod22} introduced a number of new reactions into their
chemical network involving methylene, CH$_2$. In its ground (triplet) state,
methylene is a diradical; \citeauthor{Garrod22} proposed that H-abstraction
reactions involving CH$_2$ and stable molecules on grain surfaces would lead
to production of two radicals that could immediately react with each other
to form a single product. It is unclear whether the initial abstraction
reaction between CH$_2$ and HNCO would be exothermic, but it would
almost certainly have an activation energy barrier.
However, a direct addition reaction,
\begin{equation}
{\rm CH}_2 + {\rm HNCO} \rightarrow {\rm CH}_3{\rm NCO} \label{reac4}
\end{equation}
would be strongly exothermic ($\sim$300~kJ~mol$^{-1}$). The attack
of the methylene radical on the C=N double bond in HNCO would likely have an
activation energy barrier, although the large exothermicity of the reaction
might suggest a relatively small barrier compared with, for example, the
above-mentioned addition reaction H + HNCO $\rightarrow$ NH$_2$CO (exothermic
by around 140~kJ~mol$^{-1}$). However, as the methylene reaction does not
explicitly involve an H atom, there may be no quantum tunneling mechanism
available to overcome the barrier, unlike in the case of many grain-surface
reactions involving atomic H. The determination of the efficiency of reaction
\ref{reac4} would require detailed calculations. But if CH$_2$ is a common
photodissociation product of CH$_4$, as is often assumed in the chemical
networks, then the spontaneous production of methylene in proximity to HNCO in
the bulk ice could plausibly produce methyl isocyanate in appreciable
quantities. Likewise, if the abstraction process is exothermic, then H-atom
tunneling between HNCO and CH$_2$ might allow it to proceed effectively,
producing CH$_3$NCO in a two-step process. As well as the bulk ice process,
formation of CH$_2$ on the grain/ice surface, from atomic carbon, might also
lead to reaction with HNCO.
Such a mechanism would presumably retain the observationally determined
correlation between HNCO and CH$_3$NCO \citep{Ligterink2021,Colzi2021}, once both molecules were desorbed into the gas phase at elevated temperatures.
Alternatively, as mentioned above, if CH$_3$ is formed in an excited state
through CH$_4$ photodissociation in the bulk ice, direct reaction
of the methyl radical
with abundant
HNCO might proceed efficiently.
\subsubsection{Vinyl and ethyl isocyanate}
In spite of the nondetection of vinyl and ethyl isocyanate toward Sgr B2(N1S), it is valuable to
consider their possible production mechanisms. Although we do not currently
have chemical models that include either species, we may speculate on possible
outcomes based on molecules that are presently included.
The lack of a gas-phase formation mechanism for CH$_3$NCO would also suggest
that chemistry on grains or within the ice mantles is the most plausible
scenario for the larger molecules. We may again propose radical addition
reactions involving OCN, i.e.
\begin{equation}
{\rm C}_2{\rm H}_3 + {\rm OCN} \rightarrow {\rm C}_2{\rm H}_3{\rm NCO} \label{reac5}
\end{equation}
\begin{equation}
{\rm C}_2{\rm H}_5 + {\rm OCN} \rightarrow {\rm C}_2{\rm H}_5{\rm NCO} \label{reac6}
\end{equation}
Reaction \ref{reac6} was first suggested by \citet{Rodriguez-Almeida21}.
Assuming that reaction \ref{reac3} is the most important process for
CH$_3$NCO production, the relative abundances of the radicals CH$_3$,
C$_2$H$_3$, and C$_2$H$_5$ in the ice might indicate the expected ratio of the
isocyanates (on the expectation that the photodissociation of HNCO would
provide the driving rate for all three reactions, \ref{reac3}, \ref{reac5},
and \ref{reac6}). However, in the new \citet{Garrod22} models, the ratios of
the peak abundances (during the cold collapse stage) are approximately
CH$_3$:C$_2$H$_3$:C$_2$H$_5$ = 30:1:65. While this appears consistent
with the observed ratio of the column density of methyl isocyanate to the
upper limit for vinyl isocyanate, it would also indicate that the abundance of
ethyl isocyanate should exceed that of methyl isocyanate. As noted in
Sect.~\ref{ss:nondetection_remoca}, such a large ratio is not expected.
However, reaction \ref{reac3} does not produce adequate quantities of CH$_3$NCO
in the models. If an alternative process is active at least for the production
of this molecule, while reactions \ref{reac5} and \ref{reac6} were to remain
the principal routes to vinyl and ethyl cyanide, then this would indicate that
the real column densities of the latter species should be far less than the
observational upper limits.
If the methylene reaction proposed above -- reaction \ref{reac4} -- should in
fact be the dominant production route for CH$_3$NCO, then the absence of
analogous mechanisms that form vinyl and ethyl cyanide is readily understood;
there are no equivalent chemical species that would produce C$_2$H$_3$NCO or
C$_2$H$_5$NCO by a similar reaction with HNCO. In this case, the most likely
production mechanism for ethyl isocyanate could be the photodissociaton
of, or H-abstraction from, methyl isocyanate itself, followed by the addition of
a methyl group, e.g.:
\begin{equation}
{\rm CH}_3{\rm NCO} + h\nu \rightarrow {\rm CH}_2{\rm NCO} + {\rm H}
\end{equation}
\begin{equation}
{\rm CH}_3 + {\rm CH}_2{\rm NCO} \rightarrow {\rm C}_2{\rm H}_5{\rm NCO}
\end{equation}
The H-abstraction initiated process might include the two-step insertion of
methylene into a C-H bond in methyl isocyanate. The astrochemical models
indicate that in cases in which H-abstraction/photodissociation and methyl
addition is the main formation mechanism, such as with ethylamine production
from methylamine,
the smaller homologue achieves an abundance around 10 times greater than that of the larger homologue.
Such would be in line with our observational ratio of
methyl to ethyl
isocyanate. Furthermore, \citet{Rodriguez-Almeida21} recently detected
ethyl isocyanate toward the Galactic Center cloud G+0.693-0.027, finding a
ratio CH$_3$NCO:C$_2$H$_5$NCO = $8 \pm 1$, which is also in good agreement
with the typical modeling outcome.
Once again, there is unlikely to be a comparable mechanism leading to
formation of a vinyl group, which would also tend to make the abundance of
vinyl isocyanate inferior to those of the other two.
The above ideas are, of course, highly speculative. A full chemical model
including all of these species would be desirable, although there remain a
number of poorly-defined quantities that could be important -- not least, the
possible efficiency of reaction \ref{reac4}, and the precise mechanisms
involved in experimental UV-induced production of methyl isocyanate.
\section{Conclusions}
\textbf{\label{s:conclusions}}
Laboratory rotational spectroscopy of vinyl isocyanate has been undertaken in the frequency regions 127.5--218 and 285--330~GHz. Over 1000 transition lines were assigned and measured
for the ground vibrational states of its \textit{trans} and \textit{cis} isomers. The present work provides significantly more precise values of the spectroscopic parameters which agree with those from accompanying high-level quantum-chemical computations. We report a nondetection of both vinyl and ethyl isocyanate toward
the main hot core of Sgr~B2(N) that was targeted with ALMA. We find that these
molecules are at least 11 and 3 times less abundant than methyl isocyanate,
respectively.
Despite the nondetection of vinyl isocyanate in Sgr~B2(N) the present work represents a substantial improvement on previous microwave studies below 40 GHz and meets the requirements
for further searches of this species in the interstellar space.
To this end, spectral predictions are provided in Tables~\ref{predictions-trans} and \ref{predictions-cis} and will be also available in CDMS.
\begin{acknowledgements}
The spectroscopic part of this work has been funded by the Czech Science Foundation (GACR, grant No. 19-25116Y). L.K., K.V., J.K., and K.L. gratefully acknowledge this financial support.
L.K., J.K., K.L., and P.K. thank the financial support from the Ministry of Education, Youth and Sports of the Czech Republic (MSMT) within the Mobility grant No. 8J21FR006.
Computational resources were supplied by the project "e-Infrastruktura CZ" (e-INFRA CZ LM2018140) supported by the Ministry of Education, Youth and Sports of the Czech Republic. Computational resources were provided by the ELIXIR-CZ project (LM2018131), part of the international ELIXIR infrastructure.
R.T.G. thanks E. Herbst for helpful discussions.
This paper makes use of the following ALMA data:
ADS/JAO.ALMA\#2016.1.00074.S.
ALMA is a partnership of ESO (representing its member states), NSF (USA), and
NINS (Japan), together with NRC (Canada), NSC and ASIAA (Taiwan), and KASI
(Republic of Korea), in cooperation with the Republic of Chile. The Joint ALMA
Observatory is operated by ESO, AUI/NRAO, and NAOJ. The interferometric data
are available in the ALMA archive at https://almascience.eso.org/aq/.
Part of this work has been carried out within the Collaborative
Research Centre 956, sub-project B3, funded by the Deutsche
Forschungsgemeinschaft (DFG) -- project ID 184018867.
R.T.G. acknowledges funding from the Astronomy \& Astrophysics program
of the National Science Foundation (grant No. AST 19-06489).
J.-C.G. thanks the Barrande project No. 46662VH, the Centre National d'Etudes Spatiales (CNES) and the "Programme National Physique et Chimie du Milieu Interstellaire" (PCMI) of CNRS/INSU with INC/INP co-funded by CEA and CNES for a grant.
\end{acknowledgements}
|
2,877,628,089,048 | arxiv | \section{Introduction}\label{I}
Chemical reactions have provided ubiquitous and versatile examples of
activated transitions between two metastable states, formed by the
reactants and products. In a chemical
reaction the energy necessary for the activation most often stems from the
(classical or even quantum mechanical) thermal energy that may accumulate in a single reaction
coordinate and finally enable a transition from reactants to products
\cite{P1986,PGH,HTB, PT}. In contrast to these thermally assisted escape processes other
additional
sources of energy may externally be provided for example by driving
a system with metastable states by periodic forces.
Such periodically driven stochastic systems present a particular class of
nonequilibrium processes that exhibit a broad variety of fascinating
effects \cite{JH90,JH,J} such as stochastic resonance \cite{SR}, directed
transport of Brownian particles in ratchet
type periodic potentials \cite{AH,BM,HM}
or other anomalous transport properties
as for example negative mobility \cite{NM}.
Apart from an external periodic driving, these systems typically are subject to
nonlinear dynamical laws and additionally experience
fluctuating forces describing the random impact of the environment of
the considered system \cite{Z}. Without the fluctuating forces the presence of
nonlinearities often renders these systems multistable, i.e.\ such
systems may approach different attractors \cite{P1978}, depending on
their initial
states. In combination with weak fluctuating forces these attractors
become metastable states, which means that the system will be found
most of the time in
or close to one of these states while transitions between these states
present rare events.
Each of the principal constituents of the dynamics of a periodically
driven nonlinear stochastic system is characterized by typical time
scales such as the correlation time of the fast random forces (ff),
$\tau_{\text{ff}}$, relaxation times $\tau$ of the deterministic part of the
dynamics, the period $T$ of the driving force and the times
$\tau_{\text{ms}}$ of typical sojourn within the different metastable
states (ms). In this work we will assume that the correlation
times of the fluctuating forces are much shorter than all other time
scales such that a Markovian description of the dynamics is
appropriate. Hence, we model the fluctuating forces by white noise
($\tau_{\text{ff}}=0$)
which moreover will be assumed to be Gaussian and weak. As a
consequence of these assumptions the characteristic sojourn times of the
metastable states are finite but much larger than any of the
deterministic characteristic times ($\tau_{\text{ms}} \gg \tau$) \cite{HTB}.
This time scale separation implies that the transitions between
the metastable states constitute a discrete Markovian process which
will be investigated in more detail in the present work. We will
demonstrate
that this discrete process forms the backbone of the original
continuous process on time scales that are much larger than the
deterministic relaxation times $\tau$ .
Finally, the magnitude of the driving period $T$ in relation to the
deterministic time scales $\tau$ has a decisive influence on the system's
dynamics. In the so-called semiadiabatic limit \cite{T99} the driving
period is large compared to typical deterministic relaxation
times independently of how large the driving period is compared with
the typical sojourn times. Then the time-dependent transition rates
are given by the frozen rates, i.e.\ their time dependence only results
from the slow change of those system parameters that are varied by the
driving process \cite{TL}. Within this framework stochastic resonance
\cite{TMSHL} and the dynamics of neuron models \cite{STH_2004} have
successfully been described.
Outside the regime of the so called semiadiabatic limit the escape rates no
longer instantly follow but rather lack
behind the periodic driving \cite{STH_2005}. In the present paper we
investigate this regime of intermediate to fast driving in more
detail and present effective methods to characterize the large time
behavior of periodically driven Fokker-Planck processes with
metastable states.
Previous works on periodically driven processes with metastable states
most often have been focussed on particular aspects such as on the
dependence of the average life time of a metastable state \cite{DG,PH1994},
of the exponentially leading part of escape rates within linear
response theory \cite{LMD}, or
on rates in the weak noise limit \cite{LRH,Lehmann2003}.
We close this Introduction with a short outline of the paper.
In Section~\ref{dd} we introduce some important concepts of the
deterministic dynamics of a periodically driven system with coexisting
attractors. In Section~\ref{cpdf} two alternative formulations of the
conditional probability density function are presented for events that
are separated by a time that is much larger than the characteristic
deterministic time $\tau$. The first form originates from
the Floquet representation of the
conditional probability density of a periodically driven Markov
process \cite{JH90,JH} while the second expression explicitly refers to the
dynamics of the metastable states. This second expression in particular
contains quantities that
characterize specific probability densities for each metastable state
as well as localizing functions that allocate probabilities to
the metastable states given the
state of the full continuous system. In Section~\ref{lf} we find
equations of motion both for these metastable state specific
probability densities and the localizing functions by
comparing the two
formulations of the conditional probability density at large times. In
Section~\ref{pdBbo} the theory is exemplified and numerically tested
for a bistable Brownian oscillator. Section~\ref{Con} closes with a
summary.
\section{Characterization of the deterministic dynamics}\label{dd}
In the deterministic limit the considered system is described by the
motion of a state ${\mathbf x}$ in a $d$ dimensional state space $\Sigma$ governed
by a set
of $d$ coupled differential equations
\begin{equation}
\dot{{\mathbf x}} = {\mathbf f}({\mathbf x},t)\:,
\ee{deq}
where the vector field ${\mathbf f}({\mathbf x},t)$ periodically depends on time with
period $T$, i.e.\
${\mathbf f}({\mathbf x},t+T) ={\mathbf f}({\mathbf x},t)$.
We denote the trajectory emanating at the time $s$ from the point ${\mathbf y}$ by
${\mathbf X}(t|{\mathbf y},s)$ and
assume that in the asymptotic limit of large times the motion is
bounded and characterized by a set of $n\geq 2$ different attractors
$\mathcal{A}_\alpha(t) \subset \Sigma$,
$\alpha = 1 \ldots n$, such that each trajectory approaches either of the
attractors depending on its initial state and starting time, i.e.\
${\mathbf X}(t|{\mathbf y},s) \to {\mathbf x} \in \mathcal{A}_\alpha(t)$ for $t-s$ sufficiently
large. This relaxation process happens
on a characteristic deterministic time scale of the considered system.
The attractors periodically depend on time, i.e.\
\begin{equation}
\mathcal{A}_\alpha(t+T) = \mathcal{A}_\alpha(t)\: .
\ee{At}
To each attractor a domain of attraction $\mathcal{D}_\alpha(s)$
exists
that consists of all states ${\mathbf y}$ at time $s$ from which the $\alpha^{\text{th}}$
attractor is reached.
It is formally defined as $\mathcal{D}_\alpha(s)
= \left \{{\mathbf y}| {\mathbf X}(t|{\mathbf y},s) \in \mathcal{A}_\alpha(t) \;\text{for}\; t-s
\to \infty\right \}$.
At each fixed time the domains of attraction
form a
partition of the state space into disjoint subsets, which in general
periodically depend on time
\begin{equation}
\mathcal{D}_\alpha(t+T) = \mathcal{D}_\alpha(t)\: .
\ee{Dt}
\section{Conditional probability density of time-periodic Fokker-Planck
processes with metastable states}\label{cpdf}
\subsection{Floquet representation}\label{Floq}
In many cases the description of a system in terms of deterministic
equations of motion
is sufficient in order to determine the typical behavior of the system
with sufficient accuracy. However, the presence of weak random perturbations,
which often can be modeled by Gaussian white noise, causes different
effects depending on the considered time scales: On characteristic
time scales of the deterministic motion only insignificant deviations
from the deterministic motion typically occur; those trajectories
that start close to the boundaries of the domains of attraction though
are exceptional because they may
be influenced even by small noise, cross the border of the
deterministic domain of attraction and, in this way, come close to a ``wrong''
attractor with finite probability; all other trajectories are
markedly influenced on much
longer time scales only on which transitions between the
deterministic, locally
stable states become likely. Hence, these states lose their
stability. Nevertheless, for sufficiently weak noise the system is
found most of the time close to
one of the formerly stable states. Transitions between these states do
occur with certainty even though this happens rarely. Therefore such
states can be considered as {\it metastable}.
Under the influence of Gaussian white noise the
deterministic dynamical system (\ref{deq})
becomes a Markov process that is characterized by a Fokker-Planck operator
of the following form \cite{HT,R}
\begin{equation}
L(t) = -\sum_i^d \frac{\partial}{\partial x_i} K_i({\mathbf x},t) + \sum_{i,j}^d
\frac{\partial^2}{\partial x_i \partial x_j} D_{i,j}({\mathbf x},t)\: .
\ee{FPO}
We here will restrict ourselves to periodically driven processes
where the drift $K_i({\mathbf x},t)$ and possibly also the diffusion $D_{i,j}({\mathbf x},t)$
periodically depend on time with a common period $T$. Hence, $L(t+T)
=L(t)$.
The time evolution of the system's probability density function (pdf)
$\rho({\mathbf x},t)$ is governed by the Fokker-Planck equation
\begin{equation}
\frac{\partial}{\partial t} \rho({\mathbf x},t) = L(t) \rho({\mathbf x},t)\: .
\ee{FPE}
In the deterministic limit the diffusion matrix vanishes and
the drift $K_i({\mathbf x},t)$ approaches the deterministic
drift $f_i({\mathbf x},t)$ having the properties discussed in Section~\ref{I}.
A particular solution of the Fokker-Planck equation is the conditional
pdf
$\rho({\mathbf x},t|{\mathbf y},s)$ to find the process at the state
${\mathbf x}$ at time $t$ under the condition that it was at the state ${\mathbf y}$
at time $s$.
It can formally be expressed
in terms of the Floquet representation in the following way \cite{JH90,JH,J,SR,T99}
\begin{equation}
\rho({\mathbf x},t|{\mathbf y},s) = \sum_i e^{\mu_i (t-s)} \psi_i({\mathbf x},t) \varphi_i({\mathbf y},s) \: ,
\ee{FR}
where $\psi_i({\mathbf x},t)$ and $\varphi_i({\mathbf y},s)$ are Floquet eigenfunctions and $\mu_i$ are the
corresponding Floquet exponents. They satisfy pairs of mutually adjoint Floquet
equations reading
\begin{equation}
\begin{split}
\frac{\partial}{\partial t} \psi_i ({\mathbf x},t) &= L(t) \psi_i ({\mathbf x},t) - \mu_i
\psi_i ({\mathbf x},t)\: ,\\
-\frac{\partial}{\partial t} \varphi_i ({\mathbf x},t) &= L^+(t) \varphi_i ({\mathbf x},t) - \mu_i
\varphi_i ({\mathbf x},t)\: ,
\label{FEf}
\end{split}
\end{equation}
with natural boundary conditions with respect to the state variable ${\mathbf x}$.
Moreover, both types of eigenfunctions are periodic in time
\begin{equation}
\begin{split}
\psi_{i}({\mathbf x},t+T) &= \psi_{i}({\mathbf x},t)\: , \\
\varphi_{i}({\mathbf x},t+T) &= \varphi_{i}({\mathbf x},t)\: .
\end{split}
\ee{fpsiT}
The Floquet functions $\psi_i({\mathbf x},t)$ and $\varphi_j({\mathbf x},t)$ are mutually
orthogonal for eigenvalues $\mu_i \neq \mu_j$ and can be normalized such
that
\begin{equation}
\int d{\mathbf x} \: \varphi_j({\mathbf x},t) \psi_i({\mathbf x},t) = \delta_{i,j}\: ,
\ee{BIO}
where $\delta_{i,j}$ denotes the Kronecker symbol.
The Floquet exponents $\mu_{j}$ have real parts that are negative or at most
zero.
The representation of the conditional probability in terms of the
Floquet functions further requires that these functions form a
complete set in the sense that
\begin{equation}
\sum_i \psi_i({\mathbf x},t) \varphi_i({\mathbf y},t) = \delta({\mathbf x}-{\mathbf y})\: ,
\ee{CPR}
where $\delta({\mathbf x})$ denotes the Dirac $\delta$ function.
We note that equations
(\ref{FEf}), (\ref{BIO}) and (\ref{CPR}) do not uniquely
determine
the Floquet functions
because gauge transformations of the form
\begin{equation}
\begin{split}
\bar{\psi}_j({\mathbf x},t) &= g_j(t) \psi_j({\mathbf x},t)\:, \\
\bar{\varphi}_j({\mathbf x},t) &= g_j^{-1}(t) \varphi_j({\mathbf x},t)\:,\\
\bar{\mu}_j &= \mu_j + \frac{2 \pi i}{T} n_j, \quad n_j \in \mathbb{Z}
\end{split}
\ee{ga}
with gauge factors
\begin{equation}
g_j(t) = c_j e^{2 \pi i n_j t/T}\:, \quad c_j \in \mathbb{C},\; c_j \neq 0
\ee{gj}
generate new
Floquet eigenfunctions, cf. Ref.~\cite{T_2000}. Here $\mathbb{Z}$ and
$\mathbb{C}$ denote the sets of integer and complex numbers,
respectively and $i$ the imaginary unit.
For the sake of definiteness we assume that the gauge chosen for
the Floquet representation of the conditional pdf
(\ref{FR}) is such that the Floquet
exponents assume their smallest possible absolute values.
The Floquet spectrum consisting of these Floquet exponents then
contains the value $\mu_0=0$. We assume that this Floquet exponent is
not degenerate \cite{nondeg} if the diffusion matrix is different from
zero.
The corresponding eigenfunction of
$L^{+}(t)$ is constant with respect to ${\mathbf x}$ and $t$ and can be
chosen as
$\varphi_0({\mathbf x},t) =1$; the eigenfunction $\psi_0({\mathbf x},t)$ of $L(t)$ is a
non-negative and normalized function giving the
uniquely defined asymptotic pdf. Hence, it is the unique
solution of the Fokker-Planck equation (\ref{FPE}) that is approached
at time $t$ from any initial state in the remote past at $s \to -
\infty$. As a Floquet
eigenfunction it is periodic in $t$. The normalization
\begin{equation}
\int_{\Sigma} d{\mathbf x}\: \psi_{0}({\mathbf x},t)=1
\ee{Nps0}
follows from
eq.~(\ref{BIO}) together with the fact that $\varphi_{0}({\mathbf x},t) = 1$.
For vanishing noise, the diffusion matrix $D_{i,j}({\mathbf x},t)$ vanishes
and the backward operator becomes a first order
partial differential operator
$L^+_0(t) = \sum_i f_i({\mathbf x},t) \partial/\partial x_i$
with $f_i({\mathbf x},t)$ being the components of the
deterministic vector field ${\mathbf f}({\mathbf x},t)$ governing the deterministic
motion, eq.~(\ref{deq}). For a
dynamical system with $n$ coexisting attractors the characteristic
functions of the domains of attraction represent $n$ independent
periodic solutions of the backward equation
$-\partial\varphi_0/\partial t = L_0^+(t) \varphi_0 $.
Each of the solutions is unity on one
of the domains of attraction and zero outside. All other periodic
solutions are linear combinations of these characteristic functions.
That means that a deterministic system with $n$ locally stable states
possesses an $n$-fold degenerate Floquet eigenvalue $\mu_0 =0$.
As discussed above, in the presence of
noise, the formerly locally stable states become metastable. The
$n$-fold degeneracy of $\mu_0 =0$ is
lifted, but at sufficiently weak noise there remains a group of $n$ Floquet
exponents one of which is exactly zero and the others aquire a small
negative real part. We call them the {\it slow} Floquet exponents.
For sufficiently small noise this group of slow
Floquet exponents stays well separated from all other Floquet exponents.
For large time lags, the slow Floquet exponents and the
corresponding Floquet eigenfunctions completely determine the
conditional pdf which becomes
\begin{equation}
\begin{split}
\rho({\mathbf x},t|{\mathbf y},s)& = \sum_{i =0}^{n-1} e^{\mu_i(t-s)} \psi_i({\mathbf x},t) \varphi_i({\mathbf y},s)\\
&\qquad \qquad \text{for} \; t-s \gg \tau\:,
\end{split}
\ee{rlt}
where the sum only runs over the group of $n$ slow Floquet exponents
i.e.\ over those exponents with the
smallest absolute values.
All other Floquet exponents are determined by the deterministic time
scales all of which are much shorter than those given by the slow
Floquet exponents. Here $\tau$ denotes the slowest deterministic time scale.
\subsection{Alternative representation of the conditional probability at large times}\label{acpdf}
In the presence of metastable states the process of moving
from
a state ${\mathbf y}$ at time $s$ to a state ${\mathbf x}$ at a much later time $t$
may be subdivided into three consecutive steps that correspond to three
contributions to the conditional
probability $\rho({\mathbf x},t|{\mathbf y},s)$:
Within the typical
relaxation time $\tau$, compared to which the considered time span $t-s$
is supposed to be very large, the initial state ${\mathbf y}$ will be allocated
to either of the metastable states $\beta$ with a
probability $\chi_\beta({\mathbf y},s)$;
within the remaining time $t-s -\tau \approx t-s$ the process may visit several
other metastable states and will be found in the
state $\alpha$ at the final time $t$ with a probability $p(\alpha,t|\beta,s)$.
Given the final discrete state $\alpha$, the actual continuous states are
distributed
with a pdf $\rho({\mathbf x},t|\alpha)$. For sufficiently small
noise the times within which the first and the last steps are performed
are negligibly short compared to the total time $t-s$. Therefore, the
initial allocation
to a metastable state $\alpha$ and the final allocation to a continuous
state ${\mathbf x}$ can be considered as instantaneous events. Moreover, all three
steps are independent of each other and therefore the
conditional probability $\rho({\mathbf x},t|{\mathbf y},s)$ results as
\begin{equation}
\rho({\mathbf x},t|{\mathbf y},s) = \sum_{\alpha,\beta} \rho({\mathbf x},t|\alpha) p(\alpha,t|\beta,s)
\chi_\beta({\mathbf y},s)\: .
\ee{rpc}
This particular form of the conditional pdf was
derived in the semiadiabatic limit \cite{TL} which is definded by the regime for which the driving is slow compared
to the characteristic local relaxation times but not necessarily slow
compared to the typical transition times between metastable states
\cite{T99}. We claim that this particular form of the conditional
pdf remains to hold true also beyond the
semiadiabatic limit, i.e. in
situations when the driving period is comparable or even faster than the local
relaxation times. The rare occurrence of the transitions between the
metastable states is the only condition required for eq.~(\ref{rpc})
to hold.
It implies the separation of the times needed to perform the first and
the third step compared to the much larger time of the second step and
justifies the independence of these three steps and their respective
contributions to the
conditional probability.
Below, we will infer the main properties of these three sets of
functions $\rho({\mathbf x},t|\alpha)$, $\chi_\alpha({\mathbf x},t)$ and $p(\alpha,t|\beta,s)$
from their according definitions. \\
(i) Each {\it localizing function} $\chi_\alpha({\mathbf x},t)$ assumes an almost constant
value very close to unity
within the domain of attraction $\mathcal{D}_\alpha(t)$ and vanishes
outside.
Close to the border of $\mathcal{D}_\alpha(t)$, the localizing function
$\chi_\alpha({\mathbf x},t)$
smoothly interpolates between these two values. At each point ${\mathbf x}$ all
$n$ functions $\chi_\alpha({\mathbf x},t)$ exactly add up to unity:
\begin{equation}
\sum_\alpha \chi_\alpha({\mathbf x},t) = 1\: .
\ee{cN}
(ii) Each {\it $\alpha$-specific pdf} $\rho({\mathbf x},t|\alpha)$ is a strongly
peaked function of ${\mathbf x}$ about the corresponding attractor
$\mathcal{A}_\alpha(t)$ and rapidly decays away from the attractor. As
pdf it is
normalized to unity
\begin{equation}
\int_{\Sigma} d{\mathbf x}\: \rho({\mathbf x},t|\alpha) = 1 \: ,
\ee{rN}
where the integration extends over the full state space $\Sigma$.
Within the respective domains of attraction $\mathcal{D}_\alpha(t)$ the
$\alpha$-specific pdf
almost coincides with the asymptotic
pdf $\psi_{0}({\mathbf x},t)$ up to a normalizing factor.
Property (i) of the localizing function allows one to determine the
probability $p_\alpha(t)$ of finding the metastable state $\alpha$
realized at time $t$ for a given pdf $\rho({\mathbf x},t)$ in
the following way
\begin{equation}
p_\alpha(t) = \int_{\Sigma} d{\mathbf x}\: \chi_\alpha({\mathbf x},t) \rho({\mathbf x},t)\: .
\ee{pa}
On the other hand, one can assign to a given set of probabilities
$p_\alpha(t)$ a pdf $\rho_p({\mathbf x},t)$ by decorating the
metastable states $\alpha$ with the $\alpha$-specific pdfs
yielding
\begin{equation}
\rho_p({\mathbf x},t) = \sum_\alpha \rho({\mathbf x},t|\alpha) p_\alpha(t)\: .
\ee{rp}
In order that eqs.~(\ref{pa}) and (\ref{rp}) are compatible with each
other, i.e.\ that eq.~(\ref{pa}) reproduces the prescribed probabilities
$p_\alpha(t)$ for $\rho(t) = \rho_{p}(t)$, the localizing functions and the
$\alpha$-specific pdfs must form a biorthonormal set of functions, i.e.\
\begin{equation}
\int_{\Sigma} d{\mathbf x} \: \chi_\alpha({\mathbf x},t) \rho({\mathbf x},t|\beta) = \delta_{\alpha,\beta}\: .
\ee{cr}
For a Fokker-Planck process the time evolution of a pdf $\rho({\mathbf x},t)$
is determined by the conditional pdf according to
\begin{equation}
\rho({\mathbf x},t) = \int_{\Sigma} d{\mathbf y}\:\rho({\mathbf x},t|{\mathbf y},s) \rho({\mathbf y},s)\: .
\ee{rrr}
For large time lags $t\!-\!s$ the conditional pdf can be written
as in eq.~(\ref{rpc}). Using eqs.~(\ref{rpc}), (\ref{pa}) and
(\ref{cr}) one obtains from
eq.~(\ref{rrr}) for the propagation of the
probabilities $p_\alpha(t)$
\begin{equation}
p_\alpha(t) = \sum_{\alpha,\beta} p(\alpha,t|\beta,s) p_\beta(s)\:.
\ee{ppp}
This relation expresses the occupation probabilities of the metastable
states at a time $t$ in terms of the corresponding probabilities
at an earlier time $s$. Eq.~(\ref{ppp})
hence confirms the interpretation of $p(\alpha,t|\beta,s)$ as the
conditional probability of the coarse grained process of the
metastable, discrete states $\alpha = 1 \ldots n$.
In order to derive an equation of motion for the probabilities $p_\alpha(t)$ one
differentiates both sides of eq.~(\ref{pa}) with respect to time, uses
the Fokker-Planck equation (\ref{FPE}), and expresses the pdf by
means of eq.~(\ref{cr}) in terms of the probabilities $p_\beta(t)$. In
this way one obtains
\begin{equation}
\begin{split}
\dot{p}_\alpha(t)&= \int_{\Sigma} d{\mathbf x} \: \big \{ \frac{\partial
\chi_\alpha({\mathbf x},t)}{\partial t} \rho({\mathbf x},t) \\ &\quad + \chi_\alpha({\mathbf x},t) L(t) \rho({\mathbf x},t)
\big \} \\
&=\sum_\beta k_{\alpha,\beta}(t) p_\beta(t)\:,
\end{split}
\ee{me}
where the time dependent rates $k_{\alpha,\beta}(t)$ are defined as
\begin{equation}
\begin{split}
k_{\alpha,\beta}(t)&= \int_{\Sigma} d{\mathbf x}\: \frac{\partial\chi_\alpha({\mathbf x},t)}{\partial t}
\rho({\mathbf x},t|\beta)\\
&\quad + \int_{\Sigma} d{\mathbf x}\: \chi_\alpha({\mathbf x},t) L(t) \rho({\mathbf x},t|\beta)\:.
\end{split}
\ee{k}
Eq.~(\ref{cN}) implies that the sum over the first index of the rates
vanishes, i.e.\ $\sum_\alpha k_{\alpha,\beta}(t) =0$. Therefore, eq.~(\ref{me})
can be brought into the familiar form of a master equation \cite{vK}
\begin{equation}
\dot{p}_\alpha(t) = \sum_{\beta \neq \alpha} k_{\alpha,\beta}(t) p_\beta(t) - \sum_{\beta \neq
\alpha} k_{\beta,\alpha}(t) p_\alpha(t)\:.
\ee{fme}
We expect that for sufficiently low noise the quantities
$k_{\alpha,\beta}(t)$ do not become negative for $\alpha \neq \beta$ and therefore
represent proper rates. A formal proof of the positivity
though is not available. Negative values of $k_{\alpha,\beta}(t)$ though
would indicate a breakdown
of the basic assumption that the long time behavior of the process is
described by a rate process.
\section{Localizing functions, $\alpha$-specific pdfs
and transition rates}\label{lf}
Comparing the two expressions (\ref{rlt}) and (\ref{rpc}) one finds
that the $\alpha$-specific pdfs $\rho({\mathbf x},t|\alpha)$ can be
expressed as linear combinations of the
first $n$ Floquet eigenfunctions $\psi_i({\mathbf x},t)$ and the localizing
functions $\chi_\alpha({\mathbf x},t)$ can be written in terms of $\varphi_i({\mathbf x},t)$. This leads to the
linear relations
\begin{align}
\label{rcp}
\rho({\mathbf x},t|\alpha)&= \sum_{i=0}^{n-1} C_{i,\alpha}(t) \psi_i({\mathbf x},t)\:, \\
\chi_\alpha({\mathbf x},t)&= \sum_{i=0}^{n-1} D_{\alpha,i}(t) \varphi_i({\mathbf x},t)\:,
\label{cdf}
\end{align}
where $C_{i,\alpha}(t)$ and $D_{\alpha,i}(t)$ are yet undetermined, time
dependent coefficients.
The orthogonality relations (\ref{BIO}), (\ref{cr}) and the
linear independence of the first $n$ Floquet eigenfunctions
imply the following orthogonality relations of
the coefficients $C_{i,\alpha}(t)$
and $D_{i,\alpha}(t)$:
\begin{equation}
\begin{split}
\sum_i D_{\alpha,i}(t) C_{i,\beta}(t) &= \delta_{\alpha,\beta}\:,\\
\sum_{\alpha} C_{i,\alpha}(t) D_{\alpha,j}(t) &= \delta_{i,j}\:.
\end{split}
\ee{CD}
For $i=0$ the normalization of the Floquet function $\psi_{0}({\mathbf x},t)$,
see eq.~(\ref{Nps0}),
and of the
$\alpha$-specific pdfs $\rho({\mathbf x},t|\alpha)$, see eq.~(\ref{rN}), leads to
\begin{equation}
C_{0,\alpha}(t) = 1.
\ee{C01}
Next we derive sets of coupled equations of motion for the
localizing functions and the $\alpha$-specific pdfs.
\subsection{Transition rates}\label{tr}
Using the Floquet representation of the $\alpha$-specific pdfs and
localizing functions, (\ref{rcp}) and (\ref{cdf}), in combination with the
Floquet equations (\ref{FEf})
we obtain for the rates from eq.~(\ref{k})
\begin{equation}
\begin{split}
k_{\alpha,\beta}(t)& = \sum_i \left ( \dot{D}_{\alpha,i}(t) C_{i,\beta}(t) +
D_{\alpha,i} \:\mu_i\: C_{i,\beta}(t) \right ) \\
&= \sum_i \left ( \dot{D}_{\alpha,i}(t) D^{-1}_{\beta,i}(t) +
D_{\alpha,i} \:\mu_i\;
D^{-1}_{\beta,i}(t) \right )\:,
\end{split}
\ee{kCD}
where we expressed the
coefficient matrix $C_{i,\beta}(t)$ as the inverse of $D_{\beta,i}(t)$ by means of
eq.~(\ref{CD}).
Assuming for the moment that the rates $k_{\alpha,\beta}(t)$ were known
we can rewrite
eq.~(\ref{kCD}) as of an equations of motion
for the coefficients $D_{\alpha,i}(t)$ and $C_{i,\alpha}(t)$ reading
\begin{align}
\label{dD}
\dot{D}_{\alpha,i}(t)& = \sum_{\beta} k_{\alpha,\beta}(t) D_{\beta,i}(t) -
D_{\alpha,i}(t) \mu_{i}\:,\\
-\dot{C}_{i,\alpha}(t)& = \sum_{\beta} C_{i,\beta}(t)k_{\beta,\alpha}(t) -
\mu_{i} C_{i,\alpha}(t)\:.
\label{dC}
\end{align}
It is interesting to note that these are just the Floquet equations of
the master equation (\ref{fme})
and, moreover, that the slow Floquet exponents of the Fokker-Planck
coincide with the
Floquet exponents of the master equation. This is a consequence of the
fact that the master equation specifies the transitions between the metastable
states, and, therefore, represents the backbone of the long time
evolution of the Fokker-Planck process.
With the help of eq.~(\ref{dD}) and the Floquet equations (\ref{FEf})
the following equations of motion for the $\alpha$-specific
pdfs and the localizing
functions are obtained
\begin{align}\label{er}
\frac{\partial}{\partial t} \rho({\mathbf x},t|\alpha) &= L(t) \rho({\mathbf x},t|\alpha) - \sum_{\beta}
k_{\beta,\alpha}(t) \rho({\mathbf x},t|\beta)\:, \\*[2mm]
-\frac{\partial}{\partial t} \chi_\alpha({\mathbf x},t) &= L^+(t) \chi_\alpha({\mathbf x},t) -
\sum_{\beta}k_{\alpha,\beta}(t) \chi_\beta({\mathbf x},t)\:.
\label{ec}
\end{align}
These two sets of equations for the functions $\rho({\mathbf x},t|\alpha)$ and
$\chi_{\alpha}({\mathbf x},t)$ are adjoint to each other such that the
biorthonormality of the $\alpha$-specific and the localizing functions, see
eq.~(\ref{cr}), continues to
hold for all times once it holds true at a particular instant of time.
Eqs.~(\ref{er}) and (\ref{ec}) represent a central result of this
work.
The set of coupled equations (\ref{er}) can be interpreted as the
motion of $n$ replicas of the original process. Each replica is labeled
by one of the
attractor indices $\alpha$. The corresponding processes are described by
the Fokker-Planck equation (\ref{FPE}) with additional
source and sink terms, $\sum_{\beta \neq \alpha} k_{\beta,\alpha}(t)\rho({\mathbf x},t|\alpha)$
and $-\sum_{\beta\neq \alpha} k_{\beta,\alpha}(t)\rho({\mathbf x},t|\beta)$, respectively.
This means that, say, the
$\alpha$-process dies with probability $\sum_{\beta \neq \alpha}
k_{\beta,\alpha}(t) \rho({\mathbf x}, t|\beta)$ and instantly resurrects
with probability $\sum_{\beta \neq \alpha} k_{\beta,\alpha}(t) \rho({\mathbf x},t|\alpha)$ such
that the total
probability $\int_{\Sigma} d{\mathbf x} \rho({\mathbf x},t|\alpha)$ of each replica
is conserved for all times. A natural requirement
on a process described by the set of
eqs.~(\ref{er}) is the positivity of the probabilities
$\rho({\mathbf x},t|\alpha)$. For an arbitrary choice of the rates $k_{\alpha,\beta}(t)$ this
property generally will be violated in the course of time. Only for the
correct choice of the transition rates the positivity is guaranteed to
hold. In principle, it is this requirement which determines
the rates $k_{\alpha,\beta}(t)$ on the basis of eq.~(\ref{er}).
In view of the fact that eqs.~(\ref{er}) and (\ref{ec}) are coupled
sets of equations not only for the functions $\rho({\mathbf x},t|\alpha)$ and
$\chi_\alpha({\mathbf x},t)$, respectively, but that in these equations also the
time dependent rates $k_{\alpha,\beta}(t)$ are unknown, it would be very
difficult to solve these equations exactly. Therefore
appropriate approximation schemes have to be devised. This will be
done in the remaining part of this Section.
\subsection{Absorbing boundary approximation: $\alpha$-specific pdfs}\label{spdf}
Assuming the appropriateness of the rate description, i.e.\ in
particular the
positivity of $k_{\alpha,\beta}(t)$ for all $\alpha \neq \beta$, one can decompose
the sum on the right hand
side of eq.~(\ref{er})
into a sink term $- \sum_{\beta \neq \alpha}
k_{\beta,\alpha}(t) \:\rho({\mathbf x},t|\beta)$ and a source term $\sum_{\alpha \neq \beta}
k_{\beta,\alpha}(t)\: \rho({\mathbf x},t|\alpha)$. These sink and source terms result from the
diagonal and non-diagonal parts of the rate matrix $(k_{\alpha,\beta}(t))$,
respectively.
The sink terms are linear combinations of the functions $\rho({\mathbf x},t|\beta)$,
which are strongly
concentrated about the positions of the corresponding attractors
$\mathcal{A}_\beta(t)$ with $\beta \neq \alpha$.
We approximate these narrow, even though continuously distributed
sink terms by replacing them
with sharp, absorbing states lying on
the boundaries
$\partial \mathcal{B}_\beta(t)$ of domains $\mathcal{B}_\beta(t)$. Each
domain $\mathcal{B}_\beta(t)$ contains the immediate neighborhood
of
the attractor $\mathcal{A}_\beta(t)$ in such a way that the boundary
$\partial
\mathcal{B}_\beta(t)$ separates the corresponding attractor from the
remaining state space.
Within this {\it absorbing boundary approximation} we obtain an uncoupled set of equations for
the $\alpha$-specific pdfs reading
\begin{equation}
\begin{split}
\frac{\partial}{\partial t} \bar{\rho}({\mathbf x},t|\alpha) &= L(t)
\bar{\rho}({\mathbf x},t|\alpha) + k_{\alpha}(t) \bar{\rho}({\mathbf x},t|\alpha)\:,\\
& \qquad \text{for}\;{\mathbf x} \in
\Sigma_{\alpha}(t)\:,\\
\bar{\rho}({\mathbf x},t|\alpha) &= 0\:, \quad \text{for all}\; {\mathbf x} \in
\partial \mathcal{B}_\beta(t)\;\text{with}\; \beta \neq \alpha\:,
\end{split}
\ee{ura}
where
\begin{equation}
k_{\alpha}(t) \equiv - k_{\alpha,\alpha}(t) = \sum_{\beta \neq \alpha} k_{\beta,\alpha}(t)
\ee{ka}
denotes the total decay rate of the state $\alpha$ which is the sum over
the individual rates from $\alpha$ to all other states $\beta$. The
restricted state space $\Sigma_{\alpha}(t)$ is obtained from the full state
space $\Sigma$ by excluding the immediate neighborhoods
$\mathcal{B}_{\beta}(t)$ of all metastable states $\beta$ being different
from $\alpha$. Hence, it is defined as
\begin{equation}
\Sigma_{\alpha}(t) \equiv \Sigma \smallsetminus \cup_{\beta \neq \alpha} \mathcal{B}_{\beta}(t)\:.
\ee{Sal}
On this restricted state space the function $\bar{\rho}({\mathbf x},t|\alpha)$ is
expected to represent a valid approximation of the $\alpha$-specific pdf
$\rho({\mathbf x},t|\alpha)$.
We search
for the periodic solution
of eq.~(\ref{ura}) which can be obtained in the following way. First one
numerically solves the source free problem
\begin{equation}
\begin{split}
\frac{\partial}{\partial t} \tilde{\rho}({\mathbf x},t|\alpha) &= L(t)
\tilde{\rho}({\mathbf x},t|\alpha)\:, \\
\tilde{\rho}({\mathbf x},t|\alpha) &= 0, \quad \text{for all}\; {\mathbf x} \in
\partial\mathcal{B}_\beta(t)\;\text{with}\; \beta \neq \alpha
\end{split}
\ee{tra}
with an initial condition that is positive in a small neighborhood of
the attractor $\mathcal{A}_{\alpha}(t)$ and vanishes everywhere else.
Because of the absorbing boundary conditions at $\partial
\mathcal{B}_{\beta}(t)$, with $\beta\neq \alpha$, the auxiliary function
$\tilde{\rho}({\mathbf x},t|\alpha)$ decays in time, i.e.\
\begin{equation}
N_\alpha(t) = \int_{\Sigma_{a}(t)} d{\mathbf x} \:\tilde{\rho}({\mathbf x},t|\alpha)
\ee{Na}
is a decreasing function of time. Here the integral is extended over
the restricted state space $\Sigma_{\alpha}(t)$ excluding the domains
$\mathcal{B}_{\beta}(t)$, $\beta\neq \alpha$, as defined in eq.~(\ref{Sal}). The
normalized function
\begin{equation}
\bar{\rho}({\mathbf x},t|\alpha)=\tilde{\rho}({\mathbf x},t|\alpha)/N_\alpha(t)
\ee{rxa}
then satisfies the eq.~(\ref{ura}) with the total outgoing rate given by
\begin{equation}
k_\alpha(t) = - \frac{\dot{N}_\alpha(t)}{N_{\alpha}(t)}\:.
\ee{kaN}
The such constructed solution $\tilde{\rho}({\mathbf x},t)/N_{\alpha}(t)$
approaches a
periodic function in time on the time scale of the deterministic
dynamics, and presents an
approximation to the $\alpha$-specific function $\rho({\mathbf x},t|\alpha)$.
The other rates $k_{\beta,\alpha}(t)$ leaving the metastable state $\alpha$
follow from the flux associated with
$\rho({\mathbf x},t|\alpha)$ through the boundaries $\partial \mathcal{B}_\beta(t)$
\begin{equation}
k_{\beta,\alpha}(t) = \int_{\partial \mathcal{B}_{\beta}(t)} d{\mathbf S} \cdot {\mathbf j}({\mathbf x},t|\alpha)\:,
\quad \alpha \neq \beta\:,
\ee{kj}
where $d{\mathbf S}$ denotes the surface element on $\partial \mathcal{B}_{\beta}(t)$
pointing towards the metastable state $\mathcal{A}_{\beta}(t)$, and
${\mathbf j}({\mathbf x},t|\alpha)$ the probability current carried by the pdf
$\bar{\rho}({\mathbf x},t|\alpha)$. Its components read
\begin{equation}
\begin{split}
j_{i}({\mathbf x},t|\alpha) & = K_{i}({\mathbf x},t) \bar{\rho}({\mathbf x},t|\alpha) \\
& \quad - \sum_{l}
\frac{\partial}{\partial x_{l}} D_{i,l}({\mathbf x},t) \bar{\rho}({\mathbf x},t|\alpha)\: .
\end{split}
\ee{j}
This is a generalization of the well known flux-over-po\-pu\-la\-tion
expression for the rate \cite{HTB,fop,K,schmid99}. The stationary flux
carrying pdf of the classical flux-over-population
expression is replaced by the flux carrying time-periodic pdf
$\bar{\rho}({\mathbf x},t|\alpha)$ which is normalized to one, whence also the
population is one.
The decisive difference to the
classical flux-over-population expression lies in the fact that in
eq.~(\ref{kj}) the
flux is determined as the probability flowing per time directly into the final
metastable state, which because of the surrounding absorbing boundary
acts as an outlet,
rather than through a ``saddlepoint'' or ``bottleneck'' on the common
part of the
separatrices $\partial \mathcal{D}_{\alpha}(t)$ and $\partial
\mathcal{D}_{\beta}(t)$ of the initial and the final
metastable state. In the time independent case both expressions
coincide under the
condition that a region containing the final metastable
state and the bottleneck in question is free of sources
\cite{Langer}. In contrast, in the
time-periodic case the probability current contains a periodic
contribution
which in general has a nonuniform phase, i.e.\ the phase depends on the
location ${\mathbf x}$.
Therefore, the
instantaneous probability flux through the bottleneck in general
differs from the
flux into the outlet. A large portion of probability flowing through
the bottleneck, say within the first half of the period may flow back during
the second half of the period.
Only the time averages over one period of the probabilities flowing
through the bottleneck and into the outlet do coincide.
\subsubsection{$\alpha$-Floquet functions and rates}\label{Ffr}
The functions $\tilde{\rho}({\mathbf x},t|\alpha)$ which satisfy the
Fokker-Planck equation (\ref{tra}) on the restricted state space
$\Sigma_{\alpha}(t)$ defined in eq.~(\ref{Sal})
are closely related to the Floquet functions $\psi^{\alpha}({\mathbf x},t)$ of
the Fokker-Planck operator restricted to $\Sigma_{\alpha}(t)$ with absorbing
boundaries on the surfaces of the excluded regions
$\mathcal{B}_{\beta}(t)$. These $\alpha$-Floquet functions, as we call them,
are the solutions of the
corresponding Floquet equations which read
\begin{equation}
\begin{split}
\frac{\partial}{\partial t} \psi_{i}^{\alpha}({\mathbf x},t)& =
L(t)\psi_{i}^{\alpha}({\mathbf x},t) - \mu_{i}^{\alpha} \psi_{i}^{\alpha}({\mathbf x},t)\\
& \qquad
\text{for} \;{\mathbf x} \in \Sigma_{\alpha}(t)\:,\;n = 1,2, \ldots\\
\psi_{i}^{\alpha}({\mathbf x},t)& = 0 \quad \text{for} \; {\mathbf x} \in \partial
\mathcal{B}_{\beta}(t),\; \beta \neq \alpha\:.
\end{split}
\ee{psa}
Because of the absorbing boundaries at all but one metastable states
the Floquet spectrum consisting of the $\alpha$-Floquet eigenvalues
$\mu_{i}^{\alpha}$ completely lies in the complex half plain with negative
real part. We denote the $\alpha$-Floquet eigenvalue closest to zero by
$\mu_{1}^{\alpha}$.
The absolute value of the real parts of all other $\alpha$-Floquet eigenvalues
are much larger, i.e.\ $|\mu_{1}^{\alpha}| \ll |\mu_{i}^{\alpha}| $ for all $i
\neq 1$. In the deterministic limit $\mu_{1}^{\alpha}$ approaches zero,
whereas all other $\alpha$-Floquet eigenvalues stay finite.
In terms of the $\alpha$-Floquet eigenfunctions the solution of
eq.~(\ref{tra}) becomes
\begin{equation}
\tilde{\rho}({\mathbf x},t|\alpha) = \sum_{i=1} c_{i}\: e^{\mu_{i}^{\alpha} t}\:
\psi_{i}^{\alpha}({\mathbf x},t)\:,
\ee{trpsi}
where $c_{i}$ are constant coefficients whose values depend on the choice of
the initial distribution.
For times which are large on the deterministic time scale, all terms
in the sum become negligibly small apart from the first term
corresponding to $\mu_{1}^{\alpha}$. Hence, we obtain
\begin{equation}
\tilde{\rho}({\mathbf x},t|\alpha) \propto e^{\mu^{\alpha}_{1} t} \:\psi_{1}^{\alpha}({\mathbf x},t)\:,
\ee{trps0}
and, by proper normalization
\begin{equation}
\bar{\rho}({\mathbf x},t|\alpha) = \frac{\psi_{1}^{\alpha}({\mathbf x},t)}{\int_{\Sigma_{\alpha}(t)}d{\mathbf x}
\:\psi_{1}^{\alpha}({\mathbf x},t) }\:.
\ee{rps}
With eq.~(\ref{kaN}) the total rate $k_{\alpha}(t)$ follows as the
negative logarithmic derivative of the normalization $\int_{\Sigma(t)} d{\mathbf x}\:
\psi_{1}^{\alpha} ({\mathbf x},t)$. It becomes
\begin{equation}
k_{\alpha}(t) = -\mu_{1}^{\alpha} + r_{\alpha}(t)\:,
\ee{kmr}
where
\begin{equation}
r_{\alpha}(t) = - \frac{d}{dt} \ln \int_{\Sigma_{\alpha}(t)} d{\mathbf x}\: \psi_{1}^{\alpha}({\mathbf x},t)\:.
\ee{rFp}
The average of $r_{\alpha}(t)$ over one period vanishes because
$r_{\alpha}(t)$
is the derivative of a periodic function. Hence, with eq.~(\ref{kmr})
the $\alpha$-Floquet eigenvalue $\mu_{1}^{\alpha}$ is given by the negative
averaged total rate.
If one performs the time derivative in eq.~(\ref{rFp}) one finds
\begin{equation}
\begin{split}
r_{\alpha}(t) &= - \frac{\frac{d}{dt} \int_{\Sigma_{{\alpha}}(t)} d{\mathbf x} \:
\psi_{1}^{\alpha}({\mathbf x},t)}{\int_{\Sigma_{{\alpha}}(t)} d{\mathbf x} \: \psi_{1}^{\alpha}({\mathbf x},t)} \\
&= -\frac{\int_{\Sigma_{{\alpha}}(t)}d{\mathbf x}\: \left [ L(t)\psi_{1}^{\alpha}({\mathbf x},t) -
\mu_{1}^{\alpha}\psi_{1}^{\alpha}({\mathbf x},t) \right ]} {\int_{\Sigma_{\alpha}(t)} d{\mathbf x} \:
\psi_{1}^{\alpha}({\mathbf x},t)}\\
&=\sum_{\beta \neq \alpha} \frac{\int_{\partial \mathcal{B}_{\beta}(t)} \sum_{i,j}
dS_{i}
\frac{\partial}{\partial x_{j} }D_{i,j}({\mathbf x},t)
\psi_{1}^{\alpha}({\mathbf x},t)}{\int_{\Sigma_{\alpha}(t)} d{\mathbf x} \: \psi_{1}^{\alpha}({\mathbf x},y) }\\
& \quad +
\mu_{1}^{\alpha} \\
&= \sum_{\beta \neq \alpha} k_{\beta,\alpha}(t) + \mu_{1}^{\alpha} \:.
\end{split}
\ee{rLp}
In the second equality the time derivative was performed. There, the time
dependence of the domain $\Sigma_{\alpha}(t)$ does not contribute because the
$\alpha$-Floquet function vanishes on the boundary $ \partial \Sigma_{\alpha}(t) =\cup_{\beta \neq
\alpha} \partial \mathcal{B}_{\beta}(t)$. The time derivative of
$\psi_{1}^{\alpha}({\mathbf x},t)$ was expressed by eq.~(\ref{psa}). In the next
step the
integral involving the Fokker-Planck operator was written by means of
Gauss' theorem in terms
of surface integrals over the boundary of $\Sigma_{\alpha}(t)$.
The terms in the sum on $\beta$ are the ratios of the probability fluxes
through the boundaries $\partial \mathcal{B}_{\beta}(t)$ carried by the $\alpha$-Floquet
function $\psi_{1}^{\alpha}({\mathbf x},t)$, see eq.~(\ref{j}),
and the corresponding populations
$\int_{\Sigma_{\alpha}(t)} d{\mathbf x} \: \psi_{1}^{\alpha}({\mathbf x},t)$. According to the
eqs.~(\ref{kj}) and (\ref{rps}) the terms in the sum on $\beta$ agree
with the individual rates $k_{\beta,\alpha}(t)$.
\subsection{Absorbing boundary approximations: Localizing functions}\label{Lf}
The same type of approximation as for the $\alpha$-specific pdfs
may also be applied to the equations of
motion for the localizing functions:
By neglecting those terms on the right hand side of eq.~(\ref{ec})
that are proportional
to the rates $k_{\alpha,\beta}(t)$ with $\beta\neq\alpha$ and by introducing absorbing boundary
condititions on the hypersurfaces $\partial \mathcal{B}_\beta(t))$, $\beta
\neq \alpha$ we
obtain a set of uncoupled equations for approximate $\alpha$-localizing
functions $\bar{\chi}_{\alpha}({\mathbf x},t)$ reading
\begin{equation}
\begin{split}
-\frac{\partial}{\partial t} \bar{\chi}_\alpha ({\mathbf x},t) &= L^+(t) \bar{\chi}_\alpha ({\mathbf x},t)
+k_{\alpha}(t) \bar{\chi}_{\alpha}({\mathbf x},t)\:,\\
& \qquad \text {for}\; {\mathbf x} \in \Sigma_{\alpha}(t)\:,\\
\bar{\chi}_\alpha({\mathbf x},t) &=0\:, \quad \text{for all} \;{\mathbf x} \in \partial
\mathcal{B}_\beta(t) \;\text{with}\; \beta \neq \alpha \:.
\end{split}
\ee{c0}
This absorbing boundary approximation is again justified
because the rates $k_{\alpha,\beta}(t)$ are much smaller than the inverse
time scales of the deterministic dynamics which govern the motion
within the domains of attraction.
Moreover it is consistent with the above approximation for the $\alpha$-specific
pdfs in the sense that the integrals of the products of
the $\alpha$-specific and the respective localizing
function are independent of time, i.e.\
\begin{equation}
\frac{d}{dt} \int_{\Sigma_{\alpha}(t)} d{\mathbf x} \:\chi_{\alpha}({\mathbf x},t) \rho({\mathbf x},t|\alpha)= 0\:,
\ee{kc}
as follows from eqs.~(\ref{ura}) and (\ref{c0}). Note that the
time dependence of the integration domain $\Sigma_{\alpha}(t)$ does not
contribute because the integrand vanishes at the boundary.
The biorthonormality of the localizing functions and specific pdfs
cannot be strictly maintained within this approximation. The
deviations though are expected to be exponentially small with respect
to the noise strength because of the small overlap of these
functions for different metastable states.
As in the case of the $\alpha$-specific functions the total decay rate
$k_{\alpha}(t)$ need not be known in order to determine the
$\alpha$-localizing functions. Rather one again may first determine an auxiliary
function $\tilde{\chi}_{\alpha}({\mathbf x},t)$ as the solution of
the source free equation
\begin{equation}
\begin{split}
-\frac{\partial}{\partial t} \tilde{\chi}_\alpha ({\mathbf x},t)& =
L^+(t) \tilde{\chi}_\alpha ({\mathbf x},t)\:,\\
\tilde{\chi}_{\alpha}({\mathbf x},t) &= 0, \quad \text{for all}\; {\mathbf x} \in
\partial\mathcal{B}_\beta(t)\;\text{with}\; \beta \neq \alpha\:.
\end{split}
\ee{tc}
Because of the dissipative nature of the backward operator $L^{+}(t)$
it is convenient to integrate this equation backward in time. A
forward integration easily may run into numerical problems because
unavoidable errors would grow exponentially in time.
As an appropriate final condition for $\tilde{\chi}_{\alpha}({\mathbf x},t_{0})$
one may choose a function which
is constant on the domain of attraction $\mathcal{D}_{\alpha}(t_{0})$ and
zero everywhere else. The solution of this final value problem will
approach a periodic solution on the time scale of the deterministic
dynamics. This asymptotic periodic solution must be normalized at each
instant of
time by the integral of its product with the corresponding
$\alpha$-specific function to yield the required approximation of
$\chi_{\alpha}({\mathbf x},t)$
\begin{equation}
\bar{\chi}_{\alpha}({\mathbf x},t) = \frac{\tilde{\chi}_{\alpha}({\mathbf x},t)}{Z_{\alpha}(t)}\:,
\ee{cZ}
where
\begin{equation}
Z_{\alpha}(t) = \int_{\Sigma_{\alpha}(t)} d{\mathbf x}
\:\tilde{\chi}_{\alpha}({\mathbf x},t) \bar{\rho}({\mathbf x},t|\alpha)\:.
\ee{Zt}
Using the eqs.~(\ref{ura}) and (\ref{tc}) one finds
\begin{equation}
k_{\alpha}(t) = \frac{\dot{Z}_{\alpha}(t)}{Z_{\alpha}(t)}\:.
\ee{Zk}
This relation confirms that the function given by the eqs.~(\ref{cZ})
and (\ref{Zt})
indeed is a solution of eq.~(\ref{c0}).
\section{Periodically driven Brownian bistable oscillator}\label{pdBbo}
In order to exemplify the theory developed above and to check its
consistency we consider an overdamped bistable Brownian
oscillator driven by an external force that varies
periodically in time. We choose a bistable quartic potential $V(x,t)$ that
depends periodically on time, see Fig \ref{f1}.
\begin{figure}
\includegraphics[width=8cm]{FIG1.eps}
\caption{ The bistable potential $V(x,t)$,
eq.~(\ref{Vxt}), is depicted
as a function of the position $x$ for different times $t=0$ (red, dashed
line), $t=0.2 T$ (blue, solid line), and $t=0.4 T$ (black, dotted
line) where $T$ denotes the period of the driving and for the driving
strength $A=0.1$.}
\label{f1}
\end{figure}
In conveniently chosen dimensionless
variables it reads
\begin{equation}
V(x,t) = -\frac{1}{2} x^{2} + \frac{1}{4} x^{4} - A x \sin{\Omega t}\:,
\ee{Vxt}
where $t$ is time and $x$ the position of the Brownian particle. The
strength of the periodic modulation is denoted by $A$ and its
frequency by $\Omega$.
Depending on the values of $A$ and $\Omega$ the deterministic overdamped
dynamics in this time dependent potential is either monostable or
bistable as displayed in Fig.~\ref{f2}. In the present context we are
only interested in the bistable region in which the deterministic
dynamics $\dot{x} = -V'(x,t)$ possess two stable limit cycles
$x_{-1}(t)$ and $x_{1}(t)$ and an unstable limit cycle $x_{0}(t)$
forming the separatrix between the two attractors, see Fig~\ref{f3}.
\begin{figure}
\includegraphics[width=8cm]{FIG12.eps}
\caption{The line dividing the $\log_{10} \Omega$ --
$\log_{10} A$ parameter plane into an upper monostable and a lower
bistable region of the deterministic
dynamics $\dot{x} = - V'(x,t)$ is marked
by the thick, red solid curve. The blue, thin straight
line indicates the value of the forcing strength,
$A^{\text{ad}}= 2/(3 \sqrt{3})$, below which the potential $V(x,t)$
has two minima for all times $t$.}
\label{f2}
\end{figure}
\begin{figure}
\includegraphics[width=8cm]{FIG13.eps}
\caption{ The attractors $x_{-1}(t)$, $x_{1}(t)$ and the separatrix
$x_{0}(t)$ of the deterministic dynamics $\dot{x} = - V'(x,t)$ for
the driving strength $A=0.5$ and frequency $\Omega=1$.}
\label{f3}
\end{figure}
The diffusion matrix $D$ is taken as constant. The Fokker-Planck
operator then becomes
\begin{equation}
L(t)= \frac{\partial}{\partial x} V'(x,t) + D
\frac{\partial^{2}}{\partial x^{2}}\:,
\ee{LBp}
where $V'(x,t)$ denotes the derivative of the potential with respect
to $x$. The corresponding Fokker-Planck and backward equations were numerically
solved by a collocation method based on a representation of the
solution in terms of Chebishev polynomials of degree 5 \cite{NAG}. For all
calculations a fixed number $N = 1201$ of break-points in the interval
$[-3,3]$ was used. At the ends of the interval reflecting boundary
conditions were imposed.
In the case of the forward equation an accuracy of $10^{-10}$ led to
stable results whereas for the backward equations an accuracy of
$10^{-12}$ turned
out to be necessary in order to avoid numerical artefacts. Throughout
this paper we used a fluctuation strength given by $D=1/40$. At
vanishing driving strength $A=0$ the resulting bistable symmetric
potential then possesses a barrier height per noise energy of $\Delta V
/D = \left [V(0,0)-V(1,0) \right ]/D= 10$.
\subsection{Flux-over-population rates}\label{fopr}
We first numerically determined the time dependent solution
$\tilde{\rho}(x,t|-1)$ of the
Fokker-Planck equation (\ref{tra}) on the restricted state space
$\Sigma_{-1}(t) = [-3, x_{1}(t)]$ with a reflecting boundary at $x=-3$ and an
absorbing boundary at the
the position of the right attractor $x_{1}(t)$ and
with an initial condition that is sharply
located at the position of the other attractor $x_{-1}(0)$. After a
number $n$ of periods $T=2 \pi /\Omega$ of the driving frequency $\Omega$ had
elapsed the
remaining population $N_{-1}(t)$ was identified as
\begin{equation}
N_{-1}(t) = \int_{-3}^{x_{1}(t)} dx \:\tilde{\rho}(x,t|-1),
\ee{Nm1}
see also eq.~(\ref{Na}), and the renormalized pdf
\begin{equation}
\bar{\rho}(x,t|-1) =\tilde{\rho}(x,t|-1)/N_{-1}(t)\:,
\ee{brm1}
as well as the rate
\begin{equation}
k_{1,-1}(t) = - \frac{\dot{N}_{-1}(t)}{N_{-1}(t)}
\ee{km1}
were determined. The number $n$ of transient periods was chosen
such that $k_{1,-1}(t)$
remained unchanged upon a further increase of $n$. For different values of
$\Omega$ appropriate numbers $n$ are collected in Table~\ref{t1}.
\begin{table}
\caption{Number of transient periods}\label{t1}
\begin{tabular}{ll}
$\Omega$ & $n$\\
\hline
1 & 100\\
0.5 & 50\\
0.1 & 10\\
0.01&5\\
0.001&3\\
\end{tabular}
\end{table}
In Fig.~\ref{f4} the rates $k_{1,-1}(t)$ are displayed as functions of
time for various driving frequencies. For small frequencies the time
dependent rate approaches its adiabatic form \cite{TL} that is given by the
inverse mean first time that a process needs to move from $x=x_{-1}(t)$ to $x=x_{1}(t)$
in the frozen potential. The rate then reads \cite{HTB}
\begin{equation}
k^{\text{ad}}_{1,-1}(t) = D \left [\int_{x_{-1}(t)}^{x_{1}(t)} dx \: e^{V(x,t)/D}
\int_{-3}^{x_{0}(t)} dy \:
e^{-V(y,t)/D} \right]^{-1}.
\ee{kad}
For larger frequencies the maximal value of the rate shrinks and also
becomes delayed with respect to the driving force. In the limit of
high frequencies it approches the time independent rate
$k^{\text{av}}$ of a Brownian particle moving in the
potential $\overline{V(x,t)} = T^{-1} \int_{0}^{T} dt\: V(x,t)$
averaged over one period of the driving
force.
For the potential given by eq.~(\ref{Vxt}) the average
is symmetric and given by $\overline{V(x,t)} = V(x,0)$.
Hence the rate in the limit of
high driving frequencies
coincides with the value of the adiabatic rate at $t=0$:
\begin{equation}
k^{\text{av}} = k^{\text{ad}}_{1,-1}(0)\:.
\ee{kav}
Due to the symmetry of the averaged potential, the rate
$k^{\text{av}}$ also describes the opposite transition from the state
$x_{1}(t)$ to $x_{-1}(t)$, whence we skipped the index.
At a fixed frequency the rate $k_{1,-1}(t)$ decreases with decreasing
amplitude $A$ approaching the time independent value $k^{\text{av}}$,
see Fig.~\ref{f5}
\begin{figure}
\includegraphics[width=8cm]{FIG2.eps}
\caption{ The rate $k_{1,-1}(t)$ following from eq.~(\ref{km1})
displays a maximum as a function of
$t/T$ that becomes lower and shifts towards later times within one
period if the frequency $\Omega$ increases. For the frequency
$\Omega =10^{-3}$ the rate is indistinguishable from the adiabatic rate
(\ref{kad}) (black, solid line). The other curves display the rates for
$\Omega = 10^{-2}$
(blue, dotted line), $0.1$
(red, dash-dotted line), $0.5$ (brown, dashed line) and $1$
(green, thick dots); in the
asymptotic limit $\Omega \to \infty$ the constant rate $k^{\text{av}}$
(thin solid line)
given by eq.~(\ref{kav}) is approached. In all cases the driving
strength is $A=0.1$ and the noise strength $D=0.025$.}
\label{f4}
\end{figure}
\begin{figure}
\includegraphics[width=8cm]{FIG3.eps}
\caption{ The times at which the rate $k_{1,-1}(t)$
assumes its extrema do hardly depend on the amplitude $A$.
The rate is displayed for various values of $A=0.1$ (solid,
red), $0.2$ (dotted, blue), $0.3$ (dashdotted, black), and $0.4$
(dashed, green); in all cases the frequency is
$\Omega=10$, and the noise $D=0.025$. Note that for the large amplitude
$A=0.4>A^{\text{ad}}$ the deterministic attractors $x_{\pm1}(t)$ are dynamically
stabilized, see also
Fig.~\ref{f2}.}
\label{f5}
\end{figure}
The specific pdf $\bar{\rho}(x,t|-1)$ given by
eq.~(\ref{brm1}) represents a
periodic current carrying pdf with an absorbing state
at the attractor $x_{1}(t)$. It possesses a single maximum the location of which closely
follows the deterministic motion of the attractor $x_{-1}(t)$, see
Fig.~\ref{f6}. The pdf is asymmetric about its maximum
with a breathing width that is wider if the maximum is closer to the position of
the separatrix $x_{0}(t)$.
The approximate localizing function $\bar{\chi}_{-1}(x,t)$ of the left
metastable state
$x_{-1}(t)$ on the restricted state space $\Sigma_{-1}(t)$ was obtained from
the solution $\tilde{\chi}_{-1}(x,t)$ of the backward equation
(\ref{tc}) with absorbing boundary condition at the right metastable state
$x_{1}(t)$. In order to guarantee for sufficient numerical stability,
the integration of the backward equation has to be performed backward
in time from some $t_{0}$ to times $t<t_{0}$. The final function
$\tilde{\chi}_{-1}(x,t_{0})$ was chosen such that it assumes the constant value
$1$ for all $x \in [-3,x_{-1}(t_{0})]$ then decreases monotonically
and reaches zero at the right metastable state.
After the same
number $n$ of transient periods as for the corresponding characteristic
pdf, see Table~\ref{t1}, the normalization integral (\ref{Zt})
\begin{equation}
Z_{-1}(t) = \int_{-3}^{x_{1}(t)} dx\: \tilde{\chi}_{-1}(x,t) \bar{\rho}(x,t|-1)
\ee{Zmt}
was determined.
The rates $k_{1,-1}(t)$ that follow from the logarithmic
derivative of $Z_{-1}(t)$, cf. eq.~(\ref{Zk}), were compared with the
rates obtained from eq.~(\ref{km1}). They are identical within
numerical accuracy.
Finally, the localizing function $\bar{\chi}_{-1}(x,t)$ was determined by
normalizing $\tilde{\chi}_{-1}(x,t)$ with $Z_{-1}(t)$. For an example see
Fig.~\ref{f7}. We note that the position where the localizing function
assumes the value $1/2$ coincides with the location of the separatrix at
the respective time.
\begin{figure}
\includegraphics[width=8cm]{FIG8.eps}
\caption{ The specific pdf $\bar{\rho}(x,t|-1)$ is
depicted as a function of the position $x$ for various
times $t=0.12\:T$ (red, dashed), $0.37\:T$ (blue, solid),
$0.62\:T$ (black, dotted), and $0.87\:T$ (green, dashed-dotted) for
the driving
frequency $\Omega=1$, driving amplitude $A=0.1$ and noise strength $D =
0.025$. Outside the displayed interval the specific pdf
continues to decay. It vanishes at the position of the
attractor $x_{1}(t)$. The vertical lines indicate the positions of
the attractor $x_{-1}(t)$ at the respective times. These positions
almost coincide with the maxima of the specific pdfs
at the respective times. }
\label{f6}
\end{figure}
\begin{figure}
\includegraphics[width=8cm]{FIG6.eps}
\caption{ The localizing function $\bar{\chi}_{-1}(x,t)$
interpolates between the values $1$ at the attractor $x_{-1}(t)$ and
$0$ at $x_{1}(t)$. It is displayed at various instants of time,
$t=0.12\:T$ (red, dashed), $0.37\:T$ (blue, solid), $0.62\:T$ (black,
dotted), and $0.87\:T$ (green, dash-dotted). The vertical lines denote the
positions of the separatrix of the deterministic dynamics at the
corresponding times, see Fig.~\ref{f3}. In the inset a magnification of the
center part of the plot marked by a rectangle is depicted. It
demonstrates that the localizing
functions very precisely assume the value $1/2$ (horizontal line)
at the positions of
the separatrices indicated by the vertical lines.
}
\label{f7}
\end{figure}
\subsection{Floquet approach}\label{Fa}
Here we construct the specific pdfs and the
localizing functions in terms of Floquet eiegenfunctions on the basis
of the eqs.~(\ref{rcp}) and
(\ref{cdf}). In the present case of two metastable
states these equations simplify to read
\begin{align}\label{Cr}
\rho(x,t|\pm 1)&= \psi_{0}(x,t) + C_{\pm 1}(t) \psi_{1}(x,t)\:,\\
\chi_{\pm 1}(x,t)&= \frac{C_{\mp 1}(t)}{C_{\mp 1}(t) -C_{\pm 1}(t)}
\nonumber \\
&\quad -\frac{1}{C_{\mp 1}(t)
-C_{\pm 1}(t)}\varphi_{1}(x,t)\:.\label{Cc}
\end{align}
Here we skipped the first index $i$ of $C_{i,\alpha}(t)$ since
only the values for $i=1$ are nontrivial in the case of two metastable
states. For $i=0$, $C_{0,\alpha}(t) =1$ always holds, see eq.~(\ref{C01}).
To further evaluate these equations
(i) the first two Floquet
functions of the forward and the backward equation and (ii) the
coefficients $C_{\pm 1}(t)$ were determined numerically.
The Floquet function $\psi_{0}(x,t)$ belonging to the Floquet
eigenvalue $\mu_{0}=0$ is the periodic solution of the
Fokker-Planck equation (\ref{FPE}), (\ref{LBp})
with reflecting boundary conditions at $x=\pm 3$.
As initial condition we chose
\begin{equation}
\psi_{0}(x,0) = \frac{\exp \left( - V(x,0)/D\right)}{ \int_{-3}^{3} dx \exp \left(
- V(x,0)/D\right)}\:.
\ee{ps0}
The Fokker-Planck equation was numerically solved for $n$ periods of the
driving force.
We designated this number $n$
in such a way that after subsequent $n/10$ periods the $L_{1}$-norm of
the difference of the two solutions was less than $10^{-5}$, i.e.\
\begin{equation}
||\psi_{0}(x,1.1\:n\: T) -\psi_{0}(x,n\: T)||_{1} \leq 10^{-5}\:,
\ee{L1}
where the $L_{1}$-norm of a function $f(x)$ on the interval $[-3,3]$
is defined by the integral of the its absolute value as
\begin{equation}
||f(x)||_{1} = \int_{-3}^{3} dx |f(x)| \:.
\ee{L1N}
The numbers $n$ found in this way are collected in Table~\ref{t2} for
different values of the driving frequency.
\begin{table}
\caption{Number of transient periods needed to reach convergence of
the Floquet function $\psi_{0}(x,t)$ and Floquet exponent $\mu_{1}$}\label{t2}
\begin{tabular}{lrc}
$\Omega$ & $n$ & $\mu_{1}$\\
\hline
1 & 10000&-\:4.46 $10^{-5}$\\
0.5 & 2000&- \:9.46 $10^{-5}$\\
0.1 & 1000&-\:1.54 $10^{-4}$\\
0.01&1000&-\:1.58 $10^{-4}$\\
0.001&100&-\:1.58 $10^{-4}$\\
\end{tabular}
\end{table}
The Floquet function $\psi_{1}(x,t)$ and the corresponding Floquet
exponent $\mu_{1}$ were obtained from the solution of the Fokker-Planck
equation (\ref{FPE}), (\ref{LBp}) with reflecting boundary conditions
at $x=\pm3$ and the initial condition
\begin{equation}
\tilde{\psi}_{1}(x,0) = \delta\big (x-x_{-1}(0) \big )\:.
\ee{ps1}
After a transient period of duration $n\:T$ with $n$ given by
Table~\ref{t1} the
logarithm of the $L_{1}$-norm of the difference between
$\tilde{\psi}_{1}(x,t)$ and $\psi_{0}(x,t)$ was plotted as a function of
time for several periods. Its logarithm $\ln ||\tilde{\psi}_{1}(x,t) -
\psi_{0}(x,t)||_{1}$ is the superposition of a declining
linear and a periodic function of time with period $T$ of the
driving.
The Floquet exponent $\mu_{1}$
can be read off from the inclination of the linear contribution. The results are
presented in Table~\ref{t2}. We note here that the method of the
$\alpha$-Floquet functions defined on a restricted phase space with an
absorbing state at, say $x_{1}(t)$, see Section~\ref{Ffr},
gave Floquet exponents
$\mu_{1}^{-1}$ which coincide with those based on the full state space
up to 4 or 5 digits. The same agreement was obtained from the time average
of the rates obtained by either of the methods described in the
previous Section~\ref{fopr}.
Once the Floquet exponent $\mu_{1}$ is known, the still unnormalized Floquet
eigenfunction is obtained as
\begin{equation}
\psi_{1}(x,t)= e^{-\mu_{1} t} \left (\tilde{\psi}_{1}(x,t) - \psi_{0}(x,t)\
\right)\:.
\ee{ps1n}
The first two Floquet eigenfunctions, which were normalized with
respect to the $L_{1}$-norm, are displayed in Fig.~(\ref{f8}).
\begin{figure}
\includegraphics[width=8cm]{FIG9.eps}
\caption{ The first two Floquet eigenfunctions
$\psi_{0}(x,0)$ (red, solid line)
and
$\psi_{1}(x,0)$ (blue, dashed line) of the Fokker-Planck operator
(\ref{LBp}) of a
driven Brownian oscillator in a bistable potential (\ref{Vxt}) for
the driving strengths $A=0.1$, driving frequency $\Omega=1$ and noise
strength $D=2.5\times 10^{-2}$ at $t=0$ that are displayed in panel (a)
are strongly localized in the vicinity of the two metastable states
at $x_{\pm1}(0)$. Both functions are normalized such that their
$L_{1}$-norms are one, i.e.\ $||\psi_{i}(x,t)||_{1} = \int_{-3}^{3}dx
|\psi_{i}(x,t)| = 1$. The two functions almost agree with each other up
to a change in sign close to the unstable point $x_{0}(0)$. In panel
(b), the time
dependence is indicated for the asymptotic pdf $\psi_{0}(x,t)$ for
four different times $0.12 T$ (red, dashed line), $0.37 T$ (blue,
solid line), $0.62 T$ (black, dotted line)
and $0.87T$ (green, dash-dotted line).
}
\label{f8}
\end{figure}
The Floquet eigenfunction of the backward operator
belonging to the Floquet exponent $\mu_{0}=0$
is known to be constant, i.e.\ $\varphi_{0}(x,t) = 1$. In order to determine the
Floquet eigenfunction $\varphi_{1}(x,t)$ belonging to $\mu_{1}$ we solved
the backward equation
\begin{equation}
-\frac{\partial }{\partial t} \tilde{\varphi}_{1}(x,t) =
L^{+}(t)\tilde{\varphi}_{1}(x,t)
\ee{bwe}
with the initial condition
\begin{equation}
\tilde{\varphi}_{1}(x,0)= \text{sign}(x) \cdot \left \{
\begin{array}{ll}
-1&|x|\geq 0.1\\
100 \cdot (|x|-0.1)^{2}-1 \quad&|x| \leq 0.1\:.
\end{array}
\right .
\ee{f1i}
After a transient time of duration $nT$ with $n$ given in
Table~\ref{t1} all contributions from higher Floquet functions have
become negligible and $\tilde{\varphi}_{1}(x,t)$ assumes the form
\begin{equation}
\tilde{\varphi}_{1}(x,t) = c_{0} + e^{\mu_{1}t} c_{1} \varphi_{1}(x,t)\:.
\ee{f1t}
Knowing the Floquet exponent $\mu_{1}$ we determined the constant $c_{0}$
such that $[\tilde{\varphi}_{1}(x,t) -c_{0}]\exp(-\mu_{1} t)$ becomes a
periodic function of time which is proportional to the sought-after
function $\varphi_{1}(x,t)$.
The normalization of $\varphi_{1}(x,t)$ is chosen such that
\begin{equation}
\int_{-3}^{3} dx \:\varphi_{1}(x,t) \psi_{1}(x,t) = 1\:.
\ee{fps}
The spatial and temporal dependence of $\varphi_{1}(x,t)$ is depicted in
Fig.~\ref{f9} for the same parameter values as for the periodic pdf
displayed in Fig.~\ref{f8}.
\begin{figure}
\includegraphics[width=8cm]{FIG10.eps}
\caption{The Floquet eigenfunctions $\varphi_{1}(x,t)$
of the backward operator for the times $0.12 T$ (red, dashed line),
$0.37 T$ (blue, solid line), $0.62 T$ (black, dotted line)
and $0.87T$ (green, dash-dotted line)
are almost constant apart from a narrow
region about the unstable fixed point $x_{0}(t)$. The parameters are
with $A=0.1$, $\Omega=1$ and $D=2.5\times 10^{-2}$ the same as in Fig.~\ref{f8}.
}
\label{f9}
\end{figure}
Once the Floquet functions $\psi_{i}(x,t)$ for $i=0,1$ are known the
coefficients $C_{\pm1}(t)$ can be determined from the condition that
the $\alpha$-specific pdf $\rho(x,t|\alpha)$ is negligibly small in the vicinity
of the other metastable state $x_{\beta}(t)$ ($\alpha \neq \beta$). Hence the
intergration on both sides of eq.~(\ref{Cr}) over a small neighborhood
of $x_{\mp}(t)$ gives a negligibly small contribution and thus leads to
the following expression for the coefficients $C_{\pm1}(t)$
\begin{equation}
C_{\pm1}(t) \approx -\frac{\int_{x_{\mp1}(t)-\epsilon/2}^{x_{\mp1}(t)+\epsilon/2}
dx\:\psi_{0}(x,t)}{\int_{x_{\mp1}(t)-\epsilon/2}^{x_{\mp1}(t)+\epsilon/2}
dx\:\psi_{1}(x,t)}\:.
\ee{Cpm}
As an example the coefficient $C_{-1}(t)$ is displayed in
Fig.~\ref{f10} for different values of the driving frequency. The
interval length was chosen as $\epsilon=0.1$.
\begin{figure}
\includegraphics[width=8cm]{FIG11.eps}
\caption{ The variability of the coefficient $C_{-1}(t)$
within one period $T$ of the driving decreases with increasing
frequency $\Omega= 10^{-3}$ (red, dashed), $10^{-2}$ (blue, solid),
$10^{-1}$ (black, dotted) and $1$ (green, dash-dotted). The other
parameters are with $A=0.1$ and $D=2.5\times 10^{-2}$ the same as in Fig.~\ref{f8}.
}
\label{f10}
\end{figure}
Once the first two Floquet eigenfunctions and the coefficients
$C_{\pm1}(t)$ are known, the specific pdfs $\rho(x,t|\pm1)$ and the
localizing functions $\chi_{\pm1}(x,t)$
can be calculated and compared with the results for
$\bar{\rho}(x,t|\pm 1)$ and $\bar{\chi}_{\pm1}(x,t)$, respectively, obtained by the
flux-over-population method. We here restrict ourselves to a
comparison for the specific pdf $\rho(x,t|-1)$ for fast driving with $\Omega=1$.
Fig.~\ref{f11} demonstrates the perfect agreement. Only in the immediate
vicinity of the metastable state a difference becomes visible upon
strong magnification.
\begin{figure}
\includegraphics[width=8cm]{FIG14.eps}
\caption{ The specific pdf $\rho(x,0|-1)$ was determined by
three different methods: As the flux carrying periodic pdf
$\bar{\rho}(x,t|-1)$ in the presence of a sharp absorbing boundary at
$x_{1}(t)$ (red, dashed line), and as a linear combination of the
first two Floquet eigenfunctions, see eq.~(\ref{Cr}), with coefficients
either determined by eq.~(\ref{Cpm}) (blue, solid line), or from the
solutuion of the Floquet problem of the master equation (black,
dotted line), see the discussion below. Only in the magnification
displayed in the inset a deviation of the results of these methods
becomes visible in the vicinity of the metastable state
$x_{1}(0)\approx 0.98$ where $\bar{\rho}(x_{0}(0),0|-1) =0$. We expect that these
small deviations become even smaller at smaller noise strength.
}
\label{f11}
\end{figure}
Moreover, from the coefficients $C_{\pm1}(t)$ and the Floquet exponent
$\mu_{1}$ the rate $k_{-1,1}(t)$ and $k_{1,-1}(t)$ can be determined
according to eq.~(\ref{dC}) which simplifies for $k_{1,-1}(t)$ in the case of two
metastable states to
\begin{equation}
k_{1,-1}(t) =\frac{\mu_{1} C_{-1}(t) -
\dot{C}_{-1}(t)}{C_{1}(t)-C_{-1}(t)}\: .
\ee{kpmC}
A comparison of these rates with those obtained by the reactive flux
method is presented in Fig.~\ref{f12} for different values of the
driving frequency. A qualitatively good agreement is obtained for all
frequencies whereby deviations become more visible for higher
frequencies.
\begin{figure}
\includegraphics[width=8cm]{FIG15.eps}
\caption{A comparison of the flux-over-population
rates (fop rates) (lines)
with the Floquet rate expressions (F rates) following from
eq.~(\ref{kpmC}) (symbols)
is presented for frequencies $\Omega=0.01$ (fop rates: red, solid line; F
rates: crosses) and $\Omega=0.1$ (fop rates: blue, dashed line; F rates:
circles) in panel (a), and for $\Omega=0.5$ (fop rates: red, solid line; F
rates: crosses) and $\Omega=1$ (fop rates: blue, dashed line; F rates:
circles) in panel (b). The remaining parameters are with $A=0.1$, $D=2.5\times
10^{-2}$ the same as in Fig. \ref{f8}.
}
\label{f12}
\end{figure}
\subsection{Decoration}\label{deco}
Finally, we numerically investigated the crucial assumption that
after a sufficiently large transient period the pdf $\rho({\mathbf x},t)$
takes the form of eq.~(\ref{rp}), i.e.\ it is determined
by the solutions of the master equation (\ref{fme}), $p_{\alpha}(t)$,
which are decorated by the $\alpha$-specific pdfs
$\rho({\mathbf x},t|\alpha)$. As a quantitative measure
of the distance between the numerically exact solution $\rho(x,t)$ of
the Fokker-Planck equation (\ref{FPE}), with the Fokker-Planck
operator (\ref{LBp}),
starting at the metastable state
$x_{-1}(0)$, i.e.\ with the initial condition (\ref{ps1}),
and an approximate form $\rho_{\text{a}}(x,t)$ of the pdf we employed the
$L_{1}$-norm (\ref{L1N}) of the difference of these functions.
The assumed asymptotic form
\begin{equation}
\rho_{\text{a}}(x,t) = \rho(x,t|1) p_{1}(t) + \rho(x,t|-1) p_{-1}(t)
\ee{r}
requires the knowledge of the probabilities $p_{\pm1}(t)$ which was
obtained as the solution of the master equation
\begin{equation}
\begin{split}
\dot{p}_{1}(t)& = -k_{-1,1}(t) p_{1}(t) + k_{1,-1}(t) p_{-1}(t)\\
\dot{p}_{-1}(t)& = k_{-1,1}(t) p_{1}(t) - k_{1,-1}(t) p_{-1}(t)\\
p_{1}(0)&=0\:, \quad
p_{-1}(0) =1\:,
\end{split}
\ee{mets}
where the flux-over-population expressions were taken for the rates,
see Section \ref{fopr}.
For the specific pdfs we employed three different approximations:
First we used the current carrying pdfs $\bar{\rho}(x,t|\pm 1)$ introduced in
Section~\ref{fopr}. These functions were extended onto the full state space
$[-3,3]$ by assigning the value zero beyond their respective domains of
definition, i.e.\ we defined
\begin{equation}
\begin{split}
\rho_{I}(x,t|\!-\!1)& = \left \{
\begin{array}{ll}
\bar{\rho}(x,t|\!-\!1) \;\:& \text{for}\; -3\leq x \leq x_{1}(t)\\
0& \text{for}\; x_{1}(t)\leq x \leq 3
\end{array}
\right .\\
\rho_{I}(x,t|1)& = \left \{
\begin{array}{ll}
0 & \text{for} \;-3\leq x \leq x_{-1}(t)\\
\bar{\rho}(x,t|1) \;\quad&\text{for}\; x_{-1}(t)\leq x \leq 3\:.
\end{array}
\right .
\end{split}
\ee{rI}
As a second and third approximation, in the followowing referred to as
approximation II
and III, we used the specific pdfs (\ref{Cr}) with the numerically
determined Floquet functions, see Section~\ref{Fa}, and
determined the coefficients $C_{\pm1}(t)$ in two different ways.
The approximation II was
obtained by using eq.~(\ref{Cpm}) for the coefficients $C_{\pm1}(t)$.
The approximation III is based on the fact that these coefficients
obey the Floquet equations (\ref{dC}) of the backward master
equation. We numerically solved these equations under the assumption
that the rates are given by the flux-over-population expressions.
The resulting functions $c_{\pm1}(t)$ then coincide with the
sought-after coefficients $C_{\pm1}(t) = q c_{\pm1}(t)$ up to a common
proportionality constant $q$. Finally this coefficient was determined
such that the distance between the numerical solution of the
Fokker-Planck equation and the approximation III, i.e.\ $||\rho(x,t) -
\rho_{\text{III}}(x,t)||_{1}$, became minimal at $t=n T$ with $n$ from
Table~\ref{t1}.
The coefficients $C_{\pm1}(t)$ obtained in this way are compared with
those used in the approximation II, see Fig~\ref{f13}.
\begin{figure}
\includegraphics[width=8cm]{FIG17.eps}
\caption{The comparison of the approximations II and III
for the coefficient $C_{-1}(t)$ shows perfect agreement for driving frequencies
$\Omega\leq 0.1$, see panel (a) for $\Omega=0.1$ (method II: crosses, method
III: solid line). Relatively small but on the scale of the
variability apparent deviations between the methods become visible for
$\Omega=0.5$ (red, method II: crosses, method III: solid line) and $\Omega=1$
(blue, method II: circles, method III: dashed line) in panel
(b). The remaining
parameters in both panels are with $A=0.1$, $D=2.5\times
10^{-2}$ the same as in Fig. \ref{f8}.
}
\label{f13}
\end{figure}
The relative deviation between the coefficients $C_{\pm}(t)$ resulting
from the approximations II and III were smaller than $5\times 10^{-4}$ in all
investigated cases. Clear deviations are visible only on the scale of
the variability of the coefficients for frequencies $\Omega>0.1$, see
Fig~\ref{f13}.
\begin{figure}
\includegraphics[width=8cm]{FIG16.eps}
\caption{ After a short relaxation time, the decadic
logarithm of the $L_{1}$ distance
between the numerical solution of the Fokker-Planck equation and
the proposed asymptotic form (\ref{r}) reveals a perfect
agreement with $\rho_{\text{III}}$ within the expected numerical
precision of the solution of the Fokker-Planck equation (black,
dash-dotted line). In the case
of the first method (red, solid line) which uses the decoration with
the current
carrying densities, the absorbing boundary conditions at one of the
metastable states leads to a larger distance from the asymptotic
pdf. This also happens with method II (blue, dashed line) which is
based on the estimate
(\ref{Cpm}) of the coefficients $C_{\pm 1}(t)$ which lacks a rigorous
foundation. Yet the observed agreement is very good even for rather
fast driving with the frequency $\Omega=1$.
The remaining parameters are with $A=0.1$, $D=2.5\times
10^{-2}$ the same as in Fig. \ref{f8}.
}
\label{f14}
\end{figure}
The distances between the numerically exact solution of the
Fokker-Planck equation and the pdfs obtained from the decoration of
the metastable states according to the three methods described above
are displayed in Fig.~\ref{f14}. In all cases, after a short initial
time, an exponential relaxation sets in until the pdfs obtained from
method II as well as from the
decoration with the current carrying pdfs saturate at a distance
of the order of $2\times10^{-4}$. For method III it does so at the
smaller distance of $2\times 10^{{-6}}$. This is a clear indication that
the asymptotic pdf is indeed of the form of eq.~(\ref{r}). This hence
corroborates a basic assumption of our work about the structure of the
pdf at large times.
\section{Summary}
\label{Con}
We investigated the large time stochastic dynamics of periodically
driven systems with metastable states governed by a Fokker-Planck
equation.
On time scales larger than the typical deterministic
time scale this dynamics can be completely characterized by the
localizing functions, the $\alpha$-specific pdfs and the conditional
occupation probabilities of the metastable states. The latter are
solutions of a Markovian master equation with time-dependent
rates. These rates can be expressed in terms of the localizing functions
and the $\alpha$-specific pdfs, see eq.~(\ref{k}).
Using the Floquet
representation of the conditional pdf in the large time limit we
obtained coupled equations of motion for the $\alpha$-specific densities
and an adjoint set of equations for the localizing functions. Most
interestingly, these equations of motion can be interpreted in the
spirit of Farkas' \cite{fop} and Kramers' \cite{K} idea to construct a
flux carrying stationary solution by imposing convenient sources and
sinks.
To each $\alpha$-specific density an $\alpha$-process can be assigned that
evolves according to the same
dynamical laws as
the original process with the only difference that it can instantly
be translocated in state space.
These translocations are governed by sinks and sources that cause
a sudden death of an $\alpha$-process, say, at
a point ${\mathbf x}$ and the instant resurrection of the same process at a
different point ${\mathbf y}$ in state space. The sinks are determined by the
sum of transition
rates out of the metastable state $\alpha$ multiplied by those $\beta$ specific pdfs
corresponding to states that can directly be reached from $\alpha$.
The source is given by the total rate to leave state $\alpha$ multiplied
by its specific pdf.
In this way the conservation of probability of each specific
pdf is guaranteed. Due to the resulting
intricate coupling and the dependence on the
unknown rates, an exact solution is difficult to construct and one
must rely on approximate methods to solve this set of equations of motion
for the $\alpha$-specific pdfs.
An efficient way of
approximation is based on the fact that at weak noise the
$\alpha$-specific pdfs are expected to be strongly localized in the region
of the according metastable state. This allows one to effectively
decouple the equations for the $\alpha$-specific pdfs (as well as those for
the localizing functions) and to calculate a current carrying pdf in
the presence of sharply absorbing states. The rates of all transitions
leaving the considered metastable state can then be
calculated by means of a flux-over-population expression \cite{fop,K,schmid99}. In contrast
to the case without time-dependent driving it is important to
calculate the probability flux flowing directly
into the final metastable state.
In the time independent case this flux is the same through all
hypersurfaces in state space separating
the initial from the final metastable state. In the presence of
periodic driving the total flux through a hypersurface in general
depends both on time and on the location of the chosen hypersurface.
The proper rate therefore must
be determined from the probability flux flowing directly into the final
metastable state.
We illustrated our theory with the example of a periodically
driven bistable Brownian oscillator. In contrast to a slowly driven
bistable oscillator, at finite frequencies bistability
extends to larger amplitudes of the driving force.
We found that the flux-over-population method based on the $\alpha$-specific
pdf with an absorbing boundary at the final metastable state requires
a much lesser computational effort than the direct application of the
Floquet approach. In the former case the solution of the Fokker-Planck
equation with the appropriate boundary conditions converges on the
order of the deterministic time scale, whereas for the second method
the convergence of the Floquet functions is only reached after several
transitions between the metastable states have taken place on average.
We note that based on the absorbing boundary approximation the
transition rates can also be determined by means of numerical
simulations of the Langevin equations of the considered Fokker-Planck
process \cite{TMSHL,STH_2004,STH_2005}.
We finally tested the crucial assumption of our theory saying that the
probability density resulting as the large time solution of the Fokker-Planck
equation
can be represented as the product of the probabilities of the
metastable states decorated by the specific pdfs. The time dependence
of the probabilities of the metastable states was obtained from the
solution of the master equation with the numerically determined
flux-over-population rates. The specific pdfs obtained by the
absorbing boundary approximation already lead to an excellent
agreement with the numerically exact solution of the Fokker-Planck equation
on time scales larger than a few characteristic deterministic times.
A more elaborate calculation of the specific pdfs in terms of
Floquet eigenfunctions of the Fokker-Planck operator led to a further
improvement of the agreement by two orders of magnitude confirming our
assumption.
\section*{Acknowledgments}
Two of us (P.T. P.H.) like to acknowledge innumerable
stimulating and provocative scientific discussions with Eli Pollak who
is still at an age well fitted to appreciate
and to contribute great science.
This work was supported by the DFG via research
center, SFB-486, project A10, via the project no. 1517/26-2, the
Volkswagen Foundation (project I/80424), the German Excellence
Initiative via the \textit {Nanosystems Initiative Munich} (NIM), and
Research Foundation funded by the Korean Government
(MOEHRD), Basic Research Promotion Fund Grant No.
KRF-2005-070-C00065, by the Korea Science and Engineering
Foundation Grant No. F01-2006-000-10194-0, and by
the Deutsche Forschungsgemeinschaft and the Korea Science
and Engineering Foundation in the framework of the
joint KOSEF-DFG grant no. 446 KOR 113/212/0-1.
|
2,877,628,089,049 | arxiv | \section{Introduction} \label{sec:introduction}
The ranking-based choice model is one of the most fundamental and influential discrete choice models in revenue management.
It is used by firms in a variety of industries (such as e-commerce and brick-and-mortar retail) to predict the demand for the firm's products as a function of the {subset} of products that the firm offers to their customers.
The popularity of the ranking-based choice model can be attributed to its generality: it can represent \emph{any} random utility maximization model, and thus encompasses many other popular discrete choice models such as the multinomial logit model \citep{blockmarschak,farias2013nonparametric}.
The ranking-based choice model posits that customers who visit the firm have preferences that are represented by randomly-chosen rankings, and, based on the subset of products offered by the firm,
each customer will purchase the product that is most preferred according to their personal ranking.
Unfortunately, estimating a ranking-based choice model from a firm's historical sales data is notoriously challenging in practice. The key issue is that the ranking-based choice model is comprised of around $n!$ parameters, where $n$ is the number of product alternatives that a firm can elect to offer to their customers. Because the number of past assortments $M$ that the firm has previously offered to their customers usually satisfies $M \ll n!$, many selections of these parameters can yield a ranking-based choice model that is consistent (has low or zero prediction error) with the historical sales data generated by the firm's past assortments. This leaves firms with the challenge of selecting \emph{which} ranking-based choice model, out of all of those that are consistent with their historical sales data, to use when making operational planning decisions.
The challenge of estimation in ranking-based choice models is particularly acute in the context of \emph{assortment planning}.
Here, the typical goal of a firm is to identify a new subset of products (referred to as an \emph{assortment}) to offer to their customers in order to increase the firm's expected revenue. Because the true relationship between assortments and expected revenue is unknown, firms will typically interpret an estimated ranking-based choice model as ``ground truth" and subsequently solve an optimization problem to find an assortment which maximizes the ``predicted" expected revenue \citep{aouad2018approximability,bertsimas2019exact,honhon2012optimal,van2015market,van2017expectation,feldman2019assortment,aouad2021assortment,desir2021mallows}.
This widely-used technique for identifying a new assortment is referred to in the revenue management literature as \emph{estimate-then-optimize}.
The {estimate-then-optimize} technique can be attractive from a computational standpoint, due to its decoupling of the combinatorial problems related to estimation and optimization. But the assortment which is optimal under one selection of a ranking-based choice model that is consistent with the historical sales data may be highly \emph{suboptimal} under another ranking-based choice model that is also consistent with the historical sales data.
This raises concerns about whether estimate-then-optimize
can be trusted to
identify a new assortment for the firm with an expected revenue that, at the very least, is no less than the firm’s highest expected revenue from their past assortments.
The real-world consequences of implementing a low-quality assortment
can be significant to firms.
For example, when the Super Fresh grocery chain stopped carrying many of their low-selling dry grocery items, ``their customers
took their business elsewhere, and the retailer entered bankruptcy" \citep{fisher2012mwhich}. Similarly, when Walmart
rolled out significant changes to their product offerings in 2008, the new assortments resulted in a steep decline in sales and damaged relationships between Walmart and their suppliers \citep{admag}. In fact, these negative outcomes were entirely avoidable, as Walmart eventually reinstated their previous assortments \citep{fisher2012mwhich}.
As these examples illustrate, it
can be imperative in high-stakes assortment planning problems to have guarantees that assortments produced by algorithms will
``first, do no harm" to the firm, following the code of ethics that is traditionally taught to medical students in the United States and United Kingdom \citep{smith2005origin}.
\begin{comment}
These concerns can ultimately cast doubts at firms
about whether they should implement recommended changes to their assortment offerings.
There is an abundance of real-world examples in which changes to assortment have had disastrous repercussions, ranging from Walmart's precipitous decline in store sales in 2008 due to reductions to their assortment offerings, to the Super Fresh grocery store chain which ``stopped carrying many of its low-selling dry grocery items to allow for an expansion of fresh offerings. But the eliminated products turned out to be essential to customers; when they couldn’t find them, they took their business elsewhere, and the retailer entered bankruptcy" \citep{admag,fisher2012mwhich}.
The reason that making changes to assortment offerings can have such disastrous outcomes is that making to changes to assortment offerings is often a non-trivial task which can involve high administrative costs, modifications to contractual obligations with suppliers, and active engagement by stakeholders across the firm ranging from executives to regional sales managers \citep{admag}.
The salient feature in all these examples is that implementing bad assortments is not easily reversable
Making changes to an assortment offering can thus become a slow process which is essentially irrevocable in the short run, thus elevating the stakes for switching to a new assortment.
For assortment optimization to have the greatest positive impact in high-stakes applications, it is thus imperative that we have algorithms which not only promise to improve the firm's expected revenue in the best case. We must also have guarantees that the assortments produced by our algorithm will ``first do no harm", following the adage taught to physicians in United States and United Kingdom medical schools \citep{smith2005origin}.
\end{comment}
With the above motivation,
we consider the following question:
{\emph{Is it possible to identify an assortment with an expected revenue that is {strictly greater} than the expected revenues of the firm's past assortments under {all} of the ranking-based choice models that are consistent with the historical sales data generated by the firm's past assortments?}}
The answer to this question can have considerable practical value to firms. Indeed,
an affirmative answer to this question implies that one can find a new assortment that can be trusted to improve the firm’s expected revenue in a way that is not exclusive to just one of the many ranking-based choice models that are consistent with the firm’s historical sales data.
A negative answer to this question is also practically useful,
as a negative answer warns firms that implementing an assortment found by estimate-then-optimize can result in lower expected revenues than maintaining the status quo.
In the rest of this work, we will refer to the question that is italicized at the beginning of this paragraph as the \emph{identification question} for ranking-based choice models.
Despite the practical value of the identification question, only limited progress has been made on answering it until now.
To the best of our knowledge, the only prior work that is closely related to the identification question is
that of \citet*[henceforth abbreviated as \citetalias{farias2013nonparametric}]{farias2013nonparametric}. \citetalias{farias2013nonparametric} presents algorithms for {computing} the worst-case expected revenue of a {fixed} assortment under all ranking-based choice models that are consistent with the historical sales data.
As we will see later on in \S\ref{sec:setting}, the problem of finding an assortment that {maximizes} this worst-case expected revenue turns out to be closely related to answering the identification question.
However, no algorithms to date have been developed in the literature for solving such \emph{robust assortment optimization} problems (\citet[p.867]{rusmevichientong2012robust}, \citet[p.8]{jagabathula2014assortment}),
and ``it is not clear how
one may formulate the problem of optimizing the worst-case revenue as an efficiently
solvable mathematical optimization formulation." \citep[p.118]{mivsic2016data}.
Nonetheless, we remark that algorithms and attractive structural results have been developed in recent years for other types of robust assortment optimization problems \citep{rusmevichientong2012robust,bertsimas2017robust,desir2019nonconvex,wang2020randomized}. In fact,
as a byproduct of answering the identification question, our work resolves the aforementioned gap in the literature by providing the first polynomial-time algorithms for solving robust assortment optimization problems under the data-driven uncertainty set proposed by \citetalias{farias2013nonparametric}.
In this work, we provide the first answers to the identification question by establishing a powerful structural result for a class of robust assortment optimization problems (Theorem~\ref{thm:main} in \S\ref{sec:characterization}).
Stated succinctly, our theorem establishes the first characterization of the structure of optimal assortments for robust assortment optimization problems under the data-driven uncertainty set proposed by \citetalias{farias2013nonparametric}. Surprisingly, our theorem reveals that the optimal assortments for this class of robust assortment optimization problems have a simple and interpretable structure that is similar to the structure of the widely-studied class of revenue-ordered assortments. This simple structure is important because it allows us to drastically reduce the number of candidate assortments that need to be checked in order to answer the identification question.
Our proof of Theorem~\ref{thm:main} is based on an elementary (yet intricate) analysis of reachability conditions for vertices in a data-driven class of directed acyclic graphs.
Using our structural result, we proceed to establish the first answers to the identification question. We begin by developing an \emph{impossibility result} for the case in which the collection of previously-offered assortments is equal to the
widely-studied class of {revenue-ordered assortments} (Theorem~\ref{thm:impossibility} in \S\ref{sec:revorder}). Specifically, when the historical sales data is generated from revenue-ordered assortments, our structural result reveals that it is {impossible} to have an affirmative answer to the identification question, regardless of the prices of the products and the observations of the historical sales data.
To examine the practical importance of such an impossibility result, we perform numerical experiments in which we assess the performance of assortments obtained using the estimate-then-optimize technique when the historical sales data is randomly generated from revenue-ordered assortments.
The results of our experiments are striking: in more than 98\% of the problem instances in which the estimate-then-optimize technique recommended a new assortment, the worst-case decline in expected revenue from implementing the new assortment (relative to the expected revenue from the best past assortment) exceeded the best-case increase in expected revenue. The difference in magnitude between the worst-case and best-case change in expected revenue from implementing the new assortment found from estimate-then-optimize is also significant; the average best-case improvement of the new assortment over the best previously-offered assortment is 6.56\%, while the average worst-case improvement of the new assortment over the best previously-offered assortment is -21.71\%.
Our results thus demonstrate that estimate-then-optimize can cause significant declines in a firm's expected revenue, even in settings in which the firm has previously implemented the celebrated and widely-recommended class of revenue-ordered assortments.
In view of this impossibility result, we next establish that affirmative answers can indeed exist to the identification question.
To establish this, we use our structural result to develop the first strongly polynomial-time algorithm for answering the identification question when the firm has offered two past assortments (Theorem~\ref{thm:two} in \S\ref{sec:twoassortments}). Our algorithm obtains answers to the identification question by reducing the robust assortment optimization problem to a sequence of minimum-cost network flow problems, which can be readily solved using off-the-shelf optimization software, and our algorithm has a total running time of $\mathcal{O}(n^5 \log (n r_n))$, where $r_n$ is the integral price of the most expensive product.
Using our algorithm in numerical experiments on randomly generated problem instances, we prove that affirmative answers to the identification question can exist even in the seemingly limited case of two previous assortments.
We refine the experimental findings to identify a simple example with four products in which the identification question has an affirmative answer.
At the same time, we show that our algorithm is practically tractable, as the algorithm can answer the identification question in less than 30 seconds for problem instances with $n=100$ products. These findings establish, for the first time, that there can exist tractable algorithms for answering the identification question which yield practical and actionable insights, even when a limited number of assortments have been offered in the past.
We conclude by investigating whether it can be computationally tractable to pursue answers to the identification question in general settings with more than two past assortments.
Our main results here are \emph{positive}; specifically, we establish the computational tractability of the identification question by developing an algorithm for answering it that runs in weakly polynomial time for any fixed number of past assortments (Theorem~\ref{thm:poly} in \S\ref{sec:fixed_dim}).
These contributions thus provide evidence that it can be possible to develop practical algorithms for answering the identification question for real-world problem instances, where the composition of products in the previously-offered assortments do not exhibit any convenient structure and where there may be no ranking-based choice models that have zero prediction error on the historical sales data.
As a byproduct of answering the identification question, we also obtain the first polynomial-time algorithm for solving robust assortment optimization problems under the data-driven uncertainty set proposed by \citetalias{farias2013nonparametric} for any fixed number of past assortments.
Our paper is organized as follows. In \S\ref{sec:setting}, we present our problem setting and formally state the identification question. In \S\ref{sec:characterization}, we develop our key structural result which characterizes the optimal solutions for robust assortment optimization problems under ranking-based choice models. In \S\ref{sec:applications}, we discuss three applications of our structural result to provide the first answers to the identification question.
In \S\ref{sec:conclusion}, we offer concluding thoughts and directions for future research. For brevity, all lengthy proofs of intermediary results are found in the appendices.
\paragraph{Notation and Terminology. }
We use $\R$ to denote the real numbers, $\R_+$ to denote the nonnegative real numbers, and $y^\intercal x$ to denote the inner product of two vectors. We use the phrase `{collection}' to refer to a set of sets. We let the set of all probability distributions which are supported on a finite set $\mathscr{A}$ be denoted by $\Delta_{\mathscr{A}} \triangleq \{ \lambda: \sum_{a \in \mathscr{A}} \lambda_a = 1, \; \lambda_a \ge 0 \; \forall a \in \mathscr{A} \}$. We assume throughout that a norm $\| \cdot \|$ is either the $\ell_1$-norm or $\ell_\infty$-norm, and so it follows that optimization problems of the form $\min_{x,y} \{ c^\intercal x + d^\intercal y \mid Ax + By \le b, \; \| y \| \le \eta \}$ can be referred to as linear optimization problems. We let $\mathbb{I} \{\cdot \}$ denote the indicator function, which equals one if $\cdot$ is true and equals zero otherwise.
\paragraph*{Code Availability.}
The code for conducting the numerical experiments in this paper is freely available and can be accessed at \url{https://github.com/brad-sturt/IdentificationQuestion}.
\begin{comment}
In this brief note, we provide answers to this open question by developing the first polynomial-time algorithms for robust assortment optimization under the ranking-based choice model.
Specifically, we first show under the uncertainty set of FJS that the structure of optimal assortments to these robust optimization problems is similar to the well-known `revenue-ordered assortment'. Using our characterization, we then establish that the robust assortment optimization problem can be solved in polynomial time for any fixed number of past assortments. This algorithm is thus suitable for applications in which the number of possible products that can be offered by a firm is large but the firm has previously engaged in limited experimentation due to risk aversion, regulatory constraints, or there are high industry costs associated with switching prices. Moreover, we show using our characterization to answer `revenue-improving' assortment can be found.
Our findings are based on intricate analysis of reachability conditions in a class of acyclic digraphs, which is of independent interest and offers new insights for the nascent but growing literature the analysis of robust optimization and nonparametric choice models.
\end{comment}
\begin{comment}
In this short note, we provide an affirmative answer to this open question by presenting an algorithm for solving the robust assortment optimization problem for the ranking-based choice models with runs in polynomial time for any fixed number of past assortments. Our algorithm is thus suitable for applications in which the number of possible products that can be offered by a firm is large but the firm has previously engaged in limited experimentation due to risk aversion, regulatory constraints, or there are high industry costs associated with experimentation. We develop this algorithm by establishing the first structural result for optimal solutions to these robust optimization problems. Specifically, through an intricate analysis of reachability conditions in a class of acyclic digraphs, we prove surprisingly that the optimal assortments to these robust optimization problems surprisingly have a simple structure which is closely related to the widely used class of `revenue-ordered assortments'. Apart from leading to theoretically efficient algorithms, this characterization of optimal solutions lends to novel managerial insights and theoretical impossibility of
\end{comment}
\begin{comment}
\subsection{Related Literature}
The robust optimization problem constructs the uncertainty set as all choice models whose marginals are known. Robust optimization problems with this property have appeared in the literature, such as paper of Correlation Robust Influence Maximization.
We should definitely show the MIP formulation of the robust assortment optimization problem, in which we add constraints for the MIPs.
\end{comment}
\begin{comment}
Customer behavior is \emph{complex}, and so retailers often use \emph{high-dimensional} models in revenue management to predict how their decisions will impact sales. High-dimensional models are particularly attractive in applications This is increasingly the case in assortment planning. An important example of this setting is found in {assortment planning}, in which retailers are tasked with choosing a subset of products to offer to the customers. A primary step in choosing the optimal assortment, i.e., the assortment which maximizes the retailer's revenue, is estimating a choice model that predict the sales which will be generated by offering a potential assortment. Retailers traditionally determined the assortment of products to offer to their customers by using an MNL model. However, extensive empirical and behavioral research has shown the value of incorporating models that account for behavioral phenomenon like substitutiability and canabilization. The results models are high-dimensional because the number of parameters is often much larger than the number of products. The value of these more accurate models is believed to have considerable financial revenue to retailers.
high-dimensional models are abundant in assortment planning, where efforts to faithfully capture behavioral phenemonon such as substitution and IIA have led firms to flexible choice models (such as neural networks) in which the number of parameters often exceeds the number of historical data.
However, it is widely known that estimating the parameters of a high-dimensional choice model from real-world sales data can lead to overfitting, in the form In the context of assortment planning, this list includes choice models which overcome the IIA assumption from the MNL model, and more recently, irrational models. In all cases, the flexibility of the choice models necessitates a high-dimension parameter space. Consequently, when firms aim to estimate the parameters of a family of choice models, there can be many models within the family that are consistent (i.e., perfectly fit) the data. This is problematic, however, in the context of assortment planning; when many models are consistent with the data, which should be used for evaluating a new assortment?
This paper takes a first step to solving the robust assortment optimization problem
In response to this problem, \cite*{farias2013nonparametric} (henceforth abbreviated as FJS) developed a methodology based on robust optimization. Given an assortment, they estimated the revenue by looking at the worst-case revenue prediction among all choice models that are consistent (i.e., have zero training error) with the historical data. Such a particular construction of the uncertainty is intuitive, but also has practical guarantees: namely, it follows that an assortment is guaranteed to yield a expected revenue that is strictly greater than the expected revenue of a firm's previously offered assortments (under the true choice model) if and only if the worst-case revenue is strictly greater than those of the previously offered assortments. Such a guarantee is important in practice because.
\cite*{farias2013nonparametric} estimate that their improved predictions can yield
In principle, finding the assortment which maximizes the firm's revenue under the worst-case choice model can be found by brute force. Farias et al \cite{farias2013nonparametric} do not provide any algorithms for finding the assortment which has the maximum revenue under the worst-case model from their uncertainty set. Such an algorithm is valuable because it would identify an assortment, if one exists, which is guaranteed to yield a greater expected revenue than the firm's previously assortments. However, solving the resulting robust assortment optimization is believed to be challenging. A major hurdle to overcoming it is that solving the robust evaluation problem is intractable in general and does not yield appear to offer desirable properties (e.g., supermodularity) which would allow for tractable optimization over the space of assortments.
In this paper, we develop algorithms for solving the robust assortment optimization problem over the uncertainty sets defined by \cite{farias2013nonparametric} by proving a powerful structure on teh space of uncertainty sets.
This paper makes the following contributions:
\begin{itemize}
\item We characterize the structure of optimal assortments for the robust assortment optimization problem.
\item To the best of our knowledge, this is the first algorithm with known running times for finding an optimal assortment to the robust assortment optimization problem with the uncertainty set, which has remained open since \cite{farias2013nonparametric}. The running time of the algorithm is polynomial in the number of products for any fixed number of past assortments.
\item When the number of past assortments is $M=2$, we refine our algorithm and show it runs in $\mathcal{O}(n^4 \log n)$ time.
\item We discuss the implications of our characterization on a stylied identification problem, in which a retailer aims to find an assortment that is guaranteed to outperform the previously offered assortments. Using our characterization, we prove that the identification problem is unsolvable if the retailer has only offered revenue-ordered assortments.
\end{itemize}
\end{comment}
\section{Problem Setting and the Identification Question} \label{sec:setting}
We adopt the perspective of a firm that must select a subset of products to offer to their customers. Let the universe of products available to the firm be denoted by $\mathcal{N} \triangleq \{1,\ldots,n\}$, where the no-purchase option is denoted by index $0$ and $\mathcal{N}_0 \triangleq \mathcal{N} \cup \{0\}$. The revenue generated by selling one unit of product $i \in \mathcal{N}$ is represented by $r_i > 0$, and the revenue associated with the no-purchase option is $r_0 = 0$.
An assortment is defined as any subset of products $S \subseteq \mathcal{N}_0$ that includes the no-purchase option, $0 \in S$, and we let $\mathcal{S} \triangleq \{ S \subseteq \mathcal{N}_0: 0 \in S \}$ denote the collection of all assortments.
We study a problem setting in which the underlying relationship between assortment and customer demand is unknown, and our only information on this relationship comes from historical sales data generated by the firm's previously-offered assortments. Let the previously-offered assortments be denoted by $\mathscr{M} \triangleq \{S_1,\ldots,S_M\} \subseteq \mathcal{S},$ and let the indices of these past assortments be denoted by $\mathcal{M} \triangleq \{1,\ldots,M\}$. Unless stated otherwise, we will make no assumptions on the mechanism by which the firm selected the assortments to offer in the past. That is, the firm could have chosen the previously-offered assortments by drawing products randomly; alternatively, the previously-offered assortments could have been chosen using managerial intuition or some other systematic approach.
We assume that the firm offered each past assortment $S_m \in \mathscr{M}$ to their customers for a sufficient duration to obtain an accurate estimate of the purchase frequencies, \emph{i.e.}, the fraction of customers $v_{m,i} \in [0,1]$ that purchase product $i \in S_m$ when offered assortment $S_m$. This historical sales data is assumed to be normalized such that $\sum_{i \in \mathcal{N}_0}v_{m,i} = 1$, and the purchase frequencies for products that are not in an assortment are defined equal to zero, that is, $v_{m,i} = 0$ for all $i \notin S_m$ and $m \in \mathcal{M}$.
A discrete choice model is a function that predicts purchase frequencies for the firm based on the assortment that the firm offers to their customers. A \emph{ranking-based choice model} is a type of choice model which is parameterized by a probability distribution $\lambda$ over the set of all distinct rankings of the products,
where a ranking refers to a one-to-one mapping of the form $\sigma: \{0,\ldots,n\} \to \{0,\ldots,n\}$. Specifically, a ranking $\sigma$ encodes a preference for product $i$ over product $j$ if and only if $\sigma(i) < \sigma(j)$. Let the set of all distinct rankings over the products be denoted by $\Sigma$, and we readily observe that the number of distinct rankings in this set satisfies $| \Sigma| = (n+1)!$. Given a probability distribution over rankings $\lambda \in \Delta_\Sigma$ and an assortment $S \in \mathcal{S}$, the prediction made by the ranking-based choice model for the purchase frequency of each product $i \in \mathcal{N}_0$ is given by
\begin{align*}
\mathscr{D}^\lambda_i(S) \triangleq \sum_{\sigma \in \Sigma} \mathbb{I} \left \{ i = \argmin_{j \in S} \sigma(j)\right \} \lambda_\sigma.
\end{align*}
It is straightforward to see from the above definition that a ranking-based choice model always satisfies the equality $\mathscr{D}^\lambda_i(S) = 0$ for all products $i$ that are not in the assortment $S$.
The predicted expected revenue for a firm that offers assortment $S$ under the ranking-based choice model with parameter $\lambda$ is given by
\begin{align*}
\mathscr{R}^{\lambda}(S) \triangleq \sum_{i \in \mathcal{N}_0} r_i \mathscr{D}^\lambda_i(S) = r^\intercal \mathscr{D}^\lambda(S).
\end{align*}
We say that a ranking-based choice model is \emph{consistent} with the historical sales data generated by the firm's previously-offered assortments if the difference between the predicted purchase frequency $\mathscr{D}^\lambda_i(S_m)$ and the historical sales data $v_{m,i}$ is small for each of the products $i \in S_m$ that were offered in each of the previously-offered assortments $m \in \mathcal{M}$. We define the set of all ranking-based choice models that are consistent with the historical sales data as
\begin{align*}
\mathcal{U} \triangleq \left \{\lambda \in \Delta_\Sigma: \quad \begin{aligned}
&\textnormal{there exists a vector } \epsilon \textnormal{ such that }\| \epsilon \| \le \eta \textnormal{ and } \\
&\mathscr{D}^\lambda_i(S_m) - v_{m,i} = \epsilon_{m,i} \textnormal{ for all } i \in S_m \text{ and } m \in \mathcal{M} \end{aligned} \right \},
\end{align*}
where the radius $\eta \ge 0$ of the set $\mathcal{U}$ is a parameter that is selected by the firm.
To develop an understanding for the above set of ranking-based choice models $\mathcal{U}$, let us consider the case in which the radius $\eta$ of the above set is equal to zero.
In that case, we observe that the above set contains exactly the probability distributions for which the corresponding ranking-based choice models have perfect accuracy on the historical sales data generated by the previously-offered assortments. In other words, if $\eta = 0$, then the set $\mathcal{U}$ is comprised of all of the probability distributions $\lambda \in \Delta_\Sigma$ that satisfy $\mathscr{D}^\lambda_i(S_m) = v_{m,i}$ for each of the products $i \in S_m$ that were offered in each of the previously-offered assortments $m \in \mathcal{M}$. From a theoretical perspective, it is known that the set $\mathcal{U}$ with $\eta = 0$ is guaranteed to be nonempty
if the firm's customers' behavior is captured by a random utility maximization model and if the historical sales data has been observed without noise; see \cite{blockmarschak}.
From a practical perspective, it can be reasonable to expect that the set $\mathcal{U}$ will be nonempty with $\eta = 0$ in problem instances in which the number of previously-offered assortments $M$ is much smaller than the number of parameters in the ranking-based choice model, $| \Sigma| = (n+1)!$.
Nonetheless, if there are no ranking-based choice models that have perfect accuracy on the historical sales data, then the radius $\eta$ can always be made sufficiently positive to ensure that the set of ranking-based choice models $\mathcal{U}$ is nonempty. For simplicity, we make the standing assumption throughout our paper that the set of ranking-based choice models $\mathcal{U}$ is nonempty for the firm's selection of the radius $\eta$.
\begin{comment}
In view of the above notation, we say that
\begin{question}
Does there exist a collection of previously-offered assortments $\mathscr{M}$, problem data $(r, v, \eta) \in \mathscr{P} (\mathscr{M})$, and assortment $S \in \mathcal{S}$ such that the inequality $\mathscr{R}^{\lambda}(S) > \max \limits_{m \in \mathcal{M}} r^\intercal v_m$ is satisfied by all ranking-based choice models $\lambda \in \mathcal{U}$ whtich are consistent with the problem data?
\end{question}
In words, this asks whether it is ever possible to find an assortment with an expected revenue which strictly improves over the previously-offered assortments. A negative answer to this question would imply that overcoming the issues of fidelity are ``hopeless".
\begin{question}
Does there exist a collection of previously-offered assortments $\mathscr{M}$ with $| \mathscr{M}| \ge 2$ such that, for all problem data $(r, v, \eta) \in \mathscr{P} (\mathscr{M})$ and assortments $S \in \mathcal{S}$, there exists a ranking-based choice model $\lambda \in \mathcal{U}$ consistent with the problem data which satisfies $\mathscr{R}^{\lambda}(S) \le \max \limits_{m \in \mathcal{M}} r^\intercal v_m$?
\end{question}
We focus on this formulation for a particular $\mathscr{M}$ to decouple it from the setting in $v_{1,0} = \cdots = v_{M,0} = 1$.
\end{comment}
The goal of the present work is to make progress on answering the \emph{identification question} for ranking-based choice models, which is stated formally as follows:
\vspace{1em}
\begin{center}
\fbox{\begin{minipage}{0.98\linewidth}
\begin{center}
\textbf{\underline{The Identification Question}}\\
\end{center}
\vspace{0.5em}
Given a collection of previously-offered assortments $\mathscr{M}$, historical sales data $v_1,\ldots,v_M$, and prices $r_1,\ldots,r_n$, is it possible to identify an assortment with an expected revenue that is {strictly greater} than the expected revenues of the firm's previously-offered assortments under {all} of the ranking-based choice models that are consistent with the historical sales data generated by the firm's previously-offered assortments? That is, does there exist an assortment $S \in \mathcal{S}$ that satisfies
\begin{align*}
\mathscr{R}^{\lambda}(S) > \max_{m \in \mathcal{M}} r^\intercal v_m \textnormal{ for all } \lambda \in \mathcal{U}? \label{line:outperform}
\end{align*}
\vspace{-1em}
\end{minipage}}
\vspace{0.5em}
\end{center}
In this work, we provide the first answers to the identification question by viewing the identification question through the lens of \emph{robust optimization}. Specifically, we consider the following robust assortment optimization problem, which seeks an assortment that maximizes the predicted expected revenue under the worst-case ranking-based choice model that is consistent with the historical sales data generated by the firm's previously-offered assortments:
\begin{equation} \tag{RO} \label{prob:robust}
\begin{aligned}
& \underset{S \in \mathcal{S}}{\textnormal{maximize}} && \min_{\lambda \in \mathcal{U}}\mathscr{R}^{\lambda}(S).
\end{aligned}
\end{equation}
Here, our interest in the robust assortment optimization problem~\eqref{prob:robust} is motivated by its relevance to the identification question. Specifically, we readily observe that there exists an affirmative answer to the identification question if and only if the optimal objective value of the robust assortment optimization problem~\eqref{prob:robust} is strictly greater than the highest expected revenue from the firm's previously-offered assortments, $\max_{m \in \mathcal{M}} r^\intercal v_m$. In the following section, we will develop a powerful structural result for the robust assortment optimization problem~\eqref{prob:robust} which will enable us to develop closed-form solutions and general algorithms for answering the identification question.
\section{Characterization of Optimal Assortments for \eqref{prob:robust}} \label{sec:characterization}
In this section, we establish the first characterization of the structure of optimal assortments for the robust assortment optimization problem~\eqref{prob:robust}.
In particular, we will show that there are optimal assortments for \eqref{prob:robust} with a simple structure that is closely related to the structure of revenue-ordered assortments. Recall the following definition of the collection of revenue-ordered assortments:
\begin{align*}
\bar{\mathcal{S}} \triangleq \left \{ S \in \mathcal{S}:\; \textnormal{if } i^* \in S \textnormal{ and } r_{i^*} < r_{i}, \textnormal{ then }i \in S\right \}.
\end{align*}
A fundamental result in the theory of assortment optimization is that revenue-ordered assortments are optimal under the multinomial logit choice model \citep{talluri2004revenue,gallego2004managing,rusmevichientong2014assortment}. Revenue-ordered assortments also have attractive approximation guarantees for assortment optimization problems under mixture-of-logits and ranking-based choice models with known parameters \citep{rusmevichientong2014assortment,aouad2018approximability,berbeglia2020assortment}. Due to their simplicity and strong theoretical and empirical performance, revenue-ordered assortments are widely recommended in the revenue management literature and used in industry.
In view of the above background,
we proceed to develop our main result regarding the structure of optimal assortments for the robust assortment optimization problem~\eqref{prob:robust}. To this end, we first define the following set of previously-offered assortments for each product $i \in \mathcal{N}_0$:
\begin{align*}
\mathcal{M}_i \triangleq \left \{ m \in \mathcal{M}: i \in S_m \right \}.
\end{align*}
In words, the above set contains all of the previously-offered assortments in which the firm offered product $i$ to their customers. In particular, we observe from this definition that the statement $\mathcal{M}_i \subseteq \mathcal{M}_j$ holds if, for all of the previously-offered assortments in which the firm offered product $i$, the firm also offered product $j$.
We now introduce the following new collection of assortments:
\begin{align*}
\widehat{\mathcal{S}} &\triangleq \left \{ S \in \mathcal{S}:\; \textnormal{if } i^* \in S, \; r_{i^*} < r_{i}, \textnormal{ and } \mathcal{M}_{i^*} \subseteq \mathcal{M}_i, \textnormal{ then } i \in S\right \}.
\end{align*}
The above collection has a natural interpretation as the collection of all assortments which, speaking informally, can be viewed as revenue-ordered \emph{relative to} the firm's previously-offered assortments. Indeed, consider two products $i^*, i \in \mathcal{N}_0$ for which the revenue $r_{i^*}$ from the first product $i^*$ is strictly less than the revenue $r_{i}$ from the second product $i$. Then every assortment $S \in \widehat{\mathcal{S}}$ which offers the first product must also offer the second product \emph{unless} there is historical sales data from a previously-offered assortment in which the first product $i^*$ was offered and the second product $i$ was not offered. Said another way, the collection $\widehat{\mathcal{S}}$ is comprised of all of the assortments which are revenue-ordered except on pairs of products in which the demand for the lower-revenue product has previously been observed independently of the demand for the higher-revenue product.
Our main result is the following:
\begin{theorem} \label{thm:main}
There exists an assortment $S \in \widehat{\mathcal{S}}$ that is optimal for \eqref{prob:robust}.
\end{theorem}
Our proof of Theorem~\ref{thm:main} is contained in the remainder of the present section in \S\ref{sec:prelim}-\S\ref{sec:proof}.
As an immediate consequence of Theorem~\ref{thm:main}, we will be able to develop exact algorithms for solving the robust assortment optimization problem~\eqref{prob:robust} that consist of optimizing over only the assortments in the collection $\widehat{\mathcal{S}}$. In \S\ref{sec:applications}, we develop and analyze the tractability of such exact algorithms and use our algorithms to develop the first answers to the identification question.
\subsection{Preliminary Steps} \label{sec:prelim}
We begin our proof of Theorem~\ref{thm:main} by discussing linear optimization-based techniques for computing the worst-case expected revenue $\min_{\lambda \in \mathcal{U}} \mathscr{R}^\lambda(S)$ corresponding to any assortment $S \in \mathcal{S}$. Following the notation from \S\ref{sec:setting}, we readily observe that the worst-case expected revenue for the fixed assortment $S \in \mathcal{S}$ is equal to the optimal objective value of the following linear optimization
problem:
\begin{align} \tag{WC-$S$} \label{prob:wc}
&\begin{aligned}
& \; \underset{\lambda, \epsilon}{\textnormal{minimize}} && \sum_{\sigma \in \Sigma} \sum_{i \in S} r_i \mathbb{I} \left \{ i = \argmin_{j \in S} \sigma(j)\right \} \lambda_\sigma \\
&\textnormal{subject to}&& \sum_{\sigma \in \Sigma}\mathbb{I} \left \{ i = \argmin_{j \in S_m} \sigma(j)\right \} \lambda_\sigma - \epsilon_{m,i} = v_{m,i} \quad \forall m \in \mathcal{M} \text{ and } i \in S_m\\
&&& \sum_{\sigma \in \Sigma} \lambda_\sigma = 1\\
&&& \| \epsilon \| \le \eta \\
&&& \lambda_{\sigma} \ge 0 \quad \forall \sigma \in \Sigma.
\end{aligned}
\end{align
\begin{comment}
Our proof of Theorem~\ref{thm:main} in the subsequent subsections will consist of a structural analysis of the linear optimization problem~\eqref{prob:wc}. To facilitate our subsequent analysis, the present subsection draws on reformulation techniques from \cite{jagabathula2019limit} to reduce \eqref{prob:wc} to a linear optimization problem with a reduced number of decision variables.
\end{comment}
As a preliminary step that will facilitate our developments in the subsequent subsections, the rest of \S\ref{sec:prelim} follows reformulation techniques from \citet[\S2]{jagabathula2019limit} to reduce \eqref{prob:wc} to a linear optimization problem with a smaller number of decision variables.
To construct our compact reformulation of the linear optimization problem~\eqref{prob:wc}, we introduce the following additional notation. For each assortment $S \in \mathcal{S}$ and each product in the assortment $i \in S$, let the set of rankings that prefer product $i$ to all other products in the assortment $S$ be defined as follows.
\begin{definition}\label{defn:D}
$\mathcal{D}_i(S) \triangleq \left \{ \sigma \in \Sigma: \; i = \argmin_{j \in S} \sigma (j) \right \}$
\end{definition}
\noindent Given the previously-offered assortments $S_1,\ldots,S_M$, we also define the following set of tuples of products.
\begin{definition} \label{defn:L}
$\mathcal{L} \triangleq \left \{(i_1,\ldots,i_M) \in S_1 \times \cdots \times S_M: \; \bigcap_{m \in \mathcal{M}} \mathcal{D}_{i_m}(S_m) \neq \emptyset \right \}$.
\end{definition}
\noindent To develop intuition for Definition~\ref{defn:L}, let us reflect on the relationship between the set of tuples of products $\mathcal{L}$ and the set of all distinct rankings $\Sigma$. Firstly, it follows immediately from Definition~\ref{defn:L} that each tuple of products $(i_1,\ldots,i_M) \in \mathcal{L}$ has at least one ranking $\sigma \in \Sigma$ that satisfies $\sigma \in \cap_{m \in \mathcal{M}} \mathcal{D}_{i_m}(S_m)$. Secondly, we show in the following lemma that each ranking $\sigma \in \Sigma$ satisfies $\sigma \in \cap_{m \in \mathcal{M}} \mathcal{D}_{i_m}(S_m)$ for exactly one tuple of products $(i_1,\ldots,i_M) \in \mathcal{L}$.
\begin{lemma} \label{lem:uniqueness}
For each $\sigma \in \Sigma$, there exists a unique $(i_1,\ldots,i_M) \in \mathcal{L}$ such that $\sigma \in \cap_{m \in \mathcal{M}} \mathcal{D}_{i_m}(S_m)$.
\end{lemma}
The third and final definition of \S\ref{sec:prelim}, which is presented below as Definition~\ref{defn:rho},
will play a significant role in our developments in the rest of \S\ref{sec:characterization}.
Specifically, the following definition introduces a quantity $\rho_{i_1 \cdots i_M}(S)$ for each assortment $S \in \mathcal{S}$ and each tuple of products $(i_1,\ldots,i_M) \in \mathcal{L}$. The quantity $\rho_{i_1 \cdots i_M}(S)$ can be understood as the minimum revenue among the products in the assortment $S \in \mathcal{S}$ that can be the most preferred product in $S$ under a ranking that corresponds to the tuple of products $(i_1,\ldots,i_M) \in \mathcal{L}$. It will become clear momentarily that the quantity $\rho_{i_1 \cdots i_M}(S)$ arises naturally when constructing our compact reformulation of the linear optimization problem~\eqref{prob:wc}.
\begin{definition} \label{defn:rho
$\rho_{i_1 \cdots i_M}(S) \triangleq \min \limits_{i \in S: \; \cap_{m \in \mathcal{M}} \mathcal{D}_{i_m}(S_m) \cap \mathcal{D}_i(S) \neq \emptyset} r_i$.
\end{definition}
Equipped with the above definitions, we are now ready to show that the linear optimization problem~\eqref{prob:wc} for computing the worst-case expected revenue of any fixed assortment $S \in \mathcal{S}$ can be reformulated as a linear optimization problem with a reduced number of decision variables. This compact reformulation of \eqref{prob:wc} is presented as \eqref{prob:robust_simplified} in the following Proposition~\ref{prop:reform_wc}.
\begin{proposition} \label{prop:reform_wc}
\eqref{prob:wc} is equivalent to the following linear optimization problem:
\begin{equation} \tag{WC'-$S$} \label{prob:robust_simplified}
\begin{aligned}
& \; \underset{\lambda, \epsilon}{\textnormal{minimize}} && \sum_{(i_1,\ldots,i_M) \in \mathcal{L}} \rho_{i_1 \cdots i_M}(S) \lambda_{i_1\cdots i_M}\\
&\textnormal{subject to}&& \sum_{(i_1,\ldots,i_M) \in \mathcal{L}: \; i_m = i} \lambda_{i_1 \cdots i_M} - \epsilon_{m,i}= v_{m,i} \quad \forall m \in \mathcal{M}, \; i \in S_m\\
&&& \sum_{(i_1,\ldots,i_M) \in \mathcal{L}} \lambda_{i_1 \cdots i_M} = 1 \\
&&& \| \epsilon \| \le \eta\\
&&& \lambda_{i_1 \cdots i_M} \ge 0 \quad \forall (i_1,\ldots,i_M) \in \mathcal{L}.
\end{aligned}
\end{equation}
\end{proposition}
Let us offer two remarks about the linear optimization problem~\eqref{prob:robust_simplified}. In particular, the following two remarks articulate the key properties of \eqref{prob:robust_simplified} that will be important to our developments in the subsequent subsections.
\begin{remark} \label{remark:rho}
The assortment $S$ does not appear in any of the constraints of \eqref{prob:robust_simplified}.
\end{remark}
\begin{remark}\label{remark:tractability}
\eqref{prob:robust_simplified} has $| \mathcal{L}| = \mathcal{O}(n^M)$ decision variables and $\mathcal{O}(nM)$ constraints.
\end{remark}
We note that the asymptotic upper bound in Remark~\ref{remark:tractability} on the number of decision variables in the linear optimization problem~\eqref{prob:robust_simplified} follows immediately from the fact that the set of tuples of products $\mathcal{L}$ is a subset of $S_1 \times \cdots \times S_M$; see Definition~\ref{defn:L}.
\begin{comment}
\section{Characterization of Optimal Assortments
In view of the simplified representation of the robust objective function in the previous section, we now turn to the main result of this paper.
In the previous subsection we showed that the robust objective function for any fixed assortment $S \in \mathcal{S}$ can be evaluated by solving a linear optimization problem with $\mathcal{O}(n^M)$ decision variables and $\mathcal{O}(nM)$ constraints. Hence, given any fixed $M$, the robust objective function can be evaluated in polynomial time with respect to $n$. We now shift to the problem of solving the outer-maximization problem in \eqref{prob:robust}.
In principle, \eqref{prob:robust} can be solved by brute-force, that is, by evaluating the robust objective function for all assortments $S \in \mathcal{S}$. However, this approach is clearly unattractive for our purposes, as the brute-force algorithm will require solving a total of $\left| \mathcal{S} \right| = 2^n$ linear optimization problems. We will get around this issue by proving the following result:
\begin{theorem}
There exists of collection of assortments $\widehat{\mathcal{S}}(\mathscr{M})$, depending only on the previously offered assortments $\mathscr{M}$, such that \textnormal{(1)} there exists $S \in \widehat{\mathcal{S}}(\mathscr{M})$ which is an optimal solution to \eqref{prob:robust} and \textnormal{(2)} we have $|\widehat{\mathcal{S}}(\mathscr{M})| \le \mathcal{O}(\textnormal{poly}(n))$ for fixed $M$.
\end{theorem}
In view of the above theorem, we immediately obtain a polynomial-time algorithm for solving \eqref{prob:robust} when $M$ is fixed. Indeed, we can solve the robust assortment optimization problem \eqref{prob:robust} by evaluating the robust objective function for each assortment $S \in \widehat{\mathcal{S}}(\mathscr{M})$ and returning the assortment. This is formalized below:
\begin{center}
\fbox{\begin{minipage}{0.95\linewidth}
\begin{center}
\textbf{\underline{Algorithm for Solving \eqref{prob:robust}}}\\
\end{center}
\vspace{0.5em}
Evaluate the robust objective function for each assortment $S \in \widehat{\mathcal{S}}(\mathscr{M})$, and return the assortment with the highest robust objective value.
\vspace{0.5em}
\end{minipage}}
\vspace{1em}
\end{center}
The running time of the above algorithm is polynomial in $n$, as it requires solving a polynomial number of polynomially-sized linear optimization problems.
In the remaining subsections, we develop this algorithm.
\end{comment}
\subsection{A Graphical Interpretation of Definition~\ref{defn:rho}}
\label{sec:graphical}
Equipped with
Proposition~\ref{prop:reform_wc}, we now describe our overarching strategy for our proof of Theorem~\ref{thm:main}. In a nutshell, our proof of Theorem~\ref{thm:main} will follow an exchange argument. For every arbitrary assortment $S \in \mathcal{S}$, we will show that we can construct an assortment ${S}' \in \widehat{\mathcal{S}}$ that
satisfies $\rho_{i_1 \cdots i_M}(S) \le \rho_{i_1 \cdots i_M}(S')$ for all tuples of products $(i_1,\ldots,i_M) \in \mathcal{L}$. By showing this, it will follow readily from Proposition~\ref{prop:reform_wc} and Remark~\ref{remark:rho} that the worst-case expected revenue for the new assortment $\min_{\lambda \in \mathcal{U}}\mathscr{R}^{\lambda}(S')$ will never be less than the worst-case expected revenue for the original assortment $\min_{\lambda \in \mathcal{U}}\mathscr{R}^{\lambda}(S)$.
This will prove that there always exists an assortment $S' \in \widehat{\mathcal{S}}$ that is optimal for the robust assortment optimization problem~\eqref{prob:robust}.
In view of our overarching strategy for the proof of Theorem~\ref{thm:main},
we now proceed to analyze the behavior of the functions $S \mapsto \rho_{i_1 \cdots i_M}(S)$. In particular, we will show in the rest of \S\ref{sec:graphical} that
$\rho_{i_1 \cdots i_M}(S)$ can be computed by analyzing the {reachability} of vertices in a directed acyclic graph. In the subsequent \S\ref{sec:proof}, we will use this graphical interpretation of Definition~\ref{defn:rho} to show for every arbitrary assortment $S \in \mathcal{S}$ that we can construct an assortment ${S}' \in \widehat{\mathcal{S}}$ that
satisfies $\rho_{i_1 \cdots i_M}(S) \le \rho_{i_1 \cdots i_M}(S')$ for all tuples of products $(i_1,\ldots,i_M) \in \mathcal{L}$, thereby completing the proof of Theorem~\ref{thm:main}.
To develop our alternative representation of $\rho_{i_1 \cdots i_M}(S)$, we begin by showing that the set of tuples of products $\mathcal{L}$ can be interpreted as a set of {directed acyclic graphs}.
Indeed, consider any selection of products from each of the previously-offered assortments, $(i_1,\ldots,i_M) \in S_1 \times \cdots \times S_M$. From this tuple of products, we will construct a directed graph, denoted by $\mathcal{G}_{i_1 \cdots i_M}$, in which the set of vertices in the graph is equal to $\mathcal{N}_0$, and the graph has a directed edge $(i,i_m)$ from vertex $i$ to vertex $i_m$ for each previously-offered assortment $m \in \mathcal{M}$ and each product $i \in S_m \setminus \{i_m\}$.
In Figure~\ref{fig:exhaustive}, we present visualizations of the directed graphs generated by this construction procedure. In the first intermediary result of this subsection, presented below as Lemma~\ref{lem:dag}, we show that the tuple of products $(i_1,\ldots,i_M)$ is an element of $\mathcal{L}$ if and only if the directed graph $\mathcal{G}_{i_1 \cdots i_M}$ is acyclic.
\begin{lemma} \label{lem:dag}
$(i_1,\ldots,i_M) \in \mathcal{L}$ if and only if $\mathcal{G}_{i_1 \cdots i_M}$ is acyclic
\end{lemma}
\begin{figure}[t]
\centering
\FIGURE{
\begin{minipage}{\linewidth}
\centering
\vspace{0.5em}
\subfloat[$(i_1,i_2,i_3) = (0,0,0)$]{%
\begin{tikzpicture}
\tikzset{vertex/.style = {shape=circle,draw,minimum size=1.5em}}
\tikzset{edge/.style = {->,> = latex'}}
\node[vertex] (b) at (0,0) {1};
\node[vertex] (a) at (1.4,2.8) {0};
\node[vertex] (c) at (2.8,0) {2};
\draw [-{Stealth[scale=1.25]}] (b) -- (a);
\draw [-{Stealth[scale=1.25]}] (c) -- (a);
\end{tikzpicture}%
}\qquad\quad
\subfloat[$(i_1,i_2,i_3) = (1,1,0)$]{%
\begin{tikzpicture}
\tikzset{vertex/.style = {shape=circle,draw,minimum size=1.5em}}
\tikzset{edge/.style = {->,> = latex'}}
\node[vertex] (b) at (0,0) {1};
\node[vertex] (a) at (1.4,2.8) {0};
\node[vertex] (c) at (2.8,0) {2};
\draw [-{Stealth[scale=1.25]}] (a) -- (b);
\draw [-{Stealth[scale=1.25]}] (c) -- (b);
\draw [-{Stealth[scale=1.25]}] (c) -- (a);
\end{tikzpicture}
}\qquad \quad
\subfloat[$(i_1,i_2,i_3) = (1,1,2)$]{%
\begin{tikzpicture}
\tikzset{vertex/.style = {shape=circle,draw,minimum size=1.5em}}
\tikzset{edge/.style = {->,> = latex'}}
\node[vertex] (b) at (0,0) {1};
\node[vertex] (a) at (1.4,2.8) {0};
\node[vertex] (c) at (2.8,0) {2};
\draw [-{Stealth[scale=1.25]}] (a) -- (b);
\draw [-{Stealth[scale=1.25]}] (c) -- (b);
\draw [-{Stealth[scale=1.25]}] (a) -- (c);
\end{tikzpicture}
}\\
\subfloat[$(i_1,i_2,i_3) = (2,0,2)$]{%
\begin{tikzpicture}
\tikzset{vertex/.style = {shape=circle,draw,minimum size=1.5em}}
\tikzset{edge/.style = {->,> = latex'}}
\node[vertex] (b) at (0,0) {1};
\node[vertex] (a) at (1.4,2.8) {0};
\node[vertex] (c) at (2.8,0) {2};
\draw [-{Stealth[scale=1.25]}] (a) -- (c);
\draw [-{Stealth[scale=1.25]}] (b) -- (c);
\draw [-{Stealth[scale=1.25]}] (b) -- (a);
\end{tikzpicture}
}\qquad \quad
\subfloat[$(i_1,i_2,i_3) = (2,1,2)$]{%
\begin{tikzpicture}
\tikzset{vertex/.style = {shape=circle,draw,minimum size=1.5em}}
\tikzset{edge/.style = {->,> = latex'}}
\node[vertex] (b) at (0,0) {1};
\node[vertex] (a) at (1.4,2.8) {0};
\node[vertex] (c) at (2.8,0) {2};
\draw [-{Stealth[scale=1.25]}] (a) -- (c);
\draw [-{Stealth[scale=1.25]}] (b) -- (c);
\draw [-{Stealth[scale=1.25]}] (a) -- (b);
\end{tikzpicture}
}
\vspace{0.5em}
\end{minipage}}
{Visualizations of directed graphs $\mathcal{G}_{i_1 \cdots i_M}$ corresponding to tuples of products $(i_1,\ldots,i_M) \in \mathcal{L}$.\label{fig:exhaustive}}
{Each of the five figures presents a visualization of the directed graph $\mathcal{G}_{i_1 i_2 i_3}$ corresponding to a tuple of products $(i_1,i_2,i_3) \in \mathcal{L}$ in the case where there are $M=3$ previously-offered assortments of the form $S_1 = \{0,1,2\}$, $S_2 = \{0,1\}$, and $S_3 = \{0,2\}$. We observe that there exists an incoming edge to vertex $i$ if and only if there exists a previously-offered assortment $m \in \{1,2,3\}$ that satisfies $i = i_m$.}
\end{figure}
We next introduce the definition of \emph{reachability} for vertices in the directed graph $\mathcal{G}_{i_1 \cdots i_M}$.
In particular, the following definition is standard in the study of directed graphs, and we will make extensive use of Definition~\ref{def:reachable} throughout the remainder of \S\ref{sec:characterization}.
\begin{definition} \label{def:reachable}
Let $(i_1,\ldots,i_M) \in S_1 \times \cdots \times S_M$ and $i, j \in \mathcal{N}_0$. We say that vertex $j$ is \emph{reachable} from vertex $i$, denoted by $j \prec_{i_1 \cdots i_M} i$, if there exists a directed path in $\mathcal{G}_{i_1 \cdots i_M}$ from $i$ to $j$.
\end{definition}
We adopt the convention throughout this paper that a directed path must contain at least one directed edge. Hence, we observe that if the directed graph $\mathcal{G}_{i_1 \cdots i_M}$ is acyclic, then it must be the case that a vertex may never be reachable from itself, that is, $i \nprec_{i_1 \cdots i_M} i$ for all $i \in \mathcal{N}_0$.
We now use Definition~\ref{def:reachable} to develop several intermediary results regarding the structure of the directed acyclic graph $\mathcal{G}_{i_1 \cdots i_M}$ for each tuple of products $(i_1,\ldots,i_M) \in \mathcal{L}$. We begin with a simple intermediary result, denoted below by Lemma~\ref{lem:reachable_im}, in which we characterize the vertices in the directed acyclic graph $\mathcal{G}_{i_1 \cdots i_M}$ that can be reachable from other vertices.
\begin{lemma}\label{lem:reachable_im}
Let $(i_1,\ldots,i_M) \in \mathcal{L}$ and $i,j \in \mathcal{N}_0$. If $j \prec_{i_1 \cdots i_M} i$, then there exists a previously-offered assortment $m \in \mathcal{M}$ that satisfies $j = i_m$.
\end{lemma}
\begin{proof}{Proof.}
Consider any tuple of products $(i_1,\ldots,i_M) \in \mathcal{L}$.
We recall from our construction of $\mathcal{G}_{i_1 \cdots i_M}$ that $i_1,\ldots,i_M$ are the only vertices in $\mathcal{G}_{i_1 \cdots i_M}$ that have incoming edges.
Since Lemma~\ref{lem:dag} implies that $\mathcal{G}_{i_1 \cdots i_M}$ is acyclic, we conclude that $i_1,\ldots,i_M$ are the only vertices in $\mathcal{G}_{i_1 \cdots i_M}$ that can be reachable from other vertices. \halmos
\end{proof}
In our next intermediary result, denoted by Lemma~\ref{lem:reachable}, we relate the reachability of vertices in a directed acyclic graph $\mathcal{G}_{i_1 \cdots i_M}$ to the set of rankings that correspond to $(i_1,\ldots,i_M) \in \mathcal{L}$.
\begin{lemma} \label{lem:reachable}
Let $(i_1,\ldots,i_M) \in \mathcal{L}$ and $i,j \in \mathcal{N}_0$. Then, there exists a ranking $\sigma \in \cap_{m \in \mathcal{M}} \mathcal{D}_{i_m}(S_m)$ that satisfies $\sigma(i) < \sigma(j)$ if and only if $j \nprec_{i_1 \cdots i_M} i$.
\end{lemma}
Intuitively, Lemma~\ref{lem:reachable} shows that the reachability of vertices in a directed acyclic graph $\mathcal{G}_{i_1 \cdots i_M}$ provides an encoding of the rankings that correspond to a tuple of products $(i_1,\ldots,i_M) \in \mathcal{L}$. Said another way, Lemma~\ref{lem:reachable} implies that if vertex $i$ has a directed path to vertex $j$ in a directed acyclic graph $\mathcal{G}_{i_1 \cdots i_M}$, then product $j$ is always preferred to product $i$ under all rankings that correspond to the tuple of products $(i_1,\ldots,i_M) \in \mathcal{L}$.
In our final intermediary result in \S\ref{sec:graphical}, denoted below by Lemma~\ref{lem:reachable_v2}, we develop a generalization of Lemma~\ref{lem:reachable} that relates the reachability of vertices in a directed acyclic graph $\mathcal{G}_{i_1 \cdots i_M}$ to the most preferred products in an assortment $S \in \mathcal{S}$.
\begin{lemma} \label{lem:reachable_v2}
Let $(i_1,\ldots,i_M) \in \mathcal{L}$, $S \in \mathcal{S}$, and $i \in S$. Then, there exists a ranking $\sigma \in \cap_{m \in \mathcal{M}} \mathcal{D}_{i_m}(S_m)$ that satisfies $i = \argmin_{j \in S} \sigma(j)$
if and only if $j \nprec_{i_1 \cdots i_M} i$ for all $j \in S$.
\end{lemma}
Lemma~\ref{lem:reachable_v2} establishes that a product $i \in S$ is the most preferred product from assortment $S \in \mathcal{S}$ under a ranking that corresponds to the tuple of products $(i_1,\ldots,i_M) \in \mathcal{L}$ if and only if there does not exist a directed path in the directed acyclic graph $\mathcal{G}_{i_1 \cdots i_M}$ from vertex $i$ to any vertex $j$ that satisfies $j \in S$. In other words, Lemma~\ref{lem:reachable_v2} implies that the set $\cap_{m \in \mathcal{M}} \mathcal{D}_{i_m}(S_m) \cap \mathcal{D}_i(S)$ is nonempty if and only if there is no vertex $j \in S$ which is reachable from vertex $i$.
In view of the above, we are now ready to develop our graphical interpretation of $\rho_{i_1 \cdots i_M}(S)$. This interpretation of $\rho_{i_1 \cdots i_M}(S)$, which is presented below in Proposition~\ref{prop:cost_reform}, will be instrumental to our analysis in the subsequent \S\ref{sec:proof}, where we will use this interpretation to analyze the behavior of the functions $S \mapsto \rho_{i_1 \cdots i_M}(S)$ for each tuple of products $(i_1,\ldots,i_M) \in \mathcal{L}$. Our graphical representation of $\rho_{i_1 \cdots i_M}(S)$ requires the following definition of the set $\mathcal{I}_{i_1 \cdots i_M}(S)$, which can be interpreted as the set of all vertices $i \in \mathcal{N}_0$ in the directed graph $\mathcal{G}_{i_1 \cdots i_M}$ that do not have a directed path to any of the vertices $i_1,\ldots,i_M$ that are elements of the assortment $S$.
\begin{definition} \label{defn:I}
$\mathcal{I}_{i_1 \cdots i_M}(S) \triangleq \left \{ i \in \mathcal{N}_0:\; \textnormal{for all } m \in \mathcal{M}, \; \textnormal{if } i_m \in S, \text{ then } i_m \nprec_{i_1 \cdots i_M} i \right \}$.
\end{definition}
To make sense of Definition~\ref{defn:I}, we recall from Lemma~\ref{lem:reachable_im} that a vertex $j$ in a directed acyclic graph $\mathcal{G}_{i_1 \cdots i_M}$ can be reachable from another vertex only if $j = i_m$ for some previously-offered assortment $m \in \mathcal{M}$. Therefore, it follows immediately from Lemma~\ref{lem:reachable_v2}
that $S \cap \mathcal{I}_{i_1 \cdots i_M}(S)$ is the set of products $i$ for which the set of rankings $\cap_{m \in \mathcal{M}} \mathcal{D}_{i_m}(S_m) \cap \mathcal{D}_i(S)$ is nonempty. Combining this with Definition~\ref{defn:rho}, we have concluded the proof of the following Proposition~\ref{prop:cost_reform}, which establishes our
graphical interpretation of $\rho_{i_1 \cdots i_M}(S)$.
\begin{proposition} \label{prop:cost_reform}
For all $S \in \mathcal{S}$ and $(i_1,\ldots,i_M) \in \mathcal{L}$,
$\rho_{i_1 \cdots i_M}(S) = \min_{i \in S \cap \mathcal{I}_{i_1 \cdots i_M}(S)} r_i.$
\end{proposition}
\subsection{Proof of Theorem~\ref{thm:main}} \label{sec:proof}
Equipped with
Propositions~\ref{prop:reform_wc} and \ref{prop:cost_reform}, we are now ready to present our proof of Theorem~\ref{thm:main}. In view of our overarching strategy outlined in the beginning of \S\ref{sec:graphical}, the main remaining step in the proof of Theorem~\ref{thm:main} is showing for every assortment $S \in \mathcal{S}$ that we can construct an assortment ${S}' \in \widehat{\mathcal{S}}$ that
satisfies $\rho_{i_1 \cdots i_M}(S) \le \rho_{i_1 \cdots i_M}(S')$ for each $(i_1,\ldots,i_M) \in \mathcal{L}$.
To show this, we begin by developing an intermediary result, denoted below by Lemma~\ref{lem:add_one}, that will allow us to compare the values of $\rho_{i_1 \cdots i_M}(S)$ and $\rho_{i_1 \cdots i_M}(S \cup \{i\})$ for every assortment $S \in \mathcal{S}$ and every product $i$ which is not in the assortment.
\begin{lemma} \label{lem:add_one}
For all $S \in \mathcal{S}$, $(i_1,\ldots,i_M) \in \mathcal{L}$, and $i \notin S$,
\begin{align*}
&\rho_{i_1 \cdots i_M}(S \cup \{i\}) = \begin{cases}
\rho_{i_1 \cdots i_M}(S),&\textnormal{if } i \notin \mathcal{I}_{i_1 \cdots i_M}(S),\\
\min \left \{ \min \limits_{j \in S \cap \mathcal{I}_{i_1 \cdots i_M}(S) \cap \left\{ j' \in \mathcal{N}_0: i \nprec_{i_1 \cdots i_M} j' \right\}} r_j, r_i \right \}, &\textnormal{if } i \in \mathcal{I}_{i_1 \cdots i_M}(S).\\
\end{cases}
\end{align*}
\end{lemma}
Using the above intermediary result, we show in the following Lemma~\ref{lem:plus_one_inequality} and Proposition~\ref{prop:plus_one_inequality_prop} that for each assortment $S \in \mathcal{S}$, we can construct an assortment $S' \in \widehat{\mathcal{S}}$ that
satisfies $\rho_{i_1 \cdots i_M}(S) \le \rho_{i_1 \cdots i_M}(S')$ for all tuples of products $(i_1,\ldots,i_M) \in \mathcal{L}$.
\begin{lemma}\label{lem:plus_one_inequality}
Let $S \in \mathcal{S}$ and $i \notin S$. If there exists $i^* \in S$ which satisfies $r_{i^*} < r_i$ and $\mathcal{M}_{i^*} \subseteq \mathcal{M}_i$, then $\rho_{i_1 \cdots i_M}(S) \le \rho_{i_1 \cdots i_M}(S \cup \{i\})$ for each $(i_1,\ldots,i_M) \in \mathcal{L}$.
\end{lemma}
\begin{proposition}\label{prop:plus_one_inequality_prop}
For each assortment $S \in \mathcal{S}$, there exists an assortment $S' \in \widehat{\mathcal{S}}$ which
satisfies the inequality $\rho_{i_1 \cdots i_M}(S) \le \rho_{i_1 \cdots i_M} (S' )$ for each $(i_1,\ldots,i_M) \in \mathcal{L}$.
\end{proposition}
We now complete our proof of Theorem~\ref{thm:main} by combining Proposition~\ref{prop:reform_wc} with Proposition~\ref{prop:plus_one_inequality_prop}.
\begin{proof}{Proof of Theorem~\ref{thm:main}.}
It follows immediately from Proposition~\ref{prop:reform_wc} that
\begin{align*
\eqref{prob:robust} = \max_{S \in \mathcal{S}} \left \{ \begin{aligned}
& \; \underset{\lambda,\epsilon}{\textnormal{minimize}} && \sum_{(i_1,\ldots,i_M) \in \mathcal{L}} \rho_{i_1 \cdots i_M}(S) \lambda_{i_1\cdots i_M}\\
&\textnormal{subject to}&& \sum_{(i_1,\ldots,i_M) \in \mathcal{L}: \; i_m = i} \lambda_{i_1 \cdots i_M} - \epsilon_{m,i}= v_{m,i} && \forall m \in \mathcal{M}, \; i \in S_m\\
&&& \sum_{(i_1,\ldots,i_M) \in \mathcal{L}} \lambda_{i_1 \cdots i_M} = 1 \\
&&& \| \epsilon \| \le \eta\\
&&& \lambda_{i_1 \cdots i_M} \ge 0 && \forall (i_1,\ldots,i_M) \in \mathcal{L}
\end{aligned} \right \},
\end{align*}
where the assortment $S \in \mathcal{S}$ in the maximization problem appears only in the objective of the inner minimization problem. Since any feasible solution $\lambda$ to the inner minimization problem is nonnegative, Theorem~\ref{thm:main} follows immediately from Proposition~\ref{prop:plus_one_inequality_prop}.
\halmos \end{proof}
\section{Answers to the Identification Question} \label{sec:applications}
In this section, we use Theorem~\ref{thm:main} to develop the first answers to the identification question, both for specific collections of previously-offered assortments (\S\ref{sec:revorder} and \S\ref{sec:twoassortments}) as well as algorithms for answering the identification question in general classes of problems (\S\ref{sec:twoassortments} and \S\ref{sec:fixed_dim}). We use these findings, together with concise numerical experiments, to argue that considering the identification question can be essential for finding high-quality assortments from ranking-based choice models in high-stakes assortment planning problems.
\subsection{Impossibility Result for Revenue-Ordered Assortments} \label{sec:revorder}
For our first answer to the identification question, we return to the widely-studied class of revenue-ordered assortments that was discussed in the beginning of \S\ref{sec:characterization}. Due to their simplicity and desirable theoretical guarantees, a large body of literature has advocated to firms for offering revenue-ordered assortments across numerous application domains. Equipped with Theorem~\ref{thm:main} we now show, surprisingly, that a firm which has offered the revenue-ordered assortments has, in a rigorous sense, operated under a worst-possible behavior from the perspective of obtaining affirmative answers to the identification question. For simplicity, we assume in the following Theorem~\ref{thm:impossibility} and Corollary~\ref{cor:impossibility} that the revenues of the products are distinct and satisfy $r_1 < \cdots < r_n$.
\begin{theorem}\label{thm:impossibility}
If $\mathscr{M} = \bar{\mathcal{S}}$ and $\eta = 0$, then $\max \limits_{S \in \mathcal{S}} \min \limits_{\lambda \in \mathcal{U}} \mathscr{R}^{\lambda}(S) = \max \limits_{m \in \mathcal{M}} r^\intercal v_m$. \end{theorem}
\begin{proof}{Proof.}
Let $\mathscr{M} = \bar{\mathcal{S}}$, and let the previously-offered assortments be indexed by $\mathscr{M} = \{\bar{S}_1,\ldots,\bar{S}_n \}$, whereby the $i$-th previously-offered assortment is $\bar{S}_i \triangleq \{0,i,i+1,\ldots,n-1,n\}$.
Equipped with the above notation, we first prove that the equality
$\bar{\mathcal{S}} = \widehat{\mathcal{S}}$ holds.
Indeed, choose any arbitrary assortment $S \in \widehat{\mathcal{S}}$, and let $i^* \triangleq \argmin_{j \in S: r_j > 0} r_j$ denote the product in the chosen assortment that has the smallest nonzero revenue.
It readily follows from the facts that $\mathscr{M} = \bar{\mathcal{S}}$ and $r_1 < \cdots < r_n$ that the equalities $\mathcal{M}_{i^*} = \{ m \in \mathcal{M}: m \le i^*\}$ and $\mathcal{M}_i = \{ m \in \mathcal{M}: m \le i\}$ hold for each $i \in \{i^*+1,\ldots,n \}$. Therefore, for each $i \in \{i^*+1,\ldots,n\}$, it follows from the definition of the collection $\widehat{\mathcal{S}}$, from the fact that $r_{i^*} < r_i$, and from the fact that $\mathcal{M}_{i^*} \subseteq \mathcal{M}_i$ that $i \in S$. We have thus shown that $S = \{0, i^*,i^*+1,\ldots,n-1,n\} = \bar{S}_{i^*}$, which implies that $S \in \bar{\mathcal{S}}$. Since the assortment $S \in \widehat{\mathcal{S}}$ was chosen arbitrarily, we have shown that $\widehat{\mathcal{S}} \subseteq \bar{\mathcal{S}}$. The other direction of the proof that $\bar{\mathcal{S}} = \widehat{\mathcal{S}}$ follows from the fact that the inclusion $\mathscr{M} \subseteq \widehat{\mathcal{S}}$ always holds\footnote{To see why the inclusion $\mathscr{M} \subseteq \widehat{\mathcal{S}}$ always holds, consider any previously-offered assortment $S \in \mathscr{M}$. For each product $i^* \in S$, suppose that there exists another product $i$ which satisfies $r_{i^*} < r_{i}$ and $\mathcal{M}_{i^*} \subseteq \mathcal{M}_{i}$. Since $S \in \mathcal{M}_{i^*} \subseteq \mathcal{M}_i$, we conclude that $i \in S$ must hold, which proves that $S \in \widehat{\mathcal{S}}$. } and from the fact that $\mathscr{M} = \bar{\mathcal{S}}$. Our proof that $\bar{\mathcal{S}} = \widehat{\mathcal{S}}$ is thus complete.
Using the above result, we have
\begin{align*}
\max_{S \in \mathcal{S}} \min_{\lambda \in \mathcal{U}} \mathscr{R}^{\lambda}(S) = \max_{S \in \widehat{\mathcal{S}}} \min_{\lambda \in \mathcal{U}} \mathscr{R}^{\lambda}(S) = \max_{S \in \bar{\mathcal{S}}} \min_{\lambda \in \mathcal{U}} \mathscr{R}^{\lambda}(S) = \max_{S \in \mathscr{M}} \min_{\lambda \in \mathcal{U}} \mathscr{R}^{\lambda}(S) = \max_{m \in \mathcal{M}} r^\intercal v_m,
\end{align*}
where the first equality follows from Theorem~\ref{thm:main}, the second equality holds because $\widehat{\mathcal{S}} = \bar{\mathcal{S}}$, the third equality holds because $\mathscr{M} = \bar{\mathcal{S}}$, and the final equality follows from the construction of the set of ranking-based choice models $\mathcal{U}$ (see \S\ref{sec:setting}) and from the fact that $\eta = 0$.
\halmos \end{proof}
\begin{corollary}\label{cor:impossibility}
If $\mathscr{M} = \bar{\mathcal{S}}$ and $\eta = 0$, then for each assortment $S \in \mathcal{S}$, there exists a ranking-based choice model consistent with the historical sales data $\lambda \in \mathcal{U}$ that satisfies $ \mathscr{R}^{\lambda}(S) \le \max \limits_{m \in \mathcal{M}} r^\intercal v_m$.
\end{corollary}
\begin{proof}{Proof.}
The proof of Corollary~\ref{cor:impossibility}
follows immediately from Theorem~\ref{thm:impossibility}. \halmos
\end{proof}
Stated in words, the above Theorem~\ref{thm:impossibility} and Corollary~\ref{cor:impossibility} establish that we can never have affirmative answers to the identification question when the firm has historical sales data that is generated by the revenue-ordered assortments. That is, it is impossible using the given historical sales data to identify a new assortment with a predicted expected revenue that strictly outperforms the expected revenues of the firm's past assortments under {all} of the ranking-based choice models that are consistent with the firm's historical sales data.
To understand the practical importance of a negative answer to the identification question, let us consider the estimate-then-optimize technique discussed in \S\ref{sec:introduction}. In other words, suppose that one estimates a ranking-based choice model from the historical sales data generated by the revenue-ordered assortments and then recommends that the firm implement a new assortment
which maximizes the predicted expected revenue
under the estimated ranking-based choice model. For this setting, Theorem~\ref{thm:impossibility} and Corollary~\ref{cor:impossibility}
guarantee that this estimate-then-optimize technique will never offer \emph{fidelity} to the firm, in the sense that there will always exist a ranking-based choice model which is consistent with the historical sales data for which the expected revenue for the new assortment will be less than or equal to the expected revenue of the best previously-offered assortment. Moreover, as we will further see through the following concise yet insightful numerical experiment, the expected revenue from the assortment recommended by the estimate-then-optimize technique can be {strictly} {less} than the expected revenue of the best previously-offered assortment.
To perform our numerical experiment, we begin by constructing randomly-generated problem instances. In each problem instance, the revenues for the products are drawn from the distribution $r_1,\ldots,r_n \sim \textnormal{Uniform}[0,1]$, and a base choice for the parameters $\lambda^*$ of a ranking-based choice model is drawn uniformly over the $(n+1)!$-dimensional probability simplex
\footnote{If the rankings in $\Sigma$ are indexed by $\{\sigma_1,\ldots,\sigma_{(n+1)!}\}$, then uniform sampling over the probability simplex is obtained by drawing $u_1,\ldots,u_{(n+1)!} \sim \textnormal{Uniform}[0,1]$ and setting $\lambda_{\sigma_k} \leftarrow \log(u_k) / \sum_{k' =1}^{(n+1)!}\log(u_{k'})$ for each $k=1,\ldots,(n+1)!$. } Using this base choice for the parameters, we generate historical sales data of the form $v_1,\ldots,v_n$, where each $v_m$ is the historical sales data generated by the revenue-ordered assortment $\{0,m,m+1,\ldots,n-1,n\}$ under the ranking-based choice model with the base parameter $\lambda^*$.\footnote{We sort the products in ascending order by revenue before performing our analysis, which ensures that $\{0,m,m+1,\ldots,n\}$ for each $m \in \{1,\ldots,n\}$ is a revenue-ordered assortment.} After we compute the historical sales data, we \emph{forget} the base parameters $\lambda^*$ of the ranking-based choice model and apply the estimate-then-optimize technique to obtain a new assortment. Specifically, we first estimate the parameters $\hat{\lambda}$ of a ranking-based choice model using the historical sales data; since many selections of the parameters may be consistent with the historical sales data, we choose our estimate $\hat{\lambda}$ as the optimal solution to the linear optimization problem $\min_{\lambda \in \mathcal{U}} c^\intercal \lambda$, where the cost vector $c$ is drawn uniformly over $[0,1]^{(n+1)!}$.\footnote{Alternative approaches for estimating the parameters of a ranking-based choice model from historical sales data are provided in \cite{mivsic2016data,van2015market,van2017expectation,desir2021mallows}. Our approach of estimating the parameters as $\hat{\lambda} \in \argmin_{\lambda \in \mathcal{U}} c^\intercal \lambda$ for a randomly-chosen cost vector $c$ is motivated by our desire to decouple any potential biases associated with any particular estimation procedure from an empirical assessment of the estimate-then-optimize technique. In particular, our approach is viewed as a simple way of randomly sampling the parameters from the set of all parameters of ranking-based choice models which are consistent with the historical sales data. }\footnote{An obvious downside of our simple estimation procedure for the parameters of the ranking-based choice model is that it requires solving a linear optimization problem with $\mathcal{O}(n!)$ decision variables, and, thus, our simple estimation procedure does not scale efficiently to problem instances with many products. Nonetheless, this estimation procedure is sufficiently fast for the purposes of this numerical study, where the aim is simply to assess the performance of assortments obtained by the estimate-then-optimize technique over revenue-ordered assortments. In particular, we believe it is a reasonable assumption that similar findings from our numerical experiment with $n=4$ (see Figures~\ref{fig:roa} and \ref{fig:boxplot}) would be found in experiments with larger values of $n$.} We then obtain a new assortment $S'$ as any optimal solution to the combinatorial optimization problem $\max_{S \in \mathcal{S}} \mathscr{R}^{\hat{\lambda}}(S)$ which maximizes the predicted expected revenue under the estimated ranking-based choice model.\footnote{We solve this optimization problem using the mixed-integer linear optimization formulation given by \citet[\S3.2]{bertsimas2019exact}, which is implemented using the Julia programming language with JuMP and solved using Gurobi.} Finally, we evaluate the new assortment obtained using estimate-then-optimize by computing the worst-case expected revenue of the new assortment under all ranking-based choice models that are consistent with the historical sales data, $\min_{\lambda \in \mathcal{U}} \mathscr{R}^{{\lambda}}(S')$, the best-case expected revenue of the new assortment under all ranking-based choice models that are consistent with the historical sales data, $\max_{\lambda \in \mathcal{U}} \mathscr{R}^{{\lambda}}(S')$, and the expected revenue of the best previously-offered assortment, $\max_{m \in \mathcal{M}} r^\intercal v_m$.
\begin{figure}[t]
\centering
\FIGURE{\includegraphics[width=0.8\linewidth]{figures/revenue_ordered_assortment_prev}}
{Performance of new assortments obtained using the estimate-then-optimize technique when $\mathscr{M} = \bar{\mathcal{S}}$. \label{fig:roa} }
{Each point corresponds to a randomly-generated problem instance. The $x$-axis shows the expected revenue of the best previously-offered assortment, $\max_{m \in \mathcal{M}} r^\intercal v_m$. The $y$-axis shows the worst-case expected revenue $\min_{\lambda \in \mathcal{U}} \mathscr{R}^{\lambda}(S')$ for the new assortment $S'$ obtained using the estimate-then-optimize technique. }
\end{figure}
\begin{figure}[t]
\centering
\FIGURE{\includegraphics[width=0.8\linewidth]{figures/intervals}}
{Relative improvement in expected revenue of new assortments obtained using the estimate-then-optimize technique when $\mathscr{M} = \bar{\mathcal{S}}$. \label{fig:boxplot} }
{Each of the $x$-values corresponds to one of the 137 randomly-generated problem instances from Figure~\ref{fig:roa} for which the expected revenue of the best previously-offered assortment, $\max_{m \in \mathcal{M}} r^\intercal v_m$, was strictly greater than the worst-case expected revenue $\min_{\lambda \in \mathcal{U}} \mathscr{R}^{\lambda}(S')$ for the new assortment $S'$ obtained using the estimate-then-optimize technique. The corresponding interval of $y$-values formed by the red and blue bars is the interval $[\Pi_\lambda: \lambda \in \mathcal{U}]$, where $\Pi_\lambda \triangleq 100\% \times (\mathscr{R}^{\lambda}(S') - \max_{m \in \mathcal{M}} r^\intercal v_m) / ( \max_{m \in \mathcal{M}} r^\intercal v_m)$ is the relative percentage improvement of the expected revenue of the new assortment obtained using estimate-then-optimize over the expected revenue of the firm's best previously-offered assortment for a given ranking-based choice model $\lambda \in \mathcal{U}$. For clarity, the problem instances are sorted along the $x$-axis by the endpoints of the red bars, and the dotted line shows the reflection of the endpoints of red bars over the horizontal line at zero.
\end{figure}
In Figures~\ref{fig:roa} and \ref{fig:boxplot}, we present the results of these numerical experiments for the case of $n = 4$ products over 1000 randomly-generated problem instances. In Figure~\ref{fig:roa}, we compare the expected revenue under the best previously-offered assortment, $\max_{m \in \mathcal{M}} r^\intercal v_m$, to the expected revenue of the new assortments obtained using estimate-then-optimize under the worst-case ranking-based choice model that is consistent with the historical sales data, $\min_{\lambda \in \mathcal{U}} \mathscr{R}^{{\lambda}}(S')$. We observe that the results in Figure~\ref{fig:roa} are consistent with the impossibility result from Theorem~\ref{thm:impossibility}; indeed, the worst-case expected revenues of the new assortments obtained using estimate-then-optimize never exceed the expected revenues of the best previously-offered assortments.
Furthermore, we observe for many of the problem instances that there are ranking-based choice models $\lambda \in \mathcal{U}$ that are consistent with the historical sales data for which the resulting expected revenue of the new assortment $\mathscr{R}^{\lambda}(S')$ is {strictly less} than the expected revenue under the best previously-offered assortment, $\max_{m \in \mathcal{M}} r^\intercal v_m$.
To assess whether the above findings are overly conservative from a practical standpoint, we turn to a detailed analysis of the $137$ problem instances for which the expected revenue of the best previously-offered assortment, $\max_{m \in \mathcal{M}} r^\intercal v_m$, is strictly greater than the worst-case expected revenue $\min_{\lambda \in \mathcal{U}} \mathscr{R}^{\lambda}(S')$ for the new assortment $S'$ obtained using estimate-then-optimize. In Figure~\ref{fig:boxplot} we present, for each of these 137 problem instances, a visualization of the range of relative percentage improvements of the predicted expected revenue of the new assortment, $\mathscr{R}^{\lambda}(S')$, over the expected revenue of the firm's best previously-offered assortment, $\max_{m \in \mathcal{M}} r^\intercal v_m$, which are possible to obtain under a ranking-based choice model which is consistent with the historical sales data, $\lambda \in \mathcal{U}$. Stated more precisely, each of the $x$ values in Figure~\ref{fig:boxplot} corresponds to one of these 137 problem instances, and the corresponding interval of $y$-values formed by the red and blue bars is the interval $[\Pi_\lambda: \lambda \in \mathcal{U}]$, where $\Pi_\lambda \triangleq 100\% \times (\mathscr{R}^{\lambda}(S') - \max_{m \in \mathcal{M}} r^\intercal v_m) / ( \max_{m \in \mathcal{M}} r^\intercal v_m)$ is the relative percentage improvement of the predicted expected revenue of the new assortment $S'$ over the expected revenue of the firm's best previously-offered assortment for a given ranking-based choice model $\lambda \in \mathcal{U}$. Hence, the red bars are the negative values of $\Pi_\lambda$ that can be attained under the ranking-based choice models $\lambda \in \mathcal{U}$, and the dotted line shows the reflection of the endpoints of red bars over the horizontal line at zero.
We do not assign any likelihood to the values in each interval $[\Pi_\lambda: \lambda \in \mathcal{U}]$, as there is no information available for inferring which of the ranking-based choice models are more likely to be the `truth' among the ranking-based choice models $\lambda \in \mathcal{U}$ that are consistent with the historical sales data.
The results in Figure~\ref{fig:boxplot} reveal a striking {asymmetry} between the downside and upside of implementing a new assortment found by estimate-then-optimize.
In all but two of the 137 instances, the worst-case decline in expected revenue from implementing the new assortment exceeded in magnitude the best-case increase in expected revenue. The difference in magnitude between the downside and upside is also found to be significant: the average best-case improvement of the new assortment over the best previously-offered assortment (i.e., the average of the blue endpoints) is 6.56\%, while the average worst-case improvement of the new assortment over the best previously-offered assortment (i.e., the average of the red endpoints) is -21.71\%. These numerical findings demonstrate that the downside risks to a firm from implementing a new assortment found by estimate-then-optimize can significantly exceed the potential upsides.
In conclusion, our theoretical and numerical analysis in this subsection lead to three main takeaways.
The first takeaway is that there exist collections of previously-offered assortments in which the identification question can never have an affirmative answer. This is practically useful because if such a negative answer can be established for a problem instance (either through impossibility results like Theorem~\ref{thm:impossibility} or via general algorithms like those presented in \S\ref{sec:twoassortments} and \S\ref{sec:fixed_dim}), then a firm can be encouraged to pursue additional small-scale experiments before committing to a large-scale implementation of a new assortment.
The second takeaway is that commonly-used techniques like estimate-then-optimize can lead to a strictly worse expected revenue than those of the previously-offered assortments. In fact, the numerical results in Figures~\ref{fig:roa} and \ref{fig:boxplot} show that this decline in expected revenue can be significant and outweigh the potential upside for implementing the new assortment. The third takeaway is that all of the aforementioned issues arise when the previously-offered assortments are comprised of one of the most celebrated and widely-used classes of assortments from the literature, namely, the revenue-ordered assortments. All together, these takeaways raise concerns about whether the estimate-then-optimize technique with ranking-based choice models should be trusted in high-stakes assortment planning problems.
\subsection{Affirmative Answers for Two Assortments} \label{sec:twoassortments}
In view of the impossibility result from the previous subsection, we next turn to using Theorem~\ref{thm:main} to assess whether the identification question can ever be answered affirmatively. In the subsection, we establish that the answer to this question is \emph{yes}, and in particular, we show that such an affirmative answer can be obtained even when the firm has only offered {two} past assortments.
In order to establish these results, we begin by leveraging Theorem~\ref{thm:main} to develop the first algorithm for solving the robust assortment optimization problem~\eqref{prob:robust} with running time that is polynomial in the number of products $n$. The development of such an algorithm is important because it will allow us to establish the existence of affirmative answers to the identification question through numerical experiments. Our development in this subsection of an algorithm for the case of $M = 2$ is also important because it marks
the first step towards the development of efficient \emph{general} algorithms for answering the identification question in settings with large numbers of products and relatively few past assortments. Problem instances in which there are small numbers of previously-offered assortments and large numbers of products can arise in high-stakes applications in which changing to low-quality assortments can have significant negative consequences, and so the firm has made relatively few changes to their assortments thus far.
We will show that the algorithms from this subsection extend to general settings with several past assortments in the following \S\ref{sec:fixed_dim}.
Our polynomial-time algorithm for answering the identification question when $M = 2$ is presented in \S\ref{sec:twoassortments:algorithm}, and numerical experiments using this algorithm are found in \S\ref{sec:twoassortments:numerics}.
\subsubsection{A Strongly Polynomial-Time Algorithm for Two Assortments. } \label{sec:twoassortments:algorithm}
At a high level, our algorithm for answering the identification question in the case of $M = 2$ consists of reducing the robust assortment optimization problem~\eqref{prob:robust} to solving a sequence of {minimum-cost network flow} problems.
As discussed at the end of \S\ref{sec:setting}, an algorithm for solving the robust assortment optimization problem~\eqref{prob:robust} can be immediately used to answer the identification question, since there is an affirmative answer to the identification question if and only if the optimal objective value of the robust assortment optimization problem~\eqref{prob:robust} is strictly greater than $\max_{m \in \mathcal{M}} r^\intercal v_m$. Stated formally, the main contribution of \S\ref{sec:twoassortments:algorithm} is the following:
\begin{theorem} \label{thm:two}
If $M = 2$ and $\eta = 0$, then \eqref{prob:robust} can be solved in $\mathcal{O}(n^5 \log (n r_n))$ computation time.
\end{theorem}
In the above theorem and throughout the rest of \S\ref{sec:twoassortments:algorithm}, we assume that the revenues $r_1,\ldots,r_n $ are distinct, the products are sorted in ascending order by their revenue, and the revenues are represented as nonnegative integers. We also assume without any loss of generality
that $n \in S_1 \cap S_2$ and that $S_1 \cup S_2 = \mathcal{N}_0$.\footnote{To see why the assumption that $n \in S_1 \cap S_2$ can be made without loss of generality, suppose for the sake of developing intuition that the product $n$ is not contained in $S_1 \cap S_2$. In that case, we can create a fictitious product with index $n+1$ that is associated with any arbitrary revenue in the range $r_{n+1} = (r_n,\infty)$, and we can augment the historical sales data to be $v_1' \triangleq (v_1,0) \in \R^{n+2}_+$ and $v_2' \triangleq (v_2,0) \in \R^{n+2}_+$. It is straightforward to see that any feasible solution to \eqref{prob:robust_simplified} with the augmented historical sales data will satisfy $\lambda_{n+1,i_2} = \lambda_{i_1,n+1} = 0$ for all $i_1 \in S_1$ and $i_2 \in S_2$ that satisfy $(n+1,i_2) \in \mathcal{L}$ and $(i_1,n+1) \in \mathcal{L}$. This implies that the optimal objective value of \eqref{prob:robust_simplified} will be unchanged using the augmented historical sales data for all assortments $S \in \mathcal{S}$. Hence, we have shown that we can assume without loss of generality that the previously-offered assortments satisfy $n \in S_1 \cap S_2$. The assumption that $S_1 \cup S_2 = \mathcal{N}_0$ can be made without loss of generality due to similar reasoning.} The proof of Theorem~\ref{thm:two} is found at the end of \S\ref{sec:twoassortments:algorithm}.
To prove Theorem~\ref{thm:two}, and to develop an algorithm with the desired computation time, we begin by establishing three intermediary results. The proofs of these intermediary results, denoted by Lemmas~\ref{lem:two:S}-\ref{lem:two:flow}, are relatively straightforward and can be found in Appendix~\ref{appx:two}.
In our first intermediary result, denoted by Lemma~\ref{lem:two:S}, we develop a closed-form representation of the collection of assortments $\widehat{\mathcal{S}}$. This representation will be useful in the proof of Theorem~\ref{thm:two} because it provides an efficient procedure for iterating over the assortments in $\widehat{\mathcal{S}}$. Moreover, the following Lemma~\ref{lem:two:S} is useful because it immediately implies that the number of assortments in the collection $\widehat{\mathcal{S}}$ scales quadratically in the number of products $n$.
\begin{lemma} \label{lem:two:S}
If $M = 2$, then
\begin{align}
\widehat{\mathcal{S}} = \left \{ S \in \mathcal{S}: \quad \begin{aligned} &\textnormal{there exists }i_1 \in (S_1 \setminus S_2) \cup \{n\} \textnormal{ and } i_2 \in (S_2 \setminus S_1) \cup \{n\} \textnormal{ such that}\\
&S = \left(S_1 \cap S_2 \right) \cup \left \{j \in S_1 \setminus S_2: j \ge i_1 \right \} \cup \left \{j \in S_2 \setminus S_1: j \ge i_2 \right \}\end{aligned} \right \}. \label{line:S_two}
\end{align}
\end{lemma}
Stated concretely, Lemma~\ref{lem:two:S} shows that the assortments in the collection $\widehat{\mathcal{S}}$ can be parameterized by the pairs of products from the sets $(S_1 \setminus S_2) \cup \{n \}$ and $(S_2 \setminus S_1) \cup \{n\}$. Hence, the number of assortments in the collection $\widehat{\mathcal{S}}$ is at most $(|S_1 \setminus S_2|+1) \times (|S_2 \setminus S_1|+1) = \mathcal{O}(n^2)$.
In our second intermediary result, denoted by Lemma~\ref{lem:two:L}, we develop a closed-form representation of the set of tuples of products $\mathcal{L}$. The following representation of $\mathcal{L}$ will be useful in the proof of Theorem~\ref{thm:two} because it will allow us to show that the worst-case expected revenue $\min_{\lambda \in \mathcal{U}} \mathscr{R}^{\lambda}(S)$ for each assortment $S \in \mathcal{S}$ can be computed by solving a minimum-cost network flow problem.
\begin{lemma} \label{lem:two:L}
If $M = 2$, then
$\mathcal{L} = \left((S_1 \setminus S_2) \times S_2 \right) \cup \left( S_1 \times (S_2 \setminus S_1) \right) \cup \left\{(i,i): i \in S_1 \cap S_2 \right \}$.
\end{lemma}
In our third and final intermediary result, denoted by Lemma~\ref{lem:two:flow}, we show that the worst-case expected revenue $ \min_{\lambda \in \mathcal{U}}\mathscr{R}^{\lambda}(S)$ for each assortment $S \in {\mathcal{S}}$ can be computed by solving a minimum-cost network flow problem.
\begin{lemma} \label{lem:two:flow}
If $M = 2$ and $\eta = 0$, then the following equality holds for all assortments $S \in \mathcal{S}$:
\begin{align}
\min_{\lambda \in \mathcal{U}}\mathscr{R}^{\lambda}(S) &= \sum_{i \in S_1\cap S_2} \rho_{ii}(S) v_{1,i} \;+ \notag \\%\label{line:fun_spells_fun} \\
& \quad \quad \quad \left[\begin{aligned}
& \; \underset{\lambda}{\textnormal{minimize}} && \sum_{i_1 \in S_1 \setminus S_2} \sum_{i_2 \in S_1 \cap S_2} \rho_{i_1 i_2}(S) \lambda_{i_1i_2} \\
&&&+ \sum_{i_1 \in S_1 \setminus S_2} \sum_{i_2 \in S_2 \setminus S_1} \rho_{i_1 i_2}(S) \lambda_{i_1i_2} \\
&&&+ \sum_{i_1 \in S_1 \cap S_2} \sum_{i_2 \in S_2 \setminus S_1} ( \rho_{i_1 i_2}(S) - \rho_{i_1 i_1}(S)) \lambda_{i_1i_2} \\
&\textnormal{subject to}&& \begin{aligned}[t]
& \sum_{i_2 \in S_2} \lambda_{i_1 i_2} = v_{1,i_1} && \forall i_1 \in S_1 \setminus S_2\\
& \sum_{i_1 \in S_1} \lambda_{i_1 i_2} = v_{2,i_2} &&\forall i_2 \in S_2 \setminus S_1\\
& \sum_{i_1 \in S_1 \setminus S_2} \lambda_{i_1 i} - \sum_{i_2 \in S_2 \setminus S_1} \lambda_{i i_2} = v_{2,i} - v_{1,i} &&\forall i \in S_1 \cap S_2\\
& \lambda_{i_1 i_2} \ge 0 && \forall (i_1,i_2) \in \mathcal{L}
\end{aligned}
\end{aligned}
\right]. \label{prob:flow}
\end{align}
\end{lemma}
We readily observe that the linear optimization problem on line~\eqref{prob:flow} is a minimum-cost network flow problem, where each decision variable $\lambda_{i_1 i_2}$ corresponds to the flow on a directed edge from vertex $i_1$ to vertex $i_2$ \citep[p. 296]{ahuja1988network}. In particular, we observe that the minimum-cost network flow problem on line~\eqref{prob:flow} takes place on a {complete tripartite directed acyclic graph} with $|S_1 \setminus S_2| + |S_2 \setminus S_1| + |S_1 \cap S_2| = n+1$ vertices and $|S_1 \setminus S_2|\times |S_2| + |S_2 \setminus S_1| \times |S_1 \cap S_2| = \mathcal{O}(n^2)$ directed edges. In Figure~\ref{fig:network_flow}, we present a visualization of the network corresponding to the minimum-cost network flow problem from line~\eqref{prob:flow}.
\begin{figure}[t]
\centering
\FIGURE{%
\begin{tikzpicture}
\tikzset{vertex/.style = {shape=circle,draw,minimum size=1.5em}}
\tikzset{edge/.style = {->,> = latex'}}
\node[vertex] (1) at (-0.25*360/6: 3cm) {1};
\node[vertex] (0) at (1.25*360/6: 3cm) {0};
\node[vertex] (5) at (1.75*360/6: 3cm) {5};
\node[vertex] (4) at (3.25*360/6: 3cm) {4};
\node[vertex] (3) at (3.75*360/6: 3cm) {3};
\node[vertex] (2) at (5.25*360/6: 3cm) {2};
\node[ellipse,rotate=0*360/6,line width=0.5mm,draw = red,dotted,minimum width = 3.5cm, minimum height = 2cm] (e) at (1.5*360/6: 2.897cm) {};
\node[ellipse,rotate=360/6,draw = ForestGreen,line width=0.5mm,dotted,minimum width = 3.5cm, minimum height = 2cm] (e) at (-0.5*360/6: 2.897cm) {};
\node[ellipse,rotate=2*360/6,draw = blue,line width=0.5mm,dotted,minimum width = 3.5cm, minimum height = 2cm] (e) at (3.5*360/6: 2.897cm) {};
\node[text width=3cm] at (1.5*360/6: 4.25cm) {}
\draw [-{Stealth[scale=1.25]}] (0) -- (4);
\draw [-{Stealth[scale=1.25]}] (0) -- (3);
\draw [-{Stealth[scale=1.25]}] (5) -- (3);
\draw [-{Stealth[scale=1.25]}] (5) -- (4);
\draw [-{Stealth[scale=1.25]}] (1) -- (0);
\draw [-{Stealth[scale=1.25]}] (1) -- (3);
\draw [-{Stealth[scale=1.25]}] (1) -- (4);
\draw [-{Stealth[scale=1.25]}] (1) -- (5);
\draw [-{Stealth[scale=1.25]}] (2) -- (0);
\draw [-{Stealth[scale=1.25]}] (2) -- (3);
\draw [-{Stealth[scale=1.25]}] (2) -- (4);
\draw [-{Stealth[scale=1.25]}] (2) -- (5);
\end{tikzpicture}
}
{Visualization of minimum-cost network flow problem from line~\eqref{prob:flow}.\label{fig:network_flow}}
{The figure shows a visualization of the minimum-cost network flow problem corresponding to the linear optimization problem on line~\eqref{prob:flow} for the case where the previously-offered assortments are $S_1 = \{0,1,2,5\}$ and $S_2 = \{0,3,4,5\}$. We see that there is a vertex in the graph for each product $i \in \mathcal{N}_0 \equiv \{0,1,2,3,4,5\}$. The graph is a complete tripartite directed graph, where the three partitions of vertices are denoted by the dotted ellipses and correspond to $S_1 \setminus S_2 = \{1,2\}$, $S_1 \cap S_2 = \{0,5\}$, and $S_2 \setminus S_1 = \{3,4\}$. The flow demands at each of the vertices and the flow cost for each of the directed edges can be found on line~\eqref{prob:flow}. }
\end{figure}
Using the above three intermediary results, we conclude our proof of Theorem~\ref{thm:two} by presenting an algorithm for solving the robust assortment optimization problem~\eqref{prob:robust} and establishing its running time.
\begin{proof}{Proof of Theorem~\ref{thm:two}.}
We first describe our algorithm for solving the robust assortment optimization problem~\eqref{prob:robust}, which follows a brute-force strategy. Namely, our algorithm iterates over each of the assortments $S \in \widehat{\mathcal{S}}$, and, for each such assortment, the algorithm computes the corresponding worst-case expected revenue $\min_{\lambda \in \mathcal{U}}\mathscr{R}^{\lambda}(S)$. The algorithm concludes by returning the maximum value of $\min_{\lambda \in \mathcal{U}}\mathscr{R}^{\lambda}(S)$ across the assortments $S \in \widehat{\mathcal{S}}$. The correctness of this algorithm for solving the robust assortment optimization problem~\eqref{prob:robust} follows immediately from Theorem~\ref{thm:main}.
We now analyze the running time of our algorithm by using our three intermediary results.
We assume that the two assortments $S_1$ and $S_2$ are given as sorted arrays. Under this assumption,
it is straightforward to see that the sets $S_1 \cap S_2$, $S_1 \setminus S_2$, and $S_2 \setminus S_1$ can be computed and stored as sorted arrays in $\mathcal{O}(n)$ computation time. We also require $\mathcal{O}(n)$ computation time to store copies of the sets $S_1,S_2,S_1 \cap S_2$, $S_1 \setminus S_2$, and $S_2 \setminus S_1$ in hash tables, which ensures that querying whether a given product is an element of any of these sets can be performed in $\mathcal{O}(1)$ time.
We next analyze the computation times for constructing the collection of assortments $\widehat{\mathcal{S}}$, constructing the set of pairs of products $\mathcal{L}$, and computing the quantities $\rho_{i_1 i_2}(S)$ for each assortment $S \in \widehat{\mathcal{S}}$ and each pair of products $(i_1,i_2) \in \mathcal{L}$. Indeed, using the aforementioned data structures, it follows readily from
Lemma~\ref{lem:two:S} that we can construct the collection of assortments $\widehat{\mathcal{S}}$ in $\mathcal{O}(n^3)$ computation time.\footnote{It follows from Lemma~\ref{lem:two:S} that we can efficiently iterate over the assortments in $\widehat{\mathcal{S}}$ by iterating over the pairs of products in $(S_1 \setminus S_2) \cup \{n\}$ and $(S_2 \setminus S_1) \cup \{n\}$. Constructing the collection $\widehat{\mathcal{S}}$ thus requires iterating over the $(|S_1 \setminus S_2|+1) \times (|S_2 \setminus S_1|+1) = \mathcal{O}(n^2)$ assortments, and each of the assortments is comprised of at most $\mathcal{O}(n)$ products. }
Moreover, it follows from
Lemma~\ref{lem:two:L} that the set of pairs of products $\mathcal{L}$ can be computed in $\mathcal{O}(n^2)$ time. Finally, we analyze the computation times for computing the quantities $\rho_{i_1 i_2}(S)$ for each assortment $S \in \widehat{\mathcal{S}}$ and each pair of products $(i_1,i_2) \in \mathcal{L}$. Indeed, we recall from Lemma~\ref{lem:two:S} that $| \widehat{\mathcal{S}}| = \mathcal{O}(n^2)$, and we recall from Lemma~\ref{lem:two:L} that $|\mathcal{L}| = \mathcal{O}(n^2)$. Therefore, there are $| \widehat{\mathcal{S}}| \times |\mathcal{L}| = \mathcal{O}(n^4)$ different ways of choosing an assortment $S \in \widehat{\mathcal{S}}$ and a pair of products $(i_1,i_2) \in \mathcal{L}$. For each assortment $S \in {\mathcal{S}}$ and pair of products $(i_1,i_2) \in \mathcal{L}$, it follows readily from Definition~\ref{defn:rho} and Lemmas~\ref{lem:two:S} and \ref{lem:two:L} that
\begin{align*}
\rho_{i_1 i_2}(S) &= \begin{cases}
r_{i_1},&\text{if } {\color{black}i_1, i_2 \in S_1 \cap S_2}, \; i_1 = i_2, \textnormal{ and } i_1 \in S, \\
0,&\text{if } {\color{black}i_1, i_2 \in S_1 \cap S_2}, \; i_1 = i_2, \textnormal{ and } i_1 \notin S, \\
\\[-0.5em]
r_{i_2},&\text{if } {\color{black} i_1 \in S_1 \cap S_2,\; i_2 \in S_2 \setminus S_1}, \textnormal{ and } i_2 \in S,\\
\min \left \{ r_{i_1}, \min_{j \in S \cap S_2 \setminus S_1}r_{j} \right \},&\text{if } {\color{black} i_1 \in S_1 \cap S_2,\; i_2 \in S_2 \setminus S_1},\; i_2 \notin S, \textnormal{ and } i_1 \in S, \\
0,&\text{if } {\color{black} i_1 \in S_1 \cap S_2,\; i_2 \in S_2 \setminus S_1},\; i_2 \notin S, \textnormal{ and } i_1 \notin S, \\
\\[-0.5em]
r_{i_1},&\text{if }{\color{black} i_1 \in S_1 \setminus S_2,\; i_2 \in S_1 \cap S_2}, \textnormal{ and } i_1 \in S,\\
\min \left \{ r_{i_2}, \min_{j \in S \cap S_1 \setminus S_2}r_{j} \right \},&\text{if }{\color{black} i_1 \in S_1 \setminus S_2,\; i_2 \in S_1 \cap S_2},\; i_1 \notin S, \textnormal{ and } i_2 \in S, \\
0,&\text{if }{\color{black} i_1 \in S_1 \setminus S_2,\; i_2 \in S_1 \cap S_2},\; i_1 \notin S, \textnormal{ and } i_2 \notin S, \\
\\[-0.5em]
\min \left \{ r_{i_1},r_{i_2} \right \}, &\text{if }{\color{black}i_1 \in S_1 \setminus S_2, \; i_2 \in S_2 \setminus S_1},\; i_1 \in S, \text{ and } i_2 \in S,\\
\min \left \{ r_{i_1}, \min_{j \in S \cap S_2 \setminus S_1}r_{j} \right \},&\text{if }{\color{black}i_1 \in S_1 \setminus S_2, \; i_2 \in S_2 \setminus S_1},\; i_1 \in S, \text{ and } i_2 \notin S,\\
\min \left \{ r_{i_2}, \min_{j \in S \cap S_1 \setminus S_2}r_{j} \right \},&\text{if }{\color{black}i_1 \in S_1 \setminus S_2, \; i_2 \in S_2 \setminus S_1},\; i_1 \notin S, \text{ and } i_2 \in S,\\
0,&\text{if }{\color{black}i_1 \in S_1 \setminus S_2, \; i_2 \in S_2 \setminus S_1},\; i_1 \notin S, \text{ and } i_2 \notin S.
\end{cases}
\end{align*}
\begin{comment}
\begin{align*}
\rho_{i_1 i_2}(S) &= \begin{cases}
r_{i_1},&\text{if } {\color{blue}i_1, i_2 \in S_1 \cap S_2}, \; i_1 = i_2, \textnormal{ and } i_1 \in S, \\
0,&\text{if } {\color{blue}i_1, i_2 \in S_1 \cap S_2}, \; i_1 = i_2, \textnormal{ and } i_1 \notin S, \\
\\[-0.5em]
r_{i_2},&\text{if } {\color{ForestGreen} i_1 \in S_1 \cap S_2,\; i_2 \in S_2 \setminus S_1}, \textnormal{ and } i_2 \in S,\\
\min \left \{ r_{i_1}, \min_{j \in S \cap S_2 \setminus S_1}r_{j} \right \},&\text{if } {\color{ForestGreen} i_1 \in S_1 \cap S_2,\; i_2 \in S_2 \setminus S_1},\; i_2 \notin S, \textnormal{ and } i_1 \in S, \\
0,&\text{if } {\color{ForestGreen} i_1 \in S_1 \cap S_2,\; i_2 \in S_2 \setminus S_1},\; i_2 \notin S, \textnormal{ and } i_1 \notin S, \\
\\[-0.5em]
r_{i_1},&\text{if }{\color{red} i_1 \in S_1 \setminus S_2,\; i_2 \in S_1 \cap S_2}, \textnormal{ and } i_1 \in S,\\
\min \left \{ r_{i_2}, \min_{j \in S \cap S_1 \setminus S_2}r_{j} \right \},&\text{if }{\color{red} i_1 \in S_1 \setminus S_2,\; i_2 \in S_1 \cap S_2},\; i_1 \notin S, \textnormal{ and } i_2 \in S, \\
0,&\text{if }{\color{red} i_1 \in S_1 \setminus S_2,\; i_2 \in S_1 \cap S_2},\; i_1 \notin S, \textnormal{ and } i_2 \notin S, \\
\\[-0.5em]
\min \left \{ r_{i_1},r_{i_2} \right \}, &\text{if }{\color{brown}i_1 \in S_1 \setminus S_2, \; i_2 \in S_2 \setminus S_1},\; i_1 \in S, \text{ and } i_2 \in S,\\
\min \left \{ r_{i_1}, \min_{j \in S \cap S_2 \setminus S_1}r_{j} \right \},&\text{if }{\color{brown}i_1 \in S_1 \setminus S_2, \; i_2 \in S_2 \setminus S_1},\; i_1 \in S, \text{ and } i_2 \notin S,\\
\min \left \{ r_{i_2}, \min_{j \in S \cap S_1 \setminus S_2}r_{j} \right \},&\text{if }{\color{brown}i_1 \in S_1 \setminus S_2, \; i_2 \in S_2 \setminus S_1},\; i_1 \notin S, \text{ and } i_2 \in S,\\
0,&\text{if }{\color{brown}i_1 \in S_1 \setminus S_2, \; i_2 \in S_2 \setminus S_1},\; i_1 \notin S, \text{ and } i_2 \notin S,
\end{cases}
\end{align*}
\end{comment}
We observe that the quantities $\min_{j \in S \cap S_1 \setminus S_2}r_{j}$ and $\min_{j \in S \cap S_2 \setminus S_1}r_{j} $ appear in many of the above cases, and we see that these quantities can be precomputed for each of the assortments $S \in \widehat{\mathcal{S}}$ in a total of $| \widehat{\mathcal{S}}|\times \mathcal{O}(n) = \mathcal{O}(n^3)$ computation time. Given that we have precomputed these quantities, and given the fact that our data structures allow us to query whether any product is an element of the sets $S_1 \setminus S_2$, $S_2 \setminus S_1$, and $S_1 \cap S_2$ in $\mathcal{O}(1)$ time, we conclude that all of the $\rho_{i_1 i_2}(S)$ can be computed in a total of $\mathcal{O}(n^4 + n^3) = \mathcal{O}(n^4)$ time.
In summary, we have established that constructing the collection of assortments $\widehat{\mathcal{S}}$, constructing the set of pairs of products $\mathcal{L}$, and computing the quantities $\rho_{i_1 i_2}(S)$ for each assortment $S \in \widehat{\mathcal{S}}$ and each pair of products $(i_1,i_2) \in \mathcal{L}$ can be performed in a total of $\mathcal{O}(n^4)$ computation time.
We conclude our proof of Theorem~\ref{thm:two} by establishing the total computation time of our brute-force algorithm using the information computed above. In each iteration of our algorithm, we select an assortment $S \in \widehat{\mathcal{S}}$ and compute the worst-case expected revenue $\min_{\lambda \in \mathcal{U}} \mathscr{R}^{\lambda}(S)$. As shown in Lemma~\ref{lem:two:flow}, we can compute $\min_{\lambda \in \mathcal{U}} \mathscr{R}^{\lambda}(S)$ by solving a minimum-cost network flow problem over a graph with $n+1$ vertices and $\mathcal{O}(n^2)$ edges. Using the minimum-cost network flow algorithm of \cite{orlin1997polynomial} and \cite{tarjan1997dynamic}, we observe that Problem~\eqref{prob:flow} can be solved in $\mathcal{O}(n^3 \log (n r_n))$ computation time.\footnote{The algorithm of \cite{orlin1997polynomial} and \cite{tarjan1997dynamic} computes the minimum-cost network flow on a directed graph in $\mathcal{O}( (VE \log V) \min \left \{\log( V C), E \log V \right \})$ running time, where $V$ is the number of vertices, $E$ is the number of directed edges, and $C$ is the maximum absolute value of any edge cost. The algorithm requires that $C$ is integral; for more details, see \citet[\S3]{tarjan1997dynamic}. In our case, Problem~\eqref{prob:flow} is a minimum-cost network flow problem in a directed graph where $V = n$, $E = \mathcal{O}(n^2)$, and $C = \max_{S \in \widehat{\mathcal{S}},(i_1,i_2) \in \mathcal{L}} \rho_{i_1 i_2}(S) = r_n$. } Since the worst-case expected revenue $\min_{\lambda \in \mathcal{U}} \mathscr{R}^{\lambda}(S)$ must be computed for each assortment $S \in \widehat{\mathcal{S}}$, and since it follows readily from Lemma~\ref{lem:two:S} that $| \widehat{\mathcal{S}}| = \mathcal{O}(n^2)$, we conclude that our algorithm requires a total of $\mathcal{O}(n^5 \log (n r_n))$ computation time.
\halmos \end{proof}
\subsubsection{Numerical Experiments for Two Assortments.} \label{sec:twoassortments:numerics}
We will now perform numerical experiments to show, using our polynomial-time algorithm from \S\ref{sec:twoassortments:algorithm}, that there can be affirmative answers to the identification question in the case of $M = 2$.
To perform our numerical experiments, we begin by constructing randomly-generated problem instances in a manner that is similar to that taken in \S\ref{sec:revorder}. In each randomly-generated problem instance, the revenues are drawn from the distribution $r_1,\ldots,r_n \sim \textnormal{Uniform}[0,1]$, and we sort the products such that $r_1 < \cdots < r_n$. The two past assortments $S_1,S_2 \in \mathcal{S}$ are also constructed randomly, whereby the two assortments satisfy $\{0,n\} \subseteq S_1 \cap S_2$ and, for each of the remaining products $j \in \{1,\ldots,n-1\}$, we randomly assign the product to the assortments with distribution given by $P(j \in S_1 \cap S_2) = \sfrac{1}{3}$, $P(j \in S_1 \setminus S_2) = \sfrac{1}{3}$, and $P(j \in S_2 \setminus S_1) = \sfrac{1}{3}$.
In order to generate historical sales data for these two assortments, we generate a base choice for the parameters $\lambda^*$ of the ranking-based choice model.
Because we will be performing numerical experiments on problem instances with larger values of $n$ than were considered in \S\ref{sec:revorder},
it will not be viable from a computational tractability standpoint to generate base parameters $\lambda^*$ that have nonzero values for each of the $(n+1)!$ parameters of a ranking-based choice model. To get around this, we restrict the numerical experiments to generating base parameters $\lambda^*$ which are \emph{sparse}. Specifically, we generate the base parameters in each problem instance by first randomly selecting a subset of rankings $\Sigma' \subseteq \Sigma$ of length $| \Sigma'| = K$;\footnote{We use rejection sampling to ensure that each of the $\binom{(n+1)!}{K}$ subsets of rankings is selected with equal probability.} we then assign $\lambda_\sigma^* \leftarrow 0$ for each ranking $\sigma \notin \Sigma'$, and we choose the remaining parameters $\{ \lambda_\sigma: \sigma \in \Sigma'\}$ by drawing uniformly over the $K$-dimensional probability simplex.
Using this base choice for the parameters, we generate historical sales data of the form $v_1$ and $v_2$ corresponding to the two assortments $S_1$ and $S_2$ under the ranking-based choice model with the base parameters $\lambda^*$. After we compute the historical sales data, we \emph{forget} the base parameters $\lambda^*$ of the ranking-based choice model as well as the choice of $K$, and we apply the algorithm from \S\ref{sec:twoassortments:algorithm} to obtain a new assortment, denoted by $S'$.
\begin{comment}
Because we will perform our experiments here on instances with larger values of $n$ than were considered in \S\ref{sec:revorder},
it will be computationally intractable to store all of the $(n+1)!$ parameters of a ranking-based choice model.
To get around this, we generate a base choice of the parameters $\lambda^*$ in each problem instance by first randomly selecting a subset of rankings $\Sigma' \subseteq \Sigma$ of length $| \Sigma'| = K$\footnote{In our experiments, each of the $\binom{(n+1)!}{K}$ subsets of rankings of length $| \Sigma'| = K$ have the same probability of being selected.}, setting $\lambda_\sigma^* = 0$ for each ranking not in the subset, and choosing the remaining parameters of the ranking-based choice model $\{ \lambda_\sigma: \sigma \in \Sigma'\}$ by drawing uniformly over the $K$-dimensional probability simplex
Using this base choice for the parameters, we generate historical sales data of the form $v_1$ and $v_2$ corresponding to the two assortments $S_1$ and $S_2$ under the ranking-based choice model with parameter $\lambda^*$. After we computer this historical sales data, we {forget} the parameters $\lambda^*$ of the ranking-based choice model and apply the algorithm from \S\ref{sec:twoassortments:algorithm} to obtain a new assortment, denoted by $S'$.
\end{comment}
\begin{figure}[t]
\centering
\FIGURE{\includegraphics[width=0.8\linewidth]{figures/two_assortments}}
{Performance of new assortments obtained using algorithm from \S\ref{sec:twoassortments:algorithm} when $M=2$, $K = 10$, and $n = 10$. \label{fig:twoassortments:plots}}
{Each point corresponds to a randomly-generated problem instance. The $x$-axis shows the expected revenue of the best previously-offered assortment, $\max_{m \in \mathcal{M}} r^\intercal v_m$. The $y$-axis shows the worst-case expected revenue $\min_{\lambda \in \mathcal{U}} \mathscr{R}^{\lambda}(S')$ for the new assortment $S'$ obtained by using the algorithm from \S\ref{sec:twoassortments:algorithm}. }
\end{figure}
\begin{figure}[t]
\centering
\FIGURE{\includegraphics[width=0.8\linewidth]{figures/speed_two_assortments}}
{Computation time for the algorithm from \S\ref{sec:twoassortments:algorithm} when $M=2$ and $K = 1000$. \label{fig:speed_twoassortments} }
{Results are averaged over 100 replications for each $n \in \{10,12,14,\ldots,100\}$.
\end{figure}
\begin{comment}
\begin{remark}
To show that this bound cannot be improved in general, we consider a problem in which $n$ is a power of two and $M$ is an even integer less than or equal to $2 \log_2 (n)$. For each $m \in \{1,3,\ldots,M-1\}$, we define the assortments
\begin{align*}
S_m = \{0 \} \cup \left \{ i \in \mathcal{N}: \left \lfloor \frac{i-1}{2^{m-1}} \right \rfloor \mod 2 = 0\right \}, \; S_{m+1} = \{0 \} \cup \left \{ i \in \mathcal{N}: \left \lfloor \frac{i-1}{2^{m-1}} \right \rfloor \mod 2 = 1\right \}.
\end{align*}
It can be shown for this example that $| \widehat{\mathcal{S}}| = (\frac{N}{2^{M/2}})^{2^{M/2}}$.
\end{remark}
\end{comment}
In Figures~\ref{fig:twoassortments:plots} and \ref{fig:speed_twoassortments}, we present the results of the numerical experiments conducted as described above. In Figure~\ref{fig:twoassortments:plots}, we present the results of these numerical experiments in the case of $n = 10$ products over 1000 randomly-generated problem instances and for $K = 10$. This figure compares the expected revenue under the best previously-offered assortment, $\max_{m \in \mathcal{M}} r^\intercal v_m$, to the predicted expected revenue of the new assortment under the worst-case ranking-based choice model that is consistent with the historical sales data, $\min_{\lambda \in \mathcal{U}} \mathscr{R}^{{\lambda}}(S')$. These results establish that there can be affirmative answers to the identification question in the case of $M = 2$; indeed, we observe from Figure~\ref{fig:twoassortments:plots} that there are problem instances for which the predicted expected revenue of the new assortment obtained by our algorithm from \S\ref{sec:twoassortments:algorithm} is strictly greater than the expected revenue of the best previously-offered assortment under all ranking-based choice model that are consistent with the historical sales data. In Figure~\ref{fig:speed_twoassortments}, we show the average computation times for our algorithm from \S\ref{sec:twoassortments:algorithm} on problem instances in which the number of products is varied across $n \in \{10,12,14,\ldots,100\}$ and $K = 1000$. Here, we use the larger $K = 1000$ to ensure that the sparsity of the randomly-generated base parameters $\lambda^*$ does not introduce any biases on the resulting computation times of our algorithm. The results in Figure~\ref{fig:speed_twoassortments} show that the computation time of the algorithm remains under 30 seconds even when there are one hundred products. This is viewed as promising from a practical perspective, as it shows that a general algorithm for answering the identification question for the case of $M = 2$ can scale to problem instances with realistic numbers of products.
Motivated by the numerical findings in Figure~\ref{fig:twoassortments:plots}, we further conducted an informal analysis of the randomly-generated problem instances in which the worst-case expected revenue of the new assortment was strictly greater than the expected revenue of the best previously-offered assortment. Our analysis consisted of a visual examination of the assortments that were found by the algorithm from \S\ref{sec:twoassortments:algorithm}, along with an exploratory analysis in which we manually adjusted the parameters which were used to generate the problem instances to understand their impact on the resulting assortments found by the algorithm from \S\ref{sec:twoassortments:algorithm}.
Our goal in conducting this informal analysis was to gain a qualitative understanding of the types of problem instances for which the algorithm from \S\ref{sec:twoassortments:algorithm} is able to find an affirmative answer to the identification question. We were also interested in assessing the performance of the estimate-then-optimize technique in problem instances in which there are affirmative answers to the identification question.
The main takeaway from our informal analysis is a simple but insightful example of a problem instance
for which the identification question has an affirmative answer.
The example, which is presented in Appendix~\ref{appx:simple}, is insightful for three specific reasons.
First, the example shows that there are problem instances in which the assortment $S \in \widehat{\mathcal{S}}$ that answers the identification question is not a revenue-ordered assortment. Second, the example shows that the estimate-then-optimize technique can perform poorly (that is, yield an assortment with an expected revenue which is strictly less than the best expected revenue from the previously-offered assortments) even on problem instances for which there are affirmative answers to the identification question. Third, the example shows that the aforementioned two properties can be obtained in relatively simple problem instances which are comprised of only $n=4$ products and in which the revenues and historical sales data have numerical values which are relatively easy to manipulate.
More broadly, this example illustrates that assortments which are associated with affirmative answers to the identification question can in general have a non-trivial structure, and discovering such assortments appears to be challenging without the aid of algorithms like those developed in \S\ref{sec:twoassortments:algorithm} and in the subsequent \S\ref{sec:fixed_dim}.
\subsection{Tractability of Identification Question for Fixed Number of Past Assortments} \label{sec:fixed_dim}
We conclude \S\ref{sec:applications} by studying whether it is possible to design theoretically-efficient algorithms for answering to the identification question in general settings with several previously-offered assortments. Establishing the existence of such algorithms is important because their existence suggests that practical algorithms, like that from \S\ref{sec:twoassortments:algorithm},
can be developed for answering the identification question in real-world applications with relatively small numbers of previously-offered assortments and large numbers of products. Problem instances in which there are small numbers of previously-offered assortments and large numbers of products can arise in high-stakes applications in which changing to low-quality assortments can have significant negative consequences, and so the firm has made relatively few changes to their assortments up to this point.
In this subsection, our results are {positive} and consist of developing the first polynomial-time algorithm for answering the identification question
for any fixed number of previously-offered assortments $M$.
The algorithm developed in this subsection is particularly attractive due to its generality: it does not require any assumptions on the composition of products in the previously-offered assortments. Moreover, the algorithm allows for the radius $\eta$ in the set of ranking-based choice models to be strictly positive (see \S\ref{sec:setting}); hence, the algorithm can be applied in settings in which there are no ranking-based choice models that have perfect accuracy on the firm's historical sales data. To the best of our knowledge, the general algorithm developed in this subsection is also the first for solving robust assortment optimization problems over the data-driven uncertainty set proposed by \citetalias{farias2013nonparametric} with computation time that is polynomial in the number of products $n$.
Our algorithm for answering the identification question in general settings follows the same high-level strategy as developed in \S\ref{sec:twoassortments}. Specifically, our algorithm reduces the identification question to solving the robust assortment optimization problem~\eqref{prob:robust} and checking to see if the optimal objective value of \eqref{prob:robust} is strictly greater than the best expected revenue among the previously-offered assortments. Our algorithm solves the robust assortment optimization problem by computing the worst-case expected revenue $\min_{\lambda \in \mathcal{U}} \mathscr{R}^\lambda(S)$ for each candidate assortment $S \in \widehat{\mathcal{S}}$ and outputting the assortment that has the maximum worst-case expected revenue.
The correctness of such a brute-force algorithm for solving the robust assortment optimization problem \eqref{prob:robust} follows immediately from Theorem~\ref{thm:main}. In the remainder of \S\ref{sec:fixed_dim}, we prove that there exists an implementation of the aforementioned brute-force algorithm which yields the following theoretical guarantee on its computational tractability:
\begin{theorem} \label{thm:poly}
\eqref{prob:robust} can be solved in weakly $\mathcal{O}(\textnormal{poly}(n))$ computation time for every fixed $M$.
\end{theorem}
In the above theorem and throughout the rest of \S\ref{sec:fixed_dim}, we assume that the revenues $r_1,\ldots,r_n$ are distinct. The above theorem holds for any choice of the radius $\eta \ge 0$ in the set of ranking-based choice models and any composition of products in the previously-offered assortments. We note that our algorithm is guaranteed to run in weakly, as opposed to strongly, polynomial time due to its reduction to solving linear optimization problems.
To prove Theorem~\ref{thm:poly}, we will make use of several intermediary results (Lemma~\ref{lem:fixed_dim:S:time}-\ref{lem:fixed_dim:construct_rho}), the proofs of which can be found in Appendix~\ref{appx:fixed_dim}. The primary workhorse in our proof of Theorem~\ref{thm:poly} is contained in the proof of our first intermediary result, denoted by Lemma~\ref{lem:fixed_dim:S:time}, in which we develop an algorithm for constructing the collection of assortments $\widehat{\mathcal{S}}$ from the previously-offered assortments. The algorithm which achieves the specified running time of the following lemma is based on dynamic programming over a compact graphical representation of the collection of assortments $\widehat{\mathcal{S}}$. In particular, the algorithm can be viewed as attractive due to its mild dependence on the number of assortments in the collection $\widehat{\mathcal{S}}$.
\begin{lemma} \label{lem:fixed_dim:S:time}
The collection of assortments $\widehat{\mathcal{S}}$ can be constructed in $\mathcal{O}(n^{2} (M+| \widehat{\mathcal{S}}|))$ time.
\end{lemma}
We observe that the algorithm which achieves the specified running time in Lemma~\ref{lem:fixed_dim:S:time} is efficient when the number of assortments in the collection $\widehat{\mathcal{S}}$ is small. In fact, we have seen evidence up to this point that the number of assortments in the collection $\widehat{\mathcal{S}}$ can indeed be small in special cases of problem instances: namely, we recall from \S\ref{sec:revorder} that $|\widehat{\mathcal{S}}| = \mathcal{O}(n)$ when the previously-offered assortments are the revenue-ordered assortments, and we recall from \S\ref{sec:twoassortments} that $|\widehat{\mathcal{S}}| = \mathcal{O}(n^2)$ when there are two previously-offered assortments. In the next intermediary result, denoted by Lemma~\ref{lem:fixed_dim:S}, we develop a more general result along these lines. Specifically, the following lemma establishes that the number of assortments in the collection $\widehat{\mathcal{S}}$ can be upper bounded by a polynomial of the number of products $n$ for \emph{any} fixed number of previously-offered assortments $M$. While the bound in the following lemma is not the tightest possible, the bound will be sufficient for its theoretical purpose in this subsection of proving Theorem~\ref{thm:poly}.
\begin{lemma}\label{lem:fixed_dim:S}
$| \widehat{\mathcal{S}} | \le (n+2)^{2^M}$.
\end{lemma}
The remaining intermediary results will be used in our proof of Theorem~\ref{thm:poly} to show that the worst-case expected revenue $ \min_{\lambda \in \mathcal{U}}\mathscr{R}^{\lambda}(S)$ can be computed for each assortment $S \in \widehat{\mathcal{S}}$ in weakly polynomial time for every fixed number of previously-offered assortments $M$. The following intermediary results accomplish this by showing for every fixed $M \ge 2$ and assortment $S \in \widehat{\mathcal{S}}$ that the worst-case expected revenue $ \min_{\lambda \in \mathcal{U}}\mathscr{R}^{\lambda}(S)$ can be computed by solving a linear optimization problem that can be constructed in polynomial time (Lemma~\ref{lem:fixed_dim:construct_L}-\ref{lem:fixed_dim:construct_rho}) and has a polynomial number of decision variables and constraints (see Remark~\ref{remark:tractability} from \S\ref{sec:prelim}).
\begin{lemma} \label{lem:fixed_dim:construct_L}
$|\mathcal{L}| = \mathcal{O}(n^M)$, and $\mathcal{L}$ can be constructed in $\mathcal{O}(Mn^{M+1})$ computation time.
\end{lemma}
\begin{lemma} \label{lem:fixed_dim:construct_rho}
For each assortment $S \in \widehat{\mathcal{S}}$ and each tuple of products $(i_1,\ldots,i_M) \in \mathcal{L}$, the quantity $\rho_{i_1 \cdots i_M}(S)$ can be computed in $\mathcal{O}(M^2 n)$ time.
\end{lemma}
Using the above intermediary lemmas, we conclude \S\ref{sec:fixed_dim} with our proof of Theorem~\ref{thm:poly}.
\begin{proof}{Proof of Theorem~\ref{thm:poly}.}
As described at the beginning of \S\ref{sec:fixed_dim}, we consider a brute-force algorithm for solving the robust assortment optimization problem~\eqref{prob:robust} that consists of computing the worst-case expected revenue $\min_{\lambda \in \mathcal{U}} \mathscr{R}^{\lambda}(S)$ for each candidate assortment $S \in \widehat{\mathcal{S}}$ and outputting the assortment that has the maximum worst-case expected revenue. The computation time of this brute-force algorithm is analyzed as follows. In Lemmas~\ref{lem:fixed_dim:S:time} and \ref{lem:fixed_dim:S}, we established that the collection of assortments $\widehat{\mathcal{S}}$ can be constructed in $\mathcal{O}(n^2(M + | \widehat{\mathcal{S}}|)) = \mathcal{O}(n^2(M + (n+2)^{2^M})) = \mathcal{O}(\textnormal{poly}(n))$ computation time, where the last equality holds for any fixed $M$. In Lemma~\ref{lem:fixed_dim:construct_L}, we established that the set of tuples of products $\mathcal{L}$ can be constructed in $\mathcal{O}(Mn^{M+1}) = \mathcal{O}(\textnormal{poly}(n))$ computation time, where the equality holds for any fixed $M$. Our algorithm performs an iteration for each assortment $S \in \widehat{\mathcal{S}}$, and thus our algorithm will perform $| \widehat{\mathcal{S}}| \le (n+2)^{2^M} = \mathcal{O}(\textnormal{poly}(n))$ iterations for any fixed $M$. Given any assortment $S \in \widehat{\mathcal{S}}$, we established in Lemma~\ref{lem:fixed_dim:construct_rho} that the quantities $\rho_{i_1 \cdots i_M}(S)$ for each $(i_1,\ldots,i_M) \in \mathcal{L}$ can all be computed in a total of $\mathcal{O} ( | \mathcal{L}| \times M^2 n) = \mathcal{O}(n^M \times M^2 n) = \mathcal{O}(\textnormal{poly}(n))$ computation time, where the last equality holds for any fixed $M$. Given the set of tuples of products $\mathcal{L}$ and the quantities $\rho_{i_1 \cdots i_M}(S)$ for each $(i_1,\ldots,i_M) \in \mathcal{L}$, we established in Remark~\ref{remark:tractability} from \S\ref{sec:prelim} that we can compute the worst-case expected revenue $\min_{\lambda \in \mathcal{U}} \mathscr{R}^{\lambda}(S)$ by constructing and solving a linear optimization problem with $\mathcal{O}(n^M) = \mathcal{O}(\textnormal{poly}(n))$ decision variables and $\mathcal{O}(nM) = \mathcal{O}(\textnormal{poly}(n))$ constraints for any fixed $M$. Since linear optimization can be solved in weakly polynomial time via the ellipsoid algorithm, we conclude that each iteration of our algorithm requires $\mathcal{O}(\textnormal{poly}(n))$ time for any fixed $M$. Our proof of Theorem~\ref{thm:poly} is thus complete.
\halmos \end{proof}
\section{Conclusion and Future Research} \label{sec:conclusion}
In this work, we investigated a popular class of high-dimensional discrete choice models, known as {ranking-based} choice models, in the context of assortment planning problems.
Motivated by the fact that many ranking-based choice models can be consistent with a firm's historical sales data, we considered the \emph{identification question}, which asks whether it is possible to identify a new assortment that outperforms the firm's past assortments under {all} of the ranking-based choice models that are consistent with the firm's historical sales data. By analyzing and developing algorithms for solving a class of robust assortment optimization problems, we established the existence of affirmative as well as negative answers to the identification question.
Moreover, we developed polynomial-time algorithms for answering the identification question in general settings with any fixed number of past assortments.
Together with concise numerical experiments, these findings revealed that considering the identification question can be essential to making good assortment decisions from ranking-based choice models in high-stakes assortment planning problems.
We believe this work opens up a number of promising directions for future research. First, our work showed for the first time that the identification question can be answered for one popular class of high-dimensional discrete choice models. Yet there are many other high-dimensional discrete choice models beyond the ranking-based choice model for which the identification question can be asked,
such as models for capturing irrational customer choice \citep{berbeglia2018generalized,chen2020decision,jena2021estimation}.
Second, our work showed that polynomial-time algorithms can be developed for finding assortments that answer the identification question for ranking-based choice models. However,
it may be possible in certain settings that existing algorithms for estimating high-dimensional discrete choice models such as expectation-maximization \citep{talluri2004revenue,van2017expectation,csimcsek2018expectation}, in combination with algorithms for finding assortments that maximize the predicted expected revenue,
can lead to assortments with provable performance guarantees with respect to the identification question. Establishing such guarantees would provide new assurances to firms for using estimate-then-optimize in high-stakes assortment planning problems. Finally, we believe that the present and related work ({\it{e.g.}}, \cite{kallus2018confounding,sturt2021nonparametric}) provide a starting point for using robust optimization to develop efficient algorithms that are valuable for operations management problems in which good \emph{average} performance is paramount. In particular, future work may extend the algorithms developed in the present paper to answer the identification question in numerous other revenue management problems, ranging from multi-product pricing to dynamic assortment planning.
\bibliographystyle{informs2014}
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2,877,628,089,050 | arxiv | \section{Appendix}
\input{SEBAsecPersist}
\input{SEBAproofsLassoes}
\input{SEBAproofsL12}
\input{SEBAproofsSpectral}
\bibliographystyle{plain}
\subsection{Proof of Section \ref{sec:grouped}}
\begin{proof}[Proof of Lemma \ref{lem:GroupLasso}]
a) Since \(\hat\beta\) minimizes \eqref{lassostar}, then,
\(\forall\beta\)
\eqsplit{
\sum_{ij} (Y_{ij}-x_{ij}^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}\hat\beta_i)^2+\lm\sum_\ell
\|\hat\beta_{\cdot\ell}\|_2
&\leq
\sum_{ij} (Y_{ij}-x_{ij}^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}\beta_i)^2+\lm\sum_\ell
\|\beta_{\cdot\ell}\|_2
}
Then for \(f_{ij}=Y_{ij}-\eps_{ij}\), \(\eps_{ij}\dist\normal(0,\sig^2)\)
\iid, since
\eqsplit{
\|f-\hat f\|_2^2 &= \sum_{ij}(Y_{ij}-x_{ij}^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}\beta_i-\eps_{ij})^2
\\
&= \sum_{ij} (Y_{ij}-x_{ij}^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}\beta_i)^2 + \sum_{ij} \eps_{ij}^2 +
2\sum_{ij}\eps_{ij}(x_{ij}^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}\beta_i-Y_{ij}),
}
we have
\eqsplit{
\|f-\hat f\|_2^2 + \lm\sum_\ell \|\hat\beta_{\cdot\ell}\|_2
&\leq \|f_{\beta}-f\|_2^2 + \lm\sum_{\ell}\|\beta_{\cdot\ell}\|_2
+2\sum_{ij}\eps_{ij}x_{ij}^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}(\beta_i-\hat\beta_i)
}
The last term can be bounded with high probability. Denote
\(V_{il}=\summ j1m x_{ij\ell}\eps_{ij}\dist\normal(0,\sig^2\sum_j
x_{ij\ell}^2)\). Denote the event \(\sca_\ell=\bigcap_{i=1}^n
\{|V_{i\ell}|\leq c_{il}\}\), for some \(c_{i\ell}>0\). Then
\eqsplit{
P(\sca_\ell^c) &\leq \summ i1n P(|V_{i\ell}|>c_{i\ell})
\\
&= \summ i1n 2\biggl[1-\Phi\Bigl\{c_{i\ell}/\sig\bigl(\sum_j
x_{ij\ell}^2\bigr)^{1/2}\Bigr\}\biggr]
\\
&\leq \summ i1n \exp\bigl\{- c_{i\ell}^2/2\sig^2\sum_jx_{ij\ell}^2\bigr\}
\\
&= n\exp\bigl\{-\mu^2/2\sig^2\bigr\}
}
where \(c_{il}\equiv \mu^2 \sum_j x_{ij\ell}^2\).
Now, for the event \(\sca=\bigcup_{\ell=1}^p
\sca_{\ell}=\bigcup_\ell\bigcup_i\{|V_{i\ell}|\le c_{i\ell}\}\),
\eqsplit{
P(\sca^c) &\leq \summ \ell 1p P(\sca_\ell^c) \leq npe^{-\mu^2/2\sig^2}.
}%
Thus, if \(\mu\) is large enough \(P(\sca^c)\) is small. E.g., for \(\mu=\sig A\bigl(\log(np)\bigr)^{1/2}\),
\(A>\sqrt 2\), we have \(P(\sca^c)\leq (np)^{1-A^2/2}\).
Back to the bound for \(\|f-\hat f\|+\lm
\sum_\ell\|\hat\beta_{\cdot\ell}\|^2\). We have that on \(\sca\),
\eqsplit{
\|f-\hat f\|_2^2 &\leq \|f_\beta -f\|_2^2 + \lm\sum_\ell
\bigl(\|\beta_{\cdot\ell}\|_2 - \|\hat\beta_{\cdot\ell}\|_2\bigr)
+2\sum_{i\ell}c_{il}|\beta_{il}-\hat\beta_{i\ell}|\\
&\leq \|f_\beta -f\|_2^2 + \lm\sum_\ell
\bigl(\|\beta_{\cdot\ell}\|_2 - \|\hat\beta_{\cdot\ell}\|_2\bigr)
+2\sum_{\ell} \sqrt{\sum_i c_{il}^2 }| ||\beta_{\cdot \, l}-\hat\beta_{\cdot \,
\ell}||_2,
}
since \(V_{i\ell}(\beta_{i\ell}-\hat\beta_{i\ell})\leq
c_{i\ell}|\beta_{i\ell}-\hat\beta_{i\ell}|\) on \(\sca\).
Let \(\ti\lm>0\). Add now
\(\ti\lm\sum_{\ell}\|\beta_{\cdot\ell}-\hat\beta_{\cdot\ell}\|_2\) to both
sides
\eqsplit{
\|f-\hat f\|_2^2 + \ti\lm \sum_\ell \|\beta_{\cdot\ell}
-\hat\beta_{\cdot\ell}\|_2
&\leq \|f_\scb - f\|_2^2
+ \ti\lm \sum_\ell \|\beta_{\cdot\ell}-\hat\beta_{\cdot\ell}\|_2
+\lm\sum_\ell \|\beta_{\cdot\ell}\|_2
\\
&\hspace{2em}
-\lm\sum_\ell \|\hat\beta_{\cdot\ell}\|_2
+2\max_{\ell} \sqrt{\sum_i c_{il}^2 } \,\, \sum_{\ell}||\beta_{\cdot \, l}-\hat\beta_{\cdot \,
\ell}||_2,
\\
&= \|f_\beta - f\|_2^2
+\sum_\ell S_{\beta\ell},\qquad\text{say.}
}
Now, for \(\ell\not\in J(\beta)\),
\eqsplit{
S_{\beta\ell} &= \ti\lm \|\hat\beta_{\cdot\ell} \|_2 -\lm
\|\hat\beta_{\cdot\ell}\|_2 +
2 ||\hat\beta_{\cdot \, \ell}||_2 \, \max_{\ell} \sqrt{\sum_i
c_{il}^2 }
\\
&= \|\hat\beta_{\cdot\ell}\|_2 \Bigl(\ti\lm-\lm
+ 2\max_{\ell} \sqrt{\sum_i c_{il}^2 } \Bigr)
}
This expression can be negative if \(\lm>2\max_{\ell} \sqrt{\sum_i
c_{il}^2 } \) and \(0\leq\ti\lm \leq \lm - 2\max_{\ell}
\sqrt{\sum_i c_{il}^2 } \).
On the other hand, for \(\ell\in J(\beta)\):
\eqsplit{
S_{\beta\ell} &\leq \|\beta_{\cdot\ell}-\hat\beta_{\cdot\ell}\|_2 \left( \ti \lm+\lm + 2 \max_{\ell} \sqrt{\sum_i c_{il}^2 } \right)
}
Thus,
\eqsplit{
\|\hat f- f\|_2^2 +\ti\lm \sum_\ell
\|\hat\beta_{\cdot\ell}-\beta_{\cdot\ell}\|_2
&\leq \|f_\beta - f\|_2^2 + \left(\lm + \ti\lm + 2\max_{\ell} \sqrt{\sum_i c_{il}^2 }\right) \, \sum_{\ell\in J(\beta)} \|\hat \beta_{\cdot\ell}
-\beta_{\cdot\ell}\|_2.
}
We have, therefore, the following conditions on \(\mu\) and \(\ti\lm\):
\eqsplit[nb1]{
\ti\lm &\leq \lm - 2\mu \max_{\ell} \sqrt{\sum_{i,j}
x_{ij\ell}^2}\;\;,\qquad\ell=1,\dots,p.
\\
\implies \lm &> 2\mu \max_{\ell} \sqrt{\sum_{i,j}
x_{ij\ell}^2}\;\;,\qquad\ell=1,\dots,p.
}
Recall that $\mu = A\sig\sqrt{\log(np)}, \quad A>\sqrt 2$.
Thus, for example one can take
\eqsplit{
\ti\lm &= \lm/2 = 2 \mu\max_{\ell} \sqrt{\sum_{i,j} x_{ij\ell}^2}
} which implies that
\eqsplit{
\mu&= \frac{\lm} {4\Lm_x}
\\
\Lm_x&= \max_{\ell}\sqrt{\sum_{ij} x_{ij\ell}^2}
\\
\text{i.e., } \lm &= 4 \mu \Lm_x = 4 A\sig \Lm_x \sqrt{\log(np) }.
}
Then,
\eqsplit{
\lm^2-\mu^2\max_\ell \sum_{i,j} x_{ij\ell}^2
=\frac{15}{16}\lm^2,
}
which concludes the proof of part (a).
b) Similarly, for the $\ell_1$ norm of the difference between the
betas we have, for some $\lm_1 >0$:
\eqsplit{
\|f-\hat f\|_2^2 + \lm_1 \sum_\ell \|\beta_{\cdot\ell}
-\hat\beta_{\cdot\ell}\|_1
&\leq \|f_\scb - f\|_2^2
+ \lm_1 \sum_\ell
\|\beta_{\cdot\ell}-\hat\beta_{\cdot\ell}\|_1
+\lm\sum_\ell \|\beta_{\cdot\ell}\|_2
\\
&\hspace{2em}
-\lm\sum_\ell \|\hat\beta_{\cdot\ell}\|_2
+ 2\sum_\ell \max_i c_{i\ell}\|\beta_{\cdot\ell} -
\hat\beta_{\cdot\ell}\|_1
\\
&= \|f_\beta - f\|_2^2
+\sum_\ell \ti S_{\beta\ell},\qquad\text{say.}
}
For \(\ell\not\in J(\beta)\),
\eqsplit{
\ti S_{\beta\ell} &\leqslant \lm_1 \|\hat\beta_{\cdot\ell} \|_1 -\lm
\|\hat\beta_{\cdot\ell}\|_2 + 2\sqrt{\sum_i
c_{i\ell}^2 } \|\hat\beta_{\cdot\ell}\|_2
\\
&\leq \|\hat\beta_{\cdot\ell}\|_2 \Bigl( \lm_1 \sqrt{n} -\lm
+\sqrt{\sum_i c_{i\ell}^2 } \Bigr)
}
This expression is negative if \(\lm>2\sqrt{\sum_i
c_{i\ell}^2 } + \lm_1 \sqrt{n} >0 \).
On the other hand, for \(\ell\in J(\beta)\):
\eqsplit{
\ti S_{\beta\ell} &\leq \left( \lm_1 + \lm + 2\sqrt{\sum_i
c_{i\ell}^2 }\right) \|\beta_{\cdot\ell}-\hat\beta_{\cdot\ell}\|_2
}
Thus, similarly to part a), we can take $\lm_1 \sqrt{n} = \ti\lm =
2 \mu\max_{\ell}\sqrt{\sum_{ij} x_{ij\ell}^2} = \lambda/2$ and the
same $\mu$ and $\lm$ as in a).
c) We have
\eqsplit{
\sum_j x_{ij\ell}(Y_{ij} -x_{ij}^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}\hat\beta_i) &=\lm
\frac{\hat\beta_{i\ell}} {\Bigl(\sum_k
\hat\beta_{k\ell}^2\Bigr)^{1/2}}\, ,
}
and on \sca,
\eqsplit{
\Bigl|\sum_j x_{ij\ell}(Y_{ij}-f_{ij})\Bigr| \leq c_{i\ell}=\mu \Bigl(\sum_j
x_{ij\ell}^2\Bigl)^{1/2}.
}
Thus, on \(\sca\),
\eqsplit{
\sum_{ij}\bigl(x_{ij\ell}(f_{ij} - x_{ij}^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}\hat\beta_i)\bigr)^2
&\geq \frac1m \sum_i\Bigl(\sum_j x_{ij\ell}(f_{ij}
-x_{ij}^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}\hat\beta_i)\Bigr)^2
\\
&\geq \frac1m \sum_i \Bigl(\sum_j
x_{ij\ell}(Y_{ij}-x_{ij}^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}\hat\beta_i)\Bigr)^2
-\frac1m \sum_i \Bigl(\sum_j x_{ij\ell}(f_{ij}-Y_{ij})\Bigr)^2
\\
&\geq \frac1m \sum_i\lm^2 \frac{\beta_{i\ell}^2} {\sum_i\beta_{i\ell}^2}
-\frac1m \sum_{i:\hat\beta_{i\ell}\ne 0}c_{i\ell}^2
\\
&= \frac1m\Bigl(\lm^2-\mu^2\sum_{i:\hat\beta_{i\ell}\ne 0}
x_{ij\ell}^2\Bigr).
}
Therefore
\eqsplit{
\sum_{i,j,\ell}\Bigl(x_{ij\ell}(f_{ij} -
x_{ij}^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}\hat\beta_i)\Bigr)^2
&\geq \frac1m
\sum_{\ell:\hat\beta_{i\ell}\ne0}\Bigl(\lm^2 - \mu^2\sum_{ij}x_{ij\ell}^2\Bigr)
\\
&\geq \scm(\hat\beta)\frac1m\Bigl(\lm^2-\mu^2\max_{\ell}\sum_{ij}x_{ij\ell}^2\Bigr)
\\
&= \scm(\hat\beta)\frac1m \Bigl(\lm^2 - \mu^2\max_{\ell}\sum_{ij}x_{ij\ell}^2\Bigr)
\\
&= \scm(\hat\beta) \frac1m \Bigl(\lm^2-\mu^2 \max_{\ell}\sum_{ij}x_{ij\ell}^2\Bigr)
}
On the other hand,
\eqsplit{
\sum_{ij\ell}\Bigl(x_{ij\ell}(f_{ij} -
x_{ij}^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}\hat\beta_i)\Bigr)^2
&\leq \sum_\ell \max_{ij}
x_{ij\ell}^2\|f-\hat f\|^2
}
thus implying that
\eqsplit{
\scm(\hat\beta) &\leq \|f - \hat f\|_2^2
\frac{\sum_\ell\max_{i,j}x_{ij\ell}^2} {\lm^2-\mu^2\max_\ell
\sum_{i,j}x_{ij\ell}^2}.
}
Lemma is proved.
\end{proof}
\begin{proof}[Proof of Theorem \ref{th:BRT61}]
Fix an arbitrary $\beta\in\mathbb{R}^p$ such that $\scm(\beta)\le
s$.
On event $\sca$ defined in the proof of Lemma 1, inequalities
\eqref{boundf} hold. Consider two cases.
\begin{enumerate}
\item $ 2\lambda \sum_{\ell \in J(\beta)} ||\beta_{\cdot\ell} -
\hat\beta_{\cdot\ell}||_1 \leqslant \varepsilon || f -
f_{\scb}||_2^2$. In this case, the result of the theorem follows
trivially from \eqref{boundf}.
\item $\varepsilon || f - f_{\scb}||_2^2 < 2\lambda \sum_{\ell \in
J(\beta)} ||\beta_{\cdot\ell} - \hat\beta_{\cdot\ell}||_1 $.
Denote the event defined by this inequality by $\tilde\sca$; all
subsequent inequalities hold on $\sca \cap \tilde\sca$.
On this event, we get from \eqref{boundf} that
$$
\sum_{\ell \in J^c(\beta)} ||\beta_{\cdot\ell} -
\hat\beta_{\cdot\ell}||_1 \leqslant 4\sqrt{n} (1+1/\varepsilon)
\sum_{\ell \in J(\beta)} ||\beta_{\cdot\ell} -
\hat\beta_{\cdot\ell}||_1.
$$
Assumption URE$_n(s, 4\sqrt{n} (1+1/\varepsilon))$ implies that
$$
\kappa^2 \sum_{\ell \in J(\beta)} |\beta_{i\ell} -
\hat\beta_{i\ell}|_2^2 \leqslant || X_{i\cdot\cdot}^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}
(\beta_{i\cdot } - \hat\beta_{i\cdot }) ||_2^2 = || f_{\ensuremath{\frak B}\xspace \,
i\cdot } - \hat{f}_{i\cdot } ||_2^2,
$$
where $\kappa = \kappa(s, 4\sqrt{n} (1+1/\varepsilon))$. (For
other assumptions, use the corresponding $\kappa$; for MRE2 use a)
of Lemma~\ref{lem:GroupLasso}).
Hence,
by \eqref{boundf},
\begin{eqnarray*}
\|f-\hat f\|_2^2 &\le& \|f - f_{\scb}\|_2^2
+2\lm \sum_{\ell\in J(\beta)}\|\hat\beta_{\cdot\ell}
-\beta_{\cdot\ell}\|_2\\
&\le& \|f - f_{\scb}\|_2^2 +
2\lm \sqrt{\scm(\beta)} \left[\sum_{\ell\in J(\beta)}\|\hat\beta_{\cdot\ell}
-\beta_{\cdot\ell}\|_2^2\right]^{1/2}\\
&=& \|f - f_{\scb}\|_2^2 +
2\lm \sqrt{\scm(\beta)} \left[\sum_{i=1}^n\sum_{\ell\in J(\beta)} |\hat\beta_{i \ell}
-\beta_{i \ell}|_2^2\right]^{1/2}\\
&\le& \|f - f_{\scb}\|_2^2 +
2\lm \sqrt{\scm(\beta)} \left[\sum_{i=1}^n \frac 1 {\kappa^2} \|\hat{f}_{i\cdot} - f_{\scb\, i\cdot}\|_2^2
\right]^{1/2}\\
&=& \|f - f_{\scb}\|_2^2 +
2\lm \frac{\sqrt{\scm(\beta)}}{\kappa} \|\hat{f} - f_{\scb }\|_2\\
&\le& \|f - f_{\scb}\|_2^2 +
2\lm \frac{\sqrt{\scm(\beta)}}{\kappa} \left[ \|\hat{f} - f \|_2 + \|f - f_{\scb }\|_2 \right].
\end{eqnarray*}
Using inequality $2xy \le x^2/b + b y^2$ with $b\in(0,1)$, $x =
\lm \sqrt{\scm(\beta)}/\kappa$, and $y$ being either $\|\hat{f} -
f \|_2$ or $\|f - f_{\scb }\|_2$, we can decouple the last term
to obtain
\begin{eqnarray*}
\|f-\hat f\|_2^2 &\le& \frac{1+b}{1-b} \|f - f_{\scb}\|_2^2 +
\frac{2 \lm^2\scm(\beta)}{b(1-b)\kappa^2}, \quad \forall b\in(0,1).
\end{eqnarray*}
Taking $b=\left(1+2/\varepsilon\right)^{-1}$ we obtain
\begin{eqnarray*}
\|f-\hat f\|_2^2 &\le& (1+\varepsilon) \|f - f_{\scb}\|_2^2 +
\frac{\lm^2\scm(\beta) (\varepsilon+2)^2}{\varepsilon
\kappa^2}.
\end{eqnarray*}
Substituting the value of $\lm$ and taking the infimum over all considered $\beta$ finishes the proof of the theorem.
\end{enumerate}
\end{proof}
\subsection{Proofs of Section \ref{sec:L12}}
\begin{proof}[Proof of Theorem \ref{th:LassoL1p}]
The proof follows that of Lemma 3.1 in Lounici et
al.~\cite{p:Lounici-GroupLasso}.
We start with (a) and (b). Since \(\hat\beta\) minimizes \eqref{lassoes}, then,
\(\forall\beta\)
\eqsplit{
\summ i1n ||Y_i -X_i^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}\hat\beta_i||_2^2+\lm
\summ i1n \|\hat\beta_i\|_1^{\alpha}
&\leq
\summ i1n ||Y_i -X_i^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}\beta_i||_2^2+\lm \summ i1n \|\beta_i\|_1^{\alpha},
}
and hence, for $Y_i = X_i^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em} \beta_i + \varepsilon_i$,
\eqsplit{
\summ i1n ||X_i^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}(\hat\beta_i - \beta_i)||_2^2 \leqslant \summ i1n\left[ 2\varepsilon_i^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}
X_i^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em} (\beta_i - \hat\beta_i) + \lambda (||\beta_i||_1^{\alpha} -
||\hat\beta_i||_1^{\alpha}) \right]. }
Denote \(V_{i\ell}=\summ j1m x_{ij\ell}\eps_{ij}\dist\normal(0,m
\sig^2 )\), and introduce event \(\sca_i =\bigcap_{\ell=1}^p
\{|V_{i\ell}|\leq \mu\}\), for some \(\mu>0\). Then
\eqsplit{
P(\sca_i^c) &\leq \summ \ell 1p P(|V_{i\ell}|>\mu)
\\
&= \summ \ell 1p 2\biggl[1-\Phi\Bigl\{\mu/(\sig\sqrt{m})\Bigr\}\biggr]
\\
&\leq p \exp\bigl\{- \mu^2/(2m\sig^2)\bigr\}.
}
For $\sca=\cap_{i=1}^n \sca_i$, due to independence,
$$P(\sca^c) =
\summ i1n P(\sca_i^c) \leqslant pn \exp\bigl\{-
\mu^2/(2m\sig^2)\bigr\}.
$$
Thus, if \(\mu\) is large enough, \(P(\sca^c)\) is small, e.g.,
for \(\mu=\sig A\bigl(m \log (np)\bigr)^{1/2}\), \(A>\sqrt 2\), we
have \(P(\sca^c)\leq (np)^{1-A^2/2}\).
On event $\sca$, for some $\nu>0$,
\eqsplit { &\hspace{-1em} \summ i1n \left[||X_i(\hat\beta_i - \beta_i)||_2^2 + \nu ||\beta_i -
\hat\beta_i||_1 \right]
\\
&\leqslant \summ i1n \left[
2\mu ||\beta_i -
\hat\beta_i||_1 + \lambda (||\beta_i||_1^2 -
||\hat\beta_i||_1^2)\right.
+ \left. \nu ||\beta_i - \hat\beta_i||_1 \right]
\\
&= \summ i1n\summ j1m \left[\al \, \lambda
\max(||\beta_i||_1^{\alpha-1}, ||\hat\beta_i||_1^{\alpha-1}) (|
\beta_{ij}| - |\hat\beta_{ij}|) +
(\nu+2\mu)|\beta_{ij} - \hat\beta_{ij}| \right]\\
&\leqslant \summ i1n\summ j1m \left[ \al \, \lambda
\max(B^{\alpha-1}, \hat{B}^{\alpha-1})(| \beta_{ij}| -
|\hat\beta_{ij}|) + (\nu+2\mu)|\beta_{ij} - \hat\beta_{ij}|
\right],
}
due to inequality $|x^\alpha - y^\alpha| \leqslant \alpha |x-y|
\max(|x|^{\alpha-1}, |y|^{\alpha-1})$ which holds for $\alpha
\geqslant 1$ and any $x$ and $y$. To simplify the notation, denote
$\scc =\al \, \max(B^{\alpha-1}, \hat{B}^{\alpha-1})$.
Denote $J_i = J(\beta_i) = \{j:
\,\, \beta_{ij} \neq 0 \}$, $\scm(\beta_i) = |J(\beta_i)|$. For
each $i$ and $j\in J(\beta_i)$, the expression in square brackets
is bounded above by
$$
[\lambda \scc + \nu+2\mu] \,|\beta_{ij} - \hat\beta_{ij}|,
$$
and for $j\in J^c(\beta)$, the expression in square brackets is
bounded above by $0$, as long as $\nu+2\mu \leqslant \lambda
\scc$:
$$
- \lambda \scc |\hat\beta_{ij}| +
(\nu+2\mu)|\hat\beta_{ij}| \leqslant 0.
$$
This condition is satisfied if $\nu+2\mu \leqslant \lambda \scc$.
Hence, on $\sca$, for $\nu+2\mu \leqslant \lambda \scc$,
\eqsplit{
\summ i1n \left[ ||X_i^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}(\hat\beta_i - \beta_i)||_2^2 + \nu
||\beta_i - \hat\beta_i||_1 \right] \leqslant \summ i1n [\lambda
\scc + 2\mu + \nu] ||(\beta_i - \hat\beta_i)_{J_i}||_1.
}
This implies that
\eqsplit{ \summ i1n ||X_i(\hat\beta_i - \beta_i)||_2^2 \leqslant
[\lambda \scc + \nu + 2\mu] ||(\beta -
\hat\beta )_J||_1,
}
as well as
that
\eqsplit {
||\beta - \hat\beta||_1 \leqslant
\left[ 1 + \frac{2\mu}{\nu} + \frac{\lambda}{\nu}
\scc \right] ||(\beta - \hat\beta )_J||_1.
}
Take $\nu =\lambda \scc/2$, hence we need to assume that $ 2\mu
\leqslant \lambda \scc/2$:
\eqsplit[eq:boundXdelta]{
\summ i1n ||X_i^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}(\hat\beta_i - \beta_i)||_2^2 &\leqslant
\left[ \frac {3\lambda}{2} \scc + 2\mu\right] ||(\beta -
\hat\beta )_J||_1,\\
||\beta - \hat\beta||_1 &\leqslant
\left[ 3 + \frac{4\mu} {\lambda \scc} \right] ||(\beta -
\hat\beta )_J||_1 \leqslant 4 ||(\beta - \hat\beta)_J||_1.
}
which implies
\eqsplit {
||(\beta - \hat\beta)_{J^c}||_1 \leqslant 3 ||(\beta -
\hat\beta)_J||_1.
}
Due to the generalized restricted eigenvalue assumption RE$_1(s,
3, \kappa)$, $||X^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}(\beta-\hat\beta)||_2 \geqslant \kappa \sqrt{m}
||(\beta -
\hat\beta)_J||_2$, and hence, using \eqref{eq:boundXdelta},
\eqsplit { ||X^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}(\hat\beta - \beta)||_2^2 &\leqslant \left[ \frac {3\lambda}{2}\scc + 2\mu\right]
\sqrt{n\scm(\beta)} || (\hat\beta -
\beta)_J||_2\\&\leqslant \left[ \frac {3\lambda}{2} \scc +
2\mu\right] \frac{ \sqrt{n\scm(\beta)}}{\kappa \sqrt{m} }
||X^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}(\hat\beta - \beta)||_2,
}
where $\scm(\beta)=\max_i \scm(\beta_i)$, implying that
\eqsplit { ||X^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}(\hat\beta - \beta)||_2
&\leqslant \left[ \frac {3\lambda}{2} \scc + 2\mu\right] \frac{
\sqrt{n \scm(\beta)}}{\kappa \sqrt{m} }\\ &= \frac{ \sqrt{n
\scm(\beta)}}{\kappa \sqrt{m} } \left[ \frac {3\lambda}{2} \scc +
2A \sig \sqrt{m\log(np)}\right].
}
Also, \eqsplit{ ||\beta - \hat\beta||_1 &\leqslant 4||(\beta -
\hat\beta)_J||_1 \leqslant 4\frac{\sqrt{n\scm(\beta)}}{\sqrt{m}
\kappa} ||X^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}(\beta - \hat\beta)||_2\\ &\leqslant
\frac{4n \scm(\beta)}{m \kappa^2} \left[ \frac {3\lambda}{2} \scc + 2A \sig \sqrt{m\log(np)}\right].
}
Hence, a) and b) of the theorem are proved.
(c) For $i$, $\ell$: $\hat\beta_{i\ell}\neq 0$, we have
\eqsplit{
2 X_{i\cdot \ell} (Y_i -X_i^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}\hat\beta_i) &=\lm
\alpha \sgn(\hat\beta_{i\ell}) ||\hat\beta_i||_1^{\alpha-1}\, ,
}
Hence,
\eqsplit{
\sum_{\ell: \, \hat\beta_{i\ell}\neq 0} ||X_{i\cdot \ell} X_i^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}(\beta_i-\hat\beta_i)||_2^2 &\geqslant
\sum_{\ell: \, \hat\beta_{i\ell}\neq 0} \left( ||X_{i\cdot \ell}(Y_i
-X_i^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}\hat\beta_i)||_2 - ||X_{i\cdot \ell}(Y_i
-X_i^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}\beta_i)||_2 \right)^2\\
&\geq
\sum_{\ell:\hat\beta_{i\ell}\ne0}\Bigl(\al \,\lm ||\hat\beta_i||_1^{\al - 1}/2 - \mu
\Bigr)^2
\\
&=\scm(\hat\beta_i) (\al \,\lm ||\hat\beta_i||_1^{\al - 1}/2 - \mu)^2.
}
Thus,
\eqsplit{
\scm(\hat\beta_i) &\leq \|X_i(\beta_i-\hat\beta_i)\|_2^2
\frac{m \phi_{i,\, \max}} {\left(\lm \alpha ||\hat\beta_i||_1^{\alpha-1}/2-\mu\right)^2 }.
}
Theorem is proved.
\end{proof}
\begin{proof}[Proof of Theorem \ref{th:L12merge2}.]
To satisfy the conditions of Theorem~\ref{th:LassoL1p}, we can
take $B = b$ and $\lm = \frac{4A\sig}{\al b^{\alpha-1}}\sqrt{m
\log(np)}$.
Thus, by Lemma~\ref{lem:persist},
$$
\frac{\lambda}{m\delta_n} = \frac{4A\sig}{\al b^{\al
-1}}\sqrt{\frac{\log(np)}{m}} \sqrt{\frac{ m \, \eta}{2e V \log
(n(p+1)^2)}} = C \frac{\sqrt{\eta}}{\al b^{\al -1}}\leqslant C_1,
$$
hence assumption $\lambda=\EuScript{O}(m\delta_n)$ of
Theorem~\ref{th:lassoes2} is
satisfied.
Hence, from the proof of Theorem~\ref{th:LassoL1p}, it follows
that \eqsplit{
\|\hat\beta_i\|_1 &= \EuScript{O} \left( (m\del_n/\lm_n)^{1/(\al-2)} \right) = \EuScript{O} \left( \left(\frac{b^{\al -1}}{\sqrt{\eta}}\right)^{1/(\al-2)} \right). }
Hence, we can take $B = b$ and $\hat{B}= C\left(\frac{b^{\al
-1}}{\sqrt{\eta}}\right)^{1/(\al-2)} $ for some $C>0$, and apply
Theorem~\ref{th:LassoL1p}. Then $\max(1, \hat{B}/B)$ is bounded by
$$
\max\left[1, C\frac{ b^{(\al-1)/(\al -2)-1} }{\eta^{1/(2(\al-2))}
}\right] =
\max\left[1, C\frac{ b^{1/(\al -2)} }{\eta^{1/(2(\al-2))} }
\right] = \left(\frac{C b }{\sqrt{\eta}} \right)^{1/(\al-2)},
$$
since $\frac{C b }{\sqrt{\eta}} \geqslant C_2
\frac{\eta^{1/(2(\al-1))} }{\sqrt{\eta}} \geqslant C_2
\eta^{-(\al-2)/(2(\al-1))}$ is large for small $\eta$.
Hence, \eqsplit{ &\frac {3 \alpha \lambda}{2\sqrt{ m}} \max(
B^{\alpha-1}, \hat{B}^{\alpha-1}) + 2A \sig \sqrt{ \log(np) }\\
&\leqslant 6A C \sig \sqrt{ \log(np) } \frac{ b^{(\al-1)/(\al
-2)} }{\eta^{(\al-1)/(2(\al-2))} } + 2A \sig \sqrt{ \log(np) }\\
&= 2A \sig \sqrt{ \log(np) } \left[3 C \left(\frac{ b
}{\sqrt{\eta}}\right)^{(\al-1)/(\al -2)}
+ 1 \right], } and, applying Theorem~\ref{th:LassoL1p}, we
obtain (a) and (b).
c) Apply c) in Theorem~\ref{th:LassoL1p}, summing over $i\in
\sci$:
\eqsplit{
\sum_{i\in \sci} \scm(\hat\beta_i) &\leq \|X ^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}(\beta -\hat\beta )\|_2^2
\frac{m \phi_{ \max}} {(\mu\delta)^2}\\
&\leq \frac{4 s n \phi_{ \max} }{\kappa^2 \, \delta^2 }
\left[1 + 3C \left(\frac{ b
}{\sqrt{\eta}}\right)^{(\al-1)/(\al -2)}
\right]^2. }
\end{proof}
\subsection{Proofs of Section \ref{sec:lassoes}}
\begin{proof}[Proof of Theorem \ref{th:lassoes1}]
Note that by the definition of $\dacc\ti\hat\beta_i$ and \eqref{roughapprox}.
\eqsplit[et1]{
&\hspace{-3em}mnc_n +\lm_n\summ i1n\|\dacc\ti\hat\beta_{i}\|_1^2
\\
&\leq
m\summ i1n \dacc\ti\hat\beta_i^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}\ti \Sig_i \dacc\ti\hat\beta_i+\lm_n\summ i1n \|\dacc\ti\hat\beta_i\|_1^2
\\
&\leq m\summ i1n \dacc\ti\hat\beta_i^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}\ti S_i \dacc\ti\hat\beta_i+(\lm_n+m\del_n)\summ i1n \|\dacc\ti\hat\beta_i\|_1^2
\\
&\leq m\summ i1n \ti\beta_{i0}^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}\ti S_i \ti\beta_{i0}+\lm_n\summ i1n \|\ti\beta_{i0}\|_1^2
+m\del_n\summ i1n \|\dacc\ti\hat\beta_i\|_1^2
\\
&\leq m\summ i1n \ti\beta_{i0}^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}\ti \Sig_i \ti\beta_{i0}
+ (\lm_n+m\del_n)\summ i1n \|\ti\beta_{i0}\|_1^2
+ m\del_n\summ i1n \|\dacc\ti\hat\beta_i\|_1^2
\\
&= mnC_n + (\lm_n+m\del_n)\summ i1n \|\ti\beta_{i0}\|_1^2
+ m\del_n\summ i1n \|\dacc\ti\hat\beta_i\|_1^2 .
}
Comparing the LHS with the RHS of \eqref{et1}, noting that $m\del_n\ll\lm_n$:
\eqsplit{
\summ i1n \|\dacc\ti\hat\beta_i\|_1^2
&\leq mn\frac{C_n-c_n}{\lm_n-m\del_n}
+ \frac{\lm_n+m\del_n}{\lm_n-m\del_n} \summ i1n \|\ti\beta_{i0}\|_1^2.
}
By \eqref{roughapprox} and \eqref{lassoes}:
\eqsplit[bpl1]{
\summ i1n \dacc\ti\hat\beta_{i}^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}\ti \Sig_i \dacc\ti\hat\beta_{i}
&\leq \summ i1n \dacc\ti\hat\beta_{i}^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}\ti S_i \dacc\ti\hat\beta_{i} + \del_n\summ i1n \|\dacc\ti\hat\beta_{i}\|_1^2
\\
&\leq \summ i1n \ti\beta_{i0}^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}\ti S_i \ti\beta_{i0}
+ \frac {\lm_n} m \summ i1n \|\ti\beta_{i0}\|_1^2
- \frac {\lm_n} m \summ i1n \|\dacc\ti\hat\beta_{i}\|_1^2 + \del_n\summ i1n \|\dacc\ti\hat\beta_{i}\|_1^2
\\
&\leq \summ i1n \ti\beta_{i0}^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}\ti \Sig_i \ti\beta_{i0}
+ (\frac{\lm_n} m+\del_n) \summ i1n \|\ti\beta_{i0}\|_1^2
- (\frac {\lm_n} m-\del_n) \summ i1n \|\dacc\ti\hat\beta_{i}\|_1^2
\\
&\leq \summ i1n \ti\beta_{i0}^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}\ti \Sig_i \ti\beta_{i0}
+ (\frac{\lm_n} m+\del_n) \summ i1n \|\ti\beta_{i0}\|_1^2
.
}
The result follows.
\end{proof}
\begin{proof}[Proof of Theorem \ref{th:lassoes2}]
The proof is similar to the proof of Theorem \ref{th:lassoes1}. Similar to \eqref{et1} we obtain:
\eqsplit[et2]{
&\hspace{-1em}mnc_n +\lm_n\summ i1n\|\dacc\ti\hat\beta_{i}\|_1^\al
\\
&\leq
m\summ i1n \dacc\ti\hat\beta_i^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}\ti \Sig_i \dacc\ti\hat\beta_i+\lm_n\summ i1n \|\dacc\ti\hat\beta_i\|_1^\al
\\
&\leq m\summ i1n \dacc\ti\hat\beta_i^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}\ti S_i \dacc\ti\hat\beta_i
+\lm_n\summ i1n \|\dacc\ti\hat\beta_i\|_1^\al
+m\del_n\summ i1n \|\dacc\ti\hat\beta_i\|_1^2
\\
&\leq m\summ i1n \ti\beta_{i0}^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}\ti S_i \ti\beta_{i0} + \lm_n\summ i1n \|\ti\beta_{i0}\|_1^\al
+m\del_n\summ i1n \|\dacc\ti\hat\beta_i\|_1^2
\\
&\leq m\summ i1n \ti\beta_{i0}^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}\ti \Sig_i \ti\beta_{i0}
+ \lm_n\summ i1n \|\ti\beta_{i0}\|_1^\al
+ m\del_n\summ i1n \|\ti\beta_{i0}\|_1^2
+ m\del_n\summ i1n \|\dacc\ti\hat\beta_i\|_1^2
\\
&= mnc_n + \lm_n\summ i1n \|\ti\beta_{i0}\|_1^\al
+ m\del_n\summ i1n \|\ti\beta_{i0}\|_1^2
+ m\del_n\summ i1n \|\dacc\ti\hat\beta_i\|_1^2 .
}
That is,
\eqsplit[alsqb]{
\summ i1n (\lm_n \|\dacc\ti\hat\beta_i\|_1^\al - m\del_n \|\dacc\ti\hat\beta_i\|_1^2)
&\leq \lm_n\summ i1n \|\ti\beta_{i0}\|_1^\al
+ m\del_n\summ i1n \|\ti\beta_{i0}\|_1^2
\\
&= \EuScript{O}(mn\del_n).
}
It is easy to see that the maximum of $\summ
i1n\|\dacc\ti\hat\beta_i\|_1^2$ subject to the constraint \eqref{alsqb}
is achieved when $\|\dacc\ti\hat\beta_1\|_1^2 = \dots =
\|\dacc\ti\hat\beta_n\|_1^2$. That is when $\|\dacc\ti\hat\beta_i\|_1^2$
solves $\lm_n u^\al - m\del_n u^2 = \EuScript{O}(m\del_n)$. As $\lambda_n
=\EuScript{O}(m \delta_m) $, the solution satisfies
$u=\EuScript{O}(m\del_n/\lm_n)^{1/(\al-2)}$.
Hence we can conclude from \eqref{alsqb}
\eqsplit{
\summ i1n \|\dacc\ti\hat\beta_i\|_2^2 &= \EuScript{O}(n (m\del_n/\lm_n)^{2/(\al-2)} )
}
We now proceed similar to \eqref{bpl1}
\eqsplit{
\summ i1n \dacc\ti\hat\beta_{i}^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}\ti \Sig_i \dacc\ti\hat\beta_{i}
&\leq \summ i1n \dacc\ti\hat\beta_{i}^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}\ti S_i \ti\beta_{i} + \del_n\summ i1n \|\dacc\ti\hat\beta_{i}\|_1^2
\\
&\leq \summ i1n \ti\beta_{i0}^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}\ti S_i \ti\beta_{i0}
+ \frac{\lm_n}m \summ i1n \|\ti\beta_{i0}\|_1^\al
- \frac{\lm_n}m \summ i1n \|\dacc\ti\hat\beta_{i}\|_1^\al
+ \del_n\summ i1n \|\dacc\ti\hat\beta_{i}\|_1^2
\\
&\leq \summ i1n \ti\beta_{i0}^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}\ti \Sig_i \ti\beta_{i0}
+ \frac{\lm_n}m \summ i1n \|\ti\beta_{i0}\|_1^\al
+ \del_n \summ i1n \|\ti\beta_{i0}\|_1^2
+ \del_n \summ i1n \|\dacc\ti\hat\beta_i\|_1^2
\\
&\leq \summ i1n \ti\beta_{i0}^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}\ti \Sig_i \ti\beta_{i0}
+ \EuScript{O}_p(n (m/\lm_n)^{2/(\al-2)} \del_n^{\al/(\al-2)}),
}
since $\lambda_n =\EuScript{O}(m \delta_m) $.
\end{proof}
\begin{proof}[Proof of Remark~\ref{rem:lassoes2}]
If $m \delta_m/\lambda =\ensuremath{\text{\sfa$\O$\sfb}}(1)$, then, following the proof of
Theorem~\ref{th:lassoes2}, the solution maximising $\summ
i1n\|\dacc\ti\hat\beta_i\|_1^2$ subject to the constraint \eqref{alsqb}
satisfies $\|\dacc\ti\hat\beta_i\|_1=\EuScript{O}(1)$, and hence we have
\eqsplit{
\summ i1n \dacc\ti\hat\beta_{i}^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}\ti \Sig_i \dacc\ti\hat\beta_{i}
&\leq \summ i1n \ti\beta_{i0}^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}\ti \Sig_i \ti\beta_{i0}
+ \EuScript{O}_p\left( n\lm_n/m + n\del_n\right).
}
\end{proof}
\subsection{Proofs of Section \ref{sec:spectral} }
\begin{proof}[Proof of Lemma \ref{lem:spectVsGroup}]
Let $\scb=\summ \xi1k \al_\xi \beta_\xi^* {\mathfrak{b}_\xi^*}^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}$ be the
spectral decomposition of $\scb$, where
$\beta_1^*,\dots,\beta_k^*$ are orthonormal $\R^p$ vectors,
$\mathfrak{b}_1^*,\dots,\mathfrak{b}_k^*$ are orthonormal $\R^n$ vectors,
$\al_1,\dots,\al_k\ge 0$, and $k=\min\{p,n\}$. Clearly
$|||\scb|||_1=\summ \xi1k \al_\xi$. Let $U=\summ \xi1k
e_\xi{\beta_\xi^*}^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em} $ where $e_1,\dots,e_p$ is the natural basis
of $\R^n$. Then
\eqsplit{
\|U\scb\|_{2,1} &= \| \summ\xi1k \al_\xi e_\xi{\mathfrak{b}_\xi^*}^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em} \|_{2,1} = \summ\xi1k \al_\xi = |||\scb|||_1.
}
Let $\scb=\summ \xi1k e_\xi\mathfrak{b}_\xi^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}$ where $\mathfrak{b}_1,\mathfrak{b}_2,\dots,\mathfrak{b}_k$ are orthogonal, and let $U$ be a unitary matrix. Then by Schwarz inequality
\eqsplit{
\|\scb\|_{2,1}&=
\summ j1p \|\mathfrak{b}_j\|
\\
&= \summ i1p \summ j1p U_{ij}^2 \|\mathfrak{b}_j\|\qquad&\text{since} \summ i1pU_{ij}^2=1
\\
&\le \summ i1p \sqrt {\summ j1p U_{ij}^2 \|\mathfrak{b}_j\|^2}\sqrt{\summ j1p U_{ij}^2 }\qquad&\text{by Schwarz inequality}
\\
&= \summ i1p \sqrt {\summ j1p U_{ij}^2 \|\mathfrak{b}_j\|^2} \qquad&\text{since} \summ j1pU_{ij}^2=1
\\
&= \|U\scb\|_{2,1}
}
which completes the proof of the (i).
Now, consider the $U$ defined as above for the solution of
\eqref{specPen}. Let $\ti X_i$ be the design matrices $\ti\scb$ be
the solution expressed in this basis. By the first part of the
lemma $|||\ti\scb|||_1=\|\ti\scb\|_{2,1}$. Suppose there is a
matrix $\scb\ne\ti\scb$ which minimizes the group lasso penalty.
Hence
\eqsplit{
\summ i1n \|Y_i-\ti X_i \beta_i\|^2 + \lm ||| \scb|||_1
&\leq
\summ i1n \|Y_i-\ti X_i \beta_i\|^2 + \lm \|\scb\|_{2,1}
\\
&< \summ i1n \|Y_i-\ti X_i \ti \beta_i\|^2 + \lm \|\ti\scb\|_{2,1}
\\
&= \summ i1n \|Y_i-\ti X_i \ti\beta_i\|^2 + \lm |||\ti \scb|||_1,
}
contradiction since $\ti\scb$ minimized \eqref{specPen}. Part (ii) is proved.
\end{proof}
\begin{proof}[Proof of Theorem \ref{th:sparseSpect} ]
Let $A=\summ i1n \hat\beta_i\hat\beta_i^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em} = \hat\scb \hat\scb^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}$
be of rank $s\leqslant p < n$, and hence the spectral
decomposition of $\hat\scb$ can be written as $\hat\scb=\summ
\xi1s \al_\xi \beta_\xi^{*}{\mathfrak{b}_\xi^*}^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}$, where
$\beta_1^*,\dots,\beta_s^* \in \mathbb{R}^p$ are orthonormal, and
so are $\mathfrak{b}_1^*,\dots,\mathfrak{b}_s^* \in \mathbb{R}^n$. Hence, the
rotation $U$ leading to a sparse representation $U \hat\scb$ (with
$s$ non-zero rows) is given by $U =\summ \xi1s e_\xi
{\beta_\xi^*}^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}$, where $e_1,\dots,e_p$ is the natural basis of
$\R^p$. Another way to write the rotation matrix is $U =
({\beta_1^*}^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em},\dots, {\beta_s^*}^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}, \mathbf{0}^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em},\dots,
\mathbf{0}^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em})^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}$. Denote by $U_S$ the non-zero $s\times
p$-dimensional submatrix $({\beta_1^*}^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em},\dots, {\beta_s^*}^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em})^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}$.
Let $A(t) = A+ t(\ti\beta \hat\beta_i ^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}+\hat\beta_i \ti\beta ^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em})+t^2\ti\beta \ti\beta^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}$ for some fixed $i$, with $\ti\beta \in
\fun{span}\{\hat\beta_1,\dots,\hat\beta_n\} = \fun{span}\{\hat\beta_1^*,\dots,\hat\beta_s^*\}$.
\marg{I removed the proof of the lemma on convexity of the norm,
and added this paragraph here, to replace the ref to the Lemma}
If $(x_k(t),c_k(t))$ is an eigen-pair of $A(t)$, then taking the
derivative of \(x_i^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em} x_i=1\) yields \(x_i^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}\dot x_i=0\), and
trivially, since $x_i$ is an eigenvector, also \(x_i^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em} A\dot
x_i=0\). Here $\dot{}$ and $\ddot{}$ the first and second
derivative, respectively, according to $t$. Also, we have
\eqsplit{
x_k(t) &=x_k + tu_k+\ensuremath{\text{\sfa$\O$\sfb}}(t)
\\
c_k(t) &=c_k + t\nu_k+\ensuremath{\text{\sfa$\O$\sfb}}(t)
}and
\eqsplit{
\Bigl(A +t(\ti\beta\hat\beta_i^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}+\hat\beta_i\ti\beta^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em})\Bigr)(x_k + t u_k) &= (c_k+t\nu_k)(x_k+t
u_k)+\ensuremath{\text{\sfa$\O$\sfb}}(t),
}
where $u_k\perp x_k$.
Equating the $\EuScript{O}(t)$ terms obtain
\eqsplit{
A u_k + (\ti\beta \hat\beta_i^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}+\hat\beta_i\ti\beta^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em})x_k &= c_k u_k + \nu_k x_k.
}
Take now the inner product of both sides with $x_k$ to obtain that
\eqsplit[nuk]{
\nu_k=2(\ti\beta^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em} x_k)(x_k^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}\hat\beta_i).
} Note that the null space of $A(t)$ does not depend on $t$.
Hence, if we call $\psi(\scb)=|||\scb|||_1$,
\eqsplit{
\frac{\partial}{\partial t } \psi(A(t))|_{t=0}
&= \sum_{c_k>0} \frac{\partial}{\partial t }c_k^{1/2}(t)|_{t=0}
\\
&= \frac12 \sum_{c_k> 0} \frac{\nu_k}{c_k^{1/2}}
\\
&= \ti\beta^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em} \sum_{c_k> 0} c_k^{-1/2} x_kx_k^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em} \hat\beta_i
\\
&= \ti\beta^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em} A^{+1/2}\hat\beta_i = \ti\beta^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}
(\hat\scb\hat\scb^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em})^{+1/2}\hat\beta_i\\
&= \ti\beta^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em} U_S^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em} (U_S\hat\scb\hat\scb^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em} U_S^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em})^{-1/2} U_S\hat\beta_i,
} where $A^{+1/2}$ is the generalized inverse of $A^{1/2}$.
Taking, therefore, the derivative of the target function with
respect to $\hat\beta_i$ in the directions of
$\ti\beta\in\fun{span}\{\hat\beta_1,\dots,\hat\beta_n\}$ (e.g., in
the directions $\ti\beta=\beta_{\xi}^*$, $\xi=1,\dots,s$) gives
\eqsplit{
0 &= (\beta_{\xi}^*)^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em} ( -2 X_{i}^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em} (Y_i - X_i\hat\beta_i) + \lm (\hat\scb \hat\scb
^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em})^{+1/2}\hat\beta_i), \quad \text{or, equivalently,}\\
\mathbf{0} &= U_S( 2 X_{i}^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em} (Y_i - X_i\hat\beta_i) - \lm (\hat\scb \hat\scb ^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em})^{+1/2}
\hat\beta_i).
}
Let $R= (r_1,\dots,r_p)^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}$ be the matrix of projected residuals:
\eqsplit{
R_{\ell i} = \summ j1m x_{ij\ell}(y_{ij}-x_{ij}^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}\hat\beta_i), \quad \ell=1,\dots,p\,; \; i=1,\dots,n.
}
Then
\eqsplit{
U_S R &=\frac\lm2 U_S (\hat\scb \hat\scb ^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em})^{+1/2}\hat\scb.
}
Consider again the general expansion $\hat\scb=\summ \xi1{p\wedge
n} \alpha_\xi\beta_\xi^* {\mathfrak{b}_\xi^*}^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}$. Then
$|||\hat\scb|||_1=\summ\xi1{p\wedge n} |\al_\xi|$. Taking the
derivative of the sum of squares part of the target function with
respect to $\al_\xi$ we get
\eqsplit{
\summ i1n \mathfrak{b}^*_{\xi i}{\beta^*_\xi}^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em} X_i^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}(Y_{i}-X_{i}\hat\beta_i)
&= {\beta_\xi^*}^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em} R \mathfrak{b}_\xi^*.
}
Considering the sub-gradient of the target function we obtain that
$|{\beta_\xi^*}^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em} R \mathfrak{b}_\xi^*| \leq \lm/2$, and $\al_\xi=0$ in
case of strict inequality.
\end{proof}
\begin{proof}[Proof of Theorem \ref{th:BRTspectral} ]
(a) and (b) Similarly to the proof of Theorem~\ref{th:LassoL1p},
we have
\eqsplit{
\|Y-X^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}\hat\scb\|_2^2 & = \|Y-X^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}\scb\|_2^2
+2\sum_{ij}\eps_{ij}x_{ij}^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}(\beta_i - \hat\beta_i).
}
The last term can be bounded with high probability. Introduce
matrix $M$ with independent columns $M_i = X_i \varepsilon_i \sim
\scn_p(\mathbf{0}, m\sigma^2 I_p)$, $i=1,\dots,n$, since $\sum_j
x_{ij\ell}^2 = m$. Denote $q$-Schatten norm by $|||\cdot|||_q$.
Using the Cauchy-Swartz inequality and the equivalence between
$\ell_2$ (Frobenius) and Schatten with $q=2$ norms, we obtain:
\eqsplit{
|\sum_{ij} \eps_{ij}x_{ij}^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}(\beta_i -\hat\beta_i)| &=
|\sum_{i\ell} M_{i\ell} (\beta_{i\ell} -\hat\beta_{i\ell})|
\leqslant ||\scb-\hat\scb||_2\, ||M||_{2} =
|||\scb-\hat\scb|||_2\, ||M||_{2}\\ &\leqslant
|||\scb-\hat\scb|||_1\, ||M||_{2}.
}
Now, $||M||_{2}^2 \sim m\sigma^2 \chi^2_{np}$ hence it can
be bounded by $B^2=m\sigma^2 (np + c)$ (Lemma A.1, Lounici et
al.~\cite{p:Lounici-GroupLasso}) with probability at least $1-
\exp\left(-\frac 1 8 \min(c, c^2/(np))\right)$. Denote this event
by \sca. Hence, we need to choose $c$ such that $c/\sqrt{np} \to
\en$. For example, we can take $c=A np$ with $A>1$, then $B= \sig
\sqrt{(1+A)mnp}$, and, since $\min(Anp, A^2np) = Anp$, the
probability is at least $1 - e^{-Anp/2}$.
Denote by $V$ the subspace of $\mathbb{R}^p$ corresponding to the
union of subspaces where the eigenvalues of $\scb \scb^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}$ are
non-zero, and by $P_V$ the projection on that space. Then,
$\mathbb{R}^p = V \oplus V^c$ and $\dim(V) = {\text{rank}}(\scb)\leqslant
s$.
Hence, adding $\lambda_2 |||\scb-\hat\scb|||_1$ to both sides, we
have that on \sca,
\eqsplit{
\|X^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}(\scb-\hat\scb)\|_2^2 +\lm_2 |||\scb-\hat\scb|||_1 & \leqslant \lambda |||\scb|||_1 - \lambda |||\hat\scb|||_1 +(2 B+\lm_2) |||\scb-\hat\scb|||_1\\
&\leqslant \lambda |||P_V \scb|||_1 -
\lambda \trace(P_V|\hat\scb| + (I-P_V)|\hat\scb|)\\ &+ (2 B+\lm_2)
|||P_V(\scb-\hat\scb)|||_1\\ &+(2 B+\lm_2) |||(I-P_V)(\scb-\hat\scb)|||_1\\
&\leqslant \lambda \trace(|P_V \scb|) - \lambda
\trace(P_V|\hat\scb|)
+ (2 B+\lm_2) \trace(|P_V(\scb-\hat\scb)|)\\ &+(2 B+\lm_2) \trace(|(I-P_V) \hat\scb|) - \lm \trace((I-P_V)|\hat\scb|)\\
&\leqslant (\lambda +2 B+\lm_2) \trace(|P_V(\scb-\hat\scb)|), }
if $\lambda \geqslant 2B+\lambda_2$, since $ \trace(|P_V
\hat\scb|)=\trace(|P_V|\, |\hat\scb|)=\trace( P_V \, |\hat\scb|)$.
Here $|A| = (AA^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em})^{1/2}$.
We can take, e.g.
$\lambda_2 =2B =\lambda/2$, implying that $\lm =
4\sigma\sqrt{(1+A)mnp}$.
Hence, we have that $\frac{\lm} 2 ||| \scb-\hat\scb||| \leqslant 2
\lm |||P_V(\scb-\hat\scb)|||$, i.e. $ |||(I-P_V)
(\scb-\hat\scb)||| \leqslant 3 \lm |||P_V(\scb-\hat\scb)|||$.
Thus, applying RE2$(s, 3,\kappa)$, ${\text{rank}}(\scb)\leqslant s$, we
have that
\eqsplit{
\|X^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}(\beta - \hat{\beta})\|_2^2 &\leqslant 2\lm |||P_V(\scb-\hat\scb)|||_1 \leqslant 2\lambda \sqrt{s} |||P_V(\scb-\hat\scb)|||_2\\
&= 2\lambda \sqrt{s} ||P_V(\scb-\hat\scb)||_2 \leqslant \frac{2\lambda \sqrt{s}}{\kappa\sqrt{m}} ||X^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}(\beta - \hat{\beta})||_2
}
hence
\eqsplit{
\|X^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}(\beta - \hat{\beta})\|_2 & \leqslant \frac{2\lambda
\sqrt{s}}{\kappa\sqrt{m}}.
}
Using this and the RE2 assumption,
\eqsplit{
||| \scb-\hat\scb |||_1 \leqslant 4 |||P_V( \scb-\hat\scb) |||_1
\leqslant \frac{4\sqrt{s}}{\kappa \sqrt{m}} \|X^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}(\beta -
\hat{\beta})\|_2 & \leqslant \frac{8\lambda
s}{\kappa^2 m}.
}
Substituting the value of $\lambda$, we obtain the results.
(c) Since $\hat\gamma_i = \hat{U}\hat\beta_i$ are the
solution of group lasso problem with design matrices $\ti{X}_i =
\hat{U} X_i$, for $\ell \in J(\hat\gamma)$: $\|\hat\gamma_{\cdot
\ell}\|_2 \neq 0$, $\hat\gamma_{i\ell}$ satisfies the following
equations;
$$
2 \tilde{X}_{i\cdot \ell}^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}(Y_i -X_i\hat\beta_i) =\lm
\frac{\hat\gamma_{i\ell}}{ ||\hat\gamma_{\cdot
\ell}||_2}
$$
(see also Theorem~\ref{th:sparseSpect}).
Hence,
$$
\summ i1n \left(\tilde{X}_{i\cdot \ell}^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}(Y_i -X_i\hat\beta_i)\right)^2 =\frac{\lm^2}
4.
$$
On one hand, for $\ell\in J(\hat\gamma)$, \eqsplit{ \left[ \summ
i1n \left(\tilde{X}_{i\cdot \ell} X_i^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}(\hat\beta_i
-\beta_i)\right)^2\right]^{1/2} &\geqslant \left[ \summ i1n
\left(\tilde{X}_{i\cdot \ell} (Y_i
-X_i^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}\hat\beta_i)\right)^2\right]^{1/2}\\
&- \left[ \summ i1n \left(\tilde{X}_{i\cdot
\ell} (Y_i -X_i^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}\beta_i)\right)^2\right]^{1/2}\\
&= \frac {\lm}{2} - \left( \summ i1n (U_\ell X_{i}
\varepsilon_i)^2\right)^{1/2}.}
On event \sca, \eqsplit{ \summ i1n (U_\ell X_{i} \varepsilon_i)^2
&= \summ i1n (U_\ell M_i)^2\leqslant \summ i1n ||U_\ell||_2^2
||M_i||^2 = |||M|||_2^2 \leqslant B^2 = (\lambda/4)^2. }
Summing over $\ell\in J(\hat\gamma)$, we have
\eqsplit{
\sum_{\ell\in J(\hat\gamma)} \summ i1n \left(\tilde{X}_{i\cdot
\ell} X_i^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}(\hat\beta_i -\beta_i)\right)^2 &\geqslant
\scm(\hat{\gamma}) \left( \frac{\lm}{2} - \frac{\lm}{4}\right)^2 =
\scm(\hat{\gamma}) \frac {\lm^2}{16} .}
On the other hand, \eqsplit {
\summ \ell 1 s \summ i1n \left(\tilde{X}_{i\cdot \ell}
X_i^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}(\hat\beta_i-\beta_i)\right)^2 &\leqslant \summ i1n
||\tilde{X}_{i} X_i^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}(\hat\beta_i-\beta_i)||^2_2 = \summ i1n
||X_{i} X_i^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}(\hat\beta_i-\beta_i)||^2_2\\ &\leqslant m\phi_{\max}
|| X^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}(\hat\scb-\scb)||^2_2. }
Since ${\text{rank}}(\hat\scb) = \scm(\hat{\gamma})$,
$$
{\text{rank}}(\hat\scb) \leqslant \frac{m\phi_{\max}
|| X^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}(\hat\scb-\scb)||^2_2}{(\lm/4)^2} = \frac{16 m \phi_{\max}}{\lm^2} \frac{4\lm^2 s}{m \kappa^2} = s\, \frac{64 \phi_{\max}}{\kappa^2}.
$$
\end{proof}
\begin{proof}[Proof] of Theorem~\ref{th:persist}.
Using Lemma~\ref{lem:persist}, with probability at least $1 -
\eta$,
$$
| L_{F}(\beta) - L_{\hat{F}}(\beta) | \leqslant \frac 1 {nm }
\sqrt{\frac{4e V \log (np)} {m\eta}} (n+ \summ i1n
\|\beta_i\|_1^2),
$$
since $n>1$. Note that if $n=1$, it is sufficient to replace $p$
by $p+1$ under the logarithm.
In our case, the estimators are in set $B_{n,p}$. If $\summ i1n
\beta_i \beta_i^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em} = U^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em} \Lambda U$ is the spectral decomposition,
and $\gamma_i = U \beta_i$, $\Lambda_{kk} = ||\gamma_{\cdot
k}||_2^2$, $\gamma_{\cdot k}$ are orthogonal, hence
$$
\trace\{\summ i1n \beta_i \beta_i^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em} \}^{1/2} = \summ k1p
||\gamma_{\cdot k}||_2.
$$
Thus, we need to bound $\summ i1n \|\beta_i\|_1^2 $ in terms of
$\summ k1p ||\gamma_{\cdot k}||_2$.
\marg{You have introduce a factor 2. I didn't see why, any way, the original expansion is after this one, with \% signs}\eqsplit {
\summ i1n \|\beta_i\|_1^2
&\leqslant \summ i1n M(\beta_i) \|\beta_i\|_2^2
\\
&= \max_i M(\beta_i) \summ i1n \|\gamma_i\|_2^2
\\
&= \max_i M(\beta_i) \summ \ell 1p \|\gamma_{\cdot \ell}\|_2^2
\\
&\leqslant 2
\max_i M(\beta_i) \left(\summ \ell 1p \|\gamma_{\cdot \ell}\|_2\right)^2
\\
&\leqslant \max_i M(\beta_i) b^2,
}
since $\summ \ell 1p \|\gamma_{\cdot \ell }\|_2 \leqslant b$.
Hence, with probability at least $1 - \eta$,
\eqsplit{
\sup_{F\in {\cal F} } P_{F} \left( L_{F}\left(\hat\beta\right) -
L_{F}\left(\beta^*_{F} \right)\right)
\leqslant 2\left(\frac 1
m + \frac{\max_i M(\beta_i) b^2}{nm}\right) \sqrt{\frac{4e V \log (np)}
{m\eta} }.}
Note that we can use $p$ instead of $\max_i M(\beta_i)$.
The theorem is proved.
\end{proof}
\subsection{Proofs of Section \ref{sec:spectral} }
\begin{proof}[Proof of Lemma \ref{lem:psiConvex}]
Let \(A(t) = \sum \bigl(\beta_i(t)+ t\ti\beta_i(t)\bigr)\bigl(\beta_i(t)+ t\ti\beta_i(t)\bigr)^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em} = \sum \beta_i\beta_i^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em} + t(\sum \beta_i\ti\beta_i^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em} + \sum \ti\beta_i\beta_i^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}) + t^2\sum\ti\beta_i\ti\beta_i^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}\). Let \(x_1(t),\dots,x_p(t)\) be a smooth orthonormal basis of eigenvectors of $A(t)$, and \(A(t) = \sum c_i(t)x_i(t)x_i(t)^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}\).
Denote by $\dot{}$ and $\ddot{}$ the first and second derivative, respectively, according to $t$. Then, note that taking the derivative of \(x_i^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em} x_i=1\) yields \(x_i^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}\dot x_i=0\), and trivially, since $x_i$ is an eigenvector, also \(x_i^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em} A\dot x_i=0\). Differentiating \(A(t)x_k(t) = c_k(t) x_k(t) \):
\eqsplit{
\dot A x_k + A\dot x_k &= \dot c_k x_k + c_k\dot x_k.
}
But $\dot x_k\perp x_k$, hence
\eqsplit{
\dot x_i &= -(A - c_k I)^+\dot A x_k
}
Hence, generally:
\eqsplit[eigender]{
c_k &= x_k^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em} A x_k
\\
\dot c_k &= x_k^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em} \dot A x_k \qquad\qquad\qquad\text{(Recall \(\dot x_k\perp x_k\))}
\\
\ddot c_k &= 2 x_k^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em} \dot A \dot x_k + x_k ^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}\ddot A x_k.
\\
&= -2x_k^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}\dot A (A-c_kI)^+\dot Ax_k + x_k ^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}\ddot A x_k.
}
For the particular matrices \(A(t)\):
\eqsplit{
c_k &= \sum (x_k^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}\beta_i)^2
\\
\dot c_k &= 2\sum (x_k^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}\beta_i) (x_i^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em} \ti \beta_k)
\\
\ddot c_k &= -2 x_k^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em} \bigl(\sum \beta_i\ti\beta_i^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em} + \sum \ti\beta_i\beta_i^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}\bigr) \bigr(\sum_{i\ne k} \frac1{c_i-c_k}x_ix_i^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}\bigl)\bigl(\sum \beta_i\ti\beta_i^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em} +\sum \ti\beta_i\beta_i^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}\bigr)x_k
\\
&\hspace{3em}+ 2\sum_i (x_k^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}\ti\beta_i)^2
\\
&= -2 \sum_i \sum_{j\ne k} \sum_m \frac1{c_j-c_k} (x_k^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}\beta_i)(x_j^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}\ti\beta_i)(x_j^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}\ti\beta_m)
(x_k^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}\beta_m)
\\
& \hspace{3em }-4\sum_i \sum_{j\ne k} \sum_m \frac1{c_j-c_k} (x_k^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}\beta_i)(x_j^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}\ti\beta_i)(x_j^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}\beta_m)(x_k^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}\ti\beta_m)
\\
& \hspace{3em }- 2 \sum_i \sum_{j\ne k} \sum_m \frac1{c_j-c_k} (x_k^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}\ti\beta_i)(x_j^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}\beta_i)(x_j^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}\beta_m)
(x_k^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}\ti\beta_m)
\\
&\hspace{3em}+ 2\sum_i (x_k^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}\ti\beta_i)^2
\\
&= -2\sum_{j\ne k} \frac1{c_j-c_m}\Bigl(\sum_i (x_k^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}\beta_i)(x_j^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}\ti\beta_i) + \sum_i (x_k^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}\ti\beta_i)(x_j^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}\beta_i)\Bigr)^2
\\
&\hspace{3em}+ 2\sum_i (x_k^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}\ti\beta_i)^2
}
Now,
\eqsplit{
\frac{d^2}{dt^2}\sqrt{c_k(t)}\Bigl|_{t=0}&=\frac{d}{dt}\frac{\dot c_k}{2c_k^{1/2}} = \frac{2\ddot c_k c_k - \dot c_k^2 }{4c_k^{3/2}}
}
Hence
\eqsplit{
\sum_k \frac{d^2}{dt^2}\sqrt{c_k(t)}\Bigl|_{t=0}
&= 2\sum_k \frac{\sum_i (x_k^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}\ti\beta_i)^2\sum_i (x_k^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}\beta_i)^2 - \bigl(\sum_i (x_k^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}\beta_i)(x_k^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}\ti\beta_i)\bigr)^2}{4c_k^{3/2}}
\\
&\hspace{3em} -\sum_k\sum_{j\ne k} \frac1{c_j-c_k}\frac{\Bigl(\sum_i (x_k^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}\beta_i)(x_j^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}\ti\beta_i) + \sum_i (x_k^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}\ti\beta_i)(x_j^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}\beta_i)\Bigr)^2}{c_k^{1/2}}
\\
&= 2\sum_k \frac{\sum_i (x_k^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}\ti\beta_i)^2\sum_i (x_k^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}\beta_i)^2 - \bigl(\sum_i (x_k^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}\beta_i)(x_k^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}\ti\beta_i)\bigr)^2}{4c_k^{3/2}}
\\
&\hspace{3em} +2\sum_k\sum_{j> k} \frac{\Bigl(\sum_i (x_k^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}\beta_i)(x_j^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}\ti\beta_i) + \sum_i (x_k^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}\ti\beta_i)(x_j^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}\beta_i)\Bigr)^2}{c_k^{1/2}c_j^{1/2} (c_k^{1/2}+c_j^{1/2})}
\\
&\ge 0.
}
Since \(\psi(\beta_1+t\ti\beta_1,\dots,\beta_p+t\ti\beta_p) = \sum_k\sqrt{c_k(t)}\), the lemma was proven.
\end{proof}
\begin{proof}[Proof of Theorem \ref{th:sparseSpect}]
Since the target function is invariant to rotation of $x$ and $\beta$ at the same time, we assume for simplicity that $U=I$. Suppose that \(S\) is the set of non-zero components of the estimator. Let $\hat\beta_{S}\in\R^{|S|+`}$ be the vector of nonzero entries of $\hat\beta$ in the first $|S|$ places and 0 in the $(|S|+1)$ place, let $X_{Si}$ be the corresponding columns extracted from the covariates matrix $X_i$ of the $i$-th trial. Let \(\beta_{iN}\) be \(b_i e_{|S|+1}\), where \(\ell\not\in S\), and \(e_\ell\) is the standard vector with 1 at the \(\ell\)-th position and 0 otherwise. Consider now the target function at a small neighborhood \(t=0\):
\eqsplit{
\sum_i \sum_j \|y_{ij} -x_{Sij}^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}\hat\beta_{Si}+tx_{ij\ell}b_i)\|^2 + \lm\psi (\beta_{1S}+t\beta_{1N},\dots,\beta_{nS}+t \beta_{nN}).
}
Now the spectrum is not changed by elementary operations. Hence
\eqsplit{
&\hspace{-3em}\mathfrak{S}\begin{pmatrix} \sum \hat\beta_i\hat\beta_i^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em} & t\sum b_i\hat\beta_i \\ t\sum b_i \hat\beta_i^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em} & t^2\sum b_i^2 \end{pmatrix}
\\
&= \mathfrak{S}\begin{pmatrix} \sum_i \hat\beta_i\hat\beta_i^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em} & 0 \\ 0 & t^2\sum b_i^2 - t^2\sum_i b_i\hat\beta_i^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em} \Bigl(\sum_i \hat\beta_i\hat\beta_i^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}\Bigr)^{-1} \sum_ib_i\hat\beta_i \end{pmatrix}
\\
&= \mathfrak{S}\Bigl(\sum \hat\beta_i\hat\beta_i^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}\Bigr) \union t^2 \bigl(\sum b_i^2 - \sum_i b_i\hat\beta_i^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em} \Bigl(\sum_i \hat\beta_i\hat\beta_i^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}\Bigr)^{-1} \sum_ib_i\hat\beta_i \bigr),
}
where $\mathfrak{S} A$ is the spectrum of the matrix $A$. Hence
\eqsplit{
&\hspace{-3em}\frac{\partial}{\partial t}\psi (\beta_{1S}+t\beta_{1N},\dots,\beta_{nS}+t \beta_{nN})
\\
&=\sgn(t)
\bigl(\sum b_i^2 - \sum_i b_i\hat\beta_i^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em} \Bigl(\sum_i \hat\beta_i\hat\beta_i^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}\Bigr)^{-1} \sum_ib_i\hat\beta_i \bigr)^{1/2}
}
Which means that the optimal \(t\) will be zero if for all i
\eqsplit{
|\sum_j x_{ij\ell}(y_{ij} - x_{Sij}^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em} \beta_{Si})| & \leq \lambda\bigl(1 - \hat\beta_i^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em} \Bigl(\sum_k \hat\beta_k\hat\beta_k^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}\Bigr)^{-1} \hat\beta_i \bigr)^{1/2}
}
\end{proof}
\section{Group LASSO: Bayesian perspective}
\label{sec:grouped}
Group LASSO is defined (see Yuan and Lin~\cite{p:GroupLasso-def})
by
\eqsplit[lasso]{
(\hat\beta_1,\dots,\hat\beta_n) &=
\argmin \Biggl[ \summ i1n \summ j1m (y_{ij} - x_{ij}^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}\,\beta_i)^2 + \lambda
\summ \ell1p \Bigl\{\summ i1n \beta_{i\ell}^2\Bigr\}^{1/2} \Biggr]
}
Note that \((\hat\beta_1,\dots,\hat\beta_n)\) are defined as the minimum
point of a strictly convex function, and hence they can be found by equating
the gradient of this function to 0.
Recall the notation $\scb=(\beta_1,\dots,\beta_n)=(\mathfrak{b}_1^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em},\dots,\mathfrak{b}_p^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em})^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}$. Note that \eqref{lasso} is equivalent to the mode of the
a-posteriori distribution when given $\scb$, $Y_{ij}$, $i=1,\dots,n$, $j=1,\dots,m$, are all independent, $y_{ij}\given\scb\dist
\normal(x_{ij}^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}\,\beta_i,\sig^2)$, and a-priori, $\mathfrak{b}_1,\dots,\mathfrak{b}_p$, are \iid, \eqsplit{
f_\mathfrak{b}(\mathfrak{b}_\ell) \propto \exp\bigl\{-\ti\lambda \|\mathfrak{b}_\ell\|_2\bigr\},\quad\ell=1,\dots,p,
}
where $\ti\lm ={\lm}/(2\sigma^2)$. We consider now some property
of this prior. For each $\ell$, $b_\ell$ have a spherically
symmetric distribution. In particular they are uncorrelated and
have mean 0. However, they are not independent. Change of
variables to a polar system where \eqsplit{
R_\ell &=\|\mathfrak{b}_{\ell}\|_2
\\
\beta_{\ell i} &= R w_{\ell i},\qquad w_{\ell}\in\bbs^{n-1},
}
where $\bbs^{n-1}$ is the sphere in $\R^n$. Then, clearly,
\eqsplit[fgamma]{
f(R_\ell,w_{\ell}) &= C_{n,\lm} R_\ell^{n-1} e^{-\ti\lm R_\ell}, \qquad R_\ell>0,
}
where $C_{n,\,\lm} = {\ti\lm^n \Gamma(n/2)}/{2\Gamma(n)
\pi^{n/2}}$. Thus, $R_\ell,w_{\ell}$ are independent $R_\ell\dist
\Gamma (n,\ti\lambda)$, and $w_{\ell}$ is uniform over the unit
sphere.
The conditional distribution of one of the coordinates of
$\mathfrak{b}_\ell$, say the first, given the rest has the form
\eqsplit{
f(\mathfrak{b}_{\ell 1} | \mathfrak{b}_{\ell2},\dots,\mathfrak{b}_{\ell n}, \summ i2n\mathfrak{b}_{\ell i}^2 =\rho^2)
&\propto e^{-\ti\lm\rho \sqrt{1+\mathfrak{b}_{\ell 1}^2/\rho^2}}
}
which for small $\mathfrak{b}_{\ell 1}/\rho$ looks like the normal density
with mean 0 and variance $\rho/\ti\lm$, while for large $\mathfrak{b}_{\ell
1}/\rho$ behaves like the exponential distribution with mean
$\ti\lm^{-1}$.
The sparsity property of the prior comes from the linear component
of log-density of $R$. If $\ti\lm$ is large and the $Y$s are
small, this component dominates the log-a-posteriori distribution
and hence the maximum will be at 0.
Fix now \(\ell\in\{1,\dots,p\}\), and consider the estimating equation
for $\mathfrak{b}_\ell$ --- the \(\ell\) components of the $\beta$'s. Fix the rest of the parameters and let \(\ti
Y_{ij\ell}^\scb = y_{ij}-\sum_{k\ne\ell}\beta_{ik}x_{ijk} \). Then
\(\hat\mathfrak{b}_{\ell i}\), \(i=1,\dots,n\), satisfy
\eqsplit
0&= -\summ j1m x_{ij\ell}(\ti Y^\scb_{ij\ell}-\hat\mathfrak{b}_{\ell i}x_{ij\ell}) +
\frac{\lambda \hat\mathfrak{b}_{\ell i}}{\sqrt{\sum_k \hat\mathfrak{b}_{\ell k}^2}},\qquad
i=1,\dots,n \\
&= -\summ j1m x_{ij\ell}(\ti Y^\scb_{ij\ell}-\hat\mathfrak{b}_{\ell i}x_{ij\ell})
+\lambda^*_{\ell} \hat\mathfrak{b}_{\ell i},\qquad\text{say} .
}
Hence
\eqsplit[betaell]{
\hat\mathfrak{b}_{\ell i} &= \frac{\summ j1m x_{ij\ell}\ti Y^{\scb}_{ij\ell}} {\lambda^*_{\ell}+\summ j1m
x_{ij\ell}^2}.
}
The estimator has an intuitive appeal. It is the least square estimator of
\(\mathfrak{b}_{\ell i}\), \(\summ j1m x_{ij\ell}\ti Y^{\scb}_{ij\ell}/\summ j1m
x_{ij\ell}^2 \), pulled to 0. It is pulled less to zero as the variance of
\(\mathfrak{b}_{\ell1},\dots,\mathfrak{b}_{\ell n}\) increases (and \(\lambda^*_\ell\) is
getting smaller), and as the variance of the LS estimator is lower (i.e.,
when \(\summ j1m x_{ij\ell}^2\) is larger).
If the design is well balanced, \(\summ j1m x_{ij\ell}^2\equiv
m\), then we can characterize the solution as follows. For a
fixed $\ell$, \(\hat\mathfrak{b}_{\ell 1},\cdot,\hat\mathfrak{b}_{\ell n}\) are the
least square solution shrunk toward 0 by the same amount, which
depends only on the estimated variance of \(\hat\mathfrak{b}_{\ell
1},\dots,\hat\mathfrak{b}_{\ell n}\). In the extreme case, \(\hat\mathfrak{b}_{\ell
1}=\dots =\hat\mathfrak{b}_{\ell n}=0\), otherwise (assuming the error
distribution is continuous) they are shrunken toward 0, but are
different from 0.
We can use \eqref{betaell} to solve for \(\lambda^*_{\ell}\)
\eqsplit{
\Bigl(\frac{\lambda}{\lambda^*_{\ell}}\Bigr)^2 &= \| \hat\mathfrak{b}_{\ell}\|_2^2
= \summ i1n \Biggl( \frac{\summ j1m x_{ij\ell}\ti Y^{\scb}_{ij\ell}} {\lambda^*_{\ell}+\summ j1m
x_{ij\ell}^2}\Biggr)^2.
}
Hence \(\lambda^*_{\ell}\) is the solution of
\eqsplit[lmds]{
\lambda^2 &= \summ i1n \Biggl( \frac{\lambda^*_{\ell}\summ j1m x_{ij\ell}\ti Y^{\scb}_{ij\ell}} {\lambda^*_{\ell}+\summ j1m
x_{ij\ell}^2}\Biggr)^2.
}
Note that the RHS is monotone increasing, so \eqref{lmds} has at most a
unique solution. It has no solution if at the limit
\(\lambda^*_{\ell}\to\en\), the RHS is still less than \(\lambda^2\). That is
if
\eqsplit{
\lambda^2 &> \summ i1n \Bigl( \summ j1m x_{ij\ell}\ti Y^{\scb}_{ij\ell}
\Bigr)^2
}
then \(\hat\mathfrak{b}_{\ell}=0\). In particular if
\eqsplit{
\lambda^2 &> \summ i1n \Bigl( \summ j1m x_{ij\ell} Y_{ij\ell}
\Bigr)^2,\qquad\ell=1,\dots,p
}
Then all the random effect vectors are 0. In the balanced case the
RHS is $\EuScript{O}_p(mn\log(p))$. By \eqref{fgamma}, this means that if
we want that the estimator will be 0 if the underlined true
parameters are 0, then the prior should prescribe that $\mathfrak{b}_\ell$
has norm which is $\ensuremath{\text{\sfa$\O$\sfb}}(m^{-1})$. This conclusion is supported by
the recommended value of $\lm$ given, e.g. in
\cite{p:Lounici-GroupLasso}.
Non-asymptotic inequalities and prediction properties of the group lasso
estimators under restricted eigenvalues conditions are given in \cite{p:Lounici-GroupLasso}.
\section{Introduction}
We consider the model
\eqsplit[basicModel]{
Y_i=X_i^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}\beta_i+\eps_i, \quad i=1,\dots,n,
}
or more explicitly
\eqsplit{
y_{ij} = x_{ij}^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em} \beta_i + \eps_{ij}, \quad i=1,\dots,n, \;j=1,\dots,m
}
where \(\beta_i\in\R^p\), \(X_{i}\in\R^{m\times p}\) is either
deterministic fixed design matrix, or a sample of $m$ independent \(\R^p\)
random vectors.
Generally, we think of $j$ indexing replicates (of similar items within the group) and
$i$ indexing groups (of replicates).
Finally,
$\eps_{ij}$, \(i=1,\dots,n, \;j=1,\dots,m\) are (at least
uncorrelated with the \(x\)s), but typically assumed to be \iid
sub-Gaussian random variables, independent of the regressors
$x_{ij}$. We can consider this as $n$ partially related regression models, with $m$ \iid observations on the each model. For simplicity, we assume that all variables have
expectation 0. The fact that the number of observations does not
dependent on $i$ is arbitrary and is assumed only for the sake
of notational simplicity.
The standard FDA (functional data analysis) is of this form, when
the functions are approximated by their projections on some basis. Here we have $n$ \iid random functions, and each group can be considered as $m$ noisy observations, each one is on the value of these functions at a given value of the argument. Thus, \eqsplit[fda2]{
y_{ij} = g_i(z_{ij}) + \eps_{ij},
}
where $z_{ij}\in[0,1]$. The model fits the
regression setup of \eqref{basicModel}, if $g(z)=\summ \ell1p\beta_\ell h_\ell(p)$ where $h_1,\dots,h_p$ are in $L_2(0,1)$, and $x_{ij\ell}=h_\ell(z_{ij})$.
This approach is in the spirit of the empirical Bayes approach (or compound decision theory, note however that the term ``empirical Bayes'' has a few other meanings in the literature), cf,
\cite{Robbins1,Robbins2,Zhang-EB}. The empirical Bayes to sparsity was considered before, e.g.,
\cite{Zhang-W,BG-EB, GPR, GR-EB}. However, in these discussions
the compound decision problem was within a single vector, while we
consider the compound decision to be between the vectors, where the vectors are
the basic units. The beauty of the concept of compound decision, is that we do not have to assume that in reality the units are related. They are considered as related only because our loss function is additive.
One of the standard tools for finding sparse solutions in a large
$p$ small $m$ situation is the lasso
(Tibshirani~\cite{Tibsh-Lasso}), and the methods we consider are
its extensions.
We will make use of the following notation. Introduce $l_{p,q }$
norm of a set of vectors $z_1,\dots,z_n$, not necessarily of the
same length, $z_{ij}$, $i=1,\dots,n$, $j=1,\dots,J_i$:
\begin{definition}
$||z||_{p,q} = \left[\summ i1n \left(\sum_{j\in J_i} |z_{ij}|^p
\right)^{q/p} \right]^{1/q}.$
\end{definition}
These norms will serve as a penalty on the size of the matrix $\scb=(\beta_1,\dots,\beta_n)$. Different norms imply different estimators, each appropriate under different assumptions.
Within the framework of the compound decision theory, we can have
different scenarios, and we consider three of them. In Section \ref{sec:lassoes} we investigate the situation when there is no direct relationship between
the groups, and the only way the data are combined together is via
the selection of the common penalty. In this case the sparsity
pattern of the solution for each group are unrelated.
We argue that the alternative formulation of the lasso procedure in terms
of $\ell_{2,1}$ (or, more generally, $\ell_{\alpha, 1}$) norm which we refer to as ``lassoes'' can be more natural
than the simple lasso, and this is argued from different points of view.
The motivation is as follows. The lasso method can be described in
two related ways. Consider the one group version,
$y_j=x_j^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}\beta+\eps_j$. The lasso estimator can be defined by
\eqsplit{
\text{Minimize } \summ j1m (y_j-x_j^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}\beta)^2 \quad\text{s.t.}\quad \|\beta\|_1< A.
}
An equivalent definition, using Lagrange multiplier is given by
\eqsplit{
\text{Minimize} \summ j1m (y_j-x_j^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}\beta)^2 +\lm \|\beta\|_1^\al,
}
where \al can be any arbitrarily chosen positive number. In the
literature one can find almost only $\al=1$. One exception is
Greenshtein and Ritov \cite{p:GR-persist} where $\al=2$ was found
more natural, also it was just a matter of aesthetics. We would argue that $\al>2$ may be more intuitive.
Our first algorithm generalizes this representation of the lasso directly to deal
with compound model \eqref{basicModel}.
In the framework of the compound decision problem it is possible to consider
the $n$ groups as repeated similar models for $p$ variables, and to choose the
variables that are useful for all models. We consider this in Section \ref{sec:grouped}. The relevant variation
of the lasso procedure in this case is group lasso introduced by
Yuan and Lin~\cite{p:GroupLasso-def}:
\eqsplit[lassostar]{
\text{Minimize} \summ i1n \summ j1m (y_{ij}-x_{ij}^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}\beta_i)^2 +\lm
\|\beta\|_{2,1}.
}
The authors also showed that in this case the sparsity pattern of
variables is the same (with probability 1). Non-asymptotic inequalities under restricted eigenvalue type condition for group lasso are given by Lounici et
al.~\cite{p:Lounici-GroupLasso}.
Now, the standard notion of sparsity, as captured by the $L_0$
norm, or by the standard lasso and group lasso, is basis
dependent. Consider the model of \eqref{fda2}. If, for example,
$g(z)=\ind(a<z\le b)$, then this example is sparse when
$h_\ell(z)=\ind(z>\ell/p)$. It is not sparse if
$h_\ell(z)=(z-\ell/p)^+$. On the other hand, a function $g$ which
has a piece-wise constant slope is sparse in the latter basis, but
not in the former, even though, each function can be represented
equally well in both bases.
Suppose that there is a sparse representation in some
unknown basis, but assumed common to the $n$ groups. The
question arises, can we recover the basis corresponding to the sparsest representation? We will argue that
this penalty, also known as trace norm or Schatten norm with
$p=1$, aims in finding the rotation that gives the best sparse
representation of all vectors instantaneously (Section
\ref{sec:spectral}). We refer to this method as the
rotation-invariant lasso, or shortly as the RING lasso. This is not surprising as under some
conditions, this penalty also solves the minimum rank problem (see
Candes and Recht~\cite{Candes-Recht} for the noiselss case, and
Bach~\cite{Bach-Trace} for some asymptotic results). By analogy
with the lassoes argument, a higher power of the trace norm as a
penalty may be more intuitive to a Bayesian.
For both procedures considered here, the lassoes and the
RING lasso, we present the bounds on their
persistency as well as non-asymptotic inequalities under restricted eigenvalues type condition. All the proofs are
given in the Appendix.
\subsection{Restricted eigenvalues conditions and non-asymptotic inequalities}
Before stating the conditions and the inequalities for the lassoes procedure, we
introduce some notation and definitions.
For a vector $\beta$, let $\scm(\beta)$ be the cardinality of its
support: $\scm(\beta)=\sum_i\ind(\beta_i\ne0)$. Given a matrix
$\Delta\in \mathbb{R}^{n \times p}$ and given a set $J=\{J_i\}$,
$J_i \subset \{1,\dots, p\}$, we denote $\Delta_J =
\{\Delta_{i,j}, \, i=1,\dots,n, \, j\in J_i\}$. By the complement
$J^c$ of $J$ we denote the set $\{J_1^c, \dots, J_n^c\}$, i.e. the
set of complements of $J_i$'s. Below, $X$ is $np \times m$ block
diagonal design matrix, $X = \diag (X_1, X_2, \dots, X_n)$, and
with some abuse of notation, a matrix $\Delta=(\Delta_1,\dots,\Delta_n)$ may be considered as the vector $(\Delta_1^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em},
\dots,\Delta_n^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em})^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}$. Finally, recall the notation $\scb=(\beta_1,\dots,\beta_n)$
The restricted eigenvalue assumption of Bickel et al.~\cite{p:BRT}
(and Lounici et al.~\cite{p:Lounici-GroupLasso}) can be
generalized to incorporate unequal subsets $J_i$s. In the
assumption below, the restriction is given in terms of $\ell_{q,
1}$ norm, $q\geqslant 1$.
\vspace{\bigskipamount}\par \noindent{\bf Assumption}
RE$_q(s,c_0,\kappa)$.
$$
\kappa = \min\left\{ \frac{|| X^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em} \Delta||_{2
}}{\sqrt{m}||\Delta_J||_{ 2}}: \, \max_i |J_i| \leqslant s, \,
\Delta\in \mathbb{R}^{n \times p}\setminus \{0\}, \,
||\Delta_{J^c}||_{q,1} \leqslant c_0 ||\Delta_{J}||_{q,1} \right\}
> 0.
$$
We apply it with $q=1$, and in Lounici et
al.~\cite{p:Lounici-GroupLasso} it was used for $q=2$.
We call it a {\it restricted eigenvalue assumption} to be consistent
with the literature. In fact, as stated it is a definition of
$\kappa$ as the maximal value that satisfies the condition, and
the only real assumption is that $\kappa$ is positive. However,
the larger $\kappa$ is, the more useful the ``assumption'' is.
Discussion of the normalisation by $\sqrt{m}$ can be found in
Lounici et al.~\cite{p:Lounici-GroupLasso}.
For penalty $\lambda \sum_i ||\beta_i||_1^{\alpha}$, we have the
following inequalities.
\begin{theorem}
\label{th:LassoL1p} Assume $y_{ij} \sim \scn(x_{ij}^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}\beta_i,
\sigma^2)$, and let \(\hat\beta\) be a minimizer of
\eqref{lassoes}, with
$$ \lambda \geqslant \frac{4A\sig
\sqrt{m\log(np)}}{ \alpha \max(B^{\alpha-1},
\hat{B}^{\alpha-1})},$$ where $\alpha\geqslant 1$ and
$A>\sqrt{2}$, $B \geqslant \max_i ||\beta_i||_1$ and
$\hat{B}\geqslant \max_i ||\hat\beta_i||_1$, $\max(B,\hat{B})>0$ ($B$ may depend
on $n,m,p$, and so can $\hat{B}$). Suppose that generalized
assumption RE$_1(s, 3, \kappa)$ defined above holds, $\sum_{j=1}^m
x_{ij\ell}^2 = m$ for all $i,\ell$, and $\scm(\beta_i) \leqslant
s$ for all $i$.
Then, with probability at least $1 -(np)^{1-A^2/2}$,
\begin{enumerate}
\item[(a)] The root means squared prediction error is bounded by:$$
\frac 1 {\sqrt{nm}}||X^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}(\hat\scb - \scb)||_2
\leqslant \frac{ \sqrt{s }}{\kappa \sqrt{m} }
\left[ \frac {3 \alpha \lambda}{2\sqrt{ m}} \max(B^{\alpha-1},
\hat{B}^{\alpha-1}) + 2A \sig \sqrt{ \log(np) }\right], $$
\item[(b)] The mean estimation absolute error is bounded by:
$$ \frac 1 n ||\scb - \hat\scb||_1 \leqslant
\frac{4s}{m \kappa^2} \left[ \frac {3\alpha \lambda}{2}
\max(B^{\alpha-1},
\hat{B}^{\alpha-1}) + 2A \sig \sqrt{m\log(np)}\right],$$
\item[(c)] If \, $| ||\hat\beta_i||_1^{\al - 1} -
b^{\alpha-1}/2)| \geqslant 4\delta/b^{\alpha-1}$ for some
$\delta>0$,
$$
\scm(\hat\beta_i) \leq \|X_i(\beta_i-\hat\beta_i)\|_2^2
\frac{m \phi_{i,\, \max}} {\left(\lm \alpha ||\hat\beta_i||_1^{\alpha-1}/2- A \sig \sqrt{m\log(np)}\right)^2
},
$$ where $\phi_{i, \max}$ is the maximal eigenvalue of $X_{i}^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}
X_i/m$.
\end{enumerate}
\end{theorem}
Note that for $\alpha = 1$, if we take $\lm = 2A\sig
\sqrt{m\log(np)}$, the bounds are of the same order as for the
lasso with $np$-dimensional $\beta$ ( up to a constant of 2, cf.
Theorem 7.2 in Bickel et al.~\cite{p:BRT}). For $\alpha>1$, we
have dependence of the bounds on the $\ell_1$ norm of $\beta$ and
$\hat\beta$.
We can use bounds on the norm of $\hat\beta$ given in
Theorem~\ref{th:lassoes2} to obtain the following results.
\begin{theorem}\label{th:L12merge2}
Assume $y_{ij} \sim \scn(x_{ij}^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}\beta_i, \sigma^2)$, with
$\max_{i} \|\beta_i\|_1 \leqslant b $ where $b>0$ can depend on
$n,m,p$. Take some $\eta \in (0,1)$. Let \(\hat\beta\) be a
minimizer of \eqref{lassoes}, with
$$ \lambda = \frac{4A\sig}{\al\, b^{\al-1}}\sqrt{m\log(np)},$$ $A>\sqrt{2}$,
such that $b > c \eta^{1/(2(\al-1))}$ for some constant $c>0$.
Also, assume that $C_n - c_n = \EuScript{O}(m\delta_n)$, as defined in
Theorem~\ref{th:lassoes1}.
Suppose that generalized assumption RE$_1(s, 3, \kappa)$ defined
above holds, $\sum_{j=1}^m x_{ij\ell}^2 = m$ for all $i, \, \ell$,
and $\scm(\beta_i) \leqslant s$ for all $i$.
Then, for some constant $C>0$, with probability at least $1 -
\left(\eta+ (np)^{1-A^2/2}\right)$,
\begin{enumerate}
\item[(a)] The prediction error can be bounded by:
$$ ||X^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}(\hat\scb - \scb)||_2^2
\leqslant \frac{4A^2 \sig^2 s n \log(np)} {\kappa^2 }
\left[1+3 C \left(\frac{ b }{\sqrt{\eta}
}\right)^{(\al-1)/(\al -2)}
\right]^2,$$
\item[(b)] The estimation absolute error is bounded by:
$$ ||\scb - \hat\scb||_1 \leqslant \frac{2A \sig s n
\sqrt{\log(np)}}{\kappa^2 \sqrt{m} } \left[1+3 C \left(\frac{ b
}{\sqrt{\eta} }\right)^{(\al-1)/(\al -2)}
\right].$$
\item[(c)] Average sparsity of $\hat\beta_i$:
$$\frac 1 n \summ i1n \scm(\hat\beta_i) \leqslant
\,s \,
\frac{4 \phi_{ \max}}{\kappa^2 \delta^2} \left[1+3 C \left(\frac{ b
}{\sqrt{\eta} }\right)^{1+1/(\al -2)}
\right]^2,
$$
where $\phi_{\, \max}$ is the largest eigenvalue of $X^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}
X/m$.
\end{enumerate}
\end{theorem}
This theorem also tells us how large $\ell_1$ norm of $\beta$ can
be to ensure good bounds on the prediction and estimation errors.
Note that under the Gaussian model and fixed design matrix,
assumption $C_n - c_n = \EuScript{O}(m\delta_n)$ is equivalent to
$||\scb||_2^2\leqslant C m\delta_n$.
\section{The lassoes procedure }
\label{sec:lassoes}
\subsection{Definition and persistence}
The minimal structural relationship we may assume is that the
$\beta's$ are not related, except that we believe that there is a
bound on the average sparsity of the $\beta$'s. One possible
approach would be to consider the problem as a standard sparse
regression problem with $nm$ observations, a single vector of
coefficients $\beta=(\beta_1^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em},\dots,\beta_n^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em})^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}$, and a block
diagonal design matrix $X$. This solution imposes very little on
the similarity among $\beta_1,\dots,\beta_n$.
We assume that each
vector $\beta_i$ solves a different problem, and these problems
are related only through the joint loss function, which is the sum
of the individual losses. To be clearer, we assume that for each
$i=1,\dots,n$, $z_{ij}=(y_{ij}, x_{ij}^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em})^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}$, $j=1,\dots,m$ are
\iid, sub-Gaussian random variables, drawn from a distribution
$Q_i$. Let $z_i=(y_i,x_i^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em})^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}$ be an independent sample from
$Q_i$. For any vector $a$, let $\ti a=(-1,a^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em})^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}$, and let
$\tilde\Sig_i$ be the covariance matrix of $z_i$ and
$\mathfrak{S}=(\ti\Sig_1,\dots,\ti\Sig_n)$. The goal is to find
the matrix $\hat\scb=(\hat\beta_1,\dots,\hat\beta_n)$ that
minimizes the mean prediction error:
\eqsplit[persis]{
L(\scb,\mathfrak S) &= \summ i1n \E_{Q_i}(y-x_i^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}\beta_i)^2=\summ i1n \ti\beta^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}\ti\Sig_i\ti\beta.
}
For $p$ small, the natural approach is empirical risk
minimization, that is replacing $\ti\Sig_i$ in \eqref{persis} by
$\ti S_i$, the empirical covariance matrix of $z_i$. However,
generally speaking, if $p$ is large, empirical risk minimization
results in overfitting the data. Greenshtein and Ritov
\cite{p:GR-persist} suggested minimization over a restricted set
of possible $\beta$'s, in particular, to either $L_1$ or $L_0$
balls. In fact, their argument is based on the following
simple observation
\eqsplit[roughapprox]{
\bigl| \ti\beta^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}(\ti\Sig_i - \ti S_i)\ti\beta\bigr|
&\leq \| \ti\Sig_i - \ti S_i \|_\en \|\ti\beta\|_1^2
\\
&\hspace{-10em}\text{and}
\\
\| \ti\Sig_i - \ti S_i \|_\en &=\EuScript{O}_p(m^{-1/2}\log p)
}
(see Lemma~\ref{lem:persist} in the Appendix.)
This leads to the natural extension of the single vector lasso to
the compound decision problem set up, where we penalize by the sum
of the \emph{squared} $L_1$ norms of vectors
$\ti\beta_1,\dots,\ti\beta_n$, and obtain the estimator defined
by:
\eqsplit[lassoes]{
(\dacc\ti\hat\beta_i,\dots,\dacc\ti\hat\beta_n)
&=\argmin_{\ti\beta_1,\dots,\ti\beta_n} \Bigl\{m\summ i1n \ti\beta_i^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}\ti S_i\ti\beta_i
+ \lm_n \summ i1n \|\ti\beta_i\|_1^2\Bigr\}
\\
&=\argmin_{\ti\beta_1,\dots,\ti\beta_n} \summ i1n \Bigl\{ \summ j1m (y_{ij}-x_{ij}^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}\beta_i)^2
+ \lm_n \|\ti\beta_i\|_1^2\Bigr\}.
}
\begin{theorem}
\label{th:lassoes1} Let $\beta_{i0}$, $i=1,\dots,n$ be $n$ vectors
and let $C_n=n^{-1}\summ i1n
\ti\beta_{i0}^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}\ti\Sig_i\ti\beta_{i0}$. Let $c_n=n^{-1}\summ i1n
\min_\beta\ti\beta^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}\ti\Sig_i\ti\beta$. Then
\eqsplit{
\summ i1n \dacc\ti\hat\beta_{i}^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}\ti \Sig_i \dacc\ti\hat\beta_{i}
\leq
\summ i1n \ti\beta_{i0}^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}\ti \Sig_i \ti\beta_{i0}
+ (\frac{\lm_n} m+\del_n) \summ i1n \|\ti\beta_{i0}\|_1^2
- (\frac {\lm_n} m-\del_n) \summ i1n \|\dacc\ti\hat\beta_{i}\|_1^2,
}
where $\del_n=\max_i\|\ti S_i-\it\Sig_i\|_\en$. If also $\lm_n/m\to0$ and $\lm_n/(m^{1/2}\log(np))\to\en$, then
\eqsplit[bndthb]{
\summ i1n\|\dacc\ti\hat\beta_i\|_1^2
=\EuScript{O}_p\bigl(mn\frac{C_n-c_n}{\lm_n}\bigr) + \bigl(1+\EuScript{O}(\frac{m^{1/2}}{\lm_n}\log(np))\bigr)
\summ i1n\|\ti\beta_{i0}\|_1^2
}and
\eqsplit{
\summ i1n \dacc\ti\hat\beta_{i}^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}\ti \Sig_i \dacc\ti\hat\beta_{i}
\leq
\summ i1n \ti\beta_{i0}^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}\ti \Sig_i \ti\beta_{i0}
+ \bigl(1+\ensuremath{\text{\sfa$\O$\sfb}}_p(1)\bigr)\frac{\lm_n} m \summ i1n \|\ti\beta_{i0}\|_1^2.
}
\end{theorem}
The theorem is useful when there are vectors
$\beta_{10},\dots,\beta_{n0}$ with low $L_1$ norm which are good
approximation to the minimizers of $\ti\beta^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}\ti\Sig_i\ti\beta$.
The result is meaningful, although not as strong as may be wished,
as long as $C_n-c_n\to 0$, while $n^{-1}\summ i1n
\|\ti\beta_{i0}\|_1^2=\ensuremath{\text{\sfa$\O$\sfb}}_p(m^{1/2})$. Of course, if the minimizer
of $\ti\beta^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}\ti\Sig_i\ti\beta$ itself is sparse, then by
\eqref{bndthb} $\dacc\ti\hat\beta_1,\dots,\dacc\ti\hat\beta_n$ are as sparse
as the true minimizers (sparse in the sense of $\ell_1$ norm).
Also note, that the prescription that the theorem gives for
selecting $\lm_n$, is sharp: choose $\lm_n$ as close as possible
to $m\del_n$, or slightly larger than $\sqrt m$.
\subsection{A Bayesian perspective}
The estimators $\dacc\ti\hat\beta_1,\dots,\dacc\ti\hat\beta_m$ look as if they
are the mode of the a-posteriori distribution of the $\beta_i$'s
when $y_{ij}|\beta_i\dist N(x_{ij}^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}\beta_i,\sig^2)$, the
$\beta_1,\dots,\beta_n$ are a priori independent, and $\beta_i$
has a prior density proportional to
$\exp(-\lm_n\|\ti\beta_i\|_1^2/\sig^2)$. This distribution can be
constructed as follows. Suppose $T_i\dist N(0,\lm_n^{-1}\sig^2)$.
Given $T_i$, let $u_{i1},\dots,u_{ip}$ be distributed uniformly on
the simplex $\{u_{i\ell}\geq 0, \summ \ell1n u_{i\ell}=|T_i|\}$.
Let $s_{i1},\dots,s_{ip}$ be \iid Rademacher random variables
(taking values $\pm 1$ with probabilities $0.5$), independent of
$T_i,u_{i1},\dots,u_{ip}$. Finally let $\beta_{i\ell}=u_{i\ell}
s_{i\ell} $, $\ell=1,\dots,p$.
However, this point of view is not consistent with the conditions
of Theorem \ref{th:lassoes1}. The permitted range of $\lm_n$ does
not depend on the assumed range of $\|\ti\beta_i\|$, but quite
artificially should be in the order between $m^{1/2} $ and $m$.
That is, the penalty should be increased with the number of
observations on $\beta_i$, although in a slower rate than $m$. In
fact, it goes in the `wrong' direction. As $m\to\en$, one may
wish to use weaker a-priori assumptions, and permits $T$ to have
a-priori second moment going to infinity, not to 0, as entailed by
$\lm_n\to 0$.
We would like to consider a more general penalty of the form $\summ i1n \|\beta_i\|_1^\al$. A power $\al\neq 1$ of $\ell_1$ norm of $\beta$ as a
penalty introduces a priori dependence between the variables which
is not the case for the regular lasso penalty with $\al=1$, where
all $\beta_{ij}$ are a priori independent. As $\al$ increases, the
sparsity of the different vectors tends to be the same. Note that
given the value of $\lm_n$, the $n$ problems are treated
independently. The compound decision problem is reduced to
picking a common level of penalty. When this choice is data based,
the different vectors become dependent. This is the main benefit
of this approach---the selection of the regularization is based on
all the $mn$ observations.
For a proper Bayesian perspective, we need to consider a prior with much smaller tails than the
normal. Suppose for simplicity that $c_n=C_n$ (that is, the ``true'' regressors are sparse), and
$\max_i\|\beta_{i0}\|_1<\en$.
\begin{theorem}
\label{th:lassoes2}
Let $\beta_i$ be the minimizer of $\ti\beta^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}\Sig_i\ti\beta$. Suppose $\max_i\|\beta_{i0}\|_1<\en$. Consider the estimators:
\eqsplit[lassoesAl]{
(\dacc\ti\hat\beta_i,\dots,\dacc\ti\hat\beta_n)
&=\argmin_{\ti\beta_1,\dots,\ti\beta_n} \Bigl\{m\summ i1n \ti\beta_i^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}\ti S_i\ti\beta_i
+ \lm_n \summ i1n \|\ti\beta_i\|_1^\al\Bigr\}
}
for some $\al>2$. Assume that $\lambda =\EuScript{O}(m \delta_m) =
\EuScript{O}(m^{1/2} \log p)$. Then
\eqsplit{
n^{-1}\summ i1n \|\dacc\ti\hat\beta_i\|_2^2 &= \EuScript{O}((m\del_n/\lm_n)^{2/(\al-2)} ),
}
and
\eqsplit{
\summ i1n \dacc\ti\hat\beta_{i}^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}\ti \Sig_i \dacc\ti\hat\beta_{i}
&\leq \summ i1n \ti\beta_{i0}^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}\ti \Sig_i \ti\beta_{i0}
+ \EuScript{O}_p(n (m/\lm_n)^{2/(\al-2)} \del_n^{\al/(\al-2)}).
}
\end{theorem}
\begin{remark}\label{rem:lassoes2} If the assumption $\lambda =\EuScript{O}(m \delta_m)$ does not hold, i.e. if $m \delta_m/\lambda =\ensuremath{\text{\sfa$\O$\sfb}}(1)$, then the
error term dominates the penalty and we get similar rates as in
Theorem~\ref{th:lassoes1}, i.e.
\eqsplit{
n^{-1}\summ i1n \|\dacc\ti\hat\beta_i\|_2^2 &= \EuScript{O}(1),
}
and
\eqsplit{
\summ i1n \dacc\ti\hat\beta_{i}^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}\ti \Sig_i \dacc\ti\hat\beta_{i}
&\leq \summ i1n \ti\beta_{i0}^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}\ti \Sig_i \ti\beta_{i0}
+ \EuScript{O}_p\left( n\lm_n/m\right).
}
\end{remark}
Note that we can take in fact $\lm_n\to 0$, to accommodate an increasing value of the $\dacc\ti\hat\beta_i$'s.
The theorem suggests a simple way to select $\lm_n$ based on the
data. Note that $n^{-1}\summ i1n \|\dacc\ti\hat\beta_i\|_1^2$ is a
decreasing function of $\lm$. Hence, we can start with a very
large value of $\lm$ and decrease it until $n^{-1}\summ i1n
\|\dacc\ti\hat\beta_i\|_1^2\approx \lm^{-2/\al}$.
\section{The lassoes procedure }
\label{sec:lassoes}
The minimal structural relationship we may assume is that the
$\beta's$ are not related, except that we believe that there is a
bound on the average sparsity of the $\beta$'s. One possible
approach would be to consider the problem as a standard sparse
regression problem with $nm$ observations, a single vector of
coefficients $\beta=(\beta_1^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em},\dots,\beta_n^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em})^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}$, and a block
diagonal design matrix $X$. This solution imposes very little on
the similarity among $\beta_1,\dots,\beta_n$. The lassoes
procedure discussed in this section assume that these vectors are
similar, at least in their level of sparsity.
\subsection{Prediction error minimization}
In this paper we adopt an oracle point of view. Our estimator is the empirical minimizer of the risk penalized by the complexity of the solution (i.e., by its $\ell_1$ norm). We compare this estimator to the solution of an ``oracle'' who does the same, but optimizing over the true, unknown to simple human beings, population distribution.
We assume that each
vector of $\beta_i$, $i=1,\dots,n$, solves a different problem, and these problems
are related only through the joint loss function, which is the sum
of the individual losses. To be clearer, we assume that for each
$i=1,\dots,n$, $z_{ij}=(y_{ij}, x_{ij}^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em})^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}$, $j=1,\dots,m$ are
\iid, sub-Gaussian random variables, drawn from a distribution
$Q_i$. Let $z_i=(y_i,x_i^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em})^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}$ be an independent sample from
$Q_i$. For any vector $a$, let $\ti a=(-1,a^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em})^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}$, and let
$\tilde\Sig_i$ be the covariance matrix of $z_i$ and
$\mathfrak{S}=(\ti\Sig_1,\dots,\ti\Sig_n)$. The goal is to find
the matrix $\hat\scb=(\hat\beta_1,\dots,\hat\beta_n)$ that
minimizes the mean prediction error:
\eqsplit[persis]{
L(\scb,\mathfrak S) &= \summ i1n \E_{Q_i}(y_i-x_i^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}\beta_i)^2=\summ i1n \ti\beta_i^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}\ti\Sig_i\ti\beta_i.
}
For $p$ small, the natural approach is empirical risk
minimization, that is replacing $\ti\Sig_i$ in \eqref{persis} by
$\ti S_i$, the empirical covariance matrix of $z_i$. However,
generally speaking, if $p$ is large, empirical risk minimization
results in overfitting the data. Greenshtein and Ritov
\cite{p:GR-persist} suggested (for the standard $n=1$) minimization over a restricted set
of possible $\beta$'s, in particular, to either $L_1$ or $L_0$
balls. In fact, their argument is based on the following
simple observations
\eqsplit[roughapprox]{
\bigl| \ti\beta^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}(\ti\Sig_i - \ti S_i)\ti\beta\bigr|
&\leq \| \ti\Sig_i - \ti S_i \|_\en \|\ti\beta\|_1^2
\\
&\hspace{-10em}\text{and}
\\
\| \ti\Sig_i - \ti S_i \|_\en &=\EuScript{O}_p(m^{-1/2}\log p)
}
(see Lemma~\ref{lem:persist} in the Appendix for the formal argument.)
This leads to the natural extension of the single vector lasso to
the compound decision problem set up, where we penalize by the sum
of the \emph{squared} $L_1$ norms of vectors
$\ti\beta_1,\dots,\ti\beta_n$, and obtain the estimator defined
by:
\eqsplit[lassoes]{
(\dacc\ti\hat\beta_i,\dots,\dacc\ti\hat\beta_n)
&=\argmin_{\ti\beta_1,\dots,\ti\beta_n} \Bigl\{m\summ i1n \ti\beta_i^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}\ti S_i\ti\beta_i
+ \lm_n \summ i1n \|\ti\beta_i\|_1^2\Bigr\}
\\
&=\argmin_{\ti\beta_1,\dots,\ti\beta_n} \summ i1n \Bigl\{ \summ j1m (y_{ij}-x_{ij}^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}\beta_i)^2
+ \lm_n \|\ti\beta_i\|_1^2\Bigr\}.
}
The prediction error of the lassoes estimator can be bounded in
the following way. In the statement of the theorem, $c_n$ is the minimal achievable risk, while $C_n$ is the risk achieved by a particular sparse solution.
\begin{theorem}
\label{th:lassoes1} Let $\beta_{i0}$, $i=1,\dots,n$ be $n$
arbitrary vectors and let $C_n=n^{-1}\summ i1n
\ti\beta_{i0}^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}\ti\Sig_i\ti\beta_{i0}$. Let $c_n=n^{-1}\summ i1n
\min_\beta\ti\beta^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}\ti\Sig_i\ti\beta$. Then
\eqsplit{
\summ i1n \dacc\ti\hat\beta_{i}^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}\ti \Sig_i \dacc\ti\hat\beta_{i}
\leq
\summ i1n \ti\beta_{i0}^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}\ti \Sig_i \ti\beta_{i0}
+ (\frac{\lm_n} m+\del_n) \summ i1n \|\ti\beta_{i0}\|_1^2
- (\frac {\lm_n} m-\del_n) \summ i1n \|\dacc\ti\hat\beta_{i}\|_1^2,
}
where $\del_n=\max_i\|\ti S_i-\it\Sig_i\|_\en$. If also $\lm_n/m\to0$ and $\lm_n/(m^{1/2}\log(np))\to\en$, then
\eqsplit[bndthb]{
\summ i1n\|\dacc\ti\hat\beta_i\|_1^2
=\EuScript{O}_p\bigl(mn\frac{C_n-c_n}{\lm_n}\bigr) + \bigl(1+\EuScript{O}(\frac{m^{1/2}}{\lm_n}\log(np))\bigr)
\summ i1n\|\ti\beta_{i0}\|_1^2
}and
\eqsplit{
\summ i1n \dacc\ti\hat\beta_{i}^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}\ti \Sig_i \dacc\ti\hat\beta_{i}
\leq
\summ i1n \ti\beta_{i0}^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}\ti \Sig_i \ti\beta_{i0}
+ \bigl(1+\ensuremath{\text{\sfa$\O$\sfb}}_p(1)\bigr)\frac{\lm_n} m \summ i1n \|\ti\beta_{i0}\|_1^2.
}
\end{theorem}
The result is meaningful, although not as strong as may be wished,
as long as $C_n-c_n\to 0$, while $n^{-1}\summ i1n
\|\ti\beta_{i0}\|_1^2=\ensuremath{\text{\sfa$\O$\sfb}}_p(m^{1/2})$. That is, when there is a relatively sparse approximations to the best regression functions. Here sparse means only that the $L_1$ norms of vectors is strictly smaller, on the average, than $\sqrt m$. Of course, if the minimizer
of $\ti\beta^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}\ti\Sig_i\ti\beta$ itself is sparse, then by
\eqref{bndthb} $\dacc\ti\hat\beta_1,\dots,\dacc\ti\hat\beta_n$ are as sparse
as the true minimizers .
Also note, that the prescription that the theorem gives for
selecting $\lm_n$, is sharp: choose $\lm_n$ as close as possible
to $m\del_n$, or slightly larger than $\sqrt m$.
\subsection{A Bayesian perspective}
The estimators $\dacc\ti\hat\beta_1,\dots,\dacc\ti\hat\beta_m$ look as if they
are the mode of the a-posteriori distribution of the $\beta_i$'s
when $y_{ij}|\beta_i\dist N(x_{ij}^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}\beta_i,\sig^2)$, the
$\beta_1,\dots,\beta_n$ are a priori independent, and $\beta_i$
has a prior density proportional to
$\exp(-\lm_n\|\ti\beta_i\|_1^2/\sig^2)$. This distribution can be
constructed as follows. Suppose $T_i\dist N(0,\lm_n^{-1}\sig^2)$.
Given $T_i$, let $u_{i1},\dots,u_{ip}$ be distributed uniformly on
the simplex $\{u_{i\ell}\geq 0, \summ \ell1n u_{i\ell}=|T_i|\}$.
Let $s_{i1},\dots,s_{ip}$ be \iid Rademacher random variables
(taking values $\pm 1$ with probabilities $0.5$), independent of
$T_i,u_{i1},\dots,u_{ip}$. Finally let $\beta_{i\ell}=u_{i\ell}
s_{i\ell} $, $\ell=1,\dots,p$.
However, this Bayesian point of view is not consistent with the conditions
of Theorem \ref{th:lassoes1}. An appropriate prior should express the beliefs on the unknown parameter which are by definition conceptually independent of the amount data to be collected. However, the permitted range of $\lm_n$ does
not depend on the assumed range of $\|\ti\beta_i\|$, but quite
artificially should be in order between $m^{1/2} $ and $m$.
That is, the penalty should be increased with the number of
observations on $\beta_i$, although in a slower rate than $m$. In
fact, even if we relax what we mean by ``prior'', the value of $\lm_n$ goes in the `wrong' direction. As $m\to\en$, one may
wish to use weaker a-priori assumptions, and permits $T$ to have
a-priori second moment going to infinity, not to 0, as entailed by
$\lm_n\to 0$.
We would like to consider a more general penalty of the form $\summ i1n \|\beta_i\|_1^\al$. A power $\al\neq 1$ of $\ell_1$ norm of $\beta$ as a
penalty introduces a priori dependence between the variables which
is not the case for the regular lasso penalty with $\al=1$, where
all $\beta_{ij}$ are a priori independent. As $\al$ increases, the
sparsity of the different vectors tends to be the same. Note that
given the value of $\lm_n$, the $n$ problems are treated
independently. The compound decision problem is reduced to
picking a common level of penalty. When this choice is data based,
the different vectors become dependent. This is the main benefit
of this approach---the selection of the regularization is based on
all the $mn$ observations.
For a proper Bayesian perspective, we need to consider a prior with much smaller tails than the
normal. Suppose for simplicity that $c_n=C_n$ (that is, the ``true'' regressors are sparse), and
$\max_i\|\beta_{i0}\|_1<\en$.
\begin{theorem}
\label{th:lassoes2}
Let $\beta_{i0}$ be the minimizer of $\ti\beta^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}\Sig_i\ti\beta$. Suppose $\max_i\|\beta_{i0}\|_1<\en$. Consider the estimators:
\eqsplit
(\dacc\ti\hat\beta_i,\dots,\dacc\ti\hat\beta_n)
&=\argmin_{\ti\beta_1,\dots,\ti\beta_n} \Bigl\{m\summ i1n \ti\beta_i^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}\ti S_i\ti\beta_i
+ \lm_n \summ i1n \|\ti\beta_i\|_1^\al\Bigr\}
}
for some $\al>2$. Assume that $\lambda_n =\EuScript{O}(m \delta_m) =
\EuScript{O}(m^{1/2} \log p)$. Then
\eqsplit{
n^{-1}\summ i1n \|\dacc\ti\hat\beta_i\|_1^2 &= \EuScript{O}((m\del_n/\lm_n)^{2/(\al-2)} ),
}
and
\eqsplit{
\summ i1n \dacc\ti\hat\beta_{i}^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}\ti \Sig_i \dacc\ti\hat\beta_{i}
&\leq \summ i1n \ti\beta_{i0}^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}\ti \Sig_i \ti\beta_{i0}
+ \EuScript{O}_p(n (m/\lm_n)^{2/(\al-2)} \del_n^{\al/(\al-2)}).
}
\end{theorem}
\begin{remark}\label{rem:lassoes2} If the assumption $\lambda_n =\EuScript{O}(m \delta_m)$ does not hold, i.e. if $m \delta_m/\lambda_n =\ensuremath{\text{\sfa$\O$\sfb}}(1)$, then the
error term dominates the penalty and we get similar rates as in
Theorem~\ref{th:lassoes1}, i.e.
\eqsplit{
n^{-1}\summ i1n \|\dacc\ti\hat\beta_i\|_1^2 &= \EuScript{O}(1),
}
and
\eqsplit{
\summ i1n \dacc\ti\hat\beta_{i}^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}\ti \Sig_i \dacc\ti\hat\beta_{i}
&\leq \summ i1n \ti\beta_{i0}^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}\ti \Sig_i \ti\beta_{i0}
+ \EuScript{O}_p\left( n\lm_n/m\right).
}
\end{remark}
Note that we can take in fact $\lm_n\to 0$, to accommodate an increasing value of the $\dacc\ti\hat\beta_i$'s.
The theorem suggests a simple way to select $\lm_n$ based on the
data. Note that $n^{-1}\summ i1n \|\dacc\ti\hat\beta_i\|_1^2$ is a
decreasing function of $\lm$. Hence, we can start with a very
large value of $\lm$ and decrease it until $n^{-1}\summ i1n
\|\dacc\ti\hat\beta_i\|_1^2\approx \lm^{-2/\al}$.
\subsection{General persistence result.}\label{sec:persistGen}
A sequence of estimators $\hat{\beta}^{( m,n,p)}$ is persistent
with respect to a set of distributions $\scf_{n,p}^m$ for $\beta
\in B_{n, p}$, if for any $F_{m,n,p}\in \scf_{n,p}^m$,
$$
L_{F_{m,n,p}}\left(\hat\beta^{(m,n,p)}\right) -
L_{F_{m,n,p}}\left(\beta^*_{F_{m, n,p}} \right)
\stackrel{P}{\rightarrow} 0,
$$
where $L_F(\beta) = (nm)^{-1}E_F \sum_{i=1}^n \summ j1m (Y_{ij} -
X_{ij}^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em} \beta_i)^2$, $F_{m, n, p}$ is the empirical distribution
function of $n\times (p+1)$ matrix $Z$, $Z_i = (Y_i,
X_{i1},\dots, X_{ip})$, $i=1,\dots, n$, observed $m$ times. Here
$\beta^*_{F_{m, n,p}} = \argmin_{\beta \in B_{n,p}}
L_{F_{m,n,p}}(\beta)$, and $\scf_{n,p}^m$ stands for a collection
of distributions of $m$ observations of vectors $Z_i = (Y_i,
X_{i1},\dots, X_{ip})$, $i=1,\dots,n$.
\vspace{1ex}\noindent{\bf Assumption F}. Under the distributions of random variables
$Z$ in $ \scf_{n,p}$, $\xi_{i\ell k} = Z_{i\ell} Z_{i k}$
satisfy $E\left(\max_{i=1,\dots,n}\max_{\ell,k=1,\dots p+1}
\xi_{i\ell k}^2 \right) < V$. Denote this set of distributions by
$\scf_{n,p}(V)$.\par\vspace{1ex}
This assumption is similar to one of the assumptions of
Greenshtein and Ritov (2004). It is satisfied if, for instance,
the distribution of $Z_{i\ell}$ has finite support and the
variance of $Z_{i\ell} Z_{i k}$ is finite.
\begin{lemma}\label{lem:persist}
Let $ F\in \scf_{n,p}(V)$, and denote $\Sigma_i=(\sigma_{ijk})$
and $\hat\Sigma_{i}=(\hat{\sigma}_{ik\ell})$, with $\sigma_{ijk} =
E_F Z_{ij} Z_{ik}$ and $ \hat{\sigma}_{ik\ell} = m^{-1}
\sum_{j=1}^m Z_{ik}^{(j)} Z_{i \ell}^{(j)}$, where $Z =
(Z_{i\ell}^{(j)})$ is a sample from $F^m$, $i=1,\dots,n$,
$j=1,\dots,m$, $\ell =1,\dots,p$.
Let $\hat{\beta}$ be the estimator minimising $\summ i1n \summ j1m
(Y_{ij} - X_{ij}^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}\beta_i)^2$ subject to $\beta \in B$ where $B$
is some subset of $\mathbb{R}^{n\times p}$.
Then, for any $\eta\in(0,1)$, \eqsplit{ &\text{(a)\,}
\max_{i=1,\dots, n} ||\Sigma_i - \hat{\Sigma}_i||_{\infty}
&\leqslant&
\sqrt{\frac{2e V \log (n(p+1)^2)}{ m \, \eta}},\\
& \text{(b) \,} | L_{F}(\beta) - L_{\hat{F}}(\beta) | &\leqslant&
\frac 1 {nm } \sqrt{\frac{2e V \log (n(p+1)^2)} {m\eta}} \left(
n+ \summ i1n \|\beta_i\|_1^2\right) } with probability at least $1
- \eta$.
\end{lemma}
\begin{proof}
Follows that of Theorem 1 in Greenshtein and Ritov (2004).
a) Let $\hat\sigma_{ik\ell} = \sigma_{ik\ell} +
\epsilon_{ik\ell}$, $E_i = (\epsilon_{ik\ell})$. Then, under
Assumption F and by Nemirovsky's inequality (see e.g. Lounici et
al~\cite{p:Lounici-GroupLasso}), \eqsplit{ &\hspace{-3em}P( \max_{i}||\Sigma_i -
\hat{\Sigma}_i||_{\infty} > A )\\ &\leqslant \frac 1 {A^2} E (\max_i
||\Sigma_i - \hat{\Sigma}_i||_{\infty}^2)\\ &\leqslant \frac {2e
\log (n(p+1)^2)}{mA^2} E(\max_{i=1,\dots,n}\max_{j,k=1,\dots p+1}
(Z_{ij} Z_{ik} - E(Z_{ij}
Z_{ik}))^2 )\\
& \leqslant \frac {2e V \log (n(p+1)^2) }{mA^2}. }
Taking $A = \sqrt{\frac{2e V \log (n(p+1)^2)}{ m \, \eta}}$
proves the first part of the lemma.
b) By the definition of $\hat{\beta}$ and $\beta^*_{F}$,
\eqsplit{ L_{F }(\hat{\beta} ) - L_{{F} }(\beta^*_{F }) \geqslant
0, \quad L_{\hat{F} }(\hat{\beta} ) - L_{\hat{F} }(\beta^*_{F })
\leqslant 0. }
Hence, \eqsplit{ 0 &\leqslant L_{F}\left(\hat\beta\right) -
L_{F}\left(\beta^*_{F} \right) = L_{F}\left(\hat\beta\right) -
L_{\hat{F}}\left(\hat\beta\right)\\
&+L_{\hat{F}}\left(\hat\beta\right) -
L_{{F}}\left(\hat\beta\right) +
L_{{F}}\left(\hat\beta\right)
-L_{F}\left(\beta^*_{F} \right)\\
&\leqslant 2\sup_{\beta\in B_{n,p}} |L_{F}(\beta) -
L_{\hat{F}}(\beta) |.}
Denote $\delta_i^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em} = (-1, \beta_{i,1}, \dots, \beta_{i,p})$, then
$$
L_F(\beta) = \frac 1 {nm}\summ i1n \delta_i^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em} \Sigma_{F,i}
\delta_i,
$$
where $\Sigma_{F,i}=(\sigma_{ijk})$ and $\sigma_{ijk} = E_F Z_{ij}
Z_{ik}$. For the empirical distribution function $\hat{F}_{mn}$
determined by a sample $Z_{i \ell}^{(j)}$, $i=1,\dots,n$,
$j=1,\dots,m$, $\ell =1,\dots,p$,
$\Sigma_{\hat{F},i}=(\hat{\sigma}_{ik\ell})$ and $
\hat{\sigma}_{ik\ell} = \frac 1 m \sum_{j=1}^m Z_{i k}^{(j)}
Z_{i\ell}^{(j)}. $
Introduce matrix $\hat{\sce} $ with $\hat{\sce}_{j\ell} = A$. Hence,
with probability at least $1-\eta$,
\eqsplit{ | L_{F}(\beta) - L_{\hat{F}}(\beta) | &= \left|\frac 1
{nm}\summ i1n \delta_i ^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em} (\Sigma_{F,i} - \Sigma_{\hat{F}}, i)
\delta_i \right|
\\& \leqslant \frac 1 {nm}\summ i1n |\delta_i|^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}
\hat{\sce} |\delta_i|\\ &= \frac 1 {nm } \sqrt{\frac{2e V \log
(n(p+1)^2)} {m\eta}} (n+ \summ i1n \|\beta_i\|_1^2). }
\end{proof}
\subsection{The RINGS lasso}
Also, by analogy to the lassoes procedure, we can consider the
RINGS lasso estimator:
\eqsplit[penRINGS]{
\hat\scb &= \argmin_{\scb\in\R^{p\times n}}\{\summ i1n
(y_{ij}-x_{ij}^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}\beta_i)^2+\lm|||\scb|||_1^{\al}\}.
}
Then we have the following persistency results.
\begin{theorem}\label{th:persistRINGS}
Assume that $L_F(\scb)- L_F(\scb_F^*) = \EuScript{O}\left(\frac p n
\sqrt{\frac{\log(pn)}{m}}
\right)$, $|||\scb|||_1$ is bounded. Let $\hat\scb$ - solution of the RINGS
lasso \eqref{penRINGS}.
Then, we have the following results with high probability.
1. $\alpha = 2$, $\lm \gg pm\delta_n$, then \eqsplit{
|||\hat\scb|||_1 &\leqslant \frac{\lm n}{p\delta} (L_F(\scb)-
L_F(\scb_F^*)) + \left(1 +
\EuScript{O}\left(\frac{pn\delta}{\lm}\right)\right)
|||\scb|||_1,\\
\summ i1n \dacc\ti\hat\beta_{i}^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}\ti \Sig_i \dacc\ti\hat\beta_{i}
&\leq \summ i1n \ti\beta_{i}^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}\ti \Sig_i \ti\beta_{i}
+ 2(\frac{\lm_n}m +p\del_n) ||| \scb|||_1^2
+ \lm n(1- \frac{\lm_n}{m p \del_n}) \left( L_F(\scb)-
L_F(\scb_F^*) \right)
+2n\del_n.
}
2. $\alpha > 2$, $\lm \ll pm\delta_n$. Then, \eqsplit{
|||\hat\scb|||_1 &\leqslant \min(n,p) \EuScript{O} \left(
\left(\frac{\lm}{mp\delta_n}\right)^{1/(\al-2)} \right),\\
\summ i1n \dacc\ti\hat\beta_{i}^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}\ti \Sig_i \dacc\ti\hat\beta_{i}
&\leqslant \summ i1n \ti\beta_{i}^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}\ti \Sig_i \ti\beta_{i}
+ \EuScript{O}((p+n) \del_n)+ p \del_n [\min(n,p)]^2 \EuScript{O} \left(
\left(\frac{\lm}{mp\delta_n}\right)^{2/(\al-2)} \right).
}
\end{theorem}
\end{comment}
\section{The RING lasso}
\label{sec:spectral}
The rotation invariant group (RING) lasso is suggested as a natural extension of the group lasso to the situation where the proper sparse description of the regression function within a given basis is not known in advance. For example, when we prefer to leave it a-priori open whether the function should be described in terms of the standard Haar wavelet basis, a collection of interval indicators, or a collection of step functions. All these three span the same linear space, but the true functions may be sparse in only one of them.
\subsection{Definition}
Let \(A=\sum c_i x_ix_i^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}\), be a positive semi-definite matrix,
where \(x_1,x_2,\dots\) is an orthonormal basis of eigenvectors.
Then, we define \(A^{\gamma}=\sum c_i^{\gamma} x_ix_i^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}\). We
consider now as penalty the function
\eqsplit{
|||\scb|||_1 = \trace\Bigl\{\bigl(\summ i1n
\beta_i\beta_i^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}\bigr)^{1/2}\Bigr\},
} where
$\scb=(\beta_1,\dots,\beta_n)=(\mathfrak{b}_1^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em},\dots,\mathfrak{b}_p^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em})^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}$. This
is also known as trace norm or Schatten norm with $p=1$.
Note that \(|||\scb|||_1=\sum c_i^{1/2}\) where \(c_1,\dots,c_p\)
are the eigenvalues of \(\scb\scb^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}=\summ i1n \beta_i\beta_i^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}\) (including
multiplicities), i.e. this is the $\ell_1$ norm on the singular
values of \scb. $|||\scb|||_1$ is a convex function of \scb.
In this section we study the estimator defined by
\eqsplit[specPen]{
\hat\scb&= \argmin_{\scb\in\R^{p\times n}}\{\summ i1n (y_{ij}-x_{ij}^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}\beta_i)^2+
\lm|||\scb|||_1.\}
}
We refer to this problem as RING (Rotation INvariant Group) lasso.
The lassoes penalty considered primary the columns of \scb. The
main focus of the group lasso was the rows. Penalty $|||\scb|||_1$
is symmetric in its treatment of the rows and columns since
$\mathfrak{S}\scb =\mathfrak{S}\scb^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}$, where $\mathfrak{S} A$ denotes the spectrum
of $A$. Moreover, the penalty is invariant to the rotation of the
matrix \scb. In fact, \(|||\scb|||_1 = |||T\scb U |||_1\), where
\(T\) and \(U\) are \(n\times n\) and \(p\times p\) rotation
matrices:
\eqsplit{
(T\scb U)^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em} (T\scb U) &= U^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}\scb^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}\scb U
}
and the RHS have the same eigenvalues as \(\scb^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}\scb=\sum \beta_i\beta_i^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}\).
The rotation-invariant penalty aims at finding a basis in which
$\beta_1,\dots,\beta_n$ have the same pattern of sparsity. This is
meaningless if $n$ is small --- any function is well approximated by the span of the basis is sparse in under the right rotation. However, we will argue that this can be done when $n$ is
large.
The following lemma describes a relationship between group lasso
and RING lasso.
\begin{lemma}
\label{lem:spectVsGroup}
\mbox{}\par
\begin{enumerate}[(i)]
\item $\|\scb\|_{2,1} \ge \inf_{U\in\scu} \|U \scb\|_{2,1}
=|||\scb|||_1$, where $\scu$ is the set of all unitary matrices.
\item There is a unitary matrix $U$, which may depend on the data,
such that if $X_1,\dots,X_n$ are rotated by $U^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}$, then the
solution of the RING lasso \eqref{specPen} is the solution of the
group lasso in this basis.
\end{enumerate}
\end{lemma}
\begin{comment}
\subsection{ 2D case}
\marg{Should we leave this section? It helped us a lot, but would it help the reader?}
In the 2-dimensional case ($p=2$), we can find the explicit
expression for the penalty in terms of $\beta_{\cdot \ell}$. The
eigenvalues of
\eqsplit{
\scb \scb^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em} = \begin{pmatrix} ||\beta_{\cdot 1}||_2^2 & \langle \beta_{\cdot 1}, \beta_{\cdot 2}\rangle \\ \langle \beta_{\cdot 1}, \beta_{\cdot 2}\rangle & ||\beta_{\cdot 2}||_2^2 \end{pmatrix}
}
are the solutions of $(||\beta_{\cdot 1}||_2^2 - z)(||\beta_{\cdot
2}||_2^2 - z) - \langle \beta_{\cdot 1}, \beta_{\cdot
2}\rangle^2=0$, i.e. are
$$
z_{1,2} = \frac 1 2 \left[ ||\beta_{\cdot 1}||_2^2 +
||\beta_{\cdot 2}||_2^2 \pm \sqrt{ \left(||\beta_{\cdot 1}||_2^2 -
||\beta_{\cdot 2}||_2^2\right)^2 + 4 \langle \beta_{\cdot 1},
\beta_{\cdot 2}\rangle^2} \right],
$$
and hence it can be shown that
$$
|||\scb|||_1 =z_1^{1/2}+ z_2^{1/2} = \sqrt{||\beta_{\cdot 1}||_2^2
+ ||\beta_{\cdot 2}||_2^2 + 2 \sqrt{||\beta_{\cdot 1}||_2^2
||\beta_{\cdot 2}||_2^2 - \langle \beta_{\cdot 1}, \beta_{\cdot
2}\rangle^2}}
$$
due to identity $\sqrt{A + \sqrt{A^2-4B}} + \sqrt{A -
\sqrt{A^2-4B}} = \sqrt{2A +4\sqrt{B}}$.
Hence, in the directions where $\beta_{\cdot 1}$ and $\beta_{\cdot
2}$ are orthogonal, $|||\scb|||_1$ behaves like $\ell_1$ penalty,
and in the directions they are collinear ($\langle \beta_{\cdot
1}, \beta_{\cdot 2}\rangle = \pm 1$) $|||\scb|||_1$ behaves like
$\ell_2$ penalty.
The trace norm ``ball'' (i.e., the curve $|||\beta|||_1=1$ in the
$||\beta_{\cdot 1}||_2$ and $||\beta_{\cdot 2}||_2$ plane) is
plotted in Figure \ref{figBall} for different values of
$\rho=\frac{\langle \beta_{\cdot 1}, \beta_{\cdot
2}\rangle}{||\beta_{\cdot 1}||_2 ||\beta_{\cdot 2}||_2}$, the
``correlation'' between $\beta_{\cdot 1}$ and $\beta_{\cdot 2}$.
\onefigure[0.7]{SpectBall.eps}{ The trace norm ``ball'' in the $
||\beta_{\cdot 1}||_2$ and $||\beta_{\cdot 2}||_2$ plane for
different values of $\rho$. }{figBall}
\end{comment}
\subsection{The estimator }
Let $\scb=\summ \xi1{p\wedge n}\al_\xi \beta_\xi^*{\mathfrak{b}_\xi^*}^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}$
be the singular value decomposition, or the PCA, of \scb:
$\beta_1^*,\dots,\beta_p^*$ and $\mathfrak{b}_1^*,\dots,\mathfrak{b}_n^*$ are
orthonormal sub-bases of $\R^p$ and $\R^n$ respectively,
$\al_1\geq\al_2\geq\dots$, and
$\scb\scb^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}\beta_\xi^*=\al_\xi^2\beta_\xi^*$,
$\scb^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}\scb\mathfrak{b}_\xi^*=\al_\xi^2\mathfrak{b}_\xi^*$, $\xi=1,\dots,p\wedge n$.
Let $T=\summ \xi1{p\wedge n} e_\xi{\beta_\xi^*}^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}$ (clearly,
$TT^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}=I$).
Consider the parametrization of the problem in the rotated
coordinates, $\ti x_{ij}=Tx_{ij}$ and $\ti\beta_i=T\beta_i$. Then
geometrically the regression problem is invariant:
$x_{ij}^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}\beta_i=\ti x_{ik}^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}\ti\beta_i$, and
$|||\scb|||_1=\|\ti\scb\|_{2,1} $,
up to a modified regression matrix.
The representation $\hat\scb = \summ \xi1s \al_\xi \beta_\xi^*
{\mathfrak{b}_\xi^*}^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}$ shows that the difficulty of the problem is the
difficulty of estimating $s(n+p)$ parameters with $nm$
observations. Thus it is feasible as long as $s/m\to0$ and
$sp/nm\to0$.
We have
\begin{theorem}
\label{th:sparseSpect} Suppose $p<n$. Then the solution of the
RING lasso is given by $\summ \xi1s \beta^*_\xi{\mathfrak{b}_\xi^*}^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}$,
$s=s_\lm \leq p$, and $s_\lm\dec 0$ as $\lm\to\en$. If $s=p$ then
the gradient of the target function is given in a matrix form by
\eqsplit{
-2R+\lm (\hat\scb\hat\scb^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em})^{-1/2}\hat\scb
}
where
\eqsplit{
R=\Bigl(X_1^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}(Y_1-X_1\hat\beta_1),\dots,X_n^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}(Y_n-X_n\hat\beta_n)\Bigr).
}
And hence
\eqsplit{
\hat\beta_i=\bigl( X_i^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em} X_i + \frac\lm2 (\hat\scb\hat\scb^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em})^{-1/2}\bigr)^{-1}X_i^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em} Y_i.
}
That is, the solution of a ridge regression with adaptive weight.
More generally, let $\hat\scb=\summ \xi1s \alpha_\xi
\beta_\xi^*{\mathfrak{b}_\xi}^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}$, $s<p$, where $\beta_1^*,\dots,\beta_p^*$
is an orthonormal base of $\R^p$. Then the solution satisfies
\eqsplit{
&{\beta_\xi^*}^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em} R = \frac\lm2 {\beta_\xi^*}^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em} (\hat\scb\hat\scb^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em})^{+1/2}\hat\scb, \quad \xi\leq s
\\
&|{\beta_\xi^*}^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em} R \mathfrak{b}_\xi^*| \leq \frac\lm 2, \qquad \qquad \qquad s<\xi\leq p.
}
where for any positive semi-definite matrix $A$, $A^{+1/2}$ is the Moore-Penrose generalized inverse of $A^{1/2}$.
\end{theorem}
Roughly speaking the following can be concluded from the theorem.
Suppose the data were generated by a sparse model (in \emph{some}
basis). Consider the problem in the transformed basis, and let $S$
be the set of non-zero coefficients of the true model. Suppose
that the design matrix is of full rank within the sparse model:
$X_i^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em} X_i=\EuScript{O}(m)$, and that \lm is chosen such that $\lm\gg
\sqrt{nm\log(np)}$. Then the coefficients corresponding to $S$
satisfy
\eqsplit{
\hat\beta_{Si} &= \bigl(X_i^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em} X_i + \frac\lm2 (\hat\scb_S\hat\scb_S^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em})^{1/2}\bigr)^{-1}X_i^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em} Y_i.
}
Since it is expected that $\lm(\scb_S\scb_S^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em})^{1/2}$ is only slightly larger than $\EuScript{O}(m\log(np))$, it is completely dominated
by $X_i^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em} X_i$, and the estimator of this part of the model is consistent. On the other hand, the rows of $R$ corresponding
to coefficient not in the true model are only due to noise and hence each of them is $\EuScript{O}(\sqrt {nm})$.
The factor of $\log (np)$ ensures that their maximal norm will be below $\lm/2$, and the estimator is consistent.
\subsection{Bayesian perspectives}
We consider now the penalty for $\beta_k$ for a fixed $k$. Let $A=n^{-1}\sum_{k\ne i} \beta_k\beta_k^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}$, and write the spectral value decomposition $n^{-1}\summ k1n \beta_k\beta_k^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}=\sum c_jx_jx_j^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}$ where $\{x_j\}$ is an orthonormal basis of eigenvectors. Using Taylor expansion for not too big $\beta_i$, we get
\eqsplit{
\trace\bigl( (nA+\beta_i\beta_i^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em})^{1/2}\bigr)
&\approx \sqrt{n}\trace(A^{1/2}) + \summ j1p \frac{x_j^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em} \beta_i\beta_i^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em} x_j}{2 c_j^{1/2}}
\\
&= \sqrt{n}\trace(A^{1/2}) + \frac 12 \beta_i^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em} \bigl(\sum c_j^{-1/2} x_jx_j^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}\bigr)\beta_i
\\
&= \sqrt{n}\trace(A^{1/2}) + \frac12 \beta_i^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em} A^{-1/2}\beta_i
} So, this like $\beta_i$ has a prior of $\normal(0, n \sig^2/\lm
A^{1/2})$. Note that the prior is only related to the estimated
variance of $\beta$, and $A$ appears with the power of $1/2$. Now
$A$ is not really the estimated variance of $\beta$, only the
variance of the estimates, hence it should be inflated, and the
square root takes care of that. Finally, note that eventually, if
$\beta_i$ is very large relative to $nA$, then the penalty become
$\|\beta\|$, so the ``prior'' becomes essentially normal, but with
exponential tails.
A better way to look on the penalty from a Bayesian perspective
is to consider it as prior on the \(n\times p\) matrix
\(\scb=(\beta_1,\dots,\beta_n)\). Recall that the penalty is
invariant to the rotation of the matrix \scb. In fact,
\(|||\scb|||_1 = |||T\scb U|||_1\), where \(T\) and \(U\) are
\(n\times n\) and \(p\times p\) rotation matrices. Now, this means
that if \(\mathfrak{b}_1,\dots,\mathfrak{b}_p\) are orthonormal set of eigenvectors
of \(\scb^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}\scb\) and \(\gamma_{ij}=\mathfrak{b}_j^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}\beta_i\) --- the PCA
of \(\beta_1,\dots,\beta_n\), then \(|||\scb|||_1 = \summ j1p
\bigl(\summ i1n \gamma_{ij}^2\bigr)^{1/2} \) --- the RING lasso
penalty in terms of the principal components. The ``prior'' is
then proportional to $ e^{-\lm \summ j1p \|\gamma_{\cdot
j}\|_2}$.\, which is as if to obtain a random $\scb$ from the
prior the following procedure should be followed:
\begin{enumerate
\item
Sample \(r_1,\dots,r_p\) independently from \(\Gamma(n,\lm)\) distribution.
\item
For each \(j=1,\dots,p\) sample \(\gamma_{1j},\dots,\gamma_{nj}\) independently and uniformly on the sphere with radius \(r_j\).
\item
Sample an orthonormal base \(\chi_1,\dots,\chi_p\) "uniformly''.
\item
Construct \(\beta_i = \summ j1p \gamma_{ik}\chi_k\).
\end{enumerate}
\subsection{Inequalities under an RE condition}
The assumption on the design matrix $X$ needs to be modified to
account for the search over rotations, in the following way.
\noindent{\bf Assumption RE2$(s, c_0, \kappa)$}. For some integer
$s$ such that $1 \leqslant s \leqslant p$, and a positive number
$c_0$ the following condition holds: \eqsplit{
\kappa =
\min \{ & \frac{ ||X^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em} \Delta ||_2}{ \sqrt{m} ||P_V \Delta||_2 }: \, V
\,\text{is a linear subspace of} \,\,
\mathbb{R}^p, \, \dim(V) \leqslant\, s, \,\\ & \Delta \in
\mathbb{R}^{p\times n}\setminus \{0\},
|||(I-P_{V})\Delta |||_1 \leqslant \, c_0
|||P_V \Delta|||_1 \} > 0,
} where $P_V$ is the projection on linear subspace $V$.
If we restrict the subspaces $V$ to be of the form $V =
\bigoplus_{k=1}^r \langle e_{i_k}\rangle$, $r\leqslant s$ and
$\langle e_{i}\rangle$ is the linear subspace generated by the
standard basis vector $e_i$, and change the Schatten norm to
$\ell_{2,1}$ norm, then we obtain the restricted eigen value
assumption RE$_2(s,c_0,\kappa)$ of Lounici et
al.\ \cite{p:Lounici-GroupLasso}.
\begin{theorem}
\label{th:BRTspectral}
Let \(y_{ij}\dist \normal (f_{ij},\sig^2)\) independent,
\(f_{ij}=x_{ij}^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}\beta_i\), \(x_{ij}\in \R^p\),
\(\beta_i\in\R^p\), \(i=1,\dots,n\), \(j=1,\dots,m\), $p\geqslant
2$. Assume that $\summ j1m x_{ij\ell}^2 = m$ for all $i, \, \ell$.
Let assumption RE2$(s, 3,\kappa )$ be satisfied for $X=(x_{ijl})$, where $s={\text{rank}}(\scb)$.
Consider the RING lasso estimator $\hat{f}_{ij} =
X_{ij}^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}\hat{\beta}_i$ where $\hat{\scb}$ is defined by
\eqref{specPen} with $$\lm=4 \sigma \sqrt{(A+1)m np}, \quad
\text{for some}\quad A>1.$$
Then, for large $n$ or $p$, with probability at least \(1 -
e^{-Anp/8}\),
\eqsplit{
\frac 1 {mn} \|X^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}(\scb - \hat{\scb})\|_2^2 & \leqslant
\frac{64(A+1)\sigma^2 s p}{\kappa^2\, m};\\
\frac 1 n ||| \scb-\hat\scb |||_1 & \leqslant \frac{32\sigma
\sqrt{1+A} \, s \,\sqrt{p} }{\kappa^2 \sqrt{mn}},\\
{\text{rank}}(\hat\scb) &\leqslant s\, \frac{64 \phi_{\max}
}{\kappa^2}, } where $\phi_{\max}$ is the maximal eigenvalue of
$X^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em} X/m$.
\end{theorem}
Thus we have bounds similar to those of group lasso as a function
of the threshold $\lambda$, with $s$ being the rank of $\scb$
rather than its sparsity. However, for RING lasso we need a larger
threshold compared to that of the group lasso ($\lm_{GL} = 4\sig
\sqrt{mn} \left(1+\frac{A\log p}{\sqrt{n}}\right)^{1/2}$, Lounici
et al.~\cite{p:Lounici-GroupLasso}).
\subsection{Persistence}
We discuss now the persistence of the RING lasso estimators (see
Section~\ref{sec:persistGen} for definition and a general result).
We focus on the sets which are related to the trace norm which
defines the RING lasso estimator:
$$B_{n,p} = \{\scb \in \mathbb{R}^{n\times p}: \, |||\scb|||_1 \leqslant b(n,p) \}.$$
\begin{theorem}\label{th:persist}
Assume that $n>1$. For any $ F\in \scf_{n,p}^m(V)$, $\beta\in
B_{n,p}$ and \eqsplit{\hat{\beta}^{(m,n,p)} = \argmin_{\beta\in
B_{n,p}} L_{\hat{F}}(\beta),} we have
\eqsplit{
L_{F}\left(\hat\beta\right) -
\min_{\beta\in B_{n,p}} L_{F}\left(\beta \right) \leqslant
\left(\frac 1 m + \frac{p b^2}{nm}\right) \left(16e V \frac{
\log(np) }{ m \, \eta }\right)^{1/2} }
with probability at least $1 - \eta$, for any $\eta\in (0,1)$.
\end{theorem}
\marg{I read slightly differently the theorem than you. This is
what I understand. Please check, accept or reject.}Thus, for
$\eta$ sufficiently small, the conditions $\log(np)\leqslant c_p
m^3 \eta $ and $b\leqslant c_b \sqrt{ {nm}/{p }}$, for some $c_b,
c_p >0$, imply that with sufficiently high probability, the
estimator is persistent. Roughly speaking, $b$ is the number of
components in the SVD of \scb (the rank of \scb, $\scm(\beta)$ after the proper rotation), and
if $m\gg\log n$, then what is needed is that this number will be
strictly less $ n^{1/2}m^{3/4}p^{-1/2} $. That is, if the true
model is sparse, $p$ can be almost as large as $m^{3/2}n^{1/2}$.
\marg{NB: I don't disagree but may be it is best to make it more
explicit that "roughly speaking" means treating the bound on
Schatten-$1$ norm as a bound on Schatten -0 norm?}
\subsection{Algorithm and small simulation study}
\twofigures[1.1]{NB03042009a.eps}{NB03042009b.eps}{Component
variances and eigenvalues, \(m=25\), \(n=150\) }{fig1}
A simple algorithm is the following:
\begin{enumerate
\item Initiate some small value of \(\hat
\beta_1,\dots,\hat\beta_n\). Let \(A=\summ j1n
\hat\beta_j\hat\beta_j^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}\). Fix \(\gamma\in(0,1]\), \(\eps>0\),
\(k\), and \(c>1\). \item\label{A1st1} For \(i=1,\dots,n\):
\begin{enumerate
\item Compute \(\delta_i = (X_i^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em} X_i + \lm A^{-1/2})^{-1}X_i^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}
(y_i-X_i\hat\beta_i)\). \item Update \(A\leftarrow
A-\hat\beta_i\hat\beta_i\);
\(\hat\beta_i\leftarrow\hat\beta_i+\gamma\delta_i\); \(A\leftarrow
A+\hat\beta_i\hat\beta_i\);
\end{enumerate}
\item if \(\summ j1p \ind\bigl(n^{-1}\summ
i1n\hat\beta_{ij}^2>\eps\bigr)>k \) update
\(\lambda\leftarrow\lambda c\) otherwise
\(\lambda\leftarrow\lambda/ c\). \item Return to step \ref{A1st1}
unless there is no real change of coefficients.
\end{enumerate}
To fasten the computation, the SVD was computed only every 10
values of \(i\).
As a simulation we applied the above algorithm to the following
simulated data.
We generated random \(\beta_1,\dots,\beta_{150}\in\R^{150}\) such
that all coordinates are independent, and
\(\beta_{ij}\dist\normal(0, e^{-2j/5})\). All \(X_{ij\ell}\) are
\iid \(\normal(0,1)\), and \(y_{ij} = x_{ij\cdot}^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}
\beta_i+\eps_{ij}\), where \(\eps_{ij}\) are all \iid
\(\normal(0,1)\). The true \(R^2\) obtained was approximately
0.73. The number of replicates per value of \(\beta\), \(m\),
varied between 5 to 300. We consider two measures of estimation
error:
\eqsplit{
L_{\rm par} &= \frac{\summ i1n \|\hat\beta_i-\beta_i\|_\en} {\summ i1n \|\beta_i\|_\en}
\\
L_{\rm pre} &= \frac{\summ i1n \|X_j(\hat\beta_i-\beta_i)\|_\en} {\summ i1n \|X_i\beta_i\|_\en}
}
\onefigure[1.1]{NB11Apr2009d.eps}{ Lower lip position while
repeating 32 times 'Say bob again'}{fig2}
The algorithm stopped after 30--50 iterations. Figure \label{fig1}
is a graphical presentation of a typical result. A summary is
given in Table \ref{tab1}. Note that \(m\) has a critical impact
on the estimation problem. However, with as little as \(5\)
observations per \(R^{150}\) vector of parameter we obtain a
significant reduction in the prediction error.
\begin{table}[H]
\caption{\label{tab1}The estimation and prediction error as
function of the number of observations per vector of parameters
Means (and SDK).}
\begin{center}
\begin{tabular}{|r|r|r|}
\hline
$m$ & \(L_{\rm par}\) & $L_{\rm pre}$ \\
\hline\hline
5 & 0.9530 (0.0075) & 0.7349 (0.0375) \\
\hline
25 & 0.7085 (0.0289) & 0.7364 (0.0238) \\
\hline
300 & 0.2470 (0.0080) & 0.5207 (0.0179) \\
\hline
\end{tabular}
\end{center}
\end{table}
\threefiguresV[1.5]{NB11Apr2009a.eps}{NB11Apr2009b.eps}{NB11Apr2009c.eps}{Eigenvalue,
coefficient variance and typical observed and smooth path.}{fig3}
The technique is natural for functional data analysis. We used the
data LipPos. The data is described by
Ramsay and
Silverman and can be found in
http://www.stats.ox.ac.uk/~silverma/fdacasebook/lipemg.html. The
original data is given in Figure \ref{fig2}. However we added
noise to the data as can be seen in Figure \ref{fig3}. The lip
position is measured at $m=501$ time points, with \(n=32\)
repetitions.
As the matrix \(X\) we considered the union of 6 cubic spline
bases with, respectively, 5, 10, 20, 100, 200, and 500 knots
(i.e., \(p=841\), and \(X_i\) does not depend on \(i\)). A
Gaussian noise with \(\sigma=0.001\) was added to \(Y\). The
result of the analysis is given in Figure \ref{fig3}. Figure
\ref{fig4} presents the projection of the mean path on the first
eigen-vectors of \(\summ i1n \hat\beta_i\hat\beta_i^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}\).
\onefigure[0.5]{NB11Apr2009e.eps}{Projection of the estimated mean
path on the 2 first eigen-vectors of \(\summ i1n
\hat\beta_i\hat\beta_i^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}\) and the true mean path.}{fig4}
The final example we consider is somewhat arbitrary. The data,
taken from StatLib, is of the daily wind speeds for 1961-1978 at
12 synoptic meteorological stations in the Republic of Ireland. As
the \(Y\) variable we considered one of the stations (station
BIR). As explanatory variables we considered the 11 other station
of the same day, plus all 12 stations 70 days back (with the
constant we have altogether 852 explanatory variables). The
analysis was stratified by month. For simplicity, only the first
28 days of the month were taken, and the first year, 1961, served
only for explanatory purpose. The last year was served only for
testing purpose, so, the training set was for 16 years (\(n=12\),
\(m=448\), and \(p=852\) ). In Figure \ref{fig5} we give the 2nd
moments of the coefficients and the scatter plot of predictions
vs. true value of the last year.
\twofigures[0.75]{NB14Apr2009a.eps}{NB14Apr2009b.eps}{Coefficient
2nd moment and prediction vs.true value of the test year.}{fig5}
\section{The spectral penalty}
\label{sec:spectral}
Let \(A=\sum c_i x_ix_i^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}\), be a positive semi-definite matrix, where \(x_1,x_2,\dots\) is an orthonormal basis of eigenvectors. Then, we define \(A^{\gamma}=\sum c_i^{\gamma} x_ix_i^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}\).
We consider now as penalty the function
\eqsplit{
\psi(\scb)=\psi(\beta_1,\dots,\beta_n)= \trace\Bigl\{\bigl(\summ i1n \beta_i\beta_i^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}\bigr)^{1/2}\Bigr\}.
} Note that \(\psi(\scb)=\sum c_i^{1/2}\) where \(c_1,\dots,c_p\) are the eigenvalues of \(\scb\scb^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}=\summ i1n \beta_i\beta_i^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}\) (including multiplicities).
\begin{lemma}
\label{lem:psiConvex}
\(\psi\) is a convex function.
\end{lemma}
In this section we study the estimator defined by
\eqsplit{
(\hat\beta_1,\dots,\hat\beta_n)&= \argmin_{(\beta_1,\dots,\beta_n)\in\R^{p\times n}}\{\summ i1n (y_{ij}-x_{ij}^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}\beta_i)^2+\lm\psi(\beta_1,\dots,\beta_n)\}
}
The lassoes penalty considered primary the columns of \scb. The main focus of the group lasso was the rows. The spectral penalty is symmetric in its treatment of the rows and columns since $\mathfrak{S}\scb =\mathfrak{S}\scb^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}$, where $\mathfrak{S} A$ denotes the spectrum of $A$. Moreover, the penalty is invariant to the rotation of the matrix \scb. In fact, \(\psi(\scb) = \psi(T\scb U )\), where \(T\) and \(U\) are \(n\times n\) and \(p\times p\) rotation matrices:
\eqsplit{
(T\scb U)^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em} (T\scb U) &= U^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}\scb^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}\scb U
}
and the RHS have the same eigenvalues as \(\scb^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}\scb=\sum \beta_i\beta_i^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}\).
Now, the standard notion of sparsity, as captured by the $L_0$ norm, or by the standard lasso and group lasso, is basis dependent. Consider the nonparametric setup in which $y_{ij} = g(z_{ij}) + \eps_{ij}$, where $z\in[0,1]$. The model fits the regression setup, if $x_{ij\ell}=h_\ell(z_{ij}$. If, for example, $g(z)=\ind(a<z\le b)$, then this example is sparse when $h_\ell(z)=\ind(z>\ell/p)$. It is not sparse if
$h_\ell(z)=(z-\ell/p)^*$. On the other hand a function $g$ which has a piece-wise constant slope is sparse in the latter basis, but not in the former.
The spectral penalty aims at finding a basis in which $\beta_1,\dots,\beta_n$ have the same pattern of sparsity. This is meaningless if $n$ is small --- any function is sparse in some basis. However, we will argue that this can be done when $n$ is large. We have
\begin{theorem}
\label{th:sparseSpect}
Suppose that for some rotation matrix $U$, and
\eqsplit{
\ti x_{ij\ell}=\summ l1p x_{ijl} U_{l\ell},\quad i=1,\dots,n,\; j=1,\dots,m, \;\ell=1,\dots,p,
}
there are $\hat\gamma_1,\dots,\hat\gamma_n$ such that $\hat\gamma_{i\ell}=0$ for $i=1,\dots,n$ and $\ell\not\in S\subset\{1,\dots,p\}$, and
\eqsplit{
|\sum_j \ti x_{ij\ell}(y_{ij} - \ti x_{ij}^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em} \hat\gamma_{i})|
& \leq \lambda\bigl(1 - \hat\gamma_i^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em} \Bigl(\sum_k \hat\gamma_k\hat\gamma_k^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}\Bigr)^{-1} \hat\beta_i \bigr)^{1/2}
\\
\hat\gamma_{i\ell}&=0,\quad
i=1,\dots,n,\;\ell\not\in S\subset\{1,\dots,p\},
}
\end{theorem}
Roughly speaking the theorem states that if the data were generated by a sparse model (in \emph{some} basis), that is with $\gamma$ whose most entries are very close to 0, and a few are well above the noise level, then the solution for the spectral penalty will have the same type of sparsity.
We consider now the penalty for $\beta_k$ for a fixed $k$. Let $A=n^{-1}\sum_{k\ne i} \beta_k\beta_k^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}$. Using Taylor expansion for not too big $\beta_i$, based on \eqref{eigender} we get
\eqsplit{
\trace\bigl( (nA+\beta_i\beta_i^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em})^{1/2}\bigr)
&= \sqrt{n}\trace(A^{1/2}) + \summ j1p \frac{x_j^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em} \beta_i\beta_i^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em} x_j}{2 c_j^{1/2}}
\\
&= \sqrt{n}\trace(A^{1/2}) + \frac 12 \beta_i^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em} \bigl(\sum c_j^{-1/2} x_jx_j^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}\bigr)\beta_i
\\
&= \sqrt{n}\trace(A^{1/2}) + \frac12 \beta_i^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em} A^{-1/2}\beta_i
}
So, this like $\beta_i$ has a prior of $\normal(0, n \sig^2/\lm A^{1/2})$. Note that the prior is only related to the estimated variance of $\beta$. $A$ appears with the prior of $1/2$. Now $A$ is not really the estimated variance of $\beta$, only the variance of the estimates, hence it should be inflated, and the square root takes care of that. Finally, note that eventually, if $\beta_i$ is very large relative to $nA$, then the penalty become $\|\beta\|$, so the ``prior'' becomes essentially normal, but with exponential tails.
A better way to look on the penalty from a Bayesian perspective is to consider it as prior on the \(n\times p\) matrix \(\scb=(\beta_1,\dots,\beta_n)\). Recall that the penalty is invariant to the rotation of the matrix \scb. In fact, \(\psi(\scb) = \psi(T\scb U )\), where \(T\) and \(U\) are \(n\times n\) and \(p\times p\) rotation matrices. Now, this means that if \(\chi_1,\dots,\chi_p\) are orthonormal set of eigenvectors of \(\scb^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}\scb\) and \(\gamma_{ij}=\chi_j^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}\beta_i\) --- the PCA of \(\beta_1,\dots,\beta_n\), then \(\psi(\scb) = \summ j1p \bigl(\summ i1n \gamma_{ij}^2\bigr)^{1/2} \) --- the rms penalty in terms of the principal components. The ``prior'' is then
\eqsplit{
\varpropto e^{-\lm \summ j1p \|\gamma_{\cdot j}\|_2}
}
Which is as if:
\begin{enumerate
\item
Sample \(r_1,\dots,r_p\) independently from \(\Gamma(n,\lm)\) distribution.
\item
For each \(j=1,\dots,p\) sample \(\gamma_{1j},\dots,\gamma_{nj}\) independently and uniformly on the sphere with radius \(r_j\).
\item
Sample an orthonormal base \(\chi_1,\dots,\chi_p\) "uniformly''.
\item
Construct \(\beta_i = \summ j1p \gamma_{ik}\chi_k\).
\end{enumerate}
So hopefully the rms penalty ensures that the solution will be sparse in \emph{some} basis. Or the assumption (and hope) is that the Procrustes's distance between the true \(\beta\)'s and a sparse solution is small.
\subsection{Algorithm and small simulation study}
\twofigures[1.1]{NB03042009a.eps}{NB03042009b.eps}{Component
variances and eigenvalues, \(m=25\), \(n=150\) }{fig1}
A simple algorithm is the following:
\begin{enumerate
\item Initiate some small value of \(\hat
\beta_1,\dots,\hat\beta_n\). Let \(A=\summ j1n
\hat\beta_j\hat\beta_j^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}\). Fix \(\gamma\in(0,1]\), \(\eps>0\),
\(k\), and \(c>1\). \item\label{A1st1} For \(i=1,\dots,n\):
\begin{enumerate
\item Compute \(\delta_i = (X_i^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em} X_i + \lm A^{-1/2})^{-1}X_i^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}
(y_i-X_i\hat\beta_i)\). \item Update \(A\leftarrow
A-\hat\beta_i\hat\beta_i\);
\(\hat\beta_i\leftarrow\hat\beta_i+\gamma\delta_i\); \(A\leftarrow
A+\hat\beta_i\hat\beta_i\);
\end{enumerate}
\item if \(\summ j1p \ind\bigl(n^{-1}\summ
i1n\hat\beta_{ij}^2>\eps\bigr)>k \) update
\(\lambda\leftarrow\lambda c\) otherwise
\(\lambda\leftarrow\lambda/ c\). \item Return to step \ref{A1st1}
unless there is no real change of coefficients.
\end{enumerate}
Comment: to fasten the computation, the SVD was computed only
every 10 values of \(i\).
As a simulation we applied the above algorithm to the following
simulated data (\textbf{NB05Apr2009.m}).. We generated random
\(\beta_1,\dots,\beta_{150}\in\R^{150}\) such that all coordinates
are independent, and \(\beta_{ij}\dist\normal(0, e^{-2j/5})\). All
\(X_{ij\ell}\) are \iid \(\normal(0,1)\), and \(y_{ij} =
x_{ij\cdot}^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em} \beta_i+\eps_{ij}\), where \(\eps_{ij}\) are all
\iid \(\normal(0,1)\). The true \(R^2\) obtained was
approximately 0.73. The number of replicates per value of
\(\beta\), \(m\), varied between 5 to 300. We consider two
measures of estimation error:
\eqsplit{
L_{\rm par} &= \frac{\summ i1n \|\hat\beta_i-\beta_i\|_\en} {\summ i1n \|\beta_i\|_\en}
\\
L_{\rm pre} &= \frac{\summ i1n \|X_j(\hat\beta_i-\beta_i)\|_\en} {\summ i1n \|X_i\beta_i\|_\en}
}
\onefigure[1.1]{NB11Apr2009d.eps}{ Lower lip position while
repeating 32 times 'Say bob again'}{fig2}
The algorithm stopped after 30--50 iterations. Figure \label{fig1}
is a graphical presentation of a typical result. A summary is
given in Table \ref{tab1}. Note that \(m\) has a critical impact
on the estimation problem. However, with as little as \(5\)
observations per \(R^{150}\) vector of parameter we obtain a
significant reduction in the prediction error.
\begin{table}[H]
\caption{\label{tab1}The estimation and prediction error as
function of the number of observations per vector of parameters
Means (and SDK).}
\begin{center}
\begin{tabular}{|r|r|r|}
\hline
$m$ & \(L_{\rm par}\) & $L_{\rm pre}$ \\
\hline\hline
5 & 0.9530 (0.0075) & 0.7349 (0.0375) \\
\hline
25 & 0.7085 (0.0289) & 0.7364 (0.0238) \\
\hline
300 & 0.2470 (0.0080) & 0.5207 (0.0179) \\
\hline
\end{tabular}
\end{center}
\end{table}
\threefiguresV[1.5]{NB11Apr2009a.eps}{NB11Apr2009b.eps}{NB11Apr2009c.eps}{Eigenvalue,
coefficient variance and typical observed and smooth path.}{fig3}
The technique is natural for functional data analysis. We used the
data LipPos taken James O. Ramsay and Bernard W. Silverman
http://www.stats.ox.ac.uk/~silverma/fdacasebook/lipemg.html. The
data is described in Figure \ref{fig2}. However we added noise to
the data as will be seen in Figure \ref{fig3}. The position is
measured at $m=501$ time points, with \(n=32\) repetitions.
As the matrix \(X\) we considered the union of 6 cubic spline
bases with, respectively, 5, 10, 20, 100, 200, and 500 knots
(i.e., \(p=841\), and \(X_i\) does not depend on \(i\)). A
Gaussian noise with \(\sigma=0.001\) was added to \(Y\). The
result of the analysis is given in Figure \ref{fig3}. Figure
\ref{fig4} present the projection of the mean path on the first
eigen-vectors of \(\summ i1n \hat\beta_i\hat\beta_i^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}\). (Analysis
done using NB11Apr2009.m).
\onefigure[0.5]{NB11Apr2009e.eps}{Projection of the estimated mean
path on the 2 first eigen-vectors of \(\summ i1n
\hat\beta_i\hat\beta_i^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}\) and the true mean path.}{fig4}
The final example we consider is some what arbitrary. The data,
taken from StatLib, is of the daily wind speeds for 1961-1978 at
12 synoptic meteorological stations in the Republic of Ireland. As
the \(Y\) variable we considered one of the stations (station
BIR). As explanatory variables we considered the 11 other station
of the same day, plus all 12 stations 70 days back (with the
constant we have altogether 852 explanatory variables). The
analysis was stratified by month. For simplicity, only the first
28 days of the month were taken, and the first year, 1961, served
only for explanatory purpose. The last year was served only for
testing purpose, so, the training set was for 16 years (\(N=12\),
\(m=448\), and \(P=852\) ). In Figure \ref{fig5} we give the 2nd
moments of the coefficients and the scatter plot of predictions
vs. true value of the last year. (Analysis done using
NB14Apr2009.m)
\twofigures[0.75]{NB14Apr2009a.eps}{NB14Apr2009b.eps}{Coefficient
2nd moment and prediction vs.true value of the test year.}{fig5}
\subsection{Fixed design matrix }
We start with the simple case where \(X_i=\ti X\), that is
\(y_i=X\beta_i+\eps_i\), \(i=1,\dots,n\), where \(y_i,\eps_i\in
\R^m\) and \(\beta_i\in\R^p\). The general approach for sparsity
depends heavily on representing the model in the proper base, the
one in which sparsity is attained. That is the base in which there
is a sparse approximation of the regression function. This isn't
the case with the rms penalty. The penalty function is invariant
to rotation, \(\psi(\beta_1,\dots,\beta_n)
=\psi(T\beta_1,\dots,T\beta_n)\) where \(T\) is a matrix with
orthonormal columns, \(T^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em} T=I_p\), hence, the prediction
\(X\hat\beta_i\) is invariant for a rotation of the design matrix,
such that the model is equivalent to \(Y_i=\ti X\beta_i+\eps_i\),
where \(\ti X=XT^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}\). This means that sparsity will be in the
right base.
\subsection{The spectral penalty: 2D case}
In the 2-dimensional case ($p=2$), we can find the explicit
expression for the penalty in terms of $\beta_{\cdot \ell}$. The
eigenvalues of
\eqsplit{
\scb \scb^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em} = \begin{pmatrix} ||\beta_{\cdot 1}||_2^2 & \langle \beta_{\cdot 1}, \beta_{\cdot 2}\rangle \\ \langle \beta_{\cdot 1}, \beta_{\cdot 2}\rangle & ||\beta_{\cdot 2}||_2^2 \end{pmatrix}
}
are the solutions of $(||\beta_{\cdot 1}||_2^2 - z)(||\beta_{\cdot
2}||_2^2 - z) - \langle \beta_{\cdot 1}, \beta_{\cdot
2}\rangle^2=0$, i.e. are
$$
z_{1,2} = \frac 1 2 \left[ ||\beta_{\cdot 1}||_2^2 +
||\beta_{\cdot 2}||_2^2 \pm \sqrt{ \left(||\beta_{\cdot 1}||_2^2 -
||\beta_{\cdot 2}||_2^2\right)^2 + 4 \langle \beta_{\cdot 1},
\beta_{\cdot 2}\rangle^2} \right],
$$
and hence it can be shown that
$$
\psi(\scb) =z_1^{1/2}+ z_2^{1/2} = \sqrt{||\beta_{\cdot 1}||_2^2 +
||\beta_{\cdot 2}||_2^2 + 2 \sqrt{||\beta_{\cdot 1}||_2^2
||\beta_{\cdot 2}||_2^2 - \langle \beta_{\cdot 1}, \beta_{\cdot
2}\rangle^2}}
$$
due to identity $\sqrt{A + \sqrt{A^2-4B}} + \sqrt{A -
\sqrt{A^2-4B}} = \sqrt{2A +4\sqrt{B}}$.
Hence, in the directions where $\beta_{\cdot 1}$ and $\beta_{\cdot
2}$ are orthogonal, $\psi(\scb)$ behaves like $\ell_1$ penalty,
and in the directions they are collinear ($\langle \beta_{\cdot
1}, \beta_{\cdot 2}\rangle = \pm 1$) $\psi(\scb)$ behaves like
$\ell_2$ penalty.
If the ``correlation'' between $\beta_{\cdot 1}$ and $\beta_{\cdot
2}$ is $\rho=\frac{\langle \beta_{\cdot 1}, \beta_{\cdot
2}\rangle}{||\beta_{\cdot 1}||_2 ||\beta_{\cdot 2}||_2}$, then the
penalty ``ball'' (curve as a function of $||\beta_{\cdot 1}||_2$
and $||\beta_{\cdot 2}||_2$) is plotted for different values of
$\rho$ (Figure \ref{figBall}).
\onefigure[0.7]{SpectBall.eps}{ Spectral ``ball'' as a function o
different values of $\rho$. }{figBall}
\subsection{ Equations for $\widehat{\beta}$.}
Differentiating the ``loss'' with respect to $\beta_{ij}$, we
obtain the equations for $\widehat{\beta}$:
$$
2\sum_{j} x_{ij\ell} (Y_{ij} - x_{ij\cdot }^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em} \widehat{\beta}_i) =
\lambda \, \sgn(\widehat{\beta}_{i\ell}) \left( 1 -
\widehat{\beta}_{i L}^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em} (\sum_{k} \widehat{\beta}_{k L}
\widehat{\beta}_{k L}^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em})^{-1} \widehat{\beta}_{i L} \right)^{1/2},
$$
if $\widehat{\beta}_{i\ell} \neq 0$, and if
$$
|2\sum_{j} x_{ij\ell} (Y_{ij} - x_{ij\cdot }^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em} \widehat{\beta}_i)|
\leqslant \lambda \left( 1 - \widehat{\beta}_{i L}^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em} (\sum_{k}
\widehat{\beta}_{k L} \widehat{\beta}_{k L}^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em})^{-1}
\widehat{\beta}_{i L} \right)^{1/2},
$$
$\widehat{\beta}_{i\ell} = 0$. Here $L = J(\beta) \setminus
\{\ell\}$, i.e. we assume that $\exists (\sum_{k}
\widehat{\beta}_{k L} \widehat{\beta}_{k L}^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em})^{-1}$. Does it
always? If not, how does the expression changes then?
\subsection{ Equations for $\widehat{\gamma}$.}
If we parameterize $\beta_i = U^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em} \gamma_i$, the equations for
$\gamma$ become:
$$
\sum_{jk} x_{ijk} U_{\ell k} (Y_{ij} - X_{ij}^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em} U^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em} \gamma_i ) =
\frac {\lambda}{2} \frac{\gamma_{i\ell}}{ ||\gamma_{\cdot
\ell}||_2},
$$
if $\gamma_{i\ell} \neq 0$, and
$$
|\sum_{jk} x_{ijk} U_{\ell k} (Y_{ij} - X_{ij}^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em} U^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em} \gamma_i )|
\leqslant \frac {\lambda}{2} \frac{|\gamma_{i\ell}|}{
||\gamma_{\cdot \ell}||_2},
$$
$\gamma_{i\ell} = 0$.
\subsection{The spectral penalty: inequalities}
If $\beta\beta^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em} = U C U^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}$ and $\hat\beta\hat\beta^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em} = \hat{U}
\hat{C} \hat{U}^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}$ where $\hat\beta$ is the solution of the
spectral RMS problem \eqref{rotatedlassostar}. Then $U^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}\hat\beta$
is the solution of the standard RMS (group lasso) problem
\eqref{lassostar} with $\tilde{X}_i = U^T X_i$. Hence,
Lemma~\ref{lem:GroupLasso} is satisfied for $\gamma = U^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}\beta$
and ``estimate'' $\tilde\gamma = U^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}\hat\beta$ with design
matrix $\tilde{X}$.
Note that even if $\beta$ and $\hat\beta$ are close, $\gamma$ and
$\hat\gamma = \hat{U}^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}\hat\beta$ are close only up to a
permutation of variables. Hence, we will be interested in
comparing $U^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}\hat{U}$ to a permutation matrix (e.g. testing that
all eigenvalues are close to $\pm 1$).
We may be interested in sparsity of $\beta$ and $\gamma$, as well
as the approximation error of estimating $f$ and $\beta$. Thus, we
obtain the following results.
\begin{lemma}\label{lem:RotateGroupLasso}
Let \(Y_{ij}\dist \normal (f_{ij},\sig^2)\) independent,
\(f_{ij}=x_{ij}^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}\beta_i\), \(x_{ij}\in \R^p\),
\(\beta_i\in\R^p\), \(i=1,\dots,n\), \(j=1,\dots,m\). Consider
\eqsplit[rotatedlassostar]{
(\hat\beta_1,\dots,\hat\beta_n) &=
\argmin \Biggl[ \summ i1n \summ j1m (Y_{ij} - x_{ij}^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}\,\beta_i)^2 + \lambda
\trace \Bigl\{ \summ i1n \beta_{i }\beta_i^T\Bigr\}^{1/2}
\Biggr],
}
with \(\lm=4A\sig\tilde\Lm_x\sqrt{n \log(pn)}\), where
\(\tilde\Lm_x= \max_{i}\sqrt{\sum_{\ell,j} x_{i j \ell}^2 }\) and
\(A>\sqrt2\).
Then, with probability at least \(1-(np)^{1-A^2/2}\):
\begin{enumerate}[a)]
\item \(\forall \beta\in\R^{p\times n}\):%
\eqsplit[SPECTboundf_beta1]{ %
\|f-\hat f\|_2^2 + \frac{\lm}{2\sqrt{n}} \summ {\ell}1p
\summ i1n |\gamma_{i\ell} - (U^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}\hat{U})_\ell
\hat\gamma_{i}| \le \|f - f_{\scb}\|_2^2
+ \lm (1/\sqrt{n}+1) \sum_{\ell\in J( \gamma)}\|\hat\gamma_{\cdot\ell}
- (U^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}\hat{U})_\ell \hat\gamma\|_1,
} %
where \(f_{\scb ij} = x_{ij}^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}\beta_i = x_{ij}^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em} U^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em} \gamma_i \),
and%
\eqsplit{
J(\gamma)=\Bigl\{\ell\in\{1,\dots,p\}:\summ i1n\gamma_{i\ell}^2>0\Bigr\}. %
}
\item \(\forall \beta\in\R^{p\times n}\):%
\eqsplit[SPECTboundf_beta1]{ %
\|f-\hat f\|_2^2 + \lm/2 \summ {\ell}1p
\summ i1n |\gamma_{i\ell} - (U^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}\hat{U})_\ell \hat\gamma_{i}| \le \|f - f_{\scb}\|_2^2
+ 2\lm \sum_{\ell\in J( \gamma)}\|\hat\gamma_{\cdot\ell}
- \gamma_{\cdot\ell}\|_2.
} %
\item Let $\tilde\gamma = U^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}\hat\beta$ and
\(\scm(\tilde\gamma) = |J(\tilde\gamma)|\) be the number of
non-zero coefficients of \(\tilde\gamma\),
then \eqsplit{ \scm(\tilde\gamma) \leq \|f-\hat f\|_2^2 \frac{m
p }{15\sig^2 A^2 n\log(np)}. }
\item (Do a bound on $U^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}\hat{U} - P$, $P$ - a permutation)
\end{enumerate}
\end{lemma}
Note that for any $p\geqslant 1$,
$$
||\beta_i - \hat\beta_{i}||_p \leqslant ||U||_p
||\gamma_i - U^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}\hat{U}\hat\gamma_{i}||_p,
$$
and for $p=2$,
$$
||\beta_i - \hat\beta_{i}||_2 = ||\gamma_i -
U^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}\hat{U}\hat\gamma_{i}||_2.
$$
\begin{proof}
Similarly to the proof of Lemma~\ref{lem:GroupLasso}, we have
\eqsplit{
\|f-\hat f\|_2^2 & = \|f_{\beta}-f\|_2^2
+2\sum_{ij}\eps_{ij}x_{ij}^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}(U \gamma_i - \hat{U}\hat\gamma_i).
}
The last term can be bounded with high probability. Using
inequality $|a^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em} A b| \leqslant ||A||_2\, ||a||_2\, ||b||_2$, we
obtain
\eqsplit{
&|\sum_j \eps_{ij}x_{ij}^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}(U \gamma_i -\hat{U}\hat\gamma_i)|
\leqslant \| \sum_j\eps_{ij}x_{ij}^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em} \|_2 \left[ \|U \gamma_i
-\hat{U}\hat\gamma_i\|_2 \right] \leqslant \sqrt{\sum_j c_{ij}^2
} \left[ \| \gamma_i -U^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}\hat{U}\hat\gamma_i\|_2 \right].
}
Note that on one hand,
\eqsplit{
& \sum_{ij} c_{ij} | \beta_{ij} - \hat\beta_{ij}|
= \sum_{ij} c_{ij} | U_j^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em} \gamma_{i} - \hat{U}_j^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}\hat\gamma_{i}|
\\
&\leqslant \sum_{i} \sqrt{\sum_j c_{ij}^2} \| U^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em} \gamma_{i} -
\hat{U} ^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}\hat\gamma_{i}\|_2 = \sum_{i} \sqrt{\sum_j c_{ij}^2} \|
\gamma_{i} - U\hat{U} ^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}\hat\gamma_{i}\|_2\\
& \leqslant \max_i \sqrt{\sum_j c_{ij}^2} \sum_{\ell} \|
\gamma_{\cdot \ell} - (U^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}\hat{U})_{\ell} \hat\gamma\|_1\\
&\leqslant \sqrt{n} \max_i \sqrt{\sum_j c_{ij}^2} \sum_{\ell} \|
\gamma_{\cdot \ell} - (U^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}\hat{U})_{\ell} \hat\gamma\|_2
}
On the other hand,
\eqsplit{
& \sum_{ij} c_{ij} | \beta_{ij} - \hat\beta_{ij}|
= \sum_{ij} c_{ij} | U_j^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em} \gamma_{i} - \hat{U}_j^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}\hat\gamma_{i}|
\\
&\leqslant \sum_{i} \max_j c_{ij} \| U^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em} \gamma_{i} - \hat{U}
^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}\hat\gamma_{i}\|_1 \leqslant ||U^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}||_1 \max_{ij} c_{ij}
\sum_{i} \|\gamma_{i} - U^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}\hat{U}\hat\gamma_{i}\|_1\\
&\leqslant \sqrt{p} \max_{ij} c_{ij} \sum_{\ell} \|\gamma_{\cdot
\ell} - (U^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}\hat{U})_{\ell}\hat\gamma\|_1
}
[can be also bounded by]
$$
2\sum_{i} \max_j c_{ij} \|U^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em} \gamma_{i } -
\hat{U}^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}\hat\gamma_{i}\|_1 = 2\sum_{i} \sqrt{\sum_j c_{i j}^2} \|
\gamma_{i } - U \hat{U}^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}\hat\gamma_{i}\|_2,
$$
a) We have that
\eqsplit{
\|f-\hat f\|_2^2 &+ \lm_3 \sum_{i} \|\gamma_{i} - \hat{U} U^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}\hat\gamma_{i}\|_1
\leq \|f_\scb - f\|_2^2
+ \lm_3 \sum_i
\|\gamma_{i}-\hat{U} U^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}\hat\gamma_{i}\|_1
+\lm\sum_\ell \|\gamma_{\cdot\ell}\|_2
\\
&\hspace{2em}
-\lm\sum_\ell \| (U^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}\hat{U})_\ell\hat\gamma\|_2
+ 2 \max_i \sqrt{\sum_j c_{ij}^2} \sum_{\ell i} |\gamma_{i\ell} -
(U^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}\hat{U})_{\ell}\hat\gamma_{i} |\\
&\leqslant \|f_\scb - f\|_2^2
+\lm\sum_\ell \|\gamma_{\cdot\ell}\|_2
\\
&\hspace{2em}
-\lm\sum_\ell \|U^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}\hat{U}\hat\gamma_{\cdot\ell}\|_2
+ \sum_{\ell} \left( 2\max_i \sqrt{\sum_j c_{ij}^2} + \lm_3\right) \|\gamma_{\cdot \ell} -
(\hat{U} U^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em})_{\ell}\hat\gamma \|_1\\ &= \|f_\scb - f\|_2^2 +
\sum_\ell S_\ell.
}
For $\ell \notin J(\gamma)$, we have
\eqsplit{
S_\ell &= -\lm \|(U^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}\hat{U})_\ell\hat\gamma \|_2 + \left( 2\max_i
\sqrt{\sum_j c_{ij}^2} + \lm_3\right)
\|(U^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}\hat{U})_{\ell}\hat\gamma \|_1\\ &\leqslant
\|(U^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}\hat{U})_{\ell}\hat\gamma \|_2 \left[ -\lm + \sqrt{n} \left(
2\max_i \sqrt{\sum_j c_{ij}^2} + \lm_3\right) \right] \leqslant 0
}
for $\lm \geqslant \sqrt{n} ( 2\max_i \sqrt{\sum_j
c_{ij}^2} + \lm_3)$. E.g. we can take $\lm_3 = 2\max_i
\sqrt{\sum_j c_{ij}^2} = \lm/(2\sqrt{n})$.
For $\ell \in J(\gamma)$, we have
\eqsplit{
S_\ell &\leqslant \lm \|\gamma_{\cdot\ell}-(U^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}\hat{U})_{
\ell}\hat\gamma\|_2 + ( 2\max_i \sqrt{\sum_j c_{ij}^2} + \lm_3)
\|\gamma_{\cdot \ell} - (\hat{U} U^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em})_{\ell}\hat\gamma \|_1\\
&\leqslant (\lm + 2\max_i \sqrt{\sum_j c_{ij}^2} + \lm_3)
\|\gamma_{\cdot \ell} - (\hat{U} U^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em})_{\ell}\hat\gamma \|_1 = \lm
(1+1/\sqrt{n})\|\gamma_{\cdot \ell} - (\hat{U}
U^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em})_{\ell}\hat\gamma \|_1.}
Substituting the values of $\lm$ and $\lm_3$, we obtain a).
b) We have that
\eqsplit{
\|f-\hat f\|_2^2 &+ \lm_2 \sum_{i} \|\gamma_{i} - \hat{U}
U^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}\hat\gamma_{i}\|_2
\leq \|f_\scb - f\|_2^2
+ \lm_2 \sum_i
\|\gamma_{i}-\hat{U} U^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}\hat\gamma_{i}\|_2
+\lm\sum_\ell \|\gamma_{\cdot\ell}\|_2
\\
&\hspace{2em}
-\lm\sum_\ell \|U^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}\hat{U}\hat\gamma_{\cdot\ell}\|_2
+ 2\sqrt{n} \max_i \sqrt{\sum_j c_{ij}^2} \sum_{\ell} \|
\gamma_{\cdot \ell} - (U^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}\hat{U})_{\ell} \hat\gamma\|_2
}
For $\ell \notin J(\gamma)$, we have
\eqsplit{
S_\ell &= \|U^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}\hat{U}\hat\gamma_{\cdot\ell}\|_2 [ -\lm + (
2\sqrt{n}\max_i \sqrt{\sum_j c_{ij}^2} + \lm_2)\leqslant 0 }
for $\lm \geqslant 2\sqrt{n} \max_i \sqrt{\sum_j
c_{ij}^2} + \lm_2$. E.g. we can take $\lm_2 = 2\sqrt{n}\max_i
\sqrt{\sum_j c_{ij}^2} = \lm/2$, i.e. we can use the same $\lm$.
For $\ell \in J(\gamma)$, we have
\eqsplit{
S_\ell &\leqslant \|\gamma_{\cdot\ell}-(U^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}\hat{U})_{
\ell}\hat\gamma\|_2 (\lm + 2\sqrt{n}\max_i \sqrt{\sum_j c_{ij}^2}
+ \lm_2) = 2\lm \|\gamma_{\cdot \ell} - (\hat{U}
U^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em})_{\ell}\hat\gamma \|_2.}
c) We apply Lemma~\ref{lem:GroupLasso} with design matrix
$\tilde{X}$. Then constant $\tilde{\Lm}_0$ is:
$$
\tilde{x}_{ij\ell}^2 = (U^T X_i)_{j\ell}^2 = (\sum_k x_{ijk}
U_{k\ell } )^2 \leqslant \sum_k x_{ij k}^2
\sum_k U_{k \ell}^2 = \sum_k x_{ijk}^2.
$$
Hence,
\eqsplit{
\Lm_0 (\tilde{X}) &\le \frac{\sum_\ell
\max_{ij}\tilde{x}_{ij\ell}^2} {\lm^2 - \mu^2 \max_\ell \sum_{ij}\tilde{x}_{ij\ell}^2}
\le \frac{p \max_{ij} \sum_\ell x_{ij\ell}^2} {16n\mu^2 \max_{i}
\sum_{\ell,j} x_{i j \ell}^2 - \mu^2
\sum_{ij\ell} x_{ij\ell}^2}\\
&\le \frac{p \max_{ij} \sum_\ell x_{ij\ell}^2} {15n\mu^2 \max_{i}
\sum_{\ell,j} x_{i j \ell}^2} \le \frac{p \max_{ij} \sum_\ell
x_{ij\ell}^2} {15n\mu^2 \max_{i j} \sum_{\ell} x_{i j \ell}^2}\\
&\le \frac{p }{15n\mu^2} = \frac{p }{15\sig^2 A^2 n\log(np)}
}
since \(\mu=\sig A\bigl(\log(np)\bigr)^{1/2}\).
\end{proof}
We need one of the following assumptions (repeated). Following
(e.g. Obozinski et al, 2008), denote $||x||_{p/q} = \summ i1n
\left[|| x_{\cdot i} ||_p^q \right]^{1/q}$, $x\in
\mathbb{R}^{m\times n}$.
\noindent{\bf Assumption URE2$_n(s; c_0)$}. For some integer $s$ such that
$1 \leqslant s \leqslant p$, and a positive number $c_0$ the
following condition holds:
\begin{equation}
\tilde{\kappa}_{U \, n}(s, c_0)^2 := \min_i \min_{\substack{J_0
\subseteq \{1,\dots,\,p\},\\ |J_0|\leqslant\, s}} \,
\min_{U\in \mathbb{R}^{p\times p}: \, U U^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em} = I} \min_{\substack{\delta \in \mathbb{R}^{n\times p}\setminus \{0\}, \\
||\delta_{ \cdot \, J_0^c}||_{2/1} \leqslant \, c_0
||\delta_{ \cdot \, J_0}||_{2/1} }} \frac{|| X_{i\cdot\cdot}^T U \delta_i||_2^2}{|| \delta_{i J_0}||_2^2} > 0.
\end{equation}
\noindent{\bf Assumption MURE2$_n(s; c_0)$}. For some integer $s$ such that
$1 \leqslant s \leqslant p$, and a positive number $c_0$ the
following condition holds:
\begin{equation}
\tilde{\kappa}_{U \, n}(s, c_0)^2 := \min_{\substack{J_0 \subseteq
\{1,\dots,\,p\},\\ |J_0|\leqslant\, s}} \,
\min_{U\in \mathbb{R}^{p\times p}: \, U U^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em} = I} \min_{\substack{\delta \in \mathbb{R}^{n\times p}\setminus \{0\}, \\
||\delta_{ \cdot \, J_0^c}||_{2/1} \leqslant \, c_0
||\delta_{ \cdot \, J_0}||_{2/1} }} \frac{\summ i1n ||
X_{i\cdot\cdot}^T U \delta_i||_2^2}{ \summ i1n || \delta_{i
J_0}||_2^2} > 0.
\end{equation}
\begin{theorem} Let \(Y_{ij}\dist \normal (f_{ij},\sig^2)\) independent,
\(f_{ij}=x_{ij}^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}\beta_i\), \(x_{ij}\in \R^p\),
\(\beta_i\in\R^p\), \(i=1,\dots,n\), \(j=1,\dots,m\), $p\geqslant
2$. Fix some $\varepsilon > 0$ and let Assumption MURE2$_n(s; 3
+ 4/\varepsilon)$ (or Assumption URE2$_n(s; 3 + 4/\varepsilon)$)
be satisfied for $X=(x_{ijl})$.
Consider the spectral RMS estimator $\hat{f}_{ij} =
X_{ij}^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}\hat{\beta}_i$ where $\hat{\beta}$ is defined by
\eqref{rotatedlassostar} with \(\lm=4A\sig\tilde\Lm_x\sqrt{n
\log(pn)}\), where \(\tilde\Lm_x= \max_{i}\sqrt{\sum_{\ell,j} x_{i
j \ell}^2 }\) and \(A>\sqrt2\).
Then, with probability at least \(1-(np)^{1-A^2/2}\),
\eqsplit{ %
\|f-\hat f\|_2^2 \le (1+\varepsilon)\inf_{\substack{\beta\in\mathbb{R}^p,\\ \scm(\gamma)\le s}} \left[\|f - f_{\scb}\|_2^2
+ \scm(\gamma) n\log(np) \frac{16 A^2 \sigma^2
\Lambda_{\tilde{x}}^2 C_{\varepsilon} }{\kappa_{2n}^2(s, 3+4/\varepsilon))}\right],
} %
where $C_{\varepsilon} = \frac{(\varepsilon+2)^2}{\varepsilon
(1+\varepsilon)} >0$.
Also, we can show that
\eqsplit{ %
\|f-\hat f\|_2 \le (1+\varepsilon) \inf_{\substack{\beta\in\mathbb{R}^p,\\ \scm(\gamma)\le s}} \left[\|f - f_{\scb}\|_2
+ 2\lambda \frac{\sqrt{\scm(\gamma)} }{\kappa (s,
3+4/\varepsilon) )}\right].
} %
\end{theorem}
\begin{proof}
Fix an arbitrary $\beta\in\mathbb{R}^p$ such that $\scm(\gamma)\le
s$.
On event $\sca$ defined in the proof of Lemma 1, inequalities
\eqref{boundf} hold. Consider two cases.
\begin{enumerate}
\item $2\lambda \sum_{\ell \in J(\gamma)} ||\gamma_{\cdot\ell} -
\tilde\gamma_{\cdot\ell}||_2 \leqslant \varepsilon || f -
f_{\scb}||_2^2$. In this case, the result of the theorem follows
trivially from \eqref{boundf}.
\item $\varepsilon || f - f_{\scb}||_2^2 < 2\lambda \sum_{\ell \in
J(\gamma)} ||\gamma_{\cdot\ell} - \tilde\gamma_{\cdot\ell}||_2$.
Denote the event defined by this inequality by $\tilde\sca$; all
subsequent inequalities hold on $\sca \cap \tilde\sca$.
On this event, we get from \eqref{SPECTboundf} that
$$
\sum_{\ell \in J^c(\gamma)} ||\gamma_{\cdot\ell} -
\tilde\gamma_{\cdot\ell}||_2 \leqslant (3+4/\varepsilon)
\sum_{\ell \in J(\beta)} ||\gamma_{\cdot\ell} -
\tilde\gamma_{\cdot\ell}||_2.
$$
Assumption MRE$_n(s, 3+4/\varepsilon)$ (or MURE$_n(s,
3+4/\varepsilon)$, with the corresponding $\kappa$) implies that
$$
\kappa^2 \sum_{\ell \in J(\gamma)} |\gamma_{i\ell} -
\tilde\gamma_{i\ell}|_2^2 \leqslant || X_{i\cdot\cdot}^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}
(\beta_{i\cdot } - \hat\beta_{i\cdot}) ||_2^2 = || f_{\ensuremath{\frak B}\xspace \,
i\cdot } - \hat{f}_{i\cdot } ||_2^2,
$$
where $\kappa = \kappa(s, 3+4/\varepsilon)$.
Hence, by \eqref{SPECTboundf},
\begin{eqnarray*}
\|f-\hat f\|_2^2 &\le& \|f - f_{\scb}\|_2^2
+2\lm \sum_{\ell\in J(\gamma)}\|\tilde\gamma_{\cdot\ell}
-\gamma_{\cdot\ell}\|_2\\
&\le& \|f - f_{\scb}\|_2^2 +
2\lm \sqrt{\scm(\gamma)} \left[\sum_{\ell\in J(\gamma)}\|\tilde\gamma_{\cdot\ell}
-\gamma_{\cdot\ell}\|_2^2\right]^{1/2}\\
&=& \|f - f_{\scb}\|_2^2 +
2\lm \sqrt{\scm(\gamma)} \left[\sum_{i=1}^n\sum_{\ell\in J(\gamma)} |\tilde\gamma_{\cdot\ell}
-\gamma_{\cdot\ell}|_2^2\right]^{1/2}\\
&\le& \|f - f_{\scb}\|_2^2 +
2\lm \sqrt{\scm(\gamma)} \left[\sum_{i=1}^n \frac 1 {\kappa^2} \|\hat{f}_{i\cdot} - f_{\scb\, i\cdot}\|_2^2
\right]^{1/2}\\
&=& \|f - f_{\scb}\|_2^2 +
2\lm \frac{\sqrt{\scm(\gamma)}}{\kappa} \|\hat{f} - f_{\scb }\|_2\\
&\le& \|f - f_{\scb}\|_2^2 +
2\lm \frac{\sqrt{\scm(\beta)}}{\kappa} \left[ \|\hat{f} - f \|_2 + \|f - f_{\scb }\|_2 \right].
\end{eqnarray*}
Using inequality $2xy \le x^2/b + b y^2$ with $b\in(0,1)$, $x =
\lm \sqrt{\scm(\beta)}/\kappa$, and $y$ being either $\|\hat{f} -
f \|_2$ or $\|f - f_{\scb }\|_2$, we can decouple the last term
to obtain
\begin{eqnarray*}
\|f-\hat f\|_2^2 &\le& \frac{1+b}{1-b} \|f - f_{\scb}\|_2^2 +
\frac{2 \lm^2\scm(\beta)}{b(1-b)\kappa^2}, \quad \forall b\in(0,1).
\end{eqnarray*}
Taking $b=\left(1+2/\varepsilon\right)^{-1}$ we obtain
\begin{eqnarray*}
\|f-\hat f\|_2^2 &\le& (1+\varepsilon) \|f - f_{\scb}\|_2^2 +
\frac{\lm^2\scm(\gamma) (\varepsilon+2)^2}{\varepsilon
\kappa^2}.
\end{eqnarray*}
Substituting the value of $\lm$ and taking the infimum over all considered $\beta$ finishes the proof of the theorem.
\end{enumerate}
\end{proof}
For linear regression, we obtain the following inequalities.
\begin{corollary}
\eqsplit{
\frac 1 n \sum_{i=1}^n ||X_i^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}(\beta_i -
\hat\beta_{i})||_2^2 &\leqslant \scm(\gamma) \log(np) \frac{16 A^2
\sigma^2
\Lambda_{\tilde{x}}^2 C_{\varepsilon} }{\kappa_{2n}^2(s,
3+4/\varepsilon)},\\
\frac 1 n \sum_{i=1}^n ||\beta_i - \hat\beta_{i}||_2^2 &\leqslant
C_x \scm(\gamma) \log(np) \frac{16 A^2 \sigma^2
\Lambda_{\tilde{x}}^2 (1+\varepsilon)C_{\varepsilon} }{\kappa_{2n}^2(s,
3+4/\varepsilon)},
}
where $C_x=\sum_j \max_{i\ell}x_{ij\ell}^2$.
\end{corollary}
Persistency.
\begin{proposition}\label{prop:persist}
$$
|L(\hat\beta) - L(\beta)| \leqslant C_x \sum_{\ell} \|\hat\beta_{\cdot\ell} - \beta_{\cdot\ell}\|_2^2 + \sum_{\ell}b_\ell\|\beta_{\cdot\ell}-\hat\beta_{\cdot\ell}\|_2,
$$
where $C_x=\sum_j \max_{i\ell}x_{ij\ell}^2$, $b_\ell =
2\sqrt{\sum_i c_{i\ell}^2}+1 $.
\end{proposition}
May be there is a sharper inequality?
\begin{proof} of Theorem~\ref{prop:persist}.
\begin{eqnarray*}
|L(\hat\beta) - L(\beta)| &=& \left| \sum_{i,j} (Y_{ij} - x_{ij\cdot}^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em} \hat\beta_i)^2 + \sum_{\ell} ||\hat\gamma_{\cdot \ell} ||_2 - \sum_{i,j} (Y_{ij} - x_{ij\cdot}^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em} \beta_i)^2 -\sum_{\ell} ||\gamma_{\cdot \ell} ||_2 \right| \\
&=& \left| \sum_{i,j} [ x_{ij\cdot}^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em} (\hat\beta_{i\cdot} + \beta_{i\cdot}) x_{ij\cdot}^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em} (\hat\beta_{i\cdot} - \beta_{i\cdot}) -2 Y_{ij} x_{ij\cdot}^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em} (\beta_i - \hat\beta_i)] + \sum_{\ell} ||\tilde\gamma_{\cdot \ell}-\gamma_{\cdot \ell} ||_2 \right| \\
&=& \left| \sum_{i,j} x_{ij\cdot}^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em} (\hat\beta_{i\cdot} - \beta_{i\cdot}) [x_{ij\cdot}^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em} (\hat\beta_{i\cdot} + \beta_{i\cdot}) -2 (x_{ij\cdot}^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em} \beta_i +\varepsilon_{ij} )] + \sum_{\ell} ||\tilde\gamma_{\cdot \ell}-\gamma_{\cdot \ell} ||_2 \right| \\
&=& \left| \sum_{i,j} x_{ij\cdot}^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em} (\hat\beta_{i\cdot} - \beta_{i\cdot}) [x_{ij\cdot}^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em} (\hat\beta_{i\cdot} - \beta_{i\cdot}) -2 \varepsilon_{ij} ] + \sum_{\ell} ||\tilde\gamma_{\cdot \ell}-\gamma_{\cdot \ell} ||_2 \right| \\
&\leqslant& \sum_{i,j} [x_{ij\cdot}^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em} (\hat\beta_{i\cdot} - \beta_{i\cdot})]^2 + 2 \sum_{i,j} \left|x_{ij\cdot}^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em} (\hat\beta_{i\cdot} - \beta_{i\cdot}) \varepsilon_{ij} \right| + \sum_{\ell} ||\tilde\gamma_{\cdot \ell}-\gamma_{\cdot \ell} ||_2 \\
&\leqslant& \sum_{i,j} [x_{ij\cdot}^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em} (\hat\beta_{i\cdot} - \beta_{i\cdot})]^2 + 2\sum_{i\ell}c_{il}|\beta_{il}-\hat\beta_{i\ell}| + \sum_{\ell} ||\hat\beta_{\cdot \ell}-\beta_{\cdot \ell}
||_2\\
&\leqslant& \sum_j \max_{i\ell}x_{ij\ell}^2 \sum_{i} \|\hat\beta_{i\cdot} - \beta_{i\cdot}\|_2^2 + 2\sum_{\ell}\sqrt{\sum_i c_{i\ell}^2}\|\beta_{\cdot\ell}-\hat\beta_{\cdot\ell}\|_2 + \sum_{\ell} ||\hat\beta_{\cdot \ell}-\beta_{\cdot \ell} ||_2\\
&\leqslant& \sum_j \max_{i\ell}x_{ij\ell}^2 \sum_{\ell} \|\hat\beta_{\cdot\ell} - \beta_{\cdot\ell}\|_2^2 + \sum_{\ell}\left[2\sqrt{\sum_i c_{i\ell}^2}+1\right]\|\beta_{\cdot\ell}-\hat\beta_{\cdot\ell}\|_2,
\end{eqnarray*}
where the last 4 inequalities hold on event $\sca$ defined in the
proof of Lemma~\ref{lem:RotateGroupLasso}.
\end{proof}
\section{The spectral penalty}
\label{sec:spectral}
Let \(A=\sum c_i x_ix_i^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}\), be a positive semi-definite matrix,
where \(x_1,x_2,\dots\) is an orthonormal basis of eigenvectors.
Then, we define \(A^{\gamma}=\sum c_i^{\gamma} x_ix_i^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}\). We
consider now as penalty the function
\eqsplit{
\psi(\scb)=\psi(\beta_1,\dots,\beta_n)= \trace\Bigl\{\bigl(\summ i1n \beta_i\beta_i^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}\bigr)^{1/2}\Bigr\}.
} Note that \(\psi(\scb)=\sum c_i^{1/2}\) where \(c_1,\dots,c_p\) are the eigenvalues of \(\scb\scb^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}=\summ i1n \beta_i\beta_i^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}\) (including multiplicities).
\begin{lemma}
\label{lem:psiConvex}
\(\psi\) is a convex function.
\end{lemma}
In this section we study the estimator defined by
\eqsplit[specPen]{
(\hat\beta_1,\dots,\hat\beta_n)&= \argmin_{(\beta_1,\dots,\beta_n)\in\R^{p\times n}}\{\summ i1n (y_{ij}-x_{ij}^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}\beta_i)^2+\lm\psi(\beta_1,\dots,\beta_n)\}
}
The lassoes penalty considered primary the columns of \scb. The
main focus of the group lasso was the rows. The spectral penalty
is symmetric in its treatment of the rows and columns since
$\mathfrak{S}\scb =\mathfrak{S}\scb^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}$, where $\mathfrak{S} A$ denotes the spectrum
of $A$. Moreover, the penalty is invariant to the rotation of the
matrix \scb. In fact, \(\psi(\scb) = \psi(T\scb U )\), where \(T\)
and \(U\) are \(n\times n\) and \(p\times p\) rotation matrices:
\eqsplit{
(T\scb U)^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em} (T\scb U) &= U^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}\scb^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}\scb U
}
and the RHS have the same eigenvalues as \(\scb^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}\scb=\sum \beta_i\beta_i^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}\).
Now, the standard notion of sparsity, as captured by the $L_0$
norm, or by the standard lasso and group lasso, is basis
dependent. Consider the nonparametric setup in which $y_{ij} =
g(z_{ij}) + \eps_{ij}$, where $z\in[0,1]$. The model fits the
regression setup, if $x_{ij\ell}=h_\ell(z_{ij}$. If, for example,
$g(z)=\ind(a<z\le b)$, then this example is sparse when
$h_\ell(z)=\ind(z>\ell/p)$. It is not sparse if
$h_\ell(z)=(z-\ell/p)^+$. On the other hand a function $g$ which
has a piece-wise constant slope is sparse in the latter basis, but
not in the former.
The spectral penalty aims at finding a basis in which
$\beta_1,\dots,\beta_n$ have the same pattern of sparsity. This is
meaningless if $n$ is small --- any function is sparse in some
basis. However, we will argue that this can be done when $n$ is
large.
The following lemma describes a relationship between the group lasso and the spectral norm.
\begin{lemma}
\label{lem:spectVsGroup}
\mbox{}\par
\begin{enumerate}[(i)]
\item
$\|\scb\|_{2,1} \ge \inf_{U\in\scu} \|\scb\|_{2,1} =\psi(\scb)$, where $\scu$ is the set of all unitary matrices.
\item
There is a unitary matrix $U$, which may depend on the data, such that if $X_1,\dots,X_n$ are rotated by $U^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}$, then the solution of the spectral penalty target function \eqref{specPen} is the solution of the group lasso in this basis.
\end{enumerate}
\end{lemma}
\subsection{The spectral penalty: 2D case}
In the 2-dimensional case ($p=2$), we can find the explicit
expression for the penalty in terms of $\beta_{\cdot \ell}$. The
eigenvalues of
\eqsplit{
\scb \scb^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em} = \begin{pmatrix} ||\beta_{\cdot 1}||_2^2 & \langle \beta_{\cdot 1}, \beta_{\cdot 2}\rangle \\ \langle \beta_{\cdot 1}, \beta_{\cdot 2}\rangle & ||\beta_{\cdot 2}||_2^2 \end{pmatrix}
}
are the solutions of $(||\beta_{\cdot 1}||_2^2 - z)(||\beta_{\cdot
2}||_2^2 - z) - \langle \beta_{\cdot 1}, \beta_{\cdot
2}\rangle^2=0$, i.e. are
$$
z_{1,2} = \frac 1 2 \left[ ||\beta_{\cdot 1}||_2^2 +
||\beta_{\cdot 2}||_2^2 \pm \sqrt{ \left(||\beta_{\cdot 1}||_2^2 -
||\beta_{\cdot 2}||_2^2\right)^2 + 4 \langle \beta_{\cdot 1},
\beta_{\cdot 2}\rangle^2} \right],
$$
and hence it can be shown that
$$
\psi(\scb) =z_1^{1/2}+ z_2^{1/2} = \sqrt{||\beta_{\cdot 1}||_2^2 +
||\beta_{\cdot 2}||_2^2 + 2 \sqrt{||\beta_{\cdot 1}||_2^2
||\beta_{\cdot 2}||_2^2 - \langle \beta_{\cdot 1}, \beta_{\cdot
2}\rangle^2}}
$$
due to identity $\sqrt{A + \sqrt{A^2-4B}} + \sqrt{A -
\sqrt{A^2-4B}} = \sqrt{2A +4\sqrt{B}}$.
Hence, in the directions where $\beta_{\cdot 1}$ and $\beta_{\cdot
2}$ are orthogonal, $\psi(\scb)$ behaves like $\ell_1$ penalty,
and in the directions they are collinear ($\langle \beta_{\cdot
1}, \beta_{\cdot 2}\rangle = \pm 1$) $\psi(\scb)$ behaves like
$\ell_2$ penalty.
The penalty ``ball'' (i.e., the curve $\psi=1$ in the $||\beta_{\cdot 1}||_2$
and $||\beta_{\cdot 2}||_2$ plane) is plotted in Figure \ref{figBall} for different values of $\rho=\frac{\langle \beta_{\cdot 1}, \beta_{\cdot
2}\rangle}{||\beta_{\cdot 1}||_2 ||\beta_{\cdot 2}||_2}$, the ``correlation'' between $\beta_{\cdot 1}$ and $\beta_{\cdot
2}$.
\onefigure[0.7]{SpectBall.eps}{ The spectral ``ball'' in the $ ||\beta_{\cdot 1}||_2$ and $||\beta_{\cdot 2}||_2$ plane for
different values of $\rho$. }{figBall}
\subsection{The estimator }
Let $\scb=\summ \xi1{p\wedge n}\al_\xi \beta_\xi^*{\mathfrak{b}_\xi^*}^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}$
be the spectral value decomposition, or the PCA, of \scb:
$\beta_1^*,\dots,\beta_p^*$ and $\mathfrak{b}_1^*,\dots,\mathfrak{b}_n^*$ are
orthonormal sub-bases of $\R^p$ and $\R^n$ respectively,
$\al_1\geq\al_2\geq\dots$, and
$\scb\scb^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}\beta_\xi^*=\al_\xi^2\beta_\xi^*$,
$\scb^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}\scb\mathfrak{b}_\xi^*=\al_\xi^2\mathfrak{b}_\xi^*$, $\xi=1,\dots,p\wedge n$.
Let $T=\summ \xi1{p\wedge n} e_\xi{\beta_\xi^*}^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}$ (clearly,
$TT^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}=I$).
Consider the parametrization of the problem in the rotated
coordinates, $\ti x_{ij}=Tx_{ij}$ and $\ti\beta_i=T\beta_i$. Then
geometrically the regression problem is invariant:
$x_{ij}^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}\beta_i=\ti x_{ik}^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}\ti\beta_i$, and
$\psi(\scb)=\|\ti\scb\|_{2,1} $,
up to a modified regression matrix.
The representation $\hat\scb = \summ \xi1s \al_\xi \beta_\xi^*
{\mathfrak{b}_\xi^*}^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}$ shows that the difficulty of the problem is the
difficulty of estimating $s(n+p)$ parameters with $nm$
observations. Thus it is feasible as long as $s/m\to0$ and $sp/nm\to0$
We have
\begin{theorem}
\label{th:sparseSpect}
Suppose $p<n$. Then the solution of the least squares with the spectral penalty is given by $\summ \xi1s \beta^*_\xi{\mathfrak{b}_\xi^*}^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}$, $s=s_\lm \leq p$, and $s_\lm\dec 0$ as $\lm\to\en$. If $s=p$ then the gradient of the target function is given in a matrix form by
\eqsplit{
-2R+\lm (\hat\scb\hat\scb^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em})^{-1/2}\hat\scb
}
where
\eqsplit{
R=\Bigl(X_1^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}(Y_1-X_1\hat\beta_1),\dots,X_n^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}(Y_n-X_n\hat\beta_n)\Bigr).
}
And hence
\eqsplit{
\hat\beta_i=\bigl( X_i^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em} X_i + \frac\lm2 (\hat\scb\hat\scb^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em})^{-1/2}\bigr)^{-1}X_i^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em} Y_i.
}
That is, the solution of a ridge regression with adaptive weight.
More generally, let $\hat\scb=\summ \xi1s \alpha_\xi
\beta_\xi^*{\mathfrak{b}_\xi}^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}$, $s<p$, where $\beta_1^*,\dots,\beta_p^*$
is an orthonormal base of $\R^p$. Then the solution satisfies
\eqsplit{
&{\beta_\xi^*}^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em} R = \frac\lm2 {\beta_\xi^*}^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em} (\hat\scb\hat\scb^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em})^{+1/2}\hat\scb, \quad \xi\leq s
\\
&|{\beta_\xi^*}^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em} R \mathfrak{b}_\xi^*| \leq \frac\lm 2, \qquad \qquad \qquad s<\xi\leq p.
}
where for any positive semi-definite matrix $A$, $A^{+1/2}$ is the Moore-Penrose generalized inverse of $A^{1/2}$.
\end{theorem}
Roughly speaking the following can be concluded from the theorem.
Suppose the data were generated by a sparse model (in \emph{some}
basis). Consider the problem in the transformed basis, and let $S$
be the set of non-zero coefficients of the true model. Suppose
that the design matrix is of full rank within the sparse model:
$X_i^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em} X_i=\EuScript{O}(m)$, and that \lm is chosen such that $\lm\gg
\sqrt{nm\log(np)}$. Then the coefficients corresponding to $S$
satisfy
\eqsplit{
\hat\beta_{Si} &= \bigl(X_i^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em} X_i + \frac\lm2 (\hat\scb_S\hat\scb_S^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em})^{1/2}\bigr)^{-1}X_i^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em} Y_i.
}
Since it is expected that $\lm(\scb_S\scb_S^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em})^{1/2}$ is only slightly larger than $\EuScript{O}(m\log(np))$, it is completely dominated
by $X_i^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em} X_i$, and the estimator of this part of the model is consistent. On the other hand, the rows of $R$ corresponding
to coefficient not in the true model are only due to noise and hence each of them is $\EuScript{O}(\sqrt {nm})$.
The factor of $\log (np)$ ensures that their maximal norm will be below $\lm/2$, and the estimator is consistent.
\subsection{Bayesian perspectives}
We consider now the penalty for $\beta_k$ for a fixed $k$. Let $A=n^{-1}\sum_{k\ne i} \beta_k\beta_k^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}$. Using Taylor expansion for not too big $\beta_i$, based on \eqref{eigender} we get
\eqsplit{
\trace\bigl( (nA+\beta_i\beta_i^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em})^{1/2}\bigr)
&= \sqrt{n}\trace(A^{1/2}) + \summ j1p \frac{x_j^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em} \beta_i\beta_i^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em} x_j}{2 c_j^{1/2}}
\\
&= \sqrt{n}\trace(A^{1/2}) + \frac 12 \beta_i^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em} \bigl(\sum c_j^{-1/2} x_jx_j^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}\bigr)\beta_i
\\
&= \sqrt{n}\trace(A^{1/2}) + \frac12 \beta_i^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em} A^{-1/2}\beta_i
}
So, this like $\beta_i$ has a prior of $\normal(0, n \sig^2/\lm A^{1/2})$. Note that the prior is only related to the estimated variance of $\beta$, and $A$ appears with the power of $1/2$. Now $A$ is not really the estimated variance of $\beta$, only the variance of the estimates, hence it should be inflated, and the square root takes care of that. Finally, note that eventually, if $\beta_i$ is very large relative to $nA$, then the penalty become $\|\beta\|$, so the ``prior'' becomes essentially normal, but with exponential tails.
A better way to look on the penalty from a Bayesian perspective is to consider it as prior on the \(n\times p\) matrix \(\scb=(\beta_1,\dots,\beta_n)\). Recall that the penalty is invariant to the rotation of the matrix \scb. In fact, \(\psi(\scb) = \psi(T\scb U )\), where \(T\) and \(U\) are \(n\times n\) and \(p\times p\) rotation matrices. Now, this means that if \(\mathfrak{b}_1,\dots,\mathfrak{b}_p\) are orthonormal set of eigenvectors of \(\scb^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}\scb\) and \(\gamma_{ij}=\mathfrak{b}_j^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}\beta_i\) --- the PCA of \(\beta_1,\dots,\beta_n\), then \(\psi(\scb) = \summ j1p \bigl(\summ i1n \gamma_{ij}^2\bigr)^{1/2} \) --- the rms penalty in terms of the principal components. The ``prior'' is then proportional to $ e^{-\lm \summ j1p \|\gamma_{\cdot j}\|_2}$.\, which is as if to obtain a random $\scb$ from the prior the following procedure should be followed:
\begin{enumerate
\item
Sample \(r_1,\dots,r_p\) independently from \(\Gamma(n,\lm)\) distribution.
\item
For each \(j=1,\dots,p\) sample \(\gamma_{1j},\dots,\gamma_{nj}\) independently and uniformly on the sphere with radius \(r_j\).
\item
Sample an orthonormal base \(\chi_1,\dots,\chi_p\) "uniformly''.
\item
Construct \(\beta_i = \summ j1p \gamma_{ik}\chi_k\).
\end{enumerate}
So hopefully the rms penalty ensures that the solution will be sparse in \emph{some} basis. It works when the assumption (and hope) is that the Procrustes's distance between the true \(\beta\)'s and a sparse solution is small.
\subsection{Algorithm and small simulation study}
\twofigures[1.1]{NB03042009a.eps}{NB03042009b.eps}{Component
variances and eigenvalues, \(m=25\), \(n=150\) }{fig1}
A simple algorithm is the following:
\begin{enumerate
\item Initiate some small value of \(\hat
\beta_1,\dots,\hat\beta_n\). Let \(A=\summ j1n
\hat\beta_j\hat\beta_j^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}\). Fix \(\gamma\in(0,1]\), \(\eps>0\),
\(k\), and \(c>1\). \item\label{A1st1} For \(i=1,\dots,n\):
\begin{enumerate
\item Compute \(\delta_i = (X_i^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em} X_i + \lm A^{-1/2})^{-1}X_i^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}
(y_i-X_i\hat\beta_i)\). \item Update \(A\leftarrow
A-\hat\beta_i\hat\beta_i\);
\(\hat\beta_i\leftarrow\hat\beta_i+\gamma\delta_i\); \(A\leftarrow
A+\hat\beta_i\hat\beta_i\);
\end{enumerate}
\item if \(\summ j1p \ind\bigl(n^{-1}\summ
i1n\hat\beta_{ij}^2>\eps\bigr)>k \) update
\(\lambda\leftarrow\lambda c\) otherwise
\(\lambda\leftarrow\lambda/ c\). \item Return to step \ref{A1st1}
unless there is no real change of coefficients.
\end{enumerate}
Comment: to fasten the computation, the SVD was computed only
every 10 values of \(i\).
As a simulation we applied the above algorithm to the following
simulated data.
We generated random \(\beta_1,\dots,\beta_{150}\in\R^{150}\) such that all coordinates
are independent, and \(\beta_{ij}\dist\normal(0, e^{-2j/5})\). All
\(X_{ij\ell}\) are \iid \(\normal(0,1)\), and \(y_{ij} =
x_{ij\cdot}^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em} \beta_i+\eps_{ij}\), where \(\eps_{ij}\) are all
\iid \(\normal(0,1)\). The true \(R^2\) obtained was
approximately 0.73. The number of replicates per value of
\(\beta\), \(m\), varied between 5 to 300. We consider two
measures of estimation error:
\eqsplit{
L_{\rm par} &= \frac{\summ i1n \|\hat\beta_i-\beta_i\|_\en} {\summ i1n \|\beta_i\|_\en}
\\
L_{\rm pre} &= \frac{\summ i1n \|X_j(\hat\beta_i-\beta_i)\|_\en} {\summ i1n \|X_i\beta_i\|_\en}
}
\onefigure[1.1]{NB11Apr2009d.eps}{ Lower lip position while
repeating 32 times 'Say bob again'}{fig2}
The algorithm stopped after 30--50 iterations. Figure \label{fig1}
is a graphical presentation of a typical result. A summary is
given in Table \ref{tab1}. Note that \(m\) has a critical impact
on the estimation problem. However, with as little as \(5\)
observations per \(R^{150}\) vector of parameter we obtain a
significant reduction in the prediction error.
\begin{table}[H]
\caption{\label{tab1}The estimation and prediction error as
function of the number of observations per vector of parameters
Means (and SDK).}
\begin{center}
\begin{tabular}{|r|r|r|}
\hline
$m$ & \(L_{\rm par}\) & $L_{\rm pre}$ \\
\hline\hline
5 & 0.9530 (0.0075) & 0.7349 (0.0375) \\
\hline
25 & 0.7085 (0.0289) & 0.7364 (0.0238) \\
\hline
300 & 0.2470 (0.0080) & 0.5207 (0.0179) \\
\hline
\end{tabular}
\end{center}
\end{table}
\threefiguresV[1.5]{NB11Apr2009a.eps}{NB11Apr2009b.eps}{NB11Apr2009c.eps}{Eigenvalue,
coefficient variance and typical observed and smooth path.}{fig3}
The technique is natural for functional data analysis. We used the
data LipPos. The data is described by
Ramsay and
Silverman and can be found in
http://www.stats.ox.ac.uk/~silverma/fdacasebook/lipemg.html. The
original data is given in Figure \ref{fig2}. However we added noise to
the data as can be seen in Figure \ref{fig3}. The lip position is
measured at $m=501$ time points, with \(n=32\) repetitions.
As the matrix \(X\) we considered the union of 6 cubic spline
bases with, respectively, 5, 10, 20, 100, 200, and 500 knots
(i.e., \(p=841\), and \(X_i\) does not depend on \(i\)). A
Gaussian noise with \(\sigma=0.001\) was added to \(Y\). The
result of the analysis is given in Figure \ref{fig3}. Figure
\ref{fig4} present the projection of the mean path on the first
eigen-vectors of \(\summ i1n \hat\beta_i\hat\beta_i^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}\).
\onefigure[0.5]{NB11Apr2009e.eps}{Projection of the estimated mean
path on the 2 first eigen-vectors of \(\summ i1n
\hat\beta_i\hat\beta_i^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}\) and the true mean path.}{fig4}
The final example we consider is some what arbitrary. The data,
taken from StatLib, is of the daily wind speeds for 1961-1978 at
12 synoptic meteorological stations in the Republic of Ireland. As
the \(Y\) variable we considered one of the stations (station
BIR). As explanatory variables we considered the 11 other station
of the same day, plus all 12 stations 70 days back (with the
constant we have altogether 852 explanatory variables). The
analysis was stratified by month. For simplicity, only the first
28 days of the month were taken, and the first year, 1961, served
only for explanatory purpose. The last year was served only for
testing purpose, so, the training set was for 16 years (\(N=12\),
\(m=448\), and \(P=852\) ). In Figure \ref{fig5} we give the 2nd
moments of the coefficients and the scatter plot of predictions
vs. true value of the last year.
\twofigures[0.75]{NB14Apr2009a.eps}{NB14Apr2009b.eps}{Coefficient
2nd moment and prediction vs.true value of the test year.}{fig5}
\subsection{The spectral penalty: oracle inequalities}
If $\beta\beta^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em} = U C U^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}$ and $\hat\beta\hat\beta^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em} = \hat{U}
\hat{C} \hat{U}^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}$ where $\hat\beta$ is the solution of the
spectral RMS problem \eqref{rotatedlassostar}. Then $U^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}\hat\beta$
is the solution of the standard group lasso problem
\eqref{lassostar} with $\tilde{X}_i = U^T X_i$. Hence,
Lemma~\ref{lem:GroupLasso} is satisfied for $\gamma = U^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}\beta$
and ``estimate'' $\tilde\gamma = U^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}\hat\beta$ with design
matrix $\tilde{X}$.
Note that even if $\beta$ and $\hat\beta$ are close, $\gamma$ and
$\hat\gamma = \hat{U}^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}\hat\beta$ are close only up to a
permutation of variables. Hence, we will be interested in
comparing $U^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}\hat{U}$ to a permutation matrix (e.g. testing that
all eigenvalues are close to $\pm 1$).
We may be interested in sparsity of $\beta$ and $\gamma$, as well
as the approximation error of estimating $f$ and $\beta$. Thus, we
obtain the following results.
\begin{lemma}\label{lem:RotateGroupLasso}
Let \(y_{ij}\dist \normal (f_{ij},\sig^2)\) independent,
\(f_{ij}=x_{ij}^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}\beta_i\), \(x_{ij}\in \R^p\),
\(\beta_i\in\R^p\), \(i=1,\dots,n\), \(j=1,\dots,m\). Consider
\eqsplit[rotatedlassostar]{
(\hat\beta_1,\dots,\hat\beta_n) &=
\argmin \Biggl[ \summ i1n \summ j1m (y_{ij} - x_{ij}^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}\,\beta_i)^2 + \lambda
\trace \Bigl\{ \summ i1n \beta_{i }\beta_i^T\Bigr\}^{1/2}
\Biggr].
}
Assume that $\summ j1m x_{ij\ell}^2 = m$ for all $i, \, \ell$.
Fix some $c>0$ (which may depend on $p$, $n$ and $m$), and take
$\lambda = 4 \sigma\sqrt{m (p + c)}$. Then, with probability at
least \(
1-\exp\left( -\frac n 8 \min(c, c^2/p) \right)\):
\begin{enumerate}[a)]
\item \(\forall \beta\in\R^{p\times n}\):%
\eqsplit[SPECTboundf_beta1]{ %
&\hspace{-3em}\|f_{\beta}-\hat f\|_2^2 + \frac{\lambda} 2 \sum_{\ell} ||(U^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}\hat{U} \hat\gamma)_{\cdot
\ell} - \gamma_{\cdot \ell}||_2
\\ & \leqslant 2\lambda \sum_{\ell \in J(\gamma)}
\|\gamma_{\cdot \ell}- (U^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}\hat{U} \hat\gamma)_{\cdot
\ell}\|_2 + \lambda
\sum_{\ell} (\| (U^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}\hat{U} \hat\gamma)_{\cdot
\ell}\|_2 - \|\hat\gamma_{\cdot \ell}\|_2),
}
where \(f_{\scb ij} = x_{ij}^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}\beta_i = x_{ij}^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em} U^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em} \gamma_i \),
and%
\eqsplit{
J(\gamma)=\Bigl\{\ell\in\{1,\dots,p\}:\summ i1n\gamma_{i\ell}^2>0\Bigr\}. %
}
\item $\scm(\hat\gamma)\leqslant \frac{\phi^2_{\max} (X^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}
X/m)}{\kappa^2} \, s$ \,\, [under restricted e.v. assumption - to
be moved to the theorem].
\item $||U - \hat{U} ||_2 \leqslant 2 C_{\beta, \hat\beta}
\sum_i ||\hat\beta_i - \beta_i||_2 \left(||\beta_i||_2 + \sum_i
||\hat\beta_i - \beta_i||_2 \right)$,
where $C_{\beta, \hat\beta}^2 = \sum_k \sup_{t\in(0,1)} \max_{j:
\, j\neq k} \frac 1 {(c_j(t) - c_k(t))^2}$,
$c_k(t)$ are the eigenvalues of \( A(t)=\summ i1n
\bigl( (1-t)\beta_i + t\hat\beta_i \bigr)\bigl((1-t)\beta_i +
t\hat\beta_i\bigr)^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}\).
Hence, the rotations $U$ and $\hat{U} $ are close if \scb has
distinct eigen values.
\end{enumerate}
\end{lemma}
[up to here]
We need one of the following assumptions. Following (e.g.
Obozinski et al, 2008), denote $||x||_{p/q} = \summ i1n \left[||
x_{\cdot i} ||_p^q \right]^{1/q}$, $x\in \mathbb{R}^{m\times n}$.
\noindent{\bf Assumption URE2$_n(s; c_0)$}. For some integer $s$ such that
$1 \leqslant s \leqslant p$, and a positive number $c_0$ the
following condition holds:
\begin{equation}
\tilde{\kappa}_{U \, n}(s, c_0)^2 := \min_i \min_{\substack{J_0
\subseteq \{1,\dots,\,p\},\\ |J_0|\leqslant\, s}} \,
\min_{U\in \mathbb{R}^{p\times p}: \, U U^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em} = I} \min_{\substack{\delta \in \mathbb{R}^{n\times p}\setminus \{0\}, \\
||\delta_{ \cdot \, J_0^c}||_{2/1} \leqslant \, c_0
||\delta_{ \cdot \, J_0}||_{2/1} }} \frac{|| X_{i\cdot\cdot}^T U \delta_i||_2^2}{|| \delta_{i J_0}||_2^2} > 0.
\end{equation}
\noindent{\bf Assumption MURE2$_n(s; c_0)$}. For some integer $s$ such that
$1 \leqslant s \leqslant p$, and a positive number $c_0$ the
following condition holds:
\begin{equation}
\tilde{\kappa}_{U \, n}(s, c_0)^2 := \min_{\substack{J_0 \subseteq
\{1,\dots,\,p\},\\ |J_0|\leqslant\, s}} \,
\min_{U\in \mathbb{R}^{p\times p}: \, U U^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em} = I} \min_{\substack{\delta \in \mathbb{R}^{n\times p}\setminus \{0\}, \\
||\delta_{ \cdot \, J_0^c}||_{2/1} \leqslant \, c_0
||\delta_{ \cdot \, J_0}||_{2/1} }} \frac{\summ i1n ||
X_{i\cdot\cdot}^T U \delta_i||_2^2}{ \summ i1n || \delta_{i
J_0}||_2^2} > 0.
\end{equation}
\begin{theorem}
\label{th:BRTspectral}
Let \(y_{ij}\dist \normal (f_{ij},\sig^2)\) independent,
\(f_{ij}=x_{ij}^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}\beta_i\), \(x_{ij}\in \R^p\),
\(\beta_i\in\R^p\), \(i=1,\dots,n\), \(j=1,\dots,m\), $p\geqslant
2$. Fix some $\varepsilon > 0$ and let Assumption MURE2$_n(s; 3
+ 4/\varepsilon)$ (or Assumption URE2$_n(s; 3 + 4/\varepsilon)$)
be satisfied for $X=(x_{ijl})$.
Consider the spectral RMS estimator $\hat{f}_{ij} =
X_{ij}^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}\hat{\beta}_i$ where $\hat{\beta}$ is defined by
\eqref{rotatedlassostar} with \(\lm=4A\sig\tilde\Lm_x\sqrt{n
\log(pn)}\), where \(\tilde\Lm_x= \max_{i}\sqrt{\sum_{\ell,j} x_{i
j \ell}^2 }\) and \(A>\sqrt2\).
Then, with probability at least \(1-(np)^{1-A^2/2}\),
\eqsplit{ %
\|f-\hat f\|_2^2 \le (1+\varepsilon)\inf_{\substack{\beta\in\mathbb{R}^p,\\ \scm(\gamma)\le s}} \left[\|f - f_{\scb}\|_2^2
+ \scm(\gamma) n\log(np) \frac{16 A^2 \sigma^2
\Lambda_{\tilde{x}}^2 C_{\varepsilon} }{\kappa_{2n}^2(s, 3+4/\varepsilon))}\right],
} %
where $C_{\varepsilon} = \frac{(\varepsilon+2)^2}{\varepsilon
(1+\varepsilon)} >0$.
Also, we can show that
\eqsplit{ %
\|f-\hat f\|_2 \le (1+\varepsilon) \inf_{\substack{\beta\in\mathbb{R}^p,\\ \scm(\gamma)\le s}} \left[\|f - f_{\scb}\|_2
+ 2\lambda \frac{\sqrt{\scm(\gamma)} }{\kappa (s,
3+4/\varepsilon) )}\right].
} %
\end{theorem}
For linear regression, we obtain the following inequalities.
\begin{corollary}
\eqsplit{
\frac 1 n \sum_{i=1}^n ||X_i^{\scriptstyle{\mathsf{T}}} \hspace{-0.00em}(\beta_i -
\hat\beta_{i})||_2^2 &\leqslant \scm(\gamma) \log(np) \frac{16 A^2
\sigma^2
\Lambda_{\tilde{x}}^2 C_{\varepsilon} }{\kappa_{2n}^2(s,
3+4/\varepsilon)},\\
\frac 1 n \sum_{i=1}^n ||\beta_i - \hat\beta_{i}||_2^2 &\leqslant
C_x \scm(\gamma) \log(np) \frac{16 A^2 \sigma^2
\Lambda_{\tilde{x}}^2 (1+\varepsilon)C_{\varepsilon} }{\kappa_{2n}^2(s,
3+4/\varepsilon)},
}
where $C_x=\sum_j \max_{i\ell}x_{ij\ell}^2$.
\end{corollary}
Persistency.
\begin{proposition}\label{prop:persist}
$$
|L(\hat\beta) - L(\beta)| \leqslant C_x \sum_{\ell} \|\hat\beta_{\cdot\ell} - \beta_{\cdot\ell}\|_2^2 + \sum_{\ell}b_\ell\|\beta_{\cdot\ell}-\hat\beta_{\cdot\ell}\|_2,
$$
where $C_x=\sum_j \max_{i\ell}x_{ij\ell}^2$, $b_\ell =
2\sqrt{\sum_i c_{i\ell}^2}+1 $.
\end{proposition}
May be there is a sharper inequality?
|
2,877,628,089,051 | arxiv | \section{Introduction}
In the past decade, supermassive black holes have been found to be virtually
ubiquitous at the centers of local quiescent galaxies (for reviews, see
Kormendy 2004; Ferrarese \& Ford 2005). Remarkably,
the mass of the central black hole (\ensuremath{M_\mathrm{BH}}) correlates with the global
properties of their host spheroids, primarily the stellar velocity
dispersion ($\sigma_*$; Ferrarese et al. 2000; Gebhardt et al. 2000a).
This \ensuremath{\mbh-\sigma_*}\, relation is believed to be of fundamental
importance, providing one of the strongest empirical links between
galaxy formation/evolution and nuclear activity. Measuring the
slope and scatter of the relation, and any possible dependence on
additional parameters, is essential to make progress in our
understanding of the co-evolution of galaxies and black holes.
One key open question is whether active galactic nuclei (AGNs) share
the same \ensuremath{\mbh-\sigma_*}\ relation as quiescent galaxies, as would be expected
if the relation were universal and nuclear activity were just a random
transient phase. Initial studies using several reverberation-mapped
Seyfert 1 galaxies show that the \ensuremath{\mbh-\sigma_*}\ relation of active galaxies
is consistent with that of quiescent galaxies (Gebhardt et al. 2000b;
Ferrarese et al. 2001). Other studies using larger samples with
single-epoch black hole masses suggest that the slope of the relation
may be different than that of quiescent galaxies, albeit at low
significance (e.g., Greene \& Ho 2006; Shen et al.\ 2008). However,
these later studies rely on an indirect estimator of black hole mass,
based on the kinematics and size of the broad-line region (BLR) as
inferred from single-epoch spectra. The method works as follows: the
broad-line profile gives a characteristic velocity scale $\Delta V$
(measured from either line dispersion, $\sigma_{\rm line}$, or the
full-width at half-maximum intensity, $V_{\rm FWHM}$); the average size
of the BLR ($R_{\rm BLR}$) is determined from the empirical correlation with
optical luminosity (Wandel et al. 1999; Kaspi et al. 2005; Bentz et
al.\ 2009a); and the black hole mass is obtained as
\begin{equation}
M_{\rm BH} = f \frac{(\Delta V)^2 R_\mathrm{BLR}}{G},
\label{eqn:mbh}
\end{equation}
where $G$ is the gravitational constant and $f$ is a virial
coefficient that depends on the kinematics and the geometry of the
BLR. Although single-epoch mass estimates are believed
to be accurate to $\sim 0.5$ dex (e.g., Vestergaard \& Peterson 2006),
it is hard to quantify precisely their uncertainty and any possible
biases (Marconi et al. 2008; Onken et al. 2009). Thus, the slope and
intrinsic scatter of the \ensuremath{\mbh-\sigma_*}\, relation of active galaxies remain
highly uncertain.
Multi-epoch data provide direct measurements of the BLR size via
reverberation mapping time lags ($R_{\rm BLR}= c\tau$, where $c$ is the
speed of light and $\tau$ is the measured reverberation time scale),
and more secure measurements of the broad-line kinematics as determined from the variable
component of the broad lines (Peterson et al.\ 2004). Onken et al.\
(2004) combined reverberation black hole masses with measurements of
stellar velocity dispersion for 14 objects to measure the slope
of the \ensuremath{\mbh-\sigma_*}\ relation, showing that AGNs and
quiescent galaxies lie on the same \ensuremath{\mbh-\sigma_*}\, relation for an
appropriate choice of the virial coefficient ($ f
=5.5\pm1.8$).\footnote {Note that the virial coefficient, $\langle f
\rangle$, is $5.5$ when the line dispersion ($\sigma_{\rm line}$) is
used. If $V_{\rm FWHM}$ is used, $\langle f \rangle$ has to be
properly scaled depending on the $V_{\rm FWHM}/\sigma_{\rm line}$
ratio. See Onken et al. (2004) and Collin et al. (2006) for details.}
Remarkably, the scatter in \ensuremath{M_\mathrm{BH}}\ relative to the \ensuremath{\mbh-\sigma_*}\ relation is
found to be less than a factor of 3. Unfortunately, their sample is very
small (14 objects) and covers a limited dynamic range ($6.2 < \rm log \ensuremath{M_\mathrm{BH}}\ < 8.4$),
making it difficult to simultaneously measure the intrinsic scatter
and the slope and investigate any trends with black hole mass,
or with other properties
of the nucleus or the host galaxy (cf. Collin et al.\ 2006).
Under the assumption that the relations should be the same for active
and quiescent galaxies, the Onken et al.\ (2004) study provides some
information on the geometry of the BLR, an absolute
normalization of reverberation black hole mass, and an upper limit to
the intrinsic scatter in the virial coefficient. Since the
reverberation-mapped AGN sample is the ``gold standard'' used to
calibrate all single-epoch mass estimates (e.g., Woo \& Urry 2002;
Vestergaard et al. 2002; McLure \& Jarvis 2002; Vestergaard \&
Peterson 2006; McGill et al.\ 2008; Shen et al. 2008), this comparison
is effectively a crucial link in establishing black hole masses for
all AGNs across the universe and for all evolutionary studies (e.g.,
McLure \& Dunlop 2004; Woo et al.\ 2006, 2008; Peng \etal\ 2006a,b;
Netzer \etal\ 2007; Bennert et al.\ 2010; Jahnke et al. 2009; Merloni
et al. 2010).
In this paper, we present new measurements of host-galaxy velocity
dispersion obtained from deep, high-resolution spectroscopy at the
Keck, Palomar, and Lick Observatories. The new measurements are
combined with existing data (e.g., Onken et al. 2004; Nelson et
al. 2004; Watson et al. 2008) and with recently determined black hole
masses from reverberation mapping (Bentz et al.\ 2009b) to construct
the \ensuremath{\mbh-\sigma_*}\ relation of broad-lined AGNs for an enlarged sample
of 24 objects. For the first time, we are able to determine
{\it simultaneously} the intrinsic scatter and the slope of the \ensuremath{\mbh-\sigma_*}\
relation of active galaxies. Also, by forcing the slope of the
\ensuremath{\mbh-\sigma_*}\ relation to match that of quiescent galaxies, we determine
the average virial coefficient and its nonzero intrinsic
scatter. Finally, we study the residuals from the determined \ensuremath{\mbh-\sigma_*}\
relation to investigate possible trends in virial coefficient with
properties of the active nucleus.
The paper is organized as follows. In \S 2, we describe the observations
and data reduction of the optical and the near-infrared (IR) data. New velocity
dispersion measurements are presented in \S 3, along with a summary of
previous measurements in the literature. In \S 4, we investigate the
\ensuremath{\mbh-\sigma_*}\ relation of the present-day AGNs, and determine the virial
coefficient in measuring \ensuremath{M_\mathrm{BH}}. Discussion and summary follow in \S 5
and \S 6, respectively.
\section{Observations}
The Lick AGN Monitoring Project (LAMP) was designed to determine the
reverberation time scales of a sample of 13 local Seyfert 1 galaxies,
particularly with low black hole masses ($<10^{7}$\ensuremath{{\rm M}_{\odot}}). NGC 5548,
the best-studied reverberation target, was included in our Lick
monitoring campaign to test the consistency of our results with
previous measurements. The 64-night spectroscopic monitoring
campaign, along with nightly photometric monitoring, was carried out
using the 3-m Shane reflector at Lick Observatory and other,
smaller telescopes. The
detailed photometric and spectroscopic results are described by Walsh
et al.\ (2009) and Bentz et al.\ (2009b, 2010), respectively. In
summary, nine objects including NGC 5548 showed enough variability in
the optical continuum and the H$\beta$ line to obtain the
reverberation time scales. This significantly increases the size
of the reverberation sample, particularly for this low-mass range.
For the LAMP sample of 13 objects, we carried out
spectroscopy using various telescopes, to measure stellar velocity
dispersions. Here, we describe each observation and the data-reduction
procedures. The date, total exposure time, and other parameters for
each observation are listed in Table~\ref{observationstable}.
\subsection{Optical Data}
\begin{deluxetable*}{lcccccc}
\tablecaption{Observation Log} \tablehead{ \colhead{Galaxy} &
\colhead{Telescope/} & \colhead{UT Date} & \colhead{Exposure Time}
& \colhead{PA} & \colhead{Airmass} & \colhead{S/N} \\ \colhead{} &
\colhead{Instrument} & \colhead{} & \colhead{(s)} & \colhead{(deg)} &
\colhead{} & \colhead{} }
\startdata
Arp 151 & P200/DBSP & 2003-01-28 & 5400 & 210 & 1.10 & 79 \\
IC 1198 (Mrk 871) & Keck/ESI & 2008-03-02 & 900 & 285 & 1.25 & 87\\
IC 4218 & P200/DBSP & 2003-06-02 & 3600 & 5 & 1.23 & 61\\
MCG--06-30-15 & P200/DBSP & 2003-06-01 & 5400 & 10 & 2.70 & 112\\
Mrk 1310 & Keck/ESI & 2008-03-02 & 1800 & 323 & 1.19 & 84\\
Mrk 142 & Keck/OSIRIS & 2009-05-03 & 3600 & 0 & 1.19 & 17 \\
Mrk 202 & Keck/ESI & 2008-03-02 & 1200 & 204 & 1.32 & 73\\
Mrk 290 & Lick/Kast & 2008-04-13 & 3600 & 168 & 1.07 & 27 \\
NGC 4253 (Mrk 766) & P200/DBSP & 2001-06-26 & 1200 & 220 & 1.60 & 63 \\
& Keck/OSIRIS & 2009-05-03 & 3600 & 0 & 1.03 & 67 \\
NGC 4748 & Keck/ESI & 2004-02-17 & 1200 & 25 & 1.23 & 160 \\
& Keck/OSIRIS & 2009-05-04 & 5400 & 0 & 1.37 & 61 \\
NGC 5548 & P200/DBSP & 2003-06-01 & 1800 & 59 & 1.06 & 100\\
NGC 6814 & P200/DBSP & 2003-06-01 & 3600 & 0 & 1.38 & 165 \\
SBS 1116+583A & Keck/ESI & 2008-03-02 & 1800 & 216 & 1.40 & 53 \\
\tablecomments{For OSIRIS observations, the position angle (PA)
refers to the direction of the long axis of the IFU; for
all other observations it refers to the slit PA.
The S/N refers to the signal-to-noise ratio per pixel
in the extracted spectra, at $\sim$8400-8700 \AA\
for the \ion{Ca}{2} triplet spectral region, or at
$\sim$1.47--1.61 $\mu$m for the $H$-band spectra.}
\label{observationstable}
\end{deluxetable*}
\subsubsection{Palomar Observations}
Observations of six galaxies were obtained with the Double
Spectrograph \citep[DBSP;][]{og82} at the Palomar Hale 5-m telescope
(P200).
A 2\arcsec-wide slit was used, along with the D68 dichroic. On the red side
of the spectrograph, we used a 1200 lines mm\ensuremath{^{-1}}\ grating blazed at
9400 \AA, covering the wavelength range 8330--8960 \AA\ at an
instrumental dispersion of $\sigma_i \approx 30.4$ \kms.
Each galaxy was observed with the slit oriented approximately at the
parallactic angle (Filippenko 1982)
for the midpoint of the observation. Typically two
or three exposures were taken for each galaxy to aid in cosmic-ray
removal. Flux standards and a range of velocity template stars
(primarily K-type giants) were observed during each night.
\subsubsection{Lick Observations}
Mrk 290 was observed with the Kast Double Spectrograph (Miller \& Stone
1993) at the Shane 3-m telescope at Lick Observatory on 2008 April 13
(UT dates are used throughout this paper),
during our AGN monitoring campaign. For these observations, we
used the D55 dichroic and the 830/8460 grating on the red side of the
spectrograph, covering the wavelength range 7570--9620 \AA\ at a scale
of 1.7 \AA\ pixel$^{-1}$. A 2\arcsec-wide slit was used and oriented at
the parallactic angle for the midpoint of the exposure sequence, and
the instrumental dispersion was $\sigma_i \approx 59$ \kms\ at 8600
\AA. We obtained four 900 s exposures of Mrk 290 with this setup,
along with short exposures of three velocity template stars and the
flux standard star BD+28$^\circ$4211.
\subsubsection{Keck Observations}
Five galaxies were observed with the Echellete Spectrograph and Imager
\citep[ESI;][]{sheinis02} at the Keck-II 10-m telescope. In ESI
echellette mode, the observations cover 3900--11000 \AA\ in 10
spectral orders at a scale of $\sim$11.4 km s\ensuremath{^{-1}}\ pixel\ensuremath{^{-1}}. We used
a 0\farcs75-wide slit, giving an instrumental dispersion of $\sigma_i
\approx 22$ \kms.
The slit was oriented at the parallactic angle in all observations.
During twilight of each night, we observed flux standards and several
velocity template stars with spectral types ranging from G8III to K5III.
\subsubsection{Reductions}
We reduced the DBSP and Kast data using a series of IRAF\footnote{IRAF
is distributed by the National Optical Astronomy Observatories, which
are operated by the Association of Universities for Research in
Astronomy, Inc., under cooperative agreement with the National Science
Foundation.} scripts. The standard spectroscopic data-reduction
procedure, including bias subtraction, flat fielding, wavelength
calibration, spectral extraction, and flux calibration was performed
for each data set. Optimal extraction \citep{H86} was used for obtaining
one-dimensional spectra to achieve maximal signal-to-noise ratio (S/N)
on the stellar features. The typical extraction radius was $\sim2-3''$,
which corresponds to a physical radius of $\sim1$ kpc.
The ESI observations require some special preparation and calibration
before they can be combined and used for kinematic analysis. The
calibration process undertaken includes the following steps: bias
subtraction, flat-fielding, cosmic-ray rejection, wavelength
calibration, rectification, and sky subtraction. These steps were
performed by the IRAF package {\tt EASI2D}, which was developed by
David J.~Sand and Tommaso Treu \citep{sand04} for easy extraction of
echelle orders. Approximate flux calibration was performed using ESI
response curves measured during photometric nights. During wavelength
calibration the spectra were rebinned to uniform steps in
$\log(\lambda)$ corresponding to 11.4 km s$^{-1}$, which is close to
the native pixel scale of ESI, thus minimizing covariance between
pixels and loss of resolution due to rebinning. The root-mean square
(rms) residuals in
the wavelength solution are much smaller than a pixel (typically 5\%)
and therefore negligible. Spectral resolution was measured from night-sky
lines and from wavelength-calibration lines taken in the same
configuration and reduced through the same pipeline to include all
instrumental effects.
As for the DBSP and Kast data, optimal
extraction was used for obtaining one-dimensional spectra. The
typical extraction radius of the ESI spectra was $\sim1-1.5''$, which
corresponds to a physical radius of $\sim0.5$ kpc at the mean redshift
of the observed AGNs.
\subsection{Near-Infrared Data}
\subsubsection{Keck Observations}
For three galaxies, we obtained $H$-band spectra using the
integral-field unit (IFU) OH-Suppressing Infrared Imaging Spectrograph
\citep[OSIRIS;][]{larkin06} at the Keck-II telescope, operated with the
laser guide-star adaptive optics (LGS-AO) system \citep{wiz06}. Using
the high angular resolution in the AO-corrected data, our goal was to
extract the light from an annulus around the nucleus if necessary, by
excluding light from the central, AGN-dominated region, similar to what
was done by \citet{watson08}.
The OSIRIS observations were obtained on 2009 May 3 and
4, during the first half of each night. We used the $H$ broadband
(Hbb) filter and the 0\farcs1 lenslet scale, giving a field of view of
$1\farcs6\times6\farcs4$, with spectra covering the range 1473--1803
nm at a spectral resolving power of $R\approx3800$. The long spatial
direction of the IFU was oriented north-south for all observations.
The observing sequence for each galaxy consisted of sets of four
exposures. In three exposures the nucleus of the galaxy was
placed on the IFU. Then one sky exposure with an offset of 21$\arcsec$
to the southeast was taken. The three on-source exposures in each set were
dithered northward by 0\farcs6 between exposures. Each individual
exposure had a duration of 600 s for NGC 4748, or 300 s for Mrk 142 and Mrk
766, and total on-source exposure times for each source are listed in
Table~\ref{observationstable}.
Immediately preceding and following
the observation sequence for each galaxy, we observed an A0V star for
telluric correction and flux calibration. The A0V stars were observed
using a sequence of two on-source and one off-source exposure, with
typical exposure times of 5--10 s. In order to obtain unsaturated
exposures of these bright stars, the AO loops were not closed during
the star observations. On each night, velocity template stars were
also observed, following the same observing sequence as used for the
telluric correction stars. We observed several K and M-type giant
stars as velocity templates, and A0V telluric correction stars
were also observed close in time and airmass to the velocity template
stars.
The nucleus of each galaxy was used as a tip-tilt reference source
($V\approx15-16$ mag) for the LGS-AO system, and the laser was
propagated on-axis for each dithered exposure. It is difficult to
quantify the image quality or determine the Strehl ratio in the galaxy
exposures, since the galaxy nuclei are not point sources. However, the
typical LGS-AO performance for tip-tilt point sources for these
similar magnitudes yields $K$-band Strehl ratios of 40\% to 50\%
\citep{dam06,dam07}. Therefore, given that the quality of both nights
was excellent, we conservatively estimate the $H$-band Strehl ratio to
be $\sim$15\%--20\% for our galaxy observations.
Prior to this OSIRIS run, the OSIRIS detector temperature had been
rising from its previous operating temperature of 68 K. During these
two nights, the detector temperature was steady at 74 K, which
resulted in a higher level ($\sim$40\%) of dark current than normal
for OSIRIS. The higher dark current did not significantly impact our
observations, since our exposures were not dark or background limited.
In the afternoon, multiple dark frames were taken with exposure times
matching those of our observations, and were used as part of the
sky-subtraction procedure. In addition, a new calibration file for
the observing mode with
the Hbb filter and 0\farcs1 scale was taken at a similar operating
temperature to ensure a clean spectral extraction process.
\subsubsection{Reductions}
The data were reduced using the OSIRIS data-reduction pipeline
software \citep{krabbe04}. In summary, the reduction steps included dark subtraction,
cosmic-ray cleaning and glitch identification, extraction of the
spectra into a data cube, wavelength calibration, sky subtraction
using the offset sky exposures, correction for atmospheric dispersion,
and telluric correction using the extracted spectra of the A0V stars.
The OSIRIS pipeline includes an implementation of the scaled
sky-subtraction algorithm described by \citet{davies07}, which provides a
more accurate subtraction of the strong $H$-band airglow lines than a
direct subtraction of the offset-sky exposure. However, the current
implementation of the scaled sky routine left noticeable residuals
around the wavelengths of the strongest blended OH features;
we further discuss these sky residuals below. Following the sky
subtraction and telluric correction, individual exposures were aligned
and co-added to create a final data cube for each galaxy.
We extracted 1-dimensional spectra from each data cube using
rectangular extraction regions. While we had originally planned to
extract an annular region around the nucleus in order to avoid the
central, AGN-dominated region, we found that in the reduced data the
$H$-band stellar absorption features were visible even in the central
region of the galaxy, and it did not prove necessary to exclude the
central region from the spectral extractions. We experimented with a
variety of different extraction-region sizes for each galaxy. Since
the galaxy-light profiles were very centrally concentrated, the
extracted spectra did not change significantly beyond some extraction
aperture size, and in the end we used extraction apertures that were
large enough to contain the majority of the detected galaxy light
without including unnecessarily large amounts of noise from the
surrounding regions. The extraction aperture sizes were
$1\farcs4\times2\farcs4$, $1\farcs3\times3\farcs4$, and
$1\farcs3\times3\farcs4$, respectively, for Mrk 142, Mrk 766, and NGC
4748. One-dimensional spectra were similarly extracted from the
reduced data cubes for the velocity template stars.
As a result of the imperfect sky subtraction mentioned previously,
spectral regions around the wavelengths of the brightest OH emission
blends were contaminated by noticeable residuals that appeared to
affect all pixels on the IFU in a similar fashion. In order to
correct for these additive residuals, for each galaxy we extracted
spectra from rectangular regions at the outer edges of the IFU where
there was very little galaxy continuum detected, with the same number
of pixels as the nuclear extraction, and then subtracted these nearly
blank-sky spectra from the nuclear spectra. This correction procedure
effectively removed the night-sky emission residuals from the galaxy
spectra. The procedure was applied for NGC 4748 and Mrk
142, but was not necessary for Mrk 766 since the OH emission-line
strengths appeared to be much more stable during the Mrk 766
observations, leading to a cleaner sky subtraction.
\section{Stellar Velocity Dispersions}
\begin{figure}
\epsscale{1.0}
\plotone{f1.eps}
\caption{Velocity dispersion measurements using the Palomar DBSP and
the Keck ESI spectra. The region including the main stellar
features around the \ion{Ca}{2} triplet is shown together with the
best-fit template (thick red line).}
\label{fig_dis}
\end{figure}
To measure stellar velocity dispersions, we used the \ion{Ca}{2} triplet
region (rest frame $\sim$8500\AA) in the optical spectra, and
\ion{Mg}{1} and the CO line region (rest frame $\sim$1.5--1.6 $\micron$) in
the near-IR spectra. Velocity dispersions were determined by comparing
in pixel space the observed galaxy spectra with stellar templates
convolved with a kernel representing Gaussian broadening. Velocity
template stars of various spectral types were observed with the same
instrumental setup at each telescope, minimizing any systematic errors
due to the different spectral resolution of various observing
modes. The minimum $\chi^{2}$ fit was performed using the
Gauss-Hermite Pixel Fitting software (van der Marel 1994). The merit
of fitting in pixel space as opposed to Fourier space is that typical
AGN narrow emission lines and any residuals from night-sky subtraction
can be easily masked out (e.g., Treu et al. 2004; Woo et al. 2005;
2006), and the quality of the fit can be directly evaluated.
We used low-order polynomials (order 2--4) to model the overall shape of
the continuum. Extensive and careful comparison was performed using
continua with various polynomials, and the measured velocity dispersions
based on each continuum fit with various order polynomials
are consistent within the errors (see e.g., Woo et al. 2004; 2005; 2006). Thus,
continuum fitting with low-order polynomials does not significantly affect the
velocity dispersion measurements.
\subsection{Optical Data}
For galaxies with optical spectra we mainly used the strong
\ion{Ca}{2} triplet lines (8498, 8542, 8662 \AA) to measure the
stellar velocity dispersions. We used all velocity template stars
with various spectral types (G8--K5 giants) observed with the same
instrumental setups, in order to account for possible template
mismatches. Fits were performed with each template and the mean of
these measurements was taken as the final velocity dispersion
measurement. The rms scatter around the mean of individual
measurements was added in quadrature to the mean of the individual
measurement errors from the $\chi^{2}$ fit. Given the high S/N of the
observed spectra, the uncertainties are typically $\sim$5\%.
\begin{figure}
\epsscale{1.0}
\plotone{f2.eps}
\caption{Galaxies with contaminated stellar lines. These galaxies
clearly show the \ion{Ca}{2} triplet lines; however, due to AGN
contamination, the line profiles are asymmetric. Also, the presence
of various AGN emission features complicates continuum
subtraction. The spectra for the first three objects (Mrk 871, NGC
4748, Mrk 766) are from Palomar DBSP data. The data for Mrk 290 data
were obtained at the Lick telescope, and Mrk 142 data are from
SDSS.}
\label{fig_emission}
\end{figure}
We were able to measure the stellar velocity dispersion of eight objects:
SBS 1116+538A, Arp 151, Mrk 1310, Mrk 202, IC 4218, NGC 5548, NGC
6814, and MCG--6-30-15. Among these, IC 4218 and MCG--6-30-15 were
excluded in the \ensuremath{\mbh-\sigma_*}\ relation analysis since we were not able to
measure the reverberation time scales (Bentz et al. 2009b). Figure 1
shows examples of the best-fit template along with the observed galaxy
spectra. Our measurements based on high-quality data are generally
consistent within the errors with previous measurements from the
literature for these galaxies, although many of the previously
existing measurements had very large uncertainties (see
\S\ref{previous} below).
In the case of the other five objects (Mrk 871, NGC 4748, Mrk 766, Mrk 290,
and Mrk 142), the \ion{Ca}{2} triplet lines were contaminated by AGN emission
lines as shown in Figure 2; hence, we did not obtain reliable stellar
velocity dispersion measurements. Three of these objects, that have
black hole mass measurements from reverberation mapping, were
observed in the near-IR, and we determined stellar velocity dispersions
for two of them as described in the next section.
\subsection{Infrared Data}
\begin{figure}
\epsscale{1.0}
\plotone{f3.eps}
\caption{Velocity dispersion measurements using the OSIRIS
spectra. The region including the main stellar features is shown
together with the best-fit template (thick red line). Individual
stellar lines are identified in the first panel. The regions around
narrow AGN emission lines and regions of severe template mismatch,
identified by shaded bands, are masked out before fitting.
Residuals from the fit are shown at the bottome of NGC 4748 and Mrk 766.}
\label{fig_osiris}
\end{figure}
Because of the difficulty of measuring the stellar velocity dispersion
in the optical, we obtained near-IR spectra for three galaxies with
reverberation black hole masses (NGC 4748, Mrk 766, Mrk 142) using
OSIRIS. We also observed several velocity template stars with K and M
spectral types. As noted by other studies (e.g., Watson et al. 2008),
M giant stars give better fits compared to K giants since M giants are
the representative stellar population in the near-IR bands. After
experimenting with various spectral type velocity templates, we
decided to use M1III and M2III stars for velocity dispersion
measurements and took the mean as the final measurement with an
uncertainty given by the quadrature sum of the fitting error and the
rms scatter of results obtained from different templates. Figure 3
shows the observed galaxy spectra compared with the best-fit
template. NGC 4748 shows a relatively good fit, while Mrk 766 presents
stronger residuals due to template mismatch. All major $H$-band
stellar lines, such as \ion{Mg}{1} 1.488 \micron, \ion{Mg}{1} 1.503
\micron, CO(3-0) 1.558 \micron, CO(4-1) 1.578 \micron, \ion{Si}{1}
1.589 \micron, and CO(5-2) 1.598 \micron, are present in both objects.
In the case of Mrk 142, which is fainter and at higher redshift than
the other two objects, stellar absorption lines cannot be clearly
identified and we could not obtain a reliable template fit or velocity
dispersion measurement.
\subsection{Previous Velocity Dispersion Measurements}
\label{previous}
Several objects in the LAMP sample have preexisting velocity dispersion
measurements in the literature. In most cases the published
results have fairly large uncertainties. Here we briefly summarize the
previous measurements and describe the available spectroscopic data from
the Sloan Digital Sky Survey (SDSS).
\smallskip
\textit{SBS 1116+583A:} \citet{gh06} measured $\sigma=50.3\pm18.0$
\kms\ using the SDSS spectrum. However, the velocity dispersion
was not clearly resolved by the SDSS data, which have an instrumental
dispersion of $\sim70$ \kms.
\smallskip
\textit{Arp 151:} \citet{gh06} measured $\sigma=124\pm12$ \kms\ from the
SDSS spectrum.
\smallskip
\textit{Mrk 1310:} Using the SDSS data, \citet{gh06} reported $\sigma=50\pm16$
\kms, which is below the resolution limit of the SDSS spectrum.
\smallskip
\textit{Mrk 202:} \citet{gh06} measured $\sigma=85.6\pm14$ \kms\ from the SDSS spectrum.
\smallskip
\textit{Mrk 290:} In the SDSS spectrum, the Ca II triplet lines are only weakly
detected, and they appear to be badly contaminated by Ca II emission.
The blue region of the spectrum is too dominated by the AGN for any stellar
features to be clearly visible.
\smallskip
\textit{NGC 4253 (Mrk 766):} \citet{bot05} reported a dispersion of $\sigma=81\pm17$
from the Ca II triplet lines. However, the Ca II triplet region is moderately
contaminated by emission lines. Their dispersion measurements were
carried out by cross-correlating the galaxy spectrum with that of a
template star. Without a direct fit of the broadened template star
to the galaxy spectrum, it is not clear how badly the measurement may
be affected by the emission-line contamination.
An SDSS spectrum of Mrk 766 exists, but it appears to have been taken
with the SDSS fiber positioned about 3\arcsec\ away from the galaxy
nucleus. Since the SDSS fiber missed the galaxy nucleus, the spectrum
appears similar to that of a Seyfert 2 galaxy,
dominated by starlight and with very
weak broad components to the Balmer lines. Our own data from the Lick
monitoring campaign, on the other hand, show a strongly AGN-dominated
spectrum with much more prominent broad lines. Thus, while the Ca II triplet
and other stellar absorption lines are clearly detected in the SDSS
data, the offset position of the SDSS spectrum means that it should
not be used to infer the bulge velocity dispersion.
\smallskip
\textit{NGC 5548:} The most recent
measurements of the velocity dispersion for this galaxy include
$\sigma=180\pm6$ \kms\ \citep{fer01} and $\sigma=201\pm12$ \kms\
\citep{nelson04}.
\smallskip
\textit{NGC 6814:} \citet{nw95} listed a velocity dispersion of
$115\pm18$ \kms.
\smallskip
\textit{Mrk 142}: The SDSS spectrum of Mrk 142 (SDSS
J102531.28+514034.8) shows that the object is strongly AGN dominated,
and the stellar Ca II triplet lines are severely contaminated
by Ca II emission.
In addition, the spectral region around Mg~$b$ and Fe~5270 is
completely dominated by strong \ion{Fe}{2} emission,
precluding any possibility of measuring the stellar velocity dispersion
in the optical.
\smallskip
\textit{MCG--6-30-15:} \citet{oliva99} measured $\sigma=159$ \kms\ from
template fits to the CO and Si features in the $H$ band, but they did
not list estimates of the measurement uncertainties.
\citet{mchardy05} used a spectrum around the Ca II triplet to measure $\sigma=93.5\pm8.5$
\kms, using a cross-correlation technique.
\medskip
All of the previous measurements with sufficiently high instrumental resolution
are consistent with our new measurements within the errors, except for
one object, MCG--6-30-15. The previously reported \ion{Ca}{2}-based
velocity dispersion and the $H$-band velocity dispersion of MCG--6-30-15
are dramatically different, and our measured dispersion based on the
\ion{Ca}{2} triplet is significantly lower than either previous
result. Unfortunately, we cannot determine the origin of the
discrepancy other than noting that the \ion{Ca}{2} dispersion from
\cite{mchardy05} was determined using a cross-correlation method, and
therefore it is not easy to directly compare the quality of the result
with our measurement. As an additional check, we carried out an
independent measurement of \ensuremath{\sigma_*}\ from the Palomar DBSP data
using the template-fitting code of \citet{bhs02}. This gave
$\ensuremath{\sigma_*}=75\pm5$ \kms, consistent with the value of $76\pm3$ \kms\
measured with the \citet{marel94} code. Since MCG--6-30-15 is not
included in the \ensuremath{\mbh-\sigma_*}\ relation analysis in the next section, this
does not affect our main conclusions.
\begin{deluxetable*}{lcccccc}
\tablecaption{Black hole masses and stellar velocity dispersions}
\tablehead{ \colhead{Galaxy} &
\colhead{VP} & \colhead{VP ref.} & \colhead{$\sigma_{*}$} & \colhead{error} & \colhead{$\sigma_*$ ref.} & \colhead{$\log$ \ensuremath{M_\mathrm{BH}}/\ensuremath{{\rm M}_{\odot}}} \\
\colhead{} & \colhead{($\sigma_{\rm line}^2 R_{\rm BLR}/G$)} & \colhead{} & \colhead{} & \colhead{} & \colhead{} \\
\colhead{} & \colhead{$10^{6} \ensuremath{{\rm M}_{\odot}}$} & \colhead{} & \colhead{\kms} & \colhead{} & \colhead{}
}
\startdata
\multicolumn{7}{c}{LAMP Sample} \\
\hline
Arp 151 & $1.22^{+0.16}_{-0.22}$ & 5 & 118 & 4 & this work & $6.81\pm0.12$\\
IC 4218 & \nodata & & 93 & 4 & this work & \nodata\\
MCG--6-30-15 & \nodata & & 76 & 3 & this work & \nodata\\
Mrk 1310 & $0.41^{+0.12}_{-0.13}$ & 5 & 84 & 5 & this work & $6.33\pm0.17$\\
Mrk 142 & $0.40^{+0.12}_{-0.14}$ & 5 & \nodata & \nodata & & $6.32\pm0.17$ \\
Mrk 202 & $0.26^{+0.13}_{-0.10}$ & 5 & 78 & 3 & this work & $6.13\pm0.22$\\
NGC 4253 (Mrk 766) & $0.32^{+0.21}_{-0.20}$ & 5 & 93 & 32 & this work & $6.23\pm0.30$\\
NGC 4748 & $0.47^{+0.16}_{-0.21}$ & 5 &105 &13 & this work & $6.39\pm0.20$\\
NGC 5548 & $11.9^{+0.46}_{-0.46}$ & 6 & 195 &13 & this work & $7.80\pm0.10$\\
NGC 6814 & $3.36^{+0.54}_{-0.56}$ & 5 & 95 & 3 & this work & $7.25\pm0.12$\\
SBS 1116+583A & $1.05^{+0.33}_{-0.29}$ & 5 & 92 & 4 & this work & $6.74\pm0.16$\\
\hline
\multicolumn{7}{c}{Previous Reverberation Sample} \\
\hline
Ark 120 & $27.2^{+3.5}_{-3.5}$ & 1 & 221 & 17 & 1 & $8.15\pm0.11$ \\
3C 120 & $10.1^{+5.7}_ {-4.1}$ & 1 & 162 & 20 & 2 & $7.72\pm0.23$\\
3C 390.3 & $52^{+11.7}_{-11.7}$ & 1 & 273 & 16 & 1 & $8.44\pm0.14$ \\
MRK 79 & $9.52^{+2.61}_{-2.61}$ & 1 & 130 & 12 & 1 & $7.70\pm0.16$ \\
MRK 110 & $4.57^{+1.1}_{-1.1}$ & 1 & 91 & 7 & 3 & $7.38\pm0.14$\\
MRK 279 & $6.35^{+1.67}_{-1.67}$ & 1 & 197 & 12 & 1 & $7.52\pm0.15$ \\
MRK 590 & $8.64^{+1.34}_{-1.34}$ & 1 & 189 & 6 & 1 & $7.66\pm0.12$\\
MRK 871 & $8.98^{+1.4}_{-1.4}$ & 1 & 120 & 15 & 1 & $7.67\pm0.12$ \\
NGC 3227 & $7.67^{+3.9}_{-3.9}$ & 1 & 136 & 4 & 1 & $7.60\pm0.24$\\
NGC 3516 & $7.76^{+2.65}_{-2.65}$ & 1 & 181 & 5 & 1 & $7.61\pm 0.18$\\
NGC 3783 & $5.42^{+0.99}_{-0.99}$ & 1 & 95 & 10 & 4 & $7.45\pm0.13$\\
NGC 4051 & $.287^{+0.09}_{-0.12}$ & 2 & 89 & 3 & 1 & $6.18\pm 0.19$ \\
NGC 4151 & $8.31^{+1.04}_{-0.85}$ & 3 & 97 & 3 & 1 & $7.64\pm 0.11$\\
NGC 4593 & $1.78^{+0.38}_{-0.38}$ & 4 & 135 & 6 & 1 & $6.97\pm0.14$ \\
NGC 7469 & $2.21^{+0.25}_{-0.25}$ & 1 & 131 & 5 & 1 & $7.06\pm0.11$\\
PG 1426+215 & $236^{+70}_{-70}$ & 1 & 217 & 15 & 5 & $9.09\pm 0.16$\\
\label{measurementstable}
\tablecomments{
Col. (1) name.
Col. (2) virial product (\ensuremath{M_\mathrm{BH}}=$f\times$ VP) based on the line dispersion ($\sigma_{\rm line}$) from reverberation mapping.
Col. (3) reference for virial product. 1. Peterson et al.\ 2004;
2. Denney et al.\ 2009; 3. Bentz et al.\ 2006; 4. Denney et al.\ 2006;
5. Bentz et al.\ 2009b; 6. weighted mean of Bentz et al.\ 2007 and Bentz
et al.\ 2009b.
Col. (4) stellar velocity dispersion.
Col. (5) error of stellar velocity dispersion.
Col. (6) reference for stellar velocity dispersion. 1. Nelson et al. 2004;
2. Nelson \& Whittle 1995; 3. Ferrarese et al. 2001; 4. Onken et
al. 2004; 5. Watson et al. 2008.
Col. (7) black hole mass calculated assuming the virial coefficient,
$\log f=0.72\pm0.10$. The uncertainty in the black hole mass is
calculated by adding in quadrature the measurement uncertainty of the
virial product in logarithmic space and the uncertainty in the virial coefficient (0.1
dex) as $\rm \epsilon \log \ensuremath{M_\mathrm{BH}} = \sqrt {\rm \epsilon VP^{2} / (VP ~ ln 10)^{2}
+ 0.1^{2}}$.
The average of the positive and negative errors on the virial product
is taken as the symmetric error ($\epsilon \rm VP$).
}
\end{deluxetable*}
\section{The \ensuremath{\mbh-\sigma_*}\ relation of active galaxies}
The \ensuremath{\mbh-\sigma_*}\ relation is expressed as
\begin{equation}
\log (M_{\rm BH} / M_{\odot}) = \alpha + \beta \log (\sigma_{\ast} / {200\ \mathrm{\kms}}) .
\label{eq:msigma}
\end{equation}
For a sample of quiescent galaxies for which the sphere of influence
of the black hole is spatially resolved in stellar or gas kinematics
studies, Ferrarese \& Ford (2005) measured $\alpha=8.22 \pm 0.06 $ and
$\beta=4.86 \pm 0.43$ with no evidence for intrinsic scatter. Using a
larger sample, including late-type galaxies, G\"ultekin et al.\ (2009)
recently reported $\alpha=8.12\pm0.08$ and $\beta=4.24\pm0.41$ with an
intrinsic scatter $\sigma_{\rm int}=0.44\pm0.06$.
In this section, we investigate the \ensuremath{\mbh-\sigma_*}\ relation of active
galaxies, by combining our 8 new velocity measurements of the LAMP
sample with 16 existing measurements of reverberation
black hole mass and velocity
dispersion taken from the literature. The relevant properties of the
resulting sample of 24 AGNs are listed in Table~2, along with the
original references for the measurements.
Note that the selection bias investigated by Lauer et al. (2007) is not
relevant to these reverberation objects since the bias mainly affects black holes in high-mass galaxies where the host
galaxy luminosity function is steeply falling.
As a first step we determine
the slope of the \ensuremath{\mbh-\sigma_*}\ relation (\S 4.1)
assuming that the virial coefficient $f$ is unknown but does not
depend on black hole mass. We also determine the intrinsic scatter in
the relation. Then in \S~4.2 we determine the mean value and
intrinsic scatter of the virial coefficient $f$ by assuming that the
slope and zero point of the \ensuremath{\mbh-\sigma_*}\ relation are the same for active
and quiescent galaxies. Note that in the following analysis the virial
coefficient $f$ and the zero point $\alpha$ are degenerate; therefore,
additional information is needed to determine what fraction of the
intrinsic scatter is due to $f$ and how much is due to $\alpha$, as we
discuss below.
\subsection{The Slope}
Active galaxies appear to obey an \ensuremath{\mbh-\sigma_*}\ relation with a slope
consistent with that of quiescent galaxies (e.g., Gebhardt et al.\ 2000;
Ferrarese et al.\ 2001). However, the slope has not been well
determined so far, because of limitations in sample size and in the
dynamic range of \ensuremath{M_\mathrm{BH}}\ for active galaxies with reverberation and
stellar velocity dispersion measurements. The local AGN samples with
single-epoch black hole masses selected from the SDSS
are much larger than the reverberation sample (e.g., Greene \&
Ho 2006; Woo et al. 2008), but these mass estimates have larger
uncertainties ($\sim0.5$ dex) and may suffer from unknown systematic
errors (Collin et al. 2006).
By adding our LAMP galaxies to the previous reverberation sample, we
significantly increase the sample size, particularly at low mass
scales (\ensuremath{M_\mathrm{BH}}\ $< 10^{7} \ensuremath{{\rm M}_{\odot}}$), and extend the dynamic range to almost
3 decades in \ensuremath{M_\mathrm{BH}}. This enables substantial progress in determining
the slope of the \ensuremath{\mbh-\sigma_*}\ relation, independently for active galaxies.
One important caveat is that the virial coefficient $f$ may vary
across the sample. Unfortunately, with existing data we cannot
constrain the virial coefficient of each object. Therefore, in the
analysis presented here we will assume that $f$ is
independent of black hole mass, although we allow for intrinsic
scatter at fixed black hole mass. In the next section we will take the
opposite viewpoint and assume that the slope is known from quiescent
samples when we investigate the intrinsic scatter in $f$ and its
dependence on \ensuremath{M_\mathrm{BH}}\ and AGN properties.
In practice, our observable is the so-called virial product VP$\equiv
\sigma_{\rm line}^2 R_\mathrm{BLR}/G$, which is assumed to be related to the
black hole mass by a constant virial coefficient $M_{\rm
BH}$=$f\times$VP. Substituting into Equation~\ref{eq:msigma}, one obtains
\begin{equation}
\log {\rm VP} = \alpha + \beta\ \log\left (\frac{\sigma_{\ast}}{200\ \mathrm{\kms}} \right ) - \log f.
\label{eq:VPsigma}
\end{equation}
In order to determine the slope $\beta$ of the relation, we compute
the posterior distribution function $P$. Assuming uniform priors on
the parameters $\alpha'=\alpha - \log f$, $\beta$, and intrinsic
scatter $\sigma_{\rm int}$, the posterior is equal to the likelihood
given by
\begin{equation}
-2\ln P \equiv \sum_{i=1}^{N} \frac{\left [\log \mathrm{VP}_i - \alpha' - \beta
\log (\frac{\sigma_{\ast}}{200})_i \right ]^2}{\epsilon_{{\rm
tot},i}^2} + 2 \ln \sqrt{2\pi \epsilon_{{\rm tot},i}^2},
\label{eq:likeli}
\end{equation}
\noindent
where
\begin{equation}
\epsilon_{{\rm tot},i}^2 \equiv \epsilon_{\log \mathrm{VP}_i}^2 + \beta^{\,2} \epsilon_{\log
\sigma_*i}^2 + \sigma_{\rm int}^2,
\end{equation}
\noindent
and $\epsilon_{\log \mathrm{VP}}$ is the error on the logarithm of the virial product
($= \rm \epsilon_{\rm VP} / (\rm VP ~ ln 10))$ using the average of the positive and negative errors
on the virial product as $\rm \epsilon_{\rm VP}$,
$\rm \epsilon_{\rm \log \sigma_*}$ is the error on the logarithm of the stellar velocity
dispersion ($= \rm \epsilon_{\sigma_*} / (\rm \sigma_* ~ ln 10$)),
and $\rm \sigma_{\rm int}$ is the intrinsic scatter, which is
treated as a free parameter.
Note that our adopted likelihood is
equivalent to that adopted as default by G\"ultekin et al.\ (2009) for
their maximum likelihood analysis (see their Appendix A.2), with the
inclusion of an additional Gaussian term to describe the uncertainties
in velocity dispersion. After marginalizing over the other variables,
the one-dimensional posterior distribution functions yield the
following estimates of the free parameters: $\alpha'=7.28\pm0.14$,
$\beta= 3.55 \pm 0.60$, and $\sigma_{\rm int}=0.43\pm 0.08$.
To facilitate comparison with earlier studies (e.g., Tremaine et
al. 2002), we also determine the slope with a ``normalized''
$\chi^{2}$ estimator defined as
\begin{equation}
\chi^{2} \equiv \sum_{i=1}^{N} \frac{\left [\log \mathrm{VP}_i - \alpha' - \beta
\log (\frac{\sigma_{\ast}}{200})_i \right ]^2}{\epsilon_{{\rm
tot},i}^2},
\end{equation}
where the intrinsic scatter $\sigma_{\rm int}=0.43$ is set to yield
$\chi^{2}$ per degree of freedom equal to unity. The best estimate of
the slope based on this normalized $\chi^{2}$ estimator is
$\beta=3.72^{+0.40}_{-0.39}$. If we assume no intrinsic scatter, then
the best-fit slope becomes slightly steeper as
$\beta=3.97^{+0.13}_{-0.13}$. However, confirming our previous result
based on a full posterior analysis, the intrinsic scatter cannot be
negligible since the $\chi^{2}$ per degree of freedom would be much larger
than unity ($\sim$14) if the intrinsic scatter is assumed to be zero.
These results indicate that the slopes of the \ensuremath{\mbh-\sigma_*}\ relation for
active and quiescent galaxies are consistent within the
uncertainties. The total intrinsic scatter is $0.43\pm0.08$ dex, which
is a combination of the scatter of the \ensuremath{\mbh-\sigma_*}\ relation and that of
the virial coefficient. Without additional information, we cannot
separate the contribution of the intrinsic scatter on the virial
coefficient (i.e. the diversity of broad line region kinematic
structure amongst AGN) from the intrinsic scatter in black hole mass
at fixed velocity dispersion (i.e., the diversity of black hole masses
amongst galaxies). We will address this issue in \S 4.2.
\subsection{The Virial Coefficient}
In order to transform the virial product into an actual black hole
mass, we need to know the value of the virial coefficient $f$. Our
current understanding of the geometry and kinematics of the BLR is
insufficient to compute $f$ from first principles. This creates a
systematic uncertainty in the determination of black hole mass from
reverberation mapping. As an example, if a spherical isotropic
velocity distribution is assumed, then the velocity dispersion of the
gas is a factor of $\sqrt{3}$ larger than the measured line-of-sight
velocity dispersion ($\sigma_{\rm line}$), hence a value of $f=3$ is
often assumed for a spherically symmetric BLR
\citep[e.g.,][]{wandel99, kaspi00}. If, instead, the BLR is described
by a circular rotating disk, then the value of $f$ can be
a few times larger than the isotropic case, depending on the
inclination angle and the scale height of the disk (see Collin et al.\
2006 for more details).
As mentioned above, determining the virial coefficient for individual
AGNs is not feasible due to the limited spatial resolution of the
current and foreseeable future instruments since the angular size of
the BLR is microarcseconds. Therefore, in practice an average virial
coefficient is usually determined by assuming that local Seyfert
galaxies and quasar host galaxies follow the same \ensuremath{\mbh-\sigma_*}\ relation
(Onken et al.\ 2004) and deriving the appropriate value for $f$.
Using this method, Onken et al. (2004) measured $\langle f
\rangle=5.5\pm1.8$ based on a sample of 14 reverberation-mapped
Seyfert galaxies with measured stellar velocity dispersions.
In this section we apply the same methodology to our enlarged sample
to determine the average value of $f$ as well as the intrinsic
scatter, assuming the intercept ($\alpha$) and slope ($\beta$) of the
\ensuremath{\mbh-\sigma_*}\ relation for quiescent galaxies. In practice, we modify the
likelihood given in Equation~\ref{eq:likeli} by fixing the slope and
intercept to the values obtained by two independent groups, Ferrarese
\& Ford (2005) and G\"ultekin et al. (2009). Figures 4 and 5 show the
best-fit \ensuremath{\mbh-\sigma_*}\ relations for the enlarged reverberation sample,
respectively assuming the quiescent \ensuremath{\mbh-\sigma_*}\ relation from Ferrarese
\& Ford (2005) and that from G\"ultekin et al. (2009). Adopting the
\ensuremath{\mbh-\sigma_*}\ relation of Ferrarese \& Ford (2005; $\alpha=8.22$ and
$\beta=4.86$), we determine $\log f=0.71 \pm 0.10$ and $\sigma_{\rm int} =
0.46^{+0.07}_{-0.09}$ dex. Adopting the \ensuremath{\mbh-\sigma_*}\ relation from
G\"ultekin et al (2009; $\alpha=8.12$ and $\beta=4.24$), we obtain
$\log f=0.72^{+0.09}_{-0.10}$ and $\sigma_{\rm int}= 0.44\pm0.07$. The
average values are in good agreement with $f=5.5\pm1.8$ (i.e., $\log
f=0.74^{+0.10}_{-0.13}$) as found by Onken et al.\ (2004). The average
value is inconsistent with that expected for a spherical BLR ($f=3$),
and closer to that expected for a disk-like geometry, although there
may be a diversity of geometries and large-scale kinematics (e.g.,
Bentz et al.\ 2009b; Denney et al.\ 2009).
\begin{figure}
\epsscale{1.8}
\plottwo{f4a.eps}{f4b.eps}
\caption{{\it Top:} \ensuremath{\mbh-\sigma_*}\ relation of the combined reverberation sample,
adopting the slope from Ferrarese \& Ford (2005) and the best-fit
virial coefficient from this work. {\it Bottom:} Posterior probability
contour levels (68\% and 95\%) for the virial coefficient $f$ and
its intrinsic scatter, as determined assuming the slope from
Ferrarese \& Ford (2005). The horizontal lines represent the
previous estimate from Onken et al. (2004; dashed line) with its
68\% uncertainties (dotted lines).}
\label{fig:FF05}
\end{figure}
As discussed at the end of \S~4.1, the intrinsic scatter determined in
this section is a combination of intrinsic scatter in $f$ and in zero
point of the \ensuremath{\mbh-\sigma_*}\ relation (which are degenerate in Equation
4). We need additional information to break this degeneracy. If the
intrinsic scatter of the active galaxy \ensuremath{\mbh-\sigma_*}\ relation were close to
0.31 dex (as determined for elliptical galaxies by G\"ultekin et al.\
2009), then the residual intrinsic scatter of the virial coefficient would be
0.31 dex. In contrast, if the scatter of the \ensuremath{\mbh-\sigma_*}\ relation of
active galaxies were closer to 0.44 dex (as found for all galaxies by
G\"ultekin et al.\ 2009), then the intrinsic scatter in $f$ would be
close to zero. Although we cannot break this degeneracy with current
data, the bottom line is the same as far as determining black hole
masses from reverberation mapping data is concerned (and from
single-epoch data, as a consequence). Since for the purpose of the
calibration of $f$ the two sources of scatter are indistinguishable,
the uncertainty in our calibration of $f$ (whether it is due to
diversity in BLR physics or diversity in the galaxy-AGN
connection) is the largest remaining uncertainty in black hole mass
determinations that rely on this technique.
\begin{figure}
\epsscale{1.8}
\plottwo{f5a.eps}{f5b.eps}
\caption{As in figure~\ref{fig:FF05}, but adopting the slope from
G\"ultekin et al.\ (2009).}
\label{fig:G09}
\end{figure}
To investigate any dependence of the virial coefficient on AGN
properties, we collected from the literature the H$\beta$ line widths
($V_{\rm FWHM}$ and $\sigma_{\rm line}$) and the optical AGN
luminosities for reverberation-mapped AGNs, corrected for the
host-galaxy contribution (Peterson et al. 2004; Bentz et al. 2009a). When
multiple luminosity measurements are available for given objects, we
used the weighted mean of the luminosity (Table 9 of Bentz et
al. 2009a). For the seven LAMP objects, with \ensuremath{M_\mathrm{BH}}\ and velocity
dispersion measurements, {\it Hubble Space Telescope}
imaging is not yet available for
measuring the starlight correction to the spectroscopic luminosity.
Thus we inferred the AGN luminosity from the measured time lag using
the size-luminosity relation (Bentz et al. 2009a). The bolometric
luminosity was calculated by multiplying the optical AGN luminosity by
a factor of ten (Woo \& Urry 2002), and we assumed 0.3 dex error in
the Eddington ratio.
In Figure 6, we plot the residuals from the \ensuremath{\mbh-\sigma_*}\ relation
with $\alpha=8.12$, $\beta=4.24$ taken from G\"ultekin et al.\ (2009),
and $\log f =0.72$ as determined in \S 6.2.
The residuals were determined as $\Delta \log \ensuremath{M_\mathrm{BH}} = \log VP + \log f
- (\alpha + \beta \log \sigma_*)$. To test for potential correlations
between residuals and line properties or Eddington ratios, we fit each
data set with a straight line by minimizing the normalized $\chi^2$
estimator. We find no dependence of the residuals on the line width
(either $V_{\rm FWHM}$ or $\sigma_{\rm line}$). In the case of the
Eddington ratio, we find a weak dependence with a non-zero slope
$(-2.3^{+0.6}_{-1.1}$), significant at the 95\% level; however this
weak trend is mainly due to one object with the lowest Eddington ratio
in the sample. When we remove that object, the best fit slope becomes
consistent with no correlation. Further investigation using a larger
sample, evenly distributed over the range of the Eddington ratio, is
required to better constrain the residual dependence on the Eddington
ratio. In the case of the line width, it is not straightforward to
test the dependence since the line width of most of our objects is
relatively small, $\sigma_{\rm line} < 2000$ \kms. By dividing our
sample at $\sigma_{\rm line}= 1500$ \kms\ into two groups of similar
sample size, we separately measured the virial coefficient for
narrower-line and broader-line AGNs. The difference in the virial
coefficient is $\Delta \log f = 0.1-0.2$, which is not significant
given the uncertainty of 0.15 dex on the virial coefficient.
\begin{figure}
\epsscale{1.2}
\plotone{f6.eps}
\caption{Dependence of residuals from the \ensuremath{\mbh-\sigma_*}\ relation ($\Delta
\log \ensuremath{M_\mathrm{BH}} = (\log \mathrm{VP} + \log f) - (\alpha +\beta \log
\sigma/(200~ \mathrm{\kms})$) on parameters related to the accretion
state: $V_{\rm FWHM}/\sigma_{\rm line}$ (top left); Eddington ratio
(top right); $V_{\rm FWHM}$ (bottom left); line dispersion
$\sigma_{\rm line}$ (bottom right) of the H$\beta$ line.
In this plot we adopt the local relation with $\alpha=8.12$,
$\beta=4.24$ taken from G\"ultekin et al.\ (2009),
and $\log f =0.72$ as determined in \S 6.2.}
\label{fig:residual_F_g09}
\end{figure}
\section{Discussion}
\begin{figure*}
\epsscale{1.0}
\plotone{f7.eps}
\caption{The \ensuremath{\mbh-\sigma_*}\ relation of active galaxies with reverberation
black hole masses (blue), compared with non-active galaxies with
dynamical black hole masses. The reverberation masses were
determined assuming the virial coefficient, $\log f=0.72\pm0.10$.
The solid line is the best-fit slope of the active galaxies while
the dashed line is the best fit to the inactive
galaxy samples from G\"ultekin et al. (2009).}
\label{fig_msigma}
\end{figure*}
We present the \ensuremath{\mbh-\sigma_*}\ relation of the reverberation sample in Figure
7, using the slope, $\beta=3.55\pm0.60$, determined in \S 4.1 and the
average virial coefficient, $\langle\log f\rangle=0.72\pm0.10$,
determined in \S 4.2.
Compared to the local quiescent galaxies, active galaxies follow a
consistent \ensuremath{\mbh-\sigma_*}\ relation with a similar slope and scatter. Note
that the mean black hole mass of the reverberation sample
($\langle\log \ensuremath{M_\mathrm{BH}}/\ensuremath{{\rm M}_{\odot}} \rangle=7.3\pm0.74$) is an order of magnitude
smaller than that of quiescent galaxies ($\langle\log
\ensuremath{M_\mathrm{BH}}/\ensuremath{{\rm M}_{\odot}} \rangle=8.2\pm0.79$). The slightly shallower slope of the
reverberation sample is consistent with the trend in the quiescent
galaxies that the slope is shallower for galaxies with lower velocity
dispersion ($\sigma_* < 200$ \kms); see G\"ultekin et al. (2009). In
contrast, the slope of late-type quiescent galaxies ($4.58\pm1.58$)
seems higher than the slope of our active galaxies, which are mainly
late-type galaxies. However, given the uncertainty in the slope
($\beta=3.55\pm0.60$) of the active galaxy \ensuremath{\mbh-\sigma_*}\ relation, the
difference between quiescent and active galaxies is only marginal. We
did not attempt to divide our sample into various morphology groups or
a few mass bins to test the dependence of the slope, since the sample
size is still small and biased toward lower mass objects.
The smaller mean \ensuremath{M_\mathrm{BH}}\ of the reverberation sample is due to the
relative difficulty of measuring the stellar velocity dispersions
for the more luminous and higher redshift quasars that have the
largest black hole masses.
Although there are 17 quasars with measured reverberation
black hole masses, the stellar velocity dispersions have not been measured
for most of those high-luminosity objects
(see Watson et al. 2008; Dasyra et al. 2007). In our current sample,
there are only three objects above $3\times 10^{8}$ \ensuremath{{\rm M}_{\odot}}. In order
to better constrain the \ensuremath{\mbh-\sigma_*}\ relation of active galaxies, it will
be necessary to pursue further measurements of velocity dispersions
for reverberation-mapped AGNs at high masses. Additional stellar
velocity dispersion measurements will be available based on Keck
OSIRIS spectra in the near future (Woo \& Malkan, in preparation).
The key assumption in determining the slope of the \ensuremath{\mbh-\sigma_*}\ relation
is that the virial coefficient does not systematically vary as a
function of black hole mass. Currently, we do not have sufficient
information to test whether this assumption is valid. In an empirical
study of the systematic effects of emission-line profiles, Collin et al. (2006) showed that it
was necessary to use a smaller value of the virial coefficient for
AGNs with larger $V_{\rm FWHM}$. In contrast, the change of the
virial coefficient was not required if the line dispersion
($\sigma_{\rm line}$) was used instead. These results reflect the
fact that there is a large range of $V_{\rm FWHM}/\sigma_{\rm line}$
ratio, and that systematic uncertainties are higher if $V_{\rm FWHM}$
is used.
With the enlarged sample of 24 AGNs, we investigated the dependence of
the virial coefficient on the line properties and the Eddington
ratio. However, we did not clearly detect any increasing trend of the
intrinsic scatter with the line profile, line width, or Eddington
ratio, implying that the \ensuremath{M_\mathrm{BH}}\ based on the mean virial coefficient is
not systematically over- or under-estimated as a function of line width or
the Eddington ratio. We note that since our sample is still small and
biased to narrower-line and lower Eddington ratio objects, further
investigation with a larger sample, particularly at higher mass
scales, is required to better constrain the systematic errors.
For single-epoch black hole mass determination, we suggest using $\log f=
0.72\pm0.10$ when the gas velocity is determined from the line dispersion
($\sigma_{\rm line}$).
If the FWHM of an emission line ($V_{\rm FWHM}$) is
measured instead of the line dispersion, then additional
information on the $V_{\rm FWHM}/\sigma_{\rm line}$ ratio is needed in
the black hole mass calculation.
The value of $V_{\rm FWHM}/\sigma_{\rm line}$ is typically
assumed to be 2 (Netzer 1990), although there is a range of
values. For example, the mean of $V_{\rm FWHM}/\sigma_{\rm line}$ of our reverberation
sample is $2.09$ with a standard deviation of $0.45$ (see also Peterson et
al. 2004 and McGill et al.\ 2008).
We did not attempt to determine an average virial coefficient using
the virial product based on $V_{\rm FWHM}$.
A correction factor depending on $V_{\rm FWHM}/\sigma_{\rm line}$ should be applied in addition to
the virial coefficient, $\log f= 0.72$, when gas velocity is determined from $V_{\rm FWHM}$.
Our empirical measurement of the mean virial coefficient
is not very different from that of Onken et al. (2004) although the sample size
and the dynamical range of the reverberation sample is significantly improved.
We note that our results on the mean virial coefficient are based on broad-line
dispersions measured from the rms spectra,
which are not generally available for single-epoch black hole mass determinations.
Detailed comparison of broad-line profiles and widths between rms and single-epoch spectra
is necessary to understand the additional uncertainty of single-epoch black hole mass estimates
(see Denney et al. 2009), and we will pursue further investigations of
this issue using LAMP data in the future.
\section{Summary}
We summarize the main results as follows.
\smallskip
(1) Using the high S/N optical and near-IR spectra obtained at
the Keck, Palomar, and Lick Observatories, we measured the stellar
velocity dispersion of 10 local Seyfert galaxies, including 7 objects
(SBS 1116+ 583A, Arp 151, Mrk 1310, Mrk 202, NGC 4253, NGC 4748, NGC
6814) with newly determined reverberation black hole masses from the
LAMP project (and NGC 5548, IC 4218 and MCG--6-30-15).
\smallskip
(2) For a total sample of 24 local AGNs, combining our new
stellar velocity dispersion measurements of the LAMP sample and the
previous reverberation sample with the stellar velocity dispersion
from the literature, we determined the slope and the intrinsic scatter
of the \ensuremath{\mbh-\sigma_*}\ relation in the range of black hole mass
$10^{6}<\ensuremath{M_\mathrm{BH}}/\ensuremath{{\rm M}_{\odot}}<10^{9}$. The best-fit slope is
$\beta=3.55\pm0.60$, consistent within the uncertainty with the slope
of quiescent galaxies. The intrinsic scatter, $0.43\pm 0.08$ dex, of
active galaxies is also similar to that of quiescent galaxies. Thus,
we find no evidence for dependence of the present-day \ensuremath{\mbh-\sigma_*}\ relation
slope on the level of activity of the central black hole.
\smallskip
(3) We determined the virial coefficient using the slope and the
intercept of the \ensuremath{\mbh-\sigma_*}\ relation of quiescent galaxies, taken from
two groups (Ferrarese \& Ford 2005; G\"ultekin et al.\ 2009). The
best-fit virial coefficient ($\log f$) is $0.71\pm0.10$
($0.72^{+0.09}_{-0.10}$) with an intrinsic scatter,
$0.46^{+0.07}_{-0.09}$ ($0.44\pm0.07$) with the slope, 4.86 (4.24)
taken from Ferrarese \& Ford (G\"ultekin et al. 2009). We take
$f=5.2$ (i.e., $\log f=0.72$) as the best value of the mean virial
coefficient, which is a factor of $\sim$1.7 larger than the standard
factor obtained for isotropic spatial and velocity distribution. The
substantial intrinsic scatter indicates that the virial coefficient is
the main source of uncertainties in determining black hole masses,
either using reverberation mapping or single-epoch spectra.
\acknowledgments
J.H.W. acknowledges support from Seoul National University by the
Research Settlement Fund for new faculty, and support from NASA
through Hubble Fellowship grant HF-51249 awarded by the Space
Telescope Science Institute. T.T. acknowledges support from the NSF
through CAREER award NSF-0642621, by the Sloan Foundation through a
Sloan Research Fellowship, and by the Packard Foundation through a
Packard Fellowship. Research by A.J.B. and A.V.F. is supported by NSF
grants AST-0548198 and AST-0908886, respectively. The work of
D.S. was carried out at Jet Propulsion Laboratory, California
Institute of Technology, under a contract with NASA.
Research by G. C. is supported by NSF grant AST-0507450.
We thank the referee for useful suggestions.
Data presented herein were obtained at the W. M. Keck Observatory,
which is operated as a scientific partnership among Caltech, the
University of California, and NASA; it was made possible by the
generous financial support of the W. M. Keck Foundation. The authors
wish to recognize and acknowledge the very significant cultural role
and reverence that the summit of Mauna Kea has always had within the
indigenous Hawaiian community; we are most fortunate to have the
opportunity to conduct observations from this mountain. We are
grateful for the assistance of the staffs at the Keck, Palomar, and
Lick Observatories. We thank Kartik Sheth for assisting with some of
the Palomar observations. We acknowledge the usage of the HyperLeda
database (http://leda.univ-lyon1.fr). This research has made use of
the NASA/IPAC Extragalactic Database (NED) which is operated by the
Jet Propulsion Laboratory, California Institute of Technology, under
contract with the National Aeronautics and Space Administration.
|
2,877,628,089,052 | arxiv | \section{Introduction}\label{sec1}
In \cite{Zha1996}, Zha presents a bidiagonalization algorithm for computing a few extreme generalized singular values and vectors of a large sparse or structured matrix pair $\{A,L\}$ \cite{Paige1981, Van1976, Van1985}, where $A\in\mathbb{R}^{m\times n}$ and $L\in\mathbb{R}^{p\times n}$. Based on Zha's algorithm, Kilmer \textit{et al.} \cite{Kilmer2007} develop a joint bidiagonalization process that successively reduces $\{A,L\}$ to lower and upper bidiagonal forms simultaneously, which can be used to
compute extreme generalized singular values and vectors of a matrix pair or solve large scale ill-posed problems with general form Tikhonov regularization \cite{Hansen1998,Kilmer2007,Hansen2010,Jia2018}.
Consider the compact $QR$ factorization of the stacked matrix:
\begin{equation}\label{1.1}
\begin{pmatrix}
A \\
L
\end{pmatrix} = QR =
\begin{pmatrix}
Q_{A} \\
Q_{L}
\end{pmatrix}R ,
\end{equation}
where $Q \in \mathbb{R}^{(m+p)\times n}$ is column orthonormal and $R\in \mathbb{R}^{n\times n}$ is upper triangular. We partition $Q$ such that $Q_{A}\in\mathbb{R}^{m\times n}$ and $Q_{L}\in\mathbb{R}^{p\times n}$, and thus we have $A= Q_{A}R$ and $L = Q_{L}R$. Applying the BIDIAG-1 algorithm and BIDIAG-2 algorithm \cite{Paige1982}, which correspond to the lower and upper Lanczos bidiagonalizations, to $Q_{A}$ and $Q_{L}$, respectively, we can reduce $Q_{A}$ and $Q_{L}$ to the following lower and upper bidiagonal matrices respectively:
\begin{equation}\label{1.2}
B_{k}=\begin{pmatrix}
\alpha_{1} & & & \\
\beta_{2} &\alpha_{2} & & \\
&\beta_{3} &\ddots & \\
& &\ddots &\alpha_{k} \\
& & &\beta_{k+1}
\end{pmatrix}\in \mathbb{R}^{(k+1)\times k},
\widehat{B}_{k}=\begin{pmatrix}
\hat{\alpha}_{1} &\hat{\beta}_{1} & & \\
&\hat{\alpha}_{2} &\ddots & \\
& &\ddots &\hat{\beta}_{k-1} \\
& & &\hat{\alpha}_{k}
\end{pmatrix}\in \mathbb{R}^{k\times k} .
\end{equation}
The processes produce four column orthonormal matrices, that is
\begin{equation}\label{1.3}
U_{k+1}=(u_{1},\cdots,u_{k+1}) \in \mathbb{R}^{m\times (k+1)} , \ \
V_{k}=(v_{1},\cdots,v_{k}) \in \mathbb{R}^{n\times k}
\end{equation}
computed by the BIDIAG-1 algorithm, and
\begin{equation}\label{1.4}
\widehat{U}_{k}=(\hat{u}_{1},\cdots,\hat{u}_{k}) \in \mathbb{R}^{p\times k}, \ \
\widehat{V}_{k}=(\hat{v}_{1},\cdots,\hat{v}_{k}) \in \mathbb{R}^{n\times k}
\end{equation}
computed by the BIDIAG-2 algorithm.
In order to join BIDIAG-1 and BIDIAG-2, the starting vector of BIDIAG-2 is
chosen to be $\hat{v}_{1}= v_{1}$ and the upper bidiagonalization of $Q_{L}$ continues. It has been proved in \cite{Zha1996,Kilmer2007} that the Lanczos vector $\hat{v}_{i}$ and the element $\hat{\beta}_{i}$ of $\widehat{B}_{k}$ can be computed by using the following relations:
\begin{equation}\label{1.5}
\hat{v}_{i+1} = (-1)^{i}v_{i+1} , \ \
\hat{\beta}_{i} = \alpha_{i+1}\beta_{i+1}/\hat{\alpha}_{i} .
\end{equation}
For large scale matrices $A$ and $L$, the explicitly $QR$ factorization \eqref{1.1} can be avoided by iteratively solving a least squares problem with $(A^{T}, L^{T})^{T}$ as the coefficient matrix at each iteration. Through the above modifications, we
obtain the JBD process which can efficiently reduce a large scale matrix pair $\{A,L\}$
to a bidiagonal matrix pair $\{B_{k}, \widehat{B}_{k}\}$. For details of the derivation of the algorithm, see \cite{Zha1996,Kilmer2007}. For the $k$-step JBD process, it explicitly computes three orthonormal matrices $U_{k+1}$, $\widetilde{V}_{k}$, $\widehat{U}_{k}$, a lower bidiagonal matrix $B_{k}$ and an upper bidiagonal matrix $\widehat{B}_{k}$. The two column orthonormal matrices $V_{k}$ and $\widehat{V}_{k}$ can be obtained from $\widetilde{V}_{k}$ implicitly by letting $V_{k}=Q^{T}\widehat{V}_{k}$ and $\widehat{V}_{k}=V_{k}P$, where $ P=diag(1,-1,\dots ,(-1)^{k-1})_{k \times k}$.
In exact arithmetic, the JBD process of $\{A, L\}$ is equivalent to the two joint lower and upper Lanczos bidiagonalizations. The process of computing $U_{k+1}$, $V_{k}$ and $B_{k}$ is actually the lower Lanczos bidiagonalization of $Q_{A}$, while the process of computing $\widehat{U}_{k}$, $\widehat{V}_{k}$ and $\widehat{B}_{k}$ is the upper Lanczos bidiagonalization of $Q_{L}$. Therefore, $B_{k}$ is the Ritz-Galerkin projection of $Q_{A}$ on the subspaces $span(U_{k+1})$ and $span(V_{k})$, while $\widehat{B}_{k}$ is the Ritz-Galerkin projection of $Q_{L}$ on the subspaces $span(\widehat{U}_{k})$ and $span(\widehat{V}_{k})$, where $span(\cdot)$ denotes the subspace spanned by the columns of a matrix. Therefore, the extreme generalized singular values and vectors of $\{A,L\}$ can be approximated by using the singular value decomposition(SVD) of $B_{k}$ or $\widehat{B}_{k}$, which can be achieved by a direct method since the two matrices are small scale.
Denote the roundoff unit by $\epsilon$. In the presence of rounding errors, an important problem is whether the process of computing $U_{k+1}$, $V_{k}$ and $B_{k}$ is equivalent to the lower Lanczos bidiagonalization of $Q_{A}$ with rounding error $O(\epsilon)$, and whether the process of computing $\widehat{U}_{k+1}$, $\widehat{V}_{k}$ and $\widehat{B}_{k}$ is equivalent to the upper Lanczos bidiagonalization of $Q_{L}$ with rounding error $O(\epsilon)$. Up to now, there is no research on the finite precision behavior of the JBD process. One aim of this paper is to make a rounding error analysis of the JBD process and give some important properties of it in finite precision arithmetic. We will show that the above equivalence will not hold in finite precision arithmetic any longer, and investigate what a role the rounding errors play in the loss of this equivalence and how
the rounding errors be amplified.
On the other hand, due to the influence of rounding errors, the orthogonality of Lanczos vectors computed by the JBD process will be gradually lost. This is a typical phenomenon appeared in the Lanczos-type algorithms, which is first observed in the symmetric Lanczos process \cite{Lanczos1950}. The loss of orthogonality of Lanczos vectors will lead to a delay of convergence in the computation of some extreme eigenvalues \cite{Paige1971, Paige1972, Paige1980, Meurant2006}, and sometimes it is also difficult to determine whether some computed approximations are additional copies or genuine close eigenvalues \cite{Paige1971, Paige1972, Paige1976, Paige1980}. Some reorthogonalization strategies are proposed to maintain some level of orthogonality in order to preserve convergence of Ritz values \cite{Parlett1979,Simon1984a,Simon1984b}. Especially, Simon \cite{Simon1984a} proves that semiorthogonality of Lacnzos vectors is enough to guarantee the accuracy of the computed quantities up to $O(\epsilon)$ and avoid spurious eigenvalues from appearing. The above results of the symmetric Lanczos process have been adapted to handle the Lanczos bidiagonalization by Larsen, and he proposes an efficient partial reorthogonalization strategy for the Lanczos bidiagonalization \cite{Larsen1998}. Later in \cite{Simon2000}, Simon and Zha propose a one-sided reorthogonalization strategy. A backward error analysis of the Lanczos bidiagonalization with one-sided reorthogonalization has been made by Barlow in \cite{Barlow2013}, which shows that the process of a matrix $C$ in finite precision arithmetic produces Krylov subspaces generated by a nearby matrix $C+\delta C$.
The above results can be used to analyze the convergence and accuracy of the approximate generalized values computed by using the JBD process. First, we investigate the loss of orthogonality of computed $U_{k+1}$, $\widetilde{V}_{k}$ and $\widehat{U}_{k}$ in finite precision arithmetic. We will show that the orthogonality levels of $U_{k+1}$, $\widetilde{V}_{k}$ and $\widehat{U}_{k}$ are closely related to each other. Especially, we derive an upper bound for the orthogonality level of $\widehat{U}_{k}$, which depends on not only the orthogonality levels of $\widetilde{V}_{k}$ and $\widehat{U}_{k}$, but also a gradually growing quantity $\|\widehat{B}_{k}^{-1}\|$. This result implies that as long as $\widehat{B}_{k}$ is not very ill-conditioned, the orthogonality of $\widehat{U}_{k}$ will not be too bad if we can maintain some levels of orthogonality of $U_{k+1}$ and $\widetilde{V}_{k}$. Therefore, when designing a reorthogonalization strategy for the JBD process, one only need to reorthogonalize $u_{i}$ and $\tilde{v}_{i}$ but not $\hat{u}_{i}$, which can save much reorthogonalization work.
Second, we make a detailed investigation on the convergence and accuracy of the approximate generalized values that can be computed by using the SVD of $B_{k}$ or $\widehat{B}_{k}$. Similar to the Lanczos bidiagonalization, the loss of orthogonality of Lanczos vectors computed by the JBD process leads to a delay of the convergence of Ritz values and the appearance of spurious generalized singular values. We show that we can always approximate the extreme generalized singular values by using the SVD of $B_{k}$ and semiorthogonality of Lacnzos vectors is enough to avoid spurious generalized singular values from appearing. This result is a guidance for designing an efficient semiorthogonalization strategy for the JBD process, which will be proposed in our following work. Besides, we make a investigation on the residual norm proposed by Zha \cite{Zha1996} for measuring the accuracy of approximated generalized singular values and vectors. In the presence of rounding errors, we give an upper bound of the residual norm and we show that this upper bound can be used as a stopping criterion for computing generalized singular values and vectors. Moreover, we make a brief discussion about the accuracy of the computed generalized singular vectors.
The paper is organized as follows. In Section \ref{sec2}, we review the GSVD of $\{A,L\}$ and describe the JBD process with some basic properties in exact arithmetic. In Section \ref{sec3}, we make a rounding error analysis of the JBD process in finite precision arithmetic. We establish connections between the JBD process and the two joint Lanczos bidiagonalizations, and we also investigate the loss of orthogonality of the computed Lanczos vectors. In Section \ref{sec4}, we show how to use the JBD process to compute some extreme generalized singular values and vectors of $\{A, L\}$ and discuss the convergence, accuracy and stopping criterion of the GSVD computation. In Section \ref{sec5}, we use some numerical examples to illustrate our results. Finally, we conclude the paper in Section \ref{sec6}.
Throughout the paper, we denote by $I_k$ the identity matrix of order $k$, by $0_k$ and $0_{k\times l}$ the zero matrices of order $k$ and $k\times l$, respectively. The subscripts are omitted when there is no confusion. The transpose of a matrix $C$ is denoted by $C^{T}$. The roundoff unit is denoted by $\epsilon$. The norm $\|\cdot \|$ always means the spectral or 2-norm of a matrix or vector.
\section{GSVD, joint bidiagonalization and Lanczos bidiagonalization}\label{sec2}
In this section, we provide some necessary background. We describe the GSVD, the joint bidiagonalization process and their basic properties. We also review some important properties of Lanczos bidiagonalization.
We first look at the GSVD of the matrix pair $\{A, L\}$. The compact $QR$ factorization of $(A^{T}, L^{T})^{T}$ is defined as in \eqref{1.1}. Let
\begin{equation}\label{2.1}
Q_{A} = P_{A}C_{A}W^{T} , \ \ Q_{L} = P_{L}S_{L}W^{T}
\end{equation}
be the $CS$ decomposition of the matrix pair $\{Q_{A}, Q_{L} \}$ \cite[\S 2.5.4]{Golub2013}, where $P_{A}\in \mathbb{R}^{m\times m}$, $P_{L}\in \mathbb{R}^{p\times p}$ and $W\in\mathbb{R}^{n\times n}$
are orthogonal matrices, and $C_{A}\in\mathbb{R}^{m\times n}$ and $S_{L}\in\mathbb{R}^{p\times n}$ are diagonal matrices(not necessarily square) satisfying
$C_{A}^{T}C_{A}+S_{L}^{T}S_{L}=I_{n}$.
Suppose that $Rank((A^{T}, L^{T})^{T})=r$ where $Rank(\cdot)$ denotes the rank of a matrix. It is shown by Paige and Saunders in \cite{Paige1981} that we can write $C_{A}$ and $S_{L}$ as
\begin{equation*}
C_{A} =
\bordermatrix*[()]{%
\Sigma_{A}, & 0 & m \cr
r & n-r \cr
} \ , \ \ \ \
S_{L} = \bordermatrix*[()]{%
\Sigma_{L}, & 0 & p \cr
r & n-r \cr
} \ ,
\end{equation*}
where
\begin{equation*}
\Sigma_{A} =
\bordermatrix*[()]{%
I_{q} & & & q \cr
& C_{l} & & l \cr
& & O & m-q-l \cr
q & l & r-q-l
} \ , \ \ \ \
\Sigma_{L} =
\bordermatrix*[()]{%
O & & & p-r+q \cr
& S_{l} & & l \cr
& & I_{t} & r-q-l \cr
q & l & r-q-l
} \ .
\end{equation*}
If we write $C_{l}=diag(c_{q+1}, \dots, c_{q+l})$, \ $c_{q+1}\geq \cdots \geq c_{q+l}>0$ and $S_{l}=diag(s_{q+1}, \dots, s_{q+l})$, \ $0<s_{q+1}\leq \cdots \leq s_{q+l}$, then $c_{i}^{2}+s_{i}^{2}=1, \ i=q+1, \dots, q+l$, and the generalized singular values of $\{A, L\}$ are
\begin{equation*}
\underbrace{\infty, \dots, \infty}_{q}, \ \ \underbrace{c_{q+1}/s_{q+1}, \dots, c_{q+l}/s_{q+l}}_{l}, \ \ \underbrace{0, \dots, 0}_{t} \ ,
\end{equation*}
where $t=r-q-l$.
To simplify the presentation, throughout this paper we assume that $(A^{T}, L^{T})^{T}$ is of full column rank, and thus the GSVD of $\{A, L\}$ is
\begin{equation}\label{2.2}
A = P_{A}C_{A}G^{-1} , \ \ L = P_{L}S_{L}G^{-1}
\end{equation}
with $G=R^{-1}W\in\mathbb{R}^{n\times n}$, where the invertibility of $G$ follows from
the assumption that $(A^{T}, L^{T})^{T}$ has full rank.
\begin{algorithm}[htb]
\caption{$k$-step Joint bidiagonalization(JBD) process }
\begin{algorithmic}[1]\label{alg1}
\STATE {Choosing a nonzero starting vector $b \in \mathbb{R}^{m}$,
let $\beta_{1}u_{1}=b,\ \beta_{1}=\| b\|$ }
\STATE {$\alpha_{1}\tilde{v}_{1}=QQ^{T}\begin{pmatrix}
u_{1} \\
0_{p}
\end{pmatrix} $}
\STATE { $\hat{\alpha}_{1}\hat{u}_{1}=\tilde{v}_{1}(m+1:m+p) $}
\FOR{$i=1,2,\ldots,k,$}
\STATE $\beta_{i+1}u_{i+1}=\tilde{v}_{i}(1:m)-\alpha_{i}u_{i} $
\STATE $ \alpha_{i+1}\tilde{v}_{i+1}=
QQ^{T}\begin{pmatrix}
u_{i+1} \\
0_{p}
\end{pmatrix}-\beta_{i+1}\tilde{v}_{i} $
\STATE $\hat{\beta}_{i}=(\alpha_{i+1}\beta_{i+1})/\hat{\alpha}_{i} $
\STATE $\hat{\alpha}_{i+1}\hat{u}_{i+1}=
(-1)^{i}\tilde{v}_{i+1}(m+1:m+p)-\hat{\beta}_{i}\hat{u}_{i} $
\ENDFOR
\end{algorithmic}
\end{algorithm}
We now review the joint bidiagonalization process, which is described in Algorithm \ref{alg1}. At each iteration $i=1,2,\ldots,k+1$, Algorithm~\ref{alg1} needs to compute $QQ^{T} \begin{pmatrix}
u_i \\ 0_p
\end{pmatrix}$, which is not accessible since $Q$ is not available. Let $\tilde{u}_i=\begin{pmatrix}
u_i \\ 0_p
\end{pmatrix}$. Notice that $QQ^T\tilde{u}_i$ is nothing but the orthogonal projection
of $\tilde{u}_i$ onto the column space of
$\begin{pmatrix} A \\ L
\end{pmatrix}$, which means that $QQ^T\tilde{u}_i=\begin{pmatrix}
A \\ L
\end{pmatrix}\tilde{x}_i$, where
\begin{equation}\label{2.3}
\tilde{x}_i=\arg\min_{\tilde{x}\in \mathbb{R}^n}
\left\|\begin{pmatrix}
A \\ L
\end{pmatrix}
\tilde{x}-\tilde{u}_i\right\|.
\end{equation}
The large scale least squares problem \eqref{2.3} can be solved by an iterative solver, e.g., the most commonly used LSQR algorithm \cite{Paige1982}.
In exact arithmetic, the $k$-step JBD process produces two bidiagonal matrices $B_{k}$, $\widehat{B}_{k}$ and three column orthonormal matrices $U_{k+1}$, $\widehat{U}_{k}$ and
\begin{align}\label{2.4}
\widetilde{V}_{k}=(\tilde{v}_{1},\cdots,\tilde{v}_{k}) \in \mathbb{R}^{n\times k}
\end{align}
satisfying $\tilde{v}_{i}=Qv_{i}$. The vectors $v_{i}=Q^{T}\tilde{v}_{i}$ and $\hat{v}_{i}=(-1)^{i-1}v_{i}$ can be obtained implicitly from $\tilde{v}_{i}$. The $k$-step JBD process can be written in matrix form:
\begin{align}
& (I_{m},0_{m\times p})\widetilde{V}_{k}=U_{k+1}B_{k} \label{2.5} , \\
& QQ^{T}
\begin{pmatrix}
U_{k+1} \\
0_{p\times (k+1)}
\end{pmatrix}
=\widetilde{V}_{k}B_{k}^{T}+\alpha_{k+1}\tilde{v}_{k+1}e_{k+1}^{T} \label{2.6} , \\
& (0_{p\times m},I_{p})\widetilde{V}_{k}P=\widehat{U}_{k}\widehat{B}_{k} \label{2.7} ,
\end{align}
where $ P=diag(1,-1,1,\dots ,(-1)^{k-1})_{k \times k}$, and $e_{k+1}$ is the $(k+1)$-th column of the identity matrix of order $k+1$.
It is shown in \cite{Kilmer2007} that in exact arithmetic the $k$-step JBD process satisfies
\begin{equation}\label{2.8}
AZ_{k} = U_{k+1}B_{k} , \ \ LZ_{k}=\widehat{U}_{k}\bar{B}_{k} ,
\end{equation}
where $Z_{k}=R^{-1}V_{k}=(z_{1},\dots,z_{k})$ and $\bar{B}_{k}=\widehat{B}_{k}P$. Therefore, one can use the SVD of $B_{k}$ and $\widehat{B}_{k}$ to approximate the generalized singular values and vectors of $\{A, L\}$. A detailed investigation on the GSVD computation of $\{A, L\}$ is made in Section \ref{sec4}. Also in \cite{Kilmer2007}, the authors prove that
\begin{equation}\label{2.9}
B_{k}^{T}B_{k}+P\widehat{B}_{k}^{T}\widehat{B}_{k}P=I_{k}.
\end{equation}
From \eqref{2.9} we know that the singular values of $\widehat{B}_{k}$ is determined by that of $B_{k}$. Therefore, if only some extreme generalized singular values of $\{A, L\}$ are needed, one only needs to compute the SVD of one of $B_{k}$ and $\widehat{B}_{k}$.
We mention that the above method for computing the GSVD of $\{A, L\}$ is actually an indirect procedure to compute the $CS$ decomposition \eqref{2.1} of $\{Q_{A},Q_{L}\}$, where the computation of $QR$ factors $Q$ and $R$ is avoided and all we need is an approximation to the orthogonal projection $QQ^{T}$, which can be accessed by solving \eqref{2.3} iteratively. In sparse matrix computation, if $R$ is computationally viable, there are other ways to deal with the pair $\{Q_{A},Q_{L}\}$, e.g., it is possible to use a multifrontal representation of
$Q=\begin{pmatrix}
Q_{A} \\ Q_{L}
\end{pmatrix}$
that is nearly as efficient as the representation for $R$ \cite{Lu1996}.
In exact arithmetic, one can verify that
\begin{align}
&Q_AV_k=U_{k+1}B_k,\ \ Q_A^TU_{k+1}=V_kB_k^T+\alpha_{k+1}v_{k+1}e_{k+1}^T,\label{2.10}\\
&Q_L\widehat{V}_k=\widehat{U}_k\widehat{B}_k,\ \ Q_L^T\widehat{U}_k
=\widehat{V}_k\widehat{B}_k^T+\hat{\beta}_k\hat{v}_{k+1}e_k^T , \label{2.11}
\end{align}
where $e_{k}$ the $k$-th column of the identity matrix of order $k$. Therefore, the JBD process of $\{A, L\}$ is equivalent to the combination of the lower and upper Lanczos bidiagonalizations of $Q_{A}$ and $Q_{L}$.
We now review some important properties of the (lower) Lanczos bidiagonalization of $Q_{A}$. The properties of the upper Lanczos bidiagonalization of $Q_{L}$ are the same and thus we omit them. For the $m\times n$ matrix $Q_{A}$, the Lanczos bidiagonalization process is described in Algorithm \ref{alg2}.
\begin{algorithm}[htb]
\caption{$k$-step Lanczos bidiagonalization of $Q_{A}$}
\begin{algorithmic}[1]\label{alg2}
\STATE {Choosing a starting vector $b\in \mathbb{R}^{m}$, let
$\beta_{1}=\| b \|, u_{1}=b/\beta_{1}$}
\STATE $r_{1}=Q_{A}^{T}u_{1}$, \ $\alpha_{1} = \| r_{1} \|$, \
$v_{1} = r_{1} / \alpha_{1}$
\FOR{$i=1,2,\cdots,k$}
\STATE $p_{i+1}=Q_{A}v_{i}-\alpha_{i}u_{i}$
\STATE $\beta_{i+1}=| p_{i+1}\|$, \
$u_{i+1}=p_{i+1}/\beta_{i+1}$
\STATE $r_{i+1}=Q_{A}^{T}u_{i+1}-\beta_{i+1}v_{i}$
\STATE $\alpha_{i+1}=\|r_{i+1} \|$, \
$v_{i+1}=r_{i+1}/\alpha_{i+1}$
\ENDFOR
\end{algorithmic}
\end{algorithm}
After $k$ steps, Algorithm \ref{alg2} reduces $Q_A$ to the bidiagonal matrix $B_{k}$, and it produces two orthonormal matrices $U_{k+1}$ and $V_{k}$. The $k$-step lower Lanczos bidiagonalization of $Q_{A}$ can be written in matrix form of \eqref{2.10}, while the $k$-step upper Lanczos bidiagonalization of $Q_{L}$ can be written in matrix form of \eqref{2.11}.
In finite precision arithmetic, the Lanczos vectors computed by the Lanczos bidiagonalization gradually lose their mutual orthogonality as the iteration number $k$ increases \cite{Golub1965,Larsen1998}. Following \cite{Larsen1998,Simon2000}, we can define the orthogonality level of Lanczos vectors as follows.
\begin{Def}\label{def2.1}
For a matrix $W_{k}=(w_{1}, \dots, w_{k})\in \mathbb{R}^{r\times k}$ with $\lVert w_{j}\lVert=1$, $j=1,\dots,k$, we call $\xi_{ij}^{w}=|w_{i}^{T}w_{j}|$ the orthogonality level among $w_{i}$ and $w_{j}$. We give two measures of the orthogonality level of $\{w_{1}, \dots, w_{k}\}$ or $W_{k}$:
\begin{align}
& \xi(W_{k})=\max_{1\leq i\neq j \leq k}\xi_{ij}^{w} , \label{2.12} \\
& \eta(W_{k}) = \| I_{k}-W_{k}^{T}W_{k}\| . \label{2.13}
\end{align}
\end{Def}
Notice that $\xi(W_{k})\leq\eta(W_{k})\leq k\xi(W_{k})$. In most occasions, these two quantities can be used interchangeably to measure the orthogonality level of Lanczos vectors. Let $\sigma_{i}(\cdot)$ and $\lambda_{i}(\cdot)$ be the $i$-th largest singular value and eigenvalue of a matrix respectively, then
$$\sigma_{1}^{2}(W_{k})
= \lambda_{1}(W_{k}^{T}W_{k})
= 1 + \lambda_{1}(W_{k}^{T}W_{k}-I_{k})
\leq 1 + \lVert I_{k}-W_{k}^{T}W_{k}\lVert ,$$
which leads to
\begin{equation}\label{2.14}
\|W_{k}\| \leq \sqrt{1 + \eta(W_{k})} .
\end{equation}
For Lanczos bidiagonalization, the loss of orthogonality of the Lanczos vectors will lead to appearance of spurious singular values and a delay of the convergence of Ritz values \cite{Larsen1998}. Therefore, reorthogonalization strategies are necessary to maintain some level of orthogonality in order to preserve convergence of Ritz values, see \cite{Parlett1979,Simon1984a,Simon1984b,Larsen1998,Simon2000,Barlow2013} for a few types of reorthogonalization strategies and related analysis.
\section{Joint bidiagonalization in finite precision arithmetic}\label{sec3}
When it is carried in finite precision arithmetic, by the influence of rounding errors, the behavior of the JBD process will deviate far from the ideal case in exact arithmetic. First, the JBD process of $\{A, L\}$ is not equivalent to the combination of the two Lanczos bidiagonalizations any longer. Second, orthogonalities of the three matrices $U_{k+1}$, $\widetilde{V}_{k}$ and $\widehat{U}_{k}$ will be lost gradually. In this paper, we do not consider the solution accuracy of the inner least squares problems \eqref{2.3}, although it has an important influence on the accuracy and efficiency of the algorithm. This issue is a bit more complicated, and we will study it in our future research work. In the following analysis, we always assume that \eqref{2.3} is solved accurately and thus the computed
$\begin{pmatrix}
A \\ L
\end{pmatrix}\tilde{x}_i$
is equal to the value of
$QQ^{T} \begin{pmatrix}
u_i \\ 0_p
\end{pmatrix}$
computed by explicitly using the strictly column orthonormal matrix $Q$.
First of all, we state a set of assumptions on the behavior of the rounding errors occurring in the JBD process. These assumptions apply to Lanczos-type algorithms, which are mainly from the results of rigorous analysis of the symmetric Lanczos process and Lanczos bidiagonalization. They constitute a model for the actual computation and include essential features while discard irrelevant ones. From now on, quantities such as $\alpha_{i}, \ \beta_{i}, \ u_{i}$ denote the computed ones in finite precision arithmetic. In the presence of rounding errors, the matrix form relations \eqref{2.5}-\eqref{2.7} should be rewritten by adding
rounding error terms \cite[\S 13.4]{Parlett1980}:
\begin{align}
& (I_{m},0_{m\times p})\widetilde{V}_{k}=U_{k+1}B_{k} + \widetilde{F}_{k}, \label{3.1} \\
& QQ^{T}
\begin{pmatrix}
U_{k+1} \\
0_{p\times (k+1)}
\end{pmatrix}=\widetilde{V}_{k}B_{k}^{T}+\alpha_{k+1}\tilde{v}_{k+1}e_{k+1}^{T}+\widetilde{G}_{k+1}, \label{3.2} \\
& (0_{p\times m},I_{p})\widetilde{V}_{k}P=\widehat{U}_{k}\widehat{B}_{k}+\bar{F}_{k} , \label{3.3}
\end{align}
where $ \widetilde{F}_{k}=(\tilde{f}_{1}, \dots, \tilde{f}_{k}), \ \widetilde{G}_{k+1} = (\tilde{g}_{1}, \dots, \tilde{g}_{k+1})$ and $\bar{F}_{k}=(\bar{f}_{1}, \dots, \bar{f}_{k})$ satisfying $\|\widetilde{F}_{k}\|, \|\widetilde{G}_{k+1}\|, \|\bar{F}_{k}\|=O(\epsilon)$ and $\tilde{f}_{i}, \ \tilde{g}_{i+1}$ and $\bar{f}_{i}$ are rounding error terms appeared in Algorithm \ref{alg1}. Second, the property of local orthogonality of $u_{i}$ and $\hat{u}_{i}$ holds, that is, locally the orthogonality levels of $u_{i}$ and $\hat{u}_{i}$ satisfy the following relations respectively \cite{Paige1976,Simon1984a}:
\begin{align}
& \beta_{i+1}|u_{i+1}^{T}u_{i}| = O(c_{1}(m,n)\epsilon) , \label{3.4} \\
& \hat{\alpha}_{i+1}|\hat{u}_{i+1}^{T}\hat{u}_{i}| = O(c_{2}(p,n)\epsilon) ,\label{3.5}
\end{align}
where $c_{1}(m,n)$ and $c_{2}(p,n)$ are moderate constants depend on $m$, $n$ and $p$. Finally, we assume that
\begin{equation}\label{3.6}
no \ \ \alpha_{i}, \ \beta_{i+1}, \ \hat{\alpha}_{i} \ and \ \hat{\beta}_{i} \ ever \ become \ negligible ,
\end{equation}
which is almost always true in practice, and the rare cases where $\alpha_{i}$, $\beta_{i+1}$, $\hat{\alpha}_{i}$ or $\hat{\beta}_{i}$ do become small are actually the lucky ones, since then the algorithm should be terminated, having found an invariant singular subspace. Besides, for simplicity, in our rounding error analysis, we always assume that the computed Lanczos vectors are of unit length.
\subsection{Connections between the JBD process and Lanczos bidiagonalization in finite precision arithmetic}
We now show that in the presence of rounding errors, the JBD process of $\{A, L\}$ is not equivalent to the combination of the lower and upper Lanczos bidiagonalizations of $Q_{A}$ and $Q_{L}$ any longer. We first present the following lemma.
\begin{Lem}\label{lem3.1}
Let $v_{i}=Q^{T}\tilde{v}_{i}$, $V_{k}=(v_{1},\dots,v_{k})$ and $\underline{B}_{k}=\begin{pmatrix}
B_{k-1}^{T} \\
\alpha_{k}e_{k}^{T}
\end{pmatrix}\in \mathbb{R}^{k\times k}$. Then
\begin{equation}\label{3.7}
\lVert \widetilde{V}_{k} - QV_{k} \lVert \leq \lVert \widetilde{G}_{k}\underline{B}_{k}^{-1}\lVert
= O(\lVert \underline{B}_{k}^{-1}\lVert\epsilon) .
\end{equation}
\end{Lem}
\begin{proof}
Let $Z=\begin{pmatrix}
A \\ L
\end{pmatrix}$. Then we have $QQ^{T}=ZZ^{\dag}$ and $QQ^{T}
\begin{pmatrix}
U_{k} \\
0
\end{pmatrix}=ZX_{k}$, where $``\dag"$ denotes the Moore-Penrose inverse of a matrix and $X_{k}=Z^{\dag}\begin{pmatrix}
U_{k} \\
0
\end{pmatrix}$.
From \eqref{3.2} we have $ ZX_{k} = \widetilde{V}_{k}\underline{B}_{k} + \widetilde{G}_{k}$ and thus $\widetilde{V}_{k}=ZX_{k}\underline{B}_{k}^{-1}-\widetilde{G}_{k}\underline{B}_{k}^{-1}$. Therefore,
\begin{align*}
\begin{split}
\widetilde{V}_{k} - QV_{k}
= \widetilde{V}_{k}-QQ^{T}QRX\underline{B}_{k}^{-1}+QQ^{T}\widetilde{G}_{k}\underline{B}_{k}^{-1}
= (QQ^{T}-I_{m+p})\widetilde{G}_{k}\underline{B}_{k}^{-1} ,
\end{split}
\end{align*}
and we finally obtain \eqref{3.7}.
\end{proof}
This lemma shows that $\widetilde{V}_{k}$ gradually deviates from the column space of $Q$ as the iterations progress, which is different from the ideal case in exact arithmetic that $\widetilde{V}_{k}=QV_{k}$. We can rewrite \eqref{3.1} as
\begin{align*}
\begin{split}
(I_{m},0_{m\times p})Q{V}_{k}
= U_{k+1}B_{k}+ F_{k} ,
\end{split}
\end{align*}
where $F_{k}=\widetilde{F}_{k}-(I_{m},0_{m\times p})(\widetilde{V}_{k} - QV_{k})$ and $\|F_{k}\|=O(\lVert \underline{B}_{k}^{-1}\lVert\epsilon)$. Multiply \eqref{3.2} by $Q^{T}$ from the left and combine with the above relation, we obtain the following result.
\begin{theorem} \label{thm3.1}
Suppose the inner least squares problem \eqref{2.3} is solved accurately. In finite precision arithmetic, we have
\begin{align}
& Q_{A}V_{k} = U_{k+1}B_{k}+ F_{k} , \label{3.8} \\
& Q_{A}^{T}U_{k+1}=V_{k}B_{k}^{T}+\alpha_{k+1}v_{k+1}e_{k+1}^{T}+G_{k+1} , \label{3.9}
\end{align}
where $F_{k}=(f_{1},\dots,f_{k})$, $G_{k+1}=(g_{1}, \dots, g_{k+1})=Q^{T}\widetilde{G}_{k+1}$, and $\lVert F_{k}\lVert= O(\lVert \underline{B}_{k}^{-1}\lVert\epsilon)$, $\lVert G_{k+1} \lVert = O(\epsilon)$.
\end{theorem}
This theorem implies that the error term $F_{k}$ is amplified gradually if $\lVert \underline{B}_{k}^{-1}\lVert$ grows bigger and bigger as the iterations progress. Therefore,
the process of computing $U_{k+1}$, $V_k$ and $B_k$ is not equivalent to the lower Lanczos
bidiagonalization of $Q_{A}$ with rounding error $O(\|Q_{A}\|\epsilon)=O(\epsilon)$ any longer.
In finite precision, the relation of the two computed quantities $B_{k}$ and $\widehat{B}_{k}$ is similar to \eqref{2.9}. Before giving the result, we first establish upper bounds of $\|B_{k}\|$ and $\|\widehat{B}_{k}\|$. From \eqref{3.1}, at the $i$-th step we have
\begin{align}
& \tilde{v}_{i}(1:m)=\alpha_{i}u_{i}+\beta_{i+1}u_{i+1}+\tilde{f}_i , \label{3.10} \\
& (-1)^{i-1}\tilde{v}_{i}(m+1:m+p)= \hat{\alpha}_{i}\hat{u}_{i}
+\hat{\beta}_{i-1}\hat{u}_{i-1}+\bar{f}_{i} . \label{3.11}
\end{align}
Thus $\| \alpha_{i}u_{i}+\beta_{i+1}u_{i+1}\|^{2} = \| \tilde{v}_{i}(1:m)-\tilde{f}_i\|^{2}$,
which leads to
\begin{align}\label{3.12}
\begin{split}
\alpha_{i}^{2}+\beta_{i+1}^2
&=\|\tilde{v}_{i}(1:m)\|^{2}+\| \tilde{f}_i\|^{2}-2\tilde{f}_i^{T}\tilde{v}_{i}(1:m)-2\alpha_{i}\beta_{i+1}u_{i+1}^{T}u_{i} \\
&\leq 1+O(c_{1}(m,n)\epsilon),
\end{split}
\end{align}
where we have used \eqref{3.4}. Similarly, we can get
\begin{equation}\label{3.13}
\hat{\alpha}_{i}^2 + \hat{\beta}_{i-1}^{2}\leq 1+O(c_{2}(p,n)\epsilon) .
\end{equation}
Therefore, we could obtain the upper bounds of $\|B_{k}\|$ and $\|\widehat{B}_{k}\|$\footnote{Here we use the result of an exercise from Higham's book \cite[Chapter6, Probelms 6.14]{Higham2002}, which gives the upper bound of the $p$-norm of a row/column sparse matrix.}:
\begin{align}
& \|B_{k}\| \leq \sqrt{2}\max_{1\leq i \leq k}(\alpha_{i}^{2}+\beta_{i+1}^{2})^{1/2}
\leq \sqrt{2}+O(c_{1}(m,n)\epsilon), \label{3.14} \\
& \|\bar{B}_{k}\| \leq \sqrt{2}\max_{1\leq i \leq k}(\hat{\alpha}_{i}^{2}+\hat{\beta}_{i-1}^{2})^{1/2}
\leq \sqrt{2}+O(c_{2}(p,n)\epsilon) \label{3.15} .
\end{align}
We are now ready to give the relation of $B_{k}$ and $\widehat{B}_{k}$ in finite precision arithmetic.
\begin{theorem}\label{thm3.2}
Following the hypothesis of Theorem \ref{thm3.1}. In finite precision arithmetic, we have
\begin{equation}\label{3.16}
B_{k}^{T}B_{k}+P\widehat{B}_{k}^{T}\widehat{B}_{k}P=I_{k} + E_{k } ,
\end{equation}
where $E_{k}$ is a symmetric tridiagonal matrix with bandwidth $1$, and the nonzero elements of $E_{k}$ are of $O(c_{3}(m,n,p) \epsilon)$ with $c_{3}(m,n,p)=c_{1}(m,n)+c_{2}(p,n)$.
\end{theorem}
\begin{proof}
Since
\begin{equation*}
B_{k}^{T}B_{k}=\begin{pmatrix}
\alpha_{1}^{2}+\beta_{2}^{2} &\alpha_{2}\beta_{2} & & \\
\alpha_{2}\beta_{2} &\alpha_{2}^{2}+\beta_{3}^{2} &\ddots & \\
&\ddots &\ddots & \alpha_{k}\beta_{k}\\
& &\alpha_{k}\beta_{k} &\alpha_{k}^{2}+\beta_{k+1}^{2}
\end{pmatrix},
\end{equation*}
\begin{equation*}
\widehat{B}_{k}^{T}\widehat{B}_{k}=\begin{pmatrix}
\hat{\alpha}_{1}^{2}&\hat{\alpha}_{1}\hat{\beta}_{1} & & \\
\hat{\alpha}_{1}\hat{\beta}_{1}&\hat{\alpha}_{2}^{2}+\hat{\beta}_{1}^{2} &\ddots & \\
&\ddots &\ddots &\hat{\alpha}_{k-1}\hat{\beta}_{k-1} \\
& &\hat{\alpha}_{k-1}\hat{\beta}_{k-1} &\hat{\alpha}_{k}^{2}+\hat{\beta}_{k-1}^{2}
\end{pmatrix},
\end{equation*}
nonzero elements in the left side of \eqref{3.16} are contained only in
the diagonal and subdiagonal parts.
For diagonal part, in finite precision arithmetic, we have $\hat{\beta}_{i}=(\alpha_{i+1}\beta_{i+1}/\hat{\alpha}_{i})
(1+\rho)$, where $|\rho|\leq \epsilon$
\cite[\S 2.2]{Higham2002}, and thus
$$\alpha_{i+1}\beta_{i+1} =\hat{\alpha}_{i}\hat{\beta}_{i}-\alpha_{i+1}\beta_{i+1}\rho .$$
From \eqref{3.12} we have
$$\alpha_{i+1}\beta_{i+1} \leq \frac{\alpha_{i+1}^{2}+\beta_{i+1}^2}{2}
\leq \frac{2[1+O(c_{1}(m,n)\epsilon)]}{2}
=1+O(c_{1}(m,n)\epsilon).$$
Therefore, we obtain
\begin{equation}\label{3.17}
\alpha_{i+1}\beta_{i+1}=\hat{\alpha}_{i}\hat{\beta}_{i} +\gamma_{i} ,
\end{equation}
where $|\gamma_{i}| \leq [1+O(c_{1}(m,n)\epsilon)]\epsilon = O(\epsilon)$.
For subdiagonal part, by taking norms of \eqref{3.10} and \eqref{3.11} we have
\begin{align*}
& \ \ \ \ \|\tilde{v}_{i}(1:m) \|^{2}+\|\tilde{v}_{i}(m+1:m+p)\|^{2} \\
&= \|\alpha_{i}u_{i}+\beta_{i+1}u_{i+1}+\tilde{f}_i\|^{2}+\|\hat{\alpha}_{i}\hat{u}_{i}
+\hat{\beta}_{i-1}\hat{u}_{i-1}+\bar{f}_{i}\|^{2} \\
&= \alpha_{i}^{2}+\beta_{i+1}^{2}+2\alpha_{i}\beta_{i+1}u_{i}^{T}u_{i+1}
+2\alpha_{i}u_{i}^{T}\tilde{f}_{i}+2\beta_{i+1}u_{i+1}^{T}\tilde{f}_{i}+\|\tilde{f}_{i}\|^{2} \\
&\ \ \ \ +\hat{\alpha}_{i}^{2}+\hat{\beta}_{i-1}^{2}
+2\hat{\alpha}_{i}\hat{\beta}_{i-1}\hat{u}_{i}^{T}\hat{u}_{i-1}
+2\hat{\alpha}_{i}\hat{u}_{i}^{T}\bar{f}_{i}
+2\hat{\beta}_{i-1}\hat{u}_{i-1}^{T}\bar{f}_{i}+\|\bar{f}_{i}\|^{2} .
\end{align*}
From \eqref{3.12} and \eqref{3.13} we have
\begin{align*}
& \alpha_{i}+\beta_{i+1} \leq \sqrt{2(\alpha_{i}^{2}+\beta_{i+1}^2)}\leq\sqrt{2}+O(c_{1}(m,n)\epsilon) , \\
& \hat{\alpha}_{i}+\hat{\beta}_{i-1} \leq \sqrt{2(\hat{\alpha}_{i}^{2}+\hat{\beta}_{i-1}^{2})}\leq\sqrt{2}+O(c_{2}(p,n)\epsilon) ,
\end{align*}
and thus
\begin{align*}
& \ \ \ \big|2\alpha_{i}u_{i}^{T}\tilde{f}_{i}+2\beta_{i+1}u_{i+1}^{T}\tilde{f}_{i}+\|\tilde{f}_{i}\|^{2} + 2\hat{\alpha}_{i}\hat{u}_{i}^{T}\bar{f}_{i}
+2\hat{\beta}_{i-1}\hat{u}_{i-1}^{T}\bar{f}_{i}+\|\bar{f}_{i}\|^{2} \big| \\
& = O(2(\alpha_{i} + \beta_{i+1})\epsilon)+
O(2(\hat{\alpha}_{i} + \hat{\beta}_{i-1})\epsilon) = O(\epsilon) ,
\end{align*} where we neglect high order terms of $\epsilon$.
Using the property of local orthogonality of $u_{i}$ and $\hat{u}_{i}$, we have
$$ \|2\alpha_{i}\beta_{i+1}u_{i}^{T}u_{i+1}+ 2\hat{\alpha}_{i}\hat{\beta}_{i-1}\hat{u}_{i}^{T}\hat{u}_{i-1}\|
= O(c_{3}(m,n,p) \epsilon) $$
with $c_{3}(m,n,p)=c_{1}(m,n)+c_{2}(p,n)$. Since
$$1 = \|\tilde{v}_{i}\|^{2} = \|\tilde{v}_{i}(1:m) \|^{2}+\|\tilde{v}_{i}(m+1:m+p)\|^{2},$$
we get
\begin{equation}\label{3.18}
\alpha_{i}^{2}+\beta_{i+1}^{2}+\hat{\alpha}_{i}^{2}+\hat{\beta}_{i-1}^{2}
= 1 + O(c_{3}(m,n,p) \epsilon) .
\end{equation}
Combining \eqref{3.17} and \eqref{3.18}, we finally obtain \eqref{3.16}.
\end{proof}
Similar to \eqref{2.9}, \eqref{3.16} plays an important role in the GSVD computation based on the JBD process, which implies that if only some extreme generalized singular values of $\{A, L\}$ are needed, one only needs to compute the SVD of one of $B_{k}$ and $\widehat{B}_{k}$. A detailed investigation on it is made in Section \ref{sec4}.
Since $B_{k}$ and $\bar{B}_{k}$ are of full column rank, from \eqref{3.16} we have
\begin{align*}
(B_{k}^{T})^{\dag}(B_{k}^{T}B_{k}+\bar{B}_{k}^{T}\bar{B}_{k})\bar{B}_{k}^{-1}
&= (B_{k}^{T})^{\dag}\bar{B}_{k}^{-1}+(B_{k}^{T})^{\dag}E_{k}\bar{B}_{k}^{-1} \\
&= (\bar{B}_{k}B_{k}^{T})^{\dag}+(B_{k}^{T})^{\dag}E_{k}\bar{B}_{k}^{-1} .
\end{align*}
which leads to
$$ (\bar{B}_{k}B_{k}^{T})^{\dag} = (B_{k}^{T})^{\dag}B_{k}^{T}B_{k}\bar{B}_{k}^{-1}+(B_{k}^{T})^{\dag}\bar{B}_{k}^{T}
- (B_{k}^{T})^{\dag}E_{k}\bar{B}_{k}^{-1} . $$
Notice that
$\|(B_{k}^{T})^{\dag}B_{k}^{T}\|\leq 1$, and $B_{k}^{T}=\begin{pmatrix}
\underline{B}_{k}, \beta_{1}e_{k}
\end{pmatrix}$ and thus $\|(B_{k}^{T})^{\dag}\|\leq \|\underline{B}_{k}^{-1}\|$. Using \eqref{3.14} and \eqref{3.15}, we obtain
$$\|(\bar{B}_{k}B_{k}^{T})^{\dag} \| \leq \sqrt{2}(\|\bar{B}_{k}^{-1}\|+\|\underline{B}_{k}^{-1}\|)+c_{0}(\epsilon) ,$$
where $$c_{0}(\epsilon)=\|\bar{B}_{k}^{-1}\|O(c_{1}(m,n)\epsilon)+
\|\underline{B}_{k}^{-1}\|O(c_{2}(p,n)\epsilon)+
\|\bar{B}_{k}^{-1}\|\|\underline{B}_{k}^{-1}\|O(c_{3}(m,n,p) \epsilon) .$$
Since $\bar{B}_{k}B_{k}^{T}=\begin{pmatrix}
\bar{B}_{k}\underline{B}_{k}, \beta_{1}\bar{B}_{k}e_{k}
\end{pmatrix}$, by the interlacing property of singular values, we have
$\sigma_{k}(\bar{B}_{k}\underline{B}_{k})\leq \sigma_{k}(\bar{B}_{k}B_{k}^{T})$, where $\sigma_{i}(\cdot)$ denotes the $i$-th largest singular value of a matrix, and thus
$\|(\bar{B}_{k}B_{k}^{T})^{\dag} \| \leq \| (\bar{B}_{k}\underline{B}_{k})^{-1}\|$. Fortunately, in practice, we find that it is always that $ \| (\bar{B}_{k}\underline{B}_{k})^{-1}\|\leq c\|(\bar{B}_{k}B_{k}^{T})^{\dag} \|$ with the constant $c$ not much bigger than $1$ (numerical experiments show that it is usually $1\leq c \leq 5$). Therefore, if $\|\bar{B}_{k}^{-1}\|$ and $\|\underline{B}_{k}^{-1}\|$ are not too big, we can give the following upper bound of $\|(\bar{B}_{k}\underline{B}_{k})^{-1}\|$:
\begin{equation}\label{3.19}
\|(\bar{B}_{k}\underline{B}_{k})^{-1}\|\leq c\sqrt{2}(\|\bar{B}_{k}^{-1}\|+\|\underline{B}_{k}^{-1}\|).
\end{equation}
Inequality \eqref{3.19} will be used in the proof of the following theorem.
We now show the connection between the process of computing $\widehat{B}_{k}$ and the upper Lanczos bidiagonalization of $Q_{L}$ in finite precision arithmetic. Let $\hat{v}_{i}=(-1)^{i-1}v_{i}$ and $\widehat{V}_{k}=(\hat{v}_{1},\dots,\hat{v}_{k})=V_{k}P$. Then we can rewrite \eqref{3.3} as
$$(0_{p\times m},I_{p})Q\widehat{V}_{k}=\widehat{U}_{k}\widehat{B}_{k}+\widehat{F}_{k} $$
with $\widehat{F}_{k}=\bar{F}_{k}-(0_{p\times m},I_{p})(\widetilde{V}_{k}-QV_{k})P$, which leads to
\begin{equation}\label{3.20}
Q_{L}\widehat{V}_{k}=\widehat{U}_{k}\widehat{B}_{k}+\widehat{F}_{k} ,
\end{equation}
and $\|\widehat{F}_{k} \|=O(\lVert \underline{B}_{k}^{-1}\lVert\epsilon)$ due to Lemma \ref{lem3.1}. Furthermore, we have the following result.
\begin{theorem}\label{thm3.3}
Following the hypothesis of Theorem \ref{thm3.1}. In finite precision arithmetic, we have
\begin{equation}\label{3.21}
Q_{L}^{T}\widehat{U}_{k}=\widehat{V}_{k}\widehat{B}_{k}^{T} +\hat{\beta}_{k}\hat{v}_{k+1}e_{k}^{T}+\widehat{G}_{k}
\end{equation}
with
\begin{equation}\label{3.22}
\|\widehat{G}_{k}\|
= O(c_{4}(m,n,p,k)\epsilon) ,
\end{equation}
where $c_{4}(m,n,p,k)=\|\underline{B}_{k}^{-1}\|+c_{3}(m,n,p)\|\widehat{B}_{k}^{-1}\|$.
\end{theorem}
\begin{proof}
From \eqref{3.8} and \eqref{3.9}, we get
\begin{align*}
Q_{A}^{T}Q_{A}V_{k} &=Q_{A}^{T}U_{k+1}B_{k}+Q_{A}^{T}F_{k} \\
&= (V_{k}B_{k}^{T}+\alpha_{k+1}v_{k+1}e_{k+1}^{T}+G_{k+1})B_{k}+Q_{A}^{T}F_{k} \\
&= V_{k}B_{k}^{T}B_{k}+\alpha_{k+1}\beta_{k+1}v_{k+1}e_{k}^{T}+G_{k+1}B_{k}+
Q_{A}^{T}F_{k} .
\end{align*}
Then multiply \eqref{3.20} by $Q_{L}^{T}$, we get
$$ Q_{L}^{T}Q_{L}V_{k}=(Q_{L}^{T}\widehat{U}_{k}\widehat{B}_{k}+Q_{L}^{T}\widehat{F}_{k})P .$$
Adding the above two equalities, we obtain
\begin{align*}
V_{k} &=(Q_{A}^{T}Q_{A}+Q_{L}^{T}Q_{L})V_{k} \\
&= V_{k}B_{k}^{T}B_{k}+Q_{L}^{T}\widehat{U}_{k}\widehat{B}_{k}P+\alpha_{k+1}\beta_{k+1}v_{k+1}
e_{k}^{T} +(G_{k+1}B_{k}+Q_{A}^{T}F_{k}+Q_{L}^{T}\widehat{F}_{k}P) \\
&= V_{k}(I_{k}-P\widehat{B}_{k}^{T}\widehat{B}_{k}P+E_{k})
+Q_{L}^{T}\widehat{U}_{k}\widehat{B}_{k}P+\alpha_{k+1}\beta_{k+1}v_{k+1}e_{k}^{T} \\
&\ \ \ \ +(G_{k+1}B_{k}+Q_{A}^{T}F_{k}+Q_{L}^{T}\widehat{F}_{k}P) ,
\end{align*}
which leads to
\begin{align*}
\widehat{V}_{k}\widehat{B}_{k}^{T}\widehat{B}_{k}
&= Q_{L}^{T}\widehat{U}_{k}\widehat{B}_{k}+\alpha_{k+1}\beta_{k+1}v_{k+1}e_{k}^{T}P+
(G_{k+1}B_{k}+Q_{A}^{T}F_{k} +Q_{L}^{T}\widehat{F}_{k}P+V_{k}E_{k})P \\
&= Q_{L}^{T}\widehat{U}_{k}\widehat{B}_{k}-(\hat{\alpha}_{k}\hat{\beta}_{k}+\gamma_{k})\hat{v}_{k+1}
e_{k}^{T}+(G_{k+1}B_{k}+Q_{A}^{T}F_{k}+Q_{L}^{T}\widehat{F}_{k}P+V_{k}E_{k})P
\end{align*}
and thus
$$\widehat{V}_{k}\widehat{B}_{k}^{T}
=Q_{L}^{T}\widehat{U}_{k}-\hat{\beta}_{k}\hat{v}_{k+1}e_{k}^{T}+ E_{1} + E_{2},$$
where
$$E_{1} = [(G_{k+1}B_{k}+V_{k}E_{k})P- \gamma_{k}\hat{v}_{k+1}e_{k}^{T}]\widehat{B}_{k}^{-1} , \ \
E_{2} = (Q_{A}^{T}F_{k}P + Q_{L}^{T}\widehat{F}_{k})\widehat{B}_{k}^{-1} .$$
Using the upper bounds of $\|B_{k}\|$, $\|E_{k}\|$, $|\gamma_{k}|$, and noticing that $\|V_{k}\| \leq \sqrt{1+\eta(V_{k})}$, we get
$$\|E_{1}\|=O(\bar{c}_{1}(m,n,p,k)\epsilon) $$ with $\bar{c}_{1}(m,n,p,k)=(\sqrt{2}+c_{3}(m,n,p))\|\widehat{B}_{k}^{-1}\|$. For $E_{2}$, from the expressions of $F_{k}$, $\widehat{F}_{k}$ and $\widetilde{V}_{k}-QV_{k}$, we have
\begin{align*}
Q_{A}^{T}F_{k}P + Q_{L}^{T}\widehat{F}_{k}
&= \begin{pmatrix}
Q_{A}^{T} & Q_{L}^{T}
\end{pmatrix}
\begin{pmatrix}
F_{k}P \\ \widehat{F}_{k}
\end{pmatrix} \\
&= Q^{T}\Big[
\begin{pmatrix}
\widetilde{F}_{k}P \\ \bar{F}_{k}
\end{pmatrix} -
\begin{pmatrix}
I_{m} & 0_{m\times p} \\
0_{p\times m} & I_{p}
\end{pmatrix}
(\widetilde{V}_{k}-QV_{k})P
\Big] \\
&= Q^{T}\Big[
\begin{pmatrix}
\widetilde{F}_{k}P \\ \bar{F}_{k}
\end{pmatrix} + (I_{m+p}-QQ^{T})\widetilde{G}_{k}\underline{B}_{k}^{-1}P \Big] .
\end{align*}
Using \eqref{3.19}, we have
$$\|\underline{B}_{k}^{-1}P\widehat{B}_{k}^{-1}\|=\|(\bar{B}_{k}\underline{B}_{k})^{-1}\|\leq c\sqrt{2}(\|\bar{B}_{k}^{-1}\|+\|\underline{B}_{k}^{-1}\|). $$
Thus, noticing that $\|\bar{B}_{k}^{-1}\|=\|\widehat{B}_{k}^{-1}\|$, we obtain the upper bound of $\|E_{2}\|$:
$$\|E_{2}\|=O(\bar{c}_{2}(k)\epsilon) $$
with $\bar{c}_{2}(k)=c\sqrt{2}(\|\widehat{B}_{k}^{-1}\|+\|\underline{B}_{k}^{-1}\|)
+\|\widehat{B}_{k}^{-1}\|$. By letting $\widehat{G}_{k}=-E_{1}-E_{2}$, we finally obtain the desired result.
\end{proof}
Notice that \eqref{3.20} together with \eqref{3.21} is the corresponding version of \eqref{2.11} in the presence of rounding errors. Therefore, this theorem implies that the process of computing $\widehat{U}_{k}$, $\widehat{V}_{k}$ and $\widehat{B}_{k}$ is not equivalent to the upper Lanczos bidiagonalization of $Q_{L}$ with rounding error $O(\|Q_{L}\|\epsilon)=O(\epsilon)$ any longer, since the error terms $\widehat{F}_{k}$ and $\widehat{G}_{k}$ are amplified gradually if $\|\underline{B}_{k}^{-1}\|$ and $\|\widehat{B}_{k}^{-1}\|$ grow bigger and bigger as the iterations progress.
Theorem \ref{thm3.1} and Theorem \ref{thm3.3} show that in the presence of rounding errors, the JBD process of $\{A, L\}$ is not equivalent to the combination of the lower and upper Lanczos bidiagonalizations of $Q_{A}$ and $Q_{L}$ with rounding error $O(\epsilon)$ any longer. On the other hand, for the aim of analysis, we can treat the process of computing $B_{k}$ and $\widehat{B}_{k}$ as lower and upper Lanczos bidiagonalization with gradually growing error terms respectively, where the growth rate of error terms are affected by $\|\underline{B}_{k}^{-1}\|$ and $\|\widehat{B}_{k}^{-1}\|$.
\begin{remark}\label{rem3.1}
The growth speed of $\|\underline{B}_{k}^{-1}\|$ can be controlled. In the GSVD computation problems, usually at least one matrix of $\{A, L\}$ is well conditioned, which results to that at least one of $\{Q_{A}, Q_{L}\}$ is well conditioned. If $Q_{A}$ is the well conditioned one, we implements the JBD process on $\{A, L\}$, which leads $\underline{B}_{k}$ to be a well conditioned matrix; otherwise if $Q_{L}$ is the well conditioned one, we implements the JBD process on $\{L, A\}$, which still leads $\underline{B}_{k}$ to be a well conditioned matrix. By this modification, we could always make sure that $\|\underline{B}_{k}^{-1}\|$ will not grow too big.
\end{remark}
\subsection{Loss of orthogonality of the Lanczos vectors}
It is well known that for the Lanczos bidiagonalization, orthogonality of Lanczos vectors is completely destroyed due to the influence of rounding errors, that is, once the orthogonality is destroyed at one step, the errors will propagate to future steps, which results to the loss of orthogonality of subsequent computed Lanczos vectors \cite{Simon1984a,Larsen1998}. For the JBD process, the loss of orthogonality of Lanczos vectors is similar to that of Lanczos bidiagonalization. Furthermore, the orthogonality levels of $U_{k+1}$, $\widetilde{V}_{k}$ and $\widehat{U}_{k}$ are closely related. Here we show that the orthogonality level of $\widehat{U}_{k}$ is affected by the orthogonality levels of both $U_{k+1}$ and $\widetilde{V}_{k}$.
\begin{theorem}\label{thm3.4}
Following the hypothesis of Theorem \ref{thm3.1}. In finite precision arithmetic, we have
\begin{equation}\label{3.23}
\eta(\widehat{U}_{k}) \leq \|\widehat{B}_{k}^{-1}\|^{2}\big[\eta(\widetilde{V}_{k})+2\eta(U_{k+1})+O(c_{3}(m,n,p)\epsilon)\big] .
\end{equation}
\end{theorem}
\begin{proof}
From \eqref{3.1} and \eqref{3.3}, we have
$$\widetilde{V}_{k}=
\begin{pmatrix}
U_{k+1}B_{k} \\\widehat{U}_{k}\bar{B}_{k}
\end{pmatrix} +
\begin{pmatrix}
\widetilde{F}_{k} \\ \bar{F}_{k}P
\end{pmatrix} ,$$
which leads to
$$\widetilde{V}_{k}^{T}\widetilde{V}_{k}=B_{k}^{T}U_{k+1}^{T}U_{k+1}B_{k}+
\bar{B}_{k}^{T}\widehat{U}_{k}^{T}\widehat{U}_{k}\bar{B}_{k}+E_{3} ,$$
where
$$ E_{3}= B_{k}^{T}U_{k+1}^{T}\widetilde{F}_{k}+\bar{B}_{k}^{T}\widehat{U}_{k}^{T}\bar{F}_{k}P
+\widetilde{F}_{k}^{T}U_{k+1}B_{k}+P\bar{F}_{k}^{T}\widehat{U}_{k}\bar{B}_{k}
+\widetilde{F}_{k}^{T}\widetilde{F}_{k}+P\bar{F}_{k}^{T}\bar{F}_{k}P . $$
Using \eqref{3.16}, we obtain
$$I_{k}-\widetilde{V}_{k}^{T}\widetilde{V}_{k}= B_{k}^{T}(I_{k+1}-U_{k+1}^{T}U_{k+1})B_{k}+
\bar{B}_{k}^{T}(I_{k}-\widehat{U}_{k}^{T}\widehat{U}_{k})\bar{B}_{k}-E_{k}-E_{3} ,$$
and thus
\begin{equation}\label{3.24}
I_{k}-\widehat{U}_{k}^{T}\widehat{U}_{k}= \bar{B}_{k}^{-T} \big[(I_{k}-\widetilde{V}_{k}^{T}\widetilde{V}_{k})-B_{k}^{T}(I_{k+1}-U_{k+1}^{T}U_{k+1})B_{k}+E_{k}+E_{3} \big]\bar{B}_{k}^{-1} .
\end{equation}
Using the bounds of $\|B_{k}\|$ and $\|\widehat{B}_{k}\|$ and noticing that $\|U_{k+1}\|\leq(1+\eta(U_{k+1}))^{1/2}$, $\|\widehat{U}_{k}\|\leq(1+\eta(\widehat{U}_{k}))^{1/2}$, with a simple calculation, we can get
$$\|E_{3}\|=O(\epsilon) .$$
Using the bound of $\|B_{k}\|$, we get
\begin{align*}
\|B_{k}^{T}(I_{k+1}-U_{k+1}^{T}U_{k+1})B_{k}\|
&\leq \|B_{k}\|^{2}\|I_{k+1}-U_{k+1}^{T}U_{k+1}\| \\
&\leq 2\|I_{k+1}-U_{k+1}^{T}U_{k+1}\|+O(c_{1}(m,n)\epsilon) .
\end{align*}
By taking norms of \eqref{3.24}, we finally obtain the desired result.
\end{proof}
The above theorem indicates that as long as $\widehat{B}_{k}$ is not very ill-conditioned, the orthogonality of $\widehat{U}_{k}$ will not be too bad if we can maintain some levels of orthogonality of $U_{k+1}$ and $\widetilde{V}_{k}$, which can be achieved by reorthogonalization strategies.
\begin{remark}\label{rem3.2}
From Lemma \ref{lem3.1}, we have
\begin{align*}
\widetilde{V}_{k}^{T}(I_{m+p}-QQ^{T})\widetilde{V}_{k}
&= \widetilde{V}_{k}^{T}(\widetilde{V}_{k} - QV_{k})
= \widetilde{V}_{k}^{T}(QQ^{T}-I_{m+p})\widetilde{G}_{k}\underline{B}_{k}^{-1}\\
&=-[(I_{m+p}-QQ^{T})\widetilde{V}_{k}]^{T}\widetilde{G}_{k}\underline{B}_{k}^{-1} \\
&= \underline{B}_{k}^{-T}\widetilde{G}_{k}^{T}(I_{m+p}-QQ^{T})\widetilde{G}_{k}\underline{B}_{k}^{-1} ,
\end{align*}
and thus
\begin{align*}
\begin{split}
I_{k} - V_{k}^{T}V_{k}
&= I_{k} - \widetilde{V}_{k}^{T}QQ^{T}\widetilde{V}_{k} =
I_{k} - \widetilde{V}_{k}^{T}\widetilde{V}_{k} + \widetilde{V}_{k}^{T}(I_{m+p}-QQ^{T})\widetilde{V}_{k} \\
&= (I_{k} - \widetilde{V}_{k}^{T}\widetilde{V}_{k}) + \underline{B}_{k}^{-T}\widetilde{G}_{k}^{T}(I_{m+p}-QQ^{T})\widetilde{G}_{k}\underline{B}_{k}^{-1} .
\end{split}
\end{align*}
Therefore, we obtain
\begin{equation}\label{3.25}
|\eta(\widetilde{V}_{k})-\eta(V_{k})|\leq\lVert \underline{B}_{k}^{-T}\bar{G}_{k}^{T}\lVert \lVert\bar{G}_{k}\underline{B}_{k}^{-1}\lVert
= O(\lVert \underline{B}_{k}^{-1}\lVert^{2}\epsilon^{2}) .
\end{equation}
Note that it is usually that the machine precision $\epsilon$ is about $10^{-16}$ or even smaller. For most occasions, $\|\underline{B}_{k}^{-1}\|$ is not too big (e.g. $\|\underline{B}_{k}^{-1}\|<10^{8}$). Therefore, by \eqref{3.25}, we have $|\eta(\widetilde{V}_{k})-\eta(V_{k})|=O(\epsilon)$, and thus $\eta(V_{k})$, $\eta(\widetilde{V}_{k})$ and $\xi(V_{k})$, $\xi(\widetilde{V}_{k})$ can be used interchangeably.
\end{remark}
\section{Applications to the GSVD computation}\label{sec4}
The algorithm for computing a few extreme generalized singular values and corresponding vectors of $\{A,L\}$ using the JBD process is proposed by Zha in \cite{Zha1996}. Here we make a detailed investigation on the convergence and the accuracy of the approximate generalized singular values by using the results of our rounding error analysis of the JBD process.
Following \eqref{2.2}, the $i$-th generalized singular value of $\{A,L\}$ is $c_{i}/s_{i}$, while the $i$-th corresponding generalized singular vectors are $g_{i}$, $p^{A}_{i}$ and $p_{i}^{L}$, which are the $i$-th columns of $G$, $P_{A}$ and $P_{L}$, respectively. We call $g_{i}$ the $i$-th right generalized singular vector, $p^{A}_{i}$ and $p_{i}^{L}$ the $i$-th left generalized singular vectors corresponding to $A$ and $L$ respectively. Since $c_{i}/s_{i}=\infty$ when $s_{i}=0$, we use the number pair $\{c_{i}, s_{i}\}$ to denote $c_{i}/s_{i}$.
Here we do not take account into rounding errors for a moment. Let us assume that we have computed the compact SVD of $B_{k}$:
\begin{equation}\label{4.1}
B_{k} = P_{k}\Theta_{k}W_{k}^{T}, \ \ \Theta_{k}=diag(c_{1}^{(k)}, \dots, c_{k}^{(k)}), \ \
1 \geq c_{1}^{(k)} > \dots > c_{k}^{(k)} \geq 0 \ ,
\end{equation}
where $P_{k}=(p_{1}^{(k)}, \dots, p_{k}^{(k)})\in\mathbb{R}^{(k+1)\times k}$ and $W_{k}=(w_{1}^{(k)}, \dots, w_{k}^{(k)})\in\mathbb{R}^{k\times k}$ are column orthonormal, and $\Theta_{k}\in\mathbb{R}^{k\times k}$. The decomposition \eqref{4.1} can be achieved by a direct method since $B_{k}$ is a matrix of small scale. Then the approximate generalized singular value of $\{A, L\}$ is $\{c_{i}^{(k)}, (1-(c_{i}^{(k)})^{2})^{1/2}\}$, while the approximate right generalized singular vector is $x_{i}^{(k)}=R^{-1}V_{k}w_{i}^{(k)}$ and the approximate left generalized singular vector corresponding to $A$ is $y_{i}^{(k)} = U_{k+1}p_{i}^{(k)}$.
If we also want to compute the approximations of the left generalized singular vectors corresponding to $L$, we need to compute the SVD of $\widehat{B}_{k}$. Let us assume that the SVD of $\widehat{B}_{k}$ is
\begin{equation}\label{4.2}
\widehat{B}_{k} = \widehat{P}_{k}\Psi_{k}\widehat{W}_{k}^{T}, \ \
\Psi_{k}=diag(\hat{s}_{1}^{(k)}, \dots, \hat{s}_{k}^{(k)}), \ \
0 \leq \hat{s}_{1}^{(k)} < \dots < \hat{s}_{k}^{(k)} \leq 1 \ ,
\end{equation}
where $\widehat{P}_{k}=(\hat{p}_{1}^{(k)}, \dots, \hat{p}_{k}^{(k)})\in\mathbb{R}^{k\times k}$ and $\widehat{W}_{k}=(\hat{w}_{1}^{(k)}, \dots, \hat{w}_{k}^{(k)})\in\mathbb{R}^{k\times k}$ are orthogonal. Then the approximation of $p_{i}^{L}$ is $z_{i}^{(k)}= \widehat{U}_{k}\hat{p}_{i}^{(k)}$. The approximate generalized singular values and the corresponding right vectors can also be computed from the SVD of $\widehat{B}_{k}$, which are $\{(1-(\hat{s}_{i}^{(k)})^{2})^{1/2},\hat{s}_{i}^{(k)}\}$ and $R^{-1}\widehat{V}_{k}\hat{w}_{i}^{(k)}$, respectively.
For computing $x_{i}^{(k)}$, it is shown in \cite{Zha1996} that the explicit computation of $R^{-1}$ can be avoided. Notice that
\begin{equation*}
\begin{pmatrix}
A \\ L
\end{pmatrix}x_{i}^{(k)}=QRR^{-1}V_{k}w_{i}^{(k)}=\widetilde{V}_{k}w_{i}^{(k)} ,
\end{equation*}
hence by solving a least squares problem, we can obtain $x_{i}^{(k)}$ from $\widetilde{V}_{k}w_{i}^{(k)}$.
\subsection{Convergence, accuracy and reorthogonalization}
In the presence of rounding errors, the behavior of the algorithm may deviate far from the ideal case. For simplicity, we only consider the rounding errors appeared in the JBD process, while the procedures of computing the SVD of $B_{k}$ and $\widehat{B}_{k}$, and the computation of $x_{i}^{(k)}$, $y_{i}^{(k)}$ and $z_{i}^{(k)}$ are assumed to be implemented in exact arithmetic.
First, we investigate the convergence of the computed generalized singular values using the SVD of $B_{k}$. Since $c_{i}^{2}+s_{i}^{2}=1$, in order to compute the generalized singular value $\{c_{i}, s_{i}\}$, we only need to compute $c_{i}$, which is a singular value of $Q_{A}$. Note that $c_{i}^{(k)}$, which is the singular value of $B_{k}$, is actually a Ritz value of $Q_{A}$, due to that the process of computing $B_{k}$ can be treated as the Lanczos bidiagonalization of $Q_{A}$ with rounding error $O(\lVert \underline{B}_{k}^{-1}\lVert\epsilon)$. The convergence of Ritz values of $Q_{A}$ is mainly impacted by two factors: (1) the Ritz values approximating the extreme singular values of $Q_{A}$ will converge rapidly, while the interior singular values will converge more slowly; (2) if the singular value is well separated from others, then the corresponding Ritz values converge more rapidly, otherwise converge more slowly. Hence, if we compute $c_{i}$ by the SVD of $B_{k}$, the Ritz values approximating the extreme generalized singular values of $\{A,L\}$ will converge rapidly, while the interior ones will converge more slowly.
\begin{remark}\label{rem4.1}
The convergence behavior of Ritz values of $Q_{A}$ can be derived from \cite[Theorem 2]{Saad1980}, which describes the convergence of Ritz values of the symmetric Lanczos process of a symmetric matrix $\bar{C}$. It says that we can expect rapid convergence of the Ritz values approximating the extreme eigenvalues of $\bar{C}$, and the convergence rate also depends on how well an eigenvalue separated from others. Since the Lanczos bidiagonalization of $Q_{A}$ with staring vector $b$ is mathematically identical to the symmetric Lanczos process of $Q_{A}^{T}Q_{A}$ with starting vector $Q_{A}^{T}b$, and $(c_{i}^{(k)})^{2}$ are eigenvalues of $B_{k}^{T}B_{k}$, we can conclude the above convergence behavior of Ritz values of $Q_{A}$.
\end{remark}
In finite precision arithmetic, when we use the SVD of $B_{k}$ to compute approximate generalized singular values of $\{A,L\}$, the singular values of $B_{k}$ will contain false multiple copies of converged Ritz values as the iteration number $k$ increases, which is the so called ghost singular values. Thus the generalized singular values of $\{A,L\}$ will be approximated by false multiple copies of converged Ritz values and it leads to a delay of the convergence of Ritz values. Moreover, sometimes it is difficult to determine whether some computed approximations are spurious copies or genuine close or multiple generalized singular values. This problem is closely related to the loss of orthogonality of Lanczos vectors, which can be avoided by using some types of reorthogonalization strategies, such as full reorthogonalization or the more efficient one-sided reorthogonalization \cite{Simon2000}.
By Theorem \ref{thm3.1}, the process of computing $B_{k}$ can be treated as the Lanczos bidiagonalization of $Q_{A}$ with rounding error $O(\|\underline{B}_{k}^{-1}\|\epsilon)$ that will not deviated far from $O(\epsilon)$ (see Remark \ref{rem3.1}). If the JBD process is implemented with one-sided reorthogonalization of $\tilde{v}_{i}$ such that the orthogonality level of $\widetilde{V}_{k}$ is kept around $O(\epsilon)$, by using the results of backward error analysis of the Lanczos bidiagonalization with one-sided reorthogonalization \cite[Theorem 5.2 and Corollary 5.1]{Barlow2013}, we can conclude that the computed $B_{k}$ is the exact one generated by the Lanczos bidiagonalization of a nearby matrix $Q_{A}+\delta X$, where $\|\delta X\|=O(\|\underline{B}_{k}^{-1}\|\epsilon)$. Therefore, by the perturbation theory of the singular values \cite[Corollary 8.6.2]{Golub2013}, the singular values of $Q_{A}$ can be computed with accuracy $O(\|\underline{B}_{k}^{-1}\|\epsilon)$, while the appearance of ghost singular values can be avoided and the multiple singular values can be found one by one. Since the generalized singular values are determined by the singular values of $Q_{A}$, the above assertion of the convergence and accuracy also holds for the approximate generalized singular values.
Another efficient reorthogonalization strategy is the semiorthogonalization strategy, which is first proposed by Simon for the symmetric Lanczos process \cite{Simon1984a} and then adapted by Larsen for the Lanczos bidiagonalization \cite{Larsen1998}. By Theorem \ref{thm3.1} and \cite[Theorem 5]{Larsen1998}, we have the following result.
\begin{theorem}\label{thm4.1}
Assume that the compact $QR$ factorizations of $U_{k+1}$ and $V_{k}$ are $U_{k+1}=M_{k+1}R_{k+1}$ and $V_{k}=N_{k}S_{k}$, where the diagonals of upper triangular matrices are nonnegative. Let $\delta = O(\|\underline{B}_{k}^{-1}\|\epsilon)$. If the orthogonality levels of $U_{k+1}$ and $V_{k}$ satisfy
\begin{equation}\label{4.3}
\xi(U_{k+1}), \ \xi(V_{k}) \leqslant \sqrt{\delta/(2k+1)}
\footnote{In \cite[Theorem 5]{Larsen1998}, the right-hand term is $\sqrt{\delta/k}$ instead of $\sqrt{\delta/(2k+1)}$, while the author does not prove it rigorously. In fact, this result is a corresponding version of \cite[Theorem 4]{Simon1984a}. Since the $k$-step Lanczos bidiagonalization of $Q_{A}$ with starting vector $b$ is equivalent to the $(2k+1)$-step symmetric Lanczos process \cite[\S 7.6.1]{Bjorck1996} of
$\bar{C} = \begin{pmatrix}
O & Q_{A} \\
Q_{A}^{T} & O
\end{pmatrix}$ with starting vector
$\bar{p}_{0} = \begin{pmatrix}
p_{0} \\ 0
\end{pmatrix}$(which holds not only in exact arithmetic, but also in finite arithmetic), the denominator in \eqref{4.3} should be $2k+1$.},
\end{equation}
then
\begin{equation}\label{4.4}
M_{k+1}^{T}Q_{A}N_{k}=B_{k}+X_{k} ,
\end{equation}
where the elements of $X_{k}$ are of $O(\delta)=O(\|\underline{B}_{k}^{-1}\|\epsilon)$.
\end{theorem}
Notice that $\xi(V_{k})\approx\xi(\widetilde{V}_{k})$ when $\|\underline{B}_{k}^{-1}\|$ is not too big (see Remark \ref{rem3.2}). Theorem \ref{thm4.1} indicates that the orthogonality levels of $U_{k}$ and $\widetilde{V}_{k}$ are only needed to be maintained under $(\delta/(2k+1))^{1/2}$, in order to obtain approximate singular values of $Q_{A}$ from the computed $B_{k}$ with accuracy $O(\|\underline{B}_{k}^{-1}\|\epsilon)$ and avoid ghost singular values from appearing. Following \cite{Simon1984a,Larsen1998}, we call the Lanczos vectors with orthogonality level below $(\delta/(2k+1))^{1/2}$ semiorthogonal. We have also made a detailed investigation on the JBD process with semiorthogonalization strategy and proposed an efficient partial reorthogonalization strategy, which will be presented in a forthcoming paper \cite{Li2019}.
\begin{remark}\label{rem4.2}
For $\widehat{B}_{k}$, there is also a corresponding version of Theorem \ref{thm4.1}. Let $\hat{\delta} =O(c_{4}(m,n,p,k)\epsilon)$ with $c_{4}(m,n,p,k)=\|\underline{B}_{k}^{-1}\|+c_{3}(m,n,p)\|\widehat{B}_{k}^{-1}\|$. The compact $QR$ factorizations of $\widehat{U}_{k}$ and $\widehat{V}_{k}$ are $\widehat{U}_{k}=\widehat{M}_{k}\widehat{R}_{k}$ and $\widehat{V}_{k}=\widehat{N}_{k}\widehat{S}_{k}$, where the diagonals of $\widehat{R}_{k}$ and $\widehat{S}_{k}$ are nonnegative. If the orthogonality levels of $\widehat{U}_{k}$ and $\widehat{V}_{k}$ satisfy
$$ \xi(\widehat{U}_{k}), \ \xi(\widehat{V}_{k}) \leqslant \sqrt{\hat{\delta}/(2k+1)},$$
then
$$ \widehat{M}_{k}^{T}Q_{L}\widehat{N}_{k}=\widehat{B}_{k}+\widehat{X}_{k}, $$
where the elements of $\widehat{X}_{k}$ are of $O(\hat{\delta})$.
\end{remark}
If we use the SVD of $\widehat{B}_{k}$ to compute the generalized singular values, by Remark \ref{rem4.2}, the singular values of $Q_{L}$ can be approximated by $\hat{s}_{i}^{(k)}$ with accuracy $O(\hat{\delta})$,
while the ghost singular values of $Q_{L}$ can be avoided from appearing if the orthogonaltity levels of $\widehat{U}_{k}$ and $\widetilde{V}_{k}$ are maintained under $(\hat{\delta}/(2k+1))^{1/2}$. Furthermore, from Theorem \ref{thm3.2} we have
\begin{equation}\label{4.5}
\widehat{B}_{k}^{T}\widehat{B}_{k} - PE_{k}P = (PW_{k})(I_{k}-\Theta_{k}^{2})(PW_{k})^{T},
\end{equation}
which is just the eigendecomposition of $\widehat{B}_{k}^{T}\widehat{B}_{k}$ with perturbation $- PE_{k}P$. Notice that $\|E_{k}\|=O(c_{3}(m,n,p)\epsilon)$. Therefore, the singular values of $\widehat{B}_{k}$ are determined by that of $B_{k}$ with a small perturbation, and we can hope that $\hat{s}_{i}^{(k)}$ will approximate the singular values of $Q_{L}$ with high accuracy even if $\|\widehat{B}_{k}^{-1}\|$ will grow too big, and the convergence behavior of $\hat{s}_{i}^{(k)}$ is similar to that of $c_{i}^{(k)}$.
Finally, let us make a brief discussion about the accuracy of the computed generalized singular vectors. If both $A$ and $L$ are well-conditioned, which lead to that both $\delta$ and $\hat{\delta}$ do not deviate too far from $\epsilon$, we can hope that both the left and right generalized singular vectors of $\{A,L\}$ can be computed with a high accuracy by using the SVD of $B_{k}$ and $\widehat{B}_{k}$. On the other hand, if one of $A$ and $L$ is ill-conditioned, then one of $\|\underline{B}_{k}^{-1}\|$ and $\|\widehat{B}_{k}^{-1}\|$ may grow very big, and thus we should implement the JBD process with a slight modification (see Remark \ref{rem4.2}). Suppose that we implement the JBD of $\{A,L\}$, and $\|\underline{B}_{k}^{-1}\|$ grows extremely slowly while $\|\widehat{B}_{k}^{-1}\|$ grows rapidly, then $\tilde{\delta}$ will become much bigger than $\epsilon$, and the computed left generalized singular vectors corresponding to $L$ may have a poor accuracy. We will later use some numerical examples to show the accuracy of the computed generalized singular vectors. A rigorous analysis of the accuracy of computed generalized singular vectors is certainly complicated, and we will focus on this issue in our future work.
\subsection{Residual norm and stopping criterion}
Now we discuss about the stopping criterion of the GSVD computation based on the JBD process. For simplicity, we only consider rounding errors in the JBD process, and the subsequent steps to compute approximate generalized singular values and vectors are assumed to be implemented in exact arithmetic. The following analysis focus on computing the GSVD using the SVD of $B_{k}$.
It is well known that the generalized eigenvalues of the following symmetric generalized eigenvalue problem
\begin{equation*}
A^{T}Ax = \lambda L^{T}Lx
\end{equation*} are
\begin{equation*}
\underbrace{\infty, \dots, \infty}_{q}, \ \ \underbrace{(c_{q+1}/s_{q+1})^{2}, \dots, (c_{q+l}/s_{q+l})^{2}}_{l}, \ \ \underbrace{0, \dots, 0}_{t},
\end{equation*}
and the corresponding generalized eigenvectors are the right generalized singular vectors of $\{A, L\}$. Based on this property, it is shown in \cite{Zha1996} by Zha, that one can use the residual norm
$$\|r_{i}^{(k)} \| = \| ((s_{i}^{(k)})^{2}A^{T}A-(c_{i}^{(k)})^{2}L^{T}L)x_{i}^{(k)}\|$$
as a measure for the accuracy of approximate generalized singular value pair $\{c_{i}^{(k)}, s_{i}^{(k)}\}$ and the corresponding right vector $x_{i}^{(k)}$, where $s_{i}^{(k)}=(1-(c_{i}^{(k)})^{2})^{1/2}$. In exact arithmetic, Zha has proved that
\begin{equation}\label{4.6}
\|r_{i}^{(k)} \| \leq \| R \|\alpha_{k+1}\beta_{k+1}|e_{k}^{T}w_{i}^{(k)}| ,
\end{equation}
and the quantity $\| R \|\alpha_{k+1}\beta_{k+1}|e_{k}^{T}w_{i}^{(k)}|$ can be used as a stopping criterion if $\|R\|$ or its accurate estimate is available. Considering rounding errors in the JBD process, we can obtain the following upper bound of $\|r_{i}^{(k)}\|$.
\begin{theorem}\label{thm4.2}
Following the hypothesis of Theorem \ref{thm3.1}. In finite precision arithmetic, we have
\begin{equation}\label{4.7}
\big\| [(s_{i}^{(k)})^{2}A^{T}A-(c_{i}^{(k)})^{2}L^{T}L]x_{i}^{(k)}\big\| \leq
\| R\|[\alpha_{k+1}\beta_{k+1}|e_{k}^{T}w_{i}^{(k)}|+O(\|\underline{B}_{k}^{-1}\|\epsilon)]
\end{equation}
\end{theorem}
\begin{proof}
From the proof of Theorem \ref{thm3.3} we have
$$ Q_{A}^{T}Q_{A}V_{k} = V_{k}B_{k}^{T}B_{k}+\alpha_{k+1}\beta_{k+1}v_{k+1}e_{k}^{T}+G_{k+1}B_{k}+
Q_{A}^{T}F_{k} .$$
From \eqref{4.1}, we have
$$B_{k}^{T}B_{k}=W_{k}\Theta_{k}^{2}W_{k}^{T}.$$
Using the above two equalities, and noticing that $(s_{i}^{(k)})^{2}+(c_{i}^{(k)})^{2}=1$ and $x_{i}^{(k)}=R^{-1}V_{k}w_{i}^{(k)}$, we have
\begin{align*}
& \ \ \ \ [(s_{i}^{(k)})^{2}A^{T}A-(c_{i}^{(k)})^{2}L^{T}L]x_{i}^{(k)} \\
&= [A^{T}A-(c_{i}^{(k)})^{2}(A^{T}A+L^{T}L)]R^{-1}V_{k}W_{k}e_{i} \\
&= R^{T}[Q_{A}^{T}Q_{A}V_{k}W_{k}-(c_{i}^{(k)})^{2}V_{k}W_{k}]e_{i} \\
&= R^{T}[\alpha_{k+1}\beta_{k+1}v_{k+1}e_{k}^{T}w_{i}^{(k)}+(G_{k+1}B_{k}+
Q_{A}^{T}F_{k})w_{i}^{(k)}] ,
\end{align*}
where $e_{i}$ is the $i$-th column of the identity matrix of order $k$. From Theorem \ref{thm3.1} we have
$\|(G_{k+1}B_{k}+Q_{A}^{T}F_{k})w_{i}^{(k)} \|=O(\|\underline{B}_{k}^{-1}\|\epsilon)$. Therefore, by taking norms of the above equality, the desired result is obtained.
\end{proof}
Since $Z=\begin{pmatrix}
A \\ L
\end{pmatrix} = QR$, we have
$\| R \| = \|Z\|=\sigma_{1}(Z)$. If we perform the Lanczos bidiagonalization on $Z$, then the largest Ritz value will converge to $\sigma_{1}(Z)$ in not too many iterations, so we can have an accurate estimation of $\|Z\|$. This method of estimating $\|Z\|$ has been discussed in detail and implemented with MATLAB codes in \cite{Larsen1998}. Notice that $e_{k}^{T}w_{i}^{(k)}$ is available from the SVD of $B_{k}$. As is mentioned in Remark \ref{rem3.1}, we can make sure that $\|\underline{B}_{k}^{-1}\|$ does not grow very big if we implement the JBD process with a slight modification, and thus the upper bound of $\|r_{i}^{(k)}\|$ in \eqref{4.7} does not deviate far from that in \eqref{4.6}. Therefore, the quantity $\| R \|\alpha_{k+1}\beta_{k+1}|e_{k}^{T}w_{i}^{(k)}|$ can still be used as a stopping criterion. We will use a numerical example to illustrate this in the following section.
\section{Numerical examples}\label{sec5}
We now provide several numerical examples to illustrate the results given in the previous sections, including the numerical behavior of the JBD process and the numerical performance of the GSVD computation based on the JBD process.
Throughout this section, all the numerical experiments are performed on an Intel (R) Core (TM) i7-7700 CPU 3.60GHz with the main memory 8GB using the Matlab R2017a with the machine precision $\epsilon = 2.22 \times 10^{-16}$ under the Windows 10 operating system. For each matrix pair $\{A, L\}$, we use $b=(1,\dots,1)^{T}\in\mathbb{R}^{m}$ as the starting vector of the JBD process, where $m$ is the row number in $A$. We mention that our results are based on the assumption that the inner least squares problem \eqref{2.3} is solved accurately at each step. Therefore, for the implementation of the JBD process in the numerical experiments, the $QR$ factorization of
$\begin{pmatrix}
A \\ L
\end{pmatrix}$
is computed, and $QQ^{T}\tilde{u}_{i}$ is computed explicitly using $Q$ at each step.
\subsection{Examples for the numerical behavior of the JBD process}
In this subsection, we use some examples to illustrate the numerical behavior of the JBD process in finite precision arithmetic. We choose four matrix pairs. For the first pair, the matrices $A$ and $L$, which are denoted by $A_{c}$ and $L_{s}$ respectively, are constructed by ourselves. Let $n=800$ and $C=diag(c)$, where $c=(\dfrac{3n}{2}, \dfrac{3n}{2}-1, \dots, \dfrac{n}{2}+1)/2n$. Then let $s = (\sqrt{1-c_{1}^{2}}, \dots, \sqrt{1-c_{n}^{2}})$ and $S=diag(s)$. Let $D$ be the matrix generated by the MATLAB built-in function $\texttt{D=gallery(`orthog',n,2)}$, which means that $D$ is a symmetric orthogonal matrix. Finally, let $A=CD$ and $L=SD$. By the construction, we know that the generalized singular value of $\{A, L\}$ is $\{c_{i}/s_{i}\}$ and the corresponding right vector $g_{i}$ is the $i$-th column of $D^{T}$, where $i = 1, \dots, n$. The remaining three pairs use sparse matrices taken from \cite{Davis2011}, where
\begin{equation}\label{l1}
L_1 = \left(
\begin{array}{ccccc}
1 & -1 & & & \\
& 1 & -1 & & \\
& & \ddots & \ddots & \\
& & & 1 & -1\\
\end{array}
\right)\in \mathbb{R}^{(n-1)\times n}
\end{equation}
with $n=712$, which is the discrete approximation of the first order derivative operator, and $L_{m}=diag(l)$ with $l=(2m, 2m-1, \dots, m+1)/1000$, $m=3969$. The properties of our test matrices are described in table \ref{tab1}, where $\kappa(\cdot)$ is the condition number of a matrix.
\begin{table}[htp]
\centering
\caption{Properties of the test matrices.}
\begin{tabular}{|l|l|l|l|l|l|}
\hline
$A$ &$m\times n$ &$\kappa(A)$ &$L$ &$p\times n$ &$\kappa(L)$
\\ \hline
{\sf $A_{c}$} &$800\times 800$ &2.99 &{\sf $L_{s}$} &$800\times 800$ &1.46 \\ \hline
{\sf well1850} &$1850\times 712$ &111.31 &{\sf $L_{1}$} &$711\times 712$ &453.27
\\ \hline
{\sf rdb2048} &$2048\times 2048$ &2026.80 &{\sf dw2048} &$2048\times 2048$ &5301.50 \\ \hline
{\sf c-23} &$3969\times 3969$ &22795.9 &{\sf $L_{m}$} &$3969\times 3969$ & 1.9995 \\ \hline
\end{tabular}
\label{tab1}
\end{table}
\begin{figure}[htp]
\begin{minipage}{0.48\linewidth}
\centerline{\includegraphics[width=6.0cm,height=4cm]{figs/qa_1.eps}}
\centerline{(a)}
\end{minipage}
\hfill
\begin{minipage}{0.48\linewidth}
\centerline{\includegraphics[width=6.0cm,height=4cm]{figs/qa_2.eps}}
\centerline{(b)}
\end{minipage}
\vfill
\begin{minipage}{0.48\linewidth}
\centerline{\includegraphics[width=6.0cm,height=4cm]{figs/qa_3.eps}}
\centerline{(c)}
\end{minipage}
\hfill
\begin{minipage}{0.48\linewidth}
\centerline{\includegraphics[width=6.0cm,height=4cm]{figs/qa_4.eps}}
\centerline{(d)}
\end{minipage}
\caption{ Estimated error bound of $\|F_{k}\|$: (a) {\sf \{$A_{c}$, $L_{s}$\}}; (b) {\sf \{well1850, $L_{1}$\}}; (c) {\sf \{rdb2048, dw2048\}}; (d) {\sf \{c-23, $L_{m}$\}}.}
\label{fig1}
\end{figure}
Figure \ref{fig1} depicts the growth of $\|{F}_{k}\|$ and $\|G_{k+1}\|$ appeared in \eqref{3.8} and \eqref{3.9} as the iteration number $k$ increases from $1$ to $150$. By Theorem \ref{thm3.3}, we use $10\lVert \underline{B}_{k}^{-1}\lVert\epsilon$ as the estimated upper bound of $\|F_{k}\|$. For examples (a) and (c), $\|F_{k}\|$ grow very slightly, while for examples (b) and (d), $\|F_{k}\|$ grow gradually as $k$ increases. From the four examples, we can see that $O(\lVert \underline{B}_{k}^{-1}\lVert\epsilon)$ is indeed an upper bound of $\|F_{k}\|$, and the trends of the growth of $\|F_{k}\|$ and $\lVert \underline{B}_{k}^{-1}\lVert$ are of high similarity. This implies that the growth of $\|F_{k}\|$ is mainly impacted by the growth of $\lVert \underline{B}_{k}^{-1}\lVert$. Since $QQ^{T}\tilde{u}_{i}$ is explicitly computed at each step in our experiments, $\|G_{k+1}\|=O(\epsilon)$ and it remains almost a constant.
\begin{figure}[htp]
\begin{minipage}{0.48\linewidth}
\centerline{\includegraphics[width=6.0cm,height=4cm]{figs/b_1.eps}}
\centerline{(a)}
\end{minipage}
\hfill
\begin{minipage}{0.48\linewidth}
\centerline{\includegraphics[width=6.0cm,height=4cm]{figs/b_2.eps}}
\centerline{(b)}
\end{minipage}
\vfill
\begin{minipage}{0.48\linewidth}
\centerline{\includegraphics[width=6.0cm,height=4cm]{figs/b_3.eps}}
\centerline{(c)}
\end{minipage}
\hfill
\begin{minipage}{0.48\linewidth}
\centerline{\includegraphics[width=6.0cm,height=4cm]{figs/b_4.eps}}
\centerline{(d)}
\end{minipage}
\caption{ Estimated error bound of $\|E_{k}\|$: (a) {\sf \{$A_{c}$, $L_{s}$\}};
(b) {\sf \{well1850, $L_{1}$\}}; (c) {\sf \{rdb2048, dw2048\}}; (d) {\sf \{c-23, $L_{m}$\}}.}
\label{fig2}
\end{figure}
Figure \ref{fig2} depicts the the growth of $\|E_{k}\|=\|I_{k}-B_{k}^{T}B_{k}-P\widehat{B}_{k}^{T}\widehat{B}_{k}P\|$. Since the nonzero elements of $E_{k}$ are of $O(c_{3}(m,n,p)\epsilon)$, we use $100\epsilon$ as the upper bound of $\|E_{k}\|$. From the four examples, we find that as the iteration number $k$ increases from $1$ to $150$, $\|E_{k}\|$ grows very slightly, which is due to that the nonzero elements of $E_{k}$ increase as $k$ increases.
\begin{figure}[htp]
\begin{minipage}{0.48\linewidth}
\centerline{\includegraphics[width=6.0cm,height=4cm]{figs/ql_1.eps}}
\centerline{(a)}
\end{minipage}
\hfill
\begin{minipage}{0.48\linewidth}
\centerline{\includegraphics[width=6.0cm,height=4cm]{figs/ql_2.eps}}
\centerline{(b)}
\end{minipage}
\vfill
\begin{minipage}{0.48\linewidth}
\centerline{\includegraphics[width=6.0cm,height=4cm]{figs/ql_3.eps}}
\centerline{(c)}
\end{minipage}
\hfill
\begin{minipage}{0.48\linewidth}
\centerline{\includegraphics[width=6.0cm,height=4cm]{figs/ql_4.eps}}
\centerline{(d)}
\end{minipage}
\caption{ Estimated error bound of $\|\widehat{F}_{k}\|$ and $\|\widehat{G}_{k}\|$: (a) {\sf \{$A_{c}$, $L_{s}$\}}; (b) {\sf \{well1850, $L_{1}$\}}; (c) {\sf \{rdb2048, dw2048\}}; (d) {\sf \{c-23, $L_{m}$\}}.}
\label{fig3}
\end{figure}
Figure \ref{fig3} depicts the growth of $\widehat{F}_{k}$ and $\widehat{G}_{k}$ appeared in \eqref{3.20} and \eqref{3.21} as the iteration number $k$ increases from $1$ to $150$. By Theorem \ref{thm3.3}, we use $10\lVert \underline{B}_{k}^{-1}\lVert\epsilon$ and $10(\lVert \underline{B}_{k}^{-1}\lVert + \|\widehat{B}_{k}^{-1}\|)\epsilon$ as the estimated upper bounds of $\widehat{F}_{k}$ and $\widehat{G}_{k}$ respectively. From the four examples, we can see that $O(\lVert \underline{B}_{k}^{-1}\lVert\epsilon)$ and $O((\lVert \underline{B}_{k}^{-1}\lVert + \|\widehat{B}_{k}^{-1}\|)\epsilon)$ are indeed upper bounds of $\|\widehat{F}_{k}\|$ and $\widehat{G}_{k}$ respectively, and the growth of $\|\widehat{F}_{k}\|$ and $\|\widehat{G}_{k}\|$ are mainly impacted by the growth of $\lVert \underline{B}_{k}^{-1}\lVert$ and $(\lVert \underline{B}_{k}^{-1}\lVert + \|\widehat{B}_{k}^{-1}\|)$ respectively. For the four examples, both of $\lVert \underline{B}_{k}^{-1}\lVert$ and $\|\widehat{B}_{k}^{-1}\|$ do not grow very rapidly, and thus by Theorem \ref{thm4.1} and Remark \ref{rem4.2}, we can expect that the convergence and accuracy of the approximate generalized singular values of $\{A, L\}$ computed by the SVD of $B_{k}$ or $\widehat{B}_{k}$ will not deviate far from the ideal case as long as the Lanczos vectors are kept semiorthogonal.
\begin{figure}[htp]
\begin{minipage}{0.48\linewidth}
\centerline{\includegraphics[width=6.0cm,height=4cm]{figs/orth_1.eps}}
\centerline{(a)}
\end{minipage}
\hfill
\begin{minipage}{0.48\linewidth}
\centerline{\includegraphics[width=6.0cm,height=4cm]{figs/orth_2.eps}}
\centerline{(b)}
\end{minipage}
\vfill
\begin{minipage}{0.48\linewidth}
\centerline{\includegraphics[width=6.0cm,height=4cm]{figs/orth_3.eps}}
\centerline{(c)}
\end{minipage}
\hfill
\begin{minipage}{0.48\linewidth}
\centerline{\includegraphics[width=6.0cm,height=4cm]{figs/orth_4.eps}}
\centerline{(d)}
\end{minipage}
\caption{Orthogonality level of $\widehat{U}_{k}$: (a) {\sf \{$A_{c}$, $L_{s}$\}}; (b) {\sf \{well1850, $L_{1}$\}}; (c) {\sf \{rdb2048, dw2048\}}; (d) {\sf \{c-23, $L_{m}$\}}.}
\label{fig4}
\end{figure}
Figure \ref{fig4} depicts the growth of the orthogonality level of $\widehat{U}_{k}$ measured by $\eta(\widehat{U}_{k})$ as the iteration number $k$ increases from $1$ to $150$. The estimated upper bound of $\eta(\widehat{U}_{k})$ is described in \eqref{3.23}, and here we use $\epsilon$ as the value of $O(c_{3}(m,n,p)\epsilon)$ appeared in the upper bound. From the four examples, we find that the orthogonality of $\widehat{U}_{k}$ will be gradually lost if we implement the JBD process without any reorthogonalization. The growth trends of the paired red and blue curves in the four pictures are of high similarity, which implies that the orthogonality level of $\widehat{U}_{k}$ is affected not only by $\eta(U_{k})$ and $\eta(\tilde{V}_{k})$, but also by $\|\widehat{B}_{k}^{-1}\|$. For examples (a) and (c), the two paired red and blue curves are very close at some points, which implies that our estimated upper bound of $\eta(\widehat{U}_{k})$ is tight.
\subsection{Examples for the GSVD computation}
In this subsection, we use some examples to illustrate the numerical performance of the algorithm for computing the GSVD of $\{A,L\}$ based on the JBD process. We show the convergence behavior of the Ritz value and the final accuracy of the computed generalized singular values and vectors. We also illustrate the result of Theorem \ref{thm4.1}, and show that the residual norm and its upper bound in \eqref{4.7} can be used as a stopping criterion.
\textbf{Example 1.} In this example, we show the convergence of Ritz value computed by the SVD of $B_{k}$ or $\widehat{B}_{k}$. The matrix pair $\{A,L\}$ is constructed as follows. Let $m=n=p=500$. We first construct a row vector $c$ such that $c(1) = c(2) = 0.99,\ c(3) = c(4) = 0.95, \ c(5) = 0.90,\ c(6) = 0.85,\ c(495) = 0.25,\ c(496) = 0.20,\ c(497) = 0.15,\ c(498) = 0.10,\ c(499) = c(500) = 0$ and $ \texttt{c(7:495) = linspace(0.80,0.30,489)}$ generated by the MATLAB built-in function \texttt{linspace()}, and then let $s = (\sqrt{1-c_{1}^{2}}, \dots, \sqrt{1-c_{n}^{2}})$. Let $C=diag(c)$, $S=diag(s)$ and $\texttt{D = gallery(`orthog',n,2)}$, which means that $D$ is a symmetric orthogonal matrix. Finally let $A=CD$ and $L=SD$. By the construction, we know that the $i$-th generalized singular value pair of $\{A, L\}$ is $\{c_{i},s_{i}\}$, and the multiplicities of generated singular value pairs $\{0.99, \sqrt{1-0.99^{2}}\}$, $\{0.95, \sqrt{1-0.95^{2}}\}$, $\{0, 1\}$ are 2.
\begin{figure}[htp]
\begin{minipage}{0.48\linewidth}
\centerline{\includegraphics[width=6.0cm,height=4cm]{figs/ghost_1.eps}}
\centerline{(a)}
\end{minipage}
\hfill
\begin{minipage}{0.48\linewidth}
\centerline{\includegraphics[width=6.0cm,height=4cm]{figs/ghost_2.eps}}
\centerline{(b)}
\end{minipage}
\vfill
\begin{minipage}{0.48\linewidth}
\centerline{\includegraphics[width=6.0cm,height=4cm]{figs/ghost_3.eps}}
\centerline{(c)}
\end{minipage}
\hfill
\begin{minipage}{0.48\linewidth}
\centerline{\includegraphics[width=6.0cm,height=4cm]{figs/ghost_4.eps}}
\centerline{(d)}
\end{minipage}
\caption{ Convergence of Ritz values computed by the SVD of $B_{k}$: (a) {the first six largest Ritz values, without reorthogonalization}; (b) {the first six largest Ritz values, with full reorthogonalization}; (c) {the first six smallest Ritz values, without reorthogonalization}; (d) {the first six smallest Ritz values, with full reorthogonalization}.}
\label{fig5}
\end{figure}
Figure \ref{fig5} depicts the convergence of the first six largest and smallest Ritz values computed by the SVD of $B_{k}$, where the JBD process is implemented without reorthogonalization or with full reorthogonalization. The right horizontal line indicates the value of $c_{i}$ for $i=1,\dots, 500$. In the left panel, which shows the convergence behavior without reorthogonalization, we see the interesting phenomenon that some of the converged Ritz values suddenly ``jump" to become a ghost and then converge to the next larger or smaller singular values after a few iterations, which results to many unwanted spurious copies of generalized singular values and make it difficult to determine whether these spurious copies are real multiple generalized singular values. From subfigure (a), we find that as $k$ increases, the largest Ritz value will eventually converge to a value slightly larger than $c_{1}$. This is because that the process of computing $B_{k}$ is the Lanczos bidiagonalizatioin of $Q_{A}$ with a growing error $F_{k}$. Therfore, it is necessary to implement the JBD process with reorthogonalization such that the convergence and accuracy of the Ritz values are similar to the ideal case in exact arithmetic. In the right panel, where full reorthogonalization is used, the convergence behavior is much simpler and it is similar to the ideal case in exact arithmetic. It can be found from subfigures (b) and (d), that a simple generalized singular value can be approximated by Ritz values with no ghosts appearing, while a multiple generalized singular value can be approximated one by one by the Ritz values. Notice in both panels that the extreme Ritz values converge more quickly than the interior Ritz values, and the well separated Ritz values converge more rapidly than the dense ones.
\begin{figure}[htp]
\begin{minipage}{0.48\linewidth}
\centerline{\includegraphics[width=6.0cm,height=4cm]{figs/ghost_5.eps}}
\centerline{(a)}
\end{minipage}
\hfill
\begin{minipage}{0.48\linewidth}
\centerline{\includegraphics[width=6.0cm,height=4cm]{figs/ghost_6.eps}}
\centerline{(b)}
\end{minipage}
\caption{Convergence of Ritz values computed by the SVD of $\widehat{B}_{k}$: (a) {the first six smallest Ritz values, without reorthogonalization}; (b) {the first six smallest Ritz values, with full reorthogonalization}.}
\label{fig6}
\end{figure}
Figure \ref{fig6} depicts the convergence of the first six smallest Ritz values computed by the SVD of $\widehat{B}_{k}$, which corresponding to the first largest generalized singular values of $\{A,L\}$. The convergence behavior of the largest Ritz values are similar and thus we omit it. The right horizontal line indicates the value of $s_{i}$ for $i=1,\dots, 500$. From subfigure (a), which shows the convergence of Ritz values without reorthogonalization, we find the ``ghost" phenomenon that some converged Ritz values suddenly jump and then converge to the next smaller singular values after a few iterations. Notice that as $k$ increases, the smallest Ritz value will eventually converge to a value slightly smaller than $s_{1}$, which is similar to the case in subfigure (a) of Figure \ref{fig5}. In subfigure (b), which shows the convergence of Ritz values with full reorthogonalization, the spurious copies are prohibited from appearing, and the multiplicities of the generalized singular values can be determined correctly from the convergence of Ritz values. Comparing figure \ref{fig6} and figure \ref{fig5}, we find that the convergence behavior of the smallest Ritz values computed by the SVD of $\widehat{B}_{k}$ is very similar to that of the largest Ritz values computed by the SVD of $B_{k}$, which is due to that both of them are used to approximate the largest generalized singular values of $\{A,L\}$ and the singular values of $B_{k}$ and $\widehat{B}_{k}$ are closely related, see \eqref{4.5}.
\textbf{Example 2.} In this example, we investigate the final accuracy of the approximate generalized singular values and vectors of $\{A,L\}$, which are computed by using the SVD of $B_{k}$ and $\widehat{B}_{k}$ generated by the JBD process. We choose two matrix pairs. The first pair is $\{A_{c}, L_{s}\}$, and the second pair $\{A_{500}, L_{500}\}$ is constructed as follows. Let $m=n=p=500$. We first construct a row vector $c$ such that $\texttt{c(1:4)=linspace(1.0,0.7,4)}, \ \texttt{c(5:498)=linspace(0.65,0.15,494)}$ and $c(499) = 0.10, \ c(500) = 0.05$. Then let $s = (\sqrt{1-c_{1}^{2}}, \dots, \sqrt{1-c_{n}^{2}})$, $C=diag(c)$, $S=diag(s)$ and $\texttt{D=gallery(`orthog',n,2)}$, which means that $D$ is a symmetric orthogonal matrix. Finally let $A=CD$ and $L=SD$. By the construction, the $i$-th generalized singular value is $\{c_{i},s_{i}\}$, the corresponding right vector $g_{i}$ is the $i$-th column of $D^{T}$, and both the corresponding left vectors of $A$ and $L$ are $e_{i}$, which is the $i$-th column vector of the identity matrix of order $500$.
We use the JBD process with full reorthogonalization to compute the largest generalized singular value and corresponding vectors. We use $\{c_{1}^{(k)}, \hat{s}_{1}^{(k)}\}$ to approximate $\{c_{1},s_{1}\}$, where $c_{1}^{(k)}$ is the largest singular value of $B_{k}$ and $\hat{s}_{1}^{(k)}$ is the smallest singular value of $\widehat{B}_{k}$. The right generalized singular vectors and left vectors corresponding to $A$ are computed from the SVD of $B_{k}$, while the left vectors corresponding to $L$ are computed from the SVD of $\widehat{B}_{k}$. We use the angle error
$$ |\sin\theta_{k}| = |s_{1}^{(k)}c_{1}-s_{1}c_{1}^{(k)}|$$
to measure the error between $\{c_{1}^{(k)}, \hat{s}_{1}^{(k)}\}$ and $\{c_{1},s_{1}\}$ \cite{Sun1983}. For the corresponding generalized singular vectors, we use
$$\sin\angle(g_{1}, x_{1}^{(k)}), \ \ \sin\angle(p_{1}^{A}, y_{1}^{(k)}), \ \
\sin\angle(p_{1}^{L}, z_{1}^{(k)})$$
to measure the errors between the approximated ones and the real ones.
\begin{figure}[htp]
\begin{minipage}{0.48\linewidth}
\centerline{\includegraphics[width=6.0cm,height=4cm]{figs/bnorm_1.eps}}
\centerline{(a)}
\end{minipage}
\hfill
\begin{minipage}{0.48\linewidth}
\centerline{\includegraphics[width=6.0cm,height=4cm]{figs/bhnorm_1.eps}}
\centerline{(b)}
\end{minipage}
\vfill
\begin{minipage}{0.48\linewidth}
\centerline{\includegraphics[width=6.0cm,height=4cm]{figs/bnorm_2.eps}}
\centerline{(c)}
\end{minipage}
\hfill
\begin{minipage}{0.48\linewidth}
\centerline{\includegraphics[width=6.0cm,height=4cm]{figs/bhnorm_2.eps}}
\centerline{(d)}
\end{minipage}
\caption{ Growth of $\|\underline{B}_{k}^{-1}\|$ and $\|\widehat{B}_{k}^{-1}\|$: (a),(b) {\sf \{$A_{c}$, $L_{s}$\}}; (c),(d) {\sf \{$A_{500}$, $L_{500}$\}}.}
\label{fig7}
\end{figure}
\begin{figure}[h]
\begin{minipage}{0.48\linewidth}
\centerline{\includegraphics[width=6.0cm,height=4cm]{figs/gsvd_1.eps}}
\centerline{(a)}
\end{minipage}
\hfill
\begin{minipage}{0.48\linewidth}
\centerline{\includegraphics[width=6.0cm,height=4cm]{figs/gsvd_2.eps}}
\centerline{(b)}
\end{minipage}
\caption{Accuracy of the approximated GSVD components: (a) {\sf \{$A_{c}$, $L_{s}$\}}; (b) {\sf \{$A_{500}$, $L_{500}$\}}.}
\label{fig8}
\end{figure}
Figure \ref{fig8} depicts final accuracy of the approximate generalized singular values and vectors, while figure \ref{fig7} depicts the growth of $\|\underline{B}_{k}^{-1}\|$ and $\|\widehat{B}_{k}^{-1}\|$. For the matrix pair $\{A_{c}, L_{s}\}$, the process of computing $B_{k}$ or $\widehat{B}_{k}$ does not deviate far from the Lanczos bidiagonalization of $Q_{A}$ or $Q_{L}$ with rounding error $O(\epsilon)$, respectively, and the final accuracy of the approximate generalized singular values and vectors is very high, about $O(\epsilon)$. For $\{A_{500}, L_{500}\}$, the final accuracy of the largest generalized singular value, corresponding right vector and left vector of $A$ is about $O(\epsilon)$, while the corresponding left vector of $L$ can not be approximated well. The reason is that for $\{A_{500}, L_{500}\}$, $\|\widehat{B}_{k}^{-1}\|$ grows too big and thus the process of computing $\widehat{B}_{k}$ deviates too far from the Lanczos bidiagonalization of $Q_{L}$ with rounding error $O(\epsilon)$, which leads to that $p_{1}^{L}$ can not be well approximated by using the SVD of $\widehat{B}_{k}$.
\textbf{Example 3.} In this example, we illustrate the residual norm and its upper bound in \eqref{4.7}. We also show the convergence history of the approximate generalized singular values by using both the angle error and relative error. The matrix pair $\{A,L\}$ is chosen to be $\{A_{c}, L_{s}\}$, and we use the SVD of $B_{k}$ to approximate the largest generalized singular value. From the construction of $A_{c}$ and $L_{s}$, we have $\|(A_{c}^{T},L_{s}^{T})^{T}\|=1$, and the largest generalized singular value is $\{c_{1},s_{1}\}$, where $c_{1}=0.75$ and $s_{1}=\sqrt{1-c_{1}^{2}}$. Since $s_{1}\neq 0$, the relative error
$$ |c_{1}^{(k)}/s_{1}^{(k)}-c_{1}/s_{1}|/(c_{1}/s_{1})$$
can also be used as a measure of accuracy of the approximate generalized singular values.
\begin{figure}[htp]
\begin{minipage}{0.48\linewidth}
\centerline{\includegraphics[width=6.0cm,height=4cm]{figs/res.eps}}
\centerline{(a)}
\end{minipage}
\hfill
\begin{minipage}{0.48\linewidth}
\centerline{\includegraphics[width=6.0cm,height=4cm]{figs/error.eps}}
\centerline{(b)}
\end{minipage}
\caption{ Convergence history of the approximate largest generalized singular value of $\{A_{c}, L_{s}\}$: (a) {residual norm and its upper bound}; (b) {angle error and relative error}.}
\label{fig9}
\end{figure}
The convergence history of the approximate largest generalized singular value and the residual norm are shown in figure \ref{fig9}. From subfigure (b), we find that the approximate largest generalized singular value $c_{1}^{(k)}/s_{1}^{(k)}$ converges ultimately to the real one $c_{1}/s_{1}$, and the relative error curve shows that the approximation accuracy of $c_{i}^{(k)}/s_{i}^{(k)}$ to $c_{i}/s_{i}$ is $O(\epsilon)$. From subfigure (a), we find that the residual norm and its upper bound are almost the same as the iteration number $k$ increases. The real residual norm continues descending until about $\epsilon$ while the estimated upper bound finally stagnates at a level little higher than $\epsilon$, since the upper bound of $\|r_{i}^{(k)}\|$ has a term $O(\|\underline{B}_{k}^{-1}\|\epsilon)$ which grows slightly. For the case that $\|\underline{B}_{k}^{-1}\|$ does not grow rapidly, we can omit the term $O(\|\underline{B}_{k}^{-1}\|\epsilon)$ in the upper bound. Moreover, the descending trends of the residual norm and relative error is very similar. Therefore, the residual norm or its upper bound $\| R \|\alpha_{k+1}\beta_{k+1}|e_{k}^{T}w_{i}^{(k)}|$ can be used as a stopping criterion for computing approximate generalized singular values based on the JBD process.
\section{Conclusion}\label{sec6}
We have made a rounding error analysis of the JBD process of $\{A,L\}$ in finite precision arithmetic. The results establish connections between the JBD process of $\{A,L\}$ and the two Lanczos bidiagonalizations of $Q_{A}$ and $Q_{L}$, which shows that $k$-step process of computing $U_{k+1}$, $V_{k}$ and $B_{k}$ is equivalent the lower Lanczos bidiagonalization of $Q_{A}$ with errors $\delta=O(\lVert \underline{B}_{k}^{-1}\lVert\epsilon)$, while the $k$-step process of computing $\widehat{U}_{k+1}$, $\widehat{V}_{k}$ and $\widehat{B}_{k}$ is equivalent to the upper Lanczos bidiagonalization of $Q_{L}$ with errors $\delta=O(\lVert \underline{B}_{k}^{-1}\lVert\epsilon)$ and $\hat{\delta}=O((\|\underline{B}_{k}^{-1}\|+\|\widehat{B}_{k}^{-1}\|)\epsilon)$. We have investigated the loss of orthogonality of the computed Lanczos vectors. We give an upper bound of the orthogonality level of $\eta(\widehat{U}_{k})$, which shows that the orthogonality level of $\widehat{U}_{k}$ is affected by those of $U_{k+1}$, $\widetilde{V}_{k}$ and the quantity $\|\widehat{B}_{k}^{-1}\|$.
We have shown how to use the JBD process to compute some extreme generalized singular values and vectors of $\{A,L\}$ and investigated the convergence and accuracy of the approximate generalized singular values. The results show that generalized singular values of $\{A,L\}$ can be approximated with high accuracy by using the SVD of $B_{k}$, and the orthogonality levels of $U_{k+1}$ and $\widetilde{V}_{k}$ are only needed to be maintained under $\sqrt{\delta/(2k+1)}$, in order to obtain approximate generalized singular values with high accuracy and avoid ghosts from appearing. We have also analyzed the residual norm $\|r_{i}^{(k)}\|$ appeared in the GSVD computation and shown that we can use the residual norm or its upper bound $\| R \|\alpha_{k+1}\beta_{k+1}|e_{k}^{T}w_{i}^{(k)}|$ as a stopping criterion. Moreover, we make a brief discussion about the accuracy of the computed generalized singular vectors, while a detailed investigation on this issue will be followed in our future work. Finally, we use several numerical examples to illustrate our results.
|
2,877,628,089,053 | arxiv | \subsection{Details of the bias in the MNIST experiment} \label{appendix:mnist_bias}
Recall that in \secref{sec:mnist_bias} we generate biased pairs of MNIST digits by defining
\beqn
P(l) = 0.1 \;\; \text{and} \;\; P(r | l) =
\begin{cases}
0 & \text{if } l \in \{0, \ldots, 4\} \text{ and } r \in \{0, \ldots, 4\} \\
0.2 & \text{if } l \in \{0, \ldots, 4\} \text{ and } r \in \{5, \ldots, 9\} \\
0.1 & \text{if } l \in \{5, \ldots, 9\} \\
\end{cases}
\eeqn
and sampling left then right digits. To show that this creates a larger bias on the right
than on the left, we show there is more uncertainty about left digits given right ones than
right ones given left ones. That is, we show the conditional entropy $H(l | r)$ is greater
than $H(r | l)$.
To compute the conditional entropies, we first derive
\beqn
P(r) = \sum_l P(r | l) P(l) =
\begin{cases}
0.05 & \text{if } r \in \{0, \ldots, 4\} \\
0.15 & \text{if } r \in \{5, \ldots, 9\} \\
\end{cases}
\eeqn
and
\beqn
P(l | r) = \frac{P(r | l) P(l)}{P(r)} =
\begin{cases}
0 & \text{if } l \in \{0, \ldots, 4\} \text{ and } r \in \{0, \ldots, 4\} \\
\frac{2}{15} & \text{if } l \in \{0, \ldots, 4\} \text{ and } r \in \{5, \ldots, 9\} \\
\frac{3}{15} & \text{if } l \in \{5, \ldots, 9\} \text{ and } r \in \{0, \ldots, 4\} \\
\frac{1}{15} & \text{if } l \in \{5, \ldots, 9\} \text{ and } r \in \{5, \ldots, 9\} \\
\end{cases}.
\eeqn
Using the convention $0 \log{0} = 0$, we can now compute
\beqn
H(l | r) = - \sum_r P(r) \sum_l P(l | r) \log{P(l | r)} \approx 2.0868 \\
H(r | l) = - \sum_l P(l) \sum_r P(r | l) \log{P(r | l)} \approx 1.9560 \\
\eeqn
\section{Approach: \DeCov Loss} \label{sec:approach}
\vspace{\sectionReduceBot}
To express our notion of redundant or co-adapated features, we impose a loss on the activations
of a chosen hidden layer.
In a manner similar to Dropout, our proposed Decov loss may be applied to a single layer or multiple
layers in a network. For simplicity, let us focus on a single layer.
Let $\mathbf{h}^n \in \mathbb{R}^d$
denote the activations at the chosen hidden layer,
where $n \in \{1, \ldots, N\}$ indexes one example from a batch of size $N$.
The covariances between all pairs of activations
$i$ and $j$ form a matrix $C$:
\beqn
C_{i, j} = \frac{1}{N} \sum_n (h_i^n - \mu_i) (h_j^n - \mu_j)
\eeqn
where $\mu_i = \frac{1}{N}\sum_n h_i^n$ is the sample mean of activation $i$ over the batch.
We want to minimize covariance between different features, which corresponds to penalizing
the norm of $C$. However, the diagonal of $C$ contains the variance of each hidden
activation and we have no reason to require the dynamic range of activations
to be small, so we subtract this term from the matrix norm to get our final \DeCov loss
\beqn
\mathcal{L}_{\DeCov} = \frac{1}{2} \left( \| C \|_F^2 - \| diag(C) \|_2^2 \right)
\eeqn
where $\| \cdot \|_F$ is the frobenius norm, and the $diag(\cdot)$ operator extracts the main diagonal
of a matrix into a vector.
In our experiments, subtracting the diagonal made little difference for small
networks, but led to increased stability for larger networks.
Perhaps the best quality of this loss is that it requires no supervision, so
it can be added to any set of activations.
In a manner similar to Dropout, our experiments typically apply Decov loss to
fully connected layers towards the deep end of a network (\eg, fc6 and fc7 for AlexNet).
However, note that Decov affects \emph{all parameters} up to the layer where it
is applied (and not just the parameters in the specific layer).
At first glance, one seeming peculiarity about this loss is that its global minimum can be found
by setting all weights for $\mathbf{h}$ to 0. This is similar to an
L$_2$ regularizer in that both encourage weights to tend toward 0, but
one important difference between these two regularizers is that
$\mathcal{L}_{\DeCov}$ depends on input data and is not a function
purely of a weight vector like one might find in a classical regularizer such as $L_2$ or $L_1$.
To understand this further, consider the gradient
of the loss with respect to a particular
activation $a$ for a particular example $m$
\beqn
\frac{\partial \mathcal{L}_{\DeCov}}{\partial h_a^m} =
\frac{1}{N} \sum_{j \ne a} \left[ \frac{1}{N} \sum_n (h_a^n - \mu_a) (h_j^n - \mu_j) \right] (h_j^m - \mu_j).
\eeqn
Let us denote the rightmost term in this expression by $I(j, m) = (h_j^m - \mu_j)$.
This term is large (in absolute value) when feature $j$
is discriminative for example $m$ \wrt the mean of the batch.
If $j$ were not discriminative for $m$ then
$h_j^m$ would be close to $\mu_j$.
Hence, we can consider $I$ as an ``importance'' term, corresponding to
a notion of how significant feature $j$ is for example $m$.
Also notice that the term on the left in the gradient expression is simply the covariance
between feature $a$ and feature $j$.
Thus, the gradient can be re-written as
\beqn
\frac{\partial \mathcal{L}_{\DeCov}}{\partial h_a^m} = \frac{1}{N} \sum_{j \ne a} C_{a, j} \cdot I(j, m).
\eeqn
\textbf{Interpretation.}
Intuitively, the covariance term can be thought of as measuring (linear) redundancy:
features $a$ and $j$ are redundant if they vary together.
Thus, the \DeCov loss tries to prevent features from being redundant,
but redundancy is weighted by importance ($I$).
Specifically, a feature $j$ contributes towards a large gradient of feature $a$ on example $m$
if $j$ is important for $m$ \emph{and} correlated with $a$.
This means important features correlated with $a$ (\eg, $j$) contribute to a large
gradient of $a$, suppressing the activation $h_a^m$.
A feature which fires only in specialized situations (\eg, a cat's ear) will likely
be nearly identical or noisy for most other examples (\eg, non-cats) and will not contribute towards
gradients of other specialized features.
\section{Discussion and Conclusion}
\vspace{\sectionReduceBot}
\textbf{Fine-tuning.}
In the experiments we presented, networks were always trained from scratch,
but we also tried fine-tuning networks in different scenarios. During our
ImageNet experiments we fine-tuned both the Network in Network and AlexNet
architectures initialized with parameters that weren't trained with a \DeCov loss,
but were trained with Dropout. In both cases performance either stayed where it
was at fine-tuning initialization or it decreased slightly (within statistical significance).
We found similar results when fine-tuning for other tasks like attribute classification (fine-tuning
AlexNet) and object detection (Fast RCNN~\citep{girshick2015fast}).
This, along with some cases where combining Dropout and \DeCov decreases
performance slightly suggest that the \DeCov loss may possibly be acting adversarially to activations
learned by Dropout.
Fine-tuning with \DeCov is an interesting direction for future work.
\textbf{Trends. }
All of our experiments strongly indicate two clear trends:
\begin{compactenum}
\item
\DeCov reduces overfitting as measured by the gap between train and test performance.
\item
\DeCov acts as a regularizer: performance with \DeCov is always better than
performance without either \DeCov or Dropout.
\end{compactenum}
To be clear, the results do not support that Dropout can be completely replaced by DeCov, but simply that in a number of scenarios DeCov is a useful alternative and their combination almost always works the best.
Our loss clearly has desirable regularization
properties at the expense of one extra hyper-parameter to tune.
In this work, we proposed a new \DeCov loss which explicitly penalizes the covariance between the
activations in the same layer of a neural network in an unsupervised fashion.
This loss acts as a strong
regularizer for deep neural networks, where overfitting
is a major problem and Dropout has been required to get large models to generalize
well. We show that \DeCov competes well against Dropout over a range
of experiments which investigate different scales, datasets and architectures.
\textbf{Acknowledgements. }
This work was supported in part by the following awards to DB: National Science Foundation CAREER award, Army Research Office YIP award, Office of Naval Research grant N00014-14-1-0679, AWS in Education Research Grant, and GPU support by NVIDIA. The views and conclusions contained herein are those of the authors and should not be interpreted as necessarily representing the official policies or endorsements, either expressed or implied, of the U.S. Government or any sponsor.
\section{Experiments}
\vspace{\sectionReduceBot}
We begin with a synthetic dual ``modality'' experiment, which
serves as a testbed
for measuring improvement due to decorrelation.
Next, we use an autoencoder (as in \cite{dropout})
to contrast \DeCov and Dropout. Finally, we use a variety of experiments to report
Image Classification performance on CIFAR10/100 and ImageNet, noticing
significant improvement in \emph{all cases}.
Note that we set the Dropout rate to 0.5 as suggested by ~\cite{dropout}.
\subsection{Dual modality experiments with MNIST: Predicting Side-by-Side Digits}
\label{sec:mnist_bias}
We propose a synthetic dual ``modality'' task on MNIST -- simultaneously predict the class labels
for two digits placed adjacent in an image.
We created a dataset where each example consists
of two MNIST digit images horizontally concatenated and
separated by 16 black pixels (to prevent interference between feature maps in the first layers).
\figref{fig:sbs} shows a few examples.
\begin{figure}[h]
\centering
\includegraphics[scale=0.4]{images/SBS.pdf}
\caption{We consider the task of simultaneously predicting two MNIST digits placed side by side.
By biasing right digits more than left digits at train time, we create a
controlled scenario with the type of problem we expect \DeCov to solve.}
\vspace{-5pt}
\label{fig:sbs}
\end{figure}
The important detail of this experiment is the particular bias we inject into the distribution
of left and right digits. Let
\beqn
P(l) = 0.1 \;\; \text{and} \;\; P(r | l) =
\begin{cases}
0 & \text{if } l \in \{0, \ldots, 4\} \text{ and } r \in \{0, \ldots, 4\} \\
0.2 & \text{if } l \in \{0, \ldots, 4\} \text{ and } r \in \{5, \ldots, 9\} \\
0.1 & \text{if } l \in \{5, \ldots, 9\} \\
\end{cases}.
\eeqn
To generate one example we first sample the left digit using $P(l)$ then the right using $P(r | l)$.
As shown in Appendix \ref{appendix:mnist_bias}, we can compute the conditional entropies
of one digit given the other to get $H(l | r) = 2.0868$ and $H(r | l) = 1.9360$. Since
$H(l | r) > H(r | l)$, the left digit is more informative of the right than the right is
of the left. There is no cross-digit signal at test time, so features for the right and
left digits should be completely decorrelated to generalize, but learned features
will have some correlation between left and right. Intuitively, \DeCov should help
generalization in this scenario. Our experiments support this.
We use Caffe's~\citep{caffe} reference version of LeNet~\citep{lenet}.
It has two convolution layers, each followed by pooling, then a fully connected
layer with 500 hidden units which are shared between the two softmax layers.
We apply \DeCov and/or Dropout to the 500 hidden units of the fully connected layer.
\begin{table}[h]
\vspace{10pt}
\small
\setlength{\tabcolsep}{4pt}
\begin{center}
\begin{tabular}{@{} l l c c c c c c @{}}
\toprule
& & \multicolumn{3}{c}{Left Digit} & \multicolumn{3}{c}{Right Digit} \\
\cmidrule[0.75pt](l){3-5}
\cmidrule[0.75pt](lr){6-8}
\DeCov & Dropout & train & test & train - test & train & test & train - test \\
\midrule
no & no & 99.98 $\pm$ 0.01 & 97.94 $\pm$ 0.18 & 2.05 $\pm$ 0.19 & 100.00 $\pm$ 0.00 & 96.75 $\pm$ 0.24 & 3.25 $\pm$ 0.24 \\
no & yes & 99.99 $\pm$ 0.00 & 98.45 $\pm$ 0.04 & 1.54 $\pm$ 0.04 & 99.99 $\pm$ 0.00 & 97.39 $\pm$ 0.20 & 2.61 $\pm$ 0.20 \\
yes & yes & 99.97 $\pm$ 0.01 & 98.59 $\pm$ 0.12 & 1.38 $\pm$ 0.12 & 99.99 $\pm$ 0.00 & 97.81 $\pm$ 0.07 & 2.18 $\pm$ 0.06 \\
yes & no & 99.99 $\pm$ 0.00 & \textbf{98.74} $\pm$ \textbf{0.03} & \textbf{1.25} $\pm$ \textbf{0.04} & 99.99 $\pm$ 0.00 & \textbf{97.99} $\pm$ \textbf{0.12} & \textbf{2.00} $\pm$ \textbf{0.12} \\
\midrule
\multicolumn{2}{l}{weight decay} & 99.97 & 97.86 & 2.11 & 99.97 & 96.21 & 3.76 \\
\bottomrule
\end{tabular}
\caption{\textbf{MNIST side by side results.}
As expected, biasing right digits at train time so that they are weakly informed
by left digits leads to lower performance on an unbiased test set.
More importantly, \DeCov provides greater improvements
over the baselines on the right, confirming that it leads
to better features when decorrelation is extremely likely to improve performance.
}
\label{tab:mnist_sbs}
\end{center}
\end{table}
\textbf{Results.}
\tableref{tab:mnist_sbs} reports the accuracy of left and right digit
classifiers. Our injected dataset bias can be clearly seen in the lower test accuracy
and higher train-test gap of the right classifier, indicating that all
of our networks incorporate the train time bias into their predictions.
We report mean accuracies across 4 trials, along with the standard
deviation. We also compare the effect of Dropout.
The main result is that the gaps between the performance of \DeCov and the baselines
are larger for the biased right digit (\eg, right digit test accuracy shows a $\sim$0.6\%
improvement when switching from Dropout-alone to \DeCov-alone
while the improvement for left digits is just $\sim$0.3\%).
This suggests that the baselines pick up
on the false bias and that \DeCov does the best job of correcting for it.
\DeCov also improves generalization for both classifiers since test accuracy is higher
in the bottom two rows and the train - test gap is lower in those rows.
Combining Dropout with our \DeCov loss
hurts slightly, but we note that the error bars overlap in some cases,
so this is not a statistically significant difference.
One skeptical hypothesis is that the \DeCov loss is simply enforcing something akin to
an L2 penalty on the weights. The experiments with \DeCov and Dropout already
use an L2 penalty, so this is unlikely, but a grid search over weights on
this term shows it makes little difference.
The best accuracies are reported in the last row of \tableref{tab:mnist_sbs}.
\subsection{MNIST Autoencoder}
To offer a more qualitative point of comparison, we visualized learned features using
the 2 layer autoencoder experiment
from~\citep{dropout} (section 7). In this experiment an autoencoder
is trained on raw pixels of single MNIST digits
using an encoder with 1 layer of 256 ReLU units and a decoder (untied weights)
that produces 784 ($28 \times 28$) ReLU outputs.
\figref{fig:autoencoder} shows the weights learned by the autoencoder (reshaped to align with
the input image) and mean-square reconstruction errors.
Weight initialization turned out to be an important factor for the visualizations.
Initializing all weights by sampling from $U[-\sqrt\frac{3}{n}, \sqrt\frac{3}{n}]$
(based on~\cite{glorot2010understanding}; as implemented in Caffe) led to visualizations as seen in
\cite{dropout} (the baseline looks like noise), but sampling weights from a
Gaussian with mean 0 and standard deviation 0.001 led to baseline
visualizations with faint digit outlines. The latter initialization was
used in \figref{fig:autoencoder}.
One take-away is that MSE is significantly lower for \DeCov than others.
However, the key take-away is the qualitative difference between representations learned
with Dropout and those learned with \DeCov.
Recall from \secref{sec:intro} that Dropout reduces cross-covariance while \DeCov
explicitly minimizes it. Despite this intuitive similarity, the two lead to
different learned representations.
\begin{figure}[t]
\centering
\begin{subfigure}[b]{0.30\textwidth}
\includegraphics[width=\textwidth]{{images/autoencoder/gauss.001_base_crop}.jpg}
\caption{Baseline with train MSE = 1.47 and test MSE = 1.47}
\label{fig:auto_base}
\end{subfigure}
\begin{subfigure}[b]{0.30\textwidth}
\includegraphics[width=\textwidth]{{images/autoencoder/gauss.001_xcov_xw.1_crop}.jpg}
\caption{\DeCov with train MSE = 0.98 and test MSE = .98}
\label{fig:auto_decov}
\end{subfigure}
\begin{subfigure}[b]{0.30\textwidth}
\includegraphics[width=\textwidth]{{images/autoencoder/gauss.001_drop_crop}.jpg}
\caption{Dropout with train MSE = 3.08 and test MSE = 3.03}
\label{fig:auto_drop}
\end{subfigure}
\caption{Weights learned by the first layer of a 2 layer autoencoder are
reshaped into images and visualized for a model with no \DeCov or Dropout (\figref{fig:auto_base}),
a model with \DeCov (\figref{fig:auto_decov}), and a model with Dropout (\figref{fig:auto_drop}).}
\vspace{-10pt}
\label{fig:autoencoder}
\end{figure}
\subsection{Image Classification}
\subsubsection{CIFAR10}
CIFAR10 contains 60,000 32x32
images sorted into 10 distinct categories \citep{cifar10}.
We training on the 50,000 given training
examples and testing on the 10,000 specified test samples.
Hyper-parameters (loss weights for \DeCov and weight decay) are chosen by
grid search on the standard train/val split.
We use Caffe's quick CIFAR10 architecture, which has 3 convolutional layers
followed by a fully connected layer with 64 hidden units and a softmax layer.
The hidden fully connected layer is not followed by a non-linearity.
The \DeCov loss is added only to the 64 hidden units in the hidden fully connected layer.
All reported results are average performance over 4 trials with the standard deviation indicated alongside.
\begin{table}[h]
\small
\setlength{\tabcolsep}{13pt}
\begin{center}
\begin{tabular}{@{} l l c c c @{}}
\toprule
\DeCov & Dropout & train & test & train - test \\
\midrule
no & no & 100.0 $\pm$ 0.00 & 75.24 $\pm$ 0.27 & 24.77 $\pm$ 0.27 \\
no & yes & 99.10 $\pm$ 0.17 & 77.45 $\pm$ 0.21 & 21.65 $\pm$ 0.22 \\
yes & yes & 87.78 $\pm$ 0.08 & \textbf{79.75} $\pm$ \textbf{0.17} & \textbf{8.04} $\pm$ \textbf{0.16} \\
yes & no & 88.78 $\pm$ 0.23 & 79.72 $\pm$ 0.14 & 9.06 $\pm$ 0.22 \\
\midrule
\multicolumn{2}{c}{weight decay} & 100.0 & 75.29 & 24.71 \\
\bottomrule
\end{tabular}
\caption{CIFAR10 Classification.
We can see that \DeCov with Dropout leads to the highest test performance and the lowest train-test gap.}
\label{tab:cifar10}
\end{center}
\end{table}
\paragraph{Results.}
In Table \ref{tab:cifar10}, we again
observe significant improvements when using the \DeCov loss --
there is a $\sim$4.5\% improvement in test accuracy (over no regularization).
Moreover, the \DeCov loss reduces the gap between train and val accuracies by $\sim$15\% (without Dropout)
and $\sim$16\% (with Dropout)!
Comparing the four combinations, we see that using \DeCov alone provides a larger
improvement than using Dropout. Using both \DeCov
and Dropout further improves the generalization (as measured by the gap in train and test
accuracies), but the improvement in absolute test performance does not seem statistically significant.
We again test if L2 weight decay can provide similar improvements and
find once again that the best setting gives little improvement over the baseline.
One promise of regularization is the ability to train larger networks,
so we increase the size of our CIFAR10 network. We
add another fully connected layer to the network used in the previous experiment,
double the number of filters in each convolutional layer, and double the
number of units in the fully connected layers. This larger network performs better
than the smaller version -- all accuracies are higher than corresponding entries in
\tableref{tab:cifar10}. However, there are the stronger indications of
overfitting in this network -- specifically, the train accuracies are much higher than test accuracies
(when compared to the previous network).
\tableref{tab:cifar10_bigger} shows the results.
We observe similar trends as the previous experiment --
there are significant gains from using \DeCov alone compared to Dropout alone,
and there is a further slight improvement in combining both.
Using Dropout alone gives a $\sim$1.5\% boost in test accuracy, while using \DeCov alone
provides a $\sim$4\% increase in test accuracy.
Using both yields roughly the same test performance, but the trainval and test gap is further reduced.
\begin{table}[h]
\small
\setlength{\tabcolsep}{5.5pt}
\begin{center}
\begin{tabular}{@{} l l c c c @{}}
\toprule
\DeCov & Dropout & (train+val) & test & (train+val) - test \\
\midrule
no & no & 100.00 & 77.38 & 22.62 \\
no & yes & 100.00 & 79.93 & 20.07 \\
yes & yes & 96.76 & \textbf{81.68} & \textbf{15.08} \\
yes & no & 98.15 & 81.63 & 16.52 \\
\bottomrule
\end{tabular}
\caption{CIFAR10 Classification with a bigger version of the base network}
\label{tab:cifar10_bigger}
\end{center}
\end{table}
\subsubsection{CIFAR100}
To scale up our experiments, we move to CIFAR100~\citep{cifar10}. We use the
same architecture as the base architecture for CIFAR10 and hold out the
last 10,000 of the 50,000 train examples for validation.
Table \ref{tab:cifar100}
shows that Dropout alone highest higher test performance than \DeCov alone,
but \DeCov leads to a smaller train-test gap.
Using both regularizers not only achieves the highest test accuracy, but also the smallest
train-test gap ($\sim$34\% smaller than using neither regularizer).
This suggests that the two regularizers may have complementary effects.
\begin{table}[h]
\small
\setlength{\tabcolsep}{9pt}
\begin{center}
\begin{tabular}{@{} l l c c c @{}}
\toprule
\DeCov & Dropout & train & test & train - test \\
\midrule
no & no & 99.77 & 38.52 & 61.25 \\
no & yes & 87.35 & 43.55 & 43.80 \\
yes & yes & 72.53 & \textbf{45.10} & \textbf{27.43} \\
yes & no & 77.92 & 40.34 & 37.58 \\
\bottomrule
\end{tabular}
\caption{CIFAR100 Classification Accuracies}
\label{tab:cifar100}
\end{center}
\end{table}
One more problem comes with the question of
how to weight the \DeCov loss. All of our experiments use grid search to pick this hyper-parameter.
The optimal weight varies across datasets, but
we have found consistency across variations in architecture.
We varied both the \DeCov weight and the number of hidden units in the
fully connected layer to which \DeCov is applied, training a new network for
each setting.
The best \DeCov weight (0.1)
is consistent for a range of hidden activation sizes in this dataset,
though it is different in other experiments.
\subsubsection{ImageNet}
Now we explore results for networks trained for ImageNet classification, starting by applying \DeCov to fc6 and fc7 in AlexNet~\citep{krizhevsky_nips12}.
The last 50,000 of the ILSVRC 2012 train images are held out for validation.
Our implementation comes from Caffe. In particular, it uses a fixed schedule that multiplies the learning
rate by 1/10 every 100,000 iterations (see jumps in \figref{fig:imagenet_losses}). We do not use early stopping and do not perform color augmentation.
In \figref{fig:imagenet_losses} we notice that when neither of the two regularizers -- Dropout or \DeCov -- are applied (blue line), the network overfits (it even gets 100\% train accuracy), and
the \DeCov loss (hidden activation redundancy) is higher than with any other combination of the regularizers. Applying either of the regularizers also causes a synchronous drop in both losses.
Explicitly minimizing the \DeCov loss naturally leads to much lower \DeCov losses, and we notice that this coincides with
significantly reduced overfitting. Interestingly, Dropout results in relatively lower \DeCov loss too, even when \DeCov is not optimized
for. This is further indication of the link between redundant activations and overfitting.
\begin{figure*}[h]
\vspace{-5pt}
\includegraphics[scale=0.5]{images/loss_plot/loss_plot.pdf}
\centering
\vspace{1pt}
\caption{Cross Entropy and \DeCov losses over the course of training AlexNet with 256x256 images. Note that the \DeCov val
curves are hidden by the train curves.
Interestingly, \DeCov is reduced even by Dropout, though not nearly as much as when it is explicitly minimized.}
\label{fig:imagenet_losses}
\end{figure*}
\figref{fig:alexnet} shows accuracies across different image resolutions we used to train AlexNet.
AlexNet is typically trained with 256x256 images, but training with smaller images is faster
\footnote{
Using CuDNNv3, AlexNet with 128x128 inputs takes 103ms averaged over 50 runs to compute a forward and backward pass.
For 256x256 images this time is 449ms.}
\emph{and}
reduces the number of parameters in the network. Smaller images (we use 128x128, 160x160, 192x192, and 224x224) lead to smaller feature maps output by pool5, so
the dense connection between pool5 and fc6 has fewer parameters, the model has less
capacity, and it's less likely to overfit. For example, images scaled to 256x256 (taking 227x227 crops
\footnote{At train time crops are sampled and mirrored randomly. At test time only the center 227x227 crop is used.})
lead to a weight matrix with 38 million parameters while 128x128 images (with 99x99 crops)
result in a 4 million parameter matrix.
Generally, accuracies (left plots) and the train-val gap (right plots) have a slight positve slope,
confirming that performance and overfitting increase with resolution and model capacity.
Note that the \DeCov loss weight was tuned using grid search at each resolution both with and without Dropout.
\begin{figure*}[h]
\includegraphics[scale=0.5]{images/alexnet_resolutions/top1_plot.pdf}
\includegraphics[scale=0.5]{images/alexnet_resolutions/top5_plot.pdf}
\centering
\vspace{1pt}
\caption{ImageNet classification performance using AlexNet. Plots on the left show training and validation (ILSVRC 2012 validation set)
accuracy at different resolutions.
Note how all curves have a much lower train-val gap than the (blue) baseline.}
\label{fig:alexnet}
\end{figure*}
We see that Dropout alone (green) usually has the best val accuracy, which is slightly
higher than the two losses combined (purple) and a couple points higher than \DeCov alone (red)
at higher resolutions. At the lowest resolution Dropout alone is tied with \DeCov alone.
Dropout also reduces overfitting more than \DeCov, though both independently reduce
overfitting by a large margin -- from 59.35\% to 14.7\% in the case of \DeCov @ 128x128.
Finally, we test our new regularizer on ILSVRC 2012 with one more architecture --
the Network in Network \citep{lin2013nin}.\footnote{This is the model provided in the Caffe Model Zoo: \url{https://gist.github.com/mavenlin/d802a5849de39225bcc6}}
This architecture is fully convolutional: it contains 4 convolutional layers,
with 96, 256, 384, and 1024 feature maps, respectively. Between each of these
layers and after the last are two convolutional layers which have 1x1 kernels,
which further process each feature map output by the main convolutional layers
before being fed into the next layer. To produce 1000 softmax activations,
1000 feature maps are averaged over spatial locations to produce one feature
vector. We applied \DeCov to these average pooled feature vectors.
Interestingly, this architecture has much less overfitting than AlexNet.
However, adding a \DeCov loss still decreases overfitting substantially
and improves validation accuracy. There is a small boost in performance on
validation accuracies and a significant decrease of $\sim$3\% (for top 1)
and $\sim$2\% (for top 5) in the train - val gap.
\begin{table}[h]
\small
\setlength{\tabcolsep}{5.5pt}
\begin{center}
\begin{tabular}{@{} l l c c c @{}}
\toprule
\DeCov & Dropout & ILSVRC 2012 train top 1 & ILSVRC 2012 val top 1 & train - val \\
\midrule
no & no & 71.68 & 58.67 & 13.01 \\
no & yes & 71.32 & 58.95 & 12.37 \\
yes & yes & 68.28 & \textbf{59.08} & \textbf{9.20} \\
yes & no & 68.33 & 58.85 & 9.48 \\
\midrule
\DeCov & Dropout & ILSVRC 2012 train top 5 & ILSVRC 2012 val top 5 & train - val \\
\midrule
no & no & 89.91 & 81.18 & 8.73 \\
no & yes & 89.63 & 81.53 & 8.10 \\
yes & yes & 87.99 & \textbf{81.94} & \textbf{6.05} \\
yes & no & 87.88 & 81.57 & \textbf{6.05} \\
\bottomrule
\end{tabular}
\caption{ImageNet Classification Accuracies with Network in Network}
\label{tab:nin}
\end{center}
\end{table}
\section{Introduction}
\label{sec:intro}
Deep Neural Networks (DNNs) have recently achieved remarkable success on
a wide range of tasks -- \eg, image classification on ImageNet~\citep{krizhevsky_nips12},
scene recognition on MIT Places~\citep{zhou_nips14}, image captioning with MS COCO~\citep{coco, show_and_tell, chen2015mind},
and visual question answering~\citep{VQA}.
One significant reason for improvement of these methods over their predecessors has to do with scale.
Faster computers coupled with optimization improvements such Batch Normalization, Adaptive SGD, and
ReLus let us quickly train wider and deep networks. Access to large annotated datasets and
regularizers such as Dropout has provided significant reduction in the amount of
overfitting in these large networks,
thus enabling the performance we see today.
In this paper, we focus on the problem of overfitting, which is observed when a high capacity model (such as a DNN)
performs very well on training data but poorly on held out data.
Even when trained on large annotated datasets (such as ImageNet~\citep{imagenet} or Places~\citep{zhou_nips14}, containing
millions of labelled images), deep networks are susceptible to overfitting. This problem
is further exacerbated when moving to new domains and tasks -- since DNNs tend not to
generalize with a few examples, each new task tends to require curating and annotating
a new large dataset. While there has been some success with transfer learning~\citep{girshick2014rcnn, donahue_icml14, yosinski2014transferable},
networks still overfit.
A promising alternative to creating even larger datasets is to apply different forms of regularization
to the network while training to avoid overfitting.
These methods include regularizing the norm of the weights~\citep{tikhonov_regular},
Lasso~\citep{tibshirani_lasso}, Dropout~\citep{dropout}, DropConnect~\citep{wan2013dropconnect},
Maxout~\citep{goodfellow2013maxout}, etc.
One particular regularizer of interest to DNNs is Dropout~\citep{dropout},
which attempts to prevent co-adaptation of neuron activations.
Co-adaptation occurs when two or more hidden units rely on one another to perform some function
which helps fit training data, thus becoming highly correlated.
Co-adaptation is reduced by Dropout using an approximate model averaging technique
that sets a randomly selected set of activations to zero at training time.
\cite{dropout} show that this has a regularizing effect, leading
to increased generalization and sparser, less correlated features. Notice that this
is without \emph{explicitly} encouraging decorrelation in hidden activations.
To further investigate the relationship between hidden activation correlations and overfitting,
we show in \figref{fig:teaser} two quantities from a CNN
trained for image classification on CIFAR100~\citep{cifar10} -- (1) the
amount of overfitting in the model (as measured by the gap between train and val accuracy),
and (2) the amount of correlation in hidden activations (as measured by the Frobenius
norm of the sample cross-covariance matrix computed from vectors of hidden activations;
details in \secref{sec:approach}). Both these quantities of interest are reported as a function
of amount of training data (x-axis) and with/without Dropout (left/right subplot). As
expected, both increased training data and Dropout have a regularizing
effect and lead to reduced overfitting.
The figure also shows an interesting novel trend -- as the amount of overfitting reduces, so
does the degree of correlation in hidden activations. In essence, overfitting and co-adaptation
seem to be correlated. The open question of course is -- is the relationship causal?
\begin{figure}[!h]
\centering
\vspace{-15pt}
\begin{subfigure}[b]{0.49\textwidth}
\vspace{10pt}
\includegraphics[width=\columnwidth]{images/teaser/teaser_plot_nodrop.pdf}
\caption{Without Dropout}
\label{fig:teaser_nodrop}
\end{subfigure}
\begin{subfigure}[b]{0.49\textwidth}
\vspace{10pt}
\includegraphics[width=\columnwidth]{images/teaser/teaser_plot_drop.pdf}
\caption{With Dropout}
\label{fig:teaser_drop}
\end{subfigure}
\caption{Two principal ways to prevent overfitting in deep models
are to train with more data (x axis) and to train with Dropout (right plot).
As expected, both of these decrease validation error (left axis), but they
also happen to decrease hidden activation cross-covariance (right axis).
We investigate whether explicitly minimizing cross-covariance can lead to reduced overfitting. }
\label{fig:teaser}
\end{figure}
This leads to the principal questions of this paper --
Is it possible to bias networks towards decorrelated representations by directly
reducing correlation between hidden units? And do such decorrelated representations generalize better?
\textbf{Overview and Contributions.}
The goal of this paper is to learn DNNs with decorrelated activations and
study the effect of this decorrelation on their generalization performance.
Towards this end, we propose a fairly natural
loss called \DeCov, which explicitly
encourages decorrelation between the activations in a deep neural network.
This loss requires no additional supervision, so it can be added to any existing network.
In addition to the link discussed above, our motivation also comes
from the classical literature on bagging and ensemble averaging~\citep{hansen1990neural, Perrone93whennetworks, Breiman1996},
which suggests that decorrelated ensembles perform better than correlated ones.
Our experiments encompass a range of datasets (MNIST~\citep{lenet}, CIFAR10/100~\citep{cifar10},
ImageNet~\citep{imagenet}),
and different kinds of network architectures (Caffe implementations of
LeNet~\citep{lenet}, AlexNet~\citep{krizhevsky_nips12}, and Network in Network~\citep{lin2013nin}).
All cases suggest that \DeCov acts as a novel and useful regularizer.
\section*{Appendices}
\addcontentsline{toc}{section}{Appendices}
\renewcommand{\thesubsection}{\Alph{subsection}}
\input{appendix_mnist.tex}
\end{document}
\section{Related Work}
\vspace{\sectionReduceBot}
\paragraph{Redundancy Based Representations.}
The idea of using low redundancy to learn representations
has been around for decades. In an early attempt to
model human perception, \cite{barlow1961} lists
3 possible learning principles, the 3rd being the
notion that representations should not be redundant.
Later work continued to investigate this intuition in the context
of unsupervised feature learning.
Three objectives emerged, each of which formalize the notion differently.
(1) An information theoretic view is expressed
by~\cite{linsker1988}. The main idea is to maximize information gained
by predicting the next representation/layer between input and output.
(2)
The closest objective to ours is cross-correlation (not cross-covariance), which appears
in~\citep{bengio2009slow} and complements a temporal coherence objective.
It also appears in~\citep{pearlmutter1986gmax} where it complements an objective which encourages
units to capture higher order input statistics.
(3)
Finally, redundancy minimization is realized through
predictability minimization in~\citep{schmidhuber1992} for the purpose
of learning factorial codes (representations whose units are independent).
This objective says that one unit should not be predictable
given \emph{all} of the others in its layer as input.
All of these works focus on unsupervised feature learning and do not experiment
with supervised models. Furthermore, these early pioneering works were
limited by data and evaluated small networks without many of the
modern design choices and features (e.g. ReLus, Dropout, SGD instead of Hebb's update
rule, batch-normalization, \etc).
We propose redundancy minimization for a new purpose (regularization),
evaluate it using modern techniques such as end-to-end learning using SGD
with respect to a supervised objective, and do this in the context of
harder challenges presented by modern datasets.
To the best of our knowledge, such a setting has not been considered before.
\paragraph{Correlation/Covariance Losses in Other Settings.}
Other works have used similar penalties, but in different settings and to different effects.
Deep Canonical Correlation Analysis (Deep CCA)~\citep{andrew2013deep}
and Correlational Neural Networks (CorrNets)~\citep{chandar2015corrnet} apply a similar
loss which \emph{maximizes} correlation, unlike our \emph{minimization} of cross-covariance.
Both methods are used to learn better features in the presence of multiple
views or modalities.
They embed inputs to a common space and maximize correlation between
aligned pairs.
Another idea similar to ours is that of~\cite{hidden_factors},
which aims to discover and disentangle hidden factors.
The goal is to separate supervised factors of variation (\eg, class of MNIST digits)
from unsupervised factors of variation (\eg, handwriting style).
In order to achieve this goal, they impose a covariance (not correlation) loss
between (1) the softmax outputs of a neural network trained to recognize digits and
(2) a hidden representation which is used in conjunction with (1) to reconstruct
the input (via an auto-encoder).
These two works suggest that correlation losses significantly
impact learned representations in the context of modern networks.
One key difference between these two approaches and ours is that
while their formulations decorrelate~\citep{hidden_factors} and disregard~\citep{andrew2013deep,chandar2015corrnet} parts
of \emph{different} representations, our approach tries to decorrelate parts of the
\emph{same} representation.
Moreover, the ultimate goals are different. Unlike these approaches,
our goal is simply to improve supervised classification performance by reducing overfitting,
and not to reconstruct the original data.
\textbf{Dropout and Batch Normalization.}
Two recent approaches to regularization
in deep neural networks are Dropout~\citep{dropout} and to some extent
Batch Normalization~\citep{ioffe2015batch}.
Dropout aligns with our intuition and goals more closely as it
aims to improve classification performance by reducing co-adaptation
of activations.
On the other hand, Batch Normalization focuses on faster optimization by
reducing \emph{internal co-variate shift}, which is the constant variation
of a layer's input as it learns. Some Batch Normalization results indicate it could
act as a regularizer, but this
has not been exhaustively verified yet.
Our approach is similar to Batch Normalization
due to its use of mini-batch statistics.
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